Robust Stability and 𝐻_∞ Stabilization of Switched Systems with Time-Varying Delays Using Delta Operator Approach

Abstract

This paper considers the problems of the robust stability and robust 𝐻_∞ controller design for time-varying delay switched systems using delta operator approach. Based on the average dwell time approach and delta operator theory, a sufficient condition of the robust exponential stability is presented by choosing an appropriate Lyapunov-Krasovskii functional candidate. Then, a state feedback controller is designed such that the resulting closed-loop system is exponentially stable with a guaranteed 𝐻_∞ performance. The obtained results are formulated in the form of linear matrix inequalities (LMIs). Finally, a numerical example is provided to explicitly illustrate the feasibility and effectiveness of the proposed method.

1. Introduction

Switched systems are a kind of hybrid systems consisting of a set of discrete event dynamic subsystems or continuous variable dynamic subsystems and a switching rule which defines a particular subsystem working during a certain interval of time. Switched systems have numerous applications in network control systems [1], robot control systems [2], intelligent traffic control systems [3], chemical industry control systems [4], and many other areas [5, 6]. Many important achievements on stability and stabilization of switched systems have been developed [7–10]. It was shown in the literature that the average dwell time (ADT) method is a powerful tool to deal with the stability of switched systems.

The delta operator which is a novel approach with good finite word length performance under fast sampling rates has been investigated by many researchers due to their extensive applications [11–13], for instance, optimal filtering [14], signal processing [15], robust control [16], system identification [17], and so forth. As stated in [15], the standard shift operator was mostly adopted in the study of control theories for discrete-time systems. However, the dynamic response of a discrete system does not converge smoothly to its continuous counterpart when the sampling period tends to zero; namely, data are taken at high sampling rates. The delta operator method can solve the above problem. In addition, it was shown in [15] that delta operator requires smaller word length when implemented in fixed-point digital control processors than shift operator does. So far, some useful results on delta operator systems have been formulated in [18–21]. As is well known, time delay phenomena which often cause instability or undesirable performance in control systems are involved in a variety of real systems, such as chaotic systems, and hydraulic pressure systems [22]. In the past years, a mass of results on delta operator systems with time delay have appeared [23–27]. The delta operator is defined by (𝑡)=(𝑡)𝛿𝑥𝑑𝑥(𝑥()(𝑡𝑑𝑡,𝑇=0,𝑡+𝑇−𝑥))𝑇,𝑇≠0, (1) where 𝑇 is a sampling period. When 𝑇→0, the delta operator model will approach the continuous system before discretization and reflect a quasicontinuous performance [28].

It should be noted that external disturbances are generally inevitable, and the output will be subsequently affected by disturbances in the system. Some results on 𝐻_∞ control were developed by many researchers to restrain the external disturbances [29–33]. The 𝐻_∞ control problem for a class of discrete systems was solved by using delta operator approach [34]. Low order sampled data 𝐻_∞ control using the delta operator was reported in the literature [35]. Robust 𝐻_∞ control for a class of uncertain switched systems using delta operator was investigated [36]. However, few results on the issues of robust stability and 𝐻_∞ controller design for delta operator switched systems with time-varying delay are presented, which motivates the present investigations.

In this paper, we concentrate our interest on investigating the stability and 𝐻_∞ controller design problems for delta operator switched systems with time-varying delay. The main contributions of this paper can be summarized as follows: (1) by constructing a new Lyapunov-Krasovskii functional candidate and using the average dwell time approach, an exponential stability criterion for the considered system is proposed and (2) a state feedback controller design scheme is developed such that the corresponding closed-loop system is exponentially stable with a guaranteed 𝐻_∞ performance.

The remainder of the paper is organized as follows. The formulation of the considered systems and some corresponding definitions and lemmas is given in Section 2. In Section 3, the exponential stability analysis and 𝐻_∞ control for the underlying system are developed. A numerical example is given to illustrate the feasibility and effectiveness of the proposed method in Section 4. Finally, concluding remarks are presented in Section 5.

Notations. ‖⋅‖₂ denotes the Euclidean norm. 𝜆_min(⋅) and 𝜆_max(⋅) denote the minimum and maximum eigenvalues of a matrix, respectively; 𝐴^𝑇 means the transpose of matrix 𝐴; 𝑅 denotes the set of all real numbers; 𝑅^𝑛 represents the 𝑛-dimensional real vector space; 𝑅^𝑚×𝑛 is the set of all (𝑚×𝑛)-dimensional real matrices. The notation 𝐴>0(≥0)   means that the matrix 𝐴 is positive (nonnegative) definite; diag{⋯} refers to the block-diagonal matrix; 𝐼 is the identity matrix of appropriate dimension. 𝑙₂[𝑘₀,∞) stands for the space of square summable functions on [𝑘₀,∞).

2. Problem Formulation

Consider the following delta operator switched system with time-varying delay: 𝛿𝑥(𝑘)=𝐴_𝜎(𝑘)𝑥(𝑘)+𝐴_{𝑑𝜎(𝑘)}𝑥(𝑘−𝜏(𝑘))+𝐷_𝜎(𝑘)𝑤(𝑘),𝑧(𝑘)=𝐶_𝜎(𝑘)𝑥(𝑘)+𝐺_𝜎(𝑘)𝑤(𝑘),𝑥𝑘₀+𝜃=𝜙(𝜃),𝜃=−𝜏,−𝜏+1,…,0, (2) where 𝑥(𝑘)∈𝑅^𝑛 is the state vector, 𝑧(𝑘)∈𝑅^𝑙 denotes the controlled output, and 𝑤(𝑘)∈𝑅^𝑤 represents the disturbance input belonging to 𝑙₂[𝑘₀,∞). 𝑘 means the time 𝑡=𝑘𝑇 and 𝑇>0 is the sampling period; 𝑘₀ is the initial instant. 𝜎(𝑘)∶[𝑘₀,∞)→𝑁={1,2,…,𝑁} is the switching signal with 𝑁 being the number of subsystems. 𝜏(𝑘) is the time-varying delay satisfying 0≤𝜏≤𝜏(𝑘)≤𝜏 for known constants 𝜏 and 𝜏. 𝜙(𝜃) is the discrete vector-valued initial function. 𝐶_𝑖, 𝐷_𝑖, and 𝐺_𝑖 are constant matrices with proper dimensions. 𝐴_𝑖 and 𝐴_𝑑𝑖 are uncertain real-valued matrices with appropriate dimensions and have the following form: 𝐴_𝑖𝐴_𝑑𝑖=𝐴_𝑖𝐴_𝑑𝑖+𝐻_𝑖𝐹_𝑖(𝑘)𝐸_𝑎𝑖𝐸_𝑎𝑑𝑖, (3) where 𝐴_𝑖, 𝐴_𝑑𝑖, 𝐻_𝑖, 𝐸_𝑎𝑖, and 𝐸_𝑎𝑑𝑖 are known real constant matrices of suitable dimensions and 𝐹_𝑖(𝑘) is an unknown time-varying matrix which satisfies 𝐹^𝑇_𝑖(𝑘)𝐹_𝑖(𝑘)≤𝐼. (4)

To obtain the main results, we first give some definitions and lemmas which will be essential in our later development.

