Energy Conversion and Management 46 (2005) 335–349www.elsevier.com/locate/enconman

Speed sensorless direct torque control of IMswith rotor resistance estimation

Murat Barut a, Seta Bogosyan b,*, Metin Gokasan a

a Department of Electrical and Electronic Engineering, Faculty of Engineering, Istanbul Technical University,

Maslak, 34390 Istanbul, Turkeyb Department of Electrical and Computer Engineering, University of Alaska, P.O. Box 750145,

Fairbanks, AK 997750145, USA

Received 17 January 2004; accepted 29 April 2004

Available online 2 July 2004

Abstract

Direct torque control (DTC) of induction motors (IMs) requires an accurate knowledge on the ampli-

tude and angular position of the controlled flux in addition to the information related to angular velocity

for velocity control applications. However, unknown load torque and uncertainties related to stator/rotor

resistances due to operating conditions constitute major challenges for the performance of such systems.The determination of stator resistance can be performed by measurements, but methods must be developed

for estimation and identification of rotor resistance and load torque. In this study, an EKF based solution

is sought for determination of the rotor resistance and load torque as well as the above mentioned states

required for DTC. The EKF algorithm used in conjunction with the speed sensorless DTC is tested under

eleven scenarios comprised of various changes made in the velocity reference beside the load torque and

rotor resistance values assigned in the model. With no a priori information in the estimated states and

parameters, it has been demonstrated that the EKF estimation and sensorless DTC perform quite well in

spite of the uncertainties and variations imposed on the system.� 2004 Elsevier Ltd. All rights reserved.

Keywords: Induction motor; Extended Kalman filter; Sensorless direct torque control; Load torque and rotor resistance

estimation

* Corresponding author. Tel.: +1-907-475-2755; fax: +1-907-475-5135.

E-mail address: s.bogosyan@uaf.edu (S. Bogosyan).

0196-8904/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2004.04.002

336 M. Barut et al. / Energy Conversion and Management 46 (2005) 335–349

1. Introduction

High efficiency control and estimation techniques related to induction motors (IMs) have beenfinding more and more application with Blaschke’s well-known field oriented control (FOC)method, established in 1971. There has been an extensive amount of work to improve the dynamicresponse and reduce the complexity of FOC methods. One such technique is the direct torquecontrol (DTC) method developed by Takahashi in 1984 [1], which has been getting increasedattention due to the improved dynamic performance and simplified control strategy that it offerswith respect to the FOC methods.

The DTC method involves the direct choice of the appropriate/optimum switching modes, inorder to keep the flux and torque errors within a predetermined band limit (in a hysteresis band)[2]. The errors are defined as the difference between the reference and the measured/estimatedvalues of flux and torque. Unlike FOC methods, DTC techniques require utilization of hysteresisband comparators instead of flux and torque controllers. To replace the coordinate transfor-mations and pulse width modulation (PWM) signal generators of FOC, DTC uses look-up tablesto select the switching procedure based on the inverter states. However, both methods requireaccurate knowledge of the amplitude and angular position of the controlled flux (with respect tothe stationary stator axis) in addition to the angular velocity for velocity control applications.

As is well known, speed sensors like tachometers or incremental encoders increase the size andcost of systems unnecessarily. Similar problems arise with the addition of search coils or Halleffect sensors to the motor for measurement of the flux, hindering functionality in terms ofimplementation. Thus, to improve overall system performance, state estimators or observers areusually more preferable than physical measurements. However, the fifth order and nonlinearstructure of the IM model [3], in addition to the sensitivity of the system parameters to tem-perature [4] and frequency [5], makes the design of observers for IMs a challenge. In this regard,for high performance sensorless vector control of IMs, it is essential to know the temperature andfrequency dependent variation of the stator and rotor resistances in addition to the load torque. Ina study addressing this issue [6], it has been stated that simultaneous estimation of the stator androtor resistances gives rise to instability. Moreover, while the value of the stator resistance couldbe obtained by measuring the stator temperature, there are physical difficulties in determining therotor resistance in a squirrel cage IM. Thus, estimating the rotor resistance and the load torqueappears to be a reasonable approach.

