Observer-Based Robust 𝐻_∞ Control for Switched Stochastic Systems with Time-Varying Delay

Abstract

This paper investigates the problem of observer-based robust 𝐻_∞ control for a class of switched stochastic systems with time-varying delay. Based on the average dwell time method, an exponential stability criterion for switched stochastic delay systems is proposed. Then, 𝐻_∞ performance analysis and observer-based robust 𝐻_∞ controller design for the underlying systems are developed. Finally, a numerical example is presented to illustrate the effectiveness of the proposed approach.

1. Introduction

Switched systems are a kind of hybrid dynamical systems composed of a set of continuous-time subsystems or discrete-time subsystems and a switching law that orchestrates the switching between them. Switched systems have attracted increasing attention during the past decades because of their wide applications in real-world systems, such as robot control systems [1], networked control systems [2, 3]. Many useful results on stability analysis and control synthesis for such systems have been reported in [4–8]. For example, 𝐻_∞ control of switched linear discrete-time systems with polytopic uncertainties was investigated in [8].

It is well known that the time delay phenomenon is frequently encountered in engineering and social systems, and the existence of which may cause instability or undesirable system performance. Therefore, many research efforts have been devoted to the study of switched time delay systems [9–15]. On the other hand, stochastic systems have attracted considerable attention during the past several decades. Early results can be found in [16], and the 𝐻_∞ control problem of stochastic systems with time delay was investigated in [17, 18]. The study on 𝐻₂/𝐻_∞ control of stochastic system was developed in [19]. Stability analysis on stochastic system with multiple delays was proposed in [20]. Moreover, some results on switched stochastic systems with and without time delay have been obtained (see [21–25] and the references cited therein).

In many real-world systems, state feedback control will fail to guarantee the stabilization because the states of the systems are not all measurable [26]. One of the key approaches to solve the problem is to reconstruct the states of the systems and realize the required feedback control. Hence, the observer-based control has been an interesting topic in control theory. Some results on observer-based control for stochastic delay systems or Markovian jump systems have been presented in [27–29]. However, to the best of our knowledge, the problem of observer-based robust 𝐻_∞ control for switched stochastic systems with time delay has not been fully studied, which motivates the present study.

In this paper, we aim to design an observer-based robust 𝐻_∞ controller for switched stochastic systems with time delay such that the closed-loop system is mean-square exponentially stable with 𝐻_∞ performance. The major contributions of the work can be summarized as follows: (1) a new Lyapunov-Krasovskii functional candidate is introduced to derive the exponential stability of switched stochastic systems with time delay, and the free-weighting matrix method is employed to reduce the conservatism; (2) an observer-based robust 𝐻_∞ controller design scheme for the underlying systems 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, the main results are presented. 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. In this paper, the superscript “T” denotes the transpose, and the symmetric term in a matrix is denoted by ∗. The notation 𝑋>𝑌   (𝑋≥𝑌) means that 𝑋−𝑌 is positive definite (positive semidefinite, respectively). 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space. ‖𝑥(𝑡)‖ denotes the Euclidean norm. |𝑎| denotes the absolute value of 𝑎. 𝐿₂[𝑡₀,∞) is the space of square integrable functions on [𝑡₀,∞), and 𝑡₀ is the initial time. 𝜆_max(𝑃) and 𝜆_min(𝑃) denote the maximum and minimum eigenvalues of 𝑃, respectively. 𝐴⁺ denotes the Moore-Penrose pseudoinverse of 𝐴. 𝐼 is the identity matrix. diag{𝑎_𝑖} denotes a diagonal matrix with the diagonal elements 𝑎_𝑖,𝑖=1,2,…,𝑛.

2. Problem Formulation and Preliminaries

Consider the following switched stochastic system with time delay:𝑑𝑥(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝜏𝜎(𝑡)}𝑥(𝑡−𝜏(𝑡))+𝐵_𝜎(𝑡)𝑢(𝑡)+𝐺_𝜎(𝑡)𝑣(𝑡+𝐷)𝑑𝑡_𝜎(𝑡)𝑥(𝑡)𝑑𝑤(𝑡),𝑦(𝑡)=𝐶_𝜎(𝑡)𝑥(𝑡),𝑧(𝑡)=𝐽_𝜎(𝑡)𝑥(𝑡),𝑥(𝑡)=𝜑(𝑡),𝑡∈𝑡₀−𝜏,𝑡₀, (1a)(1b)(1c)(1d)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 to belong to 𝐿₂[𝑡₀,∞], 𝑦(𝑡)∈𝑅^𝑟 is the measurable output, 𝑧(𝑡)∈𝑅^𝑞 is the controlled output, and 𝑤(𝑡) is a one-dimensional zero-mean Wiener process on a probability space (Ω𝐹𝑃) and satisfies 𝐸{𝑑𝑤(𝑡)}=0,𝐸𝑑𝑤²(𝑡)=𝑑𝑡, (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. 𝜏(𝑡) is the time delay satisfying 0≤𝜏(𝑡)≤𝜏,𝜏̇(𝑡)≤𝜏_𝑑<1, (3) where 𝜏 and 𝜏_𝑑 are known constants.

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, where 𝑡₀ is the initial instant and 𝑡_𝑘 denotes the 𝑘th switching instant. Moreover,  𝜎(𝑡)=𝑖 means that the 𝑖th subsystem is activated. For all 𝑖∈𝑁, 𝐵_𝑖, 𝐶_𝑖, 𝐽_𝑖, and 𝐺_𝑖 are known real-value matrices with appropriate dimensions, 𝐴_𝑖, 𝐴_𝜏𝑖, and 𝐷_𝑖 are uncertain real matrices with appropriate dimensions and can be written as 𝐴_𝑖𝐴_𝜏𝑖𝐷_𝑖=𝐴_𝑖+Δ𝐴_𝑖𝐴_𝜏𝑖+Δ𝐴_𝜏𝑖𝐷_𝑖+Δ𝐷_𝑖, (4) where Δ𝐴_𝑖Δ𝐴_𝜏𝑖Δ𝐷_𝑖=𝐻_𝑖𝐹_𝑖(𝑡)𝐸_1𝑖𝐸_2𝑖𝐸_3𝑖, 𝐴_𝑖, 𝐴_𝜏𝑖, 𝐷_𝑖, 𝐻_𝑖, 𝐸_1𝑖, 𝐸_2𝑖, and 𝐸_3𝑖 are known real-value matrices with appropriate dimensions, and 𝐹_𝑖(𝑡) is an unknown time-varying matrix that satisfies 𝐹^𝑇_𝑖(𝑡)𝐹_𝑖(𝑡)≤𝐼. (5)

The state feedback controller is designed as 𝑢(𝑡)=𝐾_𝜎(𝑡)𝑥(𝑡). In actual operation, however, the states of the systems are not all measurable. The following switched system is constructed to estimate the state of system (1a), (1b), (1c), and (1d):𝑑𝑥(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝜏𝜎(𝑡)}𝑥(𝑡−𝜏(𝑡))+𝐵_𝜎(𝑡)𝑢(𝑡)+𝐿_𝜎(𝑡)𝑦(𝑡)−𝑦(𝑡)𝑑𝑡,𝑦(𝑡)=𝐶_𝜎(𝑡)𝑥(𝑡),𝑥(𝑡)=𝜙(𝑡),𝑡∈𝑡₀−𝜏,𝑡₀, (6a)(6b)(6c)where 𝑥(𝑡)∈𝑅^𝑛 is the estimation of 𝑥(𝑡), 𝑦(𝑡)∈𝑅^𝑟 is the observer output, and 𝜙(𝑡)∈𝑅^𝑛 is the initial observer state function. The real state feedback controller becomes 𝑢(𝑡)=𝐾_𝜎(𝑡)𝑥(𝑡). 𝐿_𝑖 and 𝐾_𝑖 are the observer gains and controller gains to be determined, respectively.

