Finite-Time Stability Analysis for a Class of Continuous Switched Descriptor Systems

Abstract

Finite-time stability has more practical application values than the classical Lyapunov asymptotic stability over a fixed finite-time interval. The problems of finite-time stability and finite-time boundedness for a class of continuous switched descriptor systems are considered in this paper. Based on the average dwell time approach and the multiple Lyapunov functions technique, the concepts of finite-time stability and boundedness are extended to continuous switched descriptor systems. In addition, sufficient conditions for the existence of state feedback controllers in terms of linear matrix inequalities (LMIs) are obtained with arbitrary switching rules, which guarantee that the switched descriptor system is finite-time stable and finite-time bounded, respectively. Finally, two numerical examples are presented to illustrate the reasonableness and effectiveness of the proposed results.

1. Introduction

Switched systems are a special class of hybrid systems, which consist of a collection of continuous or discrete-time subsystems together with a switching rule that orchestrates switching between these subsystems to achieve the control objectives [1]. Descriptor systems are also referred to as singular systems, implicit systems, or differential-algebraic systems, which are also a natural representation of dynamic systems and can describe physical systems better than the normal linear systems. Descriptor systems have been widely applied in many practical systems such as networks, power systems, electrical circuits, economics mathematical modeling, and many other fields [2, 3]. In actual control systems, switching phenomenon of descriptor systems is ubiquitous. Nevertheless, because of the switching between multiple descriptor subsystems and the algebraic constraints in descriptor model, it is inevitably difficult to analyze and synthesize for switched descriptor systems.

Up to now, much attention has been mainly focused on system stability and reliability [2–5], 𝐻_∞ control [6–8], cost-guaranteed control [9–11], system controllability, and reachability [12, 13] for switched descriptor systems. Generally, most of existing results related to stability and performance criteria of switched descriptor systems are based on the classical Lyapunov asymptotic stability, which is defined as an infinite time interval. However, in many actual systems, such as network control systems, the practical system state does not exceed some bound during some time interval and we need to avoid saturations and the excitation of nonlinear dynamics. In this case, the asymptotic stability is not enough for practical applications, because the system could be Lyapunov asymptotically stable but it possesses undesirable transient performances. Then the concept of finite-time stability was firstly put forward in [14], which concerns the boundedness of the system state over a fixed finite-time interval. To a certain degree, the development of the finite-time stability theory is parallel with the development of Lyapunov asymptotic stability.

In recent years, the abundant studies on finite-time stability of switched systems [15–20] and descriptor systems [21–24] have been obtained. Lin et al. [15] gave some results on finite-time boundedness and finite-time weighted 𝐿₂-gain for a class of switched delay systems with time-varying exogenous disturbances. Sufficient conditions which ensure that the switched system with time-time is finite-time bounded and has finite-time weighted 𝐿₂-gain were proposed based on the average dwell time approach and the multiple Lyapunov functions technique. In [23], the issue of robust finite-time stabilization of descriptor stochastic systems with time-varying norm-bounded disturbance and parametric uncertainties via static output feedback was discussed. Suppose that the state vector is not available for feedback; a static output feedback controller in terms of restricted LMIs was provided to guarantee the underlying closed-loop descriptor stochastic system finite-time stabilization with a prescribed 𝐻_∞𝑟-disturbance attenuation over the given finite-time interval. Meanwhile, an illustrative example was employed to verify the efficiency of the proposed method. In view of the importance of practical application, we need to pay great attention to the research on finite-time stability for switched descriptor systems compared with Lyapunov asymptotic stability. Few works that deal with the finite-time stability for this type of systems have been reported. Based on the different multiple Lyapunov functions, the papers [25, 26] focused on the discrete-time switched descriptor systems and switched descriptor systems with time-varying delay, respectively.

The paper is organized as follows. Firstly, the concepts of finite-time stability and finite-time boundedness for normal systems are expanded to continuous switched descriptor systems. Then, based on the state transfer matrix method, the sufficient and necessary condition of finite-time stability for this kind of system is given. Moreover, we tackle the problems of state feedback finite-time stabilization and finite-time boundedness; the sufficient conditions for the existence of controllers are obtained with arbitrary switching rules, which guarantee that the closed-loop systems are finite-time stable and finite-time bounded, respectively. Detailed proofs are presented by using the multiple Lyapunov functions and the average dwell-time approach. Finally, two examples are presented to show the validity of the developed methodology. Our research results are totally different from those previous results and important supplements for stability study for switched descriptor systems.

2. Problem Description and Preliminaries

Consider a class of switched descriptor system as follows: 𝐸𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐵_𝜎(𝑡)𝑢_𝜎(𝑡)(𝑡)+𝐺_𝜎(𝑡)𝑤(𝑡),𝑤̇(𝑡)=𝐹_𝜎(𝑡)𝑤(𝑡), (1) where 𝑥(𝑡)∈𝑅^𝑛 is the system state, 𝑢_𝜎(𝑡)∈𝑅^𝑝 is the control input, and 𝑤(𝑡)∈𝑅^𝑞 is the exogenous disturbance signal and satisfies the constraint 𝑤^𝑇(0)𝑤(0)≤𝑑, 𝑑≥0; the switching signal 𝜎(𝑡)∶𝑍⁺→{1,2,…,𝑚} is a piecewise constant and right continuous function; 𝑚 is the number of subsystems; 𝜎(𝑡)=𝑖 represents that the 𝑖th subsystem is activated. 𝐸, 𝐴_𝑖, 𝐵_𝑖, 𝐺_𝑖, and 𝐹_𝑖 are known constant matrices with appropriate dimension, and it is assumed that rank𝐸=𝑟≤𝑛.

