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Sensorless speed control of permanent-magnetmotors with nonsinusoidal EMF waveform
C.H. De Angelo, G.R. Bossio, J.A. Solsona, G.O. Garc!ıa and M.I. Valla
Abstract: A novel sensorless speed-control strategy for permanent-magnet ACmotors (PMACMs)is presented. A nonlinear reduced-order observer and a high-gain observer are designed to estimatethe induced EMF, without mechanical position or speed sensors. In contrast with previousproposals, the EMF estimation is performed without previous knowledge of its waveform. Fromthe estimated EMF, the rotor speed and the position derivatives of flux are calculated and used forthe implementation of a PMACM speed control without mechanical position sensor. The proposedstrategy is not limited to PMACM with sinusoidal or trapezoidal EMF waveform. Moreover, itallows the PMACM ripple torque and copper losses to be minimised. The performance of theproposal is validated through simulation and experimental results.
1 Introduction
Permanent-magnet AC-motor (PMACM) drives are widelyused in high-performance applications. These motors arepreferred because of the absence of rotor windings, theirhigh efficiency, their high torque-to-inertia ratio and theirpower density.
However, an important disadvantage of PMACMs istheir torque pulsations. This pulsating torque is producedby different reasons [1, 2]:
(a) distortion of the actual airgap-flux waveform withregard to the ideal waveform due to the motor geometry;
(b) distortion of the excitation currents due to the invertorcommutation and the current controller;
(c) low resolution or offset in the current measurement;
(d) delay in the synchronisation between the excitationcurrents and the actual rotor position; and
(e) magnetic reluctance variation due to stator slots andother irregularities.
The pulsating torque produced by the interaction of stator-current magnetomotive forces and rotor-magnet fluxdistribution or the angular variation in the rotor magnetreluctance is called ‘ripple torque’. The pulsating torquegenerated by the interaction of the rotor magnetic flux andangular variations in the stator magnetic reluctance is
known as ‘cogging torque’ [1]. Recently, different techni-ques have been proposed to reduce torque pulsations. Thesetechniques can be divided into two groups [1–3]:
(i) motor-design techniques; and
(ii) active control techniques.
Motor-design techniques allow an important reduction ofcogging torque, though its complete elimination is notpossible. Within the second group, several techniques, suchas preprogrammed stator-current excitations to cancel thetorque harmonic components [4–6], torque estimation basedon model reference adaptive schemes [7] and iterativelearning control [3], have been proposed.
In [8], a programmed current-waveform control strategyfor ripple torque and ‘copper-loss’ minimisation is pro-posed. In this proposal, excitation currents, synchronisedwith rotor position, are calculated on the basis of theinstantaneous-reactive-power theory [9]. The implementa-tion of this strategy requires a previous knowledge of theposition derivative of the linked-flux waveform (inducedelectromotive force divided by speed), and the use of aposition sensor to generate the excitation-current references.The use of a position sensor increases the drive costs andreduces its mechanical robustness. This is the main reasonwhy several PMACM control techniques are used whichavoid the use of mechanical position or speed sensors. Theseproposals are applicable to PMACMs whose inducedelectromotive force (EMF) is perfectly sinusoidal ortrapezoidal [10], but they do not take into considerationthe case of PMACMs with other EMF waveforms.
In this work, a novel sensorless speed-control strategy forPMACM is presented. A nonlinear reduced-order observerand a high-gain observer are designed to estimate theinduced EMF, without mechanical position or speedsensors. From the estimated EMF, the rotor speed andthe position derivatives of flux are calculated, and used inthe implementation of a PMACM speed control withoutmechanical position sensor. The main advantage of thestrategy proposed in this paper is that the controller doesnot need information about the motor-EMF waveform, asis usually required when an observer for high performance
C.H. De Angelo, G.R. Bossio and G.O. Garc!ıa are with CONICET and alsowith Grupo de Electr!onica Aplicada GEA, Facultad de Ingenier!ıa, UNRC,Ruta Nacional #36 Km. 601, X5804BYA, R!ıo Cuarto, Argentina
J.A. Solsona is with CONICET and also with Departamento de Ingenier!ıaEl!ectrica y de Computadoras, Instituto de Investigaciones en Ingenier!ıaEl!ectrica Alfredo Desages, UNS, Av. Alem 1253 (8000), Bah!ıa Blanca,Argentina
M.I. Valla is with CONICET and also with Departamento de Electrotecnia,Laboratorio de Electr!onica Industrial, Control e Instrumentaci!on (LEICI),Facultad de Ingenier!ıa, UNLP, 1900, La Plata, Argentina
E-mail: [email protected]
r IEE, 2005
IEE Proceedings online no. 20050023
doi:10.1049/ip-epa:20050023
Paper first received 25th January and in final revised form 26th April 2005
IEE Proc.-Electr. Power Appl., Vol. 152, No. 5, September 2005 1119
is to be designed. This waveform does not necessarily needto be sinusoidal or trapezoidal, as other proposals reportedin the literature do; indeed it can take any shape. As aconsequence, the strategy is very robust in the face of theuncertainty in the EMF waveform. This waveform does notalways coincide with that reported by the manufacturer; it iseven possible to replace a motor having a different EMFwaveform, without performing a laboratory or a self-commissioning test to know the new EMF waveform. Inaddition, this proposal does not need to use a mechanicalposition or speed sensor, allowing PMACM ripple-torqueand copper-loss minimisation.
