PMSM Rotor Speed and Position Estimator.
UMPEDAC Postgraduate Srudents Renewable Energy Symposium (201
l)
PMSM Rotor Speed and
Position Estimator
Mohamed Azmi Said, W.p. Hew, N.A. Rahim
IJM Power Energy Dedicated Advanced Centre (UMpEDAC)
Level 4, Engineering Tower, University of Malaya
50603 Kuala Lumpur
azmisaid @siswa.um.edu.my
Abstract
Estimation techniques have been developed for flux
- rotor
observer,
position, and speed. The techniques have a
common problem of the offset in voltage_sensor and current-
sensor outputs increasing during integration,
)"r=vrt-R,
The
advantages
and drawbacks of
are calculated
each
From (5), rotor position
in a hostile environment, is fragile [1]. Its
is thus desired. Figure 1 is a block dfagram of a
noise, and,
sensorless
0
sampting
e)
(3)
(4)
canbe calculated as:
INTRoDUCTION
Permanent magnet synchronous motor (PMSM) is
popular for its high efficiency, high power density, very
low
inertia, and high reliability. In field_oriented control (FOC)
or vector control schemes, continuous rotor information is
mandatory. To ensure a pMSM,s high performance and
precise speed control, a position encodei is required;
it
senses and continuously feeds back both position and
speed.
The position encoder, though, imposes drawbacks on a
system: it is costly, it complicates circuit, is sensitive to
absence
tth
x'o(k) + lLG)
).,1k1 =
flux obsemer, SMO.
I.
(l)
)o(k)=1o,0_r*(V,u-r-R"ir)2"
1o&)= Xsrr,-t*(Verr,t-R,lr)4
Keywords - PMSM motor drives, sensorless, cascaded LpF,
adaptive rotor
1o&) aad )r(k)at the
from stator voltage and current:
Rotor flux linkage
(SMO). Simulation on Matlab/Simulink validaled the proposed
method.
+ l,tt=o
causing
instability. This paper describes uo att"-pito overcome the
problem through estimations of rotor posiiion and speed
of a
permanent magnet synchronous motor (pMSM) by three
methods: with cascaded low pass filter (LpF), with adaptive
sensorless rotor flux observer, and with sliding mode
observer
implementation are discussed.
fi,at
PMSM vector control drive.
o=tan,(+9)
(5)
\xD(k) )
Rotor position angle 0 depends on estimated stator flux,
which is determined by integrating the difference between
input voltage and voltage drop across resistance, as Equation
(l) shows. There is an offset error produced by current and
voltage sensors. When stator flux is indirectly integrated,
any dc offset error is also integrated. This eventuallyiauses
a large drift in stator flux estimation. The offset error is non_
periodical and unidirectional. It can not be rectified by
calculation alone. It must be overcome by some means of
compensation (see Figure 7 for offset of flux locus).
A.
Flw estimator with a programmable cascaded low_pass
filter
Figure 2 is a block diagram of the three_stage
programmable cascaded LPF [3]. The discrete transfer
characteristic of an LPF is:
Y
Figure 1: Sensorless vector control pMSM drive
Direct torque control (DTC) [2] is less
_
dependent,
X
parameter_
=-
1T
1-,-t
1+r' '
,'C(7-Z-t)
's'
(6)
T,
has fast dynamic response, and does not need a
mechanical rotor position sensor for its control loop.
Problems exist, though, namely, drift in stator_flux_linkage
estimation, owing to measurement offset error and varyiig
where 7 is the sampling time and ?
is the time
constant.
The corresponding difference equation is:
stator resistance.
