Speed and Position Estimation for PM Synchronous Motor
using self-compensated Back-EMF Observers
Marco Tursini, Roberto Petrella, Alessia Scafati
Department of Electrical Engineering - University of L'Aquila
Roio Monteluco, I-67040, Italy
tursini@ing.univaq.it, petrella@ing.univaq.it, scafati@ing.univaq.it
Abstract - The paper deals with the self-compensation of the
intrinsic estimation error in back-EMF based rotor position
observers for PM synchronous motors. The self-compensation is
based on the analytical calculation of the rotor position
estimation error for two types of popular back-EMF observers,
such as the standard-linear Luenberger Observer and the nonlinear Sliding Mode Observer. Once the compensation
characteristics are derived, they are included in the observer
itself by a proper mechanism in order to cancel the position
error affecting the estimation, thus providing the real-time selfcompensation scheme. As a consequence, the performance and
the robustness of the transducer-less drive can be improved,
both at steady state and transient operations. Test results are
presented to verify the effectiveness of the method in several
operating conditions: both simulation results using a timecontinuous Matlab/ Simulink model, and experimental results
using a DSP based transducer-less drive.
I. INTRODUCTION
Permanent magnet (PM) synchronous motor drives
represent the preferred solution in industrial automation and
control systems. The motors employed in these applications
have usually a cylindrical iron-laminated rotor, with the
permanent magnets displaced on the surface, and a threephase stator similar to that of standard induction machines.
Control of PM synchronous motors requires the
knowledge of the rotor magnet axis position (briefly “rotor
position”). For this reason they are equipped with some kind
of transducer, such as encoders or resolvers, able to provide
that information. This additional component involves cost,
encumbrance, wiring, alignment procedures, and others
tedious disadvantages. Thereafter, researchers working in
this area are focused on the removal of the position
transducer and the replacement of its function by certain
detection method, i.e. a “transducer-less” strategy.
As from theory, the counter electromotive force induced
by the rotor magnets in the stator phases, namely “backEMF”, contains the information about both speed and rotor
position. Although this information decades at zero speed,
transducer-less strategies based on back-EMF detection are
suitable for an important class of medium-high speed drives
such as cooling fans and compressors, embracing household
and automotive appliance [1][2][3].
The authors have developed a solution based on state
observer for back-EMF estimation, [4][5]. The approach
requires the simply introduction of a software algorithm in a
standard DSP based vector control scheme, it can be applied
both to sinusoidal and non-sinusoidal motors and it needs the
use of a specific starting procedure.
1-4244-0136-4/06/$20.00 '2006 IEEE
An advantage of this proposal is the availability of the
analytical solution of the observer, which allows to find out
the (theoretical) speed and position estimation errors, and the
influence of parameters and system deviations, both in
dynamic and steady state conditions [6].
This paper presents the transducer-less control scheme for
PM synchronous motors based on back-EMF observer, in
which self-compensation of the intrinsic rotor position
estimation error is implemented. Sections are organized as
follows: based on the voltage model for the PM synchronous
motor, the popular standard-linear (Luenberger) and nonlinear (Sliding Mode) back-EMF observers are briefly
recalled (Section II); the transducer-less strategy and the
drive scheme are introduced (Section III); calculation of the
self-compensation law is developed by an unified approach
for both the observers (Section IV); simulation and
experimental results are presented to validate the method,
using the former a Matlab/Simulink model, the latter a
TMS320F240 DSP based transducer-less drive (Section V
and VI).
II. BACK-EMF OBSERVERS
The PM synchronous motor can be represented in term of
two-phases-equivalent stator-fixed αβ windings as follows:
i = [ A] i + [B] v i − [B] v
where i = [i α iβ ] , v = [vα vβ ]
T
T
and v i = [v iα viβ ]
(1)
T
are the
vectors of the current, voltage, and back-EMF components,
and [A], [B] are the matrices of the system parameters; in
purely sinusoidal machines, the back-EMF components
depend on the rotor electric speed (ωr) and the magnet axis
position (θr) through the relations:
v iα (θr ) = − k e ω r sin θ r ; v iβ (θr ) = k e ω r cos θr
(2)
where ke is the back-EMF constant.
By extending the model (1) with a proper (fictitious)
estimation equation for the back-EMFs ( v i = 0 ), the
Luenberger Observer (LO) and the Sliding Mode Observer
(SMO) are respectively build as follows:
~
~
(3)
xˆ = [ A] xˆ + [ B] v + [ K] ( i − i )
~
~
xˆ = [ A] xˆ + [ B] v + [K ] sgn (i − iˆ)
(4)
where xˆ = [iˆ, vˆ i ]T is the state vector of the extended model,
[ A~] , [B~] the matrices of the extended system parameters,
5087
vα
Luenberger/
Sliding Mode
Observer
vβ
θ̂r
eq(5)
vˆ iα
iα
ω *r
Rvel
iq*
Riq
iβ
Rid
id* = 0
vˆ iβ
ω̂ (r1)
Kalman
Filter
iˆd
ω̂ r
Fig. 1. The transducer-less strategy
PWM
iβ
αβ
ia
αβ
ib
3
(5)
This approach leads to calculate the estimated speed as the
derivative of the estimated position. Unfortunately, making a
derivative has certain number of disadvantages from the
implementation point of view.
Therefore, the authors have presented an alternative
method, which employs an extended Kalman filter for the
detection of both the speed and the position “sine” and
“cosine”, using a “first-harmonic” model [5]. Hereafter, the
transducer-less strategy has been generalized as shown in
Fig. 1, where the superscript (1) identifies the outputs of the
Kalman filter. In case of SMO, the presence of the filter
permits to remove the noisy effects of chattering, while in
case of LO, it allows the application of the strategy to PM
motors with distorted (i.e. non-sinusoidal) back-EMF
waveforms.
The drive scheme implementing the transducer-less
strategy is represented in Fig. 2. The structure is that of a
standard AC brush-less control scheme arranged in the
synchronously rotating reference frame d-q, with PI speed
and current regulation, and inverter modulation through
Space Vector (SV) technique. The speed feedback and the
ic
Observer
vβ*
iα
iβ
PMSM
Fig. 2. PM synchronous motor transducer-less drive scheme
position “sine/cosine” functions needed for the Park
transformations are provided according to the transducer-less
strategy.
IV. DEVELOPMENT OF THE COMPENSATION LAW
In order to find the exact compensation law for the magnet
axis position estimation, we proceed to solve the error
equations associated to the back-EMF observer, which are
obtained from (3) as follows:
e i = [A] e i + [B] e e − k e i
e e = −v i − l k e i
According to (2), upon the estimation of the back-EMF
components, the magnet axis position could be obtained by a
reverse trigonometric formula such as:
vˆ i2α + vˆ i2β 