2 Permanent-magnet AC-motor model andcontrol strategy
2.1 ModelIn order to develop the control strategy, the dynamic modelof a PMACM is written in a stationary reference framea–b–0 [11]:
diadt¼ �R
Lia �
1
Lea þ
1
Lna
dibdt¼ �R
Lib �
1
Leb þ
1
Lnb
0 ¼ �e0 þ v0
9>>>>=>>>>;
ð1Þ
dydt¼ o
dodt¼ 1
JTe �
BJo� 1
JTL
9>>=>>; ð2Þ
where, ia, ib, ea, eb, e0, va, vb, v0 represent the current, EMFand voltage components, respectively, in the stationaryreference frame a–b–0; R and L are the stator resistance andinductance, respectively; y, o and Te represent the rotorposition and speed, and the electromagnetic torqueproduced by the motor; J and B are inertia and viscosity,respectively; and TL is the load torque.
Despite the fact that the motor has no connection to theneutral point, and therefore no zero-current component canexist, the equation corresponding to the zero-voltagecomponent is included in the motor model (1). This permitsthe extraction of some additional information about theharmonic components that exist in the phase-to-neutralEMF waveform.
The EMF induced into the stator windings can beobtained as the time derivative of the linked flux:
ea ¼qlaqy
dydt¼ faðyÞo
eb ¼qlbqy
dydt¼ fbðyÞo
e0 ¼ql0qy
dydt¼ f0ðyÞo
9>>>>>>=>>>>>>;
ð3Þ
where, la, lb and l0 are the linked-flux components. Theposition derivatives of the linked flux, fiðyÞði ¼ a; b; 0Þ, arefunctions of the rotor position and depend on the motorconstruction.
The electromagnetic torque can be expressed by [11]
Te ¼ faðyÞia þ fbðyÞib ð4Þ
Therefore it is evident that, to obtain a constant torque,currents whose waveform depends on the functions fa andfb must be imposed.
2.2 Minimum-ripple-torque-controlstrategyThe ripple-torque-minimisation strategy presented in thiswork is based on that proposed in [8]. The motor control iscarried on by imposing stator currents whose waveforms,synchronised with the rotor position, are calculated on thebasis of the instantaneous-reactive-power theory [9]. Con-sidering that the torque developed by the motor depends onthe instantaneous active power, the motor torque iscontrolled by manipulating the active-current components.On the other hand, instantaneous reactive power does notcontribute to torque development, but produces reactive-current circulation. Therefore, instantaneous-reactive-cur-rent components are set to zero to minimise copper losses.
For the implementation of this strategy, stator-currentreferences are calculated by the expressions
i�a ¼T �e fa
f2a þ f2
b
i�b ¼T �e fb
f2a þ f2
b
9>>>>=>>>>;
ð5Þ
where variables x* represent the reference of x. A moredetailed development of this expression can be found in [8].
To implement this control scheme, it is necessary to knowthe position derivatives of the linked flux fi ðyÞði ¼ a; b; 0Þ.Using a position sensor, fi (y) can be determined ‘offline’ inan experimental way for each particular motor, and storedin a look-up table to permit easy calculation of the currentreferences according to (5). However, the goal of the presentproposal is to design a control law that avoids the use oflinked flux and rotor-position sensors. For this reason, anobserver is used for obtaining estimates for those variables.These estimates replace the position derivatives of linkedflux and speed values in the control law.
