Research Article Windowed Least Square Algorithm Based PMSM … · 2019. 7. 31. · PMSM Parameters Estimation of Windowed Least Square Algorithm. When , ,and are xedvalue,thePMSM

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 131268 11 pageshttpdxdoiorg1011552013131268

Research ArticleWindowed Least Square Algorithm Based PMSMParameters Estimation

Song Wang

School of Mechanical Electrical amp Information Engineering Shandong University Weihai China

Correspondence should be addressed to Song Wang wangsong sdu163com

Received 4 June 2013 Revised 26 July 2013 Accepted 26 July 2013

Academic Editor Juan J Nieto

Copyright copy 2013 Song Wang This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Stator resistance and inductances in 119889-axis and 119902-axis of permanent magnet synchronous motors (PMSMs) are importantparameters Acquiring these accurate parameters is usually the fundamental part in driving and controlling system design toguarantee the performance of driver and controller In this paper we adopt a novel windowed least algorithm (WLS) to estimatethe parameters with fixed value or the parameter with time varying characteristic The simulation results indicate that the WLSalgorithm has a better performance in fixed parameters estimation and parameters with time varying characteristic identificationthan the recursive least square (RLS) and extended Kalman filter (EKF) It is suitable for engineering realization in embeddedsystem due to its rapidity less system resource possession less computation and flexibility to adjust the window size according tothe practical applications

1 Introduction

The high-field-strength neodymium-iron-boron (NdFeB)magnets have become commercially available with afford-able prices so the permanent magnet synchronous motor(PMSM) is receiving increasing attention due to its highspeed high power density and high efficiency It is very suit-able for some high-performance requirement applicationsfor example robotics aerospace electric ship propulsionsystems and wind power generation systems [1ndash3] It hasbeen shown that PMSM can provide significant performanceimprovement in many variable speed applications [4] Thecommonly used control method in motor control is vectorcontrol The method has a requirement of obtaining relatedparameters of the motor Therefore acquiring accurateparameters of the motor is usually the fundamental partin driving system design We cannot measure the motorparameters with normal no-load test and locked rotor test inthe work site Moreover with the increasing working time ofthe motor and the surrounding environment changes someparameters of the motor will be changed Therefore servodrivers usually have the function of parameters identificationand self-tuning [3]

Stator resistance and inductances in 119889-axis and 119902-axis areimportant parameters of motor model which are consideredas constants usually However these parameters vary withdifferent operation conditions when motor is running [5]The study object of this paper was permanent magnet ser-vomotor produced by Huada Company in Wuhan of ChinaThe experiment data showed that stator resistance valueranged from 119877

119904to 13119877

119904and that inductance value ranged

from 119871119889to 1004119871

119889when the temperature ranged from 20

degrees to 80 degrees Thus the temperature of motor had agreat influence on stator resistance and inductance in 119889-axisand 119902-axis When these parameters are treated as constantsthe stability and control performance of the system will beaffectedTherefore the realization of parameters (119877

119904 119871119889 and

119871119902) identification is essential for motion control of PMSMModel identification and parameter estimation tech-

niques have becomemature after years of development Fromthe least square estimation theory [6ndash14] and its variousimproved algorithm [15ndash23] to Kalman filter algorithm [24ndash27] neural network [28 29] genetic algorithm [30ndash32]and so forth they can serve as parameter estimation toolsHowever these methods have their own characteristics andapplicability

2 Mathematical Problems in Engineering

The least square estimation is one of the most simple andmost mature parameter estimation methods However theamount of calculation of the traditional least square methodwill increase with time sequence increase It is hard to realizein embedded chip due to the large amount of calculationand there is a problem of data saturation Kalman filteringalgorithm is put forward for system identification by Kalmanin 1960 and there is a wide range of use However it issensitive to the initial conditions [26] and its performanceis poor for time varying parameter identification [33 34]With the development of artificial intelligence technologyneural network [28 29] and genetic algorithm [30ndash32] areused to the parameter estimation These intelligent methodscan get the identification results with high accuracyHoweverit is hard to apply them in practical parameter estimationdue to the large amount of calculation and complexity ofthe algorithm Therefore least square algorithm is also acommonly parameter estimationmethod Kinds of improvedalgorithms are proposed to promote the identification per-formance of traditional least square algorithm [18ndash22 35ndash41] For example the forgetting factor is introduced in therecursive least square estimation and the past time of datawill be forgotten by index rate [42 43] However it stillcannot discard the past time of data [16] but just weakensthe impact of the past time of data for the current parameterestimation Another method is to use window method forthe time series data [16 23 44] This method can discard thepast time of data flexibly and eliminate the impact of the pasttime of data for future parameter estimationThewindow sizecan be set flexible according to practical application In thispaper we adopt windowed least square algorithm for statorresistance 119877

119904 119871119889 and 119871

119902inductance estimation and make a

comparison with recursive least square and extended kalmanFilter (EKF) From the simulation result we can see thatwindowed least square algorithm has a better performancein convergence speed and identification precision for fixedparameters and parameters with time varying characteristicsFrom the view of algorithm complexity the windowed leastsquare algorithm is suitable for engineering realization inembedded chip such as DSP and ARM

This paper is consisted of the five sections Section 2describes the principle of least square theory and the recur-sive least square algorithm Section 3 illustrates thewindowedleast square algorithm Section 4 does some simulations forPMSM parameter estimation Section 5 analyses the simula-tion results and shows some conclusions

