Aiming at the nonlinear dynamic mathematical model of permanent magnet synchronous motor, the input-output model of closed-loop system is established by direct feedback linearization control, and the controller is designed by linearization model. This method is simple and practical. At the same time, in order to overcome the shortage that feedback linearization control needs accurate model, an uncertainty predictor based on grey theory is proposed, which can predict the uncertain factors of permanent magnet synchronous motor online and adjust the feedback linearization control law accordingly, thus improving the dynamic performance of the system. The simulation results show that this method has good tracking performance and robustness for the speed control of permanent magnet synchronous motor.
Keywords: Grey Theory Predicting Feedback Linearization Permanent Magnet Synchronous Motor
Permanent magnet synchronous motor (PMSM) has been widely used in servo control system for its excellent performance. In the control of permanent magnet synchronous motor, due to the nonlinear coupling of rotor speed and stator current, the system has strong nonlinearity, especially when the system is uncertain, which makes it difficult for the system to achieve high-precision servo. During the operation of permanent magnet synchronous motor, the stator resistance, viscous friction coefficient and load torque of the motor may change greatly, and the change of these parameters will inevitably affect the servo accuracy of the system. In order to solve the problem of precise servo control of permanent magnet synchronous motor, the nonlinear control methods adopted at present mainly include variable structure control, differential geometry and passivity theory.
In recent ten years, the nonlinear control theory based on feedback linearization has made great progress. Through coordinate transformation and state feedback, a nonlinear system can be transformed into a linear system. Direct feedback linearization (DFL) is a feedback linearization method based on system input-output description, which has successfully solved many nonlinear control problems. The advantages of direct feedback linearization are that the mathematical tools used are simple, the physical concepts are clear and easy to master. However, it has an obvious deficiency. When the system parameters change, the nonlinearity of the system cannot be completely transformed into linearity, resulting in errors. 1982, Professor Deng Julong put forward the grey theory [1], which was successfully applied to many production processes. With the improvement of grey theory and the development of microprocessor, the application of grey theory in control field is more and more extensive. In this paper, a grey uncertainty predictor is proposed to predict the uncertain factors of permanent magnet synchronous motor online and adjust the feedback linearization control law accordingly, thus improving the performance of the system. This method overcomes the deficiency of feedback linearization on model accuracy, suppresses the interference of uncertain factors on the system, and achieves the expected control effect.
1, feedback linearization control of permanent magnet synchronous motor
1. 1 mathematical model of permanent magnet synchronous motor
Using surface permanent magnet synchronous motor, its model [2] based on synchronous rotating rotor coordinate system is as follows:
These include:
Where: shaft stator voltage; Is the shaft stator current; R is stator resistance; L is the stator inductance; TL is the load torque; J is the moment of inertia; B is the coefficient of viscous friction; P is a polar logarithm; ω is the mechanical angular velocity of the rotor; φ f is the permanent magnetic flux.
1.2 feedback linearization control
In order to decouple the system and avoid the problem of zero dynamic system [3], ω and ω, id are selected as the output of the system, and the new system output variable is defined as:
Derive formula (2) to obtain:
At that time, the linear control law was:
Where is the input vector of the new linear system, which can be designed according to the pole assignment theory of the linear system as follows:
Feedback linearization control obtains the required coordinate transformation and nonlinear system state feedback through Lie differentiation of output variables, and realizes decoupling of nonlinear system of permanent magnet synchronous motor. The controller is designed by linear theory, with simple design parameters and certain speed tracking performance. At the same time, it can be seen from the above derivation that feedback linearization is a kind of feedback linearization based on accurate mathematical model. When the system parameters change or the load is uncertain, the nonlinear factors of the system can not be completely eliminated, which may cause errors. In reference [8], a load observer combined with feedback linearization control is proposed to compensate the influence of load change on the system. In the next section, the uncertain factors such as stator resistance, viscous friction coefficient and load change of permanent magnet synchronous motor are predicted online by combining grey prediction, and the feedback linearization control law is adjusted to improve the control accuracy of the system.
2. Grey prediction model
2. 1GM modeling method
The grey model modeling theory is different from the conventional modeling method. It does not treat the data sequence generated by random process according to statistical law or transcendental law, but regards it as a grey quantity that changes in a certain range and time period. By sorting out the original data (also called the generation of numbers), we can find the law of numbers. Therefore, the grey model (GM) actually aims to model the generated sequence. Steps of GM modeling
The prediction model adopts the first-order univariate GM (1, 1) model, and its whitening equation is:
Where a is the development coefficient, grey input and identification parameters of the model. The basic idea is: firstly, the collected original sequence is accumulated (AGO) to get a generated sequence with regular exponential growth. Using the generated sequence, the parameters a and u are identified by least square method, and the predicted value of the generated sequence is obtained. In this way, the predicted value of the original sequence can be obtained by IAGO. The prediction algorithm is:
The accuracy of GM( 1, 1) model is related to the selection of original sequence used in modeling. In order to continuously consider the disturbance entering the system, GM( 1, 1) needs to send each newly obtained data into X(0), rebuild GM( 1, 1) and re-predict, which is the innovation model, but this innovation model has more and more information as time goes on. Therefore, every time a new piece of information is added, an old piece of information is removed, so that the number of data remains unchanged during rolling modeling, which is the rolling model of equal dimension and equal information.
