Bounded Real Criteria For Linear Time-delay Systems - ECE UNM

1016

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 7, JULY 1998

[4] R. Kumar and V. K. Garg, “State avoidance control for infinite state systems using assignment program model,” to be published. [5] R. Kumar, V. K. Garg, and S. I. Marcus, “Predicates and predicate transformers for supervisory control of discrete event dynamical systems,” IEEE Trans. Automat. Contr., vol. 38, pp. 232–247, Feb. 1993. [6] Y. Li and W. M. Wonham, “Control of vector discrete-event systems —The base model,” IEEE Trans. Automat. Contr., vol. 38, pp. 1214–1227, Aug. 1993. [7] F. Lin and W. M. Wonham, “On observability of discrete-event systems,” Inform. Sci., vol. 44, no. 3, pp. 173–198, 1988. [8] Y. Park and E. K. P. Chong, “Sensor assignment for invertibility in interruptive timed discrete event systems,” in Proc. IEEE Int. Symp. Intelligent Contr., 1994, pp. 207–212. [9] P. J. Ramadge, “Observability of discrete event systems,” in Proc. 25th IEEE Conf. Decis. Contr., 1986, pp. 1108–1112. [10] P. J. Ramadge and W. M. Wonham, “Supervisory control of a class of discrete-event processes,” SIAM J. Contr. Optim., vol. 25, no. 1, pp. 206–230, 1987. , “Modular feedback logic for discrete event systems,” SIAM J. [11] Contr. Optim., vol. 25, no. 5, pp. 1202–1218, 1987. [12] S. Takai, T. Ushio, and S. Kodama, “Static-state feedback control of discrete-event systems under partial observation,” IEEE Trans. Automat. Contr., vol. 40, pp. 1950–1954, Nov. 1995. [13] S. D. Young and V. K. Garg, “Optimal sensor and actuator choices for discrete event systems,” in Proc. 31st Ann. Allerton Conf. Communication, Control, and Computation, 1993, pp. 687–696.

I

H1

One way to cope with the control of systems with delay is the method of Smith-predictor [5], which has some merits but also suffers from a few drawbacks when it deals with model uncertainties. Another way, which is more systematic on one hand, but it also leads to more complicated controllers on the other hand, is to apply the approach of [6] to construct estimators that include a finite interval memory. Recently, an alternative approach to the control of timedelay systems has been proposed [7], [8] that applies the operator Riccati approach. Systems with state delay appear in various problems of process control [5], especially in chemical processes with recycling. They also appear in many other control problems where the time delay is encountered in the measurement. If, due to simplicity reasons, memoryless controllers are required for the latter problems, the analysis of the resulting closed loop involves stability considerations and norm analysis of state-delayed systems. For example, consider the system

H1

x _ (t)

=

Ax(t)

where w(t) and u(t) are the disturbance input and control input, respectively, and for which a controller has to be designed that is based on the delayed measurement, C x(t 0 d), where d is the time delay. More specifically, consider a controller of the type u(t)

Bounded Real Criteria for Linear Time-Delay Systems U. Shaked, I. Yaesh, and C. E. de Souza

H1

Abstract—This paper considers the problem of finding the -norm of a linear time-varying system having a delayed state and with an output that comprises a linear combination of the “current” state as well as of the “delayed” state. Both delay-independent and delay-dependent bounded real criteria are derived which provide sufficient conditions for the system to possess an -norm which is less than or equal to a prescribed bound. The stationary case for time-invariant systems is also tackled, and in this situation the proposed bounded real criteria are given in terms of linear matrix inequalities. Two examples are presented which compare the guaranteed bounds that are achieved by our results with the actual -norm of the system.

H1

H1

Index Terms—Bounded real lemma,

H1

H1 , time-delay systems.

I. INTRODUCTION

The theory of control [1], [2] has been recently applied in the field of process control [3]. Chemical and industrial processes are often characterized by transport delays which are in fact reasonable approximations to systems that have a large number of cascaded time lags (see, e.g., [4] and the references therein). Manuscript received January 4, 1996; revised July 29, 1996. This work was supported by the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico-CNPq, Brazil, under Grant 301653/96-8/NV/PQ. U. Shaked is with the Department of Electrical Engineering–Systems, TelAviv University, Tel-Aviv 69978, Israel. I. Yaesh is with Israel Military Industries, Advanced Systems Div., Ramat Hasharon, Israel. C. E. de Souza is with the Department of Systems and Control, Lab. Nacional de Computacao Cientifica-LNCC/CNPq, Petropolis, Rio de Janiero, Brazil. Publisher Item Identifier S 0018-9286(98)03596-X.