2 Least Square Estimation and Recursive LeastSquare Estimation

21 The Principle of Least Square Estimation The earlieststimulus for the development of the least square estimationtheory was apparently provided by astronomical studiesin which planet and comet motions were studied usingtelescopic measurement data The principle of the parameterestimation is simple and does not need any statistical charac-teristics of the variables It is used in system identification andparameter estimation widelyThe least square estimation still

can provide an accurate solution when other identificationmethods lose efficacy

Supposing 119910(119894) and 1199091(119894) 1199092(119894) sdot sdot sdot 119909

119899(119894) are the observa-

tion sequences of 119910 and 119909 at 1199051 1199052sdot sdot sdot 119905119898 The relationship of 119910

and 119909 is expressed

[[[[

[

119910 (1)

119910 (2)

119910 (119898)

]]]]

]

=

[[[[

[

1199091(1) sdot sdot sdot 119909

119899(1)

1199091(2) sdot sdot sdot 119909

119899(2)

1199091(119898) sdot sdot sdot 119909

119899(119898)

]]]]

]

[[[[

[

1205791

1205792

120579119899

]]]]

]

(1)

where 120579 = (1205791 1205792 120579119899) is the measured parameter set

and 119899 is the number of parameters We hope to estimatetheir values by the observation value of 119910 and 119909 at differenttime sequences 119898 is the time sequences to estimate the 119899

parameters 120579119894119898 ge 119899 is required and if119898 = 119899 we can get the

single solution from (1) as (2)

120579 = 119883minus1

119910 (2)

where 120579 is the estimation value of 120579 and inverse matrix 119883minus1

of119883 is required

120576 = 119910 minus 119883120579 (3)

where 120576 = (1205761 1205762sdot sdot sdot 120576119898)119879 is the error vector

The target function is shown in the following

119869 =

119898

sum

119894=1

1205762

119894= 120576119879

120576 (4)

Obtaining 120579 to make 119869minimum

120597119869

120597120579

10038161003816100381610038161003816100381610038161003816120579=120579= minus2119883

119879

119910 + 2119883119879

119883120579 = 0 (5)

The result is

120579 = (119883119879

119883)minus1

119883119879

119910 (6)

120579 is the least square estimate LS of 120579

22 Recursive Least Square Estimation In practical parame-ter estimation the data is always constantly to be refreshedTherefore we can further deduce (2) to a recursion algorithmThis algorithm does not need to compute the inverse matrixcalculation repeatedly and reduces the time-consuming andsystem resources occupation

The least square estimation using 119898 groups of data isshown as follows

120579 (119898) = (119883119879

119898119883119898)minus1

119883119879

119898119884119898 (7)

The119898+1moment data is (119909(119898+ 1) 119910(119898+ 1)) and then

119884119898+1

= 119883119898+1

120579 (119898 + 1) (8)

Mathematical Problems in Engineering 3

where

119884119898+1

=

[[[[[[

[

119910 (1)

119910 (119898)

119910 (119898 + 1)

]]]]]]

]

= [

[

119884119898

119910 (119898 + 1)

]

]

119883119898+1

=

[[[[[[

[

1199091(1) 119909

119899(1)

1199091(119898) 119909

119899(119898)

1199091(119898 + 1) 119909

119899(119898)

]]]]]]

]

= [

[

119883119898

119909119879

(119898 + 1)

]

]

(9)

The new least estimation equation is shown as follows

120579 (119898 + 1) = (119883119879

119898+1119883119898+1

)minus1

119883119879

119898+1119884119898+1

(10)

In order to get out of inverse matrix calculation of119883119879119898+1

sdot

119883119898+1

120579(119898 + 1) is deduced as follows

120579 (119898 + 1) = 120579 (119898) + 119875 (119898) 119909 (119898 + 1)

times [1 + 119909119879

(119898 + 1) 119875 (119898) 119909 (119898 + 1)]minus1

sdot [119910 (119898 + 1) minus 119909119879

(119898 + 1) 120579 (119898)]

(11)

Equation (11) is the recursive least square (RLS) estima-tion Recursive least square algorithm is called the general-ization Kalman filter algorithm [45 46] It is the engineeringrealization method of the least square estimation theory [17]

From the calculation of least square estimation andrecursive least square estimation we can see that the past timeof data has a big effect to future parameter estimation and alarge number of data calculations have occupied the systemresources seriously Therefore it is difficult to be realized inembedded systems

3 Windowed Least Square Estimation

Recursive least square algorithm can be used to real-timeparameter estimation However the algorithm uses the pasttime of data and the past time of data has the same impor-tance as the present data in the algorithm It weakened theimportance of current data caused a lot of system resourcespossession and affected the estimation speed and precision[44] In order to guarantee the instantaneity of parameterestimation the paper adopts windowed least squares (WLS)to estimate the parameters of PMSMThealgorithm simulatesthe window processing function of communication signalThe time series data used for parameter estimation are addedwindow handle to reduce the calculation of estimation andsystem resources possession making the algorithm easy forengineering realization

Suppose that 119875(119898) = (119883119879

119898119883119898)minus1 the parameter estima-

tion by (119909(119896) 119910(119896)) 119896 = 1 2 119898 is as follows

120579 (119898) = 119875 (119898)119883119879

119898119884119898 (12)

where

119883119898

=

[[[[

[

119909119879

(1)

