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A Self-Learning Solution for Torque Ripple Reductionfor Nonsinusoidal Permanent-Magnet Motor Drives

Based on Artificial Neural NetworksDamien Flieller, Ngac Ky Nguyen, Patrice Wira, Guy Sturtzer, Djaffar Ould

Abdeslam, Jean Mercklé

To cite this version:Damien Flieller, Ngac Ky Nguyen, Patrice Wira, Guy Sturtzer, Djaffar Ould Abdeslam, et al.. A Self-Learning Solution for Torque Ripple Reduction for Nonsinusoidal Permanent-Magnet Motor DrivesBased on Artificial Neural Networks. IEEE Transactions on Industrial Electronics, Institute of Electri-cal and Electronics Engineers, 2014, 61 (2), pp.655-666. 10.1109/TIE.2013.2257136. hal-01019390

1

A Self-Learning Solution for Torque RippleReduction for Non-Sinusoidal Permanent Magnet

Motor Drives Based on Artificial Neural NetworksDamien Flieller, Ngac Ky Nguyen, Member, IEEE, Patrice Wira, Member, IEEE, Guy Sturtzer, Djaffar Ould

Abdeslam, Member, IEEE, and Jean Merckle

Abstract—This paper presents an original method, based onartificial neural networks, to reduce the torque ripple in apermanent-magnet non-sinusoidal synchronous motor. Solutionsfor calculating optimal currents are deduced from geometricalconsiderations and without a calculation step which is generallybased on the Lagrange optimization. These optimal currents areobtained from two hyperplanes. The study takes into account thepresence of harmonics in the back-EMF and the cogging torque.New control schemes are thus proposed to derive the optimalstator currents giving exactly the desired electromagnetic torque(or speed) and minimizing the ohmic losses. Either the torqueor the speed control scheme, both integrate two neural blocks,one dedicated for optimal currents calculation and the otherto ensure the generation of these currents via a voltage sourceinverter. Simulation and experimental results from a laboratoryprototype are shown to confirm the validity of the proposedneural approach.

Index Terms—Permanent Magnet Synchronous Motor, TorqueRipple, Cogging Torque, Homopolar Current, Neuro-controller,Adaline.

NOMENCLATURE

i Stator current vectore Back-EMF vectorn Number of phases of PMSMp Number of pole pairsK1 Speed normalized back-EMF vectorK0 Vector containing only fundamental components

of K1

K2 Vector does not containing the components ofrank nq, q = 1, 2, · · · which exist in K1

θ Mechanical rotor angleΩ Mechanical rotor speedCtotal(pθ) Total torqueCem(pθ) Electromagnetic torqueCcog(pθ) Cogging torqueCref Desired torque

Manuscript received June 5, 2012. Revised October 19, 2012 and January21, 2013. Accepted for publication February 13, 2013.

Copyright (c) 2009 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

D. Flieller and G. Sturzer are with the GREEN Laboratory, INSA ofStrasbourg, France, e-mail: [damien.flieller, guy.sturtzer]@insa-strasbourg.fr.

N. K. Nguyen is with the L2EP Laboratory, Arts et Metiers ParisTech, LilleCedex, France, email: NgacKy.NGUYEN@ensam.eu.

P. Wira, D. Ould Abdeslam and J. Merckle are with the MIPS Labora-tory, University of Haute Alsace, France, email: [patrice.wira, djaffar.ould-abdeslam, jean.merckle]@uha.fr.

Ωref Desired speedkopt−i Optimal function by strategy i = 0, 1, 2iopt−i Stator current vector corresponding to strategy

i = 0, 1, 2

I. INTRODUCTION

PERMANENT MAGNET SYNCHRONOUS MOTORS(PMSMs) are widely used in many industrial production

systems due to their attractive features which are theircompactness, high torque mass ratio, high efficiency andease to be controlled. There are mainly two principaltypes of PMSMs which are characterized by sinusoidal ornon-sinusoidal back electromotive forces (back-EMF) [1].

Whatever the PMSMs, torque ripples come from variouscauses [1]–[5]. The cogging torque is generated by the inter-action between the rotor magnetic field and the stator teetheven if there is no current in the stator. Another source oftorque pulsation results from the interaction between the statorcurrents and the rotor magnetic field. Generally, this torque canbe divided into two terms. The first one is due to the rotor’smagnets and the stator currents. The second one is due tothe saillance. In this paper, only the first term is consideredbecause our interest points to the non-saillance motor. Worksrelated to saillance motors can be found in [6]–[8].

A constant torque is highly required in many applications.Therefore, various works were proposed to minimize theundesirable torque ripple [9]–[11]. These works can be dividedinto two categories. The techniques from the first one consist indeveloping the machine’s stator and rotor design to cancel theundesirable torque ripple. The magnets’s properties and howto distribute them optimally on the rotor’s surface are studiedin [5], [12]–[14]. The influence of the slots/poles ratio of themachine on the torque ripple is specifically studied in [13].In [15], four analytical models for predicting the coggingtorque in surface-mounted Permanent Magnet (PM) machines.This work focus on the influence of design parameters, suchas the slot-depth-to-slot-opening ratio and optimal pole-arc-to-pole-pitch and slot-opening-to-slot-pitch ratios, on the coggingtorque. The techniques from the second category are basedon the stator current control. An analyze of the torque’sexpression is used to calculate the best currents that cancelthe torque ripple. The literature provides various techniquesfor the optimal current determination according to adequatetransformations. For example, the individual harmonics of the

