## Appendix A DFIG Control Model Decoupling
\begin{array}{c}\dot{\textbf{{I}}}=\textbf{{A}}\textbf{{I}}+\textbf{{B} }\textbf{{U}}\\ \textbf{{Y}}=\textbf{{C}}\textbf{{I}}\end{array} \right.\] (A2)  
In Equation (A2), \(\textbf{{A}}\) is the system matrix, \(\textbf{{B}}\) is the control matrix, \(\textbf{{I}}\) is the state vector, \(\textbf{{U}}\) is the input vector, \(\textbf{{Y}}\) is the output vector and \(\textbf{{C}}\) is the output matrix. \(\textbf{{B}}=\frac{1}{\sigma L_{\text{r}}} \begin{bmatrix}1&0\\ 0&1\end{bmatrix} \), \(\textbf{{A}}= \begin{bmatrix}-\frac{R_{\text{r}}}{\sigma L_{\text{r}}}&\omega_ {\text{s}}\\ -\omega_{\text{s}}&-\frac{R_{\text{r}}}{\sigma L_{\text{r}}}\end{bmatrix} \), \(\textbf{{C}}= \begin{bmatrix}1&0\\ 0&1\end{bmatrix} \), \(\textbf{{I}}= \begin{bmatrix}\widetilde{i}_{\text{rd}}\\ \widetilde{i}_{\text{rq}}\end{bmatrix} \) and \(\textbf{{U}}= \begin{bmatrix}u_{\text{rd}}-\frac{\omega_{\text{s}}L_{\text{ m}}U_{\text{s}}}{\omega_{\text{l}}L_{\text{s}}}\\ u_{\text{rq}}\end{bmatrix} \). Equation (A1) can be decoupled with the state feedback method [28]:  
(1) First, \(d_{\text{i}}\) (\(i=1,2\)) is calculated:  
\[c_{\text{1}}\textbf{{A}}^{\text{0}}\textbf{{B}}= \begin{bmatrix}\frac{1}{C}{u_{ \text{r}}}&0\end{bmatrix} \] (A3)The minimum \(l\) value that makes \(c_{1}\mathbf{A}^{l}\mathbf{B}\
eq\mathbf{0}\) is 0, so \(d_{1}=0\). In the same way, \(d_{2}=0\) can be obtained. (2) Then, we define the following matrix according to \(d_{\mathrm{i}}\):  
\[\mathbf{D}= \begin{bmatrix}c_{1}\mathbf{A}^{d_{1}}\\ c_{2}\mathbf{A}^{d_{2}}\end{bmatrix} = \begin{bmatrix}c_{1}\mathbf{A}^{0}\\ c_{2}\mathbf{A}^{0}\end{bmatrix} = \begin{bmatrix}1&0\\ 0&1\end{bmatrix}  \tag{10}\]  
\[\mathbf{E}= \begin{bmatrix}c_{1}\mathbf{A}^{d_{1}}\mathbf{B}\\ c_{2}\mathbf{A}^{d_{2}}\mathbf{B}\end{bmatrix} = \begin{bmatrix}\frac{1}{\sigma L_{\mathrm{ r}}}&0\\ 0&\frac{1}{\sigma L_{\mathrm{r}}}\end{bmatrix}  \tag{11}\]  
\[\mathbf{L}=\mathbf{D}\mathbf{A}= \begin{bmatrix}-R_{\mathrm{r}}/(\sigma L_{\mathrm{r}})& \omega_{\mathrm{s}}\\ -\omega_{\mathrm{s}}&-R_{\mathrm{r}}/(\sigma L_{\mathrm{r}})\end{bmatrix}  \tag{12}\]  
Since the matrix \(\mathbf{E}\) is a non-singular matrix, the above equation of state can be decoupled. (3) Finally, we decouple the equation of state  
\[\dot{\mathbf{X}}=(\mathbf{A}+\mathbf{B}\mathbf{K})\mathbf{X}+\mathbf{B}\mathbf{F}\mathbf{U}=\sigma L_{\mathrm{ r}} \begin{bmatrix}u_{\mathrm{rd}}-\frac{\omega_{\mathrm{s}}L_{\mathrm{m}}L_{ \mathrm{s}}}{\omega_{\mathrm{s}}^{2}L_{\mathrm{s}}}\\ u_{\mathrm{rq}}\end{bmatrix}  \tag{13}\]  
In the equation, the matrix \(\mathbf{F}\) is the input transformation matrix, and \(\mathbf{F}=\mathbf{E}^{-1}\); the matrix \(\mathbf{K}\) is the state feedback matrix, and \(\mathbf{K}=-\mathbf{E}^{-1}\mathbf{L}\). The DFIG control model decoupling can be obtained from Equation (13), as shown in Figure 20. The dashed box in Figure 20 is the target model that needs to be decoupled. Accordingly, Equation (13) can be rewritten as:  
\[\left\{ \begin{array}{c}\dot{\dot{\widetilde{i}}}_{\mathrm{rd}}=(\omega_{ \mathrm{r}}-\omega_{1})\frac{\sigma L_{\mathrm{r}}L_{\mathrm{m}}\underline{u} L_{\mathrm{s}}}{\omega_{\mathrm{r}}L_{\mathrm{s}}}+\sigma L_{\mathrm{r}}u_{ \mathrm{rd}}^{*}\\ \dot{\widetilde{i}}_{\mathrm{rq}}=\sigma L_{\mathrm{r}}u_{\mathrm{rq}}^{*}\\ \dot{\omega}_{\mathrm{r}}=-\frac{\eta_{\mathrm{r}}L_{\mathrm{m}}u_{\mathrm{s}} }{\int\omega_{\mathrm{r}}\omega_{\mathrm{s}}\widetilde{i}}_{\mathrm{rd}}- \frac{D}{T}\omega_{\mathrm{r}}+\frac{T_{\mathrm{r}}}{\dot{T}}\end{array} \right. \tag{14}\]  
where \(u_{\mathrm{rd}}^{*},u_{\mathrm{rq}}^{*}\) represent the input quantity of the control decoupling model. As can be seen from Equation (14), the state equation of the DFIG system changes from a third-order model to a model composed of a first-order and a second-order model after decoupling through state feedback. In the case of S29 SO, based on the decoupling model, the control ratios \(u_{\mathrm{rd}}^{*}\) and \(u_{\mathrm{rq}}^{*}\) of the rotor current were designed with the back-stepping control method.