Stability Analysis and Robust 𝐻_∞ Control of Switched Stochastic Systems with Time-Varying Delay

Abstract

The problems of mean-square exponential stability and robust 𝐻_∞ control of switched stochastic systems with time-varying delay are investigated in this paper. Based on the average dwell time method and Gronwall-Bellman inequality, a new mean-square exponential stability criterion of such system is derived in terms of linear matrix inequalities (LMIs). Then, 𝐻_∞ performance is studied and robust 𝐻_∞ controller is designed. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.

1. Introduction

Switched systems, a special hybrid system, are composed of a set of continuous-time or discrete-time subsystems and a rule orchestrating the switching between the subsystems. In the last two decades, there has been increasing interest in the stability analysis and control design for such switched systems since many real-world systems such as chemical systems [1], robot control systems [2], traffic systems [3], and networked control systems [4, 5] can be modeled as such systems. The past decades have witnessed an enormous interest in the stability analysis and control synthesis of switched systems [6–11].

It is well known that time delay phenomenon exists in many engineering systems such as networked systems and long-distance transportation systems. Such phenomenon may cause the system unstable if it cannot be handled properly, which motivates many scientists to involve themselves in researching switched systems with time delay. Many results have been reported for stability analysis of switched systems with time delay [12, 13], where the asymptotical stability criteria are given by using common Lyapunov function approach in [12], and the exponential stability criteria under average dwell time switching signals are proposed in [13]. Moreover, the problem of delay-dependent global robust asymptotic stability of switched uncertain Hopfield neural networks with time delay in the leakage term is discussed in [14]. 𝐻_∞ control of continuous-time switched systems with time delay and discrete-time switched systems with time delay are investigated in [15, 16], respectively.

On the other hand, stochastic systems have attracted considerable attention during the past decades because stochastic disturbance exists in many actual operations. Many useful results on the stability analysis of stochastic systems are reported in [17–21]. The problem of robust 𝐻_∞ control for nonlinear stochastic systems with Markovian jump parameters and interval time-varying delays is considered [22]. Based on the results of stochastic systems and switched systems, the stability analysis and stabilization of switched stochastic systems are investigated in [23, 24]. Furthermore, the problems of reliable control and reliable 𝐻_∞ control for switched stochastic systems under asynchronous switching are studied in [25, 26], respectively. Recently, these results are extended to stochastic switched systems with time delay, and the exponential stability criteria are addressed [27, 28]. However, these results are very complex, which make it more difficult for us to solve many issues such as controller design under asynchronous switching and actuator failures. Therefore, there is a lot of work to do in such field. This motivates the present study.

In this paper, we focus on the mean-square exponential stability analysis and robust 𝐻_∞ control of switched stochastic systems with time-varying delay. Based on the average dwell time method and Gronwall-Bellman inequality, a new mean-square exponential stability criterion is derived. Moreover, 𝐻_∞ performance is studied and 𝐻_∞ state feedback controller is proposed. The remainder of the paper is organized as follows. In Section 2, problem statement and some useful lemmas are given. In Section 3, based on the average dwell time method and Gronwall-Bellman inequality, the mean-square exponential stability and 𝐻_∞ performance of the switched stochastic systems with time delay are investigated. Then, robust 𝐻_∞ controller is designed. In Section 4, a numerical example is given to illustrate the effectiveness of the proposed approach. Finally, concluding remarks are provided in Section 5.

Notation. Throughout this paper, the superscript ‘‘𝑇’’ denotes the transpose, and the symmetric terms in a matrices are denoted by ∗. The notation X>𝑌(X≥Y) means that matrix 𝑋−𝑌 is positive definite (positive semidefinite, resp.). 𝑅^𝑛 denotes the 𝑛 dimensional Euclidean space. ‖𝑥(𝑡)‖ denotes the Euclidean norm. 𝐿₂[𝑡₀,∞) is the space of square integrable functions on [𝑡₀,∞). 𝜆_max(𝑃) and 𝜆_min(𝑃) denote the maximum and minimum eigenvalues of matrix 𝑃, respectively. 𝐼 is an identity matrix with appropriate dimension. diag{𝑎_𝑖} denotes diagonal matrix with the diagonal elements 𝑎_𝑖,𝑖=1,2,…,𝑛.

