# Online SoC Estimation of Lithium-Ion Batteries Using a New Sigma Points Kalman Filter
## 3 Proposed Algorithm Design and Flow
\right],u=I_{L}.\]  
(3) Updating the Sigma points sets. a. Sigma points set 1 was updated. \[\chi c_{k|k-1}=\Big{[}\overline{\chi}c_{k|k-1},\overline{\chi}c_{k|k-1}+h\sqrt{ Pc_{k|k-1}},\overline{\chi}c_{k|k-1}-h\sqrt{Pc_{k|k-1}}\Big{]} \tag{36}\]  
b. Sigma points set 2 was updated. \[\chi u_{k|k-1}=\Big{[}\overline{\chi}u_{k|k-1},\overline{\chi}u_{k|k-1}+\sqrt{ (L+\lambda)Pu_{k|k-1}},\overline{\chi}u_{k|k-1}-\sqrt{(L+\lambda)Pu_{k|k-1}} \Big{]} \tag{37}\]  
(4) The measurement equation was updated. a. The state estimation and measurement equation update based on the Sigma point set 1. \[\overline{y}c_{k|k-1}=H(\overline{x}c_{k|k-1},r_{k}) \tag{38}\]  
\[Pc_{\overline{y}c_{k}}=\sum\limits_{i=1}^{L}\left\{ \begin{array}{c}W_{c1}(i )\Big{[}(H_{k}(\overline{\chi}c_{k|k-1}+h\sqrt{Pc_{k|k-1}})-H_{k}(\overline{ \chi}c_{k|k-1}-h\sqrt{Pc_{k|k-1}})\Big{]}\\ \Big{[}(H_{k}(\overline{\chi}c_{k|k-1}+h\sqrt{Pc_{k|k-1}})-H_{k}(\overline{ \chi}c_{k|k-1}-h\sqrt{Pc_{k|k-1}})\Big{]}^{T}\end{array} \right\}+R_{k} \tag{39}\]  
\[ \begin{array}{c}Pc_{\overline{x}c_{k},\overline{y}c_{k}}=\sum\limits_{i=1}^ {L}\sqrt{W_{c1}(i)}\Big{[}(\overline{\chi}c_{k|k-1}+h\sqrt{Pc_{k|k-1}})- \overline{\chi}c_{k|k-1}\Big{]}\\ \Big{[}(h_{k}(\overline{\chi}c_{k|k-1}+h\sqrt{Pc_{k|k-1}})-h_{k}(\overline{ \chi}c_{k|k-1}-h\sqrt{Pc_{k|k-1}})\Big{]}^{T}\end{array}  \tag{40}\]  
\[ \begin{array}{c}\text{where, }H(k)=\Big{[}\begin{array}{c}\frac{\partial U_{OCV}(k)} {\partial s\partial c(k)}-I_{L}(k)\times\frac{\partial R_{0}(k)}{\partial s \partial c(k)}-1&-1\end{array}\Big{]}\end{array} \]  
Calculation of the Kalman gain:  
\[Kc_{k}=Pc_{\overline{x}c_{k},\overline{y}c_{k}}/Pc_{\overline{y}c_{k}} \tag{41}\]  
Status estimation update:  
\[\overline{x}c_{k}=\overline{x}c_{k|k-1}+Kc_{k}(yc_{k}-\overline{y}c_{k|k-1}) \tag{42}\]  
Covariance update:  
\[Pc_{k}=Pc_{k|k-1}-Kc_{k}Pc_{\overline{y}c_{k}}{Kc_{k}}^{T} \tag{43}\]  
\[Xc(:,t)=\overline{x}c_{k} \tag{44}\]  
b. State estimation and measurement equation update based on the Sigma points set 2. \[\overline{y}u_{k|k-1}=H(\overline{x}u_{k|k-1},r_{k}) \tag{45}\]  
\[ \begin{array}{c}Pu_{\overline{y}c_{k}}=\sum\limits_{i=0}^{2L}\left\{\begin{array} []{c}W_{c}(i)\Big{[}H_{k}(\overline{\chi}u_{k-1}+\sqrt{(L+\lambda)Pu_{k-1}})- \overline{y}u_{k|k-1}\Big{]}\\ \Big{[}(H_{k}(\overline{\chi}u_{k-1}+\sqrt{(L+\lambda)Pu_{k-1}})-\overline{y} u_{k|k-1}\Big{]}^{T}\end{array}\right\}+R_{k}\end{array}  \tag{46}\]  
\[ \begin{array}{c}Pu_{\overline{x}c_{k},\overline{y}c_{k}}=\sum\limits_{i=0}^ {2L}W_{c}(i)\Big{[}(\overline{\chi}u_{k-1}+\sqrt{(L+\lambda)Pu_{k-1}})- \overline{\chi}u_{k|k-1}\Big{]}\\ \Big{[}H_{k}(\overline{\chi}u_{k-1}+\sqrt{(L+\lambda)Pu_{k-1}})-\overline{y}u_ {k|k-1}\Big{]}^{T}\end{array} \right\}\\ \end{array} \tag{47}\]Calculation of the Kalman gain:  
\[Ku_{k}=Pu_{\overline{x}c_{k},\overline{y}c_{k}}/Pu_{\overline{y}c_{k}} \tag{48}\]  
Status estimation update:  
\[\overline{x}u_{k}=\overline{x}u_{k|k-1}+Ku_{k}(yu_{k}-\overline{y}u_{k|k-1}) \tag{49}\]  
Covariance update:  
\[Pu_{k}=Pu_{k|k-1}-Ku_{k}Pu_{\overline{y}c_{k}}{Ku_{k}}^{T} \tag{50}\]  
\[Xu(:,t)=\overline{x}u_{k} \tag{51}\]  
(5) Determining whether the termination conditions are met and correcting the initial error of the SoC estimate. Taking the first row of the two matrices, \(Xc(:,t)\) and \(Xu(:,t)\) at time \(t\):  
\[Xc(1,t),\ Xu(1,t) \tag{52}\]  
\[\text{If}\ \Big{|}\frac{\big{|}Xc(1,t)-Xu(1,t)\big{|}}{Xu(1,t)}\Big{|}>\delta \tag{53}\]  
\[\text{Then}\ Xc(:,t)=\overline{x}u_{k} \tag{54}\]  
\[\text{Else}\ \text{If}\ \Big{|}\frac{\big{|}Xc(1,t)-Xu(1,t)\big{|}}{Xu(1,t)} \Big{|}\leq\delta \tag{55}\]  
Here, \(\delta\) is the threshold value. Set \(\delta\) to 0.0000015. This circle was ended, and the time \(t\) was printed. Then, \(M=t\). S4: The second cycle, time \(t\) from \(M\) to \(N\) seconds. (1) Determination of the Sigma points set 3. Subject to (27)  
(2) Time update and calculation of the prior mean and covariance. \[\chi c^{i}{}_{k|k-1}=A\overline{\chi}c_{k-1}+Bu(k-1) \tag{56}\]  
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