New Tools forR.obustness ofLimear Systerns

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B. Ross BarnnishDepartrnent of Electrical andCornputer EngineeringU nivers ity of Wisco nsin

Copyright O 1994 by Macrnlllan Publshtng Company' a divlsion of Macmlllan' Inc'

hnted trr the United States of Amertca

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Preface ix

of colleagues who volunteered to serve as "readers," f soon realizedthat there were considerable improvements to be made. In this re-gard, I extend my thanks to Ted Djaferis, Faryar Jabbari and LahcenSaydy. Kris Hollot, Ian Petersen, Roberto Tempo, Boris Polyak andTom Higgins also served as readers, but they need to be mentionedin categories of their own. Kris, Ian and Roberto not only provided

me with extensive technical and editorial comrnents, but also en-gaged me in lengthy discussions about the "philosophical aspects"

good friends. Over the last few months, Boris Polyak played a, verysimilar role. As far as Tom Higgins is concerned, I am most gratefulfor our frequent interactions over the last eight years. He is a truescholar who constantly made me aware of results in the literaturer,('hich were both important and unknown to me.

Throughout the entire course of this project, the support pro-vided by Macmillan Publishing Company was excelient. First andforemost, I wish to express my sincere thanks to John Griffin. Inhis capacity as Editor, he helped me in a number of ways from startto finish. As the project was drawing to a close, Leo Malek becameinvolved and provided excellent support on many issues associatedwith final production. Of particular note, Leo connected, me with anoutstanding copy editor-Lilian Brady. \

Finally, my greatest personai note of thanks goes to my imme-diate family-Marlene, Lara and Sybil. The manuscript could nothave been written without their tolerance during my frequent pe-riods of enlistment in the space cadets. This book is dedicated tothem.

B. Ross BarmishMadison, Wisconsin

xii Table of Contents

2.6 lJncertain Functions Versus Farnilies

2.7 Convention: Real Versus Complex Coefficients

2.8 Consoiidation of Notation '

2.9 Conciusion

Chapter 3. Case Study: The Fiat Dedra Engine

3.1 Introduction3.2 Control of the Fiat Dedr a Engine

22z;)

z,)

27

2929.)u

30J J

J . *

J O

. f /

38J Y

Contents

Part I. Prelirninaries

1. A GlobalIntroduction

Overview

Robustness ProblemsSome Historical Perspective .Refinement of the ScopeKharitonov's Theorem: The SparkThe Issue of Uncertainty StructureTire lvlathematical Programming Approach .

The Toolbox Philosophy .The Value Set ConceptN,Iathernatical lvlodei Versus the Tlue SystemFamily ParadigrnRobustness Analysis ParadigmRobustness Nlargin ParadigmRobust Synthesis ParadigrnTesting Sets and Computational CompiexityConclusion

2. Notation for Uncertain SystemsIntroductionNotation for Uncertain Parameters .

Subscripts and SuperscriptsUncertainty Bounding Sets and NormsNotation for Families

3.3 Discussion of the Engine Model

3.4 Tlansfer Function N4atrix for the Engine

3.5 IJncertain Parameters in the Engine IVIodel

3.6 Discussion of the Controller Modei

3.7 The Closed LooP PolYnomial

3.8 Is Symbolic Computation Really Needed?

3.9 Symbolic Computation with TYansfer Functions

3.10 Conclusion

Part II. From Robust Stability to the Value Set

Chapter 4. Robust Stabitity with a Single Parameter

4.I Stabil ity and Robust Stabil itv

4.2 Basic Definit ions and ExamPles

4.3 Root Locus AnalYsis4.4 Generalization of Root Locus

Nyquist Analysis .The Invariant Degree ConcePt

Eigenvalue Criteria for Robustness

4.8 Nlachinerv for the Proof .

Proof of the TheoremConvex Combinations and Directions

The Theorem of BialasThe Matrix Case .Introduction to Robust D-Stability

Robust D-Stabil ity GeneraiizationsExtreme Point ResultsConclusion

Chapter 5. The Spar-k: Kharitonov's Theorem

5.1 Int roduct ion5.2 Independent Uncertainty Structures

5.3 Interval Polynomial FamilY

5.4 Shorthand Notation

Chapter1 . 11 . 21 . 37 . 4

1 . 61 . .71 . 81 . 91 . 1 01 . 1 1r . 7 2r . 1 31 1 ^

1 . 1 5r . l_b

Chapter2 . r2 . 2

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xi

Table of Contents

5.5 The Kharitonov Polynomials

5.6 Kharitonov's Theoremn r^ ̂ L : . -^- - - f^- the PrOOfo. I l vadu l l r r rs lJ tu r

5.8 Proof of Kharitonov's Theorem .

5.9 Formula for the Robustness Margin

5.10 Robust Stabil ity Testing via Graphics

5.11 Overbounding via Interval Polynornials

5.12 Conclus ion

xl.tl

686 9707779I Y

8 183

86868788899 19496

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1 1 8127

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n v l a u l c u r u u r l L t r r r b

Chapter 9. The Edge Theorem I49

9.1 Introduction . I49

9.2 Lack of Extreme Point Results for Polytopes . 150

9.3 Heuristics Underlying the Edge Theorem . 150

9.4 The Edge Theorem . 752

9.5 Fiat Dedra Engine Revisited . I549.6 Root Version of the Edge Theorem . . 160

9.7 Conclusion . 162

Chapter6 . 16 . 26 . 3o . +6 . 56 . 66 . 76 . 86 . 9o . 1 u

Chapter7 . L7 . 2a . )

7 . 4

/ . o

7 . 7

6. Embellishments of Kharitonov's TheoremIntroductionLow Order fnterval PolynomialsExtensions with Degree DroppingInterval Plants with Unity FeedbackFrequency Sweeping Function 11(c..,)Robustness Margin GeometryThe Tsypkin-Polyak FunctionComplex Coefficients and TlansformationsKharitonov's Theorem with Complex CoefficientsConclusion

7. The Value Set ConceptIntroductionThe Value SetThe Zero Exclusion ConditionZero ExcLusion Condition for Robust D-Stabiliiy .Boundary Sweeping FunctionsMore General Value SetsConclusion

Chapter1 0 . 1r0.21 0 . 370.41 0 . 51 0 . 6ro.710 .8

Chapter1 1 . 17r.21 1 . 371..4I I - . J

1 1 . 6II.T1 1 . 8

Chapter1 2 . I1 2 . 212.31 t A

12.51 2 . 6L2.7

Chapter1 3 . 11 .3 .2l - J . . )

73.4

10. Distinguished Edges 764Introduction . 164Parallelotopes . 165Parpolygons . 167

Setup with an Intervai Plant . 167The Thirty-Two Edge Theorem . 168Octagonality of the Value Set . . 169Proof of Thirty-Two Edge Theorem . I75

Conclusion . 176

11. The Sixteen Plant Theorem I78

Introduction . 178

Setup with an Interval Plant . 779

Sixteen Distinguished Plants . 180

The Sixteen Plant Theorem . . 182

Controller Synthesis Technique . 182Machinery for Proof of Sixteen Plant Theorern . 186Proof of the Sixteen Plant Theorem . 792conclusion . 193

12. Rantzer's Growth Condition 196

Introduction . 196

Convex Directions . 797

Rantzer's Growth Condition . 200N4achinery for Rantzer's Growth Condition . 202Proof of the Theorem . 212Diamond Families: An Iilustrative Application . 2I4

conclusion . 21,6

13. Schur Stability and Kharitonov Regions 218In t roduct ion .2I8Low Order Coefficient UncertaintyLow Order Polynomials . 223Weak and Strong Kharitonov Regions . 224

Part III. The Polfedral TheorY

Chapter B. Polytopes of Polynomials

8.1 Introduction8.2 Affine Linear Uncertainty Structures

8.3 A Primer on Polytopes and Polygons

8.4 Introduction to Polytopes of Polynomials

8.5 Generators fol a Pol5.'tope of Polynomials

8.6 Polygonal Value Sets

8.7 Value Set for a Polytope of Polynomials .

8.8 Improvement over Rectangular Bounds

8.9 Concl.usion

ChapterL 4 . Tr4.274.3r4.4r4 .574.614.71 4 . 81 4 . 91 4 . 1 01 4 . 1 I

Part fV.

Chapterr 5 . 115.2l - c . J

15.4

r , l . o

15.71 5 . 81 5 . 9

Chapter1 6 . 116.21 6 . 3L6.4t D . o

1 6 . 616.71 6 . 81 6 . 9

Part V.

Chapter1 7 . I77.21 7 . 317.4

1 7 . 617.71 7 . 81 7 . 977.I0

Chapter1 8 . 178.21 8 . 31 8 . 41 8 . 51 8 . 6

Table of Contents

13.5 Characterization of Weak Kharitonov Regions'I 3 6 N,f nchinerw for Proof of the Theorema Y a @ u l r r r r v r J r v r

13.7 Proof of the Theorem13.8 Conclusion

xv

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246247248250252o < Q

xvi Table of Contents

Some Happenings at the Frontier

17. An Intloduction to Guardian iVlapsIntroductionOverviewTopological Preliminaries for Guardian lMapsThe Guardian MapSome Useful Guardian MapsFamilies with One Uncertain ParameterPolynomic DeterminantsThe Theorem of Saydy, Tits and AbedSchur StabilitvConclusion

18. The Arc Convexity TheoremIntroductionDefinitions for Frequency Response ArcsThe Arc Convexity Theorem

297

298298299300302JU' )

305306307t 1 1d f r

37431,4

3 1 6

L4. Multitinear Uncertainty StructuresIntroductionMore Cornplicated Uncertainty Structures

\4ultilinear and Polynomic UncertaintyInterval Matrix FarnilvLack of Extreme Point and Edge ResultsThe Mapping TheoremGeometric Interpr*et ationValue Set InterpretationNfachinery for Proof of the N{apping Theorern

Proof of the Nlapping Theorern

Conclusion

The Spherical Theory 257

'1 5 Snherica.l Polwnornial Families 258v y f r v r r v Q r r v r J /

Intr-oduction . 259

Boxes Versus Spheres . 259

Spherical Polynomial Families .260o A e

Lumping .rur

The Soh-Berger'-Dabke Theorem .267

The Vaiue Set for a Spherical Polynomial Family . . 270

Proof of the Soh-Berger*Dabke Theorem .272

Overboundi.ng via a Spherical Famil5' ' 273

corrclusion . 276

16. Embellishments for Spherical Families 278

Introdr.rction . 279

The Spectra l Set .279

Formula and Theorem of Barmish and Tempo . 280

Affine Linear Uncertainty Structures . 284

The Testing Function for Robust Stabitity 285

Machiner5' for Proof of the Theorem . 289

Proof of the Theorem .292o n o

Sonre Refinements . 'zgJ

conclusion ' 294

Nlachinery 1: Chords and Derivatives ' 3LT

Nlachinery 2: FlowMachinery 3: The poruiaa"o C""; . . . . .

18.7 Proof of the Arc Convexity Theorem18.8 Robustness Connections18.9 Conclusion

? 1 0

J Z O

JZ '+

328

Chapter 19. Five Easy Problems 33019.1 Int roduct ion . 33079.2 Problem Area 1: Generating Mechanisms . 33119.3 Problem Area 2: Conditioning of Margins . 338

19.4 Probiem Area 3: Parametric Lyapunov Theory . 34219.5 Problenr Alea 4: Polytopes of Matrices . 34719.6 Problem Area 5: Robust Performance . 351

79'7 conclusio' ' 355

Appendix A. Symbolic Cornputation for

Bibliography

Index

Fiat Dedra 358

J O . L

384

New Tools forR.ohrrstness ofLireear Systems

Part I

Chapter 1

A Global Overvlew

Synopsis

The main objective of this chapter is to provide some genetaJi-

ties about the scope'of this book. To this end, control problems

involving uncertainty are subdivided into different areas via' a'

Problem TTee. This tree is not to be taken too seriousJy-itonly sewes as a metaphor for better understanding where thefocus of this furt ljes. A secondary objective of this chapter is

to set the stage for the technical exposition to follow. To this

end, basic paradigms for robustness are described.

1.1 Int roduct ion

Much of modern control theory addresses problems involving llncer-

tainty. A typical scenario begins with a system to be controlled and a

mathematical model which includes uncertain quantities. For exam-ple, the mathematical model might involve various physical param-

eters,whose values are specified only within given bounds. In order

to provide a global overview of this text, we imprecisely subdividecontrol problems with uncertainty into three types:

r Adaptive Problemsr Stochastic Problemso Robustness Problems

o

Prelirninaries

1.2 ,/ Robustness Problems 3

This subdivision should not be interpreted too literally. Obvi-ously, many problem formulations with uncertainty do not fit neatlyinto any of the three categories; e.g., fuzzy control problems and sin-gular perturbation problems. We assume a demarcation between thethree problem areas above solely for pedagogical purposes. In addi-tion, the reader famiiiar with control theory also understands thatnany control problems with uncertainty may involve ideas from morethan one of the three areas above. For example, one can considera robust adaptive control problem or a stochastic adaptive controlproblem. Having provided our disclaimers, we stick witir our threeploblem idealization and begin this overview by simply declaring thefocal point of this textbook to be a class of robustness problems.

L.2 Robustness Problerns

Given that the field of robust control has experienced a large num-ber of breakthroughs over the last two decades, an uninitiated readermight find it quite overwhelming when first attempting to gain someperspective. Since the word "robust" appears in l i terally dozens ofdifferent contexts, it is quite natural to divide the robustness areainto a number of different subareas. We indicate such a division viatbe Problem Tree in Figure 1.2.1. As stated in the chapter synop-sis, this problem tree is only meant to be interpreted in the senseof a metaphor. The distinction between problem areas can be quite

unclear. For example, in Figure 1.2.1, we separate real and com-plex parametric uncertainty problems knowing fuli q'ell that in manycases, one encounters a mixture of both types of uncertainty. Sim-ilarly, we separate linear and nonlinear pr-oblems knowing that weoften treat nonlinearities as uncertainties in a linear system. The listof qualif ications in Figure L.2.I can occupy several pages. Neverthe-less, in order to get some perspective on the scope of this book, weignor-e these subtleties.

Note that it is possible to more fully develop the branches ofthe tree. For example, within the class of nonlinear robustness prob-iems, one can further subdivide into many different areas. To createa better focal point, however, v/e have deliberately pruned varioustree branches. \4/hen concentrating on the side of the tree dealingwith linear time-invariant robustness problems, we are not rulingout solutions which rnight involve nonlinear control or uncertaintiesentering nonlinearly into the model. Our point of view is that themathematical model of the plant includes a "nomina|" 5ystem which

robustness problems

4 Ch. I , / A Global Overrnerv

all otherrobustness problems

(noniinear, time-varying,distr ibuted, etc.)

robustness--nl-rl om c fnr

linear timeinvariant systems

analysisproblems

synthesisproblems

structured real unstructuredstructured real unstructuredand complex complex and complex cornplexuncertainty uncertainty uncertainty uncertaintl'

problems problems problems problems

FIGURE 1.2.1 A Probiem Tlee for Robust S)'stems

is linear and time-invariant. However, this does not rule out the

possibiiity that uncertainties enter nonlinearly into the plant. For

exanpie, consid.er a transfer function wllose description includes un-

cer ta in pa,rameters Qt , Q2, . . . , Qt .For f ixed q; , the p lant is l i [ear and

time-invar.iant but the q,; are permitted to enter transfer function

coefficients in a nonlinear manner; in some cases, the nominal plant

rnight be obtained with all qr set to zero

At the bottom of the linear time-invariant side of the Problem

Tlee, we see four types of robustness problems. They involve robust-

ness analysis with structured real and complex uncertainty, robust-

ness analysis with unstructured complex uncertainty, robust synthe-

sis with structured real uncertainty and robust synthesis problems

involving unstructured cornplex uncertainty' In some cases, these

uncertainties represent system parameters and in other cases, un-

modeled dynamics is the basic concern. We are now prepared to

d.escribe the scope of this text. To this end, we fi.rst provide some

historical context.

1.3 ,/ Some Historical Perspective 5

1.3 Sorne Historical PersPective

the rnost fundamental research questions are as yet unresolved, par-

and the focus on parametric uncertainty and related Popov ancl Lure-

type problems in sit jak (1969). After a hiatus of at least a decade,

we see the revitalization of interest in real parametric uncertainty in

Ackerrnann (1980). The approach taken in this text relies heavily

on a certairt ,,zero Exclusion condition." A version of this condi-

tiorl can be traced. back at least as far as Frazer and Duncan (1929)'

Perhaps it can even be argued that the paper by Frazer and Duncan

is one of the pioneering works on robustness of systerns'

As far as synthesis with complex uncertainty is concerned, we

have recently wi.tnessed some important breakthroughs for cla,sses

of synthesis problems with unmocleled dynamics; e'g',.one highlight

is the -tI€ controi problem as formulated by Zames (1981) and el-

egantly solved via Riccati equations in Doyle, Glover, Khargonekar

and Frlarrcis (1989). In a sense, these breakthroughs culminate three

decades of change which occurred in the field-from preoccupation

with optimality in the sixties to emphasis on robustness and MI\4O

problems in the eighties.To conclude this section, we mention the large body of literature

on robust stabilization in state space pioneered in the seventies b}'

t l tkin and Leitmann; e.g., see utkin (1977) and Leitmann (1979).

In retrospect, their- robust stabilization theories for systems with

time-varying uncertainty which satisfies "rriatching conditions" was a

natura,l precursor for the rnole genelal -Fl@ framework of the eigirties

and nineties.

L.4 Reflnernent of the ScoPe

By and large, the results in this book apply most directly to the anal-

ysis or the root locations of polynomiais whose coefficients depend

e Ch. 1/ AGlobalOveruew

on uncertain parameters. Frorn a control theoretic point of view'

the motivation for studying these new tools is the fact that many

robust performanc" problu-. for feedback systems can be massaged

into equivalent probiems involving a poiynomial's roots' Using this

,"d.r.tion, the iroof of many robustness results given in this book

involves eipioitation of classical theoiems on the $eometry of poly-

nomials; e-g., see Marden (1966)' In marked contrast to classical

literature on polynomials, however, the total emphasis here is on

robustness issues.withthe explosive growth of the "new tools litera,tulle" following

publication of the seminal stability theorern of Kharitonov (1978a)'

there has been increasing recognition of the fact that many of the

new results can actually be applied to a variety of robust performance

problems-not jr-rst the probiem of r-obust stabilitl'' To illustrate this

point, imagine tirat a ,tftnrur" package is available which finds the

ioots of a poiynornial' The cluestion is: How might 'r'e apply this

package to iire seemingly unrelated performance ploblem of checking

if u pttnt has f{- ,torit"lu.. than unity? Indeed, we consider a fixed

plant expressed as a quotient of polynomials

N ( s )P ( s ) : , 6

with D(s) being stable (all roots in the open left half ptane)' We-

assurne that this plant is strictly proper and' want to determine if

the ,E1- norrrr

l lP l l - : :18 IPU') l

is less than unitY.To reformulate this problem in a polyngmial frarnervork' 'we

make one observation: Since P(j') --+ 0 as (t + @1 continuity

of P(ja) dictates that the onlv *ay that l lPll." ) 1 can occur is if

V(j;\ i : 1 fo, some frequency a* > 0' This implies the existence

of an angle 0 < 0 < 2n' such that

I ' l ( j . - ) - ieD ( i . ) :

e - '

These observations lead us to study the d-parameterized polvnomial

P ( s , 0 ) : N ( s ) - e i o D G ) '

Now for the punchline: It can be shown that stabil ity of p(s, d) for

all d € lO,2Ti is equivalent to satisfaction of the r-equiled inequality

L5 ,z lftaritonor"s Theorem: The Spark 7

l lPil." ( 1; e.g., see Barrnish and Khargonekar (1990) for the defini-t iorr of p(s,0) and Chapellat, Dahleh and Bhattacharyya (f 0SO) forthe stronger stabil ity result. In summary, the -I1- problem has beenreduced to finding the roots of a family of poiynomials parameterizedby 0 e IO ,2 r l .

This exarnple is intended to illustrate one of the possible waysthat fr-rndamental polynornial problems arise from control systemperformauce problems. In a robustness context, similar ideas applywith uncertainty entering transfer function coeffrcients.

' 1.5 l(haritonov's Theorern: The Spark

lVlany of the questions addressed in this book are not new-whatis new is the machinely introduced and its ability to solve a num-

bel of problems which heretofore seemed intractabie. The fbrmu-

Iation of basic robustness problenrs and their solution for various

special cases goes back a long way. For example, in the early rvork

of Neirnark (1949), we see effective techniques for robust stability

analysis which work rvell with a small number of uncertain palame-

ters. In the book by Sil jak (1969), special classes of robust stabil ity

analysis problems for systems with structured real parametric un-

certainty are considered. For problems involving robustness analysis

with uncertain parameters entering multilinearly into transfer func.

tion coefficients, a polverful tool is the Mapping Theorem given in

the book by Zadeh and Desoer (1963). On the synthesis side, we

aheady mentioned the importance of the book by Horowitz (1963)

in bringing robustness issues to the fore.In the mid to late eighties, we see a new "explosion" of research

involving structured real parametric uncertainty. In large measure,the reason for this resrugence of interest in the area is the serninaltheorem of Kharitonov (I978a); this theorem is the takeoff point

for much of the technical exposition in this book. After the controlclothing is removed, many of the basic problems which we address

can be viewed in the context of an age-old question: How do the

roots of a polynomial depend on its coefncients? Aithough the l iter-

ature contains a rvealth of information on this question (for example,

see Gantmacher (1959) and Marden (1966)), the robustness context

is missing. In fact, a number of the results in this book can be viewed

as robustified versions of classical results.

o u ( r . \ / ^ v r v o 4

1.6 The fssue of lJncertainty Structure

After massaging a robustness analysis problem into a poiynorlial

problem with coefficients depending on uncertain parameters, the

issue of uncertainty structure arises. In the formalism of the chapters

to follow, we deal with a vector of uncertain parameters q and a

polynomial which is expressed as

n

p ( s , q ) : l a r ( q ) s i .t :0

The uncertainty structure is manifested via the coefficient ftinctions

eo(q) , . . . ,an(q) . In the ideal ized f ramework of I ihar i tonov (1978a) '

each component q; of q enters into only one coefficient. Tiris same

"independent" uncertainty structure is also exploited in the impor-

tant work of Soh, Berger and Dabke (1985), rvhich is described in

Chapter 15. In contrast to Kharitonov's framework where a box is

used to bound q, Soh, Berger and Dabke use a sphere. The obvious

point to note is that an independent uncertainty structure is highly

idealized; uncertain parameters of a system generally enter into urore

than one coeffrcient of p(s,q), and in many cases, the a4(q) are often

nonlirrear functions.lVhen dealing s'ith uncertainty structure, this text follows a nat-

ural progression. First, we explain basic ideas without significant

machinery. This is accomplished by studying robust stability for the

case of a single uncertain parameter. Subsequently, we deal with in-

dependent uncertainty structures as explained above. The next level

of complication is the affrne linear uncertainty structure. In this

case, coef[cients depend affine linearly on q and various results on

polytopes of poiynomials are deveioped. For affi.ne linear uncertainty

structures, the highlight is the Edge Theorem of Bartlett, Holiot and

Huang (1988) given in Chapter 9.To deal with more realistic robust control problems, multilinear

and nonlinear uncertainty structures are of paramount impor-tance;

e.g., consider p(s. q) above with each coeffi.cient function a1 (q) being

multilinear. At this higher level of difficulty, we see a bifurcation

in the robustness literature. That is, some authors deal with these

more difficult uncertainty structures by restricting their attention

to analytically tractable special cases, while othel authors resort

to mathematical programming. This textbook concentrates on a

selected number of analybical r-esults which are available; a highlight

is the Mapping Theorem, which is covered in Chapter 14. Although

1.7 / The MathemadcalProgrammingApproach g

we d.o not cover the mathematical programming approach in this

text, we include the section below as a gateway to this other body

of literatr.rre.

L.T The Mathernatical Prograrnrning Approach

The rnotivation for- a large body of literature is derived from the fact

that rnany robustness problems can be reformulated as rnathemati-

cal programming problenrs. Subsequerrtl-1'. olre has available a u'ide

variety of softrvare tools to accontplisll the lecluired optirnization. To

illustr-ate at the sirrrplest of levels, tve continue to 1et q represent an

uncertain parameter-vector and take Q to be a closed and bounded

restraint set for q. w-e consider a robust stability problem for the

poiynomial

p ( s , q ) : s 3 - F o r ( q ) t ' + a r ( q ) s + " o ( q ) '

Namely, cletermine if a1i roots of p(s, q) lie in the strict left half plane

for all S e Q. The appiication of the Routh-Hurwitz stability criteria

leads to the following conclusion: Robust stability is guaranteed if

ancl only if, for each q € Q, the conditions a6(q) > 0, oz(q) > 0 and

a{il&2(d - "o(q)

) 0 are satisfied. Hence' a solution to the trio of

optimization problerns

- i - - ^ 1 - \ 'q e L /

p6 "r(q); n1'6 lal(q)or(q) -

"o(q)]

Leads to a solution of the robust stability problem.

As a second. example, we note that mathernatical programming

problems arise quite naturally in the muitivariable stability margin

theories of Doyle (1982) and Safonov (1982). In many cases, they

take A to be a block diagonal uncertai'nty matrix, and for a fixed

matrix JVI, one seeks a minimum norm A such that

d e t ( - I * M A ) : 9 .

This is generally called the p, probtern. once again, it is straightfor-

ward. to reformulate this problem as a mathematical program'

The mathematica.l prograrnming approach to robustness prob-

lems has one clear advantage over more analy'tical approaches. The

forrnulation easiJ.y accommodates large classes of robust performance

criteria for systems which can have rather complicated uncertainty

structures. Many examples iilustrating the power of the approach

are given in the book by Boyd and Barratt (1990).

10 Ch.1 / A Global Overuew

by checking only four distinguished extremes'

l-.8 The Toolbox PhilosoPhy

7 .9 / The Va lue Se t Concep t l 1

is only one of many tools rvhich can be brought to bear. There aremany other specific results in modern controi theory for which thesame argument can be made. For example, the Hurwitz stabiiitycriberion and the srnall Gain Theorem should rightfully be viewedas two tools which can be brought to bear in appropriate situations.

1.9 The Value Set Concept

To conclude the overview of this book, we briefly mention one pieceof technicai machinery which we use to unifi' many results in theliterature-the so-called ualue set. An understanding of trre valueset enabies us to appreciate many robustness developments from asingle perspective. In many places throughout the book, we stateresults from the recent literature but provide new proofs-proofs viathe value set approach.

We now provide a rough explanation of the value set concept:The rnain point to note is that we can reformulate many robustnessproblems in terms of a two-dimensional set which we temporarilycallV(5); this set l ies in the complex plane. Notice that V(6) is pa-rarneterized via a real scalar 5, which we call a general'ized frequencyuariable. Now, as 6 increases, the set I,/(6) typically moves aroundthe cornplex plane. For many problems, it turns out that fhe zeroerclusion condition

o / v(6)for all 5 is both necessary and suf;frcient for satisfaction of the statedrobustness specifi cation.

As seen in the sequel, many robustness problems lead to verysimple value set geometries which can be exploited to obtain stronganalytical results. For exampie, when dealing with the intervai poly-nomial framework of Kharitonov, the generalized frequency variable6 arises via the substitution s : j6, and the value set V(6) turns outto be a rectangle whose sides are paraliel to the real and imaginaryaxes-a level rectangle.

In large measure, the power of the value set approach is derivedfrom the fact that it is a two-dimensional set, whereas the uncer-tain parameter set is typically of higher dimension; i.e., although arobustness problem r.vith an /-dimensional uncertain paramerer vec-tor q is initially formulated over Rl, we need only manipulate thetwo-dirnensional value set l,(6). Furthermore, since z(6) is onry two-dimensional, we obtain a second advantage. For cases when V(6) isreadily cbnstructable, we obtain solutions to robustness problems

which Iend themselves to implementation in graphics. That is, once

we have an analytical description of the value set y(6), it is often

convenient to simply generate this set on a computer and provide a

visual dispiay of its motion with respect to 6.

1.1o Mathernatical Model Versus the Tbue Systern

In the remaincler of this chapter, or-rr objecti.r'e is to create the appro-

priate ,,mind set" for the technical exposition to follow. we begin

ty noting that in classical control theory, one generally begins with

a mathematical model of the system and a design is car-ried out

which is aimed at satisfaction of some given set of performance spec-

ifications. Examples of such specifications involve prescriptions on

damping, overshoot, tracking and fiequency lesponse. Once the de-

sign is complete, the follo.rving question is of critical importance: If

we use an inexact mathematical model to derive the controller, will

the systerrr perforrn satisfactorily? In other words, a fundamental

concern is that a design based on an inexact mathematicai model

may result in unacceptable performance when implemented on the

trul system-the behavior of the true system may be quite different

from the behavior predicted by the mathematical model'

To iilustrate these ideas, imagine a vehicle dynamics system

which is to be d,esigned assuming a certain coefficient of friction

p : 0.5. I{owever, a fundamental concern is that p rn^y vary de-

pend.ing on whether the road surface is slippery or dry; e'g', consider

a rainy day versus a dry day. For example, suppose L]na.t 20%; varia-

tions in the coefficient p are possible. Hence, the true coefficient of

friction l ies between 0.4 and 0.6 and the question at hand is whether

the system will pelform satisfactorily if pt : 0'5 is assumed in the

design. This is the issue of robustness with respect to uncertainty in

the coefficient p.There are two important points to note about the discussion

above. First, the uncertainty in p illustrates only one of the rnany

types of uncertainties one may encounter in a control theoretic con-

text. In addition to structured real parametric uncertainty as illus-

trated by p, one may encounter uncertainty due to factors such as

unmodeled dynamics, delays and nonlinearities. These factors may

also lead to a response of the system which may be radically different

from the predictions of the mathematical model. The second point

to note is that there is typically more than one uncertain param-

eter to be considered. For example, for the vehicle system above,

1.11 / FamilYParadigm 13

one might enterlain additional uncertaint; ' in mass loading, various

spr-ing constants and damping coefiicients.

1.11 Farnily Paradigrn

Th.roughout this text, we work with bounds on uncertain quantities

without assuming a statistical description of any sort' Within this

frarnework, the notion of a family f is fundamental. For example, in

the vehicle dynamics problem described above, we adopt the point

of view that each admissible vaiue of pr between 0.4 and 0.6 defines a

different system. Hence, we have a fam'i'ly of sgstems f rather than

a fi.xed system. In this book, the word "family" is used in a wide

variety of contextsl e.g., we refer to a family of polynomials, a family

of transfer functions or a familv of matrices.

L.l2 Robustness Analysis Paradigrn

Given a faurily F and some property P, when we say that -F is

robust, we mean the following: Every member f e f has property

P. For example, we are often interested in a, famiiy of systems F

with P being some aspect of performance. Then, when we say that

tlre fam.ily f is robust, the understanding is that the perfornance

specification is satisfied for every system in the familv; i-e., for all

.f e f . Ther-e are nurnerous other possibilities for F and P. For

exanrple, F can be a family of polynomials and P might denote

stability, or .F might denote a family of transfer functions and P can

be a specification on the frequency response.One of the nain objectives of this text is to develop machinery

which can be used to ascertain whether or not a given family .F is

robust. In this regard, families of polynomials and robustness with

respect to their root locations occupy much of our attention.

L.LT.L lJncertain System Versus Family of Systems

Throughout the robustness literature, the expression uncertain sgs-

tem is often used interchangeably with fami'IE of sEstems. For the

sake of precision, however, this text makes a distinction between

these two expressions. Namely, when we refer to an uncertain sys-

tem, apriori bounds on uncertain quantities are not included. For

exarrrple, recall the vehicle dynamics system in Section 1.10 with un-

certain coefficient of friction p. When we wish to discuss the model

without specifying apriori bounds on pr, we say that we have an un-

14 Cin.7 / A Global Overvierv

:t of saYing that we have a familY

on the' uncertailr Parameter' our

incomPletel)' specified ln oLher'

' iori uncertaintl' bounds defines a

family of systems' A simila'r distinction is mad'e in a number of other

situations; e.g.' an.tt"e'iui'l polynomial plus r"rncertainty bounds de-

l inesafarn i lyofpolynomials ,oru ' ' ' ' t t ' ' ' " " ' tu int ransfer funct ionplusuncertainty to.tn.l, i"fi""t a family of transfer functions'

1-.13 Robustness Margin Paradigm

Fot- cases rvhen bounds otl the utlce

often consider th.e so-called Robust

is to fi.nd the maximal uncertainty

mance sPecification is satisfied' T

d.ynamics problem of Section 1'10

coefficient of friction p is uncertar

u,hat variations in pr might be encountered' Norv' if the perf-orru'ance

specification is satisfi 'ei at 1' ' :0'5' one can consicler the following

problem: Replace pt ' :0 '5 b 'y 1 ' :0 '5 + Apl and determine l low large

Ap can be while ptese"'in! satisfaction of the perforrnance specl-

fication. Thi.s maximal vaiie, call it r*o*' is cailed the robustness

mar..9u1.

ExERCTsE L.13.1 (Multiple Uncertainties) : Consid'er the vehicle

dynamics problem di'"t-tt'La in Section 1'10' Nor'v' however' suppose

that there are trvo unce-r-tain coefficients of friction p'1 and 1-r'2' For

exarnple, ima,grne fr:ont ancl leal tires mad'e of d'ifferent materials'

Propose various toi"t*"""t margin definitions t'hich rnake sense frorr

an aPPlicar,ions Point of view'

EXEncrsE L-L3.2 (Robustness Analysis Versus Robustness N'Iargin):

Provide an interprltation of the foilowing informal staternent: The

robustness margin paradigm """o*put'Js

the robustness a'nalt'sis

paradigm.

hpxpncrsE 1.13.3 (Recursive Calculation of Robustness Margin):

An engineer wrrtes a computer p'og'u* to solve a robustness anai-

ysis problem irr.'ott'i"g t*L """"'tu'i"

f i"tio" parameters pt ar'd P2

satisfying pi+ l l i <.-"' ttt" user specifies r ) 0' and the program

indicates whether or not robustness is guaranteea' The next d"y' T:

bosssu rp r i ses t i r eeng inee rby ind i ca t i r r g tha ta robus tnessma lg i r }

t,iii ii iiliii i

ii

['rI t

I 14 ,z Robusr S),nlhesis Paradigm 15

is demanded instead. Using the existing computer code, suggest arecllrsive plocedure for finding the desired robustness margin.

a.L4 Robust Synthesis Paradigm

Althor.rgh this book deals primarily with robustness analysis pr-ob-lerns and robustness margin problems, there are a few occasionswhere the new tools which we describe are readily applicable in arobust synthesis context. A distinguishing feature of the robust syn-thesis problem is the presence of adjustable design parameters whichneed to be selected. That is, the faraily of st'sterns clescription isexpanded to include design pararneters which ar-e chosen so as togualantee that the subsequent robustness anaiysis succeeds.

, Frorn a control theoretic point of view, the adjustable designparameters above are associated with a colnpensator in a feedbacksystem. In sorne cases, the number of such parameters is specified.For example, a PI controller C(s) : I1 -l l{zls is parameterized bythe pair of gains (Kr,Kz).The robust design problem is to pick 1(1and I(2 so that performance specifications are met for all admissiblevalues of the uncertain pa,rameters.

IVIore abstractly, suppose that we are given some desired perfor-llrance specification P, a set C which we call the class of admi,ssibLecompensators and a mapping taking each element c € C to a fam-ily of systerns f". Then the Robust Sgnthes'is Problern is to picksome c* e C such that every member f e f.- satisfies the givenperformance specifi.cation P.

1.15 Testing Sets and Cornputational Cornplexity

One of the highlights of this text is the emphasis on testing sefs. Toconvey the meaning of a testing set, we provide an example using thenotation of Section 1.12. Indeed, let -F be a given farnily and takeP to be a desired property representing some robust performancespecification. In many instances throughout this text, we identifya fi.nite subset F* : {f i, f;,.. ., f i}

"f F and prove a result of the

following sort: Property P holds for atl f e F if and only if PropertyP holds for f i , f ; , . . . , , f f . When the number of d is t inguished / i issmall, such a result often irnpiies a dramatic reduction in the com-putational complexity associated with the solution of the robustnessproblem at hand.

To illustrate the notion of testing sets in a more concrete way,imagine a system with performAnce specification P and two uncer-

l b L n . I / / - 1 1 u l u u d r v v L r l r L Y t

wi th the pa i r (q1 , q2) .

1.16 Conclus ion

In summa.ry, one of the prima.ry objectives of this text is to deveiop

robustness criteria which can be executed with "reasonable" com-

putational, effort. The first step in this direction is to develop a

notational system which facilitates exposition of technical results.

This is the focal point of the next chapter'

Notes and Related Literature 17

Notes and Related Literature

NRL 1.1 For paits of the problem tree not covered in this text; good starting

references are Rugh (1981), Vidyasagar (1985), Is idor i (1985) and KhaI i I (1992)

for nonlinear systems, Astrcjrn and wittenmark (1989) for adaptive systems and

Kumar and varaiya (1986) for stochastic systerns. A good reference overviewing

many aspects of robust control is the text by Weinrnann (1991)'

NII-L 1.2 In Section 1.3, we do not mean to irnply that the paper by Doyle'

Glover, Khargonekar and Flancis (1989) was the first to solve the -I1- control

problem. what distinguishes Doyle, Glover, Khargonekar and Flancis (1989)

from other work is the elegant mechanization of the solution via Riccati equations.

In fact, earlier papers such as chang and Pearson (1984) and Flancis, Helton and

Zames (1984) contain solutions which are less accessible to the control community

because of their reliance on more abstract interpolation concepts'

NRL 1.3 Ideas which are central to the work of soh, Berger and Dabke (1985)

can be traced back to Falrr and Meditch (1978)'

NRL 1.4 At the levei of muitilinear uncertainty structures, impetus for much

work along both vaiue set lines and rnathematical prograrrrning lines v/as pro-

vided by saeki (1986) in his revival of the Mapping Theorem, which lay dormant

since its 1963 publication in the book by Zadeh and Desoer (1963)

NRL 1.5 Formulation and solution of optimiza.tion problems obtained frorn the

Hurwitz minors is found in the work of Sideris and sanchez Pera (1989)

NRL 1.6 FYorn a mathematical point of view, the value set corresponds to the

range of acomplex-walued function. For example, if X C R" and f : X - C'

then the ralge /(x) : {f (") : r e X} is the set of complex va}ues which can be

assurned by /.

NILL 1.2 Value sets associated with rational functions are called templates in

the original work of Horowitz; e.g., see Horowitz (1963) and Horowitz (1982).

NRL 1.8 The contributions of Dasgupta (1988) and tvlin-nichelli, Anagnost and

Desoer (1989) set the stage for a number of ideas in this text' These papers

simply explain Kharitonov,s Theorern in terms of value set rectangulality and

motiwate the following question: To what extent is the value set concept useful in

the attainment of robustness results for more complicated uncertainty structures

beyond those considered by Kharitonov?

I{FLL 1.g One of the first demonstrations of the power of Kharitonov's Theorem

is given in Barrnish (1983); the result of Guiver and Bose (1983) is extended

frorn quartics to pol5'nomials of arbitrary degtee'

18 Ch . 1 / AG loba lOve rv i ew

N R L l ' l o D a v i s o u i s o f b e n c r e d i t e d a s b e i n g t h e f i r s t t o r e g u l a r l y e m p l o y t h e

word "robust" in the control iiterature; e g'' see Davison (1973)'

N R L l ' l l l n t h e s i x t i e s a n d s e v e n t i e s ' w e s e e a l i n e o f r e s e a r c h a t t a c k i n g r o -

;;;;.;";.;Iems via set propagation using the svstem dvnamics The textbook

by Schweppe (1973) consolidates much of this literature Exam'ples of more work

c o n t i n u i n g a l o n g t h i s l i n e i n c l u d . e t h e p a p e r s b y K u r z h a n s k i i ( 1 9 8 0 ) a , n d F o g e i

and Huang (1982).

NRL 1.1-2 A reconciiiation between the mathematical programming and ana-

l y - t i ca l app roaches to robus tness i s roo ted in theno t i ono fa ta i l o reda lgo r i ' t hm '

To explain this idea' we consider a robustness problem and ewaluate two choices

for its solutio rt. choice l: Reformulate the problem in mathematical program-

ming and "blindly" apply some standard software package hoping that some

reasonable engineenng solution is obtained' The word "blindly" is used above

because apriori results about Iocal versus global extrema are not establisired and

thesys tems t ruc tu re (unde r l y i ng to theop t im i za t i onp rob lem) i sno texp io i t ed .

choice 2: create a ratrrer speciaiized mathematicar prograrnming code which

expl ic i t ly takesadvantageofsysterrStructureoranalyt ical resul tsthatrnayon]y

hold for some idealized version of the problem at hand' The next two notes

ela.boratt: on these choices

NRL l . lSTop rov ideanexamp ]e i l l u s t r a t i ng thed i s t i nc t i onbe tweenCho i ces l

and 2 above, we mention the theory in Chapter 14 clealing with systerns having

a rnultilinear uncertainty structure. Making choice 1 in analyzing such systems

amounts to application of a rnathematical programrning package rvithout specif-

ica.lly exploiting the multilinear structure' On the other hand' if we are aware of

the Mapping Theorem, we can exerclse Choice 2; i'e'' we develop a specialized

cotle which exploits the convex hull approximation which the Mapping Theorem

provides. This is n'hat is meant by a ta'ilored algorithm"

NRL 1.14 A seconcl exarrrple illustrating the notion of a tailored algorithm

arises in pr theory' By using convexity properties associated wittr bounds for

the p problem, one ofben obtains a better numerical algorithm than would be

poss ib l ew i t hou t t heexp lo i t a t i ono f t h i s i n f o rma t i on . I nsummary , i t i s f e l t t ha t

onefru i t fu]d i rectronof futureworkinvo]vestheintegrat ionof themathemat ical

prograrn-rning approach and the analytical approach One incorporates analytical

results into the algorith:rric steps with the goal of improving the efficiency of

comPutation.

: i

, i

II

ii l

iIl,tiI

t l' I

i ir ll 1

! j

i i

Ir il . il i

i ti !1 l

r 1i : l. l

l ir i

' l li l

:: L

' i. l

: : i

r , lIx

r ' li 1

i j i! : lI iLt .

Chapter 2

Notation for IJncertain Svsterns

Dynopsls

This chapter introduces a rather minimal set of notation whichis carried throughout the text. Exantp)es a"re given to illus-trate the process of reformulating robustness prob)ems usingthe given notatton.

2.I Introduct ion

The primary focai point of this text is robustness problems involv-ing real pararnetric uncertainty. our objective in this chapter is toprovide a notational system which facilitates mathematical manipu-lations involving frrnsf,l6as of uncertain parameters. In most books

2.2 Notation for lJncertain pararneters

we use the notation q to represent a vector of real uncertain pa-rameters with z-th component q. '\A/s

often refer to q simply as theuncerta'inty. If the uncertainty is /-dimensional, it is often convenientto descr ibe q by wri t ing q : (qt ,qz,. . . ,qr) , whereas in other cases!

20 Ch 2 / Notarion for Uncertain Systems

we take q to be a column vector' In either event' we write q € Rl and

it is clear from the context whether an /-tupie or a column vector is

intended.Throughout thetextweencountervar iousuncerta inquant i t ies

which depend on q. To emphasize the dependence on q' we include

n u" * arg]r*e.rt-of various functions of interest' For example' as

mentioned above, to represent a transfer function with uncertain pa-

rameters, we write P(",s) instead' of the usual P(s)' If numerator

anddenominatorof th is t ransfer funct ionareofconceln 'weempha-size the dePendence on q bY writ ing

P(s . o ) : { ! ' ' qJ .

r \ u ' Y . / D ( r , q ) ,

where l/(", q) and D(s, q) are polynomials in s with coeffi'cients which

depend on f. In many cases' rve break things down to arl even

finer level. For exampl", to de"ote dependence of the coefficients of

l/(", q) and D(s, q) on q, we can write

l I ( " , q) : ion(q) ' "i :Q

I

l

iItiiI i

iliii i

i;i[11tiiil[fiiiiii;tj{

and

D ( " , q ) : f a ; ( q ) s ' 'L : V

Literally dozens of additional examples iliustrating q notation

can be drawn from linear systems theory; e.g., if a linear system

has a trad.itional state space representation i(i) : An(t)' we can

emphasize the dependence on g by writing

i ( t ) : A ( q ) t ( t ) .

Finally, note that we generally append the word "uncertain" to var-

ious quantities which-depend on g' For example' we refer to an

.un""riuin' plant P(s, S), to uncertain poilmornial N("' q) or an un-

certain rnatrix -A(q).On some occasions, it is convenient to introduce a secon'd uec-

tor of uncer-ta'i'n parameters r which is distinctly different from q'

To ill.ustrate' suppose that we wish to di-fferentiate between uncer-

,u,in p^rt-eiers-which enter the numerator of the plant versus the

denoLinator. In such a situation, we write

P(",q, t ) : #3,where N(s, q) and D(s, r) are uncertain polynomials'

I 9

2.3 ,/ Subscripts and Supericr ipts 2l

/2 .3 Subscr ipts and Superscr ipts

When working with the uncertain pararneter vectors q and r, it is of-ten cor:.venient to include zeroth components qs and r0, respectively.For example, we might focus attention on the uncertain polynomialp(s,q): eo * qts * qzs2 with q € R3 or the uncertain plant

in,,'P(t , q , r ) : o io

L'otni :0

with q 6 11m+r and r € 11n*1.If we want to designate a vector as being distinguished in some

sense, we use superscripts. For example, we might refer to the un-certain parameter vector q* having i-th component qi.

2.4 LJncertainty Bounding Sets and Norrns

For robustness problems) we often assume an apriori bound Q fortlre vector of uncertain parameters q. We call Q t};.e uncertaintybounding sef. Nlotivated by classical engineering considerations, wegenerally take Q to be a ball in some appropriate norm-usually(but not necessarily) centered at q - 0. The two most importantnorrns we consider are L- and 12. In the (* case, we consider them.aT, norTrl

l lq l l * : max lq i l .

We refer to a ball in this norrn as a bor. For exarnple, to describea box of unit radius with center e*, wa write l lq - ?-l l- < 1. Oftenwe lvant to describe such a box via componentwise boundsi e.8.,consider

Q : { q . € R l : a i < q t ( q o + f o . i : 1 , 2 , . . . , t 1 ,

where a1 and q{ are the specified bounds for the z-th component q.;o f q .

For the (2 case, we consider the standard euclidean norrn

22 Ch.2 / Noution for ljncertain Systems

Hence, a ball of unit radius and center q* is d'escribed by the inequal-

itv l lq - q. l iz ( 1 and is referred to as a splt 'ere' On a few occasions

t'ir,oj. ;;;-*; exPloit the /i norm

!.

l lq i l ' : I icu lZ: I

and refer to a ball in this norm as a d''inmond" Analogous to the l@

ard 12 cases' tf'" uufi of unit radius u'"J """te'

q* is described by

lls - qllr 5 lu.l.o

"o"'ider weighted' versions.of

^t:I^,:i-:ne norms

above. For example, if ur1 J-t)2 j-. . .rt)t are positive weights, then we

can describe the unit ball in the associated euclidean norm by the

inequalitY t

L,-7q? s tx : I

and refer to this set as an ellipsoid'

2'5 Notation for Farnilies

2'6 lJncertain l\-rnctions Versus Farnilies

l t ^ i l ^ -| \ 1 | Z

-

2.7 / Convention: ReaiVersusComplex Coefficients 23

F r c unn 2. 8. 1 i:"-rr!J,k""J^^.frt "rrcuit

for Exampre 2. 8. 1

2.7 Convention: Real Versus Cornplex Coefficients

Throughout this book, the standing assumption is that all polyno-

mials have real coefficients. On those occasions when we wish to

assurne complex coeffi.cients, we state this assumption explicitly. For

example, we say, Iet p(s, q) be an uncerta'in polynornial tuith compler

coeffici,ents. In this regard, our point of view is as follows: In rno-"t

cases, the results given here are easily modified to obtain complex

coefficient analogues. There are, however, some exceptions, aud in

addition, there are some resuits which holcl for: tlie complex case but

not the real case-and vice versa.

2.8 Consol idat ion of Notat ion

We now provide examples and exercises illustrating the transcription

of robustness problems into the notation of this text.

EXAMeLE 2.8.1 (Torque Control of a DC Xulotor): To i l lustrate the

use of g and Q notation, we consider a DC motor driving a viscously

damped inertial load as described in Bailey, Panzer and Gu (1988);

see Figure 2.8.1. The uncertain parameters in the model come from

two sources. Namely, the rnotor constant 1{ (in volts/rps) and load

moment of inertia J1 $n kg-*3) are imprecisely known; i.e., we take

0 . 2 < K < 0 . 6 a n d l 0 - 5 < J r , < 3 x 1 0 - 5 . A s s o c i a t e d w i t h t h eelectromechanicai circuit in the figure is the transfer function from

the armature voltage to the load torque, which is given by

P ( s ) :K(J1s a Ar)

(Ls + R)(Ji-s * J;s * B^ * B7) + K2

: .F( and ez : Jt and fi.xed pa-for the motor mornent of inertia,motor damping, ,L : 10-2 H for

24 Ch.2 / Notation for Uncertain Systems

1 0-5

I 0 -5

FrcunB 2.8.2 The Bounding Set Q for Example 2.8.1

the armature inductance, R : 1Q for the armature resistance andBr:2 x 10-5 N-m/rps for the load damping, we obtain a family oft ransfer funct ions P: {P( . ,q) : q e Q} descr ibed by

O.Sqtqzs * 10-5q1P(s , q ) :

(10-5 + 0.005q2)s2 + (0.00102 * 0.5q2)s * (2 x 10-5 + O.Sq?)

and uncertainty bounds 0.2 < q1 ( 0.6 and 10-5 1 qz < 3 x 10-5.

In addition, note that the uncertainty bounding set Q is a rectangle

in R2 as shown in Figure 2.8.2.

ExAt\4pLE 2.8.2 (Performance Specifications): We consider a fam-

ily of transfer functions P : {P(',q) | S € Q} and turn our atten-

tion to the frequency lesponse. For the uncertain transfer- fulction

P(t,q), the uncertain log ga'in in decibels is given by

LG(w' a) : Z1loglP ( j a ' q) l '

Now, a typical perforrnance specification P might be as follows:

Giveii a band of frequencies [r.,,-, cu1] and two functions of frequency

LG-(':) and LGa(u), we seek to guarantee that

Taking uncertain parameters q1rame te rs J^ :2 x 10 -3 kg - . t t 3B^ : 2 x 10-5 N-m/rps for the

..tij

l i

i : l , r ; i

[,,'iii.'itii.tf' il

LG-(u) < LG(u, q) < LG+(a)

2.8 ,/ Consolidation of Notation 25

Log gain

LGa@)

allowable band for LG(t't,q)

Ftcunp 2.8.3 Allowable Band for Example 2'8'2

for all q € Q and all a € la-,c., '1]; see Figure 2'B'3' When these

perforniance inequalities are met, we can say that the performance

specification P is robustlE satisfied'

ExEF-crsE 2.8.3 (State Variable Formulation) : Consider the un-

certain state variable system described by

i l ( t ) : ( r + q1)r1( t ) + (qr i qz + a) t2( t ) ;

r 2Q) : ( 3+q2 ) r1 ( t ) + (q t - ez *z ) r2 ( t ) + (2+ q t )u ( t )

with uncertainty bounds lqtl < 1 and lq2l < 1'

(a) Describe the uncertain matrix A(q) and uncertain vector b(g)

associated with the state variable representation

h ( t ) : A (q ) r ( t ) + b (q )u ( t ) '

(b) Consider the outPut

Y(t ) : c(q)r ( t )

With c(q) : [0 l* qz), f ind the uncertain transfer

P ( t , q) : c(q) ls l - A(q) l - \b(q)

26 Cjn.2 / Nolation for Uncertain Systems

for the system.

EXERCTSE 2.8.4 (Electrical circuit): consider the circuit of Pe-

tersen (1988) shown in Figure 2'8'4 with 0 : I and state variables

Frcunp 2.8.4 Electrical Circuit for Exercise 2'8'4

r1 ( t ) : u1 ( t ) , r 2 ( t ) : uz ( t ) - r r ( t ) ' 4 ( t ) : u3 ( t ) and ra ( t ) : i 1 ( t ' ) '

Generate state equations which include the tu'o resistances as uncer-

tain parameters' IJse qt: I lRt and q2 : l lRz and show that the

uncertain PolYnomial

p(s,q): s4-l- (3 - 2q't + q2)s3 + (4 - 7qt * 4qz - Aq,qz)t"

+ (5 - lrqr - 6qz - Sqsz)s * (1 - 3qt +Zqz - qtqz)

dictates the poles of the sYstern'

Wenowconsiderasecondorderai ' rcraf t t ransferJunct ' ion

P( .s ) : , = , 5 . " ( t9^ " . * t ) ,

Pt - 7s2f uZ) + Q€slu,) + 1

subject to the following uncertainty description: 1{o varies from 0'02

at medium weight "r,ri'."

to 0.20 at lightweight cruise with nominalfor this system.function

s 2 + 6 0 s + 2 5 0 0 0 . 1 6 s 2 * 0 . 2 4 s t 1

(s + 2)2

( s + 0 . 1 ) ( s + 1 0 )

Actuator

2.9 / Conciusion 27

Aircraft

n(")

Filter Rate gyro

FIGURE 2.8.5 Block Diasram for Exercise 2.8-5

value 1{o - 0.1, the natural frequencycu' varies by i70% about the

norninal value cur : 2.5 and the damping { varies by f10% about

the norninal value { : 0.30. The performance specification P is to

have a closed loop system with a pair of dominant poles at natural

frequency close to 2.5 rad/sec and darnping ciose to O.707.(a) Find a setting for the rate gyro gain -I(n which leads to satisfaction

of the performance specification with r'tn, ( and 1{" fixed at their

nominal values. Note that this is a simple root locus problenr whose

solution can be easily obtained using a standard sofbware package.

(b) For the obvious choice Qt : I{o, qz : € and q3 : an, cornpute

the aircraft transfer function P(",q) and describe the uncertaintybounding set Q.(c) For the less obvious choice Qt : I {o - 0.11, qz :2€an - I .5

arrd q3 - a2^- 6.25, compute the uncertain aircraft transfer function

P(",5) and describe the uncertainty bounding set Q.(d) Argue tirat the resulting family of aircraft transfer functions is

the same in parts (b) and (c) above.(e) Is there any reason to prefer the parameterization given in part (c)

to the one given in part (b)? Explain.

2.9 Conclus ion

The notation in this chapter faciLitates presentation of technical re-

sults in the sequei. Of particular importance is the fact that ourpolynomiais and rational functions include the extra argument q to

emphasize dependence on the uncertainty. Using this extra argu-

28 Ch.2 / Noradon forUncertajn Systems

ment, we can describe many quantities of interest with notationwhich is both compact and aesthetically pleasing. For exarnple,given a bounding set Q, a fixed frequency c..,* € R and an uncer-tain polynomial p(s, q) with coefficients depending continuously onq, the quantity

pna t : ma+ lp ( j u ,q ) l

represents the ma,:<imum attainable magnitude.The next chapter contains no nev/ theoretical concepts. Our

primary purpose is to expose some of the issues which arise whensetting up robustness problems in a real-world applications context.Hence, the reader who is interested primarily in theoretical funda-mentals can proceed directly to Chapter 4 without concern for lossof continuity in the exposition.

Notes and Related Literature

NRL 2.1 The origins of the notational system in this text are rooted in the

quadratic stabilizability literature of the seventies. For exarnple, in the work of

Leitmann (1979), uncertainty in a state space pair is denoted by writing simply

(A, (r), A-B(s)). To allow for cross coupling between the state and input matri-

ces, the notat ion (AA(q), AA(q)) was introduced; " .9. ,

see Barmish (1985).

NRL 2.2 Since the proof modifications for the complex coefficient case are usu-

aliy straightforward, this text avoids double citations of the literature; i.e., one

citation for the first paper to provide a result and a second citation for a rninor

extension to the cornplex case. In fact, a number of the routine extensions to the

complex coefficient case are relegated to the exercises.

Case Study: The Fiat Dedra Engine

30 C]n.3 / Case Study: The Fiat Dedra Engine

3,2 Control of the Fiat Dedra Engine

We consider a model for the Fiat Dedra engine given by Abate and

mance wheh idlingTwo common methods to achieve iow fuel consumption involve

as well as nonLinear effects in the engine which tend to get larger as

Lhe irlle velocity setpoint diminishes. This hampers global stability

of the system. In short, low fuel consumption optimization moves

the engine to an operating point whicir is less stable. This, together

rvith the fact that spark ignition engine models have many unceltain

parameters (such as engine temperature, fuel composition, lubricant

type and atrrrospheric pressure), serves as motivation to carry out

robustness analysis.

3.3 Discussion of the Engine Model

This section can be skipped by the reader who is not interested in en-

gineering details associated with the derivation of the engine model'

in this discussion, we deal r.vith an engine model '"r'hich has been

linearized about the idle operating point. Referring to Figure 3'3i1'

the main control inputs are the spark advance A(s) and tire throttle

valve opening (duty cycle) D(s). The output variables are the rnan'-

ifoid piessure P(s) ancl the engine speed N(s). In addition, the fuel

injection command /(s) is viewed as a reference input.

The rnanifold chamber has a control input D(s) which regulates

the incoming air-flow via a bypass valve. Neglecting any small leak-

age as well as the small fuel flow in the manifold chamber leads to a

si*pie "filling dynamics" equation: The clerivative of the manifold

pr"rrlrr" P(t) is propor:tional to a difference of two air- mass flows;

the incoming air mass flow is denoted by A'["(L) and the outgojng

flow is denoted by IuI6(t).The constant of proportionality -I{- depends on factors such as

manifold volume, atrnospheric pressure, air temperature, gas molec-

Chapter 3i , r

I

SynoPsis

To consolidate tlte concepts and notation of the two preced-

ing chapters, tve ptesettt a case study involving a model of a

Fiat Declra engitte. We also see that a symbdic nanipulation

prograin can be cluite usefd in robustness applicatiotts'

3.1 Int roduct ion

an academic "toy" model.

I-l: l

l

l

23

32 Cil'. 3 / Case Study: The Fiat Dedra Engine

y ' / c \

D(s )

MANIFOLD CHA]VIBER ROTATIONAL

DYNAMICS

N( " )

Frcunp 3.3.1 Block Diagram of Fiat Dedra trngine

ular weights and specific heat parameters. We have

P(t ) : K- l .NI" ( t ) - Mu(t ) )

The incorning flow A:4"(t) is regulated by D(s), whereas the out-going flow lUIb(t) is taken to be proportional to the engine speed^7/^\ mL:^ :^ :--^!:f ied if one thinks of the cylinders as exerting ar Y ( J J . I l u J l D J U b L r

continuous pumping action, and is actually referred to in mechan-ical jargon as a pump'ing feedback Finally, the integration processrelating P(t) to P(t) is taken to have a time constant LlK2 (depen-dent on I{^); and the model weights the effects of D(t) and N(t) onM"(t) and M6(t) by Kt and I{s, respectively. We end up with theexpressron

P(s) : , _ - r [1{1D(s)

- Ksl / (s) ] .

As far as the cornbustion process is concerned, a rather simpletorque production model is assumed. This is justified in large mea-sure by the fact that rotational dynamics filter high-frequency effectsfrom the combustion process. Thus, the variables P(s), -A/(s), /(s)and A(s) are assumed to contribute linearly to the output torque Qaccording to the equation

T.(s) : I { 6A(s) * e- sra lK aP (s) + / (s l l (s) + 1(yF(s)1,

where I{q,, I{s and 1{7 are the constants of proportionality and e-"d"represents the induction-to-power-stroke delay. This delay is asso-ciated with the fact that changes in the rnanifold get reflected in

3.3 ,/ Discussion of the Engine Model 3l

A(")COMBUSTION

p1"1 Fuel" i

'Control

P ( " )

^r(")s ) lD (

Ftcunp 3.3.2 Two-Input/Two-Output Motor Configuration

the combustion chamber oniy after the cylinder goes from the intakestroke to the power stroke. An approxirnate expression for z4 is givenin Hazell and Flower (1971): It is

120' a -

r , ,n,- ly

where r4 is the delay in seconds, n. is the number of independentlyfired cylinders and l/ is the idle setpoint speed in rpm. In this casestudy, we concentrate on the "high idle" setpoint and consider theiase when rd + 0. The remaining input in the combustion block isthe spark advance; its action is viewed as instantaneoris since it actsdirectly at the cylinders.

The rotational d;'namics are described by the basic equation

t/(s) : =:--r-tr"(") - "2,(s)1,J A - 1 _ I \ 7

where J is the moment of inertia, ft,(s) is an external torque loadand K7 is a viscous friction attenuation constant that depends onengine temperature, lubricant type and wear of the engine.

Finally, to complete this discussion, we note that the modelassumes a consf,anr air-to-fuel ratio fixed at the stoichiometric walue.This leads to setting KJ :0 and a final model with only two inputsA(s) and D(s) and two outputs P(s) and } / (s) ; see Figure 3.3.2.Note that the four transfer functions associated with G(s) are notgiven in explicit form. We now concentrate on this point.

Motor G(s)

\

3.4 / Transfer Funclion Matrix for the Engine 33

3.4 TYansfer Function Matrix for the Engine

The objective in this section is to derive a transfer function matrix

representing the engine model' To this end, we write

34 Ch.3 / Case Study: The Fiat Dedra Engine

uncertain parameters. To summatize, the transfer

of the engine is a 2 x 2 matrix G(t,d with entries

-q3q6

function matrix

ezs2'1(qrq, + qs)" -F (szq++ q2qs) '

qr(q7s + q5),

q6(s + s2) ,

qrq4

In accordance with Fiat specifications, there are various oper-

ating conditions of interest which can be expressed in terms of the

uncertain parameters qi. The description of these operating points is

given in Chapter 10 when we return to this example to demonstrate

robustness analysis techniques.

3.6 Discussion of the Contro l ler Model

The controller used here was originally designed by linear quadratic

speed error is used for steady-state purposes and the "derivator"

in ttre spark advance path guarantees no steady-state offset' The

elimination of this offset is a design requirement.

We describe the controller by a 2 x 2 transfer function model;

i.e., we write

[ ' t" l ILP( " ) l

[4' ' (") g'r(")I

Lgzr(") srr3) )

9r r (s , q ) :

9n(s , q1 :

g n ( s , q ) :

9 z z ( s , q ) :and exploit Figure 3.3.1 to obtain the g;i(s) above' Indeed, four

Iengthy but straightforward applications of Mason's Rule leads to

P / " \ '

s,.r(r) : -#

lr("1:od t J t '

-K{{a: t

P / c \ '

s r2 (s ) : * l r ( " t : ou \ 5 ) '

: ,

, \ N(" ) t9zt1s1 : -77:

lD(s):04 \ D J '

s22(s) :l /(r) rf f i lat" l :o

K6(s -t K2)

lt' + OX, -l Kz - f<s)" + (KzK+ * KzKz - KzKs)

Kt Ka.

Js2 + (JKz + I{7 - K5)s * (K3Ka -t K2K7 - KzKs)

For this engine mod.el, the gains Kt, Kz,' ' ' , Kr and the inertia '-I

are viewed. as the uncertain parameters which affect the robustness

of the system.

3.5 IJncertain Pararneters in the Engine Model

To conform with our standard q notation for uncertain parameters,

we take qr : Kt, Qz -- Kz, Q3 : K3, Q+ : K4, Qs : I{z - Ks,

qa : I{a'arld q7 : ,/. we take advantage of the fact that l(5 and

k7 **iys enter G(s) as a difference; this reduces the number of

To find the entries h;i@) of -EI(s), let kii d'enote the (i, j)-th entry of

the 2 x 4 gain matrix K6 in Figure 3.6.1. From the block diagrarn,

r - l f

I a ( , ) | - - , . I p ( , )| | : f r ( s J ILr(") l L ' (")

xo(s)

36 Cln.3 / Case Study: The Fiat Dedra Engine

3.6 / Discussion of the Controller Model 35

n ( s ) . Afunction

kes * (ktzkza + 0.05kr2 - Irukzz) .s * k 2 a + 0 . 0 5 'Ar'r0Ds h21(s ) :

hzz(s) _ krss * (ksk2a- kM,kzs + 0.05k13 + krr)s * k z + + 0 . 0 5

(knkzq - kuka. + 0.05,k11)

s ( s * k z a + 0 . 0 5 )

Finally, to complete the description of the controller, we provide theo q i n m n f r i v

I o .ooai 0 .1586 o.o8T2 -0.1202- lK c : I I '

10.0187 0.0848 0.1826 -0.0224 )

Now, by substituting for the k6r above, we obtain the entries heiG)

in the controller transfer function matrix fl(s). We obtain

FIGURE 3.6.1 Controller for the Fiat Dedra Engine

we obtain the relationships

k.D(s) : Tr(" )

- t k12p(s) * f t13n (s) * kyXp(s) ;J

t-"uo(") : Y!n@) * k22p(s) * k23n(s) -t k2axp(s)

D

: " f o ' o 5 r / " 1 .

s ' ^ \ " . / t

1xp(") : - ir(").

We now solve for A(s) and D(s) in terms of p(s) andIengthy but straightforward computation leads to transfermatrix entries

r:i' (,-\.-.1 h";, I d,ii \ :

(r, ' \ o

{ .,J

or;'l1'

, t t :I t r* r:-.,f & .7-tF-.

h 1 1 ( s ) :

h 1 2 ( s ) :

kzzss l k z + + 0 . 0 5 '

kzzs * kzt

s * k z + + 0 . 0 5 '

0.1586s + 0 .0145703hz t ( s )

s * 0 .0276

0.0872 s2 + 0.0324552 s + 0.00247 13h22(s ) : s2 + o.o276s

3-7 The Closed Loop PolYnornial

From linear systems theory (for example' see Chen (1984))' closed

loop stability considerations lead us to study the numerator of the

uncertain rational function

R( t , s ) : de t l l * G (s , s ) f / ( " ) ]

Now, using the expressions found for gai(s,q) and hii@) above, a

lengthy computation is required to obtain the desired closed loop

polynomial. This is readily accomplished via symbolic computation.

The end result is the desired numerator polynomial written in the

38 Cin. g / Case Study: The Fiat Dedra Engine

3.8 / ls S)rynbolic Computation Really Needed? 37

form p(s, q) : DLo a;(q)si.For the reader interested in verif ication,

the extremely lengthy formulas for the ai(q) are given in Appendix A'

The conclusion to be drawn from the appendix is that without a

symbolic computation package, the task of generating a closed form

for p(s, q) would be monumental.

3.8 Is Syrnbolic Cornputation Really Needed?

As dernonstrated above and in examples and exercises in chapter 2,

the uncertain functions which we wish to study are often derived

frorrr ll.- ore basic uncertainty descriptions. For example, when a state

variable system is described via an uncertain matrix A(q), then cal-

culation of an explicit expression for the characteristic polynomial

p(s,q) : det (s I - A(q)) involves considerable a lgebra. . In fact , for

practical applications of interest, the derivation of explicit expres-

sions for the required uncertain polynomiais can involve literally

hundreds of terms-often thousands. A dramatic illustration of this

point is provided by our case study involving a Fiat Dedra engine'

As seen in Appendix A, the uncertain polynomial of interest is quite

complicated.This leads us to consider the followi[g c$restiorr: In order to

carry orit a robustness analysis, do we reaIIS' need to obtain explicit

expressions for the uncertain functions of interest? For example, in

arralyzing the stability of a famity of matrices {A(q) ' q € Q} via

its uncertain characteristic polynomiai p(s, g), do we really need to

carry out the determinant calculations symbolically in (s, q)? The

reader who is unconvinced about the importance of this question

is urgecl to consider an uncertain 4 x 4 matrix ,4(q) with a simpie

uncertainty structure-say each entry of -4(q) depends linearly on

three uncertain parameters q1 , Q2 and q3 and the objective is to

derive a closed form for the characteristic polynomial'

In principle, the application of the robustness tools in the chap-

ters to follow do not require explicit expressions for the uncertain

functions of interest. what matters is the ability to carry out re-

peated evaluations. To illustrate, suppose that we are interested

in performing a robustness analysis involving the n x rz uncertain

rnatrix ,4(q) and the tool being applied requires evaluation of the

characteristic pol5'nomial p(s, Q) : det (sI - A(q)) for

( " , g ) : ( " r , q t ) , ( t r , q ' ) , . . . , ( " , n r , q t ) -

Not ice that evaluat ion of p(s,q) for (s ,q) : (s , ,q ' ) does not nec-

essarily impty that an explicit expression for p(s, q) is needed' For

Ftcuns 3.9.1 Standard Feedback Configuration

each (s;, q") combination, one can compute det (sa-I-- A(qn)) without

going through the intermed'iate step of symbolically computing the

lo"d"i".tt, l tpi",q). This argument' however, should not be erro-

neously construed to mean that a symbolic manipulation package is

not hclpful. It may weII be the case that the required determinant

calculations above are ,,expbnsive" for certain combinations of n and

l/. For such cases, an overall com

working with a sYmbolic rePresenl

bwen if the initial computational c'

the savings accrued when evaluati

effort worthwhile.

3.9 Syrnbolic Cornputation with tansfer F\rnctions

Recaii that the starting point in many robustness problems is an un-

cer ta in p lant descr i ; ; ; ;v PG,s) : 'A/ (s , q) lD(s,s-) ' *1" . t " N( t ' s)

and D(s, q) are uncertain poiynomials' S"ppose that this piant is

connected in a feedback configuration with u "orl1putt".tor

denoted

u"Zf"l : Ns@)lD6'(s), wheie Nc(") and D5:(s) are fi;<ed polvno-

mials; see Figure 3.9-' i. 'Now, in order to provide a closed form for

the closed looP transfer function

i ,Lii'i,ii:,,i1

[.iiit,,,1 i: ,r i , t i;,:, ,j

i.' iit,i.;,il : ,: Illo,i

P( t , q )

Pcr(s, q) :1 + P(s, q)C(t) l / (s, q)-n/6'(s) + D(s, q)DcG)'

a syrnbolic computation in (s, q) is required. In some cases, only partof P6y(s, q) might need to be computed. For example, rvhen studyingstability of the feedbacl< system above, the uncertain polynomial

p ( s , q ) : N ( s , q ) i / c ( s ) * D ( s , q ) D c G )

is the synbolicaily computed quantity of interest.

ExpRCrsE 3.9.1 (From Control System to Polynomial): Recallth.e torque control problem for the DC motor in Example 2.8.1 andconsider a controller given by C(s) - K1 * K2f s and nurnericalvalues g iven by J^ : 2x 10-3, B* :2x 10-s, L : I0-2, ,R : 1 andBL:2 x 10-5. Obtain explicit formulas for the coefficients of theuncertain polynomial p(s, q) which determines the closed loop poles.Express this uncertain polynomial in the form

p(s , q ) : oz (q ) t2 + a1 (q )s + ao (q )

and note that the formulas for the a;(q) are parameterized in thetwo gains Ifi and K2.

3.1O Conclus ion

This chapter cornpletes Part I of this text. We are now well pre-pared to proceed toward the attainment of technical results. Afterintroducing the notion of robust stability in Chapter 4, we reachthe first mountaintop in Chapter 5-Kharitonov's seminal theoremfor robust stability of interval polynomials. Through understand-ing of the key ideas associated with Kharitonov's simple but elegantinterval polynomial framework, it becomes possible to introduce anumber of important technical ideas which are essential in the laterchapters.

40 Ch. 3 / Case Study: The Fiat Dedra Engine

Notes and Related Literature

NRL 3.1 A more detailed discussion on the derivation of the engine model

is given by Powell (1979) and Dobner (1980) where much ernphasis is piaced

on obtaining a complete nonlinear formulation. work aimed at finding a more

simple linear model can be found in the papers by Dobner and Fluechte (1983)

and Powell, Cook and Gtizzle (1987).

NRL 3.2 The problems encountered in idle speed engine control are explained

more extensively by washino, Nishiyama and ohkubo (1986), Nishimura and

Ishii (1986), Yamagushi, Takizawa, Sanbuichi and Ikeura (1986)' and Ando and

Iv lotomochi (1987).

NRL 3.3 The paper bv olbrot and Powell (1989) provides a nice analysis of the

stability problem associated with idle speed control'

P ( t , q )

3.10 / Conc lus ion 39

N ( s , q ) D 6 ( s )

l r l

i; ii:r l

i,"] : l : ]

t , , .' 1

i , 't .l , l

i ' . 1l : , 1

i r i , i1 r ' l

t i ' li . ; i li lr' i '

l i , '

Part II

Chapter 4

Robust Stability with a Single Pararneter

SynoPsisFYom Robust StabilitY to the

Value Set

4.a Stability and Robust Stability '-l

In this chaPter, we begin exPositir

trating on the robust stability prot

obtained with onlY one uncerllaln

of reasons for restricting attention

setting before Proceeding to the I

gogicJ point of view, it is felt tha

ir.iHttt". understanding of the r

common sense takes us a Iong wq

involved. Second, for the case of a

stronger results than in a more ge

mostlmPortant reason is the fac

of robusi stabilitY Problems with

can be reduced to the single ParaI

arises when the uncertaintY boun

cients of an uncertain PolYnomial

42

4.2 / Basic Definitions and Examples 43

under these conditions, the Edge Theorem (see chapter g) enablesus to reduce the rnultiple-parameter problem to a fi.nite number ofsingle-parameter problems.

4.2 Basic Definit ions and Examples

For the sake of completeness) we now provide two basic definitions.

DEFrNrrroN 4.2.1 (Stabil lty): A fixed polynomial p(s) is said tobe stable if all its roots lie in the strict ieft half plane.

DEFrNrTror-t 4.2.2 (Robust Stabil ity): A given family of polyno-mials P : {p(.,q) : q e Q} is said to be robustly stable if., for allq € Q, pG,q) is s table; that is , for a l l q € e, a l l roots of p(s,q) t iein the strict left half plane.

ExAI\rpLE 4.2.3 (Commonsense Analysis): A family of f i.rst orderplants, characterized by uncertainty in the location of a simple pole,is descr ibed by P(s, q) :7 / (s-q) and uncer ta inty bounding lq l < 2.When the system is compensated with a unity feedback C(s) : 1, *"obtain the closed loop polynomiaip(s, q) : s+1-q. For this systern,p(s,q) has a single root sr(q) : -1+ q. Clearly, the resulting familyof polynornials P is not robustly stable because for q ) 1, st (g) Iies inthe right half plane. More generaliy, with uncertainty bound lql < ,,it is easy to see that P is robustly stable if and only if r < 1.

ExArvTpLE 4.2.4 (Slightly Nlore Complicated): For the second. orderuncertain polynomial p(s,q) : s2 * (2 - q)" + (3 - q), coefficientpositivity considerations lead us to conclude that robust stability isguaranteed if and only if the uncertainty bounding set e is containedin (-oo,2). For i l lustrative pu.rposes, we also study robust stabil ityusing the quadratic formula. Indeed, we obtain two roots.

r| ( - r + s / 2 ) + j ( , / 8 - / 2 ) i f 0 < q < 2 J z ;s , t ( o \ : {

'

|. (-t * s/2) + (t/P - s/z) ir zrt I q I 4,

which are plotted for Q : (0,4] in Figure 4.2.1. Flom the figure, it isobvious that the resulting family of polynomials p : {p(. , q) : q e e}is not robustiy stable.

trxERCrsE 4.2.5 (Robustness Margin): Consider the uncertain

44 Ch. 4 / Robust Stabiliry with a Single Parameter

Ftcune 4.2.1 Root Locations for Example 4'2'4

polynomial

p(s, q) : s3 - l - (2 - q) t ' + (3 - q)s + 4

without any specified uncertainty bound for g' Lettin1 Q, - l-','1,

show that the robustness margin

rman: sup{r : p(s,q) is s table for a l l S € Q"}

is given by r*o, = 0.43.

4.3 Root Locus AnalYsis

texts, involves creation of a fi,ctitious plant'

EXAMPLE 4.3.1 (Root Locus): we consider the uncertain poly-

nomial p(s,q) : s2 I (2 - q) t + (3 - q) in Example 4 '2 '4 and not ice

that p(s, q l . t " be wr i t ten as p(s, q) : ( t2 *2s*3) -q(" - l -1) ' Hence,

- ( s + 1 )

,D T J T d

46 Ch. 4 / Robust Stability with a Single Parameter

4.4 / Generaliz-ation of Root Locus 45

FIcunn 4.3.1 Unity Feedback System for Example 4.3.1

if we consider a fictitious plant

P(s) : ----gl-L-\ / s z t 2 s l 3

with gain q and unity feedback as shown in Figure 4.3.1, robust sta-bility analysis is accoraplished by generation of a classical root locus.If t lre uncertainty bounding set Q is an interval lq-,q+), then we re-strict out attention to the portion of the root locus correspondingto gain q between q- and q+. Note that this unity feedback sys-tern is fictitious in the following sense: The underiying systern whichgives rise to p(s, q) may be quite different from the one depicted inFigr-rre 4.3.1. For example, the polynomial p(s, q) above arises whenanalyzing stability of the unity feedback system with uncertain plant

n , , ( 2 - q ) "r ' \s ' q)

Notice that the fictitious plant P(s) does not correspond to an eval-uat ion P( t ,q) for some speci f ic S e Q.

4.4 Generalization of Root Locus

We now generalize the root locus ideas introduced via an example inthe preceding section. Namely, suppose that p(s, g) has degree n foraII q e Q and uncertain coefficients which are affine linear functionsof a single uncertain parameter q; i.e., the coefficient of s' in p(s, q)has the form

o ; ( q ) : a t q l 0 ; ,

where a4 and B; are real. In this section and the next, we view thedegree requirement on p(s,q) as a mathematical condition; further

i n te rp re ta t i ono f th i s . . i nva r i an tdeg ree , ' cond i t i on i s re lega ted toSection 4.6.

In view of the a.ssumed form for a;(q)' the uncertain polynomial

p(s,q) admits a decomposition of the form

p(s , q ) : po (s ) + qp t ( " ) ,

where ps(s) and p1 (s) are fixed polynomials' -We

can now study

robust stabiiity r.t.irtg u' root locus plot for the fictitious piant

' ? r ( s )P(") : po6

with unity feedback and gain q; see Figure 4'4'L Clearlv' a rrecessary

and suffi.cient condition flr robust stability is tirat the distinguished

portion of the root locus corresponding to q € Q remains within the

strict left half Piane.For the more general case involving more complicated coefficient

dependence on g' q7e can still view the robust stability problem in

a root locus context; i 'e', while sweeping q betwett ?- and q+' one

.o*pl , l t " , roots s1(q) , r r (S) , . . . ,s ' (q) of .p(s, q) Again ' a necessa' ry

and su_fficient condi#n1# rob,rst'rtability is that e3_ch.of these root

b ranches rema insw i th in thes t r i c t l e f t ha , l f p l ane .No t i ce ,howeve t 'that this interpretation is not as nice as the one obtained with affine

Iinear coefficient dependence on q; i 'e', i f the dependence on q 1s

nonlinear, there is nt feedback system associated with the problem

and the classical rules of thumb for root iocus generation do not

appIy.

pr (s )

FrcunB 4.4.1 A Unity Feedback System for the Fictitious Plant

4.5 ,z NyquistAnalysis 47

4.5 Nyquist Analysis

We now describe an approach to robust stability analysis based onNyquist plots rather than root locus plots. As in the precedingsection, we consider the family of polynomials P which is describedby p(", q) : po(s) + qp1(s) and Q : le-,q+]. For simplicity, we take

e- : I and q+ : 1 and make the apriori assumptions that ps(s)is stable and p(s,q) has degree n for ail S e Q. Since stabil ity ofpo(s) is necessary for robust stabil ity of P, the stabil ity conditionwe imposed is nonrestrictive; recall that the discussion of the fixeddegree requirerrrent on p(s, q) is relegated to the next section.

ExERCTsE 4.5.1 (Relationship with Nyquist PIot): With the setupand notation above, consider the fictit ious plant P(s) : 11G)/po(s)and argue that the family of polynomials P is robustly stable if andonly if

P( j . ) / ( - * , - r l u [1 , co)

for all frequencies r,.: € R. In other words, robust stability of P isequivalent to an appropriately constructed Nyquist plot not inter-secting a forbidden portion of the real a;<is. Now, derive'a similarresult for the general uncertainty bounding set Q : lq-',q*1.

ExER-crsE 4.5.2 (Root I ocus and Nyquist): For the uncertainplant

D l ^ ^ \ -r \ D t V )

-s 2 + ( 4 * q ) s * ( 3 + s )

sa + (3 + s)s3 + (5 + q)tz * (2 + q)s + 4

with unity feedback compensator and uncertainty bound lql < f,analyze robust stability by generating appropriate root locus andNyquist plots. Verify that both methods lead to the same conclusion.Subsequently, consider a variable uncertainty bound Q, : l-r, r] anduse your graphical output to compute a robustness margin

rmar : sup{r : p(s,q) is stable for all S € Q"}

for the closed loop system.

4.6 The Invariant Degree Concept

In the preceding two sections, we assumed that p(s,q) has degree nfor all S € Q. Sirnilarly, in almost ail chapters to follow, we impose an

48 Ch. 4 / Robust Stability with a Sinele Paramerer

{ ( i - - , - - ; ^ - + - l ^ - - ^ ^ ) , ^rravar rarru ucbrEs .ssumption rrrnJn"rr", convenient. The ob;ectiveof th.is section is to dernonstrate that such an assumption is ratherbenign. In a feedback control context, we see that an invariant degreeassumption amounts to appropriate properness of the loop functionP(s,q)C(s). We also explain the technical reason for imposition ofinvariant degree conditions.

DEFrNrrroN 4.6.1 (Invariant Degree): A farnily of polynomialsgiven by P : {p(.,q) t q € Q} is said to have .inuariant degree if thefollowing condition holds: Given any gl ,q2 € Q, it fotlows that

deg p(s, gt ) : deg p(" , q2) .

I f , for a l l q € Q, deg p(s,q) : n , then we cal l P a fami ly of n- thorder polynomials. Finaily, if 2 does not have invariant degree, wesay that degree droppi,ng occurs.

REMART{s 4.6.2 (Highest Order Coefficient): If we begin withthe uncertain polynomial p(s, q) : DT:ooo(q)t', notice that 2 hasinvariant degree if and only if a.(q) * 0 for a1l q € Q. We seebelow that the invariant degree assumption is intimately related toproperness of feedback control probiems.

ExERCTsE 4.6.3 (Invariant Degree in Feedback Systerns): Considela family of plants 2 described b1'

n, . l / (s , q)P \ s , q ) : ; *D l t ' q )

with uncertain;' f6gttd q € Q and l/(s,q) and D(t,q) being uncer-tain polynomiais rvith D(t,q) monic. Now, given a compensator

^, \ l /c(s)u (s, : D;6

which is connected in a feedback configuration, prove that if the un-certain loop function P(s, q)C (s) is strictly proper, then the resultingfarnily of closed loop polynomials described by

p(s, q) : l / (s , c t )Nc(s) - t D(s, q)DcG)

and q € Q has invariant degree. For the case when P(s, g) andC(s) are both proper but not strictly proper, and D(s, q) is notnecessar i ly monic, le t a, , (q) , b-(q) ,

"* and d- denote the h ighest

order coef f ic ients of l / (s , q) , D(s, S) , .Nc(s) and Do(s) , respect ive ly .

',ijri,j. . 1' :1

lI

il: r 1

'.1

4.6 / -the

Invariant Degree Concept 49

Now, assuming a'(q) and b.(q) depend continuously on q and Q is

closed and bounded, argue that the resulting famiiy of closed J,oop

poiynomials has invariant degree if and only if

[ r :6 la- (q)c*- t b-(q)d* l > 0 '

REMARKS 4.6.4 (Technical Role of Invariant Degree): We con-

sider an uncertain polynomial p(s,q), and, for i l lustrative purposes'

suppose that Q : [0, 1] is the uncertainty bounding set' Assuming

p(s,O) is stable, we imagine q increasing from 0 to 1 and obtain a

root locus which begins in the strict left half plane at the roots of

p(",0). If no degree dropping occurs, the nrrmber of branches of this

root locus r-emains constant. Hence, the only way that we can ex-

perience a transition from stability to instability is by having one

o.r nlore br:anches cross the imaginary axis. On the other hand, if

p(s,q) experiences degree dropping, then as q varies from 0 to 1, the

branches of the root locus can "leapfrog" from the strict left haif

plane into the strict right half plane rvithout crossing the imaginary

axis. This phenomenon is illustrated below'

cu € IL; i.e., p(ja,q) is nonvanishing on the irnaginary axis'

ExERCTsE 4.6.6 (Robustness Margin with Degreie Dropping): To

further demonstrate the subtleties involved in robust stability anal-

ysis with degree clropping, consider the uncertain polynornial given

by p(", q) : (2 + qt)s2 + (5 + aflz)s+ (3 + qr + q2) with uncertaintv

btunding set Q' descr ibed by lq , ; l ( r for i : I ,2 ' Let t ing the

robustness margin for degree k be given by

T ' m a r , k : s u p { r : d e g p ( s , a ) : k f o r a l i Q € Q ' } '

fi.nd r-or.g t Tmat; attd T*or,2.

50 Cin. 4 / Robust Stability rvith a Single Parameler

4.7 Eigenvalue Criteria for Robustness

DEFTNTTToN 4.7.1 (subfamilies): considel the uncertain polyno-

* i r f o t " ,q) : po(") t qpt(s) wi th Po(s) assumed stable -and

the

uncerta inty bounding set Q: [q- 'q* l * i in q- < 0 and q+ > 0 ' We

defi.ne t\e subfamilies

P ( q + ) : { P ( ' , q ) ' 0 < s < s + }

andP ( q - ) : { P ( ' , q ) : q - < s < 0 }

of the original polynomial farnily P : {p(', q) : q e Q}'

DEFrNrrroN 4.7-2 (N4aximai Stabil ity Interval): Associated with

the subfarnlly P(q+) is the right-sided robustness rnargxn

e|o, : sup{q+ , P(q+) is robust l l ' s table} ,

and associatecl rnith the subfamlIV 1Z(q-) is the le,ft-sided' robustness

nLaT-qXTL

Q,nin, : inf {q- , P (q-) is robustly stable} '

SubsequentlY, we call

Q^o r : ( q * ;n ,q . I " r )

ttre man'imal'interual for robust stability'

DEFrNrTrorv 4.7.3 (The Hurwitz Matrix): For a' fixed polynomial

; l :

: . r l r : :

rr. l;l

p ( s ) : a n s n + Q n - r s n - r + " ' + a r s + a o

4.7 / Engenva'lue Criteria fbr Robustness 5l

r,r'itlr ar. > 0, the n x n array

n " nw l a - I q 1 a - , 7 a n - D

A . n A n - 2 & n . 4 " '

0 a r - 1 c l n - 3 c l n - b ' ' '

Q a n ( L n - 2 c l n - 4 - - '

0

0

0 0 ( t r n

0 0 0 " ' a o

is called tlne Hurutitz lv[atrin associated with p(s).

REMAR-KS 4.7.4 (Hurrvitz Stabli ity Criterion): We recall the clas-

s ica l HurwiLz stabi l i t l ' c r i ter ion: A polynomial p(s) above is s table

if and only if a1l principal minors of H(p) are positive. For example,

the first principal minor is A1 : a.-1, the second principal minor is

Ar : det

and the last principal minol is A' : det H (p).

DEFrnIrrroN 4.7.5 (Ah""Q/t) and.\-n,(j1,1)): Given arrraxrl matrix

-Af, we defi.ne Af,-."Q[) to be the maximum positive real eigenvalue

of IuL When ,4.1 does not have any positive real eigenvalues, we take

AL",(A,I) : 0*. Simiiarly, we define \^..(at|) to be the minimum

negative leal eigenvalue of .4r1. When iz1 does not have any negative

real e igenvalues, lve take ) -n, ( ,4[ ) :0- .

THEoREM 4.7.6 (Eigenvalue Criterion): Consider the uncerta'in

polEnon-r , ' ia l p(s,q) : pg(s) + qpt(s) wi ' th p(s,0) : ps(s) s table and

hau'ing pos'it i ,ue coefficients and deg pg(s) > deg p1(s). Then the

man'irnal' interual for robust stabi' l i ' ty i 's descri 'bed by

-fn :lmaI

^h""( - H -1 (po) H (pr) )

52 Cl:,. 4 / Robust Snbility with a Single Parameter

tahere, for tlte purpose of conforrnabi,li,tg of matrir rnultiplication,

H(pr) is an n x n matrir obtained by treating p1(s) as an n-th order

polynom'ial.

