Piecewise lyapunov functions for switched systems with average dwell time

The stability properties of linear switched systems consisting of both Hurwitz stable and unstable subsystems are investigated by using piecewise .Lyapunov functions incorporated with an average dwell time approach. It is shown that if the average dwell time is chosen sufficiently large and the total activation time ratio between Hurwitz stable and unstable subsystems is not smaller than a specified constant, then exponential stability of a desired degree is guaranteed. The above result is also extended to the case where nonlinear norm-bounded perturbations exist.