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Vector control of Brushless DC motor (4): sensorless control based on sliding mode observer

2022-06-13 06:23:00 【Forster-C】

In more and more application scenarios , Brushless DC motor began to adopt the control mode without position sensor . Brushless DC motor operates at medium and high speed , The back EMF signal can be used to estimate the rotor position , There is more than one way to implement it , The sliding mode observer method is widely used .

In this paper, the basic principles of this method are sorted out , This paper introduces the in MATLAB/Simulink Modeling and simulation process in , Finally, a complete model file is attached .

One 、 The basic principle

Sliding mode observer （ Hereinafter referred to as" SMO） It is a state observer based on sliding mode variable structure control method . The realization process of rotor position estimation for brushless DC motor is as follows .

Suppose a brushless DC motor with vector control d Shaft inductance and q The shaft inductance is equal （ about PMSM This assumption holds ）, namely Ld=Lq=L, Can write brushless DC motor in α-β Voltage equation in coordinate system ：

$\frac{di_{\alpha }}{dt}=-\frac{R}{L}i_{\alpha }+\frac{1}{L}(u_{\alpha }-e_{\alpha })$

$\frac{di_{\beta }}{dt}=-\frac{R}{L}i_{\beta }+\frac{1}{L}(u_{\beta }-e_{\beta })$

In style ,uα、uβ Respectively α-β In a coordinate system α Axis 、β Axis voltage component ,iα、iβ Is the current component ,eα、eβ Is the back EMF , The expression is

$e_{\alpha }=-\psi _{f}\omega _{e}sin(\theta _{e})$

$e_{\beta }=\psi _{f}\omega _{e}cos(\theta _{e})$

In style ,ψf Magnitude of the flux linkage generated for the magnetic field of the rotor permanent magnet ,ωe Is the electrical angular velocity of the rotor ,θe Is the electrical angle of the rotor .

According to the expression of back EMF , Back EMF eα、eβ Contains rotor position information . According to the voltage equation , utilize SMO Get an estimate of the back EMF , Then the estimated value of rotor position can be obtained from the expression of back EMF , This is it. SMO The whole idea of rotor position estimation of Brushless DC motor .

First , The sliding mode observer is designed according to the voltage equation ：

$\frac{d\hat{i}_{\alpha }}{dt}=-\frac{R}{L}\hat{i}_{\alpha }+\frac{1}{L}(u_{\alpha }-z_{\alpha })$

$\frac{d\hat{i}_{\beta }}{dt}=-\frac{R}{L}\hat{i}_{\beta }+\frac{1}{L}(u_{\beta }-z_{\beta })$

In style , $\hat{i}_{\alpha }$ 、 $\hat{i}_{\beta }$ Is the estimated value of the current component ,zα、zβ Is the sliding mode control function .

Select sliding mode switching surface ：

$s_{\alpha }=\hat{i}_{\alpha }-i_{\alpha }$

$s_{\beta }=\hat{i}_{\beta }-i_{\beta }$

Constructing sliding mode control function ：

$z_{\alpha }=k\cdot sgn(\hat{i}_{\alpha }-i_{\alpha })$

$z_{\beta }=k\cdot sgn(\hat{i}_{\beta }-i_{\beta })$

k Is the coefficient of a symbolic function ,sgn It's a symbolic function .

According to the voltage equation and the sliding mode observer equation ：

$\frac{d\tilde{i}_{\alpha }}{dt}=-\frac{R}{L}\tilde{i}_{\alpha }+\frac{1}{L}(e_{\alpha }-z_{\alpha })$

$\frac{d\tilde{i}_{\beta }}{dt}=-\frac{R}{L}\tilde{i}_{\beta }+\frac{1}{L}(e_{\beta }-z_{\beta })$

In style ：

$\tilde{i}_{\alpha }=\hat{i}_{\alpha }-i_{\alpha }$

$\tilde{i}_{\beta }=\hat{i}_{\beta }-i_{\beta }$

If k Select the appropriate value , When SMO When entering the sliding surface , Yes

$\tilde{i}_{\alpha }=0,\frac{d\tilde{i}_{\alpha }}{dt}=0$

$\tilde{i}_{\beta }=0,\frac{d\tilde{i}_{\beta }}{dt}=0$

Then we can get the following relationship ：

$e_{\alpha }=z_{\alpha }$

$e_{\beta }=z_{\beta }$

namely zα、zβ Infinite approximation eα、eβ, Thus, the back EMF value can be estimated . After proper filtering （ Because of what you get zα、zβ High frequency switching signal ）, The estimated value of the back EMF can be obtained ：

$\hat{e}_{\alpha }=\frac{\omega _{c}}{s+\omega _{c}}z_{\alpha }$

$\hat{e}_{\beta }=\frac{\omega _{c}}{s+\omega _{c}}z_{\beta }$

Get the estimated value of rotor electric angle ：

$\hat{\theta }=arctan(-\frac{\hat{e}_{\alpha }}{\hat{e}_{\beta }})$

In addition, the delay of the filter is compensated ：

$\Delta \theta =arctan(\frac{\omega _{c}}{\omega _{e}})$

Finally, the estimated value of rotor electric angle and speed can be obtained ：

$\hat{\theta }_{e}=\hat{\theta }+\Delta \theta$

$\hat{\omega }=\frac{d\hat{\theta }_{e}}{dt}$

That's all SMO The overall process of , You can use pictures 1 schematic . The process is sorted out , Then we can MATLAB/Simulink Build the corresponding model in .

chart 1 SMO Flow chart

Two 、 Modeling process

The main circuit topology is shown in the figure 3 Shown , The DC power supply drives the brushless DC motor through a three-phase full bridge converter , use FOC Control algorithm . The load torque of Brushless DC motor can be adjusted dynamically .

chart 2 Main circuit topology

The control circuit is shown in the figure 3 Shown , Use traditional FOC Control logic ,q The outer ring of the shaft controls the speed , The inner loop controls the current ,d Shaft for id=0 control . The sensor signal is directly used in the low-speed stage , Perform sensory control ; The high-speed phase is switched to SMO, Pictured 4 Shown .SMO Use the logic described above , Estimate the motor position .