2. Problem Formulation and Preliminaries

Consider the following stochastic switched systems with time-delay:𝑑𝑥(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐵_𝜎(𝑡)𝑥(𝑡−ℎ(𝑡))+𝐶_𝜎(𝑡)𝑢(𝑡)+𝐺_𝜎(𝑡)𝑣(𝑡)𝑑𝑡+𝐷_𝜎(𝑡)𝑥(𝑡)(𝑡),𝑥(𝑡)(𝑡)𝑡𝑑𝑤=𝜑,𝑡∈₀−ℎ,𝑡₀,𝑧(𝑡)=𝑀_𝜎(𝑡)𝑥(𝑡), (2.1) where 𝑥(𝑡)∈𝑅^𝑛 is the state vector, 𝜑(𝑡)∈𝑅^𝑛 is the initial state function, 𝑢(𝑡)∈𝑅^𝑙 is the control input, 𝑣(𝑡)∈𝑅^𝑝 is the disturbance input which is assumed belong to 𝐿₂[𝑡₀,∞], 𝑧(𝑡)∈𝑅^𝑞 is the signal to be estimated, 𝑤(𝑡)∈𝑅 is a zero-mean Wiener process on a probability space (Ω𝐹𝑃) satisfying𝐸{𝑑𝑤(𝑡)}=0,𝐸𝑑𝑤²(𝑡)=𝑑𝑡, (2.2) where Ω is the sample space, 𝐹 is 𝜎-algebras of subsets of the sample space, 𝑃 is the probability measure on 𝐹, and 𝐸{⋅} is the expectation operator. ℎ(𝑡) is the system state delay satisfying0≤ℎ(𝑡)≤ℎ,ℎ̇(𝑡)≤ℎ_𝑑<1, (2.3) where ℎ_𝑑 is a known constant. The function 𝜎(𝑡)∶[𝑡₀,∞]→𝑁={1,2,…,𝑁} is a switching signal which is deterministic, piecewise constant, and right continuous. The switching sequence can be described as 𝜎∶{(𝑡₀,𝜎(𝑡₀)),(𝑡₁,𝜎(𝑡₁)),…,(𝑡_𝑘,𝜎(𝑡_𝑘))}, 𝜎(𝑡_𝑘)∈𝑁,𝑘∈𝑍, where 𝑡₀ is the initial time and 𝑡_𝑘 denotes the 𝑘th switching instant. Moreover 𝜎(𝑡)=𝑖 means that the 𝑖th subsystem is activated.

For each for all 𝑖∈𝑁, 𝐶_𝑖, 𝐺_𝑖, and 𝑀_𝑖 are known real-value matrices with appropriate dimensions, and 𝐴_𝑖,𝐵_𝑖, and 𝐷_𝑖 are uncertain real matrix with appropriate dimensions, which can be written as𝐴_𝑖𝐵_𝑖𝐷_𝑖=𝐴_𝑖𝐵_𝑖𝐷_𝑖+𝐻_𝑖𝐹_𝑖(𝑡)𝐸_1𝑖𝐸_2𝑖𝐸_3𝑖, (2.4) where 𝐴_𝑖, 𝐵_𝑖, and 𝐷_𝑖 are known real-value matrices with appropriate dimensions, and 𝐹_𝑖(𝑡) is unknown time-varying matrix that satisfies𝐹^𝑇_𝑖(𝑡)𝐹_𝑖(𝑡)≤𝐼. (2.5)

Definition 2.1. System (2.1) is said to be mean-square exponentially stable with under switching signal 𝜎(𝑡), if there exist scalars 𝜅>0 and 𝛼>0, such that the solution 𝑥(𝑡) of system (2.1) satisfies 𝐸{‖𝑥(𝑡)‖²}≤𝜅𝑒^{−𝛼(𝑡−𝑡0)}sup_{𝑡0−ℎ≤𝜃≤𝑡0}𝐸{‖𝜑(𝜃)‖²}, for all 𝑡>𝑡₀. Moreover, 𝛼 is called the decay rate.

Definition 2.2. For any 𝑇₂>𝑇₁≥𝑡₀, let 𝑁_𝜎(𝑇₁,𝑇₂) denote the switching number of 𝜎(𝑡) on an interval (𝑇₁,𝑇₂). If 𝑁_𝜎𝑇₁,𝑇₂≤𝑁₀+𝑇₂−𝑇₁𝑇_𝛼 (2.6) holds for given 𝑁₀≥0, 𝑇_𝛼>0, then the constant 𝑇_𝛼 is called the average dwell time. As commonly used in the literature, we choose 𝑁₀=0.

Definition 2.3. Let 𝛾>0 be a positive constant, for system (2.1), if there exists a controller 𝑢(𝑡) and a switching signal 𝜎(𝑡), such that(1)when 𝑣(𝑡)=0, system (2.1) is mean-square exponentially stable;(2)under zero initial condition 𝑥(𝑡)=0, 𝑡∈[𝑡₀−ℎ,𝑡₀], the output 𝑧(𝑡) satisfies𝐸^∞_𝑡0𝑒^{−𝜆(𝑠−𝑡0)}𝑧^𝑇(𝑠)𝑧(𝑠)𝑑𝑠≤𝛾^2∞_𝑡0𝑣^𝑇(𝑠)𝑣(𝑠)𝑑𝑠,∀𝑣(𝑡)∈𝐿₂𝑡₀,∞. (2.7) Then system (2.1) is said to be robustly exponentially stabilizable with a prescribed weighted 𝐻_∞ performance, where 𝜆>0.

The following lemmas play an important role in the later development.

Lemma 2.4 (see [29] (Gronwall-Bellman Inequality)). Let 𝑥(𝑡) and 𝑦(𝑡) be real-valued nonnegative continuous functions with domain {𝑡∣𝑡≥𝑡₀}; 𝑎 is a nonnegative scalar; if the following inequality 𝑥(𝑡)≤𝑎+^𝑡_𝑡0𝑥(𝑠)𝑦(𝑠)𝑑𝑠 (2.8) holds, for 𝑡≥𝑡₀, then 𝑥(𝑡)≤𝑎exp(∫^𝑡_𝑡0𝑦(𝑠)𝑑𝑠).

Lemma 2.5 (see [30]). Let 𝑈, 𝑉, 𝑊, and 𝑋 be constant matrices of appropriate dimensions with 𝑋 satisfying 𝑋=𝑋^𝑇, then for all 𝑉^𝑇𝑉≤𝐼, 𝑋+𝑈𝑉𝑊+𝑊^𝑇𝑉^𝑇𝑈^𝑇<0, if and only if there exists a scalar 𝜀>0 such that 𝑋+𝜀𝑈𝑈^𝑇+𝜀⁻¹𝑊^𝑇𝑊<0.