The stator flux linkage in DTC scheme is estimated by
integrating the difference between input voltage and voltagl
y(kT,) =
drop across stator resistance:
85
{;tr"*(kT")
+ ly(kT, -7"))
(1)
Energy Symposium (2011)
IIMPEDAC Postgraduate Students Renewable
x(kT,) '+
arefunctions offrequency or the rotor speed'
at steady state, and are calculated as follows:
X
and
G
(v* =uo'ioR)o'lva =u,-
inR)
(so')
tanl
\,J
-
I
(8)
a
(e)
n is the number of filters cascaded' When the signal
cannot
is dc, C arrd G become infinite' thus the filter
where
the cascaded
oerform the integration. Implementation of
the origin'
on
,t e flrix locus to rimain centered
[Fi
AmplindeComPensation
"ii"*t
Gyr(kT")
knowledge of the
The integration process however, requires
initial-position.information
iJ,iJ rtir"t flui position' If the
i,
tt
.orrt otter
ii
inaccurate' the motor may initially rotate
depends heavily too
At zero and very low speeds' its
y (kT,) -+
" *ro.rg direction. Cascaded LPF
tfr"
-rno,ot
i,
programmable LPF
Figure 2: Block diagram of a thrce-stage
i"qu"r"y.
or,
).oor).,
implementation draws a disadvantage'
B. Adaptive sensorless rotorflw observer
be used to
Adaptive sensorless rotor flux observer can
(1)'
Equation
From
speed
and
[4]'
rotor position
the following
by
given
"uf"of*"
is
motor
PMSM
a
oi
,h" ;;;;"t
statoivoltage and flux linkage equation:
+=v"
clt
-
R,i,
)',=L,ir+1,
ie
^ = Lu€'
^
A,
where
(10)
(
Figure 3: Signal flow graph of the rotor angle observer
11)
(t2)
0 is the rotor angle' Electrical rotor speed is
Figure 4: SPeed estimator
expressed from mechanical rotor speed:
49=r=p@*""h
of the
Figure 3 shows how this is obtained by a feedback
estimated error .2,,
- n*. For a known magnitr'rde of
(13)
,,",
the
requires an open integration of the
hence
voltage equation. The offset contained in the inputs
Estimation
of 2,
makes the output drift. The offset
Estimator for the rotor flux is:
oi
dt
= v,
-
is modeled as ) ',
R,i, + r)ou
l, = lr"it
'
lr,uf = X* is
used'
The
estimator'
observer may then be seen as a rotor field angle
of
information
no
gives
BEMF
speeds,
low
very
and
At zero
situation
the
to
principle is introduced
,tL n"iA"rgf".
i,t"*
by impresslng a current in the direction of the default
anlle. rhe current then forces the permanent
"rti*ui"a
ai"g, to that angle; this way, the estimated angle
t"
;rg*
U""L"t conlct
(14)
only witli the error caused by friction and
load.
Fizure 4 shows the signal flow graph for the rotor speed
(15)
esimator. The
*gGtt)
uro"rt solves the
modulus
problem. The algorithm for the estimator then becomes:
where fiouhas to be designed in a way that leads to a flux
estimate with constant amplitude
permanent magnet,
l2.l'
+
l.rLtu
i, = i, - L,i,
=v" - R,i" + rrbr,,"r -
0 = arg( i,)
86
(16)
(17)
(18)
IIMPEDAC Postgraduate Students Renewable Energy Symposium (2011)
0)=
ei,-or)
",(,-i)*
d0,
current
sliding mode, the observers are insensitive to
t-+ o.l
methods,
:
lu
the observer is more robust to operating conditions and
A=l
parameter uncertainties [8].
To estimate rotor position and angular speed of a
PMSM, the motor is generally modeled in stationary
reference frame. The angular speed and position information
are ready to be extracted in this frame.
In
stationary condition,
aB
Setting
reference frame can be
u
"n
R.
I
T'" T""*T'"
;=
dio
dt
r.*!r.
=-R,.-l
LU LP LP
(4 )
(4 )
e
o = ),0a" cos
eo = -2oo)" sin
The motor speed changes slowly, implying that
the model for the induced back EMF is:
io = o)"eO; ip =
ip =
0
(30)
glves:
"
(3 1)
o
: (lrrign (i,)),, = ,,
(32)
(2t)
To extract eoand
(22)
(23)
from the corresponding equivalent
Zo,
zO
as
filter outputs:
(24)
A=0
e,
control values, a low pass filter is used with
,
Zp
A"eo
interval
t-trL)
o = (lrsign (i,))"n =
u"qo
I
a finite time
*l''=l-',',i)
-r)
i*=0,
written as:
di_
is aefinea according to the
From Equations (21) and(22),
parameoor variations and disturbance. Sliding mode observer
has been presented for robust estimation ofthe state variable
of controlled objects [5-7]. Compared with other
, =Vo ;rf
sliding mode occurs after
io=0and lp=0.
