Research Article Direct Torque Control of Sensorless ...

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Research Article Direct Torque Control of Sensorless Induction Machine Drives: A Two-Stage Kalman Filter Approach Jinliang Zhang, 1 Longyun Kang, 1 Lingyu Chen, 1 Boyu Yi, 1 and Zhihui Xu 2 1 School of Electric Power, South China University of Technology, Guangzhou, Guangdong 510640, China 2 Sunwoda Electronic Corporation Limited, Shenzhen 518108, China Correspondence should be addressed to Jinliang Zhang; [email protected] Received 27 May 2015; Revised 27 August 2015; Accepted 27 August 2015 Academic Editor: Mohamed Djemai Copyright © 2015 Jinliang Zhang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Extended Kalman filter (EKF) has been widely applied for sensorless direct torque control (DTC) in induction machines (IMs). One key problem associated with EKF is that the estimator suffers from computational burden and numerical problems resulting from high order mathematical models. To reduce the computational cost, a two-stage extended Kalman filter (TEKF) based solution is presented for closed-loop stator flux, speed, and torque estimation of IM to achieve sensorless DTC-SVM operations in this paper. e novel observer can be similarly derived as the optimal two-stage Kalman filter (TKF) which has been proposed by several researchers. Compared to a straightforward implementation of a conventional EKF, the TEKF estimator can reduce the number of arithmetic operations. Simulation and experimental results verify the performance of the proposed TEKF estimator for DTC of IMs. 1. Introduction High performance control and estimation techniques for induction machines (IMs) have been finding more and more applications with Blaschke’s well-known field oriented control (FOC) method [1]. To improve the dynamic response of instantaneous electromagnetic torque and simplicity in control structure, one such technique for induction machine control is that the direct torque control (DTC) method can provide accurate fast torque control [2]. is method has become increasingly popular for industrial applications due to the simplified control strategy and lower parameter dependence, in comparison with the FOC methods [3, 4]. For DTC of IMs, the method requires information on the position and amplitude of the controlled stator flux for speed control applications. In the conventional approach, the stator flux is obtained utilizing a search coil or through Hall effect sensors, whilst speed sensors like incremental encoders or resolvers are used to monitor rotor velocity [2]. ese unnecessarily increase hardware costs and the size of the control systems and degrade the reliability of the systems when encountering defective environments. So, sensorless DTC strategy has become the hot issue in research and drawn many researchers and engineers’ attention. Conventional approaches to sensorless DTC of IMs employ the method of stator flux and rotor velocity estima- tion by using a stator voltage model [5, 6]. is method has a large error in rotor velocity estimation, particularly in the low-speed operation range. Some recent studies conducting simultaneous stator flux and rotor velocity estimation for sensorless DTC technology include model reference adap- tive system (MRAS) [7], artificial neural networks (ANN) [8], sliding mode control (SMC) [9], extended Luenberger observer [10], and extended Kalman filter (EKF) [2, 11]. e model uncertainties and nonlinearities inherent to induction motors are well suited to the EKF’s stochastic nature [2]. Using this method, it is possible to make estimation of states whilst simultaneously performing identification of parameters in a short time [12–14], even taking measurement and system noises directly into system model. is explains why the EKF estimator is widely applicable in the sensor- less DTC of IMs. However, the EKF may suffer numerical problems and computational burden due to the high order of the mathematical models. is has generally limited Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 609586, 17 pages http://dx.doi.org/10.1155/2015/609586

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Research ArticleDirect Torque Control of Sensorless Induction Machine DrivesA Two-Stage Kalman Filter Approach

Jinliang Zhang1 Longyun Kang1 Lingyu Chen1 Boyu Yi1 and Zhihui Xu2

1School of Electric Power South China University of Technology Guangzhou Guangdong 510640 China2Sunwoda Electronic Corporation Limited Shenzhen 518108 China

Correspondence should be addressed to Jinliang Zhang boyuyi2108hotmailcom

Received 27 May 2015 Revised 27 August 2015 Accepted 27 August 2015

Academic Editor Mohamed Djemai

Copyright copy 2015 Jinliang Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

ExtendedKalman filter (EKF) has beenwidely applied for sensorless direct torque control (DTC) in inductionmachines (IMs) Onekey problem associated with EKF is that the estimator suffers from computational burden and numerical problems resulting fromhigh order mathematical models To reduce the computational cost a two-stage extended Kalman filter (TEKF) based solution ispresented for closed-loop stator flux speed and torque estimation of IM to achieve sensorless DTC-SVM operations in this paperThe novel observer can be similarly derived as the optimal two-stage Kalman filter (TKF) which has been proposed by severalresearchers Compared to a straightforward implementation of a conventional EKF the TEKF estimator can reduce the numberof arithmetic operations Simulation and experimental results verify the performance of the proposed TEKF estimator for DTC ofIMs

1 Introduction

High performance control and estimation techniques forinduction machines (IMs) have been finding more andmore applications with Blaschkersquos well-known field orientedcontrol (FOC) method [1] To improve the dynamic responseof instantaneous electromagnetic torque and simplicity incontrol structure one such technique for induction machinecontrol is that the direct torque control (DTC) methodcan provide accurate fast torque control [2] This methodhas become increasingly popular for industrial applicationsdue to the simplified control strategy and lower parameterdependence in comparison with the FOC methods [3 4]

For DTC of IMs the method requires information onthe position and amplitude of the controlled stator flux forspeed control applications In the conventional approach thestator flux is obtained utilizing a search coil or through Halleffect sensors whilst speed sensors like incremental encodersor resolvers are used to monitor rotor velocity [2] Theseunnecessarily increase hardware costs and the size of thecontrol systems and degrade the reliability of the systemswhen encountering defective environments So sensorless

DTC strategy has become the hot issue in research and drawnmany researchers and engineersrsquo attention

Conventional approaches to sensorless DTC of IMsemploy the method of stator flux and rotor velocity estima-tion by using a stator voltage model [5 6] This method hasa large error in rotor velocity estimation particularly in thelow-speed operation range Some recent studies conductingsimultaneous stator flux and rotor velocity estimation forsensorless DTC technology include model reference adap-tive system (MRAS) [7] artificial neural networks (ANN)[8] sliding mode control (SMC) [9] extended Luenbergerobserver [10] and extended Kalman filter (EKF) [2 11] Themodel uncertainties and nonlinearities inherent to inductionmotors are well suited to the EKFrsquos stochastic nature [2]Using this method it is possible to make estimation ofstates whilst simultaneously performing identification ofparameters in a short time [12ndash14] even taking measurementand system noises directly into system model This explainswhy the EKF estimator is widely applicable in the sensor-less DTC of IMs However the EKF may suffer numericalproblems and computational burden due to the high orderof the mathematical models This has generally limited

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 609586 17 pageshttpdxdoiorg1011552015609586

2 Mathematical Problems in Engineering

the applicability of the EKF to real-time signal processingproblems

In order to reduce the conventional EKF computationalalgorithm complexity the main objective of this paper is topresent a two-stage extended Kalman filter (TEKF) for statorflux rotor speed and electromagnetic torque estimation of asensorless direct torque controlled IM drive The proposedestimator is an effective implementation of EKF Followingthe two-stage filtering technique as given in [15] the TEKFcan be decomposed into two filters such as the modified biasfree filter and the bias filter Compared to the conventionalEKF the main advantage of the TEKF is the ability to reducethe computational complexity whilst maintaining the samelevel of performance

The paper is organized as follows In Section 2 thesensorless DTC-SVM strategy of IMs is introduced brieflyIn Section 3 according to the discrete model of IM a con-ventional EKF algorithm for estimating stator flux rotorspeed and position is designed In Section 4 TEKF are devel-oped by the two-stage filtering approach and its stability isanalyzed In Section 5 simulation and experimental resultsare discussed Finally a conclusion wraps up the paper

2 Principle of Sensorless DTC-SVM

As elaborated in [12] a dynamic mathematical model foran IM in the stationary (120572120573) reference frame is obtained asfollows

[[[[[[[[[[[[[[[[

[

∙

119868119904120572

∙

119868119904120573

∙

120595119904120572

∙

120595119904120573

∙

120579

∙

120596119903

]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[

[

minus(119877119904

120590119871119904

+1

120590119879119903

) minus119901120596119903

1

120590119871119904

119879119903

119901120596119903

120590119871119904

0 0

119901120596119903

minus(119877119904

120590119871119904

+1

120590119879119903

) minus119901120596119903

120590119871119904

1

120590119871119904

119879119903

0 0

minus119877119904

0 0 0 0 0

0 minus119877119904

0 0 0 0

0 0 0 0 0 1

0 0 0 0 0 0

]]]]]]]]]]]]]]

]

[[[[[[[[[[[

[

119868119904120572

119868119904120573

120595119904120572

120595119904120573

120579

120596119903

]]]]]]]]]]]

]

+

[[[[[[[[[[[[[

[

1

120590119871119904

0

01

120590119871119904

1 0

0 1

0 0

0 0

]]]]]]]]]]]]]

]

[119880119904120572

119880119904120573

] (1)

where 119868119904120572

119868119904120573

120595119904120572

120595119904120573

119880119904120572

and 119880119904120573

are the stator currentsflux linkages and voltages in the stationary reference frame119877119904

and 119871119904

are the stator winding resistance and inductancerespectively 120590 is the leakage or coupling factor (where 120590 =

1minus1198712119898

119871119903

119871119904

) 119871119898

and 119871119903

are themutual inductance and rotorinductance 119879

119903

is the rotor time constant (where 119879119903

= 119871119903

119877119903

)and 119877

119903

is the rotor resistance The rotor angular velocity120596119903

is measured in mechanical radians per second 120579 is themechanical rotor position and 119901 is the number of pole pairs

The behavior of an IM inDTC technique can be describedin terms of space vectors by the following equations writtenin the stator stationary reference frame

119904

= 119877119904

119904

+119889119904

119889119905

119903

= 119877119903

119903

+ 119895120596119903

119903

+119889119903

119889119905= 0

119904

= 119871119904

119904

+ 119871119898

119903

119903

= 119871119904

119903

+ 119871119898

119904

119879119890

=3

2119901

119871119898

119871119898

2 minus 119871119904

119871119903

100381610038161003816100381611990410038161003816100381610038161003816100381610038161003816119903

1003816100381610038161003816 sin 120575

(2)

where 120575 is known as load angle which is the angle betweenrotor flux

119903

and stator flux 119904

|119904

| and |119903

| are amplitudesof 119904

and 119903

respectively From (2) it can be seen that theinstantaneous electromagnetic torque control of IMs in DTC

is determined by changing the values of load angle 120575 while119904

and 119903

maintain the constant amplitude Accelerating thestator flux with respect to the rotor flux vector will increasethe electromagnetic torque and decelerating the same vectorwill decrease the electromagnetic torque [16]

The basic idea of DTC technique of IM is to controland acquire accurate knowledge on the stator flux and elec-tromagnetic torque to achieve high dynamic performanceDTC technique involves stator flux electromagnetic torqueestimators hysteresis controllers and a simple switchinglogic (switching tables) in order to reduce the electromagnetictorque and stator flux errors rapidly [17 18] Due to the factthat the universal voltage inverter has only eight availablebasic space vectors and only one voltage space vector ismaintained for the whole duration of the control period theconventional approach causes high ripples in stator flux cur-rent and electromagnetic torque accompanied by acousticalnoise To reduce the ripples of the stator flux linkage currentand electromagnetic torque in IM drives a modified DTCusing Space Vector Modulation (SVM) method called DTC-SVM is proposed in this paper The main difference betweenconventional DTC and DTC-SVM is that DTC-SVM has aSVMmodel and two PI controllers instead of switching tableand hysteresis controllers [19 20] The system structure ofDTC-SVM can be built and shown in Figure 1 This systemoperates at constant stator flux (below rated speed) FromFigure 1 the reference torque 119879lowast

119890

is generated from regulatedspeed proportional integral (PI) Δ119879

119890

is the torque errorbetween the reference torque 119879lowast

119890

and estimated torque 119890

Mathematical Problems in Engineering 3

120596rlowast

+ + +

++

+

minus minus

minus

minus

PIPITelowast

ΔTe Δ120575

Te

s

120596r

s120572

s120573

120579slowast

P

R

120595s120572

lowast

120595s120573

lowast

Δ120595s120572

lowast

Δ120595s120573

lowast

RsIs120572 RsIs120573

Is120572

Is120573

Vs120572lowast

Vs120573lowast

SVM

abc

120572120573

A

B

C

Pulses

IM

Stator flux speed and

torque estimator

Statorvoltage

cal

IU

IV

IW

Incremental encoder

r

Arctans120572

s120573

s120572

s120573

||120595lowasts

Figure 1 System diagram of the DTC-SVM scheme

In order to compensate this error the angle of stator fluxvector must be increased from 120579

119904

to 120579119904

+ Δ120575 as shown inFigure 2 where 120579

119904

is the phase angle of stator flux vectorthat can be obtained by the flux estimator and Δ120575 is theincrement of stator flux in the next sampling timeThereforethe required reference stator flux in polar form is given bylowast

119904

= |119904

|ang120579lowast119904

Define the stator flux deviations between

lowast

119904

and 119904

asΔ119904

then

Δ120595119904120572

=10038161003816100381610038161003816lowast

119904

10038161003816100381610038161003816cos (120579lowast

119904

) minus 119904120572

Δ120595119904120573

=10038161003816100381610038161003816lowast

119904

10038161003816100381610038161003816sin (120579lowast

119904

) minus 119904120573

(3)

where Δ119904120573

and Δ119904120572

are the stationary axis components ofstator fluxΔ

119904

and 119904120572

and 119904120573

are the stator flux componentsestimation In order tomake up for stator flux deviationsΔ120595

119904120572

and Δ120595119904120573

the reference stator voltages119880lowast119904120572

and119880lowast119904120573

should beapplied on the IM which can be expressed by

119880lowast

119904120572

= 119877119904

119868119904120572

+Δ120595119904120572

119879119904

119880lowast

119904120573

= 119877119904

119868119904120573

+Δ120595119904120573

119879119904

(4)

Substituting (3) into (4) (5) can be acquired

119880lowast

119904120572

= 119877119904

119868119904120572

+(10038161003816100381610038161003816lowast

119904

10038161003816100381610038161003816cos (120579lowast

119904

) minus 119904120572

)

119879119904

119880lowast

119904120573

= 119877119904

119868119904120573

+(10038161003816100381610038161003816lowast

119904

10038161003816100381610038161003816sin (120579lowast

119904

) minus 119904120573

)

119879119904

(5)

120573

Δ120575

120575

120579r120579s

120572

120579lowast

s

120595lowast

sΔ120595

s

120595s

120595r

Figure 2 Control of stator flux linkage

Based on the reference stator voltage components 119880lowast119904120572

and 119880lowast119904120573

the drive signal for inverter IGBTs can be obtainedthrough SVMmoduleThen both the electromagnetic torqueand the magnitude of stator flux are under control therebygenerating the reference stator voltage components

3 Conventional EKF Theory

By choosing the system state vector and estimated parametervector as 119883(119905) = [119868

119904120572

119868119904120573

120595119904120572

120595119904120573]119879and 119903(119905) = [120579 120596

119903]119879

respectively 119906(119905) = [119880119904120572

119880119904120573]119879 as the input vector and

4 Mathematical Problems in Engineering

119884(119905) = [119868119904120572

119868119904120573]119879 as the output vector the IM model is

described by the general nonlinear state space model

∙

119883 (119905) = 119860 (119905)119883 (119905) + 119861 (119905) 119906 (119905) + 119863 (119905) 119903 (119905)

∙

119903 (119905) = 119866 (119905) 119903 (119905)

119884 (119905) = 119862 (119905)119883 (119905)

(6)

with

119860 (119905)

=

[[[[[[[[[

[

minus(119877119904

120590119871119904

+1

120590119879119903

) minus119901120596119903

1

120590119871119904

119879119903

119901120596119903

120590119871119904

119901120596119903

minus(119877119904

120590119871119904

+1

120590119879119903

) minus119901120596119903

120590119871119904

1

120590119871119904

119879119903

minus119877119904

0 0 0

0 minus119877119904

0 0

]]]]]]]]]

]

119861 (119905) =[[[

[

1

120590119871119904

0 1 0

01

120590119871119904

0 1

]]]

]

119879

119863 (119905) = 0

119862 (119905) = [1 0 0 0

0 1 0 0]

119866 (119905) = [0 1

0 0]

(7)

Remark 1 Matrices 119862(119905) and 119866(119905) are not affected by uncer-tainties

Remark 2 Matrix119860(119905) is time-varying because it depends onthe rotor speed 120596

119903

For digital implementation of estimator on a microcon-

troller a discrete timemathematicalmodel of IMs is requiredThese equations can be obtained from (6)

119883119896+1

= 119860119896

119883119896

+ 119861119896

119906119896

+ 119863119896

119903119896

119903119896+1

= 119866119896

119903119896

119884119896

= 119862119896

119883119896

(8)

The solution of nonhomogenous state equations (6) sat-isfying the initial condition119883(119905)|

119905=1199050= 119883(119905

0

) is

119883 (119905) = 119890119860(119905minus1199050)119883(119905

0

) + int119905

1199050

119890119860(119905minus120591)

119861119906 (120591) 119889120591 (9)

Integrating from 1199050

= 119896119879119904

to 119905 = (119896 + 1)119879119904

we can obtain that

119883((119896 + 1) 119879119904

) = 119890119860119879119904119883(119896119879

119904

)

+ int(119896+1)119879119904

119896119879

119890119860((119896+1)119879119904minus120591)119861119889120591119906 (119896119879

119904

)

(10)

The above equations lead to

119860119896

= 119890119860119879119904

119861119896

= 119860minus1

(119890119860119879119904 minus 119868) 119861

(11)

In the same way

119866119896

= 119890119866119879119904 (12)

Tolerating a small discretization error a first-order Taylorseries expansion of the matrix exponential is used

119860119896

= 119890119860119879119904 asymp 119860119879

119904

+ 119868

119866119896

= 119890119866119879119904 asymp 119866119879

119904

+ 119868

119861119896

= 119860minus1

(119890119860119879119904 minus 119868) 119861 asymp 119879

119904

119861

119863119896

= 0

(13)

with

119860119896

=

[[[[[[[[[

[

minus(119877119904

119879119904

120590119871119904

+119879119904

120590119879119903

) + 1 minus119901120596119903

119879119904

119879119904

120590119871119904

119879119903

119901120596119903

119879119904

120590119871119904

119901120596119903

119879119904

minus(119877119904

119879119904

120590119871119904

+119879119904

120590119879119903

) + 1 minus119901120596119903

119879119904

120590119871119904

119879119904

120590119871119904

119879119903

minus119877119904

119879119904

0 1 0

0 minus119877119904

119879119904

0 1

]]]]]]]]]

]

G119896

= [1 119879119904

0 1]

Mathematical Problems in Engineering 5

119861119896

=[[[

[

119879119904

120590119871119904

0 119879119904

0

0119879119904

120590119871119904

0 119879119904

]]]

]

119879

119862119896

= [1 0 0 0

0 1 0 0]

119863119896

= [0 0

0 0]

(14)

Based on discretized IM model a conventional EKFestimator is designed for estimation of stator flux currentelectromagnetic torque and rotor speed of IM for sensorlessDTC-SVM operations Treating119883

119896

as the full order state and119903119896

as the augmented system state the state vector is chosen tobe 119883119886119896

= [119883119896

119903119896]119879 119906119896

= [119880119904120572

119880119904120573]119879 and 119884

119896

= [119868119904120572

119868119904120573]119879 are

chosen as input and output vectors because these quantitiescan be easily obtained from measurements of stator currentsand voltage construction usingDC link voltage and switchingstatus Considering the parameter errors and noise of systemthe discrete time state space model of IMs in the stationary(120572120573) reference frame is described by

119883119886

119896+1

= 119860119896

119883119886

119896

+ 119861119896

119906119896

+ 119908119896

119884119896+1

= 119862119896

119883119886

119896

+ V119896

(15)

with

119860119896

= [119860119896

119863119896

0 119866119896

]

119861119896

= [119861119896

0]

119862119896

= [119862119896

0]

119879

119908119896

= [119908119909119896

119908119903119896

]

(16)

The system noise119908119896

and measurement noise V119896

are whiteGaussian sequence with zero-mean and following covariancematrices

119864[[[

[

[[

[

119908119909

119896

119908119903119896

V119896

]]

]

[[[

[

119908119909119895

119908119903119895

V119895

]]]

]

119879

]]]

]

=[[

[

119876119909119896

0 0

0 119876119903119896

0

0 0 119877119896

]]

]

120575119896119895

(17)

where 119876119909119896

gt 0 119876119903119896

gt 0 119877119896

gt 0 and 120575119896119895

is theKronecker delta The initial states 119883

0

and 1199030

are assumed tobe uncorrelated with the zero-mean noises 119908119909

119896

119908119903119896

and V119896

The initial conditions are assumed to be Gaussian randomvariables119883

0

and 1199030

that are defined as follows

119864((1198830

minus 119883lowast

0

) (1198830

minus 119883lowast

0

)119879

) = 119875119909

0

119864 (1198830

) = 119883lowast

0

119864 (1199030

) = 119903lowast

0

119864((1199030

minus 119903lowast

0

) (1199030

minus 119903lowast

0

)119879

) = 119875119903

0

119864 ((1198830

minus 119883lowast

0

) (1199030

minus 119903lowast

0

)119879

) = 119875119909119903

0

(18)

The overall structure of the EKF is well-known byemploying a two-step prediction and correction algorithm[13] Hence the application of EKF filter to the state spacemodel of IM (15) is described by

119883119886

119896|119896minus1

= 119860119896minus1

119883119886

119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

(19)

119875119896|119896minus1

= 119865119896minus1

119875119896minus1|119896minus1

119865119879

119896minus1

+ 119876119896minus1

(20)

119870119896

= 119875119896|119896minus1

119867119879

119896

(119867119896

119875119896|119896minus1

119867119879

119896

+ 119877)minus1

(21)

119875119896|119896

= 119875119896|119896minus1

minus 119870119896

119867119896

119875119896|119896minus1

(22)

119883119886

119896|119896

= 119883120572

119896|119896minus1

+ 119870119896

(119884119896

minus 119862119896

119883119886

119896|119896minus1

) (23)

with

119865119896minus1

= [119865119896minus1

119864119896minus1

0 119866119896minus1

]

119876 (sdot) = [119876119909 (sdot) 0

0 119876119903 (sdot)]

119865119896minus1

=120597

120597119883(119860119896minus1

119883119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1

) = 119860119896

119864119896minus1

=120597

120597119903(119860119896minus1

119883119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1

)

= [0 0 0 0

1199011 1199012 0 0]

119879

6 Mathematical Problems in Engineering

1199011 = (minus119901119894119904119887[119896minus1|119896minus1]

+119901

120590119871119904

120595119904120573[119896minus1|119896minus1]

)119879119904

1199012 = (119901119894119904120572[119896minus1|119896minus1]

minus119901

120590119871119904

120595119904120572[119896minus1|119896minus1]

)119879119904

119867119896

= [1198671

119896

1198672

119896

] =120597

120597119883119886(119862119896

119883119886

)

1198671

119896

= 119862119896

1198672

119896

= 0

119875 (sdot) = [119875119909 (sdot) 119875119909119903 (sdot)

(119875119909119903 (sdot))119879

119875119903 (sdot)]

(24)

4 The Two-Stage Extended Kalman Filter

41The TEKFAlgorithm Asmentioned in conventional EKFestimator previously the memory and computational costsincrease with the augmented state dimension Consideringsampling time is very small only high performance micro-controller can qualify for this work Hence the conventionalEKF algorithm may be impractical to implement The extracomputation of 119875

119909119903(sdot) terms leads to this computationalcomplexityTherefore we can reduce the computational com-plexity from application point of view if the 119875119909119903(sdot) terms canbe eliminated In this section a two-stage extended Kalmanfilter without explicitly calculating 119875119909119903(sdot) terms is discussed

Following the same approach as given in [15] the TEKFis decomposed into two filters such as the modified bias freefilter and the bias filter by applying the following two-stage119880-119881 transformation

119883119886

119896|119896minus1

= 119879 (119880119896

)119883119886

119896|119896minus1

119875119896|119896minus1

= 119879 (119880119896

) 119875119896|119896minus1

119879 (119880119896

)119879

119870119896

= 119879 (119881119896

)119870119896

119875119896|119896

= 119879 (119881119896

) 119875119896|119896

119879 (119881119896

)119879

119883119886

119896|119896

= 119879 (119881119896

)119883119886

119896|119896

(25)

where

119883119886

119896(sdot)

= [119883119896(sdot)

119903119896(sdot)

]

119870119896

= [119870119909

119896

119870119903

119896

]

119875119896(sdot)

= [

[

119875119909

119896(sdot)

0

0 119875119903

119896(sdot)

]

]

119879 (119880119896

) = [119868 119880119896

0 119868]

(26)

The main advantage of using the 119879(119880119896

) transformation isthat the inverse transformation 119879

minus1

(119880119896

) = 119879(minus119880119896

) involvesonly a change of sign Two blending matrices 119880

119896

and 119881119896

are defined by 119880119896

= 119875119909119903119896|119896minus1

(119875119903119896|119896minus1

)minus1 and 119881119896

= 119875119909119903119896|119896

(119875119903119896|119896

)minus1respectively Using characteristic of 119879(119880

119896

) (25) become

119883119886

119896|119896minus1

= 119879 (minus119880119896

)119883119886

119896|119896minus1

119875119896|119896minus1

= 119879 (minus119880119896

) 119875119896|119896minus1

119879 (minus119880119896

)119879

119870119896

= 119879 (minus119881119896

)119870119896

119875119896|119896

= 119879 (minus119881119896

) 119875119896|119896

119879 (minus119881119896

)119879

119883119886

119896|119896

= 119879 (minus119881119896

)119883119886

119896|119896

(27)

And the following relationships are obtained from (25)

119875119909

119896|119896minus1

= 119875119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

(28)

119875119903

119896|119896minus1

= 119875119903

119896|119896minus1

(29)

119875119909119903

119896|119896minus1

= 119880119896

119875119903

119896|119896minus1

(30)

119875119909

119896|119896

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

(31)

119875119903

119896|119896

= 119875119903

119896|119896

(32)

119875119909119903

119896|119896

= 119881119896

119875119903

119896|119896

(33)

Based on two-step iterative substitution method of [15]the transformed filter expressed by (27) can be recursivelycalculated as follows

119883119886

119896|119896minus1

= 119879 (minus119880119896

)

sdot (119860119896minus1

119879 (119881119896minus1

)119883119886

119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

)

(34)

119875119896|119896minus1

= 119879 (minus119880119896

)

sdot (119876119896minus1

+ 119865119896minus1

119879 (119881119896minus1

) 119875119896minus1|119896minus1

119879 (119881119896minus1

)119879

119865119879

119896minus1

)

sdot 119879 (minus119880119896

)119879

(35)

119870119896

= 119879 (119880119896

minus 119881119896

) 119875119896|119896minus1

119879 (119880119896

)119879

sdot 119867119879

119896

(119867119896

119879 (119880119896

) 119875119896|119896minus1

119879 (119880119896

)119879

119867119879

119896

+ 119877)minus1

(36)

119875119896|119896

= (119879 (119880119896

minus 119881119896

) minus 119870119896

119867119896

119879 (119880119896

))

sdot 119875119896|119896minus1

119879 (119880119896

minus 119881119896

)119879

(37)

119883119886

119896|119896

= 119870119896

(119884119896

minus 119862119896

119879 (119880119896

)119883119886

119896|119896minus1

) + 119879 (119880119896

minus 119881119896

)

sdot 119883119886

119896|119896minus1

(38)

Mathematical Problems in Engineering 7

Using (35) (37) and the block diagonal structure of 119875(sdot)

thefollowing relations can be obtained

0 = 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

minus 119880119896

119876119903

119896minus1

minus 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

0 = 119880119896

minus 119881119896

minus 119870119909

119896

119878119896

(39)

where 119880119896

and 119878119896

are defined as

119880119896

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119866minus1

119896minus1

(40)

119878119896

= 1198671

119896

119880119896

+ 1198672

119896

(41)

The above equations lead to

119880119896

119875119903

119896|119896minus1

= 119880119896

(119875119903

119896|119896minus1

minus 119876119903

119896

) (42)

119880119896

= 119880119896

(119868 minus 119876119903

119896

(119875119903

119896|119896minus1

)minus1

) (43)

119881119896

= 119880119896

minus 119870119909

119896

119878119896

(44)

Define the following notation

119860119896minus1

119879 (119881119896minus1

) = [119860119896minus1

119860119896minus1

119881119896minus1

+ 119864119896minus1

0 119866119896minus1

] (45)

The equations of themodified bias free filter and the bias filterare acquired by the next steps

Expanding (34) we have

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119898119896minus1

(46)

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

(47)

where

119898119896minus1

= (119860119896minus1

119881119896minus1

+ 119863119896minus1

minus 119880119896

119866119896minus1

) 119903119896minus1|119896minus1

(48)

Expanding (35) we have

119875119909

119896|119896minus1

= (119864119896minus1

+ 119860119896minus1

119881119896minus1

minus 119880119896

119866119896minus1

) 119875119903

119896minus1|119896minus1

lowast (119864119896minus1

+ 119860119896minus1

119881119896minus1

minus 119880119896

119866119896minus1

)119879

+ 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

+ 119880119896

119876119903

119896

119880119879

119896

(49)

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896

(50)

Then using (40) (43) and (47) (49) can be written as

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

(51)

where

119876119909

119896

= 119876119909

119896

+119872119896

(119872119896

119876119903

119896

)119879

(52)

Expanding (38) and using (41) and (44) we have

119883119896|119896

= 119883119896|119896minus1

+ (119880119896

minus 119881119896

) 119903119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

)

119903119896|119896

= 119903119896|119896minus1

+ 119870119903

119896

(119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

)

(53)

Then

119883119896|119896

= 119870119909

119896

120578119909

119896

+ 119883119896|119896minus1

(54)

where

119878119896

= 119862119896

119880119896

(55)

120578119909

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

+ (119878119896

minus 119878119896

) 119903119896|119896minus1

(56)

Expanding (36) and using (41) we have

119870119903

119896

= 119875119903

119896|119896minus1

119878119879

119896

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)minus1

119870119909

119896

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)

= 119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

(57)

Then

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(58)

Expanding (37) we have

119875119903

119896|119896

= (119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119875119909

119896|119896

= 119875119909

119896|119896minus1

+ (119880119896

minus 119881119896

) 119875119903

119896|119896minus1

(119880119879

119896

minus 119881119879

119896

)

minus (119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

(119880119879

119896

minus 119881119879

119896

))

(59)

Then using (41) and (44)

119875119909

119896|119896

= (119868 minus 119870119909

119896

1198671

119896

)119875119909

119896|119896minus1

(60)

Finally using (25) the estimated value of original state(119894119904120572

119904120573

119904120572

119904120573

) can be obtained by sum of the state119883withthe augmented state 119903

119896|119896minus1

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

(61)

119896|119896

= 119883119896|119896

+ 119881119896

119903119896|119896

(62)

Moreover the unknown parameter 119903( 119903

) is defined as

119903119896|119896minus1

= 119903119896|119896minus1

(63)

119903119896|119896

= 119903119896|119896

(64)

8 Mathematical Problems in Engineering

Based on the above analysis the TEKF can be decoupledinto two filters such as the modified bias free filter and biasfilter The modified bias filter gives the state estimation 119883

119896|119896

and the bias filter gives the bias estimate 119903

119896|119896

The correctedstate estimate 119883119886

119896|119896

(119896|119896

119903119896|119896

) of the TEKF is obtained fromthe estimates of the two filters and coupling equations119880

119896

and119881119896

[21] The modified bias free filter is expressed as follows

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119898119896minus1

(65)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

(66)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(119873119896

)minus1

(67)

119875119909

119896|119896

= (119868 minus 119870119909

119896

1198671

119896

)119875119909

119896|119896minus1

(68)

120578119909

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

+ (119878119896

minus 119878119896

) 119903119896|119896minus1

(69)

119883119896|119896

= 119870119909

119896

120578119909

119896

+ 119883119896|119896minus1

(70)

and the bias filter is

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

(71)

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896

(72)

119870119903

119896

= 119875119903

119896|119896minus1

119878119879

119896

(119873119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)minus1

(73)

119875119903

119896|119896

= (119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

(74)

120578119903

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

(75)

119903119896|119896

= 119903119896|119896minus1

+ 119870119903

119896

120578119903

119896

(76)

with the coupling equations

119878119896

= 1198671

119896

119880119896

+ 1198672

119896

(77)

119880119896

= 119880119896

(119868 minus 119876119903

119896

(119875119903

119896|119896minus1

)minus1

) (78)

119880119896

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119866minus1

119896minus1

(79)

119881119896

= 119880119896

minus 119870119909

119896

119878119896

(80)

119898119896minus1

= (119860119896minus1

119881119896minus1

+ 119863119896minus1

minus 119880119896

119866119896minus1

) 119903119896minus1|119896minus1

(81)

119876119909

119896

= 119876119909

119896

+ 119880119896

(119880119896

119876119903

119896

)119879

(82)

119873119896

= 1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

(83)

The initial conditions of TEKF algorithm are establishedwith the initial conditions of a classical EKF (119883

0|0

1199030|0

1198751199090|0

1198751199091199030|0

1198751199030|0

) so that

1198810

= 119875119909119903

0|0

(119875119903

0|0

)minus1

1198830|0

= 1198830|0

minus 1198810

1199030|0

1199030|0

= 1199030|0

119875119909

0|0

= 119875119909

0|0

minus 1198810

119875119903

0|0

119881119879

0

119875119903

0|0

= 119875119903

0|0

(84)

According to variables of full order filter119883 (1198831

1198832

1198833

1198834

)the stator flux and torque estimators for DTC-SVM ofFigure 1 are then given by

119904120572

= 1198831

119904120573

= 1198832

119904120572

= 1198833

119904120573

= 1198834

100381610038161003816100381610038161003816120595119904

100381610038161003816100381610038161003816= radic

2

119904120572

+ 2

119904120573

119904

= arctan119904120573

119904120572

119890

=3

2119901 (119904120572

119868119904120573

minus 119904120573

119868119904120572

)

(85)

where 119901 is the pole pairs of IM The estimated speed andelectromagnetic torque obtained from the TEKF observer areused to close the speed and torque loop to achieve sensorlessoperations

42 The Stability and Parameter Sensitivity Analysis ofthe TEKF

Theorem 3 The discrete time conventional extended Kalmanfilter (19)ndash(23) is equivalent to the two-stage extern Kalmanfilter (see (61)sim(83))

Proof Before proving the theorem the following five rela-tionships are needed

(1) Using (72) and (78)

119880119896+1

119866119896

119875119903

119896|119896

119866119879

119896

= 119880119896

119875119903

119896|119896minus1

(86)

(2) Using (67) and (73)

119870119909

119896

119872119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

(87)

119870119903

119896

119872119896

= 119875119903

119896|119896minus1

(119878119896

)119879

(88)

where

119872119896

= 1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

+ 119877119896

(89)

Mathematical Problems in Engineering 9

(3) Using (20) we have

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896minus1

(90)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119864119896minus1

(119875119909119903

119896minus1|119896minus1

)119879

119860119879

119896minus1

+ 119860119896minus1

119875119909119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119876119909

119896minus1

(91)

119875119909119903

119896|119896minus1

= 119860119896minus1

119875119909119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

(92)

(4) Using (21)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(93)

119870119903

119896

= (119875119909119903

119896|119896minus1

)119879

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(94)

(5) Using (22)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

(95)

119875119909119903

119896|119896

= 119875119909119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909119903

119896|119896minus1

(96)

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896

1198671

119896

119875119909119903

119896|119896minus1

(97)

By inductive reasoning suppose that at time 119896 minus 1the unknown parameter

119896minus1

and estimated state 119896minus1

areequal to the parameter 119903

119896minus1

and state 119883119896minus1

of the controlsystem respectively we show that TEKF is equivalent to theconventional EKF because these properties are still true attime 119896

Assume that at time 119896 minus 1

119883119896minus1|119896minus1

= 119896minus1|119896minus1

119903119896minus1|119896minus1

= 119903119896minus1|119896minus1

119875119909

119896minus1|119896minus1

= 11987511

119896minus1|119896minus1

119875119909119903

119896minus1|119896minus1

= 11987512

119896minus1|119896minus1

119875119903

119896minus1|119896minus1

= 11987522

119896minus1|119896minus1

(98)

where [ 119875119909119875

119909119903

(119875

119909119903)

119879119875

119903 ] and [ 11987511119875

12

(119875

12)

119879119875

22 ] represent the variance-covariance matrices of the system and estimated variablesrespectively

From (19) we have

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119863119896

119903119896minus1|119896minus1

+ 119861119896

119906119896minus1

(99)

Then using (98) (41) (62) (79) (81) (71) and (61)

119883119896|119896minus1

= 119860119896minus1

(119883119896minus1|119896minus1

+ 119881119896minus1

119903119896minus1|119896minus1

)

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

minus 119898119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

= 119896|119896minus1

(100)

Using (19) (71) (98) (63) and (64) we have

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119903119896|119896minus1

(101)

Using (91) (98) (78) (66) (79) (82) (86) and (72) we obtain

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896minus1

+ 119880119896

119866119896

119875119903

119896minus1|119896minus1

119866119879

119896

119880119879

119896

= 119875119909

119896|119896minus1

+ 119880119896

(119880119896

119875119903

119896|119896minus1

minus 119880119896

119876119903

119896minus1

)119879

= 119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

= 11987511

119896|119896minus1

(102)

Using (90) (98) (72) (32) (71) (29) and (97) we obtain

119875119903

119896|119896minus1

= 11987522

119896minus1|119896minus1

(103)

Using (92) (98) (33) (32) (79) (86) and (91)

119875119909119903

119896|119896minus1

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896minus1

119875119903

119896minus1|119896minus2

= 11987512

119896|119896minus1

(104)

Using (93) (101) (55) (73) (67) (80) and (87)

119870119909

119896

= (119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

) (1198671

119896

)119879

sdot (1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ (119881119896

+ 119870119909

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

(1198671

119896

)119879

)119880minus1

119896

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

)119880minus1

119896

+ 119881119896

119875119903

119896|119896minus1

119878119879

119896

119880minus1

119896

= 119870119909

119896

+ 119881119896

119870119903

119896

(105)

Using (94) (30) and (88) we obtain

119870119903

119896

= (119875119903

119896|119896minus1

)119879

119878119879

119896

119880minus1

119896

= 119870119903

119896|119896minus1

(106)

10 Mathematical Problems in Engineering

Next wewill show that (98) holds at time 119896 From (23)we have

119883119896|119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

)

= 119883119896|119896minus1

+ 119870119909

119896

119903119896

(107)

Then using (61) and (105) the above equation can be writtenas

119883119896|119896

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

+ (119870119909

119896

+ 119881119896

119870119903

119896

) 119903119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 1198671

119896

119883119896|119896minus1

)

+ (119880119896

minus 119870119909

119896

119878119896

) 119903119896|119896minus1

+ 119881119896

119870119903

119896

119903119896

= 119883119896|119896

+ 119881119896

119903119896|119896

= 119896|119896

(108)

Using (95) (105) (102) and (77)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

(109)

Then using (80) (68) (74) and (31) we obtain

119875119909

119896|119896

= 119875119909

119896|119896

+ 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

+ 119881119896

(119875119903

119896|119896

119878119879

119896

(119870119909

119896

)119879

minus 119870119903

119896

1198671

119896

119875119909

119896|119896minus1

)

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

= 11987511

119896|119896

(110)

Using (96) (30) (28) (105) and (80)

119875119909119903

119896|119896

= 119880119896

119875119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

1198671

119896

119880119896

minus 119881119896

119870119903

119896

1198671

119896

119880119896

)119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

119875119903

119896|119896

= 11987512

119896|119896

(111)

Using (97) (106) (95) (29) and (30) we obtain

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896|119896minus1

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119868 minus 119870119903

119896|119896minus1

119878119896

) 119875119903

119896|119896minus1

= 11987522

119896|119896

(112)

Table 1 Kalman estimation arithmetic operation requirement forthe conventional EKF structure

Number of multiplications(119899 = 6119898 = 2 and 119902 = 2)

Number of additions(119899 = 6119898 = 2 and 119902 = 2)

119860119896

119861119896

and 119862

119896

Function of system (9) Function of system (3)

119883119886

119896|119896minus1

1198992 + 119899119902 (48) 1198992 + 119899119902 minus 119899 (42)119875119896|119896minus1

21198993 (432) 21198993 minus 1198992 (396)119883119886

119896|119896minus1

2119899119898 (24) 2119899119898 (24)

119870119896

1198992

119898 + 21198991198982

+ 1198983 (168) 119899

2

119898 + 21198991198982

+ 1198983

minus 2119899119898

(104)119875119896|119896

1198992119898 + 1198993 (288) 1198992119898 + 1198993 minus 1198992 (252)Total 960 818

Finally we show that (98) holds at time 119896 = 0 This can beverified by the initial conditions of TEKF algorithm

43 Numerical Complexity of the Algorithm Tables 1 and2 show the computational effort at each sample time bythe conventional EKF algorithm and TEKF (where roughmatrix-based implementation is used) in which as definedabove 119899 is the dimension of the state vector 119883

119896

119898 is thedimension of the measurement 119884

119896

119902 is the input vector 119880119896

and 119901 is the dimension of the parameter 119903

119896

The total numberof arithmetic operations (additions and multiplications) persample time of the TEKF is 1314 compared with 1778 for arough implementation of a conventional EKF which meansthe operation cost can reduce by 26

5 Simulation and Experimental Results

51 Simulation Results To test the feasibility and perfor-mance of the TEKF method the sensorless DTC-SVM tech-nique for IM drives described in Section 2 is implementedin MATLABSIMULINK environment The values of theinitial state covariance matrices 119875

0

119876 and 119877 have a greatinfluence on the performance of the estimation methodThe diagonal initial state covariance matrix 119875

0

representsvariances or mean-squared errors in the knowledge of theinitial conditions Matrix 119876 gives the statistical descriptionof the drive system Matrix 119877 is related to measured noiseThey can be obtained by considering the stochastic propertiesof the corresponding noises However a fine evaluation ofthe covariance matrices is very difficult because they areusually not known In this paper tuning the initial values ofcovariance matrices 119875

0

119876 and 119877 is using particular criteria[22] to achieve steady-state behaviors of the relative estimatedstates as given by

119876 = diag 20 20 1119890 minus 6 1119890 minus 6 10 10

1198750

= diag 01 01 05 05 1 1

119877 = diag 01 01

(113)

Mathematical Problems in Engineering 11

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000Sp

eed

(rpm

)

(a) Speed estimation

02 04 06 08 10Times (s)

minus002

0

002

004

006

008

01

012

014

016

018

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF

Theta (real)Theta (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7Po

sitio

n (r

ad)

(d) TEKF rotor position error

Current (real)Current (TEKF)

Curr

entIs120572

(A)

02 04 06 08 10Times (s)

minus12

minus8

minus4

0

4

8

12

(e) Real stator current 119868119904120572

and estimation (TEKF)

Current (real)Current (TEKF)

minus12

minus8

minus4

0

4

8

12

Curr

entIs120573

(A)

02 04 06 08 10Times (s)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 3 Continued

12 Mathematical Problems in Engineering

Flux (EKF)Flux (TEKF)

minus05 0 05 1minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(g) Stator flux estimation by TEKF and EKF

02 04 06 08 10Times (s)

minus2

0

2

4

6

8

10

12

14

16

Ψs

(Wb)

times10minus3

(h) Difference Stator flux estimation between TEKF and EKF

Figure 3 Simulation results for parameters estimation

Table 2 Kalman estimation arithmetic operation requirement for the TEKF structure

Number of multiplications(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

Number of additions(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

119860119896

119862119896

119864119896

1198671119896

1198672119896

119861119896

and119863119896

Function of system (25) Function of system (11)119883119896|119896minus1

1198992 + 119899119902 (24) 1198992 + 119899119902 (24)119875119909

119896|119896minus121198993 (128) 21198993 minus 1198992 (112)

119870119909

119896|1198961198992

119898 + 1198991198982 (48) 119899

2

119898 + 1198991198982

minus 2119899119898 (32)119875119909

119896|1198961198992

119898 + 1198993 (96) 119899

2

119898 + 1198993

minus 1198992 (80)

119883119896|119896

2119899119898 + 119898119901 (20) 2119899119898 + 2119898119901 (24)119903119896|119896minus1

1199012 (4) 1199012 minus 119901 (2)119875119903

119896|119896minus121199013 (16) 21199013 minus 1199012 (12)

119870119903

119896|11989631199012119898 + 1199011198982 (32) 31199012119898 + 1199011198982 + 1198982 minus 4119901119898 (20)

119875119903

119896|1198961199013 + 1199012119898 (16) 1199013 + 1199012119898 minus 1199012 (12)

