## 5 Simulation Verification  
Simulation results of the altitude control of an 8-degree-of-freedom powered parafoil were used to verify the validity of the presented method. The main structure parameters of the powered parafoil are shown in Table 1. Simulation results are compared with LADRC and SMC in [13]. The bandwidth parameterization method was used to design the gain of the LESO. \[l_{1}=90,\ l_{2}=2700,\ l_{3}=2700.\]  
The time constant of the filter was selected to be \(T=0.025\) s.  
The parameters of the designed fractional sliding mode backstepping controller were designed to be  
\[\lambda_{1}=0.16,\ k_{1}=0.02,\ k=0.015,\ \alpha=0.82,\ \beta=0.36,\ \epsilon=0.01.\]  
The saturation of the control input was considered. The maximum thrust provided by the engine was limited to 400 N.  
The initial altitude of the powered parafoil was set to be 2000 m and the desired altitude \(H_{d}\) was set to be 1970 m. A gust disturbance (2 m/s) along the negative direction of the Z axis was added to the system during 100-115 s, which is shown in Figure 3.  
Figure 4 shows the observed results of the LESO. The outputs of the observer can effectively observe each state of the system. For comparison, simulations for the LADRC and the SMC in [13] were also carried out. The design of the LADRC and the SMC was the same as that in [13]. Figure 5 shows the attitude angles of the powered parafoil in the longitudinal plane for the three methods. The pitch angles remained at 9\({}^{\circ}\) after the powered parafoil entered the steady-state. The pitch angles oscillated violently when the parafoil suffered from the gust disturbance. It can be seen that the transient performance is better for the parafoil with FSMBC than for that with other two methods. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value/Unit** \\ \hline Span & 10.5/m \\ Chord & 3.1/m \\ Aspect ratio & 3 \\ Area of canopy & 33/m\({}^{2}\) \\ Length of lines & 6.8/m \\ Rigging angle & 10/deg \\ Mass of canopy & 10/kg \\ Mass of payload & 80/kg \\ Characteristic area of drag of payload & 0.6/m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structure parameters of the powered parafoil. Figure 4: Observed results of LESO. Figure 3: The gust disturbance. Figure 6 illustrates the simulation results of the altitude control. The altitude of the powered parafoil with the FSMBC stabilized at 1970 m after 13 s. The convergence time was less than that of the LADRC and SMC. Moreover, the recovery speed against the disturbance is faster for the FSMBC than for the LADRC and SMC. Compared with other two control methods, there exists about 1.4 m steady-state error for the SMC due to the lack of ESO. It can be seen from Figure 7 that the thrust input for the FSMBC stabilizes at 18 s, and the convergence speed is faster than that for the LADRC and SMC. It is obvious that the thrust input for the FSMBC is smoother than that for the LADRC and SMC. This is beneficial to energy saving and system stability. The SMC used a traditional integer order sliding surface, which caused the thrust input fluctuated. The specific performance indexes are shown in Table 2.  
Figure 5: Simulation results of the pitch angle. Figure 6: Simulation results of altitude control by LADRC, SMC, and FSMBC. Figure 10: Simulation results of altitude cotrol by LADRC, SMC, and FSMBC for variable altitude control. Figure 9: Simulation results of the pitch angle for variable altitude control. Figure 8 shows the observed results of the LESO for variable altitude control. The simulation results of pitch angles are shown in Figure 9. When the desired altitude changed to 1960 m, the altitude of the powered parafoil decreased accordingly and the pitch angle deviated from the steady-state value. When the altitude approached to the desired value, the pitch angle returned to the previous steady-state value. Figure 10 shows that the transient performance is better for the FSMBC than for the LADRC and SMC. The settling time is 8 s for the FSMBC and 11 s for the LADRC and SMC.