## 3 System Modeling and Mathematical Formulation  
EKF technology stands out as a compelling option for estimating EV battery SoH owing to several inherent advantages. Firstly, EKF, rooted in Bayesian estimation principles, can assimilate prior knowledge of battery behavior and dynamics into the estimation process. This renders EKF particularly adept at modeling nonlinear and time-varying systems, traits commonly encountered in the degradation processes of EV batteries. Through the utilization of a mathematical battery behavior model, EKF effectively monitors changes in battery health over time, thus enhancing accuracy and reliability. Moreover, EKF offers computational efficiency, making it suitable for real-time applications such as onboard battery health monitoring in EVs. Unlike ML and DL techniques, which often demand substantial data volumes for training and complex computational resources, EKF can operate efficiently even with limited computational resources. This characteristic renders it an appealing choice for embedded systems in EVs, where computational constraints may be prevalent. In contrast to data-driven ML and DL approaches that heavily rely on training data to generalize patterns, EKF employs a deterministic model of battery behavior to estimate SoH. This can prove advantageous in scenarios with limited historical data availability or when acquiring accurate ground truth labels for training ML/DL models proves challenging. Additionally, the transparency and interpretability of the mathematical model employed in EKF foster a deeper understanding of the underlying physical processes governing battery degradation. This, in turn, aids in the development of more robust and interpretable battery health estimation algorithms. This study presents a method and mathematical model for estimating the SoH of electric vehicle batteries using EKF technology. Within the EKF framework's state space representation, we define a state vector, denoted by \(\mathbf{x}\), which represents the battery's relevant characteristics. The state vector \(\mathbf{x}\) is formally expressed as Equation (1), where \(V\) is the terminal voltage of the battery, \(I_{x}\) is the current flowing through the battery which represents the system's internal state at a given time, \(T\) is the temperature of the battery, \(SoC\) is the state of charge of the battery, representing the remaining capacity of the battery relative to its fully charged state, and \(C\) is the capacity of the battery, reflecting its ability to store and deliver charge effectively over time. \[\mathbf{x}=[V,I_{x},T,SoC,C]^{T} \tag{1}\]  
The input vector \(\mathbf{u}\) comprises factors that influence battery behavior. In the context of estimating the SoH of electric vehicle batteries, typical factors affecting battery behavior include charging/discharging current (\(I\)), representing the current flowing into or out of the battery during charging or discharging processes, and temperature (\(T\)), which denotes the temperature of the battery itself. Temperature significantly impacts battery performance and degradation. Therefore, the input vector \(\mathbf{u}\) can be defined as follows (Equation (2)):  
\[\mathbf{u}=[I,T]^{T} \tag{2}\]  
Let \(\mathbf{z}\) represent a measurement vector that includes observed battery parameters such as voltage (\(V_{observed}\)), current (\(I_{observed}\)), and temperature (\(T_{observed}\)). Then, \(\mathbf{z}\) is defined as follows (Equation (3)):  
\[\mathbf{z}=[V_{observed},I_{observed},T_{observed}]^{T} \tag{3}\]  
Then, we establish a measurement model that correlates the observed measurements to the battery's internal state, which can be defined as Equation (4), where \(h(X)\) is the measurement model function and \(V\) denotes measurement noise. The measurement model function, \(h(\mathbf{x})\), relates the battery's internal state vector, \(\mathbf{x}\), to the observed measurements.