Remark 1. It is noted that the observer-based 𝐻_∞ control for stochastic systems or Markovian jump systems was considered in [27–29]. However, the results in the aforementioned papers cannot be directly applied to the switched stochastic system considered in the paper. This motivates our study. Also, the proposed observer in (6a), (6b), and (6c) is a switching observer, which is different from the existing ones given in [27–29].

From systems (1a), (1b), (1c), and (1d) and (6a), (6b), and (6c), we can obtain the following augmented closed-loop system:𝑑𝜉(𝑡)=𝐴_𝜎(𝑡)𝜉(𝑡)+𝐴_{𝜏𝜎(𝑡)}𝜉((𝑡+𝑡−𝜏))𝐺_𝜎(𝑡)𝑣(𝑡)𝑑𝑡+𝐷_𝜎(𝑡)𝜉(𝑡)𝑑𝑤(𝑡),𝑧(𝑡)=𝐽_𝜎(𝑡)𝜉(𝑡),𝜉(𝑡)=𝜑^𝑇(𝑡)𝜑^𝑇(𝑡)−𝜙^𝑇(𝑡)^𝑇,𝑡∈𝑡₀−𝜏,𝑡₀, (7a)(7b)(7c)where 𝜉(𝑡)=𝑥^𝑇(𝑡)𝑒^𝑇(𝑡)^𝑇,  𝑒(𝑡)=𝑥(𝑡)−𝑥(𝑡) denotes the state estimated error.

For 𝜎(𝑡)=𝑖, the parameters of system (7a), (7b), and (7c) are given as follows: 𝐴_𝑖=𝐴_𝑖+Δ𝐴_𝑖,𝐴_𝑖=𝐴_𝑖+𝐵_𝑖𝐾_𝑖−𝐵_𝑖𝐾_𝑖0𝐴_𝑖−𝐿_𝑖𝐶_𝑖,Δ𝐴_𝑖=𝐻_𝑖𝐹_𝑖𝐸_1𝑖,𝐴_𝜏𝑖=𝐴_𝜏𝑖+Δ𝐴_𝜏𝑖,𝐴_𝜏𝑖=𝐴_𝜏𝑖00𝐴_𝜏𝑖,Δ𝐴_𝜏𝑖=𝐻_𝑖𝐹_𝑖𝐸_2𝑖,𝐷_𝑖=𝐷_𝑖+Δ𝐷_𝑖,𝐷_𝑖=𝐷_𝑖0𝐷_𝑖0,Δ𝐷_𝑖=𝐻_𝑖𝐹_𝑖𝐸_3𝑖,𝐺_𝑖=𝐺_𝑖=𝐺^𝑇_𝑖𝐺^𝑇_𝑖^𝑇,𝐽_𝑖=𝐽_𝑖̃=𝐽_𝑖0,𝐻_𝑖=𝐻_𝑖00𝐻_𝑖,𝐹_𝑖=𝐹_𝑖00𝐹_𝑖,𝐸_𝑔𝑖=𝐸_𝑔𝑖0𝐸_𝑔𝑖0,𝑔=1,2,3. (8)

Assumption 2. 𝐵_𝑖 is full row rank, for all 𝑖∈𝑁.

Definition 3. System (1a), (1b), (1c), and (1d) with 𝑣(𝑡)=0 is said to be mean-square exponentially stable under the switching signal 𝜎(𝑡) if there exist scalars 𝜅>0 and 𝛼>0, such that the solution 𝑥(𝑡) of the system satisfies 𝐸‖𝑥(𝑡)‖²≤𝜅𝑒^{−𝛼(𝑡−𝑡0)}sup_{−𝜏≤𝜃≤0}𝐸𝑥𝑡₀+𝜃²,∀𝑡≥𝑡₀. (9)

Definition 4 (see [24]). 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 (10) holds for given constants 𝑁₀≥0 and 𝑇_𝛼>0, then the constant 𝑇_𝛼 is called the average dwell time. As commonly used in the literature, some chooses 𝑁₀=0.

Definition 5 (see [30]). For any 𝜆>0 and 𝛾>0, system (7a), (7b), and (7c) is said to be mean-square exponentially stable with a prescribed weighted 𝐻_∞ performance level 𝛾 if the following conditions are satisfied:(1)when 𝑣(𝑡)=0, system (7a), (7b), and (7c) is mean-square exponentially stable;(2)under the zero initial condition, the output 𝑧(𝑡) satisfies 𝐸^∞_𝑡0𝑒^{−𝜆(𝑠−𝑡0)}𝑧^𝑇(𝑠)𝑧(𝑠)𝑑𝑠≤𝛾^2∞_𝑡0𝑣^𝑇(𝑠)𝑣(𝑠)𝑑𝑠,∀𝑣(𝑡)∈𝐿₂𝑡₀,∞. (11)

Lemma 6 (see [31]). For any positive symmetric constant matrix 𝑀∈𝑅^𝑛×𝑛 and a scalar 𝑟>0, if there exists a vector function𝑔∶[0,𝑟]→𝑅^𝑛  such that integrations in the following are well defined, then the following inequality holds 𝑟^𝑟₀𝑔^𝑇(𝑠)(𝑠)≥𝑀𝑔𝑑𝑠_𝑟0𝑔(𝑠)𝑑𝑠^𝑇𝑀^𝑟₀𝑔(𝑠)𝑑𝑠. (12)

Lemma 7 (see [32]). 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.

The objective of this paper is to design an observer-based robust 𝐻_∞ controller for switched stochastic delay system (1a), (1b), (1c), and (1d) such that the augmented closed-loop system (7a), (7b), and (7c) is mean-square exponentially stable with a prescribed weighted 𝐻_∞ performance level 𝛾.