2.2 Equal-dimensional information rolling model
Let the sampling value of the system at time h be, and form a sequence with the previous m- 1 sampling data, so that the m data can be predicted by the grey prediction model:
K 1 step prediction is:
Then:
The above formula is a rolling prediction algorithm with equal dimension and new information, where h is the sampling time, m is the modeling dimension, a and u are the parameters obtained by time identification, and k 1 is the prediction step. Generally speaking, the modeling dimension is selected as m=5.
3. Grey predictive feedback linearization control.
3. 1 linear algorithm of grey predictive feedback for permanent magnet synchronous motor
Considering the uncertainty of the system, rewrite the equation (1).
This is another article.
Aiming at the nonlinear dynamic mathematical model of permanent magnet synchronous motor, the input-output model of closed-loop system is established by using direct feedback linearization control, and the controller is designed by using linearization model. This method is simple and practical. At the same time, in order to overcome the shortage that feedback linearization control needs accurate model, an uncertainty predictor based on grey theory is proposed, which can predict the uncertain factors of permanent magnet synchronous motor online and adjust the feedback linearization control law accordingly, thus improving the dynamic performance of the system. The simulation results show that this method has good tracking performance and robustness for the speed control of permanent magnet synchronous motor.
Keywords: Grey Theory Predicting Feedback Linearization Permanent Magnet Synchronous Motor
Nonlinear speed control of PMSM based on grey prediction
Liang 1, 2, Zhou 1, Kan 2( 1. School of Electrical Engineering, Zhejiang University, Hangzhou 3 10027. Wolong Holding Group Co., Ltd., Shangyu, China 3 12300)
Abstract: A direct feedback linearization control for PMSM nonlinear dynamic mathematical model is introduced. A closed-loop input-output system is established. The controller is designed according to the linearization model. The above design method is simple and practical. But they require the model to be accurate, so a grey uncertainty predictor is proposed. It can adjust the centralized uncertainty in PMSM to feedback linearization control law online, and improve the dynamic performance of the system. The simulation results show that the control scheme has good tracking performance and robustness to uncertainty.
Keywords: grey theory, prediction, feedback linearization, PMSM
Permanent magnet synchronous motor (PMSM) has been widely used in servo control system for its excellent performance. In the control of permanent magnet synchronous motor, due to the nonlinear coupling of rotor speed and stator current, the system has strong nonlinearity, especially when the system is uncertain, which makes it difficult for the system to achieve high-precision servo. During the operation of permanent magnet synchronous motor, the stator resistance, viscous friction coefficient and load torque of the motor may change greatly, and the change of these parameters will inevitably affect the servo accuracy of the system. In order to solve the problem of precise servo control of permanent magnet synchronous motor, the nonlinear control methods adopted at present mainly include variable structure control, differential geometry and passivity theory.
In recent ten years, the nonlinear control theory based on feedback linearization has made great progress. Through coordinate transformation and state feedback, a nonlinear system can be transformed into a linear system. Direct feedback linearization (DFL) is a feedback linearization method based on system input-output description, which has successfully solved many nonlinear control problems. The advantages of direct feedback linearization are that the mathematical tools used are simple, the physical concepts are clear and easy to master. However, it has an obvious deficiency. When the system parameters change, the nonlinearity of the system cannot be completely transformed into linearity, resulting in errors. 1982, Professor Deng Julong put forward the grey theory [1], which was successfully applied to many production processes. With the improvement of grey theory and the development of microprocessor, the application of grey theory in control field is more and more extensive. In this paper, a grey uncertainty predictor is proposed to predict the uncertain factors of permanent magnet synchronous motor online and adjust the feedback linearization control law accordingly, thus improving the performance of the system. This method overcomes the deficiency of feedback linearization on model accuracy, suppresses the interference of uncertain factors on the system, and achieves the expected control effect.
1, feedback linearization control of permanent magnet synchronous motor
1. 1 mathematical model of permanent magnet synchronous motor
Using surface permanent magnet synchronous motor, its model [2] based on synchronous rotating rotor coordinate system is as follows:
These include:
Where: shaft stator voltage; Is the shaft stator current; R is stator resistance; L is the stator inductance; TL is the load torque; J is the moment of inertia; B is the coefficient of viscous friction; P is a polar logarithm; ω is the mechanical angular velocity of the rotor; φ f is the permanent magnetic flux.
