A current prediction correction control algorithm for permanent magnet synchronous linear motor

When a permanent magnet synchronous linear motor (PMSLM) is used in a direct drive system, the current loop of drive control system is required to have a high response speed in order to achieve a high dynamic response of the system. With the traditional PI control strategy, the bandwidth of the system’s current loop is limited. In this paper, a predictive correction control algorithm is used in the current loop of permanent magnet synchronous linear motor drive control system. The algorithm is to predict the instruction voltage vector required by the next switching cycle so that the target current of the control system can reach the instruction reference current value within the next switching cycle. It is used to improve the response speed of the system current loop and realize the precise control of the single PWM cycle of the system current loop.


Introduction
The PMSLM directly uses electric energy to produce linear motion, which makes the linear actuator system simpler in structure and lighter in weight.Direct drive using a linear motor can eliminate the influence of elastic deformation and improve the motion stability of a linear actuator system.As there is no transmission intermediate link in the direct drive system, thrust fluctuation will directly act on the object and directly affect the control performance of the system.Therefore, the dynamic system often needs to adopt a high-dynamic response servo control strategy to ensure that the motor system has a strong resistance to load disturbance.At the same time, due to the strong coupling and nonlinear between the electromagnetic and the PMSLM, in order to meet the performance requirements of the servo control system, the control strategy must be used with strong robustness.The control strategies studied by many scholars in the field of PMSLM control mainly include the traditional PID control method [1][2], robust control method [3], parameter adaptive control [4], sliding mode control, variable structure control [5], and so on [6].These control methods either have poor control effect or require high precision of model parameters, which is relatively difficult to implement.
In this paper, according to the control characteristics of PMSLM, in order to improve the response speed of the current loop of the direct drive system, the current prediction correction control algorithm under the modern control theory system is selected to design the system's current loop.This algorithm does not require any simplification of the model.Its design parameters of high usability and control performance particularly good, can achieve single PWM cycle modulation.Firstly, the mathematical model of PMSLM is derived in natural ABC coordinate system and rotating dq0 coordinate system.
Then according to the analysis of the current control algorithm of the prediction correction, the current prediction correction controller is designed.Finally, the simulation model of the system's current controller is established, and the feasibility of this algorithm is analyzed by simulation.

Mathematical model of permanent magnet synchronous linear motor
Because PMSLM has the basic characteristics of a rotary permanent magnet synchronous motor (PMSM) in a direct drive system, it is a high order, nonlinear, strong coupling multi-variable drive control system.So, its actual dynamic mathematical model can be expressed as a complex high-order nonlinear differential system.In practice, it is often simplified to a simple system of second-order differential equations through a series of assumptions.
The mathematical model of PMSLM in ABC three-phase stationary coordinate system is expressed as follows: where It can be seen that there is a serious coupling between the three phases of the PMLSM under the ABC three-phase static coordinate system.Therefore, Clarke transform and PARK transform are often used to transform it to dq0 rotating coordinate system to realize the basic decoupling of PMLSM electrical parameters.The Clarke transformation matrix under the flux linkage invariance principle is expressed as follows: 2 4 1 cos( ) cos( ) 2 3 3 2 4 3 0 sin( ) sin( ) 3 3 where i α and i β are the current components in stationary Cartesian coordinates; i u , i v and i w are actual currents of three-phase winding.
The Park transformation matrix is expressed as follows: cos sin sin cos where i d and i q are the direct and ac-axis current; θ is the angle of the mover magnetic field.Thus, the mathematics of PMLSM in dq0 coordinate system is obtained as follows: where u d and u q are the direct and ac-axis voltage; L d and L q are the direct and ac-axis inductance; R a is phase resistance; ψ f is the permanent magnet flux; v m is the mechanical speed of the mover.
The magnetic co-energy of a PMLSM can be defined as follows: where, i ABC is the three-phase current vector of PMSLM.Ψ ABC is the flux vector of PMSLM; W m (Ψ ABC ,s) is the magnetic energy storage of PMSLM.
The differential of magnetic co-energy of PMLSM can be expressed as follows: ) Because the independent variable current i ABC and position s in the PMLSM are basically independent of each other, the flux vector and electromagnetic thrust can be expressed as follows: According to the above mathematical equations, the mathematical model control diagram of PMLSM can be obtained, as shown in Figure 1.Where F l is the load resistance, M is the mass of the actuator, and B is the viscous friction coefficient.According to the analysis of Figure 1, although the system i d and i q are decoupled by the change of Park transformation, there is still coupling between voltage u d of d-axis and voltage u q of q-axis.The product of velocity v m and dq-axis current appears in the model, which makes it impossible to simplify the system using the traditional linear control theory.

