Exponential Admissibility and Dynamic Output Feedback Control of Switched Singular Systems with Interval Time-Varying Delay

Abstract

This paper is concerned with the problems of exponential admissibility and dynamic output feedback (DOF) control for a class of continuous-time switched singular systems with interval time-varying delay. A full-order, dynamic, synchronously switched DOF controller is considered. First, by using the average dwell time approach, a delay-range-dependent exponential admissibility criterion for the unforced switched singular time-delay system is established in terms of linear matrix inequalities (LMIs). Then, based on this criterion, a sufficient condition on the existence of a desired DOF controller, which guarantees that the closed-loop system is regular, impulse free and exponentially stable, is proposed by employing the LMI technique. Finally, some illustrative examples are given to show the effectiveness of the proposed approach.

1. Introduction

The past decades have witnessed an enormous interest in switched systems, due to their powerful ability in modeling of event-driven systems, logic-based systems, parameter- or structure-varying systems, and so forth; for details, see [1–4] and the references therein. Switched systems are a class of hybrid systems, which consist of a collection of continuous- or discrete-time subsystems and a switching rule specifying the switching between them. When focusing on the classification problems in switched systems, it is commonly recognized that there exist three basic problems [1]: (i) finding conditions for stability under arbitrary switching; (ii) identifing the limited but useful class of stabilizing switching signals, and (iii) construct a stabilizing switching signal. Many effective methods have been presented to tackle these three basic problems such as the multiple Lyapunov function approach [5], the piecewise Lyapunov function approach [6], the switched Lyapunov function approach [7], the convex combination technique [8], and the dwell time or average dwell time scheme [9–12]. On the other hand, time-delay is very common in engineering systems and is frequently a source of instability and poor performance [13]. Therefore, control of switched time-delay systems has received more and more attention in the past few years; see [14–23] and the references therein.

As far as we know, singular systems (known also as descriptor, implicit or differential algebraic systems) also provide a natural framework for modeling of dynamic systems and describe a larger class of systems than the regular system models [24]. Switched singular systems have strong engineering background such as electrical networks [25], economic systems [26]. Recently, many results have been obtained in the literature for switched singular systems, such as stability and stabilization [27–30], reachability [31], 𝐻_∞ control and filtering problems [32]. For switched singular time-delay (SSTD) systems, due to the coupling between the switching and the time-delay and because of the algebraic constraints in singular model, the behavior of such systems is much more complicated than that of regular switched time-delay systems or switched singular systems, and thus, to date, only a few results have been reported in the literature. In [33], the robust stability and 𝐻_∞ control problems for discrete-time uncertain SSTD systems under arbitrary switching were discussed by using switched Lyapunov functions. In [34], a switching signal was constructed to guarantee the asymptotic stability of continuous-time SSTD systems. However, the aforementioned results are focused on the basic problem (i), see [33], and problem (iii), see [34], for SSTD systems. Problem (ii) is to identify stabilizing switching signals on the premise that all the individual subsystems are stable. Basically, we will find that stability is ensured if the switching is sufficiently slow [1], and it is well known that dwell time and average dwell time are two effective tools to define slow switching signals. In [9], it was shown that if all the individual subsystems are exponentially stable and that the dwell time of the switching signal is not smaller than a certain lower bound, then the switched systems is exponentially stable. This result was extended to both continuous-time switched linear time-delay systems [16] and discrete-time cases [17]. Unfortunately, so far, to the best of the authors' knowledge, the problem of solving the basic problem (ii) for SSTD systems via the dwell time or average dwell time scheme remains open and unsolved. On the other hand, the results in [33] are derived based on the state feedback controller. In fact, in many practical systems, state variables are not always available. In this case, the design of a controller that does not require the complete access to the state vector is preferable. An important example of such controller is the dynamic output feedback (DOF) controller. However, little attention has been paid to the DOF controller design problem for SSTD systems. This forms the motivation of this paper.

In this paper, we are concerned with the problems of exponential admissibility and DOF control for a class of continuous-time switched singular systems with interval time-varying delay. A full-order, dynamic, synchronously switched DOF controller is designed. First, by using the average dwell time approach and the piecewise Lyapunov function technique, a delay-range-dependent exponential admissibility criterion is derived in terms of LMIs, which guarantees the regularity, nonimpulsiveness, and exponential stability of the unforced system. A estimation of the convergence of the system is also explicitly given. Then, the corresponding solvability condition for the desired DOF controller is established by employing the LMI technique. Finally, some illustrative examples are given to show the effectiveness of the proposed approach.

Notation 1. Throughout this paper, 𝐑^𝑛 denotes the 𝑛-dimensional Euclidean space and 𝐑^𝑛×𝑛 is the set of all 𝑛×𝑛 real matrices. 𝑃>0 (𝑃≥0) means that matrix 𝑃 is positive definite (semipositive definite). 𝜆_min(𝑃) (𝜆_max(𝑃)) denotes the minimum (maximum) eigenvalue of symmetric matrix 𝑃. 𝒞_𝑛,𝑑∶=𝒞([−𝑑,0],𝐑^𝑛) denotes the Banach space of continuous vector functions mapping the interval [−𝑑,0] to 𝐑^𝑛. Let 𝑥_𝑡∈𝒞_𝑛,𝑑 be defined by 𝑥_𝑡∶=𝑥(𝑡+𝜃), 𝜃∈[−𝑑,0]. ‖⋅‖ denotes the Euclidean norm of a vector and its induced norm of a matrix and ‖𝑥_𝑡‖_𝑑=sup_{−𝑑≤𝜃≤0}‖𝑥(𝑡+𝜃)‖. The superscript “𝑇” represents matrix transposition, the symmetric terms in a matrix are denoted by “∗”. diag{⋯} stands for a block-diagonal matrix and Sym{𝐴} is the shorthand notation for 𝐴+𝐴^𝑇.

2. Preliminaries and Problem Formulation

Consider a class of SSTD systems of the form𝐸𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝑑𝜎(𝑡)}𝑥(𝑡−𝑑(𝑡))+𝐵_𝜎(𝑡)𝑢(𝑡),𝑦(𝑡)=𝐶_𝜎(𝑡)𝑥(𝑡)+𝐶_{𝑑𝜎(𝑡)}𝑥((𝑡,𝑥(𝑡)(𝑡)𝑡−𝑑))=𝜙,𝑡∈−𝑑₁−𝑑₂,0, (2.1) where 𝑥(𝑡)∈𝐑^𝑛 is the system state, 𝑢(𝑡)∈𝐑^𝑚 is the control input, and 𝑦(𝑡)∈𝐑^𝑝 is the measured output; 𝜎(𝑡)∶[0,+∞)→ℐ={1,2,…,𝑁} with integer 𝑁>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, 𝐶_𝑖 and 𝐶_𝑑𝑖 are constant real matrices with appropriate dimensions; 𝜙(𝑡)∈𝒞_{𝑛,𝑑1+𝑑2} is a compatible vector valued initial function; 𝑑(𝑡) is an interval time-varying delay satisfying𝑑₁≤𝑑(𝑡)≤𝑑₁+𝑑₂,𝑑̇(𝑡)≤𝜇, (2.2) where 𝑑₁≥0, 𝑑₂>0 and 0≤𝜇<1 are constants.

Remark 2.1. Model (2.1) can describe many practical time-delay systems (e.g., chemical engineering systems, lossless transmission lines, partial element equivalent circuit, etc.) with time-varying parameters or structures, which may be caused by random failures and repairs of the components, sudden environment changes, and varying of the operating point of a system [13, 35]. In real application, the importance of the study of controller design problem for model (2.1) also arises from the extensive applications in networked control [36].

