It[formula omitted]-Taylor-based square-root unscented Kalman filtering methods for state estimation in nonlinear continuous-discrete stochastic systems

Abstract

This paper addresses the problem of square-rooting in the Unscented Kalman Filtering (UKF) methods rooted in the Ito^-Taylor approximation of the strong order 1.5. Since its discovery the UKF has become one of the most powerful state estimation means because of its outstanding performance in numerous stochastic systems of practical value, including continuous-discrete ones. Besides, the main shortcoming of this technique is the need for the Cholesky decomposition of covariance matrices derived in its 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 computation, which can result in losing the covariance’s positivity and, hence, failing the Cholesky decomposition. The latter problem is usually overcome by means of square-root filter implementations, which propagate not the covariance itself but its square root (Cholesky factor), only. Unfortunately, negative weights arising in applications of the UKF to high-dimensional stochastic systems preclude from designing conventional square-root UKF methods. We resolve it with low-rank Cholesky factor update procedures or with hyperbolic QR transforms used for yielding J-orthogonal square roots. Our novel square-root filters are justified theoretically and examined and compared numerically to the existing UKF in a flight control scenario.