Abstract: For the nonlinear dynamic mathematical model of a permanent magnet synchronous motor (PMSM), direct feedback linearization control is adopted to establish the input-output model of the closed-loop system. The controller is designed through the linearized model, which is simple and applicable. Meanwhile, to overcome the requirement for high model accuracy in this feedback linearization control, an uncertainty predictor based on grey theory is proposed. This predictor can predict the uncertainties of the PMSM online and adjust the feedback linearization control law accordingly, thereby improving the dynamic performance of the system. Simulation results show that this method has good tracking performance and robustness for the speed control of the PMSM. Keywords: Gray theory prediction feedback linearization permanent magnet synchronous motor Nonlinear Speed ​​Control of PMSM based on Gray Prediction LIU Dong-liang1,2, Zhao Guang-zhou1, Yan Wei-can2 (1. College of Electrical Engineering, Zhejiang University, Hangzhou 310027, China2. Wolong holding group co., LTD, Shangyu 312300, China) Abstract : A direct feedback linearization control with regard to PMSM nonlinear dynamic mathematical model is introduced in the paper. And a closed loop input-output system is built. A controller is designed according to linearization model. The design methods mentioned above are simple and applicable. But they require the model must be accurate, so that the gray uncertainty predictor is brought forward. It can adjust the lumped uncertainty existed in PMSM into a feedback linearization control law on line and improve the system's dynamic performance. Key words : Gray Theory, Prediction, Feedback Linearization, PMSM (Permanent Magnet Synchronous Motor): PMSMs have gained widespread application in servo control systems due to their excellent performance. In the control of PMSMs, the nonlinear coupling between rotor speed and stator current results in strong nonlinearity, especially when system uncertainties exist. This nonlinearity makes achieving high-precision servo control difficult. During the operation of a PMSM, the stator resistance, viscous friction coefficient, and load torque can change significantly, inevitably affecting the system's servo accuracy. To solve the problem of precise servo control of PMSMs, current nonlinear control methods mainly include variable structure control, differential geometry, and passive theory. In the past decade or so, nonlinear control theory based on feedback linearization has made significant progress. Through coordinate transformation and state feedback, nonlinear systems can be transformed into linear systems. Direct feedback linearization (DFL) is a feedback linearization method based on the system's input-output description and has successfully solved various nonlinear control problems. The advantages of DFL are the simplicity of the mathematical tools used, the clarity of the physical concepts, and ease of understanding. However, it has a significant drawback: when system parameters change, the system's nonlinearity cannot be completely converted to linearity, thus introducing errors. In 1982, Professor Deng Julong proposed the grey theory[1], which was successfully applied in many production processes. With the continuous improvement of grey theory and the continuous development of microprocessors, the application of grey theory in the field of control has become more and more widespread. This paper proposes a grey uncertainty predictor to predict the uncertainty factors of permanent magnet synchronous motors online and adjusts the feedback linearization control law accordingly, thereby improving the performance of the system. This method overcomes the shortcomings of feedback linearization in terms of model accuracy requirements and suppresses the interference of uncertainty factors on the system, achieving the expected control effect. 1. Feedback linearization control of permanent magnet synchronous motor 1.1 Mathematical model of permanent magnet synchronous motor The surface-type permanent magnet synchronous motor is adopted, and its model based on the synchronous rotating rotor coordinates[2] is as follows: Where: Where, is the shaft stator voltage; is the shaft stator current; R is the stator resistance; L is the stator inductance; TL 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; Φ[sub]f[/sub] is the permanent magnet flux. 1.2 Feedback Linearization Control In order to achieve system decoupling and avoid the zero dynamic system problem [3], ω,i[sub]d[/sub] is selected as the system output, and the new system output variable is defined as: Taking the derivative of equation (2), we get: When, the linear control law is: Where, is the input vector of the new linear system, which can be designed according to the pole placement theory of linear system as: Feedback linearization control obtains the required coordinate transformation and nonlinear system state feedback by performing Lie differential on the output variable, and realizes the decoupling of the nonlinear system of permanent magnet synchronous motor. The controller is designed by linear theory, the design parameters are simple, and it has certain speed tracking performance. At the same time, it can be seen from the above derivation that feedback linearization is a feedback linearization based on an accurate mathematical model. When the system parameters change or the load is uncertain, the nonlinear factors of the system cannot be completely eliminated, which may cause errors. Reference [8] proposes a load observer to combine with feedback linearization control to compensate for the influence of load changes on the system in response to the uncertainty of the load. The next section combines grey prediction to predict uncertainties such as stator resistance, viscous friction coefficient, and load changes of permanent magnet synchronous motors online, adjusts the feedback linearization control law, and improves the accuracy of system control. 