Stability analysis and controller synthesis for a class of piecewise smooth systems

摘要

This thesis deals with the analysis and synthesis of piecewise smooth (PWS) systems. In general, PWS systems are nonsmooth systems, which means their vector fields are discontinuous functions of the state vector. Dynamic behavior of nonsmooth systems is richer than smooth systems. For example, there are phenomena such as sliding modes that occur only in nonsmooth systems. In this thesis, a Lyapunov stability theorem is proved to provide the theoretical framework for the stability analysis of PWS systems. Piecewise affine (PWA) and piecewise polynomial (PWP) systems are then introduced as important subclasses of PWS systems. The objective of this thesis is to propose efficient computational controller synthesis methods for PWA and PWP systems. Three synthesis methods are presented in this thesis. The first method extends linear controllers for uncertain nonlinear systems to PWA controllers. The result is a PWA controller that maintains the performance of the linear controller while extending its region of convergence. However, the synthesis problem for the first method is formulated as a set of bilinear matrix inequalities (BMIs), which are not easy to solve. Two controller synthesis methods are then presented to formulate PWA and PWP controller synthesis as convex problems, which are numerically tractable. Finally, to address practical implementation issues, a time-delay approach to stability analysis of sampled-data PWA systems is presented. The proposed method calculates the maximum sampling time for a sampled-data PWA system consisting of a continuous-time plant and a discrete-time emulation of a continuous-time PWA state feedback controller.