`Abstract and Applied AnalysisVolume 2012 (2012), Article ID 560897, 11 pageshttp://dx.doi.org/10.1155/2012/560897`
Research Article

## Delay-Independent Stability of Switched Linear Systems with Unbounded Time-Varying Delays

School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China

Received 6 March 2012; Accepted 20 April 2012

Copyright © 2012 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

This paper is focused on delay-independent stability analysis for a class of switched linear systems with time-varying delays that can be unbounded. When the switched system is not necessarily positive, we first establish a delay-independent stability criterion under arbitrary switching signal by using a new method that is different from the methods to positive systems in the literature. We also apply this method to a class of time-varying switched linear systems with mixed delays.

#### 1. Introduction

The theory of switched systems has historically assumed a position of great importance in systems theory and has been studied extensively in recent years [16]. A switched system is a type of hybrid dynamic system that consists of a family of continuous-time (discrete-time) subsystems and a switching signal, which determines the switching between subsystems. The stability of switched linear systems under arbitrary switching signal is a very important problem, which is usually studied by a common Lyapunov functional approach, especially a common quadratic Lyapunov functional approach [710].

Very recently, the stability of positive switched linear system has attracted a lot of attention [1116]. As usual, a system is said to be positive if its state and outputs are nonnegative whenever the initial condition and inputs are nonnegative. For stability of positive switched linear system under arbitrary switching signal, a common linear copositive Lyapunov function is usually applied [1720]. A switched linear copositive Lyapunov function has been used in discrete-time positive switched systems in [21]. When the positive switched linear system involves multiple time-varying delays that can be unbounded, it has been proved in [22] that the stability of such systems under any switching signal does not depend on delays if the switched system shares a common linear copositive Lyapunov function, which generalizes the early results in [23, 24].

For the general switched linear systems with unbounded time-varying delays, it is necessary to consider whether the similar delay-independent stability criterion under arbitrary switching signal can also be derived. Note that the system is not necessarily positive; the methods to positive systems in [22] usually do not hold. Consequently, to answer this problem, we need a new approach that is different from those methods to positive systems in the literature.

The main purpose of this paper is to establish a delay-independent stability criterion under arbitrary switching signal for the general switched linear systems with time-varying delays that can be unbounded. Since the switched systems are not necessarily positive, a new method based on some smart techniques of real analysis is proposed. By using this method, we not only present a delay-independent stability criterion for the system, but also extend the main result to a class of time-varying switched system with mixed delays, where one kind of delays is time-varying state delay that can be unbounded and the others are bounded time-varying distributed delay. Another advantage of the new method used in this paper lies in that it imposes less constraint on unbounded state delays than that given in [22] (see the corresponding discussion in Section 2).

Notations. Say if all elements of matrix are nonnegative (negative). We write if and only if . Denote by the the set of Metzler matrices whose off-diagonal entries are nonnegative. is an -dimensional real vector space, is the set of positive vectors, and is the set of real -dimensional matrices. For , denote . For positive integers , , , and , denote , , , and .

#### 2. Problem Statements and Preliminaries

Consider the following switched linear system with time-varying delays: where is the state; the piecewise continuous function is the switching signal; are constant matrices for and ; time-varying delays , , , are continuous on ; is the continuous vector-valued initial function on with .

Unlike the assumptions on the system matrices [22], we here do not require and for and . What is more, we make a less restrictive assumption on time-varying delays as follows:(H1), , .

We recall to introduce another assumption on in [22] as follows:() there exist and a scaler such that .

We show that (H1) is less constrained than (). In fact, it is not difficult to see that () implies (H1). However, (H1) does not yield (). For example, let for .

Since we see that (H1) holds while () does not hold.

In the sequel, we say system (2.1) is asymptotically stable under arbitrary switching signal, if for any , there exists such that any solution of system (2.1) under arbitrary switching signal satisfies when , and .

Generally speaking, due to the less constraint on delays and system matrices for and , the methods for positive systems in the literature usually become invalid. Consequently, a new method should be introduced to analyze the delay-independent stability for system (2.1) under arbitrary switching signal.

