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

Motivated by the current limitations of the existing algorithms for robustness analysis, in this paper we take a different direction which follows the so-called probabilistic approach. That is, we aim to estimate the probability that a control system with uncertain parameters q restricted to a box Q attains a certain level of performance γ. Since this probability depends on the underlying density function f(q), we study the following question: What is a “reasonable” density function so that the estimated probability makes sense? To answer this question, we define two new worst-case criteria and prove that the uniform density function is optimal in both cases. In the second part of the paper, we turn our attention to a subsequent problem. That is, taking f(q) as the uniform density function, we estimate the size of the so-called “good” and “bad” sets. Roughly speaking, the good set contains the parameters q E Q that have performance level better than or equal to γ while the bad set is the set of parameters q ∈ Q that have performance level worse than γ. To estimate the size of both sets, sampling is required. Then, we give bounds on the minimum sample size to attain a given accuracy and confidence