Novemb
ber 66-10, 2
2006 – Pariis, FRANC
CE
32nd IEEE Indu
ustrial Ele
ectronics Conferen
nce
PF-014893
Speed and Position Estimation for PM
Synchronous Motor using
SelfS lf-Compensated
Self
C
t dB
Back
Backk-EMF Observers
Ob
Marco TURSINI,
TURSINI Roberto PETRELLA,
PETRELLA, Alessia SCAFATI
University of L
L’Aquila
Aquila
Dept. of Electrical and Information Engineering (DEIE)
University of Udine
Dept. of Electrical, Managmt and Mechanical Eng. (DIEGM)
SS16--1 Advanced
SS16
Ad
d control
t l and
d observation
b
ti
off AC drives
d i
Thursday, November 9th, 2006, 8:00 AM
PF-014893
Aim of the paper
- provide full analytical description of previously-proposed speed and
position observers for PM synchronous motors, based on back-EMF
estimation
- calculate the (steady-state) rotor position estimation errors
s
- propose a real-time compensation strategy for rotor position estimation
error in a transducer-less PMSM drive
y
results by
y means of simulation analysis
y
and compare
p
- validate analytical
them with experiments
PF-014893
Transducer--less drive scheme
Transducer
ω *r
Rvel
iq*
Rid
id* = 0
vα
Luenberger/
Slidingg Mode
Observer
vβ
θ̂r
*
vˆ iα
cos θ̂r(1)
ω̂ (r1)
θ̂r = arccos ⎛⎜ vˆ iβ
⎝
iˆq
iβ
iˆd
ω̂ r
Kalman
Filter
vˆ i2α + vˆ i2β ⎞⎟
⎠
vα*
dq
vd*
iα
vˆ iβ
sin θ̂r(1)
Riq
vq*
αβ
iα
dq
αβ
sinθ̂r
cosθ
θ̂r
vβ*
SV
PWM
ia
αβ
iβ
ib
3
ic
Observer
vα*
vβ*
iα
iβ
PMSM
PF-014893
Permanent Magnet Synchronous Motor
βs
ωr
d
symmetricaly
-sinusoidal machine
- symmetrical
iβ
q
β
Hypothesis:
- non salient rotor
vβ
two--phase
θr - represented by two
N
S
equivalent statorstator-fixed αβ
windings
α
vα
αs
iα
Dynamical model:
i& = [A] i + [B] v i − [B] v
i = [iα , iβ ]T
v i = [ viα , viβ ]T
v = [ vα , vβ ]T
[A] = − RLss [I ]
[B] = − L1s [I ]
PF-014893
Back--EMF and PMSM Extended Model
Back
The backback-EMF components contain the information on the rotor position:
v iα (θr ) = − k e ω r sin θr
v iβ (θr ) = k e ω r cos θr
In order to arrange backback-EMF observer, backback-EMF components are added
to the state through a couple of fictitious dynamic equations:
i& = [A]i + [B]v i − [B]v
v& i = 0
x = [iα , iβ , viα , viβ ]T
Extended state
PF-014893
State and disturbance observers
&
iˆ = [A]iˆ + [B] vˆ i − [B] v + z
v&ˆ = [L] z
i
Luenberger observer (LO)
Sliding mode observer (SMO)
z = [K1 ] ⋅ (i − iˆ)
z = [K 1 ] ⋅ sgn (i − iî )
)
~
~
&xˆ = [ A
] xˆ + [B] v + [K ] (i − i )
~
~
xˆ& = [ A] xˆ + [ B] v + [K ] sgn
g (i − iˆ)
[K ] = [ k [I ]
k, l
l k [I ] ]T
Gain matrix
Gain coefficients
PF-014893
Why and how to develop a compensation law ?
Rotor position estimation error (difference between estimated and actual
position) dynamics:
e& i = [A] e i + [B] e e − k e i
e& e = −v& i − l k e i
If known, they it can be onon-line compensated !
Solution of the problem is difficult in the timetime-domain.