In DTC, the flux is conventionally obtained from the stator voltage model, using the measuredstator voltages and currents. This method, utilizing open loop pure integration suffers from thewell known problems of integration effects in digital systems, especially in the low speed operationrange [7], even with the correct knowledge of the stator resistance. Moreover, it will require therotor angular velocity for velocity control applications. Among the recent studies conductingsimultaneous flux and velocity estimation for DTC, Ref. [8] studied the sensitivity to parametervariations with an artificial neural network approach, and a robust performance to 50% variationsin the stator resistance has been obtained in Ref. [9] with a sliding mode approach, while theadaptive flux observer in Ref. [10], the extended Luenberger observer in Ref. [11] and the non-linear observer in Ref. [12] demonstrate robustness to step shaped load torque variations. Amongstudies using model reference adaptive laws, in Ref. [13], the flux and speed have been estimated,but the system response to load torque variations was not tested. In Ref. [14], the rotor velocity,

M. Barut et al. / Energy Conversion and Management 46 (2005) 335–349 337

xm, stator resistance, Rs and rotor resistance R0r are individually estimated, and good results are

obtained. However, in one of the trials where xm together with R0r and in another where xm, R0

r

and Rs are estimated together, it has been stated that the resistances converge to inaccurate values.Moreover, no tests have been performed to test the effects of the load torque variations.

Finally, in Ref. [6], the angular velocity and slip frequency, xr (reflecting the effect of the loadtorque) in addition to the rotor resistance have been taken into account starting with the initialvalue of R0

rð0Þ ¼ 0:85R0rn.

There are also extended Kalman filter (EKF) applications in the literature, taking a stochasticapproach to solution of the problem. Unlike the other methods, the model uncertainties andnonlinearities inherent in IMs are well suited to the stochastic nature of EKFs [15]. With thismethod, it is possible to make an online estimation of states while simultaneously performingidentification of parameters in a relatively short time interval [16–18], also while taking system/process and measurement noises directly into account. This is the reason why the EKF has foundwide application in sensorless control of IM’s, in spite of its computational complexity. In theEKF based previous DTC studies, Ref. [19] estimates the stator flux components and velocityunder the assumption of a known load, while in Ref. [20], the velocity is estimated as a constantparameter, avoiding the use of the equation of motion. In spite of an improved performance in thesteady state, this approach has given rise to a significant observer error in the velocity during thetransient state.

The major contribution of this study is the development of an EKF based speed sensorlessDTC system that achieves robustness to variations in rotor resistance and load torque, theuncertainties that are known to deteriorate system performance. It is the first known study toperform the estimation of load torque and rotor resistance simultaneously while also estimatingthe stator flux components, angular velocity and stator current components, also measured asoutput. The performance of the estimation and control schemes is tested with challenging vari-ations of the load torque, rotor resistance and velocity reference. The consideration of the loadtorque as a constant term in the estimation algorithm aims to capture other uncertainties besidesthe load torque that have a very slow or almost constant variation with time, i.e. viscous andCoulomb friction (in steady state). The results obtained through simulations under variouschallenging tests demonstrate the good performance of the estimation scheme requiring no apriori information on the states with their initial values taken as zero.

2. Extended mathematical model of the IM

The sensorless DTC scheme developed for an IM requires estimation of the stator flux com-ponents, wsa, wsb, angular velocity, xm and stator current components isa and isb, which are alsomeasured as output. In this study, due to the degrading effect of their unknown variations oncontrol performance, the load torque, tL and the rotor resistance, R0

r (as referred to the stator side)are also included in the extended state vector as constant states based on their slow variation intime. Thus, the so-called extended model can be obtained (as referred to the stator stationaryframe) in the following form:

_x ðtÞ ¼ f ðx ðtÞ; u ðtÞÞ þ w ðtÞ ¼ A ðx ðtÞÞx ðtÞ þ B u ðtÞ þ w ðtÞ ð1Þ

e e e e 1 e e e e e 1

338 M. Barut et al. / Energy Conversion and Management 46 (2005) 335–349

Here, the extended state vector xe, representing the estimated states and parameters, consists of isa,isb, wsa, wsb, xm, tL and R0

r; f eis a nonlinear function of the states and inputs; Ae is the system

matrix; ue is the control input vector; Be is the input matrix; and w1 is process noise. The constantstate representing the load torque is also designed to capture system uncertainties of constantnature once steady state is attained. In this study, those uncertainties are limited to viscousfriction, as this was the only uncertainty included in the model simulating the system. With theabove consideration, the extended model of an IM can be given as

