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PI and Fuzzy Estimators for Tuning the Stator Resistance in Direct TorqueControl of I nduction M achinesSay& M iP Malik E. Elbuluk* DonaldS. Zinger**

Variables in motork, +AX,= j(V-(I,+A,)R)dtte+Ate,=K(I ,+A ,)(h,+Aksm

* Dept.of Electric EngineeringUniversity of AhonAkron,OH 44325-3904, SA

Estimated control ler variablesk,+ AkSc=(V,- (I,+AI,)R,)dtte+Atec=K(I ,+AI ,)(k,+AXsc)

Abstract------Direct torque control (DTC) uses thestator resistance of the machine for the estimatlonof stator flux. The variation of stator resistance dueto temperature changes in the machine, makes theoperatlons difflcult at low speeds. A method for theestimatlon of changes In stator resistance duringthe operation of the machine Is presented. T h eestimation method I s Implemented wlth PI controland f uzzy logic control schemes. The estimatorsobserves the stator current vector of the machlne todetect the changes In stator resistance. Theperformance of the two methods is compared usingsimulation and experlmental results.1. INTRODUCTION

The variation of induction motor parameters and theireffecton the performance of the induction motor drives haslong been recognized [1,2,3]. In perfect rotor field-oriented-control methods, the controller s dependent on both rotor andstator parameters[l,2]. Direct torque control (DTC), avariation of field-oriented-control minimizes the use ofmachine parameters [4,5]. Fig. 1shows the block diagram ofthe DTC. It isessentially a sliding mode stator flux-oriented-control. The sliding mode control uses a hysteresis band todirectly control the flux and torqueof the machine. When thestator flux falls outside the hysteresis band, the inverterswitching states are changed sothat the flux takes anoptimalpath towards thedesired value [4,5]. In spite of itssimplicityit is capable of generating a fast torque response.One of the limitations of DTC is the use of statorresistance for the estimation of stator flux [4]. The variationof stator resistance due to temperature changes in themachjne, makes the controller operation difficult at lowspeeds. If the stator resistance is measured or estimatedduring the operation of the machine, the use of direct selfcontroller at low speedscan be made more reliable.This paper discusses the effects of the change in statorresistanceon the DTC. The relation between the change inthe stator current vector and change in stator resistance isderived. A method for the estimation of changes in statorresistance during the operation of the machine is presented.The estimation method is implemented with PI and fuzzylogic control schemes. The performance of the two methodsis compared using simulation and experimental results.

2. EFFECT OF VARIATION I NSTATORRESISTANCE ONDTCIn DTC, the stator flux is estimated by taking the integral

of the difference between the input voltage and the voltagedrop across thestator resistanceasThe value of stator resistance changes due to the change oftemperature during the operation of the machine. At low0-7803-1859-5/94/% 4.000 1994 IEEE

h, = (Vs-IsR&lt . . . . . . . . . (1)

** Dept. of Elect& EngineeringNorthern Illinois UniversityDekalb,IL 601152854. USA

Flux angle eFig. 1.Direct Torque control of an induction machine.

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Variabks in motorks+Aksm=

kV,-(Is+AIs)(R,+AR ,))dtte+Ate,=K(I s+AI s) (~s+A~sm

Ak,,=O J A I ~R , ~~=OAkc=O * K (A I ~h~+A IsA~sc+SA& )=0Therefore changes in the actual torque and flux from thecommand valuesare given byAXs= ~(AIsRs+IsARs) dtAt,, = K (A I ,A~sm+I sA~sm)

=K (I s+AI s)/(Is+AI s) AR s dtThese changes are same as the difference betweenestimated and actual torque and flux given by equations (4)and (5 ) . Rearranging equation ( 5 ) we have

Estimated controller variablesks +Aksc =

kvs- (1s+ds)RJ dtk+At,=K(I,+ AIs)(&+Aksc)

1!I( A- )K dt I, +A IsAR, = I, +A IS

Variables in motorAAsm=J (A I ~R,+ I~A R~+A I ~A R~)tAt,,=

K (AI ,k,+AI ,Ak,,+sAkSm)

(6). .

Estimated controller variablesAksc= lAIsRs dtAte,=

K (hI,k,+hI,Aksc+ IsAXsc 1

Equation (6) gives the change in stator resistance as afunctionof changes in stator current and electric torque.As the controller tries to keep the torque constanttherefore the stator current is not affected by the change ofload of input dc voltage. The model of the machine used inDTC is independentof al l machine parameters other than thestator resistance. Therefore the current is notaffected by thevariation of any machine parameter other than the statorresistance. The current vector is determined by the commandtorque and commands, the stator flux. This means, for aconstant f lux and torque command stator current vector isonly dependent on stator resistanceof the machine. Thereforeany change in stator current vector indicates a change in statorresistance[6].To obtain the relation between the stator current andstator resistance, a simulation was run in which the statorresistance was changed as shown in Fig. 2. Fig. 3shows thechange in stator current for this change in stator resistance.

0.245 10 15 2(TimeFig. 2 Changein stator resistance.

