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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 291624, 17 pages
http://dx.doi.org/10.1155/2012/291624
Research Article

Robust Stabilization for Stochastic Systems with Time-Delay and Nonlinear Uncertainties

1School of Electrical Engineering and Automation, Shandong Polytechnic University, Jinan 250353, China
2School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China

Received 17 September 2011; Accepted 3 November 2011

Academic Editor: Weihai Zhang

Copyright © 2012 Zhiguo Yan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with the problem of robust stabilization of stochastic systems with time-delay and nonlinear uncertainties via memory state feedback. Based on Lyapunov krasoviskii functional, some sufficient conditions on local (global) stabilization are given in terms of matrix inequalities. In particular, these stabilizable conditions for a class of nonlinear stochastic time-delay systems are derived in the form of linear matrix inequalities, which have the advantage of easy computation. Moreover, the corresponding results are further extended to the stochastic multiple time-delays systems. Finally, an example is presented to show the superiority of memory state feedback controller to memoryless state feedback controller.

1. Introduction

Stochastic differential delay equations are one of the most useful stochastic models in applications, for example, aircraft, chemical or process control system, and distributed networks. It is known that time-delay is, in many cases, a source of poor system performance or instability. Hence, the stability and stabilization of stochastic time delay systems have been recently attracting the attention of a number of researchers, see [118] and the references therein. In [2], the authors studied the stability of linear stochastic systems with uncertain time delay by generalized Riccati equation approach, while [3] extended the results of [2] to nonlinear case via linear matrix inequalities. Reference [4] investigated the problems of stabilization for a class of linear stochastic systems with norm-bounded uncertainties and state delay, and it developed two criteria for the stability analysis: delay-dependent and delay-independent. The memoryless nonfragile state feedback control law for nonlinear stochastic time-delay systems was designed in [5], in which new sufficient conditions for the existence of such controllers were presented based on the linear matrix inequalities approach. Reference [6] was concerned with the stability analysis and proposed improved delay-dependent stability criteria for uncertain stochastic systems with interval time-varying delay. The study of exponential stability of stochastic delay-differential equations was discussed in [79]. The output feedback stabilization of stochastic nonlinear time-delay systems was investigated in [10], and some stabilization criterions for nonlinear stochastic time-delay systems with state and control-dependent noise were given in [11, 12] by means of matrix inequalities.

We usually design memoryless state feedback controller for the stabilization of systems because of its advantage of easy implementation. However, its performance, for time-delay systems, cannot be better than a memory state feedback controller which utilizes the available information of the size of delay. Reference [19] has given a general form of a memory state feedback (delayed feedback) controller: 𝑢(𝑡)=𝐺𝑥(𝑡)+𝑡𝑡𝜏𝐺1(𝑠)𝑥(𝑠)𝑑𝑠.(1.1) But the task of storing all the previous states 𝑥() and computing the values of time-varying gain matrices 𝐺1() makes the practical realization of infinite-dimensional controller (1.1) very difficult. For these reasons, the controller 𝑢(𝑡)=𝐺𝑥(𝑡)+𝐺2𝑥(𝑡𝜏)(1.2) could be considered as a compromise between the performance improvement and the implementation simplicity. Reference [20] gave the sufficient conditions for the stabilization of deterministic state-delayed systems. References [21] and [22] designed a memory state feedback controller for neutral time-delay systems and singular timedelay systems, respectively. Reference [23] studied the stabilization problem for a class of discrete-time Markovian jump linear systems with time-delays both in the system state and in the mode signal via time-delayed controller and obtained a sufficient condition. What [13] actually studied is the stabilization problem of linear stochastic time-delay systems using generalized Riccati equation method. Up to now, to the best of the authors’ knowledge, the issue on memory state feedback stabilization of stochastic systems with time-delay and nonlinear uncertainties has not been fully investigated in previous literatures.

In this paper, we consider the problem on robust stabilization for stochastic systems with time-delay and nonlinear uncertainties via memory state feedback. This problem contains three inevitable aspects of practical application: timedelay, nonlinear uncertainties and more effective controller, which is more complex than the stabilization of pure stochastic systems via memoryless control. These complexities result in some difficulties of memory stabilizing controller design. By the Itô formula, mathematical expectation properties, and matrix transformation, some sufficient conditions are obtained on locally and globally asymptotic stabilization in probability by means of matrix inequalities. Especially for a class of nonlinear stochastic time-delay systems, a sufficient condition for the existence of memory state feedback stabilizing controller is obtained in terms of LMIs, which has the advantage of easy computation. Meanwhile, a memoryless state feedback controller is also given as a special case of memory state feedback controller. Moreover, the robust stabilization problem for stochastic multiple time-delays systems is further studied and a general sufficient condition is derived.