119909119879

(2)

119909119879

(119898)

]]]]

]

119884119898

=

[[[[

[

119910 (1)

119910 (2)

119910 (119898)

]]]]

]

(13)

Consider increasing a group of data (119909(119898+ 1) 119910(119898+ 1))and then the parameter estimation is shown as follows [16 2344]

120579 (119898 + 1) = (119883119879

119898+1119883119898+1

)minus1

119883119879

119898+1119884119898+1

= 120579 (119898) + 119875 (119898) 119909 (119898 + 1)

times [1 + 119909119879

(119898 + 1) 119875 (119898) 119909 (119898 + 1)]minus1

times [119910 (119898 + 1) minus 119909119879

(119898 + 1) 120579 (119898)]

119875 (119898 + 1) = (119883119879

119898+1119883119898+1

)minus1

= 119875 (119898) minus 119875 (119898) 119909 (119898 + 1)

times [1 + 119909119879

(119898 + 1) 119875 (119898) 119909 (119898 + 1)]minus1

times 119909119879

(119898 + 1) 119875 (119898)

(14)

where

119883119898+1

=

[[[[[[[

[

119909119879

(1)

119909119879

(2)

119909119879

(119898)

119909119879

(119898 + 1)

]]]]]]]

]

119884119898+1

=

[[[[[[

[

119910 (1)

119910 (2)

119910 (119898)

119910 (119898 + 1)

]]]]]]

]

(15)

Therefore119883119898+1

= [ 119909119879(1)

119883

]Consider eliminating a group of data (119909(1) 119910(1)) and

then the parameter estimation is shown as follows where

119875 (119898 + 1) = (119883119879

119883)minus1

119883 =

[[[[

[

119909119879

(2)

119909119879

(119898)

119909119879

(119898 + 1)

]]]]

]

4 Mathematical Problems in Engineering

119883119879

119898+1119883119898+1

= 119883119879

119883 + 119909 (1) 119909119879

(1)

119875 (119898 + 1)

= (119883119879

119898+1119883119898+1

minus 119909 (1) 119909119879

(1))minus1

= ((119875 (119898 + 1))minus1

+ (minus119909 (1)) 119909119879

(1))minus1

= 119875 (119898 + 1) minus 119875 (119898 + 1) (minus119909 (1))

times [1 + 119909119879

(1) 119875 (119898 + 1) (minus119909 (1))]minus1

119909119879

(1) 119875 (119898 + 1)

= 119875 (119898 + 1) + 119875 (119898 + 1) 119909 (1)

times [1 minus 119909119879

(1) 119875 (119898 + 1) 119909 (1)]minus1

119909119879

(1) 119875 (119898 + 1)

(16)

Therefore the estimation result is shown as follows

120579 (119898 + 1) = (119883119879

119883)minus1

119883119879

119884 = 119875 (119898 + 1)119883119879

119884 (17)

where

(119909 (119896) 119910 (119896)) 119896 = 2 119898 + 1 119884 =

[[[[

[

119910 (2)

119910 (119898)

119910 (119898 + 1)

]]]]

]

119883119879

119898+1119884119898+1

= 119883119879

119884 + 119910 (1) 119909 (1)

(18)

120579 (119898 + 1)

= 119875 (119898 + 1) (119883119879

119898+1119884119898=1

minus 119910 (1) 119909 (1))

= [119875 (119898 + 1) + 119875 (119898 + 1) 119909 (1)

times [1 minus 119909119879

(1) 119875 (119898 + 1) 119909 (1)]minus1

times 119909119879

(1) 119875 (119898 + 1)]

times [119883119879

119898+1119884119898=1

minus 119910 (1) 119909 (1)]

= 119875 (119898 + 1)119883119879

119898+1119884119898=1

minus 119910 (1) 119875 (119898 + 1) 119909 (1)

times [1 minus 119909119879

(1) 119875 (119898 + 1) 119909 (1)]minus1

times 119909119879

(1) 119875 (119898 + 1) 119909 (1)

minus 119910 (1) 119875 (119898 + 1) 119909 (1) + 119875 (119898 + 1) 119909 (1)

times [1 minus 119909119879

(1) 119875 (119898 + 1) 119909 (1)]minus1

times 119909119879

(1) 119875 (119898 + 1)119883119879

119898+1119884119898+1

= 120579 (119898 + 1) minus119910 (1) 119875 (119898 + 1) 119909 (1)

1 minus 119909119879 (1) 119875 (119898 + 1) 119909 (1)

times (119909119879

(1) 119875 (119898 + 1) 119909 (1) + 1

minus119909119879

(1) 119875 (119898 + 1) 119909 (1))

+ 119875 (119898 + 1) 119909 (1) [1 minus 119909119879

(1) 119875 (119898 + 1) 119909 (1)]minus1

times 119909119879

(1) 120579 (119898 + 1)

= 120579 (119898 + 1) minus119910 (1) 119875 (119898 + 1) 119909 (1)

1 minus 119909119879 (1) 119875 (119898 + 1) 119909 (1)

+119875 (119898 + 1) 119909 (1) 119909

119879

(1) 120579 (119898 + 1)

1 minus 119909119879 (1) 119875 (119898 + 1) 119909 (1)

= 120579 (119898 + 1) +119875 (119898 + 1) 119909 (1)

1 minus 119909119879 (1) 119875 (119898 + 1) 119909 (1)

times (119909119879

(1) 120579 (119898 + 1) minus 119910 (1))

