## 3 LPV Control Design  
In this section, we present an LMIs-based LPV control design for position tracking of the quadrotor. The goal is that the state variables \([x,\ \ y,\ \ z]\) follow the desired reference trajectories \([x_{ref},\ \ y_{ref},\ \ z_{ref}]\), while the yaw angle \(\psi\) is required to remain at \(\angle 0^{0}\). All the state variables \([x,\ \ y,\ \ z,\ \dot{\phi},\ \theta,\ \psi,\ \dot{\phi},\ \dot{\theta},\ \dot{\psi},\ \dot{x},\ \dot{y},\ \dot{z}]\) and the water level in the liquid tank are considered as measurable. Based on the system dynamics, the control system is split into subsystems. 1. The fully actuated subsystem dynamics that consist of \(\ddot{z}\) and \(\ddot{\psi}\). 2. The under-actuated subsystem dynamics made up of \(\dot{\phi},\dot{\theta},\ \dot{x},\ \dot{y}\). The LPV control system configuration is depicted in Figure 5. \(P(\rho)\) shows the generalized LPV plant and \(K(\rho)\) depicts the LPV controller gains. The generalized LPV plant's state-space realization is as follows [26; 27]:  
\[ \begin{cases}\hat{\chi}=\mathcal{A}(\rho)\chi+\mathcal{B}_{1}(\rho)w+\mathcal{ B}_{2}(\rho)u\\ z=\mathcal{C}_{z}(\rho)\chi+\mathcal{D}_{11}(\rho)w+\mathcal{D}_{12}(\rho)u\\ v=\mathcal{C}_{v}(\rho)\chi+\mathcal{D}_{21}(\rho)w+\mathcal{D}_{22}(\rho)u \end{cases}  \tag{16}\]  
where  
1. \(\chi\in R^{n}\), \(w\in R^{n_{w}}\), \(u\in R^{n_{u}}\), \(z\in R^{n_{z}}\) and \(v\in R^{n_{v}}\) are the system states, exogenous inputs (i.e., reference signals and disturbances), control signals, exogenous outputs, and sensed outputs, respectively. 2. \(\mathcal{A}(\rho)\), \(\mathcal{B}_{1}(\rho)\), \(\mathcal{B}_{2}(\rho)\), \(\mathcal{C}_{z}(\rho)\), \(\mathcal{D}_{11}(\rho)\), \(\mathcal{D}_{12}(\rho)\), \(\mathcal{C}_{v}(\rho)\), \(\mathcal{D}_{21}(\rho)\) and \(\mathcal{D}_{22}(\rho)\) are parameter dependent state-space matrices of appropriate dimensions. 3. \(\rho\) is a time-varying parameters vector that varies in a convex polytope of vertices \(\rho_{vi}(i=1,2,\ldots,N)\). \(N\) represents the number of vertices. The LPV control law is given as: \[u(t)=K(\rho)v(t)\] (17)  
In the LPV polytopic form, the matrices of \(P(\rho)\) can be written as  
\[ \begin{bmatrix}\mathcal{A}(\rho)&\mathcal{B}_{1}(\rho)&\mathcal{B}_{2}(\rho) \\ \mathcal{C}_{z}(\rho)&\mathcal{D}_{11}(\rho)&\mathcal{D}_{12}(\rho)\\ \mathcal{C}_{v}(\rho)&\mathcal{D}_{21}(\rho)&\mathcal{D}_{22}(\rho)\\ \end{bmatrix} =\sum_{i=1}^{N}\alpha_{i} \begin{bmatrix}A_{i}&B_{1i}&B_{2i}\\ C_{zi}&D_{11i}&D_{12i}\\ C_{vi}&D_{21i}&D_{22i}\\ \end{bmatrix}  \tag{18}\]  
In the polytopic form, LPV controller gains \(K(\rho)\) can be expressed as  
\[K(\rho)=\sum_{i=1}^{N}\alpha_{i}K_{i} \tag{19}\]  
\(\alpha_{i}(i=1,2,\ldots,N)\) represent the barycentric weights (for details, see [28; 29]). The time-varying parameter vector \(\rho\) can be expressed as  
\[\rho=\left\{\alpha_{i}\rho_{vi},\quad\alpha_{i}\geq 0,\quad\sum_{i=1}^{N} \alpha_{i}=1\right\}\]  
To design an LPV controller, a mixed sensitivity control approach is adopted [30; 31]. The weighted closed-loop system for mixed sensitivity control design is shown in Figure 6.  
Figure 5: The structure of closed-loop LPV system. \(G_{P}(\rho)\) shows the LPV model of the dynamic system to the control signal \(u\) and the disturbance signal \(w_{2}\), \(W_{e}\) is the weighting filter that is chosen to impose meaningful requirements on the closed-loop system, such as trajectory tracking and disturbance rejection, \(V_{u}\) and \(V_{d}\) are the static weights that represent the maximum values of control signal and disturbance signal, respectively. The state-space realization of \(G_{P}(\rho)\) and \(W_{e}\) is given by  
\[\left[ \begin{array}{c}\dot{x}_{p}\\ y_{p}\end{array} \right]=\left[ \begin{array}{c|cc}A_{p}(\rho)&B_{w}(\rho)V_{d} &B_{p}(\rho)\\ \hline C_{p}(\rho)&0&0\end{array} \right]\left[ \begin{array}{c}x_{p}\\ w_{2}\\ u\end{array} \right] \tag{20}\]  
\[\left[ \begin{array}{c}\dot{x}_{We}\\ z_{1}\end{array} \right]=\left[