Research Article Sensorless Control of Nonsinusoidal Permanent...

Research ArticleSensorless Control of Nonsinusoidal Permanent MagnetBrushless Motor Using Selective Torque Harmonic EliminationControl Method Based on Full-Order Sliding Mode Observer

Abolfazl Halvaei Niasar1 Marzieh Ahmadi2 and Sayyed Hossein Edjtahed1

1Department of Electrical amp Computer Engineering University of Kashan Kashan Iran2Department of Electrical amp Computer Engineering Islamic Azad University of Dolatabad Dolatabad Iran

Correspondence should be addressed to Abolfazl Halvaei Niasar halvaeikashanuacir

Received 22 June 2016 Accepted 2 November 2016

Academic Editor Antonio J Marques Cardoso

Copyright copy 2016 Abolfazl Halvaei Niasar et alThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Nowadays due to excellent advantages of permanent magnet brushless (PMBL) motors such as high efficiency and hightorquepower density they are used in many industrial and variable-speed electrical drives applications If the fabricated PMBLmotor has neither ideal sinusoidal nor ideal trapezoidal back-EMF voltages it is named nonideal (or nonsinusoidal) PMBLmotorEmploying conventional control strategies of PMSMs and BLDCMs lowers the efficiency and leads to unwanted torque ripplevibration and acoustic noises Moreover in many applications to reduce the cost and enhance the reliability of drive sensorlesscontrol techniques are usedThis paper proposes a novel sensorless control for a nonsinusoidal PMBLmotor withminimum torqueripple To develop smooth torque the selected torque harmonic elimination strategy is employed Furthermore to estimate the rotorposition and speed a novel full-order sliding mode observer is designed Proposed observer estimates the position and speed ofmotor from standstill to final speed The proposed observer is robust to uncertainty of harmonic contents in phase back-EMFvoltage and able to run the motor from standstill with closed-loop control scheme The capabilities of torque ripple minimizationand sensorless strategies are demonstrated with some simulations

1 Introduction

In two past decades and with reducing the price of perma-nent magnets material design and manufacture of perma-nent magnet brushless (PMBL) motors have developed inindustrial and nonindustrial different applications Superiorfeatures such as high efficiency high power and torquedensity low maintenance cost simple structure and easeof control are the reasons for tendency to these motorsDue to mentioned reasons PMBL motors are consideredin high performance and accurate applications as electrictransportation and aerospace and military industries or evennewly in domestic and consumer applications [1]

The PMBL motors include two main categories of ACbrushless (PMSM or BLAC) and DC brushless (BLDC)depending on the shape of phase back-EMF voltage of motor(sinusoidal or trapezoidal) The induced back-EMF voltages

in stator windings of PMSMs are quite sinusoidal whereasfor BLDCMs they are trapezoidal waveformswith flat portionover a range of 120 degrees as shown in Figures 1(a) and 1(b)This difference is due to type of stator windings that in PMSMmotor is sinusoidally distributed whereas in BLDC motor itis distributed as uniform or centralized distribution [2 3]The difference of back-EMF voltage waveforms causes theemployed control methods to be so different [4]

To develop constant instantaneous torque for PMSMsvector based control such as field oriented control (FOC) ordirect torque control (DTC) in two-axis reference frames isusually used But for BLDCMs using of vector basedmethodsis not common and their utilization leads to lots of torque rip-ple Therefore simple quasi-square (six-step) current meth-ods are employed The main advantages of six-step currentmethods for BLDCMs are hardwaresoftware simplicity andease of implementation as same as DC motors

Hindawi Publishing CorporationAdvances in Power ElectronicsVolume 2016 Article ID 9358604 13 pageshttpdxdoiorg10115520169358604

2 Advances in Power Electronics

0 05 1 15 2 25 3 35 4 45 5

0200

PMSM

Time (Sec) times10minus3

minus200

(a)

0 05 1 15 2 25 3 35 4 45 5

0200

BLD

C


minus200

(b)

50 05 1 15 2 25 3 35 4 45

0200

PMBL


minus200

(c)

Figure 1 The phase back-EMF voltage waveforms induced in the stator of various types of permanent magnet brushless (PMBL) motors

However there are some fabricated PMBL motors whosephase back-EMF voltages are neither ideal trapezoidal likeBLDCMs nor sinusoidal like PMSMs It is due to imprecisedesign or restrictions during fabrication of PMBL motorsThemain reasons for this issue are inappropriate distributionof the statorwindings improper form and span of permanentmagnet and saturation effects These motors are brieflynamed as nonsinusoidal PMBL motor in this paper Fig-ure 1(c) shows a typical back-EMF voltage of nonideal PMBLmotor Employing conventional control methods of PMSMsand BLDCMs such as vector control or quasi-square currentcontrol for nonsinusoidal PMBL motors causes significantinstantaneous torque ripple that depends directly on theharmonic contents of phase back-EMF voltage rather thanideal sinusoidal or trapezoidal shapes [5 6] In someof specialapplications such as military underwater vehicles existenceof torque ripple leads to mechanical vibration or acousticnoise that is ineligible

On the other hand closed-loop control of all types ofPMBL motors needs electronic or electromechanical sensorsfor measuring of rotor speed and position Depending onthe kind of motor and also control method type varioustypes of positionspeed sensors are used For instance quasi-square current control method of BLDCMs needs threecheap Hall-effect sensors whereas in field oriented controlof PMSMs optical encoders or resolvers are required thatare expensive Regarding nonsinusoidal PMBL motors tohave instantaneous constant in most of presented methodsthat are reviewed afterwards exact instantaneous value ofthe rotor position is essential It means that exact positionsensors must be used that are costly In addition the use ofpositionspeed sensors leads to drive hardware complexityand decreases reliability of the system Moreover in someapplications it is not possible to install position sensors suchas small motors and high-speed applications where themotorshaft is not available Also using of such sensorsmay not haveeconomic justification in low power drives Consequentlyusing of adequate positionspeed estimators can reduce costand enhances the system reliability

This paper proposes a suitable closed-loop control ofspeed with minimum torque ripple for given nonsinusoidal

PMBL motors It briefly examines previously presented con-trol methods for PMSMs and BLDCMs and introduces thesuperior controlmethod Also in conjunctionwith developedcontrol method a novel rotor positionspeed estimationbased on sliding mode observer is introduced This paperis organized as follows in Section 2 dynamic model ofnonsinusoidal PMBLmotor is presented In Section 3 variouscontrol methods for PMBL motor are briefly introduced anda superior method is presented based on selective torqueharmonic elimination method The capability of proposedcontrol method rather than other methods is endorsedby some simulations In Section 4 after briefly exploringthe sensorless control methods of PMBL motor variousproposed sliding mode observers are described and thena new full-order sliding mode observer is suggested fornonsinusoidal PMBL motor In Section 5 some simulationsare presented in various conditions using selective torque har-monic elimination controlmethodwhile the estimated rotorrsquospositionspeed is used and finally conclusions are given inSection 6

2 Dynamic Model ofNonsinusoidal PMBL Motor

The nonsinusoidal PMBL motor unlike PMSMs has nosinusoidal flux distribution in the air gap and using two-axisdq reference frame leads to lots of errors due to existenceof higher harmonics components [7] There are two mainapproaches for dynamic modeling of these motors (1) mod-eling inmultiple dq reference frames (MRF) [8] (2)modelingin stationery three-axis abc reference frame In the modelingbased on multiple reference frames according to harmoniccontents of back-EMF voltage multiple dq reference framesare considered with speeds equal to available harmonics andmotor quantities including voltages currents and fluxes aretransferred to these multiple dq reference frames [9 10]For example if the phase back-EMF voltage contains theharmonics with order 119899 = 1 3 5 and 7 then three dqreference frames are considered with synchronous speed fiveand seven times the synchronous speed There is no need to

Advances in Power Electronics 3

ea

eb

ec

MM

MLs

Ls

Ls

Rs

Rs

Rs

ia

ib

ic

n

an

bn

cn

Figure 2 Electrical equivalent circuit of dynamic model of three-phase nonsinusoidal PMBL motor in abc reference frame

consider the third harmonic because it has no role in torquedevelopment This type of modeling can be useful whenthe vector control based methods are used for the PMBLmotor This method has many computations and complexcalculations and needs powerful processors If the number ofharmonics is more complexity also increases

Another modeling method of nonsinusoidal PMBLmotor is the manner that is used for modeling of BLDCmotor [11] If the employed control method is not dependenton the model parameters this modeling approach will besuitable On this way each phase of stator is modeled asseries connection of stator resistance and inductance witha voltage source dependent on the actual phase back-EMFvoltage waveform Figure 2 shows the electrical equivalentcircuit of dynamicmodel for nonsinusoidal PMBLmotor It isassumed that the stator winding is symmetrical and the corelosses and the armature are negligible

The voltage equations of three-phase nonsinusoidalPMBL motor are expressed as follows

[[[V119886119899V119887119899V119888119899

]]]= [[[119877119904 0 00 119877119904 00 0 119877119904

]]]times [[[119894119886119894119887119894119888]]]

+ [[[119871 119904 minus119872 0 0

0 119871 119904 minus119872 00 0 119871 119904 minus119872

]]]119889119889119905 [[[

119894119886119894119887119894119888]]]

+ [[[119890119886119890119887119890119888]]]

(1)

where V119886119899 V119887119899 and V119888119899 are the stator terminal voltages tonatural point 119894119886 119894119887 119894119888 are three-phase currents of the motor119890119886 119890119887 119890119888 are the phase back-EMF voltages and 119877119904 119871 119904 and119872are stator resistance self-inductance and mutual inductanceper phase Electromagnetic torque is developed from thefollowing

119879119890 = 119890119886119894119886 + 119890119887119894119887 + 119890119888119894119888120596119898 (2)

where 120596119898 is mechanical speed of rotor shaft and obeys thefollowing rotational motion equation

119879119890 = 119879119871 + 119869119889120596119898119889119905 + 119861119891120596119898 (3)

where 119879119871 is load torque and 119861119891 and 119869 are load frictioncoefficient and moment of inertia referred to rotor shaftrespectively