Since rank𝐸=𝑟≤𝑛, there exist nonsingular transformation matrices 𝑃, 𝑄∈𝐑^𝑛×𝑛 such that 𝑃𝐸𝑄=diag{𝐼_𝑟,0}. In this paper, without loss of generality, let𝐸=diag𝐼_𝑟,0. (2.3) 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 2.2 (see [16, 37]). For the switching signal 𝜎(𝑡) and any delay 𝑑(𝑡) satisfying (2.2), the unforced part of system (2.1) 𝐸𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐴_{𝑑𝜎(𝑡)}𝑥((𝑡,𝑥𝑡−𝑑))_𝑡0(𝜃)=𝑥𝑡₀+𝜃,𝜃∈−𝑑₁−𝑑₂,0 (2.4) is said to be(1)regular if det(𝑠𝐸−𝐴_𝑖) is not identically zero for each 𝜎(𝑡)=𝑖, 𝑖∈ℐ, (2)impulse free 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 (2.4) satisfies ‖𝑥(𝑡)‖≤𝜄𝑒^{−𝜆(𝑡−𝑡0)}‖𝑥_𝑡0‖_𝑑1+𝑑2, for all 𝑡≥𝑡₀, where 𝜆>0 and 𝜄>0 are called the decay rate and decay coefficient, respectively;(4)exponentially admissible if it is regular, impulse free and exponentially stable under the switching signal 𝜎(𝑡).

Definition 2.3 (see [9]). 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, we choose 𝑁₀=0.

This paper considers the full-order DOF controller of the following form:𝐸_𝑐𝑥̇_𝑐(𝑡)=𝐴_{𝑐𝜎(𝑡)}𝑥_𝑐(𝑡)+𝐵_{𝑐𝜎(𝑡)}𝑦(𝑡),𝑢(𝑡)=𝐶_{𝑐𝜎(𝑡)}𝑥_𝑐(𝑡)+𝐷_{𝑐𝜎(𝑡)}𝑦(𝑡), (2.5) where 𝑥_𝑐(𝑡)∈𝐑^𝑛 is the controller state vector, and 𝐸_𝑐, 𝐴_𝑐𝑖, 𝐵_𝑐𝑖, 𝐶_𝑐𝑖, and 𝐷_𝑐𝑖, 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 SSTD system (2.1), identify a class of switching signal 𝜎(𝑡) and design a DOF controller of the form (2.5) such that the resultant closed-loop system is exponentially admissible under the switching signal 𝜎(𝑡).

Before ending this section, we introduce the following lemma, which is essential for the development of our main results.

Lemma 2.4. For any constant matrix 𝑍∈R^𝑛×𝑛, 𝑍=𝑍^𝑇>0, positive scalar 𝛼, and vector function 𝑥̇∶[−𝜏,∞)→R^𝑛 such that the following integration is well defined, then 𝑒^𝛼𝜏−1𝛼^𝑡_{(𝑡)𝑡−𝑑}𝑒^{𝛼(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑍𝐸𝑥̇(𝑠)𝑑𝑠≥^𝑡_{(𝑡)𝑡−𝑑}𝐸𝑥̇(𝑠)𝑑𝑠^𝑇𝑍^𝑡_{(𝑡)𝑡−𝑑}𝐸𝑥̇(𝑠)𝑑𝑠,𝑡≥0, (2.6) where 0≤𝑑(𝑡)≤𝜏.

Proof. The proof is almost the same as Lemma 1 in [20]. Using Schur complement, we have 𝑒^{𝛼(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑍𝐸𝑥̇(𝑠)𝑥̇^𝑇(𝑠)𝐸^𝑇∗𝑒^{𝛼(𝑡−𝑠)}𝑍⁻¹≥0. (2.7) Integrating it from 𝑡−𝑑(𝑡) to 𝑡, we get ^𝑡_{𝑡−𝑑(𝑡)}𝑒^{𝛼(𝑠−𝑡)}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑍𝐸𝑥̇(𝑠)𝑑𝑠^𝑡_{𝑡−𝑑(𝑡)}𝑥̇^𝑇(𝑠)𝐸^𝑇∗𝑒𝑑𝑠_𝛼𝜏−1𝑍⁻¹𝛼≥0. (2.8) Using Schur complement again, we find that Lemma 2.4 holds.

3. Main Results

In this section, we first apply the average dwell time approach to investigate the exponential admissibility for SSTD system (2.4), and give the following result.

