Robust H control for uncertain discrete stochastic time-delay systems

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Abstract

This paper deals with the problems of robust stabilization and robust H control for discrete stochastic systems with time-varying delays and time-varying norm-bounded parameter uncertainties. For the robust stabilization problem, attention is focused on the design of a state feedback controller which ensures robust stochastic stability of the closed-loop system for all admissible uncertainties, while for the robust H control problem, a state feedback controller is designed such that, in addition to the requirement of the robust stochastic stability, a prescribed H performance level is also required to be satisfied. A linear matrix inequality (LMI) approach is developed to solve these problems, and delay-dependent conditions for the solvability are obtained. It is shown that the desired state feedback controller can be constructed by solving certain LMIs. An example is provided to demonstrate the effectiveness of the proposed approach.

Introduction

Control of time-delay systems is a subject of both practical and theoretical importance which has received much attention in the past years. A great number of results on system analysis and control synthesis of such systems have been proposed [5], [9], [13], [18]. Recently, the problem of robust H control for uncertain time-delay systems has been investigated by many researchers. For example, in terms of a modified algebraic Riccati equation, sufficient conditions for the existence of robust H state feedback controllers were presented in [8], while in [3], [19], observer-based H controllers were designed via an algebraic Riccati equation approach and a linear matrix inequality (LMI) approach, respectively. It is noted that the results in [3], [8], [19] are delay-independent, which are usually conservative, especially in the case when delays are small. The delay-dependent results on robust H control problem were reported in [2], [4], respectively, where an LMI approach was developed. Similar results on robust H control via state feedback controllers for uncertain discrete time-delay systems can be found in [12], and the references therein.

On the other hand, the study of stochastic systems with time delays has been of great interest since stochastic modeling has come to play an important role in many branches of science and engineering applications. A lot of results related to robust stability analysis and stabilization for stochastic time-delay systems have been reported in the literature. In [10], [11], several sufficient conditions ensuring mean square exponential stability for stochastic systems were given. The robust stabilization problem was studied in [15], where memoryless stabilizing state feedback controllers were designed via an LMI approach. Very recently, the problems of robust stabilization and robust H control for uncertain stochastic systems with time-varying delays have been studied in [16], in which sufficient conditions for the solvability of this problem were obtained and an LMI approach was proposed. It is worth pointing out that all the above mentioned results were obtained in the context of continuous stochastic time-delay systems. When discrete stochastic systems with time delays and parameter uncertainties are concerned, no results on robust H control for such systems have been available in the literature so far, which motivates the present study.

In this paper, we are concerned with the problems of robust stabilization and robust H control for uncertain discrete stochastic systems with time delay. The parameter uncertainties are assumed to be time-varying norm-bounded, and the system delay is supposed to be time-varying but bounded. The purpose of the robust stabilization is the design of a state feedback controller such that the resulting closed-loop system is robustly stochastically stable for all admissible uncertainties. For the robust H control problem, the purpose is the design of a state feedback controller such that, in addition to the requirement of the robust stochastic stability of the closed-loop system, a specified H performance level is also required to be achieved. Based on certain LMIs, sufficient conditions for the solvability of these problems are obtained, which are dependent on the upper bound of the time delay. When these LMIs are feasible, a desired state feedback controller gain can be constructed.

Section snippets

Definitions and problem formulation

Consider the following discrete stochastic system with time-varying delay and parameter uncertainties:(Σ):x(k+1)=A(k)x(k)+Ad(k)x(k−τ(k))+B1(k)u(k)+Dv(k)+[E(k)x(k)+Ed(k)x(k−τ(k))+B2(k)u(k)]ω(k),z(k)=Cx(k),x(i)=φ(i),i=−τ2,−τ2+1,…,0,where x(k)∈Rn is the state, u(k)∈Rm is the control input, z(k)∈Rl is the controlled output, ω(k) is a zero-mean real scalar process on a probability space (Ω,F,P) relative to an increasing family (Fk)k∈N of σ-algebras FkF generated by (ω(k))k∈N, where N is the set of

Robust stabilization

In this section, an LMI approach [1] will be developed to solve the robust stabilization problem formulated in the previous section. We first introduce the following lemma which will be used in the proof of our main results.

Lemma 1

Wang et al. [14]

Let A, D, S, W and F be real matrices of appropriate dimensions such that W>0 and FTFI. Then, for any scalar ε>0 such that W−1−ε−1DDT>0,(A+DFS)TW(A+DFS)AT(W−1−ε−1DDT)−1ASTS.

The result on the robust stochastic stability analysis for system (Σ) is provided in the following

Robust H control

In this section, attention will be focused on the design of a state feedback controller such that the resulting closed-loop system is robustly stochastically stable with disturbance attenuation level γ>0. We first present the following robust H performance analysis result.

Theorem 4

Given a constant scalar γ>0, the uncertain discrete stochastic time-delay system (Σ) with u(k)=0 is robustly stochastically stable with disturbance attenuation level γ if there exist matrices P>0, Q>0, and scalars ε1>0 and ε2

Numerical example

In this section, we shall present an example to demonstrate the effectiveness and applicability of the proposed method.

Consider the uncertain discrete stochastic time-delay system (Σ) with parameters as follows:A=−10.50.20.3,Ad=0.2−0.100.1,D=0.20.1,B1=2.2−0.800.5,E=00.10.20.1,Ed=−0.10.2−0.1−0.4,B2=0.81−0.62.5,M=0.10.2,N1=[0.10],N2=[0.20.1],N3=[00.1],N4=[0.20],N5=[0.10.1],N6=[0.10.3],C=[0.20.1].In this example, we suppose σ=1 and the time-varying delay τ(k) satisfies2⩽τ(k)⩽5.The noise

Conclusions

The problems of robust stabilization and robust H control for uncertain discrete stochastic systems with time-varying delay and time-varying norm-bounded parameter uncertainties have been studied. An LMI approach has been developed to design linear state feedback controllers such that both the robust stochastic stability and a prespecified H performance level of the closed-loop system are guaranteed for all admissible uncertainties. Some delay-dependent conditions for the solvability of the

Acknowledgements

The first and the second authors’ work was supported by RGC HKU 7103/01P. The third author's work was supported by the Natural Sciences and Engineering Research Council of Canada.

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