Fakham H, Djemai M, Busawon K - Design and practical implementation of a back-EMF sliding-mode observer for a brushless DC motor - article.pdf

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Published in IET Electric Power ApplicationsReceived on 23rd May 2007Revised on 30th August 2007doi: 10.1049/iet-epa:20070242

ISSN 1751-8660

Design and practical implementation of aback-EMF sliding-mode observer for abrushless DC motorH. Fakham1 M. Djemai2 K. Busawon3

1Robotic Laboratory of Versaille, 10-12, Avenue de l’Europe, Velizy 78000, France2Control System Laboratory, ENSEA, 6 Avenue du Ponceau, Cergy-Pontoise 95014, France3Northumbria University, School of Computing, Engineering and Information Sciences, Ellison Building,Newcastle upon Tyne, NE1 8ST, UKE-mail: [email protected]

Abstract: A sliding-mode observer is proposed in order to estimate the phase-to-phase trapezoidal back-EMF in abrushless DC motor by using only the measurements of the stator currents and voltages. The main feature of theproposed observer is that it is not sensitive to the switching noise and no filtering is required. The back-EMFestimate was then used to deduce the six rotor positions of the motor. In addition, a method to obtain anestimate of the rotor speed of the motor, by exploiting the mathematical relationship between the speed andthe back-EMF, is presented. The observer of the trapezoidal back-EMF is implemented practically on a DSPboard. Simulation and experimental results are given to show the performance of the observer.

1 IntroductionIn recent years, there has been a steady rise in the demand forbrushless DC (BLDC) motors for industrial and domesticapplications. Their growing popularity is mainly due totheir high efficiency and reliability and other attractivefeatures such as long lifetime, reduced noise and overallreduction of electromagnetic interference. In general, thefact that they are more efficient than brushed DC motorsoutweigh their main disadvantage of being costly. ABLDC motor is a type of synchronous motors, havingpermanent magnets on the rotor and trapezoidal shapeback-EMF.

The traditional way to control a BLDC motor is viavoltage-source current-controlled inverters as shown inFig. 1. The inverter must supply a rectangular currentwaveform whose magnitude is proportional to the motor’sshaft torque. Three Hall-effect sensors are usuallyemployed as position sensors to perform currentcommutations. The knowledge of the six rotor positions isrequired since the main objective is to obtain quasi-squarecurrent waveforms, with dead time periods of 608. The

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main drawback of this control strategy is that the sensorsincrease the cost and render the overall system quitecumbersome. This, in turn, reduces the reliability of thetotal system.

As a result, a great deal of attention has been given tosensorless control of BLDC motor in recent years, that is,control of the BLDC motor without using the positionsensors. In that context, a number of methods to obtain orestimate the rotor positions and speed of BLDC motorshave been proposed in the literature (see references herein).In most of the existing methods, the rotor position isdetected every 608, which is necessary to ensure currentcommutations. These methods are based on a variety ofstrategies, including back-EMF voltage sensing in theundriven coils [1, 2], detection of the freewheeling diodesconduction [3], back-EMF integration [4], flux estimation[5] and the motor modification technique [6] to infer therotor position. However, the mentioned strategies workwell only over a limited range of speed. A strategy forsensorless control of BLDC motors that is suitable for awide range of rated-speed and low-cost applications is theso-called observer-based control method. As a matter of

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fact, this was the main motivation in [7] for employing theextended kalman filter (EKF) to estimate the instantaneousrotor position and speed of the BLDC motor.Unfortunately, for that particular case, it was shown thatthe speed estimation accuracy was reduced, especially at lowspeeds. In addition, the EKF has some inherentshortcomings in that it is computationally tedious toimplement and is sensitive to the influence of noise.

There is therefore a real incentive to find alternativeobserver design methods that are easy to implement andthat are robust with respect to noise, for estimating therotor position of the BLDC motor. In this respect, sliding-mode observers constitute an attractive alternative sincethey are robust with respect to measurement noise andparametric uncertainties of the system.

