A Quantitative Relationship between Non-Linearity and Unscented Kalman Filter Accuracy

Abstract

The relationship between a quantitative measure of non-linearity and the Unscented Kalman Filter (UKF) performance relative to the Extended Kalman Filter (EKF) is presented. The Non-linearity Index of an n-dimensional dynamic system is expressed as the supremum Hilbert-Schmidt (HS) norm of the change in the state transition matrix, normalized by the HS norm of the true state transition matrix. The maximum possible difference in the EKF and the UKF error depends on the sum of the Non-linearity Indices of the system and the measurement models. A launch vehicle trajectory estimation problem, a satellite orbit estimation problem and a re-entry vehicle position estimation problem are examined to verify this relation. It is observed that the difference in the EKF and the UKF errors remain within a limited range related to the Nonlinearity Indices, which corresponds to the theoretical derivation. Using this relation, it is possible to forecast the estimation performance improvement offered by the UKF relative to the EKF for a given non-linear system and measurement. This provides a quantitative approach to selecting between the UKF and the EKF.

Publication
(submitted) Automatica
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