Input-Output Stability Theory - An Improved Formulation

7/30/2019 Input-Output Stability Theory - An Improved Formulation

1/14

150 IEEE TRANSACTIONS ON A U T O M A ~ Cc o m o ~ ,OL. AC-23, NO. 2, APRIL 1978

Input-Output Stability of InterconnectedSystems Using Decompositions: AnImprovedFormulationFRANK M. CALLIER, MEMBER, IEEE, WAN S. CHAN, MEMBER, IEEE, ANDCHARLES A. DESOER, FELLOW, IEEE

Abstmc-We study the inpnt-output s t a b i i of an arbitnuy intercon-nection ofmnlti-input, multi-output subsystems which may be eithercontinww-time or discretelime. We consider, throughout, three types ofdynamics: nonlinear time-varying, linear timeinvariant disbiiuted, andlinear lime-iivariant umped. First, we use the strongly conneded compo-nent decomposition to aggregate the subsystem into strongly-connected-and ISS are then aggregated into column-snbsystems (a s ) so that theoverall system becomes a hiem chy of CSs. me basic structural resultstates that the overall system s stable if and only if every CS is stable. W ethen use the minimum-essential-set decamposition on each SCS o that itcan be viewed as a eedback nterconnection of aggreKated subsystemswhere one of them is i t se l f a hierarchy of subsysteors. Based on thisdecomposition, we present results which lead to sufficient conditio- forthe Stability of an SCS. For linear h e m v a r i a n t (transfer function)dynamics, we obtain a characteristic function which gives the necessaryand sufficient condition for the overall system Stability.We point oot thecomputational saving due to the decompositions in calculating this char-acteristicfmction. We believe that decomposition techniques, o u p l e d withotherechniques such as model reduction, aggregation, singular, andnominguhr perhubations,wplay key roles in largescale system design.

~ubsystetnsSCSS) and int~~~~~eeti~n-S~~LBYSteobsystemsISS). SCSS

I. INTRODUCTION

THIS PAPER considers the input-output stability of anarbitrarynterconnection of multi-input multi-outputsubsystems. This problem can be viewed as a generahza-tion of that dealing with the feedback interconnection ofmulti-input multi-output subsystems [1H6]. On the otherhand, since an arbitrary interconnection can always, bysuitable reformulation, be viewed as a single overall con-stant output feedback system (as is done in Fig. 2 below),the task of thispaper s to analyze the details of theinterconnections usingraph theoretic decompositiontechniques and o bring them to bear on the stabilitystudy.

was supported by the National Science Foundation under GrantManuscript received April 6 , 1977; revised July 22, 1977. This workENG74-06651-A01 and the Joint Senices Electronics Program underContract F44620-71-C-0087. The work of F. M. Callier was supported inpart by a grant from the Belgian National Fund for ScientificResearch,Brussels, Belgium.versitaires de Namur, Namur, Belgium.F. M. Callier is with the Department of Mathematics, Faculttk Uni-Computer Sciences, and the Electronics Research Laboratory, UniversityW. S. Chan was with the Department of Electrical Engineering andof California, Berkeley, CA 94720. He is now with Bell Laboratories,Holmdel, NJ 07733.Computer Sciences and the Electromcs Research Laboratory, Umverslty

C. A. Desoer is with the De p e e n t of Electrical Engineer@g+dof Cali fornia, Berkeley, CA 94720.

Basically, there are two types of stability: Lyapunovstability and input-output stability. For he Lyapunovstability, the system dynamics are restricted to ordinaryand functional-differential equations [7]. The input-out-put stability studied in this paper, allowsmuchmoregeneral types of dynamics [1H3].For work on Lyapunovor input-output stability of an interconnection of stablesubsystems,based on exploiting the relative weight ofsome terms in the system equations, see [8]-[17]. Thecrucial difference between this paper and [8H17] is thatwe use graph theoretic decomposition techniques, whichhave been used in [18H241, to exploit the structure of theinterconnection. Furthermore, we need not assume thatevery subsystem is stable. In [18H22], the stability prob-lem was not considered. In [23] Mayeda and Wax consid-ered the exponential stability of systems of ordinary d i f -ferential equations. In contrast to [8]-[13], [21H23], weuse the general input-output description for our subsys-tems; thus, our theory covers both linear and nonlinear,time-invariant and time-varying, lumped and distributedsubsystems, as well as the continuous-time and discrete-timecases [lH3]. Acomparisonbetweenour previouspaper [24] and the present paper is relegated to SectionX:Conclusions, so hat we can makespecific reference toresults of the present paper.We study the stability using four levels of aggregation.At the lowest level, we have the multi-input multi-outputsubsystemswhich are arbitrarily interconnected throughsummingnodes to form the overall system. By usingstrongly-connected-component (SCC) decomposition, weaggregate the subsystems into strongly-connected-subsys-tems ( scs s ) and intercomection-subsystems(Iss). TheseSCSs and Iss are then aggregated into column-subsys-tems (CSs) so that the overallsystemwhich s the toplevel aggregation, becomes a hierarchy of CSs.

The contents of this paper are as follows:

Section I:ntroductionSection 11: PreliminariesSection 111: System Description and Assumptions

tronic circuits with op amps, and chemical processes[25 ] .Someopen-loopunstablesystems occur in practice: ockets,elec-

0018-9286/78/0400-0150$00.75 01978 IEEE


2/14

et aL: INPWT-OUTP~~STABILITY OF INTERCONNECTED SYSTEMS

Section IV:Section V:Section VI:Section VII:Section VIII:Section IX:Section X:Appendix:

Overall System Stability WithoutUsing DecompositionSCC Decomposition of the overall Sys-temStructural ResultMES Decomposition of an SCSSufficient Conditions for the C-Stabil-ity of an SCSSimplifying Characteristic FunctionsUsing DecompositionsConclusionsProofs

The reader is urged to give particular attention to someand linguistic conventions: (i) on map and11; (ii) on the dimension of subsystems(iii) on the relabeling due to the SCCSection V; and (iv) on the relabelingMES decomposition in Section VII.

11. PRELIMINARIESIn this paper we consider an interconnection of subsys-three types of dynamics: nonlinear time-varyingsystems are described by operators be-

systems are described by their transfer functions.use NTV and LTI tooperator dynamics, and lin-transfer function dynamics, respectively.

