Finite-Time Stabilization of Switched Systems with Unstable Modes

In this paper, we study finite-time stability and stabilization of switched systems in the presence of unstable modes. In contrast to asymptotic or exponential stability where the system trajectories reach the equilibrium point as time tends to infinity, the notion of finite-time stability requires the trajectories to reach the equilibrium within a finite amount of time. We show that even if the value of the Lyapunov function increases in between two switches, i.e., if there are unstable modes in the system, finite-time stability can still be guaranteed if the finite-time convergent mode is active for a long enough cumulative time duration. Then, we present a method for the synthesis of a finite-time stabilizing switching signal. As a case study, we design a finite-time stable, output-feedback controller for a linear switched system, in which only one of the modes is both controllable and observable.