(19)

Therefore for any moment

120579 (119898 + 119896) = 120579 (119898 + 119896)

+119875 (119898 + 119896) 119909 (119896)

1 minus 119909119879 (119896) 119875 (119898 + 119896) 119909 (119896)

times (119909119879

(119896) 120579 (119898 + 119896) minus 119910 (119896))

(20)

where119898 is the window size and

119875 (119898 + 119896) = 119875 (119898 + 119896)119883119898+119896

times [1 minus 119883119879

119898+119896119875 (119898 + 119896)119883

119898+119896]minus1

119883119879

119898+119896119875 (119898 + 119896)

(21)

The window size is adjustable according to actual needsbased on the data length of regulation This can guaranteethe speed of calculation and can reduce the system resourcespossession too At this time the parameter estimation isrelated to the current 119898 data sample the past time of datahas no effect on parameter estimation and this can ensurethe instantaneity and accuracy of the parameter estimation

4 Simulations

41 PMSM Model The voltage equations flux linkage equa-tions and electromagnetic torque equations of PMSM in 119889 119902frames are as follows [47 48]

Mathematical Problems in Engineering 5

Pulse

Powergui

Continuous

4

p

4

Nana motor

ABC

PIPIPI

0dq2alfabeta

Universal bridge

g

To file2

To file

Switch

Step1Step

Scope5

Scope

PIDPID

Multimeter

3

000621000424

Iabc5

Generator

Pulses

plusmn

+

++

minus

minus

+minus Discrete SV PWM

Rs

Ld

Lq

Tm

Tm

m

RLdLq

⟨iq⟩

⟨id⟩⟨uq⟩

⟨ ⟩

⟨ud⟩

⟨ ⟩

Outputmat

RS kalmanmat

120579

Jixie rotor speed wm

wr

(rads)Jixie rotor angle 120579m (rad)

ABC

p1

uq ref

ud ref

u120573 u120573

u120572u120572

Figure 1 MATLAB motor simulation model

119906119902= 119877119904119894119902+ 119871119902119901119894119902+ 120596119903119871119889119894119889+ 120596119903120595119891

119906119889= 119877119904119894119889+ 119871119889119901119894119889minus 120596119903119871119902119894119902

(22)

In the steady state

119906119902= 119877119904119894119902+ 120596119903119871119889119894119889+ 120596119903120595119891

119906119889= 119877119904119894119889minus 120596119903119871119902119894119902

(23)

Flux linkage equations are as follows

120595119889= 119871119889119894119889+ 120595119891

120595119902= 119871119902119894119902

(24)

where119906119902and119906119889are voltages in 119902-axis and119889-axis respectively

119894119902and 119894

119889are currents in 119902-axis and 119889-axis 119877

119904is phase

resistance of stator 119871119889and 119871

119902are inductances in 119889-axis and

119902-axis 120596119903is rotor velocity 120595

119891is flux linkage established by

magnets and 119901 is the differential operatorThe mathematical model of PMSM is discretized to

estimate parameters (119877119904 119871119889 and 119871

119902) The discrete model of

PMSM is as follows

119906119902(119896) = 119877

119904119894119902(119896) + 119901120595

119902(119896) + 120596

119903120595119889(119896)

119906119889(119896) = 119877

119904119894119889(119896) + 119901120595

119889(119896) + 120596

119903120595119902(119896)

(25)

where 120595119889and 120595

119902are as follows

120595119889(119896) = 119871

119889119894119889(119896) + 120595

119891

120595119902(119896) = 119871

119902119894119902(119896)

(26)

The PMSM simulation model is established by MAT-LABSIMULINK and the PMSM running data is obtainedby the model The simulation model is shown in Figure 1

In MATLABSIMULINK the 119877119904 119871119889 and 119871

119902parameters

of PMSM are fixed in simulation We cannot simulate thetime varying characteristic of 119877

119904 119871119889 and 119871

119902 Therefore we

design a motor simulation model according to the require-ment The 119877

119904 119871119889 and 119871

119902can be changed flexibly in the

simulation

42 PMSM Parameters Estimation of Windowed Least SquareAlgorithm When 119877

119904 119871119889 and 119871

119902are fixed value the PMSM

simulation data is obtained by MATLAB The windowedleast square algorithm is used to identify the parametersThe algorithm with different window sizes is used for 119877

119904

119871119889 and 119871

119902identification From the identification result

(Figures 2 3 and 4) we can see that bigger window sizehas a better identification result However the window sizedoes not obviously have an effect on the promotion ofparameter identification precision when 119877

119904 119871119889 and 119871

119902are

fixedDifferent window sizes have big effect on the results of

parameter estimation when the estimated parameters havetime varying characteristicThemotor parameters119877

119904119871119889 and

119871119902are measured at 234∘C 30∘C 40∘C 50∘C 60∘C 70∘C and

80∘C Using piecewise linear method to simulate the timevarying of the three parameters the motor running data isobtained by MATLAB

The windowed least square is used to identify the 119877119904 119871119889

and 119871119902 The identification result is shown in Figures 5 6 and

7 when the 119877119904 119871119889and 119871

119902are changed at the same time From

the Figures 5ndash7 we can see that shorter window size has lowereffect on the promotion of identification precision Howeverthe window size is too big to improve the identificationprecisionThe identification result is better when the windowsize is 300ndash400 In the motor model 120595

