research article parameter estimation of permanent magnet synchronous...
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Research ArticleParameter Estimation of Permanent MagnetSynchronous Motor Using Orthogonal Projection andRecursive Least Squares Combinatorial Algorithm
Iman Yousefi and Mahmood Ghanbari
Department of Electrical Engineering Aliabad Katoul Branch Islamic Azad University Aliabad Katoul Iran
Correspondence should be addressed to Mahmood Ghanbari ghanbarialiabadiauacir
Received 7 January 2015 Revised 23 June 2015 Accepted 25 June 2015
Academic Editor Tomasz Kapitaniak
Copyright copy 2015 I Yousefi and M Ghanbari This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
This paper presents parameter estimation of Permanent Magnet Synchronous Motor (PMSM) using a combinatorial algorithmNonlinear fourth-order space state model of PMSM is selected This model is rewritten to the linear regression form withoutlinearization Noise is imposed to the system in order to provide a real condition and then combinatorial Orthogonal ProjectionAlgorithm and Recursive Least Squares (OPAampRLS) method is applied in the linear regression form to the system Results of thismethod are compared to the Orthogonal Projection Algorithm (OPA) and Recursive Least Squares (RLS) methods to validate thefeasibility of the proposed method Simulation results validate the efficacy of the proposed algorithm
1 Introduction
Permanent Magnet Synchronous Motor (PMSM) is a goodchoice to use in robotics industries electric vehicles petro-chemical industries sea and so forth due to large torqueto inertia ratio and high power density [1ndash5] and is aserious opponent to the induction motor [6 7] Regardingwidespread application of such motors design of high speedand high accuracy controllers is of great concern for engi-neers [8] The selected model and knowing the parameters ofthe system are important factors in design of such controllersThe pertaining model has to be accurate enough to definethe physical process of the system [4] PMSM is a nonlinearsystem If the linearization in the operating point by choosingimproper nonlinear model does not provide appropriatedynamics the speed and response speed of the controllerwould be slowed down [9] Sensitivity of the machinersquosparameter depends on various factors such as high temper-ature mechanical vibrations loading condition increase inthe service time of the motor and environmental factorsthat would cause parametric uncertainty [3 8 10] and hasconsiderable effect on the static and dynamic performanceof the system [11] Robustness to parametric uncertainty has
to be imposed by correct estimation of PMSM parametersGenerally there are two kinds of estimation to find outunknown parameters online and offline estimation Offlineestimation is carried out whenmachine is not performing butonline estimation is carried out when machine is operatingand in steady-state [12] Online estimation is a very importantnecessity for such systems Extended Kalman Filter Model-Reference method Recursive Least Squares method neuralnetworks adaptive algorithms and decoupling control algo-rithms are of the online methods to estimate the parametersof PMSM [2 10 13ndash19]
In this paper we have presented combinatorial estimationmethods that is Orthogonal Projection Algorithm (OPA)and Recursive Least Squares (RLS) OPA is presented wellin [20] to estimate the parameters of PMSM which is veryefficient for noise-free systemsThe combinatorial OPAampRLSmethod would be a good candidate for systems with highnoise which are more like real systems This method canbe very helpful for online estimation In this paper anonlinear fourth-order PMSM model is chosen and loadangle is selected as accessible output and voltages V119889 V119902 areselected as input (framed voltages of Park transformation)This nonlinear model is rewritten to linear regression form
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 418207 7 pageshttpdxdoiorg1011552015418207
2 Mathematical Problems in Engineering
without linearization or simplification Both estimators canbe applied independently but the combinatorial method hasthe features of both methods OPA leaps quickly toward theaim parameter with each estimation it does and after that RLSadapts the estimated parameter to the real parameter due toits robustness toward noise
Section 2 describes state space model transformation ofPMSM In Section 3 the proposed estimator is presentedSimulation results are illustrated in Section 4 and finally thepaper is concluded in Section 5
2 Transforming the Nonlinear Model toLinear Regression Form
To achieve high performance for PMSM the fourth-orderstate space model of the system is selected under Parktransformation [21 22]
= 119860119909+119861119906+119891 (119909)
119909 = [1199091 1199092 1199093 1199094]119879= [
120575 120596119890 119894119902 119894119889]119879
119906 = [1199061 1199062]119879= [
V119902 V119889]119879
119860 =
[
[
[
[
[
[
[
[
[
[
[
0 1 0 0
0 015119901120582119891119869
0
0 minus
minus120582119891119901
119871119902
minus
minus119877
119871119902
0
0 0 0 minus
119877
119871119889
]
]
]
]
]
]
]
]
]
]
]
119861 =
[
[
[
[
[
[
[
[
[
0 00 01119871119902
0
0 1119871119889
]
]
]
]
]
]
]
]
]
119891 =
[
[
[
[
[
[
[
[
[
[
[
[
0
15119901 (119871119889 minus 119871119902) 119894119902119894119889119869
minus
119871119889119901120596119890119894119889
119871119902
119871119902119901120596119890119894119902
119871119889
]
]
]
]
]
]
]
]
]
]
]
]
(1)
System inputs are V119889 V119902 Park transformation voltages 119894119902 119894119889120596119890 and 120575 state variables of the systemwhich are rotor angularposition rotor angular speed and Park transformation cur-rents respectively
Physical parameters of the system are 119877 stator resistance119901 pole-pairs 120582119891 magnet flux linkage 119869 inertia coefficientand 119871119889 119871119902 direct and quadrature inductances of Parktransformation respectively
The aim of this section is to transform the nonlinearstate space model of PMSM to the linear regression form
without linearization or simplification techniques As thelinear regression form is expressed in terms of an output thepertaining output is rewritten in terms of state variables Theselected output is power angle (120575 = 1199091)
Equation (3) is obtained by considering 119894119889 119894119902 according to(2) [23] and discretization based on ( = [119909(119905+119879119904)minus119909(119905)]119879119904)in which 119879119904 is the sampling time Consider
119894119889 = 119894119898 cos (120575) = 119894119898 cos (1199091)
119894119902 = 119894119898 sin (120575) = 119894119898 sin (1199091) (2)
1199091 [119896 +119879119904] = 1199091 [119896] +1198791199041199092 [119896]
1199092 [119896 +119879119904] = 1199092 [119896] +119879119904 (15119875120582119891119869
)1199093 [119896]
+119879119904(
15 (119871119889 minus 119871119902)119869
)1199093 [119896] 1199094 [119896]
1199093 [119896 +119879119904] = 119879119904 (minus120582119891119875
119871119902
)1199092 [119896]
+119879119904 (
minus119877
119871119902
) 119894119898sin (1199091 [119896]) +119879119904
119871119902
1199061 [119896]
+119879119904 (minus119901
119871119889
119871119902
) 119894119898cos (1199091 [119896]) 1199092 [119896]
+ 119894119898sin (1199091 [119896])
1199094 [119896 +119879119904] = 119894119898cos (1199091 [119896]) +119879119904 (minus119877
119871119889
) 119894119898cos (1199091 [119896])
+
119879119904
119871119889
1199062
+119879119904 (119875
119871119902
119871119889
) 119894119898sin (1199091 [119896]) 1199092 [119896]
(3)
Rewriting (3) to the linear regression form is needed in away that 1199091 is written in terms of all state variables Afterapplying estimator on linear regression form and estimationof the parameters complicated nonlinear equations have tobe solved to obtain physical parameters (119877 119901 120582119891 119869 119871119902 and119871119889) To solve this problem two linear regression forms arerequired by a heuristic method and defining new variables in(3) and (2)Therefore physical parameters can be solved fromlinear regression without solving nonlinear equation
21 Model Number 1 The second state variable of (3) can berewritten by using (2) as below
1199092 [119896 +119879119904]
= 1199092 [119896] +119879119904 (15119875120582119891119869
)1199093 [119896]
+119879119904(
15 (119871119889 minus 119871119902)119869
)
119894
2119898
2sin 2 (1199091 [119896])
(4)
Mathematical Problems in Engineering 3
By shifting forward in (4) and putting 1199093[119896 + 1] in it from (3)and substituting 119879119904 = 1
1199092 [119896 + 2]
= 1199092 [119896 + 1]
+(
15 (119871119889 minus 119871119902) 1198942119898
2119869) sin 2 (1199091 [119896 + 1])
+(
15119875120582119891119869
)(
minus120582119891119875
119871119902
)1199092 [119896]
+(
15119875120582119891119869
)(
minus119877
119871119902
) sin (1199091 [119896])
+(
15119875120582119891119871119902119869
) 1199061 [119896]
+(
15119875120582119891119869
)(minus119901
119871119889
119871119902
) cos (1199091 [119896]) 1199092 [119896]
+(
15119875120582119891119894119898119869
) sin (1199091 [119905])
(5)
Equation (5) is rewritten in terms of load angle by using therelation between the first and second state variables
1199091 [119896 + 3] minus 1199091 [119896 + 2] = 1199091 [119896 + 2] minus 1199091 [119896 + 1]
+(
15119875120582119891119869
)(
minus120582119891119875
119871119902
) (1199091 [119896 + 1] minus 1199091 [119896])
+(
15119875120582119891119869
)(
minus119877
119871119902
) sin (1199091 [119896]) +(15119875120582119891119871119902119869
)
sdot 1199061 [119896] +(15119875120582119891119869
)(minus119901
119871119889
119871119902
) cos (1199091 [119896])
sdot (1199091 [119896 + 1] minus 1199091 [119896]) +(15119875120582119891119894119898
119869
) sin (1199091 [119905])
+(
15 (119871119889 minus 119871119902) 1198942119898
2119869) sin 2 (1199091 [119896 + 1])
(6)
The above equation is rewritten to the below form by definingnew variables
1199101 [119896] = 119860111991011 +119860211991012 +119860311991013 +119860411991014 +119860511991015 (7)
1198601 = (15119875120582119891119869
)(
minus120582119891119875
119871119902
)
1198602 = (15119875120582119891119869
)(
minus119877
119871119902
)+(
15119875120582119891119894119898119869
)
1198603 = (15119875120582119891119871119902119869
)
1198604 = (15119875120582119891119869
)(minus119901
119871119889
119871119902
)
1198605 = (15 (119871119889 minus 119871119902) 119894
2119898
2119869)
11991011 = (1199091 [119896 + 1] minus 1199091 [119896])
11991012 = sin (1199091 [119896])
11991013 = 1199061 [119896]
11991014 = cos (1199091 [119896]) (1199091 [119896 + 1] minus 1199091 [119896])
11991015 = sin 2 (1199091 [119896 + 1])
1199101 [119896] = 1199091 [119896 + 3] minus 21199091 [119896 + 2] + 1199091 [119896 + 1] (8)
Finally a nonlinear model of PMSM is written to commonlinear regression form considering (7) without any lineariza-tion in which 1198601 1198607 are factors that are obtained frommultiplying physical parameters of the system and 11991011 11991015are variables that are composed of nonlinear functions inputand output
Equation (7) can be written to the linear regressionmatrix
In the following equation 1206011 is estimator vector for thefirst regression form and 1205791 parameter vector related to thisform
1199101 [119896] = 12060111198791205791 (9)
1206011 = [11991011 11991012 11991013 11991014 11991015]119879
1205791 = [1198601 1198602 1198603 1198604 1198605] (10)
22 Model Number 2 The second state variable of (3) can berewritten by using (2) as below
1199092 [119896 +119879119904]
= 1199092 [119896] +119879119904 (15119875120582119891119894119898
119869
) sin (1199091 [119905])
+119879119904(
15 (119871119889 minus 119871119902) 119894119898119869
) sin (1199091 [119896]) 1199094 [119905]
(11)
By shifting forward in (11) and putting 1199093[119896 + 1] in it from (3)and substituting 119879119904 = 1
1199092 [119896 + 2] = 1199092 [119896 + 1] +(15119875120582119891119894119898
119869
) sin (1199091 [119896 + 1])
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
) sin (1199091 [119896 + 1])
sdot cos (1199091 [119905]) +(15 (119871119889 minus 119871119902) 119894119898
119871119889119869
)1199062
4 Mathematical Problems in Engineering
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
)(
minus119877
119871119889
)
sdot sin (1199091 [119896 + 1]) cos (1199091 [119905])
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
)(119875
119871119902
119871119889
)
sdot sin (1199091 [119896 + 1]) sin (1199091 [119896]) 1199092 [119896] (12)
Equation (12) is rewritten in terms of load angle by using therelation between the first and second state variables
1199091 [119896 + 3] minus 1199091 [119896 + 2] = 1199091 [119896 + 2] minus 1199091 [119896 + 1]
+(
15119875120582119891119894119898119869
) sin (1199091 [119896 + 1])
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
) sin (1199091 [119896 + 1]) cos (1199091 [119896])
+(
15 (119871119889 minus 119871119902) 119894119898119871119889119869
) sin (1199091 [119896 + 1]) 1199062
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
)(
minus119877
119871119889
) sin (1199091 [119896 + 1])
sdot cos (1199091 [119896]) +(15 (119871119889 minus 119871119902) 119894
2119898
119869
)(119875
119871119902
119871119889
)
sdot sin (1199091 [119896 + 1]) sin (1199091 [119896]) (1199091 [119896 + 1] minus 1199091 [119896])
(13)
The above equation is rewritten to the below form by definingnew variables
1199102 [119896] = 119861111991021 +119861211991022 +119861311991023 +119861411991024 (14)
1198611 = (15119875120582119891119894119898
119869
)
1198612 = (15 (119871119889 minus 119871119902) 119894
2119898
119869
)(1+(minus119877119871119889
))
1198613 = (15 (119871119889 minus 119871119902) 119894119898
119871119889119869
)
1198614 = (15 (119871119889 minus 119871119902) 119894
2119898
119869
)(119875
119871119902
119871119889
)
1199102 [119896] = 1199091 [119896 + 3] minus 21199091 [119896 + 2] + 1199091 [119896 + 1]
11991021 = sin (1199091 [119896 + 1])
11991022 = sin (1199091 [119896 + 1]) cos (1199091 [119896])
11991023 = sin (1199091 [119896 + 1]) 1199062 [119896]
11991024
= sin (1199091 [119896 + 1]) sin (1199091 [119896]) (1199091 [119896 + 1] minus 1199091 [119896]) (15)
Finally a nonlinearmodel of PMSM iswritten to common lin-ear regression form like form number 1 considering equation(14) without any linearization in which 1198611 1198617 are factorsthat are embedded in the physical parameters of the systemand 11991021 11991024 are variables that are composed of nonlinearfunctions input and output
Equation (14) can be written to the linear regressionmatrix like form number 1
1199102 [119896] = 12060121198791205792 (16)
1206012 = [11991021 11991022 11991023 11991024]119879
1205792 = [1198611 1198612 1198613 1198614] (17)
In (13) 1206012 is estimator vector and 1205792 parameter vector relatedto form number 2
3 OPA and RLS Estimator
RLS is a powerful tool in system identification which isrobust against noise while OPA is an interesting methodwith high convergence speed with poor performance in noisyenvironment [24 25] Both advantages of these two methodscan be used simultaneously if they are combined Reference[26] is a good reference to understand the efficacy of thecombinatorial method which was applied on synchronousgenerator and good results were obtained The aim of thispaper is to estimate the physical parameter of the PMSMusingOPAampRLSmethod in noisy environmentwith arbitraryinitial values of physical parameters It is noted that nonlinearPMSM system is extremely sensitive to physical parametersvariations hence estimation of parameters of such systems isvery important by this algorithm
