on the generalized nyquist stability criterion

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-25, NO. 2, APRIL 1980 187

On the Generalized Nyquist Stability Criterion CHARLES A. DESOER, FELLOW, IEEE, AND YUNG-TERNG WANG, MEMBER, IEEE

I. INTRODUCTION

T HE NYQUIST stability criterion has been generalized in several ways for the multiinput multioutput case

(see, e.g., [1]-[7D. The concept of plotting eigenloci of the open-loop transfer function matrix is particularly attrac- tive because it allows one to check the closed-loop stability for a farnib of gain parameters by inspection. A rigorous theory of such a test was first given in [6] where the rational transfer function case was treated in detail- and much use was made of algebraic function theory, branch cuts, etc. [6]. Later MacFarlane et al. (see [71) used Riemann surfaces to discuss this stability test.

In this paper we present first a rigorous, straightfonvurd, and self-contained derivation of the stability test based on eigenloci without using either branch cuts or Riemann surjuces; only standard facts of elementary mathematical analysis and of analytic function theory are used (e.g., as in [8, ch. IX and X]). Second, we derive the stability test for distributed systems; more precisely, for those plants whose transfer function matrices belong to the recently proposed algebra 3 ( u ~ ) ~ ~ " ' [9], [14].

Section I1 derives the generalized Nyquist stability criterion for the lumped case. Section I11 develops the counterpart of Section I1 for the distributed case. To our knowledge this is the first treatment of a Nyquist test based on eigenloci which considers unstable distributed open-loop transfer functions. The recent summary [17] considers &-stable transfer functions. Unfortunately, the

Manuscript received February 12, 1979; revised September 24, 1979. Paper recommended by B. Francis, Chairman of the Linear Systems Committee. This work was supported in part by the National Science Foundation under Grant ENG76-84522 and in part by the Joint Services Electronics Program under Contract F44620-76-C-0100.

C. A. Desoer is with the Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory, University of California, Berkeley, CA 94720.

Computer Sciences and the Electronics Research Laboratory, University Y.-T. Wang was with the Department of Electrical Engineering and

of California, Berkeley, CA 94720. He is now with Bell Laboratories, Holmdel, NJ 07733.

result in [17] is incorrect: consider the following counter- example; let

and let the corresponding L &w) be defined by the Hilbert transform of I&jw)I. Note that L &w) is well-defined since l&w)l satisfies the Paley-Wiener criterion (see, e.g., [18, p. 4231, [20]). Note that I r f ( .)I E L , n L, and is discontinuous. Now w ~ l & " o ) ( e ' ~ ~ ~ ' ' ) constitutes the boundary value of a causal transfer function $(s) which is analytic in the open-right-half complex plane. Moreover, the map umh*u maps L, into L, and

Ih*uI2< sup Ii(jo)l-IuIz.

However, a Nyquist-type graphical test cannot be applied since the image of the imaginary axis under the map $(a)

consists of disconnected curves.

oER

Preliminaries

Let R(@) denote the field of real (complex, respectively) numbers. Let denote Cu { m}. Let R+ denote the nonnegative real line. Let @.,+(eo-) denote the closed- half complex plane {sECIRes>u} (open-half complex plane { s E CIRes <a}). C0+, eo- are abbreviated as @+,e--, respectively. Let R[s](R(s)) be the set of all polynomials (rational functions, respectively) in s with real coefficients. Let Z[+(s)] denote the set of zeros of the polynomial +(s) E R[s]. Let 9 [+(s)] denote the set of poles of the rational functions G(s) ER(s). Let G(s)ER(s)Px4; N,D,-' is said to be a right coprime factorization of G(s) iff G(s)= N,D,-' and (N,,D,) is right coprime 141. For (I E R, L,,o denote the set {f(.)l f ( . ) : R+ +C, jrl f(t)le-"'dt < XI}. Let u E R, the convolution algebra &(u) consists of the elements of the form

I i = O

where l)f,(.)EL,,,; 2) t,=O and ti>O, for i = 1,2;.. ; 3) f ; . E C and 6(! - ti) is the Dirac delta distribution applied at ti; and 4) Z7=olf;.le-o'i< 03. &(O) is abbreviated as &. f ( .) is said to belong to &-(a) iff there exists ul E R, u1 <a such that f ( . ) E &(a,). Let f denote the Laplace transform

0018-9286/80/0400-0187$00.75 01980 IEEE

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188 IBEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL AC-25, NO. 2, APRIL 1980

Fig. 1. Feedback system S under consideration.

off(-). Let &(u) denote the set {flfE&(u)}. Let &'?(a) denote the set { f l fE &- (u), f^ is bounded away from zero at infinity in @,,+ }. Let 3 (u) be the convolution ggebra errespopding to the pcintwise product algebra %((a)= [$-(u)~[&T(o)]-', i.e., %((a) is -the algebra of quotients f = A/d with ri E &-(a) and a E @?(a) [9], [ 141.

We consider the feedback system S shown in Fig. 1 where

6(s) E ~ ( s ) " x is proper; (1) k > 0. (2)

Let fir&-' be a right coprime factorization of &); then the closed-loop system transfer function matrix r-y is given by

fiy,.(s): = ke(s)[ I+ k6(s)]-'

=kfi , (s)[~, . (s)+ki '? , (s)]- ' . (3)

Since f i r and f i r + kfi,. are also right coprime, det (3,. + kg,.) is the closed-loop characteristic polynomial. Come- quentlyLthe ciosed-!oop syste,m is exp. stable if and only if %[det(D, +kc,)] c C- and H,, is proper. (Note that since 6 is proper, Hyr is proper if and only if det[I+ k6(m)] # 0; the latter condition will be guaranteed by the condition i) of Theorems L1 and L3 below.) By substituting 6(s)= l'?,.D;', we have

det[ 5, + kr",] detD,.(s)

det[ I+ k 6 ( s ) ] = A (4)

Since det[l_+ k6(s)]ER(s) is analytic on @, except at the poles of G(s), we can apply the argument principle 18, pp. 246-2471 to det[I+ k&)] to determine whether Z[det(s, + kN,)] c e-. For this purpose, we construct the Nyquist path pw as shown in Fig. 2,' where fw includes the poAhts + j m and - jm . ( d e +indentations guarantee that G(S) is analytic on Ha). The following result is well-known (see, e.g., [lo, p. 1411).