_isa_isb_wsa_wsb

_xm

_tL_R0r

26666666664

37777777775

|fflfflffl{zfflfflffl}_xe

¼

� Rs

Lrþ R0

rLsL0rLr

� ��ppxm

R0r

L0rLr

ppxm

Lr0 0 0

ppxm � Rs

Lrþ R0

rLsL0rLr

� �� ppxm

Lr

R0r

L0rLr0 0 0

�Rs 0 0 0 0 0 0

0 �Rs 0 0 0 0 0

� 32

ppJLwsb

32

ppJLwsa 0 0 0 � 1

JL0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

266666666664

377777777775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Ae

isaisbwsa

wsb

xm

tLR0r

2666666664

3777777775

|fflfflffl{zfflfflffl}xe

þ

1Lr

0

0 1Lr

1 0

0 1

0 0

0 0

0 0

2666666664

3777777775

|fflfflfflfflffl{zfflfflfflfflffl}Be

vsavsb

� �|fflffl{zfflffl}

ue

þw1ðtÞ

ð2Þ

ZðtÞ ¼ heðxeðtÞÞ þ w2ðtÞ ðmeasurement equationÞ ¼ H exeðtÞ þ w2ðtÞ

¼ 1 0 0 0 0 0 0

0 1 0 0 0 0 0

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

He

isaisbwsa

wsb

xm

tLR0r

2666666664

3777777775þ w2ðtÞ ð3Þ

where he is a function of the outputs; H e is the measurement matrix; w2 is measurement noise; pp isthe number of pole pairs; Lr ¼ rLs is the stator transient inductance; r is leakage or couplingfactor; Ls and Rs are the stator inductance and resistance, respectively; L0

r and R0r are the rotor

inductance and resistance, referred to the stator side, respectively; vsa and vsb are the stator sta-tionary axis components of the stator voltages; wsa and wsb are the stator stationary axis com-ponents of stator flux; and xm is the angular velocity.

3. Development of the EKF algorithm

An EKF algorithm is developed for estimation of the states in the extended IM model given inEqs. (2) and (3) to be used in the sensorless direct torque control of the IM. The Kalman filter(KF) is a well known recursive algorithm that takes the stochastic state space model of the systeminto account, together with measured outputs, to achieve the optimal estimation of states [21] inmulti-input, multi-output systems. The system and measurement noises are considered to be in the

M. Barut et al. / Energy Conversion and Management 46 (2005) 335–349 339

form of white noise. Optimality of the state estimation is achieved with minimization of thecovariance of the estimation error. For nonlinear problems, the KF is not strictly applicable sincelinearity plays an important role in its derivation and performance as an optimal filter. The EKFattempts to overcome this difficulty by using a linearized approximation where the linearization isperformed about the current state estimate [22]. This process requires the discretization of Eqs. (2)and (3)

xeðk þ 1Þ ¼ feðxeðkÞ; ueðkÞÞ þ w1ðkÞ ð4Þ

ZðkÞ ¼ H exeðkÞ þ w2ðkÞ ð5Þ

As mentioned before, the EKF involves the linearized approximation of the nonlinear model(Eqs. 4 and 5) and uses the current estimation of states xeðkÞ and inputs ueðkÞ in the linearizationby using

F eðkÞ ¼of

eðxeðkÞ; ueðkÞÞoxeðkÞ

����xeðkÞ;ueðkÞ

ð6Þ

F uðkÞ ¼of

eðxeðkÞ; ueðkÞÞoueðkÞ

����xeðkÞ;ueðkÞ

ð7Þ

Thus, the EKF algorithm can be given in the following recursive relations:

NðkÞ ¼ F eðkÞPðkÞF eðkÞT þ F uðkÞDuF uðkÞ

T þ Q ð8aÞ

Pðk þ 1Þ ¼ NðkÞ � NðkÞHTe ðDn þ H eNðkÞHT

e Þ�1H eNðkÞ ð8bÞ

xeðk þ 1Þ ¼ feðxeðkÞ; ueðkÞÞ þ Pðk þ 1ÞHT

eD�1n ðZðkÞ � H exeðkÞÞ ð8cÞ

Here, Q is the covariance matrix of the system noise, namely model error; Dn is the covariancematrix of the output noise, namely measurement noise; Du is the covariance matrix of the controlinput noise (vsa and vsb), namely input noise; and P and N are the covariance matrices of stateestimation error and extrapolation error, respectively.