Definition 1 (see [36]). Consider system (2) with 𝑤(𝑘)=0. It is said to be exponentially stable under a switching signal 𝜎(𝑘) if, for the initial condition 𝑥(𝑘₀+𝜃)=𝜙(𝜃), 𝜃=−𝜏,−𝜏+1,…,0, there exist constants 𝛼>0 and 𝛽>0 such that the solution 𝑥(𝑘) satisfies ‖𝑥(𝑘)‖≤𝛼𝑥𝑘_0𝑐𝑒^{−𝛽(𝑘−𝑘0)},∀𝑘≥𝑘₀, (5) where ‖𝑥(𝑘₀)‖_𝑐=sup_{−𝜏≤𝑘≤0}‖𝑥(𝑘₀+𝜃)‖.

Definition 2. For given 0<𝛼<1/𝑇 and 𝛾>0, system (2) is said to have an 𝐻_∞ performance level 𝛾 if there exists a switching signal 𝜎(𝑘) such that the following conditions are satisfied:(1)system (2) is exponentially stable when 𝑤(𝑘)=0;(2)under the zero-initial condition, that is, 𝜙(𝜃)=0, 𝜃=−𝜏,−𝜏+1,…,−1,0, system (2) satisfies ^∞_𝑘=𝑘0(1−𝑇𝛼)^{(𝑘−𝑘0)}‖𝑧(𝑘)‖²≤𝛾^2∞_𝑘=𝑘0‖𝑤(𝑘)‖²,∀𝑤(𝑘)∈𝑙₂𝑘₀,∞. (6)

Definition 3 (see [37]). For any switching signal 𝜎(𝑘) and any 𝑘₂>𝑘₁≥0, let 𝑁_𝜎(𝑘)(𝑘₁,𝑘₂) denote the number of switching of 𝜎(𝑘) over the interval [𝑘₁,𝑘₂). For given 𝜏_𝑎>0 and 𝑁₀≥0, if the inequality 𝑁_𝜎(𝑘)𝑘₁,𝑘₂≤𝑁₀+𝑘₂−𝑘₁𝜏_𝑎 (7) holds, then the positive constant 𝜏_𝑎 is called the average dwell time and 𝑁₀ is called the chattering bound. As commonly used in the literature, we choose 𝑁₀=0 in this paper.

Lemma 4 (see [20]). For a given matrix 𝑆=^𝑆11𝑆12_{𝑆𝑇12𝑆22}, where 𝑆₁₁ and 𝑆₂₂ are square matrices, the following conditions are equivalent:(i)𝑆<0;(ii)𝑆₁₁<0, 𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂<0;(iii)𝑆₂₂<0, 𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0.

Lemma 5 (see [20]). Let 𝑈, 𝑉, 𝑊, and 𝑋 be real matrices of appropriate dimensions with 𝑋 satisfying 𝑋=𝑋^𝑇; then, for all 𝑉^𝑇𝑉≤𝐼, 𝑋+𝑈𝑉𝑊+𝑊^𝑇𝑉^𝑇𝑈^𝑇<0 (8) if and only if there exists a scalar 𝜀 such that 𝑋+𝜀𝑈𝑈^𝑇+𝜀⁻¹𝑊^𝑇𝑊<0. (9)

Lemma 6 (see [28]). For any time function 𝑥(𝑡) and 𝑦(𝑡), the following equation holds: 𝛿𝑥(𝑡)𝑦(𝑡)(𝑥(𝑡𝑦(𝑡)(𝑡)𝛿𝑦(𝑡)(𝑥(𝑡𝛿𝑦(𝑡),=𝛿))+𝑥+𝑇𝛿)) (10) where 𝑇 is the sampling period.

The objectives of the paper are (1) to find a class of switching signal 𝜎(𝑘) such that system (2) is exponentially stable with a guaranteed 𝐻_∞ performance and (2) to determine a class of switching signal and design a state feedback controller 𝑢(𝑘)=𝐾_𝜎(𝑘)𝑥(𝑘) for the following delta operator switched system with time-varying delay: 𝛿𝑥(𝑘)=𝐴_𝜎(𝑘)𝑥(𝑘)+𝐴_{𝑑𝜎(𝑘)}𝑥((𝑘+𝐵𝑘−𝜏))_𝜎(𝑘)𝑢(𝑘)+𝐷_𝜎(𝑘)𝑤(𝑘),𝑧(𝑘)=𝐶_𝜎(𝑘)𝑥(𝑘)+𝐺_𝜎(𝑘)𝑤(𝑘),𝑥𝑘₀+𝜃=𝜙(𝜃),𝜃=−𝜏,−𝜏+1,…,0 (11) such that the corresponding closed-loop system is exponentially stable with a guaranteed 𝐻_∞ performance.

3. Main Results

3.1. Robust Stability Analysis

In this section, we will focus on the stability of system (2) with 𝑤(𝑘)=0.

Theorem 7. For a given positive constant 0<𝛼<1/𝑇, if there exist scalars 𝜀_𝑖 and positive definite symmetric matrices 𝑋_𝑖 and 𝑄_𝑖, 𝑖∈𝑁, with appropriate dimensions, such that 𝐸_𝑖𝐴_𝑑𝑖𝑋_𝑖𝑇𝑋_𝑖𝐴^𝑇_𝑖𝜀_𝑖𝐻_𝑖𝑋_𝑖𝐸^𝑇_𝑎𝑖𝑋_𝑖𝐴^𝑇_𝑑𝑖−(1−𝑇𝛼)^(𝜏+1)𝑄_𝑖𝑇𝑋_𝑖𝐴^𝑇_𝑑𝑖0𝑋_𝑖𝐸^𝑇_𝑎𝑑𝑖𝑇𝐴_𝑖𝑋_𝑖𝑇𝐴_𝑑𝑖𝑋_𝑖−𝑇𝑋_𝑖𝜀_𝑖𝑇𝐻_𝑖0𝜀_𝑖𝐻^𝑇_𝑖0𝜀_𝑖𝑇𝐻^𝑇_𝑖−𝜀_𝑖𝐸𝐼0_𝑎𝑖𝑋_𝑖𝐸_𝑎𝑑𝑖𝑋_𝑖00−𝜀_𝑖𝐼<0, (12) where 𝐸_𝑖=𝐴_𝑖𝑋_𝑖+𝑋_𝑖𝐴^𝑇_𝑖+𝛼𝑋_𝑖+(1−𝑇𝛼)(𝜏−𝜏+1)𝑄_𝑖, then system (2) with 𝑤(𝑘)=0 is exponentially stable for any switching signal 𝜎(𝑘) with the following average dwell time scheme: 𝜏_𝑎>𝜏^∗_𝑎=−ln𝜇(),ln1−𝑇𝛼 (13) where 𝜇≥1 satisfies 𝑋_𝑖≤𝜇𝑋_𝑗,𝑄_𝑖≤𝜇𝑄_𝑗,∀𝑖,𝑗∈𝑁. (14)