3 Control of Nonsinusoidal PMBL Motor withMinimum Torque Ripple

There is not any unique method to control nonsinusoidalPMBL motors unlike PMSMs and BLDCMs due to differentharmonic contents of the back-EMFvoltage in PMBLmotorsIf the back-EMF voltage is similar to ideal trapezoidal one sothe control methods of BLDCMs such as quasi-square cur-rent control are mostly used Also if the back-EMF is similarto sinusoidal one then control methods of the PMSMs suchas vector control are used For example if the back-EMFvoltage includes third harmonic and a very small percentageof higher multiple harmonics the vector control providesappropriate performance because the third harmonic isnot involved in the torque development In this section ashort review on potential methods to control nonsinusoidalPMBL motor is performed and the superior method isintroduced for a typical motor

31 A Review on Previous Control Methods of PMBL MotorsDue to the source of torque ripple in PMBLmotors that can becaused by current commutation tooth grooves or nonidealwaveform of back-EMF voltage various methods have beensuggested that can be grouped into five major categories [5]reference current shaping using estimators and observersimprovement of commutation operation disturbance rejec-tion of speed loop and high-speed flow regulator saturationThe category of methods based on reference current shapingis the most common and adequate control method for torqueripple elimination In this method a programmed currentwaveform (not as sinusoidal) is injected into motor phaseThe block diagram of this method is shown in Figure 3wherein the phase reference current waveforms are createdbased on the rotor position reference torque value andavailable harmonic of back-EMF voltage and then they areapplied to current-controlled VSI inverter In this methodnecessary information should be known about torque ripplesource

The reference current shaping can be carried out byanalyzing the components of phase back-EMF voltage in dqrotating reference frames In this way based on the compo-nents of back-EMF voltage (119890119889 and 119890119902) the reference values ofdifferent harmonics of current are made Afterwards currentregulation for each harmonic is carried out in correspondingdq reference frame and finally their outputs are combinedtogether to make the output voltages In [12] this manner hasbeen done for a PMSMmotor with nonsinusoidal back-EMFvoltage with fifth harmonic order The PI current regulatorshave been used for controlling the d and q components of


Current feedback Position sensor

Reference current waveform generator

Reference torque PMBLmotor

Current-controlledVSI inverter

ilowastA

ilowast

ilowast

B

ilowastC

Tlowaste

120579

120579r

120579r

iABC

Figure 3 The general block diagram of nonsinusoidal PMBL motor control using reference current shaping method

current in fundamental and fifth times speed dq rotatingreference frames

The vector control of nonideal PMSMs has beenimproved bymodifying the q component of reference currentin [13] In other words the reference component 119894119902119904 is modi-fied by back-EMF voltage component 119890119902 that changes due tohigher order harmonics using air-gap power relationship indq reference frame Also to decrease RMS value of currentthe reference current component 119894119889119904 is put to zero Thismethod has less calculation than the previous method butit needs Park transformations (abc to dq) and its inverse APark-like transformation has been used in [9] to reduce thetorque ripple and control of a nonsinusoidal BLDC motorFurthermore a nonlinear state feedback linearization controllaw has been used in the BLDC motor model to control pre-cisely the electromagnetic torque Proposed control methodhas a lot of computations and dependency to the motorrsquosmodel Another method was proposed in [14] that used anextension of Parkrsquos transformation to model nonsinusoidalPMSM by means of the so-called pseudo-dq axes referenceframe The proposed vector control algorithm is derived bydecomposing the motor current into two components onebeing linked to the torque and the other one to its flux In[15] an alternative approach called ldquopseudovector controlrdquo(PVC) is to reduce torque ripple in BLDCM Instead ofconventional square-wave current control it has used theprinciple of vector control to optimally design the waveformof reference current in such a way that the torque rippleis minimal The advantage of proposed method rather than[12] is that the flux weakening for constant-power high-speed mode can be achieved by injecting a negative d-axiscurrent into the control system just like PMSMs Similarattempts have been proposed in [16ndash18] Two major issuesof mentioned methods are high computational complexityand dependency on the motor parameters in which mostof them are open loop and do not include the variationsof motor parameters into control law Furthermore thevoltage controlled-voltage source inverter (VSI) that is oftenused in vector and pseudovector control methods needsvoltage decoupling It increases dependency on parameters ofmodel In next section selective torque harmonic elimination

method is used for the nonsinusoidal PMBL motor as well ascurrent-controlled VSI

32 Selective Torque Harmonic Elimination Control of Non-sinusoidal PMBL Motor If the harmonic contents of phaseback-EMF voltage are available it is possible to eliminatesome arbitrary harmonics of torque waveform by imposingof desired reference current This method has been appliedfor the BLDCmotors and is called selective torque harmonicelimination or harmonic current injection [19 20] It is brieflyexpressed for nonsinusoidal PMBL motor with phase back-EMF voltage shown in Figure 1(c) that contains the harmon-ics of order 119899 = 1 3 5 and 7with the harmonics percentage of100 33 20 and 14 respectively Suppose that the back-EMF voltage of phase ldquoardquo can be rewritten as

119890119886 (119905) = 1198641 sin120596119905 + 1198643 sin 3120596119905 + 1198645 sin 5120596119905+ 1198647 sin 7120596119905 (4)

To gain of maximum powertorque it is desired that phasecurrent ldquoardquo is in phase with phase back-EMF voltage as

119894119886 (119905) = 1198681 sin120596119905 + 1198685 sin 5120596119905 + 1198687 sin 7120596119905 (5)

This is because in the configuration assumed for given PMBLmotor the neutral connection is not used so that currentharmonics of order multiple of three cannot exist Moreoverhigher order harmonic currents than seven are not imposedbecause they only cause more stator copper losses Theinstantaneous air-gap power of phase ldquoardquo includes an averagecomponent and even higher order harmonics until 14th orderas

119875119886 (119905) = 119890119886119894119886= 1198750 + 1198752 sin 2120596119905 + 1198754 sin 4120596119905 + 1198756 sin 6120596119905 + sdot sdot sdot+ 11987514 sin 14120596119905

(6)

Considering the symmetry for phase voltages and currents ofdifferent phases the currents and voltages of two phases ldquobrdquoand ldquocrdquo have the phase shift minus120∘ and +120∘ degrees relative


+

Speedcontroller

+

+

+

Current controllerspwm

Determinationof

three-phasereferenceharmoniccurrents

pwm

pwm

Determination ofharmonic

contents of phaseback-EMF

voltage

120596lowastm

120596m

minus

E1 E5 E7

minus

minus

minus

120579r

PMBLmotor

Tlowaste

+ Vdc minus

R

ilowastas

ilowastbs

ilowastcs

PWMinverter

Figure 4 The block diagram of nonideal PMBL motor drive by selective torque harmonic method

to phase ldquoardquo respectively So the total air-gap power willcontain an average component and only harmonics of ordermultiple of six as

119875119892 (119905) = 31198750 + 31198756 sin 6120596119905 + 311987512 sin 12120596119905 (7)

and the other harmonics are eliminated Therefore theinstantaneous torque can be written as

119879119890 (119905) = 119875119892120596119903 = 1198790 + 1198796 sin 6120596119905 + 11987912 sin 12120596119905 (8)

where

1198790 = 32120596119903 [11986411198681 + 11986451198685 + 11986471198687] 1198796 = 32120596119903 [1198681 (1198647 minus 1198645) minus 11986851198641 + 11986871198641] 11987912 = 32120596119903 [minus11986851198647 minus 11986871198645]

(9)

Since the torque is proportional to the product of the back-EMF and the feed current it is possible to determine anappropriate combination of 119890 and 119894 that reduce the torqueripple to a minimum value for a given average torque 1198790Therefore only the harmonic order multiples 5 and 7 (1198685 1198687)are added to fundamental harmonic in which the mostimportant torque harmonics 1198796 and 11987912 are cancelled outSo for the given average torque of 1198790 and 1198641 1198645 and 1198647 bysolving the algebraic equation

[[[

1198641 1198645 11986471198647 minus 1198645 minus1198641 11986410 1198647 1198645]]]times [[[119868111986851198687]]]= 21205961199033 [[

[119879000]]] (10)

Table 1 Equivalent circuit parameters of employed non-sinusoidalPMBL motor

Quantity Symbol ValueResistance per phase 119877119904 02ΩSelf-inductance per phase 119871 119904 08mHMutual inductance 119872 035mHNumber of poles 119875 12Constant of back-EMF voltage 119870119890 015 V(radsec)Moment of inertia 119869 0015Nsdotms2

DC link voltage 119881dc 300VRated load torque 119879119899 15Nsdotm

the feed current harmonics are obtained from

[[[119868111986851198687]]]= [[[10063minus004700235

]]]21205961199033 1198790 (11)

By imposing three-phase reference currents with the firstfifth and seventh harmonics amplitudes as obtained thetorque ripple due to harmonics of phase back-EMF voltagewill be exactly cancelled

Figure 4 shows the block diagram of speed closed-loopsystem of nonsinusoidal PMBL motor controlled by usingselective torque harmonics elimination method The usedmotor has rated power and speed of 25 kW and 1500 rpmwhose equivalent circuit parameters are listed in Table 1

Figure 5 shows the simulation results of motor behaviorby this method The reference speed reaches its rated valuewithin tenth of a second and the motor actual speed tracksit as well The reference current which is determined bythe control system is not sinusoidal and includes harmonicswith order multiples five and seven Three hysteresis currentcontrollers are employed for tracking of reference currents


0 005 01 0150

1000

Speed response

0 005 01 015

020

Phase current

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

010

0 005 01 0150

50Electromagnetic torque

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

12141618

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

minus20

minus10

i a(A

)Zo

om o

fia

ia

Zoom

of

Wre

famp

Wm

(rpm

)

iaref

Tem

Tem

(Nmiddotm

)

Figure 5 Simulation results of the nonsinusoidal PMBL motorcontrol by the selective torque harmonic elimination method

and the results show the current response is satisfactoryThe electromagnetic torque reaches its determinedmaximumvalue of 40Nsdotmduring the startup and at final speed it settlesto the load torque NsdotmThe torque ripple peak-to-peak valueis 27Nsdotm or 16 at final speed which shows a significantimprovement rather than other suggested methods

