Bridging a gap in Kalman filtering output estimation with correlated noises or with direct feed-through from process noise into measurements

TL;DR

This paper identifies and corrects a limitation in traditional Kalman filter output estimation when noises are correlated or have direct feed-through, improving robustness and estimation accuracy in practical applications.

Contribution

It introduces a correction term for Kalman filter output estimation under correlated noises or direct feed-through, enhancing existing methods and enabling better disturbance estimation.

Findings

Correction term improves output estimation accuracy.

Method reduces estimator complexity and enhances robustness.

Implemented correction is available in MATLAB 2016.

Abstract

Traditional statements of the celebrated Kalman filter algorithm focus on the estimation of state, but not the output. For any outputs, measured or auxiliary, it is usually assumed that the posterior state estimates and known inputs are enough to generate the minimum variance output estimate, given by yn|n = Cxn|n + Dun. Same equation is implemented in most popular control design toolboxes. It will be shown that when measurement and process noises are correlated, or when the process noise directly feeds into measurements, this equation is no longer optimal, and a correcting term is needed in above output estimation. This natural extension can allow designer to simplify noise modeling, reduce estimator order, improve robustness to unknown noise models as well as estimate unknown input, when expressed as an auxiliary output. This is directly applicable in motion control applications which exhibits such feed-through, such as estimating disturbance thrust affecting the accelerometer measurements. Based on a proof of suboptimality [1], this correction has been accepted and implemented in Matlab 2016 [2].