## 5 Simulation Results
### Simulation Results of the Perturbed System with Disturbance Observer  
Under the same disturbance conditions, as listed in Section 5.2, simulation for the perturbed system with the controller blended with the disturbance observers based on (78) (proposed), (79) (Zong [28,29]), and (80) (Levant [30,31]) is conducted to evaluate the estimation performance. The simulation results are shown in Figure 5, where the 3D tracking result is presented in Figure 4a and the tracking errors in the \(x,y,z\), and \(\psi\)-axes are shown in Figure 5b-e. The results indicate that the tracking results of the control system with the proposed observer exhibit a more efficient performance than that of the controller with the Levant observer. The estimates of the observer states and disturbances of the proposed observer are shown in Figure 5f-i. Estimates of the observer states and disturbances of the Levant observer are shown in Figure 5j-m. Finally, estimates of the observer states and disturbances of the Levant observer are shown in Figure 5n-q. The simulation results indicate that the estimate performance of the proposed observer exceeds  
Figure 3: Simulation results of the nominal hexacopter system. (**a**) The 3D tracking result. (**b**) Output of \(x\)-axis. (**c**) Output of \(y\)-axis. (**d**) Output of \(z\)-axis. (**e**) Output of \(\psi\)-axis. (**f**) Output of \(\phi\)-axis. (**g**) Output of \(\theta\)-axis. (**h**) Tracking error of \(x\)-axis. (**i**) Tracking error of \(y\)-axis. (**j**) Tracking error of \(z\)-axis. (**k**) Tracking error of \(\psi\)-axis. (**l**) Control input of latitude controller. (**m**) Control input of heading controller. that of the conventional Levant and Zong observers. Table 5 shows the root mean square (RMS) value of the tracking error of each system, where it is shown that the RMS value of the proposed system decreases until an average 7% of that of Levant's system. Figure 4: Simulation results of the perturbed hexacopter system. (**a**) 3D tracking result. (**b**) Tracking error for \(x\)-axis. (**c**) Tracking error for \(y\)-axis. (**d**) Tracking error for \(z\)-axis. (**e**) Tracking error for \(\psi\)-axis. (**f**) Tracking error for \(\  
Figure 5: _Cont._  
Finally, under the disturbance condition with the 300% increase in the mass and the same other condition, simulation for the perturbed system is conducted to evaluate the estimation performance with the disturbance observers based on (78) (proposed), (79) (Zong [28,29]), and (80) (Levant [30,31]). The desired 8-shape command trajectory is described as  
\[\begin{array}{cccc}\mathbf{P}_{d}(t)=[ \begin{array}{cc}x_{d}&y_{d}&z_{d}\end{array}  ]^{T}=[ \begin{array}{cc}0&0&-4(1-e^{-0.3t})]^{T}&(m),&t\leq 4s,\\ \mathbf{P}_{d}(t)=\left[\begin{array}{c}x_{d}\\ y_{d}\\ z_{d}\end{array}\right]=\left[\begin{array}{c}8-8\cos\left(\frac{2\pi(t-4)}{ 12}\right)\\ 4\sin\left(\frac{4\pi(t-4)}{12}\right)\\ -4(1-e^{-0.3t})\end{array}\right],&t\geq 4s,\end{array} \]  
under other rotation commands set to be zero value. In the first stage, the hexacopter UAV climbs vertically for 4 s to simulate the take-off flight. Next, it follows an 8-shaped path while continuing to lift. In the second stage, the lift, sideslip, and forward performances of the hexacopter UAV are evaluated comprehensively. The simulation results are shown in Figure 6, where the 3D tracking result is presented in Figure 6a, and the tracking output and errors in the \(x,y,z\) axes are shown in Figure 6b-g. The simulation results indicate that the tracking results of the control system with the proposed observer exhibit a more efficient performance than that of the controller with the Levant observer, like the previous results. Table 6 shows the root mean square (RMS) value of the tracking error of each system, where it is shown that the RMS value of the proposed system decreases until an average 27% of that of [PERSON]'s system. Figure 6: Simulation results of the perturbed hexacopter system for 300% increase in the mass with the controller combined with the disturbance observer.