## 3 LPV Control Design
### LPV Control for Under-Actuated Subsystem
+\rho_{2} \begin{bmatrix}0\\ 0\end{bmatrix} +\rho_{3} \begin{bmatrix}0\\ -\frac{g}{6}\end{bmatrix} \]  
The weight functions are chosen as:  
\[\left[ \begin{array}{c|c}\dot{x}_{W\epsilon\phi}\\ \hline z_{1}\end{array} \right]=\left[ \begin{array}{c|c}-7.5\times 10^{-4}&1.936 \\ \hline 3.873&0.5\end{array} \right]\left[ \begin{array}{c}x_{W\epsilon\phi}\\ \hline e_{\phi}\end{array} \right] \tag{40}\]  
\[\left[ \begin{array}{c|c}\dot{x}_{W\epsilon\phi}\\ \hline z_{1\theta}\end{array} \right]=\left[ \begin{array}{c|c}-7.5\times 10^{-4} &7.45\\ \hline 1&0.5\end{array} \right]\left[ \begin{array}{c}x_{W\epsilon\phi}\\ \hline e_{\phi}\end{array} \right] \tag{41}\]  
Weighting filters \(W_{\epsilon\phi}\), \(W_{\epsilon\phi}\), \(W_{\epsilon x}\), and \(W_{\epsilon y}\) are chosen as follows:  
\[\left[ \begin{array}{c|c}\dot{x}_{W\epsilon\phi}\\ \hline z_{1\psi}\end{array} \right]=\left[ \begin{array}{c|c}-7.5\times 10^{-4} &7.45\\ \hline 1&0.5\end{array} \right]\left[ \begin{array}{c}x_{W\epsilon\phi}\\ \hline e_{\phi}\end{array} \right] \tag{42}\]  
\[\left[ \begin{array}{c|c}\dot{x}_{W\epsilon\theta}\\ \hline z_{1\theta}\end{array} \right]=\left[ \begin{array}{c|c}-7.5\times 10^{-4} &7.45\\ \hline 1&0.5\end{array} \right]\left[ \begin{array}{c}x_{W\epsilon\theta}\\ \hline e_{\theta}\end{array} \right] \tag{43}\]  
\[\left[ \begin{array}{c|c}\dot{x}_{W\epsilon x}\\ \hline z_{1x}\end{array} \right]=\left[ \begin{array}{c|c}-9.5\times 10^{-5} &0.95\\ \hline 1&0.83\end{array} \right]\left[ \begin{array}{c|c}x_{W\epsilon x}\\ \hline e_{x}\end{array} \right] \tag{44}\]  
\[\left[ \begin{array}{c|c}\dot{x}_{W\epsilon y}\\ \hline z_{1y}\end{array} \right]=\left[ \begin{array}{c|c}-9.1\times 10^{-5} &0.90\\ \hline 1&0.83\end{array} \right]\left[ \begin{array}{c|c}x_{W\epsilon y}\\ \hline e_{y}\end{array} \right] \tag{45}\]  
The values of weights \(V_{u\psi}=1\), \(V_{u\theta}=1\), \(V_{ux}=\dfrac{\pi}{3}\), and \(V_{uy}=\dfrac{\pi}{3}\) are considered. The values of static weights \(V_{dx}=0.5\) and \(V_{dy}=0.5\) are assumed. The values of the parameters associated with the LMI regions are:  
\[\alpha_{\phi}=-5,\ \ \beta_{\phi}=-30,\ \angle\rho_{\phi}=50^{\circ} \tag{46}\]  
\[\alpha_{\theta}=-5,\ \ \beta_{\theta}=-30,\ \angle\rho_{\theta}=50^{\circ} \tag{47}\]  
\[\alpha_{x}=-0.5,\ \ \beta_{x}=-4.5,\ \angle\sigma_{x}=50^{\circ} \tag{48}\]  
\[\alpha_{y}=-0.5,\ \ \beta_{y}=-4.5,\ \angle\rho_{y}=50^{\circ} \tag{49}\]The following are the gains of the LPV controllers designed using conditions (24)-(26):  
\[K_{\phi}\big{(}\rho_{\phi}\big{)} =\sum_{i=1}^{3}\alpha_{\phi i}K_{\phi i} \tag{50}\] \[K_{\theta}(\rho_{\theta}) =\sum_{i=1}^{3}\alpha_{\theta_{i}}K_{\theta i}\] (51) \[K_{x}(\rho_{x}) =\sum_{i=1}^{8}\alpha_{x_{i}}K_{x_{i}}\] (52) \[K_{y}\big{(}\rho_{y}\big{)} =\sum_{i=1}^{4}\alpha_{yi}K_{yi} \tag{53}\]  
where \(K_{\phi_{i}}(i=1,2,3)\), \(K_{\theta_{i}}(i=1,2,3)\), \(K_{x_{i}}(i=1,2,\ldots,8)\), and \(K_{y_{i}}(i=1,2,\ldots,4)\) represent the state feedback gains, \(\alpha_{\phi_{i}}(i=1,2,3)\), \(\alpha_{\theta_{i}}(i=1,2,3)\), \(\alpha_{x_{i}}(i=1,2,\ldots,8)\), and \(\alpha_{y_{i}}(i=1,2,\ldots,4)\) represent the barycentric weights, and \(\rho_{\phi}=[\rho_{1},\rho_{2}]\), \(\rho_{\theta}=[\rho_{1},\rho_{2}]\), \(\rho_{x}=[\rho_{2},\rho_{3},\rho_{4}]\), and \(\rho_{y}=[\rho_{2},\rho_{3}]\) are the scheduling parameters. The values of \(t_{min}\) and \(\gamma\) achieved using the MATLAB functions for the under-actuated subsytem are given in Table 5.  
The quantity \(t_{min}<0\), which ensures the closed-loop system's robust stability.