The paper is organized as follows. Some preliminaries and problem formulations are presented in Section 2. In Section 3, main results are given. Section 4 presents one example to illustrate the effectiveness of our developed results. Section 5 concludes this paper.

Notation 1. 𝐴: the transpose of matrix 𝐴; 𝐴0(𝐴>0): 𝐴 is positive semidefinite (positive definite) symmetric matrix; 𝐼: identity matrix; : Euclidean norm; 𝐿2([0, ), 𝐑𝑙): space of nonanticipative stochastic process 𝑦(𝑡)𝐑𝑙 with respect to an increasing 𝜎-algebra 𝑡(𝑡0) satisfying 𝐸0𝑦(𝑡)2𝑑𝑡<. 𝐼𝑛×𝑛: 𝑛×𝑛 identity matrix.

2. Preliminaries and Problem Statement

Consider the following continuous nonlinear stochastic time-delay systems: 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡𝜏)+𝐵1𝑢(𝑡)+𝐻0+(𝑥(𝑡),𝑥(𝑡𝜏),𝑢(𝑡))𝑑𝑡𝐶𝑥(𝑡)+𝐷𝑥(𝑡𝜏)+𝐷1𝑢(𝑡)+𝐻1(𝑥(𝑡),𝑥(𝑡𝜏),𝑢(𝑡))𝑑𝑤(𝑡),𝑥(𝑡)=(𝑡)𝐿2𝜔,0[],𝐶(𝜏,0,𝑅𝑛)[],,𝑡𝜏,0(2.1) where 𝑥(𝑡)𝐑𝑛 and 𝑢(𝑡)𝐑𝑚 are system state and control input, respectively; 𝑤(𝑡) is 1-dimensional standard Wiener process defined on the probability space (Ω, , 𝑡, 𝑃) with 𝐹𝑡=𝜎{𝑤(𝑠)0𝑠𝑡}; 𝐻𝑖(0,,)=0, 𝑖=0,1; 𝐴, 𝐵, 𝐵1, 𝐶, 𝐷, and 𝐷1 are constant matrices; 𝜏>0 is a certain timedelay. Under very mild conditions on 𝐻𝑖(0,,), 𝑖=0, 1, (2.1) exists a unique global solution [1]. It should be pointed out that any general nonlinear stochastic system which is sufficiently differentiable can take the form of (2.1) via Taylor’s series expansion at the origin.

Next, we give the following definitions essential for the paper.

Definition 2.1 (see [1]). System (2.1) with 𝑢(𝑡)=0 is said to be stable in probability, if for any 𝜖>0, lim𝑥0𝑃sup𝑡0𝑥(𝑡)>𝜖=0.(2.2) Additionally, if we also have lim𝑥00𝑃lim𝑡𝑥(𝑡)=0=1,(2.3) then system (2.1) with 𝑢(𝑡)=0 is said to be locally asymptotically stable in probability.
If (2.2) holds and𝑃lim𝑡𝑥(𝑡)=0=1(2.4) for all 𝑥0𝐑𝑛, then system (2.1) with 𝑢(𝑡)=0 is said to be globally asymptotically stable in probability.

Definition 2.2. If there exists a constant memory state feedback control law 𝑢(𝑡)=𝐾1𝑥(𝑡)+𝐾2𝑥(𝑡𝜏),(2.5) such that the equilibrium point of the closed-loop system 𝑑𝑥(𝑡)=𝐴+𝐵1𝐾1𝑥(𝑡)+𝐵+𝐵1𝐾2𝑥(𝑡𝜏)+𝐻0𝑥(𝑡),𝑥(𝑡𝜏),𝐾1𝑥(𝑡)+𝐾2𝑥+(𝑡𝜏)𝑑𝑡𝐶+𝐷1𝐾1𝑥(𝑡)+𝐷+𝐷1𝐾2𝑥(𝑡𝜏)+𝐻1𝑥(𝑡),𝑥(𝑡𝜏),𝐾1𝑥(𝑡)+𝐾2𝑥[](𝑡𝜏)𝑑𝑤(𝑡),𝑥(𝑡)=(𝑡),𝜏,0(2.6) is asymptotically stable in probability [1] for all 𝜏>0, then stochastic time-delay differential system (2.1) is called locally robustly stabilizable. If (2.6) is robustly stable [2], that is, the equilibrium point of (2.6) is asymptotically stable in the large [1] for all 𝜏>0, (2.1) is globally robustly stabilizable.

