Robust decentralised control using outputfeedback

Prof. 6. Hiiseyin, Ph.D., Prof. M.E. Sezer, Ph.D., and Prof. D.D. Siljak, Dr.Sci.

Indexing terms: Control theory, Feedback, Decentralised control, Robustness

Abstract: A decentralised control scheme is proposed for stabilisation of large-scale systems using dynamiccontrollers with local output feedback. The scheme is robust with respect to uncertainties in parameters andnonlinearities in the interactions among the subsystems.

1 Introduction

For either conceptual or computational reasons, large-scalesystems are modelled as interconnections of low-order sub-systems. The decomposition approach is a natural modellingscheme when the subsystems can be identified as physicalunits. It is also very effective in handling the modelling uncer-tainties by confining them to interconnections among thesubsystems. Decentralised control schemes, which are chosento fit the information structure constraints imposed by thedecompositions, can then be made sufficiently robust totolerate a wide range of nonlinearities and parameter changesin the interconnection structure.

So far, the decentralised control schemes for interconnectedsystems have been designed using static controllers with localstate feedback [1—7]. It is quite common to expect, however,that the entire state of each subsystem is not accessible forcontrol, and the control objective has to be achieved usinglocal outputs. Furthermore, the decentralised schemes, whichprovide local state estimators for the subsystems [5], may beeither infeasible (e.g. communication between subsystems isimpossible) or uneconomical (e.g. communication betweensubsystems is too costly) owing to the requirement that thelocal estimators exchange their computed outputs. Under suchdesign constraints, it is reasonable to consider the methods[8, 9] available for decentralised design using dynamic outputfeedback. These methods, however, are not suitable forstabilisation of large-scale systems because they result in closed-loop systems that are not robust; i.e. the systems cannottolerate either modelling errors or nonlinearities unless theyare infinitesimally small. Furthermore, the resulting controllersmay have the same size as the overall system, thus giving riseto dimensionality problems in the control design.

In this paper, we identify a class of interconnected systemswhich can always be stabilised by decentralised dynamicoutput feedback. The proposed control strategy is to stabiliseeach decoupled subsystem using a local dynamic output feed-back and, at the same time, make the gains of the loops of theinterconnected system sufficiently small so that stability isretained in the presence of interconnections. This strategy,which is based on the small-gain theorem [10], has been usedin the context of static state feedback [11], where it producedrobust closed-loop systems. The basic configuration of thecompensated system in this work consists of a linear time-invariant system in the forward path, which corresponds tothe decoupled subsystems, and a memoryless nonlinearity inthe feedback path, which corresponds to the interconnections.Except for the block-diagonal structure of the forward-pathsystem, the configuration is the standard multivariable Lur'e-

Paper 2149D, received 18th June 1982Prof. Huseyin is with the Department of Electrical Engineering, MiddleEast Technical University, Ankara, Turkey, Prof. Sezer is with theDepartment of Systems Engineering, University of Petroleum andMinerals, Dhahran, Saudi Arabia, and Prof. Sljakis with the School ofEngineering, University of Santa Clara, Santa Clara, CA 95053, USA

Postnikov system. In proving stability of the compensatedsystem, we use a small-gain version of the circle criterion[10,12].

2 Problem statement

We consider an interconnected system S7 composed of Nlinear time-invariant subsystems 37{ described as

S7X\ xt =

W; =

(1)

where * f ( f ) E ^ B | is the state, M,(f)E &\s the control input,yi(t)E.S%'\s the measured output of 37 {, and Jr= {1 ,2 , . . . ,N}. The matrices At, bit P,-, ct and Q{ are constant and ofappropriate dimensions. In eqn. 1, fl,-(f)E^m« and w,(f)G^'« are the interaction inputs and outputs associated withS?i, which represent, respectively, the effect of other sub-systems onS^i and the effect of 37{ on the other subsystems.The interaction inputs and outputs are related by

(2)

where the nonlinear function is continuous inN

condition

'»•••> WN) , I — h> a n d satifies the conicali=l

(3)

where K,- is a positive number.We assume that there exists a subset

= b^

Qi =

that

{Ad)

(4b)

In other words, we assume that for each subsystem 37 { eitherthe interaction input vt has the same effect on 37 { as thecontrol input ut (when i E.^f ), or the interaction output wt

is a reproduction of the measured output^,- (when / E JT —J?).It should be emphasised that this assumption about the inter-connection structure of 37 is essential in the following devel-opment.

