Research on Fuzzy PID Control of Electron Beam 3D Printing Power Supply

1. Introduction
Electron beam 3D printing technology is widely used in high-end manufacturing fields (such as aerospace and medical implants) due to its advantages of high energy density and high forming accuracy. As the core of electron beam energy control, the stability of the output voltage and current of the high-voltage power supply directly affects the molten pool shape, cladding layer quality, and final printed part performance. Although the traditional PID control has a simple structure and is easy to implement, in the electron beam 3D printing process, due to the nonlinear and time-varying characteristics of the load (the molten pool generated by the electron beam bombarding the workpiece) (such as the change of load impedance when printing different materials and different layer thicknesses), the traditional PID control parameters are difficult to adjust in real time, and problems such as overshoot and response lag are prone to occur, leading to a decrease in output accuracy. Fuzzy PID control combines the nonlinear processing ability of fuzzy control and the steady-state accuracy advantage of PID control, which can effectively solve the above problems and improve the power supply control performance.
2. Design of Fuzzy PID Control System
(1) System Control Objectives and Variable Definition
The core control objectives of the electron beam 3D printing power supply are: when the load changes or is subjected to external interference, quickly adjust the output voltage (control range 0kV~60kV) and current (control range 0mA~50mA), so that the output deviation (the difference between the actual output value and the set value) and the deviation change rate are controlled within a very small range, ensuring the stability of electron beam energy.
The input variables of the fuzzy PID control are defined as the output deviation e (unit: kV or mA) and the deviation change rate ec (unit: kV/s or mA/s), and the output variables are the three parameter adjustment amounts of the PID controller: ΔKp, ΔKi, ΔKd (proportional coefficient adjustment amount, integral coefficient adjustment amount, differential coefficient adjustment amount). According to the actual working conditions of electron beam 3D printing, the fuzzy universe of discourse of the input variables is determined: e ∈ [-5,5], ec ∈ [-5,5]; the fuzzy universe of discourse of the output variables: ΔKp ∈ [-0.5,0.5], ΔKi ∈ [-0.1,0.1], ΔKd ∈ [-0.05,0.05] (the specific values are determined according to the rated parameters of the power supply and the control accuracy requirements).
(2) Design of Fuzzy Membership Function
The triangular membership function is used to fuzzify the input and output variables, which is suitable for real-time control scenarios due to its simple calculation and high sensitivity. The fuzzy language variables of the input variables e and ec are divided into 7 levels: Negative Big (NB), Negative Medium (NM), Negative Small (NS), Zero (ZO), Positive Small (PS), Positive Medium (PM), Positive Big (PB); the fuzzy language variables of the output variables ΔKp, ΔKi, ΔKd are also divided into 7 levels, consistent with the input variables.
Taking the input variable e as an example, when e = -5, the degree of belonging to NB is 1; when e is between -5 and -3, the degree of belonging to NB decreases gradually, and the degree of belonging to NM increases gradually; when e = 0, the degree of belonging to ZO is 1, so as to realize the fuzzy conversion of variables.
(3) Construction of Fuzzy Control Rule Base
The formulation of fuzzy control rules is based on the control characteristics and engineering experience of the electron beam 3D printing power supply, following the principle of "adjusting the proportional coefficient first to speed up the response when the deviation is large, adjusting the integral coefficient to reduce the steady-state error when the deviation is small, and adjusting the differential coefficient to suppress overshoot when the deviation change rate is large". For example:
1.When e = PB (positive big output deviation) and ec = NB (negative big deviation change rate), it indicates that the current output is much higher than the set value, and the deviation has a decreasing trend. It is necessary to reduce the proportional coefficient and differential coefficient, and adjust the integral coefficient appropriately. The corresponding rule is: IF e = PB AND ec = NB THEN ΔKp = NS, ΔKi = ZO, ΔKd = NS;
1.When e = NS (negative small output deviation) and ec = PS (positive small deviation change rate), it indicates that the output is slightly lower than the set value, and the deviation has an increasing trend. It is necessary to increase the proportional coefficient, appropriately increase the integral coefficient, and reduce the differential coefficient. The corresponding rule is: IF e = NS AND ec = PS THEN ΔKp = PS, ΔKi = PS, ΔKd = NS.
