非线性系统卡尔曼滤波的二阶近似方法改进

发布时间：2018-02-03 08:41

本文关键词： 卡尔曼滤波二阶非加性噪声非线性系统　出处：《哈尔滨工业大学》2015年硕士论文　论文类型：学位论文

【摘要】：卡尔曼滤波自诞生以来就有很多应用,但典型的卡尔曼滤波只用于线性系统。扩展卡尔曼滤波(Extended Kalman Filter(EKF))是解决非线性系统的一种方法,因为其展开点实时更新使其对非线性系统有良好的估计效果,从而EKF应用广泛。然而因为EKF在线性化时是将非线性函数Taylor展开到一阶项,忽略高阶项,所以会带来误差。尽管有二阶扩展卡尔曼滤波产生,但是文献中二阶扩展卡尔曼滤波仅应用于噪声是仿射非线性的系统,即系统关于噪声是加性的(已有文献将EKF推广到含有非加性噪声系统)。事实上带非加性噪声尤其是乘性噪声的非线性系统在实际情景中是很常见的,带非加性噪声系统的状态最优估计相应有着重要的应用价值。本文主要研究扩展卡尔曼滤波问题,给出对于一般含有非加性噪声的非线性系统的二阶扩展卡尔曼滤波的递推公式。并针对含有非加性噪声的非线性系统尤其带乘性噪声系统探讨推广的EKF与改进二阶扩展卡尔曼滤波的估计效果。第一部分主要介绍课题研究的意义背景以及研究现状。第二部分给出了高斯分布的一些性质以及最小均方误差估计的方法,在最小均方误差估计意义下一般非线性系统讨论,为给出一般非加性噪声系统的二阶扩展卡尔曼滤波公式做了准备。第三部分给出一阶扩展卡尔曼滤波对于一般非加性噪声系统的递推公式。之后提出对于非加性噪声系统的二阶扩展卡尔曼滤波公式,并给出公式推导证明,在此公式推导中应用到高斯假设以及高斯分布的高阶矩计算等众多概率理论,已经不单是如同EKF简单的Taylor展开。第四部分给出仿真算例,在之后的非加性噪声系统仿真实验中,会看到改进的二阶扩展卡尔曼滤波要优于推广的EKF。
[Abstract]:Since the birth of Kalman filter, there have been many applications. But the typical Kalman filter is only used in linear systems. Extended Kalman filter is a method to solve nonlinear systems. Because its expansion point is updated in real time, it has a good estimation effect for nonlinear system, so EKF is widely used. However, because EKF expands the nonlinear function Taylor to the first order when linearization. Although the second order extended Kalman filter is produced, the second order extended Kalman filter is only applied to noise systems which are affine nonlinear. That is, the system is additive about noise (EKF has been extended to systems with non-additive noise). In fact, nonlinear systems with non-additive noise, especially multiplicative noise, are very common in actual situations. The state optimal estimation of systems with non-additive noise has important application value. In this paper, the extended Kalman filtering problem is studied. The recurrence formula of second order extended Kalman filter for nonlinear systems with general nonadditive noise is given. The generalized EKF and its modification are discussed for nonlinear systems with non-additive noise, especially for systems with multiplicative noise. The estimation effect of progressive second order extended Kalman filter. The first part mainly introduces the significance of the research background and research status. The second part gives some properties of Gao Si distribution and the method of minimum mean square error estimation. General nonlinear systems are discussed in the sense of minimum mean square error estimation. In order to give the second order extended Kalman filter formula for the general non-additive noise system, the recursive formula of the first order extended Kalman filter for the general non-additive noise system is given in the third part, and then for the non-additive system, the recursive formula for the first order extended Kalman filter is given. Second order extended Kalman filter formula for noise systems. The formula derivation proves that this formula is applied to many probability theories such as Gao Si hypothesis and the calculation of the higher-order moments of Gao Si distribution and so on. It is not just like the simple Taylor expansion of EKF. 4th gives a simulation example, in the subsequent non-additive noise system simulation experiments. It is shown that the improved second order extended Kalman filter is superior to the extended EKF.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：TN713

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