Minimax Controllers for LTI Plants Under l1-Bounded Disturbances

Abstract

A currently popular controller design technique for linear time-invariant (LTI) systems is H∞-optimization, which involves the minimization of the H∞-norm of a certain transfer function matrix. When formulated in the time domain, using state space description, this optimum design problem can be shown to be equivalent to a linear-quadratic differential game, and this formulation provides a convenient set-up for also studying finite-horizon versions of the original infinite-horizon problem. In this differential game, the minimizer is the controller and the maximizer is the unknown disturbance which is subject to an L2energy constraint. The performance of interest is the upper value of the differential game, with the controller achieving that value called the minimax controller.In this paper, we formulate such a differential game problem which arises in the context of disturbance rejection, but in the discrete time and under l1-bounded disturbances. We study the derivation of the minimax controller associated with the game, as well as the characterization of the worst-case disturbance. We show, in the context of a two-stage design problem, that the saddle point of the game involves a random disturbance, unless the initial state exceeds a certain threshold. Another feature of the solution is that the minimax controller is generally not unique, with the linear feedback controller being outperformed by nonlinear and/or memory controllers, locally or regionally.