Control Engineering Practice 13 (2005) 81–94

ARTICLE IN PRESS

*Correspondi

2093073.

E-mail addre

0967-0661/$ - see

doi:10.1016/j.con

A new adaptive approach for on-line parameter and stateestimation of induction motors

Paolo Castaldi, Walter Geri, Marcello Montanari, Andrea Tilli*

Department of Electronics, Computer Science and Systems, University of Bologna, Viale Risorgimento 2, Bologna 40136, Italy

Received 21 May 2002; accepted 23 February 2004

Abstract

A novel adaptive observer is proposed to perform on-line estimation of both state and parameters of the electrical part of

induction motors. The rotor mechanical speed and the stator currents/voltages are assumed to be measured. A particular non-

minimal state representation of the machine electrical model is derived and exploited to build the adaptive observer, based on a

series–parallel architecture. Lyapunov design is used to develop the adaptation law of the proposed solution. Both simulation results

and experimental tests confirm the good performances of the technique.

r 2004 Elsevier Ltd. All rights reserved.

Keywords: Induction motor; Adaptive observer; Parameter identification; Non-minimal realization

1. Introduction

A good knowledge of the induction motor (IM)electrical parameters is a key point for achieving high-performance motion control using this kind of electricmachine.

The IM model is multivariable, nonlinear andstrongly coupled. The most widespread control metho-dology applied in commercial drives is based on theconcept of field-orientation (Direct or Indirect, DFOCor IFOC), which guarantees torque and flux controldecoupling (Blaschke, 1972; Leonhard, 1995; Bose,1997). Another control technique which is becominginteresting for industrial applications is direct torquecontrol (DTC). This approach directly takes intoaccount the switching behavior of the power converter(namely, the inverter) used to supply the motor (Buja,Casadei, & Serra, 1998). Both previous approaches relyon a perfect knowledge of the parameters of thecontinuous-time electrical model of the IM. In nonlinearand adaptive control literature many efforts have beendevoted to developing other control algorithms for IM.Although different approaches have been used (seePeresada & Tonielli, 2000; Marino, Peresada, & Tomei,

ng author. Tel.: +39-051-2093024; fax: +39-051-

ss: atilli@deis.unibo.it (A. Tilli).

front matter r 2004 Elsevier Ltd. All rights reserved.

engprac.2004.02.008

2000, for an extensive overview), only partial and quiteweak results have been obtained in terms of robustnesswith respect to parameter uncertainties. On the otherhand, it is well known that imperfect knowledge ofmotor parameters affects the controller performance interms of achievable dynamical performance and energyefficiency (Peresada, Tilli, & Tonielli, 2003).

In order to estimate the continuous-time electricalparameters of IM, many approaches have been proposedin applicative and theoretical literature. Solutions whichallow in system identification of the motor are particularlyinteresting for industrial applications. ‘In system identifi-cation’ means that the estimation procedure can beapplied while the IM is already connected to themechanical load using the standard equipment of a typicalcommercial drive (with possible minor and inexpensivemodifications). Electric drives provided with such featuresare usually indicated as ‘self commissioning drives’.

In system identification techniques of electrical para-meters of IM can be divided in two main classes: off-and on-line methods.

The off-line methods perform the parameter identifi-cation while the motor is at standstill and no torque isrequired from the mechanical load. This kind ofprocedure is usually performed during the initializationof the motion control process.

Differently, the on-line approaches execute the IMidentification while the electric drive is operating

ARTICLE IN PRESSP. Castaldi et al. / Control Engineering Practice 13 (2005) 81–9482

normally, i.e. while motion control is performed. Themain advantage of this kind of solution with respect toprevious ones is the capability of tracking the slowvariation of motor parameters during operating condi-tions. In fact, electric machine heating produces asignificant variation of both stator and rotor windingresistance values (up to 100% for rotor resistances). This‘tracking feature’ allows control parameter adaptationin a sort of self-tuning regulator scheme. On the otherhand, on-line approaches usually demonstrate somedisadvantages. From a theoretical point of view it isusually very difficult to verify stability and convergenceproperties of the parameter estimation algorithmcombined with the adopted motor controller (typicallydesigned according to the certainty equivalence principle,Narendra & Annaswamy, 1989). In addition, from apractical point of view, on-line identifiers are usuallycharacterized by a significant computational burden.

In Sangwongwanich and Okuma (1991), differenttheoretically rigorous methods are used to identifystator and rotor resistance during normal workingoperation, but filtered derivatives of the measurementsare required. In Stephan, Bodson, and Chiasson (1994),an on-line method, based on recursive least-square(RLS), is presented to identify the electrical andmechanical IM parameters. Scaled-version of themagnetic flux is also estimated, but the derivatives ofmeasurements are assumed to be known and thecomputational load is quite heavy. In Moons and DeMoor (1995), the generalized total least-square (GTLS)technique is adopted. Filtered derivatives of themeasured signals are still needed, but particular atten-tion is paid to the reduction of noise effects. Aconstrained identification procedure is proposed to dealwith low signal–to–noise ratio conditions. In Holtz andThimm (1991), the least-square (LS) procedure has beenapplied in an original way to obtain an estimate of thestator and rotor resistances and reactances. No deriva-tives are required, but the proposed method is notstrictly recursive and the computational burden is notnegligible. In Pappano, Lyshevski, and Friedland(1998a, b, c), a theoretically elegant solution is presentedto identify all the IM drive parameters, but theknowledge of all the state variables and their derivativesis required. In Iwasaky and Kataoka (1989), Zai, DeMarco, and Lipo (1992), an extended Kalman filter(EKF) has been used to identify the machine para-meters; in Iwasaky and Kataoka (1989) particularattention has been paid to the selection of noisecovariance matrices and initial states. In Coirault,Trigeassou, K!erignard, and Gaubert (1996), a sophisti-cated method, based on nonlinear programming, isproposed. In Valdenebro, Hern!andez, and Bim (1999), aneuro-fuzzy technique is applied for on-line identifica-tion of the rotor time-constant. In Marino et al. (2000),a very interesting technique for tuning the stator and

rotor resistances in normal operating condition ispresented. The stability characteristics of the proposedmethod are formally proved and experimentally testedand no derivative of measurements is required.

