## 4 Experimental Results
### Real-Time Experiments Using a Load on the AC Motor  
In this section, load experiments are presented considering a 5.25 kg metallic piece on the motor shaft. Both controllers are used to conduct the experiments. Figure 13a shows the metal piece. The metallic piece coupled to the motor shaft is shown in Figure 13b. Figure 14 shows the angular velocity when a load is coupled to the motor shaft. When the loaded motor shaft is at rest and a set point of 400 RPM is requested, the optimal PI response presents an important overshoot, in contrast to non-linear suboptimal control. Figure 15 shows the error of the angular velocity for both controllers when a load on the motor shaft is considered. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Controller** & **IAE \(\omega\)** & **IAE \(I_{sa}\)** & **IAE \(\Psi_{ra}\)** & **Energy (Wh)** \\ \hline Suboptimal & 13.37 & 8.85 & 7.50 & 2.08 \\ Optimal PI & 15.47 & 30.91 & 46.85 & 2.12 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: IAE of \(\omega\), \(I_{sa}\), and \(\Psi_{ra}\): energy consumption. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Operation Region** & **Controller** & \(M_{p}/U_{s}\) **(\%)** & \(T_{s}\) **(s)** \\ \hline
800 RPM & Suboptimal & \(-/-\) & 1.41 \\  & Optimal PI & \(5.00/-\) & 1.71 \\ \hline
600 RPM & Suboptimal & \(-/0.83\) & 0.6 \\  & Optimal PI & \(-/4.33\) & 1.1 \\ \hline
400 RPM & Suboptimal & \(-/2.00\) & 0.7 \\  & Optimal PI & \(-/6.75\) & 1.3 \\ \hline
200 RPM & Suboptimal & \(-/-\) & 0.5 \\  & Optimal PI & \(-/11.75\) & 1.0 \\ \hline
400 RPM & Suboptimal & \(-/-\) & 0.9 \\  & Optimal PI & \(5.5/-\) & 1.3 \\ \hline
600 RPM & Suboptimal & \(-/-\) & 0.6 \\  & Optimal PI & \(3.3/-\) & 1.2 \\ \hline
800 RPM & Suboptimal & \(-/-\) & 0.5 \\  & Optimal PI & \(3.12/-\) & 1.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Overshoot, undershoot, and settling time for each of the setpoints: 800 RPM, 600 RPM, 400 RPM, and 200 RPM. Figure 16 shows the currents for both controllers when a load is considered in the motor shaft. An important overshoot of current is seen when the motor is in rest and starts the motor shaft spin. Figure 17 shows the estimate of the flux rotor for both controllers when a load is considered. The calculated control signals are shown in Figure 18.  
Figure 14: Angular velocity for both controllers with loads (sampling time: 50 μs). Figure 13: Metal piece coupled to the motor shaft. (**a**) Metallic piece weighing 5.25 kg. (**b**) Metallic piece coupled to the motor shaft. Figure 15: Error for the angular velocity for both controllers with loads (sampling time: 50 μs). In general, the control signal presents higher energy consumption when an optimal PI is used. Table 7 shows the performance comparison of both controllers when a sampling time of 50 \(\upmu\)s is considered. Table 8 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 experiments. Figure 19 shows the angular velocity when a sampling time of 100 \(\upmu\)s and a load of 5.25 kg are considered. Figure 16: Current for both controllers with loads on the motor shafts (sampling time: 50 \(\upmu\)s). Figure 17: Estimated fluxes for both controllers with loads on the motor shafts (sampling time: 50 \(\upmu\)s). Figure 18: Control signals for both controllers with loads on the motor shafts (sampling time: 50 \(\upmu\)s). A clear advantage can be appreciate when the SNDC is used. Figure 20 shows the error of the angular velocity for both controllers. The error is greater when there is a change in the set point. Figure 21 shows the currents for both controllers when a load is considered in the motor shaft.