Variable Structure Control with Hybrid-Type Input

This paper proposes a variable structure (VS) control algorithm, with which a system can be stabilized by switching the system control input among a set of simple feedback control laws. Each feedback control law is easy to be designed because it uses only one element of the state variable with a fixed positive or negative feedback gain. For an n-order system, 2n feedback control laws are needed to realize the proposed VS control. The concept of the sliding sector is used in the paper to show that there exists an area for each feedback control law, inside which a Lyapunov function decreases with the control law although the system may not be stabilized by it alone. It is shown that a common Lyapunov function for all control laws exists for a controllable system, i.e., for any state in the state space, there exists at least one control law among those laws mentioned above such that the system state is inside the sliding sector of the system with the corresponding control law. Thus a hybrid-type control rule of the variable structure control system is designed to switch the control input to the control law, which ensures the decrease of the common Lyapunov function. The proposed VS control system is quadratically stable. Simulation results are given to show the convergence of the proposed VS control system