Wei Peng and Zhou Wei from the Maglev Train and Maglev Technology Research Institute of Southwest Jiaotong University first established a mathematical model of the magnetic levitation system at the equilibrium point based on the physical model of a single electromagnet, and then adopted a pseudo-differential feedback control (PDF) strategy to control the levitation system. Theoretical analysis and computer simulations proved that the PDF levitation control system has advantages such as high stability, high precision, strong robustness, and good controllability. 1 Introduction Magnetic levitation systems are inherently nonlinear and unstable systems. There are many methods for controlling levitation, but most are complex and costly, and therefore rarely used in practice. Pseudo-differential feedback control (PDF) is a relatively new control method, and it has been successfully applied in temperature control, robotic arms, and electro-hydraulic servo systems. Both theory and practice show that PDF control has advantages such as fast response speed, strong anti-interference ability, and good robustness. This paper uses a single electromagnetic levitation system for modeling, linearization of the model, theoretical analysis, and simulation to demonstrate that the PDF control strategy has good control performance. 2 Establishment of a single electromagnet suspension system model Figure (1) shows the principle structure of a suspension system composed of a single electromagnet and a guide rail: [IMG=Figure 1 Dynamic model of single electromagnet suspension]/uploadpic/THESIS/2007/11/20071113114512952913.jpg[/IMG] Figure 1 Dynamic model of single electromagnet suspension mg: gravity of electromagnet □d: external disturbance F(i,t): electromagnetic attraction z(t): distance between the surface of the magnetic pole and the reference plane h(t): distance between the surface of the guide rail and the reference plane c(t): air gap between the magnetic pole and the guide rail i(t): control coil current u(t): voltage of the winding circuit □□ΦT: main pole flux Φm: air gap flux ΦL: leakage flux The dynamic model of the single electromagnet suspension system is mathematically modeled by assuming the winding leakage flux (i.e., ΦL＝0). The distance between the rail and the magnetic pole is c(t). Ignoring leakage flux and the magnetic reluctance of the rail and the magnetic core, the magnetic reluctance of the magnetic circuit formed by the electromagnet and the rail is mainly concentrated in the air gap between them. Therefore, the effective air gap reluctance RT is: The electromagnet winding inductance is: In the above formula: L is the inductance value, N is the number of turns of the electromagnet winding, ΦT is the main pole flux, RT is the magnetic circuit reluctance, A is the core pole area, μ0 is the air permeability, and i(t) is the control coil current. μ0＝4π＊10-7H/m is the air permeability. The air gap flux density is: The magnetic field energy is: The relationship between electromagnetic attraction and magnetic field energy is: Analysis of the voltage equation of the electromagnet winding circuit yields: Combining the force on the electromagnet in the vertical direction, the dynamic model equations of the magnetic levitation system are as follows (with downward as the positive direction): where fd is the external disturbance. Considering the change of the orbital action surface, then: z(t) = h(t) + c(t) In summary: (1) The equilibrium equation at the equilibrium point is: (2) 3 System model linearization and establishment of state equations Linearize the system near the equilibrium point (i0, c0), and expand equation (2-5) into a Taylor series at the equilibrium point: (3) Wherein, the linearized equation of the voltage equation at the equilibrium point is: (4) Wherein: The system equations for establishing the linearized system state equations (1) After linearization at the equilibrium point, we select the following state equations as state variables: (5) (6) The open-loop system structure diagram can be obtained from the system equations of the magnetic levitation system, as shown in Figure 2. For the convenience of the study, we assume Δh(t) = 0, then Δc(t) = Δz(t), that is, we do not consider the change of the orbital action surface. From the open-loop structure diagram of the magnetic levitation system, we can obtain the transfer function of the levitation air gap (position) with respect to the input voltage: The characteristic equation of the system is: (7) From this expression, we know that the coefficient of the first power term of s is missing from the characteristic equation, and the coefficient of the last term is negative. According to the Routh criterion, the characteristic equation is unstable. Moreover, the open-loop system is a third-order unstable system. Therefore, this system needs to add a feedback controller to ensure the stable levitation of the system. 4 Theory and Simulation of Pseudo-differential Feedback Controller 4.1 Origin of PDF Control Strategy When performing feedback control on the most common first-order and second-order objects in engineering, in order to ensure the stability of the system, first-order and second-order differential links should be added. However, in actual systems, it is very difficult to directly measure differential signals above the second order. If the controlled variable or its differential is used as the feedback signal after differential processing, it will cause troublesome noise problems. If the signal is first differentiated in the feedback channel, and then integrated and "restored" after passing through the comparison point, it is better to make the signal take a shortcut to the input of the final stage control element. This method can obtain differential feedback signals without differentiation, or only low-order differentiation, hence the name pseudo-differential feedback control, which greatly facilitates engineering. 4.2 Basic Idea of ​​PDF Control Strategy PDF control, first proposed in 1971 by Professor Phelan of Cornell University, is a highly practical engineering control theory. Professor Chen Liu of the School of Mechanical Engineering at Southwest Jiaotong University, during his time in the United States, collaborated with Professor Phelan to further refine and develop this theory, achieving excellent results in first-order, second-order, and higher-order systems. Currently, PDF control has been widely applied in temperature control, robotic arms, and electro-hydraulic servo systems. The basic idea of ​​the pseudo-differential feedback control strategy is: in a closed-loop error control system, the controller's role is to calculate the input error signal to generate output commands to adjust the output power of the power drive element, ultimately causing the controlled object to move in the direction of reducing the error. The ideal control action should achieve the following three objectives: (1) It should respond to any error (constant or time-varying); [IMG=Fig. 