119891is bigger than

6 Mathematical Problems in Engineering

09665

0966

09655

0965

09645

09640 01 02 03 04 05 06 07 08 09 1

Time (s)

Rs

Window size = 100Window size = 200Window size = 300

Window size = 400Window size = 500Actual data

Figure 2 Estimation result of 119877119904of WLS with different window

sizes when the parameters are fixed

0 01 02 03 04 05 06 07 08 09 1Time (s)

4256

4254

4252

425

4248

4246

4244

4242

424

Ld

times10minus3

Window size = 100Window size = 200Window size = 300

Window size = 400Window size = 500Actual data

Figure 3 Estimation result of 119871119889of WLS with different window

sizes when the parameters are fixed

119871119889119894119889 so the change of 119871

119889has little effect on the model

output Therefore the identification result of 119871119889is not very

well However the algorithm can also identify the parametercorrectly

43 PMSM Parameters Estimation of Extended Kalman FilterThe Kalman filter is a common parameter identification

6215

6214

6213

6212

6211

621

6209

6208

6207

6206

6205

Lq

0 01 02 03 04 05 06 07 08 09 1Time (s)

Window size = 100Window size = 200Window size = 300

Window size = 400Window size = 500Actual data

times10minus3

Figure 4 Estimation result of 119871119902of WLS with different window

sizes when the parameters are fixed

0 01 02 03 04 05 06 07 08 09 1Time (s)

Window size = 50Window size = 100Window size = 150Window size = 200Window size = 250Window size = 300

Window size = 350Window size = 400Window size = 450Window size = 500Actual data

125

12

115

11

105

1

095

Rs

Figure 5 Estimation result of 119877119904of WLS with different window

sizes when the parameters have time varying characteristic

method It is proposed in 1960 by Kalman [49] The theoryis applied to practical engineering immediately when it isput forward The Apollo program and C-5 plane navigationsystem design are the most successful application examplesExtended Kalman filter (EKF) is an improved model of theKalman filter which is one of the most widely applied innonlinear system filter

Mathematical Problems in Engineering 7

Discrete system state equation of EKF is

X (119896) = A (119896 minus 1)X (119896 minus 1) + B (119896 minus 1)U (119896 minus 1)

+ C (119896 minus 1) + w (119896 minus 1)

Z (119896) = H (119896 minus 1)X (119896) + k (119896)

(27)

where

A (119896 minus 1) =120597119891 (XU)

120597X119879119904

10038161003816100381610038161003816100381610038161003816X=X(119896minus1)

B (119896 minus 1) =120597119891 (XU)

120597U119879119904

10038161003816100381610038161003816100381610038161003816U=U(119896minus1)

119862 (119896 minus 1) = [119891 (XU) 119879119904minus

120597119891 (XU)

120597XX119879119904]

10038161003816100381610038161003816100381610038161003816X=X(119896minus1)

H (119896) =120597ℎ (X)

120597X

10038161003816100381610038161003816100381610038161003816X=X(119896minus1)

(28)

where X(119896) is the system state vector U(119896) is the systeminput vector Z(119896) is the system observation vector w(119896) isthe system random noise vector k(119896) is the system randomobservation noise vector w(119896) and k(119896) are noise sequenceswith zero mean and the covariance matrices are Q(119896) andR(119896)

119879119904is the sampling period the discrete linear state space

equation (29) of PMSM is established by discretization andlinearization of the model (30)

119889

119889119905=

119894119889=

119906119889

119871119889

minus119877119904

119871119889

119894119889+

120595119902

119871119889

120596119890

119894119902=

119906119902

119871119902

minus119877119904

119871119902

119894119902minus

120595119889

119871119902

120596119890

120595119889= 0

120595119902= 0

119877119904= 0

(29)

[[[[[[[

[

119894119889(119896)

119894119902(119896)

120595119889(119896)

120595119889(119896)

119877119904(119896)

]]]]]]]

]

=

[[[[[[[[[[[[

[

1 minus119904(119896 minus 1)

119871119889

119879119904

0 0119890(119896 minus 1)

119871119889

119879119904

minus119889(119896 minus 1)

119871119889

119879119904

0 1 minus119904(119896 minus 1)

119871119902

119879119878

minus119890(119896 minus 1)

119871119902

119879119904

0 minus119902(119896 minus 1)

119871119902

119879119904

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

]]]]]]]]]]]]

]

times

[[[[[[[

[

119894119889(119896 minus 1)

119894119902(119896 minus 1)

120595119889(119896 minus 1)

120595119889(119896 minus 1)

119877119904(119896 minus 1)

]]]]]]]

]

+

[[[[[[[[[

[

119879119904

119871119889

0

0119879119904

119871119902

0 0

0 0

0 0

]]]]]]]]]

]

[119906119889(119896 minus 1)

119906119902(119896 minus 1)

] + 119879119904

[[[[[[[[[[[[[

[

119889(119896 minus 1)

119904(119896 minus 1)

119871119889

119902(119896 minus 1)

119904(119896 minus 1)

119871119902

0

0

0

]]]]]]]]]]]]]

]

+ w (119896 minus 1)

(30)

[119894119889(119896)

119894119902(119896)

] = [1 0 0 0 0

0 1 0 0 0]

[[[[[[[

[

119894119889(119896)

119894119902(119896)

120595119889(119896)

120595119889(119896)

119877119904(119896)

]]]]]]]