For a N TV system, we adopt the following standard[3, S e c . 111.11, namely, let 3be the time set of5=R + for continuous-time case,+ for discrete-timecase), Y bea normed space withI I (typically 1r=R or Cn), nd 5 e the set of allS nto V. The function space 3 is

R (or C) nder pointwise addition anda norm

* 11 on 5,we obtain a normed linear subspace e of the9, iven byc A {f:S+YI 11f 1T. We say that f, is obtained by truncating f at T.C is the extended spacedefined by11f (= instead of I l f , I I . From now on

e take Y = R . A NTV systemwith n, inputs and nodescribed by an input-output operator H :Q4An operator H is said to be causal iff for all inputsE E for all T E ET the corresponding output H u satis-( H U T )==(Hu).. An operator H is said to be e-stablef there exists constants /3, y in R+ such that V u @ ,

151

I IH~I l7-G + Y l l U l l r . ( 1 )It is well known that when H is causal, then H is C-stableif and only if there exists constants ,8, y in R , such thatvu E eq

IlHull G P+YIIuII* (2)Throughout this paper, we consider only causal operators(seeAssumption1 in Section IIq. The smallest y forwhich (2) holds is called the gain of H, and is denoted byy[HI, i-e.,y [ HI i n f { y ) 3 P E R + 3 V u E C q , llHull Q p+yl lu l l } .

(3)The incremental gain of H denoted by ?[HI is defined asfollows:?[HI ~ i n f { - i ; l V ~ , , ~ 2 E C Q ,~ H u ~ - H u ~ ~ ~ G ? I I u , - u * ~

(4)Remark I : (i) Note that the bias ,8 in (1) and (2) whichis restricted to be zero in the definition of gain in [l], [2],

[14]-[17] is allowed to be nonzero in our definition as in[3]. This not only allows us to consider a more generalclass of operators, but also simplifies the stability analysisof the overall system (e.g., see Proof of Theorem III) andsharpens theconclusions (eg , see [ l , p. 2321). (ii) TheC-stability of H not only requires that H takes an input in&space into an output in C-space, but also requires thaty[H] be finite. (E) H is E-stable, or equivalently y [HI 9, [27, p. 1501, [3, p. 2491, (ii) H EP X nas an inverse in& ( i P x n ) is a commutative (noncommutative, resp.) alge-bra over the field R [27], [3]. ALTI distributed systemdescribed by its trapsfer function H :C+-+Cnoxis saidto @--stable ff H E 69 q. It is well known that if a systemis @--stable,hen: (i) for a n y p E [1, co], it takes an h-input

@nxn if and only if infsEc+ldetH(s)) O [3], and (iii)


3/14

into an ?,-output with a finite gain, i.e., it is E-stable forE =4, and (ii) it takes continuous and bounded inputs(periodic inputs, almost periodic inputs, resp.) into out-puts belonging to the same classes resp. [3], [28].Let R (s) denote the field of rational functions with realcoefficients.Let R ( S ) % ~ +enote the ring of noX n,matrices whose elements are inR(s) .By definition, a LTIlumped systems described by a transfer function inR ( s p x & . A LTI lumped system described by its transferfunction H ( s )E R ( S ) % ~ +s said to be exponentially stable(abbreviated exp-stable) i f f : (i) H ( s ) is proper (i.e.,bounded at infinity) and (ii) H ( s ) has all its poles in theopen left-half plane (i.e., H has no C+-pole). It is easy tosee that R,(s), the class of all scalar exp-stable transferfunctions is an algebra over R, in fact a subalgebra of &.

In either the lumped or distributed case, if the transferfunction H has a domain of convergence which includessome right-half plane and if, for large Re s, it is boundedby some polynomial in s, then it is causal. (The secondrequirement is indispensable: viz. e s ) [3, Th. B.3.41.Convention on Mapand Stable

Throughout his paper, (i) by a m p H , we mean anoperator H for the NTVdynamicscase, and a transferfunction H for the LTI dynamics case, and (ii) when wesay that a system described by the map H is stable, wemean the operator H is E-stable in NTV dynamics, thetransfer function H is @-stable in LTI distributed dy-namics, and the transfer function H is exp-stable in LTIlumped 0Note that when a system is described by the map H , thespecification of H prescribes the inputs and the outputs ofthe system?

Using (l), the definition of E-stability, one can easilyprove that the composition and pddition of C-stable oper-ators are again E-stable. Since ti?, Re(s)are algebras, theyare closedunder multiplication and addition. Thus wehaveLemma 1 (NTV, LTI):4 Every series-parallel connec-tion of stable subsystems is stable.

111. SYSTEM DESCRIPTIONND ASSUMFTIONSIn this paper we consider an overall system S consisting

of an arbitrary interconnection of subsystems. The subsys-tems are specified by an input-output map: they may beMIMO (multi-input multi-output) or SISO (single-inputsingle-output), unstable or stable, nonlinear or linear,time-varying or time-invariant, and continuous-time or

Fig. 1. A typical subsystem Gv of the overall system.

discrete-time. The interconnections are realized through msumming nodes as indicated by Fig. 1 . The subsystem fromnode j to node i is described by the map G,. Eachsumming node j is fed by an external input uj and by theoutputs of the subsystems ql; .. %m. The output ej ofthe ummingnode j is the input to the subsystemsGI,; . ,Gm,. In practice, a sigmficant portion of the sub-systems Ggs are absent, and hence, are represented byzero maps.

Convention on Dimension of Su bsystem sTo alleviate burdensome notations which are peripheralto the main ideas of the paper, we treat each subsystem asif it were SISO (i.e., for NTV dynamics, G,: Ce+C,; forLTI dynamics, G is a scalar transfer function); the resultspresented in this paper still hold for MIMO subsystems(i.e., for NTV dynamics, G v :EP+E:; for LTI dynamics,Gv is a matrix transfer function) by modifying the di-mension of the product spaces a~ co rd ingl y. ~ 0Assumption 1: CausalityThroughout this paper, we assume that all the subsys-tems Go are causal. 0The summing-node equations read

mei=ui+ x Gvq f o r i = l , - - - , m . (10)j = 1

In matrix notation, we have

D

- Gm, ...fine

...

usual notation of A because some of our resultshold for all three types of2We do not distinguish a transfer function from an operator by thedynamics with correspondingdefinition s of stability.3A swbe seen F l o w in the definition of the overall system stability,it makes a lot of Mference whether H is taken to be d ( I - G ) - [ or

We use NTV, LTI following each lemma, theorem, and corollaries toG < I - G ) - .indicate the type of dynam ics for which the statement holds.

