## 3 Laser Processing Control Method for MCR Frequency Modulation
### Workflow of Laser Processing Control System for MCR Frequency Modulation
\begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Parameter** \\ \hline Motor mover mass & \(M=21.3\ kg\) \\ Motor force constant & \(K_{m}=72.8\ kg\) \\ Back EMF coefficient of motor & \(K_{e}=36\ V/(m/s)\) \\ Motor viscous friction coefficient & \(B=0.01\ N\cdot s/m\) \\ Motor inductance & \(L=24\ mH\) \\ Stator winding resistance & \(R=3\ \Omega\) \\ Motor polar distance & \(\tau=36\) mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter table of X-shaft linear motor. Figure 5: Simulation model of linear motor. The position loop PID control transfer function of the linear motor servo system is as follows. \[\mathrm{G_{P}(s)=K_{P}(1+\frac{1}{T_{i}\cdot s}+T_{d}\cdot s)=K_{P}+\frac{K_{i}}{ s}+K_{d}\cdot s} \tag{9}\]  
where \(K_{P}\) is the proportional coefficient of the position loop, \(K_{i}\) is the integral coefficient of the position loop and \(K_{d}\) is the differential time constant of the position loop. The servo control model of the linear motor based on the PID control method is depicted in Figure 6.  
The proportional, integral and differential coefficients of the position loop controller are adjusted according to the PID parameter tuning principle. The initial control coefficients of the adjusted position loop are \(K_{p}=31.24\), \(K_{i}=3.45\) and \(K_{d}=15.78\). In the current industrial application, the fuzzy PID control method has never been applied to the MCR frequency modulation carrier related equipment. Because of the short moving distance and high precision of the MCR frequency modulation carrier platform, the control method based on fuzzy PID can respond in time, which is more suitable for the production requirements of MCR laser frequency modulation equipment to solve the lag effect of traditional PID control [22]. Take the motion error and the motion error change rate of the platform as the error input and the two-dimensional fuzzy controller of fuzzy PID control as the controller of the platform. The controller structure is shown in Figure 7. The error input in the fuzzy controller is the trajectory error of the actual motion point of the MCR carrier platform, and the error differential input is the variation of the trajectory error. The fuzzy subsets of input and output variables are set to seven types, namely: NB (negative big), NM (negative middle), NS (negative small), NO (zero), PS (positive small), PM (positive middle) and PB (positive big). The motion trajectory error and the motion trajectory error variation of the actual motion point of the MCR carrier platform, e and ec, respectively, are taken as input. \(\Delta K_{P}\), \(\Delta K_{i}\) and \(\Delta K_{d}\) are taken as output. The domain range is determined as \([-6,6]\). The 'Mamdani' reasoning method and the triangular membership function are used as the fuzzy controller to deal with the fuzzy subset transformed from the input and output variables. In the movement of the carrier platform, there is a certain deviation between the actual trajectory point and the expected trajectory point. The absolute value of the dif  
Figure 6: Servo control model of linear motor based on PID control method. Figure 7: Fuzzy controller structure. ference between the actual trajectory points and the expected trajectory point is used as the error input to directly obtain the position difference between the current position and the expected position of the MCR carrier platform. The error variation is calculated with the difference between the absolute values of the error values of the trajectory in the two moments. Through the error variation, it can be judged whether the difference between the actual trajectory and the expected trajectory will continue to increase or decrease the next time.