Mathematical Problems in Engineering

Volume 2014, Article ID 979130, 9 pages

http://dx.doi.org/10.1155/2014/979130

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

Key Laboratory of Advanced Process Control for Light Industry, Jiangnan University, Ministry of Education, Wuxi 214122, China

Received 9 September 2013; Revised 3 November 2013; Accepted 26 December 2013; Published 8 January 2014

Academic Editor: Shihua Li

Copyright © 2014 Pan Tinglong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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], 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 -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 -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 -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:
where is the system state, is the control input, and is the exogenous disturbance signal and satisfies the constraint , ; the switching signal 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 .

*Remark 1. *In view of the special structure of descriptor systems, the initial condition is given as . Corresponding to the switching signal , the switching sequence is defined as follows:
which means that the th subsystem is activated when and , are the initial state and initial time, respectively. In addition, we can transform the descriptor matrix of different form into the singular matrix “” 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
is said to be finite-time bounded with respect to , if the following formula:
holds, where , , , and .

*Definition 4. *Continuous switched descriptor system
is said to be finite-time stable with respect to , if the following formula:
holds, where , , and .

*Definition 5 (see [15]). *For any , if holds for given , , where denotes the switching number of over , then the constant is called the average dwell time and is the flutter bound. As commonly used in the previous literature, we choose .

*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 , , and , the system (5) is finite-time stable with respect to , if and only if
**
where is the state transition matrix.*

*Proof. *The following proof can be divided into two cases.*(a) Sufficiency*. Since is the state transition matrix of system (5), then
where . In view of , one obtains
On the other hand, from (7) and , we have
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 . By using the reduction to absurdity, if there exists , , such that
Let , ; then we get . 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 , 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 , if there exist nonsingular matrix , matrices , , and scalars , such that
**
And the average dwell time of the switching signal satisfies
**
where , , and ; then system (3) is regular, pulse-free, and finite-time bounded with respect to .*

*Proof. *First, from (15) we have

Considering and assuming that there exist invertible matrices , , such that , then it follows from (14) and (20) that
where , , .

According to (21), we can obtain as follows:
Correspondingly, suppose that , 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 ; namely, is nonsingular. Then, there exists a scalar such that and, for , 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 . Choose the multiple Lyapunov function as
Then, let and . By virtue of (14) and switched sequence (2), when , taking derivative of with respect to along the trajectory of the system (3) yields
Then, it follows from (15) that
Integrating both sides of (27) from to gives
Then, together with (17), at the switched moment we derive

For any , assuming that is the switched number of systems between , one can obtain . Considering and using (17) together with (29), based on iterative method, we have
Noticing that , then
On the other hand, it follows from (16) that
when ; putting together (31)-(32), it can easily be verified that
According to (18), we get
It follows from (19) and (34) that
By virtue of (33) and (35), we can obtain
then, considering and , we have
Thus, the system (3) is finite-time bounded with respect to , 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 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 and the exogenous disturbance , then will be a negative definite function. In this case, the system (1) is asymptotically stable on the infinite interval . 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 , if there exist nonsingular matrix , matrices , , and scalars , such that (18) and (19) hold and
**
then system (3) is regular, pulse-free, and finite-time bounded with respect to .*

*Proof. *Multiply both sides of (38) separately by and . Let ; we can obtain that (14) holds. Similarly, multiply both sides of (39) separately by and . Let ; then (15) holds. On the other hand, let , it follows from (38) and (41) that
Now, by virtue of , it follows from (42) that , . 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 .

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 .

Theorem 12. *For any , if there exist nonsingular matrices , , matrices , , and scalars , such that (18), (19), and (38)–(41) hold and
**
then the system (1) is regular, pulse-free, and finite-time bounded with respect to 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 , there exist invertible matrices , such that . Let and from (38) is given as
where , , and is a matrix with appropriate dimension. In addition, let and we can obtain . Based on the above discussion, the following equation holds:
where and . It is obvious that satisfies (41) and (43). Let , and then one obtains that (40) holds.