^
Sliding mode observer
In
errors S =
sliding mode observer theory. The system behavior can be
examined by applying equivalent control method, when the
(20)
;=
C.
with /, > ** (r,;,lrrl). The sliding hyper plane on the stator
(re)
(2s)
car.
zo(t) = eoQ) + A, (/)
(33)
zo1)=eoU)+L"(t)
(34)
not be used directly. Equations (33) and (34) are
used for better filtering and estimate the rotor angular speed
and position. The observer undertaking the filtering task is:
The sliding mode observer uses only motor elechical
equation. Its main equations are:
di,__R" _1
I
/i\
,=-r,"
Tr"*-stgnud)
dfu R. I
1 /r\
i=- ,,0 Trr*-sign\iB)
bo = -fo"eo
- L(ap - ,)
ir" = (0, - z,hp - @o - zpb*
bo = fo"eo
(26)
(27)
where
,
--1,"0
*1"*('o)
-lr(0r- e")
io = d)"eo - @"€o - L(oB "r)
ir" = (ao - e,Yp -(ao "pb-
mismatch dynamics is:
;= -i,',
observer gain. Mismatches
to = -6"0p + a"ep
motor parameters are identical with those in the model, the
*=-lr"-+",+l,ignQ.)
/, is a constant
(35)
(36)
e7)
in
the
back EMF equations are:
l, is constant observer gain, and io = io - io
ip = tf - lp arc observer mismatches. Assuming that the
where
- L(A, - z,)
(28)
(38)
(3e)
(40)
Ap=Ap-ep
where Vo=0o-eo
d" = 6" - 0)"are observer errors. According to relations in
(2e)
(21) and(22),
The dynamics are distributed by the unknown induced
EMF components. However, the back EMF components are
bounded, so they may be suppressed by continuous input,
oo =
87
-loo)"ri, (4)
(41)
Energy Symposium (2011)
UMPEDAC Postgraduate Students Renewable
ao = -)"oo)"sin
sin
(4)
uoa
cos(4)
(4
,0"
)
Lo*,".o
"""beobtainedas:
.ir(ti)=
"o'(4)=
As Figure 9 shows, adaptive sensorless rotor flux observer
UJop"t","d from zero speed. The switch from open-loop
"un
tL start up to closed loop can be full of noise and
""rt
give "f
extreme speed transients. Adaptive sensorless operates
(43)
+?
speed without changing the control structure'
SMO simulation results show accurate speed and position
Figures 10 and 1l show the actual and the
"rGuti*.anglis. The position angle error as shown in Figure
"rti*ut"a
l2 is under
+?
0.015p.u, below 5 degrees of electrical angle'
@)
)'o,wb
can be
The signal flow diagram for sliding mode observer
5'
implemented as shown in Figure
Back EMF
Cumrt
Obsfls
LPF
Obcarus
(24,(28)
Z5
rslrn0.)-
Eq.
Eq. (36),(37),(38)
(41),(42),(43),(44)
Figure 5: Block Dagram of SMO
-0.25
II.
RESULTS
in Matlab/Simulink
the PMSM model
of
Parameters
tool.
simulation
rofi**"
are as tabulated in Table I.
The observer behaviors were tested
TABLEIV.
0.25
LD'wb
DATA OF PMSM
Figure
P
8
Armature resistance R
t.t
Number of Pole
Magnet flux
d-axis
o-axis
Pairs
linkage )'
inductance
inductance
Inertia
factor
Friction
)Lo,Wb
0.28 Wb
8.65 mH
8.65 mH
J
0.0051kg.m'
F
0.0050 N.m.s
In sensorless drive operation, the estimated stator flux is
dt'ive
used to calculate torque, flux, and rotor position' DTC
implementation requires all three observer outputs as
feJJbacts to closed-loop control. Meanwhile, FOC drive
need position feedback to transform the stator current
measuied from stationary reference aB frame to rotating
reference frame f,q
-0.25
0.25
lD,w
.