A total of 49 fuzzy control rules (7×7) are formulated, covering all fuzzy combinations of input variables, ensuring that the control system can achieve accurate adjustment under different working conditions.
(4) Fuzzy Reasoning and Defuzzification
The Mamdani fuzzy reasoning algorithm is adopted. According to the fuzzification results of the input variables and the control rule base, the fuzzy set of each output variable is obtained through the "min-max" operation. For example, for the input e = PS and ec = PM, first, all rules containing e = PS or ec = PM are matched in the rule base, the trigger strength of each rule (taking the minimum value of the input variable membership degree) is calculated, and then the maximum value of the output fuzzy set corresponding to each rule is taken to obtain the fuzzy set of the output variable.
The centroid method is used for defuzzification to convert the output fuzzy set into an accurate value. The calculation formula is: \(u = \frac{\sum_{i=1}^{n} \mu_A(u_i) \cdot u_i}{\sum_{i=1}^{n} \mu_A(u_i)}\), where u is the accurate value after defuzzification, \(\mu_A(u_i)\) is the membership degree of the output fuzzy set at u_i, and n is the number of discrete points. The centroid method can ensure the smoothness and accuracy of the defuzzification result and avoid sudden changes in the output parameters.
(5) Online Adjustment of PID Parameters
The ΔKp, ΔKi, ΔKd obtained by defuzzification are superimposed with the initial PID parameters (Kp0, Ki0, Kd0, obtained by offline tuning) to obtain real-time control parameters: Kp = Kp0 + ΔKp, Ki = Ki0 + ΔKi, Kd = Kd0 + ΔKd. The control system dynamically adjusts the PID parameters according to the real-time load change to achieve accurate control of the output voltage and current.
3. System Simulation and Experimental Verification
(1) Simulation Analysis
A simulation model of the electron beam 3D printing power supply fuzzy PID control system was built based on MATLAB/Simulink. A load disturbance scenario (simulating the load impedance changing from 1MΩ to 0.8MΩ suddenly during the printing process) was set up to compare the response characteristics of traditional PID control and fuzzy PID control. The simulation results show that the overshoot of traditional PID control is 8%, and the adjustment time is 0.5s; the overshoot of fuzzy PID control is reduced to 2%, the adjustment time is shortened to 0.2s, and the steady-state error is reduced from 0.5% to 0.1%, indicating that the fuzzy PID control has significant advantages in anti-interference ability and response speed.
(2) Experimental Verification
An experimental platform was built, with the electron beam 3D printing equipment (printing material is TC4 titanium alloy, layer thickness 0.1mm) as the control object. Printing experiments were carried out using traditional PID and fuzzy PID control methods respectively. The experimental results show that when fuzzy PID control is adopted, the output voltage ripple coefficient of the high-voltage power supply is controlled within 0.3%, and the current fluctuation range is ≤0.2mA; the uniformity error of the cladding layer thickness of the printed part is reduced from 5% (traditional PID control) to 2%, the internal porosity of the printed part is reduced by 30%, and the mechanical performance (tensile strength) is increased by 5%, which fully verifies the effectiveness of the fuzzy PID control.
4. Conclusion
The fuzzy PID control system of the electron beam 3D printing power supply effectively solves the limitations of traditional PID control under time-varying loads by fuzzifying the load nonlinear characteristics and dynamically adjusting the PID parameters, significantly improving the output stability and response speed of the power supply. This control scheme provides a key control guarantee for the high-precision forming of electron beam 3D printing technology. In the future, it can be combined with the adaptive fuzzy control algorithm to further optimize the rule base and realize accurate control under more complex working conditions.