Remark 1. In view of the special structure of descriptor systems, the initial condition is given as 𝐸𝑥(𝑡₀)=𝐸𝑥₀. Corresponding to the switching signal 𝜎(𝑡), the switching sequence is defined as follows: 𝑥₀;𝑖₀,𝑡₀,…,𝑖_𝑘,𝑡_𝑘,…∣𝑖_𝑘∈{1,2,…,𝑚},𝑘=0,1,…, (2) which means that the 𝑖_𝑘th subsystem is activated when 𝑡∈[𝑡_𝑘,𝑡_𝑘+1) and 𝑥₀, 𝑡₀ are the initial state and initial time, respectively. In addition, we can transform the descriptor matrix of different form into the singular matrix “𝐸” of system (1) by nonsingular transformation.

Assumption 2. The initial state of system (1) discussed is the consistent initial state, and the system state does not jump at the switching moment.
Now, we give the definitions of finite-time stability and finite-time boundedness for the continuous switched descriptor systems.

Definition 3. Continuous switched descriptor system 𝐸𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐺_𝜎(𝑡)𝑤(𝑡),𝑤̇(𝑡)=𝐹_𝜎(𝑡)𝑤(𝑡),𝑤^𝑇(0)𝑤(0)≤𝑑 (3) is said to be finite-time bounded with respect to (𝑐₁,𝑐₂,𝑑,𝑅,𝑇_𝑓,𝜎), if the following formula: 𝑥^𝑇(0)𝐸^𝑇𝑅𝐸𝑥(0)≤𝑐₁⟹𝑥^𝑇(𝑡)𝐸^𝑇𝑅𝐸𝑥(𝑡)≤𝑐₂,∀𝑡∈0,𝑇_𝑓, (4) holds, where 𝑐₂>𝑐₁>0, 𝑑≥0, 𝑇_𝑓>0, and 𝑅>0.

Definition 4. Continuous switched descriptor system 𝐸𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡) (5) is said to be finite-time stable with respect to (𝑐₁,𝑐₂,𝑅,𝑇_𝑓,𝜎), if the following formula: 𝑥^𝑇(0)𝐸^𝑇𝑅𝐸𝑥(0)≤𝑐₁⟹𝑥^𝑇(𝑡)𝐸^𝑇𝑅𝐸𝑥(𝑡)≤𝑐₂,∀𝑡∈0,𝑇_𝑓, (6) holds, where 𝑐₂>𝑐₁>0, 𝑇_𝑓>0, and 𝑅>0.

Definition 5 (see [15]). For any 𝑡_𝑓≥𝑡₀≥0, if 𝑁_𝜎(𝑡)(𝑡₀,𝑡_𝑓)≤𝑁₀+(𝑡_𝑓−𝑡₀)/𝜏_𝑎 holds for given 𝑁₀≥0, 𝜏_𝑎>0, where 𝑁_𝜎(𝑡)(𝑡₀,𝑡_𝑓) denotes the switching number of 𝜎(𝑡) over [𝑡₀,𝑡_𝑓), then the constant 𝜏_𝑎 is called the average dwell time and 𝑁₀ is the flutter bound. As commonly used in the previous literature, we choose 𝑁₀=0.

Remark 6. Finite-time stability for norm switched descriptor systems refers to the fact that the state of slow subsystem is less than a given upper bound. According to regularity of systems, the state of fast subsystem is also less than a given upper bound.

3. Main Results

Firstly, the sufficient and necessary condition of finite-time stability for system (5) is given by applying the state transition matrix method.

Theorem 7. Given positive constants 𝑐₁, 𝑐₂, and 𝑇_𝑓, the system (5) is finite-time stable with respect to (𝑐₁,𝑐₂,𝑅,𝑇_𝑓,𝜎), if and only if Φ^𝑇_𝜎(𝑡,0)𝐸^𝑇𝑅𝐸Φ_𝜎()≤𝑐𝑡,0²𝑐₁𝐸^𝑇𝑅𝐸,∀𝑡∈0,𝑇_𝑓, (7) where Φ_𝜎(𝑡,0)=Φ_𝑖𝑘(𝑡,𝑡_𝑘)Φ_𝑖𝑘−1(𝑡_𝑘,𝑡_𝑘−1)⋯Φ_𝑖0(𝑡₁,0) is the state transition matrix.

