Research ArticlePractical Modeling and Comprehensive SystemIdentification of a BLDC Motor

Changle Xiang Xiaoliang Wang Yue Ma and Bin Xu

Vehicle Research Center School of Mechanical Engineering Beijing Institute of Technology Beijing 100081 China

Correspondence should be addressed to Bin Xu xubinbitgmailcom

Received 11 December 2014 Accepted 6 March 2015

Academic Editor Alfonso Banos

Copyright copy 2015 Changle Xiang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The aim of this paper is to outline all the steps in a rigorous and simple procedure for system identification of BLDC motorA practical mathematical model for identification is derived Frequency domain identification techniques and time domainestimationmethod are combined to obtain the unknown parametersThemethods in time domain are founded on the least squaresapproximation method and a disturbance observer Only the availability of experimental data for rotor speed and armature currentare required for identification The proposed identification method is systematically investigated and the final identified model isvalidated by experimental results performed on a typical BLDC motor in UAV

1 Introduction

Brushless direct current (BLDC) motors are prevailinglyused in high performance drive applications such as machinetools robotics space crafts and medical applications owingto their superior speed-torque characteristics high efficiencyless maintenance and wide operating speed range [1] Partic-ularly for small-scaled UAVs (Unmanned Aerial Vehicles)the BLDCmotor are receiving an increasing number of atten-tions with the advantages of small size high power densityand easy control and operationThe BLDCmotors which areused to drive rotor or propeller pursue abilities of rapid speedresponse and disturbance reject Control techniques researchincluding torque speed andposition has been intensively car-ried out such as torque controlmethod for torque ripplemin-imization [2] and sensorless control algorithms for low costapplications [3] There is a desire for algorithm validationswhen the controller is designed If the difference between sim-ulation model and the practical motor plant cannot beignored the designed controller in simulation is questionableIn this case to build an accurate mathematical model of theBLDC motor is of greater importance

The system dynamic model structure is the first part ofthe simulation environment to take into consideration In[4 5] phase commutation is considered however complexityof mathematical model is greatly enhanced It is shown in [6]

that the cogging torque is the main source of torque ripple inBLDC motor control Therefore besides the parameters inclassic linear model [1] cogging torque effect is also essentialfor a practical model as a nonlinear factor especially for thelow speed mode Meanwhile the second important issue isthe modelrsquos parameters determination method There hasbeen a large amount ofwork on the field of BLDCmotor iden-tification where time domain identification dominates [7]proposes an available method using by step voltage responseto estimate BLDCmotor parameters and load torque disturb-ance The method has some shorts because only the speedchannel transfer function using speed step voltage input isidentified with the armature current ignored and the stepinput cannot sufficiently stimulate system dynamics An inte-gral measurement steps for determining the permanent mag-net synchronous machine (PMSM) model parameters isdetailed in [8] However its implementation with a torque-sensor results in extra costs A classic least square approxima-tion method a recursive least square algorithm and an on-line batch least squares algorithm are respectively appliedin [9ndash11] However these methods require large numbers ofdata

The main focus of this paper will be on combing fre-quency identification techniques and timedomain estimationmethod in order to identify the practical mathematicalmodel which thoroughly catches the dynamic responses

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 879581 11 pageshttpdxdoiorg1011552015879581

2 Mathematical Problems in Engineering

ia

Ra La

aeb+

minus

Rotor

J 120596

PWM input

Terminal voltageESC

BLDC motor

uaElectromagnetic torque

Power supply

Figure 1 BLDC motor system equivalent configuration

features and motor torque-ripple Most of the parameterswill benefit the advantages of frequency identification in alow cost The remaining parameters on cogging torque effectdeduced in [12]will be estimated in an off-linemethod in timedomain

The paper is organized as follows First the practicalmodelling of the integrated BLDC motor is summarized InSection 3 frequency identification techniques are systemat-ically investigated and a parameter determination methodis proposed to identify first seven parameters lie in linearterm of the integrated BLDC motor model The least squaremethod is detailed in Section 4 to calculate the coggingtorque coefficients The cogging torque values are calculatedby a disturbance observer The identification experiment ona typical BLDC motor used in a UAV is given in Section 5Finally in Section 6 we will draw some concluding remarks

2 Problem Formulation

21 The Nonlinear BLDC Motor Model The comprehensivemodel integrated by bare BLDC dynamics and electronicspeed controller (ESC) model is depicted in Figure 1 TheBLDC dynamics is presented first

BLDG motor model consists of two parts One is anelectrical part which calculates electromagnetic torque andcurrent of motor The other one is a mechanical part whichBLDC Motor and Inverter Equivalent Circuit generates rev-olution of rotor Assuming the variations of the stator self-inductance with rotor position and the mutual inductancebetween the stator windings are negligible the electricaldynamics of the BLDC motor can be described as [1]

119871119886

119889119894119886

119889119905+ 119877119886119894119886+ 119890119887= V119886

119890119887= 119870119890120596

(1)

where V119886is the motor terminal voltages 119894

119886is the motor

armature current 119871119886is armature inductance 119877

119886is the

armature resistance 119890119887is the motor back-EMF and 119870

119890is the

back-EMF constant in units of V(rads)When there is no externally applied torque and only a

constant inertial load is considered the mechanical equationis given as

119869119889120596

119889119905+ 119870119891120596 = 119879 (2)

where 119879 is electromagnetic motor torque 119869 is motor loadinertia and119870

119891is damping coefficient

From the virtual work theory the electromagnetic motortorque 119879 can be derived [12] as

119879 = 119879119889+ 119879cogg + 119879ex (3)

The first term 119879119889in (3) is developed torque caused by inter-

action between the magnet field and the armature currentThe second term119879cogg known as cogging torque is due to theattraction of the permanent magnets (PMs) mounted on therotor with the slots in the stator The cogging torque is onlycorresponding to rotor position and will not vanish even inthe absence of the armature current

The developed torque 119879119889and cogging torque 119879cogg are

given as

119879119889= 119870119905119894119886

119879cogg =119870

sum

119896=1

119879119888119896cos (119896119901120579)

(4)

where 119870119905is the torque constant in units of NmA 119901 is the

number of slots 119879119888119896

is the amplitude of the cogging torquecomponent at 119896th cogging frequency and 120579 is the mechanicalangular position of the rotor

The third term is exogenousmechanical disturbances [9]which can be calculated as

119879ex = 119879119886 sin (120579) + 119879119887 cos (120579) (5)

where 119879119886and 119879

119887are the Fourier coefficients of the exogenous

mechanical disturbances

22 ESC Model The BLDC motors for UAVs are driven byESC (electronic speed controller) The ESC receives pulse-width modulation (PWM) input from some digital radiocontrol units and powers the applied terminal voltage forthe BLDC In order to avoid big current caused by violentincrease or decrease ofmotor speed a linear unit is embeddedinmost ESCTherefore such a linear dynamic mapping fromPWM input to the applied terminal voltage is proposedwhich can be represented as

119866119890=

V (119904)119906 (119904)

=119870119889

120591119904 + 1 (6)

Mathematical Problems in Engineering 3

where 119906(119904) is PWM input 119870119889is the gain from PWM to the

applied voltage and is determined by ESC parameters andpower supply 120591 is time constant of the ESC model

3 Parameters Identification Method ofLinear Terms in Frequency Domain

The parameters of the BLDCmotor provided in the previoussection are 119870

119905 119870119890 119870119891 119877119886 119871119886 119869 120591 and 119879

1198881 1198791198882 119879

119888119870

Note that the first seven parameters lie in linear term of theBLDC motor model and the last term corresponds to non-linear term which describes cogging torque All these param-eters are all assumed time-invariant This section presents anoff-line method for identifying these unknown parametersThe identificationmethod depends on given experiment datafor control PWM input and the corresponding rotor speedand armature current responses There will be two stepsto complete identification Firstly an open-loop frequencydomain method is used to identify these parameters in linearterms on the operation in high speed where cogging torquecan be ignored In the second step the cogging torque signalconsidered as a disturbance torque is estimated by usinga closed up observer Then the least square approximationmethod is applied in order to identify these nonlinear termparameters Both steps are described in detail in the followingsubsections The frequency domain identification methoddetermines most of the parameters therefore this method isbriefly described first

For the BLDC model proposed in Section 2 the non-linearity is produced by cogging torque which is especiallyprominent at lower speeds However at high speed the effectof cogging torque can be filtered out by the motor momentof inertia thus cogging torque can be ignored Thereforethese parameters in linear terms can be identified at highspeed when the BLDC motor is approximately considered asa linear systemThe block diagram formotor linear dynamicsis depicted in Figure 2

The transfer functions 120596(119904)119906119886(119904) = 119866

120596(119904) 119894119886(119904)119906119886(119904) =

119866119894(119904) and 119906

119886(119904)V119886(119904) = 119866

119890(119904) are given as

119866120596(119904) =

119870119905119871119886119869

1199042 + (119870119891119869 + 119877

119886119871119886) 119904 + 119877

119886119870119891119871119886119869 + 119870119890119870119905119871119886119869

119866119894(119904) =

(1119871119886) 119904 + 119870

119891119871119886119869

1199042 + (119870119891119869 + 119877

119886119871119886) 119904 + 119877

119886119870119891119871119886119869 + 119870119890119870119905119871119886119869

119866119890(119904) =

002187

120591119904 + 1

(7)

Note that 119866119890(119904) represents ESC dynamics according to

Section 22 and the gain value 002187 is fitted by throttlecurves from the ESC manufacturer when power supply is setin 26V output

As for identifying these foregoing parameters althoughtime-domain techniques can still be adopted it is stronglyrecommended in this case to leave the time domain in favorof the frequency domain Indeed in the frequency domain

ua(t)

G120596(s)

Gi(s)

Ge(s)

120596(t)

ia(t)

Terminalvoltage

ESC delays

Rotor speed

Armaturecurrent

BLDC linear model

a(t)

PWM input

Figure 2 Block diagram for motor linear dynamics

one has several features not found in the time domain [13](1) unbiased frequency-response estimates when experimen-tal data contain process and output measurement noise (2)access to the coherence function as an unbiased measure ofnonparametric identification accuracy and system responselinearity (3) elimination of biases and reference shifts asidentification parametersTherefore frequency identificationmethod is introduced to identify parameters in linear terms

Then the identification procedure for the unknownparameters will be addressed in detail A simple meaningfuland comprehensive method will be proposed Correspond-ingly the identification procedure is separated into five steps

Step 1 (input sweep signal design) In [13] an automaticsweep signal generating is designed as

119906 (119905) = 1199060+ 119906sweep (119905) = 1199060 + 119860 sin (120579 (119905))

120579 (119905) = [120596min minus 00187 (120596max minus 120596min)] 119905

+ 000467119879rec (120596max minus 120596min) (119890(4119905119879rec) minus 1)

(8)

where 1199060is trim value of input determining the trim values of

rotor speed and armature current 119906sweep is dynamic value ofinput119860 is sweepmagnitude 120579(119905) is signal phase [120596min 120596max]is sweep signal range and 119879rec is sweep record length

Step 2 (collection of time-history data) The sweep signal 119906acts as an input on the integrated BLDC motor plant androtor speed 120596 and armature current 119894

119886are collected as the

output for system identificationThe outputs are also dividedinto two terms as follows

120596 (119905) = 1205960+ 120596sweep (119905)

119894119886(119905) = 119894

1198860+ 119894119886sweep (119905)

(9)

where 1205960and 1198940are trim values of output and 120596sweep(119905) and

119894119886sweep(119905) are dynamic value of outputThe dynamic termswillbe used to obtain frequency responses

4 Mathematical Problems in Engineering

Step 3 (frequency response calculation) The methods tocalculate spectrum function are briefly introduced andwith greater detail in [14] First a Chirp-Z transform isapplied to initially transform time-domain response data tothe frequency-domain Then a technique of combining aweighted average of multiple windows known as CompositeWindowing is used to improve the frequency of accuracyThe results of this calculation in a high quality are inputautospectrum 119866

119909119909(119891) output autospectrum 119866

119910119910(119891) and

cross-spectrum 119866119909119910(119891) Then Frequency response function

is obtained by

(119891) =

119866119909119910(119891)

119866119909119909(119891)

(10)

Step 4 (coherence function calculation) Coherence functionis defined as

120574119909119910(119891) =

10038161003816100381610038161003816119866119909119910(119891)

10038161003816100381610038161003816

2

1003816100381610038161003816119866119909119909 (119891)1003816100381610038161003816

10038161003816100381610038161003816119866119910119910(119891)

10038161003816100381610038161003816

(11)

Coherence function can be interpreted physically as the frac-tion of the output spectrum that is linearly attributable to theinput spectrum at any frequency It can effectively and rapidlyassess the accuracy of the frequency response identificationIn particular the frequency response identification resultsare acceptable as long as the coherence function satisfies thecondition

120574119909119910ge 06 (12)

Step 5 (transfer function fitting) A numerical optimizationalgorithm is used to determine the unknown parameters byminimizing the magnitude and phase errors between thedesired BLDCmotor transfermodels119867(119904) and the associatedfrequency responses (119904) The practical cost function is

min 119869 = 20

119899119908

120596119899119908

sum

1205961

119882120574[119882119892(|119867| minus

1003816100381610038161003816100381610038161003816100381610038161003816) + 119882

119875(ang119867 minus ang)

2

]

(13)

where 119899119908is the number of frequency points and 120596

1and 120596

119899119908

are starting and ending frequencies of fit 119882120574is weighting

function dependent on the value of the coherence functionat each frequency 119882

119892and 119882

119901are the relative weights for

magnitude and phase squared errors One can obtain detailedselection of these weighing functions in [13]

All these forgoing methods to frequency system iden-tification have been incorporated into the known soft-ware CIFER (Comprehensive Identification from FrequencyResponses) Unlimited to its original application in rotorcraftCIFER is also suitable for other plant such as BLDC motorsystem for linear system identification

Step 6 (system parameters determination) Based on thesefive steps for frequency identification the transfer functionfrom PWM input to speed channel and armature channel areobtained However these parameters that is 119870

119905 119870119890 119870119891 119877119886

119871119886 119869 120591 in mechanism are still unknown Hence a method is

proposed as followsThe identified transfer functions in lumped parameters

are

119867120596(119904) =

1198701

1199043 + 119886111199042 + 119886

12119904 + 11988613

(14)

119867119894(119904) =

1198702(1198872119904 + 1)

1199043 + 119886211199042 + 119886

22119904 + 11988623

(15)

Theoretically both denominators of (14) and (15) are equalNevertheless small allowable deviation is unavoidable inpractice resulting in very close but strictly different denom-inators in these two equations In that case these unknownparameters cannot be obtained by classic undetermined coef-ficient method A feasible idea is to find suitable parametersthat ensure theoretical model to approximate identificationresult as much as possible

The bias between theoretical model and identificationresult should be defined first In [15] the bias between twotransfer functions 119866

1and 119866

2can be written as

10038171003817100381710038171198661 minus 11986621003817100381710038171003817infin

(16)

Hence the parameter determination problem can betransferred to the following optimization problem by 119867

infin

norm

min bias = 119891 (119870119905 119870119891 119870119890 119869 119877119886 119871119886 120591)

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[

[

119882120596(119867120596minus 119866120596119866119890)

119882119894(119867119894minus 119866119894119866119890)

]

]

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[[[[[[

[

119882120596(119867120596minus002187

120591119904 + 1sdot

119870119905119871119886119869

1199042 + (119870119891119869 + 119877

119886119871119886) 119904 + 119877

119886119870119891119871119886119869 + 119870119890119870119905119871119886119869

)

119882119894(119867119894minus002187

120591119904 + 1sdot

(1119871119886) 119904 + 119870

119891119871119886119869

1199042 + (119870119891119869 + 119877

119886119871119886) 119904 + 119877

119886119870119891119871119886119869 + 119870119890119870119905119871119886119869

)

]]]]]]

]

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin

(17)

Mathematical Problems in Engineering 5

0 05 1 15 20

20

40

60

80

100

Time (s)

Roto

r spe

ed (r

ads

)

Figure 3 Cogging torque effect in low speed

Equation (17) alone is probably not sufficient to find theonly solution Another two equation constraints on systemstatic step response are

1205960=

002187119870119905

119877119886119870119891+ 119870119890119870119905

1198941198860=

002187119870119891

119877119886119870119891+ 119870119890119870119905

(18)

where 1205960and 1198941198860

are respectively measured static value ofrotor speed and armature on step response experiment 119882

120596

and119882119894areweighting coefficients and119867

120596and119867

119894are identified

transfer function As a typical constrained optimizationproblem this problem is not complicated and can be readilysolved by Matlab optimization toolbox

4 Cogging Torque Parameters IdentificationMethod in Time Domain

Parameters in linear terms can be estimated by frequencyidentification method during high speed operation wherethe cogging torquersquos influence could be neglected But whenmotor speed approaches to a low operation as shown inFigure 3 such perfectly constant speed cannot be achievedIn this case cogging torque effect should be taken intoaccount Since cogging torque is a nonlinear compact on theBLDC motor model assumptions for frequency identifica-tion method is ruined Hence aiming at estimating parame-ters in cogging torque term a nonlinear system identificationmethod is necessaryThe cogging torque cannot bemeasuredor calculated by direct means In this subsection the coggingtorque considered as a disturbance will be estimated by aclosed up observer The least square method (LSM) is usedto obtain cogging torque parameters 119879

1198881 1198791198882 119879

119888119870

41 Disturbance Observer Design A general block diagramfor disturbance observer [9] shown in Figure 4 can beused to approximate disturbance signal 119906dist If inaccuraciesin the plant model are negligible and the compensator isable to ensure convergence of the error signal 119890 towards

Plant

Ideal model+

+

+

minus

minus

minus

GP(s)

Compensator

udist

udist

y

y

e

Gc(s)

u

Figure 4 Disturbance observer block diagram

zero then the compensator output dist is able to estimatedisturbance signalThus a similar way can be used to estimatenon-modelled dynamics in the case of the linear modeldiscussed in frequency domain on condition that exogenousdisturbances are ignored

Based on superposition principle for linear system thetransfer function119867

119900(119904) fromestimated to applied disturbance

can be deduced as

119867119900(119904) =

dist119906dist

=119866119875(119904) 119866119888(119904)

1 + 119866119875(119904) 119866119888(119904)

(19)

The compensator 119866119888(119904) should be chosen so that the denom-

inator of the polynomial in (19) is Hurwitz stable Thenthe system in Figure 4 is asymptotically stable Thereforethe output of the compensator dist succeeds in observingdisturbance signal 119906dist asymptotically

Depended on the previously discussed disturbanceobserver the cogging torque coefficients identification inan off-line process is realized as shown in Figure 5 Theideal model comes from mechanical dynamics in the linearBLDC motor model and therersquos no load external torqueA 119875119868 controller designed by following the two principalsintroduced in Section 41 works as a compensator

The cogging torque is equivalent to a disturbance signalof electromagnetic torque If a 119875119868 compensator is chosen thecompensator plantmodel and observer transfer functionwillbe

119866119888(119904) = 119875 +

119868

119904

119866119875(119904) =

1

119869119904 + 119870119891

119867119900(119904) =

119866119875(119904) 119866119888(119904)

1 + 119866119875(119904) 119866119888(119904)

=119875119904 + 119868

1198691199042 + (119870119891+ 119875) 119904 + 119868

(20)

Two principles will ensure the accuracy of the observersystem shown in Figure 4

6 Mathematical Problems in Engineering

++ minus

minus

Experimental data

Rotor speed120596

iaKt

Developedtorque

Electromagnetictorque

Td T

Assumedspeed

TcoggP +

I

s

Estimated coggingtorque

Rotor speederrore

Armature current 1

Js + Kf

Ideal model

PI compensator

Figure 5 Cogging torque estimation

Principle 1 Since the observation function is a typical secondorder system the system is stable if

120585 =

119870119891+ 119875

2radic119869119868gt 0 (21)

Equation (21) means that positive 119875 and 119868 sufficiently assurethe stability of the system

Principle 2 In order to improve the accuracy of the observingsignal dist the compensator should minimize the maximumgain from in whole history data that is

min 1198902=1003817100381710038171003817119910 minus 119910

10038171003817100381710038172=

119899

sum

119894

(119910 (119894) minus 119910 (119894))2

(22)

where 119899 is the number of history data points and 120576 is mini-mum energy of error signal that allowed

42 Cogging Torque Parameters Identification Based on LeastSquares Approximation Method In BLDC motor systemsince the convergence of the assumed speed towards themeasured speed is ensured by the resulting compensator theestimated cogging torque dist is able to approximate the truenonlinear dynamics well

The least squares method is considerably used in practicedue to convenient implementation The objective of thisapproximating method is to minimize the ldquoenergy of errorrdquonamely the squared difference between the measured signalsand the estimated signals Considering a vector problem theestimated output is determined by a signal model whichrelates to the unknown parameter vector 120573 = [120573

1 120573

119896]

119908 and 119911 are respectively the input of the model and thepractical measured output The least squares estimator (LSE)is to find the value of 120573 which minimizes the squares errorcriterion given as

min 119869 = 119911 minus 2=

119873

sum

119899=1

(119911 (119899) minus (119899))2

(23)

The observation interval is assumed to be 119899 = 1 2 119873If the linear least squares problem is taken into consideration

the signalmodel is linearwith respect to the unknownparam-eter therefore the output signal can be

119911 = Φ120573 (24)where Φ is the observation matrix in [16] the least squareestimator and average minimum least square error are

120573 = (120601119879

120601)minus1

120601119879

(25)

119869119886min =

( minus 120601120573)119879

( minus 120601120573)

119873

(26)

Considering the unbalanced mechanical system effectinto the estimation process [9] the estimated disturbancetorque is

dist = 119879cogg

=

119870

sum

119896=1

119879119888119896cos (119896119901120579) + 119879

119886sin (120579) + 119879

119887cos (120579)

(27)

The rotor position 120579 is obtained by integration of the time-history data of rotor speed 120596 In this way the term in (27)cos(119896119901120579) is known

Then the observation matrix of LSE is therefore writtenas

Φ = [120601119894119895 120601119904

119894119895120601119888

119894119895]

120601119894119895= cos (119895 sdot 119901120579 (119894)) 119894 = 1 119873 119895 = 1 119870

120601119904

119894119895= sin (120579 (119894)) 119894 = 1 119873 119895 = 1

120601119888

119894119895= cos (120579 (119894)) 119894 = 1 119873 119895 = 1

(28)

Then the parameter vector 120573 = [11987911988811198791198882sdot sdot sdot 119879119888119870 119879119886 119879119887] is

calculated by (25) An essential parameter 119870 the minimalnumber of relevant harmonics which gives the simplestsignal model is necessarily determined by average minimumleast square error 119869

119886min A suitable119870 is obtained to adequatelydescribe the cogging torque characteristics For this sakeof determination method we will increase the number ofharmonics119870 unless the minimum least squares error is smallenough

Mathematical Problems in Engineering 7

BLDC motorESC

Hall sensor

Rotary encoder

Digital oscilloscope

Stabilized voltage supply

Control signal generator

PWM generator

Figure 6 BLDC motor identification experimental bench setup

Table 1 BLDC motor mechanical parameters

Description Value UnitNumber of stator arms 12 Magnet poles 14 Max continuous power 3900 WRotor inertia (evaluated) 0000549 kgsdotm2

Rotary encoder inertia 000001 kgsdotm2

5 System Identification Experiment

System identification experiments are designed and carriedout to obtain data for comprehensive identification Thissection is divided into three parts First we will detail theexperimental plant in BIT Then frequency identificationresults and system main parameter determination will bedescribed Finallywewill discuss nonlinear identification andestimate cogging torque coefficients

51 Experimental Setup The experimental bench setup isshown in Figure 6 including BLDC motor driving unitcontrol unit and measure unit Each of the subsystem willbe briefly introduced by the following subsystems of theexperimental setup

511 BLDC Motor The motor to be identified is popularBLDC motor for UAVs scorpion SII-6530 Considering thesimple configuration of this motor the rotor inertia can beevaluated by Solidworks which is important for the valida-tion of the identification results Somemotor parameters andthe evaluated inertia of the rotary encoder and coupling arelisted in Table 1

512 Driving Unit The driving unit consists of a ESC and aPWM generator The ESC produced by Hobyywing similarto a motor controller and three-phase bridge receives thesignal produced by the PWM generator A stabilized voltagesource powers the ESC to drive the motor The PWMgenerator actually a servo controller is mainly responsible fordecoding control commands and generating PWM wave

0 20 40 60 80 100 1201360

1380

1400

1420

1440

Time (s)

PWM

wav

e dut

y cy

cle

Figure 7 Automatic sweep input signal ranging from 001sim10Hz

513 Control Unit The control unit is an ordinary PCinstalled by Linux operating systemThePCandPWMgener-ator are connected by RS-232 serial ports All the designedcontrol commands are sent from control unit to the PWMgenerator

514 Measure Unit In order to measure these two systemstates rotor speed and armature current a rotary encoder andhall current sensor are respectively included inmeasure unitBoth of output signal of these two sensors are recorded by thedigital oscilloscope in high sample rates The measurementsampling time of the digital oscilloscope could be set from1 120583s up to 50120583s and the memory amount is available forrecording data length of 10Mb Then the rotor speed andarmature current can be deciphered from the measured data

52 Frequency Identification Results When the power supplyis set at 23V the rotary encoder wasmounted onto the BLDCmotor rotor and the armature current wire passes throughthe hall current sensor The encoder and the hall sensorrsquosoutputs are recorded simultaneously during the experimentwith sampling frequency 119891

119904= 100Hz The rotor speed and

armature are then calculated off-lineComplying with (8) the input signal shown in Figure 7

covers frequencies ranging from 001 to 10HzThis frequencysweep input signal is utilized to excite the dynamics of our

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120240

250

260

270

280

290

300

Time (s)

Roto

r spe

ed (r

ads

)

(a) Rotor speed response

0 20 40 60 80 100 120

0

05

1

15

2

25

3

Time (s)