The algorithm involves two main stages: prediction and filtering. In the prediction stage, thenext predicted states f

eð�Þ and predicted state error covariance matrices, Pð�Þ and Nð�Þ, are pro-

cessed, while in the filtering stage, the next estimated states, xeðk þ 1Þ, obtained as the sum of thenext predicted states and the correction term (2nd) term in Eq. (8c)) are calculated. The schematicrepresentation of the algorithm is given in Fig. 1.

The algorithm utilizes the extended or augmented model in Eqs. (2) and (3) to generate alloutput states required by the sensorless direct torque control scheme, in addition to the rotorresistance and the load torque, using measured phase currents and voltages.

IM

z/1

( )kfe

ˆ

z/1

mux.

savsbvscv

outputs

( )kZ

abc

αβ

αβ

abc

( )kF e

( )kF u

uDQ, )1(ˆ +kxe

1−ξD

( )1+kP( )kN

[ ]'ˆˆˆˆˆˆˆrLmssss Rtii ωψψ βαβα

ξD Tee HH ,

Voltage sensors Current sensors

EKF Algorithm

Fig. 1. Structure of the EKF algorithm.

340 M. Barut et al. / Energy Conversion and Management 46 (2005) 335–349

4. Speed sensorless DTC system

Fig. 2 demonstrates the speed sensorless DTC system. Here, hrf stands for the sector position ofthe flux with reference to the stationary axis. The velocity controller given in the diagram is aconventional proportional integral derivative (PID) controller. The development of the sectorselector and the switching table is based on Takahashi’s study presented in Ref. [1].

5. Simulation results and observations

To test the performance of the estimation method, simulations were performed on an IM withthe rated parameters given in Table 1.

The values of the system parameters and covariance matrix elements are very affective on theperformance of the EKF estimation. In this study, to avoid computational complexity, thecovariance matrix of the system noise Q is chosen in diagonal form, also satisfying the conditionof positive definiteness. According to the KF theory, Q, Dn (measurement error covariance ma-trix) and Du (input error covariance matrix) have to be obtained by considering the stochasticproperties of the corresponding noises [7]. However, since these are usually not known, in mostcases, the covariance matrix elements are used as weighting factors or tuning parameters. In thisstudy, tuning the initial values of P and Q is done by trial and error to achieve a rapid initialconvergence and the desired transient and steady state behaviors of the estimated states and

Table 1

The nominal values and parameters of the induction motor used in the tests

P[KW]

f[Hz]

JL[kgm2]

BL

[Nm/

(rad/s)]

pp V [V] I [A] Rs [X] R0r [X] Ls [H] Lr [H] Lm [H] Nm

[rpm]

Te[Nm]

3 50 0.006 0.001 2 380 6.9 2.283 2.133 0.2311 0.2311 0.22 1430 20

EKF basedestimator

Sector Selector

SwitchingTable

Inverter

abc

abc

Fluxcomp.

Torquecomp.

Velocitycont.

IM

pulses

Fig. 2. The speed sensorless DTC system.

M. Barut et al. / Energy Conversion and Management 46 (2005) 335–349 341

parameters, while Dn and Du are determined taking into account the measurement errors of thecurrent and voltage sensors and the quantization errors of the ADCs, as given below:

Q¼diagf10�6 ½A2� 10�6 ½A2� 10�6 ½Wb2� 10�6 ½Wb2� 10�4 ½ðrad=sÞ2� 10�5 ½ðNmÞ2� 10�7 ½X2�g

P ¼ diagf 9 ½A2� 9 ½A2� 9 ½Wb2� 9 ½Wb2� 9 ½ðrad=sÞ2� 9 ½ðNmÞ2� 9 ½X2� g

Dn ¼ diagf 10�6 ½A2� 10�6 ½A2� g

Du ¼ diagf 10�3 ½V2� 10�3 ½V2� g and sampling time T ¼ 100 ls:

The bandwidth ðbwÞ of the flux comparator is chosen as 0.02 [Wb], while that of the torquecomparator ðbteÞ is 0.01 [Nm].

0 1 2 3 4 5 6 7 8 9 10 11 12 13-1500

-1000

-500

0

500

1000

1500

t[s]

n mR

epr[ .f

m]

1500

-1500

20 20 0

1500 1500 1500

(a)

(b)0 1 2 3 4 5 6 7 8 9 10 11 12 13

-20

-15

-10

-5

0

5

10

15

20

t[s]

t LN[

]m

20 20 20

-20

20 20 20

0

-20

Fig. 3. Variation of the reference speed value, nrefm , and applied load torque tL: (a) Variation of the reference speed

value, nrefm ; (b) Variation of the applied load torque, tL.

