Continuous-Discrete Derivative-Free Extended Kalman Filter Based on Euler-Maruyama and Itô-Taylor Discretizations

Core Concepts

The author explores continuous-discrete DF-EKF methods based on different discretization schemes, aiming to preserve information about stochastic processes while providing stable estimation procedures.

Abstract

The content discusses the development of derivative-free extended Kalman filtering methods for state estimation in nonlinear stochastic systems. Two discretization schemes, Euler-Maruyama and Itô-Taylor, are compared for their effectiveness in handling highly nonlinear systems. The novel DF-EKF methods aim to maintain information about stochastic processes while ensuring stable estimation procedures. The article highlights the trade-offs between accuracy and computational demands associated with different discretization error control strategies. Additionally, the importance of numerical stability in derivative-free Bayesian filters is emphasized, leading to the development of stable square-root implementation methods.
The paper presents detailed derivations of continuous-discrete DF-EKF algorithms within both discretization schemes. The IT-1.5 DF-EKF method is shown to outperform the EM-0.5 DF-EKF in terms of accuracy due to its higher-order integration scheme. The discussion also touches upon the challenges posed by numerical instability in traditional implementations and proposes solutions through stable square-root techniques derived using Cholesky factorization and singular value decomposition.

Stats

α = 1000 suggested in [22]
δ = ∆/L
Roundoff errors may destroy theoretical properties of filter covariance matrices.
Cholesky decomposition used for finding square-root factors.
Numerical instability in finite precision arithmetic discussed.
Stable square-root techniques derived within Cholesky factorization and SVD.

Quotes

"The new algorithms keep information about underlying stochastic processes."
"Square-root techniques derived within Cholesky factorization and SVD."
"Numerical instability may interrupt filtering process."

Key Insights Distilled From

Continuous-discrete derivative-free extended Kalman filter based on Euler-Maruyama and Itô-Taylor discretizations

by Maria V. Kul... at arxiv.org 03-08-2024

https://arxiv.org/pdf/2403.04448.pdf

Deeper Inquiries

How can numerical stability be improved in derivative-free Bayesian filters

Numerical stability in derivative-free Bayesian filters can be improved by implementing stable square-root techniques. These techniques help mitigate the effects of round-off errors that can arise during computations, especially when dealing with covariance matrix operations. By using methods like Cholesky decomposition or singular value decomposition (SVD) to find the square-root factors of matrices, numerical stability is enhanced. This ensures that the filter covariance matrix remains accurate and reliable throughout the filtering process, reducing the risk of computational errors.

What are the implications of using higher-order integration schemes for state estimation accuracy

Using higher-order integration schemes for state estimation can lead to improved accuracy in estimating system states. Higher-order schemes, such as the Itô-Taylor expansion of strong order 1.5, provide a more precise approximation of stochastic processes compared to lower-order schemes like Euler-Maruyama method (order 0.5). The increased accuracy from higher-order integration helps capture complex dynamics and nonlinearities present in stochastic systems more effectively, resulting in more accurate state estimates.

How do stable square-root techniques impact computational efficiency

Stable square-root techniques have a significant impact on computational efficiency in Bayesian filters by improving numerical stability while maintaining accuracy. By utilizing methods like Cholesky decomposition or SVD to handle square-root operations within the filtering algorithms, computational errors due to round-off are minimized. This not only enhances the reliability of filter calculations but also reduces the likelihood of unexpected interruptions during computations caused by numerical instability issues. As a result, stable square-root techniques contribute to smoother and more efficient filtering processes overall.