Approximation of Failure Probability Using Conditional Sampling

Abstract

In analyzing systems which depend on uncertain parameters, one technique is to partition the uncertain parameter domain into a failure set and its complement, and judge the quality of the system by estimating the probability of failure. If this is done by a sampling technique such as Monte Carlo and the probability of failure is small, accurate approximation can require so many sample points that the computational expense is prohibitive. Previous work of the authors has shown how to bound the failure event by sets of such simple geometry that their probabilities can be calculated analytically. In this paper, it is shown how to make use of these failure bounding sets and conditional sampling within them to substantially reduce the computational burden of approximating failure probability. It is also shown how the use of these sampling techniques improves the confidence intervals for the failure probability estimate for a given number of sample points and how they reduce the number of sample point analyses needed to achieve a given level of confidence.

Figures of Merit for Control Verification

“…In the context of control analysis, a particularly attractive feature of the hybrid method is that its efficiency and accuracy does not depend on the robustness of the controller. 25 This sharply contrasts with the case in Monte Carlo-based methods, where accurate robustness assessments (i.e., those based on statistics having small confidence intervals) demand a number of simulations that grow exponentially with the robustness of the controller. In other words, the better the robustness, the smaller the P [F] and the larger the number of samples required to estimate this probability accurately.…”

“…The main advantage of using Equation (25) is that the MS, and therefore the PSM, are independent of the probabilistic uncertainty model assumed. This implies that bounds corresponding to arbitrary uncertainty models are trivial to evaluate since they will only require changing F p in Equation (25). Tighter upper bounds are obtained if the reference set chosen leads to a larger P [M].…”

“…In other words, the better the robustness, the smaller the P [F] and the larger the number of samples required to estimate this probability accurately. Details on the implementation of the hybrid method and its analysis are available in References [13,25].…”

A Verification-Driven Approach to Control Analysis and Tuning

“…The resulting controller not only minimizes the failure probability but also attains a RI larger or equal to the value of δ used. The hybrid method of Reference [16] is best suited for the estimation of this probability since the upper bound to P [F] corresponding to M u = X u holds for all design points in R(d,αn).…”

Ares I static tests design

“…By contrast, probabilistic engineering design addresses uncertainties explicitly [1], [2], [3]. Probability distributions are estimated for uncertain input variables.…”