## 4 Dual Filtering  
This section formulates the DEKF and applies it to simultaneously estimate the states and parameters during battery aging. Among the parameters, the present capacity was solely estimated using a separate optimization technique due to its poor observability in the developed filter. Fundamentally, the KF serves as the optimal state estimator, if the considered system is linear and the uncertainties are represented by Gaussian random variables. If the system is not linear, a linearization process is performed at every time step to approximate the nonlinear system as a linear time-varying system. The linearized system can then be utilized within the KF framework, generating the EKF that operates on the original nonlinear system. The original nonlinear system is in the form of a state-space model that describes the dynamics of states:  
\[x_{k}=f(x_{k-1},u_{k-1})+\omega_{k-1} \tag{3}\]  
\[y_{k}=g(x_{k},u_{k})+v_{k} \tag{4}\]  
where \(\omega_{k}\) is the process noise with the covariance matrix \(\Sigma_{\omega}\), and \(v_{k}\) is the measurement noise with the covariance matrix \(\Sigma_{v}\). Here, \(f\) is a nonlinear state transition function, and \(g\) is a nonlinear measurement function. The EKF uses this system model in a recursive process to update its estimate of the present state. The two steps, time update (prediction) and measurement update, are  
Figure 5: Variations in terminal voltage predictions over cycles under **(a)** City, **(b)** Highway, and **(c)** High-speed profiles. performed repeatedly, as follows. Note that \(x^{-}\) denotes a predicted value, \(x^{+}\) denotes an updated value, \(\hat{x}\) denotes an estimated value, and \(\Sigma_{x}\) denotes the covariance of \(\hat{x}\). Initially, the present state is predicted given the past state estimate and the system input:  
\[\hat{x}_{k}^{-}=f\Big{(}\hat{x}_{k-1}^{+},u_{k-1}\Big{)} \tag{5}\]  
Next, the uncertainty of the state estimate is predicted:  
\[\Sigma_{x,k}^{-}=A_{k-1}\Sigma_{x,k-1}^{+}A_{k-1}^{T}+\Sigma_{\omega} \tag{6}\]  
From here, the measurement update is followed by the time update. The measurements of the system output are taken and compared to the predictions derived from the state estimate:  
\[\hat{x}_{k}^{+}=\hat{x}_{k}^{-}+L_{k}\big{(}y_{k}-g\big{(}\hat{x}_{k}^{-},u_{k }\big{)}\big{)} \tag{7}\]  
The state estimate is updated based on the prediction error and the Kalman gain. The Kalman gain is computed based on the sensitivity of the output to various states (represented by the Jacobian matrix in the EKF) and the uncertainty of the states (represented by the error covariance matrix):  
\[L_{k}=\Sigma_{x,k}^{-}C_{k}^{T}\Big{(}C_{k}\Sigma_{x,k}^{-}C_{k}^{T}+\Sigma_{ \varphi}\Big{)}^{-1} \tag{8}\]  
Subsequently, the uncertainty of the state estimate is updated:  
\[\Sigma_{x,k}^{+}=(I-L_{k}C_{k})\Sigma_{x,k}^{-} \tag{9}\]  
The original nonlinear system within the EKF refers to the cell model developed in Section 3. The states in the model, SOC as an example, can change rapidly over time, spanning its entire range within several minutes. In contrast, the model parameters, such as capacity, change slowly. The SOC generally denotes the residual capacity with respect to the present capacity, leading to a strong correlation between them. Therefore, we devised the DEKF based on two separate but cooperating filters: the state filter and parameter filter [39; 40]. While the state filter remains consistent with Equations (3) and (4), the parameter filter introduces a new state-space model that describes the dynamics of parameters:  
\[\theta_{k}=\theta_{k-1}+r_{k-1} \tag{10}\]  
\[d_{k}=g(x_{k},u_{k},\theta_{k})+e_{k} \tag{11}\]  
where \(r_{k}\) is the process noise and \(e_{k}\) is the measurement noise. According to Equation (10), the parameters are essentially constant, but they may change slowly with respect to time and are modeled by a fictitious process noise.