In this paper, a novel on-line technique is proposedfor IM electrical parameter and state estimation. Underthe hypothesis of linear magnetic circuit and balancedoperating conditions, the classical fifth-order IM modelis bilinear. In particular, considering the rotor speedmeasurable, the electromagnetic motor model can beassumed to be linear and time varying (LTV) with aknown dependence on time (Leonhard, 1995). Follow-ing the same concepts developed in Kreisselmeier (1977)for LTI systems, a novel special non-minimal staterepresentation for the LTV model of the IM is derived,by applying an ad hoc method. Its main feature is thatthe dynamical equations do not depend on the motorparameters, which appear linearly only in the expressionof the measurable outputs (the stator currents). Startingfrom this representation, the model reference adaptivesystem (MRAS) approach in a deterministic frameworkis exploited. A series–parallel adaptive observer (SPAO)(Narendra & Annaswamy, 1989), based on statorcurrent and mechanical speed measurements, is de-signed. A novel adaptation law is developed using aLyapunov-like technique to guarantee exponential con-vergence of the estimates under conditions of persistencyof excitation.

The proposed solution (like others where observerschemes are adopted, Raina & Toliyat, 2001), besidesperforming parameter identification, also provides anestimate of rotor fluxes (non-measurable state compo-nents). This feature leads to the following two relevantissues:

(a)

The magnetic flux can be monitored in order toavoid saturation phenomena. This feature is usefulsince the linear magnetic circuit hypothesis isadmissible only if the magnetic core is not insaturated conditions, i.e. the flux amplitude is lowerthan the nominal one calculable from the nameplatedata of the IM (Klaes, 1993; Ruff & Grotstollen,1996).

(b)

The estimated rotor flux can be used in the controlalgorithm adopted for motion control. For exam-ple, direct field-oriented control requires an explicitestimate of the rotor magnetic flux.

Note that the proposed approach does not require anyextra hardware devices with respect to commercial high-performance encoder-based drives, hence it is suitablefor self-commissioning purposes.

The paper is organized as follows. The non-minimalrepresentation of the IM is introduced in Section 2,starting from the classical IM model. In Section 3, theadaptive observer for parameter and state estimation ispresented, devoting particular attention to the design of

ARTICLE IN PRESSP. Castaldi et al. / Control Engineering Practice 13 (2005) 81–94 83

the adaptation law based on the Lyapunov-like techni-que. The parameter tuning procedure is also discussed.In Section 4, the actual digital implementation of theproposed scheme on a DSP control-board is discussed.Simulation and experimental results are reported toshow the performance of the adaptive observer pro-posed in different operating conditions. Conclusions arereported in Section 5.

2. Non-minimal representation for IM

Assuming linear magnetic circuits and balanced three-phase windings, the classical IM model, expressed in astationary two-phase reference frame ða � bÞ is thefollowing (Leonhard, 1995):

’oðtÞ ¼ m=Jðjaib � jbiaÞ � TL=J;

’jaðtÞ ¼ � aja � ojb þ aLmia;

’jbðtÞ ¼oja � ajb þ aLmib;

’iaðtÞ ¼ abja þ bojb � gia þ 1=sua;

’ibðtÞ ¼ � boja þ abjb � gib þ 1=sub; ð1Þ

where o is the rotor speed, ðja;jbÞ are the rotor fluxes(not measurable), ðia; ibÞ are the stator currents (measur-able), ðua; ubÞ are the applied stator voltages, J is thetotal rotor inertia and TL is the load torque. The IMmodel parameters are defined as

s ¼ Ls 1 �L2

m

LsLr

� �; b ¼

Lm

sLr

; m ¼3

2

Lm

Lr

;

a ¼Rr

Lr

; g ¼Rs

sþ aLmb;

where Rs;Rr;Ls;Lr are the stator/rotor resistancesand inductances, and Lm is the magnetizing in-ductance.

Remark 1. Assuming that rotor speed is measurable, theelectromagnetic dynamics of the IM (given by the lastfour equations in (1)) can be seen as a linear time-varying (LTV) system with the time-varying part exactlyknown.

In the following the analysis will be focused on theLTV electromagnetic dynamics of the IM.

Defining xðtÞ ¼ ½jaðtÞ;jbðtÞ; iaðtÞ; ibðtÞ�T and indicating

the measurable state components yðtÞ ¼ ½iaðtÞ; ibðtÞ�T asoutput,

yðtÞ ¼0 0 1 0

0 0 0 1

" #xðtÞ

the following coordinate transformation is introduced:

%xðtÞ ¼ TaxðtÞ; Ta ¼

0 0 1 0

0 0 0 1

b 0 1 0

0 b 0 1

26664

37775:

Making the dependence on y explicit in the %x statedynamics, the following model is obtained:

’%xðtÞ ¼ F ðtÞ %xðtÞ þ KF ðtÞyðtÞ þ BF uðtÞ; %xð0Þ ¼ %x0;

yðtÞ ¼ CF %xðtÞ; ð2Þ

where

F ðtÞ ¼

l 0 a o

0 l �o a

0 0 0 0

0 0 0 0

26664

37775; %x0 ¼

%x01

%x02

%x03

%x04

26664

37775;

CF ¼1 0 0 0

0 1 0 0

" #;

KF ðtÞ ¼

�ðlþ aþ gÞ �o

o �ðlþ aþ gÞ

abLm � g 0

0 abLm � g

26664

37775;

BF ¼

1=s 0

0 1=s

1=s 0

0 1=s

26664

37775:

Note that the term l½iaðtÞ; ibðtÞ�T with l negative and

constant is added and subtracted to the first twoequations of (2) in order to have a degree of freedomin selecting the first two elements of the diagonal ofmatrix F ðtÞ:

Starting from (2), the non-minimal parameterizationused to develop the new parameter and state estimator isstated in the following theorem.