Withhese definitions, the system equations (lo), orequivalently ( 1 l), become

validity of the formulas below. Th e simplified descnption avods theSThis simplifieddescription of the ubsystems does not affect themessy bookkeeplng of four levels of aggregation of subsystems.


4/14

( I - G ) e = u. (12)Assumption 2: Unique SohabilityThroughout this paper, we assume that(a) for NTV dynamics, (I- G ) - ' : w e is a map from(b) for LTI distributed dynamics, V sequences (si)?='c(c)for LTI lumpeddynamics, det (I- G ) ( m ) # O . 0Conditions under which Assumption 2(a) ssatisfied

into I and is a causal map,,where Isil+.oo, liminf,,ldet(I- G)(s,)l>O

[3, Sec. 111.51.of the Overall System Stability

The overall system S is said to be stable iff V i , j =, . . m , the maps u w G J ( I - G ) - ' u I j are stable.Remark 2: (i) At first sight, one might want to choosetion of overallystem stability: the mapI- G ) - ':u b e e stable. Note that without requiring G

G ) ( I - G ) - ,+G( I - G)-'=(I- G ) - ' . (13)( I - G ) - ' s stable if and only if the map

is stable, i.e., for i = l ; - . , m the maps u w?! G u q are stable. By definition the overall system S isif f V i j = 1, - - ,my w G e q are stable. Hence, by1, f the overall system S is stable, then the map

I- G ) - ' is stable.(ii) The converse of the last statement is not true be-two unstable terms, say Gilerand Gjkek may cancelgive a stable sum 27- Geq. An example tom = 2 and all subsystems be SISOlumped. It is easy to check that if G,,(s) =(2s +3)/+ ), G, , ( s )=0 Vs , G2&) =- s -4)/(2s - ) and G&)

-l), then a) ( I - G ) - ' and G ( I - G ) - ' areu I ~ G 2 , e ,ndIb G 2 , e 2 are both unstable (pole at s=0.5); but theirof course, stable since it is the (2,l) element ofIn order to formulate the definition of overall system

G,. denote thejth column of G . Let

Let I be the m X m identity matrix and let f R m x d begiven by

f & p ; .. i I ] . (15)Let

Note that from (14) and (15)G = K 6 . (17)Then the overall system S can be viewed as a constant

output feedback system as shown in Fig.2. Thus, theoverall system S is stable if and only if the map c ( I -k6)-'=6(I-G)-':ut+~7isstable,_

By Assumptions1 and 2, ( I - K G ) - ' : is a well-defined causal map.For theoretical development, Fig. 2is convenient. How-ever, it does not take advantage of the particular structureof the interconnection, namely that a number of G;s arezeromaps. Our objective s to take the structure into

consideration using graph theoretic decomposition tech-niques.

IV. OVERALLYSTEMTABILITYITHOUT SINGDECOMPOSITION

In this section we consider the overall system stabilitywithout using decomposition. Theorem 1 gives a sufficientcondition for the overall system stability; it is quite gen-eral since it holds for NTV dynamics with &stability, LTIdistributed dynamics with &-stability, and LTI lumpeddynamics with exp-stability.

Theorem I (NW, TI)Consider the overall system S described by (11) andsatisfying Assumptions 1 and 2. If: (a) Gv is stable V i # j ,i j = 1, ,m y (b) for every unstable G,,,(I- Gc)- s sta-ble, and (c) ( I - G ) - ' is stable, then the overall system Sis stable.Remark 3: (i) By Theorem I, under its assumptions (a)and (b), the stability of ( I - G ) - ' is equivalent to theoverall system stability. (ii) If every G,, i , j = 1 , . - - , m isstable, then Assumptions (a) and (b)are satisfied. 0For the remaining part of this section we study the&-stability (exp-stability, resp.) of the overall system S inLTI distributed (lumped, esp.)dynamics. Due to thesymmetry between the distributed and lumped cases, theresults are presented in pairs: we use D ( L , resp.) todenote distributed (lumped, resp.) case.


5/14

154 IEEE TRANSACTlONS ON AUTOMATIC CONlXOL, V O L AC-23, NO. 2,APRE 1978

Let R[s] denote the commutative ring of polynomialswith real coefficients. Let H ( s )E R(s)""", N,(s) ER [ s ] " . X " , D ( s ) ER [ s ] " X " , N , ( s ) ER [ s ] " X ~ ;hen(N,,D,N,)is said to be a right left coprime factorization (r.1.c.f.) of Hiff: (i) H=N,D- 'N,, (ii) there exist U , ( s ) R [ s ] R " % , ,(s)ER[ s ] qXn ,V , ( s ) ,V,(s)ER[s]"""such that det[U,N,+V,.D](s)#O, VsEC and det[N,U,+DV,](s)#O, V s E C .Let H ( s ) R ( s ) a X q ,N , ( s ) ~ R [ s ] " " " z ,D,(s)ER[s]""q;then (N,,D,) s said to be a rightcoprime factorization(r.c.f.) of H if f (N, , D,,I) is a r.1.c.f. of H . A left coprimefactorization (l.c.f.) is similarly defined.then ( N r ,D, N ,) is said to be a pseu do right left coprimefactorizatioc (p.r.1.c.f.) qf H iff: (i) HyiV,D -IN,, (ii) thereexist U,.E@'xn~,U , E @ ? " " , V,,V,Q""" such thatdet[ U r N r +V,.D](s)#O, VsEC, and det[N,U,+DV,](s)#O,Vs E C,, (iii) V sequences (si)?= c C+where lsilA+w, liminfi,,ldetD(si)l>O.Let H:C++C"oXq,N,E@'oX",DlE &oxno ; then (N, ,Dl) is said to be a pseudo left coprimefactorization (p.1.c.f.) of H iff ( I ,D,,N,) is a p.r.1.c.f. of H .A pseudo right coprime factorization (p.r.c.f.) is similarlydefined.It is easy to see that the definitions of coprime factori-zations given above are equivalent to those defined in theliterature (e.g., [3], [5], [6 ] , [24], [29]). The reason forintroducing new definitions is to achieve Jymmetry be-tween the distributed and the lumped cases.

