Low-rank approximated Kalman filter using Oja's principal component flow for discrete-time linear systems

TL;DR

This paper introduces a low-rank Kalman filter for discrete-time systems using Oja's principal component flow, reducing computational complexity while maintaining bounded error properties, validated through numerical simulations.

Contribution

It develops a discrete-time low-rank Kalman filter based on Oja's flow, extending continuous-time approaches and analyzing its system properties for bounded error performance.

Findings

Effective reduction in computational complexity

Bounded mean square error under certain conditions

Numerical simulations confirm practical viability

Abstract

The Kalman filter is indispensable for state estimation across diverse fields but faces computational challenges with higher dimensions. Approaches such as Riccati equation approximations aim to alleviate this complexity, yet ensuring properties like bounded errors remains challenging. Yamada and Ohki introduced low-rank Kalman-Bucy filters for continuous-time systems, ensuring bounded errors. This paper proposes a discrete-time counterpart of the low-rank filter and shows its system theoretic properties and conditions for bounded mean square error estimation. Numerical simulations show the effectiveness of the proposed method.