Arm

atur

e cur

rent

(A)

minus05

(b) Armature current response

Figure 8 Rotor speed and armature current responses

0

10

20

Frequency (rads)

Mag

nitu

de (d

B)

0

Frequency (rads)

Phas

e (de

g)

Experimental dataTransfer function fit

0

05

1

Frequency (rads)

Coh

eren

ce

100 101

100 101

100 101

minus150

minus100

minus50

(a) Rotor speed channel

Frequency (rads)

0

50

Frequency (rads)

Experimental dataTransfer function fit

Mag

nitu

de (d

B)Ph

ase (

deg)

Coh

eren

ce

0

05

1

Frequency (rads)

10minus1 100 101

100 101

102

100 101 102

minus100

minus50

minus15

minus10

minus5

(b) Armature channel

Figure 9 Rotor speed channelarmature channel identification

plant over a broad frequency range Figure 8 shows the rotorspeed and armature current extracted from experimentaldata Note that rotor speed response amplitude diminishesat higher frequencies as expected from the bandwidth ofthe corresponding channel However the armature currentresponse amplitude will increase with the even very highfrequency suggesting a very high bandwidth of the armaturecurrent channel which is considerably over the max sweepfrequency

The data obtained from this test will be applied toextract frequency responses used for themodel identification

The estimated frequency responses are calculated using spec-tral density functions with CIFER Figure 9 shows frequencyidentification of rotor speed channel and armature channelGood coherence is achieved in the required frequency rangeindicating good quality experimental data and the linearmodel assumption With respect to coherence rotor speedchannel identification result is better than that of currentchannel because the hall current sensor yields a pooraccuracy compared to rotary encoder The identified transferfunctions represented by red dashed lines show good fits ofthe magnitude and phase responses of each channel

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12900

950

1000

1050

1100

1150

1200

1250

Time (s)

PWM

wav

e dut

y cy

cle

Figure 10 Input doublet signal

Table 2 Estimated BLDC motor parameters

System parameters Symbol Value UnitRotor inertia 119869 000061 kgsdotm2

Voltage constant 119870119890

00637 V(rads)Torque constant 119870

11990502663 NmA

Armature resistance 119877119886

06187 Ω

Armature inductance 119871119886

262 mHFriction coefficient 119870

119891000291 N(rads)

Time constant of the ESC model 120591 00781 s

The two transfer functions of two channels based on (13)and (14) identified are

119867120596(119904) =

2057342

1199043 + 18951199042 + 13412119904 + 142834

119867119894(119904) =

21749 (02015119904 + 1)

1199043 + 1961199042 + 14695119904 + 149431

(29)

The identified transfer functions are taken into account todetermine BLDCmotormechanismparameters as suggestedin Section 3 Then the problem in (17) and (18) is solvedby Matlab optimization toolbox The bias between theoret-ical model and identification result is 041 The estimatedBLDC motor parameters are listed in Table 2 Note thatthe identified rotor inertia 119869 = 000061 is very closeto the evaluated value in Table 1 indicating a satisfactoryidentification accuracy of this parameter

The accuracy of the comprehensive BLDC motor modelmust be further verified by tine history data The inputdoublet signal shown in Figure 10 is used to stimulate theBLDC motor dynamics The validation results are shownin Figure 11 Both rotor speed and armature current aremeasured during the input doublet signal It is shown that theidentifiedmodel perfectly predicts the rotor speed responseshowever the true armature current response exhibits a slightdeviation from the responses predicted by the identifiedmodel This mismatch can be explained by the relatively lowcoherence in high frequency of the armature channel identifi-cationThe close agreement between the simulation response

Table 3 Estimation results for the mechanical parameters

System parameters 119909 (mean values) 120590 (standard deviations)1198791198881

293 451198791198882

527 511198791198883

3706 4461198791198884

156 411198791198885

843 2241198791198886

5612 433119879119886

1103 125119879119887

1620 332

and experimental data indicates the identified model isaccurate enough to reflect the BLDC motorrsquos dynamics

53 Cogging Torque Coefficients Identification Results Sincethe cogging torque effect exists significantly in low speed theterminal voltage of the motor is set at 65 V The disturbanceobserver is applied to calculate cogging torque as stated inSection 42 In order to eliminate the stochastic error theexperiment is repeated by 10 repetitions with single testtime of 20 s The minimum least squares error obtained by(26) versus the number of cogging torque harmonics areshown in Figure 12 with the estimated disturbance signalapproximated using LSE It is obvious that the sixth coggingtorque harmonic is highly dominant from the diagram and aslight of third harmonics is also present [17] has pointed outthat the cogging torque harmonics for three-phase motorsusually are multiples of six times the fundamental commuta-tion frequency

A simulation of the identified model is performed inorder to verify the results of the proposed parameters iden-tification method The cogging torque coefficients in Table 3and the estimated linear terms by frequency identification inSection 52 are utilized in the comprehensive BLDC motormodel A comparison betweenmeasured and estimated rotorspeed is formed to highlight the verification result (seeFigure 13) Note that there is a small phase deviation ofbetween experimental and estimated cogging torque thiscould be interpreted by the unmodeled dynamics Matchingof the simulated and experimental data is ensured Hence theproposed identification method is applicable and sufficient

6 Conclusion

In this paper a comprehensive procedure for system iden-tification applicable to BLDC motor used in UAV has beensystematically presented The procedure aims to serve asa guide for the steps necessary to determine an accurateBLDC motor model Considering the nonlinear character-istics cogging torque effect of the BLDC motor system apractical mathematical model is introduced for identifica-tion Preliminarily frequency domain identification theoryis applied to estimate the first seven parameters in linearterms of the model at a high motor speed region where thenonlinear term can be neglected Then an off-line methodfor identifying the cogging torque coefficients is performed at

10 Mathematical Problems in Engineering

0 2 4 6 8 10 12240

260

280

300

320

Time (s)

Roto

r spe

ed (r

ads

)

Experimental dataBLDC motor model

(a) Rotor speed validation of the BLDC motor dynamics

2 4 6 8 10 12

1

2

3

4

5

Time (s)

Arm

atur

e cur

rent

(A)

Experimental dataBLDC motor model

(b) Armature current validation of the BLDC motor dynamics

Figure 11 Time domain validation of the BLDC motor identified dynamics

2 3 4 5 6 7 8 90

10

20

30

Number of harmonics (K)

J min

Figure 12 The minimum least squares error versus the number of cogging torque harmonics

05 1 15 275

80

85

90

95

100

Time (s)

Nonlinear model Experimental data

02 025 03

90

92

94

96

Time (s)

Nonlinear model Experimental data

Roto

r spe

ed (r

ads

)

Roto

r spe

ed (r

ads

)

Figure 13 Validation of identified comprehensive BLDC motor model

Mathematical Problems in Engineering 11

very low speedwhen cogging torque effect exists significantlyThis method is founded on the least squares approximationmethod a disturbance observer and the identified parametersin frequency identification The results from the simulationand the real-time experiments matched closely which isconsidered as a validation of the obtained mathematicalmodel

The BLDC model identifications are conditional onlyon the availability of experimental data for rotor speed andarmature current therefore our method without sensors todetect torque or back-EMF is easy to realize and is practicaland effectiveThis research will provide greater opportunitiesfor future controller design of the BLDCmotorThe extractedlinear model can be utilized directly for linear controllerdesign and the accurately identified cogging torque effect isbeneficial to reduce the speed ripple of the motor at some lowspeed conditions using some advanced control theory suchas robust control and nonlinear control

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Shanmugasundram K M Zakaraiah and N YadaiahldquoModeling simulation and analysis of controllers for brushlessdirect current motor drivesrdquo Journal of Vibration and Controlvol 19 no 8 pp 1250ndash1264 2013

[2] C Xia Y Wang and T Shi ldquoImplementation of finite-statemodel predictive control for commutation torque ripple min-imization of permanent-magnet brushless DC motorrdquo IEEETransactions on Industrial Electronics vol 60 no 3 pp 896ndash905 2013

[3] S-Y Chang P-H Yu and W-C Su ldquoPre- and post-filteringapproach to sensorless VSS speed control of a permanentmagnet DC motor subject to Hammerstein and Wiener non-linearitiesrdquoControl Engineering Practice vol 21 no 11 pp 1577ndash1583 2013

[4] B-K Lee and M Ehsani ldquoAdvanced simulation model forbrushless dc motor drivesrdquo Electric Power Components andSystems vol 31 no 9 pp 841ndash868 2003

[5] I Kim N Nakazawa S Kim C Park and C Yu ldquoCompensa-tion of torque ripple in high performance BLDCmotor drivesrdquoControl Engineering Practice vol 18 no 10 pp 1166ndash1172 2010

[6] H Lu L Zhang and W Qu ldquoA new torque control method fortorque rippleminimization of BLDCmotors with un-ideal backEMFrdquo IEEE Transactions on Power Electronics vol 23 no 2 pp950ndash958 2008

[7] W Wu ldquoDC motor parameter identification using speed stepresponsesrdquo Modelling and Simulation in Engineering vol 2012Article ID 189757 5 pages 2012

[8] J-H Choi S-H You J Hur and H-G Sung ldquoThe design andfabrication of BLDCMotor andDrive for 42V automotive appli-cationsrdquo in Proceedings of the IEEE International Symposium onIndustrial Electronics (ISIE rsquo07) pp 1086ndash1091 June 2007

[9] A Kapun M Curkovic A Hace and K Jezernik ldquoIdentifyingdynamic model parameters of a BLDC motorrdquo Simulation

Modelling Practice and Theory vol 16 no 9 pp 1254ndash12652008

[10] A J Blauch M Bodson and J Chiasson ldquoHigh-Speed parame-ter estimation of stepper motorsrdquo IEEE Transactions on ControlSystems Technology vol 1 no 4 pp 270ndash279 1993

[11] R Patankar and L Zhu ldquoReal-time multiple parameter estima-tion for voltage controlled brushless DC motor actuatorsrdquo inProceedings of the American Control Conference (AAC rsquo04) vol4 pp 3851ndash3856 July 2004

[12] M S Islam R Islam T Sebastian A Chandy and S AOzsoyluldquoCogging torque minimization in PM motors using robustdesign approachrdquo IEEE Transactions on Industry Applicationsvol 47 no 4 pp 1661ndash1669 2011

[13] M B Tischler and R K Remple Aircraft and Rotorcraft SystemIdentification AIAA Education Series American Institute ofAeronautics and Astronautics 2006

[14] M Petra and M Utting CZT A Framework for Z Tools FormalSpecification and Development in Z and B Springer BerlinGermany 2005

[15] K Zhou and C D John Essentials of Robust Control PrenticeHall Upper Saddle River NJ USA 1998

[16] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Higher Education New York NY USA 2010

[17] J Y Hung and Z Ding ldquoDesign currents to reduce torque ripplein brushless permanent magnet motorsrdquo IEE Proceedings BElectric Power Applications vol 140 no 4 pp 260ndash266 1993

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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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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

ia

Ra La

aeb+

minus

Rotor

J 120596

PWM input

Terminal voltageESC

BLDC motor

uaElectromagnetic torque

Power supply

Figure 1 BLDC motor system equivalent configuration

features and motor torque-ripple Most of the parameterswill benefit the advantages of frequency identification in alow cost The remaining parameters on cogging torque effectdeduced in [12]will be estimated in an off-linemethod in timedomain

The paper is organized as follows First the practicalmodelling of the integrated BLDC motor is summarized InSection 3 frequency identification techniques are systemat-ically investigated and a parameter determination methodis proposed to identify first seven parameters lie in linearterm of the integrated BLDC motor model The least squaremethod is detailed in Section 4 to calculate the coggingtorque coefficients The cogging torque values are calculatedby a disturbance observer The identification experiment ona typical BLDC motor used in a UAV is given in Section 5Finally in Section 6 we will draw some concluding remarks

2 Problem Formulation

21 The Nonlinear BLDC Motor Model The comprehensivemodel integrated by bare BLDC dynamics and electronicspeed controller (ESC) model is depicted in Figure 1 TheBLDC dynamics is presented first

BLDG motor model consists of two parts One is anelectrical part which calculates electromagnetic torque andcurrent of motor The other one is a mechanical part whichBLDC Motor and Inverter Equivalent Circuit generates rev-olution of rotor Assuming the variations of the stator self-inductance with rotor position and the mutual inductancebetween the stator windings are negligible the electricaldynamics of the BLDC motor can be described as [1]

119871119886

119889119894119886

119889119905+ 119877119886119894119886+ 119890119887= V119886

119890119887= 119870119890120596

(1)

where V119886is the motor terminal voltages 119894

119886is the motor

armature current 119871119886is armature inductance 119877

119886is the

armature resistance 119890119887is the motor back-EMF and 119870

119890is the

back-EMF constant in units of V(rads)When there is no externally applied torque and only a

constant inertial load is considered the mechanical equationis given as

119869119889120596

119889119905+ 119870119891120596 = 119879 (2)

where 119879 is electromagnetic motor torque 119869 is motor loadinertia and119870

119891is damping coefficient

From the virtual work theory the electromagnetic motortorque 119879 can be derived [12] as

119879 = 119879119889+ 119879cogg + 119879ex (3)

The first term 119879119889in (3) is developed torque caused by inter-

action between the magnet field and the armature currentThe second term119879cogg known as cogging torque is due to theattraction of the permanent magnets (PMs) mounted on therotor with the slots in the stator The cogging torque is onlycorresponding to rotor position and will not vanish even inthe absence of the armature current

The developed torque 119879119889and cogging torque 119879cogg are

given as

119879119889= 119870119905119894119886

119879cogg =119870

sum

119896=1

119879119888119896cos (119896119901120579)

(4)

where 119870119905is the torque constant in units of NmA 119901 is the

number of slots 119879119888119896

is the amplitude of the cogging torquecomponent at 119896th cogging frequency and 120579 is the mechanicalangular position of the rotor

The third term is exogenousmechanical disturbances [9]which can be calculated as

119879ex = 119879119886 sin (120579) + 119879119887 cos (120579) (5)

where 119879119886and 119879

119887are the Fourier coefficients of the exogenous

mechanical disturbances

22 ESC Model The BLDC motors for UAVs are driven byESC (electronic speed controller) The ESC receives pulse-width modulation (PWM) input from some digital radiocontrol units and powers the applied terminal voltage forthe BLDC In order to avoid big current caused by violentincrease or decrease ofmotor speed a linear unit is embeddedinmost ESCTherefore such a linear dynamic mapping fromPWM input to the applied terminal voltage is proposedwhich can be represented as

119866119890=

V (119904)119906 (119904)

=119870119889

120591119904 + 1 (6)

Mathematical Problems in Engineering 3

where 119906(119904) is PWM input 119870119889is the gain from PWM to the

applied voltage and is determined by ESC parameters andpower supply 120591 is time constant of the ESC model

3 Parameters Identification Method ofLinear Terms in Frequency Domain

The parameters of the BLDCmotor provided in the previoussection are 119870

119905 119870119890 119870119891 119877119886 119871119886 119869 120591 and 119879

1198881 1198791198882 119879

119888119870

Note that the first seven parameters lie in linear term of theBLDC motor model and the last term corresponds to non-linear term which describes cogging torque All these param-eters are all assumed time-invariant This section presents anoff-line method for identifying these unknown parametersThe identificationmethod depends on given experiment datafor control PWM input and the corresponding rotor speedand armature current responses There will be two stepsto complete identification Firstly an open-loop frequencydomain method is used to identify these parameters in linearterms on the operation in high speed where cogging torquecan be ignored In the second step the cogging torque signalconsidered as a disturbance torque is estimated by usinga closed up observer Then the least square approximationmethod is applied in order to identify these nonlinear termparameters Both steps are described in detail in the followingsubsections The frequency domain identification methoddetermines most of the parameters therefore this method isbriefly described first

For the BLDC model proposed in Section 2 the non-linearity is produced by cogging torque which is especiallyprominent at lower speeds However at high speed the effectof cogging torque can be filtered out by the motor momentof inertia thus cogging torque can be ignored Thereforethese parameters in linear terms can be identified at highspeed when the BLDC motor is approximately considered asa linear systemThe block diagram formotor linear dynamicsis depicted in Figure 2

The transfer functions 120596(119904)119906119886(119904) = 119866

120596(119904) 119894119886(119904)119906119886(119904) =

119866119894(119904) and 119906

119886(119904)V119886(119904) = 119866

119890(119904) are given as

119866120596(119904) =

119870119905119871119886119869

1199042 + (119870119891119869 + 119877

119886119871119886) 119904 + 119877

119886119870119891119871119886119869 + 119870119890119870119905119871119886119869

119866119894(119904) =

(1119871119886) 119904 + 119870

119891119871119886119869

1199042 + (119870119891119869 + 119877

119886119871119886) 119904 + 119877

119886119870119891119871119886119869 + 119870119890119870119905119871119886119869

119866119890(119904) =

002187

120591119904 + 1

(7)

Note that 119866119890(119904) represents ESC dynamics according to

Section 22 and the gain value 002187 is fitted by throttlecurves from the ESC manufacturer when power supply is setin 26V output

As for identifying these foregoing parameters althoughtime-domain techniques can still be adopted it is stronglyrecommended in this case to leave the time domain in favorof the frequency domain Indeed in the frequency domain

ua(t)

G120596(s)

Gi(s)

Ge(s)

120596(t)

ia(t)

Terminalvoltage

ESC delays

Rotor speed

Armaturecurrent

BLDC linear model

a(t)

PWM input

Figure 2 Block diagram for motor linear dynamics

one has several features not found in the time domain [13](1) unbiased frequency-response estimates when experimen-tal data contain process and output measurement noise (2)access to the coherence function as an unbiased measure ofnonparametric identification accuracy and system responselinearity (3) elimination of biases and reference shifts asidentification parametersTherefore frequency identificationmethod is introduced to identify parameters in linear terms

Then the identification procedure for the unknownparameters will be addressed in detail A simple meaningfuland comprehensive method will be proposed Correspond-ingly the identification procedure is separated into five steps

Step 1 (input sweep signal design) In [13] an automaticsweep signal generating is designed as

119906 (119905) = 1199060+ 119906sweep (119905) = 1199060 + 119860 sin (120579 (119905))

120579 (119905) = [120596min minus 00187 (120596max minus 120596min)] 119905

+ 000467119879rec (120596max minus 120596min) (119890(4119905119879rec) minus 1)

(8)

where 1199060is trim value of input determining the trim values of

rotor speed and armature current 119906sweep is dynamic value ofinput119860 is sweepmagnitude 120579(119905) is signal phase [120596min 120596max]is sweep signal range and 119879rec is sweep record length

Step 2 (collection of time-history data) The sweep signal 119906acts as an input on the integrated BLDC motor plant androtor speed 120596 and armature current 119894

119886are collected as the

output for system identificationThe outputs are also dividedinto two terms as follows

120596 (119905) = 1205960+ 120596sweep (119905)

119894119886(119905) = 119894

1198860+ 119894119886sweep (119905)

(9)

where 1205960and 1198940are trim values of output and 120596sweep(119905) and

119894119886sweep(119905) are dynamic value of outputThe dynamic termswillbe used to obtain frequency responses

4 Mathematical Problems in Engineering

Step 3 (frequency response calculation) The methods tocalculate spectrum function are briefly introduced andwith greater detail in [14] First a Chirp-Z transform isapplied to initially transform time-domain response data tothe frequency-domain Then a technique of combining aweighted average of multiple windows known as CompositeWindowing is used to improve the frequency of accuracyThe results of this calculation in a high quality are inputautospectrum 119866

119909119909(119891) output autospectrum 119866

119910119910(119891) and

cross-spectrum 119866119909119910(119891) Then Frequency response function

is obtained by

(119891) =

119866119909119910(119891)

119866119909119909(119891)

(10)

Step 4 (coherence function calculation) Coherence functionis defined as

120574119909119910(119891) =

10038161003816100381610038161003816119866119909119910(119891)

10038161003816100381610038161003816

2

1003816100381610038161003816119866119909119909 (119891)1003816100381610038161003816

10038161003816100381610038161003816119866119910119910(119891)

10038161003816100381610038161003816

(11)

Coherence function can be interpreted physically as the frac-tion of the output spectrum that is linearly attributable to theinput spectrum at any frequency It can effectively and rapidlyassess the accuracy of the frequency response identificationIn particular the frequency response identification resultsare acceptable as long as the coherence function satisfies thecondition

120574119909119910ge 06 (12)

Step 5 (transfer function fitting) A numerical optimizationalgorithm is used to determine the unknown parameters byminimizing the magnitude and phase errors between thedesired BLDCmotor transfermodels119867(119904) and the associatedfrequency responses (119904) The practical cost function is

min 119869 = 20

119899119908

120596119899119908

sum

1205961

119882120574[119882119892(|119867| minus

1003816100381610038161003816100381610038161003816100381610038161003816) + 119882

119875(ang119867 minus ang)

2

]

(13)

where 119899119908is the number of frequency points and 120596

1and 120596

119899119908

are starting and ending frequencies of fit 119882120574is weighting

function dependent on the value of the coherence functionat each frequency 119882

119892and 119882

119901are the relative weights for

magnitude and phase squared errors One can obtain detailedselection of these weighing functions in [13]

All these forgoing methods to frequency system iden-tification have been incorporated into the known soft-ware CIFER (Comprehensive Identification from FrequencyResponses) Unlimited to its original application in rotorcraftCIFER is also suitable for other plant such as BLDC motorsystem for linear system identification

Step 6 (system parameters determination) Based on thesefive steps for frequency identification the transfer functionfrom PWM input to speed channel and armature channel areobtained However these parameters that is 119870

119905 119870119890 119870119891 119877119886

119871119886 119869 120591 in mechanism are still unknown Hence a method is

proposed as followsThe identified transfer functions in lumped parameters

are

119867120596(119904) =

1198701

1199043 + 119886111199042 + 119886

12119904 + 11988613

(14)

119867119894(119904) =

1198702(1198872119904 + 1)

1199043 + 119886211199042 + 119886

22119904 + 11988623

(15)

Theoretically both denominators of (14) and (15) are equalNevertheless small allowable deviation is unavoidable inpractice resulting in very close but strictly different denom-inators in these two equations In that case these unknownparameters cannot be obtained by classic undetermined coef-ficient method A feasible idea is to find suitable parametersthat ensure theoretical model to approximate identificationresult as much as possible

The bias between theoretical model and identificationresult should be defined first In [15] the bias between twotransfer functions 119866

1and 119866

2can be written as

10038171003817100381710038171198661 minus 11986621003817100381710038171003817infin

(16)

Hence the parameter determination problem can betransferred to the following optimization problem by 119867

infin

norm

min bias = 119891 (119870119905 119870119891 119870119890 119869 119877119886 119871119886 120591)

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[

[

119882120596(119867120596minus 119866120596119866119890)

119882119894(119867119894minus 119866119894119866119890)

]

]

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[[[[[[

[

119882120596(119867120596minus002187

120591119904 + 1sdot

119870119905119871119886119869

1199042 + (119870119891119869 + 119877

119886119871119886) 119904 + 119877

119886119870119891119871119886119869 + 119870119890119870119905119871119886119869

)

119882119894(119867119894minus002187

120591119904 + 1sdot

(1119871119886) 119904 + 119870

119891119871119886119869

1199042 + (119870119891119869 + 119877

119886119871119886) 119904 + 119877

119886119870119891119871119886119869 + 119870119890119870119905119871119886119869

)

]]]]]]

]

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin

(17)

Mathematical Problems in Engineering 5

0 05 1 15 20

20

40

60

80

100

Time (s)

Roto

r spe

ed (r

ads

)

Figure 3 Cogging torque effect in low speed

Equation (17) alone is probably not sufficient to find theonly solution Another two equation constraints on systemstatic step response are

1205960=

002187119870119905

119877119886119870119891+ 119870119890119870119905

1198941198860=

002187119870119891

119877119886119870119891+ 119870119890119870119905

(18)

where 1205960and 1198941198860

are respectively measured static value ofrotor speed and armature on step response experiment 119882

120596

and119882119894areweighting coefficients and119867

120596and119867

119894are identified

transfer function As a typical constrained optimizationproblem this problem is not complicated and can be readilysolved by Matlab optimization toolbox

4 Cogging Torque Parameters IdentificationMethod in Time Domain

Parameters in linear terms can be estimated by frequencyidentification method during high speed operation wherethe cogging torquersquos influence could be neglected But whenmotor speed approaches to a low operation as shown inFigure 3 such perfectly constant speed cannot be achievedIn this case cogging torque effect should be taken intoaccount Since cogging torque is a nonlinear compact on theBLDC motor model assumptions for frequency identifica-tion method is ruined Hence aiming at estimating parame-ters in cogging torque term a nonlinear system identificationmethod is necessaryThe cogging torque cannot bemeasuredor calculated by direct means In this subsection the coggingtorque considered as a disturbance will be estimated by aclosed up observer The least square method (LSM) is usedto obtain cogging torque parameters 119879

1198881 1198791198882 119879

119888119870

41 Disturbance Observer Design A general block diagramfor disturbance observer [9] shown in Figure 4 can beused to approximate disturbance signal 119906dist If inaccuraciesin the plant model are negligible and the compensator isable to ensure convergence of the error signal 119890 towards

Plant

Ideal model+

+

+

minus

minus

minus

GP(s)

Compensator

udist

udist

y

y

e

Gc(s)

u

Figure 4 Disturbance observer block diagram

zero then the compensator output dist is able to estimatedisturbance signalThus a similar way can be used to estimatenon-modelled dynamics in the case of the linear modeldiscussed in frequency domain on condition that exogenousdisturbances are ignored

Based on superposition principle for linear system thetransfer function119867

119900(119904) fromestimated to applied disturbance

can be deduced as

119867119900(119904) =

dist119906dist

=119866119875(119904) 119866119888(119904)

1 + 119866119875(119904) 119866119888(119904)

(19)

