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 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-δ)r₁, 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 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 links with μ-analysis are pursued