Minimization of the worst case peak-to-peak gain via dynamic programming: State feedback case

In this paper, we consider the problem of designing a controller that minim izes the worst case peak-to-peak gain of the closed-loop system. In particu lar, we concentrate on the case where the controller has access to the stat e of a linear plant and it possibly knows the maximal disturbance input amp litude. We apply the principle of optimality and derive a dynamic programmi ng formulation of the optimization problem. Under mild assumptions, we show that, at each step of the dynamic program, the cost to go has the form of a gauge function and can be recursively determined through simple transform ations. We study both the finite horizon and the infinite horizon case unde r different information structures. The proposed approach allows us to enco mpass and improve the recent results based on viability theory. In particul ar, we present a computational scheme alternative to the standard bisection algorithm, or gamma iteration, that allows us to compute the exact value o f the worst case peak-to-peak gain for any finite horizon. We show that the sequence of finite horizon optimal costs converges, as the length of the h orizon goes to infinity, to the infinite horizon optimal cost. The sequence of such optimal costs converges from below to the optimal performance for the infinite horizon problem. We also show the existence of an optimal stat e feedback strategy that is globally exponentially stabilizing and derive s uboptinal globally exponentially stabilizing strategies from the solutions of finite horizon problems.