Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 427893, 5 pages

http://dx.doi.org/10.1155/2014/427893

## On Exponential Stabilizability for a Class of Switched Nonlinear Systems with Mixed Time-Varying Delays

^{1}School of Mathematics, University of Jinan, Jinan, Shandong 250022, China^{2}School of Automation and Electrical Engineering, University of Jinan, Jinan, Shandong 250022, China

Received 9 June 2014; Revised 2 September 2014; Accepted 2 September 2014; Published 4 December 2014

Academic Editor: Hamid R. Karimi

Copyright © 2014 Jie Qi and Yuangong Sun. 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

We study the exponential stabilizability for a class of switched nonlinear systems with mixed time-varying delays. By using a new technique developed for positive systems, we design the average dwell time switching under which the switched nonlinear system is exponentially stable for any bounded delays. Finally, numerical examples are worked out to illustrate the main theoretical result.

#### 1. Introduction

A switched system which consists of a series of dynamical subsystems and a switching signal is a type of hybrid dynamical systems. Switched systems can be used to model many phenomena which cannot be described by purely continuous or purely discrete processes. Due to its broad applications in traffic control, chemical processing, switching power converters, and network control, the theory of switched systems has historically a position of great importance in systems theory and has been studied extensively in recent years [1–5].

Up to now, the stability of switched systems has attracted many researchers’ attention. For stability issues, two main problems have been investigated in the literature. One is to find conditions that guarantee asymptotic stability of the switched system under arbitrary switching. For this case, the common Lyapunov function is required for all subsystems [6, 7]. The other is to identify those switching signals for which the switched system is asymptotically stable, that is, stability of switched systems under constrained switching. For this case, the multiple-Lyapunov functions are a powerful and effective tool, and average dwell time (ADT) approaches have been used to investigate the stability and stabilization problems in [8].

Recently, positive switched system receives much attention. In the theory of positive switched systems, the stability problem is investigated extensively by many researchers [9–13], especially for the stability under arbitrary switching. It is well known that a common linear copositive Lyapunov function (CLCLF) is usually applied to the asymptotic stability of positive switched systems under arbitrary switching. Recently, necessary and sufficient conditions for the existence of CLCLFs were established in [13, 14]. For the case when the positive switched system does not share a CLCLF, a multiple linear copositive Lyapunov functional method was used in [15]. Some other methods to stability of switched nonlinear systems were proposed in [16–21].

In this paper, we study the exponential stabilizability for a class of switched nonlinear systems with mixed time-varying delays. Due to the the existence of both discrete and distributed time-varying delays and the assumption that the system is not necessarily positive, a new technique developed for positive systems is employed to the exponential stability under ADT switching for a class of switched nonlinear systems with mixed time-varying delays.

*Notation*. Throughout this paper, is the set of integers for any positive integer . Say a real vector if all entries of are positive (negative). The norm of the vector is defined to be . Say a square matrix is Metzler if its off-diagonal entries are nonnegative. Say a matrix is nonnegative if all its entries are nonnegative.

#### 2. Problem Statements and Preliminaries

Consider the following switched nonlinear systems with mixed time-varying delays: where ; and are piecewise continuous time-delays satisfying , , and and are constants; is a piecewise constant function of time, called a switching signal, that takes its values in the finite set ; , , are constant matrices; with for .

For the particular case when and , system (1) reduces to the following switched linear delay system: Under the assumption that is Metzler and is nonnegative for , it was proved in [11] that system (2) is asymptotically stable under arbitrary switching if there exists a common vector such that

If we do not assume that is Metzler and and are nonnegative for , set with and for , with , and with . It was proved in [22] that system (2) is asymptotically stable under arbitrary switching if there exists a common vector such that

However, in many cases, condition (3) or (4) may not hold. It may be natural to assume that there exist a set of vectors such that In this paper, we will study the exponential stability of system (1) under the milder condition of form (5).

Let be the Banach space of all continuous functions on with values in normed by the maximum norm , where and .

For a switching signal and any , let denote the number of discontinuities of in the open interval . We say that has an ADT if satisfies

Throughout this paper, system (1) is said to be exponentially stabilizable via ADT switching, if for any initial function there exist positive constants , , and (which are usually relative to the given initial function ) such that the corresponding solution of system (1) under any switching with ADT satisfies for .

