ARTICLE IN PRESS

0967-0661/$ - se

doi:10.1016/j.co

$This paper�CorrespondE-mail addr

(M. Menaa), t

com (R. Ibtiou

Control Engineering Practice 16 (2008) 67–77

www.elsevier.com/locate/conengprac

Sensorless direct vector control of an induction motor$

M. Menaaa,�, O. Touhamib, R. Ibtiouenb, M. Fadelc

aLaboratoire de Robotique Parallelisme et Electro-energetique, Departement d’Electrotechnique,

Universite des Sciences et de la Technologie Houari Boumediene BP 32 EL Allia, Alger 16111, AlgeriabDepartement de Genie Electrique, Ecole Nationale polytechnique, 10 AV Pasteur, El-Harrach, Alger, Algeria

cLaboratoire d’Electrotechnique et d’Electronique Industrielle ENSEEIHT, Toulouse, 2 Rue Camichel BP 7122-31071 Cedex7, France

Received 21 May 2006; accepted 2 April 2007

Available online 16 May 2007

Abstract

In high-performance variable speed AC drive systems it is often important to accurately determine machine parameters and rotor

speed values.

An extended complex Kalman filter (ECKF) is used to estimate rotor resistance and mutual inductance without the use of a speed

sensor. Such estimates are important in air gap flux orientation control. The ECKF is chosen over the more common extended Kalman

filter based on the real-valued model as it is not as computationally intensive. The model of an induction machine obtained by the spiral

vector theory and the ECKF are used to develop a new adaptive direct air gap flux orientation control with two sensors: one for the

stator voltage and one for the stator current without Park transformation. The simulation tests show the effectiveness of the proposed

adaptive direct air gap flux orientation control method.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Induction machine; Extended Kalman filter; Extended complex Kalman filter; Spiral vector theory; Parameter estimation; Sensorless control

and sonsorless direct air gap flux orientation control

1. Introduction

Vector control allows high-performance control oftorque, speed, or position to be achieved from an inductionmachine. Induction machines have been the most widelyused machines in fixed-speed applications for reasons ofcost, size, weight, reliability, ruggedness, simplicity, effi-ciency, and ease of manufacture. In variable-speed, high-performance drives, the DC machine and permanentmagnet synchronous machine have prevailed as theinduction machine requires more complex methods ofcontrol and more expensive, higher-rated inverters. Thedisadvantages of the induction machine becoming lessemphasized as the computing power of the microprocessorsand digital signal processors is increased, with reducing

e front matter r 2007 Elsevier Ltd. All rights reserved.

nengprac.2007.04.002

was not presented at any IFAC meeting.

ing author. Tel.: +213 50130729; fax: +213 21522528.

esses: m_menaa@yahoo.fr, mmenaa@usthb.dz

ouhom@hotmail.com (O. Touhami), ribtiouen@yahoo.

en), maurice.fadel@leei.enseeiht.fr (M. Fadel).

costs and size, together with improvements in invertertechnology (Holtz, 2002; Vas, 1990; Wade, Dunnigon, &Williams, 1997).The method of vector control can provide at least the

same performance from an inverter-driven inductionmachine as is obtainable from a separately excited DCmachine. Vector control provides decoupled control of theflux magnitude and the torque producing current.The fast torque response obtained using vector control is

achieved by estimating, measuring, or calculating themagnitude and position of the flux in the machine. Themeasurement of the flux can be made by the search coils orHall effect devices, inserted into the air gap of the machine.Alternatively, the air gap flux can be determined analyti-cally by stator field integration. However, due to the statorresistance voltage drop, the result may be biased. Thiscould affect the performance of the drive at low speed(Holtz, 2002; Ouhrouche, 2000). To overcome this draw-back, the flux must be estimated using observer. Theestimation of the flux using a state observer involves therotor resistance and mutual inductance, which are known

ARTICLE IN PRESS

Nomenclature

va stator voltage, Via, ir stator, rotor current, Alga, lgr stator, rotor leakage flux, Wbm transformation ratioR1, R2 stator, rotor resistance, O‘1, ‘2 stator, rotor leakage inductance, HLa, Lr stator, rotor main inductance, HMs, Mr mutual inductance between stator windings and

mutual inductance between rotor windings, HMrs mutual inductance between stator and rotor, HR02 rotor resistance in stator reference frame, O

‘02 rotor leakage inductance in stator referenceframe, H

om, osl, Om electrical, slip, rotor speed, rad/sF0 air gap flux, Wbe1, e2 stator, rotor dynamic e.m.f.P pole pair numberTem electromagnetic torque, NmT sampling time, sW, V state, measured noisesm, sv variance of state noise, measured noisey measured currentI unit matrix* conjugate

Re

t=0

A

Im

Fig. 1. Spiral vector in the complex plane.