119903119896|119896

2119898119901 + 119899119898 (12) 2119898119901 + 119899119898 (16)119878119896

119898119899119901 (16) 119899119898119901 (16)119880119896minus1

1198992119901 + 1198991199012 (56) 1198992119901 + 1198991199012 minus 119899119901 (48)119881119896

119899119901119898 (16) 119899119901 (8)119880119896minus1

21198991199012 (32) 21198991199012 (32)119898119896minus1

1198992

119901 + 1198991199012

+ 119899119901 (56) 1198992

119901 + 1198991199012

+ 119899119901 minus 119899 (52)119876119909

119896minus121198992

119901 (64) 21198992

119901 + 1198991199012 (64)

119878119896

119898119899119901 (16) 119898119899119901 minus 119898119901 (12)119873119896

21198981198992 (32) 21198981198992 minus 1198982 (60)Total 688 626

In the simulation a comparison is made to verify theequivalence of EKF and TEKF Real-time parameters esti-mated by TEKF are used to formulate the closed loop suchas rotor speed stator flux and electromagnetic torque Theestimations obtained by EKF algorithm are not included in

the sensorless DTC-SVM strategy and only evaluated in openloop A step reference speed was applied to the simulation

The machine is accelerated from 0 rpm to 1000 rpm at0 s and the torque load is set to 4N The simulation resultsof parameter estimation are shown in Figure 3 Figures 3(a)

Mathematical Problems in Engineering 13

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000

Spee

d (r

pm)

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus03

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Theta (real)Theta (TEKF)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

02 04 06 08 10Times (s)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)

Figure 4 Simulation results with parameter variation (119877119903

= 05119877119903nom)

and 3(c) represent the performance of the speed and rotorposition tracking capabilities of the control system withTEKF and EKF respectively Figures 3(e) 3(f) and 3(g)represent the estimated stator current and flux they showthat ripples are significantly suppressed due to the SVMmodulation scheme Figures 3(b) 3(d) and 3(h) show thatfor variations of speed reference the rotor speed rotorposition and stator flux errors between the two observers arevery little It is verified that the two observers are equivalentThe difference between two estimators is caused by accuracyloss in TEKF which uses more calculation steps

In order to further verify the performance of TEKFagainst model-plant parameter mismatches and the equiv-alence of two observers the change in rotor resistance isconsidered Rotor resistance will increase due to temperaturerise while the motor is running To simulate this conditionthe rotor resistance in TEKF and EKF is increased to 200comparedwith the normal value which is equivalent to a 50decrease in the actual rotor resistance The machine is stillaccelerated from 0 rpm to 1000 rpm at 0 s and the torque is setto 4N Figure 4 shows that for variation of rotor resistancethe steady-state speed and rotor position errors are negligible

14 Mathematical Problems in Engineering

(a)

Computer

380V

TXRX

120596r

Incrementalencoder

Udc

MicrocomputerDSP TMS3206713

Drive signalCurrentsensor

Voltagesensor

Voltagesensor

IM MC

ISU ISV

VSU VSW VSV

(b)

Figure 5 Complete drive system (a) Picture of experimental setup (b) Functional block diagram of the experimental setup

and the difference of the speed and rotor position estimationsbetween the two observers is rather null

52 Experimental Results The overall experimental setup isshown in Figure 5 and the specifications and rated parametersof the IM controller and inverter are listed in Table 3 Inthe experimental hardware an Expert3 control system fromMyway company and a three-phase two-pole 15 kW IM areappliedThe IM is mechanically coupled to a magnetic clutch(MC) which provides rated torque even at very low speedThe main processor in Expert3 control system is a floatingpoint processor TMS320C6713 with a max clock speed of225MHz All the algorithms including TEKF EKF DTCalgorithm and some transformation modules are imple-mented in TMS320C6713 with 100120583s sampling time and dataacquisition of the parameter estimations measured variablesand their visualization are realized on the cockpit provided byPEView9 software Insulated Gate Bipolar Transistor (IGBT)module is driven by the PWM signal with a switchingfrequency of 10 kHz and 2 120583s dead time The stator currentsare measured via two Hall effect current sensors The rotorangle and speed of IM are measured from an incrementalencoder with 2048 pulses per revolution

This experiment test is here to testify the performanceof TEKF and demonstrate that the two estimators aremathematically equivalent The machine is accelerated from600 rpm to 1000 rpm and 4N torque load is set Theexperimental results of parameter estimation based on twoobservers are given in Figures 6 and 7 Figures 6(a) and 6(c)show that the TEKF still has a good tracking performance ofthe speed and rotor position in experiment Figures 6(d) 6(e)and 6(f) illustrate stator flux and stator current estimationrobustness Figures 6(b) 6(g) and 6(f) referring to thedifference in speed and stator current estimations given by

Table 3 Specification of induction motor and inverter

Induction motor ValueNominal torque 10NmNominal voltage 380VRotor resistance 119877

119903

25ΩStator resistance 119877

119904

36ΩStator inductances 119871

119904

0301HRotor inductances 119871

119903

0302HMutual inductances 119871

119898

0273HPole pairs 2Invertercontroller ValueSwitching device 1000V 80A IGBTControl cycle time 100 120583sMain CPU DSP TMS320C6713 225MHz

the two observers are still small These experiment resultsprove that the two estimators are mathematically equivalentFigure 7 shows the speed and rotor position estimationsbased onTEKFandEKF for a 50decrease of rotor resistance(the same as the simulation) As expected the steady errorof the TEKF and the difference in speed and rotor positionestimations are still tiny Robustness of TEKF is verified

6 Conclusion

Themajor shortcoming of the conventional EKF is numericalproblems and computational burden due to the high orderof the mathematical models This has generally limited thereal-time digital implementation of the EKF for industrialfield So in this study a novel extended Kalman filter

Mathematical Problems in Engineering 15

Speed (real)Speed (EKF)Speed (TEKF)

Spee

d (r

min

)

02 04 06 08 10t (02 sgrid)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimationSp

eed

(rm

in)

minus005

0

005

01

015

02

025

03

035

04

045

02 04 06 08 10t (02 sgrid)

(b) Difference of speed estimation between EKF and TEKF

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation

Flux (EKF)Flux (TEKF)

minus08 minus04 1minus02 0 02 04 06 08minus06minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(d) Stator flux estimation by TEKF and EKF

Is120572 (real)Is120572 (TEKF)

Curr

entIs120572

(A)

minus6

minus4

minus2

0

2

4

6

02 04 06 08 10t (02 sgrid)

(e) Real stator current 119868119904120572

and estimation (TEKF)

Is120573 (real)Is120573 (TEKF)

minus6

minus4

minus2

0

2

4

6

Curr

entIs120573

(A)

02 04 06 08 10t (02 sgrid)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 6 Continued

16 Mathematical Problems in Engineering

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Erro

r cur

rentIs120572

(A)

02 04 06 08 10t (02 sgrid)

(g) TEKF stator current 119868119904120572

error

minus02

minus015

minus01

minus005

0

005

01

015

Erro

r cur

rentIs120573

(A)

02 04 06 08 10t (02 sgrid)

(h) TEKF stator current 119868119904120573

error

Figure 6 Experimental results for parameters estimation

Speed (real)Speed (EKF)

Speed (TEKF)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

0

005

01

015

02

025

03

035

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)Figure 7 Experimental results with parameter variation (119877

119903

= 05119877119903nom)

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article Direct Torque Control of Sensorless ...

2 Mathematical Problems in Engineering

the applicability of the EKF to real-time signal processingproblems

In order to reduce the conventional EKF computationalalgorithm complexity the main objective of this paper is topresent a two-stage extended Kalman filter (TEKF) for statorflux rotor speed and electromagnetic torque estimation of asensorless direct torque controlled IM drive The proposedestimator is an effective implementation of EKF Followingthe two-stage filtering technique as given in [15] the TEKFcan be decomposed into two filters such as the modified biasfree filter and the bias filter Compared to the conventionalEKF the main advantage of the TEKF is the ability to reducethe computational complexity whilst maintaining the samelevel of performance

The paper is organized as follows In Section 2 thesensorless DTC-SVM strategy of IMs is introduced brieflyIn Section 3 according to the discrete model of IM a con-ventional EKF algorithm for estimating stator flux rotorspeed and position is designed In Section 4 TEKF are devel-oped by the two-stage filtering approach and its stability isanalyzed In Section 5 simulation and experimental resultsare discussed Finally a conclusion wraps up the paper

2 Principle of Sensorless DTC-SVM

As elaborated in [12] a dynamic mathematical model foran IM in the stationary (120572120573) reference frame is obtained asfollows

[[[[[[[[[[[[[[[[

[

∙

119868119904120572

∙

119868119904120573

∙

120595119904120572

∙

120595119904120573

∙

120579

∙

120596119903

]]]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[

[

minus(119877119904

120590119871119904

+1

120590119879119903

) minus119901120596119903

1

120590119871119904

119879119903

119901120596119903

120590119871119904

0 0

119901120596119903

minus(119877119904

120590119871119904

+1

120590119879119903

) minus119901120596119903

120590119871119904

1

120590119871119904

119879119903

0 0

minus119877119904

0 0 0 0 0

0 minus119877119904

0 0 0 0

0 0 0 0 0 1

0 0 0 0 0 0

]]]]]]]]]]]]]]

]

[[[[[[[[[[[

[

119868119904120572

119868119904120573

120595119904120572

120595119904120573

120579

120596119903

]]]]]]]]]]]

]

+

[[[[[[[[[[[[[

[

1

120590119871119904

0

01

120590119871119904

1 0

0 1

0 0

0 0

]]]]]]]]]]]]]

]

[119880119904120572

119880119904120573

] (1)

where 119868119904120572

119868119904120573

120595119904120572

120595119904120573

119880119904120572

and 119880119904120573

are the stator currentsflux linkages and voltages in the stationary reference frame119877119904

and 119871119904

are the stator winding resistance and inductancerespectively 120590 is the leakage or coupling factor (where 120590 =

1minus1198712119898

119871119903

119871119904

) 119871119898

and 119871119903

are themutual inductance and rotorinductance 119879

119903

is the rotor time constant (where 119879119903

= 119871119903

119877119903

)and 119877

119903

is the rotor resistance The rotor angular velocity120596119903

is measured in mechanical radians per second 120579 is themechanical rotor position and 119901 is the number of pole pairs

The behavior of an IM inDTC technique can be describedin terms of space vectors by the following equations writtenin the stator stationary reference frame

119904

= 119877119904

119904

+119889119904

119889119905

119903

= 119877119903

119903

+ 119895120596119903

119903

+119889119903

119889119905= 0

119904

= 119871119904

119904

+ 119871119898

119903

119903

= 119871119904

119903

+ 119871119898

119904

119879119890

=3

2119901

119871119898

119871119898

2 minus 119871119904

119871119903

100381610038161003816100381611990410038161003816100381610038161003816100381610038161003816119903

1003816100381610038161003816 sin 120575

(2)

where 120575 is known as load angle which is the angle betweenrotor flux

119903

and stator flux 119904

|119904

| and |119903

| are amplitudesof 119904

and 119903

respectively From (2) it can be seen that theinstantaneous electromagnetic torque control of IMs in DTC

is determined by changing the values of load angle 120575 while119904

and 119903

maintain the constant amplitude Accelerating thestator flux with respect to the rotor flux vector will increasethe electromagnetic torque and decelerating the same vectorwill decrease the electromagnetic torque [16]

The basic idea of DTC technique of IM is to controland acquire accurate knowledge on the stator flux and elec-tromagnetic torque to achieve high dynamic performanceDTC technique involves stator flux electromagnetic torqueestimators hysteresis controllers and a simple switchinglogic (switching tables) in order to reduce the electromagnetictorque and stator flux errors rapidly [17 18] Due to the factthat the universal voltage inverter has only eight availablebasic space vectors and only one voltage space vector ismaintained for the whole duration of the control period theconventional approach causes high ripples in stator flux cur-rent and electromagnetic torque accompanied by acousticalnoise To reduce the ripples of the stator flux linkage currentand electromagnetic torque in IM drives a modified DTCusing Space Vector Modulation (SVM) method called DTC-SVM is proposed in this paper The main difference betweenconventional DTC and DTC-SVM is that DTC-SVM has aSVMmodel and two PI controllers instead of switching tableand hysteresis controllers [19 20] The system structure ofDTC-SVM can be built and shown in Figure 1 This systemoperates at constant stator flux (below rated speed) FromFigure 1 the reference torque 119879lowast

119890

is generated from regulatedspeed proportional integral (PI) Δ119879

119890

is the torque errorbetween the reference torque 119879lowast

119890

and estimated torque 119890

Mathematical Problems in Engineering 3

120596rlowast

+ + +

++

+

minus minus

minus

minus

PIPITelowast

ΔTe Δ120575

Te

s

120596r

s120572

s120573

120579slowast

P

R

120595s120572

lowast

120595s120573

lowast

Δ120595s120572

lowast

Δ120595s120573

lowast

RsIs120572 RsIs120573

Is120572

Is120573

Vs120572lowast

Vs120573lowast

SVM

abc

120572120573

A

B

C

Pulses

IM

Stator flux speed and

torque estimator

Statorvoltage

cal

IU

IV

IW

Incremental encoder

r

Arctans120572

s120573

s120572

s120573

||120595lowasts

Figure 1 System diagram of the DTC-SVM scheme

In order to compensate this error the angle of stator fluxvector must be increased from 120579

119904

to 120579119904

+ Δ120575 as shown inFigure 2 where 120579

119904

is the phase angle of stator flux vectorthat can be obtained by the flux estimator and Δ120575 is theincrement of stator flux in the next sampling timeThereforethe required reference stator flux in polar form is given bylowast

119904

= |119904

|ang120579lowast119904

Define the stator flux deviations between

lowast

119904

and 119904

asΔ119904

then

Δ120595119904120572

=10038161003816100381610038161003816lowast

119904

10038161003816100381610038161003816cos (120579lowast

119904

) minus 119904120572

Δ120595119904120573

=10038161003816100381610038161003816lowast

119904

10038161003816100381610038161003816sin (120579lowast

119904

) minus 119904120573

(3)

where Δ119904120573

and Δ119904120572

are the stationary axis components ofstator fluxΔ

119904

and 119904120572

and 119904120573

are the stator flux componentsestimation In order tomake up for stator flux deviationsΔ120595

119904120572

and Δ120595119904120573

the reference stator voltages119880lowast119904120572

and119880lowast119904120573

should beapplied on the IM which can be expressed by

119880lowast

119904120572

= 119877119904

119868119904120572

+Δ120595119904120572

119879119904

119880lowast

119904120573

= 119877119904

119868119904120573

+Δ120595119904120573

119879119904

(4)

Substituting (3) into (4) (5) can be acquired

119880lowast

119904120572

= 119877119904

119868119904120572

+(10038161003816100381610038161003816lowast

119904

10038161003816100381610038161003816cos (120579lowast

119904

) minus 119904120572

)

119879119904

119880lowast

119904120573

= 119877119904

119868119904120573

+(10038161003816100381610038161003816lowast

119904

10038161003816100381610038161003816sin (120579lowast

119904

) minus 119904120573

)

119879119904

(5)

120573

Δ120575

120575

120579r120579s

120572

120579lowast

s

120595lowast

sΔ120595

s

120595s

120595r

Figure 2 Control of stator flux linkage

Based on the reference stator voltage components 119880lowast119904120572

and 119880lowast119904120573

the drive signal for inverter IGBTs can be obtainedthrough SVMmoduleThen both the electromagnetic torqueand the magnitude of stator flux are under control therebygenerating the reference stator voltage components

3 Conventional EKF Theory

By choosing the system state vector and estimated parametervector as 119883(119905) = [119868

119904120572

119868119904120573

120595119904120572

120595119904120573]119879and 119903(119905) = [120579 120596

119903]119879

respectively 119906(119905) = [119880119904120572

119880119904120573]119879 as the input vector and

4 Mathematical Problems in Engineering

119884(119905) = [119868119904120572

119868119904120573]119879 as the output vector the IM model is

described by the general nonlinear state space model

∙

119883 (119905) = 119860 (119905)119883 (119905) + 119861 (119905) 119906 (119905) + 119863 (119905) 119903 (119905)

∙

119903 (119905) = 119866 (119905) 119903 (119905)

119884 (119905) = 119862 (119905)119883 (119905)

(6)

with

119860 (119905)

=

[[[[[[[[[

[

minus(119877119904

120590119871119904

+1

120590119879119903

) minus119901120596119903

1

120590119871119904

119879119903

119901120596119903

120590119871119904

119901120596119903

minus(119877119904

120590119871119904

+1

120590119879119903

) minus119901120596119903

120590119871119904

1

120590119871119904

119879119903

minus119877119904

0 0 0

0 minus119877119904

0 0

]]]]]]]]]

]

119861 (119905) =[[[

[

1

120590119871119904

0 1 0

01

120590119871119904

0 1

]]]

]

119879

119863 (119905) = 0

119862 (119905) = [1 0 0 0

0 1 0 0]

119866 (119905) = [0 1

0 0]

(7)

Remark 1 Matrices 119862(119905) and 119866(119905) are not affected by uncer-tainties

Remark 2 Matrix119860(119905) is time-varying because it depends onthe rotor speed 120596

119903

For digital implementation of estimator on a microcon-

troller a discrete timemathematicalmodel of IMs is requiredThese equations can be obtained from (6)

119883119896+1

= 119860119896

119883119896

+ 119861119896

119906119896

+ 119863119896

119903119896

119903119896+1

= 119866119896

119903119896

119884119896

= 119862119896

119883119896

(8)

The solution of nonhomogenous state equations (6) sat-isfying the initial condition119883(119905)|

119905=1199050= 119883(119905

0

) is

119883 (119905) = 119890119860(119905minus1199050)119883(119905

0

) + int119905

1199050

119890119860(119905minus120591)

119861119906 (120591) 119889120591 (9)

Integrating from 1199050

= 119896119879119904

to 119905 = (119896 + 1)119879119904

we can obtain that

119883((119896 + 1) 119879119904

) = 119890119860119879119904119883(119896119879

119904

)

+ int(119896+1)119879119904

119896119879

119890119860((119896+1)119879119904minus120591)119861119889120591119906 (119896119879

119904

)

(10)

The above equations lead to

119860119896

= 119890119860119879119904

119861119896

= 119860minus1

(119890119860119879119904 minus 119868) 119861

(11)

In the same way

119866119896

= 119890119866119879119904 (12)

Tolerating a small discretization error a first-order Taylorseries expansion of the matrix exponential is used

119860119896

= 119890119860119879119904 asymp 119860119879

119904

+ 119868

119866119896

= 119890119866119879119904 asymp 119866119879

119904

+ 119868

119861119896

= 119860minus1

(119890119860119879119904 minus 119868) 119861 asymp 119879

119904

119861

119863119896

= 0

(13)

with

119860119896

=

[[[[[[[[[

[

minus(119877119904

119879119904

120590119871119904

+119879119904

120590119879119903

) + 1 minus119901120596119903

119879119904

119879119904

120590119871119904

119879119903

119901120596119903

119879119904

120590119871119904

119901120596119903

119879119904

minus(119877119904

119879119904

120590119871119904

+119879119904

120590119879119903

) + 1 minus119901120596119903

119879119904

120590119871119904

119879119904

120590119871119904

119879119903

minus119877119904

119879119904

0 1 0

0 minus119877119904

119879119904

0 1

]]]]]]]]]

]

G119896

= [1 119879119904

0 1]

Mathematical Problems in Engineering 5

119861119896

=[[[

[

119879119904

120590119871119904

0 119879119904

0

0119879119904

120590119871119904

0 119879119904

]]]

]

119879

119862119896

= [1 0 0 0

0 1 0 0]

119863119896

= [0 0

0 0]

(14)

Based on discretized IM model a conventional EKFestimator is designed for estimation of stator flux currentelectromagnetic torque and rotor speed of IM for sensorlessDTC-SVM operations Treating119883

119896

as the full order state and119903119896

as the augmented system state the state vector is chosen tobe 119883119886119896

= [119883119896

119903119896]119879 119906119896

= [119880119904120572

119880119904120573]119879 and 119884

119896

= [119868119904120572

119868119904120573]119879 are

chosen as input and output vectors because these quantitiescan be easily obtained from measurements of stator currentsand voltage construction usingDC link voltage and switchingstatus Considering the parameter errors and noise of systemthe discrete time state space model of IMs in the stationary(120572120573) reference frame is described by

119883119886

119896+1

= 119860119896

119883119886

119896

+ 119861119896

119906119896

+ 119908119896

119884119896+1

= 119862119896

119883119886

119896

+ V119896

(15)

with

119860119896

= [119860119896

119863119896

0 119866119896

]

119861119896

= [119861119896

0]

119862119896

= [119862119896

0]

119879

119908119896

= [119908119909119896

119908119903119896

]

(16)

The system noise119908119896

and measurement noise V119896

are whiteGaussian sequence with zero-mean and following covariancematrices

119864[[[

[

[[

[

119908119909

119896

119908119903119896

V119896

]]

]

[[[

[

119908119909119895

119908119903119895

V119895

]]]

]

119879

]]]

]

=[[

[

119876119909119896

0 0

0 119876119903119896

0

0 0 119877119896

]]

]

120575119896119895

(17)

where 119876119909119896

gt 0 119876119903119896

gt 0 119877119896

gt 0 and 120575119896119895

is theKronecker delta The initial states 119883

0

and 1199030

are assumed tobe uncorrelated with the zero-mean noises 119908119909

119896

119908119903119896

and V119896

The initial conditions are assumed to be Gaussian randomvariables119883

0

and 1199030

that are defined as follows

119864((1198830

minus 119883lowast

0

) (1198830

minus 119883lowast

0

)119879

) = 119875119909

0

119864 (1198830

) = 119883lowast

0

119864 (1199030

) = 119903lowast

0

119864((1199030

minus 119903lowast

0

) (1199030

minus 119903lowast

0

)119879

) = 119875119903

0

119864 ((1198830

minus 119883lowast

0

) (1199030

minus 119903lowast

0

)119879

) = 119875119909119903

0

(18)

The overall structure of the EKF is well-known byemploying a two-step prediction and correction algorithm[13] Hence the application of EKF filter to the state spacemodel of IM (15) is described by

119883119886

119896|119896minus1

= 119860119896minus1

119883119886

119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

(19)

119875119896|119896minus1

= 119865119896minus1

119875119896minus1|119896minus1

119865119879

119896minus1

+ 119876119896minus1

(20)

119870119896

= 119875119896|119896minus1

119867119879

119896

(119867119896

119875119896|119896minus1

119867119879

119896

+ 119877)minus1

(21)

119875119896|119896

= 119875119896|119896minus1

minus 119870119896

119867119896

119875119896|119896minus1

(22)

119883119886

119896|119896

= 119883120572

119896|119896minus1

+ 119870119896

(119884119896

minus 119862119896

119883119886

119896|119896minus1

) (23)

with

119865119896minus1

= [119865119896minus1

119864119896minus1

0 119866119896minus1

]

119876 (sdot) = [119876119909 (sdot) 0

0 119876119903 (sdot)]

119865119896minus1

=120597

120597119883(119860119896minus1

119883119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1

) = 119860119896

119864119896minus1

=120597

120597119903(119860119896minus1

119883119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1

)

= [0 0 0 0

1199011 1199012 0 0]

119879

6 Mathematical Problems in Engineering

1199011 = (minus119901119894119904119887[119896minus1|119896minus1]

+119901

120590119871119904

120595119904120573[119896minus1|119896minus1]

)119879119904

1199012 = (119901119894119904120572[119896minus1|119896minus1]

minus119901

120590119871119904

120595119904120572[119896minus1|119896minus1]

)119879119904

119867119896

= [1198671

119896

1198672

119896

] =120597

120597119883119886(119862119896

119883119886

)

1198671

119896

= 119862119896

1198672

119896

= 0

119875 (sdot) = [119875119909 (sdot) 119875119909119903 (sdot)

(119875119909119903 (sdot))119879

119875119903 (sdot)]

(24)

4 The Two-Stage Extended Kalman Filter

41The TEKFAlgorithm Asmentioned in conventional EKFestimator previously the memory and computational costsincrease with the augmented state dimension Consideringsampling time is very small only high performance micro-controller can qualify for this work Hence the conventionalEKF algorithm may be impractical to implement The extracomputation of 119875

119909119903(sdot) terms leads to this computationalcomplexityTherefore we can reduce the computational com-plexity from application point of view if the 119875119909119903(sdot) terms canbe eliminated In this section a two-stage extended Kalmanfilter without explicitly calculating 119875119909119903(sdot) terms is discussed

Following the same approach as given in [15] the TEKFis decomposed into two filters such as the modified bias freefilter and the bias filter by applying the following two-stage119880-119881 transformation

119883119886

119896|119896minus1

= 119879 (119880119896

)119883119886

119896|119896minus1

119875119896|119896minus1

= 119879 (119880119896

) 119875119896|119896minus1

119879 (119880119896

)119879

119870119896

= 119879 (119881119896

)119870119896

119875119896|119896

= 119879 (119881119896

) 119875119896|119896

119879 (119881119896

)119879

119883119886

119896|119896

= 119879 (119881119896

)119883119886

119896|119896

(25)

where

119883119886

119896(sdot)

= [119883119896(sdot)

119903119896(sdot)

]

119870119896

= [119870119909

119896

119870119903

119896

]

119875119896(sdot)

= [

[

119875119909

119896(sdot)

0

0 119875119903

119896(sdot)

]

]

119879 (119880119896

) = [119868 119880119896

0 119868]

(26)

The main advantage of using the 119879(119880119896

) transformation isthat the inverse transformation 119879

minus1

(119880119896

) = 119879(minus119880119896

) involvesonly a change of sign Two blending matrices 119880

119896

and 119881119896

are defined by 119880119896

= 119875119909119903119896|119896minus1

(119875119903119896|119896minus1

)minus1 and 119881119896

= 119875119909119903119896|119896

(119875119903119896|119896

)minus1respectively Using characteristic of 119879(119880

119896

) (25) become

119883119886

119896|119896minus1

= 119879 (minus119880119896

)119883119886

119896|119896minus1

119875119896|119896minus1

= 119879 (minus119880119896

) 119875119896|119896minus1

119879 (minus119880119896

)119879

119870119896

= 119879 (minus119881119896

)119870119896

119875119896|119896

= 119879 (minus119881119896

) 119875119896|119896

119879 (minus119881119896

)119879

119883119886

119896|119896

= 119879 (minus119881119896

)119883119886

119896|119896

(27)

And the following relationships are obtained from (25)

119875119909

119896|119896minus1

= 119875119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

(28)

119875119903

119896|119896minus1

= 119875119903

119896|119896minus1

(29)

119875119909119903

119896|119896minus1

= 119880119896

119875119903

119896|119896minus1

(30)

119875119909

119896|119896

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

(31)

119875119903

119896|119896

= 119875119903

119896|119896

(32)

119875119909119903

119896|119896

= 119881119896

119875119903

119896|119896

(33)

Based on two-step iterative substitution method of [15]the transformed filter expressed by (27) can be recursivelycalculated as follows

119883119886

119896|119896minus1

= 119879 (minus119880119896

)

sdot (119860119896minus1

119879 (119881119896minus1

)119883119886

119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

)

(34)

119875119896|119896minus1

= 119879 (minus119880119896

)

sdot (119876119896minus1

+ 119865119896minus1

119879 (119881119896minus1

) 119875119896minus1|119896minus1

119879 (119881119896minus1

)119879

119865119879

119896minus1

)

sdot 119879 (minus119880119896

)119879

(35)

119870119896

= 119879 (119880119896

minus 119881119896

) 119875119896|119896minus1

119879 (119880119896

)119879

sdot 119867119879

119896

(119867119896

119879 (119880119896

) 119875119896|119896minus1

119879 (119880119896

)119879

119867119879

119896

+ 119877)minus1

(36)

119875119896|119896

= (119879 (119880119896

minus 119881119896

) minus 119870119896

119867119896

119879 (119880119896

))

sdot 119875119896|119896minus1

119879 (119880119896

minus 119881119896

)119879

(37)

119883119886

119896|119896

= 119870119896

(119884119896

minus 119862119896

119879 (119880119896

)119883119886

119896|119896minus1

) + 119879 (119880119896

minus 119881119896

)

sdot 119883119886

119896|119896minus1

(38)

Mathematical Problems in Engineering 7

Using (35) (37) and the block diagonal structure of 119875(sdot)

thefollowing relations can be obtained

0 = 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

minus 119880119896

119876119903

119896minus1

minus 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

0 = 119880119896

minus 119881119896

minus 119870119909

119896

119878119896

(39)

where 119880119896

and 119878119896

are defined as

119880119896

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119866minus1

119896minus1

(40)

119878119896

= 1198671

119896

119880119896

+ 1198672

119896

(41)

The above equations lead to

119880119896

119875119903

119896|119896minus1

= 119880119896

(119875119903

119896|119896minus1

minus 119876119903

119896

) (42)

119880119896

= 119880119896

(119868 minus 119876119903

119896

(119875119903

119896|119896minus1

)minus1

) (43)

119881119896

= 119880119896

minus 119870119909

119896

119878119896

(44)

Define the following notation

119860119896minus1

119879 (119881119896minus1

) = [119860119896minus1

119860119896minus1

119881119896minus1

+ 119864119896minus1

0 119866119896minus1

] (45)

The equations of themodified bias free filter and the bias filterare acquired by the next steps

Expanding (34) we have

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119898119896minus1

(46)

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

(47)

where

119898119896minus1

= (119860119896minus1

119881119896minus1

+ 119863119896minus1

minus 119880119896

119866119896minus1

) 119903119896minus1|119896minus1

(48)

Expanding (35) we have

119875119909

119896|119896minus1

= (119864119896minus1

+ 119860119896minus1

119881119896minus1

minus 119880119896

119866119896minus1

) 119875119903

119896minus1|119896minus1

lowast (119864119896minus1

+ 119860119896minus1

119881119896minus1

minus 119880119896

119866119896minus1

)119879

+ 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

+ 119880119896

119876119903

119896

119880119879

119896

(49)

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896

(50)

Then using (40) (43) and (47) (49) can be written as

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

(51)

where

119876119909

119896

= 119876119909

119896

+119872119896

(119872119896

119876119903

119896

)119879

(52)

Expanding (38) and using (41) and (44) we have

119883119896|119896

= 119883119896|119896minus1

+ (119880119896

minus 119881119896

) 119903119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

)

119903119896|119896

= 119903119896|119896minus1

+ 119870119903

119896

(119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

)

(53)

Then

119883119896|119896

= 119870119909

119896

120578119909

119896

+ 119883119896|119896minus1

(54)

where

119878119896

= 119862119896

119880119896

(55)

120578119909

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

+ (119878119896

minus 119878119896

) 119903119896|119896minus1

(56)

Expanding (36) and using (41) we have

119870119903

119896

= 119875119903

119896|119896minus1

119878119879

119896

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)minus1

119870119909

119896

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)

= 119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

(57)

Then

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(58)

Expanding (37) we have

119875119903

119896|119896

= (119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119875119909

119896|119896

= 119875119909

119896|119896minus1

+ (119880119896

minus 119881119896

) 119875119903

119896|119896minus1

(119880119879

119896

minus 119881119879

119896

)

minus (119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

(119880119879

119896

minus 119881119879

119896

))

(59)

Then using (41) and (44)

119875119909

119896|119896

= (119868 minus 119870119909

119896

1198671

119896

)119875119909

119896|119896minus1

(60)

Finally using (25) the estimated value of original state(119894119904120572

119904120573

119904120572

119904120573

) can be obtained by sum of the state119883withthe augmented state 119903

119896|119896minus1

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

(61)

119896|119896

= 119883119896|119896

+ 119881119896

119903119896|119896

(62)

Moreover the unknown parameter 119903( 119903

) is defined as

119903119896|119896minus1

= 119903119896|119896minus1

(63)

119903119896|119896

= 119903119896|119896

(64)

8 Mathematical Problems in Engineering

Based on the above analysis the TEKF can be decoupledinto two filters such as the modified bias free filter and biasfilter The modified bias filter gives the state estimation 119883

119896|119896

and the bias filter gives the bias estimate 119903

119896|119896

The correctedstate estimate 119883119886

119896|119896

(119896|119896

119903119896|119896

) of the TEKF is obtained fromthe estimates of the two filters and coupling equations119880

119896

and119881119896

[21] The modified bias free filter is expressed as follows

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119898119896minus1

(65)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

(66)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(119873119896

)minus1

(67)

119875119909

119896|119896

= (119868 minus 119870119909

119896

1198671

119896

)119875119909

119896|119896minus1

(68)

120578119909

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

+ (119878119896

minus 119878119896

) 119903119896|119896minus1

(69)

119883119896|119896

= 119870119909

119896

120578119909

119896

+ 119883119896|119896minus1

(70)

and the bias filter is

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

(71)

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896

(72)

119870119903

119896

= 119875119903

119896|119896minus1

119878119879

119896

(119873119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)minus1

(73)

119875119903

119896|119896

= (119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

(74)

120578119903

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

(75)

119903119896|119896

= 119903119896|119896minus1

+ 119870119903

119896

120578119903

119896

(76)

with the coupling equations

119878119896

= 1198671

119896

119880119896

+ 1198672

119896

(77)

119880119896

= 119880119896

(119868 minus 119876119903

119896

(119875119903

119896|119896minus1

)minus1

) (78)

119880119896

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119866minus1

119896minus1

(79)

119881119896

= 119880119896

minus 119870119909

119896

119878119896

(80)

119898119896minus1

= (119860119896minus1

119881119896minus1

+ 119863119896minus1

minus 119880119896

119866119896minus1

) 119903119896minus1|119896minus1

(81)

119876119909

119896

= 119876119909

119896

+ 119880119896

(119880119896

119876119903

119896

)119879

(82)

119873119896

= 1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

(83)

The initial conditions of TEKF algorithm are establishedwith the initial conditions of a classical EKF (119883

0|0

1199030|0

1198751199090|0

1198751199091199030|0

1198751199030|0

) so that

1198810

= 119875119909119903

0|0

(119875119903

0|0

)minus1

1198830|0

= 1198830|0

minus 1198810

1199030|0

1199030|0

= 1199030|0

119875119909

0|0

= 119875119909

0|0

minus 1198810

119875119903

0|0

119881119879

0

119875119903

0|0

= 119875119903

0|0

(84)

According to variables of full order filter119883 (1198831

1198832

1198833

1198834

)the stator flux and torque estimators for DTC-SVM ofFigure 1 are then given by

119904120572

= 1198831

119904120573

= 1198832

119904120572

= 1198833

119904120573

= 1198834

100381610038161003816100381610038161003816120595119904

100381610038161003816100381610038161003816= radic

2

119904120572

+ 2

119904120573

119904

= arctan119904120573

119904120572

119890

=3

2119901 (119904120572

119868119904120573

minus 119904120573

119868119904120572

)

(85)

where 119901 is the pole pairs of IM The estimated speed andelectromagnetic torque obtained from the TEKF observer areused to close the speed and torque loop to achieve sensorlessoperations

42 The Stability and Parameter Sensitivity Analysis ofthe TEKF

Theorem 3 The discrete time conventional extended Kalmanfilter (19)ndash(23) is equivalent to the two-stage extern Kalmanfilter (see (61)sim(83))

Proof Before proving the theorem the following five rela-tionships are needed

(1) Using (72) and (78)

119880119896+1

119866119896

119875119903

119896|119896

119866119879

119896

= 119880119896

119875119903

119896|119896minus1

(86)

(2) Using (67) and (73)

119870119909

119896

119872119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

(87)

119870119903

119896

119872119896

= 119875119903

119896|119896minus1

(119878119896

)119879

(88)

where

119872119896

= 1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

+ 119877119896

(89)

Mathematical Problems in Engineering 9

(3) Using (20) we have

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896minus1

(90)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119864119896minus1

(119875119909119903

119896minus1|119896minus1

)119879

119860119879

119896minus1

+ 119860119896minus1

119875119909119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119876119909

119896minus1

(91)

119875119909119903

119896|119896minus1

= 119860119896minus1

119875119909119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

(92)

(4) Using (21)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(93)

119870119903

119896

= (119875119909119903

119896|119896minus1

)119879

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(94)

(5) Using (22)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

(95)

119875119909119903

119896|119896

= 119875119909119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909119903

119896|119896minus1

(96)

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896

1198671

119896

119875119909119903

119896|119896minus1

(97)

By inductive reasoning suppose that at time 119896 minus 1the unknown parameter

119896minus1

and estimated state 119896minus1

areequal to the parameter 119903

119896minus1

and state 119883119896minus1

of the controlsystem respectively we show that TEKF is equivalent to theconventional EKF because these properties are still true attime 119896

Assume that at time 119896 minus 1

119883119896minus1|119896minus1

= 119896minus1|119896minus1

119903119896minus1|119896minus1

= 119903119896minus1|119896minus1

119875119909

119896minus1|119896minus1

= 11987511

119896minus1|119896minus1

119875119909119903

119896minus1|119896minus1

= 11987512

119896minus1|119896minus1

119875119903

119896minus1|119896minus1

= 11987522

119896minus1|119896minus1

(98)

where [ 119875119909119875

119909119903

(119875

119909119903)

119879119875

119903 ] and [ 11987511119875

12

(119875

12)

119879119875

22 ] represent the variance-covariance matrices of the system and estimated variablesrespectively

From (19) we have

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119863119896

119903119896minus1|119896minus1

+ 119861119896

119906119896minus1

(99)

Then using (98) (41) (62) (79) (81) (71) and (61)

119883119896|119896minus1

= 119860119896minus1

(119883119896minus1|119896minus1

+ 119881119896minus1

119903119896minus1|119896minus1

)

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

minus 119898119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

= 119896|119896minus1

(100)

Using (19) (71) (98) (63) and (64) we have

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119903119896|119896minus1

(101)

Using (91) (98) (78) (66) (79) (82) (86) and (72) we obtain

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896minus1

+ 119880119896

119866119896

119875119903

119896minus1|119896minus1

119866119879

119896

119880119879

119896

= 119875119909

119896|119896minus1

+ 119880119896

(119880119896

119875119903

119896|119896minus1

minus 119880119896

119876119903

119896minus1

)119879

= 119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

= 11987511

119896|119896minus1

(102)

Using (90) (98) (72) (32) (71) (29) and (97) we obtain

119875119903

119896|119896minus1

= 11987522

119896minus1|119896minus1

(103)

Using (92) (98) (33) (32) (79) (86) and (91)

119875119909119903

119896|119896minus1

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896minus1

119875119903

119896minus1|119896minus2

= 11987512

119896|119896minus1

(104)

Using (93) (101) (55) (73) (67) (80) and (87)

119870119909

119896

= (119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

) (1198671

119896

)119879

sdot (1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ (119881119896

+ 119870119909

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

(1198671

119896

)119879

)119880minus1

119896

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

)119880minus1

119896

+ 119881119896

119875119903

119896|119896minus1

119878119879

119896

119880minus1

119896

= 119870119909

119896

+ 119881119896

119870119903

119896

(105)

Using (94) (30) and (88) we obtain

119870119903

119896

= (119875119903

119896|119896minus1

)119879

119878119879

119896

119880minus1

119896

= 119870119903

119896|119896minus1

(106)

10 Mathematical Problems in Engineering

Next wewill show that (98) holds at time 119896 From (23)we have

119883119896|119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

)

= 119883119896|119896minus1

+ 119870119909

119896

119903119896

(107)

Then using (61) and (105) the above equation can be writtenas

119883119896|119896

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

+ (119870119909

119896

+ 119881119896

119870119903

119896

) 119903119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 1198671

119896

119883119896|119896minus1

)

+ (119880119896

minus 119870119909

119896

119878119896

) 119903119896|119896minus1

+ 119881119896

119870119903

119896

119903119896

= 119883119896|119896

+ 119881119896

119903119896|119896

= 119896|119896

(108)

Using (95) (105) (102) and (77)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

(109)

Then using (80) (68) (74) and (31) we obtain

119875119909

119896|119896

= 119875119909

119896|119896

+ 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

+ 119881119896

(119875119903

119896|119896

119878119879

119896

(119870119909

119896

)119879

minus 119870119903

119896

1198671

119896

119875119909

119896|119896minus1

)

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

= 11987511

119896|119896

(110)

Using (96) (30) (28) (105) and (80)

119875119909119903

119896|119896

= 119880119896

119875119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

1198671

119896

119880119896

minus 119881119896

119870119903

119896

1198671

119896

119880119896

)119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

119875119903

119896|119896

= 11987512

119896|119896

(111)

Using (97) (106) (95) (29) and (30) we obtain

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896|119896minus1

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119868 minus 119870119903

119896|119896minus1

119878119896

) 119875119903

119896|119896minus1

= 11987522

119896|119896

(112)

Table 1 Kalman estimation arithmetic operation requirement forthe conventional EKF structure

Number of multiplications(119899 = 6119898 = 2 and 119902 = 2)

Number of additions(119899 = 6119898 = 2 and 119902 = 2)

119860119896

119861119896

and 119862

119896

Function of system (9) Function of system (3)

119883119886

119896|119896minus1

1198992 + 119899119902 (48) 1198992 + 119899119902 minus 119899 (42)119875119896|119896minus1

21198993 (432) 21198993 minus 1198992 (396)119883119886

119896|119896minus1

2119899119898 (24) 2119899119898 (24)

119870119896

1198992

119898 + 21198991198982

+ 1198983 (168) 119899

2

119898 + 21198991198982

+ 1198983

minus 2119899119898

(104)119875119896|119896

1198992119898 + 1198993 (288) 1198992119898 + 1198993 minus 1198992 (252)Total 960 818

Finally we show that (98) holds at time 119896 = 0 This can beverified by the initial conditions of TEKF algorithm

43 Numerical Complexity of the Algorithm Tables 1 and2 show the computational effort at each sample time bythe conventional EKF algorithm and TEKF (where roughmatrix-based implementation is used) in which as definedabove 119899 is the dimension of the state vector 119883

119896

119898 is thedimension of the measurement 119884

119896

119902 is the input vector 119880119896

and 119901 is the dimension of the parameter 119903

119896

The total numberof arithmetic operations (additions and multiplications) persample time of the TEKF is 1314 compared with 1778 for arough implementation of a conventional EKF which meansthe operation cost can reduce by 26

5 Simulation and Experimental Results

51 Simulation Results To test the feasibility and perfor-mance of the TEKF method the sensorless DTC-SVM tech-nique for IM drives described in Section 2 is implementedin MATLABSIMULINK environment The values of theinitial state covariance matrices 119875

0

119876 and 119877 have a greatinfluence on the performance of the estimation methodThe diagonal initial state covariance matrix 119875

0

representsvariances or mean-squared errors in the knowledge of theinitial conditions Matrix 119876 gives the statistical descriptionof the drive system Matrix 119877 is related to measured noiseThey can be obtained by considering the stochastic propertiesof the corresponding noises However a fine evaluation ofthe covariance matrices is very difficult because they areusually not known In this paper tuning the initial values ofcovariance matrices 119875

0

119876 and 119877 is using particular criteria[22] to achieve steady-state behaviors of the relative estimatedstates as given by

119876 = diag 20 20 1119890 minus 6 1119890 minus 6 10 10

1198750

= diag 01 01 05 05 1 1

119877 = diag 01 01

(113)

Mathematical Problems in Engineering 11

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000Sp

eed

(rpm

)

(a) Speed estimation

02 04 06 08 10Times (s)

minus002

0

002

004

006

008

01

012

014

016

018

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF

Theta (real)Theta (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7Po

sitio

n (r

ad)

(d) TEKF rotor position error

Current (real)Current (TEKF)

Curr

entIs120572

(A)

02 04 06 08 10Times (s)

minus12

minus8

minus4

0

4

8

12

(e) Real stator current 119868119904120572

and estimation (TEKF)

Current (real)Current (TEKF)

minus12

minus8

minus4

0

4

8

12

Curr

entIs120573

(A)

02 04 06 08 10Times (s)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 3 Continued

12 Mathematical Problems in Engineering

Flux (EKF)Flux (TEKF)

minus05 0 05 1minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(g) Stator flux estimation by TEKF and EKF

02 04 06 08 10Times (s)

minus2

0

2

4

6

8

10

12

14

16

Ψs

(Wb)

times10minus3

(h) Difference Stator flux estimation between TEKF and EKF

Figure 3 Simulation results for parameters estimation

Table 2 Kalman estimation arithmetic operation requirement for the TEKF structure

Number of multiplications(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

Number of additions(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

119860119896

119862119896

119864119896

1198671119896

1198672119896

119861119896

and119863119896

Function of system (25) Function of system (11)119883119896|119896minus1

1198992 + 119899119902 (24) 1198992 + 119899119902 (24)119875119909

119896|119896minus121198993 (128) 21198993 minus 1198992 (112)

119870119909

119896|1198961198992

119898 + 1198991198982 (48) 119899

2

119898 + 1198991198982

minus 2119899119898 (32)119875119909

119896|1198961198992

119898 + 1198993 (96) 119899

2

119898 + 1198993

minus 1198992 (80)

