Automatic Control, IEEE Transactions on

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Abstract-The notion of local I, -stability iis defined. The relationship ..... notions correspond to the usual reachability and observability, respec- tively (recall that in ...
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 41, NO. 6, JUNE 1996

903

it is not hard to see that given any U E 1- there is a nontrivial system

[19] J. S. Shamma and R. Zhao, “Fading-ml-mory feedback systems and

G E BTV such that Gu = 0.

robust stability,” Automatica, vol. 29, pp 191-200, 1993. [20] R. S. Smith and M. Dahleh, Eds., “The modeling of uncertainty in control systems,” in Proc. 1992 Santa li’arbara Workshop. London: Springer-Verlag, 1994. [21] R. S. Smith and J. C. Doyle, “Model validlation-A connection between robust-control and identification,” IEEE Trans. Automat. Contr., vol. 37, pp. 942-952, 1992. [22] A. Tikku and K. Poolla, “On the worst-case identification of slowlyvarying systems,” Prepr. 10th IFAC Symp. Syst. Identijication, Copenhagen, vol. 2, pp. 127-132, 1994. [23] G. Zames and L. Y. Wang, “Local-global (double-algebrasfor slow H” adaptation: Part I: Inversion and stability,” IEEE Trans. Automat. Contr., vol. 36, pp. 130-151, 1991. [24] T. Zhou and H. Kimura, “Time domain identification for robust control,” Syst. Contr. Lett. vol. 20, pp. 167-178, 1993.

V. CONCLUSION We have studied various classes of LTV BIBO-stable systems from the point of view of approximate modeling and proved some characterizations of them. These have included fading memory systems, strongly fading memory systems, almost periodic systems, and asymptotically periodic systems. Of these, only the space of BIBOstable asymptotically periodic systems is separable and thus allows systematic model parameterizations. It would be important to find other natural separable subspaces of LTV systems. LTV systems exhibit fundamental limitations as to how accurately they can be identified from input-output data, cf. Tikku and Poolla [22]. Thus they provide a setting in which to understand the importance of realistic assumptions and a priori information about the system to be identified to obtain satisfactory modeling results.

Local Z,-Stability and Locad Small Gain Theorem for Discrete-Time Systems

REFERENCES [ l ] L. Amerio and G. Prouse, Almost-Periodic Functions and Functional Equations. New York: Van Nostrand, 1971. [2] S. Boyd and L. 0. Chua, “Fading memory and the problem of approximating nonlinear operators with Volterra series,’’IEEE Trans. Circ. Syst., vol. CS-32, pp. 1150-1161, 1985. 131 M. A. Dahleh, “Asymptotic worst-case identification with hounded noise,” in The Modeling of Uncertainty in Control Systems, R. S. Smith and M. Ddhleh, Eds. New York Springer-Verlag, 1994, pp. 157-170. [4] N. Dunford and J. T. Schwarz, Linear Operators Part I: General Theory. New York: Wiley, 1957. [5] C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties. New York: Academic, 1975. [6] M. Gevers, “Toward a joint design of identification and control,” in

Essays on Control: Perspectives in the Theory and Its Applications, H. L. Trentelman and J. C. Willems, Eds. Boston: Birkhauser, 1993. 171 T. K. Gustafsson and P. M. Makila, “Modeling of uncertain systems via linear programming,” in Proc. 12th IFAC World Congr., 1993, Sydney, pp. 293-298.

(81 Y. Katznelson, An Introducrion to Harmonic Analysis. New York Wiley, 1968. [9] R. L. Kosut, G. C. Goodwin, and M. P. Polis, Guest Eds., Special issue on system identification for robust control design, IEEE Trans. Automat. Control, vol. 37, pp. 899-1008, 1992. [IO] L. Lin, L. Y. Wang, and G. Zames, “Uncertainty principles and identification n-widths for LTI and slowly varying systems,” in Proc. Amer. Contr. Con$, Chicago, 1992. [ l I] P. M. Makila, “Ha-optimization and optimal rejection of persistent disturbances,” Automatica, vol. 26, pp. 617418, 1990. “Robust identification and Galois sequences,” Int. J. Contr., vol. [I21 -, 54, pp. 1189-1200, 1991. 113) P. M. Makill and J. R. Partington, “Robust approximate modeling of stable linear systems,” Int. J. Contr., vol. 58, pp. 665-683, 1993. [ 141 P. M. Makill, J. R. Partington, and T. K. Gnstafsson, “Robust identification,’’ Prepr. 10th IFAC Symp. Syst. Identijication, Copenhagen, 1994, vol. 1, pp. 45-63. [15] B. M. Ninness and G. C. Goodwin, “Estimation of model quality,” Prepr. 10th IFAC Symp. Syst. Identijication, Copenhagen, 1994, vol. 1, pp. 25-44. [I61 J. R. Partington, “Interpolation in normed spaces from the values of linear functionals,” Bull. London Math. Soc., vol. 26, pp. 165-170, 1994. [17] K. Poolla, P. Khargonekar, A. Tikku, J. Krause, and K. Nagpal, “A timedomain approach to model validation,” IEEE Trans. Automat. Contr., vol. 39, pp. 951-959, 1994; also in Proc. 1992 Amer. Contr. Con$, Chicago. 1181 J. S. Shamma, “The necessity of the small-gain theorem for timevarying and nonlinear systems,” IEEE Trans. Automat. Contr., vol. 36, pp. 1138-1147, 1991.

Henri Bourlks

Abstract-The notion of local I , -stability iis defined. The relationship between this notion and Lyapunov stability is clarified. A local version of the small gain theorem is then established in the case of discrete-time systems. These results are applied to stabiliity analysis of a nonlinear discrete-time delay system.

