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Low-rank approximated Kalman-Bucy filters using Oja's principal component flow for linear time-invariant systems

Daiki Tsuzuki, Kentaro Ohki

TL;DR

This work extends the low-rank approximated Kalman--Bucy filter by removing the requirement that the system matrix $A$ be symmetric, and provides a rigorous analysis of the Oja flow for general real matrices. By deriving explicit equilibrium points and their local stability, as well as a domain-of-attraction estimate, the paper establishes conditions under which the LRKB estimator maintains a bounded error covariance for controllable and observable LTI systems. A central result is a rank condition: if the reduced rank $r$ is at least the number of unstable eigenvalues of $A$ (i.e., $r \ge r'$), then the LRKB filter yields a stable closed-loop and bounded estimation error; otherwise, the error may diverge. The findings offer theoretical guarantees for scalable Kalman filtering in large-scale settings and suggest directions for extending the approach to time-varying, discrete-time, and nonlinear scenarios.

Abstract

The Kalman-Bucy filter is extensively utilized across various applications. However, its computational complexity increases significantly in large-scale systems. To mitigate this challenge, a low-rank approximated Kalman--Bucy filter was proposed, comprising Oja's principal component flow and a low-dimensional Riccati differential equation. Previously, the estimation error was confirmed solely for linear time-invariant systems with a symmetric system matrix. This study extends the application by eliminating the constraint on the symmetricity of the system matrix and describes the equilibrium points of the Oja flow along with their stability for general matrices. In addition, the domain of attraction for a set of stable equilibrium points is estimated. Based on these findings, we demonstrate that the low-rank approximated Kalman--Bucy filter with a suitable rank maintains a bounded estimation error covariance matrix if the system is controllable and observable.

Low-rank approximated Kalman-Bucy filters using Oja's principal component flow for linear time-invariant systems

TL;DR

This work extends the low-rank approximated Kalman--Bucy filter by removing the requirement that the system matrix be symmetric, and provides a rigorous analysis of the Oja flow for general real matrices. By deriving explicit equilibrium points and their local stability, as well as a domain-of-attraction estimate, the paper establishes conditions under which the LRKB estimator maintains a bounded error covariance for controllable and observable LTI systems. A central result is a rank condition: if the reduced rank is at least the number of unstable eigenvalues of (i.e., ), then the LRKB filter yields a stable closed-loop and bounded estimation error; otherwise, the error may diverge. The findings offer theoretical guarantees for scalable Kalman filtering in large-scale settings and suggest directions for extending the approach to time-varying, discrete-time, and nonlinear scenarios.

Abstract

The Kalman-Bucy filter is extensively utilized across various applications. However, its computational complexity increases significantly in large-scale systems. To mitigate this challenge, a low-rank approximated Kalman--Bucy filter was proposed, comprising Oja's principal component flow and a low-dimensional Riccati differential equation. Previously, the estimation error was confirmed solely for linear time-invariant systems with a symmetric system matrix. This study extends the application by eliminating the constraint on the symmetricity of the system matrix and describes the equilibrium points of the Oja flow along with their stability for general matrices. In addition, the domain of attraction for a set of stable equilibrium points is estimated. Based on these findings, we demonstrate that the low-rank approximated Kalman--Bucy filter with a suitable rank maintains a bounded estimation error covariance matrix if the system is controllable and observable.
Paper Structure (9 sections, 16 theorems, 23 equations)

This paper contains 9 sections, 16 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminary: Kalman--Bucy filter and its low-rank approximation
  3. Kalman--Bucy filter
  4. Low-Rank approximated Kalman--Bucy filter
  5. Equilibrium points and convergence of the Oja flow
  6. Equilibrium points of the Oja flow
  7. Convergence and domain of attraction of the Oja flow
  8. Boundedness of estimation error
  9. Conclusion

Key Result

Proposition 1

If the system $(A, G, C)$ is controllable and observable, the following algebraic Riccati equation yields a unique positive definite solution $\hat{P}_s$ and $A-\hat{P}_sC^{\top}{(HH^{\top})}^{-1}C$ is stable. Furthermore, the solution $\hat{P}(t)$ to Equation eq_hatP converges to $\hat{P}_s$.

Theorems & Definitions (32)

  • Proposition 1: bucy1987fsp
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • Proposition 3
  • proof
  • ...and 22 more