## 3 Iterative Learning Control Based on the Open-Loop PD Regulation
### Robustness and Practicability of the PD-Based Iterative Learning Algorithm  
As mentioned above, the iterative algorithm combined with the PD parameter adjustment had significant advantages for both robustness and engineering practicability. In the present study, the dynamic adjustment of the PD parameters could be realized through the built-in PID tuning control (BPTC) panel in the STEP-7 MicroWin Software of S7-200 PLC. Because many cluster PD parameters can meet the aforementioned algorithm conditions, the BPTC panel can dynamically adjust and test the multi-cluster parameters. Of course, different PD parameters have different control effects, which is also a manifestation of the robustness of the algorithm. To test the robustness, another set of PD parameters with \(K_{\mathrm{P}}=1\), \(K_{\mathrm{d}}=3.5\), were used in the same controlled object as the processing case, and the results are shown in Figures 6 and 7.  
The controlled displacements three are illustrated by the simulation results for 100 iterations by using the second set of PD parameters with \(K_{\mathrm{P}}=1\), \(K_{\mathrm{d}}=3.5\) in Figure 6.  
Compared with the results in Figures 4 and 5, in general, the results in Figures 6 and 7 maintain a similar control effect. For position variable \(x_{1}\), although the latter is somewhat larger than the former in the controlled amplitudes, it still remains within the effective control range, and the fluctuation is more stable, which is another acceptable controlled effect.  
As for the error margin of the PD-based iterative learning control for the LSJ vibration, it varied with the structural parameters of the system structure in Figure 1 and the parameters of the PD controller, but we could verify the rationality of error fluctuations using the limit range and standard deviation of errors during the iteration process. Table 2 shows the minimum, maximum, and standard deviation of the three types of errors in the two cases. The standard deviation of the first type of error in the first case is 0.02013 m, which is within the standard deviation range [0, 5%] of motion control errors for conventional yaw systems, reflecting a good LSJ control performance. The standard deviation values of the other five errors values are all very small, further verifying the effectiveness of the two sets of PD parameters and the robustness of the proposed control algorithms.  
Figure 6: The three controlled displacements illustrated by the simulation results for 100 iterations by using the second set of PD parameters with \(K_{\mathrm{P}}=1\), \(K_{\mathrm{d}}=3.5\): (a) LSJ displacement \(x_{1}=s\); (b) yaw angle \(x_{2}=\theta\); (c) another compound variable with no physical meaning, \(x_{3}\).  
As for the engineering practicability, because the PID parameters have a certain physical meaning of engineering regulation, the realization of the PID control process can be realized through many ready-made commercial controller hardware items, such as the built-in PID controllers of the aforementioned PLCs.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Cases** & **Errors** & **Minimum** & **Maximum** & **Standard Deviation** \\ \hline \(K_{\text{P}}=20\), & Error 1 (m) & 0.01644 & 0.08312 & 0.02013 \\ \(K_{\text{d}}=0.95\) & Error 2 (rad) & 0.02013 & 0.02355 & 0.00081 \\  & Error 3 (rad-s) & 0.01518 & 0.01521 & \(1.405\times 10^{-16}\) \\ \hline \(K_{\text{P}}=1\), & Error 1 (m) & 0.01292 & 0.01894 & 0.001579 \\ \(K_{\text{d}}=3.5\) & Error 2 (rad) & 0.02103 & 0.02244 & 0.000406 \\  & Error 3 (rad-s) & 0.01478 & 0.01491 & \(4.18\times 10^{-16}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: The minimum, maximum, and standard deviation of the three types of errors.  
Figure 7: Illustrations created by using the second set of PD parameters with \(K_{\text{P}}=1\), \(K_{\text{d}}=3.5\): (**a**) results of three displacements in the last iteration process; (**b**) control input signals (\(u_{1}\) and \(u_{2}\)); (**c**) changes in the maximum absolute values of errors with iterative times.