The J-Orthogonal Square-Root Euler-Maruyama-Based Unscented Kalman Filter for Nonlinear Stochastic Systems

Abstract

This paper addresses the issue of square-rooting in the Unscented Kalman Filtering (UKF) methods. Since their discovery the UKF is considered to be among the most valued state estimation algorithms because of its outstanding performance in numerous real-world applications. However, the main shortcoming of such a technique is the need for the Cholesky decomposition of predicted and filtering covariances derived in all time and measurement update steps. Such a factorization is time-consuming and highly sensitive to round-off and other errors committed in the course of calculation, which can result in losing the covariance’s positivity and, hence, in failing the Cholesky decomposition. The latter problem is usually overcome via square-root filtering implementations, which propagate not the covariance itself but only its square root (Cholesky factor). Unfortunately, negative weights arising in applications of the UKF schemes to large stochastic systems preclude from designing conventional square-root UKF methods. So, we resolve it with a hyperbolic QR factorization used for yielding J-orthogonal square roots. Our novel square-root filter is grounded in the Euler-Maruyama discretization of order 0.5. It is justified theoretically and examined and compared numerically to the conventional (non-square-root) UKF in an aircraft’s coordinated turn scenario with ill-conditioned measurements.