## 2 The Second Order Dynamic Model of the Cell and the Model of the LiBP  
### The Second Order Dynamic Model of the Cell Influenced by Voltage Noise, Noise and Current Bias  
Based on our previous work [6], the second order equivalent circuit model of the cell is shown in Figure 2.  
For the state estimation problem of the cell, we consider the states of cell to be two currents through two dynamic branches of cell \(i_{R_{1}},i_{R_{2}}\), the hysteresis \(h\), SoC \(z\) and the current bias \(b\). The input and output of the model are current though the cell \(i(t)\), and voltage \(v(t)\), respectively. The input \(i(t)\) and output \(v(t)\) are measurable and influenced by noises which have Gaussian distribution with zero mean. Set the state vector, input vector and output at the sample time \(k\) as Equation (1).  
\[\underline{x}_{k+1}=\left[\begin{array}{c}i_{R_{1}k+1}\\ i_{R_{2}k+1}\\ i_{R+1}\\ z_{k+1}\\ b_{k+1}\end{array}\right],\underline{u}_{k}=\left[\begin{array}{c}i_{k}\\ \text{sgn}(i_{k})\end{array}\right],y_{k}=v_{k} \tag{1}\]  
The state model of the cell on the discrete time domain is written in the Equation (2) as follows.  
\[\left[\begin{array}{c}i_{R_{1}k+1}\\ i_{R_{2}k+1}\\ h_{k+1}\\ z_{k+1}\\ b_{k+1}\end{array}\right]=\left[\begin{array}{cccc}A_{R_{1}C_{1}}&0&0&0&0&0\\ 0&A_{R_{2}C_{2}}&0&0&0\\ 0&0&A_{h}&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\end{array}\right]\left[\begin{array}{c}i_{R_{1}k}\\ i_{R_{2}k}\\ i_{R_{2}}\\ z_{k}\\ b_{k}\end{array}\right]+\left[\begin{array}{c}\left(1-A_{R_{1}C_{1}}\right) &0\\ \left(1-A_{R_{2}C_{2}}\right)&0\\ 0&\left(1-A_{h}\right)\\ \frac{-\eta_{h}\Delta t}{Q}&0\\ 0&0\end{array}\right]\left[\begin{array}{c}i_{k}\\ \text{sgn}(i_{k})\end{array}\right]+\left[\begin{array}{c}w_{i_{R_{1}k}}\\ w_{i_{R_{2}k}}\\ 0\\ w_{b,k}\end{array}\right] \tag{2}\]  
The matrices in the Equation (2) are calculated as  
\[\begin{array}{c}A_{R_{1}C_{1}}=\exp\left(\frac{-\Delta t}{R_{1}C_{1}} \right),A_{R_{2}C_{2}}=\exp\left(\frac{-\Delta t}{R_{2}C_{2}}\right),B_{R_{1}C _{1}}=1-A_{R_{1}C_{1}},B_{R_{2}C_{2}}=1-A_{R_{2}C_{2}}\\ A_{h}=\exp\left(-\left\lceil\frac{\eta_{h}\Delta t}{Q}\right\rceil\right);B_{h}= 1-A_{h};\underline{w}=\left[\begin{array}{cccc}w_{i_{R1}k}&w_{i_{R2}k}&0&0& 0&w_{b,k}\end{array}\right]^{T}\end{array} \tag{3}\]  
with \(\underline{w}\) is the system disturbance vector, including the noises influencing the currents through two dynamic branches of cell \(w_{i_{R_{1}k}},w_{i_{R_{2}k}}\) and current bias noise through cell \(w_{b,k}\). The parameters \(\eta_{k},\gamma_{k}\) are the columbic efficiency and positive constant; they all depend on the operating temperature of cell. The output equation of the model describing the relationship of the voltage drop on two electrodes of  
Figure 2: The second order equivalent circuit model of a lithium-ion battery.  
cell and the current, SoC--the hysteresis and currents through two dynamic branches--is written as follows.  
\[v_{k}=OCV(z_{k})+Mh_{k}-R_{1}i_{R_{1}k}-R_{2}i_{R_{2}k}-R_{0}i_{k}+\zeta_{k} \tag{4}\]  
In short, the state space model of cell related to the voltage noise, noise and current bias of the current through the cell is  
\[\left\{\begin{array}{l}\underline{x}_{k+1}=A_{k}\underline{x}_{k}+B_{k} \underline{u}_{k}+\underline{w}_{k}\\ y_{k}=\mathrm{OC}V(z_{k})+\mathbb{C}_{k}\underline{x}_{k}+D_{k}\underline{u} _{k}+\zeta_{k}\end{array}\right. \tag{5}\]  
in which, the matrices of Equation (5) are formed as follows.  
\[A_{k}=\left[\begin{array}{cccc}A_{R_{1}C_{1}}&0&0&0&0\\ 0&A_{R_{2}C_{2}}&0&0&0\\ 0&0&A_{h}&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\end{array}\right],B_{k}=\left[\begin{array}{cc}\left(1-A_{R_{1}C_ {1}}\right)&0\\ \left(1-A_{R_{2}C_{2}}\right)&0\\ 0&\left(1-A_{h}\right)\\ \frac{-\eta_{h}\Delta t}{Q}&0\\ 0&0\end{array}\right] \tag{6}\] \[C=\left[\begin{array}{cccc}-R_{1}&-R_{2}&M&0&0\end{array} \right],D=\left[\begin{array}{cc}-R_{0}&0\end{array}\right]\]  
with \(\zeta_{k}\) is the voltage noise of cell.