## 4 Results and Discussions
### Stator Current and Voltage Comparison  
The three-phase stator currents of the PMSM are plotted in Figure 4, the current spike in the start shows the inherent nature of the motor's inrush current. However, the current stabilizes in less than 1 millisecond. A second spike occurs when the speed changes from 300 rad/s to 600 rad/s. The second spike is less aggressive because high-speed performance is more accurate compared to starting/low speed performance. Ripples in stator current waveforms are not so deviating due to the accurate prediction behavior of the UKF. Similarly, the three-phase voltages of the PMSM are plotted in Figure 5. Similar behavior is observed in three-phase voltages as observed in three-phase stator currents. The switching states signal of the inverter's IGBTs are generated by the MPC, as shown in Figure 5. When the PWM signals are concentrated, it represents very low duty cycle that is adjusted based on harmonics and disturbances involved in the system. When process noise or measurement noise is low, the duty cycle considerably increases. The state space vectors controlled by the MPC generate the currents and voltages shown in Figures 4 and 5, respectively. Park's transformation is used to convert three-phase _abc_ to _dq_ axis coordinates, and the resultant plots of currents and voltages are shown in Figures 6 and 7, respectively. The back EMF generated by the PMSM is sinusoidal; therefore, after conversion from three-phase _abc_ to _dq_, switching states of the PWM are converted to sinusoidal waveforms. Moreover, the currents after transformation remain the same. Current peaks are the same in _dq_ coordinates as they are in three-phase coordinates. The MPC-UKF shows promising results because of less ripples in currents and voltages. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Symbol** & **Description** & **Value** \\ \hline \(\phi_{f}\) & amplitude of flux linkages & \(7.8\times 10^{-2}\) Wb \(-t\) \\ \(B_{\omega}\) & viscous friction coefficient & \(3.023\times 10^{-3}\) Nms/rad \\ \(B_{\omega}\) & col2 text & col3 text \\ \(L_{d}\) & direct axis inductance & \(1.68\times 10^{-2}\) H \\ \(L_{q}\) & quadrature axis inductance & \(3.48\times 10^{-2}\) H \\ \(\rho\) & number of poles & 4 \\ \(R_{s}\) & resistance of the stator & \(5\ \Omega\) \\ \(T_{s}\) & sampling time & \(2\times 10^{-4}\) s \\ \(J\) & moment of inertia & \(2.3\times 10^{-5}\) kg cm\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters Selected for PMSM System. Figure 4: Three-phase stator currents for MPC-UKF. Figure 5: Three-phase stator voltages for MPC-UKF. Figure 6: \(dq\)-reference stator currents for MPC-UKF. Stator voltages are sinusoidal because the PMSM has sinusoidal back EMF. Stator voltages are stable and do not have peak values because VSI converts from current switching states to voltage switching states to be measured and used for the estimation of the states (speed/position) directly using UKF algorithm. In Figure 1, the error between the predicted speed from UKF observer and reference speed acts as an input for the PI controller, and the PI controller generates a proportional current for the MPC so that the states can be generated according to the difference between the reference speed and the estimated speed. It is represented in Figure 8 that the reference current generated by the PI controller is very smooth except at the starting and where the input reference step changes. In Figure 8, the peaks are generated to counter the effect of inrush current behavior. Similarly, Figure 9 shows the measured current after \(dq\) transformation, and this current is compared with the reference current for the PI controller. The difference of both currents determines the frequency of switching states accordingly. The proposed hybrid MPC-UKF scheme is more robust for parametric variation and noise. However, an overshoot is observed due to aggressive achievement of steady-state.