342 M. Barut et al. / Energy Conversion and Management 46 (2005) 335–349

Eleven different scenarios are created to test the performance of the estimation and controlalgorithm in the time interval of 06 t6 13 s.

The first 10 scenarios are developed with simultaneous changes of the velocity reference(Fig. 3a) and the load torque value (Fig. 3b) used in the extended model.

The last scenario (scenario 11) is created by giving R0r in the model a step change to twice its

original value, R0r ¼ 2R0

rn.The estimation of all the states and parameters is started with an initial value of zero.The resulting system performance for all scenarios is given with Fig. 4a representing the

velocity estimate, nm, Fig. 4b depicting the velocity error, ðnrefm � nmÞ, and Fig. 4c giving theestimation error, nm � nm. The variations of the applied and estimated load torque are given inFig. 5a, with Fig. 5b representing the estimation error, ðtL � tLÞ, for this variable. The variationsrelated to the rotor resistance, R0

r, are given in Fig. 6a and b, with the former plot representing theactual and estimated variation of R0

r with the initial value of the estimate taken as zero, while thelatter plot represents the estimation error, R0

r � R0r. Finally, Fig. 7a–c represent the estimated flux

magnitude, jwsj, the error between the reference and actual (estimated) flux magnitude,jwsj

ref � jwsj, and the flux estimation error, ðjwsj � jwsjÞ, respectively. Fig. 8 shows the trajectoryof wsa and wsb.

0 1 2 3 4 5 6 7 8 9 10 11 12 13-1500

-1000

-500

0

500

1000

1500

t[s]

n mE

s.t

r[ pm

]

1500.2

-1499.9

18.3 19.9 -0.0308

1499.8 1501.4 1500

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13-40

-30

-20

-10

0

10

20

30

40

50

t[s]

n me

R f.-

n mE

sr[ .tpm

]

0.4694 -0.2129

-0.3923 -1.3825

-0.1443

1.4121

1.7113

0.1481

0.0308

1.6995

0.1903

-1.4442

0.0042

(b)

0 1 2 3 4 5 6 7 8 9 10 11 12 13-60

-40

-20

0

20

40

60

t[s]

em

nr[ pm

]

-0.1238

7.4*10-5

0.0404 0.1296

1.2*10-6

[-0.96~, -12.5~]

0.0006

-0.5618 3.3*10-6

-0.0122

-7.3*10-5

[-0.56~, -16.06~]

[-1.73~, -7.6~]

(c)

Fig. 4. Simulation results related to velocity of the EKF based estimator and the DTC: (a) Variation of the nm;(b) Variation of the velocity controller input; (c) Variation of the estimation error of nm, enm .

M. Barut et al. / Energy Conversion and Management 46 (2005) 335–349 343

5.1. Observations

5.1.1. Operation under constant tL and constant/linear velocity referencesIn intervals where the load torque in the model is given a constant value (20 Nm), the esti-

mation and control algorithms perform very well under both step type and linear variations of thevelocity reference.

With no a priori information on the load torque (and an initial value of 0 Nm), the EKFalgorithm and the control achieve a low velocity error, ranging between 0.011% and 0.031%, as

0 1 2 3 4 5 6 7 8 9 10 11 12 13

-30

-20

-10

0

10

20

30

t[s]

t Lt

& L

E s

.t]

mN[

20.1443

20.1571

20.0164

-20.1443

-20.1571

20.0023

-0.1357

20.1571

20.1573

20.1571

tL Est.

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13-50

-40

-30

-20

-10

0

10

20

30

t[s]

eLt

mN[

] -0.1443

-0.1571

-0.0164

0.1443

0.1571

[1.20~,1.63~]

[1.32~,2.24~]

-0.0023

0.1357[0.14~1.48~]

-0.1571

-0.1573

-0.1571

(b)

Fig. 5. Simulation results related to load torque of the EKF based estimator: (a) Variation of the tL and tL;(b) Variation of the estimation error of tL, etL .

