## 4 Simulation Results and Discussion  
This section presents the simulations to validate the proposed approach. The numerical simulations were carried out using the quadrotor nonlinear model represented in the Equation (9). Multiple scenarios were considered during simulations. In the first scenario, the proposed LPV algorithm was tested for reference tracking in the presence of full payload, time-varying payload, and an empty payload. Wind disturbance was applied in the second scenario, while noise was introduced to the system states in the third. Moreover, \(\mathcal{H}_{\infty}\) controller with regional pole constraints was designed for the quadrotor's linear model that was obtained using small angle assumption and the average values of the mass and inertia parameters [7]. The results of the LPV scheme were compared with those attained with the LTI control design. Case 1: The quadrotor's position is required to track a unit step signal, while the yaw angle needs to be maintained at \(\angle 0^{0}\). The reference signals and the quadrotor's position for full payload, time-varying payload, and an empty payload are depicted in Figure 10. Figure 11 depicts the corresponding attitude angles. The control inputs and motor commands are shown in Figures 12 and 13, respectively. The tracking performance was examined in terms of system rise time (\(t_{r}\)), system settling time (\(t_{s}\)), system overshoot (OS), and system MSE. The values of the performance parameters and percentage improvements are listed in Table 6, Table 7 and Table 8, respectively. The actuator will become saturated if the motor command value is higher than 500 rad/s. Control inputs were used to calculate motor commands using (8). Effective position tracking and yaw angle stabilization were accomplished in the presence of variable mass and variable MOI using the LPV scheme as compared to the LTI controller. \begin{table}
\begin{tabular}{c c c c c} \hline
**Name** & **Roll Angle** & **Pitch Angle** & **x-Position** & **y-Position** \\ \hline \(t_{min}\) & \(-0.04\) & \(-0.032\) & \(-0.1042\) & \(-0.0433\) \\ \(\gamma\) & \(1.0\) & \(1.0\) & \(1.718\) & \(1.292\) \\ \hline \end{tabular}
\end{table} 
Table 5: \(t_{min}\) and \(\gamma\) values of under-actuated subsytem. Figure 11: **(a)** The attitude angles \([\phi,\theta,\psi]\) with full payload; **(b)** The attitude angles \([\phi,\theta,\psi]\) with time-varying payload; **(c)** The attitude angles \([\phi,\theta,\psi]\) with empty payload. Figure 10: **(a)** The positions \([x,y,z]\) with full payload; **(b)** The positions \([x,y,z]\) with time-varying payload; **(c)** The positions \([x,y,z]\) with empty payload. Figure 12: (**a**) The control inputs \([U_{1},U_{2},U_{3},U_{4}]\) with full payload; (**b**) The control inputs \([U_{1},U_{2},U_{3},U_{4}]\) with time-varying payload; (**c**) The control inputs \([U_{1},U_{2},U_{3},U_{4}]\) with empty payload. Case 2: The inertia parameters may vary due to the tilting of the quadrotor. To check the performance of the proposed control scheme in the presence of inertia parameter variation, we have considered a 20% variation in the inertia. Figure 14 depicts the variation in the water level considered over time. Figure 15 depicts the variation in the inertia with the mass. The reference signals and the quadrotor's position are depicted in Figure 16. Figure 17 depicts the corresponding attitude angles. The control inputs and motor commands are shown, respectively, in Figures 18 and 19. The simulation results indicate that the LPV scheme gives good position tracking and yaw angle stabilization in the presence of variation in the inertia parameters as compared to the LTI controller.