3 Proposed observer
To avoid the use of linked flux and mechanical-positionsensors, it is necessary to estimate the rotor speed and theposition derivatives of flux from the electrical measurablevariables. The rotor speed is needed to close the speed-control loop, and position derivatives of flux are needed togenerate the current references to minimise the ripple torqueand copper losses (5).
As it can be seen in (3), the information about rotorspeed and position derivatives of flux (f functions) can beobtained from the induced EMF [12]. To do that, a novelnonlinear observer is proposed in this paper. This proposaluses a nonlinear reduced-order observer and a high-gainobserver to estimate the induced EMF without previousknowledge of its waveform, measuring motor currents andvoltages. Once the EMF is estimated, the positionderivatives of the linked flux and the rotor speed arecalculated.
The time derivatives of the induced EMF componentsa–b in (3) are first calculated to implement the observer:
deadt¼ dfa
dtoþ fa
dodt
debdt¼
dfb
dtoþ fb
dodt
9>>=>>; ð6Þ
1120 IEE Proc.-Electr. Power Appl., Vol. 152, No. 5, September 2005
From (6), the following EMF observer can be designed:
deadt¼ dfa
dtoþ fa
dodtþ g L
diadt� L
diadt
� �
debdt¼
dfb
dtoþ fb
dodtþ g L
dibdt� L
dibdt
� �9>>>=>>>;
ð7Þ
where (4) stands for estimated variables and g is a cons-tant value. The observer copies the EMF dynamics, andemploys the error between estimated and actual currentderivatives as a correction term. Note that e0 is notestimated, since it can be calculated on the basis of voltagemeasurements.
The time derivatives of the estimated currents, used inthe correction term of the observer, can be calculatedfrom (1):
diadt¼ �R
Lia �
1
Lea þ
1
Lva
dibdt¼ �R
Lib �
1
Leb þ
1
Lvb
9>>=>>; ð8Þ
The time derivative of the estimated speed (acceleration, a)can be obtained from (2) and (4), assuming that no loadtorque exists (TL¼ 0):
a ¼ dodt¼ 1
Jfaia þ fbib� �
� BJo ð9Þ
The calculus of the time derivative of the measured currents,used in the correction term (7), could introduce estimationerrors, due to the measured current ripple and measurementnoise. To reduce this error, the following change of variableis proposed:
xa ¼ ea þ gLia
xb ¼ eb þ gLib
)ð10Þ
where xa, xb are auxiliary variables.Differentiating (10), and substituting the derivatives
defined in (7) and (8), the proposed observer results in
dxadt¼ dfa
dtoþ faaþ g �Ria � ea þ vað Þ
dxbdt¼
dfb
dtoþ fbaþ g �Rib � eb þ vb
� �
9>>>=>>>;
ð11Þ
To obtain the time derivatives of f, a reduced-order high-gain observer is proposed, which is essentially an approx-imate ‘differentiator’ [13]. It is designed by employing the
variable ~f as an estimate of f, and using its derivative
ðd ~f=dtÞ as the estimate of ðdf=dtÞ. The reader can refer toAppendix 1 (Section 9.1) for a detailed development of thefollowing expressions:
dwadt¼ � 1
ewa þ
1
efa
� �dwbdt¼ � 1
ewb þ
1
efb
� �9>>=>>; ð12Þ
where Xa, Xb are auxiliary variables, and e is a small positiveparameter. Then,
dfa
dt� d ~fa
dt¼ wa þ
1
efa
dfb
dt�
d ~fb
dt¼ wb þ
1
efb
9>>>=>>>;
ð13Þ
Thus, the EMF observer will be given by
dxadt¼ wa þ
1
efa
� �oþ faaþ g �Ria � ea þ vað Þ
dxbdt¼ wb þ
1
efb
� �oþ fbaþ g �Rib � eb þ vb
� �9>>>=>>>;ð14Þ
and the estimated EMF can be obtained by solving (10):
ea ¼ xa � gLia
eb ¼ xb � gLib
)ð15Þ
From some experiments performed with several PMACMs[14, 15], it has been determined that the induced EMFcomponents obey to the approximation
e2a þ e2b � 4e20 ¼ o2 f2a þ f2
b � 4f20
� �’ o2K ð16Þ
where K is a coefficient that depends on the motorcharacteristics. Based on this approximation, the rotorspeed can be calculated from (16):
o ¼ 1
Ke2a þ e2b � 4e20� �q
ð17Þ
As it can be seen in (17), it is not possible to determine thesign of the estimated speed from this expression. However,it can be demonstrated that the sign of the estimated speedcan be calculated by using the expression
sign oð Þ ’ sign fadfb
dt� fb
dfa
dt
!