Theorem 3.1. For prescribed scalars 𝛼>0, 𝑑₁≥0, 𝑑₂>0, and 0≤𝜇<1, if for each 𝑖∈ℐ, there exist matrices 𝑄_𝑖𝑙>0, 𝑍_𝑖𝑙>0, 𝑙=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_𝑖=Φ_𝑖11𝑃^𝑇_𝑖𝐴_𝑑𝑖+𝐴^𝑇_𝑖𝑈_𝑖𝐴_𝑑𝑖𝑐₁𝐸^𝑇𝑍_𝑖1𝐸∗Φ_𝑖22𝑐₂𝐸^𝑇𝑍_𝑖2𝐸∗∗Φ_𝑖33<0, (3.2) where Φ_𝑖11=Sym{𝑃^𝑇_𝑖𝐴_𝑖}+∑²_𝑙=1𝑄_𝑖𝑙+𝛼𝐸^𝑇𝑃_𝑖−𝑐₁𝐸^𝑇𝑍_𝑖1𝐸+𝐴^𝑇_𝑖𝑈_𝑖𝐴_𝑖, Φ_𝑖22=−(1−𝜇)𝑒^{−𝛼(𝑑1+𝑑2)}𝑄_𝑖2−𝑐₂𝐸^𝑇𝑍_𝑖2𝐸+𝐴^𝑇_𝑑𝑖𝑈_𝑖𝐴_𝑑𝑖, Φ_𝑖33=−𝑒^−𝛼𝑑1𝑄_𝑖1−𝑐₁𝐸^𝑇𝑍_𝑖1𝐸−c₂𝐸^𝑇𝑍_𝑖2𝐸, 𝑐₁=(𝛼𝑑₁)/(𝑒^𝛼𝑑1−1), 𝑐₂=(𝛼𝑑₂)/(𝑒^𝛼𝑑2−1) and 𝑈_𝑖=𝑑²₁𝑍_𝑖1+𝑑²₂𝑒^𝛼𝑑1𝑍_𝑖2. Then, system (2.4) 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_𝑖𝑙≤𝛽𝑍_𝑗𝑙,𝑙=1,2,∀𝑖,𝑗∈ℐ. (3.3) 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; and (iii) to show the exponential stability of the algebraic subsystem.
Part (i) regularity and nonimpulsiveness. According to (2.3), for each 𝑖∈ℐ, denote𝐴_𝑖=𝐴_𝑖11𝐴_𝑖12𝐴_𝑖21𝐴_𝑖22,𝑍_𝑖1=𝑍_𝑖111𝑍_𝑖112𝑍^𝑇_𝑖112𝑍_𝑖122, (3.4) where 𝐴_𝑖11∈𝐑^𝑟 and 𝑍_𝑖111∈𝐑^𝑟. From (3.2), it is easy to see that Φ_𝑖11<0, 𝑖∈ℐ. Noting 𝑄_𝑖𝑙>0 and 𝑍_𝑖𝑙>0, 𝑙=1,2, we get Sym{𝑃^𝑇_𝑖𝐴_𝑖}+𝛼𝐸^𝑇𝑃_𝑖−𝑐₁𝐸^𝑇𝑍_𝑖1𝐸<0. Substituting 𝑃_𝑖, 𝐴_𝑖, 𝑍_𝑖1 and 𝐸 given as (3.1), (3.4) and (2.3) into this inequality and using Schur complement, we have Sym{𝐴^𝑇_𝑖22𝑃_𝑖22}<0, which implies that 𝐴_𝑖22, 𝑖∈ℐ, is nonsingular. Then by [24] and Definition 2.2, system (2.4) is regular and impulse free.
Part (ii) exponential stability of the differential subsystem. Define the piecewise Lyapunov functional candidate for system (2.4) as follows𝑉𝑥_𝑡=𝑉_𝜎(𝑡)𝑥_𝑡=𝑥^𝑇(𝑡)𝐸^𝑇𝑃_𝜎(𝑡)𝑥(𝑡)+^𝑡_𝑡−𝑑1𝑒^{𝛼(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄_𝜎(𝑡)1𝑥(𝑠)+𝑑𝑠_{𝑡(𝑡)𝑡−𝑑}𝑒^{𝛼(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄_𝜎(𝑡)2𝑥(𝑠)𝑑𝑠+𝑑₁⁰_−𝑑1^𝑡_𝑡+𝜃𝑒^{𝛼(𝑠−𝑡)}(𝐸𝑥̇(𝑠))^𝑇𝑍_𝜎(𝑡)1(𝐸𝑥̇(𝑠))𝑑𝑠𝑑𝜃+𝑑₂^−𝑑1_{−𝑑1−𝑑2}^𝑡_𝑡+𝜃𝑒^{𝛼(𝑠−𝑡+𝑑1)}(𝐸𝑥̇(𝑠))^𝑇𝑍_𝜎(𝑡)2(𝐸𝑥̇(𝑠))𝑑𝑠𝑑𝜃. (3.5) As mentioned earlier, the 𝑖_𝑘th subsystem is activated when 𝑡∈[𝑡_𝑘,𝑡_𝑘+1). Then, along the solution of system (2.4) under the switching sequence 𝒮, for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), we have 𝑉̇_𝑖𝑘𝑥_𝑡+𝛼𝑉_𝑖𝑘𝑥_𝑡≤2𝑥^𝑇(𝑡)𝑃^𝑇_𝑖𝑘𝐸𝑥̇(𝑡)+𝑥^𝑇(𝑡)𝑄_𝑖𝑘1𝑥(𝑡)−𝑒^−𝛼𝑑1𝑥^𝑇𝑡−𝑑₁𝑄_𝑖𝑘1𝑥𝑡−𝑑₁+𝑥^𝑇(𝑡)𝑄_𝑖𝑘2𝑥(𝑡)−1−𝜇𝑒^{−𝛼(𝑑1+𝑑2)}𝑥^𝑇(𝑡−𝑑(𝑡))𝑄_𝑖𝑘2𝑥((𝑡+(𝐸𝑥̇(𝑡𝑡−𝑑))))^𝑇𝑑²₁𝑍_𝑖𝑘1+𝑑²₂𝑒^𝛼𝑑1𝑍_𝑖𝑘2(𝐸𝑥̇(𝑡))+𝛼𝑥^𝑇(𝑡)𝐸^𝑇𝑃_𝑖𝑘𝑥(𝑡)−𝑑₁^𝑡_𝑡−𝑑1𝑒^{𝛼(𝑠−𝑡)}(𝐸𝑥̇(𝑠))^𝑇𝑍_𝑖𝑘1(𝐸𝑥̇(𝑠))𝑑𝑠−𝑑₂^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑒^{𝛼(𝑠−𝑡+𝑑1)}(𝐸𝑥̇(𝑠))^𝑇𝑍_𝑖𝑘2(𝐸𝑥̇(𝑠))𝑑𝑠. (3.6) By replacing 𝐸𝑥̇(𝑡) with 𝐴_𝑖𝑘𝑥(𝑡)+𝐴_𝑑𝑖𝑘𝑥(𝑡−𝑑(𝑡)) and using Lemma 2.4 and Schur complement, LMI (3.2) yields 𝑉̇_𝑖𝑘𝑥_𝑡+𝛼𝑉_𝑖𝑘𝑥_𝑡<0. (3.7) Integrating (3.7) from 𝑡_𝑘 to 𝑡_𝑘+1 gives 𝑉_𝜎(𝑡)𝑥_𝑡=𝑉_𝑖𝑘𝑥_𝑡≤𝑒^{−𝛼(𝑡−𝑡𝑘)}𝑉_𝑖𝑘𝑥_𝑡𝑘,𝑡∈𝑡_𝑘,𝑡_𝑘+1. (3.8) Let 𝑥(𝑡)=[^𝑥1_𝑥^(𝑡)_2(𝑡)], where 𝑥₁(𝑡)∈𝐑^𝑟 and 𝑥₂(𝑡)∈𝐑^𝑛−𝑟. From (2.3) and (3.1), it can be seen that for each 𝑖, 𝑖∈ℐ, 𝑥^𝑇(𝑡)𝐸^𝑇𝑃_𝑖𝑥(𝑡)=𝑥^𝑇₁(𝑡)𝑃_𝑖11𝑥₁(𝑡). Noting this, and using (3.3) and (3.5), at switching instant 𝑡_𝑘, we have 𝑉_𝑖𝑘𝑥_𝑡𝑘≤𝛽𝑉_𝜎(𝑡⁻_𝑘)𝑥_𝑡⁻_𝑘=𝛽𝑉_𝑖𝑘−1𝑥_𝑡⁻_𝑘,𝑘=1,2,…, (3.9) where 𝑡⁻_𝑘 denotes the left limitation of 𝑡_𝑘. Therefore, it follows from (3.8), (3.9) 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⁻_𝑘)𝑥_𝑡⁻_𝑘≤⋯≤𝑒^{−𝛼(𝑡−𝑡0)}𝛽^𝑘𝑉_𝜎(𝑡0)𝑡₀≤𝑒^{−(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑉_𝜎(𝑡0)𝑥_𝑡0. (3.10) According to (3.5) and (3.10), we obtain 𝜆₁𝑥₁(𝑡)²≤𝑉_𝜎(𝑡)𝑥_𝑡,𝑉_𝜎(𝑡0)𝑥_𝑡0≤𝜆₂𝑥_𝑡0²_𝑑1+𝑑2 (3.11) where 𝜆₁=min_{∀𝑖∈ℐ}𝜆_min(𝑃_𝑖11), and 𝜆₂=max_{∀𝑖∈ℐ}𝜆_max(𝑃_𝑖11)+(1/𝛼)(1−𝑒^−𝛼𝑑1)max_{∀𝑖∈ℐ}𝜆_max(𝑄_𝑖1)+(1/𝛼)(1−𝑒^{−𝛼(𝑑1+𝑑2)})max_{∀𝑖∈ℐ}𝜆_max(𝑄_𝑖2)+(𝑑₁/𝛼²)(𝛼𝑑₁−1+𝑒^−𝛼𝑑1)max_{∀𝑖∈ℐ}(2𝜆_max(𝑍_𝑖1)(‖𝐴_𝑖‖+‖𝐴_𝑑𝑖‖))+(1/𝛼²)(−𝑑₂+𝛼𝑑²₂𝑒^𝛼𝑑1+𝑑₂𝑒^−𝛼𝑑2)max_{∀𝑖∈ℐ}(2𝜆_max(𝑍_𝑖2)(‖𝐴_𝑖‖+‖𝐴_𝑑𝑖‖)). Then, combining (3.10) with (3.11) yields 𝑥₁(𝑡)≤𝜆₂𝜆₁𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}. (3.12)
