Optimal Switching For Linear Quadratic Problem Of Switched System

Switched system is an important class of hybrid system, which is composed of mul-tiple subsystems and a switching law among them. The research of optimization problem for switched system awlays focused on the continuous variables such as control variables and switching times, where the switching sequence is supposed to be given. However, the switching sequence will have a great influence on the performance of the system, and the optimal switching sequence is not known in advance in most of the real applications. Hence, it’s necessary to consider the problem of optimal switching sequence. Since the switching sequence is a class of discrete variables and the number of all feasible switch-ing sequences grows exponentially with respect to the number of switchings, it’s not easy to find the optimal solution. There are few method which can deal with this discrete opti-mization problem. For this, we develop a new method to search for the optimal switching sequence efficiently and effectively.The main work of this paper is the study of optimal switching for a special and impor-tant linear quadratic problem of switched system, where the switched system is governed by multiple linear system and the cost functional is quadratic. First, we consider the opti-mal switching of the linear quadratic problem for switched system in discrete time. Note that the switching sequence is the only variable to be optimized. We develop a branch and bound method to obtain the global optimal switching sequence, by constructing the lower bound dynamic system to compute the lower bound for any current switching sequence. Next, we consider the optimal switching of the linear quadratic problem for switched system in continuous time. Note that there two kinds of variables in this problem, we decompose this problem into two subproblems. The first subproblem is the optimization of switching time for a given switching sequence, which can be solved by any gradient based algorithm such as sequential quadratic programming. The second subproblem is the optimization of the switching sequence for a given switching times, where the golbal solution can be solved by constructing the lower bound system and applying the branch and bound method. Then, we solve the optimal switching problem by applying the alter- native method to two subproblems. The optimal solution can be obtained as the solution series converges to a stationary point. For these two problems, numerical examples in two cases of time invariant and time variant are given to show that our proposal methods are very efficient and effective.This thesis is organized as follow. In chapter 1, we first introduce the concept of switched system, then we review the history and development of switched system and summarize the content and novelty of this thesis. In chapter 2, we consider the optimal switching for linear quadratic problem of switched system in discrete time. In chapter 3, we consider the optimal switching for linear quadratic problem of switched system in continuous time. For these two chapters, we apply many methods such as sequential quadratic programming, control parameterization and branch and bound to solve these problems. Numerical experiments are illustrated to verify the efficiency and effectiveness of these methods. Finally, we summarize the research in this thesis and make a prospect of the research in the future.