## 4 Description of Search Algorithms
### Overview of Moth-Flame Optimization Algorithm
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Comparison** & **Difference in** & **Standard** & **DRS/SE** & **Critical Q Value** & **Conclusions** \\ \hline WOA VS CSA & 2171 & 217.22684 & 9.99416094 & 3.658 & Reject the null hypothesis \(H_{0}\) \\ WOA VS MOA & 1206 & 217.22684 & 5.55180014 & 3.658 & Reject the null hypothesis \(H_{0}\) \\ WOA VS DA & 2810 & 217.22684 & 12.9357864 & 3.658 & Reject the null hypothesis \(H_{0}\) \\ WOA VS FPA & 702 & 217.22684 & 3.23164486 & 3.658 & Retain de null hypothesis \(H_{0}\) \\ WOA VS MOD-FPA & 2486 & 217.22684 & 11.442485 & 3.658 & Reject the null hypothesis \(H_{0}\) \\ CSA VS MOD-FPA & \(-\)965 & 217.22684 & 4.4423068 & 3.658 & Retain de null hypothesis \(H_{0}\) \\ CSA VS DA & 639 & 217.22684 & 2.94162545 & 3.658 & Retain de null hypothesis \(H_{0}\) \\ CSA VS FFA & \(-\)1469 & 217.22684 & \(-\)6.7625161 & 3.658 & Retain de null hypothesis \(H_{0}\) \\ CSA VS MOD-FPA & 315 & 217.22684 & 1.45009705 & 3.658 & Retain de null hypothesis \(H_{0}\) \\ MOA VS DA & 1604 & 217.22684 & 7.3898625 & 3.658 & Reject the null hypothesis \(H_{0}\) \\ MOA VS FFA & \(-\)504 & 217.22684 & \(-\)3.2301553 & 3.658 & Retain de null hypothesis \(H_{0}\) \\ MOA VS MOD-FPA & 1280 & 217.22684 & 5.89245786 & 3.658 & Reject the null hypothesis \(H_{0}\) \\ DA VS FFA & \(-\)2108 & 217.22684 & \(-\)9.701415 & 3.658 & Retain de null hypothesis \(H_{0}\) \\ DA VS MOD-FPA & \(-\)324 & 217.22684 & \(-\)1.4915284 & 3.658 & Retain de null hypothesis \(H_{0}\) \\ FFA VS MOD-FPA & 1784 & 217.22684 & 8.21261314 & 3.658 & Reject the null hypothesis \(H_{0}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 17: Multiple comparisons through Nemenyi Test. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Comparison** & **R+** & **R\(-\)** & **\(p\)-Value** & **Conclusions** \\ \hline MOD-FPA VS WOA & 0 & 325 & 1.2290 \(\times\) 10\({}^{-5}\) & Reject the null hypothesis \(H_{0}\) \\ MOD-FPA VS CSA & 106 & 219 & 1.2900 \(\times\) 10\({}^{-1}\) & Retain de null hypothesis \(H_{0}\) \\ MOD-FPA VS MOA & 2 & 323 & 1.5705 \(\times\) 10\({}^{-5}\) & Reject the null hypothesis \(H_{0}\) \\ MOD-FPA VS DA & 240 & 85 & 3.7000 \(\times\) 10\({}^{-2}\) & Reject the null hypothesis \(H_{0}\) \\ MOD-FPA VS FFA & 0 & 325 & 1.2300 \(\times\) 10\({}^{-5}\) & Reject the null hypothesis \(H_{0}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Wilcoxon signed rank test results with a level of significance \(\propto\) = 0.05.

a statistical analysis could be performed on data. The performance of the optimized PI controller was evaluated for trajectory tracking control in terms of integral squared error (JISE), integral absolute error (JIAE), integral time absolute error (JITAE), and integral time squared error (JITSE). The effectiveness and performance of these algorithms were tested under nominal parameters with constant load torque and external load disturbances. The results of these algorithms were statistically analyzed in order to get the best algorithm. The analysis of the statistical results shows that MOD-FPA is better than CSA, WOA, MOA, FPA, DA in terms of the fitness value, JISE, JIAE, JITAE, and JITSE metrics. It should be mentioned that the six algorithms present with excellent performance, so there would be no preference or problem of using each of them indistinctly in the process of tuning the PI controllers. Conceptualization, O.A.-M., H.M.-P. and R.T.-O. ; performed the experiments O.A.-M. and R.T.-O. ; analyzed the data, H.M.-P.; Writing--Original draft preparation, O.A.-M. and H.M.-P.; Writing--Review and Editing, O.A.-M., H.M.-P. and R.T.-O. ; Funding acquisition, O.A.-M. and H.M.-P. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The authors thanks the support from the project CEMIE-Redes CONACYT-SENER Mexico, B-S-50730. [PERSON] thanks the financial support Program for Research of UPAEP. The authors declare that there is no conflict of interest. ## References

* [PERSON] et al.