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

Daiki Tsuzuki, Kentaro Ohki

TL;DR

This work tackles the computational burden of Kalman filtering in large-scale linear systems by introducing a discrete-time low-rank Kalman filter built on the Oja flow to maintain the dominant subspace. It represents the prior covariance as $P_{k|k-1} \approx U_k \tilde{R}_{k|k-1} U_k^T$ with $U_k$ on the Stiefel manifold and evolves state estimates with a low-rank Riccati update. A rank-based stability condition is established: if the system has $r'$ unstable eigenvalues and $r \ge r'$, the closed-loop dynamics are Schur stable, ensuring bounded mean-square error, while reducing per-step complexity to $O(n^2)$ via the Sherman–Morrison–Woodbury formula. Numerical simulations validate boundedness and demonstrate that increasing $r$ improves steady-state estimation accuracy, highlighting the method's scalability and practical impact for large-scale state estimation.

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.

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

TL;DR

This work tackles the computational burden of Kalman filtering in large-scale linear systems by introducing a discrete-time low-rank Kalman filter built on the Oja flow to maintain the dominant subspace. It represents the prior covariance as with on the Stiefel manifold and evolves state estimates with a low-rank Riccati update. A rank-based stability condition is established: if the system has unstable eigenvalues and , the closed-loop dynamics are Schur stable, ensuring bounded mean-square error, while reducing per-step complexity to via the Sherman–Morrison–Woodbury formula. Numerical simulations validate boundedness and demonstrate that increasing improves steady-state estimation accuracy, highlighting the method's scalability and practical impact for large-scale state estimation.

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.
Paper Structure (12 sections, 19 equations, 3 figures, 1 table)

This paper contains 12 sections, 19 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. previous work
  3. Kalman filter
  4. Oja flow
  5. main results
  6. Low-rank Kalman filter for discrete-time systems
  7. Stability of low-rank Kalman filter
  8. numerical simulation
  9. Verification of the bounded estimations
  10. Impact of the choice of $r$
  11. conclusion
  12. Acknowledgement

Figures (3)

  • Figure 1: The comparison about the time complexity between KF and LKF. The upper region of each curve means that LKF has less time complexity.
  • Figure 2: Time evolution of the mean square one-step ahead estimation errors ${\mathrm{Tr}} [ \tilde{V}_{k+1|k} ]$
  • Figure 3: ${\mathrm{Tr}} [ \tilde{V}_{\infty} ]/\mathrm{Tr}[\hat{P}]$ versus $r$

Theorems & Definitions (6)

  • proof
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