Robust stability of systems with application to singular perturbations

0005 1098 79~0401 0467 $2 0O~0

4utomama Vol 15 pp 467~,70 Pergamon Press Ltd 1979 Pnnted m Great Britain © International Federation of Automatic Control

Brief Paper Robust Stability of Systems with Application to Singular Perturbations*I" NILS R S A N D E L L , JR +

Key Word I n d e x ~ o n t r o l theory, stabdlty, control nonhnearltles, feedback, frequency response, functional analysis, sensitivity analysis, singular perturbations, system order reduction

cessible software to compute singular values (Garbow et al, 1977), this characterization is of great practical value The present paper is prompted by two observations First, the use of singular values to characterize robustness is suggestive of connections with numerical analysis, but these connections are not clear from Doyle's approach utilizing the multlvariable Nyquist theorem Second, a specific instance of the robustness question arises when a system is approximated by making a singular perturbation to reduce its order A related motivation, although not discussed in this paper, arises from the desire to use multi-model techniques in the design of decentralized controllers for large scale-systems (Khahl and Kokotovic, 1978, Sandell et al, 1978) The structure of this paper is as follows In Section 2 we consider the robust stability of M I M O feedback systems using standard results from i n p u t - o u t p u t stability theory When speciahzed to the linear time m v a n a n t case, with rational transfer function matrix perturbations, Doyle's characterization results In Section 3 we will apply the results of Section 2 to a specific robustness question arising in singular perturbation theory Section 4 contains the s u m m a r y and conclusions

Abstract--In this paper we will give a simple approach to determining conditions for stability of feedback systems subject to perturbations in the operators describing these systems The approach is based on techniques used in functional analysis, and provides an alternative development and generahzatlon of some conditions for the linear t i m e - m v a n a n t case that have appeared m the hterature very recently As an example of the apphcation of the conditions, we consider the determination of finite regions of stablhty for singularly perturbed systems

1 lntroductmn AN IMPORTANTtheme in system theory and apphcations is the robustness of various system theoretic properties to variations in the system model In this paper we will focus on the robustness of the stability property of multlvarIable feedback systems This subject is of special interest, since stablhty ms the most basic system theoretic issue and since practical feedback systems must remain stable m the face of large parameter variations In classical single-input, single-output (SISO) control system design, system robustness is measured in terms of such quantities as gain and phase margins However, these single loop quantities do not completely characterize the robustness properties of multiple-input, multiple-output (MIMO) systems since simultaneous model variations in several loops cannot be dealt with Moreover, these quantities are not defined for nonlinear and time-varying systems Very recently there has been some important work addressing the multivariable robustness issue In his thesis Safonov (1977) (Safonov and Athans, 1977), gives a powerful approach, based on a multivarlable sector stability theorem, that can characterize robustness for very general nonlinear M I M O feedback systems In a recent paper Doyle (1978) develops a robustness characterization for the linear time lnvanant M I M O case (it can be shown that Doyle's results can also be obtained by Safonov s approach {Safonov, 1978) Doyle's characterization involves computing the m i n i m u m singular value of a certain transfer function matrix, and this computation essentially determines the m m l m u m simultaneous variation of the system frequency responses that leads to mstabihty Since there is sophisticated and widely ac-

Notatmn We will use the standard notation of input output stability theory see Wlllems (1971, p 13-23), or Desoer and Vldyasagar (1975 p 38 39) = s o m e Banach space of functions ~ T - , A wlth norm II II T = s u b s e t of the real numbers X =finlte dimensional vector space f ~ = I x P ~ E ~ for all z ~ T I

(p ~c)(t)=(x(t) tO

tz

L/'.'-spa~.e ol m-*ector functions on I with Euchdean norm X~--*Y. = identity operator ".# '~~---'ee is {ausal if P ~ P ~ = P ~ for all r e T 4* =conjugate transpose of a complex matrix A

II~L= sup

lnlegrable

[I e , ~ , l - P,~x2 I[

-

~cF P x x i ~. P r x

g X~x *Received August 24, 1978, revised December 4, 1978 The original version of this paper was presented at the IRIA/IFAC W o r k s h o p of Singular Perturbations which ~ a s held in Paris France during June 1978 lntormatton about the published Proceedings of this IFAC Meeting may be obtained from Pergamon Press Ltd, Headmgton Hill Hall, Oxford OX3 0BW England This paper was recommended for pubhcation in revised form b~ associate editor Brian Anderson tThis research was supported by the Department of Energy under grant ERDA-E 149-18)-2087 and by O N R under contract N00014-76-C-0346 :~Laboratory for Information and Dectsion Systems Room 35-213, M I T , Cambridge, MA 02139, U S A

g~=[(g~).]=

[

2 Robust stabtltt3~ oJ systems We consider the feedback system depicted in Fig 1 Here the causal operator ~ ~'~'~---5°7~ represents the plant plus