We apply to the interconnected system 37 a decentraliseddynamic control:

= FiZiZi (5)

where Zi(t)E;&ri j s the state of controller9%, and Fit gi}

h( and kt are constant and of appropriate dimensions. Thecompensated subsystemj?",- consisting of^7,- and 9s7,• can be

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represented as

where jq

1

Pi =

Qi -

*~

• \A

' ~

• \ : \

- [Qi

AiXi+PiV

,zf)T,and

bikicj -

•cT-

0]

<

bthj

Fi

(6)

(7)

Our goal is to choose the order rt and the parameters Ft,gt, h{ and kt of the controllers Wt such that the null solutionxt = 0, / <E fr, of the compensated interconnected system^,consisting of the compensated subsystems S" t of eqn. 6 andthe interconnections in eqn. 2, is asymptotically stable in thelarge for all nonlinearities 0,- satisfying eqn. 3; i.e. <? is absol-utely stable. For this purpose, we first investigate the effect ofproper choice of each individual controller ^,- on the sub-system S^i-

3 Subsystem analysis

In order to simplify the notation in this Section, we dropthe subscript / from the subsystem Sf ^ of eqn. 1 so that

x = Ax + bu + pvT

y = ex

w = qTx

(8)

where x G ^ 1 " . Furthermore, we assume that the interactioninput v and interaction output w of 5ft are scalars; thusu, y, w G ^ , We assume also that yt is controllable by thecontrol input u and observable from the measured output^;i.e. the triple (A, b, cT) is controllable and^observable.

Similarly, we represent a typical controller 9^ of eqn. 5associated with J ,- as

i = Fz + gy

u = — hTz —ky(9)

where z e ^ r . The compensated subsystem J 7,-, consistingt in eqn. 8 and "g7,- in eqn. 9, can be described by

cf,-: Jc = Ax + py

w = ^TJc

where Jc = ( ^ T , z T ) T , a n d

~A-bkcT -bhT

A = '

(10)

P =0

01)

f = tiT 0]

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We now define the following transfer functions:

Gyu(s) = cT(sI-A)lb

Gyv(s) = cT{sI-A)~xp

Gwu(s) = qT(sI-A)lb

H(s) = k + hT(sI-F)1g

Ws) = qT(sI-A)lp

lip = b in eqn. 8, then, by direct computation, we obtain

(12)

Gyu(s)H(s)(13)

Similarly, if q =c in eqn. 8, then W(s) can be computed as

W(s) =Gyu(s)H(s)

(14)

Since the transfer functions in eqns. 13 and 14 are of the sameform (in fact, they have the same poles), the following argu-ment applies to both of them equally well. Therefore, in theremainder of this Section, we shall assume that p = b in eqn.8,so that W(s) is as given in eqn. 13.

We begin our development by writing Gyu(s) and Gwu(s)in eqn. 12 explicity as

— a s + 05)

where 0 < v < n — 1, and

w u S a(s)

(16)

In writing eqn. 15 we assume that Gyu(s) has n — v — 1 zeros.We shall further assume that these zeros lie in the open left-half complex p laned" . We choose the order of ^ t o be v;i.e. r = v. Then H{s) in eqn. 12 has the form

« = V. ^o(s)

= 7?o-jr (17)

Finally, using eqns. 15—17 we can express W(s) in eqn. 13 as

W(s) =oc(s)a(s) (18)

We now choose the parameters of the controller ^"as follows:(a) The coefficients i?y, / = 1, 2 , . . . , v, in eqn. 17 are

chosen arbitrarily, subject to the condition that rj(s) has allits zeros inJ^~.

(b) The coefficients oh j = 1, 2 , . . . , v, and 17O in eqn. 17are chosen as

(19)Oj = Ojp' j = \ , 2 , . . . , v 1

M o = ov + 1pv + 1 J

where p > 0 is a parameter to be specified, and ahj = 1,2,

311

. . . , v + 1, are such that the polynomial

a(s) = s + ajS + a2s + . . .

has distinct zeros all i

sup(20).

Denoting byJ2'+ the closed right-half complex plane, we nowstate the following lemma, whose proof is given in the Appendix.

Lemma 1Under the conditions (a) and (b) above, for any e > 0 thereexists a p > 0 such that, whenever p> p, the following condi-tions are satisfied:

(j)A has all eigenvalues i

max|Xfc[H/T(-/co)R/(/w)]k

(25)

The properties of the closed-loop system transfer functionW(s) stated in the above lemma enable us to develop a stabil-isation procedure for the interconnected system £? describedin eqn. 1. This we consider in the following Section.

4 Main result

The preliminaries in the previous Section have already givena clue about the mechanism of achieving stability of the closed-loop interconnected system: a small-gain version of the circlecriterion [10, 12], which we summarise below.