3. Main Results

3.1. Stability Analysis

In this subsection, in order to obtain the main results, we first focus on the problem of stability analysis for the following switched stochastic systems with time delay𝑑𝑥(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝜏𝜎(𝑡)}𝑥(𝑡−𝜏(𝑡))𝑑𝑡+𝐷_𝜎(𝑡)𝑥(𝑡)𝑑𝑤(𝑡),𝑥(𝑡)=𝜑(𝑡),𝑡∈𝑡₀−𝜏,𝑡₀. (13a)(13b)

Theorem 8. For a given scalar 𝛼>0, if there exist symmetric positive definite matrices 𝑃_𝑖,  𝑄_𝑖, and 𝑅_𝑖 and any matrices 𝑆_𝑖 such that ∑^𝑖₁₁𝑃_𝑖𝐴_𝜏𝑖𝑆^𝑇_𝑖𝑆^𝑇_𝑖𝐷^𝑇_𝑖𝑃_𝑖∗−1−𝜏_𝑑𝑄_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖0∗∗−𝑆_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖0∗∗∗−𝜏⁻¹𝑅_𝑖0∗∗∗∗−𝑃_𝑖<0,∀𝑖∈𝑁 (14) then system (13a) and (13b) is mean-square exponentially stable under arbitrary switching signal with the average dwell time 𝑇_𝛼>𝑇^∗_𝛼=𝜏+ln𝜒𝜇𝜆, (15) where 𝜇, 𝜒, and 𝜆 satisfy 𝑃_𝑖≤𝜇𝑃_𝑗,𝑄_𝑖≤𝜇𝑄_𝑗,𝑅_𝑖≤𝜇𝑅_𝑗,𝑄_𝑖≤𝛽_𝑖𝑃_𝑖,𝑅_𝑖≤𝛽_𝑖𝑃_𝑖,∀𝑖,𝑗∈𝑁,𝜆+𝛽_𝑖(1+𝜏)𝑒^𝜆𝜏−1≤𝛼,𝜒=max_𝑖∈𝑁𝜒_𝑖,𝜒_𝑖=1+𝜏𝛽_𝑖(1+𝜏)𝑒^𝜆𝜏−1_𝑖11,=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝑄_𝑖+𝜏𝑅_𝑖+𝛼𝑃_𝑖. (16)(17)