M. Menaa et al. / Control Engineering Practice 16 (2008) 67–7768

to vary with frequency, temperature and saturation. Thus,these machine parameters must be jointly estimated withflux components and their actual values must be fedforward to the speed, flux and torque controllers to makethem robust. For this purpose many researchers haveapplied the extended Kalman filter (EKF) with satisfactoryresults to estimate flux, rotor speed and rotor resistancesimultaneously (Barut, Bongosyan, & Gokasan, 2005;ElMoucary, Garcia Soto, & Mendes, 1999; Kim, Sul, &Park, 1994; Ouhrouche, 2000; Shi, Chan, Wong, & Ho,2002; Zai, DeMarco, & Lipo, 1992), but few of them haveapplied the EKF to estimate flux, rotor speed, rotorresistance and mutual inductance simultaneously. How-ever, all these works estimate the rotor flux. In this paperthe authors propose a simultaneous estimation of the airgap flux, rotor resistance and mutual inductance withoutany need to a speed sensor by extended complex Kalmanfilter which is a new variant of the EKF in the air gap fluxoriented control.

This paper will be divided into five parts. In the first one,the spiral vector theory is developed. The second part isdevoted to modelling induction machine by spiral vectortheory. In the third part, air gap flux orientation controlbased on the spiral vector theory as well as the problem ofparameter variation is discussed. The fourth part treatsthe application of the ECKF to estimate the air gap flux,rotor resistance and mutual inductance without speedsensor and without Park transformation; only one voltagesensor and one current sensor are used. Finally, computersimulations are given to show the effectiveness of theproposed method.

2. Spiral vector theory

The spiral vector theory is an exponential time functionwith a complex index. It can express almost all kinds ofstate variables that appear in electrical engineering, and isthe most natural form solution of circuit equations ofelectric circuits and performance equations of electricalmachines. Both the steady and the transient state solutionscan be expressed by spiral vector. When spiral vector

expressions are used, the steady state theory and transientstate theory of AC circuit, which have been separatedbecause of different expressions of state variables, areunified. Both steady state and transient state analysesof AC machines become simpler and easier (Menaa,1997; Touhami, Menaa, Ibtiouen, & Benhadadi, 1997;Yamamura, 1992).The spiral vector variable can be expressed in the

following form:

i ¼ Aedt with d ¼ �lþ jo and lX0,

oX0; A ¼ jAjejj. ð1Þ

In the complex plane, the variable ‘‘i’’ depicts a spiral,rotating counter-clock wise, as shown in Fig. 1.When l ¼ 0, ‘‘i’’ represents the steady state operation of

an alternating value and describes a circle in the complexplane; ‘‘i’’ is thus a circular vector. When o ¼ 0, ‘‘i’’becomes equal to Ae�lt. It represents in this case thetransient operation of a continuous value. When d ¼ 0, ‘‘i’’becomes a constant. It represents steady-state operation ofa continuous value.

ARTICLE IN PRESSM. Menaa et al. / Control Engineering Practice 16 (2008) 67–77 69

3. Induction machine model by spiral vector theory

Generally, the spiral vector theory is used to describe themodel by transfer function (De Aguiar & Cad, 2000; Muni,Pillai, & Saxena, 1996; Yamamura, 1992). In this paper, theauthors use this theory to develop a state space model.

The first step, in the estimation process is the definition of asuitable machine model. When the spiral vector theory is usedfor modelling the induction machine, it can be represented bya nonlinear third-order differential equation. This modeltakes into account both electrical and mechanical transients.Only the variables and the parameters of one stator phaseand one rotor phase are considered.

By separating the slow mechanical modes from the fastelectrical ones, linear time varying electrical modes can bededuced.

