Abstract: Permanent magnet synchronous motors (PMSMs) are complex coupled nonlinear systems. This paper applies direct feedback linearization theory, and by performing Lie differential on the output variables, obtains the required coordinate transformation and nonlinear system state feedback, thus realizing the input-output linearization of the PMSM system and simultaneously achieving system decoupling. Simulation results show that it has excellent speed tracking performance compared with traditional PID control, verifying the feasibility and effectiveness of the system design. Keywords: Permanent magnet synchronous motor; Direct feedback linearization (DFL); Lie derivatives; Linear control Abstract: A permanent magnet synchronous motor (PMSM) is a nonlinear system with significant coupling. This paper applies the direct feedback linearization (DFL) theory. The appropriate coordinate transformation and nonlinear state feedback are obtained through Lie deriving the output variable, with which the PMSM system is input-output linearized. Furthermore, this achieves complete dynamic decoupling. Simulation results show that the proposed method has better speed tracking performance compared to normal PID control, and proves the efficiency and feasibility of the designed system. Keywords: PMSM; Direct feedback linearization (DFL); Lie derivatives; Decoupling control 1. Introduction With the development of permanent magnet magnetic materials, semiconductor power devices, and control theory, permanent magnet synchronous motors (PMSMs) are playing an increasingly important role in current medium and low power motion control. They possess advantages such as compact structure, high power density, high air gap flux, and high torque-to-inertia ratio. Therefore, they are increasingly widely used in servo systems. In addition, the permanent magnet synchronous motor is a nonlinear system containing the product of angular velocity ω and current id or iq. Therefore, to obtain accurate control performance, the angular velocity and current must be decoupled. In the past ten years, nonlinear control theory based on the idea of ​​feedback linearization has made great progress [1]. Through coordinate transformation and state feedback, nonlinear systems can be transformed into linear systems. References [2,3] studied this problem using inverse system theory, but it is currently limited to motors powered by current source inverters; References [4,5] applied differential geometry theory to conduct preliminary research on the control problem of induction motors. The theory used is relatively complex and the physical concepts are not clear enough; Reference [6] conducted preliminary research on the control of induction motors using direct feedback linearization theory. Reference [7] applied feedback linearization theory to conduct speed tracking control of linear permanent magnet synchronous motors. Direct feedback linearization (DFL) is a feedback linearization method based on the system input-output description. It has successfully solved a variety of nonlinear control problems [8-9]. The advantage of direct feedback linearization is that the mathematical tools used are simple, the physical concepts are clear, and it is easy to master. This paper applies the principle of direct feedback linearization, starting from the output equation of the system, and performs coordinate transformation and state feedback to decouple the current and angular velocity of the permanent magnet synchronous motor, thereby realizing the linearization of the motor control system. The speed tracking control of the permanent magnet synchronous motor is simulated using linear control theory. The results show that direct feedback linearization control can realize speed tracking control, and at the same time, the system reduces the impact on speed when the load is suddenly added or removed. 2. Principle of Direct Feedback Linearization This section first takes a single-input single-output (SISO) system as an example to briefly introduce the principle of direct feedback linearization. Starting from the output equation of the system, the required coordinate transformation and state feedback law are obtained, and the linearization of the system is realized. The multi-input multi-output system will be introduced in the next section in conjunction with the permanent magnet synchronous motor. There is an nth-order nonlinear system as follows: where f(x) and g(x) are vector functions, and the relative degree of system (1) is p, which reflects the number of integrators between the system output and input. By the definition of Lie differential of relative degree: Now, the derivative with respect to the output y is taken: The specific methods for realizing direct feedback linearization are discussed in two cases below. (1) p=n At this time, the following coordinate transformation T(x) can be chosen: Then the original nonlinear