## 3 System Modeling and Mathematical Formulation
In the context of electric vehicle battery SoH estimation, this function relates the battery's internal state, such as voltage, current, and temperature, to the observed measurements. \(h(\mathbf{x})\) can be described as a mapping from the battery's internal state vector \(\mathbf{x}\) to the observed measurements \(\mathbf{z}\). This mapping considers the relationship between the battery's internal state and the measurements obtained from sensors. The specific mathematical expression for \(h(\mathbf{x})\) is given by Equation (5):  
\[\mathbf{z}=h(\mathbf{x})+\mathbf{v} \tag{4}\]  
\[\mathbf{z}=\left[ \begin{array}{c}V_{\text{observed}}\\ I_{\text{observed}}\\ T_{\text{observed}}\end{array} \right]=h(\mathbf{x})+\mathbf{v}=\left[ \begin{array} []{c}V\\ I\\ T\end{array} \right]+\mathbf{v} \tag{5}\]  
We define a dynamic model function \(f(x,u)\) to forecast the evolution of battery characteristics over time based on system dynamics, as shown in Equation (6). Here, \(\mathbf{x}^{\prime}\) represents the derivative of the state vector with respect to time, \(\mathbf{u}\) is the control input vector, and \(\mathbf{w}\) represents the process noise. The dynamic model elucidates the battery's voltage, current, and temperature response during charge and discharge cycles. The dynamic model function, based on system dynamics, incorporates the effects of charge and discharge current, as well as temperature, to forecast the evolution of battery characteristics over time, as described in Equation (7), where \(I_{u}\) denotes the current that is used as a control input (e.g., during charging or discharging operations). \[\mathbf{x}^{\prime}=f(\mathbf{x},\mathbf{u})+\mathbf{w} \tag{6}\]  
\[\frac{d}{dt}\left[ \begin{array}{c}V\\ I_{u}\\ T\\ SoC\\ C\end{array} \right]=f\left(\left[ \begin{array}{c}V\\ I_{u}\\ T\\ SoC\\ \end{array} \right],\left[ \begin{array}{c}I\\ T\end{array} \right]\right)+\mathbf{w} \tag{7}\]  
In the state prediction steps, our proposed EKF employs a dynamic model to predict the battery's internal state at the next time step based on the preceding state and system dynamics. It adjusts the state covariance matrix \(\mathbf{P}\) to accommodate uncertainty in the prediction. The prediction process is represented by Equations (8) and (9), where \(\mathbf{F}\) denotesthe Jacobian matrix of the dynamic model, and \(\mathbf{Q}\) denotes the covariance matrix of the process noise. \[\hat{\mathbf{x}}(k+1)=\mathbf{f}(\hat{\mathbf{x}}(k),\mathbf{u}(k)) \tag{8}\]  
\[\mathbf{P}(k+1)=\mathbf{F}\mathbf{P}(k)\mathbf{F}^{\mathrm{T}}+\mathbf{Q} \tag{9}\]  
In the subsequent measurement update step, the EKF computes the Kalman gain using Equation (10) and revises the predicted state and covariance matrix based on the difference between the predicted and observed measurements, as outlined in Equations (11) and (12). Here, \(\mathbf{H}\) represents the Jacobian matrix of the measurement model, and \(\mathbf{R}\) denotes the covariance matrix of the measurement noise. \[\mathbf{K}(k+1)=\mathbf{P}(k+1)\mathbf{H}^{T}\Big{(}\mathbf{H}\mathbf{P}(k+1) \mathbf{H}^{T}+\mathbf{R}\Big{)}^{-1} \tag{10}\]  
\[\hat{\mathbf{x}}(k+1)=\hat{\mathbf{x}}(k+1)+\mathbf{K}(k+1)(\mathbf{z}(k+1)- \mathbf{h}(\hat{\mathbf{x}}(k+1))) \tag{11}\]  
\[\mathbf{P}(k+1)=(I-\mathbf{K}(k+1)\mathbf{H})\mathbf{P}(k+1) \tag{12}\]  
\[SoH=\frac{C}{C_{i}} \tag{13}\]  
To calculate the SoH based on the estimated parameters obtained from the EKF, we further define a degradation model that reflects the relationship between the battery's internal state and its health status. The parameters are estimated utilizing the EKF framework. The process and measurement models are iteratively employed in the prediction and update steps of the EKF algorithm to estimate the internal state of the battery. The sampling rate represents the time interval between consecutive measurements of the battery's characteristics (voltage, current, temperature) during the estimation process.