119883119896|119896

2119899119898 + 119898119901 (20) 2119899119898 + 2119898119901 (24)119903119896|119896minus1

1199012 (4) 1199012 minus 119901 (2)119875119903

119896|119896minus121199013 (16) 21199013 minus 1199012 (12)

119870119903

119896|11989631199012119898 + 1199011198982 (32) 31199012119898 + 1199011198982 + 1198982 minus 4119901119898 (20)

119875119903

119896|1198961199013 + 1199012119898 (16) 1199013 + 1199012119898 minus 1199012 (12)

119903119896|119896

2119898119901 + 119899119898 (12) 2119898119901 + 119899119898 (16)119878119896

119898119899119901 (16) 119899119898119901 (16)119880119896minus1

1198992119901 + 1198991199012 (56) 1198992119901 + 1198991199012 minus 119899119901 (48)119881119896

119899119901119898 (16) 119899119901 (8)119880119896minus1

21198991199012 (32) 21198991199012 (32)119898119896minus1

1198992

119901 + 1198991199012

+ 119899119901 (56) 1198992

119901 + 1198991199012

+ 119899119901 minus 119899 (52)119876119909

119896minus121198992

119901 (64) 21198992

119901 + 1198991199012 (64)

119878119896

119898119899119901 (16) 119898119899119901 minus 119898119901 (12)119873119896

21198981198992 (32) 21198981198992 minus 1198982 (60)Total 688 626

In the simulation a comparison is made to verify theequivalence of EKF and TEKF Real-time parameters esti-mated by TEKF are used to formulate the closed loop suchas rotor speed stator flux and electromagnetic torque Theestimations obtained by EKF algorithm are not included in

the sensorless DTC-SVM strategy and only evaluated in openloop A step reference speed was applied to the simulation

The machine is accelerated from 0 rpm to 1000 rpm at0 s and the torque load is set to 4N The simulation resultsof parameter estimation are shown in Figure 3 Figures 3(a)

Mathematical Problems in Engineering 13

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000

Spee

d (r

pm)

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus03

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Theta (real)Theta (TEKF)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

02 04 06 08 10Times (s)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)

Figure 4 Simulation results with parameter variation (119877119903

= 05119877119903nom)

and 3(c) represent the performance of the speed and rotorposition tracking capabilities of the control system withTEKF and EKF respectively Figures 3(e) 3(f) and 3(g)represent the estimated stator current and flux they showthat ripples are significantly suppressed due to the SVMmodulation scheme Figures 3(b) 3(d) and 3(h) show thatfor variations of speed reference the rotor speed rotorposition and stator flux errors between the two observers arevery little It is verified that the two observers are equivalentThe difference between two estimators is caused by accuracyloss in TEKF which uses more calculation steps

In order to further verify the performance of TEKFagainst model-plant parameter mismatches and the equiv-alence of two observers the change in rotor resistance isconsidered Rotor resistance will increase due to temperaturerise while the motor is running To simulate this conditionthe rotor resistance in TEKF and EKF is increased to 200comparedwith the normal value which is equivalent to a 50decrease in the actual rotor resistance The machine is stillaccelerated from 0 rpm to 1000 rpm at 0 s and the torque is setto 4N Figure 4 shows that for variation of rotor resistancethe steady-state speed and rotor position errors are negligible

14 Mathematical Problems in Engineering

(a)

Computer

380V

TXRX

120596r

Incrementalencoder

Udc

MicrocomputerDSP TMS3206713

Drive signalCurrentsensor

Voltagesensor

Voltagesensor

IM MC

ISU ISV

VSU VSW VSV

(b)

Figure 5 Complete drive system (a) Picture of experimental setup (b) Functional block diagram of the experimental setup

and the difference of the speed and rotor position estimationsbetween the two observers is rather null

52 Experimental Results The overall experimental setup isshown in Figure 5 and the specifications and rated parametersof the IM controller and inverter are listed in Table 3 Inthe experimental hardware an Expert3 control system fromMyway company and a three-phase two-pole 15 kW IM areappliedThe IM is mechanically coupled to a magnetic clutch(MC) which provides rated torque even at very low speedThe main processor in Expert3 control system is a floatingpoint processor TMS320C6713 with a max clock speed of225MHz All the algorithms including TEKF EKF DTCalgorithm and some transformation modules are imple-mented in TMS320C6713 with 100120583s sampling time and dataacquisition of the parameter estimations measured variablesand their visualization are realized on the cockpit provided byPEView9 software Insulated Gate Bipolar Transistor (IGBT)module is driven by the PWM signal with a switchingfrequency of 10 kHz and 2 120583s dead time The stator currentsare measured via two Hall effect current sensors The rotorangle and speed of IM are measured from an incrementalencoder with 2048 pulses per revolution

This experiment test is here to testify the performanceof TEKF and demonstrate that the two estimators aremathematically equivalent The machine is accelerated from600 rpm to 1000 rpm and 4N torque load is set Theexperimental results of parameter estimation based on twoobservers are given in Figures 6 and 7 Figures 6(a) and 6(c)show that the TEKF still has a good tracking performance ofthe speed and rotor position in experiment Figures 6(d) 6(e)and 6(f) illustrate stator flux and stator current estimationrobustness Figures 6(b) 6(g) and 6(f) referring to thedifference in speed and stator current estimations given by

Table 3 Specification of induction motor and inverter

Induction motor ValueNominal torque 10NmNominal voltage 380VRotor resistance 119877

119903

25ΩStator resistance 119877

119904

36ΩStator inductances 119871

119904

0301HRotor inductances 119871

119903

0302HMutual inductances 119871

119898

0273HPole pairs 2Invertercontroller ValueSwitching device 1000V 80A IGBTControl cycle time 100 120583sMain CPU DSP TMS320C6713 225MHz

the two observers are still small These experiment resultsprove that the two estimators are mathematically equivalentFigure 7 shows the speed and rotor position estimationsbased onTEKFandEKF for a 50decrease of rotor resistance(the same as the simulation) As expected the steady errorof the TEKF and the difference in speed and rotor positionestimations are still tiny Robustness of TEKF is verified

6 Conclusion

Themajor shortcoming of the conventional EKF is numericalproblems and computational burden due to the high orderof the mathematical models This has generally limited thereal-time digital implementation of the EKF for industrialfield So in this study a novel extended Kalman filter

Mathematical Problems in Engineering 15

Speed (real)Speed (EKF)Speed (TEKF)

Spee

d (r

min

)

02 04 06 08 10t (02 sgrid)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimationSp

eed

(rm

in)

minus005

0

005

01

015

02

025

03

035

04

045

02 04 06 08 10t (02 sgrid)

(b) Difference of speed estimation between EKF and TEKF

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation

Flux (EKF)Flux (TEKF)

minus08 minus04 1minus02 0 02 04 06 08minus06minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(d) Stator flux estimation by TEKF and EKF

Is120572 (real)Is120572 (TEKF)

Curr

entIs120572

(A)

minus6

minus4

minus2

0

2

4

6

02 04 06 08 10t (02 sgrid)

(e) Real stator current 119868119904120572

and estimation (TEKF)

Is120573 (real)Is120573 (TEKF)

minus6

minus4

minus2

0

2

4

6

Curr

entIs120573

(A)

02 04 06 08 10t (02 sgrid)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 6 Continued

16 Mathematical Problems in Engineering

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Erro

r cur

rentIs120572

(A)

02 04 06 08 10t (02 sgrid)

(g) TEKF stator current 119868119904120572

error

minus02

minus015

minus01

minus005

0

005

01

015

Erro

r cur

rentIs120573

(A)

02 04 06 08 10t (02 sgrid)

(h) TEKF stator current 119868119904120573

error

Figure 6 Experimental results for parameters estimation

Speed (real)Speed (EKF)

Speed (TEKF)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

0

005

01

015

02

025

03

035

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)Figure 7 Experimental results with parameter variation (119877

119903

= 05119877119903nom)

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article Direct Torque Control of Sensorless ...

Mathematical Problems in Engineering 3

120596rlowast

+ + +

++

+

minus minus

minus

minus

PIPITelowast

ΔTe Δ120575

Te

s

120596r

s120572

s120573

120579slowast

P

R

120595s120572

lowast

120595s120573

lowast

Δ120595s120572

lowast

Δ120595s120573

lowast

RsIs120572 RsIs120573

Is120572

Is120573

Vs120572lowast

Vs120573lowast

SVM

abc

120572120573

A

B

C

Pulses

IM

Stator flux speed and

torque estimator

Statorvoltage

cal

IU

IV

IW

Incremental encoder

r

Arctans120572

s120573

s120572

s120573

||120595lowasts

Figure 1 System diagram of the DTC-SVM scheme

In order to compensate this error the angle of stator fluxvector must be increased from 120579

119904

to 120579119904

+ Δ120575 as shown inFigure 2 where 120579

119904

is the phase angle of stator flux vectorthat can be obtained by the flux estimator and Δ120575 is theincrement of stator flux in the next sampling timeThereforethe required reference stator flux in polar form is given bylowast

119904

= |119904

|ang120579lowast119904

Define the stator flux deviations between

lowast

119904

and 119904

asΔ119904

then

Δ120595119904120572

=10038161003816100381610038161003816lowast

119904

10038161003816100381610038161003816cos (120579lowast

119904

) minus 119904120572

Δ120595119904120573

=10038161003816100381610038161003816lowast

119904

10038161003816100381610038161003816sin (120579lowast

119904

) minus 119904120573

(3)

where Δ119904120573

and Δ119904120572

are the stationary axis components ofstator fluxΔ

119904

and 119904120572

and 119904120573

are the stator flux componentsestimation In order tomake up for stator flux deviationsΔ120595

119904120572

and Δ120595119904120573

the reference stator voltages119880lowast119904120572

and119880lowast119904120573

should beapplied on the IM which can be expressed by

119880lowast

119904120572

= 119877119904

119868119904120572

+Δ120595119904120572

119879119904

119880lowast

119904120573

= 119877119904

119868119904120573

+Δ120595119904120573

119879119904

(4)

Substituting (3) into (4) (5) can be acquired

119880lowast

119904120572

= 119877119904

119868119904120572

+(10038161003816100381610038161003816lowast

119904

10038161003816100381610038161003816cos (120579lowast

119904

) minus 119904120572

)

119879119904

119880lowast

119904120573

= 119877119904

119868119904120573

+(10038161003816100381610038161003816lowast

119904

10038161003816100381610038161003816sin (120579lowast

119904

) minus 119904120573

)

119879119904

(5)

120573

Δ120575

120575

120579r120579s

120572

120579lowast

s

120595lowast

sΔ120595

s

120595s

120595r

Figure 2 Control of stator flux linkage

Based on the reference stator voltage components 119880lowast119904120572

and 119880lowast119904120573

the drive signal for inverter IGBTs can be obtainedthrough SVMmoduleThen both the electromagnetic torqueand the magnitude of stator flux are under control therebygenerating the reference stator voltage components

3 Conventional EKF Theory

By choosing the system state vector and estimated parametervector as 119883(119905) = [119868

119904120572

119868119904120573

120595119904120572

120595119904120573]119879and 119903(119905) = [120579 120596

119903]119879

respectively 119906(119905) = [119880119904120572

119880119904120573]119879 as the input vector and

4 Mathematical Problems in Engineering

119884(119905) = [119868119904120572

119868119904120573]119879 as the output vector the IM model is

described by the general nonlinear state space model

∙

119883 (119905) = 119860 (119905)119883 (119905) + 119861 (119905) 119906 (119905) + 119863 (119905) 119903 (119905)

∙

119903 (119905) = 119866 (119905) 119903 (119905)

119884 (119905) = 119862 (119905)119883 (119905)

(6)

with

119860 (119905)

=

[[[[[[[[[

[

minus(119877119904

120590119871119904

+1

120590119879119903

) minus119901120596119903

1

120590119871119904

119879119903

119901120596119903

120590119871119904

119901120596119903

minus(119877119904

120590119871119904

+1

120590119879119903

) minus119901120596119903

120590119871119904

1

120590119871119904

119879119903

minus119877119904

0 0 0

0 minus119877119904

0 0

]]]]]]]]]

]

119861 (119905) =[[[

[

1

120590119871119904

0 1 0

01

120590119871119904

0 1

]]]

]

119879

119863 (119905) = 0

119862 (119905) = [1 0 0 0

0 1 0 0]

119866 (119905) = [0 1

0 0]

(7)

Remark 1 Matrices 119862(119905) and 119866(119905) are not affected by uncer-tainties

Remark 2 Matrix119860(119905) is time-varying because it depends onthe rotor speed 120596

119903

For digital implementation of estimator on a microcon-

troller a discrete timemathematicalmodel of IMs is requiredThese equations can be obtained from (6)

119883119896+1

= 119860119896

119883119896

+ 119861119896

119906119896

+ 119863119896

119903119896

119903119896+1

= 119866119896

119903119896

119884119896

= 119862119896

119883119896

(8)

The solution of nonhomogenous state equations (6) sat-isfying the initial condition119883(119905)|

119905=1199050= 119883(119905

0

) is

119883 (119905) = 119890119860(119905minus1199050)119883(119905

0

) + int119905

1199050

119890119860(119905minus120591)

119861119906 (120591) 119889120591 (9)

Integrating from 1199050

= 119896119879119904

to 119905 = (119896 + 1)119879119904

we can obtain that

119883((119896 + 1) 119879119904

) = 119890119860119879119904119883(119896119879

119904

)

+ int(119896+1)119879119904

119896119879

119890119860((119896+1)119879119904minus120591)119861119889120591119906 (119896119879

119904

)

(10)

The above equations lead to

119860119896

= 119890119860119879119904

119861119896

= 119860minus1

(119890119860119879119904 minus 119868) 119861

(11)

In the same way

119866119896

= 119890119866119879119904 (12)

Tolerating a small discretization error a first-order Taylorseries expansion of the matrix exponential is used

119860119896

= 119890119860119879119904 asymp 119860119879

119904

+ 119868

119866119896

= 119890119866119879119904 asymp 119866119879

119904

+ 119868

119861119896

= 119860minus1

(119890119860119879119904 minus 119868) 119861 asymp 119879

119904

119861

119863119896

= 0

(13)

with

119860119896

=

[[[[[[[[[

[

minus(119877119904

119879119904

120590119871119904

+119879119904

120590119879119903

) + 1 minus119901120596119903

119879119904

119879119904

120590119871119904

119879119903

119901120596119903

119879119904

120590119871119904

119901120596119903

119879119904

minus(119877119904

119879119904

120590119871119904

+119879119904

120590119879119903

) + 1 minus119901120596119903

119879119904

120590119871119904

119879119904

120590119871119904

119879119903

minus119877119904

119879119904

0 1 0

0 minus119877119904

119879119904

0 1

]]]]]]]]]

]

G119896

= [1 119879119904

0 1]

Mathematical Problems in Engineering 5

119861119896

=[[[

[

119879119904

120590119871119904

0 119879119904

0

0119879119904

120590119871119904

0 119879119904

]]]

]

119879

119862119896

= [1 0 0 0

0 1 0 0]

119863119896

= [0 0

0 0]

(14)

Based on discretized IM model a conventional EKFestimator is designed for estimation of stator flux currentelectromagnetic torque and rotor speed of IM for sensorlessDTC-SVM operations Treating119883

119896

as the full order state and119903119896

as the augmented system state the state vector is chosen tobe 119883119886119896

= [119883119896

119903119896]119879 119906119896

= [119880119904120572

119880119904120573]119879 and 119884

119896

= [119868119904120572

119868119904120573]119879 are

chosen as input and output vectors because these quantitiescan be easily obtained from measurements of stator currentsand voltage construction usingDC link voltage and switchingstatus Considering the parameter errors and noise of systemthe discrete time state space model of IMs in the stationary(120572120573) reference frame is described by

119883119886

119896+1

= 119860119896

119883119886

119896

+ 119861119896

119906119896

+ 119908119896

119884119896+1

= 119862119896

119883119886

119896

+ V119896

(15)

with

119860119896

= [119860119896

119863119896

0 119866119896

]

119861119896

= [119861119896

0]

119862119896

= [119862119896

0]

119879

119908119896

= [119908119909119896

119908119903119896

]

(16)

The system noise119908119896

and measurement noise V119896

are whiteGaussian sequence with zero-mean and following covariancematrices

119864[[[

[

[[

[

119908119909

119896

119908119903119896

V119896

]]

]

[[[

[

119908119909119895

119908119903119895

V119895

]]]

]

119879

]]]

]

=[[

[

119876119909119896

0 0

0 119876119903119896

0

0 0 119877119896

]]

]

120575119896119895

(17)

where 119876119909119896

gt 0 119876119903119896

gt 0 119877119896

gt 0 and 120575119896119895

is theKronecker delta The initial states 119883

0

and 1199030

are assumed tobe uncorrelated with the zero-mean noises 119908119909

119896

119908119903119896

and V119896

The initial conditions are assumed to be Gaussian randomvariables119883

0

and 1199030

that are defined as follows

119864((1198830

minus 119883lowast

0

) (1198830

minus 119883lowast

0

)119879

) = 119875119909

0

119864 (1198830

) = 119883lowast

0

119864 (1199030

) = 119903lowast

0

119864((1199030

minus 119903lowast

0

) (1199030

minus 119903lowast

0

)119879

) = 119875119903

0

119864 ((1198830

minus 119883lowast

0

) (1199030

minus 119903lowast

0

)119879

) = 119875119909119903

0

(18)

The overall structure of the EKF is well-known byemploying a two-step prediction and correction algorithm[13] Hence the application of EKF filter to the state spacemodel of IM (15) is described by

119883119886

119896|119896minus1

= 119860119896minus1

119883119886

119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

(19)

119875119896|119896minus1

= 119865119896minus1

119875119896minus1|119896minus1

119865119879

119896minus1

+ 119876119896minus1

(20)

119870119896

= 119875119896|119896minus1

119867119879

119896

(119867119896

119875119896|119896minus1

119867119879

119896

+ 119877)minus1

(21)

119875119896|119896

= 119875119896|119896minus1

minus 119870119896

119867119896

119875119896|119896minus1

(22)

119883119886

119896|119896

= 119883120572

119896|119896minus1

+ 119870119896

(119884119896

minus 119862119896

119883119886

119896|119896minus1

) (23)

with

119865119896minus1

= [119865119896minus1

119864119896minus1

0 119866119896minus1

]

119876 (sdot) = [119876119909 (sdot) 0

0 119876119903 (sdot)]

119865119896minus1

=120597

120597119883(119860119896minus1

119883119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1

) = 119860119896

119864119896minus1

=120597

120597119903(119860119896minus1

119883119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1

)

= [0 0 0 0

1199011 1199012 0 0]

119879

6 Mathematical Problems in Engineering

1199011 = (minus119901119894119904119887[119896minus1|119896minus1]

+119901

120590119871119904

120595119904120573[119896minus1|119896minus1]

)119879119904

1199012 = (119901119894119904120572[119896minus1|119896minus1]

minus119901

120590119871119904

120595119904120572[119896minus1|119896minus1]

)119879119904

119867119896

= [1198671

119896

1198672

119896

] =120597

120597119883119886(119862119896

119883119886

)

1198671

119896

= 119862119896

1198672

119896

= 0

119875 (sdot) = [119875119909 (sdot) 119875119909119903 (sdot)

(119875119909119903 (sdot))119879

119875119903 (sdot)]

(24)

4 The Two-Stage Extended Kalman Filter

41The TEKFAlgorithm Asmentioned in conventional EKFestimator previously the memory and computational costsincrease with the augmented state dimension Consideringsampling time is very small only high performance micro-controller can qualify for this work Hence the conventionalEKF algorithm may be impractical to implement The extracomputation of 119875

119909119903(sdot) terms leads to this computationalcomplexityTherefore we can reduce the computational com-plexity from application point of view if the 119875119909119903(sdot) terms canbe eliminated In this section a two-stage extended Kalmanfilter without explicitly calculating 119875119909119903(sdot) terms is discussed

Following the same approach as given in [15] the TEKFis decomposed into two filters such as the modified bias freefilter and the bias filter by applying the following two-stage119880-119881 transformation

119883119886

119896|119896minus1

= 119879 (119880119896

)119883119886

119896|119896minus1

119875119896|119896minus1

= 119879 (119880119896

) 119875119896|119896minus1

119879 (119880119896

)119879

119870119896

= 119879 (119881119896

)119870119896

119875119896|119896

= 119879 (119881119896

) 119875119896|119896

119879 (119881119896

)119879

119883119886

119896|119896

= 119879 (119881119896

)119883119886

119896|119896

(25)

where

119883119886

119896(sdot)

= [119883119896(sdot)

119903119896(sdot)

]

119870119896

= [119870119909

119896

119870119903

119896

]

119875119896(sdot)

= [

[

119875119909

119896(sdot)

0

0 119875119903

119896(sdot)

]

]

119879 (119880119896

) = [119868 119880119896

0 119868]

(26)

The main advantage of using the 119879(119880119896

) transformation isthat the inverse transformation 119879

minus1

(119880119896

) = 119879(minus119880119896

) involvesonly a change of sign Two blending matrices 119880

119896

and 119881119896

are defined by 119880119896

= 119875119909119903119896|119896minus1

(119875119903119896|119896minus1

)minus1 and 119881119896

= 119875119909119903119896|119896

(119875119903119896|119896

)minus1respectively Using characteristic of 119879(119880

119896

) (25) become

119883119886

119896|119896minus1

= 119879 (minus119880119896

)119883119886

119896|119896minus1

119875119896|119896minus1

= 119879 (minus119880119896

) 119875119896|119896minus1

119879 (minus119880119896

)119879

119870119896

= 119879 (minus119881119896

)119870119896

119875119896|119896

= 119879 (minus119881119896

) 119875119896|119896

119879 (minus119881119896

)119879

119883119886

119896|119896

= 119879 (minus119881119896

)119883119886

119896|119896

(27)

And the following relationships are obtained from (25)

119875119909

119896|119896minus1

= 119875119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

(28)

119875119903

119896|119896minus1

= 119875119903

119896|119896minus1

(29)

119875119909119903

119896|119896minus1

= 119880119896

119875119903

119896|119896minus1

(30)

119875119909

119896|119896

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

(31)

119875119903

119896|119896

= 119875119903

119896|119896

(32)

119875119909119903

119896|119896

= 119881119896

119875119903

119896|119896

(33)

Based on two-step iterative substitution method of [15]the transformed filter expressed by (27) can be recursivelycalculated as follows

119883119886

119896|119896minus1

= 119879 (minus119880119896

)

sdot (119860119896minus1

119879 (119881119896minus1

)119883119886

119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

)

(34)

119875119896|119896minus1

= 119879 (minus119880119896

)

sdot (119876119896minus1

+ 119865119896minus1

119879 (119881119896minus1

) 119875119896minus1|119896minus1

119879 (119881119896minus1

)119879

119865119879

119896minus1

)

sdot 119879 (minus119880119896

)119879

(35)

119870119896

= 119879 (119880119896

minus 119881119896

) 119875119896|119896minus1

119879 (119880119896

)119879

sdot 119867119879

119896

(119867119896

119879 (119880119896

) 119875119896|119896minus1

119879 (119880119896

)119879

119867119879

119896

+ 119877)minus1

(36)

119875119896|119896

= (119879 (119880119896

minus 119881119896

) minus 119870119896

119867119896

119879 (119880119896

))

sdot 119875119896|119896minus1

119879 (119880119896

minus 119881119896

)119879

(37)

119883119886

119896|119896

= 119870119896

(119884119896

minus 119862119896

119879 (119880119896

)119883119886

119896|119896minus1

) + 119879 (119880119896

minus 119881119896

)

sdot 119883119886

119896|119896minus1

(38)

Mathematical Problems in Engineering 7

Using (35) (37) and the block diagonal structure of 119875(sdot)

thefollowing relations can be obtained

0 = 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

minus 119880119896

119876119903

119896minus1

minus 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

0 = 119880119896

minus 119881119896

minus 119870119909

119896

119878119896

(39)

where 119880119896

and 119878119896

are defined as

119880119896

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119866minus1

119896minus1

(40)

119878119896

= 1198671

119896

119880119896

+ 1198672

119896

(41)

The above equations lead to

119880119896

119875119903

119896|119896minus1

= 119880119896

(119875119903

119896|119896minus1

minus 119876119903

119896

) (42)

119880119896

= 119880119896

(119868 minus 119876119903

119896

(119875119903

119896|119896minus1

)minus1

) (43)

119881119896

= 119880119896

minus 119870119909

119896

119878119896

(44)

Define the following notation

119860119896minus1

119879 (119881119896minus1

) = [119860119896minus1

119860119896minus1

119881119896minus1

+ 119864119896minus1

0 119866119896minus1

] (45)

The equations of themodified bias free filter and the bias filterare acquired by the next steps

Expanding (34) we have

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119898119896minus1

(46)

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

(47)

where

119898119896minus1

= (119860119896minus1

119881119896minus1

+ 119863119896minus1

minus 119880119896

119866119896minus1

) 119903119896minus1|119896minus1

(48)

Expanding (35) we have

119875119909

119896|119896minus1

= (119864119896minus1

+ 119860119896minus1

119881119896minus1

minus 119880119896

119866119896minus1

) 119875119903

119896minus1|119896minus1

lowast (119864119896minus1

+ 119860119896minus1

119881119896minus1

minus 119880119896

119866119896minus1

)119879

+ 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

+ 119880119896

119876119903

119896

119880119879

119896

(49)

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896

(50)

Then using (40) (43) and (47) (49) can be written as

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

(51)

where

119876119909

119896

= 119876119909

119896

+119872119896

(119872119896

119876119903

119896

)119879

(52)

Expanding (38) and using (41) and (44) we have

119883119896|119896

= 119883119896|119896minus1

+ (119880119896

minus 119881119896

) 119903119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

)

119903119896|119896

= 119903119896|119896minus1

+ 119870119903

119896

(119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

)

(53)

Then

119883119896|119896

= 119870119909

119896

120578119909

119896

+ 119883119896|119896minus1

(54)

where

119878119896

= 119862119896

119880119896

(55)

120578119909

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

+ (119878119896

minus 119878119896

) 119903119896|119896minus1

(56)

Expanding (36) and using (41) we have

119870119903

119896

= 119875119903

119896|119896minus1

119878119879

119896

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)minus1

119870119909

119896

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)

= 119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

(57)

Then

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(58)

Expanding (37) we have

119875119903

119896|119896

= (119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119875119909

119896|119896

= 119875119909

119896|119896minus1

+ (119880119896

minus 119881119896

) 119875119903

119896|119896minus1

(119880119879

119896

minus 119881119879

119896

)

minus (119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

(119880119879

119896

minus 119881119879

119896

))

(59)

Then using (41) and (44)

119875119909

119896|119896

= (119868 minus 119870119909

119896

1198671

119896

)119875119909

119896|119896minus1

(60)

Finally using (25) the estimated value of original state(119894119904120572

119904120573

119904120572

119904120573

) can be obtained by sum of the state119883withthe augmented state 119903

119896|119896minus1

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

(61)

119896|119896

= 119883119896|119896

+ 119881119896

119903119896|119896

(62)

Moreover the unknown parameter 119903( 119903

) is defined as

119903119896|119896minus1

= 119903119896|119896minus1

(63)

119903119896|119896

= 119903119896|119896

(64)

8 Mathematical Problems in Engineering

Based on the above analysis the TEKF can be decoupledinto two filters such as the modified bias free filter and biasfilter The modified bias filter gives the state estimation 119883

119896|119896

and the bias filter gives the bias estimate 119903

119896|119896

The correctedstate estimate 119883119886

119896|119896

(119896|119896

119903119896|119896

) of the TEKF is obtained fromthe estimates of the two filters and coupling equations119880

119896

and119881119896

[21] The modified bias free filter is expressed as follows

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119898119896minus1

(65)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

(66)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(119873119896

)minus1

(67)

119875119909

119896|119896

= (119868 minus 119870119909

119896

1198671

119896

)119875119909

119896|119896minus1

(68)

120578119909

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

+ (119878119896

minus 119878119896

) 119903119896|119896minus1

(69)

119883119896|119896

= 119870119909

119896

120578119909

119896

+ 119883119896|119896minus1

(70)

and the bias filter is

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

(71)

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896

(72)

119870119903

119896

= 119875119903

119896|119896minus1

119878119879

119896

(119873119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)minus1

(73)

119875119903

119896|119896

= (119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

(74)

120578119903

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

(75)

119903119896|119896

= 119903119896|119896minus1

+ 119870119903

119896

120578119903

119896

(76)

with the coupling equations

119878119896

= 1198671

119896

119880119896

+ 1198672

119896

(77)

119880119896

= 119880119896

(119868 minus 119876119903

119896

(119875119903

119896|119896minus1

)minus1

) (78)

119880119896

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119866minus1

119896minus1

(79)

119881119896

= 119880119896

minus 119870119909

119896

119878119896

(80)

119898119896minus1

= (119860119896minus1

119881119896minus1

+ 119863119896minus1

minus 119880119896

119866119896minus1

) 119903119896minus1|119896minus1

(81)

119876119909

119896

= 119876119909

119896

+ 119880119896

(119880119896

119876119903

119896

)119879

(82)

119873119896

= 1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

(83)

The initial conditions of TEKF algorithm are establishedwith the initial conditions of a classical EKF (119883

0|0

1199030|0

1198751199090|0

1198751199091199030|0

1198751199030|0

) so that

1198810

= 119875119909119903

0|0

(119875119903

0|0

)minus1

1198830|0

= 1198830|0

minus 1198810

1199030|0

1199030|0

= 1199030|0

119875119909

0|0

= 119875119909

0|0

minus 1198810

119875119903

0|0

119881119879

0

119875119903

0|0

= 119875119903

0|0

(84)

According to variables of full order filter119883 (1198831

1198832

1198833

1198834

)the stator flux and torque estimators for DTC-SVM ofFigure 1 are then given by

119904120572

= 1198831

119904120573

= 1198832

119904120572

= 1198833

119904120573

= 1198834

100381610038161003816100381610038161003816120595119904

100381610038161003816100381610038161003816= radic

2

119904120572

+ 2

119904120573

119904

= arctan119904120573

119904120572

119890

=3

2119901 (119904120572

119868119904120573

minus 119904120573

119868119904120572

)

(85)

where 119901 is the pole pairs of IM The estimated speed andelectromagnetic torque obtained from the TEKF observer areused to close the speed and torque loop to achieve sensorlessoperations

42 The Stability and Parameter Sensitivity Analysis ofthe TEKF

Theorem 3 The discrete time conventional extended Kalmanfilter (19)ndash(23) is equivalent to the two-stage extern Kalmanfilter (see (61)sim(83))

Proof Before proving the theorem the following five rela-tionships are needed

(1) Using (72) and (78)

119880119896+1

119866119896

119875119903

119896|119896

119866119879

119896

= 119880119896

119875119903

119896|119896minus1

(86)

(2) Using (67) and (73)

119870119909

119896

119872119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

(87)

119870119903

119896

119872119896

= 119875119903

119896|119896minus1

(119878119896

)119879

(88)

where

119872119896

= 1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

+ 119877119896

(89)

Mathematical Problems in Engineering 9

(3) Using (20) we have

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896minus1

(90)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119864119896minus1

(119875119909119903

119896minus1|119896minus1

)119879

119860119879

119896minus1

+ 119860119896minus1

119875119909119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119876119909

119896minus1

(91)

119875119909119903

119896|119896minus1

= 119860119896minus1

119875119909119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

(92)

(4) Using (21)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(93)

119870119903

119896

= (119875119909119903

119896|119896minus1

)119879

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(94)

(5) Using (22)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

(95)

119875119909119903

119896|119896

= 119875119909119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909119903

119896|119896minus1

(96)

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896

1198671

119896

119875119909119903

119896|119896minus1

(97)

By inductive reasoning suppose that at time 119896 minus 1the unknown parameter

119896minus1

and estimated state 119896minus1

areequal to the parameter 119903

119896minus1

and state 119883119896minus1

of the controlsystem respectively we show that TEKF is equivalent to theconventional EKF because these properties are still true attime 119896

Assume that at time 119896 minus 1

119883119896minus1|119896minus1

= 119896minus1|119896minus1

119903119896minus1|119896minus1

= 119903119896minus1|119896minus1

119875119909

119896minus1|119896minus1

= 11987511

119896minus1|119896minus1

119875119909119903

119896minus1|119896minus1

= 11987512

119896minus1|119896minus1

119875119903

119896minus1|119896minus1

= 11987522

119896minus1|119896minus1

(98)

where [ 119875119909119875

119909119903

(119875

119909119903)

119879119875

119903 ] and [ 11987511119875

12

(119875

12)

119879119875

22 ] represent the variance-covariance matrices of the system and estimated variablesrespectively

From (19) we have

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119863119896

119903119896minus1|119896minus1

+ 119861119896

119906119896minus1

(99)

Then using (98) (41) (62) (79) (81) (71) and (61)

119883119896|119896minus1

= 119860119896minus1

(119883119896minus1|119896minus1

+ 119881119896minus1

119903119896minus1|119896minus1

)

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

minus 119898119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

= 119896|119896minus1

(100)

Using (19) (71) (98) (63) and (64) we have

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119903119896|119896minus1

(101)

Using (91) (98) (78) (66) (79) (82) (86) and (72) we obtain

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896minus1

+ 119880119896

119866119896

119875119903

119896minus1|119896minus1

119866119879

119896

119880119879

119896

= 119875119909

119896|119896minus1

+ 119880119896

(119880119896

119875119903

119896|119896minus1

minus 119880119896

119876119903

119896minus1

)119879

= 119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

= 11987511

119896|119896minus1

(102)

Using (90) (98) (72) (32) (71) (29) and (97) we obtain

119875119903

119896|119896minus1

= 11987522

119896minus1|119896minus1

(103)

Using (92) (98) (33) (32) (79) (86) and (91)

119875119909119903

119896|119896minus1

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896minus1

119875119903

119896minus1|119896minus2

= 11987512

119896|119896minus1

(104)

Using (93) (101) (55) (73) (67) (80) and (87)

119870119909

119896

= (119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

) (1198671

119896

)119879

sdot (1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ (119881119896

+ 119870119909

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

(1198671

119896

)119879

)119880minus1

119896

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

)119880minus1

119896

+ 119881119896

119875119903

119896|119896minus1

119878119879

119896

119880minus1

119896

= 119870119909

119896

+ 119881119896

119870119903

119896

(105)

Using (94) (30) and (88) we obtain

119870119903

119896

= (119875119903

119896|119896minus1

)119879

119878119879

119896

119880minus1

119896

= 119870119903

119896|119896minus1

(106)

10 Mathematical Problems in Engineering

Next wewill show that (98) holds at time 119896 From (23)we have

119883119896|119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

)

= 119883119896|119896minus1

+ 119870119909

119896

119903119896

(107)

Then using (61) and (105) the above equation can be writtenas

119883119896|119896

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

+ (119870119909

119896

+ 119881119896

119870119903

119896

) 119903119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 1198671

119896

119883119896|119896minus1

)

+ (119880119896

minus 119870119909

119896

119878119896

) 119903119896|119896minus1

+ 119881119896

119870119903

119896

119903119896

= 119883119896|119896

+ 119881119896

119903119896|119896

= 119896|119896

(108)

Using (95) (105) (102) and (77)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

(109)

Then using (80) (68) (74) and (31) we obtain

119875119909

119896|119896

= 119875119909

119896|119896

+ 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

+ 119881119896

(119875119903

119896|119896

119878119879

119896

(119870119909

119896

)119879

minus 119870119903

119896

1198671

119896

119875119909

119896|119896minus1

)

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

= 11987511

119896|119896

(110)

Using (96) (30) (28) (105) and (80)

119875119909119903

119896|119896

= 119880119896

119875119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

1198671

119896

119880119896

minus 119881119896

119870119903

119896

1198671

119896

119880119896

)119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

119875119903

119896|119896

= 11987512

119896|119896

(111)

Using (97) (106) (95) (29) and (30) we obtain

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896|119896minus1

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119868 minus 119870119903

119896|119896minus1

119878119896

) 119875119903

119896|119896minus1

= 11987522

119896|119896

(112)

Table 1 Kalman estimation arithmetic operation requirement forthe conventional EKF structure

Number of multiplications(119899 = 6119898 = 2 and 119902 = 2)

Number of additions(119899 = 6119898 = 2 and 119902 = 2)

119860119896

119861119896

and 119862

119896

Function of system (9) Function of system (3)

119883119886

119896|119896minus1

1198992 + 119899119902 (48) 1198992 + 119899119902 minus 119899 (42)119875119896|119896minus1

21198993 (432) 21198993 minus 1198992 (396)119883119886

119896|119896minus1

2119899119898 (24) 2119899119898 (24)

119870119896

1198992

119898 + 21198991198982

+ 1198983 (168) 119899

2

119898 + 21198991198982

+ 1198983

minus 2119899119898

(104)119875119896|119896

1198992119898 + 1198993 (288) 1198992119898 + 1198993 minus 1198992 (252)Total 960 818

Finally we show that (98) holds at time 119896 = 0 This can beverified by the initial conditions of TEKF algorithm

43 Numerical Complexity of the Algorithm Tables 1 and2 show the computational effort at each sample time bythe conventional EKF algorithm and TEKF (where roughmatrix-based implementation is used) in which as definedabove 119899 is the dimension of the state vector 119883

119896

119898 is thedimension of the measurement 119884

119896

119902 is the input vector 119880119896

and 119901 is the dimension of the parameter 119903

119896

The total numberof arithmetic operations (additions and multiplications) persample time of the TEKF is 1314 compared with 1778 for arough implementation of a conventional EKF which meansthe operation cost can reduce by 26

5 Simulation and Experimental Results

51 Simulation Results To test the feasibility and perfor-mance of the TEKF method the sensorless DTC-SVM tech-nique for IM drives described in Section 2 is implementedin MATLABSIMULINK environment The values of theinitial state covariance matrices 119875

0

119876 and 119877 have a greatinfluence on the performance of the estimation methodThe diagonal initial state covariance matrix 119875

0

representsvariances or mean-squared errors in the knowledge of theinitial conditions Matrix 119876 gives the statistical descriptionof the drive system Matrix 119877 is related to measured noiseThey can be obtained by considering the stochastic propertiesof the corresponding noises However a fine evaluation ofthe covariance matrices is very difficult because they areusually not known In this paper tuning the initial values ofcovariance matrices 119875

0

119876 and 119877 is using particular criteria[22] to achieve steady-state behaviors of the relative estimatedstates as given by

119876 = diag 20 20 1119890 minus 6 1119890 minus 6 10 10

1198750

= diag 01 01 05 05 1 1

119877 = diag 01 01

(113)

Mathematical Problems in Engineering 11

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000Sp

eed

(rpm

)

(a) Speed estimation

02 04 06 08 10Times (s)

minus002

0

002

004

006

008

01

012

014

016

018

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF

Theta (real)Theta (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7Po

sitio

n (r

ad)

(d) TEKF rotor position error

Current (real)Current (TEKF)

Curr

entIs120572

(A)

02 04 06 08 10Times (s)

minus12

minus8

minus4

0

4

8

12

(e) Real stator current 119868119904120572

and estimation (TEKF)

Current (real)Current (TEKF)

minus12

minus8

minus4

0

4

8

12

Curr

entIs120573

(A)

02 04 06 08 10Times (s)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 3 Continued

12 Mathematical Problems in Engineering

Flux (EKF)Flux (TEKF)

minus05 0 05 1minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(g) Stator flux estimation by TEKF and EKF

02 04 06 08 10Times (s)

minus2

0

2

4

6

8

10

12

14

16

Ψs

(Wb)

times10minus3

(h) Difference Stator flux estimation between TEKF and EKF

Figure 3 Simulation results for parameters estimation

Table 2 Kalman estimation arithmetic operation requirement for the TEKF structure

Number of multiplications(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

Number of additions(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

119860119896

119862119896

119864119896

1198671119896

1198672119896

119861119896

and119863119896

Function of system (25) Function of system (11)119883119896|119896minus1

1198992 + 119899119902 (24) 1198992 + 119899119902 (24)119875119909

119896|119896minus121198993 (128) 21198993 minus 1198992 (112)

119870119909

119896|1198961198992

119898 + 1198991198982 (48) 119899

2

119898 + 1198991198982

minus 2119899119898 (32)119875119909

119896|1198961198992

119898 + 1198993 (96) 119899

2

119898 + 1198993

minus 1198992 (80)

119883119896|119896

2119899119898 + 119898119901 (20) 2119899119898 + 2119898119901 (24)119903119896|119896minus1

1199012 (4) 1199012 minus 119901 (2)119875119903

119896|119896minus121199013 (16) 21199013 minus 1199012 (12)

119870119903

119896|11989631199012119898 + 1199011198982 (32) 31199012119898 + 1199011198982 + 1198982 minus 4119901119898 (20)

119875119903

119896|1198961199013 + 1199012119898 (16) 1199013 + 1199012119898 minus 1199012 (12)

119903119896|119896

2119898119901 + 119899119898 (12) 2119898119901 + 119899119898 (16)119878119896

119898119899119901 (16) 119899119898119901 (16)119880119896minus1

1198992119901 + 1198991199012 (56) 1198992119901 + 1198991199012 minus 119899119901 (48)119881119896

119899119901119898 (16) 119899119901 (8)119880119896minus1

21198991199012 (32) 21198991199012 (32)119898119896minus1

1198992

119901 + 1198991199012

+ 119899119901 (56) 1198992

119901 + 1198991199012

+ 119899119901 minus 119899 (52)119876119909

119896minus121198992

119901 (64) 21198992

119901 + 1198991199012 (64)

119878119896

119898119899119901 (16) 119898119899119901 minus 119898119901 (12)119873119896

21198981198992 (32) 21198981198992 minus 1198982 (60)Total 688 626

In the simulation a comparison is made to verify theequivalence of EKF and TEKF Real-time parameters esti-mated by TEKF are used to formulate the closed loop suchas rotor speed stator flux and electromagnetic torque Theestimations obtained by EKF algorithm are not included in

the sensorless DTC-SVM strategy and only evaluated in openloop A step reference speed was applied to the simulation

The machine is accelerated from 0 rpm to 1000 rpm at0 s and the torque load is set to 4N The simulation resultsof parameter estimation are shown in Figure 3 Figures 3(a)

Mathematical Problems in Engineering 13

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000

Spee

d (r

pm)

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus03

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Theta (real)Theta (TEKF)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

02 04 06 08 10Times (s)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)

Figure 4 Simulation results with parameter variation (119877119903

= 05119877119903nom)

and 3(c) represent the performance of the speed and rotorposition tracking capabilities of the control system withTEKF and EKF respectively Figures 3(e) 3(f) and 3(g)represent the estimated stator current and flux they showthat ripples are significantly suppressed due to the SVMmodulation scheme Figures 3(b) 3(d) and 3(h) show thatfor variations of speed reference the rotor speed rotorposition and stator flux errors between the two observers arevery little It is verified that the two observers are equivalentThe difference between two estimators is caused by accuracyloss in TEKF which uses more calculation steps

In order to further verify the performance of TEKFagainst model-plant parameter mismatches and the equiv-alence of two observers the change in rotor resistance isconsidered Rotor resistance will increase due to temperaturerise while the motor is running To simulate this conditionthe rotor resistance in TEKF and EKF is increased to 200comparedwith the normal value which is equivalent to a 50decrease in the actual rotor resistance The machine is stillaccelerated from 0 rpm to 1000 rpm at 0 s and the torque is setto 4N Figure 4 shows that for variation of rotor resistancethe steady-state speed and rotor position errors are negligible

14 Mathematical Problems in Engineering

(a)

Computer

380V

TXRX

120596r

Incrementalencoder

Udc

MicrocomputerDSP TMS3206713

Drive signalCurrentsensor

Voltagesensor

Voltagesensor

IM MC

ISU ISV

VSU VSW VSV

(b)

Figure 5 Complete drive system (a) Picture of experimental setup (b) Functional block diagram of the experimental setup

and the difference of the speed and rotor position estimationsbetween the two observers is rather null

52 Experimental Results The overall experimental setup isshown in Figure 5 and the specifications and rated parametersof the IM controller and inverter are listed in Table 3 Inthe experimental hardware an Expert3 control system fromMyway company and a three-phase two-pole 15 kW IM areappliedThe IM is mechanically coupled to a magnetic clutch(MC) which provides rated torque even at very low speedThe main processor in Expert3 control system is a floatingpoint processor TMS320C6713 with a max clock speed of225MHz All the algorithms including TEKF EKF DTCalgorithm and some transformation modules are imple-mented in TMS320C6713 with 100120583s sampling time and dataacquisition of the parameter estimations measured variablesand their visualization are realized on the cockpit provided byPEView9 software Insulated Gate Bipolar Transistor (IGBT)module is driven by the PWM signal with a switchingfrequency of 10 kHz and 2 120583s dead time The stator currentsare measured via two Hall effect current sensors The rotorangle and speed of IM are measured from an incrementalencoder with 2048 pulses per revolution

This experiment test is here to testify the performanceof TEKF and demonstrate that the two estimators aremathematically equivalent The machine is accelerated from600 rpm to 1000 rpm and 4N torque load is set Theexperimental results of parameter estimation based on twoobservers are given in Figures 6 and 7 Figures 6(a) and 6(c)show that the TEKF still has a good tracking performance ofthe speed and rotor position in experiment Figures 6(d) 6(e)and 6(f) illustrate stator flux and stator current estimationrobustness Figures 6(b) 6(g) and 6(f) referring to thedifference in speed and stator current estimations given by

Table 3 Specification of induction motor and inverter

Induction motor ValueNominal torque 10NmNominal voltage 380VRotor resistance 119877

119903

25ΩStator resistance 119877

119904

36ΩStator inductances 119871

119904

0301HRotor inductances 119871

119903

0302HMutual inductances 119871

119898

0273HPole pairs 2Invertercontroller ValueSwitching device 1000V 80A IGBTControl cycle time 100 120583sMain CPU DSP TMS320C6713 225MHz

the two observers are still small These experiment resultsprove that the two estimators are mathematically equivalentFigure 7 shows the speed and rotor position estimationsbased onTEKFandEKF for a 50decrease of rotor resistance(the same as the simulation) As expected the steady errorof the TEKF and the difference in speed and rotor positionestimations are still tiny Robustness of TEKF is verified

6 Conclusion

Themajor shortcoming of the conventional EKF is numericalproblems and computational burden due to the high orderof the mathematical models This has generally limited thereal-time digital implementation of the EKF for industrialfield So in this study a novel extended Kalman filter

Mathematical Problems in Engineering 15

Speed (real)Speed (EKF)Speed (TEKF)

Spee

d (r

min

)

02 04 06 08 10t (02 sgrid)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimationSp

eed

(rm

in)

minus005

0

005

01

015

02

025

03

035

04

045

02 04 06 08 10t (02 sgrid)

(b) Difference of speed estimation between EKF and TEKF

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation

Flux (EKF)Flux (TEKF)

minus08 minus04 1minus02 0 02 04 06 08minus06minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(d) Stator flux estimation by TEKF and EKF

Is120572 (real)Is120572 (TEKF)

Curr

entIs120572

(A)

minus6

minus4

minus2

0

2

4

6

02 04 06 08 10t (02 sgrid)

(e) Real stator current 119868119904120572

and estimation (TEKF)

Is120573 (real)Is120573 (TEKF)

minus6

minus4

minus2

0

2

4

6

Curr

entIs120573

(A)

02 04 06 08 10t (02 sgrid)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 6 Continued

16 Mathematical Problems in Engineering

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Erro

r cur

rentIs120572

(A)

02 04 06 08 10t (02 sgrid)

(g) TEKF stator current 119868119904120572

error

minus02

minus015

minus01

minus005

0

005

01

015

Erro

r cur

rentIs120573

(A)

02 04 06 08 10t (02 sgrid)

(h) TEKF stator current 119868119904120573

error

Figure 6 Experimental results for parameters estimation

Speed (real)Speed (EKF)

Speed (TEKF)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

0

005

01

015

02

025

03

035

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)Figure 7 Experimental results with parameter variation (119877

119903

= 05119877119903nom)

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

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Page 4: Research Article Direct Torque Control of Sensorless ...

