Controllability Implies Stabilizability for Discrete-Time Switched Linear Systems

Abstract

A switched linear system is said to be controllable, if for any given initial state and terminal state, one can find a switching sequence and corresponding input such that the system can be driven from the initial state to the terminal state. Necessary and sufficient condition on the controllability of switched linear systems has been established, and a single switching sequence can be constructed to realize the controllability. In this paper, the stabilizability problem for switched linear systems is formulated and we show that controllability implies stabilizability for switched linear systems. In our framework, we using periodically switching sequence and piecewise constant feedback controller. Two stabilization design methods, the pole assignment method and the linear matrix inequality method are proposed. Furthermore, if a switched linear system is both controllable and observable, then an observer-based output feedback controller can be constructed when the system state is not available. In this case, the well-known Separation Principle is shown to still hold. All these results are built upon our previously established fundamental geometric properties of controllability realization as well as the important fact that both controllability and observability can be realized through a single switching sequence.