It is assumed that the stator and motor windings areelectrically and geometrically symmetrical, the air gap isuniform, the field distribution is uniform and that theeffects of field saturation, eddy currents and hysteresis arenegligible. In that case the induction motor may berepresented by the model of the three-phase stator androtor windings shown in Fig. 2 (instead of the D–Q model),and the following model per phase can be derived:

va ¼ R1ia þ ‘1dia

dtþ

dlga

dt,

0 ¼ R2ir þ ‘2dir

dtþ

dlgr

dt. ð2Þ

The equations describing the stator and the rotor fluxesequations are as follows:

lga ¼ Laia þMsib cos2p3

� �þMsic cos �

2p3

� �

þMrsir cos yð Þ þMrsis cos yþ2p3

� �

þMrsit cos y�2p3

� �, ð3Þ

Fig. 2. Three phase stator and rotor windings distribution of an induction

machine.

lgr ¼ Lrir þMris cos2p3

� �þMrit cos �

2p3

� �

þMrsia cos yð Þ þMrsib cos yþ2p3

� �

þMrsic cos y�2p3

� �. ð4Þ

The stator and rotor currents of symmetrical spiralvector are:

ia ¼ A1edt; ib ¼ A1e

dt�jð2p=3Þð Þ; ic ¼ A1edtþjð2p=3Þð Þ, (5)

ir ¼ A2ed0t; is ¼ A2e

d0t�jð2p=3Þð Þ; it ¼ A2ed0tþjð2p=3Þð Þ. (6)

Inserting (5) and (6) into the expressions for stator androtor flux (3) and (4), the following expressions areobtained:

lga ¼ La þMs

2

� �ia þ

3

2Mrsire

jy, (7)

lgr ¼ Lr þMr

2

� �ir þ

3

2Mrsiae

�jy. (8)

Introducing (7) and (8) in (2) gives:

va ¼ R1ia þ ‘1d

dtia þ La þ

1

2Ms

� �d

dtia

þ3

2Mrs

d

dtire

jy� �,

0 ¼ R2ir þ ‘2d

dtir þ Lr þ

1

2Mr

� �d

dtir

þ3

2Mrs

d

dtiae�jy� �

, ð9Þ

with y ¼ omt.The obtained model (9) by spiral vector theory can

represent the induction machine in any reference frame.This model contains only variables and parameters of onestator phase ‘‘a’’ and one rotor phase ‘‘r’’, which aresegregated from the other phases. In opposition to Parkmodel where the voltage and the current of two phaseshave to be known, the spiral vector model only needs thevoltage and the current of one phase to achieve a vectorcontrol (Menaa, Touhami, & Ibtiouen, 2001, 2003; Menaa,Touhami, Ibtiouen, & Iung, 1998).

4. Air gap flux orientation control based on the spiral vector

model

The air gap flux orientation control based on the spiralvector model consists of choosing a model in the referenceframe attached to the rotating field, the air gap flux vectorF0 is aligned with the real axis in order to achieve, as in aseparately excited DC machine, decoupling control be-tween the flux and the produced torque (Menaa, 1997;Menaa et al., 1998).The torque is then controlled by the imaginary

part of the stator current of the phase ‘‘a’’. At the

ARTICLE IN PRESSM. Menaa et al. / Control Engineering Practice 16 (2008) 67–7770

same time, the flux is controlled by the real part of thestator current of the phase ‘‘a’’, called the flux pro-ducing current (Menaa, 1997; Menaa et al., 1998). Toobtain the model in the reference frame fixed in therevolving field, set:

va ¼ v1ejys ,

ia ¼ i1ejys , ð10Þ

ir ¼ i2ejysl , (11)

with ysl ¼ ys � y.Replacing (10) and (11) in (9), the obtained model

referred to the revolving field is as follows:

v1 ¼ R1i1 þ ‘1 þ3

2M

� �d

dtþ jos

� �i1 þ

3

2

d

dtþ jos

� �i2p,

0 ¼ R02i2p þ ‘02 þ3

2M

� �d

dtþ josl

� �i2p

þ3

2M

d

dtþ josl

� �i1, ð12Þ

with M ¼Mrs/m.Then, the T type transient state equivalent circuit of the

induction machine in the revolving field is deduced, andrepresented in Fig. 3.

e1 ¼ �jos ‘1 þ3

2M

� �i1 þ

3

2Mi2p

� �, (13)

e2 ¼ �josl ‘02 þ3

2M

� �i2p þ

3

2Mi1

� �. (14)

The electromagnetic torque is given by:

Tem ¼94

PM Im i1i�2p

� . (15)

