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

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 [1–18] 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 [7–9]. 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)+^{𝑡𝑡−𝜏}𝐺₁(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(𝑡−𝜏)(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; 𝐴≥0(𝐴>0): 𝐴 is positive semidefinite (positive definite) symmetric matrix; 𝐼: identity matrix; ‖⋅‖: Euclidean norm; 𝐿^2ℱ([0, ∞), 𝐑^𝑙): space of nonanticipative stochastic process 𝑦(𝑡)∈𝐑^𝑙 with respect to an increasing 𝜎-algebra ℱ_𝑡(𝑡≥0) satisfying 𝐸∫^∞0‖𝑦(𝑡)‖²𝑑𝑡<∞. 𝐼_𝑛×𝑛: 𝑛×𝑛 identity matrix.

2. Preliminaries and Problem Statement

Consider the following continuous nonlinear stochastic time-delay systems: 𝑑𝑥(𝑡)=+𝐴𝐶𝑥𝑥(𝑡()𝑡+)+𝐵𝐷𝑥(𝑥𝑡(−𝑡−𝜏)𝜏+)+𝐵₁𝐷𝑢₁(𝑡)+𝐻₀(𝑥(𝑡),𝑥(𝑡−𝜏),𝑢(𝑡))𝑑𝑡𝑥(𝑡)=(𝑡)∈𝐿₂𝑢(𝑡)+𝐻₁(𝑥(𝑡),𝑥(𝑡−𝜏),𝑢(𝑡))𝑑𝑤(𝑡),𝜔,ℱ₀,𝐶([−𝜏,0],𝑅^𝑛),𝑡∈[−𝜏,0],(2.1) where 𝑥(𝑡)∈𝐑^𝑛 and 𝑢(𝑡)∈𝐑^𝑚 are system state and control input, respectively; 𝑤(𝑡) is 1-dimensional standard Wiener process defined on the probability space (Ω, ℱ, ℱ_𝑡, 𝑃) with 𝐹_𝑡=𝜎{𝑤(𝑠)∶0≤𝑠≤𝑡}; 𝐻_𝑖(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 𝐻_𝑖(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, lim_𝑥→0𝑃sup_𝑡≥0‖𝑥(𝑡)‖>𝜖=0.(2.2) Additionally, if we also have lim_𝑥0→0𝑃lim_𝑡→∞𝑥(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 and𝑃lim_𝑡→∞𝑥(𝑡)=0=1(2.4) for all 𝑥₀∈𝐑^𝑛, 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(𝑡−𝜏),(2.5) such that the equilibrium point of the closed-loop system 𝑑𝑥(𝑡)=+𝐴𝐶++𝐵𝐷₁𝐾₁₁𝑥(𝑡)+𝐵+𝐵₁𝐾₂𝑥(𝑡−𝜏)+𝐻₀𝑥(𝑡),𝑥(𝑡−𝜏),𝐾₁𝑥(𝑡)+𝐾₂𝑥(𝑡−𝜏)𝑑𝑡+𝐻₁𝐾₁𝑥(𝑡)+𝐷+𝐷₁𝐾₂𝑥(𝑡−𝜏)𝑥(𝑡)=(𝑡),[−𝑥𝜏(𝑡,)0,]𝑥(𝑡−𝜏),𝐾₁𝑥(𝑡)+𝐾₂𝑥(𝑡−𝜏)𝑑𝑤(𝑡),(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(𝑡−𝜏), 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 𝐻_𝑖.

