The Level Set Kalman Filter for State Estimation of Continuous-discrete Systems

TL;DR

The paper introduces the Level Set Kalman Filter (LSKF), a novel approach for nonlinear state estimation in continuous-discrete systems that improves accuracy and simplifies implementation by reformulating the Fokker-Planck equation as an ODE.

Contribution

The LSKF extends Kalman filtering by reformulating the Fokker-Planck equation as an ODE, enhancing the time-update step and reducing implementation complexity compared to existing methods.

Findings

LSKF outperforms CD-CKF in numerical experiments.

LSKF simplifies implementation without requiring timestep subdivision.

LSKF does not need explicit spatial derivatives of the drift function.

Abstract

We propose a new extension of Kalman filtering for continuous-discrete systems with nonlinear state-space models that we name as the level set Kalman filter (LSKF). The LSKF assumes the probability distribution can be approximated as a Gaussian, and updates the Gaussian distribution through a time-update step and a measurement-update step. The LSKF improves the time-update step when compared to existing methods, such as the continuous-discrete cubature Kalman filter (CD-CKF) by reformulating the underlying Fokker-Planck equation as an ordinary differential equation for the Gaussian, thereby avoiding expansion in time. Together with a carefully picked measurement-update method, numerical experiments show that the LSKF has a consistent performance improvement over CD-CKF for a range of parameters, while also simplifies the implementation, as no user-defined timestep subdivision between measurements is required, and the spatial derivatives of the drift function are not explicitly needed.