Controllability and reachability criteria for switched linear systems☆
Introduction
During the last decade, hybrid and switched systems have attracted considerable attention (Chase, Serrano, & Ramadge, 1993; Branicky, 1998; Wicks, Peleties, & DeCarlo, 1998; Ye, Michel, & Hou, 1998; Liberzon & Morse, 1999). Basically, a switched system consists of continuous-time/discrete-time dynamical subsystems and a rule (supervisor) that determines the switching among them.
Switched systems deserve investigation for theoretical reasons as well as for practical reasons. Switching among different system structures is an essential feature of many engineering control applications including power systems and power electronics (Williams & Hoft, 1991; Sira-Ramirez, 1991), and switched systems have numerous applications in control of mechanical systems, air traffic control, aircrafts and satellites and many other fields (Li, Wen, & Soh, 2001). Control techniques by switching among different controllers have been applied extensively in recent years. Indeed, a switched controller can provide a performance improvement over a fixed controller (Morse, 1996; Narendra & Balakrishnan, 1997; Savkin, Skafidas, & Evans, 1999). The switched controller architecture was proven to be a rigorous design framework for general nonlinear systems (Kolmanovsky & McClamroch, 1996; Caines & Wei, 1998; Leonessa, Haddad, & Chellaboina, 2001). A switched controller can also achieve certain control object which cannot be accomplished by conventional methods, such as pure feedback stabilization of nonholonomic systems (Brockett, 1983; Kolmanovsky & McClamroch, 1995).
A fundamental pre-requisite for the design of feedback control systems is full knowledge about the structural properties of the switched systems under consideration. These properties are closely related to the concepts of controllability, observability and stability which are of fundamental importance in the literature of control. There have been a lot of studies for switched systems, primarily on stability analysis and design (Branicky, 1998; Dayawansa & Martin, 1999; Liberzon & Morse, 1999). As for controllability and reachability, studies for low-order switched linear systems have been presented in Loparo, Aslanis, and IIajek (1987) and Xu and Antsaklis (1999). Some sufficient conditions and necessary conditions for controllability were presented in Ezzine and Haddad (1989) and Szigeti (1992) for switched linear control systems under the assumption that the switching sequence is fixed a priori. The complexity of stability and controllability of hybrid systems was addressed in Blondel and Tsitsiklis (1999).
For controllability analysis of switched linear control systems, a much more difficult situation arises since both the control input and the switching rule are design variables to be determined, and thus the interaction between them must be fully understood. For a switched linear discrete-time control system, the controllable set is not a subspace but a countable union of subspaces in general case (Stanford & Conner, 1980; Conner & Stanford, 1987; Ge, Sun, & Lee, 2001). For a switched linear continuous-time control system, the controllable set is an uncountable union of subspaces (Sun & Zheng, 2001).
In this paper, we investigate the controllability and reachability issues for switched linear control systems in detail. We prove that, both the controllable set and the reachable set are subspaces of the total space, and the two sets always coincide with each other. Verifiable geometric characterization is presented for the controllable subspace. Dualistic criteria for observability and determinability are also presented.
The paper is organized as follows. In Section 2, we present the definitions of controllable and reachable notions. Preliminary results are given in Section 3. A complete characterization for the controllability and reachability sets is presented in Section 4. In Section 5, we briefly address the observability and determinability issues. An illustrative example is presented in Section 6. Finally, some concluding remarks are made in Section 7.
Section snippets
Definitions
Consider a switched linear control system given bywhere are the states, , k=1,…,m are piecewise continuous input functions, is the switching path to be designed, and matrix pairs (Ak,Bk) for k∈M are referred to as the subsystems of (1).
Given a switching path , suppose its discontinuous (jump) time instants are t1<t2<⋯<ts, we refer to the sequence t0,t1,…,ts as switching time sequence, and the sequence σ(t0),σ(t1),…,σ(ts) as
Elementary analysis
Given an initial state x(t0)=x0, inputs , and a switching path , the solution of state equation (1) is given bywhere t0,t1,…,ts is the switching time sequence of σ, ts+1=tf, and i0=σ(t0),…,is=σ(ts) is the switching index sequence of σ.
The reachable set of system (1) is given by
Geometric criteria
In this subsection, we shall identify the controllable set and the reachable set for switched linear systems. Theorem 1 For switched linear system (1), the reachable set is Proof We are to design a switching path σ such that each state in can be reached from the origin via this switching path. Assume that the switching index sequence of σ is periodic, i.e.,The switching time sequence t0,…,tl and the number l are to be designed later. Let tf>tl. From (4), the
Observability and determinability
In the above analysis, reference is made to reachability and controllability only. It should be noticed that the observability and determinability counterparts can be addresses dualistically. In this section, we outline the relevant concepts and the corresponding criteria.
