Response of linear dynamic systems to non-Erlang renewal impulses: Stochastic equations approach

Abstract

Linear dynamical systems under random trains of impulses driven by a class of non-Erlang renewal processes are considered. The class considered is the one where the renewal events are selected from an Erlang renewal process. The original train of impulses is recast, with the aid of an auxiliary stochastic variable, in terms of two independent Poisson processes. Thus, by augmenting the state vector of the dynamic system with the auxiliary stochastic variables, the original non-Markov problem is converted to a Markov one. The differential equations for the response statistical moments can then be derived from the generalised Ito's differential rule. Numerical results obtained for a few different models and various sets of parameters, show that the present approach allows to account for a variety of inter arrival time's probability distributions. Transient mean value and variance of the response of a linear oscillator have been obtained from the equations for moments.