Elsevier

Automatica

Volume 43, Issue 4, April 2007, Pages 662-668
Automatica

Brief paper
Design of switching sequences for controllability realization of switched linear systems

https://doi.org/10.1016/j.automatica.2006.10.010Get rights and content

Abstract

In this paper, we study the problem of designing switching sequences for controllability of switched linear systems. Each controllable state set of designed switching sequences coincides with the controllable subspace. Both aperiodic and periodic switching sequences are considered. For the aperiodic case, a new approach is proposed to construct switching sequences, and the number of switchings involved in each designed switching sequence is shown to be upper bounded by d(d-d1+1). Here d is the dimension of the controllable subspace, d1=dimi=1mAi|Bi, where (Ai,Bi) are subsystems. For the periodic case, we show that the controllable subspace can be realized within d switching periods.

Introduction

As a special class of hybrid control systems, switched systems have attracted considerable attention during the last decade because of its importance from both theoretical and practical points of view (Cheng et al., 2005, Sun and Ge, 2005). In spite of some existing results, the study of the stability analysis and design of switched systems is just under way and the theoretical framework is far from complete. In particular, the switching mechanism has not been fully understood, and a closely related problem is to design switching sequences with switching times as less as possible. There are excellent survey papers such as Sun and Ge (2005), Liberzon and Morse (1999) and DeCarlo, Branicky, Pettersson, and Lennartson (2000).

In this paper, we are interested in designing switching sequences to achieve the controllability for switched linear systems, i.e., each controllable state set of designed switching sequences coincides with the controllable subspace of switched linear systems. We call this problem a controllability realization problem. Our goal is to design switching sequences to realize the controllability with the number of switchings as small as possible. To this end, a constructive approach is proposed for the design purpose. We will show that the upper bound for the length of each designed switching sequence is d(d-d1+1). Here d is the dimension of controllable subspace, d1=dimi=1mAi|Bi, where (Ai,Bi) are subsystems. For periodic switching sequences, we will prove that the controllability can always be realized within d switching periods. Comparison of our results with existing ones verifies the advantage of our design approach.

The controllability realization problem was formerly studied by Sun, Ge, and Lee (2002) and Xie and Wang (2003). A common feature for the switching sequences designed by them is that the number of switchings increases rapidly with the number of subsystems and dimensions. In this paper, we reduce the required number of switchings significantly to an acceptable level for both cases of aperiodic and periodic switchings. Furthermore, we will show that for the design of aperiodic switching sequences, the maximum number of switchings is only determined by the dimension of the controllable subspace.

Besides controllability realization, the design of switching sequences/rules has been widely employed for stabilization problems. Wicks, Peleties, and DeCarlo (1998) firstly developed an elegant construction of a stabilizing switching rule. The result therein was later used for switched controller design and robustness study by Sun (2005), Zhao and Dimirovski (2004), Ji, Wang, Xie, and Hao (2004), Ji, Wang, and Xie (2005), Zhai, Lin, and Antsaklis (2003), Bacciotti (2004), King and Shorten (2005) and Ishii, Basar, and Tempo (2005). Another stabilization method is to constrain switching sequences to satisfy a dwell or an average-dwell-time condition. This method was firstly proposed by Morse (1997), and further developed by Hespanha and Morse (1999), Hespanha (2004) and Zhai, Hu, Yasuda, and Michel (2001). Recently, Cheng et al. (2005) put forward a concept of switching frequency which can be deemed as a development of average-dwell-time method. Our proposed approach enriches constructive methods for switching sequences.

Controllability and observability play a fundamental role in the design and synthesis of linear control systems. For switched linear systems, complete geometric criteria for controllability and reachability were established by Sun et al. (2002) and Xie and Wang (2003). The controllability of switched bilinear systems was investigated by Cheng (2005) using Lie algebraic technique. Other necessary and/or sufficient conditions for controllability and reachability were presented by Ezzine and Haddad (1989), Krastanov and Veliov (2005), Petreczky (2006), Yang (2002), Stikkel, Bokor, and Szabó (2004), Xie and Wang (2004), Meng and Zhang (2006), Sun (2004), Ge, Sun, and Lee (2001), Stanford and Conner (1980) and Conner and Stanford (1987), etc. In our opinion, although necessary and sufficient conditions on controllability and reachability were established, the switching behaviors for controllability has not been fully investigated. This is our main purpose in the paper.

The paper is organized as follows. In Section 2, we present preliminaries including system description, definitions and supporting lemmas. Main results and comparisons with existing ones are presented in Section 3. An illustrative example is included in Section 4. Finally, Section 5 briefly concludes the work.