Theorem 2. The state %xðtÞ of the IM model (2) is

expressed, with an exponential decaying error with

arbitrary rate jlj; as

%xðtÞ ¼

zT1 ðtÞ

zT2 ðtÞ

zT3 ðtÞ

zT4 ðtÞ

26664

37775pM þ

�c5ðtÞ

c4ðtÞ

0

0

26664

37775; ð3Þ

where

zT1 ðtÞ ¼ ½0; 1;c1; 0;�c2;c7; ðc8 þ c11Þ;

ðc2 � Z1Þ; ðc8 � Z3Þ�;

zT2 ðtÞ ¼ ½�c1; 0; 0; 1;�c3;�c6; ðc9 � c10Þ;

ðc3 � Z2Þ; ðc9 � Z4Þ�;

ARTICLE IN PRESSP. Castaldi et al. / Control Engineering Practice 13 (2005) 81–9484

zT3 ðtÞ ¼ ½1; 0; 0; 0; 0; Z1; Z3; 0; 0�;

zT4 ðtÞ ¼ ½0; 0; 1; 0; 0; Z2; Z4; 0; 0�;

pTM ¼ %x03;�

al%x03; %x04;�

al%x04; ðlþ aþ gÞ; ðabLm � gÞ;

�1

s;alðabLm � gÞ;

als

�ð4Þ

and

’Z1ðtÞ ¼ y1ðtÞ; Z1ð0Þ ¼ 0;

’Z2ðtÞ ¼ y2ðtÞ; Z2ð0Þ ¼ 0;

’Z3ðtÞ ¼ u1ðtÞ; Z3ð0Þ ¼ 0;

’Z4ðtÞ ¼ u2ðtÞ; Z4ð0Þ ¼ 0;

’c1ðtÞ ¼ lc1ðtÞ þ oðtÞ; c1ð0Þ ¼ 0;

’c2ðtÞ ¼ lc2ðtÞ þ y1ðtÞ; c2ð0Þ ¼ 0;

’c3ðtÞ ¼ lc3ðtÞ þ y2ðtÞ; c3ð0Þ ¼ 0;

’c4ðtÞ ¼ lc4ðtÞ þ y1ðtÞoðtÞ; c4ð0Þ ¼ 0;

’c5ðtÞ ¼ lc5ðtÞ þ y2ðtÞoðtÞ; c5ð0Þ ¼ 0;

’c6ðtÞ ¼ lc6ðtÞ þ Z1ðtÞoðtÞ; c6ð0Þ ¼ 0;

’c7ðtÞ ¼ lc7ðtÞ þ Z2ðtÞoðtÞ; c7ð0Þ ¼ 0;

’c8ðtÞ ¼ lc8ðtÞ þ u1ðtÞ; c8ð0Þ ¼ 0;

’c9ðtÞ ¼ lc9ðtÞ þ u2ðtÞ; c9ð0Þ ¼ 0;

’c10ðtÞ ¼ lc10ðtÞ þ Z3ðtÞoðtÞ; c10ð0Þ ¼ 0;

’c11ðtÞ ¼ lc11ðtÞ þ Z4ðtÞoðtÞ; c11ð0Þ ¼ 0:

ð5Þ

Proof. See Appendix A.

The state variables of the proposed non-minimal 15thorder representation are Zi; i ¼ 1::4 and cj ;j ¼ 1::11: &

Remark 3. All the model parameters of the non-minimalrepresentation (3)–(5) are grouped together in the vectorpM ; while the dynamical part is constituted by para-meter-free or arbitrarily-parameterized differential equa-tions with known initial conditions. This key feature isexploited to design the adaptive observer as reported inSection 3. The equivalent model is similar to the onereported in Kreisselmeier (1977) for single-input single-output LTI systems, but its derivation from the multi-variable LTV IM model requires an ad hoc procedure.

Corollary 4. Given the IM model (2), the output yðtÞ is

expressed, with an exponential decaying error with

arbitrary rate jlj; as

yðtÞ ¼zT1 ðtÞ

zT2 ðtÞ

" #pM þ

�c5

c4

" #: ð6Þ

Proof. It is straightforward to observe that yðtÞ coin-cides with the first two components of %xðtÞ: &

3. Adaptive observer based on the new parameterization

In this section a state and parameter estimator isdefined on the basis of the result reported in Theorem 2and adopting a deterministic approach.

The SPAO scheme, defined in the MRAS framework(Landau, 1979; Narendra & Annaswamy, 1989), isexploited. In particular the observer dynamics andoutput expressions are defined by

* the dynamics (5);* the definition of the vectors zi; i ¼ 1::4 reported in (4);* the following two expressions for output and state

estimation:

#yðtÞ ¼zT1 ðtÞ

zT2 ðtÞ

" ##pMðtÞ þ

�c5

c4

" #; ð7Þ

#%xðtÞ ¼

zT1 ðtÞ

zT2 ðtÞ

zT3 ðtÞ

zT4 ðtÞ

26664

37775 #pM ðtÞ þ

�c5

c4

0

0

26664

37775;

#xðtÞ ¼ T�1a ð #pMðtÞÞ #%xðtÞ; ð8Þ

where #yðtÞ; #%xðtÞ and #xðtÞ are the estimates of the vectorsyðtÞ; %xðtÞ and xðtÞ respectively, whereas #pM ðtÞ representsthe estimate of the parameter vector pM whose dynamicswill be defined in the following.

Note that the zi; i ¼ 1::4 vectors used in the proposedestimator can be assumed to be equal to the ‘actual’ onesof the non-minimal parameterization owing to the exactknowledge of dynamics (5) (differential equations andinitial conditions).