The compensator 119866119888(119904) should be chosen so that the denom-

inator of the polynomial in (19) is Hurwitz stable Thenthe system in Figure 4 is asymptotically stable Thereforethe output of the compensator dist succeeds in observingdisturbance signal 119906dist asymptotically

Depended on the previously discussed disturbanceobserver the cogging torque coefficients identification inan off-line process is realized as shown in Figure 5 Theideal model comes from mechanical dynamics in the linearBLDC motor model and therersquos no load external torqueA 119875119868 controller designed by following the two principalsintroduced in Section 41 works as a compensator

The cogging torque is equivalent to a disturbance signalof electromagnetic torque If a 119875119868 compensator is chosen thecompensator plantmodel and observer transfer functionwillbe

119866119888(119904) = 119875 +

119868

119904

119866119875(119904) =

1

119869119904 + 119870119891

119867119900(119904) =

119866119875(119904) 119866119888(119904)

1 + 119866119875(119904) 119866119888(119904)

=119875119904 + 119868

1198691199042 + (119870119891+ 119875) 119904 + 119868

(20)

Two principles will ensure the accuracy of the observersystem shown in Figure 4

6 Mathematical Problems in Engineering

++ minus

minus

Experimental data

Rotor speed120596

iaKt

Developedtorque

Electromagnetictorque

Td T

Assumedspeed

TcoggP +

I

s

Estimated coggingtorque

Rotor speederrore

Armature current 1

Js + Kf

Ideal model

PI compensator

Figure 5 Cogging torque estimation

Principle 1 Since the observation function is a typical secondorder system the system is stable if

120585 =

119870119891+ 119875

2radic119869119868gt 0 (21)

Equation (21) means that positive 119875 and 119868 sufficiently assurethe stability of the system

Principle 2 In order to improve the accuracy of the observingsignal dist the compensator should minimize the maximumgain from in whole history data that is

min 1198902=1003817100381710038171003817119910 minus 119910

10038171003817100381710038172=

119899

sum

119894

(119910 (119894) minus 119910 (119894))2

(22)

where 119899 is the number of history data points and 120576 is mini-mum energy of error signal that allowed

42 Cogging Torque Parameters Identification Based on LeastSquares Approximation Method In BLDC motor systemsince the convergence of the assumed speed towards themeasured speed is ensured by the resulting compensator theestimated cogging torque dist is able to approximate the truenonlinear dynamics well

The least squares method is considerably used in practicedue to convenient implementation The objective of thisapproximating method is to minimize the ldquoenergy of errorrdquonamely the squared difference between the measured signalsand the estimated signals Considering a vector problem theestimated output is determined by a signal model whichrelates to the unknown parameter vector 120573 = [120573

1 120573

119896]

119908 and 119911 are respectively the input of the model and thepractical measured output The least squares estimator (LSE)is to find the value of 120573 which minimizes the squares errorcriterion given as

min 119869 = 119911 minus 2=

119873

sum

119899=1

(119911 (119899) minus (119899))2

(23)

The observation interval is assumed to be 119899 = 1 2 119873If the linear least squares problem is taken into consideration

the signalmodel is linearwith respect to the unknownparam-eter therefore the output signal can be

119911 = Φ120573 (24)where Φ is the observation matrix in [16] the least squareestimator and average minimum least square error are

120573 = (120601119879

120601)minus1

120601119879

(25)

119869119886min =

( minus 120601120573)119879

( minus 120601120573)

119873

(26)

Considering the unbalanced mechanical system effectinto the estimation process [9] the estimated disturbancetorque is

dist = 119879cogg

=

119870

sum

119896=1

119879119888119896cos (119896119901120579) + 119879

119886sin (120579) + 119879

119887cos (120579)

(27)

The rotor position 120579 is obtained by integration of the time-history data of rotor speed 120596 In this way the term in (27)cos(119896119901120579) is known

Then the observation matrix of LSE is therefore writtenas

Φ = [120601119894119895 120601119904

119894119895120601119888

119894119895]

120601119894119895= cos (119895 sdot 119901120579 (119894)) 119894 = 1 119873 119895 = 1 119870

120601119904

119894119895= sin (120579 (119894)) 119894 = 1 119873 119895 = 1

120601119888

119894119895= cos (120579 (119894)) 119894 = 1 119873 119895 = 1

(28)

Then the parameter vector 120573 = [11987911988811198791198882sdot sdot sdot 119879119888119870 119879119886 119879119887] is

calculated by (25) An essential parameter 119870 the minimalnumber of relevant harmonics which gives the simplestsignal model is necessarily determined by average minimumleast square error 119869

119886min A suitable119870 is obtained to adequatelydescribe the cogging torque characteristics For this sakeof determination method we will increase the number ofharmonics119870 unless the minimum least squares error is smallenough

Mathematical Problems in Engineering 7

BLDC motorESC

Hall sensor

Rotary encoder

Digital oscilloscope

Stabilized voltage supply

Control signal generator

PWM generator

Figure 6 BLDC motor identification experimental bench setup

Table 1 BLDC motor mechanical parameters

Description Value UnitNumber of stator arms 12 Magnet poles 14 Max continuous power 3900 WRotor inertia (evaluated) 0000549 kgsdotm2

Rotary encoder inertia 000001 kgsdotm2

5 System Identification Experiment

System identification experiments are designed and carriedout to obtain data for comprehensive identification Thissection is divided into three parts First we will detail theexperimental plant in BIT Then frequency identificationresults and system main parameter determination will bedescribed Finallywewill discuss nonlinear identification andestimate cogging torque coefficients

51 Experimental Setup The experimental bench setup isshown in Figure 6 including BLDC motor driving unitcontrol unit and measure unit Each of the subsystem willbe briefly introduced by the following subsystems of theexperimental setup

511 BLDC Motor The motor to be identified is popularBLDC motor for UAVs scorpion SII-6530 Considering thesimple configuration of this motor the rotor inertia can beevaluated by Solidworks which is important for the valida-tion of the identification results Somemotor parameters andthe evaluated inertia of the rotary encoder and coupling arelisted in Table 1

512 Driving Unit The driving unit consists of a ESC and aPWM generator The ESC produced by Hobyywing similarto a motor controller and three-phase bridge receives thesignal produced by the PWM generator A stabilized voltagesource powers the ESC to drive the motor The PWMgenerator actually a servo controller is mainly responsible fordecoding control commands and generating PWM wave

0 20 40 60 80 100 1201360

1380

1400

1420

1440

Time (s)

PWM

wav

e dut

y cy

cle

Figure 7 Automatic sweep input signal ranging from 001sim10Hz

513 Control Unit The control unit is an ordinary PCinstalled by Linux operating systemThePCandPWMgener-ator are connected by RS-232 serial ports All the designedcontrol commands are sent from control unit to the PWMgenerator

514 Measure Unit In order to measure these two systemstates rotor speed and armature current a rotary encoder andhall current sensor are respectively included inmeasure unitBoth of output signal of these two sensors are recorded by thedigital oscilloscope in high sample rates The measurementsampling time of the digital oscilloscope could be set from1 120583s up to 50120583s and the memory amount is available forrecording data length of 10Mb Then the rotor speed andarmature current can be deciphered from the measured data

52 Frequency Identification Results When the power supplyis set at 23V the rotary encoder wasmounted onto the BLDCmotor rotor and the armature current wire passes throughthe hall current sensor The encoder and the hall sensorrsquosoutputs are recorded simultaneously during the experimentwith sampling frequency 119891

119904= 100Hz The rotor speed and

armature are then calculated off-lineComplying with (8) the input signal shown in Figure 7

covers frequencies ranging from 001 to 10HzThis frequencysweep input signal is utilized to excite the dynamics of our

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120240

250

260

270

280

290

300

Time (s)

Roto

r spe

ed (r

ads

)

(a) Rotor speed response

0 20 40 60 80 100 120

0

05

1

15

2

25

3

Time (s)

Arm

atur

e cur

rent

(A)

minus05

(b) Armature current response

Figure 8 Rotor speed and armature current responses

0

10

20

Frequency (rads)

Mag

nitu

de (d

B)

0

Frequency (rads)

Phas

e (de

g)

Experimental dataTransfer function fit

0

05

1

Frequency (rads)

Coh

eren

ce

100 101

100 101

100 101

minus150

minus100

minus50

(a) Rotor speed channel

Frequency (rads)

0

50

Frequency (rads)

Experimental dataTransfer function fit

Mag

nitu

de (d

B)Ph

ase (

deg)

Coh

eren

ce

0

05

1

Frequency (rads)

10minus1 100 101

100 101

102

100 101 102

minus100

minus50

minus15

minus10

minus5

(b) Armature channel

Figure 9 Rotor speed channelarmature channel identification

plant over a broad frequency range Figure 8 shows the rotorspeed and armature current extracted from experimentaldata Note that rotor speed response amplitude diminishesat higher frequencies as expected from the bandwidth ofthe corresponding channel However the armature currentresponse amplitude will increase with the even very highfrequency suggesting a very high bandwidth of the armaturecurrent channel which is considerably over the max sweepfrequency

The data obtained from this test will be applied toextract frequency responses used for themodel identification

The estimated frequency responses are calculated using spec-tral density functions with CIFER Figure 9 shows frequencyidentification of rotor speed channel and armature channelGood coherence is achieved in the required frequency rangeindicating good quality experimental data and the linearmodel assumption With respect to coherence rotor speedchannel identification result is better than that of currentchannel because the hall current sensor yields a pooraccuracy compared to rotary encoder The identified transferfunctions represented by red dashed lines show good fits ofthe magnitude and phase responses of each channel

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12900

950

1000

1050

1100

1150

1200

1250

Time (s)

PWM

wav

e dut

y cy

cle

Figure 10 Input doublet signal

Table 2 Estimated BLDC motor parameters

System parameters Symbol Value UnitRotor inertia 119869 000061 kgsdotm2

Voltage constant 119870119890

00637 V(rads)Torque constant 119870

11990502663 NmA

Armature resistance 119877119886

06187 Ω

Armature inductance 119871119886

262 mHFriction coefficient 119870

119891000291 N(rads)

Time constant of the ESC model 120591 00781 s

The two transfer functions of two channels based on (13)and (14) identified are

119867120596(119904) =

2057342

1199043 + 18951199042 + 13412119904 + 142834

119867119894(119904) =

21749 (02015119904 + 1)

1199043 + 1961199042 + 14695119904 + 149431

(29)

The identified transfer functions are taken into account todetermine BLDCmotormechanismparameters as suggestedin Section 3 Then the problem in (17) and (18) is solvedby Matlab optimization toolbox The bias between theoret-ical model and identification result is 041 The estimatedBLDC motor parameters are listed in Table 2 Note thatthe identified rotor inertia 119869 = 000061 is very closeto the evaluated value in Table 1 indicating a satisfactoryidentification accuracy of this parameter

The accuracy of the comprehensive BLDC motor modelmust be further verified by tine history data The inputdoublet signal shown in Figure 10 is used to stimulate theBLDC motor dynamics The validation results are shownin Figure 11 Both rotor speed and armature current aremeasured during the input doublet signal It is shown that theidentifiedmodel perfectly predicts the rotor speed responseshowever the true armature current response exhibits a slightdeviation from the responses predicted by the identifiedmodel This mismatch can be explained by the relatively lowcoherence in high frequency of the armature channel identifi-cationThe close agreement between the simulation response

Table 3 Estimation results for the mechanical parameters

System parameters 119909 (mean values) 120590 (standard deviations)1198791198881

293 451198791198882

527 511198791198883

3706 4461198791198884

156 411198791198885

843 2241198791198886

5612 433119879119886

1103 125119879119887

1620 332

and experimental data indicates the identified model isaccurate enough to reflect the BLDC motorrsquos dynamics

53 Cogging Torque Coefficients Identification Results Sincethe cogging torque effect exists significantly in low speed theterminal voltage of the motor is set at 65 V The disturbanceobserver is applied to calculate cogging torque as stated inSection 42 In order to eliminate the stochastic error theexperiment is repeated by 10 repetitions with single testtime of 20 s The minimum least squares error obtained by(26) versus the number of cogging torque harmonics areshown in Figure 12 with the estimated disturbance signalapproximated using LSE It is obvious that the sixth coggingtorque harmonic is highly dominant from the diagram and aslight of third harmonics is also present [17] has pointed outthat the cogging torque harmonics for three-phase motorsusually are multiples of six times the fundamental commuta-tion frequency

A simulation of the identified model is performed inorder to verify the results of the proposed parameters iden-tification method The cogging torque coefficients in Table 3and the estimated linear terms by frequency identification inSection 52 are utilized in the comprehensive BLDC motormodel A comparison betweenmeasured and estimated rotorspeed is formed to highlight the verification result (seeFigure 13) Note that there is a small phase deviation ofbetween experimental and estimated cogging torque thiscould be interpreted by the unmodeled dynamics Matchingof the simulated and experimental data is ensured Hence theproposed identification method is applicable and sufficient

6 Conclusion

In this paper a comprehensive procedure for system iden-tification applicable to BLDC motor used in UAV has beensystematically presented The procedure aims to serve asa guide for the steps necessary to determine an accurateBLDC motor model Considering the nonlinear character-istics cogging torque effect of the BLDC motor system apractical mathematical model is introduced for identifica-tion Preliminarily frequency domain identification theoryis applied to estimate the first seven parameters in linearterms of the model at a high motor speed region where thenonlinear term can be neglected Then an off-line methodfor identifying the cogging torque coefficients is performed at

10 Mathematical Problems in Engineering

0 2 4 6 8 10 12240

260

280

300

320

Time (s)

Roto

r spe

ed (r

ads

)

Experimental dataBLDC motor model

(a) Rotor speed validation of the BLDC motor dynamics

2 4 6 8 10 12

1

2

3

4

5

Time (s)

Arm

atur

e cur

rent

(A)

Experimental dataBLDC motor model

(b) Armature current validation of the BLDC motor dynamics

Figure 11 Time domain validation of the BLDC motor identified dynamics

2 3 4 5 6 7 8 90

10

20

30

Number of harmonics (K)

J min

Figure 12 The minimum least squares error versus the number of cogging torque harmonics

05 1 15 275

80

85

90

95

100

Time (s)

Nonlinear model Experimental data

02 025 03

90

92

94

96

Time (s)

Nonlinear model Experimental data

Roto

r spe

ed (r

ads

)

Roto

r spe

ed (r

ads

)

Figure 13 Validation of identified comprehensive BLDC motor model

Mathematical Problems in Engineering 11

very low speedwhen cogging torque effect exists significantlyThis method is founded on the least squares approximationmethod a disturbance observer and the identified parametersin frequency identification The results from the simulationand the real-time experiments matched closely which isconsidered as a validation of the obtained mathematicalmodel

The BLDC model identifications are conditional onlyon the availability of experimental data for rotor speed andarmature current therefore our method without sensors todetect torque or back-EMF is easy to realize and is practicaland effectiveThis research will provide greater opportunitiesfor future controller design of the BLDCmotorThe extractedlinear model can be utilized directly for linear controllerdesign and the accurately identified cogging torque effect isbeneficial to reduce the speed ripple of the motor at some lowspeed conditions using some advanced control theory suchas robust control and nonlinear control

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Shanmugasundram K M Zakaraiah and N YadaiahldquoModeling simulation and analysis of controllers for brushlessdirect current motor drivesrdquo Journal of Vibration and Controlvol 19 no 8 pp 1250ndash1264 2013

[2] C Xia Y Wang and T Shi ldquoImplementation of finite-statemodel predictive control for commutation torque ripple min-imization of permanent-magnet brushless DC motorrdquo IEEETransactions on Industrial Electronics vol 60 no 3 pp 896ndash905 2013

[3] S-Y Chang P-H Yu and W-C Su ldquoPre- and post-filteringapproach to sensorless VSS speed control of a permanentmagnet DC motor subject to Hammerstein and Wiener non-linearitiesrdquoControl Engineering Practice vol 21 no 11 pp 1577ndash1583 2013

[4] B-K Lee and M Ehsani ldquoAdvanced simulation model forbrushless dc motor drivesrdquo Electric Power Components andSystems vol 31 no 9 pp 841ndash868 2003

[5] I Kim N Nakazawa S Kim C Park and C Yu ldquoCompensa-tion of torque ripple in high performance BLDCmotor drivesrdquoControl Engineering Practice vol 18 no 10 pp 1166ndash1172 2010

[6] H Lu L Zhang and W Qu ldquoA new torque control method fortorque rippleminimization of BLDCmotors with un-ideal backEMFrdquo IEEE Transactions on Power Electronics vol 23 no 2 pp950ndash958 2008

[7] W Wu ldquoDC motor parameter identification using speed stepresponsesrdquo Modelling and Simulation in Engineering vol 2012Article ID 189757 5 pages 2012

[8] J-H Choi S-H You J Hur and H-G Sung ldquoThe design andfabrication of BLDCMotor andDrive for 42V automotive appli-cationsrdquo in Proceedings of the IEEE International Symposium onIndustrial Electronics (ISIE rsquo07) pp 1086ndash1091 June 2007

[9] A Kapun M Curkovic A Hace and K Jezernik ldquoIdentifyingdynamic model parameters of a BLDC motorrdquo Simulation

Modelling Practice and Theory vol 16 no 9 pp 1254ndash12652008

[10] A J Blauch M Bodson and J Chiasson ldquoHigh-Speed parame-ter estimation of stepper motorsrdquo IEEE Transactions on ControlSystems Technology vol 1 no 4 pp 270ndash279 1993

[11] R Patankar and L Zhu ldquoReal-time multiple parameter estima-tion for voltage controlled brushless DC motor actuatorsrdquo inProceedings of the American Control Conference (AAC rsquo04) vol4 pp 3851ndash3856 July 2004

[12] M S Islam R Islam T Sebastian A Chandy and S AOzsoyluldquoCogging torque minimization in PM motors using robustdesign approachrdquo IEEE Transactions on Industry Applicationsvol 47 no 4 pp 1661ndash1669 2011

[13] M B Tischler and R K Remple Aircraft and Rotorcraft SystemIdentification AIAA Education Series American Institute ofAeronautics and Astronautics 2006

[14] M Petra and M Utting CZT A Framework for Z Tools FormalSpecification and Development in Z and B Springer BerlinGermany 2005

[15] K Zhou and C D John Essentials of Robust Control PrenticeHall Upper Saddle River NJ USA 1998

[16] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Higher Education New York NY USA 2010

[17] J Y Hung and Z Ding ldquoDesign currents to reduce torque ripplein brushless permanent magnet motorsrdquo IEE Proceedings BElectric Power Applications vol 140 no 4 pp 260ndash266 1993

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

where 119906(119904) is PWM input 119870119889is the gain from PWM to the

applied voltage and is determined by ESC parameters andpower supply 120591 is time constant of the ESC model

3 Parameters Identification Method ofLinear Terms in Frequency Domain

The parameters of the BLDCmotor provided in the previoussection are 119870

119905 119870119890 119870119891 119877119886 119871119886 119869 120591 and 119879

1198881 1198791198882 119879

119888119870

Note that the first seven parameters lie in linear term of theBLDC motor model and the last term corresponds to non-linear term which describes cogging torque All these param-eters are all assumed time-invariant This section presents anoff-line method for identifying these unknown parametersThe identificationmethod depends on given experiment datafor control PWM input and the corresponding rotor speedand armature current responses There will be two stepsto complete identification Firstly an open-loop frequencydomain method is used to identify these parameters in linearterms on the operation in high speed where cogging torquecan be ignored In the second step the cogging torque signalconsidered as a disturbance torque is estimated by usinga closed up observer Then the least square approximationmethod is applied in order to identify these nonlinear termparameters Both steps are described in detail in the followingsubsections The frequency domain identification methoddetermines most of the parameters therefore this method isbriefly described first

For the BLDC model proposed in Section 2 the non-linearity is produced by cogging torque which is especiallyprominent at lower speeds However at high speed the effectof cogging torque can be filtered out by the motor momentof inertia thus cogging torque can be ignored Thereforethese parameters in linear terms can be identified at highspeed when the BLDC motor is approximately considered asa linear systemThe block diagram formotor linear dynamicsis depicted in Figure 2

The transfer functions 120596(119904)119906119886(119904) = 119866

120596(119904) 119894119886(119904)119906119886(119904) =

119866119894(119904) and 119906

119886(119904)V119886(119904) = 119866

119890(119904) are given as

119866120596(119904) =

119870119905119871119886119869

1199042 + (119870119891119869 + 119877

119886119871119886) 119904 + 119877

119886119870119891119871119886119869 + 119870119890119870119905119871119886119869

119866119894(119904) =

(1119871119886) 119904 + 119870

119891119871119886119869

1199042 + (119870119891119869 + 119877

119886119871119886) 119904 + 119877

119886119870119891119871119886119869 + 119870119890119870119905119871119886119869

119866119890(119904) =

002187

120591119904 + 1

(7)

Note that 119866119890(119904) represents ESC dynamics according to

Section 22 and the gain value 002187 is fitted by throttlecurves from the ESC manufacturer when power supply is setin 26V output

As for identifying these foregoing parameters althoughtime-domain techniques can still be adopted it is stronglyrecommended in this case to leave the time domain in favorof the frequency domain Indeed in the frequency domain

ua(t)

G120596(s)

Gi(s)

Ge(s)

120596(t)

ia(t)

Terminalvoltage

ESC delays

Rotor speed

Armaturecurrent

BLDC linear model

a(t)

PWM input

Figure 2 Block diagram for motor linear dynamics

one has several features not found in the time domain [13](1) unbiased frequency-response estimates when experimen-tal data contain process and output measurement noise (2)access to the coherence function as an unbiased measure ofnonparametric identification accuracy and system responselinearity (3) elimination of biases and reference shifts asidentification parametersTherefore frequency identificationmethod is introduced to identify parameters in linear terms

Then the identification procedure for the unknownparameters will be addressed in detail A simple meaningfuland comprehensive method will be proposed Correspond-ingly the identification procedure is separated into five steps

Step 1 (input sweep signal design) In [13] an automaticsweep signal generating is designed as

119906 (119905) = 1199060+ 119906sweep (119905) = 1199060 + 119860 sin (120579 (119905))

120579 (119905) = [120596min minus 00187 (120596max minus 120596min)] 119905

+ 000467119879rec (120596max minus 120596min) (119890(4119905119879rec) minus 1)

(8)

where 1199060is trim value of input determining the trim values of

rotor speed and armature current 119906sweep is dynamic value ofinput119860 is sweepmagnitude 120579(119905) is signal phase [120596min 120596max]is sweep signal range and 119879rec is sweep record length

Step 2 (collection of time-history data) The sweep signal 119906acts as an input on the integrated BLDC motor plant androtor speed 120596 and armature current 119894

119886are collected as the

output for system identificationThe outputs are also dividedinto two terms as follows

120596 (119905) = 1205960+ 120596sweep (119905)

119894119886(119905) = 119894

1198860+ 119894119886sweep (119905)

(9)

where 1205960and 1198940are trim values of output and 120596sweep(119905) and

119894119886sweep(119905) are dynamic value of outputThe dynamic termswillbe used to obtain frequency responses

4 Mathematical Problems in Engineering

Step 3 (frequency response calculation) The methods tocalculate spectrum function are briefly introduced andwith greater detail in [14] First a Chirp-Z transform isapplied to initially transform time-domain response data tothe frequency-domain Then a technique of combining aweighted average of multiple windows known as CompositeWindowing is used to improve the frequency of accuracyThe results of this calculation in a high quality are inputautospectrum 119866

119909119909(119891) output autospectrum 119866

119910119910(119891) and

cross-spectrum 119866119909119910(119891) Then Frequency response function

is obtained by

(119891) =

119866119909119910(119891)

119866119909119909(119891)

(10)

Step 4 (coherence function calculation) Coherence functionis defined as

120574119909119910(119891) =

10038161003816100381610038161003816119866119909119910(119891)

10038161003816100381610038161003816

2

1003816100381610038161003816119866119909119909 (119891)1003816100381610038161003816

10038161003816100381610038161003816119866119910119910(119891)

10038161003816100381610038161003816

(11)

Coherence function can be interpreted physically as the frac-tion of the output spectrum that is linearly attributable to theinput spectrum at any frequency It can effectively and rapidlyassess the accuracy of the frequency response identificationIn particular the frequency response identification resultsare acceptable as long as the coherence function satisfies thecondition

120574119909119910ge 06 (12)

Step 5 (transfer function fitting) A numerical optimizationalgorithm is used to determine the unknown parameters byminimizing the magnitude and phase errors between thedesired BLDCmotor transfermodels119867(119904) and the associatedfrequency responses (119904) The practical cost function is

min 119869 = 20

119899119908

120596119899119908

sum

1205961

119882120574[119882119892(|119867| minus

1003816100381610038161003816100381610038161003816100381610038161003816) + 119882

119875(ang119867 minus ang)

2

]

(13)

where 119899119908is the number of frequency points and 120596

1and 120596

119899119908

are starting and ending frequencies of fit 119882120574is weighting

function dependent on the value of the coherence functionat each frequency 119882

119892and 119882

119901are the relative weights for

magnitude and phase squared errors One can obtain detailedselection of these weighing functions in [13]

All these forgoing methods to frequency system iden-tification have been incorporated into the known soft-ware CIFER (Comprehensive Identification from FrequencyResponses) Unlimited to its original application in rotorcraftCIFER is also suitable for other plant such as BLDC motorsystem for linear system identification

Step 6 (system parameters determination) Based on thesefive steps for frequency identification the transfer functionfrom PWM input to speed channel and armature channel areobtained However these parameters that is 119870

119905 119870119890 119870119891 119877119886

119871119886 119869 120591 in mechanism are still unknown Hence a method is

proposed as followsThe identified transfer functions in lumped parameters

are

119867120596(119904) =

1198701

1199043 + 119886111199042 + 119886

12119904 + 11988613

(14)

119867119894(119904) =

1198702(1198872119904 + 1)

1199043 + 119886211199042 + 119886

22119904 + 11988623

(15)

