Optimal damping of forced oscillations in discrete-time systems

In this paper we consider a linear discrete-time control system affected by an additive sinusoidal disturbance with known frequencies but unknown amplitudes and phases. The problem is to damp this forced oscillation in an optimal fashion. We show that the natural solution from the point of view of optimal control is neither robust with respect to errors in the frequencies, and thus not optimal in practice, nor independent of the unknown amplitudes and phases. The main result of this paper concerns the existence and design of a realizable, robust optimal regulator, which is universal in the sense that it does not depend on the unknown amplitudes and phases and is optimal for all choices of such parameters. The regulator allows for a considerable degree of design freedom to satisfy other design specifications. Finally, it is shown that this regulator is optimal also for a wide class of stochastic control problems.