+ B1 w (t) + B2 u(t)

=

K C x(t

0

d)

that should minimize the H -norm of the transfer function Tzw , from w to the controlled output z , defined by T z (t) = C1 x(t) + D12 u(t) with D12 [D12 C1 ] = [I 0]:

1

We readily obtain that the closed loop is given by

0 )+ ( 0 )

+ Ad x(t

x _ (t)

=

Ax(t)

z (t)

=

C1 x(t)

+ C2 x

d

t

B1 w (t)

d

where Ad = B2 K C and C2 = D12 K C . A result which is analogous to the bounded real lemma (see, e.g., [1]) is then required if one wants to determine the H -norm of Tzw . The problem of bounded realness for linear systems with a delayed state has been recently investigated in [10]–[12], which proposed delay-independent bounded real lemmas. Reference [10] applies a frequency domain approach, while [11] and [12] use a time-domain approach. Since for these results the time delay is allowed to be arbitrarily large, these versions of the bounded real lemma are, in general, conservative for many important applications. Very recently, a delay-dependent bounded real lemma, i.e., one that depends on the size of the time delay, was proposed in [13]. A common feature of the above results is that the output is in terms of the “current” state only and thus cannot be applied to the closed-loop system as above. The purpose of the present paper is to develop both delayindependent and delay-dependent bounded real lemmas for linear systems with a delayed state and where the output comprises a linear combination of the “current” state as well as of the “delayed” state. Bounded real lemmas for finite-time horizon and infinite-time horizon problems will be developed.

1

II. FINITE HORIZON BOUNDED REAL CRITERIA We consider the following linear time-varying system: x _ (t)

=

0018–9286/98$10.00  1998 IEEE

A(t)x(t)

+ Ad (t)x(t

0

d(t))

+ B1 (t)w (t)

(1)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 7, JULY 1998

where x(t) 2 Rn ; w(t) 2 Rm ; and A(t); Ad (t) and B1 (t) are bounded real matrix functions of appropriate dimensions that are piecewise continuous over [0; T ]. The scalar d(t) is a bounded varying time delay. Given a scalar > 0 and assuming that w 2 L2 [0; T ], we consider the following performance index: J = xT (T )PT x(T ) + kzk22 0 2 kwk22 ; PT = PTT  0 (2)

1017

where

y(t) Denoting



(3)

where C1 (t) and C2 (t) are bounded real, piecewise continuous, timevarying matrices. The problem is to find conditions which will ensure that J  0; 8 w 2 L2 [0; T ], subject to

8  0

x( ) = 0;

and

w( ) = 0;

8  < 0:

For simplicity of notation, we omit in the sequel the dependence of the various matrices and vectors on t, and we only keep the dependence on t 0 d. Our first result provides a bounded real criterion that is independent of the size of the time delay but depends on an upper bound for the rate of change of the delay. A delay-independent criterion is surely more conservative than a criterion that is tuned to the specific delay length in the system. It will be seen below, however, that due to its simplicity it may be preferred over a delay-dependent criterion in cases where Ad is of small norm. This simplicity is most advantageous when it is desired to perform a parametric search over a given structure of the controller. A delay-independent criterion also may be proven useful in cases with large uncertainty in the delay length. We assume that the time-delay d(t) is bounded and that it satisfies

d_(t)  < 1;

8 t 2 [0; T ]

(4)

where is a given scalar. We obtain the following result. Lemma 2.1: The performance index J of (2) is nonpositive for all w 2 L2 [0; T ], and for all bounded time-delay d(t) that satisfies (4), if there exists a symmetric positive definite matrix Q(t) that allows a symmetric positive semidefinite solution P (t) over [0; T ], with P (T ) = PT , to the following differential linear matrix inequality (DLMI):

