Abstract
Bilinear systems are ubiquitous in dynamics and control literature. The concept of bilinear systems is attributed to input-output coupling terms. Stochastically influenced bilinear systems are described via bilinear stochastic differential equations. Bilinear systems are attractive and popular for two reasons: (i) first, they offer closed-form solutions for time-varying as well as time-invariant settings (ii) they preserve some of the qualitative characteristics of non-linear stochastic systems. In this paper, we construct a mathematical theory of a scalar time-varying bilinear ‘Stratonovich’ stochastic differential equation with a vector random input by deriving its closed-form solution and related results. Secondly, the analytic result of the paper is applied to a series RL circuit Stochastic Differential Equation (SDE), a bilinear stochastic dynamic circuit. This paper accomplishes the noise analysis of the bilinear dynamic circuit by deriving the mean and variance equations as well. The theory of this paper hinges on the ‘Stratonovich calculus’, and characteristic function of the vector Brownian motion. This paper will be of interest to dynamists, stochasticians looking for advances in bilinear systems and their control. More precisely, this paper opens up research directions in stability and control of bilinear stochastic systems by exploiting the analytic results of this paper.