#### 3. Main Result

In the sequel, we denote and for and , where It is easy to see that and for .

We now present the main result of this paper.

Theorem 3.1. Assume that (H1) holds. If there exists a vector such that where , then system (2.1) is asymptotically stable under arbitrary switching signal.

Proof. Denote and where for   by (3.2). The remaining proof is divided into two parts.
(i) For any constant , there exists a constant such that when . In the sequel, we denote the th element of the solution of system (2.1) by for .
In fact, for any given , let , where When , we prove that Note that for ; then By the continuity of the solution of system (2.1), we have that there exists such that
We further show that (3.5) holds if . Otherwise, there exists and at least one index such that which implies , where means the left derivative. Set the left limitation . By (2.1), (3.1), (3.3), and (3.8), we get From (3.9), we get a contradiction with the fact . Therefore, for any , by choosing and using (3.5), we have that if . This completes the proof of part (i).
(ii) For any solution of system (2.1), .
Let for . Denote the upper limitation of by and the lower limitation of by for . Set for some and
We first show that . Assume to the contrary that . Choose a sufficiently small satisfying By the definition of , we have that , , hold for sufficiently large . Since for and , we have that there exists sufficiently large such that where , and .
On the other hand, by the assumption that and the choice of , there exists a sufficiently large such that where means the right derivative. Otherwise, we have or eventually, which contradicts with the assumption . By (3.3), it is easy to see that . Denote the right limitation . By (3.3), (3.13), and (3.14), we get where by (3.12). This is a contradiction with the fact that . Therefore, .
Next, we show that . Otherwise, . Then, for sufficiently small satisfying (3.12), there exists such that (3.13) holds, and Here, (3.16) is concluded from the property of the lower limitation . Similar to the above analysis, we have Integrating (3.17) from to on both sides, we get the following contradiction: Thus, . By the choice of and the definition of , we have that for , which implies that .
By (i) and (ii), system (2.1) is asymptotically stable under arbitrary switching signal. This completes the proof of Theorem 3.1.

Remark 3.2. For the particular case when , condition (3.2) holds if and only if is a Hurwitz matrix [25]. When , it requires that all , , share a common such that . This problem has been studied in [20], where necessary and sufficient conditions for the existence of such a vector were established.

#### 4. Extension to Time-Varying Switched Systems with Mixed Delays

We now extend Theorem 3.1 to a class of time-varying switched system with mixed delays: where , are continuous matrix function on , delays are continuous on , and .

Assume that(H2) there exist constants such that for and ;(H3) there exist constant matrices and such that, for and , When , a straightforward computation based on (4.2) yields that where . Then, similar to the analysis in Theorem 3.1, it is not difficult to get the following stability criterion for system (4.1).

Theorem 4.1. Assume that (H1)–(H3) hold. If there exists a vector such that where , then system (4.1) is asymptotically stable under arbitrary switching signal.

Consider the following uncertain switched system: where and are uncertain matrices satisfying Set Then, based on the same analysis as above, we have the following result for the uncertain system (4.5).

Theorem 4.2. Assume that (H1) and (H2) hold. If there exists a vector such that (4.4) holds, then system (4.5) is asymptotically stable under arbitrary switching signal.

#### 5. A Numerical Example

To illustrate Theorem 3.1, we present a simple numerical example of system (2.1) with and for . By (3.1), we have that It is not difficult to verify that (H1) holds and there exists a vector such that for . Therefore, by Theorem 3.1, we know that system (2.1) is asymptotically stable under arbitrary switching signal. Since system (2.1) is not positive, Theorem  2 in [22] is invalid for this case. The state of the system is given in Figure 1.

Figure 1: The state of the system under stochastic initial condition and switching.

It is not difficult to work out an example of Theorem 4.1. We omit it here due to the similarity with the above example.

#### 6. Conclusion

In this paper, we investigate the delay-independent stability of the nonpositive switched linear systems with time-varying delays. By using a new method that is different from those methods to positive systems, we show that the stability of the system is also independent of delays if the switched system shares a common linear copositive Lyapunov function. We also apply this method to a class of time-varying switched linear systems with mixed delays, which generalizes some existing results in the literature.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants nos. 60704039 and 61174217 and the Natural Science Foundation of Shandong Province under Grants nos. ZR2010AL002 and JQ201119.