Proof. Let 𝑌(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝜏𝜎(𝑡)}𝑥(𝑡−𝜏(𝑡)), then (13a) can be described as 𝑑𝑥(𝑡)=𝑌(𝑡)𝑑𝑡+𝐷_𝜎(𝑡)𝑥(𝑡)𝑑𝑤(𝑡). (18)
Choose the following Lyapunov functional candidate for the 𝑖th subsystem 𝑉_𝑖(𝑡,𝑥(𝑡))=𝑉_1,𝑖(𝑡,𝑥(𝑡))+𝑉_2,𝑖(𝑡,𝑥(𝑡))+𝑉_3,𝑖(𝑡,𝑥(𝑡)), (19) where 𝑉_1,𝑖(𝑡,𝑥(𝑡))=𝑥^𝑇(𝑡)𝑃_𝑖𝑥(𝑡),𝑉_2,𝑖(𝑡,𝑥(𝑡))=^𝑡_{(𝑡)𝑡−𝜏}𝑥^𝑇(𝑠)𝑄_𝑖𝑥(𝑠)𝑉𝑑𝑠,_3,𝑖(𝑡,𝑥(𝑡))=^𝜏₀^𝑡_𝑡−𝜃𝑥^𝑇(𝑠)𝑅_𝑖𝑥(𝑠)𝑑𝑠𝑑𝜃. (20)
For the sake of simplicity, 𝑉_𝑖(𝑡,𝑥(𝑡)) is written as 𝑉_𝑖(𝑡) in this paper. According to Itô’s formula, along the trajectory of the 𝑖th subsystem, we have 𝑑𝑉_𝑖(𝑡)=³_𝑔=1𝑑𝑉_𝑔,𝑖, (21) where 𝑑𝑉_1,𝑖(𝑡)=ℒ𝑉_1,𝑖(𝑡)𝑑𝑡+2𝑥^𝑇(𝑡)𝑃_𝑖𝐷_𝑖𝑥(𝑡)𝑑𝑤(𝑡),ℒ𝑉_1,𝑖(𝑡)=2𝑥^𝑇(𝑡)𝑃_𝑖𝐴_𝑖𝑥(𝑡)+𝐴_𝜏𝑖𝑥(𝑡−𝜏(𝑡))+𝑥^𝑇(𝑡)𝐷^𝑇_𝑖𝑃_𝑖𝐷_𝑖𝑥(𝑡),𝑑𝑉_2,𝑖(𝑡)=𝑥^𝑇(𝑡)𝑄_𝑖𝑥(𝑡)−(1−𝜏̇(𝑡))𝑥^𝑇(𝑡−𝜏(𝑡))×𝑄_𝑖𝑥((𝑡≤𝑥𝑡−𝜏))𝑑𝑡_𝑇(𝑡)𝑄_𝑖𝑥(𝑡)−1−𝜏_𝑑𝑥^𝑇(𝑡−𝜏(𝑡))×𝑄_𝑖𝑥(𝑡−𝜏(𝑡))𝑑𝑡,𝑑𝑉_3,𝑖(𝑡)=𝜏𝑥^𝑇(𝑡)𝑅_𝑖𝑥(𝑡)−^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝑅_𝑖𝑥(𝑠)𝑑𝑠𝑑𝑡. (22)
According to Lemma 6, we have𝑑𝑉_3,𝑖(𝑡)≤𝜏𝑥^𝑇(𝑡)𝑅_𝑖𝑥(𝑡)−𝜏^−1𝑡_{𝑡−𝜏(𝑡)}𝑥(𝑠)𝑑𝑠^𝑇×𝑅_𝑖^𝑡_{(𝑡)𝑡−𝜏}𝑥(𝑠)𝑑𝑠𝑑𝑡. (23)
Integrating both sides of (18) from 𝑡−𝜏(𝑡) to 𝑡, we have 𝑥(𝑡)−𝑥(𝑡−𝜏(𝑡))=^𝑡_{𝑡−𝜏(𝑡)}𝑌(𝑠)+𝑑𝑠_{𝑡𝑡−𝜏(𝑡)}𝐷_𝜎(𝑠)𝑥(𝑠)𝑑𝑤(𝑠). (24)
Thus, 2^𝑡_{(𝑡)𝑡−𝜏}𝑥(𝑠)𝑑𝑠+^𝑡_{(𝑡)𝑡−𝜏}𝑌(𝑠)𝑑𝑠^𝑇𝑆_𝑖𝜂(𝑡)𝑑𝑡=0, (25) where 𝜂(𝑡)=𝑥(𝑡)−𝑥(𝑡−𝜏(𝑡))−∫^𝑡_{𝑡−𝜏(𝑡)}𝑌(𝑠)𝑑𝑠−∫^𝑡_{𝑡−𝜏(𝑡)}𝐷_𝜎(𝑠)𝑥(𝑠)𝑑𝑤(𝑠).
Combining (21)–(25) leads to 𝑑𝑉_𝑖(𝑡)≤ℒ𝑉_𝑖(𝑡)𝑑𝑡+𝑊_𝑖(𝑡), (26) where ℒ𝑉_𝑖(𝑡)=𝜍^𝑇(𝑡)Θ_𝑖𝜍(𝑡),𝑊_𝑖(𝑡)=2𝑥^𝑇(𝑡)𝑃_𝑖𝐷_𝑖𝑥(𝑡)𝑑𝑤(𝑡)−2^𝑡_{𝑡−𝜏(𝑡)}𝑥(𝑠)𝑑𝑠+^𝑡_{𝑡−𝜏(𝑡)}𝑌(𝑠)𝑑𝑠^𝑇×𝑆_𝑖^𝑡_{(𝑡)𝑡−𝜏}𝐷_𝜎(𝑠)𝑥(𝑠)(𝑠𝜍(𝑡)=𝑥𝑑𝑤)𝑑𝑡,_𝑇(𝑡)𝑥^𝑇(𝑡−𝜏(𝑡))∫^𝑡_{(𝑡)𝑡−𝜏}𝑌^𝑇(𝑠)𝑑𝑠∫^𝑡_{(𝑡)𝑡−𝜏}𝑥^𝑇(𝑠)𝑑𝑠^𝑇,Θ_𝑖=Θ^𝑖₁₁𝑃_𝑖𝐴_𝜏𝑖𝑆^𝑇_𝑖𝑆^𝑇_𝑖∗−1−𝜏_𝑑𝑄_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖∗∗−𝑆_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖∗∗∗−𝜏⁻¹𝑅_𝑖,Θ^𝑖₁₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝑄_𝑖+𝐷^𝑇_𝑖𝑃_𝑖𝐷_𝑖+𝜏𝑅_𝑖. (27)
By using the Schur complement, we obtain from (14) that ℒ𝑉_𝑖(𝑡)<−𝛼𝑉_1,𝑖(𝑡)<0. (28)
According to (26), one obtains that 𝑑𝑉_𝑖(𝑡)≤ℒ𝑉_𝑖(𝑡)𝑑𝑡+𝑊_𝑖(𝑡)<−𝛼𝑉_1,𝑖(𝑡)𝑑𝑡+𝑊_𝑖(𝑡). (29)
Then, taking mathematical expectation, we have 𝐸𝑑𝑉_𝑖(𝑡)𝑑𝑡≤𝐸ℒ𝑉_𝑖(𝑡)<−𝛼𝐸𝑉_1,𝑖(𝑡)<0. (30)
From (16) and (19), we obtain that 𝑉_𝑖(𝑡)≤𝑉_1,𝑖(𝑡)+^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝑄_𝑖𝑥(𝑠)𝑑𝑠+𝜏^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝑅_𝑖𝑥(𝑠)𝑑𝑠≤𝑉_1,𝑖(𝑡)+𝛽_𝑖(1+𝜏)^𝑡_𝑡−𝜏𝑉_1,𝑖(𝑠)𝑑𝑠. (31)
Let 𝜎(𝑡_𝑘)=𝑖. Then, using Ito′s formula, we can obtain that for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝐸𝑒^𝜆𝑡𝑉_𝑖(𝑡)−𝐸𝑒^𝜆𝑡𝑘𝑉_𝑖𝑡_𝑘=𝐸^𝑡_𝑡𝑘ℒ𝑒^𝜆𝑠𝑉_𝑖(𝑠)𝑑𝑠≤𝐸^𝑡_𝑡𝑘𝑒^𝜆𝑠𝜆𝑉_1,𝑖(𝑠)+𝜆𝛽_𝑖(1+𝜏)×^𝑠_𝑠−𝜏𝑉_1,𝑖(𝜗)𝑑𝜗−𝛼𝑉_1,𝑖(𝑠)𝑑𝑠. (32)
Notice that ^𝑡_𝑡𝑘𝑒^𝜆𝑠𝑑𝑠^𝑠_𝑠−𝜏𝑉_1,𝑖(𝜗)𝑑𝜗=^𝑡_{𝑡𝑘−𝜏}𝑑𝜗_𝜗^𝜗+𝜏𝑒^𝜆𝑠𝑉_1,𝑖(𝜗)=1𝑑𝑠𝜆𝑒^𝜆𝜏−1^𝑡_{𝑡𝑘−𝜏}𝑒^𝜆𝑠𝑉_1,𝑖(𝑠)𝑑𝑠. (33)
Thus, it can be obtained that 𝐸𝑒^𝜆𝑡𝑉_𝑖(𝑡)−𝐸𝑒^𝜆𝑡𝑘𝑉_𝑖𝑡_𝑘≤𝐸^𝑡_𝑡𝑘𝑒^𝜆𝑠𝜆+𝛽_𝑖(1+𝜏)𝑒^𝜆𝜏−1−𝛼𝑉_1,𝑖(𝑠)Ω𝑑𝑠+𝐸_𝑖𝑡_𝑘, (34) where Ω_𝑖𝑡_𝑘=𝛽_𝑖(1+𝜏)𝑒^𝜆𝜏−1^𝑡𝑘_{𝑡𝑘−𝜏}𝑒^𝜆𝑠𝑉_1,𝑖(𝑠)𝑑𝑠≤𝛽_𝑖(1+𝜏)𝑒^𝜆𝜏−1𝑒^{𝜆𝑡𝑘𝑡𝑘}_{𝑡𝑘−𝜏}𝑉_1,𝑖(𝑠)𝑑𝑠≤𝜏𝛽_𝑖(1+𝜏)𝑒^𝜆𝜏−1𝑒^𝜆𝑡𝑘sup_{−𝜏≤𝜃≤0}𝑉_1,𝑖𝑡_𝑘+𝜃≤𝜏𝛽_𝑖(1+𝜏)𝑒^𝜆𝜏−1𝑒^𝜆𝑡𝑘sup_{−𝜏≤𝜃≤0}𝑉_𝑖𝑡_𝑘+𝜃. (35)
Noticing that 𝐸{𝑉(𝑡)}≤sup_{−𝜏≤𝜃≤0}𝐸{𝑉(𝑡+𝜃)}, one gets 𝐸𝑉_𝑖(𝑡)≤𝜒_𝑖𝑒^{−𝜆(𝑡−𝑡𝑘)}sup_{−𝜏≤𝜃≤0}𝐸𝑉_𝑖𝑡_𝑘+𝜃≤𝜒𝑒^{−𝜆(𝑡−𝑡𝑘)}sup_{−𝜏≤𝜃≤0}𝐸𝑉_𝑖𝑡_𝑘+𝜃. (36)
For any 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), from (16) and (36), it follows that 𝐸{𝑉(𝑡)}=𝐸𝑉_𝑖(𝑡)≤𝜒𝑒^{−𝜆(𝑡−𝑡𝑘)}sup_{−𝜏≤𝜃≤0}𝐸𝑉_{𝜎(𝑡𝑘)}𝑡_𝑘+𝜃≤𝜒𝜇𝑒^{−𝜆(𝑡−𝑡𝑘)}sup_{−𝜏≤𝜃≤0}𝐸𝑉_𝜎(𝑡⁻_𝑘)𝑡_𝑘+𝜃≤𝜒²𝜇𝑒^{−𝜆(𝑡−𝑡𝑘)}𝑒^{−𝜆(𝑡𝑘+𝜃−𝑡𝑘−1)}×sup_{−𝜏≤𝜃≤0}𝐸𝑉_{𝜎(𝑡𝑘−1)}𝑡_𝑘−1+𝜃≤𝜒²𝜇𝑒^𝜆𝜏𝑒^{−𝜆(𝑡−𝑡𝑘−1)}sup_{−𝜏≤𝜃≤0}𝐸𝑉_{𝜎(𝑡𝑘−1)}𝑡_𝑘−1+𝜃≤𝜒³𝜇𝑒^𝜆𝜏2𝑒^{−𝜆(𝑡−𝑡𝑘−2)}sup_{−𝜏≤𝜃≤0}𝐸𝑉_{𝜎(𝑡𝑘−2)}𝑡_𝑘−2+𝜃≤⋯≤𝜒𝜒𝜇𝑒^{𝜆𝜏𝑁𝜎(𝑡0,𝑡)}𝑒^{−𝜆(𝑡−𝑡0)}sup_{−𝜏≤𝜃≤0}𝐸𝑉_𝜎(𝑡0)𝑡₀+𝜃. (37)
When (15) holds, noticing that  𝑁_𝜎(𝑡₀,𝑡)≤(𝑡−𝑡₀)/𝑇_𝛼, one has 𝐸‖𝑥(𝑡)‖²≤𝑏𝑎𝜒𝑒^{−(𝜆−(𝜆𝜏+ln(𝜒𝜇))/𝑇𝛼)(𝑡−𝑡0)}×sup_{−𝜏≤𝜃≤0}𝐸𝑥𝑡₀+𝜃², (38) where 𝑎=min_{∀𝑖∈𝑁}𝜆_min(𝑃_𝑖), 𝑏=max_{∀𝑖∈𝑁}𝜆_max(𝑃_𝑖)+𝜏max_{∀𝑖∈𝑁}𝜆_max(𝑄_𝑖)+(𝜏²/2)max_{∀𝑖∈𝑁}𝜆_max(𝑅_𝑖).
The proof is completed.

