# Failure Warning at the End of Service-Life of Lead-Acid Batteries for Backup Applications
## 2 Linear Superposition-Voltage Aging Model
### Linear Aging Filter of Parameters to Counter Hysteresis  
The accelerating aging is periodic and consistent in experiments. If the measured parameters (\(V_{OCP}\), \(c_{bulk}\), \(\xi_{\Omega}\)) appear hysteresis, then the aging rates are adjusted according to the higher value of the two beginning points in Equation (36). This guarantees the prediction to decrease at average rates. If the measured parameters (\(V_{OCP}\), \(c_{bulk}\), \(\xi_{\Omega}\)) do not appear hysteresis, the aging rates are adjusted according to the last value as a starting point in Equation (37). This guarantees the prediction to decline at the latest rates. Two aging rates are compared in Equation (38); the lower one is to be used in prediction:  
\[x_{k}=\ \beta_{k,1}(k\text{-}o)+x_{o},\ x_{o}=max\{x_{1},x_{2}\},\ o\ =\ 1\ or\ 2. \tag{36}\]  
\[x_{k}=\ \beta_{k,2}\text{-}1+x_{(k-1)} \tag{37}\]  
\[\beta_{k}=\min\{\beta_{k,1},\ \beta_{k,2}\} \tag{38}\]  
Step 1. Initialization:  
The initial value of the parameters (\(V_{OCP}\), \(c_{bulk}\), \(\xi_{\Omega}\)) requires three measurements because of bias points in Equations (36) and (37). In Equation (39), \(\beta_{3}\) is the linear aging rate between continuous three accelerated experiments, \(x_{o,3}\) is bias point according to least square method. In Equation (40), \(\beta_{2}\) is the linear aging rate between the beginning two accelerated experiments, \(x_{o,2}\) is bias point according to least square method. In Equation (41), \(\hat{x}_{1}\), \(\hat{x}_{2}\) and \(\hat{x}_{3}\) are estimations of the parameters. In Equation (42), \(P_{3}^{-}\) is the error covariance of estimation. In Equation (43), \(M_{x}\) is the covariance matrix of measurement error. In Equation (44), \(H_{x}\) is the transformation matrix between measurement and state vector. The \(\left[ \begin{array}{c}x_{k}\\ \beta_{k}\end{array} \right]\) is state vector and \(\left[ \begin{array}{c}\hat{x}_{3}\\ \hat{\beta}_{3}\end{array} \right]\) is initializing as \(\left[ \begin{array}{c}\hat{x}_{3}\\ \beta_{3}\end{array} \right]\). In Equation (45), \(K_{3}\) is the Kalman gain. In Equation (46), \(P_{3}\) is the error covariance for updated estimation. \[\left[ \begin{array}{c}\beta_{3}\\ x_{o,3}\end{array} \right]=\left[ \begin{array}{cc}1&1\\ 2&1\\ 3&1\end{array} \right]\left[ \begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\end{array} \right] \tag{39}\]  
\[\left[ \begin{array}{c}\beta_{2}\\ x_{o,2}\end{array} \right]=\left[ \begin{array}{cc}1&1\\ 2&1\end{array} \right]\left[ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right] \tag{40}\]  
\[P_{x,3}^{-}=\left[\begin{array}{cc}E\big{(}(x_{1}\text{-}\hat{x}_{1})^{2}+( x_{2}\text{-}\hat{x}_{2})^{2}+(x_{3}\text{-}\hat{x}_{3})^{2}\big{)}&0\\ 0&E\bigg{(}\beta_{3}-\frac{\beta_{3}+\beta_{2}}{2}\bigg{)}^{2}\bigg{\}}\\ M_{V_{OCP}}=\left[ \begin{array}{cc}\big{(}10^{-3}\big{)}^{2}&0\\ 0&\big{(}10^{-4}\big{)}^{2}\end{array} \right]\ M_{\zeta_{\Omega}}=\left[ \begin{array}{cc}\big{(}10^{-1}\big{)}^{2}&0\\ 0&\big{(}10^{-2}\big{)}^{2}\end{array} \right]\ M_{\xi_{\Omega}}=\left[ \begin{array}{cc}\big{(}10^{-1}\big{)}^{2}&0\\ 0&\big{(}10^{-2}\big{)}^{2}\end{array} \right] \tag{43}\]  
\[H_{x}=\left[ \begin{array}{cc}1&0\\ 0&1\end{array} \right] \tag{44}\]\[K_{x,3} =P_{x,3}^{-}H_{x}^{T}\Big{[}H_{x}P_{x,3}^{-}H_{x}^{T}+M_{x}\Big{]}^{-1} \tag{45}\] \[P_{3} =[I-K_{3}H_{x}]P_{3}^{-} \tag{46}\]  
Step 2. State prediction:  
In Equation (47), \(\Phi_{x,k}\) is the state transition matrix, to pass incremental aging round. In Equation (48), \(\hat{\xi}_{k}^{-}\) is prediction of aging parameter at the moments of the next deep-discharge measurement, \(\hat{\beta}_{k}^{-}\) is the same as \(\hat{\beta}_{(k-1)}\). In Equation (49), \(P_{k}^{-}\) is the error covariance for following estimation. In Equation (50), \(K_{k}\) is updated before jumping to Step 3. In Equation (51), prediction parameters (\(\hat{V}_{OCP,k}\), \(\hat{c}_{bulk,k}\), \(\hat{\xi}_{\Omega,k}\) and \(\hat{r}_{\Omega,k}\)) are applied to approximate \(E\hat{\Omega}C_{j,k}\) by table look-up method.