Control Synthesis Of Piecewise Linear Systems

The piecewise linear systems (PLS) can be used to analyze and design theuncertain systems and nonlinear systems with the existed results in linear systems,and it has been widely studied and used among control engineering field.The global quadratic Lyapunov function did not use any information about thepartition of the state space, so it became very conservative in the analysis of PLS. Theconcept of continuity matrix and polyhedral cell bounding is introduced, based onwhich a piecewise quadratic Lyapunov function is constructed whos positivity isguaranteed by S-procedure. With piecewise quadratic Lyapunov function the PLS canbe analyzed effectively. All analysis condition will be formulated as convexoptimization problems, allowing PLS to be analyzed using efficient numericalcomputations.The optimal control of PLS can be characterized in terms of solution to theHamilton-Jacobi-Bellman (H-J-B) equation. However, the H-J-B equation will sufferfrom combinatorial explosion and is notoriously hard to solve in general. Based onH-J-B inequalities and piecewise quadratic Lyapunov function, the optimal control ofPLS is converted to the problem of seeking upper and lower bounds of the costfunction. The design of upper bound can be cast as a bilinear matrix inequalities (BMI)problem in which the feedback gain is a set of optimization parameters, and the lowerbound computation can be solved as a semidefinite programming problem based onlinear matrix inequalities (LMI). Thus we can avoid sloving the H-J-B equation.The BMI is a NP hard problem which can be solved effectively by geneticalgorithm. A mixed algorithm that combines genetic algorithm and interior-pointmethod is designed for solving the BMI problem. The proposed method is simple andcan be easy to carry out.With the same method in optimal control of PLS, the optimal control of piecewiselinear differential inclusions and uncertain PLS and nolinear systems is discussed, andgot corresponding results. The results from numerical examples illustrate theeffectiveness of the design and calculation method for the control law.