## 3 Identification of SSO Parameters of the Fitting-Based ERA
### The Basic Idea of the Fitting-Based ERA  
#### 3.3.1 Solving the Angular Frequencies \(\omega_{0}\) and \(\omega_{\mathrm{s}}\)  
Based on the characteristic polynomial coefficients of the system matrix, the basic idea of the fitting-based ERA proposed in this paper is to establish the angular frequency fitting equation corresponding to the fundamental frequency and oscillation frequency. The spectral decomposition of \(\mathbf{Y}_{0}\) reveals four eigenvalues. According to the principle of matrix decomposition, the four eigenvalues can be transformed into a characteristic polynomial equation. Two fundamental modes and two oscillation modes are considered, which ensures accuracy when solving \(\omega_{0}\) and \(\omega_{\mathrm{s}}\). Moreover, the flow chart of the algorithm in this paper is shown in Figure 1.  
Based on the properties of the characteristic polynomial, the characteristic polynomial equation of the system characteristic matrix \(\mathbf{Y}_{0}\) can be obtained as follows:  
\[\mu_{0}+\mu_{1}\gamma_{i}+\mu_{2}\gamma_{i}^{2}+\mu_{3}\gamma_{i}^{3}+\mu_{4} \gamma_{i}^{4}=0 \tag{18}\]  
where \(\mu_{i}\) represents the characteristic polynomial coefficients of the system matrix. Because the characteristic polynomial coefficients \(\mu_{i}\) can be solved by the eigenvalue \(\gamma_{i}\) of the system matrix \(\mathbf{Y}_{0}\), they are equivalent, and thus \(\mu_{i}\) can be used instead of \(\gamma_{i}\) to establish the subsequent angular frequency fitting equation. Moreover, in the case of model errors, \(\mu_{i}\) calculated by \(\gamma_{i}\) has calculation errors. At this time, if the correct angular frequency \(\omega_{0}\) or \(\omega_{\mathrm{s}}\) is substituted into Equation (18), the equation will no longer hold, and there will be residuals. Therefore, the angular frequency \(\omega_{0}\) or \(\omega_{\mathrm{s}}\) can be accurately solved by searching for the smallest residuals. Therefore, after obtaining the system matrix \(\mathbf{Y}_{0}\), we first calculate the characteristic polynomial coefficient \(\mu_{i}\) of the system matrix \(\mathbf{Y}_{0}\) and then establish the fitting equation of \(\omega_{0}\) and \(\omega_{\mathrm{s}}\) according to \(\mu_{i}\) as follows:  
\[ \begin{bmatrix}\mu_{0}+\mu_{1}e^{\mathrm{i}\omega_{0}1}+\mu_{2}e^{\mathrm{i} \omega_{0}2}+\mu_{3}e^{\mathrm{i}\omega_{0}3}+\mu_{4}e^{\mathrm{i}\omega_{0}4 }\\ \mu_{0}+\mu_{1}e^{-\mathrm{i}\omega_{0}1}+\mu_{2}e^{-\mathrm{i}\omega_{0}2}+ \mu_{3}e^{-\mathrm{i}\omega_{0}3}+\mu_{4}e^{-\mathrm{i}\omega_{0}4}\end{bmatrix}  + \begin{bmatrix}\delta(\omega_{0})\\ \delta(-\omega_{0})\end{bmatrix} = \begin{bmatrix}0\\ 0\end{bmatrix}  \tag{19}\]  
Figure 1: Flowchart for fitting-based ERA. \[ \begin{bmatrix}\mu_{0}+\mu_{1}e^{\mathrm{i}\omega_{\mathrm{s}}1}+\mu_{2}e^{ \mathrm{i}\omega_{\mathrm{s}}2}+\mu_{3}e^{\mathrm{i}\omega_{\mathrm{s}}3}+\mu_ {4}e^{\mathrm{i}\omega_{\mathrm{s}}4}\\ \mu_{0}+\mu_{1}e^{-\mathrm{i}\omega_{\mathrm{s}}1}+\mu_{2}e^{-\mathrm{i} \omega_{\mathrm{s}}2}+\mu_{3}e^{-\mathrm{i}\omega_{\mathrm{s}}3}+\mu_{4}e^{- \mathrm{i}\omega_{\mathrm{s}}4}\end{bmatrix} + \begin{bmatrix}\delta(\omega_{ \mathrm{s}})\\ \delta(-\omega_{\mathrm{s}})\end{bmatrix} = \begin{bmatrix}0\\ 0\end{bmatrix}  \tag{20}\]  
where \(\delta(\omega_{0})\), \(\delta(-\omega_{0})\), \(\delta(\omega_{\mathrm{s}})\) and \(\delta(-\omega_{\mathrm{s}})\) are residuals of the fitting equations. The minimum value of the residual \(||\delta(\omega_{0})||_{2}+||\delta(-\omega_{0})||_{2}\) is found by fitting Equation (19). Thus, the value of the angular frequency \(\omega_{0}\) is solved. Similarly, the minimum value of the residual \(||\delta(\omega_{\mathrm{s}})||_{2}+||\delta(-\omega_{\mathrm{s}})||_{2}\) is found by fitting Equation (20) so as to find the value of the angular frequency \(\omega_{\mathrm{s}}\). Then, the values of four constant coefficients \(R_{m}\) can be obtained with Equation (9), and Equation (10) can be utilized to additionally derive the specific frequencies, amplitudes and phases associated with the oscillation component.