Let us consider the equivalent s
s--domain transfer functions:
s E i (s ) − e i 0 = [A] E i (s ) + [B] E e (s ) − k E i (s )
s E (s ) − e = −V& (s ) − l k E (s )
e
e0
i
i
Af
After
some calculations:
l l i
1 Ls
V& i (s )
E i (s ) =
(s − s1 )(s − s2 )
s1,2 = −h ± Δ = −h ± j Δ
E e (s ) = −
h=
1
2
(k + Rs
s + 2h
V& i (s )
(s − s1 )(s − s2 )
Ls ); Δ =
1
4
(k + Rs
Ls )2 − mk
m = − l Ls
PF-014893
Back--EMF estimation error
Back
The asymptotic stability is a sufficient condition for the existence of the
sinusoidal steadysteady-state response to sinusoidal excitation:
⎡ k e ω r2 cos (ω r t + φ F )⎤
v
e e (t ) = Fv ( jω ) ω =ω ⎢
⎥
2
r k ω sin ( ω t + φ
)
⎢⎣ e r
r
Fv ⎥
⎦
⎧⎪ e eα (t ) = Av cos (ω r t + φ Fv )
⎨
⎪⎩ e eβ (t ) = Av sin (ω r t + φ Fv )
Fv ( jω ) ω =ω =
r
φ Fv
ω =ω r
( 2h ( h 2 + Δ )) 2 + ω r2 ( Δ − ω r2 − 3h 2 ) 2
[h − (ω r − Δ )(ω r + Δ )] + (2hω r )
2
2
⎛ ω r ( Δ − ω r2 − 3h 2 ) ⎞
⎟
= arctang ⎜
2
⎜
2h ( h + Δ ) ⎟⎠
⎝
2
Av = Fv ( jω r ) k e ω r2
ω r Actual rotor speed
PF-014893
Calculation of the rotor position estimation error
αβ components of the estimated backback-EMF:
v̂iα = viα + eeα (t ) = − k e ω r sinω r t + Av cos ( ω r t + φ Fv )
v
v̂iβ = viβ + eeβ (t ) = k e ω r cosω r t + Av sin ( ω r t + φ Fv )
Rotor position estimation errors:
⎛
Av cos φFv
ˆθ̂ − θ = arctg ⎜ −
r
r
⎜ k e ω r + Av sin φF
v
⎝
θ̂r − θr = arctg (ω r Ls l )
⎞
⎟
⎟
⎠
Luenberger observer
Sliding mode observer
For
F a given
i
sett off motor
t parameters,
t
the
th (steady(steady
( t d -state)
t t ) rotor
t position
iti
estimation error depends on rotor speed, but it is independent on feeding
current (i.e. load torque)
PF-014893
Simulation results
Luenberger
g
800
SM O
1600
16000
∞
←k
Steady--state estimation errors
Steady
-7.41
-6.22
-5.19
-5.08
1000 rpm -16.97
-13.22
-10.37
-10.08
2000 rpm
-32.63
-21.99
-15.52
-14.93
3000 rpm
Comments:
- LO rotor
t position
iti estimation
ti ti
errors tend to SMO ones for high
values of gain k
Position error ( θ̂ r − θ r ) [d egrees]
- for the same value of gain l,
SMO has smaller estimation
error with respected to LO
Position estim
P
mation error [d
degrees]
0
-5
-5.078
-6.22
-10
- estimation error reaches non
non-negligible values at high speed
-10.079
-14.928
LO
-15
SMO
LO
-20
SMO
-25
0
l = −28.28; k = 1600
p
is p
possible by
y
- compensation
considering the analytical error
as a first attempt
-13.22
l = −28.28
l = −56.56; k = 1600
-21.99
l = −56.56
500
1000
1500
2000
Rotor speed [RPM]
2500
3000
- actual implementation could
lead to different results due to
un--modelled effects
un
PF-014893
Simulation results
L u e n b e rg e r O b s e rv e r
1
Transient estimation errors
Position estimatio
on error [degrees]
0
0 .5 e -3
-1
-2
P o s itio n e s tim a tio n e rro r
C o m p e n s a te d p o s itio n e s tim a tio n e rro r
-3
Conditions:
-4
- offoff-line operation of the
observer
-5
-6
lo a d in s e r tio n
-6 .2 2
-7
-8
0 .0 5
0 .1
0 .1 5
0 .2
0 .2 5
0 .3
l = − 2 8 .22 8
k = 1600
0 .3 5
0 .4
0 .4 5
0 .5
T im e [s ]
S lid in g M o d e O b s e rv e r
- 1.2 Nm (60% rated) load torque
insertion at 0.35s
1
