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

In this paper we study a probabilistic 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 parameters q restricted to a box I! attains a given level of performance gamma. 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 worst-case 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 parameters q is an element of Q with a performance level better than or equal to gamma while the bad set is the set of parameters q is an element of Q with a performance level worse than gamma. We give bounds on the minimum sample size to attain a good estimate of these sets in a certain probabilistic sense.