’ sign fawb � fbwa� �
9>>>=>>>;
ð18Þ
This can easily be proved when f are sinusoidal functions.Then, the estimated speed can be calculated as
o ¼ 1
Ksign fawb � fbwa
� �e2a þ e2b � 4e20� �q
ð19Þ
From the estimated speed, the estimates of the positionderivatives of flux, necessary to implement the torquecontrol law (5), are calculated as follows:
fa ¼eao
fb ¼ebo
9>>=>>; ð20Þ
As can be seen in (20), this observer cannot be used at verylow speed, just like other proposals based on the estimationof the induced EMF. However, a slight change should beincluded in the algorithm to ensure its proper operationduring a speed reversal. If low-speed or standstill operationis needed, the proposed observer should be combined withsome open-loop starting procedures or signal-injectiontechniques, as proposed in the literature [16].
Note that the proposed observer and the sensorlesscontrol strategy are based on the model given by (1)–(4),where a constant inductance, a linear friction and inertiawere assumed. Parameter uncertainties and nonlinearitiescan cause the estimation error to converge asymptotically toa small nonzero value. However, it is possible to reduce theasymptotic estimation error by increasing the value of g[17]. In addition, it is well known that mismatches betweenthe motor and the model can affect the drive performanceadversely. To overcome this drawback, a PI controller isincluded in the speed-control loop.
IEE Proc.-Electr. Power Appl., Vol. 152, No. 5, September 2005 1121
4 Application in a sensorless speed control fordifferent motors
Figure 1 shows a simplified scheme of the proposedsensorless speed control for PMACMs. It has an externalspeed-control loop fed back by the estimated speed, andan internal torque-control loop. Current references forthe invertor current-control loop are generated through(5) by employing the functions estimated by the observer.The implemented algorithm is shown in Appendix 2(Section 9.2).
In the following paragraphs, some results obtained bynumerical simulation of the system shown in Fig. 1 arepresented. Three types of motors, with different inducedEMF waveforms, were simulated:
(a) PMACM with sinusoidal EMF (ideal);
(b) PMACM with trapezoidal EMF (ideal); and
(c) PMACM which EMF is neither sinusoidal nortrapezoidal (Axial Flux PMACM with experimentallydetermined waveform).
The same parameters were used for all the motors (seeTable 1); the induced EMF waveform being the onlychange.
The motor is supposed to be driven by a current-controlled PWM voltage-source invertor as shown in Fig. 1.The current references are calculated with the proposedalgorithm.
Figures 2–4 show simulation results of the steady-stateoperation of the three motors under consideration. Figure 2corresponds to the sinusoidal-EMF PMACM; Fig. 3 to thetrapezoidal-EMF PMACM; and Fig. 4 to the axial-fluxPMACM. For each motor, the waveform of the positionderivative of flux, corresponding to stator phase a, is shownin Figs. 2a, 3a and 4a. The actual and estimatedcomponents of the position derivative of flux in a stationaryreference frame are shown in Figs. 2b and c, Figs. 3b and cand Figs. 4b and c, respectively. It can be seen from thesefigures that the proposed observer estimates correctly theposition derivative of flux for each motor, using only thevoltage and current measurements.
In Figs. 2d, 3d and 4d, the simulated electromagnetictorque for the same operating condition is shown. It can beseen that the motor torque has only a high-frequency rippleproduced by the switched currents; but the undesired torqueharmonics (6th, 12th etc.) [1] have been eliminated.
Figures 5 and 6 show the transient behaviour duringstart-up, application of unmodelled load torque and speedreversal. Figure 5 shows the actual speed (solid line) andreference speed (broken line) for each motor, when a stepspeed reference of 250 rev/min is applied to the unloaded
Table 1: Motor and observer parameters
Three-phase permanent-magnet motor
16 poles, 2000rev/min 15KW
R¼ 10mO L¼100mH
J¼ 0.78Kgm2 B¼0.0015Kgm2/s
K¼ 0.5021
Observer parameters
g¼400 e¼ 0.0001
−+ω∗
ω�
αβ
�α� �β
�
PITe
*iabc
iab
*
vabc
iβ*
iα*
torquecontrol
(5)
proposedobserver
abc
αβ
abc current-controlledinvertor
R
S
T
PMACM
Fig. 1 Block diagram of the sensorless speed control
−0.6
−0.3
0
0.3
0.6
phas
e-a
func
tion
�, V
s/r
ad
0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80−0.6
−0.3
0
0.3
0.6
time, s
actu
al �
�, �
� an
d�
0, V
s/r
ad
b
0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80time, s
a
0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80−0.6
−0.3
0
0.3
0.6
time, s
estim
ated
��
and
��, V
s/r
ad
c
0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.800
5
10
15
20
actu
al to
rque
T e, N
m
time, s
d
Fig. 2 Steady-state performance of PMACM with sinusoidalEMF (simulation results)a Actual phase-a function fb Actual fa (solid), fb (broken) and f0 (chain dotted)
c Estimated fa (solid) and fb (broken)
d Actual torque Te
1122 IEE Proc.-Electr. Power Appl., Vol. 152, No. 5, September 2005
drive. At 4 s an 8Nm load torque is applied. All threemotors show a good transient response with negligible effectof the nonmodelled load torque.
