Stabilization of switched continuous-time systems with all modes unstable via dwell time switching

doi:10.1016/j.automatica.2013.12.028

Automatica

Volume 50, Issue 3, March 2014, Pages 940-945

https://doi.org/10.1016/j.automatica.2013.12.028 Get rights and content

Abstract

Stabilization of switched systems composed fully of unstable subsystems is one of the most challenging problems in the field of switched systems. In this brief paper, a sufficient condition ensuring the asymptotic stability of switched continuous-time systems with all modes unstable is proposed. The main idea is to exploit the stabilization property of switching behaviors to compensate the state divergence made by unstable modes. Then, by using a discretized Lyapunov function approach, a computable sufficient condition for switched linear systems is proposed in the framework of dwell time; it is shown that the time intervals between two successive switching instants are required to be confined by a pair of upper and lower bounds to guarantee the asymptotic stability. Based on derived results, an algorithm is proposed to compute the stability region of admissible dwell time. A numerical example is proposed to illustrate our approach.

Introduction

The stability issue is the main concern in the field of switched systems, which have been extensively studied in the literature (Branicky, 1998, Daafouz et al., 2002, Decarlo et al., 2000, Geromel and Colaneri, 2006, Lee and Dullerud, 2007, Liberzon, 2003, Lin and Antsaklis, 2009, Margaliot, 2006, Shorten et al., 2007, Sun and Ge, 2005). Most of the reported results are confined to the case where there exist stable subsystems within switched systems. In the early work, the research results mainly focused on the switched systems composed fully of stable modes (Allerhand and Shaked, 2011, Chesi et al., 2010, Hespanha et al., 1999, Morse, 1996). In recent years, some advances have been reached to deal with the case when there exist some unstable modes such as Xiang and Xiang (2009), Xiang and Xiao (2012), Xiang, Xiao, and Iqbal (2012), Zhai et al., 2000, Zhai et al., 2001, Zhai et al., 2002 and Zhang and Shi, 2009, Zhang and Shi, 2010, but it should be noted that these results also require the existence of (at least one) stable subsystem to ensure the stability of the whole switched system. The main idea of these results is to activate the stable modes for sufficiently long to absorb the state divergence made by unstable modes. But, when all the subsystems are unstable, this promising idea obviously fails, since there exists no stable period to compensate the state divergence effect.

As is well known, even if all subsystems are unstable, one may carefully switch between unstable modes to make the switched system asymptotically stable, and how to design appropriate switching laws to stabilize the switched system composed fully of unstable subsystems is one of the most interesting and serious challenges for switched systems (Decarlo et al., 2000, Liberzon, 2003, Lin and Antsaklis, 2009, Sun and Ge, 2005). This problem has been extensively studied for years, e.g. Li, Wen, and Soh (2001), Margaliot and Langholz (2003), Pettersson (2003), Pettersson and Lennartson (2001), Wicks, Peleties, and DeCarlo (1998), most of them resort to state-dependent switching strategies such as the min-projection strategy (Pettersson & Lennartson, 2001), largest region function strategy (Pettersson, 2003), etc., but very few results focus on the time-dependent switching law particularly concerned with dwell time, which motivates the present study. Since the previous idea based on the presence of a stable subsystem is not applicable for the case with all subsystems unstable, we have to find another way to establish stability. On the other hand, since an appropriate switching law can stabilize the system, even though all subsystems are unstable, this implies that switching behaviors can also contain a good characteristic of stabilization in some circumstances, e.g. see the examples in Branicky (1998) and Sun and Ge (2005).

For most of the previous results, the switching behavior has been viewed as a bad factor only destroying stability, such as the famous (average) dwell time technique (Hespanha et al., 1999, Morse, 1996). However, since an appropriate switching law can stabilize the system, it implies switching behaviors can also contain a good characteristic of stabilization in some circumstances. In this brief paper, when all modes are unstable, a sufficient condition ensuring the switched system asymptotically stable is proposed. Then, in order to derive computable ways to characterize the stabilization property of switching behavior and cover the results in the familiar conception called dwell time, the discretized Lyapunov function technique (Gu, Kharitonov, & Chen, 2003) is applied to the linear case. It is interesting to see that the time interval between two successive switching instants should be confined by a pair of upper and lower bounds to guarantee the asymptotic stability, which can be viewed as an extension of Allerhand and Shaked (2011), from the case composed of stable subsystems to the case fully composed of unstable subsystems. Finally, an algorithm is proposed to compute the admissible upper and lower bounds and determine the stability region for the dwell time.

This paper is organized as follows: Some preliminaries are introduced in Section 2. The stability analysis for a switched system with all subsystems unstable is presented in Section 3, and the main contributions, the computable condition for the switched linear system and computation on the stability region for the admissible dwell time are presented in Section 4. Then, a numerical example is provided in Section 5. Conclusions are given in Section 6.

