Two back-EMF-based position observers are compared for motion-sensorless synchronous reluctance motor drives. The reduced-order observer is of the second order, and the adaptive full-order observer is of the fourth order. The proposed design rules guarantee the stability of the adaptive full-order observer, if the parameter estimates are accurate. The observers are experimentally evaluated using a 6.7-kW synchronous reluctance motor drive in low-speed operation and under parameter errors. The gain selection of the second-order observer is easier, but the adaptive full-order observer is more robust against parameter variations and spatial harmonics.Peer reviewe

Hinkkanen, Marko

Luomi, Jorma

Tuovinen, Toni

Aaltodoc

This is an electronic reprint of the original article.
This reprint may differ from the original in pagination and typographic detail.
Author(s): Tuovinen, Toni & Hinkkanen, Marko & Luomi, Jorma
Title: A comparison of an adaptive full-order observer and a reduced-order
observer for synchronous reluctance motor drives
Year: 2011
Version: Post print
Please cite the original version:
Tuovinen, Toni & Hinkkanen, Marko & Luomi, Jorma. 2011. A comparison of an adaptive
full-order observer and a reduced-order observer for synchronous reluctance motor
drives. 2011 Symposium on Sensorless Control for Electrical Drives (SLED). 5. ISBN
978-1-4577-1853-3 (electronic). DOI: 10.1109/sled.2011.6051555.
Rights: © 2011 Institute of Electrical & Electronics Engineers (IEEE). Permission from IEEE must be obtained for all
other uses, in any current or future media, including reprinting/republishing this material for advertising or
promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of
any copyrighted component of this work in other work.
All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, and
duplication or sale of all or part of any of the repository collections is not permitted, except that material may
be duplicated by you for your research use or educational purposes in electronic or print form. You must
obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or
otherwise to anyone who is not an authorised user.
Powered by TCPDF (www.tcpdf.org)
1A Comparison of an Adaptive Full-Order Observer
and a Reduced-Order Observer for Synchronous
Reluctance Motor Drives
Toni Tuovinen, Marko Hinkkanen, and Jorma Luomi
Aalto University School of Electrical Engineering
Department of Electrical Engineering, P.O. Box 13000, FI-00076 Aalto, Finland
Abstract—Two back-EMF-based position observers are com-
pared for motion-sensorless synchronous reluctance motor drives.
The reduced-order observer is of the second order, and the
adaptive full-order observer is of the fourth order. The proposed
design rules guarantee the stability of the adaptive full-order ob-
server, if the parameter estimates are accurate. The observers are
experimentally evaluated using a 6.7-kW synchronous reluctance
motor drive in low-speed operation and under parameter errors.
The gain selection of the second-order observer is easier, but the
adaptive full-order observer is more robust against parameter
variations and spatial harmonics.
Index Terms—Observer, stability conditions, speed sensorless,
parameter uncertainties.
I. INTRODUCTION
Modern synchronous reluctance motors (SyRMs) are be-
coming interesting competitors to induction motors and
permanent-magnet synchronous motors in variable-speed
drives [1], [2]. The rotor position of a synchronous motor has
to be known with good accuracy in order to obtain stable
operation and high performance. The rotor position can be
either measured or estimated. Motion-sensorless control based
on position estimation is usually preferable: motion sensors are
expensive, they can be damaged or, in some environments and
applications, cannot be installed.
Position estimation methods based on signal injection can be
used for SyRMs. In order to avoid additional noise and losses,
it is desirable to use a method based on the back electromotive
force (EMF), and combine a signal-injection method with it
only at the lowest speeds [3], [4]. Since SyRMs are closely
related to permanent-magnet synchronous motors (PMSMs),
back-EMF based estimation methods proposed for PMSMs,
for example the observers proposed in [5], [6] and [7], can be
used for SyRMs with slight modifications.
The gain selection of back-EMF-based observers is crucial
in order to obtain good performance and robustness against
measurement errors and parameter variations. As the order
of the observer increases, the analysis becomes increasingly
complicated, and it might be difficult to derive the stability
conditions even if the parameter estimates are accurate. Hence,
a low-order observer is an attractive design goal.
In this paper, two different observers are compared using
a 6.7-kW SyRM drive. The reduced-order observer is of the
second order, and the adaptive full-order observer is of the
fourth order. For the adaptive full-order observer, design rules
are proposed, simplifying the stability analysis. The properties
evaluated experimentally are the sensitivity to parameter un-
certainties, sensitivity to harmonic noise, behavior during load
transients, and stability in low-speed operation.
II. SYRM MODEL AND ROTOR-POSITION OBSERVERS
A. Model
Real space vectors will be used here. For example, the
stator-current vector is is = [id, iq]T, where id and iq are
the components of the vector and the matrix transpose is
marked with the superscript T. The orthogonal rotation matrix
is defined as
J =
[
0 −1
1 0
]
.
The electrical position of the d axis is denoted by ϑm. The
d axis is defined as the direction of the maximum inductance
of the rotor. The position depends on the electrical angular