Theorem LI (Stability Theorem Based on det[ I + k&s)]: Lumped Case): Let E, be defined as above. Let the feedback system S of Fig. 1 satisfy (1) and (2). The

multiinput multioutput case, the closed-loop system and the open-loop 'It is good practice to take the c-indentations to be in C - since for the

system may have commonjwaxis poles: let

&(S) = 1 (s+l)-' 0 (s+l)-' s-l I ; then s = 0 is an open-loop pole as well as a closed-loop pole, Vk z 0.

closed-loop system S is exp. stable

- i i) det[I+k6(s)]fOy V S E ~ , ii) det[ I+ k6(s)]IsE~- encircles the origin

p: times in the counterclockwise sense, where p: denotes the number of @+-zeros of the open-loop system characteristic polynomial, counting multiplicities. 0

Remarks L1: a) In the single-input single-output case, i.e., m = 1, Theorem L1 reduces to the classical Nyquist stability criterion (see, e.g., [l 1, p. 3241).

b) Technically, Theorem L1 follows directly from the argument principle together with the fact that det[I+ ke(s)] tends to a constant as Is(+m.

A great virtue of the classical Nyquist criterion is that it checks the closed-loop stability for any k > 0 by inspection of d e Nyquist diagram of &), whereas Theorem L1 requires a plot of det[I+ k6(s)] for each k > 0 of interest. Our objective is to derive straightforwardly a generalization of the classical Nyquist stability criterion for multiinput multiou@ut systems which allows us to test the closed- loop system stability for any k >O by inspection. This is achieved by using Theorem L1 and establishng an iden- tity, which says that the number of encirclements of the origin by det[l+ k 6 ( ~ ) ] 1 ~ ~ ~ ~ can be determined via a family of closed paths formed by the eigenloci of 6(s) as s travels up along F, (see Lemma ~ 2 ) .

Now, for each s EN,,-&) is a well-defined Cmxm matrix; thus for each SE_N=, we have m (not necessarily distinct) eigenvalues of G(s), A,(s), h2(s), - - , a(s), which satisfy

det[h(s)I- e(,)] =0, i = 1,2; - ,m. ( 5 )

Note that

det[hl- 6(s)] =Am+gl(s)X"-'+ - - . +gm(s)

where the gi(s>'s are proper rational functions. Hence, we can write

det[Al- &s)] = - [ ~ o ( s ) h m + P , ( s ) A m - l 1 PO(S)

+..* +P, (s ) ] 1 =.-

a P0(s) P(gs) (6)

where &)E R[s] for i= 1,2, - * ,m; Po@) is the monic least common denominator polynomial of { gl(s), ~ ( s ) , - * ,g,(s)}. Since Z [ PO(s)] c 9 [&)I and since N , does not contain any pole of e(s),

P0(s)#O, VsEF,. (7)

n u s , for all s EN, - { +jm, - jm 1, is an eigenvalue of 6 ~ s ) iff ~ . ( s ) is a zero of the polynomial AHP(A,S). Now, for any labeling of the eigenvalues of &s), say,


DESOER AND WANG: GENERALIZED NYQUIST STABILITY CRITERION 189

I Fig. 3. An example which shows that each eigenlocus of the open-loop Fig. 2. The Nyquist path Pm: “X” denotes thejo-axis poles of &). transfer function matrix may not form a loop.

the loops y,*’s are circuits, it is legitimate to define the number of encirclements of some point in C by y;; and 3)

rn

det[ I+ k e ( s ) ] = a (1 + k&(s)), Vs E Fm. for any choice of (y,*)j=l,s.. . ,p, the sum of the number of

of encirclements of the origin by det[I+k&(s)]l,,~m. Thus, we can test the closed-loop stability for a y k > 0 .by inspection of the number of encirclements of - l / k by an indexed family of circuits (obtained from the eigenloci

i= 1 encirclements of - l / k by the ~ * ’ s is equal to the number Note that for each s E Fm

m L det[ I + k8(s ) ] = L (1 + k&(s))

i= 1 m A i ( N , ) ’ S ) *

= 2 L ( l +k&(s)) i= 1

= L (; ++)) i= 1

(by assumption (2), k >O). (9)

Now, as s travels up along f l m , we can label the eigenvalues of 8(s) such that, for each i , SH&(S) is a continuous function; hence we obtain m continuous eigenloci &(Fm) (this is possible because eigenvalues of a matrix are con- tinuour functions of the elements of the matrix, see. e.g., [12, p. 45D. Thus, at first sight, one might expect to count the number of encirclements of the point 0 by deqI+ k6(s)]l,,~, by summing the number of encirclements of the point - l /k by the &(Fm)’s. Unfortunately, this does not work because some curves A,(Fm) may not form a closed path as Example 1 shows.

Example 1

Consider

L J

Then det[hl- &s)] =A2 - (s - l)/(s + 1). The eigenloci X1(Fm),X2(~m) are shown in Fig. 3, where Fm is the

It turns out that 1) since the set of eigenvalues of &(-joo) is equal to the set of eigenvalues of &(+jm) counting multiplicities, we can always form an indexed family of loops (y,*>j=1,2...s from the set of eigenloci &(N,), i = 1,2; - ,m [see emm ma ~ 2 , part a)]; 2) since

jo-axis,w€[-m,+co]. 0

Notation

Let y ( - ) : [a,p]+C, a#P, [(w,/3]cR, y(-) is said to be a path in @ if y ( - ) is continuous. A path y ( - ) is said to be a loopify(a)=y(P).Apathy(.)issaidtobearoadify(.) is a primitive of a regulated function (hence y( e ) is differentiable except at a possibly countable number of points); if, in addition, y(a)= y ( p ) we say that y ( - ) is a circuit [8, pp. 217-2181.

