In this paper, we continue to study the derivative-free extended Kalman filtering (DF-EKF) framework for state estimation of continuous-discrete nonlinear stochastic systems. Having considered the Euler-Maruyama and It\^{o}-Taylor discretization schemes for solving stochastic differential equations, we derive the related filters' moment equations based on the derivative-free EKF principal. In contrast to the recently derived MATLAB-based continuous-discrete DF-EKF techniques, the novel DF-EKF methods preserve an information about the underlying stochastic process and provide the estimation procedure for a fixed number of iterates at the propagation steps. Additionally, the DF-EKF approach is particularly effective for working with stochastic systems with highly nonlinear and/or nondifferentiable drift and observation functions, but the price to be paid is its degraded numerical stability (to roundoff) compared to the standard EKF framework. To eliminate the mentioned pitfall of the derivative-free EKF methodology, we develop the conventional algorithms together with their stable square-root implementation methods. In contrast to the published DF-EKF results, the new square-root techniques are derived within both the Cholesky and singular value decompositions. A performance of the novel filters is demonstrated on a number of numerical tests including well- and ill-conditioned scenarios.

翻译：本文继续研究用于连续-离散非线性随机系统状态估计的无导数扩展卡尔曼滤波（DF-EKF）框架。在采用欧拉-丸山和伊藤-泰勒离散化方案求解随机微分方程的基础上，我们基于无导数EKF原理推导了相关滤波器的矩方程。与近期提出的基于MATLAB的连续-离散DF-EKF技术相比，新型DF-EKF方法保留了底层随机过程的信息，并在传播步骤中为固定迭代次数提供估计过程。此外，DF-EKF方法在处理具有高度非线性和/或不可微漂移函数及观测函数的随机系统时尤为有效，但代价是其数值稳定性（对舍入误差）较标准EKF框架有所降低。为消除无导数EKF方法的上述缺陷，我们开发了常规算法及其稳定的平方根实现方法。与已发表的DF-EKF结果不同，新型平方根技术同时在乔列斯基分解和奇异值分解框架下推导得出。通过一系列包含良态和病态场景的数值测试，验证了新型滤波器的性能。