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.

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 [

Up to now, much attention has been mainly focused on system stability and reliability [

In recent years, the abundant studies on finite-time stability of switched systems [

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.

Consider a class of switched descriptor system as follows:

In view of the special structure of descriptor systems, the initial condition is given as

The initial state of system (

Now, we give the definitions of finite-time stability and finite-time boundedness for the continuous switched descriptor systems.

Continuous switched descriptor system

Continuous switched descriptor system

For any

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.

Firstly, the sufficient and necessary condition of finite-time stability for system (

Given positive constants

The following proof can be divided into two cases.

Noticing that it is inconsistent with the hypothesis that the system (

From Theorem

For any

First, from (

Considering

According to (

In the following, we will prove that system (

For any

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 [

In order to design controller conveniently, the following conclusion is given to satisfy the condition of Theorem

For any

Multiply both sides of (

In the following, we give the following conclusion about the finite-time boundedness problem of system (

For any

According to the proof of Theorem

Now, in order to solve by means of the LMI toolbox conveniently, we will process (

Equation (

For any

If there exists

Consider the switched descriptor system as follows:

Consider the continuous switched descriptor system (

Then, according to Theorem

When

Trajectory of

Switched signal

Consider the continuous switched descriptor system (

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:

Trajectory of

Switched signal

Trajectory of

Switching signal

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.

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

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).