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Proceedings of the 2nd International Conference on Intelligent Systems and Image Processing 2014
Position Sensorless Control of PMSM Based on a Sliding Mode Observer
Zhang zhenga, , Narishaa, *, Wang xudonga
aHarbin University of Science and Technology, No.52 Xufu Road, Harbin 150080, China
The current state equation of surface mounted PMSM
in the synchronous rotating coordinate (d-q coordinates) of:
dn rdd
qqq n f
n r r
10 0
10
R ip ui L iui pR
pLL L
L
(1)
In the equation: id, iq, ud, uq are currents and
voltages of synchronous rotating coordinate system; R, L
are the stator resistance and inductance of motor; Pn, ψf
are pole pairs number of motor and the rotor flux, ωr are mechanical angular velocity of the motor.
In the stationary-phase permanent magnet synchronous
motor coordinates, coordinates and static two-phase model
of the two-phase synchronous rotating coordinate system is
shown in Figure 1.
A
B
C
U
U
dq
outU
r
Fig. 1. Diagram coordinates model of PMSM.
The equation (1) is converted to the current state of
the equation of stationary coordinate (α-β coordinates):
10 0
=1
0 0
α αα
β ββ
Ri ui L Li ui R
L L
α
β
e
e (2)
Where:
f e
f e
sin
cosα
β
e
e
(3)
In the equation: iα, iβ, uα, uβ are currents and voltages
of the stationary coordinate system; eα, eβ are electromotive
force of the stationary coordinate system; ωe, θ are rotor
electrical angular velocity and electrical angle of the motor.
3. Design of Control System Based On Sliding Mode Observer
By equation (3) shows the back EMF contains a rotor position signal, the observer can be extracted. The sliding mode observer which is designed in this article puts the stator current under static coordinates system as input of observer, through the observation of the motor back EMF, to extract rotor speed and position information of the measurements.
According to equation (2) given a current state
equation of permanent magnet synchronous motor under
stationary coordinate system, based on the theory of sliding
mode variable structure control, we can construct
the current sliding mode observer equation:
1ˆ
ˆ ˆsidi U lR
i gn i idt L L L
(4)
1ˆ
ˆ ˆsidi U lR
i gn idt L L L i (5)
Then, we can get current error equations as follows:
1 sidi e lR
i gdt L L L ni (6)
1 sidi e lR
i gdt L L L ni (7)
The key of sliding mode variable structure control
design is to control the function u(x) and design of
switching surface s(x), here we choose constant switch
control function u=uosgn(s(x)) as control function, uo is
taken as –l1, to ensure that the condition SS=0 of sliding
mode reaching is established, the value of l1 is related to the
stability of the system, here's an analysis of the range of the
l1.
By the previous conclusions, we have:
21
21
1, 0
1, 0
Ri e l i i
L Li iR
i e l i iL L
(8)
Similarly, we have:
21
21
1, 0
1, 0
Ri e l i i
L Li iR
i e l i iL L
(9)
Arrival condition is:
0
didididt
ss i i i idi dt dt
dt
(10)
That is to satisfy the 0di
idt
and 0di
idt
.
Visible only when 1 max( , )l e e , the above
conditions are met, so that we can guarantee the
stability of the error equation. In practice, l1can't take too
much, otherwise it will increase the chattering noise,
causing unnecessary estimation error.
Switching surface s(x) only to select current error
value, namely ,S x i i , visible when the sliding mode
motion occurs, as 0S x and , the equivalent 0S x
367
Proceedings of the 2nd International Conference on Intelligent Systems and Image Processing 2014
Fig.5. The sensorless control system simulation model based on the sliding mode observer with phase-locked loop
Fig.6. The SMO module
Fig.7. The PLL module
Fig.8. The rotational speed based on SMO control system
Fig.9. The speed based on the SMO and PLL control system
Fig.10. Rotor position based on SMO control system