1486 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006 Accessibility of Switched Linear Systems Daizhan Cheng, Yuandan Lin,and Yuan Wang Abstract—This note considers the controllability of switched linear sys- tems. The structure of accessibility Lie algebra is revealed. Some accessi- bility properties are proved. Certain necessary and sufficient conditions for (local or global, weak or normal) controllability of a large class of switched linear systems are obtained. Index Terms—Accessibility, controllability, Lie algebra, switched linear system. I. INTRODUCTION Consider a switched linear system (1) where the switching functions are piecewise con- stant, right continuous mappings with , and where controls are piecewise constant functions. We use to denote the solution of the system satisfying the initial value with the switching function and the input function . The controllability property of such switched linear systems was investigated by many authors, e.g., [3], [13], [8], [9], and [14]. The following notion of controllability was adopted in [9] and [14]. Definition 1.1: Consider system (1). A state is controllable at time , if there exist a time instant , a switching path , and an input function , such that . The set of controllable points is a vector space. The largest subspace of in which every point is controllable is called the controllable subspace. The system is controllable if . Let , the smallest space containing the column vectors of that is invariant under the transfor- mations . The following result, a significant contribution of [9], reveals the structure of the controllable subspace of the system. Theorem 1.2: [9] The controllable subspace of system (1) is . Hence, the system is controllable if and only if . It can be seen that the controllability notion given in Definition 1.1 is an analogue of the linear case. It deals only with controllability at the origin. We would like to point out that a switched linear system is essentially a nonlinear system with the switching functions acting as controls. For a nonlinear system, controllability at the origin is in general not sufficient to describe reachability or controllability at other points. The following example shows how the controllable subspace at (or the reachable subspace from) 0 and the reachable sets from other points may be unrelated. Manuscript received September 28, 2002; revised June 5, 2003, October 2, 2004, and February 6, 2006. Recommended by Associate Editor J. M. A. Scherpen. This work was supported by the National Science Foundation of China under Grants G60228003, G60274010, G60221301, and G60334040. D. Cheng is with the Institute of Systems Science, Chinese Academy of Sci- ences, Beijing 100080, China (e-mail: dcheng@iss.ac.cn). Y. Lin and Y. Wang are with the Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431 USA (e-mail: lin@fau.edu; ywang@fau.edu). Digital Object Identifier 10.1109/TAC.2006.880776 Example 1.3: Consider the following system: (2) for which and By Theorem 1.2, the controllable subspace is . However, it is easy to see that the submanifold is a controllable sub-man- ifold in the sense that every point on can be reached from any other point on : With and , a trajectory can go in the radius direction (either increasing or decreasing), and with it can go along a circular path. So, for any two points , there is a switching law that drives a trajectory from to . The previous example shows that even though the controllable sub- space is , the reachable set of a point still consists of almost every point in the state–space. For general nonlinear control systems, a main tool for investigating their controllability is the accessibility Lie algebras generated by the vector fields of the systems (cf., [10] and [12]). The purpose of this note is to apply the Lie algebra approach for controllability of general nonlinear control systems to switched linear systems. We are particu- larly interested in the case when the accessiblity rank condition fails. The rest of the note is organized as follows. In Section II, we discuss some preliminaries for the Lie algebras associated with the controlla- bility of nonlinear systems and the integrability of Lie algebras. In Sec- tion III, we study the structure of accessibility Lie algebras for switched linear systems. In Section IV, we investigate the controllability, in- cluding global, local, weak and normal types, of switched linear sys- tems. A topological structure of the controllable sub-manifolds and some necessary and sufficient conditions for certain controllability are obtained there. In Section V, we provide an illustrating example. In Section VI, we summarize the main conclusions. II. LIE ALGEBRA AND ITS INTEGRABILITY Consider a nonlinear control system (3) where is assumed to be analytic and defined on . The controls are piecewise constant, right continuous functions from to . If is affine in , (3) becomes an affine nonlinear system (4) A switched linear system as in (1) can be treated as a nonlinear system as follows: where the controls of the system are with , taking values in the set if . 0018-9286/$20.00 © 2006 IEEE