Theoretically both denominators of (14) and (15) are equalNevertheless small allowable deviation is unavoidable inpractice resulting in very close but strictly different denom-inators in these two equations In that case these unknownparameters cannot be obtained by classic undetermined coef-ficient method A feasible idea is to find suitable parametersthat ensure theoretical model to approximate identificationresult as much as possible

The bias between theoretical model and identificationresult should be defined first In [15] the bias between twotransfer functions 119866

1and 119866

2can be written as

10038171003817100381710038171198661 minus 11986621003817100381710038171003817infin

(16)

Hence the parameter determination problem can betransferred to the following optimization problem by 119867

infin

norm

min bias = 119891 (119870119905 119870119891 119870119890 119869 119877119886 119871119886 120591)

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[

[

119882120596(119867120596minus 119866120596119866119890)

119882119894(119867119894minus 119866119894119866119890)

]

]

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[[[[[[

[

119882120596(119867120596minus002187

120591119904 + 1sdot

119870119905119871119886119869

1199042 + (119870119891119869 + 119877

119886119871119886) 119904 + 119877

119886119870119891119871119886119869 + 119870119890119870119905119871119886119869

)

119882119894(119867119894minus002187

120591119904 + 1sdot

(1119871119886) 119904 + 119870

119891119871119886119869

1199042 + (119870119891119869 + 119877

119886119871119886) 119904 + 119877

119886119870119891119871119886119869 + 119870119890119870119905119871119886119869

)

]]]]]]

]

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin

(17)

Mathematical Problems in Engineering 5

0 05 1 15 20

20

40

60

80

100

Time (s)

Roto

r spe

ed (r

ads

)

Figure 3 Cogging torque effect in low speed

Equation (17) alone is probably not sufficient to find theonly solution Another two equation constraints on systemstatic step response are

1205960=

002187119870119905

119877119886119870119891+ 119870119890119870119905

1198941198860=

002187119870119891

119877119886119870119891+ 119870119890119870119905

(18)

where 1205960and 1198941198860

are respectively measured static value ofrotor speed and armature on step response experiment 119882

120596

and119882119894areweighting coefficients and119867

120596and119867

119894are identified

transfer function As a typical constrained optimizationproblem this problem is not complicated and can be readilysolved by Matlab optimization toolbox

4 Cogging Torque Parameters IdentificationMethod in Time Domain

Parameters in linear terms can be estimated by frequencyidentification method during high speed operation wherethe cogging torquersquos influence could be neglected But whenmotor speed approaches to a low operation as shown inFigure 3 such perfectly constant speed cannot be achievedIn this case cogging torque effect should be taken intoaccount Since cogging torque is a nonlinear compact on theBLDC motor model assumptions for frequency identifica-tion method is ruined Hence aiming at estimating parame-ters in cogging torque term a nonlinear system identificationmethod is necessaryThe cogging torque cannot bemeasuredor calculated by direct means In this subsection the coggingtorque considered as a disturbance will be estimated by aclosed up observer The least square method (LSM) is usedto obtain cogging torque parameters 119879

1198881 1198791198882 119879

119888119870

41 Disturbance Observer Design A general block diagramfor disturbance observer [9] shown in Figure 4 can beused to approximate disturbance signal 119906dist If inaccuraciesin the plant model are negligible and the compensator isable to ensure convergence of the error signal 119890 towards

Plant

Ideal model+

+

+

minus

minus

minus

GP(s)

Compensator

udist

udist

y

y

e

Gc(s)

u

Figure 4 Disturbance observer block diagram

zero then the compensator output dist is able to estimatedisturbance signalThus a similar way can be used to estimatenon-modelled dynamics in the case of the linear modeldiscussed in frequency domain on condition that exogenousdisturbances are ignored

Based on superposition principle for linear system thetransfer function119867

119900(119904) fromestimated to applied disturbance

can be deduced as

119867119900(119904) =

dist119906dist

=119866119875(119904) 119866119888(119904)

1 + 119866119875(119904) 119866119888(119904)

(19)

The compensator 119866119888(119904) should be chosen so that the denom-

inator of the polynomial in (19) is Hurwitz stable Thenthe system in Figure 4 is asymptotically stable Thereforethe output of the compensator dist succeeds in observingdisturbance signal 119906dist asymptotically

Depended on the previously discussed disturbanceobserver the cogging torque coefficients identification inan off-line process is realized as shown in Figure 5 Theideal model comes from mechanical dynamics in the linearBLDC motor model and therersquos no load external torqueA 119875119868 controller designed by following the two principalsintroduced in Section 41 works as a compensator

The cogging torque is equivalent to a disturbance signalof electromagnetic torque If a 119875119868 compensator is chosen thecompensator plantmodel and observer transfer functionwillbe

119866119888(119904) = 119875 +

119868

119904

119866119875(119904) =

1

119869119904 + 119870119891

119867119900(119904) =

119866119875(119904) 119866119888(119904)

1 + 119866119875(119904) 119866119888(119904)

=119875119904 + 119868

1198691199042 + (119870119891+ 119875) 119904 + 119868

(20)

Two principles will ensure the accuracy of the observersystem shown in Figure 4

6 Mathematical Problems in Engineering

++ minus

minus

Experimental data

Rotor speed120596

iaKt

Developedtorque

Electromagnetictorque

Td T

Assumedspeed

TcoggP +

I

s

Estimated coggingtorque

Rotor speederrore

Armature current 1

Js + Kf

Ideal model

PI compensator

Figure 5 Cogging torque estimation

Principle 1 Since the observation function is a typical secondorder system the system is stable if

120585 =

119870119891+ 119875

2radic119869119868gt 0 (21)

Equation (21) means that positive 119875 and 119868 sufficiently assurethe stability of the system

Principle 2 In order to improve the accuracy of the observingsignal dist the compensator should minimize the maximumgain from in whole history data that is

min 1198902=1003817100381710038171003817119910 minus 119910

10038171003817100381710038172=

119899

sum

119894

(119910 (119894) minus 119910 (119894))2

(22)

where 119899 is the number of history data points and 120576 is mini-mum energy of error signal that allowed

42 Cogging Torque Parameters Identification Based on LeastSquares Approximation Method In BLDC motor systemsince the convergence of the assumed speed towards themeasured speed is ensured by the resulting compensator theestimated cogging torque dist is able to approximate the truenonlinear dynamics well

The least squares method is considerably used in practicedue to convenient implementation The objective of thisapproximating method is to minimize the ldquoenergy of errorrdquonamely the squared difference between the measured signalsand the estimated signals Considering a vector problem theestimated output is determined by a signal model whichrelates to the unknown parameter vector 120573 = [120573

1 120573

119896]

119908 and 119911 are respectively the input of the model and thepractical measured output The least squares estimator (LSE)is to find the value of 120573 which minimizes the squares errorcriterion given as

min 119869 = 119911 minus 2=

119873

sum

119899=1

(119911 (119899) minus (119899))2

(23)

The observation interval is assumed to be 119899 = 1 2 119873If the linear least squares problem is taken into consideration

the signalmodel is linearwith respect to the unknownparam-eter therefore the output signal can be

119911 = Φ120573 (24)where Φ is the observation matrix in [16] the least squareestimator and average minimum least square error are

120573 = (120601119879

120601)minus1

120601119879

(25)

119869119886min =

( minus 120601120573)119879

( minus 120601120573)

119873

(26)

Considering the unbalanced mechanical system effectinto the estimation process [9] the estimated disturbancetorque is

dist = 119879cogg

=

119870

sum

119896=1

119879119888119896cos (119896119901120579) + 119879

119886sin (120579) + 119879

119887cos (120579)

(27)

The rotor position 120579 is obtained by integration of the time-history data of rotor speed 120596 In this way the term in (27)cos(119896119901120579) is known

Then the observation matrix of LSE is therefore writtenas

Φ = [120601119894119895 120601119904

119894119895120601119888

119894119895]

120601119894119895= cos (119895 sdot 119901120579 (119894)) 119894 = 1 119873 119895 = 1 119870

120601119904

119894119895= sin (120579 (119894)) 119894 = 1 119873 119895 = 1

120601119888

119894119895= cos (120579 (119894)) 119894 = 1 119873 119895 = 1

(28)

Then the parameter vector 120573 = [11987911988811198791198882sdot sdot sdot 119879119888119870 119879119886 119879119887] is

calculated by (25) An essential parameter 119870 the minimalnumber of relevant harmonics which gives the simplestsignal model is necessarily determined by average minimumleast square error 119869

119886min A suitable119870 is obtained to adequatelydescribe the cogging torque characteristics For this sakeof determination method we will increase the number ofharmonics119870 unless the minimum least squares error is smallenough

Mathematical Problems in Engineering 7

BLDC motorESC

Hall sensor

Rotary encoder

Digital oscilloscope

Stabilized voltage supply

Control signal generator

PWM generator

Figure 6 BLDC motor identification experimental bench setup

Table 1 BLDC motor mechanical parameters

Description Value UnitNumber of stator arms 12 Magnet poles 14 Max continuous power 3900 WRotor inertia (evaluated) 0000549 kgsdotm2

Rotary encoder inertia 000001 kgsdotm2

5 System Identification Experiment

System identification experiments are designed and carriedout to obtain data for comprehensive identification Thissection is divided into three parts First we will detail theexperimental plant in BIT Then frequency identificationresults and system main parameter determination will bedescribed Finallywewill discuss nonlinear identification andestimate cogging torque coefficients

51 Experimental Setup The experimental bench setup isshown in Figure 6 including BLDC motor driving unitcontrol unit and measure unit Each of the subsystem willbe briefly introduced by the following subsystems of theexperimental setup

511 BLDC Motor The motor to be identified is popularBLDC motor for UAVs scorpion SII-6530 Considering thesimple configuration of this motor the rotor inertia can beevaluated by Solidworks which is important for the valida-tion of the identification results Somemotor parameters andthe evaluated inertia of the rotary encoder and coupling arelisted in Table 1

512 Driving Unit The driving unit consists of a ESC and aPWM generator The ESC produced by Hobyywing similarto a motor controller and three-phase bridge receives thesignal produced by the PWM generator A stabilized voltagesource powers the ESC to drive the motor The PWMgenerator actually a servo controller is mainly responsible fordecoding control commands and generating PWM wave

0 20 40 60 80 100 1201360

1380

1400

1420

1440

Time (s)

PWM

wav

e dut

y cy

cle

Figure 7 Automatic sweep input signal ranging from 001sim10Hz

513 Control Unit The control unit is an ordinary PCinstalled by Linux operating systemThePCandPWMgener-ator are connected by RS-232 serial ports All the designedcontrol commands are sent from control unit to the PWMgenerator

514 Measure Unit In order to measure these two systemstates rotor speed and armature current a rotary encoder andhall current sensor are respectively included inmeasure unitBoth of output signal of these two sensors are recorded by thedigital oscilloscope in high sample rates The measurementsampling time of the digital oscilloscope could be set from1 120583s up to 50120583s and the memory amount is available forrecording data length of 10Mb Then the rotor speed andarmature current can be deciphered from the measured data

52 Frequency Identification Results When the power supplyis set at 23V the rotary encoder wasmounted onto the BLDCmotor rotor and the armature current wire passes throughthe hall current sensor The encoder and the hall sensorrsquosoutputs are recorded simultaneously during the experimentwith sampling frequency 119891

119904= 100Hz The rotor speed and

armature are then calculated off-lineComplying with (8) the input signal shown in Figure 7

covers frequencies ranging from 001 to 10HzThis frequencysweep input signal is utilized to excite the dynamics of our

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120240

250

260

270

280

290

300

Time (s)

Roto

r spe

ed (r

ads

)

(a) Rotor speed response

0 20 40 60 80 100 120

0

05

1

15

2

25

3

Time (s)

Arm

atur

e cur

rent

(A)

minus05

(b) Armature current response

Figure 8 Rotor speed and armature current responses

0

10

20

Frequency (rads)

Mag

nitu

de (d

B)

0

Frequency (rads)

Phas

e (de

g)

Experimental dataTransfer function fit

0

05

1

Frequency (rads)

Coh

eren

ce

100 101

100 101

100 101

minus150

minus100

minus50

(a) Rotor speed channel

Frequency (rads)

0

50

Frequency (rads)

Experimental dataTransfer function fit

Mag

nitu

de (d

B)Ph

ase (

deg)

Coh

eren

ce

0

05

1

Frequency (rads)

10minus1 100 101

100 101

102

100 101 102

minus100

minus50

minus15

minus10

minus5

(b) Armature channel

Figure 9 Rotor speed channelarmature channel identification

plant over a broad frequency range Figure 8 shows the rotorspeed and armature current extracted from experimentaldata Note that rotor speed response amplitude diminishesat higher frequencies as expected from the bandwidth ofthe corresponding channel However the armature currentresponse amplitude will increase with the even very highfrequency suggesting a very high bandwidth of the armaturecurrent channel which is considerably over the max sweepfrequency

The data obtained from this test will be applied toextract frequency responses used for themodel identification

The estimated frequency responses are calculated using spec-tral density functions with CIFER Figure 9 shows frequencyidentification of rotor speed channel and armature channelGood coherence is achieved in the required frequency rangeindicating good quality experimental data and the linearmodel assumption With respect to coherence rotor speedchannel identification result is better than that of currentchannel because the hall current sensor yields a pooraccuracy compared to rotary encoder The identified transferfunctions represented by red dashed lines show good fits ofthe magnitude and phase responses of each channel

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12900

950

1000

1050

1100

1150

1200

1250

Time (s)

PWM

wav

e dut

y cy

cle

Figure 10 Input doublet signal

Table 2 Estimated BLDC motor parameters

System parameters Symbol Value UnitRotor inertia 119869 000061 kgsdotm2

Voltage constant 119870119890

00637 V(rads)Torque constant 119870

11990502663 NmA

Armature resistance 119877119886

06187 Ω

Armature inductance 119871119886

262 mHFriction coefficient 119870

119891000291 N(rads)

Time constant of the ESC model 120591 00781 s

The two transfer functions of two channels based on (13)and (14) identified are

119867120596(119904) =

2057342

1199043 + 18951199042 + 13412119904 + 142834

119867119894(119904) =

21749 (02015119904 + 1)

1199043 + 1961199042 + 14695119904 + 149431

(29)

The identified transfer functions are taken into account todetermine BLDCmotormechanismparameters as suggestedin Section 3 Then the problem in (17) and (18) is solvedby Matlab optimization toolbox The bias between theoret-ical model and identification result is 041 The estimatedBLDC motor parameters are listed in Table 2 Note thatthe identified rotor inertia 119869 = 000061 is very closeto the evaluated value in Table 1 indicating a satisfactoryidentification accuracy of this parameter

The accuracy of the comprehensive BLDC motor modelmust be further verified by tine history data The inputdoublet signal shown in Figure 10 is used to stimulate theBLDC motor dynamics The validation results are shownin Figure 11 Both rotor speed and armature current aremeasured during the input doublet signal It is shown that theidentifiedmodel perfectly predicts the rotor speed responseshowever the true armature current response exhibits a slightdeviation from the responses predicted by the identifiedmodel This mismatch can be explained by the relatively lowcoherence in high frequency of the armature channel identifi-cationThe close agreement between the simulation response

Table 3 Estimation results for the mechanical parameters

System parameters 119909 (mean values) 120590 (standard deviations)1198791198881

293 451198791198882

527 511198791198883

3706 4461198791198884

156 411198791198885

843 2241198791198886

5612 433119879119886

1103 125119879119887

1620 332

and experimental data indicates the identified model isaccurate enough to reflect the BLDC motorrsquos dynamics

53 Cogging Torque Coefficients Identification Results Sincethe cogging torque effect exists significantly in low speed theterminal voltage of the motor is set at 65 V The disturbanceobserver is applied to calculate cogging torque as stated inSection 42 In order to eliminate the stochastic error theexperiment is repeated by 10 repetitions with single testtime of 20 s The minimum least squares error obtained by(26) versus the number of cogging torque harmonics areshown in Figure 12 with the estimated disturbance signalapproximated using LSE It is obvious that the sixth coggingtorque harmonic is highly dominant from the diagram and aslight of third harmonics is also present [17] has pointed outthat the cogging torque harmonics for three-phase motorsusually are multiples of six times the fundamental commuta-tion frequency

A simulation of the identified model is performed inorder to verify the results of the proposed parameters iden-tification method The cogging torque coefficients in Table 3and the estimated linear terms by frequency identification inSection 52 are utilized in the comprehensive BLDC motormodel A comparison betweenmeasured and estimated rotorspeed is formed to highlight the verification result (seeFigure 13) Note that there is a small phase deviation ofbetween experimental and estimated cogging torque thiscould be interpreted by the unmodeled dynamics Matchingof the simulated and experimental data is ensured Hence theproposed identification method is applicable and sufficient

6 Conclusion

In this paper a comprehensive procedure for system iden-tification applicable to BLDC motor used in UAV has beensystematically presented The procedure aims to serve asa guide for the steps necessary to determine an accurateBLDC motor model Considering the nonlinear character-istics cogging torque effect of the BLDC motor system apractical mathematical model is introduced for identifica-tion Preliminarily frequency domain identification theoryis applied to estimate the first seven parameters in linearterms of the model at a high motor speed region where thenonlinear term can be neglected Then an off-line methodfor identifying the cogging torque coefficients is performed at

10 Mathematical Problems in Engineering

0 2 4 6 8 10 12240

260

280

300

320

Time (s)

Roto

r spe

ed (r

ads

)

Experimental dataBLDC motor model

(a) Rotor speed validation of the BLDC motor dynamics

2 4 6 8 10 12

1

2

3

4

5

Time (s)

Arm

atur

e cur

rent

(A)

Experimental dataBLDC motor model

(b) Armature current validation of the BLDC motor dynamics

Figure 11 Time domain validation of the BLDC motor identified dynamics

2 3 4 5 6 7 8 90

10

20

30

Number of harmonics (K)

J min

Figure 12 The minimum least squares error versus the number of cogging torque harmonics

05 1 15 275

80

85

90

95

100

Time (s)

Nonlinear model Experimental data

02 025 03

90

92

94

96

Time (s)

Nonlinear model Experimental data

Roto

r spe

ed (r

ads

)

Roto

r spe

ed (r

ads

)

Figure 13 Validation of identified comprehensive BLDC motor model

Mathematical Problems in Engineering 11

very low speedwhen cogging torque effect exists significantlyThis method is founded on the least squares approximationmethod a disturbance observer and the identified parametersin frequency identification The results from the simulationand the real-time experiments matched closely which isconsidered as a validation of the obtained mathematicalmodel

The BLDC model identifications are conditional onlyon the availability of experimental data for rotor speed andarmature current therefore our method without sensors todetect torque or back-EMF is easy to realize and is practicaland effectiveThis research will provide greater opportunitiesfor future controller design of the BLDCmotorThe extractedlinear model can be utilized directly for linear controllerdesign and the accurately identified cogging torque effect isbeneficial to reduce the speed ripple of the motor at some lowspeed conditions using some advanced control theory suchas robust control and nonlinear control

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Shanmugasundram K M Zakaraiah and N YadaiahldquoModeling simulation and analysis of controllers for brushlessdirect current motor drivesrdquo Journal of Vibration and Controlvol 19 no 8 pp 1250ndash1264 2013

[2] C Xia Y Wang and T Shi ldquoImplementation of finite-statemodel predictive control for commutation torque ripple min-imization of permanent-magnet brushless DC motorrdquo IEEETransactions on Industrial Electronics vol 60 no 3 pp 896ndash905 2013

[3] S-Y Chang P-H Yu and W-C Su ldquoPre- and post-filteringapproach to sensorless VSS speed control of a permanentmagnet DC motor subject to Hammerstein and Wiener non-linearitiesrdquoControl Engineering Practice vol 21 no 11 pp 1577ndash1583 2013

[4] B-K Lee and M Ehsani ldquoAdvanced simulation model forbrushless dc motor drivesrdquo Electric Power Components andSystems vol 31 no 9 pp 841ndash868 2003

[5] I Kim N Nakazawa S Kim C Park and C Yu ldquoCompensa-tion of torque ripple in high performance BLDCmotor drivesrdquoControl Engineering Practice vol 18 no 10 pp 1166ndash1172 2010

[6] H Lu L Zhang and W Qu ldquoA new torque control method fortorque rippleminimization of BLDCmotors with un-ideal backEMFrdquo IEEE Transactions on Power Electronics vol 23 no 2 pp950ndash958 2008

[7] W Wu ldquoDC motor parameter identification using speed stepresponsesrdquo Modelling and Simulation in Engineering vol 2012Article ID 189757 5 pages 2012

[8] J-H Choi S-H You J Hur and H-G Sung ldquoThe design andfabrication of BLDCMotor andDrive for 42V automotive appli-cationsrdquo in Proceedings of the IEEE International Symposium onIndustrial Electronics (ISIE rsquo07) pp 1086ndash1091 June 2007

[9] A Kapun M Curkovic A Hace and K Jezernik ldquoIdentifyingdynamic model parameters of a BLDC motorrdquo Simulation

Modelling Practice and Theory vol 16 no 9 pp 1254ndash12652008

[10] A J Blauch M Bodson and J Chiasson ldquoHigh-Speed parame-ter estimation of stepper motorsrdquo IEEE Transactions on ControlSystems Technology vol 1 no 4 pp 270ndash279 1993

[11] R Patankar and L Zhu ldquoReal-time multiple parameter estima-tion for voltage controlled brushless DC motor actuatorsrdquo inProceedings of the American Control Conference (AAC rsquo04) vol4 pp 3851ndash3856 July 2004

[12] M S Islam R Islam T Sebastian A Chandy and S AOzsoyluldquoCogging torque minimization in PM motors using robustdesign approachrdquo IEEE Transactions on Industry Applicationsvol 47 no 4 pp 1661ndash1669 2011

[13] M B Tischler and R K Remple Aircraft and Rotorcraft SystemIdentification AIAA Education Series American Institute ofAeronautics and Astronautics 2006

[14] M Petra and M Utting CZT A Framework for Z Tools FormalSpecification and Development in Z and B Springer BerlinGermany 2005

[15] K Zhou and C D John Essentials of Robust Control PrenticeHall Upper Saddle River NJ USA 1998

[16] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Higher Education New York NY USA 2010

[17] J Y Hung and Z Ding ldquoDesign currents to reduce torque ripplein brushless permanent magnet motorsrdquo IEE Proceedings BElectric Power Applications vol 140 no 4 pp 260ndash266 1993

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Step 3 (frequency response calculation) The methods tocalculate spectrum function are briefly introduced andwith greater detail in [14] First a Chirp-Z transform isapplied to initially transform time-domain response data tothe frequency-domain Then a technique of combining aweighted average of multiple windows known as CompositeWindowing is used to improve the frequency of accuracyThe results of this calculation in a high quality are inputautospectrum 119866

119909119909(119891) output autospectrum 119866

119910119910(119891) and

cross-spectrum 119866119909119910(119891) Then Frequency response function

is obtained by

(119891) =

119866119909119910(119891)

119866119909119909(119891)

(10)

Step 4 (coherence function calculation) Coherence functionis defined as

120574119909119910(119891) =

10038161003816100381610038161003816119866119909119910(119891)

10038161003816100381610038161003816

2

1003816100381610038161003816119866119909119909 (119891)1003816100381610038161003816

10038161003816100381610038161003816119866119910119910(119891)

10038161003816100381610038161003816

(11)

Coherence function can be interpreted physically as the frac-tion of the output spectrum that is linearly attributable to theinput spectrum at any frequency It can effectively and rapidlyassess the accuracy of the frequency response identificationIn particular the frequency response identification resultsare acceptable as long as the coherence function satisfies thecondition

120574119909119910ge 06 (12)

Step 5 (transfer function fitting) A numerical optimizationalgorithm is used to determine the unknown parameters byminimizing the magnitude and phase errors between thedesired BLDCmotor transfermodels119867(119904) and the associatedfrequency responses (119904) The practical cost function is

min 119869 = 20

119899119908

120596119899119908

sum

1205961

119882120574[119882119892(|119867| minus

1003816100381610038161003816100381610038161003816100381610038161003816) + 119882

119875(ang119867 minus ang)

2

]

(13)

where 119899119908is the number of frequency points and 120596

1and 120596

119899119908

are starting and ending frequencies of fit 119882120574is weighting

function dependent on the value of the coherence functionat each frequency 119882

119892and 119882

119901are the relative weights for

magnitude and phase squared errors One can obtain detailedselection of these weighing functions in [13]

All these forgoing methods to frequency system iden-tification have been incorporated into the known soft-ware CIFER (Comprehensive Identification from FrequencyResponses) Unlimited to its original application in rotorcraftCIFER is also suitable for other plant such as BLDC motorsystem for linear system identification

Step 6 (system parameters determination) Based on thesefive steps for frequency identification the transfer functionfrom PWM input to speed channel and armature channel areobtained However these parameters that is 119870

119905 119870119890 119870119891 119877119886

119871119886 119869 120591 in mechanism are still unknown Hence a method is

proposed as followsThe identified transfer functions in lumped parameters

are

119867120596(119904) =

1198701

1199043 + 119886111199042 + 119886

12119904 + 11988613

(14)

119867119894(119904) =

1198702(1198872119904 + 1)

1199043 + 119886211199042 + 119886

22119904 + 11988623

(15)

Theoretically both denominators of (14) and (15) are equalNevertheless small allowable deviation is unavoidable inpractice resulting in very close but strictly different denom-inators in these two equations In that case these unknownparameters cannot be obtained by classic undetermined coef-ficient method A feasible idea is to find suitable parametersthat ensure theoretical model to approximate identificationresult as much as possible

The bias between theoretical model and identificationresult should be defined first In [15] the bias between twotransfer functions 119866

1and 119866

2can be written as

10038171003817100381710038171198661 minus 11986621003817100381710038171003817infin

(16)