It is well known that if (N,,D,N,) is r.1.c.f. of H and His proper, then H is exp-stable if and only if det D (s) hasno C+-zero [29].By similar reasoning as in [ 5 ] , and usingcondition (iii) in the definition of p,r.l.c.f., it iseasy toshow that if (N,,D,N,) s a p.r.1.c.f.of H ; then H is a@-stable f and only if det0(s) has no C+-zero.The following emma in spite of its simpleproof hasfar-reaching consequences: itgives us a characteristicfunction mapping from C + into C such that the overallsystem S is @-stable (exp-stable, resp.) if and only if thatcharacteristic function has no C+-zero.

Let H:c++cnoXq,, E @ O ~ ~ , D E&"", N , E & ~ ~ ;

Lemma ZD(L) (LTI)Consider the LTI distributed (lumped, resp.) constantoutput feedback system shown in Fig. 3 where H is thetransfer function in the forward path and K is a constantmatrix. Let (N, ,D,N,) be a p.r.1.c.f. (resp. r.1.c.f.)ofH . (For the lumped case, assume that H is proper.) As-sume that V sequences (sj)zlC+ where Jsil+co, liminfi-,, ldet(1- KH)(si) l> O (det(I- K H ) ( c c ) # O , resp.).Under these conditions, H ( I - K H ) - ' : ~ Zs &-stable(exp-stable, resp.) if and only if

det (D-N,K N, )(s) has noC+-zero. (18)Remark 4: (i)Lemma 2 0 L) is proved by showingthat if ( N , , D , N , ) is a p.r.1.c.f. (resp. r.1.c.f.)of H , then(N,,D-N,KN,,N,) is a p.r.1.c.f. (resp..1.c.f.) of H ( Z -K H ) - ' . It is well known that if x is the state used in a

finite dimensional minimal SSSD (state space system de-scription) of H , then the SSSD of the constant outputfeedback system H ( I-KH)-', using the same state x ,will also be minimal, since constant output feedback pre-

serves both complete controllability and complete observ-ability when the same state is used [30,p. 3651. Since ther.1.c.f. in PMSD (polynomial matrix system description) isa counterpart of the minimal realization in SSSD [3 11, [32],Lemma 2 L can be viewed as a counterpart in PMSD ofthe above well-known fact in SSSD.(ii)ApplyingLemma 2 0 L ) to the feedbacksystemconsidered in [5] , 33], we can easily obtain all the char-acteristic functions (resp. polynomials) gven in thesepapers.(iii) Note >hat Lemma 2 0 L ) will still hold if K has

elements in & (if K is a polynomial matrix, resp.). 0Under Assumptions 1 and 2, applying Lemma 2 D ( L )to he overall systep S shown in Fig. 2, and using theparticular form of K , we obtain a characteristic function(polynomial, resp.) for the overall system S .Theorem I I D(L) (LTZ)

Consider the LTI overall system S described by(1 1)and satisfying Assumptions 1 and 2. For j = 1, - - ,m , letG, denote the jth column of G, and let ( N , , D j ) be ap.r.c.f. (r.c.f., resp.) of G ,. (For the lumped case, assumethat G is proper.) Let

N = [N , l 1 e - . N.* 1, 6 ~diag (D, ; - - ,D , ) . (20)With hese definitions, the overallsystem S is @-stable(exp-stable, resp.) if and only if

de t (5-N ) ( s ) has no+-zero. (21)Remark 5: (i) Note that G =N 6 -', but that N ,0" arenot necessarilyp.r.c. (resp. r.c.). (ii) Clearly, byusingp.f.c.f. of G,'s, one can obtain a characteristic function for

the overall ystem S . However, since the size6 of theoutput of S is m 2 , such a characteristic function will bethe determinant of an m2 m 2 matrix. The characteristicfunction given by Theorem I1 D ( L ) s the determinant ofan mX m matrix where m is the size of the input u of S(seeig. 2).In Section IX below, we will discuss how the SCC andMES decompositions to be described in Sections V andVI1 implify the necessary and sufficient stability testgiven by (21).

V. SCC DECOMPOSITIONF THE OVERALLYSTEMIn this section, we apply the ideas presented in [24, Sec.

1111 to the present formulation. Recall first the graphtheoretic terminology defined in the first paragraph of [24,Sec. 1111.Consider the overall system S described above. Theinterconnectiondigraph qintf S is defined as follows:each summing node i of S corresponds to a vertex vi ofqint,nd Gilht has a directed edgefrom vi to v i i f f the6Recall footnote 5.'Note that our adjacency matrix is the transpose of the one used bygraph theorists [34], [35].


6/14

155

Fig. 3. Constant output feedback system: u bz .

subsystem Gg is not the zero map. Since each connectedcomponent of 6Dht can be analyzedseparately,withoutloss of generality, wemsume that 91bt is a connecteddigraph.Note hatdue todifferentproblem formulations theqintefined above s different from the 91ht defined in[24, Sec. 1111.We now perform the SCC decomposition on the con-nected digraph qintsing Steps 1 - 4 described in [24, Sec.111, p. 7161. Let ea,a = 1,s. - , p be the SCC's of qint,identified by using, for example, Tarjan's algorithm [36],and reordered by Steps 2-4 in [24, Sec. 111, p. 7161.A little thought reveals hat the adjacency matrixAh t ofqintfter Step 4 will be in heblock ower triangularform:

m f m i * - * m i

where:(i) m,' is the number of vertices in ea, ii) eachdiagonal block A& is the adjacency matrix of ea, nd (iii)each off diagonal block A&, OL >3 is theadjacency matrixof eaPwhich is defined to be the bipartite digraph [34, p.1681 consisting of: (a) all the vertices of eaand e,, and(b) alledges of CDht directed from a vertex in e, to avertex in ea.Notational Conventipn

From now on, without loss ofgenerality, we assumethat we start out with the overall systemS which has beenrelabeled after the SCC decomposition. 0For a >p= 1,. -. y,we define

V: A theet of vertices in SCC ea (23)u: A the m;-vector ( u J i E vz (24)e A the mi-vector (eJiEv,. (25)

GP A thernX mp' matrix [ vz,,E vi (26)the (m,'.m;) X m; matrix: diag (columns of G& )

(27)i z i , the m, X (rni.mp') matrix: [I i - . i I ] (28)

jib them,'.mi)-vectorG,q),, vi, E vj (29)[compare ( 2 3 , (28), (29) with (14), (15), (16)]. Note that

From (32) we can write

Using (30) and (31) we have

Observe that due to block-lower-triangular form of Gafter relabeling, ei ,j & do not depend on u for j >j3.For a=1, . - - ,p, we denote by Sl the strongly-con-nected-subsystem (SCS) associatedwith he SCC ea ofqint:t is obtained from the overall systemS by removingall he ummingnodes and subsystemswhich do notappear in e,. Hence, its input is u,',its outp2t i%fia, andit is described byhe map G:a(I - K:aG:a)-l =6 t a ( I - G,'J-'. In other words, Si can be viewed as theconstant output feedback system shown & Fig. 2 with thefollowing replacements: 6 by r? by Kla, u by u:, e bye 7 by yia. Consequently, the stability definition belowfollows previous pattern.The SC S S,' isaido be stable if f the map6 i a ( I - Gta)-' is stable.