FIG 1 Basic M I M O hnear feedback system 467

468

Bne! Paper

any compensation that is used The basic leedback equation is

(¢+N)e=u

,t follows that (~/+A,-#)

IA~']t,<~

(22)

I ¢+~+A'~4lc=u

'lla<

(23)

Proof By Theorem 2 10 of Wdlems (1971) (¢ + ~ ¢ - l A ~ ' ) - ~ ex,sts, IS causal and sat,sties II : + ~ - ' a ~ l l , < )c Since 1A~/) ~/

(24)

('#x)(t)=J ~ :~(t-z),c(z)dr

(25)

(26l

2 A basic result ,n numerical analysis is that ff an n x m matr,x A is lnvertIble, then A + A A is revertible for all A4 sat,sfylng

6(AA)
H<~ I1,=11'~ II.c =~ ....

(2 12)

ama~=max max cr,(G(jto))

(2 13)

where to-O

l¢-i~m

and where a,(Glju))) denotes the ~th singular value of the transfer function matrix corresponding to fg Lemma 2 (Wfllems, 1971, p 166) Let the operator N £v~, ---,L,°~', for T = [ 0 , ~c] be defined by

(Nx)(t)=g(~(t))

12 14)

where g R~---,R ~ IS continuously dlfferentlable Then

II N II, ~ sup ~(g,(xl)

a sufficient c o n d m o n for (22) is

Jl 4-1LIIA~tl, <1

/2111

where the elements of the ~mpulse response matrix 'q(tl are assumed absolutely mtegrable on T Then

the theorem follows Q E D Remarks 1 Since

II ~ 'A~ll~--<)t~-IIl~jla~ll~

12 I0)

and c a s e ill) lS s,m,lar Q E D The practical ,mportance of Theorem 2 stems trom the tact that the incremental norm of a linear convolution operator or a memoryless nonlinear operator can be readily computed Lemma 1 (Desocr and Vldyasager, 1975 p 26) Let the operator '4 ~ ' - " W 7 for r = [ 0 , ~3 be defined b,,

l ,/ __)j~ exists, is causal and has

( ~ + A.~/)- ~ = ( / + , r g

Por case ()) wt. apply Theolein I to the t.qu,mon

(2 1)

and the bas,c stablhty question ,s whether ( J + f f ) - 1 ~,~, - + : f ~ exists, ,s causal, and ,s a bounded operator in the sense that I I ( : + N ) Ill,< ~ We will assume that the nominal system ~s stable throughout th,s sectmn We a,e interested m whether the closed loop system retains these properties when subject to additive ( N - - , ~ + A N ) or mult,phcative ( a / - - , ~ + N A N ) p e r t u r b a t , o n s representing uncertainty in the dynam,c behavior of the system t h e following theorem provides the basis for our analys~s Theorem 1 Let ~/ t~--,'/, be a linear causal operator and suppose ~ - ~ ex,sts, IS causal, and It ~/ ']I~ < x. Then if A 4 d ~ - ~ / ~ IS a causal operator satisfying [ [ k 4 j j a < : and ,f

tl "

Prool

(2 7)

Combining Theorem 2 and Lemmas 1 and 2, we obtain the following result Theorem 3 Assume that the nominal s)stem ,n F,g 1 Js time lnvarmnt, stable and that the operator (¢ +<#) ~ can be represented as a convolutmn operator whose impulse response matrix has absolutely lntegrable elements Assume that (Case Ca)) A~ is a stable convolution operator whose impulse response matrix has absolutely mtegrable elements or (Case (b)) AN is a memoryless nonhnear operator reduced by the continuously dtfferentlable function g ~ - - , ~ " Then (l) the system remains stable for addmve perturbat,ons satisfying

Case Ca) ~(AG(Je)}}
where O(AA)= ItAA ll~, ~,(A)= (11A- l lt~) ' are respectively the smallest and largest singulal values of the matrix 4 Theorem 1 ,s thus a generalization of thts classical finite d,menslonal result where, of course, boundedness and causahty are not at issue In the finite dimensional case there exists AA such that # ( A 4 ) = g ( 4 1 and 4 + A A is singular th,s is easily proved by the singular value decompositmn* The robust stability questions posed at the beginning of this section are now answered m terms of Theorem 2 Theorem 2 Assume that the basic feedback system of F,g 1 ts stable Then (,) the system remains stable for add,tl~C perturbatmns prox ided

(_~ 15)

cHAg,l~})
V¢o ~ 0 Vu) >0

V~E #"'