Remark 9. In the derivation of Theorem 8, a new Lyapunov-Krasovskii functional candidate is constructed for the stability analysis of switched stochastic systems with time delay, and it is different from the ones given in [9–15]. Also, the free-weighting matrix method is utilized to reduce the conservatism.

Remark 10. If 𝜇=1 in (15), which leads to 𝑃_𝑖=𝑃_𝑗, 𝑄_𝑖=𝑄_𝑗, and 𝑅_𝑖=𝑅_𝑗, for all 𝑖,𝑗∈𝑁, then system (13a) and (13b) possesses a common Lyapunov function, and the switching signals can be arbitrary.

When 𝑤(𝑡)=0, system (13a) and (13b) becomes the following switched system with time delay:𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝜏𝜎(𝑡)}𝑥(𝑡−𝜏(𝑡)),𝑥(𝑡)=𝜑(𝑡),𝑡∈𝑡₀−𝜏,𝑡₀. (39a)(39b)

From Theorem 8, we can readily get the exponential stability criterion for switched system (39a) and (39b).

Corollary 11. Consider system (39a) and (39b), for a given scalar 𝛼>0, if there exist symmetric positive definite matrices 𝑃_𝑖,  𝑄_𝑖, and 𝑅_𝑖 and any matrices 𝑆_𝑖 such that ∑^𝑖₁₁𝑃_𝑖𝐴_𝜏𝑖𝑆^𝑇_𝑖𝑆^𝑇_𝑖∗−1−𝜏_𝑑𝑄_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖∗∗−𝑆_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖∗∗∗−𝜏⁻¹𝑅_𝑖<0,∀𝑖∈𝑁 (40) then system (39a) and (39b) is exponentially stable under arbitrary switching signal with the average dwell time scheme (15).

3.2. 𝐻_∞ Performance Analysis

In the sequel, we will investigate the problem of 𝐻_∞ performance analysis for switched stochastic systems with time delay. Consider the following system:𝑑𝑥(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝜏𝜎(𝑡)}𝑥(𝑡−𝜏(𝑡))+𝐺_𝜎(𝑡)𝑣(𝑡)𝑑𝑡+𝐷_𝜎(𝑡)𝑥(𝑡)𝑑𝑤(𝑡),𝑧(𝑡)=𝐽_𝜎(𝑡)𝑥(𝑡),𝑥(𝑡)=𝜑(𝑡),𝑡∈𝑡₀−𝜏,𝑡₀. (41a)(41b)(41c)

Theorem 12. For a given scalar 𝛼>0, if there exist symmetric positive definite matrices 𝑃_𝑖,  𝑄_𝑖, and 𝑅_𝑖 and any matrices 𝑆_𝑖 such that ∏^𝑖₁₁𝑃_𝑖𝐴_𝜏𝑖𝑆^𝑇_𝑖𝑆^𝑇_𝑖𝑃_𝑖𝐺_𝑖𝐷^𝑇_𝑖𝑃_𝑖𝐽^𝑇_𝑖∗−1−𝜏_𝑑𝑄_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖000∗∗−𝑆_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖000∗∗∗−𝜏⁻¹𝑅_𝑖000∗∗∗∗−𝛾²𝐼00∗∗∗∗∗−𝑃_𝑖0∗∗∗∗∗∗−𝐼<0,∀𝑖∈𝑁 (42) then system (41a), (41b), and (41c) is mean-square exponentially stable with a weighted prescribed 𝐻_∞ performance level 𝛾 under arbitrary switching signal with the average dwell time 𝑇_𝛼>𝑇^∗_𝛼=𝜏+ln𝜒𝜇𝜆, (43) where 𝜇, 𝜒, and 𝜆 satisfy 𝑃_𝑖≤𝜇𝑃_𝑗,𝑄_𝑖≤𝜇𝑄_𝑗,𝑅_𝑖≤𝜇𝑅_𝑗,𝑄_𝑖≤𝛽_𝑖𝑃_𝑖,𝑅_𝑖≤𝛽_𝑖𝑃_𝑖,∀𝑖,𝑗∈𝑁,𝜆+𝛽_𝑖(1+𝜏)𝑒^𝜆𝜏−1≤𝛼,𝜒=max_𝑖∈𝑁𝜒_𝑖,𝜒_𝑖=1+𝜏𝛽_𝑖(1+𝜏)𝑒^𝜆𝜏−1_𝑖11,=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝑄_𝑖+𝜏𝑅_𝑖+𝛼𝑃_𝑖. (44)(45)

