Adaptive backsliding control method of permanent magnet synchronous motor based on RBF

Abstract

The adaptive backstepping control method of permanent magnet motor has the problems of complicated coordinate transformation process and high position tracking error. Based on this, an adaptive backstepping control method of permanent magnet synchronous motor based on RBF is proposed. According to the principle of electrical machinery, the electromagnetic wave and magnetic field data are obtained, and the mathematical model of permanent magnet synchronous motor is constructed. Under the condition of keeping the resultant magnetomotive force after coordinate transformation unchanged, the structure of motor torque neural network is established by RBF method, and the coordinate transformation process is optimized. Through the compensation control strategy, the adaptive backstepping control mode is designed to realize the adaptive backstepping control of permanent magnet synchronous motor. The simulation results show that the position tracking error of the proposed method is 4.549 mm when the running time is 7 s and 43.699 mm when the running time is 14 s, which proves that the adaptive backstepping control effect of the proposed method is better.

Keywords

RBF permanent magnet synchronous motor adaptive backward control method CNC machine tools electromagnetic wave

1. Introduction

PMSM (Permanent Magnet Synchronous Motor) is evolved from electric excitation motor. It replaces the electric excitation system with permanent magnets, thus eliminating the excitation winding, slip ring and brush, while the stator structure is basically the same as that of electric excitation motor [1, 2, 3]. Because it can generate sine wave induced electromotive force in the winding during steady-state operation, it is convenient to control the permanent magnet synchronous motor, and it is used in elevator traction system, ship system, spacecraft and other fields [4, 5]. However, elastic deformation, friction, backlash, etc., which are hard to eliminate, will occur in the movement of the transmission link, which will eventually lead to the position tracking delay and nonlinear error, which will seriously affect the adaptive backstepping control accuracy of permanent magnet synchronous motor. Under the action of force and heat, the screw will be deformed, which will lead to the increase of system order, the decrease of robustness and the weakening of servo performance. In recent years, domestic and foreign scholars have made more in-depth research on the control strategy of permanent magnet synchronous motor. Jeong et al. [6] proposed a new rotor shape control method of spoke PMSM. The field weakening control can improve the high-speed operation effect of the motor, reduce the centrifugal force during high-speed operation, and solve the problems of torque reduction and irreversible demagnetization of the motor, but the robustness of the method is poor. Sahebjam et al. [7] proposed a direct speed control method for permanent magnet synchronous motor. Direct voltage control is adopted to synthesize D-axis and Q-axis stator voltages, and the regulation effect of Q-axis current loop is optimized. According to the dynamic limiter, the unexpected current rise is limited, but the position tracking effect of the method is poor. Jiang et al. [8] proposed an adaptive fuzzy control method based on command filtering. Fuzzy logic is used to approximate unknown random nonlinear functions in PMSM random system, and obstacle functions are constructed to ensure that all states of the system do not violate their constraint boundaries. The error compensation mechanism is introduced to reduce the filtering error, and the position of permanent magnet synchronous motor random system with full state constraints is tracked and controlled. But the servo control accuracy of this method is low.Compared with the traditional control method, the backstepping method has the advantage that it can achieve higher precision servo control. The premise of control design is that the controlled object of the system is stable, and the state equation of the controlled object is a nonlinear equation. The response of the backstepping method is faster, and only three to four parameters need to be adjusted when the control parameters are adjusted. From the form of control, it can be seen that the implementation of the algorithm is very simple, only one virtual control formula, two actual control formulas and one adaptive formula need to be given, and because of the nonlinearity and strong misfortunes of the model, the nonlinear algorithm is more in line with the actual situation. The biggest advantage of RBF algorithm is that the training speed is very fast, and the parameters of the model can be directly obtained by matrix operation and random sampling, which can effectively improve the control rate of permanent magnet synchronous motor. RBF algorithm has a small support set, aiming at the selected sample point, it only responds to the input near the sample, and improves the accuracy of control. The RBF Kernel is used in the middle layer of the algorithm to transform the input nonlinearly, so that the output layer can train the linear classifier, which has strong robustness and helps the permanent magnet synchronous motor to complete the adaptive backstepping control. Therefore, this paper proposes an adaptive backstepping control method of permanent magnet synchronous motor based on RBF, which can improve the universality and portability of motor control technology and further promote the long-term development of motor control technology in China.