Proof. The following proof can be divided into two cases.
(a) Sufficiency. Since Φ_𝜎(𝑡,0) is the state transition matrix of system (5), then 𝑥(𝑡)=Φ_𝜎(𝑡,0)𝑥(0),𝑡∈0,𝑇_𝑓, (8) where Φ_𝜎(𝑡,0)=Φ_𝑖𝑘(𝑡,𝑡_𝑘)Φ_𝑖𝑘−1(𝑡_𝑘,𝑡_𝑘−1)⋯Φ_𝑖0(𝑡₁,0). In view of 𝑥^𝑇(0)𝐸^𝑇𝑅𝐸𝑥(0)≤𝑐₁, one obtains 𝑥^𝑇(𝑡)𝐸^𝑇𝑅𝐸𝑥(𝑡)=𝑥^𝑇(0)Φ^𝑇_𝜎(𝑡,0)𝐸^𝑇𝑅𝐸Φ_𝜎(𝑡,0)𝑥(0). (9) On the other hand, from (7) and 𝑥^𝑇(0)𝐸^𝑇𝑅𝐸𝑥(0)≤𝑐₁, we have 𝑥^𝑇(𝑡)𝐸^𝑇𝑅𝐸𝑥(𝑡)≤𝑥^𝑇(0)𝑐₂𝑐₁𝐸^𝑇𝑅𝐸𝑥(0)≤𝑐₂. (10) Thus, the system (5) is finite-time stable according to Definition 4.
(b) Necessity. Suppose that the system (5) is finite-time stable with respect to (𝑐₁,𝑐₂,𝑅,𝑇_𝑓,𝜎). By using the reduction to absurdity, if there exists 𝐸𝑥(𝑡^∗)≠0,  𝑡^∗∈[0,𝑇_𝑓), such that 𝑥^𝑇𝑡^∗Φ^𝑇_𝜎𝑡^∗,0𝐸^𝑇𝑅𝐸Φ_𝜎𝑡^∗,0𝑥𝑡^∗≥𝑥^𝑇𝑡^∗𝑐₂𝑐₁𝐸^𝑇𝑅𝐸𝑥𝑡^∗>0. (11) Let 𝜆=𝑐₁/𝑥^𝑇(𝑡^∗)𝐸^𝑇𝑅𝐸𝑥(𝑡^∗), 𝑥₀=𝜆𝑥(𝑡^∗); then we get 𝑥^𝑇₀𝐸^𝑇𝑅𝐸𝑥₀=𝑐₁. By virtue of (11), we have 𝑥^𝑇₀Φ^𝑇_𝜎𝑡^∗,0𝐸^𝑇𝑅𝐸Φ_𝜎𝑡^∗,0𝑥₀≥𝑥^𝑇₀𝑐₂𝑐₁𝐸^𝑇𝑅𝐸𝑥₀. (12) Meanwhile, according to (8), we can obtain 𝑥^𝑇𝑡^∗𝐸^𝑇𝑅𝐸𝑥𝑡^∗=𝑥^𝑇₀Φ^𝑇_𝜎𝑡^∗,0𝐸^𝑇𝑅𝐸Φ_𝜎𝑡^∗,0𝑥₀≥𝑐₂. (13)
Noticing that it is inconsistent with the hypothesis that the system (5) is finite-time stable with respect to (𝑐₁,𝑐₂,𝑅,𝑇_𝑓,𝜎), the proof is completed.

Remark 8. From Theorem 7 we can obtain the sufficient and necessary condition, which guarantees that the switched descriptor system (5) is finite-time stable. However, it is difficult and inconvenient to calculate state transition matrix and design controller. Thus, it is difficult to apply in the actual systems.

Theorem 9. For any 𝑖,𝑗∈{1,2,…,𝑚}, if there exist nonsingular  matrix 𝑃_𝑖, matrices 𝑄_𝑖>0, 𝑍_𝑖>0, and scalars 𝛼≥0, 𝜇≥1 such that 𝑃_𝑖𝐸=𝐸^𝑇𝑃^𝑇_𝑖𝑃≥0,_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃^𝑇_𝑖−𝛼𝑃_𝑖𝐸𝑃_𝑖𝐺_𝑖𝐺^𝑇_𝑖𝑃^𝑇_𝑖𝐹^𝑇_𝑖𝑄_𝑖+𝑄_𝑖𝐹_𝑖−𝛼𝑄_𝑖𝑃<0,_𝑖𝐸=𝐸^𝑇𝑅^1/2𝑍_𝑖𝑅^1/2𝑃𝐸,_𝑖𝐸≤𝜇𝑃_𝑗𝐸,𝑄_𝑖≤𝜇𝑄_𝑗,𝜆₁𝑐₁+𝜆₃𝑑𝜆₂𝑐₂<𝑒^{−𝛼𝑇𝑓}. (14)(15)(16)(17)(18) And the average dwell time of the switching signal 𝜎 satisfies 𝜏_𝑎>𝜏^∗_𝑎=𝑇_𝑓ln𝜇ln𝜆₂𝑐₂−ln𝜆₁𝑐₁+𝜆₃𝑑−𝛼𝑇_𝑓, (19) where 𝜆₁=max_{𝑖=1,2,...,𝑚}{𝜆_max(𝑍_𝑖)}, 𝜆₂=min_{𝑖=1,2,...,𝑚}{𝜆_min(𝑍_𝑖)}, and 𝜆₃=max_{𝑖=1,2,...,𝑚}{𝜆_max(𝑄_𝑖)}; then system (3) is regular, pulse-free, and finite-time bounded with respect to (𝑐₁,𝑐₂,𝑑,𝑅,𝑇_𝑓,𝜎).