Figure 6 shows the flux locus calculated from an ideal
inpuritator current without the -p-resence of dc offset' In
aitual implementation, with an offset measurement present'
there is u'd.ift io stator flux locus, caused by the offset error
Figure 7: Flux locus with stator cunent measurement considering
in flux linkage estimation.
Cascaded LPF
will
and voltage without
dc offset.
o
Lo
La
6: Flux locus for ideal measurement of cunent
compensate the-offset.error; see
Figure 8. A filter resets the integrator used in estimating the
*tit* nu* fnkage. The flux locus is seen to remain centered
oo th" o.igin. It-clearly starts from the origin, requiring an
initial rotoi position. The observer also depends on motor
frequency 6a, requiring the motor to run at some speed flrst
before the cascaded LFF can be implemented'
88
&
offset'
UMPEDAC Postgraduate Students Renewable Energy Symposium (2011)
)'o,wb
0.5
--t!-
0.25
0
R\
Ur
t//r'
\
\\
-0.25
/#
\
- 0.5
-*=
-0.25
0.u 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 cn
{
Tire
--#
0
(s)
Figure 1 1: SMO estimated position
0.2s
0.5
)'D,Wb
Figure 8: Flux locus calculated by using flux astimator with programrnable
cascaded low-pass fi lter
)"a,wb
Figure 12: SMO position estimation error
M.
-0.2
0 0.25 0.5
-0.25 0
-0.2s
)'D,Wb
Figure 9: Flux locus calculated based on active rotor flux observer. However,
at zero and low speeds, adaptive sensorless requires a non-
,oo
reference value.
i
CoNCTSIoNs
Offset error due to the presence of dc in stator current and
voltage measurement in PMSM motor drive causes drifts in
stator-flux calculation, in turn causing inaccurate estimation
of rotor angle position. Cascaded l,PF is able to diminish the
drift but the calculation is assumed running at steady state.
Adaptive sensorless rotor flux observer may be used at zero
or low speeds but the current reference must be set to a nonzero value initially. SMO can calculate rotor position angle
accurately, with low position-error.
ACKNoWLEDGMEI.IT
"a
The support of UM Power Energy Dedicated Advanced
Centre (UMPEDAC) is grarefully acknowledged. M.A.
Said's research study is funded by Universiti Teknikal
Malaysia Melaka.
(Text edited by wirani-umpedac@yahoo.com)
RepgneNcEs
A
[1]
0.04 0.06 0.08
0.1 0.J2 0..t4 0.16 0.18 0.2
0.22
Tme (s)
T. Lixin, Z. Limh, M.F. Rahman, and Y. Hu, "A novel
direct torque controlled interior permanent magnet
synchronous machine drive with low ripple in flux and
torque and fixed switching frequency," Power
Electronics, IEEE Transactions on, vol. 19, pp. 346354,2004.
[2] I. Takahashi
Figure l0: Actual rotor position angle
t3l
89
and T. Noguchi, "A New Quick-Response
and High-Efficiency Control Strategy of an Induction
Motor," Indu:stry Applications, IEEE Transoctions on,
vol. IA-22, pp. 820-827, 1986.
M.F. Rahman, M.E. Haque, T. Lixin, arrd Z. Limm,
"Problems associated with the direct torque control of
LJMPEDAC Postgraduate Students Renewable Energy Symposium (201l)
an interior permanent-magnet synchronous motor drive
and their remedies," Industrial Electronics, IEEE
[4]
Transactions on, vol.51, pp. 799-809,20M.
H. Rasmussen, P. Vadstrup, and H. Borsting, "Adaptive
sensorless field oriented control of PM motors including
zero speed," in Industrial Electonics, 204 IEEE
Intemational Symposium on,2004, pp. 1191-1196 vol.
2.
Changsheng and M. Elbululq "A sliding mode
observer for sensorless control of permanent magnet
synchronous motors," in Industry Applications
Conference, 2(W. Thirty-Sixth IAS Annual Meeting.
Conference Record of the 2001 IEEE,20Ol, pp. 12731278 vol.2.