iα
dq
v*α
III. TRANSDUCER-LESS SCHEME
θ̂r = arccos  vˆ iβ

SV
vβ*
αβ
sinθ̂r
cosθˆ r
and [K ] = [ k [I ] l k [I ] ]T is a gain matrix ( [I ] is the identity
matrix, k and l are gain coefficients).
According to (3), the LO estimates the back-EMF
components (and currents) from the measurement of the
motor phase voltages and currents; it uses the linear value of
the current estimation error (i.e. the difference between the
measured and the estimated one) as correction feedback, and
the observer is linear. The SMO has exactly the same
structure, but, according to (4), it uses the sign of the current
estimation error (instead of the linear value) to build the
correction feedback, and the observer is non-linear, [7][8].
As demonstrated, the SMO is capable to give same results
as the LO with smaller gains, [6]. Unfortunately, due to its
own nature, sliding mode estimates are affected by
“chattering” and a proper filtering action is needed.
vα*
dq
vd*
iˆq
sin θ̂(r1)
cos θ̂(r1)
vq*
(6)
where e i and e e are the current and back-EMF estimation
errors respectively (defined as the difference between
estimated and actual values), and v i = [ v iα , viβ ]T are the
time derivative of the back-EMF.
The solution of the differential equations (6) is quite
difficult in the time domain, thus leading to use
Laplace (L) −transform−based method. The s −domain
transfer functions can be obtained as follows:
sE i (s ) − e i 0 = [A] E i (s ) + [B] E e (s ) − k E i (s)
sE e (s ) − e e 0 = −V i (s ) − l k E i (s )
(7)
where V i (s ) is the L−transform of the back-EMF timederivative, and e e 0 , e i 0 are the initial conditions for the
back-EMF and current estimation error respectively, that
can be assumed zero without loose of generality. After some
calculations, current and back-EMF estimation errors can be
written as follow:
E i (s ) =
1 Ls
V (s )
(s − s1 )(s − s 2 ) i
E e (s ) = −
where m = − l Ls and:
5088
s + 2h
(s − s1 )(s − s 2 )
V i (s )
(8)
(9)
s1, 2 = −h ± ∆ = − h ± j ∆
h=
1
2
(k + Rs
Ls ); ∆ =
1
4
(k + Rs
(10)
Ls )2 − mk
The values of parameters m, k must be chosen in order to
assure asymptotic stability and fast response, e.g. complex
roots of (9): (h > 0) ∪ ( ∆ < 0) .
The asymptotic stability is a sufficient condition for the
existence of sinusoidal steady-state response to sinusoidal
excitation. This leads to the following expression for the
back-EMF estimation error:
 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
r
Fv ) 

