Maximally Robust Controllers for Multivariable Systems

Abstract

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.

An explicit state-space solution to the one-block super-optimal distance problem

“…Early references report applications in the areas of disturbance rejection [Kwa86], robust stabilization [KN89], [Nym95] and hierarchical H ∞ design [HJ98a], [HJW97]. Applications of super-optimization in the areas of robust stabilization and structured-singular value approximations can be found in [GHJ00] and [JHMG06].…”

“…Reference [GHJ00] applies super-optimization techniques in the area of maximal robust-stabilization of LTI systems under additive perturbations: Explicit expressions for the improved robust stability radius are derived by imposing structure on the perturbation set via a uniform frequency constraint in the mostcritical direction which is identified. The method is also used in [GHJ00], [JHMG06] to derive an upper bound on the structured singular value for multivariable systems in the case of complex structured block-diagonal perturbations, which is tighter than the convex upper bound provided by the "D-iteration". In this context, the multiplicity of the largest Hankel singular value becomes a crucial consideration, which motivates the detailed analysis of the general problem presented in this paper.…”

“…Nevertheless, in the present work and also others (e.g. [PF85], [KHJ07]) it is argued that super-optimization fits naturally within the modern robust control-theoretic framework, and can be used to define hierarchical optimization problems in which additional performance and stability objectives can be addressed [PF85], [GHJ00].…”

On the Gap Between the Complex Structured Singular Value and Its Convex Upper Bound

“…In our previous work [14] we obtained bounds on µ(M ) by embedding the underlying block-structured uncertainty set within a larger set. This was constructed by imposing the least-conservative bound on the projection of the structured uncertainty in the direction defined by the singular vectors corresponding to the smallest singular value of M .…”

“…The method has been used successfully for real and mixed-type of structured uncertainty, resulting in algorithms with excellent computational performance [15,17]. In its dynamic version, this technique was also used in [14] to identify the set of all maximally-robust controllers which guarantee robust stability for the largest possible class of unstructured additive perturbations containing the uncertainty ball of maximum radius as a subset. (In this case, directionality arises from the Schmidt vectors of a Hankel operator related to the problem; these remain invariant for all maximally-robust controllers).…”