J. l,ee, J. Son, and H. Kim, "A High Speed Sliding
Mode Observer for the Sensorless Speed Confiol of a
PMSM," In"dustrial Electronics, IEEE Transactions on,
[5] L.
t6l
vol.PP,pp.l-1.
[7] G. Foo and M.F.
Rahman, "Direct torque and flux
.
controlled IPM synchronous motor drive using a hybrid
signal injection and adaptive sliding mode observer," in
TENCON 2009 - 2009 IEEE Region 10 Conference,
[8]
J. Dong, Z. Zltengnng, and W. Fei,
2009, pp. l-7.
Observer
for PMSM
"A Sliding Mode
speed and rotor position
considering saliency," in Power Electonics Speciaksts
Conference, 2008. PESC 2008. 1888,2008, pp. 809814.
90
l)
PMSM Rotor Speed and
Position Estimator
Mohamed Azmi Said, W.p. Hew, N.A. Rahim
IJM Power Energy Dedicated Advanced Centre (UMpEDAC)
Level 4, Engineering Tower, University of Malaya
50603 Kuala Lumpur
azmisaid @siswa.um.edu.my
Abstract
Estimation techniques have been developed for flux
- rotor
observer,
position, and speed. The techniques have a
common problem of the offset in voltage_sensor and current-
sensor outputs increasing during integration,
)"r=vrt-R,
The
advantages
and drawbacks of
are calculated
each
From (5), rotor position
in a hostile environment, is fragile [1]. Its
is thus desired. Figure 1 is a block dfagram of a
noise, and,
sensorless
0
sampting
e)
(3)
(4)
canbe calculated as:
INTRoDUCTION
Permanent magnet synchronous motor (PMSM) is
popular for its high efficiency, high power density, very
low
inertia, and high reliability. In field_oriented control (FOC)
or vector control schemes, continuous rotor information is
mandatory. To ensure a pMSM,s high performance and
precise speed control, a position encodei is required;
it
senses and continuously feeds back both position and
speed.
The position encoder, though, imposes drawbacks on a
system: it is costly, it complicates circuit, is sensitive to
absence
tth
x'o(k) + lLG)
).,1k1 =
flux obsemer, SMO.
I.
(l)
)o(k)=1o,0_r*(V,u-r-R"ir)2"
1o&)= Xsrr,-t*(Verr,t-R,lr)4
Keywords - PMSM motor drives, sensorless, cascaded LpF,
adaptive rotor
1o&) aad )r(k)at the
from stator voltage and current:
Rotor flux linkage
(SMO). Simulation on Matlab/Simulink validaled the proposed
method.
+ l,tt=o
causing
instability. This paper describes uo att"-pito overcome the
problem through estimations of rotor posiiion and speed
of a
permanent magnet synchronous motor (pMSM) by three
methods: with cascaded low pass filter (LpF), with adaptive
sensorless rotor flux observer, and with sliding mode
observer
implementation are discussed.
fi,at
PMSM vector control drive.
o=tan,(+9)
(5)
\xD(k) )
Rotor position angle 0 depends on estimated stator flux,
which is determined by integrating the difference between
input voltage and voltage drop across resistance, as Equation
(l) shows. There is an offset error produced by current and
voltage sensors. When stator flux is indirectly integrated,
any dc offset error is also integrated. This eventuallyiauses
a large drift in stator flux estimation. The offset error is non_
periodical and unidirectional. It can not be rectified by
calculation alone. It must be overcome by some means of
compensation (see Figure 7 for offset of flux locus).
A.
Flw estimator with a programmable cascaded low_pass
filter
Figure 2 is a block diagram of the three_stage
programmable cascaded LPF [3]. The discrete transfer
characteristic of an LPF is:
Y
Figure 1: Sensorless vector control pMSM drive
Direct torque control (DTC) [2] is less
_
dependent,
X
parameter_
=-
1T
1-,-t
1+r' '
,'C(7-Z-t)
's'
(6)
T,
has fast dynamic response, and does not need a
mechanical rotor position sensor for its control loop.
Problems exist, though, namely, drift in stator_flux_linkage
estimation, owing to measurement offset error and varyiig
where 7 is the sampling time and ?
is the time
constant.
The corresponding difference equation is:
stator resistance.