Inserting the expression of the air gap flux F0 ¼32

Mði1 þ

i2pÞ in (12) and (15) gives:

v1 ¼ R1i1 þ ‘1d

dtþ jos

� �i1 þ

d

dtþ jos

� �F0,

0 ¼ R02i2p þ ‘02

d

dtþ josl

� �i2p þ

d

dtþ josl

� �F0, ð16Þ

Tem ¼32

P Im i1F�0� �

. (17)

Fig. 3. T-type transient state equivalen

As air gap flux orientation requires that ReðF0Þ ¼ F0

and ImðF0Þ ¼ 0, (16) and (17) become:

v1 ¼ R1i1 þ ‘1d

dtþ jos

� �i1 þ

d

dtþ jos

� �ReðF0Þ,

0 ¼ R02i2p þ ‘02

d

dtþ josl

� �i2p þ

d

dtþ josl

� �ReðF0Þ, ð18Þ

Tem ¼32

PRe F0ð ÞImði1Þ. (19)

Considering the air gap flux and the rotor speed asreference, the air gap flux orientation equations obtain thefollowing form:

Imði1Þ ¼2

3

1

P

Temref

F0ref

,

‘02R02

d

dtReði1Þ þReði1Þ ¼

sR02

d

dtF0ref

þ1

ð3=2ÞMF0refþ‘02R02

osl Imði1Þ,

osl ¼‘02ðd=dtÞ Imði1Þ þ R02 Imði1Þ

sF0ref� ‘02 Reði1Þ

, ð20Þ

os ¼ osl þ om,

Reðv1Þ ¼ ‘1d

dtReði1Þ þ

d

dtF0refþ R1 Reði1Þ � ‘1os Imði1Þ,

Imðv1Þ ¼ ‘1d

dtImði1Þ þ R1 Imði1Þ � os ‘1 Reði1Þ þ F0ref

� �,

ð21Þ

with s ¼ ð‘02=32

MÞ þ 1.The bloc diagram of direct vector control is represented

in Fig. 4.The calculation of the slip speed ‘‘osl‘‘and flux controller

parameters depend on the rotor resistance and mutualinductance that significantly can vary with temperature,frequency and saturation. Any error in the slip speedcalculation entails error in the air gap flux positionresulting in coupling between the flux and torque produ-cing currents due to axis misalignment. This results in atorque response with possible overshoot or undershootand a steady-state error. There is, therefore, a require-ment to track the changes in the rotor resistance and

t circuit of the induction machine.

ARTICLE IN PRESS

Fig. 4. Speed sensorless direct air gap flux orientation control of an induction machine.

M. Menaa et al. / Control Engineering Practice 16 (2008) 67–77 71

mutual inductance with a parameter identifier. In mostapplications, speed sensors are necessary and essential inspeed control loop. However, sensors have several dis-advantages in terms of drive cost, reliability, and noiseimmunity (Holtz, 2002).

Several papers deal with the speed and parametersestimation. The most used in the last decade is the EKF,but generally these papers treat the speed, rotor flux androtor resistance estimation. In this paper the authorspresent for the first time the simultaneous estimation of theair gap flux, rotor resistance and mutual inductancewithout speed sensor in the direct air gap flux control.

The ECKF is a variation of nonlinear Kalman filter inthe complex form. It has been shown by (Nishiyama, 1997)that the ECKF is more attractive than the real one ‘‘EKF’’from the point of view of modelling and stabilityconsideration. The ECKF is more used in the powersystem estimation (Dash, Pradhan, & Panda, 1999; Flores,Gu, & Bollen, 2003) but not in electrical machineestimation. To the best of knowledge, this paper is thefirst one that treats the parameter estimation of inductionmachine with ECKF.

The Kalman filter algorithm and its extension are aspecial kind of observer that provides optimal filtering ofthe noises in measurement and inside the system if thecovariances of these noises are known. The system and

measurement noise disturbances are white and haveGaussian probability distribution. When the Kalman filteris used the state space model of an induction machine in thestator reference frame is required; to obtain the model instator reference, setting i1 ¼ ia, i2 ¼ ire

jy and v1 ¼ va in (9),the following induction machine model in the statorreference frame is obtained:

v1 ¼ R1i1 þ ‘1 þ L1ð Þd

dti1 þ

3

2M

d

dti2,

0 ¼ R2i2 þ ‘2 þ L2ð Þd

dt� jom

� �i2

þ3

2M

d

dt� jom

� �i1. ð22Þ

After all treatment and simplification, the following statespace model of an induction machine in the stator referenceframe, which is function of the air gap flux, is obtained.