Equation (18) can be guaranteed by the following LMIs. For any , there exist scalars , such that

Theorem 13. *For any , if there exist matrices , , , and scalars , such that (19) and (46) hold and
**
then the switched descriptor system (1) is regular, pulse-free, and finite-time bounded with respect to via the action of the state feedback controller , where
**
and the matrix is nonsingular; matrices , satisfy , .*

*Remark 14. *If there exists , , we can obtain the sufficient condition of finite-time boundedness for switched systems with constant disturbance based on the above conclusion. If there exists , and 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:
**
For any , if there exist matrices , , and scalars , such that
**
and the average dwell time satisfies
**
then the switched descriptor system (49) is regular, pulse-free, and finite-time stable with respect to via the action of the state feedback controller , where is nonsingular and matrices , satisfy , .*

#### 4. Numerical Simulations

*Example 16. *Consider the continuous switched descriptor system (1) with parameters as follows:
The values of , , , , , , , and are given as follows: , , , let , , , , and .

Then, according to Theorem 13, we get
In the following, we have by calculating
Then we can obtain the designed state feedback controllers as follows:
Furthermore, we can obtain , , , and . For any switching signal with average dwell time , the switched descriptor system (1) is finite-time stable.

When , the trajectory of is presented in Figure 1 and the switching signal is shown in Figure 2.

*Example 17. *Consider the continuous switched descriptor system (49) with parameters as follows:
Choosing the matrix and letting , , , , , and , by virtue of Theorem 15, the feasible solutions are given as
Now, it can be obtained as
Then we can obtain the designed controllers as follows:
By calculating, we have , , and . For any switching signal with average dwell time , the system (49) is finite-time stable with respect to 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.