Proof. First, from (15) we have 𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃^𝑇_𝑖−𝛼𝑃_𝑖𝐸<0. (20)
Considering rank𝐸=𝑟 and assuming that there exist invertible matrices 𝑀, 𝑁, such that 𝐸=𝑀𝐸𝑁=diag{𝐼_𝑟,0}, then it follows from (14) and (20) that 𝑃_𝑖𝐸=𝐸^𝑇𝑃^𝑇_𝑖≥0,𝑃_𝑖𝐴_𝑖+𝐴^𝑇_𝑖𝑃^𝑇_𝑖−𝛼𝑃_𝑖𝐸<0, (21)(22) where 𝐸=𝑀𝐸𝑁=diag{𝐼_𝑟,0}, 𝐴_𝑖=𝑀𝐴_𝑖𝑁, 𝑃_𝑖=𝑁^𝑇𝑃_𝑖𝑀⁻¹.
According to (21), we can obtain 𝑃_𝑖 as follows: 𝑃_𝑖=𝑃_𝑖1𝑃_𝑖20𝑃_𝑖3,𝑃_𝑖1>0,det𝑃_𝑖3≠0. (23) Correspondingly, suppose that 𝐴_𝑖=^{𝐴𝑖11𝐴𝑖12}_{𝐴𝑖21𝐴𝑖22}, and it follows from (22) that ###𝑃_𝑖3𝐴_𝑖22+𝐴^𝑇_𝑖22𝑃^𝑇_𝑖3<0, (24) where we do not need to know the expression of # and it does not affect the following discussion. It follows from (24) that 𝑃_𝑖3𝐴_𝑖22+𝐴^𝑇_𝑖22𝑃^𝑇_𝑖3<0; namely, 𝐴_𝑖22 is nonsingular. Then, there exists a scalar 𝑠 such that det(𝑠𝐸−𝐴_𝑖)≠0 and, for ∀𝑠, degdet(𝑠𝐸−𝐴_𝑖)=rank𝐸 holds. Thus, the system (3) is regular and pulse-free.
In the following, we will prove that system (3) is finite-time bounded with respect to (𝑐₁,𝑐₂,𝑑,𝑅,𝑇_𝑓,𝜎). Choose the multiple Lyapunov function as 𝑉(𝑡,𝑥(𝑡),𝑤(𝑡))=𝑉_𝜎(𝑡,𝑥(𝑡),𝑤(𝑡))=𝑥^𝑇(𝑡)𝑃_𝜎𝐸𝑥(𝑡)+𝑤^𝑇(𝑡)𝑄_𝜎𝑤(𝑡). (25) Then, let 𝑉(𝑡,𝑥(𝑡),𝑤(𝑡))=𝑉(𝑡) and 𝑉_𝜎(𝑡,𝑥(𝑡),𝑤(𝑡))=𝑉_𝜎(𝑡). By virtue of (14) and switched sequence (2), when 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), taking derivative of 𝑉(𝑡) with respect to 𝑡 along the trajectory of the system (3) yields 𝑉̇(𝑡)=𝑉̇_{𝜎(𝑡𝑘)}(𝑡)=𝑥(𝑡)𝑤(𝑡)^𝑇×𝑃_{𝜎(𝑡𝑘)}𝐴_{𝜎(𝑡𝑘)}+𝐴^𝑇_{𝜎(𝑡𝑘)}𝑃^𝑇_{𝜎(𝑡𝑘)}𝑃_{𝜎(𝑡𝑘)}𝐺_{𝜎(𝑡𝑘)}𝐺^𝑇_{𝜎(𝑡𝑘)}𝑃^𝑇_{𝜎(𝑡𝑘)}𝐹^𝑇_{𝜎(𝑡𝑘)}𝑄_{𝜎(𝑡𝑘)}+𝑄_{𝜎(𝑡𝑘)}𝐹_{𝜎(𝑡𝑘)}×𝑥(𝑡)𝑤(𝑡). (26) Then, it follows from (15) that 𝑉̇<𝛼𝑉=𝛼𝑉_{𝜎(𝑡𝑘)}(𝑡). (27) Integrating both sides of (27) from 𝑡_𝑘 to 𝑡 gives 𝑉(𝑡)<𝑒^{𝛼(𝑡−𝑡𝑘)}𝑉_{𝜎(𝑡𝑘)}𝑡_𝑘. (28) Then, together with (17), at the switched moment 𝑡_𝑘we derive 𝑉(𝑡)<𝑒^{𝛼(𝑡−𝑡𝑘)}𝜇𝑉_𝜎(𝑡⁻_𝑘)𝑡⁻_𝑘. (29)
For any 𝑡∈[0,𝑇_𝑓), assuming that 𝑁 is the switched number of systems between [0,𝑇_𝑓), one can obtain 𝑁_𝜎(0,𝑡)≤𝑁. Considering 𝜇≥1 and using (17) together with (29), based on iterative method, we have 𝑉(𝑡)<𝑒^𝛼𝑡𝜇^{𝑁𝜎(0,𝑡)}𝑉_𝜎(0)(0)≤𝑒^𝛼𝑡𝜇^𝑁𝑉_𝜎(0)(0). (30) Noticing that 𝑁≤𝑇_𝑓/𝜏_𝑎, then 𝑉(𝑡)<𝑒^𝛼𝑡𝜇^{𝑇𝑓/𝜏𝑎}𝑉_𝜎(0)(0). (31) On the other hand, it follows from (16) that 𝑉(𝑡)≥𝑥^𝑇(𝑡)𝑃_𝜎𝐸𝑥(𝑡)≥𝜆_min𝑍_𝜎𝑥^𝑇(𝑡)𝐸^𝑇𝑅𝐸𝑥(𝑡)≥𝜆₂𝑥^𝑇(𝑡)𝐸^𝑇(𝑡),𝑉𝑅𝐸𝑥_𝜎(0)(0)=𝑥^𝑇(0)𝑃_𝜎(0)𝐸𝑥(0)+𝑤^𝑇(0)𝑄_𝜎(0)𝑤(0)≤𝜆_max𝑍_𝜎(0)𝑥^𝑇(0)𝐸^𝑇𝑅𝐸𝑥(0)+𝜆_max𝑄_𝜎(0)𝑑≤𝜆₁𝑥^𝑇(0)𝐸^𝑇𝑅𝐸𝑥(0)+𝜆₃𝑑, (32) when 𝑥^𝑇(0)𝐸^𝑇𝑅𝐸𝑥(0)≤𝑐₁; putting together (31)-(32), it can easily be verified that 𝑥^𝑇(𝑡)𝐸^𝑇𝑅𝐸𝑥(𝑡)<𝑒^𝛼𝑡𝜇^{𝑇𝑓/𝜏𝑎}𝜆₁𝑐₁+𝜆₃𝑑𝜆₂. (33) According to (18), we get ln𝜆₂𝑐₂−ln𝜆₁𝑐₁+𝜆₃𝑑−𝛼𝑇_𝑓>0. (34) It follows from (19) and (34) that 𝜇^{𝑇𝑓/𝜏𝑎}𝜆₁𝑐₁+𝜆₃𝑑𝜆₂<𝑒^{−𝛼𝑇𝑓}𝑐₂. (35) By virtue of (33) and (35), we can obtain 𝑥^𝑇(𝑡)𝐸^𝑇𝑅𝐸𝑥(𝑡)<𝑒^𝛼𝑡𝑒^{−𝛼𝑇𝑓}𝑐₂=𝑒^{𝛼(𝑡−𝑇𝑓)}𝑐₂; (36) then, considering 𝑡<𝑇_𝑓 and 𝛼≥0, we have 𝑥^𝑇(𝑡)𝐸^𝑇𝑅𝐸𝑥(𝑡)<𝑐₂. (37) Thus, the system (3) is finite-time bounded with respect to (𝑐₁,𝑐₂,𝑑,𝑅,𝑇_𝑓,𝜎), and the proof is completed.