To prove this claim the motor behavior is also simulatedby using three common control methods of PMBL motorsincluding three-phase quasi-square current control directtorque control and vector control The simulation results ofthese methods are shown in Figure 6The torque ripple valueof thesemethods is compared with selective torque harmonicelimination method as summarized in Table 2 Simulationresults confirm that proposed selective torque harmonicelimination method has significant advantages such as theease of implementation low calculations and less torqueripple value compared to other methods

4 Sensorless Control ofNonsinusoidal PMBL Motor

For nonsinusoidal PMBL motor control presented in previ-ous section the precise rotor position information is essentialfor generating of references currents The rotor positionis measured by accurate electromechanical sensors such as

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

010

Direct torque control method

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

101520

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

Quasi-square current control method

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

81216

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

Vector control method0

13

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

1216

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

i a(A

)i a

(A)

i a(A

)

minus10

010

minus10

010

minus10

Tem

(Nmiddotm

)Tem

(Nmiddotm

)Tem

(Nmiddotm

)

Figure 6 Simulation results for current and torque waveforms ofnonsinusoidal PMBL motor control by using other conventionalmethods (direct torque control quasi-square current control andvector control)

Table 2 Comparison of the relative torque ripple of non-sinusoidalPMBL motor by using various control methods in the final speed

Control method THD of torqueDirect torque control 80Quasi-square current control withthree-phase current feeding 45Vector control 33Selective torque harmonic eliminationcontrol 16

encoders or resolvers But using the estimator of the rotorposition and speed in the permanent magnet motor drivescontrol is also highly regarded to reduce the cost of designand construction and also enhance reliability [22] Variousmethods have been proposed to estimate the position andspeed of PMBL motors [21 23] As follows a brief overview


Referencemodel (RM)

Adaptivemodel (AM)

Correctionmechanism

Output RM

Output AM

Inputs+

_

Figure 7 Block diagram of closed-loop observers [21]

on the types of sensorlessmethods is explored and then someattempts to design of sliding mode observers are investigatedand finally a new sliding mode observer is investigated fornonsinusoidal PMBL motor

41 Review on Sensorless Control Methods of PMBL MotorsThe position estimation methods of PMBL motors canbe divided into two categories open-loop and closed-loopmethods [24] The open-loop methods obtain the rotorposition information from the motor model and directmeasurement of voltage terminals without using any internalcorrectionmechanismThese classes ofmethods are used dueto their simplicity and have various types including methodsbased on the back-EMF voltage methods based on stator fluxlinkage methods based on inductance changes and meth-ods based on high frequency signal injection Despite thesimplicity of open-loop methods they are faced with manyrestrictions including poor dynamic response sensitivity toparameters of motor and difficulty at low speeds especiallyat startup

In contrast with the open-loop methods the closed-looptechniques are based on observer schemes that use an internalcorrection mechanism These methods are mainly basedon model reference adaptive systems (MRAS) includingLuenberger observer disturbance and slidingmode observerand Kalman filter Figure 7 shows the generic block diagramof closed-loop observers composing the reference model andthe adaptive model The reference model is the motor whoseoutputs are the currents while the adaptive model is themotor model that estimates the motor currents The errorbetween the estimated currents and the measured ones isfed back to the adaptive model A simplified classificationof the closed-loop methods is the reduced-order observersand full-order observers The reduced-order observers areclosed-loop schemes that do not contain the mechanicalmotor model and so have fewer dynamic equations andcomputations They are based on two approaches currentobserver and flux observer The output of reduced-orderobservers ismainly stator currents or fluxes and to determinerotor positionspeed extra computation is needed

With respect to reduced-order observers the full-orderobservers include the mechanical model whose output is therotor position that is used to get the estimated currents viathe inverse magnetic modelThe error between the estimatedcurrents and the measured ones is fed back to the adaptive

modelThe closed-loop observers aremore accurate and haveless error but are often based on themotor dynamicmodelingand are dependent on equivalent circuit parameters So theused algorithm must be designed as robust and adaptive

Among closed-loop observers the sliding modeobservers (SMO) have satisfactory dynamic responseand good robustness to the dynamic model parameterschanges and linear and nonlinear unmodeled dynamicsMajor presented sliding mode observers estimate the phaseback-EMF voltage components in the stationary (119890120572 119890120573) orrotating (119890119889 119890119902) two-axis reference frames using measuredstator currents and voltages Then the rotorrsquos positionand speed are calculated from mathematical relations Forinstance in [25] for a given BLDCmotor the voltages valuesof 119890120572 and 119890120573 have been estimated using a second-order SMOand the rotor position is obtained from

120579119903 = 1205872 minus tanminus1 (119890120573119890120572) (12)

and to calculate of the rotorrsquos speed and to calculate thespeed of rotor the derivative of position of rotor has tobe determined Derivation may lead to significant compu-tational error due to switching noises Similar attempts havebeen presented in [26 27] where back-EMF components 119890119889and 119890119902 in rotating dq reference frame have been estimated bythe sameway To avoid errors and problems due to derivationsome researches have been suggested using of phase-lockedloop (PLL) to calculate motor speed from estimated positionthat actually increases the order of dynamic equations of theobserver [24 28]

42 Design of Full-Order Sliding Mode Observer for Nonsi-nusoidal PMBL Motor As mentioned in previous sectiondue to advantages of sliding mode observers and computa-tional errors in reduced-order observers a novel full-ordersliding mode observer is presented to estimate the rotorrsquosposition and speed of nonsinusoidal PMBLmotor as followsThe designed observer is especially for nonlinear uncertainsystems The observer inputs are stator voltage componentsin 120572120573 stationary reference frame (V119904120572 V119904120573) The system statevariables are electrical angular position and speed of rotorand two stator current components in120572120573 stationary referenceframe as follows

119909119879 = [120579119903 120596119903 119894119904120572 119894119904120573] (13)


The dynamic equations of PMBL motor expressed by (1)can be rewritten in 120572120573 reference frame as

119889120579119903119889119905 = 120596119903 (14)

119889120596119898119889119905 = 1119869 (32 (119890120572119894120572 + 119890120573119894120573120596119898 ) minus 119879119871 minus 119861120596119898) (15)

119889119894119904120572119889119905 = minus119877119904119871 119904 119894119904120572 +1119871 119904 V119904120572 minus

1119871 119904 119890120572 (16)

119889119894119904120573119889119905 = minus119877119904119871 119904 119894119904120573 +1119871 119904 V119904120573 minus

1119871 119904 119890120573 (17)

where 119890120572 and 119890120573 are components of phase back-EMF voltagethat are obtained fromClark transformation of nonsinusoidalwaveform in Figure 1(c) Also the mechanical speed 120596119898should be replaced with equivalent electrical speed 120596119903 in(15) The above state space equations can be expressed in thegeneral form as follows

= 119860119909 + 119861119906 + Φ (119909 119906) + 119863119891 (119910 119906) 119910 = 119862119909 (18)

where 119909 119906 and 119910 vectors are the state variables inputsand outputs respectively Φ(119909 119906) is the known nonlinearterm of the system and is assumed to be Lipschitz withrespect to 119909 for all 119906 [29]The function 119891(119910 119906) represents theunknown term of the systems that is bounded by the knownfunction120588(119910 119906)The input vector119906 includes the stator voltagecomponents of V119904120572 and V119904120573 the output vector 119910 includes 119894119904120572and 119894119904120573 stator current components The matrices 119860 119861 119862 and119863 and functions Φ(119909 119906) and 119891(119910 119906) in (18) are defined inAppendix AMoreover the following conditionsmust bemetfor this observer

(1) rank(CD) = rank(D)

(2) All the invariant zeros of the matrix triple 119860119863 119862 liein the left half plane

For this system the condition of rank(CD) = rank(D) =2 is satisfied Also the system has not any zero thereforecondition 2 is satisfied

Then the state space equations of system (18) can bewritten as follows

1 = 119860111199091 + 119860121199092 + 1198611119906 + Φ1 (119909 119906) 2 = 119860211199091 + 119860221199092 + 1198612119906 + Φ2 (119909 119906) + 1198632119891 (119910 119906) 119910 = 11986221199092

(19)

where Φ1 and Φ2 are the first two components and the lasttwo components of matrix Φ(119909 119906) respectively and 1198611 and1198612 are the first two rows and the last two rows of matrix 119861respectively Given that 119891(119910 119906) are the output uncertaintiesthen the first two rows which are related to 1 are zero and1198632 is the last two rows of matrix119863

Now let us to introduce a coordinate transformation119911 = 119879119909 for designing the sliding mode observer where 119879 isdefined as follows

119879 = [1198682 1198710 1198682]

119879minus1 = [1198682 minus1198710 1198682 ] (20)

Then the state equations are obtained in the new coordinatesystem as follows

1 = (11986011 + 11987111986021) 1199111+ (11986012 minus 11986011119871 + 119871 (11986022 minus 11986021119871)) 1199112+ [1198682 119871] 119861119906 + [1198682 119871]Φ (119879minus1119911 119906)

2 = 119860211199111 + (11986022 minus 11986021119871) 1199112 + 1198612119906 + Φ2 (119879minus1119911 119906)+ 1198632119891 (119910 119906)

119910 = 11986221199112

(21)

Then in the new 119911 coordinates sliding mode observerequations have the following form


1199112 = 119860211 + (11986022 minus 11986021119871) 2 + 1198612119906 + Φ2 (119879minus1 119906)+ 1198632119891 (119910 119906)

119910 = 11986222

(22)

where

V = 120588 sdot sign (119910 minus 11986222) (23)

and 120588 is positive scalar Dynamic equations of the estimationerror vector are obtained by combining (21) with (22)-(23) asfollows

1198901 = (11986011 + 11987111986021) 1198901+ [1198682 119871] (Φ (119879minus1119911 119906) minus Φ (119879minus1 119906)) (24)

119890119910 = 1198622119860211198901 + (1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870) 119890119910+ 11986221198632119891 (119910 119906) minus 1198622V+ 1198622 (Φ2 (119879minus1119911 119906) minus Φ2 (119879minus1 119906))

(25)

According to (24) the matrix 119871 should be chosen so that theterm (11986011 + 11987111986021) is stable Also the gain matrix 119870 can beconsidered as follows

