## 3 Proposed VPDPI Speed Regulator
### The Stability Analysis of Regulator  
Based on the transfer function block diagram in Figure 1 and the system identification results \(G_{obj}\) in (7), it can be seen that the open-loop transfer function \(G_{open}(s)\) of the sensorless system can be equivalent to below:  
\[G_{open}(s)=\frac{K_{p}(s+K_{i})}{s}G_{obj}(s) \tag{11}\]  
Figure 7: The structure diagram of VPDPI regulator control algorithm.  
According to (11), the driving system designed in this paper is equivalent to a sixth-order system, which has six open-loop poles and two open-loop zeros. Therefore, it is a better choice to use root locus analysis system stability. When the PI parameters are standardized, the open-loop transfer function of the system can be further written as follow:  
\[G_{open}(s)=\frac{K_{c}(8.023\times 10^{10}s^{2}+9.208\times 10^{11}s+8.406 \times 10^{11})}{s^{6}+1.4\times 10^{4}s^{5}+2.9\times 10^{6}s^{4}+2\times 10^{9}s ^{3}+3.3\times 10^{10}s^{2}+4.8\times 10^{7}s} \tag{12}\]  
In (12), \(K_{c}\) is the amplification factor.  
The root locus of the closed system is obtained by using the root trajectory drawing rule, as shown in Figure 8. Among them, Figure 8a is the overall root trajectory, and Figure 8b,c are magnified zero-pole images.  
It is well known that, to ensure the stability of the system, all root trajectories should be restricted to the left half plane of the complex frequency domain. However, it can be seen in Figure 8b that when \(K_{c}\) is greater than 0.45, the root locus crosses the imaginary axis and enters the right half-plane of the complex frequency domain. This means that the system is stable when the limit \(K_{c}\) is less than the critical value.  
When the integral coefficient \(K_{i}=1\), the Bode diagram under different proportionality coefficients is shown in Figure 9, and the corresponding amplitude-frequency margin and phase frequency margin information are shown in Table 1.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Proportionality Coefficient** & **Magnitude Margin** & **Phase Margin** & **Stability** \\ \(K_{p}\) & \(G_{m}\) & \(P_{m}\) & **(Yes/No)** \\ \hline
0 & 20.3 (at 65.8 rad/s) & 11.1 (at 18.8 rad/s) & Yes \\
0.1 & 12.9 (at 377 rad/s) & 81 (at 41.1 rad/s) & Yes \\
0.2 & 6.89 (at 378 rad/s) & 83.8 (at 85.5 rad/s) & Yes \\
0.3 & 3.39 (at 378 rad/s) & 78.8 (at 140 rad/s) & Yes \\
0.4 & 0.897 (at 379 rad/s) & 18.4 (at 349 rad/s) & Yes \\
0.5 & \(-1.04\) (at 379 rad/s) & \(-11.9\) (at 398 rad/s) & No \\ \hline \hline \end{tabular}
\end{table}
Table 1: The characteristics with different proportionality coefficients.  
Figure 8: (**a**) The overall trend of the root locus. (**b,c**) show enlarged images of different proportions of poles, respectivelyof the sensorless system.  
As above aforementioned results, \(K_{p}\) has a positive phase margin in the range of \(0\sim 0.4\), and the control system is more stable. To ensure the stability of the sensorless system, the proportionality coefficient \(K_{p1}\) and \(K_{p2}\) should be less than 0.45 when the integration coefficient \(K_{i}\) is set to 1. Therefore, When the median of this range \(K_{p}=0.2\) is selected, the different \(K_{i}\) is selected to discuss the optimal value of \(K_{i}\) in the selected data. The Bode diagram under different integral coefficients is shown in Figure 10, and the corresponding amplitude-frequency margin and phase frequency margin information are shown in Table 2.  
In terms of the above results, when the system is purely proportional control, it is an unstable system. However, with the increase of the integral coefficient from 0 to 1, the stability margin of the system decreases slightly, but it is always stable. Therefore, the influence of proportion coefficient and integral coefficient on position sensorless system is comprehensively considered in this paper, and the final control parameters are shown in Table 3.  
Figure 10: The bode diagram under the different \(K_{i}\).  
Figure 9: The bode diagram under the different \(k_{p}\).