^Let us consider the compensated interconnected systemi^, which can be described in compact form as

n - ( 2 1 )

w - Qx J

where

and

A = d\ag(AuA2,... ,AN)

P = diag(PuP2,...,PN)

Q = diag(Qx,Q2,...,QN)

with

p. = [p? o T ] T Qi = [Qi 0]

This linear part of 5? has the transfer function

W(s) = Q(sI-AylP (22)

which is in block diagonal form, i.e. W(s) = diagW2(s), . . . , JVN(s)], with the diagonal blocks.

K(s) = QiisI—A^Pf i<EjT (23)

corresponding to the compensated subsystems^,- of eqn. 26.Also, it follows from eqn. 3 that the aggregate nonlinearity

N

4>:^"-^ m ,m= 2 mh satisfies

H K K H W I I (24)

for some positive number n.We now use the following result of Reference 12 about the

absolute stability of £7 described by eqn. 21.

Lemma 2Suppose the matrix A has all its eigenvalues in^3". Then thesystem J?*in eqn. 21 is absolutely stable if

where \k denotes the eigenvalues of the indicated matrix.

Using lemmas 1 and 2, we arrive at our main result as follows.

Theorem 1Suppose that the triplets (Af, bt, c

Tt) of eqn. 1 are controll-

able and observable and, in addition, have all the zeros m^~.Suppose also that the pairs (P{, Qt) satisfy the condition givenin eqns. 4. Then there exist a set of local controllers Wi,i^Jr, of eqn. 5 such that the compensated interconnectedsystem J?7 of eqn. 21 is absolutely stable.

ProofBy the condition given in eqns. 4, the subsystem transferfunctions Wt(s) of eqn. 23 have the forms

Wt(s) =

(26a)

(26b)

where At corresponds to A in eqn. 11, and c,- and c t areobtained from b{ and c,- by augmenting with zeros. Then itfollows from lemma 1 that under the conditions of thetheorem the local controllers 9s7,- can be designed such that thematrices At have all their eigenvalues in^2", and that for anyset of €{, i Gi/f,

|| Wi(s) || < et (27)

for all sG>^+. In particular, expr. 27 holds for all s=jco,'. Thus the matrix A of eqn. 21 has all its eigenvalues

", and letting ei= i/K2, the transfer function W(s) in

eqn. 22 satisfies expr. 25. Then the proof follows from lemma 2.

We note that the conditions given in eqns. 4 have a similarstructural interpretation to those derived in the context ofdecentralised stabilisation by static state feedback [11, 13].Constraining the structure of interaction among the subsystems,the conditions given in eqns. 4 make it possible to design localcontrollers which can stabilise the interconnected system inde-pendent of the size of the interconnections so long as they arebounded. The conditions given in eqns. 4, however, are not theweakest conditions for decentralised stabilisability, and onecan expect to obtain less restrictive conditions by a moredetailed analysis of the effect of local controllers on thesubsystems and on the interconnections, as has been done inthe case of state feedback [11]. We should also note that theconditions given in eqns. 4 can be a promising starting point toattack the problem of suboptimal design of local controllers,the state feedback version of which has been considered inReference 13.

5 Acknowledgment

The research reported herein has been supported in part bythe US National Science Foundation under grant ECS-8011210,and in part by the US Department of Energy, Division ofElectric Energy Systems under contract DE-AC0377ET29138.

6 References

1 DAVISON, E.J.: The decentralized stabilization and control of aclass of unknown nonlinear time-varying systems', Automatica,1974, 10, pp. 309-316

2 IKEDA, M., UMEFUJI, O., and KODAMA, S.: 'Stabilization oflarge-scale linear systems', Syst.-Comput.-Controls, 1976, 7, pp.34-41 v ,

3 SILJAK, D.D., and VUKCEVIC, M.B.: 'Decentrally stabilizablelinear and bilinear systems', Int. J. Control, 1977, 26, pp. 289-305

4 SEZER, M.E., and HUSEYIN, 6.: 'Stabilization of linear time-

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invariant interconnected systems using local state feedback', IEEETrans., 1978, SMC-8, pp. 751-756

5 SILJAK, D.D.: 'Large-scale dynamic systems: stability and structure'(North-Holland, 1978)

6 SEZER, M.E., and HUSEYIN, 6.: 'On decentralized stabilizationof the interconnected systems', Automatica, 1980,16, pp. 205—209

7 IKEDA, M., and SILJAK, D.D.: 'On decentrally stabOizable large-scale systems', ibid., 1980, 16, pp. 331-334

8 WANG, S.H. and DAVISON, E.J.: 'On the stabilization of multi-variable decentralized systems', IEEE Trans., 1973, AC-18, pp.473-478

9 CORFMAT, J.P., and MORSE, A.S.: 'Decentalized control of linearmultivariable systems', Automatica, 1976,12, pp. 479-495

10 ZAMES, G.: 'On the input-output stability of time-varying non-linear feedback systems. Pt. I: Conditions derived using concepts ofloop gain, conicity, and positivity', IEEE Trans., 1966, AC-11,pp. 228-238

