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The set of all optimal controllers which maximize a robust stability radius for unstructured additive perturbations may be obtained using standard Hankel-norm approximation methods. These controllers guarantee robust stability for all perturbations which lie inside an open ball in the uncertainty space (say, of radius r 1 ). Necessary and sufficient conditions are obtained for a perturbation lying on the boundary of this ball to be destabilizing for all maximally robust controllers. It is thus shown that a "worst-case direction" exists along which all boundary perturbations are destabilizing. By imposing a parametric constraint such that the permissible perturbations cannot have a "projection" of magnitude larger than (1 − δ)r 1 , 0 < δ ≤ 1, in the most critical direction, the uncertainty region guaranteed to be stabilized by a subset of all maximally robust controllers can be extended beyond the ball of radius r 1 . The choice of the "best" maximally robust controller-in the sense that the uncertainty region guaranteed to be stabilized becomes as large as possible-is associated with the solution of a superoptimal approximation problem. Expressions for the improved stability radius are obtained and some interesting links with µ-analysis are pursued.

Abstract. The gap between the complex structured singular value of a complex matrix M and its convex upper bound is considered. New necessary and sufficient conditions for the existence of the gap are derived. It is shown that determining whether there exists such a gap is as difficult as evaluating a structured singular value of a reduced rank matrix (whose rank is equal to the multiplicity of the largest singular value of M ). Furthermore, if an upper bound on this reduced rank problem can be obtained, it is shown that this provides an upper bound on the original problem that is lower than the convex relaxation upper bound. An example that illustrates our procedure is given. We also give the solution of several structured-approximation problems of independent interest.

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