## 3 Simulation and Experimental Analysis
### Simulation Results and Analysis
Figure 9 shows the rotor speed waveforms of the two controllers. The standard of rotor stability is that the rotor speed reaches the set value and can be stabilized around the set value. In Figure 9a, it takes about 0.07 s for the rotor speed to reach the set value. In Figure 9b, the waveform converges faster. The new scheme eliminates the influence of internal disturbances such as harmonic components and parameter changes. The speed waveform is smoother. The new scheme is more suitable for places with high requirements for motor speed accuracy. According to the above waveforms, the control scheme of combining the traditional backstepping controller with the ESO does not eliminate the influence of the high-order harmonics and internal parameter changes of the motor. Therefore, the problem of jitter and poor sinusoidal performance will occur in this control scheme, especially at low speeds. The system always maintains an estimation error of about 0.5 rad, which affects the stable operation of the motor. Compared with the traditional scheme, the control scheme based on the integral backstepping controller and the ELESO has better tracking performance at a low speed. In the second stage of acceleration, the traditional control mode requires  
Figure 8: PMSM d and q-axis currents 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 7: PMSM rotor position tracking error 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. 0.06 s of reaction time, and the improved scheme only needs 0.01 s to reach the rated speed, which improves the reaction speed of the system. The working principle of PMSM is that the magnetic field acts on the current force to make the motor rotate. When the motor is unloaded, the electromagnetic force is far greater than the force required for rotor rotation. When the motor is loaded, the resistance of the load on the rotor requires more force to rotate. Therefore, we need to conduct the motor on load starting test to verify whether the new scheme meets the expectations. Set the motor speed to the rated speed of 1000 r/min and the load to 6 N\(\cdot\)m. The two methods are simulated, and the experimental results are analyzed. Figure 10 shows the waveforms of the PMSM stator three-phase current under load startup test. Both schemes can obtain relatively stable three-phase waveforms under load startup, but the sinusoidal property of Figure (a)a is worse than that of Figure (b)b due to the existence of harmonics. In Figure (b)b, the LESO2 timely estimated the harmonic components, and the external interference of the ELESO output no longer contains high-order harmonics, so the waveform is more sinusoidal. Figure 11 shows the rotor position estimation waveforms under load startup. As shown in Figure 11, both control schemes can finally achieve the goal of real-time tracking. Figure 10: PMSM stator three-phase current waveforms under load startup of the (**a**) Control scheme of the combining backstepping controller with the ESO and (**b**) Control Scheme of combining the integral backstepping controller with the ELESO. Figure 9: PMSM rotor speed 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. According to the error waveforms shown in Figure 12, it can be obtained that the new scheme can converge faster. Compared with Figure 12a, the integral backstepping controller eliminates the steady-state error and further improves the system performance. Figure 13 shows the d and q-axis currents of the two schemes under load startup. The higher harmonics are filtered out in Figure 13b, and the waveforms are smoother than those in Figure 13a.