## 5 Simulation Results  
The Backstepping-ADRC controller presented above was applied on the complete system model described in Section 3 in simulations. The proposed wind gust disturbances were exerted from \(t=10\) s, which is illustrated in Figure 1. The attitude command for the helicopter controlled by the proposed controller is \(\phi_{c}=\theta_{c}=\psi_{c}=5.73^{\circ}\times\sin(t+10)\). All simulation tests were carried out in the system described in Section 6. A PID controller with high integral gain was also tested on the model under the same conditions because it is capable of rejecting disturbances due to its integral term. This allows for comparison with the Backstepping-ADRC. The main parameters of the model, and controller are given in Tables 1 and 2. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Description** & **Value** \\ \hline \(I_{xx},I_{yy},I_{zz}\) & Moments of inertia & 0.0541, 0.0607, 0.1601 (kg \(\cdot\) m\({}^{2}\)) \\ \(M\) & Gross weight & 2.8(kg) \\ \(R_{m}\) & Radius of main rotor & 0.485(m) \\ (\(L_{u}\),\(L_{v}\)) & Turbulence scale length & 23.6, 23.6 \\ (\(\sigma_{u},\sigma_{v}\)) & The turbulence intensities & 0.99, 0.99 \\ \(U\) & Reference air speed & 5(m/s) \\ \(R_{12},R_{3}\) & Control input coefficients & \(\left[\begin{array}{cc}34.48&185.79\\ 150.19&-35.91\end{array}\right]\),12.48 \\ \hline \end{tabular}
\end{table} 
Table 1: Model parameters. Figure 1: Wind gust based on Dryden spectral model. (**a**) Wind gust in longitudinal direction; (**b**) wind gust in lateral direction. The simulation results are exhibited in Figures 2-5. Figure 2 presents the attitude tracking results of the proposed Backstepping-ADRC controller under the same conditions for comparison with the PID controller. Since the wind gusts were applied in the longitudinal and lateral directions, the pitch angle and roll angle should be emphasized. It is clear that the control method developed in this paper has better performance, as the tracking delay of the method is almost negligible, while the delay of PID control is nearly 1 s. Meanwhile, the amplitude of the tracking error is less than 3\({}^{\circ}\) when in the worst situation of the whole simulation progress, however, the PID's reaches 10. For the roll angle, the average value of the tracking error is 0.65\({}^{\circ}\) for the Backstepping-ADRC controller, while it is 3.28\({}^{\circ}\) for PID controller. Additionally, the standard deviations of the tracking errors are 0.66\({}^{\circ}\) and 2.33\({}^{\circ}\), respectively. For the pitch angle, the average value of the tracking error is 0.47\({}^{\circ}\) for the Backstepping-ADRC controller, while it is 2.14\({}^{\circ}\) for PID controller. Additionally, the standard deviations of the tracking errors are 0.38\({}^{\circ}\) and 1.65\({}^{\circ}\), respectively. The disturbance observation performance is illustrated in Figure 3. It is shown that the estimated value of the total disturbance is almost equal to the real one, and the delay is less than 0.2 s, which is totally acceptable. The system inputs generated by the controller are shown in Figure 4.  
The simulation results shown above demonstrate that the proposed controller is capable of accurate and agile attitude tracking under external wind gust disturbances as well as nonlinear aerodynamic forces in forward flight. The total disturbances were successfully estimated by the rapid and precise estimation of ESO. When compared to the PID controller, the attitude tracking of the Backstepping-ADRC is more accurate and has less lag. In addition, the standard deviations and means of tracking errors are much smaller. While smaller mean tracking errors indicate that the control strategy has better control accuracy, there is 80% improvement of the method proposed in this paper according to the above data. Meanwhile, smaller standard deviations mean that the control method has better stability and better capability of disturbance rejection.