9

ATd P B1T P where

9

P Ad

0(1 0 )Q(t 0 d) 0

P B1

0

0

2

I

 0; 8 t 2 [0; T ]

P_ + AT P + P A + C1T C1 + (1 0 )01 C2T C2 + Q:

(5)

(6)

Proof: For a specific time-varying d that satisfies (4), we consider the following function: t U (t) xT (t)P (t)x(t) + xT Q + (1 0 )01 C2T C2 x d: t0d(t) (7) Considering (1), it can be easily established that

dU (t) = xT P_ + AT P + P A + C1T C1 + Q dt + (1 0 )01 C2T C2 x + 2xT P Ad y 0 xT C1T C1 x 0 (1 0 d_)yT Q(t 0 d)y 0 yT C2T (t 0 d)C2(t 0 d)y _ + 2xT P B1 w 0 0 d yT C2T (t 0 d)C2 (t 0 d)y (8) 10

[xT yT wT ]T

and

where PT is a given weighting matrix for the terminal state, x(T ), and

C1 (t)x(t) z= C2 (t 0 d(t))x(t 0 d(t))

x(t 0 d(t)):

9

ATd P B1T P

M where

P Ad

0(1 0 d_)Q(t 0 d) 0

9 is as in (6), it follows

P B1

0

0

2

I

from (8) that

_ dU =  T M + 2 kwk2 0 kz k2 0 0 d yT C2T (t 0 d)C2 (t 0 d)y: dt 10 (9) Since x(t) = 0; T dU

d

0

8 t  0, it is readily obtained from (7) that

d

= xT (T )PT x(T ) +

T T 0d(T )

xT Q + (1 0 )01 C2T C2 x d:

By considering (9), this implies that T T J = T M d 0 xT Q + (1 0 )01 C2T C2 x d 0 T 0d(T ) T _ 0 0 d yT C2T ( 0 d)C2( 0 d)y d: (10) 0

10

Note that in view of (5), it results that M (t)  0; 8 t 2 [0; T ] and for all bounded d(t) that satisfies (4). Hence, it follows from (10) that J  0 for all w 2 L2 [0; T ]. Lemma 2.1 provides a bounded real criterion in terms of the DLMI of (5). Note that using Schur complements, (5) is equivalent to the Riccati differential inequality (RDI) P_ + AT P + P A + P 02 B1 B1T + (1 0 )01 Ad Q01 (t 0 d)AdT P + C1T C1 + (1 0 )01 C2T C2 + Q  0: (11) The result of Lemma 2.1 is most powerful, in the sense that it establishes a condition for the performance index J to be nonpositive independently of the delay length. Note that the DLMI, or the RDI, depends on the upper bound for the rate of change of the delay and not on the length, or an upper bound, of the delay. Such a condition should thus be, in general, conservative and will usually hold only for Ad of small norm. A less conservative condition that depends on the length of the delay is derived in the next lemma, for a specific d(t). Lemma 2.2: The performance index J of (2) is nonpositive for all w 2 L2 [0; T ], and for a given time-delay d(t), if there exist constant symmetric positive definite matrices P1 ; P2 ; and P3 ; with I 0 d(t)P3 > 0; 8 t 2 [0; T ], that allow a symmetric positive semidefinite solution P (t) over [0; T ], with P (T ) = PT , to the following RDI: P_ + (A + Ad )T P + P (A + Ad ) + d AT P1 A + ATd (t + d)P2 Ad (t + d) + P 02 B1 (I 0 dP3 )01 B1T + 0d + 02 Ad GATd P + C1T C1 + C2T C2  0 (12) where

G(t) and

0d

t t0d

B1 ( )P301 B1T ( ) d

dAd P101 + P201 ATd :

(13)

1018

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 7, JULY 1998

Proof: Considering the system of (1) subject to x( )  0 and w( ) = 0; 8  < 0, we find that for t  0

0; 8 

t

x(t 0 d) = x(t) 0 Thus, for t

t0d

0

[Ax + Ad x( 0 d)] d 0 t

x_ (t) = [A(t) + Ad (t)]x(t) 0 Ad (t)

1 d 0 Ad (t)

t t0d

B1 w d

t0d

t t0d

=

2 (t)

t0d t

3 (t)

t0d

=

0 d) d

xT (t)P (t)Ad (t)B1 ( )w( )d 0 xT (t)P (t)B1 (t)w(t):