I. INTRODUCTIO~ The small gain theorem [8], [15] plays a fundamental role in stability analysis of nonlinear systems in an input-output viewpoint. This theorem applies to discrete-time systems and to continuous-time ones as well. The type of stability which is then obtained is “ZPstability” (1 5 p 5 CO); see, e.g., [13]. ‘This approach was limited by the fact that only global results are available in the literature. This will become clear in the following discussiion; consider the standard closed-loop system in Fig. 1. Let us denote as S” the linear space of all sequences z = (z(O),z ( l ) ,. . .), where s ( t ) E R”, V t ; G 1 i s a causal input-output operator S” + S” associated with a system C1 [2]’; suppose that the Z,-gain of G I is finite and is denoted as ?,(GI). Moreover, assume that G Z is a causal memoryless operator S” -+ S”, defined by a nonlinearity @: N + R“ -+ R”, as follows:

( G z s ) ( t )= @(t.z ( t ) ) , V X E S”. Vt E N

(1)

Manuscript received March 14, 1994; revised March 17, 1995. The author is with the Electricit6 de France, 92141 Clamart Cedex, France and Ecole Normale Suptrieure de Cachan, 94230 Cachan, France. Publisher Item Identifier S 0018-9286(96)02820-6. ’This notion was defined in [2] in the continuous-time case. In the discrete-time case, let C he a system defined by a state-space realization V ( t 1) = f ( t , ~ ( t u)(,t ) ) y(t) , = g ( t . ~ ( t ~) ,( t )where ) , V ( t ) is the state, u ( t ) E R”,y(t) E Rq. Assume that ~ ( 0 )= 0 (zero initial condition at initial time t = 0). Then, it is easy to prove by induction that for any t 2 0, y(t) can be expressed in function o f t , U ( ( ) ) ,. . . u ( t ) ;in other words, there exists a causal operator G :S” -+ S Q such that y = Gu;G is called the input-output operator associated with C . If ZI is time-invariant, the initial time can be shifted without inconvenience.

+

0018-9286/96$05.00 0 1996 IEEE

.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 41, NO. 6, JUNE 1996

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B. Usefil Inequalities 5

I

Obviously, for every p E [ l , m ) . 2; C E l g , then we clearly have

I

lClm I

(U

where N denotes the set of natural integers. Suppose that @ satisfies the following condition: 3a

> 0: li+(t, 0,

3a > 0, V< E R"

+b)P

C l & . If

C E R"

114l~I I l 4 P .

ICIP,

Proposition I : For any reals a p E [ l . x )one , has Fig. 1. Standard closed-loop system C,]

CO"

20

5 2,(aP

and b

20

and

(4) and for every

+bP).

(5)

Proof Let Q and r be any positive integers. One has (by the binomial identity) in

+ b)'

Using the Minkowski inequality, one obtains (a

+ b)+

5

W(aP

+b

y '

- 24"(a*" < + Y").

(3)

Vt E

ll 0, l \ ~ l l ~ < , t E 3 (IG ~ ( l , , t 5 k ~ \ ~ If~ Kt ~ is~ nonempty, . ~ } . G is said to be locally 2,-stable (2-lp-s), and ?,i(G) = inf(I 0. In addition, note that

because G is causal [14], [Z]. Remark 2: The local eo-stability of causal operators can also be defined as follows: G: S" --t S" is locally co-stable if i) z E e: 11. PRELIMINARIES G z E e?, and ii) G is l-l,-s. The global eo-stability has been defined in [9]. A. Notation Remark 3: If G is linear, then ^j,l(G) = y, (G), where y,(G) In everything that follows, p i s equal to oc or is a real satisfying is the !,-gain of G [I51 (note that following the terminology used in p 2 1.Let E R": if p < m, then ] 0,

vu E s n f m

> 0: I I [ ( ~ t ) ~ ( e t - l ) ~ ] ~ 0, set Tq = { t > 0,3u E S'L+7n: l l ~ l l < ~ ,7~ and llel/lp.t 2 5 1 or lie211p,t 2 E Z } . Finally, let 17 be such that 0 < r j 5 min(5.p). If 8, let T = min('&). For t < T , IIul/,,t < 17 llelll,,t < E I and lle211P,t < € 2 . Hence, IlyJl,,t 5 y; llet\lp,t(i= 1 , 2 ) . One has el = U I - Y ~ , and e2 = U Z + Y I ; hence I l e i l l P , t I IIU;Ilp,t+Y311e311P,t, where i. j = 1 or 2 and i # j . Therefore, Ilezllp,t I (1 - -ylyz)-' [ / / U I l l P . t + ?3/IUJll,,tl I X;(II~lll,,t ll~ZIlP,t)I 2XZII4IP,t (by ( 5 ) for p < x,this inequality being obvious for p = m). Hence, Ileillp,t < 2X, 7 I 2X; E Ip / 4 . in particular, ~ ~ e i ~ ~ p< ,r-i p/4, so that I l e l l P , T - l < p / 2 . Therefore, II[(U.)~ (e,-~)~]*ll~ 5 l l u . l l ~ . ~ ~ ~ e ~ ~ , 0 and 7%> 0 (i = 1 . 2 ) Proof: such that IleZIIP,t < E , lIv;ll,,t 5 YzllezIl,,t, V t with YIYZ < 1. i) Reachability: Let x* belong to a neighborhood V of 0, such For i = 1 , 2 , let p i ? 0 < pt < E; be such that Vu E Snf",Vt. that V C U, and set t* = (y;?...,~;)~. Define U by l l [ ( ~ t ) ~ ( e ~ - l )