Robust performance against time-varying structured perturbations

In this paper, we consider the problem of robust performance analysis for some nominal system M(z) against bounded, linear, time-varying, structured feedback perturbations. We introduce a natural input-output notion of rate-of-variation for a linear time-varying operator. We then exhibit upper and lower bounds on the maximum rate-of-variation of these perturbations against which robust performance is achievable. Using these bounds, we show that the existence of frequency dependent D-scales that render the norm of M(z) less than one is necessary and sufficient for robust performance against arbitrarily slowly varying structured linear perturbations of norm less than one. This result suggests that it is natural and well justified to deal with the frequency-dependent upper bound for the μ “norm,” rather than μ itself. This is reassuring given that the upper bound is easily and reliably computable, while computation of complex μ appears difficult