Let y ( . ) : [a,P]+C be a circuit. Let the point a e y , where y: = y([a,p]). Then C(a; y ) denotes the number of countercZockwise encirclements of a E@ by y, i.e. [8, p. 2231

Generalized Nyquist Stability Criterion

For technical reasons, we parameterize the Nyquist path Fm shown in Fig. 2 by t E [awaq+ J, a compact interval. More precisely, we define differentiable function N,(.): [aOflq+ll+@ SO that2 fmaaoyaq+lD= f l a y N,(a,) = - j a y Nm(aq+,)= +jm. For convenience, we shall choose the parameterization Fm( .) to be a strictly increas- ing function; thus as t increases from cu, to aq+,, the point Rm(t) moves upward along Em from -joo to +joo (e.& first use the ordinate o of NmAas a parameter and then choose t =tan-’o). Note that G(s) has multiple eigenval- ues3 only at a finite number of points in @, and thus in

domain under the map f(.). *We use fl.) to denote a h t w n and f to denote the image of its

the discnrmnan %e polynomial A ~ f l R s ) in (6) has multiple zeros for some 1 EQ: iff

apolynomial in s, there are only a finite number of such 1’s. t e(s) of f l h s ) is zero at 1 113, pp. 248-2501. since e(s) is


1 9 0 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. ~ ~ 2 5 , NO. 2, APRE 1980

E, - { - j m , + Jca}. Consequently, the €-indentations of N, in Fig. 2 can be chosen so small that Vs EP, - { - jm , +jca}, 6 ( s ) has multiple eigenvalues only at a finite number of points on thejw-axis, o E (- c o , + ca). Let jbl , jb2, - - ,jbq denote such points, i.e., ecjb;), i = 1,2, - - ,q, has multiple eigenvalues. Let a], a2, - - , aq E R be such that a,<a,<.-- <aq<aq+I-and that F,(cr,)= jbi, i=1,2,-.-,q. Let Ii(-): [cr,-1,4+@, i=1,2;-.,q+l, be such that Ii(t)=&,(t), V t E[(~i-~,cr,], i.e., Fa(.) is the juxtaposition of A ( * ) , i = 1,2; - - ,q+ 1 [8, p. 2171. The images of the li( .>'s are shown in Fig. 4. Let I E {1,2;..,q+l); for each SEI, which is not an endpoint of I,, 6 ( s ) has m distinct eigenvalues X,(s), i = 1,2, . , m. Since S H ~ ( S ) is continuous on F, and since the eigenvalues of a matrix are continuous functions of the elements of the matrix [12, p. 451, we can uniquely define m continuous oriented eigenloci yi,, i = 1,2, * * , m by

Y;z(t): = X , ( W ) ) 9 t q ~ / - I , L y r ] . (1 1)

Lemma L2 shows that from (vi]) , we can

construct an indexed family of circuits whose image will be called the generalized Nyquist diagram of 6(s ) .

Lemma L.2 (Generalized Nyquist Diagram: Lumped Case): Given a proper 6(s) E R ( S ) ~ ~ " ' and the associated

; = I 2 ... , , , m / = 1 , 2 ; " , q + l

eigenloci (vir) defined by (1 l), then i = 1 , 2 , . . . , m

a) the members of (yiz) /=1,2,...,q+l

can be juxtaposed

to form an indexed family of circuts (~j*) j=l ,L. . . ,~ for some 1 < p < m ;

i = 1 , 2 , . . . , m /=1,2 , . . . , q + 1.

b) Let A ( - ) : [aO,aq+,]+@ be defined by m

A( t ) := (l+kyi/(t)), V t E [ a j - , , a , ] , i = I

I = 1,2; * ,q+ 1, (12)

then

function, and A(aJ =A(ao+ ,); i) A(.) is a circuit, i.e., A(-) is a primitive of a regulated

iii) C(0; A)=E&lC( - l /k;~j*) for any indexed family I = 1,2,-. . ,q+ 1

..., obtained in part a). Proof of Lemma L2: S e e Appendix.

Remark L2: To count the number of encirclements of (- l/k,O) by the %*Is, we actually only need that the eigenloci yil)s, thus the xi's, be continuous (see Eilenberg's method described in [8, p. 2511). Thus, the fact that &( .)'s are analytic in except at a finite number of points is not essential in proving Lemma L2.

Now we are ready to present the generalized Nyquist stability criterion for the lumped case.

Theorem L3 (Generalized Nyquist Stability Criterion: Lumped Case): Consider the feedback system S shown in Fig. 1 and described by (l), (2). Associated with &s) E

s - p l a n e

A

' 1 1 I Fig. 4. The Nyquist path Pm shown in Fig. 2 is considered as the juxtapositionofthepathsIi(.),i=1,2,...,q+l.jb,,i=1,2,...,q ,are the points on thejw-axis such that Gobi) has multiple eigenvalues.

