On Robust Stability with Structured Time-Invariant Perturbations

We consider the problem of robust stability for uncertain systems described in terms of multiple scalar structured time-invariant perturbations, which are norm bounded. The issue we address is the role that the type of perturbation norm plays in the robust stability conditions. To this end, the L p-induced norms (for 1 p 1) on the perturbations are considered, and some simple relations between the L p-induced norm of linear time-invariant systems are derived. We then show that the structured singular value is the necessary and suucient condition for robust stability for all of the L p-induced norms. We conclude that for time-invariant perturbations, in contrast to the time-varying case, robust stability conditions seem to be independent of the type of perturbation norms.