344 M. Barut et al. / Energy Conversion and Management 46 (2005) 335–349

can be seen in the time intervals 06 t6 1 s, 1 s6 t6 2 s, 2 s6 t6 3 s, 3 s6 t6 4 s, 4 s6 t6 5 s,7 s6 t6 8 s, 8 s6 t6 9 s, 10 s6 t6 11 s and so on. In the first, third and fourth of the aboveintervals, a velocity reference with a linear variation in time has been applied to the system, whilein the rest of the intervals, a constant velocity reference has been used.

However, independent of the velocity reference, a very good performance has been obtained,mainly due to the consistence between the applied and assumed load torque in the model.

5.1.2. Operation under linear tL and constant/linear velocity references

In the time interval 5 s6 t6 7 s, tL is given a variation ðtL ¼ �20þ 20ðt � 5ÞÞ, while nrefm isvaried as nrefm ¼ �1500þ 1520ðt � 5Þ between 5 s6 t6 6 s, during which the velocity error,enmð%Þ ¼ 1:4121

20� 100 ¼ 7:06% and estimation error enmð%Þ ¼ 0:96

20� 100 ¼ 4:8%.

As noted before, the increased errors in the velocity output and estimate are due to theinconsistency between the tL in the EKF model (which is constant) and the imposed variation of tL(linear) in the model representing the plant for simulation purposes.

5.1.3. Operation in the low velocity region, with no load references

In the interval 8 s6 t6 9 s, both the velocity reference and tL are made zero, giving rise to avelocity error of enm ¼ 0:0308 rpm in steady state and an estimation error of enm ¼ �0:5618 rpm,

0 1 2 3 4 5 6 7 8 9 10 11 12 130

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t[s]

Rr

& 'R

r'E st

.h

O[ m

]

2.133 2.1332.133 2.1329 2.133 2.1302 2.1332.1184 2.133 2.127 2.133

4.2656 4.266

Rr' Est.

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13-0.5

0

0.5

1

1.5

2

2.5

t[s]

er

R[ 'O

hm]

2.5*10-5

-5.2*10-7

-3.7*10-5

9.8*10-5

-6.2*10-7

0.0028

0.0146

1.7*10-6

3.3*10-6

0.006

5.1*10-7

3.9*10-4

-1.2*10-6

(b)

Fig. 6. Simulation results related to rotor resistance of the EKF based estimator: (a) Variation of the R0r and R0

r;

(b) Variation of the estimation error of R0r, eR0

r.

M. Barut et al. / Energy Conversion and Management 46 (2005) 335–349 345

which are acceptable results considering the challenge posed by the velocity region and change intL.

5.1.4. Operation with linear velocity reference and linear tLIn the interval 9 s6 t6 10 s, the variations nrefm ¼ 1500ðt � 9Þ and tL ¼ 20ðt � 9Þ give rise to a

velocity error of enmð%Þ ¼ 1:71500

� 100 ¼ 0:11% and an estimation error that varies betweenenmð%Þ ¼ 0:56

1500� 100 ¼ 0:037% and enmð%Þ ¼ 16:06

1500� 100 ¼ 1:07%. Once again, relatively higher but

still acceptable errors are caused mainly by the linear variation of tL.

5.1.5. Operation under reversal of velocity reference

In the interval 2 s6 t6 4 s, the velocity reference is reversed from 1500 rpm to )1500 rpm witha linear variation of nrefm ¼ 1500ðt � 2Þ � 1500ðt � 2Þ. During this interval, tL is also given a var-iation of tL ¼ 20sgnðnmÞ. After a brief transient while the velocity and torque pass through zero,velocity errors of enmð%Þ ¼ 0:3923

1500� 100 ¼ 0:026% and enmð%Þ ¼ 1:3825

1500� 100 ¼ 0:0923% occur at

t ¼ 2 and 4 s, respectively. The velocity estimation errors for the same instants areenmð%Þ ¼ 0:0404

1500� 100 ¼ 0:0027% and enmð%Þ ¼ 0:1296

1500� 100 ¼ 0:0086%, respectively. Considering

the error, it can be noted that the system has responded quite well to the simultaneous reversal ofnrefm and tL.

0 1 2 3 4 5 6 7 8 9 10 11 12 130

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t[s]

| ψs|E

sW[ t

b]

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13-0.2

0

0.2

0.4

0.6

0.8

1

t[s]

| ψs|r

|- ψs|E

stW[

b]

(b)

0 1 2 3 4 5 6 7 8 9 10 11 12 13-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t[s]

e | ψ|s

W[ b]

(c)

Fig. 7. Simulation results related to flux of the EKF based estimator and the DTC: (a) Variation of the jwsj;(b) Variation of the flux comparator input; (c) Variation of the estimation error of jwsj, ejws j.

