Asymptotic Properties of Linear Filter for Noise Free Dynamical System

TL;DR

This paper investigates the stability and asymptotic behavior of the Kalman-Bucy filter in noise-free dynamical systems, extending known results to cases lacking controllability and exploring limits with small system noise.

Contribution

It proves the stability of the Kalman-Bucy filter for noise-free systems and analyzes the asymptotic proximity of optimal linear filters for non-Gaussian initial conditions.

Findings

Kalman-Bucy filter is stable in noise-free systems despite lack of controllability.

Optimal linear filters for certain non-Gaussian initial conditions asymptotically approach Kalman-Bucy filter.

Filter with small system noise converges to the zero-noise filter for Gaussian initial conditions.

Abstract

It is known that Kalman-Bucy filter is stable with respect to initial conditions under the conditions of uniform complete controllability and uniform complete observability (Bishop et. al 2017, Ocone et. al 1996). In this paper, we prove the stability of Kalman-Bucy filter for the case of noise free dynamical system. The earlier stability results cannot be applied for this case, as the system is not controllable at all. We further show that the optimal linear filter for certain 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.