Assumption 3.1. There exists an 𝜖>0, such thatsup‖‖𝐻_𝑖𝑥,𝑦,𝐾₁𝑥+𝐾₂𝑦‖‖≤𝜖(‖𝑥‖+‖𝑦‖),𝑖=0,1,(3.1) holds for all 𝑥,𝑦∈𝐔 (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, 𝐾₂∈𝐑^𝑚×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 𝐑^𝑛, then system (2.1) can be globally robustly stabilized by the same controller.
In (3.2), 𝑍 and 𝑍₁ are defined by𝑍=⎡⎢⎢⎣Σ¹Σ₂∗𝐷+𝑍𝐷₁𝐾₂′𝑃𝐷+𝐷₁𝐾₂−𝑄₁⎤⎥⎥⎦,=⎡⎢⎢⎣Σ³0Σ0₄⎤⎥⎥⎦,(3.3) where Σ₁=𝑃𝐴Σ+₂𝐵₁𝐾₁+𝐴+𝐵₁𝐾₁^𝑃+𝑄+𝐶+𝐷₁𝐾₁^𝑃𝐶+𝐷₁𝐾₁,Σ₃=𝑃𝐵+𝐵₁𝐾₂+𝐶+𝐷₁𝐾₁^𝑃𝐷+𝐷₁𝐾₂,Σ=𝜖3+3‖𝐶‖+3‖‖𝐷₁‖‖⋅‖‖𝐾₁‖‖+3𝜖+‖𝐷‖+𝜖‖‖𝐷₁‖‖⋅‖‖𝐾₂‖‖‖𝑃‖𝐼,₄=𝜖3+‖𝐷‖+‖‖𝐷₁‖‖⋅‖‖𝐾₂‖‖+‖𝐶‖+‖𝐷₁‖⋅‖𝐾₁‖+2𝜖‖𝑃‖𝐼.(3.4)

