## 4 Simulation and Results Analysis
### Results  
The results are illustrated in the following figures. In Figures 3 and 4, (a) shows the time response of the quadrotor's attitude angles, (b) shows the angular velocities, (c) demonstrates the variation of the three control inputs over time and (d) displays the computation time of each controller. Figure 5 presents the white noise disturbances applied during the simulation. Figure 6 exhibits the attitude tracking curves in step inputs with disturbances.  
As shown in Figure 3a, all controllers effectively track the desired input in pitch control with an approximate 270 ms delay behind the command curve. In terms of roll control, all controllers demonstrate comparable performance, following the target command accurately within 650 ms. For yaw control, the FLMPC controller maintains a steady angle close to 0. In contrast, the LMPC controller exhibits slight fluctuations, while the AMPC controller experiences more noticeable fluctuations. Integrating these observations with the computational time in Figure 3d, it is evident that the FLMPC controller achieves superior control performance with relatively lower computational cost, requiring only one-tenth of the computational time of the AMPC.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Values** & **Unit** \\ \hline \(m\) & 0.8 & kg \\ \(I_{x}\) & \(9.7\times 10^{-3}\) & kg\(\cdot\)m\({}^{2}\) \\ \(I_{y}\) & \(9.7\times 10^{-3}\) & kg\(\cdot\)m\({}^{2}\) \\ \(I_{2}\) & \(1.7\times 10^{-2}\) & kg\(\cdot\)m\({}^{2}\) \\ \(l\) & 0.06 & m \\ \(g\) & 9.81 & m/Figure 3: Controller performance in roll oscillations and pitch step. (**a**) Angular attitudes curves. (**b**) Angular velocities curves. (**c**) Control inputs curves. (**d**) Computation time.  
Figure 4: Controller performance in roll, pitch and yaw step inputs. (**a**) Angular attitudes curves. (**b**) Angular velocities curves. (**c**) Control inputs curves. (**d**) Computation time.  
As observed in Figure 4a, all MPC controllers effectively reach and stabilize at their target values with minimal overshoot. FLMPC provides the fastest response for the pitch angle, stabilizing in approximately 0.95 s, followed by AMPC at around 1.65 s, and LMPC, which is influenced by channel coupling, taking the longest at about 2.7 s. For the roll and yaw angles, all controllers demonstrate similar settling times, though LMPC shows greater fluctuation due to coupling effects. Figure 4b shows that the variations in \(p\) and \(r\) remain well within their upper constraints, while Figure 4c illustrates that the maximum control inputs do not exceed their specified boundaries. Table 2 provides a summary of key performance metrics, including response time, overshoot and mean computation time for three controllers.  
Figure 5: Disturbances applied to each attitude angle.  
Figure 6: Attitude tracking curves in step inputs with disturbances.  
The FLMPC and AMPC controllers demonstrated superior performance in Figure 6. Both of them showing fluctuations under disturbances of less than 10%, while LMPC exhibited notable fluctuations, with a maximum overshoot of 50%, indicating a lower robustness to disturbances. These results confirm that FLMPC provides robust and stable control across all attitude angles, mitigating the impact of external disturbances effectively.