Remark 10. Since different Lyapunov functions can be constructed for different subsystems, so the multiple Lyapunov functions method is an effective and flexible design tool. Now, the multiple Lyapunov functions have been employed and discussed to study the stability and performance of switched or hybrid systems such as [27–29]. In this paper, the function 𝑉(𝑡) in the proof of Theorem 9 is taken as the multiple Lyapunov functions. Compared with the classical Lyapunov function for switched systems of asymptotical stability, there is really no requirement of negative definiteness or negative semidefiniteness on 𝑉(𝑡). Actually, if we limit the constants 𝛼<0 and the exogenous disturbance 𝑤(𝑡)=0, then 𝑉(𝑡) will be a negative definite function. In this case, the system (1) is asymptotically stable on the infinite interval [0,+∞). We can find the detailed proof in [29].
In order to design controller conveniently, the following conclusion is given to satisfy the condition of Theorem 9.

Theorem 11. For any 𝑖,𝑗∈{1,2,…,𝑚}, if there exist nonsingular matrix 𝑋_𝑖, matrices 𝑄_𝑖>0, 𝑍_𝑖>0, and scalars 𝛼≥0, 𝜇≥1 such that (18) and (19) hold and 𝐸𝑋_𝑖=𝑋^𝑇_𝑖𝐸^𝑇𝐴≥0,_𝑖𝑋_𝑖+𝑋^𝑇_𝑖𝐴^𝑇_𝑖−𝛼𝐸𝑋_𝑖𝐺_𝑖𝐺^𝑇_𝑖𝐹^𝑇_𝑖𝑄_𝑖+𝑄_𝑖𝐹_𝑖−𝛼𝑄_𝑖𝑋<0,_𝑖−𝑇𝐸=𝐸^𝑇𝑅^1/2𝑍_𝑖𝑅^1/2𝐸,1−2𝜇𝐸𝑋_𝑖+𝜇𝐸𝑋_𝑗≤0,𝑄_𝑖≤𝛼𝑄_𝑗, (38)(39)(40)(41) then system (3) is regular, pulse-free, and finite-time bounded with respect to (𝑐₁,𝑐₂,𝑑,𝑅,𝑇_𝑓,𝜎).

Proof. Multiply both sides of (38) separately by 𝑋_𝑖^−𝑇 and 𝑋_𝑖⁻¹. Let 𝑋_𝑖^−𝑇=𝑃; we can obtain that (14) holds. Similarly, multiply both sides of (39) separately by diag{𝑋_𝑖^−𝑇,𝐼} and diag{𝑋_𝑖⁻¹,𝐼}. Let 𝑋_𝑖^−𝑇=𝑃; then (15) holds. On the other hand, let 𝑋_𝑖=𝑃_𝑖^−𝑇, it follows from (38) and (41) that 𝐸𝑃_𝑖^−𝑇−𝛼𝐸𝑃_𝑖^−𝑇−𝛼𝑃_𝑖⁻¹𝐸^𝑇+𝛼𝐸𝑃_𝑗^−𝑇𝑄≤0,_𝑖≤𝛼𝑄_𝑗. (42) Now, by virtue of −𝛼𝑃_𝑖⁻¹𝑃_𝑗𝐸𝑃_𝑖^−𝑇≤−𝛼𝐸𝑃_𝑖^−𝑇−𝛼𝑃_𝑖⁻¹𝐸^𝑇+𝛼𝐸𝑃_𝑗^−𝑇, it follows from (42) that 𝐸𝑃_𝑖^−𝑇−𝛼𝑃_𝑖⁻¹𝑃_𝑗𝐸𝑃_𝑖^−𝑇≤0,  𝑄_𝑖≤𝛼𝑄_𝑗. Obviously, the previous equation is equivalent to (17), so (41) can ensure that (17) holds. From Theorem 9, it is easy to obtain that system (3) is regular, pulse-free, and finite-time bounded with respect to (𝑐₁,𝑐₂,𝑑,𝑅,𝑇_𝑓,𝜎).