3.1. Adaptation law

In order to complete the definition of the proposedscheme, the parameter estimation dynamics (the so-called adaptation law) has to be selected. The mainpurpose is to guarantee asymptotic convergence of stateand parameter estimates. From (8) it results that thestate estimation tends to correct values if the parameterestimation convergence is guaranteed. Defining theparameter estimation error as

*pM ðtÞ ¼ #pMðtÞ � pM

and the measurable output estimation error as

*yðtÞ ¼ #yðtÞ � yðtÞ ¼zT1 ðtÞ

zT2 ðtÞ

" #*pMðtÞ; ð9Þ

ARTICLE IN PRESS

1Functions lmðG0Þ and lM ðG0Þ return the eigenvalue of G0 with

minimum and maximum real part, respectively.

P. Castaldi et al. / Control Engineering Practice 13 (2005) 81–94 85

the following adaptation law is chosen:

’#pMðtÞ ¼ �GðtÞ½z1ðtÞ z2ðtÞ� *yðtÞ; #pMð0Þ ¼ #pM0;

’G�1ðtÞ ¼ aG�1ðtÞ � aG�10 þ b½z1ðtÞ z2ðtÞ�

zT1 ðtÞ

zT2 ðtÞ

" #;

G�1ð0Þ ¼ G�10 > 0; ð10Þ

where the 9th order matrix G0 is symmetric and positivedefinite and a; b are constant tuning coefficients.

In the following theorems it will be shown that thematrix GðtÞ is symmetric, 8t and that the proposedadaptation law guarantees the convergence objectivespursued, with some constraints on coefficients a and b:

Theorem 5. Consider the IM model (1), assuming that the

angular speed measurement oðtÞ is available. Consider the

related adaptive observer described by (5), (7) and (8) with

the adaptation law given by (10), where

ao0; 0pbp2:

It follows that the parameter estimation error *pM ðtÞ is

bounded.

Proof. The following candidate Lyapunov function isdefined:

W ðtÞ ¼ *pTMðtÞG�1ðtÞ *pM ðtÞ ð11Þ

for the non-autonomous system given by (10), using (9)to express *y: From the definition of the GðtÞ dynamicsin (10)

G�1ðtÞ ¼ G�10 þ

Z t

0

eaðt�tÞb½z1ðtÞ z2ðtÞ�zT1 ðtÞ

zT2 ðtÞ

" #dt; ð12Þ

where G�10 is supposed to be positive definite and the

integral term is positive semi-definite, so that G�1ðtÞ ispositive definite 8t: Hence (11) is a positive definite time-varying quadratic form. From the time derivative offunction W ð *pMÞ’WðtÞ ¼ 2 *pT

MðtÞG�1ðtÞ’*pMðtÞ þ *pTM ðtÞ ’G�1ðtÞ *pMðtÞ

¼ *pTMðtÞ ðb � 2Þ½z1ðtÞ z2ðtÞ�

zT1 ðtÞ

zT2 ðtÞ

" #(

þab

Z t

0

eaðt�tÞ½z1ðtÞ z2ðtÞ�zT1 ðtÞ

zT2 ðtÞ

" #dt

)

*pMðtÞp0 ð13Þ

it results that it represents a negative semi-definitequadratic form. Hence it follows that W ðtÞ is aLyapunov function and that the parameter estimationerror is bounded. &

Remark 6. A time-varying gain matrix GðtÞ is adopted,as in (Kreisselmeier, 1977), in order to tune the adaptivegain according to the level of persistency of excitationgiven by ½z1ðtÞ z2ðtÞ�½z1ðtÞ z2ðtÞ�T: In particular, by

means of (12), matrix ðG�1ðtÞ � G�10 Þ is positive semi-

definite, hence

G�1ðtÞXG�10 ) GðtÞpG0; 8tX0:

In particular, when b > 0 and the input functions ziðtÞare non-null, the adaptation gains have a decreasingbehavior. Differently, in the case of b ¼ 0; the adapta-tion gains are constant independently of the coefficient‘a’ and the input functions.

In the following theorem the more interesting case ofdecreasing adaptation gains is analyzed.

Theorem 7. Consider the IM model (1) assuming that the

angular speed measurement oðtÞ is available. Consider the

related adaptive observer described by (5), (7) and (8) with

the adaptation law given by (10), where

ao0; 0obp2:

If two constants h1 > 0; T > 0 exist such that

h1I9pZ tþT

t

eaðtþT�tÞ½z1ðtÞ z2ðtÞ�zT1 ðtÞ

zT2 ðtÞ

" #dt; 8tX0

ð14Þ

then the non-autonomous dynamics of *pM has a globally

exponentially stable equilibrium point at the origin. In

particular, the exponential decaying rate is greater than or

equal to le; with1

le ¼jajbh1lmðG0Þ

2ð1 þ bh1lmðG0ÞÞ: ð15Þ

Proof. From (13) it follows that

’WðtÞp ab *pTM ðtÞ

Z t

0

eaðt�tÞ½z1ðtÞ z2ðtÞ�zT1 ðtÞ

zT2 ðtÞ

" #dt

( )

*pM ðtÞ: ð16Þ

Eqs. (12) and (11) give

W ðtÞp*pT

MðtÞ *pM ðtÞlmðG0Þ

þ b *pTMðtÞ

Z t

0

eaðt�tÞ½z1ðtÞ z2ðtÞ�zT1 ðtÞ

zT2 ðtÞ

" #dt

( )

*pMðtÞ

and

W ðtÞp *pTMðtÞ

I9

lmðG0Þþ b

Z t

0

eaðt�tÞ½z1ðtÞ z2ðtÞ�zT1 ðtÞ

zT2 ðtÞ

" #dt

( )

*pMðtÞ: ð17Þ

ARTICLE IN PRESSP. Castaldi et al. / Control Engineering Practice 13 (2005) 81–9486

Moreover, using (14) in (17), 8tXT it results

W ðtÞp1 þ bh1lmðG0Þ

h1lmðG0Þ*pT

MðtÞ

Z t

0

eaðt�tÞ½z1ðtÞ z2ðtÞ�zT1 ðtÞ

zT2 ðtÞ

" #dt

( )

