Low-rank approximated Kalman filter using Oja's principal component flow for discrete-time linear systems (2407.05675v2)

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.

• The paper introduces a low-rank Kalman filter that leverages Oja's principal component flow to approximate the state covariance in a reduced-dimensional space.
• It achieves a reduction in computational complexity from O(n³) to O(n²) while maintaining bounded mean square error under specific stability conditions.
• The approach is validated through numerical simulations, demonstrating its practical effectiveness for real-time state estimation in high-dimensional systems.

Low-rank Approximated Kalman Filter Using Oja's Principal Component Flow for Discrete-time Linear Systems

The paper "Low-rank approximated Kalman filter using Oja's principal component flow for discrete-time linear systems" introduces a low-rank approximation of the Kalman filter specifically designed for discrete-time linear systems. The primary motivation for this research stems from the computational burdens associated with using Kalman filters in high-dimensional systems. The proposed method leverages the Oja principal component flow to approximate the state covariance matrix in a lower-dimensional space, ultimately improving computational efficiency.

The traditional Kalman filter, fundamental for state estimation problems across various applications, encounters significant computational challenges due to its reliance on solving the Riccati equation, particularly as system dimensionality increases. To mitigate these challenges, the authors draw inspiration from approaches that modify the Riccati equation. Notably, previous work by Yamada and Ohki established low-rank Kalman-Bucy filters for continuous-time systems, guaranteeing bounded estimation errors under certain conditions. This paper extends those ideas to discrete-time systems, addressing gaps in the current literature on low-rank filtering methods applicable to discrete-time settings.

Proposed Method and Theoretical Contributions

The novel low-rank approximated Kalman filter (hereafter LKF) proposed in this paper integrates the Oja flow for principal component approximation into the Kalman filtering framework. The primary steps of the LKF include:

1. Initialization: The state and covariance are initialized in lower dimensions using carefully chosen matrices.
2. Oja Flow Solution: The Oja flow is solved to maintain the low-rank approximation's stability. This step involves projecting the high-dimensional covariance matrix into a lower-dimensional subspace.
3. Kalman Gain Calculation: A reduced-dimensional Kalman gain is computed, significantly decreasing computational costs.
4. State Estimation Updates: The state estimation is updated using the aforementioned low-dimensional matrices.
5. Error Covariance Update: The error covariance matrix is updated using a reduced-dimensional Riccati equation.

A significant theoretical contribution is the derivation of conditions under which the mean square error (MSE) of the proposed LKF remains bounded. The authors accomplish this by proving the stability of the error covariance matrix updates and delineating conditions for choosing the dimension of the reduced subspace (r). Specifically, it is shown that for the filter to remain stable, r must be at least as large as the number of unstable eigenvalues of the system matrix.

Numerical Results and Computational Efficiency

The authors validate the effectiveness of the proposed LKF through numerical simulations, showcasing scenarios where the Kalman filter's computational burden becomes infeasible. The LKF is notably efficient, reducing the complexity from $O(n^3)$ to $O(n^2)$, an improvement particularly beneficial when dealing with large-dimensional state and observation matrices.

The algorithm's performance in terms of computational efficiency is illustrated using a detailed comparison with traditional Kalman filters (KF) and information filters (IF). The per-step computational burden of LKF shows a marked reduction, making it a practical alternative for large-scale systems. A significant observation is the demonstrated trade-off between filter accuracy and computational efficiency. The simulations reveal that the LKF's error remains within acceptable bounds while providing substantial computational savings.

Implications and Future Work

The proposed LKF has significant implications for practical applications wherein high-dimensional state-space models are typical, such as atmospheric modeling, financial systems, and power grid management. The reduction in computational complexity makes real-time state estimation feasible in such domains, potentially enabling more responsive and adaptive system designs.

From a theoretical standpoint, the use of Oja's principal component flow to maintain a low-rank structure in the error covariance matrix opens new avenues for research in approximation methods in state estimation tasks. Future research could extend this approach to more complex settings, including nonlinear systems and adaptive filtering scenarios.

The authors also suggest potential improvements and expansions of their work, mentioning the importance of developing low-rank Kalman filters for more general discrete-time linear time-varying and nonlinear systems. Notably, integrating reduced QR-decomposition-based methods for continuous-time systems might provide further insights and robustness to the proposed discrete-time approaches.

Conclusion

In summary, this paper makes a significant contribution by presenting a low-rank Kalman filter for discrete-time linear systems that effectively balances computational efficiency and estimation accuracy. By ensuring bounded estimation errors through rigorous theoretical analysis, the proposed method stands as a viable alternative for high-dimensional systems, with practical implications across diverse applications. Future work will undoubtedly build upon these foundational results, exploring further improvements and extensions to broader system classes and application domains.