In Fig. 6 the actual speed (solid line) and reference speed(broken line) during a speed reversal are presented. Thethree graphs correspond to the three motors under analysis.As stated above, the proposed observer cannot work at zerospeed, so hysteresis was added around null speed to allowspeed reversal without indetermination. Again, this resultsshow an acceptable closed-loop response even when cross-ing zero speed.
In summary, simulation results show a good closed-loopresponse in steady-state and transient operation. The torqueripple is almost negligible when the proposed observer andcontrol strategy are used.
5 Experimental results
To validate the proposed control scheme, a laboratoryprototype drive was implemented using an axial-fluxPMACM, whose parameters are shown in Table 1. ThePMACM motor was fed with a voltage-source invertorwith a fast current-control loop. Motor voltages andcurrents were measured using standard Hall-effect sensorsand 12-bit A/D convertors. The acutal speed was alsomeasured, to compare it with the estimated one. The
observer and the control strategy were implemented on aPC in ‘C’ language. Differential equations were discretisedusing the Euler method with a 200ms sampling period. Theexperimental data were captured using a digital scope andplotted with the simulation software for an easiercomparison with the simulation results. High-gain observervariables were appropriately scaled to avoid ill-conditioningproblems with very small e values. Note that fourdifferential equations must be solved to implement thisobserver; therefore it is equivalent to the implementation ofpreviously proposed full-order observers [18] as regards thecomputational time.
Figure 7 shows the actual (Fig. 7a) and estimated(Fig. 7b) position derivative of flux when the motor isrunning at 300 rev/min. As can be seen, the estimatedwaveform of the position derivative of flux is very similar tothe actual waveform. This allows the use of the estimatedwaveforms to minimise the torque ripple and copper losses.The time derivatives of fa and fb, estimated by the high-gain observer, are shown in Fig. 7c. The measured currentwaveforms in the a–b reference frame (ia, ib) are shown inFig. 7d.
To test the observer response, initially the motor-drivecontrol loop was closed using the measured variables. Alow-speed test is presented in Fig. 8. The unloaded drive isrunning at 10 rev/min when a sudden change in thereference speed occurs. The estimated and measured speeds
0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80−0.6
−0.30
0.3
0.6
time, s
phas
e-a
func
tion
�, V
s/ra
d
a
0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80−0.6−0.3
0
0.3
0.6
time, s
actu
al �
�, �
� an
d�
0, V
s/ra
d
b
0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80−0.6
−0.3
0
0.3
0.6
time, s
estim
ated
��
and
��,
Vs/
rad
c
0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.800
5
10
15
20
actu
al to
rque
T e, N
m
time, s
d
Fig. 3 Steady-state performance of PMACM with trapezoidalEMF (simulation results)a Actual phase-a function fb Actual fa (solid), fb (broken) and f0 (chain dotted)
c Estimated fa (solid) and fb (broken)
d Actual torque Te
0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80−0.6
−0.3
0
0.3
0.6
time, s
phas
e-a
func
tion
�, V
s/ra
d
a
0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80−0.6−0.3
0
0.3
0.6
time, s
actu
al �
�, �
� an
d�
0, V
s/ra
d
b
0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80−0.6
−0.3
0
0.3
0.6
time, s
estim
ated
��
and
��, V
s/ra
d
c
0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.800
5
10
15
20
actu
al to
rque
T e, N
m
time, s
d
Fig. 4 Steady-state performance of axial-flux PMACM (simula-tion results)a Actual phase-a function fb Actual fa (solid), fb (broken) and f0 (chain dotted)
c Estimated fa (solid) and fb (broken)
d Actual torque Te
IEE Proc.-Electr. Power Appl., Vol. 152, No. 5, September 2005 1123
are shown in Fig. 8a and b, respectively. A speed-reversaltest is shown in Fig. 9. It can be seen that the speed signcannot be determined precisely at low speed, owing to theabove mentioned measurement noise.
