Exponential Admissibility and 𝐻_∞ Control of Switched Singular Time-Delay Systems: An Average Dwell Time Approach

Abstract

This paper deals with the problems of exponential admissibility and 𝐻_∞ control for a class of continuous-time switched singular systems with time-varying delay. The 𝐻_∞ controllers to be designed include both the state feedback (SF) and the static output feedback (SOF). First, by using the average dwell time scheme, the piecewise Lyapunov function, and the free-weighting matrix technique, an exponential admissibility criterion, which is not only delay-range-dependent but also decay-rate-dependent, is derived in terms of linear matrix inequalities (LMIs). A weighted 𝐻_∞ performance criterion is also provided. Then, based on these, the solvability conditions for the desired SF and SOF controllers are established by employing the LMI technique, respectively. Finally, two numerical examples are given to illustrate the effectiveness of the proposed approach.

1. Introduction

Many real-world engineering systems always exhibit several kinds of dynamic behavior in different parts of the system (e.g., continuous dynamics, discrete dynamics, jump phenomena, and logic commands) and are more appropriately modeled by hybrid systems. As an important class of hybrid systems, switched systems consist of a collection of continuous-time or discrete-time subsystems and a switching rule orchestrating the switching between them and are of great current interest; see, for example, Decarlo et al. [1], Liberzon [2], Lin and Antsaklis [3], and Sun and Ge [4] for some recent survey and monographs. Switched systems have great flexibility in modeling parameter-varying or structure-varying systems, event-driven systems, logic-based systems, and so forth. Also, multiple-controller switching technique offers an effective mechanism to cope with highly complex systems and/or systems with large uncertainties, particularly in the adaptive context [5]. Many effective methods have been developed for switched systems, for example, the multiple Lyapunov function approach [6, 7], the piecewise Lyapunov function approach [8, 9], the switched Lyapunov function method [10], convex combination technique [11], and the dwell-time or average dwell-time scheme [12–15]. Among them, the average dwell-time scheme provides a simple yet efficient tool for stability analysis of switched systems, especially when the switching is restricted and has been more and more favored [16].

On the other hand, time delay is a common phenomenon in various engineering systems and the main sources of instability and poor performance of a system. Hence, control of switched time-delay systems has been an attractive field in control theory and application in the past decade. Some of the aforementioned approaches for nondelayed switched systems have been successfully adopted to hand the switched time-delay systems; see, for example, Du et al. [17], Kim et al. [18], Mahmoud [19], Phat [20], Sun et al. [21], Sun et al. [22], Wang et al. [23], Wu and Zheng [24], Xie et al. [25], Zhang and Yu [26], and the references therein.

Recently, a more general class of switched time-delay systems described by the singular form was considered in Ma et al. [27] and Wang and Gao [28]. It is known that a singular model describes dynamic systems better than the standard state-space system model [29]. The singular form provides a convenient and natural representation of economic systems, electrical networks, power systems, mechanical systems, and many other systems which have to be modeled by additional algebraic constraints [29]. Meanwhile, it endows the aforementioned systems with several special features, such as regularity and impulse behavior, that are not found in standard state-space systems. Therefore, it is both worthwhile and challenging to investigate the stability and control problems of switched singular time-delay systems. In the past few years, some fundamental results based on the aforementioned approaches for standard state-space switched time-delay systems have been successfully extended to switched singular time-delay systems. For example, by using the switched Lyapunov function method, the robust stability, stabilization, and 𝐻_∞ control problems for a class of discrete-time uncertain switched singular systems with constant time delay under arbitrary switching were investigated in Ma et al. [27]; 𝐻_∞ filters were designed in Lin et al. [30] for discrete-time switched singular systems with time-varying time delay. In Wang and Gao [28], based on multiple Lyapunov function approach, a switching signal was constructed to guarantee the asymptotic stability of a class of continuous-time switched singular time-delay systems. With the help of average dwell time scheme, some initial results on the exponential admissibility (regularity, nonimpulsiveness, and exponential stability) were obtained in Lin and Fei [31] for continuous-time switched singular time-delay systems. However, to the best of our knowledge, few work has been conducted regarding the 𝐻_∞ control for continuous-time switched singular time-delay systems via the dwell time or average dwell time scheme, which constitutes the main motivation of the present study.

In this paper, we aim to solve the problem of 𝐻_∞ control for a class of continuous-time switched singular systems with interval time-varying delay via the average dwell time scheme. Both the state feedback (SF) control and the static output feedback (SOF) control are considered. Firstly, based on the average dwell time scheme, the piecewise Lyapunov function, as well as the free-weighting technique, a class of slow switching signals is identified to guarantee the unforced systems to be exponentially admissible with a weighted 𝐻_∞ performance 𝛾, and several corresponding criteria, which are not only delay-range-dependent but also decay-rate-dependent, are derived in terms of linear matrix inequalities (LMIs). Next, the LMI-based approaches are proposed to design an SF controller and an SOF controller, respectively, such that the resultant closed-loop system is exponentially admissible and satisfies a weighted 𝐻_∞ performance 𝛾. Finally, two illustrative examples are given to show the effectiveness of the proposed approach.

Notation 1. Throughout this paper, the superscript 𝑇 represents matrix transposition. 𝐑^𝑛 denotes the real 𝑛-dimensional Euclidean space, and 𝐑^𝑛×𝑛 denotes the set of all 𝑛×𝑛 real matrices. 𝐼 is an appropriately dimensioned identity matrix. 𝑃>0 (𝑃≥0) means that matrix 𝑃 is positive definite (semi positive definite). diag{⋅,⋅,⋅} stands for a block diagonal matrix. 𝜆_min(𝑃) (𝜆_max(𝑃)) denotes the minimum (maximum) eigenvalue of symmetric matrix 𝑃, 𝐿₂[0,∞) is the space of square-integrable vector functions over [0,∞), ‖⋅‖ denotes the Euclidean norm of a vector and its induced norm of a matrix, and Sym{𝐴} is the shorthand notation for 𝐴+𝐴^𝑇. In symmetric block matrices, we use an asterisk (∗) to represent a term that is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Preliminaries and Problem Formulation

Consider a class of switched singular time-delay system of the form 𝐸𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝑑𝜎(𝑡)}𝑥(𝑡−𝑑(𝑡))+𝐵_𝜎(𝑡)𝑢(𝑡)+𝐵_{𝑤𝜎(𝑡)}𝑤(𝑡),𝑧(𝑡)=𝐶_𝜎(𝑡)𝑥(𝑡)+𝐶_{𝑑𝜎(𝑡)}𝑥(𝑡−𝑑(𝑡))+𝐷_𝜎(𝑡)𝑢(𝑡)+𝐷_{𝑤𝜎(𝑡)}𝑤(𝑡),𝑦(𝑡)=𝐿_𝜎(𝑡)𝑥(𝑡),𝑥(𝜃)=𝜙(𝜃),𝜃∈−𝑑₂,0, (2.1) where 𝑥(𝑡)∈𝐑^𝑛 is the system state, 𝑢(𝑡)∈𝐑^𝑚 is the control input, 𝑧(𝑡)∈𝐑^𝑞 is the controlled output, 𝑦(𝑡)∈𝐑^𝑝 is the measured output, and 𝑤(𝑡)∈𝐑^𝑙 is the disturbance input that belongs to 𝐿₂[0,∞); 𝜎(𝑡)∶[0,+∞)→ℐ={1,2,…,𝐼} with integer 𝐼>1 is the switching signal; 𝐸∈𝐑^𝑛×𝑛 is a singular matrix with rank𝐸=𝑟≤𝑛; for each possible value, 𝜎(𝑡)=𝑖, 𝑖∈ℐ, 𝐴_𝑖, 𝐴_𝑑𝑖, 𝐵_𝑖, 𝐵_𝑤𝑖, 𝐶_𝑖, 𝐶_𝑑𝑖, 𝐷_𝑖, 𝐷_𝑤𝑖, and 𝐿_𝑖 are constant real matrices with appropriate dimensions; 𝜙(𝜃) is a compatible continuous vector-valued initial function on [−𝑑₂,0]; 𝑑(𝑡) denotes interval time-varying delay satisfying 𝑑₁≤𝑑(𝑡)≤𝑑₂,𝑑̇(𝑡)≤𝜇<1, (2.2) where 0≤𝑑₁<𝑑₂ and 𝜇 are constants. Note that 𝑑₁ may not be equal to 0.

