Abstract: Based on the analysis of the mathematical model of a permanent magnet synchronous motor (PMSM), a speed regulator for the control system was designed using the sliding mode variable structure control method. The current loop employs current hysteresis tracking PWM, a widely used technique in engineering, to achieve precise control of the stator current. Furthermore, a model of the entire control system was established using MATLAB. Simulation results verified the feasibility of the sliding mode variable structure control method, providing a new approach for future research. Keywords: Permanent Magnet Synchronous Motor Sliding Mode Variable Structure Control Current Hysteresis Tracking PWM MATLAB [b][align=center]Simulation Research of Sliding Mode Control Based on Permanent Magnet Synchronous Motor Zhang Yongsong Shu Zhibing[/align][/b] Abstract: Through the analysis of Permanent Magnet Synchronous Motor (PMSM)'s mathematical model, the Sliding Mode Control scheme in the design of speed regulator of the control system is introduced in this paper. In order to track the stator's current precisely, we design the current regulator using Current Hysteresis Band Pulse Width Modulation (CHBPWM). On this basis, we use MATLAB to establish the model of the whole system. The simulation results prove that this kind of control scheme is feasible and this paper provides a new way of thinking for future research. Key words: Permanent Magnet Synchronous Motor (PMSM), Sliding Mode Control (SMC), Current Hysteresis Band Pulse Width Modulation (CHBPWM), MATLAB 0 Introduction In recent years, with the rapid development of microelectronics technology, power electronics technology and rare earth permanent magnet materials, permanent magnet synchronous motors (PMSMs) have been widely used in the industrial field. Compared with other types of motors, permanent magnet synchronous motors have advantages such as small size, simple structure, large output torque and high efficiency. Therefore, the study of control methods for permanent magnet synchronous motors has become a hot topic. In some situations where torque requirements are high, traditional PID control methods are difficult to meet the system requirements. For example, in robot drive devices, the moment of inertia will change with the movement of the robot arm and the change of load [1]. The introduction of sliding mode variable structure control (SMC) has solved these problems well. It can force the system state to slide according to the pre-designed switching surface, so that the system is basically unaffected by parameter changes and external disturbances, and improves the accuracy and robustness of the system. 1 Mathematical Model of Permanent Magnet Synchronous Motor Taking a three-phase star-type stator winding connection mode permanent magnet synchronous motor as an example. The following assumptions are made without affecting the control performance: (1) The saturation of the motor core is ignored, and eddy current and hysteresis losses are not considered. (2) Ignore the effects of tooth cogging, commutation process and armature reaction. (3) The three-phase windings are completely symmetrical, and the magnetic field of the permanent magnet is sinusoidally distributed around the air gap. (4) The armature winding is uniformly and continuously distributed on the inner surface of the stator. The main function of the three-phase stator AC is to generate a rotating magnetic field, so it can be equivalent to a two-phase system. In the permanent magnet synchronous motor, a reference coordinate fixed to the rotor is established, the magnetic pole axis is taken as the d axis, and the electrical angle 90 degrees ahead in the direction of rotation is taken as the q axis. The axis of the a-phase winding is taken as the reference axis, and the electrical angle between the d axis and the reference axis is θ[2]. The relationship between the three-phase abc coordinates and the dq synchronous rotating coordinates is shown in Figure 1. [align=center] Figure 1 Rotating coordinate diagram of permanent magnet synchronous motor dq[/align] From this, the voltage equation and electromagnetic torque equation of permanent magnet synchronous motor in the dq rotating coordinate system are obtained as follows: The mechanical motion equation is as follows: In the above equations (1) to (4): represents the equivalent voltage of the stator on the dq axis; represents the equivalent inductance of the stator on the dq axis; represents the equivalent flux linkage of the rotor magnetic field; R[sub]s[/sub] represents the stator resistance; T[sub]e[/sub] represents the electromagnetic torque; T[sub]L[/sub] represents the load torque; B represents the viscous friction coefficient; P[sub]n[/sub] represents the number of pole pairs of the motor; represents the mechanical angular velocity of the rotor; represents the electric angular velocity of the rotor; represents the rotational inertia of the motor rotor. Since the permanent magnet synchronous motor is a nonlinear, multivariable, strongly coupled time-varying system, it is necessary to establish its decoupling state equation. Taking the convex-mounted permanent magnet synchronous motor as the object, the state equation of the permanent magnet synchronous motor can be obtained by the vector control method, that is: Assuming that under zero initial conditions, the influence of the viscous friction coefficient is ignored, the block diagram of the permanent magnet synchronous motor system shown in Figure 2 can be obtained. [align=center] Figure 2 Block diagram of permanent magnet synchronous motor system[/align] 2 Sliding mode variable structure design method Sliding mode variable structure control is a control strategy in variable structure control. The fundamental difference between this control strategy and the conventional control strategy is the discontinuity of control, that is, a switching characteristic that makes the system "structure" change at any time. This control characteristic can force the system to