## 3 Results
### Comparison of HO-ADRC and PID Control Performance with Sinusoidal Speed Signal over Zero  
After comparing the square wave signal for input, the sinusoidal signals for input are also compared in this study. This aim is to compare the ability of the two control methods to suppress friction interference at a speed over zero. We input the sinusoidal speed signal \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Frequency Point Name** & **Frequency (rad/s)** & **Frequency (Hz)** & **Amplitude (dB)** \\ \hline A & \(237.06\) & \(37.75\) & \(2.64\) \\ B & \(900.60\) & \(143.41\) & \(-1.50\) \\ D & \(91.16\) & \(14.52\) & \(2.42\) \\ E & \(967.85\) & \(154.12\) & \(-0.82\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Closed-loop amplitude-frequency characteristic parameters. Figure 8: Comparison of tracking performance between active disturbance rejection and PID for the square wave input signal. (**a**) Contrast detail diagram; (**b**) tracking error of PID and ADRC. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Frequency Point Name** & **Frequency (rad/s)** & **Frequency (Hz)** & **Amplitude (dB)** \\ \hline C & \(552.73\) & \(88.01\) & \(-3.00\) \\ F & \(229.05\) & \(36.47\) & \(-3.00\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Closed-loop bandwidth parameters. with an amplitude of 3 rad/s and a frequency of 1 Hz to compare the control effect between the two methods. The results are shown in Figures 9 and 10. Figure 10: Tracking comparison diagram of HO-ADRC and PID at zero crossing of sinusoidal velocity signal. (**a**) Contrast detail diagram; (**b**) tracking deviation comparison diagram of HO-ADRC and PID by sinusoidal velocity signal input; (**c**) comparison of HO-ADRC and PID in disturbance suppression; (**d**) comparison of HO-ADRC and PID in noise suppression. Figure 9: Friction effect diagram at zero speed crossing. (**a**) The overall comparison diagram; (**b**) Contrast detail diagram. Figure 9 shows the friction curve at a speed over zero to compare the effects of the HO-ADRC and PID. We take Figure 9 as a reference benchmark to compare the tracking influence of the two methods, HO-ADRC and PID, in suppressing the friction-induced tracking influence when the speed passes through zero. The detailed comparison between the two methods can be seen in Figure 10. Figure 10 shows that both methods suppress the influence of the friction when the speed passes through zero. However, the effect of the HO-ADRC is better than that of PID. Figure 10b shows the comparison of tracking errors between the two methods under the sinusoidal speed signal in detail. Figure 10c,d show the comparison of the disturbance suppression and noise suppression capabilities between the HO-ADRC and PID methods. In Figure 9, the friction is obvious when the speed is over zero and the speed is reversed. This, which is called staggy phenomenon, is due to the friction. Staggy phenomenon refers to the temporary stop due to the existence of static friction moment. The rotary shaft will not work normally until the motor torque is greater than the static friction torque. Figure 10 shows the comparison of the HO-ADRC and PID tracking in the input sinusoidal velocity signal. It can be seen that HO-ADRC is better than PID when dealing with friction. As can be seen from Figure 10, both methods can track the given sinusoidal speed signal well, but when the speed exceeds zero, HO-ADRC is less affected by friction, and the impact is half the impact of the PID. Figure 10b shows the deviation comparison of the two methods by the sinusoidal velocity signal input. At a speed over zero, the PID is 0.146 rad/s, compared with 0.077 rad/s of the HO-ADRC. It can be seen that in the simulation platform, the control effect of HO-ADRC is better, and HO-ADRC is better than PID in terms of disturbance suppression ability. We have also compared the disturbance suppression and noise suppression capabilities of the HO-ADRC and PID methods. The results are shown in Figure 10c,d.