The Convergence of the Extended Kalman Filter

TL;DR

This paper proves that the extended Kalman filter converges locally for a wide class of nonlinear systems, with errors diminishing exponentially under certain smoothness and observability conditions.

Contribution

It establishes local convergence of the extended Kalman filter for nonlinear systems with specific smoothness and observability assumptions.

Findings

Error decreases exponentially over time when initial error is small.

Convergence holds for systems that are $C^2$ and uniformly observable.

Provides theoretical guarantees for the EKF's stability in nonlinear settings.

Abstract

We demonstrate that the extended Kalman filter converges locally for a broad class of nonlinear systems. If the initial estimation error of the filter is not too large then the error goes to zero exponentially as time goes to infinity. To demonstrate this, we require that the system be $C^2$ and uniformly observable with bounded second partial derivatives.