Disturbance attenuation in LTI plants with finite horizon: Optimality of nonlinear controllers

It is an established fact that search for an optimum in the problem of disturbance attenuation in linear time-invariant (LTI) plants can be restricted to linear designs when the sole criterion is the minimization of the H^∞-norm of an appropriate input-output operator. What is not known, however, is whether there exist also optimal nonlinear controllers which would lead to uniformly better performance (than linear controllers) in a neighborhood of the worst-case disturbance. It is shown in this paper that for the finite-horizon version, and with the disturbance energy fixed, there do exist optimal (minimax) controllers outside the linear class, which can oout-perform the best linear controller locally. It is also argued that a differential (dynamic) games set-up provides a natural framework for the time-domain derivation of optimal (minimax) controllers for tracking, model matching and disturbance rejection problems, in both finite and infinite horizons.