WORST-CASE PROPERTIES OF THE UNIFORM-DISTRIBUTION AND RANDOMIZED ALGORITHMS FOR ROBUSTNESS ANALYSIS

In this paper we study a probabilistic approach which is an alternativ e to the classical worst-case algorithms for robustness analysis and d esign of uncertain control systems. That is, we aim to estimate the pr obability that a control system with uncertain parameters q restricted to a box I! attains a given level of performance gamma. Since this pr obability depends on the underlying distribution, we address the follo wing question: What is a ''reasonable'' distribution so that the estim ated probability makes sense? To answer this question, we define two w orst-case criteria and prove that the uniform distribution is optimal in both cases. In the second part of the paper we turn our attention t o a subsequent problem. That is, we estimate the sizes of both the so- called ''good'' and ''bad'' sets via sampling. Roughly speaking, the g ood set contains the parameters q is an element of Q with a performanc e 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 ga mma. We give bounds on the minimum sample size to attain a good estima te of these sets in a certain probabilistic sense.