Continuity properties of the parametric stability margin

We consider linear control systems with uncertain parameters appearing linearly in the characteristic polynomial. Suppose the system is stable for some nominal parameter values. Then the robustness margin is commonly defined as the smallest norm of a destabilizing deviation from the nominal parameter values. A large number of papers have been devoted to computation and optimization of such robustness margins. The possible ill-conditioning of such computations was pointed out by the paper [1], showing that the robustness margin may very well be a discontinuous function of problem data. In the more general framework of structured singular values, [3] has shown that, under mild conditions, presence of complex uncertainty blocks is a sufficient condition for continuity. Furthermore, [2] have proved that the most common upper bound on structured singular values for mixed complex/real uncertainty always depends continuously on its argument. The purpose of the present paper is to give a complete analytical characterization of possible discontinuities in the robustness margin. In later work, our aim is to use this characterization to define a condition number for the robustness margin. We note that parallell work in this direction has been performed in [5].