_X eðtÞ ¼ AeX eðtÞ þ BeUðtÞ,

Y eðtÞ ¼ CeX eðtÞ, ð23Þ

where

X e tð Þ ¼i1

F0

" #; Y e tð Þ ¼ i1 and U tð Þ ¼ v1.

ARTICLE IN PRESSM. Menaa et al. / Control Engineering Practice 16 (2008) 67–7772

The coefficients of the matrices Ae, Be and Ce are given inAppendix A.

As the parameter estimation of an induction machine isto be implemented on a computer, so a time-discrete modelis the most appropriate. The discrete time varying model isdeduced from the continuous model by applying the Eulerformula (first order). It is given by:

X eðk þ 1Þ ¼ AeDX eðkÞ þ BeDUðkÞ,

Y eðkÞ ¼ CeDX eðk þ 1Þ. ð24Þ

The coefficients of the matrices AeD, BeD and CeD aregiven in Appendix A.

In practice, the induction machine cannot be representedperfectly by the previous deterministic model. To take intoaccount the system uncertainties and disturbances, thefollowing stochastic model is introduced:

X eðk þ 1Þ ¼ AeDX eðkÞ þ BeDUðkÞ þW ðkÞ,

Y eðkÞ ¼ CeDX eðk þ 1Þ þ V ðkÞ. ð25Þ

The state noise W and measured noise V can be anycombination of harmonics and complex Gaussian whitenoise with zero-mean and variances s2w and s2v , respectively.Real and imaginary parts of W(k) are mutually indepen-dent and have the same variance, i.e. s2w ¼ s2wr þ s2wi ands2wr ¼ s2wi. The real and imaginary parts of V(k) are alsomutually independent and have the same variance, i.e. s2v ¼s2vr þ s2vi and s2vr ¼ s2vi.

The ECKF allows simultaneous estimation of complexstates and parameters. These parameters are considered asextra states in an augmented state vector. This augmentedmodel is nonlinear because of multiplication betweenstates. Thus it must be linearized along the state trajectoryto give a linear perturbation model. The standard complexKalman filter is then applied on this linearized model. Theextended complex Kalman filter only requires machinesquantities that can be easily measured (stator voltage andstator current). In this new approach only the voltage andcurrent of one phase of stator are required, but in the DQ

model the voltage and current of two phases of stator arenecessary (Baghli, Al-Rouh, & Rezzoug, 2006; Duval,

Fig. 5. Adjustment parameter in speed control of an induction

Clerc, & Le Gorrec, 2006; Loron and Laliberte, 1993). Adynamic model of three phases induction machine, byspiral vector theory, has two complex state variablesnamely the stator current (i1) and air gap flux (F0) (inopposition to the Park model where four state variablesexist). An extended induction machine model results if therotor resistance, mutual inductance and rotor speed areincluded as additional state variables. The extended modelcan be expressed as follows:

xðk þ 1Þ ¼ f ðxðkÞ;UðkÞ; kÞ þW ðkÞ,

ZðkÞ ¼ HðkÞxðkÞ þ V ðkÞ, ð26Þ

with

f xðkÞ;UðkÞ; kð Þ ¼AeDX eðkÞ þ BedUðkÞ

YðkÞ

" #

YðkÞ ¼ R0232

M Om

h it

. ð27Þ

Then, linearizing the above system around the states andapplying an ECKF, the obtained nonlinear recursive filterbased on the ECKF for estimating a complex state andparameters of an induction machine is given by:Prediction of state

xpðk þ 1Þ ¼ f xðkÞ;UðkÞ; kð Þ. (27a)

Estimation of error covariance matrix

Ppðk þ 1Þ ¼ AðkÞPðkÞA�tðkÞ þQ. (27b)

Kalman filter gain

K ¼ Ppðk þ 1ÞH�tðkÞ HðkÞPp k þ 1ð ÞH�tðkÞ þ R ��1

. (27c)

State estimation

xðk þ 1Þ ¼ xpðk þ 1Þ þ K yðkÞ �HðkÞxðkÞ½ �. (27d)

Update of the error covariance matrix

Pðk þ 1Þ ¼ Ppðk þ 1Þ � KHðkÞPpðk þ 1Þ. (27e)

The matrices A(k) and H(k) are given in Appendix A.

machine using the direct air gap flux orientation control.