The stator flux linkage in DTC scheme is estimated by
integrating the difference between input voltage and voltagl
y(kT,) =
drop across stator resistance:
85
{;tr"*(kT")
+ ly(kT, -7"))
(1)
Energy Symposium (2011)
IIMPEDAC Postgraduate Students Renewable
x(kT,) '+
arefunctions offrequency or the rotor speed'
at steady state, and are calculated as follows:
X
and
G
(v* =uo'ioR)o'lva =u,-
inR)
(so')
tanl
\,J
-
I
(8)
a
(e)
n is the number of filters cascaded' When the signal
cannot
is dc, C arrd G become infinite' thus the filter
where
the cascaded
oerform the integration. Implementation of
the origin'
on
,t e flrix locus to rimain centered
[Fi
AmplindeComPensation
"ii"*t
Gyr(kT")
knowledge of the
The integration process however, requires
initial-position.information
iJ,iJ rtir"t flui position' If the
i,
tt
.orrt otter
ii
inaccurate' the motor may initially rotate
depends heavily too
At zero and very low speeds' its
y (kT,) -+
" *ro.rg direction. Cascaded LPF
tfr"
-rno,ot
i,
programmable LPF
Figure 2: Block diagram of a thrce-stage
i"qu"r"y.
or,
).oor).,
implementation draws a disadvantage'
B. Adaptive sensorless rotorflw observer
be used to
Adaptive sensorless rotor flux observer can
(1)'
Equation
From
speed
and
[4]'
rotor position
the following
by
given
"uf"of*"
is
motor
PMSM
a
oi
,h" ;;;;"t
statoivoltage and flux linkage equation:
+=v"
clt
-
R,i,
)',=L,ir+1,
ie
^ = Lu€'
^
A,
where
(10)
(
Figure 3: Signal flow graph of the rotor angle observer
11)
(t2)
0 is the rotor angle' Electrical rotor speed is
Figure 4: SPeed estimator
expressed from mechanical rotor speed:
49=r=p@*""h
of the
Figure 3 shows how this is obtained by a feedback
estimated error .2,,
- n*. For a known magnitr'rde of
(13)
,,",
the
requires an open integration of the
hence
voltage equation. The offset contained in the inputs
Estimation
of 2,
makes the output drift. The offset
Estimator for the rotor flux is:
oi
dt
= v,
-
is modeled as ) ',
R,i, + r)ou
l, = lr"it
'
lr,uf = X* is
used'
The
estimator'
observer may then be seen as a rotor field angle
of
information
no
gives
BEMF
speeds,
low
very
and
At zero
situation
the
to
principle is introduced
,tL n"iA"rgf".
i,t"*
by impresslng a current in the direction of the default
anlle. rhe current then forces the permanent
"rti*ui"a
ai"g, to that angle; this way, the estimated angle
t"
;rg*
U""L"t conlct
(14)
only witli the error caused by friction and
load.
Fizure 4 shows the signal flow graph for the rotor speed
(15)
esimator. The
*gGtt)
uro"rt solves the
modulus
problem. The algorithm for the estimator then becomes:
where fiouhas to be designed in a way that leads to a flux
estimate with constant amplitude
permanent magnet,
l2.l'
+
l.rLtu
i, = i, - L,i,
=v" - R,i" + rrbr,,"r -
0 = arg( i,)
86
(16)
(17)
(18)
IIMPEDAC Postgraduate Students Renewable Energy Symposium (2011)
0)=
ei,-or)
",(,-i)*
d0,
current
sliding mode, the observers are insensitive to
t-+ o.l
methods,
:
lu
the observer is more robust to operating conditions and
A=l
parameter uncertainties [8].
To estimate rotor position and angular speed of a
PMSM, the motor is generally modeled in stationary
reference frame. The angular speed and position information
are ready to be extracted in this frame.
In
stationary condition,
aB
Setting
reference frame can be
u
"n
R.
I
T'" T""*T'"
;=
dio
dt
r.*!r.
