On stochastic perturbation theory for linear systems

This paper is concerned with the dynamic response of a general class of linear systems the parameters of which undergo step perturbations. The maximum size of these perturbations are limited by the stability bounds of the perturbed system. Using the assumptions that : (a) the system´s input vector is a stationary process, random or deterministic; and (b) the step perturbations are random and statistically independent of the system input vector, a technique is developed for the exact analysis of the dynamic response of the system. It is shown that the state and output covariance matrices of the perturbed system can be represented by first-order matrix differential equations. It is indicated that a quasi-linear mathematical model of the perturbation process may be obtained for small perturbations. This model describes the dynamics of the parameter-perturbation-transmission path from the perturbations in the parameters to the dynamic response in the system covariance matrix. The theory is developed for continuous linear systems, but may be analogously extended to discrete cases.