4.8 MachinerY for the Proof

The proof of Theorem 4.7.6 is established with the aid of three tech-

nical lemmas. No proof is given for the first two of these lerrrmas

because they are standard results which can be found in books such

as Gantmacher (1959) and lVlarden (1966). The first lemma provides

Orlando's formula and the second lemma is the classical result on

continuous dependence of the roots of a poiynomial on parameters'

This continuity result is used many times in later chapters-both

explicitly and implicitlY.

LEMMA 4.8.1 (orlando's Formula): cons,ider a fixed polynornial

P(s) : ansn + an- lsn- l + " ' + r '1s + @o

with an ), O, roots sl, s2, - . - , sn and Hurwitz matrir H (p) ' Then

d,et H(p) : ( - t ; ' ( ' - t ) /2a7- tao l I (s ; - l - s t ) '1,1 i1k(n

t l t a t s1(q) , "z(q) , .

. . , s . (q) are the roots of p(s, q) '

pRoop: we establish the result for ?(q+) and note that the proof

for P(q-) is identical. Proceeding with necessity, we assume that

p(q*i is robustly stable and simply observe that nonsingularity of

" : . '

" : t r )

i': rii

ii"ij

and

Qmin\ ^o - ( -H . t ( po )H(p t ) ) '

4.8 ,/ Machinery for the Proof 53

H(p(.,q)) for all q € l0,S+] follows from the classical Hurwitz Sta-bility Criterion: see Remarks 4.7.4. Tb.at is, the last principal minoris the deterrn inant o l H(p( . ,q)) , which rnust be nonvanishing.

To estabiish sufficiency, we assume that II(p(., q)) is nonsingularfor all q € 10, q+] and must prove that P(ct+) is robustly stable.Proceeding by contladiction, suppose p(s.q-) is unstable for someq" € [0, q+]. h accordance rvith Lemma 4.8.2, there is a continuouslyvarying root s;. (q) which is jn the open left half plane rvhen q = Q(recall ps(s) is stable) and in the closed right half plane rvhen q : q".By continuity of s;-(g), there exists some 4 e (0, q*] such that s;-(ri)Iies on the imaginary axis. This situation is illustrated pictorially ir:.Figule 4.8.1; the crossing of the axis is apparent.

Ftcunn 4.8.1 Continuous Root Path Crossing Imaginary Axis

To complete the proof, we clairn that 11(p(.,, i)) is singular; thisis the contradiction we seek. This claim is easily established usingOrlando's forrrrula in Lemma 4.8.1. Indeed, there are two possibil i-t ies. The f i . rs t possib i l i ty is that s i (d) :0- In th is case, ao(q) :0,wlriclr forces det H(p(.,d)) :0 via Orlando's formula. The secondpossibil i ty is that si-@) + 0 Since the roots of p(s,Q) appear inconjugate pairs and s1.(f) is purely imaginary, there must exist adifferent root s,r,*(, i) such that s6' (Q) : -t* (ri). This implies thattlre term si"(4) + sk-@) vanishes in Orlando's formula, which again

54 Ch. 4 / Robust Stabiliry rvith a Single Paramerer

f o r c e s d e t , H ( n ( - . 6 \ \ : 0 . E\ r \ 1 a / /

4.9 Proof of the Theorern

To prove Theorem 4.7.6, tve derive the formula for q[o, and simply

note that the derivation of q-,on rcrrs along identical lines. Indeed,

for fi;<ed q+ ) O, it follows from Lemma 4.8.3 that P(q+) is robustly

stable if and only if H (p(' , q)) is nonsinguiar for alt q e [0, q*i . Noting

thatH (p( . , q)) = H (po + qpr) : H (po) + sH (pr) ,

it follows that qfio, is the largest vaiue of q+ such Lhat

de t f I l ( ps ) +qH(p t ) l *0

for a,l l q e (0, q+). Now, since p6(s) is stable, If (po) is invertible and

we can multiply by l{-t (po) lq above and characterize qfi', by the

condition r I I

det | -1

+ H- ' (po)H(pr) l * 0L S I

for all q € (0, s,!."r).There are now two cases to consider.

Case 1: The matrix -p-t(po)H(pt) has no positive real eigenval-

ues. In this case, there is no q > 0 leading to a zero deterrntnant

above. Hence, we obtain q*or: *@.Case 2: The matrix -H-7@o)H(pt) has positive real eigenwalues

0 < , \ f < I t < " < U H e n c e , t h e l a r g e s t v a l u e o f q > O l e a d i n g

to a nonvanishing determinant above is q*or:71^t.

Combining Cases 1 and 2 now yields the formula

-fo :aman Ad^""(-H-1(po)rrkr))

EXI]RCTSE 4.9.1 (Left-Sided and Right-Sided Robustness Margins) :

Find the left-sided and right-sided robustness margins for the uncer-

ta in polynomial p(s, a) : sa + (6 + q)r3 + 4s2 + (10 * q)s -F 8.

4.1o Convex Cornbinations and Directions

In the derivation of analytical results, it is often more convenient to

describe p(s, q) : p0 (s) + qpt(s), the one-parameter uncertain poly-

nomial, using the notion of conuer combinat' ions. With Q : lq-,q*1,the associated family of poll'nomials has entreme po'ints p(s, q-) and

"ii;1', I

ilIll: Il ; ' , ,fr,, Ii i , .-t :[ , , t riar.: . ri . ,f;r :' ':it :

i j ' li a . :iri.,:.' j ; ] :

i t ' ,

i ' ;f , " '[ :i ' ri: '.,8, . ' : .1

i r i i

i : , ' , il : ' i :

i,;i' )i.:l , if ! |

ii: '

l,l? ,

i;:r , :,i _ l i : , , r

1. ,: :1i : li : r i

l . ' . :

t t r . , l

' ir"i

,r !:f

4. l l / TheTheorem of Bia las 55

p(s,q+). Furthermore, given arry q € Q, we can view p(s,q) as apoint on a l ine segment joining p(s,q-) and p(s, q+) in the space ofpolynomials; we express p(s,q) as a conuer comb'ination of p(s,q-)and p(s, q*) by tak ing

!

t - Q ' -

Q

q - - q -

and writing

p \ s , q ) : A P ( s , s ) + ( r - A J p ( s , g ' ) .

Conversely, for every A e [0, 1], there corresponds some g e [s-,q+]such that p(", ,\) : p(s,q). Given this isomorphism between S € Qand A € [0, 1], it is purely a matter of convenience whether we workwith the original fa.mily of poll.nomials or we work with an equivalentfamily P : {p(.,)) : ) e [0, 1]] defined by

p(" , ) ) : )Fo(s) + (1 - ) )Fr( " ) ,

where ps(s) and p1(s) are fixed polynomials. That is, P :P. Ofcourse, the fixed polynomials associated with P are not the sameones associated with P.

ExERcrsE .LO.I (Representation Using a Direction): For thefamily of polynomials described by p(s,q) : po(s) -F gp1(s) and

Q : [ q - , q+ ] , l e t / ( " ) : po ( " ) + q -p r ( " ) and e (s ) : ( q+ - s - )p r ( s )and define

p(s, A) : " f (s) + )e(s) .

Prove that the fami ly of polynomialsP: {p( . , . \ ) : A e [0,1] ] is thesame as the original family. Hi,nt: If q € Q,Iet

A : q , - q

q + - q -

and consider ) e [0, 1].

4.71 The Theorern of Bialas

To consolidate the technical ideas associated with Theorem 4.7.6and the convex combination representation zibove, we relegate thetheorem of Bialas (1985) to an exercise. Note that the proof forthe exercise below is established via a straightforward mirnic of theproof of Theorem 4.7.6. However, care must be taken in makins thp

56 Ch. 4 / Robirsr Srabil ity with a Single Parameter

distinction between the robustness margin problem and the robuststabil ity problem.

I?ExERcrse 4.11.1 (Bialas (1985)): Consider the family of polyno-m ia i s P desc r i bed byp (s , ) ) : ) ps (s ) + (1 - , 4 )p , ( " ) and A € [ 0 ,1 ] ,where ps(s) and pr(s) are fi.xed poiynomials with ps(s) stable withpositive coefficients and n - deg po(s) > deg p1(s). Prove that P isrobustly stable if and only if the matrix H-r(po)H(p1) has no purelyreal nonpositive eigenvalues.

4. I2 The Matr ix Case

In this section, we outline the steps required for generalization ofTheorem 4.7.6 to the matrix case. To this end, we consider an un-certain matrix of the form

A ( q ) : A o * q A t ,

where -46 and ,41 are fixed n x n. matrices. Assuming ,46 is sta-b1e, 'rve define left-sided and right-sided robustness margins as in thepolynomial case; i.e., for the subfamilies of matrices $iven by

and

we define

A ( q + ) - - { e k ) : 0 ( q < s + }

A ( q - ) : { a k ) i q - < q 1 q + } ,

I . ! ' / L \ ,

Qio, : sup{g+ , A(q*) is robusrly stable}

and

Q^o, : inf {q+ : A(q-) is robustly stable}

with the understanding that a family of matrices .4 is deemed stable ifevery .4 € ,4 has all its roots in the strict left haif plane. Equivalentll',the polynornial

P a ( s ) : d e t ( s I - , 4 )

is stabie for all A e A.We are now prepared to expose the "secret" associated with the

matrix case: We convert the robust stability problem for the n x nmatrix family "4 into a robust nonsingularity problem for an n2 x n2

, -+matrlx laml1y -rl' .

in .4+ has the form

4.72 / The Matrix Case 57

Furthermore, a typical uncertain matrix A* (q)

A+(s) -- ,+[ + oA{,

58 Ch.4 / Robust Stabi l i ry rvi th a Single Parameter

T h e n - 4 I B a n d A @ B a r e 4 x 4 m a t r i c e s ' N l o r e o v e l '

A a D -n w D -

5 6 1 0 1 2

7 8 1 4 1 6

15 78 20 21

where AI e\d Ai

straightforward comPutation.are obtained via a

A ^ 1 2 -a v D -

6 2

9 0

0 9

given by the formula

A @ B = A E f n ; r + f n r & 8 ,

where 17" denotes the lc x k identity matrix' Finally' the Kronecker

d.i,fference A e B is given by the formula

A e B : A @ ( - B ) .

REMART(s 4.L2.2 (Special Case); In the results to follow' A is an

rL x rL matrix and wetfb"r, "rr"o,,nter

AO A' Using the formula

A e A : A e I n + I n & 4 ,

it is straightforward to see that A O A is fne n2 x n'2 matrix having

( 2 , j ) - t h b i o c k A l a i i l n f o r i : j a n d a i i l n f o r i f j '

ExAN4PLE 4.J'2.3 (Kronecker Operations): Suppose'

REMARKS 4.12.4 (Eigenvalues Associated lvith Kronecker): In

the sequel, it is useiullo exploit some u'ell-known facts about the

eigerr.ru.lues of l{ronecker products and sums' Indeed, Iet A e Rnl xn1

ancl B eF{n2xn2 and takl.\1(A) and )r(B) to be the i-th eigenvalue

of A and B, respectively' Then the eigenvalues of A A B consist of

lhe n1n2 proclucts of tfr" forrn );r(4)inr(B) and the eigenvalues of

Ag B consist of the 721?22 sums of the form A' ' (A) + Ai \Lt ) '

trxERCrsE 4.!2-5 (Eigenvalue Computation) : With.A and B as

in Example 4.12.3, t-,rJth" remark above to find the eigenvalues of

A @ B and A E lJ. Observe that the solution of tire two required

eigenvalue problems are easily carried. out by hand because -4'and

B are 2 x 2 tnattices. Without expioitation of the rernark, howevet,

tr,vo nruch more diflficult 4x4eigenvalue problerns need to be solved'

ExiiRcrsE 4.L2.6 (\4atrix Problem Is Nlore General): Argue that

the matrix problem of finding Qpin arrd' q*o" is a generalization of the

polynomial probleur. Hl'"t Li1iethat the embedding of polynornial

p0(s) + spl(s) into a companlJn canonical forrn leads to the desired

matrix uncertainty stru"ti.te A(q) : Ao * qAt' Furthermore' show

that this matrix uncertainty stiucture has a characteristic polyno-

mial of the for-m

p ( s , q ) : p 0 ( s ) + q p r ( " ) + q z p z ( t ) + " ' + q " p , , ( s ) ,

wl rere p0(s) , p t (s) , - . - , Pn(s) are f ixed 'o-- l ' ' .1 , B: l t u lL 3 4 l L 7 8 l

412 / The Matrix Case 59

ExERCTSE 4.I2.7 (From Stabil ity to Nonsingularity): Considerthe farn i ly of n x n matr ices A: {A(d : S e Q} wi th Q : lq- ,q+lbeing a prescribed interval and As assumed stable. Letting

A t : { A @ ) o A ( q ) : q € Q } ,

prove that A is robustly stable if and only if "4+ is robustly non-singular; i.e., every A+ € A+ is nonsingular. Hint: Speciaiize theeigenvalue characterization in Remarks 4.72.4 to A(q) O A(q).

REtr4AR.r(s 4.L2.8 (Linkage and Formulae): We are now prepared toderive the desired formulas for the left-sided and right-sided sided ro-bqstness margins qfio, and e*in for the family of matrices ,4. Indeed,in wiew of the reduction of the stability problem to a nonsingularityproblem described above, we simply mimic the arguments used toprove Theorem 4.7.6; see Section 4.9. With (Ao e Ao) + q(At @ At)replacing H(po) + qH(pt), we arrive at the formulas

60 Ch. 4 / Robust Stabil iry with a Single paramerer

which represents the pitching velocity and na(t), which represents thepitch angle. The output of the system y(t) is the forward velocity11 (t) and the input ,r,r,(l) is the elevator angle. For this system, aKalman filter design is used for stabilization. The controller is givenby

" ( t ) : [ 0 .e630 - 14.0e5e 1.1138 11.7e08 ] t ( t ) ,

where i(t) is the state estimate obtained from the equation

i1r; : (A+ BK - Lc)fr(t) + La(t)

with output rnatrix C : lI 0 0 0] , state feedback vector

K : | 0 . 9 6 3 9 - 1 4 . 0 9 s 9 1 . 1 1 3 8 1 9 . 7 9 l

and transpose of the observer gain vector

Lr : lg . r4r7 - o . t r2 l 1 .6171 1.1261 1" .

fo study robust stability with respect to changes in the longi-tudinal static stability derivative, we replace the (3,2) entry of ,4by ozz(q) : -34.72 * q; hence we wr i te ,4(g) instead of ,4 . For thecontroller observer system with closed loop matrix

a ^ - -

find the maximal interval of q for robust stability using the forrnulasfor q^on and qfio, above.

4.L3 fnt roduct ion to Robust 2-Stabi l i ty

In this section, we extend robust stability concepts to allow for amore general root location region. For example, if P is a family ofsecond order plants, a typical robustness specification might be asfollows: Each plant in P should have a damping ratio no larger thansome prescribed ( and a degree of stabil ity of at least o. LettingD(t,S) denote the uncertain denominator polynomial, this specifica-tion is tantamount to a constraint that for each q € Q, the roots ofD(", q) i ie in a region 2 of the sort given in Figure 4.13.1. In each ofthese two examples, we insist that the system has a certain robust2-stability property; see definition to foIlow. By dealing with gen-era|D regions in later chapters, we obtain a unified theory; e.g., the

+Q^ot

Qmin

, t * " , ( - (40 o ,46) -1 ( ,41 e Ar ) ) '

1\ -mG(Ao e .40) -1 (Ar e -a i ) )

ExERCTsE 4.L2.9 (Consolidation) : A state variable model for anA4D jet fighter is given by

-0 .0605 -32 .37 0 .0 32 .2 0 . 0

-0 .1064

- J J . d

0 .0

-0 .00014 - r .475 1 .0 0 .0; . ( + \ - & \ L J - T

-0 .0111 -34 .72 -2 .793 0 .0

: A r ( t ) t Bu ( t ) ,

rvhele the state z(i) has components 11(f), rvhich represents tl iefbrward veiocity, n2(t), which represents the angle of attack, U(t),

[ 'to' BK

LI L C A ( q ) + B K * L C )

u( t )

0 .0r . 00 .00 .0

F lcunn4 . l 3 . l 2Reg ion fo rDamp ingandDegreeo fS tab i l i t yCons t ra in t s

same theory applies to both continuous and discrete-time systems'

In the contrnuous case, it is convenient to take D to be a subset of

the left half plane, r.vhereas in d.iscrete-ti:IrteD is taken to be a subset

of the unit disc. We now provide some basic definitions'

D E F r N r r r o N 4 ' 1 3 . 1 - ( 2 - S t a b i l i t v ) : L e t D C C a n d t a k e p ( s ) t o b ea fixed polynomial. Then p(s) is said to be D-stable if all i ts roots

Iie in the region D.

to be robustlE Sch'ur stable'

4.13 / Introduction to Robust 2 - Stabi l i t l ' 6l

62 Cln. 4 / Robust Stabi l i ry rvi th a Single Parameter

Flcunn 4.13.2 D Region for Dominant Roots and Degree of Stabil ity

a robust D-stability specification? Explain.

4.t4 Robust 2-Stabil ity Generalizations

The main objective of this section is to point out that for polyrromials

with one uncertain parameter, the root locus and Nyquist analyses

of Sections 4.3 and 4.5 admit robust 2-stabil ity generalizations in

a rather obvious way. Indeed, Iet D be a desired root location re-

g ion and takeP : {p( ' ,q) : q e Q} to be a fami ly of polynomials

described byp ( s , q ) : p o ( s ) + q p r ( s )

with p6(s) and p1(.s) being fixed polynomiais and uncertainty bound-

ittg r"i Q' : lqt,s+] We assume that P has invariant degree' Now,

arguing as in Section 4.3' the root iocus plot for

p( " ) : P t l t lPo ( 5.)

tells us everything we wish to know about robust D-stability of P;

i.e., p is rolustly stable if and oniy if the distinguished po-rtion of

the root locus plot for p(s), corresponding to plant gain q e [q-,q+],remains in 2. As far as a Nyquist-iike criterion is concerned, slightly

more thought is required; see the exercise below.

i r l

Ii i ' i llr 'II

i ,

D - D t U D 2 U D 3

4.15 / Extreme Point Results 63

. ExERcrsr_ 4.L4.L (Nyquist-Like Result for Robust 2-stabil ity):

Consider the family of polynomials P above with associated fictitious

plant P(s). In addition, assume po(s) is D-stable and let D c C be

open with boundary 0D. Prove llnat' P is robustiy D-stable if and

only if for aII d e 0D, p(d) does not meet a subset of the real axis

deterrnined by q- and g+. Characterize this "forbidden" subset.

4.I5 Extrerne Point Results

Thus far, we hawe demonstrated that we can readily obtain solutions

to robust stabiiity problems with a single uncertain parameter en-

tering afEne linearly into a polynomial's coefficients. Given that our

criteria inwolve root locus plots, Nyquist plots and eigenvalue com-

putatibn, the following question is natural to aSk: Can we gua,rantee

iobust stability more simply by checking stability of the extreme

points? The exercise below answers this question in the negative for

the general case.

EXERCTSE 4.L5.7 (Stabitity of Extreme Points Do Not Suffice):

W i t h / ( s ) : 1 0 s 3 + s 2 + 6 s * 0 . 5 7 a n d 9 ( s ) : s 2 * 2 s l I a s s p e c i f i e d

in Bialas and Garloff (1985), consider the family of polynomials 2

which is descr ibed by p(s, ) ) : / (s) + )9(s) and . \ € [0,1] . Ver i fy

that the two extremes p(s, O) : / ( " ) and p(s,1) : 9(s) are stable

but p(s, A) is unstabie for a range of ) e [0, t] ; e.g., the intermediate

polynornial p(s,0.5) is unstable.

REMARKS 4.L5.2 (Conditions for Extremality): In view of the

exercise above, we conclude that additional conditions must be irn-

posed on the pair (/(s),g(s)) in order to infer robust stabil ity of

the entire famiiy from stability of the extremes. In fact, the issue of

extreme point results is a main focal point in many of the chapters

to foliow. As we develop new machinery, increasingly sophisticated

extreme point results are obtained. To appreciate the need for ad-

ditional machinery, the ambitious reader might attempt the exercise

below from first principles-

, , ExERCrsn 4.15.3 (For the Undergraduate): Let /(s) be a. stable

polynomial of degree trvo or more and assume that /(s) * (s -F 1) is

stable. Now Prove that

P(s, ) ) : " f (s) + A(s+ 1)

is stable for al l A e 10,11.

64 Cb. 4 / Robust Stability rvith a Single Parameter.

4.76 Conclus ion

In this chapter, we concentrated on one uncertain parameter andobtained three different solutions to the robust stability problem-asolution involving root locus plots, a soiution involving Nyquist plotsand a solution involving eigenvalues. For the more general robustZ-stability problem, the extension of the root locus and Nyquistconcepts was immediate. However, no extension of the eigenvaluecriterion of Theorem 4.7.6 was given. Althougir such an extensionexists, we do not provide it until much later in the text because itrequires a rnuch more abstract level of presentation. Although notrecommended, it is possibie to proceed directly from th.is point in thetext to Chapter 17 and read the D-stabii ity generalization; that is,the tools in Chapters 6-16 are not instrumental to the proofs givenin Chapter 17.

Notes and Related Literature

NRL 4.1 If we consider complex rather than real uncertainty, there is a srm-

ple connection between the ideas in this chapter and the classical Small Gain

Theorem. Indeed, suppose that q is a complex uncertaiu pararneter- s'ith uncer-

tainty bound lSl < 1 Take p6(s) to be a stable complex coeffrcient polynomial

and p1 (s) is another complex coefHcient polynomial ra'ith deg pr(s) 1 cleg pq(s).

Then , w i t h p ( s ,q ) : f o ( s ) + ap r ( s ) , i t i s easy t o sho rv t ha t t he resu l t i r r g f am i lS '

of polynomials P : {p(., q) , S e Q} is robustly stable if and only if the fictitious

plant P(s) : pt(s)/po(s) has 11- norm less than unity. This is the sarne con-

dition q'e obtain using the classical feedback interconnection associated s'ith the

Small Gain Theorem. That is, given the plant P(s) with complex uncertainty q

in the feedback path, the quantity I/llPll* indicates how large lql can be before

instability is encountered.

NRL 4.2 The survey paper by Bres'er (1978) provides a detailed review of

Kronecker operations.

NRL 4.3 For the matrix case, there are many other transforrnations (besides the

Kronecker sum) taking the robust stability problern into a robust nonsingularity

problem. For exarnple, with m : n(n * 1)/2, thele exist many possible linear

mappings T:P.n*n - R-"- which sene the same funct ion as the Kronecker-

sum. For such cases, the eigenvalue problem associated rvith computation ol q^ont t

and ql.. can be significantly smaller; see F\ and Barmish (1988) for details.

Chapter 5

The Spark: I(haritonov's Theorern

bynopsrs

Th ischapter isdevoted to thesemina] theoremofKhar j tonov .Tlte technica.l ideas undeilying the proof serve as a pedagogical

s tepp ings toneforc )eve lopnento f t ] len lo regenera lva ]ueset

""ii"pi u'hich urufies many results in later chaptets' In fact,

the liharitonov recta,ngle which we introduce is actudly a value

set corresponding to a rather specialized uncertainty structure.

5.1 Int roduct ion

The Irain result in this chapter, I{haritonov's Theorem, addresses a

11}ore gener.al r-obustness problems. In a sense, most of the chapters

to follow are testimo[ials to the nerv way of thinking rvhich comes

from the proof of I{haritonov's Theorem'

5.2 Independent IJncertainty Structures

In this section, we introduce the independent uncertainty structure'

Results for this highly specialized structure shor,rld not be viewed as

65

66 Ch 5 / The Spark: I iharitonov's Theorem

an end in itself. With this simpJ.er theory under our belts' however'

we are prepared to deal with more general polytopic and multilinear

uncertainty structures in the chapters to follow.

Ferhaps the rnost compelling motivation for the study of inde-

pendent uncertainty structures is derived from the follorving scenario:

An engineer generates a fixed modei for a control system and obtains

the associated characteristic polynomial p(s). Aithough the presence

of parametric uncertainty is acknowledgecl, the dependence on q is

p(s) can be tolerated.It is also worth noting that in many cases' a more complicated

true robustness margin is 16%. It can be argued that the conser-'-

vatism resulting from overbounding is not critical when the perfor-

mance specification is sti l l met.

DEFrNrrroN 5.2.1 (Independent IJncertainty $1t.t"1ltt"; '

certain polynomial +_ ,p ( s , q ) : ) a ; ( q ) s '; - n

is said to have an 'independent uncerta'inty structure if each compo-

nent q; of q enters into only one coefficient.

ExERCISE 5.2.2 ( Independent Uncerta inty Si rucLure) : Does the

uncertain poiynornial

p ( s , q ) : s 3 * ( q r + 4 q " * 6 ) s 2 * ( q t - 3 q a ) s - F ( s o + 5 )

have an independent uncertainty structure? Explain.

An un-

5.3 / Internal Polgromial Family 67

5.3 Interval PolYnornial FarniIY

In this section, we define interval polynornial families and the concept

of lumping. By lumpingr lve mean combining uncertainties so as to

obtain a description of the same family of polynomials involving a

smaller number of uncertain parameters.

DEFrNrrroN 5.3.1 (Interval Polynomial Family) : A famiiy of poly-

nomials P : {p(- ,q) : q e Q} is sa id to be an ' in terual polgnomial

famity if p(s,g) has an independent uncertainty structure, each co-

efficient depends continuously on q and Q is a box. For brevi'uy, we

often drop the word "family" and simply refer to P as an interual

polynorn'ial.

EXAMeLE 5.3.2 (Simple Interval Polynomiai): An interval poly-

nomiai family ? arises from the uncertain polynomiai described by

p (s ,q ) : ( 5 + q+ )sa + (3 + q3 )s3 + (2 + ez )sz + (4+ o r ) s + (6 + qo )

w i th unce r ta in t y bounds l q r l < 1 f o r i : 0 , I , 2 ,3 ,4 .

ExAMeLE 5.3.3 (Some Coeffrcients Fixed): Notice that the defi-

nition of interval polynomial does not rule out the possibility that

sorne coeffi.cients of p(s, q) are fixed rather than uncertain; e.g., con-

s ider p(s,s) : (5 + q+)sa+3s3 + (2 + q2)s2 + (4 + gr)" -F 6 wi th a

given box Q for the uncertainty bounding set.

EXA^TIPLE 5.3-4 (Lumping Interval Polynornials): The uncertainty

representation often involves a certain type of redundancy. For ex-

a m p l e , i f p ( s , e ) : s 3 + ( 5 + q z * 2 q z ) s 2 + ( 6 + 2 q t * \ q + ) s + ( 3 + q o )

and bounds l q l l < 0 .5 f o r i : 0 , ! , 2 ,3 ,4 , one can " l ump" t he unce r -

tainty as follor.vs: Define new uncertain parameters Q2:51q2+2q3,

4t :_6 * 2qt * Sqa and 4o : 3 * qo, a new uncertainty bounding

set Q by 2.5 < do < 3.5, 2.5 < q, < 9.5 and 3.5 < dz < 6.5 and

a new uncerta in polynomial FG,d: s3 -F qzs2 + qts * qo.We cal l

P : {p(-,i l , q e Q} a Lumped uersio_n of the original family P and

Ieave it to the reader to verify that P : P.

EXERCTsE 5.3.5 (Lumping with More Complicated Dependence):

The objective of this exercise is to demonstrate that lumping is pos-

sible rvith rrlore complicated dependence on q. To this end, consider

an interval polynomial family 2 described by

p(s,q) : (5 + ee 'cos qz)s2 + (s in(q3 + q+) i - t )s + (qsq| + "q ' )

68 Ch 5 / The Spark: Kharitonov's Theorem

a n d l q 1 j ( 1 f o r i : 1 , 2 , . . . , 7 . P r o v i d e a c h a r a c t e r i z a t i o n o f alumped version P of P.

ExERCTsE 5.3.6 (A Lumping Theorem): This exercise generalizeson the one above. Indeed, consider an interval poiynomial fam-1ly P : {p(.,q) : S € Q} with p(s,q) having coefficients depend-ing continuously on_ q. Prove that the_re exists a second intervalpolynomial farnily P : {p(., i l , 4_€ Q} with FG,q) of the formp(t, q) : DT:o Qisx and, moreover, P : P.

5.4 Shorthand Notat ion

In view of the discussion of lumping above, we henceforth work withan uncertain polynomial of the form

n

p ( s , q ) : l o o t oi :0

when dealing with an interval family. Such a family is completelydescribed by the shorthand notation

n

p ( s , q ) : L l q n , q l ) t oi :o

with [q;-,81+] denoting the bounding interval for the i-th cornponentof uncertainty qi. In the context of this convenient abrise of notation,we can refer to p(s,q) as an interual polynomial

5.5 The I{haritonov Polynornials

In order to describe Kharitonov's lfheorem for robust stability, wefirst define four fixed polynomials associated with an interval poly-nomial family P. In the definition below, note that the polynornialsare fixed in the sense that only the bounds at and Qf, entel into thedescription but not the q1 themselves. We also emphasize that thenumbel of polynomials is four-independent of the degree of p(s, g).That is, four is a magic nunrber.

DEFrNrrroN 5.5.1 (The Kharitonov Polynomials): Associated withtlre interval polynomial p(s, rl): LT:o[q;,qf,)"" are the four fixedK h arit o n o u p o lg n o mi al s

t s 2 L i - 4 - 5 - r An r ( s ) : 4 0 * Q 1 s t Q 2 s - t q j s - + Q a s ' f q s s " t 4 6 s " - f . . . i

5.6 ,/ Kharitonov's Theorem 69

Kz(r) : s ' r * s ls + q; r ' * q t " ' * q t tn * s f , " t * q ;s6 * ' ' ' ;

r ( a ( " ) : qo f * q f " * q ; " ' *n f " ' * o t t n *q ; "5 +qu "u

+ " ' ;

K+(s) : qo * q l r + n{ t ' *q t " t * qn tn +Q5+s5 + s ; - "u + " "

ExA\4pLE 5.5.2 (Construction of Kharitonov Polynomials): The

I{haritonov polynomials are easily constructed by inspection. To

illtistrate, the four Kharitonov polynomials corresponding to the in-

terval polynomial

p (s , q ) : [ 1 , 2 ] s5 * 13 ,4 )sa + 15 , 6 l s3 + [ 7 , 8 ] s2 + [ 9 , 10 ] s + [ 11 , 12 ]

d L Y

I {1 (s ) : 11 * 9s + 8s2 + 6s3 + 3s4 + s5 ;

K2(s) : 12 + 10s + 7s2 + 5s3 + 4sa + 2s5;

Ks( t ) : 12 - l9s + 7s2 + 6s3 + 4sa + s5 ;

I { a ( s ) : 1 1 * 1 0 s + 8 s 2 + 5 s 3 + 3 s 4 l - 2 s 5 '

5,6 Kharitonov's Theorern

we now present the celebrated theorem of Kharitonov (1978a) and

aLso il lustrate its application. The proof of the theorem is relegated

to Lhe next two sections.

THEOREM 5.6.1 (Khar i tonov (1978a)) : An interual polynornia l

farni,Iy P uith 'inuariant degree i's robustly stable if and only if its

four Kltaritonou polynomi'als are stable.

EXAMeLE 5.6.2 (Application of Kharitonov's Theorem): For the

interval polynomial

p (s , q ) : [ 0 .25 , 1 .25 ]s3 + 12 .75 ,3 -25 ]s2 + [ 0 .75 , 1 ' 25 ] s + [ 0 ' 25 ' 1 ' 25 ] ,

the four Kharitonov polynomials are

I {1(s) : 0 .25 + 0 '75s l - 3 '25s2 f 1 '25s3;

I {2 (s ) : I . 25 + 1 ' 25s - l 2 ' 75s2 -1 - 0 ' 25s3 ;

- l { 3 ( s ) : t . 25+0 '75s* 2 ' 75s2 * I ' 25s3 ;

I {a(s) : 0 .25 + L.25s ' t 3 '25s2 * 0 '25s3'

70 Cln.5 / The Spark: I(haritonov's Theorem

Using the classical Hurwitz criterion, it is easy to verify that all four

I{haritonov polynomials above are stable' Hence, we conclude that

the interval polynomial farnily is robustly stable.

ExERcrsE 5.6.3 (Application of I(haritonov's Theorem) : Consider

the interval polynomial farnily which is given in Example 5'5'2' Is it

robustly stable?

5.7 MachinerY for the Proof

For some readers, there is a temptation to skip sections containing

technical proofs. For the case of Kharitonov's Theorem, howeveL'

the author's advice is to continue reading. The icleas introduced in

this section and the next are at the heart of many generalizations

later chapters.

5.7. I The l (har i tonow Rectangle

describe the subset of the complex plane given by

P( jao ,Q) : {P ( j t o , q ) : q € Q} '

vertices of p(jr ' to,Q) are precisely the 1{,(jc,- '6).

To establish rectangularity, we examine tire real and irnaginary

parts of p(juo,q). Indeed, we fi.rst observe that

Rep( jas, r ) : ,L ,_oo( i ro)o:

no_ q2a3* q+at * so, ,S- l ' qaL")E - " '

and

i n"qt(i'o)iIrn p(jas, q) : : qr,,o - q3a3 + qrrS- qr r[ + qsrues -,' '

5.7 / Machinery for the proof 7I

Notice that no q1 which enters Re p(jr,_,s, g) enters Im, p(jas,g) andvice versa. fn view of this decoupling between real and imaginarypa'ts, the set p(jao, Q) consists of all cornplex nurnbers z such that

Re z : qo - qzaS -f qEut - q6u3* qsr../fi - .. .

for some admissible q € Q and

r 'm z: qreo - qsu- ,3 * ss. , . | - qr . [ - f ssL,)B - . . .

for sorne admissible S € Q.We now argue that the set of all generatable pairs (f ie z,Irn z)

above is a rectangle which is obtained by finding the minimum anclnraximura values of Re p(jroo,q) and lrn p(jws,q) with respect toS e Q. Indeed, since each q'i enters only one coefficient of p(s, q), forR.e p(jut6,g), we can minimize or rrraximize each ternr. individuallyto obtain

min Re p( iu. to.q) : qo - q{-B + q+.3 - qt rS+ st@$ -q€Q

: Re InUao)

and

nra7: Re p(juto,q) : qt - qz.2o + q[.3 - stu| f qs+rfi - .secl

- Re l{2(jus).

As far as Irn p(j-0, q) is conccrned, one must pay attention tothe sign of cu6 in deciding whether to use qt or qf when minimizingor maximizing. Keeping this issue in mind, for c,.rs ) 0, we obtain

Irn p(jes, e) : qt tto - qt.3 + s;,i3 - q{.3 + . . .

72 Cln. 5 / The Spark: I(haritonov's Theorem

For the maximization problem, the same tlpe of reasoning leads to

max r rnn( ian,ot : J l rn Kauws) i f c" 's ) 0 ;

q€Q I Int K3Uus) i f os < 0.

. F L , - - f ^ -a'uD r@r, uur orglrrrr"ot, i r tdi .ute that p(juo,Q) is bounded by

the rectangle given in Figure 5.7.1; i .e' , i f z e p(jus, Q) and c, 'rs ) 0,

Im l{a(jus)

Irn K3Uas)

Re K1( jas) Re K2(jus)

FIGURE 5.7.1 The Kharitonov Rectangie for cus ) 0

tt ' j Re l{1(j.o) S Re z I Re K2(ias);: t ii , : t j

i , ,, i , j Irn K3(ju:s) < Irn 7 1 Im l{q(iwo).

i,.:tii To complete the argument, we now claim that this boundlng rectan-

il ' . i l i gle is precisely equal Lo p(jus,Q). That is, every value in this rectan-

i":,: j gle is realizable by soffre q € Q. Indeed, by viewing Re p(jus, q) as aj : l / \ ! ^ D ^ - r T - ^ l ; , . ^ \ ^ - ^ * ^ ^ ^ . i - - f - ^ -i , : ' , ,,r i mappm[

rnlnq e Q

and similarly, for aJs ( 0,

rr l rn Inr D(iun,q) : q{.o - qi .3 + q{rf i - s?;[ + . . .q e Q * r o u

Combining these trvo cases, we arrive at \

1,,,1 (qt,qs,qs,...) to R, a simple intermediate value argument guaran-

tees that for each z satisfying the two inequalities above, there exists

min Irn p(i th. q\ :qeq/

I n" Nrli;,o1

I tr" xn1i.;1if crs > 0;

if c..,s < 0.

| : : : . ) _ ^ _ O - _ ^ -

l , ' , ir, l some uncertainty q. € Q such that p(juo,e.): z. fn summary, the

,.,1 set p(jus,Q) is precisely the rectangle depicted in Figure 1,7

1l . :,1 We now relate the vertices of the rectangle p(iuo,Q) to the

I{haritonov polynomials:

Southwest Vertex : Re Kt(iuto) + j lm Ks(jro)

p( iuo , Q)

5.7 / Nlachinery for the Proof 73

: Re Kt( j . . .o) + j r rn Kt( jao)

: Kt( j .o) ;

Northeast Vertex : Re Kz(iac) + j lrn l{a(iao)

: Re Kz( iuo) + j tm Kz( j .o)

: I{z(juto);

Southeast Vertex : Re Kz(iao) + jIm Ks(joo)

: Re I{s(j,no) + jtrn Ks(j.o)

: Ks(j 'oo);

Northwest Ver tex : Re l i t ( ju .s) + j In l . I ia( ju6)

: Re K+(jro) r jIm' Ka(jws)

: K+( jao) .

This leads to our final depiction of lhe Khari'tonou rectangle given in

Fi.gure 5.7.2. The key point to note is that each vertex is associated

Im

Kq( j .L .o ) Kz( jao)

IqUus) Kz(jro)

FIcunn 5.7.2 Simplified Kharitonov Rectangle for r.16 ) 0

with a unique Kharitonov polynomiai.

74 Cl-t.5 / The Spark: Kharitonov'sTheorem

ExERcrsE 5.7.2 (Kharitonov Rectangle for c"'6 < 0 and cr'rs:0):

Sketch the Kharito";;-;;;""tIe p(iu;'O) for c's ( 0 with vertices

carefully labeled. no. 's : 0, "notice

thai p(ju,s'q; : [q[' eoFi'

REMART(s 5.7.3 (Motion of Kharitonov Rectangle): Thus far' the

discussion of the Nr'",ito,,o., rectangle has been in the context of a

frozen frequency u) : Lr)o' We now entertain ffe

nofion ",f-"^*"^":l:i

i fr" fr"qrr"rrcy. Indeed, we begin at a:0 and imagine cu increasrng'

This results in motion of the kharitonov rectangle' That-is' we have

a rectangle moving t'olr"d the compiex plane ivith wertices l{6(j*')

obtained by evaluation of the Kharitonov polynomials' Generaily'

it" ai*"n.lons of this rectangle vary with the frequency c''''

ExAMPLE S-7.4 (Illustration of Motion): For the interval polyn'o-

rnial

p (s ,q ) : [ 0 .25 ,1 .25 ]s3 + 12 '75 ,3 ' 25 )s2 + [ 0 ' 75 ' 1 ' 25 ] s + [ 0 ' 25 ' 1 ' 25 ]

which we analyzed' in Example 5'6'2' '"ve illustrate the motion of the

Kharitonov rectangie p(jt,Q) in Figure 5'7'3 for trventy frequencies

0 .6

0 .4

i.ri"'ii , ti l l

l , r ,

i lIt'i . . l[, ';iii ,i

lii jii,ii;,"1

i,..il : , f ' ,1i . l I

i.'..r ,jtr,iir i .11

i r l , : i : , i i i

l;l.ii1llr;i i'.1'lI,.i,1.r,ii ' ; , ' 1l',,,,r.. ,i

1

0 .8

Irn 0'2

0

-0.2

-0.4

_ 0 . 6 - +-3 -2 .5 -2 -1 .5 -1 -0 '5 0 0 '5 1 1 '5

He

Flcuno 5.7.3 Motion of Kharitonov Rectangle for Example 5'7'4

p( jeo, Q)

evenly spaced between a:0 and cu:1 ' Not ice that th is rectangle

5.7 / Machinery for the proof js

begins at a : 0 as an interval on the positive real axis and thenlrlo\res from the first to the second quadrant as cr is increased.

5.7.5 Angle Considerat ions

In this subsection, we review some basic facts about the angle of a,polynornial as a function of frequency. We include the proof of thewell-known lemma beLow because the underlying ideas are useful inlater chapters. In a control setting, the lemma below is often creditedto Vlikhailov (1938).

LEMMA 5.7.6 (IVlonotonic Angle Property): Suppose that pls) isa stable polgnorn'ial. Then the angle of p(jto) is a stri,ctlg .increo,szng

furicti,on of a € R. Furtherrnore, as to uari,es frorn 0 to *a, Lp(j.)enperiences a'n increment of ntrf 2.

PRoor': First, we write p(s) : I{ lIT:r(" - tn), where ;f e R andRe zn < 0 for z : 1,2,. ,r '1. The angle of p(ja) is given by

Lp(j .) : Lr{ +i +{ i . _ "o).

i : l

Wit l r 0 t ( . ) : | -Uw - z i ) and the a id of F igure 5.7.4,we make the

Frcunp 5.7.4 0i(u) is a Strictly Increasing F\rnction of c..r

following observations, noting that zi lies in the strict left half plane:

76 Clr.5 / The Spark: Kharitonov's Theorem

Tf z- ' is nrrr-elw rea.l- then as cu varies from 0 to *co, )a(a) is str ict lyL L - X L e V u r v r . 7

incr-easing and experiences a net increment of. rf 2. If za is cornplex,

we work wtth za in combination with its conjugaLe zi. Now, as c'.r

increases from 0 to -Foo, the corresponding angles qi(a) are strictly

increasing and contribute a net increment total of zr. The proof of

the lemma is completed by summing over the 7i(u). d

ExERCTsE 5.7.7 (More General Angle Considerations): Supposep(s) is an n-th order polynomial with nl roots in the strict left halfplane and n2 roots in the strict right half plane. Assume rr1]_ r12:71

and show that as c..r varies from 0 to + co, 4-pU.) experiences

a total change in angle of (n1 -n2)r12. AIso modify the result to

aiiow ft r the case when p(s) has some roots on the imaginary axis.

5.7.8 The Zero Exclus ion Condi t ion

In this subsection, we introduce the Zero Exclusion condition. The

technical ideas associated with this condition arise time and tirne

again throughout the remainder of this text. since we a.re currently

working within the framework of interval polynomials, the lernma

below is not stated in fuil generality; the most general version which

we provid.e is given in Theorem 7.4.2. In addition to facilitating the

proof of Kharitonov's Theorem, the lemma below is also of practical

use because it suggests a simple test for robust stability which is easy

to implement in graphics.

LEI\4MA 5.7.9 (Zero Exclusion Condition) : Su'ppose that an in'terual

polynomi,al fami,ty P : {p(', q) : q € Q} has inuariant degree and at

least one stable member p(s, qo). ThenP 'is robustlE stable i ' f and only

i,J z : O is ercluded frorn the Khari,tonou rectangle at aLl nonnegatiue

frequenc'ies; ' i .e.,

o 4 PU' ,Q)

for all frequencies c,', > 0.

pRoor': we fi.rst justify the restriction to nonnegative frequencies.

To this end, note that z € p(i.,Q) if and only if z. e p(-ju-t,Q)'

TIence, q'ithout loss of generality, we restrict our attention to cl ) 0.

To establish necessity, we assume that P is robustly stable and

must prove that 0 # p(jr,Q) for all c.r € R. Proceeding by contradic-

tion, suppose that 0 e p(ia*, Q) for some frequency c..'* € R' Then

p( ju* ,g*) : 0 for some q* e Q; i .e . , the polynomial p(s,q*) has a

root at "

: jr* which contradicts robust stability of P.

5.8 ,/ Proof of I(hari tonov's Theorem 77

To establish suffi.ciency, we assume that 0 4 pU-,Q) fol allcu € R and must show that 2 is robustly stable. Proceeding bycontrac. l ic t ion, i f 2 is not robust ly s table, then p(s,g l ) is unstablefo l some qr € Q. Now. for I e 10,1] , IeL

p (s , , \ ) : p ( s ,As t + ( t - I ) qo )

and notice that l(s, ^) € ? because )ql + (1 - ))s0 € Q. Nloreover,for ) : 0, p(s, 0) : p(s, qu) has aI1 roots in the strict left half planeand for- , \ : 1 , FG,I ) : p(s,qr) has at least one root in the c losedrigirb half plane. Since the roots of p(", )) depend continuously on A(Lerrrrla 4.8.2), there exists a,l* € [0, 1] such that f(s, )*) has a rooton the imaginaly axis. Equivalently, p(ja*,.\*q1 + (1 - A-)q0) : Ifor some r.-,* € R. This impiies that 0 e p(ia*,Q), which is thecontla,diction we seek. E

REMAFLI{s 5,7.LO (Real Versus Complex Coefficients): Whenworking with the Zelo Exclusion Condition for the complex coeffi-cierrt case. we can no longer restrict attention to ut ) 0; i.e., we can-not explo i t the fact that z ep( j ro,Q) i f and only i f z . e p(- ja ,Q).

In this case, the lemma above requires a minor modification: Un-del the standing hypolheses, 2 js robust ly s table i f and only i f0 / p(j.,Q) for all a,l € R. This arises in Chapter 6 when weconsider the complex coefflcient version of Kharitonov's Theorem.

5.8 Proof of I{haritonov's Theorern

The proof of necessity is trivial; i.e., if 2 is robustly stable, it followstlrat the four Kharitonov poiynornials are stable because Ka(s) e Pfor z : L,2,3,4. To establish sufHciency, we assume that the fourKharitonor. po).ynonials a"re stable and must prove tirat P is robustlystable. Proceeding by contradiction. suppose that P is not robustlystable. Using thc standard notat ion p(s,q) : L i :o lSt , go+]s i , *econsider two cases.Case 1: 0 e [St, So+] Recall ing the invariant degree assumption,it rrrust be true Linat q- and qf have the same sign. Without lossof generality, say that the signs of qn and qf are positive. Thenit follows that at least one of the four Kharitonov polynomials, cailit l{r-(s), }ras coefficient of s', whicir is positive, and coefficient ofsu, which is nonpositive. This contradicts the assumed stabil ity ofKr- (") because a stable polynomial must have nonzero coefficientswhich all have the same sisn.

78 Ch 5 / The Spark: Kharitonov's Theorem

Without loss of generality, assume that this piercing occurs on the

southern boundary of. pQu,Q) as shown in Figure 5'8'1' Also, note

i , ii , , . 1

l . ll,l I

r i il . ' : i

Iill . ' , ,1i., .i[,,,.[ , : . ,l:r.,llrr;[ ' ' ":[.iti

,r'.i ,ij

"1l,i:i,:,:l

iili'iii-:r.1t.:rl+i;til

iill

[,;iili,,f i,ii:l

1,i l

FIcunB5 .8 . lP ie rc ing theBoundaryo f t l r e l ( ha r i t onovRec tang le

that z: 0 cannot be coincident with Kt(jq or IQQA) because

ftt"l and 1{3(s) are assurned stable' To complete the proof' we

;"oti, continuity of the l{i( j.,) and the N{onotonic Angle Property

iil;tr. 5.7.6). illu,*"]v, for 5D > 0 suitablv small, it follows that

oo < LKz ( i ( a+6d , ) ) < 900

and1800 < LK \U@ +64) ) <270 ' '

we now have the contradiction which we seek because simultaneous

satisfaction of the two angle inequalities above makes it impossible

for the southern boundary of the rectangle p(j (6 + 6A), Q) to remain

parallel to the real a;cis. E

KzUa) rnoves

this way

( *"tn'

K{jA) moves this waY

5.9 ,/ Formula for the Robusrness Margin 79

5.9 Forrnula for the Robustness Margin

For an interval polynornial family, by combining the results of thischapter with those of Chapter 4, we obtain the robustness marginformulas of Fu and Barmish (1g88). To this end, we describe a.nrz-th order interval pol1,1orr1tut family with stable nominal p6(s) andvariable uncertainty bound r > 0 by writing

p,ft, q1: po(s) * , "1'"1_.0 , rr]r i .

We view the el ) 0 above as scale factors which determine the asoectratios of the uncertainty bounding set Q,. Letting P. denote theresulting family of polynomials, our objective is to piovide a formulafor the robustness marein

rmoz : sup{r : P, is robustly stable}.

To obtain the desired formula, we fi.rst argue that Kharitonov'sTheorern enables us to reduce the robustness margin problem to fourseparate problems for the uncertain polynomials {ps(s)+qpr,l(s)}4 ,,rvhere

p r , r ( s ) - - € o - e 1 s * e z s z + e 3 s 3 - e 4 s 4 - e 5 s s * € 6 s 6 + . . . ;

h ,2G) : co - l - €1s - e2s2 - e3s3 * e4s4* e5s5 - e6s6 - . . . ;

p r , s ( s ) - € 0 e 1 s - € 2 s 2 * e 3 s 3 1 _ e 4 s 4 - e s s 5 - e 6 s 6 * . . . ;

p r ,+ (s ) - - €o l - e r s -F e2s2 - €3s3 - €4s4 + esss * e6s6 - . . . .

Now, applying Theorem 4.7.6 and taking the worst case with respectto i : L ,2 ,3 ,4 , we a r r i ve a t t he f o rmu la

' ! ' ^ ^ - : - i . I

' ;<A' h", (_ H -1 (po) H (h,i))

5.10 Robust Stabil ity Testing via Graphics

The Zero Exclusion Condition (see Lemma S.T.g) suggests a simplegraphical procedure for checking robust stability-watch the mo-tion of the Kharitonov rectangle p(ju,t,Q) as u varies frop 0 to *ooand determine by inspection if the condition 0 / p(j.,Q) is sat-isfied. This raises the following question: Can we find some finiteplecomputable cutoff frequency a. ) '0 such that O d p(j.,Q) for

80 Clt 5 / The Spark: I{raritonov's Theorem

aII u > c.,,.? That is, can we terminate the frequency sweep at the

frequency a : ac?The existence of a. is easily established using the invariant de-

gree condition. Indeecl, suppose that p(s,q) : ILoIqt ,q[]si and,

ivithout loss of generality, assume that qn > 0 for i : 0,7,- " 'r-L'

Then given any q € Q, it is easy to see that fol cu ) 0,

lp(j. , q)l >- q^ a- -T d ,'i :0

Since the right-hand side tends to *co as u --'+ -l-co, it follows that for

any prescri[ed € ) 0 t]rere exists an cu. ) 0 such that lp(.it,dl> P

f o r a l l u ) ) e c . H e n c e , 0 / p ( i . , Q ) f o r a l l u > a . '

In fact, we can easi11' compute an applopriate c''rt' For example'

one can take c..r" to be the largest real root of the polynomial

n - l

f ( r ) : q n a n - L q f , r o; - 1

m a x { a g , a r t . . . , o n - r }1 1 : f f -

an

Hence, for the interval polynomial p(s,q), rvith q; > 0,it follows

that an appropriate cutoff frequency is given by

. r n a x { q f , , q l , - . . , q I _ JU)c: I -f ----=__-

Qn

EXAMPLE 5.1O.1 (Il lustlation of Graphics Method): We consider

the in terval polynomial far i i ly P: {p( ' ,q) : q e Q} descr ibed by

p (s , q ) : s6 * 13 '95 , 4 .05 ]s5 + [ 3 ' 95 , 4 ' 05 ] sa + [ 5 ' 95 , 6 ' 05 ] s3

+ [2.95, 3 .05]s2 + [1 .95, 2 .05]s + [0 '45, 0 '55] '

In accordance with Lemma 5.7'9, the fi 'rst step in the graphical test

for robust stability requires that we guarantee that at least one poly-

nomial in P is stable. using the midpoint of each interval above, we

In t

5.11 / OverboundingviaInrerval pol l ,nomials gI

0.5

0 . 4

0 . 3

0 . 2

0 . 1

0

-0 .4-0 .3

FIGURE 5.10.1 Graphical Robust stabil ity Test for Exampre 5.10.1

obtain p(s, q0) : s6 * 4t5 + 4sa + 6s3 + Jsz +2s_F 0.5, whose roots ares r = - 3 . 2 6 8 1 , s 2 , 3 = - 0 . 1 3 2 8 + 0 . 9 4 7 3 j , s 4 , 5 = - 0 . 0 7 3 1 + O . T I g } jand s6 = -0 .3201 .

Next, ln accorda.nce with the discussion of c.toff freque'ciesabove, we compute the largest real root of the test polynomiat y1.);that is, with

82 Cln.5 / The Spark: I{-rarironov's Theorem

later clrapters. The second alternative is the so-called ouerboun,d,-ing nrcthod, which is described below. One u'arrring, lrorvever', is inorder-: Although the overbounding rnethod is easy to use, it rnaylead to r-rnduly conservative results; i.e., $.e only obtain sufficientconditions fol r-obustness. In short, associated lvith over-bounding isa tlade-off between ease of use and deglee of conset-vatisrn.

In the rernainder of this section, we tlo longer require the pol1,-nomials p(s,q) to have an iudependenL uncerta inty s t r r - rc tu le, and,in addition, Q is not necessai:i ly a box. We begin with the uncertainpolyrromial p(s,q) : DT:oot(q)s' and an uncertainty bounding setQ rvhich is closed and bounded. Assuming the coefficient functionsai(q) depend continuously on q, -,ve defi.ne the bounds

4; : rrl'ip at(q)( leQ

andqf : *u+

"r(s)' s€q

and simply observe that the famiiy of polynom.ials 2 described b1'

n\- r-- -

FG.4 ) : \ 141 . l i ) s 'i :o

is a superseL of P. Therefore, any robustness property which hoidsfor t lre intervai poI5'no-'tt family 2 must hold for P.In particular,robust stabil ity of 2 implies robust stabil ity of 2. Note, h<.rwever,that the converse is not true. Tirese points are i l lustrated via theexarnples below.

ExAMeLE 5.11.1 (Success of Overbounding) : Consider the familyof polynornials P described by

p (s , q ) : sa + (5 + 0 .2q rqz * 0 . l q1 - 0 .1q2 )s3 + (6 + 3q tqz - 4qz ) t2

+ (6 + 6qt - 8qz)s + (0.5 - 3qtqz)

and uncertainty bound lqil < 0.25 for- i : 1,2. The objective is todeterrnine whether P is robustly stable. To this end, we computebounds

Qs : rn_i4 ao(q) : ^ ̂ -n l in_^ ̂- (0.5 - Sqtqz) : 0.3125;q€Q *0 .25<q i<0 .25

Ad- : *"X ao(q) : t4a>C. -_ (0 5 - Sqtqz): 0.G875;q € Q - O . 2 5 1 q i 1 O . 2 5

5 . 1 1 Overbounding via Interval polynornials

As mentioned in the introduction to this chapter, the independentuncertainty structure is restrictive because uncertain parameters typ-ically enter into more than one coefficient. For such ,,dependent',uncertainty structures, we consider two alternatives: The fi.rst al-bernative is to develop more general results; this is the topic of

n +

: n l i n a t ( o ) :q e c l

: m a x a t ( O ) :qt\1

5 1 2 / C o n c l r r s i u i i b c

-_mi r r (6 + 6q t - 8qz) :2 .5 ;- 0 .25<q i<O .25

^ ̂ t4ax . - ̂- (6 + 6q t - Sqz) : 9 .5 .-0.25<,t ,<0.25

84 Ch 5 / The Spark: I0raritonor"s Theoretn

Notes and Related Literature

N R L S . l T h e p a p e r b y F a e d o ( 1 9 5 3 ) a p p e a r s t o h a v e p r o v i d e d i m p o r t a n t m o t i _

vation for Kharitonov's work.

NRL 5.2 Kharitonov,s original proof is based on the Herrnite-Biehler Theorern;

e.g. , see Gantmacher (1959). Indeed, consider a polynomial p(s) decornposed

into even and odd parts p(s) : p","n(s2) I sp"6a(s2)' Then, according to the

Hermite-Biehler Theorem, p(s) is stable if and only if p"u.n(r) and po2a(r) have

h ighes to rde rcoe f f r c i en t so f t hesames ignandnega t i ve rea ]d i s t i nc t i n t e r l ac i ng

roo t s ; e . g . , i f po l ynom ia l p ( s ) has odc i deg ree and c . , r 1 re ,2 1 " ' ( r " ' - and

roJ I r o , 2

ro L 1 r e , r 1 : Lo .2 < ' r e ,2 1 " ' 1 r o ,n , 1 r " ' * ' The key i dea beh ind t he o r i g i na i

these intervals are assocrated with the Kharitonov polynomials. subsequently, it

i sa rgued tha t sa t i s f ac t i ono f t he roo t i n t e r ] ac i ngcond i t i on fo reachKha r i t onov

polynomial irrrplies satisfaction of the root interlacing condition for the entire

f a rn i l y .TheHerm i t e_B ieh le r } i neo fa t t ack i sno tpu rs r red i r r t h i s t ex tbecause

we want to explain as mary results as possibie within the unifying framework of

value sets. The l{haritonov rectangle is in fact an example iilustrating the rnore

general value set concept of Chapter 7'

NILL 5.3 The key ideas underlying our proof of Kharitonov's Theorem corne from

Dasgupta (1988) and Nlinnichelli, Anagnost and Desoer (1989). ivlore specifically,

we note that Dasgupta (1988) exposes the rectangular geometry of p( ju 'Q) and

Minniclrelli, Anagnost and Desoer (1989) exploits rectangularity and Lhe Zero

Exclusion Condition to obtain a sirnple proof of the theorem'

NRL 5.4 The paper by Flazer and Duncan (f929) appears to be the first to use

fne Zero Exclusion Condition in a robust stability context'

NRLS'SFormorecompl icateduncerta intystructures, lAreiandYedaval l i (1989)

The potential for further research involving such methods is illustrated by the

family P in Exercise 5.11.2. A robust stability test based on overbounding by

an interval polynornial is inconclusive but multiplication of the even patt'by 7f q

and the odd part by unity leads to an intervai poiynomial whose robust stability

p G , q )

ExERcrsE 5.L1,.2 (Failure of Overbounding): In this exercise, theobjective is to i l lustlate how overboundil lg can fail. To this end, con-sider the family of polynomials 2 given in Wei and Yedavall i (1g8g);i.e., the farnily 2 is described by

P ( s , q ) : s 4 - F s 3 + 2 q s 2 * s * q

rvit lr uncertainty bounding set I : [1.5,4] . Argue thatP is r.obustlystabie but the overbounding family

pG,4 ) : sa t - s3 + 13 , S ]s2 + s + [1 .5 , ] ]

has an unstable Kharitonov polvnomial.

5 . 7 2 Conclusion

In a sense, Kharitonov's Theorem raises more questions than it an-swels. To illustrate the type of questions suggested by Kharitonov,sTheorem, we consider the robust Schur stability problem for an inter-val. polynomial family 2: Indeed, if the associated four Kharitonovpolynomials have all their roots in the interior of the unit disc, doesit follow tlnat P is robustly Schur stable? If not, does it suffice totest polynomials associated with all the vertices of Q? Nlore gener-ally, for what type of root iocation regions does a Kharitonov-likeextrerne point result hold? The list of possible questions seems end-less. In Chapter 13, rve characterize classes of 2 regi.ons for which2-stability of the polynomials associated rvith the extreme pointsof the Q box irnplies robust Z-stability of the associated intervalpolynomial family.

Notes and Related Li terature g5

is easily verified by Kharitonov's Theorem.

NRL 5.6 There are a number of papers in the literature involving transfor-mations aimed at facilitating robust stability anall'sis. For example, using theshifted circles in Petersen (1989), one can deai with the so-called Delta trans-form for a discrete-time system; for similar extreme point results involving Deltatransformation) see also soh (1gg1) The paper by vaidyanathan (1990) providesanother example of a transformation used for discrete-time problems.

NILL 5.7 some alternatives to the technique described in section 5 11 are givenin papers by Djaferis (1991) and Pujara (1990). These papers describe differentoverbounding families which are sometimes useful.

NRL 5.8 Rather than working with the original coefHcients, one can consider aboundi'g box B in the space of Markov parameters. By breaking an n-th orderp(s) into i ts even and odd parts as p(s) : p"""n(s2) ! sp"aa(s2), a cont inuedfraction expansion for p"aa(r)fp"u"n(r) ieads to the set of ]vlarkov pararnerers;see Gan tmache r ( 1959 ) . I f n : 2 rn , we ob ta i n pa ramc te r s ( bs .b r , . . , bz ,n_ r ) ,and i f n : 2rn - l , we obtain (b_1, bo, . . . ,bzu,_r) . \ \ r i th th is . representatror ,robust stability is guaranteed if and only if two distinguished polynomials ar.estable. For example, if. n : 2m. ard the box B is described by bl < b1 < bf, fort : 0, 1, 2, . . . ,2rn - 1, the f i rs t d ist inguished polynomial has \ {arkov parame-ters (bo , b! ,b; , . . . ,b[^-r ) and the second dist iuguished polynomial has N,Iarkovpararneters (bf , b i , bt , . . . ,br*_r) ; see Hol lot ( f989) for fur ther e laborat ion. Ofcourse' a fuDdamental limitation of these results is that the relationship betweenthe Nlarkov parameters and the original parameters is gener-ally quite compli-cated. This complication motivates interesting research problems involving sys-terr identification for robust control.

NRL 5.9 we me'tion a body of work aimed at generalization of Kharitonov,sTireorenr to scattering Hurwtz polynomials For example, in papers by Bose (1ggg),I { i rn and Bose (1988) and BeLsu (1989), the uncerta in polynomial p(s,g) is re_placed by a mui t iva.r iate uncerta in polynomial p(sr ,sz, . . ,sn,q) and intervalbounds on the coefficients are irnposed.

Chapter 6

Ernbellishrnents of l{haritonov's Theorern

bynopsts

For interval polynomial families, this chapter provides a nun)-

ber of exfensions, refinements and alternatives to Khadtono'v's

Theorem. Of particdar note is the Tsypkin-Pdyak p)ot fot

easy visualization of robustness marg'lns.

6.1 Int roduct ion

Once one is familiar with the technical ideas associated with the

Kharitonov rpctangle, it becomes possible to develop many exten-

sions and refinements of tire results in chapter 5. This chapter

concentrates on an important subset of these extensions and refine-

ments. In particular, when considering the robust stability ploblem

for low order interval polynomials. we see that fewer than four

Kharitonov polynomials need only be tested. The chapter also in-

cludes extensions of Kharitonov's Theorem for problems involving

degree dropping and probiems involving complex coefficients. Fi-

nally, two "alternative" tobust stability tests are described' The

first involves plotting a scalar function of frequency and the second

involves a Nyquist-like plot in the complex plane'

86

6.2 / Lot't Order Interval Poll,nomials Bj

6.2 Low Order Interval Polynornials

The objective in this section is to establish the fact that less thanfour Kharitonov polynomials are needed for robust stability testingwhen an interval polynomiai has degree five or less; this is the resultof Anderson, Jury and Mansour (1987) .

THEoR-EM 6.2.1 (Simplif ied Kharitonov Theorem for Degree n :3):

Consider an 'interual polynorn'ial family P witlt 'inuariant degree n : 3and lotoest order coefficient bound eo: > 0. ThenP is robustly stablei,f and only i,f the single Kharitonou polynom'ial Ks(") is stable.

Pnoor': As in the proof of Kharitonov's Theorem, necessity followsimrnediately from the fact that 1{3(s) e P. To establish sufficiency,we now assume that f{3 (s) is stable and must prove that P is robustlystable. In view of the Zero Exclusion Condition (Lemma 5.7.9), wemust show that 0 / p(jr,Q) for ail o ) 0. Indeed, as r,r ranges from0 to -1*oo, the Monotonic Angle Property (Lemma 5.7.6) indicatesthat [I{3(jcu) increases monotoriicaliy from O to 3112. A typicaltrajectory of Ks(ja), obtained by varying o from 0 to *oo, is shownin Fieure 6.2.1.

FtcunB 6.2.1 Typical Tbajectory of K3Qu) for a.' ) 0

The proof is now completed by noting that all points in theKharitonov rectangle p(ja,Q) Iie northwest of IQ(ju) for u.r ) 0.Therefore, if {Ks(ju;) increases monotonically from O to 3r f 2, avoid-

88 C]n.6 / Embellishments of Kharitonov's Theorem

ing z :0, it is impossible for zero to enter p(ju, Q) at any frequency

, > O. Hence, P is robustlY stable. E

ExERCTsE 6.2.2 (Tihe Role of q; > 0): Does Theorem 6'2'1 hold

without the assumption qs ) 0? Explain'

EXERCTSE 6.2.3 (Result for n : 4 and rz : 5); Consider an interval

polynomial family ? with invariant degree n : 4 and lowest older

coemcient bound St > 0.(a) Prove that P is robustly stable if and only if the tv.'o I(haritonov

polynomials 1(2(s) and /{3(s) are stable.(b) Now, establish a three-polynomial result for an intervai polyno-

mial family with invariant degree n:5. Note that the statement of

the result no longer requires the assumption qs- > 0'

6.3 Extensions with Degree Dropping

The anaiysis of singular control problems plovides motivation for

asking if Kiraritonov's Theoreln can be extended to accommodate

deglee dropping. To illustrate how degree dropping arises, suppose

that one begins with the standard singular systeln description

Eft)n(t) : An(t)

with c(f) € R. and matrlx E(e) being singular for some values of

the real parameter e. For exarnple, it is often the case that E(e) is

singular for e : 0. we observe that the characteristic polynornial

P (s , e ) : de t ( sE (e ) - A )

can exhibit degree dropping for values of e r-endering E(e) singular.

As a simple i l lustration, if t(e) : e and A : -1, the polynornial

p(s, e) : €5 J- 1 drops in degree from one to zer-o when e : 0'

we consider n ) 2 and first dispose with trivial cases for which

p(s,q) : LT:olq.i ,qf,)si car- drop in degree with coefficients of op-

p..i[" .ieiioi .o-""nember of the family. For example, if qf, > 0

and q*1 ( 0, then the PolYnomial

n -2

p * ( s ) : q I t ' t q n - ' s n - 7 + I c f , " o

is a rnember of the family but cannot be stable because its coefficients

do not , a l l have the same s ign.

I{z(j.z)

Ih(j.t)

6.4 / Interva-l Plants rvith Unity Feedback 89

In view of the argument above, henceforth we concentrate onthe case when gn ) 0 for i : 0, 1, ... jTz. First, we consider the casewhen the degree can drop by three or more.

ExERcrsE 6.3.1 (Degree Drop of Three or Nzfore): Consider theinterwal polynomial p(s,q) : DT:olqt,qf,l"o with degree n ) 4,q ; : Q n - t : Q n - z : 0 a n d s i * > 0 f o r i : 0 , L , 2 , 3 , 4 . P r o v e t h a tthis family of polynornials is not robustly stable; this is the result ofMeerov (1947). Hi,nt: Wifn /(") : LT:to{ s'and e(s) : DT.f qf

"i,relate the robust stabil ity problem to a root locus problem for thefictit ious plant P(s) : g(s)/f (s). Now study the asymptotes associ-ated with this root locus problem

ExER-cISE 6.3.2 (Degree Drop of One) : In this exercise, we ex-arnine some of the issues addressed in N,Iori and Kokame (1992).The starting point is an interval polynomial family 2 described byp(s, q) : L?:o[qr , s [ l t t .(a) Suppose that n I 4 and assume a degree drop of at most one;say q; :0, qI ) 0 and a.t> 0. Show that robust stabil ity of P isequivalent to stability of the four Kharitonov polynomials.(b) For degree drops of one at most and n ) 5, show that robuststability 2 is equivalent to stability of six polynomials-the four fullorder Kharitonov polynomials plus the two auxiliary polynorrrials

I {s(" ) : qn_fn-L * qn_ztn-2 + qI_s"n-3 + q[_nsn-a +- . .

and

I{a(t) - ql-fn-r -l qn_ztn-2 + qn_ztn-3 + q[_nsn-a + . . .

associated with degree dropping.

6.4 Interval Plants with Unity Feedback

We now develop an extension of Kharitonov's Theorem to a class ofunity feedback control systems. To this end, a definition is required.

DEFrNrrroN 6.4.1 (Interval Plants) : An interual plant farnzly P isdescribed by

P ( o - - \ . N ( " , s )

D ( s , r )

with an uncertain numerator polvnomial 1\/(", q) : I3o q1si, uncer-tain denominator polynomial D(s,r) : ILo rlsz and boxes Q and

90 C!l.. 6 / Embellishments of l{haritonov's Theorem

-R as uncertainty bounding sets for q and r, respectively; i.e., P isa quotient of interval polynomiai families. For norational sirrrplicitv,rve can write

i [nn,n{ ) , 'D r - ^ ^ \ - i : o

f \ D , q , t ) - - a - - .

\ - i . - ,+1"ifi"

'

We use the no ta t i onP : {P ( - , q , r ) : q € Q ; r e f i } and o f t en r -e fe rto 2 simply as an interual plant.

REIvTARKS 6.4.2 (Lurnping for Interval Plants): For a proper in-terval plant defined by the condition n ) m, obsen'e that if unityfeedback is used, we can lump uncertainties in the closed loop poly-nomial ; i .e . ,

m n

p(s,q, r ) : t [s t ,q f , ] " i +L l r i , r l ls io-: i:o

n: I l n f +11 ,o l + ru+ ls i + t [ r , , r f ] s i .

i :O i :n+l

In other words, the imposition of unity feedback preserves the in-terval polynomiai structure. This fact is slightiy generalized in theexercise below.

IrxER-crsE 6.4.3 (Pure Gain Compensator): In this exercise, theresult of Ghosh (1985) is established. Indeed, consider an intervalplant with pur-e gain compensator C(s) : 1( and give conditions onthe uncertainty bounds under which the closed loop polynornial hasinvariant degree. Under such conditions, consider I{ ) 0 and 1{ < 0as separate cases and argue that the family of closed loop polyno-mials is robustly stable if and only if four I{halitonov poiynomialsassociated with this family are stable. Describe the four Kltaritonouplants associated with the four Kharitonov polynomials. Notice thatthe sign of 1( is instrumental to the selection of the four relevantp lants.

ExERCTsE 6.4.4 (Interval Plant Calculation) ; Considel a unityfeedback control system rvith interval plant

Ii .I

i , ,

10.75,)..251s + [0.75, 1.25]P ( s , q ) :

ss + 12.75,3.251s2 + [8 .75,9.25]s + [0 .2b, g2S1

a t r / E-^^ ' -^*^. ' Srveeping Funct ion F1 (ro) 91

(a) Using Theorem 6.2.1, determine if the resulting family of closedloop polynomials is robustly stable.(b) Suppose that instead of using a unity feedback, a compensatorC(s) : I /s is used. Use the resul t in Exerc ise 6.2.3 to determine i fthe resr-rlting family of closed loop polynomiais is robustly stable.

ExERCTsE 6.4.5 (Nonunity Feedback): Consider the interval plant

farnily of Hollot and Yang (1990) described by

P ( t , q ) :(s + 0.1) (s + 0.2)

and Q : [1,5000] with compensator

(s + 3)(s + a)s ( s + 2 5 ) ( s + 7 5 )

connected in a feedback confi.guration. Using the uncertain closedloop polynomial

p ( s , s ) : l / ( s , q ) l / c ( s ) - r D ( s , q ) D c ( s ) '

velify that the family P : {p(.,q) : q e Q} has irrvaliant degree's table extremes p(s,1) and p(s,5000) but p(s,2) is unstable.

6.5 FYequency Sweeping F\rnction H(co)

This embellisirment of Kharitonov's Theorem can be contrasted to

tire graphical test for robust stabil ity given in Section 5.10. Re-call that the graphical test for robust stability involves checkingzero exclusion from the Kharitonov rectangle as cu is varied fron0 to *'oo. In this section, we see that we can study robust stabil ityby checking for positivity of a speciaily constructed scalar functionof frequencv H(a). Hence, instead of generating two-dimensionalKharitonov rectangles, we can examine the plot of the scalar func-tion H (u) to determ.ine if the family of polynomials P is robustiystable. The theorem below can be viewed as a frequencv domainalternative to Kharitonov's Theorem.

Tr{EoREM 6.5.1 (Barmish (1989)) : Let P be an ' in terual polyno'rnzal family uti,th inuariant degree, at least one stable rnernber andassoc'iated Kharitonou polynom'ials 1(1 (s), Kz(s), l(e(s) and I(a(s).Then, wi,th

H Qo) : nrax{J?e K t ( j . ) , - Re I {2( j a) , I m I f i ( j w) , - I rn I { s( j - ) } ,

92 Ch.6 / Embellishments of I(haritonov'sTheorem

it follous that P is robustlg stable if and only i,f

H(u) > 0

for aII frequenc'ies t't ) 0.

PRoop: Letting p(ja,Q) denote the Kharitonov rectangie as defi.nedin Section 5.7.1, recall the Zero Exclusion Condition (Lemma 5.7.9)indicates that P is robustly stable if and only if 0 4 pU.,Q) forall frequericies o ) 0. Now, we refer to the Kharitonov rectangle inFigure 5.7.2 and note that the argument to foilor'v does not requirepU.,Q) to be in the fir 'st quadrant. Indeed, at frequency c, > 0,zero is excluded from the Kharitonov rectangle p(jr,Q) if and onlyif one ol more of the following conditions holds: lfhe point z : Olies to the left of the western boundary of p(ja,Q) in the sense thatRe I(1Ur) > 0; the point z : 0 l ies to the right of the easternboundary of p(ju, Q) in the sense that Re K2(jut) ( 0; the pointe:0 l ies below the southern boundary of p( iw,Q) in the sense thatIm, Ks(ju) > 0; the point z : 0 l ies above the northern boundary ofp(i,a, Q) in the sense that Irn l{a(ju) ( 0. Equivalently, o y' p(i., Q)if and only if Re Ky(j..,) ) 0 or -Re l{2(ju) > 0 or lrn l{3(ju,,) > Oot -Im K+(jr) > 0. The "or" of these four conditions is equivalentto the requirement that

max{Re I {1( ju) , -Re K2( ja) , I rn l {3( j " l ) , - I rn Ka( ju: ) } > O.

That is, O / p(jr,Q) for all ru ) 0 if and only if H(-) > 0 for allfrequencies cu > 0. I

ExAMeLE 6.5.2 (Plottirrg H(u)): To i l lustrate the use of H(a),we consider the interval polynornial

p (s , q ) : [ 0 .75 , 1 .25 ]s3 + 12 .75 ,3 .25 )s2 + [ 0 .75 , 1 .25 ]s + [ 0 .75 , 1 .25 ]

which was already analyzed using Kharitonov's Theorem in Exam-ple 5.6.2. To apply Theorem 6.5.1, we first verify that the farn-ily of polynornials has at least one stable member. Indeed, withqo : q r : Q t : 1 .25 and qz :3 .25 , s tab i l i t y i s t r i v i a l l y ve r i f i ed .Next, by a straightforward substitution into the 11(c,,') formula above,we obtain

H ( . ) : max{0. 75- 3 .25u2, -7.25*2.7 5u12, 0.7 5a *7.25u3, - 1 . 25, - 0 .7 5" ,3 |

^ - r^r ^ t u l t - -^-sus cu is ind icated in F igure 6.5.1. Since th isn P r u u u r 1 1 l ! @ / v E r

function remains positive for all d ) 0, Theorem 6.5.1 guarantees

: l,itrj

6.5 ,/ Frequency Srveeping Function 11 (co) 93

5

4 .5

4

3 . 5

H (ut)' 2 . 5

0.5

0

u)

Ftcuns 6.5.1 A Plot of H@) Versus c.-,

robust stability. Note that we do not carr'v the frequerlcv sweep

all the way out to ti : -1-oo because \ re can guarantee apriori that

H (-) > 0 beyoud sorne cutoff frecluency ut : LDc, see the discussion

in Section 5.10 and the exercise belorn,.

ExEFLCTsE 6.5.3 (High-Frequency Behavior): Given an interval

polynomial family with invariant degree, prove that the condition

lirn I1(a.') : -l-co

is satisfied.

ExEFLCTsE 6.5.4 (Relationship with Kharitonov Rectangle): For

corrrplex numbers z € C, consider the classical mar norm defined by

l l z l l " " : max{ lRe z l , l lm z l } '

Uncler the hypotheses of Theorem 6.5.1, prove that if 0 # p(ir,Q)at sorle flequency c,,' ) 0, then

H ( u ) : _ T i n ^ . l l " l l * .zep\Ja,\1 )

- l l r q t i c : i n r r i vpn f r pn r rencw r , r ) 0 H ( r , t \ can he v i ewed as t heI I I @ U I D r 4 u 4 E ) r v u r r r r u Y u l r a L J w ' v 1

disbance in max norm frorn the origin to the closest point (in max

norrrr) to the Kharitonov rectangle p(i.,Q).

94 Cln.6 / Embellishmenrs of Khar-itonov's Theorem

6.6 Robustness Margin Geornetry

Fbr a robustly stable interwal polynomial family, there is a temp_tation to associate the distance of the l{haritonov rectangle to theorigin z : 0 with the robustness margin. Said another .w.ay, whenp(jr't,Q) remains "far away" from z : 0 for all c..r ) 0, there is a temp-tation to conclude that there is a "significant" robustness rnargin;when p(ju:, Q) is "close" to zero for some frequencies, the tempta_tion is to conclude that the robustness margin is smari. The mainobjective of this section is to show that such reasoning can be falla-cious. This provides motivation for the Tsypkin*polyak analysis inthe next section.

To quantify the idea above, we study the beha'ior of the distancebetween the origin z:0 and the Kharitonov rectangle p(jut, e). Wewant to find the minimum of this distance with respect to frequency;i.e., we want to calculate the closest distance between ail possibleKharitonov rectangles and the origin. For ex'ample, two naturalmeasures for minimum distance are

2

1 . 5

1

1 . 4t . 20 .80 . 60 .40 . 2

and

d* in :m in{ l l z l l - : z € p ( ja ,Q) ; c . , , > 0 }

di^t- : min{ lz l : z € p( ju: , Q); . > O}.

In the e>ra"mple below, we work with d-an noting that the conclusionswhich we draw are also valid for other minimum distance ,''easuressuclr as dlnn above.

ExAMeLE 6.6.1 (Inadequacy of the Distance Nleasure): We con_sider the e-paraineterized uncertain polynomial

P ' ( s ' q ) : s 2 * e s + ( 1 0 0 * q )

with uncertainty bounding set Q, : l-r,rl . Since the resulting fam_ily of polynomials is monic and second order, stability is equivarentto positivity of coefficients. Hence, iobust stability is guaranteed ifand only if r < 100. In other words, a natural ,obusiness marginbased on parameter space considerations leads to r^o,: 100. Theimportant point to note is that this robustness margin is invariantwith respect to e ) 0.

We now compute the distance measure dmin(€) as a functionof the parameter e ) 0. Indeed, given any frequency @ ) 0, weobtain the Kharitonov rectangle by first noting that the value setir

ill

6.6 ,/ Robustness Margin Geometry 95

FIcunB 6.6.1 Line Segments for Example 6.6.1

is a straight line segment with constant imaginary part; i.e., withRe p(iut, a) : 100 + q - ,2 and Irn p(i., q) : €cd? we obtain

p . ( j c , t , Q " ) : { z e C : 1 0 0 - u 2 - r l R e z / - 7 0 0 - t t 2 + r ; f n t ' z : e t r }

rvlrich is depicted in Figure 6.6.1. Note that p,(ju,Q,) is generatedusing the trvo l(haritonov polynornials

1(r ( " ) : I {n(" ) : s2 tes * (100 - r )

to describe the lefi endpoint and the two Kharitonov polynomials

Ih(s) : Ks (s) : s2 * es * (100 * r )

to describe the right endpoint.Now, with the help of the Figure 6.6.1, for r < 100 and fi.xed

frequency u ) 0, it is easy to see that

_g in^ , l l r l l - : max { tOO - .2 - r , - 100 * a2 - r , eu } .zep\JU,c4 )

Next, we compute the frequency at which p(jr,Q) is closest to z :0

by setting ea; : 100 - u2 - 11 . This leads to

)€-_ _ I

z

L 0 0 - u 2 - r \ 0 0 - a 2 + r R e

d* i n (e ) :

96 Ch.6 / Embellishments of Kharitonov's Theorem

Hence, we conclude that d*i-(e) * 0 as e ---+ 0' Notice that this

conclusion holds even if the uncertainty bound r is small. The main

point to note is that smallness of d*in(e) for small r is inconsistent

with parameter space considerationsl i.e., even if e is small, the ro-

bustness margin is r*o': 100. For small r ) 0, the fact that d^';n(e)

can get arbitrarily small as e - 0 tells us t/nat d^rn is not a good

robustness indicator.

REMAFxS 6.6.2 (Understanding the Example Above): To en-

hance our understanding of the example above, we study the roots

of p, (s ,q) as a funct ion of e ) 0. By set t ing p, (s ,q. ) :0, we f ind the

pa,ir of roots

" r , z ( q ) :

Hence, for lql < 100 and small e ) 0, we obtain a l ightly dampecl

root pair which approaches the irnaginary axis as 6 + 0. This rnan-

ifests itself via smallness of d-1.(e) and closeness of the Kharitonov

rectangle to the origin. In other words, smallness Qf d^n. goes hand

in hand with closeness of the roots of p(s, q) to the imaginary axis.

As e - 0, the roots approach the imaginary axis even when the

ro-bustness margin for q.s ta-Ec-

6.7 The TsYPkin-PolYak F\rnction

Motivated by the comparison between d^;n ar-.d r-o" above, the

main objective of this section is to describe a technique for graphicai

visualization of the robustness margin. We want to generate a plot

which, upon inspection by eye, provides easily understood inforrra-

tion about the robustness margin for stabilitl,. Recognizing that a

plot of the Kharitonov rectangle does not explicitly provide such in-

{brmation, we now proceed to construct the r'obust stability testing

function described by Tsypkin and Polyak (1991).

In the analysis to follow, we use the sarne notational conwention

as in Section 5.9. That is, we emphasize the dependence on the

uncertainty bound r > 0 by writing

n - 7

p,@, q) : po(") * " E l -e; , ea)six : u

and interpret the e, ) 0 above as scale factors which determine the

aspect ratios of the uncertainty bounding set Q". We call p6(s) the

- - ) - ; (100 + q) -

6.7 / The Tsypkin-Pol1'akFuncrion 97

Ftcunp 6.7.1 Uncertainty Bounding Set for Example 6.7.1

nomi,nal polynomial and study the interval polynomial family

P , : { p , ( . , q ) , q e Q , } .

To avoid obfuscating technicalities invoiving degree dropping, noticethat uncertain parameters only enter coefHcients up to order n - 1above. In Theorem 6.7.2 to follow, we make use of the max norm onz e C. We recal l that l lz l l - : max{ l /?e z l , l l rn z l } .

ExAMeLE 6.7.1 (Uncertainty Bounds): For the interval polynomialf am i l y desc r i bed by p " ( s , q ) : ( s2 * 10s * 5 ) + r ( [ - 3 ,3 ] s + [ - 1 ,1 ] ) ,the uncertainty bounding set Q" is shown in Figure 6.7.1. Noticehow the weights €o : 1 and e1 : 3 are used for shaping Q" and tjrescalar r ) 0 is used for magnification.

TFTEoREM 6.7.2 (Tsypkin and Polyak (1991)): For fired r ) O,consider the 'interual polynomial farni,ly P, wi,th order n ) 2, positiuetueights €o,€1, . . . t€n-L and stable norn i ,nal ps(s) . Let

G- - ( , , , - f i " Po ( i a ) , . I r n po ( i t ' t )- 1 r \ d ) :

\ - . . _ / r j

a - . - .

, t i - - ' * ,L*o 'Th,en, w'ith man nornl, on z € C, i,t follows that P, is robustly stable

98 Ch.6 / Embeil ishmens of Kharitonov's Theorem

i,f and only i,f the zero frequency condi,t'ion

is satisfied and

lpo(r0)l > ".0

l lc re(w) l l - > "

for all frequenci,es a ) O.

Pnoop: Since p6(s) is assumed stable, application of the Zero Exclu-sion Condition (Lernrna 5.7.9) indicates that P, is robustly stable ifand only if 0 / p,(ju, Q) for all r,,' ) 0. First, for c,t : 0, the nec-essary and sufficient condition for zero exclusion is |po(f0) I > r.0.Now, using the formula for p,(s,q), it is easy to see that for f ixedc.,' ) 0, the Kharitonov rectangle pr(i,,:, Q) is centeled at ps(7u), haswidth given by

dn@) :2 r \ - e i a "' ,7 .n

in the real coordinate direction and has heisht

d 1 @ ) : 2 r ) , . e ; u '

in the imaginarlr coordinate dit"",i;: This rectangle is depictedin Figure 6.7.2. From th is descr ipt ion of p, ( ju ,Q), i r is obviousthat 0 / p , ( j r ,Q) i f and only i f e i ther lRe ps( je) l > dp(r" t ) /2 or

l l rn ps( ja) l > d1@)/2. Equivalent ly ,0 / p , ( ju t ,Q) i f and onlyif either lRe G7p(co)l > r or l lrn Grp(a)l > r. That is, robuststabil ity is guaranteed if and only if the condition l lGTp(c.. ') l l- > tis satisfied for all frequencies u > 0. f,

REMARKS 6.7.3 (Graphical Visualization): The theorem of Tsyp-kin and Polyak suggests a natural procedure for graphical visualiza-tion of robustness margin

rmar : suP{r : P, is robustly stable}.

That is, we generate a Nyquist-like piot of the complex functionGrp(r) and imagine a square box centered at z: 0. The radius rof this box is initiallv small so that the box fits entirely 'finside" theGrp(.) plot. Next, we let the radius of the box expand unti l a firstcontact is made with the Grp@) plot. Now, in accordance with thetheorem, we denote the radius of the box associated with this firstcontact as rmar (see Figure 6.7.3). Now, taking the zero frequencl'

, ,: I

j j , ]

j, 1r lIl

I

I

lI

I, i

l,

f,t

a

po( j0 , )

6.7 / ^fh.e Tsypkin-Polyak Function 99

r00 Ch.6 / Embellishments of Kharitonov's Theorem

Frctrns 6.7.3 First Contact rvith the G7p(tr) Plot

norninal model described in Nise (1992). The plant transfel function,

given in Figure 6.7.5, represents the aggregation of eievatoI. actuatol,

vehicie dynamics and pitch rate sensor.(a) Verif}, that the nominal closed loop system is stable for 1i : 4'

(b) With a nominal closed loop polynomial pe(s) : DT:oa4st ob-

tained in (a), take uncertainty weights e,: based on percentages of

t l r e a i ; i . e . , w i t h e i : a i f o r i : 0 ,1 ,2 , . . . ' r L - 1 , f i nd t he robus tness

margin 'I '*o, by plotting the function Grp(.) '

(c) Using the percentage rveighting scheme in (b), studl' the effect

of increasing the loop gain 1( ort rmar; Dote that instability occurs

when tlre gain satisfies K > 25.9.

ExERCISE 6.7.6 (Extension) : In the setup for Theoreur 6.7.2, nor-e

that all rveights €i were assumed positive. Thi.s exerci.se is concerned

with the case when only a subset of the weights is positive. In other

words, only a subset of coefflcients is unceltain. Indeed, we now

assume that e,; ) 0 for zl : 0, 1, 2, . . . ,n-1, allowing for the possibil i ty

that e1 : Q.(a) For the rn eakened hypotheses that er : 0 for all z odd and ,o I O

dr(r)

I

Ftcunp 6.7.2 The Kharitonov Rectangle p,(ja,Q)

condition into account we let rs: lpo(j})l/e and obtain

rma t : m in { rg , r [ ^ " r ] .

E)cAtr4pLE 6.7.4 (Application of l lheorern 6.7.2): We consider theinterval polynomial family P" with nominal

po(s) : s6 + 15s5 -| 104sa I 420s3 * 1019s2 -F 1365s * 676: ( s + 1 ) ( s + a ) ( s + 2 + 3 j ) ( s + 2 - 3 j ) ( s + 3 + 2 j ) ( s + 3 - 2 j )

a n d s c a l i n g f a c t o r s e o : 6 7 6 , e r : 6 8 2 . 5 , € 2 : 5 0 9 . 5 , e s : 2 1 0 ,e4:52, €s : 15 and e6 : 1 . Now, the funct ion to be p lot ted is

Gre(-) :-a6 +ro4aa - ro:^gtP +676 l S u L a - 4 2 0 u : 2 + 1 3 6 5

,,t6 + 52ua -l 509.5a2 + 676 r 5 . , 4 * 2 r 0 a 2 + 6 8 2 . 5

By examination of Figure 6.7.4, L}re radius of the largest inscribedbox is rlro, x 0.2227. Hence, r*o, N min{l, 0.2227} : 0.2227.

