This paper addresses the problem of square-rooting in the Unscented Kalman Filtering (UKF) methods rooted in the It$\widehat{\mathrm{o}}$-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.

本文针对以It为根的Unscented Kalman滤波（UKF）方法中的平方根问题$\widehat{\mathrm{Ø}}$-泰勒强阶近似1.5。自从发现以来，UKF已经成为最强大的状态估计手段之一，因为它在众多具有实用价值的随机系统（包括连续离散系统）中均具有出色的性能。此外，该技术的主要缺点是需要在其时间和测量更新步骤中导出协方差矩阵的Cholesky分解。这种分解非常耗时，并且对舍入和计算过程中发生的其他错误非常敏感，这可能导致失去协方差的正性，从而使Cholesky分解失败。通常通过平方根滤波器实现来克服后一个问题，该实现不传播协方差本身，而仅传播其平方根（Cholesky因子）。很遗憾，UKF在高维随机系统中的应用产生的负权重无法设计常规的平方根UKF方法。我们使用低阶Cholesky因子更新程序或双曲线来解决它用于产生J正交平方根的QR变换。我们在理论上证明了我们新颖的平方根滤波器，并在飞行控制场景中对现有的UKF进行了数值检验和比较。