Maximally Robust Controllers for Multivariable Systems

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 $(1-\delta ) r_1,\;0<\delta\leq1$, 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.