Off-axis multivariable circle stability criterionProf. C.J. Harris., B.Sc, M.A., Ph.D., C.Eng., F.I.M.A., M.I.E.E., and R.K. Husband., B.Sc.

Indexing terms: Nonlinear systems, Multivariable control systems, Stability, Feedback

Abstract: In 1968, the scalar circle stability criterion was extended by Cho and Narendra to the off-axis caseby the use of multipliers Q, Q'1 which have an RL or RC realisable structure, so that the product of thelinear portion of the feedback system and the multiplier was positive-real. Fabl, Freedman and Zamesproduced a multivariable on-axis circle criterion for systems whose linear part is normal for all frequencies.As yet, the off-axis circle criterion has not yet been established for the multivariable case, although Cook(1976) has shown that a criterion of this type does provide the conditions for the absence of limit cycles.Utilising the loop transformation theorem and the passivity theorem, an off-axis multivariable circle stabilitycriterion is established via the method of multipliers for nonlinear feedback systems with a normal linearoperator. The criterion is shown to have a simple graphical interpretation based on the Nyquist plots of theeigenvalues of the system.

1 Introduction

An area of major interest in modern control theory has beenthe determination of the stability of feedback nonlinearsystems. The study of input/output stability of dynamicalsystems was originated [1—3] as an alternative to the dif-ferential-equation approach based on Lyapunov's directmethod [4, 5] . In this early work, the now classical on-axiscircle stability criterion for nonlinear single -input/single -outputsystems was established. In 1968, these scalar results wereextended by Cho and Narendra to the off-axis case by the useof multipliers Q,Q~l which have an RL or RC realisablestructure, so that the product q(s)g(s) of the linear portion ofthe feedback system and the multiplier is positive-real. Formultivariable systems, Sandberg [2,3,7] first introduced theconcept of extended spaces and derived stability conditionsfor multivariable feedback systems in the L2,LP and Lspaces, but gave no graphical interpretation. Falb, Freedmanand Zames [8] produced in 1969 a multivariable on-axiscircle stability criterion with the restriction that the linearportion of the nonlinear feedback system was normal for allfrequencies. In 1973, Rosenbrock and Cook [9,10] presenteda slightly different multivariable circle stability criterion fordiagonal nonlinearities based on the diagonal dominance ofthe linear part of the feedback system. More recently, Meesand Rapp [11] (1978) and Valenca and Harris [12] (1979)have extended multivariable circle criteria to account forlinear operators that are non-normal. As yet, the less conserva-tive off-axis circle criterion has not been extended to themultivariable case, although Cook [13] has shown that acriterion of this form does provide the conditions for theabsence of limit cycles, which of course allows the possibilityof unbounded responses.

In this short paper, we derive an Li stability result for anonlinear multivariable feedback system with a normal linearoperator. The approach adopted is based on the loop-trans-formation and passivity theorems; the stability theorem isshown to have a simple off-axis circle-type graphical interpret-ation that involves the Nyquist plots of the eigenvalues of thesystem together with a circle determined by the bounds onthe nonlinearity. Consider the class of nonlinear feedbacksystems described by the following equation:

y = G{u-N(y)} 0)where the nonlinearity N():Rn -+Rn is assumed to beboth causal and anticausal, time invariant (except on sets of

Paper 1450D, first received 12th March and in revised form Sth June1981Prof. Harris is with the Department of Electrical & Electronic Engin-eering, Royal Military College of Science, Shrivenham, SwindonSN6 9LA, England, and Mr. Husband is with the Department of Engin-eering Science, University of Oxford OX1 3PJ, England

zero measure), unbiased and sector bound by the followinglipschitz-type condition:

<*(y -

where a and 0 are constants in R, for all y,y' GRn and £is a constant \nR+.

G is a linear time-invariant operator on L\ which can becharacterised by a matrix of stable impulse response functionswith each element in the algebra .4 [1, 14].

Let the convolution algebra A{a),a&R have elements ofthe form

g(t) =i=0

t-tt) t>0

0 t<0

where ga&L la for L la defined as the set

\g(t)\ exp(-at)dt

and where t0 = 0 and tt > 0 for i = 1, 2 , . . . ;&• GR and

2 \g(\ exp(— ati)<°°. g(t) is said to belong to A .(a) if,i = 0

and only if, there exists a cxj Gi? such t h a t # ( ) GA(ox). Thealgebra A _(a) will be referred to as A.