Since rank𝐸=𝑟≤𝑛, there exist nonsingular matrices 𝑃, 𝑄∈𝐑^𝑛×𝑛 such that 𝐼𝑃𝐸𝑄=^𝑟0.00 (2.3) In this paper, without loss of generality, let 𝐼𝐸=^𝑟0.00 (2.4) Corresponding to the switching signal 𝜎(𝑡), we denote the switching sequence by 𝒮∶={(𝑖₀,𝑡₀),…,(𝑖_𝑘,𝑡_𝑘)∣𝑖_𝑘∈ℐ,𝑘=0,1,…} with 𝑡₀=0, which means that the 𝑖_𝑘 subsystem is activated when 𝑡∈[𝑡_𝑘,𝑡_𝑘+1). To present the objective of this paper more precisely, the following definitions are introduced.

Definition 2.1 (see [2]). For any 𝑇₂>𝑇₁≥0, let 𝑁_𝜎(𝑇₁,𝑇₂) denote the number of switching of 𝜎(𝑡) over (𝑇₁,𝑇₂). If 𝑁_𝜎(𝑇₁,𝑇₂)≤𝑁₀+(𝑇₂−𝑇₁)/𝑇_𝑎 holds for 𝑇_𝑎>0, 𝑁₀≥0, then 𝑇_𝑎 is called average dwell time. As commonly used in the literature [21, 26], we choose 𝑁₀=0.

Definition 2.2 (see [21, 29, 32]). For any delay 𝑑(𝑡) satisfying (2.2), the unforced part of system (2.1) with 𝑤(𝑡)=0𝐸𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝑑𝜎(𝑡)}𝑥((𝑡,𝑥𝑡−𝑑))_𝑡0(𝜃)=𝑥𝑡₀+𝜃=𝜙(𝜃),𝜃∈−𝑑₂,0 (2.5) is said to be(1)regular if det(𝑠𝐸−𝐴_𝑖) is not identically zero for each 𝜎(𝑡)=𝑖, 𝑖∈ℐ, (2)impulse if deg(det(𝑠𝐸−𝐴_𝑖))=rank𝐸 for each 𝜎(𝑡)=𝑖, 𝑖∈ℐ, (3)exponentially stable under the switching signal 𝜎(𝑡) if the solution 𝑥(𝑡) of system (2.5) satisfies ‖𝑥(𝑡)‖≤𝜄𝑒^{−𝜆(𝑡−𝑡0)}𝑥_𝑡0𝑐,∀𝑡≥𝑡₀, (2.6) where 𝜆>0 and 𝜄>0 are called the decay rate and decay coefficient, respectively, and ‖𝑥_𝑡0‖_𝑐=sup_{−𝑑2≤𝜃≤0}{‖𝑥(𝑡₀+𝜃)‖}, (4)exponentially admissible under the switching signal 𝜎(𝑡) if it is regular, impulse free, and exponentially stable under the switching signal 𝜎(𝑡).

Remark 2.3. The regularity and nonimpulsiveness of the switched singular time-delay system (2.5) ensure that its every subsystem has unique solution for any compatible initial condition. However, even if a switched singular system is regular and causal, it still has inevitably finite jumps due to the incompatible initial conditions caused by subsystem switching [33]. For more details about the impulsiveness effects on the stability of systems, we refer readers to Chen and Sun [34], Li et al. [35], and the references therein. In this paper, without loss of generality, we assume that such jumps cannot destroy the stability of system (2.1). Nevertheless, how to suppress or eliminate the finite jumps in switched singular systems is a challenging problem which deserves further investigation.

Definition 2.4. For the given 𝛼>0 and 𝛾>0, system (2.1) is said to be exponentially admissible with a weighted 𝐻_∞ performance 𝛾 under the switching signal 𝜎(𝑡), if it is exponentially admissible with 𝑢(𝑡)=0 and 𝑤(𝑡)=0, and under zero initial condition, that is, 𝜙(𝜃)=0, 𝜃∈[−𝑑₂,0], for any nonzero 𝑤(𝑡)∈𝐿₂[0,∞), it holds that ^𝑡₀𝑒^−𝛼𝑠𝑧^𝑇(𝑠)𝑧(𝑠)𝑑𝑠≤𝛾^2𝑡₀𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠. (2.7)

Remark 2.5. For switched systems with the average dwell time switching, the Lyapunov function values at switching instants are often allowed to increase 𝛽 times (𝛽>1) to reduce the conservatism in system stability analysis, which will lead to the normal disturbance attenuation performance hard to compute or check, even in linear setting [15, 36]. Therefore, the weighted 𝐻_∞ performance criterion (2.7) [15, 21, 24] is adopted here to evaluate disturbance attenuation while obtaining the expected exponential stability.
This paper considers both SF control law 𝑢(𝑡)=𝐾_𝜎(𝑡)𝑥(𝑡) (2.8) and SOF control law 𝑢(𝑡)=𝐹_𝜎(𝑡)𝑦(𝑡), (2.9) where 𝐾_𝑖 and 𝐹_𝑖, 𝜎(𝑡)=𝑖, 𝑖∈ℐ, are appropriately dimensioned constant matrices to be determined.
Then, the problem to be addressed in this paper can be formulated as follows. Given the switched singular time-delay system (2.1) and a prescribed scalar 𝛾>0, identify a class of switching signal 𝜎(𝑡) and design an SF controller of the form (2.8) and an SOF controller of the form (2.9) such that the resultant closed-loop system is exponentially admissible with a weighted 𝐻_∞ performance 𝛾 under the switching signal 𝜎(𝑡).

3. Exponential Admissibility and 𝐻_∞ Performance Analysis

First, we apply the average dwell time approach and the piecewise Lyapunov function technique to investigate the exponential admissibility for the switched singular time-delay system (2.5) and give the following result.

Theorem 3.1. For prescribed scalars 𝛼>0, 0≤𝑑₁≤𝑑₂ and 0<𝜇<1, if for each 𝑖∈ℐ, there exist matrices 𝑄_𝑖𝑙>0, 𝑙=1,2,3, 𝑍_𝑖𝑣>0, 𝑀_𝑖𝑣, 𝑁_𝑖𝑣, 𝑆_𝑖𝑣, 𝑣=1,2, and 𝑃_𝑖 of the following form: 𝑃_𝑖=𝑃_𝑖110𝑃_𝑖21𝑃_𝑖22, (3.1) with 𝑃_𝑖11∈𝐑^𝑟, 𝑃_𝑖11>0, and 𝑃_𝑖22 being invertible, such that Φ_𝑖11Φ_𝑖12Φ_𝑖13−𝑆_𝑖1𝐸𝑐₁𝑁_𝑖1𝑐₁₂𝑆_𝑖1𝑐₁₂𝑀_𝑖1𝐴^𝑇_𝑖𝑈_𝑖∗Φ_𝑖22Φ_𝑖23−𝑆_𝑖2𝐸𝑐₁𝑁_𝑖2𝑐₁₂𝑆_𝑖2𝑐₁₂𝑀_𝑖2𝐴^𝑇_𝑑𝑖𝑈_𝑖∗∗Φ_𝑖3300000∗∗∗Φ_𝑖440000∗∗∗∗−𝑐₁𝑍_𝑖1000∗∗∗∗∗−𝑐₁₂𝑍_𝑖200∗∗∗∗∗∗−𝑐₁₂𝑍_𝑖20∗∗∗∗∗∗∗−𝑈_𝑖<0, (3.2) where Φ_𝑖11=Sym𝑃^𝑇_𝑖𝐴_𝑖+𝑁_𝑖1𝐸+³_𝑙=1𝑄_𝑖𝑙+𝛼𝐸^𝑇𝑃_𝑖,Φ_𝑖12=𝑃^𝑇_𝑖𝐴_𝑑𝑖+𝑁_𝑖2𝐸^𝑇+𝑆_𝑖1𝐸−𝑀_𝑖1𝐸,Φ_𝑖13=𝑀_𝑖1𝐸−𝑁_𝑖1Φ𝐸,_𝑖22=−1−𝜇𝑒^−𝛼𝑑2𝑄_𝑖3+Sym𝑆_𝑖2𝐸−𝑀_𝑖2𝐸,Φ_𝑖23=𝑀_𝑖2𝐸−𝑁_𝑖2𝐸,Φ_𝑖33=−𝑒^−𝛼𝑑1𝑄_𝑖1,Φ_𝑖44=−𝑒^−𝛼𝑑2𝑄_𝑖2,𝑐₁=1𝛼𝑒^𝛼𝑑1−1,𝑐₁₂=1𝛼𝑒^𝛼𝑑2−𝑒^𝛼𝑑1,𝑑₁₂=𝑑₂−𝑑₁,𝑈_𝑖=𝑑₁𝑍_𝑖1+𝑑₁₂𝑍_𝑖2. (3.3) Then, system (2.5) with 𝑑(𝑡) satisfying (2.2) is exponentially admissible for any switching sequence 𝒮 with average dwell time 𝑇_𝑎≥𝑇^∗_𝑎=(ln𝛽)/𝛼, where 𝛽≥1 satisfies 𝑃_𝑖11≤𝛽𝑃_𝑗11,𝑄_𝑖𝑙≤𝛽𝑄_𝑗𝑙,𝑍_𝑖𝑣≤𝛽𝑍_𝑗𝑣,𝑙=1,2,3,𝑣=1,2,∀𝑖,𝑗∈ℐ. (3.4) Moreover, an estimate on the exponential decay rate is 𝜆=(1/2)(𝛼−(ln𝛽)/𝑇_𝑎).