ExERCTsE 6.7.5 (Pitch Control Loop): In this exercise, we considera robust stability problem associated with a pitch control loop foran unrnanned free-swimming submersible vehicle. We begin'vith the

- L o

Grp(r)

dn(r)

Irn

-1

Ftcuns 6.7.4 Plot of Gre(-) for Example 6.7.4

for at least one i even, argue that Theorem 6.7.2 rernains valid using

(b) For the case rvhen e.; : 0 for all i even and e6 I 0 for all at leastone zl odd, describe the appropriately modified Grp(r) function.

1 . 5

0 . 5

- 0 . 5

6.7 / 'fhe Tslpkin-PolyakFunclion l0l

102 Ch.6 / Embellishments of Kharitonov's Theorem

ExERCTsE 6.7.7 (Reconcil iation): The objective of this exercise isto utilize the Tsypkin-Polyak function to carry out a robust stabil-ity analysis for the e-pararneterized uncertain polynomial given byp.(s, q) : s2 * es * (100 -F q) and studied by commonsense analysis inExample 6.6.1. Using the Tsypkin-Polyak framework, show that aformal calculation produces the correct result rmar :100 even whenthe parameter e ) 0 is small.

6.8 Cornplex Coefficients and T\:ansformations

The final set of embellishments which we consider involves uncertainpolynomials u'ith complex coemcients. Before proceeding, the ob-vious question to ask is how complex coefficient polynomials mightarise. Although the complex coefficient case can be motivated frornmodeliing considerations, more direct motivation is derived from thefact that the solution of many complex variable problems is oftenfacilitated via transformation. fn some cases, a real coefficient poly-nomial is transformed into a relrl coefficient poiynomial (for example,a bil inear transformation). rn'hile rn other cases, a transformation of areal coeffi.cient polynomial leads to a complex coeffi.cient polynomial.This is illustrated via the exarnple below.

ExAMeLE 6.8.1 (How Complex Coefficients Arise): We begin rvitha family of polynornials 2 : {p(.,q) : q e Q} having real coef-ficients, and suppose that we are dealing rvith a control problemwhere damping is of concern. For example, say that the dampingcone D for the loots of p(s,q) is given in Figure 6.8.1. We nowargue that we can transform tire real coefficient robust D-stabilityproblem at hand to a (strict left half plane) robust stabil ity prob-lem with complex coefHcients. Indeed, we first express 2 as an in-tersection of two half plaines; i.e., we write D : D+ aD-, whereD- -

{z e C i r - O < fz < Z" - d] is the lower halfplane andD + : { z e C : d < L z < a * / } i s t h e u p p e r h a l f p l a n e . T h e s e t r v oregions are shown in Figure 6.8.2.

We norv obselve that P is robustly D-stabie if and only if 2 isrobustly D+-stable anJ P is robustly D--stable. Therefore, we canstudy the robust ?l+-stabil ity problem and the robust Z--stabil ityproblem separately. We illustrate for 2+ noting that an identicalargument is used for D- . Indeed, i f p(s, q) : ILo a i (q)s i is a realcoeffrcient polynomial and we introduce the change of variables

; e i

Ii i' r i ,'1'4

i

.i't i

. -',..]i

-0 .5 0 0 .5 1Re

III

0.25K(s + 0.435)

s4 + 3 .456s3 * 3 .457 s2 * 0 .719s + 0 .0416

Frcunn 6.7.5 Nominal Pitch Control Svstem for Exercise 6.2.5

2 : t " - i ( [ - i l ,

6.8 ,/ Complex Coefficients and Transformadons 103

\.ve obtain t}re transforrned polgnornial

p ( " , q ) : p ( ze i ( t - 6 ) , n1

Notice that the original family of polynomrals P is 2+-stable if andonly if the transformed family of polynomials P : {p(., q) : q e Q} isrobustly stable; i.e., u'e LakeD to be the strict ieft haif piane for thetransformed problem. It is also apparent that fi(z,g) has complexcoefEcients but p(s, q) does not. In fact, we can write

n

p(" , q) : \ lw(d + i Fr (q) l r i ,; -n

where

a i (q ) : aa (q )cos

0;@) : aa(s) s in

104 Cjn.6 / Embell ishmens of I(har- i tonov'sTheorem

FIGURE 6.8.2 The Flalfplanes 2- andD+ for Example 6'8' l

formations for which affrne linearity is preserved'

6.9 l{haritonov's Theorern with cornplex coefficients

Given an uncertain polynomial with complex coefficients, we use real

uncertain parameters Qi and r; to denote uncertainty in the real and

imaginary parts of the coefficients of si, respectively; i'e', we write

p(s, q, 11 : ilou * jr.i) si .; -n

shorthand notation

: no(q) - t a.(q)zei( i -A) + . . . - t a.(q)zn"in([-Q): a o ( q ) + a t ( q ) f s i n @ * 7 c o s 0 ] " + . . -

+ a-(q) [.". (T -

"o) -F 7 sin (+ -

"r)],

' ( i - r ) ''G - r ) n

- \ - / 1 . .

"'-n

FIGURE 6.8.1 Damping Cone D tbr ExarnpLe 6.8.1 f r , i

f o r i : 0 , 1 , . . . , n .- ( o n r \y \ v ) Y t I ,s[] + j lr; ,r l l)s'

6.9 / I{haritonov's Theorem with Complex Coefficients 105

Fol brevity, we can simply refer to p(s,q,r) above as a comTtle-r co-effi c'i ent int eru al p o Ig n o mi aI.

fn contrast to the real coefficient case, robust stability analysisfo'the complex coefftcient case requires eight Kharitonov polynorni-als rather than four. To provide some insight as to why four ,,extra',

Kharitonov polynornials are required, we mention a fundamental dif-ference between the real and complex cases. Namely, if p(s) is a realcoefficient poJ.ynomial, lve can restrict fr-equency sweeps to c.,' ) Q;that is, fol cu ) 0 and p. (j-) denoting the complex conjugate ofp(j-) ' t 'e han'e

pG ju) : p*(j,,.t).

In contrast, if p(s) is a complex coeffi.cient polynomial, the equalityabove typically does not hold. In the proof of l{iraritonov's Theoremfor the real coeffi.cient case in Section 5.8, we exploited tihis conj,u-gacA propertg itt restricting the analysis to frequencies a-l > 0; i.e.,if p(jut,Q) denotes the Khalitonov rectangle at a given frequencycr € R, the conjugacypropertytells us that z ep(ju.t,Q) if and onlyi f z * € p ( - j u , Q ) .

The rnajor difference in the complex coefficient case is that weneed to consider both cu ) 0 and a I 0 separately. Althoughexploitation of the conjugacy property is no longer valid, we canstill rvork with a Kharitonov rectangle taking care to discriminatebetween vertices for c,r < 0 versus o ) 0. In other wor.d.s, fourKharitonov polynomials is used for cu ) 0 and a "different,' set offour Kharitonov polynomials is used for cu ( 0.

ExERCTsE 6.9.1 (Lack of Conjugacy Property): For the complexcoefficient interval poiylottriut family P : {p(.,q) : q e e} describedby p ( " , ? ) : s3 + (5 + j q ) t 2 +4s+ 5 and e : [ - 1 ,1 ] , show tha r f o re : L , p ( - j a , Q ) I p . ( j a , Q ) .

DEFrNrrroN 6'9.2 (co'rplex coefficient I{haritonov polynornials) :Associated rvith the complex coefficient interval poiynomial given asp(s,q," ) : ILo( lqo ,qf l + j l r ; , r {Dsi are e ighr f lxed Khar i tonoupolynomi,als. The first four polynomials

1{r+(s) : (q i + j r i )+@l + j r [ )s+(q[ + j r f ) " '+(q3++7r1)s '+. . . ;

N{ ( ' ) : (qoF + j , t )+fu{ +iry) '+Gi- t j r l )s2 +(q + j r { )"r+. . . ;r<r* (") : (q0+ + j r; ) + @r + j r r ) s + (qz + j r[ ) t2 + (q{ + j"{)"t + . . . ;xf (") : (st + j r t ) + @{ +jr{) ' + (q[ - t j r2)s2 + (qi +jr i )"r+. . .

106 Ch. 6 / Embellishments of Kharitonov'sTheorem

are associated with ut ) 0, and the second four polynomials

Kf ( " ) : ka + j r ; )+(q[ + j r l ) "+(qz+ + j r [ )s2 +(q i +, j r t )s3+. . . ;

Kz G) : ( sd -+ r r .F ) +@l + j r { ) " - (q i + j r 2 )s2 + (q { + j r ; ) s3+ . . . :/ I - \

^ s ( s J : \ Q [ t j r o l + ( q [ + 7 r f ) s + ( q , + j r [ ) s 2 + ( q i + j r r ) s s + . . ' :

K+ ( t ) : (qo +:ro ' ' ) + (q, + j r l )s+ (qz+ t j12 )s2 + (qr+ +7r j - ) " t +. . .

are associated 'with c.., ( 0.

REMARKS 6.9.3 (Coefficient Pattern): Analogous to the real coef-ficient case, we see a basic pattern in the reai and imaginary parts ofthe coeffi.cients-two lower bounds followed by two lower bounds fo1-lowed by two upper bounds, etc. For the sake of completeness, it isalso important to mention that for complex coefficient poiynomials,rve use the same definition of invariant degree as in the real coefficientcase; i .e . , 2 has invar iant degree i f deg p(s, qL, 11) : deg p(s, q2,12)for all pairs (q1,11) and (q',r ') in the bounding set Q x E.

TITEoREM 6.9.a (Kharitonov (1978b)): ,4 cornpler coeffici.ent i.n-terual polgnomial fami ly P : {p( . ,q , r ) : q e Q;r e R} u i th i .n-uariant degree is robustlg stable if and only if i ts eight Kharitonoupolynom'ials are stable.

PRoop: We only sketch the proof because it is conceptually' identicalto the one used for the real coefficient case; see Section 5.8. \!e beginby noting that both the Zero Exclusion Condition (Lemma 5.7.9) forrobust stabil ity and the lVJonotonic Angle Property (Lemma 5.7.6)for stable polynomials remain valid in the cornplex coefficieut case.Now, to study the Kharitonov rectangle, we fix a frequenc! a : utgand seek a description of the set

p ( j , ' ; o ,Q , R ) : { p ( j . o , q , r ) : q e Q ; r € ,9 }

For c..rs ) 0, arguing as in Subsection 5.7.1, the seL p(jas,Q) is seento be a rectangle rvith southwest vertex K{ (jro), northeast vertext t - l - r \ , 1 , r , - 1 ,t\ i \Jtio)j sourneasr vercex Kt(j. i l and northwest vertex K[(j.o);recall Figure 5.7.2. Similarly, for crs ( 0, the description of the setremains the same except Kt Uro) replaces Kf (jro)

Analogous to Section 5.8, the proof of necessity is immediateand the proof of sufficiency involves two cases. In Case 1, we assume0 € p(j), Q) and contradict the stability of at least one I{haritonovpolynomial. In Case 2, we assume that 0 / p(j}, Q) and proceed

: | j

6.10 / Conclusion I07

by contr-adiction; i.e., if P is not robustly stable, the Zero ExclusionCondition implies that 0 € p(j.*, Q) for some c.'* e R.

The cornpletion of the proof involves two subcases. In Sub-case 2A, we take d* ) 0, and, arguing as in Section 5.8, we arriveat a contradiction using /{r+(s), 1(r+(s), Kr*(") and K1(s). In Sub-case 28, we take cu* < 0 and use the l(n (s) in l ieu of the /{o+(s) toarrive at a contradiction in a similar manner. I

6 .10 Conclus ion

Although the forrnally stated objective of this chapter was to provideextensions and refinements of Kharitonov's Theorem, there was asecbnd "secret" objective-to further demonstrate the power of theI(haritonov rectangle as a technical device. We are now well preparedto proceed with some generalizations. To this end, the next chapterdeals with more general uncertainty structures; we see that Lhe ualueset plays exactly the same role as played by the Kharitonov rectangle.\zlore specifically, instead of working with the zero exclusion from theKharitonov rectangie, we work with zero exclusion from the value set.

Another important point to note is that the value set formulationis rvell suited for the more general robust D:stabiiity framework. In-stead of the sweeping frequency cd, .we sweep a ge,neralized frequencyvariable 5; the sweeping of d is seen to correspond with a sweep ofthe boundary of D. For each 6, we obtain a vaiue set, call iLV(5),and derive a general zero exclusion condition

o / v ( 6 )for robust 2-stability.

Notes and Related Literature

NRL 6.1 The comparison between r^o, an:rd d^;^ jn Section 6.6 raises concern

about the robustness margin definition in parameter space. In addition to ex-

posing the fact that r*.. is ulinforrnative about root locations, the literature

also describes ill-conditioning problerns associated with numerical compuralron

of , r^o"; e.9. , see Barmish, Khargonekar, Shi and Tempo (1990).

NRL 6.2 As pointed out in Tsypkin and Polyak (1991), the function G7p(ut) in

Theorem 6.7.2 has a number of important properties which are'consequences of

being a N{ikhailov-type function; see jvlikhailov (1938) and Lemrna 5.7.6. For in-

stance, with po(s) assumed stable, G1p(cr) has a monotonically increasing phase

whiclr moves through n quadrants in turn with total phase increase of ntr/2.

108 Ch.6 / Embel l ishments of Khar i tonov'sTheorert

NRL 6.3 In Tsypkin and Polyak (1991), a more general set t ing is considered

tlran that providli in Theorern 6.7.2; t1o.e uncertainty bou'ding set Q- is a ball

i.,. .ty p."r".ibed lp norm. For example, fbr the case p: 2' if one begins with

nominals i

p . ( s , S ) : p o ( s ) + r ) c t s

r - 0

and the testing function to be plotted turns out to be

By specia"lizing to the case when all weights are equal, the formula above leads im-

mediately to the solution given in soh, Berger and Dabke (1985) and its control

theoretic version given by Biernacki, Huang and Bhattacharyya (1987)' How-

ever, in both of these papers, the graphical visualization of robustness margin

is missing. The distinguishing feature of the Tsypkin Polyak function is that

it is complex-valued, whereas, in the papers cited above, a real-valued testing

function is used.

NRL 6.4 F\rther motivation for studying the complex coefficient case is given in

the paper by Bose and Shi (1987). Whirling shafts, vibrational systerns and fiIters

are mentioned as examples of systems whose models involve complex coefficient

polynomials

NRL 6.5 The transformation of the damping cone problem into a strict left

half plane problem in Exarnple 6.8.1 is only one of many possible ways by which

conformal mappings induce complex coefificient polynomials. In the robustness

analysis of sondergeld (1983), a table ofother useful transformations is provided.

t,, ;I

i r ,t , :l : r r

l l ., ,t .

i ,i'.:.i,1 . , ' .L - l

i ' . l, : l ri r : ' ii ' l I

i . i , i 1i,,l,jr:i rl]l ' , ' lf ,i ii , i

i , , jr ' 1l r , ll , . lt . t : j

i r , . , ]i . , ' ii . i l i: ; : . I

i , : il , ll . ti::i iitr,l ' ji . ll t : i j l

l:.;: 1i i

; l j-lj : ,.]

Chapter 7

The Value Set Concept

Synopsjs

frr flris chapter, tlte value set is defined and seen to be a gener-alizatiott of tlte l{ltaritonov rectang}e. Sweeping the irnaginaryaxjs js replaced by sweeping the boundary of a desired root lo-cation region D. Subsequently, a link is establislted beh.veenrobust D-stability and the value set; i.e., a more general zeroexclusion condition is established.

7.1 Int roduct ion

When we exposed I{haritonov's Theorem and its embeliishmentsin Chapters 5 and 6, the Zero Exclusion Condition (Lemma S.T.g)played a critical role. That is, for an interval polynomial family withinvariant degree and at least one stable member, robust stability isequivalent to zer-o exclusion from the Kharitonov rectangie p(j.,Q)at all frequencies cl ) 0.

In this chapter, we no ionger restrict our attention to intervalpolynomials. For rather general uncertainty structures, we definethe value set which is seen to be a generalization of the Kharitonovrectangle. Subsequently, we see that the Zero Exclusion Condition isstill meaningful in this more general setting. In fact, as the chapterprogresses, we also dispense with left half plane stability and considerthe value set concept in the more general framework of robust D-stabil ity as defined in Section 4.13.

110 Ch. 7 / The Value Set Concept

It is not our intention to solve the robust 2-stability problern

in this chapter. Our primary objective is to demonstrate that therobust D-stability problem is equivalent to a zero exclusion frorrr anappropriately constructed value set. This sets the stage for laterchapters which address the problem of value set characterizationand subsequent solution of the lobust 2-stability problern. In thischapter, only relatively simpie value sets are characterized.

In the remainder of this text, there are two contexts within whichvalue sets arise. First, in many situations '!ve use the value set asa technical Stepping stone within a proof. For such cases, under-standing the final result does not require any knorvledge about value

sets-the value set can be hidden from the user. Kharitonov's The-

orem exemplif ies this situation; i.e., from a useris point of view, one

does not need to know about the Kharitonov rectangle to use the

theorem in testing for robust stability.The second context within which the value set arises: Tltere are

many results whicb. are communicated in a computer-aided graphics

framern'ork. For such cases, we provide a recipe for construction ofthe value set, and, the robustness test amounts to displaying this set

as a function of a generalized frequency variable. In this context,

the value set is on "center stage" in the sense that the user rnustunderstand the meaning of the value set in order to generate the

appropriate graphical display.

7.2 The Value Set

Roughly speaking, given an uncertain polynomial p(s,q) and an un-certainty bounding set Q, then, at a fured frequency t..r € R, thevalue set is the subset of the complex plane consisting of all values

which can be assumed by p(ir,q) as q ranges over Q. Said another

way,p( jut ,Q) is the range of p( i r , ' ) .This idea is s tated formal ly inthe definit ion below.

DEFTNTTIoN 7.2.1 (The Value Set): Given a family of polynomials

P : {p(- ,q) : q € Q}, the ualue set at f requency u, ' € R is g iven by

p ( j a , Q ) : { p ( i u , q ) : ( t e Q } .

That is , p( j . ,Q) is the image of Q under p( ja , ' ) .

ExAMPLE 7.2-2 (Yalue Set as a Straight Line Segment): We con-

sider the uncertain polynomial p(s,S) : s2 I (2 - q)t + (3 - q) anduncertainty bounding set Q: [0,4] . Notice that for f ixed cu € R,

1 0 9

7.2 / The Value Set f t l

Rep( ju: ,e) : 3 - u2 - q and l rnp( j " . , ) : (2 -q)cu. Hence, for each.r € R, the value set p(jtt,Q) is a straight I ine segment joiningp ( j u t , 0 ) : ( 3 - - 2 ) + 2 ju and p ( j u , ) : - ( 1 + .2 ) - 2 j . .Th i s i si l lustrated in Figure 7.2.7 for 0 { a < 4. Notice that p(s,q) f its ex-

4

2

Ini 0

- 4

-6

-8-18 -16 - r4 -L2 -10 - t * " -U -4 -2 0 2 4

F'IGURE 7.2.1 The Motion of the Value Set i l Example 7.2.2

actly into the one-parameter framework of Chapter 4. For example,with /(s) : s2+2s+3 and 9(s)

-- -4s-4, the family of polynornialsdescribed by p(s, A) : /(s) + )9(s) and A : [0, 1] is the same as theoriginal family P : {p(.,q): q e Q}. These ideas are generalized inthe lemma below; the nearly trivial proof is omitted.

LEI\4MA 7.2.3 (Yalue Set with One Parameter Entering Affinely) :Let f (s) and g(s) be fi.red polynomials. Then, for the fami,ly of poly-norn ' ia ls descr ibed by p( t , l ) : / ( " ) + )9(s) and A € A: [0 ,1] , u i th

fixed u € R, the ualue set p(ju:, 4v) is the straight line segrnenl joiningthe points p( j , ' t ,0) : f ( ju) and p( ja, I ) : f ( ja) + SUu).

ExERcrsE 7.2.4 (Yalue Set Parallelogram): Given the family ofpolynornials described by

p(s,q) : s4 -F (3sr + 4qz * ]r2)s3 + (qr - 2q2 + 6)s2

* (2qt - 3qz + 8)s + (6 + q l ) ,

112 Ch.7 / The Value Set Concept

tr lcunB 7.3.1 Pathwise Connected Set in R2

lqrl < 1 and lq2l ( 1, argue that the vaiue set p(jt,.Q) is a palallel-

.;;"- and plot it for ten frequencies evenly spaced between cu : 0

u.rrd , : 2. This problem serves as a preview of the polygoDal value

set theory in the chapter to foliow'

7.3 T\te Zero Exclusion Condition

when rve introduced. the Zero Exclusion condition for interval poly-

nomials in section 5.7.8, it was noted ttrat this condition actually

applies to much nlore general uncertainty structures. In this section,

we establislt a Zero Exclusion Condition for a family of polynomials

2 having continuous coefficient functions a;(q) and pathv"ise con-

nected uncertainty bounding set Q. For completeness, we no.w define

pathwise connectedness.

REMART(s 7.3.2 (Pathwise Connectedness): The notion of path-

wise connectedness is i l lustrated in Figure 7.3.1. From an applica-

tions point of view, assurning pathwise connectedness of the uncer-

o(0) : uo

o ( 1 ) : u 1

7 3 / The Zero Exclusion Condirion 113

tainty bounding set Q is typically quite reasonable. In this regard,note that every convex set Q (such as a sphere or a box) is pathwiseconnected. This is easily established by noting that if qo,qL € Q, thefunction @ : [0, I] - Q given by

o ( r ) : ( 7 - t ) q o + t q t

is cont inuous and sat is f i .es the condi t ions @(0): q0, O(1) : q1 andO ( f ) € Q f o r a l l l € 1 0 , 1 1 .

TrrEoREM 7.3.3 (Zero Exclusion Condition) : Suppose that a fam-iLy of Ttolynom'ials P : {p(., q) : S € Q) has inuariant degree w,ithassoc'iated uncertainty bounding set Q tthich i,s pathwi,se connected,con t i nuous coe f f i c i en t f unc t i ons a ; (q ) f o r i : 0 ,1 ,2 , . . . , n and a tleast one stable mernber p(", q0) . Th,en P ,is robustly stable i,f and,onLy i , f the or ig in, z :0, is exc luded f rorn, the ua, lue set p( ja ,Q) atall frequencies a ) 0; i.e., P is robustly stable i,f and only i.f

o f P ( j r ,Q )

for n. l . l . f rent tc.n.r . i .e.q, , r ) 0.

PRoor': To establish necessity) we assume that 2 is robustiy stableand rrrust shor.v that 0 f p(j., Q) for all c.., ) 0. Proceeding by contra-diction, suppose 0 e p(jc,;-, Q) for some a,l* ) 0. Then, p(ju*, e.) : 0for some q* € Q. This contradicts the assumed robust stabil ity of 2.

To establish sufficiency, we assume that 0 4 p(jr,Q) for allcu 2 0 and must show that 2 is robusLly stable. Proceeding bycontradiction, suppose that p(s,ql) is unstable for some qr e Q.Tlten, by pathwise connectedness of Q, there exists a continuousfunct ion Q : [0 ,7] - Q such that O(0) : g0 and O(1) : qr . Now, inaccordance wi th Lemma 4.8.2, Iet s1(g) , s2(S), . . . , s . (q) denote rootfunctions for p(s,q) which vary continuously witb. respect to q e Q.S ince , p (s , q1 ) i s uns tab le , i t f o i l ows tha t f o r some i * e { I , 2 , . . . , n } ,

"o-(qt) is in the right half plane. On the other hand, we know that

sr-(q0) Iies in the strict left half plane. Next, for t € [0,1], notice thatsi"(O(t)) describes a continuously varying root of p(s, O(i)) whichbegins in the strict left half plane at s;" (O(0)) and terminates in thel ig l r t hal f p lane ab s1" (O(t ) ) . In v iew of cont inui ry of s1. (O(f ) ) wi threspect Lo f , there musL exis t some f* € (0, 1] such thar s i . (O(f - ) ) l ieson the imaginary axis. Takin1 e* : O(l-) and noting that p(s, q*)has an imaginary root, it follows that p(jut" , A.) : O for some o* ) 0.Hence, 0 € p(je*, Q), which is the contradiction we seek. f

114 Clt 7 / The Value Set Concept

ExERCTsE 7.3.4 (Finite Union of Pathwise Connected Sets): Givean example showing that Theorem 7.3.3 no longer remains validwhen Q is a finite union of pathwise connected sets.

ExERCTsE 7.3.5 (Cutoff Frequency): In addition to the hypothe-ses associated with the Zero Exciusion Condition, assume that Q isbounded. Prove that there efsts some frequerrcy @c > 0 having theproperty that 0 / p(j.,Q) for a ) ac. H,int: Consider the function

n- - -

.f (.) : (ry'6 1"" k)0." - f

,m pr(q)Dui.

7.4 Zero F.xclusion Condition for Robust 2-Stabii ity

The objective of this section is to show that the zero exclusion con-cept can easiiy be exlended to the more general robust 2-stabil ityframework. To this end, we first provide a more general definition ofthe value set. Instead of evaluating an uncertain polynomial alongthe irnaginary axis, we consider an arbitrary evaluation point z e C.Furthermore, instead of sweeping the imagjnary axis, we sweep rheboundary of 2. The fi.nal result, Theorem 7.4.2, provides the mostgeneral version of the Zero Exclusion Condition, which is given inthis book. In later chapters, this theorem is frequently invoked.

DEFTNT:rroN 7.4.1 (The Value Set at z € C): Given a famity ofpo l ynomia l sP : {p ( . , q ) | q € Q} , t i ne ua , l ue se ta t z € C i s g i ven by

p ( z , Q ) : { p ( z , q ) , q e Q }

That is , p(" ,Q) is the image o l Q under p( t , ' ) .

THEoREM 7.4.2 (Zero Exclusion Condition): Let D be arr. operr.subse t o f t he comp le rp lane and suppose tha tP : {p ( . , q ) : q e Q} i ta fami,ly of polEnomials with'inuo,riant degree, uncertainty boundzngset Q which zs pathwise connected. Furthermore, assLlrrLe that thecoefficient functions a;(q) are cont,inuous anC that P has at least oneD-stable member p( t ,qo) . Then P is robust ly D-stabte i f and only i f

o 4 p Q , Q )

for all z e 0D, tuhere 0D denotes the boundary ofD.

PRooR: We omit the proof since it is nearly identical to that givenfor Theorem 7.3.3. !

Fil;li1,,I lii,i:liir, I

[t'i[,i rit,,'li ' if t

[': ilir:ill . . r

i',i,j1.,i,ii.iii . i ' li t r , ' ; j

I 16 Cb.7 / The Va lue Set Concept

7 4 / Zero Exclusion Condirion for Robust D - Stabil itv l lE

ExER-crsE 7.4.3 (Bounded D and Invariant Degree): Urrder thestrengthened h5,pothesis that D is bounded, argue that the invariantdegree assumption is no longer needed in Theorem 7.4.2.

ExAr\4rpLE 7.4.4 (Robust Schur Stabil ity): We take D to be theinterior of the unit disc and consider- the family of polynomials Pd e s c r i b e d b y p ( s , Q ) : s 3 + ( 0 . 5 + q ) s 2 - ( 0 . 2 5 + q ) a n d Q : [ - 0 . 1 , 0 . 1 ] .To apply tt.e Zero Exclr-rsion Condition, we first identifli one stabletnernbet' of 2. Indeed, by taking q : qo : 0 and computing lootss r , z : - 0 . 5 + 7 0 . 5 a n d s s : 0 . 5 , w e s e e t h a t p ( s , q 0 ) i s S c l r u r s t a b l e .Next, rve sweep z around the unit circle while plotting the value setp(z,Q). Since q enters affine Iinearly into the coefficients, a minorextension of Lernma 7.2.3 (wirh z replacing 7cu) enables us to as-ser t that p(z,Q) is the st ra ight l ine segment jo in ing p(2,-0.1) andp(2,0. I ) . In F igure 7.4. I , the value set p(z,Q) is shown using 500

1 . 5

0 . 5

Irn 0

-0 .5

- 1

Re

Frcunn 7.4.1 The Value Set p(2, Q) for Example 7.4.4 witin lSl < 0.1

evaluation points z : zi evenly spaced around the unit circle. It isclear by inspection that 0 # p(t,Q) for a}l z. Hence, we concludethat P is robustly 7)-stable.

\Me now entertain an increase in the uncertainty bound. Taking

Q: l -0.3,0.3] , the value set is p lot ted again us ing 500 evaluat ionpoints and P is sti l l seen to be robustly stable; see Figure 7.4.2.By further increasing the uncertainty bound, we can determine the

D

o <

Ftcunp 7.4.2 The Valu€ Set p(2, Q) for Example 7'4'4 v"ith lql < 0'3

robustness margin; i.e.' i f we take Q: l-,,r] , r 've seek the supremal

value of r, cali iL r*o,, for which r-obust 2-stability is guaranteed. By

gradually increasing r while checking the zero Exclusion condition,

we obtain robustness margin rmo' x 0.38.

ExERcrsE 7.4.5 (CtLoff for Boundary Sweep): In this exelcise,

we consider the 2-stability analogue of the cutoff frequency concept

discussed in Section 5.10. Indeed, suppose 2 is unbounded, Q is

bounded. and.P : {p(',q): q e Q} is a family of polynomials with

invariant degree and with continuous coeffrcient functions ai(q) for' i : 0 ,L,2, . . . , n . Now, prove there ex is ts a bounded subset Dg of

2 such tha t i f z € D /Ds : { z € D "

z # Do } , t hen 0 4 pQ,Q) '

Hdnt'. Corstruct a function /(z) such that 1irn1,1-* f ("): 1co and

0 4 pQ, Q) whenever / (z) > 0.

7.5 BoundarY SweePing Functions

This section introduces the notion of a boundary su'eeping function.

we see tirat such a function facilitates calculations associated with

value set generation and zero exclusion testing. Indeed, since testing

for satisfaction of the zero Exclusion condition involves sweeping

the boundary DD of D, tl is convenient to have a scalar parametel

-2 -1 .5 -1 -0 .5 0 0 .5 1Re

1 . 5

1

0 . 5

[m. 0

-0 .5

-1

1 . 50 . 5- u . 5- 1

7.5 / Boundary Srveeping Funcrions Ll7

6 whiclr can be used to parameterize motion along 0D. For robustD-stability analysis, this scalar 6 plays a role which is analogous totlre fi 'equency c,r in ordinary r.obust stabil ity analysis.

DEFINTTIoN 7.5.1 (Boundar-y Sweeping Function): Suppose thatD is an open subset of the complex plane with boundary 0D. Tinen,given an interval (perhaps semi-infinite or infinite) 1 C R, a mappingQp : I ---+ 0D is said to be a boundarg sweeping Junction for D if Qpis continuous and onto; i.e., (D2 is continuous and for each pointz € 0D, there exists some 6 € -I such that

a D @ ) : z '

The scalar 6 is called a generalized frequency variable for D.

ExAMeLE 7.5.2 (Ha"lfplanes): When D is the strict left half nlane

a 6 @ ) : j 6

corresponds to setting s: ja.Notice that there are infinitely manyother possibilities for a boundary sweeping function for the strictI e f t l r a l f p l ane ; e .g . , t a " ke 1 : ( - oo ,co ) and Ao@) :7ds in 6 . Now,suppose D is a haifplane reflecting our concern about the degree ofstabi l i ty ; that say D : { " € C: Re" 1-o} , where a > 0 is g iven.Then we can take ,I : (-oo, co) and

O p ( 6 ) : - o l j 5

as a boundary sweeping funclion.

ExAMeLE 7.5.3 (Unit Disc): When D is the interior of the unitdisc, we obtain a boundary sweeping function with I : 10, 1] and

az(6) : cos 2tr6 * j sin 216.

ExAMPLE 7.5.4 (Damping): For a robust Z-stabil ity problem lviththedarrrp ingcone D : {z € C : r -0 < Lz < r - t?} and} < 0 < r f2,a suitable boundary sweeping function is described by 1 : (--, -) 'and

118 Ch. 7 / The Vah.re Set Concept

ExERcrsE 7.5.5 (Zero Exclusion via Boundary Sweeping Functions):In addition to assumptions associated with the Zero Exclusion Con-dition in Theorem 7.4.2. suppose thar D has boundary sweepingfunction Q7t : I r+ AD. Prove that P is robustJ.y stable if and onl'' if

o { p @ p ( 6 ) , Q )

for all generalized frequencies 5 e L

ExERCTsE 7.5.6 (N4ore General Boundary Sweeping Functions):Consider the desired root location region given by D : D1l)D2,where Dl and D2 arethe strips Dl : {z € C : -1 < Re z < 0}and Dz : {z € C : Re z I -7;-1 < Irn z < 1}. Tlne ZeroExclusion Condition, as giwen in Exercise 7.5.5, cannot be appliedin the obvious wa)r because the domain 1 of Q2 must be an interval.while Q2(5) is continuous. Generalize the statement of the ZeroExclusion Condition to allow for D regions of the sort describedabove.

ExERCTsE 7.5.7 (More General Interpretation of Zero Exclusion):Let D1,D2,. . .,D- be disjoint open subsets of the cornplex plane andsuppose P: {p(- ,q) : q e Q} is a fami ly of poiy 'norn ia ls wi th invar i -ant degree, uncertainty bounding set Q which is pathwise connectedand con t i nuous coe f f i c i en t f unc t i ons a i (q ) f o r i : O ,L ,2 , . . . , n . Fo reac l r q € Q and i € {1 ,2 , . . . , r n } , I e t n ; (q ) deno te t he number o froots of p(s,q) inDi . F inal ly , assume that for some q0 e Q, pG,qo)has no roo ts i n t he bounda ry o f D :D tUDzU ' . .UD- . Now, p rovethat each of the root indice" A(S) remains invariant over Q if andonly if tLre Zero Exciusion Condition O / p(r, Q) is satisfied for allpoints z e AD.

7.6 More General Value Sets

In this section, we provide further testirnony to the power of the valueset point of vierv. To this end, we begin by noting that the definitionof the value set is trivialiy generaiized to uncertain functions whichare not necessarily polynomials. After all, the value set is nothingmore tiran the range of a function. However, the value set is spe-cial in the sense that the range of concern is only two-dimensionai.This fact drives much of the theory in this book; i.e., we replace amultidimensional robustness problem over the uncertainty boundingset Q with a two-dimensional geometry problem over the value set.With these ideas in mind, the definition below is natural for func-

[ 6 . o . g + j 6 s i n 0 i f 5 < 0 ;\f - 6 c o s 0 + j S s i n d i f 6 > 0 .

Q p ( 6 ) :

7.6 / More General Value Sets I l9

Frcune 7.6.1 Uncertain De1a1. System lbr- Example 7.6.2

tions which are not necessarily polynomic. Using the more generalvalue set definition below, we are equipped to enlarge the class ofrobustness problems which we can address.

DEFrNrrroN 7.6.1 (Value Set at z e C): Given an uncertaintybounding set Q and an uncertain function defi.ned by a mappingF : C x Q - C, the aalue set at z €C is g iven by

F ( " , Q ) : { F ( z , q ) : q e Q } .

That is , F(r ,Q) is the image of Q under F(r , . ) .

ExAMeLE 7.6.2 (Delay Systems) : For the uncertain delay systemdepicted in Figure 7.6.1, the closed loop uncertain quas'ipolynornialis given by

p "@,q1 : l \ r ( s , q )Ns (s )e - ' " + D (s , q )Dc (s ) .

Now, given an uncertainty bounding set Q and a fixed delay r > 0,tlre associated value set at s: iu is

P' ( ju , Q) : {P,( ja , q) : s e Q}.

Using this definition, there are large classes of robust stability prob-Iems for delay systems which can be attacked using the Zero Exclu-sion Condition 0 ( p,(ju, Q); see the notes at the end of the chapterfor further discussiol.

120 C]n.7 / The Value Set Concept

V^(s) W-t(s)

Flcuns 7.6.2 RC Filter for Example 7.6.4

ExERcrsE 7.6.3 (Line Segment as Value Set for Delay Systems):

With the delay system selup as in the example above, suppose that

l/(", q) and D(s, q) have coefHcients depending affine linearly on a

single parameter q € [0, 1] ; i .e., there exist 6xed polynomiais lfo(s)'

Nr(" ) , Do(") and D1(s) such that

l/(s, q) : 1t/o(s) + q1llr(s)

andD ( t , q ) : D o ( s ) l - q D 1 ( s ) .

Argue that for s : ja , the value set p, ( ja ,Q) for the c losed loop

polynomial is a straight l ine segment with endpoints p"(ia,0) and

pr(jw,1). In othel words, the l inear nature of the value set segments(established in Lemma 7.2.3) is preserved in the presence of delay,

ExERCTsE 7.6.4 (RC Filter): Consider the simple RC fi lter with re-

sistance J? and capacitance C given in Figure 7.6.2. \ l i th unceltainty

ranges described by J-20% about the nominal values -R : 1000f2 and

C = l00tt'F, take g : RC and derive the uncertain transfer function

'l

r l s . o ) : -r - i _ q s

and associ.ated uncertainty bounding set I : [0.064,0.144] . For this

rational function, characterize the value set P(ja, Q) associated with

the {i-equency response and plot for representative frequencies in the

r a n g e 0 < c , . ' < 1 0 .

ExERCTsE 7.6.5 (Uncertainty in a Simple Zero): In the exercise

above, note that the value set associated with uncertainty in a pole

was nonlinear (curved) at each fixed frequency. For the case of a

_{(",s) "-""D(" , q )

7.7 ,/ Conciusion 121

simple zero, horlrever, the situation is much simpler. Indeed, considerthe uncertain transfer function

P( t , q ) :

nr u / r T - T z \, n ( s - a ) t t t s - z ; l

; - 1

ll(" - r,)i : 1

with fi:<ed gain K, all poles pi + 0 fixed and all zeros z; fi.xed exceptfor the zero at s : e. Now, using the uncertainty bounding setwhich is the interval Q: [q-,q+] , argue that at frequencies a ) 0not corresponding to a pole, the value set P(jw,Q) is the straightl ine segment joining P(j., q-) and P(jr, g+). Generalize this resultfor uncertain transfer functions of the form

p ( " - \ -- \ u , y / -

;tfu(") + ql/r(s)D o ( t )

I

where lfo("), l/r(") and D6(s) are fixed poiynomials.

ExERcrsE 7.6.6 (Conjugacy Property for Rationa,l Functions):Given an uncertain familv of plants described by

n / r N ( " , q )i \ o r 9 , / -

D C n )

and q € Q,Iet frequency c.. ' ) 0 be given such that 0 I D(ja,Q).Taking z* to be the compiex conjugate of z, prove that z e P(.ju,Q)if and only if

z * e P ( - j u , Q ) .

ExERCTsE 7.6.7 (Value Set for Rational Function): For the familyof plants described by P(s, q) : (s + q)lG + 3q) and Q : l l ,2l, gen-erate a plot of the value set p(ju, Q) for ten evenly spaced frequencypoints in the range 0 ( cu < 10.

7.7 Conclus ion

In this chapter, we reformulated a number of robustness problernsin terms of zero exclusion from an appropriate value set. This leadsus to the following question, which we begin to address in the nextchapter: Can we delinea.te irrrportant robustness problems for whichvalue set construction is computationallv tractable?

122 Ch.7 / The Value Set Concept

Notes and Related Literature

NRL 7.1 ff an uncertain polynomiaf pG,q) has continuous coefficient functionsand the uncertainty bounding set Q is "nice" (for example, say Q is closed,bounded and convex), there is a temptation to conclude that the value set is sim-ply connected. However, as pointed out in Barmish, Ackermann and Hu (1992),such a conclusion is erroneous. For example. with

p ( s , q ) : ( s + q 1 ) ( s + s 2 ) ( s + s 3 )

and uncerta inty bound lq l < r t for i : 1,2,3, the vaiue set p( j0.5,Q) has a

hole.

NRL 7,2 Rather than working with the value set, another takeoff point forrobust stability analysis involves working rvith the stability boundary in pa-rameter space. Such rvork begins rvith the so-called 2-partition technique ofNeimark (L947) and is further pursued in Ackerrnann (1980) To illustrate, bysetting real and irnaginary parts to zero, we see that given the family of polyno-m ia l s desc r i bed byp (s ,S ) : t : 0 on (q ) " t and q € Q , t he cond i t i on 0 e p ( j u ,Q )is equivalent to

0 : ao ( s ) - az (q )uL2 + a4 (q )u4 - a6 (q ) , r o + . . .

and0 : a r ( q ) u - o r ( q ) . " + a ; ( q ) e s - a 7 ( q ) c . : 7 + ' - .

for some q Q Q. Using the fact that the upper left (n - 1) x (n - l) blockn"- t (p(s,q)) of the Hurwi tz matr ix H(p(t ,q)) is the Sylvester resul tant corre-sponding to the two equations above, we obtain a description for the stabilityboundary Namely,

o o ( q ) : 0 , " , a . 1 ( q ) : g

de t I { . - r ( p ( s , s ) ) : 0 ,

where det f { - - r (p(s, g)) denotes the upper lef t (n-1) x (n.-1) b lock of FI(p(-s, q)) .

NRL 7.3 For firrther elaboration on robust stability in a delay systems context,

two basic references are Barmish and Shi (1989) and Ft, Olbrot and Polis (1989).

and

The Polyhedral Theory

Part IfI

Chapter B

Polytopes of Polynornials

Synopsis

Poffiopes of polynomials are nattllal objects to study when

dealing with affi.ne linear uncertaintTr structures. -Flom a tech-nical point of view, the most important point to note aboutpolytopes of polynomials is that they have value sets v'hich

axe convex pdygons in the comp)ex plane. This fact facilitatessolution of many types of robusf,ness probLems.

8.1 fnt roduct ion

Pr-imary motivation for this chapter is derived from the fact that

a robustness theory for independent uncertainty structures leads to

conservative results when applied to more general uncertainty struc-tures. To reinforce this point, we recall the discussion followingKharitonov's Theorem (see Section 5.11): By replacing a family ofpolynomials P with an overbounding interval polynomial farnily T,it may turn out that P is robustly stable but P is not. In otherwords, overbounding via independent uncertainty structures is con-servative in the sense that oniy sufficient conditions for robustnessare obtained.

In this chapter, we attack dependent uncertainty structures ina more direct manner. To this end, we consider the case when thesystem coefH.cients depend affine linearly on the vector of uncertain

t24

8.2 ,/ Affine Linear UncertaintyStructnres I25

parameters q. In this affine linear framework, pol1'topes of polynorni-

als are the natural objects to study when the uncertainty bounding

set Q is a box. The fact that such families of polynomials turn out

to have value sets which are convex polygons paves the way for many

results in the chapters to folIow.

8.2 Affi.ne Linear lJncertainty Structures

The main objectlve of this section is to indicate a number of ways

by which affine linear uncertainty structures arise. Of foremost im-portance is the following fact: Affine linear uncertainty structuresa.rp n lpqerved r r r r r ler la , r r 'e c lasses of feedbirck in tet 'coutrect ior ts . We

lrow llroceed to mal<e this statement more precise. Since affine linear

uncer-tainty structures have only been casually rnentioned tlrr.rs far,(for exarnple, see Section 4.4), we include a folmal definit ion for the

sake of cornpleteness.

DEFtNrrroN 8.2.1 (Af f ine L inear Uncerta inty Structu le) : An

uncertain polynomial p(s,q) :Li:oai(q)"' is said to have ai airt,ne

l"inear uncerta'inty structure if each coeffi.cient function a1 (g) is an

a fHne l i nea r f unc t i on o f q ; i . e . , f o r each i € {0 , I , 2 , . . . , r 2 } , t he re

exists a colunur vector a; and a scalar dt sucir that

o i q ) : a T q i A t

wlrere oTk) is the transpose of a;(q). Nlore generally, an uncertain

rational function, r,vhich we write as P(s, q) : lV(s, q) lD(s, q). is said

to lrave an affine Linear uncerta'it^tt'y structure if both polynornials

A1(", q) and D(s, q) have affine linear uncertainty structures.

REMARKS 8.2.2 (Feedback Interconnection) : We now consider an

uncerta in p lant P(s,Q) : N(s, q) lD(s,q) connected in the feedbackconfiguration of Figure 8.2.1 with a compensator

^/ - \ l /c(s)" \ r / : D . C )

A sirnple calculation leads to the closed loop transfer function

P ^ , ( q - \ _ ' n / ( s ' q ) D 6 : ( s )r L r - \ - , q ) : f f i .

Irr the lemma below, we see that if P (t, q) has an affine linear uncer-

tairrty structure, then so does Pg1(s,g). In other words, the afHne

126 C}r.8 / Polyropes of polynomials

FIGURE 8.2.1 Feedback connection preserving lJncertainty structure

linear uncertainty structure is preserved in going from the open loopto the closed loop.

LEMMA 8.2-3 (Affine Linear lJncertainty preservation): consirJeran uncertain plant connected, i.n a feedback configuration as i,n Fi,g-ure 8.2.1 and assurne that P(s,q) has an affine l inear uncertai,ntystructure. Then it follou-ts that the uncertain closed, Ioop transferfunctiom Pcr(s,q) also has an affine l inear uncertainty structure.

PRoor: with q € R/, the affi.ne linear uncertainty structure forP(t,q) enables us to express the numerator and denominator ofP(" ,q) in the form

i/(", q) : l/o(") + f anru,(")i : )

and t

D(t ,d : Do(s) + ) - - a ;D, i (s )Z : I

with the l[ (s) and D; (s) being fixed polynomia]s for z € {0, 1, . . . , !} .Now, it is easy to verify that the closed loop system transfer functionhas numerator

(.Nct(s,q) : l /6(s)Dc(s) + f , l ;N, (")Dc(s)

Iilr'.

8 2 / Affine Linear Uncertainty Structures 127

and denominator

Dcn(" ,q) : Afo(" )Nc(" ) +D6(s)D6(s)[.

+ t qr[ l / , (s)l /c(s) + Dr(s)rc(")]i : L

By inspection, it is obvious that l/611,(s,q) and Dcr(s,q) also haveafHne linear uncertainty structures. E

FLEMAF-r(s 8.2.4 (Variations on the Same Theme): There are manyvariations of Lernma 8.2.3; i.e., if P(s, q) has an affine l inear uncer-tainty structure, then every transfer function of prb,ctical interesthab the same structure as well. This is illustrated in the exercise be-low using the sensitivity function and the complementary sensitivity

function.

EXERCTsE 8.2.5 (Sensitivity and Complementary Sensitivity): For

the feedback interconnection in Figure 8 .2. 1 , prove that if P (" , q) has ' '

an af;0ne linear uncertainty structure, then the sensi'tiuity funct'ion

D \ D t 4 ) -

1 - l - P(s, q)C(s)

and tlre complementary sens'it'iui,ty function

m , , P ( s , q ) C ( s )- 1 \ s . s / : 1 + p ( e , q f c o

I

also have afHne linear uncertainty structures.

EXERCTsE 8.2.6 (Linear Fractional Tlansformations): Take p(s, q)to be an uncertain polynomial of order n having an affrne linearuncertainty structure and let ^lr, ^12,'y3 and 74 be real with either

% + O ot j+ # 0. Now, prove that the transformed polynomial

/ r y . c - ] - r y ^ \

p(t ,q) : p | - !# 'q ) ( lss + ta)"\?3s + 74 /

also has an affine linear uncertainty structure. As a special Jase, as-sociated with the robust stability problem for an interval polynornialfamily is a robust Schur stability problem with uncertain polynomial

having argument z (to emphasize discrete-time) and given by

128 Ch. B / Pol)'topes of Pol;T romials

Notice that p(z,q) has an affine l inear uncertainty structure.

ExERCTsE 8.2.7 (Special Case) : Suppose tl iat ,4(q) is an uncertain

matrix lvith uncertain parameter vector q entering affine linearly into

oniy one row or one column. Letting

p (s , q ) : de t ( s / - A (q ) ) ,

prove that the family of polynomials P : {p(', q) , q e Q} has an

affine linear uncertainty structure. Hint: Expand the deterrninant

via tle row or column which contains the uncertain paraneters'

-EXAMPLE 8.2.8 (Overborrnding Nonl inear Uncerbainty Structures) :

In order to deal with uncertain parameters entering noniinearly into

a system, it often suffices to generate an overbounding family which

has an affine iinear uncertainty structure. To illustrate, consider the

uncertain poiynomial

P(s, q) : s3 -l- (+q? + 2q1 -r 3q2 * 3)s2

+ @7 + zq2, + 2)s -F (6qi + az + 4)

with uncertainty bounds lq,l < 1 for ri : 7,2. By defining new

variables 4t : et,Qz : Qz, 4z : a7 and qn * q3,, we can define the

o u erb o un d'in g p olg n o mi al

p(s,4) : s3 l - (24, + 3q2 + 4q3 * 3)s2

+ (es + 2Qa-t 2)s + (64r +q2 + 4)

which has an affine linear uncertainty structure and associated un-

ce r ta in t y bounds l q r l < 1 f o r i : I , 2 ,3 ,4 -

ExERCTsE 8.2.9 (Poiytope Arising from Rank One pz Problem):

We consider the feedback interconnection which is tlsed extensively

in the p theory; see Figure 8.2.2. Furthermore) we assume that the

matrix M(s) has entries which are stable rational functions and the

systern matrix M(s) is of the form

ar t ( s ) : o ( s )b r ( s )

with a(s) and b(s) being n-dimensional vectors of rationai functions

rvith i-th component a;(s) and b1 (s), respectively. Prove that

n

d e r ( I + M ( s ) A ( q ) ) : 1 + f o ; ( s ) b 1 ( s ) q ;t - 1

';7\.

?

r-J

0 q, l

8.3 / A Primer on Polytopes and Polygons 129

u (')

130 Ch.8 / Polytopes of Polynomials

Convex Set

Ftcunn 8.3.1 Examples

Nonconvex Set

of Convex and Nonconvex Sets

Frcunp 8.2.2 Confisuration for Exercise 8.2.9

and argue that robust stability can be studied by using an uncertain

polynomial p(s, q) which has an affine linear uncertainty structure.

Hi,nt: CIear the denominators fron the entries of aa(s) and h,(s) in

the expression for the determinant above.

ExERcrsE 8.2.10 (Converse): Suppose that p(s, q) is an uncertain

polynomial having affi.ne linear uncertainty structure. Take A(q)

diagonal. (as in the exercise above) and show that there is a rnatrix

of the form n4(s) : @(s)bT(s) wi th both a(s) and b(s) being rat ional

and p (s , q ) = de t (1 + M(s )A (q ) ) .

8.3 A Prirner on Polytopes and Polygons

In order to create a. foundation for the robustness analysis to follow,

we now review some elementary material from the theory of convex

analysis. Some readers rnay opt to skip this section.

8.3.1 Convex Set and Convex I Iu I I

A set C g Ra is said. to be conuerif the line joining any two points cl

arrd c2 in C remains enlirely within C; i.e., given any cr,c2 €C and

) e [ 0 ,1 ] , i t f o l l o r vs t ha t ) c1+ (1 - \ ) . ' € C ; we ca l l Ac1+ (1 - ) ) c2

a conuer comb' inat ionof c1 and c2. In F igure 8.3.1, a convex set and

a nonconvex set in R2 are depicted. The reader can easily verify

that common multidimensional sets such as rectangles, spheres and

diamonds are convex.

Boundary of conv X

Frcunp 8.3.2 A Nonconvex Set and Its Convex HulI

Given a set C C Rk (not necessarily convex), its conuer hull,conv C, is the "srnallest" convex set which contains C. More pre-cisely, if C+ denotes the collection of all convex sets which containsthe set C, then we have

c o n v c : n C + .C+ €c+

If the given set C is already convex, it foliows that conv C : C. InFigure 8.3.2, an example of a nonconvex set with its convex hull isgiven. Notice the set inclusion conv C ) C.

Boundary of

a nonconvex set X

8.3 / A Primer on Poly'topes and Polygons l3l

8.3.2 Polytopes and Polygons

A polytope P in RA is the convex hull of a finite set of points

{p r , p2 , . . . , p * } . We r v r i t e

p : conv{p'}

and cal l {p ' ,p ' , . . . ,p*} the set of generators. Note that the set ofgenerators can be highly nonunique. For example, in Figure 8.3.3,the points p3, p5 and pr are optional for inclusion in a generating set

FIcunB 8.3.3 Polytope in R2

for P. The extreme point concept, covered in the next subsection,enables us to identify a unique set of generators.

In the sequel, it is important to make a distinction betweenpolytopes in R2 and poll4opes in Rft with k > 2. When manip-ulating value sets, we work with polytopes in the two-dimensionalcomplex plane C, u,hich we identify with R2 whenever convenient.Henceforth, we refer to a polytope in R2 as a polygon. Accordingto this convention, both polytopes and polygons are automaticaliyconvex. We make note of this point because many authors make adistinction between a polygon and a convex polygon. For example,according to some authors, a star-shaped figure can be a polygon

without necessarily being a convex polygon.

8.3.3 Extrerne Points

Suppose P: conv{pi } is apolytope in Rk. Then apointp € P issaid to be an et;treme poi,nt of P if it cannot be expressed as a convex

132 Ch.8 / PolytopesofPoiynomials

combination of two distinct points in P. That is, there does not exist

p o , p b € P w i t h p " * p b a n d ) e ( 0 , 1 ) s u c h t h a t \ p " t $ ^ - \ ) n b : n '-Foi"*u-pte, in Figure 8.3.4, the extreme points ate 7|, P2 , P3,pa and

: , j

t , ' t '

! , .i ,i , , it .i:t;.

l ' : , : . :f : r

l l : , ; 'I r ,

L'.i.i : i : . rliir .:i l "iir r.i:rr;i i l I1 l . r ' ,l i l ; ; : : it ; ,i.il r-L[ ] .1 r

t il . : .I r l ; ' '; l :,: li r i . ' ji;1, 1i : : i : i

i : t : : ; ! ii: I t1:!,. . ,i:i ..

Flcune 8.3.4 Poi'Y'toPe in R3

p5. Although the interior point p6 rnight be included in a generatipg

set, it is not an extrerue point. Given a finite set of generatols {p'}for a polytope P, the set of extreme points is a subset of the set

of gener.ators. Furthermore, the set of extreme points can be called

a minimal generating sel in the sense that any other generating set

contains the set of extremes.In many applications, generators or extrene points of a poly-

tope are specifiecl implicitly rather than explicitly. A prime example

occurs in the theory of linear programming where polytopes are de-

scribed by a set of linear inequalities in the matrix fotrn Ar 1 b'

8.3.4 Convex Cornbination Property

G iven a po l ybope P : conv {p r ,P2 , . - . ,P * } , eve ry po in t p € P can

be expressed as a conuer comb'ination of the p'; lhat is, there exist

real scalars ) r , )2, . . . , ) - ) 0 such that

m

, - \- ),rrir 1 ) ' L r

8.3 / A Primer on Polytopes and Polygons f 33

anon

\ - r . - r

In the sequel, it is sornetimes convenient to describe the constraintset for A using the notation

A : { ) € R - : A ; ) 0 f o r r i : 1 , 2 , - . 1 r r l . m d f ) r : 1 } .; - l

For such cases, A is called a unit sirnplet,To illustrate the notion of convex combinations, consider the

polygon P in Figure 8.3.3. Observe that P is the union of threetr iangies g iven by Pr : conv{pr ,p6,p8} , Pz : conv{pr ,p2,p6} andPs : conv{p2,p4,p6}. Now, any point p e Pt can be explessed as aconvex Combination )rpl * Aop6 + trspS. For example, a point suchas p7 rnight be obtained with )t : Ao - le : Lf3, a point suciras p3 is generated with )1 I 0, )s I 0 and )o : 0 and finally, anextreme point such as p6 is obtained with )6 : 1 and )r : )s - 0.To conclude, we observe that the description of a point p € P as aconvex combination of extreme points is nonunique. For example, apoirrt sucir as p7 can be expressed as a convex combinationof pr, yt6

r R a A l eano p ' o | P ' , P" and p" ., The fact that we can describe every point p e P in Figure 8.3.3

as a convex cornbination of three or less extrerle poi nts is not partic-ular to the example at hand. In fact, Carrheodory's Theorern tellsus: Every point in a polytope P c Rft is expressible as a convex com-bination of l i l-1 extreme points at most; e.g.) see Rockafellar (1970).

8.3.5 Edges of a Polytope

Given any two points ro and rb it. Rk, we denote the straight linesegment joining these points by lc",zb] . Notice that every pointr e lro,rb] catr be expressed uniquely as a convex combination of roand zD: that is.

r : ) . r o + ( 1 - ) ) z b

for sorne unique A e [0, 1]. Furthermore, if r,o : rb, [x",rb] degener-ates to a point which is viewed as a special case of a l ine.

lVe now consider l ines of the form lp'r,p""1, where p', and p'2are extlemes of a given polytope P and p'r * p'r. We say thatlmlt nzzf tq qn ainp of P if the following condition holds: Given anyW ) r I ' " " ' ^ '

p" ,pb € P w i th p " ,pb / [ p ' r , po r ] , i t f o l l ows thaL fua ,pb la lp i , , p i r ) : d . t .

134 Cln.B / Polytopes ofPoly'nomials

In two or three dimensions, the edges of a polytope are apparent by

inspection. For example, in Figure 8.3.3, edges of the poll ' tope P are

lpt ,p ' l , lp2,pnf , lpn,pul , lpu,pr) and lp8,p l ] and in F igure 8.3.4, rhe

edges of the polygon P are lp t ,p '1, lp t , p31, lp ' ,pnl , Lpr ,ps l , fp ' ,p t l ,

[p' , pt], lP3 , pnl and fpa , p5] .

8 .3.6 Operat ions on Polytopes

In this subsection, we provide a number of basic facts about opera-tions on polytopes.

LnMMA 8.3.7 (Direct Sum for Two Polytopes): Giuen tulo poly-

topes P1: conu{pf i r } andPz: conu{rz ' iz} i 'nFtk, the d i rect su 'm,

P1 * Pz : { p r + p2 , p ' €P t ;pz q Pz }

'is a polytope. Moreouer,

P 1 + P 2 : ' o n ' { P t ' i ' + P 2 ' " ' } '

REMART(s 8.3.8 (Direct Sum for Polytope and Point): For the

special case when P2 consists of a single point, P1 -F Pz correspondsto a transiation of P. In the lemrrra below, we provide another usefulcharacterization of Pr l- Pz.

LEMMA 8.3.9 (Another Direct Sum Description): Gi,uen ttuo poly-

topes P 1 : conu{pf i} and P2 ' in Rk , ' i t f ollows that

P r -F Pz : con l ) [ J (P t ' t * Pz ) 'v . .

L

ExAMeLE 8.3.10 (Il lustration of Direct Sum): To i l lustrate forma-tion of the direct sum via application of the lemma above, supposetha t P1 : conv {2 +2 i , 4 -2 i , 6 *67 } and P2 i s t he un i t squa re i ntire complex pIane. Then, the lemma leads to the direct sum, which

is shown in Figure 8.3.5.

ExERcrsE 8.3.11 (Less Restriction on P2): Argue that Lemma 8.3.9remains valid when P2 is an arbitlary convex set which is not nec-essarily polytopic. I i lustrate by considering Exampie 8.3.10 with Pz

being the unit disc rathel than the unit square. Sketch the resultingdi rect sum P1 i_Pz.

6j

p r ' r +Pz

P r * P z

'is a polytope. Moreouer,

a P : { a p : p e p }

aP : conu {ap ' } .

FrcunE, 8.3.5 Formation of Direct Sum in Example 8.3.10

LEMMA 8.3.12 ( Intersect ion of Two Polytopes) : Letp l and,p2 betwo polgtopes i,n F.k. Then it follows that Py)P2 is a polytr.rpe.

fi :ffiTTJ;1Ji"iilJff LHTI**#,1:;1'""iil:",:,'ir:?:lther P1 or P2; e.g., in Figure 8.3.6, the points p" and, pb areextremepoints of P1 1-l P2 but are not extreme points of P1 or P2.

LnT4MA 8.3.14 (Multiptication of a Scalar and a polytope) : Giuena polytope P : conu{pi} and a real scalar a, ,it follows that the set

8.3 / A Primer on Pol)topes a:rd Poly,gons 135

p l ,3 i _Pz

Boundary of

P r * P z

P 1 ' 2 + P 2

136 Ch.8 / PolytoPes of Polynomials

Ftcunr 8.3.6 New Extreme Points Created by Intersection

REMART{s 8.3.16 (Loss of Extreme Points): Given two polytopes

P1 and P2, the geometry associated with formation of conv(P1 U Pz)

is depicted in Figure 8.3.7. Note that some of the extreme points of

P1 and P2 are no longer extremes of conv(P1 UPz)'

LEMMA 8.3.17 (Affine Lirrear T\'ansformation of a Polytope) t Fup-pose that P : c)nu{pi} is a polgtope in Fck' and, T : Rft' - Rk' is-an

affine linear .transformation. Then the set TP : {Tp : p e P} i's

a, polgtope inRk2. Moreouer,

TP : conu{Tp'}

and, euery edge point of TP is the image of some edge poi'nt of P '

That 'is, i,J r is an edge poi,nt of TP, then r : Tp for some edge

p o i n t p € P .

REMARKS 8.3.18 (Edge lvlapping) : In the lemma above, note

that not ali edge points of P map into edge points of. TP ' Roughly

speal(ing, some of the edges of P can map into the "inside" of TP '

Aiso, the lemma does not rule out the possibility that points which

are not on the edge of P are mapped onto an edge of -7P' As a

simple illustration, suppose P is the unit square in R2 and take

? : R2 - R2 to be the rnapping which projects a point p € R2

onto its first component p1 ; i.e., the coordinates ol Tp are Tlp : pr

andT2p:0. Observe that every point in P is mapped into an edge

noint of 7P.LEMMA 8.3.15 (Convex Hull of a Union): Giuen any two poly-topes P1- conu{pr ' i l } andPz: conu{p2' i2} inp1k, i t fo l lows thatconu(P1UPz) i,s a polytope w,ith generating set {pr,it) l) {p2,or}.

P1 f l P2

8.4 / introducrion to Polytopes of Pol l ,nomials l}z

Frcuns 8.3.2 Formation of conv(p1 lJ p2)

ExAMeLE 8.3.19 (Tu'o-Dimensional Domain and Range): An im_portant special case of Lemrna 8.3.17 is obtaineci ruhen p is a polygonin the complex plane and we seek a description of the set

z p : { z p : p e p }

with z € C being a given complex number. Note that if p € p, wecan write

and view the formation of zp as a iinear transformation from R2to R2' Norv, Lemma 8.3.17 provides us with a description of thegenerators of ?P. In fact, if p : conv{p'} and

z : R e j o ,

we obtain

?P: conv{Rejepi}.

In other words, the z-th generator for ?p is simply obtained frorn pivia a scal.ing and a I'otation.

8.4 Introduction to polytopes of polynornials

We are now prepared to consider polytopes in the context of poly_nomials. However, before proceeding, it is important to draw the

I n" ,7,l t Re z -rrn ,l I a" ell t^ , r l : l . t ,n z Re , ) L t * o)

138 Ch. B / Poiytopes of Polynomials

reader's attention to one point in order to facil i tate understandingof the exposition to follow: Although the polytopes in this ,section

are defined abstractly in a polynomiai function space, there is a nat-

ural isomorphisrn between a poiytope of polynomials and its set of

coeffi.cients. Hence, once things are set up correctly, we can apply

all the machinery in Section 8.3. When we perform operations.on a

polytope of polynomials in a convex analysis context, there is 6ltvaysan appropriate interpretation in coefficient space.

DEFrNrrrox 8.4.1 (Polytope of Polynomiais): A family of polyno-

rn ia ls P : {p( ' ,q) , q e Q} is sa id to be a polytope of polynorn ' ia ls t f

p(s, q) has an affine linear uncertainty structure and Q is a polytope'

If Q : conv{gi}, then we call p(s, q') the i '-th generator for P'

ExAMPLE 8.a.2 (Polytope of Polynomials): If a polytope of poly-

nomials 2 is descr ibed by p(s, q) : s2 +(4qt*3qz*2)s- l (2qv-qz-15) ,

lqrl ( 1 and lqzl < 1, the uncertainty bounding set Q has four ex-

t t u * " " q r : ( - L , _ 7 ) , q 2 : ( - 1 , 1 ) , 9 3 : ( 1 , - 1 ) a n d q a : ( 1 , 1 ) '

The four associated generators are given by p(", qt) : s2 '- 5s + 4,

p ( s , q 2 ) : s 2 * s * 2 , p ( s , q 3 ) : s 2 * 3 s * 8 a n d p ( s , q a ) : s 2 * 9 s * 6 '

ExERCTsE 8.4.3 (Generators in Coefficient Space): Consider a

polytope of polynomials P : {p( ' ,s) : q e Q} wi th coef f ic ienf ' uector

a(q) for p(s,q). Prove that the coefficient set

" ( Q ) : { " @ ) ' s e Q }

is a polytope and, moreover, argue that it Q: conv{g'}, then

a ( Q ) : c o n v { a ( q " ) } .

8.5 Generators for a Polytope of Polynornials

Our objective in this section is to show that the p(s,qt) in Defini-

t ion 8.4.1rightly deserve to be cailed generators. To this end, notice

that since p it u poly'tope with z-th generator Qi, an1 q e Q can be

expressed as a convex combination

^ - \ - \ . - iLl - /r "xV

x

8.5 / Generators for a Polytope of Polynomials 139

for appropriate A in the unit simplex. Now, using the fact that thecoefficients of p(s,q) depend affine l inearly on q, we obtain

p ( s , q ) : i o n ( I r r q a ) " di :0 lc

: I Aop1", qk) .k

Hence, for each q € Q , p(s, q) is a convex combination of the p(s, qi).This justif ies call ing p(s,q') the i-th generator for P and writ ins

p : conv {p ( . , q ' ) }

with the understanding that operations (such as taking the convexhull) in the space of polynomials can be associated with operationson q or the coefficients. For example, if g'1 and qi" ane generators ofthe uncertainty bounding set Q, then the convex combination

p ( " , ) ) : Ap (s , q i ' ) + ( r - ) ) p ( s , q t ' ? )

is associated with the polynomial p(s, q)) with q^ : \qi ' -t (I - Xlrtz .Equivalently, we can associate fr(s,)) with the coeffi.cient vector

a^ : Aa(qi,) + (r - A)o(qn').

ExAMPLE 8.5.1 (Interval Polynomial as a Special Case): In thisexample, we view an interval polynomial

n

p ( s , q ) : L l q o , q [ ] t ni :0

in the polytopic framework. fndeed, if q& denotes the k-fh ertremepo'int of the associated uncertainty bounding set Q, the i-th com-ponent q! of qk is qo or 91+. Hence, this interval polynomial familycan be described using at most 2'*r generators. Associated with thek-th extreme point q* of Q is the k-th, ertrerne

n

p( s. ak\ : 5- o-f "i.U "

i :o

ExERCTSE 8.5.2 (Enumeration of Extremes): Enumerate the eightextreme polynomiais for the polytope associated with the intervalpo l ynomia l p ( s ,q ) : 2sa + [ 1 ,2 ] s3 + 5s2 + [ 3 ,4 ] s + [ 5 ,6 ] .

140 C\"r. 8 / Poiytopes of Poiynomials

REMARKS 8.5.3 (Generators Usi'ng the Unit Simplex) : In view

of the d.iscussion above, it is often convenient to describe a farniiv

PIex rather than the uncertaintY

i p , . ( r ) , p 2 ( s ) , . . . , p - ( s ) a r e f i x e d

mP1ex, we can define the familY

of aII convex combinations of the

s in P if there exists some vector

A e A such that nLp(s) = I )oPn(")

We writeP : conv{p;( ' ) }

uncertain PolYnomial'

ExERcrsE 8-5.4 (Polyhope over the Unjt Simplex): Starting with

f ixed polynomials po(") ,pr(" ) , - . ' ,wG), consider the polytope of

polynomials P described bY

!

p(s, q) : po (s) + | cr r ; (s) ,

s € Q arrd Q: "o"t{qo}' Describe this family of polynomiais 2 in

terms of the unit simPlex'

EXERCTsE 8-5.5 (Diamond Family of Polynomials): - Repeat Exer-

cise 8.5.4 with the uncertainty bounding set Q described by

t

Z-'ln'l t t'

where ut t ) 0 for i : I ,2 , " ' ,1 ' We refer to the tut as ueights for

the uncer ta in paranreters. 'The resul t ing setP: {p( ' ,q) : q e Q} is

called a d,'iamond' family of polynomiais'

nS- \ -1 t r k t i: 2- ) -^ko ' i \q- )s-i :O k

9 1 0 r r R.p(" , q ' )

Ftcunp 8.6.1 Value Set Generation for Examole g.6.1

8.6 Polygonal Value Sets

Iu this section, we concentrate on characterization of value sets for apol)/tope of polyno'ials. Since the interval polynomial framework isa special case of the polytopic fra'rework, the value set which we ob-tain is a gene'alization of the Kharitonov rectangle of section s.T.r.For polytopes of polynomials, we now argue that the relevant varuesets are poiygons in the complex plane. To see one of the many pos-sible ways by which polygonal value sets arise, we provide a simpletnotivating example.

ExAMpLE 8.6.1 (Polygonal Value Set): We consider the intervalpolynorrriai p(s,q) : s2 i [1, 2]s * [3, 4] and take z : 2 + j forconstruction of the value set p(z,Q). Indeed, by substitution, weI l ndp (2+ j ,S ) : ( 3+ ao - t2q r )+ j ( 4+a r ) and then seek to map the

p(2, Q) is a parallelogram in the complex plane r,vith edges which areobtained by mapping the edges of Q through p(2,.). For example,the edge of Q obtained by joining p(",qr) and p(2,q2) comes fromthe edge joining q\ and q2.

REMARxS 8.6.2 (Generalization): In view of the motivating ex-ample above, our goal is to characterize the value set p(z,e) for anrore general polytope of polynomials. An informal statement of the

142 Ch.8 / Polytopes of Polyromials

technicai resuit given below is as follows: Fol a pol1'tope of polyno-

mials, the value set p(2, Q) is a polygon with generating set {p(r, q')}

and edges which come from the edges of Q. The reader should be

forewarned, however, that not all edges of Q necessarily map into

edges of p(" ,Q).To i l lust rate, not ice that in F igure 8.6.2, the edge

p ( 2 , ' )

- f h i q e r l o o n r q n q t n e n e r ' l o e

FicuRE 8.6.2 N,Iapping an Edge of Q into the Interior of p(z,Q)

lq" ,q" l is mapped to the in ter ior of the set p(z,Q).

8.7 Value Set for a Polytope of Polynornials

Given that the discussion of polygonal value sets above is in the con-text of numerical examples, the objective in this section is to provide

a general result. We now provide a polygonal characterization of.value sets for a polytope of polynomials.

PRoPosrrroN 8. 7. 1 (Value Set for a Polytope of Poilmomials) : Let

P : {p(-, q) : q € Q} b" a polytope of polynornials wi,th uncer-tai,ntybounding set Q - conu{q'}. Then, for f ired z € C, the ualue setp(r ,Q) is a polggon wi th generat ing set {p(r ,q ' ) } . Th,at ' is ,

p(2, Q) : conu{p(2, q ' ) } .

Furthermore, all edges of the polggon p(z,Q) are obta'ined from' the

edges of Q in the foltoui,ng sense: If zs is a point orl arl edge of

p (z ,Q) , t hen zs : p ( z ,q \ f o r some qo on an edge o f Q .

PRoop: For fixed z € C, we note that the mapping T : Q '--+ C

defined byTq : p (2 , q )

frn

8.6 / Pol,vgonal Value Sers L4I

' :P(t, qo)

z I ? rplz, ic l " )

p ( 2 , Q )

p ( " , ' )

This edge mapsto the inside.

p ( 2 , q " )

P ( 2 , q " )

P(2 , q ' " )

B 7 / Yalue Ser fbr a Poll,tope of polinomiais 143

is affine l inear. Since Q is a polytope andTQ : p(z,Q), the propo-sition foilows immediately from Lemma 8.3.17. That is, the valueset p(2, Q) is the image of Q under T. E

ExAMeLE 8.7.2 (Polygonal Value Sets): Given the uncertain poly-nomial

p (s ,q ) : (Zq t - ez - t qe * 1 ) s3 * (3q r - 3qz+8s *3 )s2

| ( 3 q i l q z + a s * 3 ) s + ( q r - a z * 2 c . 3 a 3 1

and uncertainty bounding set Q which is described bV lqtl < 0.24bfor i : 7 ,2,3, Proposi t ion 8.7.1 indicates that the value set p(z,Q)is a polygon with eight generators. For example, corresponding tothe particular generator q5 - (0.245,-0.245,0.245), we obtain thepolynomial p(" , g t ) : 1 .98s3+4 .775s2+Z.T35s*3.98. In F igure 8.7.2,blre value set p(jt't,Q) is shown for thirty frequencies evenly spaced

J

.)

f r n 1n <

n

- u - a

- 1

Ftcunn 8.7.1 Polygonal Value Sets for Exampie 8.2.2

between u : 0 and r.., : 1.5. Notice that even though there areeight generators, the value set has only six extreme points; i.e., twoof the extreme points of the box Q are mapped into the interior ofp(jut, Q). Now, rve test for robust stability using the value set plot in

I44 Ch 8 ,z Polytopes of PolFromials

P to be robustLY stable.

EXERCISE8.7.3(Sate l l i teAt t i tudeContro l ) :T l le t ransfer f r rnct ionfor a satellite attitude control problem, as given by Franklin, Powell

and Emani-Naeini (1986), is

ffi:qJY##'D

ExERCTsE 8.7.4 (Dominant Pole Specification): A feedback s)'stem

gives lise to the polytope of polynomials 2" described by

p ( s , q ) : s 3 * ( 1 0 + s 2 ) 5 2 + ( 2 9 + q r ) s * ( 3 0 + q r * q z )

verify bhat the robustness margin

rnar: sup{r :P, has the specified root distribution}

ic oirren 6w r^-- = 0.35. Use the more general interpretaLion of

,"r:" """r"rior.'ili

Exercise 7.5.T and. generate a value set plot for an

appropriate range of frequencies'

- 6 - 4 - 2 0Re

8.7 / YaIve Set for a Polytopes of Polynomials I45

ExERCTSE 8.7.5 (DC Motor with Resonant Load): The goal in thisexercise is to apply polygonal value set theory using the mocjel oi'aDC motor in Example 2.8.1. In contrast to the analysis in Chapter 2,consider uncertain parameters 0.5 x 10-2 < Jy < 1.5 x l0-2 and2 x 10-3 181, < 4 x 10-1. The remaining parameters are fixed atL : 5 x 1 0 - 3 , R : 1 , J * : 2 x 1 0 - 3 , B * : 2 x I 0 - 3 , 1 { : 0 . 5 a n dI{, : 2 x 103. In order to study pole locations as a function of theuncertain parameters, take et : Jz and q2 - Bl and concentrateon the tlansfer function from armature voltage to shaft speed. Thistransfer function is given by

P ( r ) :K J 7 s 2 * I { B ; s + K K "

a( " ) )

where

A(s) : J*JrLsa * (B^J:,L * B:.J*L -l J^JrR)s3* (B^B1L + J^K"L i K"J1L

I B^J' ,R* Br.J-R+ K2JL)s2+ (B-I{"L I BrK"L * B*BrR

+ J^K"Rt I{"J;R+ K2B1)s+ (Bn K,R* B1K"R+ K2I{") .

(a) Verify that the uncertain denorninator polynomial is giwen by

p(s ,q ) = 10-591sa * (2 x 10- tq , + 10-5q2)s3

, ili',1f?;j#,11-,n;*:,;:o-''"'+ 5 . 0 4 x 1 0 2 + 2 x 1 0 3 q 2 .

(b) With uncertainty bounding set,

Q : { q € R 2 : 0 . 0 0 5 < q 1 < 0 . 0 1 5 ; 0 . 0 0 2 < q z < 0 . 4 }

obtained from the data above, take desired pole location region Dto be a damping cone with angle Q: 450 as in Figure 6.8.1. Bysweeping the boundary of D, generate appropriate vaLue sets p(z,e)and use the Zero Exclusion Condition (Theorem 7.4.2) to determinewhether ? is robustly D-stable; i.e., determine whether the dampingspecification is robustly satisfied.

146 Ch. B ,z Polytopes of Poly'nomials

8.8 knprowement over Rectangular Bounds

To demonstrate that a polygonal value set plot leads to better results

than overbounding vie' a Kharitonov rectangle (see Section 5.11), we

compare results using both techniques in the exarrrpLe below.

ExAMPLE 8 .8 .1 ( cOnServa l i sm 9 f Qyg l l - , n r r nd ino \ ' t r - n r t ho ^n l r ' -

tope of polynomials 2 described by

p ( s , q ) : s 4 - l - ( 2 S z + t ) s 3 + ( Z q r - q 2 * 4 ) s 2

t ( q r + 1 ) s + ( q r - 2 q z + 2 )

and Q : {q e R2 : - 0 .5 l q t / - 2 -0 .3 l qz < 0 .3 } , r ve ca r r y ou t

two robust stability analyses. First, rve replace 2 by the overbound-ing interval polynomial family 2 described by

FG,e) : s4 I [0 .4, 1 .6]s3 + 12.7,s .3]s2 + [0 .7, 1 .3]s + [0 .9, 4 .6]

and reach the conclusion that 2 is not robustly stable. That is, by a

straightforward calculation, it is easy to verify that the l{haritonovpolynomial

1 {3 (s ) : s4 -F 1 .6s3 + 2 .7s2 +0 .7s - t 4 .6

has an unstable root pai r sr ,z x 0.4099 + i1 .1106.To begin the second analysis, rve verify the critical precondition

for application of the Zero Exclusion Condition (Theorem 7 .4.2) . In-

deed, wi th q0 : 0 , p(" ,qo) : sa+s3+4s2+s+2 is a s table mernber of

2. Next, a preiiminary computation (see Section 7.3) indicates that

a suitable cutoff frequency is t. ' . : 1; i.e.' 0 / p(j.,Q) for frequency(r ) ec. Subsequently, in accordance with Proposition 8.7.1, wegenerate 40 polygonal value sets corresponding to frequencies evenly

spaced between a : 0 and c..r : 1; see Figure 8.8.1. Within cornptr-

tational l imits, u'e conclude that 0 / p(j.,Q) for all c.. ' ) 0. Hence,

by the Zero Exclusion Condition) we say that 2 is robustly stable.The conclusion to be drawn is that working rvith the overbound-

ing famiiy 2 is inconclusive but working 'with polygonal value sets

leads to the conclusion that 2 is robustiy stable.

ExERcrsE 8.8.2 (Value Set Comparison): With the same setup

as in the example above, generate value sets FUr,Q) correspondingto the overbounding interval polynomial F(ir,Q). For purposes of

comparison with Figure 8.8.1, use 40 frequencies evenly spaced be-

tr,veen u : 0 and u : 1. Finally, observe that the Zero Exclusion

8.8 ,/ Improvemenr Over Rectangular Bounds l4j

D t r

t <

z

f t n 1n q

0

, -0 .5

- 1

- 6 - 4 - 2 0 2Re

Ftcunp 8.8.1 Vajue Set for Example 8.8.1

Condition is violated for the Kh.aritonov rectaneles but holds for thepolygons p( iu,Q).

ExERCTsE 8.8.3 (Total Subtended Angle) : There are nany ap-proaches for numericai implementation of a zero exclusion test. ToiJ.lustrate, suppose that P : {p(.,q) : q e Q} is a polytope of poly-nomials with Q having set of extremes {qt}. For a given frequencyu ) 0 , I e t

/ - / . t r \

o . t . , \ - . ^ . - - 1 / t r n P \ J u 1 q " ) \UL\W) - LAL\ \E;iG-6 )

and defi.ne ltre total subtended angle by

e@) : rnax.9i(6) - rnin0a(6).

Now, at frequency a ) 0, argue that tlne Zero Exclusion ConditionO F pUu, Q) is satisfied if and only if

0(r'.t) < r.

Generalize this result to the robust 2-stabilitv problem.

f 48 Ch. B / Pol) ' topes of Pol laromials

8.9 Conclus ion

In this chapter, we introduced polytopes of polynomials. To this

end, we first reviewed some basics from convex analysis and then

characterized value sets as (convex) polygons in the complex plane.

This characterization is quite useful in many chapters to follow- In

particular, in the next chapter, we see that the edges of the value

set polygon are very important . Indeed, i f P: {p( ' ,q) : q e Q} is a

polytope of polynomials and

t : { t o : i e I }

denotes the set of edges of Q, then we see that a loss of robust

2-stability is synonymous with satisfaction of an edge penetration

cond'it ion0 e p(2, t r )

for some z € AD and some i e I. Subsequently, it becomes possi-

ble to reduce the robust D-stability problem, formuiated over the

multidimensional set Q, to a fi.nite set of single-parameter problerns.

Notes and Related Literature

NRL 8.1 With regard to Exercises 8.2.9 and 8'2.10, a more detailed discussion

on the reia.tionship between pr theory and the theory in this text is given in Chen,

Fan and Nett (1992).

NRL 8.2 For a more detailed exposition of the material on convex anaiysis in

Section 8.3, see references such as Rockafellar (1970)' Stcier and Witzgall (1970)

and Aubin and Vinter (1980).

NRL 8.3 For the DC motor with resonant load in Exercise 8.7.5, Baiiey' Panzer

and Gu (1988) concentrate on generation of the value set of the transfer func-

tion rather than the denominator polynomial. when these value sets are dis-

played on the Nichol's chart, we obtain the l{orowitz templates; see lTorowitz

and Sidi (1972).

Chapter 9

The Edge Theorern

Dy_l?opsrs

The focal point of this chapter is the celebrated Edge Theo-rem of Barilett, Hollot and Huang: Under mild conditions, apolytope of po17,t161111a1s P is robustly D-stable if and ody ifevery polynotnial on an edge of P is D-stable. Thjs has strojjg-ratnifica.tions throughout the remainder of this text. Tlte facttltat the number of edges can be prohibitively )arge paves theway for results in later chapters.

9.1 Int roduct ion

Having established the fact that every polytope of polynomials iraspolygonal value sets, we are now pr-epared to study the Edge Theo-rern and its ramifications In this chapter) rve see that the interior ofthese polygonal value sets are unimportant as far as robust stabilitvis concerned. With the machinery we have in pIace, it is easy toshow that the problem of robust stability is soived by working witha set of "edge polynomials" which can be identified with edges of thevalue set. Since our description of these edge pol;,nomials involvesonly one-parameter, we end up with a set of one-parameter. robuststabil ity pr-oblems in l ieu of the original problem. Hence, the one-parameter formuiation of Chapter 4, which may have seemed quitespecialized at the time it was introduced, is actually quite important.

150 Ch.9 / The Edge Theoren.r

The key idea is that we can reduce a problern with a multidimen-sional uncertainty set to a finite number nf nna-ne.omaiar oroblemswhose solutions are readilv available.

9.2 Lack of Extrerne Point Results for Polytopes

Perhaps the most important motivation for the analysis to followis derived from the fact that Kharitonov-like extreme point resultsdo not hold for a general polytope of poiynomials. For example,recall ing the pair of polynomials /(s) : 1Os3 + s2 + 6s * 0.57 ands(s) : E2 +2sl I analyzed in Exerc ise 4.15.1, rve found that wi th

P(s, ) ) : " f (s) + As(s) ,

both p(s,0) and p(s, 1) are stable but p(s, 0.5) is unstable.In view of such exarnples, we proceed as follorvs: In this chap-

ter, we concentrate on results which apply to all polytopes; thesieare called edge results. In later chapters, we identify rich classes of( "1(" ) ,S(") ) pai rs for which extreme point resul ts hold. That is , sra-bil i ty of p(s,0) and p(s, 1) implies stabil ity of p(s, )) for all ) e [0, 1].In other words, under strengthened hypotheses, we can do "better"than'edges. We see tirat such hypotheses have an interpretation ina feedback control context; i.e., rvhen the plant has an affrne linearuncertainty structure.

9.3 l{euristics lJnderlying the Edge Theorern

Before providing the main results of this chapter, rve give a heuristicargument which motivates the Edge Theorem. Indeed, suppose thatP : {p(-,q) : rl e Q} is a polytope of polynomials rvith at leastone stable member and the desired root location region D is anopen subset of the compiex piane. Now, with the Zero ExclusionCondition (Theorem 7.4.2) in mind, we sweep z over the boundary0D of D ard obtain the value set p(2, Q) W" are interested to seeif z :0 ever enters this set.

Suppose that init ially,0 / p(zo,Q) with z6 € AD. Now, as rvemove away from z6 along 0D, we know from Proposition 8.7.1 thatthe value set p(z,Q) is a poiygon. Furthermore, with a z movingalong 0D, either z :0 remains outside of p(z,Q)

"t we arrive at

some critical z1 e 0D for which the origin, z : 0, lies on the edge ofthe polygon p(z,Q). In other words, the situation which we associatewith a loss of robust D-stabil ity occurs when the origin, z : 0, l ieson an edge of p(4,Q) for some z1 € 0D; this situation is depicted

149

in Figure

9.3 ,/ Heuristics Underlying the Edge Theorem 151

9.3.1. Recall ing that edges of the value set are obtained

FIcunB 9.3.1 Loss of Robust 2-Stabil ity

from edges of Q (see Lemma 8.3.17) , this heuristic argument ieads usto conjecture that under suitable regularity conditions, a necessaryand sufficient condition for robust Z-stabiiity of P is 2-stability of

all edge polynomials of the form

Pt, , r . r (s , ) ) : )Pt , (s) + ( r - ) )P, ; r ( " ) ,

where p i r (s) : p(s,q i ' ) , ph(s) : p(s, q i ' ) and q i ' and q i " are extremepoints of Q having the property that the straight l ine joining qt' andq"2 is art edge of Q. For example, if Q is the three-dimensional boxsho.r'n in Figure 9.3.2, then this heuristic argument indicates thatwe need only check for robust 2-stabiiity with q restricted to one oftlre twelve edges e1 through e12.

To further illustrate the ideas above, suppose that each corrrpo-nent of q has bounds si I q; < qo+. i|hen edge polynomials of ?associated with edge e5 of Q in Figure 9.3.2 are of the form

p i , i " ( s , A ) : A p ( " , s l , s ; , s t ) + ( 1 - ) ) p ( s , q l , q i , q { ) ,

and the edge polynomials associated with ee are of the form

p41o(s , ) ) : , \ p (s , q l , qz ,q i ) + ( i - , \ ) p (s , q l , q f , q i ) .

I52 Ch.9 / TheEdeeTheorem

q'" eg q 'n

Frcuno 9.3.2 The Edges of the Box Q

REvTARKS 9.3.1 (Checking the Edges): Recall ing the analysis for

the one-parameter case in Chapter 4, u'e note that checking lo-

bust D-stability of an edge polynomial is readiiy accomplished ina number of ways ranging from root locus to Nyquist methods. An-other alternative for edge testing involves value set generation. Withpr , , , ; ' (s , ) ) : Apt , (s)+ ( i - \ )pu,(s) and ) e A : [0 , 1] , we recai l thatthe value set p(z,A) is a straight l ine segment with endpoints p(2,0)

and p(2,1); see Lernnra 7.2.3. }Jence. for each such edge, we verify2-stability of one endpoint and check for satisfaction of the ZeroExclusion Condition

0 / p(2, t \ )

f o r a l l z e 0 D .

9.4 The Edge Theorern

We now consolidate the ideas above by formal.ly stating the cele-irrated Eclge Theolem. Note that in the theorern statement below, aboundary srn'eeping function is assumed for the desired loot Iocationregion 2. Although the theorem holds under weaker assumptiont,lrat 2 is simply connected, rve include a boundary sweeping func-tion because the proof becomes rnuch sirnpler to understand withoutseriously compromising the domain of applicability. In the chaptersto follow, $/e see nulrrerous applications of the theorem below; seealso the notes at the end of this chapter for a.dditional discussion ofvarious extensions and refinements of the theoren.

p ( q , Q )

it:r,

9.4 / TheEdgeTheorem 153

THEoREM 9.4.1 (Bartlett, HoIIot and Huang (1988)): Suppose thatD is an open subset of th.e cornpler plane wi.th boundary sweeping

func t i on Qp : I , C , and l e t P : { p ( - , q ) : q e Q} be a po l y topeof polynornials with inuariant degree. Then P is robustly D-stable iJand only i,f for each pai,r of ertreme po'ints q" and q'2 correspond,i,ngto an edge of the set Q, the polynomial

P t1 ' ; ' ' ( s ' A ) : )P (s ' q " ' ) + (L - A )P (s ' q " ' )

is D-stable for a l l ) e [0, 1] .