O B ( S ) " ' ~ ~ are the eigenloci (yir) defined by

(1 1). Then the closed-loop system S is exp. stable

i = 1 , 2 , . - . , m 2=1 ,2 , . . . ,q+ l

*i P 5 ) x C(-k;Y;)=PO+ 1 J= 1

where p: denotes the number of C+-zeros of the open- loop system characteristic polynomial counting multiplicities and ( ~ j * ) j = ~ , ~ . . . , ~ denote any indexed family of circuits formed from (y i l )

i = 1 , 2 ; . . , m

Proof of Theorem L3: See Appendix. I = 1,2,. . . ,q+ 1

Remarks L3: a) Theorem L3 generalizes the well- known classical Nyquist stability criterion to the multiinput multioutput case. Note that Theorem L3 requires us to check the number of encirclements of - l /k by a finite family of cireits (Y,*),=~,~...,~ which is formed from the eigenloci of G(s) as s moves up along the Nyquist path %, of Fig. 2.

b) The logic of Theorem L3 is as follows: i) by Theo- rem L1, the system S is exp. stable iff as s moves up along z,, det[I+ k&(s)] has a change of phase of 27rp0, ; ii) for all s where 6(s) is well-defined (hence VsEF,), deqI+ k&s)] = IIy= 1 + kX,(s)); iii) by Lemma L2, as s moves up along F,, the eigedoci &(SI, i = 1,2, * - - ,m form a family of circuits %*, j = 1,2,. - - ,p; iv) therefore the determina- tion of the change of phase of det[I+ k&s)], required by i) above, reduces to counting the net number of encirclements of (- l / k , O ) by the $'s. Note that the plotting of the eigenloci is just a computational trick ~ e d to determine the net change in phase of det[l+ kG(s)]. Thus, we need not mention branch cuts and/or Riemann surfaces and/or the extended argument principle [6], [ 161, [17], ~ 9 1 .


DESOER AND WANG: GENERALIZED NYQUIST STABILITY CRlTF!RION 191

111. DISTRIBUTED CASE Now we have the following stability theorem which is the counterpart of Theorem L1 in the distributed case.

We consider the feedback shown in Fig* '7 Theorem Dl (StabiliW Theorem Based on det[Z+ where Distributed Case): Let F, be defined as above. Let the

&s) E 4 feedback system S of Fig. 1 satisfy (13)-(17). Assume that for some a, < 0 (13)

yote that any proper rational function is an element of B(u) for any a E R. Thus, the present formulation in- ckd? the lumped case in Section I1 as a special case. Let (%,,9,) be a a,-right-representation of &) [9], [14]; equivalently,

i) &s)= %,(s)@,(s)-l with %, ~ k - ( c l ~ m ~ m , 4, E k- (ao)"""; (15)

ii) therefxi$ %,~E$-(oo)mxm, '?( E $ - ( u , ) ~ " ~ such that '%, %, + 'It 9, = I ; (16)

iii) det'%,(s) is analytic and bounded away from zero at co in Coo+. (17)

Without loss of generality [ 141, $,(s) may be taken to be a proper rational function matrix with all its poles in hence by (17),

lim det 6,(s) = constant ZO. (18)

Note that the closed-loop system transfer function matrix r b y is given by

IsI--fcQ

Zjyr=k6(s) [I+kG(s) ] - '

= k%,[ G, + k%,] - l. (19)

U.t.c. the closed-loop system S is @(a)-stable, for some O < O

i) det[ Z + k 6 ( s ) ] #O, Vs E Fm; (22) ii) det[ I + k e ( s ) ] I , , ~ ~ encircles the origin

p t times in the counterclockwise sense, where p t denotes the number of @+-zeros of the ?pen-loop system characteristic function det 9,(s), counting multiplicities. (23)

Proof of Theorem Di: See Appendix. Remarks Dl: a) Assumption (21) means that the almost

periodic part of G(jo) is a constant matrix. This constant matrix is the zero matrix in many practical cases.

b) Theorem D l guarantees @(+stability, for some a < 0; this implies exp. stability: more precisely, in view of (21), fiy,(s) tends to a constant as IsI+co in @+, hence Hy,(t)=HoS(t)+H,(t)whereHoER"X" andH,(.)EL,,,. Consequently, the response y ( - ) of the feedback system S t: any step input r(t)= l(t)u, where U E R", will tend to Hy,(0)u expo2entiaZly (since there exists some p< 00 such that ly(t) - Hy,(0)ul Q plulea'). Thus HY, can be said to be exp. stable.

Sigce (%,,6,+k%,) is also a,-right-coprime, det[@,+ k9Lr] is the closed-loop characteristic function [14]. Consequentlx, the Aclosed-loop system is @-stable iff infRe,>,ldet[9, + k%,]( > 0. By substituting 6 = %, 9; ', we have

To obtain, for the distributed case, a generalized Nyquist stability criterion similar to Theorem L3, we note that the discriminant of the polynomial AHdet[AI- &)I is a polynomial in the elements of 6(s) [13, pp. 248-2501, hence

det [ &,(s) + k%,(s)] det 9,(s)

det[ Z+ k&(s)] = ,. - (20)

Since det[Z+k&)] is analytic on Coo+ except for afnite num)er of poles in Coo+ [these poles occur at the zeros of det Qr(s)], we can apply the argument principle [8, pp. 246-247J to eeqI+k&(s)] to determine whether infRes>,ldet[Qr + k%,]L>O. For this purpose, we construct a Nyquist path N , as in the lumped case (see Fig. 2). Note that 1) the €-indentations are again taken to the left of the jw-axis for the same reason as stated in the lumped case; and 2) by assumptions (13) and ( 15)-(17), 6(s) is analytic on Cue+ except for afnite nupber of poles in Cao+ [these poles are the zeros of det9,(s)]. Since a, <O, the c-indentations can be taken to the left of the jw-axis poles of &); furthermore, we choose the indenta- tion radius Q (ao( so that 4 s ) is analytic on H,.

i) it is an analytic function of s in Coo+ - Y[det 6,(s)]; (24)

ii) it can only have a finite number of zeros in any compact subset of Coo+. (25)

Thus, we can choose the radii of the +indentations so that for S E N , , e(,) has multiple eigenvalues only for some isolated points on the jo-axis, where o E[ - c o y + 001. Suppose thft there is only a finite number of values of w such that G(jw) has multiple eigenvalues. Then the construction of the generalized Nyquist diagram of Lemma L2 still holds and we are right back to the previous case. However, as showp in the Example 2, it may happen that, for some &) E 3 ( u ~ ) ~ " ~ , 6(s) has multiple eigenvalues for an infinite number of points on the jo-axis. Conse- quently, a naive generalization of Lemma L2 requires a construction of an infinite digraph (see the proof of Lemma L2).