rotor speed ωm according to
dϑm
dt
= ωm (1a)
To simplify the analysis in the following sections, the machine
model will be expressed in the estimated rotor reference frame,
whose d axis is aligned at ϑˆm with respect to the stator
reference frame. The stator inductance is
L = e−ϑ˜mJ
[
Ld 0
0 Lq
]
eϑ˜mJ (1b)
where ϑ˜m = ϑˆm − ϑm is the estimation error in the rotor
position, Ld the direct-axis inductance, and Lq the quadrature-
axis inductance. The voltage equation is
dψs
dt
= us −Rsis − ωˆmJψs (1c)
where ψs is the stator-flux vector, us the stator-voltage vector,
Rs the stator resistance, and ωˆm = dϑˆm/dt is the angular
speed of the coordinate system. The stator current is a non-
linear function
is = L
−1ψs (1d)
of the stator-flux vector and the position error ϑ˜m.
2B. Reduced-Order Observer
The reduced-order observer proposed in [7] is considered. It
is based on estimating the rotor position and the d component
ψˆd of the stator flux in the estimated rotor coordinates. For a
SyRM, the componentwise presentation of the observer is
dψˆd
dt
= ud − Rˆsid + ωˆmLˆqiq + k1(ψˆd − Lˆdid) (2a)
dϑˆm
dt
=
uq − Rˆsiq − Lˆq diqdt + k2(ψˆd − Lˆdid)
ψˆd
(2b)
where Lˆd and Lˆq are the estimated d and q axis inductances,
respectively, Rˆs is the estimated stator resistance, and k1 and
k2 are the observer gains. The observer is of the second order,
and there are only two gains.
With accurate parameter estimates, the closed-loop system
consisting of (1) and (2) is locally stable in every operating
point if the gains are given by
k1 = −b+ β(c/ωˆm − ωˆm)
β2 + 1
, k2 =
βb − c/ωˆm + ωˆm
β2 + 1
(3)
where β = iq/id, and the design parameters b > 0 and c > 0
may depend on the operating point [7]. The parameters b and
c are actually the coefficients of the characteristic polynomial,
s2 + bs + c, of the linearized system consisting of (1) and
(2). The relation between the two design parameters for the
reduced-order observer is chosen as
c =
√
3b|ωˆm|+ ωˆ2m (4)
which guarantees maximum robustness against parameter er-
rors in low-speed operation [8].
C. Adaptive Full-Order Observer
In the adaptive full-order observer [5], [6], both stator-flux
vector components are estimated,
dψˆs
dt
= us − Rˆsiˆs − ωˆmJψˆs −Ki˜s (5a)
iˆs = Lˆ
−1
ψˆs (5b)
where iˆs is the estimated stator-current vector, the estimation
error of the stator current is i˜s = iˆs − is and K is the gain
matrix.
The rotor speed is estimated with the PI mechanism
ωˆm = kpi˜s + ki
∫
i˜sdt (6)
The gain vectors kp and ki are chosen to utilize the estimation
error only in the q axis direction,
kp = [0, kp], ki = [0, ki] (7)
The proposed gain matrix is
K =
[−Lˆdk1 − Rˆs Lˆqβk1
−Lˆdk2 Lˆqβk2 − Rˆs
]
(8)
where k1 and k2 are given by (3). For convenience, the gains
kp and ki are selected according to
kp =
Lˆqd
(Lˆd − Lˆq)id
(9a)
ki =
Lˆqe
(Lˆd − Lˆq)id
(9b)
where d and e are design parameters, which may depend on
the rotor speed. With this gain selection, the characteristic
polynomial of the closed-loop system consisting of (1), (5),
(6), (7), (8) and (9) can, after linearization, be split into a
product of two second-order polynomials,
(s2 + bs+ c)(s2 + ds+ e) (10)
and the stability is guaranteed for all positive values of b, c,
d and e, if the parameter estimates are accurate. The observer
is of the fourth order, and there are four gains. It can be seen
that the characteristic polynomial (10) is closely related to the
reduced-order observer, since the characteristic polynomial of
the reduced-order observer is s2 + bs+ c.
The proposed observer design is a subset of all stable
design possibilities, since the gain matrix (8) has only two
free parameters instead of four. With the proposed design,
however, the stability analysis based on (10) is considerably
simpler than that of a more general fourth-order polynomial.
III. EXPERIMENTAL SETUP AND PARAMETERS
The motion-sensorless control system was implemented in a
dSPACE DS1104 PPC/DSP board. A 6.7-kW four-pole SyRM
was fed by a frequency converter that is controlled by the
DS1104 board. The rated values of the SyRM are: speed 3175
r/min; frequency 105.8 Hz; line-to-line rms voltage 370 V;
rms current 15.5 A; and torque 20.1 Nm. The base values for
angular speed, voltage, and current are defined as 2pi · 105.8
rad/s,
√
2/3 · 370 V, and √2 · 15.5 A, respectively.
A servo motor was used as a loading machine. The rotor
speed ωm and position ϑm were measured using an incremen-
tal encoder for monitoring purposes. The shaft torque Tm was
measured using a Dataflex 22 torque measuring shaft. The total
moment of inertia of the experimental setup is 0.015 kgm2 (2.7
times the inertia of the SyRM rotor).
The stator currents and the DC-link voltage were measured,
and the reference voltage obtained from the current controller
was used for the observer. The sampling was synchronized
to the modulation, and both the switching frequency and the
sampling frequency were 8 kHz. A simple current feedforward
compensation for dead times and power device voltage drops
was applied.
The control system was augmented with a speed controller,
whose feedback signal was the speed estimate ωˆm obtained
from the proposed observer. The bandwidth of this PI con-
troller, including active damping [9], was 2pi · 5.3 rad/s (0.05
p.u.). The estimate of the per-unit electromagnetic torque was
evaluated as Tˆe = (Ld − Lq)idiq.
The gain values were chosen based on empirical results.
The gain b = 2 was used for the reduced-order observer. For
3TABLE I
PER-UNIT PARAMETERS FOR SATURATION MODEL
Ld0 Lq0 α γ δ id0 iq0
3.15 0.685 2.24 0.353 0.085 0.2 0.2
the adaptive full-order observer, the gains were: b = 0.05,
c = 0.025|ωˆm|+ ωˆ2m, d = 2 and e = 1.
The saturation has been modeled as functions of the mea-
sured current,
Ld =


