Robust stability of discrete-time systems under parametric

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C. Wen and D. J. Hill, “Global boundedness of discrete-time adaptive control just using estimator projection,” Automatica, pp. 1!43-1157, Nov. 1992. R. H. Middleton and Y. Wang, “The intemal model principle in the adaptive control of time varying linear systems,” Tech. Rep. EE8830, Univ. Newcastle, Australia, 1988. B. Egardt, Stability of Adaptive Controllers. New York: SpringerVerlag, 1979. G. C. Goodwin and K. S. Sin, Adaptive Filtering Prediction and Control. Englewood Cliffs, NJ: Prentice-Hall, 1984. R. H. Middleton and G. C. Goodwin, “Adaptive control of time varying linear systems,” IEEE Trans. Automat. Contr., vol. 33, pp. 150-155, 1988. B. E. Ydstie, “Stability of discrete model reference control revisited,” Syst. Contr. Lett., vol. 13, pp. 429438, 1989. -, “Stability of the direct self-tuning regulator,” in Foundations of Adaptive Control, P. V. Kokotovic, Ed., 1991, pp. 201-238. S. M. Naik, P. R. Kumar, and B. E. Ydstie, “Robust continuous-time adaptive control by parameter projection,” IEEE Trans. Automat. Contr., vol. 37, pp. 182-197, 1992. H. Minc, Nonnegative Matrices. New York: Wiley, 1988.

among many others in [2]-[5] to obtain explicit robustness bounds for state-space models of continuous-time systems under additive perturbations. Some of these results have also been reproduced for discrete-time systems (see, for example [61-[81). The main objective of this paper is to link the stability robustness problem of discrete-time systems to that of continuous-time systems. We show, using two different approaches, that stability robustness of a discrete-time system can be reformulated as that of an auxiliary continuous-time system. One of these approaches makes use of Lyapunov theory and yields a sufficient condition. The second approach, which is based on the properties of Kronecker products, provides a necessary and sufficient condition at the expense of an increase in the dimensionality. This is a pleasing development, since it allows for a direct application of the known results on stability robustness bounds for continuous-time systems to discrete-time systems. The results are applied to stability analysis of interconnected systems, where the interconnections are treated as perturbations on a collection of stable subsystems. This demonstrates how a knowledge of the structure of perturbations can be exploited to obtain simple robustness bounds. 11. PROBLEM STATEMENT

Robust Stability of Discrete-Time Systems Under Parametric Perturbations

Consider a discrete-time system under additive multiparameter perturbations, which is described as

Mehmet Karan, M. Erol Sezer, and Ogan Ocali

2, : ~ ( k 1) = A ( p ) z ( k )

Absfract-Stability robustness analysis of a system under parametric perturbations is concerned with characterizing a region in the parameter space in which the system remains stable. In this paper, two methods are presented to estimate the stability robustness region of a linear, time-invariant, discrete-time system under multiparameter additive perturbations. An inherent difficulty, which originates from the nonlinear appearance of the perturbation parameters in the inequalities defining the robustness region, is resolved by transforming the problem to stability of a higher order continuous-time system. This allows for application of the available results on stability robustness of continuous-time systems to discrete-time systems. The results are also applied to stability analysis of discrete-time interconnected systems, where the interconnections are treated as perturbations on decoupled stable subsystems.

I. INTRODUCTION An essential feature of complex dynamic systems is the uncertainty in the system parameters, which may arise due to modeling errors or change of operating conditions. The analysis of stability in the presence of uncertainty is the subject of the robust stability problem. A common approach to stability robustness analysis is to model the uncertainty as perturbations on a nominal stable model. A measure of degree of stability of the nominal system can then be used to obtain bounds on the perturbations which the system can tolerate without going unstable. Lyapunov’s direct method provides a convenient way to estimate the degree of stability. It also directly yields bounds on tolerable perturbations [ 11. This feature of the Lyapunov approach has been used

Manuscript received January 4, 1991; revised June 7, 1991 and April 16, 1993. M. Karan is with the Department of Systems Engineering, Australian National University, ACT 0200, Australia. M. E. Sezer and 0. Ocali are with the Department of Electrical and Electronics Engineering, Bilkent University, Ankara 06533, Turkey. IEEE Log Number 9216453.

+

where z ( k ) E R” is the state of 2, at the discrete time instant k E p = [ P I p z ... pmIT E R” is a vector of real perturbation parameters, and

z+,

m

r=l

with A and E,, r = 1, 2 , . . . , m, being constant n x n real matrices. We assume that the matrix A(0) = A is Schur-stable, that is, has all the eigenvalues in the open unit disk in the complex plane. We would like to describe an open neighborhood of the origin in the parameter space in which 2, remains stable. More precisely, we are interested in a region

s1 = { p I A(cup)isSchur-stableforalln E [0, 11)

in the parameter space IZ”. Since, in general, it is difficult to characterize R explicitly in terms of the perturbation parameters, we aim at obtaining estimates of C2 as regular volumes embedded in R which can be characterized explicitly.