192 IEEE TRANSACnONS ON AUTOMATIC CONTROL VOL. AC-25, NO. 2, APRIL 1980

Example 2

Consider

where il2(s)= C[ gI2(t)] with

Note that 6(s) E 8- (ao)2x2 for any a,> -4; thus, G(s) satisfies assumption 113). The discriminant of the polynomial Awdet[M- G(s)] is, for this case, equal to (83- 4g, , (s) / [ (s + 8)3(s + 4)]. By direct calculation using tables [ 15, formulas 440.1 1 and 440.121, we obtain

1,2; - ,%. U.t.c. the closed-loop system S is &(IT)-stable for some u < O

P 1 5) x c( - ;Y,.)=P+ j = 1 0

where p : denotes the number 2f C+-zeros of the open- loop characteristic function det qr(s) , counting multiplicities. 0

Remark 0 3 : Note that Theorem D3 does not assert anything when (- 1/ k, 0) belongs to one of the disks 5fim,c). his represents no loss in practice, since if ( - 1 / k,O) belongs to such a disk, then det[l+ kGCjo)] = O(c) at high frequencies and clearly the system would then be very sensitive to small changes in k or in 6 at those frequencies; indeed cor such k's, the transfer function matrix, from r to e, H e r ( j w ) = O ( l / z ) at those frequencies.

Proof of Theorem 03: See Appendix.

Thus, the discriminant has an infinite number of jo-axis zeros, i.e., 6(s) has multiple eigenvalues for an infinite number of points on thejw-axis. 0

To overcome the difficulty of dealing with a graph with an infinite number of nodes, we note the following. Let hm, i = 1,2,. * - , p,, be the distinct eigenvalues of e( + j m ) with multiplicities mi, respectively. Then by continuity, for any given z > 0, there exists Q > 0 such that for i= 1,2,. - , &w) has mi eigenvalues inside the closed disk 6(&m,c), Vlwl> a. Now by (25), &w) has multiple eigenvalues at only afinite number of points within the compact interval [ - j0, +ja]. Therefore, by assigning a set V, of p,, nodes which now represent the closed disks D(Xim,z), i = 1,2, * - * , CLO, we obtain a finite digraph C3 by the construction stated in the proof of Lemma L2 and we are right back to the previous case. Now following the procedure stated in the proof of Lemma L2, we identify, in the digraph B , a finite indexed family of loops Ti , i= 1,2; - . ,p , for some p Qm. The corresponding eigenloci will then form a finite indexed family of circuits (y,?)i,=l,z...,p provided that additional mi directed roads are arbitrarily joined in eack disk Dfi.m,z), i = 1,2,- - , po from the eigenvalues of G(+jQ) to xim and then to the eigenvalues of & - j ~ ) . These (y,*)+ ..., constitute the required generalized Nyquist diagram and the generalized Nyquist stability criterion for the distributed case is stated in Theorem D3.

Theorem 0 3 (Generalized Nyquist Stability Criterion: Distributed Case): Consider the feedback system S shown in Fig. 1 and described by (13)-(17) and (21). Consider any finite indexed family of circuits ( ~ j * ) , = ~ , ~ . . . , ~ and the closed disks Eam,€), i = 1,2, * ,% as indicated above. Let k>O be such that (- l/k,0)4Bfim;c) for i =

Example 3

Consider the feedback system S shown in Fig. 1 with the open-loop transfer function matrix &s) described by (z6) of Example 2. The generalized Nyquist diagram of G(s) is shown in Fig. 5. Note that & j w ) has multiple eigenvalues at ON k2.9868, ? 15.44, * - * , etc. (which are the solutions of sin0/2 =w/2cosw/2; see (27) of Exam- ple 2). The behavior of the eigenloci A,(-), A2(.) is further magnified in Fig. 6: a simple local analysis confirms that, at w.y8.9868, the eigenloci A,(-) and A,(-) make a 90" turn.

Refemng to the generalized Nyquist diagram of &s) shown in Fig. 5, we know that, by Theorem D3 above, the closed-loop system is @(a)-stable (for some o <0) for k E (0,6] since the open-loop system has no C+-poles. [7

IV. CONCLUSION

The generalized Nyquist stability criterion based on the eigenloci of the open-loop transfer function matrix is derived for the lumped as well as the distriFuted case. First, a stability theorem based on det[l+kG(s)] is presented (see Theorem L1 and Theorem Dl). Construction of the generalized Nyquist diagram is shown in Lemma L2. The generalized Nyquist stability criterion is then presented in Theorem L3 (Theorem D3) which shows that the stability condition of Theorem L1 (Theorem Dl, respectively) can be checked via the encirclement condition of the critical point by the circuits formed by the eigenloci of the open-loop transfer function matrix.

In practice, this graphical test is applied as*follows. 1) Compute for each w ER, the eigenvalues of G ( j w ) ; as w increases, these eigenvalues describe some directed curves on the complex plane; 2) pick a starting point on any curve, say, at o = 0, and follow the curve as w increases; 3) if a point of multiple eigenvalues is reached, continue the



I 1 C - plane

Fig. 5. The generalized Nxquist diagram of 5s) considered in Exam- ple 3 [G(s) is specified by Os)].

-0.2500 -0.2450 0.2400 0 2350

~ ~ 8 . 9 6 p 8 . 9 9 6 5 ‘ -0.3300

C - plane I

Fig. 6. Blowup of the. eigenlog A,(.) and A,(-) in the neighborhood of 04.9868 r/s where WO) has a multiple eigenvalue.

process by choosing arbitrariEy a not-yet-traversed curve until a closed curve is formed; 4) repeat the procedure described in 2) and 3) until all eigenloci are exhausted; 5 ) count the sum of the number of encirclements of (- l /k, 0) by the closed courves formed and determine the closed-loop stability by conditions i) and ii) of Theorem L3 or D3.