Proof. Choose the following Lyapunov-Krasoviskii functional: 𝑉(𝑡,𝑥)=𝑥^(𝑡)𝑃𝑥(𝑡)+^𝜏0𝑥^(𝑡−𝑠)𝑄𝑥(𝑡−𝑠)𝑑𝑠,(3.5) where 𝑃>0 and 𝑄>0 are the solutions of (3.2). Let ℒ be the infinitesimal generator of the closed-loop system (2.6), then, by the Itô’s formula, we have ℒ𝑉(𝑡,𝑥(𝑡))=+𝑥^2(𝑥𝑡)_𝑃𝐴+𝐵₁𝐾₁+𝐴+𝐵₁𝐾₁^𝑃+𝑄+𝐶+𝐷₁𝐾₁^𝑃𝐶+𝐷₁𝐾₁𝑥(𝑡)⋅𝐷(+𝑡)𝐷𝑃₁𝐵+𝐵₁𝐾₂+𝐶+𝐷₁𝐾₁^⋅𝑃𝐷+𝐷₁𝐾₂𝑥(𝑡−𝜏)+𝑥^(𝑡−𝜏)+2𝐻_0𝐾₂^𝑃𝐷+𝐷₁𝐾₂−𝑄𝑥(𝑡−𝜏)+2𝐻_1𝑃𝑥(𝑡)+2𝐻^1𝑃𝐷+𝐷₁𝐾₂𝑥(𝑡−𝜏)𝑃𝐶+𝐷₁𝐾₁𝑥(𝑡)+𝐻^1𝑃𝐻₁=⎡⎢⎢⎣𝑥(𝑥𝑡(−𝑡)𝜏)⎤⎥⎥⎦^𝑍⎡⎢⎢⎣𝑥(𝑥𝑡(−𝑡)𝜏)⎤⎥⎥⎦+2𝐻^0𝑃𝑥(𝑡)+𝐻^1𝑃𝐻₁+2𝐻^1𝑃𝐷+𝐷₁𝐾₂𝑥(𝑡−𝜏)+2𝐻^1𝑃𝐶+𝐷₁𝐾₁𝑥(𝑡).(3.6) In addition, by (3.1), we obtain 2𝐻^0𝑃𝑥(𝑡)+2𝐻^1𝑃𝐷+𝐷₁𝐾₂𝑥(𝑡−𝜏)+2𝐻^1𝑃𝐶+𝐷₁𝐾₁𝑥(𝑡)+𝐻^1𝑃𝐻₁≤2𝜖‖𝑃‖1+‖𝐶‖+‖‖𝐷₁‖‖⋅‖‖𝐾₁‖‖+𝜖‖𝑥(𝑡)‖²++22𝜖𝜖‖𝑃+‖𝜖₂1+‖𝐶‖+‖‖𝐷₁‖‖⋅‖‖𝐾₁‖‖+‖𝐷‖+‖‖𝐷₁‖‖⋅‖‖𝐾₂‖‖+𝜖‖𝑥(𝑡)‖⋅‖𝑥(𝑡−𝜏)‖‖𝑃‖⋅‖𝑥(𝑡−𝜏)‖².(3.7) By inequality |𝑎𝑏|≤(1/2)(𝑎²+𝑏²), then (3.7) becomes 2𝐻^0𝑃𝑥(𝑡)+2𝐻^1𝑃𝐷+𝐷₁𝐾₂𝑥(𝑡−𝜏)+2𝐻^1𝑃𝐶+𝐷₁𝐾₁𝑥(𝑡)+𝐻^1𝑃𝐻₁≤𝜖3+3‖𝐶‖+3‖‖𝐷₁‖‖⋅‖‖𝐾₁‖‖+3𝜖+‖𝐷‖+‖‖𝐷₁‖‖⋅‖‖𝐾₂‖‖‖𝑃‖𝐼‖𝑥(𝑡)‖²+𝜖3+‖𝐷‖+‖‖𝐷₁‖‖⋅‖‖𝐾₂‖‖+‖‖𝐾₂‖‖‖𝐶‖+‖𝐷₁‖⋅‖𝐾₁‖+2𝜖‖𝑃‖𝐼‖𝑥(𝑡−𝜏)‖²=⎡⎢⎢⎣𝑥(𝑥𝑡(−𝑡)𝜏)⎤⎥⎥⎦^𝑍₁⎡⎢⎢⎣𝑥(𝑥𝑡(−𝑡)𝜏)⎤⎥⎥⎦.(3.8) Substituting (3.8) into (3.6), it follows ℒ𝑉(𝑡,𝑥(𝑡))≤⎡⎢⎢⎣𝑥(𝑥𝑡(−𝑡)𝜏)⎤⎥⎥⎦^𝑍+𝑍₁⎡⎢⎢⎣𝑥(𝑥𝑡(−𝑡)𝜏)⎤⎥⎥⎦.(3.9) According to (3.2), that is ℒ𝑉(𝑡,𝑥(𝑡))<0 in the domain {𝑡>0}×𝐔 for 𝑥≠0, 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 𝐻_𝑖≡0, 𝑖=0, 1, and the matrix inequality 𝑍<0 has solutions 𝑃>0, 𝑄>0 and 𝐾₁, 𝐾₂∈𝐑^𝑚×𝑛, then the linear stochastic time-delay system 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏)+𝐵₁𝑢(𝑡)𝑑𝑡+𝐶𝑥(𝑡)+𝐷𝑥(𝑡−𝜏)+𝐷₁𝑢(𝑡)𝑑𝑤(𝑡)(3.10) is globally robustly stabilizable. If 𝐷=0, 𝐷₁=0, and the following LMI: ⎡⎢⎢⎢⎢⎢⎢⎣𝑋𝐶Σ𝑃𝐵𝑃_51−𝑃0𝑄𝑃−0𝐶𝑃′𝐵₁00𝑋00−𝑄⎤⎥⎥⎥⎥⎥⎥⎦<0(3.11) has solutions 𝑃>0, 𝑌∈𝐑^𝑚×𝑛, 𝑋∈𝐑^𝑛×𝑚, and 𝑄>0, then 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏)+𝐵₁𝑢(𝑡)𝑑𝑡+𝐶𝑥(𝑡)𝑑𝑤(𝑡)(3.12) is globally robustly stabilizable, where Σ₅=𝐴𝑃+𝑃𝐴′+𝐵₁𝑌+𝑌′𝐵^1+𝐵𝑄𝐵′+𝐵𝑋𝐵^1+𝐵₁𝑋𝐵′, and a stabilizing feedback control law 𝑢(𝑡)=𝐾₁𝑥(𝑡)+𝐾₂𝑥(𝑡−𝜏)=𝑌𝑃⁻¹𝑥(𝑡)+𝑋𝑄⁻¹𝑥(𝑡−𝜏).

