Switched system optimal control based on parameterizations of the control vectors and switching instants

This paper presents a new approach for solving optimal control problems for switched systems with prespecified order of the sequence of subsystems. For such problems, we need to seek both the optimal switching instants and the optimal continuous inputs. The optimal control problem for hybrid systems is transcribed into an equivalent problem parameterized by the switching instants and the control vectors. Through the discretization of the control space, the control vector is approximated by the B-spline functions. In order to search for the optimal switching instants, we introduce the normalized time variable and do normalized treatment for the control arcs. The optimal switching control problem is converted to a nonlinear programming problem. The control profiles and switching instants act as decision variables. Two examples are solved using the proposed method so as to demonstrate the effectiveness of the method proposed.

I. Introduction

A switched system is a particular kind of hybrid system that consists of a number of subsystems and a switching law specifying the active subsystems at each time instant. Switched systems arise in many real world applications, such as the control of mechanical systems, the automotive industry, aircraft and air traffic control, and switching power converters. Some details on these applications can be found in [1].