Remark 2.3. Definition 2.2 gives locally (globally) robustly stabilizable of stochastic time-delay systems via memory state feedback control law 𝑢(𝑡)=𝐾1𝑥(𝑡)+𝐾2𝑥(𝑡𝜏), which is more general than that via memoryless state feedback control law [12]. This is because Definition 2.2 reduces to the corresponding definition under memoryless state feedback control law when 𝐾2=0.

The aim of this paper is to find a constant memory state feedback control law (2.5), such that the equilibrium point of (2.6) is asymptotically stable in probability for all 𝜏>0.

3. Main Results

In this section, we will give some sufficient conditions of the stabilization of system (2.1). Without loss of generality, we can give the following assumption for nonlinear function 𝐻𝑖.

Assumption 3.1. There exists an 𝜖>0, such that𝐻sup𝑖𝑥,𝑦,𝐾1𝑥+𝐾2𝑦(𝜖𝑥+𝑦),𝑖=0,1,(3.1) holds for all 𝑥,𝑦𝐔 (a neighborhood of the origin).

The following general theorem is presented, which yields several applicable corollaries.

Theorem 3.2. If (3.1) holds and 𝐾1, 𝐾2𝐑𝑚×𝑛, 𝑃>0, 𝑄>0 are the solutions of the following matrix inequality 𝑍+𝑍1<0,(3.2) then system (2.1) can be locally robustly stabilized by (2.5). If 𝐔 is replaced by 𝐑𝑛, then system (2.1) can be globally robustly stabilized by the same controller.
In (3.2), 𝑍 and 𝑍1 are defined byΣ𝑍=1Σ2𝐷+𝐷1𝐾2𝑃𝐷+𝐷1𝐾2,𝑍𝑄1=Σ300Σ4,(3.3) where Σ1=𝑃𝐴+𝐵1𝐾1+𝐴+𝐵1𝐾1𝑃+𝑄+𝐶+𝐷1𝐾1𝑃𝐶+𝐷1𝐾1,Σ2=𝑃𝐵+𝐵1𝐾2+𝐶+𝐷1𝐾1𝑃𝐷+𝐷1𝐾2,Σ3𝐷=𝜖3+3𝐶+31𝐾1𝐷+3𝜖+𝐷+𝜖1𝐾2Σ𝑃𝐼,4𝐷=𝜖3+𝐷+1𝐾2+𝐶+𝐷1𝐾1+2𝜖𝑃𝐼.(3.4)

Proof. Choose the following Lyapunov-Krasoviskii functional: 𝑉(𝑡,𝑥)=𝑥(𝑡)𝑃𝑥(𝑡)+𝜏0𝑥(𝑡𝑠)𝑄𝑥(𝑡𝑠)𝑑𝑠,(3.5) where 𝑃>0 and 𝑄>0 are the solutions of (3.2). Let be the infinitesimal generator of the closed-loop system (2.6), then, by the Itô’s formula, we have 𝑉(𝑡,𝑥(𝑡))=𝑥𝑃(𝑡)𝐴+𝐵1𝐾1+𝐴+𝐵1𝐾1𝑃+𝑄+𝐶+𝐷1𝐾1𝑃𝐶+𝐷1𝐾1𝑥(𝑡)+2𝑥𝑃(𝑡)𝐵+𝐵1𝐾2+𝐶+𝐷1𝐾1𝑃𝐷+𝐷1𝐾2𝑥(𝑡𝜏)+𝑥(𝑡𝜏)𝐷+𝐷1𝐾2𝑃𝐷+𝐷1𝐾2𝑄𝑥(𝑡𝜏)+2𝐻0𝑃𝑥(𝑡)+2𝐻1𝑃𝐷+𝐷1𝐾2𝑥(𝑡𝜏)+2𝐻1𝑃𝐶+𝐷1𝐾1𝑥(𝑡)+𝐻1𝑃𝐻1=𝑥(𝑡)𝑥(𝑡𝜏)𝑍𝑥(𝑡)𝑥(𝑡𝜏)+2𝐻0𝑃𝑥(𝑡)+𝐻1𝑃𝐻1+2𝐻1𝑃𝐷+𝐷1𝐾2𝑥(𝑡𝜏)+2𝐻1𝑃𝐶+𝐷1𝐾1𝑥(𝑡).(3.6) In addition, by (3.1), we obtain 2𝐻0𝑃𝑥(𝑡)+2𝐻1𝑃𝐷+𝐷1𝐾2𝑥(𝑡𝜏)+2𝐻1𝑃𝐶+𝐷1𝐾1𝑥(𝑡)+𝐻1𝑃𝐻1𝐷2𝜖𝑃1+𝐶+1𝐾1+𝜖𝑥(𝑡)2𝐷+2𝜖𝑃1+𝐶+1𝐾1𝐷+𝐷+1𝐾2++𝜖𝑥(𝑡)𝑥(𝑡𝜏)2𝜖+𝜖2(𝑃𝑥𝑡𝜏)2.(3.7) By inequality |𝑎𝑏|(1/2)(𝑎2+𝑏2), then (3.7) becomes 2𝐻0𝑃𝑥(𝑡)+2𝐻1𝑃𝐷+𝐷1𝐾2𝑥(𝑡𝜏)+2𝐻1𝑃𝐶+𝐷1𝐾1𝑥(𝑡)+𝐻1𝑃𝐻1𝐷𝜖3+3𝐶+31𝐾1𝐷+3𝜖+𝐷+1𝐾2(𝑃𝐼𝑥𝑡)2𝐷+𝜖3+𝐷+1𝐾2+𝐾2𝐶+𝐷1𝐾1+2𝜖𝑃𝐼𝑥(𝑡𝜏)2=𝑥(𝑡)𝑥(𝑡𝜏)𝑍1.𝑥(𝑡)𝑥(𝑡𝜏)(3.8) Substituting (3.8) into (3.6), it follows 𝑉(𝑡,𝑥(𝑡))𝑥(𝑡)𝑥(𝑡𝜏)𝑍+𝑍1𝑥(𝑡)𝑥(𝑡𝜏).(3.9) According to (3.2), that is 𝑉(𝑡,𝑥(𝑡))<0 in the domain {𝑡>0}×𝐔 for 𝑥0, so the local stabilization of Theorem 3.2 is obtained by Corollary of [1]. By the same discussion, the global stabilization conditions can be also obtained by Theorem 4.4 of [1].