Proof. If 𝐻_𝑖(⋅,⋅,⋅)=0, 𝑖=0,1, we can take 𝜖=0 in (3.1), then ℒ𝑉(𝑡,𝑥(𝑡))<0 for (𝑡,𝑥)∈𝑡>0×𝐑^𝑛, except possibly at 𝑥=0.
Thus, the first part of Corollary 3.3 is proved.
Furthermore, if 𝐷=0, 𝐷₁=0, (3.2) degenerates into𝑍=⎡⎢⎢⎣Σ⁶𝐵+𝐵₁𝑃𝐵+𝐵₁𝐾₂𝐾₂′𝑃−𝑄⎤⎥⎥⎦<0,(3.13) where Σ₆=𝑃(𝐴+𝐵₁𝐾₁)+(𝐴+𝐵₁𝐾₁)′𝑃+𝑄+𝐶′𝑃𝐶.
According to Schur’s complement, (3.13) is equivalent toΣ₆+𝑃𝐵+𝐵₁𝐾₂𝑄⁻¹𝐵+𝐵₁𝐾₂′𝑃<0.(3.14) Then, pre- and post-(3.14) by 𝑃⁻¹, we have 𝑃⁻¹Σ₆𝑃⁻¹+𝐵+𝐵₁𝐾₂𝑄⁻¹𝐵+𝐵₁𝐾₂′<0.(3.15) Setting 𝑃=𝑃⁻¹, 𝑌=𝐾₁𝑃⁻¹, 𝑄=𝑄⁻¹, and 𝑋=𝐾₂𝑄⁻¹. 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: ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣√Γ^∗1√2𝑃𝐶′𝑃𝐵′002𝑃𝐵𝐶𝑃−00𝑃−00𝐼−00𝐼√00002𝐷′𝐾^2𝐵^1000000𝐵√₁2𝐷−𝑃0𝐾₂0−𝑃⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.16) has solution 𝑃>0, 𝑌, and 𝐾₂, then the stochastic linear time-delay controlled system 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏)+𝐵₁𝑢(𝑡)𝑑𝑡+𝐶𝑥(𝑡)+𝐷𝑥(𝑡−𝜏)+𝐷₁𝑢(𝑡)𝑑𝑤(𝑡)(3.17) is globally robustly stabilizable, where Γ^∗1=𝐴′𝑃+𝑃𝐴+𝐵₁𝑌+𝑌′𝐵^1+𝑃. Moreover, the stabilizing feedback control law 𝑢(𝑡)=𝑌𝑃⁻¹𝑥(𝑡)+𝐾₂𝑥(𝑡−𝜏).(3.18)