4 Mathematical Problems in Engineering

119884(119905) = [119868119904120572

119868119904120573]119879 as the output vector the IM model is

described by the general nonlinear state space model

∙

119883 (119905) = 119860 (119905)119883 (119905) + 119861 (119905) 119906 (119905) + 119863 (119905) 119903 (119905)

∙

119903 (119905) = 119866 (119905) 119903 (119905)

119884 (119905) = 119862 (119905)119883 (119905)

(6)

with

119860 (119905)

=

[[[[[[[[[

[

minus(119877119904

120590119871119904

+1

120590119879119903

) minus119901120596119903

1

120590119871119904

119879119903

119901120596119903

120590119871119904

119901120596119903

minus(119877119904

120590119871119904

+1

120590119879119903

) minus119901120596119903

120590119871119904

1

120590119871119904

119879119903

minus119877119904

0 0 0

0 minus119877119904

0 0

]]]]]]]]]

]

119861 (119905) =[[[

[

1

120590119871119904

0 1 0

01

120590119871119904

0 1

]]]

]

119879

119863 (119905) = 0

119862 (119905) = [1 0 0 0

0 1 0 0]

119866 (119905) = [0 1

0 0]

(7)

Remark 1 Matrices 119862(119905) and 119866(119905) are not affected by uncer-tainties

Remark 2 Matrix119860(119905) is time-varying because it depends onthe rotor speed 120596

119903

For digital implementation of estimator on a microcon-

troller a discrete timemathematicalmodel of IMs is requiredThese equations can be obtained from (6)

119883119896+1

= 119860119896

119883119896

+ 119861119896

119906119896

+ 119863119896

119903119896

119903119896+1

= 119866119896

119903119896

119884119896

= 119862119896

119883119896

(8)

The solution of nonhomogenous state equations (6) sat-isfying the initial condition119883(119905)|

119905=1199050= 119883(119905

0

) is

119883 (119905) = 119890119860(119905minus1199050)119883(119905

0

) + int119905

1199050

119890119860(119905minus120591)

119861119906 (120591) 119889120591 (9)

Integrating from 1199050

= 119896119879119904

to 119905 = (119896 + 1)119879119904

we can obtain that

119883((119896 + 1) 119879119904

) = 119890119860119879119904119883(119896119879

119904

)

+ int(119896+1)119879119904

119896119879

119890119860((119896+1)119879119904minus120591)119861119889120591119906 (119896119879

119904

)

(10)

The above equations lead to

119860119896

= 119890119860119879119904

119861119896

= 119860minus1

(119890119860119879119904 minus 119868) 119861

(11)

In the same way

119866119896

= 119890119866119879119904 (12)

Tolerating a small discretization error a first-order Taylorseries expansion of the matrix exponential is used

119860119896

= 119890119860119879119904 asymp 119860119879

119904

+ 119868

119866119896

= 119890119866119879119904 asymp 119866119879

119904

+ 119868

119861119896

= 119860minus1

(119890119860119879119904 minus 119868) 119861 asymp 119879

119904

119861

119863119896

= 0

(13)

with

119860119896

=

[[[[[[[[[

[

minus(119877119904

119879119904

120590119871119904

+119879119904

120590119879119903

) + 1 minus119901120596119903

119879119904

119879119904

120590119871119904

119879119903

119901120596119903

119879119904

120590119871119904

119901120596119903

119879119904

minus(119877119904

119879119904

120590119871119904

+119879119904

120590119879119903

) + 1 minus119901120596119903

119879119904

120590119871119904

119879119904

120590119871119904

119879119903

minus119877119904

119879119904

0 1 0

0 minus119877119904

119879119904

0 1

]]]]]]]]]

]

G119896

= [1 119879119904

0 1]

Mathematical Problems in Engineering 5

119861119896

=[[[

[

119879119904

120590119871119904

0 119879119904

0

0119879119904

120590119871119904

0 119879119904

]]]

]

119879

119862119896

= [1 0 0 0

0 1 0 0]

119863119896

= [0 0

0 0]

(14)

Based on discretized IM model a conventional EKFestimator is designed for estimation of stator flux currentelectromagnetic torque and rotor speed of IM for sensorlessDTC-SVM operations Treating119883

119896

as the full order state and119903119896

as the augmented system state the state vector is chosen tobe 119883119886119896

= [119883119896

119903119896]119879 119906119896

= [119880119904120572

119880119904120573]119879 and 119884

119896

= [119868119904120572

119868119904120573]119879 are

chosen as input and output vectors because these quantitiescan be easily obtained from measurements of stator currentsand voltage construction usingDC link voltage and switchingstatus Considering the parameter errors and noise of systemthe discrete time state space model of IMs in the stationary(120572120573) reference frame is described by

119883119886

119896+1

= 119860119896

119883119886

119896

+ 119861119896

119906119896

+ 119908119896

119884119896+1

= 119862119896

119883119886

119896

+ V119896

(15)

with

119860119896

= [119860119896

119863119896

0 119866119896

]

119861119896

= [119861119896

0]

119862119896

= [119862119896

0]

119879

119908119896

= [119908119909119896

119908119903119896

]

(16)

The system noise119908119896

and measurement noise V119896

are whiteGaussian sequence with zero-mean and following covariancematrices

119864[[[

[

[[

[

119908119909

119896

119908119903119896

V119896

]]

]

[[[

[

119908119909119895

119908119903119895

V119895

]]]

]

119879

]]]

]

=[[

[

119876119909119896

0 0

0 119876119903119896

0

0 0 119877119896

]]

]

120575119896119895

(17)

where 119876119909119896

gt 0 119876119903119896

gt 0 119877119896

gt 0 and 120575119896119895

is theKronecker delta The initial states 119883

0

and 1199030

are assumed tobe uncorrelated with the zero-mean noises 119908119909

119896

119908119903119896

and V119896

The initial conditions are assumed to be Gaussian randomvariables119883

0

and 1199030

that are defined as follows

119864((1198830

minus 119883lowast

0

) (1198830

minus 119883lowast

0

)119879

) = 119875119909

0

119864 (1198830

) = 119883lowast

0

119864 (1199030

) = 119903lowast

0

119864((1199030

minus 119903lowast

0

) (1199030

minus 119903lowast

0

)119879

) = 119875119903

0

119864 ((1198830

minus 119883lowast

0

) (1199030

minus 119903lowast

0

)119879

) = 119875119909119903

0

(18)

The overall structure of the EKF is well-known byemploying a two-step prediction and correction algorithm[13] Hence the application of EKF filter to the state spacemodel of IM (15) is described by

119883119886

119896|119896minus1

= 119860119896minus1

119883119886

119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

(19)

119875119896|119896minus1

= 119865119896minus1

119875119896minus1|119896minus1

119865119879

119896minus1

+ 119876119896minus1

(20)

119870119896

= 119875119896|119896minus1

119867119879

119896

(119867119896

119875119896|119896minus1

119867119879

119896

+ 119877)minus1

(21)

119875119896|119896

= 119875119896|119896minus1

minus 119870119896

119867119896

119875119896|119896minus1

(22)

119883119886

119896|119896

= 119883120572

119896|119896minus1

+ 119870119896

(119884119896

minus 119862119896

119883119886

119896|119896minus1

) (23)

with

119865119896minus1

= [119865119896minus1

119864119896minus1

0 119866119896minus1

]

119876 (sdot) = [119876119909 (sdot) 0

0 119876119903 (sdot)]

119865119896minus1

=120597

120597119883(119860119896minus1

119883119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1

) = 119860119896

119864119896minus1

=120597

120597119903(119860119896minus1

119883119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1

)

= [0 0 0 0

1199011 1199012 0 0]

119879

6 Mathematical Problems in Engineering

1199011 = (minus119901119894119904119887[119896minus1|119896minus1]

+119901

120590119871119904

120595119904120573[119896minus1|119896minus1]

)119879119904

1199012 = (119901119894119904120572[119896minus1|119896minus1]

minus119901

120590119871119904

120595119904120572[119896minus1|119896minus1]

)119879119904

119867119896

= [1198671

119896

1198672

119896

] =120597

120597119883119886(119862119896

119883119886

)

1198671

119896

= 119862119896

1198672

119896

= 0

119875 (sdot) = [119875119909 (sdot) 119875119909119903 (sdot)

(119875119909119903 (sdot))119879

119875119903 (sdot)]

(24)

4 The Two-Stage Extended Kalman Filter

41The TEKFAlgorithm Asmentioned in conventional EKFestimator previously the memory and computational costsincrease with the augmented state dimension Consideringsampling time is very small only high performance micro-controller can qualify for this work Hence the conventionalEKF algorithm may be impractical to implement The extracomputation of 119875

119909119903(sdot) terms leads to this computationalcomplexityTherefore we can reduce the computational com-plexity from application point of view if the 119875119909119903(sdot) terms canbe eliminated In this section a two-stage extended Kalmanfilter without explicitly calculating 119875119909119903(sdot) terms is discussed

Following the same approach as given in [15] the TEKFis decomposed into two filters such as the modified bias freefilter and the bias filter by applying the following two-stage119880-119881 transformation

119883119886

119896|119896minus1

= 119879 (119880119896

)119883119886

119896|119896minus1

119875119896|119896minus1

= 119879 (119880119896

) 119875119896|119896minus1

119879 (119880119896

)119879

119870119896

= 119879 (119881119896

)119870119896

119875119896|119896

= 119879 (119881119896

) 119875119896|119896

119879 (119881119896

)119879

119883119886

119896|119896

= 119879 (119881119896

)119883119886

119896|119896

(25)

where

119883119886

119896(sdot)

= [119883119896(sdot)

119903119896(sdot)

]

119870119896

= [119870119909

119896

119870119903

119896

]

119875119896(sdot)

= [

[

119875119909

119896(sdot)

0

0 119875119903

119896(sdot)

]

]

119879 (119880119896

) = [119868 119880119896

0 119868]

(26)

The main advantage of using the 119879(119880119896

) transformation isthat the inverse transformation 119879

minus1

(119880119896

) = 119879(minus119880119896

) involvesonly a change of sign Two blending matrices 119880

119896

and 119881119896

are defined by 119880119896

= 119875119909119903119896|119896minus1

(119875119903119896|119896minus1

)minus1 and 119881119896

= 119875119909119903119896|119896

(119875119903119896|119896

)minus1respectively Using characteristic of 119879(119880

119896

) (25) become

119883119886

119896|119896minus1

= 119879 (minus119880119896

)119883119886

119896|119896minus1

119875119896|119896minus1

= 119879 (minus119880119896

) 119875119896|119896minus1

119879 (minus119880119896

)119879

119870119896

= 119879 (minus119881119896

)119870119896

119875119896|119896

= 119879 (minus119881119896

) 119875119896|119896

119879 (minus119881119896

)119879

119883119886

119896|119896

= 119879 (minus119881119896

)119883119886

119896|119896

(27)

And the following relationships are obtained from (25)

119875119909

119896|119896minus1

= 119875119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

(28)

119875119903

119896|119896minus1

= 119875119903

119896|119896minus1

(29)

119875119909119903

119896|119896minus1

= 119880119896

119875119903

119896|119896minus1

(30)

119875119909

119896|119896

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

(31)

119875119903

119896|119896

= 119875119903

119896|119896

(32)

119875119909119903

119896|119896

= 119881119896

119875119903

119896|119896

(33)

Based on two-step iterative substitution method of [15]the transformed filter expressed by (27) can be recursivelycalculated as follows

119883119886

119896|119896minus1

= 119879 (minus119880119896

)

sdot (119860119896minus1

119879 (119881119896minus1

)119883119886

119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

)

(34)

119875119896|119896minus1

= 119879 (minus119880119896

)

sdot (119876119896minus1

+ 119865119896minus1

119879 (119881119896minus1

) 119875119896minus1|119896minus1

119879 (119881119896minus1

)119879

119865119879

119896minus1

)

sdot 119879 (minus119880119896

)119879

(35)

119870119896

= 119879 (119880119896

minus 119881119896

) 119875119896|119896minus1

119879 (119880119896

)119879

sdot 119867119879

119896

(119867119896

119879 (119880119896

) 119875119896|119896minus1

119879 (119880119896

)119879

119867119879

119896

+ 119877)minus1

(36)

119875119896|119896

= (119879 (119880119896

minus 119881119896

) minus 119870119896

119867119896

119879 (119880119896

))

sdot 119875119896|119896minus1

119879 (119880119896

minus 119881119896

)119879

(37)

119883119886

119896|119896

= 119870119896

(119884119896

minus 119862119896

119879 (119880119896

)119883119886

119896|119896minus1

) + 119879 (119880119896

minus 119881119896

)

sdot 119883119886

119896|119896minus1

(38)

Mathematical Problems in Engineering 7

Using (35) (37) and the block diagonal structure of 119875(sdot)

thefollowing relations can be obtained

0 = 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

minus 119880119896

119876119903

119896minus1

minus 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

0 = 119880119896

minus 119881119896

minus 119870119909

119896

119878119896

(39)

where 119880119896

and 119878119896

are defined as

119880119896

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119866minus1

119896minus1

(40)

119878119896

= 1198671

119896

119880119896

+ 1198672

119896

(41)

The above equations lead to

119880119896

119875119903

119896|119896minus1

= 119880119896

(119875119903

119896|119896minus1

minus 119876119903

119896

) (42)

119880119896

= 119880119896

(119868 minus 119876119903

119896

(119875119903

119896|119896minus1

)minus1

) (43)

119881119896

= 119880119896

minus 119870119909

119896

119878119896

(44)

Define the following notation

119860119896minus1

119879 (119881119896minus1

) = [119860119896minus1

119860119896minus1

119881119896minus1

+ 119864119896minus1

0 119866119896minus1

] (45)

The equations of themodified bias free filter and the bias filterare acquired by the next steps

Expanding (34) we have

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119898119896minus1

(46)

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

(47)

where

119898119896minus1

= (119860119896minus1

119881119896minus1

+ 119863119896minus1

minus 119880119896

119866119896minus1

) 119903119896minus1|119896minus1

(48)

Expanding (35) we have

119875119909

119896|119896minus1

= (119864119896minus1

+ 119860119896minus1

119881119896minus1

minus 119880119896

119866119896minus1

) 119875119903

119896minus1|119896minus1

lowast (119864119896minus1

+ 119860119896minus1

119881119896minus1

minus 119880119896

119866119896minus1

)119879

+ 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

+ 119880119896

119876119903

119896

119880119879

119896

(49)

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896

(50)

Then using (40) (43) and (47) (49) can be written as

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

(51)

where

119876119909

119896

= 119876119909

119896

+119872119896

(119872119896

119876119903

119896

)119879

(52)

Expanding (38) and using (41) and (44) we have

119883119896|119896

= 119883119896|119896minus1

+ (119880119896

minus 119881119896

) 119903119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

)

119903119896|119896

= 119903119896|119896minus1

+ 119870119903

119896

(119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

)

(53)

Then

119883119896|119896

= 119870119909

119896

120578119909

119896

+ 119883119896|119896minus1

(54)

where

119878119896

= 119862119896

119880119896

(55)

120578119909

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

+ (119878119896

minus 119878119896

) 119903119896|119896minus1

(56)

Expanding (36) and using (41) we have

119870119903

119896

= 119875119903

119896|119896minus1

119878119879

119896

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)minus1

119870119909

119896

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)

= 119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

(57)

Then

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(58)

Expanding (37) we have

119875119903

119896|119896

= (119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119875119909

119896|119896

= 119875119909

119896|119896minus1

+ (119880119896

minus 119881119896

) 119875119903

119896|119896minus1

(119880119879

119896

minus 119881119879

119896

)

minus (119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

(119880119879

119896

minus 119881119879

119896

))

(59)

Then using (41) and (44)

119875119909

119896|119896

= (119868 minus 119870119909

119896

1198671

119896

)119875119909

119896|119896minus1

(60)

Finally using (25) the estimated value of original state(119894119904120572

119904120573

119904120572

119904120573

) can be obtained by sum of the state119883withthe augmented state 119903

119896|119896minus1

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

(61)

119896|119896

= 119883119896|119896

+ 119881119896

119903119896|119896

(62)

Moreover the unknown parameter 119903( 119903

) is defined as

119903119896|119896minus1

= 119903119896|119896minus1

(63)

119903119896|119896

= 119903119896|119896

(64)

8 Mathematical Problems in Engineering

Based on the above analysis the TEKF can be decoupledinto two filters such as the modified bias free filter and biasfilter The modified bias filter gives the state estimation 119883

119896|119896

and the bias filter gives the bias estimate 119903

119896|119896

The correctedstate estimate 119883119886

119896|119896

(119896|119896

119903119896|119896

) of the TEKF is obtained fromthe estimates of the two filters and coupling equations119880

119896

and119881119896

[21] The modified bias free filter is expressed as follows

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119898119896minus1

(65)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

(66)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(119873119896

)minus1

(67)

119875119909

119896|119896

= (119868 minus 119870119909

119896

1198671

119896

)119875119909

119896|119896minus1

(68)

120578119909

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

+ (119878119896

minus 119878119896

) 119903119896|119896minus1

(69)

119883119896|119896

= 119870119909

119896

120578119909

119896

+ 119883119896|119896minus1

(70)

and the bias filter is

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

(71)

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896

(72)

119870119903

119896

= 119875119903

119896|119896minus1

119878119879

119896

(119873119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)minus1

(73)

119875119903

119896|119896

= (119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

(74)

120578119903

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

(75)

119903119896|119896

= 119903119896|119896minus1

+ 119870119903

119896

120578119903

119896

(76)

with the coupling equations

119878119896

= 1198671

119896

119880119896

+ 1198672

119896

(77)

119880119896

= 119880119896

(119868 minus 119876119903

119896

(119875119903

119896|119896minus1

)minus1

) (78)

119880119896

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119866minus1

119896minus1

(79)

119881119896

= 119880119896

minus 119870119909

119896

119878119896

(80)

119898119896minus1

= (119860119896minus1

119881119896minus1

+ 119863119896minus1

minus 119880119896

119866119896minus1

) 119903119896minus1|119896minus1

(81)

119876119909

119896

= 119876119909

119896

+ 119880119896

(119880119896

119876119903

119896

)119879

(82)

119873119896

= 1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

(83)

The initial conditions of TEKF algorithm are establishedwith the initial conditions of a classical EKF (119883

0|0

1199030|0

1198751199090|0

1198751199091199030|0

1198751199030|0

) so that

1198810

= 119875119909119903

0|0

(119875119903

0|0

)minus1

1198830|0

= 1198830|0

minus 1198810

1199030|0

1199030|0

= 1199030|0

119875119909

0|0

= 119875119909

0|0

minus 1198810

119875119903

0|0

119881119879

0

119875119903

0|0

= 119875119903

0|0

(84)

According to variables of full order filter119883 (1198831

1198832

1198833

1198834

)the stator flux and torque estimators for DTC-SVM ofFigure 1 are then given by

119904120572

= 1198831

119904120573

= 1198832

119904120572

= 1198833

119904120573

= 1198834

100381610038161003816100381610038161003816120595119904

100381610038161003816100381610038161003816= radic

2

119904120572

+ 2

119904120573

119904

= arctan119904120573

119904120572

119890

=3

2119901 (119904120572

119868119904120573

minus 119904120573

119868119904120572

)

(85)

where 119901 is the pole pairs of IM The estimated speed andelectromagnetic torque obtained from the TEKF observer areused to close the speed and torque loop to achieve sensorlessoperations

42 The Stability and Parameter Sensitivity Analysis ofthe TEKF

Theorem 3 The discrete time conventional extended Kalmanfilter (19)ndash(23) is equivalent to the two-stage extern Kalmanfilter (see (61)sim(83))

Proof Before proving the theorem the following five rela-tionships are needed

(1) Using (72) and (78)

119880119896+1

119866119896

119875119903

119896|119896

119866119879

119896

= 119880119896

119875119903

119896|119896minus1

(86)

(2) Using (67) and (73)

119870119909

119896

119872119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

(87)

119870119903

119896

119872119896

= 119875119903

119896|119896minus1

(119878119896

)119879

(88)

where

119872119896

= 1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

+ 119877119896

(89)

Mathematical Problems in Engineering 9

(3) Using (20) we have

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896minus1

(90)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119864119896minus1

(119875119909119903

119896minus1|119896minus1

)119879

119860119879

119896minus1

+ 119860119896minus1

119875119909119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119876119909

119896minus1

(91)

119875119909119903

119896|119896minus1

= 119860119896minus1

119875119909119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

(92)

(4) Using (21)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(93)

119870119903

119896

= (119875119909119903

119896|119896minus1

)119879

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(94)

(5) Using (22)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

(95)

119875119909119903

119896|119896

= 119875119909119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909119903

119896|119896minus1

(96)

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896

1198671

119896

119875119909119903

119896|119896minus1

(97)

By inductive reasoning suppose that at time 119896 minus 1the unknown parameter

119896minus1

and estimated state 119896minus1

areequal to the parameter 119903

119896minus1

and state 119883119896minus1

of the controlsystem respectively we show that TEKF is equivalent to theconventional EKF because these properties are still true attime 119896

Assume that at time 119896 minus 1

119883119896minus1|119896minus1

= 119896minus1|119896minus1

119903119896minus1|119896minus1

= 119903119896minus1|119896minus1

119875119909

119896minus1|119896minus1

= 11987511

119896minus1|119896minus1

119875119909119903

119896minus1|119896minus1

= 11987512

119896minus1|119896minus1

119875119903

119896minus1|119896minus1

= 11987522

119896minus1|119896minus1

(98)

where [ 119875119909119875

119909119903

(119875

119909119903)

119879119875

119903 ] and [ 11987511119875

12

(119875

12)

119879119875

22 ] represent the variance-covariance matrices of the system and estimated variablesrespectively

From (19) we have

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119863119896

119903119896minus1|119896minus1

+ 119861119896

119906119896minus1

(99)

Then using (98) (41) (62) (79) (81) (71) and (61)

119883119896|119896minus1

= 119860119896minus1

(119883119896minus1|119896minus1

+ 119881119896minus1

119903119896minus1|119896minus1

)

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

minus 119898119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

= 119896|119896minus1

(100)

Using (19) (71) (98) (63) and (64) we have

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119903119896|119896minus1

(101)

Using (91) (98) (78) (66) (79) (82) (86) and (72) we obtain

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896minus1

+ 119880119896

119866119896

119875119903

119896minus1|119896minus1

119866119879

119896

119880119879

119896

= 119875119909

119896|119896minus1

+ 119880119896

(119880119896

119875119903

119896|119896minus1

minus 119880119896

119876119903

119896minus1

)119879

= 119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

= 11987511

119896|119896minus1

(102)

Using (90) (98) (72) (32) (71) (29) and (97) we obtain

119875119903

119896|119896minus1

= 11987522

119896minus1|119896minus1

(103)

Using (92) (98) (33) (32) (79) (86) and (91)

119875119909119903

119896|119896minus1

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896minus1

119875119903

119896minus1|119896minus2

= 11987512

119896|119896minus1

(104)

Using (93) (101) (55) (73) (67) (80) and (87)

119870119909

119896

= (119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

) (1198671

119896

)119879

sdot (1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ (119881119896

+ 119870119909

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

(1198671

119896

)119879

)119880minus1

119896

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

)119880minus1

119896

+ 119881119896

119875119903

119896|119896minus1

119878119879

119896

119880minus1

119896

= 119870119909

119896

+ 119881119896

119870119903

119896

(105)

Using (94) (30) and (88) we obtain

119870119903

119896

= (119875119903

119896|119896minus1

)119879

119878119879

119896

119880minus1

119896

= 119870119903

119896|119896minus1

(106)

10 Mathematical Problems in Engineering

Next wewill show that (98) holds at time 119896 From (23)we have

119883119896|119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

)

= 119883119896|119896minus1

+ 119870119909

119896

119903119896

(107)

Then using (61) and (105) the above equation can be writtenas

119883119896|119896

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

+ (119870119909

119896

+ 119881119896

119870119903

119896

) 119903119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 1198671

119896

119883119896|119896minus1

)

+ (119880119896

minus 119870119909

119896

119878119896

) 119903119896|119896minus1

+ 119881119896

119870119903

119896

119903119896

= 119883119896|119896

+ 119881119896

119903119896|119896

= 119896|119896

(108)

Using (95) (105) (102) and (77)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

(109)

Then using (80) (68) (74) and (31) we obtain

119875119909

119896|119896

= 119875119909

119896|119896

+ 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

+ 119881119896

(119875119903

119896|119896

119878119879

119896

(119870119909

119896

)119879

minus 119870119903

119896

1198671

119896

119875119909

119896|119896minus1

)

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

= 11987511

119896|119896

(110)

Using (96) (30) (28) (105) and (80)

119875119909119903

119896|119896

= 119880119896

119875119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

1198671

119896

119880119896

minus 119881119896

119870119903

119896

1198671

119896

119880119896

)119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

119875119903

119896|119896

= 11987512

119896|119896

(111)

Using (97) (106) (95) (29) and (30) we obtain

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896|119896minus1

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119868 minus 119870119903

119896|119896minus1

119878119896

) 119875119903

119896|119896minus1

= 11987522

119896|119896

(112)

Table 1 Kalman estimation arithmetic operation requirement forthe conventional EKF structure

Number of multiplications(119899 = 6119898 = 2 and 119902 = 2)

Number of additions(119899 = 6119898 = 2 and 119902 = 2)

119860119896

119861119896

and 119862

119896

Function of system (9) Function of system (3)

119883119886

119896|119896minus1

1198992 + 119899119902 (48) 1198992 + 119899119902 minus 119899 (42)119875119896|119896minus1

21198993 (432) 21198993 minus 1198992 (396)119883119886

119896|119896minus1

2119899119898 (24) 2119899119898 (24)

119870119896

1198992

119898 + 21198991198982

+ 1198983 (168) 119899

2

119898 + 21198991198982

+ 1198983

minus 2119899119898

(104)119875119896|119896

1198992119898 + 1198993 (288) 1198992119898 + 1198993 minus 1198992 (252)Total 960 818

Finally we show that (98) holds at time 119896 = 0 This can beverified by the initial conditions of TEKF algorithm

43 Numerical Complexity of the Algorithm Tables 1 and2 show the computational effort at each sample time bythe conventional EKF algorithm and TEKF (where roughmatrix-based implementation is used) in which as definedabove 119899 is the dimension of the state vector 119883

119896

119898 is thedimension of the measurement 119884

119896

119902 is the input vector 119880119896

and 119901 is the dimension of the parameter 119903

119896

The total numberof arithmetic operations (additions and multiplications) persample time of the TEKF is 1314 compared with 1778 for arough implementation of a conventional EKF which meansthe operation cost can reduce by 26

5 Simulation and Experimental Results

51 Simulation Results To test the feasibility and perfor-mance of the TEKF method the sensorless DTC-SVM tech-nique for IM drives described in Section 2 is implementedin MATLABSIMULINK environment The values of theinitial state covariance matrices 119875

0

119876 and 119877 have a greatinfluence on the performance of the estimation methodThe diagonal initial state covariance matrix 119875

0

representsvariances or mean-squared errors in the knowledge of theinitial conditions Matrix 119876 gives the statistical descriptionof the drive system Matrix 119877 is related to measured noiseThey can be obtained by considering the stochastic propertiesof the corresponding noises However a fine evaluation ofthe covariance matrices is very difficult because they areusually not known In this paper tuning the initial values ofcovariance matrices 119875

0

119876 and 119877 is using particular criteria[22] to achieve steady-state behaviors of the relative estimatedstates as given by

119876 = diag 20 20 1119890 minus 6 1119890 minus 6 10 10

1198750

= diag 01 01 05 05 1 1

119877 = diag 01 01

(113)

Mathematical Problems in Engineering 11

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000Sp

eed

(rpm

)

(a) Speed estimation

02 04 06 08 10Times (s)

minus002

0

002

004

006

008

01

012

014

016

018

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF

Theta (real)Theta (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7Po

sitio

n (r

ad)

(d) TEKF rotor position error

Current (real)Current (TEKF)

Curr

entIs120572

(A)

02 04 06 08 10Times (s)

minus12

minus8

minus4

0

4

8

12

(e) Real stator current 119868119904120572

and estimation (TEKF)

Current (real)Current (TEKF)

minus12

minus8

minus4

0

4

8

12

Curr

entIs120573

(A)

02 04 06 08 10Times (s)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 3 Continued

12 Mathematical Problems in Engineering

Flux (EKF)Flux (TEKF)

minus05 0 05 1minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(g) Stator flux estimation by TEKF and EKF

02 04 06 08 10Times (s)

minus2

0

2

4

6

8

10

12

14

16

Ψs

(Wb)

times10minus3

(h) Difference Stator flux estimation between TEKF and EKF

Figure 3 Simulation results for parameters estimation

Table 2 Kalman estimation arithmetic operation requirement for the TEKF structure

Number of multiplications(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

Number of additions(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

119860119896

119862119896

119864119896

1198671119896

1198672119896

119861119896

and119863119896

Function of system (25) Function of system (11)119883119896|119896minus1

1198992 + 119899119902 (24) 1198992 + 119899119902 (24)119875119909

119896|119896minus121198993 (128) 21198993 minus 1198992 (112)

119870119909

119896|1198961198992

119898 + 1198991198982 (48) 119899

2

119898 + 1198991198982

minus 2119899119898 (32)119875119909

119896|1198961198992

119898 + 1198993 (96) 119899

2

119898 + 1198993

minus 1198992 (80)

119883119896|119896

2119899119898 + 119898119901 (20) 2119899119898 + 2119898119901 (24)119903119896|119896minus1

1199012 (4) 1199012 minus 119901 (2)119875119903

119896|119896minus121199013 (16) 21199013 minus 1199012 (12)

119870119903

119896|11989631199012119898 + 1199011198982 (32) 31199012119898 + 1199011198982 + 1198982 minus 4119901119898 (20)

119875119903

119896|1198961199013 + 1199012119898 (16) 1199013 + 1199012119898 minus 1199012 (12)

119903119896|119896

2119898119901 + 119899119898 (12) 2119898119901 + 119899119898 (16)119878119896

119898119899119901 (16) 119899119898119901 (16)119880119896minus1

1198992119901 + 1198991199012 (56) 1198992119901 + 1198991199012 minus 119899119901 (48)119881119896

119899119901119898 (16) 119899119901 (8)119880119896minus1

21198991199012 (32) 21198991199012 (32)119898119896minus1

1198992

119901 + 1198991199012

+ 119899119901 (56) 1198992

119901 + 1198991199012

+ 119899119901 minus 119899 (52)119876119909

119896minus121198992

119901 (64) 21198992

119901 + 1198991199012 (64)

119878119896

119898119899119901 (16) 119898119899119901 minus 119898119901 (12)119873119896

21198981198992 (32) 21198981198992 minus 1198982 (60)Total 688 626

In the simulation a comparison is made to verify theequivalence of EKF and TEKF Real-time parameters esti-mated by TEKF are used to formulate the closed loop suchas rotor speed stator flux and electromagnetic torque Theestimations obtained by EKF algorithm are not included in

the sensorless DTC-SVM strategy and only evaluated in openloop A step reference speed was applied to the simulation

The machine is accelerated from 0 rpm to 1000 rpm at0 s and the torque load is set to 4N The simulation resultsof parameter estimation are shown in Figure 3 Figures 3(a)

Mathematical Problems in Engineering 13

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000

Spee

d (r

pm)

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus03

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Theta (real)Theta (TEKF)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

02 04 06 08 10Times (s)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)

Figure 4 Simulation results with parameter variation (119877119903

= 05119877119903nom)

and 3(c) represent the performance of the speed and rotorposition tracking capabilities of the control system withTEKF and EKF respectively Figures 3(e) 3(f) and 3(g)represent the estimated stator current and flux they showthat ripples are significantly suppressed due to the SVMmodulation scheme Figures 3(b) 3(d) and 3(h) show thatfor variations of speed reference the rotor speed rotorposition and stator flux errors between the two observers arevery little It is verified that the two observers are equivalentThe difference between two estimators is caused by accuracyloss in TEKF which uses more calculation steps

In order to further verify the performance of TEKFagainst model-plant parameter mismatches and the equiv-alence of two observers the change in rotor resistance isconsidered Rotor resistance will increase due to temperaturerise while the motor is running To simulate this conditionthe rotor resistance in TEKF and EKF is increased to 200comparedwith the normal value which is equivalent to a 50decrease in the actual rotor resistance The machine is stillaccelerated from 0 rpm to 1000 rpm at 0 s and the torque is setto 4N Figure 4 shows that for variation of rotor resistancethe steady-state speed and rotor position errors are negligible

14 Mathematical Problems in Engineering

(a)

Computer

380V

TXRX

120596r

Incrementalencoder

Udc

MicrocomputerDSP TMS3206713

Drive signalCurrentsensor

Voltagesensor

Voltagesensor

IM MC

ISU ISV

VSU VSW VSV

(b)

Figure 5 Complete drive system (a) Picture of experimental setup (b) Functional block diagram of the experimental setup

and the difference of the speed and rotor position estimationsbetween the two observers is rather null

52 Experimental Results The overall experimental setup isshown in Figure 5 and the specifications and rated parametersof the IM controller and inverter are listed in Table 3 Inthe experimental hardware an Expert3 control system fromMyway company and a three-phase two-pole 15 kW IM areappliedThe IM is mechanically coupled to a magnetic clutch(MC) which provides rated torque even at very low speedThe main processor in Expert3 control system is a floatingpoint processor TMS320C6713 with a max clock speed of225MHz All the algorithms including TEKF EKF DTCalgorithm and some transformation modules are imple-mented in TMS320C6713 with 100120583s sampling time and dataacquisition of the parameter estimations measured variablesand their visualization are realized on the cockpit provided byPEView9 software Insulated Gate Bipolar Transistor (IGBT)module is driven by the PWM signal with a switchingfrequency of 10 kHz and 2 120583s dead time The stator currentsare measured via two Hall effect current sensors The rotorangle and speed of IM are measured from an incrementalencoder with 2048 pulses per revolution

This experiment test is here to testify the performanceof TEKF and demonstrate that the two estimators aremathematically equivalent The machine is accelerated from600 rpm to 1000 rpm and 4N torque load is set Theexperimental results of parameter estimation based on twoobservers are given in Figures 6 and 7 Figures 6(a) and 6(c)show that the TEKF still has a good tracking performance ofthe speed and rotor position in experiment Figures 6(d) 6(e)and 6(f) illustrate stator flux and stator current estimationrobustness Figures 6(b) 6(g) and 6(f) referring to thedifference in speed and stator current estimations given by

Table 3 Specification of induction motor and inverter

Induction motor ValueNominal torque 10NmNominal voltage 380VRotor resistance 119877

119903

25ΩStator resistance 119877

119904

36ΩStator inductances 119871

119904

0301HRotor inductances 119871

119903

0302HMutual inductances 119871

119898

0273HPole pairs 2Invertercontroller ValueSwitching device 1000V 80A IGBTControl cycle time 100 120583sMain CPU DSP TMS320C6713 225MHz

the two observers are still small These experiment resultsprove that the two estimators are mathematically equivalentFigure 7 shows the speed and rotor position estimationsbased onTEKFandEKF for a 50decrease of rotor resistance(the same as the simulation) As expected the steady errorof the TEKF and the difference in speed and rotor positionestimations are still tiny Robustness of TEKF is verified

6 Conclusion

Themajor shortcoming of the conventional EKF is numericalproblems and computational burden due to the high orderof the mathematical models This has generally limited thereal-time digital implementation of the EKF for industrialfield So in this study a novel extended Kalman filter

Mathematical Problems in Engineering 15

Speed (real)Speed (EKF)Speed (TEKF)

Spee

d (r

min

)

02 04 06 08 10t (02 sgrid)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimationSp

eed

(rm

in)

minus005

0

005

01

015

02

025

03

035

04

045

02 04 06 08 10t (02 sgrid)

(b) Difference of speed estimation between EKF and TEKF

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation

Flux (EKF)Flux (TEKF)

minus08 minus04 1minus02 0 02 04 06 08minus06minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(d) Stator flux estimation by TEKF and EKF

Is120572 (real)Is120572 (TEKF)

Curr

entIs120572

(A)

minus6

minus4

minus2

0

2

4

6

02 04 06 08 10t (02 sgrid)

(e) Real stator current 119868119904120572

and estimation (TEKF)

Is120573 (real)Is120573 (TEKF)

minus6

minus4

minus2

0

2

4

6

Curr

entIs120573

(A)

02 04 06 08 10t (02 sgrid)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 6 Continued

16 Mathematical Problems in Engineering

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Erro

r cur

rentIs120572

(A)

02 04 06 08 10t (02 sgrid)

(g) TEKF stator current 119868119904120572

error

minus02

minus015

minus01

minus005

0

005

01

015

Erro

r cur

rentIs120573

(A)

02 04 06 08 10t (02 sgrid)

(h) TEKF stator current 119868119904120573

error

Figure 6 Experimental results for parameters estimation

Speed (real)Speed (EKF)

Speed (TEKF)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

0

005

01

015

02

025

03

035

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)Figure 7 Experimental results with parameter variation (119877

119903

= 05119877119903nom)

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article Direct Torque Control of Sensorless ...