Proof. The proof is divided into three parts: (i) to show the regularity and nonimpulsiveness; (ii) to show the exponential stability of the differential subsystem; (iii) to show the exponential stability of the algebraic subsystem.
(i) Regularity and nonimpulsiveness. According to (2.4), for each 𝑖∈ℐ, denote 𝐴_𝑖=𝐴_𝑖11𝐴_𝑖12𝐴_𝑖21𝐴_𝑖22, (3.5) where 𝐴_𝑖11∈𝐑^𝑟. From (3.2), it is easy to see that Φ_𝑖11<0, 𝑖∈ℐ. Noting 𝑄_𝑖𝑙>0, 𝑙=1,2,3, we get Sym𝑃^𝑇_𝑖𝐴_𝑖+𝑁_𝑖1𝐸+𝛼𝐸^𝑇𝑃_𝑖<0. (3.6) Substituting 𝑃_𝑖 and 𝐸 given as (3.1) and (2.4) into this inequality yields ⋆⋆⋆𝐴^𝑇_𝑖22𝑃_𝑖22+𝑃^𝑇_𝑖22𝐴_𝑖22<0, (3.7) where ⋆ denotes a matrix which is not relevant to the discussion. This implies that 𝐴_𝑖22, 𝑖∈ℐ, is nonsingular. Then, by Dai [29] and Definition 2.1, system (2.5) is regular and impulse free.
(ii) Exponential stability of differential subsystem. Define the piecewise Lyapunov functional candidate for system (2.5) as the following: 𝑉𝑥_𝑡=𝑉_𝜎(𝑡)𝑥_𝑡=𝑥^𝑇(𝑡)𝐸^𝑇𝑃_𝜎(𝑡)𝑥(𝑡)+²_𝑣=1^𝑡_{𝑡−𝑑𝑣}𝑥^𝑇(𝑠)𝑒^{𝛼(𝑠−𝑡)}𝑄_{𝜎(𝑡)𝑣}𝑥(𝑠)+𝑑𝑠_{𝑡(𝑡)𝑡−𝑑}𝑥^𝑇(𝑠)𝑒^{𝛼(𝑠−𝑡)}𝑄_𝜎(𝑡)3𝑥(𝑠)+𝑑𝑠_{0−𝑑1𝑡𝑡+𝜃}(𝐸𝑥̇(𝑠))^𝑇𝑒^{𝛼(𝑠−𝑡)}𝑍_𝜎(𝑡)1(𝐸𝑥̇(𝑠+))𝑑𝑠𝑑𝜃_{−𝑑1−𝑑2𝑡𝑡+𝜃}(𝐸𝑥̇(𝑠))^𝑇𝑒^{𝛼(𝑠−𝑡)}𝑍_𝜎(𝑡)2(𝐸𝑥̇(𝑠))𝑑𝑠𝑑𝜃. (3.8) Then, along the solution of system (2.5) for a fixed 𝜎(𝑡)=𝑖, 𝑖∈ℐ, we have 𝑉̇_𝑖𝑥_𝑡≤2𝑥^𝑇(𝑡)𝑃^𝑇_𝑖𝐸𝑥̇(𝑡)+²_𝑣=1𝑥^𝑇(𝑡)𝑄_𝑖𝑣𝑥(𝑡)−𝑥^𝑇𝑡−𝑑_𝑣𝑒^{−𝛼𝑑𝑣}𝑄_𝑖𝑣𝑥𝑡−𝑑_𝑣+𝑥^𝑇(𝑡)𝑄_𝑖3𝑥(𝑡)−1−𝜇𝑥^𝑇(𝑡−𝑑(𝑡))𝑒^−𝛼𝑑2𝑄_𝑖3𝑥(𝑡−𝑑(𝑡))+(𝐸𝑥̇(𝑡))^𝑇𝑑₁𝑍_𝑖1+𝑑₁₂𝑍_𝑖2(𝐸𝑥̇(𝑡−))_{𝑡𝑡−𝑑1}(𝐸𝑥̇(𝑠))^𝑇𝑒^{𝛼(𝑠−𝑡)}𝑍_𝑖1(𝐸𝑥̇(𝑠))𝑑𝑠−^𝑡−𝑑1_𝑡−𝑑2(𝐸𝑥̇(𝑠))^𝑇𝑒^{𝛼(𝑠−𝑡)}𝑍_𝑖2(𝐸𝑥̇(𝑠))𝑑𝑠−𝛼²_𝑣=1^𝑡_{𝑡−𝑑𝑣}𝑥^𝑇(𝑠)𝑒^{𝛼(𝑠−𝑡)}𝑄_𝑖𝑣𝜉(𝑠)𝑑𝑠−𝛼^𝑡_{(𝑡)𝑡−𝑑}𝑥^𝑇(𝑠)𝑒^{𝛼(𝑠−𝑡)}𝑄_𝑖3𝑥(𝑠)𝑑𝑠−𝛼⁰_−𝑑1^𝑡_𝑡+𝜃(𝐸𝑥̇(𝑠))^𝑇𝑒^{𝛼(𝑠−𝑡)}𝑍_𝑖1(𝐸𝑥̇(𝑠))𝑑𝑠𝑑𝜃−𝛼^−𝑑1_−𝑑2^𝑡_𝑡+𝜃(𝐸𝑥̇(𝑠))^𝑇𝑒^{𝛼(𝑠−𝑡)}𝑍_𝑖2(𝐸𝑥̇(𝑠))𝑑𝑠𝑑𝜃. (3.9) From the Leibniz-Newton formula, the following equations are true for any matrices 𝑁_𝑖𝑣, 𝑆_𝑖𝑣, and 𝑀_𝑖𝑣, 𝑣=1,2, with appropriate dimensions 2𝑥^𝑇(𝑡)𝑁_𝑖1+𝑥^𝑇(𝑡−𝑑(𝑡))𝑁_𝑖2𝐸𝑥(𝑡)−𝐸𝑥𝑡−𝑑₁−^𝑡_𝑡−𝑑1𝐸𝑥̇(𝑠),2𝑥𝑑𝑠_𝑇(𝑡)𝑆_𝑖1+𝑥^𝑇(𝑡−𝑑(𝑡))𝑆_𝑖2𝐸𝑥(𝑡−𝑑(𝑡))−𝐸𝑥𝑡−𝑑₂−^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝐸𝑥̇(𝑠),2𝑥𝑑𝑠_𝑇(𝑡)𝑀_𝑖1+𝑥^𝑇(𝑡−𝑑(𝑡))𝑀_𝑖2𝐸𝑥𝑡−𝑑₁−𝐸𝑥(𝑡−𝑑(𝑡))−^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝐸𝑥̇(𝑠)𝑑𝑠. (3.10) On the other hand, the following equation is also true: −^𝑡−𝑑1_𝑡−𝑑2(𝐸𝑥̇(𝑠))^𝑇𝑒^{𝛼(𝑠−𝑡)}𝑍_𝑖2(𝐸𝑥̇(𝑠))𝑑𝑠=−^{𝑡−𝑑(𝑡)}_𝑡−𝑑2(𝐸𝑥̇(𝑠))^𝑇𝑒^{𝛼(𝑠−𝑡)}𝑍_𝑖2(𝐸𝑥̇(𝑠))𝑑𝑠−^𝑡−𝑑1_{(𝑡)𝑡−𝑑}(𝐸𝑥̇(𝑠))^𝑇𝑒^{𝛼(𝑠−𝑡)}𝑍_𝑖2(𝐸𝑥̇(𝑠))𝑑𝑠. (3.11) By (3.8)–(3.11), we have 