Part (iii) exponential stability of the algebraic subsystem. Since 𝐴_𝑖22, 𝑖∈ℐ, is nonsingular, set 𝐺_𝑖=[^{𝐼𝑟−𝐴𝑖12𝐴−1𝑖22}_{0𝐴−1𝑖22}] and 𝐻=[^𝐼𝑟0_{0𝐼𝑛−𝑟}]. Then, it is easy to get𝐸∶=𝐺_𝑖𝐼𝐸𝐻=^𝑟0,𝐴00^𝑖∶=𝐺_𝑖𝐴_𝑖𝐴𝐻=^𝑖110𝐴_𝑖21𝐼_𝑛−𝑟,𝑃_𝑖∶=𝐺_𝑖^−𝑇𝑃_𝑖𝑃𝐻=^𝑖110𝑃_𝑖21𝑃_𝑖22, (3.13) 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_𝑖22=𝐴^𝑇_𝑖22𝑃_𝑖22. According to (3.13), denote 𝐴_𝑑𝑖∶=𝐺_𝑖𝐴_𝑑𝑖𝐴𝐻=^𝑑𝑖11𝐴_𝑑𝑖12𝐴_𝑑𝑖21𝐴_𝑑𝑖22,𝑄_𝑖𝑙∶=𝐻^𝑇𝑄_𝑖𝑙𝑄𝐻=^𝑖𝑙11𝑄_𝑖𝑙12𝑄_𝑖𝑙21𝑄_𝑖𝑙22,𝑍_𝑖𝑙∶=𝐺_𝑖^−𝑇𝑍_𝑖𝑙𝐺_𝑖⁻¹=𝑍_𝑖𝑙11𝑍_𝑖𝑙12𝑍_𝑖𝑙21𝑍_𝑖𝑙22,𝑙=1,2. (3.14) Let 𝜉(𝑡)=𝜉₁(𝑡)𝜉₂(𝑡)∶=𝐻⁻¹𝑥(𝑡)=𝑥(𝑡), (3.15) where 𝜉₁(𝑡)∈𝐑^𝑟 and 𝜉₂(𝑡)∈𝐑^𝑛−𝑟. Then, for any fixed 𝜎(𝑡)=𝑖, 𝑖∈ℐ, system (2.4) is 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)). (3.16)(3.17) By (3.2) and Schur complement, we have [^{Φ′𝑖11𝑃𝑇𝑖𝐴𝑑𝑖}_{∗Φ′𝑖22}]<0, where Φ^′_𝑖11=Sym{𝑃^𝑇_𝑖𝐴_𝑖}+∑²_𝑙=1𝑄_𝑖𝑙+𝛼𝐸^𝑇𝑃_𝑖−𝑐₁𝐸^𝑇𝑍_𝑖1𝐸 and Φ^′_𝑖22=−(1−𝜇)𝑒^{−𝛼(𝑑1+𝑑2)}𝑄_𝑖2−𝑐₂𝐸^𝑇𝑍_𝑖2𝐸. Pre- and postmultiplying this inequality by diag{𝐻^𝑇,𝐻^𝑇} and diag{𝐻,𝐻}, respectively, noting the expressions in (3.13) and (3.14), and using Schur complement, we have Sym𝑃^𝑇_𝑖22+²_𝑙=1𝑄_𝑖𝑙22𝑃^𝑇_𝑖22𝐴_𝑑𝑖22∗−1−𝜇𝑒^{−𝛼(𝑑1+𝑑2)}𝑄_𝑖222<0. (3.18) Pre- and postmultiplying this inequality by [−𝐴^𝑇_𝑑𝑖22𝐼] and its transpose, respectively, and noting 𝑄_𝑖122>0 and 0≤𝜇<1, we obtain 𝐴^𝑇_𝑑𝑖22𝑄_𝑖222𝐴_𝑑𝑖22−𝑒^{−𝛼(𝑑1+𝑑2)}𝑄_𝑖222<0. Then, according to Lemma 7 in [38], we can deduce that there exist constants ℏ_𝑖>1 and 𝜂_𝑖>0 such that 𝑒^{(1/2)𝛼(𝑑1+𝑑2)}𝐴_𝑑𝑖22^𝑙≤ℏ_𝑖𝑒^{−𝜂𝑖𝑙},𝑙=0,1,…,∀𝑖∈ℐ. (3.19)
Define𝑡⁰=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. (3.20)(3.21) As mentioned earlier, under the switching sequence 𝒮, for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), the 𝑖_𝑘th subsystem is activated. Then, from (3.17) and (3.20), we have 𝜉₂(𝑡)−𝐴_𝑖𝑘21𝜉₁𝑡⁰−𝐴_{𝑑𝑖𝑘21}𝜉₁𝑡¹−𝐴_{𝑑𝑖𝑘22}𝜉₂𝑡¹. (3.22) Similarly, it can be obtained that 𝜉₂(𝑡¹)=−𝐴_𝑖𝑘21𝜉₁(𝑡¹)−𝐴_{𝑑𝑖𝑘21}𝜉₁(𝑡²)−𝐴_{𝑑𝑖𝑘22}𝜉₂(𝑡²). Substituting this into (3.22), we get 𝜉₂(𝑡)=(−𝐴_{𝑑𝑖𝑘22})²𝜉₂(𝑡²)−∑¹_𝑗=0(−𝐴_{𝑑𝑖𝑘22})^𝑗(𝐴_𝑖𝑘21𝜉₁(𝑡^𝑗)+𝐴_{𝑑𝑖𝑘21}𝜉₁(𝑡^𝑗+1)). Continuing in the same manner and noting that 𝑡^𝑗<𝑡^𝑗−1, then there exists a finite positive integer 𝑇_𝑖𝑘 such that 𝜉₂(𝑡)=−𝐴_{𝑑𝑖𝑘22}^𝑇𝑖𝑘𝜉₂𝑡^𝑇𝑖𝑘−^{𝑇𝑖𝑘−1}_{𝑗𝑖𝑘=0}−𝐴_{𝑑𝑖𝑘22}^𝑗𝑖𝑘𝐴_𝑖𝑘21𝜉₁𝑡^𝑗𝑖𝑘+𝐴_{𝑑𝑖𝑘21}𝜉₁𝑡^{𝑗𝑖𝑘+1}, (3.23) where 𝑡^𝑇𝑖𝑘∈(𝑡_𝑘−1,𝑡_𝑘] and 𝑡^𝑇𝑖𝑘→𝑡_𝑘. When 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), the 𝑖_𝑘−1th subsystem is activated. Then, following a similar procedure as the above, there exists a finite positive integer 𝑇_𝑖𝑘−1 such that 𝜉₂𝑡^𝑇𝑖𝑘=−𝐴_{𝑑𝑖𝑘−122}^{𝑇𝑖𝑘−1}𝜉₂𝑡^{𝑇𝑖𝑘+𝑇𝑖𝑘−1}−^{𝑇𝑖𝑘+𝑇𝑖𝑘−1−1}_{𝑗𝑖𝑘−1=𝑇𝑖𝑘}−𝐴_{𝑑𝑖𝑘−122}^{𝑗𝑖𝑘−1−𝑇𝑖𝑘}𝐴_{𝑖𝑘−121}𝜉₁𝑡^{𝑗𝑖𝑘−1}+𝐴_{𝑑𝑖𝑘−121}𝜉₁𝑡^{𝑗𝑖𝑘−1+1}, (3.24) where 𝑡^{𝑇𝑖𝑘+𝑇𝑖𝑘−1}∈(𝑡_𝑘−2,𝑡_𝑘−1] and 𝑡^{𝑇𝑖𝑘+𝑇𝑖𝑘−1}→𝑡_𝑘−1. After 𝑘-times iterative manipulations, 𝑡 belongs to [𝑡₀,𝑡₁), and there exists a finite positive integer 𝑇_𝑖0 such that 𝜉₂𝑡^{𝑇𝑖𝑘+⋯+𝑇𝑖1}=−𝐴_𝑑𝑖022^𝑇𝑖0𝜉₂𝑡^{𝑇𝑖𝑘+⋯+𝑇𝑖0}−^{𝑇𝑖𝑘+⋯+𝑇𝑖0−1}_{𝑗𝑖0=𝑇𝑖𝑘+⋯+𝑇𝑖1}−𝐴_𝑑𝑖022^{𝑗𝑖0−𝑇𝑖𝑘−⋯−𝑇𝑖1}𝐴_𝑖021𝜉₁𝑡^𝑗𝑖0+𝐴_𝑑𝑖021𝜉₁𝑡^𝑗𝑖0+1, (3.25) where 𝑡^{𝑇𝑖𝑘+⋯+𝑇𝑖0}∈(−𝑑₁−𝑑₂,𝑡₀] and 𝑡^{𝑇𝑖𝑘+⋯+𝑇𝑖0}→𝑡₀. By a simple induction, we have 𝜉₂(𝑡)=^𝑘_𝑗=0−𝐴_{𝑑𝑖𝑗22}^𝑇𝑖𝑗𝜉₂𝑡^{𝑇𝑖𝑘+⋯+𝑇𝑖1+𝑇𝑖0}−^{𝑇𝑖𝑘−1}_{𝑗𝑖𝑘=0}−𝐴_{𝑑𝑖𝑘22}^𝑗𝑖𝑘𝐴_𝑖𝑘21𝜉₁𝑡^𝑗𝑖𝑘+𝐴_{𝑑𝑖𝑘21}𝜉₁𝑡^{𝑗𝑖𝑘+1}−^𝑘_𝑝=1^𝑘_𝑞=𝑝−𝐴_{𝑑𝑖𝑞22}^{𝑇𝑖𝑞𝑇𝑖𝑘+⋯+𝑇𝑖𝑝−1−1}_{𝑗𝑖𝑝−1=𝑇𝑖𝑘+⋯+𝑇𝑖𝑝}𝜑₁(𝑡)+𝜑₂(𝑡), (3.26) where 𝜑₁(𝑡)=−𝐴_{𝑑𝑖𝑝−122}^{𝑗𝑖𝑝−1−𝑇𝑖𝑘−⋯−𝑇𝑖𝑝}𝐴_{𝑖𝑝−121}𝜉₁𝑡^{𝑗𝑖𝑝−1},𝜑₂(𝑡)=−𝐴_{𝑑𝑖𝑝−122}^{𝑗𝑖𝑝−1−𝑇𝑖𝑘−⋯−𝑇𝑖𝑝}𝐴_{𝑑𝑖𝑝−121}𝜉₁𝑡^{𝑗𝑖𝑝−1+1} (3.27) Therefore, from (3.15), (3.21), and (3.26), and noting 𝑡^{𝑇𝑖𝑘+⋯+𝑇𝑖0}∈(−𝑑₁−𝑑₂,𝑡₀], we obtain 𝜉₂(𝑡)≤Δ₁+Δ₂+Δ₃+Δ₄+Δ₅, (3.28) where