From Theorem 3.2, we can derive some useful results, which can be expressed in terms of LMIs.

Corollary 3.3. If 𝐻𝑖0, 𝑖=0, 1, and the matrix inequality 𝑍<0 has solutions 𝑃>0, 𝑄>0 and 𝐾1, 𝐾2𝐑𝑚×𝑛, then the linear stochastic time-delay system 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡𝜏)+𝐵1𝑢(𝑡)𝑑𝑡+𝐶𝑥(𝑡)+𝐷𝑥(𝑡𝜏)+𝐷1𝑢(𝑡)𝑑𝑤(𝑡)(3.10) is globally robustly stabilizable. If 𝐷=0, 𝐷1=0, and the following LMI: Σ5𝑃𝑃𝐶𝐵1𝑋𝐶𝑃𝑄00𝑃0𝑃0𝑋𝐵1𝑄00<0(3.11) has solutions 𝑃>0, 𝑌𝐑𝑚×𝑛, 𝑋𝐑𝑛×𝑚, and 𝑄>0, then 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡𝜏)+𝐵1𝑢(𝑡)𝑑𝑡+𝐶𝑥(𝑡)𝑑𝑤(𝑡)(3.12) is globally robustly stabilizable, where Σ5𝑃=𝐴+𝑃𝐴+𝐵1𝑌+𝑌𝐵1+𝐵𝑄𝐵+𝐵𝑋𝐵1+𝐵1𝑋𝐵, and a stabilizing feedback control law 𝑢(𝑡)=𝐾1𝑥(𝑡)+𝐾2𝑃𝑥(𝑡𝜏)=𝑌1𝑄𝑥(𝑡)+𝑋1𝑥(𝑡𝜏).

Proof. If 𝐻𝑖(,,)=0, 𝑖=0,1, we can take 𝜖=0 in (3.1), then 𝑉(𝑡,𝑥(𝑡))<0 for (𝑡,𝑥)𝑡>0×𝐑𝑛, except possibly at 𝑥=0.
Thus, the first part of Corollary 3.3 is proved.
Furthermore, if 𝐷=0, 𝐷1=0, (3.2) degenerates intoΣ𝑍=6𝑃𝐵+𝐵1𝐾2𝐵+𝐵1𝐾2𝑃𝑄<0,(3.13) where Σ6=𝑃(𝐴+𝐵1𝐾1)+(𝐴+𝐵1𝐾1)𝑃+𝑄+𝐶𝑃𝐶.
According to Schur’s complement, (3.13) is equivalent toΣ6+𝑃𝐵+𝐵1𝐾2𝑄1𝐵+𝐵1𝐾2𝑃<0.(3.14) Then, pre- and post-(3.14) by 𝑃1, we have 𝑃1Σ6𝑃1+𝐵+𝐵1𝐾2𝑄1𝐵+𝐵1𝐾2<0.(3.15) Setting 𝑃=𝑃1, 𝑌=𝐾1𝑃1, 𝑄=𝑄1, and 𝑋=𝐾2𝑄1. Again, by Schur’s complement, (3.15) is equivalent to (3.11). Thus, the second part of Corollary 3.3 is also proved.