Proof. Choose the following Lyapunov-Krasovskii functional candidate for the 𝑖th subsystem 𝑉_𝑖(𝑘)=𝑉_𝑖1(𝑘)+𝑉_𝑖2(𝑘)+𝑉_𝑖3(𝑘),∀𝑖∈𝑁, (15) where 𝑉_𝑖1(𝑘)=𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘),𝑉_𝑖2(𝑘)=𝑇^𝑘−1_{𝑠=𝑘−𝜏(𝑘)}(1−𝑇𝛼)^{(𝑘−𝑠)}𝑥^𝑇(𝑠)𝑆_𝑖𝑥(𝑠),𝑉_𝑖3(𝑘)=𝑇^−𝜏_{𝑙=−𝜏+1}^𝑘−1_{𝑠=𝑘+𝑙}(1−𝑇𝛼)^𝑘−𝑠𝑥^𝑇(𝑠)𝑆_𝑖𝑥(𝑠). (16)
Taking the delta operator manipulations of Lyapunov functional candidate 𝑉_𝑖(𝑘) along the trajectory of system (2), by Lemma 6, we have 𝛿𝑉_𝑖1(𝑘)=𝛿𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘)=𝛿𝑥^𝑇(𝑘)𝑃_𝑖𝑥(𝑘)+𝑥^𝑇(𝑘)𝑃_𝑖𝛿(𝑥(𝑘𝑥))+𝑇𝛿^𝑇(𝑘)𝑃_𝑖𝛿(𝑥(𝑘=𝐴))_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝜏(𝑘))^𝑇𝑃_𝑖𝑥(𝑘)+𝑥^𝑇(𝑘)𝑃_𝑖𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥((𝑘𝐴𝑘−𝜏))+𝑇_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝜏(𝑘))^𝑇×𝑃_𝑖𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝜏(𝑘))=𝑥^𝑇(𝑘)𝑃_𝑖𝐴_𝑖𝑥(𝑘)+𝑥^𝑇(𝑘)𝑃_𝑖𝐴_𝑑𝑖𝑥(𝑘−𝜏(𝑘))+𝑥^𝑇(𝑘)𝐴^𝑇_𝑖𝑃_𝑖𝑥(𝑘)+𝑥^𝑇(𝑘−𝜏(𝑘))𝐴^𝑇_𝑑𝑖𝑃_𝑖𝑥(𝑘)+𝑇𝑥^𝑇(𝑘)𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑖𝑥(𝑘)+𝑇𝑥^𝑇(𝑘)𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑑𝑖𝑥(𝑘−𝜏(𝑘))+𝑇𝑥^𝑇(𝑘−𝜏(𝑘))𝐴^𝑇_𝑑𝑖𝑃_𝑖𝐴_𝑖𝑥(𝑘)+𝑇𝑥^𝑇(𝑘−𝜏(𝑘))𝐴^𝑇_𝑑𝑖𝑃_𝑖𝐴_𝑑𝑖𝑥((𝑘=𝑥(𝑘)𝑥((𝑘𝑘−𝜏))𝑘−𝜏))^𝑇×𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖+𝑇𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑖𝑃_𝑖𝐴_𝑑𝑖+𝑇𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑑𝑖𝐴^𝑇_𝑑𝑖𝑃_𝑖+𝑇𝐴^𝑇_𝑑𝑖𝑃_𝑖𝐴_𝑖𝑇𝐴^𝑇_𝑑𝑖𝑃_𝑖𝐴_𝑑𝑖×𝑥(𝑘)𝑥((𝑘,𝑘−𝜏))𝛿𝑉_𝑖2(𝑘)=1𝑇𝑉_𝑖2(𝑘+1)−𝑉_𝑖2(𝑘)=1𝑇𝑇^𝑘+1−1_{𝑠=𝑘+1−𝜏(𝑘+1)}(1−𝑇𝛼)^{(𝑘+1−𝑠)}𝑥^𝑇(𝑠)𝑆_𝑖𝑥(𝑠)−𝑇^𝑘−1_{(𝑘)𝑠=𝑘−𝜏}(1−𝑇𝛼)^{(𝑘−𝑠)}𝑥^𝑇(𝑠)𝑆_𝑖𝑥(𝑠)≤−𝑇𝛼^𝑘−1_{(𝑘)𝑠=𝑘−𝜏}(1−𝑇𝛼)^{(𝑘−𝑠)}𝑥^𝑇(𝑠)𝑆_𝑖𝑥(𝑠)+(1−𝑇𝛼)𝑥^𝑇(𝑘)𝑆_𝑖𝑥(𝑘)−(1−𝑇𝛼)^(𝜏+1)𝑥^𝑇(𝑘−𝜏(𝑘))𝑆_𝑖𝑥((𝑘+𝑘−𝜏))_{𝑘−𝜏𝑠=𝑘+1−𝜏}(1−𝑇𝛼)^{(𝑘+1−𝑠)}𝑥^𝑇(𝑠)𝑆_𝑖𝑥(𝑠),𝛿𝑉_𝑖3(𝑘)=1𝑇𝑉_𝑖3(𝑘+1)−𝑉_𝑖3(𝑘)=1𝑇𝑇^−𝜏_{𝑙=−𝜏+1}^𝑘−1+1_{𝑠=𝑘+1+𝑙}(1−𝑇𝛼)^{(𝑘+1−𝑠)}𝑥^𝑇(𝑠)𝑆_𝑖𝑥(𝑠)−𝑇^−𝜏_{𝑙=−𝜏+1}^𝑘−1_{𝑠=𝑘+𝑙}(1−𝑇𝛼)^{(𝑘−𝑠)}𝑥^𝑇(𝑠)𝑆_𝑖𝑥(𝑠)=−𝑇𝛼^−𝜏_{𝑙=−𝜏+1}^𝑘−1_{𝑠=𝑘+𝑙}(1−𝑇𝛼)^{(𝑘−𝑠)}𝑥^𝑇(𝑠)𝑆_𝑖𝑥(𝑠)+(1−𝑇𝛼)𝜏−𝜏𝑥^𝑇(𝑘)𝑆_𝑖𝑥(𝑘)−^𝑘−𝜏_{𝑠=𝑘+1−𝜏}(1−𝑇𝛼)^{(𝑘+1−𝑠)}𝑥^𝑇(𝑠)𝑆_𝑖𝑥(𝑠). (17)(18)(19) Combining (17)–(19), we have 𝛿𝑉_𝑖(𝑘)+𝛼𝑉_𝑖(𝑘)=𝑥(𝑘)𝑥(𝑘−𝜏(𝑘))^𝑇×𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖+𝑇𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑖𝑃_𝑖𝐴_𝑑𝑖+𝑇𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑑𝑖𝐴^𝑇_𝑑𝑖𝑃_𝑖+𝑇𝐴^𝑇_𝑑𝑖𝑃_𝑖𝐴_𝑖𝑇𝐴^𝑇_𝑑𝑖𝑃_𝑖𝐴_𝑑𝑖×𝑥(𝑘)𝑥(𝑘−𝜏(𝑘))+𝛼𝑃_𝑖+(1−𝑇𝛼)𝜏−𝜏+1𝑥^𝑇(𝑘)𝑆_𝑖𝑥(𝑘)−(1−𝑇𝛼)^(𝜏+1)𝑥^𝑇(𝑘−𝜏(𝑘))𝑆_𝑖𝑥((𝑘≤𝑥(𝑘)𝑥((𝑘𝑘−𝜏))𝑘−𝜏))^𝑇Ω_𝑖𝑥(𝑘)𝑥((𝑘,𝑘−𝜏)) (20) whereΩ_𝑖=𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖+𝛼𝑃_𝑖+(1−𝑇𝛼)×𝜏−𝜏+1𝑆_𝑖+𝑇𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑖𝑃_𝑖𝐴_𝑑𝑖+𝑇𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑑𝑖𝐴^𝑇_𝑑𝑖𝑃_𝑖+𝑇𝐴^𝑇_𝑑𝑖𝑃_𝑖𝐴_𝑖𝑇𝐴^𝑇_𝑑𝑖𝑃_𝑖𝐴_𝑑𝑖−(1−𝑇𝛼)^(𝜏+1)𝑆_𝑖. (21)Applying Lemma 4, we can obtain that Ω_𝑖<0 is equivalent to𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖+𝛼𝑃_𝑖+(1−𝑇𝛼)𝜏−𝜏+1𝑆_𝑖𝑃_𝑖𝐴_𝑑𝑖𝑇𝐴^𝑇_𝑖𝐴^𝑇_𝑑𝑖𝑃_𝑖−(1−𝑇𝛼)^(𝜏+1)𝑆_𝑖𝑇𝐴^𝑇_𝑑𝑖𝑇𝐴_𝑖𝑇𝐴_𝑑𝑖−𝑇𝑃_𝑖⁻¹<0. (22)Using diag𝑃_𝑖⁻¹𝑃_𝑖⁻¹𝐼 to pre- and post-multiply both sides of (22), respectively, we have 𝐸_𝑖𝐴_𝑑𝑖𝑃_𝑖⁻¹𝑇𝑃_𝑖⁻¹𝐴^𝑇_𝑖𝑃_𝑖⁻¹𝐴^𝑇_𝑑𝑖−(1−𝑇𝛼)^(𝜏+1)𝑃_𝑖⁻¹𝑆𝑃_𝑖⁻¹𝑇𝑃_𝑖⁻¹𝐴^𝑇_𝑑𝑖𝑇𝐴_𝑖𝑃_𝑖⁻¹𝑇𝐴_𝑑𝑖𝑃_𝑖⁻¹−𝑇𝑃_𝑖⁻¹<0, (23) where 𝐸_𝑖=𝐴_𝑖𝑃_𝑖⁻¹+𝑃_𝑖⁻¹𝐴^𝑇_𝑖+𝛼𝑃_𝑖⁻¹+(1−𝑇𝛼)(𝜏−𝜏+1)𝑃_𝑖⁻¹𝑆_𝑖𝑃_𝑖⁻¹.
Denote 𝑄_𝑖=𝑃_𝑖⁻¹𝑆_𝑖𝑃_𝑖⁻¹ and 𝑋_𝑖=𝑃_𝑖⁻¹; then, substituting (3) into (23) and applying Lemmas 4 and 5, we obtain that (23) is equivalent to (12). Thus, from (12), we can easily obtain 𝛿𝑉_𝑖(𝑘)+𝛼𝑉_𝑖(𝑘)≤0. (24) It follows from (24) that 𝛿𝑉_𝑖(𝑘)=𝑉_𝑖(𝑘+1)−𝑉_𝑖(𝑘)𝑇≤−𝛼𝑉_𝑖(𝑘),𝑉_𝑖(𝑘+1)−𝑉_𝑖(𝑘)≤−𝛼_𝑖𝑇𝑉_𝑖(𝑘),𝑉_𝑖(𝑘+1)≤(1−𝛼𝑇)𝑉_𝑖(𝑘). (25) Let 𝑘₁<⋯<𝑘_𝑞 denote the switching instants of 𝜎(𝑘) over the interval [𝑘₀,𝑘). Consider the following piecewise Lyapunov functional for system (2): 𝑉(𝑘)=𝑉_𝜎(𝑘)(𝑘)=𝑉_{𝜎(𝑘𝑝)}(𝑘),∀𝑘∈𝑘_𝑝,𝑘_𝑝+1,𝑝=0,1,…,𝑞. (26) From (14), we obtain 𝑉_{𝜎(𝑘𝑝)}𝑘_𝑝≤𝜇𝑉_𝜎(𝑘⁻_𝑝)𝑘⁻_𝑝,𝑝=0,1,…,𝑞. (27) It can be obtained from (24), (27), and Definition 3 that 𝑉_𝜎(𝑘)(𝑘)≤(1−𝑇𝛼)^{(𝑘−𝑘𝑞)}𝑉_{𝜎(𝑘𝑞)}𝑘_𝑞≤𝜇(1−𝑇𝛼)^{(𝑘−𝑘𝑞)}𝑉_𝜎(𝑘⁻_𝑞)𝑘⁻_𝑞≤𝜇(1−𝑇𝛼)^{(𝑘−𝑘𝑞−1)}𝑉_{𝜎(𝑘𝑞−1)}𝑘_𝑞−1≤𝜇²(1−𝑇𝛼)^{(𝑘−𝑘𝑞−1)}𝑉_𝜎(𝑘⁻_𝑞−1)𝑘⁻_𝑞−1≤⋯≤𝜇^{𝑁𝜎(𝑘)(𝑘0,𝑘)}(1−𝑇𝛼)^{(𝑘−𝑘0)}𝑉_𝜎(𝑘0)𝑘₀≤𝜇^{(𝑘−𝑘0)/𝜏𝑎}(1−𝑇𝛼)^{(𝑘−𝑘0)}𝑉_𝜎(𝑘0)𝑘₀=𝜇^1/𝜏𝑎(1−𝑇𝛼)^{(𝑘−𝑘0)}𝑉_𝜎(𝑘0)𝑘₀. (28) Considering the definition of 𝑉_𝜎(𝑘)(𝑘), it yields that 𝑉_𝜎(𝑘)(𝑘)≥𝑎‖𝑥(𝑘)‖²,𝑉_𝜎(𝑘)𝑘₀≤𝑏𝑥𝑘₀²_𝑐, (29)(30) where 𝑎=min_𝑖∈𝑁𝜆_min𝑃_𝑖,𝑏=max_𝑖∈𝑁𝜆_max𝑃_𝑖+𝑇𝜏²−𝜏𝜏+𝜏𝜆_max𝑆_𝑖,𝑥𝑘_0𝑐=sup_{−𝜏≤𝜃≤0}𝑥𝑘₀+𝜃. (31) Combining (29) and (30), we have ‖𝑥(𝑘)‖²≤𝑏𝑎𝜇^1/𝜏𝑎(1−𝑇𝛼)^{(𝑘−𝑘0)}𝑥𝑘₀². (32) Therefore, system (2) with 𝑤(𝑘)=0 is exponentially stable under the average dwell time scheme (13).
The proof is completed.