2 Open-loop structure diagram of magnetic levitation system]/uploadpic/THESIS/2007/11/2007111312515177025Y.jpg[/IMG] Fig. 2 Open-loop structure diagram of magnetic levitation system [IMG=Fig. 3 Control block diagram of PDF controller]/uploadpic/THESIS/2007/11/2007111312532911251F.jpg[/IMG] Fig. 3 Control block diagram of PDF controller [IMG=Table 1 Main parameters of magnetic levitation system model]/uploadpic/THESIS/2007/11/2007111312554326561O.jpg[/IMG] Table 1 Main parameters of magnetic levitation system model (2) The result of this control action can completely eliminate errors caused by all forms of disturbance; (3) If the controlled object does not act according to the instruction, the controller should be able to generate a control signal that makes the power drive element provide an output power that increases with time. A typical PDF control block diagram is shown in Figure 3. In the figure, R is the reference input, the amplitude of the reference input, E is the difference signal, C is the output signal, L is the disturbance signal, G(S) is the transfer function of the controlled object, the box with a transfer function of 1 represents the final stage control unit, M1 is its low-energy control signal, and M2 is its high-energy output signal. Due to the constraints of the actual control element, M2 ≤ Mmax is always true; Ki, Kp, and Kd are the integral, proportional, and derivative coefficients of the PDF controller, respectively. 4.3 Calculation of PDF control parameters The determination of control parameters is the main work of the integrated control system, and simulation is performed on this basis. From the PDF control block diagram shown in Figure (3), three control parameters Ki, Kp, and Kd need to be determined. They need to be determined by constraint optimization according to a selected objective function (such as ITAE, IAS, ISE, and minimum settling time, etc.). These methods each have their merits, but in terms of algorithm versatility and ease of use, Professor Chen Liu's "characteristic root structure theory" has greater advantages. The model parameters for the suspension system are shown in Table 1. The design specifications that the actual control system should meet are: rise time tr ≤ 0.2s, settling time ts ≤ 0.4s, and overshoot MP ≤ 0.2. Based on the above indicators and model parameters, according to the PDF control block diagram shown in Figure (3), its transfer function can be obtained as: (8) Its closed-loop characteristic equation is: (9) According to the given performance indicators and the characteristic root structure theory of Chen's method for determining the PDF control coefficients, the desired poles of the closed-loop system are selected as: s1,2=-20, s3=-200. At this time, the system has a fast response, no overshoot, and strong robustness, so the characteristic equation of the system is obtained as: □(s)=s3+240s2+8400s+80000 (10) Comparing the coefficients of equations (9) and (10), we can obtain: (11) Substituting the data in equation (11) into (8) yields the transfer function as: 4.4 Strategy Analysis and Matlab Simulation of PDF Suspension Control System 4.4.1 Stability Aspect [IMG=Figure 4 [PDF control block diagram with parameters]/uploadpic/THESIS/2007/11/2007111313052914101A.jpg[/IMG] Figure 4 PDF control block diagram with parameters [IMG=Figure 5 PDF controller signal response curve]/uploadpic/THESIS/2007/11/2007111313073497262M.jpg[/IMG] Figure 5 PDF controller signal response curve As shown in Figure 4, its closed-loop transfer function is: Obviously, the characteristic polynomial of the system is third order and can be decomposed into the form of, that is, by making the corresponding coefficients on both sides of the equation equal, we can obtain From equation (12), we can know that by appropriate adjustment, the system can achieve any desired closed-loop transfer function denominator polynomial. Therefore, the PDF has many adjustable parameters and the system has high controllability. 4.4.2 Regarding control accuracy, as shown in Figure 4, its open-loop transfer function is: Therefore, the error transfer function of the PDF control system is: For a step input signal, its steady-state error is: Thus, the PDF control has no static error for step input signals, exhibiting high control accuracy. 4.4.3 Regarding robustness, the robustness of the system mainly reflects its ability to resist external disturbances and changes in the internal parameters of the controlled object. For the PDF levitation control system, referring to Figure 4, the system's transfer function for external disturbance L input is: The steady-state error generated by the step disturbance on the output is: That is, the system has no steady-state error for step disturbance signals. When there is disturbance, the steady-state error generated on the output is: It can be seen that the steady-state error of the system for the speed signal is independent of the parameters of the controlled system, but only related to one parameter of the PDF controller. That is, changes in the controlled system have no impact on the system's disturbance performance. Therefore, the PDF control strategy has strong robustness to changes in the parameters of the controlled object. 4.4.4 Time Response Analysis The PDF controller demonstrates superior performance in both static and dynamic aspects. Figure 5 shows the simulation results of the PDF closed-loop control system under step input and step disturbance signals. As shown in Figure 5a, the PDF levitation control system is relatively stable, with zero steady-state error under step input. The system reaches stability within 0.3 seconds without overshoot, exhibiting good dynamic performance. When the input is zero, the steady-state error is almost zero under step disturbance. Figure 5b shows that the steady-state error is almost zero. The PDF controller is insensitive to changes in the controlled parameters, exhibiting strong robustness and high accuracy. This indicates that the PDF controller does not require precise mathematical models of the controlled object; certain deviations do not affect system performance. 5 Conclusion This paper first establishes a mathematical model of the suspension control system using the single electromagnet method, and then uses the PDF control strategy to control the suspension system. Through performance analysis and computer simulation analysis of the PDF control system, the following conclusions are drawn: (1) The PDF suspension control system has advantages such as fast response speed, high control accuracy, good controllability, strong anti-interference ability and robustness, and has strong practical application value and broad development prospects. (2) PDF suspension control does not require precise mathematical models of the controlled object, and certain deviations will not affect the performance of the system. Proceedings of the Second Servo and Motion Control Forum Proceedings of the Third Servo and Motion Control Forum