Proof. Applying the well-known inequality 𝑋′𝑌+𝑌′𝑋≤𝛾𝑋′𝑋+𝛾⁻¹𝑌′𝑌,∀𝛾>0,(3.19) and supposing 𝛾=1 for simplicity, we have 2𝑥′(𝑡≤)𝑃𝑥𝐵_1𝐾₂𝑥(𝑡−𝜏)+2𝑥^(𝑡)𝐶+𝐷₁𝐾₁′𝑃⋅𝐷+𝐷₁𝐾₂𝑥(𝑡−𝜏)+𝑥(_𝑡)𝑃+𝐶+𝐷₁𝐾₁^𝑃𝐶+𝐷₁𝐾₁𝑥(𝑡)(𝑡−𝜏)𝐾^2𝐵^1𝑃𝐵₁𝐾₂+𝐷+𝐷₁𝐾₂^𝑃⋅𝐷+𝐷₁𝐾₂𝑥(𝑡−𝜏).(3.20) Let Γ₁=Σ₁+𝑃+(𝐶+𝐷₁𝐾₁)′𝑃(𝐶+𝐷₁𝐾₁), Γ₂=2(𝐷+𝐷₁𝐾₂)′𝑃(𝐷+𝐷₁𝐾₂)+𝐾^2𝐵^1𝑃𝐵₁𝐾₂−𝑄. Then, 𝑍≤⎡⎢⎢⎣Γ¹∗𝑃Γ𝐵₂⎤⎥⎥⎦=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{𝑃⁻¹,𝐼}, and by Schur’s complement, we have Γ<0 is equivalent to⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣√Γ^∗1√2𝑃⁻¹𝐶′𝑃⁻¹𝐵′002𝑃𝐶𝑃⁻¹−𝑃⁻¹0000₋₁𝐵00−𝑄0⁻¹−0𝑄√002𝐷′𝐾^2𝐵^1000000𝐵√₁2𝐷−𝑃⁻¹0𝐾₂0−𝑃⁻¹⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(3.22) where Γ^∗1=𝐴𝑃⁻¹+𝑃⁻¹𝐴+𝐵₁𝐾₁𝑃⁻¹+𝑃⁻¹𝐾^1𝐵^1+𝑃⁻¹. Set 𝑃=𝑃⁻¹, 𝑌=𝐾₁𝑃⁻¹=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,𝑦∈𝐔(𝑥,𝑦∈𝐑^𝑛). If the LMIs: ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣Π₁𝑃𝑃√2𝑃𝐶′𝐵+𝐵₁𝐾₂𝑃𝑃−0𝑄−0𝛼006𝜖²√𝐼00𝐵′+2𝐾𝐶_2𝑃00−𝑃0𝐵^10𝑄0𝑃≤≤𝛼𝛼𝐼𝐼,0−𝜖²𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,((33..2234))7𝜖²0<𝛼≤1,,((33..2256)) have solutions 𝑃>0, 𝛼, 𝑄>0, 𝐾₂, and 𝑌∈𝐑^𝑚×𝑛, then system (2.1) can be locally (globally) robustly stabilized by 𝑢(𝑡)=𝑌𝑃⁻¹𝑥(𝑡)+𝐾₂𝑥(𝑡−𝜏),(3.27) where Π₁=𝐴𝑃+𝑃𝐴′+𝐵₁𝑌+𝑌′𝐵′₁+𝑃.

