A dynamic games approach to controller design: disturbance rejection in discrete time

It is shown that the discrete-time disturbance-rejection problem, formulated in finite and infinite horizons, can be solved by making direct use of the available results on linear-quadratic zero-sum dynamic games. Under perfect state measurements the approach leads to a minimax controller which achieves the best performance bound, and also to a characterization of all linear controllers under which disturbance attenuation does not exceed a prescribed bound. For the former, the worst-case disturbance turns out to be a correlated random sequence with a discrete distribution, which means that the problem (viewed as a dynamic game between the controller and the disturbance) does not admit a pure-strategy saddle point. Also formulated is a stochastic version of the problem, where the disturbance is a partially stochastic process with fixed higher order moments (other than the mean). Here the minimix controller depends on the energy bound of the disturbance, provided that it is below a certain threshold. Several numerical studies are included to illustrate the main results.