2

Fourier’s series of the back-EMF are used in order to obtainthe stator currents in [16]–[18]. The work presented in [16]optimizes the currents only for harmonics of rank 5 and 7while the method presented in [17], [18] gives the optimalcurrents by calculating a pseudo-inverse matrix containing theharmonics of the back-EMF. A formula was proposed in [19]which works in the a-b-c reference frame. In this method thehomopolar current is null and moreover, the loss by Jouleeffect is not optimal. Always in the a-b-c reference frame,the works presented in [20], [21] give the expressions of theoptimal currents. These methods are based on a Lagrange’soptimization to obtain the currents which minimize the statorohmic losses and maintain the desired torque. While workingin the d-q reference frame of Park, the authors in [22], [23]determine the expressions of the d-q optimal currents; [23] op-timizes only for one harmonic of rank 6. A direct expression ofthe a-b-c currents is given in [24], [25] by using an extensionof Park’s transformation. In [26], an adaptive process basedon the Fourier series expansion is presented for a machinecontaining 4 pairs of poles and 24 stator slots. The flow chartproposed leads to the determination of the current iq(pθ) inorder to compensate for the 6th rank harmonic of the torqueripple. The repetitive and iterative learning control schemeswere presented in [4], [27]. Based on a vectorial approach,the generation of optimal current references for multiphasePMSM in real time is reported in [28]. This approach reducesthe computing operations compared to scalar methods whichgenerally require a large amount of calculus. The performanceof the proposed method is experimentally valided in normaland fault mode (open-circuited phases). Zhao et al. in [29]recently presented an approach based on harmonic currentinjection, on which the Redundant Flux-Switching Permanent-Magnet Motor (R-FSPM) can be operated with high dynamicperformance and good behavior at steady-state. If the back-EMF of the R-FSPM is highly non-sinusoidal, then the currentcalculation with the proposed method becomes complicated.

The work presented in this paper is different from theapproaches mentioned above. A torque control scheme and aspeed control scheme are proposed to cancel the torque ripple,including the cogging torque. The design of the controllers isbased on Adaline Neural Networks (ANNs) [30], [31]. Indeed,the learning capabilities of the ANNs allow to calculate theoptimal currents which give exactly the desired torque (ora desired speed) and minimum ohmic losses. This solutioncan be applied for a multiphase PM machine under normaloperation or under open-circuit fault conditions.

This paper is organized as follows. The problem relatedto torque pulsation due to the harmonics of the back-EMFis presented in Section II. Section III presents new geo-metrical considerations leading to the optimal currents. Theyare based on the definition of hyperplanes whose equationsdepend on stator currents. Section IV proposes an originaldirect torque (or speed) controller based on an Adaline neuralnetwork. Simulation results in Section V and experimentalresults in Section VI confirm the validity of the proposedneural approach. Finally, concluding remarks are provided inSection VII.

0

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

2.0

dC

totalC

1

TK i

1e

1i

pq 2p

Fig. 1. Torque obtained in a non-sinusoidal three-phase machine in BLACmode

II. PMSM’S TORQUE RIPPLE

The stator current and the back-EMF vectors of a n phasesPMSM can be defined respectively by:

i =[i1 i2 · · · ii . . . in

]T, (1)

e =[e1 e2 · · · ei . . . en

]T. (2)

If the machine is non-sinusoidal, we can develop the back-EMF for the ith phase as:

ei =

kmax∑k=1

eak,i sin (kpθ) + ebk,i cos (kpθ), i = 1, · · · , n,

(3)and the fundamental component of ei is:

e0i = ea1,i sin (pθ) + eb1,i cos (pθ), i = 1, · · · , n, (4)

where kmax is the highest considered harmonic of the back-EMF, θ is the angular position, eak,i and ebk,i are the Fouriercoefficients. p is the number of pole pairs.

The speed normalized back-EMF vector K1 can be ex-pressed by:

K1 =1

Ω

[e1 e2 · · · ei · · · en

]T. (5)

We can define K0 by:

K0 =1

Ω

[e01 e02 · · · e0i · · · e0n

]T, (6)

where Ω = dθdt is the rotor’s speed.

The total torque of the machine is given by:

Ctotal(pθ) = Cem(pθ) + Ccog(pθ), (7)

with Cem(pθ) = KT1 i which represents the electromagnetic

torque of the interaction between the stator currents and therotor’s magnetic field. Fig. 1 shows the current and the back-EMF of phase 1 as well as the total torque of a non-sinusoidalthree-phase machine. It is obvious that the sinusoidal excita-tion currents reveal a torque pulsation. To supply the machinecorrectly, we have to determine the stator current for eachphase such that the desired torque Cref is obtained.

3

TABLE IRELATIVE ANGULAR FREQUENCIES OF THE COGGING TORQUE

PULSATIONS IN A THREE-PHASE PMSM (n = 3) [32], [34]

Winding Ns 2p angular frequencies mlp of Ccog

Distributed 18 6 18, 36, 54,. . .36 6 36, 72, 108. . .9 6 18, 36, 54,. . .

Concentrated 12 10 60, 120, 180. . .12 14 84, 168, 252,. . .

Ccog is the cogging torque which can be expressed by [12],[32], [33]:

Ccog(pθ) =

lmax∑l=1

Ccoga,ml sin (mlpθ) + Ccogb,ml cos (mlpθ)

=

lmax∑l=1

Ccog,ml cos (mlpθ + φml) , l = 1, 2, 3 . . .

(8)

where mp is the least common multiple of the number of statorslots Ns and the number of poles. We have:

Ccog,ml =√C2

coga,ml + C2cogb,ml, (9)

tanφml =− Ccoga,ml

Ccogb,ml. (10)

Tab. I gives the angular frequencies of the pulsation of thecogging torque in a three-phase machine (n = 3) for severalfeasible combinations of slot number and pole number ofPMSM [32], [34].