#### 3. Main Results

In the sequel, we assume that there exist vectors , , such that Then, we have the following global exponential stability criterion for system (1).

Theorem 1. *System (1) is exponentially stabilizable via ADT switching if there exist vectors , , such that (7) holds.*

*Proof. *For a given switching sequence , let for ; that is, the th subsystem is active on . For any given , we choose an appropriate constant such that for . Set for . Then, we have
Denote . Noting that
we can get from (7) that there exists an appropriate constant such that
Let . Then, system (1) reduces to the following system:
where and .

First, we get from (10) that there exists a constant such that, for any , , and ,
Set . Then, we have that, for any and ,
By the continuity of at , there exists such that
We now prove that (14) holds on for any given . Otherwise, there exist and at least one index such that, for and ,
It implies that . On the other hand, for and , we get from (11) that
Therefore, by (15), we have that
From (12), we have that
Noting that , we get from (17) and (18) that
It yields that from (12) with and , which contradicts the fact that . Therefore, (14) holds on for any . By letting and tend to 1 and , respectively, we have that
where . Therefore,
Since is continuous at , by repeating the above procedure, we can conclude that
By induction, we have that, for each ,
Therefore, for any , we get from (23) that
If we set and , where , and satisfies . By (24), we have that system (1) is exponentially stabilizable via ADT switching. This completes the proof of Theorem 1.

If there exists a common vector such that we have that . Then, Theorem 1 yields the following corollary.

Corollary 2. *If there exists a common vector such that (25) holds, then system (1) is globally exponentially stable under any switching.*

#### 4. Illustrative Examples

*Example 1. *Consider system (1) with , , , , and
Based on a straightforward computation, we have that
We see that there does not exist a vector such that for . Otherwise, we can get and . This is a contradiction. Therefore, condition (25) is invalid for this case. However, there exist two vectors and such that
that is, condition (7) holds. For any initial condition satisfying , we can choose . It is not difficult to verify that (10) holds for and . Therefore, by Theorem 1, all solutions satisfying the initial condition are exponentially stabilizable via the switching with ADT .

*Example 2. *Consider system (1) with , , and
For this case, we have that
It is not difficult to verify that there does not exist a vector such that for . Note that (10) reduces to
By solving the above inequality, we get a solution , , and . Therefore, by Theorem 1 and its proof, we have that system (1) is exponentially stabilizable via the switching with ADT .

#### 5. Conclusion

This paper has investigated the exponential stabilizability for a class of switched nonlinear systems with mixed time-varying delays by using a new technique developed for positive systems. By using a new method developed for positive systems, we design the appropriate ADT switching under which the system is exponentially stable. The main results generalize some existing results in the literature. Two numerical examples are also worked out to illustrate the effectiveness and sharpness of the given theoretical result. Stability analysis for the more general switched nonlinear systems with mixed time delays will be further investigated in the future.

#### Conflict of Interests

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

#### Acknowledgments

The authors thank the reviewers for their valuable comments on this paper. This work was supported by the Natural Science Foundation of Shandong Province under Grant no. JQ201119 and the National Natural Science Foundation of China under Grant nos. 61174217, 61374074, and 61473133.