Remark 8. When 𝜇=1 in (14), which leads to 𝑋_𝑖=𝑋_𝑗, 𝑄_𝑖=𝑄_𝑗, ∀𝑖,𝑗∈𝑁, and 𝜏^∗_𝑎=0 by (13), system (2) has a common Lyapunov-Krasovskii functional and the switching signal can be arbitrary.
When 𝜏(𝑘)=0, system (2) with 𝑤(𝑘)=0 becomes the following system: 𝛿𝑥(𝑘)=𝐴_𝜎(𝑘)+𝐴_{𝑑𝜎(𝑘)}𝑥(𝑘),𝑧(𝑘)=𝐶_𝜎(𝑘)𝑥(𝑘). (33) Then we have the following corollary.

Corollary 9. For a given positive constant 0<𝛼<1/𝑇, if there exist scalars 𝜀_𝑖 and positive definite symmetric matrices 𝑋_𝑖, ∀𝑖∈𝑁, of appropriate dimensions, such that𝐴_𝑖+𝐴_𝑑𝑖𝑋_𝑖+𝑋_𝑖𝐴^𝑇_𝑖+𝐴^𝑇_𝑑𝑖+𝛼𝑋_𝑖𝑇𝑋_𝑖𝐴^𝑇_𝑖+𝐴^𝑇_𝑑𝑖𝜀_𝑖𝐻_𝑖𝑋_𝑖𝐸^𝑇_𝑎𝑖+𝐸^𝑇_𝑎𝑑𝑖𝑇𝐴_𝑖+𝐴_𝑑𝑖𝑋_𝑖−𝑇𝑋_𝑖𝜀_𝑖𝑇𝐻_𝑖0𝜀_𝑖𝐻^𝑇_𝑖𝜀_𝑖𝑇𝐻^𝑇_𝑖−𝜀_𝑖𝐸𝐼0_𝑎𝑖+𝐸_𝑎𝑑𝑖𝑋_𝑖00−𝜀_𝑖𝐼<0, (34)then system (33) is exponentially stable for any switching signal 𝜎(𝑘) with average dwell time scheme (13), where 𝜇≥1 satisfies 𝑋_𝑖≤𝜇𝑋_𝑗,∀𝑖,𝑗∈𝑁. (35)

3.2. 𝐻_∞ Performance Analysis

The following theorem gives sufficient conditions for the existence of an 𝐻_∞ performance level for system (2).

Theorem 10. For given positive constants 𝛾 and 0<𝛼<1/𝑇, if there exist scalars 𝜀_𝑖 and positive definite symmetric matrices 𝑋_𝑖 and 𝑄_𝑖, 𝑖∈𝑁, of appropriate dimensions, such that Δ_𝑖𝐴_𝑑𝑖𝑋_𝑖𝐷_𝑖𝑇𝑋_𝑖𝐴^𝑇_𝑖𝑋_𝑖𝐶^𝑇_𝑖𝜀_𝑖𝐻_𝑖𝑋_𝑖𝐸^𝑇_𝑎𝑖𝑋_𝑖𝐴^𝑇_𝑑𝑖−(1−𝑇𝛼)^𝜏+1𝑄_𝑖0𝑇𝑋_𝑖𝐴^𝑇_𝑑𝑖00𝑋_𝑖𝐸^𝑇_𝑎𝑑𝑖𝐷^𝑇_𝑖0−𝛾²𝐼𝑇𝐷^𝑇_𝑖𝐺^𝑇_𝑖00𝑇𝐴_𝑖𝑋_𝑖𝑇𝐴_𝑑𝑖𝑋_𝑖𝑇𝐷_𝑖−𝑇𝑋_𝑖0𝜀_𝑖𝑇𝐻_𝑖0𝐶_𝑖𝑋_𝑖0𝐺_𝑖𝜀0−𝐼00_𝑖𝐻^𝑇_𝑖00𝜀_𝑖𝑇𝐻^𝑇_𝑖0−𝜀_𝑖𝐸𝐼0_𝑎𝑖𝑋_𝑖𝐸_𝑎𝑑𝑖𝑋_𝑖0000−𝜀_𝑖𝐼<0, (36) where Δ_𝑖=𝐴_𝑖𝑋_𝑖+𝑋_𝑖𝐴^𝑇_𝑖+𝛼𝑋_𝑖+(1−𝑇𝛼)(𝜏−𝜏+1)𝑄_𝑖, then system (2) is exponentially stable with an 𝐻_∞ performance level 𝛾 for any switching signal 𝜎(𝑘) with average dwell time scheme (13), where 𝜇≥1 satisfies (14).

