Approximations of the Mortensen observer using higher order extended Kalman filters

TL;DR

This paper introduces a polynomial approximation method for the Mortensen observer using higher order extended Kalman filters, demonstrating potential for convergence through numerical experiments.

Contribution

It develops a new approximation approach for the Mortensen observer based on higher order extended Kalman filters and tensor differential equations.

Findings

Numerical experiments with polynomials up to order eight show promising results.

The method indicates local convergence to the Mortensen observer.

The approach generalizes the extended Kalman filter for higher order approximations.

Abstract

A polynomial approximation of the minimum energy estimator, also called Mortensen observer, is discussed. The method relies on successive differentiations of an underlying value function and the Hamilton-Jacobi-Bellman equation, respectively. By means of neglecting higher order derivatives of the value function along the unknown observer trajectory, a coupled set of nonlinear tensor structured differential equations is derived. In its simplest form, the approach boils down to the well-known extended Kalman filter. Numerical experiments with polynomials up to the order eight illustrate the potential of the new approach and indicate local convergence to the Mortensen observer.