Our goal is to cancel the torque ripple and we’ll obtain:

Ctotal(pθ) = Cref . (11)

From now, a torque analysis is necessary to design anefficient controller. Indeed, we want to establish a relation-ship between the back-EMF and the torque. This allows todetermine the coefficients of the current harmonic terms andtherefore allows to eliminate several or all undesirable torqueharmonic terms. In steady state, when the stator current is:

ii =

hmax∑h=1

iah,i sin (hpθ) + ibh,i cos (hpθ), i = 1, · · · , n,

(12)the electromagnetic torque is then given by:

Cem(pθ) = C0 +

qmax∑q=1

Ca,qn′ sin (qn′pθ) +Cb,qn′ cos (qn′pθ),

(13)where n′ = 2n if the back-EMF contains only odd oreven harmonic components, hmax and qmaxn

′ are the highestharmonics of stator currents and electromagnetic torque re-spectively. In the general case, when the back-EMF containsboth odd and even harmonic terms, we have n′ = n. Thisanalyze is obtained by an extension of the work presentedin [17]. C0 is the mean value and Ca,qn′ , Cb,qn′ are the Fouriercoefficients of Cem(pθ). qmax is determined by:

qmax = integer(kmax + hmax

n′). (14)

TABLE IIRELATIVE ANGULAR FREQUENCY OF THE TORQUE PULSATIONS IN A

THREE-PHASE PMSM (n = 3) [33]

h k 1 3 5 7 9 11 13 15 17

1 0 6 6 12 12 183 0,6 6,12 12,185 6 0 12 6 187 6 12 0 18 6 249 6,12 0,18 6,2411 12 6 18 0 24 613 12 18 6 24 0 3015 12,18 24 0,3017 18 12 24 6 30 019 18 24 12 30 6 3621 18,24 12,30 6,36

We note:Cqn′ =

√C2

a,qn′ + C2b,qn′ . (15)

The compensation of the torque ripple leads to:

C0 = Cref (16)

and this condition is obtained when:qmax∑q=1

Ca,qn′ sin (qn′pθ) + Cb,qn′ cos (qn′pθ) = −Ccog(pθ).

(17)The torque pulsations produced by the stator current com-

ponents h and the back-EMF components k for a three-phasemachine are given by Tab. II [33]. It is obvious that a non-sinusoidal PMSM exited by sinusoidal currents gives a torquepulsation. In order to supply the machine correctly, the statorcurrent for each phase must be determined so that:

KT1 i = Cref − Ccog(pθ). (18)

There is an infinity of solutions i for (18) and we aimto obtain one giving a minimum ohmic losses. In [21],[22], [35], [36], solutions are obtained from an optimizationbased on Lagrange’s method to derive the stator currents. Thework presented thereafter will complete these solutions byusing a geometrical approach to deduce the optimal currents.Furthermore, the optimal currents can be easily obtained by afeed-back current control scheme based on an Adaline neuralnetwork. It should be noticed that these currents give exactlythe desired torque by taking into account the cogging torque.

III. GEOMETRICAL INTERPRETATION OF THE THREEOPTIMAL CURRENTS

(18) is an equation of a hyperplane. Let P be the hyperplanethat contains all points M whose coordinates are equal to iand which satisfies (18). K1 is thus normal to P .

In the case of a star coupled machine, the zero homopolarcurrent constraint can be introduced by:

uT1 i = 0, (19)

with u1 =[1 1 · · · 1

]T , ∥u1∥2 = n.(18) is also an equation of an other hyperplane. Let H1 be a

second hyperplane where all points M verify (19). u1 is thusnormal to H1. P and H1 are represented on Fig. 2.

4

1u

1K

1optM

-

1opt-

iX

2opt-

i2opt

M-

1H

O

P

-0optM

-0opti

0X

Fig. 2. Geometrical representation of the three optimal current vectorsiopt−0, iopt−1, iopt−2

1u

1K

O

2K

0K

1D

1optM

-

2optM

-

-0optM

2D

0D

Fig. 3. Geometrical representation of the three vectors K0, K1, and K2

First, we try to find i = iopt−0 proportional to K0:

iopt−0 = kopt−0K0. (20)

From (18) and (20), we have:

kopt−0 =Cref − Ccog(pθ)

KT1 K0

, (21)

and finally, we obtain:

iopt−0 =Cref − Ccog(pθ)

KT0 K1

K0 =Cref − Ccog(pθ)

KT1 K0

K0. (22)

The current of the ith phase is therefore:

iopt−0,i =Cref − Ccog (pθ)

n∑k=1

eke0k

Ωe0i . (23)

It should be noticed that iopt−0 always gives a zero homopolarcurrent. This solution is represented by the point Mopt−0

on Fig. 2. iopt−0 does not give the minimal ohmic losses.Let us define Mopt−1, represented on Fig. 2, that belongs tothe hyperplane P and is nearest point to the origin O. Wewill show that taking some geometrical considerations allowsto easily derive the expression of the current vector iopt−1,

corresponding to Mopt−1, which has the minimal module. Thecurrent iopt−1 is thus proportional to K1, so:

iopt−1 =

(KT

1

∥K1∥iopt−1

)K1

∥K1∥, (24)

with K1

∥K1∥ is the unit vector in the direction of K1. From (18),we have KT

1 iopt−1 = KT1 iopt−0, and (24) becomes:

iopt−1 =

(KT

1

∥K1∥iopt−0

)K1

∥K1∥, (25)

and with KT1 K1 = ∥K1∥2, we obtain:

iopt−1 =(Cref − Ccog(pθ))KT

1 K0

KT1 K0

K1

∥K1∥2

=Cref − Ccog(pθ)

KT1 K1

K1. (26)

iopt−1 gives the minimal ohmic losses and the ith phase canbe written as follow:

iopt−1,i =Cref − Ccog (pθ)

n∑k=1

ek2Ωei. (27)

In a star connected machine, we have to introduce theadditional constraint given by (19).