In the following, we give the following conclusion about the finite-time boundedness problem of system (1) via the action of the state feedback controller 𝑢_𝜎(𝑡)=𝐾_𝜎𝑥(𝑡).

Theorem 12. For any 𝑖,𝑗∈{1,2,…,𝑚}, if there exist nonsingular matrices 𝑋_𝑖, 𝐿_𝑖, matrices 𝑄_𝑖>0, 𝑍_𝑖>0, and scalars 𝛼≥0, 𝜇≥1 such that (18), (19), and (38)–(41) hold and 𝐴_𝑖𝑋_𝑖+𝑋^𝑇_𝑖𝐴^𝑇_𝑖+𝐵_𝑖𝐿_𝑖+𝐿^𝑇_𝑖𝐵^𝑇_𝑖−𝛼𝐸𝑋_𝑖𝐺_𝑖𝐺^𝑇_𝑖𝐹^𝑇_𝑖𝑄_𝑖+𝑄_𝑖𝐹_𝑖−𝛼𝑄_𝑖<0, (43) then the system (1) is regular, pulse-free, and finite-time bounded with respect to (𝑐₁,𝑐₂,𝑑,𝑅,𝑇_𝑓,𝜎) via the action of the state feedback controller 𝑢_𝑖(𝑡)=𝐿_𝑖𝑋_𝑖⁻¹𝑥(𝑡).

Proof. According to the proof of Theorem 11, first replace 𝐴_𝑖 with 𝐴_𝑖+𝐵_𝑖𝐾_𝑖 and let 𝐾_𝑖𝑋_𝑖=𝐿_𝑖, and then it is easy to obtain the condition of Theorem 12.
Now, in order to solve by means of the LMI toolbox conveniently, we will process (38) and (40). According to rank𝐸=𝑟, there exist invertible matrices 𝑀, 𝑁 such that 𝐸=𝑀𝐸𝑁=diag{𝐼_𝑟,0}. Let 𝑋_𝑖=𝑁⁻¹𝑋_𝑖𝑀^𝑇 and from (38) 𝑋_𝑖 is given as 𝑋_𝑖=𝑋_𝑖10𝑋_𝑖3𝑋_𝑖4, (44) where 𝑋_𝑖1>0, det(𝑋_𝑖4)≠0, and 𝑋_𝑖3 is a matrix with appropriate dimension. In addition, let Ψ=𝑁⁰_{𝐼𝑛−𝑟} and we can obtain 𝐸Ψ=0. Based on the above discussion, the following equation holds: 𝑋_𝑖𝑋=𝑁^𝑖10𝑋_𝑖3𝑋_𝑖4𝑀^−𝑇𝑋=𝑁^𝑖10+00𝑋00_𝑖3𝑋_𝑖4𝑀^−𝑇=𝑁Ω_𝑖𝑁^𝑇𝐸^𝑇+Ψ𝑌_𝑖𝑀^−𝑇, (45) where Ω_𝑖=diag{𝑋_𝑖1,#}>0 and 𝑌_𝑖=𝑋_𝑖3𝑋_𝑖4. It is obvious that 𝑋_𝑖=𝑁Ω_𝑖𝑁^𝑇𝐸^𝑇+Ψ𝑌_𝑖𝑀^−𝑇 satisfies (41) and (43). Let 𝑍_𝑖=𝑅^−1/2𝑀^𝑇Ω_𝑖⁻¹𝑀𝑅^−1/2, and then one obtains that (40) holds.
Equation (18) can be guaranteed by the following LMIs. For any 𝑖,𝑗∈{1,2,…,𝑚}, there exist scalars 𝜀₁, 𝜀₂ such that 𝜀₁𝐼<𝑅^1/2𝑀⁻¹Ω_𝑖𝑀^−𝑇𝑅^1/2𝑄<𝐼,_𝑖<𝜀₂𝐼,−𝑒^𝛼𝑇𝑓𝑐₂+𝜀₂𝑑−√𝑐₁−√𝑐₁−𝜀₁<0. (46)