Section snippets

Preliminaries

Let $R$ denote the field of real numbers, $R_{\geq 0}$ stand for non-negative real numbers, and $R^{n}$ be the $n$ -dimensional real vector space. $| \cdot |$ stands for the Euclidean norm. Class $K$ is a class of strictly increasing and continuous functions $[0, \infty) \to [0, \infty)$ which is zero at zero. Class $K_{\infty}$ denotes the subset of $K$ consisting of all those functions that are unbounded. The notation $P > 0 (P \geq 0)$ means $P$ is real symmetric and positive definite (semi-positive definite). $I$ stands for the identity matrix with appropriate

Stability analysis

It has been well recognized that the multiple Lyapunov function (MLF) $V_{i} (t), i \in M$ is a popular stability analysis tool for switched systems, especially under dwell time constrained switching (Hespanha et al., 1999, Morse, 1996). At each switching instant $t_{n}$ from mode $i$ to $j$ , the switching always causes a bounded increment of $V_{i} (t)$ which is described by $V_{j} (t_{n}) < μ V_{i} (t_{n}), i \neq j, \forall i, j \in M$ , where $μ > 1$ . When unstable subsystems are involved, a class of Lyapunov functions $V_{i} (t)$ are allowed to increase with

Problem description

Consider the switched linear system in the form of $\dot{x} (t) = A_{σ (t)} x (t)$ where $x (t)$ and $σ (t)$ are described the same as in (1). Without loss of generality, all the subsystem matrices $A_{i}, i \in M$ are supposed to have eigenvalues located in the right half-plane. In the framework of dwell time, the following intuitive observations can be seen.

The basic nature of a stable switched system composed fully of unstable modes is that the state trajectories driven by each activated mode diverge to different

Example

Consider system (7) composed of two subsystems as: $A_{1} = [\begin{matrix} - 1.9 & 0.6 \\ 0.6 & - 0.1 \end{matrix}], A_{2} = [\begin{matrix} 0.1 & - 0.9 \\ 0.1 & - 1.4 \end{matrix}] .$ The eigenvalues of $A_{1}$ are $λ_{1, 1} = 0.0817$ and $λ_{1, 2} = - 2.0817$ , and eigenvalues of $A_{2}$ are $λ_{2, 1} = 0.0374$ and $λ_{2.2} = - 1.3374$ . Obviously, they are unstable since both of them have eigenvalues located in the right half-plane.

(1) Given the initial state as $x (0) = {[3 5]}^{T}$ and a periodical switching sequence $S$ satisfies $t_{n + 1} - t_{n} = 2, n = 0, 1, 2, \dots$ . Therefore, we see that $τ_{\min} = τ_{\max} = 2$ . Then, by (11), (12), (13), (14), (15), if we fix $L = 1$

Conclusions

The stability problem of a switched continuous-time system with all modes unstable is addressed in this paper. By exploiting the stabilization property of switching behavior, a sufficient condition is proposed. Then, the discretized Lyapunov function technique is introduced to particularly study the linear case, a computable sufficient condition is presented which relates to a pair of upper and lower bounds of dwell time. An algorithm is provided to compute the upper and lower bounds, and the

Acknowledgments

The authors would like to thank the reviewers for their helpful comments and suggestions which have helped improve the presentation of the paper.

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Weiming Xiang received his B.S. degree from the Department of Electrical Engineering, East China Jiaotong University, Nanchang, China, in 2005 and his M.S. degree from the Department of Automation, Nanjing University of Science and Technology, Nanjing, China, in 2007. Since 2007, he has been a lecturer at the Southwest University of Science and Technology, Mianyang, China, where he teaches system and control theory. He is currently pursuing a Ph.D. degree from the Department of Transportation and Logistics, Southwest Jiaotong University, Chengdu, China. His current research interests are in the area of switched systems and control, robust control and filtering, nonlinear systems and control, fuzzy systems and transportation systems.

Jian Xiao received his B.S. and M.S. degrees in Electrical Engineering from Hunan University, China, in 1982 and his Ph.D. degree from Southwest Jiaotong University, China, in 1989, and then worked as a postdoctoral researcher at the Polytechnic University of Milan, Italy, during 1991–1993. He has been the professor of Electrical Engineering at the Southwest Jiaotong University since 1994. He is currently also a guest professor for Hunan University, East China Jiaotong University, the University of Leoben and Lanzhou Jiaotong University. Currently, he serves as Vice President of the Sichuan Society of Automation and Instrumentation, Member of the Board of Directors of the Chinese Instrument and Automation Society, Member of the Board of Directors of the Chinese Automation Association, Commissary of the Expert Consultative Committee of the Chinese Automation Association, and Chairman of the Theoretic Electrical Engineering Commission, Sichuan Society of Electrical Engineering. He is an editorial member of the Journal of Information and Electronics, and Automation Information. He has authored/co-authored over 200 papers and four books. His research interests include robust control, fuzzy systems and power electronics.

^☆: This work is supported by the National Natural Science Foundation of China (51177137, 61134001). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Constantino M. Lagoa under the direction of Editor Roberto Tempo.

¹: Tel.: +86 13990130910; fax: +86 816 6332139.

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Brief paperStabilization of switched continuous-time systems with all modes unstable via dwell time switching☆

Abstract

Introduction

Section snippets

Preliminaries

Stability analysis

Problem description

Example

Conclusions

Acknowledgments

Automatica

European Journal of Control

Journal of the Franklin Institute

Automatica

Robust stability and stabilization of linear switched systems with dwell time

IEEE Transactions on Automatic Control

Multiple Lyapunov functions and other analysis tools for switched and hybrid systems

IEEE Transactions on Automatic Control

A non-conservative LMI condition for stability of switched systems with guaranteed dwell time

IEEE Transactions on Automatic Control

Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach

IEEE Transactions on Automatic Control

Perspectives and results on the stability and stabilization of hybrid systems

Proceedings of IEEE

Stability and stabilization of discrete-time switched systems

International Journal of Control

Stability of time-delay systems

Uniformly stabilizing sets of switching sequences for switched linear systems

IEEE Transactions on Automatic Control

Stabilization of a class of switched systems via designing switching laws

IEEE Transactions on Automatic Control

Brief paper
Stabilization of switched continuous-time systems with all modes unstable via dwell time switching☆