used in the simulation experiment and the one implementedby the ABB converter are naturally different. On the otherhand, the electrical parameters are not constants anymore. Bytuning the system and noise covariance matrices diagonalvalues, the gains in the i<strong>de</strong>ntification algorithms may beadjusted either in or<strong>de</strong>r to track the respective electricalparameters following closer the instantaneous variations ofthe real parameters or, instead, to track the mean values overthe time. Therefore some non-similarities exist between thegenerated signals in Simulink and the ones generated by theABB frequency converter and, consequently, the evolutions ofthe estimated parameters in these two phases of the work area little bit different.(e)(f)(a)(g)(b)(h)(c)(d)Fig. 4. Performance of the proposed i<strong>de</strong>ntification metho<strong>do</strong>logy. (a)Measured electrical rotor speed, (b) estimated magnitu<strong>de</strong> of rotor flux, (c)estimated state x 3 - scaled inverse of rotor time constant, (d) estimated statex 4 - scaled magnetizing inductance, (e) estimated parameter θ 1 - stator scaledtransient inductance, (f) estimated parameter θ 2 - scaled stator resistance, (g)estimated and simulated rotor flux q component and (h) generated statorcurrent q component and the simulated one using the online estimatedparameters.The experimental results, presented in fig. 4, show theperformance of stator parameters estimation by using therecursive prediction error based approach and the rotor fluxcomponents and rotor parameters estimation by using theEKF.In or<strong>de</strong>r to guarantee the validity of the i<strong>de</strong>ntification454
metho<strong>do</strong>logy it is necessary to validate the i<strong>de</strong>ntifiedinduction motor mo<strong>de</strong>l through some validation test toevaluate the performance of the algorithms. The validationtest was based on the simulation of a modified inductionmotor mo<strong>de</strong>l, with the online estimated parameters, byinjection of the measured voltages and angular speed andsubsequent comparison of simulated and measured statorcurrent dq components.As can be seen, in fig. 4(g), the estimated flux, afterconverted into the stator reference frame, and the same one,simulated in the stator reference frame with the parametersestimated by the above-proposed metho<strong>do</strong>logy, are verysimilar. Like the rotor flux, the measured stator current, in thestator reference frame, closely matches the same currentsimulated with the estimated parameters, as shown in fig.4(h).VI. CONCLUSIONThe presented i<strong>de</strong>ntification metho<strong>do</strong>logy, based on theabove-proposed recursive estimators, is proposed for jointonline flux and all electrical parameters estimation of a vectorcontrolled induction motor by means of the EKF for rotorflux and rotor parameters, with in<strong>de</strong>pen<strong>de</strong>nt estimation of thestator parameters by using a recursive prediction error basedapproach, instead of a full parameter and flux estimator witha high or<strong>de</strong>r EKF. It is not restricted to steady-state operation,being capable to operate in transient conditions and thepossibility of enabling or disabling the stator parametersestimation becomes available, taking into account themomentary machine dynamics.The stator and rotor reference frames are used for,respectively, stator parameters and rotor states (flux andparameters) estimation, as being the most natural ones foreach case.The proposed metho<strong>do</strong>logy enables the estimation withthe EKF of rotor flux and parameters, at every iteration, inboth transient and steady-state operation with rough initialseeds for stator parameters. One or both stator parameters canthen be estimated, or updated from time to time, wheneverthe dynamics of the induction motor is suitable for thispurpose. The in<strong>de</strong>pen<strong>de</strong>ncy of the estimators is an advantageand is even mandatory since the normal operating conditionsare not always generating persistent excitation for parameteri<strong>de</strong>ntification. Since stator resistance has slow dynamicswhen compared with the other electrical parameters and canbe measured or even estimated from time to time, then if itsvalue is assumed to be known we can improve even more therobustness of the estimation of the other three parameters.The i<strong>de</strong>ntification metho<strong>do</strong>logy that has just beenpresented in this paper can be applied for auto-tuning andadaptive direct field-oriented induction motor control.ExˆAPPENDIXThe EKF algorithm) ) ))) T{ xe(0)} = xe(0) = [ x(k)θ(k)],E{ ( xe(0) − xe(0))( xe(0) − xe(0))}⎡ f ( xˆ(k | k),u(k),θˆ(k | k)) ⎤ ⎡rs( k)⎤( k + 1| k)=+ = f ( xˆ( k | k),u(k))e∂fF(k)=e∂heH ( k)=∂x⎢⎢⎣( x ( k),u(k))eTe∂x( k))Pˆ(k + 1| k)= F(k)P(k | k)F( x ( k)) ⎡∂h( x(k),θ(k)) ∂h( x(k),θ(k))eTe( k)K(k + 1) = Pˆ(k + 1| k)H)y(k + 1) =)xe( k + 1| k)P(k + 1| kθ(k)xˆe ( k + 1| k))xe( k)= xe( k|k)TT( k)( k)+ R= ⎢⎢⎣⎡∂f= ⎢⎢⎢⎣∂x( k)⎢r ( k)⎥⎣ θ ⎦( x(k),u(k),θ(k)) ∂f( x(k),u(k),θ(k))∂x( k)0T[ H ( k)Pˆ(k + 1| k)H ( k)+ R ]−1mh( xˆe ( k + 1| k),k ))+ 1) = xˆe ( k + 1| k)+ K(k + 1) [ y(k + 1) − y(k + 1) ]+ 1) = [ I − K(k + 1) H ( k)] Pˆ(k + 1| k)L(k + 1) = P(k)ψ(k + 1)) )θ(k + 1) = θ(k)+ L(k + 1))θ=θ(k)P(k + 1) = P(k)− L(k + 1) ψsT⎥⎥⎦eTeTe∂θ ( k)The RPEM algorithm)ψ(k + 1) = ∂y(k,θ)/∂θ , ψ − gradient⎤⎥⎥⎦e)xe( k + 1| k)T[ ψ ( k + 1) P(k)ψ(k + 1) + R ]( y(k + 1) − y(k + 1) )T( k + 1) P(k)+ RACKNOWLEDGMENT)s−1m= P(0)+ r ( k)Tse∂θ ( k)I⎤⎥⎥⎥⎦)xe( k|k)The authors would like to thank the availability of thelaboratorial resources by the Laboratório <strong>de</strong> Electrónica eInstrumentação during the preparation of this paper.REFERENCES[1] T. 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Freitas, “A Real-time Estimator of ElectricalParameters for Vector Controlled Induction Motor using a ReducedOr<strong>de</strong>r Exten<strong>de</strong>d Kalman Filter”, in Proc. EPE-PEMC’02, Cavtat &Dubrovnik, Croatia, September 9-11, 2002, paper T11-029.[6] P. Vas, “Parameter Estimation, Condition Monitoring, and Diagnosis ofElectrical Machines”, Oxford Science Publications, 1993.[7] A. J. L. Harrison, D. P. Stoten, “Generalized Finite Difference Methodsfor Optimal Estimation of Derivatives in real-Time Control Problems”,in Proc. Instn Mech Engrs, vol. 209, pp 67-78, 1995.[8] G. C. Goodwin, K. S. Sin, “Adaptive Filtering Prediction and Control”,Prentice-Hall, 1984.[9] L. Ljung, “System I<strong>de</strong>ntification, Theory for the User”, Prentice Hall,1999.455
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