Remark 3.4. Reference [13] considered the analogous problem to Corollary 3.3 by delay feedback, where the main result is expressed by means of generalized algebraic Riccati equations (GAREs) GAREs. However, Corollary 3.3 gives a sufficient condition in terms of LMIs which are easy to be solved.

Corollary 3.5. If the following LMI: Γ12𝑃𝐶𝑃𝐵002𝐶𝑃𝑃0000𝑃0𝐼000𝐵00𝐼2𝐷𝐾2𝐵10002𝐷𝑃0000𝐵1𝐾2𝑃0<0,(3.16) has solution 𝑃>0, 𝑌, and 𝐾2, then the stochastic linear time-delay controlled system 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡𝜏)+𝐵1𝑢(𝑡)𝑑𝑡+𝐶𝑥(𝑡)+𝐷𝑥(𝑡𝜏)+𝐷1𝑢(𝑡)𝑑𝑤(𝑡)(3.17) is globally robustly stabilizable, where Γ1𝑃=𝐴+𝑃𝐴+𝐵1𝑌+𝑌𝐵1+𝑃. Moreover, the stabilizing feedback control law 𝑃𝑢(𝑡)=𝑌1𝑥(𝑡)+𝐾2𝑥(𝑡𝜏).(3.18)

Proof. Applying the well-known inequality 𝑋𝑌+𝑌𝑋𝛾𝑋𝑋+𝛾1𝑌𝑌,𝛾>0,(3.19) and supposing 𝛾=1 for simplicity, we have 2𝑥(𝑡)𝑃𝐵1𝐾2𝑥(𝑡𝜏)+2𝑥(𝑡)𝐶+𝐷1𝐾1𝑃𝐷+𝐷1𝐾2𝑥(𝑡𝜏)𝑥(𝑡)𝑃+𝐶+𝐷1𝐾1𝑃𝐶+𝐷1𝐾1𝑥(𝑡)+𝑥𝐾(𝑡𝜏)2𝐵1𝑃𝐵1𝐾2+𝐷+𝐷1𝐾2𝑃𝐷+𝐷1𝐾2𝑥(𝑡𝜏).(3.20) Let Γ1=Σ1+𝑃+(𝐶+𝐷1𝐾1)𝑃(𝐶+𝐷1𝐾1), Γ2=2(𝐷+𝐷1𝐾2)𝑃(𝐷+𝐷1𝐾2)+𝐾2𝐵1𝑃𝐵1𝐾2𝑄. Then, Γ𝑍1𝑃𝐵Γ2=Γ.(3.21) Obviously, if Γ<0, then 𝑍<0. Applying the Theorem 3.2, the closed-loop system of (3.17) is robustly stable [2].
Then, pre- and post-multiplying Γ<0 by diag{𝑃1,𝐼}, and by Schur’s complement, we have Γ<0 is equivalent toΓ12𝑃1𝐶𝑃1𝐵002𝐶𝑃1𝑃1𝑃000010𝑄1000𝐵00𝑄2𝐷𝐾2𝐵10002𝐷𝑃10000𝐵1𝐾20𝑃1<0,(3.22) where Γ1=𝐴𝑃1+𝑃1𝐴+𝐵1𝐾1𝑃1+𝑃1𝐾1𝐵1+𝑃1. Set 𝑃=𝑃1, 𝑌=𝐾1𝑃1=𝐾1𝑃, 𝑄=𝐼, (3.22) is equivalent to (3.16). This ends the proof of Corollary 3.5.

Below, for 𝐷=0, 𝐷1=0, we give another sufficient condition for the local (global) stabilization of system (2.1) in the terms of LMIs.

Theorem 3.6. For 𝐷=0, 𝐷1=0 in (2.1), suppose (3.1) holds for all 𝑥,𝑦𝐔(𝑥,𝑦𝐑𝑛). If the LMIs: Π1𝑃𝑃2𝑃𝐶𝐵+𝐵1𝐾2𝛼𝑃𝑄000𝑃06𝜖2𝐼002𝐶𝑃00𝑃0𝐵+𝐾2𝐵1000𝜖2𝐼<0,(3.23)𝑃𝛼𝐼,(3.24)𝑄𝛼𝐼7𝜖2,(3.25)0<𝛼1,(3.26) have solutions 𝑃>0, 𝛼, 𝑄>0, 𝐾2, and 𝑌𝐑𝑚×𝑛, then system (2.1) can be locally (globally) robustly stabilized by 𝑃𝑢(𝑡)=𝑌1𝑥(𝑡)+𝐾2𝑥(𝑡𝜏),(3.27) where Π1=𝐴𝑃+𝑃𝐴+𝐵1𝑌+𝑌𝐵1+𝑃.

