Estimation and synthesis of reachable set for discrete systems with switching characteristics

Abstract

This thesis is concerned with the problems of reachable set estimation and synthesis for a class of dynamic systems with switching characteristics.

In terms of estimation of the reachable set, some ellipsoids and hyperpyramids are utilized to bounded the reachable set of systems with switching characteristics. Specifically, the following four types of systems are exploited: (1) For generally switched systems under arbitrarily switching signal, the reachable set bounding conditions which determine the bounding ellipsoids are derived by using a Lypunov-based inequality and the quadratic S-procedure. Usually, the bounding ellipsoids are expected to be as small as possible, three optimization approaches are developed to minimize the bounded ellipsoids. In addition, Genetic Algorithm (GA) is used to search for the optimal value of the decision variables in the obtained reachable set estimation conditions. (2) For general switched systems under periodically switching signal (also termed as periodic systems in the thesis), the lifting approach and the pseudo-periodic Lyapunov function approach are utilized to determine the bounding ellipsoids for the reachable set. GA is adopted in the pseudo-periodic Lyapunov function approach to search for the optimal value of the decision variables. (3) For switched positive systems under arbitrarily switching signal, the reachable set bounding conditions which determine the bounding hyper-pyramids are derived by utilizing two Lyapunov-based inequalities and the S-procedure technique. The bounding hyper-pyramids are minimized through two optimization methods. (4) For switched positive systems under periodically switching signal (also termed as periodic positive systems in the thesis), the lifting approach and the pseudo-periodic co-positive Lyapunov function approach are used to determine the bounding hyper-pyramids for the reachable set.

In terms of synthesis of the reachable set, the control objective is to manipulate the reachable set of closed-loop system to lie within a given region or to make the reachable set as small as possible. Based on the reachable set estimation results, sufficient conditions are established for the existence of the desired state-feedback controllers. GA is used to search for the optimal state-feedback controller that makes the reachable set of closed-loop system as small as possible.