In this paper, taking into account the above observations,we propose a sliding-mode observer to estimate thetrapezoidal back-EMF in the undriven coils of a BLDCmotor. The estimated back-EMF is in turn used to deducethe rotor position and speed of the motor. To be moreprecise, the estimated trapezoidal back-EMF obtained is, infact, the phase-to-phase back-EMF that is inducedbetween two phases. It does not depend on the statorvoltage harmonics (that are multiple of order three) or onthe commutation noise introduced by the inverter (neutralpoint cannot be connected). The estimated phase-to-phasetrapezoidal back-EMF is employed to infer the sixinstantaneous rotor position via simple zero crossingdetectors (ZCDs). In addition, a method to obtain a directestimation of the speed, by exploiting the mathematicalrelationship between the latter and the back-EMF, is given.As far as the authors are aware, sliding-mode observershave not been employed in the context of sensorless controlof BLDC motors in the literature.

Simulation results has shown the good performance of theproposed sliding-mode observer with respect to parameter

Figure 1 Sensorless control strategy using sliding-modeobserver

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variations and noise. The sliding-mode observer wasimplemented practically on a DSP board (eZdsp F2812)using a laboratory set-up. The experimental resultsobtained consolidates the good performance of the observerwith respect to noise rejection.

The outline of the paper is as follows. In the next section,the model of BLDC motor is developed on the basis of whicha sliding-mode observer of the phase-to-phase trapezoidalback-EMF is constructed. In Section 3, a ZCD of theback-EMF to obtain the six rotor positions is presented. InSection 4, a direct approach is developed to obtain thespeed from the estimated phase-to-phase back-EMF.Sections 5 and 6 are devoted to simulation andexperimental results. Finally, some conclusions are drawnon the overall design and the results obtained.

2 BLDC motor modelIn BLDC motors, there are only two of the three phasesconducting at any point of time, as the stator windingneutral point of the machine is floating and is notaccessible in general. This often makes it impossible todirectly measure phase voltages.

The BLDC motor is modelled in the stationary referenceframe abc using:

1. the subtractions of currents (Ia 2 Ib, Ib 2 Ic, Ic 2 Ia),which are measured between two phases,

2. the phase-to-phase back-EMF (Eab, Ebc, Eca)

3. and the phase-to-phase voltages (Uab, Ubc, Uca).

As a result, the following model has been derived [8, 9]

d

dtIa � Ib

� �¼ �

R

LIa � Ib

� ��

1

LEab þ

1

LUab

d

dtIb � Ic

� �¼ �

R

LIb � Ic

� ��

1

LEbc þ

1

LUbc

d

dtIc � Ia

� �¼ �

R

LIc � Ia

� ��

1

LEca þ

1

LUca

8>>>>><>>>>>:

(1)

To reduce the order of model (1), only the electricquantities of the two phases (a–b) and (b–c) are taken,respectively. The following traditional assumptions are made.

† The distribution of the phase-to-phase back-EMF istrapezoidal, and its variation is very slow.

† The motor is unsaturated.

† The armature reaction is negligible.

† The phases are balanced.

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The following model is then obtained:

d

dtIa � Ib

� �¼ �

R

LIa � Ib

� ��

1

LEab þ

1

LUab

d

dtIb � Ic

� �¼ �

R

LIb � Ic

� ��

1

LEbc þ

1

LUbc

d

dtEab ¼ 0

d

dtEbc ¼ 0

(2)

System (2) can be written in the state space as follows

_x1¼ �a1x1 � a2x3 þ a2V1

_x2¼ �a1x2 � a2x4 þ a2V2

_x3¼ 0

_x4¼ 0

y1 ¼ x1y2 ¼ x2

(3)

where x1 ¼ Ia 2 Ib, x2 ¼ Ib 2 Ic, x3 ¼ Eab, and x4 ¼ Ebc arethe state variables, V1 ¼ Uab, and V2 ¼ Ubc are the inputvariables, y1 and y2 represent the output variables anda1 ¼ (R/L) and a2 ¼ (1/L) are the system’s parameters.

Under the assumption that the system is balanced, thethird back-EMF, Eca, between the two phases (c–a) can beeasily deduced by using the formula

Eab þ Ebc þ Eca ¼ 0 (4)

In the following section, the above model (3) is used todesign a sliding-mode observer in order to estimate thephase-to-phase back-EMF Eab and Ebc.

