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

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.


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][3][4][5],  ∞ control [6][7][8], costguaranteed control [9][10][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][16][17][18][19][20] and descriptor systems [21][22][23][24] have been obtained.Lin et al. [15] gave some results on finite-time boundedness and finite-time weighted  2 -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  2 -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 timevarying 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  ∞ 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.

Problem Description and Preliminaries
Consider a class of switched descriptor system as follows: where () ∈   is the system state,   () ∈   is the control input, and () ∈   is the exogenous disturbance signal and satisfies the constraint   (0)(0) ≤ ,  ≥ 0; the switching signal () :  + → {1, 2, . . ., } is a piecewise constant and right continuous function;  is the number of subsystems; () =  represents that the th subsystem is activated.,   ,   ,   , and   are known constant matrices with appropriate dimension, and it is assumed that rank  =  ≤ .
Remark 1.In view of the special structure of descriptor systems, the initial condition is given as ( 0 ) =  0 .
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 5 (see [15]).For any where  () ( 0 ,   ) denotes the switching number of () over [ 0 ,   ), then the constant   is called the average dwell time and  0 is the flutter bound.As commonly used in the previous literature, we choose  0 = 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.

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  1 ,  2 , and   , the system (5) is finite-time stable with respect to ( 1 ,  2 , ,   , ), if and only if where Proof.The following proof can be divided into two cases.
(b) Necessity.Suppose that the system ( 5) is finite-time stable with respect to ( 1 ,  2 , ,   , ).By using the reduction to absurdity, if there exists ( * ) ̸ = 0,  * ∈ [0,   ), such that Let  = √ 1 /  ( * )  ( * ),  0 = ( * ); then we get   0    0 =  1 .By virtue of (11), we have Meanwhile, according to (8), we can obtain Noticing that it is inconsistent with the hypothesis that the system (5) is finite-time stable with respect to ( 1 ,  2 , ,   , ), 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.
Proof.First, from (15) we have Considering rank  =  and assuming that there exist invertible matrices , , such that  =  = diag{  , 0}, then it follows from ( 14) and (20) that where According to (21), we can obtain   as follows: Correspondingly, suppose that   = [  11  12  21  22 ], and it follows from (22) that where we do not need to know the expression of # and it does not affect the following discussion.It follows from (24) that  3  22 +   22   3 < 0; namely,  22 is nonsingular.Then, there exists a scalar  such that det( −   ) ̸ = 0 and, for ∀, deg det( −   ) = rank  holds.Thus, the system (3) is regular and pulse-free.
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][28][29].In this paper, the function () 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 ().Actually, if we limit the constants  < 0 and the exogenous disturbance () = 0, then () 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.
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   () =   ().
Proof.According to the proof of Theorem 11, first replace   with   +     and let     =   , 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 rank  = , there exist invertible matrices ,  such that  =  = diag{  , 0}.Let   =  −1     and from (38)   is given as where  1 > 0, det( 4 ) ̸ = 0, and  3 is a matrix with appropriate dimension.In addition, let Ψ =  [ 0  − ] and we can obtain Ψ = 0. Based on the above discussion, the following equation holds: where , and then one obtains that (40) holds.
Then we can obtain the designed controllers as follows: (59) By calculating, we have  1 = 0.0318,  2 = 0.0257, and   >  * = 1.3758.For any switching signal () with average dwell time   > 1.3758, the system (49) is finite-time stable with respect to ( 1 ,  2 , , ,   , ) via the action of the state feedback controller.Figure 3 shows the trajectory of   ()  (), 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: the whole system is not finite-time stable any more.The trajectory of   ()  () is shown in Figure 5 and the switching signal () is shown in Figure 6.

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.
Mathematical Problems in Engineering       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.