PRoon: The proof of necessity is trivial; i.e., for all extreme pointsq"L and q'" of Q and all ) e [0, 1], the polynomial ptrj"(s,)) e P isa member of P. Hence, robust Z-stabii ity of 2 implies 2-stabil ityof the polynomial p6i . , (s , A) for a l l ) e [0,1] .

To establish sufficiency, we assume D-stability of each of the edgepolynomials ph,tr(s, )) and must prove that the famiiy P is robustiyD-stable. Proceeding by contradiction, we assume 2 is not robustlyD-stable. Then, by the Zero Exclusion Condition (Theorem 7.4.2),there exists some 5o € l such that 0 e p(O2(50),Q). The proof nowbreaks down into two cases.Case 1: There ex is ts some 61 € l such that 0 / p(An@r),Q).Without loss of generaiity, say 61 ) 6s. Corresponding to extremepoint q' of Q, we let

p ; ( s ) : p ( s ,q ' )

denote the i-th generator of 2. Since Oa(6) varies continuously, itfollows that

p(Ao@), Q) : cot r , r {p"@o\)) }

varies continuously as a set. Now, using the continuity of p(@2 (6), Q),there must exist some 6* e I such that z : 0 lies on an edge of

P(or> (6. ) , Q) : conv{pa(o2(6")} .

Since the edges of p(@o(6-) ,Q) come f rom the edges of Q (s."Lerrrma 8.3.17) , it follows that there are two extreme points q'1 andq'2 corresponding to an edge of Q and some A* € [0, 1] such that

) " p i , (Qp (6 . ) ) + ( r - A * )po " (OD(6 . ) ) : g .

However, this contradicts the assumed D-stabil ity of ph,.iz(s, .\*); i.e.,Oz(5*) is a root of p; r , ;z(" , ) * ) rvh ich is not in 2.Case 2: For a l l 6 e 1, 0 € p(Ao@),Q). W" f i rs t p ick ss € C suchthat p(s6, cD + 0 for all S € Q. Note that the existence of s6 is

I54 Ch.I / The Eclse Theorem

guaranteed because we can construct a bound for the roots of p(s,q)for q € Q; see Sect ion 5.10. Hence, 0 # p( to,Q) :

"onrr {p l ( ro)} .Now, we fix any 6o € I and observe that our standing hypothesesguarantee that 0 e p@p(ls), Q).

Next, we consider a path from s6 to @2r(6s). For- o € [0, 1],l e t s (a ) : ( 1 - a ) so + aA72@s) . Now, s i nce p (so ,Q) : p ( s (0 ) ,8 ) ,0 / p(s(0) ,Q) and 0 € p(s(1) ,Q), cont inuous var- ia t ion of the setp(s(a) ,Q) :

"ot r . ' {p(" (o) ,qt ) } impl ies that there must ex is t some

"f i rs t " a* € [0, 1] such that 0 e p(s(a*) , Q). Let t ing

a * : s u p { a : 0 / p ( s ( a ) , Q ) } ,

we ciaimthat s(a*) /D. To prove this claim, we proceed by contra-diction and assume that s(a*) € 2. Hence, there exisis some d < a*such that s(a) e 0D. However, in view of our standing assumptionthat 0 e p(Ap@),Q) tor a l l 6 e 1, i t fo l lows that 0 € p(s(d) ,8) .This, however, contradicts the maximality of a*.

Our next c la im is that z : 0 l ies on the boundary of p(s(a*) ,e) ;i.e., 0 € )p(s(a.),8). Proceeding again by contradi ction, if z : 0 l iesinter ior to p(s(o*) , Q), then. by cont inui t5 ' o f p(s(a) , Q) wi th respectto a. there exists some some d < a* such that 0 € p(s(a), Q). Again,we have contradicted the maximality of a*.

The proof of Case 2 is now completed as in Case 1; i.e., s(a*)l ies in the boundaly of the polygon p(s(a*),Q) if and only if thereexists some pair of extreme points qtl and q'tt corresponding to anedge of Q and some )* € [0, 1] such that p6,6(s(a*) , A*) : 0 . Thiscontradicts the 2-stabil ity of f; i (s, )"). E

ExERCTSE 9.4.2 (Stronger Version) : N4odify the proof above toshow that the Edge Theorem remains valid under the weaker hy-potlresis that q?t and qtz are extreme points of Q corresponding tothe edges of the coefficient set a(Q). Although the number of edgesof a(Q) is typically less than the number of edges of Q, the identif i-cation of thcse edges is generally nontrivial. Hence, frorn an appli-cations point of o'is-n', it can be argued that this stronger version ofthe Edge Theorern is "less useful" than Theorem 9.4.1.

9.5 Fiat Dedra Engine Revis i ted

In this section, we il iustrate the application of the Edge Theoremusing the Fiat Dedra engine model derived in Section 8.2. The srarr-ing point for the analysis is the seventh order closed loop polynomialp(s,q) given in Appendix A. In accordance with Fiat specifications,

9.5 / Fi^t Dedra Engine Revisited 155

we consider three operating conditions. Using our standard q nota-

tion, the three operating conditions of interest are the vectors Q : Qn,q: q.B and q : gc associated with the rows given in the table below.

Qt Qz Qz Qa qb Qa Qz

A 2.1608 0.1027 0.0357 0.5607 0.0100 4.4962 1.0000

H J . + J Z J 0.L627 0 . 1 1 3 9 0.2539 0.0208 2.0247 1.0000

C 2.1608 0.1027 0.0357 0.5607 0.0208 4.4962 10.000

Three Operating Conditions of Interest in g-Space

The three operating points are interpreted as follows: Operaiing

point g/ represents the completely unloaded engine at idle speed.

Operating point qE is the most common of the three and represents a

slightly loaded engine at idle speed. It can be viewed as the norn'inal.

Finally, operating point qc represents deviations from the nominal

parameter set as in A, but with a larger inertia and different values

for the motor gains 1{5 and l(7.

9.5.1 The Polytope of Polynornia ls

Corresponding to each of the three operating points in the table, we

compute a closed loop polynomial by evaluating the uncertain poly-

nomial p(s,q) in Appendix A. After a lengthy but str-aightforwardcomputation, we obtain

p(s,qA)=s7 * 1.444309s6 +0.736t252s5 +0.1772927s4 + 0.02648999s3

+ 2.442136 x 10-3s2 + 1.13555 x 10-as + 1.73903 x 10-6;

p(s,qB)xs7 -F 1.336368s0 + 0.ozzg808s5 + 0.1802489s4- l - o-02924289s3

+ 0.002765453s2 + 1.270495 x 10-as + r .920464 x 10-6;

p(s,qc) . ' to0s7 + 68.95635s6 + 16.31662s5 ' l 2 r2o266sa + 0.1708726s3

+ 0 .008123757s2 i _1 .965509 x 10 -as + 1 .830692 x 10 -6 .

There are many possibilities for studying robust stability lvi.th

respect to transitions between operating points. In th.is iilustrative

appiication of the Edge Theorem, we consider transitions associated

with the polytope of polynomials

156 Ch.9 / TheEdgeTheorem

p(s , qc)

/ 4 \

P \ s ' q " )

FIcunn 9.5.1 A TYiangle of Poiynomials for the Fiat Dedra

Note that this family can be viewed as a triangle in the space ofpolynomials or the space of coefficients; see Figure 9.5.1.

9.5.2 Appl icat ion of the Edge Theorern

By inspection, we observe that the polytope of polynomials P sat-isfies the preconditions for application of the Edge Theorem: The

theorem indicates that we do not need to check stability of all poly-

nomials in the triangle-we need only examine the three edges de-

scribed by

pAB(s , ) ) : ) p ( " , s \ + ( l - ) )P (s , qB ) ;

pAC(s , ) ) : ) p ( s , s \ + (1 - ) )P (s , qc ) ;

pBc (s , ) ) : ) p ( s , qB ) + (1 - A )p (s ' qc )

a n d A € A : [ 0 , 1 ] .We now summarize the results of numerical computations: We

first calculate the roots of p(s, qA), p(s, qB) and p(t, qc) and find thateach of these three polynomials is stable. For example, the roots ofp( t ,qA) are found to be s1 = 0.0276, s2. ; . = -0.7047 + j0 .0041,

s4,5 = -0.0540 + j0 .1332, s6 = -0.4810 and sz = -0.5785.

The next step in the computation is to identifS' the "interest-ing" frequency ranges associated with zero exclusion testing for eachedge. By carrying out a preliminary frequency sweep for each edge,

[,

[iil

p ( s , q B )

P : corrv {p( . , qo) , p(- , qB ) , p( . , sc ) } .

i , i , j,,.i1t i . ; i : j

ii,rii,r',.i]

f",,i r , 1 :

f , , .

f58 Ch.9 / TheEdgeTheorem

9.5 / Fiar Dedra Engine Rer.isited I57

it is easily verif ied that a suitable cutoff frequency is a": 9.4.In other words, for c..r ) u., the three Zero Exclusion Conditions0 F peeU., L) , 0 / pec( j . , A) and 0 / pBc( j . , A) a l1 hold.

To compute the robust stability test, we now generate vaiue

sets for each edge. In accordance with Remarl<s 9.3.1, each of thesevalue sets is a straight l ine segment; e.g., for pap(s,A), the va,lue setp( juL,A) is the l ine segment jo in ing pta( ja,0) and p.4n( j . ,1) . Mreprovide three representat ive value set p lots in F igures 9.5.2, 9.5.3and 9.5.4 using the clit ical frequency lange in eaclt case. Since Lhe

4e-05

3e-05

2e-05

1e-05

Irn0

-1e-05

-2e-05

-3e-05-1e-05 0 1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 7e-05 Be-05 9e-05

Re

FIGURE 9.5.2 Value Sets of pa6(ja, A) for 0.1 1 u < 0.2

Zero Exclusion Condition is satisfi.ed in all ca,ses) we conclude fromTheorern 7.4.2 tinat each of the edges is stable. Hence, by the EdgeTheorem, we conclude that 2 is robustly stable.

EXEFLCTsE 9.5.3 (Nr-rmerical Conditioning) : Using the triangle ofpolynomials 2 for the Fiat Dedra engine above, verifi.cation of theZelo Exclusion Condition for edge AC required a computation atrather higir resolr-rt ion. Hence, the following qucstion arises: Givenour judgrnent cali in reaching the conclusion 0 / pAC(ja,A) for allc.., ) 0, what can be done to boost our confidence that the cornputedsolution is correct? This issue is addressed below.

FrcuRE 9.5.3 Value Sets of pag(ju,A) for e.I < u < 0.2

(a) Verify that p(s, qA) and p(s, qc) have a common root s = _0.0276.(b) Argue that s r= -0.0276 is a root of pas(s,A) for all ) € A.(c) Motivated by (b), define a nerv family of polynomials by

p a r . - ( s . t r 1 : P a c ( s ' ) )' s + 0.0276

ExERcrsE 9.b.4 (Appl icat jon of the Edge Theorem); Use the EdgeTheorem to investigate robust stability of the porytope of polynorni.-als considered in Bartlett, Hollot and Huang (f SgSl;' i .e., take

Pt ( " ) : s3 + g '77s2 - l - 30 '6s - l 18 '27;

Pz ( t ) : s3 + 15s2 * 75s * 25 ;

P r ( s ) : s3 + 8 ' 96s2 * 27 '9s - t - 15 '61 ;

0.0001

8e-05

OC-UD

4e-05

fm- " " 2e-05

0

- ze-uD

-4e-ub

-4e-05 -2e,05 2e-05 4e-05 6e-05 8e-05 0.0001

Re

IIII

_L

I

9.5 / FiarDedra Engine Revisited 159

0.006

0.005

0.004

0.003

Jrn 0.0O2

0.001

' 0

-0.001

-0.002-0.006 -0.005 -0.004 -0.003 -0.002 -0.001 0 0.001 0.002

Re

FIcunp 9 .5 .4 Va lue Sets o f . ppa l ( ju ,A) fo l 0 .3 1u < 0 .4

p4(s ) : s3 + 11 .43s2 + 20 .2 t+ 82 .5

and s tudy robus t s tab i l i t y o f P : conv {p1 ( ' ) , pz ( ' ) , ps ( ' ) , p+ ( ) }

ExERCTsE 9.5.5 (The Example of Soh and Foo (1990)):- Consider

the polytope of polynomials with generators p1 (s) : s' l2s 12,pr( t ) : s214s*5 andp3(s) : s2+2s*5. Next , wi th z6 : -1.5* j1.5

a n d z $ : - 1 . 5 - j L . 5 , l e t D : D 1 O D 2 , w h e r e

D 1 : { z e C : l z - r o l > r } ;

D 2 : { z e C : l z - " 6 1 > r } .

With c > 0 suitably small, show that the edges of P are robustiy

D-stable but P is not robustly 2-stable. Explain this pathology in

Iight of the trdge Theorem.

EXERCTsE 9.5.6 (The Roie of Invariant Degree): To demonstrate

the importance of the invariant degree requirement in the Edge

Theorem, consider the polytope of polynomials 2 of Sideris and

Barmis l r (1989) descr ibed byp(" , n1 : q1s2l (2q1tq2)s ' l (q t*qz+I) ,

160 Ch. .9 / TheEdgeTheorem

7f 4 and 0 < qz < 1. With simply connected D region as

Figure 9.5.5, argue that the four edges of P are robustlyo < q tshown tn

Frcunp 9.5.5 2 Region for Exercise 9.5.6

2-stable but wi th qr : 11128 and q2:718, p(s, i l has a root out-

s ide 2.

9.6 Root Version of the Edge Theorern

lb motivate the deveiopment below, note that tire Edge Theoren-r.

provid.es a yes or no solution to the robust 2-stability problem for

a polytope of polynomials. We now present a lesser known ver-

sion of the Edge Theorerrr which provides more detailed information

aboui the actual root locations for a polybope of polynornials. Using

the roots associated with the edge polynomials, we can obtain the

boundary of the so-called spectral set of P.

DEFrNrrroN 9.6.1 (The Spectral Set) : Given a fanily of polyno-

miais P : {p( . ,c t ) : q e Q}, we cal l

o lPl : {z e C : P(z,q) : 0 for sorne q € Q}

tJne spectral set (or root set) of P'

REMARKS 9.6.2 (Spectral Set and Robust 2-Stabil ity) : The spec-

9.6 / RootVersion of the Edge Theorem 161

tral set generation problem generalizes the robust Z-stability prob-lem in the following sense: Once we have the spectral set o[2] , wehave a complete characterization of all possible D-regions for whichrobust 2-stabil ity is guaranteedl i.e., if olPl is in hand ar'dD C Cis given, robust D-stability is guara.nteed if and only if the condition

olPl c D

is satisfied.

Et<EF.crsE 9.6.3 (Basic Properties of a[2]): Given a polytope ofpolynornials 2 with invariant degree, show that olPl is both closedand bounded.

No:rATror.I 9.6.4 (trdges and Boundary) : Given a polytope P, wetake t (P) to be i ts set of edges. Now, i f P : {p( . ,q) : q e Q}is a polytope of polynomials, we can use the natural isomorphisrnbetween polynomials and coeffi.cients to defi.ne edges of 2. Namely,if t(a(Q)) denotes the set of edges of a(Q) in R'*1, we take

t ( P ) : { p ( - , q ) : a ( q ) e t ( " ( Q ) ) }

for the set of edges of 2.

rtEtr4ARr(s 9.6.5 (Edges of Q): From an applications point of view,however, it is rnore convenient to work with the edges of Q ratherthan tlre edges of a (Q) ; recall the discussion in Exercise 9 .4.2. Ha- npif t(Q) denotes the set of edges of Q, we work with the set

tq(P) : {p( - , q) : q € t (a) }

in l ieu of t(P). In vierv of the fact that a(Q) is the image of Q underan afEne linear transformation, it follows from Lemma 8.3.17 that

t (P) c tee).

Finally, note that in the theorem to follow, we use the notation )ofPlto denote the boundary of the spectral set olPl.

THtroFtEtvI 9.6.6 (Root Version of the Edge Theorem): Giuen apolytope oJ polynomials P : {p(.,q) : S € Q} wiLh irtuariant degree,zt follous that

aofP] e o[tae))

162 Ch,g / TheEdgeTheorem

PRoor': We fix z e )olPl and must show that z e olt(P)1. Pro-ceeding by contradiction, suppose z F ol€(P)] . For each extremepoint pair (qo,,qo') defining an edge of Q, we obtain an edge poly-nomia l p i , 6 ( s , ) ) : ) p ( " , g i ' ) + (1 - ) ) p ( s , q i z ) f o r P . We knowthat 0 / ph,tr(z,A) and by closedness of o[P] , z e ofPl. Now, foreach (g ' r ,g?2) pai r def i .n ing an edge polynomial of P, cont inuib l 'o fpil jr(., )) implies that there exists some €i1,i2 ) 0 such that if z' e Cwi th lz l - z l < € i1 , i2 , then 0 / ph, t r ( " ' ,4y) and zt € o lP) .

Now, rvithe: rrt i!. e;r, in

it follows that the disc B,(z) - {"' , lr' - ,l < e } is whoily contained

in the spectral set of P; i.e., B,(z) c olPl. However, this contradictsthe standing assumption that z e ?ofPl. E

ExERCTsE 9.6.7 (Stronger Version): Modify the proof above toshow that

aopl e o[ t (P)1.

ExERCISE 9.6.8 (Spectral Set Generation) : Use the root versionof the Edge Theorem to generate the spectral set for the triangle ofpolynomials 2 associated with the Fiat Dedra engine; see Section 9.5.

9.7 Conclus ion

The Edge Theorern is the takeoff point for the generation of manyresults in the sequel. For example, when dealing with interval plantsand fi.rst order compensators in Chapter 11, the robust stability prob-Iem for the closed loop is f irst reduced to an edge probiem and sub-sequently, new machinery makes it possible to reduce the checkingof edges to the checking of extreme points. Hence, the statementof the final result makes no mention of edges-the reliance on theEdge Theorem occurs at the level of proof rather than at the level ofapplication. As far as more direct application of the Edge Theoremis concerned, a fundamental limitation is encountered beca,use thenumber of edges of a polytope can grow' exponentially fast with re-spect to the number of variables describing it. This issue is the focalpoint of the next chapter. For poll'topes generated from feedbackcontrol systems, this exponential growth problem can be overcorne.

Notes and Related Literature 169

Notes and Related Literature

NRL 9.1 The proof of the Edge Theorem by Bartlett, Hollot and Huang (t9gg)is carried out in coefficient space without recourse to the value set.

NRL 9.2 In the papers by F\.r and Barrnish (1989), Tits (1990) and Soh andFoo (1990)' refinements of the Edge Theorem are provided for more generalclasses of D regions. For example, in F\r and Barmish (1ggg), it is assumed that1) has the following property: Through every point in D'. there is an unboundedpath that remains within D'. In Tits (1990), arr even less restrictive conditionis used: Either the condition of F], and Barmish (19gg) holds or 0 € D., and.through every point in D" there is a continuous path to the origin which remarnsttl 1)- -

NRL 9.3 At a more general level, an edge theorem can be provided for delaysysterns; see Fu, olbrot and Polis (1989). In fact, it is even possible to providea sirnilar result for a polytope of analytic functions; see soh and Foo (lggg)

and Dasgupta, Parker, Anderson, Kraus and Mansour (lgg1).

NRL 9.4 The discovery of the )-invariant root s = -0.02T6 found in Exer-cise 9.5.3 motivates a number of interesting research problems associated withnumerical conditioning of robust stability cornputations. For exarnple, given apolytope of polynomials P, provide classes of isomorphisms p - p such tbatrobust stability is invariant and computations for F are "simpler" in some quan-tifiable sense.

NRL 9-5 The connection between value set theory and the root version of theEdge Theorem is recognized in the Ph.D. dissertation of Bartlett (19g0a); ourproof is an adaptation of the one given in this reference.

NRL 9.6 The topic of transitional models for robust stability analysis seerns npefor future research. To briefly illustrate what is meant, consider the following ai-ternative to the polytope of polynomials used in the Fiat Dedra analysis: Giventhe three operating points qA, qB and gc, study robust stability of the farnily ofpolynomiais descr ibed by p(s, , \ ) : p(s, , \ rgA + Arqu + )3gc) wi th ) restr ic tedto the unit simplex A. Note that this family of polynomials has coefficients de-pendilg nonlinearly on q. There are rnany interesting modeling issues rnotivatingnew research. For example, suppose that transitions bet$,een operating porntsmay be restricted; e.g , one must pass through qB in going from qA to qc Suchrestrictions can be incorporated into the formulation of new robustness problems.

Chapter 10

Distinguished Edges

DJ'nOpSIS

The ntain obstacle associated' ,ith application of tlie Edge The-

orem is computatiotTal complexjty; for arbitrary pdytopes, tlle

nunrber of edges can be excessivelJ/ )arge' In t)tis chapter, u'e

see that a "stnall" distinguished subset of ilte edges can often

be identified. These edges are distinguished in the sense fhaf

the remaining edges need tTot be checked in a robust D-stability

analysis. Of particular interest is the Thirty-Two Edge Theo-

rem of Chapellat and Bhattacharyya This result applies to

feedback systems involwng intett'al pJants-

10.1 Int roduct ion

From an applications point of view, there is one difficulty associated

with the Edge Theorem-a cornbinator-ic erplos'ion in the nurnber of

edges of Q as the number of uncertain parameters increases. For

example, if Q is an /-dimensional box, then the number of edges of

Q is given bY N"d'g., - lzt-r '

This exponential growth with respect to (. can lead to selious com-putational difficulties. For I suitably large, the required robustness

computation can easily exceed the capability of modern computers-

In order to overcome the combinatoric problem above, lve con-

sider the follorving question: From the !2t-r posslble edges of. Q, can

164

r i lt , l) i

' iI

l rI

r il li ; ii 1

l , il i

li ti r j

I

r ll i l

l

,;l

i : 1

i : lJ i

r ; ll ' :t , , li !

i " i| , j

L ltl

i ,'il : ' ]i . : li i il l i r : i1 , " 1i . i : i

I ,,]l : . : 1, " l

, , i' , : ]: ; . l

r l r r i

. i : lj r : ], , ' i

i ll , i

r , jl t

I t it . , : lr ' ' ll : - i

l i i,::j: r ' , . i

l ' : , i lI t : i ll i : . , it , r . i. t lj : r j : i

I : , . i

l t ,

" .i l

fr..l , i t

[i'l',,'i:-,

I0.2 / Parailelotopes 165

we identify a "small" distinguished subset of c,itical edges whichare important as far as robust 2-stability anaiysis is concerned? Inother words, once we have found these distinguished edges, we canignore all remaining edges in a robust stability analysis. when sucha subset is readily identifiable, then the Edge Theorem becomes moreuseful as an application tool for problems with L being large.

LO.Z Parallelotopes

St'-ong motivation for the technical discussion to follow is derivedfrorn an irnportant observation: Given a polytope of polynomialsP : {?( . ,q) : q e Q},even rhough the uncer ta inty bounding setQ C Rt has a total number of edges which increases exponentiallywith respect to (., at f ixed frequency cu € R, the value set p(ja,e)may have a number of edges which increases at a much slower ratewith respect to !.. fn fact, we see below that for the case when eis a box, we obtain a, value set which is a ',parallelotope" havingat most 2!, edges. This name is derived from the following fact:Whenever the value set has a nonempty interior, for each edge ei'there is a second edge e1" f e;, such that e1, and e;, are palailel.Before proceeding, the reader should be forewarned tirat care mustbe exer:cised in exploiting the parallelotope property in a foequencysu'eeping context. That is, t}'e 24 distinguished edges may change asa fi-rnction of frequencyl further discussion of this issue is proviciedin the notes at the end of this chapter.

DEFTNT:rroN 10.2.1 (Paral le iotopes) : Let p : {p( . ,q) : q e e}be a polytope of polynomials with Q c Rl being a box. Then p iscalled a parallelotope of polynomials.

REMARKS 70.2.2 (Boxes Versus Polytopes): Frorn an applica_tions point of vierv, working with parallelotopes rather than generarpolytopes can hardly be viewed as restrictive. Since e representsuncerta.inty bounds for underlying physical parameters, in most ap-plications, the box rnodel is cluite appropriate.

ExnRCTsE LO.2.3 (Nurnber of Edges of a Polygon) : Take p to be apolygon in the complex plane and let IUI : {rrz1, n1.2t. . . ,rnN} denotethe finite set of numbers representing the "slopes', of its edges. Arguethat the total number of edses of P is at most 2ll.

I66 Ch. i0 / Distinguished Edges

LEMMA 10 .2 .4 (Va lue Se t ) : Le tP . : { p ( ' , q ) : q e Q} b " a pa ra l '

lelotope of polynomi,als wi,th Q C Pct. Then, giuen anll z e C, the

ualue set p(z,Q) is a polygon wi ' th2[ . s ' ides at most .

PRoor: Since p(s, q) has an afflne ]inear uncertainty dtructure, we

can writeL

p(s, q) : po (s) + | o. in. i (s),i :o

whele po (s),pr (s), . . .,p^s) are fi.xed poLynomials. Now, in accordancewith Proposition 8.7.1, the value set p(2, Q) is a convex polygon with

edges which are obtained from the edges of Q. Since Q is a box, each

edge of Q is obtained by setting all but one qr to one of its extreme

values and varl.ing the rernaining nncertain parameter betlveen its

bounds; i.e., given uncertainty bounds at < qt < qf,, all edges of

pQ,Q) are generated using the expression

Fe,qi : po(z) + skpx(z) +7)ufn;?),x f K

where q! -- qf or go- for- i,+ k, S; < sk S Cf, and pnQ) * O.To complete the proof, rve observe that the edge of. p(z,Q) as-

sociated with qp has slope (perhaps infinite) given by

. : _ . _ I rn pr(z)" " * -

R e p k Q ) '

Since the slope nzk does not depend on the fixed choices of qi+ above,

every edge of p(z,Q) has a slope which assunres at most one of I

possibie values. In view of Exercise 10.2.3, rve conclude that p(z,Q)

has at rnost 2!, edees. S

REMART(s LO.2.5 (Subtlety) : Consistent with the discussion at

the beginning of this section, we make note of one subtlety a"ssoci-

ated with the practical exploitation of Lemma I0.2.4. Narnely, the

2/ distinguished edges in the lemma above may vary with z. For

example, in robust stabil ity analysis with; : ju, as c.r is swept frorn

0 to *oo, the following situation can occltr: As one increases tire

frequency from cu : ar to Lr : (D2t some of the distinguished edges

of p(ju.t,Q) can move into the inter-ior of p(ju, Q) and new edges can

emerge as members of the distinguished set.

t:t ,i i iliF ,

ii;I

i i ri.

iita

i i ,i i .l i .

! ,' ,l : 'i'r

[,t$ l

i i, l

il 1i : r l tl : , i

l i , I iI! . . i :| ' ' : i

l - ifl ril ' l i

i

10.3 Parpolygons 'o{

In this section, lve provide a minor ernbellish'rent on Lerlma I0.2.4.we see that tire value set not only has at n.rost 2L sides but aiso has r'iopposite sides which are parailel

I0.3 / Parpolygons 167

Ftcune 10 4 .1 Bas ic Setup fo r th is Chapter

DEFrrvrrroN 1O.3.1 (Parpolygon): Let P be a polygon with distinctedges e1 ,€2 , . . . , €N . \Ve ca l i P a pa rpo lggon i f e i t he r rV : 1 ( t hedegenerate case) or the following condition is satisfied: For eachi 1 € { 1 , 2 , . . . , A I } , t h e r e e x i t s s o m e i 2 e { I , 2 , . . . , 1 / } s u c h t h a tit * iz and ei, and e4, are parallel.

ExEn-crsE aO 3.2 (Value Set as a Parpolygon) : Under the hypothe-ses of Lemma70.2.4, prove that the value set p(z,Q) is a parpolygon.Hznt: Identify parallel edges of p(z,Q) using paraltel eciges of e.

Lo.4 Setup with an Interwal Plant

Throughout the remainder of this chapter, the focal point is avoid-ance of the cornbinatoric explosion problem associated with robuststabil ity testing for a polytope of polynomials; recall the discussionin Section 10.1. We now describe an important cjass of poll,topesfol which this problem can be totally eliminated. Namely, we con-sider the stability problem which arises upon interconnection of aninterval plant and a fixed compensator as shown in Figure 10.4.1.

168 Ch I0 ,/ Distinguished Edges

MIe represent the lnterval plant family P bJ'writing

. l / (s , q)P ( e n r \r \ " 1 y 1 . /

D ( s , r )

wi th m n

l r ( " ,q) : I [cn- , q{ ]s i ;nG,r ; : ! l r r , t f ] " i 'X:U

Subsequently, we exPress

C(s ) : C1 (s )C2(s )

as a quotient of PolYnornials

l/n (s)c(s) : D;(")

and obtain the family of closed loop poiynon-ria:ls Pcr described by

p ( s , q , r ) : l / ( s , q ) N c ( s ) + D ( s , r ) D s ( s ) ,

S e Q a n d r € J ? ; i . e . , P c r , : { p ( ' , q , r ) : q € Q ; r e R } ' S i n c e t h e

uncertainty bounding set Q x -R can have as nlany as

N.d.s.": (n -F rn * 2) , 2n*n*1

10.5 The ThirtY-Two Edge Theorern

The proof of the theorem below is relegated to the next two sections.

TTTEoREM 10.5.1 (Chapellat and Bhattacharyya (1989)): Con-

sid,er an ,interual ptant f amity P hauing Kh,o,ri,tonou polynom'ials 1\/r (s),

.

1 i

I

l ti 1i ll

' i

:i t l

i , ll r .i _

i ' il r , r lL : I

i : : )

l ' : i ' r i

I i t t i : t, , , . i . i

,- !: t)

cz(')Interval Plant

P : { P ( . , q , , r ) : q € Q ; r € R }

cr (s)

10.6 / Octagonaliry of the Value Set 169

l / r ( " ) , Ah(") , A/a(" ) and, D1(s) , D2(s) , Ds( t ) , D+(t ) for the numer-ator and denonz'inator, respect,iuely, and compensation blocks C1(s)and C2(,s) as indicated in Figure 10.4.1. Assum'ing the farnily ofclosed loop polynom,ials P6y has i,nuariant degree. robust stabi,ti,ty ofP67 i,s guaranteed if and only i.f all edge polynomials of tlze form

e(s, ) ) : l / , , (s) ,A16, (s) + Da, , ; " (s , ) )D6:(s)

w i t h i , 1 € { I , 2 , 3 , 4 } a n d ( i z , i z ) € { ( 1 , 3 ) , ( 1 , 4 ) , ( 2 , 3 ) , ( 2 , 4 ) } o r

e(s, ) ) : Ni t , iz(s , ) )N6r(s) * Dh(s)D6r(s)

w z t h ( i 1 , i ' z ) e { ( 1 , 3 ) , ( r , 4 ) , ( 2 , 3 ) , ( 2 , 4 ) } a n d r . 3 € { 1 , 2 , 3 , 4 } a r e s t a -b le for a l l . \ € [0, 1] .

10.6 Octagonality of the Value Set

Our proof of Th-eoren 10.5.1 expioits a characterization of the valueset p(ja,Q, R). In addition to facil i tating the proof of the theor-em,this chara.ctelization is useful in its or.vn right; it can be used togenerate a graphics display of p(ja,Q,R). Hence, satisfaction or-fai.lure of the Zero Exclusion Condition

o / p ( j w , Q , R )

can be verif ied by inspecrion.Indeed, we begin with the setup in Section 10.4. Now, associated

witlr the interval piant family P are four Kharitonov polynomials forthe numerator

Nr(" ) : q ; + s is + s{ t '+ Q3+s3 + q; tn * q is5 + Q6+so +. . .r r /z(s) : qoF l - s f r + q; t ' t q" t t + s f tn +s*r t + qt "u f . . .Ns(s) : qo* * s fs + q; t ' tq f " , + qt to +q;r t f q t "6 + . . .r u+ ( " ) : go *s f "+ q { t ' + q t " t f q ; t n *e5+s5 +s f "u + . . .

and four Kharitonov polynomials for the denorninator

, i ( s ) : r t + r , s * r { s 2 + " r + r 3 + r ; s a + r 1 s 5 * r j - " 6 + . . .Dz( t ) : r i l + r fs -F r ;s2 + r ;s3 + , [ ta + r . f , s5 + ro

"6 + . . .

D : (s ) : r0 l - + r , s - t r i s2 +"u+s3 + ' r [sa +r1s5 + r ;s6 + . . .D a ( t ) : r ; + r f s * r f s 2 +

" i " 3 + r ; s a +

" f , s 5 + . u i - " 6 + . . .

I70 Ch. 10 / Distinguished Edges

Recall ing the discussion on interval polynomiais in Section 5.3'the trvo value sets N(J.,Q) and D(j.,.R) are rectangles and obtainthe value set for the closed loop polynomial as the direct sum

p( ju, Q, R) : Nc Uu) N ( ju , Q) + D c U u) D ( i a , R) .

To characterize this set. we consider trvo cases and combine the re-sults at the end.

Case 1 (The Degenerate Case): It D6(ju) : 0, then

P( i . ,Q , R ) : Nc ( i r )N ( i . ,Q ) .

Recall ing the remarks associated with Example 8.3.19, we can view

Nc(j.) as a l inear transformation on N(j-,Q). Hence, we thenobtain p(ja,Q,ft) by rotating the Kharitonov rectangle N(ju,Q)through an angle 0: LNc(jcu) and scaling all points by l l/6'(7r,, ')1.This siLuation is depicted in Figure 10.6.1. It is plain to see that the

N2(jut)Ng ju)

N3$u)N6( ju)Na(ju)Ng(ju

N+(j.)

Nt(ju)

N1(ju)Ivs(ju)

N ( j ' , Q )

Nz( ju )

Ns(ia)

FIGURE 10.6.1 The Value Set for the Degenerate Case

value set p(ju,Q,f i) is described using the four edge polynomials

l l

i:i::

p(ju, Q, R)

e1(s, )) : [ ) .n/2(s) + (1 - ))Ar3(s)] ,Vg(s);

10.6 ,/ Ocmgonali ty of the Value Set I7l

e2(s, . \ ) : [ ) l /1(s) + (1 - ) )Af t (s) ] l r /c ' (s) ;

e3(s, A) : [ ) l /1(s) + (1 - ) ) /V i (s) ] ,V6(s) ;

ea(s, A) : [ )A!(s) + (1 - ) )Ar+(s) ]Nc(s) .

This partial result will later be combined with the results obtainedfor the nondegenerate case below.

Case 2 (The Nondegenerate Case): It D6Qtt) I 0, then division byDc( ja) Ieads to

p( j , ' t ,Q, R) : Dc( ia) [C( j " , )N( i r ,Q) + D( ia, R) ] .

Now, by viewing C(j.) as a l inear transformation on N(j.,Q),it follows that C(ju;)N(j.,Q) is simply a rectangle which is a ro-tated and scaled version of. N(ja, Q); ..u Example 8.3.19. ApplyingLemma 8.3.9, we arrive at the formula

p( j , ' t , Q, R) : D c ( j . ) . conv l ) {c ( i 4 w ( j - , Q) + D i ( j a) } .

Now, we can easily identify the edges of p(jut,Q,/?) by firstobtaining the edges of the set

P(u,') : co""[J{c( ju)N(ja,Q) + D{ju)}.x

To this end, we claim that P(c,.') is an octagon (or a degenerateoctagon). To prove this, there are four cases to consider; i.e., the'i-bh case corresponds to

0 < [ c ( j . , ' ) s z ] .z

We now concentrate on the first case, O < LC(ju) < rl2. Byviewing C(j.) as a linear transformation on N(j., Q) and treatingDtUr) as a translation, it now follows that each of the four setsC(iu-t)N(ja,Q) + Dr(jr) is a rotated rectangle with center obtainedby "applying" C(j.) to the centerpoint of l/( ju,',Q), followed bytranslation by Di(ja). Now, to complete the construction, we forrnP(c,,,) as the convex hull of the union of these four rectangles. Subse-quently, Lemma 8.3.15 leads us to conclude that P(cl) is an octagonwith eight edges as shown in Figure 70.6.2. Of course, the figuremust be interpreted in the appropriate manner because it does notrepresent all possible geometriesl e.g., if Im D2(ju:): Im, Ds(ju),'we can obtain a hexagon instead of an octagon. Nevertheless, the

172 Ch.70 / Dist lngulshed Edges

C(ju)Nz(ju) + Da(ja) C( ju )Nz( ja ) + Dz( ju )

C(j". ,)Na(ja)+Da(ju)

P a

C(ju)Na( jut)

+ D 1 ( j u )

u DC( ju )N1 ( j u ) + De ( i LD)

C( ju t )Nr ( ju ) + Dt ( ju )

FIGURE 10.6.2 Construction of P(cr)

figure gives us the complete description of the eight edges of the

value set octagon with the understanding that some of the edges

in our list may be nonunique. In other words' although we use

the expression ualue set octagon, the understanding is that in some

cases the formulas to follow can lead to degenerate octagons such as

hexagons, quadrilaterals or line segments. Nevertheless, eight is the

upper bound on the number of edges.

To summarize, associated with the eight edges e1, €2, "',Eg in

the figure are eight edge polynom'ials f.ot the value set p(ju,Q,R)'

For example, beginning with edge E1 , multiplication by D6(jr''') leads

to the edge polynomial

z1(s ' ) ) : l rz(s) l /c(s) + D2,a(s, ) )D5:(s) ,

whereDz ,a (s , ) ) : IDz (s ) + ( t - ) )Da (s ) '

In a nearly identical rrranner, we can obtain formulas for the remain-

ing seven edges of the value set octagon. The presentation of these

formulas is facilitated with the definition below.

DEFrNrTronr 1O.6.1 (Crit ical Numerator and Denominator Edges):

consider an interval plant family P with Kharitonov numerator and

C (j u) Ns(j u:)

+Dz ( j a )

C( ju- t )Nz( ju)

+Dz( icu)

10.6 / Octagonali6,of the Value Set L73

denorlinator polynomials -A[(s), t/z("), lh(") and }/a(s) and D1(s),Dz(s), rs(s) and Da(s), respectively. Then, for each pair

( i t , i z ) € { ( 1 , 3 ) , ( 1 , 4 ) , ( 2 , 8 ) , ( 2 , 4 ) } ,

we define a crit'ical numerator edge polynorn,ial

IVt , , i " (s ,^) : )1\ I , , (s) + (1 - A)A! , (s)

and a crit ical denom,inator edge polynom,ial

Dt , , . i " (s , ) ) : )Dr , (s) + (1 - ) )D; , (s) .

REMARKS 10.6.2 (Formuiae for Edges of the Value Set): UsingFigure 10.6.2 and arguments identical to the one used to obtaine1(s, A), we sti l l continue to assume that

0 < 4 C ( i , , t ) . 72

and obtain the complete set of eight edge poil,noto'uts chara.cterizingthe vaiue set octagon. Namely, with A : [0, 1] and

e1(s, . \ ) : 1 t /2(s) l /6r(s) + D2,a(s, A)Ds(s) ;

e2(s, . \ ) : Nz,s(s) l /c(s) + D2(s, . \ )D6:(s) ;

, es(s, ) ) : A[ (s)Jt /6(s) + D2,3(s, ) )Dc(s) ;' '

ea (s , ) ) : l / r , s ( " ) l / c ( " ) * D3 (s , ) )Ds (s ) ;

e5 (s , A ) : .A / i ( s ) I / c ( s ) + D1 ,3 (s , ) )Dc (s ) ;

e6 (s , ) ) : l / r , + ( s ) l / c ( s ) + D1 (s , ) )Dc (s ) ;

e7(s, ) ) : 1 \L(s) l /6(s) + D1,a(s, ) )D5:(s) ;

es(s, A) : -Ah,+(s)N6,(s) + Da(s, ) )D6:(s) ,

the value set octagon p(ja,Q,R) has i-th edge

et( ja , t \ ) : {e ; ( ju . ) ) : ) e ; \ } .

Note that these formulas hold for both the degenerate and nonde-o A n p r q f p n q c o c

ExERCTSE 10.6.3 (other Possibil i t ies for [c(iu)): For the rernain-ing cases character izedby r /2 < LC( j . ) 1T, T < |C( j . ) < Br /2

L74 Ch. 10 / Dist inguishecl Edges

and 3rf 2 < 4-C(j.) ( 2zr, describe the edge polynomials associateclrvit lr tbe value set octagon p(ju,Q,??). Note that in each case, clif-ferent (it, iz) combinations are used to describe the crit ical srlocs fnrthe plant numerator and denominator.

REMART{s 10.6.4 (Summary) : Taking the union of all four casesfor [C(jto), we obtain a rna;dmum of thirty-two (not necessarilydistinct) edges which describe the behavior of the value set at allfrequencies; these edges are l isted in Theorem 10.5.1. At any fixedfrequency, horvever, only eight edges at most are in pla1r. We nowprovide an exampJ.e illustrating how the value set octagon is gener-ated in graphics. Subsequently, we visualll, inspect the graphical plotfor satisfaction of theZero Exclusion Condition (Theorem 7.4.2).

ExAMeLE 10.6.5 (Zero Exclusion Testing): We consider the inter-val plant

P ( s , q , r ) : [4, 6]s3 + [3, 5]s2 + 12, 4ls+ [6, 8]s3 + [4 ,6]s2 + 15, 7 ls + [7 ,9]

connected in the feedback configuration of Figure 10.4.1 with com-pensators C1(s) : 1 and C2(s) : f/(s + l-). \Are analyze the ro-bust stability of this systerrr by generating the value set octagon andchecking for satisfaction of tine Zero Exciusion Condition. Indeed,we first compute the closed J.oop polynomial

p ( s , q , r ) : s a + ( 1 1 + % + r 2 ) s 3 + ( 1 5 + e z : _ r l + r 2 ) s z

+ ( I 7 - t q t + ro + r1 )s + (15 + go * ro )

and easily verify that with aII qi - 0, the nominal p(s,0,0) is sta-ble. Next, we generate tl ie value set octagon. Noting the fact that-nl2 < 4-C(j.) ( 0 for all c,. ' ) 0, the eight distinguished edges ofthe value set octagon are invariant over the entire frequency range.This fact is confirmed by the value set plot in Figure 10.6.3. Theplot'vas generated using 50 evenly spaced frequency points over thecritical range 0 ( c..' < 1.3. From the plot, rve also observe that theZero Exclusion Condition (Theorern 7.4.2) is violated. Hence, thefamily of closed loop polynomials is not robustly stable.

REMART(s 10.6.6 (Edge Reduction): The power of the octagonalvalue set characterization of this chapter is demonstrated in the ex-ample above. Suppose that instead of using the theory in this chap-ter, one attacks the example above via direct application of the Edge

70.7 / Proof of Thirty-Trvo Edge Theorem 175

(

Int 0

1 n

-10

Re

Frcune 10.6.8 Value Set Octagon for Example 10.6.b

llheorem (see section 9.3). since the number of uncertain parame-ters is ( : 7, the uncertainty bounding box e x R has l2t-L : 44gedges. without the octagonai value set characterization, direct ap.piication of the Edge Theorem dictates that each of these 44g edsesmust be individuallv tested.

ExER-crsE ro.6-7 (Robustness Margin) : For the feedback systemdescribed in Exarnple 10.6.5, repiace the uncertainty bounding setQ x R by a uniformly proportioned box witb variable radius r ) O.Now, verify that the robustness margin is r*o, = 0.2b. Furthermore,when carrying out computations, note that one needs to be carefulthat the family of closed loop polynomials pcr continues to haveinvariant degree as the uncertainty bound is increased.

70.7 Proof of Thirty-Two Edge Theorern

1 0

20

176 Ch.70 / DistinguisheclEdges

argrrment; i.e., rve begin by noting thatPcr has at least one stable

*J-bul. and at sufficiently high c.,, ) 0, satisfaction of bhe conclition

o / p ( i a , Q , R )

is guaranteed by invariant clegree oI Pct'' Stibsequently' if the Zero

Exclusion condition (Theorem 7.4.2) is violated, thele rnust be some

value set penetration flequency cu" > 0 for which z : 0 lies on an

ed,ge of p(iw" , Q, R). Howevet, in view of the thirty-t-'vo edge charac-

teiiratlon of p(ir,b, A) itt the preceding section, such a penetration

would contradict the stancling assumption th.at all of the edge poly-

nomials e(s, )) are stable. E

1O.8 Conclus ion

The results in this chapter were obtained by strengthening the hy-

potheses of chapter 9. That is, instead of allo.n'ing afbitral'v poly-

iopes of polynomials for the numer..atol and denomi'atol of the plant,

r,ve worked with interval polynomials' The main payoff rvas a dra-

rnatic reduction in computational complexity. For problems with I

uncertain parameter-s, the value set is an octagon at each frequency'

Furthermore, taking the union over all frequencies, no rnore than

thirty-two edges ever come in[o play'

I n t h e n e x t c h a p t e r , w e s p e c i a l i z e t ] r e r e s u l t e v e n f r r r t h e r . W eremain within the reaim of interval plants but restrict our atten-

tion to fi.rst order compensator-s. The benefit associated with this

restriction on the compensator* is quite simply explained: Instead of

t i r i l t y - twoedges ,u 'eob ta inanex t remepo in t r - ' esu l t i nvo l v i ngs i x -teendist i , ,gu ishedplants.Si r rcemanyindustr ia lcontro l ]ers(suchasthe classical lead, lag and PI compensators) are fir-st order, there is

strong motivation for consideration of this special case'

Noces ald Related Literature 177

Notes and Related Literature

NRL 10.1 With the goal of further exploiting the 2!. edge property for a par-allelotope, the pa.per by Djaferis and Hollot (1g89a) identifies a finite nurnber offixed frequencies for which "edge switching" can occur. The key idea is describedroughly as follows: One creates a frequency partition (0, m) : U[r(c..r;,c,ri11)with each c,,," obtained by finding the roots of an appropriately constructed fixedpolynornial. This partition has the property that on each interval (u.';,ru1_p1), thedistinguished edges of p(j.,Q) are invariant. The identification of a finite set ofdistinguished frequencies is also central to the the work of sideris (tg9t), wherea nore corrrplete analysis of computational complexity is considered .

NR-L 10.2 To address the issue of computational complexity, some authors dis-pense entirely with the Edge Theorem and work more directiy toward verifica-t ion of the Zero Exclusion condi t ion; e.g. , see Barmish (1989). In Sar iderel i

and Kern (1987) and Vicino (1989), a. frequency parameterized linear program

is used. Roughly speaking, it is possible to take the data describing the robuststability problem and create a frequencg pararneterized linear program with ob-jective cr (.)r and constraint A(uL)r < b(r.l) . Letting ,^".(-) denote the infimurn

at frequency c.r ) 0, we can compute the robustness margin r*o" by minimizingr^..(co) with respect to c.r ) Q.

NILL 1O.3 In all of the papers cited in the note above, the issue of combina-

toric explosion prevails. More effective methods for overcoming the combinatoric

explosiou problem are given in Sider is (1991) and Kraus and Tyuol (1991). In

the case of Sideris (1991), computational complexity is reduced via special com-putations between simplex steps. and in the case of Kraus and T}ucjl (lgg1), analgorithrn for value set construction is giverr rvhich does not require enumerationof either edges or extreme points.

NILL 1O.4 The paper by Rantzer (1992a) also addresses the issue of computa-tional cornplexity by identi{ying a "small" nurnber of distinguished edges. This

more general framework involves testing sets which might also include extremepoints.

Chapter 11

The Sixteen Plant Theorern

Jynopsrs

If the compensator for an interval p)ant js first order. testilTg

thirty-tw'o edges for -robust stability of the dosed loop is no

longer required. Under this strengthening of )rypotltesis, i, js

shown that stability of sixteen distinguished closed loop systernsimplies robust stabiliLy of bhe entire family. In view of Lltis

extreme point result and the fact that the nttmber of parameters

entefing into the controller is at most three (the pole, tlle zero

and Lhe gain), i t becomes possjble Lo use fhe resu/ts in t ir is

cllapter in a synthesis context.

11.1 Int roduct ion

The takeoff point for this chapter is the follorving question: Given an

interval plantP with compensator interconnected as in Figure 10.4.1,

under what conditions can we establish robust stabil ity of the closed

Ioop by testing a "small" finite subset of systems corresponding to

extrerrre members of P? In other words) lvhen can we dispense with

the thirty-t$/o edges of Theorem 10.5.1 and work solely with the set

of extreme plants?We have aiready encountered one situation which leads to ex-

treme point results of the sort described abor.e. Recalling Exer-

cise 6.4.3 for the special case when C(s) : l( is a pure gain compen-

sator, we know that robust stability of the closed loop is guaranteed

t78I

1l-2 , / Setup rv i th an Interval Planr 179

i f and only if four distinguished extrerne plants are stabil ized; weneed only eight plants if the sign of K is not specified. This resulris a simple consequence of the fact that with pule gain compensa-tion, the intervai polynomial structure is preserved in going fromtlre open loop to tl ie closed loop. In contrast, \.ve sav in Chapters g

and 10 that systems with more general compensators do not havetb.is property. That is, an intelval plant problern gets converted intoa polytopic probletl under feedback.

In this chapter, we concenrrare on a special case motivated byindustrial applications. Nameiy, we restrict our attention to clas-sical first older compensators such as the ciassical lead, lag and PIcontrollers. The main result of this chapter is the Sixteen Pjant The-orerl of Barmish, Hollot, Kraus and Tenr.po (1992). M/e see that itis necessary and sufficient to stabilize only sirteen extreme plants inorder to stabiiize the entire familv.

L7.2 Setup with an Interval Plant

We concentrate on the interval plant

P ( s , q , r ) :

n

\-- l, ,tI *il : J L a z 1 a x J -

; - n

n - L- \ - r - + l

s , . + ) l r . T - ' I- ' l J L ' x 1 ' x J

As usual, Q and -R denote the boxes bounding the uncertain pararn-eter vectors q and r, respectively, and we assume that the resultingfami ly of p lants P : {P( . ,q, r ) : q € Q;r e l?} is s t r ic t ly proper;tlrat is, rn < Iz. We consider the compensation scheme indicatediri Figure IL?.I and express C(s) : Cr(s)Cz(s) as a quotient ofpolynomials by writ ing

nt - \ i /c(s)L , \ r , / :

D c G ) .

Since our sole concern is first order compensators. we take

l / c ( s ) : I { ( s - z )

D c ( s ) : s - P

180 Ch. ).1 / The Sixteen Plant Theorem

FlcuRE 11.2.1 Basic Setup for this Chapter

i , v i t h l { l 0 a n d e i t h e r p + o o r z f 0 . G i v e n t h e c o m p e n s a t o r

c(s) above and expressing the uncertain plant as the quotient of

uncertain polynomials

. l / ( s , q )P( t , q ,a : DG; ,

the resulting closed loop polynomial is

p ( s , q , r ) : K ( s - z ) N ( s , q ) + ( s - P ) D ( s , r ) '

This leads to the family of ciosed ioop polynomiais

P c L : { p ( ' , q , r ) : q e Q ; r e R } .

11.3 Sixteen Dist inguished Plants

In this section, we make a speciai selection from the set of extreme

plants. For motivation, observe that if Q is (m * l)-dimensional and

h is n-dimensional, the uncertainty bounding set Q x R has number-

of extreme points given bY

N" ' t : 2mfn* r '

Furthermore, we can associate extreme piants with the extremes of

e * R in a simpie manner: It {qt} denotes the set of extrernes for

Q and {ri} denotes the set of extremes for R, we generate the ly'"1

extreme plants by considering aii plants of the form

N ( s , q " )Pr, jr ls) : T,i*u p , i i z ) '

and

Interval Piant

P : { P ( . , q , r ) : q e Q ; r e R }

Cr(s )

I irZ / Sixteen Distinguisl.red Planm 181

From the set of extremes above, we now seiect a distinguished subsetof sixteen plants in accordance with the definition below.

DEFTNTTToN 11.3.1 (The Sixteen Iiharitonov Plants): Given aninterval plant family P with Kharitonov polynomials l/r(s), l/z("),l fe(s) and Na(s) and D1(s) , D2(s) , D3(s) and D, t (s) fo l the nu-merator and denominator, respectivelv, (see Section 5.5), we definesirteen, Kh,o.ritonou pLants bv

P / o \ l / i ' ( " )

' 2 7 , 1 2 \ " t D i r ( S )

r , v i t h i i 1 , i , 2 € { I , 2 , 3 , 4 } .

DEFrNrrroN 11.3.2 (Associated Closed Loop Polynorniais) : Forthe feedback system under consideration (see Section 11.2). lve as-sociate a c losed loop polynomial

P6 ,6 (s ) : K (s - " )N ; , ( s )

+ ( s - p )D tG)

with each Kha"ritonov plant P,r,;r(s).

ExAN4eLE 11.3.3 (Sixteen Plants and Polynomials): We considelthe interval plant farniiy 2,

D / ^ _ \ [ 4 .5 ' 5 .5 ] s3 + [ 3 .5 ' 4 .5 )s2 + [ 2 .5 ' 3 .5 ] s + 16 .5 ' 7 .5 ]^ \ " 1 4 1 ' / s 3 + [ 4 . 5 , 5 . 5 ] s 2 + [ 5 . 5 , 6 . 5 ] s + 1 7 . 5 , 8 . 5 ]

1

and note that Exercise 10.6.7 indicates that with the first order com-pensator C (s) : L I $ +7), robust stabil ity is guaranteed. In contrastto Chaptei- 10, where an octagonal value set rvas used, the results inthis chapter enable us to reach the same conclusion more simply be-cause C(s) is fi.rst order. To this end, note that it is straightforwardto caiculate the sixteen Kharitonov plants and the associated ciosedloop polynomials. To i l lustrate, it is readily verif ied that

P ^ ̂ / " ' \ -' z , t Y " 1 -4 . 5 s 3 + 3 . 5 s 2 + 3 . 5 s * 7 . 5

i , - as r * 4 .5s2 + 5 .5s l - 8 .5

and the associated closed ioop polynomial is

p2JG) : "4

+ 1os3 + 13 .5s2 * 17 .5s * 16 .

After stating the theorem below, the robust stability anaiysis for thiss l /s tem is completed.

I82 Ch. i I / The Sixteen Plant Theorem

LL-4 The Sixteen Plant Theorern

We are now prepared to provide the main result of this chapter. Theproof is re legated to Sect ions 11.6 and 11.7.

THEoREM 17.4.I (Barmish, Hollot, Kraus and Tempo (1992)):Consider the strictly proper' 'interuo.l plant familA P u'ith fi,rst ordercompensator C(s) as descr i ,bed' in Sect ion 11.2. Then C(s) robust lystabilizes P if and only i,f it stab'il'izes eaclt, of the sirteen Kharitonouplants; 2.e., the family of closed loop polynomials Pcr is robustlys tab le i f and on l ' y ' i f p ; r , i r ( s ) i s . s tab le f o r i 1 , i , 2 e { I , 2 ,3 ,4 } .

ExERcTsE LL.4.2 (Application of the Theorem): Consider the in-

terval piant family P with compensator C(s) in Example 11.3.3. Us-ing the Sixteen Plant Theorem) show that C(s) robustiy stabil izesP .

11-.5 Controller Synthesis Technique

Although the Sixteen Plant Theorem is stated as an analysis result,a moment's reflection indicates that the theorem is quite useful ina synthesis context as well. To explain this point, consider the fol-lowing illustration: Suppose that one wants to construct a robustlystabilizing PI controller

f/

C ( s ) : K r + +

for an interval plant family 2.Then, to determine if appropriate gains I(1 and l(2 exist, we

first set up sixteen Routh tables-one for each Kharitonov plant withcompensator C(s). Noting that the first column entries of these ta-bles are functions of 1(1 and K2, the positivity requirement for stabil-ity leads to a set of inequalities. Since these inequalities only involvethe two parameters -Ff1 and K2, a graphical description of ttre set ofstabilizing gains is easily generated. In conclusion, a necessary andsufilcient condition for the existence of a robust stabilizing control]erwith the specified form is nonemptiness of the set of gains satisfyingthe inequalities associated with the Routh tables. Moreover, any fea-sible point (Ki, K;) in this set of gains is associated with a robustlystabil izing PI controller. Note that the idea is readily extended to a

11.5 / Controller Synrhesis Technique 183

more general class of first order compensators having the form

^ / \ K ( s - z )u ( s ) _

s _ e

Using the Sixteen Plant Theorem, one can characterize the set ofstabil izing triples (K, z, p).

ExAMeLE 11.5.1 (Synthesis Using the Sixteen Plant Theorem):We consider the model of an experimental oblique wing aircraft givenin Dorf (7974). In the absence of uncertainty, the aircraft rransferfunction is

64.q + 12gP ( " \' s4 * B . zs3 * 65 .ds2 - l J2s '

Now, to illustrate application of the Sixteen Plant Theorem in arobust synthesis context, we repiace P(s) by the interval plant familyP described by

P ( s , q , r ) :qts + qo

s4 a r3s3 + rzs2 -l- r'1s -F rg

and consider uncertainty bounds 90 < qo < 166, 54 < q-t < 74,-0 .1 < ro ( 0 .1 , 30 .1 < 11 < 33 .9 , 50 .4 < 12 < 80 .8 and 2 .8 ( r s (

4.6. For this interval plant family P,tlte objective is to determine ifa, robr.rstly stabilizing PI compensator

C ( s ) : K r + { Zs

exists. If we determine that a robust stabilizer exists, we also wantto compute appropriate gains .F(1 and K2. For the sake of brevity, wedo not show all numerical calculations below. However, we provideenough detail so that the reader can replicate all calculations.

The first step in the synthesis procedure is to generate each ofthe sixteen Kharitonov plants with an associated Routh table forthe resulting closed loop polynornial. Note that we must carry outcomputations parametrically in the controller gains 1(1 and 1(2. Toil lustrate, using the Kharitonov polynomials l lz(s) and D1 (s), weobtain the associated piant

74s -r 766P 2 , 1 ( s ) :

sa + 4.6s3 -F 80.8s2 t - 30.1s - 0.1

and, with PI compensator C(s), the associated closed loop polyno-miai is found to be

pzl!) : s5 * 4.6s4 + 80.8s3 + (30.1 - t T4K1)s2+ (-0.1 + 1661i -t 741{z)s + 7661<2.

184 Ch. 11 / The Sixteen Plant Theorem

Now, using pzlG), we generate the Routh table

s5 1 80.8 -t(Kt) + 74IQ

s4 4.6 30.7 -t 74Kt 766K2

s3 74.3 - 16.1r(1 -'y(I(t) + 37 '9K2 0

s2 at(Kt , K2) /a2(K1) l66lh 0

s1 B(K1, K2) lar (Kr , Kz) 0 0

so L66K2 0 0

w-flere

aJK t, I{z) : 2236.89 + 424g.gg K t - L7 4'34K z - 71il ' K21 ;

oz( I { t ) x 74.3 - 16.1, I {1;

gQ<t, Xz) = -223.689 + 370,899I{1 + 705, 617K?

Lg7,772Kl - 831, 6061{z + 529,2831{tKz

-88, 182.91{?x, - 6,607.491$.;

t Q { t ) : 0 ' 1 - 1 6 6 1 { 1 '

In a similar mannet, one can generate Routh tables for the remaining

fifteen Kharitonov plants. Using the sixteen Routh tables, the next

step in the synthesis procedure is to enforce positivity for each of the

firs1 colu'rns. This leads to inequalities involving I{1 and Ii2; e.9.,

for.the Routh table for the Kharitonov plant Pz,r(s) abot'e, positivity

of the first column leads to the stabil ity conditions

a ' ( 1 ( r . I i r ) O ( K r , K z )A-1 < J.6: /r'2 > 0;

ffi > 0; --# > u'

We can now easily display the set of gains K2,1 satisfying the

inequalit ies above; see Figure 11.5.1. In a similal manner, we calL

generate the set of stabilizing gains Kir,i" for all remaining pairs

(lr, lr) € {I,2,3,4} and display the result graphica}ly' To obtain the

final resuit, we nust enforce the requirement that (Kt, Kz) sta.bilizes

all sixteen Kharitonov plants simultaneously. Hence, the desired set

of stabilizing gains is given bY

rc : I K; , , t " .a t , x 2

Any one of a u'ide variety of two-variable graphics routines can

be used to display the set K. To illustrate, for the lange of gains

11.5 ,/ Control ler Sy.nthesis Technique 185

1 . 5

1

u . o

FIcune 11.5.1 The Region K2.1 for Example 11.5.1

O < I{r ( 2 and 0 < Kz < 1.5, the set of robust PI stabil izers isshown in Figure \L5.2. Since this set is nonempty, 2 is robustlystabilizable. When stabilizing the interval plant P, tve can selectany (1{1 , Kz) € K. For example, a robust stabilizer is given by

C(s \ :0 .9 + g? .s

Although we have already established that C(s) is a robust sta-bilizer, it is of interest to provide an independent validation of thisresult via the Zero Exclusion Condition (Theorern7.4.2). This taskis simplified by using the octagonal value set characterization whichis given in Section 10.6. Indeed, we fi.rst calculate the uncertainclosed loop polynomiaL

p (s ,q , r ) : s5 i _ r ss4 - l r 2s3 * (0 .9q1 + r t ) s2

* (0 .9q6 * 0 .2q1+ ro )s * o .2qs .

Next, we generate the value set p(ju,Q, R). An init ial fr-equencysweep for 100 evenly spaced points in the range 0 I t.t 17.5 indicatesthat a low-frequency "zoorrr" is required to determine if the ZeroExclusion Condition is satisfi.ed; see Figure 11.5.3. By concentrating

1 8 6 C h 1 l / T h e S i x t e e n P l a n r T h e o r e m

r . 2

0 .2 0 .4 0 .6 0 .8 L . 2 1 . 4 1 . 6

FIGURE 11.5.2 The Set of Robust Stabi l izers K for Example 11.5.1

our attention on the range 0 I a 1 1.2, we use the second walue setp lot in F igure 11.5.4 to conclude that 0 Q pQ.,Q,R) for a l l c . . , > 0.Hence, v'e have verified that the Zero Exclusion Condition is satisfiedand C(s) is a robust stabil izer.

11.6 Machinery for Proof of Sixteen Plant Theorern

In ttris section, rve develop some machinery to facilitate the proof ofthe Sixteen Plant Theorem. The reader interested solely in applica-tion of the theorem can proceed directlv to Section 11.8.

11.6.1 Nonincreasing Phase Property

In this subsection, we concentrate on the one-parameter family ofpolynomials described by

p(s , I ) : f ( s ) * Ae (s ) ,

where /(s) and 9(s) are fixed polynomials and ) € A : [0, 1]. Re-call ing Exercise 4.I5.7, we cannot guarantee lobust stabil ity by sim-pl ) ' checking t i re ext remes p(s,0) : / (s) and p(s, 1) : / (s) + S(s)However, under the strengthened hypothesis Lhat gQu;) has nonin-creasing phase, we see below that such an extrerne point result is ob-

7 . 4

I{r

1t{.

6 8 1 0 L 2 1 4 1 6Kz

I i 6 ,z Machiner l , for Proof of Sixreen Plarr t Theorem lST

-2,000

Im -4,0OO

-6,000

-9,000

' -10,000

-12 ,000-2,000 0 2,000 4,000 6,000 8,000 10,000 12,000

Re

trIGURE 11.5.3 Crude Frequency Sweep for Example 11.5.1

tained. To this end, lve begin byrecaLlingthat the value set p(ja,L)is a straight l ine segment rvith endporrrts p(jut,0) and p(ju,I); seeLemnta 7.2.3. Furthermore, if g(ja) f 0, notice that the slope ofp( j . ,L) is g iven by

Irn q( i.u\( t ) : # : t a n L g U . ) .ne g\Ja )

the proof of the lemma below. This iemma is due to Rantzer (1gg0)and Fu (1991).

LEMMA L1-.6.2 (Nonincreasing Phase property) : Letp be a fami,Iyof polynom,ials haui,ng ,inuariant degree and. d,escribed, by

P(s , A ) : " f ( s ) * ) e ( s ) ,

where f (s) and g(s) are fired polynomials and, A € A : 10,1]. ,4s_suming that [g(ja) ' is noni.ncreasing, it follows that p is robustLysto,b le i f and onlg i , f p(s,O) and p(s, I ) are stable.

188 Ch 1 I / TheSix teenPlan tTheorem

-60-160 -140 -120 -100 -60 -40 -20

Re

Frcunp 11.5.4 Refined Frequency Sweep for Example 11'5.1

Pn-oor': Since necessitlz is triviai, we concentrate on the proof of

sufficiency. We assume that p(s,0) and p(s, 1) are stable and m'ust

prove that the family ? is robustly stable. Proceeding by contradic-

tion, suppose P is not robustly stable. Then, by the Zero Exclusion

Condition (Theorem 7.4.2), there exists some frequency c,r* ) 0 such

i , i ra t 0 € p( ja . , t \ ) . S ince p(s,0) and p(s,1) are stable, the value

set p(jut*, A) cannot be a point. fn accordance with the discussion

above, the value set p(ju*,A) is a l ine segment (perhaps vertical or

horizontal) which includes the origin z : 0 but not as an endpoint.

We consider the case wlten p(jw*,O) and p(ja*,l) l ie in the

interior of Quadrants 1 and 3, respective).y, and sirnply note that the

other possible geometdes are handled in an identical manner. We

now define the two cones

4,000

2,000

0

L20

100

-20

-40

C s : { z e C : [ p ( j a * , O ) < + " . I ]

C 1 : { z e C : [ p ( j a * , 1 ) < * . T ]

depicted in F igure IL.6.2. Using the stabi l i ty of p(s,0) and p(s,7) ,

the lvlonotonic Angle Property (Lemma 5.7.6) and rvith continuity

of [p(ju:,0) and Lp(i., l), we arrive at the following point: For

i 1.6 , / Machinery for Proof of Sixteen Plant Theorem 189

Ch. 11 / The Sixteen PlantTheorem

FtcunB 11.6.2 Il lustration for the Proof of Lemma 11.6.2

real positiue coefficient polgnornial p(s) decomposed into euen andodd, parts as

p(s) : p .u.n(s2) I spoaa(s2) .

Then the follotuing three statem,ents o,re equi,ualent:(f l fhe real coefficient polynomial p(s) i,s stable;(ii,) The cornplex coefficient polgnornial ,

't\pr ( " ) : P" , .n( js) - t jp"aa( j " )

is stable;(iii,) Th,e compler coefficient polynom'ial

pz( t ) : P"u"n(- j s) * sp.aa(- j s)

is stabl,e.

REMARKS 11.6.6 (Extreme Point Result): In the lemma below,the usefulness of the transformation betweerr real and complex co-effi.cient polynomials is demonstrated. We obtain an extreme pointresuit which facilitates the proof of the Sixteen Plant Theorem.

LEMMA 1,I-6.7 (A Stepping Stone): Let f (s) be a polynomiat ofdegree n and suppose that g(s) 'is a polEnomi.al of the forrn

190

WhenRe e( ic. ,) : O

P\Ja , r )

When S( j . ) :0' p( j - ,n.) : { / (rc.- l )}

p P ( j - , 1 ),,.

, , /N:::',"'"/va-w /

' / "

n{a) \ ryhen tm g\Ja)z 4

. Z t p U a , O ) p ( j a , I )

p ( j . ,0 ) PUt " '0 )

FIcuns 11.6.1 Value Set Possibii i t ies for p(ja,lr)

Ao-r ) 0 sufficiently small, it must be true that p(i (u. * Ao), 0) e C6and p(j(r 't* + Ac.,),1) e C1. Notice that this condition implies thatthe slope of the value set satisfies m(u* + L..,) > ,n(.*). On the otherI rand, s ince Lg( ja) is noninc leasing, the s lope m(t . , ) : tan 4g( j . )must satisfy rn(a* + Ac., ') < m(a"). Hence, we have reached thecontrad.iction r.vhich we seek. E

ExER"crsE 11.6.3 (Results in the Literature): To recover knownresults in Lhe literatu:'e using Lcnnr.a 11.6.2, consider the tr.vo situa-tions described below a,nd ar-gue that an extreme point result holds.(a) Suppose that g(s) contains all even or all odd powers of s andspecialize the lemma to obtaiu the result in Bialas and Garloff (1985).(b) Suppose that 9(s) is antistable (aI1 r'oots in the strict right halfplane) and specialize the lemrna to obtain the extleme poini resuitin Petersen (1990).

LL.6.4 TYaresforrnations: Real Versus Cornplex

In this subsection, rve state a basic lemma from the theory of polyno-rnials. We see below that there is a fundamentai relationship betweenstable real coefficient polynomials of order n and stable complex co-efficient poiynornials of order approximately equal to nf2. A niceproof of the lenrma belorv is given in Jury (197'l).

LEI\4I\4A 1l-.6.5 (Real Versus Complex Coefficients): Consider the

p(j( . . * Ac,, , ) ,0

p( j (a* * Ac. , ' ) ,1)

s ( s ) : ( a s + B ) h ( s )

71.6 / Machinery for Proof of Sixteen Plant Theorem 191

with h(s) being a polgnomi,a.l of degree k < n - 7 ltauing onlg euenpouers of s or only odd pouers of s. Then, g,iuen any a,B €P", thefarnily of polgnomials P described bg

p(", t r ) : l (s) + )e(s)

and ), e [0, 1] zs robustlg stable if and onlg if th.e tu.to entrerrues p(s, 0)n.nll n( -q . 1) o.re. stn.hle.

PRoor': Since necessity is trivial, we proceed to establish suffrciency.Indeed, we assume that p(s,0) and p(s, 1) are stable and mirst provethat P is robustly stable. In view of Part (b) of Exercise 11.6.3, weassume that a and B are both nonzero; otherwise A enters into onlyail even order terms or all odd order terms and sufficiency followsimmediately. In addition, we assume h(s) has only even powers of sand work with p1 (s) in Lernma 11.6.5; note that a nearly identicalproof is used for the odd power case usingB2(s) rather thanpl(s) inthe lemrna. Finally, r.vithout loss of generality, we also assume p(s, 0)and p(s,1) both have all positive coefficients. In this regard, recallthat p(s,0) and p(s,1) are both assurned to be stable and have thesame coefHcient of s-. Writine

"f (") : f "u"n(s2) -t sJoaa(s2)

andh (s ) : h "u .n (s2 )

where /(.) and h(.) are polynomials, we obtain

p (s , , \ ) : f " u .n ( " ' ) * s f "aa (sz ) + ) (es + B )h " , " ^ ( s2 ): l f " , "^(s2) r A7h"u"n(s") l + tU"oo( t '1 + \oh" , "^1t '11.

Observing that p(s, )) has positive coefficients for all ,\ € [0, 1] ,in accordance with Lemma 11.6.5, it suffices to prove that the com-plex coefHcient polynomial

p1(" , , \ ) : I f " " " . ( js) I \8h" , " - ( is) l+ j l f "aa( js) + \ah" , "^( js) l: l .f """^(j

s) + j f "aa(j s)l + ^@ + aj)h.,".(j s)

is stable for all ) € [0, 1]. To this end, we already know from t]resarne lemma that pr (", 0) and p1 (s, 1) are stable.

The proof is now com.pleted by contradiction: We assume thatpr(",)) is unstable for some ) e [0, 1] and note that the Zero Ex-clusion Condition (Theorem 7.4.2) also holds for complex coefficientpolynomials; a nearly identical proof applies. Hence, there exists

192 Clt. 17 / The Sixteen PlantTheorem

some c,;* € R and )* e (0, 1) such that f1 (Jr*, A*) : 0. We first rule

out the possib i l i t ies that B1( j . " ,0) : 0 or Ft ( ja* ,1) : 0 becalse

th is would contradic t s tabi l i ty of f i1(s,0) and pr(" ,1) .

It now follows that for each c,l in some neighborhood f,) of c..r"'

the value set p(ju,A) is a l ine segment in the complex plane with

endpoints ptUr,O) and pt( ja , l ) and constant s lope m(c.r ) : a /P.

Ti; coil ipletc Lhe proof, note that the origin z :0 is not an end-point of this l ine segment and moreover, since pr(s,0) and f1 (s,1)

are stable, the angles Lpt( ja .0) and LFt( j . ,1) are increasing f r rnc-

tions of c..r; note that we ale using the fact that the \4onotonic Angle

Property (Lemma 5.7.6) rernains valid for complex coefficient poly-

nornials. The proof is now completed using an argurnent which is

nearly identical to the one given in the proof of Lemma 11.6.7; i.e.,

us i i rg the Monotonic Angle Property oI [p1( jut ,0) and Lpt( i - ,7)and the fact that O e p(ja*,A), we can create two cones Co and C1

as in Figure II.6.2 and algue that for'1.-.- | suffi,ciently smalI, the

value set has slope m(r) > *(.*). This, horn'ever, contradicts the

constancy of rn(a) for cu € fl. The proof is now complete' I

77.7 Proof of the Sixteen P1ant Theorern

Since the proof of necessity is trivial, we proceed directly to the proof

of sufficiency. Indeed, we assume that C(s) stabilizes the sixteen

Kharitonov plants and rnust show that C(s) robustl l ' stabil izes the

famil], 2. Since the closed loop polynomiai

p (s , t t , r ) : ,F r ( s - z ) . n / ( s , s ) + ( s * p )D (s , r )

has degree nlt for ail S e Q and r € fi, the Zero Exclusion

Condition (Theorem 7.4.2) applies; i.e., to estabiish robust stabil ity,

it suffices to show that 0 F pU-, Q, R) for all c"' ) 0. Furthermore, in

accordance with Theorem 10.5.1, it is suffi.cient to estabiish stabil ity

of at most thirty-tu'o edge polynomials rvith trvo possible forms. The

fi.rst form is

e (s , ) ) : K (s - z )N r , ( s ) + ( s - p )Da" , i " ( s , A )

r v i t h i 1 € { 1 , 2 , 3 , 4 } , ( i z , z 3 ) e { ( 1 , 3 ) , ( 1 , 4 ) , ( 2 , 3 ) , ( 2 , 4 ) } a n d

Db , t " ( s , A ) : AD , " ( s ) + ( t - ) )D ; . ( s ) '

The second form is

e (s , ) ) : l i ( " - z )N r . , , l , ( s ) + ( s - p )D ; . ( s , ) )

71.8 / Conclusion f93

w i t h ( 2 1 , z 2 ) e { ( 1 , 3 ) , ( 1 , 4 ) , ( 2 , 3 ) , ( 2 , 4 ) } , i s € { 7 , 2 , 3 , 4 } a n d

AI,r,;, (s, .\) : )N;, (s) + (f - ))l/,, (r)

To complete the proof of lhe theorem, we pick a typical edgepolynomial e(s, ,\) above and argue that with A : [0, 1] , the standingassumption that the sixteen Kharitonov plants are stabilized guar-antees stability for the entire edge family

E : { e ( . , ) ) : ) e A } .

For i l lustrative purposes, we take z1 : 2, ' iz: 4 and iz : 3 and notethat the proof for all other edge combinations is carried out in anidentical manner. lr[eq,, with

e (s , ) ) : K (s - z ) I ' { 2 ,a (s , ) ) + ( s - p )DsG)

: I{(s - z)l).N2(s) + (r - r) lra(") l + (s - p)Dt(s): p4s!) + ) l((s - z)lN2(s) - AIa(")1,

we make the followine identif ications with Lemma 11.6.5:

/ ( s ) - p+ ,s (s ) ;

i z ( s ) - l / z ( " ) -Na (s ) ;

) (os t - B) - AI{ (s - z) .

194 Ch. 11 / The Sixteen PlantTheorem

question is: For the case of higher order compensators, how restric-tive are the conditions under which an extrene point result holds?The results in the next chapter shed some light on this question.

Note, however, that for cases when higher order compensatorslend themselves to extreme point results, we do not obtain a "nice"synthesis theory as in the case of first order compensators; suchresults are mainly useful in an analysis context. To elaborate onthis point, note that q'hen the number of parameters entering C(s)is greater than two or three, the graphics approach described inthis chapter is no longer valid. Although one can stiil use a finitenumber of Routh tables to generate inequality constraints on thecompensator parameters, the finding of a feasible point (when oneexists) amounts to solving a potentially diff icult nonlinear program-

Notes and Related Literature

NRL 11.1 Historically, the result of Hollot and Yang (1990) paved the.rvay for

the Sixteen Plant Theorern. With the same setup as in the theorem. the following

weaker result is established: To robustly stabilize the interval plant farnily, it is

necessary and sufficient to stabilize the entire set of extreme plants. Recall that

the number of extreme plants can be as high as -l/."1 : 2m+n+r.

NRL 11.2 Lemma 11.6.7 is a minor extension of a resul t g iven in the paper by

Hollot and Yang (1990)

NRL 1l- .3 As ment ioned in Sect ion 11.8, for the case of h igher order compen-

sators, it is unclear whether an extreme point result for interval plants is usefuI

in a synthesis context. Except for some rather special cases (such as those involw-

ing minimum phase and one-sign high frequencS, gain assumptions), the robust

stabilization problem for a finite plant collection is unsolved; for example, see

Youla, Bongiorno and Lu (1974), Saeks and Nlurray (1982) and Vidyasagar and

Viswanadham (1982).

NRL 11.4 An interesting embellishrnent of the Sixteen Plant Theorern involves

further reduction in tire number of plants required in the robust stability test.

For exarnple, under the strengthened h5,pothesis that the compensator C(s) is

either lead or lag rvith gain 1{ of prescribed sign, only eight distinguished pla.nts

need to be tested; see the paper by Barmish, Hollot, Kraus and Tempo (1992)

for additional detaiis.

NRL 11.5 Although the Sixteen Plant Theorem was given in a robust stabilitycontext, an interesting robust performance result is obtainable as a byproduct:Consider a strictly proper interval plant family P as in Section 11.2 with robustlystable denominator family D- Then, using the Sixteen Plant Theorem, it is easily

fi

t ,

ill,

a '1.?

)

Since C(s) stabil izes the sixteen Kharitonov piants, it is easy to verify{that all preconditions of the lemma are satisfied. In particular, notice

'' i

that .zV2(s) - Aar(") has only even powers of s and p(s,0) : pa,s(s) -"

arrd p(s, 7) : p4$) are stable- Therefore, the lemma guaranteesthat e(s, )) is stable for all ) e A : 10,1]. The proof of the theoremis now complete. E

11.8 Conclus ion

Extreme point resuits presented in this chapter raise an interestingissue. Suppose that we remain within the realm of interval plantsbut we allow the compensator to be more genelal; i.e., we no longerrestrict C(s) to be first order. Then it is of interest to give condi-tions under which stabil ization of some distinguished subset of theextrerne plants implies stabilization of the entire interval family. Inthis regard, note that some sort of assumption must be imposed be-cause of counterexamnles of the sort siven in Exercise 6.4.5. The

P (s , q , r )

Notes and Related Literature 195

, FTcURE 11.8 1 Confisuration Associated with -EIr Problem

shown that the worst case l1- norm is attained on one ofthe Sixteen Kharitonov- l ^ - + ^ . + L ^ r i -P r @ r r u D ,

m a x l l P ( s , q . r ) l l - : . . r g ? + . l l p , , , r , ( s ) l l - .( q , r ) € e x R ( i r . ; 2 ) e { r , 2 , 3 , 4 }

This is the result given in Mori and Ba.rnett (1988) and Chapellat, Dahleh andBhattacharyya (1990). To prove this result using the Sixteen Plant Theorem,the key idea is to consider a feedback ioop with unmodeled dynamics A(s) in tirefeedback path; see Figure 11.8.1. With A(s) proper, stable and rat ional , one canrelate the i I - norrn of P(s, q,r ) to a destabi l iz ing perturbat ion A(s) which canbe interpolated at the "critical frequency" via a first order compensator.

It is interesting to note that technica.l arguments used to establish this resultdo not easily genera).ize to frequency weighted norrns; i.e., if W(s) is proper, stableand rational, it is of interest to deveiop conditions under which the maximum

uo : ro,##*^ l l la l (s)P(s ' q , r ) l l -

is attained on one of the sixteen Kharitonov plants. The lack of a generai extremepoint result in this g'eighted case is related to the fact that a generalization ofthe sixteen Piant Theorern is not immediate for higher order compensators. Forexample, t'or the interval plant

D / ̂ ^ \ [ 1 , 5000 ]r \ s , q ) : ; E 5 "

of Hollot and Yang (1990) rvith compensator

c(s) : (s + 3 ) (s + a )( s + 0 . r ) ( s + 0 . 2 ) ( s + 7 5 )

conrrected as in Figure 7I.2.7, it is easy to prove that C(s) stabilizes the two

extreme plants P(s, 1) and P(s,5000), but the c iosed loop system obtained using

P(s,2) is unstable. It is also worth noting that even for simple controllers in the

PID class, counterexamples of the sort above can be given.

Chapter 12

Rantzer's Growth Condition

Dyllopsis

In tlte prcceding chapters, we encountet-ed sonTe special cJasses

of po15,[6ps5 of polynornia]s for v'hich robust stability can be

ascertained fron stability of the extremes. Thjs tlteme is ntore

fdly developed jn tJrjs chapter. Wlten we rvor* w'rtl a typi-

ca) edge pdynomial described by p(s,f) : /( t) -F )9(s) and

,\ e [0, I], the satisfaction of Rantzer's Growth Condition on

the rate of change of the angle LgUr) enables us to test for

robust staUtity using the exLreme polynomials p(s, 0) : /(")and p(s , t ) : / ( " ) + s (s ) .

72.L Introduction

When working with an affine linear uncertainty structure, .we saw in

Chapter 9 that the Edge Theorem holds under rather weak hypothe-

ses. As \lre strengtitened the hypotheses in Chapters 10 and 11, theresults became progressivei] 'stronger. For example, given a simplefeedback system involving an interval plant, we saw that the test for

robust stabil ity invoives only tf i i t1"-two edges; see Theot'em 10.5.1.

By further restricting the compensator to be fir-st order, we saw inTheorem IL4.1, that the robust stability test turned out to invoive

only sixteen extreme plants.In making the jump from an edge result to an extreme point

result, we saw repeatediy that the following fundamental problem

196

i ,

I: l

i ltl

nrl 'i''i

i i - ( ji i r l

,?,lJ

i . ji : ' ll ii,',1 l : rt i l

i ; i

I2.2 / Convex Direcrions I97

arises: Let /(s) and g(s) be fixed polynomials and consider the one-parameter family P described by

P(s, )) : "f (s) + )e(s)

and A € [0, 1]. Give conditions under which stability of the extremesp(s,0) :

" f (s) and p(s, 1) : / (s)+g(s) impl ies robust s tabi l i ty of thefarnily P. In this chapter, we generaiize the extreme point resuitsdeveloped thus far. Under mild reguiarity conditions) we see thatif the angle LSU.) satisfies a certain growth condition, then robuststability of P is equivalent to stability of the extremes.

'J-2.2 Conwex Directions

The notion of a convex direction is instrumental to this chapter.In order to motivate this concept, we consider p(s, )) above andmake one fundamental observation about the extreme point resr.rltsattained in all plevious chapters: In every case, if '"ve reca,st theextreme point problenr in the /(s) + )g(s) setting, the crit ical as-surnptions involve g(s) but not /(s). To make this point clear-, rveconsider some exanoles and an exerclse.

ExAMpLE L2.2.1, (Kharitonov's Problem) : By reducing Kharitonov'sprobiem in Chapter 5.to an edge problem associated with uncertaintyin the coefficient of sh, it is straightforward to verify that lve obtainan edge polynomial of the form

p ( s , ) ) : / ( " ) * ) s A .

Hence, in terrns of Kharitonov's framervork, /(s) can be rather gen-eral but the polynomial

g ( " ) : sk

is lestricted to having a special form.

ExAMeLE 12.2.2 (Nonincreasing Angle Property): When we ex-posed the nonincreasing phase property in Section 11.6.1, /(s) wasrather general but a nonincreasing angle of g(j,',,) was assumed.

ExAMeLE 72.2.3 (Sixteen Plant Theorem): Lemma 11.6.2 wasfundamental to the proof of the Sixteen Plant Theorem in Sec-tion 11.4, In the lemma, /(s) was permitted to be any stable poly-nomial but g(s) was constrained to be of the for:m

198 Ch. 12 / Rarirzer's Grorvth Condition

with a, 0 e R and h(s) being a polynornial satisfying

deg h (s ) < des / ( s ) - t

and having oniy even powers of s or only odd porvers of s.

ExERCTSE L2.2.4 (Interval Plant and Compensator): Consideran interval plant family P with compensator C(s) : Nc(s)/Dc(s)connected in feedback configuration as in Figure IO.4.I.(a) For the ciosed loop system, reduce the robust stabil ity problernto an /(s) + )9(s) problern and show that 9(s) is of the form

g(s) : Z<sal;6r(s)

s(s) : xsknsG).

(b) Using Theorem 10.5.1, argue that it suffices to study no morethan thirty-tu'o y1"; + )9(s) problems with g(s) of the forrn

g (s ) : h ( s ) l / 6 ( s )

org (s ) : l t ( s )D6 f t )

u.ith h(s) having only even powers of s or only odd powers of s.(c) For the more general case when the plant numerator and denom-inator have affine linear uncertainty structures, describe the appro-pr iate poiynomial 9(s) to be studied.

REMARKS 12.2.5 (Preparing for a Definit ion): In view of theexercise above, extreme point results involving a general /(s) but aspecific class of g(s) have interpretations in a feedback context. Thatis, a condition on g(s) implicit ly describes a clasS of compensatorsfor which an extreme point result holds. In the theory to follow, weview g(s) as a direction in a space of stable polynomials and seekdirections rvhich are "nice" in the sense that an extreme point resultholds. These nice directions are now defined nrore formally.

DEFrNrrrot.r 12.2.6 (Convex Direction) : A monic polynorerial g(s)is said to be a conuen direction (for the space of stable n-th orclerpoll'nomials) if the following condition is satisfi.ecl: Given anv stabren-th order polynomial /(s) such that /(s) + g(s) is also statie anddeg ( / (s)+)g(s)) : n for a l l A e [0,1] , i t fo l lows that the polynornia l/ ( " ) + )s(s) is s table for a l l ) € 10,11.

OI

s ( s ) : ( c s * , 0 ) n G )

L2.2 / Convex Direcrions 199

REMARKS L2.2.7 (Interpretation): Note that the monicity require-rnent in the definition above is introduced solely for convenience; itfacil i tates notation in the sequel. When dealing with an edge polyno-rn ia l p ( s , ) ) : / ( s )+ )9 (s ) w i t h nonmon ic a (s ) : LLoa1s , , r obus tstabil ity can be ascertained by scaling /(s) and g(s) bv the factora* te induce monicity.

The convex direction concept is depicted graphically in Fig-nre 12.2.1. R-orn the figure, it is apparent that 91 (s) is a convex

boundary betweenstable and unstable^ ^ 1 , , - ^ * : ^ l -lJUl.y uulr l rar-

/ ( " ) + 1s2(s )

200 Cjn. 12 / Raltzer's Grorvth Condiuon

by para1le1 translation of th-e original /(") f A9(s) l ine' we obtain a

nlr,i- srable starting point /(s) for which /(") + 92(s) is stable but

/ ( r )+ )9(s) is unstable for some ) e (0 ' 1) '

ExERcrsE L2.2.8 (First order case) : Argue that a monic first

order polynomial g(s) d,efines a convex direction in the space of poly-

nornials of order n ) I. Hint: see the rnachinery associated q'ith tlie

proof of the Sixteen Plant Theorem in Section 11'7'

EXERCTSE 72 .2 -g (Second Order Case) : Using the fact that a rnonic

second order polynomial is stable if and only if all coeffic^ients have

the same sign, argue that every monic polynomial g(s) : s2 +a1s*ag

defines a convex direction.

].2.3 Rantzer's Growth Condition

In the theorem below, a complete characterization of convex direc-

tions is provided. we relegate the proof to the next two sections.