Ld0 − αid − δ
∣∣∣ iq
id0
∣∣∣ , if id ≤ id0
Ld0 − αid − δ
∣∣∣ iq
id
∣∣∣ , otherwise (11a)
Lq =


Lq0 − γ|iq| − δ
∣∣∣ id
iq0
∣∣∣ , if iq ≤ iq0
Lq0 − γ|iq| − δ
∣∣∣ id
iq
∣∣∣ , otherwise (11b)
where id0 and iq0 are transition values for id and iq to avoid
divisions by small numbers. The saturation model parameters
are given in Table I. The estimated stator resistance is Rˆs =
0.042 p.u.
IV. EXPERIMENTAL RESULTS
Fig. 1 shows the effect of the parameter errors on the
position estimation error at the speed ωˆm = 0.1 p.u. with
50% rated load torque applied. The reduced-order observer is
used in Fig. 1(a), and the adaptive full-order observer is used
in Fig. 1(b). The data is captured by varying each parameter
estimate from 90% up to 110% of the actual value in 10
seconds. It can be seen that the model parameters Rˆs and
Lˆq have only a small effect on the position error, whereas
incorrect value for Lˆd increases the estimation error rapidly in
Fig. 1(a), when the reduced-order observer is used. According
to Fig. 1(b), the adaptive full-order observer is less sensitive to
parameter errors. It should be noted that the relative errors of
Ld and Lq are defined with respect to the (original) operation
point values. As the estimation error increases, the actual
values of id and iq change, resulting changes in actual values
of Ld and Lq due to saturation.
Experimental results of a stepwise speed reversal from
ωˆm = 0.10 p.u. to ωˆm = −0.10 p.u. and back to 0.10 p.u.
with rated load torque applied are depicted in Fig. 2. The
reduced-order observer is used in Fig. 2(a), and the adaptive
full-order observer is used in Fig. 2(b). It can be seen that the
reduced-order observer amplifies the estimation noise in the
regenerating mode in Fig. 2(a). This behavior is analyzed in
[8]. With the adaptive full-order observer, the amplitude of the
estimation noise does not depend on the operating mode, as
seen in Fig. 2(b).
When the reduced-order observer is used, spatial harmonics
cause noise in the position estimate in the regenerating mode
if the machine is highly saturated [8]. Results of a slow change
of id from 0.3 p.u. to 0.5 p.u. with −50% rated load torque
applied are shown in Fig. 3. The reduced-order observer is
used in Fig. 3(a), and the adaptive full-order observer is used in
Fig. 3(b). It can be seen that as id increases in the regenerating
mode, the noise in the position estimate of the reduced-order
observer increases, but the adaptive full-order observer is not
sensitive to spatial harmonics.
Experimental results of load-torque steps when the speed
reference was kept at 0.05 p.u. are shown in Fig. 4. The
load torque was stepped to −75% of the rated load torque
at t = 2.5 s, reversed at t = 7.5 s, and removed at t =
12.5 s. The reduced-order observer is used in Fig. 4(a), and
the adaptive full-order observer is used in Fig. 4(b). It can
be seen that the observers behave well in load transients in
low-speed operation.
V. CONCLUSIONS
In this paper, two back-EMF-based observers are compared
for synchronous reluctance motor drives. The gain selection of
the observers is crucial in sustained low-speed operation. The
analysis and tuning of the reduced-order observer is consider-
ably easier than that of the adaptive full-order observer. The
adaptive full-order observer has four design parameters, and
the reduced-order observer has only one design parameter. The
proposed design rules guarantee the stability of the adaptive
