Performance Optimization in the Presence of Norm Bounded Structured Uncertainty

A performance criterion given as a bound on the H ∞ norm of some transfer function is known to be equivalent to the stability condition with respect to the auxiliary unstructured uncertainty. Therefore, robust stability and performance of the system in the presence of a block diagonal norm bounded uncertainty with (m) elements is easily transformed into a structured singular value problem with respect to the overall diagonal uncertainty with (m+1) blocks. When the norm bounds on the uncertainty and the performance condition are known and scaled to the value of one, the synthesis becomes a search for a compensator that will make the associated structured singular value smaller than one. This is usually done by the "D-K" iteration introduced by Doyle. In this paper we look at the case where the norm bound on one of the blocks in the structured uncertainty is not known a priori. The problem then becomes one of finding the extreme value of that bound such that there exist a stabilizing compensator that guarantees robust stability and performance. An recursive algorithm based on altering standard "D-K" iteration is proposed.