卡尔曼滤波器:数据融合

卡尔曼滤波器学习笔记

【卡尔曼滤波器】_DR_CAN合集学习笔记

数据融合（Data Fusion）

假设用两个测量误差不同的秤,测量同一个的物体的质量，两个秤的测量误差都符合正态分布，

数据融合就是综合两个秤的测量结果和测量误差，估计物体的真实质量。

	A秤	B秤	说明
测量结果	$z_1=30g$	$z_1=32g$
测量误差	$\sigma =2g$	$\sigma =4g$	正态分布标准差

估计真实值

令：

$$\begin{array}{rlr}\hat{Z}& =Z_1+K(Z_2-Z_1)& K\in [0,1]\\ \text{估计值}& =\text{测量结果1}+\text{系数K}(\text{测量结果2}-\text{测量结果1})\end{array}$$

计算K

要使得估计值最接近真实值,

• 即要使估计值的标准差$\sigma (\hat{Z})$最小，
• 即方差${\sigma }^2(\hat{Z})=\text{Var}(\hat{Z})$最小

Var(X) 表示随机变量X的方差

$$\begin{array}{rl}{\sigma }^2(\hat{Z})& =\text{Var}(\hat{Z})\\ & =\text{Var}(Z_1+K(Z_2-Z_1))\\ & =\text{Var}(Z_1+K{Z}_2-K{Z}_1)\\ & =\text{Var}((1-K)Z_1+K{Z}_2)\\ & =\text{Var}((1-K)Z_1)+\text{Var}(K{Z}_2)& \text{(两个随机变量相互独立)}\\ & =(1-K{)}^2\text{Var}(Z_1)+K^2\text{Var}(Z_2)\\ & =(1-K{)}^2{\sigma }^2(Z_1)+K^2{\sigma }^2(Z_2)\end{array}$$

要想知道K取何值时方差最小，即求其对k的导数=0

$$\begin{array}{rl}\frac{d({\sigma }^2(\hat{Z}))}{dK}& =0\\ \frac{d((1-K{)}^2{\sigma }^2(Z_1)+K^2{\sigma }^2(Z_2))}{dK}=0\\ -2(1-K){\sigma }^2(Z_1)+2K{\sigma }^2(Z_2)& =0\\ -(1-K){\sigma }^2(Z_1)+K{\sigma }^2(Z_2)& =0\\ K{\sigma }^2(Z_1)-{\sigma }^2(Z_1)+K{\sigma }^2(Z_2)& =0\\ K{\sigma }^2(Z_1)+K{\sigma }^2(Z_2)& ={\sigma }^2(Z_1)\\ K({\sigma }^2(Z_1)+{\sigma }^2(Z_2))& ={\sigma }^2(Z_1)\\ K& =\frac{{\sigma }^2(Z_1)}{{\sigma }^2(Z_1)+{\sigma }^2(Z_2)}\end{array}$$

所以

$$\begin{array}{rl}K& =\frac{{\sigma }^2(Z_1)}{{\sigma }^2(Z_1)+{\sigma }^2(Z_2)}\\ & =\frac{2^2}{2^2+4^2}\\ & =\frac{4}{4+16}\\ & =0.2\\ \hat{Z}& =Z_1+K(Z_2-Z_1)\\ & =30+0.2(32-30)\\ & =30+0.4\\ & =30.4\\ {\sigma }^2(\hat{Z})& =(1-K{)}^2{\sigma }^2(Z_1)+K^2{\sigma }^2(Z_2)\\ & =(1-0.2{)}^2{2}^2+{0.2}^2{4}^2\\ & =0.64\times 4+0.04\times 16\\ & =2.56+0.64\\ & =3.2\\ \sigma (\hat{Z})& =\sqrt{{\sigma }^2(\hat{Z})}\\ & =\sqrt{3.2}\\ & =1.7888\end{array}$$

	A秤	B秤	估计
结果	$z_1=30g$	$z_1=32g$	$\hat{Z}=30.4$
误差	$\sigma =2g$	$\sigma =4g$	$\sigma =1.7888$