Hence the parameter determination problem can betransferred to the following optimization problem by 119867

infin

norm

min bias = 119891 (119870119905 119870119891 119870119890 119869 119877119886 119871119886 120591)

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[

[

119882120596(119867120596minus 119866120596119866119890)

119882119894(119867119894minus 119866119894119866119890)

]

]

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin

=

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

[[[[[[

[

119882120596(119867120596minus002187

120591119904 + 1sdot

119870119905119871119886119869

1199042 + (119870119891119869 + 119877

119886119871119886) 119904 + 119877

119886119870119891119871119886119869 + 119870119890119870119905119871119886119869

)

119882119894(119867119894minus002187

120591119904 + 1sdot

(1119871119886) 119904 + 119870

119891119871119886119869

1199042 + (119870119891119869 + 119877

119886119871119886) 119904 + 119877

119886119870119891119871119886119869 + 119870119890119870119905119871119886119869

)

]]]]]]

]

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817infin

(17)

Mathematical Problems in Engineering 5

0 05 1 15 20

20

40

60

80

100

Time (s)

Roto

r spe

ed (r

ads

)

Figure 3 Cogging torque effect in low speed

Equation (17) alone is probably not sufficient to find theonly solution Another two equation constraints on systemstatic step response are

1205960=

002187119870119905

119877119886119870119891+ 119870119890119870119905

1198941198860=

002187119870119891

119877119886119870119891+ 119870119890119870119905

(18)

where 1205960and 1198941198860

are respectively measured static value ofrotor speed and armature on step response experiment 119882

120596

and119882119894areweighting coefficients and119867

120596and119867

119894are identified

transfer function As a typical constrained optimizationproblem this problem is not complicated and can be readilysolved by Matlab optimization toolbox

4 Cogging Torque Parameters IdentificationMethod in Time Domain

Parameters in linear terms can be estimated by frequencyidentification method during high speed operation wherethe cogging torquersquos influence could be neglected But whenmotor speed approaches to a low operation as shown inFigure 3 such perfectly constant speed cannot be achievedIn this case cogging torque effect should be taken intoaccount Since cogging torque is a nonlinear compact on theBLDC motor model assumptions for frequency identifica-tion method is ruined Hence aiming at estimating parame-ters in cogging torque term a nonlinear system identificationmethod is necessaryThe cogging torque cannot bemeasuredor calculated by direct means In this subsection the coggingtorque considered as a disturbance will be estimated by aclosed up observer The least square method (LSM) is usedto obtain cogging torque parameters 119879

1198881 1198791198882 119879

119888119870

41 Disturbance Observer Design A general block diagramfor disturbance observer [9] shown in Figure 4 can beused to approximate disturbance signal 119906dist If inaccuraciesin the plant model are negligible and the compensator isable to ensure convergence of the error signal 119890 towards

Plant

Ideal model+

+

+

minus

minus

minus

GP(s)

Compensator

udist

udist

y

y

e

Gc(s)

u

Figure 4 Disturbance observer block diagram

zero then the compensator output dist is able to estimatedisturbance signalThus a similar way can be used to estimatenon-modelled dynamics in the case of the linear modeldiscussed in frequency domain on condition that exogenousdisturbances are ignored

Based on superposition principle for linear system thetransfer function119867

119900(119904) fromestimated to applied disturbance

can be deduced as

119867119900(119904) =

dist119906dist

=119866119875(119904) 119866119888(119904)

1 + 119866119875(119904) 119866119888(119904)

(19)

The compensator 119866119888(119904) should be chosen so that the denom-

inator of the polynomial in (19) is Hurwitz stable Thenthe system in Figure 4 is asymptotically stable Thereforethe output of the compensator dist succeeds in observingdisturbance signal 119906dist asymptotically

Depended on the previously discussed disturbanceobserver the cogging torque coefficients identification inan off-line process is realized as shown in Figure 5 Theideal model comes from mechanical dynamics in the linearBLDC motor model and therersquos no load external torqueA 119875119868 controller designed by following the two principalsintroduced in Section 41 works as a compensator

The cogging torque is equivalent to a disturbance signalof electromagnetic torque If a 119875119868 compensator is chosen thecompensator plantmodel and observer transfer functionwillbe

119866119888(119904) = 119875 +

119868

119904

119866119875(119904) =

1

119869119904 + 119870119891

119867119900(119904) =

119866119875(119904) 119866119888(119904)

1 + 119866119875(119904) 119866119888(119904)

=119875119904 + 119868

1198691199042 + (119870119891+ 119875) 119904 + 119868

(20)

Two principles will ensure the accuracy of the observersystem shown in Figure 4

6 Mathematical Problems in Engineering

++ minus

minus

Experimental data

Rotor speed120596

iaKt

Developedtorque

Electromagnetictorque

Td T

Assumedspeed

TcoggP +

I

s

Estimated coggingtorque

Rotor speederrore

Armature current 1

Js + Kf

Ideal model

PI compensator

Figure 5 Cogging torque estimation

Principle 1 Since the observation function is a typical secondorder system the system is stable if

120585 =

119870119891+ 119875

2radic119869119868gt 0 (21)

Equation (21) means that positive 119875 and 119868 sufficiently assurethe stability of the system

Principle 2 In order to improve the accuracy of the observingsignal dist the compensator should minimize the maximumgain from in whole history data that is

min 1198902=1003817100381710038171003817119910 minus 119910

10038171003817100381710038172=

119899

sum

119894

(119910 (119894) minus 119910 (119894))2

(22)

where 119899 is the number of history data points and 120576 is mini-mum energy of error signal that allowed

42 Cogging Torque Parameters Identification Based on LeastSquares Approximation Method In BLDC motor systemsince the convergence of the assumed speed towards themeasured speed is ensured by the resulting compensator theestimated cogging torque dist is able to approximate the truenonlinear dynamics well

The least squares method is considerably used in practicedue to convenient implementation The objective of thisapproximating method is to minimize the ldquoenergy of errorrdquonamely the squared difference between the measured signalsand the estimated signals Considering a vector problem theestimated output is determined by a signal model whichrelates to the unknown parameter vector 120573 = [120573

1 120573

119896]

119908 and 119911 are respectively the input of the model and thepractical measured output The least squares estimator (LSE)is to find the value of 120573 which minimizes the squares errorcriterion given as

min 119869 = 119911 minus 2=

119873

sum

119899=1

(119911 (119899) minus (119899))2

(23)

The observation interval is assumed to be 119899 = 1 2 119873If the linear least squares problem is taken into consideration

the signalmodel is linearwith respect to the unknownparam-eter therefore the output signal can be

119911 = Φ120573 (24)where Φ is the observation matrix in [16] the least squareestimator and average minimum least square error are

120573 = (120601119879

120601)minus1

120601119879

(25)

119869119886min =

( minus 120601120573)119879

( minus 120601120573)

119873

(26)

Considering the unbalanced mechanical system effectinto the estimation process [9] the estimated disturbancetorque is

dist = 119879cogg

=

119870

sum

119896=1

119879119888119896cos (119896119901120579) + 119879

119886sin (120579) + 119879

119887cos (120579)

(27)

The rotor position 120579 is obtained by integration of the time-history data of rotor speed 120596 In this way the term in (27)cos(119896119901120579) is known

Then the observation matrix of LSE is therefore writtenas

Φ = [120601119894119895 120601119904

119894119895120601119888

119894119895]

120601119894119895= cos (119895 sdot 119901120579 (119894)) 119894 = 1 119873 119895 = 1 119870

120601119904

119894119895= sin (120579 (119894)) 119894 = 1 119873 119895 = 1

120601119888

119894119895= cos (120579 (119894)) 119894 = 1 119873 119895 = 1

(28)

Then the parameter vector 120573 = [11987911988811198791198882sdot sdot sdot 119879119888119870 119879119886 119879119887] is

calculated by (25) An essential parameter 119870 the minimalnumber of relevant harmonics which gives the simplestsignal model is necessarily determined by average minimumleast square error 119869

119886min A suitable119870 is obtained to adequatelydescribe the cogging torque characteristics For this sakeof determination method we will increase the number ofharmonics119870 unless the minimum least squares error is smallenough

Mathematical Problems in Engineering 7

BLDC motorESC

Hall sensor

Rotary encoder

Digital oscilloscope

Stabilized voltage supply

Control signal generator

PWM generator

Figure 6 BLDC motor identification experimental bench setup

Table 1 BLDC motor mechanical parameters

Description Value UnitNumber of stator arms 12 Magnet poles 14 Max continuous power 3900 WRotor inertia (evaluated) 0000549 kgsdotm2

Rotary encoder inertia 000001 kgsdotm2

5 System Identification Experiment

System identification experiments are designed and carriedout to obtain data for comprehensive identification Thissection is divided into three parts First we will detail theexperimental plant in BIT Then frequency identificationresults and system main parameter determination will bedescribed Finallywewill discuss nonlinear identification andestimate cogging torque coefficients

51 Experimental Setup The experimental bench setup isshown in Figure 6 including BLDC motor driving unitcontrol unit and measure unit Each of the subsystem willbe briefly introduced by the following subsystems of theexperimental setup

511 BLDC Motor The motor to be identified is popularBLDC motor for UAVs scorpion SII-6530 Considering thesimple configuration of this motor the rotor inertia can beevaluated by Solidworks which is important for the valida-tion of the identification results Somemotor parameters andthe evaluated inertia of the rotary encoder and coupling arelisted in Table 1

512 Driving Unit The driving unit consists of a ESC and aPWM generator The ESC produced by Hobyywing similarto a motor controller and three-phase bridge receives thesignal produced by the PWM generator A stabilized voltagesource powers the ESC to drive the motor The PWMgenerator actually a servo controller is mainly responsible fordecoding control commands and generating PWM wave

0 20 40 60 80 100 1201360

1380

1400

1420

1440

Time (s)

PWM

wav

e dut

y cy

cle

Figure 7 Automatic sweep input signal ranging from 001sim10Hz

513 Control Unit The control unit is an ordinary PCinstalled by Linux operating systemThePCandPWMgener-ator are connected by RS-232 serial ports All the designedcontrol commands are sent from control unit to the PWMgenerator

514 Measure Unit In order to measure these two systemstates rotor speed and armature current a rotary encoder andhall current sensor are respectively included inmeasure unitBoth of output signal of these two sensors are recorded by thedigital oscilloscope in high sample rates The measurementsampling time of the digital oscilloscope could be set from1 120583s up to 50120583s and the memory amount is available forrecording data length of 10Mb Then the rotor speed andarmature current can be deciphered from the measured data

52 Frequency Identification Results When the power supplyis set at 23V the rotary encoder wasmounted onto the BLDCmotor rotor and the armature current wire passes throughthe hall current sensor The encoder and the hall sensorrsquosoutputs are recorded simultaneously during the experimentwith sampling frequency 119891

119904= 100Hz The rotor speed and

armature are then calculated off-lineComplying with (8) the input signal shown in Figure 7

covers frequencies ranging from 001 to 10HzThis frequencysweep input signal is utilized to excite the dynamics of our

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120240

250

260

270

280

290

300

Time (s)

Roto

r spe

ed (r

ads

)

(a) Rotor speed response

0 20 40 60 80 100 120

0

05

1

15

2

25

3

Time (s)

Arm

atur

e cur

rent

(A)

minus05

(b) Armature current response

Figure 8 Rotor speed and armature current responses

0

10

20

Frequency (rads)

Mag

nitu

de (d

B)

0

Frequency (rads)

Phas

e (de

g)

Experimental dataTransfer function fit

0

05

1

Frequency (rads)

Coh

eren

ce

100 101

100 101

100 101

minus150

minus100

minus50

(a) Rotor speed channel

Frequency (rads)

0

50

Frequency (rads)

Experimental dataTransfer function fit

Mag

nitu

de (d

B)Ph

ase (

deg)

Coh

eren

ce

0

05

1

Frequency (rads)

10minus1 100 101

100 101

102

100 101 102

minus100

minus50

minus15

minus10

minus5

(b) Armature channel

Figure 9 Rotor speed channelarmature channel identification

plant over a broad frequency range Figure 8 shows the rotorspeed and armature current extracted from experimentaldata Note that rotor speed response amplitude diminishesat higher frequencies as expected from the bandwidth ofthe corresponding channel However the armature currentresponse amplitude will increase with the even very highfrequency suggesting a very high bandwidth of the armaturecurrent channel which is considerably over the max sweepfrequency

The data obtained from this test will be applied toextract frequency responses used for themodel identification

The estimated frequency responses are calculated using spec-tral density functions with CIFER Figure 9 shows frequencyidentification of rotor speed channel and armature channelGood coherence is achieved in the required frequency rangeindicating good quality experimental data and the linearmodel assumption With respect to coherence rotor speedchannel identification result is better than that of currentchannel because the hall current sensor yields a pooraccuracy compared to rotary encoder The identified transferfunctions represented by red dashed lines show good fits ofthe magnitude and phase responses of each channel

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12900

950

1000

1050

1100

1150

1200

1250

Time (s)

PWM

wav

e dut

y cy

cle

Figure 10 Input doublet signal

Table 2 Estimated BLDC motor parameters

System parameters Symbol Value UnitRotor inertia 119869 000061 kgsdotm2

Voltage constant 119870119890

00637 V(rads)Torque constant 119870

11990502663 NmA

Armature resistance 119877119886

06187 Ω

Armature inductance 119871119886

262 mHFriction coefficient 119870

119891000291 N(rads)

Time constant of the ESC model 120591 00781 s

The two transfer functions of two channels based on (13)and (14) identified are

119867120596(119904) =

2057342

1199043 + 18951199042 + 13412119904 + 142834

119867119894(119904) =

21749 (02015119904 + 1)

1199043 + 1961199042 + 14695119904 + 149431

(29)

The identified transfer functions are taken into account todetermine BLDCmotormechanismparameters as suggestedin Section 3 Then the problem in (17) and (18) is solvedby Matlab optimization toolbox The bias between theoret-ical model and identification result is 041 The estimatedBLDC motor parameters are listed in Table 2 Note thatthe identified rotor inertia 119869 = 000061 is very closeto the evaluated value in Table 1 indicating a satisfactoryidentification accuracy of this parameter

The accuracy of the comprehensive BLDC motor modelmust be further verified by tine history data The inputdoublet signal shown in Figure 10 is used to stimulate theBLDC motor dynamics The validation results are shownin Figure 11 Both rotor speed and armature current aremeasured during the input doublet signal It is shown that theidentifiedmodel perfectly predicts the rotor speed responseshowever the true armature current response exhibits a slightdeviation from the responses predicted by the identifiedmodel This mismatch can be explained by the relatively lowcoherence in high frequency of the armature channel identifi-cationThe close agreement between the simulation response

Table 3 Estimation results for the mechanical parameters

System parameters 119909 (mean values) 120590 (standard deviations)1198791198881

293 451198791198882

527 511198791198883

3706 4461198791198884

156 411198791198885

843 2241198791198886

5612 433119879119886

1103 125119879119887

1620 332

and experimental data indicates the identified model isaccurate enough to reflect the BLDC motorrsquos dynamics

53 Cogging Torque Coefficients Identification Results Sincethe cogging torque effect exists significantly in low speed theterminal voltage of the motor is set at 65 V The disturbanceobserver is applied to calculate cogging torque as stated inSection 42 In order to eliminate the stochastic error theexperiment is repeated by 10 repetitions with single testtime of 20 s The minimum least squares error obtained by(26) versus the number of cogging torque harmonics areshown in Figure 12 with the estimated disturbance signalapproximated using LSE It is obvious that the sixth coggingtorque harmonic is highly dominant from the diagram and aslight of third harmonics is also present [17] has pointed outthat the cogging torque harmonics for three-phase motorsusually are multiples of six times the fundamental commuta-tion frequency

A simulation of the identified model is performed inorder to verify the results of the proposed parameters iden-tification method The cogging torque coefficients in Table 3and the estimated linear terms by frequency identification inSection 52 are utilized in the comprehensive BLDC motormodel A comparison betweenmeasured and estimated rotorspeed is formed to highlight the verification result (seeFigure 13) Note that there is a small phase deviation ofbetween experimental and estimated cogging torque thiscould be interpreted by the unmodeled dynamics Matchingof the simulated and experimental data is ensured Hence theproposed identification method is applicable and sufficient

6 Conclusion

In this paper a comprehensive procedure for system iden-tification applicable to BLDC motor used in UAV has beensystematically presented The procedure aims to serve asa guide for the steps necessary to determine an accurateBLDC motor model Considering the nonlinear character-istics cogging torque effect of the BLDC motor system apractical mathematical model is introduced for identifica-tion Preliminarily frequency domain identification theoryis applied to estimate the first seven parameters in linearterms of the model at a high motor speed region where thenonlinear term can be neglected Then an off-line methodfor identifying the cogging torque coefficients is performed at

10 Mathematical Problems in Engineering

0 2 4 6 8 10 12240

260

280

300

320

Time (s)

Roto

r spe

ed (r

ads

)

Experimental dataBLDC motor model

(a) Rotor speed validation of the BLDC motor dynamics

2 4 6 8 10 12

1

2

3

4

5

Time (s)

Arm

atur

e cur

rent

(A)

Experimental dataBLDC motor model

(b) Armature current validation of the BLDC motor dynamics

Figure 11 Time domain validation of the BLDC motor identified dynamics

2 3 4 5 6 7 8 90

10

20

30

Number of harmonics (K)

J min

Figure 12 The minimum least squares error versus the number of cogging torque harmonics

05 1 15 275

80

85

90

95

100

Time (s)

Nonlinear model Experimental data

02 025 03

90

92

94

96

Time (s)

Nonlinear model Experimental data

Roto

r spe

ed (r

ads

)

Roto

r spe

ed (r

ads

)

Figure 13 Validation of identified comprehensive BLDC motor model

Mathematical Problems in Engineering 11

very low speedwhen cogging torque effect exists significantlyThis method is founded on the least squares approximationmethod a disturbance observer and the identified parametersin frequency identification The results from the simulationand the real-time experiments matched closely which isconsidered as a validation of the obtained mathematicalmodel

The BLDC model identifications are conditional onlyon the availability of experimental data for rotor speed andarmature current therefore our method without sensors todetect torque or back-EMF is easy to realize and is practicaland effectiveThis research will provide greater opportunitiesfor future controller design of the BLDCmotorThe extractedlinear model can be utilized directly for linear controllerdesign and the accurately identified cogging torque effect isbeneficial to reduce the speed ripple of the motor at some lowspeed conditions using some advanced control theory suchas robust control and nonlinear control

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Shanmugasundram K M Zakaraiah and N YadaiahldquoModeling simulation and analysis of controllers for brushlessdirect current motor drivesrdquo Journal of Vibration and Controlvol 19 no 8 pp 1250ndash1264 2013

[2] C Xia Y Wang and T Shi ldquoImplementation of finite-statemodel predictive control for commutation torque ripple min-imization of permanent-magnet brushless DC motorrdquo IEEETransactions on Industrial Electronics vol 60 no 3 pp 896ndash905 2013

[3] S-Y Chang P-H Yu and W-C Su ldquoPre- and post-filteringapproach to sensorless VSS speed control of a permanentmagnet DC motor subject to Hammerstein and Wiener non-linearitiesrdquoControl Engineering Practice vol 21 no 11 pp 1577ndash1583 2013

[4] B-K Lee and M Ehsani ldquoAdvanced simulation model forbrushless dc motor drivesrdquo Electric Power Components andSystems vol 31 no 9 pp 841ndash868 2003

[5] I Kim N Nakazawa S Kim C Park and C Yu ldquoCompensa-tion of torque ripple in high performance BLDCmotor drivesrdquoControl Engineering Practice vol 18 no 10 pp 1166ndash1172 2010

[6] H Lu L Zhang and W Qu ldquoA new torque control method fortorque rippleminimization of BLDCmotors with un-ideal backEMFrdquo IEEE Transactions on Power Electronics vol 23 no 2 pp950ndash958 2008

[7] W Wu ldquoDC motor parameter identification using speed stepresponsesrdquo Modelling and Simulation in Engineering vol 2012Article ID 189757 5 pages 2012

[8] J-H Choi S-H You J Hur and H-G Sung ldquoThe design andfabrication of BLDCMotor andDrive for 42V automotive appli-cationsrdquo in Proceedings of the IEEE International Symposium onIndustrial Electronics (ISIE rsquo07) pp 1086ndash1091 June 2007

[9] A Kapun M Curkovic A Hace and K Jezernik ldquoIdentifyingdynamic model parameters of a BLDC motorrdquo Simulation

Modelling Practice and Theory vol 16 no 9 pp 1254ndash12652008

[10] A J Blauch M Bodson and J Chiasson ldquoHigh-Speed parame-ter estimation of stepper motorsrdquo IEEE Transactions on ControlSystems Technology vol 1 no 4 pp 270ndash279 1993

[11] R Patankar and L Zhu ldquoReal-time multiple parameter estima-tion for voltage controlled brushless DC motor actuatorsrdquo inProceedings of the American Control Conference (AAC rsquo04) vol4 pp 3851ndash3856 July 2004

[12] M S Islam R Islam T Sebastian A Chandy and S AOzsoyluldquoCogging torque minimization in PM motors using robustdesign approachrdquo IEEE Transactions on Industry Applicationsvol 47 no 4 pp 1661ndash1669 2011

[13] M B Tischler and R K Remple Aircraft and Rotorcraft SystemIdentification AIAA Education Series American Institute ofAeronautics and Astronautics 2006

[14] M Petra and M Utting CZT A Framework for Z Tools FormalSpecification and Development in Z and B Springer BerlinGermany 2005

[15] K Zhou and C D John Essentials of Robust Control PrenticeHall Upper Saddle River NJ USA 1998

[16] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Higher Education New York NY USA 2010

[17] J Y Hung and Z Ding ldquoDesign currents to reduce torque ripplein brushless permanent magnet motorsrdquo IEE Proceedings BElectric Power Applications vol 140 no 4 pp 260ndash266 1993

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

0 05 1 15 20

20

40

60

80

100

Time (s)

Roto

r spe

ed (r

ads

)

Figure 3 Cogging torque effect in low speed

Equation (17) alone is probably not sufficient to find theonly solution Another two equation constraints on systemstatic step response are

1205960=

002187119870119905

119877119886119870119891+ 119870119890119870119905

1198941198860=

002187119870119891

119877119886119870119891+ 119870119890119870119905

(18)

where 1205960and 1198941198860

are respectively measured static value ofrotor speed and armature on step response experiment 119882

120596

and119882119894areweighting coefficients and119867

120596and119867

119894are identified

transfer function As a typical constrained optimizationproblem this problem is not complicated and can be readilysolved by Matlab optimization toolbox

4 Cogging Torque Parameters IdentificationMethod in Time Domain

Parameters in linear terms can be estimated by frequencyidentification method during high speed operation wherethe cogging torquersquos influence could be neglected But whenmotor speed approaches to a low operation as shown inFigure 3 such perfectly constant speed cannot be achievedIn this case cogging torque effect should be taken intoaccount Since cogging torque is a nonlinear compact on theBLDC motor model assumptions for frequency identifica-tion method is ruined Hence aiming at estimating parame-ters in cogging torque term a nonlinear system identificationmethod is necessaryThe cogging torque cannot bemeasuredor calculated by direct means In this subsection the coggingtorque considered as a disturbance will be estimated by aclosed up observer The least square method (LSM) is usedto obtain cogging torque parameters 119879

1198881 1198791198882 119879

119888119870

41 Disturbance Observer Design A general block diagramfor disturbance observer [9] shown in Figure 4 can beused to approximate disturbance signal 119906dist If inaccuraciesin the plant model are negligible and the compensator isable to ensure convergence of the error signal 119890 towards

Plant

Ideal model+

+

+

minus

minus

minus

GP(s)

Compensator

udist

udist

y

y

e

Gc(s)

u

Figure 4 Disturbance observer block diagram

zero then the compensator output dist is able to estimatedisturbance signalThus a similar way can be used to estimatenon-modelled dynamics in the case of the linear modeldiscussed in frequency domain on condition that exogenousdisturbances are ignored

Based on superposition principle for linear system thetransfer function119867

119900(119904) fromestimated to applied disturbance

can be deduced as

119867119900(119904) =

dist119906dist

=119866119875(119904) 119866119888(119904)

1 + 119866119875(119904) 119866119888(119904)

(19)

The compensator 119866119888(119904) should be chosen so that the denom-

inator of the polynomial in (19) is Hurwitz stable Thenthe system in Figure 4 is asymptotically stable Thereforethe output of the compensator dist succeeds in observingdisturbance signal 119906dist asymptotically

Depended on the previously discussed disturbanceobserver the cogging torque coefficients identification inan off-line process is realized as shown in Figure 5 Theideal model comes from mechanical dynamics in the linearBLDC motor model and therersquos no load external torqueA 119875119868 controller designed by following the two principalsintroduced in Section 41 works as a compensator

The cogging torque is equivalent to a disturbance signalof electromagnetic torque If a 119875119868 compensator is chosen thecompensator plantmodel and observer transfer functionwillbe

119866119888(119904) = 119875 +

119868

119904

119866119875(119904) =

1

119869119904 + 119870119891

119867119900(119904) =

119866119875(119904) 119866119888(119904)

1 + 119866119875(119904) 119866119888(119904)

=119875119904 + 119868

1198691199042 + (119870119891+ 119875) 119904 + 119868

(20)

Two principles will ensure the accuracy of the observersystem shown in Figure 4

6 Mathematical Problems in Engineering

++ minus

minus

Experimental data

Rotor speed120596

iaKt

Developedtorque

Electromagnetictorque

Td T

Assumedspeed

TcoggP +

I

s

Estimated coggingtorque

Rotor speederrore

Armature current 1

Js + Kf

Ideal model

PI compensator

Figure 5 Cogging torque estimation

Principle 1 Since the observation function is a typical secondorder system the system is stable if

120585 =

119870119891+ 119875

2radic119869119868gt 0 (21)

Equation (21) means that positive 119875 and 119868 sufficiently assurethe stability of the system

