Necessary and sufficient conditions for controllability of switched linear systems

We study the controllability of switched linear systems. First, the concept of controllable state set is introduced as the basic tool for evaluating the controllability. We then prove that, for a switched linear system, there exists a basic switching sequence such that the controllable state set of this basic switching sequence is equal to the controllable state set of the system. Hence, the controllability can be realized by using only one switching sequence. This striking fact reveals some intrinsic properties of switched linear systems, and is the most fundamental starting point for further development. Based on this fact, a necessary and sufficient condition for controllability of switched linear systems is established. Furthermore, these results are extended to switched linear systems with time-delay. A geometric criterion for controllability of switched time-delay systems is established as well. Illustrative examples are provided.