119870 = minus (11986022 minus 11986021119871)119862minus12 + 119862minus12 119860 119904 (26)


Current controllers

+

Speed controller

+

+

+

pwm

pwm

pwm

Determination of

contents of thephase back-EMF

voltage

Full-order sliding mode

observer

PMBLmotor

Determinationof

three-phase

referenceharmonic

currents

harmonic

120596lowastm

minus

E1 E5 E7

minus

minus

minusTlowaste

m120579r

is120572 is120573 s120572 s120573

PWMinverter

ilowastas

ilowastbs

ilowastcs

+ Vdc minus

Figure 8 Block diagram of SMO sensorless control system by selective torque harmonic eliminationmethod for nonsinusoidal PMBLmotor

where 119860 119904 is a symmetric positive definite matrix to ensurethat the following matrix is symmetric negative definite

1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870 (27)

As a result the linearized nominal system matrix of theestimation error dynamic system is stable and the estimationerror asymptotically tends to zero The linearized systemmatrix is described by

[11986011 + 11987111986021 0119862211986021 1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870] (28)

For this system the sliding surface is considered as

119878 = (1198901 119890119910) | 119890119910 = 0 (29)

In this case according to the mentioned conditions forselection of matrices 119871 and K the best response is obtainedfor 119871 = minus 1198901199011199041198682 and 119860 119904 = 20001198682 due to the givenuncertainty 119891(119910 119906) the value of 120588 is chosen to 20 The stateestimation error dynamic equations represent the slidingdynamics when it is limited to the sliding surface 119878 It is onlynecessary to ensure stability of 1198901 so that asymptotic stabilityof these equations is proved relative to the sliding surfaceFor this purpose Lyapunov function can be considered as119881 = 11989011987911198751198901 The stability proof of this 119881 function is brieflypresented in Appendix B

5 Simulation of Sensorless Control System ofNonsinusoidal PMBL Motor

In this section the closed-loop control system of the non-sinusoidal PMBL motor with selective torque harmonicelimination method is simulated by using estimated positionand speed of the rotor of introduced sliding mode observerThe block diagram of the system is shown in Figure 8

0 005 01 0150

100200300400

Rotor position estimation

0 005 01 0150

50010001500

Rotor speed estimation

0 005 01 015

05

10Speed estimation error

Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

minus10

minus5

(rpm

)W

mamp

Wmh

at120579 r

(deg

)

Figure 9 Estimated position and speed of nonsinusoidal PMBLmotor in the sensorless closed-loop control system

The estimated position and speed of the rotor arecompared to the real position and speed in Figure 9 Themaximum amount of position error is 10 degrees where itsmain reason is the use of low filtered voltages and currentsvalues of the stator as observer inputs However this amountof error does not affect the speed tracking Also the speedtracking error is less than 8 rpm or 06 of reference speedFigure 10 shows developed electromagnetic torque where thetorque ripple is about 45Nsdotm at final speed The increase oftorque ripple compared to case with sensor shown in Figure 5is due to the estimated position error It is easy to showthat the torque ripple has been mentioned far more than


0 005 01 0150

1020304050

Electromagnetic and load torque

012 0125 013 0135 014 0145 015101214161820

Time (Sec)

Time (Sec)

Teamp

Tlo

ad(N

middotm)

Zoom

(Nmiddotm

)

Figure 10 Developed electromagnetic torque in the sensorlessclosed-loop control system of nonsinusoidal PMBL motor

0 005 01 015

020

Current response

014

014

1

014

2

014

3

014

4

014

5

014

6

014

7

014

8

014

9

010

Zoom

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

0

Curr

ents

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

910111213

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

ref

minus20

minus10

10

minus10

iref

i qamp

i qre

f(A

)i a

ampI r

ef(A

)

Figure 11 Nonsinusoidal PMBL motor currents waveforms in thesensorless closed-loop control system

45Nsdotm in the event that other control methods are used bythe estimated position and speed via SMO

Figure 11 shows current response of the driving systemTracking of phase current has been performed well Alsothe reference waveform of stator current component 119902 and

008 009 01 011 012 013 014 015 016

0200

Phase back-EMF voltage

008 009 01 011 012 013 014 015 0160

200400


0 002 004 006 008 01 012 014 016 018 020

1020

Erro

r (de

g)

0 002 004 006 008 01 012 014 016 018 020

1000


008 009 01 011 012 013 014 015 016

05

Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

minus200

minus5

(rpm

)W

mamp

Wmh

ate a

(V)

120579 r(d

eg)

Figure 12 Estimated rotor position and speed of nonsinusoidalPMBLmotor during the change of harmonic contents of phase back-EMF voltage

its real value have been shown Since the reference current119894119902119904 contains the fifth and seventh harmonics it has periodicalvariationsThe real value of 119894119902119904 has high frequency oscillationdue to hysteresis current controller and chattering of SMO

To verify the robustness of sliding mode observer to vari-ations of motor parameters the harmonic contents of phaseback-EMF voltage are changed during motor operation Thephase back-EMF voltage at the time 119905 = 01 sec changesto quite sinusoidal form and then at the time 119905 = 015 secchanges to waveform with harmonics of orders 1 3 5 and 7with amplitudes of 100 33 20 and 13 of fundamentalharmonic It should be noted that this scenario may nothappen for a real motor and we want to show the capability ofslidingmode observer against uncertainmodel of the systemThe changes of the phase back-EMF voltage and trackingerror of the rotor position and speed are shown in Figure 12The tracking error has not significantly changed while theback-EMF voltage changes which means that the SMO isrobust to uncertainties Figure 13 shows the motor torqueand current waveforms during this scenario The torqueripple decreases while the back-EMFwaveform changes to besinusoidal


0 002 004 006 008 01 012 014 016 018 020

204060


008 009 01 011 012 013 014 015 01610

15

20

0 002 004 006 008 01 012 014 016 018 02

0

50Current response

008 009 01 011 012 013 014 015 016

010

Zoom

(A)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Zoom

(Nmiddotm

)

minus50

minus10

Teamp

Tlo

ad(N

middotm)

i aamp

I ref

(A)

Figure 13 The motor current and torque waveforms during thechange of harmonic contents of phase back-EMF voltage

6 Conclusion

A novel sensorless control based on full-order sliding modeobserver for closed-loop speed control of a nonsinusoidalPMBLmotor has been developed in this paper It is accompa-nied with torque ripple minimization strategy using selectiveharmonic elimination method Proposed control methoddoes not have any dependency on the motor parametersexcept harmonic contents of phase back-EMF voltage andit uses current-controlled VSI Also contrary to vectorcontrolled based method it does not need any Park trans-formations and voltage decoupling Torque ripple resultingfrom proposed method is at least among various proposedtechniques To reduce the cost and to enhance the reliabilitya four-order new sliding mode observer (SMO) has beendeveloped to estimate the instant position and speed of PMBLmotor directlyDeveloped SMOhas special features includingfast response less estimation error and robustness againstuncertainties of the motor parameters such as back-EMF orstator resistance Moreover it can run the motor as closed-loop scheme from the standstill without employing open-loop starting that is relevant in other open-loop and someclosed-loop estimators

The simulation results endorse satisfactory behavior oftorque ripple reduction control as well as speed estimator Toenhance the performance of the proposed drive it is possibleto predict the harmonic contents of phase back-EMF voltagevia suitable observers or via more calculations with proposedsliding mode observer

Appendix

A Matrices and Functions Presented in (18)

119860 =

11986011⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 10 00 00 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟11986021

11986012⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0 0

0 0minus 119877119871 119904 00 minus 119877119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

11986022

119861 =

1198611⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 01119871 119904 00 1119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198612

119862 =1198621⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[ 0 00 0

1198622⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 0

0 1 ]

119863119891 (119910 119906) =

1198631⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 0

minus01119877119904119871 119904 00 minus01119877119904119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198632

Φ (119909 119906) = [1119869 32119870119890 (1198752 )2 (119894119904120572 (sin 120579119903 + 025 sin 3120579119903

+ 01 sin 5120579119903 + 002 sin 7120579119903) minus 119894119904120573 (cos 120579119903+ 025 cos 3120579119903 + 01 cos 5120579119903 + 002 cos 7120579119903) minus 2119875sdot 119879119871 minus 119861119898 2119875120596119903) minus 119870119890119871 119904 120596119903 (sin 120579119903 + 025 sin 3120579119903+ 01 sin 5120579119903 + 002 sin 7120579119903) 119870119890119871 119904 120596119903 (cos 120579119903 + 025sdot cos 3120579119903 + 01 cos 5120579119903 + 002 cos 7120579119903)]

(A1)


B Stability Proof ofLyapunov Function 119881 = 11989011987911198751198901

From what is given function is derived as follows

= 1198901198791 ((11986011 + 11987111986021)119879 119875 + 119875 (11986011 + 11987111986021)) 1198901+ 21198901198791119875 [1198682 119871] (Φ (119879minus1119911 119906) minus Φ (119879minus1 119906))

= 1198901198791 (119860119879119875119879 + 119875119860) 1198901+ 2 (1198751198791198901) (Φ (119879minus1119911 119906) minus Φ (119879minus1 119906))

(B1)

where

119875 fl 119875 [1198682 119871] 119860 fl [1198601111986021]

(B2)

According to famousYoungrsquos inequality (2119883119879119884 le 120576119883119879119883+(1120576)119884119879119884) which exists for each scalar 120576 gt 0 inequality therelation is obtained as follows [29]

le 1198901198791 (119860119879119875119879 + 119875119860) 1198901 + 12057611989011987911198751198751198791198901+ 1120576 (Φ (119879minus1119911 119906) minus Φ (119879minus1 119906))119879sdot (Φ (119879minus1119911 119906) minus Φ (119879minus1 119906))

(B3)

As a result

le 1198901198791 (119860119879119875119879 + 119875119860) 1198901 + 12057611989011987911198751198751198791198901+ 1120576 (119871Φ)2 1003817100381710038171003817119890110038171003817100381710038172

= 1198901198791 (119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868) 1198901(B4)

So if the following relationship is established

119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868 lt minus120572119875 (B5)

in this case we have

le minus12057211989011987911198751198901 = minus120572119881 (B6)

