## 6 Conclusions  
This paper presents an effective control method for solving high-dimensional high-frequency vibration-control problems based on reinforcement learning (RL) and a finite-impulse-response (FIR) filter. First, the FIR filter was used to build a simulator for the vibration system with five inputs (the forces of the vibration exciter and the four electromagnetic actuators) and four outputs (the four acceleration signals). Then, a reinforcement-learning algorithm interacted with the simulator to find a near-optimal control policy to meet the specified goals. Finally, the controller was validated by numerical simulations and experiments. The numerical results show that the peak values of the PSD curves at specific frequencies were maximumly reduced by more than 63%. The experimental results show that the controller can effectively reduce various vibrations (the peak values of the PSD curves maximumly reduced by more than 52%) within a frequency range from 50 Hz to 60 Hz. This means that we can provide an effective method to solve high-dimensional high-frequency vibrational-control problems.  
In this work, the controller was designed and validated within a frequency range from 50 Hz to 60 Hz. In fact, the controller can be designed within a wider frequency range, such as a range from 10 Hz to 100 Hz. Generally, a larger neural network is needed for vibration control in a wider frequency range. Moreover, it will be much more difficult to train. Meanwhile, the controller hardware should be improved to guarantee the computation speed, as a larger network costs more computation resources. Our further work focuses on designing and training a complex neural-network architecture for a vibration controller in a wider frequency range. Meanwhile, we also have to update the hardware to achieve a faster computation speed.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Uncontrolled (V2/Hz)**} & \multicolumn{2}{c}{**Controlled (V2/Hz)**} & \multicolumn{2}{c}{**Reduction (\%)**} \\ \hline No. 1 Sensor & \(\updelta\)55 Hz & 0.0248 & 0.2225 & \(-\)979.05\% \\ No. 2 Sensor & \(\updelta\)55 Hz & 0.0687 & 0.1650 & \(-\)140.21\% \\ No. 3 Sensor & \(\updelta\)55 Hz & 0.007 & 0.0419 & \(-\)499.75\% \\ No. 4 Sensor & \(\updelta\)55 Hz & 0.1264 & 0.3692 & \(-\)192.00\% \\ \hline \hline \end{tabular}
\end{table}
Table 10: Peak values of PSD curves of sensor signals.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Uncontrolled (V)** & **Controlled (V)** & **Reduction (\%)** \\ \hline No. 1 Sensor & 0.0732 & 0.2140 & \(-\)192.29\% \\ No. 2 Sensor & 0.1206 & 0.1852 & \(-\)53.56\% \\ No. 3 Sensor & 0.0420 & 0.0948 & \(-\)125.43\% \\ No. 4 Sensor & 0.1624 & 0.2753 & \(-\)69.47\% \\ \hline \hline \end{tabular}
\end{table}
Table 9: RMS values of sensor signals.  
**Author Contributions:** Conceptualization, H.C. and X.F.; methodology, X.F.; software, A.Z.; validation, G.W.; formal analysis, Z.Z.; investigation, G.W.; resources, Z.Z.; data curation, G.W.; writing--original draft preparation, X.F.; writing--review and editing, H.C.; visualization, A.Z.; supervision, H.C.; project administration, H.C.; funding acquisition, H.C. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by National Natural Science Foundation of China (11991034). And The APC was funded by National Natural Science Foundation of China (11991034).  
**Acknowledgments:** The authors are thankful for the support of the Postdoctoral Research Station of Huazhong University of Science and Technology. The research was also supported by the National Natural Science Foundation of China (11991034). This financial support is gratefully acknowledged.  
**Conflicts of Interest:** The authors declare no potential conflict of interest with respect to the research, authorship and/or publication of this article.