7/14

156 I EEETRAPTSACTIONSON AUTOMATIC CONTROL, OL AC-23, NO. 2, APRIL 1978By Assumption 2 in Section 111 and block-lower-trian- VI. STRUCTURAL.RESULTgular structure in (32), ( I- G;J- is a well-defined causal

map.Hence,everySCS s described by a well-definedcausal map.For a >3=1, . - , - , we denote by S:, the intercon-nection-subsystem (IS)associatedwith ea, of $ht: it isobtained from the overall ystemby emoving all thesumming nodes and subsystems which do not appear inern,.Hence, its input is u; , its output is i:,, nd it sdescribed by the map G&.

Theorem I11 belowgives a necessary and sufficientcondition for the overallsystem stability. This result isstructural in the sense that it is based on the block-lower-triangular structure obtained by SCC decomposition. Itholds for NTV dynamics with -stability, LTI distributeddynamics with @-stabilityand LTI lumped dynamics withexp-stability.Theorem ZII ( N T K L T I )

The IS S,, is said to be stable iff the map G:b is stable. ConsiderIn view of (27)-(3), a little thought will reveal that the satisfyingthe Overall system s described by (32) andand 2- The Overall is

overall system S : u,);= I-,(y&)t,p=can be viewed as aseries-parallel connection of SCSs, ISs, and constant gainsubsystemsk&s s shown in Fig. for the case p=3.(CS) described by he maps G&(Z- Gjb ) - , a = & /3+1,. . ., n e s e maps are the contribution of # ton this section,

stable if and only if V/3=1, * .,I, S S, are stable. 0For p =1, ., , we denote by S, the column-subsystem VII. MES DECOMPOSITIONF AN SCS

we apply the ideas presented in 124, Set.( y :b)E=, whileneglecting the effect of all other inputs v] to the present

In order to avoid( U ; ) g ; . we shall develop t he concept of minimum essential set.ne cs s is said to be stable if f the maps Throughout Sections and VIII, we study the stabil-~ : , ( I - G i , ) - ~ , a = p , B + l ; . . , ~ are stable. ity of a singlemap, namely, ( I - G:J-. For convenienceA ~ ~ , ~y (2+(33), h e overall system s can be viewed and to alleviate the already burdensome notation, we willas a hierarchy of_CSs interconnected through constant drop the subscript a throughout Sections VI1 and VIII.gain subsystems K&s as shown in Fig. 4. Note that eachCS is an aggregation of one SCS and several Iss.Prior to establishing in Theorem I11 below that the e= ( I - G ) - u . (34)overall system is stable if and only if every CS is stable,we note Some relationships between the stabilities of each In addition to the graph theoretic terms defined in theCS and the corresponding SCS and ISSwhich are &rect first paragraph of 124, Set. 11117we need the followingconsequences of the definitions and of the structural de- terms. By definition, u C v is called an essential Set Of acomposition. digraph 9 (V ,E) iff the section graph ( V - U ) isFact 1 (NTK L T I ) acyclic. Given a digraph, an essential set with minimumIfS S, is stable, then SCSS i is stable. number of vertices is called a minim um essential ser (MES)Since SCs s; is stable implies that the map ( I - G i b ) - of the digraph. It should be noted that OUT definitionsis stable, we have allow an acyclic digraph to have self-loops; his followsFact 2 ( N W , LTZ) from our requirement that a circuit be of length >1.If SCS Sp is stable and V a>p, IS S:, are stable, then Consider the strongly connected subsystem S and itsCS S, is stable. 0 interconnection digraph (? (V,E) which by construc-Note that the stability of a CS does not imply -that of tion is strongly connected. Wenow perfom the M Ethe corresponding ISs. However, in view of G& =G& ( I - decomposition on e.Gib ) - (Z - G i b ) ,we have Step I : Find an essential set V 2 of (? and defineFact 3 ( N W , L T I ) V 12 V- V2 .By construction, the section graph (?(V)If Gib is stable and CS S, is stable, then V a >@,S S& is acyclic.are stable. 0 Step 2: Relabel the vertices of (? so that every vertex inSuppose cs sb is stable and for Some a>& 1s s.$. s V 1 s numbered lower than al l the vertices in v2.not stable. In LTI dynamics, this mplies that there exlStS Step 3: Relabel hevertices of e(v1) S O h a t its adja-some pole-zero cancellation between G& and ( I - G i b ) - . cency matrix A 11 is a lower triangular matfix.8Clearly, under independent parameter perturbations of A little thought reveals that the adjacency matrix A of(%p and G,&, such Pole-zero cancellation will not be (? after Step 3 willbe n the borderedower triangularpreserved. Thus for the stability of CS S, to be robust, it form:is reasonable to assume that every IS S$ is stable. Underthis assumption, from Facts 1 and 2, we have m m2

we write

Fact 4 (NTV, LTI)If every IS S&,a>3 is stable, then the stability of CSSb isquivalent to the stability of SCS S i . 0This fact emphasizes the importance of the stability

study of the SCSs (seeection VI11 below). Recall footnote 7.

(35)


8/14

Fig. 5. Flow graph associate d with (41), (42). Substitute (45) into (44); e havee l=( I -G11) - 'G12(I -6:") -1U2. (46)

(i) for i = 1,2,mi is the number of vertices in ViA is a ower triangular matrix.To exploit the structure of C? as much as possible, it isone should use a minimum essential set n theproblem of finding a minimum essen-

set has been studied by many researchers [39]-[46].a finite amount of work. However

potentiallyTo perform Step 1,we must first

sate for the fact that we allow self-loops, so weremove all the self-loops in e, then apply the algo-in [44] o find a minimum essential set and2 of the decompositiondone easily. Step 3 is carried out by using topologi-l sort described in 134, p. 4021, [37, p. 2581.

