Probabilistic robustness analysis: Explicit bounds for the minimum number of samples

In this paper, we study robustness analysis of control systems affected by bounded uncertainty. Motivated by the difficulty to perform this analysis when the uncertainty enters into the plant coefficients in a nonlinear fashion, we study a probabilistic approach. In this setting, the uncertain parameters q are random variables bounded in a set Q and described by a multivariate density function f(q). We then pose the following question: Given a performance level, what is the probability that this level is attained? The main content of this paper is to derive explicit bounds for the number of samples required to estimate this probability with a certain accuracy and confidence a priori specified. It is shown that the number obtained is inversely proportional to these thresholds and it is much smaller than that of classical results. Finally, we remark that the same approach can be followed to study several problems in a control system context. For example, we can evaluate the worst-case H_∞ norm of the sensitivity function of multi-input multi-output systems or compute μ when the robustness margin is of interest.