Optimal Linear Quadratic Regulator of Switched Systems

This paper considers the optimal control problem of linear switched systems with linear quadratic (LQ) cost or multiple LQ cost. By adopting an embedding transformation, the switching design problem is relaxed and transformed into a traditional optimal control problem. The bang-bang-type solutions of the embedded optimal control problems are obtained for both the positive definite LQ cost case and the multiple LQ cost case, which are the optimal solutions to the original problems. The switching sequence of modes and the switching instants can be calculated by solving a closed-form optimal switching condition. The optimal state feedback control law is determined simultaneously. Finally, numerical results are provided to illustrate the effectiveness of the proposed method.