## 3 Local Observability Conditions for the Extended Battery ECM
### Observability Analysis for the Extended Sub-Models  
The observability for the first sub-model depicted in Figure 5b was already studied in [31]. The conclusion was that the time varying behavior of battery current \(I_{\text{batt}}\) was of utmost importance to guarantee good observability conditions for the estimation of R\({}_{0}\), R\({}_{1}\), C\({}_{1}\). To be specific, in order to estimate the ECM parameters without affecting the battery discharging process, an additional pseudo random binary signal (PRBS) can be added to the battery constant charging current by the charging controller. The added PRBS provides a persistently exciting signal, which makes the extended system observable [31]. Therefore, only the observability analysis for the second sub-model will be developed in the following. When the current source is disconnected, \(I_{\text{batt}}=I_{\text{d}}\), where \(I_{\text{d}}\) stands for the self-discharge current. This current is naturally very small compared to the current that flows in or out of the battery when connected to the load. Therefore, the voltage across R\({}_{0}\) and the RC networks can be neglected when compared to \(V_{\text{OC}}\). Eventually, the state-space representation of this sub-model can be written as:  
\[\left\{ \begin{array}{l}\frac{\text{d}sc(t)}{\text{d}t}=\frac{-\eta I_{ \text{batt}}(t)}{\text{C}_{n}(t)}\\ \frac{\text{d}\text{C}_{n}(t)}{\text{d}t}\approx 0\end{array} \right. \tag{4}\]  
\[y_{2}=V_{\text{OC}}(soc)\]  
Besides, one can first recall that a numerical determination theorem for the local observability of a nonlinear model \(\left\{ \begin{array}{l}\dot{\mathbf{x}}=f(\mathbf{x},u)\\ y=h(\mathbf{x},u)\end{array} \right.\), where \(\mathbf{x}\) is an \(n\times 1\) state vector, \(y\) is a scalar output, \(u\) is a scalar system input, \(f(\cdot)\) and \(h(\cdot)\) are nonlinear functions, at point \(x^{0}\) is as follows [31]:  
**Theorem 1**. : _If the following Jacobian matrix \(\mathbf{J}\) has full rank, then the nonlinear system is locally observable at \(x^{0}\)._  
Figure 5: Battery ECM decomposition according to the superposition principle. \[\mathbf{J}=\frac{\partial\mathbf{Y}}{\partial\mathbf{x}}=\frac{\partial}{\partial\mathbf{x}}\left[ \begin{array}{c}h(\mathbf{x},u)\\ L_{f}(h)(\mathbf{x},\mathbf{U})\\ L_{f}(h_{1})(\mathbf{x},\mathbf{U})\\ \vdots\\ L_{f}(h_{n-2})(\mathbf{x},\mathbf{U})\end{array} \right] \tag{5}\]  
_where \(\mathbf{Y}=[y,y^{\prime},...,y^{(n-1)}]^{\mathsf{T}}\), \(\mathbf{U}=[u,u^{\prime},...,u^{(n-1)}]^{\mathsf{T}}\), and the ith derivative of the scalar output \(y\) can be represented by Lie operator \(L_{f}(h)\), which is the total differentiation of \(h\):_  
\[ \begin{split} y^{(i)}&=L_{f}(h_{i-1})(\mathbf{x},\mathbf{U})\\ &=\frac{\partial h_{i-1}}{\partial\mathbf{x}}(\mathbf{x},\mathbf{U})f(\mathbf{x},u )+\frac{\partial h_{i-1}}{\partial u}(\mathbf{x},\mathbf{U})\frac{d\mathbf{U}}{dt}\\ \textit{e.g.,}\textit{y}^{\prime}&=\frac{dy}{dt}=\frac{ \partial h}{\partial\mathbf{x}}f+\frac{\partial h}{\partial u}u^{\prime}=h_{1}( \mathbf{x},u,u^{\prime})\end{split}  \tag{6}\]  
Hence, according to Theorem 1, the Jacobian matrix \(\mathbf{J}\) for the extended sub-model (4) is:  
\[\mathbf{J}=\frac{\partial(y_{2},\dot{y}_{2})}{\partial(soc,\mathrm{C}_{n})}= \left[ \begin{array}{cc}\frac{\mathrm{d}V_{\mathrm{OC}}}{\mathrm{d}soc}&0\\ \Theta&\frac{\mathrm{d}V_{\mathrm{OC}}}{\mathrm{d}soc}\cdot(\frac{\eta\mathrm{ I}_{\mathrm{batt}}(t)}{\mathrm{C}_{n}^{2}(t)})\end{array} \right] \tag{7}\]  
Note 1: \(\dot{y}_{2}=\frac{\mathrm{d}V_{\mathrm{OC}}(soc)}{\mathrm{d}t}=\frac{\mathrm{ d}V_{\mathrm{OC}}}{\mathrm{d}soc}\cdot\frac{\mathrm{d}soc}{\mathrm{d}t}= \frac{\mathrm{d}V_{\mathrm{OC}}}{\mathrm{d}soc}\cdot(\frac{-\eta\mathrm{I}_{ \mathrm{batt}}(t)}{\mathrm{C}_{n}(t)})\)  
Note 2: Because of the zero in (7), \(\Theta\) is just a symbol to represent the corresponding term in the \(2\times 2\) matrix, and it will have no effect on the determinant of \(\mathbf{J}\).