𝑉̇_𝑖𝑥_𝑡+𝛼𝑉_𝑖𝑥_𝑡≤𝜂^𝑇(𝑡)Φ_𝑖+𝐴^𝑇_𝑖𝑑₁𝑍_𝑖1+𝑑₁₂𝑍_𝑖2𝐴_𝑖+𝑐₁𝑁_𝑖𝑍⁻¹_𝑖1𝑁^𝑇_𝑖+𝑐₁₂𝑆_𝑖𝑍⁻¹_𝑖2𝑆^𝑇_𝑖+𝑐₁₂𝑀_𝑖𝑍⁻¹_𝑖2𝑀^𝑇_𝑖𝜂(𝑡)−^𝑡_𝑡−𝑑1𝜂^𝑇(𝑡)𝑁_𝑖+(𝐸𝑥̇(𝑠))^𝑇𝑒^{𝛼(𝑠−𝑡)}𝑍_𝑖1𝑒^{𝛼(𝑡−𝑠)}𝑍⁻¹_𝑖1𝜂^𝑇(𝑡)𝑁_𝑖+(𝐸𝑥̇(𝑠))^𝑇𝑒^{𝛼(𝑠−𝑡)}𝑍_𝑖1^𝑇−𝑑𝑠_{𝑡−𝑑(𝑡)𝑡−𝑑2}𝜂^𝑇(𝑡)𝑆_𝑖+(𝐸𝑥̇(𝑠))^𝑇𝑒^{𝛼(𝑠−𝑡)}𝑍_𝑖2𝑒^{𝛼(𝑡−𝑠)}𝑍⁻¹_𝑖2𝜂^𝑇(𝑡)𝑆_𝑖+(𝐸𝑥̇(𝑠))^𝑇𝑒^{𝛼(𝑠−𝑡)}𝑍_𝑖2^𝑇−𝑑𝑠_{𝑡−𝑑1(𝑡)𝑡−𝑑}𝜂^𝑇(𝑡)𝑀_𝑖+(𝐸𝑥̇(𝑠))^𝑇𝑒^{𝛼(𝑠−𝑡)}𝑍_𝑖2𝑒^{𝛼(𝑡−𝑠)}𝑍⁻¹_𝑖2𝜂^𝑇(𝑡)𝑀_𝑖+(𝐸𝑥̇(𝑠))^𝑇𝑒^{𝛼(𝑠−𝑡)}𝑍_𝑖2^𝑇𝑑𝑠, (3.12) where 𝜂(𝑡)=[𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−𝑑(𝑡))𝑥^𝑇(𝑡−𝑑₁)𝑥^𝑇(𝑡−𝑑₂)]^𝑇, 𝐴_𝑖=[𝐴_𝑖𝐴_𝑑𝑖00], and Φ_𝑖=Φ_𝑖11Φ_𝑖12Φ_𝑖13−𝑆_𝑖1𝐸∗Φ_𝑖22Φ_𝑖23−𝑆_𝑖2𝐸∗∗Φ_𝑖330∗∗∗Φ_𝑖44,𝑁_𝑖=𝑁_𝑖1𝑁_𝑖200,𝑀_𝑖=𝑀_𝑖1𝑀_𝑖200,𝑆_𝑖=𝑆_𝑖1𝑆_𝑖200. (3.13) By Schur complement, LMI (3.2) implies Φ_𝑖+𝐴^𝑇_𝑖𝑑₁𝑍_𝑖1+𝑑₁₂𝑍_𝑖2𝐴_𝑖+𝑐₁𝑁_𝑖𝑍⁻¹_𝑖1𝑁^𝑇_𝑖+𝑐₁₂𝑆_𝑖𝑍⁻¹_𝑖2𝑆^𝑇_𝑖+𝑐₁₂𝑀_𝑖𝑍⁻¹_𝑖2𝑀^𝑇_𝑖<0. (3.14) Notice that the last three parts in (3.12) are all less than 0. So, if (3.14) holds, then 𝑉̇_𝑖𝑥_𝑡+𝛼𝑉_𝑖𝑥_𝑡<0. (3.15) For an arbitrary piecewise constant switching signal 𝜎(𝑡), and for any 𝑡>0, we let 0=𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯, 𝑘=1,2,⋯, denote the switching points of 𝜎(𝑡) over the interval (0,𝑡). As mentioned earlier, the 𝑖_𝑘th subsystem is activated when 𝑡∈[𝑡_𝑘,𝑡_𝑘+1). Integrating (3.15) from 𝑡_𝑘 to 𝑡_𝑘+1 gives 𝑉𝑥_𝑡=𝑉_𝜎(𝑡)𝑥_𝑡≤𝑒^{−𝛼(𝑡−𝑡𝑘)}𝑉_{𝜎(𝑡𝑘)}𝑥_𝑡𝑘,𝑡∈𝑡_𝑘,𝑡_𝑘+1. (3.16) Let 𝑥(𝑡)=^𝑥1_𝑥^(𝑡)_2(𝑡), where 𝑥₁(𝑡)∈𝐑^𝑟 and 𝑥₂(𝑡)∈𝐑^𝑛−𝑟. From (2.4) and (3.1), it can be deduced that for each 𝜎(𝑡)=𝑖, 𝑖∈ℐ𝑥^𝑇(𝑡)𝐸^𝑇𝑃_𝑖𝑥(𝑡)=𝑥^𝑇₁(𝑡)𝑃_𝑖11𝑥₁(𝑡). (3.17) In view of this, and using (3.4) and (3.8), at switching instant 𝑡_𝑖, we have 𝑉_{𝜎(𝑡𝑖)}𝑥_𝑡𝑖≤𝛽𝑉_𝜎(𝑡⁻_𝑖)𝑥_𝑡⁻_𝑖,𝑖=1,2,…, (3.18) where 𝑡⁻_𝑖 denotes the left limitation of 𝑡_𝑖. Therefore, it follows from (3.16), (3.18), and the relation 𝑘=𝑁_𝜎(𝑡₀,𝑡)≤(𝑡−𝑡₀)/𝑇_𝑎 that 𝑉_𝜎(𝑡)𝑥_𝑡≤𝑒^{−𝛼(𝑡−𝑡𝑘)}𝛽𝑉_𝜎(𝑡⁻_𝑖)𝑥_𝑡⁻_𝑖≤⋯≤𝑒^{−𝛼(𝑡−𝑡0)}𝛽^𝑘𝑉_𝜎(𝑡0)𝑡₀≤𝑒^{−(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑉_𝜎(𝑡0)𝑥_𝑡0. (3.19) According to (3.8) and (3.19), we obtain 𝜆₁𝑥₁(𝑡)²≤𝑉_𝜎(𝑡)(𝑡),𝑉_𝜎(𝑡0)𝑥_𝑡0≤𝜆₂𝑥_𝑡0²_𝑐, (3.20) where 𝜆₁=min_{∀𝑖∈ℐ}𝜆_min𝑃_𝑖11,𝜆₂=max_{∀𝑖∈ℐ}𝜆_max𝑃_𝑖11+1𝛼1−𝑒^−𝛼𝑑1max_{∀𝑖∈ℐ}𝜆_max𝑄_𝑖1+1𝛼1−𝑒^−𝛼𝑑2max_{∀𝑖∈ℐ}𝜆_max𝑄_𝑖2+𝜆_max𝑄_𝑖3+1𝛼²𝛼𝑑₁−1+𝑒^−𝛼𝑑1max_{∀𝑖∈ℐ}2𝜆_max𝑍_𝑖1𝐴_𝑖+𝐴_𝑑𝑖+𝛼𝑑₁₂−𝑒^−𝛼𝑑1+𝑒^−𝛼𝑑2𝛼²max_{∀𝑖∈ℐ}2𝜆_max𝑍_𝑖2𝐴_𝑖+𝐴_𝑑𝑖. (3.21) Considering (3.19) and (3.20) yields 𝑥₁(𝑡)²≤1𝜆₁𝑉_𝜎(𝑡)𝑥_𝑡≤𝜆₂𝜆₁𝑒^{−(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_𝑡0²_𝑐 (3.22) which implies 𝑥₁(𝑡)≤𝜆₂𝜆₁𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_𝑡0𝑐. (3.23)