Current prediction and correction control algorithm for PMSLM
The traditional PMSLM control system usually adopts a simple PI control algorithm [7].In the dynamic process of the current loop, the velocity v m is constant.It only takes into account the PWM modulated wave update delay and does not consider the voltage coupling between dq-axes.The bandwidth of the system may be further affected when the delay of current sampling and the inverter delay are taken into account.In order to improve the response speed of the current loop of the direct drive system, the current prediction correction control under the modern control theory system is chosen here to design the current loop of the system.
By further analyzing the mathematical model of PMSLM, the state space expression of PMSLM is obtained as follows: The control system of PMSLM is controlled by digital controller.Its control period is small enough that it can discretize the state space expression of the system.The discretization equation can be expressed as follows: The state space expression of the above system can be expressed as follows: In the direct predictive control algorithm, the current instruction signal i*(k) is used to replace the current value i(k+1) at the moment k+1 in the state space expression.
The dq-axial voltage to be applied to the stator can be obtained by calculation, and the condition that the actual current value at time of k+1 reaches the command current value at time of k is satisfied.The voltage equation of the dq-axis is as follows: ( ) The system model of PMSM direct drive control system with predictive correction control algorithm is shown in Figure 2.Under ideal conditions, the system adjusts ud and uq on the stator windings through the current prediction correction control algorithm.It can make the system current in the next PWM cycle to reach the command value, and then significantly improve the response speed of the system and current loop limit bandwidth.

Simulation of current prediction and correction control algorithm for PMSLM
The current control performance of PMSM direct drive control system under the current prediction correction algorithm is simulated and analyzed.The structure of the system under the current predictive correction control algorithm is shown in Figure 3.The prototype motor's basic parameters are shown in Table 1.When the command current of the qaxis is given as rated current 25A and the command signal of the d-axis is 0A.The response waveform of the PMSLM current is shown in Figures 4-7.q-axis current

Current of instruction Following the current
It can be seen from Figures 4-7 that the response performance of the predictive control current loop is much better than that of the current loop under PI control.As can be seen from the local magnification of the current waveform, the response time of the q-axis current under the predictive control is 0.4ms, the maximum current fluctuation is 0.8A, the steady-state current ripple is 3.2%, and the steady-state error of the current control is 0.5%.
At the same time, the following ability of the system under the current predictive correction control algorithm to the dynamic current command signal is verified.Given by the simulation system, the current command signal of the q-axis is an AC sinusoidal signal with a peak-to-peak value of 25A and a frequency of 500Hz, and the command signal of the d axis is zero.The response waveforms of the PMSLM current is shown in Figures 8-11.It can be seen from Figure 8 that under the predictive correction control of the current loop, the qaxis current of the linear motor system can well track the command current.The switching frequency of the control system is 10kHz, so the digital controller samples the system current every 10 -4 s.Choosing to observe the current entering the digital sampling, the discrete simulation waveform of the q-axis command signal and the response current of the system are shown in Figure 10.It proves that the actual q-axis current can follow the q-axis command current signal well by using predictive control.It can be seen from the local magnification that basically the actual current signal can track the upper command current signal in the next switching cycle.The system can realize the current signal adjustment in one PWM cycle.It is proved that the current prediction correction algorithm can reduce the response time of the current loop, greatly accelerate the dynamic response ability of the current loop, and improve the regulation ability of the system's current loop.It can be seen from Figure 9 that under the current prediction algorithm, the d-axis current loop of the linear motor control system also has very good control performance, and the steady-state error is only 0.09A, which can meet the needs of the actual system control operation.

Conclusions
In order to achieve a high dynamic response of PMSLM direct drive system, the current prediction correction algorithm designed by modern control theory is used to design the current loop of the system.The current prediction correction algorithm is to predict the instruction voltage vector required by the next switching cycle so that the target current of the control system can reach the instruction reference current value within the next switching cycle.In this paper, the mathematical model of PMSLM is derived, the current prediction correction control algorithm is studied, and the current prediction correction controller is designed.The simulation model of direct drive control system of PMSLM was established for system simulation.Simulation results show that the control idea is more direct and clearer.The predictive correction algorithm does not simplify the model by using various assumptions the design process, like the traditional control method, and the designed parameters have high usability.The analysis results show that the predictive correction control algorithm can effectively improve the current dynamic response ability of the system compared with the traditional PI control algorithm.The system can also realize single PWM period modulation, which improves the dynamic adjustment ability of the current loop.

Figure 1 .
Figure 1.Current loop control strategy of PMSLM control system.

Figure 2 .
Figure 2. System model of PMSM based on predictive correction control algorithm.

Figure 3 .
Figure 3. Structure drawing of PMSM control system using predictive correction control algorithm.

Figure 8 .
Figure 8. Waveform q-axis current response under the current predictive correction control algorithm.

Figure 9 .
Figure 9. Waveform d-axis current response under the current predictive correction control algorithm.
, u A , u B and u C are three-phase stator voltages of A, B, and C; i A , i B ,and i C are three-phase stator currents of A, B, and C; e A , e B , and e C are rotational electromotive force induced by permanent magnet armature windings of A, B and C; R a is the stator resistance of each phase; L A , L B , and L C are the stator winding self-inductance; M AB , M BC , and M CA are the stator winding mutual inductance; P differential operator.