3. Main Results

3.1. Stability Analysis

In this subsection, we will focus on the exponential stability analysis of switched stochastic systems with time-varying delay.

Consider the following switched stochastic system:𝑑𝑥(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐵_𝜎(𝑡)𝑥(𝑡−ℎ(𝑡))𝑑𝑡+𝐷_𝜎(𝑡)𝑥(𝑡)(𝑡),𝑥(𝑡)(𝑡)𝑡𝑑𝑤=𝜑,𝑡∈₀−ℎ,𝑡₀. (3.1)

Theorem 3.1. Considering system (3.1), for a given scalar 𝛼>0, if there exist symmetric positive definite matrices 𝑃_𝑖,𝑄_𝑖>0 satisfying 𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝛼𝑃_𝑖+𝑄_𝑖𝑃_𝑖𝐵_𝑖𝐷^𝑇_𝑖𝑃_𝑖∗−1−ℎ_𝑑𝑄_𝑖0∗∗−𝑃_𝑖<0, (3.2) for all 𝑖∈𝑁, then system (3.1) is mean-square exponentially stable under arbitrary switching signal with the average dwell time: 𝑇_𝛼≥𝑇^∗_𝛼=ln𝜇𝛼, (3.3) where 𝜇≥1 satisfies 𝑃_𝑖≤𝜇𝑃_𝑗,𝑄_𝑖≤𝜇𝑄_𝑗,∀𝑖,𝑗∈𝑁. (3.4)

Proof. Consider the following Lyapunov functional for the 𝑖th subsystem: 𝑉_𝑖(𝑡,𝑥(𝑡))=𝑉_1,𝑖(𝑡,𝑥(𝑡))+𝑉_2,𝑖(𝑡,𝑥(𝑡)), (3.5) where 𝑉_1,𝑖(𝑡,𝑥(𝑡))=𝑥^𝑇(𝑡)𝑃_𝑖𝑥(𝑡),𝑉_2,𝑖(𝑡,𝑥(𝑡))=^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)𝑄_𝑖𝑥(𝑠)𝑑𝑠. (3.6) For the sake of simplicity, 𝑉_𝑖(𝑡,𝑥(𝑡)) is written as 𝑉_𝑖(𝑡) in this paper.
According to Itô formula, along the trajectory of system (3.1), we have 𝑑𝑉_𝑖(𝑡)=ℒ𝑉_𝑖(𝑡)𝑑𝑡+2𝑥^𝑇(𝑡)𝑃_𝑖𝐷_𝑖𝑥(𝑡)𝑑𝑤(𝑡), (3.7) where ℒ𝑉_𝑖(𝑡)=2𝑥^𝑇(𝑡)𝑃_𝑖𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−ℎ(𝑡))+𝑥^𝑇(𝑡)𝑄_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝐷^𝑇_𝑖𝑃_𝑖𝐷_𝑖𝑥(𝑡)−1−ℎ̇(𝑡)𝑥^𝑇(𝑡−ℎ(𝑡))𝑄_𝑖𝑥(𝑡−ℎ(𝑡)). (3.8) According to (2.3), we can obtain that ℒ𝑉_𝑖(𝑡)≤2𝑥^𝑇(𝑡)𝑃_𝑖𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−ℎ(𝑡))+𝑥^𝑇(𝑡)𝑄_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝐷^𝑇_𝑖𝑃_𝑖𝐷_𝑖𝑥(𝑡)−1−ℎ_𝑑𝑥^𝑇(𝑡−ℎ(𝑡))𝑄_𝑖𝑥(𝑡−ℎ(𝑡))=𝜉^𝑇(𝑡)Θ_𝑖𝜉(𝑡), (3.9) where 𝜉(𝑡)=𝑥(𝑡)𝑥(𝑡−ℎ(𝑡)),Θ_𝑖=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝑄_𝑖+𝐷^𝑇_𝑖𝑃_𝑖𝐷_𝑖𝑃_𝑖𝐵_𝑖∗−1−ℎ_𝑑𝑄_𝑖. (3.10) Using Schur complement, it is not difficult to get that if inequality (3.2) is satisfied, the following inequality can be obtained: ℒ𝑉_𝑖(𝑡)<−𝛼𝑉_1,𝑖(𝑡)<0. (3.11) Combining (3.7) with (3.11) leads to 𝑑𝑉_𝑖(𝑡)=ℒ𝑉_𝑖(𝑡)𝑑𝑡+2𝑥^𝑇(𝑡)𝑃_𝑖𝐷_𝑖𝑥(𝑡)𝑑𝑤(𝑡)<−𝛼𝑉_1,𝑖(𝑡)𝑑𝑡+2𝑥^𝑇(𝑡)𝑃_𝑖𝐷_𝑖𝑥(𝑡)𝑑𝑤(𝑡). (3.12) Noticing (2.2) and taking the expectation to (3.12), we have 𝐸𝑑𝑉_𝑖(𝑡)𝑑𝑡=𝐸ℒ𝑉_𝑖(𝑡)<−𝛼𝐸𝑉_1,𝑖(𝑡)<0. (3.13) According to (3.4)–(3.6), we have 𝐸𝑉_𝑝(𝑡)≤𝜇𝐸𝑉_𝑞(𝑡)=𝜇𝐸𝑉_𝑞(𝑡⁻),𝐸𝑉_1,𝑝(𝑡)≤𝜇𝐸𝑉_1,𝑞(𝑡),∀𝑝,𝑞∈𝑁. (3.14)(3.15) Assume that the 𝑖th subsystem is activated during [𝑡_𝑘,𝑡_𝑘+1) and 𝑗th subsystem is activated during [𝑡_𝑘−1,𝑡_𝑘), respectively. Using Itô formula and according to (3.13)–(3.15), we have, for any 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝐸𝑉_1,𝑖(𝑡)≤𝐸𝑉_𝑖(𝑡)=𝐸𝑉_𝑖𝑡_𝑘+𝐸^𝑡_𝑡𝑘ℒ𝑉_𝑖(𝑠)𝑉𝑑𝑠<𝜇𝐸_𝑗𝑡⁻_𝑘−𝛼𝐸^𝑡_𝑡𝑘𝑉_1,𝑖(𝑠)𝑉𝑑𝑠=𝜇𝐸_𝑗𝑡_𝑘−1+𝜇𝐸^𝑡𝑘_𝑡𝑘−1ℒ𝑉_1,𝑗(𝑠)𝑑𝑠−𝛼𝐸^𝑡_𝑡𝑘𝑉_1,𝑖(𝑠)𝑉𝑑𝑠<𝜇𝐸_𝑗𝑡_𝑘−1−𝛼𝜇𝐸^𝑡𝑘_𝑡𝑘−1𝑉_1,𝑗(𝑠)𝑑𝑠−𝛼𝐸^𝑡_𝑡𝑘𝑉_1,𝑖(𝑠)𝑉𝑑𝑠≤𝜇𝐸_𝑗𝑡_𝑘−1−𝛼𝐸^𝑡_𝑡𝑘−1𝑉_1,𝑖(𝑠)𝑑𝑠≤⋯≤𝜇^{𝑁𝜎(𝑡0,𝑡)}𝐸𝑉𝑡₀−𝛼𝐸^𝑡_𝑡0𝑉_1,𝑖(𝑠)𝑑𝑠. (3.16) According to Lemma 2.4 and when (3.3) holds, we have 𝐸𝑉_1,𝑖(𝑡)≤𝜇^{𝑁𝜎(𝑡0,𝑡)}𝑒^{−𝛼(𝑡−𝑡0)}𝐸𝑉𝑡₀≤𝜅𝑒^{−(𝛼−(ln𝜇/𝑇𝛼))(𝑡−𝑡0)}𝐸𝑉𝑡₀. (3.17) Moreover, we can obtain 𝐸‖𝑥(𝑡)‖²≤𝜅𝑒^{−𝜆(𝑡−𝑡0)}sup_{−ℎ≤𝜃≤0}𝐸‖𝑥(𝜃)‖², (3.18) where 𝜅=max_𝑖∈𝑁(𝜆_max(𝑃_𝑖)+ℎ𝜆_max(𝑄_𝑖))/min_𝑖∈𝑁𝜆_min(𝑃_𝑖), and 𝜆=(1/2)(𝛼−(ln𝜇/𝑇_𝛼)) is the decay rate.
The proof is completed.

