Optimal linear filtering for linear systems with state-dependent noise

Abstract

The purpose of this paper is to determine the linear optimal filtering algorithm for a message generated by noisy observations of a linear dynamic system with state-dependent, stochastic disturbances. These disturbances can be considered as stochastic parameter variations. As a consequence of the state-dependent noiso the message process is non-Gaussian. Hence the filter obtained by solving the Wiener-Hopf equation is only the optimal linear operation on the data. The optimal filter is non-linear. Unfortunately the dynamical equations for optimal nonlinear filtering can only be solved approximately. We show that one approximation reduces to the linear optimal filter. As an application we determine the linear optimal filter for a second-order system. This example provides us with a comparison of the performance of the linear optimal filter with a filter designed neglecting the presence of the state-dependent disturbances.