Proof. Applying the well-known inequality (3.19) again and supposing 𝛾=1 for simplicity, we have (if 0<𝑃𝐼/𝛼 for some 𝛼>0) 2𝐻0𝑃𝑥(𝑡)+2𝐻1𝑃𝐶𝑥(𝑡)+𝐻1𝑃𝐻16𝜖2𝛼(𝑥𝑡)2+𝑥(𝑡𝜏)2+𝑥(𝑡)(𝑃+𝐶𝑃𝐶)𝑥(𝑡)(3.28) which holds because 2𝐻0𝑃𝑥=𝐻0𝑃1/2𝑃1/2𝑥+𝑥𝑃1/2𝑃1/2𝐻0𝐻0𝑃𝐻0+𝑥𝑃𝑥2𝜖2𝛼𝑥(𝑡)2+𝑥(𝑡𝜏)2𝐻+𝑥𝑃𝑥,1𝑃𝐻12𝜖2𝛼𝑥(𝑡)2+𝑥(𝑡𝜏)2,2𝐻1𝑃𝐶𝑥=𝐻1𝑃1/2𝑃1/2𝐶𝑥+𝑥𝐶𝑃1/2𝑃1/2𝐻12𝜖2𝛼(𝑥𝑡)2+𝑥(𝑡𝜏)2+𝑥(𝑡)𝐶𝑃𝐶𝑥(𝑡).(3.29) Substituting (3.28) into (3.6), it follows that 𝑉(𝑡,𝑥(𝑡))𝑥(𝑡)𝑥(𝑡𝜏)𝑍𝑥(𝑡)𝑥(𝑡𝜏),(3.30) where Π𝑍=1+6𝛼𝜖2𝐼𝑃𝐵+𝐵1𝐾26𝛼𝜖2𝐼𝑄.(3.31) Considering (3.24), (3.25), and (3.26), it follows that Π𝑍1+6𝛼𝜖2𝐼𝑃𝐵+𝐵1𝐾2𝜖2𝐼.(3.32) Let 𝑍1=Π1+6𝛼𝜖2𝐼𝑃𝐵+𝐵1𝐾2𝜖2𝐼,(3.33) where Π1=𝑃(𝐴+𝐵1𝐾1)+(𝐴+𝐵1𝐾1)𝑃+𝑄+𝑃+2𝐶𝑃𝐶.
Obviously, if 𝑍1<0, then 𝑍<0. So if (3.1) holds for all 𝑥𝐔(𝑥𝐑𝑛), and 𝑍<0, then system (2.1) can be locally (globally) robustly stabilized by 𝑢(𝑡)=𝐾1𝑥(𝑡)+𝐾2𝑥(𝑡𝜏).
Note that 𝑍1<0 is equivalent to that𝑃𝐴+𝐵1𝐾1+𝐴+𝐵1𝐾16𝑃+𝑄+𝑃+2𝐶𝑃𝐶+𝛼𝜖2𝐼+𝑃𝐵+𝐵1𝐾2𝜖2𝐼𝐾2𝐵1+𝐵𝑃<0.(3.34) Then pre- and postmultiply (3.34) by 𝑃1, we have 𝐴+𝐵1𝐾1𝑃1+𝑃1𝐴+𝐵1𝐾1+𝑃1𝑄𝑃1+𝑃1+2𝑃1𝐶𝑃𝐶𝑃1+𝑃16𝛼𝜖2𝐼𝑃1+𝐵+𝐵1𝐾2𝜖2𝐼𝐾2𝐵1+𝐵<0.(3.35) Setting 𝑃=𝑃1, 𝑌=𝐾1𝑃1=𝐾1𝑃, and 𝑄=𝑄1 by the Schur’s complement, (3.35) is equivalent to (3.23). Thus, the theorem is proved.

In the special case when 𝐾2=0, our results reduce the corresponding results in memoryless state feedback case. The following theorem gives a sufficient condition for the existence of memoryless state feedback controller of system (2.1) with 𝐷=0, 𝐷1=0.

Theorem 3.7. For 𝐷=0, 𝐷1=0 in (2.1), suppose there exists an 𝜖>0, sup𝐻𝑖𝑥,𝑦,𝐾1𝑥𝜖(𝑥+𝑦),𝑖=0,1,(3.36) holds for all 𝑥, 𝑦𝐔(𝑥,𝑦𝐑𝑛), if the LMIs Π1𝑃𝑃2𝛼𝑃𝐶𝐵𝑃𝑄000𝑃06𝜖2𝐼002𝐶𝑃00𝑃0𝐵000𝜖2𝐼<0(3.37) and (3.24), (3.25), and (3.26) have solutions 𝑃>0, 𝑄>0, 𝛼, and 𝑌𝐑𝑚×𝑛, then system (2.1) can be locally (globally) robustly stabilized by 𝑃𝑢(𝑡)=𝑌1𝑥(𝑡).(3.38)

Proof. It is derived by the same procedure as the proof of Theorem 3.6.

