## 6 Results and Discussion
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Parameter** & **Value** \\ \hline Proportional gain of the slave PI controllers for currents \(I_{q}\) and \(I_{d}\) & \(K_{ps}\) & 1.0 \\ Integral gain of the slave PI controllers for currents \(I_{q}\) and \(I_{d}\) & \(K_{is}\) & 10.0 \\ Proportional gain of the FSLC (Fourier Series Learning Controller) & \(\alpha_{n}\) & 0.037 \\ Learning gain of the FSLC & \(\gamma_{n}\) & 0.03 \\ Number of terms of the FSLC & \(N\) & 4 \\ Proportional gain of the master PI (Proportional Integral) controller & \(K_{pm}\) & 0.0477 \\ Integral gain of the master PI controller & \(K_{im}\) & 2.38 \\ Learning coefficient of the ANN (Artificial Neural Network) & \(\eta\) & 4.9 \\ Number of neurons in the intermediate layer of the ANN & \(N_{n}\) & 3 \\ Gain of the position filter & \(A\) & 5.0 \\ Sample period (s) & \(T_{s}\) & 0.005 \\ Velocity reference (rad/s) & \(\omega_{Ref}\) & 0.3142 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Gains and parameters of the control system. Figure 5 shows the control signals \(I_{qRef}\) of controllers and the measured currents \(I_{q}\) of the closed-loop systems. Figure 5b Shows the behavior of the PI controller. It can be observed that the control signal and the current measured \(I_{q}\) are apparently similar, and there are some small variations. Figure 5c shows the behavior of the ANN controller. In the learning stage, there are large variations of the control signal and also of the measured current. After the learning stage, the variations are reduced, and the control signal is similar to the measured current, which corresponds to the moment of the reduction of the velocity ripple. Figure 5a shows the behavior of the Fourier series learning controller. In the learning stage, the control signal increases to 5.5 A, and it is saturated; however, the measured current only reaches 3.5 A. Later, the signal control decreases to \(-5.5\) A, but the measured current only decrease to \(-1.7\) A. After the learning stage finishes, the measured current \(I_{q}\) tries to follow the control signal \(I_{qRef}\), and the velocity ripple is reduced. Later, when the torque disturbance is applied from 50 s, the currents are incremented up to 0.5 A in order for the change of load to be compensated. Figure 4: Estimation of the angular velocity when it is controlled to 0.3142 rad/s. (**a**) FSLC as the velocity controller; (**b**) PI as the velocity controller; and (**c**) ANN as the velocity controller. Figure 3: Angular positions when the velocity is controlled to 0.3142 rad/s. (**a**) FSLC as the velocity controller; (**b**) PI as the velocity controller; and (**c**) ANN as the velocity controller. Figure 6 shows the errors of the control systems obtained by each controller. In Figure 6b, it is observed that the PI controller cannot compensate the torque ripple, and the error varies to \(-1.2\) rad/s. Figure 6c shows the error obtained when the ANN is the velocity controller. During the first \(18.0\) s, there are variations of the error up to \(-0.4\) rad/s, but after the learning time, the error oscillates around \(0\) rad/s, between \(-0.07\) and \(0.8\) rad/s. In Figure 6a, the error obtained with the FSLC as the velocity controller is shown. During the learning time, the error reaches \(-2.5\) rad/s; however, after this time, the error converges, and it remains near zero. Thus, it is experimentally demonstrated that the FSLC achieves the error convergence of a nonlinear system. When the controllers are under a constant disturbance of torque after \(50\) s, it is shown that this disturbance only affects the ANN as the controller, since from the disturbance, the error varies up to \(-0.2\) rad/s. However, in the case of the FSLC as the controller the error continues around zero, and the variations are maintained between \(-0.05\) and \(0.05\) rad/s. Then, it is experimentally demonstrated that the FSLC can add the integral part in a control system.