Proof. We can easily obtain that (14) is satisfied if (42) holds. Thus, system (41a), (41b), and (41c) with 𝑣(𝑡)=0 is mean-square exponentially stable.
When 𝑣(𝑡)≠0, let Γ(𝑡)=𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑣^𝑇(𝑡)𝑣(𝑡). (46)
Choosing the same Lyapunov functional candidate as (19) and following the proof line of Theorem 8, we have 𝑑𝑉_𝑖(𝑡)≤ℒ𝑉_𝑖(𝑡)𝑑𝑡+𝑊_𝑖(𝑡), (47) where 𝑊_𝑖(𝑡)=2𝑥^𝑇(𝑡)𝑃_𝑖𝐷_𝑖𝑥(𝑡)𝑑𝑤(𝑡)−2^𝑡_{𝑡−𝜏(𝑡)}𝑥(𝑠)𝑑𝑠+^𝑡_{𝑡−𝜏(𝑡)}𝑌(𝑠)𝑑𝑠^𝑇×𝑆_𝑖^𝑡_{(𝑡)𝑡−𝜏}𝐷_𝜎(𝑠)𝑥(𝑠)𝑑𝑤(𝑠)𝑑𝑡, (48) and ℒ𝑉_𝑖(𝑡) satisfies ℒ𝑉_𝑖(𝑡)+Γ(𝑡)=𝜍^𝑇(𝑡)Θ_𝑖𝜍(𝑡),𝜍(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−𝜏(𝑡))∫^𝑡_{𝑡−𝜏(𝑡)}𝑌^𝑇(𝑠)𝑑𝑠∫^𝑡_{𝑡−𝜏(𝑡)}𝑥^𝑇(𝑠)𝑑𝑠𝑣^𝑇(𝑡)^𝑇,Θ_𝑖=Θ^𝑖₁₁𝑃_𝑖𝐴_𝜏𝑖𝑆^𝑇_𝑖𝑆^𝑇_𝑖𝑃_𝑖𝐺_𝑖∗−1−𝜏_𝑑𝑄_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖0∗∗−𝑆_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖0∗∗∗−𝜏⁻¹𝑅_𝑖0∗∗∗∗−𝛾²𝐼,Θ^𝑖₁₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝑄_𝑖+𝐷^𝑇_𝑖𝑃_𝑖𝐷_𝑖+𝜏𝑅_𝑖+𝐽^𝑇_𝑖𝐽_𝑖. (49)
Using the Schur complement, from (42), we get ℒ𝑉_𝑖(𝑡)+Γ(𝑡)<−𝛼𝑉_1,𝑖(𝑡)<0. (50)
It follows that 𝐸𝑑𝑉_𝑖(𝑡)≤𝐸ℒ𝑉_𝑖(𝑡)𝑑𝑡. (51)
From (44), we obtain that 𝐸𝑉_{𝜎(𝑡𝑘)}𝑡_𝑘≤𝜇𝐸𝑉_{𝜎(𝑡𝑘−1)}𝑡_𝑘. (52)
For any 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), using Ito′s formula and taking the mathematical expectation, one has 𝐸𝑉_𝜎(𝑡)(𝑡)=𝐸𝑉_{𝜎(𝑡𝑘)}𝑡_𝑘+^𝑡_𝑡𝑘𝐸𝑑𝑉_{𝜎(𝑡𝑘)}(𝑠𝑉)≤𝐸_{𝜎(𝑡𝑘)}𝑡_𝑘+^𝑡_𝑡𝑘𝐸ℒ𝑉_{𝜎(𝑡𝑘)}(𝑠)(𝑠−+Γ)𝑑𝑠_𝑡𝑡𝑘𝐸{Γ(𝑠)}𝑉𝑑𝑠<𝜇𝐸_{𝜎(𝑡𝑘−1)}𝑡_𝑘−^𝑡_𝑡𝑘𝐸{Γ(𝑠)𝑑𝑠}<𝜇²𝐸𝑉_{𝜎(𝑡𝑘−2)}𝑡_𝑘−1−𝐸𝜇^𝑡𝑘_𝑡𝑘−1Γ(𝑠)𝑑𝑠+^𝑡_𝑡𝑘Γ(𝑠)𝑑𝑠<⋯<𝜇^{𝑁𝜎(𝑡0,𝑡)}𝐸𝑉𝑡₀−𝐸𝜇^{𝑁𝜎(𝑡0,𝑡)𝑡1}_𝑡0Γ(𝑠)𝑑𝑠+𝜇^{𝑁𝜎(𝑡1,𝑡)𝑡2}_𝑡1Γ(𝑠)𝑑𝑠+⋯+^𝑡_𝑡𝑘Γ(𝑠)𝑑𝑠=𝜇^{𝑁𝜎(𝑡0,𝑡)}𝐸𝑉𝑡₀−𝐸^𝑡_𝑡0𝑒^{𝑁𝜎(𝑠,𝑡)ln𝜇}Γ(𝑠)𝑑𝑠. (53)
Under the zero initial condition, we obtain that 𝐸^𝑡_𝑡0𝑒^{𝑁𝜎(𝑠,𝑡)ln𝜇}Γ(𝑠)𝑑𝑠<0. (54)
According to (46), one has 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. (55)
Multiplying both sides of (55) 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. (56)
Noticing that  𝑁_𝜎(𝑡₀,𝑠)≤(𝑠−𝑡₀)/𝑇_𝛼 and 𝑇_𝛼≥𝜏+ln(𝜒𝜇)/𝜆≥ln𝜇/𝜆, one obtains that 𝐸^𝑡_𝑡0𝑒^{−𝜆(𝑠−𝑡0)}𝑧^𝑇(𝑠)𝑧(𝑠)𝑑𝑠<𝛾^2𝑡_𝑡0𝑣^𝑇(𝑠)𝑣(𝑠)𝑑𝑠. (57)
When 𝑡→∞, the following inequality is derived: 𝐸^∞_𝑡0𝑒^{−𝜆(𝑠−𝑡0)}𝑧^𝑇(𝑠)𝑧(𝑠)𝑑𝑠<𝛾^2∞_𝑡0𝑣^𝑇(𝑠)𝑣(𝑠)𝑑𝑠. (58)
The proof is completed.

3.3. Observer-Based Robust  𝐻_∞  Stabilization

Now, we are in a position to design an observer-based robust 𝐻_∞ controller for system (1a), (1b), (1c), and (1d) such that the augmented closed-loop system (7a), (7b), and (7c) is mean-square exponentially stable with a prescribed weighted 𝐻_∞ performance level 𝛾. Based on Theorem 12, a sufficient condition for the existence of such a controller is presented in the following theorem.

Theorem 13. Consider system (1a), (1b), (1c), and (1d), for a given scalar 𝛼>0, if there exist scalars 𝜀_𝑖>0, symmetric positive definite matrices 𝑃_𝑖, 𝑄_𝑖, and 𝑅_𝑖, and any matrices 𝑆_𝑖, 𝑌_𝑖, and 𝑍_𝑖 such that, for all 𝑖∈𝑁,Ω^𝑖₁₁𝑃_𝑖𝐴_𝜏𝑖𝑆^𝑇_𝑖𝑆^𝑇_𝑖𝑃_𝑖𝐺_𝑖𝐷^𝑇_𝑖𝑃_𝑖𝐽^𝑇_𝑖̃𝑃_𝑖𝐻_𝑖0𝜀_𝑖𝐸^𝑇_1𝑖𝜀_𝑖𝐸^𝑇_3𝑖∗−1−𝜏_𝑑𝑄_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖00000𝜀_𝑖𝐸^𝑇_2𝑖0∗∗−𝑆_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖0000000∗∗∗−𝜏⁻¹𝑅_𝑖0000000∗∗∗∗−𝛾²𝐼000000∗∗∗∗∗−𝑃_𝑖00𝑃_𝑖𝐻_𝑖00∗∗∗∗∗∗−𝐼0000∗∗∗∗∗∗∗−𝜀_𝑖𝐼000∗∗∗∗∗∗∗∗−𝜀_𝑖𝐼00∗∗∗∗∗∗∗∗∗−𝜀_𝑖𝐼0∗∗∗∗∗∗∗∗∗∗−𝜀_𝑖𝐼<0, (59)then there exists an observer-based controller such that system (7a), (7b), and (7c) is mean-square exponentially stable with a prescribed weighted 𝐻_∞ performance level 𝛾 under arbitrary switching signal with the average dwell time𝑇_𝛼>𝑇^∗_𝛼=𝜏+ln𝜒𝜇𝜆, (60) where 𝜇, 𝜒, and 𝜆 satisfy 𝑃_𝑖≤𝜇𝑃_𝑗,𝑄_𝑖≤𝜇𝑄_𝑗,𝑅_𝑖≤𝜇𝑅_𝑗,𝑄_𝑖≤𝛽_𝑖𝑃_𝑖,𝑅_𝑖≤𝛽_𝑖𝑃_𝑖,∀𝑖,𝑗∈𝑁,𝜆+𝛽_𝑖(1+𝜏)𝑒^𝜆𝜏−1≤𝛼,𝜒=max_𝑖∈𝑁𝜒_𝑖,𝜒_𝑖=1+𝜏𝛽_𝑖(1+𝜏)𝑒^𝜆𝜏,Ω−1_𝑖11=Υ^𝑇_𝑖+Υ_𝑖+𝑄_𝑖+𝜏𝑅_𝑖+𝛼𝑃_𝑖,Υ_𝑖=𝑃_𝑖𝐴_𝑖+𝑍_𝑖−𝑍_𝑖0𝑃_𝑖𝐴_𝑖−𝑌_𝑖𝐶_𝑖,𝑃_𝑖=diag𝑃_𝑖,𝑃_𝑖,𝑄_𝑖=diag𝑄_𝑖,𝑄_𝑖,𝑅_𝑖=diag𝑅_𝑖,𝑅_𝑖. (61)(62)(63)
Moreover, if the above conditions have a feasible solution, the controller gain matrices and the observer gain matrices can be obtained by 𝐾_𝑖=𝐵⁺_𝑖𝑃_𝑖⁻¹𝑍_𝑖 and 𝐿_𝑖=𝑃_𝑖⁻¹𝑌_𝑖.

