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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 134296, 7 pages
http://dx.doi.org/10.1155/2013/134296
Research Article

Stabilization of Discrete-Time Planar Switched Linear Systems with Impulse

1School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China
2School of Automation and Electrical Engineering, University of Jinan, Jinan, Shandong 250022, China

Received 20 January 2013; Accepted 3 April 2013

Academic Editor: Hua Su

Copyright © 2013 Yanli Zhu 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 stabilization problem of discrete-time planar switched linear systems with impulse. When all subsystems are controllable, based on an explicit estimation on the state transition matrix, we establish a sufficient condition such that the switched impulsive system is stabilizable under arbitrary switching signal with given switching frequency. When there exists at least one uncontrollable subsystem, a sufficient condition is also given to guarantee the stabilization of the switched impulsive system under appropriate switching signal.

1. Introduction

Recent years have witnessed a rapid progress for switched systems, for example, see monographs [13] and survey papers [4, 5]. As usual, a switched system means 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. It is well known that switched systems have a deep background in engineering such as computer disk system [6], robotics [7], power systems [8], air traffic management [9], and automated vehicles [10].

During the last three decades, there is an increasing interest on the stability analysis for switched systems. For stability issues; one important problem is to find conditions that guarantee asymptotic stability of the switched system for arbitrary switching signal. Such a problem is usually studied by using a common Lyapunov functional approach, especially by using a common quadratic Lyapunov functional approach [1114]. A multiple Lyapunov functional method was used to study the stability of switched systems with delays in [15].

For systems that switch among a finite set of controllable linear systems, the stabilization problem of continuous-time switched systems with arbitrary switching frequency was studied in [16, 17] by developing an improved estimation on transition matrices. Very recently, the results in [16, 17] were further extended to switched systems with impulses and perturbations [18]. So far, the stability and stabilization problems for switched systems were studied in [1931], to name a few.

In this paper, motivated by the work in [1618], we study the stabilization problem of discrete-time planar switched linear systems under impulse and arbitrary switching signal with given switching frequency. When all subsystems are controllable, we obtain a discrete analogue of the main result in [17]. We also consider the case when there exist both controllable subsystems and uncontrollable subsystems. Before giving our main results, we first establish an estimation on the transition matrix for each controllable subsystem, which plays a key role in the stabilization problem of the switched system. For the uncontrollable subsystems, an estimation on the solution is given by using the Lyapunov functional approach. Then, we show that the discrete-time switched impulsive system is also stabilizable under appropriate switching signal when there exist uncontrollable subsystems.

This paper is organized as follows. In Section 2, some preliminaries are formulated. The main results of this paper are given in Section 3. Two examples are worked out in Section 4 to illustrate the main results. Section 5 concludes the paper.

2. Preliminaries

Consider the following planar discrete-time linear system: where is the state, is the controlled input, and and are matrices of appropriate dimensions.

Under the following linear feedback law: the solution of the system (1) takes the form where is called the transition matrix.

When the system (1) is controllable, we first establish an estimation on the transition matrix .

Lemma 1. Let and be constant matrices such that the pair is controllable. Then, for any , there exists a matrix such that where is a constant, which is independent of and can be estimated precisely in terms of and .

Proof. First, we consider the case of single input, that is, . Noting that is controllable, we can choose a feedback matrix such that eigenvalues of satisfy
In particular, for any , we can choose
Set where , , are determined by
Let
We have that , , are also eigenvalues of , and is in controller canonical form. Let
It is not difficult to see that
that is,
It implies that
Consequently,
Let and . First, we have
Second, noting that , we get from (6) that
So,
Since substituting (15), (17), and (18) into (14) yields that where , which is independent of . Therefore, we have that Lemma 1 holds for the single input case.
For the multiple-input case, one sees that for any such that , there exists such that is itself controllable. Hence, the conclusion of the single-input case that has been proved above is applicable to the controllable pair . Therefore, for any , there exists such that
The proof of Lemma 1 is completed.

When is uncontrollable, for any given feedback , there always exist a positive-definite symmetric matrix and an appropriate constant such that which can be solved by using the GEVP solver in the LMI Toolbox of MATLAB [32].

Define the following Lyapunov function:

It is easy to see that where and denote the smallest and the largest eigenvalue of the positive definite symmetric matrix .

Lemma 2. For system (1), if the pair is uncontrollable and (21) holds, then for any given feedback matrix , there exists a constant such that

Proof. Let the Lyapunov function be defined by (22). Along the solution of system (1), we have
By (21) and (25), we obtain which implies that
By (23) and (27), we have
Thus,
By induction, we have
This completes the proof of Lemma 2.