Proof. Equation (12) in Theorem 7 can be directly derived from (36). Thus, system (2) is exponentially stable. We are now in a position to show the 𝐻_∞ performance of system (2).
Choosing the Lyapunov-Krasovskii functional candidate (15) and following the proof line of Theorem 7, we get 𝛿𝑉_𝑖1(𝑘)=𝑥^𝑇(𝑘)𝑃_𝑖𝛿𝑥(𝑘)+(𝛿𝑥(𝑘))^𝑇𝑃_𝑖𝑥(𝑘)+𝑇(𝛿𝑥(𝑘))^𝑇𝑃_𝑖𝛿𝑥(𝑘)=𝑥^𝑇(𝑘)𝑃_𝑖𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝜏(𝑘))+𝐷_𝑖𝑤(𝑘)+𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝜏(𝑘))+𝐷_𝑖𝑤(𝑘)^𝑇𝑃_𝑖𝑥(𝑘)+𝑇𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝜏(𝑘))+𝐷_𝑖𝑤(𝑘)^𝑇×𝑃_𝑖𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝜏(𝑘))+𝐷_𝑖𝑤(𝑘)=𝑥(𝑘)𝑥((𝑘𝑤(𝑘)𝑘−𝜏))^𝑇Θ_𝑖𝑥(𝑘)𝑥((𝑘𝑤(𝑘)𝑘−𝜏)), (37) where Θ_𝑖=𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖+𝑇𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑖𝑃_𝑖𝐴_𝑑𝑖+𝑇𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑑𝑖𝑃_𝑖𝐷_𝑖+𝑇𝐴^𝑇_𝑖𝑃_𝑖𝐷_𝑖𝐴^𝑇_𝑑𝑖𝑃_𝑖+𝑇𝐴^𝑇_𝑑𝑖𝑃_𝑖𝐴_𝑖𝑇𝐴^𝑇_𝑑𝑖𝑃_𝑖𝐴_𝑑𝑖𝑇𝐴^𝑇_𝑑𝑖𝑃_𝑖𝐷_𝑖𝐷^𝑇_𝑖𝑃_𝑖+𝑇𝐷^𝑇_𝑖𝑃_𝑖𝐴_𝑖𝑇𝐷^𝑇_𝑖𝑃_𝑖𝐴_𝑑𝑖𝑇𝐷^𝑇_𝑖𝑃_𝑖𝐷_𝑖,𝛿𝑉_𝑖2(𝑘)≤−𝑇𝛼^𝑘−1_{𝑠=𝑘−𝜏(𝑘)}(1−𝑇𝛼)^{(𝑘−𝑠)}𝑥^𝑇(𝑠)𝑆_𝑖𝑥(𝑠)+(1−𝑇𝛼)𝑥^𝑇(𝑘)𝑆_𝑖𝑥(𝑘)−(1−𝛼𝑇)^(𝜏+1)𝑥^𝑇(𝑘−𝜏(𝑘))𝑆_𝑖𝑥((𝑘+𝑘−𝜏))_{𝑘−𝜏𝑠=𝑘+1−𝜏}(1−𝑇𝛼)^{(𝑘+1−𝑠)}𝑥^𝑇(𝑠)𝑆_𝑖𝑥(𝑠),𝛿𝑉_𝑖3(𝑘)=−𝑇𝛼^−𝜏_{𝑙=−𝜏+1}^𝑘−1_{𝑠=𝑘+𝑙}(1−𝑇𝛼)^{(𝑘−𝑠)}𝑥^𝑇(𝑠)𝑆_𝑖𝑥(𝑠)+(1−𝑇𝛼)𝜏−𝜏𝑥^𝑇(𝑘)𝑆_𝑖𝑥(𝑘)−^𝑘−𝜏_{𝑠=𝑘+1−𝜏}(1−𝑇𝛼)^{(𝑘+1−𝑠)}𝑥^𝑇(𝑠)𝑆_𝑖𝑥(𝑠). (38)
It follows from (37)-(38) that 𝛿𝑉_𝑖(𝑘)+𝛼𝑉_𝑖(𝑘)+𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝑤^𝑇(𝑘)𝑤(𝑘)=𝛿𝑉_𝑖(𝑘)+𝛼𝑉_𝑖(𝑘)+(𝐶𝑥(𝑘)+𝐺𝑤(𝑘))^𝑇(𝐶𝑥(𝑘)+𝐺𝑤(𝑘))−𝛾²𝑤^𝑇(𝑘)𝑤(𝑘)=𝑥(𝑘)𝑥((𝑘𝑤(𝑘)𝑘−𝜏))^𝑇Θ_𝑖𝑥(𝑘)𝑥((𝑘𝑤(𝑘)𝑘−𝜏)), (39) whereΘ_𝑖=Π_𝑖𝑃_𝑖𝐴_𝑑𝑖+𝑇𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑑𝑖𝑃_𝑖𝐷_𝑖+𝑇𝐴^𝑇_𝑖𝑃_𝑖𝐷_𝑖+𝐶^𝑇_𝑖𝐺_𝑖𝐴^𝑇_𝑑𝑖𝑃_𝑖+𝑇𝐴^𝑇_𝑑𝑖𝑃_𝑖𝐴_𝑖𝑇𝐴^𝑇_𝑑𝑖𝑃_𝑖𝐴_𝑑𝑖−(1−𝑇𝛼)^𝜏+1𝑆_𝑖𝑇𝐴^𝑇_𝑑𝑖𝑃_𝑖𝐷_𝑖𝐷^𝑇_𝑖𝑃_𝑖+𝑇𝐷^𝑇_𝑖𝑃_𝑖𝐴_𝑖+𝐺^𝑇_𝑖𝐶_𝑖𝑇𝐷^𝑇_𝑖𝑃_𝑖𝐴_𝑑𝑖𝑇𝐷^𝑇_𝑖𝑃_𝑖𝐷_𝑖+𝐺^𝑇_𝑖𝐺_𝑖−𝛾²𝐼,Π_𝑖=𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖+𝑇𝐴^𝑇_𝑖𝑃_𝑖𝐴_𝑖+𝛼𝑃_𝑖+(1−𝑇𝛼)𝜏−𝜏+1𝑆_𝑖+𝐶^𝑇_𝑖𝐶_𝑖. (40)
Applying Lemma 4, we can obtain that Θ_𝑖<0 is equivalent to the following inequality:𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖+𝛼𝑃_𝑖+(1−𝑇𝛼)𝜏−𝜏+1𝑆_𝑖𝑃_𝑖𝐴_𝑑𝑖𝑃_𝑖𝐷_𝑖𝑇𝐴^𝑇_𝑖𝐶^𝑇_𝑖𝐴^𝑇_𝑑𝑖𝑃_𝑖−(1−𝑇𝛼)^𝜏+1𝑆_𝑖0𝑇𝐴^𝑇_𝑑𝑖0𝐷^𝑇_𝑖𝑃_𝑖0−𝛾²𝐼𝑇𝐷^𝑇_𝑖𝐺^𝑇_𝑖𝑇𝐴_𝑖𝑇𝐴_𝑑𝑖𝑇𝐷_𝑖−𝑇𝑃_𝑖⁻¹0𝐶_𝑖0𝐺_𝑖0−𝐼<0. (41)