346 M. Barut et al. / Energy Conversion and Management 46 (2005) 335–349

5.1.6. Operation under R0r ¼ 2R0

rn (constant velocity reference/load torque)

As mentioned before, another challenge for the control of an IM system is the uncertaintiesrelated to R0

r. The robustness of the performance of the estimation algorithm to variations is testedby increasing the value of R0

r to twice the value assigned in the model, in the interval11 s6 t6 12 s. The response of the system is noted to be quite satisfactory with a velocity error ofenmð%Þ ¼ 1:4442

1500� 100 ¼ 0:096% and velocity estimation error of enmð%Þ ¼ 0:0122

1500� 100 ¼ 0:0081%

after a brief transient state. Once again, the largest estimation error in R0r takes place in the

Fig. 8. Trajectory of wsa and wsb.

M. Barut et al. / Energy Conversion and Management 46 (2005) 335–349 347

interval 5 s6 t6 7 s where the load torque is given a linear variation, while in all other intervals,very small errors are obtained in the transient and steady-state.

5.1.7. Uncertainties captured in constant tLAs mentioned in Sections 1 and 2, the EKF scheme also facilitates the indirect evaluation of

uncertainties that have the same variation as a state or parameter that is being estimated. Con-sidering this study, in which viscous friction ðFv ¼ bLxmÞ is taken into account in the modelrepresenting the system but not in the extended model, the estimate of tL as a constant state alsoshould include the viscous friction value once steady-state is reached. This fact can be demon-strated easily with the calculations below.

In the intervals 1 s6 t6 2 s and 10 s6 t6 13 s, during which both the velocity reference andload torque are given positive constant values, the error in the torque estimation is etL ¼ �0:571,and in the interval 4 s6 t6 5 s, where both the velocity reference and torque are given negativevalues, the error is found to be etL ¼ 0:571. The angular velocity in all these intervals isenm ¼ 1500� 0:2 rpm. Thus, for the value of the viscous friction coefficient bL ¼ 0:001 used in themodel

xmð1Þ ¼ 2p� nm=60 ¼ 2pð1500:2 ½rpm� þ 7:4� 10�5 ½rpm�Þ=60 ¼ 157:1006 ½rad=s�

Fv ¼ BLxmð1Þ

Fv ¼ 0:001� 157:1006 ¼ 0:1571006 ½Nm�

which is equal to etL , as expected.

348 M. Barut et al. / Energy Conversion and Management 46 (2005) 335–349

This fact should also be taken into consideration in evaluation of the load torque estimation.By inspecting the tL estimate, it can be observed that although with the linear variations andreversals of tL, some estimation error is caused in a relatively short transient duration, in theintervals with constant velocity reference and constant tL, this error is much lower, with thesubtraction of the Fv, from tL, yielding an error approximately equal to zero.

The applied algorithm has also kept the variation of flux magnitude in all intervals within theadmissible hysteresis band. Thus, with consideration of all the results, it can be observed that theexpected performance is attained.

6. Conclusion

In this study, an extended Kalman filter (EKF) algorithm is developed for the speed sensorlessdirect torque control (DTC) of induction motors. DTC requires accurate knowledge of theamplitude and angular position of the controlled flux (with respect to the stationary stator axis) inaddition to the angular velocity for the purpose of velocity control.

The major contribution of this study is the increased robustness towards uncertainties in therotor resistance and load torque, the effects of which are known to give rise to performancedeteriorations in such systems. This is achieved by an EKF algorithm that performs simulta-neous estimation of the rotor resistance and load torque as well as the stator flux componentsand the angular velocity. The performance of the algorithm is tested with 11 scenarios devel-oped by giving step type and linear variations to the load torque and angular velocity reference,while robustness to rotor resistance, R0

r, variation is tested with step type changes imposed onR0r.The system performance is observed to be quite good under step type variations and reversals in

the load–torque and step/linear changes and reversals in the angular velocity. The system has alsodemonstrated the expected robustness to step type variations forced on the R0

r, and acceptableerrors are obtained even with the linear variations and reversals of the load torque. The estimationof the load torque estimate, tL, as a constant state in this algorithm also accounts for the viscousfriction in this case, thereby improving the estimation performance.

Acknowledgements

This work was supported in part by the Istanbul Technical University Research Foundation.

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