Convergence of discrete time Kalman filter estimate to continuous time estimate

TL;DR

This paper analyzes how the discrete time Kalman filter estimate approaches the continuous time estimate as the discretization becomes finer, providing convergence rates for various system types including infinite-dimensional and analytic semigroup cases.

Contribution

It derives convergence rate estimates for discrete Kalman filter estimates approaching continuous estimates across different system classes, including finite and infinite dimensional systems.

Findings

Convergence rates are established for finite-dimensional systems.

Convergence rates are derived for infinite-dimensional systems with bounded or unbounded observation operators.

Specialized convergence rate results are provided for systems governed by an analytic semigroup.

Abstract

This article is concerned with the convergence of the state estimate obtained from the discrete time Kalman filter to the continuous time estimate as the temporal discretization is refined. We derive convergence rate estimates for different systems, first finite dimensional and then infinite dimensional with bounded or unbounded observation operators. Finally, we derive the convergence rate in the case where the system dynamics is governed by an analytic semigroup. The proofs are based on applying the discrete time Kalman filter on a dense numerable subset of a certain time interval $[0,T]$.