## 6 Simulation Results and Discussion  
The designed controller is tested through numerical simulation in MATLAB. The following three control methods are employed to carry out comparative simulation experiments on the loaded quadrotor. The initial state selected during simulation is \( \begin{bmatrix}0&0&0\end{bmatrix} \)rad, \( \begin{bmatrix}0&0&0\end{bmatrix} \)m. The quadrotor model parameters are shown in Table 1.  
**Case Study 1. ** Quadrotor mass \(\mathrm{M}=5\) kg, load mass \(\mathrm{m}=3\) kg; given input signal: \(Z_{\mathrm{d}}=6\) m, \(\theta_{\mathrm{d}}=35\)\({}^{\circ}\), \(\phi_{\mathrm{d}}=45\)\({}^{\circ}\), \(\psi_{\mathrm{d}}=55\)\({}^{\circ}\). The simulation results of BP\(-\)ALADRC are compared with LADRC and DSADRC. Figure 4 is the output curves, and Figure 5 is the output curves and error curves. In the altitude channel, BP\(-\)ALADRC has the fastest tracking speed, BP\(-\)ALADRC and DSADRC can reach stability within 1 s and 1.2 s, respectively, while LADRC needs 2.3 s to achieve stable tracking. In the channels of pitch and row angle, while maintaining fast track speed, BP\(-\)ALADRC has significantly less overshoot than DSADRC and LADRC. In the yaw angle channel, BP\(-\)ALADRC was significantly faster to respond. In summary, the BP\(-\)ALADRC controller has the fastest tracking speed and smallest overshoot compared to DSADRC and LADRC, and can quickly and stably reach system stabilization, which proves the effectiveness of the BP\(-\)ALADRC controller. \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Value** \\ \hline \(k\) & 0.21 \(\mathrm{Ns}^{2}/\mathrm{rad}^{2}\) \\ \(M\) & 5 kg \\ \(I_{x}\) & 0.8 \(\mathrm{kg}\cdot\mathrm{m}^{2}\) \\ \(I_{y}\) & 0.8 \(\mathrm{kg}\cdot\mathrm{m}^{2}\) \\ \(I_{z}\) & 1.5 \(\mathrm{kg}\cdot\mathrm{m}^{2}\) \\ \(g\) & 9.8 \(\mathrm{m}/\mathrm{s}^{2}\) \\ \(l\) & 0.15 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the quadrotor model. Figure 4: System output: (**a**) Altitude and Roll angle output; (**b**) Pitch and Yaw angle output. Figure 5: System output errors: (**a**) Altitude and Roll angle output errors; (**b**) Pitch and Yaw angle output errors. **Case Study 2. ** In this test, we use BP\(-\)LADRC without PD adaptive law as a comparison. To simulate the external interference received by the quadrotor during the working process, we use Gaussian white noise with a variance of 2.5 and a mean value of 0 to represent the external interference. As shown in Figure 6.  
According to Figures 7 and 8, when the quadrotor is disturbed, the tracking curve fluctuation of BP\(-\)ALADRC to the reference input is the smallest; BP\(-\)ALADRC has the fastest tracking speed, while LADRC has the slowest tracking speed; the overshoot of BP\(-\)ALADRC is significantly smaller than that of BP\(-\)LADRC without PD adaptive law; to sum up, the introduction of BP can reduce the overshoot and speed up the system response speed. At the same time, the addition of the PD adaptive law further increases the stability of the system, eliminates the adverse effects of parameter deviations on the system, further reduces the system overshoot. Meanwhile, system response speed is further accelerated in the height channel, and the control effect of the controller is improved. Figure 6: Gaussian noise simulation diagram. Figure 7: System output: (**a**) Altitude and Roll angle output; (**b**) Pitch and Yaw angle output. From Figure 9, when the quadrotor is affected by external disturbances, the parameters \(k_{p}\) and \(k_{d}\) of the PD controller will be adjusted timely on the basis of the system state, which can eliminate the parameter deviation. **Case Study 3. ** The load mass carried by the quadrotor will change in real time; in this test, we make the load mass decrease continuously, and the system input signal remains unchanged. The BP\(-\)ALADRC controller is compared with mass adaptive control law and the BP\(-\)ALADRC controller without mass adaptive control law. Figure 10 is a load mass change diagram.