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On LQ control of discrete-time switched system with switching cost

We study in this paper the discrete-time linear-quadratic (LQ) control of linear switched systems with switching cost. We first introduce a corresponding auxiliary problem, which is an LQ control problem of switched systems with an upper bound on the switching number. By using dynamic programming (DP), we derive explicitly the switching region, the switching sequence, the feedback gain, and the value function. We find that, although the DP-based algorithm may not be a polynomial-time algorithm for general situations, it is indeed efficient for some special cases. While recognizing the inefficiency of such a DP algorithm for general situations, we further develop a lower bounding scheme by combining a revised DP algorithm with semidefinite programming. Integrating such a lower bounding scheme into a branch and bound framework, the auxiliary problem can be efficiently solved.