Estimation process is started with orthogonal projectionand orthogonal base vectors of estimator that are in the samedimension with the number of parameters (subspace of OPAestimation method) which are produced [20 24 25] Afterproducing estimation subspace RLS is used and last estima-tion of parameters in OPA method is applied to it which is agood initial estimation for RLS to increase the convergencespeed Then RLS continues the estimation process with agood accuracy to achieve physical parameters Stages of thecombinatorial algorithm are given in the following
Stage zero OPA is started with 119905 = 0 and 120576 = 0 andselection of initial parameters 1205790 and 1198750 covarianceidentity matrixStage one a step is added to 119905 (119905 = 119905+1) and estimatorvector 120601119905minus1 is calculatedStage two if the estimator vector of this stage is notlinearly dependent on the estimator vector of the
Mathematical Problems in Engineering 5
Table 1 Parameters of the understudy system [27]
119877 119871
119902119871
119889120582
119891119895 119901
287Ω 9mH 7mH 0175mWb 8 gm2 4
previous stage (120601119905minus1119879119901119905minus1120601119905minus1 = 0) we return to stage
three otherwise previous values of parameters aresubstituted (119901119905 = 119901119905minus1 120579119905 =
120579119905minus1) and we go to stageoneStage three 120579119905 and 119901119905 are obtained after recursiveequation of (18)Stage four if the rank of estimator matrix is equalto the number of parameters (Rank(Φ) = dim(120579))then OPA is finished in 119905OPA and we go to stage fiveotherwise we return to stage oneStage five to start the estimation with RLS method120576 = 1 and 120579119905OPA are considered as initial estimationStage six a step is added to 119905 (119905 = 119905 + 1) and estimatorvector 120601119905minus1 is calculatedStage seven 120579119905 and 119901119905 are updated using (18)Stage eight if the stop condition is not met we returnto stage six Consider the following
120579119905 =
120579119905minus1 +119875119905minus1 sdot 120601119905minus1
120576 + 120601119905minus1119879sdot 119875119905minus1 sdot 120601119905minus1
sdot [119910 [119905] minus
120579119905minus1119879
sdot 120601119905minus1]
119875119905 = 119875119905minus1 +119875119905minus1 sdot 120601119905minus1 sdot 120601119905minus1
119879sdot 119875119905minus1
120576 + 120601119905minus1119879sdot 119875119905minus1 sdot 120601119905minus1
(18)
4 Simulation
To validate the efficiency and accuracy of the combinatorialOPAampRLSmethod the proposed method is implemented ona PMSM with specifications of Table 1 [27]
The signal (PRBS) is applied to the input of system fordata production to identify and excite the dynamics and anoise with SNR = 15 is added to simulate a real conditionfor output of the system (load angle) which is shown inFigure 2 (amplitude and output) Simulation is performed inMATLAB software
To show the superiority of the proposed method it iscompared to OPA and RLS estimator The criteria for evalua-tion are the speed and accuracy estimator in converging to thereal parameters in a way that the second norm of parameterestimation error vector (1198712) is minimized
1198712 =1003817
1003817
1003817
1003817
1003817
120579 minus
120579
1003817
1003817
1003817
1003817
10038172 (19)
These criteria are evaluated and compared for the threemethods that is OPA RLS andOPAampRLSmethods for bothlinear regression forms from the second part with the sameinitial estimation and stop condition
This criterion is shown in Figure 1 for RLS andOPAampRLSmethods It is shown that this criterion is first decreased
0 2 4 6 8 10 12 14 16 18 200
5
10
15
Time
Am
plitu
de er
ror
Error OPAampRLS = 0026663mdasherror RLS = 59899
Form 1 OPAampRLSFirst OPASecond RLSForm 2 OPAampRLS
First OPASecond RLSForm 1 RLSForm 2 RLS
0 005 01 015 02 025 03 035 04 045 05012345678
Figure 1 Second norm of parameters estimation error in OPA
0 5 10 15minus50
050
Time
Am
plitu
dere
al o
utpu
t
0 5 10 15minus50
050
Time
Am
plitu
dees
timat
ion
outp
ut
0 5 10 150
0102
Time
Am
plitu
deer
ror
Figure 2 The figure shows output of real condition and estimatedmodel and their difference As shown estimated and real output
for OPA method with respect to speed which shows quickconvergence toward objective parameters but after that thistrend is not preserved and system would oscillate anddiverges due to noise It is shown from Figure 1 that thesecondnormof factors estimation error for RLS is descendingbut its slope is slight (low) which requires too many samplesfor parameters convergence However it is prominent that inOPAampRLSmethod parameters would leap greatly toward theobjective parameters with a quick decrease in initial samplesand after that it has a proper trend until the convergenceto real parameters and appropriate estimation which is veryproper for noisy condition This is because both features ofRLS and OPA are used in a single algorithm
As discussed in Section 2 to avoid nonlinear compli-cated calculation for estimating physical parameters tworegression forms are defined according to (9) and (13) Nowthe values of physical parameters are calculated with simplemathematics and given in Table 2
6 Mathematical Problems in Engineering
Table 2 Comparison between the convergence ofOPA andRLS andOPAampRLS methods with the same condition
119877 119871119902 119871119889 120582119891 119895 119901 TimeReal 2875 9 7 0175 8 4 mdashOPA minus0264 minus0331 81717 01547 minus0495 minus0180 20RLS 14925 71954 77590 01911 78661 32461 20OPAampRLS 2897 90650 69981 01744 8006 4012 20
Table 3 Investigation of normalized residue in three algorithms
The normalized residualOPA method 12023 (dB)RLS method 3023 (dB)OPAampRLS method minus3756 (dB)
Figure 2 shows output of real condition and estimatedmodel and their difference As shown estimated and realoutput are alike with agreeable accuracy
One of the criteria for evaluating the error of estimationis the normalized residue criterion according to (20) [28 29]Here this criterion is calculated and is given in Table 3 It isobserved that the proposed method is superior with respectto other methods Consider the following
119869dB = 10 log(sum
119873
1 (119910 (119905) minus (119905))2
sum
119873
1 119910 (119905)2 ) (20)
5 Conclusion
In this paper a new combinatorialmethod ofOrthogonal Pro-jection Algorithm and Recursive Least Squares (OPAampRLS)for parameter estimation of PMSM in noise-covered environ-ment is proposed To avoid solving complicated nonlinearequation two linear regression forms of fourth-order statespace PMSM were rewritten and OPAampRLS estimator wascompared to OPA and RLS estimators Speed and accuracy ofconvergence were investigated in three algorithms and it wasshown that OPAampRLS method has better results with respectto the two single methods The proposed algorithm can beapplied to any linear or nonlinear system where state spacemodel of the system is rewritten to linear regression form Itis also verified that this method is very appropriate for onlineestimation
Nomenclature
Constant
119877 Stator resistance119901 Pole-pairs120582119891 Magnet flux linkage119869 Inertia coefficient119871119889 Direct and quadrature inductances of Park
transformation
119871119902 Direct and quadrature inductances of Parktransformation
1205791 Parameter vector form 11205792 Parameter vector form 2
Variables
119894119902 Park transformation currents119894119889 Park transformation currents120596119890 Rotor angular speed120575 Rotor angular position119894119898 Nominal current1206011 Estimator vector form 11206012 Estimator vector form 21199101 Output form 11199102 Output form 2119901 Covariance matrix1198712 Second norm of parameter estimation error vector119869 Normalized residue
Indices
119896 Step sample time in discrete time119905 Time (sec)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Xu U Vollmer A Ebrahimi and N Parspour ldquoOnlineestimation of the stator resistances of a PMSM with considera-tion of magnetic saturationrdquo in Proceedings of the InternationalConference and Exposition on Electrical and Power Engineering(EPE rsquo12) pp 360ndash365 IEEE Iasi Romania October 2012
[2] K Liu Q Zhang J Chen Z Q Zhu and J Zhang ldquoOnlinemultiparameter estimation of nonsalient-pole PM synchronousmachines with temperature variation trackingrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 5 pp 1776ndash1788 2011
[3] QMeng T Zhang J-FHe J-Y Song and J-WHan ldquoDynamicmodeling of a 6-degree-of-freedom Stewart platform driven bya permanent magnet synchronous motorrdquo Journal of ZhejiangUniversity Science C vol 11 no 10 pp 751ndash761 2010
[4] E G Shehata ldquoSpeed sensorless torque control of an IPMSMdrive with online stator resistance estimation using reducedorder EKFrdquo International Journal of Electrical Power amp EnergySystems vol 47 no 1 pp 378ndash386 2013
[5] N Ozturk and E Celik ldquoSpeed control of permanent magnetsynchronous motors using fuzzy controller based on geneticalgorithmsrdquo International Journal of Electrical Power amp EnergySystems vol 43 no 1 pp 889ndash898 2012
[6] P Pillay and R Krishnan ldquoModeling of permanent magnetmotor drivesrdquo IEEE Transactions on Industrial Electronics vol35 no 4 pp 537ndash541 1988
[7] O Aydogmus and S Sunter ldquoImplementation of EKF basedsensorless drive system using vector controlled PMSM fed by amatrix converterrdquo International Journal of Electrical Power andEnergy Systems vol 43 no 1 pp 736ndash743 2012
Mathematical Problems in Engineering 7
[8] L Liu and D A Cartes ldquoSynchronisation based adaptiveparameter identification for permanent magnet synchronousmotorsrdquo IET Control Theory and Applications vol 1 no 4 pp1015ndash1022 2007
[9] C-H Lin andC-P Lin ldquoThe hybrid RFNN control for a PMSMdrive electric scooter using rotor flux estimatorrdquo ElectricalPower and Energy Systems vol 51 pp 889ndash898 2012
[10] F Aymen N Martin L Sbita and J Novak ldquoOnline PMSMparameters estimation for 32000rpmrdquo in Proceedings of theInternational Conference on Control Engineering amp InformationTechnology (CEIT rsquo13) vol 3 pp 148ndash152 Sousse Tunisia June2013
[11] G Junli and Z Yingfan ldquoResearch on parameter identificationand PI self-tuning of PMSMrdquo in Proceedings of the 2nd Inter-national Conference on Information Science and Engineering(ICISE rsquo10) pp 5251ndash5254 December 2010
[12] G Gatto I Marongiu and A Serpi ldquoDiscrete-time parameteridentification of a surface-mounted permanent magnet syn-chronousmachinerdquo IEEE Transactions on Industrial Electronicsvol 60 no 11 pp 4869ndash4880 2013
[13] K Liu and Z Q Zhu ldquoOnline estimation of the rotor fluxlinkage and voltage-source inverter nonlinearity in permanentmagnet synchronous machine drivesrdquo IEEE Transactions onPower Electronics vol 29 no 1 pp 418ndash427 2014
[14] Y Shi K Sun L Huang and Y Li ldquoOnline identification ofpermanent magnet flux based on extended Kalman filter forIPMSM drive with position sensorless controlrdquo IEEE Transac-tions on Industrial Electronics vol 59 no 11 pp 4169ndash4178 2012
[15] B Nahid Mobarakeh F Meibody-Tabar and F M Sargos ldquoOn-line identification of PMSM electrical parameters based ondecoupling controlrdquo in Proceedings of the Conference Recordof the IEEE Industry Applications Conference 36th IAS AnnualMeeting vol 1 pp 266ndash273 Chicago Ill USA September 2001
[16] T Boileau N Leboeuf B Nahid-Mobarakeh and F Meibody-Tabar ldquoOnline identification of PMSM parameters parameteridentifiability and estimator comparative studyrdquo IEEE Transac-tions on Industry Applications vol 47 no 4 pp 1944ndash1957 2011
[17] T Boileau B Nahid-Mobarakeh and F Meibody-Tabar ldquoOn-line identification of PMSM parameters model-reference vsEKFrdquo in Proceedings of the IEEE Industry Applications SocietyAnnual Meeting (IAS rsquo08) pp 1ndash8 can October 2008
[18] K Liu Z Q Zhu and D A Stone ldquoParameter estimation forcondition monitoring of PMSM stator winding and rotor per-manent magnetsrdquo IEEE Transactions on Industrial Electronicsvol 60 no 12 pp 5902ndash5913 2013
[19] C Cai F Chu ZWang andK Jia ldquoIdentification and control ofPMSM using adaptive BP-PID neural networkrdquo in Advances inNeural Networks vol 7952 of Lecture Notes in Computer Sciencepp 155ndash162 Springer 2013
[20] M Ghanbari I Yousefi and V Mossadegh ldquoOnline parameterestimation of permanent magnet synchronous motor usingorthogonal projection algorithmrdquo Indian Journal ScientificResearch vol 2 no 1 pp 32ndash36 2014
[21] A K Parvathy R Devanathan and V Kamaraj ldquoApplicationof quadratic linearization to control of permanent magnetsynchronous motorrdquo in Proceedings of the Joint InternationalConference on Power System Technology and IEEE Power IndiaConference pp 158ndash163 2008
[22] M A Hamida J de Leon A Glumineau and R BoisliveauldquoAn adaptive interconnected observer for sensorless control ofPM synchronous motors with online parameter identificationrdquo
IEEE Transactions on Industrial Electronics vol 60 no 2 pp739ndash748 2013
[23] R Krishnan Permanent Magnet Synchronous and Brushless DCMotor Drives Taylor amp Francis 2010
[24] J Zhao and I Kanellakopoulos ldquoActive identification fordiscrete-time nonlinear control I Output-feedback systemsrdquoIEEE Transactions on Automatic Control vol 47 no 2 pp 210ndash224 2002
[25] H Agahi M Karrari W Rosehart and O P Malik ldquoApplica-tion of active identification method to synchronous generatorparameter estimationrdquo in Proceedings of the IEEE Power Engi-neering Society General Meeting (PES rsquo06) Montreal CanadaJune 2006
[26] HAgahiActive identification for nonlinearmultivariable systemidentification and its application for synchronous generator iden-tification [PhD thesis] Amirkabir University of TechnologyTehran Iran 2007
[27] J ChiassonModeling and High Performance Control of ElectricMachines IEEE Press Series on Power Engineering IEEE Press2005
[28] M Ghanbari and M Haeri ldquoParametric identification offractional-order systems using a fractional Legendre basisrdquoProceedings of the Institution of Mechanical Engineers Part IJournal of Systems and Control Engineering vol 224 no 3 pp261ndash274 2010
[29] M Ghanbari and M Haeri ldquoOrder and pole locator estimationin fractional order systems using bode diagramrdquo Signal Process-ing vol 91 no 2 pp 191ndash202 2011
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
without linearization or simplification Both estimators canbe applied independently but the combinatorial method hasthe features of both methods OPA leaps quickly toward theaim parameter with each estimation it does and after that RLSadapts the estimated parameter to the real parameter due toits robustness toward noise
Section 2 describes state space model transformation ofPMSM In Section 3 the proposed estimator is presentedSimulation results are illustrated in Section 4 and finally thepaper is concluded in Section 5
2 Transforming the Nonlinear Model toLinear Regression Form
To achieve high performance for PMSM the fourth-orderstate space model of the system is selected under Parktransformation [21 22]
= 119860119909+119861119906+119891 (119909)