(iii) Exponential stability of algebraic subsystem. Since 𝐴_𝑖22, 𝑖∈ℐ, is nonsingular, we choose 𝐺_𝑖=𝐼_𝑟−𝐴_𝑖12𝐴⁻¹_𝑖220𝐴⁻¹_𝑖22𝐼,𝐻=^𝑟00𝐼_𝑛−𝑟. (3.24) Then, it is easy to get 𝐸∶=𝐺_𝑖𝐼𝐸𝐻=^𝑟0,𝐴00^𝑖∶=𝐺_𝑖𝐴_𝑖𝐴𝐻=^𝑖110𝐴_𝑖21𝐼_𝑛−𝑟,𝑃_𝑖∶=𝐺_𝑖^−𝑇𝑃_𝑖𝑃𝐻=^𝑖110𝑃_𝑖21𝑃_𝑖22, (3.25) where 𝐴_𝑖11=𝐴_𝑖11−𝐴_𝑖12𝐴⁻¹_𝑖22𝐴_𝑖21, 𝐴_𝑖21=𝐴⁻¹_𝑖22𝐴_𝑖21, 𝑃_𝑖11=𝑃_𝑖11, 𝑃_𝑖21=𝐴^𝑇_𝑖12𝑃_𝑖11+𝐴^𝑇_𝑖22𝑃_𝑖21, and 𝑃_𝑖21=𝐴^𝑇_𝑖22𝑃_𝑖22. According to (3.25), denote 𝐴_𝑑𝑖∶=𝐺_𝑖𝐴_𝑑𝑖𝐴𝐻=^𝑑𝑖11𝐴_𝑑𝑖12𝐴_𝑑𝑖21𝐴_𝑑𝑖22,𝑄_𝑖𝑙∶=𝐻^𝑇𝑄_𝑖𝑙𝑄𝐻=^𝑖𝑙11𝑄_𝑖𝑙12𝑄_𝑖𝑙21𝑄_𝑖𝑙22,𝑍_𝑖𝑣∶=𝐺_𝑖^−𝑇𝑍_𝑖𝑣𝐺_𝑖⁻¹=𝑍_𝑖𝑣11𝑍_𝑖𝑣12𝑍_𝑖𝑣21𝑍_𝑖𝑣22,𝑀_𝑖𝑣∶=𝐻^𝑇𝑀_𝑖𝑣𝐺_𝑖⁻¹=𝑀_𝑖𝑣11𝑀_𝑖𝑣12𝑀_𝑖𝑣21𝑀_𝑖𝑣22,𝑁_𝑖𝑣∶=𝐻^𝑇𝑁_𝑖𝑣𝐺_𝑖⁻¹=𝑁_𝑖𝑣11𝑁_𝑖𝑣12𝑁_𝑖𝑣21𝑁_𝑖𝑣22,𝑆_𝑖𝑣∶=𝐻^𝑇𝑆_𝑖𝑣𝐺_𝑖⁻¹=𝑆_𝑖𝑣11𝑆_𝑖𝑣12𝑆_𝑖𝑣21𝑆_𝑖𝑣22,𝑙=1,2,3,𝑣=1,2 (3.26) and let 𝜉(𝑡)=𝜉₁(𝑡)𝜉₂(𝑡)∶=𝐻⁻¹𝑥(𝑡)=𝑥(𝑡), (3.27) where 𝜉₁(𝑡)∈𝐑^𝑟 and 𝜉₂(𝑡)∈𝐑^𝑛−𝑟. Then, for any 𝜎(𝑡)=𝑖, 𝑖∈ℐ, system (2.5) is a restricted system equivalent (r.s.e.) to 𝜉̇₁(𝑡)=𝐴_𝑖11𝜉₁(𝑡)+𝐴_𝑑𝑖11𝜉₁(𝑡−𝑑(𝑡))+𝐴_𝑑𝑖12𝜉₂(𝑡−𝑑(𝑡)),−𝜉₂(𝑡)=𝐴_𝑖21𝜉₁(𝑡)+𝐴_𝑑𝑖21𝜉₁(𝑡−𝑑(𝑡))+𝐴_𝑑𝑖22𝜉₂(𝑡−𝑑(𝑡)). (3.28) By (3.2) and Schur complement, we have Φ_𝑖11Φ_𝑖12∗Φ_𝑖22<0. (3.29) Pre- and postmultiplying this inequality by diag{𝐻^𝑇,𝐻^𝑇} and diag{𝐻,𝐻}, respectively, noting the expressions in (3.25) and (3.26), and using Schur complement, we have 𝑃^𝑇_𝑖22+𝑃_𝑖22+∑³_𝑙=1𝑄_𝑖𝑙22𝑃^𝑇_𝑖22𝐴^𝑇_𝑑𝑖22∗−1−𝜇𝑒^−𝛼𝑑2𝑄_𝑖322<0. (3.30) Pre- and postmultiplying this inequality by [−𝐴^𝑇_𝑑𝑖22𝐼] and its transpose, respectively, and noting 𝑄_𝑖122>0, 𝑄_𝑖222>0, and 𝜇≥0, we obtain 𝑒^{(1/2)𝛼𝑑2}𝐴_𝑑𝑖22^𝑇𝑄_𝑖322𝑒^{(1/2)𝛼𝑑2}𝐴_𝑑𝑖22−𝑄_𝑖322<0. (3.31) Then, according to Lemma 5 in Kharitonov et al. [37], we can deduce that there exist constants ℏ_𝑖>1 and 𝜂_𝑖∈(0,1) such that 𝑒^{(1/2)𝛼𝑑2}𝐴_𝑑𝑖22^𝑙≤ℏ_𝑖𝑒^{−𝜂𝑖𝑙},𝑙=0,1,… (3.32) Define 𝑡⁰=𝑡,𝑡^𝑗=𝑡^𝑗−1−𝑑𝑡^𝑗−1𝐴,𝑗=1,2,…,₂₁=max_{∀𝑖∈ℐ}𝐴_𝑖21,𝐴_𝑑21=max_{∀𝑖∈ℐ}𝐴_𝑑𝑖21,𝐴_𝑑22=max_{∀𝑖∈ℐ}𝐴_𝑑𝑖22,∀𝑖∈ℐ. (3.33)
Now, following similar line as in Part 3 in Theorem 1 of Lin and Fei [31], it can easily be obtained that 𝜉₂(𝑡)≤𝜒₁+𝜒₂+𝜒₃+𝜒₄+𝜒₅𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_𝑡0𝑐, (3.34) where 𝜒₁=^𝑘_𝑗=0ℏ_𝑖𝑗𝑒^{−𝜂𝑖𝑗𝑇𝑖𝑗},𝜒₂=ℏ_𝑖𝑘𝐴₂₁𝜆₂𝜆₁𝑒^𝜂𝑖𝑘𝑒^𝜂𝑖𝑘,𝜒−1₃=ℏ_𝑖𝑘𝑒^{(1/2)𝛼𝑑2}𝐴_𝑑21𝜆₂𝜆₁𝑒^𝜂𝑖𝑘𝑒^𝜂𝑖𝑘,𝜒−1₄=𝐴₂₁𝜆₂𝜆₁^𝑘_𝑝=1ℏ_𝑖𝑝−1^𝑘_𝑞=𝑝ℏ_𝑖𝑞𝑒^{−𝜂𝑖𝑞𝑇𝑖𝑞}𝑒^{𝜂𝑖𝑝−1}𝑒^{𝜂𝑖𝑝−1},𝜒−1₅=𝑒^{(1/2)𝛼𝑑2}𝐴_𝑑21𝜆₂𝜆₁^𝑘_𝑝=1ℏ_𝑖𝑝−1^𝑘_𝑞=𝑝ℏ_𝑖𝑞𝑒^{−𝜂𝑖𝑞𝑇𝑖𝑞}𝑒^{𝜂𝑖𝑝−1}𝑒^{𝜂𝑖𝑝−1}.−1 (3.35)𝑇_𝑖𝑘,𝑇_𝑖𝑘−1,…,𝑇_𝑖0 are positive finite integers, respectively, satisfying 𝑡^𝑇𝑖𝑘∈𝑡_𝑘−1,𝑡_𝑘,𝑡^𝑇𝑖𝑘→𝑡_𝑘,𝑡^{𝑇𝑖𝑘+𝑇𝑖𝑘−1}∈𝑡_𝑘−2,𝑡_𝑘−1,𝑡^{𝑇𝑖𝑘+𝑇𝑖𝑘−1}→𝑡_𝑘−1,⋮𝑡^{𝑇𝑖𝑘+⋯+𝑇𝑖0}∈−𝑑₂,𝑡₀,𝑡^{𝑇𝑖𝑘+⋯+𝑇𝑖0}→𝑡₀. (3.36) Combining (3.27), (3.23) and (3.34) yields that system (2.5) is exponentially stable for any switching sequence 𝒮 with average dwell time 𝑇_𝑎≥𝑇^∗_𝑎=ln𝛽/𝛼. This completes the proof.

