## 8 Conclusions  
In this study, to suppress the nonlinear LS vibration that occurs on the basis of yaw crawling, a new type of additional motor-driving design is created. The nonlinear system based on the iterative learning control method of the time-varying continuous system are solved and investigated. Some concluding remarks can be drawn from the results.  
* In the present study, on the one hand, an additional driving torque method is used to suppress the amplitude of the LSJ, which is a new type of additional driving design based on auxiliary motors to achieve nonlinear LSJ suppression. This design scheme is rarely mentioned in the relevant literature. On the other hand, an iterative algorithm based on an open-loop PD is used to achieve jitter suppression by using MATLAB/SIMULINK and its external communication functions;
* Detailed discussions are conducted on the accuracy, speed, effectiveness, and mathematical methods of the iterative algorithm. In order to verify the feasibility of the intelligent iterative algorithm in practical applications in this study, a C-HITL experimental platform based on OPC communication and PLC controller hardware is adopted, and the consistency between the theoretical simulation and experimental results is verified through the HMI connected to the experimental platform;
* The LSJ nonlinear system is converted into an underactuated system, and then it is processed by an iterative algorithm based on open-loop PD to meet the comprehensive requirements of precision, speed, and reliability. In the C-HITL platform based on the  
Figure 12: The implementation and validation steps of the C-HITL platform.  
OPC, the iterative algorithm runs on a PC, while the PD algorithm runs on the built-in PD module of the PLC, which seems to be able to achieve more effective and faster engineering applications, providing new ideas for the engineering application of other intelligent algorithms;
* In the iterative algorithm proposed in the present study, different sampling steps based on the Solver improvement and large-step hardware sampling are used to describe engineering applications, and compared with theoretical simulation results, obtaining reliable and consistent results.
* Once again, it is emphasized that LSJ vibration control is a rarely studied topic, but its harmfulness is becoming increasingly apparent under extreme working conditions. At present, some studies on LSJ control are based on a qualitative control method established on the basis of constant-speed yaw angle control. The LSJ control in this study is a comprehensive control method based on the nonlinear LSJ displacement and nonlinear yaw angle, accompanied by a dynamic lubrication scheme synchronously driven by the auxiliary motor. With the use of a fast-iterative learning algorithm based on solver acceleration, it has more important engineering practical significance and is easy to handle in extreme working conditions.  
[PERSON] conceived the original idea and wrote the manuscript; [PERSON] and [PERSON] jointly supervised the study. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Natural Science Foundation of Shandong Province of China, grant number ZR2022 ME093.  
Not applicable.  
The data presented in this study are available in article.  
The authors declare that there is no conflicts of interests.