Planar Switched Systems Stabilizability

As an important class of hybrid dynamical systems, a switched system is composed of several continuous time subsystems or discrete time subsystems and the switching signal among them. With its own particularity, it is different from the traditional continuous time systems or discrete time systems. Due to the existence of the switching signal, its dynamics may become very complicated, for example, the state trajectory may jump at the switching point, and its stability may change. Hence, it is very important that how to choose the proper switching signal. During the last three decades of so, there is an increasing interest on the modeling, analysis, synthesis, and control of switched systems. Nowadays more and more people have paid attention to the stability analysis of the switched systems and the study of the switching control.This dissertation devotes on the study of the second-order continuous (discrete) time switched linear systems, including the design of the feedback controllers and the switching signals, the globally asymptotical stability, quadratic stability and the dynamical quality of the state response of the closed-loop systems. The main contributions of the research work presented in this dissertation are as follows:1. In the first part, the quadratic stabilizability problem of the second-order switched systems' is studied. Under the assumption of stabilization or controllability of the subsystems, the necessary and sufficient condition is proposed through designing switching state feedback controllers, and guarantees the quadratic stability of the closed-loop switched system under arbitrary switching. Meanwhile, the design algorithm for the switching state feedback controllers and the common Lyapunov function of the closed-loop switched system is given.2. In the second part, the stabilizability problem of a class of single-input switched linear systems is considered. The dimension of the system is reduced with the variable-structure control. The sufficient conditions of the uniform stabilization of the systems and the existence of the admissible stabilizing strategies o f the systems are obtained through the study of the sliding mode of the reduced systems. And the detailed admissible stabilizing strategy sets are proposed. The completely admissible stabilizing strategies for second-order switched systems are given as an application.3. In the third part, the dynamic behavior quality of planar closed-loop switched systems is analysised near the switching boundary. Motivated by the i dea o f t he v ariable-structure c ontrol, t he c onception o f a pproaching region and leaving region of the switching boundary is developed. The sufficient conditions of whether sliding mode occurs or not on the switching boundary are obtained. Under the assumption of the switching delay time being positive, the sufficient conditions of quasi-sliding-mode occurring on the switching boundary are given.In this dissertation, simulations are made for the major design schemes. Simulation results show the effectiveness of the proposed approaches.