Probabilistic enhancement of classical robustness margins: The unirectangularity concept

The focal point of this paper is a control system subjected to parametric uncertainty. Motivated by the newly emerging theory of probabilistic robustness, the risk of performance violation is assessed with uncertainty bounds which exceed classical deterministic margins. For a wide class of problems, the Uniformity Principle (UP) developed by Barmish and Lagoa (Math. Control Signals Systems 10 (1997) 203-222), makes it possible to estimate the probability of performance satisfaction with almost no a priori statistical information about the uncertainty. The application of the UP is, however, limited to problems satisfying certain convexity and symmetricity conditions. Since such conditions are violated in many practical problems, the objective in this paper is to extend the application of the UP. To this end, by working with a so-called unirectangularity condition, a procedure is implemented for computing probabilities of performance and the associated improvements of deterministic robustness margins. That is, given any robustness radius r₀ which is computable via deterministic methods, a probabilistic enhancement of this margin R₀(ε)≥r₀ with pre-specified level of risk ε>0 is provided. The radius R₀(ε) is called a risk-adjusted robustness margin.