3. Stabilization of Discrete-Time Planar Switched Impulsive Systems

Now, we study the stabilization of the following discrete-time switched linear system: where is the state, is the input, and is a switching signal for some positive integer , which is a piecewise constant function. When , system (31) switches to the th subsystem. Moreover, is a constant matrix, representing the impulse effect on the system at the switching time. Moreover,   denote the discontinuous points (or switching points) of , and denote . and , , are system matrices of appropriate dimensions.

Throughout this paper, we assume that(H1), where is a positive constant.

Under the linear feedback law for , , system (31) reduces to the following closed-loop system:

Denote the frequency of the switching signal by where is the number of activated subsystems on . If is controllable for , we have the following result.

Theorem 3. Assume that (H1) and (H2) hold and is controllable for . Then, there exist a set of feedback matrices such that the closed-loop system (32) is asymptotically stable for any switching signal with a frequency .

Proof. For any , assume that for some positive integer . Note that . By the definition of , we can choose a constant such that for sufficiently large . Without loss of generality, we assume that for . Set . It is easy to see that . Let and   be sufficiently small such that . For such a choice of , by Lemma 1, there exist a set of feedback matrices such that for any , where . For any , we have
Since , that is, , we obtain
By the analysis and (H1), we obtain
Noting that and , we have that system (31) is stabilizable under arbitrary switching signal with a frequency . This completes the proof of Theorem 3.

Next, we consider the case when there exist both controllable subsystems and uncontrollable subsystems for system (31). For the sake of convenience, we suppose that(H2) are uncontrollable subsystems and are controllable subsystems, where .

Denote the switching frequency of those controllable subsystems by where is the number of activated controllable subsystems on . Denote the total activation time for those controllable subsystems on by . In this paper, we assume that there exists a constant such that

Similar to the above analysis, for any given feedback matrices , there exist positive definite symmetric matrices and positive constants such that

Set

Theorem 4. Assume that (H2), (39), and (40) hold. Then, there exist a set of feedback matrices such that the closed-loop system (32) is asymptotically stable for any switching signal with a frequency .

Proof. For any , assume that for some positive integer . Denote the number of activated controllable subsystems in by . Since . By the definition of , there exists a constant such that for without loss of generality. Set . It is easy to see that since . Let the constant satisfies where is determined by (41) under any given feedback matrices .
For any , assume that , and . By (35), we have
If , by (34) and (H1), we have
If , by Lemma 2, (41), and (H1), we have
Based on the same analysis, there exists a feedback matrix for each such that
We now choose sufficiently small such that . Under the feedback law for , we get from (39) and (44)–(46) by induction that
Noting that and , we have that system (32) is stabilizable under arbitrary switching signal with a frequency . This completes the proof of Theorem 4.

4. Examples

In order to illustrate the theoretical result, we consider two examples.

Example 1. Consider the switched systems (31), with , and

It is not difficult to verify that is controllable for .

By Lemma 1, we get . We choose , then the closed-loop poles of (32) are and . We have the feedback matrices

Let and the switching frequency . Based on the proof of Theorem 3, we can choose that , then

We can get . By Theorem 3, the system (32) is asymptotically stable with .

Example 2. Consider the switched systems (31), with , and
It is not difficult to verify that is controllable for , and is uncontrollable.
By Lemma 1, we get . Choose , we have the feedback matrices
Let and . The switching frequency , . For given . Based on the (40), we can choose and , then
By Theorem 4, we have then
By Theorem 4, the system (32) is asymptotically stable with .

5. Conclusion

In this paper, the stabilization problem of discrete-time planar switched linear systems with impulse is investigated. When all the subsystems are controllable, we first establish an estimation on the transition matrix for each controllable subsystem, which is a discrete analogue of the corresponding result in [17]. By using such an estimation, we prove that the discrete-time switched impulsive system is stabilizable under arbitrary switching signal with a given switching frequency. When there exists at least one uncontrollable subsystem, by using a Lyapunov functional approach, we show that the stabilizability of the switched impulsive system can be retained for the appropriate switching frequency of those controllable subsystems.

Acknowledgments

The authors thank the reviewers for their helpful and valuable comments on this paper. This work was supported by the Natural Science Foundation of Shandong Province under Grant nos. JQ201119 and ZR2010AL002 and the National Natural Science Foundation of China under Grant no. 61174217.

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