6: aealy,byallowing A in (35) to be in Note that (47) generalizes for the nonlinearcase the stan-[481~we Can further reduce dard expression relating closed-loop gain and the open-size of The tradeoff n computational efficiency loop gain in the linear case.MES and the MES decomposi-emains an open question.

mII . SUFFICIENTCONDITIONS FOR THEE-STABILITY OF AN scsFrom now on, without loss of generality, we assumeout with the SCC e which has been re- In this section we present two sufficient conditions forMES decomposition. the C-stability of an SCS based on the MES decomposi-

Using (41), (42), and the SmallGain Theorem [3] at theFor i, =1,2, we define tion.V 2 the MES of (2 (36) MES decomposition level,e haveVI A V'- T/2u i A the m'-vector (uk)kE ie A the m '-vector ek)k vi

Theorem IV (NW)(37) Consider the map (I- G')-I described by41)-(42)(38) and satisfying Assumptions 1 and 2. Suppose that y [GI2],y [ ( I - GI1)-'], y [ ( I - G22)-1], y[G21(I-- G")-'] are(39) finite. Under these conditions, if

Equation (34) can now be written as two equations:(I-G1')e'-G'2e2=u' (41)-G2'e'+(I-GZ2)e2=u2. (42)

We define6 2 2 2 G22+ (321 (1-G11)-1G12 (43)

The matrix signal flowgraph [49] associated with theRemark 7 (i) When u'=8 , from (41)

(41), (42) is given in Fig. 5.

then the map ( I - GC)-' is E-stable.Remark 8: (i) Theorem I Vstill holds if the superscripts1 and 2 are interchanged throughout.(ii)Note that y[G2'], y [ ( I - G")-'] are finite implies

that y[G2'(I- GI1)-'] is finite.(iii) We shall now check that under the assumptions ofTheorem IV Y he necessary conditions for the E-stabilityof ( I-G') - given in Remark 7 (i), namely, the C-stabili-ties of ( I - and of ( I - Gl1)-'GI2(Z- G22)-1, aresatisfied. Since y[G2'(I- G")-1G'2(I- G=)-']< y[G21(I

(44) Cy's.Wote that this calculation does not require inearity in any of the


9/14

158 IEEE TRANSAC~ONS ON AUTOMATIC CONTROL VOL. AC-23, NO. 2,APRIL 1978

- G 1 1 ) - ' ] . y [ G ' 2 ] - y [ ( I - G 2 ' ) - ' ] ,y the assumption ( 48 )of Theorem IV and the Small Gain Theorem, [ I - G 2 ' ( I-G " ) - ' G 1 2 ( I - G 2 2 ) - 1 ] - 1s E-stable. In view of (47), thistogether withG-stability of ( I - G 2 2 ) - implies thatis e-stable. Since ( I - G " ) - ' , G ' 2 are s-sumed to be &table, so is ( I - G " ) - 1 G ' 2 ( I - 6=)-'.

From Theorems I and IV, we haveCorollary IV.1 ( N TV )Consider the SCS S' after MES decomposition andsatisfyingAssumptions 1 and 2 . If: (a) V i Z j , iJ E V',y [ G , ] is finite; (b) V i V', y [ ( I - G , ) - ' ] is finite; (c)V i V 2 ,y [ G i i ] is finite or y [ ( I - GJ- '1 is finite; (d)y [ ( I - G U ) - ' ] s finite; and (e) y [ G 2 ' ( I - G " ) - ' ] . y [ G 1 2 ]. y [ ( I - G = ) - l ] < 1 , then the SCS S' is &-stable. 0Theorem V ( N TV )

Consider the map (I- G')- ' described by (41) - (42)and satisfying Assumptions 1 and 2. If y [ G 2 ] ,y [ ( I - G I 1 ) - ' ] , y [ ( I - 6 = ) - ' ] , [ G 2 ' ( I - G " ) - ' ] arefinite, then themap ( I - G C ) - ' is C-stable. 0Remark 9: (i) Theorem V still holds if the superscripts1 and 2 are interchanged throughout. By the MES decom-position: (a) ( I - G 1 1) has a lower-triangular structure andthus is easy to invert; (b) ( I - (?"), which has no particu-lar structure, has a size equal to IV21and usually IV2( J , i E Yc,j E V I , -j;[G,] is finite and there exists


10/14

et al.: INPUT-OUTP~~TABILITY OF INERCQN'NE(;TED SYSTEMS 159we haveCorollay II.1 D(L) (LTI)Consider a LTI overallsystem S described by 32),

isfies Assumptions 1 and 2, and has the factori-assumeproper.)The overall system S is @-stable (exp-stable, resp.) ifif Va= 1; * * ,p

X. CONCLUSIONSThis paper has treated in a very general setting the

nput-output stability of an arbitrary interconnection ofubsystem. Four classes or results are presented: (i) forLTI dynamics, a sufficient condition for

the overall system stability without using any decomposi-tion (Theorem I); (ii) for both NTV and LTI dynamics,he structural Theorem I11 stating the equivalencebe-tween the overall system stability and those of the Cs's;iii) for NTV dynamics, sufficient conditions for the C -tability of ( I-G&)-' and of SCS S,C using the MES

decomposition m eo re ms N nd V and their corollaries);nd (iv) for LTI dynamics, both lumped and distributed, aharacteristic function for theoverall ystem stabilityTheorem I1 D (L),Corollary 11.1 D (L)].This paper uses the same graph theoretic decomposi-ions as [24], but it drops the artificial distinction between

the subsystems G and c;s of[24]. This leads to moretransparent theorem formulations. This paper uses a moreefined definition of the overallsystem stability madenecessary by the multiple-loop nature of the problem. Aore systematic use is made of the closure properties oftable maps, of the block-lower-triangular structure result-ng from the SCC decomposition, and of the speciales of the definition of stability. As a result, themportant concept of column subsystem merges andeads to necessaryand ufficient condition for stability(Theorem III), and, in the LTI case, the overall system

characteristic function reduces to the product of the col-u m n subsystem characteristic functions.

One important feature of the graph theoretic decom-is that t is independent of the dy-namics of the subsystem; hence, it leads to results forTV and LTI dynamics.