2. Adaptive backsliding control method of permanent magnet synchronous motor based on RBF

2.1 Obtaining electromagnetic wave and magnetic field data

The noise level of electromagnetic vibration noise is closely related to the amplitude of radial electromagnetic wave, force wave frequency and force wave order, and the mechanical vibration characteristics of stator system, such as natural frequency, mechanical damping and impedance. The stator of an AC permanent magnet synchronous motor is composed of an iron core, a three-phase symmetrical winding, a casing and an end cover, among which the three-phase symmetrical winding is connected in a Y-shape and embedded in the iron core tooth groove. The determination of the structure of AC permanent magnet synchronous motor not only needs to consider its operating conditions and application requirements, but also depends on the selection of permanent magnet materials. According to the direction of magnetic field, the motor can be divided into axial magnetic field rotor structure and radial magnetic field rotor structure. According to the overall structure of the motor, it can be divided into internal rotor structure and external rotor structure. According to the structure of motor stator, it can be divided into concentrated winding structure and distributed winding structure. According to the structure of motor rotor, it can be divided into convex mounting structure, embedded structure and embedded structure. For permanent magnet synchronous motor, the analysis and calculation of the magneto motive force of stator winding similar to induction motor [9]. According to the principle of electrical machinery, when sine power is supplied, the fundamental magnetic force of single-phase winding is:

$\displaystyle l({p,q})=\left\{{\begin{array}[]{l}\frac{4}{\pi}\frac{\sqrt{2}D}% {2r}\times({p,q})^{2}\\ \cos\theta\cos r\\ \end{array}}\right.$ (1)

In Eq. (1), $p$ represents the winding factor of the fundamental magnetic force, $q$ represents the winding factor of the harmonic, $D$ represents the total number of turns in series of each phase, $r$ represents the effective value of the power supply current, and $\theta$ represents the included Angle of the radial electromagnetic force of vibration. The winding factor of harmonics in the formula is the product of pitch coefficient and distribution coefficient, as shown below:

$\displaystyle S_{0}=h_{q}\times h_{\delta}$ (2) $\displaystyle S_{1}=\sin q\frac{d}{\eta}\times 90^{\circ}$ (3) $\displaystyle S_{2}=\frac{\sin q\frac{\eta}{2}}{\delta\sin q\frac{\eta}{2}}$ (4)

In Eqs (2)–(4), $h$ represents the pitch of the coil, $\delta$ represents the pole distance, $d$ represents the number of slots in each phase of each pole, and $q$ has the same meaning as Eq. (1). These harmonics will change the distribution of the air gap magnetic field of the motor, so that the air gap magnetic field is enriched with harmonics, especially around the switching frequency of the inverter. To sum up, the synthetic magnetomotive force of the stator windings is:

$\displaystyle G=E\cos({\beta-\phi})$ (5)

In Eq. (5), $E$ represents the amplitude of the stator winding magneto motive force, $\beta$ represents the Angle between the stator winding magneto motive force and the rotor magneto motive force, and $\phi$ represents the amplitude of the rotating magneto motive force. The PMSM with external structure can effectively reduce rotor diameter and rotational inertia of the motor. When the permanent magnet is directly bonded to the rotor shaft, low inductance can be obtained, which greatly improves the dynamic performance of the rotor. Permanent magnet synchronous motors with embedded structure embed permanent magnets in the core of the rotor. This structure not only improves the mechanical strength of the rotor but also has a small magnetic circuit air gap, which is more suitable for weak magnetic operation of the motor [10, 11, 12]. Ac permanent magnet synchronous motors can be divided into implicit pole synchronous motors and salient pole synchronous motors according to different rotor assembly methods. Under the condition of frequency conversion power supply, the motor is similar to the sine wave power supply, but there is time harmonic in the stator current. Based on the above description, the steps of obtaining electromagnetic wave and magnetic field data are completed.