*pMðtÞ: ð18Þ

Therefore, using (18) in (16),

’WðtÞpabh1lmðG0Þ

1 þ bh1lmðG0ÞW ðtÞ; 8tXT ð19Þ

hence

jj *pMðtÞjjp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilMðG0ÞlmðG0Þ

sjj *pMð0Þjje�leðt�TÞ; 8tXT : &

ð20Þ

Remark 8. Persistency of excitation for LTI systems canbe formulated in terms of the number of sinusoids in theinputs exploiting the frequency representation of suchmodels (Narendra & Annaswamy, 1989). In the caseconsidered an equivalent criterion cannot be derivedowing to the IM model non-linearity. However, in realimplementation the persistency of excitation condition(14) can be checked a posteriori or on line by a simplefirst-order LTI filtering of each element of½z1ðtÞ z2ðtÞ�½z1ðtÞ z2ðtÞ�T: The value of h1 in (14) caneven be estimated a priori using the coarse nominalinformation on the IM and the input voltage signalsexpected in typical working conditions.

Remark 9. By inverting the definition of pM in (4), it ispossible to calculate the IM physical parameters s; a; gand the product Lmb: In order to determine Lm and b; itis necessary to add the hypothesis Lr ¼ Ls; which isusually verified in practice. Hence it follows that

Lm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2½ðbLmÞ

2 þ ðbLmÞ�q

and it is possible to deter-

mine Lm and b separately.

3.2. Observer parameter tuning

In Section 3.1, constraints on the design parametershave been given in order to obtain asymptotic conver-gence of the estimation process, but there are no furtherindications on how these parameters can be tuned.

It is worth observing that, in order to increase theconvergence rate of the parameter estimation andguarantee the robustness of the observer, a rigorousmethodology for the optimal tuning of the designparameters is very difficult to define, since the adaptiveobserver is nonlinear and the IM-model is time-varying.In the following some guidelines to select first-attempt

parameter values (i.e. l; a; b;G0) are derived, using somequalitative considerations. In Section 4, an iterativeprocedure to improve the first-attempt tuning isdescribed.

3.2.1. Parameter l selection

The following considerations have to be taken intoaccount for l selection:

* from Theorem 2 it is known that the behavior of thenon-minimal model (3)–(5) is not equal to the IMmodel (2), even if the parameter vector is correctbecause of the possible non-null IM initial statevariables. This error exponentially converges to zerowith decaying rate jlj;

* the larger jlj; the worse the effect of noise present inthe observer input;

* in actual applications, the discrete-time version of theadaptive observer is implemented, thus jlj should besufficiently smaller than the sampling frequency.

Hence a natural choice is to set l slightly larger than themotor rated angular frequency. It represents a trade-offbetween the signal content preservation and the noisereduction.

3.2.2. Parameters a; b and G0 selection

A first practical rule in selecting a and b is to imposeabEb � 2 in order to balance the ‘‘integral’’ and‘‘proportional’’ parts in the Lyapunov function time-derivative reported in (13).

In addition, it is worth noting that both theparameters, linked by the above constraint, nonlinearlyaffect the convergence rate le as clarified in (14) and(15). In particular, as jaj tends to infinity, le tends to 0:In fact, according to (15), the negative parameter a

modulates through an exponential function the contentof the signal z1 and z2: Hence, to get sufficiently largevalues of the convergence rate it is convenient to set a

small enough, to avoid strong filtering of the harmoniccontent of the input signals.

A diagonal structure is usually adopted for the matrixG0 owing to its simplicity and numerical robustness. Theminimum eigenvalue lmðG0Þ is set using (15) andaccording to the required minimal convergence rate le;the previously selected values for a and b and anestimate of h1 (see Remark 8).

Finally, as in the case of parameter l; the convergencerate le cannot be set arbitrarily large owing to actualdigital implementation and measurement noise.

4. Simulation and experimental results

In this section considerations about the discrete-timeversion of the estimator, the persistency of excitationcondition and the parameter tuning procedure are given.

ARTICLE IN PRESSP. Castaldi et al. / Control Engineering Practice 13 (2005) 81–94 87

Simulation and experimental results performed on a1:3 kW induction motor are reported to show theperformance that can be obtained with the proposedalgorithm.

Considering the applications of the IM parameterobserver for high performance IM control algorithmsand the limitations introduced by the computationalpower of the currently available commercial DSPs, thesampling time of the discrete-time version of theadaptive observer has been chosen to be equal to Ts ¼600 ms; both for simulation and implementation for theexperimental tests. In this way, reduced performancedegradation due to discretization is introduced and thealgorithm can be implemented for on-line estimation.The dynamical part of the estimation algorithm isdiscretized as follows:

* for the state equations (5) the exponential matrixmethod is applied, leading to an exact discretizationwhen the inputs are constant during the samplinginterval;

* the discrete-time version of the adaptation law (10)is obtained by means of the Euler method, owingto its nonlinearity. In particular, a direct GðtÞmatrix dynamics is implemented in order to avoidinverse matrix calculus, as shown in the followingequation:

’GðtÞ ¼ � aGðtÞ þ aGðtÞG�10 GðtÞ � bGðtÞ½z1ðtÞ z2ðtÞ�

zT1 ðtÞ

zT2 ðtÞ

" #GðtÞ:

During simulations and experiments, in order to testthe proposed estimator without introducing additionaleffects due to involved control algorithms, a simplevoltage–frequency (v/f) control algorithm is used.Moreover, additional sinusoids are added to the mainvoltage command to guarantee the persistency ofexcitation condition. This solution leads to the followingmodified v/f control algorithm:

VaðtÞ

VbðtÞ

" #¼

V ðo�Þ cos e�

V ðo�Þ sin e�

" #þ

XN

i¼1

Vi cos ei

Vi sin ei

" #;

’e� ¼ o�;

’ei ¼ oi;

V ðo�Þ ¼ Ko�;

where o� is the speed reference, K is the v/f gainchosen as K ¼ 1:4 V=ðrad=sÞ (according to therated voltage and speed of the tested IM), N is thenumber of additional sinusoids with frequency oi andamplitude Vi:

In order to select the number of additional sinusoidsN and their frequency and amplitude, simulations inwhich the IM is required to track different constantspeed references have been performed. During these

tests, a different number of additional harmoniccomponents with various values of frequencies andamplitudes have been added to the main voltage. Thepersistency of excitation condition is evaluated throughthe computation of h1 as described in Remark 8. It isfound that a suitable choice which guarantees anacceptable level of persistency of excitation is with N ¼3 additional harmonic components at frequencies o1 ¼150 rad=s; o2 ¼ 300 rad=s; o3 ¼ 450 rad=s and withamplitudes V1 ¼ V2 ¼ V3 ¼ 10 V correspondingly. Inthe sequel, simulation and experiments are performedwith this choice for the modified v/f control.