Mathematical Problems in Engineering 5

119861119896

=[[[

[

119879119904

120590119871119904

0 119879119904

0

0119879119904

120590119871119904

0 119879119904

]]]

]

119879

119862119896

= [1 0 0 0

0 1 0 0]

119863119896

= [0 0

0 0]

(14)

Based on discretized IM model a conventional EKFestimator is designed for estimation of stator flux currentelectromagnetic torque and rotor speed of IM for sensorlessDTC-SVM operations Treating119883

119896

as the full order state and119903119896

as the augmented system state the state vector is chosen tobe 119883119886119896

= [119883119896

119903119896]119879 119906119896

= [119880119904120572

119880119904120573]119879 and 119884

119896

= [119868119904120572

119868119904120573]119879 are

chosen as input and output vectors because these quantitiescan be easily obtained from measurements of stator currentsand voltage construction usingDC link voltage and switchingstatus Considering the parameter errors and noise of systemthe discrete time state space model of IMs in the stationary(120572120573) reference frame is described by

119883119886

119896+1

= 119860119896

119883119886

119896

+ 119861119896

119906119896

+ 119908119896

119884119896+1

= 119862119896

119883119886

119896

+ V119896

(15)

with

119860119896

= [119860119896

119863119896

0 119866119896

]

119861119896

= [119861119896

0]

119862119896

= [119862119896

0]

119879

119908119896

= [119908119909119896

119908119903119896

]

(16)

The system noise119908119896

and measurement noise V119896

are whiteGaussian sequence with zero-mean and following covariancematrices

119864[[[

[

[[

[

119908119909

119896

119908119903119896

V119896

]]

]

[[[

[

119908119909119895

119908119903119895

V119895

]]]

]

119879

]]]

]

=[[

[

119876119909119896

0 0

0 119876119903119896

0

0 0 119877119896

]]

]

120575119896119895

(17)

where 119876119909119896

gt 0 119876119903119896

gt 0 119877119896

gt 0 and 120575119896119895

is theKronecker delta The initial states 119883

0

and 1199030

are assumed tobe uncorrelated with the zero-mean noises 119908119909

119896

119908119903119896

and V119896

The initial conditions are assumed to be Gaussian randomvariables119883

0

and 1199030

that are defined as follows

119864((1198830

minus 119883lowast

0

) (1198830

minus 119883lowast

0

)119879

) = 119875119909

0

119864 (1198830

) = 119883lowast

0

119864 (1199030

) = 119903lowast

0

119864((1199030

minus 119903lowast

0

) (1199030

minus 119903lowast

0

)119879

) = 119875119903

0

119864 ((1198830

minus 119883lowast

0

) (1199030

minus 119903lowast

0

)119879

) = 119875119909119903

0

(18)

The overall structure of the EKF is well-known byemploying a two-step prediction and correction algorithm[13] Hence the application of EKF filter to the state spacemodel of IM (15) is described by

119883119886

119896|119896minus1

= 119860119896minus1

119883119886

119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

(19)

119875119896|119896minus1

= 119865119896minus1

119875119896minus1|119896minus1

119865119879

119896minus1

+ 119876119896minus1

(20)

119870119896

= 119875119896|119896minus1

119867119879

119896

(119867119896

119875119896|119896minus1

119867119879

119896

+ 119877)minus1

(21)

119875119896|119896

= 119875119896|119896minus1

minus 119870119896

119867119896

119875119896|119896minus1

(22)

119883119886

119896|119896

= 119883120572

119896|119896minus1

+ 119870119896

(119884119896

minus 119862119896

119883119886

119896|119896minus1

) (23)

with

119865119896minus1

= [119865119896minus1

119864119896minus1

0 119866119896minus1

]

119876 (sdot) = [119876119909 (sdot) 0

0 119876119903 (sdot)]

119865119896minus1

=120597

120597119883(119860119896minus1

119883119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1

) = 119860119896

119864119896minus1

=120597

120597119903(119860119896minus1

119883119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1

)

= [0 0 0 0

1199011 1199012 0 0]

119879

6 Mathematical Problems in Engineering

1199011 = (minus119901119894119904119887[119896minus1|119896minus1]

+119901

120590119871119904

120595119904120573[119896minus1|119896minus1]

)119879119904

1199012 = (119901119894119904120572[119896minus1|119896minus1]

minus119901

120590119871119904

120595119904120572[119896minus1|119896minus1]

)119879119904

119867119896

= [1198671

119896

1198672

119896

] =120597

120597119883119886(119862119896

119883119886

)

1198671

119896

= 119862119896

1198672

119896

= 0

119875 (sdot) = [119875119909 (sdot) 119875119909119903 (sdot)

(119875119909119903 (sdot))119879

119875119903 (sdot)]

(24)

4 The Two-Stage Extended Kalman Filter

41The TEKFAlgorithm Asmentioned in conventional EKFestimator previously the memory and computational costsincrease with the augmented state dimension Consideringsampling time is very small only high performance micro-controller can qualify for this work Hence the conventionalEKF algorithm may be impractical to implement The extracomputation of 119875

119909119903(sdot) terms leads to this computationalcomplexityTherefore we can reduce the computational com-plexity from application point of view if the 119875119909119903(sdot) terms canbe eliminated In this section a two-stage extended Kalmanfilter without explicitly calculating 119875119909119903(sdot) terms is discussed

Following the same approach as given in [15] the TEKFis decomposed into two filters such as the modified bias freefilter and the bias filter by applying the following two-stage119880-119881 transformation

119883119886

119896|119896minus1

= 119879 (119880119896

)119883119886

119896|119896minus1

119875119896|119896minus1

= 119879 (119880119896

) 119875119896|119896minus1

119879 (119880119896

)119879

119870119896

= 119879 (119881119896

)119870119896

119875119896|119896

= 119879 (119881119896

) 119875119896|119896

119879 (119881119896

)119879

119883119886

119896|119896

= 119879 (119881119896

)119883119886

119896|119896

(25)

where

119883119886

119896(sdot)

= [119883119896(sdot)

119903119896(sdot)

]

119870119896

= [119870119909

119896

119870119903

119896

]

119875119896(sdot)

= [

[

119875119909

119896(sdot)

0

0 119875119903

119896(sdot)

]

]

119879 (119880119896

) = [119868 119880119896

0 119868]

(26)

The main advantage of using the 119879(119880119896

) transformation isthat the inverse transformation 119879

minus1

(119880119896

) = 119879(minus119880119896

) involvesonly a change of sign Two blending matrices 119880

119896

and 119881119896

are defined by 119880119896

= 119875119909119903119896|119896minus1

(119875119903119896|119896minus1

)minus1 and 119881119896

= 119875119909119903119896|119896

(119875119903119896|119896

)minus1respectively Using characteristic of 119879(119880

119896

) (25) become

119883119886

119896|119896minus1

= 119879 (minus119880119896

)119883119886

119896|119896minus1

119875119896|119896minus1

= 119879 (minus119880119896

) 119875119896|119896minus1

119879 (minus119880119896

)119879

119870119896

= 119879 (minus119881119896

)119870119896

119875119896|119896

= 119879 (minus119881119896

) 119875119896|119896

119879 (minus119881119896

)119879

119883119886

119896|119896

= 119879 (minus119881119896

)119883119886

119896|119896

(27)

And the following relationships are obtained from (25)

119875119909

119896|119896minus1

= 119875119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

(28)

119875119903

119896|119896minus1

= 119875119903

119896|119896minus1

(29)

119875119909119903

119896|119896minus1

= 119880119896

119875119903

119896|119896minus1

(30)

119875119909

119896|119896

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

(31)

119875119903

119896|119896

= 119875119903

119896|119896

(32)

119875119909119903

119896|119896

= 119881119896

119875119903

119896|119896

(33)

Based on two-step iterative substitution method of [15]the transformed filter expressed by (27) can be recursivelycalculated as follows

119883119886

119896|119896minus1

= 119879 (minus119880119896

)

sdot (119860119896minus1

119879 (119881119896minus1

)119883119886

119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

)

(34)

119875119896|119896minus1

= 119879 (minus119880119896

)

sdot (119876119896minus1

+ 119865119896minus1

119879 (119881119896minus1

) 119875119896minus1|119896minus1

119879 (119881119896minus1

)119879

119865119879

119896minus1

)

sdot 119879 (minus119880119896

)119879

(35)

119870119896

= 119879 (119880119896

minus 119881119896

) 119875119896|119896minus1

119879 (119880119896

)119879

sdot 119867119879

119896

(119867119896

119879 (119880119896

) 119875119896|119896minus1

119879 (119880119896

)119879

119867119879

119896

+ 119877)minus1

(36)

119875119896|119896

= (119879 (119880119896

minus 119881119896

) minus 119870119896

119867119896

119879 (119880119896

))

sdot 119875119896|119896minus1

119879 (119880119896

minus 119881119896

)119879

(37)

119883119886

119896|119896

= 119870119896

(119884119896

minus 119862119896

119879 (119880119896

)119883119886

119896|119896minus1

) + 119879 (119880119896

minus 119881119896

)

sdot 119883119886

119896|119896minus1

(38)

Mathematical Problems in Engineering 7

Using (35) (37) and the block diagonal structure of 119875(sdot)

thefollowing relations can be obtained

0 = 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

minus 119880119896

119876119903

119896minus1

minus 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

0 = 119880119896

minus 119881119896

minus 119870119909

119896

119878119896

(39)

where 119880119896

and 119878119896

are defined as

119880119896

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119866minus1

119896minus1

(40)

119878119896

= 1198671

119896

119880119896

+ 1198672

119896

(41)

The above equations lead to

119880119896

119875119903

119896|119896minus1

= 119880119896

(119875119903

119896|119896minus1

minus 119876119903

119896

) (42)

119880119896

= 119880119896

(119868 minus 119876119903

119896

(119875119903

119896|119896minus1

)minus1

) (43)

119881119896

= 119880119896

minus 119870119909

119896

119878119896

(44)

Define the following notation

119860119896minus1

119879 (119881119896minus1

) = [119860119896minus1

119860119896minus1

119881119896minus1

+ 119864119896minus1

0 119866119896minus1

] (45)

The equations of themodified bias free filter and the bias filterare acquired by the next steps

Expanding (34) we have

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119898119896minus1

(46)

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

(47)

where

119898119896minus1

= (119860119896minus1

119881119896minus1

+ 119863119896minus1

minus 119880119896

119866119896minus1

) 119903119896minus1|119896minus1

(48)

Expanding (35) we have

119875119909

119896|119896minus1

= (119864119896minus1

+ 119860119896minus1

119881119896minus1

minus 119880119896

119866119896minus1

) 119875119903

119896minus1|119896minus1

lowast (119864119896minus1

+ 119860119896minus1

119881119896minus1

minus 119880119896

119866119896minus1

)119879

+ 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

+ 119880119896

119876119903

119896

119880119879

119896

(49)

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896

(50)

Then using (40) (43) and (47) (49) can be written as

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

(51)

where

119876119909

119896

= 119876119909

119896

+119872119896

(119872119896

119876119903

119896

)119879

(52)

Expanding (38) and using (41) and (44) we have

119883119896|119896

= 119883119896|119896minus1

+ (119880119896

minus 119881119896

) 119903119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

)

119903119896|119896

= 119903119896|119896minus1

+ 119870119903

119896

(119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

)

(53)

Then

119883119896|119896

= 119870119909

119896

120578119909

119896

+ 119883119896|119896minus1

(54)

where

119878119896

= 119862119896

119880119896

(55)

120578119909

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

+ (119878119896

minus 119878119896

) 119903119896|119896minus1

(56)

Expanding (36) and using (41) we have

119870119903

119896

= 119875119903

119896|119896minus1

119878119879

119896

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)minus1

119870119909

119896

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)

= 119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

(57)

Then

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(58)

Expanding (37) we have

119875119903

119896|119896

= (119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119875119909

119896|119896

= 119875119909

119896|119896minus1

+ (119880119896

minus 119881119896

) 119875119903

119896|119896minus1

(119880119879

119896

minus 119881119879

119896

)

minus (119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

(119880119879

119896

minus 119881119879

119896

))

(59)

Then using (41) and (44)

119875119909

119896|119896

= (119868 minus 119870119909

119896

1198671

119896

)119875119909

119896|119896minus1

(60)

Finally using (25) the estimated value of original state(119894119904120572

119904120573

119904120572

119904120573

) can be obtained by sum of the state119883withthe augmented state 119903

119896|119896minus1

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

(61)

119896|119896

= 119883119896|119896

+ 119881119896

119903119896|119896

(62)

Moreover the unknown parameter 119903( 119903

) is defined as

119903119896|119896minus1

= 119903119896|119896minus1

(63)

119903119896|119896

= 119903119896|119896

(64)

8 Mathematical Problems in Engineering

Based on the above analysis the TEKF can be decoupledinto two filters such as the modified bias free filter and biasfilter The modified bias filter gives the state estimation 119883

119896|119896

and the bias filter gives the bias estimate 119903

119896|119896

The correctedstate estimate 119883119886

119896|119896

(119896|119896

119903119896|119896

) of the TEKF is obtained fromthe estimates of the two filters and coupling equations119880

119896

and119881119896

[21] The modified bias free filter is expressed as follows

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119898119896minus1

(65)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

(66)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(119873119896

)minus1

(67)

119875119909

119896|119896

= (119868 minus 119870119909

119896

1198671

119896

)119875119909

119896|119896minus1

(68)

120578119909

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

+ (119878119896

minus 119878119896

) 119903119896|119896minus1

(69)

119883119896|119896

= 119870119909

119896

120578119909

119896

+ 119883119896|119896minus1

(70)

and the bias filter is

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

(71)

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896

(72)

119870119903

119896

= 119875119903

119896|119896minus1

119878119879

119896

(119873119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)minus1

(73)

119875119903

119896|119896

= (119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

(74)

120578119903

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

(75)

119903119896|119896

= 119903119896|119896minus1

+ 119870119903

119896

120578119903

119896

(76)

with the coupling equations

119878119896

= 1198671

119896

119880119896

+ 1198672

119896

(77)

119880119896

= 119880119896

(119868 minus 119876119903

119896

(119875119903

119896|119896minus1

)minus1

) (78)

119880119896

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119866minus1

119896minus1

(79)

119881119896

= 119880119896

minus 119870119909

119896

119878119896

(80)

119898119896minus1

= (119860119896minus1

119881119896minus1

+ 119863119896minus1

minus 119880119896

119866119896minus1

) 119903119896minus1|119896minus1

(81)

119876119909

119896

= 119876119909

119896

+ 119880119896

(119880119896

119876119903

119896

)119879

(82)

119873119896

= 1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

(83)

The initial conditions of TEKF algorithm are establishedwith the initial conditions of a classical EKF (119883

0|0

1199030|0

1198751199090|0

1198751199091199030|0

1198751199030|0

) so that

1198810

= 119875119909119903

0|0

(119875119903

0|0

)minus1

1198830|0

= 1198830|0

minus 1198810

1199030|0

1199030|0

= 1199030|0

119875119909

0|0

= 119875119909

0|0

minus 1198810

119875119903

0|0

119881119879

0

119875119903

0|0

= 119875119903

0|0

(84)

According to variables of full order filter119883 (1198831

1198832

1198833

1198834

)the stator flux and torque estimators for DTC-SVM ofFigure 1 are then given by

119904120572

= 1198831

119904120573

= 1198832

119904120572

= 1198833

119904120573

= 1198834

100381610038161003816100381610038161003816120595119904

100381610038161003816100381610038161003816= radic

2

119904120572

+ 2

119904120573

119904

= arctan119904120573

119904120572

119890

=3

2119901 (119904120572

119868119904120573

minus 119904120573

119868119904120572

)

(85)

where 119901 is the pole pairs of IM The estimated speed andelectromagnetic torque obtained from the TEKF observer areused to close the speed and torque loop to achieve sensorlessoperations

42 The Stability and Parameter Sensitivity Analysis ofthe TEKF

Theorem 3 The discrete time conventional extended Kalmanfilter (19)ndash(23) is equivalent to the two-stage extern Kalmanfilter (see (61)sim(83))

Proof Before proving the theorem the following five rela-tionships are needed

(1) Using (72) and (78)

119880119896+1

119866119896

119875119903

119896|119896

119866119879

119896

= 119880119896

119875119903

119896|119896minus1

(86)

(2) Using (67) and (73)

119870119909

119896

119872119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

(87)

119870119903

119896

119872119896

= 119875119903

119896|119896minus1

(119878119896

)119879

(88)

where

119872119896

= 1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

+ 119877119896

(89)

Mathematical Problems in Engineering 9

(3) Using (20) we have

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896minus1

(90)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119864119896minus1

(119875119909119903

119896minus1|119896minus1

)119879

119860119879

119896minus1

+ 119860119896minus1

119875119909119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119876119909

119896minus1

(91)

119875119909119903

119896|119896minus1

= 119860119896minus1

119875119909119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

(92)

(4) Using (21)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(93)

119870119903

119896

= (119875119909119903

119896|119896minus1

)119879

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(94)

(5) Using (22)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

(95)

119875119909119903

119896|119896

= 119875119909119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909119903

119896|119896minus1

(96)

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896

1198671

119896

119875119909119903

119896|119896minus1

(97)

By inductive reasoning suppose that at time 119896 minus 1the unknown parameter

119896minus1

and estimated state 119896minus1

areequal to the parameter 119903

119896minus1

and state 119883119896minus1

of the controlsystem respectively we show that TEKF is equivalent to theconventional EKF because these properties are still true attime 119896

Assume that at time 119896 minus 1

119883119896minus1|119896minus1

= 119896minus1|119896minus1

119903119896minus1|119896minus1

= 119903119896minus1|119896minus1

119875119909

119896minus1|119896minus1

= 11987511

119896minus1|119896minus1

119875119909119903

119896minus1|119896minus1

= 11987512

119896minus1|119896minus1

119875119903

119896minus1|119896minus1

= 11987522

119896minus1|119896minus1

(98)

where [ 119875119909119875

119909119903

(119875

119909119903)

119879119875

119903 ] and [ 11987511119875

12

(119875

12)

119879119875

22 ] represent the variance-covariance matrices of the system and estimated variablesrespectively

From (19) we have

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119863119896

119903119896minus1|119896minus1

+ 119861119896

119906119896minus1

(99)

Then using (98) (41) (62) (79) (81) (71) and (61)

119883119896|119896minus1

= 119860119896minus1

(119883119896minus1|119896minus1

+ 119881119896minus1

119903119896minus1|119896minus1

)

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

minus 119898119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

= 119896|119896minus1

(100)

Using (19) (71) (98) (63) and (64) we have

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119903119896|119896minus1

(101)

Using (91) (98) (78) (66) (79) (82) (86) and (72) we obtain

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896minus1

+ 119880119896

119866119896

119875119903

119896minus1|119896minus1

119866119879

119896

119880119879

119896

= 119875119909

119896|119896minus1

+ 119880119896

(119880119896

119875119903

119896|119896minus1

minus 119880119896

119876119903

119896minus1

)119879

= 119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

= 11987511

119896|119896minus1

(102)

Using (90) (98) (72) (32) (71) (29) and (97) we obtain

119875119903

119896|119896minus1

= 11987522

119896minus1|119896minus1

(103)

Using (92) (98) (33) (32) (79) (86) and (91)

119875119909119903

119896|119896minus1

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896minus1

119875119903

119896minus1|119896minus2

= 11987512

119896|119896minus1

(104)

Using (93) (101) (55) (73) (67) (80) and (87)

119870119909

119896

= (119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

) (1198671

119896

)119879

sdot (1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ (119881119896

+ 119870119909

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

(1198671

119896

)119879

)119880minus1

119896

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

)119880minus1

119896

+ 119881119896

119875119903

119896|119896minus1

119878119879

119896

119880minus1

119896

= 119870119909

119896

+ 119881119896

119870119903

119896

(105)

Using (94) (30) and (88) we obtain

119870119903

119896

= (119875119903

119896|119896minus1

)119879

119878119879

119896

119880minus1

119896

= 119870119903

119896|119896minus1

(106)

10 Mathematical Problems in Engineering

Next wewill show that (98) holds at time 119896 From (23)we have

119883119896|119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

)

= 119883119896|119896minus1

+ 119870119909

119896

119903119896

(107)

Then using (61) and (105) the above equation can be writtenas

119883119896|119896

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

+ (119870119909

119896

+ 119881119896

119870119903

119896

) 119903119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 1198671

119896

119883119896|119896minus1

)

+ (119880119896

minus 119870119909

119896

119878119896

) 119903119896|119896minus1

+ 119881119896

119870119903

119896

119903119896

= 119883119896|119896

+ 119881119896

119903119896|119896

= 119896|119896

(108)

Using (95) (105) (102) and (77)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

(109)

Then using (80) (68) (74) and (31) we obtain

119875119909

119896|119896

= 119875119909

119896|119896

+ 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

+ 119881119896

(119875119903

119896|119896

119878119879

119896

(119870119909

119896

)119879

minus 119870119903

119896

1198671

119896

119875119909

119896|119896minus1

)

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

= 11987511

119896|119896

(110)

Using (96) (30) (28) (105) and (80)

119875119909119903

119896|119896

= 119880119896

119875119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

1198671

119896

119880119896

minus 119881119896

119870119903

119896

1198671

119896

119880119896

)119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

119875119903

119896|119896

= 11987512

119896|119896

(111)

Using (97) (106) (95) (29) and (30) we obtain

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896|119896minus1

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119868 minus 119870119903

119896|119896minus1

119878119896

) 119875119903

119896|119896minus1

= 11987522

119896|119896

(112)

Table 1 Kalman estimation arithmetic operation requirement forthe conventional EKF structure

Number of multiplications(119899 = 6119898 = 2 and 119902 = 2)

Number of additions(119899 = 6119898 = 2 and 119902 = 2)

119860119896

119861119896

and 119862

119896

Function of system (9) Function of system (3)

119883119886

119896|119896minus1

1198992 + 119899119902 (48) 1198992 + 119899119902 minus 119899 (42)119875119896|119896minus1

21198993 (432) 21198993 minus 1198992 (396)119883119886

119896|119896minus1

2119899119898 (24) 2119899119898 (24)

119870119896

1198992

119898 + 21198991198982

+ 1198983 (168) 119899

2

119898 + 21198991198982

+ 1198983

minus 2119899119898

(104)119875119896|119896

1198992119898 + 1198993 (288) 1198992119898 + 1198993 minus 1198992 (252)Total 960 818

Finally we show that (98) holds at time 119896 = 0 This can beverified by the initial conditions of TEKF algorithm

43 Numerical Complexity of the Algorithm Tables 1 and2 show the computational effort at each sample time bythe conventional EKF algorithm and TEKF (where roughmatrix-based implementation is used) in which as definedabove 119899 is the dimension of the state vector 119883

119896

119898 is thedimension of the measurement 119884

119896

119902 is the input vector 119880119896

and 119901 is the dimension of the parameter 119903

119896

The total numberof arithmetic operations (additions and multiplications) persample time of the TEKF is 1314 compared with 1778 for arough implementation of a conventional EKF which meansthe operation cost can reduce by 26

5 Simulation and Experimental Results

51 Simulation Results To test the feasibility and perfor-mance of the TEKF method the sensorless DTC-SVM tech-nique for IM drives described in Section 2 is implementedin MATLABSIMULINK environment The values of theinitial state covariance matrices 119875

0

119876 and 119877 have a greatinfluence on the performance of the estimation methodThe diagonal initial state covariance matrix 119875

0

representsvariances or mean-squared errors in the knowledge of theinitial conditions Matrix 119876 gives the statistical descriptionof the drive system Matrix 119877 is related to measured noiseThey can be obtained by considering the stochastic propertiesof the corresponding noises However a fine evaluation ofthe covariance matrices is very difficult because they areusually not known In this paper tuning the initial values ofcovariance matrices 119875

0

119876 and 119877 is using particular criteria[22] to achieve steady-state behaviors of the relative estimatedstates as given by

119876 = diag 20 20 1119890 minus 6 1119890 minus 6 10 10

1198750

= diag 01 01 05 05 1 1

119877 = diag 01 01

(113)

Mathematical Problems in Engineering 11

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000Sp

eed

(rpm

)

(a) Speed estimation

02 04 06 08 10Times (s)

minus002

0

002

004

006

008

01

012

014

016

018

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF

Theta (real)Theta (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7Po

sitio

n (r

ad)

(d) TEKF rotor position error

Current (real)Current (TEKF)

Curr

entIs120572

(A)

02 04 06 08 10Times (s)

minus12

minus8

minus4

0

4

8

12

(e) Real stator current 119868119904120572

and estimation (TEKF)

Current (real)Current (TEKF)

minus12

minus8

minus4

0

4

8

12

Curr

entIs120573

(A)

02 04 06 08 10Times (s)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 3 Continued

12 Mathematical Problems in Engineering

Flux (EKF)Flux (TEKF)

minus05 0 05 1minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(g) Stator flux estimation by TEKF and EKF

02 04 06 08 10Times (s)

minus2

0

2

4

6

8

10

12

14

16

Ψs

(Wb)

times10minus3

(h) Difference Stator flux estimation between TEKF and EKF

Figure 3 Simulation results for parameters estimation

Table 2 Kalman estimation arithmetic operation requirement for the TEKF structure

Number of multiplications(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

Number of additions(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

119860119896

119862119896

119864119896

1198671119896

1198672119896

119861119896

and119863119896

Function of system (25) Function of system (11)119883119896|119896minus1

1198992 + 119899119902 (24) 1198992 + 119899119902 (24)119875119909

119896|119896minus121198993 (128) 21198993 minus 1198992 (112)

119870119909

119896|1198961198992

119898 + 1198991198982 (48) 119899

2

119898 + 1198991198982

minus 2119899119898 (32)119875119909

119896|1198961198992

119898 + 1198993 (96) 119899

2

119898 + 1198993

minus 1198992 (80)

119883119896|119896

2119899119898 + 119898119901 (20) 2119899119898 + 2119898119901 (24)119903119896|119896minus1

1199012 (4) 1199012 minus 119901 (2)119875119903

119896|119896minus121199013 (16) 21199013 minus 1199012 (12)

119870119903

119896|11989631199012119898 + 1199011198982 (32) 31199012119898 + 1199011198982 + 1198982 minus 4119901119898 (20)

119875119903

119896|1198961199013 + 1199012119898 (16) 1199013 + 1199012119898 minus 1199012 (12)

119903119896|119896

2119898119901 + 119899119898 (12) 2119898119901 + 119899119898 (16)119878119896

119898119899119901 (16) 119899119898119901 (16)119880119896minus1

1198992119901 + 1198991199012 (56) 1198992119901 + 1198991199012 minus 119899119901 (48)119881119896

119899119901119898 (16) 119899119901 (8)119880119896minus1

21198991199012 (32) 21198991199012 (32)119898119896minus1

1198992

119901 + 1198991199012

+ 119899119901 (56) 1198992

119901 + 1198991199012

+ 119899119901 minus 119899 (52)119876119909

119896minus121198992

119901 (64) 21198992

119901 + 1198991199012 (64)

119878119896

119898119899119901 (16) 119898119899119901 minus 119898119901 (12)119873119896

21198981198992 (32) 21198981198992 minus 1198982 (60)Total 688 626

In the simulation a comparison is made to verify theequivalence of EKF and TEKF Real-time parameters esti-mated by TEKF are used to formulate the closed loop suchas rotor speed stator flux and electromagnetic torque Theestimations obtained by EKF algorithm are not included in

the sensorless DTC-SVM strategy and only evaluated in openloop A step reference speed was applied to the simulation

The machine is accelerated from 0 rpm to 1000 rpm at0 s and the torque load is set to 4N The simulation resultsof parameter estimation are shown in Figure 3 Figures 3(a)

Mathematical Problems in Engineering 13

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000

Spee

d (r

pm)

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus03

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Theta (real)Theta (TEKF)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

02 04 06 08 10Times (s)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)

Figure 4 Simulation results with parameter variation (119877119903

= 05119877119903nom)

and 3(c) represent the performance of the speed and rotorposition tracking capabilities of the control system withTEKF and EKF respectively Figures 3(e) 3(f) and 3(g)represent the estimated stator current and flux they showthat ripples are significantly suppressed due to the SVMmodulation scheme Figures 3(b) 3(d) and 3(h) show thatfor variations of speed reference the rotor speed rotorposition and stator flux errors between the two observers arevery little It is verified that the two observers are equivalentThe difference between two estimators is caused by accuracyloss in TEKF which uses more calculation steps

In order to further verify the performance of TEKFagainst model-plant parameter mismatches and the equiv-alence of two observers the change in rotor resistance isconsidered Rotor resistance will increase due to temperaturerise while the motor is running To simulate this conditionthe rotor resistance in TEKF and EKF is increased to 200comparedwith the normal value which is equivalent to a 50decrease in the actual rotor resistance The machine is stillaccelerated from 0 rpm to 1000 rpm at 0 s and the torque is setto 4N Figure 4 shows that for variation of rotor resistancethe steady-state speed and rotor position errors are negligible

14 Mathematical Problems in Engineering

(a)

Computer

380V

TXRX

120596r

Incrementalencoder

Udc

MicrocomputerDSP TMS3206713

Drive signalCurrentsensor

Voltagesensor

Voltagesensor

IM MC

ISU ISV

VSU VSW VSV

(b)

Figure 5 Complete drive system (a) Picture of experimental setup (b) Functional block diagram of the experimental setup

and the difference of the speed and rotor position estimationsbetween the two observers is rather null

52 Experimental Results The overall experimental setup isshown in Figure 5 and the specifications and rated parametersof the IM controller and inverter are listed in Table 3 Inthe experimental hardware an Expert3 control system fromMyway company and a three-phase two-pole 15 kW IM areappliedThe IM is mechanically coupled to a magnetic clutch(MC) which provides rated torque even at very low speedThe main processor in Expert3 control system is a floatingpoint processor TMS320C6713 with a max clock speed of225MHz All the algorithms including TEKF EKF DTCalgorithm and some transformation modules are imple-mented in TMS320C6713 with 100120583s sampling time and dataacquisition of the parameter estimations measured variablesand their visualization are realized on the cockpit provided byPEView9 software Insulated Gate Bipolar Transistor (IGBT)module is driven by the PWM signal with a switchingfrequency of 10 kHz and 2 120583s dead time The stator currentsare measured via two Hall effect current sensors The rotorangle and speed of IM are measured from an incrementalencoder with 2048 pulses per revolution

This experiment test is here to testify the performanceof TEKF and demonstrate that the two estimators aremathematically equivalent The machine is accelerated from600 rpm to 1000 rpm and 4N torque load is set Theexperimental results of parameter estimation based on twoobservers are given in Figures 6 and 7 Figures 6(a) and 6(c)show that the TEKF still has a good tracking performance ofthe speed and rotor position in experiment Figures 6(d) 6(e)and 6(f) illustrate stator flux and stator current estimationrobustness Figures 6(b) 6(g) and 6(f) referring to thedifference in speed and stator current estimations given by

Table 3 Specification of induction motor and inverter

Induction motor ValueNominal torque 10NmNominal voltage 380VRotor resistance 119877

119903

25ΩStator resistance 119877

119904

36ΩStator inductances 119871

119904

0301HRotor inductances 119871

119903

0302HMutual inductances 119871

119898

0273HPole pairs 2Invertercontroller ValueSwitching device 1000V 80A IGBTControl cycle time 100 120583sMain CPU DSP TMS320C6713 225MHz

the two observers are still small These experiment resultsprove that the two estimators are mathematically equivalentFigure 7 shows the speed and rotor position estimationsbased onTEKFandEKF for a 50decrease of rotor resistance(the same as the simulation) As expected the steady errorof the TEKF and the difference in speed and rotor positionestimations are still tiny Robustness of TEKF is verified

6 Conclusion

Themajor shortcoming of the conventional EKF is numericalproblems and computational burden due to the high orderof the mathematical models This has generally limited thereal-time digital implementation of the EKF for industrialfield So in this study a novel extended Kalman filter

Mathematical Problems in Engineering 15

Speed (real)Speed (EKF)Speed (TEKF)

Spee

d (r

min

)

02 04 06 08 10t (02 sgrid)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimationSp

eed

(rm

in)

minus005

0

005

01

015

02

025

03

035

04

045

02 04 06 08 10t (02 sgrid)

(b) Difference of speed estimation between EKF and TEKF

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation

Flux (EKF)Flux (TEKF)

minus08 minus04 1minus02 0 02 04 06 08minus06minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(d) Stator flux estimation by TEKF and EKF

Is120572 (real)Is120572 (TEKF)

Curr

entIs120572

(A)

minus6

minus4

minus2

0

2

4

6

02 04 06 08 10t (02 sgrid)

(e) Real stator current 119868119904120572

and estimation (TEKF)

Is120573 (real)Is120573 (TEKF)

minus6

minus4

minus2

0

2

4

6

Curr

entIs120573

(A)

02 04 06 08 10t (02 sgrid)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 6 Continued

16 Mathematical Problems in Engineering

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Erro

r cur

rentIs120572

(A)

02 04 06 08 10t (02 sgrid)

(g) TEKF stator current 119868119904120572

error

minus02

minus015

minus01

minus005

0

005

01

015

Erro

r cur

rentIs120573

(A)

02 04 06 08 10t (02 sgrid)

(h) TEKF stator current 119868119904120573

error

Figure 6 Experimental results for parameters estimation

Speed (real)Speed (EKF)

Speed (TEKF)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

0

005

01

015

02

025

03

035

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)Figure 7 Experimental results with parameter variation (119877

119903

= 05119877119903nom)

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Page 6: Research Article Direct Torque Control of Sensorless ...

6 Mathematical Problems in Engineering

1199011 = (minus119901119894119904119887[119896minus1|119896minus1]

+119901

120590119871119904

120595119904120573[119896minus1|119896minus1]

)119879119904

1199012 = (119901119894119904120572[119896minus1|119896minus1]

minus119901

120590119871119904

120595119904120572[119896minus1|119896minus1]

)119879119904

119867119896

= [1198671

119896

1198672

119896

] =120597

120597119883119886(119862119896

119883119886

)

1198671

119896

= 119862119896

1198672

119896

= 0

119875 (sdot) = [119875119909 (sdot) 119875119909119903 (sdot)

(119875119909119903 (sdot))119879

119875119903 (sdot)]

(24)

4 The Two-Stage Extended Kalman Filter

41The TEKFAlgorithm Asmentioned in conventional EKFestimator previously the memory and computational costsincrease with the augmented state dimension Consideringsampling time is very small only high performance micro-controller can qualify for this work Hence the conventionalEKF algorithm may be impractical to implement The extracomputation of 119875

119909119903(sdot) terms leads to this computationalcomplexityTherefore we can reduce the computational com-plexity from application point of view if the 119875119909119903(sdot) terms canbe eliminated In this section a two-stage extended Kalmanfilter without explicitly calculating 119875119909119903(sdot) terms is discussed

Following the same approach as given in [15] the TEKFis decomposed into two filters such as the modified bias freefilter and the bias filter by applying the following two-stage119880-119881 transformation

119883119886

119896|119896minus1

= 119879 (119880119896

)119883119886

119896|119896minus1

119875119896|119896minus1

= 119879 (119880119896

) 119875119896|119896minus1

119879 (119880119896

)119879

119870119896

= 119879 (119881119896

)119870119896

119875119896|119896

= 119879 (119881119896

) 119875119896|119896

119879 (119881119896

)119879

119883119886

119896|119896

= 119879 (119881119896

)119883119886

119896|119896

(25)

where

119883119886

119896(sdot)

= [119883119896(sdot)

119903119896(sdot)

]

119870119896

= [119870119909

119896

119870119903

119896

]

119875119896(sdot)

= [

[

119875119909

119896(sdot)

0

0 119875119903

119896(sdot)

]

]

119879 (119880119896

) = [119868 119880119896

0 119868]

(26)

The main advantage of using the 119879(119880119896

) transformation isthat the inverse transformation 119879

minus1

(119880119896

) = 119879(minus119880119896

) involvesonly a change of sign Two blending matrices 119880

119896

and 119881119896

are defined by 119880119896

= 119875119909119903119896|119896minus1

(119875119903119896|119896minus1

)minus1 and 119881119896

= 119875119909119903119896|119896

(119875119903119896|119896

)minus1respectively Using characteristic of 119879(119880

119896

) (25) become

119883119886

119896|119896minus1

= 119879 (minus119880119896

)119883119886

119896|119896minus1

119875119896|119896minus1

= 119879 (minus119880119896

) 119875119896|119896minus1

119879 (minus119880119896

)119879

119870119896

= 119879 (minus119881119896

)119870119896

119875119896|119896

= 119879 (minus119881119896

) 119875119896|119896

119879 (minus119881119896

)119879

119883119886

119896|119896

= 119879 (minus119881119896

)119883119886

119896|119896

(27)

And the following relationships are obtained from (25)

119875119909

119896|119896minus1

= 119875119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

(28)

119875119903

119896|119896minus1

= 119875119903

119896|119896minus1

(29)

119875119909119903

119896|119896minus1

= 119880119896

119875119903

119896|119896minus1

(30)

119875119909

119896|119896

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

(31)

119875119903

119896|119896

= 119875119903

119896|119896

(32)

119875119909119903

119896|119896

= 119881119896

119875119903

119896|119896

(33)

Based on two-step iterative substitution method of [15]the transformed filter expressed by (27) can be recursivelycalculated as follows

119883119886

119896|119896minus1

= 119879 (minus119880119896

)

sdot (119860119896minus1

119879 (119881119896minus1

)119883119886

119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

)

(34)

119875119896|119896minus1

= 119879 (minus119880119896

)

sdot (119876119896minus1

+ 119865119896minus1

119879 (119881119896minus1

) 119875119896minus1|119896minus1

119879 (119881119896minus1

)119879

119865119879

119896minus1

)

sdot 119879 (minus119880119896

)119879

(35)

119870119896

= 119879 (119880119896

minus 119881119896

) 119875119896|119896minus1

119879 (119880119896

)119879

sdot 119867119879

119896

(119867119896

119879 (119880119896

) 119875119896|119896minus1

119879 (119880119896

)119879

119867119879

119896

+ 119877)minus1

(36)

119875119896|119896

= (119879 (119880119896

minus 119881119896

) minus 119870119896

119867119896

119879 (119880119896

))

sdot 119875119896|119896minus1

119879 (119880119896

minus 119881119896

)119879

(37)

119883119886

119896|119896

= 119870119896

(119884119896

minus 119862119896

119879 (119880119896

)119883119886

119896|119896minus1

) + 119879 (119880119896

minus 119881119896

)

sdot 119883119886

119896|119896minus1

(38)

Mathematical Problems in Engineering 7

Using (35) (37) and the block diagonal structure of 119875(sdot)

thefollowing relations can be obtained

0 = 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

minus 119880119896

119876119903

119896minus1

minus 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

0 = 119880119896

minus 119881119896

minus 119870119909

119896

119878119896

(39)

where 119880119896

and 119878119896

are defined as

119880119896

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119866minus1

119896minus1

(40)

119878119896

= 1198671

119896

119880119896

+ 1198672

119896

(41)

The above equations lead to

119880119896

119875119903

119896|119896minus1

= 119880119896

(119875119903

119896|119896minus1

minus 119876119903

119896

) (42)

119880119896

= 119880119896

(119868 minus 119876119903

119896

(119875119903

119896|119896minus1

)minus1

) (43)

119881119896

= 119880119896

minus 119870119909

119896

119878119896

(44)

Define the following notation

119860119896minus1

119879 (119881119896minus1

) = [119860119896minus1

119860119896minus1

119881119896minus1

+ 119864119896minus1

0 119866119896minus1

] (45)

The equations of themodified bias free filter and the bias filterare acquired by the next steps

Expanding (34) we have

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119898119896minus1

(46)

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

(47)

where

119898119896minus1

= (119860119896minus1

119881119896minus1

+ 119863119896minus1

minus 119880119896

119866119896minus1

) 119903119896minus1|119896minus1

(48)

Expanding (35) we have

119875119909

119896|119896minus1

= (119864119896minus1

+ 119860119896minus1

119881119896minus1

minus 119880119896

119866119896minus1

) 119875119903

119896minus1|119896minus1

lowast (119864119896minus1

+ 119860119896minus1

119881119896minus1

minus 119880119896

119866119896minus1

)119879

+ 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

+ 119880119896

119876119903

119896

119880119879

119896

(49)

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896

(50)

Then using (40) (43) and (47) (49) can be written as

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

(51)

where

119876119909

119896

= 119876119909

119896

+119872119896

(119872119896

119876119903

119896

)119879

(52)

Expanding (38) and using (41) and (44) we have

119883119896|119896

= 119883119896|119896minus1

+ (119880119896

minus 119881119896

) 119903119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

)

119903119896|119896

= 119903119896|119896minus1

+ 119870119903

119896

(119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

)

(53)

Then

119883119896|119896

= 119870119909

119896

120578119909

119896

+ 119883119896|119896minus1

(54)

where

119878119896

= 119862119896

119880119896

(55)

120578119909

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

+ (119878119896

minus 119878119896

) 119903119896|119896minus1

(56)

Expanding (36) and using (41) we have

119870119903

119896

= 119875119903

119896|119896minus1

119878119879

119896

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)minus1

119870119909

119896

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)

= 119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

(57)

Then

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(58)

Expanding (37) we have

119875119903

119896|119896

= (119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119875119909

119896|119896

= 119875119909

119896|119896minus1

+ (119880119896

minus 119881119896

) 119875119903

119896|119896minus1

(119880119879

119896

minus 119881119879

119896

)

minus (119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

(119880119879

119896

minus 119881119879

119896

))

(59)

Then using (41) and (44)

119875119909

119896|119896

= (119868 minus 119870119909

119896

1198671

119896

)119875119909

119896|119896minus1

(60)

Finally using (25) the estimated value of original state(119894119904120572

119904120573

119904120572

119904120573

) can be obtained by sum of the state119883withthe augmented state 119903

119896|119896minus1

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

(61)

119896|119896

= 119883119896|119896

+ 119881119896

119903119896|119896

(62)

Moreover the unknown parameter 119903( 119903

) is defined as

119903119896|119896minus1

= 119903119896|119896minus1

(63)

119903119896|119896

= 119903119896|119896

(64)

8 Mathematical Problems in Engineering

Based on the above analysis the TEKF can be decoupledinto two filters such as the modified bias free filter and biasfilter The modified bias filter gives the state estimation 119883

119896|119896

and the bias filter gives the bias estimate 119903

119896|119896

The correctedstate estimate 119883119886

119896|119896

(119896|119896

119903119896|119896

) of the TEKF is obtained fromthe estimates of the two filters and coupling equations119880

119896

and119881119896

[21] The modified bias free filter is expressed as follows

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119898119896minus1

(65)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

(66)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(119873119896

)minus1

(67)

119875119909

119896|119896

= (119868 minus 119870119909

119896

1198671

119896

)119875119909

119896|119896minus1

(68)

120578119909

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

+ (119878119896

minus 119878119896

) 119903119896|119896minus1

(69)

119883119896|119896

= 119870119909

119896

120578119909

119896

+ 119883119896|119896minus1

(70)

and the bias filter is

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

(71)

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896

(72)

119870119903

119896

= 119875119903

119896|119896minus1

119878119879

119896

(119873119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)minus1

(73)

119875119903

119896|119896

= (119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

(74)

120578119903

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

(75)

119903119896|119896

= 119903119896|119896minus1

+ 119870119903

119896

120578119903

119896

(76)

with the coupling equations

119878119896

= 1198671

119896

119880119896

+ 1198672

119896

(77)

119880119896

= 119880119896

(119868 minus 119876119903

119896

(119875119903

119896|119896minus1

)minus1

) (78)

119880119896

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119866minus1

119896minus1

(79)

119881119896

= 119880119896

minus 119870119909

119896

119878119896

(80)

119898119896minus1

= (119860119896minus1

119881119896minus1

+ 119863119896minus1

minus 119880119896

119866119896minus1

) 119903119896minus1|119896minus1

(81)

119876119909

119896

= 119876119909

119896

+ 119880119896

(119880119896

119876119903

119896

)119879

(82)

119873119896

= 1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

(83)

The initial conditions of TEKF algorithm are establishedwith the initial conditions of a classical EKF (119883

0|0

1199030|0

1198751199090|0

1198751199091199030|0

1198751199030|0

) so that

1198810

= 119875119909119903

0|0

(119875119903

0|0

)minus1

1198830|0

= 1198830|0

minus 1198810

1199030|0

1199030|0

= 1199030|0

119875119909

0|0

= 119875119909

0|0

minus 1198810

119875119903

0|0

119881119879

0

119875119903

0|0

= 119875119903

0|0

(84)

According to variables of full order filter119883 (1198831

1198832

1198833

1198834

)the stator flux and torque estimators for DTC-SVM ofFigure 1 are then given by

119904120572

= 1198831

119904120573

= 1198832

119904120572

= 1198833

119904120573

= 1198834

100381610038161003816100381610038161003816120595119904

100381610038161003816100381610038161003816= radic

2

119904120572

+ 2

119904120573

119904

= arctan119904120573

119904120572

119890

=3

2119901 (119904120572

119868119904120573

minus 119904120573

119868119904120572

)

(85)

where 119901 is the pole pairs of IM The estimated speed andelectromagnetic torque obtained from the TEKF observer areused to close the speed and torque loop to achieve sensorlessoperations

42 The Stability and Parameter Sensitivity Analysis ofthe TEKF

Theorem 3 The discrete time conventional extended Kalmanfilter (19)ndash(23) is equivalent to the two-stage extern Kalmanfilter (see (61)sim(83))

Proof Before proving the theorem the following five rela-tionships are needed

(1) Using (72) and (78)

119880119896+1

119866119896

119875119903

119896|119896

119866119879

119896

= 119880119896

119875119903

119896|119896minus1

(86)

(2) Using (67) and (73)

119870119909

119896

119872119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

(87)

119870119903

119896

119872119896

= 119875119903

119896|119896minus1

(119878119896

)119879

(88)

where

119872119896

= 1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

+ 119877119896

(89)

Mathematical Problems in Engineering 9

(3) Using (20) we have

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896minus1

(90)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119864119896minus1

(119875119909119903

119896minus1|119896minus1

)119879

119860119879

119896minus1

+ 119860119896minus1

119875119909119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119876119909

119896minus1

(91)

119875119909119903

119896|119896minus1

= 119860119896minus1

119875119909119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

(92)

(4) Using (21)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(93)

119870119903

119896

= (119875119909119903

119896|119896minus1

)119879

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(94)

(5) Using (22)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

(95)

119875119909119903

119896|119896

= 119875119909119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909119903

119896|119896minus1

(96)

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896

1198671

119896

119875119909119903

119896|119896minus1

(97)

By inductive reasoning suppose that at time 119896 minus 1the unknown parameter

119896minus1

and estimated state 119896minus1

areequal to the parameter 119903

119896minus1

and state 119883119896minus1

of the controlsystem respectively we show that TEKF is equivalent to theconventional EKF because these properties are still true attime 119896

Assume that at time 119896 minus 1

119883119896minus1|119896minus1

= 119896minus1|119896minus1

119903119896minus1|119896minus1

= 119903119896minus1|119896minus1

119875119909

119896minus1|119896minus1

= 11987511

119896minus1|119896minus1

119875119909119903

119896minus1|119896minus1

= 11987512

119896minus1|119896minus1

119875119903

119896minus1|119896minus1

= 11987522

119896minus1|119896minus1

(98)

where [ 119875119909119875

119909119903

(119875

119909119903)

119879119875

119903 ] and [ 11987511119875

12

(119875

12)

119879119875

22 ] represent the variance-covariance matrices of the system and estimated variablesrespectively

From (19) we have

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119863119896

119903119896minus1|119896minus1

+ 119861119896

119906119896minus1

(99)

Then using (98) (41) (62) (79) (81) (71) and (61)

119883119896|119896minus1

= 119860119896minus1

(119883119896minus1|119896minus1

+ 119881119896minus1

119903119896minus1|119896minus1

)

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

minus 119898119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

= 119896|119896minus1

(100)

Using (19) (71) (98) (63) and (64) we have

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119903119896|119896minus1

(101)

Using (91) (98) (78) (66) (79) (82) (86) and (72) we obtain

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896minus1

+ 119880119896

119866119896

119875119903

119896minus1|119896minus1

119866119879

119896

119880119879

119896

= 119875119909

119896|119896minus1

+ 119880119896

(119880119896

119875119903

119896|119896minus1

minus 119880119896

119876119903

119896minus1

)119879

= 119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

= 11987511

119896|119896minus1

(102)

Using (90) (98) (72) (32) (71) (29) and (97) we obtain

119875119903

119896|119896minus1

= 11987522

119896minus1|119896minus1

(103)

Using (92) (98) (33) (32) (79) (86) and (91)

119875119909119903

119896|119896minus1

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896minus1

119875119903

119896minus1|119896minus2

= 11987512

119896|119896minus1

(104)

Using (93) (101) (55) (73) (67) (80) and (87)

119870119909

119896

= (119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

) (1198671

119896

)119879

sdot (1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ (119881119896

+ 119870119909

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

(1198671

119896

)119879

)119880minus1

119896

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

)119880minus1

119896

+ 119881119896

119875119903

119896|119896minus1

119878119879

119896

119880minus1

119896

= 119870119909

119896

+ 119881119896

119870119903

119896

(105)

Using (94) (30) and (88) we obtain

119870119903

119896

= (119875119903

119896|119896minus1

)119879

119878119879

119896

119880minus1

119896

= 119870119903

119896|119896minus1

(106)

10 Mathematical Problems in Engineering

Next wewill show that (98) holds at time 119896 From (23)we have

119883119896|119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

)

= 119883119896|119896minus1

+ 119870119909

119896

119903119896

(107)