system is transformed into: For the last equation, an assumed input quantity v is introduced, let: In this way, the system (8) can be transformed into a linear system, that is, its input v(t) can be designed according to the linear system theory, and then the feedback linearization control of the original nonlinear system can be obtained by equation (9): (2) p 3. Feedback Linearization of Permanent Magnet Synchronous Motor 3.1 Mathematical Model The surface-type permanent magnet synchronous motor is adopted, and its dq model based on the synchronous rotating rotor coordinates is as follows: Where u[sub]d[/sub], u[sub]q[/sub] are the stator voltages of the d and q axes; is the stator current of the d and q axes; R is the stator resistance; L is the stator inductance; T[sub]L[/sub] is the load torque; J is the moment of inertia; B is the viscous friction system; P is the number of pole pairs; ω is the rotor mechanical angular velocity; Φ[sub]f[/sub] is the permanent magnet flux. Equations (13), (14), and (15) can be simplified to: 3.2 Coordinate Transformation In order to achieve system decoupling and avoid the zero-dynamic system problem, ω and id are chosen as the system outputs, and the new output variables are defined as: Differentiating equation (17): Since the system is a three-input three-output system and its relative order is {1,1,1}, that is, its sum is equal to the order of the system, the system can be linearized with feedback and there is no zero-dynamic problem. Let the assumed control quantity be: Then the linearized system is: Thus, the state feedback control can be designed according to the pole placement theory of linear systems as: Substituting equations (13), (14), and (15) into (19) and (20), the actual control quantities uq and ud are obtained . 4. System Example Simulation The direct feedback linearization control block diagram of the permanent magnet synchronous motor system is shown in Figure 1. By adjusting parameters k[sub]1[/sub], k[sub]2[/sub], k[sub]3[/sub], the system reaches a satisfactory configuration point. The parameters of the permanent magnet synchronous motor are shown in Table 1. Table 1 Parameters of permanent magnet synchronous motor Direct feedback linearization control parameters are: (1) Assuming the reference speed is 500r/s, the simulation results of DFL control are shown in Figure 2(a), and the simulation results of general PID control are shown in Figure 3(a). The simulation results show that the DFL control of the system has a better speed tracking capability than the general PID control. Figure 1 System control block diagram (2) Assuming the reference speed is 500r/s, the system accelerates to 50r/s in 0.4 seconds and decelerates to 50r/s in 0.8 seconds. The simulation results are shown in Figure 2(b) and Figure 3(b). It shows that DFL control gives the system good dynamic performance. Figure 2. Simulation diagram of direct feedback linearization control. Figure 3. Simulation diagram of PID control. (3) Assuming the reference speed is 100 r/s, a load of 10 Nm is suddenly added at 0.25 seconds and the load is removed at 0.5 seconds. The simulation results are shown in Figure 2(c) and Figure 3(c). It shows that the DFL control system reduces the impact on speed when the load is suddenly added and removed. From the above simulation, it can be seen that compared with the general PID control, the DFL control of the system accelerates the tracking speed of the system while reducing the adjustment parameters, and has strong robustness. 5. Conclusion This paper applies direct feedback linearization control to the speed tracking of permanent magnet synchronous motor. Compared with the general PID control method, this design method reduces the adjustment parameters and simplifies the control design of the system. Through Matlab simulation and comparison with general PID control, it is shown that the system has good tracking performance, which verifies the effectiveness and feasibility of the system design. References [1] Feng Guang, Huang Lipei, Zhu Dongqi. High performance control of induction motor based on auto-distrubance refection controller [J]. Proceedings of the CSEE, 2001, 21(10): 55-58 [2] Isidori A. Nonlinear control systems [M]. 2nd Edition, Springer Verlag, 1989. [3] Cao Jianrong, et al. Inverse system-based decoupling control of induction motor [J]. Journal of Electrical Engineering, 1999, 14(1): 7-11 [4] Liu Guohai, Dai Xianzhong Xianzhong). Decoupling control of an induction motor speeding system [J]. Journal of Electrical Engineering, 2001, 16(5): 30-34 [5] Luca AD, Ulivi G.. Design of an exact nonlinear controller for induction motors [J]. IEEE Trans.AC., 1989, 34(12): 1304-1307 [6] Zhang Chunpeng, et al. 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Vol 149, No. 2, 2002. About the author: Liu Dongliang (1977-), male, PhD candidate, engaged in applied research on nonlinear control and motor servo systems.