]

+ k (119896) (31)

8 Mathematical Problems in Engineering

Window size = 50Window size = 100Window size = 150Window size = 200Window size = 250Window size = 300

Window size = 350Window size = 400Window size = 450Window size = 500Actual data

0 01 02 03 04 05 06 07 08 09 1Time (s)

445

44

435

43

425

42

Ld

times10minus3

Figure 6 Estimation result of 119871119889of WLS with different window

sizes when the parameters have time varying characteristic

Using the simulation model and getting the motor output 119894119889

119894119902 119906119889 119906119902120596119890 the initial values of119875119876 and119877 in EKF algorithm

are

119875 = diag ([01 01 00004 0002 002])

119876 = diag ([30 15 0005 003 003])

119877 = diag ([01 002])

(32)

When119877119904 119871119889 and 119871

119902of motor are fixed the identification

result table of EKF recursive least square and windowed leastsquare algorithm is shown in Table 1

From Table 1 we can see that identification result ofEKF algorithm is as good as the windowed least squarealgorithm when 119877

119904 119871119889 and 119871

119902are fixed The comparison

diagrams of identification result are shown in Figure 8 whenthe parameters are fixed

EKF and recursive least square algorithms cannot achievereasonable result when the parameters have time varyingcharacteristic or have a drastic change However windowedleast square algorithm can achieve good identification resultwhen 119877

119904 119871119889 and 119871

119902have time varying characteristic at the

same time (Figures 5ndash7)

5 Analysis and Conclusion

Through the previous different PMSM parameters identifica-tion experiments we can see the following

(1) When the parameters of PMSM have no time vary-ing characteristic three methods can achieve better

0 01 02 03 04 05 06 07 08 09 1Time (s)

655

65

645

64

635

63

625

62

Lq

Window size = 50Window size = 100Window size = 150Window size = 200Window size = 250Window size = 300

Window size = 350Window size = 400Window size = 450Window size = 500Actual data

times10minus3

Figure 7 Estimation result of 119871119902of WLS with different window

sizes when the parameters have time varying characteristic

identification result in precision and accuracy Inthe calculation and instantaneity of identificationrecursive least square algorithm has a fatal flaw ofdata saturation so the precision and accuracy of thealgorithm are hard to guarantee It is difficult torealize in embedded system the real-time parametersidentification due to the amount of calculations andsystem resources possession of EKF [28]Thewindowsize of windowed least square algorithm is flexibleso we can choose the collected data according tothe changes of the parameters It will reduce theinfluence of the past time of data to the currentparameter identification guarantee the accuracy andinstantaneity of identification and reduce the systemresources possession at the same time

(2) When the parameters of PMSM have strong timevarying characteristic the EKF and recursive leastsquare algorithms cannot guarantee the precisionand accuracy of identification However windowedleast square algorithm can get better identificationresultTherefore EKF and recursive least square algo-rithms are suitable for fixed parameters estimationor parameters with weak time varying characteristicidentification Windowed least square algorithm canget a good result both for fixed parameters and fortime varying parameters identification

Embedded technology is widely used in the motor driverand controller at present However the embedded chip(MCU DSP ARM etc) has certain restriction in computingspeed and storage space Therefore windowed least square

Mathematical Problems in Engineering 9

Table 1 Estimation result comparison of EKF RLS and WLS when 119877119904 119871119889 and 119871

119902are fixed value

Temperature (∘C) 234 30 40 50 60 70 80

119877119904

Actual data (Ω) 09664 10008 10373 10770 11245 11592 11751EKF

Estimation data09694 10036 10400 10795 11266 11612 11611

RLS 09607 10077 10271 10712 11216 11451 11771WLS 09653 09997 10362 10760 11235 11583 11742

119871119889times 10minus3

Actual data (mH) 424 426 428 430 431 434 436EKF

Estimation data42368 42882 42475 42941 43331 43755 43595

RLS 41602 43210 43506 42611 42814 42021 43807WLS 42465 42664 42861 43058 43157 4345 44365

119871119902times 10minus3

Actual data (mH) 621 626 630 634 640 644 648EKF

Estimation data62095 62604 63001 63400 64012 64395 64800

RLS 62531 62551 65172 63703 64118 64658 64727WLS 62089 62581 62982 63387 63981 64384 64780

20 30 40 50 60 70 80

ActualEKF

RLSWLS

125

12

115

11

105

1

095

Rs

(a) Estimation result of 119877119904

20 30 40 50 60 70 80

ActualEKF

RLSWLS

445

44

435

43

425

42

415

Ld

(b) Estimation result of 119871119889

20 30 40 50 60 70 80

655

65

645

64

635

63

625

62

Lq

ActualEKF

RLSWLS

(c) Estimation result of 119871119902

Figure 8 Comparison diagram of EKF RLS and WLS when 119877119904 119871119889 and 119871

119902are fixed value

10 Mathematical Problems in Engineering

Figure 9 The PMSM experiment system

Figure 10 The prototype DSP-based PMSM driver

algorithm is a better choice for PMSM parameters identi-fication of motor driver and controller This paper is thebeginning of work There are a lot of work to do such astransplant the algorithm to practical controller and controlsystem (in Figures 9 and 10) which is designed to control thePMSM in practical application

Acknowledgment

This paper is supported by the Shandong Province Scienceand Technology Development Plan of China (Grant no2011GGE27053)

