On piecewise quadratic control-Lyapunov functions for switched linear systems

In this paper, we prove that a discrete-time switched linear system is exponentially stabilizable if and only if there exists a stationary hybrid-control law that consists of a homogeneous switching-control law and a piecewise-linear continuous-control law under which the closed-loop system has a piecewise quadratic Lyapunov function. Such a converse control-Lyapunov function theorem justifies many of the earlier controller-synthesis methods that have adopted piecewise-quadratic Lyapunov functions and piecewise-linear continuous-control laws for convenience or heuristic reasons. Furthermore, several important properties of the proposed stabilizing control law are derived and their connections to other existing controllers studied in the literature are discussed.