12 16) and (u) the system remains stable for multlphcatlve perturbations sat,sfying Case Ca) "~(AO(j
~(Ag,(~))." a ( l + ( ,

w)~0

L(po)) V~o>O

V~e )A~ (217)

and (i,) the system remains stable for multlphcatlve perturbations provided

Here G(jto) and AG(po) are the transler funct,ons of 'q and AY/ and a ( 4 ) and alA) denote the m a x i m u m and m , m m u m singular values of ,t Proo/ (t) From Lemmas t and 2 and (2 16) be ha~e in either case

II ( / + ~q) 'NAN ]Is < l

IIANII~ 11(¢+<~) Iii,'

(2 IS)

[I(:+N) 'HdIANI[,- l

(219)

(29) so that

*The singular value decomposition of an n x n nonslngular complex matrix 4 is 4 = UZI~ *, where b and I, are unitary n × n m a m c e s , Z = d i a g (al, ,or) and the singular values a, are the non-negative square roots of the eigen~alues of A* 4 See Laub (1978) for references, a more general definition, and an excellent discussion of the fundamental role of the singular ~alue decomposition in linear systems theory

(If I The prool IS similar

Q ED Remalks 1 Notice the analogy between Theorem 3 concernmg lobust stablhty of hnear systems and the classical result quoted previously concerning robust lnvers,on of matrices (or bounded operators)

Brief Paper 2 Theorem 3 for the case of rational transfer functmn matrices Is the result of Doyle alluded to previously Doyle's proof is completely different, however, depending on the multlvanable Nyqmst Theorem, so that the connections with the inversion issue are only implicit 3 Note the appearance of the quanUty I+ G-t(j09) in the multlphcative case of Theorem 3, rather than the more usual return difference matrix I + G(309)

469

Figure 2 closely resembles the perturbed feedback system considered in the previous section with

G(s)=A22t A2t(sI-AIt)-1A12

(3 5)

A G(s, e) = - es(esl - A22 )- t

(3 6)

In the previous section we have discussed the robustness of the stability property of a dynamm system to model variations In this section we will consider a particular form of model variation due to a singular perturbatmn* We consider systems of the form

except that the perturbation Is post-multlphcatlve rather than premultlphcatlve However, assuming G(s) has full rank as a rational matrix, It is easily verified that the analysis of the preceding section is essentially unaffected Thus we have the following result Theorem 4 Assume that the system of Fig 2 is stable for e = 0 Then it remains stable for all e > 0 satisfying the mequahty

xl(t)=Altxt(t)+ A12x2(t)

a'(aG009, ~)) < a(l + G - ~(./to))

3 Application to singular perturbatmn theory

e'z2(t)= A21xl(t)+ A22x2(t)

(3 1)

where e > 0 is a small parameter, and it is assumed that the matrix A22 Is stable (has elgenvalues with negative real parts) We define the so-called degenerate system

Xld(t)=(All - A12A221A21)xIa(t)

Xl(S)=(sl--Alt) IA12x2(s)

for all 09 > 0 The use of Theorem 4 is illustrated by the following examples

Example 1 xt(t) =xt(t)+ 2x2(t) } gx2(t) = --x2(t)--Xl(t)

(3 2)

associated with (3 1) This system is a reduced order system that neglects certain high-frequency or parasitic effects incorporated in the model (3 1) It has been shown that the stability of (3 2) (in the sense that the elgenvalues of the system matrix have negauve real parts) Is insensitive to these effects in the sense that there exists Co>0 such that (3 1) is stable for all 0 < e < e 0 if (3 2) is (Desoer and Shensa, 1970) We propose to examine the robustness of the stability (in the input-output sense of Section 2) ot (3 2) to the parasmc effects present in (3 1) We begin by Laplace-transforming equations (3 1) (assuming zero initial condition)

G(s)-

(3 4)

To apply the input-output stablhty results of the previous section it is necessary to apply a test input to the system This is most conveniently done as illustrated in Fig 2, although other locations are posstblet

(3 8)

2

(3 9)

s-1

AG(s,e)=

gS

(3 10)

~s+l

In this case for which G and AG are scalars, the condition (3 7) is equivalent to

t l + G-1009)[ >1 aG009, ~)[

(3 11)

I 1 + G009) I> I GO09)AG009,~)l

(3 12)

or

(3 3)

x 2 ( s ) = ( e d - 422) tA21xt(s) =[ l - *s(esl- A 22)- l] ( - A221Azi )xt (s)

(37)

for all 09 > 0 The condition (3 12) has an interesting graphical mterpretauon Specifically the Nyqmst locus of G(j09) must avoid the critical point - 1 by at least the distance I G(j09)AG(309)[ It is easily verified that for e = +1 and 0 9 = x ~ we have I1 + G(/09) I =] G(j09)AG(j09, a) I so that g0 = 1 It can be directly verified that the system (3 8) becomes unstable for e = 1