In other words the designed sliding mode observeralways remains stable

Competing Interests

The authors declare that they have no competing interests

References

[1] R Krishnan Permanent-Magnet Synchronous and Brushless DCMotor Drives John Wiley amp Sons 2002

[2] J F Gieras andMWing Permanent Magnet Motor TechnologyDesign and Applications Marcel Dekker Inc New York NYUSA 2nd edition 2002

[3] D C Hanselman Brushless Permanent-Magnet Motor DesignMagna Physics Lebanon Ohio USA 2006

[4] E Klintberg Comparison of control approaches for permanentmagnet motors (Master of Science) [MS thesis] Department ofEnergy and Environment Chalmers University of TechnologyGoteborg Sweden 2013

[5] T M Jahns and W L Soong ldquoPulsating torque minimizationtechniques for permanent magnet AC motor drivesmdasha reviewrdquoIEEE Transactions on Industrial Electronics vol 43 no 2 pp321ndash330 1996

[6] ANrsquodiaye C Espanet andAMiraoui ldquoReduction of the torqueripples in brushless PMmotors by optimization of the supplymdashtheoretical method and experimental implementationrdquo in Pro-ceedings of the 2004 IEEE International Symposium on IndustrialElectronics (IEEE-ISlE rsquo04) pp 1345ndash1350 Ajaccio France May2004

[7] A Halvaei Niasar H Moghbelli and A Vahedi ldquoModeling andsimulationmethods for brushless DCmotor drivesrdquo in Proceed-ings of the 1st International Conference on Modeling Simulationand Applied Optimization (ICMSAO rsquo05) vol 5 pp 24ndash167Sharjah United Arab Emirates 2005

[8] P L Chapman S D Sudhoff and C A Whitcomb ldquoMulti-ple reference frame analysis of non-sinusoidal brushless DCdrivesrdquo IEEE Transactions on Energy Conversion vol 14 no 3pp 440ndash446 1999

[9] D Grenier and L A Dessaint ldquoA park-like transformationfor the study and the control of a non-sinusoidal brushlessDC motorrdquo in Proceedings of the 21st International Conferenceon Industrial Electronics Control and Instrumentation (IEEEIECON rsquo95 pp 837ndash843 1995

[10] H Lei and H A Toliyat ldquoBLDC motor full speed rangeoperation including the flux-weakening regionrdquo in Proceedingsof the IEEE Annual Industry Applications Conference (IAS rsquo03)vol 1 pp 618ndash624 Salt Lake City Utah USA 2003

[11] B K Lee B Fahimi and M Ehsani ldquoDynamic modeling ofbrushlessDCmotor drivesrdquo inProceedings of the EuropeanCon-ference on Power Electronics and Applications (EPE rsquo01) GrazAustria 2001

[12] F Bonvin and Y Perriard ldquoBLDC motor control in multipledq axesrdquo in Proceedings of the 8th International Conference onIEE Power Electronics and Variable Speed Drives pp 500ndash505London UK 2000

[13] S Bolognani L Tubiana and M Zigliotto ldquoSensorless controlof PM synchronous motors with non-sinusoidal back EMFfor home appliancerdquo in Proceedings of the IEEE InternationalElectricMachines andDrives Conference (IEMDC rsquo03) pp 1882ndash1888 Madison Wis USA June 2003

[14] A Lidozzi L Solero F Crescimbini and R Burgos ldquoVectorcontrol of trapezoidal back-EMF PM machines using Pseudo-Park transformationrdquo in Proceedings of the 39th IEEE AnnualPower Electronics Specialists Conference (PESC rsquo08) pp 2167ndash2171 Rhodes Greece June 2008

[15] C-M Ta ldquoPseudo-vector controlmdashAn alternative approachfor brushless DC motor drivesrdquo in Proceedings of the IEEEInternational Electric Machines and Drives Conference (IEMDCrsquo11) pp 1534ndash1539 May 2011

[16] S J Park H W Park M H Lee and F Harashima ldquoA newapproach for minimum-torque-ripple maximum-efficiency


control of BLDC motorrdquo IEEE Transactions on IndustrialElectronics vol 47 no 1 pp 109ndash114 2000

[17] A Oliveira A Monteiro M L Aguiar and D P GonzagaldquoExtendedDQ transformation for vectorial control applicationsof non-sinusoidal permanent magnet synchronous machinesrdquoin Proceedings of the IEEE 36th Annual Power ElectronicsSpecialists Conference vol 1 pp 1807ndash1812 Recife Brazil 2005

[18] P Kshirsagar and R Krishnan ldquoEfficiency improvement evalu-ation of non-sinusoidal back-EMF PMSMmachines using fieldoriented current harmonic injection strategyrdquo in Proceedings ofthe IEEEEnergyConversionCongress andExposition (ECCE rsquo10)pp 471ndash478 Atlanta Ga USA 2010

[19] J Y Hung and Z Ding ldquoDesign of currents to reduce torqueripple in brushless permanent magnet motorsrdquo IEE ProceedingsB Electric Power Applications vol 140 no 4 pp 260ndash266 1993

[20] ANrsquodiaye C Espanet andAMiraoui ldquoReduction of the torqueripples in brushless pmmotors by optimization of the supplymdashtheoretical method and experimental implementationrdquo in Pro-ceedings of the 2004 IEEE International Symposium on IndustrialElectronics IEEE-ISlE pp 1345ndash1350 May 2004

[21] R Bojoi M Pastorelli J Bottomley P Giangrande and CGerada ldquoSensorless control of PMmotor drivesmdasha technologystatus reviewrdquo in Proceedings of the 1st Workshop on ElectricalMachines Design Control and Diagnosis (WEMDCD rsquo13) pp168ndash182 IEEE Paris France March 2013

[22] A Halvaei Niasar A Vahedi and H Moghbelli ldquoSensorlesscontrol of four-switch brushless DCmotor drive without phaseshifterrdquo IEEE Transactions on Power Electronics vol 23 no 6pp 3079ndash3087 2008

[23] D Yousfi A Halelfad and M El Kard ldquoReview and evaluationof some position and speed estimation methods for PMSMsensorless drivesrdquo in Proceedings of the International Conferenceon Multimedia Computing and Systems (ICMCS rsquo09) pp 409ndash414 IEEE Ouarzazate Morocco April 2009

[24] Y Zhao CWei Z Zhang andW Qiao ldquoA Review on PositionSpeed Sensorless Control for Permanent-Magnet SynchronousMachine-Based Wind Energy Conversion Systemsrdquo IEEE Jour-nal of Emerging and Selected Topics in Power Electronics vol 1no 4 pp 203ndash216 2013

[25] G R A Markadeh S I Mousavi S Abazari and A KargarldquoPosition sensorless direct torque control of BLDC motorrdquo inProceedings of the IEEE International Conference on IndustrialTechnology IEEE (ICIT rsquo08) pp 1ndash6 April 2008

[26] M R Feyzi M Shafiei M Bahrami Kouhshahi and S A KMozaffari Niapour ldquoPosition sensorless direct torque controlof Brushless DC motor drives based on sliding mode observerusing NSGA-II Algorithm optimizationrdquo in Proceedings of the2nd Power Electronics Drive Systems and Technologies Confer-ence (PEDSTC rsquo11) pp 151ndash156 IEEE Tehran Iran February2011

[27] Freescale Semiconductor ldquoSensorless PMSM vector con-trol with a sliding mode observer for compressors usingMC56F8013rdquo Document Number DRM099 2008

[28] AMurrayM Palma andAHusain ldquoPerformance comparisonof permanent magnet synchronous motors and controlledinduction motors in washing machine applications using sen-sorless field oriented controlrdquo in Proceedings of the 2008 IEEEIndustry Applications Society Annual Meeting (IAS rsquo08) pp 1ndash6Edmonton Canada October 2008

[29] Y Shtessel Ch Edwards L Fridman and A Levant SlidingModeControl andObservation Springer Berlin Germany 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



0 05 1 15 2 25 3 35 4 45 5

0200

PMSM


minus200

(a)

0 05 1 15 2 25 3 35 4 45 5

0200

BLD

C


minus200

(b)

50 05 1 15 2 25 3 35 4 45

0200

PMBL


minus200

(c)

Figure 1 The phase back-EMF voltage waveforms induced in the stator of various types of permanent magnet brushless (PMBL) motors

However there are some fabricated PMBL motors whosephase back-EMF voltages are neither ideal trapezoidal likeBLDCMs nor sinusoidal like PMSMs It is due to imprecisedesign or restrictions during fabrication of PMBL motorsThemain reasons for this issue are inappropriate distributionof the statorwindings improper form and span of permanentmagnet and saturation effects These motors are brieflynamed as nonsinusoidal PMBL motor in this paper Fig-ure 1(c) shows a typical back-EMF voltage of nonideal PMBLmotor Employing conventional control methods of PMSMsand BLDCMs such as vector control or quasi-square currentcontrol for nonsinusoidal PMBL motors causes significantinstantaneous torque ripple that depends directly on theharmonic contents of phase back-EMF voltage rather thanideal sinusoidal or trapezoidal shapes [5 6] In someof specialapplications such as military underwater vehicles existenceof torque ripple leads to mechanical vibration or acousticnoise that is ineligible

On the other hand closed-loop control of all types ofPMBL motors needs electronic or electromechanical sensorsfor measuring of rotor speed and position Depending onthe kind of motor and also control method type varioustypes of positionspeed sensors are used For instance quasi-square current control method of BLDCMs needs threecheap Hall-effect sensors whereas in field oriented controlof PMSMs optical encoders or resolvers are required thatare expensive Regarding nonsinusoidal PMBL motors tohave instantaneous constant in most of presented methodsthat are reviewed afterwards exact instantaneous value ofthe rotor position is essential It means that exact positionsensors must be used that are costly In addition the use ofpositionspeed sensors leads to drive hardware complexityand decreases reliability of the system Moreover in someapplications it is not possible to install position sensors suchas small motors and high-speed applications where themotorshaft is not available Also using of such sensorsmay not haveeconomic justification in low power drives Consequentlyusing of adequate positionspeed estimators can reduce costand enhances the system reliability