Proof. Applying the well-known inequality (3.19) again and supposing 𝛾=1 for simplicity, we have (if 0<𝑃≤𝐼/𝛼 for some 𝛼>0) 2𝐻^0𝑃𝑥(𝑡)+2𝐻^1𝑃𝐶𝑥(𝑡)+𝐻^1𝑃𝐻₁≤6𝜖²𝛼‖𝑥(𝑡)‖²+‖𝑥(𝑡−𝜏)‖²+𝑥′(𝑡)(𝑃+𝐶′𝑃𝐶)𝑥(𝑡)(3.28) which holds because 2𝐻^0𝑃𝑥=𝐻^0𝑃^1/2⋅𝑃^1/2𝑥+𝑥′𝑃^1/2⋅𝑃^1/2𝐻^0≤≤𝐻2𝜖^0₂𝑃𝐻₀+𝑥′𝑃𝑥𝛼‖𝑥(𝑡)‖²𝐻+‖𝑥(𝑡−𝜏)‖²+𝑥′𝑃𝑥,_1𝑃𝐻₁≤2𝜖²𝛼‖𝑥(𝑡)‖²2𝐻_1+‖𝑥(𝑡−𝜏)‖²,𝑃𝐶𝑥=𝐻^1𝑃^1/2⋅𝑃^1/2𝐶𝑥+𝑥′𝐶′𝑃^1/2⋅𝑃^1/2𝐻^1≤2𝜖²𝛼‖𝑥(𝑡)‖²+𝑥′(𝑡)𝐶′𝑃𝐶𝑥+(𝑡)‖.𝑥(𝑡−𝜏)‖²(3.29) Substituting (3.28) into (3.6), it follows that ℒ𝑉(𝑡,𝑥(𝑡))≤⎡⎢⎢⎣𝑥(𝑥𝑡(−𝑡)𝜏)⎤⎥⎥⎦^𝑍⎡⎢⎢⎣𝑥(𝑥𝑡(−𝑡)𝜏)⎤⎥⎥⎦,(3.30) where 𝑍=⎡⎢⎢⎣Π^∗1+6𝛼𝜖²∗𝐼𝑃6𝐵+𝐵₁𝐾₂𝛼𝜖²𝐼−𝑄⎤⎥⎥⎦.(3.31) Considering (3.24), (3.25), and (3.26), it follows that 𝑍≤⎡⎢⎢⎣Π^∗1+6𝛼𝜖²∗𝐼𝑃𝐵−+𝜖₂𝐵₁𝐾₂𝐼⎤⎥⎥⎦.(3.32) Let 𝑍₁=⎡⎢⎢⎣Π^∗1+6𝛼𝜖²∗𝐼𝑃𝐵−+𝜖₂𝐵₁𝐾₂𝐼⎤⎥⎥⎦,(3.33) where Π^∗1=𝑃(𝐴+𝐵₁𝐾₁)+(𝐴+𝐵₁𝐾₁)′𝑃+𝑄+𝑃+2𝐶′𝑃𝐶.
Obviously, if 𝑍₁<0, then 𝑍<0. So if (3.1) holds for all 𝑥∈𝐔(𝑥∈𝐑^𝑛), and 𝑍<0, then system (2.1) can be locally (globally) robustly stabilized by 𝑢(𝑡)=𝐾₁𝑥(𝑡)+𝐾₂𝑥(𝑡−𝜏).
Note that 𝑍₁<0 is equivalent to that𝑃𝐴+𝐵₁𝐾₁+𝐴+𝐵₁𝐾₁′𝑃+𝑄+𝑃+2𝐶′𝑃𝐶+6𝛼𝜖²𝐼+𝑃𝐵+𝐵₁𝐾₂𝜖⁻²𝐼𝐾^2𝐵^1+𝐵′𝑃<0.(3.34) Then pre- and postmultiply (3.34) by 𝑃⁻¹, we have 𝐴+𝐵₁𝐾₁𝑃⁻¹+𝑃⁻¹𝐴+𝐵₁𝐾₁^+𝑃⁻¹𝑄𝑃⁻¹+𝑃⁻¹+2𝑃⁻¹𝐶′𝑃𝐶𝑃⁻¹+𝑃⁻¹6𝛼𝜖²𝐼𝑃⁻¹+𝐵+𝐵₁𝐾₂𝜖⁻²𝐼𝐾^2𝐵^1+𝐵′<0.(3.35) Setting 𝑃=𝑃⁻¹, 𝑌=𝐾₁𝑃⁻¹=𝐾₁𝑃, and 𝑄=𝑄⁻¹ 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, sup‖𝐻_𝑖𝑥,𝑦,𝐾₁𝑥‖≤𝜖(‖𝑥‖+‖𝑦‖),𝑖=0,1,(3.36) holds for all 𝑥, 𝑦∈𝐔(𝑥,𝑦∈𝐑^𝑛), if the LMIs ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣Π𝑃𝑃₁−𝑃0𝑄−𝑃0𝛼√20𝑃𝐶′𝐵06𝜖²√𝐼002𝐵𝐶′𝑃0000−0𝑃−𝜖0₂𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0(3.37) and (3.24), (3.25), and (3.26) have solutions 𝑃>0, 𝑄>0, 𝛼, and 𝑌∈𝐑^𝑚×𝑛, then system (2.1) can be locally (globally) robustly stabilized by 𝑢(𝑡)=𝑌𝑃⁻¹𝑥(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)+^𝑞_𝑗=1𝐵_𝑗𝑥𝑡−𝜏_𝑗+^𝑞_𝑗=1+𝐶𝑥(𝑡)+_𝑞𝐵_1𝑗𝑢_𝑗(𝑡)+𝐻₀𝑥(𝑡),𝑥𝑡−𝜏_𝑗,𝑢_𝑗(𝑡)𝑑𝑡_𝑗=1𝐷_𝑗𝑥𝑡−𝜏_𝑗+^𝑞_𝑗=1𝑥(𝑡)=(𝑡)∈𝐿₂𝐷_1𝑗𝑢_𝑗(𝑡)+𝐻₁𝑥(𝑡),𝑥𝑡−𝜏_𝑗,𝑢_𝑗(𝑡)𝑑𝑤(𝑡),𝜔,ℱ₀,𝐶([−ℎ,0],𝑅^𝑛),𝑡∈[−ℎ,0],(3.39) where 𝜏_𝑗>0, 𝑗=1,…,𝑞, denote the state delay; ℎ=max{𝜏_𝑗,𝑗∈[1,𝑞]}.