By using the same reasoning as in [SO], sufficient condi-tions under which the exp-stability of the overall system isobust can easily be obtained. Due to space limitation, weo not present them in ths paper.The mainchallenge in decentralized control of largecale systems is due to informational constraints: theontroller at subsystem i has no knowledge of the infor-ation at subsystem j (e.g., the output observed, the

control applied, etc.), and viceversa.However, in thepreliminaryanalysis of a large system one should firstconcentrate on the qualitative system properties (e.g., sta-bility, controllability, observability,etc.), as opposed tothe control of the large scale system; at that stage informa-tion contraints do not come into the picture. For suchpreliminary study graph theoretic decompositions tech-niques, not necessarily restricted to SC C and MES decom-positions, should be used to analyze the structure of thelarge scale system. It is clear that the structure emergingfrom such an analysis will play a crucial role in thecontrol problem.The stabilization of the overall system is decentralizedinto the stabilization of each CS and a CS will be stabi-lized as soon as the corresponding SCS and s'sarestabilized.Even though this paper considered mostly interconnec-tion of nonlinear subsystems, a key feature of its analysiswas that the interconnection occurred through summingnodes which are Zinear elements. (This type of additivenonlinear interaction is also found in most works basedon the Lyapunov function technique [13].) It is not clearat this time what will be the most successful model fortruly nonlinear interaction among subsystems. Some workhas already been reported for the feedback case [38], [51].

APPENDIXProof of Theorem I

Let U A { ( i , j ) j ) ( i= j nd Gjj is unstable}. By assumption,( I-G)- ' u b e = (q)y=, is stable. Since V ( i , j ) U , Gg isstable, the mapu t+Gijq is stable V ( i , j ) U ( A 4

Now consider some ( i , )E U . Substituting e, by (lo), wehave

mGjjej=Gjj(I-Gji)-'j # i

By (A. 1) and Lemma 1, the map

By (13) the assumed stability of ( I - GiJ- ' s equivalentto the mapGij I-Gji)- is stable. (A41

In view of (A.2), (A.3) and (A.4) together imply thatu b G j j e i is stable. (A.5)

(A.l)and (A.5) establish the overallsystem stability.Q.E.D.


11/14

1 60 IEEE TRANSACTIONS ON AUTOMATC CONTROL, OL. AC-23, NO . 2, APRIL 1978Proof of Lemma 2D(L) (si)?=lcC, where Isil+c0, limidi-,, Idet(I- ke)(si)l>0. Now we apply Lemma 2 0 to the overall systemS (Fig.We first prove the lemma for the distributed case.2) with Hte,K+i,r + ~ , D+j,,+I and with

N = kE;Theorem I I D follows.ecall the well-established identity for matrices M , N ,M(I-NM)-'=(I-MN)-'M. ( A 4 For the lumped case, the proofollowsimilarly. Q.E.D.

Substituting for H and usingA.6)eaveroof of Theorem 111H ( I - K H ) - ' = N , D - ~ N , ( I - K N , D - ' N , ) - ' By definition, Vp =1, - ,p, CS S, are stable if and

=N,D-~(I-N,KN,D-~)-'N,=N,(D-N,KN,)-'N,. (A.7)

only ifV ~ l l > = 1,- - ,p, e ( I- G )-' are stable. (A.21)

Hence, the equivalence of the lemmawbe establishedonce we show that the three matrices in (A.7) form a V a >/3 =1, . , , (u:),!= are stable. (A.22)p.r.1.c.f.of H ( I - K H ) - ' . By asyxnption (N, , D,N,) is ap.r.1.c.f.of H ; hence there exist &-matrices U,, ,, V,, V, Recallsuch thatdet [U,N,+ V,D](s) ~;,=G;,(I-G;, ) - I

By definition, the overall system is stable if and only if

=det[(U,+ V , N , K ) N , + V , ( D - N , K N , ) ] ( s ) # O ;=vsEC , ( ~ - 8 ) We proveA.21)+(A.22)by induction on p. Suppose(A.21) holds. From (33), Va>1,

anddet[N,U,+DV,](s) y:l =6;' ( I - Gtl )-luf,

=det[N,(U,+KN,K)+(D-N,KN,)V,](s)#O i.e., 6 t l ( I- Gf1)-' :u t wyZl. Thus, (A.21) mplies that(A.22) holdsVa>p=1.Vs E C,. (A.9) Suppose (A.22) holdsV a 2 /3 =1, * - , - 1. (A.23)

Since K is a constant matrix, and U,, ,, N,, U,, , , V/ From (33), VCll>y , we haveare &-matrices, by closure proJerty of algebra &, (U,+V r N / K ) , U,+KN,.VI) re also @-matrices. Now (A.lO)

det(D-N,KN,)=detD-det(I- N,KN,D-')y &=Gy (I-G;)-

=detD.det(I-KN,D-'N,) Note that byA.21) 6' ( I - G&)-' is stable V a 2 y, andby A.23) the map (ui i= I + ( ~ ~ ) ; : ~s stable. Thus,=detD.det(I- K H ) . (A'1') (A.22)holds Va2 j3= 1; - - ,y. Hence, by induction,w c Y

By definition of p.r.l.c.f., V sequences (si)?= C c, whereIsil+0o, lim inf,,,ldetD(si)l>O and by assumption limsequences,

(A.22) holds Va2 /3 =1, * - , .Weprove A.22)+(A.21)by contradiction. Supposeid,,, ldet(I- KH)(si) l>0. Hence byA.1 I), for all such (A.22) suppose^ for Of contradiction' that forsome a,P wth a>p, e& (I - GiP)-' is not stable.For LTI dynamics, consider the input (8, . . ,lim inf Idet(D- N,KNr)(si)l>O . O,u,&O,. - - ,@.-By (33) and linearity, the correspondingi+m output is

This togetherwith (A.7)-(A.10) imply that (N,,D -N,KN,,N,) is a p.r.1.c.f.f H ( I - K H ) - ' . j:,= ( I - Gib )- u;.