Remark 3.2. Theorem 3.1 provides a sufficient condition of the exponential admissibility for the switched singular time-delay system (2.5). Note that due to the existence of algebraic constraints in system states, the stability analysis of switched singular time-delay systems is much more complicated than that for switched state-space time-delay systems [21–23, 25, 38]. Note also that the condition established in Theorem 3.1 is not only delay-range-dependent but also decay-rate-dependent. The delay-range-dependence makes the result less conservative, while the decay-rate-dependence enables one to control the transient process of differential and algebraic subsystems with a unified performance specification.

Remark 3.3. Different from the integral inequality method used in our previous work [31], the free-weighting matrix method [39] is adopted when deriving Theorem 3.1, and thus no three-product terms, for example, 𝐴^𝑇_𝑖𝑍_𝑖𝑣𝐴_𝑖, 𝐴^𝑇_𝑑𝑖𝑍_𝑖𝑣𝐴_𝑑𝑖, and so forth, are involved, which greatly facilitates the SF and SOF controllers design, as seen in Section 4.

Remark 3.4. If 𝛽=1 in 𝑇_𝑎≥𝑇^∗_𝑎=(ln𝛽)/𝛼, which leads to 𝑃_𝑖11≡𝑃_𝑗11, 𝑄_𝑖𝑙≡𝑄_𝑗𝑙, 𝑍_𝑖𝑣≡𝑍_𝑗𝑣, 𝑙=1,2,3, 𝑣=1,2, for all 𝑖,𝑗∈ℐ, and 𝑇^∗_𝑎=0, then system (2.5) possesses a common Lyapunov function, and the switching signals can be arbitrary.
Now, the following theorem presents a sufficient condition on exponential admissibility with a weighted 𝐻_∞ performance of the switched singular time-delay system (2.1) with 𝑢(𝑡)=0.

Theorem 3.5. For prescribed scalars 𝛼>0, 𝛾>0, 0≤𝑑₁≤𝑑₂, and 0<𝜇<1, if for each 𝑖∈ℐ, there exist matrices 𝑄_𝑖𝑙>0, 𝑙=1,2,3, 𝑍_𝑖𝑣>0, 𝑀_𝑖𝑣, 𝑁_𝑖𝑣, 𝑆_𝑖𝑣, 𝑣=1,2, and 𝑃_𝑖 with the form of (3.1) such that Φ_𝑖=Φ_𝑖11Φ_𝑖12Φ_𝑖13−𝑆_𝑖1𝐸Φ_𝑖15𝐶^𝑇_𝑖𝑐₁𝑁_𝑖1𝑐₁₂𝑆_𝑖1𝑐₁₂𝑀_𝑖1∗Φ_𝑖22Φ_𝑖23−𝑆_𝑖2𝐸Φ_𝑖25𝐶^𝑇_𝑑𝑖𝑐₁𝑁_𝑖2𝑐₁₂𝑆_𝑖2𝑐₁₂𝑀_𝑖2∗∗Φ_𝑖33000000∗∗∗Φ_𝑖44Φ00000∗∗∗∗_𝑖55𝐷^𝑇_𝑤𝑖000∗∗∗∗∗−𝐼000∗∗∗∗∗∗−𝑐₁𝑍_𝑖100∗∗∗∗∗∗∗−𝑐₁₂𝑍_𝑖20∗∗∗∗∗∗∗∗−𝑐₁₂𝑍_𝑖2<0, (3.37) where Φ_𝑖11=Φ_𝑖11+𝐴^𝑇_𝑖𝑈_𝑖𝐴_𝑖,Φ_𝑖12=Φ_𝑖12+𝐴^𝑇_𝑖𝑈_𝑖𝐴_𝑑𝑖,Φ_𝑖15=𝑃^𝑇_𝑖𝐵_𝑤𝑖+𝐴^𝑇_𝑖𝑈_𝑖𝐵_𝑤𝑖,Φ_𝑖22=Φ_𝑖22+𝐴^𝑇_𝑑𝑖𝑈_𝑖𝐴_𝑑𝑖,Φ_𝑖25=𝐴^𝑇_𝑑𝑖𝑈_𝑖𝐵_𝑤𝑖,Φ_𝑖55=−𝛾²𝐼+𝐵^𝑇_𝑤𝑖𝑈_𝑖𝐵_𝑤𝑖, (3.38) and Φ_𝑖11, Φ_𝑖12, Φ_𝑖13, Φ_𝑖22, Φ_𝑖23, Φ_𝑖33, Φ_𝑖44, and 𝑈_𝑖 are defined in (3.2). Then, system (2.1) with 𝑢(𝑡)=0 is exponentially admissible with a weighted 𝐻_∞ performance 𝛾 for any switching sequence 𝒮 with average dwell time 𝑇_𝑎≥𝑇^∗_𝑎=(ln𝛽)/𝛼, where 𝛽≥1 satisfying (3.4).

Proof. Choose the piecewise Lyapunov function defined by (3.8). Since (3.37) implies (3.2), system (2.1) with 𝑢(𝑡)=0 and 𝑤(𝑡)=0 is exponentially admissible by Theorem 3.1. On the other hand, similar to the proof of Theorem 3.1, from (3.37), we have that for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝑉̇_𝑖𝑘𝑥_𝑡+𝛼𝑉_𝑖𝑘𝑥_𝑡+Γ(𝑡)≤0, (3.39) where Γ(𝑡)=𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑤^𝑇(𝑡)𝑤(𝑡). This implies that 𝑉_𝑖𝑘𝑥_𝑡≤𝑒^{−𝛼(𝑡−𝑡𝑘)}𝑉_𝑖𝑘𝑥_𝑡𝑘−^𝑡_𝑡𝑘𝑒^{−𝛼(𝑡−𝑠)}Γ(𝑠)𝑑𝑠. (3.40) By induction, we have 𝑉_𝑖𝑘𝑥_𝑡≤𝛽𝑒^{−𝛼(𝑡−𝑡𝑘)}𝑉_𝑖𝑘−1𝑥_𝑡𝑘−^𝑡_𝑡𝑘𝑒^{−𝛼(𝑡−𝑠)}Γ(𝑠)⋮𝑑𝑠≤𝛽^𝑘𝑒^−𝛼𝑡𝑉_𝑖0𝑥₀−^𝑡_𝑡𝑘𝑒^{−𝛼(𝑡−𝑠)}Γ(𝑠)𝑑𝑠−^𝑘−𝑝_𝑝=1𝛽^{𝑘−𝑝𝑡𝑝+1}_𝑡𝑝𝑒^{−𝛼(𝑡−𝑠)}Γ(𝑠)𝑑𝑠=𝑒^{−𝛼𝑡+𝑁𝛼(0,𝑡)ln𝛽}𝑉_𝑖0𝑥₀−^𝑡₀𝑒^{−𝛼(𝑡−𝑠)+𝑁𝛼(𝑠,𝑡)ln𝛽}Γ(𝑠)𝑑𝑠. (3.41) Under zero initial condition, (3.41) gives 0≤−^𝑡₀𝑒^{−𝛼(𝑡−𝑠)+𝑁𝛼(𝑠,𝑡)ln𝛽}Γ(𝑠)𝑑𝑠. (3.42) Multiplying both sides of (3.42) by 𝑒^{−𝑁𝛼(0,𝑡)ln𝛽} yields ^𝑡₀𝑒^{−𝛼(𝑡−𝑠)−𝑁𝛼(0,𝑠)ln𝛽}𝑧^𝑇(𝑠)𝑧(𝑠)𝑑𝑠≤𝛾^2𝑡₀𝑒^{−𝛼(𝑡−𝑠)−𝑁𝛼(0,𝑠)ln𝛽}𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠. (3.43) Noting that 𝑁_𝛼(0,𝑠)≤𝑠/𝑇_𝑎 and 𝑇_𝑎≥𝑇^∗_𝑎=(ln𝛽)/𝛼, we get 𝑁_𝛼(0,𝑠)ln𝛽≤𝛼𝑠. Then, it follows from (3.43) that ∫^𝑡₀𝑒^{−𝛼(𝑡−𝑠)−𝛼𝑠}𝑧^𝑇(𝑠)𝑧(𝑠)𝑑𝑠≤𝛾²∫^𝑡₀𝑒^{−𝛼(𝑡−𝑠)}𝑤^𝑇(𝑠)𝑤(𝑠)𝑑𝑠. Integrating both sides of this inequality from 𝑡=0 to ∞ leads to inequality (2.7). This completes the proof of Theorem 3.5.

