Speed_sensorless_direct_torque_control_of_IMs_with_rotor_resistance_estimation

Speed sensorless direct torque control of IMswith rotor resistance estimationMurat Barut a ,Seta Bogosyanb,*,Metin GokasanaaDepartment of Electrical and Electronic Engineering,Faculty of Engineering,Istanbul Technical University,Maslak,34390Istanbul,TurkeybDepartment of Electrical and Computer Engineering,University of Alaska,P.O.Box 750145,Fairbanks,AK 997750145,USAReceived 17January 2004;accepted 29April 2004Available online 2July 2004AbstractDirect torque control (DTC)of induction motors (IMs)requires an accurate knowledge on the ampli-tude and angular position of the controlled flux in addition to the information related to angular velocity for velocity control applications.However,unknown load torque and uncertainties related to stator/rotor resistances due to operating conditions constitute major challenges for the performance of such systems.The determination of stator resistance can be performed by measurements,but methods must be developed for estimation and identification of rotor resistance and load torque.In this study,an EKF based solution is sought for determination of the rotor resistance and load torque as well as the above mentioned states required for DTC.The EKF algorithm used in conjunction with the speed sensorless DTC is tested under eleven scenarios comprised of various changes made in the velocity reference beside the load torque and rotor resistance values assigned in the model.With no a priori information in the estimated states and parameters,it has been demonstrated that the EKF estimation and sensorless DTC perform quite well in spite of the uncertainties and variations imposed on the system.Ó2004Elsevier Ltd.All rights reserved.Keywords:Induction motor;Extended Kalman filter;Sensorless direct torque control;Load torque and rotor resistanceestimationEnergy Conversion and Management 46(2005)335–349/locate/enconman*Corresponding author.Tel.:+1-907-475-2755;fax:+1-907-475-5135.E-mail address:s.bogosyan@ (S.Bogosyan).0196-8904/$-see front matter Ó2004Elsevier Ltd.All rights reserved.doi:10.1016/j.enconman.2004.04.002336M.Barut et al./Energy Conversion and Management46(2005)335–3491.IntroductionHigh efficiency control and estimation techniques related to induction motors(IMs)have been finding more and more application with Blaschke’s well-knownfield oriented control(FOC) method,established in1971.There has been an extensive amount of work to improve the dynamic response and reduce the complexity of FOC methods.One such technique is the direct torque control(DTC)method developed by Takahashi in1984[1],which has been getting increased attention due to the improved dynamic performance and simplified control strategy that it offers with respect to the FOC methods.The DTC method involves the direct choice of the appropriate/optimum switching modes,in order to keep theflux and torque errors within a predetermined band limit(in a hysteresis band) [2].The errors are defined as the difference between the reference and the measured/estimated values offlux and torque.Unlike FOC methods,DTC techniques require utilization of hysteresis band comparators instead offlux and torque controllers.To replace the coordinate transfor-mations and pulse width modulation(PWM)signal generators of FOC,DTC uses look-up tables to select the switching procedure based on the inverter states.However,both methods require accurate knowledge of the amplitude and angular position of the controlledflux(with respect to the stationary stator axis)in addition to the angular velocity for velocity control applications. As is well known,speed sensors like tachometers or incremental encoders increase the size and cost of systems unnecessarily.Similar problems arise with the addition of search coils or Hall effect sensors to the motor for measurement of theflux,hindering functionality in terms of implementation.Thus,to improve overall system performance,state estimators or observers are usually more preferable than physical measurements.However,thefifth order and nonlinear structure of the IM model[3],in addition to the sensitivity of the system parameters to tem-perature[4]and frequency[5],makes the design of observers for IMs a challenge.In this regard, for high performance sensorless vector