The algebra A is a commutative Banach algebra, and thefunctiong GA is invertible in .4 if, and only if, [14, 19]

(3)Inf \g(s)\>0Re(s)>0

The importance of the above results can be seen by consider-ing the functional equation

y = —Gy + u 2e (4)

where G :Lie^-L%e is a linear time-invariant operator suchthat Gu =g® u,gGAn *n .L\e is the extended version of V\and is considered to be equipped with the topology of theprojective limit of spaces L%[0, T] for all T> 0. From eqn. 3,the functional feedback system of eqn. 4 defines either a Z,"stable system if (or an A stable system if and only if)

Inf |det[/ + £ ( s ) ] | > 0Re(s)>0

This result can be graphically tested by the generalised Nyquistcriterion [15], i.e. the feedback system of eqn. 4 is U\ stableif the contour defined by the concatenation of the eigenvaluearcs of g(s), generated when s describes the extended imagin-ary axis, does not encircle the — 1 point. The problem ofestablishing necessary and sufficient conditions for 2 " stabilityis highlighted by the observation that Anxn does not charac-

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tense all linear time-invariant operators on L2. It appears thatnecessary and sufficient conditions for input/output stabilityin the form of a Nyquist-type criterion will only be obtainableby contracting the signal space to a subset of L2 to avoidcertain pathological operators. Such an approach was proposedby Valenca and Harris [15] (1980) with the spaces X2 and

If the spaces on which the feedback system of eqn. 1 isdefined are inner-product spaces (which, if complete, areHilbert spaces H), then the passivity theorem [14,16] showsthat the following theorem holds.

Theorem 1The feedback system y = G{u —N(y)} is H stable for oper-ators G,N : He^>He if the following conditions are satisfied:

(c) the operator N is passive, i.e. <LNx, x > T > a for someconstant oc, for allx EHe and TBR

(b) the operator G is strictly passive, i.e. <Gx,x> T>(3 HJCTII

2 + a f o r / 3 > 0 a n d f o r a l l j c e # c , : r e / ? .

An important Hilbert space is L2, the set of real w-vectorvalued functions that are Lebesgue square integrable over[0, °°). Using this signal space, the following frequency-domainconclusions can be drawn about the passivity of a class oflinear time-invariant operators on U\.

L2e is the extended Hilbert space of all real-valued functionsx satisfying the condition

xT = PTx for all T E R, and PT is the truncation operator.

Lemma 1 [14]The convolution operator Gu =g ® u;g£An*n,u€:L2e ispassive if and only if {g(j'co) + g(jco)*}is positive-definite forco > 0, and strictly passive if, and only if, there exists a 5 > 0such that the least eigenvalue of {g(fco) +g(jco)*} is greaterthan or equal to 5 for co > 0 .

Note that the preceding analysis tested the operator G inthe unextended space, and so, strictly speaking, conclusionsregarding the passivity of G have to be arrived at by noting thecausality properties of the algebra A. The stringent passivityconditions on the operators of the feedback system can berelaxed by application of loop-transformation theorems andthe introduction of multipliers [15].

Let us introduce the multiplier Q

Qu*q® u uGL2

where

q(s) = d(s)In

ft=o h h

with bk>ak>0 for all k. That is, the multiplier 9(s) haspoles and zeros alternating along the negative real axis, with azero nearest to, but not at, the origin. The inverse class of thismultiplier, defined by qis)'1 =6(s)In, plays an equallyimportant role in the following analysis.

Theorem 2Under the above conditions, the feedback system y = G{u —N(y)} is L2 stable if the following conditions hold:(i) Q(I + BG ) ( / + AG y1 is strictly passive(ii) (N-A)(B-Nyx(fl is passive(iii) G^I + BG)'1 :L^^L2.

ProofAn application of the passivity theorem to Fig. Id gives con-ditions that ensure that, if u2 &L2, then;^ GL". Hence, toinfer u,y stability it only remains to note that K i and/C2 areoperators on L2 under the conditions of the theorem and thefollowing definitions of ,4 and B.

If the A and B loop-transformation matrices are chosenas A = odn and B = 0/n, it can be shown that condition (ii) isautomatically satisfied. In fact, this property is a direct resultof the nonlinearity ./V satisfying the inequality of eqn. 2.Using standard sector transformations, it can be seen that(N — A)(B—N)~l is incrementally inside the sector {0,A:}.Hence, the positivity of (N-A)(B -N)~l Q'1 follows froma vector generalisation of Zames theorem. It is also necessaryto establish that (N — A)(B — N)Q is positive for an off-axisinterpretation. This can be proved in a parallel manner.