Remark 3.2. The exponential stability criterion of stochastic switched systems with time-varying delay is given in Theorem 3.1. When 𝑤(𝑡)=0, system (3.1) is degenerated to the switched system with time-varying delay, which can be described as 𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐵_𝜎(𝑡)𝑥((𝑡,𝑥(𝑡)(𝑡)𝑡𝑡−ℎ))=𝜑,𝑡∈₀−ℎ,𝑡₀. (3.19) Using the same method, we can obtain the following exponential stability criterion of switched system (3.19).

Corollary 3.3. Considering system (3.19), for a given scalar 𝛼>0, if there exist symmetric positive definite matrices 𝑃_𝑖,𝑄_𝑖>0 satisfying 𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝛼𝑃_𝑖+𝑄_𝑖𝑃_𝑖𝐵_𝑖∗−1−ℎ_𝑑𝑄_𝑖<0, (3.20) for all 𝑖∈𝑁, system (3.19) is exponentially stable under arbitrary switching signal with average dwell time satisfying (3.3).

3.2. 𝐻_∞ Performance Analysis

In this subsection, we will investigate the 𝐻_∞ performance of switched stochastic systems with time-varying delay.

Consider the following switched stochastic system:𝑑𝑥(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐵_𝜎(𝑡)𝑥(𝑡−ℎ(𝑡))+𝐺_𝜎(𝑡)𝑣(𝑡)𝑑𝑡+𝐷_𝜎(𝑡)𝑥(𝑡)(𝑡),𝑥(𝑡)(𝑡)𝑡𝑑𝑤=𝜑,𝑡∈₀−ℎ,𝑡₀,𝑧(𝑡)=𝑀_𝜎(𝑡)𝑥(𝑡). (3.21)