The observer parameters l; a; b and G0 are firstselected according to the first-attempt tuning proceduredescribed in Section 3.2. On the other hand, in order toincrease the convergence velocity of the estimation, afiner tuning is performed. According to the coarse ratedvalues of the IM and the typical operating conditions, aset of simulations is iteratively performed and theobserver gains are selected in order to minimize theintegral performance index

Q ¼XN

i¼1

X9

j¼5

Z Tsim

0

j *pMjðtÞj

pMj

dt

" #i

;

where N is the number of simulations, i represents thesimulation index and Tsim is the length of eachsimulation. By applying a numerical optimizationprocedure, the resulting observer parameters are:

l ¼ �200; a ¼ �0:73; b ¼ 0:6;

G0 ¼ diagð104; 600; 1:2 104; 500;

1:2 105; 105; 103; 2 103; 103Þ: ð21Þ

Experimental tests were performed on a33 Hz 1:3 kW induction motor in order to test theperformance of the proposed estimator. The experi-mental tests were carried out using a rapid prototypingstation (RPS), which includes:

(1)

a Personal Computer acting as the Operator Inter-face during the experiments;

(2)

a custom floating-point digital signal processor(DSP) board (based on TMS320C32) directlyconnected to the PC bus. The DSP board performsdata acquisition (eight 12-bit A/D data channelsplus two interfaces for incremental encoder), imple-ments control algorithms and generates the PWMsignals (two symmetrical three-phase PWM mod-ulator with programmable dead time);

(3)

a 50A/380 VRMS three-phase inverter, operated at10 kHz switching frequency during experiments.Dead time of the inverter is set to 1:5 ms;

(4)

a 2-pole, 33 Hz; 1:3 kW induction motor. Name-plate data and nominal electric parameters of theIM, obtained via traditional no-load and locked-rotor tests (Vas, 1993), are listed in Table 1;

ARTICLE IN PRESSP. Castaldi et al. / Control Engineering Practice 13 (2005) 81–9488

(5)

Tabl

IM n

Rate

Rate

Rate

Mag

Rate

Rate

Roto

Pole

Lm

sbga

Fig.

axis

from

a current-controlled dc motor used to provide theload torque and acting as inertial load.

In order to filter out the modulation ripple, two statorphase currents, measured by Hall-effect zero-fieldsensors, are simultaneously and synchronously sampledat the symmetry point of the PWM signals. The motor

e 1

ameplate data and roughly estimated electric parameters

d power 1:3 kW

d speed 2000 rpm ð@33 HzÞd current 2:8 A

netizing current 1:1 A

d voltage 380 V

d torque 6 N m

r inertia 0:00225 kg m2

pair number 1

0.875

0.101

9.35

205

10.7

0 20 40 60 80 100 12010

5

0

5

10

p M1, p

M3

0 20 40 60 80 100 120100

50

0

50

100

p M5

0 20 40 60 80 100 1200

5

10

15

20

p M7

0 20 40 60 80 100 1201

0.5

0

p M9

time (s)

−

−

−

−

−

−

(a)

1. Experimental results with o� ¼ 60 rad=s and no load: (a) Estimated

current and current estimation error from t ¼ 0 s to 2 s: (d) Measured a

t ¼ 118 s to 120 s:

speed is measured by means of a 5000 pulse/revolutionincremental encoder. A simple technique based on phasecurrent sign (Jeong & Park, 1991) is used to compensatefor the effects of dead-time in voltage generation, toachieve better accuracy for the inverter output voltage.The sampling time for the adaptive observer and the v/fcontroller is set to Ts ¼ 600 ms: With the adopted DSPboard the computational burden needed for the estima-tion algorithm is about 400 ms:

In order to avoid parameter drift due to measurementnoise and model uncertainties, a standard solution basedon a dead-zone operator is applied in the actualparameter adaptation law. Another crucial point in realimplementation of the proposed scheme is the pureintegration of current and voltage signals reported in(5). To avoid drift due to measurements and numericaloffsets, low-pass filters with a time constant of 10 s areused to replace integrators. This adjustment does notsignificantly impair the estimation performances whensufficiently high-speed reference o� is imposed.

Experimental results, relative to a constant mechan-ical speed reference equal to o� ¼ 60 rad=s with no-load

0 20 40 60 80 100 12010

5

0

5

10

p M2, p

M4

0 20 40 60 80 100 120100

90

80

70

60

p M6

0 20 40 60 80 100 12020

10

0

10

20

p M8

time (s)

−

−

−

−

−

−

−

−

−

parameters #pM : (b) Estimated physical IM parameters. (c) Measured a-

-axis current, current estimation error and estimated a-axis rotor flux

ARTICLE IN PRESS

0 20 40 60 80 100 1200

0.5

1

1.5

2

L m

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

σ

0 20 40 60 80 100 1200

5

10

15

20

β

time (s)

0 20 40 60 80 100 1200

5

10

15

20

α

0 20 40 60 80 100 12050

100

150

200

250

300

γ

time (s)

(b)

0 0.5 1 1.5 2

−3

−2

−1

0

1

2

3

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3

time (s)

i a(A

)i a

(A)

∼

(c)

118 118.5 119 119.5 1203

2

1

0

1

2

3

118 118.5 119 119.5 1203

2

1

0

1

2

3

118 118.5 119 119.5 1202

1

0

1

2

time (s)

−

−

−

−

−

−

−

−

i a(A

)i a

(A)

∼φ a

(Wb)

∧

(d)

Fig. 1 (continued).

