Worst-case properties of the uniform distribution and randomized algorithms for robustness analysis

Abstract

In this paper we study aprobabilistic approach which is an alternative to the classical worst-case algorithms for robustness analysis and design of uncertain control systems. That is, we aim to estimate the probability that a control system with uncertain parametersq restricted to a boxQ attains a given level of performance γ. Since this probability depends on the underlying distribution, we address the following question: What is a “reasonable” distribution so that the estimated probability makes sense? To answer this question, we define two worstcase criteria and prove that the uniform distribution is optimal in both cases. In the second part of the paper we turn our attention to a subsequent problem. That is, we estimate the sizes of both the so-called “good” and “bad” sets via sampling. Roughly speaking, the good set contains the parametersq∈Q with a performance level better than or equal to γ while the bad set is the set of parametersq∈Q with a performance level worse than γ. We give bounds on the minimum sample size to attain a good estimate of these sets in a certain probabilistic sense.