full-order observer, if the parameter estimates are accurate.
The experimental results indicate that the adaptive full-order
observer is less sensitive to harmonic noise and parameter
uncertainties than the reduced-order observer.
ACKNOWLEDGMENT
The authors gratefully acknowledge ABB Oy for the finan-
cial support.
REFERENCES
[1] E. Capecchi, P. Guglielmo, M. Pastorelli, and A. Vagati, “Position-
sensorless control of the transverse-laminated synchronous reluctance
motor,” IEEE Trans. Ind. Appl., vol. 37, no. 6, pp. 1768–1776, Nov./Dec.
2001.
[2] H. F. Hofmann, S. R. Sanders, and A. EL-Antably, “Stator-flux-oriented
vector control of synchronous reluctance machines with maximized
efficiency,” IEEE Trans. Ind. Electron., vol. 51, no. 5, pp. 1066–1072,
Oct. 2004.
[3] A. Piippo, M. Hinkkanen, and J. Luomi, “Sensorless control of PMSM
drives using a combination of voltage model and HF signal injection,” in
Conf. Rec. IEEE-IAS Annu. Meeting, vol. 2, Seattle, WA, Oct. 2004, pp.
964–970.
[4] O. Wallmark, L. Harnefors, and O. Carlson, “An improved speed and
position estimator for salient permanent-magnet synchronous motors,”
IEEE Trans. Ind. Electron., vol. 52, no. 1, pp. 255–262, Feb. 2005.
[5] G. Yang, R. Tomioka, M. Nakano, and T. H. Chin, “Position and speed
sensorless control of brushless DC motor based on an adaptive observer,”
IEEJ Trans. Ind. Appl., vol. 113, pp. 579–586, May 1993.
[6] A. Piippo, M. Hinkkanen, and J. Luomi, “Analysis of an adaptive observer
for sensorless control of interior permanent magnet synchronous motors,”
IEEE Trans. Ind. Electron., vol. 55, no. 2, pp. 570–576, Feb. 2008.
[7] M. Hinkkanen, T. Tuovinen, L. Harnefors, and J. Luomi, “A combined
position and stator-resistance observer for salient PMSM drives: design
and stability analysis,” IEEE Trans. Power Electron., no. 99, 2011,
doi:10.1109/tpel.2011.2118232, in press.
[8] T. Tuovinen, M. Hinkkanen, and J. Luomi, “Analysis and design of a
position observer with resistance adaptation for synchronous reluctance
motor drives,” in Proc. IEEE ECCE’11, Phoenix, AZ, Sep. 2011, in press.
[9] L. Harnefors, “Design and analysis of general rotor-flux-oriented vector
control systems,” IEEE Trans. Ind. Electron., vol. 48, no. 2, pp. 383–390,
Apr. 2001.
40.9 0.95 1 1.05 1.1
−15
0
15
0.9 0.95 1 1.05 1.1
−5
0
5
0.9 0.95 1 1.05 1.1
−5
0
5
Lˆd/Ld
Lˆq/Lq
Rˆs/Rs
ϑ˜
m
(de
g)
ϑ˜
m
(de
g)
ϑ˜
m
(de
g)
(a)
0.9 0.95 1 1.05 1.1
−15
0
15
0.9 0.95 1 1.05 1.1
−5
0
5
0.9 0.95 1 1.05 1.1
−5
0
5
Lˆd/Ld
Lˆq/Lq
Rˆs/Rs
ϑ˜
m
(de
g)
ϑ˜
m
(de
g)
ϑ˜
m
(de
g)
(b)
Fig. 1. Measured errors in the position estimate at ωˆm = 0.1 p.u. with 50% rated load torque applied: (a) reduced-order observer and (b) adaptive full-order
observer. The data is captured by varying each model parameter from 90% up to 110% of the actual value in 10 seconds.
−0.15
0
0.15
 
 
0 1 2 3 4 5
−15
0
15
ω
m
(p.
u
.
)
ϑ˜
m
(de
g)
t (s)
ωref
ωˆm
ωm
(a)
−0.15
0
0.15
 
 
0 1 2 3 4 5
−15
0
15
ω
m
(p.
u
.
)
ϑ˜
m
(de
g)
t (s)
ωref
ωˆm
ωm
(b)
Fig. 2. Experimental results of a stepwise speed reversal (0.10 p.u. → −0.10 p.u. → 0.10 p.u.) with rated load torque applied: (a) reduced-order observer,
(b) adaptive full-order observer.
5−1
0
1
 
 
0 4 8 12
−15
0
15
i s
(p.
u
.
)
ϑ˜
m
(de
g)
t (s)
id
iq
(a)
−1
0
1
 
 
0 4 8 12
−15
0
15
i s
(p.
u
.
)
ϑ˜
m
(de
g)
t (s)
id
iq
(b)
Fig. 3. Experimental results of a slow change in id from 0.3 p.u. to 0.5 p.u. with −50% rated load torque applied: (a) reduced-order observer, (b) adaptive
full-order observer.
−0.2
0
0.2
 
