Time-varying vs. time-invariant compensation for rejection of persistent bounded disturbances and robust stabilization

Summary form only given, as follows. Consideration is given to time-varying compensation for linear time-invariant discrete-time plants subject to persistent bounded disturbances. In the context of certain feedback objectives, it is shown that time-varying compensation offers no advantage over time-invariant compensation. These results complement similar results for feedback systems subject to finite-energy disturbances. It is shown that time-varying compensation does not improve the optimal rejection of persistent bounded disturbances. This result is obtained by exploiting a key observation that any time-varying compensator which yields a given degree of disturbance rejection must do so uniformly over time, thereby removing any advantage of time-variation. This key observation was used to show that time-varying compensation does not improve the optimal rejection of disturbances, regardless of the norm used to measure the disturbances. Thus, absolutely summable, finite-energy, or persistent bounded disturbances can be treated in the same manner. It is shown that time-varying compensation does not help in the bounded-input-bounded-output robust stabilization of time-invariant plants with unstructured uncertainty. It is thus also shown that the small-gain theorem is both necessary and sufficient for the bounded-input-bounded-output stability of certain linear time-varying plants subject to unstructured linear time-varying perturbations.