The J-Orthogonal Square-Root MATLAB-Based Continuous-Discrete Unscented Kalman Filtering Method

Abstract

The paper suggests a general solution to the square-rooting problem existed for the Unscented Kalman Filter (UKF) since its appearance in the late 1990s. As properly noted in engineering literature, the previously suggested Cholesky-based UKF implementations are, in fact, the ‘pseudo’ square-root versions. Their key feature is the utilization of one-rank Cholesky update procedure required at each filtering step because of the possibly negative sigma points’ weights. In a finite precision arithmetic, the resulting downdated matrix might be not a positive definite matrix. This yields a failure of the UKF estimator in practice. We resolve this problem by suggesting a novel square-root approach based on the J-orthogonal matrix utilization for updating the required Cholesky factors. Additionally, we explain how the MATLAB language with built-in numerical integration schemes developed for solving ordinary differential equations can be easily and effectively used for accurate calculations when implementing the continuous-discrete UKF time update stage.