2.2 Constructing mathematical model of permanent magnet synchronous motor

The mathematical model of PMSM control object should accurately reflect the static and dynamic characteristics of the controlled system, and the accuracy of the mathematical model is the key to the performance of the control method [13, 14]. At present, vector control strategy is the most widely used control strategy for PMSM. Vector control, also known as field-oriented control, uses frequency converter to match with DC motor in speed regulation range, effectively controls the torque generated by asynchronous motor, and is suitable for AC induction motor and DC brushless motor. The vector control strategy can operate in the whole frequency range and output the rated torque when the motor is at zero speed, so as to achieve the purpose of rapid acceleration and deceleration. The development of vector control strategy depends on precise dynamic mathematical model. Under the action of the electromagnetic torque, the actuator can run in a straight line in the opposite direction to the traveling wave magnetic field. In the stationary coordinate system and the rotating coordinate system, the conversion relation of the stator current is:

$\displaystyle w=\sqrt{\frac{2}{3}}[{\cos({\theta-120^{\circ}})}]\times\frac{% \gamma}{\sqrt{2}}$ (6)

In Eq. (6), $\gamma$ represents the synchronous speed of the rotating motor, and $\theta$ has the same meaning as Eq. (1). Since the input current of a permanent magnet synchronous motor is a three-phase sine wave, the shape of the electromotive force of the motor is also sine. According to the classification of salient pole and salient pole, permanent magnet synchronous motor can be roughly divided into two types, namely salient pole permanent magnet synchronous motor and implicit pole permanent magnet synchronous motor [15]. The salient pole type permanent magnet synchronous motor is selected, and its permanent magnet is bonded on the rotor and appears tile sheet. The air gap magnetic field of the motor makes the stator core produce the electromagnetic force which varies with the position of time and space, and this electromagnetic force causes the vibration of the motor stator system. The vibration of the stator system makes the gas around it pulsate, that is, it produces radiated electromagnetic noise. The above conversion relation also applies to voltage, current and flux equations:

$\displaystyle K=\sqrt{\frac{2}{3}}\left({\varepsilon\cos\theta-\phi\sin\theta+% \frac{1}{\sqrt{2}}}\right)$ (7)

In Eq. (7), $\varepsilon$ represents the sum of stator winding pressure drops, and $\theta$ has the same meaning as Eq. (1). For the static and dynamic rotor eccentricity, winding connection, stator slotting and core saturation of permanent magnet motor, the air gap magnetic field will produce rich harmonics, resulting in alternating electromagnetic force. Its significant advantages are that it is easy to produce and process, has good speed regulation performance, and its straight axis inductance value is the same as the cross axis inductance value. In the adaptive modeling process, the controlled object contains independent internal noise and this noise can be expressed as additive zero-mean noise, which is added to the output of the controlled object. If the adaptive filter has enough weights to match the dynamic characteristics of the controlled object, the impulse response of the controlled object is the Wiener solution of the adaptive filter. Then, because the permanent magnet is bonded to the rotor, its adhesion strength is weakened in the long time high-speed operation, affecting its development. The voltage balance equation of the A-phase stator windings is:

$\displaystyle\frac{\mu}{m}+k_{v}=T$ (8)

In Eq. (8), $\mu$ represents the flux chain of the permanent magnet on the actuator, $m$ represents the motion speed of the actuator, $k$ represents the copper consumption of the armature, and $v$ represents the change rate of the electromagnetic energy storage. Since the core of permanent magnet linear synchronous motor is disconnected, it cannot form a closed magnetic circuit, which will cause edge effect, and the motor parameters will also be affected by saturated magnetic circuit and high temperature [16, 17, 18]. Previous studies show that electromagnetic vibration noise is particularly prominent in the motor vibration noise category. Compared with other noises, it is not easy to control and eliminate, and has a great impact on the vibration noise of the whole motor. Therefore, the following hypothesis is made: the saturation of the core of the PMSM is ignored. For implicit pole permanent magnet synchronous motor, the permanent magnet is loaded in the interior of the rotor, which is before the magnet steel and iron core of ferromagnetic material form the protective pole shoe, it is more suitable for need high speed situations, but also it is because of the role of the pole shoe, cause its direct axis inductance value and quadrature axis inductance values are different. It is assumed that permanent magnet linear synchronous motor has no damping winding and no damping effect, and eddy current and hysteresis loss of permanent magnet linear synchronous motor are ignored [19, 20]. Only the primary fundamental EMF is considered in the analysis, and the back EMF is sine. Based on the above considerations, this paper chooses convex pole PMSM as the actuator, because the control of convex pole PMSM is simple and easy, and the speed range is large enough to meet the task of driving the speed control system. Based on the above description, the mathematical model of PMSM is constructed.