Proof. According to Theorem 12, we get that system (7a), (7b), and (7c) is mean-square exponentially stable with a weighted 𝐻_∞ performance level 𝛾 if the following inequality is satisfied: Ψ_𝑖=Ψ^𝑖₁₁𝑃_𝑖𝐴_𝜏𝑖𝑆^𝑇_𝑖𝑆^𝑇_𝑖𝑃_𝑖𝐺_𝑖𝐷^𝑇_𝑖𝑃_𝑖𝐽^𝑇_𝑖∗−1−𝜏_𝑑𝑄_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖𝑆000∗∗−_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖000∗∗∗−𝜏⁻¹𝑅_𝑖000∗∗∗∗−𝛾²𝑃𝐼00∗∗∗∗∗−_𝑖0∗∗∗∗∗∗−𝐼<0, (64) where Ψ^𝑖₁₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝑄_𝑖+𝜏𝑅_𝑖+𝛼𝑃_𝑖, and 𝑃_𝑖, 𝑄_𝑖, and 𝑅_𝑖 are symmetric positive definite matrices with appropriate dimensions.
Then, we have Ψ_𝑖=Ψ_𝑖+ΔΨ_𝑖<0, (65) where Ψ_𝑖=Ψ^𝑖₁₁𝑃_𝑖𝐴_𝜏𝑖𝑆^𝑇_𝑖𝑆^𝑇_𝑖𝑃_𝑖𝐺_𝑖𝐷^𝑇_𝑖𝑃_𝑖𝐽^𝑇_𝑖̃∗−1−𝜏_𝑑𝑄_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖𝑆000∗∗−_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖000∗∗∗−𝜏⁻¹𝑅_𝑖000∗∗∗∗−𝛾²𝑃𝐼00∗∗∗∗∗−_𝑖0,∗∗∗∗∗∗−𝐼ΔΨ_𝑖=𝐸^𝑇_1𝑖𝐹^𝑇_𝑖𝐻^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐻_𝑖𝐹_𝑖𝐸_1𝑖𝑃_𝑖𝐻_𝑖𝐹_𝑖𝐸_2𝑖000𝐸^𝑇_3𝑖𝐹^𝑇_𝑖𝐻^𝑇_𝑖𝑃_𝑖0𝐸^𝑇_2𝑖𝐹^𝑇_𝑖𝐻^𝑇_𝑖𝑃_𝑖𝑃000000000000000000000000000_𝑖𝐻_𝑖𝐹_𝑖𝐸_3𝑖=𝑃0000000000000_𝑖𝐻_𝑖00𝑃00000000_𝑖𝐻_𝑖𝐹00^𝑖00𝐹_𝑖𝐸^𝑇_1𝑖𝐸^𝑇_3𝑖𝐸^𝑇_2𝑖00000000000^𝑇+𝐸^𝑇_1𝑖𝐸^𝑇_3𝑖𝐸^𝑇_2𝑖0𝐹0000000000^𝑖00𝐹_𝑖^𝑇𝑃_𝑖𝐻_𝑖00𝑃00000000_𝑖𝐻_𝑖00^𝑇,Ψ^𝑖₁₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+𝑄_𝑖+𝜏𝑅_𝑖+𝛼𝑃_𝑖. (66)
From Lemma 7, we have Ψ_𝑖≤Ψ_𝑖+𝜀_𝑖⁻¹𝑃_𝑖𝐻_𝑖00𝑃00000000_𝑖𝐻_𝑖𝑃00^𝑖𝐻_𝑖00𝑃00000000_𝑖𝐻_𝑖00^𝑇+𝜀_𝑖𝐸^𝑇_1𝑖𝐸^𝑇_3𝑖𝐸^𝑇_2𝑖0𝐸0000000000^𝑇1𝑖𝐸^𝑇_3𝑖𝐸^𝑇_2𝑖00000000000^𝑇<0. (67)Choose 𝑃_𝑖=𝑃_𝑖=diag{𝑃_𝑖,𝑃_𝑖}, 𝑄_𝑖=𝑄_𝑖=diag{𝑄_𝑖,𝑄_𝑖}, and 𝑅_𝑖=𝑅_𝑖=diag{𝑅_𝑖,𝑅_𝑖}, and let 𝑍_𝑖=𝑃_𝑖𝐵_𝑖𝐾_𝑖 and 𝑌_𝑖=𝑃_𝑖𝐿_𝑖. By using the Schur complement, we can obtain that (67) is equivalent to (59).
Thus, according to Theorem 12, we obtain from (59)–(62) that system (7a), (7b), and (7c) is mean-square exponentially stable with a weighted 𝐻_∞ performance level 𝛾. Moreover, we can obtain the controller gain matrices 𝐾_𝑖=𝐵⁺_𝑖𝑃_𝑖⁻¹𝑍_𝑖 and the observer gain matrices 𝐿_𝑖=𝑃_𝑖⁻¹𝑌_𝑖.
The proof is completed.

Remark 14. An observer-based 𝐻_∞ controller design scheme is proposed in the paper. Compared with the existing results presented in [27–29], a remark advantage of the work is that the proposed observer is mode-dependent, which means that each subsystem has its individual observer. Moreover, the proposed observer not only ensures the convergence of the estimated error of each subsystem, but also guarantees that the estimated error of the whole system converges to zero exponentially.

In Theorem 13, when we choose that 𝑄_𝑖=𝑅_𝑖=𝑃_𝑖, it is not difficult to get that 𝛽_𝑖=1 and 𝜒=𝜒_𝑖=1+𝜏(1+𝜏)(𝑒^𝜆𝜏−1),  for all 𝑖∈𝑁. Then, the following corollary is derived.