Then using (61) and (105) the above equation can be writtenas

119883119896|119896

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

+ (119870119909

119896

+ 119881119896

119870119903

119896

) 119903119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 1198671

119896

119883119896|119896minus1

)

+ (119880119896

minus 119870119909

119896

119878119896

) 119903119896|119896minus1

+ 119881119896

119870119903

119896

119903119896

= 119883119896|119896

+ 119881119896

119903119896|119896

= 119896|119896

(108)

Using (95) (105) (102) and (77)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

(109)

Then using (80) (68) (74) and (31) we obtain

119875119909

119896|119896

= 119875119909

119896|119896

+ 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

+ 119881119896

(119875119903

119896|119896

119878119879

119896

(119870119909

119896

)119879

minus 119870119903

119896

1198671

119896

119875119909

119896|119896minus1

)

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

= 11987511

119896|119896

(110)

Using (96) (30) (28) (105) and (80)

119875119909119903

119896|119896

= 119880119896

119875119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

1198671

119896

119880119896

minus 119881119896

119870119903

119896

1198671

119896

119880119896

)119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

119875119903

119896|119896

= 11987512

119896|119896

(111)

Using (97) (106) (95) (29) and (30) we obtain

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896|119896minus1

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119868 minus 119870119903

119896|119896minus1

119878119896

) 119875119903

119896|119896minus1

= 11987522

119896|119896

(112)

Table 1 Kalman estimation arithmetic operation requirement forthe conventional EKF structure

Number of multiplications(119899 = 6119898 = 2 and 119902 = 2)

Number of additions(119899 = 6119898 = 2 and 119902 = 2)

119860119896

119861119896

and 119862

119896

Function of system (9) Function of system (3)

119883119886

119896|119896minus1

1198992 + 119899119902 (48) 1198992 + 119899119902 minus 119899 (42)119875119896|119896minus1

21198993 (432) 21198993 minus 1198992 (396)119883119886

119896|119896minus1

2119899119898 (24) 2119899119898 (24)

119870119896

1198992

119898 + 21198991198982

+ 1198983 (168) 119899

2

119898 + 21198991198982

+ 1198983

minus 2119899119898

(104)119875119896|119896

1198992119898 + 1198993 (288) 1198992119898 + 1198993 minus 1198992 (252)Total 960 818

Finally we show that (98) holds at time 119896 = 0 This can beverified by the initial conditions of TEKF algorithm

43 Numerical Complexity of the Algorithm Tables 1 and2 show the computational effort at each sample time bythe conventional EKF algorithm and TEKF (where roughmatrix-based implementation is used) in which as definedabove 119899 is the dimension of the state vector 119883

119896

119898 is thedimension of the measurement 119884

119896

119902 is the input vector 119880119896

and 119901 is the dimension of the parameter 119903

119896

The total numberof arithmetic operations (additions and multiplications) persample time of the TEKF is 1314 compared with 1778 for arough implementation of a conventional EKF which meansthe operation cost can reduce by 26

5 Simulation and Experimental Results

51 Simulation Results To test the feasibility and perfor-mance of the TEKF method the sensorless DTC-SVM tech-nique for IM drives described in Section 2 is implementedin MATLABSIMULINK environment The values of theinitial state covariance matrices 119875

0

119876 and 119877 have a greatinfluence on the performance of the estimation methodThe diagonal initial state covariance matrix 119875

0

representsvariances or mean-squared errors in the knowledge of theinitial conditions Matrix 119876 gives the statistical descriptionof the drive system Matrix 119877 is related to measured noiseThey can be obtained by considering the stochastic propertiesof the corresponding noises However a fine evaluation ofthe covariance matrices is very difficult because they areusually not known In this paper tuning the initial values ofcovariance matrices 119875

0

119876 and 119877 is using particular criteria[22] to achieve steady-state behaviors of the relative estimatedstates as given by

119876 = diag 20 20 1119890 minus 6 1119890 minus 6 10 10

1198750

= diag 01 01 05 05 1 1

119877 = diag 01 01

(113)

Mathematical Problems in Engineering 11

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000Sp

eed

(rpm

)

(a) Speed estimation

02 04 06 08 10Times (s)

minus002

0

002

004

006

008

01

012

014

016

018

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF

Theta (real)Theta (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7Po

sitio

n (r

ad)

(d) TEKF rotor position error

Current (real)Current (TEKF)

Curr

entIs120572

(A)

02 04 06 08 10Times (s)

minus12

minus8

minus4

0

4

8

12

(e) Real stator current 119868119904120572

and estimation (TEKF)

Current (real)Current (TEKF)

minus12

minus8

minus4

0

4

8

12

Curr

entIs120573

(A)

02 04 06 08 10Times (s)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 3 Continued

12 Mathematical Problems in Engineering

Flux (EKF)Flux (TEKF)

minus05 0 05 1minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(g) Stator flux estimation by TEKF and EKF

02 04 06 08 10Times (s)

minus2

0

2

4

6

8

10

12

14

16

Ψs

(Wb)

times10minus3

(h) Difference Stator flux estimation between TEKF and EKF

Figure 3 Simulation results for parameters estimation

Table 2 Kalman estimation arithmetic operation requirement for the TEKF structure

Number of multiplications(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

Number of additions(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

119860119896

119862119896

119864119896

1198671119896

1198672119896

119861119896

and119863119896

Function of system (25) Function of system (11)119883119896|119896minus1

1198992 + 119899119902 (24) 1198992 + 119899119902 (24)119875119909

119896|119896minus121198993 (128) 21198993 minus 1198992 (112)

119870119909

119896|1198961198992

119898 + 1198991198982 (48) 119899

2

119898 + 1198991198982

minus 2119899119898 (32)119875119909

119896|1198961198992

119898 + 1198993 (96) 119899

2

119898 + 1198993

minus 1198992 (80)

119883119896|119896

2119899119898 + 119898119901 (20) 2119899119898 + 2119898119901 (24)119903119896|119896minus1

1199012 (4) 1199012 minus 119901 (2)119875119903

119896|119896minus121199013 (16) 21199013 minus 1199012 (12)

119870119903

119896|11989631199012119898 + 1199011198982 (32) 31199012119898 + 1199011198982 + 1198982 minus 4119901119898 (20)

119875119903

119896|1198961199013 + 1199012119898 (16) 1199013 + 1199012119898 minus 1199012 (12)

119903119896|119896

2119898119901 + 119899119898 (12) 2119898119901 + 119899119898 (16)119878119896

119898119899119901 (16) 119899119898119901 (16)119880119896minus1

1198992119901 + 1198991199012 (56) 1198992119901 + 1198991199012 minus 119899119901 (48)119881119896

119899119901119898 (16) 119899119901 (8)119880119896minus1

21198991199012 (32) 21198991199012 (32)119898119896minus1

1198992

119901 + 1198991199012

+ 119899119901 (56) 1198992

119901 + 1198991199012

+ 119899119901 minus 119899 (52)119876119909

119896minus121198992

119901 (64) 21198992

119901 + 1198991199012 (64)

119878119896

119898119899119901 (16) 119898119899119901 minus 119898119901 (12)119873119896

21198981198992 (32) 21198981198992 minus 1198982 (60)Total 688 626

In the simulation a comparison is made to verify theequivalence of EKF and TEKF Real-time parameters esti-mated by TEKF are used to formulate the closed loop suchas rotor speed stator flux and electromagnetic torque Theestimations obtained by EKF algorithm are not included in

the sensorless DTC-SVM strategy and only evaluated in openloop A step reference speed was applied to the simulation

The machine is accelerated from 0 rpm to 1000 rpm at0 s and the torque load is set to 4N The simulation resultsof parameter estimation are shown in Figure 3 Figures 3(a)

Mathematical Problems in Engineering 13

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000

Spee

d (r

pm)

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus03

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Theta (real)Theta (TEKF)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

02 04 06 08 10Times (s)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)

Figure 4 Simulation results with parameter variation (119877119903

= 05119877119903nom)

and 3(c) represent the performance of the speed and rotorposition tracking capabilities of the control system withTEKF and EKF respectively Figures 3(e) 3(f) and 3(g)represent the estimated stator current and flux they showthat ripples are significantly suppressed due to the SVMmodulation scheme Figures 3(b) 3(d) and 3(h) show thatfor variations of speed reference the rotor speed rotorposition and stator flux errors between the two observers arevery little It is verified that the two observers are equivalentThe difference between two estimators is caused by accuracyloss in TEKF which uses more calculation steps

In order to further verify the performance of TEKFagainst model-plant parameter mismatches and the equiv-alence of two observers the change in rotor resistance isconsidered Rotor resistance will increase due to temperaturerise while the motor is running To simulate this conditionthe rotor resistance in TEKF and EKF is increased to 200comparedwith the normal value which is equivalent to a 50decrease in the actual rotor resistance The machine is stillaccelerated from 0 rpm to 1000 rpm at 0 s and the torque is setto 4N Figure 4 shows that for variation of rotor resistancethe steady-state speed and rotor position errors are negligible

14 Mathematical Problems in Engineering

(a)

Computer

380V

TXRX

120596r

Incrementalencoder

Udc

MicrocomputerDSP TMS3206713

Drive signalCurrentsensor

Voltagesensor

Voltagesensor

IM MC

ISU ISV

VSU VSW VSV

(b)

Figure 5 Complete drive system (a) Picture of experimental setup (b) Functional block diagram of the experimental setup

and the difference of the speed and rotor position estimationsbetween the two observers is rather null

52 Experimental Results The overall experimental setup isshown in Figure 5 and the specifications and rated parametersof the IM controller and inverter are listed in Table 3 Inthe experimental hardware an Expert3 control system fromMyway company and a three-phase two-pole 15 kW IM areappliedThe IM is mechanically coupled to a magnetic clutch(MC) which provides rated torque even at very low speedThe main processor in Expert3 control system is a floatingpoint processor TMS320C6713 with a max clock speed of225MHz All the algorithms including TEKF EKF DTCalgorithm and some transformation modules are imple-mented in TMS320C6713 with 100120583s sampling time and dataacquisition of the parameter estimations measured variablesand their visualization are realized on the cockpit provided byPEView9 software Insulated Gate Bipolar Transistor (IGBT)module is driven by the PWM signal with a switchingfrequency of 10 kHz and 2 120583s dead time The stator currentsare measured via two Hall effect current sensors The rotorangle and speed of IM are measured from an incrementalencoder with 2048 pulses per revolution

This experiment test is here to testify the performanceof TEKF and demonstrate that the two estimators aremathematically equivalent The machine is accelerated from600 rpm to 1000 rpm and 4N torque load is set Theexperimental results of parameter estimation based on twoobservers are given in Figures 6 and 7 Figures 6(a) and 6(c)show that the TEKF still has a good tracking performance ofthe speed and rotor position in experiment Figures 6(d) 6(e)and 6(f) illustrate stator flux and stator current estimationrobustness Figures 6(b) 6(g) and 6(f) referring to thedifference in speed and stator current estimations given by

Table 3 Specification of induction motor and inverter

Induction motor ValueNominal torque 10NmNominal voltage 380VRotor resistance 119877

119903

25ΩStator resistance 119877

119904

36ΩStator inductances 119871

119904

0301HRotor inductances 119871

119903

0302HMutual inductances 119871

119898

0273HPole pairs 2Invertercontroller ValueSwitching device 1000V 80A IGBTControl cycle time 100 120583sMain CPU DSP TMS320C6713 225MHz

the two observers are still small These experiment resultsprove that the two estimators are mathematically equivalentFigure 7 shows the speed and rotor position estimationsbased onTEKFandEKF for a 50decrease of rotor resistance(the same as the simulation) As expected the steady errorof the TEKF and the difference in speed and rotor positionestimations are still tiny Robustness of TEKF is verified

6 Conclusion

Themajor shortcoming of the conventional EKF is numericalproblems and computational burden due to the high orderof the mathematical models This has generally limited thereal-time digital implementation of the EKF for industrialfield So in this study a novel extended Kalman filter

Mathematical Problems in Engineering 15

Speed (real)Speed (EKF)Speed (TEKF)

Spee

d (r

min

)

02 04 06 08 10t (02 sgrid)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimationSp

eed

(rm

in)

minus005

0

005

01

015

02

025

03

035

04

045

02 04 06 08 10t (02 sgrid)

(b) Difference of speed estimation between EKF and TEKF

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation

Flux (EKF)Flux (TEKF)

minus08 minus04 1minus02 0 02 04 06 08minus06minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(d) Stator flux estimation by TEKF and EKF

Is120572 (real)Is120572 (TEKF)

Curr

entIs120572

(A)

minus6

minus4

minus2

0

2

4

6

02 04 06 08 10t (02 sgrid)

(e) Real stator current 119868119904120572

and estimation (TEKF)

Is120573 (real)Is120573 (TEKF)

minus6

minus4

minus2

0

2

4

6

Curr

entIs120573

(A)

02 04 06 08 10t (02 sgrid)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 6 Continued

16 Mathematical Problems in Engineering

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Erro

r cur

rentIs120572

(A)

02 04 06 08 10t (02 sgrid)

(g) TEKF stator current 119868119904120572

error

minus02

minus015

minus01

minus005

0

005

01

015

Erro

r cur

rentIs120573

(A)

02 04 06 08 10t (02 sgrid)

(h) TEKF stator current 119868119904120573

error

Figure 6 Experimental results for parameters estimation

Speed (real)Speed (EKF)

Speed (TEKF)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

0

005

01

015

02

025

03

035

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)Figure 7 Experimental results with parameter variation (119877

119903

= 05119877119903nom)

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Direct Torque Control of Sensorless ...

Mathematical Problems in Engineering 7

Using (35) (37) and the block diagonal structure of 119875(sdot)

thefollowing relations can be obtained

0 = 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

minus 119880119896

119876119903

119896minus1

minus 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

0 = 119880119896

minus 119881119896

minus 119870119909

119896

119878119896

(39)

where 119880119896

and 119878119896

are defined as

119880119896

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119866minus1

119896minus1

(40)

119878119896

= 1198671

119896

119880119896

+ 1198672

119896

(41)

The above equations lead to

119880119896

119875119903

119896|119896minus1

= 119880119896

(119875119903

119896|119896minus1

minus 119876119903

119896

) (42)

119880119896

= 119880119896

(119868 minus 119876119903

119896

(119875119903

119896|119896minus1

)minus1

) (43)

119881119896

= 119880119896

minus 119870119909

119896

119878119896

(44)

Define the following notation

119860119896minus1

119879 (119881119896minus1

) = [119860119896minus1

119860119896minus1

119881119896minus1

+ 119864119896minus1

0 119866119896minus1

] (45)

The equations of themodified bias free filter and the bias filterare acquired by the next steps

Expanding (34) we have

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119898119896minus1

(46)

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

(47)

where

119898119896minus1

= (119860119896minus1

119881119896minus1

+ 119863119896minus1

minus 119880119896

119866119896minus1

) 119903119896minus1|119896minus1

(48)

Expanding (35) we have

119875119909

119896|119896minus1

= (119864119896minus1

+ 119860119896minus1

119881119896minus1

minus 119880119896

119866119896minus1

) 119875119903

119896minus1|119896minus1

lowast (119864119896minus1

+ 119860119896minus1

119881119896minus1

minus 119880119896

119866119896minus1

)119879

+ 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

+ 119880119896

119876119903

119896

119880119879

119896

(49)

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896

(50)

Then using (40) (43) and (47) (49) can be written as

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

(51)

where

119876119909

119896

= 119876119909

119896

+119872119896

(119872119896

119876119903

119896

)119879

(52)

Expanding (38) and using (41) and (44) we have

119883119896|119896

= 119883119896|119896minus1

+ (119880119896

minus 119881119896

) 119903119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

)

119903119896|119896

= 119903119896|119896minus1

+ 119870119903

119896

(119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

)

(53)

Then

119883119896|119896

= 119870119909

119896

120578119909

119896

+ 119883119896|119896minus1

(54)

where

119878119896

= 119862119896

119880119896

(55)

120578119909

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

+ (119878119896

minus 119878119896

) 119903119896|119896minus1

(56)

Expanding (36) and using (41) we have

119870119903

119896

= 119875119903

119896|119896minus1

119878119879

119896

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)minus1

119870119909

119896

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)

= 119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

(57)

Then

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(58)

Expanding (37) we have

119875119903

119896|119896

= (119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119875119909

119896|119896

= 119875119909

119896|119896minus1

+ (119880119896

minus 119881119896

) 119875119903

119896|119896minus1

(119880119879

119896

minus 119881119879

119896

)

minus (119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

(119880119879

119896

minus 119881119879

119896

))

(59)

Then using (41) and (44)

119875119909

119896|119896

= (119868 minus 119870119909

119896

1198671

119896

)119875119909

119896|119896minus1

(60)

Finally using (25) the estimated value of original state(119894119904120572

119904120573

119904120572

119904120573

) can be obtained by sum of the state119883withthe augmented state 119903

119896|119896minus1

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

(61)

119896|119896

= 119883119896|119896

+ 119881119896

119903119896|119896

(62)

Moreover the unknown parameter 119903( 119903

) is defined as

119903119896|119896minus1

= 119903119896|119896minus1

(63)

119903119896|119896

= 119903119896|119896

(64)

8 Mathematical Problems in Engineering

Based on the above analysis the TEKF can be decoupledinto two filters such as the modified bias free filter and biasfilter The modified bias filter gives the state estimation 119883

119896|119896

and the bias filter gives the bias estimate 119903

119896|119896

The correctedstate estimate 119883119886

119896|119896

(119896|119896

119903119896|119896

) of the TEKF is obtained fromthe estimates of the two filters and coupling equations119880

119896

and119881119896

[21] The modified bias free filter is expressed as follows

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119898119896minus1

(65)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

(66)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(119873119896

)minus1

(67)

119875119909

119896|119896

= (119868 minus 119870119909

119896

1198671

119896

)119875119909

119896|119896minus1

(68)

120578119909

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

+ (119878119896

minus 119878119896

) 119903119896|119896minus1

(69)

119883119896|119896

= 119870119909

119896

120578119909

119896

+ 119883119896|119896minus1

(70)

and the bias filter is

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

(71)

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896

(72)

119870119903

119896

= 119875119903

119896|119896minus1

119878119879

119896

(119873119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)minus1

(73)

119875119903

119896|119896

= (119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

(74)

120578119903

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

(75)

119903119896|119896

= 119903119896|119896minus1

+ 119870119903

119896

120578119903

119896

(76)

with the coupling equations

119878119896

= 1198671

119896

119880119896

+ 1198672

119896

(77)

119880119896

= 119880119896

(119868 minus 119876119903

119896

(119875119903

119896|119896minus1

)minus1

) (78)

119880119896

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119866minus1

119896minus1

(79)

119881119896

= 119880119896

minus 119870119909

119896

119878119896

(80)

119898119896minus1

= (119860119896minus1

119881119896minus1

+ 119863119896minus1

minus 119880119896

119866119896minus1

) 119903119896minus1|119896minus1

(81)

119876119909

119896

= 119876119909

119896

+ 119880119896

(119880119896

119876119903

119896

)119879

(82)

119873119896

= 1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

(83)

The initial conditions of TEKF algorithm are establishedwith the initial conditions of a classical EKF (119883

0|0

1199030|0

1198751199090|0

1198751199091199030|0

1198751199030|0

) so that

1198810

= 119875119909119903

0|0

(119875119903

0|0

)minus1

1198830|0

= 1198830|0

minus 1198810

1199030|0

1199030|0

= 1199030|0

119875119909

0|0

= 119875119909

0|0

minus 1198810

119875119903

0|0

119881119879

0

119875119903

0|0

= 119875119903

0|0

(84)

According to variables of full order filter119883 (1198831

1198832

1198833

1198834

)the stator flux and torque estimators for DTC-SVM ofFigure 1 are then given by

119904120572

= 1198831

119904120573

= 1198832

119904120572

= 1198833

119904120573

= 1198834

100381610038161003816100381610038161003816120595119904

100381610038161003816100381610038161003816= radic

2

119904120572

+ 2

119904120573

119904

= arctan119904120573

119904120572

119890

=3

2119901 (119904120572

119868119904120573

minus 119904120573

119868119904120572

)

(85)

where 119901 is the pole pairs of IM The estimated speed andelectromagnetic torque obtained from the TEKF observer areused to close the speed and torque loop to achieve sensorlessoperations

42 The Stability and Parameter Sensitivity Analysis ofthe TEKF

Theorem 3 The discrete time conventional extended Kalmanfilter (19)ndash(23) is equivalent to the two-stage extern Kalmanfilter (see (61)sim(83))

Proof Before proving the theorem the following five rela-tionships are needed

(1) Using (72) and (78)

119880119896+1

119866119896

119875119903

119896|119896

119866119879

119896

= 119880119896

119875119903

119896|119896minus1

(86)

(2) Using (67) and (73)

119870119909

119896

119872119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

(87)

119870119903

119896

119872119896

= 119875119903

119896|119896minus1

(119878119896

)119879

(88)

where

119872119896

= 1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

+ 119877119896

(89)

Mathematical Problems in Engineering 9

(3) Using (20) we have

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896minus1

(90)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119864119896minus1

(119875119909119903

119896minus1|119896minus1

)119879

119860119879

119896minus1

+ 119860119896minus1

119875119909119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119876119909

119896minus1

(91)

119875119909119903

119896|119896minus1

= 119860119896minus1

119875119909119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

(92)

(4) Using (21)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(93)

119870119903

119896

= (119875119909119903

119896|119896minus1

)119879

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(94)

(5) Using (22)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

(95)

119875119909119903

119896|119896

= 119875119909119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909119903

119896|119896minus1

(96)

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896

1198671

119896

119875119909119903

119896|119896minus1

(97)

By inductive reasoning suppose that at time 119896 minus 1the unknown parameter

119896minus1

and estimated state 119896minus1

areequal to the parameter 119903

119896minus1

and state 119883119896minus1

of the controlsystem respectively we show that TEKF is equivalent to theconventional EKF because these properties are still true attime 119896

Assume that at time 119896 minus 1

119883119896minus1|119896minus1

= 119896minus1|119896minus1

119903119896minus1|119896minus1

= 119903119896minus1|119896minus1

119875119909

119896minus1|119896minus1

= 11987511

119896minus1|119896minus1

119875119909119903

119896minus1|119896minus1

= 11987512

119896minus1|119896minus1

119875119903

119896minus1|119896minus1

= 11987522

119896minus1|119896minus1

(98)

where [ 119875119909119875

119909119903

(119875

119909119903)

119879119875

119903 ] and [ 11987511119875

12

(119875

12)

119879119875

22 ] represent the variance-covariance matrices of the system and estimated variablesrespectively

From (19) we have

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119863119896

119903119896minus1|119896minus1

+ 119861119896

119906119896minus1

(99)

Then using (98) (41) (62) (79) (81) (71) and (61)

119883119896|119896minus1

= 119860119896minus1

(119883119896minus1|119896minus1

+ 119881119896minus1

119903119896minus1|119896minus1

)

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

minus 119898119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

= 119896|119896minus1

(100)

Using (19) (71) (98) (63) and (64) we have

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119903119896|119896minus1

(101)

Using (91) (98) (78) (66) (79) (82) (86) and (72) we obtain

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896minus1

+ 119880119896

119866119896

119875119903

119896minus1|119896minus1

119866119879

119896

119880119879

119896

= 119875119909

119896|119896minus1

+ 119880119896

(119880119896

119875119903

119896|119896minus1

minus 119880119896

119876119903

119896minus1

)119879

= 119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

= 11987511

119896|119896minus1

(102)

Using (90) (98) (72) (32) (71) (29) and (97) we obtain

119875119903

119896|119896minus1

= 11987522

119896minus1|119896minus1

(103)

Using (92) (98) (33) (32) (79) (86) and (91)

119875119909119903

119896|119896minus1

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896minus1

119875119903

119896minus1|119896minus2

= 11987512

119896|119896minus1

(104)

Using (93) (101) (55) (73) (67) (80) and (87)

119870119909

119896

= (119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

) (1198671

119896

)119879

sdot (1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ (119881119896

+ 119870119909

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

(1198671

119896

)119879

)119880minus1

119896

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

)119880minus1

119896

+ 119881119896

119875119903

119896|119896minus1

119878119879

119896

119880minus1

119896

= 119870119909

119896

+ 119881119896

119870119903

119896

(105)

Using (94) (30) and (88) we obtain

119870119903

119896

= (119875119903

119896|119896minus1

)119879

119878119879

119896

119880minus1

119896

= 119870119903

119896|119896minus1

(106)

10 Mathematical Problems in Engineering

Next wewill show that (98) holds at time 119896 From (23)we have

119883119896|119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

)

= 119883119896|119896minus1

+ 119870119909

119896

119903119896

(107)

Then using (61) and (105) the above equation can be writtenas

119883119896|119896

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

+ (119870119909

119896

+ 119881119896

119870119903

119896

) 119903119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 1198671

119896

119883119896|119896minus1

)

+ (119880119896

minus 119870119909

119896

119878119896

) 119903119896|119896minus1

+ 119881119896

119870119903

119896

119903119896

= 119883119896|119896

+ 119881119896

119903119896|119896

= 119896|119896

(108)

Using (95) (105) (102) and (77)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

(109)

Then using (80) (68) (74) and (31) we obtain

119875119909

119896|119896

= 119875119909

119896|119896

+ 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

+ 119881119896

(119875119903

119896|119896

119878119879

119896

(119870119909

119896

)119879

minus 119870119903

119896

1198671

119896

119875119909

119896|119896minus1

)

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

= 11987511

119896|119896

(110)

Using (96) (30) (28) (105) and (80)

119875119909119903

119896|119896

= 119880119896

119875119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

1198671

119896

119880119896

minus 119881119896

119870119903

119896

1198671

119896

119880119896

)119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

119875119903

119896|119896

= 11987512

119896|119896

(111)

Using (97) (106) (95) (29) and (30) we obtain

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896|119896minus1

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119868 minus 119870119903

119896|119896minus1

119878119896

) 119875119903

119896|119896minus1

= 11987522

119896|119896

(112)

Table 1 Kalman estimation arithmetic operation requirement forthe conventional EKF structure

Number of multiplications(119899 = 6119898 = 2 and 119902 = 2)

Number of additions(119899 = 6119898 = 2 and 119902 = 2)

119860119896

119861119896

and 119862

119896

Function of system (9) Function of system (3)

119883119886

119896|119896minus1

1198992 + 119899119902 (48) 1198992 + 119899119902 minus 119899 (42)119875119896|119896minus1

21198993 (432) 21198993 minus 1198992 (396)119883119886

119896|119896minus1

2119899119898 (24) 2119899119898 (24)

119870119896

1198992

119898 + 21198991198982

+ 1198983 (168) 119899

2

119898 + 21198991198982

+ 1198983

minus 2119899119898

(104)119875119896|119896

1198992119898 + 1198993 (288) 1198992119898 + 1198993 minus 1198992 (252)Total 960 818

Finally we show that (98) holds at time 119896 = 0 This can beverified by the initial conditions of TEKF algorithm

43 Numerical Complexity of the Algorithm Tables 1 and2 show the computational effort at each sample time bythe conventional EKF algorithm and TEKF (where roughmatrix-based implementation is used) in which as definedabove 119899 is the dimension of the state vector 119883

119896

119898 is thedimension of the measurement 119884

119896

119902 is the input vector 119880119896

and 119901 is the dimension of the parameter 119903

119896

The total numberof arithmetic operations (additions and multiplications) persample time of the TEKF is 1314 compared with 1778 for arough implementation of a conventional EKF which meansthe operation cost can reduce by 26

5 Simulation and Experimental Results

51 Simulation Results To test the feasibility and perfor-mance of the TEKF method the sensorless DTC-SVM tech-nique for IM drives described in Section 2 is implementedin MATLABSIMULINK environment The values of theinitial state covariance matrices 119875

0

119876 and 119877 have a greatinfluence on the performance of the estimation methodThe diagonal initial state covariance matrix 119875

0

representsvariances or mean-squared errors in the knowledge of theinitial conditions Matrix 119876 gives the statistical descriptionof the drive system Matrix 119877 is related to measured noiseThey can be obtained by considering the stochastic propertiesof the corresponding noises However a fine evaluation ofthe covariance matrices is very difficult because they areusually not known In this paper tuning the initial values ofcovariance matrices 119875

0

119876 and 119877 is using particular criteria[22] to achieve steady-state behaviors of the relative estimatedstates as given by

119876 = diag 20 20 1119890 minus 6 1119890 minus 6 10 10

1198750

= diag 01 01 05 05 1 1

119877 = diag 01 01

(113)

Mathematical Problems in Engineering 11

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000Sp

eed

(rpm

)

(a) Speed estimation

02 04 06 08 10Times (s)

minus002

0

002

004

006

008

01

012

014

016

018

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF

Theta (real)Theta (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7Po

sitio

n (r

ad)

(d) TEKF rotor position error

Current (real)Current (TEKF)

Curr

entIs120572

(A)

02 04 06 08 10Times (s)

minus12

minus8

minus4

0

4

8

12

(e) Real stator current 119868119904120572

and estimation (TEKF)

Current (real)Current (TEKF)

minus12

minus8

minus4

0

4

8

12

Curr

entIs120573

(A)

02 04 06 08 10Times (s)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 3 Continued

12 Mathematical Problems in Engineering

Flux (EKF)Flux (TEKF)

minus05 0 05 1minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(g) Stator flux estimation by TEKF and EKF

02 04 06 08 10Times (s)

minus2

0

2

4

6

8

10

12

14

16

Ψs

(Wb)

times10minus3

(h) Difference Stator flux estimation between TEKF and EKF

Figure 3 Simulation results for parameters estimation

Table 2 Kalman estimation arithmetic operation requirement for the TEKF structure

Number of multiplications(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

Number of additions(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

119860119896

119862119896

119864119896

1198671119896

1198672119896

119861119896

and119863119896

Function of system (25) Function of system (11)119883119896|119896minus1

1198992 + 119899119902 (24) 1198992 + 119899119902 (24)119875119909

119896|119896minus121198993 (128) 21198993 minus 1198992 (112)

119870119909

119896|1198961198992

119898 + 1198991198982 (48) 119899

2

119898 + 1198991198982

minus 2119899119898 (32)119875119909

119896|1198961198992

119898 + 1198993 (96) 119899

2

119898 + 1198993

minus 1198992 (80)

119883119896|119896

2119899119898 + 119898119901 (20) 2119899119898 + 2119898119901 (24)119903119896|119896minus1

1199012 (4) 1199012 minus 119901 (2)119875119903

119896|119896minus121199013 (16) 21199013 minus 1199012 (12)

119870119903

119896|11989631199012119898 + 1199011198982 (32) 31199012119898 + 1199011198982 + 1198982 minus 4119901119898 (20)

119875119903

119896|1198961199013 + 1199012119898 (16) 1199013 + 1199012119898 minus 1199012 (12)

119903119896|119896

2119898119901 + 119899119898 (12) 2119898119901 + 119899119898 (16)119878119896

119898119899119901 (16) 119899119898119901 (16)119880119896minus1

1198992119901 + 1198991199012 (56) 1198992119901 + 1198991199012 minus 119899119901 (48)119881119896

119899119901119898 (16) 119899119901 (8)119880119896minus1

21198991199012 (32) 21198991199012 (32)119898119896minus1

1198992

119901 + 1198991199012

+ 119899119901 (56) 1198992

119901 + 1198991199012

+ 119899119901 minus 119899 (52)119876119909

119896minus121198992

119901 (64) 21198992

119901 + 1198991199012 (64)

119878119896

119898119899119901 (16) 119898119899119901 minus 119898119901 (12)119873119896

21198981198992 (32) 21198981198992 minus 1198982 (60)Total 688 626

In the simulation a comparison is made to verify theequivalence of EKF and TEKF Real-time parameters esti-mated by TEKF are used to formulate the closed loop suchas rotor speed stator flux and electromagnetic torque Theestimations obtained by EKF algorithm are not included in

the sensorless DTC-SVM strategy and only evaluated in openloop A step reference speed was applied to the simulation

The machine is accelerated from 0 rpm to 1000 rpm at0 s and the torque load is set to 4N The simulation resultsof parameter estimation are shown in Figure 3 Figures 3(a)

Mathematical Problems in Engineering 13

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000

Spee

d (r

pm)

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus03

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Theta (real)Theta (TEKF)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

02 04 06 08 10Times (s)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)

Figure 4 Simulation results with parameter variation (119877119903

= 05119877119903nom)

and 3(c) represent the performance of the speed and rotorposition tracking capabilities of the control system withTEKF and EKF respectively Figures 3(e) 3(f) and 3(g)represent the estimated stator current and flux they showthat ripples are significantly suppressed due to the SVMmodulation scheme Figures 3(b) 3(d) and 3(h) show thatfor variations of speed reference the rotor speed rotorposition and stator flux errors between the two observers arevery little It is verified that the two observers are equivalentThe difference between two estimators is caused by accuracyloss in TEKF which uses more calculation steps

In order to further verify the performance of TEKFagainst model-plant parameter mismatches and the equiv-alence of two observers the change in rotor resistance isconsidered Rotor resistance will increase due to temperaturerise while the motor is running To simulate this conditionthe rotor resistance in TEKF and EKF is increased to 200comparedwith the normal value which is equivalent to a 50decrease in the actual rotor resistance The machine is stillaccelerated from 0 rpm to 1000 rpm at 0 s and the torque is setto 4N Figure 4 shows that for variation of rotor resistancethe steady-state speed and rotor position errors are negligible

14 Mathematical Problems in Engineering

(a)

Computer

380V

TXRX

120596r

Incrementalencoder

Udc

MicrocomputerDSP TMS3206713

Drive signalCurrentsensor

Voltagesensor

Voltagesensor

IM MC

ISU ISV

VSU VSW VSV

(b)

Figure 5 Complete drive system (a) Picture of experimental setup (b) Functional block diagram of the experimental setup

and the difference of the speed and rotor position estimationsbetween the two observers is rather null

52 Experimental Results The overall experimental setup isshown in Figure 5 and the specifications and rated parametersof the IM controller and inverter are listed in Table 3 Inthe experimental hardware an Expert3 control system fromMyway company and a three-phase two-pole 15 kW IM areappliedThe IM is mechanically coupled to a magnetic clutch(MC) which provides rated torque even at very low speedThe main processor in Expert3 control system is a floatingpoint processor TMS320C6713 with a max clock speed of225MHz All the algorithms including TEKF EKF DTCalgorithm and some transformation modules are imple-mented in TMS320C6713 with 100120583s sampling time and dataacquisition of the parameter estimations measured variablesand their visualization are realized on the cockpit provided byPEView9 software Insulated Gate Bipolar Transistor (IGBT)module is driven by the PWM signal with a switchingfrequency of 10 kHz and 2 120583s dead time The stator currentsare measured via two Hall effect current sensors The rotorangle and speed of IM are measured from an incrementalencoder with 2048 pulses per revolution

This experiment test is here to testify the performanceof TEKF and demonstrate that the two estimators aremathematically equivalent The machine is accelerated from600 rpm to 1000 rpm and 4N torque load is set Theexperimental results of parameter estimation based on twoobservers are given in Figures 6 and 7 Figures 6(a) and 6(c)show that the TEKF still has a good tracking performance ofthe speed and rotor position in experiment Figures 6(d) 6(e)and 6(f) illustrate stator flux and stator current estimationrobustness Figures 6(b) 6(g) and 6(f) referring to thedifference in speed and stator current estimations given by

Table 3 Specification of induction motor and inverter

Induction motor ValueNominal torque 10NmNominal voltage 380VRotor resistance 119877

119903

25ΩStator resistance 119877

119904

36ΩStator inductances 119871

119904

0301HRotor inductances 119871

119903

0302HMutual inductances 119871

119898

0273HPole pairs 2Invertercontroller ValueSwitching device 1000V 80A IGBTControl cycle time 100 120583sMain CPU DSP TMS320C6713 225MHz

the two observers are still small These experiment resultsprove that the two estimators are mathematically equivalentFigure 7 shows the speed and rotor position estimationsbased onTEKFandEKF for a 50decrease of rotor resistance(the same as the simulation) As expected the steady errorof the TEKF and the difference in speed and rotor positionestimations are still tiny Robustness of TEKF is verified

6 Conclusion

Themajor shortcoming of the conventional EKF is numericalproblems and computational burden due to the high orderof the mathematical models This has generally limited thereal-time digital implementation of the EKF for industrialfield So in this study a novel extended Kalman filter

Mathematical Problems in Engineering 15

Speed (real)Speed (EKF)Speed (TEKF)

Spee

d (r

min

)

02 04 06 08 10t (02 sgrid)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimationSp

eed

(rm

in)

minus005

0

005

01

015

02

025

03

035

04

045

02 04 06 08 10t (02 sgrid)

(b) Difference of speed estimation between EKF and TEKF

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation

Flux (EKF)Flux (TEKF)

minus08 minus04 1minus02 0 02 04 06 08minus06minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(d) Stator flux estimation by TEKF and EKF

Is120572 (real)Is120572 (TEKF)

Curr

entIs120572

(A)

minus6

minus4

minus2

0

2

4

6

02 04 06 08 10t (02 sgrid)

(e) Real stator current 119868119904120572

and estimation (TEKF)

Is120573 (real)Is120573 (TEKF)

minus6

minus4

minus2

0

2

4

6

Curr

entIs120573

(A)

02 04 06 08 10t (02 sgrid)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 6 Continued

16 Mathematical Problems in Engineering

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Erro

r cur

rentIs120572

(A)

02 04 06 08 10t (02 sgrid)

(g) TEKF stator current 119868119904120572

error

minus02

minus015

minus01

minus005

0

005

01

015

Erro

r cur

rentIs120573

(A)

02 04 06 08 10t (02 sgrid)

(h) TEKF stator current 119868119904120573

error

Figure 6 Experimental results for parameters estimation

Speed (real)Speed (EKF)

Speed (TEKF)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

0

005

01

015

02

025

03

035

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)Figure 7 Experimental results with parameter variation (119877

119903

= 05119877119903nom)

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Research Article Direct Torque Control of Sensorless ...

8 Mathematical Problems in Engineering

Based on the above analysis the TEKF can be decoupledinto two filters such as the modified bias free filter and biasfilter The modified bias filter gives the state estimation 119883

119896|119896

and the bias filter gives the bias estimate 119903

119896|119896

The correctedstate estimate 119883119886

119896|119896

(119896|119896

119903119896|119896

) of the TEKF is obtained fromthe estimates of the two filters and coupling equations119880

119896

and119881119896

[21] The modified bias free filter is expressed as follows

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119898119896minus1

(65)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896

(66)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(119873119896

)minus1

(67)

119875119909

119896|119896

= (119868 minus 119870119909

119896

1198671

119896

)119875119909

119896|119896minus1

(68)

120578119909

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

+ (119878119896

minus 119878119896

) 119903119896|119896minus1

(69)

119883119896|119896

= 119870119909

119896

120578119909

119896

+ 119883119896|119896minus1

(70)

and the bias filter is

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

(71)

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896

(72)

119870119903

119896

= 119875119903

119896|119896minus1

119878119879

119896

(119873119896

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

)minus1

(73)

119875119903

119896|119896

= (119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

(74)

120578119903

119896

= 119884119896

minus 119862119896

119883119896|119896minus1

minus 119878119896

119903119896|119896minus1

(75)

119903119896|119896

= 119903119896|119896minus1

+ 119870119903

119896

120578119903

119896

(76)

with the coupling equations

119878119896

= 1198671

119896

119880119896

+ 1198672

119896

(77)

119880119896

= 119880119896

(119868 minus 119876119903

119896

(119875119903

119896|119896minus1

)minus1

) (78)

119880119896

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119866minus1

119896minus1

(79)

119881119896

= 119880119896

minus 119870119909

119896

119878119896

(80)

119898119896minus1

= (119860119896minus1

119881119896minus1

+ 119863119896minus1

minus 119880119896

119866119896minus1

) 119903119896minus1|119896minus1

(81)

119876119909

119896

= 119876119909

119896

+ 119880119896

(119880119896

119876119903

119896

)119879

(82)

119873119896

= 1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

(83)

The initial conditions of TEKF algorithm are establishedwith the initial conditions of a classical EKF (119883

0|0

1199030|0

1198751199090|0

1198751199091199030|0

1198751199030|0

) so that

1198810

= 119875119909119903

0|0

(119875119903

0|0

)minus1

1198830|0

= 1198830|0

minus 1198810

1199030|0

1199030|0

= 1199030|0

119875119909

0|0

= 119875119909

0|0

minus 1198810

119875119903

0|0

119881119879

0

119875119903

0|0

= 119875119903

0|0

(84)

According to variables of full order filter119883 (1198831

1198832

1198833

1198834

)the stator flux and torque estimators for DTC-SVM ofFigure 1 are then given by

119904120572

= 1198831

119904120573

= 1198832

119904120572

= 1198833

119904120573

= 1198834

100381610038161003816100381610038161003816120595119904

100381610038161003816100381610038161003816= radic

2

119904120572

+ 2

119904120573

119904

= arctan119904120573

119904120572

119890

=3

2119901 (119904120572

119868119904120573

minus 119904120573

119868119904120572

)

(85)

where 119901 is the pole pairs of IM The estimated speed andelectromagnetic torque obtained from the TEKF observer areused to close the speed and torque loop to achieve sensorlessoperations

42 The Stability and Parameter Sensitivity Analysis ofthe TEKF

Theorem 3 The discrete time conventional extended Kalmanfilter (19)ndash(23) is equivalent to the two-stage extern Kalmanfilter (see (61)sim(83))

Proof Before proving the theorem the following five rela-tionships are needed

(1) Using (72) and (78)

119880119896+1

119866119896

119875119903

119896|119896

119866119879

119896

= 119880119896

119875119903

119896|119896minus1

(86)

(2) Using (67) and (73)

119870119909

119896

119872119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

(87)

119870119903

119896

119872119896

= 119875119903

119896|119896minus1

(119878119896

)119879

(88)

where

119872119896

= 1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119878119896

119875119903

119896|119896minus1

119878119879

119896

+ 119877119896

(89)

Mathematical Problems in Engineering 9

(3) Using (20) we have

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896minus1

(90)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119864119896minus1

(119875119909119903

119896minus1|119896minus1

)119879

119860119879

119896minus1

+ 119860119896minus1

119875119909119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119876119909

119896minus1

(91)

119875119909119903

119896|119896minus1

= 119860119896minus1

119875119909119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

(92)

(4) Using (21)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(93)

119870119903

119896

= (119875119909119903

119896|119896minus1

)119879

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(94)

(5) Using (22)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

(95)

119875119909119903

119896|119896

= 119875119909119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909119903

119896|119896minus1

(96)

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896

1198671

119896

119875119909119903

119896|119896minus1

(97)

By inductive reasoning suppose that at time 119896 minus 1the unknown parameter

119896minus1

and estimated state 119896minus1

areequal to the parameter 119903

119896minus1

and state 119883119896minus1

of the controlsystem respectively we show that TEKF is equivalent to theconventional EKF because these properties are still true attime 119896

Assume that at time 119896 minus 1

119883119896minus1|119896minus1

= 119896minus1|119896minus1

119903119896minus1|119896minus1

= 119903119896minus1|119896minus1

119875119909

119896minus1|119896minus1

= 11987511

119896minus1|119896minus1

119875119909119903

119896minus1|119896minus1

= 11987512

119896minus1|119896minus1

119875119903

119896minus1|119896minus1

= 11987522

119896minus1|119896minus1

(98)

where [ 119875119909119875

119909119903

(119875

119909119903)

119879119875

119903 ] and [ 11987511119875

12

(119875

12)

119879119875

22 ] represent the variance-covariance matrices of the system and estimated variablesrespectively

From (19) we have

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119863119896

119903119896minus1|119896minus1

+ 119861119896

119906119896minus1

(99)

Then using (98) (41) (62) (79) (81) (71) and (61)

119883119896|119896minus1

= 119860119896minus1

(119883119896minus1|119896minus1

+ 119881119896minus1

119903119896minus1|119896minus1

)

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

minus 119898119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

= 119896|119896minus1

(100)

Using (19) (71) (98) (63) and (64) we have

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119903119896|119896minus1

(101)

Using (91) (98) (78) (66) (79) (82) (86) and (72) we obtain

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896minus1

+ 119880119896

119866119896

119875119903

119896minus1|119896minus1

119866119879

119896

119880119879

119896

= 119875119909

119896|119896minus1

+ 119880119896

(119880119896

119875119903

119896|119896minus1

minus 119880119896

119876119903

119896minus1

)119879

= 119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

= 11987511

119896|119896minus1

(102)

Using (90) (98) (72) (32) (71) (29) and (97) we obtain

119875119903

119896|119896minus1

= 11987522

119896minus1|119896minus1

(103)

Using (92) (98) (33) (32) (79) (86) and (91)

119875119909119903

119896|119896minus1

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896minus1

119875119903

119896minus1|119896minus2

= 11987512

119896|119896minus1

(104)

Using (93) (101) (55) (73) (67) (80) and (87)

