Probabilistic $\mu$-Analysis for Stability and $\mathcal{H}_{\infty}$ Performance Verification

In this paper, new μ-based algorithms are proposed in the field of probabilistic robustness analysis. The objective is to compute tight bounds on the probability for a parametrically uncertain and possibly high order system to meet some stability and performance criteria. In this approach, uncertain parameters are treated as random variables with given probability distributions. Internal stability is treated first and H
_∞
performance in the scalar case is considered next. The main contribution is to provide tight bounds on the probability measure in the latter case. The proposed algorithms are based on branch-and-bound techniques and tested on several benchmarks from the literature, demonstrating their efficiency and the potential of the probabilistic setting in reducing the conservatism of μ-analysis. They have been integrated in the SMART Robustness Analysis Library of the SMAC Toolbox developed by ONERA (http://w3.onera.fr/smac).