=-R,.-l
LU LP LP
(4 )
(4 )
e
o = ),0a" cos
eo = -2oo)" sin
The motor speed changes slowly, implying that
the model for the induced back EMF is:
io = o)"eO; ip =
ip =
0
(30)
glves:
"
(3 1)
o
: (lrrign (i,)),, = ,,
(32)
(2t)
To extract eoand
(22)
(23)
from the corresponding equivalent
Zo,
zO
as
filter outputs:
(24)
A=0
e,
control values, a low pass filter is used with
,
Zp
A"eo
interval
t-trL)
o = (lrsign (i,))"n =
u"qo
I
a finite time
*l''=l-',',i)
-r)
i*=0,
written as:
di_
is aefinea according to the
From Equations (21) and(22),
parameoor variations and disturbance. Sliding mode observer
has been presented for robust estimation ofthe state variable
of controlled objects [5-7]. Compared with other
, =Vo ;rf
sliding mode occurs after
io=0and lp=0.
^
Sliding mode observer
In
errors S =
sliding mode observer theory. The system behavior can be
examined by applying equivalent control method, when the
(20)
;=
C.
with /, > ** (r,;,lrrl). The sliding hyper plane on the stator
(re)
(2s)
car.
zo(t) = eoQ) + A, (/)
(33)
zo1)=eoU)+L"(t)
(34)
not be used directly. Equations (33) and (34) are
used for better filtering and estimate the rotor angular speed
and position. The observer undertaking the filtering task is:
The sliding mode observer uses only motor elechical
equation. Its main equations are:
di,__R" _1
I
/i\
,=-r,"
Tr"*-stgnud)
dfu R. I
1 /r\
i=- ,,0 Trr*-sign\iB)
bo = -fo"eo
- L(ap - ,)
ir" = (0, - z,hp - @o - zpb*
bo = fo"eo
(26)
(27)
where
,
--1,"0
*1"*('o)
-lr(0r- e")
io = d)"eo - @"€o - L(oB "r)
ir" = (ao - e,Yp -(ao "pb-
mismatch dynamics is:
;= -i,',
observer gain. Mismatches
to = -6"0p + a"ep
motor parameters are identical with those in the model, the
*=-lr"-+",+l,ignQ.)
/, is a constant
(35)
(36)
e7)
in
the
back EMF equations are:
l, is constant observer gain, and io = io - io
ip = tf - lp arc observer mismatches. Assuming that the
where
- L(A, - z,)
(28)
(38)
(3e)
(40)
Ap=Ap-ep
where Vo=0o-eo
d" = 6" - 0)"are observer errors. According to relations in
(2e)
(21) and(22),
The dynamics are distributed by the unknown induced
EMF components. However, the back EMF components are
bounded, so they may be suppressed by continuous input,
oo =
87
-loo)"ri, (4)
(41)
Energy Symposium (2011)
UMPEDAC Postgraduate Students Renewable
ao = -)"oo)"sin
sin
(4)
uoa
cos(4)
(4
,0"
)
Lo*,".o
"""beobtainedas:
.ir(ti)=
"o'(4)=
As Figure 9 shows, adaptive sensorless rotor flux observer
UJop"t","d from zero speed. The switch from open-loop
"un
tL start up to closed loop can be full of noise and
""rt
give "f
extreme speed transients. Adaptive sensorless operates
(43)
+?
speed without changing the control structure'
SMO simulation results show accurate speed and position
Figures 10 and 1l show the actual and the
"rGuti*.anglis. The position angle error as shown in Figure
"rti*ut"a
l2 is under
+?
0.015p.u, below 5 degrees of electrical angle'
@)
)'o,wb
can be
The signal flow diagram for sliding mode observer
5'
implemented as shown in Figure
Back EMF
Cumrt
Obsfls
LPF
Obcarus
(24,(28)
Z5
rslrn0.)-
Eq.
Eq. (36),(37),(38)
(41),(42),(43),(44)
Figure 5: Block Dagram of SMO
-0.25
II.
RESULTS
in Matlab/Simulink
the PMSM model
of
Parameters
tool.
simulation
rofi**"
are as tabulated in Table I.
The observer behaviors were tested
TABLEIV.