This paper proposes a suitable closed-loop control ofspeed with minimum torque ripple for given nonsinusoidal

PMBL motors It briefly examines previously presented con-trol methods for PMSMs and BLDCMs and introduces thesuperior controlmethod Also in conjunctionwith developedcontrol method a novel rotor positionspeed estimationbased on sliding mode observer is introduced This paperis organized as follows in Section 2 dynamic model ofnonsinusoidal PMBLmotor is presented In Section 3 variouscontrol methods for PMBL motor are briefly introduced anda superior method is presented based on selective torqueharmonic elimination method The capability of proposedcontrol method rather than other methods is endorsedby some simulations In Section 4 after briefly exploringthe sensorless control methods of PMBL motor variousproposed sliding mode observers are described and thena new full-order sliding mode observer is suggested fornonsinusoidal PMBL motor In Section 5 some simulationsare presented in various conditions using selective torque har-monic elimination controlmethodwhile the estimated rotorrsquospositionspeed is used and finally conclusions are given inSection 6

2 Dynamic Model ofNonsinusoidal PMBL Motor

The nonsinusoidal PMBL motor unlike PMSMs has nosinusoidal flux distribution in the air gap and using two-axisdq reference frame leads to lots of errors due to existenceof higher harmonics components [7] There are two mainapproaches for dynamic modeling of these motors (1) mod-eling inmultiple dq reference frames (MRF) [8] (2)modelingin stationery three-axis abc reference frame In the modelingbased on multiple reference frames according to harmoniccontents of back-EMF voltage multiple dq reference framesare considered with speeds equal to available harmonics andmotor quantities including voltages currents and fluxes aretransferred to these multiple dq reference frames [9 10]For example if the phase back-EMF voltage contains theharmonics with order 119899 = 1 3 5 and 7 then three dqreference frames are considered with synchronous speed fiveand seven times the synchronous speed There is no need to


ea

eb

ec

MM

MLs

Ls

Ls

Rs

Rs

Rs

ia

ib

ic

n

an

bn

cn





[[[V119886119899V119887119899V119888119899

]]]= [[[119877119904 0 00 119877119904 00 0 119877119904

]]]times [[[119894119886119894119887119894119888]]]

+ [[[119871 119904 minus119872 0 0

0 119871 119904 minus119872 00 0 119871 119904 minus119872

]]]119889119889119905 [[[

119894119886119894119887119894119888]]]

+ [[[119890119886119890119887119890119888]]]

(1)


119879119890 = 119890119886119894119886 + 119890119887119894119887 + 119890119888119894119888120596119898 (2)


119879119890 = 119879119871 + 119869119889120596119898119889119905 + 119861119891120596119898 (3)











ilowastA

ilowast

ilowast

B

ilowastC

Tlowaste

120579

120579r

120579r

iABC






119890119886 (119905) = 1198641 sin120596119905 + 1198643 sin 3120596119905 + 1198645 sin 5120596119905+ 1198647 sin 7120596119905 (4)


119894119886 (119905) = 1198681 sin120596119905 + 1198685 sin 5120596119905 + 1198687 sin 7120596119905 (5)



(6)



+

Speedcontroller

+

+

+


Determinationof


pwm

pwm



voltage

120596lowastm

120596m

minus

E1 E5 E7

minus

minus

minus

120579r

PMBLmotor

Tlowaste

+ Vdc minus

R

ilowastas

ilowastbs

ilowastcs

PWMinverter



119875119892 (119905) = 31198750 + 31198756 sin 6120596119905 + 311987512 sin 12120596119905 (7)


119879119890 (119905) = 119875119892120596119903 = 1198790 + 1198796 sin 6120596119905 + 11987912 sin 12120596119905 (8)

where


(9)


[[[


[119879000]]] (10)





[[[119868111986851198687]]]= [[[10063minus004700235

]]]21205961199033 1198790 (11)





0 005 01 0150

1000

Speed response

0 005 01 015

020

Phase current

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

010

0 005 01 0150


013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

12141618

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

minus20

minus10

i a(A

)Zo

om o

fia

ia

Zoom

of

Wre

famp

Wm

(rpm

)

iaref

Tem

Tem

(Nmiddotm

)






013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

010


013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

101520

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014


013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

81216

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014


13

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

1216

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

i a(A

)i a

(A)

i a(A

)

minus10

010

minus10

010

minus10

Tem

(Nmiddotm

)Tem

(Nmiddotm

)Tem

(Nmiddotm

)






Referencemodel (RM)

Adaptivemodel (AM)

Correctionmechanism

Output RM

Output AM

Inputs+

_








120579119903 = 1205872 minus tanminus1 (119890120573119890120572) (12)



119909119879 = [120579119903 120596119903 119894119904120572 119894119904120573] (13)



119889120579119903119889119905 = 120596119903 (14)

119889120596119898119889119905 = 1119869 (32 (119890120572119894120572 + 119890120573119894120573120596119898 ) minus 119879119871 minus 119861120596119898) (15)

119889119894119904120572119889119905 = minus119877119904119871 119904 119894119904120572 +1119871 119904 V119904120572 minus

1119871 119904 119890120572 (16)

119889119894119904120573119889119905 = minus119877119904119871 119904 119894119904120573 +1119871 119904 V119904120573 minus

1119871 119904 119890120573 (17)


= 119860119909 + 119861119906 + Φ (119909 119906) + 119863119891 (119910 119906) 119910 = 119862119909 (18)






1 = 119860111199091 + 119860121199092 + 1198611119906 + Φ1 (119909 119906) 2 = 119860211199091 + 119860221199092 + 1198612119906 + Φ2 (119909 119906) + 1198632119891 (119910 119906) 119910 = 11986221199092

(19)



119879 = [1198682 1198710 1198682]

119879minus1 = [1198682 minus1198710 1198682 ] (20)



2 = 119860211199111 + (11986022 minus 11986021119871) 1199112 + 1198612119906 + Φ2 (119879minus1119911 119906)+ 1198632119891 (119910 119906)

119910 = 11986221199112

(21)



1199112 = 119860211 + (11986022 minus 11986021119871) 2 + 1198612119906 + Φ2 (119879minus1 119906)+ 1198632119891 (119910 119906)

119910 = 11986222

(22)

where

V = 120588 sdot sign (119910 minus 11986222) (23)




(25)




Current controllers

+

Speed controller

+

+

+

pwm

pwm

pwm

Determination of


voltage


observer

PMBLmotor

Determinationof

three-phase

referenceharmonic

currents

harmonic

120596lowastm

minus

E1 E5 E7

minus

minus

minusTlowaste

m120579r

is120572 is120573 s120572 s120573

PWMinverter

ilowastas

ilowastbs

ilowastcs

+ Vdc minus



1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870 (27)


[11986011 + 11987111986021 0119862211986021 1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870] (28)


119878 = (1198901 119890119910) | 119890119910 = 0 (29)




0 005 01 0150

100200300400


0 005 01 0150

50010001500


0 005 01 015

05


Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

minus10

minus5

(rpm

)W

mamp

Wmh

at120579 r

(deg

)




0 005 01 0150

1020304050


012 0125 013 0135 014 0145 015101214161820

Time (Sec)

Time (Sec)

Teamp

Tlo

ad(N

middotm)

Zoom

(Nmiddotm

)


0 005 01 015

020

Current response

014

014

1

014

2

014

3

014

4

014

5

014

6

014

7

014

8

014

9

010

Zoom

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

0

Curr

ents

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

910111213

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

ref

minus20

minus10

10

minus10

iref

i qamp

i qre

f(A

)i a

ampI r

ef(A

)




008 009 01 011 012 013 014 015 016

0200


008 009 01 011 012 013 014 015 0160

200400


0 002 004 006 008 01 012 014 016 018 020

1020

Erro

r (de

g)

0 002 004 006 008 01 012 014 016 018 020

1000


008 009 01 011 012 013 014 015 016

05

Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

minus200

minus5

(rpm

)W

mamp

Wmh

ate a

(V)

120579 r(d

eg)





0 002 004 006 008 01 012 014 016 018 020

204060


008 009 01 011 012 013 014 015 01610

15

20

0 002 004 006 008 01 012 014 016 018 02

0

50Current response

008 009 01 011 012 013 014 015 016

010

Zoom

(A)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Zoom

(Nmiddotm

)

minus50

minus10

Teamp

Tlo

ad(N

middotm)

i aamp

I ref

(A)


6 Conclusion



Appendix


119860 =

11986011⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 10 00 00 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟11986021

11986012⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0 0

0 0minus 119877119871 119904 00 minus 119877119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

11986022

119861 =

1198611⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 01119871 119904 00 1119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198612

119862 =1198621⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[ 0 00 0

1198622⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 0

0 1 ]

119863119891 (119910 119906) =

1198631⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 0

minus01119877119904119871 119904 00 minus01119877119904119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198632

Φ (119909 119906) = [1119869 32119870119890 (1198752 )2 (119894119904120572 (sin 120579119903 + 025 sin 3120579119903


(A1)






(B1)

where

119875 fl 119875 [1198682 119871] 119860 fl [1198601111986021]

(B2)



(B3)

As a result

le 1198901198791 (119860119879119875119879 + 119875119860) 1198901 + 12057611989011987911198751198751198791198901+ 1120576 (119871Φ)2 1003817100381710038171003817119890110038171003817100381710038172

= 1198901198791 (119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868) 1198901(B4)


119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868 lt minus120572119875 (B5)


le minus12057211989011987911198751198901 = minus120572119881 (B6)


Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










ea

eb

ec

MM

MLs

Ls

Ls

Rs

Rs

Rs

ia

ib

ic

n

an

bn

cn





[[[V119886119899V119887119899V119888119899

]]]= [[[119877119904 0 00 119877119904 00 0 119877119904

]]]times [[[119894119886119894119887119894119888]]]

+ [[[119871 119904 minus119872 0 0

0 119871 119904 minus119872 00 0 119871 119904 minus119872

]]]119889119889119905 [[[

119894119886119894119887119894119888]]]

+ [[[119890119886119890119887119890119888]]]

(1)


119879119890 = 119890119886119894119886 + 119890119887119894119887 + 119890119888119894119888120596119898 (2)


