Maximally Robust Controllers for Multivariable Systems

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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 r1 ). 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 $(1-\delta ) r_1,\;0 < \delta\leq1$, 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 r1 . 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 $\mu$-analysis are pursued.