## 4 Experimental Results
### Implementation of the Suboptimal Control  
This section describes how signal processing was performed to implement suboptimal control. Figure 3 illustrates the proposed scheme. The following is the proposed implementation method:  
(1) The off-line part consists of considering the general equations rewritten by the Clarke transform and subsequently its discretization. Finally, the recursive least squares method exposed in Section 2.2 is applied to identify the parameters of the discretized nonlinear system. (2) In the online part of the proposed scheme, which uses the dynamic programming approach in the discrete time domain considering a quadratic performance index, a suboptimal finite-horizon control sequence is obtained using the algorithm presented in Section 3.2. Applying the inverse Clarke transform, it is possible to return to a three-phase system, and since the system is balanced, only one phase of it is considered. For the said phase (stator voltage), its effective value (RMS) is calculated, and based on the operating principle of the scalar control system, the conversion to a frequency reference signal (control signal) is performed. This control signal is sent through the myRIO board to the VFD. The system feedback is the rotor speed measured through an incremental encoder and the stator current provided by the VFD. Both signals are connected to the myRIO board (see Figure 3). \begin{table}
\begin{tabular}{c c} \hline MODEL: & VFD007 EL1A \\ \hline INPUT & 1 PH 100–120 V 50/60 Hz \\ \hline OUTPUT & 3 PH 0–240 V 4.2 A 0.75 kW/1 HP \\ \hline FREQUENCY RANGE & 0.1–600 Hz \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of the VFD Delta. Source: Delta Electronics, INC [28]. Figure 3: Proposed suboptimal control sequence. Figure 2: Hardware used for the experimental platform. Figure 4 shows the proposed control scheme. It could be described as follows. The inner PID control of an industrial VFD, usually used to regulate the velocity of the rotor of the motor, is tuned as follows: the proportional part is the unity, and the integral and derivative parts are disabled (parameters 10.02, 10.03, and 10.04 in the VFD, respectively, see Table 3). It allows an external control signal to enter to the VFD from the MyRIO card and the PC with the LabVIEW software. The setpoint of the inner PID of the VFD is set to zero by the VFD menu configuration by setting the available knob to the dispositive. The inner PID comparator of the VFD is configured with positive feedback: The part of the process variable is positive. In the only analog input available in the VFD, the suboptimal nonlinear control signal, given by the MyRIO card, is introduced. A root mean square voltage, which is equivalent to the calculated voltage for each phase, is introduced to the industrial VFD in the input terminal for the PID feedback of the VFD. The encoder signal is read through the MyRIO card and processed in LabVIEW software. To evaluate the efficiency of the proposed controller, the acceleration and deceleration times are adjusted to the minimum value available in the VFD. A parallel configuration of the inner PID was programmed in the VFD. The adjustment of the VFD Delta VFD007 EL11A parameters is shown in Table 3.  
It is important to clarify that the set points for each tested operation zone were obtained via a previous step response experiment, and they are taken as the stable state values of each state variable. The stable state values are calculated for the controllers and depend on  
Figure 4: Integration of the proposed control scheme.