Theorem 3.4. Considering system (3.21), for a given scalar 𝛼>0, if there exist symmetric positive definite matrices 𝑃_𝑖,𝑄_𝑖>0 such that 𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝛼𝑃_𝑖+𝑄_𝑖𝑃_𝑖𝐵_𝑖𝑃_𝑖𝐺_𝑖𝐷^𝑇_𝑖𝑃_𝑖𝑀^𝑇_𝑖∗−1−ℎ_𝑑𝑄_𝑖000∗∗−𝛾²𝐼00∗∗∗−𝑃_𝑖0∗∗∗∗−𝐼<0 (3.22) hold for all 𝑖∈𝑁, system (3.21) is said to have weighted 𝐻_∞ performance 𝛾 under arbitrary switching signal with the average dwell time: 𝑇_𝛼≥𝑇^∗_𝛼=ln𝜇𝛼, (3.23) where 𝜇≥1 satisfies 𝑃_𝑖≤𝜇𝑃_𝑗,𝑄_𝑖≤𝜇𝑄_𝑗,∀𝑖,𝑗∈𝑁. (3.24)

Proof. By Theorem 3.1, we can readily obtain that system (3.21) is mean-square exponential stable when 𝑣(𝑡)=0.
Assume that the 𝑖th subsystem is activated during [𝑡_𝑘,𝑡_𝑘+1). Choose the following Lyapunov functional candidate for the 𝑖th subsystem: 𝑉_𝑖(𝑡)=𝑉_1,𝑖(𝑡)+𝑉_2,𝑖(𝑡), (3.25) where 𝑉_1,𝑖(𝑡)=𝑥^𝑇(𝑡)𝑃_𝑖𝑥(𝑡),𝑉_2,𝑖(𝑡)=^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)𝑄_𝑖𝑥(𝑠)𝑑𝑠. (3.26) Using Itô formula, along the trajectory of system (3.21); we have 𝑑𝑉_𝑖(𝑡)=ℒ𝑉_𝑖(𝑡)𝑑𝑡+2𝑥^𝑇(𝑡)𝑃_𝑖𝐷_𝑖𝑥(𝑡)𝑑𝑤(𝑡), (3.27) where ℒ𝑉_𝑖(𝑡)=2𝑥^𝑇(𝑡)𝑃_𝑖𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−ℎ(𝑡))+𝐺_𝑖𝑣(𝑡)+𝑥^𝑇(𝑡)𝑄_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝐷^𝑇_𝑖𝑃_𝑖𝐷_𝑖𝑥(𝑡)−1−ℎ̇(𝑡)𝑥^𝑇(𝑡−ℎ(𝑡))𝑄_𝑖𝑥(𝑡−ℎ(𝑡)) (3.28) Let Γ(𝑡)=𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑣^𝑇(𝑡)𝑣(𝑡); we have ℒ𝑉_𝑖(𝑡)+Γ(𝑡)=𝜍^𝑇(𝑡)Φ_𝑖𝜍(𝑡), (3.29) where 𝜍(𝑡)=[𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ(𝑡))𝑣^𝑇(𝑡)]^𝑇, and Φ_𝑖=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝑄_𝑖+𝑀^𝑇_𝑖𝑀_𝑖+𝐷^𝑇_𝑖𝑃_𝑖𝐷_𝑖𝑃_𝑖𝐵_𝑖𝑃_𝑖𝐺_𝑖∗−1−ℎ_𝑑𝑄_𝑖0∗∗−𝛾²𝐼. (3.30) Combining (3.22) with (3.29)–(3.30), and using Schur complement, we have ℒ𝑉_𝑖(𝑡)+Γ(𝑡)<−𝛼𝑉_1,𝑖(𝑡)<0. (3.31) Noticing (2.2) and taking the expectation to (3.27), we have 𝐸𝑑𝑉_𝑖(𝑡)𝑑𝑡=𝐸ℒ𝑉_𝑖(𝑡). (3.32) According to (3.25)–(3.27), we have 𝐸𝑉_𝑝(𝑡)≤𝜇𝐸𝑉_𝑞(𝑡⁻),∀𝑝,𝑞∈𝑁. (3.33) Using Itô formula, we have, for any 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝐸𝑉_𝑖(𝑡)=𝐸𝑉_𝑖𝑡_𝑘+^𝑡_𝑡𝑘𝐸ℒ𝑉_𝑖(𝑠)+Γ(𝑠)𝑑𝑠−𝐸^𝑡_𝑡𝑘Γ(𝑠)𝑉𝑑𝑠<𝜇𝐸_𝑖−1𝑡⁻_𝑘−𝐸^𝑡_𝑡𝑘Γ(𝑠)𝑉𝑑𝑠=𝜇𝐸_𝑖−1𝑡_𝑘−1+𝜇𝐸^𝑡𝑘_𝑡𝑘−1ℒ𝑉_𝑖−1(𝑠)+Γ(𝑠)𝑑𝑠−𝜇𝐸^𝑡𝑘_𝑡𝑘−1Γ(𝑠)𝑑𝑠−𝐸^𝑡_𝑡𝑘Γ(𝑠)𝑉𝑑𝑠<𝜇𝐸_𝑖−1𝑡_𝑘−1−𝜇𝐸^𝑡𝑘_𝑡𝑘−1Γ(𝑠)𝑑𝑠−𝐸^𝑡_𝑡𝑘Γ(𝑠)𝑑𝑠<⋯<𝜇^{𝑁𝜎(𝑡0,𝑡)}𝐸𝑉𝑡₀−𝜇^{𝑁𝜎(𝑡0,𝑡)}𝐸^𝑡1_𝑡0Γ(𝑠)𝑑𝑠−⋯−𝐸^𝑡_𝑡𝑘Γ(𝑠)𝑑𝑠=𝜇^{𝑁𝜎(𝑡0,𝑡)}𝐸𝑉𝑡₀−𝐸^𝑡_𝑡0𝑒^{𝑁𝜎(𝑠,𝑡)ln𝜇}Γ(𝑠)𝑑𝑠. (3.34) Under zero initial condition, we can obtain 𝐸^𝑡_𝑡0𝑒^{𝑁𝜎(𝑠,𝑡)ln𝜇}Γ(𝑠)𝑑𝑠<0. (3.35) Moreover, we have 𝐸^𝑡_𝑡0𝑒^{𝑁𝜎(𝑠,𝑡)ln𝜇}𝑧^𝑇(𝑠)𝑧(𝑠)𝑑𝑠<𝛾^2𝑡_𝑡0𝑒^{𝑁𝜎(𝑠,𝑡)ln𝜇}𝑣^𝑇(𝑠)𝑣(𝑠)𝑑𝑠. (3.36) Multiplying both sides of (3.36) by 𝑒^{−𝑁𝜎(𝑡0,𝑡)ln𝜇} leads to 𝐸^𝑡_𝑡0𝑒^{−𝑁𝜎(𝑡0,𝑠)ln𝜇}𝑧^𝑇(𝑠)𝑧(𝑠)𝑑𝑠<𝛾^2𝑡_𝑡0𝑒^{−𝑁𝜎(𝑡0,𝑠)ln𝜇}𝑣^𝑇(𝑠)𝑣(𝑠)𝑑𝑠. (3.37) Noticing 𝑁_𝜎(𝑡₀,𝑠)≤((𝑠−𝑡₀)/𝑇_𝛼) and 𝑇_𝛼≥𝑇^∗_𝛼=ln𝜇/𝛼, we have 𝐸^𝑡_𝑡0𝑒^{−𝛼(𝑠−𝑡0)}𝑧^𝑇(𝑠)𝑧(𝑠)𝑑𝑠<𝛾^2𝑡_𝑡0𝑣^𝑇(𝑠)𝑣(𝑠)𝑑𝑠. (3.38) When 𝑡→∞, it leads to 𝐸^∞_𝑡0𝑒^{−𝛼(𝑠−𝑡0)}𝑧^𝑇(𝑠)𝑧(𝑠)𝑑𝑠<𝛾^2∞_𝑡0𝑣^𝑇(𝑠)𝑣(𝑠)𝑑𝑠. (3.39) The proof is completed.

