Square-root high-degree cubature Kalman filters for state estimation in nonlinear continuous-discrete stochastic systems

Abstract

This paper addresses the problem of square-rooting in the Cubature Kalman Filtering (CKF) originated from Arasaratnam and Haykin in 2009. Presently, this technique has been accommodated to various cubature rules, including high-degree ones. Since its discovery the CKF has become one of the most powerful state estimation methods because of outstanding performance and robustness in numerous engineering applications. Its high-degree versions are shown to be accurate and even comparable to particle filters, which are considered to be among the most effective algorithms for treating nonlinear stochastic systems. However, the lack of square-root implementations within high-degree CKFs makes them vulnerable to round-off and other errors committed because of a potential covariance matrix positivity loss, which may encounter in practice. This shortcoming affects severely and fails high-degree CKFs since the Cholesky factorization of predicted and filtering covariances underlying the filters in use may not be fulfilled for indefinite matrices. Here, we resolve it by means of hyperbolic QR transforms applied for yielding J-orthogonal square roots. Our novel square-root algorithms are justified theoretically and examined and compared numerically to the existing non-square-root CKF and some other available filters in a simulated flight control scenario, including that with ill-conditioned measurements.