随机变量经非线性变换后统计性质的确定

扩展卡尔曼滤波应用于非线性系统的状态估计时，首先要对系统的动力方程进行线性化，这样就给状态的估计带来一定的误差。文中通过对非线性变换的函数进行级数展开，获得了非线性变换后随机变量的真实均值和协方差的表达式，并得到一阶线性化的形式；然后提出了一种精度更高的变换算法，并从理论上证明了该变换算法较之线性化的做法能够更好地逼近非线性变换后的真实均值和协方差，数值试验也证明了这一点。

A High-Accuracy Method for Computing Statistics of Stochastic Variables after Nonlinear Transform

In this paper it is firstly ponited out that the first step of the application of the Extended Kalman Filter in state estimation of nonlinear systems is to make linearization to the system dynamic equations, which causes errors in state estimation. Then through series expansion of the nonlinear transformation function, expressions of the true mean and covariance of the stochastic variable that results from nonlinear transformation are obtained, which also result in the form of first-order linearization. Finally a transform method with higher accuracy is introduced. It is proved that the transform method can approximate the true mean and covariance better than the linearization method does. The result is confirmed through a numerical experiment.