2 0; 8 t 2 [0; T ] such that

P T (t)

8 t 2 [0; T ] (22) then it follows from (20) that J  0 for all w 2 L2 [0; T ]. Finally,

02zT y  zT X 01 z + yT Xy we find that for any n 2 n constant symmetric matrices P1 > 0 and P2 > 0, and m 2 m symmetric constant matrix P3 > 0 and P4 (t) = P4T (t) > 0; 8 t 2 [0; T ]

and

 T (t) (t) dt;

S (t)  0 and P4 (t) = I 0 d(t)P3 ;

xT (t)P (t)Ad (t)A( )x( ) d

Since for any z; y matrix X 2 0 that positive definite matrices P1 ; P2 ; and P3 , with I 0 dP allow a symmetric positive semidefinite solution P (t) over [0; T ], with P (T ) = PT , to the following RDI: P_ + (A + Ad )T P + P (A + Ad ) + d AT P1 A + ATd P2 Ad  3 )01 B1T + dA  d P101 + P201 ATd + P 02 B1 (I 0 dP + 02 Ad GATd P + C1T C1 + C2T C2  0: (23) In the case where the matrix B1 is time-invariant, using Schur complements, (12) and (23) can be easily transformed into DLMI’s. We readily obtain from Lemma 2.2 the following result. Corollary 2.2: Consider the system of (1) where the matrices Ad and B1 are assumed time-invariant. Given a scalar d > 0, the performance index J of (2) is nonpositive for all w 2 L2 [0; T ], and for all time-delay d(t) satisfying 0  d(t)  d, if there exist constant symmetric positive definite matrices P1 ; P2 ; and P3 ; with  3 > 0, and a matrix P (t) = P T (t)  0; 8 t 2 [0; T ], with I 0 dP P (T ) = PT , that solve the following DLMI:  Ad  Ad B1  Ad dP dP 8(P; P1 ; P2 ; d) dP P B1  Td P 1 dA 0 dP 0 0 0 2  Td P 0 0 dP 0 0 dA  3) B1T P 0 0 0 2 (I 0 dP 0 T T 3  0 0 0 0 2 dP dB1 Ad P  0; 8 t 2 [0; T ] (24)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 7, JULY 1998

where

8(P; P1 ; P2 ; d) = P_ + (A + Ad )T P + P (A + Ad ) + d AT P1 A + ATd P2 Ad + C1T C1 + C2T C2 : III. INFINITE HORIZON BOUNDED REAL CRITERIA This section addresses the problem of bounded realness on infinitehorizon. In this case, the matrices of the state-space model of (1) and (3) are constant; however, the time-delay d is still allowed to be timevarying. The input signal w is assumed to be an arbitrary signal in L2 [0; 1) and the performance index J of (2) is replaced by

J0 = kzk22 0 2 kwk22 (25) where is a given positive scalar and now k1k denotes the L2 [0; 1)-

norm. Throughout this section, (1) is said to be globally uniformly asymptotically stable (g.u.a.s.) if the equilibrium point x(t)  0 of the functional differential equation (1), with w(t)  0, is g.u.a.s. The problem we address here is to find conditions which will ensure that the system of (1) is g.u.a.s. and J0  0; 8 w 2 L2 [0; 1), subject to zero initial conditions and w( )  0; 8 < 0. A delay-independent criterion for J0 to be nonpositive, which is the stationary counterpart of Lemma 2.1, is as follows. Lemma 3.1: Consider the system of (1) and (3) under the assumption that all the matrices are constant. Given a scalar d > 0, this system is g.u.a.s., and the performance index J0 of (25) is nonpositive for all w 2 L2 [0; 1) and for all bounded time-delay d(t) that satisfies (4) if there exist symmetric positive definite matrices P and Q satisfying the following linear matrix inequalities (LMI’s):

90

ATd P

P Ad 0(1 0 )Q < 0

90 + C1T C1 + (1 0 )01 C2T C2

(26)

P Ad

P B1

0

0 2 I

0(1 0 )Q

ATd P B1T P

0

0 (27)

where

90

AT P

+ P A + Q:

(28)

