Fast Universal Algorithms for Robustness Analysis

TL;DR

This paper introduces fast, universal randomized algorithms for robustness analysis of uncertain systems, capable of handling arbitrary requirements without relying on deterministic robustness margins, and includes a novel, simple formula for binomial confidence intervals.

Contribution

The paper presents a new class of efficient, universal algorithms for probabilistic robustness analysis that do not depend on deterministic margin computation, with a novel explicit binomial confidence interval formula.

Findings

Algorithms are universally applicable to various robustness problems.

The explicit binomial confidence interval formula is simple and tight.

The methods outperform traditional approaches in efficiency and accuracy.

Abstract

In this paper, we develop efficient randomized algorithms for estimating probabilistic robustness margin and constructing robustness degradation curve for uncertain dynamic systems. One remarkable feature of these algorithms is their universal applicability to robustness analysis problems with arbitrary robustness requirements and uncertainty bounding set. In contrast to existing probabilistic methods, our approach does not depend on the feasibility of computing deterministic robustness margin. We have developed efficient methods such as probabilistic comparison, probabilistic bisection and backward iteration to facilitate the computation. In particular, confidence interval for binomial random variables has been frequently used in the estimation of probabilistic robustness margin and in the accuracy evaluation of estimating robustness degradation function. Motivated by the importance of fast computing of binomial confidence interval in the context of probabilistic robustness analysis, we have derived an explicit formula for constructing the confidence interval of binomial parameter with guaranteed coverage probability. The formula overcomes the limitation of normal approximation which is asymptotic in nature and thus inevitably introduce unknown errors in applications. Moreover, the formula is extremely simple and very tight in comparison with classic Clopper-Pearson's approach.