J • G(I-AG)'1

N-A

m V

U2 i n• i

- • Q (1• BG)(U

-A)(B-N)

AG)'1

' V

y 2

Fig. 1 Loop transformations

Kl =G(/ + BG)'l(B—A)K2 =(B-Ny1Q-i

2 Off-axis stability criterion

Theorem 2 has a simple graphical interpretation when thelinear operator g(jcS) is normal (i.e. g(jcS) commutes withg(jco)* for all frequencies co). With the multivariable on-axiscircle criterion [11, 12], extensions to include non-normaloperators involved either the bounding away of the eigenvaluesof some normal approximate from the critical disc or anincrease in the critical disc size to compensate for non-nor-mality. Unfortunately, the presence of the multiplier in thepassivity approach seems to rule out any parallel developmentfor the off-axis circle criterion. This possibly explains thedifficulty that Cook [13] (1976) experienced in establishing asimilar result.

Theorem 3g (/co) is normal for all co.

(a) Case 1: | 3 > a > 0 . Under the above conditions, the feed-back system described by eqn. 1 is L" stable if the eigenvaluesof g(jto) do not encircle or intersect an off-axis disc passingthrough the points (— a"1 — A, 0), (— /T1 + A, 0) in thecomplex plane A > 0.

{b) Case 2: 0 > 0 > a . The system described by eqn. 1 is L2

stable if the eigenvalues of g(jco) are wholly contained withinthe off-axis disc passing through the points (— (3"1 + A, 0),( - a ' 1 - A , 0 ) , A > 0 .

ProofTo establish theorem 3, it is sufficient to show that conditions(i) and (iii) of theorem 2 are satisfied by the conditions oftheorem 3. To invoke lemma 1, it is necessary to establisha priori that Q(I + BG)(I + AG)~l has an impulse response

216 IEEPROC, Vol. 128, Pt. D, No. 5, SEPTEMBER 1981

in An X n . Q and (/ + BG) obviously have impulse responsesin An x n , and so, by noting that An x n is closed under con-volution, it remains to show that (/ + BG)~l is An xn stable.As g(jcS) is normal, this is readily concluded [19] if theeigenvalues of g(jcS) do not encircle the point — j3~1. Thiscondition is implicit in theorem 3.

Let

F = Q[I + BG] [I + AG]~l

Lemma 1 concludes that F is passive if, and only if, thereexists a 5 > 0

Inf X [F(/co) + F{joS)* ] > 8 > 0

As i(/co) is normal, a unitary matrix E(jco) decomposesg(joS) into diagonal form. Hence

F = Q[I + EE* BGEE*][I + EE* AGEE*]'1

= EQ[I + * A [£(/«)]][I + A\[£(/w)]]~l E*

Thus the operator E also decomposes FQ'oS) into diagonalform, and similarly the Hermitian matrix [F(jcS) + F(jco)* ]is also so decomposed by E. Hence

= 2Re {6 ( /«)( / + 1?X [£( /«)] ) ( / + ^ X [£(/<o)])"J

and so the conditions of theorem 2 are met if, and only if,

Inf

for i = l , 2 , . . . , « and ( / + BG)~l :Ln2 -• L?. These con-

ditions are immediately recognisable as a multivariable off-axiscircle criterion and are satisfied by the conditions oftheorem 3.

A more conservative off-axis circle stability criterion, whichavoids the computation of the eigenvalues X,- [£(/co)], can bereadily derived by invoking Gershgorin's theorem, wherebythe eigenvalue plots of theorem 3 are replaced by the envelopeof circles of radii

fe=0or I \iMUcS)\

k=0

centred on the Nyquist plots of£w(/od) for i = 1, 2, . . . , « .We note that, in the above off-axis circle criterion, once a

circle has been selected to pass through the points (—a"1 —A,0] and (— 0"1 +A,0) for a particular Nyquist plot ofX,- [G(jcS)], it must be used for all other loop evaluations.

3 Discussion

As a numerical example, consider the feedback system ofFig. la with linear portion given by the normal matrix transferfunction

£(?) =s(s+

(s + 2) - 2

- 2 (s+5)

The eigenvalues of /(s)are given by Xx (s) = {s(s + 6)(s + 1)} l

and X2 = {s(s + I)2 j " 1 , and their Nyquist plots are shown inFig. 2. If we select an arbitrary lower limit on the nonlinearityN of a<mina,- of 0.64, the on-axis circle stability criterion

yields an upper limit of /3 > max ft to the nonlinearity of

|3 = 1.54, whereas theorem 3 gives the less conservative valueof 0 = 2.63 for closed-loop stability (see Fig. 2).