119909 = [1199091 1199092 1199093 1199094]119879= [
120575 120596119890 119894119902 119894119889]119879
119906 = [1199061 1199062]119879= [
V119902 V119889]119879
119860 =
[
[
[
[
[
[
[
[
[
[
[
0 1 0 0
0 015119901120582119891119869
0
0 minus
minus120582119891119901
119871119902
minus
minus119877
119871119902
0
0 0 0 minus
119877
119871119889
]
]
]
]
]
]
]
]
]
]
]
119861 =
[
[
[
[
[
[
[
[
[
0 00 01119871119902
0
0 1119871119889
]
]
]
]
]
]
]
]
]
119891 =
[
[
[
[
[
[
[
[
[
[
[
[
0
15119901 (119871119889 minus 119871119902) 119894119902119894119889119869
minus
119871119889119901120596119890119894119889
119871119902
119871119902119901120596119890119894119902
119871119889
]
]
]
]
]
]
]
]
]
]
]
]
(1)
System inputs are V119889 V119902 Park transformation voltages 119894119902 119894119889120596119890 and 120575 state variables of the systemwhich are rotor angularposition rotor angular speed and Park transformation cur-rents respectively
Physical parameters of the system are 119877 stator resistance119901 pole-pairs 120582119891 magnet flux linkage 119869 inertia coefficientand 119871119889 119871119902 direct and quadrature inductances of Parktransformation respectively
The aim of this section is to transform the nonlinearstate space model of PMSM to the linear regression form
without linearization or simplification techniques As thelinear regression form is expressed in terms of an output thepertaining output is rewritten in terms of state variables Theselected output is power angle (120575 = 1199091)
Equation (3) is obtained by considering 119894119889 119894119902 according to(2) [23] and discretization based on ( = [119909(119905+119879119904)minus119909(119905)]119879119904)in which 119879119904 is the sampling time Consider
119894119889 = 119894119898 cos (120575) = 119894119898 cos (1199091)
119894119902 = 119894119898 sin (120575) = 119894119898 sin (1199091) (2)
1199091 [119896 +119879119904] = 1199091 [119896] +1198791199041199092 [119896]
1199092 [119896 +119879119904] = 1199092 [119896] +119879119904 (15119875120582119891119869
)1199093 [119896]
+119879119904(
15 (119871119889 minus 119871119902)119869
)1199093 [119896] 1199094 [119896]
1199093 [119896 +119879119904] = 119879119904 (minus120582119891119875
119871119902
)1199092 [119896]
+119879119904 (
minus119877
119871119902
) 119894119898sin (1199091 [119896]) +119879119904
119871119902
1199061 [119896]
+119879119904 (minus119901
119871119889
119871119902
) 119894119898cos (1199091 [119896]) 1199092 [119896]
+ 119894119898sin (1199091 [119896])
1199094 [119896 +119879119904] = 119894119898cos (1199091 [119896]) +119879119904 (minus119877
119871119889
) 119894119898cos (1199091 [119896])
+
119879119904
119871119889
1199062
+119879119904 (119875
119871119902
119871119889
) 119894119898sin (1199091 [119896]) 1199092 [119896]
(3)
Rewriting (3) to the linear regression form is needed in away that 1199091 is written in terms of all state variables Afterapplying estimator on linear regression form and estimationof the parameters complicated nonlinear equations have tobe solved to obtain physical parameters (119877 119901 120582119891 119869 119871119902 and119871119889) To solve this problem two linear regression forms arerequired by a heuristic method and defining new variables in(3) and (2)Therefore physical parameters can be solved fromlinear regression without solving nonlinear equation
21 Model Number 1 The second state variable of (3) can berewritten by using (2) as below
1199092 [119896 +119879119904]
= 1199092 [119896] +119879119904 (15119875120582119891119869
)1199093 [119896]
+119879119904(
15 (119871119889 minus 119871119902)119869
)
119894
2119898
2sin 2 (1199091 [119896])
(4)
Mathematical Problems in Engineering 3
By shifting forward in (4) and putting 1199093[119896 + 1] in it from (3)and substituting 119879119904 = 1
1199092 [119896 + 2]
= 1199092 [119896 + 1]
+(
15 (119871119889 minus 119871119902) 1198942119898
2119869) sin 2 (1199091 [119896 + 1])
+(
15119875120582119891119869
)(
minus120582119891119875
119871119902
)1199092 [119896]
+(
15119875120582119891119869
)(
minus119877
119871119902
) sin (1199091 [119896])
+(
15119875120582119891119871119902119869
) 1199061 [119896]
+(
15119875120582119891119869
)(minus119901
119871119889
119871119902
) cos (1199091 [119896]) 1199092 [119896]
+(
15119875120582119891119894119898119869
) sin (1199091 [119905])
(5)
Equation (5) is rewritten in terms of load angle by using therelation between the first and second state variables
1199091 [119896 + 3] minus 1199091 [119896 + 2] = 1199091 [119896 + 2] minus 1199091 [119896 + 1]
+(
15119875120582119891119869
)(
minus120582119891119875
119871119902
) (1199091 [119896 + 1] minus 1199091 [119896])
+(
15119875120582119891119869
)(
minus119877
119871119902
) sin (1199091 [119896]) +(15119875120582119891119871119902119869
)
sdot 1199061 [119896] +(15119875120582119891119869
)(minus119901
119871119889
119871119902
) cos (1199091 [119896])
sdot (1199091 [119896 + 1] minus 1199091 [119896]) +(15119875120582119891119894119898
119869
) sin (1199091 [119905])
+(
15 (119871119889 minus 119871119902) 1198942119898
2119869) sin 2 (1199091 [119896 + 1])
(6)
The above equation is rewritten to the below form by definingnew variables
1199101 [119896] = 119860111991011 +119860211991012 +119860311991013 +119860411991014 +119860511991015 (7)
1198601 = (15119875120582119891119869
)(
minus120582119891119875
119871119902
)
1198602 = (15119875120582119891119869
)(
minus119877
119871119902
)+(
15119875120582119891119894119898119869
)
1198603 = (15119875120582119891119871119902119869
)
1198604 = (15119875120582119891119869
)(minus119901
119871119889
119871119902
)
1198605 = (15 (119871119889 minus 119871119902) 119894
2119898
2119869)
11991011 = (1199091 [119896 + 1] minus 1199091 [119896])
11991012 = sin (1199091 [119896])
11991013 = 1199061 [119896]
11991014 = cos (1199091 [119896]) (1199091 [119896 + 1] minus 1199091 [119896])
11991015 = sin 2 (1199091 [119896 + 1])
1199101 [119896] = 1199091 [119896 + 3] minus 21199091 [119896 + 2] + 1199091 [119896 + 1] (8)
Finally a nonlinear model of PMSM is written to commonlinear regression form considering (7) without any lineariza-tion in which 1198601 1198607 are factors that are obtained frommultiplying physical parameters of the system and 11991011 11991015are variables that are composed of nonlinear functions inputand output
Equation (7) can be written to the linear regressionmatrix
In the following equation 1206011 is estimator vector for thefirst regression form and 1205791 parameter vector related to thisform
1199101 [119896] = 12060111198791205791 (9)
1206011 = [11991011 11991012 11991013 11991014 11991015]119879
1205791 = [1198601 1198602 1198603 1198604 1198605] (10)
22 Model Number 2 The second state variable of (3) can berewritten by using (2) as below
1199092 [119896 +119879119904]
= 1199092 [119896] +119879119904 (15119875120582119891119894119898
119869
) sin (1199091 [119905])
+119879119904(
15 (119871119889 minus 119871119902) 119894119898119869
) sin (1199091 [119896]) 1199094 [119905]
(11)
By shifting forward in (11) and putting 1199093[119896 + 1] in it from (3)and substituting 119879119904 = 1
1199092 [119896 + 2] = 1199092 [119896 + 1] +(15119875120582119891119894119898
119869
) sin (1199091 [119896 + 1])
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
) sin (1199091 [119896 + 1])
sdot cos (1199091 [119905]) +(15 (119871119889 minus 119871119902) 119894119898
119871119889119869
)1199062
4 Mathematical Problems in Engineering
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
)(
minus119877
119871119889
)
sdot sin (1199091 [119896 + 1]) cos (1199091 [119905])
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
)(119875
119871119902
119871119889
)
sdot sin (1199091 [119896 + 1]) sin (1199091 [119896]) 1199092 [119896] (12)
Equation (12) is rewritten in terms of load angle by using therelation between the first and second state variables
1199091 [119896 + 3] minus 1199091 [119896 + 2] = 1199091 [119896 + 2] minus 1199091 [119896 + 1]
+(
15119875120582119891119894119898119869
) sin (1199091 [119896 + 1])
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
) sin (1199091 [119896 + 1]) cos (1199091 [119896])
+(
15 (119871119889 minus 119871119902) 119894119898119871119889119869
) sin (1199091 [119896 + 1]) 1199062
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
)(
minus119877
119871119889
) sin (1199091 [119896 + 1])
sdot cos (1199091 [119896]) +(15 (119871119889 minus 119871119902) 119894
2119898
119869
)(119875
119871119902
119871119889
)
sdot sin (1199091 [119896 + 1]) sin (1199091 [119896]) (1199091 [119896 + 1] minus 1199091 [119896])
(13)
The above equation is rewritten to the below form by definingnew variables
1199102 [119896] = 119861111991021 +119861211991022 +119861311991023 +119861411991024 (14)
1198611 = (15119875120582119891119894119898
119869
)
1198612 = (15 (119871119889 minus 119871119902) 119894
2119898
119869
)(1+(minus119877119871119889
))
1198613 = (15 (119871119889 minus 119871119902) 119894119898
119871119889119869
)
1198614 = (15 (119871119889 minus 119871119902) 119894
2119898
119869
)(119875
119871119902
119871119889
)
1199102 [119896] = 1199091 [119896 + 3] minus 21199091 [119896 + 2] + 1199091 [119896 + 1]
11991021 = sin (1199091 [119896 + 1])
11991022 = sin (1199091 [119896 + 1]) cos (1199091 [119896])
11991023 = sin (1199091 [119896 + 1]) 1199062 [119896]
11991024
= sin (1199091 [119896 + 1]) sin (1199091 [119896]) (1199091 [119896 + 1] minus 1199091 [119896]) (15)
Finally a nonlinearmodel of PMSM iswritten to common lin-ear regression form like form number 1 considering equation(14) without any linearization in which 1198611 1198617 are factorsthat are embedded in the physical parameters of the systemand 11991021 11991024 are variables that are composed of nonlinearfunctions input and output
Equation (14) can be written to the linear regressionmatrix like form number 1
1199102 [119896] = 12060121198791205792 (16)
1206012 = [11991021 11991022 11991023 11991024]119879
1205792 = [1198611 1198612 1198613 1198614] (17)
In (13) 1206012 is estimator vector and 1205792 parameter vector relatedto form number 2
3 OPA and RLS Estimator
RLS is a powerful tool in system identification which isrobust against noise while OPA is an interesting methodwith high convergence speed with poor performance in noisyenvironment [24 25] Both advantages of these two methodscan be used simultaneously if they are combined Reference[26] is a good reference to understand the efficacy of thecombinatorial method which was applied on synchronousgenerator and good results were obtained The aim of thispaper is to estimate the physical parameter of the PMSMusingOPAampRLSmethod in noisy environmentwith arbitraryinitial values of physical parameters It is noted that nonlinearPMSM system is extremely sensitive to physical parametersvariations hence estimation of parameters of such systems isvery important by this algorithm
Estimation process is started with orthogonal projectionand orthogonal base vectors of estimator that are in the samedimension with the number of parameters (subspace of OPAestimation method) which are produced [20 24 25] Afterproducing estimation subspace RLS is used and last estima-tion of parameters in OPA method is applied to it which is agood initial estimation for RLS to increase the convergencespeed Then RLS continues the estimation process with agood accuracy to achieve physical parameters Stages of thecombinatorial algorithm are given in the following
Stage zero OPA is started with 119905 = 0 and 120576 = 0 andselection of initial parameters 1205790 and 1198750 covarianceidentity matrixStage one a step is added to 119905 (119905 = 119905+1) and estimatorvector 120601119905minus1 is calculatedStage two if the estimator vector of this stage is notlinearly dependent on the estimator vector of the
Mathematical Problems in Engineering 5
Table 1 Parameters of the understudy system [27]
119877 119871
119902119871
119889120582
119891119895 119901
287Ω 9mH 7mH 0175mWb 8 gm2 4
previous stage (120601119905minus1119879119901119905minus1120601119905minus1 = 0) we return to stage
three otherwise previous values of parameters aresubstituted (119901119905 = 119901119905minus1 120579119905 =
120579119905minus1) and we go to stageoneStage three 120579119905 and 119901119905 are obtained after recursiveequation of (18)Stage four if the rank of estimator matrix is equalto the number of parameters (Rank(Φ) = dim(120579))then OPA is finished in 119905OPA and we go to stage fiveotherwise we return to stage oneStage five to start the estimation with RLS method120576 = 1 and 120579119905OPA are considered as initial estimationStage six a step is added to 119905 (119905 = 119905 + 1) and estimatorvector 120601119905minus1 is calculatedStage seven 120579119905 and 119901119905 are updated using (18)Stage eight if the stop condition is not met we returnto stage six Consider the following
120579119905 =
120579119905minus1 +119875119905minus1 sdot 120601119905minus1
120576 + 120601119905minus1119879sdot 119875119905minus1 sdot 120601119905minus1
sdot [119910 [119905] minus
120579119905minus1119879
sdot 120601119905minus1]
119875119905 = 119875119905minus1 +119875119905minus1 sdot 120601119905minus1 sdot 120601119905minus1
119879sdot 119875119905minus1
120576 + 120601119905minus1119879sdot 119875119905minus1 sdot 120601119905minus1
(18)
4 Simulation
To validate the efficiency and accuracy of the combinatorialOPAampRLSmethod the proposed method is implemented ona PMSM with specifications of Table 1 [27]
The signal (PRBS) is applied to the input of system fordata production to identify and excite the dynamics and anoise with SNR = 15 is added to simulate a real conditionfor output of the system (load angle) which is shown inFigure 2 (amplitude and output) Simulation is performed inMATLAB software
To show the superiority of the proposed method it iscompared to OPA and RLS estimator The criteria for evalua-tion are the speed and accuracy estimator in converging to thereal parameters in a way that the second norm of parameterestimation error vector (1198712) is minimized
1198712 =1003817
1003817
1003817
1003817
1003817
120579 minus
120579
1003817
1003817
1003817
1003817
10038172 (19)
These criteria are evaluated and compared for the threemethods that is OPA RLS andOPAampRLSmethods for bothlinear regression forms from the second part with the sameinitial estimation and stop condition
This criterion is shown in Figure 1 for RLS andOPAampRLSmethods It is shown that this criterion is first decreased
0 2 4 6 8 10 12 14 16 18 200
5
10
15
Time
Am
plitu
de er
ror
Error OPAampRLS = 0026663mdasherror RLS = 59899
Form 1 OPAampRLSFirst OPASecond RLSForm 2 OPAampRLS
First OPASecond RLSForm 1 RLSForm 2 RLS
0 005 01 015 02 025 03 035 04 045 05012345678
Figure 1 Second norm of parameters estimation error in OPA
0 5 10 15minus50
050
Time
Am
plitu
dere
al o
utpu
t
0 5 10 15minus50
050
Time
Am
plitu
dees
timat
ion
outp
ut
0 5 10 150
0102
Time
Am
plitu
deer
ror
Figure 2 The figure shows output of real condition and estimatedmodel and their difference As shown estimated and real output