TrrEoR-E\4 12.3.1 (Rantzer (1992a)): A polgnorni.al g(s) is a con'uer

direction if and onlg if the gro'wth cond'it'ion

fi+nu,t tl*]#9\i,s sati,sfi,ed, for aII frequencies c..r ) 0 such that S(ju) * 0

REMAR_KS 1,2.3.2 (Interpretation) : Notice that the theorem ex-

cludes consideration of frequencies r..r € R for which S(iu) :0' The

condition S(i.) * 0 also guarantees that the expression for {g(ju)

makes sense; i .e . , i f sU4:0, the angle of 9( jc" ' ) is ambiguous'

EXAMPLE L2.3.3 (checking the Growbh condition): For the poly-

nomialsG) : s4 - 2s1 - 13s2 I I4s * 24,

we first qenerate

d Ls(jr)du

-2ao - r6ua - 38@2 + 336

unstabiezz-th orderpolynomials

nonconvex

/ (s ) + gz(s )

stablen-th orderpolynomials

direction

/ (s )

/ ( ' )

/ ( s ) / ( " ) + 'YsrG)convex direction

FIGURE 12.2.1 Conex Directions in the Space of Polynomials

direction because /(") +'ySr(s) remains within the stable set fol aII

7 > 0 and all starting points /(s). On the other hand, for the caseof g2(s),

"f (") + lsz(s) can become unstable for a range of 7 > 0.

I{ence, Oz(s) ca3not be a convex direction because we can find a newstar t ing point 1(s) having the property Lhat / (s) a: rd / (s) + gz(s)

are stable but / (s) + )92(s) is unstable for some A e (0,1) . Tosee intuit ively how /(s) is obtained, let [71,?z] denote the range ofinstabil ity for 7. Notice that if ' lz I I, Ire can take /(s) :

"f(s)and induce instabil ity for ) : ?2. On the other hand, if 12 > I,

(.n + 13c..'2 + 24)2 + (2u3 +r4u)'2

2 a 6 + 4 0 c o a + 2 3 0 a 2 + 3 3 6sin 2fsl ju)D , ,L W

and

1u+ tsrz + 24)2 * (2u3 * r4a)2

12.3 / Rantzer's Groq.th Condition 20I

By plotting the trvo quantities above for c.,, ) 0, it is straightforwardto ver*if; ' t, l iat Lhe grorvtl 'L condition is satisfied. By car-rying outthe plot, it is also easy to see that rve eventuilly encounter- a cutoff{iequency d" } 0 above which

a LgU.) < od.a

Hence, the gror.vth condition is automaticaliy satisfi.ed for a ) uc.We can obtain an "estimaLe" for ,1. b)' computing the maximumpositive root of the numerator Z,sG I76aa * 3&,t2 - 336 associatedwitlr the late of change of {g(jtu). We obtain us. x I.7677 andcan simplify the computation by studying the growth condition forfrequencies c.r e [0.c.,,.).

ExnFLCTSE a2.3.4 (Checking the G::owth Condition): Determine if

s(s) : s5 - s4 + s3 + s2 + s- 1 is a convex clirection.

ExEFLcrsE 72.3.5 (Consistency with Previous Results): The objec-tive of this exercise is to demonstrate that the growth condition isconsistent with otirer results already developed in this text. In eachcase below, show that g(s) is a convex direction.(a) Reconcil iation with Kha,ritonov's Theorem: Suppose that g(s)contains either all even powers of s or all odd pov,'ers of s.(b) Reconciliation with Nonincreasing Angle Ploperty: Suppose that4SU4 is uonincreasing.(c) Reconcil iation r.vith Sixteen Plant Theorem: Suppose that g(s)is of the form g(s) : (s*a)h(s) with /z(s) containing either all evenpowers of s or all odd powers of s.

ExEn"crsE L2.3.6 (Extension of Sixteen Plant Theorem): The ob-jective of this exercise is to extend the dornain of applicability of theSixteen Plant Theorem given in Section 11.4. Indeed, consider aninterval plant family 2 satisfying the conditions of the theorem withproper compensator

n/ - \ l /c(s)u \ r / :

D"€,

which is no longer required to be fi.rst order. Instead, assr.rme thatl/c(") and Dc(s) satisfy Rantzer's Growth Condition. Now, provethat C(s) robustly stabil izes P if and only if C(s) stabil izes each oftlre sixteen Kharitonov plants. Hint: First prove that if 9(s) is aconvex dir-ection and /z(s) is a monic polynomial containing either

202 Ch. 12 / Rantzer's Gro*th Condition

all even powers of s or all odd powers of s, then

iG ) : e (s )h (s )

is also a convex direction. Subsequently, in mimicking the proofof the Sixteen Piant Theorem, observe that each of the thirty-trvodistinguished edges (see Theorem 10.5.1) are expressible in the form

p(s, ) ) : " f (s) + ) } /6(s)h(s)

OI

p(s , ) ) : / ( " ) + AD6(s )h (s )

with /(s) being stable and h(s) as above.

72.4 Machinery for Rantzer's Growth Condition

In this section, we develop some rather technicai rnachinery r.vhichfacil i tates the proof of Theorem 72.3.\. This section and the nextcan be skipped by the reader interested primarily in application ofthe result. We first introduce some notation and then provide twolemmas which conveniently describe the rate of change of the angleof a polynomial.

NorArroN 1.2.4.L (Angles and Their Derivatives): To avoid curn-bersome notation associated with the calculus of phase derivatives,we adopt the following notational convention: Given a fi.xed polyno-mial p(s) and a frequency c,r ) 0 such that p(ju) f 0, we take

oo@): Lp( i , )and

^ t / t d "

, -9 r ( u ) : ^ A n U a ) .

Having defined these quantities, .we use a compact notion for de-scription of phase derivatives evaluated at a given frequency rlrs ) 0;that is, we take

r . lo'r(uo): ;LrU,)l-:_"

to be the evaluation of ?'r(u) at a : uo.

LEMMA L2.4.2 (Rate of Change of Angle): Giuen a polynom'ialp(s) and a frequency a,, ) 0 such thatp(ju) * 0, it follaus that

f - t ( ; , d ) l

7 '^ (u t ) : Im. l * t ,' L p \Ja) )

204 Ch. 12 / Ranuer's Growth Condir ion

the convenient formI2.4 / Madninery for Rantzer's Growth Condition 203

where p'(j-) denotes the deriuatiue of p(ju); i.e.,

p, ( ja) : ! n" p( ju)) + i lm p(ja\.' d'a " dt'' /

PRoon: Using the shorthand notation

R(u) : Re p( ja)

andI(a) : Im' p(jr ' t)

witlr associated derivailves E{(cu) and It(to), respectively, we write

p ' ( j r ) : R ' (a ) + j r , ( o ) .

Next, we carry out a straightforward calculation to obtain

A I / , , \ -w P \ w

) -

R(r,t) It (a) - I (w) R' (u)

R r ( r ) + 12 (a )

s'Ua)sUu)

f'( i.)

Comparing the expressions for p,(j.) and O,o(i":) above with

p ' ( j - ) _ R t (u )+ j l ' ( u )

p( j t t ) R(e) + j I (a)

_ R(t t )R' (u) + I (w)1,(u) + j fR(a)r l (u) - r (ut )R,(u) l,

it is easy to verify that

. f n t ( i , . , \ 1

0 , ( i r ) : r n t . l " ) 1 * , r l I- L p \ J a ) )

LEMMA L2.4-3 (Angle Formula for convex combinations): Giuentwo polynomials f (s) and g(s), a scalar )* e (0, I) and, a frequencya* > O such that f ( j r . ) # 0, S( j - . ) * O, f ( j r . ) _ t g( j r " ) I O and,f ( j - . ) - t A*g( ju") :0 , ' i t fo l lows that

9 'nQ' t* ) : \ *0 ' f ( . * ) + (1 - S.)0,y*n@-).

PRoop: Let co > 0 be given such that f (j".,) # 0, g(j.) I 0 andf ( j-) + sU.) I 0. tror ) e (0,1), we can express j,1i.1yg1i.7 in

+ ( 1 - ^ )

^( f ( j . ) + s( ju) )

f ' ( j r ) + s 'Ur)( r - r ) ( / ( ru) + s( iw)) - (1 - x) f ( j , ) '

Now, specializing to Lr : s,1+ and ) : )* and invoking Lemma 12.4.2,we obtain

I a ' ( i L D \ 10 ' ^ ( u - ) : I m l : I

Ls( j . . ) )f f t ( i , , ' * \ I t f t ( i , , t * \ - t - n t ( i , , ' * \ 1

: \ l t = , " , * ! l + r r - A ) I r n l ' = \ t , * ' , ' " , ' ! * , ' l

L " f ( - r . . ) I I f ( j - . ) + g( jLD.) )

: \ *o ' f ( - * ) + ( r - \ * )o ' yan (a * ) ,

which is the stated result of the ]emma. E

L2.4.4 Two Phase Derivative Minirnization Problerns

We now formulate and solve two minirrrization problems whose so-lutions are instrumental to the proof of Theorem 12.3.1. The datadescribing the first problem consists of an integer d ) 1, an angle0 e l},2rl and a fixed frequency (, : u)o > 0. We seek

' . : ;#o'oTto)'

where the infimum above is taken over the family 2 consisting ofall stable polynomials p(s) of degree d such that' [p(jao) : L Forsimplicity, we denote this requirement in the sequel by writing

LpUro ) : 0 .

We refer to the problem of finding p" as the Phase Deriuatiue Min-im'ization Problem.

The second problem is the same as the first except for the factthat Pp consists of all polynomials of the form

P(s ) : sk r ( s )

with k > 0 being any positive integer and r(s) being a stable poly-nomial of degree d. Since the constraint set Pp rs a superset of the

12.4 / Machinery for Rantzer's Growth Condirion Z0S

constraint set P used for the first problem, we refer to this secondproblenr as the Relctred Phase Deriuatiue M'inimization Problem. WeIet 1tfi denote the infimal value for this problem and note that

t_rh. ! p".

LEIvIMA 1,2.4.5 (Solut ion for F i rs t Order Case): For d: I and1r/2 < 0 12r, the Phase Deriuat' iue Mi,nirnizat,ion Problem has nofeas ibLe po in t s . Fo r010 < r f2 , an op t ima l so lu t i on ' i s g . i uen by

P* (s) : I{ (s + u-ts cot 0)

wi.th Ii ) 0 bei,ng an arb'itrary constz,nt. Furthermore, the minimun'r,ualue is

* s i n20r 2roo

PRoor: Since a stable polynomial p(s) has an angle which satisfi.es0 < ?oQ'.,) < rrl2 for alI c,..r ) 0, the Phase Derivative NlininizationPr:oblern has no feasib le points for r f2 < 012r. We noq' take0 < d < rf 2 and consider candidate polynomials of the form

p ( s ) : 1 { ( s + a )

witlr a ) 0 for stabil ity and 1{ t ' 0 to guarantee d: I. In this ca,se.the constraint do(c-'s) : d forces

Q. : Lto COt 0.

Now, differentiating the angle of p*(jut), we obtain

p. : o'o-@o)

d . l u l l- - tan-' [-o "*

B.] l-:-.

Using the differentiation formula

d _ r l d u

E E a n ' u :

r + u 2 d r ' )

a straightforwald calculation yields

206 Ch. 12 / Ranrzer's Growth Condition

ExERCTsE L2.4.6 ([I inor Extension): For d : 1, prove that theRelaxed Phase Derivative N'Iinimization Problem has minimum valuegiven by

p.: ]+-ry1l z D o l

Describe elernents p*(s) in 2p which achieve the minimurn.

ExERCTsE 72.4.7 (Solution for Special Case): For the special casewhen 9 is an integer multiple of r f 2, show that the infimal value forthe Phase Derivative Minimization Problern and its relaxed versionis given by

l - t * : l "h :0.

H'int: Construct a sequence of stable polynomials ipt(s))t, sr-rchthat [pp(juto) : rnr12 and ]'ou(ro) --* 0.

LEMMA L2.4.8 (Lower Bound for Higher Order Case): For arbi-trary degree d > I, any admissible polynom,ial p € Pn for the RelaredP has e D eriu atiu e |V[inimizati o n Problem s ati,sfi e s

. I s i n 291o ' r l t t o ) , | , *

|

Hence, a louer bound for the infimal ualues of the Phase DeriuatiueMini,mi,zat' ion Problem and'its relared uersion is giuen by

wh'ich is neaer attained.

PRoop: In accordance with the problem formulation, we considercandidate polynomials of the form

P(s ) : sk r ( s )

with r(s) being stable with degree d > L We proceed by inductionon the degree of r(s) and use the resuits in Exercise 12.4.6 for d, : L.Therefore, to prove the lemma, we assume that the desired inequalityfor p," hoids, but not strictly, for degree d: n. We must prove thatit holds strictly for degree d: n * 1. Indeed, given any adrnissiblep(s) in 24 associated with r(s) having degree d: n * 1, we canwrite

t , "> r ,h= l#1,

, . : " ' * :u u

12.4 / Machinery for Rantzer's Growth Condition 207

r ( s ) : ( s 2 + a 1 s + a 6 ) l ( s )

with al ) 0, ae > 0 and t(s) being a stable polynomial of degreed t : n - 1 . Now, w i t h

p (s ) : " k1 "2

+ c1 , s t cs ) t ( s ) ,

we consider three cases.Case 1: If t'tfi < 40, we define a polynornial of one Iower degree by

pr (t) : "f t(o1"

* ao - , fr; t1";.

It is stralghtforward to verify that gor(.o) : 0o@o) and by straight-forward di fferentiation,

tL,: c#ffi#G+oi@s)Now, by the inductive hypothesis,

, l ' , ,(.o) - |t ' : "I" , . - , _ l Z a o I

Norv, to complete the proof for this case, it suffices to show that0'o@o) > eLr(c..'6). To this end., we compare the left and right-handsides above. Indeed, by straightforward calculation, we arrive at thedesired conclusion from the chain of inequalities

0'o@o) :a t ( a o + . 3 )

L A t ( , . , ^ \I v t \ e u )(o, -.8)" + alafi

208 Ch. 12 / Rantzer's Growtir Condition

Hence, to complete the proof for this case. it suffices to show thatat t , ,^ \ : 4 / / , ,^ \ \ Iow, us ing a chain of inequal i t ies whic i r is near lyv p \ q u ) / w . p 2 \ u u ) . ' \ r

identical to the one used in Case 1, we obtain

o'o@d:6ffifu,+oi@s)

, _ ar(aq_ a. '3)\ao - ' ' ' t6) ' + alQ + ?i@s)

: oL'@o)'

Case 2: It ufi > 40, we defi.ne the polynomial

pr(r) :"** ' [ { r

- f t l '+

o,] 21";

and it is straightforrvard to verify that 0rr(.0) : 0o@o).more' bY the inductive

;t:t:: ' : rsin 26,1lrrluo) ,-

| ,^ t

a1(wfi - as)+ 0'1Q';s)

( o o - . 3 ) ' + a l w 2 o

: e,rr(r,,0).

Case 3: If u,,fi : cvo, we wor-k with the polynomiai

p s ( " ) : c l s k + r l ( s ) ,

anc1, analogous to Case 2, we have 0or( .o) :0p(uo) . Sincep3(s) sat-

isfies the inductive hypothesis, it remains to prove that the inequality

0o(ao) > 7'r"(ao) holds. Indeed, r.rsing a straightforwarcl calculation,

we arrive at the desired conclusion by noting that

,oo\:o) -- L a o'r(ro) > o'r(.o) : opr(.o).

This compietes the proof of the lemma. E

Ln'MMA 72.4.9 (!fhe Infimai Value): The i'nfi'mal ualue for the

Phase Deri,uatiue Min'imizat'ion Problem and 'its relared uersion is

gi,uen bg* + l s i n 2 d l

lJ : l rR: | , r , I

PRoop: In view of Lemma 12.4.8, it sriff ices to establish the equal-

ity above for pt*; i.e., the result for pt'fu then follorvs automatically.

Furthermore, we already know that

l s i n 2 d lt '1" >l ,^ |

and for d : 1, we also know that the desired result holds with

equality; see Lemma 72.4.5. To complete the proof, we consider

d > 2 and our objective is to exhibit a sequence of stable polynornials

{pr(")}Pr such that 0oo(.o) : 0 for all k and

1 . ^ t , \ l s i n 2 d l,l.rrn YD,.(c.ro) : l--;-l7 c + m ' l a L D O |

Further-

12.4 / Machinery for Rantzer's Growth Condition 209

For simplicity, we break the remainder of the proof into three casesand exclude consideration of 0 : rntr/2 for nt : 0,1,2,3; see Exer-cise 72.4.7 for the analysis of this case.Case 1: If 0 < 0 < rrf2, we construct the desired polynomial se-quence by exploit ing the solution for d : 1; that is, with scalara. : uo cot, 0, we know that p(s) : s -F a is a stable polynomial sucht h a t 9 o \ ' t s ) : 0 a n d 0 ' o @ o ) : p * . N o w , t a k i n g a k : a * L f k , w edefine the sequence merrrbers

pf t (s) : (s + a6)(6, , .s + 1)d-1,

where 5r > 0 is selected (as a function of as) such that 1ou(rs): g.Note that the existence of 6g is guaranteed because the equation

4( j -o * o7,) * (d ' - L) L( i6x.o * r ) : I

Iras a solution for d > 0 suitably small. Now. since dk + a asli - oo, it follows that 56 - 0 as k --- co to preserve satisfaction ofthe requilement that 0r*(.0) : d. The proof is now compieted byexploit ing continuity of the phase derivative; i.e.,

d . . . . l l s i n 2 9 l. I I m U , " ( . r g J : , I + J L D + a ) l : l _ l .A - 6 r " ' - '

d U " " ' l - : _ o | 2 A I

Case 2: If 2 < d < 4 and r(d - I)/2 < 0 < rdf2, t}ren wiLl:.ctk : (t + \1k,, we defi.ne the sequence members

pk (s ) : ( a ; s - t - 1 ) ( s + d r )d - t ,

where 61, is selected (as a function of a7") such that gon(ro) : d. Ther-emainder of the proof now runs along the same lines as in Case l.C a s e 3 : I f d > 3 , r f 2 < e < m i n { 2 2 r , n ( d - I ) 1 2 } a n d 0 l r n t r / 2 f o rnt : 2,3,4, we defi.ne the sequence members

p t " l s ) : ( s + a ) ( s - t 1 ) " ( 6ps + I ) d - " -L ,

whe re a : ao l t an? l , I l u < d -2 and 1A +0 and d6 * 0 a reselected so rhat ?or(as) : 0 for all ,k. Once again, the proof iscompleted in the sarre nanner as in Case 1. !

12.4.LO Construction of Nonconvex Dir,ections

In this subsection, we provide a result which is used to prove that aconvex direction g(s) necessarily satisfies the growth condition.

210 Ch. 12 / Ranrzer's Growth Condition

LEMMA L2.4.LL (Perturbed Polynomial): Suppose g(s) is a poly-nom'ial of degree d > 2 such that at frequencA ao ) 0, g(juLs) I Oand

^ t , \ . i s i n 2 d " ( c u o ) lan lao) t l - l

Then, g'iuen anA n > max{4, d}, there eri,sts a polynornial f (t) ofd e g r e e n - 2 s u c h t h a t

p.(s) : (s2 + uf;\11s) * ee(s)

is stable for e sui,tably srnall.

PnooF: Considering the Phase Derivative N,Iinirnization Problernwitlr d : 0g(-o), rve note that Lemrna, 12.4.9 i.ndicates that there ex-ists astablepolynomial /(s) of degreen-2 such that 0f @o) : ?s\)o)or 01(as) : -?g(uto) and

o'n(uo) > o'.(,,o) t I ti'

?dn ('o)

I ." r 2 c i . o l

Note that if the degree of g(s) is greater than that of /(s), rve hawe0t@o) :0g(ro) -zr , and i f the degree of / (s) is greater than or equalto that of 9(s) , we have 0t@o) : 9s( .o) .

To prove that /(s) satisfies the requirements of the lemma, weuse a root locus argument; i.e., we consider a unity feedback systernwith open loop transfer function

D t ^ \ - 69 (s )r €\d./ _ ("2;arreNotice that the root locus with respect to e tells us about the de-sired behavior of pr(s). We first observe that as. r 0, the rootlocus branches are close to the open loop poles. Since /(s) is stable,there are two branches near the points s : ljao and the rernainingbranches are in the strict left half plane. Therefore, to complete theproof, we develop an €-pararneterization of the potentially bad pairof conjugate roots emanating from s : tjao; by continuous depen-dence of the roots on e (see Lemma 4.8.2), we are assured that theremaining roots are in the strict left half plane for e suitably small.

We now consider the mapping 6 H e(e) defined implicit iy by theconditions z(O) : jus and p,(r(r)): 0. Equivalentiy, lett ing

h(s) : (s + jws)f (s),

aI1d

z / (e ) :

S o f f i n o r - O i tv v w w l r 4 5 " " , . "

z ' (0 ) :

Now, to prove thaL Re z'(O)imaginary. To prove this claim,?t@o) : 0g( .0) - r ' Using thettrat

. d . lRe z t (0 ) : ] Re z (e ) l - 0

ae l e : 0

s2 IRe z" (0) : f= Re z(e)l < 0.

c lc . t ._^

12.4 / Machinery for Rantzer's Crorrth Condidon 2ll

[ he equa t i on p , ( z (e ) ) : 0 i s changed to

p , ( " ( r ) ) : ( z (e ) - j c , : s )h (z (e ) ) + es (z (e ) ) : 0 .

To complete the proof, it suffices to show that

2I2 Ch. 12 / Rantzer's Grorvth Condirion

Setting e:0 and substituting the expression for z'(0) found above,

after some straightforward algebra' we obtain

z , (o ) : r j# r# ( f f i - f f i )real part of z"(0) and invoking Lemma 12.4.2, it now

Re z',(o):## (dn@o) - en(.o))

To complete the proof of the lemma, recall that g(jus)lh(jas) is

ptrrely imaginary and observe that 9'nQoo) > 1L@o) follows from

e,o!t) > 0,7kt) and o|(uto) : 0L@). using these facts and the expres-

.tn fo, Fie z"(O) abt-re, it follows immediately that Re z"(0) < 0' E

12.5 Proof of the Theorern

We now proceed to prove Theorem 12.3-1'. To establish suffi'ciency,

we assume that the growth condition is satisfied and must prove that

g(s) is a convex direction. In view of tire fact that ali directions are

trivially convex for d:1 (Exercise 12.3.5), we assume that d > 2

and proceed by contradiction. Indeed, if g(s) is not a convex direc-

tion, then there exists a stable polynomial /(s) of order n ) d such

that /(s) + s(s) is stable, p(s, ,x) : "f (s) + Ig(s) has order n for all

) e (0,1) and p(s, ) . ) is unstable for some ) . € (0,1) . Apply ing

the zero Exclusion condition (see Theore m 7 .4.2) , there exists some

.\6 € (0, 1) and o16 ) 0 such that the condition p(iu.to, Ao) : 0 l 'rolds'

Hence, with A: [0, 1] , the value seL p(jt ' ts,A) is a stra.ight l ine seg-

rnent which includes z : 0 but not as an endpoint; see Figure l2'5'I

for the case 0 < 7n(.o) < r. Without loss of generality, we complete

the proof for the case 0 < ?n@r) ( n-, noting that a nearly identi-

cal proof is used if -r < ?n@o) ( 0. Flom the fi.gure, we see that

0 t+n ( .o ) :7g (uo ) and 0y ( ' t s ) : ?s (uo ) - T '

Applying Lemma l'2.4'8 to both /(s) and /(s) +9(s)' we obtain

o'1(uo), l l \Jff i l : l : '"#*tdll z u o l l L @ u l

Taking thefollows that

In other words, the satisfaction of these two conditions guaranteesthd,t z(e) Iies in the strict left half piane for e sufficiently smal.i.

To study the behavior of z(e), we differentiate both sides of the

equa t i on p ( . (e ) ) : 0 and ob ta in

SQG))- h ( z (e ) ) + ( z (e ) j us ) t r ' ( z (e ) ) + es ' ( z (e ) ) '

follon's that

sUuo)h( j ro ) '

: 0, we claim that z/(0) is purelywe recall that 0y(u.,s) : 9g(ao) orformula for z'(0) above, it follows

- t ( ^ \ _ ^ l g ( i . o ) l' \vr - J 2c ' ts l f ( jus) l

, t ( o \ : - o l g ( i r o l l- \ " r " 2 t ' ts l f ( jL . )s) l '

In either event, z'(0) is purely imaginary.The next objective is to prove that J?e ,"(0) < 0. To this end,

a lengthy differentiation of z'(e) yields the complicated formuia

t t , \ s ( z (e ) ) l h ' ( z (e ) ) z ' ( e ) - r z ' ( e )h ' ( z (e ) ) + ( z (e ) - j us )h " ( z (e ) ) z ' ( e ) lz \ € ) :

_ s ' (z(e))z ' (e) [n(z(e)) + (z(e) - ju6)h ' (z(e)) + ee ' (z(e)) ]

[h(z(e)) + (z(e) - jL . .6)h ' (z(e)) + es ' (z(e))12

, g(z(e)) ls 'QG)) + es" (z(e))z ' (e))- lN*)W+,/(,( i )) l ' '

o'r*n(a)rl*#Pl

I2.5 / Proof of the Theorem 213

f U-o)

FicunB 12.5.1 Value Set Geometry for Proof of Sufficiency

However, in vievr of Lemrna L2.4.3, rve also know that

9 'n@to) : s- 0 '7kuo) + ( r - ) * )0 '1 *n(a) .

combining this equality with the two inequalities above for 7'yeLo)and 0'7*n("L,6), it follows that

0, " ( -d t r . l t ' l J%-( 'o ) l+

f r - 1 -1 l s in 2dn(c ro) l :

I t i " ?dn( ro)1 .Y\ - ' | 2ao I

' ao I I 2uso I

This is the contradiction we seel<.We now proceed to establish necessity. That is, we assume that

g(s) is a convex direction and must prove that the growth conditionis satisfi.ed. As in the proof of sufficiency, we take d : deg SG) > 2.Proceeding by contradiction, we assume that

g,.(ao), ' lsin ?dn(r" 'o) |l 2 a o l

for sorne r,is ) 0 such that ?n@s) I O.

2I4 Ch. 12 / R.anuer's Grorvth Condition

Now, in accoldance with Lernma I2.4.I\, there exists a stablepolynomial /(s) of degree n - 2 with n > max{4, d} such that

p,(s) : (s2 + rofr) y (s) + ee(s)

is stable for e sufficiently srnall. selecting some fixed e ) 0 for whichthe polynornials p.(s) and p-.(s) are both stable, rve define a farnilyof poiynomials by

P . ( s , A ) :(s2 +. l1S1s) - ee(s)

+ Ae(s)

and ) € [0,1]. Observe that this family has the property that thetu'o extremes p(s,0) and p(s, 1) are both stable. Furthermore, withthe parameter e reduced further if necessary, the construction of/(s) guarantees that this famiiy of polynomials has invariant degree.Noting, however, that

2ep , ( s ,0 .5 ) : ( s2 + u2 ; y 1s1 ,

it follows that p.(s,0.5) is unstable. This contradicts the fact that9(s) is a convex direction. !

12.6 Diarnond Farnil ies: An fl lustrative Application

rn this section, we dernonstrate the power of the growth condition byconsidering a problern which can be viewed as "dual" to Kharitonov'sproblem. Motivated by classical duality between |* and, /1 for finitesequences, we endow the space of uncertain parameters with t:he l,rn o r m ; i . e . , i f q : ( q o , e t t . . . , q r ) , w e t a k e

n

l lq l l ' : f loo li :0

This choice of norm motivates the definition beiow.

DEFrnrrrroN 12.6.1 (Diamond Polynomial tramily): A d,,iarnond.poLynomial fami,ly P is described by an uncertain polynomial of theform

p(s ,q ) : eo * q1s I q2s2 + . . . + Qn_ tsn - r * qnsn

rvith coefficients q; known to iie in the (n-F 1)-dimensional diamondw i th cen te r q * : ( q6 ,q i , . . . , g f i ) and rad ius r > 0 ; i . e . , ad rn i ss ib iecoefficients q : (qo,Qt,. . ., gr) satisfy

f Uro) + sj-o)

on0.o)

lso - s6 l + lq t - s i l + ' . . * l s . - s ; l < r .

12.6 / DiamondFamilies: An IllustrativeApplication 215

Letting Q denote this uncertainty bounding set, the resulting poly-nomial family is P : {p(., q) | q e Q}.

ExERcrsE L2-6.2 (Value Set and Edges) : For the diamond poly-

nomial family P above, assume that the order n is even and prove

that for a ) 0, the value set p(jut,Q) is a diamond in the complexplane described by

p( j u , Q) : conv{ur (w) , u2(a) , uz(a) , u+(a)} ,

where

u t ( . ) : p ( j . , q * ) -F rmax {1 , r . . r ' } ;

,z( r ) : p( ju , q*) - r max{1, cu ' } ;

u3(r . , ' ) : p( ja ,q") - t j r rnax{w," . ,n-L} ;

ua(a) : p(j., q*) - jr rnax{u, *'n-r}.

Now, describe the value set for the case when the order n is odd.

ExERcrsE L2.6.3 (Eight Distinguished Edges): With setup as

in Exercise 12.6.2, use the Edge Theorem (see Section 9.3) in con-junction with the value set description for p(ju,Q) io the exercise

above to prove the result in Tempo (1990); i.e., assume that lqfi l > r

and argue that P is robustly stable if and ciniy if the eight edge

polynomials

e1 (s , A ) : p ( s , q " ) - ^ r - ( 1 - ' \ ) r s ;

e2 (s , ) ) - - p ( s ,g * ) + A r - ( 1 - . \ ) r s ;

e3(s, ) ) : p(s, q*) + Ar -F (1 - . \ ) rs ;

ea (s , ) ) : p ( s , q * ) - ) r * ( 1 - ) ) r s ;

e5 (s , ) ) : p ( s ,q * ) - ) r sn - ' - ( f - A ) r s ' ;

e6 (s , A ) : p ( s ,g * ) + ) r s ' - t - ( r - A ) r s ' ;

e7 (s , ) ) : p ( s ,g * ) + ) r s ' - t + ( f - A ) r s ' ;

es (s , A ) : p ( s ,q * ) - ) r sn - t + (1 - ) , ) r s '

are stable for a l l A € [0,1] .

ExERcrsE !2.6.4 (Extreme Point Result): Using the eight edge

polynomials in the exercise above, apply Theorem 12.3.\ and arrive

at the result of Barmish, Tempo, Hollot and Kang (1992); i.e., 2 is

216 Ch. 12 ,/ Rantzer's Grbwh Condition

robustly stable if and only if the eight extreme polynomials

P l ( 5 ) : P ( s , q " ) + r ;

p z ( s ) : p ( s , q * ) - r ;

mG) : p(s, q") + rs;p 4 ( s ) : p ( s , q * ) - r s

/ * \ n - 1p b [ s ) : p l s , q ) + r s - - ;

n - 1p 6 l s ) : P l s , Q ) - 7 - s " - ip z ( " ) : p ( s , q " ) * r s " ;

p e ( s ) : p ( s , q " ) - r s "

are stable.

ExERCTsE 12.6.5 (Extreme Points for a Diamond Family) : Con-sider the diamond family P of the fourth order polynornials describedby cen te r q * : ( q6 ,q i , q6 ,q i , q i ) : ( 3 .49 ,7 .98 ,6 .49 ,3 .00 ,1 .00 ) andradius r : 0.5. Determine if 2 is robustl l ' stable.

ExERcrsE L2-6.6 (Robustness Margin): With the setup as in Exer-cise 12.6.5, replace the radius r : 0.5 by a variable parameter r ) 0.Denoting the resulting family of polynorrrials by P,, take r*o, to bethe supremal value of the radius r for which robust stability of P, ispreserved. Verify that r-o, = 0.9467.

L2.7 Conclus ion

The main objective of this chapter was to enrich the class of poly-topes lending themselves to extreme point results for robust stability.This rvas accomplished via Rantzer's Grow-th Condition. In the nextchapter, we continue to emphasize extreme points. However, insteadof concentrating on uncertainty structure, we concentrate on the de-sired r-oot location region 2. The fundamental question is: Underwhat conditions on 2 do we obtain extreme point results for intervalpolynomid. families? We see that a convexity condition involvingthe reciprocal set IfD plays an important role.

Notes and Related Literature

NRL 12.1 lvlost of the technical concepts in this chapter associated with the

growth condition in Theorern 12.3.1 are due to Rantzer (1992a).iji1i li i

Notes and Related Literature 2I7

NRL 12.2 The study of robust stability in the diarnond framework was first

suggested by Tempo (1990).

NRL 12.3 It is interesting to note that the extreme point result of Exercise 12'6.4

does not hold for weighted d,iarnonds. To demonstrate this point, we consider

the weighted diamond family of poll'nomials described as follows: T}'e center

polynornia l is g iven byp(" , q*) : s"+2s'*2.201s14 and the uncerta inty bounding

se t i s desc r i bed by 10 iq3 - s j l +1001q2 -q ; l +1001q1 - c i l +2 .5 lqo - q f i l < 1 . I t i seasiiy werified that atl extreme polynomials p(s,q') are stable but the polynornia,l

p-(s) : 1.05s3+2s2f 2.2Ols*4.2 is an unstable mernber of th is d iamond farni ly ;

see the dissertation by Kang (1992) for further details.

NILL 12.4 Contrary to most results on robust stability, an extreme point result

does not hold for certain types of diamond families with complex poiynomials; i.e.,

real and complex coefficient farnilies are fundamentally different in this rega.rd'

I ndeed , we cons ide r p ( s ,u ,u ) : f l o (u r + i u i ) s i , whe re t r : ( uo ,u t , . ' . , un )

and u : (uo,ur , . . . , r^) l ie in uni t d iamonds U and V, respect ively ' Now, for the

second order case with diamond centers given by u6 : -4.3776, ui : 0'0717, ui :

I .2272, u6 : 1.8398, u i :15.1285 and ui :6.3118, i t is st ra ight forward to ver i fy

that there are at most 36 extreme polynomials rvirich are all stable. However,

t l r e po l ynom ia l p - ( s ) : p ( s ,u * , o * ) + 0 .5s - ( 0 .5 +7 ) i s a r nember o f t he f am i l y

and has two roots g iven by st = -2.3249- j0.0440 and s2 = 0.0002* j0.3271.

Ilence, the family is not robustly stable even though ail the extremes are stabie;

see the paper by Barrnish, Tempo, Hollot and Kang (1992) for further details.

NR-L 12.5 There is also an e;<tension of Rantzer's Growth Condition s'hich

is applicable to delay systems. In the paper by Kharitonov and Zhabko (1992),

conve). conrbinations of quasipolynomials are considered ar'd a denerali'zed growth

condition is used to characterize convex directions.

iii

i,li lt,,iiii,ii i li i i i

iiliirli i l( irx

Iiitiil:jlii rliiiirlt i liil

iil;:l

Chapter 13

Schur Stabil ity and l{haritonov Regions

Dyl?opsrs

For the problem of robust Schur stability of interval polynomi-als, there has been considerable attention devoted to the attain-ment of Kharitonov-like resu.lfs. Although it can be argued thatno rcsult to date compares v,itlt Kltaritono\rtb Theorent in itselegance and simplicity, a number of usefu] robustness ctiteriahat e nevertlteless emerged. The first part of this chapter coverssome developments dong these lines and motivates the stud5,of weak l{haritonov regions. These are regions D in the com-p)ex plane for whiclt D-stability of all the extreme polynomialsintplies robust D -stab ility.

13.1 Int roduct ion

When studying robust stability of discrete-time systems, the fol-lowing question immediately corrres to mind: Is there a sirnple andelegant robust Schur stabilitS, criterion which rnight be appropriatelycalled t}ee d'iscrete-t'ime analogue of Kharitonov's Theorem? In tleisregard, we consider the interval polynomia.l

n

p(', q) : L, ln; , q, )r it :0

with argument z instead of s to emphasize that the open unit disc isthe desired root locationD of concern. As usual, we let {q'} denote

218

13.2 / Lot^t Order Coefficient Uncertainry 219

the finite set of extreme points (2n+L at most) of the uncertaintybounding set Q and pose a basic question: Can we identify a distin-guished subset {d} of {qi} which can be used to guarantee robustSch.ur stabil ity? That is, is there a subset {d} "f {g'} having theproperty that Schur stabil ity of each p(2,() irnplies that the inter-val polynomiai family P: {p(.,q): q € Q} is robustly Schur stable?Tiris chapter begins by answering the most general version of thisquestion in the negative. Subsequently, we proceed to deal with anumber of special cases for which a positive answer can be given.

The question about an extreme point solution for the robustSchur stability problem is a special case of a more general question:Given an interval polynomial family P, for what class of 2 regionscan we establish that Z-stability of a subset of the extremes impliesrobust Z-stability? After dealing with the case rvhen 2 is the openunit disc, we consider the generalization to large classes of 2 regions.

ExAN4eLE 13.1.1 (No General Extreme Point Result): To demon-strate that Kharitonov's Theorem does not generalize to the Schurstability case in the obvious way, we consider the interval polynomial

of Bose and Zeheb (1986). For q - -I7/8, it is easy to verify that theextrerne polynorrrial p(2, -I7 l8) has four roots 21,2 = 0.786 + j0.596,

zs = 0.924 and z4 = -0.371 which are all interior to the unit disc; e.sirnilar conclusion is reached for p(2,17 l8). That is, its four roots aregiven by zr,2 = 0.786+j0.596, zs x -O.924 and z4 = 0'371. However,a fi.nal computation reveals that the intermediate polynomial p(2,0)has roots zr,2 = +j1.303, zz,4 = +0.4433 rvhich are not all inside theunit disc.

FLEMART(s 13.L.2 (Low Order Polynomials): Later in the chapter,we establish that examples of the sort given above cannot be givenfor polynomials having order n < 3; i.e., a discrete-tirne interwalpolynomiai of order n 13 is robustly Schur stable if and sllrr if rho

extrerne poiynornials are Schur stable.

73.2 Low Order Coefficient lJncertainty

Although Example 13.1.1 rules out a general Kharitonov-like resultfor the robust Schur stability problem, there is one special case forwhich strong results can be given. Namely, when the coefEcient of z'

220 Ch. 13 ,/ Schur Stabiiify and Kharitonov Regions

is fixed for rl > L?1, a,tt extreme point result emerges; Uy LEI above,

we mean the lar[est integer less than or equal to nf2. To simplify

the proof of the technical results to follow, without loss of generality,

we make a simplifying assumPtion.

DEFrNrrroN 13.2.1 (Nontriviality Condition): The discrete-time

interval polynornial

p(", q) :ilr; , nf)"ni :0

is said to satisfy a Nontriu'iality Condition tf lhe two quantities

p ; : i q ,; - n

and

t t : i q fi : 0

are either both positive or both negative.

REMARKS L3.2-2 (No Loss of Generality) : To see that there is no

loss of generality associated with the imposition of the Nontriviality

Condition, notice that if po- and pl- hat'e opposite signs, it follows

that

for some q* ' Hence, p( \ ,q*) : 0 , which ru les out robust Schur

stabil ity; i.e., z : 1 is a root of p(s, q*). Although not needed in the

sequel, a similar nontriviality condition can be enforced with respect

to z : -1. In other words, there would be no loss of generality in

assuming that the two quantities

and

^ | 1 7 7 7 1 " 3 , Ip l z , q ) : z - * L - u , 8 l '

+ t " " -

i

n\ - *

\ n - : t l

n n

, , - - \ - - - r \ - - f

u o : 2 _ q i f ) . q i

i euen i odd

n n

" d : L q f + l o ti euen i odd

are either both positive or both negative'

13.2 / Low Order Coefficient Uncertainty 221

TrrEoREM L3.2.3 (Hollot and Bartlett, (1986)): Consi,der" the'interual polynomial fami,ly P described by p(r,q) : ILo[Si ,qf]t 'with uncertainty bound'ing set Q hauing ertreme poi,nt set {q'}. Inadditior-t, assurne that the Nontriuial,ity Conditi.on is satisf,ed a.nrl

L "q t : q i f o r i : L t ) + I , l } J + 2 , . . . , n w h e r e l | ) d e n o t e s t h e l a r g e s tinteger less than or equal to ft. Th,en P ' is robustly Schur stable ifand only i,f each, of th,e ertreme polynornials p(2,q") i.s Schur stable.

PRoor': The proof of necessity is trivial because robust Schur sta-l - r i l j+r , n f '1) i - - r i - " schur s tabi l i ty of each p(2,q, ) ; i .e . , p(2,q, ) € p.

To establish sufficiency, we use the well-knou'n bilinear transforma-tion to convert a unit disc problem into a left half plane protrlem.Namely, by defining the uncertain polynomial

p ( " , q ) : ( s - r l " o ( * , n ) ,\ o - r , /

it follorvs that the original interval polynomial family 2 is robustlySchur stable if and only if the transformed family of polynomialsP : {p(.,q) : q e Q} is robustly stable (strict iefb half plane).Furthermole? \ re observe that the transformed family of polynomialsF has invariant degree because the coefficient of s' is f i ls qi; i.e.,the Nontriviality Condition guaiantees that fhis sum cannot vanish.

We now absume that each of the extreme polynomials 1r(s, g') isstable and it must be shown that P is robustly stable. Proceedingby contradiction, suppose 2 is not robustly stable. Since p(s, q)has an affine linear uncertainty structure, the Edge Theorem (seeSection 9.3) implies that there exists some k S tE) and extrernalsettings qr : (lf - e1 or qf fot i I A such t-hat the edge polynomial

p r ( s ,qx ) : qk (s+ 1 ) f t ( s - 1 )n -A + f ao+ ( " + r ) i ( s - 1 )n -?i+k

is stable for q6 - 96 and qk : qf but unstable for some qk e (qi , q[).Now, by the Zero Exclusion Condition (see Theorem 7.4.2), thereexists sonre qf; e (qa , qil) and sone cu* ) 0 such that pk(ja+, gi) : 0.Equivalently, the origin, z:0,l ies in the interior of the straight l inesegment jo in ing pn( ja* ,s f ) and pn( ja- ,q[ ) .

Witlrout loss of genera,l ity, we take 0 < |p(ja*,(I[) < rf2 andn < 4p( j r* ,e i ) < 3r f 2 as indicated in F igure I3.2. I ; the argumentto follow is not specific to these quadrants. Now, recailing Exam-ple 7 -2.2, the slope of this value set l ine seornenf is

Irn (ja + 7)k (j,,t - 1)rr-t

222 Ch. 13 / Schur Stabiliry and Kharitonov Regions

Frcunp 13.2.1 Value Set for the proof of Theorem 13.2.3

Clairn; The slope rnk(u) is nonincreasing with respect io a.r ) 0.prove this claim, we define the complex frequency function

zp(,t) : ( j. + t)k (j. - 1)n-fr

whose angle is

dn(') : ro'r": )'r-:r; i'#-

tan-lc''')

Since

rnk(u) : tan 6x@)and the tangent function is monotonic, we need only show that

dd*.(r) . o.

d,u

Indeed, differentiating above, we obtain

2 k - nddn@) _dr's

i:riili ll i

f1i1i ;[.ii1i l

IrI lIli i : lr q ?

tii ii i

/ . * - \p k \ J a , q k )

Re ( ju + I )k( . iu - 1)n- t '

I -I LD.

13.3 / Lorv Order Polynomials 223

Now, since k < LEJ, we conclude that

dd*(a) - ^d , > " '

Hence, the claim is established.To complete the proof of the theorem, we appiy the result in

the claim above to the critical frequency L,r : Lo+. To arrive atthe desired contradiction, we first exploit stabil ity of p6(s,Sft ) and

r -f rpnG,S[). That is, for A,ut ] 0 sufficiently small, the NlonoLonicAngle Property (see Lemma 5.7.6) forces pk( j (LD- * Aa, l ) ,q f ) in tothe interior of the corr.e

C, , : { z i | pk ( j a * ,S f , ) < + " <T }L

and pp(j(a* + At.u;, Sa ) into the interior of the cone

C- : {" : Lpk(jL,)",S;) < +" . #}

However, these ner.v locations for pp(ju.t,As ) and px(jr 't,gt) are in-consistent with the fact that rnkkD) is nonincreasing; i.e., t ire straightIine joining any two points in the interiors of C- and C1 has slopegreater than rng(c..,*). Having arrived at this contradiction, the proofof the theorem is now cornplete. E

13.3 Low Order Polynornials

M/ith Theorem 13.2.3 in hand, we now consider the special case ofinterval polynomials having order n ( 3. First, we address the sirn-plest cases, rt : I and n: 2, in the exercise below.

ExERcrsE 13.3.1 (F i rs t and Second Order Cases) : For monic in-tervai polynomials of or-der ri: 1 and rL:2, argue that robust Schurstability is equivalent to Schur stability of the extremes by explicitlydisplaying the roots.

Rbnaan"xs 13.3.2 (Tli ird Order Case): We nou' argue that thethird order case can be reduced to a case which can be handled byTheorem 73.2.3. Indeed, for the third order interval polynomial

p(2, q) : z3 * lq ; , q{1"2 + lq i , q{ ] , + [qt , qo ' ]

lvith uncertainty bounding set Q, we can write down conditions forSchur stabil ity expressed paran'retrically in q; e.9., by applf i11g 11't.

224 Ch.73 / Schur Stability and Kharitonov Regions

positive innerwise criterion of Jury (1974), robust Schur stabil ity is

guaranteed if and only if the inequalities

1 * q o - 1 - q r + q 2 > 0 ;

I - q r * q t - g o > 0 ;

1 - 0 3 - q r r l q o q 2 > 0 ;

1 * q s > 0 ;

1 - q e > o

are satisfied.Now, observe that if qs and Q1 are fixed and q2 is allowed to vary,

the inequalities above are affine linear with respect' to q, < q2 < s{ .

Using the basic fact that a linear function f(q2) ovet an interval

lqi,q{) remains positive if and oniy if f (qz) > 0 and /(qf ) > 0, we

arrive at the follos'ing point: Robust Schur stability is guaranteed if

and only i f p(z,q) is Schur s table for a i l q of the form q : (qo,qt ,q i )

or q: (qo,qt ,qf ) wi th qo < qo < qof and at < qt < s f . We

have now reduced the problem to a point which permits application

of Theorem 13.2.3; i.e., since only coefficients of order one and two

are uncertain, we conclude that Schur stability of the extrernes is

equivalent to robust Schur stability. There are at most eight such

extlemes.

L3.4 Weak and Strong Kharitonov Regions

Questions involving the attainrnent of extrerne point results for- r'o-

bust Schur stability of interval polynomials can be phr-ased in a rrruch

more general context: Given an interval polynomial family P with

set of extternes {p(t,qo)} and a desired root location region D, ::tn-

d.er what conditions does 2-stability for each of the extr-emes p(s, qi)

imply robust 2-stabiiity of P? To address this cluestion in a precise

manner, we require some preliminaries.

DEFrNrrroN l-3.4.1 (Weak and Strong Kharitonov Regions): An

open set D q C in the cornplex piane is said to be a weak Khtt'ritonou

region if the following condition is satisfied: For any given intervalpolynorniai family P : {p(',q) , S e Q} }raving invariant degree,

stabil ity of p(s, qt) for each. extreme point q' of Q implies ::obust

D-stabii ity of 2. We say that D is a strong l(haritonou region if

there ex is ts a f in i te index set f (D) C {1,2,3, . . . ,1 / } having the

following property: Given any interval polynomial family P with

invariant degree, there exists a labeling of the extreme Poll'11ot.t'-

ii.il

l

Ii

i

I

' ll

I: l,1rl:l

:jl

13.5 / Characterization of Weak Kharitonov Regions ZZs

als {p(s,qt)} such that D-stabii ity of p(s,qi) for i € 1(2) irnpliesrobust D-stability of P. The understanding above is that the label-ing scheme for the extreme polynomials may depend on the order nof tire interval polynornial family, but the cardinality l/ of 1(D) isindependenl of n, p( . ,q) and Q; l / on ly depends on t l te region 2.

REMARKS 73.4.2 (Elaboration) : To elaborate on the definitionabove, observe that the strict left half piane is a strong Kharitonovregion because the requirements of Definit ion 13.4.1 are satisiied bytak ing I (D) : { I ,2 ,3,4} . From the point of v iew of cornputat ionalcompiexity, it is obvious that resuits involving strong Kharitonovregions ar-e quite porverful when the cardinality of I(D) is small.In contrast, weak I{haritonov regions have the undesirable propertythat the number of extremes to be tested increases exponentiallywith respect to the degree n.

13.5 Characterization of Weak l{haritonov Regions

In this section. the main objective is to provide the rather generalcharactelization of weak Kharitonov regions due to Rantzer (1992b).

DEFrNrrroN l-3.5.1 (Regularity): A region D C C is said to beregular if it is open and sirnply connected with boundary 0D, whichcan be directed in a positively oriented manner with an associatedpiecewise C2 boundary sweeping Qp : I - 0D; see Section 7.5 foran introduction to boundary sweeping functions. The proof of thetheorem below is relegated to the next two sections.

THEoREM L3.5.2 (Rantzer (1992b)): Suppose D g C i,s regular.Then D is a weak Kh,aritonou region i,f both, D and its rec'iprocal,

: { z € C : z d : \ f o r s o m e d € D } ,

REMART(s 13.5.3 (Rea1 Versus Complex Coefficients): The readeris reminded that the standing assumption in this text is that unlessotherwise stated, all polynomials have real coefficients. We empha-size this point because the theorem above provides an example ofa resuLt for which the real and complex versions are different. Inthe real coefficient case, convexity of D and 7lD is only sufficientrn'hereas in the complex coefEcient case, this condition is both nec-

226 Ch. 13 ,/ Schur Stabiliry and Kharitonov Regions

essary and suffi.cient. Before proceeding toward the proof of thetheorem, we demonstrate the power of the convexity condition intwo ways. First, we specialize the theorem to recover known results.Second, we demonstrate how the theorem is used to determine inter-esting weak Kharitonov regions. This is accompiished via a sequenceof examples and exercises.

EXAMPLE L3.5.4 (Strict Lefi HaIf Plane): If D is the strict lefthalf plane, then 7fD : D. Hence, both 2 and I/D are convexand we arrive at a conclusion which is consistent (also weaker) thanKharitonov's Theorem. That is, the strict left half plane is a weal<Kharitonov region.

ExAMeLE 13.5.5 (Shifted Half Piane): For degree of stabil ityproblems, a scalar a > 0 is specified and the shifted half piane

D : { z € C : R e z < - o }

is the root Iocation region of interest. Now, straightforward calcula-tion indicates that z € 7/D if and oniy if

From the description ot1./D above and the sketch in Figure 13.5.1,it is again obvious that both D and If D are convex. We concludethat D is a weak Kharitonov region.

ExAMPLE 13.5.6 (Unit Disc): For robust Schur stabil ity problerns,we take 2 to be the interior of the unit disc and obtain

* : { " e C : l z l > 1 } .

The nonconvexity of this region is consistent with the absence ofextreme point results in the Schur case; see Section 13.1.

ExAMPLE L3.5.7 (Damping Cone): When both stabil ity and damp-ing is of concern, we take

D : { z € C : r - d < 4 2 < r * d } ,

where 6 e (0,rlZ) is the so-cailed damping angie. Since LfD : D,it follows that both D anclIfD are convex. Hence, we conclude thatD is a weak Kharitonov region.

| 1 t 1l z l ^ l ( ^|

'2o |

'2o

1D

13.5 / Characterizarion of Weak I{haritonov Regions Z2j

FIGURE 13.5.1 The Region 2-1 for Example 13.5_5

ExEncrsE 13.5.8 (Shifted Disc): In the theory of delta transfor-mation, it is important to know whether or not the roots of a givenpolynomial lie in the interior of a disc 2 which is wholly containedin the strict left half plane.(a) Show that such a set D is a weak Kharitonov region.(b) For the more general case when D is the interior of a disc witharbitrary center and radius, argue that 7/D is convex if and only ifthe condition

0 / i n L D

is satisfied.

EXERCTsE 13.5.9 (Intersection of Weak Kharitonov Regions) : LetDr and D2 be two open convex regions in C such that 7 /D; is convexf o r i : 1 , 2 . L e t t i n g

D : D t O D z ,

show that lfD rn:ust be convex. Use this result to prove that theregion 2 in Figure 13.5.2 is a weak Kharitonov region. This regionreflects concerns for both damping and degree of stability.

ExERcISE 13.5.10 (Generai Characterization): If. D C C is reg-ular, argue that D and 7/D are both convex if and only if D is an

228 Ch. 13 ,/ Schur Stability and Kharitonov Regions

FIGURE 13.5.2 2 Region for trxercise 13'5.9

intersection of open discs and halfpla.nes rvhich exclude zero; this

intersection is not necessarilv finite.

13.6 Machinery for Proof of the lfheorern

The objective in this section is to develop a number of tecirnical

results wirich are instrurnental to the proof of Theorem 13.5.2. Both

this section and the next can be skipped by the reader interested

primarily in application of the results.

13.6.1 Convention for Functions and Their Derivatives

Sr.rppose that D C C is regular u'ith piecewjse C2 boundary sweeping

function Qp : I + AD and p(s) is a polynomial. Then throughout

this section we encounter many complex functions of the generalized

f requency 5 e I . I f F :1 - C i s a p iecew ise C2 comp iex f unc t i on

of 6, rvhenever notationali-t' 6en."nient we use F'(6) to denote the

derivative of F(6). In the sequel, F'(6) can be taken as either the

left or right derivative whenever ambiguity arises. For example, if

6 - 6- is a generalized freqrrencl'where F/(6) does not exist' we can

13 6 / Machinery for Proof of the Theorem Z2g

take Jr'(5*) to be the left derivative

F ' (6 . ) : F ' - ( 6 . ) : l i q r F (6 . ) - F (6 . - h )

hJo h

or the right derivative

F /A* t h ) - F (6 . )F ' ( 5 * ) : F L 6 . ) : l . r m - \ "t \ - / E L o h

Finally, we draw attention to a shorthand notation for evaluation ofderivatives; i.e.,

F ' (5. ) : F ' (6) i t :a- .

In some cases, F(5) corresponds to the evaluation of a polynomialp(s) for s : Oa(d) . Wi th F(6) : 'p(OD(6)) , we use the shor thand,notation

?eQ) : 4p (oo (6 ) )

ExAMeLE 1-3.6.2 (The Generalized Flequency 6 Is an Endpoint):Suppose that D is the interior of the unit disc and we use the bound-ary sweeping function AoG) : cos 2n6 -f j sin 2n6 with 6 € [0,1].Then, for 6:0, if we use the notation Ab(O), the understanding isthat the right derivative, Of,(0) : @b.*(0) :2trj is intended. For5 : l, i f we use the notation @!(1), the understanding is that theIeft derivative O!(1) - Ab,_(I) :2ttj is intended.

EXAMeLE 13.6.3 (Nondifferentiable Point): Consider the dampingcone

2- q*D : { r € C : ; . . + r . t }

with boundary sweeping function

| _d " 'T i f d < o ;Q n ( d ) : (

l a ' i x i f 6 > o '

Then for 5 : 0, we can take A!(0) to be either the left derivai,jve

ab@: @p, - (o ) - - " j i ,

or instead we can use the right derivative,

230 Ch. 13 ,/ Schur Stability and I{raritonov Regions

ExoRCTsE J3.6.4 (Shifted Disc): Consider the shifted disc

D : { z € C : l z + 1 1 < 1 }

with boundary sweeping function O2:(6) - -1- cos 2r5 - j srn 2tt6'

With d(6) : 4Ao@), describe the one-sided derivatives di(0), 91(1)

and the total derivative d'(5) for 6 e (0,1)'

REMARKS L3.6.5 (Angle Considerations): An irnportant concept

enterirrg into the proof of Theorem I3.5.2 is a generalization of tire

lVlonotonic Angle Property; see Lemma 5.7.6. Indeed, if D C C is

regular rvith boundary sweeping function Qp : I ----' 0D and p(s) is

a D-stable polynomial, we first write

n

p ( s ) : x l l ( s - z )L : L

with I( € R and z; € D. Now, if we evaluate the rate of change of

the angle do(5) aiong 0D, we obtain

where

With the aid of Figure 13.6.f it is easy to see that each angle con-

tribution 0t;(6) above is positive for the case when D is convex.' Foi:

nonconvex D, we also see from the figure that it is possible to have

0i@) < 0 for some values of 6. The proof of the lemma below (wl-rich

we ornit) amounts to a formalization of these ideas.

LEMMA 13.6.6 (lVlonotonic Angle Property) : Suppose thatD C C,is regular and conuer with boundarg sweep'ing fu,nct'iotz Q7t : I ----' DD '

Then, g'iuen anE D-stable polynom'ial p(s), i ' t follouts that the angl,e

0o(5) : Lp@o(6)) ' is a strictly ' increasing ftmction of 6 e I. That

i s , 0 L ( 6 ) > 0 f o r a t t S e I '

ExERCTsE 1-3.6.7 (Converse): Consider the converse of the lemma

above; i.e., suppose that D C C is regular with an associated bound-

ary sweeping function Qyt : I - 0D and has the property tt,.at 0r(5)

dp@)

et@)

D convex

13.6 / Machinery for Proof of the Theorem 2Sl

Im

2 nonconvex D convex

Ftcuno 13 6.1 da(d) Increases When D Is Convex

is a strictly increasing function of 6 e I for every 2-stable polyno-mialp(s). Does it follow thatD is convex? In this regard, is there adifference between real and complex coefficient polynomials?

ExERcrsE 13.6.8 (A lvlimic: From Stabil ity to D-Stabil ity): Nlimictlre proofs of Lemmas 12.4.2 and 12.4.3 to establish the following re-sults under tb.e assumption that D C C is regular with an associatedboundary srveeping function @p : I ---+ AD.(a) Given any polynonial p(s) and a generalized frequency 6 e Isuch that p@p(6)) + 0, it follows that

f r ' / ro^ r5)'\ ' ter(6) : r* l ' -#ob!) l

L p ( o z ( a ) J - ' ' J

(b) Given two polynomials /(s) and 9(s), a scalar )* e (0,1) and ageneralized frequency 6* e I such that

f @D(5. ) ) + o ,

/ (oz(6 . ) ) + s@o(6 . ) ) *0

232 Ch. 13 ,/ Schur Stabiliry and Kharitonov Regiorts

and

f (ao(6-) ) * , \e(o2(6. ) ) :0 ,

it follows that

e 's(6*) : A"g ' f (6*) + (1 - ^ . )0 ' f +st (6*) '

LEMMA 13.6.9 (Angle Condition for the Reciprocal): Suppose that

D C C i,s regular u'ith an associated boundary sweep'ing function

Qyt : I - 0D and O I D- If L/D ' is conuer, then giuen any n-th

ord,er D-stable polgnomi'al p(s), it follows that

oL(6) > na.pQ)

for all 6 e I.

pRoor.: we first prove the lemma for all D-stable polynomials of the

form p(s) : 5 * z with z € C' Indeed, using the boundary sweeping

function rop(6) : ^ : ^a p ( d )

fo r 1 /D, s ince p : s + !- z

is 1/2-stable, Lernma 13.6.6 guarantees that for

e;@) > ofor aiL S e I. Using this inequaiity and the expression for 0'1161 \n

Exercise 13.6.8, we obtain the chain of inequalit ies

o < Imf"--9295;1r11lnlwo\o )) .l

I i = , . ^ . - lT ^ l - 6 : - l h ) l

L t t u | - 1 ' u \ - / l

laD(o) +; I

. I zap(6) . . ( ob(6) \ l: '* lortro1 * "

^ \-oe16)/l

. t -zaoQ) I: 'IT,L La;a;1o;6y * "y1r o i . ( 6 ) t _ i o i ( 6 ) l: rmla;6 i ) - tn t* ;e1

: oL@) - ob6)

Re

Hence,

IZ.7 / Proof of the Theorem 233

oe?) > a;@)Tlris completes the proof for the case n: I.

Now, to establish the desired inequality for polynomials of arbi-trary degree n > 1, we need only sum the contributions of each rootof p(s) . That is , i f

p ( s ) : 1 r ( " - r r ) ( " - , r ) . . . ( t - , - )

f o r 1 ( € R a n d z i € D f o l i : 7 , 2 , . . . , 1 2 , t h e n

234 Ch. 13 ,/ Schur Stabiliry and Kharitonov Regions

Now, using the result of Exercise 13.6.8 and the known form forpr( t ) - p1(s) , i t fo l lows that

- - - - t - , ^ , I

^"0; , (6.)1,-^ * (7-^")0e,(6.) la:0. :UlP9i : kab(6.) 'l a :o -

In view of the convex combination on the left-hand side above, thereare two possibil i t ies: Either

0;,(6.) < kab?.)

or o;,(5.) < kabu\.

Without loss of generality, we assume that the first of the two in-

equalit ies above holds for pt("). Now, since pi(s) is 2-stable, we see

that t l ie inequality 1Lr@-) < kQb$.) contradicts the requirement

of Lemma 13.6.9 that

0r(6) > nO'.p(6)

for every 2-stable polynomial p("). E

l -3.8 Conclusron

This chapter and its seven predecessors can be viewed as a sequentialdevelopment of results emanating directiy from Kharitonov's Theo-

rem in Chapter 5. In all cases, we heavily exploited the affine linear

and independent uncertainty structures in order to obtain either ex-

treme point results or edge results. In the next chapter, where more

complicated multiiinear uncertainty structures are considered, we see

a marked departure from the frarnework of this chapter. Neverthe-

less, resuits developed thus far prove to be useful in a certain "convex

hul l " context . That is , i f Q is a box andP : {p( ' ,q) : q e Q} is a

family of polynomials whose coefficients depend multilinearly on g,

we obtain a simple description of the convex hull of P; i.e-, if a(q) is

the coefficient vector and {q'} is the set of extreme points of Q,wesee that

conv a(Q) : .ott-r{a(q')}.

n ) ( / ; f \ - / A j _ ? . \

e : (5 ) : , u + \ vD | l y ) - z i ) > na^ (6 ) . E

Y ' ' d hA : I

] .3 .7 Proof of the Theorern

To sinrplify the proof of Theolem 13.5.2, we assume that ED is C2lather than piecewise C2 and note that the arguments to followcan be readily modified using left and right derivatives wheneverappropriate. Proceeding by contradiction, we assume that both 2and I /D are convex but D is not a lveak Kharitonov region. By theEdge Theorem (see Section 9.4), there exist a pair of 2-stable n-thorder polynomials p1 (s) and p2(s), corresponding to the extremes ofsome n-th order interval polynomial family P, such that

Pr (s) - Pz(s) : 11" t '

for sorne K g R and sorne nonnegative integer k I n. N'Ioreover, thepolynomial defined by

p (s , ) ) : p l ( s ) + A lpz (s ) _ p r ( r ) l

is not D-stable for some ) e (0, i). Bv the Zero Exclusion Condition(see Theorern 7.3.3), there rnust exist sorne )* e (0, 1) and 6* e Isuch that

P ( Q o ( 6 . ) , ) * ) : 0 '

That is,

p r ( @ p ( 5 . ) ) * \ . 1 p 2 ( Q p ( 6 . ) ) - p r ( 4 2 ( 5 - ) ) l : s .

Notes arci Related Literature 235

FIGURE 13.8.1 Bounding Set for the Pair (et. ,e--t)

Notes and Related Literature

NRL 13.1 For more details on extreme point results for robust Schur stability

of lower order poiynomials, see Ciesl ik (1987).

NRL 13.2 The ideas introduced in Section 6.2 are further embellished in Kraus,

Anderson, Jury and N,Iansour (1988). For example, for n:3, not a l l e ight

extrernes need to be tested. Attention can be restricted to a subset of extreme

points corresponding to "critical" constraints.

I{RL 13.3 In Kraus, Anderson and Mansour (1988), a new uncertainty modei is

considered in a robust Schur stability context. These authors begin with an un-

certain polynornial p(2, q) : LLo qr"n having independent uncertainty structure

but dispose with the box bound Q for q. Instead, they consider an uncertainty

bound of the sort shown Figure 13.8.1 for pairs of coefficients (q^_;,qo). To illus-

t rate, for the case when n is even and i : nf2, at interval lOq-,Ug*) is taken

as a bould for qe When the uncertain parameters are bounded in the manner

above, we say that Q is a product of 45o rotated, rectangles and let {qt} denote the

extreme points of 8. \,Vith this new setup, the resulting family of polynomials P

is robustly Schur stable if and only if the polynornial p(2, qn) is Schur stable for

each extreme point g' of Q.

FIRL 13.4 In a follow-up paper by Mansour, Kraus and Anderson (t988), even

stronger results than those described in the note above are established. Under

the same hypothesis, they show that a "small" subset of the extreme polynomi-

236 Ch. 13 ,/ Schur Stabiiity arrd Kharitonov Regions

als need only be considered F'urtherrnore, working with chebyshev and Jacobi

polynomials, they develop a recipe for selection of these distinguished extremes.

The reduction in the number of extremes can be quite dramatic'

NRL 13.5 In the paper by Perez, Decampo and Abdal lah (1992), a novel tech-

nical method involving barycentric coordinates leads to extreme point results for

classes of rotated rectangles with angles other than 450 For example' results are

given for angular rotat ions S el t r / ,3r /a l .

NRL 13.6 A condition for robust aperiodicity of a discrete-tirne interval polyno-

mial is given in Soh (1986); i.e., we say that a discrete-time interval polynornial

farnlly p: {p(., q) : q € Q} is robustly schur aperiodic if the following condition

holds: Given any q € Q, a l l roots of p(z,q) are real and posi t ive and l ie in [0, 1) .

subsequentl;,, it is shown that robust Schur aperiodicity can be ascertained by

testing only two distinguished extreme polynomials'

NRL 1-3.7 For the robust Schur stabitity problern, a number of authors opt

for sufficient conditions in lieu of the combinatorics associated q'ith the Edge

Theorem; e.g., see Bose, Jury and Zeheb (1986) where a set of inequalities is given

and Vaidyanathan (1990) and Bartlett and Hollot (1988) where transformations

of the original polynomial are used

i'ititi i l

ti

Chapter 14

Multi l inear Uncertainty Structures

Dyrlopsrs

The focal point of this chapter is robust D-stabihty of systemsv,itlt multilinear uncertainty sTrr"tures; i.e., we consider uncer-tain polynotnja.ls w'.hose coefficients depend mdtilineat'ly on thet ectot' of uncertain paratneters q. Using the Nlapping Theorem,r\/e can often obtain the tightest possible polSrtopic overboundfor tlte value sets and coeffi.cienf sets of interest.

]-4.1 Introduction

The basic motivation for- this chapter is the following fact: The uncer-tainty structures which arise in typical applications are more compli-cated than those which we have analyzed in the polytopic frameworkof Ch.apters 8-13. It is easy to describe applications involving highlynonlinear dependence of various system coeffi.cients on the vector- q ofuncertain paranreters. For example, recall the case study involvingthe Fiat Dedra engine in Chapter 3.

When dealing with cornplicated unceltainty structures in a ro-bustness context, ther-e are various avenues of attack. In some cases,the number of unceltain parameters is small and no formai theory isrequired-a practical solr-rtion is obtainable via some sort of griddingof the uncertainty bounding set Q. For more formal r'obustness anal-yses) we mention some alLernatives; to some extent, the discussionbelow amounts to a review of points raised in Section 1.7.

237

238 Ch. 14 ,/ Mulrilinear Uncertainty Sructures

14.2 More Cornplicated Uncertainty Structures

ln some cases, a fanr.i ly of polynomials or- raLional functions havirrga complicated uncertainty structure can be overbounded by a farnilyhaving a simpler structure for which analytical tools are readily avail-able. For example, in Sections 5.2 and 8.2, it was demonstrated thatfamilies of polynornials with complicated dependence on q can oftenbe overbounded by interval polynomials or polytopes of polynomials.Of course, the overbounding process introduces conservatism whichmay or may not be tolerable. The issue of overbounding prevails inrnuch of the robustness l iterature; it is not specific to the new toolsin this book.

A second approach to dealing rvith nonlj.near uncertaintlr struc-tures involves reforrnulation of r-obustness pr-oblerns within the frame-work of mathernatical programming. hr other words, one massages arobustness problem into an equivalent optirnization ploblem rvhosesohrtion can then be obtained using a wide variety of softr,vare tools.In some cases, the feedback control configuration induces specialproperties on the resuiting mathematical program which can be ex-ploited in a computational algorithm. On the positive side, reformu-lation of a robustness'problem as a mathematical program nakes itposslble to deal s' ith rather general situations. Complicated uncer-tainty structures and sophisticated performance specifications can behandled. On the negative side, one must deal with a host of issuesinvolving local versus giobal minima and cornputational cornplex-ity. Another negative is that a mathematical programming methodgenerally tells us very l itt le about how the solution depends on ad-justable parameters such as the compensator gains.

Given any uncertain polynomial or rational function whose coef-ficients depend nonlinearly on uncertain parameters, the attainmentof analytical results for robust stability is generaily accomplishedat the expense of restricting the class of noniinearities under con-sideration. This is the line of attack taken in this chapter. In thenext section, we define both multilinear and polynomic uncer-taintystructures. Subsequently, we describe a transformation relating thepolvnomic case to the muiti l inear case.

L4.3 Multi l inear and Polynornic lJncertainty

In this section, our objectives are twofoid. First, we formally definea muiti l inear uncertainty structure. Second, rve demonstrate howmultilinear uncertainty structures arise via exarnples and exercises.

i

t.l

14.3 / Mululinear and Polynomic Uncertainry Z3g

DEFrNrrlroN 14.3.1 (lvlultilinear and Poiynomic {Jncertainty Struc-tures): An uncertain polynomial p(s, q) : DT-ooo(q)t' is said tolrave a rnultilinear uncertai,nty structure if each of the coefficient func-tions a; (q) is multi l inear. That is, if all but one component of thevector q is fi.xed, then a1(q) is affine linear in the remaining com-ponent of q. Vlore generally, p(s, q) is said to have a polAnonri,cuncerta'intg structure if each of the coefficient functions a;(q) is amultivariable polynomial in the components of g.

REr\TAFLKS L4.3.2 (Multi l inear Functions): Although we definedmultilinearity in the context of uncertain polynornials above, in thesequel, we have occasion to work with a more general multilinearfunction f tPln -* Rk. When we call / multilinear, the understand-ing is that each component function ft : R" * R is multilinear inthe sense of the definition above.

ExA^4pLE I4.3.3 (Ivlult i l inear Uncertainty Structule): The uncer-tain polynomial

p(s, s) : s3 * (6qrqrqs -l 4qzqz - \qt + 4)s2

* ( qrqs - 6qrqz+ qs)s * (5qt - az - t 5)

has a rrrultilinear uncertainty structure. If the coefficient of s isc l ranged to a1(q) - 4qt - 6qrqz - q32, th"r , p(s,q) has a polynornicuncertainty structure.

REMAFLKS L4.3.4 (Hierarchy): Thus far, we have defi.ned fourdifferent types of uncertainty structures for polynomials. LettingPind."p, Poff, Pmuttilin arrd Ppory denol.e the set of uncertain polyno-mials p(s, q) with independent, affine linear, multilinear, and poly-nomic uncertainty structures, respectively, we draw attention to theobvious inclusion

f i - 6 - 6 - 6t x n d e p ! ' m u t t L l x n I t p o l a .

240 Ch. 14 ,/ Multilinear Uncertainty Structures

and. Ai, are located. in different positions; e'g', i f t 'he (2'2) entry of

-zl1 is iionzero, then the (2, 2) entry of '42 must be zero' This type

of uncertainty structure is synonymous lvith the well-known class of

interval matrices. Prove that the uncertain characteristic poiyriornial

p (s , q ) : de t ( s1 - e (d )

has a multilinear uncertainty structure'

EXAMPLE 14.3.6 (Property of Nlutti l inear uncertainty s,tlucture):

Suppose that an urrcertlin polynomial has coefficient of s2 given by

"r(d : lqtqzqsl3qtqz*4qzqs* 5gr + Sqz+4' If q2 and q3 are fixed'

we'can isolate an affine l inear function of q1 ; i 'e',

oz (q ) : q t f z t (qz , a t ) * gz , t l qz ' q ' z ) '

where fzl(qz, qz) : 6qzqs I 3qz -t5 and gzJ(qz, qs) : Aqzcls * Sqz -t 4'

Similarly, if q1 and q3 are fixed, we can isolate q2\ i 'e' '

oz(q) : qzfz,z(qt , qs) + gz,z(qt , q t ) ,

whe re f z , z (q t , qs ) : 6q rqe -F 3q i - l - 4qz *5 and g2 ,2 (q1 'Qz ) : \ q t r 4 '

Finally, for q1 and q2 fixed, we can isolate 93; i 'e',

oz(q) : qzfz,z(qt, qz) 't 92l(qr' q2)

wi th /2,3(91 ,qz) :6qrqz * 4qz; gz,s(qt ,qz) : 3cnqz l - 5qr + 5qz + + '

, r ( q ) : qx f ; t k *k ) + . f ; 2@+k) .

REMARKS 14.3.8 (Transformation): Further motivation for the

study of muitilinear uncertainty structures is provided by the lemma

below. we see that a rather general class of pr-oblems v,'ith poly-

nomic uncertainty structure and polytopic uncertainty bounds can

be transformed into problems with multilinear uncertainty structure

and poil'topic uncertainty bourrds'

l ; ,

ilrl : rt : :ii,,r .

ExERcTSE L4.3.5 (Matrix withpose that A(q) is an uncertain nin the form

A(q ) :

where each Aa is a fixed rL x rLFurLherrnore, assume thaL if i1 I

Independent Uncertainties): Sup-x n matrix which can be expressed

tf ,4,o,.L

rnatrix having one nonzero entry.'i2, therr the nonzero entries of Aa,

14.3 / Mulrilinear and Polynonric Uncertainry 241

LEI\4rr4A L4.3.9 (Sideris and Sanchez Per'a (198g)): Consider thefarn i , ly of polynomialsP: {p( . ,q) : q e Q} u i thp(s,q) hau, ing poly-nom'ic uncertainty structure and uncertainty bounding set Q uhi,chis a polytope. Then there etists a second farnily of polynomialsP : {p( . ,q) , 4 e Q} such that f i (s ,Q) t tas rnul t i l inear uncer ta i ,n tystructure, Q is o, polytope and

D - D

PRoor': Let qf" denote the highest power of q6 rvhich appears in thecoefficient functions. Then we make the substitution

k

nf _ ffotij : r

in p(s, q) where qtJ,dt,z,. . .,4t,t", are new variables which comprise thenew set of uncertain parameters

4 : ( q t l , Q t , 2 , . . . , Q r , h , Q 2 , r , Q 2 , 2 , . . . , e 2 , k 2 , . . . , e l , n r )

that replaces the or ig inal q e Rt in p(s,q) . Now, we take p(s,g)to be the new uncertain polynornial obtained via the substitutionsabove. Notice that d € RN. where

r u : f k ;i :o

and, rr^ oreover, p(s, q) has a multilinear uncertainty structure.To complete the proof, we now define the uncertainty bounding

set Q for f. Indeed, for i I !.,\et

and take

Now, with

Q r : { q c R f f i q i , L : e t , z : " ' : q t , x u }

Q o : { q € R N : ( g r , r , q 2 , r , . . . , q t , ) e Q } .

242 Ch. i4 / Mulrilinear Uncertainry Srmcrures

intersection of a polytope with a finite collection of l inear varietiesis a polytope. E

ExAMpLE 14-3.10 (From Polynomic to jVlult i l inear) : To i l lustr-atethe transformation associated with the lemma above, we begin withthe uncertain polynomial

p(s,q) : s3 + (eqi + q?qz + qrq2 +3qr + t0)s2+ (+q? + qZ + 15)s * (6qtqz -t. LT)

'witlr nncertainty bounds -1 ( gr ( 1 and -2 < q, ( 2. We now"expand" the uncerta^inty space by defining new variables accordi.gto tire substitutions qi :- fu|zds and q] - qqfts.This leads to a newfamily of polynomtals P described by q e R5, uncertain polynomialwith multilinear uncertainty structure given by

p(s,q) : s3 * (34r4rqs r qtLzsa * qf ia* 3qr * 10)s2

+ @qQz * qaqs + 15)s + (6fuqa + 77)

and polytopic uncertainty bounding set Q described by -1 ( d, a 1,d t : d z : d s , - 2 < q q < 2 a n d Q a : 4 r .

1-4.4 Interval Matrix Family

Robust stability analysis in a state space setting provides strongmotivation for this section. we concentrate on an zz x n uncertainmatrix A(q) with independent uncertainty structure (in the senseof matrices) and uncertainty bounding set Q which is a box; theassociated characteristic polynomial is

p ( s , q ) : d e t ( s I - a k ) ) .

DEFrNrrrorv 14.4.1 (Interval Matrix Family) : A family of n x nmatrices A: {A(q) : q e Q} is said to be an interual matrir fam.i.Igif Q is a box, the entries a;1 (q) depend continuously on q and eachcomponent qt of q enters into only one entry aalk) of A(q).

NoifATroN 14.4.2 (Lumping and Extreme lvlatrices): Analogousto tire study of interval polynomials in Chapter 5, we can Iumpuncertaintjes within any entry of A(q). ft is convenient to view qas a vector in a euclidean space of dimensionn2 or less and create a

r r l

lii,{iiil

t^ /-\ .1L { : I l e l i l

; - n

we obtain the desired family P : {p(., q) , q e 9}. SV construction,it follows that P : P. Furthermore, since Oo-is a polybope andeach Q6 is a linear variety, it fotlows that Q is a polytope; i.e., the

I4.4 / Interval Matrix Family 243

col'respondence betrveen components of q and entries aei(q) of A(q);i.e., if aij(S) is nonconstant with respect to q, we lvrite

aij (q) : qij

and consider uncertainty bounds

q i < q i i < q + -

Analogous to the case of interval polynornials, we adopt a shorthandnotation: We use a rlatrix whose entries are intervals to describe .4.Fol example, i f ,n : 2, we ur i te

In this context, we call A(q) an interual matrir. Since the uncertaintybounding set Q is a box, we can view the family of matrices .4 asa box in ttre euclidean space of at most dimension n2. The z-thextrerne point q' of Q induces art er.treme mo.trir A(q') lor A.

ExAMeLE 7-4.4.3 (Some Entries Fixed): Note that the setup abovepermits a subset of the entries of A(cl) to be fixed. For example, wecan considel the interval matrix

r - - tI 1 t O a l I

i / \ I t L ' t o J IA \ Q ) : 1 . . - . ^ |

L [ 4 ' s ] 6 I

w i t h q : ( q n , q z z ) e P t 2 .

DEFTNT:uor{ 74.4.4 (Robust Z-Stabil ity): If D C C is open, thefamily of n x n matrices A: {a@) | S € Q} is said to be robustlg D-stable if the associated polynomial family induced via the relationship

p ( s , q ) : d e t ( s I - e @ ) )

is robustly D-stable. Analogous to the case of polynomials, the termrobust stability is reserved for the case when D is the strict left halfplane and the term robust Schur stabilitu is reserved for the casewhen D is the interior of the unit disc.

ExERcTsE 1-4.4,5 (\,Iult i l inear Uncertainty Structure): Specializethe result of Exercise 14.3.5 to the interval rnatrix case; i.e., argue

244 Ch. 14 ,/ Muirilinear Uncertainry Sructures

that the uncertain characteristic poiynomials associated with an in-

terval rnatrix family has a muitilinear uncertainty structure'

ExERCTsE 14.4.6 (Homogeneity Property) : If p(s,q) is the un-

certain characteristic polynomial associated with an n x n intervai

matrix family, show that each (multilinear) product of the qiJ has at

most degree n. For example, in the 3 x 3 case, a fourth order term

such as qrTqnq2zqn cannot arise in the coefficients because this term

involves a product of four eti. For the case when all entries of A(q)

are uncertain, argue that each term entering the coefifrcients involves

exactly n uncertainties forming a multilinear product'

L4.5 Lack of Extrerne Point and Edge Results

In view of the emphasis on extreme point and edge results in the

preceding chapters, it is natural to ask whether similar results are

available for more general multiiinear uncertainty stluctules. We see

below that the answer is no. In order to obtain such lesults, rather

strong additional conclitions must be imposed on the multilinearity-

EXERCTSE L4.5.1' (Lack of Extreme Point and Edge Results): We

consider the farnily of polynomials of Barmish, Fr.r and saleh (1988)

described by

p(s, q) : sa l - (qt + qz -F 2.56)s3

t (qt qz - t 2 .06q1 - l 7 .56rq2 * 2 '877)s '

- l - ( 1 .06q rqz * 4 .841q1* r . 56 rq2 * 3 .164 )s

+ (4.032q1q2 - t 3 .773q1* 1.985q2 + 1 '853)

TLEMARKS 14.5-2 (The Realizability Issue) : The next issue which.rve acldress is whether examples of the sort above can be ruled out

via imposition of a generating mechanisrn for p(s,q)' For exarnple,

it is of interest to know whether p(s,q) is realizable as the character-

istic polynomial of some state space system. In the exercise below,

we see that this is indeed the case. By starting with a lathel sirnpie

l - , - L r | , ' l

I Lqr r , q i r ) Lq tz ,q iz l IA \ q ) : | |

Llqit,qtl lqiz,q[z] l

I4.5 / l-ack of Extreme Point and Edge Results 245

state space descl ip t ion n( t ) : A(q)r ( t ) , the character is t ic polyno-mial p(s, g) : det(sI - l(q)) turns out to be the "nasty" polynomialn( .c o\ in Evele isc 14.51 above.( \ " i a )

ExEFLCTsE 14-5.3 (Realization by an Interval Matrix): Verify thatthe characteristic polynomial p(s,q) for the intervaL matrix

f - ] 5 - o 5 l - 1 ' 0 6 - 0 . 0 6 0L r . u ) " . " 1

-0 .25 -0 .03 1 .0 0 .5A ( - \ -

0 . 2 5

0

-4 .0 -1 .03 0

0 . 5 o [ - 4 . 0 , - 1 . 0 ]

is the same as the one given in Exercise 14.5.1 above. Hence, evenunder the strengthened hypothesis that the multilinear uncertaintystructure is 'induced by an interval matrix family, we still cannotguarantee robust stability by restricting our a.ttention to extremen n i n f q n r o d o o q

REMARKS L4.5.4 (Lower Order Polynomials): Since the two ex-ercises above involve fourth order polynomials, it is natural to askwhether some sort of extreme point or edge result is possible forthe lower order cases. Via the pair of exercises below, we see thatan extrerne point result emerges for rt : 2. In preparation for thesecond order analysis, we provide a lemma which is fundamental tomathemaLical programming.

I-EI\,II\zA L4.5.5 (Ivlultilinear Function on a Box): Suppose Q is aborin Rt wi,th. set of ertreme po'ints {qi} and f , Q --- R is multi-linear. Then both tlte mari.mum and min'imum of f (q) are attainedat ertrerne points. That i,s,

rna-x /(q) : mqx f (q')q € Q " " i

andmi+ / (s) : min . f (q ' ) .q e Q " ' - ' i "

PRoor': We establish the maximization result and simply note that anearly identical proof can be used for minimization. Indeed, supposethat bounds for the components qt of q are given by q.i t q6 < qf andlet q € Q be arbitrarily selected. Assume that for sorre componentqx of q, the strict inequality q; < qk < ef holds.

246 Ch. 14 / lr{ulri l inear lJncertainq'Structures

Tc complete the proof, it suffices to show that there is a vectorq* € Q wi th q l : q6 for i .+ k , qI : q i or qf r and f (q. ) > f (q) . I "this r.vay, we can eliminate all nonextrerrre components of g withoutdecreasing /(q) it order to establisir the existence of q*, we use afactorization as in Exercise L4.3.7: i.e., we express "f (q) u"

f (q) : q^f t (q**) + f r (q*k) ,

where ft(q+*) and f2(q*k) are multi l inear functions on Rl-l. Wenow consider three possibil i t ies. First, if ft(q+*) : 0, it does notmat ter i f rve take q i : q i or qt : O[ . In e i ther event , f (q-) : f (q) .The second possibil i ty is that h(q+*) ) 0. Norv, by taking qi : q[,i t f o l l o r vs t ha t / ( q . ) > f ( q ) . F ina l l y . i f f r f u+ r ' 1 < 0 .by bak ingqi : qL, we again conclude that / (q-) > / (S) H

nxERCTsE L4.5.6 (Extreme Point Result fol Robust Nonsingular-ity): Let A: {A(q) : q € Q} be an interval matrix family. Takingq' to be the i-th extreme point of Q, prove that .4 is robustly non-singuiar (that is, ,4(q) is nonsingular for all s € Q) if and only ifeach of the deterrninants A(q') has the same sign. Hint: ApplyLemma 14.5.5 to " f (S) : aet A(q) .

ExERcrsE 1.4.5.7 (Second Order Case): Consider the uncertainpo l ynomia l p ( s ,q ) : s2 +o t (q ) " +ao (q ) w i t h ae (q ) and a1 (q ) hav inga multilinear uncertainty structure.(a) Prove that the family of polynomials 2 : {p(',q) : S e Q} isrobustly stable if and only if the set of extremes {p(-, q')} is stable.H'int: Consider the resuit in Lemma I4.5.5 in conjunction with thefact that in the second order case, coefficient positivity is equivalentto stability.(b) Using the result in (a), prove that a 2 x 2 interval rnatrix isrobustly stable if and only if each member of the set of extremematrices {A(q")} is stable.

14.6 The Mapping Theorern

For uncertain polynomiais with multilinear uncertainty structures,robust stability analysis can be quite complicated. Horvever, rviththe aid of the Nlapping Theorem below, we can often establish ro-bust stability using a "special" overbounding family of polynomials.The power of the Xzlapping Theorern is derived from the fact that thisoverbounding family turns out to be the convex hull of the originalfamily. Nloreover, this convex hull family is seen to be a polytope

14.7 / Geometric Interpretation 247

of polynomials. Hence, it becomes possible to apply many of theresults covered in the last several chapters while recognizing the factthat u,e are working in a sufficiency context. In contrast to otheroverbounding methods, the fact that we are working with the con-vex hull lends credibility to the "tightness" of our approximation.After stating the theorem below, we move immediately to interpre-tation and illustrative examples. Tire proof of the theorem below isre legated to Sect ions 14.9 and 14.10.

TIlnoREM L4.6.L (The Mapping Theorem) : Suppose Q cRt is abor utith ertreme points {qi} and f t Q --- Rk is multilinear. Let

f ( A ) : { f ( q ) , q e e }

denote the range of f , Then i,t follows that

cona f (Q) : conu{f (q ' ) } .

14.7 Geornetric Interpretation

The geornetrl' r..o.trted with the Nlapping Theorem is illustratedirr Fignre 14.7.1 for k:2. Notice that we obtain the tightest pos-

f @')

\ conv / (Q)

Flcunn 14.7.1 Geometry Associated with the Mapping Theorem

sible polygon bounding the range set /(Q). In view of the NlappingTheorem, we can rule out a number of possible shapes fbr the range

/(Q). Roughly speaking, any /(Q) which "curves outward" rathertlran "inward" is not realizable. For example, since taking the con-vex hull of the f (qo) in Figure I4.7.2 does not yield the convex hullof f (Q), it follows that /(Q) cannot be the range of sbme multi l inearfunction on a box. In particular, the arc joining /(q1) and f (q2) i,inconsistent with the requirement of inward curvature.

There are also other inconsistencies in the figure. For example,since the straight line joining q2 and q3 defi.nes an edge of Q, it must

I

;I

248 Ch. 14,/ Mult i l inearUncertaintyStructures

f (qr )f (q2) f (q3)

f (q ' )

/ ( q8 )

FIcunp 14.7.2 A Geometry Which Is Not Realizable

be the case that /(Q) includes the straight l ine joining /(q2) and

f (q3); notice that this line is missing. A similar comment applies to

i t - t " pr i t . (q ' , q") , (qu , q") and (q1, q5 ) .

ExERCTsE J4.7.L (The Range of /(Q)): Suppose that Q is the unit

square in R2 and / (q) is the two-dimensional multilinear function

with components /1 (q) : qrq2 and fz(q) : qt* qz' Describe the

range / (Q) and sketch i t . Compare f (Q) and conv / (Q) '

REMART(s L4.7.2 (Coefficient Interpretation) : Suppose that p(s, q)

is an uncertain polynomial with coefEcient vector a(q) depending

multi l inearly on q. Then, if Q is a box with set of extremes {qi}, the

Mapping Theorem provides a simple description of the convex hull

o f the coef f ic ient se[ ; i .e . ,

conv a(Q) : "ot . t {o(q ' ) } '

14.8 Value Set InterPretation

The ideas in the preceding section have interesting and useful in-

terpretations in a value set context. Indeed, suppose Q c Rl is a

box with extreme points {qii and p(s,q) is an uncertain poivnomial

having a multilinear uncertainty structure. Then, given any z € C,

we consider the mapping f : Q --* R2 described by

(qn)

f @ ) f ( f )

q v - -+ (Re p(2 , q ) , Im p(2 , q ) ) .

