| This dissertation reports some progresses in the analysis and design of switched systems which is an important class of hybrid systems. The dissertation consists of three parts.; In Part I, we motivate the studies in this dissertation and introduce some preliminaries of switched systems. In particular, a formal modeling framework for switched systems is developed which will be used throughout the dissertation.; In Part II, we develop an approach for the asymptotic stabilization of second-order linear time invariant switched systems. The main features of the results are the necessary and sufficient conditions for asymptotic stabilizability of such systems and the construction of stabilizing control laws. The approach is derived through a careful study of system vector fields by utilizing the geometric properties of the two dimensional Euclidean space. This result is among the very few necessary and sufficient results as opposed to the many sufficient only results derived from variations of the Lyapunov's second method.; In Part III, we report progresses in optimal control of switched systems. Unlike classical optimal control problems, such problems require not only the solutions of optimal continuous inputs but also the solutions of optimal mode switching sequences since the system dynamics vary before and after each switching instant. Such problems are carefully formulated in the dissertation and difficulties in solving them are pointed out. Methodologies for dealing with some of the difficulties are discussed. We then propose a two stage optimization methodology for solving such optimal control problems. Since many practical problems only concern Stage 1 optimization where the number of switchings and the sequence of active subsystems are given, we then concentrate on Stage 1 optimization problems and propose two approaches for solving them. The first approach is via nonlinear optimization and based on direct differentiations of value functions. The second approach is based on parameterization of the switching instants. These two approaches are also shown to be especially efficient for general switched linear quadratic problems. |