Theorem 13. For any 𝑖,𝑗∈{1,2,…,𝑚}, if there exist matrices Ω_𝑖=diag{𝑋_𝑖1,#}>0, 𝑄_𝑖>0, 𝐿_𝑖, 𝑌_𝑖 and scalars 𝛼≥0, 𝜇≥1 such that (19) and (46) hold and 1−2𝜇𝐸𝑁Ω_𝑖𝑁^𝑇𝐸^𝑇+𝜇𝐸𝑁Ω_𝑗𝑁^𝑇𝐸^𝑇𝑄≤0,_𝑖≤𝛼𝑄_𝑗,Ξ_𝑖𝐺_𝑖𝐺^𝑇_𝑖𝐹^𝑇_𝑖𝑄_𝑖+𝑄_𝑖𝐹_𝑖−𝛼𝑄_𝑖<0, (47) then the switched descriptor system (1) is regular, pulse-free, and finite-time bounded with respect to (𝑐₁,𝑐₂,𝑑,𝑅,𝑇_𝑓,𝜎) via the action of the state feedback controller 𝑢_𝑖(𝑡)=𝐿_𝑖𝛽⁻¹(Ω_𝑖,𝑌_𝑖)𝑥(𝑡), where Ξ_𝑖=𝐴_𝑖𝛽Ω_𝑖,𝑌_𝑖+𝛽^𝑇Ω_𝑖,𝑌_𝑖𝐴^𝑇_𝑖+𝐵_𝑖𝐿_𝑖+𝐿^𝑇_𝑖𝐵^𝑇_𝑖−𝛼𝐸𝛽Ω_𝑖,𝑌_𝑖, (48) and the matrix 𝛽(Ω_𝑖,𝑌_𝑖)=𝑁Ω_𝑖𝑁^𝑇𝐸^𝑇+Ψ𝑌_𝑖𝑀^−𝑇 is nonsingular; matrices 𝑀, 𝑁 satisfy 𝑀𝐸𝑁=diag{𝐼_𝑟,0}, Ψ=𝑁0𝐼_𝑛−𝑟^𝑇.

Remark 14. If there exists 𝐸=𝐼, 𝐹_𝑖=0, we can obtain the sufficient condition of finite-time boundedness for switched systems with constant disturbance based on the above conclusion. If there exists 𝑤(𝑡)=0, and 𝑉(𝑡) is taken as common Lyapunov function, then the system is of consistent finite-time boundedness with arbitrary switching signals.

Theorem 15. Consider the switched descriptor system as follows: 𝐸𝑥̇(𝑡)=𝐴_𝜎(𝑡)𝑥(𝑡)+𝐵_𝜎(𝑡)𝑢_𝜎(𝑡)(𝑡). (49) For any 𝑖,𝑗∈{1,2,…,𝑚}, if there exist matrices Ω_𝑖=diag{𝑋_𝑖1,#}>0, 𝐿_𝑖, 𝑌_𝑖 and scalars 𝛼≥0, 𝜇≥1 such that 1−2𝜇𝐸𝑁Ω_𝑖𝑁^𝑇𝐸^𝑇+𝜇𝐸𝑁Ω_𝑗𝑁^𝑇𝐸^𝑇𝐴≤0,_𝑖𝛽Ω_𝑖,𝑌_𝑖+𝛽^𝑇Ω_𝑖,𝑌_𝑖𝐴^𝑇_𝑖+𝐵_𝑖𝐿_𝑖+𝐿^𝑇_𝑖𝐵^𝑇_𝑖−𝛼𝐸𝛽Ω_𝑖,𝑌_𝑖𝜀<0,₁𝐼<𝑅^1/2𝑀⁻¹Ω_𝑖𝑀^−𝑇𝑅^1/2<𝜀₂𝜀𝐼,₂𝑐₁<𝜀₁𝑐₂ (50) and the average dwell time satisfies 𝜏_𝑎>𝜏^∗_𝑎=𝑇_𝑓ln𝜇ln𝜆₂𝑐₂−ln𝜆₁𝑐₁−𝛼𝑇_𝑓, (51) then the switched descriptor system (49) is regular, pulse-free, and finite-time stable with respect to (𝑐₁,𝑐₂,𝑑,𝑅,𝑇_𝑓,𝜎) via the action of the state feedback controller 𝑢_𝑖(𝑡)=𝐿_𝑖𝛽⁻¹(Ω_𝑖,𝑌_𝑖)𝑥(𝑡), where 𝛽(Ω_𝑖,𝑌_𝑖)=𝑁Ω_𝑖𝑁^𝑇𝐸^𝑇+Ψ𝑌_𝑖𝑀^−𝑇 is nonsingular and matrices 𝑀, 𝑁 satisfy 𝑀𝐸𝑁=diag{𝐼_𝑟,0}, Ψ=𝑁0𝐼_𝑛−𝑟^𝑇.

4. Numerical Simulations

Example 16. Consider the continuous switched descriptor system (1) with parameters as follows: 𝐴₁=,𝐵−2.11.0−0.3−0.8−0.40.40.8−0.61.0₁=11−1,𝐺₁=,𝐹0.20.1−0.10.30.1−0.1₁=,𝐴00.1−0.10₂=,𝐵−2.6−0.50.50.60.5−0.8−0.50.21.0₂=11−1,𝐺₂=,𝐹−0.10.20.2−0.1−0.10.3₂=.00.1−0.10 (52) The values of 𝐸, 𝛼, 𝜇, 𝑐₁, 𝑐₂, 𝑑, 𝑇_𝑓, and 𝑅 are given as follows: 𝛼=0.3, 𝜇=1.2, 𝐸=diag{1,1,0}, let 𝑐₁=2, 𝑐₂=10, 𝑑=3, 𝑇_𝑓=1, and 𝑅=𝐼.
Then, according to Theorem 13, we get Ω₁=,Ω0.52490.026600.02660.46350000.4749₂=,𝑌0.53530.027500.02750.47870000.4749₁=,𝑌−0.5388−0.2418−0.5585₂=,𝐿1.2591−0.2093−1.1868₁=,𝐿0.22300.4453−0.2745₂=,𝑄−0.0427−1.17100.3543₁=,𝑄2.1001002.1001₂=.2.1001002.1001 (53) In the following, we have by calculating 𝛽Ω₁,𝑌₁=,𝛽Ω0.54530.026600.02660.46350−0.5388−0.2418−0.5585₂,𝑌₂=.0.53530.027500.02750.478701.2591−0.2093−1.1868 (54) Then we can obtain the designed state feedback controllers as follows: 𝑢₁(𝑡)=𝐿₁𝛽⁻¹Ω₁,𝑌₁𝑥(𝑡)=𝑥(𝑡),𝑢0.83761.16910.4914₂(𝑡)=𝐿₂𝛽⁻¹Ω₂,𝑌₂𝑥(𝑡)=0.7566−2.6204−0.2985𝑥(𝑡). (55) Furthermore, we can obtain 𝜆₁=2.1949,  𝜆₂=1.8078,  𝜆₃=2.1001, and 𝜏_𝑎>𝜏^∗=0.2517. For any switching signal 𝜎(𝑡) with average dwell time 𝜏_𝑎>0.2517, the switched descriptor system (1) is finite-time stable.
When 𝑤(𝑡)=^sin(0.1𝑡)_cos(0.1𝑡), the trajectory of 𝑥^𝑇(𝑡)𝐸^𝑇𝑅𝐸𝑥(𝑡) is presented in Figure 1 and the switching signal is shown in Figure 2.

