In order to solve the problem of large torque and flux ripple in the traditional direct torque control of permanent magnet synchronous motor, Super-Twisting sliding mode variable structure control strategy is adopted. Under the dl coor-dinate system, the torque and flux controller is designed. In the direct torque control, the problem of speed overshoot and considerable fluctuation during PI control is adopted. The synovial speed controller is designed, and the space vector pulse width modulation technology (SVPWM) is adopted to stabilize the inverter switching frequency. The simulation results demonstrate that the method can effectively reduce the torque and flux linkage and accelerate the system response.
Permanent magnet synchronous motor has the characteristics of small size, lightweight, high power factor and high efficiency, and is widely used in the field of robots and electric vehicles [
A lot of research has been conducted by domestic and foreign scholars on the problems existing in traditional direct torque control. Literature [
In this paper, the super-spiral sliding mode variable structure control is used to design the flux linkage and torque controller. The velocity controller is designed by using the sliding film control method based on the approach law. The system designed in this paper is compared with the traditional direct torque control system. It shows that the method adopted in this paper can effectively reduce the ripple of flux linkage and torque, and accelerate the speed response and anti-disturbance capability.
In order to simplify the analysis, the following assumptions are often made in the mathematical modeling of permanent magnet synchronous motors:
1) Ignore the saturation of the iron core inside the motor;
2) Excluding the eddy current and hysteresis loss when the motor is running;
3) The stator winding current is a three-phase sinusoidal current, and the induced electromotive force of the stator armature winding is also a sine wave [
The stator voltage equation is:
{ u α = L s d i α d t + R s i α − ω r ψ f sin θ r u β = L s d i β d t + R s i β + ω r ψ f cos θ r (1)
where: u α , u β , i α , i β are components of the stator voltage vector and the current vector on the α and β axes;
R s , L s are stator resistance and stator inductance;
ω r is the motor angular velocity;
ψ f is a permanent magnet flux linkage;
θ r is the rotor position angle.
The stator flux linkage equation is [
{ ψ α = ∫ ( u α − R s i α ) d t ψ β = ∫ ( u β − R s i β ) d t (2)
The electromagnetic torque equation is:
T e = 3 2 p ( ψ α i β − ψ β i α ) (3)
The motor motion equation is:
d ω m d t = p J ( T e − T L ) (4)
where: p is the pole number of the motor;
J is the moment of inertia;
TL is the load torque.
The stator voltage equation is:
{ u d = d ψ d d t + R s i d − ω r ψ q u q = d ψ q d t + R s i q + ω r ψ d (5)
The stator flux linkage equation is:
{ ψ d = L d i d + ψ f ψ q = L q i q (6)
The electromagnetic torque equation is [
T e = 3 2 p i q [ i d ( L d − L q ) + ψ f ] (7)
where: u d , u q are the components of the stator voltage on the dq axis;
i d , i q are the components of the stator current on the dq axis;
ψ d , ψ q are the components of the stator flux linkage on the dq axis;
L d , L q are the inductances on the dq axis;
ψ f is the stator flux linkage.
Sliding mode variable structure control has discontinuity, the system structure follows the switching characteristics of the time change, the system parameters are independent of the outside world, the response speed is fast, and it has good robustness. The super-spiral algorithm-based synovial controller can effectively eliminate the pulsation. The formula of the super-helical control algorithm is as follows:
{ u = K P | s | r sgn ( s ) + u 1 d u 1 d t = K I sgn ( s ) (8)
K P and K I are gains.
The sufficient conditions for the stability of the control system are:
{ K P ≥ A M B m K I ≥ 4 A M B m 2 ⋅ B M ( K P + A M ) B m ( K P − A M ) 0 ≤ r ≤ 0.5 (9)
Among them: B m ≤ B ≤ B M , A M ≥ | A | .
And A, B meets:
d 2 y d t 2 = A ( x , t ) + B ( x , t ) d u d x (10)
Sliding mode variable structure control has discontinuity, the system structure follows the switching characteristics of the time change, the system parameters are independent of the outside world, the response speed is fast, and it has good robustness. The super-spiral algorithm-based synovial controller can effectively eliminate the pulsation. The formula of the super-helical control algorithm is as follows [
Magnetic chain controller:
{ u d = K P 1 | S ψ | r sgn ( S ψ ) + u d 1 d u d 1 d t = K I 1 sgn ( S ψ ) (11)
Torque controller:
{ u q = K P 2 | S T e | r sgn ( S T e ) + u q 1 d u q 1 d t = K I 2 sgn ( S T e ) (12)
where: S ψ is the difference between a given flux linkage and the actual flux linkage;
S T e is the difference between the given torque and the actual torque;
K P 1 , K I 1 , K P 2 , K I 2 are gains.
