## 3 Control Strategy of Motor Control System
### Closed Loop Regulator Design
#### 3.2.1 Automatic Speed Regulator (ASR) Design  
Since the speed loop is designed with fuzzy PI control, it is necessary to design the speed PI regulator first and calculate the theoretical value of the initial PI parameter. The simplified current loop can be treated as an intermediate link of the speed loop, and its closed-loop transfer function can be simplified as:  
\[G_{\text{ACR}}(s)=\frac{1}{3T_{\text{s}}s+1} \tag{10}\]  
According to Equation (10), the structure of the PMSM vector control system can be simplified as shown in Figure 5.  
Figure 4: The dynamic structure diagram of the vector control system of PMSM.  
Figure 5: The simplified structure of the vector control system of PMSM.  
From Figure 5, the open-loop transfer function of the speed loop is:  
\[\begin{array}{ll}G_{\text{ASR}}(s)&=\frac{K_{\text{p}}(\tau_{\text{n}s}+1)}{ \tau_{\text{n}s}}\frac{1}{3T_{\text{s}}s+1}\frac{1}{t_{\text{d}}s+1}\cdot\frac{3 }{2}n_{\text{p}}\psi_{\text{f}}\cdot\frac{30}{\pi\gamma t/s}\\ &=\frac{45n_{\text{p}}\psi_{\text{f}}K_{\text{p}}}{\pi\gamma t_{\text{n}}}\cdot \frac{(\tau_{\text{n}s}+1)}{s^{2}(4T_{\text{s}}s+1)}=\frac{K_{\text{ASR}}( \tau_{\text{n}s}+1)}{s^{2}(4T_{\text{s}}s+1)}\end{array} \tag{11}\]  
where \(K_{\text{ASR}}=\frac{45n_{\text{p}}\psi_{\text{f}}K_{\text{p}}}{\pi\gamma t_{ \text{n}}}\).  
According to the Equation (11), the speed loop is a third order system. The system block diagram is shown in Figure 6.  
According to the actual object system response and anti-disturbance requirements on the speed, the speed loop should be equivalent to a typical II-type system. As shown in Figure 6, the system has two characteristics:  
* The system contains two pure integration loops, the initial slope of the amplitude-frequency characteristic is \(-40\) dB, and the initial phase angle is \(-180^{\circ}\);
* The corner frequency of the system is \(1/(4T_{\text{s}})\) in the inertia loop and \(1/\tau_{n}\) in the differential loop.  
The intermediate frequency bandwidth is within the range of the differential loop and the integral loop. The intermediate frequency bandwidth determines the system response speed, and the phase margin of the cut-off frequency determines the system stability. For the electric motor control system, the parameter design process is to optimize these two indexes.  
The highest point of the phase-frequency curve is the point with the largest phase margin. Meanwhile, this point is also the midpoint of the two-corner frequency. Thus, the calculation formula of the phase margin is:  
\[\text{lg}w_{\text{c}}=\text{lg}\frac{1}{4T_{\text{s}}}-\frac{h}{2} \tag{12}\]  
where \(h\) is the intermediate frequency bandwidth.  
Then the phase margin is:  
\[w_{\text{c}}=\frac{1}{4T_{\text{s}}\cdot 10^{\frac{h}{2}}} \tag{13}\]  
Because  
\[20\text{lg}K_{\text{N}}-40\text{lg}\frac{1}{\tau_{\text{n}}}-20(\text{lg}w_{ \text{c}}-\text{lg}\frac{1}{\tau_{\text{n}}})=0 \tag{14}\]  
Then  
\[w_{\text{c}}=K_{\text{N}}\cdot\tau_{\text{n}} \tag{15}\]  
Based on the previous analysis, the PI parameter of the speed regulator can be obtained as:  
\[\left\{\begin{aligned} K_{\text{p}}(ASR)&=\frac{\pi \gamma t}{45T_{\text{sm}}\cdot 10^{\frac{h}{2}}P_{\text{n}}\psi_{\text{f}}}\\ K_{\text{i}}(ASR)&=\frac{\pi\gamma t}{45T_{\text{sm}}^{2} \cdot 10^{\frac{h}{2}}P_{\text{n}}\psi_{\text{f}}}\end{aligned}\right. \tag{16}\]  
where \(T_{\text{sm}}=4T_{\text{s}}\).  
Figure 6: The block diagram of speed loop for open-loop system.