 e r
(11)
where ω r is the actual angular rotor speed and
Fv ( jω ) ω =ω =
r
φ Fv
ω =ω r
2
( 2h ( h + ∆ ))
2
+ ω r2
(
∆ − ω r2
− 3h
2
 ω ( ∆ − ω r2 − 3h 2 ) 

= arctang  r

2 h (h 2 + ∆ ) 

TABLE I
POSITION ERRORS
2 2
)
[h − (ω r − ∆ )(ω r + ∆ )] + (2hω r )2
2
value of gain k (which affects the LO only). Numerical
results are resumed in TABLE I. Parameters from a
commercial PM synchronous motor available in laboratory
have been considered (see the Table in Appendix), aiming at
comparing simulation and experimental verifications.
For the same gain (l) value, the SMO has a smaller
position error with respect to the LO. An increment of the
gain l reduces the position error, which nevertheless reaches
non-negligible values at high speed. Thereafter a
compensation mechanism of the position error with the speed
is necessary. The same relations (15) for LO and (16) for
SMO can be assumed as first-attempt (theoretical)
compensation laws. Obviously, discrepancies and nonmodelled effects could lead to correct the theoretical
compensation in the actual implementation.
Luenberger
(12)
800
Splitting the vector equation (11) into its components gives:
 e eα (t ) = Av cos (ω r t + φ Fv )

 e eβ (t ) = Av sin (ω r t + φ Fv )
∞
←k
-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
(All results are obtained with gain l = −28.28)
0
Position estimation error [degrees]
Equation (13) provides a mean to calculate the αβ
estimated components of the back-EMF as follows:
vˆ iα = viα + e eα (t )
= − k e ω r sinω r t + Av cos (ω r t + φ Fv )
vˆ iβ = viβ + e eβ (t)
(14)
= k e ω r cosω r t + Av sin (ω r t + φ Fv )
From (14) it is straightforward to calculate the estimation
error for the magnet axis position. After some simplifications
one obtains:




-5
-5.078
-6.22
-10.079
-10
-14.928
LO
-15
SMO
LO
-20
SMO
-25
0
l = −28.28; k = 1600
-13.22
l = −28.28
l = −56.56; k = 1600
-21.99
l = −56.56
500
1000
1500
2000
2500
3000
Rotor speed [RPM]
(15)
Assuming a “high-current-gain” LO ( k → ∞ ) the ideal
SMO is achieved for the current estimation (that means
e i = e i = 0 ). Equation (15) is then obtained as follows:
θ̂r − θr = arctg (ω r Ls l )
16000
Position error (θ̂r − θr ) [degrees]
(13)
where Av = Fv ( jω r ) k e ω 2r .

