Abstract : This paper takes a permanent magnet synchronous linear motor as the research object, and designs speed controllers for the linear motor control system using fuzzy PD control and fuzzy PID control methods respectively, and compares and analyzes them with traditional PID speed controllers. Furthermore, addressing some unique problems of linear motors, such as adverse effects of frictional disturbances, the two controllers are optimized to weaken and suppress the adverse effects of disturbances, achieving a fast and accurate response to the given speed signal, thereby meeting the design requirements.
Keywords : Permanent magnet synchronous linear motor (PMSLM); speed controller; fuzzy PID control; Matlab
Keyword : PMSLM; speedcontroller; fuzzyPIDcontrol; Matlab
[Chinese Library Classification Number] TS49 [Document Identification Code] B
1 Introduction
Linear drive technology takes linear motors as the main research object, electromagnetic induction principle as the theoretical basis, and integrates electromagnetics, power electronics, intelligent control, control engineering, signal processing, mechanics, dynamics and other disciplines into a new technology [1]. It has shown great practical value and played an extremely important role in both civilian and military fields. This paper takes practical application as the background, according to the corresponding design indicators, and studies the control strategy of the system based on the actual problems in the linear drive control system, and designs a controller that can achieve the system performance indicators.
2 Mathematical Model of Linear Motor System
2.1 dq model of permanent magnet synchronous linear motor
The mathematical model of the permanent magnet synchronous linear motor is basically the same as that of the permanent magnet rotary synchronous motor. In the derivation process, some assumptions are made first: 1. Core saturation is ignored; 2. Eddy current and hysteresis losses are ignored; 3. There is no damping winding on the mover and the permanent magnet has no damping effect; 4. The back electromotive force is sinusoidal.
Considering only the fundamental component and using the dq-axis model, the flux linkage equation for a permanent magnet linear synchronous motor is:
(1)
( (2)
((3)
Where are the flux linkages of the d-axis and q-axis of the permanent magnet, respectively; and are the inductances and currents of the d-axis and q-axis of the permanent magnet, respectively. For a surface-mounted motor with a permanent magnet, , , and are constants.
The voltage equation for the dq-axis model is
(4)
(5)
Where, and are the d-axis and q-axis mover voltages, respectively; and are the d-axis and q-axis mover flux linkages, respectively; is the mover resistance; is the linear velocity; is the pole moment; and is the differential operator.
The expression for electromagnetic thrust is:
(6)
In field-oriented control, if the d-axis current is not zero and the direction is controlled, it can demagnetize or magnetize the permanent magnet. This demagnetizing effect enables field weakening control of the permanent magnet synchronous linear motor. If the d-axis current is zero, then only the magnetic field generated by the permanent magnet exists along the d-axis. Typically, permanent magnet linear synchronous motors operate in the constant thrust range, therefore a control method with zero d-axis current is used. Using a control strategy with the excitation component in the inner current loop, the q-axis voltage-current equation is:
(7)
electromagnetic thrust expression:
(8)
Where is the electromagnetic thrust coefficient, is the pole moment, and is the permanent magnet flux linkage. The mechanical equation of motion is:
(9)
(10)
(11)
Where v is the mover velocity, is the viscous friction coefficient, is the total mass of the mover and the load it drives, is the total resistance, is the load resistance, is the end effect force, is the sliding friction force between the mover and the guide rail, and is the linear displacement of the mover. Equations (7) and (8) describe the mathematical model of the permanent magnet linear motor. A Laplace transform of the two equations yields...
(12)
(13)
From equations (12) and (13), the model block diagram of the permanent magnet synchronous linear motor can be obtained, as shown in Figure 1 below.
Figure 1. Block diagram of permanent magnet synchronous linear motor model
The current loop control is analyzed, and its structure is a power amplifier driver stage with current negative feedback. In practical applications, the driver配套 with the motor contains such a power amplifier driver circuit with current negative feedback. Figure 2.6 shows a block diagram of a motor model with a current feedback power amplifier stage.
For a given current control signal, the parameters of the Automatic Current Regulator (ACR) in the current loop are determined during driver design and are generally not adjustable. Its power amplifier driver stage acts as a current source, controlling the motor's armature current.
Directly controlled by the input signal of the power amplifier stage, the influence of the back EMF is suppressed by the negative feedback current loop. In addition, because the mechanical inertia of the motor is much greater than the electromagnetic inertia of the armature winding, the response speed of the current loop is much faster than that of the motor speed loop. Therefore, the influence of the back EMF loop can be ignored in practice.