2. Grey Prediction Model 2.1 GM Modeling Method Grey model modeling theory is different from conventional modeling methods. It does not process the data sequence generated by the random process according to statistical laws or a priori laws, but regards it as a grey quantity that changes within a certain amplitude range and a certain time zone. It finds the laws of numbers by organizing the original data (also known as number generation). Therefore, the grey model (GM) is actually modeling the generated number sequence. The steps of GM modeling [sup][4][/sup] are as follows: The first-order, single-variable GM(1,1) model is used as the prediction model, and its whitening equation is: where a is the development coefficient of the model, u is the grey input, and is the identification parameter. The basic idea is as follows: First, the collected original data sequence is accumulated (AGO) to obtain a regularly exponentially increasing generated data sequence. Using the generated data sequence, the least squares method is used to identify the parameters a and u, and the predicted value of the generated data sequence can be obtained. In this way, inverse accumulation (IAGO) can be performed to obtain the predicted value of the original data sequence. The prediction algorithm is as follows: The accuracy of the GM(1,1) model is related to the selection of the original data sequence used for modeling. In order to continuously take into account the disturbances that enter the system one after another, GM(1,1) needs to send each newly obtained data into X[sup](0)[/sup], reconstruct GM(1,1), and re-predict. This is the innovation model. However, as time goes by, the information in this innovation model increases, the storage capacity increases, and the amount of computation also increases. This is not suitable for the real-time and speed requirements of industrial process control. Moreover, the information of old data will decrease over time, and may even submerge new effective information. Therefore, by adding new information while removing old information to maintain a constant number of data points during rolling modeling, this is the equal-dimensional information rolling model. 2.2 Equal-Dimensional Information Rolling Model Let the sampled value at time h be and form a sequence with the previous m-1 sampled data. From these m data, a one-step prediction formula is obtained through the grey prediction model: k[sub]1[/sub] steps of prediction is: Then: The above formula is the equal-dimensional information rolling prediction algorithm, where h is the sampling time, m is the modeling dimension, a and u are the parameters identified at time h, and k[sub]1[/sub] is the number of prediction steps. Generally, the modeling dimension is chosen as m=5. 3. Grey Predictive Feedback Linearization Control 3.1 PMSM Grey Predictive Feedback Linearization Algorithm Considering the uncertainties of the system, the equation (1) is rewritten. Where: In the formula, is the parameter under normal conditions. The uncertainties are defined as: Similarly, ω,i[sub]d[/sub] is chosen as the output of the system. Then, the actual control quantity is obtained by the direct feedback linearization control law: In the formula, is the uncertainty block. From equations (12) and (13), it can be seen that if the value of the uncertainty block can be predicted and the feedback linearization control law is adjusted in real time, the nonlinear system can be completely converted into a linear system, and the system decoupling can be achieved. Discretized by equations (10) and (11), we get: The prediction sequence in equations (14, 15) can be obtained by the grey equal-dimensional information rolling model (9): Where: a,u is the parameter obtained by speed identification at time k; aa,uu is the parameter obtained by current identification at time k; 3.2 System Simulation Results The block diagram of the grey predictive feedback linearization control of the permanent magnet synchronous motor is shown in Figure 1. By adjusting parameters k1, k2, and k3, the system reaches a satisfactory configuration point. The parameters of the permanent magnet synchronous motor are stator resistance R = 0.56Ω, stator inductance L = 0.0153H, permanent magnet flux Φf = 0.82Wb, and number of pole pairs P = 3. Figure 1 shows the system control block diagram. Direct feedback linearization (i.e., W = 0) is used as a comparison in the simulation. (1) At t = 5s, the load disturbance is: ; As shown in Figure 2, the upper part of the figure represents the speed tracking given square wave speed n, and the lower part of the figure represents the speed tracking error E. Figure 2 shows the feedback linearization tracking response and error curve of load change. (2) At t = 5s, the parameter changes are: ; As shown in Figure 3. Figure 3 shows the tracking response and error curves of the feedback linearization for motor parameter changes. As can be seen from Figures 2 and 3, the tracking performance of the system deteriorates when there are uncertainties such as load changes or motor parameter changes. Now, under the same conditions, the gray feedback linearization control method is used to control the speed. The simulation results are shown in Figures 4 and 5. Figure 4 shows the speed response and tracking error curves achieved by the gray feedback linearization control method when the load changes. From the figure, it can be seen that at t=5s, the motor speed fluctuates slightly, but the motor can quickly track the given speed again. Figure (5) shows the speed response and tracking error curves achieved by the gray feedback linearization control method when the motor parameters change. It can also be seen from the figure that the tracking error is reduced by using the gray feedback linearization method. Therefore, the gray feedback linearization control method has robust performance against uncertainties such as system parameters and load. Figure 4 shows the tracking response and error curves of the gray feedback linearization for load changes. Figure 5 shows the tracking response and error curves of the gray feedback linearization for motor parameter changes. 4. Conclusion The gray predictive feedback linearization control algorithm proposed in this paper has certain robustness and fast tracking ability, and reduces the complexity of the algorithm. In addition, grey theory can be combined with algorithms such as fuzzy control and neural network control to improve system performance and increase control accuracy. References [1] Deng Julong. Grey Control System [M]. Wuhan: Huazhong University of Science and Technology Press, 1987. [2] PRAGASAN PILLAY, and R. KRISHNAN. Modeling of Permanent Magnet Motor Drives. IEEE Transactions on Industrial Electronics. Vol.35. No.4,1988 [3] J. Zhou and Y. Wang. Adaptive backstepping speed controller design for a permanent magnet synchronous motor. IEE Proc. Electr. Power Appl. Vol 149,No. 2, 2002 [4] Ching-Chang Wong, Chia-Chong Chen, Design of Fuzzy control systems with a switching Grey Prediction, IEEE Trans.1998 [5] Liu Guohai, Dai Xianzhong. Decoupling control of induction motor speed regulation system [J]. Journal of Electrical Engineering, 2001, 16(5): 30-34 [6] Zhang Chunpeng, Lin Fei, Song Wenchao et al. Nonlinear control of asynchronous motor based on direct feedback linearization [J]. Proceedings of the CSEE, 2003, 23(2): 99-102 [7] Zou Jian, Yang Yingchun, Zhu Jing. Predictive fuzzy control strategy based on grey model and its application [J]. Proceedings of the CSEE, 2002, 22(9): 12-14 [8] Liu Dongliang, Yan Weican, Zhao Guangxiu. Motor feedback linearization control based on torque disturbance estimation [J]. Journal of Electric Power System and Automation, 2005(5): 60-63