Proof of Theorem 11D(L)Let E A diag(N,,; - - ,N ,m) .Thus, C=%E and since (Nj,D,)js_a p.r.c.f. of,, ;=for j= 1; , m , it is easy to see that (N,D) is a p.r.c.f. of6,or equivalently, (E,fi , I) is a p.r.1.c.f. of e. From (13,G = kg.nus,Assumption 2 is equivalent to v sequences


12/14


13/14

162 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, OL AC-23, NO. 2, A P R IL 1978

Hence, ( I - G)- : u I , u 2 ) b ( e , e 2 )s E-stable. Q.E.D.Proof of Corollav V. l

It follows from Assumption 2 that (I- G i i ) - is a well-defined map; by Assumption @) of the corollary, V i VI ,there xists .Zi,EC such that GiiCiiEC; hence, Uii 2( I- G,,);,, EC and, consequently, ( I- G,)- Uii e . Thus,by Remark l(iv), Assumptions a) and (b) respectivelyimply that

V i > j , i~ V,jE VI,y [ G,]s finite, A.31)V i VI,y [ (I- G i i ) - ] isinite.A.32)

Assumption (a) implies that?[ G ] isinite.A.33)

Assumptions a), (b) and the ower-triangular orm of( I - G ) together imply that

f [ I - G)-l] isinite.A.34)Similarly, (A.31), (A.32) and the lower-triangular form of(I-G I ) together imply that

y [ ( I - G)- l ] is finite.A.35)Assumption (d) implies that

y [G 1 2 ] is finite.A.36)By Theorem V, (A.33)-(A.36) and Assumptione)together imply that

( I - G)- is-stable.A.37)By Theorem I , (A.31), (A.33, Assumptions(c) and (d)togethermply that the SC S S is C-stable. Q.E.D.

REFERENCES[11 G.Zames On he input-output stability of time-v.arying nonlinearfeedback systems-Part I: Conchons derived usrng he conceptsof loop gain, conicity, and positivity, IEEETram.Automat.

Contr., vol. AC-11, pp. 228-238, Apr. 1966.[2] J. C. Willem, The Analysis of Feedback System. Cambridge,MA: M.I.T. Press, 1971.[3] C.A. Desoer and M. Vidyasagar,Feedback Systems, Input-OulputProperties. New York:Academic, 1975.[4] F. M. Callier and C. A. Desoer,LP-stability (1


14/14

TRANSACTIONS ON AUTOMATIC CONTROL,OL. AC-23,NO. 2, APRIL 197863

Zem, D.M. Himmelblau, Ed. NewYork: North Holland/American Elsevier, 1973.L. K. Cheung and E. S. Kuh, The bordered triangular matrix andminimum essential sets of a digraph, IEE E Trans. Circuits Syst.,minimal feedback vertex set of a directed graph, IEEE Trans.G. W. Smith, Jr. and R B. Walford, The identification of aCirnrits Syst., vol. CAS-22, pp. 9-14, Jan. 1975.F. Harary, Onminimalfeedback vertex sets of a digraph, IEEER. M. Karp, On the computational complexity of combinatorialTrans. Circuits Syst., vol. CAS-22, pp. 839-840, Oct. 1975.problems, Networks, vol. 5, pp. 45-68, 1975.G. Guardabassi and A. Sangiovanni-Vincentelli, A two levelsalgorithm for tearing, IEEE Trans. Circuits @st, vol.CAS-23,D.E. Riegle and P. M. Lin, Matrix signal flow graphs and anoptimum opological method for evaluating heir gain, IEEEC.A. Desoer, F.M. M e r , and W. S. Chan, Robustness ofTrans. Circuit Theoy, vol. CT-19, pp. 427-434, Sept.1972.IEEE Tram. Automat. Contr., ol. AC-23, pp. 586-590, Aug. 1977.stability conditions for linear time-invariant eedbacksystems,IEEE Trans. Automat.Contr., vol. AC-24 pp. 792-795, Dec. 1975.D. Carluci and F. Donati, Control of norm uncertain systems,

VO~.CAS-21, pp. 633-689, S v t . 1974.

pp. 783-791, Dw. 1976.

Frank M. Callier (S70-M72)was born onNovember 27,1942 in Antwerp, Belgium. Hereceived the electrical engineers degree from theUniversity of Ghent, Ghent, Belgium, in 1966, acertificate in NuclearEngineeringrom theSwiss Federal Institute of Technology, Ziirich,Switzerland, in 1967, and the M.S. and Ph.D.degrees in electrical engineering and computersciencesrom the University of California,Berkeley, CA in 1970 and 1972, respectively.From September 1969 to July 1973, he was

wth the Department of Electrical Engineering and Computer Sciences,University of California, at Berkeley, where he was an acting AssistantProfessor during the academic year 1972-1973. From July 1973 toAugust 1974, he was with the Laboratory for Automatic Control, Uni-versity of Ghent, as a qualified Researcherof the Belgian National Fundfor Scientific Research, Brussels, Belgium. n September 1974, he joinedthe Department of Mathematics, Facultis Universitaires de Namur,Namur, Belgium, where he is currently an Associate Professor.His mainresearch nterest is in the area of control and system heorywithemphasis on multivariable and distributed system stability. He also hasinterests in optimization.

Charles A. Desoer (S50-A53-SM57-F64), for a photograph and biog-raphy see page 2 of the February 1978 issue of this TRANSACTIONS.

TheStabilization of Digraphs of VariableParameterSystemsPETER E. CAINES, MEMBER, IEEE, AND ROBERT S . PRINTIS, MEMBER, IEEE

Abstruct-Directed acyclic graphsof dynamical systems are cousidered.uch graphs possess a natural and unique layering of their nodes.Each

is taken to be associated with a dynamical system in such a way thatparameter of a given system is some nonlinear functionof the statesf s y s tem connected to it and lying in higher layers of the network. Sucharrangement we call a digraph of variableparameter systems. It isthat digmphs of a large classof variable parameter systems may beby associating a compensator (of observer-linear feedback type)able parameter system.Each compensatorneed only observe

the output an d input of the variable parameter system with which it isassodated; however, an argument is presented to show that the perfor-mance of the eontrolleddigraph is enhaneed f the compensators s i g n a l toeacb other in a manner consistent with the flow of the parametr ic dis-~ ~ ~ t h e n e ~ o ~ I n t h i s s e n s e , i t i s s h o w n t h a t a n ~variable parameter systems is efficientlystabilized by amultilevel, orhidcalo n t r o l system.

I. INTRODUCTION

0018-9286/78/0400-0163$00.75 0 1978 IEEE

Input-Output Stability Theory - An Improved Formulation

Documents

Transcript of Input-Output Stability Theory - An Improved Formulation