APPENDIX

Proof of Lemma L2: a) To show that the members of ( Y J can be juxtaposed to form an indexed

i =1 ,2 , . . . ,m I = 1,2,. . . ,q+ 1.

family of circmts (vj*)+ ,,2 ,..., for some 1 < p <my we construct a digraph 4 associated with (yir)

i= l .2-... m I = 1,2,. . . ,p+ 1

, , ~

To each point jb, E where &bl) has multiple elgen- values I = 1,2, * - , q, we assign a set V, %f pl nodes where pl =number of distinct eigenvalues of Gijb]); thus each node of VI represents a distinct eigenvalue of &&). We further assign a set V, of p,, nodes to represent the distinct eigenyalues of 6L- j~ ) as well as those of 6( +jm), (note that G( + j m ) = G( - jm), hence they have the same set of eigenvalues counting multiplicities). NOW for I = 1,2, * - * , q, we assign m directed branches from to V,, each such branch represents one of the m continuous eigenloci yil,

“0 “I

Fig, 7. A typical digraph 9 corresponding to a 4X4&): i (+ j? )=

has two distinct eigenvalues with multiplicities 2, respechvely (repre- G(- jw) has four distinct eigenvalues (represented by Vq) and m) sented by V,).

i = 1,2,. - , m. Similarly, we assign m directed branches from Vq to V,, each such branch represents one of the m continuous eigenloci yi,q+ I , i = 1,2, - - - , m. A typical digraph corresponding to a !X 4 6(s) is :hewn in Fig. 7. Note that in this exampleYAG( + j m ) and G( - j m ) has four distinct eigenvalues and GijO) has two distinct eigenvalues of multiplicities 2, respectively; hence q= 1 and we have only two set of nodes, Vo and VI. Note that, by construction, the number of the directed branches entering (or leaving) any node of 4 is equal to the multiplicity of the corresponding eigenvalue. Hence, the digraph 4 has the following property:

for any node of 9 , the number of branches entering that node is equal to the number of branches leaving that node.

(All We now construct a family of circuits by suitable juxtaposition of the members of (yi,) . We

select an arbitrav node of 9. We move along any directed branch to some other node. As we repeat this process, by property (Al) of 4 and the fact that 4 has afinite number of nodes, we eventually reach an already traversed node; thus this process identifies a simple loop r, in the digraph 4 . Corresponding to each branch of rl is one member of

i=1,2,....m I = 1,2,.. . ,q+ 1

the family (Yi/) . By the construction of the i =1 ,2 , . . . ,m

digraph 9 , it follows that the eigenloci yi,’s corresponding to branches of rl form a not-necessarily-simple loop which we call y:. Now we delete from 4 the branches associated with r,. Note that the remaining digraph still has the property (Al). Now if the remaining digraph contains no branch, there is only one simple loop rl in 4 and the eigenloci (yil) form a single loop vj*

i=1,2,...,m

in C and we are done. Otherwise, we select an arbitrary I= 1,2,.. . ,q+ 1

node of 9 which has at least one outgoing branch and repeat the above procedure until all branches of 4 are exhausted. The result is a finite indexed family of loops (%*)i., 1,2,...,p in the complex plane @, where the integer pE{1,2;-- , m } . To see that the yj+’s, j= 1 , 2 , - - - , p are circuits, we note that by the implicit function theorem [S,

1=1 ,2 ; . . ,q+ l


194 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL Ac-25, NO. 2, APRIL 1980

pp. 270-2731, the &(.)'s are differentiable between con- secutive points where &u) has multiple eigenvalues. Now since the I,( -)'s are chosen to be differentiable on (a,- 1, a/), we have, by (1 l), the yir(.)'s are differentiable on (a,- 1, a!). Therefore the y7's are circuits.

b) i) Since yiI(-), i=1,2, .-- ,m; 1=1,2;.. , q + l are

roads on [aI-l,aI] and

m m

IT (1 + k ~ , ( a / ) ) = II (1 + kyi./+ l(aI)), i=l i = l

A ( - ) is differentiable on (ao,aq+ except at a1,a2,- - - , aq. Furthermore,

A(aJ= II (1+kyiI(a0))= II (l+kh-") m m

i = 1 i = 1 m m

= I I ( l + k h + w ) = I I ( 1 + k ~ i , q + , ( a 4 + , ) ) = ~ ( a q + , ) i= 1 i= 1

where A.-w,&+m denote the eigenvalues of &( - jco) and 6( +jca), respectively. Hence, A( .) is a circuit.

ii) Follows from (1 1). iii)

Note that we can relabel the yi;s into yp,'s so that for each ~ ~ ( 1 , 2 , ~ ~ ~ , m } , t h e j u x t a p o s i t i o n o f y ~ ~ , 1 = 1 , 2 ; ~ ~ , q + 1 form a continuaus eigenlocus, say yS. Thus, (A21 becomes

(by (12) and the fact that

% + I d(A( t ) ) =Jq A(t)

Z

=27rjC(O; A). Q.E.D.

Proof of Theorem L3: The closed-loop system is exp. stable

i) det[ I + k d ( s ) ] zO, VSEN,; ii) C(O;det[Z+k&(s)]I,,,_)=pO,

(by Theorem L1)

det[ I + k 6 ( I , ( f ) ) ] Z O , Vt E [ al-lya,] , V I = 1,2; - - ,q+ 1

I = 1,z.. . , 4+ 1

(parameterization of N,) m

V t € [ a14,aI], I = 42,. - - ,q+ 1

1 I = 1,r.. . ,q+ 1 J

(by definition of yir)

lii) c( - 1 ;A) =Po,

-1 1 i) - e(Yi*>j=l,*,. . . , p

P 1 (by Lemma L2) ii) x c( -k;Y;)=P:.

j- 1

Q.E.D. Proof of Theorem Dl: We will show, in two steps, that

under the assumptions (1?)-(17) and (21), the closed-loop transfer function fiYr E &(a) for some (T <O (thus the closed-loop system is exp. stable) iff (22) and (23) hold.