Section snippets

Preliminaries

Consider a switched linear control system given byx˙(t)=Aσx(t)+Bσu(t),where x(t)Rn is the state, u(t)Rp is the input, and σ(t):[t0,){1,,m} is the switching sequence to be designed. Moreover, σ(t)=i implies that the ith subsystem (Ai,Bi) is activated.

Given a switching sequence σ:[t0,tf]{1,,m}, suppose its discontinuous (jump) time instants are t0<t1<<ts-1. We refer to the sequence t0,t1,,ts-1 as switching time sequence, and the sequence σ(t0)=i0,σ(t1)=i1,,σ(ts-1)=is-1 as switching

Aperiodic switching sequences

Let L(π) be the length of switching sequence π. Obviously, L(π) indicates the number of switchings involved in π. Let μ=min{j|dimWj=dimWj+1,j=1,2,}, and d1=dimW1. The following result reveals the relationship between the required number of switchings and the dimension of controllable state subspace C.

Theorem 1

For system(1), there exist switching sequencesπk,k=1,2,,with length of each one satisfyingL(πk)d(d-d1+1),such that for eachπk,C(πk)=Wn=C,where d and n are the dimensions ofWnand the state,

An illustrative example

Example 1

Consider the switched system (1) with m=n=3, andA1=010000000,b1=03×1;A2=03×3,b2=[0,1,0]T,A3=03×3,b3=[0,0,1]T.It can be verified that d1=dimW1=2,W2=W3=R3. Hence, the controllable subspace C is R3,μ=2, and d=3. After some computations, we haveW2=span010,001,-h10h>0=span{ξ1,ξ2,ξ3;h>0}.Sinceξ1A2|b2C(π1),π1={(2,h)},ξ2A3|b3C(π2),π2={(3,h)},ξ3e-A1hA2|b2C(π3),π3={(1,h)(2,h)},it follows from the proof of Theorem 1 that the desired switching sequence is π(h)=π1π2π3={(2,h)(3,h)(1,h)(2,h)}.

Conclusion

An approach is proposed in this paper to design aperiodic switching sequences with much less switchings than the existing approaches in literature. The result shows a relationship between the number of switchings and the dimension of the controllable subspace. We also show that it is sufficient to employ periodic switchings to realize controllability within d switching periods, where d is the dimension of the controllable subspace. The present results bring a deep understanding on switching

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 60604032, 60674050, 10601050, 60528007) and National 973 Program (2002CB312200). The authors would like to thank the three reviewers for their constructive and insightful suggestions for further improving the quality and presentation of this paper.

Zhijian Ji received the M.S. degree in Applied Mathematics from Ocean University of China in 1998, and the Ph.D. degree in system theory from Intelligent Control Laboratory, Center for Systems and Control, Department of Mechanics and Engineering Science, Peking University, Beijing, China in 2005. He is currently an Associate Professor with School of Automation Engineering, Qingdao University. His current research interests are in the fields of nonlinear control systems, switched and hybrid

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  • Cited by (0)

    Zhijian Ji received the M.S. degree in Applied Mathematics from Ocean University of China in 1998, and the Ph.D. degree in system theory from Intelligent Control Laboratory, Center for Systems and Control, Department of Mechanics and Engineering Science, Peking University, Beijing, China in 2005. He is currently an Associate Professor with School of Automation Engineering, Qingdao University. His current research interests are in the fields of nonlinear control systems, switched and hybrid systems, networked systems, and swarm dynamics. He was the winner of the First-class Scholarship of China Petrol in 2003, the Academic Innovation Award and the May Fourth Scholarship of Peking University in 2004.

    Long Wang was born in Xian, China on Feb. 13, 1964. He received his Bachelor's, Master's, and Doctor's degrees in Dynamics and Control from Tsinghua University and Peking University in 1986, 1989, and 1992, respectively. He has held research positions at the University of Toronto, Canada, and the German Aerospace Center, Munich, Germany. He is currently Cheung Kong Chair Professor of Dynamics and Control and Director of Center for Systems and Control of Peking University. He is also Vice-Director of National Key Laboratory of Complex Systems and Turbulence. He is a panel member of the Division of Information Science, National Natural Science Foundation of China. He is on the editorial boards of Progress in Natural Science, Acta Automatica Sinica, Journal of Control Theory and Applications, Control and Decision, Information and Control, etc. His research interests are in the fields of networked systems, hybrid systems, swarm dynamics, cognitive science, collective intelligence, and bio-mimetic robotics.

    Xiaoxia Guo received her M.S. degree in Applied Mathematics from Ocean University of China in 1997, and the Ph.D. degree in Computational Mathematics from Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Science in 2005. She is now with Department of Mathematics, Ocean University of China. Her research interests are in the fields of Numerical Computation, and matrix equations in control theory.

    This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor teD Iwasaki under the direction of Editor R. Tempo.

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