By the above discussion about stochastic systems with single delay (2.1), we further study robust stabilization for the following stochastic systems with multiple delays𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝑞𝑗=1𝐵𝑗𝑥𝑡𝜏𝑗+𝑞𝑗=1𝐵1𝑗𝑢𝑗(𝑡)+𝐻0𝑥(𝑡),𝑥𝑡𝜏𝑗,𝑢𝑗+(𝑡)𝑑𝑡𝐶𝑥(𝑡)+𝑞𝑗=1𝐷𝑗𝑥𝑡𝜏𝑗+𝑞𝑗=1𝐷1𝑗𝑢𝑗(𝑡)+𝐻1𝑥(𝑡),𝑥𝑡𝜏𝑗,𝑢𝑗𝑥(𝑡)𝑑𝑤(𝑡),(𝑡)=(𝑡)𝐿2𝜔,0([],𝐶,0,𝑅𝑛)[],,𝑡,0(3.39) where 𝜏𝑗>0, 𝑗=1,,𝑞, denote the state delay; =max{𝜏𝑗,𝑗[1,𝑞]}.

For system (3.39), the following memory state feedback control law is adopted:𝑢𝑗(𝑡)=𝐾𝑗1𝑥(𝑡)+𝐾𝑗2𝑥𝑡𝜏𝑗,𝑗=1,,𝑞.(3.40) Applying control law (3.40) to system (3.39), the resulting closed-loop system is given by𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝑞𝑗=1𝐵𝑥𝑡𝜏𝑗+𝐻0𝑥(𝑡),𝑥𝑡𝜏𝑗,𝐾𝑗1𝑥(𝑡)+𝐾𝑗2𝑥𝑡𝜏𝑗+𝑑𝑡𝐶𝑥(𝑡)+𝑞𝑗=1𝐷𝑥𝑡𝜏𝑗+𝐻1𝑥(𝑡),𝑥𝑡𝜏𝑗,𝐾𝑗1𝑥(𝑡)+𝐾𝑗2𝑥𝑡𝜏𝑗𝑥𝑑𝑤(𝑡),(𝑡)=(𝑡)𝐿2𝜔,0([],𝐶,0,𝑅𝑛)[],,𝑡,0(3.41) where 𝐴=𝐴+𝑞𝑗=1𝐵1𝑗𝐾𝑗1, 𝐵=𝐵𝑗+𝐵1𝑗𝐾𝑗2, 𝐶=𝐶+𝑞𝑗=1𝐷1𝑗𝐾𝑗1, 𝐷=𝐷𝑗+𝐷1𝑗𝐾𝑗2.

By the same analysis as Theorem 3.2, we obtain the following theorem which gives a general sufficient condition for the robust stabilization of stochastic multiple time-delays system (3.39).

Theorem 3.8. If (3.1) holds, and 𝐾𝑗1, 𝐾𝑗2, 𝑃>0, and 𝑄>0 are the solutions of the following matrix inequality 𝑍0+𝑍1<0,(3.42) then system (3.39) can be locally robustly stabilized by 𝑢𝑗(𝑡)=𝐾𝑗1𝑥(𝑡)+𝐾𝑗2𝑥(𝑡𝜏𝑗). Especially if 𝐔 is replaced by 𝐑𝑛, then system (3.39) can be globally robustly stabilized by the same controller.
In (3.42), 𝑍0 and 𝑍1 are defined by𝑍0=𝑍011𝑍012𝑍022,𝑍1=𝑍11100𝑍122,(3.43) where 𝑍011=𝑃𝐴+𝑞𝑗=1𝐵1𝑗𝐾𝑗1+𝐴+𝑞𝑗=1𝐵1𝑗𝐾𝑗1+𝑃+𝑄𝐶+𝑞𝑗=1𝐷1𝑗𝐾𝑗1𝑃𝐶+𝑞𝑗=1𝐷1𝑗𝐾𝑗1,𝑍012=𝑃𝑞𝑗=1𝐵𝑗+𝐵1𝑗𝐾𝑗2+𝐶+𝑞𝑗=1𝐷1𝑗𝐾𝑗1𝑃𝑞𝑗=1𝐷𝑗+𝐷1𝑗𝐾𝑗2,𝑍022=𝑞𝑗=1𝐷𝑗+𝐷1𝑗𝐾𝑗2𝑞𝑗=1𝐷𝑗+𝐷1𝑗𝐾𝑗2𝑍𝑄,111=𝜖3+3𝐶+3𝑞𝑗=1𝐷1𝑗𝑞𝑗=1𝐾𝑗1++3𝜖𝐷+𝜖𝑞𝑗=1𝐷1𝑗𝑞𝑗=1𝐾𝑗2𝑍𝑃𝐼,122=𝜖3+𝐷+𝑞𝑗=1𝐷1𝑗𝑞𝑗=1𝐾𝑗2++𝐶𝑞𝑗=1𝐷1𝑗𝑞𝑗=1𝐾𝑗1+2𝜖𝑃𝐼.(3.44)

