## 6 Conclusions  
In this article, a VPDPI control method is creatively proposed to eliminate the integral saturation phenomenon of PMSM sensorless system, and its original contributions can be extracted as follows:  
1. The essence of integral saturation in traditional PI controller is unexpectedly revealed based on the classical control theory, and the corresponding control scheme is further developed from the perspective of dynamic response characteristic. Consequently, the novel VPDPI contains two interconnected regulators with independent effects: (a) a DPI regulator stably eliminates integral saturation by utilizing feedback compensation coefficients \(\gamma\), and (b) a VP regulator ensures the system response speed by switching the proportional coefficient between \(k_{p1}\) and \(k_{p2}\).
2. According to the distribution of root trajectory curve, the stability of the system is analyzed. Meanwhile, the parameters range of speed regulator is given by using frequency domain analysis method based on the identified model. Thereinto, the system has a higher stability margin when the proportional coefficient is selected in the range of 0\(\sim\)0.4. When the integral coefficients are set as 0\(\sim\)1, it can be found that the system is always in a stable state, but the amplitude-frequency margin and phase-frequency margin are slightly weakened.  
In addition, the simulation and hardware experiments with different speed regulators are tested under the same control parameters. On the one hand, the simulation results can demonstrate the effectiveness of the new regulator under random operation conditions. Compared with the traditional PI regulator, the adjusting time of the proposed regulator is about 0.05 s under rated speed with no\(-\)load torque, which is less than 44.2% of the tradition PI regulator. More importantly, the overshoot is totally eliminated, which is about 461 r/min in traditional PI regulator. When the system exists a low load disturbance, the dynamic landing is reduced about 16% than that of traditional PI regulator. Moreover, the observed speed error fluctuation is also decreased about 42.9%. Similarly, the control performance remains consistent under variable speed. On the other hand, the experimental results further verify the dynamic performance, accuracy and stability of the novel regulator under multiple working conditions, especially in the suppression abilities of overshoot and the estimated speed ripple. In summary, the effectiveness and reasonability of the proposed method are both amply verified. In the future, some intelligent algorithms will be combined with the proposed method to achieve automatic adjustment of threshold parameters.  
[PERSON] wrote this article, designed the control method, implemented the hardware platform, drew the figures, and performed the simulation as well as the experiment. [PERSON] supervised the study, coordinated the investigations, and checked the manuscript's logical structure. [PERSON] validated the simulation and hardware experimental data and reviewed the writing and editing. [PERSON] provided practical suggestions and evaluated the feasibility of the application. All authors have read and agreed to the published version of the manuscript.  
This work was Supported in part by the Natural Science Foundation of China under Grant No. 61563045, and in part by the International Cooperation Projects of Shihezi University under Grant GJHZ202003.  
Not applicable.  
We sincerely thank Shanghai Yuankuan Energy Technology Co., Ltd. for providing technical guidance for the use of the hardware experimental platform.  
The authors declare no conflict of interest.