14.8 ,/ Value Set Interpretation 249

Since -R'e p(z,q) and Irn p(z,q) are multi l inear rvith respect to q, theMapping Theorem teils us that

conv p(2,8) : conv{p(r ,S ' ) } .

Having this convex hull description available, it is now easy tostate a suffi.cient condition for robust 2-stability. The lemma be-Iow is an immediate consequence of the Zero Exclusion Condition(Theorem 7.4.2) in conjunction with the Mapping Theorem above.

LEI\4MA L4.8.L (Robust D-Stabil ity Criterion): Cons,ider a fami,lyof polynomialsP: {p( . ,q) : q e Q} wi th inuar i ,ant degree, rnul t i l in-ear uncerta'inty structure and at least one D-stable member p(s,qo).In addition, asslrrne that Q 'is a bor with ertreme points {q'} and thedes'ired root location region D 'is open. Th,en P 'is robustlg D-stable'if the Zero Erclusion Condition

0 / conu{p(z,q") }

is sat'isfied for aII z € AD.

REI;IARKS 14.8.2 (Conservatism) : In vierv of the lemma above,we now provide a value set interpretation for the fact that the NIap-ping Theorem only leads to a sufficient condition for robust stability.Suppose Q C Rt is a box and the uncertain poiynomial p(s, q) hasa multilinear uncertainty structure. Then, in a robust 2-stabilityanalysis, it is possible that at some frequency u* > 0, we have0 € conv p( ju. t * ,Q) but 0 / p( j r . ,Q) . This s i tuat ion is depictedin Fip ' r r re . l .1 .8.1 Not ice that we nrust deem the robust s la, f i l i tv te<finconclusive. Said another way, if we are applying the polytope sta-bility theory to the overbounding family obtained via the NlappingTheorem, we do not know if stability is lost when z : 0 penetratesthe set conv p(ja, Q). Roughly speaking, the true value set p(ja,Q)is unobservable through the eyes of the lvlapping Theorem. In theexarrrple below, we illustrate this undesirable phenomenon.

ExAN4eLE 14.8.3 (Nasty Value Set): For the family of polynomialsdescribed in Exercise 74.5.I, we begin with the set of extreme points

9 1 : ( 0 , 0 ) , q ' : ( 0 , 3 ) , q 3 : ( 1 , 0 ) a n d q 4 : ( 1 , 3 ) a n d g e n e r a f e

p (s ,q r ) : . s4 * 2 .56s3 + 2 .B7 rsz * 3 .164s * 1 .g53 ;

p(" , q2) : s4 * 5.56s3 t 7 .544s2 * 7.847s * 7.808;

p(s, q3) : s4 - i - 3 .56s3 t - 5 .931s2 * 8.065s * 5.885;

p(s,q4) : s4 t 6 .56s3 - t r2.674s2 * 15.868s + 23.677.

250 Ch. 14 ,/ Mr.rltilinear Uncertainty Structures

FrcuRE 14.8.1 Inconclusive Robust Stabil itv Test

Now, motivated by the remarks above, r,ve examine both the ,,true',

value set p(j-,Q) and its convex hull conv{pt(j.)} at the frequencyc. . , : 1 .5; see F ' igure I4.8.2. Not ice that z : 0 l ies ins ide conv p( jw,e)but does not i ie inside p(j.,Q) itself. In other rvords, the nasty vaiueset geornetry discussed in Remarks 14.8.2 above is realizable.

I4.9 Machinery for Proof of the Mapping Ttreorern

rn this section, we develop some basic machinery for the pr:oof of theNfapping Theorem. To this end, we provicie a quick primer on somebasics from convex analysis; the reader is also referred to Sectiori g.3rvhere more elernentary concepts from con\/ex analysis are coverecl.

If X C R'. tlien the sultport function h, : Rn ---+ R on X isdefined by

l t( f i : sup srr,u e 7 \

where z and g are viewed as column vectors above and yT denotes thetranspose of 3r. The first point to note is that the support functronon X can be identified with hyper-planes supporting X. fn otherwords, for a fixed y € P.n, X is contained in the closed halfspaceciescribed bY

aj r < h(u).

This situation is depicted in Figure 74.9.7. The second point to note

p ( j u . , Q )

fm.D

A

-6

14.9 / Machinery for Proof of the Mapping Theorem 251

252 Ch. 14,/ Mult i l inearUncertainryStruc[ures

I

I r r ' / \r A ' r : n \ A )

ar r < h(v) ar" > h(a)

Flcunp 14.9.1 Supporting Hyperplane for X

This fact is basic to the theory of linear programming. It is really just

another way of saying that a linear function on a polytope achievesits maximum (and minimum) at an extreme point. The converseis a lso t rue. That is , i f { r1, n2, . . . ,2P} is a set of points for whichequality holds in the equation above for all A € H:n, then it foliowsthat X is a polybope wi th {* r , r2, . . . ,nP} as a generat ing set . Inthe language of convex analysis, we say that the support functionis fi.nitely generated. In the proof of the Mapping Theorem in thesection to follow, we make use of the lemma below.

LEMMA L4.9-L (Equal Convex HuIIs): If X1 and' X2 are two closed,

conu Xy : conu X2.

L4.Lo Proof of the Mapping Theorern

o o l- 5 - + - J - L - I

Re

FIGURE 14.8-2 Value Set and Its Convex Hull for Example 14-8'3

is that if X is closed, then the convex hull of X, denoted conv X,

is obtained by intersecting aII possible closed halfspaces as indicated

above with E ranging over FL'; that is,

convX : n T lu ,v€R'

rvhereH o : { r : A T r 1 n @ ) } .

The formation of conv X in this manner is depicted in Figure 14.9.2.

On the left side of Figure L4.g-2, an approximation to conv X is

shown using hnrte.tY manY supPor

side of Figure 14.9.2, the true convex huII is shown'

A funclamental difference between general convex sets and poly-

topes is the nature of their support functions. For the poiytope

X : conv { r l , f r 2 , . . . , rP } ,

it is wetl known that for any given A €. P.n, we can compute the

support function for X by using only the extreme points of X. Since

the generat ing set { r r , r2, . . . , rp} conta ins aI I ext reme points, i t

follows thatl,(il : '1'9* YT 'i '

x \ p

the finite point setF : {f (q')}

14.II / Conclusion 253

FIcuRE 14.9.2 Finite Approximation to conv X and the Ttue conv X

254 Cl-r. l4 ,/ Multilinear Uncertainty Structures

classes of pollmomial families ior which the Zera Exclusion Condition0 ( conv p(ja,Q) is both necessary and suffi.cient for robust stabil-ity? Another direction for further work involves algorithm develop-ment. Essentially, one reformulates the robust stability problem in amathematical programming context and proceeds toward a solutionvia some iteration process which exploits the lVlapping Theorem; forfurther discussion see Sections 1.7. 1.8. 14.1 and the notes to follow.

have the sane support function. To this end, let A e R' be arbitrar-ily fi.xed and let h1(9) and hrfu) denote the support functions on

/(Q) and .F, respectively. Now, by definition of the support function,

we know thathr(v) : sup vr,f (s).

- q € Q

Since 3r € Rk is fixed, we can view the computation of hy(y) as aproblem of maximizing the mu,ltilinear function

J (q ) : a r f @)

on the box Q. Now, applying Lemma I4.5.5, we conclude that

ur f G.i) - h7(f i . 5lt' 1 (v) : max'L

L4.LL Conclusion

As shown in this chapter, robust stability analysis based on the NIap-ping Theorem leads to sufficient but not necessary conditions. How-

ever, the fact that we use a "tight" overbound (the convex hull of

the original family) raises an interesting question: Are there large

maxL

. Nores ard Related Literarure ZEb

Notes and Related Literature

NFLL 14.1 The issue of overbounding arises in many places in the robustness

literature- For example, in the line of research on robustness margins (see

Doyle (1982) and Safonov (1982)), the quality of the bornds obtained for the

structured singular value are of paramount importance. A similar issue arises

in fI- theory; e.g., if one wants to use the Riccati equations in Doyle, Glovbr,

Khargoneka.r and FYancis (1989) for systems with structured real uncertainty,

one approach involves overbounding of real parametric uncertainty by discs in

the complex plane.

NFLL 14.2 In the recent robustness literature, a line of research popularizing the

mathernatical programming approach begins with Kiendl (1985), Kiendl (1987),

de Gaston and Safonor ' (1988), Sider is and Sanchez Pena (1989), Chang and

Ekdal (1989), Ossadnik and Kiendl (1990) and Vic ino, Tesi and lv l i lanese (1990).

In the fi.rst six of these papers, various schemes are proposed for partitioning

of the upcertainty bounding set Q. Subsequently, stability (or instability) is

estdblished on local subdomajns and the issue of covering Q is addressed. For

a nice exposition of the branch and bound techniques associated with srrch a,n

approaiil, sbe the textbook by Boyd and Barratt (1990).

NFLL 14-3 In view of the transforrnation from polynomic to multilinear uncer-

tainty structures in Section 14 3, an important question arises: Does the Nlapping

Theorem hold if Q is a polytope instead of a box? Since the answer to this ques-

tion is no, a future breakthrough for multilinear uncertainty structures may have

lirnited applicability to more general polynomic uncertainty structures.

NFLL 14.4 A nurnber of authors obtain results for special cases of the robust sta-

bility problem by imposing stronger assumptions on the multilinear uncertainty

structure. For example, in l{haritonov (1979), no uncertainty entering even order

coeffrcients is allowed to enter into odd order coefEcients and vice versa. Sub-

sequently, an extreme point result for robust stability is attained; see Panier,

Fan and Tits (1989) for further extensions. Slight generalizations of this type of

el.en-odd decoupling result are given in papers by Djaferis (1988) and Djaferis

and Hol lot (1989b).

NRL 14.5 Another special class of the multilinbar uncerta.inty structures is stud-

ied in Barmish and Shi (1990) The uncertain polvnorniai p(s, q) is assumed to

lrave the form XY* [/V where X, Y , U and V, correspond to interval polynomial

farnilies. Subsequently, it is shorvn that tire satisfaction of a "covering condition"

is both necessary and sufficient for robust stabilit-v.

NRL 14.6 In the interest ing Ph.D. d issertat ion of Zong (1990), an " inner inter-

section exclusion" condition for robust stability is given in the context of multi-

256 Ch. 14 / Mukilinear Uncertainty Structures

Iinear uncertainty structures The use of this condition is roughly analogous to

the way the Mapping Theorem is exploited to establish a suffrcient condition for

robust stability- The dissertation also includes interesting results for the 3 x 3

inl,erval matrix Problem.

NRL 14.7 In Wei and Yedavalli (1989), nonlinear uncertainty structures are

treated by transforming real and imaginary parts of p(s,q). However, no system-

atic method for constructing the desired transformation is given'

NRL 14.8 In the control literature, the Mapping Theorern appears at least as

ear ly as 1963 in the book by Zadeh and Desoer (1963). The paper by saeki (1986)

is credited with revival of these ideas. since the Mapping Theorem leads to con-

servative results, the papers by de Gaston and Safonov (1988) and Sideris and

sanchez Pe[a (1989) are relevant In both cases) a domain-splitting algorithm

for robust stability is proposed. Motivated by computational inefliciencies asso-

ciated with the frequency sweep used in these two papers, sideris and sanchez

Pena (1989) provide a diferent algorithm which is based on the Routh table.

NRL 14.9 With the goal of eliminating conservatism associated with application

of the lvlapping Theorem, a number of authors have concentrated on the charac-

terization of polynomiais p(s, q) having rnultiiinear uncertainty structure, poly-

topic unccrtainty bound Q and satisfying the following condition: At each fre-

quency cu ) 0, condi t io l -p( ja,Q) : "ont ,

p( ju,Q) 'For such cases) the value set

is a poly.tope and a complete solution of the robust stability probiem is straight-

forward. This line of research begins with the paper by Hollot and xu (1989)

r,r,here a conjecture is given involving the image of a polytope under a multilinear

function; see also Polyak (1992) and Tsing and Tits (1992) for further work in

this direction.

iil

[,ii:r;

l:t

Part IV

The Spherical Theory

Chapter 15

Spherical Polynornial Farnilies

Synopsrs

In this chapLer, Tlte 12 analogue of Kharitouov's problem isconsidered. We work with a familv of oolvnomials P withindependent uncefiainty structure and uncertainty boundingset Q which .is a sp/:ere. The highlight of the chapter is theSoh-Berger-Dabke Theorem. This theorem provides a simpJemethod for robust stability testing using a frequency dependentscalar function.

lS.L fnt roduct ion

To motivate the technical exposition in this chapter, we begin witha family of polynomials P described by

n/ \ S - /p(s ,q ) : l a r (o )s '

and q e Q. We pose two questions which are important to answerwhen working in an applications context: First, what uncertaintystructure is being assumed for the uncertain coefficient functionsai(d? This question has already occupied much of our attentionin the earlier chapters. Second, what type of uncertainty boundingset Q is being assumed? Given that most results in the robustness

258

15.2 / BoxesVersusSpheres 259

literature involve sets Q which are either boxes or spheres, somecomments are in order.

The point of view in this text is that in most applications, it

is not worth agonizing whether to use a box or a spherical repre-

sentation for Q. Imprecision in the engineering problem formulationenables us to use either model. We elaborate on this point. For iI-lustrative purposes, suppose that we are dealing with two uncertainpararneters q1 and Qz (mass and coefficient of friction, respectively)and we ask the engineer what bounds should be assumed. A typ-ical answer might be: The coefficient of friction qr can experiencevariations up to 20 or 30 percent about its nomina.l value qf - 0.4,and the mass 92 can vary up to 10 or 20 percent about its nominalvahe q$: 56.8 kilograms. To further embell ish this scenario, sup-pose that we ask the engineer whether we should assume a sphere ora box for the uncertain parameter vector g. The answer rve receiveis: Of course, we must assume a box because the mass and frictionvariations are independent; it does not make sense to use spheres.

L5.2 Boxes Versus Spheres

Continuing with the hypothetical scenario above, suppose that we go

into our rnatherrratical toolbox and find that the only theoretical tool

available for the problem at hand requires an assumption that Q is a

sphere. Or more precisely, application of the availabie tool requires

Q to be an ellipsoid rvhich we view as a sphere using an appropriatelyweighted norm. Should we ignore the engineer's advice and applythe available spherical theory?

To decide whether to ignore the engineer, we sketch the un-certainty bounding set Q as shown in Figure 15.2.1. In view ofthe irnprecise description of the uncertainty bounds, we know thatthe "true" bounding set Q lies between Q*i, and Q*or; that is,

Q*n C Q C Q^or. After a few mornents' reflection, we concludethat we can take advantage of this latitude in the description of Q.By appropriate choice of weights w1 ) 0 and w2 ) 0 and a radiusof uncertainty r ) 0, we can approximately represent Q via theinequality

*?(qr - s?)2 + *22@z - s|)z s 12

and then go ahead and apply our spherically based theory to theproblem at hand. In conclusion, an engineer's insistence on the useof boxes versus spheres is not really justifiable when the uncertaintybounds are not t ightly specified.

260 Ch. 15 ,/ Spherical Pol1'nomial Families

FIGURE 15.2.1 Q^o, and Q^;n

The choice of sphere versus box shoulcl be dictated by the avail-

able toois for solution of the problem at hand. This point serves as

our launching point for this chapter and the next. By wolking with

a spher-ical uncertainty structure, we obtain some polvelful new tools

to facilitate lobustness analysis.

15.3 Spherical Polynornial Farnil ies

In this section, rve provide the formai definit ion of a spherical poly-

nomial fantily. If we associate an interval pol1'no*tut family with

ilhe {* norm (Q is a box), then it is natural to think of the theory

in this chapter as the /2 analogue of the theory in Chapter 5' To

this end, we now provide a stightly more genelal definition of t]rre 12

norm than that given in Section 2'4.

DEFrNrrroN 15.3.1 (Weighted /2 Norm and Eltipsoid): Given a

k x k positive-definite symmetric rnatrix I4z and z € RE, line weighted

eucl'idean norm of z is given bY

l l " l l z , ,v : ( r rw,7h '

15.3 / Spherical Polynomial Families 261

r ,9+

r ? +

FIGURE 15.3.1 Ell ipsoidal Geometry for Weighted Euclidean Norm

We call trV the weighting rnatrir. Furthermore, given r ) 0 andz0 e Rft, we define an ell'ipso'idir_Ftk centered ateo by the inequality

( n - r o ) r w ( r - r o ) < r 2 .

F . n r r i r r q l o n f l - rs r v r r u f J I

l l r - r o l l 2 s r y < r .

REI\rrARr{s 15.3.2 (Weighted Euclidean Norms) : The standard def-initions above can be easily visualized. For example, if k : 2 andVIz : diag{*?,*Z}, we obtain the ell ipsoid depicted in Figure 15.3.1.

DEFrNrrror.I 15.3.3 (Spherical Polynomiai Family): A family ofpolynomialsP : {p( . ,q) | S € Q} is sa id to be a spher ica l polynomzal

farniLy if p(s, q) has an independent uncerta,inty structure and Q isan ell ipsoid.

DEFrNrrroN 15.3.4 (Spherical Plant tramily) : A set of rationalfunct ions p : {P(s,e, r ) : e e Q;r e R} is ca l led a spherzcal p lant

farnily if P(s, q,r) can be expressed as the quotient of uncertainpolynomials,

n l - , . l / ( " , 9 )/ \ D , 9 , ' ) - D G J ) ,

262 Ch. 15 ,z Sphericai Poll'nomial Families

wi th Q and .R being e l l ipsoids and N(t , q) and D(s, r ) having inde-pendent uncertainty structures.

REMARKS 15.3.5 (Alternative Definit ion): In some cases (for ex-

ample, see the latter part of Chapter 16), it is convenient to definea spherical plant family using a joint bound for (q, r) rather thanindividual bounds for q and r. If V\ and W2 are square weightingmatrices with dimension n1 : dim q and ??z: dim r, respectivelylwe can work with the weishted norm

l l l ^ - \ l t - - . -i l \ v . , . / i l 1 1 ' -

and bounding set

( Q , R ) : { ( s , r ) ' l l ( q , r ) l l r . r , S 1 }

when describing the unit sphere.

ExAMbLE 15.3.6 (Centering) : We consider a spherical polynomialfamily 2 described by

p (s , s ) : ( 4 + q3 )s3 + (2 + qz )s2 + (1 + s1 )s + (0 .5 + qo )

with ell ipsoidal uncertainty bound llqllr,* ( 1 and weiglrt ing matrix

i 4z : d i ag {2 ,5 ,3 ,1 } f o r Q : (eo ,Qr ,e2 ,q3 ) . No t i ce t ha t by cen te r i ngthis family on the vector 4o : (0.5, I,2,4), we obtain an equivalentdescription of 2. That is, we can wor-k with the uncertain polynomialp(t,q) : (lo I qts * qzs2 * 4st3 and uncertainty bounding set Qdescr ibed Uy l l4 - 40l lz ,* < t .

REMARKS L5.3.7 (Representations) : In view of the fact that aspherical polynomial family can be centered, we often begin rvith

p ( s , q ) : i n r 'i : 0

and uncertainty bound

l l q - q o l l r , * < ,

with I4l being a positive-defi.nite s)rmmetric matrix, g0 repr-esentingLhe nominul and r ) 0 being tl ie radius of uncertainty. An equiva-lent representation is obtained by extracting the nominal polynornial

+ ll"l l7r,r

15.4 / Lumping 263

ps(s) : p(s,qo) and using the representation

p ( s , q ) : p o ( s ) + i o , r oi :o

with l lqllz,w < r. With this representation, we can adopt the pointof vierv that the uncertainty bounding set Q is centered at zet:o.

ExAMPLE 15.3.8 (Not All Coefficients Uncertain): Analogous tothe case of interval polynomials, in some cases we might only havea strict subset of the coemcients being uncertain. For example, con-s ider p(s, e) : s3 + (2 + qz)s2 + 4s I (3 * qo) .

15.4 Lurnping

To further extend the analogy between spherical and interval poly-nornial families, we now consider the issue of lumping; recail Exer-cise 5.3.5. For the case of spherical polynornial families, however,the lumping process is somewhat rnore subtle.

ExAMPLE 15.4.1 (lvlotivation): Consider the uncertain po}ynomial

p ( s , q ) : ( 3 + q z ) s 2 + ( 4 + 3 q t - 5 q : ) s + ( 2 + 2 q o + 6 q s )

with uncertainty bound llqllz < 2. Our intuit ion tells us that variousuncertainties can be combined; i.e., Iump q6 and qa together, lump q1and q3 together. To carry out this lurnping process precisely, we needsome machinery. A standard result from matrix algebra is stated asa lemma below.

LEMMA L5.4.2 (ldinimum Norm): Consider R" with l l . l l2,s andIet A be a fi,retl real n x m matrir hauing rank n. Th.en, giue'n anyb e R'", the minirnum norrn solution of

i.s giuen by

A r : b

r^,1 N _ lv-r AT (AW-t 4r yt 6.

NorATroN 15.4.3 (Subvectors and lvlatrices): Given the fact thatwe want to allow for the possibility that only a subset of the co-effi.cients of p(s, q) are uncertain, it is convenient to introduce thenot ion of subuectors. Indeed, suppose , : (n0, f r1, . . . , zr ) and index

264 Ch. 15 ,/ Spherical Polynomiai Families

se t 1C {0 , 1 ,2 , . . . i TL_ l , n } i s nonempty ' Then we l e t z1 deno te t he

subvector of r obtained by retaining components nt fot i € 1 and

deleting components na fot i / I. To avoid permutations among the

" .*p" i " " t , o f

" t , i t is a lways assumed that i f I : {h , iz , " ' ' i ' } '

then i1 < xz < "' f i ,-t < 2". l lhis convention makes 11 uniquely

def ined. To i l lust rat e, i f r : ( ro, rL,Tz,rs , ra) and 1 : {2 ' 4} ' then

we obta in aI : ( r2,na) .

REMARKS L5'4.4 (Interpretation for Uncertain Poiynomials): We

now interpret the subvector notation above in terms of uncertain

fotyno-iutr. If p(s, q) : LT:ooo(q)tn,is an uncertain polynomial

i"tro." coeffici 'ent ,rrto, it ,,.tqt, then o'l(q) is the subvecto-r of a(q)

generated. Lsing the component. ol(q) with z € /' Note that for a

sphericai potynlmial family with q € Rl, the independent uncer-

tairrty structure enables us to express any subvector or(q) as

a I ( q ) : A t q + b I ,

where A7 \s amatrixhaving dimal(q) rows and' I columns and br is

a column vector having d'im aI (q) entries' Finally, if we aliow some

of the coeffi.cients ar(qJ to be fixed, say aa(q) : a,i for i f I, it ts

often convenient to write

p(s, q) : I a ; (q) s i + l a ;s ' .ie l ie l

In this way, it is easy to emphasize which coeffrcients are fi.xed and

which coeffi.cients are uncertain.

ExAMPLE 15.4.5 (Subvectors and Matrix Representation): For the

uncertain poll'nomiai

p (s, q) : (2 + q4) s4 +( 3 +qs * 2 qs ) s3 + (2 - q2 a4q6 ) s2 + ( 6+ 2 qr ) s * ( qo + 4)

wi th I : {0 ,2,4} , we have

ii.l

i.'jril

iiiiiilt i l

il1iiiil

a r ( q ) :

15.4 /

Furthermore, we can write o1(q) - Ate l b1, u,hele

1 0 0 0 0 0 0

0 0 - 1 0 0 0 4

0 0 1 0 0

Lumping 265

l '

i

t :i , il i! ' l

!,'i, : l

r ii : ll !

i il l

iii ii ii'ii : ii i

l 1t i it ii i jir l1 : li.r1

t i! i! , 1' ) i

266 Ch. 15 / Spherical Polgromial Families

Adding A1q0 on each side above, it follows that q € aI (Q) if and onlyif there exists 4 e Rl such that ll|llr,w ( r and ArQ : S-So. Now, animportant observation to make is that the independent uncertaintystructure (each qi enters only one coefficient) guarantees that eachcolumn of ,47 has exactly one nonzero entrl'. Hence, ,47 has ranlt

equal to n1, \ts number of rows. Therefore, using Lemma 75.4.2,an appropriate f exists if and only if the minimum norm solution ofthe equation AtQ : A - g0 has norm less than or equal to r. llhisminimum norm solution is given by

Q^,rN _ W-Le! 9,r147-t477-t (q -- q0),Tr{EoF-EM t'5.4.6 (Lumping for a Spherical Polynornial Family):Suppose thatP : {p( . ,q) , q e Q} i t a spherzcal polgnom, ia l fami lydescri,bed by

p ( s , q ) : I " o ( s ) " i + I o n ' i

and

Q : {q€ Rr : I lq - qol lr,w 3 r}-

[]sing the representation

a I G ) : A r q - t b

for the n1-di,rnens,ional subuector of a(q) whose rows are not constantwttl t respect to q,

D ( s . q \ : f o ' " t + f a , s te ^ u * ,

a n . r l Q € Q , w h , e r e

Q : {q€ R" r , l l 4 - l o l l z , v < r } ,

q o : A r r l L ' + b I

and,vv : (Ar6r-t aryt .

Thrcn it follows thatn ^

P p n o n . T ) e r r n t i n o i h n , ' a n o a a l n l ( n \ l ^ . ,

" r ( Q ) : { A r q t t t : q € Q } ,iL su fHces to show tha t a1(q : Q. Indeec i , nor ice tha t q e

" I1gS i f

and only if thele exists some q € Q such Lhat

The proof is concluded by observing that the condition

llqno*ll, < ,

is equivalent to the condition S e Q. W

oxAMeLE L5.4.7 (Lumping): Returning to Example 15.4.1, forthe uncertain polynorniai

p(s,q): (3 + qz)s" + (4 + 3qt - 5qs)s + (2 + Zqo -t 6qt)

uncertainties from five to three. In accordance with Theorern 15.4.6,we take index set I : {0, 1,2}, center q0 : 0, identity weightingrnatrix IM : I and radius r : 2. We use the representation

ar (s) -

Qo

v t

qJ

Q+

- A r q l b r

; d l

and compute W : diag{1/40,7/34,L}. Now, we calculate center

f : Arqo lb :_12 4 3 ] " and obta in thelumped spher ica l fami lyof polynomials P described by p(t, d = dzs2 * 4rs * {6 and

*,* - 42 + fr@

- +)' + Gz - B)2 s +.

A r q t b : q .

15.5 / The Soh-Berger-Dabke l'lteorern 267

ExERCTSE 15.4.8 (Lumping): Appty Theorem 15.4.6 to the spher-ical family of polynomials described by

p(s, q) : ss + (4* q+- 3 qs) sa + (Z+qs ) s3 + (6+ Sq2 - J q6) s2+6s + ( go * 6qr )

and l l q - so l l 2 ,w < 10 . Take nomina l q0 - ( 1 ,1 ,1 ,0 ,0 ,0 ,1 ) andweight ing matr ix I rZ : d iag{1,2, 6, ! ,20, 5, 10} .

R,EMARKs 15.4.9 (Lumping for a Spherical Plant Family): Wenote that Theorem 15.4.6 is readily adapted to handle a sphericalp lant farn i ly P : {P(- ,q, r ) : q e Q;r € R}. Indeed, i f we expressP(",q,r) as the quotient of uncertain polynomials, we simply appiyTheorern 75.4.6 to the nurnerator and denominator separately.

15.5 The Soh-Berger-Dabke Theorern

The focal point of this section is the Soh-Berger-Dabke Theorem.This theorem provides a complete solution for the /2 analogue ofKharitonov's problem. We consider the robust stability problem fora spherical family of polynornials and generate a special scalar func-tion of frequency. This function is used to compute the robustnessmargin-the largest uncertainty bound r for which robust stability isguaranteed. For simplicity of exposition, we state the theorem belowwith all coefficients equally weighted. In the exercises following thetheorern, we indicate modifications in the theory which are neededto handie the cases where the description of Q includes a weightingmatrix W or only a subset of the coefficients are uncertain. Theproof of the theorem is relegated to Sections 15.6 and 15.7,

THEon-rM l-5.5.1 (Soh, Berger and Dabke (1985)); Consider thespheri,cal family of polynorni,als P wi.th i,nuariant degree n ) 7 de-scribed by

p ( s , q ) : p o ( s ) + i a , r oi :0

wi,th nominaln

p o ( s ) : D _ q t n?:U

and uncertai,ntE bounding set l lSll2 < r. For u ) O, let

Gsn o(.) : lRe?q(i '-))2 * lL" po(i?)12 .

Z- u)-- L r'oi euen t odd,

268 Ch. 15 / Spherical Polynomial Families

Then .P is robustly stable if and onIE i.f po(s) is stable, the zero fre-quencA condit ' ion

l o o l > "

is sat'isfi,ed and

for aII frequencies o > 0.

GsBD(a) > 12

R-EMART{s 1,5.5.2 (Robustness Margin): The Soh-Berger-Dabke

theorem suggests a simple method to compute a robustness margin.

I ndeed , i f we use the no ta t i o lQ r : { g e R ' * t ' l l s l l 2 < r } andP , ' :

{p(-,q) | a € Q,} to emphasize the dependence on the uncertainty

bound r ) 0, the quantity of interest is

Tmat: sup{r :P, has invariant degree and is robustly stable}.

Now, if we defi.ne

and

+ / \t t , t t :' m a z \ ' , /

r|o, : i \fo

,-""(r),

we ob ta in Tmar :m in { l a6 l , l on l , r * . o , }

as the robustness margin.

EXAMPLE 15.5.3 (Computation of r*or)i To compute a robustness

margin for the uncertain polynomial

p (s ,q ) : ( 1+ q3 )s3 + (1 .5 + qz )s2 + (1 .4+ q r ) s * ( 2 + qo ) ,

we consider nominal po(s) : s3 + 1.5s2 -t 1.4s -F 2 and generate

(2 - r.5a2.)2 (7.4a - ":3)2G s a o :

L + a 4 - - 2 + u s

From the plot of ,I.,(.) versus c,, ' > 0 in Figure 15'5'1' the minimum

vaiue is approximately r f io , = 0.0011. With oo:2 and a3:1, we

compu te l t na r : m in { l as l , l o - l , r | " ' } = m in {2 ,1 ,0 ' 0011 } : 0 ' 0011 '

3 . 2 5 u a - 8 . 8 c . . , 2 + 5 . 9 6L i _ a a

GsBn(u)

- l l \r"^or\u )

0.0022

0.002

0.0018

0.0016

0.0014

0.0012

15.5 / The Soh-Berger-Dabke Theorem 269

270 Ch.75 / Spherical Pol;T romial Famil ies

weiglr ted norm l lq l l2,r / with I4z : diag{*fr ,onl , . . . , \u7}, show thatthe Soh-Berger-Dabke Theorem remains valid with the zero fre-quency condition replaced by

lwoo.ol > r

and with modified testing function

0.0011 . 1 5 1 . 1 6 1 . 1 6 5

u)

FIGURE 15.5.1 Plot of ,*",(.) for Example 15.5.3

a( t ) : q ( t ) ,

where 11(l) is the transverse velocity, fr2(t) is the angular rate of theship's coordinate frame relative to its response frame, r3(l) is thedeviation distance on an axis perpendicular to the track and ca(f) isthe deviation angle.(a) Using a l inear feedback control u(t) : I{p1(t) * K34(t), deter-mine the set K of gain pairs (Kr, Ks) for which closed loop stabilityof the system is guaranteed.(b) Fix some 'ocentrally located" stabilizing gain pair (Ifr, I{s) e Kand let ps(s) denote the resulting closed loop poiynomial. With thesegains fixed, consider a spherical family of polynomials with nominalp6(s) and compuLe lhe associated robustness margin r-o, using Lheformula in Remarks 15.5.2 above.

ExER-crsE 15.5.5 (Weighted Norm) : For the case when n haq

G s a o ( r ) : lRe ps(iw)

ExERcrsE 1-5.5.6 (Subset of Coefficients Fixed): Consider the un-certain polynomial

et associated with those coefHcients which;tatement of the Soh-Berger-Dabke The-

b- a SBherical Folynornib'l Familf

can be skipped by the reader interestedthe results. Our objective is to provide

alue set associated rvith a spherical poly-I S € Q\. The proof of the Soh-Berger-

:he next section, exploits this charaqteri-:he next section, exploits this charaqteri-zation. To this end, we now show that at each frequency e > 0, thevalue set p(j.,Q) is an ell ipse in the complex plane. Throughoutthis section, to avoid triviaiit ies, we assume that deg p(s,q) > 7.In acidition, we work with an uncertain polynomial of the formp(s, q) : p0(s) + ILo gis? with uncertainty bound llqll, < ,.

To begin the analysis, we hold the frequency e > 0 fi-:<ed andobserve that z ep(ju,Q) if and only if thele exists sorne q € Q suchthat

Z: A(a)q * b(c, . ' ) ,

ililri

fittl

where

272 Ch. f 5 ,/ Spherical Poll'nomial Famiiies

15.6 /

A(u) :

Tlre Vaiue Set for a Spherical Poly.nomial Family 271

0a 1-' ' l

L o o 0 . l

andr - l

, , I f i" po(j.) |o \ a ) : I l .l r rn ps( ja) )

Next, observing that rank A(-) : 2 when u )> 0, it foliows thatz € p(jut,Q) if and only if the minimum norrn solution qL[N(.) ofthe equatiort i : A(-)q + b(c..') has norm iess than or equal to r.Using Lemma 75.4.2, we generate

' qn * (-) : Ar Qi) (A(,') Ar (u))-r (z - u1u11

and enforce the condition l lqMtrl l, < , to arrive at the followingconclusion: For c..r > 0, the value set p(j.,Q) is the ell ipse in thecomplex plane described by

(z - b(a)) r rv (ut ) (Z - b(u)) < r2 ,

wherew(r) : (A@)Ar ( r , t ) ) - r .

ifo further simplify the description of the ellipse p(j u , Q) , we nowdevelop a closed form description of the weighting matrix W(a). In-deed, using the formulas for A(a.') andW (a) above, a straightforwardcornputation yields

r . )w(u) :d i "c1+, ,+ l

| 07"- fio )

Now, substituting for i, bQt) andW(t';) in the ell iptical value setdescr ipt ion (Z-b(co)) rVr@)( i - b(r ) ) < 12,we reach the conclus ionthat z € p( ja ,Q) i f and only i f

lRe z - Re po( jo)12 - [ I rn z - I rn po( ju) ]z . _z5- -"

-r \-.r2t

: 'Z r l - -

i euen r odd

The set of z € C satisfying the inequality above is clearly an eilipse;see Figure 15.6.1. This ell ipse is centered at the nominal po(j.)

FtcunB 15.6.1 Value Set Eli ipse p(i ' ,Q) for o > 0

with major axis in the real direction having length

/ \ r / 2R o : 2 r { f , ' n }-

\n1i^ /

and major axis in the imaginary direction having length

/ \ t / zI n : 2 r { f r 2 o I"

\7oo /

For the case when u) : O, the analysis above does not hold

because rank,4(0) :1' Nevertheless, for this degenerate case, the

value set is easy to d,escribe. Indeed, notice that for Q:0, the value

set is the real interval given bY

p( j ) , Q) : fao - r , as I r ) .

By combining the two analyses for the cases r,; ) 0 and c'; : 0, we

have a compiete description of the value set.

l-. io.7 Proof of the Soh-Berger-Dabke Theorern

First, it is easy to see that stabii ity of pe(s) is necessary for robust

stability of 2 becaus e Q : 0 is admissible' Also' when cu : 0, in order

au) '

0

Irn ps(iLD)

15.8 ,/ Overbounding r.ia a Spherical Family 273

to avoid the possibility of a root at s : 0, it is also necessary to have

looi > r1 equiwalentJ.y ,0 / p( j } ,Q). Therefore, in the rema" inder ofthe proof, we assume that p6(s) is sta"ble and lasl > r. \\re must provethat GsBn(.) > ,' for all o,, ) 0 is both necessary and sufficient forrobust stability. Indeed, since all the preconditions for applicationof t lre Zero Exclusion Condition (Theoren 7.4.2) are satisfied, itfollows that P is robustly stable if and only if 0 / p(j.,Q) for allu: > O. To complete the proof, we use bhe inequality characterizingz e p(ja,Q); i.e., in accordance witb. the conclusion reached in theprececling section, it follows that 0 /p(j.,Q) if and oniy if

\ ] ,L irn po\Jur ))- > , 2 .

\ - , , , 2 'L t *i odd

Hence, robust stabil ity of P is guararrtced if aucl only if

Gspp (u ) > r2

for all frequencies cu > 0. E

15.8 Overbounding via a Spherical Farnily

In this section, we see that a family of polynomials which is non-spherical can often be overbounded by a spherical family. The readershouLd recall, however, that overbounding can lead to a conservativeresult: see the discussion in Section 5.11. To describe the key ideainvolved in the overbounding process, we begin with a family ofpolynornials 2 described by

p ( s , q ) : p o ( s ) + l a ; ( q ) s 'i € I

and g € Q w i th ps (s ) be ing the nomina l and 1 C {0 , 1 ,2 , . . . , n }denoting the index set describing the coeffrcients which are uncertain.Suppose that this family does not have an independent uncertaintystructure and the uncertainty bounding Q is not necessarily a sphere.

As a first step, we compute a bound for the norm of or (q). Thatis, we find f ) 0 such that

f ) m_ax l l " t@)yr .q e q

Subsequently, it follows that P is a subset of the overbounding familyP described by

p (s ,Q) : po (s ) + ld . i s i * I a i s i ,i e I i { I

274 Ch 15 ,/ Spherical Poly,nomial Families

a n d q € Q , w h e r e

A : { q € R - r , l l 4 l l z < ' } .

In view of the set inciusion P C P , any robustness criterion which is

satisfied by the ouerbound'ing family P is automatically satisfied forthe original family 2.

ExAMPLE 15.8.1 (Centering): In some cases, we can improve uponthe bound P above by "centering" the Q set. We now ii iustrate thistechnique via an example which is considered in two different ways.

Strppose that p(s, a) : s2 * (3cos2q)s * 5 and Q : [0 , 5] . We f i rs t

overbound without centering Q. Using the prescription above, wetake 1: {1} and obta in

f : r na r a t l i : m -ax - 3cos2q - 3 .q € Q q € [ 0 , 5 ]

Tlr is leads to p(s, Q) : s2 * hs * 5 and 0 : [ -3,3] . We nowprovide a second solution which involves centering. After lvrit ingp(s, q) : s2 I (3 cosz q - 1.5)s i * 1 .5s -F 5, we take the nom' inal to bepo(s) : s2 - 1.5s -F 5 and obta in

r : r y . ? T , l 3 c o s 2 s - 1 . 5 1 - 1 . 5 ,ge lu ra l

FG,4 ) : s2 l ( 1 .5 + 4 r ) s * 5 and Q : l - 1 .5 , 1 .5 ] . Obse rwe tha t t i r i s

family of polynomials is a strict subset of the one obtained rvithout

centering. In conclusion, centerin1 of Q can be effective in tighten-

ing the overbound P. In many cases, there are other commonsense

considerations which can be used to improve the bounding process.

ExERCTsE 15.8.2 (Vlaximum Eigenvalue Ovelbound) : Consider

the fami ly of polynomialsP descr ibed byp(s, S) : po(s)+Lrer a; (q)s 'and l lq l l2 < r . Assume that aI (q) is l inear wi th respect to g; i .e . ,

one can write ar(g) : Are, where ,47 is an n1 x !. matrix. Now, if

rar.k 41 : l, show that an overbounding spherical familv of polyno-

mials 2 is described by F(s, q) : Lrct Qisx and llS-l lr < r where,

r : r)il""(aT,qt)

and A*n"(Af,4.7) denotes the largest eigenvalue of. A!A1.

EXAMrLE 15.8.3 (Conservatism in Overbounding): To see that

the overbounding procedure (using the maximum eigenvalue) above

15.8 / Overbounding via a Spherical Family 27b

submalineactuatol transfer function

FtcunB 15.8.1 Depth Control System for Example 15.8.4

can be conservative, consider the uncertain polynomial

p ( s , q ) : s 3 J - ( q + 3 ) s 2 r 4 s * q * 2

wi th uncer ta inty bounding set Q: [ -1.5, 1.5] . F i rs t , we generalethe Routh table:

s 3 r 4

s 2 q l 3 q * 2

" 1+30s o q * 2

Since there are no sign changes in the first column of the table forall q € Q, this family of poll.nomials is robustly stable. On theother hand, if we overbound as prescribed in Exercise 15.8.2. we u.qe1 : {0,2}, Ar : [1 1]1 and r : 1.5 and obtain an overboundingfamily 2 described by

FG, d : s3 -t- (42 + s1t' * 4s -f (ao + z1

and ll4ll2 < 7.5rt. It is now easy to see that P is not robustly stable.For example, to induce instability, it suffices to take 4z: 0 for anyqs satisf,/ing -I.5\/2 < 4o < -2.

ExAt\,{pLE 15.8.4 (Depth Control System) : To demonstrate theoverbounding process in the context of rational functions, we con-sider the rnodel of a submarine depth control systern in Dorf Q97a);see Figure 15.8.1. A pressure transducer is used to measure thedepth, and the gain of the stem plane actuator is set at I{ : O.2.

276 Ct-t.75 / Spherical Poly'nomial Families

It is straightforward, to verifi' that the closed loop system is stable'

we now examine the effect on stability for uncertainty up to 2O7o in

the poie and zero locations in the approximate submarine transfer

function. Hence,,in l ieu of P(s), we take

. (s * 0.2 + qs)zP ( t , q , t ) :

" 2 + g f i + " 0w i t h - 0 . 0 4 ( g o < 0 . 0 4 a n d - 0 . 0 0 2 ( r 9 ( 0 . 0 0 2 . N o t i c e t l r . a t t h edenominato, of P1t, q, r) is a spherical familv of polynomials but the

numerator

lr/(", s) : s2 I (0.4 + 2qs)s * (q3 + O'Aqo + 0'04)

is not. To obtain a spherical overbound for the numerator, we take

nominal No(") : s2 -l- 0'4s -F 0'04, index set I : {0, 1} and corp'pute

ilrax l lar(q) l l : maxlqel<0.04 " lqo l<0.04

Hence, we end up with a numerator overbounding family which is

described by

1 r / ( r , 4 ) : s2 * ( 0 .4+ 4 r ) s + (0 .04 + so )

and ll4ll < 0.082. Now' to study robustness of closed loop stabil itv''we can use the overbounding polynomial

F (s , 4 , r ) : I { N ( s , q ) + sD(s , r )

: s3 * 0.2s2 + (0.Zqr + ro -F 0.09)s + 0.2q0 + 0 '008'

ExERcrsE 15.8.5 (Robustness Margin): For the depth control sys-

tem in the example above, use the formula for r*o" in Section 15.5

to compute a robustness margin for the overbounding farnily of poly-

nomials which was obtained'

15.9 Conclus ion

In this chapter, t:he 12 analogue of Kharitonov's interval polynomial

problem was studied. The highlight was the soh-Berger-Dabke The-

orem. Modu]o some minor techlical assurnptions about c,.r : 0 and

avoidance of degree dropping, the theorem tells us that robust sta-

bility of a spherical polynomial family can be checked by generating

the frequency function GsBo(r) and finding its infimum'

4qA + @3 + O.+qo)2 x 0.082'

/ c - .1 - O ? \2

s2 + o .01

Notes and Relared Literature 277

Notes and Related Literature

NRL 15.1 The important paper by Fam and ivleditch (1978) appears to have

motivated the work of Soh, Berger and Dabke (1985). In a sense, it is reasonable

to say that Soh, Berger and Dabke were the first to realize that the key ideas of

Farn and lvleditch had important rarnifications in robustness theory.

NRL 15.2 For spherical polynomial families, thr: rJevelupment of robust stability

concepts in a feedback control context are at[ributabie to Biernacki, Hwang and

Bhattacharrgra (1987). In the next chapter, we describe their control theoretic

extension ofthe Soh-Berger-Dabke Theorern. In Hinrichsen and Pritchard (1988)

and Keel, Bhattacharryga and Howze (1988), additional results are given.

NR-L 15.3 Tine 1.2 version of the result of Tsypkin and Polyak (1991) is directlyrelated to the result given in the Soh-Berger-Dabke Theorern. Namely, thecornplex frequency function

v r r \ w J -

of Tsypkin and Polyak (see Section 6.7) is sinply related to the scalar functionof Soh, Berger and Dabke; i .e. ,

r r r 2( J s B D \ u ) : l \ r T P \ a ) l

Chapter 16

Ernbellishrnents for Spherical Farnilies

bynopsls

In thjs cl)apter, v,e continue to concentrate on sp)terical families'

In the first part of tlle chaptet, we provide the spectril set

cltaracterization of Batmish and Tempo- In the second part

of the chapter, a sphericd plant farnily and a compensator are

connected in a feedback loop. With this control system setup,

the resuJts of Biernacki, Hwang and Bhattacharyya facil,itate

robust stability analYsis.

16.1 Int roduct ion

central to the technical development of the last chapter is the rhin-

imum norm solution to a set of linear equations Ar : b- In the first

part of this chapter, we use the same set of technical ideas to develop

a cllaracterization of the spectral set for a spherical polynomial fam-

ily. With the spectral set in hand, we have complete knowledge of

aII'possible 2 regions for which robust stabiiity is guaranteed. In

the second part of this chapter, we concentlate on lobust stability of

feedback systems involving a spherical plant family and sorne given

compensator. Recalling the analysis of interval plants in chapter 10,

the reader is reminded that the independent uncertainty structure

which is present in the plant is no longer present in the closed loop

polynor4ial p(s,i l. Instead, p(s,q.) has an affine l inear uncertainty

structure; e.g., see Lemma 8.2.3. For this reason' the theory in

278

76.2 / The Spectral Set 279

Chapter 15 (which applies to independent uncertainty structures) isextended later in the chaoter.

L6.2 The Spectral Set

F 'or a fami ly of polynomials P: {p( . , q) : q e 8} , io Sect ion 9.6 webriefly introduced the spectral set

o[Pl : {z e C : p(z,q) : 0 for some q € Q}

in the context of the root version of the Edge Theorem. We nowstudy the spectral set in greater detail. For the case of sphericalpolynornial farnilies, our goai is to obtain a useful charactelization ofthe boundary of o[P]. One obvious motivation for the study of olP)is a desire to know somethine more about the distribution of the roots

cletailed information about the distribution of the system's poles is i"importanl from a performance point of view. q:r

To illustrate the usefulness of a spectral set characterization in b'comparison to a yes-no soiution by robust 2-stability analysis, sup-, {pose that we seek a description of the "minimal" damping cone whichi n-contains all the roots of p(s,q) for q € Q. Notice that a candidate' -$=

cone is parameterized by an angle 0; e.9., for 0 < 0 <rf2, lcl-

D 6 : { z € C : r - 0 < [ z < n * 0 ] .

Now, by gradually decreasing d from rf 2 to 0 and applying a robustD-stability test for each d, we obtain a practical solution to theproblern. Ifowever, this type of solution neethod involves some sortof recursive scheme requiring iterative adjustment of d. We arriveat the minimal damping cone but gain little insight into the rootdistribution. For more complicated root Iocation regions, a sirnilariterative scherne can also be developed. Once the shape of the 2region is specified, we can carry out iterations involving expansionand contraction of D.

In the next section, our main objective is to avoidD region itera-tion completely. To this end, we develop formulas which describe theboundary of ofP). With these formulas, we can display olPl graph-icaily and imrnediately know the solution to the robust 2-stability

' q ,

280 Ch. 16 ,/ Embeltishments for Spherical Families

problem for every possible 2 region. That is' P is robustly Z-stabie

it ana only if olP) c 2' This situation is depicted in Figure L6'2'I '

Robust Z-StabilitY

F tcuns 16 .2 .1

Lack of Robust 2-StabilitY

Relationship Between o(P) and D

iii li t

l ii l

; : l

16.3 Forrnula and Theorern of Barrnish and Ternpo

Given a spherical polynornial family 2 having invaliant degree, we

no* de.,elop a formula which characterizes the boundary ol ofPl.

To this end, we coustmct a function O(z) having the property that

z e alPl if and onl1.' if aQ) < 0. The justificati.on of the construction

below is relegated to the proof of Theorem 16'3'4'

DEFrNrrroN 16.3.1 (The Spectral Set Weighting \4atrix): Given

a spherical polynomial family P : {p(', S) : q e 8} with invariant

degree n) l,the spectral set wei,ght' ing matrir is defined by

where

r r l t - \ - ( l ( , \ . a T ( r - 1

\ ' ^ \ - / ' - " ) )

)

I t n " z " ' R e z nA ( z ) : 1

l0 Irn z " ' I rn zn

ExERCTsE 16.3.2 (Closed Form Descr ipt ion otW(z)) : Show that

16.3 ,/ Formula and Theorem of Barmish and Tempo

a closed form for the spectral set weighting matrix above is

i1r,,- ,')' _f;ro" zi11r,n "i7

282 Ch. 16 ,/ Embellishments for Spherical Families

for some q e Q. Equivalently, z € olP) if and only if every minimumnorm solutton qtutN (z) of the equation above satisfies

llqoo* (")ll, < ,.

\\re now consider two cases.Case 1: Irn z I O. Then rank A(z) : 2 and in accordance rvithLemma I5.4.2, the minimum norrn solution is unique and given bv

sL'IN Q) : -A1: G) (e1r7er 1"1)- ' FoQ) : -Ar (r)w(")Fo(.),

rvhere W(z) ts the spectral set weighting matrix. Now, by enforcingthe conditior l lq^{N (")l lz 3 r, it follows that z e olPl if and only if

p[ 1z;ttt 1z1po(") S "' .

Equivalently, z € olPl if and only if O(z) < 0.Case 2z lrn z : 0. In this case, fo(z) is real, Re z' : z' fori , : 0 , I , 2 , . . . , r t a n d I m z ' : 0 f o r i : L , 2 , . . - , f l . H e n c e , t h espectrai set rlrembership condition A(z)q: -FoQ) becornes

[ i " " 2 " n f

q : - p o ( z ) .L_

Again invoking Lemma 15.4.2, the minimum norrrr solution for thiscquation is uniquely given by

2 8 1

w @ : ; a )n

- \ - r p - , i \ ( r - . i \

; - l

n

\ - / p - , i \ 2/ . - g L v

2 /

where

n n

A ( z \ : \ - / p o " ' : ) 2

. \ - r T - , i \ 2

; - n ; - l

DEFr^IrrroN 16.3.3 (Spectral Set Boundary Function O(z)): Givena spherical polynomial farnily P: {p(.,q): q e 8} with invariantdegree n ) 7, uncertainty bound r ) 0 and norninal p6(s), thespectral set boundary function O : C ---+ R2 is defined by

- (I,"" "o11r,n "1)"

/ ^I p t r ' ( " )W(r)p iQ) - 12 i r Im z l0 ;

9 ( Z l : 1

I p3(t) - r 'LT:o"'n i t Irn z : o.

where

i , \ l R e p s ( " )P o \ 2 1 : ,

l l rn ps(z)

is the vector representation for po(").

THEoREM L6.3.4 (Barmish and Tempo (f 991)): G,iuen a spheri,calpo l ynom ' i a l f am i l yP : {p ( . , q ) , q e Q} w i t h , i nua r i an t deg reen } L ,i,t follous that z e olP) i,f and only zf

@(z ) < 0 .

PRoor: Given a candidate point z € C, observe that z e olPl ifand only i f p(z,S) :0 for some S e Q. Now, wi th A(z) as g ivenin Definit ion 16.3.1 and p(s, q) : po(s) + Ito qisxl we express thecondi t ion p(z,q) :0 in matr ix form; i .e . , z e o lPl i f and onJy i f

Enforcing the condition l lqMN(z)ll, < r, a straightforward calcula-tion leads to the conclusion that z e olPl if and only if O(z) < O. E

REI4ARKS 16.3.5 (Computational Aspects); The characterizationof the spectral set via the inequality AQ) ( 0 makes it possibleto use a contour plotting routine to display lesults in an easy-to-understand manner. In this legard, notice that @(z) is rational inz off the reai axis and polynomic in z on the leal axis. lVloreover,

T^ l v I N t - , , - rY \ - / n

\ - " 2 i/--; - n

1

z

)z-

zn

P o \ z ) .

I i[1

iii i

A( r )q : - po (z )

16.3 / Forrnula and Theorem of Barmish and Ternpo 283

0 . 5

Irn

- 1

Re

Ftcun-p 16.3.1 Spectral Set for Example 16.3.6

since z is only two-dimensional, we obtain a graphical display. Thisis illustrated via an example below.

ExAI\4PLE 16.3.6 (Spectral Set Generation): For the sphericaipolynomial famity 2 with nominal po(s) : s6 + 2s5 -l- 3sa + 4s3 -l-

5s2 + 6s -t 7 and uncertainty bound r : 0.55, a contour plot forthe function A(z) 'was generated. The resulting spectral seL olP),charactefized b1' AQ) < 0, is displayed in Figure 16.3.1.

ExERCTSE 16.3.7 (Coalescence of Root Clouds): For the exampleabove, generate olP) for smaller values of the uncertainty bound rand verify that the six distinct root clouds are obtained when theuncertainty bound r is small enough.

ExERCTSE 16.3.8 (Recovery of Soh-Berger-Dabke Theorern) : Theobjective of this exercise is to develop a concrete connection betweenthe robust stability problem and the spectral set generation problem.To tlr is end, suppose that P is a spherical poiynomial family withinvariant degree n ) L and stabie nominal p0(s). Argue that robuststabil ity is guaranteed if and olly if the spectral set does not crossthe imaginary axis. Now, restricting attention to the imaginary a;cis,take z : ja and shorv that for c't ) 0, the spectral set rveighting

284 Ch. 16 / Embellishments for Spherical Families

matrix is given by

r . rl r r l

t\' '(a): dias J -.;F , 5i.r, I

| / w / . * I

I tT". i-odd )

Subsequently, by invoking Theorem 16.3.4, conclude that ju e olP)if and only if

(Re po(jtt))2 , (Irn po(ju,,))z . -2\ - - ,

- \ - " l t

- '

,?".* ,Ao*EquivalentJ.y, ja / olPl it and only if

GsBp(t ) > 12,

where G snn@) is the Soh-Berger-Dabke function given in the state-ment of Theorem 15.5.1.

L6.4 Affine Linear lJncertainty Structures

Our objective in this section is to refine the analysis of Chapter 15 toaccount for affi.ne linear uncertainty structures which arise from feed-back interconnections involving a spherical plant famii)/ and a con-pensator; see Section 8.2 where this issue is f irst discussed. In thisregard, note that the Soh-Berger-Dabke Theorem (see Section 15.5)does not apply because the closed loop polynomial does not inheritthe independent uncertainty structure of the plant.

The analysis begins with the spherical plant farnily ? describedby the quotient of uncertain polynomials,

A r r c q )P ( s . q , r ' ) - a' D ( s , r ) '

anrl uncerbainty bound (Q, J?) which is a sphere of radius p > O;recall ing the notational convention in Section 15.3, (q, r) e (Q,R) ifand only if l l(s,r)l lz < p. To make the uncertainty structure nroreexplicit, we write

n^ r / \ A r Z f , \ - ;l v { S , q l : l v o ( 5 1 + ) Q ; S '

L ' "

z :o

1 . 5

-0 .5

0.5-0 .5- 1-2o <

-\

IIt o

16.5 / The Tesring Function for Robust Stability 285

and n

D ( s , r ) : D o ( s ) * f r ; s di:0

with Afo(s) and Ds(s) representing tlne norninal rrurrterator and de-nominator, respectively. Finally, we also assume that a compensator

C(s) is given. Expressing the compensator as the quotient of coprime

Poll'1o*t"t.ra( ̂ \ - Nc(s)v \ D . / _

D . @ ,

the resulting closed loop polynomial is

p ( s , q , r ) : l / ( s , q ) N c G ) t D ( s , r ) D s ( s ) .

16.5 The Testing F\rnction for Robust Stabil ity

With the setup above, our first objective is to describe the construc-tion of the robust stability testing function Genn(c.,') of Biernacki,

Hwa.ng and Bhattachargra (1987) . Analogous to the function given

in the Soh-Berger-Dabke Theorem (see Section 15.5), the infimum

of. GpsB(o) is central to the robust stabil ity test.There are two basic ingredients involved in the recipe for the

firnntiorr (t louo(r,t\ We first introduce the vector notationv D n D \ * )

I n e o ( l . . o . r \ fP ' ( j t ,q , r ) : I

l lm" p( ja , q , r ) )

to represent t lre closed loop polynomial p(s, q,r) a,t frequency u ) 0.Particularizing below to g - 0 and r : 0, the first ingredient in theGaan(d formula is

F(i., 0,0) : I ne ('n/6 (7c' ' ')1v c (i ') + D s(j a) D c U'D

]I Im.(Ns(jLD),n/c(rr) + Ds(ju) D c0.)) )

The second ingredient in the GBns(r) formula is a 2 x 2 symmetricweighting matrk W(a) whose inverse W-t(r) has entries

[I , r-1(. , ) ]1,1 : (R" Nc(ir)) ' ,*_r 'u

t (Re Ds(jwD'

" f ^r"m n

+ (Im Nc(ir)) ' L r" + ( Im Dc(jr)) ' L r ' ' ,i od.d. i o d d

286 Ch. I6 ,/ Embellishmenrs for Spherical Families

lW -' (.)1t,, : lI,V

- | (t..))z,t/ m .

: (Re Ns(ju))(Irn Nc(jr)( I c.,, 'o -\;i^

+ (Re D6;( ju))( Im Dc(jr)( f , , , ,\

and

r r f l - l l . . \ ' r t r . ' . ^ r ^ ( . ; , , , \ 1 , $ , . , 2 i t ( r t m

1 w - ' ( r ' t ) l z , z : ( [ r t N c U t t ) ) - , * ^ . ' "

+ ( R e N 6 ( j - ) ) , \ - -

+ (Inz p"fi.)t), f, 12, + (R" n.1i.1i'i"{ .rr.'i odd.

Using the notation above, .we now define

G s n n (r) : f ( j,,), O, 0)W (c,L)f(j w, 0, O).

we are now prepared to state the robustness criterion. To avoiddegenerate cases, we assume below that the prant nurnerator anddenominator are at ieast first order; the degenerate zero order casesare analyzed in the exercises of section 16.8. The proof of theorembelow is relegated to Sections 16.6 and 16.2.

THEoREM 16.5.1 (Biernacki, Hwang and Bhatta.charylra (1997)):Consi,der a spherical family of ptants which .is d,escri,bed. bg p :{P ( . , q , r ) , ( q , r ) e (Q ,R) } w i t h unce r ta in t y bound , l l ( q , r ) l l z < pand, uith cornpensator C(s) : Nc(s)lDs(s) with ,A/c(") and, Dc(s)coprirne- Assurne that the assoc'iated fami,Ig of closed loop polEnoini-a ls P61 : {p(- ,q, r ) : (q, r ) e (Q,R)} has, inuar iant d,egree ancl thenominal plant numerator Ns(s) and denom,inator Do(s) are polyno_m'i 'als of ord,er one or rnore. ThenPgT'is robustlE stable if and. ontyi f

p ( s ,0 ,0 ) : A I s ( s )Nc , ( s ) + D6 (s ) Dc (s )

i,s stable, the zero frequency condition

i o d d

i c a n + ; e f r P r l n - )

for all frequenci,es u; ) 0.

16.5 / The Testing Function for Robust Stability 287

REMArlr{s 16.5.2 (Associated Robustness Margin) : Before plov-

ing the theorem, we complete our generalization of the theory in

Chapter 15 by developing robustness margin formulas. The standing

assumptions are those given in Theorem 16.5.1 above. To emphasize

the dependence on the uncertainty bound p, we use the notation

(Q , R) o instead of (8, R) and take

P c t , p : { p ( ' , q , r ) : ( q , r ) e ( Q , R ) o }

to be the associated family of closed loop polynomials' Hence, the

robustness margin of interest is

pnac : sup{p : PCz,p has invariant degree and is robustly stable}.

Analogous to Section 15.5, we are going to describe the robust-

ness margin as the minimum of three quantities lpol, lp-l and pfio,.

Beginning with the enforcement of the zero frequency condition in

Theorern 16.5.1, q'e obtain

lp (70,0,0) i l 'n /o( j0)1t ic(0) + Ds(70)rc f f0) |t x t : Ir v

l C ( : O ) l l c ( i ] ) I

Next, for frequencies o ) 0, we define

I

p*..*(r) : GEnB@)

and obtainp|no, : inf p[,,(c'-,).

Finally, to enforce the invariant degree requirement, we considertwo cases. In the analysis below, we use the notation {L^, bn, c^. anddn. to denote the highest order coefficients of 1t/6(s), Do("), l ic(")and D6(s), respectively.Case 1: I f P(s, q, r )C(s) is s t r ic t ly proper for a l l (q , r ) e (Q,R),then the family Pc1 inas invariant degree if and only if the plant

denominator has invariant degreel i.e., lr. l { bn. Hence, in thiscase' u'e bake

P. : lb-r.

Case 2 : I f P (s ,q , r )C (s ) i s p rope r f o r a l l ( q , r ) e (Q ,R) (bu t no t

strictly proper), a straightforrvard calculation leads to the invariantdegree condition lo*c*. * bndn"l ) q^"*" + rndn". Hence, by max-irnizing the right-hand side above with respect to all pairs (q^,rn)

288 Ch. 16 ,/ Embellishments for Spherical Families

FIGURE 16.5.1 Nominal Svstem for Example 16'5 3

satisfying l l(q,")l l , < P,we conclude that invarlant degree of the

family Pcr,p is guaranteed if and only if P 1 Pn, where

"m. t -n.

In summ.ary, from the analysis above, it follows tha't

pmaz : rnin{pg, prr, pf,nor}.

EXAMPLE 16.5.3 (Computation of Robustness lvlargin) : For the

first order unstable systetTl controlled via an integrator in Figure 16.5.1'

the nominal closed loop polynomialp(s) : "2

+ s-F 2 is stable' we

now compute the robustness margin P^o, fot the uncertain plant

P ( s ' o ) - ( q r + 1 ) s + ( q o + 1 ) '\ u ' Y ' l -

( r 1 + I ) s + ( r s - 1 ) '

Using the formulas above, we obtain Po : I and pn: 1 by inspec-

tion. Now, to generate pL.,(-), we r-equire the weighting matrix

W(u). Using the formulas in Section 16.5, rr 'e compute the entries

[ r4 i -1(Lr) ] r , , : r t + 4; l l4 ' -1( . )1r , , : l lv ' -L1t . ,1)2, , : O; [ t l '

11( ' ; ' ] , "

:

tta + 4. Next. ure calculate

dmcm. * bndn.

F U a , o , o ) :

Ph. . \a )

1

0 .95

0 . 9

0 .85

0 . 8

0.75

0 . 7

0.65

0 . 6

0 .55

0 . 5

16.6 ,/ Machinery for Proof of the Theorem 289

u - - J u - t 4

a a + 4

290 Ch. 16 ,/ Embellishments for Spherical Families

standing assurnptions in Theorem 16.5.1. Indeed, for f ixed c.., > 0and z € C, we begin by noting Lhat z € p(j-, Q, R) if and only if

z : p( jL , ) ,0 ,0) + inn,Low" l j . ) + i ro l jn"1 i -7i :O i :O

fo r some (q , . ) c (Q,R) . Equ iva len t ly , t t s ing vec lo r no ta t io r , i tfollows that z e p(ja, Q, R) if and only if

, _ [ Re Ng(ju) -ulm Ng(j. l ) -u2Re Nc(j.) ,3rrn Ns(i-) - l

"" -

t Inr Ns(ju) utRe Ns(ja) -u2frn Nc(j-) -rtRe lvc(3o) ' l

r* | " "

Dc( j r r ) -a l rn D5( ju ) -u2Re Dc( i . ) . ' I rn Da;1 i - ) I

"' l r r " DcU-) aRe Da; ( ju ) - -2 I ^ Dc( j . ) - .3Re t l " ( j - ) " ' )

+ FUa,0 ,0)for some (q,r) e (Q,R).Factor ing the expressions above. i t fo l lorvstha t z + p ( ju ,Q,R) i f and on ly i f

T - A / ^ / ; , . , ) - / - A I ^ / ; , . , \ " 1 r - l

. l R . N 1 U . ) - l r n N q Q u ) l l t O - . . , ' 0 " r -z : | | I - t q

I r r" Ng1i. . ) Re Nc(ja) I [0, 0 -c,- ,3 J

, I o" Dc(i.) -Irn D6Qwl I [ t o -u2-

l r* Dc(i",) Re D6(ju) I Lo , o

0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5Lt)

FIGURtr 16 .5 .2 P lo t o f p l " " ( r ) fo r Example 16 .5 .3

and

G a n B @) : f ( j,D, 0, 0)W (t '-,)f(j u:, 0, 0)

I

II

which leads to

I+ / \ n i / \

P ^ o r \ a ) : \ j h H g \ Q ) :

The plot of pfi"r(to) over tb.e relevant frequency range is indicatedin Figure 16.5.2. From the graph, we see that the minirnum occurs

at r't* x 1.41 with associated valwe pfio, = 0.50. Hence, we concludethat prno, : min{ lp6l , lp* | , ph"r} = min{1, 1, 0.50} : 0 '50.

16.6 Machinery for Proof of the Theorern

We now develop some technicai machinery associated with the proof

of Theorem 16.5.1. Paralleling the development in Section 15.6, we

proceed to characterize the value set

p ( j a , Q , R ) : { p ( j w , e , r ) : ( q , r ) e ( Q , R ) }

for the family of closed loop polynornials. Throughout this section,

we work with the feedback svstem described in Section 16.4 and the

+ f ( iu ,0 ,O)

for some (q,r) e (Q,R).We def ine the 2 x 2 compensator matr i ,ces

A r ^ t , , \ -J \ C \ w ) -

l

D g ( w ) :

(rn + I) matrix

IA^r(u) : I

LI 0 - u 2 0 t t 4 0 - c . r 6

\ tu 0 -c . r3 O u5 0

t h e 2 x

16.6 ,/ Machineryfbr Proof of the Theorem 291

for the numerator and the 2 x (n * 1) matrix An(r) for the denom-inator having the same form as Atr(a).Now, with

{tc(.) : [Ns(w) Dc(.)l

and . . r - l, l / r y ( . ) - r

A \ a ) : | | '

L o A o @ ) )

it follows that z e p(ja, Q, R) if and only if

. r l

i : a6@t )A( . ) l q | + f ( i t o ,o ,o )L r l

f o r so rne (q , r ) e (Q ,R) .The arguments to follow parallel those given in Section 15.6. We

fi.rst observe that the condition z € p(ja,Q,-R) is equivalent to the

following: Any minimum norm solution (q* N (.),rMN @D of the /

equation above satisfies

l i (s - t ( r ) , , * * ( - ) ) l l , < p.

We now claim that the minimum norm solution is unique. To prove

this clairn, we recall Lemrna 15.4.2; i.e., it sufices to show that the

standing assumptions guarantee

rank f,)5:(u,') A(t';) : 2.

Indeed, notice that the presence of the diagonal subblocks diag{1, o}in both Ax(.) and Ap(r':) assure that rank AQ';) : 4. Further-more, expioit ing coprimeness of l/5:(s) and D6r(s), it foilows thateither det Ng(u) I 0 or det D6;(a) I 0. Hence, we are sure thatrank f25r( t . , ) :2 , which, when combined wi th rank,4(r , . , ) :4 , guar-

antees that rank Q6(a)A(co) :2.

Having established the rank condition above, we now obtain theunique minimum norm solution. To this end we call

vrt (a) : la6 @) A(u) Ar @)aT @\-'

the ualue set ueighting matrir and generate

2gZ Ch 16 / Embellishments for Spherical Farnilies

Furthermore, a straightforward calculation using the expression for

(qnu* ( . ) , r tu IN @)) above leads to

l l (q"r(r), ,rIN @Dll l : l i - b-ur,o,0)lrvl(u)lz - p*(ju),0,0)l '

In summary, given an)' frequenc.y.y,> 0 and z € C' it foiiows

trrat z € p(ja,"Q,-R) If u..td ottly if (qMN(c"'), 'MN@)) has at rrost

norm p. Equivalently, z lies in tire ellipse described by

l i - F(jr,0, 0)l?lY(r.,) lz - F(j ' ,0, 0)l < P' '

ExERCTsE 16.6.1 (The Weighting Nlatrix) : After expressing the

inverse of the ualue set ue'ighting matrit as

w -' (.) : lNc @) A x @) AT @) N3 @) + D 6 Q') A n G't) AT UL)D[ (',)]-','

verify tirat for cu ) 0, the entries of 2 x 2 inverse 14t-t(u) are those

given in Section 16.5.

ExEn-CTsE L6-6-2 (Zero Frequency): For frequency u :0' show

that the value set is the real interval

p ( j o , Q , R ) : l p ( j 0 , 0 ' 0 ) - p l c ( i o ) l , p ( j 0 , 0 , 0 ) + p l C ( j o ) l l

L6.7 Proof of the Theorern

zero frequencY condition

I P( jO, 0 ' 0 ) Ip < l-Co) I

i1

ii1I

[qnut( ( , ) " t t t ( ( . ' ) ] : ar p1a[@)w(a) lz - p-( jc . . ' ,0 ,0) ]

E

1 294 ChJ6 ,z Embellishrlre+rts for Spherical Families

16.8 / Some Refinements 293

the description of the value set ellipse given in Section 16.6 above,

we see immediately that 0 / p(jr,Q, R) it and only if

pq (i a, o, o)w (u)f( j u, o, o) > p2 .

Using the expressions for the entries of the weighting matrix found in

Exercise 16.6.1, we recognize that the inequality above is equivalentto Gsap@) > F . E

16.8 Sorne Refinernents

In the preceding analysis, we made a number of assumptions forpedagogical purposes. The objective in this section is provide someexercises aimed at various special cases which we excluded for sirn-p l ic i ty . Throughout th is sect io l , P : {P( ' ,q , r ) t (q , r ) e (Q,R)}

is taken to be a spherical plant farniiy with compensaior C(s) as

descr ibed in Sect ion 16.4 and Theorem 16.5.1.

ExERCTsE 16.8.1 (Zero Order Numerator and Denominator) : One

degenerate case which we omitted in Theorem 16.5.1 is characterized

by rn : TL : 0. If the family of closed loop polynomials Pcr h.as

invariant degree and p(s, 0, 0) is stable, verify that robust stabiiityof Pcr is equivalent to satisfaction of tlne Zero Exclusion Conditlon

o / f p ( j . , 0 , 0 ) - p l c ( j u ) l , p ( j . , 0 . 0 ) + p l c ( t . ) l l

at all frequencies a-' ) 0.

ExER.crsE 76.8.2 (Other (-,n) Cornbinations): With rn : 0 andn )> 7, assume llnat Pcr has invariant degree and p(s, 0, 0) is stable.Now, refine the Gp11 s(..,) formula in Theorem 16.5.1 and observethat the zero frequency condition remains unchanged.

ExERCTsE 16.8.3 (On1y a Subset of Coefficients Uncertain) : Let-

ting 1ry C {0, 1, 2,. . . ,nz} denote an index set for the plant numera-

tor, we takeN ( " , q ) : A I o ( r ) + \ o r s i .

i€17'r

Simi lar ly , le t t ing Ip C {0, I ,2 , . . - , rn} denote an index set for theplant denominator. we take

D ( s , r ) : D o G ) * l n s i .i € I p

For tlre nondegenerate cases when Iry and I p each have card,inalitytwo or rnore) provide a refinement of Theolem 16.5.1.

ExERCTsE 16.8.4 (Weighted Norm) : Provide a refinement of The-orem 16.5.1 for the case when a weighted euclidean norm ll (q, i l lr,wis used in i ieu of the standard euclidean norrn l] (q,r)l lr.

16 .9 Conc lus ion

This chapter completes Part fV of this text. The remaining chaptersprovide a sampling of results which may be of interest to the rnoreadvanced reader. Some of the technical deweloprnents are slightlymore abstract than those given in the earlier- chapters and some ofthe results are probably of more interest to the researcher than thepractit ioner. In particular, Chapter 19 includes detailed descriptionsof a nurnber of open problems.

Notes and Related Literature 295

Notes and Related Literature

NRL 16.1 The results given on the spectral set_aiso have an obvious root locus

interpretation. Namely, one can introduce a gain -K into p(s, q) and generate

spectral sets for different values ofl(; see the paper by Barrnish and Tempo (f990)

for further details.

I$RL 16.2 As early as the fifties; authors have forrnulated multivariable root

locus problems which are similar to the spectral set generation problem; e.g., see

the textbook by Tbr.r-xal (1955). In the later work by Zeheb and Waiach (1977),

a two-parameter root locus problem is considered and rather specific assump-

tions (rnotivated by circuit theory) are made about the uncertainty structure.

A number of papers following Zeheb and Walach's work deal with the so-called

zero set concept; e.9., see Zeheb and Walach (1981) and Fluchter, Srebro and

Zeheb (1987). It is seen that the zero set provides a lathel general framework

for dealing with rnultivariable root loci. In practice, however, the cornpulaLiorrai

complexity associated with this approach is high.

I\[RL 16.3 An extension of the spectral set theory of Sections 16.2 and 16.3 is

pursued in a paper by }donov (1992). The author develops a cha,racterization of

the spectral set for a fa,mily of matrices ,4. This set is described by an uncertain

n x n ma t r i x A (q ) : Ao + t : : o g tA r w i t h A t € R* , " - f i xed f o r i , : 0 , I , 2 , . . . , ! .

and a spherical ulcertainty bounding set Q for q. In addition, it is assrrmed

that the Ai car\ be simultaneously upper triangularized; i.e., there exists a fixed

matr ix t rz such t ]na ' : tV- \AiV is upper t r iangular for i : O, t ,2, . . . ,1. Artexample

o.f'such a set of At is a com:rrutative family; i.e., if A;rAar: AtzA4 for all

.z t , iz € {0,L,2, . . . ,1} . lhe,rr V caq be lakel tq be uqiary; e.g.} sqe l {sr4 ar td

Johnson (1988) for a more compiete description of sets of rnatrices which can be

sirnultaneously upper triangularized,

296 Ch. 16 ,/ Embellishments for Spherical Families

admissible perturbations are real nxnm'atrices with llAAll : o(AA) and a(AA)

denotes the largest singular value of A-A. Note that no "clean" formula fof T^o,

has been given in the literature to date.

NRL 16.6 For the matrix problem in the note above, the lower bound of Qiuand Davison (1992) is interesting because it may turn out to be sharp' Theyshow that

r^o, 2 rnin{o(A), B(A),}

where a(A) is the smallest shgular value of A,

I a - , , , T 1B(A) : i n f sup d2n -1 |

' ' . - |' u > o o < " k 1 l . - i l A )

arrd 6k(IvI) denotes the ,k-th largest singular value of a matrix -flzl.

NRL 16.4 Our analysis in this cha,pter appiies to the class of ^ffine linear uncer-

tainty structures generatable from the feedback system setup described irr Sec-

tion 16,4. A nrrrnber of authors (for example, see Hbrrichsen and Prichard (1989)

and Tsyrkin and Polyak (1991)) begin at the more basic level of polyno-ials and

derive similar results. To this end. these authors work with an uncertain

nom ia l o f t he f o r rn p ( s . g ) po ( " ) t

fixed polvnomials. Now. s'ith uncertainty bounding set taken to be a sphere, ro-

bust stability criteria are developed. With this setup, hol'ever, a characterizatiorr

o[ the spectra l set hes not been given.

NRL 16.5 Some of the robustness margin problems of Chapters 15 and 16 also

liave matrix versions- For example, if A is a stable real n x n matrix, it is

of interest to find r-mat : sup{r : A + LA is stable for all llAAll { r}, rvhere

a iu ( s ) , whe re pe (s ) , p r ( s ) , . . . , p r ( s ) a re

Part V

Chapter 17

An Introduction to Guardian Maps

Synopsis

In Chapter 4, we developed eigenvalue criteria for robust sta-

bility problems with one uncertain parameter entering affine

Iinea,rJy into the coefficients of the poJynomial af interest' This

Chapter ci,oncentrates on the more general framtework of Saydy,

Tits and Abed. Using the guatdian map concept, it becornes

possible to deal with a general root location regionD and classes

of uncertainty structwes which are not necessadJy affine linear.

17.7 Introduction

To motivate the technical exposition of this chapter, we review the

Bialas criterion given in Section 4.11. Namely, given a family'of

polynornials P witin invariant degree described by ) e [0,1] and

p (s , ) ) : ) po (s ) + (1 - ) )P r ( " )

withps(s) stable, with positive coeffi.cients and degpe(s) > degpl (s),

robust stabilit-y is assured if and only if the matrix H*t(po)H(pt)

has no purely real nonpositive eigenvaluesl recall that H(p4,) is the

Hurwitz matrix for p;(s).The criterion of Bialas above has some attractive features. First,

it is quite general in the sense that no stringent assumptio4s on pe(s)

and p1 (s) are required. In contrast to the theory in Chapter 72, we

298

Sorne Fdappenings at theFbontier

IIII

IIIi lIII i

h,il i ll i l[1 ill

fifl['lll l l" lii:lf

77.2 / Overvierv 299

are not insisting on an extreme point result. Recali ing Cirapter 12,the class of po15'1e16ials for which extreme point results apply isl imited by the requirement that Ranrzer's Growth Condition is sat-isfied. The second attractive feature of the Bialas criterion is moreconceptual in nature. Since it is based on coefHcient manipulationlather than frequency sweeping, we gain a degree of insight aboutthe cornputational complexity associated with robust stability test-ing. The existence of a nonpositive eigenvalue of H-r(po)H(p1) isdecidable via a finite number of arithmetic operations; this issue isfurther pursued in the notes at the end of this chapter.

The third attractive feature of the Bialas criterion which we men-tion serves as the takeoff point for this chapter-its generalizabilitr'.W.e see in this chapter that eigenvalue criteria for robust stabilitycan be derived for rather general one-parameter uncertainty struc-tures. In fact, many of the ideas in the sections to follow admitgeneralizations involving more than one uncertain parameter.

a7.2 Overv iew

Since this chapter is somewhat more technical and abstract thanmany of its predecessors, we prpvide a brief overview of the exposi-tion to follow: Given a desired root location region D, we plan toconstruct a guardian rnap u. When we want to solve stability prob-lems involving polynomials, the domain of u is the set of n-th orderpolynomials; when we want to solve matrix stability problems, thedomain of ru is the ser of n x n matrices. In either evenl, the rangeof z is the reals and the construction of u involves the desired rootlocation region D and the declared order n of the poiynomial.

Once u is determined, we generate a repre.sentationwith lespectto the family of polynornials or matrices under consideration. For ex-ample, for a family of matrices A: {A(^) : A e [0, 1]], we constructa polynomia] matrix function

N

"f()) : Ier'i :0

which represents the action of u on A(A) in the sense that

u ( A ( A ) ) : d e t / ( ) ) .

The final step of the analysis involves construction of a rnatrtx M(depending on .Fo, Ft,. . ., F1,') whose eigenvalues tell us whether thefamily. -4 is robustly D-stabie.

300 Ch. 17 / An Inroduction to Guardia-n N'Iaps

1,7.3 Topological Prelirninaries for Guardian Maps

The technicai exposition to follorv makes use of a number of basic

topological concepts. We now consolidate this critical material.

DEFrNrrroN 17.3.1 (Open Neighborhood and Open Sets) : Let

Pn d.enote the set of rz-th order polynomials. To each n-th order

po l ynomia l p ( s ) : LT :oa i s i , we take a : ( ao ,a r , " ' , 4 . ) t o be i t s

coefficient representation and note that an * 0: In addition, rve

assume that some norm ll ' l l is declared on R'+1 ' Now, given any

e > 0 and some p € Pn, we call

n

B,(p) : {p' , p'(t) : la' ;s" and l lo' - " l l S t}

i.:o

is wholly contained in 2.

REMART(S 17.g.2 (Characterizations) : Note that an open neigh-

borhood P in Pn can be associated with an open neighborhood in

11n-1-1 . Furthermore, such a neighborhood has the following pfop-

erty: If {pn}fl, is a sequence of polynomials converging to p (in the

sense that their coefflcient representations {ol}Er converge to a) ,

then there exists a positive integer l/ such that p; € P for all 'k > l/'

EXERCTSE 17.3.3 (Association with Polynomial coefrcients): Ar-

gue tirat every open set ,4 c gn+1 which does not contain points of

i hu f o t ' - a : ( ao ,a r , . . . t dn - r , 0 ) can be assoc ia ted w i th an open se t

P 1n the space of n-th order polynomials Pt'

ExEnCTsE 1'7.3.4 (Open Sets in the Cornplex Plane): Let 2 be an

open subset of the complex plane. using continuous dependence of

roots on coeffi.cients (Lemma 4.8.2), argue that the set of n-th order

polynomials P having its roots in 1) is open'

DEFTNT:I'rON 17.3.5 (Closure): Let P be a set of n-th order poly-

nomials. Then a polynomial p" e Pn is said to be a point of closure

iir

17.3 / Topological Preliminaries for Guardian Irdaps 301

of P i:f the follorving condition is satisfied: Every open neighborhoodof p* contains points in 2. We use

c IP : { n ep " : p i s a po in t o f c l osu re o f P }

to denote t]ne closure of P.

REMARKS L7.3.6 (Alternative Definit ion): The closure of P canalso be defined via sequences. Indeed, if the family P lnas coefficientset -4 c R'*1, t l ien a polynomial p-(s) : L!:ooiti is a point ofclosure of P tf a; + 0 and Lhere exists a sequence {a6}p., in "4conve rg ing to o * : ( a6 ,a i , . . . , a ; ) . Tha t i s , l l ox -o * l l - O as k + oo .

ExERCTsE L7.3.7 (Closure): Let P be a set of n-th order polyno-mials which ale ail stable. Using continuous dependence of roots oncoefFr . ier r ts lT,ernnrn, . . l 8 2) arorre tha. t c l 2 consists of a i l n- th orderpoll'nomials having all its roots in the closed left half plane. Thisset includes poly.nornia.ls with roots on the imaginary' axis.

DEFlnIrrroN l-7.3.8 (Boundary of a Set of Polynornials): Let Pbe a set of n-th order polynomials and take p* (s) to be an n-th orderpolynomial (not necessarily in 2) having associated coeffi.cient vectorgiven by o*. Then we say that p* Ii,es on tlze boundary of P if everyneighborhood of p* contains n-th order polynomials both in 2 andin P", the complement of 2. Equivalently,

AP : { p " €P" ' B , (p * ) aP * S and B , (p * ) r \P ' + g f o r a I I € > 0 }

denotes the set of boundary points of P.

ExERCTsE 17.3.9 (Boundary) : If P is the set of stabie n-th orderpolynomials) argue tlnat 0P consists of all n-th order polynomialsIraving one or more roots on the imaginary axis. Hint: First expressthe polynomial as p(s) : KIIT:,@ -t z;).

REMART{s 17.3.10 (Topological Considerations for Matrices): Wenow proceed to develop matrix analogues for some of the defi.nitionsabowe. To this end, we view an n x Tn matrix A as an elementin Rmxn and define the relevant topological concepts in a mannerentirely analogous to the polynomial case. For example, if l i . l l is adeclared norm on the space of n x rn matrices and e ) 0 is given,the ball of radius e centered at A is given by

B"(A) : {A ' e P.nxn , l lA ' - A l l < , } .

3O2 Ch. 17 ,/ An Introduclion to Guardian Maps

Similarly, a set of matrices A a R *n is said to be open if givenany A € lt, there exists an open neighborhood of A which is whollycontained in "zt. Finally, if -4 is a set of rn x n matrices, we definepoints of closure of .4 and the set cl "4 exactly as in the polynomialcase. That is, .4 is a point of closure of A if every baII B.(A) aboutA contains points in both A and cl -4. As in the polynornial case,the boundary of A. denoted by 0A, consists of all rn x n matrices ,4having the property that every open neighborhood of ,4. meets both.A and A', the complement of A. Finally, we note that all of thetopologicai concepts above can be interpreted i.n terms of sequencesas in the polynomial case.

ExERCTsE 1-7.3.LL (Topological Concepts for X,Iatr-ices): If "4 isthe set of nonsingular n x n matrices, argue that cl A: Rt*' andd-A consists of all n x n sinsular matrices.

I7.4 The Guardian Map

In this section, we provide the formal definition of the guardian mapand some simple exercises i l lustrating its construction. In order toconstruct other useful guardian maps, we introduce some additionalmathematical machinery in the next section.

DEFTNTTToN 17.4.1 (Guardian Map): Let u : Pn - R be a givenmapping. Then u is said to guard an open set of n-th order poly-n o m i a l s P i f v ( p ) l 0 f o r p € P a n d u ( p ) : 0 f o r p e A P . F o rsuch a case, we call u a guardian map for P. For matrices, a sirnilardefinit ion applies. That is, if u:P-nxn + R, then z is said to guardan open set of n x n matr ices,4 i f

" (A) +0 for ,4 e A and u(A). : g

for A € 0"4. For such cases, we call u a guardian rTtap for A.

ExERCTsE 1,7.4.2 (Guardian Map for Nonsingularity) : Argue thatthe map defined by

"(A) : det A

guards the set of n x n nonsingular matrices.

ExERCTsE 17.4.3 (Polynomials Nonvanishing at Zero): Argue tl iatthe map defined by

u(p) : p(0)

guards the set of n-th order polynomials p(s) which do not vanishat the point z : 0.

304 CJn. 17 / An h-rrroduction to Guardian Maps

guards the set of n x n Schur stable matrices; i 'e', the set of rL x IL

matrices having eigenvalues in the interior of the r-rnit disc'

ExAMeLE 77.5.4 (Guardian Nlap for Schur Stabil ity of Poiynomi-

als): To construct a guardian map associated with the Schur sta-

bility problem for polynomiais' we begin with a polynomial

n/ \ \ - - ip ( s J :

h " n ,

and form the (n - 1) x (tt - 1) matrix

77.5 / Sorne Useful Guardian IVIaps 303

L7.S Sorne Useful Guardian Maps' fhe nhioni irra ^{ this section is to construct and catalos a nurrr-r r o u v u v

ber of guardian maps associated with robust D-stabil ity problerns.To construct the relevant maps, we exploit Kronecker products anclI{ronecker sums. These notions have already been covered in Sec-tion 4.L2 in the context of affine Iinear uncertainty struct'res. If .Ais an n x n matrix, we see below that the reiationship between theeigenvalues of ,4. and A @ A proves to be quite usefui.

ExAMeLE L7-5-I (Guardian NIap for Stable Matrices): We claimthat the rnap u ' Rnxn ---+ R described by

" ( A ) : d e t A e Afor ,4 g P.nxn is a guardian map for the set .4 of stabie n x n matrices.To justify this claim, we first observe that if A e 0A, then, usingtlre result in Exercise 17.3.9, it is easy to see that .A has one ormore eigenvalues on the irnaginary axis. Now, we must show thatu(A) : 0. Indeed, let .\,, (.4) be any eigenvalue on the imaginary axisand let )or(A) be an eigenvalue of ,4 which is the complex conjugateo f )1 , (A ) ; i f A i r ( , 4 ) : 0 , t hen take )1 , (A ) : A t , (A ) . Now, i n v i ewof Remarks 4.12.4, \r(A) -t Atr(A): 0 is an eigenvaiue of ,4. e ,4.Hence, A @ A is si.ngular and it follows that u(A) : A.

To complete the justification, we now assume that A € ,4 is suchthat u(A) : 0; rve must show tinal A € AA. Indeed, since ) : 0 isan eigenvalue of -zl O,4, using a similar argument to the one above,,4 must have an eigenvalue on the imaginary axis; Hence, it followsthat ,4 € AA.

ExERcTSE L7.5.2 (Guardian Map for Stable polynomials): Asgiwen in section 4.7, take H(p) to be the Hurwitz '''atrix associatedwith a polynorniai p(s) : LT:ooi"i with a, ) 0. Show that

,(P) : det II(P)

guards the set of n-th order stable polynomials with an ) O.

