Abstract

This paper focuses on the optimal minimum mean square error estimation of a nonlinear function of state (NFS) in linear Gaussian continuous-time stochastic systems. The NFS represents a multivariate function of state variables which carries useful information of a target system for control. The main idea of the proposed optimal estimation algorithm includes two stages: the optimal Kalman estimate of a state vector computed at the first stage is nonlinearly transformed at the second stage based on the NFS and the minimum mean square error (MMSE) criterion. Some challenging theoretical aspects of analytic calculation of the optimal MMSE estimate are solved by usage of the multivariate Gaussian integrals for the special NFS such as the Euclidean norm, maximum and absolute value. The polynomial functions are studied in detail. In this case the polynomial MMSE estimator has a simple closed form and it is easy to implement in practice. We derive effective matrix formulas for the true mean square error of the optimal and suboptimal quadratic estimators. The obtained results we demonstrate on theoretical and practical examples with different types of NFS. Comparison analysis of the optimal and suboptimal nonlinear estimators is presented. The subsequent application of the proposed estimators demonstrates their effectiveness.

中文翻译：

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线性高斯连续动力系统中状态向量非线性函数的两阶段估计算法

摘要

本文重点研究线性高斯连续时间随机系统中非线性状态函数（NFS）的最优最小均方误差估计。NFS代表状态变量的多元函数，该函数携带目标系统的有用信息进行控制。所提出的最优估计算法的主要思想包括两个阶段：在第一阶段计算的状态向量的最优卡尔曼估计在第二阶段基于NFS和最小均方误差（MMSE）准则进行非线性变换。最佳MMSE估计的解析计算中一些具有挑战性的理论方面，通过使用特殊NFS的多元高斯积分来解决，例如欧几里得范数，最大值和绝对值。对多项式函数进行了详细研究。在这种情况下，多项式MMSE估计器具有简单的封闭形式，并且在实践中很容易实现。我们得出了最佳和次优二次估计量的均方根误差的有效矩阵公式。我们在不同类型的NFS的理论和实践示例中证明了获得的结果。给出了最优和次优非线性估计量的比较分析。拟议估计量的后续应用证明了其有效性。给出了最优和次优非线性估计量的比较分析。拟议估计量的后续应用证明了其有效性。给出了最优和次优非线性估计量的比较分析。拟议估计量的后续应用证明了其有效性。