Let us define Mopt−2, represented on Fig. 2, that belongsto the intersection of H1 and P and is nearest to the originO. Once again, we will show that taking some geometricalconsiderations allows to easily derive the expression of thecurrent vector iopt−2, corresponding to Mopt−2, which has theminimal module.

The current iopt−2 is proportional to K2:

iopt−2 =

(KT

2

∥K2∥iopt−2

)K2

∥K2∥, (28)

where K2 is determined by (Fig. 3):

K2 = K1 −KT

1 u1

∥u1∥2u1 = K1 − uT

1 K1u1

n. (29)

It can be seen from (29) that K2 does not contain harmonicscomponents of rank nq, q = 1, 2, · · · which exist in the vectorK1. K2 can thus be expressed by:

K2 =1

Ω

[e′1 e′2 · · · e′i · · · e′n

]T, (30)

where:

e′i =ei −1

n

kmax∑k=1

ek

=

k′max∑

k=1,k =nq

eak,i sin (kpθ) + ebk,i cos (kpθ), i = 1, . . . , n.

(31)

k′max is the highest component of e′i and k′max = kmax whenkmax = nq.

From (18), we have:

KT1 iopt−2 = KT

1 iopt−0 (32)

5

where:

KT1 iopt−2 =

(K2 +

KT1 u1

∥u1∥2u1

)T

iopt−2 = KT2 iopt−2 (33)

KT1 iopt−0 =

(K2 +

KT1 u1

∥u1∥2u1

)T

iopt−0 = KT2 iopt−0 (34)

We can thus deduce:

KT2 iopt−2 = KT

2 iopt−0. (35)

From (22) and (35), the expression (28) becomes:

iopt−2 =

(KT

2

∥K2∥iopt−0

)K2

∥K2∥

=

(KT

2 K0

KT2 K2

Cref − Ccog(pθ)

KT1 K0

)K2. (36)

From (29), we obtain:

KT2 K0 = KT

1 K0. (37)

Moreover, from the Pythagorean theorem:

KT2 K2 = KT

1 K1 −(KT

1 u1

)2∥u1∥2

= KT1 K2 = KT

2 K1. (38)

According to (37) and (38), iopt−2 is obtained as follow:

iopt−2 =Cref − Ccog(pθ)

KT1 K2

K2. (39)

The expression for the ith phase current iopt−2 is given by:

iopt−2,i =Cref − Ccog (pθ)

n∑k=1

(ek − 1

n

n∑l=1

el

)2

(ei −

1

n

n∑l=1

el

)Ω

=Cref − Ccog (pθ)

nn∑

k=1

ek2 −(

n∑k=1

ek

)2

(nei −

n∑l=1

el

)Ω. (40)

iopt−2 is the current vector which offers less ohmic losses withzero homopolar current.

This section has developed the expressions of optimalcurrents which are already presented in various works byusing different approaches. Indeed, iopt−0 is identical to thesolution given in [19]. Based on the Lagrange’s optimization,the currents iopt−1 and iopt−2 are also obtained in [21],[22], [37]. Our solution leads to the same results. We give acompleted geometrical representation on which we can easilydetermine these different optimal currents.

Expressions (22) (26) and (39) can be generalized to faultyconditions. In the faulty phases, the current can be null (opencircuit), equal to a saturation value, or be a short-circuitcurrent. The current of the healthy phases can be calculatedif the defect conditions are known by imposing the desiredtorque, minimizing Joule losses, and fulfilling the constraintson the homopolar current (free or null). But, it is difficult todetermine in real time the cogging torque and the nature ofthe fault. A learning schemes based on an Adaline network isthus proposed as a torque controller to provide the Fourier’s

0.16

0.18

0.20

0.22

0.24

0.26

180° 360°pq

0optk

-

2optk

-

1optk

-

A

Wb

Fig. 4. Three optimal functions kopt−i in non-sinusoidal back-EMF vectors

kopt-1

y1N=1

y1

N=2,3,4,5,6»

270° 360°pq

0.16

0.18

0.20

0.22

0.24

0.26

A

Wb

Fig. 5. Optimal function kopt−1 obtained for different values of N(mlmax = 12, qmaxn′ = 12, qmaxn′ = 18 is negligible)

coefficients of an optimal function. This technique will thusbe able to calculate the desired optimal currents on real time.This main contribution is presented in the following section.

IV. TORQUE AND SPEED CONTROLLERS BASED ONADALINE NEURAL NETWORKS

A. Principal ideas for the torque and speed control

The optimal current is a product between kopt−i (pθ) withthe vector Ki, i = 0, 1, 2. The optimal currents iopt−i can berewritten as:

iopt−0 =Cref − Ccog (pθ)

KT1 K0

K0 = kopt−0 (pθ)K0, (41)

iopt−1 =Cref − Ccog (pθ)

KT1 K1

K1 = kopt−1 (pθ)K1, (42)

iopt−2 =Cref − Ccog (pθ)

KT1 K2

K2 = kopt−2 (pθ)K2. (43)

Fig. 4 gives the shapes of the three optimal functions kopt−i

corresponding to a cogging torque and to a non-sinusoidalback-EMF (the Fourier’s components of the back-EMF aregiven in section V-A).

