Design of complex vector regulator for Linear Induction Motor

The edge-end effect in the operation of a Linear Induction Motor (LIM) will lead to great changes in motor parameters, and the d and q axes of LIM have cross-coupling terms. The PI regulator usually used in traditional vector control has a high requirement on the accuracy of parameters, and the PI regulator cannot realize the complete axis decoupling control of d and q, which will lead to the decline of the control accuracy. In this paper, the mathematical model of the LIM considering side effects is adopted, and a complex vector regulator is used to replace the traditional PI regulator, which realizes decoupling control of the d-q axes and eliminates the influence of coupling terms. Finally, this paper is verified on dSPACE semi-physical platform. Results show that the decoupling effect of d-q axes using complex vector regulator is due to the PI regulator.


Introduction
In recent years, with the aggravation of urban traffic congestion, urban economic development has been limited to a certain extent.A very important means to solve this problem is to develop urban rail transit.LIM has attracted wide attention because of its advantages of producing linear motion without any intermediate conversion mechanism.LIM used in rail transit have been widely used due to their advantages of small turning radius, strong climbing ability, and low noise generation [1].
However, LIM will be affected by the edge effect during operation, which leads to a change in the LIM.Therefore, the control algorithm of LIM must consider the influence of the edge effect on LIM parameters.Some scholars have derived the influence of the end-effect on LIM parameters and obtained the correction result of the equivalent circuit of LIM [2].
LIM vector control is equivalent to LIM as a DC motor through coordinate transformation to achieve secondary flux decoupling from electromagnetic thrust [3].The PI regulator parameters used in vector control are usually fixed.However, LIM will change its parameters due to the existence of the side effect.With the increase of LIM running speed, LIM parameters will change more and more, which will lead to the decline of the control precision [4].Moreover, the LIM model after coordinate transformation still has coupling terms, which cannot achieve complete decoupling [5].
In this paper, a complex vector regulator is used to replace PI regulator in the traditional Field Oriented Control (FOC).Considering that the parameters of LIM will change with the change of running speed, edge end effect influence is taken into account when deducing the formula.Firstly, the mathematical model of LIM is derived based on complex vector theory, and then a controller is designed to cancel the complex vector poles of LIM, to improve current dynamic LIM performance.After LIM mathematical model is built, it is downloaded to the dSPACE semi-physical platform as the controlled object.Finally, DSP is used as a controller to control the dSPACE semi-physical platform.

Mathematical model of LIM
where sd u and sq u are primary d-q axis voltages, sd i and sq i are primary d-q axis currents, sd ψ and sq ψ are primary d-q axis flux, rd i and rq i are secondary d-q axis currents, rd ψ and rq ψ are secondary d-q axis flux, s R is primary resistance, r R is secondary resistance, s L is primary inductance, r L is secondary inductance, m L is mutual inductance, e ω is the synchronous angular velocity, r ω is the secondary angular velocity of motion, r1 where τ is pole distance.
where M is primary mass, v is primary speed, and L F is the load.Relation between primary velocity and secondary angular velocity is

Complex vector regulator
The traditional current controller is the PI regulator in the synchronous rotation d-q coordinate system.The current excitation component and thrust component in the synchronous rotation d-q coordinate system obtained after the coordinate transformation of the LIM mathematical model are cross-coupled.
With the increase of LIM running speed, the influence of the cross-coupling term will be more and more significant.Therefore, it is necessary to eliminate the effects of coupling terms.
In order to solve these problems, the complex vector current regulator is used to replace the traditional PI regulator.According to complex vector theory, the complex vector mathematical model of LIM and the complex vector poles of the motor were derived.And the complex vector current regulator was used to offset the complex vector poles of the motor to realize the decoupling of the current excitation component and the thrust component [6].The specific derivation process is as follows: According to the voltage equation and flux equation of linear induction motor, the state equation with sd i and sq i , sd ψ and sq ψ as state variables can be solved.It can be calculated from the third and fourth lines of Equation (1).
where s is the differential operator.
When the secondary magnetic field orientation is controlled, secondary q-axis flux can be ignored, and secondary d-axis flux is approximately equal to the secondary flux.
According to the complex vector theory, the following vectors are defined, sdq sd sq u ju   u is the primary voltage vector; sdq sd sq i ji   i is the primary current vector.Therefore, the linear induction motor model can be expressed in complex vector form as The following variables are defined as By substituting Equation (9) into Equation (8), the primary voltage equation of linear induction motor can be obtained In Equation (10), there is no cross-coupling problem between sdq Ri and sdq Lsi .
According to Formula (11), the pole of the LIM can be obtained as To offset poles of LIM, and eliminate the influence of the coupling term, the method of pole-zero cancellation can be adopted in designing the complex vector current regulator, for its transfer function e i ( ) where k is expected bandwidth of current loop.

