Lagrangian Grid-based Estimation of Nonlinear Systems with Invertible Dynamics

TL;DR

This work tackles Bayesian state estimation for nonlinear, non-Gaussian discrete-time systems by developing a nonlinear Lagrangian grid-based filter (LGbF) that respects invertible state dynamics. By aligning the prediction grid with the inverse dynamics and decomposing updates into advection and diffusion, the LGbF reduces computational complexity from $O(N^2)$ to $O(N\\log N)$ while maintaining robustness and determinism. The approach is validated on the Hénon map and a 5D coordinated turn/tracking scenario, showing comparable accuracy to particle and Eulerian GbFs but with superior stability and reduced computational burden in many settings. The method provides a practical, open-source solution for high-dimensional, nonlinear filtering in safety-critical applications. It generalizes the GbF framework to nonlinear invertible dynamics and preserves the deterministic nature advantageous for real-time implementations.

Abstract

This paper deals with the state estimation of non-linear and non-Gaussian systems with an emphasis on the numerical solution to the Bayesian recursive relations. In particular, this paper builds upon the Lagrangian grid-based filter (GbF) recently-developed for linear systems and extends it for systems with nonlinear dynamics that are invertible. The proposed nonlinear Lagrangian GbF reduces the computational complexity of the standard GbFs from quadratic to log-linear, while preserving all the strengths of the original GbF such as robustness, accuracy, and deterministic behaviour. The proposed filter is compared with the particle filter in several numerical studies using the publicly available MATLAB\textregistered\ implementation\footnote{https://github.com/pesslovany/Matlab-LagrangianPMF}.

Figures (3)

• Figure 1: Probability density function and its approximation by point-mass density.
• Figure 2: Illustration of steps of LGbF prediction using the Hénon model.
• Figure 3: Performance illustration of tracking using 5D model.