 
−2
0
2
 
 
0 5 10 15
−15
0
15
T
(p.
u
.
)
ω
m
(p.
u
.
)
ϑ˜
m
(de
g)
t (s)
ωˆm
ωm
Tˆe
Tm
(a)
−0.2
0
0.2
 
 
−2
0
2
 
 
0 5 10 15
−15
0
15
T
(p.
u
.
)
ω
m
(p.
u
.
)
ϑ˜
m
(de
g)
t (s)
ωˆm
ωm
Tˆe
Tm
(b)
Fig. 4. Experimental results showing load-torque steps (0 → −75% rated → 75% rated) when the speed reference is kept at 0.05 p.u.: (a) reduced-order
observer, (b) adaptive full-order observer.


English

A comparison of an adaptive full-order observer and a reduced-order observer for synchronous reluctance motor drives

Toni Tuovinen

Marko Hinkkanen

Jorma Luomi

Crossref

Aaltodoc Publication Archive

This is an electronic reprint of the original article.This reprint may differ from the original in pagination and typographic detail.Author(s): Tuovinen, Toni & Hinkkanen, Marko & Luomi, JormaTitle: A comparison of an adaptive full-order observer and a reduced-orderobserver for synchronous reluctance motor drivesYear: 2011Version: Post printPlease cite the original version:Tuovinen, Toni & Hinkkanen, Marko & Luomi, Jorma. 2011. A comparison of an adaptivefull-order observer and a reduced-order observer for synchronous reluctance motordrives. 2011 Symposium on Sensorless Control for Electrical Drives (SLED). 5. ISBN978-1-4577-1853-3 (electronic). DOI: 10.1109/sled.2011.6051555.Rights: © 2011 Institute of Electrical & Electronics Engineers (IEEE). Permission from IEEE must be obtained for allother uses, in any current or future media, including reprinting/republishing this material for advertising orpromotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse ofany copyrighted component of this work in other work.All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, andduplication or sale of all or part of any of the repository collections is not permitted, except that material maybe duplicated by you for your research use or educational purposes in electronic or print form. You mustobtain permission for any other use. Electronic or print copies may not be offered, whether for sale orotherwise to anyone who is not an authorised user.Powered by TCPDF (www.tcpdf.org)1A Comparison of an Adaptive Full-Order Observerand a Reduced-Order Observer for SynchronousReluctance Motor DrivesToni Tuovinen, Marko Hinkkanen, and Jorma LuomiAalto University School of Electrical EngineeringDepartment of Electrical Engineering, P.O. Box 13000, FI-00076 Aalto, FinlandAbstract—Two back-EMF-based position observers are com-pared for motion-sensorless synchronous reluctance motor drives.The reduced-order observer is of the second order, and theadaptive full-order observer is of the fourth order. The proposeddesign rules guarantee the stability of the adaptive full-order ob-server, if the parameter estimates are accurate. The observers areexperimentally evaluated using a 6.7-kW synchronous reluctancemotor drive in low-speed operation and under parameter errors.The gain selection of the second-order observer is easier, but theadaptive full-order observer is more robust against parametervariations and spatial harmonics.Index Terms—Observer, stability conditions, speed sensorless,parameter uncertainties.I. INTRODUCTIONModern synchronous reluctance motors (SyRMs) are be-coming interesting competitors to induction motors andpermanent-magnet synchronous motors in variable-speeddrives [1], [2]. The rotor position of a synchronous motor hasto be known with good accuracy in order to obtain stableoperation and high performance. The rotor position can beeither measured or estimated. Motion-sensorless control basedon position estimation is usually preferable: motion sensors areexpensive, they can be damaged or, in some environments andapplications, cannot be installed.Position estimation methods based on signal injection can beused for SyRMs. In order to avoid additional noise and losses,it is desirable to use a method based on the back electromotiveforce (EMF), and combine a signal-injection method with itonly at the lowest speeds [3], [4]. Since SyRMs are closelyrelated to permanent-magnet synchronous motors (PMSMs),back-EMF based estimation methods proposed for PMSMs,for example the observers proposed in [5], [6] and [7], can beused for SyRMs with slight modifications.The gain selection of back-EMF-based observers is crucialin order to obtain good performance and robustness againstmeasurement errors and parameter variations. As the orderof the observer increases, the analysis becomes increasinglycomplicated, and it might be difficult to derive the stabilityconditions even if the parameter estimates are accurate. Hence,a low-order observer is an attractive