An Alternative Method for the Optimal Switching Problem of Linear Quadratic Switched System

Optimal switching is a special class of optimal control problems for hybrid dynamic systems. In this paper, we consider the optimal switching problem of linear-quadratic switched systems. The aim is to design a suitable switching strategy with the constraint on the number of switchings so that the quadratical performance achieves the minimum value. This problem is difficult to be solved because of the tight coupling between the continuous switching time and the discrete switching sequence. In our method, we first divide this hybrid optimization problem into two subproblems. In each of them, only one type of variable is considered. Then, we develop a gradient-based method with the time-scaling transformation to process the optimal switching time problem and a branch and bound method based on a series of exact lower bounds to handle the optimal switching sequence problem, respectively. By solving these two subproblems alternatively, the optimal switching strategy satisfying the constraint on the number of switchings can be obtained. Numerical examples are given to demonstrate the efficiency of the proposed method.