## 4 Experimental Results
### Real Time Experiments Using Nonlinear Discrete Control and Optimal PI Control  
Three experiments were conducted using the inner PID of the industrial VFD configured with the proportional gain as one and the integral and derivative parts set to zero:  
(1) The suboptimal nonlinear discrete control signal plus offset was sent to the VFD using an analog input and the software LabVIEW (without and with a load). (2) An optimal PI control was implemented using the virtual instrument PID of LabVIEW, and its control signal was sent to the VFD via an analog input without and with a load. (3) A constant signal was introduced to the industrial VFD. This constant represents the RMS voltage in a stable state (the offset of the first experiment described here) to maintain the velocity of the motor shaft at a specific setpoint. First, all experiments were carried out without a load on the shaft, and we selected four different setpoints: 800 RPM, 600 RPM, 400 RPM, and 300 RPM. These setpoint changes were made to test the performance of both controllers in different operating zones and to observe possible overshoots and undershoots in each change and the settling time of the process variable. Table 4 shows the setpoint values used in the experiments and the stable-state stator voltage for each setpoint of the rotor speed. The horizon of the performance index given by (7) was fixed at \(N=140,000\), which corresponds to a time of 140 s, with a sampling time of 0.001 s. The following numerical values were chosen for the matrices \(Q\), \(H\), and \(R\):  
\[H=\left[ \begin{array}{cccc}10&0&0&3&1\\ 0&10&0&1&1\\ 0&0&10&0&0\\ 3&1&0&10&0\\ 1&1&0&0&10,\\ \end{array} \right],\]  
\[Q=\left[ \begin{array}{cccc}10&0&0&1&1\\ 0&10&0&1&1\\ 0&0&10&0&0\\ 1&1&0&10&0\\ 1&1&0&0&10\end{array} \right],\]  
and  
\[R=\left[ \begin{array}{cc}0.009&0\\ 0&0.009\end{array} \right]. \tag{11}\] \begin{table}
\begin{tabular}{l c c c} \hline \hline
**SP (Rotor Speed)** & **SP Stator Current (\(I_{sa}\)** & **SP Flux (\(Y_{ra}\)** & **Stable State Stator** \\ Three-Phase System)** & **Three-Phase System)** & **Voltage (\(V_{sa}\)** \\ \hline
800 RPM & 0.78 A & 0.013 Wb & 97.5 VAC\({}_{r}ms\) \\ \hline
600 RPM & 0.81 A & 0.019 Wb & 73.2 VAC\({}_{r}ms\) \\ \hline
400 RPM & 0.82 A & 0.028 Wb & 49 VAC\({}_{r}ms\) \\ \hline
200 RPM & 0.85 A & 0.059 Wb & 24 VAC\({}_{r}ms\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Set points (SP) and stable state values for the controllers without a load. A profile of the rotor speed was previously established and is shown in Figure 5. To compare the performance of the controllers given by (8), an optimal PI was implemented (see ref. [35]) considering a first-order plant for each set point of the rotor speed. Based on this strategy, the penalization matrices \(Q_{He}\) and \(R_{He}\) were chosen according to the desired performance of the system response (for some maximum overshoot and settling time). The gains obtained were introduced into the PID block function of the LabVIEW software. These gains were \(K_{p}=0.1\) and \(K_{i}=0.08\). Figure 5 shows the response of the rotor speed when both schemes (suboptimal nonlinear control and optimal PI) were used. A smooth response can be seen when the suboptimal nonlinear control was used: no overshoot was shown, and both controllers displayed an undershoot when the rotor speed setpoint was lower in the time instants of 20, 40, and 60 s. The undershoot can be explained, because the motor does not have a break system. However, the suboptimal control presents less undershoot compared to the optimal PI. Figure 6 shows the control signal \(V_{sa}\) calculated using the LabVIEW software and sent to the VFD. A smoother control signal can be seen when the suboptimal nonlinear control is applied; this implies a behavior with fewer oscillations in the rotor speed (see Figure 5). Figure 7 depicts the error signal of the rotor speed. A light advantage of the suboptimal nonlinear control can be seen; this advantage is greater with the changes in the setpoint.