Minimax controllers for LTI plants under 1-bounded disturbances

A currently popular controller design technique for linear time-invariant (LTI) systems is H^∞-optimization, which involves the minimization of the H^∞ norm of 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 L energy 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 l-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.