ExAMpLE 77.5-3 (Guardian NIap for Schur Stabil ity of Matrices):Notice that if A is an n x rL matrix and )r is one of its eigenvarues,

t l ren,4 e A-181 has lA; i t - 1 as an e igenvalue; see Remarks 4. I2.4.From this observation, it is easy to see that the map defined. by

A n A n _ 1 d n - 2 " ' A 3 A 2 - A O

0 a n c t r n - r " ' a 4 - a o a J - a t

0 - ao - - ' a r ' ' CLn - ( 1n -4 c l n - ! - c l n -3

-aO -Af -A2 ' ' ' -CLn-3 dn - an-2

of Jury and Pavlidis (1963). The proposed guardian z is now given

by the formula

"(p) : p(r )p(O)det .9(p) .

verification that u(p) satisfies the requirements for guarding the set

of Schur stable polynornials is relegated to the exercise below'

ExERcrsE L7.5.5 (Verif i.cation): With setup as in Example 17'5'4,

IeL 21,22, . . . ,2 ' denote the roots of p(s) . Using the fact that

det S(P) : oT-' I l (7 - za,z;,) 'i z > . i l . - 1

prove that u(p) is indeed an appropriate guardian rnap for the set of

Schur stable polynomials,

Exnn-crsE L7.5.6 (Guardian Map for a Damping Region): Given

any angle 0 e ftrlZ,zr), prove that the map defined by

guards the set of n x n matrices having eigenvalues which lie in the

damping cone given by

C : { z € C : 0 < f z < 2 r - 0 ] .

u ( A ) : d e t ( A 8 A - l a I )

306 Ch. i7 ,/ An lntroducdon to Guardian lvlaps

oolvnomial p(s) : "-

+ Il--01 oisi into the companion form rnatrix77.6 / Familieswith One fJncertain Para:rcter 305

No-rv construct a guardian map for the set of polynornials having allroots in this cone.

ExEn-crsE L7.5.7 (Guardian NIap for a Strip): Given any B > O,prove that the map defined by

u(A) : det[(,a + j Br) e @ - j Br)]

grrards the set of n x n matrices with eigenvalues in the strip

S : { z e C : l l r n " l < p } .

Now construct a guardian map for the set of polynomials having allroots in this strip.

L7.6 Families with One lJncertain Pararneter

wb now proceed toward a generalization of the result of Bialas; seesection 4.11. Instead of working with the strict left haif piane, 'veallow for rather general 2 regions. In addition, we work in a moregeneral setting allowing pol;momic dependence on the uncertain pa-lameter. For comparison purposes. note tha.t in Chapter 4, affineIinear dependence was assumed.

For the remainder of this chapter, we work with either an un-certain polynomial of the form

t

p(s , ) ) : f . l i pa(s )i :0

or an uncertain matrix of the form

[.A ( ^ ) : D ^ 0 A , ,

i :0

where the set of polynomials ps(s) , p t ( " ) , . . . ,p^s) and matr ices ,46,At , . . . ,A! above are f ixed. Note that for i :0 , we obta in the nom-inals p(s,0) : ps(s) and ,4(0) - As. Without loss of generality,we take the uncertainty bounding set to be A : [0, i] and consiclerthe family of polynomials P : {p(., A) : ) € A} and the farnily ofmatrices a : {A(A) : ) e A}.

Taking note of the natural embedding of the coeffi.cients of a

0 1

0 0

0 0

-ao -a t

0 0

1 0

0

0

I^ c \ P ) -

0 0- A 2 - C 1 3 " ' - a n - l

we henceforth restrict our attention to the family of matrices A'

once themat r i x resu ] t i sg i ven ,an in te rp re ta t i on fo r thepo l ynomia lcase is readil;r available'

L7.7 PolYnornic Deterrninants

In preparation for the main result of this chapter' rve need one more

technical concept' fJ*oti"ute the d'efinition to follow' notice tb'at in

the examples of the freceding section'.each of the guardian map eval-

uations v(A) isg"tJtli"Ju'itaking the- deterrrinant of some matrix

functicn /(A). nurih;-;tt;' obsen'e that /(A) depends polvnomi-

ally on the entries of A'

DEFrNrrroN 17.7'l- (Polynomial Determinant Representation): Let

,4 be an open set of'"' ""t'

matrices with guardian map 1' We sav

f,hat u admits a polynomial d'eterm'inant representat' ion'if ' there ex-

ists a marrix f*r"t;:;"; ' ,-11'"' --*_Rk*k such that /(A) depends

fotyno*i"tly on the entries of A and

u (A ) : de t / (A )

for all A effxn '

DEFrNrrro N 17 '7 '2 (Ivlatrix Representation) : Suppose that

L

A(r) : Lx"+,; - f i

is an uncertain n x n matrix and / : Rnxn ---+ p'bx'k i" rnatrix function

such that i (/) d;;;;;; p"ivryTl3llv on the entries of A' Then'

excruding the trr.ri i i -rppi"i F(A):0, the unique set of matrices

77 .B / The Theorem of Saydy, Tits and Abed 307

Fo, Ft , . . . ,F1, . such that Fry l0 and

^// / 1 / \ \ \ s - t

J @ Q ) ) : l A ' F ii :o

for all ) is called tine matrir representation of J(A(A)).

ExERcrsE L7.7.3 (Existence and Uniqueness): Under the condi-tions of the deflnition above, argue that the matrices F, exist and areunique. Hint: To estabiish uniqueness, notice that two polynomials

with the same values rnust have the same coefficients.

ExAMeLE L7.7-4 (Derivation of lvlatrix Representation): To i l-lustrate the concepts associated with the defi.nition above, consider-4(^) : ,40 -F AAr with

Now, with f (A) : Az , a straightforrn'ard computatiori yields

/ ( ,4())) :

Hence, t1r.e matlix representation of /(A(A)) is described by ,A/ : 2and

308 Ch. 17 / An Introduction to Guardian Maps

TITEoREM 17.8.1 (Saydy, T i ts and Abed (1990)) : Let D C C be

open and , l e tA - - { , 4 ( ) ) : ) e A } be a fam i l y o f nxn ma t r i ces

descri,bed bE!.

A(I) : Lsoe

is D-stable and' the matrir

I/I(A,D) : -F;r Fr

has no real eigenualues in the interual [1,+oo) ' More generallg' for

l/ > 1, the same result holds tai'th

0

0

0

-Fo tF

r 2 l [ r z ' ]l ; A t : l I- 1 3 l L 4 0 . 1

8) I 2 ) ,2+ ro r r a . l1 5 . \ - 4 8 ^ 2 + 6 ^ + 7 1

T -| g . \ ' ,+t ^l 4 A ' +

[ - r a lt lL - 4 7 1

A/r l ^ ' f ' t \ -r v ! \ ^ t u J -

1 0 0

0 . r 0

: :

0 0 0- - 1 - n - 1 n n - l n- f o ^ f

r u - r - I ' O - - t ' N - 2 - r o , c 1 ' , - 3

. 0

I- - t -

" - f o

- t t

From the expansion above, it is obvious that Fo, Ft and Fz provide

a unique representation.

L7.8 The Theorern of Saydy, Tits and Abed

We are now prepared to present the main result of this chapter. Thetheorem below applies to a large class of robust 2-stability problemsrvith one uncertain parameter.

* ^ [ t ' o l * ^ , [ n ' l1 1 5 6 . 1 L 4 8 J

[ - i e . l I s r oF n : l l ; F : l-

l - n z ) f r s ol s z l

; F z : l IL 4 8 l

17 .B / The Theorem of Saydy, Tits and Abed 309

not lraving any real eigenvalues in the intervai [t, +oo). Since ,46 isD-stable, we need only establish this condition for A e (0, 1J. NTow,since the guarding property of z implies that

"(Ao) : det F6 does

not vanish, we use the matrlx representation for /(A())) and write

/ N \

/ (1( ) ) ) : ro ( r + I s in ; I& l .\ "::r /

Hence,/ ! \

v (A( ) ) ) : de t Fo de t { r+ L ) 'Fb- '4 l ,\ = /

and it is apparent tb.at u(AQ)) + 0 for all ) e (0,1] if and only if

/ N \

der [ 1+ | r ;4 - i rn l # o\ = /

f o r a l l ) e ( 0 , 1 ] .To complete the proof, we interpret the nonvanishing determi-

nant condition above in terrns of the characteristic polynomial

/ N - l \

det(),r - I I(A,D)) :aet ( .rNr + t )tFb-lf . ,-N ) .\ ? : 0 /

Noting that the nonvanishing property of this determinant is invari-ant to the change of variable ):1/,\, i t follows that z(,4())) * 0for ) e (0,1] if and only if

/ - , \ + ode t ( )1 - M(A ,D) / ,

for ail ) e [1,oo). That ts, Iu[(A,2) has no real eigenvalues in thereal intervat 11, +co). E

ExERcrsE LT .8.2 (General Uncertainty Bounds): Nlodify the state-ment of Theorem 17.8.1 to handle the more general interval of the

I \ - \ + 1 . , r ^ i lIOrm /\ : L^ , /\ ' l lnsteao OI -1\ : LU, 11 .

ExAI\,{rLE 17.8.3 (Application of the Theorem) : We consider thefarnily of matrices .,4. described by

310 Ch. 17 / An Introduction to Guardian l\{aps

with

| - ^ - l [ . . ] r - - l. l - 5 u l , a , : I t

t l , o r : l - l

- j I ,a o : I o

L o - 4 ) L r 1 l L - 1 - 1 1

. \ e [ 0 , 1 ] a n d D r e g i o n w h i c h i s t h e s t r i c t l e f t h a l f p l a n e ' T o c r e a t elrrpii i" i Theorern iz's.t, we use the guardian map

"(A) : det AoA

of Example 17.5.1 and form

A(I) @ A())

This Lcads to poiynornial determinant representation

/(A(A)) : det u(A()) ) : Fo + AFr + \2 Fz '

where the F, are obtained by inspection of the Iironecker sum above;

we obtain

F N :

To apply the theorem, we veriSr that As is stab

f o r m t h e S x 8 m a t r i x

le and we need to

M(A,r) : l -0 , - - i , - IL-rt 'r, -Ft 'F l

A straightforward calculation yields

-0 .6667 -0 .75 -0 .75 -0 .8333

-0.3889 -0.6528 -0.4306 -0.6944

-0 .3889 -0 .4306 -0 .6528 -O-6944

-0.1944 -0.3958 -0.3958 -0.5972

o

- J

0

A ( ) ) : A o * A A t + A 2 A 2

F;LF1 x

17.9 ./ Schr.r Stabilitv 311

and

Fo rF2 x

An eigenvalue computation now leads to \,2(M) = 1.06tj0.99,

\3(NI) = 0.11 + j0 .46, Asp(M) = 0.11 + j0 .46 arrd ) .7,s(M) = 0.Since none of the eigenvalues above are approaching any purely real) e [1, co), we deem -4 to be robustly stable.

ExERCTsE 17.8.4 (Alternative Method): For the family of matrices-4 in the example above, show that the characteristic polynomial hast l re form p(s,A) : s2 I a1() )s + as() ) . F ind expressions for ae() )and a1 ()) Bv exarnining the roots of the equations a6(,\):0 anda1 ()) : 0, conclude that A is robustly stable.

L7.9 Schur Stabil ity

To dernonstrate the generality of Theoren 17.8.1, we see below thatwe can recover a known result for robust Schur stability. Subse-quentl}.. we deal with a more general robust Schur stability problem

involving quadratic dependence on ).

ExEFLcrsE 77.9.L (The Result of Ackermann and Barmish (1988)):Consider the family of uncertain pollmomiafs P : {p( , A) : ) e A}described by

p (s , A ) : ) po (s ) + (1 - ) ) p r ( " ) ,

where ps(s) and pr(s) are fixed polynomials and p6(s) is assumedto be Schur stable with deg po(s) > d"gpr(s). Using the guardianrnap u(p) : p(1)p(0)det ,5(p) given in Exercise 17.5.5, argue thatP is robustly Schur stable if and only if .9-t(po).9(pr) has no realeigenvalues in (-oo,0] . in the expression above, pi(s) is viewed asa polynomial with the same degree as ps(s) for conformability ofrnatrix multiplication.

ExER-crsE LT ,9.2 (Quadratic Dependence): Consider the uncertainfamily of matrices "4 described b1'

3LZ Ch. 17 / An IntroducLion to Guardian Maps

wi th,4s, At ,Az e R'* ' , -As Schur s table and A € [0, 1] . Let t ingAt : At-46 and taking Fo, Ft and F2 to be a matrix representationfor

f ( A ( ^ ) ) : A o & A o - I & I + ) [ 1 0 e A t + A t g , 4 o ] + \ 2 A t & A r ,

conclu.ie that with 2 being the interior of the unit disc, "4 is robustlySchur stable if and oniy if the matrix

M ( A , D ) :

has no real eigenvaiues in the interval [1, +oo).

L7.LO Conclus ion

For robust D-stability problems involving families of polynomials andmatrices having a singie uncertain parameter, the theoly of guardianmaps appears to be quite powerful in a robust 2-stability context.The ideas in this chapter can be extended to address more general Dregions which admit semi.guardi.a?z maps rather than guardian mapsand systems with rnore than one uncertain parameter; see the notesto follow for further discussion. IIowever, the computational corn-plexity associated with the multiparameter case can be prohibitive.

, 4 ( ) ) : A o * A A t + ^ 2 A z

Notes and Related Literature 313

Notes and Related Literature

I{FLL 17.1 In Section 17.1, we mentioned that robust stability is often decid-

able in a finite number of steps. To elaborate, suppose p(s, q) is an uncertain

polynomial with coefficients depending polynornially on the components et of q

and the uncertainty bounding set Q is a box. Since the leading principle minors

A1 (q) of the associated Hurwitz matrix also depend polynomially on g, the ro-

bust stability problem boils down to a positiwity problem; i.e., deterrnine if the

set of multivariable polynomials {Ai(q)} is positive on the box Q. This problem

is solwabie in a f in i te number of steps; e.g. , see Bose (1982) for a n ice exposi t ion

I.IFLL 17.2 In the feedback control literature, the issue of decidability was first

raised in the context ofoutput stabilization by Anderson, Bose and Jury (1975).

At the heart of their theory is a tree-branch-type algorithm motivated by the

decision calculus of Tarski (1951). The practical applicability of these ideas is

severely inhibited by the fact that the number of tree-branch contingencies grows

intoierably fast as a" function of data dirnension. Although less explicit about the

use of Tarski-like ideas, the zero set theory in papers such as Hertz, Jury and

Zeheb (1987) and Zeheb and Walach (1981) is iirnited in the same way.

NRL 17.3 In Theorem 17,8.1, the hlpothesis that z(A) is a polynomic deter-

minant map can be rveakened. If u(A) is simply polynomic but not necessarily

"realizable" via a determinant operation, a robust 2-stability criterion may still

be achievable via the theory of Sturrn sequences; e.g , see lVlarden (1966).

NRL 17.4 By using the slightly more technical definition of a sem.iguardian rnap,

an even richer theory results; see Saydy. Tits and Abed (1990). These a.uthors

also provide some interesting extensions of the theory for systems with more than

one uncertain parameter. For example, for the case of a bivariate polynomial

desc r i bed byp (s , ) r , ) z ) : L ' , n r \ \ ' ) ?p r r , ; " ( s ) w i t h bound ing i n t e r va l s A r and

A2 for )1 and )2 r'espectively, under rather weak hJpotheses onD, it is possible to

reduce the robust D-stability problem to a. finite set of single-variable problerns

NRL 17.5 For a fuller exposition of many fundamental concepts underlying

the theory in this chapter, see the paper on root clustering by Gutman and

Jury (1981). For a more complete t leat ise, see the book by Gutman (1990).

NRL 17.6 In some cases, the guardian maps corlstructed in this chapter are

obtained from matrices rvhose dimension is not necessarily rninirnal. For example,

instea.d of using the n2 x n2 matrix A @ A, one can work the lower Sclilaflian

rnatr ix A[2] ; see Brocket t (1973). I t is a lso possib le to use the theory of b ia l ternate

products, Bezoutia,ns and Sylvester resultants to construct a nurnber of novel

guardian and semiguardian maps. This topic is pursued in the paper bv Saydy,

Ti ts and Abed (1990).

Chapter 18

ifhe Arc Convexity Theorern

I

lii ll l

iii,1l i

ili.,,1i : . 'I

:i '

SynoPsls

Associated with a po)ynomia) p(s) and an interval Q e R is a

frequency T-esponse a;c' Tltis arc'is obtained by sweeping the

frequency Q over Q and ptotting p(ja) in the complex pla'ne'

In this chaptet, Ive esta;ljsh tie Art Convexity Theorem of

Harnann and Barmislt' Tltat is' jf p(s) is stable tvith net phase'iong"

of 780 degrees or less' as a .increases

o.rre-r Q' tJre as-

sociated rr. ,rru"i- be convex. The cbapter also includes some

exfensions and ramifications'

18.1 Int roduct ion

Throughout this text, we have seen that the proof of robustness re-

Sul ts involv inguncerta inpolynomialsof tenle] . iesonclassica]p lop-erties of fixed polynomiis-properties that have been known for

decades. For example, in the ptott "t

the Zero Exclusion Condition

in Section 7-4,weex;loited' the r'vell-known fact that the roots of a

O.ft".-t"f depend continuously on its coemcients' A second exam-

pie is provided by the proof of l{haritonov's Theorem' which involved

i"oi"ir"ri." of tte Minotonic Angle Property; see Lemma 5'7'6' In

this chapter., we establish a fundamental convexity property which

i. l;d* than the Monotonic Angle Property' That is'- given a

,r"irL" p"olynomial p(s), if we plot p(ju) fot r"' Z- !' we obtain an

arc which has a speci;l convexity propertyl roughly speaking' arc

314

316 Ch. 18 / TheArcConvexityTheorem

IB.2 / Definirions for Frequency Response Arcs 3I5

convexity implies a monotonic angle but not conversely. A secondobjective of this chapter is to raise possibilities for the application ofarc convexity in a robustness context. It is felt that further researchalong these lines would be fruitful.

1.8.2 Definit ions for Fbequency Response Arcs

In this section, we provide the basic definitions which are essentialfor the er<position of the main result of this chapter. In the definitionbelow, the "arcs" n'hic]r we describe are in fact portions of the wellknown lVlikhailov piot; see lVlikhailov (1938).

DEFrrvrrroN 18.2.1 (Frequency Response Arc): Given a poiyno-mial p(s) and an interval f,) C R, the plot of p(jc,L) for o increasingover f) is called a frequency response a,rc or simply an arc.

DEFnvrrroN 18.2.2 (Properness): Given a polynomiat p(s) and aninterval f) C IL, the associated frequency response arc is said to beproper if it does not pass through the origin and the net change inthe phase of p(ju) is no more than 180 degrees as c.; increases overO; otherwise, the arc is said Lo be unproper.

DEFn\r:froN 18.2.3 (Convexity): Given a polyrromial p(s) and aninterval f) C R, the associated frequency response arc is said tobe conuer if the following condition holds: Given any two distinctfrequencies u)t,u)2 € f), the arc does not intersect the interior of the" t r iangle" wi th ver t ices z0 - 0, z t : p( jc . t ) and z2: p( juz) . Thisdefinition also applies to the degenerate cases where zo1 zr arrd z2 arecollinear. Wlien this condition fails, the arc is said to be nonconuex.

REMARKS 78.2.4 (Geometry for Convexity Definition): Takingp(s), ,'tt and t't2 as in the definition above, the notion of arc con-vexity is easily understood with the aid of Figure 18.2.1. We seethat arc A is convex; notice that triangle zs,z1,z2 is not penetrated.On the other- hand, arc B is nonconvex because there is a frequencyrange (rot,r"*) for rvhich the arc is interior to triangle zg,z1,z2. An-other important point to observe is that arc B can be identifiedwith a polynomial having monotonically increasing angle but whichis nonconvex. In the theorem to follow, we see that such behavioris irnpossible if p(s) is stable. In other words, arc convexity is notimplied by monotonicity' of the angle; a deeper level of analysis isrequired.

z o : 0 R e

FIcunB 18'2'1 Convex and Nonconvex Arcs

ExAMPLE 18.2.5 (Four Basic Arc Types): To i l lustrate the basic

ia"u, rU"t", notice it*t l" Figure 18'22: arc A is proper and convex'

a,rc B is proper uod oo"toot'"x) arc C is improper and convex and

arc D is imProPer and nonconvex'

18.3 'Ihe Arc ConvexitY Theorern

The proof of the theorem below is relegated to the next two sections'

TrlnoR-EM 18-3.1 (Hamann and Barmish (1992)): AII proper arcs

associated. with the JrequencE response of a stable polynomial are con-

UCT.

ExERcrsE 18.3.2 (strict conve:

response arc associated with a Po

R ts strictlE convex if, given aq

O)r,Lr2 € f), the arc intersects the tr.

,r ' : iur) and z2 : PUaz) at onlY.the

p.fvtt"Li"is of degree ri > z, prove that all proper frequency response

arcsarestr ic t lyconvex.-H, inf :Sinceconvexi ty isestabi ishedint i retheorem above, rule out the existence of l inear sections of an arc; i 'e' '

i f for some .orrrtoni, o,b € R we have Re p(ju) : alm p(ju) -f b on

l',ji i

i iI . i

i l! t li 1

i,. I[ i ,, itr:'.1'i1i ', i,,,1l ' , ; ii , ' " ' l

nonconvex arc z 2 : p ( j u 2 )convex arc

z 2 : p ( j u t 2 ) '

Int

IB.4 / Machinery 1: Chords and Derivatives 317

-250 -200 -150 -100

FIGURE 18.2.2 Four Basic Arc TYPes

an interval of frequency having positive length, argue that analyticity

of p(ju-,) demands thaL Re p(io:) : alm' p(ju:) l_ b for ali cu e R'

18.4 Machinery 1: Chords and Deriwatives

This section and the next two are devoted to the developrnent of the

technical machinery which is used in the proof of the Arc Convexity

Theorem. The leader interested soiely in application of the theorem

can proceecl directlv to Section 18.8. We begin the exposition by

nraking a distinction bett'een properness and strict properness of an

arc. An obvious proper:ty reiating strictly proper arcs and chords is

also given.

DEFrNrrroN 18.4.1 (Strictly Proper Arc): Given a polynomialp(s)

arnd a frequency inter-val S-t: ["rr,cu2] , we say that the associated arc

is strictly proper if, as cu increases from o1 to a2, p(ju) I 0 and the

rret angle change of p(ju) is less than 180 degrees.

DEFrNrrroIv 18.4.2 (Chord): Given a polynorrial p(s) and a fre-

quency interval f) : lu.,r, c,,r2], consider the straight line segment join-

ing tlre endpoints p(i.,;_) and p(jut2) of the associated arc. We call

this iine the chord associated with this arc.

100

-50

- r00

_ I D U

-50 0 50 100 150Rc

arc A

3f8 Ch. 18 / TheArcConvexiryTheorem

LEMMA 18.4.3 (Chord of a Strictly Proper Arc): If a c-hord asso-

c'iated, w'ith a strictly proper arc has rLorLzero enrlpoints' then it does

not pass t'l'trough Lhe origin'

PRoor: The conclusion of this lemma follows easily from the fact

that a chord passing through the origin would correspond to a net

angle change of at least 180 degrees' E

REMARKS 18.4.4 (Derivatives): We now provide a lernma which

g"rr"rutiru. the fact ihtt "t''bility

of a poiynomial p(s) implies stabil-

ity of its derivative y1"1. Vf"tg!"tr"rttt ' , we reiate the roots of p(s)

lo tfro." of p'(s)' fhu f"*ma betow is due to Lucas; e'g' ' see Mar-

den (1966 ) .

LEMMA 18.4.5 (Roots of a Derivative Polynomial): G'iuen a poly-

nomial p(s), the rools of i ' ts deriuatiue

/ , \ dP l s )v \ " / d s

tie within the conuer hull of the set of roots of p(s)

18.4.6 (Proof): Prove the lemma above' Hi'nt: F.:,rst'

: 1{fl}r(s - sa) and then show that

n 1

p ' ( r ) : P ( " ) E *

Subsequently,it z is a root of p'(s)' argue that tf z y'conv{s;}' then

f t + 0 .Z 2 ' - c . u xZ : L

REMART(s :18.4-7 (D-StabIe Case): Notice that if p(s) is a D-stable

potvno-iut and'D is convex, it folloyl.that the convex huII of its root

set must be a polygon which is wholly contained in D' I{ence' D-

.i"Uifi,v of p(s) l*it iu. Z-stabil ity of p'(s)' In the. sequel' we take

D to be the stricb left half plane, noting that a similar arc convexitv

r e s u l t c a n a l s o b e e s t a b i i s h e d f o r t h e m o r e g e n e r a l c a s e w h e n 2 i sconvex; see 'che notes at the end of this chapter'

EXERCISE18.4.8(Curvature) :Giventr ,v ice-d i f ferent iablefunct iorrsoift"ot,"n.y X(.) 'and Y(o-'), Iet cu € R be a frequency at which at

EXERCISE

write p(s)

1B'5 'z MachinerY ?: Flow 319

least one of these functions has a nonvanishins derivative. For suchc,.r, the curvature of the curve associated with

Z ( a ) : X ( , t ) + i Y ( . )

is gi.ven bydX d2Y d2X dYil-tr

- wil

lrgl' * (#)213/2(a) Taking p(s) to be a stable polynornial, X(r) : Re p(ja) anc)Y (u) : Irn p(j u), prove that

C2(a) > 0

for a1l c.., € R. Hi,nt: Taking note of Lemrna 18.4.b. exploit the factthat stability of p(s) implies stability of the derivative polynomialp'(s). Hence, the angle of p'(j.) :Y'(,,:) - jX'(.) is increasing.(b) Prowide an example of a polynomial p(s) which has the propertythat the associated curvature of p(ja) is positive over a proper arcwhich is nonconvex.

18.5 Machinery 2: F low

For a polynomial p("), we now develop some ideas involving thetrajectory of the frequency response as c,r is increased. Associatedrvitlr each frequency o € R is the fl,ou of the response which onemight liken to a velocity of p(s) at the point s : ju.

DEFrl.rrrroN 18.5.1 (FIow u1 5 : ju): Given a polynomial p(s)and a frequency c,-r e R, the flow at s : jcu is given by

F,(,):e#D*iL!!#!Y)

LEMI\nA 18.5.2 (Nlonotonic Angle of Flow): Giuen a stable poly-r-romialp(s), i,ts f i,ow Fr(ut) has a angle u-thich is a continuo,us, rlon-decreasing function of the frequency c.., € R.

PRoop: By applying the chain rule, it is apparent that

3?0 Ch. l8 / The Arc ConvexitY Theorem

Since p/(s) is stable (see Lemma 18'4'5)' i ts angle is monotonically

increasing (Lemma S)i 'ol-Hence, angle monotonicity for Fo(r'r) fol-

lows immediatety from tire equality above' H

NorA:uoN 18.5.3 (Some llalfplanes): We now introduce some s1m-

pf" g".*"tric notation to aid in the convexity analysis of frequency

response arcs. IndeeJ, gi.r"n any poiynomial p(s) and a frequency

;; e R such that, (; ;'. i j o, "oosa"'ii;:,::'fr}Lil x"fr:::*:fi

the complement of Ro (cl.)' Thes'

in F igure 18 '5 '1.

f .t

I

Ll ,i )

!' ' ', 'I

ILt :)l ,I

t ,t

i.II .t

it .It :L : ,L

i ' 'i ' 1 ,

i -

FiGURE 18'5'1 The Halfplanes 7lf, and72-'

Associated with a polynomial p(t) lt a second set of halfplanes

obtained from chorcit-"1;;g u'tt u'" ' ' IncLeed' if p(s) is a polynomial

trictly proper arc for frequencies

o2) to be the closed haifPlane con-

re origin of the comPlex Plane' and

halfplane which is the complement

rsider the two auxiliary halfplanes

ich are simply parallel translates

ith their separati lrg plane passing'e 18 .5 ' 2 .

Fo@): 4?1=*f1":,- : jp',(j-)

78.6 / Machinery 3: The Forbidden Cone g2l

p( jaz)

Re

=-- PU*i^ t + ' t

, f t r . g \ L D t , a Z ) -. / L n n4 t - /t - L o p \ u l a 2 ) ' , - - - !1 [ ( - ' ' " )

- ( o \ t ' t t , a z ) - - -

322 Ch. 18 / The ArcConvexiryTheorem

Pnoop: By Lemma 18.6.1, the flow must obey the beginning con-

straint Fo(r) e RI (u) and endpoint constraint Fo(u,2) e rzf G:z)

Furthermore, using Lemma 18.4'5' the angle of the flow is contin-

uous and nondecreasing in r..r. Therefore, as cu increases from cu1

to u2, the flow (viewed as a vector) can only rotate continuously

couoterclockwise. Notice ihat Fo(ar) begins inRI(o;t), and by strict

properness of the arc, it follows that f|(c..') reaches RI(i'z) before

"*iiitrg RI (rr) Hence, Fr(r) remains within RI (c'tt) u 7Z; (u2) for

a l l o € l u 1 , a 2 ) - 3

REMARKS 18.6.3 (Describing the Forbidden core): As a conse-

quence of Lemma 18.6.2, we have established the existence of a for-

i i l ,a"n coneinthe complex plane; i.e., a nonempty cone rn'ithin rvhich

the flow for a strictly proper alc may not reside; see Figure 18.6.1.

T h i s c o n c e p t i s s u m m a r i z e d i n t h e f o l l o w i n g l e m m a . N o p r o o f i sgiven because the result follows immediately from Lemma 18.6.2.

FIGURE 18.5.2 The Halfplanes HI,?1; aud Their Translates

18.6 Machinery 3: The Forbidden Cone

In this section, rve establish the existence of a special cone in thecorrrplex plane which cannot be penetrated bv the flow associatedwith a strictly proper arc of a stable polynoraial.

LE^4^4A 18.6.1 (Stable Fiorv Restriction): G'iuen a stable polgno-mial p(s) and a frequen,cg t- '* € R, its f lnu satisfi.es the cond,it ionFo(.") e R[(u-). Furth.errnore, there etists an e > 0 such thatp( i . ) e RI(u. ) for a l l f requenc' ies a e (u* ,a* I e) .

PRoop: Viewing the flow Fr(..) as a derivative vector for p(ja) at4,1 : L4+ as in Lemma. 18.4.5, the monotonically increasing angJ.e ofp(ja) dictates that Fo(a.,-) e RI(u-). Now, this sarne flow condi-tion forces p(ja) in nf@.) for aI] o in some sufrciently smalr rightneighborhood of a'". F

LEh4r\4A L8.6.2 (Stable Flow Further Restricted) : Suppose that p(s)is a stabLe polgnomial whose ualues p(ju) generate a stri.ctly properarc for cu € f) : l ,,tt,a2l. Then the assoc,iated fl,ou sat.isfies

Fo@) e RI (a) u RI (toz)

for all frequenc'ies uL € 9.

Frcunn 18.6.1 The Forbidden Cone R;(ar) oRo @tz)

LEMMA 1g.6.4 (Forbidden cone): suppose that p(s) z.s a stable

polynom'ial whose ualues p(ju;) generate 7 strictly proper arc for fre'' quJn r i " ro

€ o : f a t , a2 l . ThenR; (u t )nRo@z) f A ' Moreoue r ,

Fo@) #R;(uL)nRo@z) for a l l f requen'c ies cu e f ) '

Irnp ( i " 2 )

RI(o,t)

R; (t'tt)

R; ("Lr) aRl @u)

[@r)',n; @r)

lB.7 / Proof of the Arc Convexity Theorem 323

L8.7 Proof of the Arc Convexity Theorem

For a polynornial of order rL : 0, the result is trivial since the fre-quency response arc is a point. The analysis to foilow thereforeassumes that n ) 1. Indeed, given a stable polynomial p(s) and afrequency interval Q: [or, a2] C R such that the arc d.escribed byp(ja) for cu € f,) is proper, to establish convexity it suffi.ces to showthat the arc l ies in cIH[(u1,t,,2). To this end we consider two cases.Case 1: The arc is proper but not strictiy p-roper. In this case, thenet angle increment is exactly 180 degrees, and the chord joiningthe endpoints of the arc passes through the origin. Therefore, thearc cannot intersect the chord for any ut* e (a1,u2); i.e., if p(jut.)intersects the chord for some us" € (tt1,c..,2), then either LpU.t):|p(j.") or {p(ju2) : Lp(ju"), which contradicts the increasingphase of p(jt'.,).Case 2: The arc is strictly proper. We now claim that

Fo@r) e clTtf,s(ut, ..,z),

from Lemma 18.6.1 we know that Fr(u,t1) eR[(tt). Letting int Zdenote the interior of a set z , to prove'the clairi,'we'must thein show ittthat a florv in RI (ut) n intHo,s@r1.,,2) is not possible. proceeding E fby contradic t ion, assume Fo@r) eRI(a)n intHo,o@1,c, . r2) . Now, I i

?o

by contradic t ion, assume Fo@r) eRI(a)n intHo,o@1 ,c, . r2) . Now,by Lemma 18.5.2, the phase of the flow is nondecteJsing and hence is f. Icapable of only counterclockr.vise rotation. Notice that the condition i, fFr(.r) e RI (o;t)ninr']1o,o\tt, cu2) requires that there exists an € > 0 ]&such that p(ju) e RI (a1)aifiH, (a1, r,.t2) for to € (t t1, utrtel. Hence,in order for the arc to return to the chord at p(j-z), the flow tnustenter 71[,o(ur,@2). However, this wou]d require that en route, theflow must enter the forbidden cone Ro (a) a?-i (u,,2), contradictingLemrna 18.6.4. Thus the claim is established.

We are now ready to complete the proof. We need to prove thatp(iu) e clH[ (at,us2) for aI] r,.r € f.r,rz). Since rve already knorv thatp(.iL'.,r) e clH[(ttr,r,-t2) and p(ju:z) e clH[(ur,tt2), we proceed bycontradiction. Suppose that for u* € (w1,..,z), p(j.) € cll l [\LDL,L1)2)for a l l w € 1u1,r , . , * ] but p( j (a* + e)) e in tH,(u: t ,az) for a l l e ) 0sufficiently srnall. This implies that there exists some €* ) 0 suchthaL Fo(a) € inrHr,o@1,u-t2) for all cu € (w*,r* -F e*]. Now, similarto the argument iri the claim above, in order for the arc to returnto the chord at p(ju;z), the flow must enter H[,o@t,r,r2), However,this would require that en route, the flow musi'enter the forbidden

324 Ch. 1B ,z The Arc Convexity Theorem

cone Ro@L) nR;(':), contrad'icting Lemma 18'6'4' The proof of

Theorem 18.3.1 is norv comPlete. B

18 .8 Robus tnessConnec t i ons

To stimulate further work on the connections between arc convexity

and robustness theory, we begin with a definition'

DEFrNrrrorrr 18.8.1 (Inner Frequency Response Set): Given a

stable polynornial p(s), the ' inner frequenca Tesponse set, denoted

IFRSlpl , li defined to be the open connected subset of the corrplex

plrn"-*hich contains the origin and is bounded by the curve obtained

by plotting p(jr,-.') rvith cu varying from -co to *co'

ExAMPLE 18.8.2 (Gener-ation of Inner Frequency Response): For

the hlgh order polynomial

p(s) : s12 + 2.53s11 + 25.404s 'o + 47 'g24se + 224.75s8 + 311'96"7

-F 856.72s6 -F 846.82s5 t 1388.22sa -F 920.54s3

)- 770. I0s2 * 303'08s + 36.713,

the plot of p(ja) is generated and the inner frequency lesponse set

is indicated in Figure 18'8.1. We also use this example to motivate

the anzrlysis to follow. The lbllorn'ing observation is critical: Imagine

a squar.e of radius r 2 0 (centered at the orgin) inscribed inside

the inner frequericl.r 'esponse set. For small values of r, this square is.\n ho11), contained within the set. As r increases, the square eventually

contacts the boundary of the inner frequenc)/ Iesponse set a! an

extreme point.

REMART(S 18.8.3 (Convexity of the Innel Frequenclr l lssp.nse Set):

In the example above, it is no coincidence that the inscribed squale

contacts the inner frecluency respollse set at an extrerne point rather

than along an edge. This type of contact is a consequence of the

Arc Convexity Theorem; i.e., since p(s) is stable, it can be shown

that its inner frequency response set is convex; see llamann anci

Barmish (1992) for a formal justification of this statement. The

next exercise elaborates on these comments'

ExERCTsE 18.8.4 (Inscribed Poiygon): Let p(s) be a stable poly-

nomial rvith a convex inner frequency response set and let A e C be

,ii

lB.8 ,/ Robusmess Connections 325

BO

60

40

20

Irn 0

_20

-40

-60

-80-200 -150 -100 -50 0 50 100

Re

Ftcuno 18.8.1 The Inner Frequency Response Set for Example 18.8.2

any (convex) polygon rvhich contains z:0 in its interior. Let

.ynaa : sup{l, : p(j,,;) I 1t\ for att r,.' e R}.

Beginning at ^/ - 0, argue that as 7 increases, first contact betweenp(jru) and 7A occurs at an extreme point- That is, when 7 : -fmaz,one of the extreme points of the polygon ^f^orL contacts the arcassocjated with the polynorrrial p(s).

ExER-crsE 18.8.5 (Stabitity-Preserving Complex Gains): For agiver rational function fi '(s), define the notions of arc, proper arc)convex arc and inner frequency response set IFRS['R] by mimickingthe definitions given in the polynomial case. Note that r't(s) is notnecessarily assumed to be proper.(a) Argue that if R(s) is strictly proper, irs inner lrequency responses e l , i s e i f h c r c m n t v n r f h e s i n o l p f n . 1 O 1

(b) If I i(s) is either proper or improper without zeros a.long theirnaginary axis, ar-gue that its inner frequency response set is anopen set containing z : 0 as an interior point.(c) Suppose that P(s) is proper and stable and consider the classicalinterconnection shown in Figure 18.8.2. Allow 1{ to be a complexgain and Iet K,no, C C denote the largest pathr.vise connected set ofgains which contains 1{ : 0 as an interior point and preserves closed

32C) Ch. i8 / - fheArc ConvexityTheorem

FIGURE 18.8.2 Feedback Interconlection for Exercise 18.8.5

Ioop stabil ity. Prove that

t r_lK - o ' : I F R S l E lL t ,

Hint: IJse the Zero Exclusion Condition (see Theolern7-4.2).

REMART{s 18.8.6 (Extreme Point Destabil ization) : Let the plant

P(") : Np(s) / D p(s) be proper and stable and consider the feedbacl<

system in Figure 18.8.2. Taking the set of admissible gains to be the

polygonK : conv{ I ( t , I {2 , . . . , I { t } ,

we define the robustness margin in the usual way; i.e.,

rmar: sup{r :ciosed loop stabil ity is assumed for all K e rK}-

Now, observe that if the inner frequency response set for IIP(s) is

convex, by invoking the result in the exercise above, we arrive at the

following conclusion: As r increases from O to r*or, the first loss of

stability occurs at one of the extreme points r-o"Ii; of r*orK. In

other words, to study closed loop stability, lve need only consider the

ertrerne potEnom'iat;;Tl:?,or"

r"l * D p (s)

w i t h i e { I , 2 , . . . , ( . } . L e t t i n g

rmat. i : in f { r : V; , ' (s) is unstable} ,

it follows that

1B.B ,u Robus[ress Connections ZZ7

f'maa : pf;r*"", t .

REMARKS 18.8.7 (Unrnodeled Dynamics): The ideas above canbe readily extended to deal with classes of unmodeled dynamics. Toelaborate, in Figure 18.8.2, suppose that we replace the pure gain1{ in the feedback path by a proper stabie rational function A(s) ofthe form

A ( s ) : 5 \ 1 ' - ( s ) ,

wirere I4l(s) : Nyy(s)lDpy(s) is a weighting function describing theshape of the frequency response and 6 is a complex gain. TakingA*o, C C to be the lar-gest pathwise connected set of gains whichcontains 5 : 0 as an interior point and preserves closed ioop stabiiity,the ar-guments given above are then easiiy modified to arrive at theconclusion thaL

A.*o,: IFRS f_LlI W P )

Ifence, if 7/[I,I/(s)P(s)] has a convex inner frequency response setand if the admissible set of gains is a polygon

A : c o n v { 5 r , 6 2 , . . . , 5 e ) ,

then robust stability can be studied using the extreme porynomials

Vr, , (s) : r6; l /p(s) j \ I rv(" ) + D p(s)Dry(s)

v, ' i th i - - 7 ,2, . . . ,1 . For example, i f A is the uni t square, we needonly consicler the four extreme poiynomials

Vr, , (s) : r (1 +7) ivp(s)1t f t4 , (s) + Dp(s)Dys(s) ;Vz, . (s) : r (1 -7)1t /p(s) t f ta , (s) + Dp(s)Dq,(s) ;

Vs , , ( s ) : r ( - I - t j )Np (s )Ny , ( s ) + Dp (s )Dv r ( r ) ;Va , , ( s ) : r ( - L - j )Np (s )Nyy (s ) + D i " ( s ) DwG) .

In view of the discussion above, if rR(s) is a proper rationalfunction, it is of interest to give conditions under which its inverset/R(s) has a convex inner frequency response set. One class ofrational functions having this property is described berow. This isthe class considered by Tesi, Vicino and Zappa (fgg2); see the notesat the end of the chapter for further discussion.

928 Ch. 18 / The Arc Convexity Theorem

EXERCTSE 18.8.8 (stabie Ait-Poie Tlansfer Functi 'ons) :

stable all-pole transfer function described by

I{P (s ) : = - - ^ -

D p l s )

with l{ € R, prove that IFRS [+]

tt convex'

For the

ii

18.9 Conclus ion

suppose thatZ ( a ) : X ( u ) + j Y ( . )

is a complex frequency function with convex arcs and tlt(u) is a posi-

t ivefunct ionof f requency.Theni t isof in terest todevelopcondi t ionson Q@) under which tine scaled funct'ion

. z(.)7 ( , , t \"v \ * l , ! ( r )

Notes and Relatecl Literature 329

Notes and Related Literature

NILL 18.1 A more generai form of the Arc convexity Theorem is given inH@ana ard BdEish (19921. By sftepic the h.,'rdary lla dlqrr.i r^-r rc&tion reqion r, we obtzil octoeralized rretu Chapter 19cation region D, we obtain generalized lrequency response arcs For a D-stable Unap

polynomia"l, these arcs are again convex. A primary source of motivation forthis more general framework is the schur stability problem; i.e., since the unitdisc is convex, all proper arcs associated with the frequency response of a schurpolynomia,i must be convex.

^IRL 18.2 In Exercise 18.4.8, the relationship between arc convexity and curva-ture rx'as explored. In this regard. it is important to mention the body of ]iteraturedeaiing with the clockwise property of the Nyquist locus; see Horowitz and Ben-Adam (1989), Bartlett (1990b) and Tesi, Vicino and Zappa (1992). The departurepoint for corlparison with the Arc Convexity llheorem is the interesting resultirr Tesi, Vicino and Zappa (Igg2): i.e., if p(s) : I{/Dp(s) is a stable all_pole trans_fer ftinction, the clockq'ise property for the Nyquist prot of p(jut) is guaranteed ift l r e r oo t s o f p ( s ) a l l l i e w i t h i n a damp ing cone C : { z e C : n -0 < az < r - 1 0 }rvith ldl < l. For the same transfer furction, the Arc co'vexity Theorern leadsto a rather different resu.lt. Namely, the Nyquist piot of the inverseplant p-1(s)

always has the clockwise property-even if the dampi'g cone requirement aboveis violated. Another important point to note is that arc convexity is a more strin-gent requirement than the clockwise property; i.e., arc convexity irnplies that theclockwise property is satisfled but the converse does not hold. one critical dif-ference is the "centering" about z :0 in the arc convexity framework, For thisreason, the positive curvature result in Exercise 18.4.8 is not easilv modified tovie ld a proof of Theorem 18-3.1

19.1 Int roduct ion

The title of this chapter is a deliberate misnomer' A more appropri-

ate chapter title would be "Five Problem Areas of Potential Inter-

est." The choice of chapter title was dictated by two considelations:

First, the use of the rather flippant title "Five Easy Problems" is

a way of signaling that the style of presentation in this chapter is

intended to be less formal than its predecessors. our goai is to stim-

ulate new research directions rather than providing "mainstream"

results which are intended to be cast in stone. In a sense, the reader

should view this chapter as a biased survey of a subset of recent de-

the index under the letter N.

330

Five Easy Problerns

Synopsts

Tltis final chapter provides an overview of five research direc-

tions involving robustness of systems with real patametric un'

certainty. \\re intmediately make the disclaimer that the five

areas being described represent the author's bias' If a later edi-

tion of this text emerges, )t is quite possible tltat thjs chapter

will look dramaticallY diffetent.

19 2 / Problem Area 1: Generating Mechanisms 331

l-9.2 Problern Area 1: Generating Mechanisrns

Let p(s, g) be an uncertain polynomial q,ith some compricated uncer-

be given? In the current l i terature, there is sti l l not a clear un-derstanding rvhat constitutes a "strong" robustness result versus a"weak" one. For example, is a robustness result considered ,,strong',if its irnplementation requires solution of a nonlinear program with-out a guarantee that a global optimurn is attainable? rf the answeris no to this question, then we can consider the question again withthe r.vord t 'convex" replacing,,nonlinear.',

With regard to the questions above, it is felt that an impor_tant line of research involves studying how the structure of feedbackcontrol systems gives rise to special mathematical properties whichfacilitate solution of the robustness probrem at hand. For example,can we describe rich classes of feedback systems which give rise toan uncertain closed loop polynornial p(s, q) having special propertieswhich can be exploited to facilitate robust stability analysis?

we use the rvords "generating rnechanism" in connlction r.vith

not arbitrary. To illustrate, if two uncertain branch gains q1 and, q2appear in touching loops, then the product terrn q1q2 cannot appearwhen Mason's rule is applied. we now describe orr" lin" of researchrnotivated by these ideas.

The ideas belo.v come from Barmish, Ackermann and Hu (1gg2).Note that the uncertain parameters qi can be taken as either realor complex. we now concentrate on a special class of uncertainpolynorlials q'hich has a speciar property-polynomials in this crassadmit a decomposition which can be described via a tree diasram.This idea is i i lustrated via examples.

33? Ci'. ' ,9 / Five EasY Problems

irg.2.1 The Tbee-Structured Decornposi t ion (TSD)

The objective of this subsection is to describe some special plop-

erties associated with classes of uncertain polynomials obtained via

application of Nrlason's rule. In the exposition below, if q e Rz and

I C {7 ,2 , . . . , 1 } , t hen q r deno tes t he subvec to r ob ta ined f rom q by

deleting components qa for i / I.

DEFr lyr : r roN 19.2.2 (Pa"r t i t ions) : Suppose I i c I : {1 ,2, " ' ,1}

fo|i : 1, 2. We call {ft ' 12} a nontriuial partitio'n of the inder 'get I

if .I1 and 12 are both nonempty, 1r O f2 :0 and 110 12 : I '

DEFTNTTToN 19.2.3 (Decomposabil ity): The uncertain polynornial.

p(s, q) is said to be sum decornposable if there exists a nontrivial par-

i i t i " " {h, I r } o f 1 : { I ,2 , . . . ,7} and uncer ta in polynomials p(s, qr ' )

and p(s, qI') such that

p (s , q ) = p (s , q I ' ) + p (s , q I ' ) .

Sirnilarly, p(s, q) is said to be product decornposable 1f there exists a

nontr iv ia l par t i t ion { I r , Iz) of I : {1 ,2, . . . , ! } , uncel ta in polynomi-

als p(s, q1') and p(",q1") and. a fixed polynomial ps(s) such that

p ( s , q ) = p ( s , q l ' ) p ( s , q I \ + p o ( " ) .

Finally, p(s,q) is said. to be decomposable if i t is either sum decom-

posable or product decomposable. In this case, the uncertain poly-

nomials p(s,qI ' ) , p(" ,q/ t ) and p6(s) involved in the decomposi t ion

are called ch'i ld.ren of p(s, q). If p(s,q) is not decomposable, then we

say that iL is indecornPosable.

DEFrNrrroN 19.2.4 (Descendents and k-Decomposabil ity): Su-p-

pose that p(r, q) is decomposable and has children p(t, qI'), p(", qr')

and p1.6(s) wi thp(s,q1 ' )bei t tg fur ther decomposable for e i ther i : 1

o. 2 : '2 . Let p(s, 'q i r1) ' , p( r ,nrr .z) and pr ,o(s) denote the chj ldren of

such p(s, g1u) . Then p(s,qI ' ' ' ) , p(" , n ' t t ' r ) and ps(s) are cal led grand'

children of p(s, q). By continuing the decornposition process, it rnay

that p(s, q) is k-d.ecomposable if a sequence of decompositions can be

19.2 / Problem Area 1: Generaring lvlechanisms 333

applied so that all indecomposable descendents, of p(s, g) depend onat most A cornponents q; of q. For k : 7 ,we say that p(s, q) is to ta l lydecomposable.

REMART(S L9.2.5 (Nonuniqueness and Tlee Structure): Notice thatthe process o1 generatirrg descendenrs of p(s,q) is highly nonunique.Although there may be many ways to decompose p(s,q), the defi-nit ion of k-decomposabil ity requires one sequence of decompositionsleading to indecomposable descendents, each depending on at rnostk components q1 of q. AIso note that p(s,q) can have no more than2!. - L indecomposable descendents.

There is an obvious tree structure which can be identified with ak-decomposable uncertain polynomial p(s, S). We now il lustrate thernethod of tlee construction via an example.

ExAMeLE L9.2.6 (Il lustration of TSD): Suppose that p(s, q) is sumdecomposable as

p(s, q) = p(s, qr \ + p(s, q. I r ) ,

where p(s,qI') adrnits a further product decomposition

p(s, qI , ) - p(" , q l r ' r )p( t , q I ' , ' )

and p(s,q1') admits a further sum decomposition

p(s, qI ' ) = p(s, ntz t ) - l p(s, qrr , r ) .

At the l rer<t s ten srnnose t l ra . t n l ' .s . a l t . t ) and n( s .oI r , r ) are indecom-posable but p(s, qrr'z) admits a sum decomposition

p( t , qrr , " ) = ?(s, nh,z,1 + pG, nrr ,z ,z1

and p(s, q12,,) adrnits a procluct decomposition

p(s, qI ' ' " ) = P(s, qr" '2 ' ' )p(s, n lz ,z ,z1 .

Then, the associated TSD is seen in Figure 19.2.1.

REI\4ARr(s L9.2.7 (Algebra of Sets in the Complex Plane): TheTSD motivates some interesting research problems involving alge-braic operations on sets in the complex plane. For example, in thework of Polyak, Scherbakov and Shmuiyian (1993), interesting ge-ometrical objects (such as the product of two discs in the comnlcv

334 cln. 19 /

^ ( o n I t \/ f - \

P \ s , q - ' )

p( t ' q I ' ' ' ) p(t , q I ' , ' ) p(s, qr ' , ' ) p(s, qI ' ' ' )

+--l i--\p(s, qI ' ' ' ' ' ) p(s, qI ' ' ' ' ' ) p(s, qI ' ' ' ' ' ) p(" , qrz 'z 'z)

FIcunB 19.2.1 Generation of a TSD Using Indecomposable Descendents

plane) are represented analytically. One appealing aspect of such

work is the t.wo-dimensional nature of value sets. This enables us to

display results graphically. We now elaborate on this point.

It is felt that an interesting line of research involves the algebra

of sets in the complex plane. For example, suppose that D1, D2 and

D3 are three discs in the complex plane with O / Dz. Then it is of

interest to find ways to characterize or display combinations of 21 ,D2 and Dsi e.8., consider the problern of constructing sets such as

n nU 1 U '^

D3

orD : DtDzDz.

Returning to p(s, q) in Figure 19.2.L, notice that the final walue

set p(jut,Q) is described via the eiernentary operations {-F, -, '} ot

pairwise combinations of two-dimensional sets in the complex plane.

To further illustrate, suppose that the indecomposable descendents

of p(s, q) have independent or affrne linear uncertainty stmctures'

Then, associated with the leaves of the TSD in Figure I9.2.I ate

the polygonal or rectangular value sets p(jw,QI' '" ' '), p(j-,gh'z'27,

p ( j a ,Q I ' , , , ' ) , p ( j . , q l z , z , z \ , p ( j r ,Q r ' l ) and p ( j a ,Qhr ) , wh i ch can

19.2 / ProblemArea 1: GeneradngMechanisn,rs 335

easily be computed and stored. Subsequently, the TSD indicatesthat one can begin at the bottom of the tree and perform pairwisecornbinations of sets to build up the value set p(j.,Q).

ExAIr,lPLE 19.2.8 (Affine Linear Uncertainty Structures): To seethat the TSD concept handles affine linear uncertainty structuresas a special case, say p(s, q) : po (s) + If:r etpt(s), where pr (s) arefixed polynomials. Noting that we can express p(s,q) in the nestedform

p(s, q) : ( ( ( (po(") + qrpr(s)) + qzpz(s)) + qspa(") ) + qqpa.(s)) ,

it folows that p(s, q) is totally decomposabie. The associated TSDis i l lr.rstrated in Figure 19.2.2.

P (s , q )

po(s) + sLhG) s2p2(s)

Ftcune 19.2.2 A TSD for Polytopes of Polynomials

ExERcrsE 7,9.2.9 (Uncertainty in Poles and Zeros) : Suppose thatthe uncertain transfer function

D t - ^ - , - K ( " - s1 ) ( s - s2 ) " ' ( t - q * )t \ D r Y : ' ) - @

is connected in a unity feedback configuration. show that the result-ing closed loop polynomial is totaliy decornposable.

,___f_r - - lpo(") + qrpr(s) + qzpz(s) -F qspe(s) q+pa.(s)

+po(s) + arnr(s) + qzpz(s) qsps(s)

,_r__r - l

il

i l

336 Ch. i9 / Fi. . 'e Easl 'Problems

Frcunn 19.2.3 Interconnection for Exercise 19.2.10

ExERCTsE 19.2.10 (Cascade Combination rvith Feedback) : For the

system in Figure 79.2.3, assume that each uncertain parametel'enter-s

into e i ther one numerator A/ , (s , ' ) or one denominator D;(s, ' ) . Ful ' -

t he rmore , f o r i : I , 2 , . . . , t u , f l _1_ 1 , assume tha t .AL (s , ' ) and D6(s , ' )

are R;4-decomposable. Letting

k: o2ffikr,pror.e that the closed'loop polynomial is fu-decomposable.

L9.2.L-l ' The TSD for Rational Functions

The decomposability concept, introduced in the context of polyno-

mials, generalizes to rational functions. In view of the fact that the

definitions are nearly identical to those used in the polynomial case,we only sketch the key ideas. Indeecl , i f z : ( " r ,

"2, . . ' , z t ) € Cl and

R ( z ) : R ( 2 1 , 2 2 , . . . , z t )

is a rnultiwariable polynornial, the various notions of decomposabil-ity of R(z) are defined in the natural way; e.g., R(") : zt I zz

is trivially sum decomposable and J?(z) : zLzz is trivially product

decomposable. This rnore general TSD framework is useful for ratio-

na1 fractions. For example, suppose that the z-th block in a feedback

loop is a transfer function P'(s) which includes some unmodeled dy-

namics Aa(s) entering additively. In this case, at fixed frequency

r,., € R we identify zt witln P"(jr) + A.i(ja) ' Now, if we have apriori

bounds for L.i(jw), an appropriate choice of R(z) enables us to studyvalue sets for various loop functions of interest.

l /z (s , . )

D z G , ' )

l / ' (s , )

D' (s , ' )1\/r (s, .)

D r ( s , ' )

1fr+r (s, ')

D r+ r ( s , ' )

Pr(s , ' )

I ) / - \1 4 \ r r ' /

19.2 / Problem Area l: Generarins Mechanisms 337

FIGURE 19.2.4 Feedback Interconlection for Example Ig.2. l2

To iilustrate the idea above, we represent a parallel connectionof I blocks via

R ( z ) : z t * z z I z s + " ' I z t '

Since R(z) is totally decomposable, we see that a value set ciescrip-tion for the overall transfer function is obtained by performing I - 1set additions.

ExANdr'LE L9.2.I2 (Robust Closed Loop Stabil ity): For the uncer_tain feedback system in Figure 19.2.4, robust stability is governedby the zeros of the uncertain rational function

P ( " , . ) : 1 * P r ( " , ) P s ( s , . ) P a ( s , . ) + P 2 ( s , . ) p s ( s , ) p a ( " , . )

To make a connection witb. TSD theorv. we take

R ( r ) : I * z 1 z 3 z 4 l z 2 z s z 4

and notice that this function is totally decomposable; i.e.,

R ( z ) : I * 4 z a ( 2 1 + , 2 ) .

The resulting TSD is shown in Figure Ig.2.S.

REMARKS L9.2.73 (New Direcrions): For the example above, theexistence of a TSD was demonstrated by performing a factortzation.

338 Ch. 19 / Five Easy Problems

l l z j z a ( z 1 t z 2 )

z324

FrcunB 19.2.5 TSD for Example Ig.2. I2

In some cases, however, such a factorization may not be transparent.For tlre case when a simple factorization is not available, the theoryof unique factorization domains may prove to be useful; e.g., seeLang (1965). Another set of research probiems arises by consideringthe mathematical equations describing the dynamics of the physicalsystem under consideration. rn some cases, the interconnection ofphysical components suggests a natural rsD. A nice i l lustration ofthis concept is given by the ma,ss-spring-damper systern consideredby Ackermann and Sienel (1990).

19.3 Problern Area 2: Condi t ion ing of Margins

when carrying out robustness computations on a digital cornputer,the following fundamental question arises: wiil small chanqes in theinput data lead to small changes in calculated robustness kargins?our focal point in this section is the issue of continuity of cornputedquantities as a function of the input data. In this regard, .we notethat there is a fundamentai difference between rear and complexuncertainties. when the uncertain parameters gi are complex, dis-continuity can be ruled out under rather mild regularity cond.itions.rrowever, if the g; are real, then the possibility exists for discontin-uous dependence of the robust stability margin on the input data.vlatters are further complicated by the fact that at the point of dis-continuity in the space of problem data, the robustness rnargin r*o,may be much smaller than at neighboring points. This may lead topotentially deceptive conclusions.

z4

19 3 / Problem Area 2: Condirioning of Margins 339

Altirough disconlinuity of the robustness nrargi n may be non-generic, motivation fol further r-esearch is provided by the followingfact: In regions of data space close to the discontinuity setl rlnarcan be highly ill-conditioned. rt is felt that analysis of conditioningproperties of the robust stabil ity problem is an important topic area.

To substantiate many of the remarks above, the remainder ofthis section pr:ocecds aloirg the i ines in Barrnish, l{hargonekar., Shiand Terrrpo (1990). The example rvhich we provide illustrating thediscontinuity of the robust.ess margin is based on a unity feedbacksystem-the plant has uncertain pararneters entering affine linearlyinto nurnerator and denominator- coefficients. using d to representthe data describing the systen, the robustness ''.argin is written ex-plicitiy as r^or(d). Subsequently, rve see that there exists a sequenceof data (d(A))8. converging to some d* such that

] \ r * " "1d (k ) ) > r ^o , (d " ) .

That is, if one solves the sequence of robustness margin problemscorresponding to d(A). the margins r-",(d(k)) may differ consider-ably from r^o"(d*).This happens even as rl(k) gets arbitrari ly closeto the I imi t point d- .

IvorATroN 19.3.1 (Uncertain Polynornials and Data):witli the uncertain polynomial

We work

p ( s , q ) : s ' * L " o ( n ) t oi.:o

with coefficient functions ai(q) which are affrne linear in q. Since v'eare working with a variable uncertainty bound r ) 0, the dependenceon r is emphasized by writ ing Q, for Lhe uncertainty bounding set.That is, we consider th.e box

Q , : { q € R l : l q t l < , f o r i : I , 2 , . . . , 1 }

and, as usual, the robustness margin is given by

rmar: sup{r : p(s,q) is s table for a l l S € er} .

Within this frarner,vork, the problem data consists of inteqers0 - A i - - ^ ^ - A ^ - - ) ^ * - / ^ - \ ^ - - r a r , , . m . , .r : urlr q a\o n : deg p(s,q) and the set of coefflcient functionsao (. ),ar (.),. . .,a.-r (. ). To i l lustrate the discontinuity phenomenon,we use a finite-dimensional space for this problem data. That is, each

340 Ch. 19 / Five EasyProblems

aa(.) is viewed as a mapping on data uectors d e Rp to continuous

functions of q. For example, a fami' ly of problems might be described

b y p : 6 , ( . : 2 , n : 2 a n d

p(s,q) : s2 * (dt. -t dzqr t fuq2)s + (ds+ dsqt + daqz)'

A specificrobustness margin problem is obtained with d1 : 2, d2 : 1,

d t : 4 , d+ : 3 , ds :6 and d6 : 12 . Th i s Leads to

p(s, q) : s2 I (2 + qt r 4q2)s + (3 + 6cn + 12qz)'

Within this data space context, two robustness margin problems

are deemed to be "close together" if their associated data vectors

(call them dr and d2) are close together in some arbitrary but fixed

norm on Rp; i.e., l ldr - d2ll is small. To denote dependence on d,

we henceforth write paG,q) and r^o,(d) in l ieu of p(s, q) and r^o,,

respectively.

L9.3.2 Exarnple Il lustrating Discontinuity

Before ploceeding, it is important to note that it is easy to construct

relatively trivial examples for which discontinuity of r-o, can'easily

be demonstrated. Such examples involve cases when there is only

one uncertain parameter, cases when the uncertainty structure is

trighly nonlinear, cases when the l imiting polynomial pd(s, g) is only

marginally stable and cases when p4*(s,q) is structurally different

from p41;r1 (", g); ".g.,

pa*(s,q) has lower degree or a smaller nurnber

of uncertainties than pd-(s,q). In contrast, the example below is

simple 1'et nontrivial.We consider a unity feedback system with open loop transfer

function denoted by

^ , \ r, l /a(s, q)r d \ s , q ) : n o

D o G , d ,

where l\/a(s, q) and D2(s, g) are uncertain polynomials, Ks is a fi;<ed

gain and Q- is the box given above. In this example, the limiting

system is described by I{a" : a, N4* (", q) : 4a I l1aqv and

Da- (s,q) : sa + (20 - 20q2)ss + (44 + 2a * 10qt - 4}q2)s2

F (20 + 8a. -l20aq1 - 2}q2)s + a2 ,

where a : 3 + zJt. Using our data notation, we write I{d. : do,

l/a(t, q) : dt * d2e1 and

DaG, q) : s4 -l- (dz -l dsqz)st * (du 't daQt * d'zqz)s2

+ (ds + dsqr * drcqz)s -t dn.

19.3 / Problem Area e: Conditioning of lr4argins 341

By comparing the expressions for I{a", l/4-(s,q) and Da"(s,q) withI{a, Na(s,q) and Da(t,q), respectively, it is clear that the di arereadi ly avai lable; e.g. , d6 : a , d l . : 4a, d i - IOa, dI :20, etc .

Now, we consider the data sequence (d(,k))P, described by

where ak: a- I/k, This sequence corresponds to the case wherethe plant data is fi.xed and the gain a6 is converging to a.

In order to obtain the robustness margin along the d(k) se-quence, we study the closed loop pollmomial

Pa i r y ( s , q ) : Ka ( t " )Na (k ) ( s ,g ) t Da11"1@,q )

: s4 t (20 - 20c12)s3 + 1++a 2a, -t I0r11- 40q2)s2

+ (20-F 8a r- 2oaq1 - 20q2).s-t a(Ea - I "

ro1r, - f )ar)

and for the limiting case, we study the closed loop polynomial

pa* (s ,q ) : a I ' ' l a - ( " , q ) + Da ( " ,q )

: s4 -t- (20 - 20q2)s3 + 1++ * 2o, -t 10qr - 40q2)s2

+ (20 * 8a l- 20aqy - 2oq2)s + (5r-2 * tOa2q1).

By a lengthy computation whose details are described in Barmish,Khargonekar, Shi and Tempo (1990), it can be verif ied that

0.417 =1 - #

: a t1

r^ . , (d(k) ) > r*o"(d*) : =

= 0.234.

In other words, r*or(d) depends discontinuously on the data d.

REMARKS 19.3.3 (Interpretation and Further Research): The ex-ampie above illustrates the "false sense of security" associated withthe robustness margin. To further elaborate, if qi : qt x 0.234, twoof the roots of the closed loop polynom:al pag"1 (",g*) approach theimaginary axis as k ---+ co. That is, na61(s,q*) is "nearly" destabi-Iized by an uncertainty vector q* whose norm is 0.234 despite thefact that the predicted margin is approximately 0.477.

For the case of affi.ne linear uncertainty structures, the papers byTesi and Vicino (1991) and Rantzer (1992c) provide conditions underwhich the robustness margin depends continuously on the data; the

342 Ch. 19 / Five EasyProblems

paper by Packard and Pandev (1991) aims to regularize the compu-tat ion of r -o, (d) by adding "smal l " f ic t i r ious complex per turbat ions.

At the heart of the discontinuity problem is a certain rank-dropping phenomenon. That is. with q € Rt , r\re can aiways write

with A(cr) E 112x1 and b(r,.,) a 112x1. It turns out that discontinuityin the margin is accompanied by A(.o) losing rank at sorne cr,,e ) Q.

The example by Ackermann, Hu and I{aesbauer (1990), however,indicates that the problem of regularity of robust stability computa-tions is a lot more subtle than simply detecting whether r*o,(d) iscontinuous;' i.e., for the uncertain polynomial

p(s, q) : s3-F(qr - tqz+t)s2 - t (qr+ qz+3)s+ (1+€2 i6qr t lqzt2qtqz) ,

it is straightforrvard to verify that for small e ) 0, there is an "island"of instabil ity described by

( q t - I ) 2 + ( q z - 7 ) ' < r ' .

However, for any fixed e > 0, the robust stability computations arecontinuous with respect to the data. Many methods of computationwill "miss" the instability when e is suitabl5' small.

1-9.4 Problern Area 3: Pararnetric Lyapunov Theory

Throughout this text, we have worked almost exclusively with'un-certain polyrromials rather than uncertain rnatrices. For exarnple, ina robust stability analysis involving an uncertain state space system

i ( t ) : A (q ) r ( t ) ,

we indicated that it can be addressed using the uncertain character-istic polynomial

p ( s , q ) : d e t ( s / - e ( q ) )

In this section, we raise the possibility that in many cases theremay often be an advantage to working directly with matrices. Tomotivate further research at the matrix level, we survey a numberof existing results. Unless otherwise stated, throughout this section,the uncertainty bounding set Q is taken to be a box.

I Re nQu' t ) l : A@)q+ b(u. , )

l f rn p( ju t , q) )l a ; f o r i l o ;d i ( k ) : <

l o* foll : 0,

79.4 / Problem Area 3: Parametric Lyapunov Theory 343

To begin, we first mention some matrix results which can betriviaily obtained by "lifting" the polynomial theory to the matrixlevel. For example, suppose that A(q) is the companion canonicalform

0 1 0

0 0 1 0

A ( n \ -

0 1

qo qr q2 " ' Qn- t

Then we obtain a simple matrix analogue of Kharitonov's Theorem:A 'is robustly stable if and only if four distinguished ertrerne matri,-ces ar'e stable. Of course, the four distinguished matrices to whichwe refer are obtained from the Kharitonov polynomials associatedwith the interval polynomial farnily with the characteristic given bypo l yno rn ia l p ( s ,q ) : de t ( s I - e (S ) ) .

There are also some trivial rnatrix ana.logues of robust stabilityresults for polytopes of polynornials. For example, if q enters affinelinearly into only a single row or column of A(q), then the resultingcharacteristic polynomiai p(s, q) turns out to have an afftne linearuncertainty structure and the many results in this text on polybopesof polynornials are applicabte. More generally, if A(q) has rank onedependence on q, the same result holds.

In fact, in the dissertation of EI Ghaoui (1990), there is dis-cussion of the class of matrix uncertainty structures which permitlinkage with the theory for poll,topes of polynornials: Indeed, if

[.A ( o \ : A n + Y A ; q i_ _ \ Y

/

v, ' i th -4r € R'xt f ixed for i , : 0 , I ,2 , . . . , ( . , i t is of in terest to pro-vide conditions on the rnatrices ,4s, At, Az, ..., A2 for which thecharacteristic polynornial p(s, S) : det(s/ - l(q)) has affi.ne linearuncertaintlr structure.

A9.4.7 Polytopes of Matrices and Lyapunov Functions

If t lre entries of A(q) depend affine l ineariy on q, rhen rhe natrixfamily A : {A(q) : q € Q} is called a polytope of matrices or apolgtop,ic matrir family. The fact tinat A is polytopic is explained

344 Ch. 19 / Five Easy Problems

by noting that if qi is the 'i-th extreme point of Q and we take

At: A(q ' ) , thenA: conv{Ar}.

In Chapter 74, we already discussed a special subclass of rnatrix

poly-topes. Namely, we considered the robust stability problem for

an 1nter.'al matrix family. In view of the lirnited results available

for this special case, we expect the more genelal matrix polytope

problem to be difficult at the level of 4 x 4 and above. we now

mention some approaches to this problem and special cases which

have been solved.There are a number of papers in the literature rvhich aim to es-

tablish robust stability using a so-called con'L|non Lgapunou function.

For example, in the paper by Horisberger and-Belanger (1976)' the

following idea is ""tttu.l,

Suppose that P : PT > 0 is such that

l l e + P A t < o

for all generators A; of A. Then it foliows that

A T P + P A < O

for all A e A. To see that this conclusion is correct, notice that

any A € A car_ be expressed as a convex combination A: Di' \;At

with all .\t > 0 and fl )l : 1' Now, using the fact that posi-

tively weighted sums of negative-definite matrices are still negative-

definite, we obtain the desired result by noting that

Ar P + PA: lx6 ! e + PAi ) .x

An interesting generalizaLior- of the ideas above is given by Garofalo,

celentano and Glielmo (1992): If eact't entry of A(q) i,s a quotient

of multi l inear functions of Q, then once again, a cornrnon Lyapunou

matrir for the ertremes of Q serues as a corrlrrlon Lyapunou rnatrit

Jor a l l o f A.The work of Shi and Gao (1986) provides a concrete exarnple for

which a common Lyapunov function is readily available: If A is a

polytope of symmetric rnatrices, then stabili,ty o.f the set oJ generators

{Ao} u equiualent to robust stabi'Ii,ty of A. T]his is easily explained

by noting that A;+ AT < 0 implies that P: l serves as a cornmon

Lyapunov function.More generally, the problem of finding a corrimon Lyapunov

function does not appear to admit a simple analytical solution. Frorn

0 0

19.4 / Problem Area 3: Parametric Lyapunov Theory 345

a practical point of view, however, this presents no major obstaclebecause this problem can be cast in a convex programming frame-work. To see this, we provide a convexity argument in the style ofBoyd and Barratt (1990). Indeed, if we let P denote the set of nxnpositive-definite symmetric matrices and define

J i (P) : A^^, |ATP + PAi l

for each generator Aa and

J(P) : nax J6(P),

it is easy to see that the existence of a cornmon Lyapunov functionis equivalent to

inf J(P) < o.

We now claim that this in-fimum problem is a convex program. In-deed, convexity of the set ? is immediate; in fact, P is a convex cone.To see that J(P) is a convex function, we use the well-known fact(for example, see Rockafeilar (1970)) that the pointwise supremumof an indexed collection of convex functions {J, : r € X} is stillconvex. Subsequently, to establish convexity of Ji(P), we take

J,(P): r r lATP * PAlr ,

X : { r : l l r l l < r }

a,nd observe that J"(P) is linear (hence convex) in the entries of thernatrix P. Hence, by viewing J(P) as the pointwise supremum ofthe fi.nite collection of the Jt(P), we conclude that J(P) is convex.

RENTART(s L9.4.2 (Parameterized Lyapunov Function): Furtherresearch is motivated by the following simple fact: It is easy to con-struct polytopes of matrices which are robustly stable but do notadmit a comrnon Lyapunov function. In other words, solution ofrobust stability problems via the cornmon Lyapunov function ap-proach is inherently conservative. Roughly speaking, if the "spread"of the uncertainty is large, it is unreasonable to expect the sameLyapunov function to work for all A e A. These comments motivatethe search for parameterized Lyapunou functions. The takeoff pointfor work along these lines is the following obvious fact; The polytopzc

farn i ly of nxn matr ices A: {A(q) | q e Q} is robust ly s table i . f

346 Ch. 19 / F:ie EasyProblems

and onlg i,f there ex:ists a posi,tiue-defi.nite symmetric matrir functionP : Q -'-+ Rnxn such that

,sr (q)P(q) + P(q)A(q) < o

for all q € Q. A fundamental research problem involves identificationof classes of polytopes -4 for which the existence of an appropriateP(q) can be ascertained. Only a few papers have been written alongthose lines. We mention a sampling of the rather specialized resultsobtained to date.

In f he nrnpr h. Barmish and DeNln.rcn (1c lR6) the r rncer ta inP 4 } , v L \ f u v v l )

parameter vector q is expressed as the convex combination

: 4^ono ,

where q' denotes the z-th extreme point of q. Subsequently, if P, isa Lyapunov matrix for the 'j-th generaLor 44, a parameterization

P(A) : r r ,Rx

is proposed and conditions are given under rvhich P()) proves thestability of .4. We refer to P()) above as a.ffinely pararneterized andnote that a different class of affine parametelizations is pursued inLeal and Gibson (1990). In both papers, stringent side conditionsmust be satisfied in order to prove robust stability of -4.

For the important special case when A(q) arises by embeddinga stable interval polynornial farnily into a matr-ix cornpanion forrn,the paper by lVlansour and Anderson (1992) establishes that thereis a bilinearly parameterized Lyapunov rnatrix P(q) which can beused to prove robust stabil ity. In concir,rsion. the theory of para-metric Lyapunow functions is only in its infancy-sorre of the mostfundamental questions are yet to be answered. For example, withthe same setup as in Nlansour and Anderson (1992), can one provethe robust stability using a parametric Lyapunov fr-rnction with P(q)having affine l inear dependence on q? Anot her interesting area of re-search involves the lelationship between the classical Popov criterionand the parametric Lyapunov function; for example, see the paper

by Haddad and Bernste in (1992).

19.4.3 A Conjecture

To further emphasize the fact that there is a, wealth of open researchproblems involving parametric Lyapunov functions, we conclude thisi1

i l: l

i1i i l

19.5 / Problem Area 4: Polyropes of Matrices 347

section with a conjecture involving one of the most basic problemswhich one might address.

We concentrate on the special case when ,4 is simply the con-vex lruli of two reai n x n matrices -4s and ,4.1. Assuming that Ais robustly stable, we conjecture that ,4 admits an affine l inearlvparameterized quadratic Lyapunov function. That is, if

, 4 ( ) ) : ( 1 - ) ) , 4 0 + ) A 1

is stable for all ) e [0, 1], there exist n x zz symmetlic matrices P6and P1 such that

P ( . \ ) : P o * ) P i

is positive-definite and

AT(^)P(A) +P() ) ,4( ) ) < o

f o r a l l A € [ 0 , 1 ] .

19.5 Problern Area 4: Polytopes of Matr ices

In this section, we continue to focus on the robust stability problemfor a polytope of matrices. In contrast to the preceding section,we no longer concentrate on Lyapunov theory. Instead, we considerproblerns whose solutions shed light on the computational complexityof the robust stabil ity problem.

Nlotivated by the many robust stability results available for poly-topes of polynomials, it is natural to ask whether resu,lts along thesame lines are possible for a polytope of matrices. Recalling theexample of Barmish, Fu and Saleh (1988) in Exercise I4.S.I, we al-ready know that even for the special case of interval matrices, neitherextreme point resuJ.ts nor edge results are possible. The interestingresult of Cobb and DeMarco (1989) tells us even more: IJ A is apolgtope of n x n matrices wzth, n ) 3, then stabi,Iity of atl facesof dimension 2n - 4 is sufficient to guarantee robusl. stabil ity of A.Moreouer, zf the d'imens,ion of A (uieued. as a subset of Rn') i,s 2n-4or greater, there are eramples of matrir polytopes which are unstablebut haue the property that all faces of d,imension 2n - 5 are stable.

One pathway to the study of computational complexity is moti-vated by a certain relationship between robust stabilitv and robustnonsingularity. To clearly explain this linkage, we consider the fam-1 Iy A : { -a (S ) | S € Q} o f nxn ma t r i cesw i th A (q ) depend ingcontinuously on q. Under rather rnild conditions, there is a family

348 Ch. 19 / Five EasYProblems

of li.near transfbrmations (for example, see Bialas (1985) and Fu and

Barmish (1988)) mapping,4 into a new family "4 having the follow-

itlg prop"rty, fh" fimt'Ia-A'is robustlg stabte i'f and onlE i'f the farnily

-A,"1, ,iUutily nonsinguior, "".

Section 4'12 and also note that this

idea is used many tiles in Chapter 17. As an example, using the

linear mapT A : A Q A ,

the robust stability problem is readily transformed into a robust

nonsingularity Problem.A n i m p o r t a n t p o i n t t o n o t e i s t h a t a n y l i n e a r t r a n s f o r m a t i o n

on -4 preseives the affine linear dependence of matrix entries o. the

uncertain parameters. Hence, the resulting nonsingularity problem

a lso invo i vesapo l y topeo fma t r i ces .Howeve r , i t i sa l so impor tan tto mention that an independent uncertainty structure in A(q) gets

transformed into an affrne linear uncerta'inty for A(q) O A(q)' Said

another way, a linear transformation taking the robust stability prob-

lem to a robust nonsingularity problem does not pleserve the interval

matrix structure; i.e., ive begin with an interval matrix family -4 and

end up with a pol1'bope of matrices TA' To illustrate this' observe

that tle transformation 7,4(q) : A(q) e A(q) maps

into

r - l

l Q r r - L IA \ q ) : I I

L 3 azz j

2 q t t

.f

,l

0

, 1 - A ( ^ \ -L ^ \ q ) -

1

Qr * qzz

0

0 1

Qvt_|' qzz 1

3 2qn

19'5.1 Robust NonsingularitY

Motivated by the discussion above, we now concentrate on the robust

nonsingularity problem for a polytope of matrices' For the special

19.5 / ProblemArea4: PolytopesofMatrices 349

case of an interval matrix family A : {A(q) | q € Q}, we recail Exer-

cise 14.5.6: A is robustly nonsingular if and only if for each ertremepoint qi of Q, det A(qi) has tl ' t e sarne sign. From an applicationpoint of view, a weakness of this result is that as the dimension ofrnatrix A(q) in ,4 increases, there is a combinatoric explosion in the

number of extreme points. Notice that if ,4 is an rL x rL interval

matrix family, there can be as many as 2n- extreme points. Thisimmediately suggests a nurnber of basic questions about the compu-

tational complexitv of the robust stability and robust nonsingularityproblems.

Under strengthened hlpotheses, however, special classes of ro-

bust stabil ity problems can be solved. For example, suppose that

,4 is an rL x l'L matrix with nonnegative off-diagonal entries. Then,

Le t t i ng . 4 i - deno te t he uppe r A ' x k b lock o f , 4 f o r A ; : 1 ,2 , . - . , n , i n

accoldance with classical results from tnatrix algebra (for example,

see Gantmacher (1959)), it follows that A is stable if and only if

( - r ) k d e t ( f t . - A r ) > 0

for k : I ,2 , . . . ,?2, rv l tere -16 denotes the k x l , ident i ty matr ix . A

sirnilar result holds for Schur stability with the added restriction

that the diagonal entries of A are nonnegative and the cleterminant

condition above is replaced by det(ft, - '4,r) > 0. If we consider an

it x Tr interval matrix family A : {A(q) : q € Q} wit}r nolrnegative

off-diagonal entries, we can exploit the result above in combination

with the fact that a muitilinear function on a box achieves both its

minirrrurn and ma-ximum at an extreme point; e-g., see Lernma 14-5-5'

Now, we arrive at t ire foliowing extreme point result: A is robustly

stable zf and onlg if

( lL aetQp - ,qx@')) > o

fo r aLL e r t ren te po in t s q i o f Q and a l l k e {1 ,2 , . . . , n } . Fo r t he

case of robust Schur stability, rninor modifications of the arguments

above lead to a similar result given by Shafai, Perev, Cowley and

Chehab (1991). For further extensions involving irreducible interval

rnatrices, see NIayer (1984).To reduce the number of extreme points to be tested, the impor-

tant paper by Rohn (1989) begins rvith a nonsingular n x rL rnatrix

,46 and given bounds r;r') 0 for the entries Q;i of an interval rnatrix

LA(q). Defining the family of matrices

350 Ch. 19 / Five EasyProblems

with variable magn'ifi,cation factor r )> 0, the objective is to obtain

the robustness margin

T'maz: sup{r '. A, is robustiy nonsingular}'

Taking /? to be the n x n matrix having (i',j)-th entry r;7, rve

provide some standard terminology which is needed in order to de-

scribe Rohn's result; see also Demrnel (1988) . Indeed, if IVI is a

square matrix, let

po(M): ma-x{l)l : ) is a real eigenvaiue of IUI}

w i t h p g ( M ) : 0 i f n o e i g e n v a l u e s o f A [ a r e r e a ] ' A S q u a r e r n a t r i x5 is said to be a signature matrir if it is diagonal with all diagonal

entries equal to either *1 or -1. Now,'"ve let S be the set of rt ' x rL

signature matrices and observe that 5 has 2t members. we are now

prepared to present Rohn's Theorern.

TITEoREM Lg.5.2 (Rohn (1989)): G,iuen r ) 0, the interuaL matrir

fami,ty A, is robustly nons'ingular i,f and only if cl'et Ao and the 4"d.eter in inants

d 'et (As+rStRSz), obta ined w' i th 51,52 €5, haue the

sarrte nonzero s'ign. Moreouer,

1Tmat

max5, €s ps (,SrA;1'SzR)

REMART(S 19.5.3 (Connections with trr Theory and Complexity) :

It is interesting to note that the theorem above has a bearing on

the computation of the real structured singuiar value (real p) for the

case whe.I the /- norm is used for the uncertain parameter vector.

To this end, we begin with a real square rank one matrix r1'1 and the

goal is to comPute

1M: in f { l l q l l . " : de t (1+ &1A(q ) ) : o l .

lt'6\tut )

where l lqll* denotes the l- norm of g and

A ( q ) : d i a g { g 1 , q z , . " , Q t } '

In the numerical computation of pr,r"(IuI) , the following lorver bound

has traditionallY been used:

A, . : {As+eAA(q) : 0 ( e ( r and q e Q}

p*(M) > T;lJ po(sM).

79.6 / ProblemArea5: RobustPerformance gEL

In view of the extreme point theory of Rohn (1989), it can be shownthat this lower bound is sharp. For further discussion of extremepoint results in this framework, see also Holohan and Safonov (1989),El Ghaoui (1990) and Chen, Fan and Nett (1992).

To conclude this section, we note that the line of research abovehas resulted in control system researchers devoting attention to is-sues of computational complexity as defined in the field of computerscience; e.g., see Garey and Johnson (1979). Some init ial results inthis direction are given in the paper by Rohn and Poljak (1992) wherea class of robust nonsingularity problems are shorvn to be NP-hard.Coxson and DeMarco (1992) establish that the problem of decidingif the real structured singular value of a matrix is bounded above bya given constant is NP-hard, and Nemirovskii (1992) addresses thecomputational complexity of a class of robust stabilitl' problems.

19.6 Problern Area 5: Robust Perforrnance

Given the degree to which this text has emphasized extreme pointresults for robust stability, it is natural ask: What type of robustperforrnance criteria can be addressed in an extreme point context?Over the last few years, we see the beginning of a new line of researchin this direction. In the subsections to follow, we overview some ofthe recent developments and briefly mention some of tire interestingopen problems.

19.6.1 Pararnetr ic H- Norrn

The first result which we mention involves -tf@ analysis with struc-tured real uncertainty. For completeness, recail that if P(s) i:s proper,stable and rational. then the -I/rc norm is giwen by

l lP l l - : sup lP( i r )1.

In the theorem below, we see that for a stable interval plant family,the worst-case -E1- norm is attained by one of tire sixteen KharitonovpJ.ants; see Chapter 11 for further details.

THEoFLEI\zI L9.6.2 (See Mori and Barnett (1988) and Chapellat,Dahleh and Bhattacharyya (1990)): Consider a robustlg stablepropeT' interual plant fam;ila P : {P(",q,r) t q € Q,r e R} uit l-tmonic denorninator and sirteen associated Kh,aritonou plants Pt, j"(s)

352 Ch. i9 / FiYe EasY Problems

fo r i f i z : ! , 2 ,3 ,4 . Then i t f o l l ows tha t

max l lP ( ' , q , t ) l l - : r r l qx l lP ' , , ; r l l - 'q€Q,reR x7 , t2

REMART{s 19.6.3 (Extensions) : The ideas central to the proof of

Theorem 19.6.2 enter into the proof of a number of closely related re-

sults. For exarnple, in Chapellat, Dahleh and Bhattacharllra (1990)'

a real parameter version of tn" smali Gain Theorem is established

and in-Mori and Barnett (1988)' a robust Popov-like criterion is

p rov ided in the in te rva lp lan tcon tex t .These resu l t sa reex t ,endedin pu,p"r, by Dahleh, Tesi and Vicino (1991), Vicino and Tesi (1991)

rnd Rrntrur (1992a). It is also interesting to point out that for the

frequency-weighted version of the problem above, severe restrictions

on ih" *Ligtttittg function are required in order to obtain an extreme

point result; see Hollot, Tempo and BIondeI (1992)'

t9.6.4 Positive-Realness

The issue of positive-realness (addressed in the classical Popov the-

ory) is studied in an extreme point context in the paper by Dasgupta

and Bhagwat (1987). If

. 1 / ( s , q )P ( t ' q ) : a i " , n ) ,

is an uncertain plant' then for fixed Q, we lecall that P(s' q) is

strictly positive-real (SPR) if both J\/(" 'q) and D(s'q) are stable

and -Re -r(ir,S)

> 0 ior all o € R' We say tirat a family of plants

P : {P( . , 'a) , i ' e Q} ts robust lv SPR i f P(s ' q) is SPR for a l l s € Q'

In tlre theorern below, the problem of guaranteeing that an interval

p }an t fam i l y i s robus t l ySPR isadd ressed ; theexe rc i se fo l l ow ing thetL"or"rn allows for a more general class of plants'

THEoREM 19.6-5 (Dasgupta and Bhagwat (1987)): -Cons'ider the

interual plant famitaP with fired' numerator N(s,q) : N(s) ' Letti 'ng

Dr(s) , i " ( t ) ,Ds(") and' Da(s) d 'enote the four I {har i tonou denomi-

,iio, potgn'om'i,als, i't follows that P 'is robustly SPR if and only i'f

N ( s ' ) l D ; ( s ) i s S P R f o r i : 1 , 2 , 3 , 4 '

ExERcrsE 19.6.6 (l{ore General Interval Plant): Consider- an in-

terval plant family 2 with its four Kharitonov numerators r^v/t(')'

Ab(s) , -Ns(s) and Na(s) and i ts four denominators Dr(s) ' Dz(s) '

,j:1

19.6 / ProblemArea5: RobustPerformance 353

De(") and Da(s). Prove that P is robustly SPR if and only if eachof the extreme plants

^ r . / s )P . / " \ - , . ? r \_ z r , L2 \ - . / _

D ; r ( r )

i s SPR. H in t : W i t ] n P ( " ,q , r ) : l / ( s ,q ) lD (s , r ) and f i xed o € R 'instead of studying the condition

m in Re P ( j a ,Q , r ) ) 0 ,q€Q, reR

work withm in Re N( ju ,q )D . ( j u , r ) > 0 ,

9 € Q ' r a R

where D*( j . , r ) denotes the complex conjugate of D( ju,z-) . Nuw,this problem can be analyzed by noting that it involves minimizationof a multilinear function on a box.

RENnARKS 19.6.7 (Extensions): In their paper, Dasgupta andBhagwat (1987) also point out the applicability of robust SPR results

in an adaptive output error identification context. There are also anurnber of papers dealing with extensions and variations on the SPRLhen: .e. In papers by Dasgupta (1987) , Bose and Delansky (1989) '

Chapellat, Dahleh and Bhattacharl.ya (1991) and Shi (1991), the

SPR problem is studied under the weaker hypothesis that the nu-'--^-^+^- Ar/- ^\ ^^'.. be uncertain as in the exercise above: in someI I I E I 4 U U r r Y \ ' ) , V / ! 4 1 r U s u r r u s r u a r r r @ o r ( r

cases) plants with complex coeffrcieuts are considered. It is also worth

noting that the l ine of proof used in Exercise 19.6.6 above also worksfor SPR problems with multilinear uncertainty structures. However,instead of a four-plant result as in Theorem 19.6.5, the piants associ-ated with all of the extreme q' come into play: see the recent papers

by Dasgupta, Parker, Anderson, l{r 'aus and -\rlansour (1991), Boseand Delansky (1989) and Shi (199i). Finally, rve mention extremepoint results for the case when the plant is not necessarily SPR but

can be rendered SPR by iuitable addition of a positive constantl see

Chapellat, Dahleh and Bhattacharyya (1991).

