Spatial discretization error in Kalman filtering for discrete-time infinite dimensional systems

TL;DR

This paper develops a reduced-order Kalman filter for infinite-dimensional systems, providing a bound on the discretization error using Riccati equations and sensitivity analysis.

Contribution

It introduces an optimal finite-dimensional state estimator for infinite-dimensional systems and derives error bounds related to spatial discretization.

Findings

Derived Riccati difference equation for error covariance

Established bounds on discretization-induced estimation errors

Provided a framework for finite element-based state estimation

Abstract

We derive a reduced-order state estimator for discrete-time infinite dimensional linear systems with finite dimensional Gaussian input and output noise. This state estimator is the optimal one-step estimate that takes values in a fixed finite dimensional subspace of the system's state space --- consider, for example, a Finite Element space. We then derive a Riccati difference equation for the error covariance and use sensitivity analysis to obtain a bound for the error of the state estimate due to the state space discretization.