0.25
LD'wb
DATA OF PMSM
Figure
P
8
Armature resistance R
t.t
Number of Pole
Magnet flux
d-axis
o-axis
Pairs
linkage )'
inductance
inductance
Inertia
factor
Friction
)Lo,Wb
0.28 Wb
8.65 mH
8.65 mH
J
0.0051kg.m'
F
0.0050 N.m.s
In sensorless drive operation, the estimated stator flux is
dt'ive
used to calculate torque, flux, and rotor position' DTC
implementation requires all three observer outputs as
feJJbacts to closed-loop control. Meanwhile, FOC drive
need position feedback to transform the stator current
measuied from stationary reference aB frame to rotating
reference frame f,q
-0.25
0.25
lD,w
.
Figure 6 shows the flux locus calculated from an ideal
inpuritator current without the -p-resence of dc offset' In
aitual implementation, with an offset measurement present'
there is u'd.ift io stator flux locus, caused by the offset error
Figure 7: Flux locus with stator cunent measurement considering
in flux linkage estimation.
Cascaded LPF
will
and voltage without
dc offset.
o
Lo
La
6: Flux locus for ideal measurement of cunent
compensate the-offset.error; see
Figure 8. A filter resets the integrator used in estimating the
*tit* nu* fnkage. The flux locus is seen to remain centered
oo th" o.igin. It-clearly starts from the origin, requiring an
initial rotoi position. The observer also depends on motor
frequency 6a, requiring the motor to run at some speed flrst
before the cascaded LFF can be implemented'
88
&
offset'
UMPEDAC Postgraduate Students Renewable Energy Symposium (2011)
)'o,wb
0.5
--t!-
0.25
0
R\
Ur
t//r'
\
\\
-0.25
/#
\
- 0.5
-*=
-0.25
0.u 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 cn
{
Tire
--#
0
(s)
Figure 1 1: SMO estimated position
0.2s
0.5
)'D,Wb
Figure 8: Flux locus calculated by using flux astimator with programrnable
cascaded low-pass fi lter
)"a,wb
Figure 12: SMO position estimation error
M.
-0.2
0 0.25 0.5
-0.25 0
-0.2s
)'D,Wb
Figure 9: Flux locus calculated based on active rotor flux observer. However,
at zero and low speeds, adaptive sensorless requires a non-
,oo
reference value.
i
CoNCTSIoNs
Offset error due to the presence of dc in stator current and
voltage measurement in PMSM motor drive causes drifts in
stator-flux calculation, in turn causing inaccurate estimation
of rotor angle position. Cascaded l,PF is able to diminish the
drift but the calculation is assumed running at steady state.
Adaptive sensorless rotor flux observer may be used at zero
or low speeds but the current reference must be set to a nonzero value initially. SMO can calculate rotor position angle
accurately, with low position-error.
ACKNoWLEDGMEI.IT
"a
The support of UM Power Energy Dedicated Advanced
Centre (UMPEDAC) is grarefully acknowledged. M.A.
Said's research study is funded by Universiti Teknikal
Malaysia Melaka.
(Text edited by wirani-umpedac@yahoo.com)
RepgneNcEs
A
[1]
0.04 0.06 0.08
0.1 0.J2 0..t4 0.16 0.18 0.2
0.22
Tme (s)
T. Lixin, Z. Limh, M.F. Rahman, and Y. Hu, "A novel
direct torque controlled interior permanent magnet
synchronous machine drive with low ripple in flux and
torque and fixed switching frequency," Power
Electronics, IEEE Transactions on, vol. 19, pp. 346354,2004.
[2] I. Takahashi
Figure l0: Actual rotor position angle
t3l
89
and T. Noguchi, "A New Quick-Response
and High-Efficiency Control Strategy of an Induction
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[4]
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sensorless field oriented control of PM motors including
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Changsheng and M. Elbululq "A sliding mode
observer for sensorless control of permanent magnet
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[5] L.
t6l
vol.PP,pp.l-1.
[7] G. Foo and M.F.
Rahman, "Direct torque and flux
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controlled IPM synchronous motor drive using a hybrid
signal injection and adaptive sliding mode observer," in
TENCON 2009 - 2009 IEEE Region 10 Conference,
[8]
J. Dong, Z. Zltengnng, and W. Fei,
2009, pp. l-7.
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for PMSM
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90