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Robust stabilisation of multivariable systems under co-prime factor perturbations: Directionality and super-optimisation

Robust stabilisation of MIMO LTI systems under normalised co-prime factor unstructured uncertainty is considered. The maximal robust stability radius ε* is derived and a “worst-case” direction is identified, along which all boundary uniformly-destabilising perturbations are shown to lie, i.e. all perturbations of norm ε* which destabilise the closed-loop system for every optimal (maximally robust) controller. By imposing a parametric constraint on the projection of admissible perturbations along this direction (uniformly in frequency), it is shown that it is possible to extend the robust stability radius in every other direction, using a subset of all optimal (maximally-robust) controllers, by solving a super-optimal Nehari extension problem. A closed-form expression is obtained for the constrained robust stability radius, μ*(δ) which depends on the first two-superoptimal levels of the closed-loop system, while the identified “worst-case” direction corresponds to the maximal Schmidt pair of a Hankel operator related to the problem. The paper extends the results of [GHJ00] for additive uncertainty models to the co-prime uncertainty case.