control of IMs,it is essential to know the temperature and frequency dependent variation of the stator and rotor resistances in addition to the load torque.In a study addressing this issue[6],it has been stated that simultaneous estimation of the stator and rotor resistances gives rise to instability.Moreover,while the value of the stator resistance could be obtained by measuring the stator temperature,there are physical difficulties in determining the rotor resistance in a squirrel cage IM.Thus,estimating the rotor resistance and the load torque appears to be a reasonable approach.In DTC,theflux is conventionally obtained from the stator voltage model,using the measured stator voltages and currents.This method,utilizing open loop pure integration suffers from the well known problems of integration effects in digital systems,especially in the low speed operation range[7],even with the correct knowledge of the stator resistance.Moreover,it will require the rotor angular velocity for velocity control applications.Among the recent studies conducting simultaneousflux and velocity estimation for DTC,Ref.[8]studied the sensitivity to parameter variations with an artificial neural network approach,and a robust performance to50%variations in the stator resistance has been obtained in Ref.[9]with a sliding mode approach,while the adaptiveflux observer in Ref.[10],the extended Luenberger observer in Ref.[11]and the non-linear observer in Ref.[12]demonstrate robustness to step shaped load torque variations.Among studies using model reference adaptive laws,in Ref.[13],theflux and speed have been estimated, but the system response to load torque variations was not tested.In Ref.[14],the rotor velocity,x m,stator resistance,R s and rotor resistance R0r are individually estimated,and good results areobtained.However,in one of the trials where x m together with R0r and in another where x m,R0rand R s are estimated together,it has been stated that the resistances converge to inaccurate values. Moreover,no tests have been performed to test the effects of the load torque variations. Finally,in Ref.[6],the angular velocity and slip frequency,x r(reflecting the effect of the load torque)in addition to the rotor resistance have been taken into account starting with the initialvalue of R0r ð0Þ¼0:85R0r n.There are also extended Kalmanfilter(EKF)applications in the literature,taking a stochastic approach to solution of the problem.Unlike the other methods,the model uncertainties and nonlinearities inherent in IMs are well suited to the stochastic nature of EKFs[15].With this method,it is possible to make an online estimation of states while simultaneously performing identification of parameters in a relatively short time interval[16–18],also while taking system/ process and measurement noises directly into account.This is the reason why the EKF has found wide application in sensorless control of IM’s,in spite of its computational complexity.In the EKF based previous DTC studies,Ref.[19]estimates the statorflux components and velocity under the assumption of a known load,while in Ref.[20],the velocity is estimated as a constant parameter,avoiding the use of the equation of motion.In spite of an improved performance in the steady state,this approach has given rise to a significant observer error in the velocity during the transient state.The major contribution of this study is the development of an EKF based speed sensorless DTC system that achieves robustness to variations in rotor resistance and load torque,the uncertainties that are known to deteriorate system performance.It is thefirst known study to perform the estimation of load torque and rotor resistance simultaneously while also estimating the statorflux components,angular velocity and stator current components,also measured as output.The performance of the estimation and control schemes is tested with challenging vari-ations of the load torque,rotor resistance and velocity reference.The consideration of the load torque as a constant term in the estimation algorithm aims to capture other uncertainties besides the load torque that have a very slow or almost constant variation with time,i.e.viscous and Coulomb friction(in steady state).The results obtained through simulations under various challenging tests demonstrate