Remark 3.5. When 𝑑𝑤(𝑡)=0, system (3.21) is reduced to a switched delay system, which can be described as 𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐵_𝜎(𝑡)𝑥(𝑡−ℎ(𝑡))+𝐺_𝜎(𝑡)𝑣(𝑡),𝑥(𝑡)=𝜑(𝑡),𝑡∈𝑡₀−ℎ,𝑡₀. (3.40) Using the method proposed in Theorem 3.4, we can obtain the following conclusion.

Corollary 3.6. Considering system (3.40), for a given scalar 𝛼>0, if there exist symmetric positive definite matrices 𝑃_𝑖,𝑄_𝑖>0 such that 𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝛼𝑃_𝑖+𝑄_𝑖𝑃_𝑖𝐵_𝑖𝑃_𝑖𝐺_𝑖𝑀^𝑇_𝑖∗−1−ℎ_𝑑𝑄_𝑖00∗∗−𝛾²𝐼0∗∗∗−𝐼<0 (3.41) hold for all 𝑖∈𝑁, system (3.40) is said to have weighted 𝐻_∞ performance 𝛾 under arbitrary switching signal with the average dwell time scheme (3.23).

3.3. Design of Robust 𝐻_∞ Controller

In this subsection, the following robust 𝐻_∞ controller𝑢(𝑡)=𝐾_𝜎(𝑡)𝑥(𝑡) (3.42) will be designed for system (2.1). Then the corresponding closed-loop system can be described as𝑑𝑥(𝑡)=𝐴_𝜎(𝑡)+𝐶_𝜎(𝑡)𝐾_𝜎(𝑡)𝑥(𝑡)+𝐵_𝜎(𝑡)𝑥(𝑡−ℎ(𝑡))+𝐺_𝜎(𝑡)𝑣(𝑡)𝑑𝑡+𝐷_𝜎(𝑡)𝑥(𝑡)(𝑡),𝑥(𝑡)(𝑡)𝑡𝑑𝑤=𝜑,𝑡∈₀−ℎ,𝑡₀,𝑧(𝑡)=𝑀_𝜎(𝑡)𝑥(𝑡). (3.43)