design goal.In this paper, two different observers are compared usinga 6.7-kW SyRM drive. The reduced-order observer is of thesecond order, and the adaptive full-order observer is of thefourth order. For the adaptive full-order observer, design rulesare proposed, simplifying the stability analysis. The propertiesevaluated experimentally are the sensitivity to parameter un-certainties, sensitivity to harmonic noise, behavior during loadtransients, and stability in low-speed operation.II. SYRM MODEL AND ROTOR-POSITION OBSERVERSA. ModelReal space vectors will be used here. For example, thestator-current vector is is = [id, iq]T, where id and iq arethe components of the vector and the matrix transpose ismarked with the superscript T. The orthogonal rotation matrixis defined asJ =[0 −11 0].The electrical position of the d axis is denoted by ϑm. Thed axis is defined as the direction of the maximum inductanceof the rotor. The position depends on the electrical angularrotor speed ωm according todϑmdt= ωm (1a)To simplify the analysis in the following sections, the machinemodel will be expressed in the estimated rotor reference frame,whose d axis is aligned at ϑˆm with respect to the statorreference frame. The stator inductance isL = e−ϑ˜mJ[Ld 00 Lq]eϑ˜mJ (1b)where ϑ˜m = ϑˆm − ϑm is the estimation error in the rotorposition, Ld the direct-axis inductance, and Lq the quadrature-axis inductance. The voltage equation isdψsdt= us −Rsis − ωˆmJψs (1c)where ψs is the stator-flux vector, us the stator-voltage vector,Rs the stator resistance, and ωˆm = dϑˆm/dt is the angularspeed of the coordinate system. The stator current is a non-linear functionis = L−1ψs (1d)of the stator-flux vector and the position error ϑ˜m.2B. Reduced-Order ObserverThe reduced-order observer proposed in [7] is considered. Itis based on estimating the rotor position and the d componentψˆd of the stator flux in the estimated rotor coordinates. For aSyRM, the componentwise presentation of the observer isdψˆddt= ud − Rˆsid + ωˆmLˆqiq + k1(ψˆd − Lˆdid) (2a)dϑˆmdt=uq − Rˆsiq − Lˆq diqdt + k2(ψˆd − Lˆdid)ψˆd(2b)where Lˆd and Lˆq are the estimated d and q axis inductances,respectively, Rˆs is the estimated stator resistance, and k1 andk2 are the observer gains. The observer is of the second order,and there are only two gains.With accurate parameter estimates, the closed-loop systemconsisting of (1) and (2) is locally stable in every operatingpoint if the gains are given byk1 = −b+ β(c/ωˆm − ωˆm)β2 + 1, k2 =βb − c/ωˆm + ωˆmβ2 + 1(3)where β = iq/id, and the design parameters b > 0 and c > 0may depend on the operating point [7]. The parameters b andc are actually the coefficients of the characteristic polynomial,s2 + bs + c, of the linearized system consisting of (1) and(2). The relation between the two design parameters for thereduced-order observer is chosen asc =√3b|ωˆm|+ ωˆ2m (4)which guarantees maximum robustness against parameter er-rors in low-speed operation [8].C. Adaptive Full-Order ObserverIn the adaptive full-order observer [5], [6], both stator-fluxvector components are estimated,dψˆsdt= us − Rˆsiˆs − ωˆmJψˆs −Ki˜s (5a)iˆs = Lˆ−1ψˆs (5b)where iˆs is the estimated stator-current vector, the estimationerror of the stator current is i˜s = iˆs − is and K is the gainmatrix.The rotor speed is estimated with the PI mechanismωˆm = kpi˜s + ki∫i˜sdt (6)The gain vectors kp and ki are chosen to utilize the estimationerror only in the q axis direction,kp = [0, kp], ki = [0, ki] (7)The proposed gain matrix isK =[−Lˆdk1 − Rˆs Lˆqβk1−Lˆdk2 Lˆqβk2 − Rˆs](8)where k1 and k2 are given by (3). For convenience, the gainskp and ki are selected according tokp =Lˆqd(Lˆd − Lˆq)id(9a)ki =Lˆqe(Lˆd − Lˆq)id(9b)where d and e are design parameters, which may depend onthe rotor speed. With this gain selection, the characteristicpolynomial of the closed-loop system consisting of (1), (5),(6), (7), (8) and (9) can, after linearization, be split into aproduct of two second-order polynomials,(s2 + bs+ c)(s2 + ds+ e) (10)and the stability is guaranteed for all positive values of b, c,d and e, if the parameter estimates are accurate. The observeris of the fourth order, and there are four gains. It can be seenthat the characteristic polynomial (10) is closely related to thereduced-order observer, since the characteristic polynomial ofthe reduced-order observer is s2 + bs+ c.The proposed observer design is a subset of all stabledesign possibilities, since the gain matrix (8) has only twofree parameters instead of four. With the proposed design,however, the stability analysis based on (10) is considerablysimpler