Claim I: As a consequence of (20) and of assumption (21), conditions (22) and (23) are equivalent to

inf I i ( s ) l>O and inf IR(s)l > O (A3) Res > 0 S €@ =

where i ( s ) : =det[@,(s)+ k(%,(s)]. Note that by (15), i ( s ) is analytic in Ca0+ and hence

analytic on gm; furthermore, by (18) and (21), i ( s ) tends to a constant, say R(co), as IsI+co in @+. Thus, the argument principle together with (20) show that (22) and (23) are equivalent to (A3).

Claim 2: Equation (A3) is equivalent to

inf I i(s)l> 0, for some a < 0. 644)

It is clear that (A4) implies (A3). Thus, we only have to show that (A3) implies (A4). To see this, we note that assumption (15) and the closure of the algebra &-(ao) p d e r addition and multiplication imply that ii;(s)E &- (ao); thus there is some a1 <ao such that i ( s ) E &(al). Therefore, X(t)e-‘l‘E&, where ~ ( t ) : = e-’K(s)]. This implies that fpr somep2~(ul,a0), tX(t)e-‘2‘ E&, i.e., E[rx(t)] = %(s) E &(aJ c &(ao). Thus, ?(s) is analytic and bounded in e,,+. Now let p: =inf,,, Ix(s)l and p’:= supsECO0+~x’(s)~. Then v E > o , ~ ~ > O (Zi., take a = ( p - 2r)/p‘) such that VsoEFm, Vls-sol<S

Ix(s)l> Ix(s0)l- $Is - sol > P - P f b - sol >.5

Res>o

(by the mean value theorem [ 8, Theorem (8.5.4)]).

Hence, there is a u > 0 such that (A4) holds. Thus, conditions (22) and (23) are equivalent-to iA4)

wgch, by (19) and the ao-right-coprimeness of (GJr,Gj),+

~ G J ~ ) , is equivalent to H,,,E $(a)mXm, for some o<o. Q.E.D.

Proof of Theorem 03: We prove Theorem D3 in four steps.

Step 1: We show that the closed-loop system S is &(a)-stable, for some a > 0 if and only if (22) and (23) are satisfied with fm being replaced by the standard Nyquist D-contour.

Using the notation above, &jm) E RmX“ has p,, distinct eigenvalues km with multiplicities tni, respectively. By assumptions (13), (15), (le), and Pl), G ( . ) is continuous in @,,+ and, as Is1+=00 in Cue+, G(s)+S(jco). Hence, VE >0, 3Q>O such that a) all the @+-zeros of det Gj),(s) have magnitude smaller than a; and b) Vs E@,~+ with Is1 > Q, &(s) has mi eigenvalues inside the disk @&m,~),

for i= 1,2; - - ,p,,, hence

infIdet[I+k&(s)]I>O

where the infimum is taken over s E@,,+ with Is1 >Q. Next we consider the standard Nyquist D-contour No formed by Ea n [ - jQ, + jQ] and the right semicircle r: = {SI Is1 = Q } n @+ . We apply the principle of the argument [S, p. 247, Theorem (9.17.2)] to det[I+k&(s)] with the D-contour No and thus conclude that under the assumptions of Theorem D3, the closed-loop system S is &(u)- stable for some a<O if and only if (22) and (23) are satisfied with fia being replaced by No.

Step 2: We show that C(0, deqI + k&(s)]lsEND) = 22;- ,C( - l/k, ygj) where the circuits y&’s are formed by the eigenloci Ai(. ) when s travels along No.

Without loss of generality, we assume that Q is chosen so that &(s) has no multi@e eigenvalues on the semicircle r. Furthermore, by (25), G(jw) has multiple eigenvalues at only a finite number of points, say jb,Jb,,. * ,jb,, within the compact interval [ - jQ, + jQ]. Thus, following the proof of Lemma L2 with gm being replaced by N,, and yi,, I = 1, q + 1 being modified accordingly, we have that the number of encirclements of the origin by det[l+ k&(s)]ISEND is equal to Z$,C(- l/k,y&), where the circuits yzj’s are formed by the eigenloci Ai( -) when s travels along No.

Step 3: We show that Zj.= ,C( - 1/ k , y?j) = C( - 1 /k, ygJ), where the yg;s denote one particular

assignment of the indexed family of circuits (%*)j=l,s...,p specified in Theorem D3.

Note that the y$’s constructed in step 2 are identical to one particular assignment of the generalized Nyquist dia- g a m (yj*)j=1,2,...,p except the roads inside the disks D(hm, E), i = 1,2, - , m. Now, by [8, Theorem (9.6.4)] and the proof of the part biii) of Lemma 2, we conclude that E;= , C( - 1 / k, y$_= Z;= C( - 1 / k, y z J ) since by assumption, (-l/k,O)@D(hp“,~), Vi=1,2,--.,m.

Step 4: Following the proof of the part biii) of Lemma L2, since Z;= ,C( - l /k, y;) is invariant under the assignment scheme used in the proof of the part a) Lemma L2, we have that

where ( ~ j * ) ~ = ~ , ~ . . . , ~ denote any indexed family of the generalized Nyquist diagram. Therefore, the closed-loop system S is (+stable for some o<O if and only if i) and ii) of Theorem (D3) are satisfied. Q.E.D.

ACKNOWLEDGMENT

The authors wish to thank Prof. E. Pol&, Dr. Edward 0. Lo, and Mr. V. H. L. Cheng for stimulating discus- sions. The comments of a reviewer were particular helpful in improving the original draft.

REFERENCES

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[2] J. H. Davis, “Encirclement conditions for stability and instability of feedback systems with delays,” Int. J. Contr., vol. 15,. pp.

