## 4 Experimental Results
### Real Time Experiments Using Nonlinear Discrete Control and Optimal PI Control
Figure 8 shows the tracking of the stator current \(I_{sa}\) given for the VFD. A clear advantage can be observed when using the suboptimal nonlinear control. In fact, the convergence of all state variables to the set points is better, in the sense that they do not present large oscillations. This implies energy savings, resulting in less deterioration of the rotor windings [36]. Figure 5: Rotor speed \(\omega(k)\). Figure 6: Stator voltage \(V_{sd}(k)\). The estimated flux \(\Psi_{ra}\) of the three-phase system is shown in Figure 9. Similar to the stator current \(I_{sa}\), a significant advantage is shown when the suboptimal control is used compared to the optimal PI control. On the one hand, notice that if an overflux occurs, it may cause saturation of the magnetic core of the stator and rotor. Saturation causes overheating and can lead to motor failure [37]. On the other hand, when the magnetic flux is reduced, it reduces the torque capability and affects the motor's ability to handle the load [36]. Finally, another experiment without a load is conducted. In fact, the stable-state voltage for each setpoint of rotor speed is introduced in the analog input of the VFD via the software LabVIEW and the MyRIO card, which means that the scheme is in am \"open loop\". However, the industrial VFD has an inner V/F control. The results of this comparison with respect to the rotor speed are shown in Figure 10. Clearly, the suboptimal control outweigh  
Figure 8: Stator electric current \(I_{sa}(k)\). Figure 7: Rotor speed error. Figure 9: Estimated rotor flux \(\Psi_{ra}\). Figure 11 shows the stator current for the three-phase system. Clearly, the initial transitory peak of the current is greater than the one produced by the suboptimal control and the optimal PI. Figure 12 shows the flux for the three-phase system. Important variations occur when the \"open loop\" signal is applied, in contrast to the suboptimal control, which is smoother. **Remark 3**. : _The use of modern control along with industrial VFDs for AC motors is not common in the specialized literature (please see the cited references in the Introduction). Due to the good performance of the proposed control scheme, this could be a potential complement to a variable frequency driver (VFD) with inner V/F control, which allows for suboptimal regulation of the all states (currents, fluxes, and angular velocity), which is not possible with the PID available in the industrial VFDs._  
Figure 10: Rotor speed \(\omega(k)\) with external control and with suboptimal control. Table 5 compares the overshoot (\(M_{p}\)), undershoot (\(U_{s}\)), and the settlement time for the setpoints of 800 RPM, 600 RPM, and 400 RPM between both controllers, the suboptimal nonlinear discrete control and optimal PI control, in the time interval [0, 80] corresponding to decreasing set points and (80, 140] corresponding to increasing set points of the programmed trajectory tracking task. According to the results given in Table 5, the suboptimal discrete nonlinear control (SDNC) provides a better performance compared to the optimal PI (OPI). This is because SDNC does not produce overshoots, and it presents a response with lower undershoots than those produced by OPI. Table 6 shows the integral absolute errors (IAE) of the angular velocity of the motor shaft (\(\omega\)), the stator current (\(I_{sa}\)), and the rotor flux (\(\Psi_{ra}\)). The energy consumption is calculated based on the current measurements \(I_{sa}\) and the voltage \(V_{sa}\) during all the experiments and is shown in Figure 5.0  
The experiments described in this section were carried out on a three-phase motor with a free load and a sampling time of 1 ms. In the next section, load experiments are presented. In these new tests, the sampling time is fixed to 50 us and 100 us to guarantee precise sampling of the measurements.