Solving The Global Optimal Solution Of A Class Of Optimal Switching Problem Via Relaxation Method

Switched system have wide applications in many area,such as manufacturing control,traffic control,sensor data collection,power converter,and so on.How to control the subsys-tems according to the dynamic characteristics of switched systems,such that the performance of the system achieve the optimal under a switching law.This has become the research top-ic nowadays.The optimal switching problem is to find an appropriate switching sequence at the appropriate time so that the the given objective function value is the best.Because the switching sequence is discrete valued,the optimal switching problem is a discrete optimiza-tion problem,and is NP hard to find the global optimal solution.Complete enumeration search is always required to find the global optimal solution,which is very expensive.In this paper,we propose the relaxation method to find the global optimal solution of a class of optimal switching problem.This thesis is organized as follows.In chapter 1,we introduce concept and computation method of optimal control problem.Then,the switched system and optimal switching problems are introduced.In chapter 2,we review the methods for saving the optimal switching problem,including enumeration method,discrete filled function method and relaxation method.In chapter 3,we consider the global optimal solution for the optimal switching prob-lem in discrete time,where these subsystems and the cost functional is linear.The optimal switching problem is a discrete optimization problem.By introducing weighted functions,the optimal switching problem is transformed into a relaxed optimization problem.The global optimal solution is analyzed and the equivalence between the optimal switching problem and the corresponding relaxed problem is established.Then,the global optimal solution can be obtained by solving the relaxed problem.Numerical examples are demonstrated to show that the relaxation method is efficient and effective to obtain the global optimal solution.In chapter 4,we consider the global optimal solution for the optimal switching problem in discrete time,where these subsystems are linear and the cost functional is quadratic.The general relaxation method fail to find the global optimal solution.Then,we transform the optimal switching problem into an equivalent problem and apply a modified relaxation to find the global optimal solution.Two numerical examples are computed by the optimal control software MISER3,which shows the modified relaxation method is efficient and effective to obtain the global optimal solution.In chapter 5,we summarize the research in this thesis and make a prospect for future research.