[3] F. M. Callier and C. A. Desoer, “A graphical test for checking the stability of a linear timeinvariant feedback system,” IEEE Truns. Automat. Contr., vol. AC-17, pp. 713-780, Dec. 19’72; and “On simplifying a graphical stability criterion for linear distributed feedback systems,” IEEE Trans. Automat. Contr., vol. AC-21, pp. 128-129, Feb. 19’76.

[4] C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Ourput Properties. New York: Academic, 1975, ch. 4.

[5] R. Saeks, “On the encirclement condition and its generalization,” IEEE Trans. Circuits Syst., vol. CAS-22, no. 10, pp. 780-785, 1975.

[6] J. F. Barman and J. Katzenelson, “A generaked Nyqust-type stability criterion for multivariable feedback systems,” Int. J.

[7l A. G. J. MacFarlane and I. Postlethwaite, “Generalized Nyquist Contr., vol. 20, no. 4, pp. 593-622, 1974.

V O ~ . 118, pp. 1291-1297, 1971.

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stability criterion and multivariable root loci,” Znr. J. Con@., vol. 25, no. 1, pp. 81-127, 1977. J. Dieudonni, Foundations of Modern Anaiysis. New York: Academic, 2nd printing, 1969. F. M. M e r and C. k Desoer, “An algebra of transfer functions for distributed linear time-invariant systems,” ZEEE Trans. Circuits @st., vol. CAS-21, no. 9, pp. 65 1-662, 1978.

York: Academic, 1974. H. H. Rosenbrock, Computer-Aided Connol @stem Design. New

T. E. Fortmann and K. L. Hitz, An infroduction to Linear Control @stems. New York: Dekker, 1977.

Academic, 1972. J. M. Ortega, Numerical A M ~ s ~ s , A Second Course. New York:

M. Bikher, Zntroduction to Higher Algebra New York: Dover, 1964.

disturbance rejection in multivariable distributed systems,” in F. M. M e r and C. A. Desoer, “Stabilization, tracking and

Jan. 1979, pp. 5 13-5 I4 (complete version available as ERL Memo. Proc. 17th ZEEE Conf. on Decision and Control, San Diego, CA,

M78/83, Univ. of California, Berkeley). H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. New York: Macmillan, 1961. A. G. J. MacFarlane and I. Postlethwaite, “Extended principle of

J. M. E. Valenca and C. J. Harris, “A Nyquist type criterion for the argument,” Znt. J. Contr., vol. 27, no. 1, pp. 49-55, 1978.

the stability of multivariable linear systems,” presented at the 17th IEEE Cod. on Decision and Control, San Diego, CA, Jan. 1979. L. Zadeh and C. A. Desoer, Linear @stem Theory. New York: McGraw-Hill, 1963.

Approach to the Anabsis of Linear Multiwriable Feedback System I. Postlethwaite and A. G. J. MacFarlane, A Complex Variable

(Lecture Notes on Control and Znfonnation Sciences, no. 12). New York: Springer-Verlag, 1979.

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ERE TRANSAC”I0NS ON AUTOMATIC CONTROL, VOL. AC-25, NO. 2, APRIL 1980

Charles A. Desoer (S50-A’53-SM’57-F‘64) was born in Brussels, Belgium, on January 11, 1926. He received the Radio Engineer’s degree from the University of Liege, Liege, Belgium, in 1949, and the Sc.D. degree from the Massachusetts Institute of Technology, Cambridge, in 1953.

From 1953 to 1958 he was with Bell Labora- tories, Murray Hill, NJ. Since 1958 he has been with the Department of Electrical Engineering and Computer Science and the Electronics R e search Laboratory, University of California,

Berkeley, where he is currently Professor of Electrical Engineering and Computer Sciences. During the academic year 1%3-1964 he was on industrial leave at Bell Laboratories. During the academic year 1967-1968 he held a Research Professorship at the Miller Institute. From 1965 to 1972 he was the Associate Editor of SZAM Journal on Control. He was the Guest Editor of the November 1972 issue of the IEEE TRANSACTIONS ON C h c m THEORY (Special Issue on Nonlinear Circuits). His interest is in the area of system theory, with an emphasis on nonlinear systems and circuits. He is the author, with L. A. Zadeh, of Linear System Theory (New York: McGraw-Hill, 1963); with E. S. Kuh, of Basic Circuit Themy (New York: M e w - H i l l , 1969); of Notes for a Second Course on Linear @stems (New York: Van Nostrand-Reinhold, 1970); and with M. Vidyasagar, of Feedback System; Znpur-Oir$ut Propertia (New York: Academic, 1975). In addition, he has authored numerous technical papers on circuits and systems.

Dr. Desoer is a member of the American Association for the Advance ment of Science, the Society of Industrial and Applied Mathematics, the American Mathematical Society, the Mathematical Association of America, and Sigma Xi. His awards include: Best Paper Prize (with J. Wing), Joint Automatic Control Conference, 1962; Medal of the Univer- sity of Liige, 1970; Guggenheim Fellowship, 1970-1971; Distingushed Teaching Award, University of California, Berkeley, 1971, and IEEE Education Medal, 1975.

Ymg-Terng Wang (S’76-M’79) was born in Taipei, Taiwan, China, on December 27, 1949. He received the B.S. degree in electrical engineering from National Taiwan University in 1972 and the MS. and Ph.D. degrees in electrical engineering and computer sciences from the University of California, Berkeley, in 1977 and 1978, respectively.

From 1972 to 1974, he was a Second Lieute nant Communication Officer, operating and maintaining telecommunication equipment in

the Chinese Army. From 1975 to 1979, he was with the Electronics Research Laboratory, University of California, Berkeley. In 1979, he was a lecturer in the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley. He is now with the Bell Laboratories, Holmdel, NJ. His research interests are in the areas of circuits and systems, particularly in nonlinear systems and feedback theory.


on the generalized nyquist stability criterion

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