随机变量非线性变换统计性质的高精度逼近算法

Approximating statistics of stochastic variables after non-linear transformation with high-accuracy

应用扩展卡尔曼滤波对非线性系统进行状态估计时 ,要对系统的动力方程进行线性化 ,这就为状态的估计带来一定的误差。本文首先对非线性变换的函数进行级数展开 ,获得了随机变量经非线性变换后的真实均值和协方差表达式 ,并得到一阶线性化的均值和协方差 ,然后提出了一种精度更高的变换算法用以逼近非线性变换后的真实均值和协方差 ,理论分析和数值试验都表明新算法不仅具有更高的精度 。

This paper first pointed out that the first step of the application of the Extended Kalman Filter in state estimation of non-linear systems is to make linearization to the system dynamic equations,which cause errors in state estimation. Then through series expansion of the non-linear transformation function,formulae of the true mean and covariance of the stochastic variable resulting from the non-linear transformation are obtained,which also lead to the form of the first-order linearization. Finally a transform method with higher accuracy is introduced. And it is proved that the transform method can approximate the true mean and covariance better than the linearization method does. Both theoretical analysis and numerical experiment prove that the new method is not only more accurate than linearization but also easier to implement.