P. Castaldi et al. / Control Engineering Practice 13 (2005) 81–94 89

torque are reported in Fig. 1. In Fig. 1(a) transients ofthe estimated parameter vector #pMðtÞ are reported,confirming the exponential convergence of the para-meter estimation. In Fig. 1(b) estimated physicalparameters obtained from vector #pMðtÞ are shown. Aconvergence rate of about t ¼ 120 s can be achievedwith the chosen observer gains. Steady-state estimated

IM parameters are similar to rated IM parameters. Amismatching is present in the estimation of parametersLm and s obtained using the adaptive observer withrespect to the rough off-line estimation method. This ismainly due to different levels of the rotor flux amplitudein the identification procedures and to the fact thatthe magnetic circuit of the IM is not perfectly linear. In

ARTICLE IN PRESS

Table 2

Estimated IM parameters during experimental tests at different speed reference and load torque

o% ðrad=sÞ 40 60 80 100

TL ðN mÞ 0 0 2 0 2 0 2

Lm 1.19 1.16 1.12 1.15 1.14 1.15 1.17

s 0.0876 0.0878 0.0881 0.0891 0.0878 0.0872 0.0866

b 11.1 10.9 10.9 10.7 10.9 11.0 11.1

g 230 221 240 216 237 232 241

a 11.1 10.9 9.3 10.6 9.4 9.5 9.1

176 177 178 179 1800

20406080

100120

ω, ω

*

176 177 178 179 1800

0.5

1

1.5

2

L m

176 177 178 179 1800

0.05

0.1

0.15

0.2

σ

time (s)

176 177 178 179 1800

5

10

15

20

β

176 177 178 179 1800

5

10

15

20α

176 177 178 179 18050

100

150

200

250

300

γ

time (s)

Fig. 2. IM parameter estimation during rotor speed transient.

P. Castaldi et al. / Control Engineering Practice 13 (2005) 81–9490

Fig. 1(c) the current estimation error both at thebeginning and at the end of the estimation process isshown, enlightening its convergence to zero after theinitial transient. Only the ia current is reported, since theib current behavior is similar. In Fig. 1(d) the a-axiscomponent of the estimated rotor flux, obtained fromthe parameter vector #pMðtÞ and the observer state vector#%xðtÞ; is reported.

In order to test the sensitivity of the observer todifferent operating conditions of the IM (i.e. differentrotor speed and load torque), various experiments havebeen performed, imposing a constant speed referencebetween 40 and 100 rad=s without load and withconstant load equal to 2:0 N m: During tests, motortemperature is maintained constant, in order not to havedrift in the values of rotor and stator resistance. Steady-state estimated physical IM parameters (obtained after a120 s estimation process) are listed in Table 2. Theestimated parameters are close to the nominal parameter

values obtained with standard off-line identificationtechnique. Reduced sensitivity with respect to theoperating conditions of the IM is enlightened. Smallvariations of the estimated parameters depending on theworking condition are present due to current measure-ment noise and imperfect knowledge of the appliedstator voltage, due to inverter nonlinearities andimprecise compensation of the dead-time effect. More-over, since the sampling time is quite long, thediscretization method used to implement the algorithmaffects the accuracy of the estimation. Performedexperiments confirm that the adopted observer gainsguarantee correct parameter estimation in a wide rangeof operating conditions with estimation convergencerate compatible with the thermal drift of the parameters.

In Fig. 2 an experimental test with variable rotorspeed is reported. During the test the v/f controlled IMis required to track a trapezoidal speed reference signalranging from 40 to 100 rad=s with maximum time

ARTICLE IN PRESS

0 100 200 300 400 500 6000

0.5

1

1.5

2

L m

0 100 200 300 400 500 6000

0.05

0.1

0.15

0.2

σ

0 100 200 300 400 500 6000

5

10

15

20

β

time (s)

0 100 200 300 400 500 6000

5

10

15

20

α

0 100 200 300 400 500 60050

100

150

200

250

300

γ

time (s)

Fig. 3. Simulation results of the physical IM parameters estimation during rotor and stator resistance variation. Actual IM parameters are indicated

with dashed lines.

P. Castaldi et al. / Control Engineering Practice 13 (2005) 81–94 91

derivative equal to ’o� ¼ 200 rad=s2: In Fig. 2 theparameter estimation behavior is shown, after atransient in the parameter estimation of 178 s: It canbe noted that during rotor speed transient, the estimatedIM parameters are close to those estimated in previousreported tests and that negligible variation of theestimated parameters is obtained.

In order to show the parameter tracking performanceof the proposed adaptive observer while a variation ofthe IM parameters occurs, a simulation with rotor andstator resistance variation equal to DRr ¼ 50% andDRs ¼ 20% correspondingly is performed, with constantspeed reference equal to 60 rad=s and no applied loadtorque. Simulation results, reported in Fig. 3, show thatthe estimated parameters tend to the real ones and thatthe estimator is capable of tracking slow variations ofIM parameters, e.g. due to thermal drift.

5. Conclusions

A new method for on-line identification of theelectrical parameters and estimation of the electromag-netic state of induction motors has been presented. Itrequires the stator currents and mechanical angularspeed measurements. The state estimation allows thecontinuous monitoring of the rotor flux. This is useful toavoid magnetic saturation and to realize a full-statefeedback controller for the induction motor. Theproposed adaptive observer is based on a non-minimal

representation of the IM. Exponential convergenceproperties of both parameters and states estimationhas been formally proved.

Simulation and experimental results have confirmedthe effectiveness of the proposed method. In particular,the estimated parameters obtained in different operatingconditions are close to the nominal IM parametersestimated with standard off-line procedure. Parameterestimation is not affected by operating conditions of theIM, such as speed variation and applied load torque.The rate of convergence of the estimation allows for thecompensation of resistance variations due to thermaldrift.