Using diag𝑃_𝑖⁻¹𝑃_𝑖⁻¹𝐼𝐼𝐼 to pre- and post-multiply both sides of (41), respectively, we have 𝑀_𝑖𝐴_𝑑𝑖𝑃_𝑖⁻¹𝐷_𝑖𝑇𝑃_𝑖⁻¹𝐴^𝑇_𝑖𝑃_𝑖⁻¹𝐶^𝑇_𝑖𝑃_𝑖⁻¹𝐴^𝑇_𝑑𝑖−(1−𝑇𝛼)^𝜏+1𝑃_𝑖⁻¹𝑆_𝑖𝑃_𝑖⁻¹0𝑇𝑃_𝑖⁻¹𝐴^𝑇_𝑑𝑖0𝐷^𝑇_𝑖0−𝛾²𝐼𝑇𝐷^𝑇_𝑖𝐺^𝑇_𝑖𝑇𝐴_𝑖𝑃_𝑖⁻¹𝑇𝐴_𝑑𝑖𝑃_𝑖⁻¹𝑇𝐷_𝑖−𝑇𝑃_𝑖⁻¹0𝐶_𝑖𝑃_𝑖⁻¹0𝐺_𝑖0−𝐼<0, (42) where 𝑀_𝑖=𝐴_𝑖𝑃_𝑖⁻¹+𝑃_𝑖⁻¹𝐴^𝑇_𝑖+𝛼𝑃_𝑖⁻¹+(1−𝑇𝛼)(𝜏−𝜏+1)𝑃_𝑖⁻¹𝑆_𝑖𝑃_𝑖⁻¹.
Set 𝑄_𝑖=𝑃_𝑖⁻¹𝑆_𝑖𝑃_𝑖⁻¹ and 𝑋_𝑖=𝑃_𝑖⁻¹; then, substituting (3) into (42) and applying Lemmas 4 and 5, we can obtain that (42) is equivalent to (36).
Therefore, one has, for 𝑘∈[𝑘_𝑝,𝑘_𝑝+1), 𝑉(𝑘)≤(1−𝑇𝛼)^{(𝑘−𝑘𝑝)}𝑉_{𝜎(𝑘𝑝)}𝑘_𝑝−^𝑘−1_{𝑠=𝑘𝑝}(1−𝑇𝛼)^{(𝑘−1−𝑠)}Λ(𝑠), (43) where Λ(𝑠)=𝑇‖𝑧(𝑠)‖²−𝛾²𝑇‖𝑤(𝑠)‖².
Following the proof line of (28), we obtain 𝑉_𝜎(𝑘)(𝑘)≤𝜇(1−𝑇𝛼)^{(𝑘−𝑘𝑞)}𝑉_𝜎(𝑘⁻_𝑞)𝑘⁻_𝑞−^𝑘−1_{𝑠=𝑘𝑞}(1−𝑇𝛼)^{(𝑘−1−𝑠)}Λ(𝑠)≤𝜇(1−𝑇𝛼)^{(𝑘−𝑘𝑞−1)}𝑉_{𝜎(𝑘𝑞−1)}𝑘_𝑞−1−𝜇^𝑘𝑞−1_{𝑠=𝑘𝑞−1}(1−𝑇𝛼)^{(𝑘−1−𝑠)}Λ(𝑠)−^𝑘−1_{𝑠=𝑘𝑞}(1−𝑇𝛼)^{(𝑘−1−𝑠)}Λ(𝑠)=𝜇^{𝑁𝜎(𝑘)(𝑘𝑞−1,𝑘)}(1−𝑇𝛼)^{(𝑘−𝑘𝑞−1)}𝑉_{𝜎(𝑘𝑞−1)}𝑘_𝑞−1−𝜇^{𝑁𝜎(𝑘)(𝑘𝑞−1𝑘,𝑘)𝑞−1}_{𝑠=𝑘𝑞−1}(1−𝑇𝛼)^{(𝑘−1−𝑠)}Λ(𝑠)−^𝑘−1_{𝑠=𝑘𝑞}(1−𝑇𝛼)^{(𝑘−1−𝑠)}Λ(𝑠)≤⋯≤𝜇^{𝑁𝜎(𝑘)(𝑘0,𝑘)}(1−𝑇𝛼)^{(𝑘−𝑘0)}𝑉_𝜎(𝑘0)𝑘₀−𝜇^{𝑁𝜎(𝑘)(𝑘0𝑘,𝑘)1−1}_𝑠=𝑘0(1−𝑇𝛼)^{(𝑘−1−𝑠)}Λ(𝑠)−𝜇^{𝑁𝜎(𝑘)(𝑘1𝑘,𝑘)2−1}_𝑠=𝑘1(1−𝑇𝛼)^{(𝑘−1−𝑠)}Λ(𝑠)−⋯−^𝑘−1_{𝑠=𝑘𝑞}(1−𝑇𝛼)^{(𝑘−1−𝑠)}Λ(𝑠)=𝜇^{𝑁𝜎(𝑘)(𝑘0,𝑘)}(1−𝑇𝛼)^{(𝑘−𝑘0)}𝑉_𝜎(𝑘0)𝑘₀−^𝑘−1_𝑠=𝑘0𝜇^{𝑁𝜎(𝑘)(𝑠,𝑘)}(1−𝑇𝛼)^{(𝑘−1−𝑠)}Λ(𝑠). (44)
Under the zero initial condition, we get 0≤−^𝑘−1_𝑠=𝑘0𝜇^{𝑁𝜎(𝑘)(𝑠,𝑘)}(1−𝑇𝛼)^{(𝑘−1−𝑠)}Λ(𝑠). (45) Namely, ^𝑘−1_𝑠=𝑘0𝜇^{𝑁𝜎(𝑘)(𝑠,𝑘)}(1−𝑇𝛼)^{(𝑘−𝑠)}‖𝑧(𝑠)‖²≤𝛾^2𝑘−1_𝑠=𝑘0𝜇^{𝑁𝜎(𝑘)(𝑠,𝑘)}(1−𝑇𝛼)^{(𝑘−𝑠)}‖𝑤(𝑠)‖². (46)
Multiplying both sides of (46) by 𝜇^{−𝑁𝜎(𝑘)(𝑘0,𝑘)} leads to ^𝑘−1_𝑠=𝑘0𝜇^{−𝑁𝜎(𝑘)(𝑘0,𝑠)}(1−𝑇𝛼)^{(𝑘−𝑠)}‖𝑧(𝑠)‖²≤𝛾^2𝑘−1_𝑠=𝑘0𝜇^{−𝑁𝜎(𝑘)(𝑘0,𝑠)}(1−𝑇𝛼)^{(𝑘−𝑠)}‖𝑤(𝑠)‖². (47)
From Definition 3 and (13), we have 𝜇^{−𝑁𝜎(𝑘)(𝑘0,𝑠)}≤(1−𝑇𝛼)^𝑠−𝑘0. (48)
Combining (47) and (48) leads to ^𝑘−1_𝑠=𝑘0(1−𝑇𝛼)^{(𝑠−𝑘0)}(1−𝑇𝛼)^{(𝑘−𝑠)}‖𝑧(𝑠)‖²≤𝛾^2𝑘−1_𝑠=𝑘0(1−𝑇𝛼)^{(𝑘−𝑠)}‖𝑤(𝑠)‖². (49)
Then, summing both sides of (49) from 𝑘₀ to ∞ leads to ^∞_𝑘=𝑘0(1−𝑇𝛼)^{(𝑘−𝑘0)}‖𝑧(𝑘)‖²≤𝛾^2∞_𝑘=𝑘0‖𝑤(𝑘)‖². (50) According to Definition 2, we can conclude that the theorem is true.
The proof is completed.