Figures 10–12 show the closed-loop-drive performancewhen the estimated variables are used in the controlalgorithm. A high-speed test is shown in Fig. 10, when the
reference speed (broken line) goes from 1000 rev/min to2000 rev/min. The low-speed test is shown in Fig. 11, whena reference speed profile (100–500–100 rev/min) is applied.In these Figures, a good closed-loop response can beobserved, even at low speed and with braking action.
In Fig. 12, an 8Nm constant load torque is applied at 8 s,in order to test the sensorless-drive response with non-modelled load torque. A small steady-state error appears in
0 1 2 3 4 5 60
100
200
300
time, s
actu
al s
peed
and
refe
renc
e sp
eed,
re
v/m
in
actu
al s
peed
and
refe
renc
e sp
eed,
re
v/m
in
actu
al s
peed
and
refe
renc
e sp
eed,
re
v/m
in
a
0 1 2 3 4 5 60
100
200
300
time, s
b
0 1 2 3 4 5 60
100
200
300
time, s
c
Fig. 5 Start-up and nonmodelled load-torque test (simulationresults)Reference speed o� (broken) and actual speed o (solid)a PMACM with sinusoidal EMFb PMACM with trapezoidal EMFc Axial-flux PMACM
0 1 2 3 4 5 6 7 8−250−125
0125250
time, sa
0 1 2 3 4 5 6 7 8−250−125
0125250
time, sb
0 1 2 3 4 5 6 7 8−250−125
0125250
time, s
actu
al s
peed
and
refe
renc
e sp
eed,
rev/
min
actu
al s
peed
and
refe
renc
e sp
eed,
rev/
min
actu
al s
peed
and
refe
renc
e sp
eed,
rev/
min
c
Fig. 6 Speed-reversal test (simulation results)Reference speed o� (broken) and actual speed o (solid)a PMACM with sinusoidal EMFb PMACM with trapezoidal EMFc Axial-flux PMACM
−0.8
−0.4
0
0.4
0.8
−0.8
−0.4
0
0.4
0.8
mea
sure
d �
� and
��, V
s/r
ad
estim
ated
�� a
nd
��, V
s/r
ad
−200
−100
0
100
200
estim
ated
d�
� /d
t
and
d��
/dt,
V/r
ad
−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05−15
−10
−5
0
5
10
15
time, s
mea
sure
d cu
rren
ts
i � and
i �, A
d
−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05
time, s
c
−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05
time, s
a
−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05
time, s
b
Fig. 7 Experimental results (variables in a stationary referenceframe)a Position derivatives of flux fa (solid) and fb (broken), measuredb Position derivatives of flux fa (solid) and fb (broken), estimatedc Time derivatives of fa (solid) and fb (broken), estimatedd Currents waveforms, ia (solid) and ib (broken), measured
−0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0
50
100
time, s
mea
sure
d sp
eed,
rev/
min
a
−0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0
50
100
time, s
estim
ated
spe
ed,
rev/
min
b
Fig. 8 Low-speed estimation (experimental results)a Measured speed ob Estimated speed o
1124 IEE Proc.-Electr. Power Appl., Vol. 152, No. 5, September 2005
the motor speed. This small error is produced by theunknown load torque that is not included in the observermodel. The observer converges to a small bounded error. Ifthis small error is unacceptable for the drive performance,it can be eliminated by extending the observer as proposedin [19].
6 Conclusions
A sensorless speed-control strategy for permanent-magnetAC motors with nonsinusoidal EMF waveform waspresented in this work. The proposed strategy uses anonlinear reduced-order observer and a high-gain observerto estimate the induced EMF, measuring only the statorcurrents and voltages. It also allows the reduction of theripple torque produced by nonsinusoidal EMF waveforms.
Simulation results demonstrate that it is feasible to applythe proposed strategy to PMACMs with almost any EMFwaveforms. The proposed strategy was implemented withan axial-flux PMACM whose EMF waveform is neithersinusoidal nor trapezoidal. Experimental results demon-strate a high performance of the observer and a goodclosed-loop response of the complete sensorless controlstrategy.