than that of a more general fourth-order polynomial.III. EXPERIMENTAL SETUP AND PARAMETERSThe motion-sensorless control system was implemented in adSPACE DS1104 PPC/DSP board. A 6.7-kW four-pole SyRMwas fed by a frequency converter that is controlled by theDS1104 board. The rated values of the SyRM are: speed 3175r/min; frequency 105.8 Hz; line-to-line rms voltage 370 V;rms current 15.5 A; and torque 20.1 Nm. The base values forangular speed, voltage, and current are defined as 2pi · 105.8rad/s,√2/3 · 370 V, and √2 · 15.5 A, respectively.A servo motor was used as a loading machine. The rotorspeed ωm and position ϑm were measured using an incremen-tal encoder for monitoring purposes. The shaft torque Tm wasmeasured using a Dataflex 22 torque measuring shaft. The totalmoment of inertia of the experimental setup is 0.015 kgm2 (2.7times the inertia of the SyRM rotor).The stator currents and the DC-link voltage were measured,and the reference voltage obtained from the current controllerwas used for the observer. The sampling was synchronizedto the modulation, and both the switching frequency and thesampling frequency were 8 kHz. A simple current feedforwardcompensation for dead times and power device voltage dropswas applied.The control system was augmented with a speed controller,whose feedback signal was the speed estimate ωˆm obtainedfrom the proposed observer. The bandwidth of this PI con-troller, including active damping [9], was 2pi · 5.3 rad/s (0.05p.u.). The estimate of the per-unit electromagnetic torque wasevaluated as Tˆe = (Ld − Lq)idiq.The gain values were chosen based on empirical results.The gain b = 2 was used for the reduced-order observer. For3TABLE IPER-UNIT PARAMETERS FOR SATURATION MODELLd0 Lq0 α γ δ id0 iq03.15 0.685 2.24 0.353 0.085 0.2 0.2the adaptive full-order observer, the gains were: b = 0.05,c = 0.025|ωˆm|+ ωˆ2m, d = 2 and e = 1.The saturation has been modeled as functions of the mea-sured current,Ld =Ld0 − αid − δ∣∣∣ iqid0∣∣∣ , if id ≤ id0Ld0 − αid − δ∣∣∣ iqid∣∣∣ , otherwise (11a)Lq =Lq0 − γ|iq| − δ∣∣∣ idiq0∣∣∣ , if iq ≤ iq0Lq0 − γ|iq| − δ∣∣∣ idiq∣∣∣ , otherwise (11b)where id0 and iq0 are transition values for id and iq to avoiddivisions by small numbers. The saturation model parametersare given in Table I. The estimated stator resistance is Rˆs =0.042 p.u.IV. EXPERIMENTAL RESULTSFig. 1 shows the effect of the parameter errors on theposition estimation error at the speed ωˆm = 0.1 p.u. with50% rated load torque applied. The reduced-order observer isused in Fig. 1(a), and the adaptive full-order observer is usedin Fig. 1(b). The data is captured by varying each parameterestimate from 90% up to 110% of the actual value in 10seconds. It can be seen that the model parameters Rˆs andLˆq have only a small effect on the position error, whereasincorrect value for Lˆd increases the estimation error rapidly inFig. 1(a), when the reduced-order observer is used. Accordingto Fig. 1(b), the adaptive full-order observer is less sensitive toparameter errors. It should be noted that the relative errors ofLd and Lq are defined with respect to the (original) operationpoint values. As the estimation error increases, the actualvalues of id and iq change, resulting changes in actual valuesof Ld and Lq due to saturation.Experimental results of a stepwise speed reversal fromωˆm = 0.10 p.u. to ωˆm = −0.10 p.u. and back to 0.10 p.u.with rated load torque applied are depicted in Fig. 2. Thereduced-order observer is used in Fig. 2(a), and the adaptivefull-order observer is used in Fig. 2(b). It can be seen that thereduced-order observer amplifies the estimation noise in theregenerating mode in Fig. 2(a). This behavior is analyzed in[8]. With the adaptive full-order observer, the amplitude of theestimation noise does not depend on the operating mode, asseen in Fig. 2(b).When the reduced-order observer is used, spatial harmonicscause noise in the position estimate in the regenerating modeif the machine is highly saturated [8]. Results of a slow changeof id from 0.3 p.u. to 0.5 p.u. with −50% rated load torqueapplied are shown in Fig. 3. The reduced-order observer isused in Fig. 3(a), and the adaptive full-order observer is used inFig. 