The optimal functions kopt−i(pθ) with i = 0, 1, 2 can bewritten by a sum of Fourier’s terms as follow:

kopt−i = k0−i+

N∑q=1

(kaqn′−i sin(qn′pθ) + kbqn′−i cos(qn

′pθ)) .

(44)The functions kopt−i (i = 0, 1, 2) will be learned and synthe-sized by an Adaline network.

The value of N is important in order to compensate forthe torque ripple and has to be determined correctly. This

6

refC

Voltage Source Inverter

+

torquecomputation

emC

timer

mq

motori

K

iy

torque controllerspeed controller

load

controlstrategies

1S

3S

5S

dcV

2S

4S

6S

M

PMSM

PW

M g

ener

ato

r

1 6S S-

PLL

refW

å+

mW

s=0

s=1

switch

estimation

0,1,2i=K

currentcontrollers

opt i-i

measure

multiplication

S

å

(s=1)(s=0)

2 +1 termsN

e

Fig. 6. Torque or speed control scheme for a PMSM based on an Adaline neural network

value can be obtained after a mathematical decomposition ofthe optimal function kopt−i. We present the development ofkopt−1. Based on the value of N for kopt−1, we can determinethe ones for kopt−0 and kopt−2. Indeed, the product of KT

1 K1

can be expressed as:

KT1 K1 = A0 +

qmax∑q=1

Aqn′ cos(qn′pθ + ψqn′) (45)

where qmax = int(2kmax

n′

).

When the coefficients Aqn′ << A0, we can write:

1

KT1 K1

≈ 1

A0

(1−

qmax∑q=1

Aqn′

A0cos(qn′pθ + ψqn′)

). (46)

Finally, we obtain:

kopt−1 =Cref − Ccog(pθ)

KT1 K1

=C0

A0

(1−

lmax∑l=1

Ccog,ml

C0cos(mlpθ + φml)

−qmax∑q=1

Aqn′

A0cos(qn′pθ + ψqn′)

). (47)

By observing (47), we notice that the Adaline can estimatecorrectly the function kopt−1 if the input vector x contains allthe terms presented in kopt−1. So, the highest rank of Adaline’sinput (Nn′) has to be:

Nn′ = max

(mlmax, int

(2kmax

n′

)n′). (48)

With the same development, we determine N for the kopt−0:

Nn′ = max

(mlmax, int

(kmax + 1

n′

)n′), (49)

and for the kopt−2:

Nn′ = max

(mlmax, int

(kmax + k′max

n′

)n′). (50)

Fig. 5 shows the estimation of kopt−1 with different valuesof N (the machine’s parameters are given in section V). Wecan see that a good estimation is obtained when N ≥ 2.

For the three strategies, the vector containing the Fouriercoefficients must be determined:

w∗i =

[k0−i kan′−i kbn′−i .. kaNn′−i kbNn′−i

]T.

(51)Based on this, a torque control and a speed control can besynthesized in one single scheme represented by Fig. 6. Theswitch s allows to choose between the torque control or thespeed control (s = 1 or s = 0 respectively) according to theinverter’s characteristics and to the machine’s coupling. Thefunctions kopt−i (i = 0, 1, 2) will be learned and synthesizedby the Adaline network of the controller based on the threepossible strategies:

• Strategy 0: Only the fundamental component of the back-EMF is used. This strategy can be used for all machinesand all couplings with or without a neutral line.

• Strategy 1: The back-EMF is used completely, with allits harmonic components. This strategy can be used forstar-coupled machines without a neutral line, which doesnot have harmonic components that are multiple of threein their back-EMFs, or all the machines that support ahomopolar current.

• Strategy 2: The back-EMF is used without its harmoniccomponent of rank multiple of n. This strategy can beused for all star-coupled machines without a neutral line.

The Adaline controller takes the place of the traditional torqueor speed controller for ensuring the convergence of the motor’storque or speed toward the desired ones. All the torque ripplesare compensated. The details about the Adaline are presentedin the next section (Ki can be obtained off-line by anotherAdaline, see Section VI-A).

B. Adaline-based controller designAdaline’s structure is shown in Fig. 7. The rule of the torque

or the speed controller based on the Adaline is to provide

7

0 kw

-an i

w¢-

sinm

n pq¢

sinm

Nn pq¢

cosm

Nn pq¢

Må

0 kw

-1w

0 iw

-0x

M

LMSxe

iy

+

-

å refW

mW

cosm

n pq¢bn i

w¢-

aNn iw

¢-

bNn iw

¢-

iw

M

M

Fig. 7. Adaline-based torque or speed control

the optimal functions kopt−i with i = 0, 1, 2. The same inputvector of the Adaline is used for the three strategies:

x =[x0 sin(n′pθ) cos(n′pθ) · · ·

sin(Nn′pθ) cos(Nn′pθ)]T (52)

where θ is supposed to be close to θm which is measured byan encoder.

For each control strategy, the weights of Adaline are definedby:

wi =[w0−i wan′−i wbn′−i .. waNn′−i wbNn′−i

]T.

(53)The output of Adaline is then:

yi = wiTx

=w0−ix0 +N∑q=1

(waqn′−i sin(qn′pθ) + wbqn′−i cos(qn

′pθ)) .

