First Excursion Probabilities of Non-Linear Dynamical Systems by Importance Sampling

This paper suggests a procedure to estimate first excursion probabilities for non-linear dynamical systems subjected to Gaussian excitation. The approach is based on the mean up-crossing rate and importance sampling method. Firstly, by using of Poisson assumption and Rice formula, the equivalent linear system is carried out. The linearization principle is that non-linear and linear systems have the same up-crossing rate for a specified threshold. Secondly, an importance sampling technique is used in order to estimate excursion probabilities for the equivalent linear system. The variance of the failure probability estimates, the number of samples and the computational time are reduced significantly compared with direct Monte Carlo simulations.