Remark 3.9. From Theorem 3.7, some useful results can be easily derived for stochastic multiple time-delays systems (3.39), which are similar to the results obtained for stochastic single time-delay systems (2.1).

4. Numerical Example

Now, we present one example to illustrate the effectiveness of our developed result (Theorem 3.6) in testing the stabilization of nonlinear stochastic time-delay system (2.1).

In (2.1), take 𝐷=0, 𝐷1=0, and,𝐵𝐴=5.002.231.562.15,𝐵=0.240.891.220.761=,𝐻2.254.48,𝐶=0.050.150.150.100=sin𝑢(𝑡)𝑥2𝑥(𝑡𝜏)1(𝑡)cos𝑢(𝑡)𝑥1𝑥(𝑡𝜏)2,𝐻(𝑡)1=exp𝑢(𝑡)+𝑥1(𝑡𝜏)+𝑥2(𝑡𝜏)2𝑥2(𝑢𝑡)exp2(𝑡)𝑥21𝑥(𝑡𝜏)1,(𝑡)𝜙(0)=108,𝜏=0.5.(4.1) Obviously, (2.1) holds for all 𝑥𝐑𝑛 with 𝜖=1.

Case 1 (Memory State Feedback Stabilization). Substituting all the above data into (3.23) and then solving the LMIs (3.23), (3.24), (3.25), and (3.26) by LMIs Toolbox, we can obtain solutions ,𝐾𝑃=0.46250.06260.06260.3383>0,𝛼=0.9015,𝑌=0.13770.66742=,0.23910.2149𝑄=0.11410.00520.00520.0974>0.(4.2) So by Theorem 3.6, system (2.1) can be globally robustly stabilized by 𝑃𝑢(𝑡)=𝑌1𝑥(𝑡)+𝐾2𝑥(𝑡𝜏)=0.5790𝑥1(𝑡)2.0795𝑥2(𝑡)0.2391𝑥1(𝑡𝜏)+0.2149𝑥2(𝑡𝜏).(4.3) The state trajectories of close-loop system (2.6) and control curve in memory state feedback case are illustrated as Figure 1, from which, we see that the closed-loop system (2.6) takes only one second to have been stable.

fig1
Figure 1: State trajectories and control input in memory state feedback case.

Case 2 (Memory-Less State Feedback Stabilization). Solving LMIs (3.37), (3.24), (3.25), and (3.26), we obtain ,𝑃=0.56090.19640.19640.4117>0,𝑌=0.06020.5168𝑄=0.09230.00170.00170.0790>0,𝛼=0.9243.(4.4) So by Theorem 3.7, system (2.1) can be globally robustly stabilized by 𝑃𝑢(𝑡)=𝑌1𝑥(𝑡)=0.4449𝑥1(𝑡)1.5770𝑥2(𝑡).(4.5) The state trajectories of close-loop system (2.6) and control curve in memoryless state feedback case are illustrated as Figure 2, from which, it can be seen that the closed-loop system (2.6) takes 1.5 seconds to have been stable.
From the two simulation results, the time needed to stabilize system using memory state feedback controller is less than that using memory-less state feedback controller, which shows the advantage of memory state feedback control.

fig2
Figure 2: State trajectories and control input in memoryless state feedback case.

5. Conclusions

This paper has discussed memory state feedback stabilization of stochastic systems with time-delay and nonlinear uncertainties. Some sufficient conditions have been given for the existence of a memory state feedback stabilizing control law in terms of linear matrix inequalities, which have the advantage of easy computation. The corresponding results to stochastic single time-delay systems have been further extended to the stochastic multiple time-delays systems. The results obtained in this paper can be reduced to the corresponding results in memoryless state feedback case and may also be extended to other stochastic system model.

Acknowledgment

This work is supported by the National Natural Science Foundation of China under Grant 61074088, and the Starting Research Foundation of Shandong Polytechnic University under Grant 12045501. The authors thank the reviewers and editors for their very helpful comments and suggestions which have improved the presentation of this paper.

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