Energy analysis of a class of state-dependent switched systems with all unstable subsystems

In this paper, we investigate the stability and periodicity of a class of state-dependent switched systems with all unstable subsystems by means of energy analysis. We firstly transform the unstable subsystems reversibly into the form of second order mechanical systems, and then construct energy functions by calculating the sum of kinetic and potential energies of each subsystem. After that, two switching lines, derived from the lines with the largest and smallest energy drops, make the stable phase trajectory approach to the equilibrium point at the fastest speed. In addition, we explore possible dynamic behaviors of the switched system under a pair of switching line including asymptotic stability, instability and periodicity. Furthermore, based on the bisection method and nested intervals theorem, we design a state-dependent switching law, which makes the switched system periodic initiated from a stable switching law. Finally, numerical simulation examples are provided to illustrate the effectiveness and less conservativeness of the proposed method with practical significance.