Note that g(s) is not strictly diagonally dominant, and sothe envelope of the Gershgorin circles of g(s) centred oni i iO 6 ^) a nd £22(/w) (see Fig. 3) will lead to far more con-servative values for the upper limits for ./V than for the eigen-value plots of Fig. 2. Fig. 3 shows that the off-axis circlecriterion based on Gershgorin circles is almost as good asthe on-axis circle criterion based on eigenvalue plots for asystem which is marginally diagonally dominant. This con-

off-axis circle

Fig. 2 Off-axis circle criterion based on eigenvalue plots

- -1

Fig. 3 Off-axis Gershgorin circle criterion— Gershorgin circle envelope

1EEPROC, Vol. 128, Pt. D, No. 5, SEPTEMBER 1981 217

stitutes a considerable practical computational saving in deter-mining closed-loop stability.

In many dynamical systems, the linear operator G is non-normal, and it would seem appropriate to adopt the tech-nique of Valenca and Harris [12] whereby G is approximatedby some normal operator <£, and the radius of the criticalcircle in the Nyquist plot is enlarged by a function of the normof the error of approximation (G — <£). Indeed, using thismethod, condition (iii) of theorem 2 becomes

-N)'1 ~H] Q~l is passivewhere

H = (B -A){I + B&Y1 (G -

The normal operator 3> is selected to minimise the norm of//.Unfortunately, in the case of non-normal G, condition (ii) oftheorem 2 is now dependent on the sector bounds of themultiplier Q, and a simple off-axis multivariable circle criterionfor feedback systems with non-normal operators is not possiblewithout further restrictions on the multiplier Q. An avenue ofinvestigation currently being pursued is the use of a noncausalmultiplier to contract the sector of the multiplier while leavingits phase characteristics unaltered.

4 Acknowledgment

The authors would like to thank GEC Turbine Generators,for supporting this work.

5 References

1 ZAMES, G.: 'On the input-output stability of time varying non-linear feedback systems. Pts. I, II', IEEE Trans., 1966, AC-11,pp. 228-238,465-476

2 SANDBERG, I.W.: 'On the L2 boundedness of solutions of non-linear functional equations', Bell Syst. Tech. J., 1964, 43, pp.1581-1599

3 SANDBERG, I.W.: 'A frequency domain condition for the stabilityof feedback systems containing a single time varying nonlinearelement', ibid., 1964,43, pp. 1601-1608

4 BROCKETT, R.W., and WILLEMS, J.L.: 'Frequency domain stab-ility cirteria Pts. I, II', IEEE Trans., 1965, AC-10, pp. 255-261,407-413

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7 SANDBERG, I.W.: 'Some results on the theory of physical systemsgoverned by nonlinear functional equations', Bell Syst. Tech. J.,1965,44, pp. 871-898

8 FALB, P.L., FREEDMAN, M.I., and ZAMES, G.: 'Input-outputstability - a general viewpoint'. 4th World IFAC Congress, Warsaw,1969,4.1, pp. 3-15

9 ROSENBROCK, H.H.: 'Multivariable circle theorems' in 'Recentmathematical developments in control' (Academic Press, 1973)

10 COOK, P.A.: 'Modified multivariable circle theorems' Ibid.11 MEES, A.I., and RAPP, P.E.: 'Stability criteria for multiloop

nonlinear feedback systems' in 'Proceedings IFAC MVTS, 1977,Fredericton, Canada' (Pergamon Press, 1978)

12 VALENCA, J.M.E., and HARRIS, C.J.: 'Stability criteria fornonlinear multivariable systems', Proc. IEE, 1979, 126, (6), pp.623-627

13 COOK, P.A.: 'Conditions for the absence of limit cycles', IEEETrans., 1976, AC-21,pp. 339-345

14 DESOER, C.A., and VIDYASAGAR, M.: 'Feedback systems: input-output properties' (Academic Press, New York, 1975)

15 VALENCA, J.M.E.,and HARRIS, C.J.: 'Nyquist criterion for input-output stability of multivariable systems', Int. J. Control, 1980,31, pp. 917-935

16 HARRIS, C.J., and VALENCA, J.M.E.: 'Extended space theoryin the study of system operators'. RMCS technical report EEED,1980

17 ZAMES, G., and FALB, P.L.: 'Stability conditions for systems with-monotone and slope restricted nonlinearities', SIAM J. Control,1980, 6, pp. 89-108

18 MacFARLANE, A.G.J., and POSTLETHWAITE, I.: 'A complexvariable approach to the analysis of linear multivariable feedbacksystems' (Springer-Verlag, LNCIS-12, Berlin, 1979)

19 DESOER, C.A., and WANG, Y.T.: 'On the generalized Nyquiststability criterion',IEEE Trans., 1980, AC-25, pp. 187-196

20 MEES, A.I., and ATHERTON, D.P.: 'The Popov criterion for multi-loop feedback systems', ibid., 1980, AC-25, pp. 924-928

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