基于Euler-Maruyama和Itô-Taylor离散化的连续-离散无导数扩展卡尔曼滤波：常规实现与平方根实现
Continuous-discrete derivative-free extended Kalman filter based on Euler-Maruyama and Itô-Taylor discretizations: Conventional and square-root implementations
AI总结：
针对连续-离散非线性随机系统状态估计，本文基于Euler-Maruyama和Itô-Taylor离散化推导新型DF-EKF方法，并开发Cholesky与奇异值分解的平方根实现以提升数值稳定性，经良态和病态场景测试验证了性能。
摘要：
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。在考虑用于求解随机微分方程的Euler-Maruyama和Itô-Taylor离散化方案的基础上，我们依据无导数EKF原理推导了相关滤波器的矩方程。与近期推导的基于MATLAB的连续-离散DF-EKF技术不同，新型DF-EKF方法保留了底层随机过程的信息，并为传播步骤中的固定迭代次数提供了估计流程。此外，DF-EKF方法在处理具有高度非线性和/或不可微漂移项与观测函数的随机系统时尤为有效，但需要付出的代价是其数值稳定性（抗舍入误差能力）相较于标准EKF框架有所下降。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成果不同，新型平方根技术是在Cholesky分解和奇异值分解两种框架下推导得到的。我们通过多项数值测试（包括良态和病态场景）验证了新型滤波器的性能。
英文摘要：
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 Itô-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.
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