## 4 New Cascaded Framework  
Since the battery state and parameter estimations are usually carried out in the discrete time domain inside BMS [33], the classical estimation methods will be reviewed rapidly based on the discrete time model (9) before presenting the new cascaded estimation framework. The discrete time model (9) is obtained with the zero-order hold approximation under the assumption that the current \(I_{\text{batt}}\) is constant between two adjacent sampling points. \(k\) is the time index.  
In addition, considering the uncertainty of the modeling and data acquisition process, the stochastic model is employed instead of a deterministic one. Hence, \(\omega_{1},\omega_{2},\omega_{2}\) are additive model errors. Using the common random walk model [19], \(\omega_{4},\omega_{5},\omega_{6},\omega_{7}\) are added to the slowly varying parameters. \(v\) is measurement noise. The sampling period is \(T\). It is assumed that all noises are additive white Gaussian (AWG).  
\[\left\{\begin{array}{l}V_{1}(k)=\mathrm{e}^{\frac{-T}{R_{1}(k-1)\zeta_{1}(k -1)}}V_{1}(k-1)+\mathrm{R}_{1}(k-1)\cdot\left(1-\mathrm{e}^{\frac{-T}{R_{1}(k -1)\zeta_{1}(k-1)}}\right)\cdot I_{\text{batt}}(k-1)+\omega_{1}(k-1)\\ V_{2}(k)=\mathrm{e}^{\frac{-T}{R_{2}\zeta_{2}}}V_{2}(k-1)+\mathrm{R}_{2}\cdot \left(1-\mathrm{e}^{\frac{-T}{R_{2}\zeta_{2}}}\right)\cdot I_{\text{batt}}(k -1)+\omega_{2}(k-1)\\ soc(k)=soc(k-1)-\frac{\eta T}{\mathrm{C}_{\text{n}}(k-1)}\cdot I_{\text{batt }}(k-1)+\omega_{3}(k-1)\\ \mathrm{R}_{0}(k)=\mathrm{R}_{0}(k-1)+\omega_{4}(k-1)\\ \mathrm{R}_{1}(k)=\mathrm{R}_{1}(k-1)+\omega_{5}(k-1)\\ \mathrm{C}_{1}(k)=\mathrm{C}_{1}(k-1)+\omega_{6}(k-1)\\ \mathrm{C}_{\text{n}}(k)=\mathrm{C}_{\text{n}}(k-1)+\omega_{7}(k-1)\\ \end{array}\right. \tag{9}\]  
\[V_{\text{batt}}(k)=V_{\text{OC}}(soc)-V_{1}(k)-V_{2}(k)-\mathrm{R}_{0}(k)\cdot I _{\text{batt}}(k)+v(k)\]  
Based on (9), one way to execute the state and parameter estimation is to use a single observer, which is called the joint estimation method. Another way that resorts to two parallel observers is named as the dual estimation method, where parameter estimation can be isolated. Thus, it makes the multi-time scale estimation possible [17]. Although employing more advanced observers can improve the monitoring result based on these two classical estimation structures, the improvement is limited due to the incorrect parameter grouping.  
Figure 6: Schematic resume of the necessary observability conditions for extended ECM.  
After obtaining the necessary observability conditions, it shows that battery usable capacity estimation has a higher demand for the observability condition. Especially, its inherent weak observability caused by the term \(\left(\frac{dV_{\text{OC}}}{d\text{soc}}\right)^{2}\) brings the challenge. It is better to estimate the battery usable capacity separately. Hence, the new proposed cascaded framework for battery state and parameter estimation is shown in Figure 7, where \(\widehat{X}\) (\(X=\text{SOC},\text{R}_{0},\text{R}_{1},\text{C}_{1},V_{\text{OC}},\text{C}_{n}\)) stands for the estimated vector. Three estimation modules can be clearly observed from this schematic representation, namely the battery capacity estimation, the battery OCV estimation, and the battery SOC and ECM parameter estimation. Besides, the aforementioned three parts are connected together by some key state variables. Details for each part will be introduced in the following.