References

[1] M A Rahman and P Zhou ldquoAnalysis of brushless permanentmagnet synchronous motorsrdquo IEEE Transactions on IndustrialElectronics vol 43 no 2 pp 256ndash267 1996

[2] MOoshimaA Chiba A Rahman andT Fukao ldquoAn improvedcontrol method of buried-type IPM bearingless motors consid-ering magnetic saturation and magnetic pull variationrdquo IEEETransactions on Energy Conversion vol 19 no 3 pp 569ndash5752004

[3] K Liu Z Q Zhu Q Zhang and J Zhang ldquoInfluence ofnonideal voltage measurement on parameter estimation inpermanent-magnet synchronous machinesrdquo IEEE Transactionson Industrial Electronics vol 59 no 6 pp 2438ndash2447 2012

[4] F Caricchi F Crescimbini and O Honorati ldquoLow-cost com-pact permanent magnet machine for adjustable-speed pumpapplicationrdquo IEEETransactions on IndustryApplications vol 34no 1 pp 109ndash116 1998

[5] P Milanfar and J H Lang ldquoMonitoring the thermal conditionof permanent-magnet synchronous motorsrdquo IEEE Transactions

onAerospace and Electronic Systems vol 32 no 4 pp 1421ndash14291996

[6] T Kailath ldquoAn innovations approach to least-squares estima-tionmdashpart I linear filtering in additive white noiserdquo IEEETransactions on Automatic Control vol 13 pp 646ndash655 1968

[7] D G Robertson and J H Lee ldquoA least squares formulation forstate estimationrdquo Journal of Process Control vol 5 no 4 pp291ndash299 1995

[8] J S Gibson G H Lee and C F Wu ldquoLeast-squares estimationof inputoutput models for distributed linear systems in thepresence of noiserdquo Automatica vol 36 no 10 pp 1427ndash14422000

[9] S Tunali and I Batmaz ldquoDealing with the least squares regres-sion assumptions in simulation metamodelingrdquo Computers ampIndustrial Engineering vol 38 no 2 pp 307ndash320 2000

[10] R M Fernandez-Alcala J Navarro-Moreno and J C Ruiz-Molina ldquoLinear least-square estimation algorithms involvingcorrelated signal and noiserdquo IEEE Transactions on Signal Pro-cessing vol 53 no 11 pp 4227ndash4235 2005

[11] V Kratschmer ldquoLeast-squares estimation in linear regressionmodels with vague conceptsrdquo Fuzzy Sets and Systems vol 157no 19 pp 2579ndash2592 2006

[12] M J Garcıa-Ligero A Hermoso-Carazo and J Linares-PerezldquoLeast-squares linear estimation of signals from observationswith Markovian delaysrdquo Journal of Computational and AppliedMathematics vol 236 no 2 pp 234ndash242 2011

[13] S Ma C Quan R Zhu C J Tay L Chen and Z GaoldquoApplication of least-square estimation in white-light scanninginterferometryrdquo Optics and Lasers in Engineering vol 49 no 7pp 1012ndash1018 2011

[14] Q Wang and L Zhang ldquoLeast squares online linear discrimi-nant analysisrdquo Expert Systems with Applications vol 39 no 1pp 1510ndash1517 2012

[15] C J Demeure and L L Scharf ldquoSliding windows and latticealgorithms for computing QR factors in the least squares theoryof linear predictionrdquo IEEE Transactions on Acoustics Speechand Signal Processing vol 38 no 4 pp 721ndash725 1990

[16] K Zhao L Fuyun H Lev-Ari and J G Proakis ldquoSlidingwindow order-recursive least-squares algorithmsrdquo IEEE Trans-actions on Signal Processing vol 42 no 8 pp 1961ndash1972 1994

[17] H Liu and Z He ldquoA sliding-exponential window RLS adaptivefiltering algorithm properties and applicationsrdquo Signal Process-ing vol 45 no 3 pp 357ndash368 1995

[18] K Yoo and H Park ldquoFast residual computation for slidingwindow recursive least squares methodsrdquo Signal Processing vol45 no 1 pp 85ndash95 1995

[19] Y Xia M S Kamel and H Leung ldquoA fast algorithm for ARparameter estimation using a novel noise-constrained least-squares methodrdquo Neural Networks vol 23 no 3 pp 396ndash4052010

[20] A Aknouche E M Al-Eid and A M Hmeid ldquoOffline andonline weighted least squares estimation of nonstationarypower119860119877119862119867 processesrdquo Statistics amp Probability Letters vol 81no 10 pp 1535ndash1540 2011

[21] L Xie H Yang and F Ding ldquoRecursive least squares parameterestimation for non-uniformly sampled systems based on thedata filteringrdquo Mathematical and Computer Modelling vol 54no 1-2 pp 315ndash324 2011

[22] J Oliver R Aravind and K M M Prabhu ldquoImproved leastsquares channel estimation for orthogonal frequency divisionmultiplexingrdquo IET Signal Processing vol 6 no 1 pp 45ndash532012

Mathematical Problems in Engineering 11

[23] T Sadiki M Triki and D T M Slock ldquoWindow optimizationissues in recursive least-squares adaptive filtering and trackingrdquoin Proceedings of the 38th IEEE Annual Asilomar Conferenceon Signals Systems and Computers pp 940ndash944 Pacific GroveCalif USA November 2004

