A Frequency-Domain Characterization of Optimal Error Covariance for the Kalman-Bucy Filter

TL;DR

This paper provides a frequency-domain integral characterization of the optimal error covariance in the Kalman-Bucy filter, linking it to plant dynamics and noise statistics through algebraic Riccati equations and Bode integrals.

Contribution

It introduces a novel frequency-domain framework for analyzing the algebraic Riccati equation in Kalman-Bucy filtering, connecting it to classical control theory concepts.

Findings

Explicit frequency-domain expression for error covariance trace.

Relation between Riccati equation and Bode integral.

Reduction to existing results through the new framework.

Abstract

In this paper, we discover that the trace of the division of the optimal output estimation error covariance over the noise covariance attained by the Kalman-Bucy filter can be explicitly expressed in terms of the plant dynamics and noise statistics in a frequency-domain integral characterization. Towards this end, we examine the algebraic Riccati equation associated with Kalman-Bucy filtering using analytic function theory and relate it to the Bode integral. Our approach features an alternative, frequency-domain framework for analyzing algebraic Riccati equations and reduces to various existing related results.