Av cos φFv
θ̂r − θr = arctg  −
 k e ω r + Av sin φF
v

1600
SMO
(16)
For a given set of motor parameters, the (steady state)
position error depends of the rotor speed, while it is
independent of the feeding current that means, according to
the vector controller set-up, independency of the load torque.
Fig. 3 shows such a behaviour vs. speed for both LO (15) and
ideal SMO (16), for two different values of gain l and a given
Fig. 3. Position estimation error vs. speed.
V. SIMULATION RESULTS
For completeness of analysis, a Matlab/Simulink timecontinuous model of the transducer-less drive in Fig. 2 has
been developed, with the magnet axis position computed by
(9) in order to avoid the influence of Kalman filtering.
Results in Figs. 4 and 5 refer to the no-load transient from
0 to 1000 RPM, “off-line” LO and ideal SMO respectively,
and 1.2 Nm load torque (60% of the rated one) inserted at
0.35s. Gain values compatible for an actual implementation
are assumed. The position errors obtained from LO and SMO
and those with compensation, achieved subtracting the
(respective) analytical values from the previous ones, are
5089
Luenberger Observer
1
Position estimation error [degrees]
0
0.5e-3
-1
-2
Position estimation error
Compensated position estimation error
-3
-4
-5
-6
load insertion
-6.22
-7
-8
0.05
0.1
0.15
0.2
0.25
0.3
l = −28.28
k = 1600
0.35
0.4
0.45
0.5
Time [s]
Fig. 4. Position estimation error with LO.
Sliding Mode Observer
1
Position estimation error [degrees]
0
-5.5e-3
-1
-2
Position estimation error
Compensated position estimation error
-3
Fig. 6. Experimental drive system.
-4
-5
-5.08
load insertion
-6
l = −28.28
-7
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Time [s]
Fig. 5. Position estimation error with SMO.
reported. The numerical values of the position errors shown
on the plots (uncompensated case) can be found to match
those reported in the second and fourth columns of TABLE I
respectively. The independence of the position error from the
load conditions is confirmed at steady state. It is interesting
to notice that compensation, although carried out by the
steady state laws, gives quite good results also during
transient operation.
VI. EXPERIMENTAL RESULTS
The performance of the proposed transducer-less backEMF based strategy has been investigated using the
experimental drive system shown in Fig. 6, in order to
evaluate the discrepancies between theory and practice due to
the un-modelled effects in the real system.
The experimental drive consists of a single board drive
unit, the (already mentioned) commercial PM synchronous
motor and the necessary development and testing tools. The
drive unit includes the control hardware and an integrated
IGBT based Intelligent Power Module (IPM). The phase
currents are measured on the three lower branches of the
inverter by means of resistive sensors integrated in the IPM.
The control hardware makes use of a fixed point Digital
Signal Processor (DSP) TMS320F240. Set points and main
parameters of the control scheme can be changed in real-time
by means of a host PC linked to the DSP through a standard
RS-232 interface. The host PC is also used to run the DSP
development and debugger tools. A scope has been used to
display in real-time the variables calculated inside the control
algorithm by means of a 2 channels digital-to-analog
interface, which is mapped on the I/O addressing space of the
µC DSP. The execution of the control routine (including the
back-EMF observer) is synchronized to the modulation
carrier. The PWM period has been fixed to 150 µs, resulting
in 6.6 kHz switching frequency. At each PWM period the
current control loop, the observer with Kalman filter and the
speed regulator are executed.
During the development of the control program, a
4096 ppr incremental encoder is employed to measure the
actual position and speed and compare them with the
estimated ones. As mentioned, computation of the estimated
position is not necessary in the transducer-less strategy,
which makes use of the estimated “sine/cosine” functions
provided by the Kalman filter. Thereafter, the estimated
position has been computed for monitoring and analysing
purposes only, applying an inverse trigonometric function to
one output of the filter:
θ̂r ,mon = arccos ( cosθ̂(r1) )
(17)
The LO and SMO algorithms are obtained through
discretization of the time-continuous relations (3) and (4): in
comparing experimental and theoretical results one can
consider the gains of the actual implementation equal to:
5090
k =1600 and l = −28.28 (these are the gains of the equivalent
time-continuous observers).
In the first test case, the steady state operation at different
speeds (and no-load) is investigated: the speed set-point is
changed in sequence from 500 to 2000 rpm (with 500 rpm
step) using a slow ramp-varying reference.
Figures from 7 to 9 refer to the SMO operation, they show
the traces of the position error (in electrical degrees) in
comparison with the actual speed, and particularly:
- Fig. 7 shows the case without any compensation: one
notices that the mean value of the position error is a function