for OPA method with respect to speed which shows quickconvergence toward objective parameters but after that thistrend is not preserved and system would oscillate anddiverges due to noise It is shown from Figure 1 that thesecondnormof factors estimation error for RLS is descendingbut its slope is slight (low) which requires too many samplesfor parameters convergence However it is prominent that inOPAampRLSmethod parameters would leap greatly toward theobjective parameters with a quick decrease in initial samplesand after that it has a proper trend until the convergenceto real parameters and appropriate estimation which is veryproper for noisy condition This is because both features ofRLS and OPA are used in a single algorithm
As discussed in Section 2 to avoid nonlinear compli-cated calculation for estimating physical parameters tworegression forms are defined according to (9) and (13) Nowthe values of physical parameters are calculated with simplemathematics and given in Table 2
6 Mathematical Problems in Engineering
Table 2 Comparison between the convergence ofOPA andRLS andOPAampRLS methods with the same condition
119877 119871119902 119871119889 120582119891 119895 119901 TimeReal 2875 9 7 0175 8 4 mdashOPA minus0264 minus0331 81717 01547 minus0495 minus0180 20RLS 14925 71954 77590 01911 78661 32461 20OPAampRLS 2897 90650 69981 01744 8006 4012 20
Table 3 Investigation of normalized residue in three algorithms
The normalized residualOPA method 12023 (dB)RLS method 3023 (dB)OPAampRLS method minus3756 (dB)
Figure 2 shows output of real condition and estimatedmodel and their difference As shown estimated and realoutput are alike with agreeable accuracy
One of the criteria for evaluating the error of estimationis the normalized residue criterion according to (20) [28 29]Here this criterion is calculated and is given in Table 3 It isobserved that the proposed method is superior with respectto other methods Consider the following
119869dB = 10 log(sum
119873
1 (119910 (119905) minus (119905))2
sum
119873
1 119910 (119905)2 ) (20)
5 Conclusion
In this paper a new combinatorialmethod ofOrthogonal Pro-jection Algorithm and Recursive Least Squares (OPAampRLS)for parameter estimation of PMSM in noise-covered environ-ment is proposed To avoid solving complicated nonlinearequation two linear regression forms of fourth-order statespace PMSM were rewritten and OPAampRLS estimator wascompared to OPA and RLS estimators Speed and accuracy ofconvergence were investigated in three algorithms and it wasshown that OPAampRLS method has better results with respectto the two single methods The proposed algorithm can beapplied to any linear or nonlinear system where state spacemodel of the system is rewritten to linear regression form Itis also verified that this method is very appropriate for onlineestimation
Nomenclature
Constant
119877 Stator resistance119901 Pole-pairs120582119891 Magnet flux linkage119869 Inertia coefficient119871119889 Direct and quadrature inductances of Park
transformation
119871119902 Direct and quadrature inductances of Parktransformation
1205791 Parameter vector form 11205792 Parameter vector form 2
Variables
119894119902 Park transformation currents119894119889 Park transformation currents120596119890 Rotor angular speed120575 Rotor angular position119894119898 Nominal current1206011 Estimator vector form 11206012 Estimator vector form 21199101 Output form 11199102 Output form 2119901 Covariance matrix1198712 Second norm of parameter estimation error vector119869 Normalized residue
Indices
119896 Step sample time in discrete time119905 Time (sec)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Xu U Vollmer A Ebrahimi and N Parspour ldquoOnlineestimation of the stator resistances of a PMSM with considera-tion of magnetic saturationrdquo in Proceedings of the InternationalConference and Exposition on Electrical and Power Engineering(EPE rsquo12) pp 360ndash365 IEEE Iasi Romania October 2012
[2] K Liu Q Zhang J Chen Z Q Zhu and J Zhang ldquoOnlinemultiparameter estimation of nonsalient-pole PM synchronousmachines with temperature variation trackingrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 5 pp 1776ndash1788 2011
[3] QMeng T Zhang J-FHe J-Y Song and J-WHan ldquoDynamicmodeling of a 6-degree-of-freedom Stewart platform driven bya permanent magnet synchronous motorrdquo Journal of ZhejiangUniversity Science C vol 11 no 10 pp 751ndash761 2010
[4] E G Shehata ldquoSpeed sensorless torque control of an IPMSMdrive with online stator resistance estimation using reducedorder EKFrdquo International Journal of Electrical Power amp EnergySystems vol 47 no 1 pp 378ndash386 2013
[5] N Ozturk and E Celik ldquoSpeed control of permanent magnetsynchronous motors using fuzzy controller based on geneticalgorithmsrdquo International Journal of Electrical Power amp EnergySystems vol 43 no 1 pp 889ndash898 2012
[6] P Pillay and R Krishnan ldquoModeling of permanent magnetmotor drivesrdquo IEEE Transactions on Industrial Electronics vol35 no 4 pp 537ndash541 1988
[7] O Aydogmus and S Sunter ldquoImplementation of EKF basedsensorless drive system using vector controlled PMSM fed by amatrix converterrdquo International Journal of Electrical Power andEnergy Systems vol 43 no 1 pp 736ndash743 2012
Mathematical Problems in Engineering 7
[8] L Liu and D A Cartes ldquoSynchronisation based adaptiveparameter identification for permanent magnet synchronousmotorsrdquo IET Control Theory and Applications vol 1 no 4 pp1015ndash1022 2007
[9] C-H Lin andC-P Lin ldquoThe hybrid RFNN control for a PMSMdrive electric scooter using rotor flux estimatorrdquo ElectricalPower and Energy Systems vol 51 pp 889ndash898 2012
[10] F Aymen N Martin L Sbita and J Novak ldquoOnline PMSMparameters estimation for 32000rpmrdquo in Proceedings of theInternational Conference on Control Engineering amp InformationTechnology (CEIT rsquo13) vol 3 pp 148ndash152 Sousse Tunisia June2013
[11] G Junli and Z Yingfan ldquoResearch on parameter identificationand PI self-tuning of PMSMrdquo in Proceedings of the 2nd Inter-national Conference on Information Science and Engineering(ICISE rsquo10) pp 5251ndash5254 December 2010
[12] G Gatto I Marongiu and A Serpi ldquoDiscrete-time parameteridentification of a surface-mounted permanent magnet syn-chronousmachinerdquo IEEE Transactions on Industrial Electronicsvol 60 no 11 pp 4869ndash4880 2013
[13] K Liu and Z Q Zhu ldquoOnline estimation of the rotor fluxlinkage and voltage-source inverter nonlinearity in permanentmagnet synchronous machine drivesrdquo IEEE Transactions onPower Electronics vol 29 no 1 pp 418ndash427 2014
[14] Y Shi K Sun L Huang and Y Li ldquoOnline identification ofpermanent magnet flux based on extended Kalman filter forIPMSM drive with position sensorless controlrdquo IEEE Transac-tions on Industrial Electronics vol 59 no 11 pp 4169ndash4178 2012
[15] B Nahid Mobarakeh F Meibody-Tabar and F M Sargos ldquoOn-line identification of PMSM electrical parameters based ondecoupling controlrdquo in Proceedings of the Conference Recordof the IEEE Industry Applications Conference 36th IAS AnnualMeeting vol 1 pp 266ndash273 Chicago Ill USA September 2001
[16] T Boileau N Leboeuf B Nahid-Mobarakeh and F Meibody-Tabar ldquoOnline identification of PMSM parameters parameteridentifiability and estimator comparative studyrdquo IEEE Transac-tions on Industry Applications vol 47 no 4 pp 1944ndash1957 2011
[17] T Boileau B Nahid-Mobarakeh and F Meibody-Tabar ldquoOn-line identification of PMSM parameters model-reference vsEKFrdquo in Proceedings of the IEEE Industry Applications SocietyAnnual Meeting (IAS rsquo08) pp 1ndash8 can October 2008
[18] K Liu Z Q Zhu and D A Stone ldquoParameter estimation forcondition monitoring of PMSM stator winding and rotor per-manent magnetsrdquo IEEE Transactions on Industrial Electronicsvol 60 no 12 pp 5902ndash5913 2013
[19] C Cai F Chu ZWang andK Jia ldquoIdentification and control ofPMSM using adaptive BP-PID neural networkrdquo in Advances inNeural Networks vol 7952 of Lecture Notes in Computer Sciencepp 155ndash162 Springer 2013
[20] M Ghanbari I Yousefi and V Mossadegh ldquoOnline parameterestimation of permanent magnet synchronous motor usingorthogonal projection algorithmrdquo Indian Journal ScientificResearch vol 2 no 1 pp 32ndash36 2014
[21] A K Parvathy R Devanathan and V Kamaraj ldquoApplicationof quadratic linearization to control of permanent magnetsynchronous motorrdquo in Proceedings of the Joint InternationalConference on Power System Technology and IEEE Power IndiaConference pp 158ndash163 2008
[22] M A Hamida J de Leon A Glumineau and R BoisliveauldquoAn adaptive interconnected observer for sensorless control ofPM synchronous motors with online parameter identificationrdquo
IEEE Transactions on Industrial Electronics vol 60 no 2 pp739ndash748 2013
[23] R Krishnan Permanent Magnet Synchronous and Brushless DCMotor Drives Taylor amp Francis 2010
[24] J Zhao and I Kanellakopoulos ldquoActive identification fordiscrete-time nonlinear control I Output-feedback systemsrdquoIEEE Transactions on Automatic Control vol 47 no 2 pp 210ndash224 2002
[25] H Agahi M Karrari W Rosehart and O P Malik ldquoApplica-tion of active identification method to synchronous generatorparameter estimationrdquo in Proceedings of the IEEE Power Engi-neering Society General Meeting (PES rsquo06) Montreal CanadaJune 2006
[26] HAgahiActive identification for nonlinearmultivariable systemidentification and its application for synchronous generator iden-tification [PhD thesis] Amirkabir University of TechnologyTehran Iran 2007
[27] J ChiassonModeling and High Performance Control of ElectricMachines IEEE Press Series on Power Engineering IEEE Press2005
[28] M Ghanbari and M Haeri ldquoParametric identification offractional-order systems using a fractional Legendre basisrdquoProceedings of the Institution of Mechanical Engineers Part IJournal of Systems and Control Engineering vol 224 no 3 pp261ndash274 2010
[29] M Ghanbari and M Haeri ldquoOrder and pole locator estimationin fractional order systems using bode diagramrdquo Signal Process-ing vol 91 no 2 pp 191ndash202 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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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
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
By shifting forward in (4) and putting 1199093[119896 + 1] in it from (3)and substituting 119879119904 = 1
1199092 [119896 + 2]
= 1199092 [119896 + 1]
+(
15 (119871119889 minus 119871119902) 1198942119898
2119869) sin 2 (1199091 [119896 + 1])
+(
15119875120582119891119869
)(
minus120582119891119875
119871119902
)1199092 [119896]
+(
15119875120582119891119869
)(
minus119877
119871119902
) sin (1199091 [119896])
+(
15119875120582119891119871119902119869
) 1199061 [119896]
+(
15119875120582119891119869
)(minus119901
119871119889
119871119902
) cos (1199091 [119896]) 1199092 [119896]
+(
15119875120582119891119894119898119869
) sin (1199091 [119905])
(5)
Equation (5) is rewritten in terms of load angle by using therelation between the first and second state variables
1199091 [119896 + 3] minus 1199091 [119896 + 2] = 1199091 [119896 + 2] minus 1199091 [119896 + 1]
+(
15119875120582119891119869
)(
minus120582119891119875
119871119902
) (1199091 [119896 + 1] minus 1199091 [119896])
+(
15119875120582119891119869
)(
minus119877
119871119902
) sin (1199091 [119896]) +(15119875120582119891119871119902119869
)
sdot 1199061 [119896] +(15119875120582119891119869
)(minus119901
119871119889
119871119902
) cos (1199091 [119896])
sdot (1199091 [119896 + 1] minus 1199091 [119896]) +(15119875120582119891119894119898
119869
) sin (1199091 [119905])
+(
15 (119871119889 minus 119871119902) 1198942119898
2119869) sin 2 (1199091 [119896 + 1])
(6)
The above equation is rewritten to the below form by definingnew variables
1199101 [119896] = 119860111991011 +119860211991012 +119860311991013 +119860411991014 +119860511991015 (7)
1198601 = (15119875120582119891119869
)(
minus120582119891119875
119871119902
)
1198602 = (15119875120582119891119869
)(
minus119877
119871119902
)+(
15119875120582119891119894119898119869
)
1198603 = (15119875120582119891119871119902119869
)
1198604 = (15119875120582119891119869
)(minus119901
119871119889
119871119902
)
1198605 = (15 (119871119889 minus 119871119902) 119894
2119898
2119869)
11991011 = (1199091 [119896 + 1] minus 1199091 [119896])
11991012 = sin (1199091 [119896])
11991013 = 1199061 [119896]
11991014 = cos (1199091 [119896]) (1199091 [119896 + 1] minus 1199091 [119896])
11991015 = sin 2 (1199091 [119896 + 1])
1199101 [119896] = 1199091 [119896 + 3] minus 21199091 [119896 + 2] + 1199091 [119896 + 1] (8)
Finally a nonlinear model of PMSM is written to commonlinear regression form considering (7) without any lineariza-tion in which 1198601 1198607 are factors that are obtained frommultiplying physical parameters of the system and 11991011 11991015are variables that are composed of nonlinear functions inputand output
Equation (7) can be written to the linear regressionmatrix
In the following equation 1206011 is estimator vector for thefirst regression form and 1205791 parameter vector related to thisform
1199101 [119896] = 12060111198791205791 (9)
1206011 = [11991011 11991012 11991013 11991014 11991015]119879
1205791 = [1198601 1198602 1198603 1198604 1198605] (10)
22 Model Number 2 The second state variable of (3) can berewritten by using (2) as below
1199092 [119896 +119879119904]
= 1199092 [119896] +119879119904 (15119875120582119891119894119898
119869
) sin (1199091 [119905])
+119879119904(
15 (119871119889 minus 119871119902) 119894119898119869
) sin (1199091 [119896]) 1199094 [119905]
(11)
By shifting forward in (11) and putting 1199093[119896 + 1] in it from (3)and substituting 119879119904 = 1
1199092 [119896 + 2] = 1199092 [119896 + 1] +(15119875120582119891119894119898
119869
) sin (1199091 [119896 + 1])
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
) sin (1199091 [119896 + 1])
sdot cos (1199091 [119905]) +(15 (119871119889 minus 119871119902) 119894119898
119871119889119869
)1199062
4 Mathematical Problems in Engineering
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
)(
minus119877
119871119889
)
sdot sin (1199091 [119896 + 1]) cos (1199091 [119905])
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
)(119875
119871119902
119871119889
)
sdot sin (1199091 [119896 + 1]) sin (1199091 [119896]) 1199092 [119896] (12)
Equation (12) is rewritten in terms of load angle by using therelation between the first and second state variables
1199091 [119896 + 3] minus 1199091 [119896 + 2] = 1199091 [119896 + 2] minus 1199091 [119896 + 1]
+(
15119875120582119891119894119898119869
) sin (1199091 [119896 + 1])
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
) sin (1199091 [119896 + 1]) cos (1199091 [119896])
+(
15 (119871119889 minus 119871119902) 119894119898119871119889119869
) sin (1199091 [119896 + 1]) 1199062
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
)(
minus119877
119871119889
) sin (1199091 [119896 + 1])
sdot cos (1199091 [119896]) +(15 (119871119889 minus 119871119902) 119894
2119898
119869
)(119875
119871119902
119871119889
)
sdot sin (1199091 [119896 + 1]) sin (1199091 [119896]) (1199091 [119896 + 1] minus 1199091 [119896])
(13)
The above equation is rewritten to the below form by definingnew variables
1199102 [119896] = 119861111991021 +119861211991022 +119861311991023 +119861411991024 (14)
1198611 = (15119875120582119891119894119898
119869
)
1198612 = (15 (119871119889 minus 119871119902) 119894
2119898
119869
)(1+(minus119877119871119889
))
1198613 = (15 (119871119889 minus 119871119902) 119894119898
119871119889119869
)
1198614 = (15 (119871119889 minus 119871119902) 119894
