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

We consider the problem of designing a controller that minimizes the worst-case peak to peak gain of the closed loop system. We concentrate on the case where the controller has access to the state of a linear plant and it possibly knows the maximal disturbance input amplitude. We apply the principle of optimality and derive a dynamic programming formulation of the optimization problem. 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 transformations. We study both the finite horizon and the infinite horizon cases. The proposed approach allows one to encompass and improve the recent results based on viability theory. The formulation presented allows one to consider, together with worst case inputs, fixed known inputs, and it can naturally incorporate actuator saturation constraints