# Research on Backstepping Linear Active Disturbance Rejection Control of Hypersonic Vehicle
## 5 Simulation and Analysis  
In order to avoid a given mutation in height and speed, the prefilters are used to generate the velocity and altitude commands of the hypersonic vehicle in this paper:  
\[V_{d}=(V_{c}-V_{0})\times\frac{\lambda^{2}}{s^{2}+2\mu\lambda s+\lambda^{2}}+ V_{0} \tag{54}\]  
\[h_{d}=(h_{c}-h_{0})\times\frac{\lambda^{2}}{s^{2}+2\mu\lambda s+\lambda^{2}}+ h_{0} \tag{55}\]  
where \(\mu\) and \(\lambda\) are the filter parameters; \(V_{0}\) and \(h_{0}\) are the initial velocity and altitude of the aircraft; \(V_{c}\) and \(h_{c}\) are the given step reference. The function of the prefilter is to modulate the step \(V_{c}\), \(h_{c}\) into \(V_{d}\), \(h_{d}\). The initial simulation values of the hypersonic vehicle are shown in Table 2 below. **Example 1**. : _In order to verify the effectiveness of the algorithm proposed in this paper, the speed and altitude reference tracking is carried out without interference. The control strategy requires a speed of 50 m/s and a height of 200 m. The controller parameters are shown in the following Table 3. In order to verify the superior performance of the controller designed in this paper, the classical PID controller is adopted to track the speed and altitude of the aircraft, which is the same as above. The \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Description** & **Value** & **Unit** \\ \hline \(V_{0}\) & Initial value of velocity & 4590 & m/s \\ \(\gamma_{0}\) & Initial value of track angle & 0 & deg \\ \(h_{0}\) & Initial value of altitude & 33,528 & m \\ \(\alpha_{0}\) & Initial value of angle of attack & 2.745 & deg \\ \(Q_{0}\) & Initial value of pitch angular & 0 & deg/s \\  & rate & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Initial simulation values of hypersonic vehicle. It can be seen that the speed and altitude can track the given speed and altitude stably without interference. It can be seen from the height tracking curve in Figure 3 that the two controllers have reached the preset height at last, but BSLADRC obviously has a better tracking effect than PID. Before 90 s, the PID tracking effect is not as good as the BSLADRC tracking effect and has obvious error, and BSLADRC has smaller overshoot than PID, almost no overshoot and can quickly reach a stable state. From the speed tracking curve in Figure 4, we can see the two controllers have reached the preset velocity at last, but BSLADRC obviously has a better tracking effect than PID. Before 90 s, the PID tracking effect is not as good as the BSLADRC tracking effect and has obvious error, and BSLADRC has smaller overshoot than PID, almost no overshoot and can quickly reach a stable state. Figure 5 shows that the tracking curve of the track angle converges to the steady state value in about 1.2 s after the given speed change and altitude change instructions, which shows that the controller can control and track the track angle quickly and reach the steady state quickly. Figure 6 shows the tracking curve of the angle of attack. The curve shows that it converges to the steady state value in about 2.7 s; that is, the aircraft can stabilize its attitude in about 2.7 s so that it can fly stably; that is, it can quickly reach the steady state of flight. Figure 7 shows that both controllers can finally reach a steady state, but the error of BSLADRC can converge to 0 faster. The fluctuation before 40 s of BSLADRC indicates that the aircraft will be affected by some disturbances inside the aircraft during the ascent stage, so it will lead to fluctuations. After reaching the steady state value, the stable flight of the aircraft can be realized; that is, the designed controller can make the aircraft rise to the preset altitude. However, the time for PID to reach stability is too long, and the error is also very large, so BSLADRC has better performance and stability than PID control. Figure 8 shows the tracking curve of speed error.