of actual rotor speed, thus in good agreement with theoretical
expression (16) and the simulation analysis. One notices also
that the computed error is affected by a noise band whose
amplitude in the order of 3 degrees around the mean value.
This noise can be attributed to the residual “chattering” of
SMO, which is not completely eliminated by the Kalman
filter;
- Fig. 8 shows the case when self-compensation is
implemented according to the (theoretical) expression (16):
one notices that the adopted formula gives appreciable results
but not an exact compensation over the whole speed range, as
the mean value of the (absolute) error is reduced to 3.5
degrees in the worst high-speed case (2000 rpm), with
respect to 7 degrees without compensation;
- Fig. 9 shows the case when an empirical self-compensation
is implemented, according to the experimental measurement
of the position error in Fig. 7. In this case, the same linear
compensation strategy is adopted but with a
different value of the slope. One can notice that the mean
value of the error is zero over the whole speed range, thus
confirming the validity of the proposed theoretical analysis.
1 [V] → 2.25 [degrees]
1 [V] → 600 [RPM ]
ωr
(θ̂ r , mon − θ r )
1 [V] → 2.25 [degrees]
1 [V] → 600 [R PM ]
ωr
( θ̂ r , mon − θ r )
Fig. 7. Position estimation error with SMO
at different speed values (from 500 to 2000 rpm).
1 [V] → 2.25 [degrees]
1 [V] → 600 [RPM]
ωr
( θ̂ r , mon − θ r )
Fig. 8. Position estimation with SMO and theoretical compensation
at different speed values (from 500 to 2000 rpm).
Fig. 9. Position estimation with SMO and empirical compensation
at different speed values (from 500 to 2000 rpm).
The same kind of test with step variation of the speed setpoint is repeated for the LO (Figs. 10 and 11):
- Fig. 10 shows the case without any compensation: one
notices that the mean value of the position error increases
with the speed similarly but being greater than the one with
SMO at the same speed. The results are in good agreement
with theoretical and simulation analyses. The noise affecting
the error signal is less important than with SMO: it appears
concentrated in particular zones and is related to the low
accuracy of the actual implementation of the inverse
trigonometric function (17);
- Fig. 11 shows the case when the empirical selfcompensation is implemented, according to the experimental
measurement of the position error in Fig. 10: same as for
SMO, the mean value of the error is forced to zero at each
operating speed.
The influence of the load torque on the performance of the
back-EMF observer has been investigated too, both without
and with self-compensation. Results in the case of the SMO
are presented: the overall behaviour of the rotor position
error vs. the load torque is summarized in Fig. 12, over a
large torque-speed range (tests cover the variation of the load
torque from zero to 75% of the rated one, with 0.5 Nm step,
and speed from 500 to 2500 rpm). Differently from theory
(where the independence of the load torque is found) one
notices an effect of the load on the position error, although
quite small (not greater than two degrees in all the tested
conditions); compensation by the theoretical law is right in
5091
direction but the quantity is excessive for increasing speed:
nevertheless the maximum (absolute) error is limited to about
5 degrees at the maximum tested speed.
exact compensation requires a proper adjustment by
experimental tests.
1 [V ] → 2.25 [degrees]
1 [V ] → 600 [R PM ]
ωr
( θ̂ r , mon − θ r )
Fig. 12. Position estimation error vs. speed with SMO
as a function of load torque.
Fig. 10. Position estimation error with LO
at different speed values (from 500 to 2000 rpm).
APPENDIX
MOTOR PARAMETERS
rated power /current
rated speed/torque
pole pairs
no-load back-EMF @ rated speed
stator resistance
stator inductance
rotor inertia
inverter DC voltage
1 [V ] → 2.25 [degrees]
1 [V ] → 600 [R PM ]
ωr
630 W - 2.5 A rms (*)
3000 rpm (*) - 2.0 Nm (*)
4
82.72 Vrms (*)
1.9 Ω
6 mH
22⋅10-5 kgm2
300 V
(*): base values assumed for displaying results
VIII. REFERENCES
( θ̂ r , mon − θ r )
[1]
[2]
Fig. 11. Position estimation with LO and empirical compensation
at different speed values (from 500 to 2000 rpm).
VII. CONCLUSIONS
[3]
The possibility to compensate the intrinsic position
estimation error in a transducer-less control scheme for PM
synchronous based on back-EMF observers and Kalman
filter has been analysed. Basic results are as follows:
- 1st: it is possible to evaluate a theoretical compensation law
for the steady-state operation, both with Luenberger and
Sliding Mode observers;
- 2nd: such laws yield satisfactory compensation also during
fast transients;
- 3rd: 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-state, and negligible one
during transients if slow ramp-varying speed reference is
used;
- 4th: the un-modelled effects in the actual drive, motor and
real-time system introduce some discrepancies from theory:
[4]
[5]
[6]
[7]
[8]
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