3 Sliding-mode observer designFirst of all, note that system (3) is, in fact, a linear system withparametric uncertainty. By using the measurements of thestator currents and the input phase-to-phase voltage, theobserver therefore takes the following form

x:

1 ¼ �a1x1 � a2 x3þa2V1þK1I s

x:

2 ¼ �a1x2 � a2 x4þa2V2þK2I s

x:

3 ¼ K3Is

x:

4 ¼ K4Is

(5)

where K1 ¼ (k11, k12), K2 ¼ (k21,k22), K3 ¼ (k31, k32), K4 ¼

(k41, k42) are the gains of the observer, and the sliding

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surface S is given by

S ¼s1

s2

� �¼

x1 � x1

x2 � x2

� �¼

0

0

� �

Is ¼sign(s1)

sign(s2)

� � :

More precisely, the observer has the following form

x:

1 ¼ �a1x1 � a2 x3 þa2V1 þ k11 sign(x1 � x1 )þk12 sign(x2 � x2 )

x:

2 ¼ �a1x2 � a2 x4 þa2V2 þ k21 sign(x1 � x1 )þk22 sign(x2 � x2 )

x:

3 ¼ k31 sign(x1 � x1 )þ k32 sign(x2 � x2 )

x:

4 ¼ k41 sign(x1 � x1 )þ k42 sign(x2 � x2 )

(6)

The above observer requires the selection of eight gains whichhas to be chosen appropriately to ensure the convergence ofthe estimated state to the real state. However, this task canbe tedious, even though not impossible. For the sake ofsimplicity, we shall employ a simpler version of the aboveobserver with a reduced number of gains. More specifically,we set k12 ¼ k21 ¼ k32 ¼ k41 ¼ 0 and employ the followingobserver throughout the rest of the paper

x:

1 ¼ �a1x1 � a2 x3 þa2V1þ k11 sign(x1 � x1 )

x:

2 ¼ �a1x2 � a2 x4 þa2V2þ k22 sign(x2 � x2 )

x:

3 ¼ k31 sign(x1 � x1 )

x:

4 ¼ k42 sign(x2 � x2 )

(7)

We shall show that in order to ensure the convergence of theobserver, the gains should satisfy the following conditions

(C1) k11 . a2 e3

�� max

and k22 . a2 e4

�� max

(C2) (k31=k11) , 0 and (k42=k22) , 0

More precisely, we state the following proposition.

Proposition 1: Assume that system (3) is uncoupled (i.e. x1

does not depend on x2), then system (7) is an asymptoticobserver for system (3) provided that its gains satisfyconditions (C1) and (C2).

Proof: By setting e ¼ x� x, it can be shown that the errordynamics is given by

_e1 ¼ �a2e3 � k11 sign(e1)_e2 ¼ �a2e4 � k22 sign(e2)_e3 ¼ �k31 sign(e1)_e4 ¼ �k42 sign(e2)

(8)

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Consider the following candidate Lyapunov function

V (e) ¼1

2e21 þ

1

2e22

The time derivative of V (e) along the trajectories of e isgiven by

_V (e) ¼ e1 _e1 þ e2 _e2

¼ e1 �a2e3 � k11 sign(e1)� �

þ e2 �a2e4 � k22 sign(e2)� �

We need to find conditions on the observer gains, in orderto force _V (e) to be negative. These conditions are

† if e1 . 0, the switching gain k11 . 2a2e3;

† if e1 , 0, the switching gain k11 . a2e3;

† if e2 . 0, the switching gain k22 . 2a2e4;

† if e2 , 0, the switching gain k22 . a2e4.

As the vector e(t) is bounded for all time t, the aboveconditions can be reduced to condition (C2) given above;that is, k11 . a2 e3

�� max

and k22 . a2 e4

�� max

. Consequently,the vector e1,2(t) ¼ (e1(t), e2(t)) is globally asymptoticallystable. We therefore have _V (e) , 0 for e = 0 and _V (e) ¼ 0when e ¼ 0.