1019

The time-derivative of V (t) along the state trajectory of system (1) is

V_ (t) = xT (t)(AT P + P A + Q)x(t) + 2xT (t)P Ad x(t 0 d) 0 (1 0 d_)x(t 0 d)T Qx(t 0 d)  [xT (t) xT (t 0 d)] A9T0P 0(1P0Ad )Q x(xt (0t)d) < 0 d where 90 is as in (28) and the last inequality follows from the LMI of (26). This, together with (29), implies that the system of (1) is g.u.a.s.; see e.g., [9]. In view of the global uniform asymptotic stability of (1), it follows that z belongs to L2 [0; 1) and then J0 is well defined. Thus, the proof that J0  0 can be carried out similarly to that of Lemma 2.1. Similarly to Lemma 2.1, the result of Lemma 3.1 is independent of the size of the time delay, i.e., the delay is allowed to be any bounded piecewise continuous function of time with a rate of change satisfying (4). The criterion of Lemma 3.1 is given in terms of LMI’s and thus has the following advantages: 1) it does not involve any parameter tuning and 2) it can be tested numerically very efficiently using interior point algorithms that have been recently developed for solving linear matrix inequalities; see, e.g., [14] and [15]. Remark 3.1: We note that the problem of finding the smallest

> 0, namely 0 , can be computed by solving the following convex optimization problem in P; Q; and  = 2 : minimize  subject to P > 0;

t

0

t d

xT ()Qx() d

 Ad  Ad dP 80 (P; P1 ; P2 ; d) dP  Td P 1 0 dA 0dP  Td P 2 dA 0 0dP

0 I 0 dP

where P and Q are symmetric positive definite matrices to be chosen. Note that

min (P )kx(t)k2  V (t)  [max (P ) + dmax(Q)]

p

The bound 0 is given by 0 = 3 , where  3 is the optimal value of the optimization problem. Note that this optimization problem can be solved numerically very efficiently (see [14] and [15]). A delay-dependent bounded real criterion is presented in the next lemma. Lemma 3.2: Consider the system of (1) and (3) under the assumption that all the matrices are constant. Given a scalar d > 0, this system is g.u.a.s. and the performance index J0 of (25) is nonpositive for all w 2 L2 [0; 1), and for all constant time-delay d satisfying 0  d  d, if there exist symmetric positive definite matrices P; P1 ; P2 ; and P3 satisfying the following LMI’s:

Proof: Let the following Lyapunov functional candidate exist for the system of (1):

V (t) = xT (t)P x(t) +

Q > 0;  > 0; (26); and (27):

(32)

where

sup kx()k

2 0

 [t d;t]

2

:

(29)

80 (P; P1 ; P2 ; d) = (A + Ad )T P + P (A + Ad ) + d AT P1 A + ATd P2 Ad :

 Ad dP  Ad  Ad B1 dP 80 (P; P1 ; P2 ; d) + C1T C1 + C2T C2 dP P B1 dATd P  0dP1 0 0 0  Td P 2 dA 0 0dP 0 0 0  3) 0 0 0 2 (I 0 dP 0 B1T P 3  1T ATd P 0 0 0 0 2 dP dB

(33)

(31)

1020

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 7, JULY 1998

Fig. 1. The amplitude Bode plot of

Tzw

for the first example.

Proof: First, note that by [13, Th. 3.1], the global uniform asymptotic stability of system (1) follows immediately from (30). This implies that z belongs to L2 [0; 1) and J0 is well defined. The proof that J0  0 then follows, similarly to that of Lemma 2.2. It should be noted that the assumption of a constant time delay is only required for the stability proof. Similarly to Lemma 2.1, the part of the proof of Lemma 3.2 related to the condition J0  0 also holds for a time-varying delay, as long as (1) is guaranteed to be g.u.a.s. In view of this, we have the following corollary. Corollary 3.1: Consider the system of (1) and (3) under the assumption that all the matrices are constant and (1) is g.u.a.s. Given a scalar d > 0, the performance index J0 of (25) is nonpositive for all w 2 L2 [0; 1) and for all time-delay d(t) satisfying 0  d(t)  d if there exist a symmetric matrix P  0 and symmetric positive definite matrices P1 ; P2 ; and P3 satisfying the LMI’s of (31) and (32). Remark 3.2: Similarly to Lemma 3.1, the criteria of Lemma 3.2 and Corollary 3.1 are given in terms of linear matrix inequalities. They have the advantages that no parameter tuning is required and can be tested numerically very efficiently using recently developed algorithms for solving linear matrix inequalities [14] and [15]. It should be observed that the problem of finding the largest d for a fixed , or the smallest for a fixed d, can easily solved without the need of tuning any parameter. Indeed, the largest value of d for a fixed > 0 can be computed by solving the following quasiconvex  optimization problem in P; P1 ; P2 ; P3 ; and d: maximize subject to