Consider a switched linear control system with outputs given bywhere , and are the states, inputs and outputs, respectively, is the switching path to be
An illustrative example
Example 2 Consider the switched systems given bywhere ej, 1⩽j⩽n is the unit column vector with the jth entry equal to one.
To compute the controllable subspace , we follow the procedure presented in Section 4.3.
It can be readily seen thatBy searching the independent vectors inwe obtain thatContinue this process, we havefor k=2,…,m−1, and
Conclusion
In this paper, detailed controllability and reachability analysis has been carried out for switched linear control systems. It has been proven that, both the controllable and reachable sets are subspaces of the total space, and the two sets always coincide with each other. The controllable subspace is exactly the minimal Ak-invariant subspace for k∈M which contains . Criteria for observability and determinability have also been obtained by duality. These results generalize Wonham's
Acknowledgements
The authors would like to thank the anonymous reviewers for their constructive and insightful comments for further improving the quality of this work. Z. Sun was partially supported by the National Science Foundation of China under Grant 60104002 and partially supported by the National Key Basic Research Development Project (973) of China under Grant G1998020309.
Zhendong Sun received the B.S. degree in Applied Mathematics from Ocean University of Qingdao (China) in 1990, the M.S. in Systems Science from Xiamen University (China) in 1993, and the Ph.D. degree in Electrical Engineering from Beijing University of Aeronautics and Astronautics (China) in 1996.
During 1996–1998, he was a postdoctoral research associate in Department of Automation, Tsinghua University, China. In 1998, he joined the faculty of Science, Beijing University of Aeronautics and
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Zhendong Sun received the B.S. degree in Applied Mathematics from Ocean University of Qingdao (China) in 1990, the M.S. in Systems Science from Xiamen University (China) in 1993, and the Ph.D. degree in Electrical Engineering from Beijing University of Aeronautics and Astronautics (China) in 1996.
During 1996–1998, he was a postdoctoral research associate in Department of Automation, Tsinghua University, China. In 1998, he joined the faculty of Science, Beijing University of Aeronautics and Astronautics, as an associate professor. From 2000 to 2001, he had been with the National University of Singapore as a research fellow.
Dr. Sun's current research interests are in the fields of nonlinear control systems, switched and hybrid systems, and sampled data systems. He was the winner of the Guan Zhao-Zhi Award at the Chinese Control Conference (HongKong, China) in 2000.
S.S. Ge received the B.Sc. degree from Beijing University of Aeronautics and Astronautics (BUAA), Beijing, China, in 1986, and the Ph.D. degree and the Diploma of Imperial College (DIC) from Imperial College of Science, Technology and Medicine, University of London, in 1993. From May 1992 to June 1993, he did his postdoctoral research at Leicester University, England. He has been with the Department of Electrical & Computer Engineering, the National University of Singapore since 1993, and is currently as an Associate Professor. He was a visiting staff in Laboratoire de'Automatique de Grenoble, France in 1996, the University of Melbourne, Australia in 1998, 1999, University of Petroleum and Shanghai Jiaotong University, China in 2001.
Dr. Ge has authored and co-authored over 100 international journal and conference papers, two monographs and co-invented two patents. He served as an Associate Editor on the Conference Editorial Board of the IEEE Control Systems Society in 1998 and 1999, has been serving as an Associate Editor, IEEE Transactions on Control Systems Technology since June 1999, and a Member of the Technical Committee on Intelligent Control of the IEEE Control System Society since 2000. He was the winner of the 1999 National Technology Award, Singapore. He serves as a technical consultant local industry. He is currently a Senior Member of IEEE. His current research interests are Control of nonlinear systems, Neural Networks and Fuzzy Logic, Robot Control, Real-Time Implementation, Path Planning and Sensor Fusion.
T.H. Lee received the B.A. degree with First Class Honours in the Engineering Tripos from Cambridge University, England, in 1980; and the Ph.D. degree from Yale University in 1987. He is a tenured Professor in the Department of Electrical and Computer Engineering at the National University of Singapore. He is also currently Head of the Drives, Power and Control Systems Group in this Department, and the Vice-Dean (Research) in the Faculty of Engineering.
Dr. Lee's research interests are in the areas of adaptive systems, knowledge-based control and intelligent mechatronics. He has published extensively in these areas, and currently holds Associate Editor appointments in Automatica; the IEEE Transactions in Systems, Man and Cybernetics; Control Engineering Practice (an IFAC journal); the International Journal of Systems Science (Taylor and Francis, London); and Mechatronics journal (Oxford, Pergamon Press). Dr. Lee was a recipient of the Cambridge University Charles Baker Prize in Engineering.
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Ian Petersen under the direction of Editor Roberto Tempo.
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Present address: The Seventh Research Division, Beijing University of Aeronautics and Astronautics, Beijing 100083, China.