## 3 Simulation and Experimental Analysis
### Simulation Results and Analysis  
Table 1 shows the parameters of the PMSM. According to the flow chart of the FOC control of PMSM, a simulation model is built in Matlab/Simulink for experimental verification. The backstepping controllers replace the traditional PI controllers and add an integral term to them. The ELESO obtains back EMF. In this paper, the control scheme combining the backstepping controller and the ESO is regarded as the old scheme, and the control scheme combining the integral backstepping controller and the ELESO is regarded as the new scheme. The new scheme is compared with the old scheme to verify the effectiveness of the new control scheme. According to the actual situation of the motor parameters, the characteristics of current tracking performance and anti-interference, the bandwidth of the LESO1 is selected as 500 Hz, and the bandwidth of the LESO2 is selected as 3000 Hz in the ELESO. In the backstepping controller, \(K_{1}=4000,K_{\omega}=200,K_{2}=13,000,K_{q}=350,K_{3}=8000,k_{d}=830\). Set the step size as \(1\times 10^{-5}\) s. The motor accelerates from the static state to 500 rad/s. The rotor speed is raised to 1000 r/min at 0.5 s, and 6 N\(\cdot\)m loads are applied to the rotor. Then, the motor runs at the rated speed. Figure 4 shows the estimated back EMF under the old scheme. Due to the existence of harmonics, the estimated value of back EMF is not ideal. The influence of high-order harmonics on estimation accuracy is especially obvious at the low speed. According to the definition of the PLL and Equation (13), the back EMF information is finally converted into the estimated position information of the rotor. Therefore, this paper mainly shows the advantages of the new scheme from the position and speed of information. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Stator resistance/\(\Omega\) & 1.5 \\ Rated torque/N\(\cdot\)m & 6 \\ d-axis inductance/mH & 2.48 \\ q-axis inductance/mH & 2.95 \\ Rated speed/rpm & 1000 \\ Moment of inertia/kg\(\cdot\)m\({}^{2}\) & 0.0014 \\ Flux linkage/wb & 0.07 \\ Viscous friction/B & \(7.2\times 10^{-4}\) \\ Pole pairs & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: PMSM parameters. Figure 4: PMSM estimated back EMF of the control scheme of combining the backstepping controller with the ESO. Figure 5 shows the PMSM three-phase current waveforms of the two control schemes under no-load startup. Comparing the two waveforms, it can be concluded that the scheme based on the integral backstepping controller and the ELESO has a higher sinusoidal current waveform because it filters out the higher harmonics. Figure 6 shows the waveforms comparison between the estimated rotor position and the actual value under the two schemes. Combined with the rotor position error waveforms shown in Figure 7, it is obvious that the traditional scheme always has an estimation error of about 0.5 rad under no-load startup. The improved scheme will have fewer errors only at the moment of state change. With the stable operation of the motor, the position estimation error under the improved scheme is always stable at about 0 rad. Figure 5: PMSM stator three-phase current waveforms under no-load startup of the (**a**) Control scheme of combining the backstepping controller with the ESO and (**b**) Control Scheme of combining the integral backstepping controller with the ELESO. Figure 6: PMSM rotor position tracking under no-load startup of the (**a**) Control scheme of combining the backstepping controller with the ESO and (**b**) Control Scheme of combining the integral backstepping controller with the ELESO. Figure 8 shows the d and q-axis currents of the two controllers. Due to the existence of higher harmonics, the waveforms in Figure 8a are not smooth. The high-order harmonics are filtered out in Figure 8b, so the d and q-axis currents are more stable. In addition, when the state changes in 0 s and 0.5 s, the jump amplitude of the current in Figure 8b is smaller.