d P > 0; P1 > 0; P2 > 0; P3 > 0; d > 0; and (30)–(32).

On the other hand, the smallest > 0, namely 0 , for a fixed d > 0 can be found by solving the following quasiconvex optimization problem in P; P1 ; P2 ; P3 ; and  = 2 : minimize subject to

 P > 0; P1 > 0; P2 > 0; P3 > 0;  > 0; and (30)–(32):

p

The bound 0 is given by 0 = 3 , where  3 is the optimal value of the optimization problem. Note that the above optimization problems have the form of a generalized eigenvalue problem, which

is known to be solvable numerically very efficiently; see, e.g., [14] and [15]. IV. EXAMPLES We give two examples that demonstrate the use of the above theory. The first example compares the delay-independent result with the corresponding delay-dependent solution. It shows that for Ad of small norm, the former result is not much more conservative than the latter and may thus be preferred due to its simplicity. In the second example, we treat a case that cannot be solved by a delay-independent method. Example 4.1: Consider the system of (1)–(3) with a constant time delay and the following constant matrices (this is a modified version of an example of [3]):

A= B1 =

0

1

02 03

0 ; 1

;

Ad =

C1 = [1 0];

0

0:1

00:2 00:3 C2 = 0;

d = 0:4:

Applying Lemma 3.1 to this system we find that the minimum achievable value of is 0 = 0:556. To compare this 0 with the actual H1 -norm of the system, we depicted the Bode-plot (the system is single-input/single-output) of the transfer function

Tzw (s) = C1 (sI 0 A 0 Ad e0sd )01 B1 : The result for jT (jw)j; w 2 0 and Q = QT > 0 such that the conditions of Lemma 3.1 are satisfied for any  0. If we apply, however, Lemma 3.2 for d = 0:846 (which is very close to the maximum delay of 0.8571 for which [17] predicted global uniform asymptotic stability of the systems), the minimum achievable is 0 = 2. If we repeat the calculations for = 3, we find that the largest d for which the conditions of Lemma 3.2 are satisfied is 0.893, and for = 100 the largest achievable d is 0.98. The bound 0 and the corresponding Bode plot of Tzw for d = 0:846 are depicted in Fig. 2, where it is seen that in this example the bound of 0 is less tight than the one achieved in Example 4.1. We only note here that one of the reasons for the large difference between 0 and 0.2364, the actual peak of jTzw (j!)j, is the fact that as the system is guaranteed to be global uniform asymptotic stable, Corollary 3.1 guarantees the bound of 0 also for arbitrarily fast time-varying delays. V. CONCLUSION

Two lemmas have been introduced for evaluating the H -norm of linear time-varying systems with time delays. The first provides a sufficient condition for a given system to possess an H -norm that is less than or equal to a prescribed value. This condition guarantees, in fact, that in the time-invariant case the system will satisfy the required H -norm bound irrespective of the delay length. It would thus be satisfied only in cases where the state matrix Ad for the delayed state is of small norm.