(54)

The Adaline weights are solved using an iterative linear LeastMean Squares (LMS) algorithm [38] in order to minimize thetorque or speed error. The weights of the Adaline can be inter-preted, giving thus a non negligible advantage to the Adalineover other types of ANNs. The Adaline is well-suited andideal for approximating and learning linear relations. It willthus be used to learn the expressions previously developed.The Adaline weights are adjusted at each sampled time k:

wi(k + 1) =wi(k) + ηϵx

xTx=

wi(k) + ηϵx

x20 +N, (55)

where η is a learning rate [38], ϵ is the torque or speed error.On each iteration, the weights of Adaline are enforced to

converge toward the amplitudes of current corresponding tothe control strategy. After convergence,

wi(k)−−−→k→∞w∗i , (56)

where w∗i is the optimal solution given by (51). Finally, the

output of Adaline yi converges toward kopt−i.

-0.5

0

0.5

-10

-5

0

5

10

0.05

0.10

0.15

0.20

0.25

0.30

a)

b)

c)

d)

1e

W

0optk

- 1optk

- 2optk

-

0opti

-

1opti

-

totalC

2opti

-

cogC

-0.5

0

0.5

1.0

1.5

2.0

0(rad)

3e

W

2e

W

8 10642pq

Fig. 8. Simulation results of a three-phase non-sinusoidal machine: a)Φ′

1, Φ′2, Φ′

3 (Wb); b) kopt−0, kopt−1 and kopt−2 (A/Wb) obtained bythe Adaline controller; c) optimal currents (one phase) obtained with threeapproaches (A); d) torque developed (N.m)

V. SIMULATION RESULTS

In this section, simulation results are presented to evaluateand to compare the performances of the approaches previouslydeveloped. A non-sinusoidal three-phase PMSM (n = 3) isconsidered and a constant torque of 1.5 N.m is desired.

A. Case without a cogging torque

The following numerical example is taken. The non-sinusoidal three-phase PMSM has no cogging torque and aspeed normalized back-EMF for the first phase expressed by:

e1 (pθ)

Ω=pϕf1 sin(pθ) + 3pϕf3 sin(3pθ) + 5pϕf5 sin(5pθ)

+7pϕf7 sin(7pθ) + 9pϕf9 sin(9pθ) (57)

with ϕf1 = 0.1223 Wb, ϕf3 = 0.0258 Wb, ϕf5 = 0.0027Wb, ϕf7 = −0.0049 Wb, and ϕf9 = −0.0054 Wb. In thiscase, kmax = 9 (according to (3)) and k′max = 7 according

8

0.98

1.00

1.02

1.04

1.062

0opt-

i

2

1opt-

i

2

2opt-

i

5.2%4.9%

0 1 2control strategies

ohmic losses(pu)

Fig. 9. Ohmic losses (in pu) given by the three torque control strategies

0 0.02 0.04 0.06

-10

-5

0

5

10a)

b)

c)

theorical currents currents obtained by Adaline

time (s)

-10

-5

0

5

10

-10

-5

0

5

10

Fig. 10. a) iopt−0 obtained with (41) learned by an Adaline ; b) iopt−1

obtained with (42) learned by an Adaline; c) iopt−2 obtained with (43) learnedby an Adaline (Ω = 314rad/s and η = 0.1)

to (31). e1 contains only odd terms, the ranks of torque rippleis thus 2qn = 6q and q = 1, 2, 3.

According to (49), (48), (50), we chose 6N = 18 to reach anoptimal solution. Fig. 8 shows the performances of the neuraltorque control with 6N = 18. More precisely, Fig. 8 a) showsthe flux linkage, Fig. 8 b) shows the three optimal functionskopt−0, kopt−1 and kopt−2 obtained by the Adaline controller.Fig. 8 c) gives the shapes of the three optimal currents obtainedwith three strategies and Fig. 8 d) shows the total torquedeveloped. It can be seen that the approach based on iopt−0

gives the same results than the one based on iopt−2 since thesetwo currents are close.

1.44

1.46

1.48

1.5

1.52

1.54

1.56

6 6N

totalC =6 12N

totalC

= 3.3%total

CD =

0

(rad)

1 2 3 4 5 6

pq

Fig. 11. Torques Ctotal (N.m) obtained when 6N = 6 and 6N = 12 withstrategy iopt−2

The stator’s ohmic losses have been estimated for thethree approaches. Fig. 9 confirms that the ohmic losses ob-tained with iopt−1 are minimum. The currents iopt−0 present5.2 % more ohmic losses and the losses of iopt−2 are lightlysmaller. iopt−2 are thus the optimal solution for a machinewhich does not have a homopolar current. Fig. 10 showsthe currents iopt−0, iopt−1 and iopt−2 obtained by learningrespectively (41), (42) and (43) with an Adaline. We can seethat the convergence of all strategies is obtained approximatelyin one round.

B. Case with a cogging torque

To clarify the performance of the proposed control strategy,a cogging torque is now introduced under the same previousconditions. Let us thus consider the following cogging torque:

Ccog(pθ) = 0.06 sin(6pθ) + 0.03 sin(12pθ). (58)

The simulated results with a cogging torque are shown byFig. 11 which compares the total torques obtained for thestrategy based on iopt−2 with two cases, i.e., 6N = 6 and6N = 12. This figure shows that for 6N = 6, the torque ofrank 6 is completely compensated but the torques of ranks 12,18 and 24 are only partially eliminated. Finally, the torqueripple results in ∆Ctotal = 3.3 %. In the second case, it canbe seen from the same figure that the torque ripples of rank6, 12 and 18 are perfectly eliminated for 6N = 12. For thisvalue of N = 2, Fig. 12 shows the optimal functions andthe optimal currents, kopt−i and iopt−i with i = 0, 1, 2, andthe resulting torques. It can be seen that all torque ripple,including the cogging torque, is elimined. Finally, we haveCtotal = Cref = 1.5 N.m.