11 SEZER, M.E., and SILJAK, D.D.: 'On decentralized stabilizationand structure of linear large scale systems', Automatica, 1981, 17,pp. 641—644

12 MOLANDER, P.: 'Stabilization of uncertain systems'. Departmentof Automatic Control, Lund Institute of Technology, Lund^Sweden,report CODEN: LUFTD2/TFRT-1020/1 -11, 1979

13 SEZER, M.E., and SILJAK, D.D.: 'Robustness of suboptimalcontrol: gain and phase margin', IEEE Trans., 1981, AC-26, pp.907-911

14 INAN, K.: 'Asymptotic root loci in linear multivariable controlsystems'. Dissertation, Middle East Technical University, Ankara,Turkey, 1978

7 Appendix: Proof of lemma 1

Let W(s) in eqn. 18 be written as

W(s) =6(s)

(28)

where \pj and 0y, / = 1, 2, . . . , « + v, are functions of theparameter p . Comparing eqns. 18 and 28, and using eqn. 19,we have

00) = pvova(s)

(29)

The proof of lemma 1 is based on the following variation of aresult by Inan [14].

Lemma 3As p -*00, k + 1 zeros of 0(s) approach p times the zeros ofo(s) in eqn. 20, while the remaining n — 1 zeros of 0(s)approach the zeros of (5(s)r](s) in the following sense:

(a) Let fi be a zero of o(s) (which is, by assumption,simple). Then there exist Mx > 0 and px > 0 such that anexact zero n(p) of 0 (s) satisfies

\ld(p)-pfi\<Mi for all p>p!

(b) Let \ be a zero of 0(S)T?(S) with multiplicity /. Thenthere exist M2 > 0 and p2 > 0 such that exactly / zeros,Xy(p),/ = 1 ,2 , . . . , / , of 6(s) satisfy

\\j(p)-\\<M2p-1/l for all p>p2

Now consider 0(s) in eqn. 28, where 0(s) has its zeros iby assumption, and 17(5) and o(s) have their zeros i n ^ ~ bythe choice of the parameters of the controller 9s'. Then lemma3 ensures that there exists p~3 > 0 such that 6 (s) has all itszeros in^2" for all p> p3. Since the triple (A, b, cT) is con-trollable and observable, the proof of the first part of lemma 1is complete. To prove the second part, let us denote the zerosof a (s) by jUj, j' - 1, 2 , . . . , v + 1, and those of /3(S)T?(S) by\ h j=\, 2 , . . . , n - \ . Also let / i y ( p ) , / = l , 2 , . . . , v+l,and Xy(p), / = 1, 2 , . . . , n — 1, denote the exact zeros of

0(s) which approach p/zy and Xy, respectively, as p -> °°. Let usrewrite W(s) in eqn. 28 as

./, /wW(s) =

"n [s-nj(p)] V [s-Xj

j=1 j=1

a(s)

v + \ n-1n (s-pnj) n (s-\j)

n - 1 o —(30)

S i n c e 7 ( 5 ) h a s d e g r e e n — \, a n d Ay S ^ 1 , / = 1 , 2 , . . . ,n — 1 , w e h a v e f o r s o m e M3 > 0

y(s) (31)

On the other hand, using the result of lemma 3 it is easy toshow that there exists p4 > 0 such that, for p > p 4 ,

< 2 for all s

/ = 1 ,2 , . . . , (32a)

and

< 2 for all s

/ = 1,2, . . . , « — 1 (32ft)

Now consider the first term on the right-hand side of eqn. 30,which can be expanded into partial fractions as

o(s) Rj(p)

n (s — PUJ

where

(33)

i = 1 , 2 , . . . , * >v+ 1I]

(34)

Using eqn. 19,/?y(p) in eqn. 34 can be computed as

Rj(p) -

II

IEEPROC, Vol. 129, Pt. D, No. 6, NOVEMBER 1982 313

.V I -X ..V- 1

so that

Sinee Hj G ^

S — PjJLj

= M4j

~, eqn

P

<oo j -

. 36 implies

1 Re Hj |

= 1,2,

that

7 !

.,v+\

for all s G^ + , so that from eqn. 33

(35)

(36)

o(s)

n (s-(37)

From eqn. 30 and exprs. 31, 32 and 37 it follows that, ifp > max (p~3, p~4 }, then

M/(s)|<-2"+ i ;M3M4P

(38)

for all s e^+. Now, for any e > 0,lettingps = (2n+vM3M4)/eand p~= max {p~3, p~4, Ps}, it follows that

I W(s) I < e (39)

for all p > p and for all s G ^ + , thus completing the proof ofthe lemma.

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