Widths and Uncertainty Principles for LTI and Slowly Varying Systems

The optimal worst-case uncertainty that can be achieved by identification depends on the observation time. In the first part of the paper, this dependence is evaluated for selected linear time invariant systems in the 2’ and H” norms and shown to be derivable from a monotonicity principle. The minimal time required is shown to depend on the metric complexity of the a priori information set. Two notions of n-width (or metric dimension) are introduced to characterize this complexity. In the second part of the paper, the results are applied to systems in which the law governing the evolution of the uncertain elements is not time invariant. Such systems cannot be identified accurately. The inherent uncertainty is bounded in the case of slow time variation. The n-widths and related optimal inputs provide benchmarks for the evaluation of actual inputs occurring in adaptive feedback systems.