## 6 Conclusions  
An evolutionary computation-based FOFPID controller is developed in this study to control a two-axis gantry platform. The fractional-order controller benefits from the flexibility of the parameters, and consequently outperforms conventional PID variants. Incorporating the fuzzy logic control structure helps increase design flexibility in the control law. Several evolutionary computation algorithms are tested and compared in simulation and experiment: PSO, IWO, GWO, and BBO. These are used to optimize the FOFPID control parameters while minimizing the tracking error index as measured by the ITAE for four contours, circle, heart, star, and bowtie. The optimization processes are implemented in MATLAB/Simulink simulations and the acquired parameters are then examined in a series of experiments using a practical gantry platform.  
The simulation and experimental results verify that the control design could cope with the synchronous control issues and mitigate contour tracking errors. The GWO algorithm outperforms other optimization methods using ATE 11.38172 \(\upmu\)m, TESD 16.21718 \(\upmu\)m, and ITAE 990.97163 when measured using the average of all contours. Compared to PSO, BBO, and IWO, the GWO algorithm showed a performance advantage of 1.6%, 1.9%, and 2.4% in ITAE, respectively. The demonstrated control scheme effectively compensates for synchronous tracking errors, which subsequently shows no significant peaks in turning, and non-smooth positions of the contours. The contribution of this research is to integrate the evolutionary computation algorithms with the FOFPID, optimize the controller parameters, and compare their performance via simulations and experiments. The control scheme is implemented on a PMLSM-actuated gantry platform to tackle the synchronous error and achieve superior tracking results.  
Conceptualization, W.-L.M.; methodology, W.-L.M. and C.-Y.K.; software, C.-Y.K.; validation, W.-L.M. and C.-Y.K. and S.-H.C.; formal analysis, C.-Y.K.; investigation, C.-Y.K.; resources, W.-L.M.; data curation, C.-Y.K.; writing--original draft preparation, S.-H.C. and C.-Y.K.; writing--review and editing, S.-H.C.; visualization, C.-Y.K.; supervision, W.-L.M. and S.-H.C.; project administration, W.-L.M.; funding acquisition, W.-L.M. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Ministry of Science and Technology, grant number 111-2221-E-224-057-.  
No new data were created or analyzed in this study. Data sharing is not applicable to this article.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **ATE (\(\upmu\)m)** & **TESD (\(\upmu\)m)** & **ITAE** \\ \hline \multicolumn{4}{c}{Bow contour} \\ \hline PSO & 16.95719 & 28.80572 & 1018.65117 \\ IWO & 23.79299 & 42.70955 & 1018.99118 \\ BBO & 23.44690 & 39.53237 & 1014.40448 \\ GWO & 14.36346 & 24.53757 & 976.54349 \\ \hline \multicolumn{4}{c}{Average} \\ \hline PSO & 16.38123 & 23.06968 & 1007.07463 \\ IWO & 23.54277 & 37.092195 & 1015.26410 \\ BBO & 19.25410 & 26.46550 & 1010.29870 \\ GWO & 11.38172 & 16.21718 & 990.97163 \\ \hline \hline \end{tabular}
\end{table}
Table 6: _Cont._