## 3 LPV Control Design
### LPV Control for Fully Actuated Subsystem  
LPV controller is devised for the fully actuated subsystem dynamics by solving the LMIs given in (24)-(26) at each vertex of the rectangular polytope using the LMI control toolbox [34].  
The descriptor LPV form of altitude dynamics is  
\[\left\{\begin{aligned} & E_{z}(\rho_{z})\begin{bmatrix}z\\ z\end{bmatrix}=\begin{bmatrix}0&1\\ 0&-\mu t\end{bmatrix}\begin{bmatrix}z\\ z\end{bmatrix}+B_{z}(\rho_{z})u_{4}\\ \\ & y_{z}=\begin{bmatrix}1&0\end{bmatrix}\begin{bmatrix}z\\ z\end{bmatrix}\end{aligned}\right. \tag{28}\]  
where  
\[E_{z}(\rho_{z})=\begin{bmatrix}1&0\\ 0&0\end{bmatrix}+\rho_{1}\begin{bmatrix}0&0\\ 0&1\end{bmatrix}+\rho_{3}\begin{bmatrix}0&0\\ 0&0\end{bmatrix}+\rho_{4}\begin{bmatrix}0&0\\ 0&0\end{bmatrix}\]  
\[B_{z}(\rho_{z})=\begin{bmatrix}0\\ -1\end{bmatrix}+\rho_{3}\begin{bmatrix}0\\ \frac{1}{2}\end{bmatrix}+\rho_{4}\begin{bmatrix}0\\ \frac{1}{2}\end{bmatrix}\]  
The descriptor LPV form of yaw dynamics is  
\[\left\{\begin{aligned} & E_{\psi}(\rho_{\psi})\begin{bmatrix}\psi\\ \psi\end{bmatrix}=\begin{bmatrix}0&1\\ 0&-g_{1}\mu\end{bmatrix}\begin{bmatrix}\psi\\ \psi\end{bmatrix}+\begin{bmatrix}0\\ 1\end{bmatrix}U_{3}\\ \\ & y_{\psi}=\begin{bmatrix}1&0\end{bmatrix}\begin{bmatrix}\psi\\ \psi\end{bmatrix}\end{aligned}\right. \tag{29}\]  
where  
\[E_{\psi}(\rho_{\psi})=\begin{bmatrix}1&0\\ 0&g_{2}\end{bmatrix}+\rho_{1}\begin{bmatrix}0&0\\ 0&g_{1}\end{bmatrix}\]  
The weighting filters \(W_{ez}\) and \(W_{e\psi}\) are selected as follows,  
\[\left[\begin{aligned} &\frac{x_{Wez}}{z_{1z}}\end{aligned} \right]=\left[\begin{aligned} & -5.6\times 10^{-6}&0.56\\ & 1&0.5\end{aligned}\right]\left[\begin{aligned} & x_{Wez}\\ e_{z}\end{aligned}\right] \tag{30}\]  
\[\left[\begin{aligned} &\frac{x_{We\psi}}{z_{1\psi}}\end{aligned} \right]=\left[\begin{aligned} & -4.5\times 10^{-4}&0.45\\ & 1& 1\end{aligned}\right]\left[\begin{aligned} & x_{Web}\\ e_{\psi}\end{aligned}\right] \tag{31}\]  
The static weights \(V_{uz}\) and \(V_{up}\) are expected to be 0.5 and 1, respectively. The static weight \(d_{z}=0.5\) is taken into consideration. The bounded disturbance \(d_{r}\) is assumed as \(3.2\times 10^{-3}\) kg/s. The parameters associated with the desired LMI regions are:\[\alpha_{z}=-0.5,\ \ \ \beta_{z}=-30,\ \ \angle\varphi_{z}=50^{\circ} \tag{32}\]  
\[\alpha_{\psi}=-0.5,\ \ \ \beta_{\psi}=-5,\ \ \angle\varphi_{\psi}=50^{\circ} \tag{33}\]  
The LPV controller gains are:  
\[K_{z}(\rho_{z})=\sum_{i=1}^{8}\alpha_{z_{i}}K_{z_{i}} \tag{34}\]  
\[K_{\psi}(\rho_{\psi})=\sum_{i=1}^{2}\alpha_{\psi_{i}}K_{\psi_{i}} \tag{35}\]  
where \(K_{z_{i}}(i=1,2,\ldots,8)\) and \(K_{\psi_{i}}(i=1,2)\) represent the state feedback gains, \(\alpha_{z_{i}}(i=1,2,\)\(\ldots,8)\) and \(\alpha_{\psi_{i}}(i=1,2)\) represent the barycentric weights, and \(\rho_{z}=[\rho_{1},\rho_{3},\rho_{4}]\) and \(\rho_{\psi}=\rho_{1}\) are the time-varying parameters.  
The robust closed-loop system stability and quadratic \(\mathcal{H}_{\infty}\) performance \(\gamma\) of the LPV system are tested by using the functions \(pdlstab\) and \(quadperf\) from the MATLAB LMI Toolbox's. The LPV system will be stable, if the quantity \(t_{min}\) returned by the \(pdlstab\) function is negative (i.e., \(t_{min}<0\)). The values achieved using the functions are given in Table 4.  
The quantity \(t_{min}<0\), which ensures the closed-loop system's robust stability.