ARTICLE IN PRESS

Table 1

Rated values and parameters of the induction machine

Rated power 2 kW

Rated speed 1425 rpm

Rated frequency 50Hz

Rated voltage 220/380V

Rated current 4/7A

Pole pair number 2

Rotor inertia 0.0164 kgm2

Stator resistance, R1 4.1049ORotor resistance, R02 5.253OMutual inductance, (3/2)M 0.2145H

Stator leakage inductance, ‘1 0.0132H

Rotor leakage inductance in stator reference frame, ‘02 0.0132H

M. Menaa et al. / Control Engineering Practice 16 (2008) 67–77 73

5. Simulation results

The simulation tests were carried out using the direct airgap flux orientation control scheme of Fig. 5 without anyspeed measurement. The control is realized when thevoltage and current of one phase of stator are trackedwithout any geometric transformation, whereas in the caseof the vector control based on the DQ model, it is necessaryto track two voltages (vas and vbs) and two currents (ias andibs) when applying Park transformation (Barut, Bongos-yan, & Gokasan, 2002; Castaldi, Geri, Montanari, & Tilli,2005; Loron and Laliberte, 1993; Mueller, 1999; Poyhonen,Arkiko, Jover, & Hyotyniemi, 2005; Vas, 1990; Yazid,Menaa, Touhami, & Ibtiouen, 2005).

To test the performance of the proposed control, thesimulations were performed on a 50Hz, 2 kW inductionmachine with the rated parameters listed in Table 1. The

initial values are x0 ¼ 0þ 0j 0þ 0j R0s32

M 0h it

and

the sampling time T ¼ 0.1ms.The performance of the ECKF estimation is very

sensitively influenced by the values of the system para-meters and covariance matrix elements. In this study, toavoid computational complexity, the covariance matrix ofthe state noise Q, the covariance of the measured noise R

and the error covariance matrix P are chosen in diagonalform, also satisfying the condition of positive definiteness.According to the Kalman filter theory, Q and R have to beobtained by considering the stochastic properties of thecorresponding noises (Nishiyama, 1997; Vas, 1998); how-ever, since these are usually not known, in most cases, thecovariance matrix elements are used as weighting factors ortuning parameters. In this study tuning the initial values ofP, Q and R is done by trial and error to achieve a rapidinitial convergence. The values of P, Q and R are:

P0 ¼ diag 103 103 1022 3:51017 314104 �

,

Q ¼ diag 108 108 5108 51013 1:571020 �

,

R ¼ 51020.

Figs. 6 and 7 show different scenarios used to test theperformances of the estimation and control algorithm

under load torque, noise and rotor resistance increase by50% and mutual inductance decrease by 20% to their ratedvalues.Fig. 6(a–d) show in the first part (0 spto2.5 s) the

simulation test between zero speed and 125 rad/s under aload torque of 10Nm from standstill to 1.5 s. It can beobserved that the rotor speed and air gap flux are notaffected by the load torque change and the rotor speederror and air gap flux error are small. In the second part(2.5 spto5.3 s), the simulation test in the field weakeningrange is shown and for the time range (3.75 sptp4.75 s) aload torque of 5Nm is applied to the shaft, the rotor speedand air gap flux are not disturbed a lot by the torquechange and their errors are low. In the third part the speedreference goes down to 125 rad/s with no load torque andthe estimated rotor speed and air gap flux track theirreferences with a weak error. In the last part the rotorspeed is inversed (�125 rad/s) and at 8 s a negative loadtorque (�10Nm) is applied, remarking that the speed errorand flux error are not important. The rotor speedestimation error (Omachine�Om) is not more than 0.5% inthe steady state, which is the same value for the speedcontroller input (Omref�Om). For the air gap flux estima-tion error ðF0machine � F0Þ the maximum value is 0.5% inthe steady state and for the air gap flux controller inputðF0ref � F0Þ it is almost equal to zero in the steady state.Fig. 6e and f show the variation of the estimated rotor

resistance and estimated mutual inductance. Let us notethat the estimated rotor resistance converges quickly to itsreference and the estimated mutual inductance decreasesfrom initial value to its true value with no significant error.The estimated mutual inductance is scarcely affected by thespeed inversion. The rotor resistance error is not over0.95% in steady state and the mutual inductance error isless than 8% in steady state.Fig. 7(a–d) show in the first part the machine start-up