the good performance of the estimation scheme requiring no a priori information on the states with their initial values taken as zero.2.Extended mathematical model of the IMThe sensorless DTC scheme developed for an IM requires estimation of the statorflux com-ponents,ws a ,ws b,angular velocity,x m and stator current components i s a and i s b,which are alsomeasured as output.In this study,due to the degrading effect of their unknown variations oncontrol performance,the load torque,t L and the rotor resistance,R0r (as referred to the stator side)are also included in the extended state vector as constant states based on their slow variation in time.Thus,the so-called extended model can be obtained(as referred to the stator stationary frame)in the following form:_x e ðtÞ¼feðx eðtÞ;u eðtÞÞþw1ðtÞ¼A eðx eðtÞÞx eðtÞþB e u eðtÞþw1ðtÞð1ÞM.Barut et al./Energy Conversion and Management46(2005)335–349337Here,the extended state vector x e ,representing the estimated states and parameters,consists of i s a ,i s b ,w s a ,w s b ,x m ,t L and R 0r ;f e is a nonlinear function of the states and inputs;A e is the system matrix;u e is the control input vector;B e is the input matrix;and w 1is process noise.The constant state representing the load torque is also designed to capture system uncertainties of constant nature once steady state is attained.In this study,those uncertainties are limited to viscous friction,as this was the only uncertainty included in the model simulating the system.With the above consideration,the extended model of an IM can be given as_i s a _i s b _w s a _w s b _x m _t L _R 0r 2666666666437777777775|fflfflffl{zfflfflffl}_xe ¼ÀR s r þR 0r L s 0r r Àp p x m R 0r0r r p p x m r 000p p x m ÀR s L r þR 0r L s L 0r L r Àp p x m L r R 0r L 0r L r 000ÀR s 0000000ÀR s 00000À3p p L w s b 3p p L w s a 000À1L 000000000000000266666666664377777777775|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}A ei s a i s b w s a w s b x m t L R 0r 26666666643777777775|fflfflffl{zfflfflffl}x e þ1r 001r 100100000026666666643777777775|fflfflfflfflffl{zfflfflfflfflffl}Bev s a v s b |fflffl{zfflffl}u e þw 1ðt Þð2ÞZ ðt Þ¼h e ðx e ðt ÞÞþw 2ðt Þðmeasurement equation Þ¼H e x e ðt Þþw 2ðt Þ¼10000000100000 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}H ei s a i s b w s a w s b x m t L R 0r26666666643777777775þw 2ðt Þð3Þwhere h e is a function of the outputs;H e is the measurement matrix;w 2is measurement noise;p p is the number of pole pairs;L r ¼r L s is the stator transient inductance;r is leakage or coupling factor;L s and R s are the stator inductance and resistance,respectively;L 0r and R 0r are the rotor inductance and resistance,referred to the stator side,respectively;v s a and v s b are the stator sta-tionary axis components of the stator voltages;w s a and w s b are the stator stationary axis com-ponents of stator flux;and x m is the angular velocity.3.Development of the EKF algorithmAn EKF algorithm is developed for estimation of the states in the extended IM model given in Eqs.(2)and (3)to be used in the sensorless direct torque control of the IM.The Kalman filter (KF)is a well known recursive algorithm that takes the stochastic state space model of the system into account,together with measured outputs,to achieve the optimal estimation of states [21]in multi-input,multi-output systems.The system and measurement noises are considered to be in the338M.Barut et al./Energy Conversion and Management 46(2005)335–349form of white noise.Optimality of the state estimation is achieved with minimization of the covariance of the estimation error.For nonlinear problems,the KF is not strictly applicable since linearity plays an important role in its derivation and performance as an optimalfilter.The EKF attempts to overcome this difficulty by using a linearized approximation where the linearization is performed about the current state estimate[22].This process requires the discretization of Eqs.