## 3 LPV Control Design
### LPV Control for Under-Actuated Subsystem  
To address the under-actuation, a multi-loop control strategy is adopted. An inner-loop LPV scheme is suggested for roll and pitch dynamics, while an outer-loop LPV strategy is proposed for the y-position and x-position dynamics to generate the roll and pitch commands. The inner-loop receives the roll and pitch commands \([\phi_{ref},\ \ \rho_{ref}]\) from the outer-loop to track the desired reference positions \([y_{ref},\ \ x_{ref}]\). The scheduling parameters \(\rho_{1}\) and \(\rho_{2}\) are not independent, which leads to conservatism. To reduce conservatism, we use a triangle polytope, BED, rather than a rectangular polytope, ABCD, as shown in Figure 9 (for details, see [35; 36]). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **z-Position** & **Yaw Angle** \\ \hline \(t_{min}\) & \(-0.0512\) & \(-0.0158\) \\ \(\gamma\) & \(1.95\) & \(0.02\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: \(t_{min}\) and \(\gamma\) values of fully actuated subsytem. Figure 9: The structure of the Polytopes. The roll dynamics in descriptor LPV form is  
\[ \begin{cases}E_{\phi}(\rho_{\phi})\begin{bmatrix}\dot{\phi}\\ \dot{\phi}\end{bmatrix}=A_{\phi}(\rho_{\phi})\begin{bmatrix}\dot{\phi}\\ \dot{\phi}\end{bmatrix}+B_{\phi}(\rho_{\phi})U_{1}\\ y_{\phi}=\begin{bmatrix}1&0\end{bmatrix}\begin{bmatrix}\dot{\phi}\\ \dot{\phi}\end{bmatrix}\end{cases}  \tag{36}\]  
where  
\[E_{\phi}(\rho_{\phi}) = \begin{bmatrix}1&0\\ 0&e_{2}\end{bmatrix} +\rho_{1} \begin{bmatrix}0&0\\ 0&e_{1}\end{bmatrix} +\rho_{2} \begin{bmatrix}0&0\\ 0&e_{3}\end{bmatrix} \] \[A_{\phi}(\rho_{\phi}) = \begin{bmatrix}0&1\\ 0&-2e_{1}m\end{bmatrix} +\rho_{1} \begin{bmatrix}0&0\\ 0&0\end{bmatrix} +\rho_{2} \begin{bmatrix}0&0\\ 0&-e_{2}m\end{bmatrix} \] \[B_{\phi}(\rho_{\phi}) = \begin{bmatrix}0\\ 0\end{bmatrix} +\rho_{1} \begin{bmatrix}0\\ 0\end{bmatrix} +\rho_{2} \begin{bmatrix}0\\ 1\end{bmatrix} \]  
The descriptor LPV form of pitch dynamics is  
\[ \begin{cases}E_{\theta}(\rho_{\theta})\begin{bmatrix}\dot{\theta}\\ \dot{\theta}\end{bmatrix}=A_{\theta}(\rho_{\theta})\begin{bmatrix}\theta\\ \dot{\theta}\end{bmatrix}+B_{\theta}(\rho_{\theta})U_{2}\\ y_{\theta}=\begin{bmatrix}1&0\end{bmatrix}\begin{bmatrix}\dot{\theta}\\ \dot{\theta}\end{bmatrix}\end{cases}  \tag{37}\]  
where  
\[E_{\theta}(\rho_{\theta}) = \begin{bmatrix}1&0\\ 0&f_{2}\end{bmatrix} +\rho_{1} \begin{bmatrix}0&0\\ 0&f_{1}\end{bmatrix} +\rho_{2} \begin{bmatrix}0&0\\ 0&f_{3}\end{bmatrix} \] \[A_{\theta}(\rho_{\theta}) = \begin{bmatrix}0&1\\ 0&-2f_{1}m\end{bmatrix} +\rho_{1} \begin{bmatrix}0&0\\ 0&0\end{bmatrix} +\rho_{2} \begin{bmatrix}0&0\\ 0&-f_{2}m\end{bmatrix} \] \[B_{\theta}(\rho_{\theta}) = \begin{bmatrix}0\\ 0\end{bmatrix} +\rho_{1} \begin{bmatrix}0\\ 0\end{bmatrix} +\rho_{2} \begin{bmatrix}0\\ 1\end{bmatrix} \]  
The LPV form of x-position dynamics is  
\[ \begin{cases}\begin{bmatrix}\dot{x}\\ \dot{x}\end{bmatrix}=A_{x}(\rho_{x})\begin{bmatrix}x\\ \dot{x}\end{bmatrix}+B_{x}(\rho_{x})u_{x}\\ y_{x}=\begin{bmatrix}1&0\end{bmatrix}\begin{bmatrix}x\\ \dot{x}\end{bmatrix}\end{cases}  \tag{38}\]where  
\[A_{x}(\rho_{x})= \begin{bmatrix}0&1\\ 0&0\end{bmatrix} +\rho_{2} \begin{bmatrix}0&0\\ 0&-m_{t}\end{bmatrix} +\rho_{3} \begin{bmatrix}0&0\\ 0&0\end{bmatrix} +\rho_{4} \begin{bmatrix}0&0\\ 0&0\end{bmatrix} \]  
\[B_{x}(\rho_{x})= \begin{bmatrix}0\\ -g\end{bmatrix} +\rho_{2} \begin{bmatrix}0\\ 0\end{bmatrix} +\rho_{3} \begin{bmatrix}0\\ g\end{bmatrix} +\rho_{4} \begin{bmatrix}0\\ g\end{bmatrix} \]  
The descriptor LPV form of y-position dynamics is  
\[ \begin{cases}\begin{bmatrix}y\\ \dot{y}\end{bmatrix}=A_{y}\big{(}\rho_{y}\big{)}\begin{bmatrix}y\\ \dot{y}\end{bmatrix}+B_{y}\big{(}\rho_{y}\big{)}u_{y}\\ \\ y_{y}=\begin{bmatrix}1&0\end{bmatrix}\begin{bmatrix}y\\ \dot{y}\end{bmatrix}\end{cases}  \tag{39}\]  
where  
\[A_{y}\big{(}\rho_{y}\big{)}= \begin{bmatrix}0&1\\ 0&0\end{bmatrix} +\rho_{2} \begin{bmatrix}0&0\\ 0&-m\end{bmatrix} +\rho_{3} \begin{bmatrix}0&0\\ 0&0\end{bmatrix} \]  
\[B_{y}\big{(}\rho_{y}\big{)}= \begin{bmatrix}0\\ g\end{bmatrix}