Response of dynamic systems to renewal impulse processes: Generating equation for moments based on the integro-differential Chapman–Kolmogorov equations

In the present paper the method is developed for the derivation of differential equations for statistical moments of the state vector (response) of a non-linear dynamic system subjected to a random train of impulses. The arrival times of the impulses are assumed to be driven by a non-Poisson counting process. The state vector of the dynamic system is then a non-Markov process and no method is directly available for the derivation of the equations for response moments. 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. Hence the conversion is carried out at the expense of augmentation of the state space of the dynamic system by auxiliary Markov states. For the augmented problem the sets of forward and backward integro-differential Chapman–Kolmogorov equations are formulated. The general, generating equation for moments is obtained with the aid of the forward and backward integro-differential Chapman–Kolmogorov operators. The developed method is illustrated by the examples of several renewal impulse processes.