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.

中文翻译：

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精确混合型连续离散扩展和五度培养皿卡尔曼滤波器的平方根方法

近来已经为离散和连续时间非线性随机系统开发了五度库曼卡尔曼滤波器（5D-CKF）。在已发表的著作中，已经提到了数值稳定的平方根5D-CKF实现方法，尽管对于三阶CKF来说很容易获得，但不可行。基本的数学推导的关键问题是在五度定律中出现了负权系数，这阻止了过滤器方程的平方根分解。作者通过利用双曲线变换代替了工程文献中传统的用于平方根的常用变换，解决了5D-CKF方法中存在的这个基本问题。二维码以及SVD转换。理论结果在连续离散混合型估计器ACD-EKF-5DCKF的情况下得到了说明，并且可以应用于任何其他5D-CKF策略。还提供了数值实验。