C. Case with a cogging torque and a fault on one phase

Simulations with a faulty machine are performed to confirmthe validity of the proposed method. An additional pulsationof the electromagnetic torque always appears when one of themachine’s phases is faulty. The compensation of this pulsationis ensured by the healthy phases. In the following, the thirdphase of machine is faulty and the results are presented byFig. 13. In spite of a faulty phase, the control based on the

9

0

0.15

0.20

0.25

-10

-5

0

5

10

-0.5

0

0.5

1.0

1.5

2.0

0.30

1optk

- 2optk

-0optk

-

2cog

C totalC

(rad)

a)

b)

c)

108642pq

0opt-

i1opt-

i2opt-

i

1with

opt-

i

0with

opt-

i

2with

opt-

i

Fig. 12. Control performance with a cogging torque; a) kopt−0, kopt−1

and kopt−2 (A/Wb) obtained by the Adaline controller; b) optimal currentsobtained with three approaches (A); c) torque developed (N.m) and coggingtorque (N.m)

three strategies is able to maintain the torque as a constant.Comparatively to the results obtained in Fig. 12 (with acogging torque but no faulty phase), it can be seen that thereare more harmonics in the optimal functions kopt−i in thepresence of a fault. According to the training capabilities of theAdaline, these harmonics are adjusted in real time in order toproduce the currents that compensate for the torque pulsationcreated by the faulty phase. Finally, a desired constant torqueis obtained.

The proposed neural approaches are also compared to aPI (Proportional Integrator) controller under the same faultyconditions. Fig. 13 c) shows the irregular behavior of thetorque obtained by the PI controller. The neuronal controllerbased on the optimal strategies is more efficient. Maintaininga constant torque under faulty phase conditions remains adifficult task.

VI. EXPERIMENTAL RESULTS

A three-phase non-sinusoidal machine PMSM with Rs = 3Ω, Ls = 12.25 mH and a pair poles number p = 3 is connectedto a three-phase Voltage Source Inverter (VSI). The rotor’sposition as well as the stator currents are measured in real-time by using an incremental coder and currents sensors. Themeasures are sent to a process running on a dSPACE DS1104board hosted by a PC. The process consists of the control

0

0.2

0.4

0.6

-20

-10

0

10

20

-0.5

0

0.5

1.0

1.5

2.0

0(rad)

1with

opt-

i

0with

opt-

i

2with

opt-

i

totalC

2cog

C

1optk

-2opt

k-0opt

k-

a)

b)

c)

with PIcontroller

108642pq

0opt-

i1opt-

i2opt-

i

Fig. 13. Control performance with a cogging torque and a fault on one phase;a) kopt−0, kopt−1 and kopt−2 (A/Wb) obtained by the Adaline controller;b) optimal currents obtained with three approaches (A); c) torque developed(N.m) and cogging torque (N.m)

Dspace1104 current

sensor

inverterincrementalencoder

PMSM& load

PC

transformer

Fig. 14. Experimental platform setup

algorithm based on the proposed strategies which provides thesignals sent to a PWM generator for the control the inverter.

A. Speed normalized back-EMF estimation

In practice, the PMSM is actuated by an induction machinewhich has no cogging torque in order to precisely estimate thespeed normalized back-EMF. In this experiment, the speed andthe back-EMF are measured by standard sensors. The speednormalized back-EMF depends on the internal structure (rotorand stator) of the machine and its amplitude is not related tothe rotor speed.

10

0 0.05 0.01 0.15

-1.0

-0.5

0

0.5

1.0

(s)

measure

estimation

Fig. 15. Estimation of the speed normalized back-EMF by an Adaline(experimentation)

The optimal functions are learned with an Adaline networkwith an input vector composed of cos kpθ and sin kpθ termswith k varies from 0 to 15. After convergence, the signal k1 =e1Ω is estimated with:

k1 =− 0.234 cos (pθ) + 0.563 sin (pθ)

+ 0.123 cos (3pθ) + 0.060 sin (3pθ)

− 0.003 cos (5pθ) + 0.007 sin (5pθ)

+ 0.001 cos (7pθ)− 0.003 sin (7pθ)

+ 0.002 cos (9pθ) + 0.005 sin (9pθ)

+ 0.003 cos (11pθ) + 0.003 sin (11pθ) + ... (59)

In (59), terms of ranks 13 and 15 are negligible and aretherefore not written down. The simultaneous presence ofcosine and sine terms clearly indicates the dissymmetry of theback-EMF. Moreover, there are no even terms in the back-EMF, the torque is thus Cqn′ with n′ = 6. The estimatedspeed normalized back-EMF is given by Fig. 15.

B. Neural speed control

The torque control scheme can not be carried out becausethe plate-form must be equipped with a torque meter. Thus,only the speed control scheme is presented.

The test machine is a star coupled PMSM. Therefore, onlythe strategy with iopt−0 or with iopt−2 can be employed. Wechose the strategy based on iopt−2 because of its lower ohmiclosses. In (59), we note: k′max = 11. By taking the highestangular frequency of Ccog is mlmax = 18, we chose N = 3to compensate all the significant torque harmonics.

A constant reference speed has been fixed at Ωref = 70rad/s. Results are presented by Fig. 16. Fig. 16 a) shows theelectromagnetic torque Cem. We can notice that the expres-sion (17) is justified. The pulsation of Cem compensates forthe cogging torque that exists in the motor. This compensationleads to a smooth speed which is shown in Fig. 16 c). Thereference currents obtained by the optimal solution and themeasured currents are shown by Fig. 16 b). It can be seenthat the resulting currents provided by the neural currentcontroller are close to their references. The non-sinusoidalcurrents obtained with the Adaline controller compensate forall the torque ripple.