119870119909

119896

= (119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

) (1198671

119896

)119879

sdot (1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ (119881119896

+ 119870119909

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

(1198671

119896

)119879

)119880minus1

119896

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

)119880minus1

119896

+ 119881119896

119875119903

119896|119896minus1

119878119879

119896

119880minus1

119896

= 119870119909

119896

+ 119881119896

119870119903

119896

(105)

Using (94) (30) and (88) we obtain

119870119903

119896

= (119875119903

119896|119896minus1

)119879

119878119879

119896

119880minus1

119896

= 119870119903

119896|119896minus1

(106)

10 Mathematical Problems in Engineering

Next wewill show that (98) holds at time 119896 From (23)we have

119883119896|119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

)

= 119883119896|119896minus1

+ 119870119909

119896

119903119896

(107)

Then using (61) and (105) the above equation can be writtenas

119883119896|119896

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

+ (119870119909

119896

+ 119881119896

119870119903

119896

) 119903119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 1198671

119896

119883119896|119896minus1

)

+ (119880119896

minus 119870119909

119896

119878119896

) 119903119896|119896minus1

+ 119881119896

119870119903

119896

119903119896

= 119883119896|119896

+ 119881119896

119903119896|119896

= 119896|119896

(108)

Using (95) (105) (102) and (77)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

(109)

Then using (80) (68) (74) and (31) we obtain

119875119909

119896|119896

= 119875119909

119896|119896

+ 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

+ 119881119896

(119875119903

119896|119896

119878119879

119896

(119870119909

119896

)119879

minus 119870119903

119896

1198671

119896

119875119909

119896|119896minus1

)

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

= 11987511

119896|119896

(110)

Using (96) (30) (28) (105) and (80)

119875119909119903

119896|119896

= 119880119896

119875119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

1198671

119896

119880119896

minus 119881119896

119870119903

119896

1198671

119896

119880119896

)119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

119875119903

119896|119896

= 11987512

119896|119896

(111)

Using (97) (106) (95) (29) and (30) we obtain

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896|119896minus1

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119868 minus 119870119903

119896|119896minus1

119878119896

) 119875119903

119896|119896minus1

= 11987522

119896|119896

(112)

Table 1 Kalman estimation arithmetic operation requirement forthe conventional EKF structure

Number of multiplications(119899 = 6119898 = 2 and 119902 = 2)

Number of additions(119899 = 6119898 = 2 and 119902 = 2)

119860119896

119861119896

and 119862

119896

Function of system (9) Function of system (3)

119883119886

119896|119896minus1

1198992 + 119899119902 (48) 1198992 + 119899119902 minus 119899 (42)119875119896|119896minus1

21198993 (432) 21198993 minus 1198992 (396)119883119886

119896|119896minus1

2119899119898 (24) 2119899119898 (24)

119870119896

1198992

119898 + 21198991198982

+ 1198983 (168) 119899

2

119898 + 21198991198982

+ 1198983

minus 2119899119898

(104)119875119896|119896

1198992119898 + 1198993 (288) 1198992119898 + 1198993 minus 1198992 (252)Total 960 818

Finally we show that (98) holds at time 119896 = 0 This can beverified by the initial conditions of TEKF algorithm

43 Numerical Complexity of the Algorithm Tables 1 and2 show the computational effort at each sample time bythe conventional EKF algorithm and TEKF (where roughmatrix-based implementation is used) in which as definedabove 119899 is the dimension of the state vector 119883

119896

119898 is thedimension of the measurement 119884

119896

119902 is the input vector 119880119896

and 119901 is the dimension of the parameter 119903

119896

The total numberof arithmetic operations (additions and multiplications) persample time of the TEKF is 1314 compared with 1778 for arough implementation of a conventional EKF which meansthe operation cost can reduce by 26

5 Simulation and Experimental Results

51 Simulation Results To test the feasibility and perfor-mance of the TEKF method the sensorless DTC-SVM tech-nique for IM drives described in Section 2 is implementedin MATLABSIMULINK environment The values of theinitial state covariance matrices 119875

0

119876 and 119877 have a greatinfluence on the performance of the estimation methodThe diagonal initial state covariance matrix 119875

0

representsvariances or mean-squared errors in the knowledge of theinitial conditions Matrix 119876 gives the statistical descriptionof the drive system Matrix 119877 is related to measured noiseThey can be obtained by considering the stochastic propertiesof the corresponding noises However a fine evaluation ofthe covariance matrices is very difficult because they areusually not known In this paper tuning the initial values ofcovariance matrices 119875

0

119876 and 119877 is using particular criteria[22] to achieve steady-state behaviors of the relative estimatedstates as given by

119876 = diag 20 20 1119890 minus 6 1119890 minus 6 10 10

1198750

= diag 01 01 05 05 1 1

119877 = diag 01 01

(113)

Mathematical Problems in Engineering 11

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000Sp

eed

(rpm

)

(a) Speed estimation

02 04 06 08 10Times (s)

minus002

0

002

004

006

008

01

012

014

016

018

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF

Theta (real)Theta (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7Po

sitio

n (r

ad)

(d) TEKF rotor position error

Current (real)Current (TEKF)

Curr

entIs120572

(A)

02 04 06 08 10Times (s)

minus12

minus8

minus4

0

4

8

12

(e) Real stator current 119868119904120572

and estimation (TEKF)

Current (real)Current (TEKF)

minus12

minus8

minus4

0

4

8

12

Curr

entIs120573

(A)

02 04 06 08 10Times (s)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 3 Continued

12 Mathematical Problems in Engineering

Flux (EKF)Flux (TEKF)

minus05 0 05 1minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(g) Stator flux estimation by TEKF and EKF

02 04 06 08 10Times (s)

minus2

0

2

4

6

8

10

12

14

16

Ψs

(Wb)

times10minus3

(h) Difference Stator flux estimation between TEKF and EKF

Figure 3 Simulation results for parameters estimation

Table 2 Kalman estimation arithmetic operation requirement for the TEKF structure

Number of multiplications(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

Number of additions(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

119860119896

119862119896

119864119896

1198671119896

1198672119896

119861119896

and119863119896

Function of system (25) Function of system (11)119883119896|119896minus1

1198992 + 119899119902 (24) 1198992 + 119899119902 (24)119875119909

119896|119896minus121198993 (128) 21198993 minus 1198992 (112)

119870119909

119896|1198961198992

119898 + 1198991198982 (48) 119899

2

119898 + 1198991198982

minus 2119899119898 (32)119875119909

119896|1198961198992

119898 + 1198993 (96) 119899

2

119898 + 1198993

minus 1198992 (80)

119883119896|119896

2119899119898 + 119898119901 (20) 2119899119898 + 2119898119901 (24)119903119896|119896minus1

1199012 (4) 1199012 minus 119901 (2)119875119903

119896|119896minus121199013 (16) 21199013 minus 1199012 (12)

119870119903

119896|11989631199012119898 + 1199011198982 (32) 31199012119898 + 1199011198982 + 1198982 minus 4119901119898 (20)

119875119903

119896|1198961199013 + 1199012119898 (16) 1199013 + 1199012119898 minus 1199012 (12)

119903119896|119896

2119898119901 + 119899119898 (12) 2119898119901 + 119899119898 (16)119878119896

119898119899119901 (16) 119899119898119901 (16)119880119896minus1

1198992119901 + 1198991199012 (56) 1198992119901 + 1198991199012 minus 119899119901 (48)119881119896

119899119901119898 (16) 119899119901 (8)119880119896minus1

21198991199012 (32) 21198991199012 (32)119898119896minus1

1198992

119901 + 1198991199012

+ 119899119901 (56) 1198992

119901 + 1198991199012

+ 119899119901 minus 119899 (52)119876119909

119896minus121198992

119901 (64) 21198992

119901 + 1198991199012 (64)

119878119896

119898119899119901 (16) 119898119899119901 minus 119898119901 (12)119873119896

21198981198992 (32) 21198981198992 minus 1198982 (60)Total 688 626

In the simulation a comparison is made to verify theequivalence of EKF and TEKF Real-time parameters esti-mated by TEKF are used to formulate the closed loop suchas rotor speed stator flux and electromagnetic torque Theestimations obtained by EKF algorithm are not included in

the sensorless DTC-SVM strategy and only evaluated in openloop A step reference speed was applied to the simulation

The machine is accelerated from 0 rpm to 1000 rpm at0 s and the torque load is set to 4N The simulation resultsof parameter estimation are shown in Figure 3 Figures 3(a)

Mathematical Problems in Engineering 13

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000

Spee

d (r

pm)

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus03

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Theta (real)Theta (TEKF)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

02 04 06 08 10Times (s)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)

Figure 4 Simulation results with parameter variation (119877119903

= 05119877119903nom)

and 3(c) represent the performance of the speed and rotorposition tracking capabilities of the control system withTEKF and EKF respectively Figures 3(e) 3(f) and 3(g)represent the estimated stator current and flux they showthat ripples are significantly suppressed due to the SVMmodulation scheme Figures 3(b) 3(d) and 3(h) show thatfor variations of speed reference the rotor speed rotorposition and stator flux errors between the two observers arevery little It is verified that the two observers are equivalentThe difference between two estimators is caused by accuracyloss in TEKF which uses more calculation steps

In order to further verify the performance of TEKFagainst model-plant parameter mismatches and the equiv-alence of two observers the change in rotor resistance isconsidered Rotor resistance will increase due to temperaturerise while the motor is running To simulate this conditionthe rotor resistance in TEKF and EKF is increased to 200comparedwith the normal value which is equivalent to a 50decrease in the actual rotor resistance The machine is stillaccelerated from 0 rpm to 1000 rpm at 0 s and the torque is setto 4N Figure 4 shows that for variation of rotor resistancethe steady-state speed and rotor position errors are negligible

14 Mathematical Problems in Engineering

(a)

Computer

380V

TXRX

120596r

Incrementalencoder

Udc

MicrocomputerDSP TMS3206713

Drive signalCurrentsensor

Voltagesensor

Voltagesensor

IM MC

ISU ISV

VSU VSW VSV

(b)

Figure 5 Complete drive system (a) Picture of experimental setup (b) Functional block diagram of the experimental setup

and the difference of the speed and rotor position estimationsbetween the two observers is rather null

52 Experimental Results The overall experimental setup isshown in Figure 5 and the specifications and rated parametersof the IM controller and inverter are listed in Table 3 Inthe experimental hardware an Expert3 control system fromMyway company and a three-phase two-pole 15 kW IM areappliedThe IM is mechanically coupled to a magnetic clutch(MC) which provides rated torque even at very low speedThe main processor in Expert3 control system is a floatingpoint processor TMS320C6713 with a max clock speed of225MHz All the algorithms including TEKF EKF DTCalgorithm and some transformation modules are imple-mented in TMS320C6713 with 100120583s sampling time and dataacquisition of the parameter estimations measured variablesand their visualization are realized on the cockpit provided byPEView9 software Insulated Gate Bipolar Transistor (IGBT)module is driven by the PWM signal with a switchingfrequency of 10 kHz and 2 120583s dead time The stator currentsare measured via two Hall effect current sensors The rotorangle and speed of IM are measured from an incrementalencoder with 2048 pulses per revolution

This experiment test is here to testify the performanceof TEKF and demonstrate that the two estimators aremathematically equivalent The machine is accelerated from600 rpm to 1000 rpm and 4N torque load is set Theexperimental results of parameter estimation based on twoobservers are given in Figures 6 and 7 Figures 6(a) and 6(c)show that the TEKF still has a good tracking performance ofthe speed and rotor position in experiment Figures 6(d) 6(e)and 6(f) illustrate stator flux and stator current estimationrobustness Figures 6(b) 6(g) and 6(f) referring to thedifference in speed and stator current estimations given by

Table 3 Specification of induction motor and inverter

Induction motor ValueNominal torque 10NmNominal voltage 380VRotor resistance 119877

119903

25ΩStator resistance 119877

119904

36ΩStator inductances 119871

119904

0301HRotor inductances 119871

119903

0302HMutual inductances 119871

119898

0273HPole pairs 2Invertercontroller ValueSwitching device 1000V 80A IGBTControl cycle time 100 120583sMain CPU DSP TMS320C6713 225MHz

the two observers are still small These experiment resultsprove that the two estimators are mathematically equivalentFigure 7 shows the speed and rotor position estimationsbased onTEKFandEKF for a 50decrease of rotor resistance(the same as the simulation) As expected the steady errorof the TEKF and the difference in speed and rotor positionestimations are still tiny Robustness of TEKF is verified

6 Conclusion

Themajor shortcoming of the conventional EKF is numericalproblems and computational burden due to the high orderof the mathematical models This has generally limited thereal-time digital implementation of the EKF for industrialfield So in this study a novel extended Kalman filter

Mathematical Problems in Engineering 15

Speed (real)Speed (EKF)Speed (TEKF)

Spee

d (r

min

)

02 04 06 08 10t (02 sgrid)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimationSp

eed

(rm

in)

minus005

0

005

01

015

02

025

03

035

04

045

02 04 06 08 10t (02 sgrid)

(b) Difference of speed estimation between EKF and TEKF

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation

Flux (EKF)Flux (TEKF)

minus08 minus04 1minus02 0 02 04 06 08minus06minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(d) Stator flux estimation by TEKF and EKF

Is120572 (real)Is120572 (TEKF)

Curr

entIs120572

(A)

minus6

minus4

minus2

0

2

4

6

02 04 06 08 10t (02 sgrid)

(e) Real stator current 119868119904120572

and estimation (TEKF)

Is120573 (real)Is120573 (TEKF)

minus6

minus4

minus2

0

2

4

6

Curr

entIs120573

(A)

02 04 06 08 10t (02 sgrid)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 6 Continued

16 Mathematical Problems in Engineering

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Erro

r cur

rentIs120572

(A)

02 04 06 08 10t (02 sgrid)

(g) TEKF stator current 119868119904120572

error

minus02

minus015

minus01

minus005

0

005

01

015

Erro

r cur

rentIs120573

(A)

02 04 06 08 10t (02 sgrid)

(h) TEKF stator current 119868119904120573

error

Figure 6 Experimental results for parameters estimation

Speed (real)Speed (EKF)

Speed (TEKF)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

0

005

01

015

02

025

03

035

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)Figure 7 Experimental results with parameter variation (119877

119903

= 05119877119903nom)

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Research Article Direct Torque Control of Sensorless ...

Mathematical Problems in Engineering 9

(3) Using (20) we have

119875119903

119896|119896minus1

= 119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119876119903

119896minus1

(90)

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119864119896minus1

(119875119909119903

119896minus1|119896minus1

)119879

119860119879

119896minus1

+ 119860119896minus1

119875119909119903

119896minus1|119896minus1

119864119879

119896minus1

+ 119876119909

119896minus1

(91)

119875119909119903

119896|119896minus1

= 119860119896minus1

119875119909119903

119896minus1|119896minus1

119866119879

119896minus1

+ 119864119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

(92)

(4) Using (21)

119870119909

119896

= 119875119909

119896|119896minus1

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(93)

119870119903

119896

= (119875119909119903

119896|119896minus1

)119879

(1198671

119896

)119879

(1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

(94)

(5) Using (22)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

(95)

119875119909119903

119896|119896

= 119875119909119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909119903

119896|119896minus1

(96)

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896

1198671

119896

119875119909119903

119896|119896minus1

(97)

By inductive reasoning suppose that at time 119896 minus 1the unknown parameter

119896minus1

and estimated state 119896minus1

areequal to the parameter 119903

119896minus1

and state 119883119896minus1

of the controlsystem respectively we show that TEKF is equivalent to theconventional EKF because these properties are still true attime 119896

Assume that at time 119896 minus 1

119883119896minus1|119896minus1

= 119896minus1|119896minus1

119903119896minus1|119896minus1

= 119903119896minus1|119896minus1

119875119909

119896minus1|119896minus1

= 11987511

119896minus1|119896minus1

119875119909119903

119896minus1|119896minus1

= 11987512

119896minus1|119896minus1

119875119903

119896minus1|119896minus1

= 11987522

119896minus1|119896minus1

(98)

where [ 119875119909119875

119909119903

(119875

119909119903)

119879119875

119903 ] and [ 11987511119875

12

(119875

12)

119879119875

22 ] represent the variance-covariance matrices of the system and estimated variablesrespectively

From (19) we have

119883119896|119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119863119896

119903119896minus1|119896minus1

+ 119861119896

119906119896minus1

(99)

Then using (98) (41) (62) (79) (81) (71) and (61)

119883119896|119896minus1

= 119860119896minus1

(119883119896minus1|119896minus1

+ 119881119896minus1

119903119896minus1|119896minus1

)

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

= 119860119896minus1

119883119896minus1|119896minus1

+ 119861119896minus1

119906119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

minus 119898119896minus1

+ 119863119896minus1

119903119896minus1|119896minus1

+ 119860119896minus1

119881119896minus1

119903119896minus1|119896minus1

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

= 119896|119896minus1

(100)

Using (19) (71) (98) (63) and (64) we have

119903119896|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119866119896minus1

119903119896minus1|119896minus1

= 119903119896|119896minus1

(101)

Using (91) (98) (78) (66) (79) (82) (86) and (72) we obtain

119875119909

119896|119896minus1

= 119860119896minus1

119875119909

119896minus1|119896minus1

119860119879

119896minus1

+ 119876119909

119896minus1

+ 119880119896

119866119896

119875119903

119896minus1|119896minus1

119866119879

119896

119880119879

119896

= 119875119909

119896|119896minus1

+ 119880119896

(119880119896

119875119903

119896|119896minus1

minus 119880119896

119876119903

119896minus1

)119879

= 119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

= 11987511

119896|119896minus1

(102)

Using (90) (98) (72) (32) (71) (29) and (97) we obtain

119875119903

119896|119896minus1

= 11987522

119896minus1|119896minus1

(103)

Using (92) (98) (33) (32) (79) (86) and (91)

119875119909119903

119896|119896minus1

= (119860119896minus1

119881119896minus1

+ 119864119896minus1

) 119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896

119866119896minus1

119875119903

119896minus1|119896minus1

119866119879

119896minus1

= 119880119896minus1

119875119903

119896minus1|119896minus2

= 11987512

119896|119896minus1

(104)

Using (93) (101) (55) (73) (67) (80) and (87)

119870119909

119896

= (119875119909

119896|119896minus1

+ 119880119896

119875119903

119896|119896minus1

119880119879

119896

) (1198671

119896

)119879

sdot (1198671

119896

119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119877119896

)minus1

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ (119881119896

+ 119870119909

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

(1198671

119896

)119879

)119880minus1

119896

= (119875119909

119896|119896minus1

(1198671

119896

)119879

+ 119870119909

119896

119878119896

119875119903

119896|119896minus1

119878119879

119896

)119880minus1

119896

+ 119881119896

119875119903

119896|119896minus1

119878119879

119896

119880minus1

119896

= 119870119909

119896

+ 119881119896

119870119903

119896

(105)

Using (94) (30) and (88) we obtain

119870119903

119896

= (119875119903

119896|119896minus1

)119879

119878119879

119896

119880minus1

119896

= 119870119903

119896|119896minus1

(106)

10 Mathematical Problems in Engineering

Next wewill show that (98) holds at time 119896 From (23)we have

119883119896|119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

)

= 119883119896|119896minus1

+ 119870119909

119896

119903119896

(107)

Then using (61) and (105) the above equation can be writtenas

119883119896|119896

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

+ (119870119909

119896

+ 119881119896

119870119903

119896

) 119903119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 1198671

119896

119883119896|119896minus1

)

+ (119880119896

minus 119870119909

119896

119878119896

) 119903119896|119896minus1

+ 119881119896

119870119903

119896

119903119896

= 119883119896|119896

+ 119881119896

119903119896|119896

= 119896|119896

(108)

Using (95) (105) (102) and (77)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

(109)

Then using (80) (68) (74) and (31) we obtain

119875119909

119896|119896

= 119875119909

119896|119896

+ 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

+ 119881119896

(119875119903

119896|119896

119878119879

119896

(119870119909

119896

)119879

minus 119870119903

119896

1198671

119896

119875119909

119896|119896minus1

)

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

= 11987511

119896|119896

(110)

Using (96) (30) (28) (105) and (80)

119875119909119903

119896|119896

= 119880119896

119875119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

1198671

119896

119880119896

minus 119881119896

119870119903

119896

1198671

119896

119880119896

)119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

119875119903

119896|119896

= 11987512

119896|119896

(111)

Using (97) (106) (95) (29) and (30) we obtain

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896|119896minus1

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119868 minus 119870119903

119896|119896minus1

119878119896

) 119875119903

119896|119896minus1

= 11987522

119896|119896

(112)

Table 1 Kalman estimation arithmetic operation requirement forthe conventional EKF structure

Number of multiplications(119899 = 6119898 = 2 and 119902 = 2)

Number of additions(119899 = 6119898 = 2 and 119902 = 2)

119860119896

119861119896

and 119862

119896

Function of system (9) Function of system (3)

119883119886

119896|119896minus1

1198992 + 119899119902 (48) 1198992 + 119899119902 minus 119899 (42)119875119896|119896minus1

21198993 (432) 21198993 minus 1198992 (396)119883119886

119896|119896minus1

2119899119898 (24) 2119899119898 (24)

119870119896

1198992

119898 + 21198991198982

+ 1198983 (168) 119899

2

119898 + 21198991198982

+ 1198983

minus 2119899119898

(104)119875119896|119896

1198992119898 + 1198993 (288) 1198992119898 + 1198993 minus 1198992 (252)Total 960 818

Finally we show that (98) holds at time 119896 = 0 This can beverified by the initial conditions of TEKF algorithm

43 Numerical Complexity of the Algorithm Tables 1 and2 show the computational effort at each sample time bythe conventional EKF algorithm and TEKF (where roughmatrix-based implementation is used) in which as definedabove 119899 is the dimension of the state vector 119883

119896

119898 is thedimension of the measurement 119884

119896

119902 is the input vector 119880119896

and 119901 is the dimension of the parameter 119903

119896

The total numberof arithmetic operations (additions and multiplications) persample time of the TEKF is 1314 compared with 1778 for arough implementation of a conventional EKF which meansthe operation cost can reduce by 26

5 Simulation and Experimental Results

51 Simulation Results To test the feasibility and perfor-mance of the TEKF method the sensorless DTC-SVM tech-nique for IM drives described in Section 2 is implementedin MATLABSIMULINK environment The values of theinitial state covariance matrices 119875

0

119876 and 119877 have a greatinfluence on the performance of the estimation methodThe diagonal initial state covariance matrix 119875

0

representsvariances or mean-squared errors in the knowledge of theinitial conditions Matrix 119876 gives the statistical descriptionof the drive system Matrix 119877 is related to measured noiseThey can be obtained by considering the stochastic propertiesof the corresponding noises However a fine evaluation ofthe covariance matrices is very difficult because they areusually not known In this paper tuning the initial values ofcovariance matrices 119875

0

119876 and 119877 is using particular criteria[22] to achieve steady-state behaviors of the relative estimatedstates as given by

119876 = diag 20 20 1119890 minus 6 1119890 minus 6 10 10

1198750

= diag 01 01 05 05 1 1

119877 = diag 01 01

(113)

Mathematical Problems in Engineering 11

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000Sp

eed

(rpm

)

(a) Speed estimation

02 04 06 08 10Times (s)

minus002

0

002

004

006

008

01

012

014

016

018

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF

Theta (real)Theta (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7Po

sitio

n (r

ad)

(d) TEKF rotor position error

Current (real)Current (TEKF)

Curr

entIs120572

(A)

02 04 06 08 10Times (s)

minus12

minus8

minus4

0

4

8

12

(e) Real stator current 119868119904120572

and estimation (TEKF)

Current (real)Current (TEKF)

minus12

minus8

minus4

0

4

8

12

Curr

entIs120573

(A)

02 04 06 08 10Times (s)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 3 Continued

12 Mathematical Problems in Engineering

Flux (EKF)Flux (TEKF)

minus05 0 05 1minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(g) Stator flux estimation by TEKF and EKF

02 04 06 08 10Times (s)

minus2

0

2

4

6

8

10

12

14

16

Ψs

(Wb)

times10minus3

(h) Difference Stator flux estimation between TEKF and EKF

Figure 3 Simulation results for parameters estimation

Table 2 Kalman estimation arithmetic operation requirement for the TEKF structure

Number of multiplications(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

Number of additions(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

119860119896

119862119896

119864119896

1198671119896

1198672119896

119861119896

and119863119896

Function of system (25) Function of system (11)119883119896|119896minus1

1198992 + 119899119902 (24) 1198992 + 119899119902 (24)119875119909

119896|119896minus121198993 (128) 21198993 minus 1198992 (112)

119870119909

119896|1198961198992

119898 + 1198991198982 (48) 119899

2

119898 + 1198991198982

minus 2119899119898 (32)119875119909

119896|1198961198992

119898 + 1198993 (96) 119899

2

119898 + 1198993

minus 1198992 (80)

119883119896|119896

2119899119898 + 119898119901 (20) 2119899119898 + 2119898119901 (24)119903119896|119896minus1

1199012 (4) 1199012 minus 119901 (2)119875119903

119896|119896minus121199013 (16) 21199013 minus 1199012 (12)

119870119903

119896|11989631199012119898 + 1199011198982 (32) 31199012119898 + 1199011198982 + 1198982 minus 4119901119898 (20)

119875119903

119896|1198961199013 + 1199012119898 (16) 1199013 + 1199012119898 minus 1199012 (12)

119903119896|119896

2119898119901 + 119899119898 (12) 2119898119901 + 119899119898 (16)119878119896

119898119899119901 (16) 119899119898119901 (16)119880119896minus1

1198992119901 + 1198991199012 (56) 1198992119901 + 1198991199012 minus 119899119901 (48)119881119896

119899119901119898 (16) 119899119901 (8)119880119896minus1

21198991199012 (32) 21198991199012 (32)119898119896minus1

1198992

119901 + 1198991199012

+ 119899119901 (56) 1198992

119901 + 1198991199012

+ 119899119901 minus 119899 (52)119876119909

119896minus121198992

119901 (64) 21198992

119901 + 1198991199012 (64)

119878119896

119898119899119901 (16) 119898119899119901 minus 119898119901 (12)119873119896

21198981198992 (32) 21198981198992 minus 1198982 (60)Total 688 626

In the simulation a comparison is made to verify theequivalence of EKF and TEKF Real-time parameters esti-mated by TEKF are used to formulate the closed loop suchas rotor speed stator flux and electromagnetic torque Theestimations obtained by EKF algorithm are not included in

the sensorless DTC-SVM strategy and only evaluated in openloop A step reference speed was applied to the simulation

The machine is accelerated from 0 rpm to 1000 rpm at0 s and the torque load is set to 4N The simulation resultsof parameter estimation are shown in Figure 3 Figures 3(a)

Mathematical Problems in Engineering 13

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000

Spee

d (r

pm)

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus03

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Theta (real)Theta (TEKF)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

02 04 06 08 10Times (s)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)

Figure 4 Simulation results with parameter variation (119877119903

= 05119877119903nom)

and 3(c) represent the performance of the speed and rotorposition tracking capabilities of the control system withTEKF and EKF respectively Figures 3(e) 3(f) and 3(g)represent the estimated stator current and flux they showthat ripples are significantly suppressed due to the SVMmodulation scheme Figures 3(b) 3(d) and 3(h) show thatfor variations of speed reference the rotor speed rotorposition and stator flux errors between the two observers arevery little It is verified that the two observers are equivalentThe difference between two estimators is caused by accuracyloss in TEKF which uses more calculation steps

In order to further verify the performance of TEKFagainst model-plant parameter mismatches and the equiv-alence of two observers the change in rotor resistance isconsidered Rotor resistance will increase due to temperaturerise while the motor is running To simulate this conditionthe rotor resistance in TEKF and EKF is increased to 200comparedwith the normal value which is equivalent to a 50decrease in the actual rotor resistance The machine is stillaccelerated from 0 rpm to 1000 rpm at 0 s and the torque is setto 4N Figure 4 shows that for variation of rotor resistancethe steady-state speed and rotor position errors are negligible

14 Mathematical Problems in Engineering

(a)

Computer

380V

TXRX

120596r

Incrementalencoder

Udc

MicrocomputerDSP TMS3206713

Drive signalCurrentsensor

Voltagesensor

Voltagesensor

IM MC

ISU ISV

VSU VSW VSV

(b)

Figure 5 Complete drive system (a) Picture of experimental setup (b) Functional block diagram of the experimental setup

and the difference of the speed and rotor position estimationsbetween the two observers is rather null

52 Experimental Results The overall experimental setup isshown in Figure 5 and the specifications and rated parametersof the IM controller and inverter are listed in Table 3 Inthe experimental hardware an Expert3 control system fromMyway company and a three-phase two-pole 15 kW IM areappliedThe IM is mechanically coupled to a magnetic clutch(MC) which provides rated torque even at very low speedThe main processor in Expert3 control system is a floatingpoint processor TMS320C6713 with a max clock speed of225MHz All the algorithms including TEKF EKF DTCalgorithm and some transformation modules are imple-mented in TMS320C6713 with 100120583s sampling time and dataacquisition of the parameter estimations measured variablesand their visualization are realized on the cockpit provided byPEView9 software Insulated Gate Bipolar Transistor (IGBT)module is driven by the PWM signal with a switchingfrequency of 10 kHz and 2 120583s dead time The stator currentsare measured via two Hall effect current sensors The rotorangle and speed of IM are measured from an incrementalencoder with 2048 pulses per revolution

This experiment test is here to testify the performanceof TEKF and demonstrate that the two estimators aremathematically equivalent The machine is accelerated from600 rpm to 1000 rpm and 4N torque load is set Theexperimental results of parameter estimation based on twoobservers are given in Figures 6 and 7 Figures 6(a) and 6(c)show that the TEKF still has a good tracking performance ofthe speed and rotor position in experiment Figures 6(d) 6(e)and 6(f) illustrate stator flux and stator current estimationrobustness Figures 6(b) 6(g) and 6(f) referring to thedifference in speed and stator current estimations given by

Table 3 Specification of induction motor and inverter

Induction motor ValueNominal torque 10NmNominal voltage 380VRotor resistance 119877

119903

25ΩStator resistance 119877

119904

36ΩStator inductances 119871

119904

0301HRotor inductances 119871

119903

0302HMutual inductances 119871

119898

0273HPole pairs 2Invertercontroller ValueSwitching device 1000V 80A IGBTControl cycle time 100 120583sMain CPU DSP TMS320C6713 225MHz

the two observers are still small These experiment resultsprove that the two estimators are mathematically equivalentFigure 7 shows the speed and rotor position estimationsbased onTEKFandEKF for a 50decrease of rotor resistance(the same as the simulation) As expected the steady errorof the TEKF and the difference in speed and rotor positionestimations are still tiny Robustness of TEKF is verified

6 Conclusion

Themajor shortcoming of the conventional EKF is numericalproblems and computational burden due to the high orderof the mathematical models This has generally limited thereal-time digital implementation of the EKF for industrialfield So in this study a novel extended Kalman filter

Mathematical Problems in Engineering 15

Speed (real)Speed (EKF)Speed (TEKF)

Spee

d (r

min

)

02 04 06 08 10t (02 sgrid)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimationSp

eed

(rm

in)

minus005

0

005

01

015

02

025

03

035

04

045

02 04 06 08 10t (02 sgrid)

(b) Difference of speed estimation between EKF and TEKF

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation

Flux (EKF)Flux (TEKF)

minus08 minus04 1minus02 0 02 04 06 08minus06minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(d) Stator flux estimation by TEKF and EKF

Is120572 (real)Is120572 (TEKF)

Curr

entIs120572

(A)

minus6

minus4

minus2

0

2

4

6

02 04 06 08 10t (02 sgrid)

(e) Real stator current 119868119904120572

and estimation (TEKF)

Is120573 (real)Is120573 (TEKF)

minus6

minus4

minus2

0

2

4

6

Curr

entIs120573

(A)

02 04 06 08 10t (02 sgrid)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 6 Continued

16 Mathematical Problems in Engineering

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Erro

r cur

rentIs120572

(A)

02 04 06 08 10t (02 sgrid)

(g) TEKF stator current 119868119904120572

error

minus02

minus015

minus01

minus005

0

005

01

015

Erro

r cur

rentIs120573

(A)

02 04 06 08 10t (02 sgrid)

(h) TEKF stator current 119868119904120573

error

Figure 6 Experimental results for parameters estimation

Speed (real)Speed (EKF)

Speed (TEKF)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

0

005

01

015

02

025

03

035

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)Figure 7 Experimental results with parameter variation (119877

119903

= 05119877119903nom)

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Research Article Direct Torque Control of Sensorless ...

10 Mathematical Problems in Engineering

Next wewill show that (98) holds at time 119896 From (23)we have

119883119896|119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 119862119896

119883119896|119896minus1

)

= 119883119896|119896minus1

+ 119870119909

119896

119903119896

(107)

Then using (61) and (105) the above equation can be writtenas

119883119896|119896

= 119883119896|119896minus1

+ 119880119896

119903119896|119896minus1

+ (119870119909

119896

+ 119881119896

119870119903

119896

) 119903119896

= 119883119896|119896minus1

+ 119870119909

119896

(119884119896

minus 1198671

119896

119883119896|119896minus1

)

+ (119880119896

minus 119870119909

119896

119878119896

) 119903119896|119896minus1

+ 119881119896

119870119903

119896

119903119896

= 119883119896|119896

+ 119881119896

119903119896|119896

= 119896|119896

(108)

Using (95) (105) (102) and (77)

119875119909

119896|119896

= 119875119909

119896|119896minus1

minus 119870119909

119896

1198671

119896

119875119909

119896|119896minus1

+ (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

(109)

Then using (80) (68) (74) and (31) we obtain

119875119909

119896|119896

= 119875119909

119896|119896

+ 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

119880119879

119896

minus 119881119896

119870119903

119896

1198671

119896

119875119909

119896|119896minus1

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

+ 119881119896

(119875119903

119896|119896

119878119879

119896

(119870119909

119896

)119879

minus 119870119903

119896

1198671

119896

119875119909

119896|119896minus1

)

= 119875119909

119896|119896

+ 119881119896

119875119903

119896|119896

119881119879

119896

= 11987511

119896|119896

(110)

Using (96) (30) (28) (105) and (80)

119875119909119903

119896|119896

= 119880119896

119875119903

119896|119896minus1

minus 119870119909

119896

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

1198671

119896

119880119896

minus 119881119896

119870119903

119896

1198671

119896

119880119896

)119875119903

119896|119896minus1

= (119880119896

minus 119870119909

119896

119878119896

minus 119881119896

119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

(119868 minus 119870119903

119896

119878119896

) 119875119903

119896|119896minus1

= 119881119896

119875119903

119896|119896

= 11987512

119896|119896

(111)

Using (97) (106) (95) (29) and (30) we obtain

119875119903

119896|119896

= 119875119903

119896|119896minus1

minus 119870119903

119896|119896minus1

1198671

119896

119880119896

119875119903

119896|119896minus1

= (119868 minus 119870119903

119896|119896minus1

119878119896

) 119875119903

119896|119896minus1

= 11987522

119896|119896

(112)

Table 1 Kalman estimation arithmetic operation requirement forthe conventional EKF structure

Number of multiplications(119899 = 6119898 = 2 and 119902 = 2)

Number of additions(119899 = 6119898 = 2 and 119902 = 2)

119860119896

119861119896

and 119862

119896

Function of system (9) Function of system (3)

119883119886

119896|119896minus1

1198992 + 119899119902 (48) 1198992 + 119899119902 minus 119899 (42)119875119896|119896minus1

21198993 (432) 21198993 minus 1198992 (396)119883119886

119896|119896minus1

2119899119898 (24) 2119899119898 (24)

119870119896

1198992

119898 + 21198991198982

+ 1198983 (168) 119899

2

119898 + 21198991198982

+ 1198983

minus 2119899119898

(104)119875119896|119896

1198992119898 + 1198993 (288) 1198992119898 + 1198993 minus 1198992 (252)Total 960 818

Finally we show that (98) holds at time 119896 = 0 This can beverified by the initial conditions of TEKF algorithm

43 Numerical Complexity of the Algorithm Tables 1 and2 show the computational effort at each sample time bythe conventional EKF algorithm and TEKF (where roughmatrix-based implementation is used) in which as definedabove 119899 is the dimension of the state vector 119883

119896

119898 is thedimension of the measurement 119884

119896

119902 is the input vector 119880119896

and 119901 is the dimension of the parameter 119903

119896

The total numberof arithmetic operations (additions and multiplications) persample time of the TEKF is 1314 compared with 1778 for arough implementation of a conventional EKF which meansthe operation cost can reduce by 26

5 Simulation and Experimental Results

51 Simulation Results To test the feasibility and perfor-mance of the TEKF method the sensorless DTC-SVM tech-nique for IM drives described in Section 2 is implementedin MATLABSIMULINK environment The values of theinitial state covariance matrices 119875

0

119876 and 119877 have a greatinfluence on the performance of the estimation methodThe diagonal initial state covariance matrix 119875

0

representsvariances or mean-squared errors in the knowledge of theinitial conditions Matrix 119876 gives the statistical descriptionof the drive system Matrix 119877 is related to measured noiseThey can be obtained by considering the stochastic propertiesof the corresponding noises However a fine evaluation ofthe covariance matrices is very difficult because they areusually not known In this paper tuning the initial values ofcovariance matrices 119875

0

119876 and 119877 is using particular criteria[22] to achieve steady-state behaviors of the relative estimatedstates as given by

119876 = diag 20 20 1119890 minus 6 1119890 minus 6 10 10

1198750

= diag 01 01 05 05 1 1

119877 = diag 01 01

(113)

Mathematical Problems in Engineering 11

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000Sp

eed

(rpm

)

(a) Speed estimation

02 04 06 08 10Times (s)

minus002

0

002

004

006

008

01

012

014

016

018

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF

Theta (real)Theta (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7Po

sitio

n (r

ad)

(d) TEKF rotor position error

Current (real)Current (TEKF)

Curr

entIs120572

(A)

02 04 06 08 10Times (s)

minus12

minus8

minus4

0

4

8

12

(e) Real stator current 119868119904120572

and estimation (TEKF)

Current (real)Current (TEKF)

minus12

minus8

minus4

0

4

8

12

Curr

entIs120573

(A)

02 04 06 08 10Times (s)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 3 Continued

12 Mathematical Problems in Engineering

Flux (EKF)Flux (TEKF)

minus05 0 05 1minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(g) Stator flux estimation by TEKF and EKF

02 04 06 08 10Times (s)

minus2

0

2

4

6

8

10

12

14

16

Ψs

(Wb)

times10minus3

(h) Difference Stator flux estimation between TEKF and EKF

Figure 3 Simulation results for parameters estimation

Table 2 Kalman estimation arithmetic operation requirement for the TEKF structure

Number of multiplications(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

Number of additions(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

119860119896

119862119896

119864119896

1198671119896

1198672119896

119861119896

and119863119896

Function of system (25) Function of system (11)119883119896|119896minus1

1198992 + 119899119902 (24) 1198992 + 119899119902 (24)119875119909

119896|119896minus121198993 (128) 21198993 minus 1198992 (112)

119870119909

119896|1198961198992

119898 + 1198991198982 (48) 119899

2

119898 + 1198991198982

minus 2119899119898 (32)119875119909

119896|1198961198992

119898 + 1198993 (96) 119899

2

119898 + 1198993

minus 1198992 (80)

119883119896|119896

2119899119898 + 119898119901 (20) 2119899119898 + 2119898119901 (24)119903119896|119896minus1

1199012 (4) 1199012 minus 119901 (2)119875119903

119896|119896minus121199013 (16) 21199013 minus 1199012 (12)

119870119903

119896|11989631199012119898 + 1199011198982 (32) 31199012119898 + 1199011198982 + 1198982 minus 4119901119898 (20)

119875119903

119896|1198961199013 + 1199012119898 (16) 1199013 + 1199012119898 minus 1199012 (12)

119903119896|119896

2119898119901 + 119899119898 (12) 2119898119901 + 119899119898 (16)119878119896

119898119899119901 (16) 119899119898119901 (16)119880119896minus1

1198992119901 + 1198991199012 (56) 1198992119901 + 1198991199012 minus 119899119901 (48)119881119896

119899119901119898 (16) 119899119901 (8)119880119896minus1

21198991199012 (32) 21198991199012 (32)119898119896minus1

1198992

119901 + 1198991199012

+ 119899119901 (56) 1198992

119901 + 1198991199012

+ 119899119901 minus 119899 (52)119876119909

119896minus121198992

119901 (64) 21198992

119901 + 1198991199012 (64)

119878119896

119898119899119901 (16) 119898119899119901 minus 119898119901 (12)119873119896

21198981198992 (32) 21198981198992 minus 1198982 (60)Total 688 626

In the simulation a comparison is made to verify theequivalence of EKF and TEKF Real-time parameters esti-mated by TEKF are used to formulate the closed loop suchas rotor speed stator flux and electromagnetic torque Theestimations obtained by EKF algorithm are not included in

the sensorless DTC-SVM strategy and only evaluated in openloop A step reference speed was applied to the simulation

The machine is accelerated from 0 rpm to 1000 rpm at0 s and the torque load is set to 4N The simulation resultsof parameter estimation are shown in Figure 3 Figures 3(a)

Mathematical Problems in Engineering 13

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000

Spee

d (r

pm)

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus03

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Theta (real)Theta (TEKF)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

02 04 06 08 10Times (s)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)

Figure 4 Simulation results with parameter variation (119877119903

= 05119877119903nom)

and 3(c) represent the performance of the speed and rotorposition tracking capabilities of the control system withTEKF and EKF respectively Figures 3(e) 3(f) and 3(g)represent the estimated stator current and flux they showthat ripples are significantly suppressed due to the SVMmodulation scheme Figures 3(b) 3(d) and 3(h) show thatfor variations of speed reference the rotor speed rotorposition and stator flux errors between the two observers arevery little It is verified that the two observers are equivalentThe difference between two estimators is caused by accuracyloss in TEKF which uses more calculation steps

In order to further verify the performance of TEKFagainst model-plant parameter mismatches and the equiv-alence of two observers the change in rotor resistance isconsidered Rotor resistance will increase due to temperaturerise while the motor is running To simulate this conditionthe rotor resistance in TEKF and EKF is increased to 200comparedwith the normal value which is equivalent to a 50decrease in the actual rotor resistance The machine is stillaccelerated from 0 rpm to 1000 rpm at 0 s and the torque is setto 4N Figure 4 shows that for variation of rotor resistancethe steady-state speed and rotor position errors are negligible

14 Mathematical Problems in Engineering

(a)

Computer

380V

TXRX

120596r

Incrementalencoder

Udc

MicrocomputerDSP TMS3206713

Drive signalCurrentsensor

Voltagesensor

Voltagesensor

IM MC

ISU ISV

VSU VSW VSV

(b)

Figure 5 Complete drive system (a) Picture of experimental setup (b) Functional block diagram of the experimental setup

and the difference of the speed and rotor position estimationsbetween the two observers is rather null

52 Experimental Results The overall experimental setup isshown in Figure 5 and the specifications and rated parametersof the IM controller and inverter are listed in Table 3 Inthe experimental hardware an Expert3 control system fromMyway company and a three-phase two-pole 15 kW IM areappliedThe IM is mechanically coupled to a magnetic clutch(MC) which provides rated torque even at very low speedThe main processor in Expert3 control system is a floatingpoint processor TMS320C6713 with a max clock speed of225MHz All the algorithms including TEKF EKF DTCalgorithm and some transformation modules are imple-mented in TMS320C6713 with 100120583s sampling time and dataacquisition of the parameter estimations measured variablesand their visualization are realized on the cockpit provided byPEView9 software Insulated Gate Bipolar Transistor (IGBT)module is driven by the PWM signal with a switchingfrequency of 10 kHz and 2 120583s dead time The stator currentsare measured via two Hall effect current sensors The rotorangle and speed of IM are measured from an incrementalencoder with 2048 pulses per revolution

This experiment test is here to testify the performanceof TEKF and demonstrate that the two estimators aremathematically equivalent The machine is accelerated from600 rpm to 1000 rpm and 4N torque load is set Theexperimental results of parameter estimation based on twoobservers are given in Figures 6 and 7 Figures 6(a) and 6(c)show that the TEKF still has a good tracking performance ofthe speed and rotor position in experiment Figures 6(d) 6(e)and 6(f) illustrate stator flux and stator current estimationrobustness Figures 6(b) 6(g) and 6(f) referring to thedifference in speed and stator current estimations given by

Table 3 Specification of induction motor and inverter

Induction motor ValueNominal torque 10NmNominal voltage 380VRotor resistance 119877

119903

25ΩStator resistance 119877

119904

36ΩStator inductances 119871

119904

0301HRotor inductances 119871

119903

0302HMutual inductances 119871

119898

0273HPole pairs 2Invertercontroller ValueSwitching device 1000V 80A IGBTControl cycle time 100 120583sMain CPU DSP TMS320C6713 225MHz

the two observers are still small These experiment resultsprove that the two estimators are mathematically equivalentFigure 7 shows the speed and rotor position estimationsbased onTEKFandEKF for a 50decrease of rotor resistance(the same as the simulation) As expected the steady errorof the TEKF and the difference in speed and rotor positionestimations are still tiny Robustness of TEKF is verified

6 Conclusion

Themajor shortcoming of the conventional EKF is numericalproblems and computational burden due to the high orderof the mathematical models This has generally limited thereal-time digital implementation of the EKF for industrialfield So in this study a novel extended Kalman filter

Mathematical Problems in Engineering 15

Speed (real)Speed (EKF)Speed (TEKF)

Spee

d (r

min

)

02 04 06 08 10t (02 sgrid)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimationSp

eed

(rm

in)

minus005

0

005

01

015

02

025

03

035

04

045

02 04 06 08 10t (02 sgrid)

(b) Difference of speed estimation between EKF and TEKF

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation

Flux (EKF)Flux (TEKF)

minus08 minus04 1minus02 0 02 04 06 08minus06minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(d) Stator flux estimation by TEKF and EKF

Is120572 (real)Is120572 (TEKF)

Curr

entIs120572

(A)

minus6

minus4

minus2

0

2

4

6

02 04 06 08 10t (02 sgrid)

(e) Real stator current 119868119904120572

and estimation (TEKF)

Is120573 (real)Is120573 (TEKF)

minus6

minus4

minus2

0

2

4

6

Curr

entIs120573

(A)

02 04 06 08 10t (02 sgrid)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 6 Continued

16 Mathematical Problems in Engineering

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Erro

r cur

rentIs120572

(A)

02 04 06 08 10t (02 sgrid)

(g) TEKF stator current 119868119904120572

error

minus02

minus015

minus01

minus005

0

005

01

015

Erro

r cur

rentIs120573

(A)

02 04 06 08 10t (02 sgrid)

(h) TEKF stator current 119868119904120573

error

Figure 6 Experimental results for parameters estimation

Speed (real)Speed (EKF)

Speed (TEKF)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

0

005

01

015

02

025

03

035

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)Figure 7 Experimental results with parameter variation (119877

119903

= 05119877119903nom)

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article Direct Torque Control of Sensorless ...