without load torque, the rotor speed and air gap flux tracktheir references with weak error. In the second part(0.6 spto3.4 s), the simulation test in low speed is shownand between (1.7 spto2.75 s) a load torque of 5Nm isapplied to the shaft, remarking that the estimated rotorspeed and air gap flux are not affected by torque change.The rotor speed error and the air gap flux error are weak. Inthe last part, the simulation test at zero speed is shown. Theestimated rotor speed and air gap flux track their referenceswith a very small error. The rotor speed estimation error(Omachine�Om) is not more than 1.8% in the steady state.The maximum value of the speed controller input(Omref�Om) is 1.5%. For the air gap flux estimation error(F0machine�F0) the maximum value is 0.8% in the steadystate and for the air gap flux controller input (F0ref�F0) it isalmost equal to zero in the steady state.Fig. 7e and f depict the variation of the estimated rotor

resistance and mutual inductance, which converge to theirtrue values with small error. The rotor resistance error isless than 0.65% in steady state and the mutual inductanceerror is not more than 0.6% in steady state.

ARTICLE IN PRESS

0 8 10

time (s)

-200

-100

0

100

200

300

-150

-50

50

150

250

Roto

r sp

eed

(ra

d/s

)

0 6 10

-4

-2

0

2

-5

-3

-1

1

3

Sp

eed

est

imati

on

err

or

(rad

/s)

10 8 4

time (s)

0

40

80

120

160

-20

20

60

100

140

Sp

eed

con

troll

er i

np

ut

(rad

/s)

0 6 10

time (s)

0

0.2

0.4

0.6

0.8

1

0.1

0.3

0.5

0.7

0.9

Air

gap

flu

x (

Wb

)

0 10

-0.008

-0.004

0

0.004

-0.006

-0.002

0.002

0.006A

irgap

flu

x e

stim

ati

on

err

or

(Wb

)

10 8

time (s)

0

0.4

0.8

0.2

0.6

1

Air

gap

flu

x c

on

troll

er i

np

ut

(Wb

)

0 4 10

0

4

8

12

10

2

6

Roto

r re

sist

an

ce (

Oh

m)

10 8 6 0

time (s)

-4

0

4

8

-2

2

6

Roto

r re

sist

an

ce e

rror

(Oh

m)

0 2 10

0.17

0.18

0.19

0.2

0.21

0.22

0.165

0.175

0.185

0.195

0.205

0.215

Mu

tual

ind

uct

an

ce (

Hen

ry)

10 8

time (s)

-0.04

-0.02

0

0.02M

utu

al

ind

uct

an

ce e

rror

(Hen

ry)

2 4 6 2 4 8

2 06

2 4 8 2 4 6 8

6 4 2 0

4

6

6 8

4 2 0

2 6 8

4 2

Fig. 6. Simulation tests results, (a) speed variation, (b) error of speed variation, (c) air gap flux variation, (d) error of air gap flux variation, (e) rotor

resistance variation and rotor resistance error, (f) mutual inductance variation and mutual inductance error.

M. Menaa et al. / Control Engineering Practice 16 (2008) 67–7774

These results show that the ECKF and the direct air gapflux orientation control achieve a low speed error and alsogive stable air gap flux response which is almost unaffectedby the load torque change and give a good estimation ofmachine parameters. The obtained results show that the

expected performance is attained. In summary, let us notethat:

�

The spiral vector theory permits to reduce the modelorder, which reduces the order of the covariance

ARTICLE IN PRESS

0 2 5

time (s)

-20

20

60

100

140

0

40

80

120

Roto

r sp

eed

(ra

d/s

)

0 2 4 5

0

2

4

Sp

eed

est

imati

on

err

or

(rad

/s)

5 3 1 0

time (s)

0

40

80

120

-20

20

60

100

140

Sp

eed

con

troll

er i

np

ut

(rad

/s)

0 3

time (s)

0

0.4

0.8

1.2

0.2

0.6

1

Air

gap

flu

x (

Wb

)

0 4

-0.008

-0.004

0

0.004

0.008

-0.006

-0.002

0.002

0.006A

irgap

flu

x e

stim

ati

on

err

or

(Wb

)

5 1

time (s)