make small-amplitude, high-frequency up and down movements along a specified state trajectory under certain conditions, that is, "sliding mode". The system in the sliding mode has the advantages of fast response, strong robustness and simple physical implementation[3]. Applying sliding mode variable structure control to permanent magnet synchronous motor control has become a research hotspot at home and abroad. From the perspective of control strategy, sliding mode variable structure can be combined with vector control or with direct torque control. This paper applies the former, using the sliding mode variable structure strategy to design the speed loop of the system, while the current loop still adopts the currently widely used current hysteresis control. The basic requirements of variable structure control are: (1) Existence: that is, select the sliding mode function so that the motion of the control system on the switching surface is gradually stable and the dynamic quality is good. (2) Reachability: that is, determine the control action so that all motion trajectories reach the switching surface in a finite time. The state trajectory of the second-order system is shown in Figure 3. [align=center] Figure 3 State trajectory of the second-order system[/align] Let the state description of the second-order system be: Then the parameters of the sliding mode variable structure regulator of the second-order system should satisfy: 3 Simulation of the sliding mode variable structure control system of permanent magnet synchronous motor 3.1 Structure of the sliding mode variable structure control system of permanent magnet synchronous motor The structure of the sliding mode variable structure control system of permanent magnet synchronous motor in this paper is shown in Figure 4. It can be seen from the figure that the whole system is mainly composed of a speed sliding mode variable structure controller, a dq-abc coordinate transformation unit, a current hysteresis controller, a PWM inverter and the permanent magnet synchronous motor body. It can be seen that this is a typical double-loop system, with the inner loop composed of the current hysteresis controller and the PWM inverter bridge. The current hysteresis controller can accurately track the reference current and is a widely used control method in engineering. The outer loop of the system uses a speed sliding mode variable structure controller to control the rotational speed. Since a vector control method for the motor is adopted, a dq-abc coordinate transformation unit is required. [align=center] Figure 4 Block diagram of the permanent magnet synchronous motor sliding mode variable structure system[/align] 3.2 Simulation Model The sliding mode variable structure control system model of the permanent magnet synchronous motor can be built using the Simulink toolbox in MATLAB 7.1, as shown in Figure 5. A modular design method was adopted for modeling; the built modules can be replaced at any time to verify the feasibility of other control methods. The modeling methods for each module are described below. [align=center] Figure 5 Model of sliding mode variable structure control system for permanent magnet synchronous motor[/align] 3.2.1 Speed ​​sliding mode variable structure controller module Using the design method of the sliding mode variable structure controller given above, the speed regulator of the system can be designed accordingly. Let the output of the state variable regulator, i.e., the current setpoint, be , and ignore viscous friction to obtain the mathematical model of the system in phase space as: Considering the system speed limitation, the sliding mode switching function is taken as s=cx[sub]1[/sub]+x[sub]2[/sub], where c is a constant. Let the output of the sliding mode variable structure controller be: Let , and then we can obtain that the parameters of the speed sliding mode variable structure regulator should satisfy: In order to reduce the jitter of the sliding mode control in the design, an integral element can be added to the output of the regulator to convert the switching signal output by the sliding mode variable structure controller into an average torque command signal. Then, the model of the speed sliding mode variable structure controller can be established as shown in Figure 6: [align=center] Figure 6 Speed ​​sliding mode controller module[/align] 3.2.2 dq-abc The coordinate transformation module employs vector control to reduce the order and decouple the nonlinear, multivariable, and strongly coupled permanent magnet synchronous motor system. The basic idea of ​​vector control is to transform the stator AC current in the three-phase stationary coordinate system into an equivalent AC current in the two-phase stationary coordinate system through a three-phase/two-phase transformation (Clark transformation), and then into an equivalent current in the two-phase rotating coordinate system through a rotor field-oriented rotational transformation (Parker transformation). When using control, the magnitude of i[sub]q[/sub] directly determines the electromagnetic torque of the motor. The reverse process of the above transformation is the dq-abc coordinate transformation module used in this system. That is, the dq-abc transformation is a transformation that converts the current, voltage, and flux linkage on the rotating axis (rotor) to the fixed axis (stator). A set of isolated zero-sequence systems needs to be added during the transformation. The transformation formula is given below: The dq-abc coordinate transformation module is then established as shown in Figure 7. Due to the use of control, the input of i[sub]d[/sub] is always 0, and the input of is also always 0. [align=center] Figure 7 dq-abc coordinate transformation module[/align] 3.2.3 The current hysteresis controller module generates PWM waveforms mainly through three methods: calculation method, modulation method and tracking method[4]. The current hysteresis controller used in this paper belongs to the third method. It implements closed-loop