For system (3.39), the following memory state feedback control law is adopted:𝑢_𝑗(𝑡)=𝐾_𝑗1𝑥(𝑡)+𝐾_𝑗2𝑥𝑡−𝜏_𝑗,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)+^𝑞_𝑗=1+𝐵𝑥𝑡−𝜏_𝑗+𝐻₀𝑥(𝑡),𝑥𝑡−𝜏_𝑗,𝐾_𝑗1𝑥(𝑡)+𝐾_𝑗2𝑥𝑡−𝜏_𝑗𝑑𝑡𝐶𝑥(𝑡)+^𝑞_𝑗=1𝑥(𝑡)=(𝑡)∈𝐿₂𝐷𝑥𝑡−𝜏_𝑗+𝐻₁𝑥(𝑡),𝑥𝑡−𝜏_𝑗,𝐾_𝑗1𝑥(𝑡)+𝐾_𝑗2𝑥𝑡−𝜏_𝑗𝑑𝑤(𝑡),𝜔,ℱ₀,𝐶([−ℎ,0],𝑅^𝑛),𝑡∈[−ℎ,0],(3.41) where 𝐴=𝐴+∑^𝑞𝑗=1𝐵_1𝑗𝐾_𝑗1, 𝐵=𝐵_𝑗+𝐵_1𝑗𝐾_𝑗2, 𝐶=𝐶+∑^𝑞𝑗=1𝐷_1𝑗𝐾_𝑗1, 𝐷=𝐷_𝑗+𝐷_1𝑗𝐾_𝑗2.