Figure 1 

Trajectory of 𝑥𝑇(𝑡)𝐸𝑇𝑅𝐸𝑥(𝑡).

Figure 2 

Switched signal 𝜎(𝑡).

Example 17. Consider the continuous switched descriptor system (49) with parameters as follows: 𝐴₁=−1−212−1301−2,𝐵₁=1,𝐴−1−9₂=−21−11−102−2−1,𝐵₁=3−8.−1 (56) Choosing the matrix 𝐸=diag{1,1,0} and letting 𝛼=0.1,  𝜇=1.2, 𝑐₁=1, 𝑐₂=5, 𝑇_𝑓=6, and 𝑅=𝐼, by virtue of Theorem 15, the feasible solutions are given as Ω₁=,Ω33.4670−2.15490−2.154937.288400036.9815₂=,𝑌31.46900.543500.543538.903100036.9815₁=,𝑌6.844421.344915.4656₂=,𝐿−1.3833−31.484332.2021₁=,𝐿3.70615.11685.1099₂=11.9850−1.8186−5.5726. (57) Now, it can be obtained as 𝛽Ω₁,𝑌₁=,𝛽Ω33.4670−2.15490−2.154937.284406.844421.344915.4656₂,𝑌₂=.31.46900.543500.543538.90310−1.3833−31.484332.2021 (58) Then we can obtain the designed controllers as follows: 𝑢₁(𝑡)=𝐿₁𝛽⁻¹Ω₁,𝑌₁𝑥(𝑡)=𝑥(𝑡),𝑢0.0400−0.04960.3304₂(𝑡)=𝐿₂𝛽⁻¹Ω₂,𝑌₂𝑥(𝑡)=0.3766−0.1921−0.1731𝑥(𝑡). (59) By calculating, we have 𝜆₁=0.0318,  𝜆₂=0.0257, and 𝜏_𝑎>𝜏^∗=1.3758. For any switching signal 𝜎(𝑡) with average dwell time 𝜏_𝑎>1.3758, the system (49) is finite-time stable with respect to (𝑐₁,𝑐₂,𝑑,𝑅,𝑇_𝑓,𝜎) via the action of the state feedback controller. Figure 3 shows the trajectory of 𝑥^𝑇(𝑡)𝐸^𝑇𝑅𝐸𝑥(𝑡), and Figure 4 shows the switching signal.
If the switching is too frequent, it is possible that the whole system is not finite-time stable. For instance, given the switching signal as follows: 𝜎(𝑡)=1,𝑥₂𝑥₃−𝑥₁𝑥₂<0,2,𝑥₂𝑥₃−𝑥₁𝑥₂≥0, (60) the whole system is not finite-time stable any more. The trajectory of 𝑥^𝑇(𝑡)𝐸^𝑇𝑅𝐸𝑥(𝑡) is shown in Figure 5 and the switching signal 𝜎(𝑡) is shown in Figure 6.

Figure 3 

Trajectory of 𝑥𝑇(𝑡)𝐸𝑇𝑅𝐸𝑥(𝑡).

Figure 4 

Switched signal 𝜎(𝑡).

Figure 5 

Trajectory of 𝑥𝑇(𝑡)𝐸𝑇𝑅𝐸𝑥(𝑡).

Figure 6 

Switching signal 𝜎(𝑡).

5. Conclusion

In this paper, the issues of finite-time stability and finite-time boundedness for a class of continuous switched descriptor systems have been studied. The sufficient and necessary condition of finite-time stability for switched descriptor systems is presented by applying the state transition matrix method. The obtained condition has certain theoretical value, but it also has two disadvantages in practical application. First, it is difficult to calculate the state transition matrix; on the other hand, it is inconvenient to design controller. In order to solve these problems, based on the average dwell time approach and the multiple Lyapunov function technique, the existence of state feedback controllers is proposed with arbitrary switching rules, which guarantee that the switched descriptor systems are finite-time stable and finite-time bounded, respectively. More possible future works are to consider output feedback stabilization for the uncertain switched descriptor systems with time-varying delay.

Conflict of Interests

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

Acknowledgments

This study was supported by the National Natural Science Foundation of China (Grant no. 61174032), the Natural Science Foundation of Jiangsu Province (Grant no. BK2012550), and the Fundamental Research Funds for the Central University (Grant no. JUDCF11040).