Variable Structure Control with Sliding Sector for Hybrid Systems

This paper proposes a variable structure (VS) controller with sliding sector for hybrid systems where subsystems do not need to be stable and may be unknown. The controller is designed to switch among subsystems to stabilize the hybrid system by a variable structure switching law. The concept of the sliding sector is used in the paper to show that there exists an area, inside which a Lyapunov function decreases, for any system no matter whether it is stable or unstable. A Lyapunov function is designed by assuming that a stable convex combination of the subsystems exists. Then the VS controller is designed such that the Lyapunov function decreases in every period for the hybrid system to be switched once through all subsystems. The extremum seeking control algorithm is used to determined the switching rule such that the Lyapunov function track a pre-determined decrease signal. The proposed VS system is quadratically stable. Simulation results are given to show the convergence of the proposed VS control system