PID Brushless DC Motor Control Based on Fuzzy Optimization

Brushless DC motors overcome the mechanical commutation of ordinary DC motors and are particularly suitable for flexible control using electronic control devices. They are currently widely used in high-precision automation instruments such as robot joint control. A typical control algorithm is to use the traditional proportional-integral-differential (PID) controller for control. However, the performance of the PID controller depends entirely on the adjustment of its gain parameters. In recent years, people have also proposed to use many artificial intelligence controls such as neural network algorithms, genetic algorithms, and fuzzy logic control to design PID controllers. Among them, fuzzy logic control is known for its good handling ability for nonlinear and uncertain parameters, and is particularly suitable for controlling systems with highly nonlinear performance and a large number of random disturbances such as DC brushless motors. This paper will introduce a control method for brushless DC motors based on fuzzy logic optimization and perform simulation.

1 Brushless DC motor and its mathematical modeling
Brushless DC motor is a typical mechatronics product, which consists of a motor body, a position detector, an inverter and a controller. The following will take the two-phase conduction star three-phase six-state mode as an example to analyze the mathematical model of the brushless DC motor.
1.1 Three-phase winding terminal voltage equation
Since the rotor's magnetic resistance does not change with the position of the rotor, the self-inductance and mutual inductance of the stator winding are constants. Considering that the three-phase winding is star-connected ia+ib+ic=0, Lmia+L-mib+Lmic=0; the terminal voltage balance equation of the three-phase winding:

In the formula, ua, ub, uc are the stator phase winding voltages, V; ia, ib, ic are the stator phase winding currents, A; ea, eb, ec are the stator phase winding back electromotive force, V; r is the resistance of each phase winding Ω; Ls is the inductance of each phase winding, H; Lm is the mutual inductance between each two phase windings, H; uN is the neutral point voltage of the motor system.
The equivalent circuit of BLDCM is shown in Figure 1. In the figure, Ud is the DC side voltage, VT1~VT6 are power switching devices, and VD1~VD6 are freewheeling diodes.

1.2 Winding back electromotive force equation
Ignoring the core saturation and slot effect, the resistance and inductance of each phase of the stator are equal, there is no damping winding on the rotor, and the induced electromotive force of the stator winding is a typical 120° trapezoidal wave. Therefore, the function expression of the back electromotive force of the stator A phase winding in the range of 0 to 2π can be obtained Where

ωr is the mechanical angular velocity of the rotor; ke is the back electromotive force coefficient. Similarly, the function expressions of eb and ec can be obtained. 1.3 Torque equation and motion equation 2 Fuzzy PID control method In order to achieve real-time and high-accuracy control and thus improve the output performance of the DC brushless motor, we will design a fuzzy PID controller to adjust the gain parameter of the PID controller in the future. 2.1 PID control Mathematical expression of continuous PID control Where e(t) is the difference between the input and output of the system, u(t) is the control signal generated by the PID controller, Kp is the proportional gain, T1 is the integral time constant, and TD is the differential time constant. Mathematical expression of discrete PID control In the formula, KI=KPT/TI, KD=KPTD/T, T is the sampling period; KP, KI and KD are three adjustable parameters. The task of the PID controller is to determine the values ​​of these parameters. 2.2 Fuzzy PID controller Figure 2 shows the structure diagram of a typical fuzzy PID controller, in which nr is the speed of the given motor, n represents the actual motor speed, e is the system error, and ec is the differential value of e. When the motor is working, in each sampling period, through the fuzzy control instruction, the fuzzy reasoning mechanism detects the rate of change of e and ec, thereby generating their fuzzy variables E and Ec respectively. Then, the controller will immediately adjust the original three parameters of KP, KI and KD of the PID controller, so that the PID controller can always generate the optimal control signal for the DC brushless motor.

In this system, KPf, KIf and KDf represent the increments of KP, KI and KD logical variables. According to the values ​​of E(k) and Ec(k), their fuzzy semantic values ​​E and Ec are shown in Table 1.

Fuzzy control rules are a collection of fuzzy conditional statements that are derived from the experience of experts and the skills of operators. In the fuzzy PID control method of this paper, the fuzzy library is described by the following 49 fuzzy statements:

Where KP0, KI0 and KD0 are the original PID controller parameters, which are generally given by the Ziegler-Nichols debugging equation. D[x] is the defuzzification process.
The fuzzy logic of E, Ec, KPf, KIf and KDf are all defined as: {NB, NM, NS, ZO, PS, PM, PB}, representing: large negative, medium negative, small negative, zero, small positive, medium positive, large positive. In addition, the domains of these variables are all defined in the integer range of -6 to +6, and the membership functions are triangular functions, as shown in Figure 3.

3 Establishment of simulation model
This paper uses Matlab/Simulink to build a simulation model to realize the whole system control of BLDCM. This paper will establish the simulation module of BLDCM based on the above motor mathematical model.
The BLDCM control system adopts speed and current loop dual closed-loop speed regulation. The speed outer loop is optimized and controlled by fuzzy PID regulator, and the current inner loop adopts triangle wave comparison regulation instead of hysteresis comparison control, so as to suppress the large amount of switching noise generated by the non-constant switching frequency. The whole system includes BLDCM body module, voltage inverter module, speed PI control module, current control and PWM signal generation module.

The whole system simulation frame is shown in Figure 4, where the fuzzy control is designed using the fuzzy control module contained in Simulink. The PID module part will perform simulation experiments on the PID controller with and without fuzzy.

4 Simulation results and analysis
To verify the correctness of the model, the simulation model will be simulated. The parameters of BLDCM are as follows: rated voltage ucd=450V, moment of inertia J=8.0×10-4N·m2, stator resistance r=2.8 75 Ω, stator inductance Ls=8.5x10-3H, mutual inductance Lm=0.37×10-3H, pole pair number nb=4, back electromotive force coefficient ke=0.1805 V/(rad·s-1).
In order to verify the static and dynamic performance of the simulation model of the control system of the designed BLDCM, a load is added to the motor at 0.3s, and the simulation curves of the A-phase speed, torque and current under fuzzy PID control and general PID control under stable speed are measured as shown in Figure 5.

It can be seen from the simulation diagram that the system with the fuzzy PID controller responds quickly and smoothly at the reference speed compared with the general PID controller system, and the speed overshoot is significantly reduced; the torque pulsation is relatively small after the load is added, and the time to return to the normal speed is also short; the waveform of the phase current is also relatively ideal.

5 Conclusion
Based on the analysis of the mathematical model of the brushless DC motor, a control system simulation modeling method based on the fuzzy PID controller is proposed. The modeling method is tested using the speed and speed double closed-loop control method. The simulation test results show that compared with the DC brushless motor system controlled by the general PID controller, the system controlled by the fuzzy PID controller has faster response ability, higher adjustment accuracy, and better stability. In addition, this simulation experiment also shows that this control method is suitable for the accuracy and precision required for robot joint control, which lays the foundation for the author's next step of designing a DSP robot joint controller based on TI's TMS28 series.