2119898
119869
)(119875
119871119902
119871119889
)
1199102 [119896] = 1199091 [119896 + 3] minus 21199091 [119896 + 2] + 1199091 [119896 + 1]
11991021 = sin (1199091 [119896 + 1])
11991022 = sin (1199091 [119896 + 1]) cos (1199091 [119896])
11991023 = sin (1199091 [119896 + 1]) 1199062 [119896]
11991024
= sin (1199091 [119896 + 1]) sin (1199091 [119896]) (1199091 [119896 + 1] minus 1199091 [119896]) (15)
Finally a nonlinearmodel of PMSM iswritten to common lin-ear regression form like form number 1 considering equation(14) without any linearization in which 1198611 1198617 are factorsthat are embedded in the physical parameters of the systemand 11991021 11991024 are variables that are composed of nonlinearfunctions input and output
Equation (14) can be written to the linear regressionmatrix like form number 1
1199102 [119896] = 12060121198791205792 (16)
1206012 = [11991021 11991022 11991023 11991024]119879
1205792 = [1198611 1198612 1198613 1198614] (17)
In (13) 1206012 is estimator vector and 1205792 parameter vector relatedto form number 2
3 OPA and RLS Estimator
RLS is a powerful tool in system identification which isrobust against noise while OPA is an interesting methodwith high convergence speed with poor performance in noisyenvironment [24 25] Both advantages of these two methodscan be used simultaneously if they are combined Reference[26] is a good reference to understand the efficacy of thecombinatorial method which was applied on synchronousgenerator and good results were obtained The aim of thispaper is to estimate the physical parameter of the PMSMusingOPAampRLSmethod in noisy environmentwith arbitraryinitial values of physical parameters It is noted that nonlinearPMSM system is extremely sensitive to physical parametersvariations hence estimation of parameters of such systems isvery important by this algorithm
Estimation process is started with orthogonal projectionand orthogonal base vectors of estimator that are in the samedimension with the number of parameters (subspace of OPAestimation method) which are produced [20 24 25] Afterproducing estimation subspace RLS is used and last estima-tion of parameters in OPA method is applied to it which is agood initial estimation for RLS to increase the convergencespeed Then RLS continues the estimation process with agood accuracy to achieve physical parameters Stages of thecombinatorial algorithm are given in the following
Stage zero OPA is started with 119905 = 0 and 120576 = 0 andselection of initial parameters 1205790 and 1198750 covarianceidentity matrixStage one a step is added to 119905 (119905 = 119905+1) and estimatorvector 120601119905minus1 is calculatedStage two if the estimator vector of this stage is notlinearly dependent on the estimator vector of the
Mathematical Problems in Engineering 5
Table 1 Parameters of the understudy system [27]
119877 119871
119902119871
119889120582
119891119895 119901
287Ω 9mH 7mH 0175mWb 8 gm2 4
previous stage (120601119905minus1119879119901119905minus1120601119905minus1 = 0) we return to stage
three otherwise previous values of parameters aresubstituted (119901119905 = 119901119905minus1 120579119905 =
120579119905minus1) and we go to stageoneStage three 120579119905 and 119901119905 are obtained after recursiveequation of (18)Stage four if the rank of estimator matrix is equalto the number of parameters (Rank(Φ) = dim(120579))then OPA is finished in 119905OPA and we go to stage fiveotherwise we return to stage oneStage five to start the estimation with RLS method120576 = 1 and 120579119905OPA are considered as initial estimationStage six a step is added to 119905 (119905 = 119905 + 1) and estimatorvector 120601119905minus1 is calculatedStage seven 120579119905 and 119901119905 are updated using (18)Stage eight if the stop condition is not met we returnto stage six Consider the following
120579119905 =
120579119905minus1 +119875119905minus1 sdot 120601119905minus1
120576 + 120601119905minus1119879sdot 119875119905minus1 sdot 120601119905minus1
sdot [119910 [119905] minus
120579119905minus1119879
sdot 120601119905minus1]
119875119905 = 119875119905minus1 +119875119905minus1 sdot 120601119905minus1 sdot 120601119905minus1
119879sdot 119875119905minus1
120576 + 120601119905minus1119879sdot 119875119905minus1 sdot 120601119905minus1
(18)
4 Simulation
To validate the efficiency and accuracy of the combinatorialOPAampRLSmethod the proposed method is implemented ona PMSM with specifications of Table 1 [27]
The signal (PRBS) is applied to the input of system fordata production to identify and excite the dynamics and anoise with SNR = 15 is added to simulate a real conditionfor output of the system (load angle) which is shown inFigure 2 (amplitude and output) Simulation is performed inMATLAB software
To show the superiority of the proposed method it iscompared to OPA and RLS estimator The criteria for evalua-tion are the speed and accuracy estimator in converging to thereal parameters in a way that the second norm of parameterestimation error vector (1198712) is minimized
1198712 =1003817
1003817
1003817
1003817
1003817
120579 minus
120579
1003817
1003817
1003817
1003817
10038172 (19)
These criteria are evaluated and compared for the threemethods that is OPA RLS andOPAampRLSmethods for bothlinear regression forms from the second part with the sameinitial estimation and stop condition
This criterion is shown in Figure 1 for RLS andOPAampRLSmethods It is shown that this criterion is first decreased
0 2 4 6 8 10 12 14 16 18 200
5
10
15
Time
Am
plitu
de er
ror
Error OPAampRLS = 0026663mdasherror RLS = 59899
Form 1 OPAampRLSFirst OPASecond RLSForm 2 OPAampRLS
First OPASecond RLSForm 1 RLSForm 2 RLS
0 005 01 015 02 025 03 035 04 045 05012345678
Figure 1 Second norm of parameters estimation error in OPA
0 5 10 15minus50
050
Time
Am
plitu
dere
al o
utpu
t
0 5 10 15minus50
050
Time
Am
plitu
dees
timat
ion
outp
ut
0 5 10 150
0102
Time
Am
plitu
deer
ror
Figure 2 The figure shows output of real condition and estimatedmodel and their difference As shown estimated and real output
for OPA method with respect to speed which shows quickconvergence toward objective parameters but after that thistrend is not preserved and system would oscillate anddiverges due to noise It is shown from Figure 1 that thesecondnormof factors estimation error for RLS is descendingbut its slope is slight (low) which requires too many samplesfor parameters convergence However it is prominent that inOPAampRLSmethod parameters would leap greatly toward theobjective parameters with a quick decrease in initial samplesand after that it has a proper trend until the convergenceto real parameters and appropriate estimation which is veryproper for noisy condition This is because both features ofRLS and OPA are used in a single algorithm
As discussed in Section 2 to avoid nonlinear compli-cated calculation for estimating physical parameters tworegression forms are defined according to (9) and (13) Nowthe values of physical parameters are calculated with simplemathematics and given in Table 2
6 Mathematical Problems in Engineering
Table 2 Comparison between the convergence ofOPA andRLS andOPAampRLS methods with the same condition
119877 119871119902 119871119889 120582119891 119895 119901 TimeReal 2875 9 7 0175 8 4 mdashOPA minus0264 minus0331 81717 01547 minus0495 minus0180 20RLS 14925 71954 77590 01911 78661 32461 20OPAampRLS 2897 90650 69981 01744 8006 4012 20
Table 3 Investigation of normalized residue in three algorithms
The normalized residualOPA method 12023 (dB)RLS method 3023 (dB)OPAampRLS method minus3756 (dB)
Figure 2 shows output of real condition and estimatedmodel and their difference As shown estimated and realoutput are alike with agreeable accuracy
One of the criteria for evaluating the error of estimationis the normalized residue criterion according to (20) [28 29]Here this criterion is calculated and is given in Table 3 It isobserved that the proposed method is superior with respectto other methods Consider the following
119869dB = 10 log(sum
119873
1 (119910 (119905) minus (119905))2
sum
119873
1 119910 (119905)2 ) (20)
5 Conclusion
In this paper a new combinatorialmethod ofOrthogonal Pro-jection Algorithm and Recursive Least Squares (OPAampRLS)for parameter estimation of PMSM in noise-covered environ-ment is proposed To avoid solving complicated nonlinearequation two linear regression forms of fourth-order statespace PMSM were rewritten and OPAampRLS estimator wascompared to OPA and RLS estimators Speed and accuracy ofconvergence were investigated in three algorithms and it wasshown that OPAampRLS method has better results with respectto the two single methods The proposed algorithm can beapplied to any linear or nonlinear system where state spacemodel of the system is rewritten to linear regression form Itis also verified that this method is very appropriate for onlineestimation
Nomenclature
Constant
119877 Stator resistance119901 Pole-pairs120582119891 Magnet flux linkage119869 Inertia coefficient119871119889 Direct and quadrature inductances of Park
transformation
119871119902 Direct and quadrature inductances of Parktransformation
1205791 Parameter vector form 11205792 Parameter vector form 2
Variables
119894119902 Park transformation currents119894119889 Park transformation currents120596119890 Rotor angular speed120575 Rotor angular position119894119898 Nominal current1206011 Estimator vector form 11206012 Estimator vector form 21199101 Output form 11199102 Output form 2119901 Covariance matrix1198712 Second norm of parameter estimation error vector119869 Normalized residue
Indices
119896 Step sample time in discrete time119905 Time (sec)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Xu U Vollmer A Ebrahimi and N Parspour ldquoOnlineestimation of the stator resistances of a PMSM with considera-tion of magnetic saturationrdquo in Proceedings of the InternationalConference and Exposition on Electrical and Power Engineering(EPE rsquo12) pp 360ndash365 IEEE Iasi Romania October 2012
[2] K Liu Q Zhang J Chen Z Q Zhu and J Zhang ldquoOnlinemultiparameter estimation of nonsalient-pole PM synchronousmachines with temperature variation trackingrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 5 pp 1776ndash1788 2011
[3] QMeng T Zhang J-FHe J-Y Song and J-WHan ldquoDynamicmodeling of a 6-degree-of-freedom Stewart platform driven bya permanent magnet synchronous motorrdquo Journal of ZhejiangUniversity Science C vol 11 no 10 pp 751ndash761 2010
[4] E G Shehata ldquoSpeed sensorless torque control of an IPMSMdrive with online stator resistance estimation using reducedorder EKFrdquo International Journal of Electrical Power amp EnergySystems vol 47 no 1 pp 378ndash386 2013
[5] N Ozturk and E Celik ldquoSpeed control of permanent magnetsynchronous motors using fuzzy controller based on geneticalgorithmsrdquo International Journal of Electrical Power amp EnergySystems vol 43 no 1 pp 889ndash898 2012
[6] P Pillay and R Krishnan ldquoModeling of permanent magnetmotor drivesrdquo IEEE Transactions on Industrial Electronics vol35 no 4 pp 537ndash541 1988
[7] O Aydogmus and S Sunter ldquoImplementation of EKF basedsensorless drive system using vector controlled PMSM fed by amatrix converterrdquo International Journal of Electrical Power andEnergy Systems vol 43 no 1 pp 736ndash743 2012
Mathematical Problems in Engineering 7
[8] L Liu and D A Cartes ldquoSynchronisation based adaptiveparameter identification for permanent magnet synchronousmotorsrdquo IET Control Theory and Applications vol 1 no 4 pp1015ndash1022 2007
[9] C-H Lin andC-P Lin ldquoThe hybrid RFNN control for a PMSMdrive electric scooter using rotor flux estimatorrdquo ElectricalPower and Energy Systems vol 51 pp 889ndash898 2012
[10] F Aymen N Martin L Sbita and J Novak ldquoOnline PMSMparameters estimation for 32000rpmrdquo in Proceedings of theInternational Conference on Control Engineering amp InformationTechnology (CEIT rsquo13) vol 3 pp 148ndash152 Sousse Tunisia June2013
[11] G Junli and Z Yingfan ldquoResearch on parameter identificationand PI self-tuning of PMSMrdquo in Proceedings of the 2nd Inter-national Conference on Information Science and Engineering(ICISE rsquo10) pp 5251ndash5254 December 2010
[12] G Gatto I Marongiu and A Serpi ldquoDiscrete-time parameteridentification of a surface-mounted permanent magnet syn-chronousmachinerdquo IEEE Transactions on Industrial Electronicsvol 60 no 11 pp 4869ndash4880 2013
[13] K Liu and Z Q Zhu ldquoOnline estimation of the rotor fluxlinkage and voltage-source inverter nonlinearity in permanentmagnet synchronous machine drivesrdquo IEEE Transactions onPower Electronics vol 29 no 1 pp 418ndash427 2014
[14] Y Shi K Sun L Huang and Y Li ldquoOnline identification ofpermanent magnet flux based on extended Kalman filter forIPMSM drive with position sensorless controlrdquo IEEE Transac-tions on Industrial Electronics vol 59 no 11 pp 4169ndash4178 2012
[15] B Nahid Mobarakeh F Meibody-Tabar and F M Sargos ldquoOn-line identification of PMSM electrical parameters based ondecoupling controlrdquo in Proceedings of the Conference Recordof the IEEE Industry Applications Conference 36th IAS AnnualMeeting vol 1 pp 266ndash273 Chicago Ill USA September 2001
[16] T Boileau N Leboeuf B Nahid-Mobarakeh and F Meibody-Tabar ldquoOnline identification of PMSM parameters parameteridentifiability and estimator comparative studyrdquo IEEE Transac-tions on Industry Applications vol 47 no 4 pp 1944ndash1957 2011
[17] T Boileau B Nahid-Mobarakeh and F Meibody-Tabar ldquoOn-line identification of PMSM parameters model-reference vsEKFrdquo in Proceedings of the IEEE Industry Applications SocietyAnnual Meeting (IAS rsquo08) pp 1ndash8 can October 2008
[18] K Liu Z Q Zhu and D A Stone ldquoParameter estimation forcondition monitoring of PMSM stator winding and rotor per-manent magnetsrdquo IEEE Transactions on Industrial Electronicsvol 60 no 12 pp 5902ndash5913 2013
[19] C Cai F Chu ZWang andK Jia ldquoIdentification and control ofPMSM using adaptive BP-PID neural networkrdquo in Advances inNeural Networks vol 7952 of Lecture Notes in Computer Sciencepp 155ndash162 Springer 2013
[20] M Ghanbari I Yousefi and V Mossadegh ldquoOnline parameterestimation of permanent magnet synchronous motor usingorthogonal projection algorithmrdquo Indian Journal ScientificResearch vol 2 no 1 pp 32ndash36 2014
[21] A K Parvathy R Devanathan and V Kamaraj ldquoApplicationof quadratic linearization to control of permanent magnetsynchronous motorrdquo in Proceedings of the Joint InternationalConference on Power System Technology and IEEE Power IndiaConference pp 158ndash163 2008
[22] M A Hamida J de Leon A Glumineau and R BoisliveauldquoAn adaptive interconnected observer for sensorless control ofPM synchronous motors with online parameter identificationrdquo
IEEE Transactions on Industrial Electronics vol 60 no 2 pp739ndash748 2013
[23] R Krishnan Permanent Magnet Synchronous and Brushless DCMotor Drives Taylor amp Francis 2010
[24] J Zhao and I Kanellakopoulos ldquoActive identification fordiscrete-time nonlinear control I Output-feedback systemsrdquoIEEE Transactions on Automatic Control vol 47 no 2 pp 210ndash224 2002
[25] H Agahi M Karrari W Rosehart and O P Malik ldquoApplica-tion of active identification method to synchronous generatorparameter estimationrdquo in Proceedings of the IEEE Power Engi-neering Society General Meeting (PES rsquo06) Montreal CanadaJune 2006