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 𝐾_𝑗1, 𝐾_𝑗2, 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 𝑢_𝑗(𝑡)=𝐾_𝑗1𝑥(𝑡)+𝐾_𝑗2𝑥(𝑡−𝜏_𝑗). Especially if 𝐔 is replaced by 𝐑^𝑛, then system (3.39) can be globally robustly stabilized by the same controller.
In (3.42), 𝑍⁰ and 𝑍¹ are defined by𝑍⁰=⎡⎢⎢⎣𝑍⁰¹¹𝑍⁰¹²∗𝑍⁰²²𝑍₁⎤⎥⎥⎦,=⎡⎢⎢⎣𝑍¹¹¹0𝑍0₁₂₂⎤⎥⎥⎦,(3.43) where 𝑍⁰¹¹=𝑃𝐴+^𝑞_𝑗=1𝐵_1𝑗𝐾_𝑗1+𝐴+^𝑞_𝑗=1𝐵_1𝑗𝐾_𝑗1^+𝐶+_𝑞𝑃+𝑄_𝑗=1𝐷_1𝑗𝐾_𝑗1^𝑃𝐶+^𝑞_𝑗=1𝐷_1𝑗𝐾_𝑗1𝑍₀₁₂,=𝑃^𝑞_𝑗=1𝐵_𝑗+𝐵_1𝑗𝐾_𝑗2+𝐶+^𝑞_𝑗=1𝐷_1𝑗𝐾_𝑗1^𝑃^𝑞_𝑗=1𝐷_𝑗𝑍₀₂₂+𝐷_1𝑗𝐾_𝑗2,=^𝑞_𝑗=1𝐷_𝑗+𝐷_1𝑗𝐾_𝑗2^^𝑞_𝑗=1𝑍𝐷_𝑗+𝐷_1𝑗𝐾_𝑗2−𝑄,₁₁₁=𝜖3+3‖𝐶‖+3‖‖‖‖^𝑞_𝑗=1𝐷_1𝑗‖‖‖‖⋅‖‖‖‖^𝑞_𝑗=1𝐾_𝑗1+‖𝐷‖+𝜖‖‖‖‖_𝑞‖‖‖‖+3𝜖_𝑗=1𝐷_1𝑗‖‖‖‖⋅‖‖‖‖^𝑞_𝑗=1𝐾_𝑗2𝑍‖‖‖‖‖𝑃‖𝐼,₁₂₂=𝜖3+‖𝐷‖+‖‖‖‖^𝑞_𝑗=1𝐷_1𝑗‖‖‖‖⋅‖‖‖‖^𝑞_𝑗=1𝐾_𝑗2+‖‖‖‖‖‖‖‖+‖𝐶‖_𝑞_𝑗=1𝐷_1𝑗‖‖‖‖⋅‖‖‖‖^𝑞_𝑗=1𝐾_𝑗1‖‖‖‖+2𝜖‖𝑃‖𝐼.(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, and𝐴𝐵=⎡⎢⎢⎣−−51..050622..2135⎤⎥⎥⎦,𝐵=⎡⎢⎢⎣−10.2.224−00.8.796₁⎤⎥⎥⎦,𝐻=⎡⎢⎢⎣−42.4.285⎤⎥⎥⎦,𝐶=⎡⎢⎢⎣−00.1.055−−00..110₀5⎤⎥⎥⎦,=⎡⎢⎢⎣sin𝑢(𝑡)𝑥²cos𝑢(𝑡)𝑥₁(𝑡−𝜏)𝑥₁(𝑡)𝐻(𝑡−𝜏)𝑥₂(𝑡)₁⎤⎥⎥⎦,=⎡⎢⎢⎣exp−𝑢(𝑡)+𝑥¹exp−𝑢₂(𝑡−𝜏)+𝑥₂(𝑡−𝜏)²𝑥²(𝑡)𝜙(0)=108(𝑡)_𝑥²¹(𝑡−𝜏)𝑥₁(𝑡)⎤⎥⎥⎦,,𝜏=0.5.(4.1) Obviously, (2.1) holds for all 𝑥∈𝐑^𝑛 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 𝑃𝑌𝐾==⎡⎢⎢⎣−0−0.4.𝛼006.62=1253607.79−00−0.13.0503.,68623667⎤⎥⎥⎦4₂>,0,𝑄==⎡⎢⎢⎣00..−1010045.21239001..0009057.2241⎤⎥⎥⎦4>90,.(4.2) So by Theorem 3.6, system (2.1) can be globally robustly stabilized by 𝑢(𝑡)==𝑌−0𝑃.57⁻¹𝑥90(𝑡𝑥)₁+𝐾₂𝑥(𝑡−𝜏)−0.2391𝑥₁(𝑡)−2.0795𝑥₂(𝑡)(𝑡−𝜏)+0.2149𝑥₂(𝑡−𝜏).(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.

(a)

(b)

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𝑄=𝑌=⎡⎢⎢⎣=−⎡⎢⎢⎣0000.05...𝛼100.609900=66219043702.9−00−02..000.404.0731.1195.9170167⎤⎥⎥⎦648⎤⎥⎥⎦>>,0,0,(4.4) So by Theorem 3.7, system (2.1) can be globally robustly stabilized by 𝑢(𝑡)=𝑌𝑃⁻¹𝑥(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.

(a)

(b)

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.