## 3 Results  
### Parameters Setting of Comparison Experiment between HO-ADRC and PID  
In order to make the comparison results more convincing, the various parameters of the HO-ADRC and PID in the simulation platform are adjusted. The regulation of the parameters optimizes the control system stability error. The table of parameter values is shown in Table 1.  
In practice, PID usually does not use differential signals. This is because differential signals are hardly obtained well in the actual system. Often, it will amplify the noise. The discrete step length is a fixed value whose size depends on the sampling frequency of the system.  
The simulation parameters are set according to Table 1. The Bode diagram of the closed loop of PI and HO-ADRC is shown in Figure 7. As can be seen from Figure 7, the two control methods have achieved the best control performance according to the simulation parameters.  
In Figure 7a, the closed-loop curve has two resonant peaks: A and B. Generally speaking, the amplitude of point A should not exceed 3 dB, otherwise the bandwidth of the system (\(\pm 3\) dB) will be too low, leading to the poor control effect of the system. It is also required that the amplitude of point B should not exceed 0 dB, otherwise the noise influence will be increased. The same index requirements are still applicable to points D and E in Figure 7b. When calculating the system bandwidth, we should find the frequency point of the amplitude-frequency characteristic curve at \(-3\) dB, as shown in C and F in Figure 7. In the conventional range, the larger the system bandwidth, the better the system control effect.  
The frequencies A, B, C, D, E, and F are shown in Tables 2 and 3.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable Name** & **Variable Symbol** & **Variable Value** \\ \hline fast factor of the tracking differential & \(r\) & 7000 \\ simulation time interval & \(h\) & 0.001 \\ weight factor of \(z_{1}\) & \(\beta_{01}\) & 1000 \\ weight factor of \(z_{2}\) & \(\beta_{02}\) & 3000 \\ weight factor of \(z_{3}\) & \(\beta_{03}\) & 5000 \\ weight factor of \(z_{4}\) & \(\beta_{04}\) & 10,000 \\ weight factor of \(z_{5}\) & \(\beta_{05}\) & 100,000 \\ weight factor of error signal & \(b_{01}\) & 4 \\ weight factor of stability error integrated signal & \(b_{05}\) & 1000 \\ weight factor of stability error differential signal & \(b_{02}\) & 0.01 \\ nonlinear saturation factor of the stability error signal & \(\alpha_{1}\) & 0.6 \\ nonlinear saturation factor for the stability error & \(\alpha_{2}\) & 1.2 \\ nonlinear saturation factor of the stability error & \(\alpha_{3}\) & \(-0.6\) \\ \multicolumn{2}{c}{integral signal switching threshold} & \(\delta\) & 0.8 \\ weight factor of PID stability error signal & \(k_{p}\) & 3 \\ weight factor of PID stability error integral signal & \(k_{i}\) & 220 \\ weight factor of PID stability error differential signal & \(k_{d}\) & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary table of simulation parameters.  
Figure 7: Bode diagram of the closed-loop system with HO-ADRC and PID. (**a**) Bode diagram of the closed-loop system with HO-ADRC; (**b**) Bode diagram of the closed-loop system with PID.  
As can be seen from Table 2, the amplitude of point A and point D is similar, but the frequency of point A is greater than that of point D. At bandwidth points C and F, it can be seen that the closed-loop bandwidth of the system using HO-ADRC is greater than that using PID. Point E, where the amplitude is \(-0.82\) dB, is closer to \(0\) dB than point B, where the amplitude is \(-1.5\) dB. This means that the parameters of the PID are closer to the \"maximum ability\" than the HO-ADRC. Through the closed-loop frequency characteristic curve, both methods achieve the optimization of their control, and the comparative test has a certain reference value.