control of the current to ensure its sinusoidal waveform. The specific implementation method is to use the desired output current waveform as the command signal and the actual current waveform as the feedback signal. The on/off state of each power switching device in the inverter circuit is determined by comparing the instantaneous values ​​of the two, so that the actual output tracks the change of the command signal. [align=center] Figure 8 Current hysteresis controller module[/align] As shown in Figure 8, the input is the three-phase reference current represented by a vector. The phase difference between the three-phase currents is 120 degrees, as well as the three-phase current fed back from the motor side. The loop width of the hysteresis comparator is 2h. When the actual current is lower than the reference current and the deviation is greater than h, the corresponding positive phase is turned on and the negative phase is turned off[5]. When the actual current is higher than the reference current and the deviation is greater than h, the corresponding positive phase is turned off and the negative phase is turned on. Therefore, the choice of hysteresis width is very important. When the hysteresis width 2h is larger, the switching frequency of the power device can be reduced, but the current distortion is greater and the harmonic components are higher. If the hysteresis width is too small, the current waveform is better but the switching frequency is increased [6] [7]. In the simulation, we can ignore the dead time of the power switch, that is, we assume that the “on” and “off” of the devices on the same bridge arm are completed instantaneously. Therefore, there is still a certain gap between the simulation system and the actual system. 3.2.4 Inverter Module The general inverter module provided by SimPowerSystem4.1.1 in MATLAB7.1 is used for modeling. Three pairs of IGBT power devices are used, and the freewheeling diodes are connected in reverse parallel. The current hysteresis controller controls the conduction and turn-off of the switching devices, thereby generating three-phase terminal voltage output. The specific module is shown in Figure 9. [align=center] Figure 9 Inverter Module[/align] In this model, the m terminal of the IGBT model can output the current and voltage of the IGBT model, which is convenient for detecting the power consumption of the IGBT. Here, the signal is terminated by the Terminator. 3.2.5 The motor module uses the Permanent Magnet Synchronous Machine from SimPowerSystem/Machines as the permanent magnet synchronous motor module. This motor model is encapsulated based on the direct-axis and quadrature-axis flux linkage theory. We can set its parameters. The A, B, and C input terminals are used to connect to the inverter module, the Tm input terminal is used to connect to the load, and the m output terminal is used to output various indicators of the motor during operation, such as: three-phase current, dq voltage, mechanical angular velocity, electromagnetic torque Te, rotor angle, etc. During the simulation, the permanent magnet synchronous motor parameters were set as follows: permanent magnet flux 0.175Wb, number of pole pairs 4, moment of inertia, stator resistance 4Ω, stator inductance 7mH, and viscous friction coefficient 0. A Bus Selector module is added to the m output terminal of the motor to select the motor parameters to be extracted, specifically including: the three-phase current signal of the stator, the electromagnetic torque signal, the rotor mechanical angular velocity signal, and the rotor angle signal. A step input signal is added to the Tm input terminal to simulate the load conditions of the motor at different times. 3.3 Simulation and Result Analysis In Simulink, the simulation time was set to 0.18s. The ode45 variable step size algorithm was selected. ode45 is based on the explicit fourth and fifth order Runge-Kutta method and the Dormand-Prince formula, and this algorithm is effective for most systems. The system started with a load of 3 N·m and a given angular velocity of 600 rad/s. At 0.04 seconds, the load suddenly increased to 10 N·m. Using the Scope (oscilloscope) module, various parameters of the motor operation were measured, and the waveforms are shown in Figures 10-12. From the simulation waveforms, it can be seen that when the speed is given as 600 rad/s, the speed follows the given value in about 0.02s. When the load suddenly increases to 10 N·m at 0.04s, the speed experiences a pulsation with a maximum amplitude of 5 rad/s. After an adjustment time of about 0.01s, the speed again follows the given value. The waveform of the stator three-phase current is basically sinusoidal with some distortion. The current tracking performance can be improved by reducing the current hysteresis width (2h) or by using other PWM control methods. The amplitude of the three-phase current shows a significant jump after a sudden load change, which is in line with expectations. The torque waveform shows significant jumps during load changes, mainly due to frequent switching of the current hysteresis controller, but torque tracking is basically achieved. [align=center] Figure 10 Electromagnetic Torque Waveform[/align] [align=center] Figure 11 Speed ​​Waveform Figure 12 Three-Phase Current Waveform[/align] 4 Conclusion Based on the analysis of the mathematical model of the permanent magnet synchronous motor, this paper designs the system's speed regulator using the sliding mode variable structure control method and employs a current hysteresis controller to achieve tracking control of the three-phase sinusoidal current. Then, the entire system model is built using MATLAB software, and simulation results prove the feasibility of this control method. Due to the modular modeling method, the system control scheme can be flexibly replaced. For example, the inner current loop can also use the sliding mode variable structure control method. Therefore, this system has significant room for improvement, laying the foundation for future research.