Time-Varying Versus Time-Invariant Compensation for Rejection of Persistent Bounded Disturbances and Robust Stabilization

This paper considers time-varying compensation for linear time-variant 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 existing results for feedback systems subject to finite-energy disturbances. First, 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 is further exploited 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 may be treated in the same manner. It is then shown that time-varying compensation does not help in the bounded-input bounded-output robust stabilization of time-invariant plants with unstructured uncertainty. In doing so, it is 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.