Principle 2 In order to improve the accuracy of the observingsignal dist the compensator should minimize the maximumgain from in whole history data that is

min 1198902=1003817100381710038171003817119910 minus 119910

10038171003817100381710038172=

119899

sum

119894

(119910 (119894) minus 119910 (119894))2

(22)

where 119899 is the number of history data points and 120576 is mini-mum energy of error signal that allowed

42 Cogging Torque Parameters Identification Based on LeastSquares Approximation Method In BLDC motor systemsince the convergence of the assumed speed towards themeasured speed is ensured by the resulting compensator theestimated cogging torque dist is able to approximate the truenonlinear dynamics well

The least squares method is considerably used in practicedue to convenient implementation The objective of thisapproximating method is to minimize the ldquoenergy of errorrdquonamely the squared difference between the measured signalsand the estimated signals Considering a vector problem theestimated output is determined by a signal model whichrelates to the unknown parameter vector 120573 = [120573

1 120573

119896]

119908 and 119911 are respectively the input of the model and thepractical measured output The least squares estimator (LSE)is to find the value of 120573 which minimizes the squares errorcriterion given as

min 119869 = 119911 minus 2=

119873

sum

119899=1

(119911 (119899) minus (119899))2

(23)

The observation interval is assumed to be 119899 = 1 2 119873If the linear least squares problem is taken into consideration

the signalmodel is linearwith respect to the unknownparam-eter therefore the output signal can be

119911 = Φ120573 (24)where Φ is the observation matrix in [16] the least squareestimator and average minimum least square error are

120573 = (120601119879

120601)minus1

120601119879

(25)

119869119886min =

( minus 120601120573)119879

( minus 120601120573)

119873

(26)

Considering the unbalanced mechanical system effectinto the estimation process [9] the estimated disturbancetorque is

dist = 119879cogg

=

119870

sum

119896=1

119879119888119896cos (119896119901120579) + 119879

119886sin (120579) + 119879

119887cos (120579)

(27)

The rotor position 120579 is obtained by integration of the time-history data of rotor speed 120596 In this way the term in (27)cos(119896119901120579) is known

Then the observation matrix of LSE is therefore writtenas

Φ = [120601119894119895 120601119904

119894119895120601119888

119894119895]

120601119894119895= cos (119895 sdot 119901120579 (119894)) 119894 = 1 119873 119895 = 1 119870

120601119904

119894119895= sin (120579 (119894)) 119894 = 1 119873 119895 = 1

120601119888

119894119895= cos (120579 (119894)) 119894 = 1 119873 119895 = 1

(28)

Then the parameter vector 120573 = [11987911988811198791198882sdot sdot sdot 119879119888119870 119879119886 119879119887] is

calculated by (25) An essential parameter 119870 the minimalnumber of relevant harmonics which gives the simplestsignal model is necessarily determined by average minimumleast square error 119869

119886min A suitable119870 is obtained to adequatelydescribe the cogging torque characteristics For this sakeof determination method we will increase the number ofharmonics119870 unless the minimum least squares error is smallenough

Mathematical Problems in Engineering 7

BLDC motorESC

Hall sensor

Rotary encoder

Digital oscilloscope

Stabilized voltage supply

Control signal generator

PWM generator

Figure 6 BLDC motor identification experimental bench setup

Table 1 BLDC motor mechanical parameters

Description Value UnitNumber of stator arms 12 Magnet poles 14 Max continuous power 3900 WRotor inertia (evaluated) 0000549 kgsdotm2

Rotary encoder inertia 000001 kgsdotm2

5 System Identification Experiment

System identification experiments are designed and carriedout to obtain data for comprehensive identification Thissection is divided into three parts First we will detail theexperimental plant in BIT Then frequency identificationresults and system main parameter determination will bedescribed Finallywewill discuss nonlinear identification andestimate cogging torque coefficients

51 Experimental Setup The experimental bench setup isshown in Figure 6 including BLDC motor driving unitcontrol unit and measure unit Each of the subsystem willbe briefly introduced by the following subsystems of theexperimental setup

511 BLDC Motor The motor to be identified is popularBLDC motor for UAVs scorpion SII-6530 Considering thesimple configuration of this motor the rotor inertia can beevaluated by Solidworks which is important for the valida-tion of the identification results Somemotor parameters andthe evaluated inertia of the rotary encoder and coupling arelisted in Table 1

512 Driving Unit The driving unit consists of a ESC and aPWM generator The ESC produced by Hobyywing similarto a motor controller and three-phase bridge receives thesignal produced by the PWM generator A stabilized voltagesource powers the ESC to drive the motor The PWMgenerator actually a servo controller is mainly responsible fordecoding control commands and generating PWM wave

0 20 40 60 80 100 1201360

1380

1400

1420

1440

Time (s)

PWM

wav

e dut

y cy

cle

Figure 7 Automatic sweep input signal ranging from 001sim10Hz

513 Control Unit The control unit is an ordinary PCinstalled by Linux operating systemThePCandPWMgener-ator are connected by RS-232 serial ports All the designedcontrol commands are sent from control unit to the PWMgenerator

514 Measure Unit In order to measure these two systemstates rotor speed and armature current a rotary encoder andhall current sensor are respectively included inmeasure unitBoth of output signal of these two sensors are recorded by thedigital oscilloscope in high sample rates The measurementsampling time of the digital oscilloscope could be set from1 120583s up to 50120583s and the memory amount is available forrecording data length of 10Mb Then the rotor speed andarmature current can be deciphered from the measured data

52 Frequency Identification Results When the power supplyis set at 23V the rotary encoder wasmounted onto the BLDCmotor rotor and the armature current wire passes throughthe hall current sensor The encoder and the hall sensorrsquosoutputs are recorded simultaneously during the experimentwith sampling frequency 119891

119904= 100Hz The rotor speed and

armature are then calculated off-lineComplying with (8) the input signal shown in Figure 7

covers frequencies ranging from 001 to 10HzThis frequencysweep input signal is utilized to excite the dynamics of our

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120240

250

260

270

280

290

300

Time (s)

Roto

r spe

ed (r

ads

)

(a) Rotor speed response

0 20 40 60 80 100 120

0

05

1

15

2

25

3

Time (s)

Arm

atur

e cur

rent

(A)

minus05

(b) Armature current response

Figure 8 Rotor speed and armature current responses

0

10

20

Frequency (rads)

Mag

nitu

de (d

B)

0

Frequency (rads)

Phas

e (de

g)

Experimental dataTransfer function fit

0

05

1

Frequency (rads)

Coh

eren

ce

100 101

100 101

100 101

minus150

minus100

minus50

(a) Rotor speed channel

Frequency (rads)

0

50

Frequency (rads)

Experimental dataTransfer function fit

Mag

nitu

de (d

B)Ph

ase (

deg)

Coh

eren

ce

0

05

1

Frequency (rads)

10minus1 100 101

100 101

102

100 101 102

minus100

minus50

minus15

minus10

minus5

(b) Armature channel

Figure 9 Rotor speed channelarmature channel identification

plant over a broad frequency range Figure 8 shows the rotorspeed and armature current extracted from experimentaldata Note that rotor speed response amplitude diminishesat higher frequencies as expected from the bandwidth ofthe corresponding channel However the armature currentresponse amplitude will increase with the even very highfrequency suggesting a very high bandwidth of the armaturecurrent channel which is considerably over the max sweepfrequency

The data obtained from this test will be applied toextract frequency responses used for themodel identification

The estimated frequency responses are calculated using spec-tral density functions with CIFER Figure 9 shows frequencyidentification of rotor speed channel and armature channelGood coherence is achieved in the required frequency rangeindicating good quality experimental data and the linearmodel assumption With respect to coherence rotor speedchannel identification result is better than that of currentchannel because the hall current sensor yields a pooraccuracy compared to rotary encoder The identified transferfunctions represented by red dashed lines show good fits ofthe magnitude and phase responses of each channel

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12900

950

1000

1050

1100

1150

1200

1250

Time (s)

PWM

wav

e dut

y cy

cle

Figure 10 Input doublet signal

Table 2 Estimated BLDC motor parameters

System parameters Symbol Value UnitRotor inertia 119869 000061 kgsdotm2

Voltage constant 119870119890

00637 V(rads)Torque constant 119870

11990502663 NmA

Armature resistance 119877119886

06187 Ω

Armature inductance 119871119886

262 mHFriction coefficient 119870

119891000291 N(rads)

Time constant of the ESC model 120591 00781 s

The two transfer functions of two channels based on (13)and (14) identified are

119867120596(119904) =

2057342

1199043 + 18951199042 + 13412119904 + 142834

119867119894(119904) =

21749 (02015119904 + 1)

1199043 + 1961199042 + 14695119904 + 149431

(29)

The identified transfer functions are taken into account todetermine BLDCmotormechanismparameters as suggestedin Section 3 Then the problem in (17) and (18) is solvedby Matlab optimization toolbox The bias between theoret-ical model and identification result is 041 The estimatedBLDC motor parameters are listed in Table 2 Note thatthe identified rotor inertia 119869 = 000061 is very closeto the evaluated value in Table 1 indicating a satisfactoryidentification accuracy of this parameter

The accuracy of the comprehensive BLDC motor modelmust be further verified by tine history data The inputdoublet signal shown in Figure 10 is used to stimulate theBLDC motor dynamics The validation results are shownin Figure 11 Both rotor speed and armature current aremeasured during the input doublet signal It is shown that theidentifiedmodel perfectly predicts the rotor speed responseshowever the true armature current response exhibits a slightdeviation from the responses predicted by the identifiedmodel This mismatch can be explained by the relatively lowcoherence in high frequency of the armature channel identifi-cationThe close agreement between the simulation response

Table 3 Estimation results for the mechanical parameters

System parameters 119909 (mean values) 120590 (standard deviations)1198791198881

293 451198791198882

527 511198791198883

3706 4461198791198884

156 411198791198885

843 2241198791198886

5612 433119879119886

1103 125119879119887

1620 332

and experimental data indicates the identified model isaccurate enough to reflect the BLDC motorrsquos dynamics

53 Cogging Torque Coefficients Identification Results Sincethe cogging torque effect exists significantly in low speed theterminal voltage of the motor is set at 65 V The disturbanceobserver is applied to calculate cogging torque as stated inSection 42 In order to eliminate the stochastic error theexperiment is repeated by 10 repetitions with single testtime of 20 s The minimum least squares error obtained by(26) versus the number of cogging torque harmonics areshown in Figure 12 with the estimated disturbance signalapproximated using LSE It is obvious that the sixth coggingtorque harmonic is highly dominant from the diagram and aslight of third harmonics is also present [17] has pointed outthat the cogging torque harmonics for three-phase motorsusually are multiples of six times the fundamental commuta-tion frequency

A simulation of the identified model is performed inorder to verify the results of the proposed parameters iden-tification method The cogging torque coefficients in Table 3and the estimated linear terms by frequency identification inSection 52 are utilized in the comprehensive BLDC motormodel A comparison betweenmeasured and estimated rotorspeed is formed to highlight the verification result (seeFigure 13) Note that there is a small phase deviation ofbetween experimental and estimated cogging torque thiscould be interpreted by the unmodeled dynamics Matchingof the simulated and experimental data is ensured Hence theproposed identification method is applicable and sufficient

6 Conclusion

In this paper a comprehensive procedure for system iden-tification applicable to BLDC motor used in UAV has beensystematically presented The procedure aims to serve asa guide for the steps necessary to determine an accurateBLDC motor model Considering the nonlinear character-istics cogging torque effect of the BLDC motor system apractical mathematical model is introduced for identifica-tion Preliminarily frequency domain identification theoryis applied to estimate the first seven parameters in linearterms of the model at a high motor speed region where thenonlinear term can be neglected Then an off-line methodfor identifying the cogging torque coefficients is performed at

10 Mathematical Problems in Engineering

0 2 4 6 8 10 12240

260

280

300

320

Time (s)

Roto

r spe

ed (r

ads

)

Experimental dataBLDC motor model

(a) Rotor speed validation of the BLDC motor dynamics

2 4 6 8 10 12

1

2

3

4

5

Time (s)

Arm

atur

e cur

rent

(A)

Experimental dataBLDC motor model

(b) Armature current validation of the BLDC motor dynamics

Figure 11 Time domain validation of the BLDC motor identified dynamics

2 3 4 5 6 7 8 90

10

20

30

Number of harmonics (K)

J min

Figure 12 The minimum least squares error versus the number of cogging torque harmonics

05 1 15 275

80

85

90

95

100

Time (s)

Nonlinear model Experimental data

02 025 03

90

92

94

96

Time (s)

Nonlinear model Experimental data

Roto

r spe

ed (r

ads

)

Roto

r spe

ed (r

ads

)

Figure 13 Validation of identified comprehensive BLDC motor model

Mathematical Problems in Engineering 11

very low speedwhen cogging torque effect exists significantlyThis method is founded on the least squares approximationmethod a disturbance observer and the identified parametersin frequency identification The results from the simulationand the real-time experiments matched closely which isconsidered as a validation of the obtained mathematicalmodel

The BLDC model identifications are conditional onlyon the availability of experimental data for rotor speed andarmature current therefore our method without sensors todetect torque or back-EMF is easy to realize and is practicaland effectiveThis research will provide greater opportunitiesfor future controller design of the BLDCmotorThe extractedlinear model can be utilized directly for linear controllerdesign and the accurately identified cogging torque effect isbeneficial to reduce the speed ripple of the motor at some lowspeed conditions using some advanced control theory suchas robust control and nonlinear control

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Shanmugasundram K M Zakaraiah and N YadaiahldquoModeling simulation and analysis of controllers for brushlessdirect current motor drivesrdquo Journal of Vibration and Controlvol 19 no 8 pp 1250ndash1264 2013

[2] C Xia Y Wang and T Shi ldquoImplementation of finite-statemodel predictive control for commutation torque ripple min-imization of permanent-magnet brushless DC motorrdquo IEEETransactions on Industrial Electronics vol 60 no 3 pp 896ndash905 2013

[3] S-Y Chang P-H Yu and W-C Su ldquoPre- and post-filteringapproach to sensorless VSS speed control of a permanentmagnet DC motor subject to Hammerstein and Wiener non-linearitiesrdquoControl Engineering Practice vol 21 no 11 pp 1577ndash1583 2013

[4] B-K Lee and M Ehsani ldquoAdvanced simulation model forbrushless dc motor drivesrdquo Electric Power Components andSystems vol 31 no 9 pp 841ndash868 2003

[5] I Kim N Nakazawa S Kim C Park and C Yu ldquoCompensa-tion of torque ripple in high performance BLDCmotor drivesrdquoControl Engineering Practice vol 18 no 10 pp 1166ndash1172 2010

[6] H Lu L Zhang and W Qu ldquoA new torque control method fortorque rippleminimization of BLDCmotors with un-ideal backEMFrdquo IEEE Transactions on Power Electronics vol 23 no 2 pp950ndash958 2008

[7] W Wu ldquoDC motor parameter identification using speed stepresponsesrdquo Modelling and Simulation in Engineering vol 2012Article ID 189757 5 pages 2012

[8] J-H Choi S-H You J Hur and H-G Sung ldquoThe design andfabrication of BLDCMotor andDrive for 42V automotive appli-cationsrdquo in Proceedings of the IEEE International Symposium onIndustrial Electronics (ISIE rsquo07) pp 1086ndash1091 June 2007

[9] A Kapun M Curkovic A Hace and K Jezernik ldquoIdentifyingdynamic model parameters of a BLDC motorrdquo Simulation

Modelling Practice and Theory vol 16 no 9 pp 1254ndash12652008

[10] A J Blauch M Bodson and J Chiasson ldquoHigh-Speed parame-ter estimation of stepper motorsrdquo IEEE Transactions on ControlSystems Technology vol 1 no 4 pp 270ndash279 1993

[11] R Patankar and L Zhu ldquoReal-time multiple parameter estima-tion for voltage controlled brushless DC motor actuatorsrdquo inProceedings of the American Control Conference (AAC rsquo04) vol4 pp 3851ndash3856 July 2004

[12] M S Islam R Islam T Sebastian A Chandy and S AOzsoyluldquoCogging torque minimization in PM motors using robustdesign approachrdquo IEEE Transactions on Industry Applicationsvol 47 no 4 pp 1661ndash1669 2011

[13] M B Tischler and R K Remple Aircraft and Rotorcraft SystemIdentification AIAA Education Series American Institute ofAeronautics and Astronautics 2006

[14] M Petra and M Utting CZT A Framework for Z Tools FormalSpecification and Development in Z and B Springer BerlinGermany 2005

[15] K Zhou and C D John Essentials of Robust Control PrenticeHall Upper Saddle River NJ USA 1998

[16] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Higher Education New York NY USA 2010

[17] J Y Hung and Z Ding ldquoDesign currents to reduce torque ripplein brushless permanent magnet motorsrdquo IEE Proceedings BElectric Power Applications vol 140 no 4 pp 260ndash266 1993

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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

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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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

++ minus

minus

Experimental data

Rotor speed120596

iaKt

Developedtorque

Electromagnetictorque

Td T

Assumedspeed

TcoggP +

I

s

Estimated coggingtorque

Rotor speederrore

Armature current 1

Js + Kf

Ideal model

PI compensator

Figure 5 Cogging torque estimation

Principle 1 Since the observation function is a typical secondorder system the system is stable if

120585 =

119870119891+ 119875

2radic119869119868gt 0 (21)

Equation (21) means that positive 119875 and 119868 sufficiently assurethe stability of the system

Principle 2 In order to improve the accuracy of the observingsignal dist the compensator should minimize the maximumgain from in whole history data that is

min 1198902=1003817100381710038171003817119910 minus 119910

10038171003817100381710038172=

119899

sum

119894

(119910 (119894) minus 119910 (119894))2

(22)

where 119899 is the number of history data points and 120576 is mini-mum energy of error signal that allowed

42 Cogging Torque Parameters Identification Based on LeastSquares Approximation Method In BLDC motor systemsince the convergence of the assumed speed towards themeasured speed is ensured by the resulting compensator theestimated cogging torque dist is able to approximate the truenonlinear dynamics well

The least squares method is considerably used in practicedue to convenient implementation The objective of thisapproximating method is to minimize the ldquoenergy of errorrdquonamely the squared difference between the measured signalsand the estimated signals Considering a vector problem theestimated output is determined by a signal model whichrelates to the unknown parameter vector 120573 = [120573

1 120573

119896]

119908 and 119911 are respectively the input of the model and thepractical measured output The least squares estimator (LSE)is to find the value of 120573 which minimizes the squares errorcriterion given as

min 119869 = 119911 minus 2=

119873

sum

119899=1

(119911 (119899) minus (119899))2

(23)

The observation interval is assumed to be 119899 = 1 2 119873If the linear least squares problem is taken into consideration

the signalmodel is linearwith respect to the unknownparam-eter therefore the output signal can be

119911 = Φ120573 (24)where Φ is the observation matrix in [16] the least squareestimator and average minimum least square error are

120573 = (120601119879

120601)minus1

120601119879

(25)

119869119886min =

( minus 120601120573)119879

( minus 120601120573)

119873

(26)

Considering the unbalanced mechanical system effectinto the estimation process [9] the estimated disturbancetorque is

dist = 119879cogg

=

119870

sum

119896=1

119879119888119896cos (119896119901120579) + 119879

119886sin (120579) + 119879

119887cos (120579)

(27)

The rotor position 120579 is obtained by integration of the time-history data of rotor speed 120596 In this way the term in (27)cos(119896119901120579) is known

Then the observation matrix of LSE is therefore writtenas

Φ = [120601119894119895 120601119904

119894119895120601119888

119894119895]

120601119894119895= cos (119895 sdot 119901120579 (119894)) 119894 = 1 119873 119895 = 1 119870

120601119904

119894119895= sin (120579 (119894)) 119894 = 1 119873 119895 = 1

120601119888

119894119895= cos (120579 (119894)) 119894 = 1 119873 119895 = 1

(28)

Then the parameter vector 120573 = [11987911988811198791198882sdot sdot sdot 119879119888119870 119879119886 119879119887] is

calculated by (25) An essential parameter 119870 the minimalnumber of relevant harmonics which gives the simplestsignal model is necessarily determined by average minimumleast square error 119869

119886min A suitable119870 is obtained to adequatelydescribe the cogging torque characteristics For this sakeof determination method we will increase the number ofharmonics119870 unless the minimum least squares error is smallenough

Mathematical Problems in Engineering 7

BLDC motorESC

Hall sensor

Rotary encoder

Digital oscilloscope

Stabilized voltage supply

Control signal generator

PWM generator

Figure 6 BLDC motor identification experimental bench setup

Table 1 BLDC motor mechanical parameters

Description Value UnitNumber of stator arms 12 Magnet poles 14 Max continuous power 3900 WRotor inertia (evaluated) 0000549 kgsdotm2

Rotary encoder inertia 000001 kgsdotm2

5 System Identification Experiment

System identification experiments are designed and carriedout to obtain data for comprehensive identification Thissection is divided into three parts First we will detail theexperimental plant in BIT Then frequency identificationresults and system main parameter determination will bedescribed Finallywewill discuss nonlinear identification andestimate cogging torque coefficients

51 Experimental Setup The experimental bench setup isshown in Figure 6 including BLDC motor driving unitcontrol unit and measure unit Each of the subsystem willbe briefly introduced by the following subsystems of theexperimental setup

511 BLDC Motor The motor to be identified is popularBLDC motor for UAVs scorpion SII-6530 Considering thesimple configuration of this motor the rotor inertia can beevaluated by Solidworks which is important for the valida-tion of the identification results Somemotor parameters andthe evaluated inertia of the rotary encoder and coupling arelisted in Table 1

512 Driving Unit The driving unit consists of a ESC and aPWM generator The ESC produced by Hobyywing similarto a motor controller and three-phase bridge receives thesignal produced by the PWM generator A stabilized voltagesource powers the ESC to drive the motor The PWMgenerator actually a servo controller is mainly responsible fordecoding control commands and generating PWM wave

0 20 40 60 80 100 1201360

1380

1400

1420

1440

Time (s)

PWM

wav

e dut

y cy

cle

Figure 7 Automatic sweep input signal ranging from 001sim10Hz

513 Control Unit The control unit is an ordinary PCinstalled by Linux operating systemThePCandPWMgener-ator are connected by RS-232 serial ports All the designedcontrol commands are sent from control unit to the PWMgenerator

514 Measure Unit In order to measure these two systemstates rotor speed and armature current a rotary encoder andhall current sensor are respectively included inmeasure unitBoth of output signal of these two sensors are recorded by thedigital oscilloscope in high sample rates The measurementsampling time of the digital oscilloscope could be set from1 120583s up to 50120583s and the memory amount is available forrecording data length of 10Mb Then the rotor speed andarmature current can be deciphered from the measured data

52 Frequency Identification Results When the power supplyis set at 23V the rotary encoder wasmounted onto the BLDCmotor rotor and the armature current wire passes throughthe hall current sensor The encoder and the hall sensorrsquosoutputs are recorded simultaneously during the experimentwith sampling frequency 119891

119904= 100Hz The rotor speed and

armature are then calculated off-lineComplying with (8) the input signal shown in Figure 7

covers frequencies ranging from 001 to 10HzThis frequencysweep input signal is utilized to excite the dynamics of our

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120240

250

260

270

280

290

300

Time (s)

Roto

r spe

ed (r

ads

)

(a) Rotor speed response

0 20 40 60 80 100 120

0

05

1

15

2

25

3

Time (s)

Arm

atur

e cur

rent

(A)

minus05

(b) Armature current response

Figure 8 Rotor speed and armature current responses

0

10

20

Frequency (rads)

Mag

nitu

de (d

B)

0

Frequency (rads)

Phas

e (de

g)

Experimental dataTransfer function fit

0

05

1

Frequency (rads)

Coh

eren

ce

100 101

100 101

100 101

minus150

minus100

minus50

(a) Rotor speed channel

Frequency (rads)

0

50

Frequency (rads)

Experimental dataTransfer function fit

Mag

nitu

de (d

B)Ph

ase (

deg)

Coh

eren

ce

0

05

1

Frequency (rads)

10minus1 100 101

100 101

102

100 101 102

minus100

minus50

minus15

minus10

minus5

(b) Armature channel

Figure 9 Rotor speed channelarmature channel identification

plant over a broad frequency range Figure 8 shows the rotorspeed and armature current extracted from experimentaldata Note that rotor speed response amplitude diminishesat higher frequencies as expected from the bandwidth ofthe corresponding channel However the armature currentresponse amplitude will increase with the even very highfrequency suggesting a very high bandwidth of the armaturecurrent channel which is considerably over the max sweepfrequency

The data obtained from this test will be applied toextract frequency responses used for themodel identification

The estimated frequency responses are calculated using spec-tral density functions with CIFER Figure 9 shows frequencyidentification of rotor speed channel and armature channelGood coherence is achieved in the required frequency rangeindicating good quality experimental data and the linearmodel assumption With respect to coherence rotor speedchannel identification result is better than that of currentchannel because the hall current sensor yields a pooraccuracy compared to rotary encoder The identified transferfunctions represented by red dashed lines show good fits ofthe magnitude and phase responses of each channel

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12900

950

1000

1050

1100

1150

1200

1250

Time (s)

PWM

wav

e dut

y cy

cle

Figure 10 Input doublet signal

Table 2 Estimated BLDC motor parameters

System parameters Symbol Value UnitRotor inertia 119869 000061 kgsdotm2

Voltage constant 119870119890

00637 V(rads)Torque constant 119870

11990502663 NmA

Armature resistance 119877119886

06187 Ω

Armature inductance 119871119886

262 mHFriction coefficient 119870

119891000291 N(rads)

Time constant of the ESC model 120591 00781 s

The two transfer functions of two channels based on (13)and (14) identified are

119867120596(119904) =

2057342

1199043 + 18951199042 + 13412119904 + 142834

119867119894(119904) =

21749 (02015119904 + 1)