1

1

1

1021

The second lemma introduces another sufficient condition which depends linearly on three positive definite matrix parameters that have to be found. Unlike the first lemma, the condition of this second lemma depends strongly on the length of the time delay, and for a delay that tends to zero this condition coincides with the standard bounded real lemma that has been derived for systems without delay. Both lemmas are given in terms of an RDI. The criteria that have been obtained for the finite horizon case reduce to linear matrix inequalities in the infinite horizon time-invariant case. An additional linear strict inequality should be solved, however, to secure the global uniform asymptotic stability of the system. REFERENCES [1] J. Doyle, K. Glover, P. Khargonekar, and B. Francis, “State-space solutions to standard 2 and control problems,” IEEE Trans. Automat. Contr., vol. 34, pp. 831–846, 1989. [2] M. Green and D. J. N. Limebeer, Linear Robust Control. Englewood Cliffs, NJ: Prentice-Hall, 1995. [3] M. Morari and E. Zafiriou, Robust Process Control. Englewood Cliffs, NJ: Prentice-Hall, 1989. [4] M. Malek-Zavarei and M. Jamshidi, Time Delay Systems: Analysis Optimization and Applications. Amsterdam, The Netherlands: North Holland, 1987. [5] O. J. M. Smith, “Closer control of loops with dead time,” Chem. Eng. Progress, vol. 53, no. 5, pp. 217–219, 1957. [6] A. W. Pila, U. Shaked, and C. E. de Souza, “ filtering for continuous-time linear systems with delay,” in Proc. 13th IFAC World Congr., San Francisco, CA, July 1996, vol. J, pp. 49–54. [7] A. Kojima and S. Ishijima, “Explicit formulas for operator Riccati equations arising in control with delays,” in Proc. 34th IEEE Conf. Decision and Control, New Orleans, LA, Dec. 1995, pp. 4175–4181. [8] G. Tadmor, “ control in systems with a single input delay,” in Proc. Amer. Control Conf., Seattle, WA, May 1995, pp. 312–325. [9] J. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993. [10] J. H. Lee, S. W. Kim, and W. H. Kwon, “Memoryless controllers for delayed systems,” IEEE Trans. Automat. Contr., vol. 39, pp. 159–162, 1994. [11] S. I. Niculescu, C. E. de Souza, J. M. Dion, and L. Dugard, “Robust memoryless control for uncertain linear systems with time-varying delay,” in Proc. European Control Conf., Rome, Italy, Sept. 1995, pp. 1814–1818. [12] L. Xie and C. E. de Souza, “Output feedback control of uncertain time-delay systems,” in Proc. European Control Conf., Gr¨oningen, The Netherlands, July 1993.

H

H1

H1

H1

H1

H1

H1

1022

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 7, JULY 1998

H1

[13] X. Li and C. E. de Souza, “Robust stabilization and control of uncertain linear time-delay systems,” in Proc. 13th IFAC World Congr., San Francisco, CA, July 1996, vol. H, pp. 113–118. [14] S. Boyd, L. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15. SIAM: Philadelphia, PA, 1994. [15] Y. Nesterov and A. Nemirovsky, Interior Point Polynomial Methods in Convex Programming, vol. 13. SIAM: Philadelphia, PA, 1994. [16] S. I. Niculescu, L. Dugard, and J. M. Dion, “Stabilit´e et stabilization robustes des syst`emes a` retard,” in Proc. Journ´ees Robustesse, Toulouse, France, Feb. 1995. [17] C. E. de Souza and X. Li, “Delay dependent stability of linear timedelay systems: An LMI approach,” in Proc. 3rd IEEE Mediterranean Symp. New Directions in Control and Automation, Limassol, Cyprus, July 1995, vol. 2, pp. 1–5.

tackled in [10]. The design of robust filters and controllers for both uncertain linear and nonlinear continuous-time systems using sampled measurements has been investigated in [11] and [12]. In this paper, we investigate an -filtering problem for a class of uncertain continuous-time systems under sampled measurements. The class of uncertain systems is described by a state-space model with real time-varying norm-bounded parameter uncertainties, unknown time delays in the state equation. Here attention is focused on the design of linear dynamic filters using local sampled measurements which guarantee the robust stability as well as a prescribed performance for the filtering error dynamics, irrespective of parameter uncertainties and unknown time delays. The performance measure we use is defined directly in the continuous-time context. It is assumed that the initial state of the uncertain system is unknown and both the cases of finite and infinite horizon filtering are considered. We show that the robust sampled-data filtering problems can be solved via scaled output feedback sampled-data control problems that incorporate unknown initial states and involve neither parameter uncertainties nor unknown time delays. Therefore, existing results on sampled-data control such as those in [6], [13], and [10] can be applied to obtain solutions to the problem of robust sampleddata filtering for the uncertain systems. Throughout this paper