Stabilization of hybrid system with unstabilizable subsystems

This paper proposes a variable structure (VS) control algorithm for a hybrid system where all of subsystems are unstabilizable, i.e. there exists at least one uncontrollable and unstable subspace for each subsystem. It has been shown that there exists a norm decrease subspace called sliding sector inside which a Lyapunov function decreases without any control effort even if the system is unstable. In this paper, the sliding sector is used to find a norm decrease subspace for each subsystem and a feedback law is designed so that each point in the state space is inside at least a sliding sector of some subsystem. Therefore the VS control law of the hybrid system is to switch the hybrid system among unstabilizable subsystems to ensure the decrease of the Lyapunov function in the state space. Simulation results are given to show the efficiency of the proposed hybrid control algorithm.