Optimal control of discrete-time switched linear systems

This paper considers the optimal control problems of switched autonomous linear systems and switched non-autonomous linear systems in discrete-time, where the objective is that of minimizing a linear quadratic performance index function or multiple linear quadratic performance index function. The subsystems switching sequence and the number of switchings are free in this paper. By employing an embedding technique, we embed the origin switched autonomous or non-autonomous linear systems into a larger family of systems. Owing to the embedding technique, the optimal control problem of the origin switched linear systems can be relaxed into a traditional optimal control problem for the larger family systems. We thus derive the optimal switching control law and the optimal control input for the bang-bang type optimal solution of the relaxed optimal control problem simultaneously. The optimal solution of the origin switched optimal control problem can be obtained by the bang-bang type optimal solution of the relaxed optimal control problem. Finally, results of numerical simulation examples are given to demonstrate the remarkable effectiveness and superiority of the developed method.