When sliding-mode regime occurs, that is, when system’sbehaviour is on the sliding surface S, we have(e1(t), e2(t)) ¼ ( _e1(t), _e2(t)) ¼ 0. In such a case, we have

0 ¼ �a2e3 � k11 sign(e1)0 ¼ �a2e4 � k22 sign(e2)

or

�I s ¼sign(e1)sign(e2)

� �¼

a2e3

k11a2e4

k22

0B@

1CA (9)

The vector �I sDI s represents the equivalent output errorinjection term necessary to maintain a sliding-mode on S.

Using the equivalent output (9), the reduced-ordersliding-mode is governed by

_e3 ¼ �k31=a2e3ðk11Þ

_e4 ¼ �k42=a2e4ðk22Þ(10)

Under constraints (C2), the tracking error e(t) convergesto zero exponentially. This completes the proof ofProposition 1. A

Remark 1: Note that the estimation error can be made toconverge to zero arbitrarily fast by appropriate choice of theobserver gains satisfying conditions C1 and C2.


4 Rotor position detectionThe sensorless control scheme, in which the above sliding-mode observer is employed, is illustrated in Fig. 1.Generally, the BLDC motor is controlled through voltage-source current-controlled inverters whereby the driveemploys an inner current loop with an outer speed loop.The inverter must supply a rectangular current waveformwhose magnitude, IMAX, is proportional to the machineshaft torque. The equivalent DC current is obtainedthrough the sensing of two of the three armature currents.From these currents, the absolute value is taken, and a DCcomponent – which corresponds to the amplitude, IMAX,of the original phase currents – is obtained. This DCcomponent is compared with a reference coming from theoutput of the speed regulator 1 [PI(1)], and the error signalis processed through a PI controller 2 [PI(2)]. The outputof the PI controller 2 is compared with a saw-tooth carriersignal, to generate the PWM for the power transistors. Atthe same time, the estimated position determines whichcouple among the six transistors of the inverter shouldreceive this PWM signal.

In this scheme, the outputs of the sliding-mode observerprovide an estimate of the phase-to-phase back-EMF. Theestimate is then fed to the ZCDs to estimate thecommutation points.

As shown in Fig. 2, the detection of the six rotor positions(ur) of the motor can be easily determined from the estimatedphase-to-phase back-EMFs. For instance, during the switchcommutation ‘a ’ of the transistor ‘Ta ’, one can deduce thatthe phase-to-phase back-EMF Eab is always positive andthat the phase-to-phase back-EMF Eca is always negative.

Figure 2 Phases currents (Ia, Ib, Ic) and phase-to-phaseback-EMFs (Eab, Ebc, Eca)


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This means that it is possible to obtain a sufficient conditionin order to extract the state switch commutation ‘a ’ of thetransistor ‘Ta ’; that is, it is enough to have Eab . 0 andEca , 0. For the other switch commutations, the sameanalysis logic is employed.

5 Rotor speed estimationIn order to obtain the instantaneous rotor speed of the motor,we employ the method given in [10]. The method usesthe mathematical relation between the magnitudeEmax(phase-to-phase) and rotor speed defined by

vr ¼Emax(phase-to-phase)

2KEMF

(11)

with

Emax(phase-to-neutral) ¼ KEMFvr

where KEMF is the back-EMF constant. In other words

Emax(phase-to-neutral) ¼Emax(phase-to-phase)

2

The magnitude Emax(phase-to-phase) can be determined usingthe estimated phase-to-phase back-EMF and sixcommutations points. Using the phase-to-phase back-EMFestimate provided by sliding-mode observer and the sixcommutations points determined through the ZCD block,the Emax amplitude is obtained as depicted in Fig. 3.

6 Simulation resultsIn order to study the performance of the proposed sliding-mode observer for both the position and speed estimationsin relation to the sensorless BLDC motor drive, a detaileddigital computer program using simulation packageMATLAB/Simulink was developed. The motor

Figure 3 Determination of Emax of the phase-to-phase back-EMFs


parameters are given in Section 7. The robustness of theproposed observer with respect to noise and parametervariations was tested. We have carried out a series ofsimulations whereby we have fixed the speed at 1000 rpm.The first set of simulations analyses the performance ofthe observer under the effect of a measurement noise in thecurrent. A 10% uniform random noise was added to thecurrent. As it can be seen in Fig. 4, the observer is robustwith respect to measurement noise. Even though notshown here, a similar performance was observed for theestimation of Ebc.