#### References

- D. Liberzon,
*Switching in Systems and Control*, Birkhäuser, Boston, Mass, USA, 2003. - G. Xie and L. Wang, “Periodic stabilizability of switched linear control systems,”
*Automatica*, vol. 45, no. 9, pp. 2141–2148, 2009. View at Google Scholar - B. Du and X. Zhang, “Delay-dependent stability analysis and synthesis for uncertain impulsive switched system with mixed delays,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 381571, 9 pages, 2011. View at Publisher · View at Google Scholar - M. De la Sen and A. Ibeas, “Stability results of a class of hybrid systems under switched continuous-time and discrete-time control,”
*Discrete Dynamics in Nature and Society*, vol. 2009, Article ID 315713, 28 pages, 2009. View at Publisher · View at Google Scholar - R. Guo, “Stability analysis of a class of switched nonlinear systems with an improved average dwell time method,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 214756, 8 pages, 2014. View at Google Scholar - R. Shorten, K. S. Narendra, and O. Mason, “A result on common quadratic Lyapunov functions,”
*IEEE Transactions on Automatic Control*, vol. 48, no. 1, pp. 110–113, 2003. View at Google Scholar - A. Ibeas and M. de la Sen, “Exponential stability of simultaneously triangularizable switched systems with explicit calculation of a common Lyapunov function,”
*Applied Mathematics Letters*, vol. 22, no. 10, pp. 1549–1555, 2009. View at Google Scholar - M. S. Branicky, “Multiple Lyapunov functions and other analysis tools for switched and hybrid systems,”
*IEEE Transactions on Automatic Control*, vol. 43, no. 4, pp. 475–482, 1998. View at Google Scholar - L. Gurvits, R. Shorten, and O. Mason, “On the stability of switched positive linear systems,”
*IEEE Transactions on Automatic Control*, vol. 52, no. 6, pp. 1099–1103, 2007. View at Google Scholar - L. Fainshil, M. Margaliot, and P. Chigansky, “On the stability of positive linear switched systems under arbitrary switching laws,”
*IEEE Transactions on Automatic Control*, vol. 54, no. 4, pp. 897–899, 2009. View at Google Scholar - X. Liu and C. Dang, “Stability analysis of positive switched linear systems with delays,”
*IEEE Transactions on Automatic Control*, vol. 56, no. 7, pp. 1684–1690, 2011. View at Google Scholar - J. Qi and Y. Sun, “Global exponential stability of certain switched systems with time-varying delays,”
*Applied Mathematics Letters*, vol. 26, no. 7, pp. 760–765, 2013. View at Google Scholar - E. Fornasini and M. E. Valcher, “Linear copositive Lyapunov functions for continuous-time positive switched systems,”
*IEEE Transactions on Automatic Control*, vol. 55, no. 8, pp. 1933–1937, 2010. View at Google Scholar - Z. Wu and Y. Sun, “On easily verifiable conditions for the existence of common linear copositive Lyapunov functions,”
*IEEE Transactions on Automatic Control*, vol. 58, no. 7, pp. 1862–1865, 2013. View at Google Scholar - X. Zhao, L. Zhang, P. Shi, and M. Liu, “Stability of switched positive linear systems with average dwell time switching,”
*Automatica*, vol. 48, no. 6, pp. 1132–1137, 2012. View at Google Scholar - C. Qin, Z. R. Xiang, and H. R. Karimi, “Robust ${H}_{\infty}$ reliable control of delta operator switched systems with time-varying delays under asynchronous switching,”
*Transactions of the Institute of Measurement and Control*, 2014. View at Publisher · View at Google Scholar - Z. Wu, M. Cui, P. Shi, and H. R. Karimi, “Stability of stochastic nonlinear systems with state-dependent switching,”
*IEEE Transactions on Automatic Control*, vol. 58, no. 8, pp. 1904–1918, 2013. View at Google Scholar - Y. Kao, C. Wang, H. R. Karimi, and R. Bi, “Global stability of coupled Markovian switching reaction-diffusion systems on networks,”
*Nonlinear Analysis: Hybrid Systems*, vol. 13, pp. 61–73, 2014. View at Publisher · View at Google Scholar - H. Zhang and J. Wang, “Combined feedback-feedforward tracking control for networked control systems with probabilistic delays,”
*Journal of the Franklin Institute: Engineering and Applied Mathematics*, vol. 351, no. 6, pp. 3477–3489, 2014. View at Google Scholar - H. Zhang, Y. Shi, and J. Wang, “On energy-to-peak filtering for nonuniformly sampled nonlinear systems: a markovian jump system approach,”
*IEEE Transactions on Fuzzy Systems*, vol. 22, no. 1, pp. 212–222, 2014. View at Google Scholar - H. Zhang, Y. Shi, and J. Wang, “Observer-based tracking controller design for networked predictive control systems with uncertain Markov delays,”
*International Journal of Control*, vol. 86, no. 10, pp. 1824–1836, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. Sun, “Delay-independent stability of switched linear systems with unbounded time-varying delays,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 560897, 11 pages, 2012. View at Publisher · View at Google Scholar