State dependent Girsanov’s controls in time variant reliability estimation in randomly excited dynamical systems

Abstract The problem of time variant reliability estimation of structural dynamical systems subjected to nonstationary, Gaussian, random excitations is considered. The system equations are cast in the form of Ito’s stochastic differential equations and the problem of reliability estimation is tackled based on Monte Carlo simulations with a Girsanov transformation based sampling variance reduction scheme. The problem of time variant reliability analysis is first cast as an equivalent problem in series system reliability analysis. Novel contribution of the work lies in proposing procedures to arrive at state dependent (closed loop) Girsanov’s controls. Suboptimal Girsanov’s controls for estimating the time variant reliability are derived based on component level ideal controls, which are exactly obtainable for linear systems, and, via a local linearization step for nonlinear systems. It is shown that a simplified version of the above closed loop controls, that avoids linearization step for nonlinear systems, can be deduced by minimizing a distance measure similar to what has been done for arriving at open loop controls. Illustrations on multi-degree of freedom linear/nonlinear systems demonstrate the superior performance of the proposed method vis-a-vis the existing open loop control based methods. Limited largescale Monte Carlo simulations are used to verify the acceptability of solutions based on the proposed scheme.