The second set of simulation is carried out when a smallerror of 20% in the value of stator inductances is made.Fig. 5 shows that this has little influence on theconvergence of the observer. However, one can observe theappearance of oscillations and peaks during the transientperiod and which diminish during steady state.

The rotor speed and position estimation is done via thezero detectors as mentioned previously. The profile of themotor speed vr and its estimate vr is shown in Fig. 6.

Figure 5 Estimation of Eab under a variation in Lcs

Figure 4 Estimation of Eab under noisy measurement

Figure 6 Measured speed vr and observed speed vr

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Here again the good tracking performance of the observer canbe observed. Finally, the rotor position ur and its estimate ur

are shown in Fig. 7.

7 Experimental resultsThe proposed estimation method was implementedexperimentally by using a laboratory set-up, which is shownin Fig. 8. It comprises a BLDC motor with trapezoidalback-EMF as illustrated in Fig. 9. The motor rating is:220 V, Imax ¼ 30A, 1800 rpm, Cn ¼ 12 N m. The motorparameters were Rs ¼ 0.966 V, Lcs ¼ 11.592 mH, p ¼ 2and Ke ¼ 0.665 V/rad/s.

The principal elements of the test drive block are:

† the voltage-source inverter with insulated gate bipolartransistor,

† LEMs (sensors currents and terminal voltages)

† and the IC30S circuit switch which isolate the IGBT fromthe control circuit.

The control and estimation algorithm has beenimplemented on a eZdspF2812 evaluation board asdepicted in Fig. 8. In the control algorithm, we employ aninner current loop without the outer speed loop. Thesliding-mode observer, measurements currents and theterminal voltages are implemented with the same samplingperiod (100 s). The algorithm has been generated by codecompose studio (CCS) version 2.2. This extends the basic

Figure 8 Experimental set-up

Figure 7 Rotor position ur and its estimate ur


code generation tools with a set of debugging and real-timeanalysis capabilities.

After having generated the control algorithm code of theBLDC motor, one can observe that the shape of the motorphase current resembles the ideal 1208 step waveform. Acomparison of the results obtained in sumulation andexperimentally is shown in Figs. 10 and 11.

One can observe that the profile of both currents is similar.Also, as is illustrated in Fig. 12, the motor phase current ispulsating, which is provided in real time by the computerthrough the CCS.

It is well known that the technique of the slipping modesgenerates undesirable chattering; this problem can becircumvented by replacing the switching sign function by acontinuous function which is smoothed in the vicinity ofthe slipping surface [11]. In our case, we chose thefollowing function

sign(si) ¼

1 if si . m3

1� n2

m3 � m2

(si � m2)þ n2 if m2 � si � m3

n2 � n1

m2 � m1

(si � m1)þ n1 if m1 � si � m2

n1 � n0

m1 � m0

(si � m0)þ n0 if m0 � si � m1

si

m0

if si

�� , m1

n1 � n0

m1 � m0

(si þ m0)� n0 if �m1 � si � �m0

n2 � n1

m2 � m1

(si þ m1)� n1 if �m2 � si � �m1

1� n2

m3 � m2

(si þ m2)� n2 if �m3 � si � �m2

�1 if si , �m3

8>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>:

Figure 9 Measured actual trapezoidal phase-to-phaseback-EMF of BLDCM


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where m0 ¼ 0:1, m1 ¼ 1, m2 ¼ 1:6 and n0 ¼ 0:01, n1 ¼ 0:5and n2 ¼ 0:8.

The above sign function is depicted in Fig. 13.

Fig. 14 shows the profile of trapezoidal phase-to-phaseback-EMF as estimated by the sliding-mode observer. Thetop trace shows the trapezoidal phase-to-phase back-EMFEab. The bottom trace depicts the trapezoidal phase-to-phase back-EMF Ebc.