1.2 feedback linearization control
In order to decouple the system and avoid the problem of zero dynamic system [3], ω and ω, id are selected as the output of the system, and the new system output variable is defined as:
Derive formula (2) to obtain:
At that time, the linear control law was:
Where is the input vector of the new linear system, which can be designed according to the pole assignment theory of the linear system as follows:
Feedback linearization control obtains the required coordinate transformation and nonlinear system state feedback through Lie differentiation of output variables, and realizes decoupling of nonlinear system of permanent magnet synchronous motor. The controller is designed by linear theory, with simple design parameters and certain speed tracking performance. At the same time, it can be seen from the above derivation that feedback linearization is a kind of feedback linearization based on accurate mathematical model. When the system parameters change or the load is uncertain, the nonlinear factors of the system can not be completely eliminated, which may cause errors. In reference [8], a load observer combined with feedback linearization control is proposed to compensate the influence of load change on the system. In the next section, the uncertain factors such as stator resistance, viscous friction coefficient and load change of permanent magnet synchronous motor are predicted online by combining grey prediction, and the feedback linearization control law is adjusted to improve the control accuracy of the system.
2. Grey prediction model
2. 1 GM modeling method
The grey model modeling theory is different from the conventional modeling method. It does not treat the data sequence generated by random process according to statistical law or transcendental law, but regards it as a grey quantity that changes in a certain range and time period. By sorting out the original data (also called the generation of numbers), we can find the law of numbers. Therefore, the grey model (GM) actually aims to model the generated sequence. In step [4] of GM modeling, the first-order univariate GM( 1, 1) model is adopted as the prediction model, and its whitening equation is:
Where a is the development coefficient of the model and u is the grey input, which is the identification parameter. The basic idea is: firstly, the collected original sequence is accumulated (AGO) to get a generated sequence with regular exponential growth. Using the generated sequence, the parameters a and u are identified by least square method, and the predicted value of the generated sequence is obtained. In this way, the predicted value of the original sequence can be obtained by IAGO. The prediction algorithm is:
The accuracy of GM( 1, 1) model is related to the selection of original sequence used in modeling. In order to continuously consider the disturbance entering the system, GM( 1, 1) needs to send each newly obtained data into X(0), rebuild GM( 1, 1) and re-predict, which is the innovation model, but this innovation model has more and more information as time goes on. Therefore, every time a new piece of information is added, an old piece of information is removed, so that the number of data remains unchanged during rolling modeling, which is the rolling model of equal dimension and equal information.
2.2 Equal-dimensional information rolling model
Let the sampling value of the system at time h be, and form a sequence with the previous m- 1 sampling data, so that the m data can be predicted by the grey prediction model:
K 1 step prediction is:
Then:
The above formula is the rolling prediction algorithm with equal dimension and new information, where h is the sampling time, m is the modeling dimension, a and u are the parameters identified at h time, and k 1 is the prediction steps. Generally speaking, the modeling dimension is selected as m=5.
3. Grey predictive feedback linearization control.
3. 1 PMSM grey prediction feedback linearization algorithm
Considering the uncertainty of the system, rewrite the equation (1).
These include:
Where: parameters defining uncertain factors under normal conditions:
Similarly, if ω and id are selected as system outputs, then
The actual control quantity of direct feedback linearization control law;
Where is the uncertain factor block, it can be seen from formulas (12) and (13) that if the value of the uncertain factor block can be predicted and the feedback linearization control law can be adjusted in real time, then the nonlinear system can be completely transformed into a linear system and the system can be decoupled.
Discretization is carried out by equations (10) and (1 1), and the following results are obtained:
The prediction sequence in the formula (14, 15) can be obtained by the grey rolling model (9):
Where: A and U are the parameters obtained by velocity identification at K; Aa and UU are the parameters obtained by current identification at time k;
3.2 System simulation results
The block diagram of grey predictive feedback linear control for permanent magnet synchronous motor is shown in figure 1. By adjusting the parameters K 1, K2 and K3, the system can reach a satisfactory configuration point. The parameters of permanent magnet synchronous motor are stator resistance R = 0.56Ω, stator inductance L=0.0 153H, permanent magnet flux φ f = 0.82 WB, and pole number P=3.
Figure 1 system control block diagram
Direct feedback linearization (that is, W=0) is used for comparison in simulation.
(1) When t=5s, load interference:
; As shown in fig. 2, the upper part of the graph shows the speed tracking of a given square wave speed n, and the lower part of the graph shows the speed tracking error e.
Fig. 2 Feedback linearization tracking response and error curve of load change
(2) When t=5s, the parameters change:
; As shown in figure 3.
Fig. 3 Response and error curves of feedback linearization tracking motor parameter changes.
As can be seen from Figure 2 and Figure 3, when the system is affected by uncertain factors such as load change or motor parameter change, the tracking performance of the system becomes worse. Now, under the same conditions mentioned above, the speed control adopts the gray feedback linearization control method. The simulation results are shown in Figures 4 and 5. Among them, Figure 4 shows the speed response and tracking error curve realized by the gray feedback linearization control method when the load changes. As can be seen from the figure, the motor speed fluctuates slightly when t=5s, but soon the motor can track the given speed again. Figure (5) is the speed response and tracking error curve realized 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 grey feedback linearization control method is robust to uncertain factors such as system parameters and load.
Fig. 4 Grey feedback linearization tracking response and error curve of load change
Fig. 5 Grey feedback linearization tracking response and error curve of motor parameter change
4. Conclusion
The grey 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, the grey theory can be combined with fuzzy control, neural network control and other algorithms to improve the system performance and control accuracy.