  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.
