Probabilistic µ-Analysis Using Mapped Uncertainties

This paper proposes a new probabilistic µ-analysis approach to compute the probability for stability of LTI systems with normally-distributed uncertainties and can set the base for probabilistic µ-analysis of uncertain systems with arbitrary continuous probability distributions. In contrast to Monte Carlo methods, which require a high number of samples over the entire uncertainty space to approximate the probability accurately, the proposed algorithm gives a guaranteed probability for stability. The probability for stability is maximized within a bi-level optimization. Mapped uncertainties are introduced and are transformed into the true uncertainty space. The inner level of the bi-level optimization calculates the singular structure value µ, which is used for the extension of the allowed uncertainty space. At the outer level of the bi-level optimization the transformation parameters are adapted such that the probability is maximized. The proposed algorithm is not limited to uniform or truncated probability distributions. The algorithm is applied to the closed-loop system of a highly-agile aircraft in the longitudinal motion for illustration.