Reliability analysis of polynomial systems subject to p-box uncertainties

This paper proposes a reliability analysis framework for systems subject to multiple design requirements that depend polynomially on uncertain parameters. The values these polynomials take at a given realization of the uncertain parameters dictate whether that realization is a failure or a success point. In this paper, reliability analysis refers to the estimation or bounding of the probability of failure for a given model of the uncertainty. The probability distributions of the uncertain parameters are presumed to belong to a given probability box (also known as a p-box). This does not give sufficient information to determine the failure probability of such a system exactly, but does limit the range of values it might take. Two techniques for bounding this range are proposed herein. In the first approach, we calculate the p-box of the requirements functions by propagating all the hyper-rectangles defined by the p-box of the uncertain parameters. In the second approach, we find inner bounding sets of the safe and failure domains and search for the elements of the p-box that minimize and maximize the probability of such sets. Iterative refinement of the bounding sets allows tightening arbitrarily closely the offset between the actual failure probability range and the calculated outer bound. In both techniques, bounds of the functions describing the design requirements over hyper-rectangular sets are calculated and iteratively refined by expanding them using Bernstein bases.