[26] HAgahiActive identification for nonlinearmultivariable systemidentification and its application for synchronous generator iden-tification [PhD thesis] Amirkabir University of TechnologyTehran Iran 2007
[27] J ChiassonModeling and High Performance Control of ElectricMachines IEEE Press Series on Power Engineering IEEE Press2005
[28] M Ghanbari and M Haeri ldquoParametric identification offractional-order systems using a fractional Legendre basisrdquoProceedings of the Institution of Mechanical Engineers Part IJournal of Systems and Control Engineering vol 224 no 3 pp261ndash274 2010
[29] M Ghanbari and M Haeri ldquoOrder and pole locator estimationin fractional order systems using bode diagramrdquo Signal Process-ing vol 91 no 2 pp 191ndash202 2011
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
4 Mathematical Problems in Engineering
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
)(
minus119877
119871119889
)
sdot sin (1199091 [119896 + 1]) cos (1199091 [119905])
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
)(119875
119871119902
119871119889
)
sdot sin (1199091 [119896 + 1]) sin (1199091 [119896]) 1199092 [119896] (12)
Equation (12) is rewritten in terms of load angle by using therelation between the first and second state variables
1199091 [119896 + 3] minus 1199091 [119896 + 2] = 1199091 [119896 + 2] minus 1199091 [119896 + 1]
+(
15119875120582119891119894119898119869
) sin (1199091 [119896 + 1])
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
) sin (1199091 [119896 + 1]) cos (1199091 [119896])
+(
15 (119871119889 minus 119871119902) 119894119898119871119889119869
) sin (1199091 [119896 + 1]) 1199062
+(
15 (119871119889 minus 119871119902) 1198942119898
119869
)(
minus119877
119871119889
) sin (1199091 [119896 + 1])
sdot cos (1199091 [119896]) +(15 (119871119889 minus 119871119902) 119894
2119898
119869
)(119875
119871119902
119871119889
)
sdot sin (1199091 [119896 + 1]) sin (1199091 [119896]) (1199091 [119896 + 1] minus 1199091 [119896])
(13)
The above equation is rewritten to the below form by definingnew variables
1199102 [119896] = 119861111991021 +119861211991022 +119861311991023 +119861411991024 (14)
1198611 = (15119875120582119891119894119898
119869
)
1198612 = (15 (119871119889 minus 119871119902) 119894
2119898
119869
)(1+(minus119877119871119889
))
1198613 = (15 (119871119889 minus 119871119902) 119894119898
119871119889119869
)
1198614 = (15 (119871119889 minus 119871119902) 119894
2119898
119869
)(119875
119871119902
119871119889
)
1199102 [119896] = 1199091 [119896 + 3] minus 21199091 [119896 + 2] + 1199091 [119896 + 1]
11991021 = sin (1199091 [119896 + 1])
11991022 = sin (1199091 [119896 + 1]) cos (1199091 [119896])
11991023 = sin (1199091 [119896 + 1]) 1199062 [119896]
11991024
= sin (1199091 [119896 + 1]) sin (1199091 [119896]) (1199091 [119896 + 1] minus 1199091 [119896]) (15)
Finally a nonlinearmodel of PMSM iswritten to common lin-ear regression form like form number 1 considering equation(14) without any linearization in which 1198611 1198617 are factorsthat are embedded in the physical parameters of the systemand 11991021 11991024 are variables that are composed of nonlinearfunctions input and output
Equation (14) can be written to the linear regressionmatrix like form number 1
1199102 [119896] = 12060121198791205792 (16)
1206012 = [11991021 11991022 11991023 11991024]119879
1205792 = [1198611 1198612 1198613 1198614] (17)
In (13) 1206012 is estimator vector and 1205792 parameter vector relatedto form number 2
3 OPA and RLS Estimator
RLS is a powerful tool in system identification which isrobust against noise while OPA is an interesting methodwith high convergence speed with poor performance in noisyenvironment [24 25] Both advantages of these two methodscan be used simultaneously if they are combined Reference[26] is a good reference to understand the efficacy of thecombinatorial method which was applied on synchronousgenerator and good results were obtained The aim of thispaper is to estimate the physical parameter of the PMSMusingOPAampRLSmethod in noisy environmentwith arbitraryinitial values of physical parameters It is noted that nonlinearPMSM system is extremely sensitive to physical parametersvariations hence estimation of parameters of such systems isvery important by this algorithm
Estimation process is started with orthogonal projectionand orthogonal base vectors of estimator that are in the samedimension with the number of parameters (subspace of OPAestimation method) which are produced [20 24 25] Afterproducing estimation subspace RLS is used and last estima-tion of parameters in OPA method is applied to it which is agood initial estimation for RLS to increase the convergencespeed Then RLS continues the estimation process with agood accuracy to achieve physical parameters Stages of thecombinatorial algorithm are given in the following
Stage zero OPA is started with 119905 = 0 and 120576 = 0 andselection of initial parameters 1205790 and 1198750 covarianceidentity matrixStage one a step is added to 119905 (119905 = 119905+1) and estimatorvector 120601119905minus1 is calculatedStage two if the estimator vector of this stage is notlinearly dependent on the estimator vector of the
Mathematical Problems in Engineering 5
Table 1 Parameters of the understudy system [27]
119877 119871
119902119871
119889120582
119891119895 119901
287Ω 9mH 7mH 0175mWb 8 gm2 4
previous stage (120601119905minus1119879119901119905minus1120601119905minus1 = 0) we return to stage
three otherwise previous values of parameters aresubstituted (119901119905 = 119901119905minus1 120579119905 =
120579119905minus1) and we go to stageoneStage three 120579119905 and 119901119905 are obtained after recursiveequation of (18)Stage four if the rank of estimator matrix is equalto the number of parameters (Rank(Φ) = dim(120579))then OPA is finished in 119905OPA and we go to stage fiveotherwise we return to stage oneStage five to start the estimation with RLS method120576 = 1 and 120579119905OPA are considered as initial estimationStage six a step is added to 119905 (119905 = 119905 + 1) and estimatorvector 120601119905minus1 is calculatedStage seven 120579119905 and 119901119905 are updated using (18)Stage eight if the stop condition is not met we returnto stage six Consider the following
120579119905 =
120579119905minus1 +119875119905minus1 sdot 120601119905minus1
120576 + 120601119905minus1119879sdot 119875119905minus1 sdot 120601119905minus1
sdot [119910 [119905] minus
120579119905minus1119879
sdot 120601119905minus1]
119875119905 = 119875119905minus1 +119875119905minus1 sdot 120601119905minus1 sdot 120601119905minus1
119879sdot 119875119905minus1
120576 + 120601119905minus1119879sdot 119875119905minus1 sdot 120601119905minus1
(18)
4 Simulation
To validate the efficiency and accuracy of the combinatorialOPAampRLSmethod the proposed method is implemented ona PMSM with specifications of Table 1 [27]
The signal (PRBS) is applied to the input of system fordata production to identify and excite the dynamics and anoise with SNR = 15 is added to simulate a real conditionfor output of the system (load angle) which is shown inFigure 2 (amplitude and output) Simulation is performed inMATLAB software
To show the superiority of the proposed method it iscompared to OPA and RLS estimator The criteria for evalua-tion are the speed and accuracy estimator in converging to thereal parameters in a way that the second norm of parameterestimation error vector (1198712) is minimized
1198712 =1003817
1003817
1003817
1003817
1003817
120579 minus
120579
1003817
1003817
1003817
1003817
10038172 (19)
These criteria are evaluated and compared for the threemethods that is OPA RLS andOPAampRLSmethods for bothlinear regression forms from the second part with the sameinitial estimation and stop condition
This criterion is shown in Figure 1 for RLS andOPAampRLSmethods It is shown that this criterion is first decreased
0 2 4 6 8 10 12 14 16 18 200
5
10
15
Time
Am
plitu
de er
ror
Error OPAampRLS = 0026663mdasherror RLS = 59899
Form 1 OPAampRLSFirst OPASecond RLSForm 2 OPAampRLS
First OPASecond RLSForm 1 RLSForm 2 RLS
0 005 01 015 02 025 03 035 04 045 05012345678
Figure 1 Second norm of parameters estimation error in OPA
0 5 10 15minus50
050
Time
Am
plitu
dere
al o
utpu
t
0 5 10 15minus50
050
Time
Am
plitu
dees
timat
ion
outp
ut
0 5 10 150
0102
Time
Am
plitu
deer
ror
Figure 2 The figure shows output of real condition and estimatedmodel and their difference As shown estimated and real output
for OPA method with respect to speed which shows quickconvergence toward objective parameters but after that thistrend is not preserved and system would oscillate anddiverges due to noise It is shown from Figure 1 that thesecondnormof factors estimation error for RLS is descendingbut its slope is slight (low) which requires too many samplesfor parameters convergence However it is prominent that inOPAampRLSmethod parameters would leap greatly toward theobjective parameters with a quick decrease in initial samplesand after that it has a proper trend until the convergenceto real parameters and appropriate estimation which is veryproper for noisy condition This is because both features ofRLS and OPA are used in a single algorithm
As discussed in Section 2 to avoid nonlinear compli-cated calculation for estimating physical parameters tworegression forms are defined according to (9) and (13) Nowthe values of physical parameters are calculated with simplemathematics and given in Table 2
6 Mathematical Problems in Engineering
Table 2 Comparison between the convergence ofOPA andRLS andOPAampRLS methods with the same condition
119877 119871119902 119871119889 120582119891 119895 119901 TimeReal 2875 9 7 0175 8 4 mdashOPA minus0264 minus0331 81717 01547 minus0495 minus0180 20RLS 14925 71954 77590 01911 78661 32461 20OPAampRLS 2897 90650 69981 01744 8006 4012 20
Table 3 Investigation of normalized residue in three algorithms
The normalized residualOPA method 12023 (dB)RLS method 3023 (dB)OPAampRLS method minus3756 (dB)
Figure 2 shows output of real condition and estimatedmodel and their difference As shown estimated and realoutput are alike with agreeable accuracy
One of the criteria for evaluating the error of estimationis the normalized residue criterion according to (20) [28 29]Here this criterion is calculated and is given in Table 3 It isobserved that the proposed method is superior with respectto other methods Consider the following
119869dB = 10 log(sum
119873
1 (119910 (119905) minus (119905))2
sum
119873
1 119910 (119905)2 ) (20)
5 Conclusion
In this paper a new combinatorialmethod ofOrthogonal Pro-jection Algorithm and Recursive Least Squares (OPAampRLS)for parameter estimation of PMSM in noise-covered environ-ment is proposed To avoid solving complicated nonlinearequation two linear regression forms of fourth-order statespace PMSM were rewritten and OPAampRLS estimator wascompared to OPA and RLS estimators Speed and accuracy ofconvergence were investigated in three algorithms and it wasshown that OPAampRLS method has better results with respectto the two single methods The proposed algorithm can beapplied to any linear or nonlinear system where state spacemodel of the system is rewritten to linear regression form Itis also verified that this method is very appropriate for onlineestimation
Nomenclature
Constant
119877 Stator resistance119901 Pole-pairs120582119891 Magnet flux linkage119869 Inertia coefficient119871119889 Direct and quadrature inductances of Park
transformation
119871119902 Direct and quadrature inductances of Parktransformation
1205791 Parameter vector form 11205792 Parameter vector form 2
Variables
119894119902 Park transformation currents119894119889 Park transformation currents120596119890 Rotor angular speed120575 Rotor angular position119894119898 Nominal current1206011 Estimator vector form 11206012 Estimator vector form 21199101 Output form 11199102 Output form 2119901 Covariance matrix1198712 Second norm of parameter estimation error vector119869 Normalized residue
Indices
119896 Step sample time in discrete time119905 Time (sec)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Xu U Vollmer A Ebrahimi and N Parspour ldquoOnlineestimation of the stator resistances of a PMSM with considera-tion of magnetic saturationrdquo in Proceedings of the InternationalConference and Exposition on Electrical and Power Engineering(EPE rsquo12) pp 360ndash365 IEEE Iasi Romania October 2012
[2] K Liu Q Zhang J Chen Z Q Zhu and J Zhang ldquoOnlinemultiparameter estimation of nonsalient-pole PM synchronousmachines with temperature variation trackingrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 5 pp 1776ndash1788 2011
[3] QMeng T Zhang J-FHe J-Y Song and J-WHan ldquoDynamicmodeling of a 6-degree-of-freedom Stewart platform driven bya permanent magnet synchronous motorrdquo Journal of ZhejiangUniversity Science C vol 11 no 10 pp 751ndash761 2010
[4] E G Shehata ldquoSpeed sensorless torque control of an IPMSMdrive with online stator resistance estimation using reducedorder EKFrdquo International Journal of Electrical Power amp EnergySystems vol 47 no 1 pp 378ndash386 2013
[5] N Ozturk and E Celik ldquoSpeed control of permanent magnetsynchronous motors using fuzzy controller based on geneticalgorithmsrdquo International Journal of Electrical Power amp EnergySystems vol 43 no 1 pp 889ndash898 2012
[6] P Pillay and R Krishnan ldquoModeling of permanent magnetmotor drivesrdquo IEEE Transactions on Industrial Electronics vol35 no 4 pp 537ndash541 1988
[7] O Aydogmus and S Sunter ldquoImplementation of EKF basedsensorless drive system using vector controlled PMSM fed by amatrix converterrdquo International Journal of Electrical Power andEnergy Systems vol 43 no 1 pp 736ndash743 2012
Mathematical Problems in Engineering 7
[8] L Liu and D A Cartes ldquoSynchronisation based adaptiveparameter identification for permanent magnet synchronousmotorsrdquo IET Control Theory and Applications vol 1 no 4 pp1015ndash1022 2007
[9] C-H Lin andC-P Lin ldquoThe hybrid RFNN control for a PMSMdrive electric scooter using rotor flux estimatorrdquo ElectricalPower and Energy Systems vol 51 pp 889ndash898 2012
[10] F Aymen N Martin L Sbita and J Novak ldquoOnline PMSMparameters estimation for 32000rpmrdquo in Proceedings of theInternational Conference on Control Engineering amp InformationTechnology (CEIT rsquo13) vol 3 pp 148ndash152 Sousse Tunisia June2013
[11] G Junli and Z Yingfan ldquoResearch on parameter identificationand PI self-tuning of PMSMrdquo in Proceedings of the 2nd Inter-national Conference on Information Science and Engineering(ICISE rsquo10) pp 5251ndash5254 December 2010
[12] G Gatto I Marongiu and A Serpi ldquoDiscrete-time parameteridentification of a surface-mounted permanent magnet syn-chronousmachinerdquo IEEE Transactions on Industrial Electronicsvol 60 no 11 pp 4869ndash4880 2013
[13] K Liu and Z Q Zhu ldquoOnline estimation of the rotor fluxlinkage and voltage-source inverter nonlinearity in permanentmagnet synchronous machine drivesrdquo IEEE Transactions onPower Electronics vol 29 no 1 pp 418ndash427 2014
[14] Y Shi K Sun L Huang and Y Li ldquoOnline identification ofpermanent magnet flux based on extended Kalman filter forIPMSM drive with position sensorless controlrdquo IEEE Transac-tions on Industrial Electronics vol 59 no 11 pp 4169ndash4178 2012