Simulation and dSPACE semi-physical experiment results
To verify the correctness of the complex vector regulator method used in the paper, LIM model was built by Simulink for simulation.Then, the mathematical model of LIM was downloaded to dSPACE, DSP of TI company (model: TMS320F28335) was used as the controller, and dSPACE as a controlled object of closed-loop control system for validation.Parameters of LIM used in this paper are shown in Table 1.The sampling frequency of the controller is 5 kHz.

Simulation results and analysis
The simulation conditions in Figure 2 and Figure 3 are as follows: LIM starts at 0 s, gradually accelerates to 20 km/h, and gradually accelerates to 50 km/h 10 s later.The velocity reference is given by the ramp function.It is shown from the figure that both PI regulator and complex vector regulator can make LIM run in a wide speed range in vector control.And two methods both can make d axis and q axis current feedback value rapidly following the given values.Simulation conditions of Figure 4 and Figure 5 are as follows: the no-load steady-state running speed of the LIM is 20 km/h before 5 s, and the reference value of q axis current jumps from 0 A to 20 A at 5 s.The current loop in Figure 4 uses a PI regulator and the current loop in Figure 5 uses a complex vector regulator.We can see the q axis current step, using complex vector controller d shaft current changes significantly less than when using PI regulator d shaft current of variations.Simulation results show that d-q axis current decoupling degree is better when a complex vector regulator is used, and better decoupling performance can be obtained.Figure 5. d-q axis current waveform at Q-axis current step (complex vector regulator).Figure 7. Vector control experiment (using complex vector regulator).

Experimental results and analysis
Figure 6 and Figure 7 show the waveforms when PI regulator and complex vector regulator are used respectively.LIM runs at a steady speed of 20 km/h at the beginning, and then gradually accelerates to 50 km/h.It can be seen that both methods can make the linear induction motor run stably in a large speed range.The feedback current of LIM q axis can always follow the given value during operation.The experimental conditions in Figure 8 and Figure 9 are that the reference value of q axis current jumps from 0 A to 20 A when the motor runs steady-state at 20 km/h.The d-axis feedback current in Figure 8 is slightly larger than that in Figure 9.The final steady-state error of both methods is zero.Experimental results and simulation analysis are the same.Therefore, the decoupling performance of the complex vector regulator is better than PI regulator. .d-q axis current feedback when q axis current step (using PI regulator).
Figure 9. d-q axis current feedback during qaxis current step (using complex vector regulator).

Conclusion
To solve the poor decoupling performance problem caused by the coupling term of d-q axis in vector control of LIM, the paper uses the complex vector regulator to replace PI regulator used in the current loop of traditional vector control.Firstly, the mathematical model of LIM is given.complex vector mathematical model of LIM is given by complex vector theory and the model of complex vector regulator is derived by using the method of pole-zero cancellation.Finally, simulation and dSPACE experiment results show that d-q axis decoupling performance of proposed method is superior to vector control, and the correctness of complex vector regulator method is verified.
e sdq jLω i is the current cross-coupling term.E is the back electromotive force term, related to LIM speed and flux, which change much slower than the current change rate, and are generally regarded as disturbances.The model transfer function of the linear induction motor can be obtained without considering the disturbance.

Figure 4 .
Figure 4. d-q axis current waveform at Q-axis current step (PI regulator).

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Figure8.d-q axis current feedback when q axis current step (using PI regulator).
With primary d-q axis current sd