Further results concerning delay-dependent [H.sub.[infinity]] control for uncertain discrete-time systems with time-varying delay.(Research Article)(Report)

1. Introduction

During the past decades, considerable attention has been paid to the problems of stability analysis and control synthesis of time-delay systems. Many methodologies have been proposed and a large number of results have been established (see, e.g., [1-4] and the references therein). All these results can be generally divided into two categories: delay-independent stability conditions [5, 6] and delay-dependent stability conditions [7-12]. The delay-independent stability condition does not take the delay size into consideration, and thus is often conservative especially for systems with small delays, while the delay-dependent stability condition makes fully use of the delay information and is usually less conservative than the delay-independent one.

Up to now, the most important approach to deal with delay in the states of the systems is the use of Lyapunov-Krasovskii functionals, which has been largely employed to obtain convex conditions mainly for continuous-time systems subjected to retarded states. However, discrete-time systems with state delay have received little attention. This mainly because that for precisely known discrete-time systems with constant delay, it is always possible to derive a delay-free system by state augmentation [10, 11]. Although, such an approach is valid for the system with constant delays, it fails to deal with timevarying delay case, which is more frequently encountered than the constant case in practice. Recent results on discrete time-delay systems can be found in [13] where delay-dependent stability criteria were considered using a sum inequality. In [14], stability conditions for discrete time-delay systems were presented, while less conservative results were given in [15] by using a more general Lyapunov-Krasovskii functional than that in [14]. In [16], the authors summarized the recent results concerning robust stabilization of discrete-time systems with state delay. Sufficient LMI conditions were presented checking the robust stability for a class of linear discrete-time systems with time-varying delay and polytopic uncertainties; robust state feedback gains with memory were also designed. These results were mainly with the stability analysis and state feedback controller design. Very few people have investigated the delay-dependent [H.sub.[infinity]] control problem of discrete time-delay systems. In [17], the authors proposed an exponential output feedback [H.sub.[infinity]] controller. Delay-dependent robust [H.sub.[infinity]] control conditions for uncertain linear systems with lumped delays were given in [18], which were proved to be less conservative than some previous results. Also delay-dependent results were derived in [19] by combining a descriptor model transformation approach with Moon's bounding technique [9]. Very recently, in order to reduce the conservatism of the result in [19], a finite sum inequality approach was proposed in [20] and some less conservative [H.sub.[infinity]] control condition was derived. Although the result in [20] is superior to that in [19], it is still a sufficient condition and has conservatism to some extent, which leaves open room for further improvement.

Naturally, one may say that whether we can employ the similar Lyapunov functional, fewer variables, and reduced complexity of the algorithm to obtain less conservatism than the existing results. In this paper, we will further study the robust [H.sub.[infinity]] control problem for uncertain discrete-time system with time-varying delays. By introducing some slack matrix variables, new delay-dependent conditions for [H.sub.[infinity]] control problem are proposed in terms of LMI form, while no model transformation and bounding technique are employed. It is also shown that the complexity of the algorithm is considerably reduced and the result in this paper is less conservative than that in [18-20]. Numerical examples are finally provided to demonstrate the effectiveness of the main results.

Notations

Throughout this paper, [R.sup.n] represents the n-dimensional Euclidean space; [R.sup.mxn] is the set of all m x n real matrices. For real symmetric matrices X and Y, the notation X [greater than or equal to] Y (resp., X > Y) means that the matrix X - Y is positive semidefinite (resp., positive definite). The superscript "T" denotes the transpose. I is an identity matrix with appropriate dimension. [Z.sup.+] denotes the set of {0, 1, 2, ...}. [L.sub.2] refers to the space of square summable infinite vector sequences. In symmetric block matrices, we use an asterisk "*" to represent a term that is induced by symmetry. Matrices, if not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

2. Problem Formulation

Consider the following uncertain discrete-time systems with time-varying delay [20]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

where x(k) [member of] [R.sup.n], z(k) [member of] [R.sup.m], and z(k) [member of] [R.sup.p] are the state, control input, and controlled output, respectively; [omega](k) [member of] [R.sup.q] is the exogenous disturbance input, which belongs to [L.sup.2]. [phi](k) is the initial condition; [A.sup.0], [A.sup.1], [B.sup.1], [B.sup.2], [C.sub.0], [C.sup.1], [D.sup.11], and [D.sup.12] are known real constant matrices. The time-varying parameter uncertainties are norm-bounded and meet with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.2)

where F(k) is an unknown real time-varying matrix and satisfies the following bound condition:

FT (k)F(k) [greater than or equal to] I. (2.3)

[D.sub.1],[D.sub.2] ang [e.SUB.e] (l = 1,2,3,4) are known constant matrices of appropriate dimensions describing how the uncertainty F(k) enters the nominal matrices …