Linear Stochastic Dynamical System under Uncertain Load: Inverse Reliability Analysis

The reliability of a linear dynamical system driven by a partially known Gaussian load process, specified only through its total average energy, is studied. A simple dynamic parallel and series system reliability network is investigated for the failure analysis using the crossing theory of stochastic processes. The critical input power spectral density of the load process which maximizes the mean crossing rate of a parallel or series system network emerges to be fairly narrow banded and hence fails to represent the erratic nature of the random input realistically. Consequently, a tradeoff curve between the maximum mean crossing rate of the reliability network and the disorder in the input, quantitatively measured through its entropy rate, is generated. Mathematically, the generation of the tradeoff curve of nondominated solutions, known as the Pareto optimal set, leads to a nonlinear, nonconvex, and multicriteria optimization problem with conflicting objectives. A recently developed Pareto optimization technique, implemented through genetic algorithm, has been successfully exploited to solve the optimization problem. A suitable exploitation of stochastic process theory circumvents the task of handling an apparent robust optimization problem (i.e., optimization under uncertainty) and at the same time, reduces the dimensionality of the multiobjective optimization scheme in view of both the multiobjectivity and constraints in the aforementioned methodology. The optimally disordered inputs which simulate the worst performance of the system of a spring-supported coupled rod assembly, has been studied as a numerical illustration of the mathematical formulation.