电动汽车用锂离子电池<strong>SOC</strong>估算方法综述

电动汽车用锂离子电池SOC估算方法综述

付诗意, 吕桃林, 闵凡奇, 罗伟林, 罗承东, 吴磊, 解晶莹

Review of estimation methods on SOC of lithium-ion batteries in electric vehicles

Shiyi FU, Taolin LYU, Fanqi MIN, Weilin LUO, Chengdong LUO, Lei WU, Jingying XIE

表1 EKF、UKF和CKF的递推过程

Table 1 Recursive process of EKF、UKF and CKF

对非线性系统

\{\begin{array}{l} x_{k} = f (x_{k - 1}, u_{k}) + w_{k} \\ z_{k} = g (x_{k}, u_{k}) + v_{k} \end{array}

给定初始值

x_{0}

、

P_{0}

、

Q

、

R

。

EKF：

计算雅克比矩阵： $A_{k} = {\frac{? f (x_{k}, u_{k})}{x_{k}}}_{|x_{k} = {\overset{?}{x}}_{k}}$ ； $C_{k} = {\frac{? g (x_{k}, u_{k})}{x_{k}}}_{|x_{k} = {\overset{?}{x}}_{k}}$

计算状态变量的一部预测及协方差矩阵： ${\overset{?}{x}}_{k}^{-} = f ({\overset{?}{x}}_{k - 1}, u_{k})$ ； $P_{k}^{-} = A_{k} P_{k - 1} A_{k}^{T} + Q$

计算卡尔曼增益： $K_{k} = P_{k}^{-} C_{k}^{T} {(C_{k} P_{k}^{-} C_{k}^{T} + R)}^{- 1}$

计算观测值： ${\overset{?}{z}}_{k} = g (x_{k - 1}, u_{k})$

状态和协方差矩阵更新： ${\overset{?}{x}}_{k} = {\overset{?}{x}}_{k}^{-} + K_{k} (z_{k} - {\overset{?}{z}}_{k})$ ； $P_{k} = (I - K_{k} C_{k}) P_{k}^{-}$

UKF：

构造sigma点集与权值，对于n维状态向量，一共有2n+1个sigma点

$x_{k - 1}^{i} = \{\begin{array}{l} {\overset{?}{x}}_{k - 1} ??? (i = 0) \\ {\overset{?}{x}}_{k - 1} + {(\sqrt[]{λ + n} P_{x})}_{i} ??? (i = 1, ?, n) \\ {\overset{?}{x}}_{k - 1} - {(\sqrt[]{λ + n} P_{x})}_{i} ??? (i = n + 1, ?, 2 n) \end{array}$ ；

$\{\begin{array}{l} ω_{m}^{0} = \frac{λ}{λ + n} \\ ω_{c}^{0} = \frac{λ}{λ + n} + 1 - α^{2} + β \\ ω_{m}^{i} = ω_{c}^{i} = \frac{1}{2 (λ + n)} ??? (i = 1, ?, 2 n) \end{array}$

计算sigma数据点集的一步预测值： $x_{k}^{-, i} = f (x_{k - 1}^{i}, u_{k}) ??? (i = 0, ?, 2 n)$

计算状态变量的一步预测及协方差矩阵： $x_{k}^{-} = \sum_{i = 0}^{2 n} ω_{m}^{i} x_{k}^{-, i}$ ； $P_{k}^{-} = \sum_{i = 0}^{2 n} ω_{m}^{i} (x_{k}^{-} - x_{k}^{-, i}) {(x_{k}^{-} - x_{k}^{-, i})}^{T} + Q$

计算新sigma点集： $X_{k}^{-, i} = \{\begin{array}{l} x_{k}^{-} ??? (i = 0) \\ x_{k}^{-} + {(\sqrt[]{λ + n} P_{k}^{-})}_{i} ??? (i = 1, ?, n) \\ x_{k}^{-} - {(\sqrt[]{λ + n} P_{k}^{-})}_{i} ??? (i = n + 1, ?, 2 n) \end{array}$ 其中 ${(P)}_{i}$ 表示矩阵 $P$ 的第i列

利用新sigma点集代入观测方程： ${\overset{?}{z}}_{k}^{i} = g (X_{k}^{-, i}, u_{k}) ??? (i = 0, ?, 2 n)$

