## 3 Iterative Learning Control Based on the Open-Loop PD Regulation
### Discussion of the Iterative Learning Algorithm  
Figure 5 illustrates the results of the three displacements in the last iteration process, the driving torques of the lead gear (\(u_{2}\)) and auxiliary drive gear (\(u_{1}\)), and the changes in the maximum absolute values of error \(i(i=1,2,3)\) with the iterative times during the 40 iterations for the three controlled displacements. As mentioned above, the auxiliary driving torque, \(u_{1}=0.2u_{2}-T\Delta\mu\), fluctuates within a reasonable range and is physically realizable.  
From the perspective of maximum errors, during 40 iterations, the errors of the yaw angles were basically kept near 0.02 rad, which reflected the effectiveness of yaw tracking and the stability of the yaw process. The tracking errors of the composite variable, \(x_{3}\), were basically stable at 0.15 rad\(\cdot\)s. This meant that, although the time component in the composite variable increased linearly, the composite variable remained stable as time  
Figure 5: Illustrations: (**a**) results of the three displacements in the last iteration process; (**b**) control input signals (\(u_{1}\) and \(u_{2}\)); (**c**) changes in maximum absolute values of errors with iterative times.  
progressed, which further proved the stability and convergence of the yaw motion, and further demonstrated the robustness of the iterative algorithm. The tracking errors of the LS vibration fluctuated between 0.082 m and 0.084 m after 25 iterations, indicating that the LS vibration was basically stable and no large jump occurred, thus achieving the goal of chatter control.  
In the present study, the details of the iterative algorithm based on the PD controller needed to be elaborated further:  
* The target value was set as a sine function with a certain frequency, so the results of the iterative algorithm were universal. In other words, if the target value was set as a constant value, the convergence control of the controlled displacement could be completely realized through the iterative control function;
* Formula (2) actually represents a nonlinear system, which is integrated into a formal linear system after variable replacement, as shown in Formula (11). The governing system in Equation (11) was actually an under-actuated system. The iterative algorithm based on PD control, whether dealing with nonlinear systems or under-actuated systems, presented unique advantages, which was exactly the reason why iterative the learning control algorithm was used instead of other control algorithms in this design. At the same time, the iterative algorithm combined with the PD parameter adjustment had significant advantages for both robustness and engineering practicability;
* In Figure 2a, the yaw drive auxiliary motor acted on the position of object 2, and the direction of the driving torque was perpendicular to the direction of \(O-O_{1}\), so that the free vibration equation of Formula (2) would not be changed. As for the setting of control input signal \(u_{1}\) as 0.2 times the value of the input signal, \(u_{2}\), this occurred because an input signal that was too large, \(u_{1}\), would have a reverse reaction, and an input signal that was too small, \(u_{1}\), would have also weakened the control effect;
* Formula (13) was written as follows: \(u_{k+1}(t)=u_{k}(t)+\left(\Gamma\frac{\mathrm{d}}{\mathrm{d}t}+L\right)e_{k+1 }(t)\), that is, a closed-loop PD control with higher accuracy. We used an open-loop PD instead of a closed-loop PD here to speed up the controller hardware as much as possible on the premise of ensuring the control effect. This was also the reason why we adopted the PD control instead of a conventional PID control, which could greatly improve the running speed and considerably shorten the scanning cycle of the PLC system mentioned later on in the study.