## 6 Results and Discussion  
The reference velocity for controllers and the master PI gains tested in the experimentation, were the lowest velocity and the same gains studied by [32]. In Table 1 are shown the gains and parameters of the control system. Note that the two slave PI controllers for currents \(I_{q}\) and \(I_{d}\) have the same proportional and integral gains. The learning coefficient \(\eta\) and the number of neurons in the intermediate layer \(N_{n}\) for the ANN were chosen by observing the dynamics performance of the closed loop system. In the same way, the number of terms \(N\) and the gains \(\alpha_{n}\) and \(\gamma_{n}\) of the FSLC were chosen caring the stability conditions (40) and (41) and observing the dynamics performance of the control system. Note that, if \(\gamma_{n}\) is bigger than \(\alpha_{n}\), the stability conditions are not satisfied, and the stability cannot be assured; which is an example of the wrong selection of gains. First, the angular position \(\theta\) of the PMSM is shown in order to observe the different performances of the three controllers. A constant velocity must produce a straight line of the angular position from zero to \(2\pi\). Figure 3 shows the behavior of controllers when the velocity of PMSM is controlled. The PI controller cannot reduce the torque ripple, as can be observed in Figure 3b, where a distortion of angular position produced by the torque ripple is shown. The ANN reduces the torque ripple after a learning time of 18.0 s approximately, as is shown in Figure 3c. However, the Fourier series learning controller reduce the torque ripple faster than the ANN in 2.7 s approximately, as is shown in Figure 3a. Additionally, a change of load is applied after 50 s. A mass of 1.2 kg is loaded by the PMSM. According to the rotor size of the motor, the torque generated by this mass is 0.0824 Nm. It is observed that the three controllers can compensate the constant disturbance; however, the torque ripple continues affecting the PI as the velocity controller. Figure 4 shows the estimation of the velocities \(\theta\) of the closed-loop systems for the three controllers. In Figure 4b is shown the estimation of velocity of the closed-loop system with a PI as the velocity controller. It can be observed that the torque ripple affects the velocity, and the PI cannot compensate it. In Figure 4c is shown the estimation of velocity of the closed-loop system with the ANN as the velocity controller. It can be observed that the speed varies around the desired velocity while the ANN learns, but only after 18.0 s, approximately, the torque ripple is reduced. In Figure 4a is shown the estimation of velocity of the closed-loop system with the FSLC as the velocity controller. In the first 2.7 s, during the learning process, a peak of velocity occurs, which is compensated in 0.16 s, approximately, and later, the velocity ripple is reduced. When the disturbance is applied from 50 s, the three controllers can compensate it, in 1.14 s, approximately. After the disturbance, the FSLC and the PI controllers are not affected. However, in Figure 4c, it is observed that the ANN is affected before 50 s; namely, the variations of the velocity are bigger than the variations before the disturbance.