Example 2 Xl

=

~2

= - - X l - - 112

--XI

G(s) = FIG 2 Singular perturbation m the frequency domain

AG(s, e ) -

4-'~2

~

1 s+l ES

l

+es

(3 13) (3 14)

(3 15)

As m the previous example, we will check (3 12) We have *See Kokotovlc, O'Malley and Sannutl (1976) for an excellent survey of results in singular perturbation theory -?To insure the equivalence of the input-output stability analysis with the condition that the system matrix of (3 1) has elgenvalues with negative real parts, it is necessary to have the conditions At I

AI2

A21

A22

[0

I]

I 4 ~ 21

[

A.

controllable,

AI;]

A21e A~2-J

observable

~/2 + tn2 ->2> 1 ~ e09/.v,' 1 +F2(D 2 so that the system ts stable for all ~>01

4 Summary and concluswns In this paper we have considered the robustness of the stability property of a multxvanable feedback system to varlauons in the system model The approach was based on standard techniques of input-output stablhty theory In the tlme-mvanant case, this approach speclahzes to give sufficient conditions for stablhty under additive and multlphcatlve perturbations that are easily verified by computing the singular values of certain transfer function matrices We then apphed the robustness condition to the analysis of singularly perturbed systems We were able to give an exphclt, readily

47(I

Brlel Papel

computable bound on the magnitude ol the perturbation parameter ~ that can be tolerated and still have a stabihtv analysis of the reduced system ,~ahd for the lull system The results of this paper are felt to be of interest for two reasons F~rst the characterization and design ol robustly stable M I M O feedback systems is a fundamental problem in control theory that has yet to be completel'v lesolved Second as has been emphasized by many authors l e g Zames 1976 Safono,~, 1977) a fundamental problem in large-scale system theory is to give conditions for the success of designs based on multiple aggregate models of a single large sv,,tem this )s essentially a robustness problem

A(kno~ledgements The ideas in this paper arose during the course of dlscussmns with M Athans, J C Doyle, P Kokotowc, A J Laub, M Safono', and G Stein Part of the work for this paper was accomphshed during a visit to the Centro dl Studm dei Slsteml dl Controllo e Calcolo Automatlca of the Umversity of Rome, the author gratefully acknowledges the hospltahty of Professor P Nlcolo

ReJerenceb Safonov, M G 11977) Robustness and stability aspects of stochastic mulm, arlable feedback system design P h D Thesis Massachusetts Institute of Technology Safonov, M G and M Athans (1977) Gain and phase margins for multdoop L Q G regulators IEEE Trans Aut Control AC-22, 173

l)o~lc I ( 11978) Robuxtne~ ol muhfloop hneal Iccdha
Salonox M (J (197,~) Personal et)mmunlcatlOll (,arbow B S I M Bo,,le J [ Donganna and ( B M o k n {1977) ,Xlutrtx ft~t'ns~st~m Routines L I ' ~ P 4 ( K (,utdt L xtenston Lecture Notes m ( o m p u t e r St.Jence \ o [ q l Sprmger-Verlag, Ne~ ~ ork Sandelt N R Jr P Varaiya M Athans and M (J Salono~ (1978) Sur~e3 of decentralized control methods for large scale systems IEEE Trans 4ut Control AC-23, 108 Khahl H k and P ~, Kokoto~lc (1978) (.ontrol ~ll itegies for decision maker,, using different models of the same system IEEE Tran~ 4ut Control AC-23, 289 Wlllem J ( 11971) lh( lnull~t~ o! t'~tdha(!, ~l~tt'ms M I T Press Cambridge Mass De~o~r (. A and M ~ld,¢asagar 11975) teedha~h b~,sl(.nt~ Input-Output fh opertte ~ Academic Press, New York Laur, A J (1978) Linear multl;arlable contiol Numerical considerations In,nted paper 4melt(an ~Iathematital Societ$ Short Course on Control Theor~ Providence R I Also ESL-P-833 Electronic Systems Laboratory Massachusetts fnstitute of Technology Kokoto,,Ic P V R [- O Malle~ Jr and P Sanntm 11976) Singular perturbatlons and order reduction m contlol theory an oxervie~ 4utomatita 12, 123 DesoeI ( ~ and M 1 Shensa 119701 Netv, orkb v, lth ~er), small and ver) large parasitics natural frequencies and stablhD Pto~ IEEE 12. 1933 Zame,,, G 11976) On feedback hierarchies and complexit,, (~,~ information) /976( D( Clearwater, Florida