3.3. 𝐻_∞ Controller Design

In this section, a state feedback controller 𝑢(𝑘)=𝐾_𝜎(𝑘)𝑥(𝑘) will be designed for system (11) such that the corresponding closed-loop system (51) is exponentially stable and satisfies an 𝐻_∞ performance. Consider 𝛿𝑥(𝑘)=𝐴_𝜎(𝑘)+𝐵_𝜎(𝑘)𝐾_𝜎(𝑘)𝑥(𝑘)+𝐴_{𝑑𝜎(𝑘)}𝑥(𝑘−𝜏(𝑘))+𝐷_𝜎(𝑘)𝑤(𝑘),𝑧(𝑘)=𝐶_𝜎(𝑘)𝑥(𝑘)+𝐸_𝜎(𝑘)𝑤(𝑘),𝑥𝑘₀+𝜃=𝜑(𝜃),𝜃=−𝜏,−𝜏+1,…,−1,0, (51) where 𝐾_𝑖, 𝑖∈𝑁, are the controller gains to be determined. 𝐵_𝑖 are uncertain real-valued matrices with appropriate dimensions and have the following form𝐵_𝑖=𝐵_𝑖+𝐻_𝑖𝐹_𝑖(𝑘)𝐸_𝑏𝑖. (52)

Theorem 11. Consider system (11). For given positive constants 𝛾 and 0<𝛼<1/𝑇, if there exist scalars 𝜀_𝑖, positive definite symmetric matrices 𝑄_𝑖 and 𝑋_𝑖, and any matrices 𝑊_𝑖, 𝑖∈𝑁, of appropriate dimensions, such thatΥ_𝑖𝐴_𝑑𝑖𝑋_𝑖𝐷_𝑖𝑇𝐴_𝑖𝑋_𝑖+𝐵_𝑖𝑊_𝑖^𝑇𝑋_𝑖𝐶^𝑇_𝑖𝜀_𝑖𝐻_𝑖𝐸_𝑎𝑖𝑋_𝑖+𝐸_𝑏𝑖𝑊_𝑖^𝑇𝑋_𝑖𝐴^𝑇_𝑑𝑖−(1−𝑇𝛼)^𝜏+1𝑄_𝑖0𝑇𝑋_𝑖𝐴^𝑇_𝑑𝑖00𝑋_𝑖𝐸^𝑇_𝑎𝑑𝑖𝐷^𝑇_𝑖0−𝛾²𝐼𝑇𝐷^𝑇_𝑖𝐺^𝑇_𝑖𝑇𝐴00_𝑖𝑋_𝑖+𝐵_𝑖𝑊_𝑖𝑇𝐴_𝑑𝑖𝑋_𝑖𝑇𝐷_𝑖−𝑇𝑋_𝑖0𝜀_𝑖𝑇𝐻_𝑖0𝐶_𝑖𝑋_𝑖0𝐺_𝑖𝜀0−𝐼00_𝑖𝐻^𝑇_𝑖00𝜀_𝑖𝑇𝐻^𝑇_𝑖0−𝜀_𝑖𝐸𝐼0_𝑎𝑖𝑋_𝑖+𝐸_𝑏𝑖𝑊_𝑖𝐸_𝑎𝑑𝑖𝑋_𝑖0000−𝜀_𝑖𝐼<0, (53)where Υ_𝑖=(𝐴_𝑖𝑋_𝑖+𝐵_𝑖𝑊_𝑖)+(𝐴_𝑖𝑋_𝑖+𝐵_𝑖𝑊_𝑖)^𝑇+𝛼𝑋_𝑖+(1−𝑇𝛼)(𝜏−𝜏+1)𝑄_𝑖, then under the state feedback controller 𝑢(𝑘)=𝐾_𝜎(𝑘)𝑥(𝑘),𝐾_𝑖=𝑊_𝑖𝑋_𝑖⁻¹ (54) and the average dwell time scheme (13), the closed-loop system (51) is exponentially stable with a prescribed 𝐻_∞ performance level 𝛾, where 𝜇≥1 satisfies (14).