Figure 2. Linear motor model with current negative feedback
Typically, the bandwidth of the current loop is more than five times that of the speed loop. Furthermore, the lag in current filtering and inverter control can be equivalent to a small inertial element, and can be combined into a single small inertial element using the same methods. Therefore, when designing speed and position loop controllers, the current loop is often treated as a small inertial element, and sometimes its transfer function is even directly equivalent to 1. If the current loop is treated as a small inertial element, its time constant is empirically around 1/2, further simplifying the model as shown in Figure 3.
Figure 3. Block diagram of the simplified permanent magnet synchronous motor model.
2.2 Mathematical Model of the Entire Control System
In practical systems, linear encoders are used as displacement sensors to detect the position of the motor mover in real time, and the mover's speed is indirectly obtained through corresponding calculations by the controller. During system modeling and simulation, the position sensor can also be treated as a unity feedback element. Similarly, in speed closed-loop control, the speed sensor is also treated as a unity feedback element. In this design, speed feedback is indirectly obtained through the position sensor; if the control chip's processing speed is fast enough, the speed feedback can be considered real-time.
This project primarily studies the linear motor model problem, specifically the process by which a linear drive control system pushes a load, operates stably at a given speed within its effective stroke, and ultimately ejects the load from the track at a fixed speed. Simultaneously, the design process must ensure both fast and error-free speed response, while maintaining system stability and speed under certain internal and external disturbances (model parameter perturbations, friction).
To improve the system's ability to withstand external disturbances and to ensure the system's response speed, the entire system is designed as a cascade control system with dual closed loops of speed loop and current loop.
Based on the above analysis and simplification, the mathematical model of the entire system is shown in Figure 4. The current loop has been simplified to a small inertial element.
Figure 4 Simplified system block diagram
Figure 4 below shows the system block diagram of the speed feedback control of the linear motor. Its relevant parameters are:
The system open-loop transfer function is:
3. Establishment of Speed Control Scheme
In linear motor servo control systems, the linear motor mover directly drives the load. Changes in the load and external disturbances will directly affect the performance of the servo system. At the same time, the end effect of the linear motor, changes in system parameters (mover mass, viscous friction coefficient, etc.), and state observation noise will all reduce the servo performance of the system, making traditional PID control inadequate.
Therefore, this paper adopts fuzzy control for design. The essence of a fuzzy controller is a fuzzy PD controller, which uses the error signal and its rate of change in the feedback system to calculate the control quantity.
The basic principle of fuzzy control is to generate a control output through fuzzy inference based on an expert knowledge base generated from existing expert knowledge. Figure 5 shows a block diagram of fuzzy control of the speed of a permanent magnet synchronous linear motor.
Figure 5. Structure of the fuzzy speed control system for permanent magnet synchronous linear motor.
In fuzzy quantization, the input fuzzy variables are the velocity error E and the error variation law EC, and the output fuzzy variable is U. This fuzzy inference model is established with dual inputs and a single output. The universes of discourse for variables E, EC, and U are set as follows: each of the fuzzy variables, and v, has seven fuzzy linguistic values: PB, PM, PS, ZO, NS, NM, and MB. Quantities E, EC, and U are represented by fuzzy sets according to certain membership relationships. The membership functions for each variable in this paper are triangular membership functions, as shown in Figure 6.
Figure 6 Membership functions of each variable and .
After fuzzifying the input and output variables of the system simulation, fuzzy rules are established using expert experience to perform fuzzy inference. Once the fuzzy inference rules are established, a fuzzy rule display image can be obtained, as shown in Figure 7.
(a) Fuzzy rule display Figure 1
(b) Fuzzy rule display Figure 2
Figure 7 Fuzzy rule display diagram
4 System Simulation
(1) Traditional PI control
Figure 8 Simulation diagram of traditional PI control
(2) Fuzzy PD control
Figure 9 Simulation diagram of fuzzy PD control
(3) Fuzzy PID control
Figure 10 Simulation diagram of fuzzy PID control
A comparative analysis of traditional PID control, fuzzy control, and fuzzy PID control is presented in Table 1 below.
Table 1 Performance Comparison of Traditional PID, Fuzzy PID, and Fuzzy PID Controllers
5. Conclusion
This paper combines the advantages of traditional PID control (strong anti-interference capability) and fuzzy controller (fast response speed) to improve the dynamic performance of the system while making the system error-free, thereby improving the steady-state performance. It has strong practicality, achieving a fast and accurate response to a given speed signal, thus meeting design requirements.
About the author: Qin Ling (1990-), male, is a master's student at the School of Control Science and Engineering, Shandong University. His main research direction is computer control.