Corollary 15. Consider system (1a), (1b), (1c), and (1d), for a given scalar 𝛼>0, if there exist scalars 𝜀_𝑖>0, symmetric positive definite matrices 𝑃_𝑖>0, and matrices 𝑆_𝑖, 𝑌_𝑖, and 𝑍_𝑖 such that, for all 𝑖∈𝑁,Ξ^𝑖₁₁𝑃_𝑖𝐴_𝜏𝑖𝑆^𝑇_𝑖𝑆^𝑇_𝑖𝑃_𝑖𝐺_𝑖𝐷^𝑇_𝑖𝑃_𝑖𝐽^𝑇_𝑖̃𝑃_𝑖𝐻_𝑖0𝜀_𝑖𝐸^𝑇_1𝑖𝜀_𝑖𝐸^𝑇_3𝑖∗−1−𝜏_𝑑𝑃_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖00000𝜀_𝑖𝐸^𝑇_2𝑖0∗∗−𝑆_𝑖−𝑆^𝑇_𝑖−𝑆^𝑇_𝑖0000000∗∗∗−𝜏⁻¹𝑃_𝑖0000000∗∗∗∗−𝛾²𝐼000000∗∗∗∗∗−𝑃_𝑖00𝑃_𝑖𝐻_𝑖00∗∗∗∗∗∗−𝐼0000∗∗∗∗∗∗∗−𝜀_𝑖𝐼000∗∗∗∗∗∗∗∗−𝜀_𝑖𝐼00∗∗∗∗∗∗∗∗∗−𝜀_𝑖𝐼0∗∗∗∗∗∗∗∗∗∗−𝜀_𝑖𝐼<0, (68) then there exists an observer-based robust 𝐻_∞ controller such that system (7a), (7b), and (7c) is mean-square exponentially stable with a prescribed weighted 𝐻_∞ performance level 𝛾 under arbitrary switching signal with the average dwell time 𝑇_𝛼>𝑇^∗_𝛼=𝜏+ln𝜒𝜇𝜆, (69) where 𝜇, 𝜒, and 𝜆 satisfy 𝜒=1+𝜏(1+𝜏)𝑒^𝜆𝜏,()𝑒−1𝜆+1+𝜏^𝜆𝜏𝑃−1≤𝛼,_𝑖≤𝜇𝑃_𝑗,∀𝑖,𝑗∈𝑁,Ξ^𝑖₁₁=Λ^𝑇_𝑖+Λ_𝑖+𝑃_𝑖+𝜏𝑃_𝑖+𝛼𝑃_𝑖,Λ_𝑖=𝑃_𝑖𝐴_𝑖+𝑍_𝑖−𝑍_𝑖0𝑃_𝑖𝐴_𝑖−𝑌_𝑖𝐶_𝑖,𝑃_𝑖=diag𝑃_𝑖,𝑃_𝑖. (70)
Moreover, the controller gain matrices are 𝐾_𝑖=𝐵⁺_𝑖𝑃_𝑖⁻¹𝑍_𝑖, and the observer gain matrices are 𝐿_𝑖=𝑃_𝑖⁻¹𝑌_𝑖.

4. Numerical Example

Consider system (1a), (1b), (1c), and (1d) with the following parameters 𝐴₁=310−3,𝐴_𝜏1=,𝐵−0.100.10.1₁=1211,𝐶₁=,𝐽2003₁=3014,𝐷₁=,𝐺0.100.10.6₁=1.500.10.4,𝐻₁=,𝐸0.1000₁₁=0.1000,𝐸₂₁=,𝐸00.100₃₁=,𝐴000.10₂=400−2,𝐴_𝜏2=,𝐵−0.10.10−0.2₂=1121,𝐶₂=,𝐽3021₂=2114,𝐷₂=,𝐺0.100.30.5₂=0.50.200.6,𝐻₂=,𝐸0000.1₁₂=0000.1,𝐸₂₂=,𝐸000.10₃₂=,𝐹00.100₁(𝑡)=sin𝑡00sin𝑡,𝐹₂(𝑡)=.cos𝑡00cos𝑡 (71)

The disturbance input 𝑣(𝑡)=15𝑒^−0.2𝑡12𝑒^{−0.3𝑡𝑇}, and 𝜏(𝑡)=0.3+0.2sin𝑡; by calculation, we can obtain that 𝜏̇(𝑡)≤𝜏_𝑑=0.2 and 𝜏(𝑡)≤𝜏=0.5. Choosing 𝛼=1.4,  𝛾=1.0 and solving the LMIs in Corollary 15, we have 𝑃₁=,𝑌4.0319−0.7058−0.705813.0192₁=,𝑍82.457919.7441−15.992625.9295₁=,𝑃−98.8224−4.9793−5.0165−48.5838₂=,𝑌4.1782−0.5687−0.568711.9156₂=,𝑍91.2715−45.8464−48.736499.0875₂=,𝑆−105.7969−5.8641−5.1805−64.8133₁=,𝑆1.9716−0.2408−0.08090.0342−0.23015.22800.0555−0.4124−0.08220.05532.0559−0.30490.0286−0.4044−0.30176.0017₂=,𝜀1.5102−0.0781−0.07300.1131−0.07964.65230.0254−0.6518−0.07920.01192.1184−0.23290.1161−0.6486−0.24135.2970₁=54.6307,𝜀₂=51.0439. (72)

Then, we can obtain the following observer gain matrices: 𝐿₁=,𝐿20.43015.2958−0.12092.2787₂=21.4270−9.9052−3.06757.8431 (73) and the controller gain matrices 𝐾₁=,𝐾21.3520−5.7638−23.08251.9288₂=.23.8921−3.3845−49.43821.2267 (74)

Moreover, we have 𝜇=4.3051, 𝜆=0.7, 𝜒=1.3143, and 𝑇^∗_𝛼=𝜏+ln(𝜒𝜇)/𝜆=2.9759. Taking 𝑇_𝛼=3>𝑇^∗_𝛼, and letting 𝑥(𝑡)=00^𝑇, 𝑡∈[−0.5,0), 𝑥(0)=2−2^𝑇, and 𝑒(0)=11^𝑇, and simulation results are shown in Figures 1–5.

Figure 1 

Switching signal.

Figure 2 

State x 1 of the closed-loop system.

Figure 3 

State x 2 of the closed-loop system.

Figure 4 

The estimated error e 1.

Figure 5 

The estimated error e 2.

Figure 1 shows the switching signal of the switched system with the average dwell time 𝑇_𝛼=3. Figures 2 and 3 illustrate the state trajectories of the closed-loop system. The estimated errors are plotted in Figures 4 and 5, respectively. We can see from Figures 2–5 that the proposed observer can guarantee the convergence of the estimated error and the designed controller can guarantee the stability of the corresponding closed-loop system. This demonstrates the effectiveness of the proposed method.

In addition, some observer-based controller design approaches proposed in the existing literature [27–29] are only applicable to stochastic systems or Markovian jump systems and they cannot be used to stabilize the system considered in this section, which also shows the advantage of the proposed method.

5. Conclusions

In this paper, the problem of observer-based robust 𝐻_∞  stabilization for stochastic switched systems with time delay has been investigated. By using the average dwell time method, sufficient conditions which guarantee the mean-square exponential stability of switched stochastic systems with time delay are derived. Then, 𝐻_∞ performance analysis and observer-based 𝐻_∞ control for such systems are developed. Finally, a numerical example is given to demonstrate the effectiveness of the proposed approach.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant No. 61273120.