I

0.2 0.3 0.4 0.5 0.6 0.7 0.8Stator resi stance

Fig. 3 Graph between themagnitude of stator currentvector and stator resistance

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The value of the stator current vector simulated has beenfiltered toeliminate any ripple. It can beseen from the figurethat the trajectory of the current is not the same for theincrease and decrease in the stator resistance. Thus therelation between thetwovariables is nonlinear.3. CORRECTION OF THE STATOR RESISTANCEAn estimator hasbeen designed toestimate the change instator resistance of the machine during the operation. Theestimator observes the stator current vector, and if achange isdetected, a corresponding changes in the stator resistance ismade. The estimator hasbeen designed based on PI controland fuzzy logic control schemes.a) F uzzyResistance Estimator:Fig.4 shows the resistance estimator based on fuzzylogic control. Stator current vector error and change in statorcurrent vector error are used as inputs to the estimator [6].The magnitudeof the stator current vector is obtained fromthe measured stator currents given by:Is&) =4&-2 . . . . . . . . 0This is filtered and fedtothe fuzzy resistance estimator.

Current vector error and change in current vector error aredehedas:e&) = I~&)*-I ,&) . . . . . . . . (8)Ae&) =e@) e&-1) . . . . . . . . (9)Where I, is the current vector corresponding to the fluxand torque commands and Is is the measured stator currentvector.Each of the two input variables, e@) and Ae&), andoutput variable ARs is divided into five fuzzy segmentsnamely NL , NS, ZE, PS and PL. The range of control (oruniverse of discourse) for stator current vector error is chosenbetween - lA and 1A and a triangular function wasused for themembership distribution. Fig. 5 shows the membershipdistribution of the stator current error. The universe of

distribution for the change in current error. The universeofdiscourse for change instator resistance is chosen between-0.008ohmsand 0.016 ohms and a triangular function isusedfor membership distribution. Fig. 7shows the membershipdistribution for the change in stator resistance, These valuesare chosen for a 3 hp induction machine with values ofcommand torque and command f lux of 11.9 (rated) and0.342Volt-sec respectively.The control rule applied can bedescribed in termsof theinput variables e and Ae and the output variable AR,. The

ithrule Ri: can bewritten asRi: If e is Ai, Ae is Bi then ARs is Ciwhere Ai, Bi and Ci are the fuzzy segments of e, AeandARs respectively.The control rules are formulated using the stator currentresponse of the induction motor for a change in statorresistance as shown in Figs. 2,3 and 8. There are total oftwenty five rules as shown in Fig. 9. The distribution of therules in the table shows symmetry.Product operation rule of fuzzy implication is used asinference method. The firing strengtha1anda2 of first andsecondrules may be expressedas [81:a1=PNLW .P a ( A d

a2=CL * CrNS(WSimilarly the fi ring strength of ith rule isgiven byai =&$e * WAe(Ae)Where ki (e) isthe grade of membership of ei segment ofcurrent error e and PA ei is the grade of membership of Aeisegment of change in current error Ae.

discourse for changeinstator current vector error is chosenbetween -0.4A and 0.4 A and a triangular function isused forthe membership distribution. Fig. 6shows the membership

Fig. 5 M embership distribution of stator current vector error.

.I NS ZE PS PL

Fig. 6 Membership distribution of change in stator currentvector error.

NS ZE ps PL

-0.008 0003 0 0.007 0.016Fig. 7 Membership distribution of change in stator resistance

Fig. 4 DTCwith f uzzy resistance estimator.

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b) PI Resistance Estimator:

m.oJ I5 10 1s 2(TimFig.8 Error in thestator current vector for linearchange in thestator resistance.

FigAccording to the product operation rule, as a fuzzypARnRulei(ARs)=suP(ai wARn'(ARs))Where pARnRulei(ARs) is the grade of membership ofthe control decision A Rn in the ith control rule, and~AR;(AR) s the grade of membershipof the ARn' segment

of the output variableAR in the ith control rule.The membership function p m n of thecontrol decisionARn is pointwise given as:

implication th rule leadsto the control decision [81:

* (10)5j =1pARn = maxPARiRulej(ms) . *The fuzzy output obtained after the inference needstobedefuzzified to get the actual output. The center averagemethod is used for the defuzzification. According to thismethod the output in a discrete universe is given by [81:

.=1 . . . . . (11)25j=l

AR, =pms(A Rsj)

where ~A R ~( A R , ~)s the grade of membership of thefuzzy output segment ARsj given by jth rule and C(ARsj) isvalue of output fuzzy segment in the actual universe ofdiscourse at the maximum membership point. ARs is theactual valueof the change in stator resistance of the motor.

Fig. 10 shows the resistance estimator based on PIcontrol. The error in the stator current vector is used as aninput to the estimator. The equation for the PI resistanceestimator is given by:. . . . . (12)ISARs =KpAI, +Ki-

where Kp and Ki are the proportional and integral gainsof the PI estimator.Gradient decent method was used for tuning theestimator. (See Appendix) [91. An emor was inserted in thestator resistance of the motor and this error was used to tunethe gains of the controller. Fig. 11shows the block diagramof the tuning system. The gains are updated using thefollowing equation:and AK i =C (AR* - AR) P I s dtwhereC is a constant.The gains are adjusted iteratively till their valuesconverge. For the motor specifications given earlier thevalues of the gains obtained after tuning are:

.S

AKp =C (AR* - AR) AIS

Kp =0.00923K i =0.000876

TV

Fig.10 DTC with PI resistance estimatorIrcf

A I S

Fig. 11Block diagramof the tuning setup forthe PI resistance estimator.

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4. SIMULATIONRESULTS 15-hGE 10:.z 5 -3 :H i9