Invariance Of Control Systems And Switching Design Via Nonsmooth Method

Invariance analysis and design of control systems are the core issues in the field of control, and nonsmooth analysis method in their research plays a very important role. In this thesis, the invariance analysis and switching design of control systems are investigated by employing non-smooth method. The invariance analysis problems are studied according to the linear control system as well as the nonlinear control systems with uncertainties. The stability design prob-lems are mainly studied for switched systems with polytopic uncertainties. The main research results can be summarized as follows:(1) In chapter2, convex set theory and viability theory are applied to study the problems of unbounded polyhedral invariant sets for linear control systems. A method is obtained for determining an unbounded invariant set of the system. When the extreme directions of the unbounded polyhedron satisfy certain conditions, a method is obtained to verify whether the polyhedron is the weak invariant set for a class of linear control systems. A sufficiency condition is also obtained such that the unbounded polyhedron is the strong invariant set for more general linear control systems. Two illustrating examples are presented.(2) Chapter3studies the invariant set and region-of-attraction of the polytopic uncertainty systems by utilizing the piecewise smooth Lyapunov functions. Some sufficient condi-tions for determining the invariant set of the robust region-of-attraction are obtained. By the method of enlarging attraction region, an optimization model is formulated for calcu-lating the invariant set of the systems. By polynomial theory the optimization model is transformed into the problem which can be easily solved by optimization method. Some special piecewise smooth functions, i.e., the sum of Max type and Min type functions and the Minmax type functions are considered, and the corresponding sufficient condi-tions for invariance and robust attractive region of polytopic uncertainty systems are also proposed.(3) Chapter4studies the invariant set and region-of-attraction of the uncertainty systems with parametric bounds. The sufficient conditions are obtained for determining the invariant set of the robust region-of-attraction. The determination method for robust region-of-attraction of the systems is obtained for two piecewise smooth functions, i.e, the Max type and Min type functions. The optimization models are formulated for calculating the invariant set of the systems by employing polynomial theory and the method of enlarg-ing attraction region. Then, the optimization models are transformed to ones which can be easily solved by optimization method. The efficiency of the proposed determination method is validated by an illustrating example.(4) In chapter5, the exponential stability design of switched nonlinear systems with polytopic uncertainties is studied by employing Min type nonsmooth function. The switching law among subsystems depends on the directional derivative along the extreme directions of subsystems. A sufficient condition for exponential stability design of the switched systems is established with careful considerations of the sliding modes and the directional derivatives along sliding modes.(5) In chapter6, the composite quadratic Lyapunov functions are used to design the linear switched systems with polytopic uncertainties. The switching law is designed and the matrix conditions of stability design are derived as an optimization problem which is for-mulated with bilinear matrix inequalities (BMIs). The numerical examples are given and the simulation results demonstrate the efficiency of the proposed stability design methods.