[24] G Welch and G Bishop ldquoAn introduction to the Kalman filterrdquo1997

[25] P J Hargrave ldquoA tutorial introduction to Kalman filteringrdquo inProceedings of the IEE Colloquium on Kalman Filters Introduc-tion Applications and Future Developments pp 11ndash16 1989

[26] M Gautier and P Poignet ldquoExtended Kalman filtering andweighted least squares dynamic identification of robotrdquo ControlEngineering Practice vol 9 no 12 pp 1361ndash1372 2001

[27] H M Al-Hamadi and S A Soliman ldquoKalman filter foridentification of power system fuzzy harmonic componentsrdquoElectric Power Systems Research vol 62 no 3 pp 241ndash248 2002

[28] 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

[29] A Bechouche H Sediki D O Abdeslam and S HaddadldquoIdentification of induction motor at standstill using artificialneural networkrdquo in Proceedings of the 36th Annual Conferenceon IEEE Industrial Electronics Society (IECON rsquo10) pp 2908ndash2913 Glendale Ariz USA 2010

[30] F Alonge F DrsquoIppolito and FM Raimondi ldquoLeast squares andgenetic algorithms for parameter identification of inductionmotorsrdquo Control Engineering Practice vol 9 no 6 pp 647ndash6572001

[31] S Mishra ldquoA hybrid least square-fuzzy bacterial foragingstrategy for harmonic estimationrdquo IEEE Transactions on Evo-lutionary Computation vol 9 no 1 pp 61ndash73 2005

[32] R Liao H Zheng S Grzybowski and L Yang ldquoParticleswarm optimization-least squares support vector regressionbased forecasting model on dissolved gases in oil-filled powertransformersrdquo Electric Power Systems Research vol 81 no 12pp 2074ndash2080 2011

[33] R A Zadeh A Ghosh and G Ledwich ldquoCombination ofKalman filter and least-error square techniques in powersystemrdquo IEEE Transactions on Power Delivery vol 25 no 4 pp2868ndash2880 2010

[34] S Bolognani R Oboe andM Zigliotto ldquoSensorless full-digitalPMSM drive with EKF estimation of speed and rotor positionrdquoIEEE Transactions on Industrial Electronics vol 46 no 1 pp184ndash191 1999

[35] M Haardt ldquoStructured least squares to improve the per-formance of ESPRIT-Type algorithmsrdquo IEEE Transactions onSignal Processing vol 45 no 3 pp 792ndash799 1997

[36] M Ghogho A Swami and A K Nandi ldquoNon-linear leastsquares estimation for harmonics in multiplicative and additivenoiserdquo Signal Processing vol 78 no 1 pp 43ndash60 1999

[37] J Angeby ldquoEstimating signal parameters using the nonlinearinstantaneous least squares approachrdquo IEEE Transactions onSignal Processing vol 48 no 10 pp 2721ndash2732 2000

[38] J F Weng and S H Leung ldquoNonlinear RLS algorithm foramplitude estimation in class a noiserdquo IEE ProceedingsmdashCommunications vol 147 no 2 pp 81ndash86 2000

[39] D Zachariah M Sundin M Jansson and S ChatterjeeldquoAlternating least-squares for low-rank matrix reconstructionrdquoIEEE Signal Processing Letters vol 19 no 4 pp 231ndash234 2012

[40] R Montoliu and F Pla ldquoGeneralized least squares-based para-metric motion estimationrdquo Computer Vision and Image Under-standing vol 113 no 7 pp 790ndash801 2009

[41] Z Yingjie and G Liling ldquoImproved moving least squares algo-rithm for directed projecting onto point cloudsrdquoMeasurementvol 44 no 10 pp 2008ndash2019 2011

[42] S Seongwook J-S Lim S J Baek and K-M Sung ldquoVariableforgetting factor linear least squares algorithm for frequencyselective fading channel estimationrdquo IEEE Transactions onVehicular Technology vol 51 no 3 pp 613ndash616 2002

[43] S MorimotoM Sanada and Y Takeda ldquoMechanical sensorlessdrives of IPMSM with online parameter identificationrdquo IEEETransactions on Industry Applications vol 42 no 5 pp 1241ndash1248 2006

[44] H Sakai and H Nakaoka ldquoFast sliding window QRD-RLSalgorithmrdquo Signal Processing vol 78 no 3 pp 309ndash319 1999

[45] S Reece and S Roberts ldquoAn introduction to Gaussian processesfor the Kalman filter expertrdquo in Proceedings of the 13th Confer-ence on Information Fusion (FUSION rsquo10) pp 1ndash9 2010

[46] A Giorgano F M Hsu and J Wiley ldquoBook review least-square estimationwith applications to digital signal processingrdquoIEE Proceedings FmdashCommunications Radar and Signal Process-ing vol 132 no 7 1985

[47] S Wang R Zhao W Chen G Li and C Liu ldquoParameter iden-tification of PMSMbased on windowed least square algorithmrdquoin Proceedings of the Manufacturing Science and Technology(ICMST rsquo11) pp 5940ndash5944 Singapore 2012

[48] S Wang S Shi C Chen G Yang and Z Qu ldquoIdentificationof PMSM based on EKF and elman neural networkrdquo inProceedings of the IEEE International Conference on Automationand Logistics (ICAL rsquo09) pp 1459ndash1463 Shenyang China 2009

[49] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Transactions of the ASMEmdashJournal of Basic Engi-neering D vol 82 pp 35ndash45 1960

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