## 4 Conclusions  
Throughout this research, we developed a control strategy with the aim of increasing the accuracy of a commercial PEA. After an analysis of the previous investigations from other authors and based on the study that we made, we found that the hysteresis was the main non-linear phenomenon to be counteracted. First, we made an analysis of the properties of a commercial PEA PK4 FYC2 from Thorlabs. According to the manufacturer, the maximum error is 15%, which is a considerable value for applications where high precision is required. As the main reference, we used a triangular wave because it is a complex signal to be tracked due to the high-frequency harmonics and sudden slope changes. Secondly, we proposed a robust sliding controller due to the advantages studied in the related works from the introduction. Thus, we chose to use a QCSMC, which belongs to the HOSMC, so that the chattering is reduced in comparison to classic SMCs. Commonly, sliding controllers have two terms where one of them is achieved through a mathematical model, but instead, a distinctive feature of our controller is the use an ANN. We contrasted the proposed design with a conventional PID in terms of several metrics, such as IAE (that was also used to tune the gains of each framework in experiments), MSE and chattering. In regards to the results, at first, we analysed the performance of the ANN, which provided a suitable RMSE and fitting to the hysteresis. Later, we implemented the QCSMC-ANN and the PID in a dSpace platform for a real-time experiment. The results showed that the QCSMC-ANN generated a lower tracking error, which oscillated around the null value. The PID displayed a slow compensation and even with a disparity that had a variation with a higher amplitude that reached the 0.1 \(\upmu\)m. Nevertheless, both schemes showed suitable control signals in the analysed graphs. Additionally, because we wanted to test the flexibility of both structures against reference changes, we used a sine wave and a triangular reference with variable amplitude. In this case, the first case exhibited a demeanour that was similar in the error and control signal because the reference was softer \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Reference** & \multicolumn{3}{c}{**IAE**} & \multicolumn{3}{c}{**RMSE (\(\upmu\))**} & \multicolumn{3}{c}{**Chatt(u) in 4 s**} \\ \hline  & **QCSMC-ANN** & **PID** & **Diff (\%)** & **QCSMC-ANN** & **PID** & **Diff (\%)** & **QCSMC-ANN** & **PID** & **Diff (\%)** \\ \hline Triangle & 0.1314 & 0.28 & 53.07 & 0.0404 & 0.0756 & 46.5 & 412.75 & 640.97 & 35.6 \\ Sine wave & 0.0518 & 0.28 & 81.5 & 0.0161 & 0.0795 & 79.7 & 108.8 & 600.8 & 81.8 \\ Var. Amp. & 0.2067 & 0.33 & 33.6 & 0.0318 & 0.0519 & 38.5 & 514.88 & 659.4 & 22 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: A comparison of the different metrics. than the previous one. In regards to the variable amplitude signal, the QCSMC-ANN still provided a superior performance, which was verified in graphical and numerical analysis. Finally, we calculated the mentioned metrics, which showed concrete values of the performance from the comparisons made. Thus, in terms of the IAE and RMSE, the QCSMC-ANN had a important distinction favourable for the QC-SMC in the proposed reference signals. The calculated chattering also showed a significant difference, where the QC-SMC carried the leading trend. As future related works, we expect to study assorted options, including the usage of other ANNs configurations with several training algorithms to improve the accuracy. Certainly, this also implies taking into account the computational resources in terms of the training time. The QCSMC can also be improved through the usage of an adaptive algorithm for gain scheduling. Conceptualization, O.B. and C.N. ; methodology, O.B., M.D. and C.N. ; software, C.N. ; validation, C.N. ; formal analysis, O.B., M.D. and C.N. ; investigation, O.B. and C.N. ; resources, O.B. ; writing--original draft preparation, C.N. and M.D. ; writing--review and editing, O.B., C.N.