(2) and(3)x e ðkþ1Þ¼feðx eðkÞ;u eðkÞÞþw1ðkÞð4ÞZðkÞ¼He xeðkÞþw2ðkÞð5ÞAs mentioned before,the EKF involves the linearized approximation of the nonlinear model(Eqs.4and5)and uses the current estimation of states^xe ðkÞand inputs^u eðkÞin the linearizationby usingF e ðkÞ¼o feðx eðkÞ;u eðkÞÞo x eðkÞ^xeðkÞ;^ueðkÞð6ÞF u ðkÞ¼o feðx eðkÞ;u eðkÞÞo u eðkÞ^xeðkÞ;^ueðkÞð7ÞThus,the EKF algorithm can be given in the following recursive relations:NðkÞ¼FeðkÞPðkÞF eðkÞTþF uðkÞD u F uðkÞTþQð8aÞPðkþ1Þ¼NðkÞÀNðkÞH Te ðD nþH e NðkÞH TeÞÀ1H e NðkÞð8bÞ^x e ðkþ1Þ¼^feðx eðkÞ;^u eðkÞÞþPðkþ1ÞH TeDÀ1nðZðkÞÀH e^x eðkÞÞð8cÞHere,Q is the covariance matrix of the system noise,namely model error;D n is the covariance matrix of the output noise,namely measurement noise;D u is the covariance matrix of the control input noise(v s a and v s b),namely input noise;and P and N are the covariance matrices of state estimation error and extrapolation error,respectively.The algorithm involves two main stages:prediction andfiltering.In the prediction stage,thenext predicted states^fe ðÁÞand predicted state error covariance matrices,PðÁÞand NðÁÞ,are pro-cessed,while in thefiltering stage,the next estimated states,^xe ðkþ1Þ,obtained as the sum of thenext predicted states and the correction term(2nd)term in Eq.(8c))are calculated.The schematic representation of the algorithm is given in Fig.1.The algorithm utilizes the extended or augmented model in Eqs.(2)and(3)to generate all output states required by the sensorless direct torque control scheme,in addition to the rotor resistance and the load torque,using measured phase currents and voltages.M.Barut et al./Energy Conversion and Management46(2005)335–349339340M.Barut et al./Energy Conversion and Management46(2005)335–3494.Speed sensorless DTC systemFig.2demonstrates the speed sensorless DTC system.Here,^h rf stands for the sector position of theflux with reference to the stationary axis.The velocity controller given in the diagram is a conventional proportional integral derivative(PID)controller.The development of the sector selector and the switching table is based on Takahashi’s study presented in Ref.[1].5.Simulation results and observationsTo test the performance of the estimation method,simulations were performed on an IM with the rated parameters given in Table1.The values of the system parameters and covariance matrix elements are very affective on the performance of the EKF estimation.In this study,to avoid computational complexity,the covariance matrix of the system noise Q is chosen in diagonal form,also satisfying the condition of positive definiteness.According to the KF theory,Q,D n(measurement error covariance ma-trix)and D u(input error covariance matrix)have to be obtained by considering the stochastic properties of the corresponding noises[7].However,since these are usually not known,in most cases,the covariance matrix elements are used as weighting factors or tuning parameters.In this study,tuning the initial values of P and Q is done by trial and error to achieve a rapid initial convergence and the desired transient and steady state behaviors of the estimated states andparameters,while D n and D u are determined taking into account the measurement errors of the current and voltage sensors and the quantization errors of the ADCs,as given below:Q ¼diag f 10À6½A 2 10À6½A 2 10À6½Wb 2 10À6½Wb 2 10À4½ðrad =s Þ2 10À5½ðNm Þ2 10À7½X 2 g P ¼diag f 9½A 29½A 29½Wb 2 9½Wb 2 9½ðrad =s Þ29½ðNm Þ29½X 2 gD n ¼diag f 10À6½A 2 10À6½A 2 g D u ¼diag f 10À3½V 210À3½V 2 gand sampling time T ¼100l s :The bandwidth ðb w Þof the flux comparator is chosen as 0.02[Wb],while that of the torque comparator ðb t e Þis 0.01[Nm].Table 1The nominal values and parameters of the induction motor used in the tests P [KW]f [Hz]J L[kg m 2]B L [Nm/(rad/s)]p pV [V]I [A]R s [X ]R 0r [X ]L s [H]L r [H]L m [H]N m[rpm]T e[Nm]3500.0060.0012380 6.9 2.283 2.1330.23110.23110.22143020M.Barut et al./Energy Conversion and Management 46(2005)335–349341Eleven different scenarios are created to test the performance of the estimation and control algorithm in the time interval of06t613s.Thefirst10scenarios are developed with simultaneous changes of the velocity reference (Fig.3a)and the load torque value(Fig.3b)used in the extended model.The last scenario(scenario11)is created by giving R0r in the model a step change to twice itsoriginal value,R0r ¼2R0r n.The estimation of all the states and parameters is started with an initial value of zero.The resulting system performance for all scenarios is given with Fig.4a representing thevelocity estimate,^n m,Fig.4b depicting the velocity error,ðn refm À^n mÞ,and Fig.4c giving theestimation error,n mÀ^n m.The