Δ₁=^𝑘_𝑗=0𝐴_{𝑑𝑖𝑗22}^𝑇𝑖𝑗𝑥_{𝑡0𝑑1+𝑑2},Δ₂=𝐴^𝑇₂₁^𝑖𝑘−1_{𝑗𝑖𝑘=0}𝐴_{𝑑𝑖𝑘22}^𝑗𝑖𝑘𝜉₁𝑡^𝑗𝑖𝑘,Δ₃=𝐴^𝑇_𝑑21^𝑖𝑘−1_{𝑗𝑖𝑘=0}𝐴_{𝑑𝑖𝑘22}^𝑗𝑖𝑘𝜉₁𝑡^{𝑗𝑖𝑘+1},Δ₄=𝐴^𝑘_21𝑝=1^𝑘_𝑞=𝑝𝐴_{𝑑𝑖𝑞22}^{𝑇𝑖𝑞𝑇𝑖𝑘+⋯+𝑇𝑖𝑝−1−1}_{𝑗𝑖𝑝−1=𝑇𝑖𝑘+⋯+𝑇𝑖𝑝}𝜑₁(𝑡,Δ₅)=𝐴^𝑘_𝑑21𝑝=1^𝑘_𝑞=𝑝𝐴_{𝑑𝑖𝑞22}^{𝑇𝑖𝑞𝑇𝑖𝑘+⋯+𝑇𝑖𝑝−1−1}_{𝑗𝑖𝑝−1=𝑇𝑖𝑘+⋯+𝑇𝑖𝑝}𝜑₂(𝑡). (3.29) Note 𝑡₀≥𝑡^{𝑇𝑖𝑘+⋯+𝑇𝑖0}=𝑡−^{𝑇𝑖𝑘+⋯+𝑇𝑖0−1}_𝑗=0𝑑𝑡^𝑗≥𝑡−𝑇_𝑖𝑘+⋯+𝑇_𝑖0𝑑₁+𝑑₂. (3.30) Using (3.19) and the relation 𝑇_𝑎≥𝑇^∗_𝑎=(ln𝛽)/𝛼, the first term in (3.28) can be estimated as Δ₁=^𝑘_𝑗=0𝑒^{(1/2)𝛼(𝑑1+𝑑2)}𝐴_{𝑑𝑖𝑗22}^𝑇𝑖𝑗𝑒^{−(1/2)𝛼(𝑇𝑖𝑘+⋯+𝑇𝑖0)(𝑑1+𝑑2)}𝑥_{𝑡0𝑑1+𝑑2}≤^𝑘_𝑗=0ℏ_𝑖𝑗𝑒^{−𝜂𝑖𝑗𝑇𝑖𝑗}𝑒^{−(1/2)𝛼(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}≤^𝑘_𝑗=0ℏ_𝑖𝑗𝑒^{−𝜂𝑖𝑗𝑇𝑖𝑗}𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}∶=𝜒₁𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}. (3.31) By (3.15), (3.12), (3.21), (3.20) and (3.19), we get 𝐴_{𝑑𝑖𝑘22}^𝑗𝑖𝑘𝜉₁𝑡^𝑗𝑖𝑘≤𝐴_{𝑑𝑖𝑘22}^𝑗𝑖𝑘𝜆₂𝜆₁𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡𝑗𝑖𝑘−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}≤𝜆₂𝜆₁𝑒^{(1/2)𝛼(𝑑1+𝑑2)}𝐴_{𝑑𝑖𝑘22}^𝑗𝑖𝑘𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡𝑗𝑖𝑘−1−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}≤⋯≤𝜆₂𝜆₁𝑒^{(1/2)𝛼(𝑑1+𝑑2)}𝐴_{𝑑𝑖𝑘22}^𝑗𝑖𝑘𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡0−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}≤𝜆₂𝜆₁ℏ_𝑖𝑘𝑒^{−𝜂𝑖𝑘𝑗𝑖𝑘}𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}. (3.32) Then, the second term in (3.28) can be estimated as Δ₂≤𝐴₂₁𝜆₂𝜆₁^{𝑇𝑖𝑘−1}_{𝑗𝑖𝑘=0}ℏ_𝑖𝑘𝑒^{−𝜂𝑖𝑘𝑗𝑖𝑘}𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}≤ℏ_𝑖𝑘𝐴₂₁𝜆₂𝜆₁𝑒^𝜂𝑖k𝑒^𝜂𝑖𝑘𝑒−1^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}∶=𝜒₂𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}. (3.33) Similarly, the third term in (3.28) can be bounded by Δ₃≤ℏ_𝑖𝑘𝑒^{(1/2)𝛼(𝑑1+𝑑2)}𝐴_𝑑21𝜆₂𝜆₁𝑒^𝜂𝑖𝑘𝑒^𝜂𝑖𝑘𝑒−1^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}∶=𝜒₃𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}. (3.34) In addition, following a similar deduction as that in (3.32), we obtain 𝜑₁(𝑡)≤𝜆₂𝜆₁𝑒^{(1/2)𝛼(𝑑1+𝑑2)𝑇𝑖𝑘+⋯+𝑇𝑖𝑝}ℏ_𝑖𝑝−1𝑒^{−𝜂𝑖𝑝−1(𝑗𝑖𝑝−1−𝑇𝑖𝑘−⋯−𝑇𝑖𝑝)}𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}. (3.35) Then, considering this and (3.19), the fourth term in (3.28) can be estimated as Δ₄≤𝐴₂₁𝜆₂𝜆₁^𝑘_𝑝=1^𝑘_𝑞=𝑝𝑒^{(1/2)𝛼(𝑑1+𝑑2)}𝐴_{𝑑𝑖𝑞22}^{𝑇𝑖𝑞𝑇𝑖𝑘+⋯+𝑇𝑖𝑝−1−1}_{𝑗𝑖𝑝−1=𝑇𝑖𝑘+⋯+𝑇𝑖𝑝}ℏ_𝑖𝑝−1𝑒^{−𝜂𝑖𝑝−1(𝑗𝑖𝑝−1−𝑇𝑖𝑘−⋯−𝑇𝑖𝑝)}×𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}≤𝐴₂₁𝜆₂𝜆₁^𝑘_𝑝=1ℏ_𝑖𝑝−1^𝑘_𝑞=𝑝ℏ_𝑖𝑞𝑒^{−𝜂𝑖𝑞𝑇𝑖𝑞}𝑒^{𝜂𝑖𝑝−1}𝑒^{𝜂𝑖𝑝−1}𝑒−1^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}∶=𝜒₄𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}. (3.36) Similarly, the fifth term in (3.28) can be bounded by Δ₅≤𝑒^{(1/2)𝛼(𝑑1+𝑑2)}𝐴_𝑑21𝜆₂𝜆₁^𝑘_𝑝=1ℏ_𝑖𝑝−1^𝑘_𝑞=𝑝ℏ_𝑖𝑞𝑒^{−𝜂𝑖𝑞𝑇𝑖𝑞}𝑒^{𝜂𝑖𝑝−1}𝑒^{𝜂𝑖𝑝−1}𝑒−1^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}∶=𝜒₅𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}. (3.37) Therefore, using (3.31) and (3.33)–(3.37), ‖𝜉₂(𝑡)‖ can be estimated as 𝜉₂(𝑡)≤𝜒₁+𝜒₂+𝜒₃+𝜒₄+𝜒₅𝑒^{−(1/2)(𝛼−(ln𝛽)/𝑇𝑎)(𝑡−𝑡0)}𝑥_{𝑡0𝑑1+𝑑2}. (3.38) Combining (3.15), (3.12) and (3.38) yields that system (2.4) 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. In terms of LMIs, Theorem 3.1 presents a delay-range-dependent exponential admissibility condition for the switched singular systems with interval time-varying delay. It is noted that this condition is obtained by using the integral inequality (Lemma 2.4); no additional free-weighting matrices are introduced to deal with the cross-term. Therefore, the condition proposed here involves much less decision variables than those obtained by using the free-weighting matrices method [16, 19, 21, 22] if the same Lyapunov function is chosen.

