Asymptotic properties of linear filter for deterministic processes

Abstract

It is known that Kalman–Bucy filter is stable with respect to initial conditions under the assumptions of uniform complete controllability and uniform complete observability. In this paper, we prove the stability of Kalman-Bucy filter for the case of noise free dynamical system, i.e., for deterministic processes. The earlier stability results cannot be applied for this case, as the system is not controllable. We further show that the optimal linear filter for a general class of non-Gaussian initial conditions is asymptotically proximal to Kalman–Bucy filter. It is also shown that the filter corresponding to non-zero system noise in the limit of small system noise approaches the filter corresponding to zero system noise in the case of Gaussian initial conditions.