3(b). It can be seen that as id increases in the regeneratingmode, the noise in the position estimate of the reduced-orderobserver increases, but the adaptive full-order observer is notsensitive to spatial harmonics.Experimental results of load-torque steps when the speedreference was kept at 0.05 p.u. are shown in Fig. 4. Theload torque was stepped to −75% of the rated load torqueat t = 2.5 s, reversed at t = 7.5 s, and removed at t =12.5 s. The reduced-order observer is used in Fig. 4(a), andthe adaptive full-order observer is used in Fig. 4(b). It canbe seen that the observers behave well in load transients inlow-speed operation.V. CONCLUSIONSIn this paper, two back-EMF-based observers are comparedfor synchronous reluctance motor drives. The gain selection ofthe observers is crucial in sustained low-speed operation. Theanalysis and tuning of the reduced-order observer is consider-ably easier than that of the adaptive full-order observer. Theadaptive full-order observer has four design parameters, andthe reduced-order observer has only one design parameter. Theproposed design rules guarantee the stability of the adaptivefull-order observer, if the parameter estimates are accurate.The experimental results indicate that the adaptive full-orderobserver is less sensitive to harmonic noise and parameteruncertainties than the reduced-order observer.ACKNOWLEDGMENTThe authors gratefully acknowledge ABB Oy for the finan-cial support.REFERENCES[1] E. Capecchi, P. Guglielmo, M. Pastorelli, and A. Vagati, “Position-sensorless control of the transverse-laminated synchronous reluctancemotor,” IEEE Trans. Ind. Appl., vol. 37, no. 6, pp. 1768–1776, Nov./Dec.2001.[2] H. F. Hofmann, S. R. Sanders, and A. EL-Antably, “Stator-flux-orientedvector control of synchronous reluctance machines with maximizedefficiency,” IEEE Trans. Ind. Electron., vol. 51, no. 5, pp. 1066–1072,Oct. 2004.[3] A. Piippo, M. Hinkkanen, and J. Luomi, “Sensorless control of PMSMdrives using a combination of voltage model and HF signal injection,” inConf. Rec. IEEE-IAS Annu. Meeting, vol. 2, Seattle, WA, Oct. 2004, pp.964–970.[4] O. Wallmark, L. Harnefors, and O. Carlson, “An improved speed andposition estimator for salient permanent-magnet synchronous motors,”IEEE Trans. Ind. Electron., vol. 52, no. 1, pp. 255–262, Feb. 2005.[5] G. Yang, R. Tomioka, M. Nakano, and T. H. Chin, “Position and speedsensorless control of brushless DC motor based on an adaptive observer,”IEEJ Trans. Ind. Appl., vol. 113, pp. 579–586, May 1993.[6] A. Piippo, M. Hinkkanen, and J. Luomi, “Analysis of an adaptive observerfor sensorless control of interior permanent magnet synchronous motors,”IEEE Trans. Ind. Electron., vol. 55, no. 2, pp. 570–576, Feb. 2008.[7] M. Hinkkanen, T. Tuovinen, L. Harnefors, and J. Luomi, “A combinedposition and stator-resistance observer for salient PMSM drives: designand stability analysis,” IEEE Trans. Power Electron., no. 99, 2011,doi:10.1109/tpel.2011.2118232, in press.[8] T. Tuovinen, M. Hinkkanen, and J. Luomi, “Analysis and design of aposition observer with resistance adaptation for synchronous reluctancemotor drives,” in Proc. IEEE ECCE’11, Phoenix, AZ, Sep. 2011, in press.[9] L. Harnefors, “Design and analysis of general rotor-flux-oriented vectorcontrol systems,” IEEE Trans. Ind. Electron., vol. 48, no. 2, pp. 383–390,Apr. 2001.40.9 0.95 1 1.05 1.1−150150.9 0.95 1 1.05 1.1−5050.9 0.95 1 1.05 1.1−505Lˆd/LdLˆq/LqRˆs/Rsϑ˜m(deg)ϑ˜m(deg)ϑ˜m(deg)(a)0.9 0.95 1 1.05 1.1−150150.9 0.95 1 1.05 1.1−5050.9 0.95 1 1.05 1.1−505Lˆd/LdLˆq/LqRˆs/Rsϑ˜m(deg)ϑ˜m(deg)ϑ˜m(deg)(b)Fig. 1. Measured errors in the position estimate at ωˆm = 0.1 p.u. with 50% rated load torque applied: (a) reduced-order observer and (b) adaptive full-orderobserver. The data is captured by varying each model parameter from 90% up to 110% of the actual value in 10 seconds.−0.1500.15  0 1 2 3 4 5−15015ωm(p.u.)ϑ˜m(deg)t (s)ωrefωˆmωm(a)−0.1500.15  0 1 2 3 4 5−15015ωm(p.u.)ϑ˜m(deg)t (s)ωrefωˆmωm(b)Fig. 2. Experimental results of a stepwise speed reversal (0.10 p.u. → −0.10 p.u. → 0.10 p.u.) with rated load torque applied: (a) reduced-order observer,(b) adaptive full-order observer.5−101  0 4 8 12−15015i s(p.u.)ϑ˜m(deg)t (s)idiq(a)−101  0 4 8 12−15015i s(p.u.)ϑ˜m(deg)t (s)idiq(b)Fig. 3. Experimental results of a slow change in id from 0.3 p.u. to 0.5 p.u. with −50% rated load torque applied: (a) reduced-order observer, (b) adaptivefull-order observer.−0.200.2  −202  0 5 10 15−15015T(p.u.)ωm(p.u.)ϑ˜m(deg)t (s)ωˆmωmTˆeTm(a)−0.200.2  −202  0 5 10 15−15015T(p.u.)ωm(p.u.)ϑ˜m(deg)t (s)ωˆmωmTˆeTm(b)Fig. 4. Experimental results showing load-torque steps (0 → −75% rated → 75% rated) when the speed reference is kept at 0.05 p.u.: (a) reduced-orderobserver, (b) adaptive full-order observer.

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A comparison of an adaptive full-order observer and a reduced-order observer for synchronous reluctance motor drives

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