Appendix

Proof of Theorem 2. Let Fðt; 0Þ be the state transitionmatrix of the homogeneous non-stationary linear systemderived from (2). It is known that Fðt; 0Þ is the solutionof the following differential matrix equation

’Fðt; 0Þ ¼ F ðtÞFðt; 0Þ; Fð0; 0Þ ¼ I4 ðA:1Þ

hence, by definitions (5) it follows that

Fðt; 0Þ ¼

elt 0 �al þ

ale

lt c1

0 elt �c1 �al þ

ale

lt

0 0 1 0

0 0 0 1

266664

377775: ðA:2Þ

ARTICLE IN PRESSP. Castaldi et al. / Control Engineering Practice 13 (2005) 81–9492

The motion of the linear model (2) can be separatedin two terms: the free motion %xLðtÞ and the forcedmotion %xF ðtÞ:

The free motion is defined by the following expres-sion:

%xLðtÞ ¼ Fðt; 0Þ %x0:

The motion %xLðtÞ is constituted by the sum of a transientpart %xLT ðtÞ and a steady-state part %xLSðtÞ: In particular

%xLðtÞ ¼ %xLT ðtÞ þ %xLSðtÞ;

where, after some simple computation the two addendscan be represented as follows:

%xLT ðtÞ ¼

elt 0 ale

lt 0

0 elt 0 ale

lt

0 0 0 0

0 0 0 0

266664

377775 %x0; ðA:3Þ

%xLSðtÞ ¼

c1 %x04 � al %x03

�c1 %x03 � al %x04

%x03

%x04

26664

37775

¼

0 1 c1 0

�c1 0 0 1

1 0 0 0

0 0 1 0

26664

37775

%x03

�al %x03

%x04

�al %x04

26664

37775: ðA:4Þ

Owing to the negativity of l the transient component

%xLT is exponentially vanishing, hence %xL tends to %xLS

exponentially with rate jlj:Considering the forced motion, it is straightforward

to verify the following equality:

½KF ðtÞ BF �yðtÞ

uðtÞ

" #

¼

�ðlþ aþ gÞ 0 1s 0 0 �1

0 �ðlþ aþ gÞ 0 1s 1 0

ðabLm � gÞ 0 1s 0 0 0

0 ðabLm � gÞ 0 1s 0 0

26664

37775

y1ðtÞ

y2ðtÞ

u1ðtÞ

u2ðtÞ

y1ðtÞoðtÞ

y2ðtÞoðtÞ

26666666664

37777777775

hence, %xF can be expressed as

%xF ðtÞ ¼Z t

0

Fðt; tÞ

�ðlþ aþ gÞ 0 1s 0 0 �1

0 �ðlþ aþ gÞ 0 1s 1 0

ðabLm � gÞ 0 1s 0 0 0

0 ðabLm � gÞ 0 1s 0 0

26664

37775

y1ðtÞ

y2ðtÞ

u1ðtÞ

u2ðtÞ

y1ðtÞoðtÞ

y2ðtÞoðtÞ

26666666664

37777777775

dt: ðA:5Þ

It is now possible to split the state transition matrix,and the forced motion, into two parts

Fðt; tÞ ¼ F1ðt; tÞ þ F2ðt; tÞ ) %xF ðtÞ ¼ %xF1ðtÞ þ %xF2ðtÞ

with

F1ðt; tÞ ¼

elðt�tÞ 0 0 ctðtÞ

0 elðt�tÞ �ctðtÞ 0

0 0 1 0

0 0 0 1

266664

377775;

F2ðt; tÞ

¼

0 0 �alð1 � elðt�tÞÞ 0

0 0 0 �alð1 � elðt�tÞÞ

0 0 0 0

0 0 0 0

266664

377775; ðA:6Þ

’ctðtÞ ¼ lctðtÞ þ oðtÞ; ctðtÞ ¼ 0:

The first part of the forced motion is described by thefollowing differential equation with initial condition

%xF1ð0Þ ¼ 0:

’%xF1 ¼

l 0 0 o

0 l �o 0

0 0 0 0

0 0 0 0

26664

37775 %xF1

þ

�ðlþ aþ gÞ 0 1=s 0 0 �1

0 �ðlþ aþ gÞ 0 1=s 1 0

ðabLm � gÞ 0 1=s 0 0 0

0 ðabLm � gÞ 0 1=s 0 0

26664

37775

y1

y2

u1

u2

y1o

y2o

26666666664

37777777775: ðA:7Þ

ARTICLE IN PRESSP. Castaldi et al. / Control Engineering Practice 13 (2005) 81–94 93

It is easy to verify that the solution is given by

%xF1ðtÞ ¼

�c2 c7 ðc8 þ c11Þ

�c3 �c6 ðc9 � c10Þ

0 i1 i30 i2 i4

26664

37775

lþ aþ g

abLm � g

1=s

264

375

þ

�c5

c4

0

0

26664

37775; ðA:8Þ

where ciðtÞ are defined according to (5).By means of definitions (A.5) and (A.6), the second

part of the forced motion is equivalent to the followingexpression:

%xF2ðtÞ ¼Z t

0

�alð1 � elðt�tÞÞ 0

0 �alð1 � elðt�tÞÞ

0 0

0 0

266664

377775

ðabLm � gÞ 0 1=s 0

0 ðabLm � gÞ 0 1=s

" #

y1ðtÞ

y2ðtÞ

u1ðtÞ

u2ðtÞ

26664

37775 dt;

hence

%xF2ðtÞ ¼

ðc2 � Z1Þ ðc8 � Z3Þ

ðc3 � Z2Þ ðc9 � Z4Þ

0 0

0 0

26664

37775

alðabLm � gÞ

als

" #; ðA:9Þ

where ciðtÞ and ZiðtÞ are defined according to (5).Adding (A.4), (A.8) and (A.9), Eqs. (3), (4) and (5) are

obtained. Hence by taking into account the vanishingproperties of %xLT proof is completed. &

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