Theorem 3.7. Considering system (2.1), for given scalars 𝛼, 𝜀_𝑖>0, ℎ_𝑑<1, if there exist matrix 𝑍_𝑖, symmetric positive definite matrices 𝑋_𝑖, 𝑌_𝑖> 0 such that Σ^𝑖₁₁𝐵_𝑖𝑋_𝑖𝐺_𝑖𝑋_𝑖𝐷^𝑇_𝑖𝑋_𝑖𝑀^𝑇_𝑖𝑋_𝑖𝐸^𝑇_1𝑖𝑋_𝑖𝐸^𝑇_3𝑖∗Σ^𝑖₂₂000𝑋_𝑖𝐸^𝑇_2𝑖0∗∗−𝛾²𝐼0000∗∗∗Σ^𝑖₄₄000∗∗∗∗−𝐼00∗∗∗∗∗−𝜀_𝑖𝐼0∗∗∗∗∗∗−𝜀_𝑖𝐼<0 (3.44) holds for all 𝑖∈𝑁, with the average dwell time: 𝑇_𝛼≥𝑇^∗_𝛼=ln𝜇𝛼, (3.45) where 𝜇≥1 satisfies 𝑋_𝑗≤𝜇𝑋_𝑖,𝑋_𝑖⁻¹𝑌_𝑖𝑋_𝑖⁻¹≤𝜇𝑋_𝑗⁻¹𝑌_𝑗𝑋_𝑗⁻¹,∀𝑖,𝑗∈𝑁, (3.46) then, there exists a robust 𝐻_∞ controller: 𝑢(𝑡)=𝐾_𝜎(𝑡)𝑥(𝑡),𝐾_𝑖=𝑍_𝑖𝑋_𝑖⁻¹, (3.47) which can render the corresponding closed-loop system (3.43) mean-square exponentially stable with weighted 𝐻_∞ performance 𝛾, where ^𝑖₁₁=𝑋_𝑖𝐴^𝑇_𝑖+𝑍^𝑇_𝑖𝐶^𝑇_𝑖+𝐴_𝑖𝑋_𝑖+𝐶_𝑖𝑍_𝑖+𝛼𝑋_𝑖+𝑌_𝑖+𝜀_𝑖𝐻_𝑖𝐻^𝑇_𝑖,_𝑖22=−1−ℎ_𝑑𝑌_𝑖,_𝑖44=𝜀_𝑖𝐻_𝑖𝐻^𝑇_𝑖−𝑋_𝑖. (3.48)

Proof. By Theorem 3.4, system (3.43) is mean-square exponentially stable with weighted 𝐻_∞ performance 𝛾 if the following inequalities are satisfied: Λ^𝑖₁₁𝑃_𝑖𝐵_𝑖𝑃_𝑖𝐺_𝑖𝐷^𝑇_𝑖𝑃_𝑖𝑀^𝑇_𝑖∗−1−ℎ_𝑑𝑄_𝑖000∗∗−𝛾²𝐼00∗∗∗−𝑃_𝑖0∗∗∗∗−𝐼<0, (3.49) where Λ^𝑖₁₁=(𝐴_𝑖+𝐶_𝑖𝐾_𝑖)^𝑇𝑃_𝑖+𝑃_𝑖(𝐴_𝑖+𝐶_𝑖𝐾_𝑖)+𝛼𝑃_𝑖+𝑄_𝑖.
Then using Λ_𝑃𝑖=diag{𝑃_𝑖⁻¹,𝑃_𝑖⁻¹,𝐼,𝑃_𝑖⁻¹,𝐼} to pre- and postmultiply Λ^𝑖, we have 𝑇^𝑖=Λ_𝑃𝑖Λ^𝑖Λ_𝑃𝑖<0. (3.50)
Furthermore, 𝑇^𝑖=𝑇^𝑖₁₁𝐵_𝑖𝑋_𝑖𝐺_𝑖𝑋_𝑖𝐷^𝑇_𝑖𝑋_𝑖𝑀^𝑇_𝑖∗𝑇^𝑖₂₂000∗∗−𝛾²𝐼00∗∗∗−𝑋_𝑖0∗∗∗∗−𝐼<0, (3.51) where 𝑇^𝑖₁₁=𝑋_𝑖(𝐴_𝑖+𝐶_𝑖𝐾_𝑖)^𝑇+(𝐴_𝑖+𝐶_𝑖𝐾_𝑖)𝑋_𝑖+𝛼𝑋_𝑖+𝑌_𝑖, 𝑇^𝑖₂₂=−(1−ℎ_𝑑)𝑌_𝑖, 𝑋_𝑖=𝑃_𝑖⁻¹, and 𝑌_𝑖=𝑃_𝑖⁻¹𝑄_𝑖𝑃_𝑖⁻¹.
Combining (3.51) with (2.4), we have 𝑇^𝑖=𝑇^𝑖+Δ𝑇^𝑖, (3.52) where 𝑇^𝑖=𝑇^𝑖₁₁𝐵_𝑖𝑋_𝑖𝐺_𝑖𝑋_𝑖𝐷^𝑇_𝑖𝑋_𝑖𝑀^𝑇_𝑖∗𝑇^𝑖₂₂000∗∗−𝛾²𝐼00∗∗∗−𝑋_𝑖0∗∗∗∗−𝐼<0,Δ𝑇^𝑖=𝑋_𝑖𝐸^𝑇_1𝑖𝐹^𝑇_𝑖𝐻^𝑇_𝑖+𝐻_𝑖𝐹_𝑖𝐸_1𝑖𝑋_𝑖𝐻_𝑖𝐹_𝑖𝐸_2𝑖𝑋_𝑖0𝑋_𝑖𝐸^𝑇_3𝑖𝐹^𝑇_𝑖𝐻^𝑇_𝑖0𝑋_𝑖𝐸^𝑇_2𝑖𝐹^𝑇_𝑖𝐻^𝑇_𝑖𝐻000000000_𝑖𝐹_𝑖𝐸_3𝑖𝑋_𝑖𝑇000000000_𝑖11=𝑋_𝑖𝐴_𝑖+𝐶_𝑖𝐾_𝑖^𝑇+𝐴_𝑖+𝐶_𝑖𝐾_𝑖𝑋_𝑖+𝛼𝑋_𝑖+𝑌_𝑖,and𝑇^𝑖₂₂=−1−ℎ_𝑑𝑌_𝑖. (3.53)
Moreover, Δ𝑇^𝑖=𝐻_𝑖000000𝐻_𝑖𝐹00^𝑖00𝐹_𝑖𝑋_𝑖𝐸^𝑇_1𝑖𝑋_𝑖𝐸^𝑇_3𝑖𝑋_𝑖𝐸^𝑇_2𝑖0000000^𝑇+𝑋_𝑖𝐸^𝑇_1𝑖𝑋_𝑖𝐸^𝑇_3𝑖𝑋_𝑖𝐸^𝑇_2𝑖0𝐹000000^𝑖00𝐹_𝑖^𝑇𝐻_𝑖000000𝐻_𝑖00^𝑇. (3.54) According to Lemma 2.5, we have Δ𝑇^𝑖≤𝜀_𝑖𝐻_𝑖000000𝐻_𝑖𝐻00^𝑖000000𝐻_𝑖00^𝑇+𝜀_𝑖⁻¹𝑋_𝑖𝐸^𝑇_1𝑖𝑋_𝑖𝐸^𝑇_3𝑖𝑋_𝑖𝐸^𝑇_2𝑖0𝑋000000^𝑖𝐸^𝑇_1𝑖𝑋_𝑖𝐸^𝑇_3𝑖𝑋_𝑖𝐸^𝑇_2𝑖0000000^𝑇. (3.55) Substituting (3.55) to (3.52), and using Schur complement, we can obtain that (3.52) is equivalent to (3.44). Denoting 𝑋_𝑖=𝑃_𝑖⁻¹, and 𝑌_𝑖=𝑃_𝑖⁻¹𝑄_𝑖𝑃_𝑖⁻¹, it is easy to get that (3.46) is equivalent to (3.24).
The proof is completed.

