## 2 System Dynamic Model
### Variation in the System's Mass and Moments of Inertia Parameters
\begin{cases}\dot{m}=a_{1}\dot{\ell}\\ I_{x}=2b_{1}mn+b_{2}\dot{m}\\ \dot{I}_{y}=2c_{1}m\dot{m}+c_{2}\dot{m}\\ I_{z}=d_{1}\dot{m}\end{cases}  \tag{12}\]  
Table 3 shows the parameter values in Equation (11). The LPV model for a dynamic system is not unique. Multiple LPV models can exist for a dynamic system. The time-varying parameters chosen in the LPV model determine the complexity and effectiveness of the control system. The following assumptions are taken into account in order to establish an affine LPV model of quadrotor with time-varying payload [11; 16]. * The sine and cosine of tilt angles are approximated by the second-order Taylor's expansion. * The input \(U_{4}\approx mg\), \(sin\psi\approx 0\), and \(cos\psi\approx 1\) are considered to make the dependency between system states \(\left( \begin{matrix}i.e.,&y\end{matrix} \right)^{T}\)_on_\(\left[ \begin{matrix}\phi&\theta\end{matrix} \right]^{T}\right)\) explicitly. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline \(a_{1}\) & \(-3.1967\times 10^{-2}\) & kg \\ \(a_{2}\) & \(1.0501\times 10^{-1}\) & kg \\ \(b_{1}\) & \(-2.3181\times 10^{-2}\) & m\({}^{2}\)/kg \\ \(b_{2}\) & \(5.3709\times 10^{-3}\) & m\({}^{2}\) \\ \(b_{3}\) & \(-1.9308\times 10^{-4}\) & kg m\({}^{2}\) \\ \(c_{1}\) & \(-2.3731\times 10^{-2}\) & m\({}^{2}\)/kg \\ \(c_{2}\) & \(5.1061\times 10^{-3}\) & m\({}^{2}\) \\ \(c_{3}\) & \(-1.6102\times 10^{-4}\) & kg m\({}^{2}\) \\ \(d_{1}\) & \(5.7349\times 10^{-4}\) & m\({}^{2}\) \\ \(d_{2}\) & \(6.5828\times 10^{-5}\) & kg m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter values. Using the above assumptions, and considering the small gyro effect, the simplified nonlinear model is obtained, which is as follows:  
\[ \begin{cases}I_{x}\ddot{\phi}=U_{1}-l_{x}\dot{\phi}\\ I_{y}\dot{\theta}=U_{2}-l_{y}\dot{\phi}\\ I_{z}\ddot{\psi}=U_{3}-l_{z}\dot{\psi}\\ \ddot{x}=-\left(\theta-\frac{\theta^{3}}{6}\right)\left(1-\frac{\phi^{2}}{2} \right)g-\frac{\dot{m}}{\dot{x}}-\frac{W_{fx}}{m}\\ \ddot{y}=\left(\phi-\frac{\phi^{3}}{6}\right)g-\frac{\dot{m}}{\dot{m}}\dot{y}- \frac{W_{fy}}{m}\\ m\ddot{z}=-\left(1-\frac{\theta^{2}}{2}\right)\left(1-\frac{\phi^{2}}{2} \right)\left(mg+u_{4}\right)+mg-\dot{m}\dot{z}-W_{fz}\end{cases}  \tag{13}\]  
By using Equations (11)-(13), the LPV model of the quadrotor is obtained, which is given as  
\[ \begin{cases}(b_{1}\rho_{1}+b_{2}+b_{3}\rho_{2})\ddot{\phi}=\rho_{2}U_{1}-(2b_ {1}\dot{\phi}+b_{2}\rho_{2}\dot{\phi})d_{r}\\ (c_{1}\rho_{1}+c_{2}+c_{3}\rho_{2})\ddot{\theta}=\rho_{2}U_{2}-\left(2c_{1} \dot{\theta}+c_{2}\rho_{2}\dot{\theta}\right)d_{r}\\ (d_{1}\rho_{1}+d_{2})\ddot{\psi}=U_{3}-d_{1}d_{r}\dot{\psi}\\ \ddot{x}=-g\theta+\frac{g}{2}\rho_{3}\theta+\frac{g}{6}\rho_{4}\theta-d_{r} \rho_{2}\dot{x}-\rho_{2}d_{x}\\ \dot{y}=g\phi-\frac{g}{6}\rho_{3}\phi-d_{r}\rho_{2}\dot{y}-\rho_{2}d_{y}\\ \rho_{1}\ddot{z}=\frac{1}{2}\rho_{3}u_{4}+\frac{1}{2}\rho_{4}u_{4}-u_{4}-d_{r} \dot{z}-d_{z}\end{cases}  \tag{14}\]  
where \(\rho=[\rho_{1}\ \rho_{2}\ \rho_{3}\ \rho_{4}]^{T}\) is a time-varying parameters vector. The terms \(\frac{\phi^{2}\theta^{2}}{4}\) and \(\frac{\phi^{2}\theta^{3}}{12}\) are neglected because of their high power and relatively modest contribution. \( \begin{bmatrix}d_{x}&d_{y}&d_{z}&d_{r}\end{bmatrix} ^{T}\) denote the bounded disturbances and are given as  
\[ \begin{cases}d_{x}=W_{fx}\\ d_{y}=W_{fy}\\ d_{z}=W_{fz}-\frac{mg\phi^{2}}{2}-\frac{mg\theta^{2}}{2}\\ d_{r}=\dot{m}\end{cases}  \tag{15}\]  
The scheduling parameters of the LPV model are  
\[