Multidirectional optimal robustness under gap and coprime factor-uncertainties

In the multivariable case optimal stability boundaries obtained by conventional ℋ^∞-methods are in general conservative with respect to perturbations occurring off the the “weakest” direction, that is, off the direction in which the gain of the closed-loop transfer matrix is maximal. By choosing a compensator that minimizes all closed-loop singular values, a robustness criterion may be derived in which this directional conservatism is removed by maximally expanding the stability boundary in all principal gain directions. This is carried out in the context of gap-metric by introducing an appropriate weighted directed gap that strengthens the optimal robustness criterion of the ordinary gap in the above sense. Moreover, it is shown that multidirectional robustness optimization in the weighted directed gap is equivalent to multidirectional robustness optimization under weighted coprime factor uncertainties