Remark 3.8. Theorem 3.4 presents the sufficient conditions which could guarantee that the switched stochastic delay system is stable with 𝐻_∞ performance; when the robust 𝐻_∞ control problem is considered, we can solve the problem by substituting the closed-loop uncertain parameters 𝐴_𝑖+𝐶_𝑖𝐾_𝑖, 𝐵_𝑖, 𝐷_𝑖 to Theorem 3.4.

4. Numerical Example

In this section, a numerical example is given to illustrate the effectiveness of the proposed approach. Consider system (2.1) with the following parameters:𝐴₁=20.91.4−2,𝐵₁=−0.300.1−0.5,𝐶₁=2132,𝐷₁=,𝐺−0.300.10.2₁=−0.800−0.1,𝑀₁=21−13,𝐸₁₁=0.700.90,𝐸₂₁=,𝐸0.2000.8₃₁=0.6000,𝐻₁=0.4000.8,𝐴₂=−210−3,𝐵₂=,𝐶−0.40.10−0.8₂=1213,𝐷₂=−0.20.050−0.4,𝐺₂=−0.100−0.9,𝑀₂=,𝐸1−231₁₂=0.2000,𝐸₂₂=0.300.10,𝐸₃₂=00.400.5,𝐻₂=,𝐹0.5000.3₁=sin𝑡00sin𝑡,𝐹₂=,cos𝑡00cos𝑡 (4.1) the disturbance input 𝑣(𝑡)=[_50𝑒^−0.5𝑡_10𝑒^−0.5𝑡]^𝑇.

Let 𝛼=0.6, ℎ(𝑡)=1+0.5sin𝑡, 𝜀₁=𝜀₂=1, 𝛾=1; then solving the LMIs in Theorem 3.7, we have𝑋₁=1.05270.03920.03921.0410,𝑌₁=6.01160.06590.06596.5879,𝑍₁=,𝐾−17.083730.073113.0224−37.1398₁=−17.329529.540113.7192−36.1921,𝑋₂=0.9153−0.0358−0.03581.0266,𝑌₂=,𝑍6.06510.06400.06405.4975₂=−26.15264.006413.4430−5.1025,𝐾₂=−28.45822.909914.5120−4.4641,𝜇=8.3261. (4.2) Then we obtain that 𝑇^∗_𝛼=ln𝜇/𝛼=3.5323. Thus, under the average dwell time 𝑇_𝛼>𝑇^∗_𝛼, the designed controller can guarantee that the corresponding closed-loop system is mean-square exponentially stable with weighted 𝐻_∞ performance.

Simulation results are shown in Figures 1–3, where the initial state 𝑥(𝑡)=[0,0]^𝑇, 𝑡∈[−ℎ,0), and 𝑥(0)=[2,−2]^𝑇. The switching signal with the average dwell time 𝑇_𝛼=4 is shown in Figure 1, and Figures 2 and 3 show state 𝑥₁ and 𝑥₂ of the closed-loop system, respectively.

Figure 1 

Switching signal.

Figure 2 

State 𝑥1 of the closed-loop system.

Figure 3 

State 𝑥2 of the closed-loop system.

5. Conclusions

In this paper, the exponential stability analysis and robust 𝐻_∞ control for switched stochastic time delay systems have been investigated. Based on the average dwell time method and Gronwall-Bellman inequality, a new mean-square exponential stability criteria and 𝐻_∞ performance analysis are presented. Furthermore, robust 𝐻_∞ controller is designed to guarantee that the corresponding closed-loop system is mean-square exponentially stable. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach. The proposed method provides a powerful tool to solve many other problems such as controller design under asynchronous switching and actuator failures. These problems are the topics of the future research.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. 60974027 and NUST Research Funding (2011YBXM26).