19.6.8 Steady State, Overshoot and Nyquist

In this final subsection, we review a number of resuits which seemto raise at least as many questions as they answer. In the work of

Bartlett (1990c), an uncertain plant P(t,q) with multi l inear uncer-

tainty structule is considered and steady-state error for a unit step

354 Ch. 19 ,/ Five EasY Problems

input is the prime consideration. A robust stability assurnption is

imposed and it is shown that both the rnaTintum and the m,inirnum

of the steady-state er'r-or occur on one of the ettreme plants P(s,q') '

Bartlett then goes on to provide a counterexample to the ternpting

conjecture that the maximal peak overshoot is also attained at an

extreme. To this end, he considers the farniiy of plants w-ith affine

Iinear uncertainty structure described by

j \ D t V , /

-

Q.aq + 0 . 1 )s2 + (7 .7 q + 0 .8 )s -F 1

and g € [0,1]. It turns out that the peak overshoot corresponding to

a step input is not maximtzed at the extremes q - 0 or q : 1; e.8.,

4 : 0.5 leads to a higher overshoot value. A similar example is also

given for discrete-time sYstems.In the work of Hollot and Tempo (1991), the frequency response

of an interval plant is considered. To briefly overview sorne of their

resul ts , suppose that P : {P(s, q, r ) : q € Q,r Q R} is a s t r ic t ly

proper interval plant farnily rvith monic denominator and, to keep

the exposition simple, we also assume that for all r € R the plant

denorninator D(s,r) has no roots on the imaginary a;ris' Then the

Ngquist se, associated with this family of plants is defined by

N : { e ( i u . t , q , r ) : c u € R ; q € Q ; r € R } .

The boundary 0N of the Nyquist set, called the Nyqu'ist enuelope, is

the focal point and the following question is addressed: what points

z e 0N on the Nyquist envelope are Kharitonov points? By this, we

nean points z e 0N such that z : Pil,iz(jc.,') for one of the sixteen

Kharitonov plants P',.,r(.s) and some o € R.

In this regard. it is already l<nown frorn the work of Fu (1991)

that all points on the Nyquist envelope come from the edges of

the uncertainty bounding set Q x R. In fact, one can restrict at-

tention to the thirty-two edges identified bv Chapellat and Bhab-

taclrarl 'ya (1989). If P is robustly stabil ized by unity feedback, then

the I{haritonov points on the envelope include the rninimal gain mar-

gin points and minimal phase margin points. If P(s, q) is (open loop)

stable for all q € Q, the points associated rvith the maxirnum -F{-

norm also lie on the Nyquist envelope. Such a point is identified

with a q* € Q and an u* € R such that

Il

r , l

i 1

lili ir ii i li]i,ii

i,ri r l

i lt it : l

l r'lt' ̂l r i : l , :

. . ,r1 t ' i t t ,' ' . ,1: .,i ,r':::;

' , , r : : j ; : 1 1 ; , , .: , : i ! : t : : : t .

i - - i r . : t i

lP ( j r . , q.) l : 2?f lP ( ia, t )1.

19.7 / Conclusion 355

356 Ch. 19 / Five EasY Problems

FIGURE 19.6.1 Kharitonov Points on the Nyquist Envelope

Hollot and Tempo (1991) also establish that there are rnanyother interesting Kharitonov points. For example, they prove thatthe critical points associated with minimurrr sensitivity and comple-mentary sensitivity are Kharitonov points. Along these same lines,we a,lso mention the paper by Kimura and Hara (1991) where a morerestrictive class of interval plants yith fixed nr:rnerator is considered.To further clarify the geometry associated with the discussion above,we refer to Figure 19.6.1 *heie three distinguiahed points on theNyquist envelope are depicted.

L9.7 Conclusion

suggest topics for further research. At a number of selected pointswithin the chapter, rather specific research problems were suggested.

Fbom a control theoretic point of view, it can be argued that the"most important probler-n" was omitted from the list of five given.That is, this chapter made no mention of the robust synthesis prob-

l ; i :, '

Appendix A

Syrnbolic Cornputation for Fiat Dedra

Appendix andThe closed loop poll'aomial for the Fiat Dedra case study (see Chap-ter 3) was obtained using a symbolic manipulator. Note that the for-mulas given below have been simplified; before cornbining like terms,the formulas for the ai(q) are much more complicated.

oo(q) : 6.82079 x 1O-sqrqsq'n + a.azors x I0-5 qtqzqqqs;

o{q) : 7.61760 x t}*a qZqS + 7.61760 x to-a q|q?+

+ 4.0274L x I\-aq1q2qr2 + 0.003367}6q1%qf,

+ 6.82079 x 10-sqtqqqs -F 5.16120 x t}-aq|qsqa

+ 0.00336706qq2q+qs + 6.82079 x 70-" qtqzq+qr

+ 6.28987 x 7O-sqrqzqsq1 + 4.O2I4I x L}-aqrq1q4qs

+ 6.28987 x \O-sqtqtq+q6 -F 0.00152352q2qqaq5

+ 5.16120 x lo-aqzqsqqqai

or(q) : 4.0214\ x 70-aqtq? -l0.00r52352q2q!

+ o.o5szqf;q! + o.osszqlql+ 0.0189477 qtqzqZ -t 0,034862q1qq!a

+ 0.00336706qtqq,qs + 6.82079 x 70-5qtq+qt

+ 6.28987 x 10-sqrqsqe * 0.00152352qqaqg

+ 5.16120 x I}-aqqaqa - 0.00234048q1qaq6

358

Bibliography

Appendix A: Syrnbolic Conrpuadon fbr Fiat Dedra

+ 0.034862 qrq2q4qs + 0.0237398q?rqrqu

+ 0.00152352s2quq', + 5.16120 x t \ -aqZqaqr

+ 0.003367o6qtqzqsqz + 0'00287 4T6qtqzqsqe

+ 8.04282 x 7}-aqtqzqsqr + 6.28987 x 10-5q1q2q1q1

+ 0.0189477 qtqsq+qs + 0.0028741 \qtqsq+qa

+ 4.02147 x 7O-aqtqsq+qz l- 0.110Aqzqsq+qs

+ 0.023739 Sqzqsqaqs + 0.00152352qzqsqqqz- 0.00234048qzqsqsqa + 0.0010322 qzqsqai

al(q) : 0.018,9477 qtq? + o.tto+qzq?,

+ 5.16120 x ]ro-aqsqa + q\q? + 7.67760 x tO-aqZq?

+ siql* 0.1586q1q2q1" + +.ozt4r x ro-aq1q2ql

* 0.0872q1qq2 + 0.034862q1qaq5

+ 0.00336706qtq+qz + 0.00287 4l6hqsqa

+ 6.28987 x 10-sqrgoqz -F 0.00103224q2q6q7

l- 0.1104q3 Q+Qs 't 0.0237398q2q+qa

+ 0.00152352q2qEqr - 0.00234048qsq.sqe

+ 0.I826qlQsQa * 0.LI04qlq5q7

+ 0.0237398q\qas, - 0.0848q!qaq6

* 0.087 2q1qze+es -t 0.034862q1q2qaq7

+ 0.0215658qtqzqsqa + 0.0378954qrqzqsq7

+ 0.00287 4l6qfizqaqz * 0.I586q1qsqaq5

+ 0.021565 Sqtqzqtqe + 0.0189477 qtqsq+qr

* 2qzqsq+gs --F 0.1$26qzqyq.4q6

* O.77o4q2qq4qz - 0.0848q2qsq5q6- 0.00234048qzqsqaqr +7.61760 x to-aq|

+ O.O4747g5qzqsqa + 8.04282 x 70-aqtqsqt

+ 0.00304704q2qsq2;

Appendix A: S1'rnbolic Computation for Fiat Dedra

+ 0.021565 Sqtqsqa + O'00287 4|6qtqaqz

+ 0.O4747g5qzqaqt * 2qzqqqs -t 0'I826qsqaq6

* 0.I7}4qsqaqr - 0'0848qsqsqa - 0'00234048clzqaqz

* 2qSssqt + o'7826qlqe qz * 0'0872qqzq&r

* 0'377 2qlqzqsqt * 0' 021 5658q 7q2q6 q7

* 0.1586q1qzQa,Qz * 2qzqsq+qz - 0'0848c12qsq6q7

+ 0-0552q! -t O'3652q2qsqo l- 0'0378954qtqsqz

* 0.2208q2q-"q7;

o"(q) : 0 -0189477 q1q? + o'tto+s2s? + 0'1'826q5q6

-F 0.1104q5qt * 0 '0237398qaqt

+ qZqT -F 0.1586q1qrq4 + o'oez2qq+qz

+ 0.0215658qtqaq2 1= 0'3652q2q6q7

I 2qzq+qz - O'OgnSn.n aSz * q? + 7'61760 x 1'0-aq]

* 0.3772q1q5q7 I 4qzqsQti

oo(s) : 0.1586qrq? + zqzq? + 2qsqz + 0'1826q6q7

+ o'O552qi i

a t@) : q7 .

oa(q) :0.1586q1q1 + 4.02147 x t \ -aqtq?

+ 2qzq? + 0.00152352q2q? * 0.0237398q5q6

+ 0.001523S2qsqt + 5.16120 x ] 'o-aqaqz

+ 0.0552q1q7 + o.orsg4TTqtqzq?* O.0872q1QsQs * 0.034862q1qaq7

Bibliography

Abate, NtI. and V. Di Nunzio, "Idle speed conbrol using opti-mal regulation," XXIII FISITA Congress, Associazione Tecnicadell 'Automobile, Technical paper, vol. 1, no. 905008, Torino, Italy,1990 .

Abate, NI., B. R. Barmish, C. E. Nl-rri l lo-Sanchez and R. Tempo,"Robust pelformance analysis of the idle speed control of a sparkigrrit ion engine," Proceedings of th,e American Control Conference,Chicago, I11. , 1992.

Ackermann. J. E., "Parameter space design of robust controlsystems," IEEE Transactions on Automat'ic Control, vol. AC:25,pp. 1058-1072, 1980.

Ackermann, J. E. and B. R. Barmish, "Robust Schur stabii ity of apolytope of polynomials," IEEE Transactions on Automatic Control,vo l . AC-33, pp. 984-986, 1988.

Ackermann, J. E., H. Z. Hu and D. Kaesbauer, "Robustness analysis:A case study, " IEEE Transact'ions on Autornat'ic Control, vol. AC-35 , pp . 352*356 , 1990 .

Acke r rna .nn .T E and \M S ienp l " \ vVha i , i s a " l a . r se " t r t tmbe t O fL t u . u . v . u r e r f v r t

pararneters in robust systems," Proceed'ings of the IEEE Conferenceof Decision and Cpntrol, I{onolulu, Hawaii, 1990.

362 BibiiographY

Anderson, B. D. O., N. K' Bose and E' I ' Jury, "Output feedback

stabil ization and related problems-solution via decision methods,"

IEEE Transactions on Autornatic control, vol. AC-20, pp. 53-66,

1975.

Anderson, B.polynornials,"pp. 909-913,

D. O., E. I. Jur:y and M. IVlansour, "On robust Hurwitz

IEEE Transactions on Automatzc Control, vol' AC-32,

1987.

Ando, H. and M. Motomochi, "contribution of fuel transport lag

and statistical perturbation in combustion to oscillation of SI engine

speed. at id.le," soci,ety of Automotiue Eng'ineers Techn'ico.l Paper se-

r ies, no.870545, 1987-

Argoun, M. 8., ,,Allowable coefficient perturbations with preserved

sta:UiUty of a Hurwitz polynomi a\," Internat'ional Journal of ControL'

vol. 44, pp.927-934, 1986.

Argoun, N4. 8., "stability of a Hurwitz polynomial under coeffi-

cient perturbations: Necessary and sufficient conditions, Interna-

tional Journal of Control, vol' 45, pp' 739-744, Ig87 '

Astrom, K. J. ancl B. Wittenmark, Adaptiue Control, Addison-

Wesley, Reading, Mass., 1989.

Aubin, J. P. and R. B. Vinter (editors) , Conuet Analys'is and Op'

timizalion, (Research notes in mathematics; 57), Pitman Advanced

Publishing Program, Boston, Ivlass', 1980'

Bailey, F. N., D. Panzer and G. Gu, "Two algorithms for frequency

domain design of robust control systems," Internat'ional Journol of

Control, vol. 48' pp. 1787-1806' 1988'

1984.

Barmish, B. R., "Necessary anci sufficient conditions for quadratic

stabilizability of an uncertain system," Journal of optimi'zation The'

orA and, Appli 'cati 'ons, vo1. 46, pp' 399-408, 1985'

Barrnish, B. R., "New tools for robustness anaiysis," Proceedings of

the IEEE conference on Dec,is,ion and control, Austin, Tex., 1988.! ;

!

Ii

Bibliograpl.ry 363

Barmish,8.R., "A generalization of Kharitonov's four polynomialconcept for robust stability problems with linearly dependent co-efficient perturbations," IEEE Transact,ions on Automatic Control,vo l . AC-34, pp. 157-165, 1989.

Barmish, B.R., J. E. Ackermann and H. Z. IHu, "The tree struc-tured decomposition: A new approach to robust stabil ity analysis,',Proceedings of the Conference on Inforrnat'ion Sciences and Systems,Johns Hopkins University, Baltimore, NId., 1989.

Barmish, B. R. and C. L. DeMarco, "A new method for improvementof robustness bounds for linear state equatiens," Proceedi,ngs of theConJerence on Information Sci.ences o,nd Systems, Johns HopkinsUniversity, Baltimore, Md., 1986.

Barmish, B. R., M. Fu and S. Saleh, "Stabil ity of a polytope of matri-ces; Counterexamples," fEEE Transactions on Autornatic Control,vo l . AC-33, pp. 569-572, 1988.

Barmish, B. R. and C. V. Hollot, "Counter-example to a recentresult on the stability of interval matrices by S. Bialas ," InternationalJournal of Control, vol. 39, pp. 1103-1104, 1984.

Barmish, B. R., C. V. Hollot, F. J. Kraus and R. Tempo, "Extremepoint results for robust stabilization of interval plants with first ordercompensators," IEEE Transactions on Autornat,ic Control, vot. AC-37, pp. 707-7L4, 1992.

Barmish, B. R. and P. P. Khargonekar, "Robust stability of feedbackcontrol systems with uncertain parameters and unmodelled dynam-ics," Afathematics of Control, S,ignals and Systems, vol. 3, pp. 197-2 1 0 , 1 9 9 0 .

Barmish, B. R., P. P. Khargonekar, Z. Shi and R. Tempo, ,,Robust-

ness margin need not be a continuous function of the problem data,"Systerns and Control Letters, r 'ol. 15, pp. g1-98, 1gg0.

Barmish, B. R. and Z. Shi, "Robust stability of perturbed systemswith tim.e delays," Autornat' ica, vol. 25, pp. 371-381, 1989.

Barmish, B. R. and Z. Shi, "Robust stability of a class of polyno-mials with coefficients depending multilinearly on pertu.rbations,"IEEE Transactions on Automat'ic Control, vol. AC-35, pp. 1040-1043 , 1990 .

364 Bibliographl,

Barmish, B. R. and R. Ternpo, "The robust root locus," Automatica,

vol. 26, pp. 283-292, 1990.

Barmish, B. R. and R. Tempo, "On the spectral set for a family of

polynomials," IEEE Transacti,ons on AutonTat'ic Control, vol. AC-36,

p p . 1 1 1 - 1 1 5 , 1 9 9 1 .

Barrnisir, B. R., R. Tempo, C. V. Hollot and H. I. Kang, "An ex-

treme point result for robust stability of a diamond of polynomials,"

IEEE Transact' ions on Automatic Control, vol' AC-37, pp. 1460-

1462 , 1992 .

Barmish, B. R. and K. H. Wei, "Simultaneous stabil izabil ity of single

input-single output systems," Proceedings of the Internat'ional Sym-

posium on Mathematical Theory of Networks and Systems, Stock-

holm, 1985.

Bartlett, A. C., "Veltex and Edge Theorems Which Simplify Classi-

cal Analyses of Linear Systerns with Uncertain Parametets," Ph.D'

Dissertati.on, Department of Electrical and Computer Engineering,

University of lVlassachusetts, Amherst, lVlass., 1990a.

Bartlett, A. C., "Counter-example to 'Clockwise nature of Nyquist

]ocus of stable transfer functions,' " fnternational Journo'L of Control,

vol. 51, pp. l,479-7483, 1990b.

Bartlett, A. C., "Vertex results for the steady state anall'sis of un-

certain systems," Proceedings of the IEEE Conference of Dec'isiort

and Control, 'Honoluiu, Hawaii, 1990c.

Bartlett, A. C. and C. V. Hollot, "A necessary and sufficient con-

dition for Schur invariance and generalized stability of poly-topes of

polynomials," IEEE Transact'ions on Automatic Control, vol AC-33,

pp.575-578, 1988.

Bartlett, A. C., C. V. Hotlot and L. Huang, "Root locations of an

entire polytope of polynornials: It sffices to check the edges," Math-

ematics of Control, Signals and System.s, vol. 1, pp.61-71, 1988-

Berge, C., Topological Spaces, Oliver and Boyd, London, England,

1963 .

Bialas, S., "A necessary and suficient condition for stability of inter-

val matrices," International Journal of Control, voi. 37' pp' 717-722,

1983 .

Bibliography 365

Bialas, S., "A necessary and sufficient condition for the stal,.iiitr. ofconvex conrbinations of stable polynomials or matrices,' ' Bulletin ofthe PoLish Aca,demy of Sciences, Tech,n'icaL Sc'iences, vol. 33, pp.473-480 , 1985 .

Bialas. S. and J. Garloff, "Convex combinations of stable polynomi-oro ' ) T^. , - - . t ^ t +h. Frankl in Inst i tu te, vo l .319, pp. 373-377, 1985.c ! 1 u t

Bierrracki, R. XL, H. Hwang and S. P. Bhattacharyya, ' 'Robust sta-L : r : r - - - . - : ! L ^ - - - - ^ - - . - . r l r ea . l nn . ra . r ne l , e r ne r t t r t ha t i ons . t ' IEEE T rans -u I I r u . y w l L l l J U I t l u u u l c u r L o r 1 r @ r @ r r r s u u r l J L

act' ions on AutorrLati,c Control, voi. AC-32. pp. 495-506, 1987.

Bose, N. K., Appli,ed tr[ult idirnens'ional Systerns Theory, Van Nos-trand Reinhold, New York, 1982.

Bose, N. K. and J. F. Delansky, "Boundary implications for inter-val positive rational functions," IEEE Transactions on Circuzts andSystems, voi. CAS-36, pp. 454-458, 1989.

Bose, N. I{., E. I. Jury and E. Zeheb, "On robust Hurwitz and Schurpolynomials," Proceedings of the IEEE Conference on Decision andControl. Athens, Greece, 1986.

Bose, N. K. and Y. Q. Shi, "A simple general proof of Kharitonov'sgeneralized stability criterion," IEEE Transactions on Circuits andSystems, vol. CAS-3.1, pp. 1233-1237, L987.

Bose, N. K. and E. Zeheb, "Kharitonov's theorem and stabil itytest of rnultidimensional digitai fiIters," IEE Proceedings, ParL G,vol . 133, pp. 187-190, 1986.

Boyd, S. P. and C. H. Barratt, L' inear Cont'roller Des'ign: Limi.ts ofPerJormance, Prentice-Hal1, Englewood CIiffs, N.J., 1990.

Brewer, J. W., "Kronecker products and matrix calculus in systemtheory," IEEE TrcLnsact'ions on C'ircuits and Systems, vol. CAS-25,pp.772-781, 1978.

Brockett, R.W., "Lie algebras and Lie groups in control theory,"in Geometric Methods in Sgstem Theory, D. Q. Nlayne and R. W.Brockett, eds., Reidel, Dordrecht, Netherlands, pp. 32-82, 1973.

Chang, B. C. and O. Ekdal, "Calculation of the real structured sin-gular value via poJ.ytopic poiynomia\s," Proceedi,ngs of the AmericanControl Conference, Pittsburgh, Pa., 1989.

366 Bibliography

Chang, B. C. and J. B. Pearson, Jr., "Optimal disturbance reduction

in linear multivariable systems," IEEE Transact'ions on Automatic

Control,, vol. AC-29, pp. 880-887, 7984-

Cbapeliat, H. and S. P. Bhattacharyya, !(A generaiizabion of

Kharitonov's theorem: Robust stability of interval plants," IEEE

Transact' ions on Automat'ic Control, voI. AC-34, pp. 306-311, 1989.

Chapellat, H., IvI. Dahleh and S. P. BhattacharJ/T/a' "Robust stabil ity

under structured and unstructured perturbations," IEEE Transac-

t ' ions on Automat ic Contro l , vo l . AC-35, pp ' 1100-1108, 1990.

Chapellat, H., NI. Dahleh and S. P. BhattacharJrya' "On robust non-

linear stability of interval control s)'stems," IEEE Transactzons on

Automat ' ic Contro l , vo l . AC-36, pp. 59-67, 1991'

Chen, C. T., Linear System Theory and Des'ign, HoIt, Rinehart and

Winston, New York, 1984.

Chen, J., lvl. K. H. Fan and C. N. Nett, "The structured singular

value and stabitity of uncertain polynomiais: A missing iiuk," Proc-

cedings of the American Control Conference, Chicago, III., 1992.

Cieslik, J., "On possibil i t ies of the extension of Kharitonov's stabil-

ity test for interval polynomials to the discrete-time case," IEEE

Transactions on Au,tornati,c Control, vol. AC-32, pp. 237-238, L987-

Cobb, J. D. and C. L. Delvlarco, "The minirnal dj.mension of stable

faces to guarantee stability of a matrix polytope," IEEE Tran'.sac-

t' ions of Automati,c Control, vol. AC-34, pp. 990-992' 1989.

Coxson, G. E. and C. L. DeMarco, "Computing the structured real

singular value is NP-hard," Technical Report ECF' 92-4, Depart-

ment of Electrical and Computer Engineering, University of Wis-

consin, Madison, Wis., 1992.

Dahleh, M., A. Tesi and A. Vicino, "On the robust Popov criterion

for intervai Lur'e systems)" Technical Report RT 24/97, Diparti-

mento di Sistemi e Informatica, IJniversita di Firenze, Itaiy, 1991.

Dasgupta, S., "A Kharitonov-like theorem for systems under non-

linear parameter feedback," Proceed'ings of the IEEE Conference on

Decision and Control, Los Angeles, Calif., 1987.

Dasgupta, S., "Kharitonov's theorem revisited," Systerns and Co'n-

t ro l Let ters, vo l . 11, pp. 381-384, 1988.

Bibliographl, 367

Dasgupta, S. and A. S. Bhagwat, "Conditions for designing strictlypositiwe r-eal transfer functions for adaptive output error identifica-tior1," IEEE Transact'ions on Circuits and Systems, vol. CAS-34,pp . 731 -736 , 1987 .

Dasgupta, S., P. J. Parker, B. D. O. Anderson, F. J. Kraus ancl N4.N,Iansour, "Frequency domain conditions for robust stability verifica-tion of linear and nonlinear dynamical systems," IEEE Transactionson C,ircu'its and Systems, vol. CAS-38, pp. 38gJg7, 1991.

Davison, E. J., "The robust control of a servomechan.ism problemfor linear time-invariant multivariable systems," Proceedings of theAllerton Conference, !973.

de Gaston, R. R. E. and N4. G. Safonov, "Exact calcuiation of therrrultiioop stability margin," IEEE Transactions on Automatic Con-t ro l , vo l . AC-33, pp. 156-171, 1988.

Demmel, J., "On structured singular values," Proceedi,ngs of theIEEE ConJerence on Decis,ion and, Control, Austin, Tex., 1988.

Desoer, C. A. and \,1. Vidyasagar, Feedback Sgstems: Input*OutputProperties, Academic Press, New York, 1975.

Djaferis, T. E., "Shaping conditions for the robust stabil ity of poly-nornials with multilinear parameter uncertainty," Proceedi,ngs o.f theConference on Decition and Control, Los Angeles, Calif., 1988.

Djaferis, T.E., "To stabilize an interval plant family, it suffices tostabil ize 64 polynomials," Technical Report ECE-DEC-g0-1, Depart-ment of Electr-ical and Computer Engineering, University of lvlas-sachuset ts , i r4ass. , 1991.

Djaferis, T. tr. and C. V. Hollot, "The stability of a farnill' efpolynomials can be deduced from a finite number of O(k3) of fre-quency checks," IEEE Transact'ions on Automatic Control. vol. AC-34, pp. 982-986, 1989a.

Djaferis, T. E. and C. V. Hollot, "Parameter partitioning via shapingconditions for the stability of farnilies of polynomials," IEEE Trans-act' ions of Autornatic Control, vol. AC-34, pp. 1205-1209, 1g8gb.

Dobner, D. J., "A mathematical engine modei for development ofdynamic engine control," Transactions of the Soc,iety of AutomotiueEngineers, vo l . 89, no. 800054, 1g80.

368 BibliograPhl'

Dobner, D. J. ancl R. D. Fruechte, "An engine model for dynarnic

engine control development," Proceed'i'ngs of tlt'e American Control

Conference, San Francisco, Calif., 1983'

Dorf, R. C., Mocl'ern Control Systems, Addison-Wesley' Reading'

Mass. , 1974.

D o y l e ' J . C ' , . . A n a l y s i s o f f e e d b a c k s y s t e m w i t h s t r u c t u r e d u n c e r -tainty," IEE Proceed'ings,:ro|. 129, Part D, pp' 242-250' 1982'

Doyle, J. C., K. Glover, P. P. Ithargonekar and B' A' Francis"'State

space solutions to standard H2 and. -E1- control problerrrs," IEEE

Transactions on Automatic control, voi. AC-34, pp' 831-847, 1989'

Eggleston, H. G., Conuet'ity, (Cambridge Ttacts in lVlathematics and

M1,-thematical physics; No. 47), Cambridge U'iversity Press, Carn-

br idge, England, 1958.

El Ghaoui, L., ..Robustness of Linear Systerns,', Ph.D. dissertatiorr,

Departrnent of Aeronautics and Astronautics, Stanfor-d univelsitv'

Stanford, Calif., 1990.

F a e d o , S . , . . A n e w s t a b i j i t y p r o b l e m f o r p o l y n o r r r i a ] s w i t l r r e a l c o _efficients,'; Ann. St:uola Norm' Sup' Pisa Scz' Fzs' Mat' Ser' 3-7'

pp. 53-63, 1953.

Fam, A. T. and J. s. N{editch, "A canonicai par-anreter space for

i inear systems,,, IEEE Transactions on Automatic control, vol. Ac-

23, pp. 454-458,1978-

Fogel, E. and Y. F. Huang, "On the vahre of information in systerrr

identifi,catiort," Att'tomatica, vol. 18, pp' 229-238, 7982'

Francis, B. A.' A Course i,n Hoo Control Theory (Lecture notes

in control and information sciences, no. 88), springer-ver1ag, New

York ,1987 .

Francis, B. A., J' W. Helton and G' Zames, "1{€ optimal feedback

controllers for linear multivariable systems," IEEE Transact'ions on

Autornatic Control, vol. AC-29' pp' 888-900, 1984'

FYankJ ' i n ,G .F . ' J .D .Powe l l andA .Eman i -Nae in t ,FeedbackCon t ro lof Dynami'c Systems, Addison-Wesley, New York, 1986'

i5. : . , .^li" F.l;il v{*

i I \i q d! f i

,,i i"', t -ll r

I i 'I ii:l i li;l ijl.l i.,

il ri i.

Bibliography 369

Frazer, R. A. and W. J. Duncan, "On the criteria for the stabil ity ofsmail motion," Proceedi"ngs of the Rogal Soc'iety. A, vol. L24, pp. 642654, London, England, 1929.

Fruchter, G., U.Srebro and E. Zeheb, "On sevetal variable zerosets and application to NzIIIVIO robust feedback stabilization," IEEETransact' ions on C'ircu'its and Systems, vo1. CAS-34, pp. L270-I220,1987 .

Fu, N'I., "Comments on'A necessary and sufficient condition for thepositive-definiteness of interval symrnetric matrices, "' Internat' ionalJournal of ControL, vol. 46, p. 1485, 1987.

Fu, lvl., "A Class of weak Kharitonov regions for robust stability ofIinear uncertain systems,') IEEE Tro,nsactions on Autornatic Control,vo l . AC-36, pp. 975-978, 1991.

Fti, N,I. and B. R. Barmish, "Nlaximal unidirectional perturbationborrnds for stabilitl. of polynomiais and tratrices," Systerns o,nd Con-t ro l LeLters, vo l . 11, pp. 173-179. 1988.

Fu, NI. and B. R. Ba.rmish, "Polytopes of polynomials q'ith zeros in aprescribed set," IEEE Transact'ions on Automatic Control, vol. AC-34, pp. 544-546, 1989.

Fu, NI., A. W. Olbrot and NI. P. Polis, " Robust stabil ity for t imedelay systems: The edge theorem and graphical tests," IEEE Trans-act' ions on Autom,at' ic Control, vol. AC-34, pp. 813-820, 1989.

Gantmacher, F. R., Tlte Theory of Matri,ces, vols. I and II, Chelsea,New York, 1959.

Garey, ivl. R. and D. S. Johnson, Computers and Intractabi' l i ' ty: A

Gui,de to the Th.eory of NP-Cornpleteness, W. H. Freeman, San Fran-c i sco , Ca l i f . , 1979 .

Garofalo, F., G. Celentano and L. Glielmo, "Stabil ity robustness ofinterval matrices via Lyapunov quadratic fotms," to appear in IEEETransacti,ons on Autornatic Control, 1992.

Girosh, B. K., "Some new results on the simultaneous stabii izabii ityof a family of single input, single output systems," Systerns andControl Letters, vol 6, pp. 39-45, 1985.

370 BibiiograPhY

Guiver, J. P. and N. K' Bose' "strictly Hurrvitz property invariance

of quartics und.er coefficient perturbation"' IEEE Transactions on

Auiornati 'c Control, vol' AC-28, pp' 106-107' 1983'

Gntman. 5.. Root Clustering ' in Parameter Space' Springer-Verlag'

i leq ' - iork, 1990'

G u t m a n , s . a n d E . I . J u r y , . . A g e n e r a l t h e o r . y f o r m a t r i x r o o t c l r r s -tering in subregiorrs of the complex plane"' IEEE Transactions on

Automati,c Coitrol, vol' AC-26, pp' 853-863' 1981'

Haddad, W. Nl. and D' S' Bernstein, "Paramete'--dependent Lya-

p l l nov func t i onsand thed i sc re te - t imePopo \ ' c r i t e r i o r r f o r robus tlnu.lysi" and synthe sis," Proceed'i'ngs of the American Control Con-

ference, Chicago, I i l ' , 1992'

Hamann, J. C. and B' R' Barmish' "Convexity of frequency response

o.", ,rrro"iu.ted with a stable poly-nomial ," Proceedi'ngs of the Amer-

ican Control Conference, Chicago, III ' ' 1992'

Hara, S., T. Kimura and R' Kondo, "Ilto conttol system design bv

a, pararnetel space approach"' Proceed"ings of the Sgrnposium on the

Mathemat,ical Theory- of Networks and' systems, Kobe, Japan, 1991'

Hazell, P. A. and J. o. Flower, "sampled-data theory applied to the

modei l ingandcontro lanalys isofcompressionigni t ionengines-Partf," Int"ri 'otional Journal of Control' vol' 13' pp' 549-562' l97L'

Hertz, D., E. I. Jury and E' Zeheb' "Root exclusion from com-

plex polydomains u.,ti "o"'"

of its applications,'' Automati'ca' vol' 23'

pp. 399-404, L987'

H i n r i c h s e n . D . a n d A . J . P r i t c h a r d , . . A r o b u s t n e s s m e a s u r e f o r l i n -

" ' ,syste* . t rnderst ructuredrealparameterper turbat ions, ' ,Reportf ' Io. iS+, Institut fur Dynamische Systeme' Bremen' Germany' 1988'

Hinrichsen, D. and A. J' Pritchard, "An appiication of state space

methoclstoobta inexpl ic i t formulaeforrobustnesslT leasulesofpoly.nomials," in Robustness i 'n ld'enti ' f icati 'on and Control' M' Milanese'

R. Tempo and A- Vicino, ed"s., Plenum, New York, pp' 183-206'

1989 .

Hollot, C. V., "Kharitonov-Iike results in the space of Markov pa-

rameters," IEEE Transactions on Automatic Control' vol' LC-34'

pp . 536 -538 , 1989 '

Bibliographl, 37L

Hoilot, C. V. and A. C. Bartlett, "Some discrete-time counterparts toI(haritonov's stabiiity criterion fbl uncertain systems," IEEE Trans-act ions on Automat ic Contro l , vo i . AC-31, pp. 355-356, 1986.

Hollot, C. V. and R. Tempo, "On the Nyquist envelope of an inter-val plant farnily," Proceedings of t lze American ControL Conference,Boston. Vlass. . 1991.

Hollot, C. V., R. Tempo and V. Blondel, "11- performance of inter-val plants and interval feedback systems," in Robustness of DynamicSysterns uLith Parameter Uncerta,inty, M. Nfansour, S. Balemi andW. Tbuol , eds. . B i rkhauser, Basei , Swi tzet land,7992.

Hc;llot, C. V. and F. Yang, "Robust stabil ization of interval plants us-ing lead or lag compensators," Systerns and Control Letters, vol. l+,pp . 9 -12 , 1990 .

Hollot, C. V. and Z. L. X:u, "\,Vhen is the imagc of a rnullilinear iurrc-tion a polytope? A conjecture," Proceed'ings of the IEEE Conferenceon Decis ion and Contro l ,Tarnpa, F1a. , 1989.

Holohan, A. N4. and j\tl. G. Safonov, "On computing the N4I1VIO realstructured stability margin," Proceedi,ngs of the IEEE Conference onDecision and Control, Tampa, Fla., 1989.

ITnricl-.o.oo- I{ P. and P. R. Belanger, ,,Regulators for linear time:--'^-:^-r -1^-t^ with uncertain parameters," IEEE Transactions onr ! v d r I4 I IU P l a l f L 5

Autornatic Control, vol. AC-21, pp. 705-708, 1976.I

f "\Horn, R. and C. Johnson, Matri:r Analysis, Cambridge UniversityPress, Cambridge, England, 1988.

Horowitz, I. M., Synthesis of Feedback Sgstems, Academic Press,New York, 1963.

Horowitz, I. M., "Feedback systems wiLh nonlinear uncertain plants,"fnternat' ional Journal of Control, vol. 36, pp. 155-171, 1982.

Horowitz, I. M. and S. Ben-Adam, "Clockwise nature of Nyquistlocus of stable transfer functions," Internat'ional Journal of Control,vo l . 49, pp. 1433-1436, 1989.

Horowitz, I. M. and Nf. Sidi, "synthesis of feedback systems withlarge plant ignorance for prescribed time-domain tolerances," Inter-nati,ono,I Journal of Control, vol. 16, pp. 287-309, 1972.

372 BibliograPhl'

isidori. A., Nonlinear control systerns, springer-verlag, New York,

1985 .

Jury, E. I., Inners and' Stabili,ty of Dynamic Systems, Wiley' New

York ,1974 .

Jury, E. I. and T. Pavlidis, "stabii ity and aperiodicity constraints

forLystems design," IEEE Transactions on Circuit Theorg' vol' 10'

pp. 137-141, 1963'

Kang, H. I., "Extreme Point Results for Robustness of Control Svs-

t"-{; ' Ph.D. Dissertation' Department of Electrical and Computer

Engineering, University of Wisconsin, Madison, Wis' ' 1992'

Karl, W. C., J. P. Greschak and G' C' Verghese, "Comments on 'A

necessary and sufficient condition for the stability of interval rnatri-

ces,' " Internat' ional Journal oJ Control, vol' 39, pp 849-851 ' 7984'

KeeI, L. H., S. P. Bhattacharyya and J' W' Howze' "Robust control

.,rrith stluctured perturbations," IEEE Transact'ions on Automatic

Control, vol. AC-33, PP. 68-78, 1988'

Khali l, H. K', Nonli 'near Systems, Macrnil lan' New York' 1992'

Kharitonov, V. L., "Asyrnptotic stability of an equilibrium posi-

tion of a family of systems of linear differential equations,,' Di'ffer-

ents'ial 'nye [Jrauneniya, vol. 14, pp' 2086-2088, 1978a'

Kharitonov, v. L., "on a generalization of a stabil ity criterion,"

I zu estiy a Akademii N auk K az akhskoi, s s R's eriy a Fi'zika M atemat'ik a,

no. 1, pp. 33-57. 1978b'

I{haritonov, V. L., "The Routh-Hurwitz problem for families of poly-

nomials and quasipolynomials," Izuetiy Akademi'i Nauk Kazakh'skoi'

SSR, Seria fi,zikomatemat'icheska'ia, no 26, pp' 69-79' 1979'

Kha r i t onov ,V .L .andA .P .Zhabko , . .S tab i l i t yo f convexhu l i o fquas ipolynornials," in Robustness of Dynamic Sgstems with Pararneter

(Jn"certa'int1,7, M. Ivlansour' S' Balemi and W' Tbuol' eds'' Birkhauser'

Basel, Switzeriand, 1992.

Kiend. l ,H. , . .Tota lestabi l i ta tvon] inearenregelungssystemenbeiun.genau bekannten parameteln der regelstrecke," Automat'isieTung-

stechn'ik, vol. 33' pp. 379-386, 1985'

Bibliography 373

Kiendl, H., "Robustheitanalyse von regelungssystemen mit dernretlrode del konvexen zerlegun g," Autornatisierung stech.nik, vol. 35,pp. 192-202, 1987.

Kinl, K. D. and N. K. Bose, "Invariance of the strict Hurwitz prop-erty for bivariate polynomials under coefficient perturbations," IEEETransactions on Automatic Conh^ol, vo).. AC-33, pp. \\72-\174,1988 .

Kirnura, T. and S. Hara, "A r-obust control systern design by aparameter space approach based on sign definite condition," Pro-ceed'ings of the Korean Automati,c Control Con.ference, Seoul, Korea,1 9 9 1 .

Kraus, F. J., B. D. O. Anderson, E. I. Jury and VI. N{ansour, "Ontl.r.e lobustness of low order Schur polynornials," IEEE Transact'ionson Circuits and Systems, vol. CAS-35, pp. 570-577, 1988.

Kraus, F. J., B. D. O. Anderson and IvI. lvlansour, ' iRobust Schurpolynomial stability and I{haritonov's theor ern," Internati o nal J our-nal oJ Control, vol. 47, pp. I2I3-I225, 1988.

Kraus, F. J. and W. Tfuol, "Robust stability of control systemswith polytopical uncertainty: A Nyquist approach," Internati,onalJournal of Control, voI. 53, pp. 967-983, 1991.

1;1 I{urnar, P. R. and P. Varaiya, Stochasti,c Systems: Estimation, Iden-Lifi.cati,on and Adapti,ue Control, Prentice-Hal}, Englewood Cliffs,N . J . , 1 9 9 6 .

I{ur-zhanskii, A. 8., "Dynamic control system identif ication underuncerta.inty conditions.'' Problems of ControL and Informat'ion The-o ry , vo l . 9 , no . 6 . pp . 395 406 , 1980 .

I{wakernaak, H., "A condition for robust stabilizabllity," Systernsand Contro l Let ters,vol 2, pp. 1-5, 1982.

iI.!tog, 5., Algebra, Addison-Wesley, New York, 1965.

Leal, M. A. and J. S. Gibsor, "A first-order Lyapunov robustnessrriethod for linear systems with uncertain parameters," IEEE Trans-actions on Autornati,c Control, vol. AC-35, pp. 1068-1070, 1990.

Leitrnann, G., "Guaranteed asymptotic stabil ity for some linear sys-tenrs with bounded uncerLainties," Journal of Dynamic Systems,fufeasurement and ControL, vo l . 101, pp. 272-216,7979.

374 BibliograPhY

Luenberger, D' G', Introd'uction t-o-Linear and Nonlinear Program-

'*i"g, l6aison-Wes1ey, Reading' Mass' 1973'

Nzlansour, IVI., ..Robust stability of interval matrices,,' Proceedings of

the IEEE Conference on Dec'is' ion and' Control' Tampa' Fla' ' 1989'

Nlansour, Nl. and B' D' O' Anderson' "Kharitonov's Theorem and

the second. *"tnoa ot Lyup..,'o-,,,, tn Robustness of Dynanlic Sustems

wzth pararn"t", Un""rioi,ntgM. Nlansour, S' Balemi and W' Ttuoi'

eds., Birkhauser' Basel, Switzerland' 1992'

/ lulutt.onr, }r 'I., F' J' Kraus and B' D' O' Anderson' "Strong

t- ffiffi;*" irt"o,"* for d'iscrete systems"' Proceedings of the IEEE-Con\"r"rr"

on Decis'ion and' Control' Austin' Tex'' 1988'

- {N{arden, M', Geometrg of Polynomi'als' American lvlathernatical So--

. i " t , r , Providence, R' I ' , 1966'

l,.rl \,Ity"r, G., "On the convergence of"

Linear Algebra and Its Applicati 'ons'

powers of interval matrices,"

vol . 58, pP. 201-216, 1984'

I): l1IxI

,1:ll,j't, l

.,]l r ll , i

,:'l':'l

r,iir 1l : r i

' ' ,1

1,,,!

:;1,i,,1t1:;\r ia i ; : i

j;i ) ili , , : / l.)'r i.1

.a'.1']

1 , , : J

",i,i

\{eerov, NI. V', "Aubomatic control systems with indefinitely large

;;;;tJ'

'Automatik;a i Telem'ekhan'ika' vol' 8' pp' 152-167 ' 1947'

Nlicidreton, R. FI. and G. c. Goodwin, , ' Improved finite word length

characteristi"s r' aigitJ control using delta operators!" IEEP Trans-

act' ions on Autornatl i ' Cont'ol ' vot' AC-gt' pp' 1015-1021' 1986'

Nlilihailov, A. W', "Ivlethod of harmonic analysis in control theory"'

Automat ika ' i Telemekhan' ika ' vo l ' 3 ' pp ' 27-81' 1938'

lvlinnichelii, R. J., J' J' Anagnost and C' A' Desoer' "An elernentarv

proof of Kharitonov's theorem with extensions"' IEEE Transactions-on

Auro*otirc Control, vol' AC-34' pp' 995-998' 1989'

lVlonov, V' V', "On the spectral set of a famiiy of matrices"' to

appear, 1992.

Nlori, T- and S' Barnett, "On stability. tests for some classes of dy-

namical systems *ilft p"ttttbed coeffi'ci enls"' IMA Journal of lulath-

ematical Control and"Informat'ion' voI' 5' pp' 117-123' 1988'

Nlori, T. ancl H' Kokame, "Convergence property of jnterval matrices

and interval poiynomials," Internat'ional Journal of Control' vol' 45'

pp . 481 -484 , 1987 '

Bibliography 375

N{ori, T. and H. Kokame, "stability of interval polynomials with

vanislring extreme coefHcients," Recent Aduances in lu[athemat'ical

Theory of Systems, Control, Networks and Signal Process'ing I, pp.

409-414, N4ita Press, Tokyo, Japan, 1992.

Neimark, Y. I., "On the problem of the distribution of the roots of

polynornials," Dokl. Akad. Nauk, vol. 58, 1947.

Neimark, Y. I., Stabil i.ty of Linearized SEstems, Leningrad Aeronau-

tical Engineering Aca.demy, Leningrad, USSR, 1949'

Nemirokvskii, A., "Several NP-hard problern arise in robust stability

analysis," to appear, 1992.

Nise, N. 5., Control Systerns Engineering, Benjarnin Cumrnings, New

York, 1992.

Nishlmura, Y. and K. Ishii, "Engine idle stability analysis and

control," Soci,ety of Automoti,ue Eng'ineers Tech'n'ical Paper Series,

no . 860415 , 1986 .

Ossadnik, H. and H. I{iendl, "Robuste quadratische Ljapunov-funktionen," Autornatisierungstechnik, vol. 38, pp. 174-182, I99O-

Olbrot, A. \4/. and B. I{. Powell, " Robust design and analysis of third

and fourth order time delay systems with applications to automoti.veidle speed control," Proceedings of the American Control Conference,P i t t qh r r r r r l r . Pa . - . 1 989 .

Packard, A. A., IAh.o.t's Neu taith 1t,?, Department of Nlechanical

Engineering, University of California, Berkeley, Calif. ' 1987.

Packard, A. A. and A. P. Pandey, "Continuity properties of thereal/corrrplex structured singular va1ue," Technicai report, Depart-ment of lvlechanical Engineering, University of California, Berkeley,Cal i f . , 1991.

Panier, E. R., N4. K. FL Fan and A. L. Tits, "On the robust stabil-

it5' 66 polynomials rvith no cross-coupling between the perturbationsin tlre coefficients of even and odd powers," Systems and ControlLet ters, voI . L2, pp. 29I 299, 1989.

Perez, F., D. Decarnpo and C. Abdallah, "New extrerne-poini robusLstability results for discrete-time polynontials," Proceedings of theAmerican Control Con.ference, Chicago, I l l ., 1992.

376 BibliograPhl'

Petersen' I. R', "A collection of results on the stability of families

of polynomials with multilinear parameter depenclence,'' Technical

Report EE8801' D"p;t;;;t of Electrical Engineering' Universitv

Coll,ege, University 5ifo"?^d""th Wales' Australian Defence Force

AcadImY, Canberra, Australia' 1988'

Petersen, I' R', "A class of stability regions for which a Kharitonov-

Iike theorem holds,; IEEE TYansact'ions on Automat'ic Control'

vo l . AC-34, PP' 1111-1115' 1989'

Petersen, I. R., "A new extensron to Kharitonov's theorem'" |EEE

Transact' ion, on 'Su;*"tl '"-C""t-l ' vo}' AC-35' pp' 825-828' 1990'

Polyak, B' T', "Robustness analysis for multilinear per-turbations"'

tn Robustness of Dynam'ic SA'jl^I w.ith Parct'meter Uncerto'inty'

NI. Mansou,, S' gui";i and"W' TYuol' eds'' Birkhauser' Basel'

Switzerland, 7992'

Polyak, B. T., P' S' Scherbakov and' S' R' Shmulyian' "Construction

of value set foru oUt't""tt analysis via circular arithmetic"' to appear

irt Internat,on"t l"lu""t- oJ nolu't and' Nonli'near Control' 1993'

Z. Tsypkin, "Frequency criteria of robus,t stabil-

i ii.Jt" ."stems,'; Autlmatika i Telernekhct'nika'

Powell, B. K'' "A dynamic model for automotive engine control anal-

-,,sis.,, proceed,ings "l;;"

IEEE Conference on Deci's'i'on and Control'

Fort Lauderdale, Fla', 1979'

Powell, B. K.' J. A' Cook and J W' Grizzlet "Nilodeling and anzr' lysi 's

of an inrrer"rrtty *rrti;ru,te sampling fuer injected engi.e idle speed

control loop," p'"'"""i1''i'-of the- Arnerican Control ConJerence' IVIin-

neapolis, \ ' ' I inn', 1987'

Pujartr.. L. R', "On the stabii ity of uncertain polynomials with de-

oerrclent coefficients'' IEEE Transactions on Auton'tatic Control'

vol. AC-35, PP' 756-759' 1990'

Qiu, L. and E' J Davison' "Bounds .ln

the real stabil ity radius"'

rn Robustness of Dyna'mic Sgstems with Paro'tneter Uncerta'intzes'

\4. _ N4 ansour; s. B;i"#t"a#aw.-rroo[- e'cis., Biikhtubei, BAsel,

Srvi tzer land , lgg '2 '

Polyak, B. T' and Y

ity and aPeriodicitYpp. 45-54, 1990'

Bibl iography 377

Rantzer, A., "Hutwitz testing sets for parallel polytopes of polyno-

rnials," System.s and Control Letters, vol. 15, pp' 99-104, 1990.

Rantzer, A., "stabil ity conditions for polybopes of polynomials,"; .

t-rr-ErJ7 transz,cLzons on Automat'ic Control, vol' AC-37, pp' 79-89,

L992a .

Rantzer, A.. "Kharitonov's weak bheorem holds if and only if the

stability region and its reciprocal are convex," to appear in l'n'te-rna-

t' ional Journal ctf Nonl' inear and Robust Control, 1992b.

Rantzer, A., "Continuity properties of the parameter stability mar-

ojn 11 Prnrced.i.n.ns nf tl'te American Control Conference, Chicago, I11.,6 r r r t

t r v v v v v u ' u y u v J

1 c |o tn

Rantzer, A. and A. Magretsky, "A convex parameterization of ro-

bustly stabil izing controllers," in Robustness of Dyno'mic Systems

uith Parameter (Jncerta'inty, NI. Vlansour, S. Balemi and W' Truol,

eds., Birkhauser, Basel, Switzerland, 7992.

Rockafellar, R. T., Conuet Analysis, Princeton University Press,

Pr inceton, N.J. , 1970.

Rohn, J., "systems of l inear interval equations," Linear Algebra and

Applicati,ons, vol. 126, pp. 39-78' 1989.

Rohn, J. and S. Poijak, "Checking robust nonsingularity is NP-

hard," to appear in Systems and Control Letters, 1992.

Ruclin, W., Real and Complet Analys'is, NlcGraw-Hiil, New York,

1968 .

Rtrgh, W. J., Nonl' inear Systern Theory, Johns Hopkins University

Press, Bal t imore, NId. , 1981.

Saeki, NI., "Nzlethod of robust stability analysis with highly struc-

tured uncertainties," IEEE Transact'iori,s on Automatic Control,

vo l . AC-31, pp. 935-940, 1986'

Saeks, R. and J. N4urray, "Fractional representation, algebraic ge-

ornetry and the simultaneous stabilization problem," IEEE Trarts-

act' ions on Automat'ic Control, voI. AC-27, pp. 859-903' 1982.

Safonov, IvI. G., Stabi,I ' i tA and Robustness of Mult ' iuariable Feedba'ck

Systems, MIT Press, Cambridge' \4ass., 1980.

378 Bibliographl,

Safonor', NI. G., "Stability margins of diagonally perturbed rnul-Livariable feedback systems." IEE Proceedings, vol. 129, Part D,pp.257-256, 1982.

Sarldereli, IvI. K. and F. J. Kern, "The stabil ity of polynornials under correlated coefficient perturbations," Proceedings of the IEEEConference on Decis'ion and Control, Los Angeles, Calif., 1987.

Saydy, L., A. L. Tits and E. H. Abed, "Guardian maps and the gen-eralized stability of parametrized families of matrices and polynorni-als," Mathemat'ics of Control, Si,gnals and Systems, vol. 3, pp. 345-371 , 1990 .

Schweppe, F. C. lJncertainty Dynamical Systerns, Prentice Hall, En-glewood C1i f fs , N.J. , 1973.

Shafa i . 8 . , K. Perev, D. Cowley and Y. Chehab, "A necessary andsufficient condition for the stability of nonnegative interval discretesysbems," IEEE Transactions on Autornatic Control, vol. AC-36,pp.742-746, 1991.

Shi, Y. Q., "Robust (strictly) positive interval rational functions,"IEEE Transactions on C'ircur,ts and Systems, vol. CAS-38, pp. 552-5 5 4 , 1 9 9 1 .

Shi, Z. and W. B. Gao, "A necessary and suffi.cient condition for thepositive-definiteness of interval symmetric natrices," Internati,onalJournal of Control, vol. 43, pp. 325-328, 1986.

Sideris, A., ' 'An efficient algorithm for checli ing Lhe robusL stabil ityof a polytope of poiynomials," t\[athemat'ics of Control, Si,gnals, andSyslems. vol . 4 . pp. 315-337, 1991.

Sidel is , A. and B. R. Barnr ish. ' 'An edge theorem for poly topes ofpoilrn6*1u1s which can drop in degree," Systerns and Control Letters,vol . 13. pp. 233-238, 1989.

Sideris, A. and R. S. Sanchez Pena, "Fast computa,tion of the multi-variable stability margin for real interrelated unceltain parameters,"IEEE Transact' ions on Automatic Control, vol. AC-34, pp. 1272-1 2 7 6 , 1 9 8 9 .

Sil jak, D. D., Nonli,near Systerns, M/i1ey, New York, 1969.

Bibliograpl:i' 379

Soh, C. 8., "Robust stabil ity of discrete-time systems using deltaoperators," IEEE Transact'ions on Automat'ic Control, vol. AC-36,pp. 377-380, 1991.

Soh, C. B. and C. S. Berger, "Damping margins of polynomials withperturbed coefficients," IEEE Transactions on Autornati'c Control,vo i . AC-33, pp. 509-511, 1988.

Soh, C. 8., C. S. Berger and K. P. Dabke, "On the stabil ity propertiesof polynornials with perturbed coefficients," IEEE Transactions onAutomat ic Contro l , vo l . AC-30, pp. 1033-1036, 1985.

Soh, Y. C. and Y. K. Foo, "Generalized edge tlteorem," Systems andControl Letters, vol. 12, pp. 219-224, 1989.

Soh, Y. C. and Y. K. Foo, "A note on the edge theorent," Systemsand Control Letters, vol. 15, pp. 41-43, 1990.

Sondergeld, K. P., "A generalization of the Routh-Hurwitz stabilitycriteria and an application to a problem in robust controller design,"IEEE Transact' ions on Automat'ic Control, vol. AC-28, pp. 965-970,1983 .

Stoer, J. and C. Witzgall, Conueri,ty and Optim'izat'ion 'in FiniteD'imens'ions, Springer-Verlag, Berlin, Germany, 1970.

Tarski. A., A Dec'is' ion Method for Elementary Algebra and Geotne-fry, IJnivelsity of California Press, Berkeley, Calif., 1951.

Tempo, R., "A dual result to I(haritonov's theorent," fEEE Trans-act' ions on Automatic Control, vol. AC-35, pp. 195-198, 1990

Tesi A. and A. Vicino, "Design of optimally robust controllers withfew degrees of freedom," Technical Report RT 5/91, Dipartimentodi Sistemi e Informatica, IJniversita di Firenze,Firenze,Italy, 1991.

Tesi, A., A. Vicino and G. Zappa, "Clockwise property of the Nyquistpiot with implications for absolute stabiiity," Automat'ica, vol. 28,pp. 71-80, 1992.

Tits, A., "Comments on 'Polytopes of polynornials with zeros in aprescribed set'by N4. Fu and B. R. Barnrish," fEEE Transact' ionson Automatzc Control, vol. AC-35, pp. 7276-1277,I99O.

380 Bibliography

l'sing, i i . K. and A. L. I its, "On the multi l inear image of a cube,"

in Robustness of Dynam'ic Systems u'ith Pararneter UncertaintE,

M. Mansour, S. Ba1emi and W. tucii, eds., Birkhauser' Basel,

Switzerland, 1992.

Tiuxal, J. G., Autornatic Feedback Control System Synthesis,

McGraw-Hill, New York, 1955.

Tsypkin, Y. Z. and B. T. Polyak, "Frequency domain criterion for the

/P-robust stability of continuous linear systems," IEEE Transact'ions

on Autornatic Control, vol. AC-36, pp. 1464*1469, 1991.

Utkin, V. I., "Variable structure systems with sliding modes," IEEE

Transactions on Automatic Control, vol. AC-22, pp' 272-222, 1977-

4;(Vaidyanathan, P. P., "Derivation of new and existing discrete-t-'

\i-" Kharitonov theorems based on discrete-time reactances") IEEE

Transactions on Acousti'cs, Speech and S'ignal Process'ing, vol. ASSP-

38, pp. 277-285,7990.

Vicino, A., "Robustness of pole locations in perturbed systerns,"

Automat ica, vo l . 25, pp. 109-114, 1989.

Vicino. A.. A. Tesi and M. Milanese, "Computation of nonconser-

vative stability perturbation bounds for systems with nonlinearly

correlatecl uncertaintie s," fE EE Trans act'ions on Autonlat'ic C ontrol,

vo l . AC-35, pp. 835-841, 1990.

Vicino, A. and A. Tesi, "strict positive realnessl New results for in-

terval piants plus controller families," Proceedings of the IEEE Con'

ference on Decision and Control, Brighton, England, 1991.

'*1vitlyu.^gar, NI., Control SEstem Sgnthesis: A Factorization Ap-

proach, MIT Press, Boston, Mass., 1985.

Vidyasagar, M. and N. Viswanadham, "Algebraic design techniques

fbr reliable stabilizatior^," fEEE Transacti,ons on Autornat'ic Control,

vo l . AC-27, pp. 1085-1095, 1982.

Washino, S., R. Nishiyama and S. Ohkubo, "A fundamental study for

the control of periodic oscil lations of SI engine revolutions," Societg

of Automot'iue Engineers Technical Paper Series, no. 860411, 1986.

Wei, K. and R. K. Yedavalli, "Robust stabilizability for }inear sys-

tems with both paraneter variation and unstructured uncertainty,"

Bibl iography 3Bl

Transactions on Autornati 'c Control, vol ' AC-34, pp' 1'19-150,IEEEr q R q

l l iWeinmann, A., Uncerta'in Models and

Verlag, New York, 1991.

Yamagushi, H., S. Takizawa, H. Sanbuichi and K' Ikeura' "Analy-

sis on idle speecl stabil ity in port fuel inject ion engines," Society of

Automotiue Engineers Technical Paper SeTies, no' 861389, 1986'

Youla. D. C. , J . J . Bongiorno, Jr ' and C' N' Lu, "Single- loop

feedback-stabllization of linear multivariable dynamical plants," Az-

tom.a t i cc t , vo l . 10 , pp . 159 -173 , I974 '

c,(Youla, D. C., H. A. Jabr and J. J. Bongiorno, Jr', "Nlodern Wiener-'Hopf

design of optirnal controllers, Part II: The multivariable case,"

IEEE Transact' ioTt's on Automati 'c Control, vol' AC-21, pp' 319-338,

L976 .

Zadeh., L. A. and C. A. Desoer, L'ineo'r Sgstem Theory-A State-

Space Approach, NIcGraw-HilI, New York, 1963'

Zarrres, G., ,,Feedback and optimal sensitivity: Nlodel reference

transformations, multiplicative seminorms and approximate in-

verses,)) IEEE Transaction's on Automat'ic Control, vol' AC-26,

p p . J 0 I - 3 2 0 , 1 9 8 1

Zarrres, G. arld B. A. Francis, "A nel approach to classical fre-

quenc}; methods: FeecLbacl< ancl minimax sensitivity," Proceed'in'gs

if tn" nnn Conference on Decision, and Control, San Diego, Calif.,

r981 .

Zeheb, E. and E. Walach, "2-Palameter Root-Loci Concept and

sorne Applicatiorts," IEEE International Journal of circu.zt Theory

ctn,d AppLicat i 'ons, vo l . 5 , pp 305-315,1977'

Zeheb, E. and E. M/alach, "Zero sets of ru.ult idimensi.onal f irnctions

arrcl stability of multidimensional systems," IEEE Transa,ctions on

Acoustics. Speech' et'nd Signal Processinq,vol ASSP-2g, pp' 197-206,

1 9 8 1 .

zong, s. Q., ,,Robust Stabil ity Analysis of control systr:ms u'ith

Structur-ed P:rrarnetric Uncertainties and an Application to a Robot

CorLl r .o l Syst .em." Ph.D. d issel tat ion. Dcpart rnel t oF Pl rvs ics arrd

Electrical Engineering, IJnivelsity of Bremen, Germa'ny, 1990'

Robust ConLrol, Springer-

A Bialas $eoreh (c'irerioD) of'

AdaPrive Probleru' 2 5' l? 5t56 298-99 505

,[ n d e X ar6ne unear uncerointv smrcmres BGmac]ri' Hwans dd

sa U ncerdnlv s@crures - 'Jine En-a@c}larvva {hmrem of'

lmer r08 t85 93

Aircra.ft, e*p.nmerE, ob,que wins. ra? P lil ?lt l-'i -"1 ----, "." ",Aircnft transt.r tinctio., ,- 189 ' mrhnr^ fo' P-foof !80-s2

d8'csmonoronic, ?5-?6,1a8 r9:230, Boundarv fscdon sP<bAl sd 2a1

Boundc-'v oi se! ofPolvnomials lnl

r B o u n d 2 r s h e e P n g l l c 1 8 2 1 5 3 2 9special notation for, 202- -

Bound iuS S"Ur ( e ' @na bo 'Lno ing

APenodn i r , 'oous . !3CA'c conve{in Theo-.m.5rrF2! u'l:i.-tj--,..-- .- - ^- .,^

Proofof,523-24 muldlineetuncdon on '45-16

" n&lid€t7 for Prool 3l ? 2! sPhcresverslN' 259 60

stateFe.r ol 316

Autosati. shiP sFering, sEte ldiabl€

node l fo r .269 CCbords ,3 l t 18 ,320

B Clded looP' robs!3hbilirY of' 90

Bmish and TcmPo, fonnula and 178_79 337

tlreorem of, 28N1 Closed looP Porynomials S"

Batdeit, gollot dd Hmns, Edg€ ?olvnodills-closed looP

Tr ' - ^ * - " r ' { -1 i -

C lo :eo ooP nd ' t l r r fu r r ' t io " l / t -2 r i

Index

384

Closur-e of sets, 300-1Combinatoric explosion, 16+.65,

t67 , r77 , 349Comrnon Lyapunov function, 34+45f n m n a n c a t n r c

f i rst - orcler, 176, 178-80, 183, f 95h i - L - . ^ - , t ^ . I a d

pur e gain, 90, 179

Cor-r-rplementaw sensitir,iry, I 27

Complex coef f ic ients, 108

K]:aritonor,'s Theor-em with, I 04-7uotat ion for , 23

real versus, 23, 77, lB9-92, 2I7,

225-26transfornrations of, 1 09-4, I BB-92

Complex plane

algebra of sets i r - r , 333-35^ n P ' i c P r e ; n l O O

ovelbounding by discs in, 356Compr r t a t i on

discont inui ty rv i rh respect to datain, 338-42

syrnbolic, 38-39

Corrrputzrrional complexity, 1 76,

1 7 7 , 2 2 5 , 2 3 8 , 3 1 ? , 3 5 1polvropes oI merr ices and, 347--18

tes t i ng se t s and , l 5 - I 6Concorde SST, 26-27

Cor i jugacv propertv, 105, 121

Cont inuous root dependence, 52

Cor r t r o l l e r s y r r r hes i s t ec l r n i que . 182 -86

Control problern rv i th uncerta int l ' ,

rhree t,vpes of, 2-3/ - n n t - n l t h a n n ,

nathemat ical r lodels in, 12-13

sl ,nthesis in. 355-56

Convex combinat ions, 5+-55, I29,

132 -33 ,138 -39 , 346

angle fomula for, 203-4generali zccl grolvth condition

and , 217

Convex di rect ion, 198-200, 209-12Convex funct ion, 345

Index 385

Convex hu l l s , 130 , 13 I , 135 ,234 ,246 ,e47 o4q_^9 e53 3 lB

Convexiq ' , 315

Convex sets, I29-30

CON\,TX software program, \r1

Cr i t ical edges. See Dist ingrr ished

edges

Cutof f f requency, 79, 1 14, 116, 201

Dl -par- t i t ion technique, I2 ' )

/ -stabi l i ry . See Robnst / - s tabi l i tv

Damp ing cones , 102 -3 , 108 , 117 , 145 ,99R 99q ?9q

guardian maps for, 304-5m in i r na l , 279

DC motorpolygonal lalr-re set theory

appl ied ro, 145torque control of , 23-24

Decis ion calculus, 313Decomposi t ion, r ree-structured,

332-38Degree dropping, 49, 88-89Delav systems, 119-20, 122, 163

gro\r ' th condiLion applied to,277Delta rransform, 85Depth control system, 275-76Determinanc representa[ion. 306,

3 0 8 , 3 1 0 , 3 1 3Diamond, in s[andard eucl idean

n o r m , 2 2Diamond famill, of polynomials,

140,214-16,217Discon t inr.r i ty ph enomenon, 33842Dist inguished eclges, 164-77, 215Dominant pole specif icat ions, 144

386 Index

EEdge penetrar ion condi t ion, 148

Edge resul ts, 150

Edges of polytopes, 1 33-34 , 136, 142

distinguished, 16+-77, 275

exponent ia l growth of , 162, 164

Sce alsoPolynomials - edge of family

Edge s'rvitching, 177F,. r - - 'T-r . -^-A'- /q r do_63, 215

for affine Iinear r-rncertainrv

stnrctures, 196

formal statement of, 752-54

octagonal value set cornpared to,

1,74-75

root version of, 160-62

Eigenvalue criteria for robnstness,

50-52, 7 9, 27 4, 298-99, 303,

3 0 8 , 3 1 1 - 1 2Eigenvalue Criterion

matrix case for, 56-60, 64

proof of , 54machinery for proof, 52-54

Elecrr ic motor, See DC motor

El l ipsoid, 22, 260-61, 270

Eucl idean norms

standard, 2 l -22rve igh ted ,26 l

Extrerrre marrix, 243

Extreme point resul ts

for diamond famlly, 27 6, 277

feedback, 326lack o f , 150 , 219 , 24M6 ,35 , \

for mr.r l t i l in car t rncerta inty

structures, 244-46

on Nyqr:ist enr,elope, 354-56

o f po l y t opes , f 31 -32 , i 36 , 139 , 196

posi t ive realness in context oF,352

Rantzer's Grou'th Condition and,

1 9 6 - 9 7 , 2 1 6in robust nonsingrLlar i ry tesr ing.

349-50robusr Schur stabi l in ' encl , 219, 221

robust srabi l i ry and. 54, 63, 83,

2 1 5 - 1 6 , 2 5 5 , 3 2 7

Sixteen Planr Theorem and,1 7 8 - 8 1 , 1 9 0 , 1 9 3 - 9 4

i ' . ( P I l n - ^ l - , 1 - - ? R 4

sfead,v-sare error and, 353-54support fr-rnction and 251,

FFamilies

as funclamenral not ion, 13interval matrix, 24245, 344,

348-49interval plant. Sea Inten'al

planm interval polynon-rial. SeaPolynomials interval

- ^ . - . : ^ - f ^ - o o

with one uncerta in parameter,305-1 2

polytope of pol l ,nomials, 138pol,v-topic matrix, 3 4g.6robustness detlned in terrrrs of, 13

trncerta in funct ions versus, 22See al.so Spherical pol1'no-rn i a l f a rn i l i e s ' Suh fan r i l i e s

Family of svstemsd e f i n i r i n n n f I ?

"uncertain systern" as interchange-able l ' i th , 13-14

Feedbackin aJf i r re l i r rear nncerraint t

strlrctlrres, 125-27in . { rc Conr e-r ig ' Theorem, : i25-26for automat ic ship steer ing, 269

cascade combinat ion rv i t l - r , 336decidabi l i tv issr . re in, 313dominant pole speci f icat ion in, 144exponent ia l gro\vth problem in,

1 6 2

interval p lants ancl ,89-91, I64,167-68, 179, 196-98

invar iant degree and, 48robust performance problems for ,

6 , 331

of

standard configuration for, 38in theorerl of Biernacki, Hrvang

and Bharacharyya, 289-90uni ty, 45, 46, 89-91, 2I0, 335, 339, 940

Fiat Dedra engine, 29-39,154-57,1 6 3 , 3 5 8 - 6 0

Ficritious piants, 44-47Forbidden cone,327-22Frequency

cutof f , 79, 114, f I6, 20igeneral ized, 17, 107, 177

Frequency, parameterized linearprogram, 177

Frequertcy response arcs, 314definirions of, 315general ized, 329

Frequency respol)se set , inner, 325-28Frequency srveeping, 91-93, 256Fu and Barmish, formulae of, 79

Fuel economy, 30

GGeneral ized f requency, 11, i07, I l7Generat ing mechanisms, 33 l -38Growtl-r condidon. See Rantzer's

Grorvth Condi t ionGua rd ian maps ,298 -313

catalog of, 303-305def in i r ion of ,3O2matr ices of , 299, 301-304, 310, 3f4

topological preliminaries for, 300-302

HH - rtreory, 5-7, 16,64, Lgb,2bb,

35r-52Halfplanes, 302-21Hermite-Biehler Theorem, 84Homogenei ty,244Huru'itz rnarix, 50-52, I22, 309, 313Hunvitz sabiliry criterion, I I

I { l ,perplanes ,251-52

Index 387

IIII conditionin g, 107, 33842Improper arc, 315lnner f requenc), response ser, 325-26Interval matrix, 24244, 344, 34849Interval plants

compensators for , 167-68, 178, 198feedback for , 89-9i , I64, 167-68,

1 7 9 , 1 9 6 - 9 8frequency response of, 354-55robust stabilization of, 194

Interval polynomials. SeePolynomials - interval

Invariant degree, 1749, 115, 287

Edge Theorem and, I59-62, 175

KKharitonov plants, 90, I81. See also

Sixteen Plant TheoremKharitonov points on Nyquist

envelope, 354-55Kharitonov rectangle, 7 O-7 5, 86,

91 -93 ,96 -98 , I 07 ,146value set as generalization of, 109

Kharitonov regions, 225-34robust Z-stabiliry and, 218theorem of

machinerJ for proof of, 228-33p roo fo f , 233 -34statement of,225

weak and srong, 22+-28Kharitonov's Theorem, 65-85, 197

with complex coefFrcients, 104-7degree dropping ald, 88-89discrete-rime analogue of, 218-19frequency domain alternative of

9r-93importance of ,7, 65/" analogue of,258,267matrix anaiogue of, 343

388 I ndex

proof of , 77-78' 84

machinen' for Proof' 70-77

publication of, iii, r't

s impl i f ied, for c legree rr : 3 '

87-88

statemetlt of, 69

in uniq ' feedback, 89-91

I{r-onecker operatiolls, 57-58' 64' 303

LLaplace transform, L44

Left deriyariye, 929

Lineal f ract ional u-atrsformadorr ,

127-28

Lrrn.rping, 67-68, 90' 24243

for spherical polylomial families'

263-61

L,vapunov fun ctions, 3 4247

MMACSYi\'IA softrvare, 29

Ii,Iappinge d g e , 1 3 6 , 1 4 2

See nlso Guardian maPs;

Semiguardian maPs

Mapping Theorem, 7 ' 237,24H8'9 6 6 9 6 6

geomeu)/ of, 247-48

proofof ,252-53macirinera for Proof, 250-52

statement of, 248

lVlarkov parameters, 85

Nlason's Rule, 33, 332

I4ass - spl rug - damper systerrr, 338

lvlathematical programning, 9-1 0, 1 7'

1 8 , 2 3 8 , 2 5 5 , 3 4 5

Itlathematica software, 29

Matricescol lpensator , 290-91

ex feme ,243 ,245

of gr.rardiar-r maps, 299, 301-4, 310,

J I J

Hunvitz, 50-b2, I22, 303, 313'rvitl-r inclependen t Llncertalntres'

qqo_ln 9J9-+-+

irrLerlal, 242-,41, 3'14, 348-49

for pol1'nonrial coeffrcienrs' 305-8

poli'topes of, 343-51

represer'tt^tlon of, 30G-7, 309-i 0

Schlaflian, 3l3

s.ignature, 350

uncertain, 25

for" uncerlaiu poll'non' als' 264-66'o ^ 6 ^ -

rr'eighting, 261

specu-al set , 279-80

symnett-ic, 284-85, 344' 346

vaLue set , 291-92

I\4atrix case for Eigenvalue Criterion,

5b-tlu

I\4atrix lepleserltatloll, 306-7, 309-1 0

N' Iax norm, 21, 93

Ivlikhailov frtnctiou, 7 -5,

I 07

MII\IO (multi-input multi-ourput)

systems, 5

N,Iinin-ral generatillg set, 13?

N,Iinimization problerls. in proof ol

Ranzer''s Growth Condition,

204- l 0

Nlirrinrr-rnr nor-rn, 263, 27I' 278'282'

2 9 1

Nlonotonic angles, 75-76, 188' 192,

2 3 0 , 3 1 3 - 1 4 , 3 1 9 - 2 0

! r t heo ry , 10 , I 28 , 148 , 350 -51

Nlr,rltilinear funcdon on a box, 24546

N{ultilinear urlcertai11$' stmctul.es'

Sar Llncertarnty

stl-Llc tlues - multilinear

NNiclrolsorr,Jack, Fiue Eaq Pieces

(mov ie ) , 330

Noncor-rvex arcs, 315

Nonconvex di rect ions, 209-1 2

: !

.j

Nonconvex sets. 129-30Nonincreasing phase property,

186-89, r97 ,20rN o n s i n o r r l a r i t v_ - - _ - _ _ _ D - _ * _ _ - /

guardian maps for, 302robust , 52-53,59

extreme point resul t for ,246NP-hard, 351robust stabilit,v transformed into,

347-5tRohn's Theorem ou, 350

Nontrivialiq, Condition, 220, 221Norrns, 27-22 Sce also Er-rclidean

norms; Max norm; Ir{inimumnorrn; Parametr ic 11- normWeighted norms

Notat ion, l9-28for angles and their der ivat ives,

2024Fo r i n t ena l r na r r i ces e43

for interlai polynomials, 68Nvquist analysis, 10. 4749,62-63, 98,

329Nyquist envelope, 353-54

oOpen neighborhood, 300Open sets, 300Orlando's formula, 52-54n . , - - ! - ^ . , , - - l i . - -

b1' discs in cornplex plahe, 356b1 i : r dependcn t r r nce r t a i nq ,

structures, 66b,v inten,al pol)nomials, 8I-83, 84,

r24, 146by N{apping Theorem, 24V+7of rntr l t i l inear urr ce r ta int l ' s tnr l l l res,

238of nonlinear Lrncertainw structures,

I 2 8

f ia spherical farnilies, 274-76

Index 389

PParallelotopes, I 65-67Parameterized Lyapunov functions,

34547Parametr ic -FI norm, 351-52Parpoll,gons, 167Pathrv ise connecredness, I 12-14Perfornance Sre Robusr perfomancePhase der ivat ive minimizat ion

n r n h l e m e 9 O 4 - Or ^ - - - ' _ - _ - t - - _ -

PI contro l lers. robust iy stabiJ iz ing,182-86

Pitch contro l loop, 99-I00Polygonal value sets, I41-49Polygons

nlrmber of edges of , . l65

P d r P U r / B U r 1 5 , r u /

polytopes as, 1 31 , I37 , 14243Polynornials

angles of, as funcLions of frequencv,t 5 - / o

boundan of set of . 30Ic i ass i c ; t l l i t e ra t t r r e o r r . 6closed loop, 144, 169

r l e c o n r n o s a h l e ' i ? i - 3 6

in Fiat Dedra case, 36-37, 358Routh table for , 182-86

seventh order, analysis oi 154-55

special mathematical propertiesoi 331

complex coefficients. See Complexcoefficients

deterrninant representation of, 306,3 0 8 , 3 1 0 , 3 1 3

edge of fami ly of , 149, 151-152,1 6 9 , 2 1 5

cr i t ical numerator anddenominator , 172-73

i n t c r \ , 1 1 6 6 G 7 - 6 R

complex coel f ic ient , 104-5

extreme poinr resul ts for ' , 216I ihar i tonor ' 's Theorem on, 69-70Iorv order, 87-88, 200,219,223-

1J 9,f-\ 94G

390 Index

overbounding v ia, 81-83, 84' 124'

146

in polytopic framert'ork, 139

robLrst Schur stabiligv oF, 218

I{.haritonov, 68-69, 89, 168-69

complex coefficients, 1 05-6

sixteen Kharitonov Plants, 181

theorem of, 106-7

nom ina l , 9T

nonvanishing ^t zero. 302, 309

paral le lotopes of , 165-67

root locus analYsis of, 5-6, 4'{-{6

roo|s dePendence on coef f f ic ients

o r , t

roots for derivatives of' 318

Arc Convexity Theorem and, 328

guarclian maPS for, 303, 304

Polvtopesof poll'nomials, 124-48

def in i t ion of , I38

cl iauroncl fami l l 'o f , 140' 214- l6

er lges, 133-34, I36, t '12

Edge Theorem aPPl icat ions, 155-

62extrenle Points of , 131-32. 136,

1 3 9 , 1 9 6

generators for, 1 38-40

operat ions on, I34-37

as polygons, 131

value set for, I42-45

of marr ices, 343-5l

Popov cr i ter ion, 346, 352

Positive realness, 352-53

Problem Tree,34

Prope r a r c , 315

RRantzer's Grot'th Condition, 196-2 I 6'

299

Proofof,212-74rnachinen' for proof, 202-72

sraLement of, 200

RC f i l ter , 120

Rectangle s

Kharitonor'. .See Khari tonov

rectangle

ro ta red ,235 ,236

Regtrlar regior-I, definition of , 225P e n r c s e n t e r i o r r 9 Q Q

marix,306-7, 309-10pol).nomial cleterminant, 306' 308,

3 1 0 , 3 I 3Riccari equadons, 5, 255, 356I J i - l . t . l - r i . : r r i v e 9 9 Q

Robr.rst, cleFurition of, 13

Robust l-stabilitt,, 60-62

in b ivar iate polynomial case, 3I3

cr i ter ion for ,249

def in i t ion of , 61, 243

edge penetral ion condi t ion arrc l ,

148muhilinear uncertainty strllctllres

ancl,237-55

spectral set and analysis of, 279-80

theorern of Saydy, Tits ancl Abecl.

307-1 1

value set for , 107, 109-10

rveak I {har i rouov regions and, 21S

zero exclusirt.u conclitior-r for,

1 14-1 6'\ee a.lso Ecige Theoretn

Robustness nrargin. I+-I-D, 4341' 47,9*t_RR qct6

for- compl icaLec[ l t I ]cerLain Lv

srr l lc t l r res.66

def in i t ion of , 326

rvith clegree droppitrg, 49

for c l ianond fami l l ' , 216

discontinuity of, in calculated clata,

338-42geometry of, 94-96inten'al polynomial, 79

octagonal vz l l l re set ancl , 175

in pol;'gonal value set, 144

r ight - ancl lef t - s ided. 54

scalar f t t r rct ion oF f i -equerrcy for '

cornpurar ion of , 267 -68, 27 6,277

Tsypkir-r-Pol1'ak fur-r cti on forv isual izar ion oi 9G-102

Robrrstr ress problems, 2-3, 17, 2ZBProblem Tree for, 3--1

Robust nonsingr-rlari tv. SaaNonsingnlar i t r ' - robust

Robust perforlttancecr i ter izr for- , 351-55noCat ior-) for , 2f25

Robust Schur a1:er iocl ic in ' , 236Robust Schur- stabitirl, Sze Schr-u-

stabi l i tY - robustRcrbtrst SPR propern/ , 3b2-SsRobust stabi l i t ; , , 7

of c losed lool : , 90, 778-79, 3Zjdefinition of, 43of d iscrete- t ime s1,stems, 21Bextr-ente point ancl edge resul ts in,

24-1-.+6finite nlrrnber: of sreps in cleterni-

n l r i on o f , 312grapl i ical resr i r - rg of ,7g-81, g l

Lr apunov funcr ions in proof of ,34+-17

marr-ix analogr-res of, 343marrix case and, 56-60numerical condi t ioning of compu-

tar ions of , 157-58, 163single paramercr, 42-64. 186-88,

299 ,305 - r?s rab i l i t l bou r r c l a r l ' . ana l vs i s o f , I 22ur state space serr ing, 5,242,24btransfornted into robusr

nonsingular i t l ' , 347-b Itransition;ll ntodels for analysis of,

163See a ls o I'l:^arft ort o\,'s Thcorem;

Nlapping Theorem; Robusr2- stabi l iq , ; Sixteen plant

Theorem; rhirq.Tiuo p4*"

Theorern

Index

Robust synrhesis, 5, 15, t82-86, 356Rohn's Theorern, 350Roo t c l us te r i ng ,3 lSRoot locus, 5-6

generalizadon of, 45-46Nyquisr analysis and, 47robr.rst stabilis' ar-rd, 49, 62, 89s r r e r t r e l c e t o n r l 9 O (- t - _ _ _ ' _ '

Root se[ , I60-62Rotated rectangles, 235, 36

Routh tables, 182-86

SSatel l i te ar t i rude connol , 144Sa1dy, Ti ts and Abed. theorem of ,

307 - I tScalar, mulriplication of polytope

and, I35S c e l e d f r r n r t i n n ? 9 R

Schlaflian r-r-ratrix, 3I3Schur stabilin'

generalized frequency responsearcs, 329

guardian maps for , 303-4, 311-12inrerval lnau^ices and, 349robus t , 6 l , 83 , l i 5 -16 , 218 -36

r f e f i n i r i n r r a f 9 4 ?

Iiharitonov regions and, 2 1 Ine rv t r nce r ra i n t y mode l t o r , 235

Semiguardian maps, 312, 313Sensitivity, 127Ship steer ing, state var iable

rnodel for , 269Signature matrix, 350Sixteen Plant Theorem, 178-95,

197-98

parametric 11- norm and, 35I-52proof of, 192-93

rnachinel' for proof, 186-92reduct ion in nurnber of p lanrs

for, I 94

392 Index

statement of , I82

Sma l l Ga i r - r Theo rem, 11 ,64 ,352

SIvIP sofrware, 29

Sofnvare, iv , 9, 29, 238

Soh-Berger-Dabke Theorem, B, I08,

258,267-77

p roo fo f , 272 -73spectral set generatior-l and, 283-84

srarement of 267-68

Specral sets

in root versior-r of Edge Theoreu,

I 60-62for- spher ical polynomial fani l ies,

??9-84. 295

Spher ical p lant fami l ies

affi ne linear Lulcertarng/ str uctlrres

ancl , 284-85, 295

in theorem of Biernacki, Hrvang

and Bhattaciraqrya, 285-94

def in i t ion of , 261-62

lurnping for,267

Spher ical pol)momial fami l ies, 258-84

. l et lnl f r on OI, Z5:,-OU

lurnping for, 263-67

overbounding v ia, 2 73-76

value seLs for ,27O-72

SPR(srr ic t ly Posi t ive real) , 352-53

Stability/ - , c lef in i t ion of , 61

defrn i t ion oi 43

Stabili 11' Theolem. See Kharitonor"s

Theoretn

Stable all-pole trallsfer fr-rnction'

328-29

Steacll^stare error-, extreme Plantsand, 353-54

Stoclrast ic problems, 2-3, 17

Strictly positive real (SPR), 352-53

Strictly' proper arc, 317

Strip, guardian maPs for, 305

Sructr.rred singular value, 255, 350- See

also P theory'

Srurm sequences, 313

Subfamilies, definition of, 50

Subnrers ib le vehic les (submarines)

depth control s)rstelrl for, 274-'76

pi tch contro l looP for , 99- i00

Subscr ipts,2 lSubvectors, 263-65

Superscr ipts,2 l

Support funcdolr , 250-52

S)'lvester resuitant, 122

Synnbolic colTrplrtadon, 37-39

TI a l lo rec l a lgo l . l ln rn , I /

Testing funcLions

Go", (r, :) ,285-89Gro, (co) , 267-70C.,, (a),97-102] : I (o r ) ,91-94

, l e snng se t s , r 5 - lD . r / /

Thirty-nto Edge Theolem, 164,

I 68-7 6

proofof , 175-76

I oo tbox PJ . l l l o soP l IY . r u - l I

Transfer funcdons

ai lcraf i ,26, 183

closed loop, 125-26

performance sPecifi cations and,o , r 9 i

sable all-pole , 328,329

unce r t a i n , 121 ,335

Transformation

of conrplex coefFrc ienrs, 102-4,

188-92fol compl icared nncerta inr l '

structures, 84

of clarnping cone Problem, 108

Lap lace ,144linear fraction al, I27 -28

nonl inear, 256

froln pol1'nomic to multilinear,

24042,255

ofrobust stabiliw to robust

nonsingularitY, 347 -5I

3 9 1

Index 395

Tree - stmctLrred decorlposit ion spectral plant Famil ies ar-rd,( T S D ) , 3 3 2 - 3 8 2 3 + _ 8 5 , 2 9 5

Tsl,pkir-Pol1,ak function, 9,1, 96-102, TSD ancl. 335277,323 depenclent, 121-25

independenr, 3, 65-66, 234.239U definirior-r of, 66

l,ncertain functions, 24. 11g in t-narr ir transformation tOfamil ies 'ersus, 22 roblrst lronsingulanrr ' , 348

{-.1ncerrair-r pararneters, 4, I 2IJncerta iu state I 'ar iable sysrem, 25lJncertajlt sysre rrrs

de f i n i t i on o f . 13 -14no tadon f o r . 19 -23

lJr-rcertzr in t ranster f iurcr ion, I21, 335LIncerta int l ,

lorr ' -o lc ieL cocf l lc ier t r . 1 l 9 . : :J

l r ; r t i remar ic: r l moclels zrncL, I 2- l 3

If arant e Lncin Fiat Decire engine mocleJ,

J.i--1+

i n < l e p e r L r l e n t r r r r c e r t l i n L r

s t r l rc tu res and. 66

n-rarrix, 3-{3mr-r l t i l inear 8, 17, 239-57

dellnit ion of, 2{lSPR problems rvirh, 353

s r e : r d v s ' r r e e r r n r i n ' 1 5 3 - 5 - 1

r e l r r e s e t i r ' l t e r I J l e t a t i o l l o f ,

243-50, 256,Sr rL lso \ Iapping Theorem

uon l i ne r r . 3 , ( i 6 , 238overborrncl inq of , I28

poJvnon i c , 212 .255clefinitior:r of. 239

LTnique tac tor izat ion c lornair rs, 338

Ur-r i t s inplex, 133, 140

vValne sets, 109-22

def in i t ions oi I10, 118-19explanation of, ll-12as gencral izat- ion of X}rar iLonov

t 'ecrangle, 109importance of idea of, rvNlapping Theorem interpreted b1',

249-50octagonal , 169-75rvi th one parameter enter ing

affinely, 111as parallelograms, 11 1-12as parpolygons, 167

394 Index

pol ,Ygonal , I4 l -49

of pol1 ' topes of polynomials, 124,

142-45in r obns t l - s t ab i l i t y , 107 , 109 -10

for spherical poll,norlial families,270-t2

as stra ight l ine segmen ts, I 10-1 1 ,

1 2 0

wWeak Khariconov regions, 218,

22+-28Weighted norns, I95, 269-7 A, 294

ZZero L,xcl r . rs lon Uoncl i t ion, 5, I l , 76-

77 , Jg ,84 , r07 , r 09 ,249bourclary srveeping fr.rnctions and,

1 1 8F ^ - - ^ , - ^ 1 - - - ^ - F n ^ ; ̂ 1 1 O O l V n O r n i _

a ls , i91-92Edge T l reorem and, I50 , 157pathlvise connecredness and,

1 1 2 - 1 4for polygonal value sets, 144-47proof o f , I I3 , 31-+for robust l-stabi l i ry, 114-16,

, 4 q 9 F ' 4

s t 2 f e m e n r q o F l l q 1 l a

Thirq.Trvo Edge Theorern and, I

1 6 9 , 7 4 , 1 7 6 , 1 8 5 - 8 67 P r ^ . P t r h e n ^ r ? l ?

lireratu-c on,5, ?notaLion for , 19-20. 23-2-1synth csis problent u ' i r l - r , 355-56

rezr l versus cornpler , 338in s imple zero, 120-21

IJncertainrl, l:or,rncling se rs, 2 I-22,2 4 - 1 J . 1 i 9 , I : : . 1 S 0

of c l iamond pol l 'nomial fani ly ,2 14 -15

rrrul tilinear uncertainty strltctr-rresand , 237

N) ' qu i s r cnvc lo l r c e r rd , 35 f -55par t i t ioning of , 255as sphe re o r -box , 258 -60I s \ l ) k i l l - [ r o l \ : r l < t r r nc r i o r r a r rd , 97 -98

[Jncerhir r tv st ructur-es, 8-vaf f ine l inear, 3, 45, 104, 12.1-29,

138 ,234 ,239 ,278del ln i r ion of , l2 i rd iscont inr-r i t1 ' , 341-42Edge Theorer l for , 196

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