Posiition estimation errror [degrees]
0
-5 .5 e -3
-1
-2
- position estimation error is
compensated by means of
analytical values
P o s iti
itio n e s ti
tim a ti
tio n e rro r
C o m p e n s a te d p o s itio n e s tim a tio n e rro r
-3
-4
-5
- 5 .0 8
lo a d in s e r tio n
-6
l = − 2 8 .2 8
-7
0
0 .0 5
0 .1
0 .1 5
0 .2
0 .2 5
T im e [s ]
0 .3
0 .3 5
- nono-load speed transient from 0
to 1000 RPM
0 .4
0 .4 5
0 .5
PF-014893
Experimental results
The drive system
Braking test bench
IEE--488 controlled
IEE
t ll d h
hysteresis
t
i b
brake
k
(Magtrol HDHD-710
710--8NA8NA-0040)
PMSM under test
rated power /current
rated speed/torque
pole pairs
no load back-EMF
no-load
back EMF @ rated speed
stator resistance
stator inductance
rotor inertia
inverter DC voltage
630 W - 2.5 A rms (*)
3000 rpm (*) - 2.0 Nm (*)
4
82 72 Vrms (*)
82.72
1.9 Ω
6 mH
22⋅10-5 kgm2
300 V
Control & power electronics
16--bit fixed
16
fixed--point DSP microcontroller
IGBT intelligent power module
150 μs control and modulation cycles
PF-014893
uncompensated
500 RPM
ωr
Experimental results
SMO operation
Conditions:
(θˆ r ,mon − θr )
9°
- no
no--load
- 500 to 2000 RPM
theoretical compensation
- 500 RPM slow varying ramp
increments
Comments:
- noise is related to residual
“chattering” on the backback-EMF
estimates
empirical
p
compensation
p
- theoretical compensation is not
the exact one, but provides good
results
- empirical
i i l compensation
ti is
i
obtained by varying the slope of
the compensating angle
PF-014893
uncompensated
500 RPM
ωr
Experimental results
LO operation
Conditions:
(θˆ r ,mon − θr )
9°
- no
no--load
- 500 to 2000 RPM
- 500 RPM slow varying ramp
increments
- same gains as SMO
Comments:
- position error has different slope
empirical
p
compensation
p
- noise is lower
PF-014893
Experimental results
SMO torque/speed analysis
Comments:
- load torque affects position
estimation error (differently
from theoretical results))
- the effect of the load torque
is quite small (less than 2
degrees in every condition)
- overover-compensation is
obtained if theoretical laws
are adopted (dotted lines)
- compensation with
theoretical values leads to a
maximum error of 5 degrees
PF-014893
Conclusions
The possibility to compensate the intrinsic rotor position estimation error in a
transducer--less control scheme for PM synchronous motors based on backtransducer
backemf observers has been analysed
analysed::
- it is possible to evaluate a theoretical compensation law for the steady
steady-p
both with Luenberger
g and Sliding
g Mode observers
state operation,
- such laws yield satisfactory compensation also during fast transients
- it has been proven that the presence of a Kalman filter in cascade to the
back--EMF observer does not introduce additional position error at steady
back
steady-state, and negligible one during transients if slow ramp
ramp--varying speed
reference is used
un--modelled effects in the actual drive, motor and real
real--time system
- the un
introduce some discrepancies from theory
theory:: exact compensation requires
a proper adjustment by experimental tests