#### 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).

#### References

- S. Shi, Q. Zhang, Z. Yuan, and W. Liu, “Hybrid impulsive control for switched singular systems,”
*IET Control Theory & Applications*, vol. 5, no. 1, pp. 103–111, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - G. Xie and L. Wang, “Stability and stabilization of switched descriptor systems under arbitrary switching,” in
*Proceedings of the IEEE International Conference on Systems, Man and Cybernetics (SMC '04)*, pp. 779–783, Hague, The Netherlands, October 2004. View at Scopus - Y. J. Yin and J. Zhao, “Stability of switched linear singular systems with impulsive effects,”
*Acta Automatica Sinica*, vol. 33, no. 4, pp. 446–448, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - T. C. Wang and Z. R. Gao, “Asymptotic stability criterion for a class of switched uncertain descriptor systems with time-delay,”
*Acta Automatica Sinica*, vol. 34, no. 8, pp. 1013–1016, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - J.-X. Lin and S.-M. Fei, “Robust exponential admissibility of uncertain switched singular time-delay systems,”
*Acta Automatica Sinica*, vol. 36, no. 12, pp. 1773–1779, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - S. P. Ma, C. H. Zhang, and Z. Wu, “Delay-dependent stability of ${H}_{\infty}$ control for uncertain discrete switched singular systems with time-delay,”
*Applied Mathematics and Computation*, vol. 206, no. 1, pp. 413–424, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - H. J. Gao and Y. Q. Wu, “A design scheme of variable structure $H$-infinity control for uncertain singular Markov switched systems based on linear matrix inequality method,”
*Nonlinear Analysis: Hybrid Systems*, vol. 1, no. 3, pp. 306–316, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - D. Koenig and B. Marx, “${H}_{\infty}$-filtering and state feedback control for discrete-time switched descriptor systems,”
*IET Control Theory & Applications*, vol. 3, no. 6, pp. 661–670, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - Z.-R. Gao and Z.-C. Ji, “Optimal guaranteed cost control for a class of switched singular systems with time-delay,”
*Systems Engineering and Electronics*, vol. 33, no. 6, pp. 1358–1361, 2011. View at Publisher · View at Google Scholar · View at Scopus - Z.-Q. Gu, H.-P. Liu, F.-C. Liao, and Y.-J. Wang, “Guaranteed cost control for uncertain time-delay switched singular systems based on LMI,”
*Systems Engineering and Electronics*, vol. 32, no. 1, pp. 147–151, 2010. View at Google Scholar · View at Scopus - K. Yang, Y. X. Shen, and Z. C. Ji, “Robust ${H}_{\infty}$ Guaranteed cost control for uncertain switched descriptor delayed systems with nonlinear disturbance,”
*Proceedings of the IMechE—part I: Journal of Systems and Control Engineering*, vol. 227, no. 9, pp. 686–691, 2013. View at Publisher · View at Google Scholar - B. Meng and J.-F. Zhang, “Reachability conditions for switched linear singular systems,”
*IEEE Transactions on Automatic Control*, vol. 51, no. 3, pp. 482–488, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - B. Meng and J. Zhang, “Reachability analysis of switched linear discrete singular systems,”
*Journal of Control Theory and Applications*, vol. 4, no. 1, pp. 11–17, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Dorato, “Short time stability in linear time-varying systems,” in
*Proceedings of the IRE International Convention Record*, pp. 83–87, New York, NY, USA, 1961. - X. Z. Lin, H. B. Du, and S. H. Li, “Finite-time boundedness and ${L}_{2}$-gain analysis for switched delay systems with norm-bounded disturbance,”
*Applied Mathematics and Computation*, vol. 217, no. 12, pp. 5982–5993, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - G. L. Zhao and J. C. Wang, “Finite time stability and ${L}_{2}$-gain analysis for switched linear systems with state-dependent switching,”
*Journal of the Franklin Institute*, vol. 350, no. 5, pp. 1075–1092, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - X. Lin, H. Du, S. Li, and Y. Zou, “Finite-time stability and finite-time weighted ${L}_{2}$-gain analysis for switched systems with time-varying delay,”
*IET Control Theory & Applications*, vol. 7, no. 7, pp. 1058–1069, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - H. Liu, Y. Shen, and X. Zhao, “Finite-time stabilization and boundedness of switched linear system under state-dependent switching,”
*Journal of the Franklin Institute*, vol. 350, no. 3, pp. 541–555, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Lin, H. Du, S. Li, and Y. Zou, “Finite-time boundedness and finite-time ${l}_{2}$ gain analysis of discrete-time switched linear systems with average dwell time,”
*Journal of the Franklin Institute*, vol. 350, no. 4, pp. 911–928, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Xiang, C. Qiao, and M. S. Mahmoud, “Finite-time analysis and ${H}_{\infty}$ control for switched stochastic systems,”
*Journal of the Franklin Institute*, vol. 349, no. 3, pp. 915–927, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-E. Feng, Z. Wu, and J.-B. Sun, “Finite-time control of linear singular systems with parametric uncertainties and disturbances,”
*Acta Automatica Sinica*, vol. 31, no. 4, pp. 634–637, 2005. View at Google Scholar · View at Scopus - S. Zhao, J. Sun, and L. Liu, “Finite-time stability of linear time-varying singular systems with impulsive effects,”
*International Journal of Control*, vol. 81, no. 11, pp. 1824–1829, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Zhang, C. Liu, and X. Mu, “Robust finite-time ${H}_{\infty}$ control of singular stochastic systems via static output feedback,”
*Applied Mathematics and Computation*, vol. 218, no. 9, pp. 5629–5640, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Su, Q. L. Zhang, and J. Ai, “Practical and finite-time fuzzy adaptive control for nonlinear descriptor systems with uncertainties of unknown bound,”
*International Journal of Systems Science*, vol. 44, no. 12, pp. 2223–2233, 2013. View at Publisher · View at Google Scholar - Z. R. Gao, Y. X. Shen, and Z. C. Ji, “Uniform finite-time stability of discrete-time switched descriptor systems,”
*Acta Physica Sinica*, vol. 61, no. 12, Article ID 120203, 2012. View at Publisher · View at Google Scholar - K. Yang, Y. X. Shen, and Z. C. Ji, “Uniform finite-time stability for a class of continuous-time switched descriptor systems with time-varying delay,”
*Control and Decision*. In press. - L. Long and J. Zhao, “${H}_{\infty}$ control of switched nonlinear systems in $p$-normal form using multiple Lyapunov functions,”
*IEEE Transactions on Automatic Control*, vol. 57, no. 5, pp. 1285–1291, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - M. A. Müller and D. Liberzon, “Input/output-to-state stability and state-norm estimators for switched nonlinear systems,”
*Automatica*, vol. 48, no. 9, pp. 2029–2039, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X.-M. Sun, J. Zhao, and D. J. Hill, “Stability and ${L}_{2}$-gain analysis for switched delay systems: a delay-dependent method,”
*Automatica*, vol. 42, no. 10, pp. 1769–1774, 2006. View at Publisher · View at Google Scholar · View at MathSciNet