[15] B Nahid Mobarakeh F Meibody-Tabar and F M Sargos ldquoOn-line identification of PMSM electrical parameters based ondecoupling controlrdquo in Proceedings of the Conference Recordof the IEEE Industry Applications Conference 36th IAS AnnualMeeting vol 1 pp 266ndash273 Chicago Ill USA September 2001
[16] T Boileau N Leboeuf B Nahid-Mobarakeh and F Meibody-Tabar ldquoOnline identification of PMSM parameters parameteridentifiability and estimator comparative studyrdquo IEEE Transac-tions on Industry Applications vol 47 no 4 pp 1944ndash1957 2011
[17] T Boileau B Nahid-Mobarakeh and F Meibody-Tabar ldquoOn-line identification of PMSM parameters model-reference vsEKFrdquo in Proceedings of the IEEE Industry Applications SocietyAnnual Meeting (IAS rsquo08) pp 1ndash8 can October 2008
[18] K Liu Z Q Zhu and D A Stone ldquoParameter estimation forcondition monitoring of PMSM stator winding and rotor per-manent magnetsrdquo IEEE Transactions on Industrial Electronicsvol 60 no 12 pp 5902ndash5913 2013
[19] C Cai F Chu ZWang andK Jia ldquoIdentification and control ofPMSM using adaptive BP-PID neural networkrdquo in Advances inNeural Networks vol 7952 of Lecture Notes in Computer Sciencepp 155ndash162 Springer 2013
[20] M Ghanbari I Yousefi and V Mossadegh ldquoOnline parameterestimation of permanent magnet synchronous motor usingorthogonal projection algorithmrdquo Indian Journal ScientificResearch vol 2 no 1 pp 32ndash36 2014
[21] A K Parvathy R Devanathan and V Kamaraj ldquoApplicationof quadratic linearization to control of permanent magnetsynchronous motorrdquo in Proceedings of the Joint InternationalConference on Power System Technology and IEEE Power IndiaConference pp 158ndash163 2008
[22] M A Hamida J de Leon A Glumineau and R BoisliveauldquoAn adaptive interconnected observer for sensorless control ofPM synchronous motors with online parameter identificationrdquo
IEEE Transactions on Industrial Electronics vol 60 no 2 pp739ndash748 2013
[23] R Krishnan Permanent Magnet Synchronous and Brushless DCMotor Drives Taylor amp Francis 2010
[24] J Zhao and I Kanellakopoulos ldquoActive identification fordiscrete-time nonlinear control I Output-feedback systemsrdquoIEEE Transactions on Automatic Control vol 47 no 2 pp 210ndash224 2002
[25] H Agahi M Karrari W Rosehart and O P Malik ldquoApplica-tion of active identification method to synchronous generatorparameter estimationrdquo in Proceedings of the IEEE Power Engi-neering Society General Meeting (PES rsquo06) Montreal CanadaJune 2006
[26] HAgahiActive identification for nonlinearmultivariable systemidentification and its application for synchronous generator iden-tification [PhD thesis] Amirkabir University of TechnologyTehran Iran 2007
[27] J ChiassonModeling and High Performance Control of ElectricMachines IEEE Press Series on Power Engineering IEEE Press2005
[28] M Ghanbari and M Haeri ldquoParametric identification offractional-order systems using a fractional Legendre basisrdquoProceedings of the Institution of Mechanical Engineers Part IJournal of Systems and Control Engineering vol 224 no 3 pp261ndash274 2010
[29] M Ghanbari and M Haeri ldquoOrder and pole locator estimationin fractional order systems using bode diagramrdquo Signal Process-ing vol 91 no 2 pp 191ndash202 2011
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
Mathematical Problems in Engineering 5
Table 1 Parameters of the understudy system [27]
119877 119871
119902119871
119889120582
119891119895 119901
287Ω 9mH 7mH 0175mWb 8 gm2 4
previous stage (120601119905minus1119879119901119905minus1120601119905minus1 = 0) we return to stage
three otherwise previous values of parameters aresubstituted (119901119905 = 119901119905minus1 120579119905 =
120579119905minus1) and we go to stageoneStage three 120579119905 and 119901119905 are obtained after recursiveequation of (18)Stage four if the rank of estimator matrix is equalto the number of parameters (Rank(Φ) = dim(120579))then OPA is finished in 119905OPA and we go to stage fiveotherwise we return to stage oneStage five to start the estimation with RLS method120576 = 1 and 120579119905OPA are considered as initial estimationStage six a step is added to 119905 (119905 = 119905 + 1) and estimatorvector 120601119905minus1 is calculatedStage seven 120579119905 and 119901119905 are updated using (18)Stage eight if the stop condition is not met we returnto stage six Consider the following
120579119905 =
120579119905minus1 +119875119905minus1 sdot 120601119905minus1
120576 + 120601119905minus1119879sdot 119875119905minus1 sdot 120601119905minus1
sdot [119910 [119905] minus
120579119905minus1119879
sdot 120601119905minus1]
119875119905 = 119875119905minus1 +119875119905minus1 sdot 120601119905minus1 sdot 120601119905minus1
119879sdot 119875119905minus1
120576 + 120601119905minus1119879sdot 119875119905minus1 sdot 120601119905minus1
(18)
4 Simulation
To validate the efficiency and accuracy of the combinatorialOPAampRLSmethod the proposed method is implemented ona PMSM with specifications of Table 1 [27]
The signal (PRBS) is applied to the input of system fordata production to identify and excite the dynamics and anoise with SNR = 15 is added to simulate a real conditionfor output of the system (load angle) which is shown inFigure 2 (amplitude and output) Simulation is performed inMATLAB software
To show the superiority of the proposed method it iscompared to OPA and RLS estimator The criteria for evalua-tion are the speed and accuracy estimator in converging to thereal parameters in a way that the second norm of parameterestimation error vector (1198712) is minimized
1198712 =1003817
1003817
1003817
1003817
1003817
120579 minus
120579
1003817
1003817
1003817
1003817
10038172 (19)
These criteria are evaluated and compared for the threemethods that is OPA RLS andOPAampRLSmethods for bothlinear regression forms from the second part with the sameinitial estimation and stop condition
This criterion is shown in Figure 1 for RLS andOPAampRLSmethods It is shown that this criterion is first decreased
0 2 4 6 8 10 12 14 16 18 200
5
10
15
Time
Am
plitu
de er
ror
Error OPAampRLS = 0026663mdasherror RLS = 59899
Form 1 OPAampRLSFirst OPASecond RLSForm 2 OPAampRLS
First OPASecond RLSForm 1 RLSForm 2 RLS
0 005 01 015 02 025 03 035 04 045 05012345678
Figure 1 Second norm of parameters estimation error in OPA
0 5 10 15minus50
050
Time
Am
plitu
dere
al o
utpu
t
0 5 10 15minus50
050
Time
Am
plitu
dees
timat
ion
outp
ut
0 5 10 150
0102
Time
Am
plitu
deer
ror
Figure 2 The figure shows output of real condition and estimatedmodel and their difference As shown estimated and real output
for OPA method with respect to speed which shows quickconvergence toward objective parameters but after that thistrend is not preserved and system would oscillate anddiverges due to noise It is shown from Figure 1 that thesecondnormof factors estimation error for RLS is descendingbut its slope is slight (low) which requires too many samplesfor parameters convergence However it is prominent that inOPAampRLSmethod parameters would leap greatly toward theobjective parameters with a quick decrease in initial samplesand after that it has a proper trend until the convergenceto real parameters and appropriate estimation which is veryproper for noisy condition This is because both features ofRLS and OPA are used in a single algorithm
As discussed in Section 2 to avoid nonlinear compli-cated calculation for estimating physical parameters tworegression forms are defined according to (9) and (13) Nowthe values of physical parameters are calculated with simplemathematics and given in Table 2
6 Mathematical Problems in Engineering
Table 2 Comparison between the convergence ofOPA andRLS andOPAampRLS methods with the same condition
119877 119871119902 119871119889 120582119891 119895 119901 TimeReal 2875 9 7 0175 8 4 mdashOPA minus0264 minus0331 81717 01547 minus0495 minus0180 20RLS 14925 71954 77590 01911 78661 32461 20OPAampRLS 2897 90650 69981 01744 8006 4012 20
Table 3 Investigation of normalized residue in three algorithms
The normalized residualOPA method 12023 (dB)RLS method 3023 (dB)OPAampRLS method minus3756 (dB)
Figure 2 shows output of real condition and estimatedmodel and their difference As shown estimated and realoutput are alike with agreeable accuracy
One of the criteria for evaluating the error of estimationis the normalized residue criterion according to (20) [28 29]Here this criterion is calculated and is given in Table 3 It isobserved that the proposed method is superior with respectto other methods Consider the following
119869dB = 10 log(sum
119873
1 (119910 (119905) minus (119905))2
sum
119873
1 119910 (119905)2 ) (20)
5 Conclusion
In this paper a new combinatorialmethod ofOrthogonal Pro-jection Algorithm and Recursive Least Squares (OPAampRLS)for parameter estimation of PMSM in noise-covered environ-ment is proposed To avoid solving complicated nonlinearequation two linear regression forms of fourth-order statespace PMSM were rewritten and OPAampRLS estimator wascompared to OPA and RLS estimators Speed and accuracy ofconvergence were investigated in three algorithms and it wasshown that OPAampRLS method has better results with respectto the two single methods The proposed algorithm can beapplied to any linear or nonlinear system where state spacemodel of the system is rewritten to linear regression form Itis also verified that this method is very appropriate for onlineestimation
Nomenclature
Constant
119877 Stator resistance119901 Pole-pairs120582119891 Magnet flux linkage119869 Inertia coefficient119871119889 Direct and quadrature inductances of Park
transformation
119871119902 Direct and quadrature inductances of Parktransformation
1205791 Parameter vector form 11205792 Parameter vector form 2
Variables
119894119902 Park transformation currents119894119889 Park transformation currents120596119890 Rotor angular speed120575 Rotor angular position119894119898 Nominal current1206011 Estimator vector form 11206012 Estimator vector form 21199101 Output form 11199102 Output form 2119901 Covariance matrix1198712 Second norm of parameter estimation error vector119869 Normalized residue
Indices
119896 Step sample time in discrete time119905 Time (sec)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Xu U Vollmer A Ebrahimi and N Parspour ldquoOnlineestimation of the stator resistances of a PMSM with considera-tion of magnetic saturationrdquo in Proceedings of the InternationalConference and Exposition on Electrical and Power Engineering(EPE rsquo12) pp 360ndash365 IEEE Iasi Romania October 2012
[2] K Liu Q Zhang J Chen Z Q Zhu and J Zhang ldquoOnlinemultiparameter estimation of nonsalient-pole PM synchronousmachines with temperature variation trackingrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 5 pp 1776ndash1788 2011
[3] QMeng T Zhang J-FHe J-Y Song and J-WHan ldquoDynamicmodeling of a 6-degree-of-freedom Stewart platform driven bya permanent magnet synchronous motorrdquo Journal of ZhejiangUniversity Science C vol 11 no 10 pp 751ndash761 2010
[4] E G Shehata ldquoSpeed sensorless torque control of an IPMSMdrive with online stator resistance estimation using reducedorder EKFrdquo International Journal of Electrical Power amp EnergySystems vol 47 no 1 pp 378ndash386 2013
[5] N Ozturk and E Celik ldquoSpeed control of permanent magnetsynchronous motors using fuzzy controller based on geneticalgorithmsrdquo International Journal of Electrical Power amp EnergySystems vol 43 no 1 pp 889ndash898 2012
[6] P Pillay and R Krishnan ldquoModeling of permanent magnetmotor drivesrdquo IEEE Transactions on Industrial Electronics vol35 no 4 pp 537ndash541 1988
[7] O Aydogmus and S Sunter ldquoImplementation of EKF basedsensorless drive system using vector controlled PMSM fed by amatrix converterrdquo International Journal of Electrical Power andEnergy Systems vol 43 no 1 pp 736ndash743 2012
Mathematical Problems in Engineering 7
[8] L Liu and D A Cartes ldquoSynchronisation based adaptiveparameter identification for permanent magnet synchronousmotorsrdquo IET Control Theory and Applications vol 1 no 4 pp1015ndash1022 2007
[9] C-H Lin andC-P Lin ldquoThe hybrid RFNN control for a PMSMdrive electric scooter using rotor flux estimatorrdquo ElectricalPower and Energy Systems vol 51 pp 889ndash898 2012
[10] F Aymen N Martin L Sbita and J Novak ldquoOnline PMSMparameters estimation for 32000rpmrdquo in Proceedings of theInternational Conference on Control Engineering amp InformationTechnology (CEIT rsquo13) vol 3 pp 148ndash152 Sousse Tunisia June2013
[11] G Junli and Z Yingfan ldquoResearch on parameter identificationand PI self-tuning of PMSMrdquo in Proceedings of the 2nd Inter-national Conference on Information Science and Engineering(ICISE rsquo10) pp 5251ndash5254 December 2010
[12] G Gatto I Marongiu and A Serpi ldquoDiscrete-time parameteridentification of a surface-mounted permanent magnet syn-chronousmachinerdquo IEEE Transactions on Industrial Electronicsvol 60 no 11 pp 4869ndash4880 2013
[13] K Liu and Z Q Zhu ldquoOnline estimation of the rotor fluxlinkage and voltage-source inverter nonlinearity in permanentmagnet synchronous machine drivesrdquo IEEE Transactions onPower Electronics vol 29 no 1 pp 418ndash427 2014
[14] Y Shi K Sun L Huang and Y Li ldquoOnline identification ofpermanent magnet flux based on extended Kalman filter forIPMSM drive with position sensorless controlrdquo IEEE Transac-tions on Industrial Electronics vol 59 no 11 pp 4169ndash4178 2012
[15] B Nahid Mobarakeh F Meibody-Tabar and F M Sargos ldquoOn-line identification of PMSM electrical parameters based ondecoupling controlrdquo in Proceedings of the Conference Recordof the IEEE Industry Applications Conference 36th IAS AnnualMeeting vol 1 pp 266ndash273 Chicago Ill USA September 2001
[16] T Boileau N Leboeuf B Nahid-Mobarakeh and F Meibody-Tabar ldquoOnline identification of PMSM parameters parameteridentifiability and estimator comparative studyrdquo IEEE Transac-tions on Industry Applications vol 47 no 4 pp 1944ndash1957 2011
[17] T Boileau B Nahid-Mobarakeh and F Meibody-Tabar ldquoOn-line identification of PMSM parameters model-reference vsEKFrdquo in Proceedings of the IEEE Industry Applications SocietyAnnual Meeting (IAS rsquo08) pp 1ndash8 can October 2008
[18] K Liu Z Q Zhu and D A Stone ldquoParameter estimation forcondition monitoring of PMSM stator winding and rotor per-manent magnetsrdquo IEEE Transactions on Industrial Electronicsvol 60 no 12 pp 5902ndash5913 2013
[19] C Cai F Chu ZWang andK Jia ldquoIdentification and control ofPMSM using adaptive BP-PID neural networkrdquo in Advances inNeural Networks vol 7952 of Lecture Notes in Computer Sciencepp 155ndash162 Springer 2013
[20] M Ghanbari I Yousefi and V Mossadegh ldquoOnline parameterestimation of permanent magnet synchronous motor usingorthogonal projection algorithmrdquo Indian Journal ScientificResearch vol 2 no 1 pp 32ndash36 2014
[21] A K Parvathy R Devanathan and V Kamaraj ldquoApplicationof quadratic linearization to control of permanent magnetsynchronous motorrdquo in Proceedings of the Joint InternationalConference on Power System Technology and IEEE Power IndiaConference pp 158ndash163 2008
[22] M A Hamida J de Leon A Glumineau and R BoisliveauldquoAn adaptive interconnected observer for sensorless control ofPM synchronous motors with online parameter identificationrdquo
IEEE Transactions on Industrial Electronics vol 60 no 2 pp739ndash748 2013
[23] R Krishnan Permanent Magnet Synchronous and Brushless DCMotor Drives Taylor amp Francis 2010