1199043 + 1961199042 + 14695119904 + 149431

(29)

The identified transfer functions are taken into account todetermine BLDCmotormechanismparameters as suggestedin Section 3 Then the problem in (17) and (18) is solvedby Matlab optimization toolbox The bias between theoret-ical model and identification result is 041 The estimatedBLDC motor parameters are listed in Table 2 Note thatthe identified rotor inertia 119869 = 000061 is very closeto the evaluated value in Table 1 indicating a satisfactoryidentification accuracy of this parameter

The accuracy of the comprehensive BLDC motor modelmust be further verified by tine history data The inputdoublet signal shown in Figure 10 is used to stimulate theBLDC motor dynamics The validation results are shownin Figure 11 Both rotor speed and armature current aremeasured during the input doublet signal It is shown that theidentifiedmodel perfectly predicts the rotor speed responseshowever the true armature current response exhibits a slightdeviation from the responses predicted by the identifiedmodel This mismatch can be explained by the relatively lowcoherence in high frequency of the armature channel identifi-cationThe close agreement between the simulation response

Table 3 Estimation results for the mechanical parameters

System parameters 119909 (mean values) 120590 (standard deviations)1198791198881

293 451198791198882

527 511198791198883

3706 4461198791198884

156 411198791198885

843 2241198791198886

5612 433119879119886

1103 125119879119887

1620 332

and experimental data indicates the identified model isaccurate enough to reflect the BLDC motorrsquos dynamics

53 Cogging Torque Coefficients Identification Results Sincethe cogging torque effect exists significantly in low speed theterminal voltage of the motor is set at 65 V The disturbanceobserver is applied to calculate cogging torque as stated inSection 42 In order to eliminate the stochastic error theexperiment is repeated by 10 repetitions with single testtime of 20 s The minimum least squares error obtained by(26) versus the number of cogging torque harmonics areshown in Figure 12 with the estimated disturbance signalapproximated using LSE It is obvious that the sixth coggingtorque harmonic is highly dominant from the diagram and aslight of third harmonics is also present [17] has pointed outthat the cogging torque harmonics for three-phase motorsusually are multiples of six times the fundamental commuta-tion frequency

A simulation of the identified model is performed inorder to verify the results of the proposed parameters iden-tification method The cogging torque coefficients in Table 3and the estimated linear terms by frequency identification inSection 52 are utilized in the comprehensive BLDC motormodel A comparison betweenmeasured and estimated rotorspeed is formed to highlight the verification result (seeFigure 13) Note that there is a small phase deviation ofbetween experimental and estimated cogging torque thiscould be interpreted by the unmodeled dynamics Matchingof the simulated and experimental data is ensured Hence theproposed identification method is applicable and sufficient

6 Conclusion

In this paper a comprehensive procedure for system iden-tification applicable to BLDC motor used in UAV has beensystematically presented The procedure aims to serve asa guide for the steps necessary to determine an accurateBLDC motor model Considering the nonlinear character-istics cogging torque effect of the BLDC motor system apractical mathematical model is introduced for identifica-tion Preliminarily frequency domain identification theoryis applied to estimate the first seven parameters in linearterms of the model at a high motor speed region where thenonlinear term can be neglected Then an off-line methodfor identifying the cogging torque coefficients is performed at

10 Mathematical Problems in Engineering

0 2 4 6 8 10 12240

260

280

300

320

Time (s)

Roto

r spe

ed (r

ads

)

Experimental dataBLDC motor model

(a) Rotor speed validation of the BLDC motor dynamics

2 4 6 8 10 12

1

2

3

4

5

Time (s)

Arm

atur

e cur

rent

(A)

Experimental dataBLDC motor model

(b) Armature current validation of the BLDC motor dynamics

Figure 11 Time domain validation of the BLDC motor identified dynamics

2 3 4 5 6 7 8 90

10

20

30

Number of harmonics (K)

J min

Figure 12 The minimum least squares error versus the number of cogging torque harmonics

05 1 15 275

80

85

90

95

100

Time (s)

Nonlinear model Experimental data

02 025 03

90

92

94

96

Time (s)

Nonlinear model Experimental data

Roto

r spe

ed (r

ads

)

Roto

r spe

ed (r

ads

)

Figure 13 Validation of identified comprehensive BLDC motor model

Mathematical Problems in Engineering 11

very low speedwhen cogging torque effect exists significantlyThis method is founded on the least squares approximationmethod a disturbance observer and the identified parametersin frequency identification The results from the simulationand the real-time experiments matched closely which isconsidered as a validation of the obtained mathematicalmodel

The BLDC model identifications are conditional onlyon the availability of experimental data for rotor speed andarmature current therefore our method without sensors todetect torque or back-EMF is easy to realize and is practicaland effectiveThis research will provide greater opportunitiesfor future controller design of the BLDCmotorThe extractedlinear model can be utilized directly for linear controllerdesign and the accurately identified cogging torque effect isbeneficial to reduce the speed ripple of the motor at some lowspeed conditions using some advanced control theory suchas robust control and nonlinear control

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Shanmugasundram K M Zakaraiah and N YadaiahldquoModeling simulation and analysis of controllers for brushlessdirect current motor drivesrdquo Journal of Vibration and Controlvol 19 no 8 pp 1250ndash1264 2013

[2] C Xia Y Wang and T Shi ldquoImplementation of finite-statemodel predictive control for commutation torque ripple min-imization of permanent-magnet brushless DC motorrdquo IEEETransactions on Industrial Electronics vol 60 no 3 pp 896ndash905 2013

[3] S-Y Chang P-H Yu and W-C Su ldquoPre- and post-filteringapproach to sensorless VSS speed control of a permanentmagnet DC motor subject to Hammerstein and Wiener non-linearitiesrdquoControl Engineering Practice vol 21 no 11 pp 1577ndash1583 2013

[4] B-K Lee and M Ehsani ldquoAdvanced simulation model forbrushless dc motor drivesrdquo Electric Power Components andSystems vol 31 no 9 pp 841ndash868 2003

[5] I Kim N Nakazawa S Kim C Park and C Yu ldquoCompensa-tion of torque ripple in high performance BLDCmotor drivesrdquoControl Engineering Practice vol 18 no 10 pp 1166ndash1172 2010

[6] H Lu L Zhang and W Qu ldquoA new torque control method fortorque rippleminimization of BLDCmotors with un-ideal backEMFrdquo IEEE Transactions on Power Electronics vol 23 no 2 pp950ndash958 2008

[7] W Wu ldquoDC motor parameter identification using speed stepresponsesrdquo Modelling and Simulation in Engineering vol 2012Article ID 189757 5 pages 2012

[8] J-H Choi S-H You J Hur and H-G Sung ldquoThe design andfabrication of BLDCMotor andDrive for 42V automotive appli-cationsrdquo in Proceedings of the IEEE International Symposium onIndustrial Electronics (ISIE rsquo07) pp 1086ndash1091 June 2007

[9] A Kapun M Curkovic A Hace and K Jezernik ldquoIdentifyingdynamic model parameters of a BLDC motorrdquo Simulation

Modelling Practice and Theory vol 16 no 9 pp 1254ndash12652008

[10] A J Blauch M Bodson and J Chiasson ldquoHigh-Speed parame-ter estimation of stepper motorsrdquo IEEE Transactions on ControlSystems Technology vol 1 no 4 pp 270ndash279 1993

[11] R Patankar and L Zhu ldquoReal-time multiple parameter estima-tion for voltage controlled brushless DC motor actuatorsrdquo inProceedings of the American Control Conference (AAC rsquo04) vol4 pp 3851ndash3856 July 2004

[12] M S Islam R Islam T Sebastian A Chandy and S AOzsoyluldquoCogging torque minimization in PM motors using robustdesign approachrdquo IEEE Transactions on Industry Applicationsvol 47 no 4 pp 1661ndash1669 2011

[13] M B Tischler and R K Remple Aircraft and Rotorcraft SystemIdentification AIAA Education Series American Institute ofAeronautics and Astronautics 2006

[14] M Petra and M Utting CZT A Framework for Z Tools FormalSpecification and Development in Z and B Springer BerlinGermany 2005

[15] K Zhou and C D John Essentials of Robust Control PrenticeHall Upper Saddle River NJ USA 1998

[16] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Higher Education New York NY USA 2010

[17] J Y Hung and Z Ding ldquoDesign currents to reduce torque ripplein brushless permanent magnet motorsrdquo IEE Proceedings BElectric Power Applications vol 140 no 4 pp 260ndash266 1993

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

BLDC motorESC

Hall sensor

Rotary encoder

Digital oscilloscope

Stabilized voltage supply

Control signal generator

PWM generator

Figure 6 BLDC motor identification experimental bench setup

Table 1 BLDC motor mechanical parameters

Description Value UnitNumber of stator arms 12 Magnet poles 14 Max continuous power 3900 WRotor inertia (evaluated) 0000549 kgsdotm2

Rotary encoder inertia 000001 kgsdotm2

5 System Identification Experiment

System identification experiments are designed and carriedout to obtain data for comprehensive identification Thissection is divided into three parts First we will detail theexperimental plant in BIT Then frequency identificationresults and system main parameter determination will bedescribed Finallywewill discuss nonlinear identification andestimate cogging torque coefficients

51 Experimental Setup The experimental bench setup isshown in Figure 6 including BLDC motor driving unitcontrol unit and measure unit Each of the subsystem willbe briefly introduced by the following subsystems of theexperimental setup

511 BLDC Motor The motor to be identified is popularBLDC motor for UAVs scorpion SII-6530 Considering thesimple configuration of this motor the rotor inertia can beevaluated by Solidworks which is important for the valida-tion of the identification results Somemotor parameters andthe evaluated inertia of the rotary encoder and coupling arelisted in Table 1

512 Driving Unit The driving unit consists of a ESC and aPWM generator The ESC produced by Hobyywing similarto a motor controller and three-phase bridge receives thesignal produced by the PWM generator A stabilized voltagesource powers the ESC to drive the motor The PWMgenerator actually a servo controller is mainly responsible fordecoding control commands and generating PWM wave

0 20 40 60 80 100 1201360

1380

1400

1420

1440

Time (s)

PWM

wav

e dut

y cy

cle

Figure 7 Automatic sweep input signal ranging from 001sim10Hz

513 Control Unit The control unit is an ordinary PCinstalled by Linux operating systemThePCandPWMgener-ator are connected by RS-232 serial ports All the designedcontrol commands are sent from control unit to the PWMgenerator

514 Measure Unit In order to measure these two systemstates rotor speed and armature current a rotary encoder andhall current sensor are respectively included inmeasure unitBoth of output signal of these two sensors are recorded by thedigital oscilloscope in high sample rates The measurementsampling time of the digital oscilloscope could be set from1 120583s up to 50120583s and the memory amount is available forrecording data length of 10Mb Then the rotor speed andarmature current can be deciphered from the measured data

52 Frequency Identification Results When the power supplyis set at 23V the rotary encoder wasmounted onto the BLDCmotor rotor and the armature current wire passes throughthe hall current sensor The encoder and the hall sensorrsquosoutputs are recorded simultaneously during the experimentwith sampling frequency 119891

119904= 100Hz The rotor speed and

armature are then calculated off-lineComplying with (8) the input signal shown in Figure 7

covers frequencies ranging from 001 to 10HzThis frequencysweep input signal is utilized to excite the dynamics of our

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120240

250

260

270

280

290

300

Time (s)

Roto

r spe

ed (r

ads

)

(a) Rotor speed response

0 20 40 60 80 100 120

0

05

1

15

2

25

3

Time (s)

Arm

atur

e cur

rent

(A)

minus05

(b) Armature current response

Figure 8 Rotor speed and armature current responses

0

10

20

Frequency (rads)

Mag

nitu

de (d

B)

0

Frequency (rads)

Phas

e (de

g)

Experimental dataTransfer function fit

0

05

1

Frequency (rads)

Coh

eren

ce

100 101

100 101

100 101

minus150

minus100

minus50

(a) Rotor speed channel

Frequency (rads)

0

50

Frequency (rads)

Experimental dataTransfer function fit

Mag

nitu

de (d

B)Ph

ase (

deg)

Coh

eren

ce

0

05

1

Frequency (rads)

10minus1 100 101

100 101

102

100 101 102

minus100

minus50

minus15

minus10

minus5

(b) Armature channel

Figure 9 Rotor speed channelarmature channel identification

plant over a broad frequency range Figure 8 shows the rotorspeed and armature current extracted from experimentaldata Note that rotor speed response amplitude diminishesat higher frequencies as expected from the bandwidth ofthe corresponding channel However the armature currentresponse amplitude will increase with the even very highfrequency suggesting a very high bandwidth of the armaturecurrent channel which is considerably over the max sweepfrequency

The data obtained from this test will be applied toextract frequency responses used for themodel identification

The estimated frequency responses are calculated using spec-tral density functions with CIFER Figure 9 shows frequencyidentification of rotor speed channel and armature channelGood coherence is achieved in the required frequency rangeindicating good quality experimental data and the linearmodel assumption With respect to coherence rotor speedchannel identification result is better than that of currentchannel because the hall current sensor yields a pooraccuracy compared to rotary encoder The identified transferfunctions represented by red dashed lines show good fits ofthe magnitude and phase responses of each channel

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12900

950

1000

1050

1100

1150

1200

1250

Time (s)

PWM

wav

e dut

y cy

cle

Figure 10 Input doublet signal

Table 2 Estimated BLDC motor parameters

System parameters Symbol Value UnitRotor inertia 119869 000061 kgsdotm2

Voltage constant 119870119890

00637 V(rads)Torque constant 119870

11990502663 NmA

Armature resistance 119877119886

06187 Ω

Armature inductance 119871119886

262 mHFriction coefficient 119870

119891000291 N(rads)

Time constant of the ESC model 120591 00781 s

The two transfer functions of two channels based on (13)and (14) identified are

119867120596(119904) =

2057342

1199043 + 18951199042 + 13412119904 + 142834

119867119894(119904) =

21749 (02015119904 + 1)

1199043 + 1961199042 + 14695119904 + 149431

(29)

The identified transfer functions are taken into account todetermine BLDCmotormechanismparameters as suggestedin Section 3 Then the problem in (17) and (18) is solvedby Matlab optimization toolbox The bias between theoret-ical model and identification result is 041 The estimatedBLDC motor parameters are listed in Table 2 Note thatthe identified rotor inertia 119869 = 000061 is very closeto the evaluated value in Table 1 indicating a satisfactoryidentification accuracy of this parameter

The accuracy of the comprehensive BLDC motor modelmust be further verified by tine history data The inputdoublet signal shown in Figure 10 is used to stimulate theBLDC motor dynamics The validation results are shownin Figure 11 Both rotor speed and armature current aremeasured during the input doublet signal It is shown that theidentifiedmodel perfectly predicts the rotor speed responseshowever the true armature current response exhibits a slightdeviation from the responses predicted by the identifiedmodel This mismatch can be explained by the relatively lowcoherence in high frequency of the armature channel identifi-cationThe close agreement between the simulation response

Table 3 Estimation results for the mechanical parameters

System parameters 119909 (mean values) 120590 (standard deviations)1198791198881

293 451198791198882

527 511198791198883

3706 4461198791198884

156 411198791198885

843 2241198791198886

5612 433119879119886

1103 125119879119887

1620 332

and experimental data indicates the identified model isaccurate enough to reflect the BLDC motorrsquos dynamics

53 Cogging Torque Coefficients Identification Results Sincethe cogging torque effect exists significantly in low speed theterminal voltage of the motor is set at 65 V The disturbanceobserver is applied to calculate cogging torque as stated inSection 42 In order to eliminate the stochastic error theexperiment is repeated by 10 repetitions with single testtime of 20 s The minimum least squares error obtained by(26) versus the number of cogging torque harmonics areshown in Figure 12 with the estimated disturbance signalapproximated using LSE It is obvious that the sixth coggingtorque harmonic is highly dominant from the diagram and aslight of third harmonics is also present [17] has pointed outthat the cogging torque harmonics for three-phase motorsusually are multiples of six times the fundamental commuta-tion frequency

A simulation of the identified model is performed inorder to verify the results of the proposed parameters iden-tification method The cogging torque coefficients in Table 3and the estimated linear terms by frequency identification inSection 52 are utilized in the comprehensive BLDC motormodel A comparison betweenmeasured and estimated rotorspeed is formed to highlight the verification result (seeFigure 13) Note that there is a small phase deviation ofbetween experimental and estimated cogging torque thiscould be interpreted by the unmodeled dynamics Matchingof the simulated and experimental data is ensured Hence theproposed identification method is applicable and sufficient

6 Conclusion

In this paper a comprehensive procedure for system iden-tification applicable to BLDC motor used in UAV has beensystematically presented The procedure aims to serve asa guide for the steps necessary to determine an accurateBLDC motor model Considering the nonlinear character-istics cogging torque effect of the BLDC motor system apractical mathematical model is introduced for identifica-tion Preliminarily frequency domain identification theoryis applied to estimate the first seven parameters in linearterms of the model at a high motor speed region where thenonlinear term can be neglected Then an off-line methodfor identifying the cogging torque coefficients is performed at

10 Mathematical Problems in Engineering

0 2 4 6 8 10 12240

260

280

300

320

Time (s)

Roto

r spe

ed (r

ads

)

Experimental dataBLDC motor model

(a) Rotor speed validation of the BLDC motor dynamics

2 4 6 8 10 12

1

2

3

4

5

Time (s)

Arm

atur

e cur

rent

(A)

Experimental dataBLDC motor model

(b) Armature current validation of the BLDC motor dynamics

Figure 11 Time domain validation of the BLDC motor identified dynamics

2 3 4 5 6 7 8 90

10

20

30

Number of harmonics (K)

J min

Figure 12 The minimum least squares error versus the number of cogging torque harmonics

05 1 15 275

80

85

90

95

100

Time (s)

Nonlinear model Experimental data

02 025 03

90

92

94

96

Time (s)

Nonlinear model Experimental data

Roto

r spe

ed (r

ads

)

Roto

r spe

ed (r

ads

)

Figure 13 Validation of identified comprehensive BLDC motor model

Mathematical Problems in Engineering 11

very low speedwhen cogging torque effect exists significantlyThis method is founded on the least squares approximationmethod a disturbance observer and the identified parametersin frequency identification The results from the simulationand the real-time experiments matched closely which isconsidered as a validation of the obtained mathematicalmodel

The BLDC model identifications are conditional onlyon the availability of experimental data for rotor speed andarmature current therefore our method without sensors todetect torque or back-EMF is easy to realize and is practicaland effectiveThis research will provide greater opportunitiesfor future controller design of the BLDCmotorThe extractedlinear model can be utilized directly for linear controllerdesign and the accurately identified cogging torque effect isbeneficial to reduce the speed ripple of the motor at some lowspeed conditions using some advanced control theory suchas robust control and nonlinear control

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Shanmugasundram K M Zakaraiah and N YadaiahldquoModeling simulation and analysis of controllers for brushlessdirect current motor drivesrdquo Journal of Vibration and Controlvol 19 no 8 pp 1250ndash1264 2013

[2] C Xia Y Wang and T Shi ldquoImplementation of finite-statemodel predictive control for commutation torque ripple min-imization of permanent-magnet brushless DC motorrdquo IEEETransactions on Industrial Electronics vol 60 no 3 pp 896ndash905 2013

[3] S-Y Chang P-H Yu and W-C Su ldquoPre- and post-filteringapproach to sensorless VSS speed control of a permanentmagnet DC motor subject to Hammerstein and Wiener non-linearitiesrdquoControl Engineering Practice vol 21 no 11 pp 1577ndash1583 2013

[4] B-K Lee and M Ehsani ldquoAdvanced simulation model forbrushless dc motor drivesrdquo Electric Power Components andSystems vol 31 no 9 pp 841ndash868 2003

[5] I Kim N Nakazawa S Kim C Park and C Yu ldquoCompensa-tion of torque ripple in high performance BLDCmotor drivesrdquoControl Engineering Practice vol 18 no 10 pp 1166ndash1172 2010

[6] H Lu L Zhang and W Qu ldquoA new torque control method fortorque rippleminimization of BLDCmotors with un-ideal backEMFrdquo IEEE Transactions on Power Electronics vol 23 no 2 pp950ndash958 2008

[7] W Wu ldquoDC motor parameter identification using speed stepresponsesrdquo Modelling and Simulation in Engineering vol 2012Article ID 189757 5 pages 2012

[8] J-H Choi S-H You J Hur and H-G Sung ldquoThe design andfabrication of BLDCMotor andDrive for 42V automotive appli-cationsrdquo in Proceedings of the IEEE International Symposium onIndustrial Electronics (ISIE rsquo07) pp 1086ndash1091 June 2007

[9] A Kapun M Curkovic A Hace and K Jezernik ldquoIdentifyingdynamic model parameters of a BLDC motorrdquo Simulation

Modelling Practice and Theory vol 16 no 9 pp 1254ndash12652008

[10] A J Blauch M Bodson and J Chiasson ldquoHigh-Speed parame-ter estimation of stepper motorsrdquo IEEE Transactions on ControlSystems Technology vol 1 no 4 pp 270ndash279 1993

[11] R Patankar and L Zhu ldquoReal-time multiple parameter estima-tion for voltage controlled brushless DC motor actuatorsrdquo inProceedings of the American Control Conference (AAC rsquo04) vol4 pp 3851ndash3856 July 2004

[12] M S Islam R Islam T Sebastian A Chandy and S AOzsoyluldquoCogging torque minimization in PM motors using robustdesign approachrdquo IEEE Transactions on Industry Applicationsvol 47 no 4 pp 1661ndash1669 2011

[13] M B Tischler and R K Remple Aircraft and Rotorcraft SystemIdentification AIAA Education Series American Institute ofAeronautics and Astronautics 2006

[14] M Petra and M Utting CZT A Framework for Z Tools FormalSpecification and Development in Z and B Springer BerlinGermany 2005

[15] K Zhou and C D John Essentials of Robust Control PrenticeHall Upper Saddle River NJ USA 1998

[16] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Higher Education New York NY USA 2010

[17] J Y Hung and Z Ding ldquoDesign currents to reduce torque ripplein brushless permanent magnet motorsrdquo IEE Proceedings BElectric Power Applications vol 140 no 4 pp 260ndash266 1993

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

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120240

250

260

270

280

290

300

Time (s)

Roto

r spe

ed (r

ads

)

(a) Rotor speed response

0 20 40 60 80 100 120

0

05

1

15

2

25

3

Time (s)

Arm

atur

e cur

rent

(A)

minus05

(b) Armature current response

Figure 8 Rotor speed and armature current responses

0

10

20

Frequency (rads)

Mag

nitu

de (d

B)

0

Frequency (rads)

Phas

e (de

g)

Experimental dataTransfer function fit

0

05

1

Frequency (rads)

Coh

eren

ce

100 101

100 101

100 101

minus150

minus100

minus50

(a) Rotor speed channel

Frequency (rads)

0

50

Frequency (rads)

Experimental dataTransfer function fit

Mag

nitu

de (d

B)Ph

ase (

deg)

Coh

eren

ce

0

05

1

Frequency (rads)

10minus1 100 101

100 101

102

100 101 102

minus100

minus50

minus15

minus10

minus5

(b) Armature channel

Figure 9 Rotor speed channelarmature channel identification

plant over a broad frequency range Figure 8 shows the rotorspeed and armature current extracted from experimentaldata Note that rotor speed response amplitude diminishesat higher frequencies as expected from the bandwidth ofthe corresponding channel However the armature currentresponse amplitude will increase with the even very highfrequency suggesting a very high bandwidth of the armaturecurrent channel which is considerably over the max sweepfrequency

The data obtained from this test will be applied toextract frequency responses used for themodel identification

The estimated frequency responses are calculated using spec-tral density functions with CIFER Figure 9 shows frequencyidentification of rotor speed channel and armature channelGood coherence is achieved in the required frequency rangeindicating good quality experimental data and the linearmodel assumption With respect to coherence rotor speedchannel identification result is better than that of currentchannel because the hall current sensor yields a pooraccuracy compared to rotary encoder The identified transferfunctions represented by red dashed lines show good fits ofthe magnitude and phase responses of each channel

Mathematical Problems in Engineering 9

0 2 4 6 8 10 12900

950

1000

1050

1100

1150

1200

1250

Time (s)

PWM

wav

e dut

y cy

cle

Figure 10 Input doublet signal

Table 2 Estimated BLDC motor parameters

System parameters Symbol Value UnitRotor inertia 119869 000061 kgsdotm2

Voltage constant 119870119890

00637 V(rads)Torque constant 119870

11990502663 NmA

Armature resistance 119877119886

06187 Ω

Armature inductance 119871119886

262 mHFriction coefficient 119870

119891000291 N(rads)

Time constant of the ESC model 120591 00781 s

The two transfer functions of two channels based on (13)and (14) identified are

119867120596(119904) =

2057342

1199043 + 18951199042 + 13412119904 + 142834

119867119894(119904) =

21749 (02015119904 + 1)

1199043 + 1961199042 + 14695119904 + 149431

(29)

The identified transfer functions are taken into account todetermine BLDCmotormechanismparameters as suggestedin Section 3 Then the problem in (17) and (18) is solvedby Matlab optimization toolbox The bias between theoret-ical model and identification result is 041 The estimatedBLDC motor parameters are listed in Table 2 Note thatthe identified rotor inertia 119869 = 000061 is very closeto the evaluated value in Table 1 indicating a satisfactoryidentification accuracy of this parameter

The accuracy of the comprehensive BLDC motor modelmust be further verified by tine history data The inputdoublet signal shown in Figure 10 is used to stimulate theBLDC motor dynamics The validation results are shownin Figure 11 Both rotor speed and armature current aremeasured during the input doublet signal It is shown that theidentifiedmodel perfectly predicts the rotor speed responseshowever the true armature current response exhibits a slightdeviation from the responses predicted by the identifiedmodel This mismatch can be explained by the relatively lowcoherence in high frequency of the armature channel identifi-cationThe close agreement between the simulation response