119879119890 = 119879119871 + 119869119889120596119898119889119905 + 119861119891120596119898 (3)











ilowastA

ilowast

ilowast

B

ilowastC

Tlowaste

120579

120579r

120579r

iABC






119890119886 (119905) = 1198641 sin120596119905 + 1198643 sin 3120596119905 + 1198645 sin 5120596119905+ 1198647 sin 7120596119905 (4)


119894119886 (119905) = 1198681 sin120596119905 + 1198685 sin 5120596119905 + 1198687 sin 7120596119905 (5)



(6)



+

Speedcontroller

+

+

+


Determinationof


pwm

pwm



voltage

120596lowastm

120596m

minus

E1 E5 E7

minus

minus

minus

120579r

PMBLmotor

Tlowaste

+ Vdc minus

R

ilowastas

ilowastbs

ilowastcs

PWMinverter



119875119892 (119905) = 31198750 + 31198756 sin 6120596119905 + 311987512 sin 12120596119905 (7)


119879119890 (119905) = 119875119892120596119903 = 1198790 + 1198796 sin 6120596119905 + 11987912 sin 12120596119905 (8)

where


(9)


[[[


[119879000]]] (10)





[[[119868111986851198687]]]= [[[10063minus004700235

]]]21205961199033 1198790 (11)





0 005 01 0150

1000

Speed response

0 005 01 015

020

Phase current

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

010

0 005 01 0150


013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

12141618

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

minus20

minus10

i a(A

)Zo

om o

fia

ia

Zoom

of

Wre

famp

Wm

(rpm

)

iaref

Tem

Tem

(Nmiddotm

)






013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

010


013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

101520

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014


013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

81216

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014


13

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

1216

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

i a(A

)i a

(A)

i a(A

)

minus10

010

minus10

010

minus10

Tem

(Nmiddotm

)Tem

(Nmiddotm

)Tem

(Nmiddotm

)






Referencemodel (RM)

Adaptivemodel (AM)

Correctionmechanism

Output RM

Output AM

Inputs+

_








120579119903 = 1205872 minus tanminus1 (119890120573119890120572) (12)



119909119879 = [120579119903 120596119903 119894119904120572 119894119904120573] (13)



119889120579119903119889119905 = 120596119903 (14)

119889120596119898119889119905 = 1119869 (32 (119890120572119894120572 + 119890120573119894120573120596119898 ) minus 119879119871 minus 119861120596119898) (15)

119889119894119904120572119889119905 = minus119877119904119871 119904 119894119904120572 +1119871 119904 V119904120572 minus

1119871 119904 119890120572 (16)

119889119894119904120573119889119905 = minus119877119904119871 119904 119894119904120573 +1119871 119904 V119904120573 minus

1119871 119904 119890120573 (17)


= 119860119909 + 119861119906 + Φ (119909 119906) + 119863119891 (119910 119906) 119910 = 119862119909 (18)






1 = 119860111199091 + 119860121199092 + 1198611119906 + Φ1 (119909 119906) 2 = 119860211199091 + 119860221199092 + 1198612119906 + Φ2 (119909 119906) + 1198632119891 (119910 119906) 119910 = 11986221199092

(19)



119879 = [1198682 1198710 1198682]

119879minus1 = [1198682 minus1198710 1198682 ] (20)



2 = 119860211199111 + (11986022 minus 11986021119871) 1199112 + 1198612119906 + Φ2 (119879minus1119911 119906)+ 1198632119891 (119910 119906)

119910 = 11986221199112

(21)



1199112 = 119860211 + (11986022 minus 11986021119871) 2 + 1198612119906 + Φ2 (119879minus1 119906)+ 1198632119891 (119910 119906)

119910 = 11986222

(22)

where

V = 120588 sdot sign (119910 minus 11986222) (23)




(25)




Current controllers

+

Speed controller

+

+

+

pwm

pwm

pwm

Determination of


voltage


observer

PMBLmotor

Determinationof

three-phase

referenceharmonic

currents

harmonic

120596lowastm

minus

E1 E5 E7

minus

minus

minusTlowaste

m120579r

is120572 is120573 s120572 s120573

PWMinverter

ilowastas

ilowastbs

ilowastcs

+ Vdc minus



1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870 (27)


[11986011 + 11987111986021 0119862211986021 1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870] (28)


119878 = (1198901 119890119910) | 119890119910 = 0 (29)




0 005 01 0150

100200300400


0 005 01 0150

50010001500


0 005 01 015

05


Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

minus10

minus5

(rpm

)W

mamp

Wmh

at120579 r

(deg

)




0 005 01 0150

1020304050


012 0125 013 0135 014 0145 015101214161820

Time (Sec)

Time (Sec)

Teamp

Tlo

ad(N

middotm)

Zoom

(Nmiddotm

)


0 005 01 015

020

Current response

014

014

1

014

2

014

3

014

4

014

5

014

6

014

7

014

8

014

9

010

Zoom

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

0

Curr

ents

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

910111213

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

ref

minus20

minus10

10

minus10

iref

i qamp

i qre

f(A

)i a

ampI r

ef(A

)




008 009 01 011 012 013 014 015 016

0200


008 009 01 011 012 013 014 015 0160

200400


0 002 004 006 008 01 012 014 016 018 020

1020

Erro

r (de

g)

0 002 004 006 008 01 012 014 016 018 020

1000


008 009 01 011 012 013 014 015 016

05

Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

minus200

minus5

(rpm

)W

mamp

Wmh

ate a

(V)

120579 r(d

eg)





0 002 004 006 008 01 012 014 016 018 020

204060


008 009 01 011 012 013 014 015 01610

15

20

0 002 004 006 008 01 012 014 016 018 02

0

50Current response

008 009 01 011 012 013 014 015 016

010

Zoom

(A)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Zoom

(Nmiddotm

)

minus50

minus10

Teamp

Tlo

ad(N

middotm)

i aamp

I ref

(A)


6 Conclusion



Appendix


119860 =

11986011⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 10 00 00 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟11986021

11986012⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0 0

0 0minus 119877119871 119904 00 minus 119877119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

11986022

119861 =

1198611⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 01119871 119904 00 1119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198612

119862 =1198621⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[ 0 00 0

1198622⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 0

0 1 ]

119863119891 (119910 119906) =

1198631⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 0

minus01119877119904119871 119904 00 minus01119877119904119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198632

Φ (119909 119906) = [1119869 32119870119890 (1198752 )2 (119894119904120572 (sin 120579119903 + 025 sin 3120579119903


(A1)






(B1)

where

119875 fl 119875 [1198682 119871] 119860 fl [1198601111986021]

(B2)



(B3)

As a result

le 1198901198791 (119860119879119875119879 + 119875119860) 1198901 + 12057611989011987911198751198751198791198901+ 1120576 (119871Φ)2 1003817100381710038171003817119890110038171003817100381710038172

= 1198901198791 (119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868) 1198901(B4)


119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868 lt minus120572119875 (B5)


le minus12057211989011987911198751198901 = minus120572119881 (B6)


Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














ilowastA

ilowast

ilowast

B

ilowastC

Tlowaste

120579

120579r

120579r

iABC






119890119886 (119905) = 1198641 sin120596119905 + 1198643 sin 3120596119905 + 1198645 sin 5120596119905+ 1198647 sin 7120596119905 (4)


119894119886 (119905) = 1198681 sin120596119905 + 1198685 sin 5120596119905 + 1198687 sin 7120596119905 (5)



(6)



+

Speedcontroller

+

+

+


Determinationof


pwm

pwm



voltage

120596lowastm

120596m

minus

E1 E5 E7

minus

minus

minus

120579r

PMBLmotor

Tlowaste

+ Vdc minus

R

ilowastas

ilowastbs

ilowastcs

PWMinverter



119875119892 (119905) = 31198750 + 31198756 sin 6120596119905 + 311987512 sin 12120596119905 (7)


119879119890 (119905) = 119875119892120596119903 = 1198790 + 1198796 sin 6120596119905 + 11987912 sin 12120596119905 (8)

where


(9)


[[[


[119879000]]] (10)





[[[119868111986851198687]]]= [[[10063minus004700235

]]]21205961199033 1198790 (11)





0 005 01 0150

1000

Speed response

0 005 01 015

020

Phase current

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

010

0 005 01 0150


013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

12141618

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

minus20

minus10

i a(A

)Zo

om o

fia

ia

Zoom

of

Wre

famp

Wm

(rpm

)

iaref

Tem

Tem

(Nmiddotm

)






013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

010


013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

101520

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014


013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

81216

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014


13

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

1216

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

i a(A

)i a

(A)

i a(A

)

minus10

010

minus10

010

minus10

Tem

(Nmiddotm

)Tem

(Nmiddotm

)Tem

(Nmiddotm

)






Referencemodel (RM)

Adaptivemodel (AM)

Correctionmechanism

Output RM

Output AM

Inputs+

_








120579119903 = 1205872 minus tanminus1 (119890120573119890120572) (12)



119909119879 = [120579119903 120596119903 119894119904120572 119894119904120573] (13)



119889120579119903119889119905 = 120596119903 (14)

119889120596119898119889119905 = 1119869 (32 (119890120572119894120572 + 119890120573119894120573120596119898 ) minus 119879119871 minus 119861120596119898) (15)

119889119894119904120572119889119905 = minus119877119904119871 119904 119894119904120572 +1119871 119904 V119904120572 minus

1119871 119904 119890120572 (16)

119889119894119904120573119889119905 = minus119877119904119871 119904 119894119904120573 +1119871 119904 V119904120573 minus

1119871 119904 119890120573 (17)


= 119860119909 + 119861119906 + Φ (119909 119906) + 119863119891 (119910 119906) 119910 = 119862119909 (18)






1 = 119860111199091 + 119860121199092 + 1198611119906 + Φ1 (119909 119906) 2 = 119860211199091 + 119860221199092 + 1198612119906 + Φ2 (119909 119906) + 1198632119891 (119910 119906) 119910 = 11986221199092

(19)



119879 = [1198682 1198710 1198682]

119879minus1 = [1198682 minus1198710 1198682 ] (20)



2 = 119860211199111 + (11986022 minus 11986021119871) 1199112 + 1198612119906 + Φ2 (119879minus1119911 119906)+ 1198632119891 (119910 119906)