Figure 10 Phase a current Ia obtained experimentally


Fig. 15 shows the plot of the rotor position. The topfigure shows the rotor position measured through theposition sensor placed on the shaft of motor. The bottomfigure shows the estimated rotor position provided by theobserver.

It can be observed that the estimated rotor position is inagreement with what is measured directly by the positionsensor. These experimental results therefore confirm thoseobtained in simulation. They make it possible to appreciate

Figure 11 Phase a current Ia obtained in simulation

Figure 12 Measured current of BLDC motor (time scale 100! 10 ms)

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Figure 13 Sign function

Figure 14 Estimated phase-to-phase back-EMF (time scale100! 10 ms)

Figure 15 Estimated and actual rotor position (time scale100! 10 ms)


the static and dynamic performances of the observer andshow its capability to filter the noises coming from thecurrent sensors.

8 ConclusionsIn this paper, a sliding-mode observer design is proposed inorder to estimate the trapezoidal back-EMF of a BLDCmotor. The estimated back-EMF is, in turn, used toprovide an estimation of the rotor position via ZCDs. Anestimation of the rotor speed is also provided by exploitingthe mathematical relationship between the back-EMF andthe rotor speed. The main feature of the proposed observeris that it allows to obtain good estimates of the back-EMFeven in the presence of noise and parametric uncertainties;hence attributing to the observer reasonable robustness andfiltering characteristics. Simulation and experimental resultshave confirmed the good convergence and robustnessperformance of the observer. It also shows the applicabilityof the observer without inducing any discontinuity in thereconstruction of the state.

9 References

[1] BECERRA R.C., JAHNS T.M., EHSANI M.: ‘Four quadrantsensorless brushless ECM drive’. IEEE Applied PowerElectronics Conf. and Exposition, 1991, pp. 202–209

[2] SHAO J.: ‘Direct back EMF detection method forsensorless brushless DC motor drives’, MS thesis,Department of Electrical Engineering, Virginia Polytechnicinstitute Virginia, September 2003

[3] OGASAWARA S., AKAGI H.: ‘An approach to positionsensorless drive for brushless dc motors’, IEEE Trans. Ind.Appl., 1991, 27, (5), pp. 928–933

[4] MOREIRA J.: ‘Indirect sensing for rotor flux position ofpermanent magnet AC motors operating in a wide speedrange’. IEEE Industry Application Society Annual Meeting,1994, pp. 401–407

[5] BEJERKE S.: ‘Digital signal processing solutions for motorcontrol using the TMS320F240DSP-controller’. ESIEE, Paris,1996

[6] SIMON D., FEUCHT D.: ‘Synchronous motor phase controlby vector addition of induced winding voltages’, IEEETrans. Ind. Electron., 2004, 51, (3), pp. 537–544

[7] TERZIC B., JADRIC M.: ‘Design and implementation of theextended Kalman filter for the speed and rotor positionestimation of brushless DC motor’, IEEE Trans. Ind.Electron., 2001, 48, (6), pp. 1065–1073

[8] KENJO T., NAGAMORI S.: ‘Permanent magnet and brushlessDC motors’ (Clarendon Press, Oxford, 1985)


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[9] MILLER T.J.E.: ‘Brushless permanent magnet andreluctance motor drives’ (Clarendon Press, Oxford, 1989)

[10] FAKHAM H., DJEMAI M., BLAZEVIC P.: ‘Sliding modes observerfor position and speed estimations in brushless DC motor(BLDC motor)’. IEEE ICIT, Tunisia, 2004

Electr. Power Appl., 2008, Vol. 2, No. 6, pp. 353–361i: 10.1049/iet-epa:20070242

[11] DJEMAI M., BARBOT J.P., GLUMINEAU A., BOISLIVEAU R.:‘Observateur a modes glissants pour la machineasynchrone’. APII-JESA-36, Commande et Observation,2002, pp. 657–667

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Fakham H, Djemai M, Busawon K - Design and practical implementation of a back-EMF sliding-mode observer for a brushless DC motor - article.pdf