Theory of Stochastic Linear Systems with Gaussian Parameter Variations

Abstract

A theory of linear systems with randomly varying parameters is developed under the assumption that the parameter processes are Gaussian processes obtained by linear filtering of white noise. By this device, the restriction to white-noise parameter processes of previous studies has been removed. By a method related somewhat to a technique used in the theory of the multiple scattering of light, a system of linear integral equations for the determination of various second-order moments of the system output is derived. This system can be solved explicitly in some interesting cases. A theory of stability in mean and in mean square is given for random systems. It was found that mean square stability depended only on the values of the auto- and cross-correlation functions of the parameter processes at the origin and not on the detailed structure of these functions. The stability theory is applied to an RLC circuit with randomly varying capacitance. The possibility of stabalizing unstable deterministic systems with random noise is discussed.