0

0.4

0.8

0.2

0.6

1

Air

gap

flu

x c

on

troll

er in

pu

t (W

b)

0 5

5

6

7

8

5.5

6.5

7.5

Roto

r re

sist

an

ce (

Oh

m)

5

time (s)

0

1

2

0.5

1.5

2.5

Roto

r re

sist

an

ce e

rror

(Oh

m)

0

0.17

0.18

0.19

0.2

0.21

0.175

0.185

0.195

0.205

0.215

Mu

tual

ind

uct

an

ce (

Hen

ry)

5

time (s)

-0.04

-0.03

-0.02

-0.01

0

-0.045

-0.035

-0.025

-0.015

-0.005

0.005

Mu

tual

ind

uct

an

ce e

rror

(Hen

ry)

1 3 4 1 3

4 2

1 2 4 5 1 2 3 5

4 3 2 0

1 2 3 4

4 3 2 1 0

1 2 3 4 5

4 3 2 1 0

Fig. 7. Simulation tests results, (a) speed variation, (b) error of speed variation, (c) air gap flux variation, (d) error of air gap flux variation, (e) rotor

resistance variation and rotor resistance error, (f) mutual inductance variation and mutual inductance error.

M. Menaa et al. / Control Engineering Practice 16 (2008) 67–77 75

matrices. In this case, the error covariance matrix P andthe covariance matrix of state noise Q have (5� 5)elements each instead of (7� 7) elements in theconventional theory (Barut et al., 2005; Vas, 1998;

Yazid et al., 2005). The covariance of the measurednoise R is a scalar instead of (2� 2) elements in theconventional theory (Barut et al., 2005; Vas, 1998; Yazidet al., 2005).

ARTICLE IN PRESSM. Menaa et al. / Control Engineering Practice 16 (2008) 67–7776

�

In this new method R is not a matrix, so, the inversionmatrix in the ECKF algorithm (27c) is avoided in theopposite of the EKF based on the conventional model.This will reduce the computational time. � The spiral vector model permits to reduce the sensor

numbers, leading to decreasing noise effect.

6. Conclusion

This paper has presented a novel model for inductionmachine associated to a new nonlinear filter ‘‘ECKF’’ forspeed sensorless direct air gap flux orientation control withparameter estimation.

The major contribution of this study is the reduction of thenumber of sensors from four or six in DQ model to twosensors using this new technique. The simultaneous estimationof the rotor resistance and mutual inductance as well as the airgap flux components and the rotor speed are accomplished.

The performances of the ECKF algorithm and direct airgap flux orientation control are tested for differentscenarios obtained by applying step and linear variationto the rotor speed reference with load torque, rotorresistance and mutual inductance variation.

The performances observed of the system are satisfac-tory under step type variation and reversal of the loadtorque and step/linear changes and reversal in the rotorspeed. The system has also demonstrated the expectedrobustness to step variations forced on the rotor resistanceand mutual inductance.

These observations and the obtained results motivate furtherexperimental implementation of the developed method.

Acknowledgment

This paper enters in the setting of a cooperation agreementinternational CMEP-Tassili under the 05MDU662 code.

Appendix A

The coefficients of systems (23) are as follows:

Ae ¼Ae11 Ae12

Ae21 Ae22

" #; Be ¼

1=‘T

‘02=‘Ts

" #and

Ce ¼ 1 0 �

,

Ae11 ¼� R1sþ R02 � j‘02om

� �‘1sþ ‘02

,

Ae12 ¼ðR02=

32

MÞ � jsom

‘1sþ ‘02,

Ae21 ¼R02s�‘02RT

s‘T

þ j‘02s�1þ

‘02‘Ts

� �om,

Ae22 ¼ðR02=

32

MÞ � jsom

s�1þ

‘02‘Ts

� �,

‘T ¼ ‘1 þ‘02s

and RT ¼ R1 þR02s

.

The coefficients of systems (24) are as follows:

AeD ¼ eðAe�TÞ � I þ AeT ; CeD ¼ Ce,

BeD ¼ A�1e AeD � Ið ÞBe � BeT ; tk ¼ kT .

The matrices of A(k) and H(k) are given by

AðkÞ ¼qf

qx¼

AeDðkÞq AeDX eðkÞþBeDUðkÞð Þ

qY

0 I

" #,

HðkÞ ¼ CeDqqY CeDX eðkÞð Þ

h i.

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