Remark 3.6. Note that when 𝛽=1, which is a trivial case, system (2.1) with 𝑢(𝑡)=0 achieves the normal 𝐻_∞ performance 𝛾 under arbitrary switching.

4. Controller Design

In this section, based on the results of the previous section, we are to deal with the design problems of both SF and SOF controllers for the switched singular time-delay system (2.1).

4.1. SF Controller Design

Applying the SF controller (2.8) to system (2.1) gives the following closed-loop system: 𝐸𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝑑𝜎(𝑡)}𝑥(𝑡−𝑑(𝑡))+𝐵_{𝑤𝜎(𝑡)}𝑤(𝑡),𝑧(𝑡)=𝐶_𝜎(𝑡)𝑥(𝑡)+𝐶_{𝑑𝜎(𝑡)}𝑥(𝑡−𝑑(𝑡))+𝐷_{𝑤𝜎(𝑡)}𝑤(𝑡),𝑥(𝜃)=𝜙(𝜃),𝜃∈−𝑑₂,0, (4.1) where 𝐴_𝜎(𝑡)=𝐴_𝜎(𝑡)+𝐵_𝜎(𝑡)𝐾_𝜎(𝑡),𝐶_𝜎(𝑡)=𝐶_𝜎(𝑡)+𝐷_𝜎(𝑡)𝐾_𝜎(𝑡). (4.2) The following theorem presents a sufficient condition for solvability of the SF controller design problem for system (2.1).

Theorem 4.1. For prescribed scalars 𝛼>0, 𝛾>0, 0≤𝑑₁≤𝑑₂, and 0<𝜇<1, if for each 𝑖∈ℐ, and given scalars 𝜖_𝑖𝑓, 𝑓=1,2,…,6, 𝜖_𝑖7>0, and 𝜖_𝑖8>0, there exist matrices 𝑅_𝑖𝑙>0, 𝑙=1,2,3, 𝑍_𝑖𝑣>0, 𝑇_𝑖, and 𝑋_𝑖 of the following form: 𝑋_𝑖=𝑋_𝑖110𝑋_𝑖21𝑋_𝑖22, (4.3) with 𝑋_𝑖11∈𝐑^𝑟, 𝑋_𝑖11>0, and 𝑋_𝑖22 being invertible, such that Ψ_𝑖11Ψ_𝑖12Ψ_𝑖13Ψ_𝑖14𝐵_𝑤𝑖Ψ_𝑖16Ψ_𝑖17𝑐₁₂𝜖_𝑖5𝐼𝑐₁₂𝜖_𝑖3𝐼Ψ_𝑖110Ψ_𝑖111Ξ_𝑖∗Ψ_𝑖22Ψ_𝑖23Ψ_𝑖240Ψ_𝑖26Ψ_𝑖27𝑐₁₂𝜖_𝑖6𝐼𝑐₁₂𝜖_𝑖4𝐼Ψ_𝑖210Ψ_𝑖2110∗∗Ψ_𝑖33000000000∗∗∗Ψ_𝑖4400000000∗∗∗∗−𝛾²𝐼𝐷^𝑇_𝑤𝑖000𝑑₁𝐵^𝑇_𝑤𝑖𝑑₁₂𝐵^𝑇_𝑤𝑖0∗∗∗∗∗−𝐼000000∗∗∗∗∗∗−𝑐₁𝑍_𝑖100000∗∗∗∗∗∗∗−𝑐₁₂𝑍_𝑖20000∗∗∗∗∗∗∗∗−𝑐₁₂𝑍_𝑖2000∗∗∗∗∗∗∗∗∗Ψ_𝑖101000∗∗∗∗∗∗∗∗∗∗Ψ_𝑖11110∗∗∗∗∗∗∗∗∗∗∗−Γ_𝑖<0, (4.4) where Ψ_𝑖11=Sym𝐴_𝑖𝑋_𝑖+𝐵_𝑖𝑇_𝑖+𝜖_𝑖1𝐸𝑋_𝑖+𝛼𝑋^𝑇_𝑖𝐸^𝑇,Ψ_𝑖12=𝐴_𝑑𝑖𝑅_𝑖3+𝜖_𝑖2𝑋^𝑇_𝑖𝐸^𝑇+𝜖_𝑖5𝐸𝑅_𝑖3−𝜖_𝑖3𝐸𝑅_𝑖3,Ψ_𝑖13=𝜖_𝑖3𝐸𝑅_𝑖1−𝜖_𝑖1𝐸𝑅_𝑖1,Ψ_𝑖14=−𝜖_𝑖5𝐸𝑅_𝑖2,Ψ_𝑖16=𝑇^𝑇_𝑖𝐷^𝑇_𝑖+𝑋^𝑇_𝑖𝐶^𝑇_𝑖,Ψ_𝑖17=𝑐₁𝜖_𝑖1Ψ𝐼,_𝑖110=𝑑₁𝑇^𝑇_𝑖𝐵^𝑇_𝑖+𝑑₁𝑋^𝑇_𝑖𝐴^𝑇_𝑖,Ψ_𝑖111=𝑑₁₂𝑇^𝑇_𝑖𝐵^𝑇_𝑖+𝑑₁₂𝑋^𝑇_𝑖𝐴^𝑇_𝑖,Ψ_𝑖22=−1−𝜇𝑒^−𝛼𝑑2𝑅_𝑖3+Sym𝜖_𝑖6𝐸𝑅_𝑖3−𝜖_𝑖4𝐸𝑅_𝑖3,Ψ_𝑖23=𝜖_𝑖4𝐸𝑅_𝑖1−𝜖_𝑖2𝐸𝑅_𝑖1,Ψ_𝑖24=−𝜖_𝑖6𝐸𝑅_𝑖2,Ψ_𝑖26=𝑅_𝑖3𝐶^𝑇_𝑑𝑖,Ψ_𝑖27=𝑐₁𝜖_𝑖2𝐼,Ψ_𝑖210=𝑑₁𝑅_𝑖3𝐴^𝑇_𝑑𝑖,Ψ_𝑖211=𝑑₁₂𝑅_𝑖3𝐴^𝑇_𝑑𝑖,Ψ_𝑖33=−𝑒^−𝛼𝑑1𝑅_𝑖1,Ψ_𝑖44=−𝑒^−𝛼𝑑2𝑅_𝑖2,Ψ_𝑖1010=−2𝑑₁𝜖_𝑖7𝐼+𝑑₁𝜖²_𝑖7𝑍_𝑖1,Ψ_𝑖1111=−2𝑑₁₂𝜖_𝑖8𝐼+𝑑₁₂𝜖²_𝑖8𝑍_𝑖2,Ξ_𝑖=𝑋^𝑇_𝑖𝑋^𝑇_𝑖𝑋^𝑇_𝑖,Γ_𝑖=diag𝑅_𝑖1,𝑅_𝑖2,𝑅_𝑖3. (4.5) Then, there exists an SF controller (2.8) such that the closed-loop system (4.1) with 𝑑(𝑡) satisfying (2.2) is exponentially admissible with a weighted 𝐻_∞ performance 𝛾 for any switching sequence 𝒮 with average dwell time 𝑇_𝑎≥𝑇^∗_𝑎=ln𝛽/𝛼, where 𝛽≥1 satisfies 𝑋_𝑖11≥𝛽⁻¹𝑋_𝑗11,𝑅_𝑖𝑙≥𝛽⁻¹𝑅_𝑗𝑙,𝑍_𝑖𝑣≤𝛽𝑍_𝑗𝑣,𝑙=1,2,3,𝑣=1,2,∀𝑖,𝑗∈ℐ. (4.6) Moreover, the feedback gain of the controller is 𝐾_𝑖=𝑇_𝑖𝑋_𝑖⁻¹,𝑖∈ℐ. (4.7)