111. ESTIMATION OF ROBUSTNESS REGION VIA LYAPUNOV THEORY Our first approach to estimating s1 is through Lyapunov theory. Let V(z) = zT Pr be a Lyapunov function for the nominal system corresponding to p = 0, where P E R n X n is the unique, symmetric, positive-definite solution of the discrete-time Lyapunov equation

A ~ PA P = -Q

(3.1)

for some symmetric, positive-definite matrix Q E R n x n .The difference of V(z) along the solutions of the perturbed system 23 in (2.1) is computed as

0018-9286/94$04.00 0 1994 IEEE

1

(2.3)

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Corollary 1: The discrete-time perturbed system 2) is stable if the following bounds are satisfied

Where m

=&

r ~ - 1 / 2 ( ~ , T ~+~A * P E , ) Q - ' / ~

~

r=l

m

0ClprlcM(cr) :

+ ~ ~ ~ r p ~ Q - 1 ~ z E ~ p E , Q - l(3.3) /2. r=ls=l

RP : -

From (3.2), we observe that a sufficient condition for p E R is that Q ( p ) - I be negative definite or, equivalently,

~M[Q(P)I < 1

det[Q(p) - I]= -l"detF(p)

lgymIprI< nii1 (FIGTI) r=l

as:

(3.4)

where U M ( - ) denotes the maximum singular value of the indicated matrix. For the single-parameter perturbation case, i.e., when m = 1, (3.4) can be reduced, using a majorization on u ~ [ Q ( p ) to ], a quadratic inequality in lpl, from which an explicit bound for Ipl can be obtained. In the multiparameter perturbation case ( m > 1) , however, the left-hand side of (3.4) becomes a quadratic polynomial in { Ip,.l}, and characterization of R in terms of 1p.I is not as easy. The only available result [8] in this case involves the calculation of singular values of an m n x m n matrix. To avoid the difficulty, we try to obtain an alternative to the condition in (3.4), which guarantees negative definiteness of Q ( p ) - I , and thus stability of 2). For this purpose, we observe that

0, then p* E R. Otherwise, there exists a* > 0 such that A(ap*) is Schur-stable for all 0 5 a < a*, and A(a*p*) has an eigenvalue on the unit circle. Then A(a*p*)8 A(a*p*)has an eigenvalue at s = 1, and by (4.7), F ( a * p * ) has an eigenvalue at s = 0. Thus, a* > 1, and therefore, p* E R. This completes the proof. Like Theorem 1, Theorem 2 also allows us to use continuoustime results to obtain stability robustness bounds for D. Moreover, since it is directly based on the eigenvalues of A @ ) , it provides a necessary and sufficient condition for stability of D.However, since the dimension of the auxiliary system S is higher F a n that of further majorizations involved in the estimation of R may result in more conservative robustness bounds than Theorem 1. RC

s,

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NO. 5, MAY 1994

Although standard Lyapunov theory can be used to obtainperturbation bounds for S, since the nominal system matrix F of S is not we would like to describe a method to a simple one as that of generate a suitable Lyapunov function for F. LRmma: Let A1 and A2 be Schur-stable matrices, for which there exist positive-definite matrices PI and P2 that satisfy

s,

i =i,2

A?P,A, - P, = - Q ~ ,

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v.

APPLICATION TO INTERCONNECTED SYSTEMS

Consider a discrete-time interconnected system which consists of N subsystems described as N

V :zZ(k+ 1) = A Z z t ( k )+ E Z J A z 3 x , ( k ) , ,=1

i = 1, 2 , . . . ,ib- (5.1)

(4.8)

where ~ ( k E )72"' is the state of the ith isolated subsystem for some positive-definite matrices Q1 and

Q2.

Then, the matrix (4.9)

V D , : ~ : , ( k + l ) = A , ~ , ( ki =) ,l , 2 , . . . , N

which is assumed to be stable, A,, are constant real matrices, and p,, are interconnection gains, which are treated as perturbation parameters. Letting z = ... T : ~ ] ~A, = diag{Al, A 2 , . . . , A ~ } and E,, = ( E ~ ~ ) N with ~ N ,

.

[TT ZT

is Hurwitz-stable, and

(5.2)

(4.10) is a Lyapunov matrix for F which satisfies

p'Tp+ p

p = -Q

- pTpp

the interconnected system in (5.1) can be described in a compact form as (4.1 1) V:z(k+l)=

where (4.12)

Choosing V ( x ) = z T P z as a Lyapunov function for V ,where P is the solution of (3.1) with Q = I, Corollary 1 gives the following robustness regions in the parameter space of 2):

Prooj? Follows directly from

N N

FTP+PP= (P+I)'P(P+I)-P-P=TPP

+

+

(5.4)

(4.13)

i=l j=1

-6.

on noting that ( F I ) ~ P ( P I ) - P = Applying the result of the Lemma to F in (4.2) with Q1 = Q 2 = I , we obtain

where where P is the solution of

A T P A - P = -I.

v(i) iTP$

(4.15)

Now, choosing = as a Lyapunov function for 3 of (4.4), and using the result of [lo], we obtain the following perturbation bounds for stability. Corollary 2: The discrete-time perturbed system V is stable if the following bounds are satisfied:

From the block-diagonal structure of A, it follows that P = diag{ P I , P2, . . . ,P N } ,where P, are solutions of

A T P ~ A ~- P, = - I .

(5.7)

This block-diagonal structure of P, together with the special structures of the perturbation matrices E,,, allows for obtaining explicit expressions for ah-r(GZ3) appearing in (5.5). For example,

m

r=l

where GZ = [A;Pz1/' 0 M ( ~ 2=, u )

w(~,,I ) ~ ~ ~ ' ( A ~ P , A , , ) ~ . ~ '(5.9) (P~).

Thus, the stability region N

A;P,A,], so that

f i in~ (5.5) includes the region

N

where (see (4.17) at the bottom of the page).

(4.17)

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a ~ ,

which is smaller than but easier to compute. Note that it is also E, EzGz,) possible to give an explicit expression for 0n;1’/~(C~ to obtain an approximation to in (5.5). Alternatively, applying Corollary 2 to the interconnected system V in (5.1), we get the robustness regions N

fiD :

N

Czlp,,

VI. EXAMPLE To illustrate the application of our results, we consider an interconnection of three subsystems, with