2.3 RBF optimizing the coordinate transformation process

The proposal of RBF neural network is inseparable from biology [21]. In the human brain, local adjustment and mutual coverage of receptive fields are the characteristics of human brain response. Based on this characteristic of receptive fields, Moody and Darken proposed a neural network structure, namely RBF network [22]. The working principle of permanent magnet synchronous motor and synchronous motor is roughly the same, is to input three-phase AC to the motor stator, and ac motor as the implementation of the control method. Several important criteria to evaluate the vibration noise of the motor are: low air gap magnetic field harmonic distortion rate, small radial force amplitude and high force wave order, and avoid the force wave frequency of the electromagnetic dynamic wave close to the natural frequency of the stator system. For permanent magnet synchronous motors, the electromagnetic force generated by air gap magnetic field on the inner surface of stator core is divided into radial component and tangential component. Three-phase permanent magnet synchronous motor stator winding A, B, C by three-phase sinusoidal current, produce the rotation of the synthesis of magnetic potential, sine distribution in space, there is mutual inductance between facies, and in the two-phase system, phase, which are perpendicular to each other, there is no this kind of phenomenon of mutual inductance between windings, which simplifies the equations, usually take the zero sequence current component. So that the same speed regulation characteristics can be obtained with DC machine. The principle of coordinate transformation is that the torque and output electromagnetic power of the motor remain unchanged before and after coordinate transformation, that is, the synthetic magneto-motive force generated after coordinate transformation remains unchanged. Radial component is the fundamental source of stator core vibration and radiation noise. The energy of the motor is transmitted through the magnetic field, so the principle of coordinate transformation is: the magnetic potential generated in different coordinate systems is the same, and the coordinate remains unchanged. Replace the original three-phase current with a new current and voltage vector, and the expression formula is as follows:

$\displaystyle\left[{\begin{array}[]{l}t_{1}\\ t_{2}\\ t_{3}\\ \end{array}}\right]=R\left[{\begin{array}[]{l}t_{a}\\ t_{b}\\ t_{c}\\ \end{array}}\right]$ (9)

In Eq. (9), $t_{1},t_{2},t_{3}$ represents the new current and $t_{a},t_{b},t_{c}$ represents the original current. The external electronic voltage and current should also be sinusoidal wave, generally provided by AC PWM variable voltage converter. Vector control of permanent magnet synchronous motor is essentially to control the output value of the motor to make their values equal to the given value [23]. In order to achieve the same dc motor control effect of torque, further control of the motor speed. Then, the coordinate transformation formula can be obtained through Eq. (10):

$\displaystyle\left\{{\begin{array}[]{l}n_{x}=\frac{2}{3}\left[{-\frac{1}{2}({n% _{y}+n_{z}})}\right]\\ \\ n_{y}=\frac{({t_{b}-t_{c}})}{\sqrt{3}}\\ n_{z}=\frac{1}{3}({n_{x}+n_{y}+n_{z}})\\ \end{array}}\right.$ (10)

In Eq. (10), $x, y, z$ respectively represents the stationary coordinate system, and $n$ represents the polar logarithm of the motor. Therefore, to study the electromagnetic vibration and noise of the motor, we should start with the magnetomotive force and air gap magnetic conductivity of the motor system, and then analyze the established magnetic field and the radial electromagnetic force that has an impact on the vibration and noise. The speed controller adopts the control strategy that the value in the vector control is equal to 0, so it only needs to control the vector control value to realize the control of the electromagnetic torque of the permanent magnet synchronous motor. It is a reduced order, variable reduction and decoupling transformation. 3s/2s transformation and inverse transformation are the basis of realizing asynchronous motor vector control. The duty cycle of the vector synthesized by the voltage is calculated and output to the inverter. Finally, the sine wave current with variable frequency and amplitude is generated as the input current of the permanent magnet synchronous motor to realize the real-time control of the AC speed regulation system and ensure that the system has good static and dynamic control performance. Based on the above description, complete the steps of optimizing the coordinate transformation process.

