## 4 Simulation Verification and Result Analysis
### Performance Testing under Multiple Operating Conditions
#### 4.2.2 Performance Verification at Various Speed  
The running state of the rated speed is the pursuit of the most driver systems. Due to the disturbance, it will need the sensorless system to run smoothly around the rated speed. Therefore, testing near the rated speed will be more comprehensive to verify the performance of the proposed regulator. (1) Performance verification at 80% rated speed:  
When the speed is set at 800 r/ min, the estimated speed and speed error curves under two regulator are shown in Figures 18 and 19, respectively. Figures 20a,b record the response of position Angle and the response error under VPDPI regulator, respectively. The response of position Angle and the response error under VPDPI regulator are presented in Figures 21a,b, respectively. Moreover, the output torque and three-phase current state of PMSM are shown in Figures 22 and 23, respectively. The operation results in Figures 18 and 19 can reveal the control performance of the proposed control method from two perspectives. From the perspective of sensorless observation: the estimation errors generally exists in the starting stage. After a period of adjustment, the speed estimation errors tend to be stable. When the liability torque is added to the original system, the speed estimation error of VPDPI regulator changes little, and it soon achieves stable again. However, the control system using PI regulator once again experience an oscillation process, and it finally reaches a steady state process. From Figures 18 and 19, the proposed method controls the error range within \(\pm\)1.4 r/min, and it improves the error observation accuracy by 64.1% than the traditional PI regulator (\(\pm\)3.9 r/min). Therefore, the dynamic performance and steady-state performance of the traditional PI regulator have improved in the observation process, especially the suppression of overshoot. From the perspective of speed response curve performance: the traditional PI control system has a large overshoot since the integral saturation influence. Through a adjusting process, the traditional PI control system is gradually stabilize within the error band at 0.13 s (\(\Delta\) = 5%). On the contrary, the proposed VPDPI method takes into account the influence of integral saturation. The VPDPI control system is more faster and more stable entering the error band, which greatly weakens the influence of overshoot on system balance. After the load is added to the sensorless system, the dynamic landing under the  
Figure 21: The estimation of rotor position Angle using PI. (**a**) Observed and actual rotor position Angle. (**b**) Rotor position Angle observation error. Figure 22: Phase current response process under two control methods. (**a**) Current response under the proposed VPDPI control. (**b**) Current response under the PI control method. traditional PI regulator reaches 290 r/min, which go through a long period of disturbance rejection process. Unexpectedly, the control system under the VPDPI regulator only has a dynamic landing of 32 r/min, and the response quickly recovered to a stable state. Therefore, it is worth noting that the proposed method has more advantages in response curve performance analysis. In addition to the emphasis of the observation effect, the accuracy of the observation is also an important indicator of the control of the sensorless system. As shown in Figures 20 and 21, the system estimation under the control of steady state time is similar. Nonetheless the position Angle error of the VPDPI regulator is more smoother. Meanwhile, the dynamic landing of VPDPI system is more smaller when the load is added. In Figures 22 and 23, it can be seen that the VPDPI regulator converges to the steady-state value with a more shorter adjustment time than the traditional PI regulator. In conclusion, when the speed is set at 800 r/min, the sensorless system using VPDPI regulator has a better dynamic performance, steady-state performance and observation effect than the traditional PI system.