Proof. Replacing 𝐴_𝑖 in (36) with 𝐴_𝑖+𝐵_𝑖𝐾_𝑖, we getΘ_𝑖=𝜑_𝑖𝐴_𝑑𝑖𝑋_𝑖𝐷_𝑖𝑇𝑋_𝑖𝐴_𝑖+𝐵_𝑖𝐾_𝑖^𝑇𝑋_𝑖𝐶^𝑇_𝑖𝜀_𝑖𝐻_𝑖𝑋_𝑖𝐸_𝑎𝑖+𝐸_𝑏𝑖𝐾_𝑖^𝑇𝑋_𝑖𝐴^𝑇_𝑑𝑖−(1−𝑇𝛼)^𝜏+1𝑄_𝑖0𝑇𝑋_𝑖𝐴^𝑇_𝑑𝑖00𝑋_𝑖𝐸^𝑇_𝑎𝑑𝑖𝐷^𝑇_𝑖0−𝛾²𝐼𝑇𝐷^𝑇_𝑖𝐺^𝑇_𝑖𝑇𝐴00_𝑖+𝐵_𝑖𝐾_𝑖𝑋_𝑖𝑇𝐴_𝑑𝑖𝑋_𝑖𝑇𝐷_𝑖−𝑇𝑋_𝑖0𝜀_𝑖𝑇𝐻_𝑖0𝐶_𝑖𝑋_𝑖0𝐺_𝑖𝜀0−𝐼00_𝑖𝐻^𝑇_𝑖00𝜀_𝑖𝑇𝐻^𝑇_𝑖0−𝜀_𝑖𝐸𝐼0_𝑎𝑖+𝐸_𝑏𝑖𝐾_𝑖𝑋_𝑖𝐸_𝑎𝑑𝑖𝑋_𝑖0000−𝜀_𝑖𝐼<0, (55)where 𝜑_𝑖=(𝐴_𝑖+𝐵_𝑖𝐾_𝑖)𝑋_𝑖+𝑋_𝑖(𝐴_𝑖+𝐵_𝑖𝐾_𝑖)^𝑇+𝛼𝑋_𝑖+(1−𝑇𝛼)(𝜏−𝜏+1)𝑄_𝑖.
Denoting 𝑊_𝑖=𝐾_𝑖𝑋_𝑖, (53) is directly obtained.
The proof is completed.

We are now in a position to give an algorithm for determining 𝐾_𝑖 and 𝜏^∗_𝑎.

Algorithm 12. Step 1. Input the system matrices.

Step 2. Choose the parameters 0<𝛼<1/𝑇 and 𝛾>0. By solving (53), one can obtain the solutions of 𝜀_𝑖, 𝑊_𝑖, 𝑋_𝑖, and 𝑄_𝑖.

Step 3. By (54), with the obtained 𝑊_𝑖 and 𝑋_𝑖, one can compute the gain matrices 𝐾_𝑖.

Step 4. Compute 𝜇 and 𝜏^∗_𝑎 by (13)-(14).

4. Numerical Example

Consider system (11) with parameters as follows: 𝐴₁=0.5−0.700.4,𝐴_𝑑1=,𝐵0.200−0.4₁=0.1200.4−1,𝐶₁=,𝐷0.2−0.18₁=00.01,𝐺₁𝐻=0.01,₁=0.05−0.05,𝐸_𝑎1=−0.110.03^𝑇,𝐸_𝑎𝑑1=0.03−0.1^𝑇,𝐸_𝑏1=0.02−0.01^𝑇,𝐴₂=1.2−1.31.2−0.8,𝐴_𝑑2=,𝐵00.30.10₂=−0.1000.3,𝐶₂=,𝐷0.1−0.18₂=0.02−0.01,𝐺₂𝐻=0.05,₂=0.07−0.1,𝐸_𝑎2=0.06−0.13^𝑇,𝐸_𝑎𝑑2=0.01−0.03^𝑇,𝐸_𝑏2=−0.240.01^𝑇,𝐹₁(𝑘)=𝐹₂(𝑘)=sin(𝑘). (56) Choosing 𝜏=1, 𝜏=0, 𝛼=0.9, 𝛾=2, and 𝑇=0.25 and solving (53) in Theorem 11, we obtain 𝑋₁=,𝑄48.993628.906328.906357.2489₁=,𝑋28.348216.594016.594039.3879₂=,𝑄7.764612.210312.210335.4768₂=,4.35239.91879.918735.9927 (57) and the state feedback gain matrices are as follows: 𝐾₁=,𝐾−25.260614.3786−11.31177.9375₂=.24.9665−8.260616.0614−12.3235 (58)

According to (14), we have 𝜇=6.5134. Then, from (13), we get 𝜏_𝑎>𝜏^∗_𝑎=7.3516. Choosing 𝜏_𝑎=7.5, the simulation results are shown in Figures 1 and 2, where the initial conditions are 𝑥(0)=−11^𝑇, 𝑥(𝑘)=00^𝑇, and  𝑘∈[−1,0) and the exogenous disturbance input is 𝑤(𝑘)=0.05𝑒^−0.5𝑘. The switching signal with average dwell time 𝜏_𝑎=7.5 is shown in Figure 1 and the state responses of the corresponding closed-loop system are given in Figure 2.

Figure 1 

Switching signal.

Figure 2 

State responses of the closed-loop system.

From Figures 1 and 2, it is easy to see that the designed controller can guarantee that the resulting closed-loop system is exponentially stable. This demonstrates the effectiveness of the proposed method.

5. Conclusions

In this paper, the robust stability and 𝐻_∞ controller design problems for time-varying delay switched system using delta operator approach have been investigated. By using the average dwell time approach and constructing a Lyapunov-Krasovskii functional candidate, sufficient conditions for the existence of a state feedback 𝐻_∞ controller are presented. Finally, a numerical example is given to illustrate the feasibility of the proposed approach. In our future work, we will study the problem of robust 𝐻_∞ filtering for delta operator switched systems with uncertainties and time-varying delays.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

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