According to the stator flux vector reference system, ψ s = ψ d , the stator flux linkage is derived [
d ψ s d t = u d − R s i d (13)
Secondary derivation of the flux linkage:
d 2 ψ s d t 2 = R s 2 L s i d − p R s ω i q − R s L s u d + u ˙ d (14)
In the above formula, R s , L s , i d , p , ω are bounded values, and therefore the formula (14) satisfies the stable condition of the formula (9).
For the hidden pole permanent magnet synchronous motor, assuming that the stator flux linkage amplitude is constant, the torque is also derived:
d T e d t = 3 2 p ψ s d i q d t (15)
Continue to guide the torque:
d 2 T e d t 2 = 3 2 p ψ f [ − p 2 ψ f J i d i q + p B J i d ω + ( R s 2 L s 2 − p 2 ψ f 2 J L s − p 2 ω 2 ) i q + 2 p R s L s ω i d − p ω L s u s d + ( p ψ f B J L s + p ψ f R s L s 2 ) p ψ f B J L s ω − R s L s 2 u s q + p ψ f T L J L s + u ˙ s q L s ] (16)
It can be seen that the parameters of the second derivative of the torque are bounded values, and the analysis method is the same as the stator flux linkage, which also satisfies the stable condition of Equation (9).
The motor motion equation is:
d ω m d t = p J ( T e − T L ) (17)
The design switching function is:
s = ω m ∗ − ω m (18)
where: ω m ∗ is the given speed of the motor;
ω m is the actual speed of the motor.
According to the concept of the approach law of Gao Weibing [
s ˙ = − ε | s | α sgn ( s ) − k s (19)
Among them: 1 > α > 0 , ε > 0 , k > 0 .
Available from Equation (18):
s ˙ = − ω ˙ m = p J ( T L − T e ) = − ε | s | α sgn ( s ) − k s (20)
Further, the slip film controller output expression is:
T e = J p [ ε | s | α sgn ( s ) + k s ] + T L (21)
Select the Lyapunov function as:
V = 1 2 s 2 (22)
Derivation of Equation (22) yields:
V ˙ = s ˙ s = s [ − ε | s | α sgn ( s ) − k s ] = − ( ε | s | α + 1 + k s 2 ) < 0 (23)
This formula satisfies the stability conditions of Lyapulov, which proves the stability and accessibility of the system.
In this paper, the simulation model is built by matlab/simulink. The system includes multiple subsystem modules such as flux linkage and torque estimation module, coordinate transformation module and speed adjustment module. The improved system model is shown in
The parameters of permanent magnet synchronous motor are set as follows: the stator winding resistance is R s = 1.2 Ω , stator inductance L d = L q = 0.0085 H , rotor flux linkage ψ f = 0.175 Wb , moment of inertia J = 0.0008 kg ⋅ m 2 , pole logarithm p = 4, the system friction coefficient B = 0. The system simulation parameters are: given speed is ω m ∗ = 600 rad / min , given flux linkage is ψ * = 0.3 Wb , load torque T L = 1.5 N ⋅ m is applied in 0.2 seconds. Simulation time is 0.4 s. The simulation results are shown in Figures 2-7.
It can be seen from Figures 2-7 that when the motor speed increases from 0 to 600 r/min, compared with
In this paper, based on the sliding mode theory, the flux linkage and torque controller and the sliding mode variable speed controller are designed. Compared with traditional direct torque control simulation, the method used in this paper can effectively reduce the flux linkage and torque ripple. The system has strong anti-interference and robustness.
The authors declare no conflicts of interest regarding the publication of this paper.
Luo, Y.F., Tan, G.Q. and Su, C. (2019) Direct Torque Control of Permanent Magnet Synchronous Motor Based on Sliding Mode Variable Structure. Open Access Library Journal, 6: e5758. https://doi.org/10.4236/oalib.1105758