variations of the applied and estimated load torque are given in Fig.5a,with Fig.5b representing the estimation error,ðt LÀ^t LÞ,for this variable.The variationsrelated to the rotor resistance,R0r ,are given in Fig.6a and b,with the former plot representing theactual and estimated variation of R0r with the initial value of the estimate taken as zero,while thelatter plot represents the estimation error,R0r À^R0r.Finally,Fig.7a–c represent the estimatedfluxmagnitude,j^ws j,the error between the reference and actual(estimated)flux magnitude,j w s j refÀj^w s j,and theflux estimation error,ðj w s jÀj^w s jÞ,respectively.Fig.8shows the trajectoryof ws a and ws b.342M.Barut et al./Energy Conversion and Management46(2005)335–349M.Barut et al./Energy Conversion and Management46(2005)335–3493435.1.Observations5.1.1.Operation under constant t L and constant/linear velocity referencesIn intervals where the load in the model is given a constant value(20Nm),the esti-mation and control algorithms perform very well under both step type and linear variations of the velocity reference.With no a priori information on the load torque(and an initial value of0Nm),the EKF algorithm and the control achieve a low velocity error,ranging between0.011%and0.031%,ascan be seen in the time intervals 06t 61s,1s 6t 62s,2s 6t 63s,3s 6t 64s,4s 6t 65s,7s 6t 68s,8s 6t 69s,10s 6t 611s and so on.In the first,third and fourth of the above intervals,a velocity reference with a linear variation in time has been applied to the system,while in the rest of the intervals,a constant velocity reference has been used.However,independent of the velocity reference,a very good performance has been obtained,mainly due to the consistence between the applied and assumed load torque in the model.5.1.2.Operation under linear t L and constant/linear velocity referencesIn the time interval 5s 6t 7s,t L is given a variation ðt L ¼À20þ20ðt À5ÞÞ,while n ref m isvaried as n refm ¼À1500þ1520ðt À5Þbetween 5s 6t 66s,during which the velocity error,e nm ð%Þ¼1:412120Â100¼7:06%and estimation error ^e nm ð%Þ¼0:9620Â100¼4:8%.As noted before,the increased errors in the velocity output and estimate are due to the inconsistency between the t L in the EKF model (which is constant)and the imposed variation of t L (linear)in the model representing the plant for simulation purposes.5.1.3.Operation in the low velocity region,with no load referencesIn the interval 8s 6t 69s,both the velocity reference and t L are made zero,giving rise to avelocity error of e nm ¼0:0308rpm in steady state and an estimation error of ^e nm ¼À0:5618rpm,344M.Barut et al./Energy Conversion and Management 46(2005)335–349which are acceptable results considering the challenge posed by the velocity region and change in t L.5.1.4.Operation with linear velocity reference and linear t LIn the interval9s6t610s,the variations n refm ¼1500ðtÀ9Þand t L¼20ðtÀ9Þgive rise to avelocity error of e nmð%Þ¼1:71500Â100¼0:11%and an estimation error that varies between^e nmð%Þ¼0:56Â100¼0:037%and^e nmð%Þ¼16:06Â100¼1:07%.Once again,relatively higher but still acceptable errors are caused mainly by the linear variation of t L.5.1.5.Operation under reversal of velocity referenceIn the interval2s6t64s,the velocity reference is reversed from1500rpm to)1500rpm witha linear variation of n refm ¼1500ðtÀ2ÞÀ1500ðtÀ2Þ.During this interval,t L is also given a var-iation of t L¼20sgnðn mÞ.After a brief transient while the velocity and torque pass through zero,velocity errors of e nmð%Þ¼0:39231500Â100¼0:026%and e nmð%Þ¼1:38251500Â100¼0:0923%occur att¼2and4s,respectively.The velocity estimation errors for the same instants are^e nmð%Þ¼0:04041500Â100¼0:0027%and^e nmð%Þ¼0:12961500Â100¼0:0086%,respectively.Consideringthe error,it can be noted that the system has responded quite well to the simultaneous reversal ofn ref m and t L.M.Barut et al./Energy Conversion and Management46(2005)335–3493455.1.6.Operation under R0r ¼2R0r n(constant velocity reference/load torque)As mentioned before,another challenge for the control of an IM system is the uncertaintiesrelated to R0r .The robustness of the performance of the estimation algorithm to variations is testedby increasing the value of R0r to twice the value assigned in the model,in the interval11s6t612s.The response of the system is noted to be