Mathematical Problems in Engineering 11

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000Sp

eed

(rpm

)

(a) Speed estimation

02 04 06 08 10Times (s)

minus002

0

002

004

006

008

01

012

014

016

018

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF

Theta (real)Theta (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation (TEKF)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7Po

sitio

n (r

ad)

(d) TEKF rotor position error

Current (real)Current (TEKF)

Curr

entIs120572

(A)

02 04 06 08 10Times (s)

minus12

minus8

minus4

0

4

8

12

(e) Real stator current 119868119904120572

and estimation (TEKF)

Current (real)Current (TEKF)

minus12

minus8

minus4

0

4

8

12

Curr

entIs120573

(A)

02 04 06 08 10Times (s)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 3 Continued

12 Mathematical Problems in Engineering

Flux (EKF)Flux (TEKF)

minus05 0 05 1minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(g) Stator flux estimation by TEKF and EKF

02 04 06 08 10Times (s)

minus2

0

2

4

6

8

10

12

14

16

Ψs

(Wb)

times10minus3

(h) Difference Stator flux estimation between TEKF and EKF

Figure 3 Simulation results for parameters estimation

Table 2 Kalman estimation arithmetic operation requirement for the TEKF structure

Number of multiplications(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

Number of additions(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

119860119896

119862119896

119864119896

1198671119896

1198672119896

119861119896

and119863119896

Function of system (25) Function of system (11)119883119896|119896minus1

1198992 + 119899119902 (24) 1198992 + 119899119902 (24)119875119909

119896|119896minus121198993 (128) 21198993 minus 1198992 (112)

119870119909

119896|1198961198992

119898 + 1198991198982 (48) 119899

2

119898 + 1198991198982

minus 2119899119898 (32)119875119909

119896|1198961198992

119898 + 1198993 (96) 119899

2

119898 + 1198993

minus 1198992 (80)

119883119896|119896

2119899119898 + 119898119901 (20) 2119899119898 + 2119898119901 (24)119903119896|119896minus1

1199012 (4) 1199012 minus 119901 (2)119875119903

119896|119896minus121199013 (16) 21199013 minus 1199012 (12)

119870119903

119896|11989631199012119898 + 1199011198982 (32) 31199012119898 + 1199011198982 + 1198982 minus 4119901119898 (20)

119875119903

119896|1198961199013 + 1199012119898 (16) 1199013 + 1199012119898 minus 1199012 (12)

119903119896|119896

2119898119901 + 119899119898 (12) 2119898119901 + 119899119898 (16)119878119896

119898119899119901 (16) 119899119898119901 (16)119880119896minus1

1198992119901 + 1198991199012 (56) 1198992119901 + 1198991199012 minus 119899119901 (48)119881119896

119899119901119898 (16) 119899119901 (8)119880119896minus1

21198991199012 (32) 21198991199012 (32)119898119896minus1

1198992

119901 + 1198991199012

+ 119899119901 (56) 1198992

119901 + 1198991199012

+ 119899119901 minus 119899 (52)119876119909

119896minus121198992

119901 (64) 21198992

119901 + 1198991199012 (64)

119878119896

119898119899119901 (16) 119898119899119901 minus 119898119901 (12)119873119896

21198981198992 (32) 21198981198992 minus 1198982 (60)Total 688 626

In the simulation a comparison is made to verify theequivalence of EKF and TEKF Real-time parameters esti-mated by TEKF are used to formulate the closed loop suchas rotor speed stator flux and electromagnetic torque Theestimations obtained by EKF algorithm are not included in

the sensorless DTC-SVM strategy and only evaluated in openloop A step reference speed was applied to the simulation

The machine is accelerated from 0 rpm to 1000 rpm at0 s and the torque load is set to 4N The simulation resultsof parameter estimation are shown in Figure 3 Figures 3(a)

Mathematical Problems in Engineering 13

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000

Spee

d (r

pm)

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus03

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Theta (real)Theta (TEKF)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

02 04 06 08 10Times (s)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)

Figure 4 Simulation results with parameter variation (119877119903

= 05119877119903nom)

and 3(c) represent the performance of the speed and rotorposition tracking capabilities of the control system withTEKF and EKF respectively Figures 3(e) 3(f) and 3(g)represent the estimated stator current and flux they showthat ripples are significantly suppressed due to the SVMmodulation scheme Figures 3(b) 3(d) and 3(h) show thatfor variations of speed reference the rotor speed rotorposition and stator flux errors between the two observers arevery little It is verified that the two observers are equivalentThe difference between two estimators is caused by accuracyloss in TEKF which uses more calculation steps

In order to further verify the performance of TEKFagainst model-plant parameter mismatches and the equiv-alence of two observers the change in rotor resistance isconsidered Rotor resistance will increase due to temperaturerise while the motor is running To simulate this conditionthe rotor resistance in TEKF and EKF is increased to 200comparedwith the normal value which is equivalent to a 50decrease in the actual rotor resistance The machine is stillaccelerated from 0 rpm to 1000 rpm at 0 s and the torque is setto 4N Figure 4 shows that for variation of rotor resistancethe steady-state speed and rotor position errors are negligible

14 Mathematical Problems in Engineering

(a)

Computer

380V

TXRX

120596r

Incrementalencoder

Udc

MicrocomputerDSP TMS3206713

Drive signalCurrentsensor

Voltagesensor

Voltagesensor

IM MC

ISU ISV

VSU VSW VSV

(b)

Figure 5 Complete drive system (a) Picture of experimental setup (b) Functional block diagram of the experimental setup

and the difference of the speed and rotor position estimationsbetween the two observers is rather null

52 Experimental Results The overall experimental setup isshown in Figure 5 and the specifications and rated parametersof the IM controller and inverter are listed in Table 3 Inthe experimental hardware an Expert3 control system fromMyway company and a three-phase two-pole 15 kW IM areappliedThe IM is mechanically coupled to a magnetic clutch(MC) which provides rated torque even at very low speedThe main processor in Expert3 control system is a floatingpoint processor TMS320C6713 with a max clock speed of225MHz All the algorithms including TEKF EKF DTCalgorithm and some transformation modules are imple-mented in TMS320C6713 with 100120583s sampling time and dataacquisition of the parameter estimations measured variablesand their visualization are realized on the cockpit provided byPEView9 software Insulated Gate Bipolar Transistor (IGBT)module is driven by the PWM signal with a switchingfrequency of 10 kHz and 2 120583s dead time The stator currentsare measured via two Hall effect current sensors The rotorangle and speed of IM are measured from an incrementalencoder with 2048 pulses per revolution

This experiment test is here to testify the performanceof TEKF and demonstrate that the two estimators aremathematically equivalent The machine is accelerated from600 rpm to 1000 rpm and 4N torque load is set Theexperimental results of parameter estimation based on twoobservers are given in Figures 6 and 7 Figures 6(a) and 6(c)show that the TEKF still has a good tracking performance ofthe speed and rotor position in experiment Figures 6(d) 6(e)and 6(f) illustrate stator flux and stator current estimationrobustness Figures 6(b) 6(g) and 6(f) referring to thedifference in speed and stator current estimations given by

Table 3 Specification of induction motor and inverter

Induction motor ValueNominal torque 10NmNominal voltage 380VRotor resistance 119877

119903

25ΩStator resistance 119877

119904

36ΩStator inductances 119871

119904

0301HRotor inductances 119871

119903

0302HMutual inductances 119871

119898

0273HPole pairs 2Invertercontroller ValueSwitching device 1000V 80A IGBTControl cycle time 100 120583sMain CPU DSP TMS320C6713 225MHz

the two observers are still small These experiment resultsprove that the two estimators are mathematically equivalentFigure 7 shows the speed and rotor position estimationsbased onTEKFandEKF for a 50decrease of rotor resistance(the same as the simulation) As expected the steady errorof the TEKF and the difference in speed and rotor positionestimations are still tiny Robustness of TEKF is verified

6 Conclusion

Themajor shortcoming of the conventional EKF is numericalproblems and computational burden due to the high orderof the mathematical models This has generally limited thereal-time digital implementation of the EKF for industrialfield So in this study a novel extended Kalman filter

Mathematical Problems in Engineering 15

Speed (real)Speed (EKF)Speed (TEKF)

Spee

d (r

min

)

02 04 06 08 10t (02 sgrid)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimationSp

eed

(rm

in)

minus005

0

005

01

015

02

025

03

035

04

045

02 04 06 08 10t (02 sgrid)

(b) Difference of speed estimation between EKF and TEKF

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation

Flux (EKF)Flux (TEKF)

minus08 minus04 1minus02 0 02 04 06 08minus06minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(d) Stator flux estimation by TEKF and EKF

Is120572 (real)Is120572 (TEKF)

Curr

entIs120572

(A)

minus6

minus4

minus2

0

2

4

6

02 04 06 08 10t (02 sgrid)

(e) Real stator current 119868119904120572

and estimation (TEKF)

Is120573 (real)Is120573 (TEKF)

minus6

minus4

minus2

0

2

4

6

Curr

entIs120573

(A)

02 04 06 08 10t (02 sgrid)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 6 Continued

16 Mathematical Problems in Engineering

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Erro

r cur

rentIs120572

(A)

02 04 06 08 10t (02 sgrid)

(g) TEKF stator current 119868119904120572

error

minus02

minus015

minus01

minus005

0

005

01

015

Erro

r cur

rentIs120573

(A)

02 04 06 08 10t (02 sgrid)

(h) TEKF stator current 119868119904120573

error

Figure 6 Experimental results for parameters estimation

Speed (real)Speed (EKF)

Speed (TEKF)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

0

005

01

015

02

025

03

035

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)Figure 7 Experimental results with parameter variation (119877

119903

= 05119877119903nom)

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Research Article Direct Torque Control of Sensorless ...

12 Mathematical Problems in Engineering

Flux (EKF)Flux (TEKF)

minus05 0 05 1minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(g) Stator flux estimation by TEKF and EKF

02 04 06 08 10Times (s)

minus2

0

2

4

6

8

10

12

14

16

Ψs

(Wb)

times10minus3

(h) Difference Stator flux estimation between TEKF and EKF

Figure 3 Simulation results for parameters estimation

Table 2 Kalman estimation arithmetic operation requirement for the TEKF structure

Number of multiplications(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

Number of additions(119899 = 4119898 = 2 119902 = 2 and 119901 = 2)

119860119896

119862119896

119864119896

1198671119896

1198672119896

119861119896

and119863119896

Function of system (25) Function of system (11)119883119896|119896minus1

1198992 + 119899119902 (24) 1198992 + 119899119902 (24)119875119909

119896|119896minus121198993 (128) 21198993 minus 1198992 (112)

119870119909

119896|1198961198992

119898 + 1198991198982 (48) 119899

2

119898 + 1198991198982

minus 2119899119898 (32)119875119909

119896|1198961198992

119898 + 1198993 (96) 119899

2

119898 + 1198993

minus 1198992 (80)

119883119896|119896

2119899119898 + 119898119901 (20) 2119899119898 + 2119898119901 (24)119903119896|119896minus1

1199012 (4) 1199012 minus 119901 (2)119875119903

119896|119896minus121199013 (16) 21199013 minus 1199012 (12)

119870119903

119896|11989631199012119898 + 1199011198982 (32) 31199012119898 + 1199011198982 + 1198982 minus 4119901119898 (20)

119875119903

119896|1198961199013 + 1199012119898 (16) 1199013 + 1199012119898 minus 1199012 (12)

119903119896|119896

2119898119901 + 119899119898 (12) 2119898119901 + 119899119898 (16)119878119896

119898119899119901 (16) 119899119898119901 (16)119880119896minus1

1198992119901 + 1198991199012 (56) 1198992119901 + 1198991199012 minus 119899119901 (48)119881119896

119899119901119898 (16) 119899119901 (8)119880119896minus1

21198991199012 (32) 21198991199012 (32)119898119896minus1

1198992

119901 + 1198991199012

+ 119899119901 (56) 1198992

119901 + 1198991199012

+ 119899119901 minus 119899 (52)119876119909

119896minus121198992

119901 (64) 21198992

119901 + 1198991199012 (64)

119878119896

119898119899119901 (16) 119898119899119901 minus 119898119901 (12)119873119896

21198981198992 (32) 21198981198992 minus 1198982 (60)Total 688 626

In the simulation a comparison is made to verify theequivalence of EKF and TEKF Real-time parameters esti-mated by TEKF are used to formulate the closed loop suchas rotor speed stator flux and electromagnetic torque Theestimations obtained by EKF algorithm are not included in

the sensorless DTC-SVM strategy and only evaluated in openloop A step reference speed was applied to the simulation

The machine is accelerated from 0 rpm to 1000 rpm at0 s and the torque load is set to 4N The simulation resultsof parameter estimation are shown in Figure 3 Figures 3(a)

Mathematical Problems in Engineering 13

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000

Spee

d (r

pm)

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus03

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Theta (real)Theta (TEKF)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

02 04 06 08 10Times (s)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)

Figure 4 Simulation results with parameter variation (119877119903

= 05119877119903nom)

and 3(c) represent the performance of the speed and rotorposition tracking capabilities of the control system withTEKF and EKF respectively Figures 3(e) 3(f) and 3(g)represent the estimated stator current and flux they showthat ripples are significantly suppressed due to the SVMmodulation scheme Figures 3(b) 3(d) and 3(h) show thatfor variations of speed reference the rotor speed rotorposition and stator flux errors between the two observers arevery little It is verified that the two observers are equivalentThe difference between two estimators is caused by accuracyloss in TEKF which uses more calculation steps

In order to further verify the performance of TEKFagainst model-plant parameter mismatches and the equiv-alence of two observers the change in rotor resistance isconsidered Rotor resistance will increase due to temperaturerise while the motor is running To simulate this conditionthe rotor resistance in TEKF and EKF is increased to 200comparedwith the normal value which is equivalent to a 50decrease in the actual rotor resistance The machine is stillaccelerated from 0 rpm to 1000 rpm at 0 s and the torque is setto 4N Figure 4 shows that for variation of rotor resistancethe steady-state speed and rotor position errors are negligible

14 Mathematical Problems in Engineering

(a)

Computer

380V

TXRX

120596r

Incrementalencoder

Udc

MicrocomputerDSP TMS3206713

Drive signalCurrentsensor

Voltagesensor

Voltagesensor

IM MC

ISU ISV

VSU VSW VSV

(b)

Figure 5 Complete drive system (a) Picture of experimental setup (b) Functional block diagram of the experimental setup

and the difference of the speed and rotor position estimationsbetween the two observers is rather null

52 Experimental Results The overall experimental setup isshown in Figure 5 and the specifications and rated parametersof the IM controller and inverter are listed in Table 3 Inthe experimental hardware an Expert3 control system fromMyway company and a three-phase two-pole 15 kW IM areappliedThe IM is mechanically coupled to a magnetic clutch(MC) which provides rated torque even at very low speedThe main processor in Expert3 control system is a floatingpoint processor TMS320C6713 with a max clock speed of225MHz All the algorithms including TEKF EKF DTCalgorithm and some transformation modules are imple-mented in TMS320C6713 with 100120583s sampling time and dataacquisition of the parameter estimations measured variablesand their visualization are realized on the cockpit provided byPEView9 software Insulated Gate Bipolar Transistor (IGBT)module is driven by the PWM signal with a switchingfrequency of 10 kHz and 2 120583s dead time The stator currentsare measured via two Hall effect current sensors The rotorangle and speed of IM are measured from an incrementalencoder with 2048 pulses per revolution

This experiment test is here to testify the performanceof TEKF and demonstrate that the two estimators aremathematically equivalent The machine is accelerated from600 rpm to 1000 rpm and 4N torque load is set Theexperimental results of parameter estimation based on twoobservers are given in Figures 6 and 7 Figures 6(a) and 6(c)show that the TEKF still has a good tracking performance ofthe speed and rotor position in experiment Figures 6(d) 6(e)and 6(f) illustrate stator flux and stator current estimationrobustness Figures 6(b) 6(g) and 6(f) referring to thedifference in speed and stator current estimations given by

Table 3 Specification of induction motor and inverter

Induction motor ValueNominal torque 10NmNominal voltage 380VRotor resistance 119877

119903

25ΩStator resistance 119877

119904

36ΩStator inductances 119871

119904

0301HRotor inductances 119871

119903

0302HMutual inductances 119871

119898

0273HPole pairs 2Invertercontroller ValueSwitching device 1000V 80A IGBTControl cycle time 100 120583sMain CPU DSP TMS320C6713 225MHz

the two observers are still small These experiment resultsprove that the two estimators are mathematically equivalentFigure 7 shows the speed and rotor position estimationsbased onTEKFandEKF for a 50decrease of rotor resistance(the same as the simulation) As expected the steady errorof the TEKF and the difference in speed and rotor positionestimations are still tiny Robustness of TEKF is verified

6 Conclusion

Themajor shortcoming of the conventional EKF is numericalproblems and computational burden due to the high orderof the mathematical models This has generally limited thereal-time digital implementation of the EKF for industrialfield So in this study a novel extended Kalman filter

Mathematical Problems in Engineering 15

Speed (real)Speed (EKF)Speed (TEKF)

Spee

d (r

min

)

02 04 06 08 10t (02 sgrid)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimationSp

eed

(rm

in)

minus005

0

005

01

015

02

025

03

035

04

045

02 04 06 08 10t (02 sgrid)

(b) Difference of speed estimation between EKF and TEKF

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation

Flux (EKF)Flux (TEKF)

minus08 minus04 1minus02 0 02 04 06 08minus06minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(d) Stator flux estimation by TEKF and EKF

Is120572 (real)Is120572 (TEKF)

Curr

entIs120572

(A)

minus6

minus4

minus2

0

2

4

6

02 04 06 08 10t (02 sgrid)

(e) Real stator current 119868119904120572

and estimation (TEKF)

Is120573 (real)Is120573 (TEKF)

minus6

minus4

minus2

0

2

4

6

Curr

entIs120573

(A)

02 04 06 08 10t (02 sgrid)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 6 Continued

16 Mathematical Problems in Engineering

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Erro

r cur

rentIs120572

(A)

02 04 06 08 10t (02 sgrid)

(g) TEKF stator current 119868119904120572

error

minus02

minus015

minus01

minus005

0

005

01

015

Erro

r cur

rentIs120573

(A)

02 04 06 08 10t (02 sgrid)

(h) TEKF stator current 119868119904120573

error

Figure 6 Experimental results for parameters estimation

Speed (real)Speed (EKF)

Speed (TEKF)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

0

005

01

015

02

025

03

035

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)Figure 7 Experimental results with parameter variation (119877

119903

= 05119877119903nom)

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 13: Research Article Direct Torque Control of Sensorless ...

Mathematical Problems in Engineering 13

Speed (real)Speed (EKF)Speed (TEKF)

02 04 06 08 10Times (s)

0

200

400

600

800

1000

Spee

d (r

pm)

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus03

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

Spee

d (r

pm)

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Theta (real)Theta (TEKF)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

02 04 06 08 10Times (s)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10Times (s)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)

Figure 4 Simulation results with parameter variation (119877119903

= 05119877119903nom)

and 3(c) represent the performance of the speed and rotorposition tracking capabilities of the control system withTEKF and EKF respectively Figures 3(e) 3(f) and 3(g)represent the estimated stator current and flux they showthat ripples are significantly suppressed due to the SVMmodulation scheme Figures 3(b) 3(d) and 3(h) show thatfor variations of speed reference the rotor speed rotorposition and stator flux errors between the two observers arevery little It is verified that the two observers are equivalentThe difference between two estimators is caused by accuracyloss in TEKF which uses more calculation steps

In order to further verify the performance of TEKFagainst model-plant parameter mismatches and the equiv-alence of two observers the change in rotor resistance isconsidered Rotor resistance will increase due to temperaturerise while the motor is running To simulate this conditionthe rotor resistance in TEKF and EKF is increased to 200comparedwith the normal value which is equivalent to a 50decrease in the actual rotor resistance The machine is stillaccelerated from 0 rpm to 1000 rpm at 0 s and the torque is setto 4N Figure 4 shows that for variation of rotor resistancethe steady-state speed and rotor position errors are negligible

14 Mathematical Problems in Engineering

(a)

Computer

380V

TXRX

120596r

Incrementalencoder

Udc

MicrocomputerDSP TMS3206713

Drive signalCurrentsensor

Voltagesensor

Voltagesensor

IM MC

ISU ISV

VSU VSW VSV

(b)

Figure 5 Complete drive system (a) Picture of experimental setup (b) Functional block diagram of the experimental setup

and the difference of the speed and rotor position estimationsbetween the two observers is rather null

52 Experimental Results The overall experimental setup isshown in Figure 5 and the specifications and rated parametersof the IM controller and inverter are listed in Table 3 Inthe experimental hardware an Expert3 control system fromMyway company and a three-phase two-pole 15 kW IM areappliedThe IM is mechanically coupled to a magnetic clutch(MC) which provides rated torque even at very low speedThe main processor in Expert3 control system is a floatingpoint processor TMS320C6713 with a max clock speed of225MHz All the algorithms including TEKF EKF DTCalgorithm and some transformation modules are imple-mented in TMS320C6713 with 100120583s sampling time and dataacquisition of the parameter estimations measured variablesand their visualization are realized on the cockpit provided byPEView9 software Insulated Gate Bipolar Transistor (IGBT)module is driven by the PWM signal with a switchingfrequency of 10 kHz and 2 120583s dead time The stator currentsare measured via two Hall effect current sensors The rotorangle and speed of IM are measured from an incrementalencoder with 2048 pulses per revolution

This experiment test is here to testify the performanceof TEKF and demonstrate that the two estimators aremathematically equivalent The machine is accelerated from600 rpm to 1000 rpm and 4N torque load is set Theexperimental results of parameter estimation based on twoobservers are given in Figures 6 and 7 Figures 6(a) and 6(c)show that the TEKF still has a good tracking performance ofthe speed and rotor position in experiment Figures 6(d) 6(e)and 6(f) illustrate stator flux and stator current estimationrobustness Figures 6(b) 6(g) and 6(f) referring to thedifference in speed and stator current estimations given by

Table 3 Specification of induction motor and inverter

Induction motor ValueNominal torque 10NmNominal voltage 380VRotor resistance 119877

119903

25ΩStator resistance 119877

119904

36ΩStator inductances 119871

119904

0301HRotor inductances 119871

119903

0302HMutual inductances 119871

119898

0273HPole pairs 2Invertercontroller ValueSwitching device 1000V 80A IGBTControl cycle time 100 120583sMain CPU DSP TMS320C6713 225MHz

the two observers are still small These experiment resultsprove that the two estimators are mathematically equivalentFigure 7 shows the speed and rotor position estimationsbased onTEKFandEKF for a 50decrease of rotor resistance(the same as the simulation) As expected the steady errorof the TEKF and the difference in speed and rotor positionestimations are still tiny Robustness of TEKF is verified

6 Conclusion

Themajor shortcoming of the conventional EKF is numericalproblems and computational burden due to the high orderof the mathematical models This has generally limited thereal-time digital implementation of the EKF for industrialfield So in this study a novel extended Kalman filter

Mathematical Problems in Engineering 15

Speed (real)Speed (EKF)Speed (TEKF)

Spee

d (r

min

)

02 04 06 08 10t (02 sgrid)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimationSp

eed

(rm

in)

minus005

0

005

01

015

02

025

03

035

04

045

02 04 06 08 10t (02 sgrid)

(b) Difference of speed estimation between EKF and TEKF

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation

Flux (EKF)Flux (TEKF)

minus08 minus04 1minus02 0 02 04 06 08minus06minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(d) Stator flux estimation by TEKF and EKF

Is120572 (real)Is120572 (TEKF)

Curr

entIs120572

(A)

minus6

minus4

minus2

0

2

4

6

02 04 06 08 10t (02 sgrid)

(e) Real stator current 119868119904120572

and estimation (TEKF)

Is120573 (real)Is120573 (TEKF)

minus6

minus4

minus2

0

2

4

6

Curr

entIs120573

(A)

02 04 06 08 10t (02 sgrid)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 6 Continued

16 Mathematical Problems in Engineering

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Erro

r cur

rentIs120572

(A)

02 04 06 08 10t (02 sgrid)

(g) TEKF stator current 119868119904120572

error

minus02

minus015

minus01

minus005

0

005

01

015

Erro

r cur

rentIs120573

(A)

02 04 06 08 10t (02 sgrid)

(h) TEKF stator current 119868119904120573

error

Figure 6 Experimental results for parameters estimation

Speed (real)Speed (EKF)

Speed (TEKF)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

0

005

01

015

02

025

03

035

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)Figure 7 Experimental results with parameter variation (119877

119903

= 05119877119903nom)

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 14: Research Article Direct Torque Control of Sensorless ...

14 Mathematical Problems in Engineering

(a)

Computer

380V

TXRX

120596r

Incrementalencoder

Udc

MicrocomputerDSP TMS3206713

Drive signalCurrentsensor

Voltagesensor

Voltagesensor

IM MC

ISU ISV

VSU VSW VSV

(b)

Figure 5 Complete drive system (a) Picture of experimental setup (b) Functional block diagram of the experimental setup

and the difference of the speed and rotor position estimationsbetween the two observers is rather null

52 Experimental Results The overall experimental setup isshown in Figure 5 and the specifications and rated parametersof the IM controller and inverter are listed in Table 3 Inthe experimental hardware an Expert3 control system fromMyway company and a three-phase two-pole 15 kW IM areappliedThe IM is mechanically coupled to a magnetic clutch(MC) which provides rated torque even at very low speedThe main processor in Expert3 control system is a floatingpoint processor TMS320C6713 with a max clock speed of225MHz All the algorithms including TEKF EKF DTCalgorithm and some transformation modules are imple-mented in TMS320C6713 with 100120583s sampling time and dataacquisition of the parameter estimations measured variablesand their visualization are realized on the cockpit provided byPEView9 software Insulated Gate Bipolar Transistor (IGBT)module is driven by the PWM signal with a switchingfrequency of 10 kHz and 2 120583s dead time The stator currentsare measured via two Hall effect current sensors The rotorangle and speed of IM are measured from an incrementalencoder with 2048 pulses per revolution

This experiment test is here to testify the performanceof TEKF and demonstrate that the two estimators aremathematically equivalent The machine is accelerated from600 rpm to 1000 rpm and 4N torque load is set Theexperimental results of parameter estimation based on twoobservers are given in Figures 6 and 7 Figures 6(a) and 6(c)show that the TEKF still has a good tracking performance ofthe speed and rotor position in experiment Figures 6(d) 6(e)and 6(f) illustrate stator flux and stator current estimationrobustness Figures 6(b) 6(g) and 6(f) referring to thedifference in speed and stator current estimations given by

Table 3 Specification of induction motor and inverter

Induction motor ValueNominal torque 10NmNominal voltage 380VRotor resistance 119877

119903

25ΩStator resistance 119877

119904

36ΩStator inductances 119871

119904

0301HRotor inductances 119871

119903

0302HMutual inductances 119871

119898

0273HPole pairs 2Invertercontroller ValueSwitching device 1000V 80A IGBTControl cycle time 100 120583sMain CPU DSP TMS320C6713 225MHz

the two observers are still small These experiment resultsprove that the two estimators are mathematically equivalentFigure 7 shows the speed and rotor position estimationsbased onTEKFandEKF for a 50decrease of rotor resistance(the same as the simulation) As expected the steady errorof the TEKF and the difference in speed and rotor positionestimations are still tiny Robustness of TEKF is verified

6 Conclusion

Themajor shortcoming of the conventional EKF is numericalproblems and computational burden due to the high orderof the mathematical models This has generally limited thereal-time digital implementation of the EKF for industrialfield So in this study a novel extended Kalman filter

Mathematical Problems in Engineering 15

Speed (real)Speed (EKF)Speed (TEKF)

Spee

d (r

min

)

02 04 06 08 10t (02 sgrid)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimationSp

eed

(rm

in)

minus005

0

005

01

015

02

025

03

035

04

045

02 04 06 08 10t (02 sgrid)

(b) Difference of speed estimation between EKF and TEKF

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation

Flux (EKF)Flux (TEKF)

minus08 minus04 1minus02 0 02 04 06 08minus06minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(d) Stator flux estimation by TEKF and EKF

Is120572 (real)Is120572 (TEKF)

Curr

entIs120572

(A)

minus6

minus4

minus2

0

2

4

6

02 04 06 08 10t (02 sgrid)

(e) Real stator current 119868119904120572

and estimation (TEKF)

Is120573 (real)Is120573 (TEKF)

minus6

minus4

minus2

0

2

4

6

Curr

entIs120573

(A)

02 04 06 08 10t (02 sgrid)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 6 Continued

16 Mathematical Problems in Engineering

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Erro

r cur

rentIs120572

(A)

02 04 06 08 10t (02 sgrid)

(g) TEKF stator current 119868119904120572

error

minus02

minus015

minus01

minus005

0

005

01

015

Erro

r cur

rentIs120573

(A)

02 04 06 08 10t (02 sgrid)

(h) TEKF stator current 119868119904120573

error

Figure 6 Experimental results for parameters estimation

Speed (real)Speed (EKF)

Speed (TEKF)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

0

005

01

015

02

025

03

035

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)Figure 7 Experimental results with parameter variation (119877

119903

= 05119877119903nom)

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 15: Research Article Direct Torque Control of Sensorless ...

Mathematical Problems in Engineering 15

Speed (real)Speed (EKF)Speed (TEKF)

Spee

d (r

min

)

02 04 06 08 10t (02 sgrid)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimationSp

eed

(rm

in)

minus005

0

005

01

015

02

025

03

035

04

045

02 04 06 08 10t (02 sgrid)

(b) Difference of speed estimation between EKF and TEKF

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation

Flux (EKF)Flux (TEKF)

minus08 minus04 1minus02 0 02 04 06 08minus06minus1

Ψs120572 (Wb)

minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1Ψs120573

(Wb)

(d) Stator flux estimation by TEKF and EKF

Is120572 (real)Is120572 (TEKF)

Curr

entIs120572

(A)

minus6

minus4

minus2

0

2

4

6

02 04 06 08 10t (02 sgrid)

(e) Real stator current 119868119904120572

and estimation (TEKF)

Is120573 (real)Is120573 (TEKF)

minus6

minus4

minus2

0

2

4

6

Curr

entIs120573

(A)

02 04 06 08 10t (02 sgrid)

(f) Real stator current 119868119904120573

and estimation (TEKF)

Figure 6 Continued

16 Mathematical Problems in Engineering

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Erro

r cur

rentIs120572

(A)

02 04 06 08 10t (02 sgrid)

(g) TEKF stator current 119868119904120572

error

minus02

minus015

minus01

minus005

0

005

01

015

Erro

r cur

rentIs120573

(A)

02 04 06 08 10t (02 sgrid)

(h) TEKF stator current 119868119904120573

error

Figure 6 Experimental results for parameters estimation

Speed (real)Speed (EKF)

Speed (TEKF)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

0

005

01

015

02

025

03

035

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)Figure 7 Experimental results with parameter variation (119877

119903

= 05119877119903nom)

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 16: Research Article Direct Torque Control of Sensorless ...

16 Mathematical Problems in Engineering

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Erro

r cur

rentIs120572

(A)

02 04 06 08 10t (02 sgrid)

(g) TEKF stator current 119868119904120572

error

minus02

minus015

minus01

minus005

0

005

01

015

Erro

r cur

rentIs120573

(A)

02 04 06 08 10t (02 sgrid)

(h) TEKF stator current 119868119904120573

error

Figure 6 Experimental results for parameters estimation

Speed (real)Speed (EKF)

Speed (TEKF)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

300

400

500

600

700

800

900

1000

1100

1200

(a) Speed estimation (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

Spee

d (r

min

)

0

005

01

015

02

025

03

035

(b) Difference of speed estimation between EKF and TEKF (119877119903=

05119877119903nom)

Position (real)Position (TEKF)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(c) Real rotor position and estimation by TEKF (119877119903= 05119877

119903nom)

02 04 06 08 10t (02 sgrid)

minus1

0

1

2

3

4

5

6

7

Posit

ion

(rad

)

(d) TEKF rotor position error (119877119903= 05119877

119903nom)Figure 7 Experimental results with parameter variation (119877

119903

= 05119877119903nom)

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 17: Research Article Direct Torque Control of Sensorless ...

Mathematical Problems in Engineering 17

called two-stage extended Kalman filter is proposed In theparticular case of parameter estimations for sensorless DTC-SVM this novel extended Kalman filter can reduce the arith-metic operations effectively compared to the traditional EKFThe simulation and experimental results show that TEKFhas good dynamic characteristics and stability The TEKFprovides a new idea to reduce computational complexityand allows us to use cheaper microcontroller in practicalapplication

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol IA-22 no 5 pp 820ndash827 1986

[2] D Casadei F Profumo G Serra and A Tani ldquoFOC and DTCtwo viable schemes for induction motors torque controlrdquo IEEETransactions on Power Electronics vol 17 no 5 pp 779ndash7872002

[3] L Zhong M F Rahman W Y Hu and K W Lim ldquoAnalysis ofdirect torque control in permanent magnet synchronous motordrivesrdquo IEEE Transactions on Power Electronics vol 12 no 3pp 528ndash536 1997

[4] YWang and Z Deng ldquoImproved stator flux estimationmethodfor direct torque linear control of parallel hybrid excitationswitched-flux generatorrdquo IEEE Transactions on Energy Conver-sion vol 27 no 3 pp 747ndash756 2012

[5] J Faiz andM B B Sharifian ldquoDifferent techniques for real timeestimation of an induction motor rotor resistance in sensorlessdirect torque control for electric vehiclerdquo IEEE Transactions onEnergy Conversion vol 16 no 1 pp 104ndash110 2001

[6] T Ohtani N Takada and K Tanaka ldquoVector control ofinduction motor without shaft encoderrdquo IEEE Transactions onIndustry Applications vol 28 no 1 pp 157ndash164 1992

[7] T Iwasaki and T Kataoka ldquoApplication of an extended Kalmanfilter to parameter identification of an induction motorrdquo inProceedings of the Industry Applications Society Annual Meetingvol 1 pp 248ndash253 October 1989

[8] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[9] A Kheloui K Aliouane M Medjaoui and B Davat ldquoDesignof a stator flux sliding mode observer for direct torque controlof sensorless induction machinerdquo in Proceedings of the IEEEIndustrial Applications Conference Annual Meeting vol 3 pp1388ndash1393 2000

[10] D Casadei G Serra A Tani L Zarri and F ProfumoldquoPerformance analysis of a speed-sensorless induction motordrive based on a constant-switching-frequency DTC schemerdquoIEEE Transactions on Industry Applications vol 39 no 2 pp476ndash484 2003

[11] W Zhang and J J Luo ldquoSpeed and rotor flux estimationof induction motors based on extended kalman filterrdquo in

Proceedings of the 6th International Conference on NetworkedComputing and Advanced Information Management (NCM rsquo10)pp 157ndash160 August 2010

[12] M Barut S Bogosyan and M Gokasan ldquoSpeed sensorlessdirect torque control of IMs with rotor resistance estimationrdquoEnergy Conversion andManagement vol 46 no 3 pp 335ndash3492005

[13] T J Vyncke R K Boel and J A A Melkebeek ldquoOn the statorflux linkage estimation of an PMSM with extended Kalmanfiltersrdquo in Proceedings of the 5th IET International Conferenceon Power Electronics Machines and Drives (PEMD rsquo10) pp 1ndash6April 2010

[14] A Arias L Romeral E Aldabas and M Jayne ldquoStator fluxoptimised Direct Torque Control system for induction motorsrdquoElectric Power Systems Research vol 73 no 3 pp 257ndash265 2005

[15] C-S Hsieh and F-C Chen ldquoOptimal solution of the two-stageKalman estimatorrdquo IEEE Transactions on Automatic Controlvol 44 no 1 pp 194ndash199 1999

[16] J Rodrıguez J Pontt C Silva R Huerta and H MirandaldquoSimple direct torque control of induction machine using spacevector modulationrdquo Electronics Letters vol 40 no 7 pp 412ndash413 2004

[17] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988

[18] I Takahashi and T Noguchi ldquoA new quick-response andhigh-efficiency control strategy of an induction motorrdquo IEEETransactions on Industry Applications vol 22 no 5 pp 820ndash827 1986

[19] S Sayeef and M F Rahman ldquoComparison of propor-tional+integral control and variable structure control of interiorpermanent magnet synchronous motor drivesrdquo in Proceedingsof the IEEE Power Electronics Specialists Conference (PESC rsquo07)pp 1645ndash1650 June 2007

[20] D Swierczynski M P Kazmierkowski and F Blaabjerg ldquoDSPbased direct torque control of permanent magnet synchronousmotor (PMSM)using space vectormodulation (DTC-SVM)rdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo02) vol 3 pp 723ndash727 July 2002

[21] K H Kim J G Lee and C G Park ldquoAdaptive two-stageextended kalman filter for a fault-tolerant INS-GPS looselycoupled systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 45 no 1 pp 125ndash137 2009

[22] S Bolognani L Tubiana and M Zigliotto ldquoExtended kalmanfilter tuning in sensorless PMSM drivesrdquo IEEE Transactions onIndustry Applications vol 39 no 6 pp 1741ndash1747 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 18: Research Article Direct Torque Control of Sensorless ...

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Torque Ripple Reduction Based Direct Torque Control for … · 2019-07-01 · Torque Ripple Reduction Based Direct Torque Control for Induction Motor Drive . Abstract—Direct Torque

Torque Ripple Reduction Based Direct Torque Control for … · 2019-07-01 · Torque Ripple Reduction Based Direct Torque Control for Induction Motor Drive . Abstract—Direct Torque

Sensorless DTC of an Induction Motor based on …...observer is used for our sensorless direct torque control (DTC) of a three-phase squirrel cage induction motor drive (SCIM), combining

Sensorless DTC of an Induction Motor based on …...observer is used for our sensorless direct torque control (DTC) of a three-phase squirrel cage induction motor drive (SCIM), combining

Sensorless Direct Torque Control of Induction Motor …jarguz/papers/EPE_PEMC_2012.pdf · Sensorless Direct Torque Control of Induction Motor Drive with LC Filter Jaroslaw Guzinski

Sensorless Direct Torque Control of Induction Motor …jarguz/papers/EPE_PEMC_2012.pdf · Sensorless Direct Torque Control of Induction Motor Drive with LC Filter Jaroslaw Guzinski

Optimal Sensorless Four Switch Direct Power Control of ...

Optimal Sensorless Four Switch Direct Power Control of ...

A Robust Sensorless Direct Torque Control of Induction ...ljs.academicdirect.org/A12/035_056.pdf · In recent years significant advances have been made on the sensorless control ...

A Robust Sensorless Direct Torque Control of Induction ...ljs.academicdirect.org/A12/035_056.pdf · In recent years significant advances have been made on the sensorless control ...

Mechanical Sensorless LPV torque ripple control of Hybrid ...

Mechanical Sensorless LPV torque ripple control of Hybrid ...

Direct Back EMF Detection Method for Sensorless Brushless ...

Direct Back EMF Detection Method for Sensorless Brushless ...

Sliding mode approach applied to sensorless direct torque ...

Sliding mode approach applied to sensorless direct torque ...

Hybrid Sensorless Field Oriented and Direct Torque Control ...

Hybrid Sensorless Field Oriented and Direct Torque Control ...

High Performance Sensorless Vector Control of · PDF fileHigh Performance Sensorless Vector Control of ... 3 Sensorless Vector Control of ... Figure 3.4 IRFO full load torque transient

High Performance Sensorless Vector Control of · PDF fileHigh Performance Sensorless Vector Control of ... 3 Sensorless Vector Control of ... Figure 3.4 IRFO full load torque transient

Improved Torque Control of High Speed Shaft-Sensorless ...

Improved Torque Control of High Speed Shaft-Sensorless ...

Speed sensorless direct torque control of IMs with rotor resistance estimationread.pudn.com/downloads133/doc/comm/567381/Speed... ·  · 2007-01-21Speed sensorless direct torque

Speed sensorless direct torque control of IMs with rotor resistance estimationread.pudn.com/downloads133/doc/comm/567381/Speed... ·  · 2007-01-21Speed sensorless direct torque

System Automation Drive - EFES · PDF fileSystem Automation Drive ... the Torque Current Limit Control and the Zero Torque. (Torque Setpoint ... Sensorless Vector Control

System Automation Drive - EFES · PDF fileSystem Automation Drive ... the Torque Current Limit Control and the Zero Torque. (Torque Setpoint ... Sensorless Vector Control

Very High Torque Direct Drive motor: 4.5kg.cm Unbreakable ... · Professional High Torque Direct Drive turntable. Very High Torque Direct Drive motor: 4.5kg.cm Extremely fast start,

Very High Torque Direct Drive motor: 4.5kg.cm Unbreakable ... · Professional High Torque Direct Drive turntable. Very High Torque Direct Drive motor: 4.5kg.cm Extremely fast start,

Neural network-based sensorless direct power control of ...publications.lib.chalmers.se/records/fulltext/244773/local_244773.pdf · Neural network-based sensorless direct power control

Neural network-based sensorless direct power control of ...publications.lib.chalmers.se/records/fulltext/244773/local_244773.pdf · Neural network-based sensorless direct power control

FLC for DTC of induction motor driven by MC with Speed …jestr.org/downloads/Volume9Issue1/fulltext91272016.pdf ·  · 2016-04-23Fuzzy Logic Controller for Sensorless Direct Torque

FLC for DTC of induction motor driven by MC with Speed …jestr.org/downloads/Volume9Issue1/fulltext91272016.pdf ·  · 2016-04-23Fuzzy Logic Controller for Sensorless Direct Torque

portal.utfpr.edu.brportal.utfpr.edu.br/documentos/conselhos/cogep/... · P. Sensorless vector and direct torque control (Monographs in Elecand Elec Eng v 4?) Oxtòrd. Oxford Univ.

portal.utfpr.edu.brportal.utfpr.edu.br/documentos/conselhos/cogep/... · P. Sensorless vector and direct torque control (Monographs in Elecand Elec Eng v 4?) Oxtòrd. Oxford Univ.

Sensorless Vector Control and Implementation: Why · PDF filesensorless vector control and how to implement it using Renesas ... Direct torque control ... Sensor less Vector Control

Sensorless Vector Control and Implementation: Why · PDF filesensorless vector control and how to implement it using Renesas ... Direct torque control ... Sensor less Vector Control

Sensorless Speed and Position Control with DTFC of ... · PDF filework deals a sensorless direct torque control for induction motor drives, and in particular the ... space vector from

Sensorless Speed and Position Control with DTFC of ... · PDF filework deals a sensorless direct torque control for induction motor drives, and in particular the ... space vector from

PWM BASED SENSORLESS DIRECT TORQUE CONTROL FOR … · 2018-11-03 · gain better torque and current control i.e with minimum torque and current pulsations. Most of the proposed techniques

PWM BASED SENSORLESS DIRECT TORQUE CONTROL FOR … · 2018-11-03 · gain better torque and current control i.e with minimum torque and current pulsations. Most of the proposed techniques