Abstract: A dimension-reduced linear Luenberger observer is constructed using the stator quadrature-axis current and speed equations of a permanent magnet synchronous motor to obtain the motor speed. The observer is simple and easy to implement, and fast convergence speed can be obtained by configuring the eigenvalues. A direct feedback linearization control strategy is adopted to design the system controller, enabling the system to have good speed tracking and torque response. The effectiveness and feasibility of the system design are verified by Matlab simulation. Keywords: Permanent magnet synchronous motor, sensorless, direct feedback linearization control, observer , feedback linearization control research of speed sensorless permanent magnet synchronous motor Abstract The speed of permanent magnet synchronous motor was acquired using reduced order Luenberger observer constructed from stator q-axis current and speed equation. The observer is simple and achieves fast convergence with the eigenvalues ​​setting. The system controller was designed using direct feedback linearization control. The system features fast speed tracking and torque response. Matlab simulation result proves the efficiency and feasibility of the system design. Key words PMSM speed sensorless feedback linearization control observer 1 Introduction With the development of permanent magnet materials, semiconductor power devices and control theory, permanent magnet synchronous motors (PMSM) are playing an increasingly important role in current medium and small power motion control. Permanent magnet synchronous motors have advantages such as compact structure, high power density, high air gap flux and high torque-to-inertia ratio [1]. In traditional permanent magnet synchronous motor servo control, the most common method is to install sensors (such as encoders, solvers, tachogenerators, etc.) on the rotor shaft to obtain the speed and position. However, these sensors increase the cost of the system (the price of some high-precision sensors can even be compared with the price of the motor itself), reduce the reliability of the system, and their application is limited by conditions such as temperature, humidity and vibration, making the system not widely applicable to various occasions. In order to overcome the shortcomings of using sensors, many scholars have carried out research on sensorless permanent magnet synchronous motor control systems [2]. This paper uses the stator quadrature shaft current and speed equation of permanent magnet synchronous motor to construct a dimension-reduced linear Luenberger observer to obtain the speed of the motor. By configuring the eigenvalues, a fast convergence speed can be obtained. The direct feedback linearization control strategy is applied to the design of the system controller, so that the system has fast speed tracking and torque response. Direct feedback linear control obtains the required coordinate transformation and nonlinearization by differentiating the output variables, and also realizes the decoupling of the system. Finally, the effectiveness and feasibility of the system design are verified by Matlab simulation. 2 Mathematical Model of Permanent Magnet Synchronous Motor Based on the surface-mounted permanent magnet synchronous motor, its dp model based on the synchronous rotating rotor coordinates[3] is as follows (its quadrature and direct axis inductances are approximately equal, i.e., L[sub]d[/sub]= L[sub]q[/sub]= L): Where: u[sub]d[/sub] and u[sub]q[/sub] are the d-axis and q-axis stator voltages, i[sub]d[/sub] and i[sub]q[/sub] are the d-axis and q-axis stator currents, 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 coefficient, P is the number of pole pairs, ω is the rotor mechanical angular velocity, and is the permanent magnet flux. 3 Design of Feedback Linearization Controller Feedback linearization control is a feedback linearization based on an accurate model. By transforming coordinates, nonlinear factors are eliminated and the system is converted into a linear system. The controller is designed using linear theory. To achieve system decoupling and avoid the zero-dynamic system problem, ω and i[sub]d[/sub] are chosen as the system outputs, and the new system output variables are defined as: 4 Design of the speed observer For the mathematical model of the permanent magnet synchronous motor, the new input is assumed to be: Substituting equations (7) to (8) into equations (1) to (2), the new state equation of the permanent magnet synchronous motor is obtained as: In this way, the system becomes a linear system, and a linear observer can be constructed to estimate it; in addition, it can be seen from equation (11) that the speed ω is only related to i[sub]q[/sub], so a dimension-reduced observer can be constructed to estimate the speed. The speed equation (11) and the quadrature axis current equation of the permanent magnet synchronous motor are used to form the following dimension-reduced matrix equation (the load torque is assumed to be zero to simplify the system equation). For the above matrix equation, a linear Luenberger observer is designed: where are the estimated values ​​of the quadrature axis current and the speed, respectively. Subtracting the two matrix equations above, we can obtain: From matrix equation (12), it can be seen that by adjusting the parameter γ, the speed observation error can be made to approach zero, and the eigenvalues ​​of the configuration matrix can be adjusted by adjusting the parameter γ to make the system converge quickly. From equation (12), it can be seen that the observer proposed by the system is only of the first order, so the computational burden is significantly reduced. 5 System Simulation and Experimental Waveforms A dimension-reduced linear Luenberger observer is used to obtain the motor speed. Feedback linearization control is used to design the speed and current controllers of the system, as shown in Figure 1. By adjusting the parameters k[sub]1[/sub], k[sub]2[/sub], and 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 Permanent Magnet Synchronous Motor Parameters Figure 1 System Control Structure Block Diagram The initial setting tracking speed of the motor is 500 r/min, and the speed drops to 100 r/min at 0.5s. The initial load torque of the motor is 5 N•m, which increases to 10 N•m at 0.25s. The simulation parameters are as follows: The simulation results are shown in Figure 2. The simulation results show that the system's observer can estimate the motor speed in a timely manner, enabling the system to have fast speed tracking and torque response. Its design advantages include fewer adjustment parameters and a smaller computational load for speed observation. When the motor speed changes, the observer can quickly converge to the given speed. The control strategy proposed in this paper was experimentally studied on an AC servo system experimental platform. Figure 3.1 shows the voltage waveform corresponding to the motor speed of 450 r/min, Figure 3.2 shows the two-phase current waveform at this time, and Figure 3.3 shows the waveform for position estimation. 6. Conclusion This paper uses a linearly reduced-dimensional Luenberger observer to obtain the motor speed. Fast convergence speed can be obtained through the configuration of eigenvalues. A direct feedback linearization control strategy is used to design the system controller, enabling the system to have good speed tracking and torque response. Furthermore, the system has few adjustment parameters, making it easy to implement in engineering. References [1] PILLA YP, KRISHNAN R. Application characteristics of permanent magnet synchronous and brushless DC motor for servo drives [J]. IEEE Transaction on Industry Application, 1991, 27 (5): 986-996. [2] Liang Yan, Li Yongdong. Overview of sensorless permanent magnet synchronous motor vector control system [J]. Electrical Drive, 2003 (4): 4-9. [3] PILLAYP, KRISHNAN R. Modeling of permanent magnet motor drives [J]. IEEE Transactions on Industry Electronics, 1988, 35 (4): 537-541. [4] Cao Jianrong, Wei Zeguo. Research on decoupling control of induction motor based on inverse system theory [J]. Journal of Electrical Engineering, 1999, 14 (1): 7-11. [5] Zhang Chunpeng, Lin Fei. Nonlinear control of asynchronous motor based on direct feedback linearization [J]. Proceedings of the CSEE, 2003, 23 (2): 99-102. [6] MA IDU M, BOSE B K. Rotor position estimation scheme of a permanent magnet synchronous machine for high performance variable speed drive[J]. IEEE IA S An2nual Meeting, 1992 (1): 48-53. [7] SOL SONA J, VALLA MI, MU RAVCHIK C. A nonlinear reduced observer order for permanent magnet synchronous motor [J]. IEEE Trans. Ind, 1996, 43 (4): 492-497.