7 Acknowledgments
This work was supported by Universidad Nacional de R!ıoCuarto (UNRC), Universidad Nacional de La Plata(UNLP), Universidad Nacional del Sur (UNS), FON-CyT-ANPCyT and CONICET.
8 References
1 Jahns, T., and Soong, W.: ‘Pulsating torque miniminzation techniquesfor permanent magnet AC motor drives – a review’, IEEE Trans.,1996, IE-43, (2), pp. 321–330
2 Hanselman, D.: ‘Effect of skew, pole count and slot count onbrushless motor radial force, cogging torque and back EMF’, IEEProc. Electr. Power Appl,. 1997, 144, (5), pp. 325–330
3 Lam, B., Panda, S., and Xu, J.: ‘Periodic torque ripples minimisationin permanent magnet synchronous motor drives using iterativelearning control’. Proc. 9th European Conf. Power Electronics andApplications, EPE 2001, Graz, Austria, 2001
4 Holtz, J., and Springob, L.: ‘Identification and compensation oftorque ripple in high-precision permanent magnet motor drives’, IEEETrans., 1996, IE-43, (2), pp. 309–320
5 Cho, K.-Y., Bae, J.-D., Chung, S.-K., and Youn, M.-J.: ‘Torqueharmonics minimisation in permanent magnet synchronous motorwith back EMF estimation’, IEE Proc. Electr. Power Appl., 1994, 141,(6), pp. 323–330
6 Brunsbach, B.-J., Henneberger, G., and Klepsch, T.: ‘Compensationto torque ripple’. Proc. Sixth IEE Int. Conf. Electrical Machines andDrives, Oxford, UK, 1993, pp. 588–593
7 Chung, S.-K., Kim, H.-S., Kim, C.-G., and Youn, M.-J.: ‘A newinstantaneous torque control of PM synchronous motor for high-performance direct-drive applications’, IEEE Trans., 1998, PE-13, (3),pp. 388–400
8 Leidhold, R., Garc!ıa, G., and Watanabe, E.: ‘PMAC Motor controlstrategy, based on the instantaneous active and reactive power, forripple-torque and copper-losses minimization’. Proc, IEEE 26thAnnual Conf. Industrial Electronics Soc. (IECON’00), Nagoya,Japan, 2000, Vol. 2, pp. 1401–1405
9 Akagi, H., Kanazawa, Y., and Nabae, A.: ‘Generalized theory of theinstantaneous reactive power in three-phase circuits,’. Proc. Int. PowerElectronics Conf. (IPEC’83), Tokyo, Japan, 1983, pp. 1375–1386
10 Johnson, J., Ehsani, M., and G.uzelg.unler, Y.: ‘Review of sensorlessmehtods for brushless DC’. Conf. Record 1999 IEEE IndustryApplications Conf., 34th IAS Annual Meeting, Phoenix, USA, 1999,Vol. 1, pp. 143–150
11 Krishnan, R.: ‘Electric motor drives: modelling, analysis and control’(Prentice Hall, 2001)
12 Solsona, J., Valla, M.I., and Muravchik, C.: ‘On speed and rotorposition estimation in permanent-magnet AC drives’, IEEE Trans.,2000, IE-47, (5), pp. 1176–1180
13 Dabroom, A., and Khalil, H.K.: ‘Numerical differentiation usinghigh-gain observers’. Proc. 36th IEEE Conf. Decision and Control,San Diego, USA, 1997, Vol. 5, pp. 4790–4795
14 De Angelo, C., Bossio, G., Solsona, J., Garc!ıa, G., and Valla, M.:‘Sensorless speed control of permanent magnet motors with torqueripple minimization’ Proc. 28th Annual Conf. IEEE IndustrialElectronics Soc., IECON’02, Seville, Spain, 2002, Vol. 1, pp. 680–685
15 De Angelo, C.H.: ‘Control para m!aquinas de CA de imanespermanents con FEM arbitraria, sin sensores mec!anicos’. Doctoralthesis, Departamento de Electrotecnia, Facultad de Ingenier!ıa,
mea
sure
d sp
eed,
rev/
min
estim
ated
spe
ed,
rev/
min
−2 0 2 4 6 8 10 12
−500
0
500
time, sa
−2 0 2 4 6 8 10 12
−500
0
500
time, s
b
