Probability distribution and moments of the first-excursion time for dynamic systems under non-Poisson impulse processes

Abstract

The present paper deals with the problem of the first-excursion time for the response of dynamic systems to non-Poisson (renewal) impulse process excitations. As the state vector of the dynamic system is a non-Markov process, no method is directly available for the derivation of the equations for the cumulative distribution function or moments of the first-excursion time. The original non-Markov problem is converted into a Markov one by recasting the excitation process with the aid of an auxiliary, pure-jump stochastic process characterized by a Markov chain, leading to the augmentation of the state space of the dynamic system by auxiliary Markov states. The conditional probability density—discrete probability distribution function characterizing jointly the augmented state vector, as well as the cumulative distribution function of the first-excursion time (defined as a mixed-type, joint cumulative distribution-discrete probability function), satisfy the set of backward integro-differential Chapman-Kolmogorov equations. The recursive integro-differential equations for statistical moments of the first-excursion time are also obtained. For four example non-Poisson impulse processes the explicit equations are obtained.