2.4 Designing adaptive backstepping control mode

Backstepping control is widely used in motor control system design because of its unique structure design process and good ability to deal with uncertainty. Moreover, backstepping control is a recursive design method of nonlinear systems. Its main feature is that it is easy to deal with the uncertainty and unknown parameters in the system. The basic principle of adaptive backstepping control is to eliminate the nonlinear characteristic quantity in the nonlinear system by using the full state feedback quantity in the control method, so that the system input and output have linear relationship characteristics. Finally, the new pseudo linear system is synthesized. In this new system, the inverse system with order integral is realized by feedback method, and a certain compensation control strategy is applied to transform the original nonlinear system into a standard normalized system with linear relation and decoupling. The nonlinearity here refers to a control method that starts from the equation farthest from the control input and moves backward toward the control input without linear constraints. The basic backward design was first applied to deterministic systems and then extended to uncertain structures. The development of backsliding control is briefly summarized: first, the most basic integrator backsliding is studied, and then the system for strict feedback is developed. Adaptive control and regular feedback control and optimal control, is also a kind of control method based on mathematical model, the difference is adaptive control is based on a priori knowledge about the model and the disturbance is less, can be extracted in the process of the operation of the system to keep the information on the model, the model gradually perfected. The structure of the position loop adaptive inverse controller of the system is shown in Fig. 1.

Figure 1.

Adaptive inverse control structure diagram.

As can be seen from Fig. 1, the position loop adopts adaptive inverse controller, and the system adopts disturbance eliminator to eliminate disturbance, realize overall control and disturbance elimination, and make both of them reach the optimal at the same time, which is also the unique advantage of adaptive inverse control. Specifically, the model parameters can be identified continuously according to the input and output data of the object. This process is called online identification of the system. Permanent magnet synchronous motor speed control system is a highly nonlinear system, and general linear control methods are not ideal [24, 25]. In order to solve its control problems, the nonlinear control methods are mainly variable structure control, passive control, etc., but in these nonlinear control design methods, the stability of the system is largely determined by the parameters of the system design, which limits the scope of application. In practical applications, the stability of the system is a prerequisite. With the continuous operation of the system, through online identification, the model will become more and more accurate, more and more close to the reality. As the model continues to improve, it is clear that the control functions integrated based on this model will also continue to improve. Adaptive backsliding control is used in strict feedback nonlinear systems to make the system globally stable and asymptotically tracking. The basic idea of backward control is to deduce the stable control law by constructing Lyapunov function step by step, and finally obtain the ideal controller. In this sense, the control method has certain adaptability. That is, by on-line identification and change of controller parameters, the system gradually ADAPTS to external and internal interference and keeps its own good performance. The backward controller can ensure the stability of the system, realize the global adjustment or tracking of the system, and achieve the desired performance index. However, the backward control is less robust because it depends on the parameters of the control object. Different from other nonlinear control methods, adaptive backsliding control does not depend on solving the value of nonlinear system, and does not even need to analyze the stability of the system. Adaptive backsliding control only cares about the change of system feedback, so its application scope is wider. Feedback linearization originally evolved gradually from differential geometry theory. Some states of the system are selected to form subsystems, lyapunov functions are constructed, and virtual control functions are designed to make subsystems stable. The dynamic inverse is different from the traditional local linearization using Taylor series expansion, which takes into account any cautionary nonlinearity. Therefore, adaptive backsliding control is not only global and accurate, but also applicable to the whole region. According to the virtual control in the previous step, the virtual error variable is designed, the new subsystem is composed of the virtual error variable and the previous subsystem, the lyapunov function of the new subsystem is constructed, and the virtual control is designed to make the new subsystem stable. If the actual control of the system has not been obtained, then return to the first step to continue the design, if the actual control is obtained, then continue to design down. Based on this, the steps of designing adaptive backsliding control mode are completed.

3. Simulation test

3.1 Simulation environment

Matlab/Simulink is used to simulate the system under the condition that the main frequency is 1. According to the backstepping adaptive control of permanent magnet synchronous motor, the number of agent is set, and the parameters under static and dynamic characteristics are set.

