Probabilistic enhancement of classical robustness margins: a class of nonsymmetric distributions

The focal point of this paper is a control system subject to parametric uncertainty. Motivated by recent results in the newly emergent area of probabilistic robustness, we address the problem of risk assessment when the classical robustness margin is exceeded, without a priori knowledge of the distribution of the uncertain parameters. The only assumption is that the distribution belongs to a class F_r^a introduced in this paper. In contrast to previous work, the class F_r^a contains both symmetric and nonsymmetric distributions; only unimodality is required. For this class, we provide a new version of the truncation principle; i.e., under mild conditions on the performance specifications, the assessment of risk of performance violation can be done using only a subset of the admissible distributions, which we call truncated uniform distributions. Also, if the set of uncertainties that verify the performance specifications is convex, then it is proven that the risk can be assessed using only the “extremes” of the class of truncated uniform distributions; i.e., the assessment of the risk can be done using only a finite subset of the admissible distributions. These results are then applied in the linear matrix inequality context. Finally, a way of estimating risk is provided for the nonconvex case. The procedure presented enables the enhancement of robustness margins provided by a deterministic method