# Inductance Estimation Based on Wavelet-GMDH for Sensorless Control of PMSM
## 5 Simulation Results
### Simulation Results  
Figure 9 presents the experimental results of the \(q\)-axis inductance, speed error, and theta error caused by the load torque compensation current, \(i_{\delta}\), during constant speed operation at 1000 rpm in a motor drive system. From the results in Figure 9b, it can be observed that the inductance variation during magnetic saturation due to periodic loads showed significant fluctuations of approximately \(\pm\)50% compared to the normal condition. In addition, as seen in Figure 9c,d, in the condition of magnetic saturation, the speed error ripple reached a maximum of 120 rpm to \(-\)150 rpm, and the angle error was over 25 degrees. Based on the experimental results, the simulation modeling of sensorless motor control was conducted to correlate with actual signals such as the magnetic saturation magnitude and speed ripple according to the load torque compensation current. Figure 8: Simulation model for magnetic saturation compensation of sensorless control system. Figure 10 shows the simulation results of the torque compensation current, inductance, and speed performed by simulating the equivalent load conditions as the experimental results. When the \(\delta\)-axis current is controlled with a peak-to-peak amplitude of 30 A in Figure 10a, the actual inductance fluctuates, as depicted in Figure 10b. Conducting control without compensating for the inductance variations due to magnetic saturation leads to significant ripple in the estimated speed, as shown in Figure 10c. Figure 11 represents the comparison of the estimated speed between no compensation, GMDH based compensation, and wavelet-GMDH based compensation in the time domain when operating at a constant speed of 1000 rpm under conditions of magnetic saturation. It shows the results obtained when basic sensorless control is performed without compensation, as well as when magnetic saturation compensation control is applied based on the GMDH algorithm and wavelet-GMDH algorithm, respectively. The estimated speed error displays peak-to-peak ripples of over \(\pm\)100 rpm when compensation control is not applied. On the other hand, when GMDH based compensation is applied, the ripple is reduced by half, and when wavelet-GMDH based compensation is applied, it is reduced to less than \(\pm\)30 rpm. Figure 9: Experimental results when the motor operated at 1000 rpm with a periodic load torque for (**a**) \(\delta\)-axis current, (**b**) \(q\)-axis inductance, (**c**) speed error, and (**d**) theta error. Furthermore, in the frequency domain comparison results, as depicted in Figure 12, prior to the application of compensation control, the amplitude of the first harmonic component, which has the greatest influence on the speed ripple, was significantly large at about 80 rpm. In the case of GMDH, the amplitude of the first harmonic of the speed reduced to 1/7 th compared to the uncompensated condition. On the other hand, after applying the proposed compensation control, it was observed that the amplitude decreased to less than 10 rpm, representing a significant improvement over GMDH. Particularly, as the frequency increases, the amplitudes of the second and third harmonic frequency components are nearly eliminated. Figure 11: Comparison of estimated speed between no compensation, GMDH based compensation, and wavelet-GMDH based compensation in time domain. Figure 10: Simulation results when the motor operated at 1000 rpm with a periodic load torque for (**a**) \(\delta\)-axis current, (**b**) \(q\)-axis inductance, and (**c**) estimated speed. the training accuracy, it was confirmed that high-precision control can be achieved even in situations with high noise and severe load conditions. The proposed methodology was evaluated through offline testing using simulated data obtained by replicating the measured load conditions.