Fig. 9 Speed-reversal test (experimental results)a Measured speed ob Estimated speed o
−2 0 2 4 6 8 10 12 14 16 18
1000
1500
2000
time, s
refe
renc
e sp
eed
and
mea
sure
d sp
eed,
rev/
min
Fig. 10 Sensorless closed-loop test (high-speed) (experimentalresults)Reference speed (broken line) and actual speed (solid line)
refe
renc
e sp
eed
and
mea
sure
d sp
eed,
rev/
min
−2 0 2 4 6 8 10 12 14 16 180
100200300400500600
time, s
Fig. 11 Sensorless closed-loop test (low-speed) (experimentalresults)Reference speed (broken line) and actual speed (solid line)
refe
renc
e sp
eed
and
mea
sure
d sp
eed,
rev/
min
−2 0 2 4 6 8 10 12 14 16 180
100200300400500600
time, s
Fig. 12 Sensorless closed-loop test (low-speed with load torque)(experimental results)Reference speed (broken line) and actual speed (solid line)
IEE Proc.-Electr. Power Appl., Vol. 152, No. 5, September 2005 1125
Universidad Nacional de La Plata, La Plata, Argentina 2004, (inSpanish). Available: http://sedici.unlp.edu.ar/search/request.php?id_document¼ARG-UNLP-TPG-0000000061
16 Linke, M., Kennel, R., and Holtz, J.: ‘Sensorless speed and positioncontrol of synchronous machines using alternating carrier injection’.Proc. IEEE Int. Electric Machines and Drives Conf. (IEMDC’03),2003, Vol. 2, pp. 1211–1217
17 Solsona, J.A., and Valla,M.I.: ‘Disturbance and nonlinear Luenbergerobservers for estimating mechanical variables in permanent magnetsynchronous motors under mechanical parameters uncertainties’,IEEE Trans., 2003, IE-50, (4), pp. 717–725
18 Zhu, G., Kaddouri, A., Dessaint, L.-A., and Akhrif, O.: ‘A nonlinearstate observer for the sensorless control of a permanent-magnet ACmachine’, IEEE Trans., 2001, IE-48, (6), pp. 1098–1108
19 Solsona, J., Valla, M.I., and Muravchik, C.: ‘Nonlinear control of apermanent magnet synchronous motor with disturbance torqueestimation’, IEEE Trans., 2000, EC-15, (2), pp. 163–168
9 Appendixes
9.1 Reduced-order high-gain observerThe reduced-order high-gain observer used to obtain the
time derivatives of f can be constructed as follows [13].Setting
x1 ¼ f
x2 ¼dx1dt¼ df
dt
9=; ð21Þ
where f is known (sub-indexes corresponding to a–bcomponents have been suppressed for clarity), and defining
x1 ¼ ~f
x2 ¼dx1dt¼ d ~f
dt
9=; ð22Þ
it is desired to estimate x2 without estimating x1. Then thefollowing observer is proposed:
dx2dt¼ 1
edx1dt� dx1
dt
� �ð23Þ
where e is a small positive parameter. To avoid thecalculation of dx1/dt, a variable change is proposed:
X ¼ x2 �1
ex1 ð24Þ
Taking the time derivative of (24), and replacing from (23),the reduced-order observer results in
dwdt¼ � 1
eX þ 1
ex1
� �ð25Þ
and
x2 ¼ wþ 1
ex1 ð26Þ
Returning to the original variables, the reduced-order high-gain observer results in
dwdt¼ � 1
ewþ 1
ef
� �ð27Þ
and
d ~fdt¼ wþ 1
ef ð28Þ
9.2 Proposed observer algorithm
(a) Initial conditions: oð0Þ; fað0Þ; fbð0Þ; wað0Þ; wbð0Þ(b) Inputs: ia; ib; ua; ub; u0(c) Algorithm:
a ¼ dodt¼ 1
Jfaia þ fbib� �
� BJo ð29Þ
dxadt¼ wa þ
1
efa
� �oþ faaþ g �Ria � ea þ vað Þ
dxbdt¼ wb þ
1
efb
� �oþ fbaþ g �Rib � eb þ vb
� �
9>>>>=>>>>;ð30Þ
dwadt¼ � 1
ewa þ
1
efa
� �dwbdt¼ � 1
ewb þ
1
efb
� �9>>>=>>>;
ð31Þ
ea ¼ xa � gLia
eb ¼ xb � gLib
)ð32Þ
o ¼ 1
Ksign fawb � fbwa
� �e2a þ e2b � 4e20� �q
ð33Þ
fa ¼eao
fb ¼ebo
9>>=>>; ð34Þ
1126 IEE Proc.-Electr. Power Appl., Vol. 152, No. 5, September 2005