[24] J Zhao and I Kanellakopoulos ldquoActive identification fordiscrete-time nonlinear control I Output-feedback systemsrdquoIEEE Transactions on Automatic Control vol 47 no 2 pp 210ndash224 2002
[25] H Agahi M Karrari W Rosehart and O P Malik ldquoApplica-tion of active identification method to synchronous generatorparameter estimationrdquo in Proceedings of the IEEE Power Engi-neering Society General Meeting (PES rsquo06) Montreal CanadaJune 2006
[26] HAgahiActive identification for nonlinearmultivariable systemidentification and its application for synchronous generator iden-tification [PhD thesis] Amirkabir University of TechnologyTehran Iran 2007
[27] J ChiassonModeling and High Performance Control of ElectricMachines IEEE Press Series on Power Engineering IEEE Press2005
[28] M Ghanbari and M Haeri ldquoParametric identification offractional-order systems using a fractional Legendre basisrdquoProceedings of the Institution of Mechanical Engineers Part IJournal of Systems and Control Engineering vol 224 no 3 pp261ndash274 2010
[29] M Ghanbari and M Haeri ldquoOrder and pole locator estimationin fractional order systems using bode diagramrdquo Signal Process-ing vol 91 no 2 pp 191ndash202 2011
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
6 Mathematical Problems in Engineering
Table 2 Comparison between the convergence ofOPA andRLS andOPAampRLS methods with the same condition
119877 119871119902 119871119889 120582119891 119895 119901 TimeReal 2875 9 7 0175 8 4 mdashOPA minus0264 minus0331 81717 01547 minus0495 minus0180 20RLS 14925 71954 77590 01911 78661 32461 20OPAampRLS 2897 90650 69981 01744 8006 4012 20
Table 3 Investigation of normalized residue in three algorithms
The normalized residualOPA method 12023 (dB)RLS method 3023 (dB)OPAampRLS method minus3756 (dB)
Figure 2 shows output of real condition and estimatedmodel and their difference As shown estimated and realoutput are alike with agreeable accuracy
One of the criteria for evaluating the error of estimationis the normalized residue criterion according to (20) [28 29]Here this criterion is calculated and is given in Table 3 It isobserved that the proposed method is superior with respectto other methods Consider the following
119869dB = 10 log(sum
119873
1 (119910 (119905) minus (119905))2
sum
119873
1 119910 (119905)2 ) (20)
5 Conclusion
In this paper a new combinatorialmethod ofOrthogonal Pro-jection Algorithm and Recursive Least Squares (OPAampRLS)for parameter estimation of PMSM in noise-covered environ-ment is proposed To avoid solving complicated nonlinearequation two linear regression forms of fourth-order statespace PMSM were rewritten and OPAampRLS estimator wascompared to OPA and RLS estimators Speed and accuracy ofconvergence were investigated in three algorithms and it wasshown that OPAampRLS method has better results with respectto the two single methods The proposed algorithm can beapplied to any linear or nonlinear system where state spacemodel of the system is rewritten to linear regression form Itis also verified that this method is very appropriate for onlineestimation
Nomenclature
Constant
119877 Stator resistance119901 Pole-pairs120582119891 Magnet flux linkage119869 Inertia coefficient119871119889 Direct and quadrature inductances of Park
transformation
119871119902 Direct and quadrature inductances of Parktransformation
1205791 Parameter vector form 11205792 Parameter vector form 2
Variables
119894119902 Park transformation currents119894119889 Park transformation currents120596119890 Rotor angular speed120575 Rotor angular position119894119898 Nominal current1206011 Estimator vector form 11206012 Estimator vector form 21199101 Output form 11199102 Output form 2119901 Covariance matrix1198712 Second norm of parameter estimation error vector119869 Normalized residue
Indices
119896 Step sample time in discrete time119905 Time (sec)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Xu U Vollmer A Ebrahimi and N Parspour ldquoOnlineestimation of the stator resistances of a PMSM with considera-tion of magnetic saturationrdquo in Proceedings of the InternationalConference and Exposition on Electrical and Power Engineering(EPE rsquo12) pp 360ndash365 IEEE Iasi Romania October 2012
[2] K Liu Q Zhang J Chen Z Q Zhu and J Zhang ldquoOnlinemultiparameter estimation of nonsalient-pole PM synchronousmachines with temperature variation trackingrdquo IEEE Transac-tions on Industrial Electronics vol 58 no 5 pp 1776ndash1788 2011
[3] QMeng T Zhang J-FHe J-Y Song and J-WHan ldquoDynamicmodeling of a 6-degree-of-freedom Stewart platform driven bya permanent magnet synchronous motorrdquo Journal of ZhejiangUniversity Science C vol 11 no 10 pp 751ndash761 2010
[4] E G Shehata ldquoSpeed sensorless torque control of an IPMSMdrive with online stator resistance estimation using reducedorder EKFrdquo International Journal of Electrical Power amp EnergySystems vol 47 no 1 pp 378ndash386 2013
[5] N Ozturk and E Celik ldquoSpeed control of permanent magnetsynchronous motors using fuzzy controller based on geneticalgorithmsrdquo International Journal of Electrical Power amp EnergySystems vol 43 no 1 pp 889ndash898 2012
[6] P Pillay and R Krishnan ldquoModeling of permanent magnetmotor drivesrdquo IEEE Transactions on Industrial Electronics vol35 no 4 pp 537ndash541 1988
[7] O Aydogmus and S Sunter ldquoImplementation of EKF basedsensorless drive system using vector controlled PMSM fed by amatrix converterrdquo International Journal of Electrical Power andEnergy Systems vol 43 no 1 pp 736ndash743 2012
Mathematical Problems in Engineering 7
[8] L Liu and D A Cartes ldquoSynchronisation based adaptiveparameter identification for permanent magnet synchronousmotorsrdquo IET Control Theory and Applications vol 1 no 4 pp1015ndash1022 2007
[9] C-H Lin andC-P Lin ldquoThe hybrid RFNN control for a PMSMdrive electric scooter using rotor flux estimatorrdquo ElectricalPower and Energy Systems vol 51 pp 889ndash898 2012
[10] F Aymen N Martin L Sbita and J Novak ldquoOnline PMSMparameters estimation for 32000rpmrdquo in Proceedings of theInternational Conference on Control Engineering amp InformationTechnology (CEIT rsquo13) vol 3 pp 148ndash152 Sousse Tunisia June2013
[11] G Junli and Z Yingfan ldquoResearch on parameter identificationand PI self-tuning of PMSMrdquo in Proceedings of the 2nd Inter-national Conference on Information Science and Engineering(ICISE rsquo10) pp 5251ndash5254 December 2010
[12] G Gatto I Marongiu and A Serpi ldquoDiscrete-time parameteridentification of a surface-mounted permanent magnet syn-chronousmachinerdquo IEEE Transactions on Industrial Electronicsvol 60 no 11 pp 4869ndash4880 2013
[13] K Liu and Z Q Zhu ldquoOnline estimation of the rotor fluxlinkage and voltage-source inverter nonlinearity in permanentmagnet synchronous machine drivesrdquo IEEE Transactions onPower Electronics vol 29 no 1 pp 418ndash427 2014
[14] Y Shi K Sun L Huang and Y Li ldquoOnline identification ofpermanent magnet flux based on extended Kalman filter forIPMSM drive with position sensorless controlrdquo IEEE Transac-tions on Industrial Electronics vol 59 no 11 pp 4169ndash4178 2012
[15] B Nahid Mobarakeh F Meibody-Tabar and F M Sargos ldquoOn-line identification of PMSM electrical parameters based ondecoupling controlrdquo in Proceedings of the Conference Recordof the IEEE Industry Applications Conference 36th IAS AnnualMeeting vol 1 pp 266ndash273 Chicago Ill USA September 2001
[16] T Boileau N Leboeuf B Nahid-Mobarakeh and F Meibody-Tabar ldquoOnline identification of PMSM parameters parameteridentifiability and estimator comparative studyrdquo IEEE Transac-tions on Industry Applications vol 47 no 4 pp 1944ndash1957 2011
[17] T Boileau B Nahid-Mobarakeh and F Meibody-Tabar ldquoOn-line identification of PMSM parameters model-reference vsEKFrdquo in Proceedings of the IEEE Industry Applications SocietyAnnual Meeting (IAS rsquo08) pp 1ndash8 can October 2008
[18] K Liu Z Q Zhu and D A Stone ldquoParameter estimation forcondition monitoring of PMSM stator winding and rotor per-manent magnetsrdquo IEEE Transactions on Industrial Electronicsvol 60 no 12 pp 5902ndash5913 2013
[19] C Cai F Chu ZWang andK Jia ldquoIdentification and control ofPMSM using adaptive BP-PID neural networkrdquo in Advances inNeural Networks vol 7952 of Lecture Notes in Computer Sciencepp 155ndash162 Springer 2013
[20] M Ghanbari I Yousefi and V Mossadegh ldquoOnline parameterestimation of permanent magnet synchronous motor usingorthogonal projection algorithmrdquo Indian Journal ScientificResearch vol 2 no 1 pp 32ndash36 2014
[21] A K Parvathy R Devanathan and V Kamaraj ldquoApplicationof quadratic linearization to control of permanent magnetsynchronous motorrdquo in Proceedings of the Joint InternationalConference on Power System Technology and IEEE Power IndiaConference pp 158ndash163 2008
[22] M A Hamida J de Leon A Glumineau and R BoisliveauldquoAn adaptive interconnected observer for sensorless control ofPM synchronous motors with online parameter identificationrdquo
IEEE Transactions on Industrial Electronics vol 60 no 2 pp739ndash748 2013
[23] R Krishnan Permanent Magnet Synchronous and Brushless DCMotor Drives Taylor amp Francis 2010
[24] J Zhao and I Kanellakopoulos ldquoActive identification fordiscrete-time nonlinear control I Output-feedback systemsrdquoIEEE Transactions on Automatic Control vol 47 no 2 pp 210ndash224 2002
[25] H Agahi M Karrari W Rosehart and O P Malik ldquoApplica-tion of active identification method to synchronous generatorparameter estimationrdquo in Proceedings of the IEEE Power Engi-neering Society General Meeting (PES rsquo06) Montreal CanadaJune 2006
[26] HAgahiActive identification for nonlinearmultivariable systemidentification and its application for synchronous generator iden-tification [PhD thesis] Amirkabir University of TechnologyTehran Iran 2007
[27] J ChiassonModeling and High Performance Control of ElectricMachines IEEE Press Series on Power Engineering IEEE Press2005
[28] M Ghanbari and M Haeri ldquoParametric identification offractional-order systems using a fractional Legendre basisrdquoProceedings of the Institution of Mechanical Engineers Part IJournal of Systems and Control Engineering vol 224 no 3 pp261ndash274 2010
[29] M Ghanbari and M Haeri ldquoOrder and pole locator estimationin fractional order systems using bode diagramrdquo Signal Process-ing vol 91 no 2 pp 191ndash202 2011
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
Mathematical Problems in Engineering 7
[8] L Liu and D A Cartes ldquoSynchronisation based adaptiveparameter identification for permanent magnet synchronousmotorsrdquo IET Control Theory and Applications vol 1 no 4 pp1015ndash1022 2007
[9] C-H Lin andC-P Lin ldquoThe hybrid RFNN control for a PMSMdrive electric scooter using rotor flux estimatorrdquo ElectricalPower and Energy Systems vol 51 pp 889ndash898 2012
[10] F Aymen N Martin L Sbita and J Novak ldquoOnline PMSMparameters estimation for 32000rpmrdquo in Proceedings of theInternational Conference on Control Engineering amp InformationTechnology (CEIT rsquo13) vol 3 pp 148ndash152 Sousse Tunisia June2013
[11] G Junli and Z Yingfan ldquoResearch on parameter identificationand PI self-tuning of PMSMrdquo in Proceedings of the 2nd Inter-national Conference on Information Science and Engineering(ICISE rsquo10) pp 5251ndash5254 December 2010
[12] G Gatto I Marongiu and A Serpi ldquoDiscrete-time parameteridentification of a surface-mounted permanent magnet syn-chronousmachinerdquo IEEE Transactions on Industrial Electronicsvol 60 no 11 pp 4869ndash4880 2013
[13] K Liu and Z Q Zhu ldquoOnline estimation of the rotor fluxlinkage and voltage-source inverter nonlinearity in permanentmagnet synchronous machine drivesrdquo IEEE Transactions onPower Electronics vol 29 no 1 pp 418ndash427 2014
[14] Y Shi K Sun L Huang and Y Li ldquoOnline identification ofpermanent magnet flux based on extended Kalman filter forIPMSM drive with position sensorless controlrdquo IEEE Transac-tions on Industrial Electronics vol 59 no 11 pp 4169ndash4178 2012
[15] B Nahid Mobarakeh F Meibody-Tabar and F M Sargos ldquoOn-line identification of PMSM electrical parameters based ondecoupling controlrdquo in Proceedings of the Conference Recordof the IEEE Industry Applications Conference 36th IAS AnnualMeeting vol 1 pp 266ndash273 Chicago Ill USA September 2001
[16] T Boileau N Leboeuf B Nahid-Mobarakeh and F Meibody-Tabar ldquoOnline identification of PMSM parameters parameteridentifiability and estimator comparative studyrdquo IEEE Transac-tions on Industry Applications vol 47 no 4 pp 1944ndash1957 2011
[17] T Boileau B Nahid-Mobarakeh and F Meibody-Tabar ldquoOn-line identification of PMSM parameters model-reference vsEKFrdquo in Proceedings of the IEEE Industry Applications SocietyAnnual Meeting (IAS rsquo08) pp 1ndash8 can October 2008
[18] K Liu Z Q Zhu and D A Stone ldquoParameter estimation forcondition monitoring of PMSM stator winding and rotor per-manent magnetsrdquo IEEE Transactions on Industrial Electronicsvol 60 no 12 pp 5902ndash5913 2013
[19] C Cai F Chu ZWang andK Jia ldquoIdentification and control ofPMSM using adaptive BP-PID neural networkrdquo in Advances inNeural Networks vol 7952 of Lecture Notes in Computer Sciencepp 155ndash162 Springer 2013
[20] M Ghanbari I Yousefi and V Mossadegh ldquoOnline parameterestimation of permanent magnet synchronous motor usingorthogonal projection algorithmrdquo Indian Journal ScientificResearch vol 2 no 1 pp 32ndash36 2014
[21] A K Parvathy R Devanathan and V Kamaraj ldquoApplicationof quadratic linearization to control of permanent magnetsynchronous motorrdquo in Proceedings of the Joint InternationalConference on Power System Technology and IEEE Power IndiaConference pp 158ndash163 2008
[22] M A Hamida J de Leon A Glumineau and R BoisliveauldquoAn adaptive interconnected observer for sensorless control ofPM synchronous motors with online parameter identificationrdquo
IEEE Transactions on Industrial Electronics vol 60 no 2 pp739ndash748 2013
[23] R Krishnan Permanent Magnet Synchronous and Brushless DCMotor Drives Taylor amp Francis 2010
[24] J Zhao and I Kanellakopoulos ldquoActive identification fordiscrete-time nonlinear control I Output-feedback systemsrdquoIEEE Transactions on Automatic Control vol 47 no 2 pp 210ndash224 2002
[25] H Agahi M Karrari W Rosehart and O P Malik ldquoApplica-tion of active identification method to synchronous generatorparameter estimationrdquo in Proceedings of the IEEE Power Engi-neering Society General Meeting (PES rsquo06) Montreal CanadaJune 2006
[26] HAgahiActive identification for nonlinearmultivariable systemidentification and its application for synchronous generator iden-tification [PhD thesis] Amirkabir University of TechnologyTehran Iran 2007
[27] J ChiassonModeling and High Performance Control of ElectricMachines IEEE Press Series on Power Engineering IEEE Press2005
[28] M Ghanbari and M Haeri ldquoParametric identification offractional-order systems using a fractional Legendre basisrdquoProceedings of the Institution of Mechanical Engineers Part IJournal of Systems and Control Engineering vol 224 no 3 pp261ndash274 2010
[29] M Ghanbari and M Haeri ldquoOrder and pole locator estimationin fractional order systems using bode diagramrdquo Signal Process-ing vol 91 no 2 pp 191ndash202 2011
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
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