Gaussian Bayesian Networks for Estimating Stiff Continuous-Discrete Stochastic Systems with Ill-Conditioned Measurements

TL;DR

This work tackles robust state estimation for stiff continuous–discrete stochastic systems with ill-conditioned measurements by introducing a Gaussian Bayesian Network–based Extended Kalman Filter (GBN-EKF). By recasting the EKF measurement update as BN-based inference, the method avoids explicit matrix inversions, preserving positive semi-definiteness and enhancing numerical stability in ill-conditioned scenarios while maintaining EKF-like performance in stiff regimes. Numerical experiments on Dahlquist-type SDEs and Van der Pol oscillators show that CD-GBN-EKF matches CD-EKF in well-conditioned settings and consistently reduces ARMSE under ill-conditioning, outperforming sigma-point methods that may fail due to propagation instability. The approach demonstrates the potential of BN-inspired filtering to deliver robust, inversion-free updates for challenging continuous–discrete systems with practical impact for control and signal processing applications.

Abstract

This paper introduces a Gaussian Bayesian Network-based Extended Kalman Filter (GBN-EKF) for non-linear state estimators on stiff and ill-conditioned continuous-discrete stochastic systems, with a further analysis on systems with ill-conditioned measurements. For most nonlinear systems, the Unscented Kalman Filter (UKF) and the Cubature Kalman Filter (CKF) typically outperform the Extended Kalman Filter (EKF). But, in state estimation of stochastic systems, the EKF outperforms the CKF and UKF. This paper aims to extend the advantages of the EKF by applying a Gaussian Bayesian Network approach to the EKF (GBN-EKF), and analyzing its performance against all three filters. The GBN-EKF does not utilize any matrix inversions. This makes the GBN-EKF stable with respect to ill-conditioned matrices. Further, the GBN-EKF achieves comparable accuracy to the EKF in stiff and ill-conditioned stochastic systems, while having lower root mean squared error (RMSE) under these conditions.

Figures (8)

• Figure 1: Gaussian Bayesian Network for discrete-time filtering. Each of these nodes is assumed to be normally distributed, taking the form $x \sim \mathcal{N}(\mu,\, \Sigma)$. The blue nodes are deterministic, and the pink nodes are stochastic
• Figure 2: This figure plots ARMSE vs $\delta$ for the Dahlquist SDE with $j = 1$
• Figure 3: This figure plots ARMSE vs $\delta$ for the Dahlquist SDE with $j = 3$
• Figure 4: This figure plots ARMSE vs $\delta$ for the Van der Pol oscillator SDE with well-conditioned measurements
• Figure 5: This figure plots ARMSE vs $\delta$ for the Van der Pol oscillator SDE with ill-conditioned measurements, $\sigma = 10^{-2}$