Remark 3.3. Equation (3.26) plays an important role in analyzing the exponential stability of the algebraic subsystem, which can be seen as a generalization of the iterative equation in [37] for nonswitched singular time-delay system to switched case.

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_𝑖𝑙≡𝑍_𝑗𝑙, 𝑙=1,2, for all 𝑖,𝑗∈ℐ, and 𝑇^∗_𝑎=0, then system (2.4) possesses a common Lyapunov function and the switching signals can be arbitrary.

In the following, we are to deal with the design problem of DOF controller for the SSTD system (2.1). Applying the DOF controller (2.5) to system (2.1) gives the following closed-loop system𝐸𝜂̇(𝑡)=𝐴_𝜎(𝑡)𝜂(𝑡)+𝐴_{𝑑𝜎(𝑡)}𝜂(𝑡−𝑑(𝑡)), (3.39) where 𝜂(𝑡)=[𝑥^𝑇(𝑡)𝑥^𝑇_𝑐(𝑡)]^𝑇, and𝐸=𝐸00𝐸_𝑐,𝐴_𝜎(𝑡)=𝐴_𝜎(𝑡)+𝐵_𝜎(𝑡)𝐷_{𝑐𝜎(𝑡)}𝐶_𝜎(𝑡)𝐵_𝜎(𝑡)𝐶_{𝑐𝜎(𝑡)}𝐵_{𝑐𝜎(𝑡)}𝐶_𝜎(𝑡)𝐴_{𝑐𝜎(𝑡)}𝐴_{𝑑𝜎(𝑡)}=𝐴_{𝑑𝜎(𝑡)}+𝐵_𝜎(𝑡)𝐷_{𝑐𝜎(𝑡)}𝐶_{𝑑𝜎(𝑡)}0𝐵_{𝑐𝜎(𝑡)}𝐶_{𝑑𝜎(𝑡)}0. (3.40) The following Theorem presents a sufficient condition for solvability of the DOF controller design problem for system (2.1).

Theorem 3.5. For prescribed scalars 𝛼>0, 𝛾>0, 𝑑₁≥0, 𝑑₂>0 and 0≤𝜇<1, if for each 𝑖∈ℐ, and given scalars 𝜉_𝑖1 and 𝜉_𝑖2, there exist matrices 𝑌_𝑖, 𝐴_𝑐𝑖, 𝐵_𝑐𝑖, 𝐶_𝑐𝑖, 𝐷_𝑐𝑖, 𝑄_𝑖𝑙11, 𝑄_𝑖𝑙12, 𝑄_𝑖𝑙22, 𝑍_𝑖𝑙11, 𝑍_𝑖𝑙12, 𝑍_𝑖𝑙22, 𝑙=1,2, and 𝑅_𝑖 and 𝑈_𝑖 of the following form 𝑅_𝑖=𝑅_𝑖110𝑅_𝑖21𝑅_𝑖22,𝑈_𝑖=𝑈_𝑖110𝑈_𝑖21𝑈_𝑖22 (3.41) with 𝑅_𝑖11∈𝐑^𝑟×𝑟, 𝑈_𝑖11∈𝐑^𝑟×𝑟, 𝑅_𝑖11>0, 𝑈_𝑖11>0, 𝑅_𝑖22∈𝐑^{(𝑛−𝑟)×(𝑛−𝑟)}, 𝑈_𝑖22∈𝐑^{(𝑛−𝑟)×(𝑛−𝑟)}, and 𝑅_𝑖22 and 𝑈_𝑖22 being invertible, such that Υ_𝑖11Υ_𝑖12Υ_𝑖13Υ_𝑖14Υ_𝑖150𝑐₁𝐸^𝑇𝑍_𝑖111𝐸𝑐₁𝐸^𝑇𝑍_𝑖112𝐸∗Υ_𝑖22Υ_𝑖23Υ_𝑖24Υ_𝑖250𝑐₁𝐸^𝑇𝑍^𝑇_𝑖112𝐸𝑐₁𝐸^𝑇𝑍_𝑖122𝐸∗∗Υ_𝑖33Υ_𝑖34Υ_𝑖35000∗∗∗Υ_𝑖44Υ_𝑖45000∗∗∗∗Υ_𝑖55Υ_𝑖56𝑐₂𝐸^𝑇𝑍_𝑖211𝐸𝑐₂𝐸^𝑇𝑍_𝑖212𝐸∗∗∗∗∗Υ_𝑖66𝑐₂𝐸^𝑇𝑍^𝑇_𝑖212𝐸𝑐₂𝐸^𝑇𝑍_𝑖222𝐸∗∗∗∗∗∗Υ_𝑖77Υ_𝑖78∗∗∗∗∗∗∗Υ_𝑖88<0,𝑄_𝑖𝑙=𝑄_𝑖𝑙11𝑄_𝑖𝑙12∗𝑄_𝑖𝑙22>0,𝑍_𝑖𝑙=𝑍_𝑖𝑙11𝑍_𝑖𝑙12∗𝑍_𝑖𝑙22>0,𝑙=1,2, (3.42)(3.43) where Υ_𝑖11=Sym{𝜉_𝑖1𝐴_𝑖+𝜉_𝑖1𝐵_𝑖𝐷_𝑐𝑖𝐶_𝑖+𝐵_𝑐𝑖𝐶_𝑖}+∑²_𝑙=1𝑄_𝑖𝑙11+𝛼𝐸^𝑇𝑅_𝑖−𝑐₁𝐸^𝑇𝑍_𝑖111𝐸, Υ_𝑖12=𝐴_𝑐𝑖+𝜉_𝑖1𝐵_𝑖𝐶_𝑐𝑖+𝜉_𝑖2𝐴^𝑇_𝑖+𝜉_𝑖2𝐶^𝑇_𝑖𝐷^𝑇_𝑐𝑖𝐵^𝑇_𝑖+𝐶^𝑇_𝑖𝐵^𝑇_𝑐𝑖+∑²_𝑙=1𝑄_𝑖𝑙12−𝑐₁𝐸^𝑇𝑍_𝑖112𝐸, Υ_𝑖13=−𝜉_𝑖1𝐼+𝑅^𝑇_𝑖+𝜉_𝑖1𝐴^𝑇_𝑖+𝜉_𝑖1𝐶^𝑇_𝑖𝐷^𝑇_𝑐𝑖𝐵^𝑇_𝑖+𝐶^𝑇_𝑖𝐵^𝑇_𝑐𝑖, Υ_𝑖14=−𝑌_𝑖+𝜉_𝑖2𝐴^𝑇_𝑖+𝜉_𝑖2𝐶^𝑇_𝑖𝐷^𝑇_𝑐𝑖𝐵^𝑇_𝑖+𝐶^𝑇_𝑖B^𝑇_𝑐𝑖, Υ_𝑖15=Υ_𝑖35=𝜉_𝑖1𝐴_𝑑𝑖+𝜉_𝑖1𝐵_𝑖𝐷_𝑐𝑖𝐶_𝑑𝑖+𝐵_𝑐𝑖𝐶_𝑑𝑖, Υ_𝑖22=Sym{𝐴_𝑐𝑖+𝜉_𝑖2𝐵_𝑖𝐶_𝑐𝑖}+∑²_𝑙=1𝑄_𝑖𝑙22+𝛼𝐸^𝑇𝑈_𝑖−𝑐₁𝐸^𝑇𝑍_𝑖122𝐸, Υ_𝑖23=−𝜉_𝑖2𝐼+𝐴^𝑇_𝑐𝑖+𝜉_𝑖1𝐶^𝑇_𝑐𝑖𝐵^𝑇_𝑖, Υ_𝑖24=−𝑌_𝑖+𝐴^𝑇_𝑐𝑖+𝜉_𝑖2𝐶^𝑇_𝑐𝑖𝐵^𝑇_𝑖+𝑈^𝑇_𝑖, Υ_𝑖25=Υ_𝑖45=𝜉_𝑖2𝐴_𝑑𝑖+𝜉_𝑖2𝐵_𝑖𝐷_𝑐𝑖𝐶_𝑑𝑖+𝐵_𝑐𝑖𝐶_𝑑𝑖, Υ_𝑖33=−2𝜉_𝑖1𝐼+𝑑²₁𝑍_𝑖111+𝑑²₂𝑒^𝛼𝑑1𝑍_𝑖211, Υ_𝑖34=−𝜉_𝑖2𝐼−𝑌_𝑖+𝑑²₁𝑍_𝑖112+𝑑²₂𝑒^𝛼𝑑1𝑍_𝑖212, Υ_𝑖44=−Sym{𝑌_𝑖}+𝑑²₁𝑍_𝑖122+𝑑²₂𝑒^𝛼𝑑1𝑍_𝑖222, Υ_𝑖55=−(1−𝜇)𝑒^{−𝛼(𝑑1+𝑑2)}𝑄_𝑖211−𝑐₂𝐸^𝑇𝑍_𝑖211𝐸, Υ_𝑖56=−(1−𝜇)𝑒^{−𝛼(𝑑1+𝑑2)}𝑄_𝑖212−𝑐₂𝐸^𝑇𝑍_𝑖212𝐸, Υ_𝑖66=−(1−𝜇)𝑒^{−𝛼(𝑑1+𝑑2)}𝑄_𝑖222−𝑐₂𝐸^𝑇𝑍_𝑖222𝐸, Υ_𝑖77=−(1−𝜇)𝑒^−𝛼𝑑1𝑄_𝑖111−𝑐₁𝐸^𝑇𝑍_𝑖111𝐸−𝑐₂𝐸^𝑇𝑍_𝑖211𝐸, Υ_𝑖78=−(1−𝜇)𝑒^−𝛼𝑑1𝑄_𝑖112−𝑐₁𝐸^𝑇𝑍_𝑖112𝐸−𝑐₂𝐸^𝑇𝑍_𝑖212𝐸, Υ_𝑖88=−(1−𝜇)𝑒^−𝛼𝑑1𝑄_𝑖122−𝑐₁𝐸^𝑇𝑍_𝑖122𝐸−𝑐₂𝐸^𝑇𝑍_𝑖222𝐸. Then, there exists a DOF controller in the form of (2.5), such that system (3.39) is exponentially admissible for any switching sequence 𝒮 with average dwell time 𝑇_𝑎≥𝑇^∗_𝑎=ln𝛽/𝛼, where 𝛽≥1 satisfies 𝑅_𝑖11≤𝛽𝑅_𝑗11,𝑈_𝑖11≤𝛽𝑈_𝑗11,𝑄_𝑖𝑙≤𝛽𝑄_𝑗𝑙,𝑍_𝑖𝑙≤𝛽𝑍_𝑗𝑙,𝑙=1,2,∀𝑖,𝑗∈ℐ. (3.44) Moreover, a desired DOF controller realisation is given by 𝐴_𝑐𝑖=𝑌_𝑖⁻¹𝐴_𝑐𝑖,𝐵_𝑐𝑖=𝑌_𝑖⁻¹𝐵_𝑐𝑖,𝐶_𝑐𝑖=𝐶_𝑐𝑖,𝐷_𝑐𝑖=𝐷_𝑐𝑖,∀𝑖∈ℐ. (3.45)