3.2 Simulation parameter setting

Taking an 8-pole 48 slot permanent magnet synchronous motor as an example, its power supply frequency is 172 Hz. The effects of slot width, pole arc coefficient, pole slot fit, rotor eccentricity and no-load and responsible conditions on the air gap magnetic field and electromagnetic vibration noise of the motor are calculated and analyzed. The initial speed is 400 R/min. after reaching the stable state, the given sudden change is 600 R/min. In the opposite traction loading mode adopted in this experiment, because the centers on both sides of the coupling cannot be fully aligned, the mechanical angle of the motor will rotate 360, and the resistance will be different at different angles. The higher the speed, the greater the resistance. The main structural parameters of the prototype are shown in Table 1.

Table 1
Main structural parameters of the prototype

Serial number	Name	Parameter
1	Rated power/kw	1.05
2	Rated voltage/V	380
3	Stator outer diameter/mm	122
4	Stator Bore/mm	80
5	Polar logarithm	4
6	Moment of inertia N/ms ${}^{2}$	0.000529
7	Permanent magnet remanence/T	1.16
8	Magnetization length of permanent magnet/mm	3.19

3.3 Simulation test results

On the basis of Table 1, the adaptive backsliding control method of permanent magnet synchronous motor designed this time is tested experimentally. The adaptive backsliding control method of PMSM based on association rules and the adaptive backsliding control method of PMSM based on data mining are selected and compared with the adaptive backsliding control method of PMSM in this paper. The position tracking errors of the three methods were tested under different running time. The smaller the error, the better the control effect. Figures 2 and 3 show the test results.

Figure 2.

Running time 7 s position tracking error (mm).

Figure 3.

Running time 14 s position tracking error (mm).

According to Fig. 2, when the permanent magnet synchronous motor runs for 7 s, the position tracking errors of the adaptive backward control method of the permanent magnet synchronous motor in this paper and the other two methods are 4.549 mm, 17.615 mm, 18.016 mm respectively. This is because the proposed method optimizes the coordinate transformation process by using RBF neural network, simulates the mathematical model of AC motor as a mathematical model similar to that of DC motor, and obtains the same speed regulation characteristics as that of DC motor, thus reducing the position tracking error under short-time operation.

According to Fig. 3, when the permanent magnet synchronous motor runs for 14 s, the mean position tracking errors of the adaptive backward control method of the permanent magnet synchronous motor in this paper and the other two methods are 43.699 mm, 61.592 mm and 62.566 mm, respectively, indicating that the position tracking errors of the adaptive backward control method of the permanent magnet synchronous motor designed this time are smaller. This is because the proposed method builds the mathematical model of permanent magnet synchronous motor through vector control strategy, balances the voltage of A-phase stator winding, reduces the motor running loss, and then reduces the position tracking error under long-term running.

4. Conclusion

This paper expounds the control method of permanent magnet synchronous motor and analyzes its working principle. By using coordinate transformation, the mathematical models of permanent magnet synchronous motor in three different coordinate systems are established, and many factors affecting motor speed are analyzed, which facilitates the adaptive backstepping design of permanent magnet synchronous motor.

The proposed method has high dynamic response capability and good adaptability. RBF neural network can well estimate the position of the motor rotor at high and low speed, thus realizing sensorless control of permanent magnet linear synchronous motor.

The proposed method has good stability and dynamic response capability, and can achieve high adaptive backstepping control accuracy of permanent magnet synchronous motor.

The proposed method has the characteristics of fast response, small overshoot and strong robustness, and can keep low position tracking error in both short-time and long-time operation.

However, due to the limited research time and research conditions, the selection of experimental scope is not wide enough, and the research results of saturation and end effect of permanent magnet linear synchronous motor core still have limitations. Therefore, in the following experimental selection, we can further study the problem of core saturation and end effect, so as to consolidate the research results of adaptive backstepping control of permanent magnet synchronous motor and provide theoretical support for the design of adaptive backstepping control system in the future.

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Adaptive backsliding control method of permanent magnet synchronous motor based on RBF

Abstract

Keywords

1. Introduction

2. Adaptive backsliding control method of permanent magnet synchronous motor based on RBF

2.1 Obtaining electromagnetic wave and magnetic field data

3.1 Simulation environment

3.2 Simulation parameter setting

Table 1 Main structural parameters of the prototype

References

Table 1
Main structural parameters of the prototype