#### References

1. D. Liberzon and A. S. Morse, “Basic problems in stability and design of switched systems,” IEEE Control Systems Magazine, vol. 19, no. 5, pp. 59–70, 1999.
2. D. Liberzon, Switching in Systems and Control, Birkhäuser, Boston, Mass, USA, 2003.
3. Z. Sun and S. S. Ge, Switched Linear Systems: Control and Design, Springer, New York, NY, USA, 2005.
4. Z. Sun and S. S. Ge, Stability Theory of Switched Dynamical Systems, Springer, London, UK, 2011.
5. Y. G. Sun, L. Wang, G. Xie, and M. Yu, “Improved overshoot estimation in pole placements and its application in observer-based stabilization for switched systems,” IEEE Transactions on Automatic Control, vol. 51, no. 12, pp. 1962–1966, 2006.
6. Y. Sun, “Stabilization of switched systems with nonlinear impulse effects and disturbances,” IEEE Transactions on Automatic Control, vol. 56, no. 11, pp. 2739–2743, 2011.
7. D. Liberzon, J. P. Hespanha, and A. S. Morse, “Stability of switched systems: a Lie-algebraic condition,” Systems & Control Letters, vol. 37, no. 3, pp. 117–122, 1999.
8. A. A. Agrachev and D. Liberzon, “Lie-algebraic stability criteria for switched systems,” SIAM Journal on Control and Optimization, vol. 40, no. 1, pp. 253–269, 2001.
9. D. Cheng, L. Guo, and J. Huang, “On quadratic Lyapunov functions,” IEEE Transactions on Automatic Control, vol. 48, no. 5, pp. 885–890, 2003.
10. 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.
11. 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.
12. O. Mason and R. Shorten, “On linear copositive Lyapunov functions and the stability of switched positive linear systems,” IEEE Transactions on Automatic Control, vol. 52, no. 7, pp. 1346–1349, 2007.
13. M. Margaliot, “A counterexample to a conjecture of Gurvits on switched systems,” IEEE Transactions on Automatic Control, vol. 52, no. 6, pp. 1123–1126, 2007.
14. 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.
15. H. Alonso and P. Rocha, “A general stability test for switched positive systems based on a multidimensional system analysis,” IEEE Transactions on Automatic Control, vol. 55, no. 11, pp. 2660–2664, 2010.
16. X. Xue and Z. Li, “Asymptotic stability analysis of a kind of switched positive linear discrete systems,” IEEE Transactions on Automatic Control, vol. 55, no. 9, pp. 2198–2203, 2010.
17. F. Knorn, O. Mason, and R. Shorten, “On linear co-positive Lyapunov functions for sets of linear positive systems,” Automatica, vol. 45, no. 8, pp. 1943–1947, 2009.
18. O. Mason and R. Shorten, “The geometry of convex cones associated with the Lyapunov inequality and the common Lyapunov function problem,” Electronic Journal of Linear Algebra, vol. 12, pp. 42–63, 2005.
19. S. Bundfuss and M. Dür, “Copositive Lyapunov functions for switched systems over cones,” Systems & Control Letters, vol. 58, no. 5, pp. 342–345, 2009.
20. 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.
21. X. Liu, “Stability analysis of switched positive systems: a switched linear copositive Lyapunov function method,” IEEE Transactions on Circuits and Systems II, vol. 56, no. 5, pp. 414–418, 2009.
22. 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.
23. X. Liu, W. Yu, and L. Wang, “Stability analysis for continuous-time positive systems with time-varying delays,” IEEE Transactions on Automatic Control, vol. 55, no. 4, pp. 1024–1028, 2010.
24. X. Liu, W. Yu, and L. Wang, “Stability analysis of positive systems with bounded time-varying delays,” IEEE Transactions on Circuits and Systems II, vol. 56, no. 7, pp. 600–604, 2009.
25. L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications, Wiley-Interscience, New York, NY, USA, 2000.