Proof. According to Theorem 3.5, the closed-loop system (4.1) is exponentially admissible with a weighted 𝐻_∞ performance 𝛾 if for each 𝑖∈ℐ, there exist matrices 𝑄_𝑖𝑙>0, 𝑙=1,2,3, 𝑍_𝑖𝑣>0, 𝑀_𝑖𝑣, 𝑁_𝑖𝑣, 𝑆_𝑖𝑣, 𝑣=1,2, and 𝑃_𝑖 with the form of (3.1) such that inequality (3.37) holds with 𝐴_𝑖 and 𝐶_𝑖 instead of 𝐴_𝑖 and 𝐶_𝑖, respectively. By Schur complement, (3.37) is equivalent to Φ^′_𝑖11Φ_𝑖12Φ_𝑖13−𝑆_𝑖1𝐸𝑃^𝑇_𝑖𝐵_𝑤𝑖𝐶^𝑇_𝑖𝑐₁𝑁_𝑖1𝑐₁₂𝑆_𝑖1𝑐₁₂𝑀_𝑖1𝑑₁𝐴^𝑇_𝑖𝑑₁₂𝐴^𝑇_𝑖∗Φ_𝑖22Φ_𝑖23−𝑆_𝑖2𝐸0𝐶^𝑇_𝑑𝑖𝑐₁𝑁_𝑖2𝑐₁₂𝑆_𝑖2𝑐₁₂𝑀_𝑖2𝑑₁𝐴^𝑇_𝑑𝑖𝑑₁₂𝐴^𝑇_𝑑𝑖∗∗Φ_𝑖3300000000∗∗∗Φ_𝑖440000000∗∗∗∗−𝛾²𝐼𝐷^𝑇_𝑤𝑖000𝑑₁𝐵^𝑇_𝑤𝑖𝑑₁₂𝐵^𝑇_𝑤𝑖∗∗∗∗∗−𝐼00000∗∗∗∗∗∗−𝑐₁𝑍_𝑖10000∗∗∗∗∗∗∗−𝑐₁₂𝑍_𝑖2000∗∗∗∗∗∗∗∗−𝑐₁₂𝑍_𝑖200∗∗∗∗∗∗∗∗∗−𝑑₁𝑍⁻¹_𝑖10∗∗∗∗∗∗∗∗∗∗−𝑑₁₂𝑍⁻¹_𝑖2<0, (4.8) where Φ_𝑖12, Φ_𝑖13, Φ_𝑖22, Φ_𝑖23, Φ_𝑖33, and Φ_𝑖44 are defined in (3.2), and Φ^′_𝑖11=Sym𝑃^𝑇_𝑖𝐴_𝑖+𝑁_𝑖1𝐸+³_𝑙=1𝑄_𝑖𝑙+𝛼𝐸^𝑇𝑃_𝑖. (4.9) Since 𝑃_𝑖11>0 and 𝑃_𝑖22 is invertible, then 𝑃_𝑖 is invertible. Let 𝑋_𝑖=𝑃_𝑖⁻¹,𝑅_𝑖1=𝑄⁻¹_𝑖1,𝑅_𝑖2=𝑄⁻¹_𝑖2,𝑅_𝑖3=𝑄⁻¹_𝑖3. (4.10) By (3.1), 𝑋_𝑖 has the form of (4.3). Pre- and postmultiplying (4.8) by diag{𝑋^𝑇_𝑖,𝑅_𝑖3,𝑅_𝑖1,𝑅_𝑖2,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼} and its transpose, respectively, and noting (4.10), we obtain Φ^′′_𝑖11Φ^′′_𝑖12Φ^′′_𝑖13Φ^′′_𝑖14𝐵_𝑤𝑖Φ^′′_𝑖16Φ^′′_𝑖17Φ^′′_𝑖18Φ^′′_𝑖19Φ^′′_𝑖110Φ^′′_𝑖111∗Φ^′′_𝑖22Φ^′′_𝑖23Φ^′′_𝑖240Φ^′′_𝑖26Φ^′′_𝑖27Φ^′′_𝑖28Φ^′′_𝑖29Ψ_𝑖210Ψ_𝑖211∗∗Ψ_𝑖3300000000∗∗∗Ψ_𝑖440000000∗∗∗∗−𝛾²𝐼𝐷^𝑇_𝑤𝑖000𝑑₁𝐵^𝑇_𝑤𝑖𝑑₁₂𝐵^𝑇_𝑤𝑖∗∗∗∗∗−𝐼00000∗∗∗∗∗∗−𝑐₁𝑍_𝑖10000∗∗∗∗∗∗∗−𝑐₁₂𝑍_𝑖2000∗∗∗∗∗∗∗∗−𝑐₁₂𝑍_𝑖200∗∗∗∗∗∗∗∗∗−𝑑₁𝑍⁻¹_𝑖10∗∗∗∗∗∗∗∗∗∗−𝑑₁₂𝑍⁻¹_𝑖2<0, (4.11) where Φ^′′_𝑖11=Sym𝐴_𝑖𝑋_𝑖+𝑋^𝑇_𝑖𝑁_𝑖1𝐸𝑋_𝑖+𝑋^𝑇_𝑖³_𝑙=1𝑄_𝑖𝑙𝑋_𝑖+𝛼𝑋^𝑇_𝑖𝐸^𝑇,Φ^′′_𝑖12=𝐴_𝑑𝑖𝑅_𝑖3+𝑋^𝑇_𝑖𝑁_𝑖2𝐸^𝑇𝑅_𝑖3+𝑋^𝑇_𝑖𝑆_𝑖1𝐸𝑅_𝑖3−𝑋^𝑇_𝑖𝑀_𝑖1𝐸𝑅_𝑖3,Φ^′′_𝑖13=𝑋^𝑇_𝑖𝑀_𝑖1𝐸−𝑁_𝑖1𝐸𝑅_𝑖1,Φ^′′_𝑖14=−𝑋^𝑇_𝑖𝑆_𝑖1𝐸𝑅_𝑖2,Φ^′′_𝑖16=𝑋^𝑇_𝑖𝐶^𝑇_𝑖,Φ^′′_𝑖17=𝑐₁𝑋^𝑇_𝑖𝑁_𝑖1,Φ^′′_𝑖18=𝑐₁₂𝑋^𝑇_𝑖𝑆_𝑖1,Φ^′′_𝑖19=𝑐₁₂𝑋^𝑇_𝑖𝑀_𝑖1,Φ^′′_𝑖110=𝑑₁𝑋^𝑇_𝑖𝐴^𝑇_𝑖,Φ^′′_𝑖111=𝑑₁₂𝑋^𝑇_𝑖𝐴^𝑇_𝑖,Φ^′′_𝑖22=−1−𝜇𝑒^−𝛼𝑑2𝑅_𝑖3+Sym𝑅_𝑖3𝑆_𝑖2𝐸𝑅_𝑖3−𝑅_𝑖3𝑀_𝑖2𝐸𝑅_𝑖3,Φ^′′_𝑖23=𝑅_𝑖3𝑀_𝑖2𝐸−𝑁_𝑖2𝐸𝑅_𝑖1,Φ^′′_𝑖24=−𝑅_𝑖3𝑆_𝑖2𝐸𝑅_𝑖2,Φ^′′_𝑖26=𝑅_𝑖3𝐶^𝑇_𝑑𝑖,Φ^′′_𝑖27=𝑐₁𝑅_𝑖3𝑁_𝑖2,Φ^′′_𝑖28=𝑐₁₂𝑅_𝑖3𝑆_𝑖2,Φ^′′_𝑖29=𝑐₁₂𝑅_𝑖3𝑀_𝑖2. (4.12) Now, introducing change of variables 𝑁_𝑖1=𝜖_𝑖1𝑋_𝑖^−𝑇,𝑁_𝑖2=𝜖_𝑖2𝑅⁻