Square-rooting approaches to accurate mixed-type continuous-discrete extended and fifth-degree cubature Kalman filters

• Published in: IET radar, sonar & navigation sonar & navigation, 2020-11, Vol.14 (11), p.1671-1680
• Main Authors: Kulikova, Maria V, Kulikov, Gennady Yu
• Format: Article
• Language: English
• Subjects: 5D‐CKF methodology; 5D‐CKF strategy; authors resolve this essential problem; fifth‐degree cubature Kalman filter; fifth‐degree cubature rule; filters; gradient methods; Kalman filters; mixed‐type estimator ACD‐EKF‐5DCKF; nonlinear filters; numerically stable square‐root 5D‐CKF implementations; required square‐root factorisation; Research Article; singular value decomposition; square‐rooting approaches; stochastic processes; stochastic systems; third‐degree CKF; underlying mathematical derivation
• ISSN: 1751-8784; 1751-8792
• DOI: 10.1049/iet-rsn.2020.0161

The fifth-degree cubature Kalman filter (5D-CKF) has been recently developed for both the discrete- and continuous-time non-linear stochastic systems. In the published works, it has been mentioned that numerically stable square-root 5D-CKF implementations are not feasible to derive, although they are easily obtained for the third-degree CKF. The key problem of the underlying mathematical derivation is negative weight coefficients appeared in the fifth-degree cubature rule, which prevents the required square-root factorisation of the filters’ equations. The authors resolve this essential problem existed for the 5D-CKF methodology by utilising hyperbolic transformations instead of the usual ones traditionally used for square-rooting in the engineering literature. The authors’ solution is given within both the Cholesky and singular value decomposition (SVD) and is based on matrix calculus with the hyperbolic QR and SVD transformations involved. The theoretical results are illustrated in a case of the continuous-discrete mixed-type estimator ACD-EKF-5DCKF and can be applied to any other 5D-CKF strategy. Numerical experiments are also provided.