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estimated values of stator parameters and then, a RPEM isused to estimate the parameter vector θ ) , which corresponds'to the scaled parameters L s (or L s ) and R s . The state vector xand the parameter θ are scaled by constants K i ’s, fornumerical reasons, as follows:) ) ) ) ) )x(k,θ−) ==[ x ( k)x ( k)x ( k)x ( k)]1rr−1[ k ψ ( k)k ψ ( k)k τ ( k)k L ( k)]1t krd12rq2) )'θ(k,xt k) = k5Ls( k)or) )θ( k,x ) k6R( k)ort k=3s'[ k L ( k)k R ( )]) )θ( k,xt k) = s 635 s k44M=(3)(4a)(4b). (4c)For the EKF algorithm, the model structure is formed by anonlinear discrete state space 4 th -order model, obtained bydiscretization of state equation in (1), and the output equationin (2), as presented in [5] and results as follows, where T s isthe sampling period:⎡r⎤ ⎡−1ψrd( k + 1) 1−Tsτr0⎢ r ⎥ = ⎢⎢⎣ψrq(k + 1) ⎥⎦⎢⎣0 1−Tsτ⎡−1τ 0 ⎤ ⎡rT( ) ⎤sLMrisdk+ ⎢−1⎥⎢ r ⎥⎢⎣0 TsLMτr⎥⎦⎢⎣isq(k)⎥⎦ursd= −τ)( k)− R i−1rψrrdrs sd)'( k)− Ls( k)− ω(k)ψ−1r⎤ ⎡ψ⎥ ⎢⎥⎦⎢⎣ψrrdrrq( k)⎤⎥ +( k)⎥⎦( i&rr( t ) − ω(k)i ( k))sdrrqk( k)+ Lsq−1rM τrisd=( k)As we can see, the state equation in (5) is independent ofrotor speed and stator parameters. Moreover, the matrices arediagonal and, therefore, it becomes very simple to get higherorder approximation in the discretization process. The rotorspeed and stator parameters only appear in the outputequation in (6) where the measured output is a function of thestator parameters which are estimated by using the modelstructure described bellow.For the recursive prediction error based method, a linearregression model structure is used, which is derived from thestator voltage equation, expressed in the stator referenceframe, as follows:ussd(5)(6)s ' ss( k)= R i ( k)+ L ( k)i& ( t ) + ψ) & ( t ) . (7)s sdObviously, instead of the d components in (7), the q onescould be selected. The equation above can then be rewrittenas a general linear regression as follows:)y( k,x)= θ(k)u(k)(8)ssdkrdkwhere, for the case of full estimation, like in (4c), we have:andssy k,x) ) = u ( k)− ψ) & ( t )(9)( sd rd ks ' sθ ( k ) u(k)= R i ( k)+ L ( k)i&( t ) . (10)s sdAs we can verify the equations (9) and (10), described asa linear regression like in (8), do not depend on neither therotor parameters nor the rotor speed. However, they dependon rotor flux d (or q) component’s first derivative. The firstderivatives in (9) and (10) are computed by the followinggeneral recursive filter that can be found in [7], in order toobtain better results than by using Euler’s formula,n∑ − 1s i=0s1x& = dx / dt ≈ C ( − )t=ti x tkiTs(11)k TThe weights C i can also be found in [7] and aredetermined after Taylor series expansion of the equationabove to m+1 terms, with m = { 1,2, L,n}, m being the orderof the filter and n the number of points. For systemidentification purposes, as in this case, an important aspect totake into account is the delay introduced by the filter. Thedelay must be the same as the one introduced by the Euler’sformula that is implicit in the discretization process of sateequation in (1) if the linear terms of the Taylor’s developmentare adopted as in (5). For sampling frequencies below 5kHz,the set of coefficients [ 11 − 18 9 − 2] / 6 , has producedthe best results. This is an unusual discretization process.Indeed, this strategy presented by the authors in [4, 5] hasbeen proven to give better results.This methodology is similar to an adaptive state estimatorfor nonlinear systems described, in general terms, in [8] andapplied here for joint state and parameter estimation. The setupin fig. 1 is a very natural and simple way to achieve thisobjective. Beyond the advantage of separating the estimationof τ r , L M and the rotor flux components from the statorparameter, by adapting the estimator to the machine operatingpoint or, in other words, to the information contained in themeasured signals, it permits to overcome many of thedisadvantages associated with the EKF, namely, a strongcomputational effort, eventually biased estimates, and notguaranteed convergence [8]. Furthermore, this can be a goodalternative to the extended Luenberger observer suggested in[1] to solve the steady-state bias problem detected in the jointrotor flux and rotor time constant estimation, and analternative to [2, 3], in terms of computational effort.As far as the induction motor is concerned the mainadvantage of this methodology is that it enables thesimultaneous estimation of rotor flux components and rotorparameters by using the EKF, and the estimation of statorparameters by using a RPEM based approach for jointestimation of stator parameters or even two RPEMs forindependent estimation of these parameters.sdk450