quite satisfactory with a velocity error ofe nmð%Þ¼1:44421500Â100¼0:096%and velocity estimation error of^e nmð%Þ¼0:01221500Â100¼0:0081%after a brief transient state.Once again,the largest estimation error in R0r takes place in the346M.Barut et al./Energy Conversion and Management46(2005)335–349interval5s6t67s where the load torque is given a linear variation,while in all other intervals, very small errors are obtained in the transient and steady-state.5.1.7.Uncertainties captured in constant t LAs mentioned in Sections1and2,the EKF scheme also facilitates the indirect evaluation of uncertainties that have the same variation as a state or parameter that is being estimated.Con-sidering this study,in which viscous frictionðF v¼b L x mÞis taken into account in the model representing the system but not in the extended model,the estimate of t L as a constant state also should include the viscous friction value once steady-state is reached.This fact can be demon-strated easily with the calculations below.In the intervals1s6t62s and10s6t613s,during which both the velocity reference andload torque are given positive constant values,the error in the torque estimation is e tL ¼À0:571,and in the interval4s6t65s,where both the velocity reference and torque are given negativevalues,the error is found to be e tL ¼0:571.The angular velocity in all these intervals ise nm¼1500Æ0:2rpm.Thus,for the value of the viscous friction coefficient b L¼0:001used in the modelx mð1Þ¼2pÂn m=60¼2pð1500:2½rpm þ7:4Â10À5½rpm Þ=60¼157:1006½rad=sF v¼B L x mð1ÞF v¼0:001Â157:1006¼0:1571006½Nmwhich is equal to e tL ,as expected.This fact should also be taken into consideration in evaluation of the load torque estimation. By inspecting the t L estimate,it can be observed that although with the linear variations and reversals of t L,some estimation error is caused in a relatively short transient duration,in the intervals with constant velocity reference and constant t L,this error is much lower,with the subtraction of the F v,from t L,yielding an error approximately equal to zero.The applied algorithm has also kept the variation offlux magnitude in all intervals within the admissible hysteresis band.Thus,with consideration of all the results,it can be observed that the expected performance is attained.6.ConclusionIn this study,an extended Kalmanfilter(EKF)algorithm is developed for the speed sensorless direct torque control(DTC)of induction motors.DTC requires accurate knowledge of the amplitude and angular position of the controlledflux(with respect to the stationary stator axis)in addition to the angular velocity for the purpose of velocity control.The major contribution of this study is the increased robustness towards uncertainties in the rotor resistance and load torque,the effects of which are known to give rise to performance deteriorations in such systems.This is achieved by an EKF algorithm that performs simulta-neous estimation of the rotor resistance and load torque as well as the statorflux components and the angular velocity.The performance of the algorithm is tested with11scenarios devel-oped by giving step type and linear variations to the load torque and angular velocity reference,while robustness to rotor resistance,R0r ,variation is tested with step type changes imposed onR0 r .The system performance is observed to be quite good under step type variations and reversals inthe load–torque and step/linear changes and reversals in the angular velocity.The system has alsodemonstrated the expected robustness to step type variations forced on the R0r ,and acceptableerrors are obtained even with the linear variations and reversals of the load torque.The estimation of the load torque estimate,t L,as a constant state in this algorithm also accounts for the viscous friction in this case,thereby improving the estimation performance.AcknowledgementsThis work was supported in part by the Istanbul Technical University Research Foundation.References[1]Takahashi I,Noguchi T.A mew quick-response and high-efficiency control strategy of an induction motor.IEEETrans Ind Appl1986;IA-22(5):820–7.[2]Casadei D,Profumo F,Tani A.FOC and DTC:two viable schemes for induction motors torque control.IEEETrans Power Electron2002;17(5):779–87.348M.Barut et al./Energy Conversion and Management46(2005)335–349。