Proof. From Theorem 3.1, we known that system (3.39) is exponentially admissible for any switching sequence 𝒮 with average dwell time 𝑇_𝑎≥𝑇^∗_𝑎=(ln𝛽)/𝛼, where 𝛽≥1 satisfying (3.3), if for each 𝑖∈ℐ, there exist matrices 𝑄_𝑖𝑙>0, 𝑍_𝑖𝑙>0, 𝑙=1,2, and 𝑃_𝑖 with the form of (3.1) such that the inequality (3.2) with 𝐸, 𝐴_𝑖 and 𝐴_𝑑𝑖 instead of 𝐸, 𝐴_𝑖 and 𝐶_𝑖, respectively, holds. By decomposing Φ_𝑖 in (3.2), we obtain that for each 𝑖∈ℐΦ_𝑖=Π_𝑖Λ_𝑖Π^𝑇_𝑖<0, (3.46) where 𝐽_𝑖 is any invertible matrix with compatible dimension, and Π_𝑖=𝐼00𝐴_𝑖𝐴_𝑑𝑖00𝐼000𝐼^𝑇,Λ_𝑖=Λ_𝑖11𝑃^𝑇_𝑖−𝐽_𝑖+𝐴^𝑇_𝑖𝐽^𝑇_𝑖𝐽_𝑖𝐴_𝑑𝑖𝑐₁𝐸^𝑇𝑍_𝑖1𝐸∗−𝐽_𝑖−𝐽^𝑇_𝑖+𝑈_𝑖𝐽_𝑖𝐴_𝑑𝑖0∗∗Λ_𝑖33𝑐₂𝐸^𝑇𝑍_𝑖2𝐸∗∗∗Λ_𝑖44 (3.47) with Λ_𝑖11=Sym{𝐽_𝑖𝐴_𝑖}+∑²_𝑙=1𝑄_𝑖𝑙+𝛼𝐸^𝑇_𝑖𝑃_𝑖−𝑐₁𝐸^𝑇𝑍_𝑖1𝐸, Λ_𝑖33=−(1−𝜇)𝑒^{−𝛼(𝑑1+𝑑2)}𝑄_𝑖2−𝑐₂𝐸^𝑇𝑍_𝑖2𝐸, and Λ_𝑖44=−𝑒^−𝛼𝑑1𝑄_𝑖1−𝑐₁𝐸^𝑇𝑍_𝑖1𝐸−𝑐₂𝐸^𝑇𝑍_𝑖2𝐸. Hence, Φ_𝑖<0 holds if Λ_𝑖<0. (3.48)
Let 𝐸=diag{𝐸,𝐸}. For each 𝑖∈ℐ, define𝑃_𝑖=diag𝑅_𝑖,𝑈_𝑖. (3.49) By (2.3) and (3.41), we have 𝐸^𝑇𝑅_𝑖=𝑅^𝑇_𝑖𝐸≥0,𝐸^𝑇𝑈_𝑖=𝑈^𝑇_𝑖𝐸≥0, (3.50) which combining (3.49) yields 𝐸^𝑇𝑃_𝑖=𝑃^𝑇_𝑖𝐸≥0. (3.51) Then, from (3.41), (3.44) and (3.51), it can be deduced that 𝐸^𝑇𝑃_𝑖≤𝛽𝐸^𝑇𝑃_𝑗,∀𝑖,𝑗∈ℐ. (3.52)
Denote𝐽_𝑖=𝜉_𝑖1𝐼𝑌_𝑖𝜉_𝑖2𝐼𝑌_𝑖. (3.53) Substituting (3.43), (3.49) and (3.53) into (3.48), and defining 𝐴_𝑐𝑖=𝑌_𝑖𝐴_𝑐𝑖,𝐵_𝑐𝑖=𝑌_𝑖𝐵_𝑐𝑖,𝐶_𝑐𝑖=𝐶_𝑐𝑖,𝐷_𝑐𝑖=𝐷_𝑐𝑖,∀𝑖∈ℐ (3.54) we can easily obtain (3.42). This completes the proof.

