## 4 Simulation Verification and Result Analysis
### Performance Testing under Multiple Operating Conditions
#### 4.2.1 Performance Verification at Rated Speed  
As shown in Table 4, all the comparative experiments are conducted on the same control parameters, and the performance can be fairly compared with different control algorithms. Meanwhile, all the simulation results are obtained under the same running operations, which refers to the load that changes from no-load to 2 N\(\cdot\)m at 0.4 s.  
When the speed is set at rated speed (1000 r/min), the speed response and the system estimated speed error using the proposed VPDPI regulator are shown in Figure 12a,b, respectively. Figure 13a,b show the speed response and the estimated speed error using the traditional PI regulator, respectively. Figures 14a and 15a record PMSM rotor position Angle information under VPDPI regulator and PI regulator, respectively. The Angle tracking error of the sensorless system under the above two regulators are shown in Figures 14b and 15b, respectively. Moreover, the output torque and three-phase current state of PMSM are shown in Figures 16 and 17, respectively.  
Figure 14: The estimation of rotor position Angle using VPDPI. (**a**) Observed and actual rotor position Angle. (**b**) Rotor position Angle observation error.  
Figure 13: The speed response of PI regulator under 1000 r/min. (**a**) Observed speed and actual speed. (**b**) Estimation error curve.  
Figure 15: The estimation of rotor position Angle using PI. (**a**) Observed and actual rotor position Angle. (**b**) Rotor position Angle observation error.  
From Figures 12 and 13, It is worth noting that the VPDPI regulator has a only 0.05 s adjusting time, whereas the adjustment time of the traditional PI regulator is about 0.113 s. Meanwhile, the sensorless system is no overshoot or steady state error under the control of the VPDPI regulator. However, the overshoot can reach 461 r/min using the traditional PI regulator. On the other hand, the traditional PI regulator experiences a 280 r/min dynamic landing when the load changed, and a 45 r/min dynamic landing is happening in the VPDPI regulator. The adjustment process of sensorless system can also be further seen from the current and torque response results in Figures 16 and 17, respectively.  
In the observation accuracy, the observation speed controlled by the VPDPI is about \(\pm\)1.2 r/min. On the contrary, the observation speed of the traditional PI regulator has a more higher fluctuation, which even can reach \(\pm\)2.8 r/min. Meanwhile, the observation error curves in Figures 14 and 15 show that the Angle tracking error on the basic of VPDPI regulator is more stable, which ensures the observation accuracy of PMSM sensorless system. In summary, the performance advantages of VPDPI regulator can be effectively verified at rated speed.  
There is no doubt that dynamic performance and steady-state performance are important performance indicators of the regulating ability of the reaction regulator. However, in order to meet the complexity and uncontrollability of the actual field conditions, robustness has gradually become an important reference in measuring the performance of the regulator. Therefore, Appendix A tests the robustness of the proposed regulator at rated speed when parameters \(R_{s}\) or \(J\) do not match. Since \(R_{s}\) is configured as 0.011 in Table 4,  
Figure 16: Three phase current response process under two control methods. (**a**) Current response under the proposed VPDPI control. (**b**) Current response under the PI control method  
Figure 17: Torque response process under two control methods. (**a**) Torque response under the proposed VPDPI control. (**b**) Torque response under the PI control method  
VPDPI regulation performance is tested when \(R_{s}\) is 0.0011, 0.021 and 0.031, respectively. On the other hand, since \(J\) is configured as 0.0008 in Table 4, VPDPI regulation is tested when \(J\) is 0.0006, 0.001 and 0.0012, respectively.