Table 3 Estimation results for the mechanical parameters

System parameters 119909 (mean values) 120590 (standard deviations)1198791198881

293 451198791198882

527 511198791198883

3706 4461198791198884

156 411198791198885

843 2241198791198886

5612 433119879119886

1103 125119879119887

1620 332

and experimental data indicates the identified model isaccurate enough to reflect the BLDC motorrsquos dynamics

53 Cogging Torque Coefficients Identification Results Sincethe cogging torque effect exists significantly in low speed theterminal voltage of the motor is set at 65 V The disturbanceobserver is applied to calculate cogging torque as stated inSection 42 In order to eliminate the stochastic error theexperiment is repeated by 10 repetitions with single testtime of 20 s The minimum least squares error obtained by(26) versus the number of cogging torque harmonics areshown in Figure 12 with the estimated disturbance signalapproximated using LSE It is obvious that the sixth coggingtorque harmonic is highly dominant from the diagram and aslight of third harmonics is also present [17] has pointed outthat the cogging torque harmonics for three-phase motorsusually are multiples of six times the fundamental commuta-tion frequency

A simulation of the identified model is performed inorder to verify the results of the proposed parameters iden-tification method The cogging torque coefficients in Table 3and the estimated linear terms by frequency identification inSection 52 are utilized in the comprehensive BLDC motormodel A comparison betweenmeasured and estimated rotorspeed is formed to highlight the verification result (seeFigure 13) Note that there is a small phase deviation ofbetween experimental and estimated cogging torque thiscould be interpreted by the unmodeled dynamics Matchingof the simulated and experimental data is ensured Hence theproposed identification method is applicable and sufficient

6 Conclusion

In this paper a comprehensive procedure for system iden-tification applicable to BLDC motor used in UAV has beensystematically presented The procedure aims to serve asa guide for the steps necessary to determine an accurateBLDC motor model Considering the nonlinear character-istics cogging torque effect of the BLDC motor system apractical mathematical model is introduced for identifica-tion Preliminarily frequency domain identification theoryis applied to estimate the first seven parameters in linearterms of the model at a high motor speed region where thenonlinear term can be neglected Then an off-line methodfor identifying the cogging torque coefficients is performed at

10 Mathematical Problems in Engineering

0 2 4 6 8 10 12240

260

280

300

320

Time (s)

Roto

r spe

ed (r

ads

)

Experimental dataBLDC motor model

(a) Rotor speed validation of the BLDC motor dynamics

2 4 6 8 10 12

1

2

3

4

5

Time (s)

Arm

atur

e cur

rent

(A)

Experimental dataBLDC motor model

(b) Armature current validation of the BLDC motor dynamics

Figure 11 Time domain validation of the BLDC motor identified dynamics

2 3 4 5 6 7 8 90

10

20

30

Number of harmonics (K)

J min

Figure 12 The minimum least squares error versus the number of cogging torque harmonics

05 1 15 275

80

85

90

95

100

Time (s)

Nonlinear model Experimental data

02 025 03

90

92

94

96

Time (s)

Nonlinear model Experimental data

Roto

r spe

ed (r

ads

)

Roto

r spe

ed (r

ads

)

Figure 13 Validation of identified comprehensive BLDC motor model

Mathematical Problems in Engineering 11

very low speedwhen cogging torque effect exists significantlyThis method is founded on the least squares approximationmethod a disturbance observer and the identified parametersin frequency identification The results from the simulationand the real-time experiments matched closely which isconsidered as a validation of the obtained mathematicalmodel

The BLDC model identifications are conditional onlyon the availability of experimental data for rotor speed andarmature current therefore our method without sensors todetect torque or back-EMF is easy to realize and is practicaland effectiveThis research will provide greater opportunitiesfor future controller design of the BLDCmotorThe extractedlinear model can be utilized directly for linear controllerdesign and the accurately identified cogging torque effect isbeneficial to reduce the speed ripple of the motor at some lowspeed conditions using some advanced control theory suchas robust control and nonlinear control

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Shanmugasundram K M Zakaraiah and N YadaiahldquoModeling simulation and analysis of controllers for brushlessdirect current motor drivesrdquo Journal of Vibration and Controlvol 19 no 8 pp 1250ndash1264 2013

[2] C Xia Y Wang and T Shi ldquoImplementation of finite-statemodel predictive control for commutation torque ripple min-imization of permanent-magnet brushless DC motorrdquo IEEETransactions on Industrial Electronics vol 60 no 3 pp 896ndash905 2013

[3] S-Y Chang P-H Yu and W-C Su ldquoPre- and post-filteringapproach to sensorless VSS speed control of a permanentmagnet DC motor subject to Hammerstein and Wiener non-linearitiesrdquoControl Engineering Practice vol 21 no 11 pp 1577ndash1583 2013

[4] B-K Lee and M Ehsani ldquoAdvanced simulation model forbrushless dc motor drivesrdquo Electric Power Components andSystems vol 31 no 9 pp 841ndash868 2003

[5] I Kim N Nakazawa S Kim C Park and C Yu ldquoCompensa-tion of torque ripple in high performance BLDCmotor drivesrdquoControl Engineering Practice vol 18 no 10 pp 1166ndash1172 2010

[6] H Lu L Zhang and W Qu ldquoA new torque control method fortorque rippleminimization of BLDCmotors with un-ideal backEMFrdquo IEEE Transactions on Power Electronics vol 23 no 2 pp950ndash958 2008

[7] W Wu ldquoDC motor parameter identification using speed stepresponsesrdquo Modelling and Simulation in Engineering vol 2012Article ID 189757 5 pages 2012

[8] J-H Choi S-H You J Hur and H-G Sung ldquoThe design andfabrication of BLDCMotor andDrive for 42V automotive appli-cationsrdquo in Proceedings of the IEEE International Symposium onIndustrial Electronics (ISIE rsquo07) pp 1086ndash1091 June 2007

[9] A Kapun M Curkovic A Hace and K Jezernik ldquoIdentifyingdynamic model parameters of a BLDC motorrdquo Simulation

Modelling Practice and Theory vol 16 no 9 pp 1254ndash12652008

[10] A J Blauch M Bodson and J Chiasson ldquoHigh-Speed parame-ter estimation of stepper motorsrdquo IEEE Transactions on ControlSystems Technology vol 1 no 4 pp 270ndash279 1993

[11] R Patankar and L Zhu ldquoReal-time multiple parameter estima-tion for voltage controlled brushless DC motor actuatorsrdquo inProceedings of the American Control Conference (AAC rsquo04) vol4 pp 3851ndash3856 July 2004

[12] M S Islam R Islam T Sebastian A Chandy and S AOzsoyluldquoCogging torque minimization in PM motors using robustdesign approachrdquo IEEE Transactions on Industry Applicationsvol 47 no 4 pp 1661ndash1669 2011

[13] M B Tischler and R K Remple Aircraft and Rotorcraft SystemIdentification AIAA Education Series American Institute ofAeronautics and Astronautics 2006

[14] M Petra and M Utting CZT A Framework for Z Tools FormalSpecification and Development in Z and B Springer BerlinGermany 2005

[15] K Zhou and C D John Essentials of Robust Control PrenticeHall Upper Saddle River NJ USA 1998

[16] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Higher Education New York NY USA 2010

[17] J Y Hung and Z Ding ldquoDesign currents to reduce torque ripplein brushless permanent magnet motorsrdquo IEE Proceedings BElectric Power Applications vol 140 no 4 pp 260ndash266 1993

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 9

0 2 4 6 8 10 12900

950

1000

1050

1100

1150

1200

1250

Time (s)

PWM

wav

e dut

y cy

cle

Figure 10 Input doublet signal

Table 2 Estimated BLDC motor parameters

System parameters Symbol Value UnitRotor inertia 119869 000061 kgsdotm2

Voltage constant 119870119890

00637 V(rads)Torque constant 119870

11990502663 NmA

Armature resistance 119877119886

06187 Ω

Armature inductance 119871119886

262 mHFriction coefficient 119870

119891000291 N(rads)

Time constant of the ESC model 120591 00781 s

The two transfer functions of two channels based on (13)and (14) identified are

119867120596(119904) =

2057342

1199043 + 18951199042 + 13412119904 + 142834

119867119894(119904) =

21749 (02015119904 + 1)

1199043 + 1961199042 + 14695119904 + 149431

(29)

The identified transfer functions are taken into account todetermine BLDCmotormechanismparameters as suggestedin Section 3 Then the problem in (17) and (18) is solvedby Matlab optimization toolbox The bias between theoret-ical model and identification result is 041 The estimatedBLDC motor parameters are listed in Table 2 Note thatthe identified rotor inertia 119869 = 000061 is very closeto the evaluated value in Table 1 indicating a satisfactoryidentification accuracy of this parameter

The accuracy of the comprehensive BLDC motor modelmust be further verified by tine history data The inputdoublet signal shown in Figure 10 is used to stimulate theBLDC motor dynamics The validation results are shownin Figure 11 Both rotor speed and armature current aremeasured during the input doublet signal It is shown that theidentifiedmodel perfectly predicts the rotor speed responseshowever the true armature current response exhibits a slightdeviation from the responses predicted by the identifiedmodel This mismatch can be explained by the relatively lowcoherence in high frequency of the armature channel identifi-cationThe close agreement between the simulation response

Table 3 Estimation results for the mechanical parameters

System parameters 119909 (mean values) 120590 (standard deviations)1198791198881

293 451198791198882

527 511198791198883

3706 4461198791198884

156 411198791198885

843 2241198791198886

5612 433119879119886

1103 125119879119887

1620 332

and experimental data indicates the identified model isaccurate enough to reflect the BLDC motorrsquos dynamics

53 Cogging Torque Coefficients Identification Results Sincethe cogging torque effect exists significantly in low speed theterminal voltage of the motor is set at 65 V The disturbanceobserver is applied to calculate cogging torque as stated inSection 42 In order to eliminate the stochastic error theexperiment is repeated by 10 repetitions with single testtime of 20 s The minimum least squares error obtained by(26) versus the number of cogging torque harmonics areshown in Figure 12 with the estimated disturbance signalapproximated using LSE It is obvious that the sixth coggingtorque harmonic is highly dominant from the diagram and aslight of third harmonics is also present [17] has pointed outthat the cogging torque harmonics for three-phase motorsusually are multiples of six times the fundamental commuta-tion frequency

A simulation of the identified model is performed inorder to verify the results of the proposed parameters iden-tification method The cogging torque coefficients in Table 3and the estimated linear terms by frequency identification inSection 52 are utilized in the comprehensive BLDC motormodel A comparison betweenmeasured and estimated rotorspeed is formed to highlight the verification result (seeFigure 13) Note that there is a small phase deviation ofbetween experimental and estimated cogging torque thiscould be interpreted by the unmodeled dynamics Matchingof the simulated and experimental data is ensured Hence theproposed identification method is applicable and sufficient

6 Conclusion

In this paper a comprehensive procedure for system iden-tification applicable to BLDC motor used in UAV has beensystematically presented The procedure aims to serve asa guide for the steps necessary to determine an accurateBLDC motor model Considering the nonlinear character-istics cogging torque effect of the BLDC motor system apractical mathematical model is introduced for identifica-tion Preliminarily frequency domain identification theoryis applied to estimate the first seven parameters in linearterms of the model at a high motor speed region where thenonlinear term can be neglected Then an off-line methodfor identifying the cogging torque coefficients is performed at

10 Mathematical Problems in Engineering

0 2 4 6 8 10 12240

260

280

300

320

Time (s)

Roto

r spe

ed (r

ads

)

Experimental dataBLDC motor model

(a) Rotor speed validation of the BLDC motor dynamics

2 4 6 8 10 12

1

2

3

4

5

Time (s)

Arm

atur

e cur

rent

(A)

Experimental dataBLDC motor model

(b) Armature current validation of the BLDC motor dynamics

Figure 11 Time domain validation of the BLDC motor identified dynamics

2 3 4 5 6 7 8 90

10

20

30

Number of harmonics (K)

J min

Figure 12 The minimum least squares error versus the number of cogging torque harmonics

05 1 15 275

80

85

90

95

100

Time (s)

Nonlinear model Experimental data

02 025 03

90

92

94

96

Time (s)

Nonlinear model Experimental data

Roto

r spe

ed (r

ads

)

Roto

r spe

ed (r

ads

)

Figure 13 Validation of identified comprehensive BLDC motor model

Mathematical Problems in Engineering 11

very low speedwhen cogging torque effect exists significantlyThis method is founded on the least squares approximationmethod a disturbance observer and the identified parametersin frequency identification The results from the simulationand the real-time experiments matched closely which isconsidered as a validation of the obtained mathematicalmodel

The BLDC model identifications are conditional onlyon the availability of experimental data for rotor speed andarmature current therefore our method without sensors todetect torque or back-EMF is easy to realize and is practicaland effectiveThis research will provide greater opportunitiesfor future controller design of the BLDCmotorThe extractedlinear model can be utilized directly for linear controllerdesign and the accurately identified cogging torque effect isbeneficial to reduce the speed ripple of the motor at some lowspeed conditions using some advanced control theory suchas robust control and nonlinear control

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Shanmugasundram K M Zakaraiah and N YadaiahldquoModeling simulation and analysis of controllers for brushlessdirect current motor drivesrdquo Journal of Vibration and Controlvol 19 no 8 pp 1250ndash1264 2013

[2] C Xia Y Wang and T Shi ldquoImplementation of finite-statemodel predictive control for commutation torque ripple min-imization of permanent-magnet brushless DC motorrdquo IEEETransactions on Industrial Electronics vol 60 no 3 pp 896ndash905 2013

[3] S-Y Chang P-H Yu and W-C Su ldquoPre- and post-filteringapproach to sensorless VSS speed control of a permanentmagnet DC motor subject to Hammerstein and Wiener non-linearitiesrdquoControl Engineering Practice vol 21 no 11 pp 1577ndash1583 2013

[4] B-K Lee and M Ehsani ldquoAdvanced simulation model forbrushless dc motor drivesrdquo Electric Power Components andSystems vol 31 no 9 pp 841ndash868 2003

[5] I Kim N Nakazawa S Kim C Park and C Yu ldquoCompensa-tion of torque ripple in high performance BLDCmotor drivesrdquoControl Engineering Practice vol 18 no 10 pp 1166ndash1172 2010

[6] H Lu L Zhang and W Qu ldquoA new torque control method fortorque rippleminimization of BLDCmotors with un-ideal backEMFrdquo IEEE Transactions on Power Electronics vol 23 no 2 pp950ndash958 2008

[7] W Wu ldquoDC motor parameter identification using speed stepresponsesrdquo Modelling and Simulation in Engineering vol 2012Article ID 189757 5 pages 2012

[8] J-H Choi S-H You J Hur and H-G Sung ldquoThe design andfabrication of BLDCMotor andDrive for 42V automotive appli-cationsrdquo in Proceedings of the IEEE International Symposium onIndustrial Electronics (ISIE rsquo07) pp 1086ndash1091 June 2007

[9] A Kapun M Curkovic A Hace and K Jezernik ldquoIdentifyingdynamic model parameters of a BLDC motorrdquo Simulation

Modelling Practice and Theory vol 16 no 9 pp 1254ndash12652008

[10] A J Blauch M Bodson and J Chiasson ldquoHigh-Speed parame-ter estimation of stepper motorsrdquo IEEE Transactions on ControlSystems Technology vol 1 no 4 pp 270ndash279 1993

[11] R Patankar and L Zhu ldquoReal-time multiple parameter estima-tion for voltage controlled brushless DC motor actuatorsrdquo inProceedings of the American Control Conference (AAC rsquo04) vol4 pp 3851ndash3856 July 2004

[12] M S Islam R Islam T Sebastian A Chandy and S AOzsoyluldquoCogging torque minimization in PM motors using robustdesign approachrdquo IEEE Transactions on Industry Applicationsvol 47 no 4 pp 1661ndash1669 2011

[13] M B Tischler and R K Remple Aircraft and Rotorcraft SystemIdentification AIAA Education Series American Institute ofAeronautics and Astronautics 2006

[14] M Petra and M Utting CZT A Framework for Z Tools FormalSpecification and Development in Z and B Springer BerlinGermany 2005

[15] K Zhou and C D John Essentials of Robust Control PrenticeHall Upper Saddle River NJ USA 1998

[16] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Higher Education New York NY USA 2010

[17] J Y Hung and Z Ding ldquoDesign currents to reduce torque ripplein brushless permanent magnet motorsrdquo IEE Proceedings BElectric Power Applications vol 140 no 4 pp 260ndash266 1993

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

10 Mathematical Problems in Engineering

0 2 4 6 8 10 12240

260

280

300

320

Time (s)

Roto

r spe

ed (r

ads

)

Experimental dataBLDC motor model

(a) Rotor speed validation of the BLDC motor dynamics

2 4 6 8 10 12

1

2

3

4

5

Time (s)

Arm

atur

e cur

rent

(A)

Experimental dataBLDC motor model

(b) Armature current validation of the BLDC motor dynamics

Figure 11 Time domain validation of the BLDC motor identified dynamics

2 3 4 5 6 7 8 90

10

20

30

Number of harmonics (K)

J min

Figure 12 The minimum least squares error versus the number of cogging torque harmonics

05 1 15 275

80

85

90

95

100

Time (s)

Nonlinear model Experimental data

02 025 03

90

92

94

96

Time (s)

Nonlinear model Experimental data

Roto

r spe

ed (r

ads

)

Roto

r spe

ed (r

ads

)

Figure 13 Validation of identified comprehensive BLDC motor model

Mathematical Problems in Engineering 11

very low speedwhen cogging torque effect exists significantlyThis method is founded on the least squares approximationmethod a disturbance observer and the identified parametersin frequency identification The results from the simulationand the real-time experiments matched closely which isconsidered as a validation of the obtained mathematicalmodel

The BLDC model identifications are conditional onlyon the availability of experimental data for rotor speed andarmature current therefore our method without sensors todetect torque or back-EMF is easy to realize and is practicaland effectiveThis research will provide greater opportunitiesfor future controller design of the BLDCmotorThe extractedlinear model can be utilized directly for linear controllerdesign and the accurately identified cogging torque effect isbeneficial to reduce the speed ripple of the motor at some lowspeed conditions using some advanced control theory suchas robust control and nonlinear control

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Shanmugasundram K M Zakaraiah and N YadaiahldquoModeling simulation and analysis of controllers for brushlessdirect current motor drivesrdquo Journal of Vibration and Controlvol 19 no 8 pp 1250ndash1264 2013

[2] C Xia Y Wang and T Shi ldquoImplementation of finite-statemodel predictive control for commutation torque ripple min-imization of permanent-magnet brushless DC motorrdquo IEEETransactions on Industrial Electronics vol 60 no 3 pp 896ndash905 2013

[3] S-Y Chang P-H Yu and W-C Su ldquoPre- and post-filteringapproach to sensorless VSS speed control of a permanentmagnet DC motor subject to Hammerstein and Wiener non-linearitiesrdquoControl Engineering Practice vol 21 no 11 pp 1577ndash1583 2013

[4] B-K Lee and M Ehsani ldquoAdvanced simulation model forbrushless dc motor drivesrdquo Electric Power Components andSystems vol 31 no 9 pp 841ndash868 2003

[5] I Kim N Nakazawa S Kim C Park and C Yu ldquoCompensa-tion of torque ripple in high performance BLDCmotor drivesrdquoControl Engineering Practice vol 18 no 10 pp 1166ndash1172 2010

[6] H Lu L Zhang and W Qu ldquoA new torque control method fortorque rippleminimization of BLDCmotors with un-ideal backEMFrdquo IEEE Transactions on Power Electronics vol 23 no 2 pp950ndash958 2008

[7] W Wu ldquoDC motor parameter identification using speed stepresponsesrdquo Modelling and Simulation in Engineering vol 2012Article ID 189757 5 pages 2012

[8] J-H Choi S-H You J Hur and H-G Sung ldquoThe design andfabrication of BLDCMotor andDrive for 42V automotive appli-cationsrdquo in Proceedings of the IEEE International Symposium onIndustrial Electronics (ISIE rsquo07) pp 1086ndash1091 June 2007

[9] A Kapun M Curkovic A Hace and K Jezernik ldquoIdentifyingdynamic model parameters of a BLDC motorrdquo Simulation

Modelling Practice and Theory vol 16 no 9 pp 1254ndash12652008

[10] A J Blauch M Bodson and J Chiasson ldquoHigh-Speed parame-ter estimation of stepper motorsrdquo IEEE Transactions on ControlSystems Technology vol 1 no 4 pp 270ndash279 1993

[11] R Patankar and L Zhu ldquoReal-time multiple parameter estima-tion for voltage controlled brushless DC motor actuatorsrdquo inProceedings of the American Control Conference (AAC rsquo04) vol4 pp 3851ndash3856 July 2004

[12] M S Islam R Islam T Sebastian A Chandy and S AOzsoyluldquoCogging torque minimization in PM motors using robustdesign approachrdquo IEEE Transactions on Industry Applicationsvol 47 no 4 pp 1661ndash1669 2011

[13] M B Tischler and R K Remple Aircraft and Rotorcraft SystemIdentification AIAA Education Series American Institute ofAeronautics and Astronautics 2006

[14] M Petra and M Utting CZT A Framework for Z Tools FormalSpecification and Development in Z and B Springer BerlinGermany 2005

[15] K Zhou and C D John Essentials of Robust Control PrenticeHall Upper Saddle River NJ USA 1998

[16] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Higher Education New York NY USA 2010

[17] J Y Hung and Z Ding ldquoDesign currents to reduce torque ripplein brushless permanent magnet motorsrdquo IEE Proceedings BElectric Power Applications vol 140 no 4 pp 260ndash266 1993

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 11

very low speedwhen cogging torque effect exists significantlyThis method is founded on the least squares approximationmethod a disturbance observer and the identified parametersin frequency identification The results from the simulationand the real-time experiments matched closely which isconsidered as a validation of the obtained mathematicalmodel

The BLDC model identifications are conditional onlyon the availability of experimental data for rotor speed andarmature current therefore our method without sensors todetect torque or back-EMF is easy to realize and is practicaland effectiveThis research will provide greater opportunitiesfor future controller design of the BLDCmotorThe extractedlinear model can be utilized directly for linear controllerdesign and the accurately identified cogging torque effect isbeneficial to reduce the speed ripple of the motor at some lowspeed conditions using some advanced control theory suchas robust control and nonlinear control

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] R Shanmugasundram K M Zakaraiah and N YadaiahldquoModeling simulation and analysis of controllers for brushlessdirect current motor drivesrdquo Journal of Vibration and Controlvol 19 no 8 pp 1250ndash1264 2013

[2] C Xia Y Wang and T Shi ldquoImplementation of finite-statemodel predictive control for commutation torque ripple min-imization of permanent-magnet brushless DC motorrdquo IEEETransactions on Industrial Electronics vol 60 no 3 pp 896ndash905 2013

[3] S-Y Chang P-H Yu and W-C Su ldquoPre- and post-filteringapproach to sensorless VSS speed control of a permanentmagnet DC motor subject to Hammerstein and Wiener non-linearitiesrdquoControl Engineering Practice vol 21 no 11 pp 1577ndash1583 2013

[4] B-K Lee and M Ehsani ldquoAdvanced simulation model forbrushless dc motor drivesrdquo Electric Power Components andSystems vol 31 no 9 pp 841ndash868 2003

[5] I Kim N Nakazawa S Kim C Park and C Yu ldquoCompensa-tion of torque ripple in high performance BLDCmotor drivesrdquoControl Engineering Practice vol 18 no 10 pp 1166ndash1172 2010

[6] H Lu L Zhang and W Qu ldquoA new torque control method fortorque rippleminimization of BLDCmotors with un-ideal backEMFrdquo IEEE Transactions on Power Electronics vol 23 no 2 pp950ndash958 2008

[7] W Wu ldquoDC motor parameter identification using speed stepresponsesrdquo Modelling and Simulation in Engineering vol 2012Article ID 189757 5 pages 2012

[8] J-H Choi S-H You J Hur and H-G Sung ldquoThe design andfabrication of BLDCMotor andDrive for 42V automotive appli-cationsrdquo in Proceedings of the IEEE International Symposium onIndustrial Electronics (ISIE rsquo07) pp 1086ndash1091 June 2007

[9] A Kapun M Curkovic A Hace and K Jezernik ldquoIdentifyingdynamic model parameters of a BLDC motorrdquo Simulation

Modelling Practice and Theory vol 16 no 9 pp 1254ndash12652008

[10] A J Blauch M Bodson and J Chiasson ldquoHigh-Speed parame-ter estimation of stepper motorsrdquo IEEE Transactions on ControlSystems Technology vol 1 no 4 pp 270ndash279 1993

[11] R Patankar and L Zhu ldquoReal-time multiple parameter estima-tion for voltage controlled brushless DC motor actuatorsrdquo inProceedings of the American Control Conference (AAC rsquo04) vol4 pp 3851ndash3856 July 2004

[12] M S Islam R Islam T Sebastian A Chandy and S AOzsoyluldquoCogging torque minimization in PM motors using robustdesign approachrdquo IEEE Transactions on Industry Applicationsvol 47 no 4 pp 1661ndash1669 2011

[13] M B Tischler and R K Remple Aircraft and Rotorcraft SystemIdentification AIAA Education Series American Institute ofAeronautics and Astronautics 2006

[14] M Petra and M Utting CZT A Framework for Z Tools FormalSpecification and Development in Z and B Springer BerlinGermany 2005

[15] K Zhou and C D John Essentials of Robust Control PrenticeHall Upper Saddle River NJ USA 1998

[16] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Higher Education New York NY USA 2010

[17] J Y Hung and Z Ding ldquoDesign currents to reduce torque ripplein brushless permanent magnet motorsrdquo IEE Proceedings BElectric Power Applications vol 140 no 4 pp 260ndash266 1993

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