Remark 3.6. Note that condition Φ_𝑖 of Theorem 3.1 involves some product terms between the Lyapunov matrices and the system matrices, which complicates the DOF control synthesis problem. To solve this problem, in the proof of Theorem 3.5, we have made a decoupling between the Lyapunov matrices and the system matrices by introducing a slack matrix 𝐽_𝑖 in condition Λ_𝑖. Compared with the variable change method used in [39, 40], the decoupling technique proposed here simplifies the DOF controller design problem greatly, which decreases the conservatism in some sense.

Remark 3.7. Scalars 𝜉_𝑖1 and 𝜉_𝑖2, 𝑖∈ℐ, in Theorem 3.5 are tuning parameters which need to be specified first. The optimal values of these parameters can be found by applying some optimization algorithms such as the program fminsearch in the optimization toolbox of MATLAB, the branch-and-band algorithm [41].

Remark 3.8. It is noted that in this paper, the derivative matrix 𝐸 is assumed to be switch-mode-independent. If 𝐸 is also switch-mode-dependent, then 𝐸 is changed to 𝐸_𝑖, 𝑖∈ℐ. In this case, the transformation matrices 𝑃 and 𝑄 should become 𝑃_𝑖 and 𝑄_𝑖 so that 𝑃_𝑖𝐸_𝑖𝑄_𝑖=diag{𝐼_𝑟𝑖,0}, and the state of the transformed system becomes 𝑥(𝑡)=𝑄_𝑖⁻¹(𝑡)=[𝑥^𝑇_𝑖1(𝑡)𝑥^𝑇_𝑖2(𝑡)]^𝑇 with 𝑥^𝑇_𝑖1(𝑡)∈𝐑^𝑟𝑖 and 𝑥^𝑇_𝑖1(𝑡)∈𝐑^{𝑛−𝑟𝑖}, which implies that there does not exist one common state space coordinate basis for all subsystems. Then, some assumptions for 𝐸_𝑖 (e.g., 𝐸_𝑖, 𝑖∈ℐ, have the same right zero subspace [27]) should be made so that 𝑄_𝑖 remains the same; in this case, the method presented in this paper is also valid. How to investigate the general SSTD system with 𝐸 being switch-mode-dependent is an interesting problem for future work via other approaches.

4. Numerical Examples

In this section, some numerical examples are presented to demonstrate the effectiveness of the proposed methods.

Example 4.1. Consider the switched system (2.4) with 𝐸=𝐼, 𝑁=2 (e.g., there are two subsystems) and the following parameters, which are borrowed from [21]: 𝐴₁=−200−0.9,𝐴_𝑑1=−100−1,𝐴₂=−200−0.7,𝐴_𝑑2=−10−1−1. (4.1) For 𝜇=0.4, 𝛼=0.5 and 𝛽=1.1, employing the LMIs in [21, 22] and those in Theorem 3.1 yields an allowable upper bound 𝑑₂ (in this paper 𝑑₂=𝑑₁+𝑑₂) of the delay 𝑑(𝑡) that guarantees the stability of system (2.4). Table 1 shows the values of the upper bound for various 𝑑₁ and the number of involved variables by using different methods. It is easily seen from Table 1 that Theorem 3.1 of this paper not only provides better results than those criteria in [21, 22] but also reduces the computational overhead to some extent.

Table 1 

Comparison of allowable upper bound 𝑑₂ for different 𝑑₁ in Example 4.1.

Example 4.2. Consider the switched system (2.4) with 𝑁=2 and the related parameters are given as follows: 𝐸=1000,𝐴₁=0.7300−1,𝐴_𝑑1=,𝐴−1.1100.5₂=0.40−0.1−1,𝐴_𝑑2=,−10.100.1 (4.2) and 𝑑₁=0.1, 𝑑₂=0.2, 𝜇=0.4 and 𝛼=0.5. It can be verified that both of the above two subsystems are stable. Let 𝛽=1; it can be found that there is no feasible solution to this case, which implies that there is no common Lyapunov function for the above two subsystems (see Remark 3.4). Now, we consider the average dwell time scheme, and set 𝛽=1.2. Solving the LMIs (3.2) gives the following solutions: 𝑃₁=19.7719017.9611819.3011,𝑄₁₁=0.7438−1.2478−1.247868.3107,𝑄₁₂=10³×,𝑍0.00020.01020.01021.0735₁₁=624.44250.88100.8810379.5848,𝑍₁₂=,𝑃332.4246−0.4902−0.4902373.4311₂=23.37110−26.9375642.4532,𝑄₂₁=0.7501−0.8269−0.826966.5062,𝑄₂₂=,𝑍0.22018.93168.9316917.3497₂₁=560.83840.76430.7643379.5975,𝑍₂₂=294.2097−0.4003−0.4003373.4858 (4.3) which means that the above switched system is exponentially admissible. Moreover, by further analysis, it can be found that the allowable minimum of 𝛽 is 𝛽_min=1.046 when 𝛼=0.5; in this case 𝑇^∗_𝑎=(ln𝛽_min)/𝛼=0.0899.

Example 4.3. Consider the switched system (2.1) with 𝑁=2 and ,𝐴𝐸=1000₁=0.901−5,𝐴_𝑑1=,𝐵0.50.110.1₁=0.30.1,𝐶₁=0.30.1,𝐶_𝑑1=,𝐴0.10.1₂=0.50.12−5,𝐴_𝑑2=0.20.51.50.1,𝐵₂=,𝐶0.40.1₂=0.10.3,𝐶_𝑑2=0.10.1, (4.4) and 𝑑(𝑡)=0.3+0.2sin(1.5𝑡). A simple calculation yields 𝑑₁=0.1, 𝑑₂=0.4 and 𝜇=0.3. By simulation, it can be checked that both of the above two subsystems with 𝑢(𝑡)=0 are unstable, and the state responses of the corresponding open-loop systems are shown in Figures 1 and 2, respectively, with the initial condition given by 𝜙(𝑡)=[12]^𝑇, 𝑡∈[−0.5,0]. In view of this, our goal is to design a DOF control 𝑢(𝑡) in the form of (2.5) such that the closed-loop system is exponentially admissible.

Figure 1 

State trajectories of the open-loop subsystem 1.

Figure 2 

State trajectories of the open-loop subsystem 2.

Set 𝛼=0.5, 𝛽=1.05 (thus 𝑇_𝑎≥𝑇^∗_𝑎=(ln𝛽)/𝛼=0.0976), and choose 𝜉₁₁=0.9255, 𝜉₁₂=0.0067, 𝜉₁₃=0.9811, 𝜉₁₄=0.0016. Solving the LMIs (3.41)–(3.44), the corresponding gain matrices of the DOF controller are computed as𝐴_𝑐1=−26.3829−0.4700−0.8920−0.8529,𝐴_𝑐2=−67.74800.64239.5961−0.8803,𝐵_𝑐1=,𝐵0.6811−0.2272_𝑐2=4.0654−1.1668,𝐶_𝑐1=33.63910.1324,𝐶_𝑐2=,𝐷23.4740−0.5180_𝑐1=−22.4825,𝐷_𝑐2=−17.9093 (4.5) To show the effectiveness of the obtained DOF controller, giving a random switching signal with the average dwell time 𝑇_𝑎≥0.13 as shown in Figure 3, we get the state trajectories of the closed-loop system as shown in Figure 4, for the given initial condition 𝜙(𝑡)=[12]^𝑇, 𝑡∈[−0.5,0]. It is clear that the designed controller is feasible and ensures the stability of the closed-loop system despite the switching and the time-varying delay.

Figure 3 

Switching signal with the average dwell time 𝑇𝑎>0.13.

Figure 4 

State trajectories of the closed-loop system under DOF control.

5. Conclusions

In this paper, the problems of exponential admissibility and DOF control for a class of continuous-time switched singular systems with interval time-varying delay have been investigated. A class of switching signals has been identified for the switched singular time-delay systems to be exponentially admissible under the average dwell time scheme. The DOF controller has been designed, and the corresponding solvability condition has been established by using the LMI technique. Numerical examples have been provided to illustrate the effectiveness of the proposed methods.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (60904020, 61004032), the Research Fund for the Doctoral Program of Higher Education of China (20090092120027), and the Natural Science Foundation of Jiangsu Province (BK2010200).