计算观测估计值的均值： ${\overset{?}{z}}_{k} = \sum_{i = 0}^{2 n} ω_{m}^{i} {\overset{?}{z}}_{k}^{i}$

计算自相关矩阵： $P_{z z, k} = \sum_{i = 0}^{2 n} ω_{c}^{i} ({\overset{?}{z}}_{k} - {\overset{?}{z}}_{k}^{i}) {({\overset{?}{z}}_{k} - {\overset{?}{z}}_{k}^{i})}^{T} + R$

计算互相关矩阵： $P_{x z, k =} \sum_{i = 0}^{2 n} ω_{c}^{i} (x_{k}^{-} - X_{k}^{-, i}) {({\overset{?}{z}}_{k} - {\overset{?}{z}}_{k}^{i})}^{T}$

计算卡尔曼增益： $K_{k} = P_{x z, k} P_{z z, k}^{- 1}$

状态和协方差矩阵更新： ${\overset{?}{x}}_{k} = x_{k}^{-} + K_{k} (z_{k} - {\overset{?}{z}}_{k})$ ； $P_{k} = P_{k}^{-} - K_{k} P_{z z, k} K_{k}^{T}$

CKF：

构造容积点集，对于n维状态向量，一共有m=2n个容积点： $x_{k - 1}^{-, i} = S_{k - 1}^{-} ξ_{i} + {\overset{?}{x}}_{k - 1}$

其中 $S_{k - 1}$ 通过对协方差矩阵进行cholesky分解获取，如式： $P_{k - 1} = S_{k - 1} S_{k - 1}^{T}$

$ξ_{i} = \sqrt[]{\frac{m}{2}} [\begin{matrix} (\begin{matrix} 1 \\ 0 \\ ? \\ 0 \end{matrix}) & (\begin{matrix} 0 \\ 1 \\ ? \\ 0 \end{matrix}) & \begin{matrix} ? & \begin{matrix} (\begin{matrix} 0 \\ 0 \\ ? \\ 1 \end{matrix}) & (\begin{matrix} - 1 \\ 0 \\ ? \\ 0 \end{matrix}) & (\begin{matrix} 0 \\ - 1 \\ ? \\ 0 \end{matrix}) & \begin{matrix} ? & (\begin{matrix} 0 \\ 0 \\ ? \\ - 1 \end{matrix}) \end{matrix} \end{matrix} \end{matrix} \end{matrix}]$

传播容积点： $x_{k}^{-, i} = f (x_{k - 1}^{-, i}, u_{k})$

计算状态一步预测值： $x_{k}^{-} = \frac{1}{m} \sum_{i = 1}^{m} x_{k}^{-, i}$

计算状态预测方差矩阵： $P_{k}^{-} = \frac{1}{m} \sum_{i = 1}^{m} x_{k}^{-, i} {(x_{k}^{-, i})}^{T} - x_{k}^{-} {(x_{k}^{-})}^{T} + Q$

计算方差矩阵平方根 $S_{k}^{-}$ ： $P_{k}^{-} = S_{k}^{-} {(S_{k}^{-})}^{T}$

构造新容积点集： $X_{k - 1}^{-, i} = S_{k}^{-} ξ_{i} + x_{k}^{-}$

传播新容积点： $z_{k}^{-, i} = h (X_{k - 1}^{-, i}, u_{k})$

计算观测估计值： ${\overset{?}{z}}_{k} = \frac{1}{m} \sum_{i = 1}^{m} z_{k}^{-, i}$

计算自相关矩阵： $P_{z z, k} = \frac{1}{m} \sum_{i = 1}^{m} z_{k}^{-, i} {(z_{k}^{-, i})}^{T} - {\overset{?}{z}}_{k} {\overset{?}{z}}_{k}^{T} + R$

计算互相关矩阵： $P_{x z, k} = \frac{1}{m} \sum_{i = 1}^{m} X_{k - 1}^{-, i} {(z_{k}^{-, i})}^{T} - x_{k}^{-} {\overset{?}{z}}_{k}^{T}$

计算卡尔曼增益： $K_{k} = P_{x z, k} {(P_{z z, k}^{-})}^{- 1}$

状态和协方差矩阵更新： ${\overset{?}{x}}_{k} = x_{k}^{-} + K_{k} (z_{k} - {\overset{?}{z}}_{k})$ ； $P_{k} = P_{k}^{-} - K_{k} P_{z z, k} K_{k}^{T}$