Finally, Fig. 17 compares the speed obtained by our ap-proach to the one obtained when the motor is feed by thesinusoidal currents. It clearly shows that the speed pulsation

69

70

71

0.1

0.2

0.3

0.4

-0.5

0

0.5

10 11 12 13 14 15( )pq

70 (rad/s)ref

W =

mW

2opti

- motori

(rad/s)

(A)

a)

b)

c)

(N.m)

Fig. 16. Experimental performance of the PMSM speed control: a) elec-tromagnetic torque, b) iopt−2 obtained and currents consumed by PMSMimotor , c) rotor speed obtained

66

68

72

74

(rad/s)

Wobtained by the sinusoidal currents

Wobtained by2opt-

i

70

76

6410 11 12 13 14 15

( )pq

Fig. 17. Experimental speed obtained by the currents iopt−2 compared tothe one derived by the sinusoidal currents

obtained by the neural approach is eliminated. When themachine is supplied by the sinusoidal currents, it is obviousthat there is a torque ripple which leads to the speed pulsation.By using the proposed approach, the problem of torque andspeed pulsation that is present in the control of conventionalPMSMs have been solved.

VII. CONCLUSION

A new solution to calculate the optimal currents for non-sinusoidal multiphase PMSMs and particularly for three-phasenon-sinusoidal machines has been presented in this paper.These optimal currents are derived from control strategieswhich depend on machine’s structure (for example, with orwithout the neutral current). The optimization criterion con-sists of a desired constant torque and minimal ohmic losses.

11

These optimal currents are directly deduced from a geometri-cal development instead of calculations based on the Lagrangeoptimization. A new torque (or speed) control scheme hasbeen proposed. According to the learning capability of Adalineneural network, the optimal currents are obtained in real time.In each control scheme, the Adaline controller takes the placeof the traditional torque or speed controller for ensuring theconvergence of the motor’s torque or speed toward the desiredone. The torque (or speed) ripple is efficiently compensated.This has been verified by various tests and comparisons underdifferent conditions, even when one of the machine’s phasesis faulty. The proposed simulations and experiments clearlyconfirm the high performances of our approaches.

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12

Damien Flieller received the M.Sc. degree inElectrical Engineering from the Ecole NormaleSuperieure (Cachan), France, in 1988 and the Ph.D.degree in Electrical Engineering from the Universityof Paris, France, in 1995. Since 1995, he is anAssociate Professor in the Department of ElectricalEngineering, INSA of Strasbourg, France. He is nowdirector of the ERGE Group (Electrical EngineeringResearch Team). His research’s field are modelingand control of synchronous motors, active filter, andinduction heating converters.

Ngac Ky Nguyen received his B.Sc degree inElectrical Engineering from the Ho Chi Minh CityUniversity of Technology (HCMUT), Viet Namin 2005. In France, he received his M.Sc. fromPoly’Tech Nantes, in 2007 and his Ph.D. from theUniversity of Haute Alsace (UHA), in 2010, bothin Electrical and Electronic Engineering. From 2011to 2012, he was with the Electrical EngineeringDepartment, INSA of Strasbourg, France. SinceSeptember 2012, he is Associate Professor with theLaboratoire d’Electrotechnique et d’Electronique de

Puissance de Lille (L2EP), Arts et Metiers ParisTech, Lille Cedex, France.His research interests are modeling and control of synchronous motors andpower converters.

1501

Patrice Wira (M’04) received the M.Sc. degreeand the Ph.D. degree in Electrical Engineering fromthe University of Haute Alsace, Mulhouse, France,in 1997 and 2002, respectively. He received theAccreditation to Supervise Research (the FrenchHabilitation a Diriger des Recherches) in ComputerSciences from the University of Haute Alsace in2009. He was an Associate Professor with the MIPSLaboratory (Laboratoire Modelisation, Intelligence,Processus, Systeme) at the University of Haute Al-sace. Since 2011, he is a Full Professor. He is author

or coauthor of more than 20 technical papers covering his research interestsfrom artificial neural networks for the modeling and simulation of complexautomation systems, neuro-control approaches, to adaptive control systems.

Guy Sturtzer received the Ph.D. degree in ElectricalEngineering from the Ecole Normale Superieure,Cachan, France, in 2001. He joined the Depart-ment of Electrical Engineering, INSA de Strasbourg,France, where he is an Associate Professor. Hismain interests include electrical machines, powerconverters and renewable energies.

414

Djaffar Ould Abdeslam received the Ing. degree inElectronics Engineering from the University of Tizi-Ouzou, Algeria, in 2000, M.Sc. degree in ElectricalEngineering from the University of Franche-Comte,Besancon, France, in 2002 and the Ph.D. degree inele ctrical engineering from the University of HauteAlsace (UHA), Mulhouse, France, in 2005. Since2005, he is with the MIPS Laboratory, UHA, asan Associate Professor. His work concerns artificialneural networks, fuzzy logic and advanced controlapplied to power active filters and power electronics.

338

Jean Merckle received the M.Sc. and Ph.D.degreesin Electrical from University Nancy I, Nancy,France, in 1982 and 1988, respectively. In 1988,he joined the MIPS Laboratory, University of HauteAlsace, Mulhouse, France, where he participated inseveral adaptive signal processing projects. From1991 to 1993, he was with the department ofElectrical and Computer Engineering, Universityof California and San Diego, contributing to a 3-D optoelectronic neural architecture with efficientlearning. He is currently a professor of Electrical

and Computer Engineering. His research interests include adaptive neuralcomputation with application to power electronic systems control.