119910 = 11986221199112

(21)



1199112 = 119860211 + (11986022 minus 11986021119871) 2 + 1198612119906 + Φ2 (119879minus1 119906)+ 1198632119891 (119910 119906)

119910 = 11986222

(22)

where

V = 120588 sdot sign (119910 minus 11986222) (23)




(25)




Current controllers

+

Speed controller

+

+

+

pwm

pwm

pwm

Determination of


voltage


observer

PMBLmotor

Determinationof

three-phase

referenceharmonic

currents

harmonic

120596lowastm

minus

E1 E5 E7

minus

minus

minusTlowaste

m120579r

is120572 is120573 s120572 s120573

PWMinverter

ilowastas

ilowastbs

ilowastcs

+ Vdc minus



1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870 (27)


[11986011 + 11987111986021 0119862211986021 1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870] (28)


119878 = (1198901 119890119910) | 119890119910 = 0 (29)




0 005 01 0150

100200300400


0 005 01 0150

50010001500


0 005 01 015

05


Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

minus10

minus5

(rpm

)W

mamp

Wmh

at120579 r

(deg

)




0 005 01 0150

1020304050


012 0125 013 0135 014 0145 015101214161820

Time (Sec)

Time (Sec)

Teamp

Tlo

ad(N

middotm)

Zoom

(Nmiddotm

)


0 005 01 015

020

Current response

014

014

1

014

2

014

3

014

4

014

5

014

6

014

7

014

8

014

9

010

Zoom

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

0

Curr

ents

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

910111213

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

ref

minus20

minus10

10

minus10

iref

i qamp

i qre

f(A

)i a

ampI r

ef(A

)




008 009 01 011 012 013 014 015 016

0200


008 009 01 011 012 013 014 015 0160

200400


0 002 004 006 008 01 012 014 016 018 020

1020

Erro

r (de

g)

0 002 004 006 008 01 012 014 016 018 020

1000


008 009 01 011 012 013 014 015 016

05

Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

minus200

minus5

(rpm

)W

mamp

Wmh

ate a

(V)

120579 r(d

eg)





0 002 004 006 008 01 012 014 016 018 020

204060


008 009 01 011 012 013 014 015 01610

15

20

0 002 004 006 008 01 012 014 016 018 02

0

50Current response

008 009 01 011 012 013 014 015 016

010

Zoom

(A)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Zoom

(Nmiddotm

)

minus50

minus10

Teamp

Tlo

ad(N

middotm)

i aamp

I ref

(A)


6 Conclusion



Appendix


119860 =

11986011⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 10 00 00 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟11986021

11986012⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0 0

0 0minus 119877119871 119904 00 minus 119877119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

11986022

119861 =

1198611⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 01119871 119904 00 1119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198612

119862 =1198621⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[ 0 00 0

1198622⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 0

0 1 ]

119863119891 (119910 119906) =

1198631⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 0

minus01119877119904119871 119904 00 minus01119877119904119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198632

Φ (119909 119906) = [1119869 32119870119890 (1198752 )2 (119894119904120572 (sin 120579119903 + 025 sin 3120579119903


(A1)






(B1)

where

119875 fl 119875 [1198682 119871] 119860 fl [1198601111986021]

(B2)



(B3)

As a result

le 1198901198791 (119860119879119875119879 + 119875119860) 1198901 + 12057611989011987911198751198751198791198901+ 1120576 (119871Φ)2 1003817100381710038171003817119890110038171003817100381710038172

= 1198901198791 (119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868) 1198901(B4)


119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868 lt minus120572119875 (B5)


le minus12057211989011987911198751198901 = minus120572119881 (B6)


Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










+

Speedcontroller

+

+

+


Determinationof


pwm

pwm



voltage

120596lowastm

120596m

minus

E1 E5 E7

minus

minus

minus

120579r

PMBLmotor

Tlowaste

+ Vdc minus

R

ilowastas

ilowastbs

ilowastcs

PWMinverter



119875119892 (119905) = 31198750 + 31198756 sin 6120596119905 + 311987512 sin 12120596119905 (7)


119879119890 (119905) = 119875119892120596119903 = 1198790 + 1198796 sin 6120596119905 + 11987912 sin 12120596119905 (8)

where


(9)


[[[


[119879000]]] (10)





[[[119868111986851198687]]]= [[[10063minus004700235

]]]21205961199033 1198790 (11)





0 005 01 0150

1000

Speed response

0 005 01 015

020

Phase current

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

010

0 005 01 0150


013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

12141618

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

minus20

minus10

i a(A

)Zo

om o

fia

ia

Zoom

of

Wre

famp

Wm

(rpm

)

iaref

Tem

Tem

(Nmiddotm

)






013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

010


013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

101520

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014


013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

81216

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014


13

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

1216

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

i a(A

)i a

(A)

i a(A

)

minus10

010

minus10

010

minus10

Tem

(Nmiddotm

)Tem

(Nmiddotm

)Tem

(Nmiddotm

)






Referencemodel (RM)

Adaptivemodel (AM)

Correctionmechanism

Output RM

Output AM

Inputs+

_








120579119903 = 1205872 minus tanminus1 (119890120573119890120572) (12)



119909119879 = [120579119903 120596119903 119894119904120572 119894119904120573] (13)



119889120579119903119889119905 = 120596119903 (14)

119889120596119898119889119905 = 1119869 (32 (119890120572119894120572 + 119890120573119894120573120596119898 ) minus 119879119871 minus 119861120596119898) (15)

119889119894119904120572119889119905 = minus119877119904119871 119904 119894119904120572 +1119871 119904 V119904120572 minus

1119871 119904 119890120572 (16)

119889119894119904120573119889119905 = minus119877119904119871 119904 119894119904120573 +1119871 119904 V119904120573 minus

1119871 119904 119890120573 (17)


= 119860119909 + 119861119906 + Φ (119909 119906) + 119863119891 (119910 119906) 119910 = 119862119909 (18)






1 = 119860111199091 + 119860121199092 + 1198611119906 + Φ1 (119909 119906) 2 = 119860211199091 + 119860221199092 + 1198612119906 + Φ2 (119909 119906) + 1198632119891 (119910 119906) 119910 = 11986221199092

(19)



119879 = [1198682 1198710 1198682]

119879minus1 = [1198682 minus1198710 1198682 ] (20)



2 = 119860211199111 + (11986022 minus 11986021119871) 1199112 + 1198612119906 + Φ2 (119879minus1119911 119906)+ 1198632119891 (119910 119906)

119910 = 11986221199112

(21)



1199112 = 119860211 + (11986022 minus 11986021119871) 2 + 1198612119906 + Φ2 (119879minus1 119906)+ 1198632119891 (119910 119906)

119910 = 11986222

(22)

where

V = 120588 sdot sign (119910 minus 11986222) (23)




(25)




Current controllers

+

Speed controller

+

+

+

pwm

pwm

pwm

Determination of


voltage


observer

PMBLmotor

Determinationof

three-phase

referenceharmonic

currents

harmonic

120596lowastm

minus

E1 E5 E7

minus

minus

minusTlowaste

m120579r

is120572 is120573 s120572 s120573

PWMinverter

ilowastas

ilowastbs

ilowastcs

+ Vdc minus



1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870 (27)


[11986011 + 11987111986021 0119862211986021 1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870] (28)


119878 = (1198901 119890119910) | 119890119910 = 0 (29)




0 005 01 0150

100200300400


0 005 01 0150

50010001500


0 005 01 015

05


Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

minus10

minus5

(rpm

)W

mamp

Wmh

at120579 r

(deg

)




0 005 01 0150

1020304050


012 0125 013 0135 014 0145 015101214161820

Time (Sec)

Time (Sec)

Teamp

Tlo

ad(N

middotm)

Zoom

(Nmiddotm

)


0 005 01 015

020

Current response

014

014

1

014

2

014

3

014

4

014

5

014

6

014

7

014

8

014

9

010

Zoom

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

0

Curr

ents

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

910111213

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

ref

minus20

minus10

10

minus10

iref

i qamp

i qre

f(A

)i a

ampI r

ef(A

)




008 009 01 011 012 013 014 015 016

0200


008 009 01 011 012 013 014 015 0160

200400


0 002 004 006 008 01 012 014 016 018 020

1020

Erro

r (de

g)

0 002 004 006 008 01 012 014 016 018 020

1000


008 009 01 011 012 013 014 015 016

05

Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

minus200

minus5

(rpm

)W

mamp

Wmh

ate a

(V)

120579 r(d

eg)





0 002 004 006 008 01 012 014 016 018 020

204060


008 009 01 011 012 013 014 015 01610

15

20

0 002 004 006 008 01 012 014 016 018 02

0

50Current response

008 009 01 011 012 013 014 015 016

010

Zoom

(A)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Zoom

(Nmiddotm

)

minus50

minus10

Teamp

Tlo

ad(N

middotm)

i aamp

I ref

(A)


6 Conclusion



Appendix


119860 =

11986011⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 10 00 00 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟11986021

11986012⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0 0

0 0minus 119877119871 119904 00 minus 119877119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

11986022

119861 =

1198611⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 01119871 119904 00 1119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198612

119862 =1198621⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[ 0 00 0

1198622⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 0

0 1 ]

119863119891 (119910 119906) =

1198631⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 0

minus01119877119904119871 119904 00 minus01119877119904119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198632

Φ (119909 119906) = [1119869 32119870119890 (1198752 )2 (119894119904120572 (sin 120579119903 + 025 sin 3120579119903


(A1)






(B1)

where

119875 fl 119875 [1198682 119871] 119860 fl [1198601111986021]

(B2)



(B3)

As a result

le 1198901198791 (119860119879119875119879 + 119875119860) 1198901 + 12057611989011987911198751198751198791198901+ 1120576 (119871Φ)2 1003817100381710038171003817119890110038171003817100381710038172

= 1198901198791 (119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868) 1198901(B4)


119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868 lt minus120572119875 (B5)


le minus12057211989011987911198751198901 = minus120572119881 (B6)


Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 005 01 0150

1000

Speed response

0 005 01 015

020

Phase current

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

010

0 005 01 0150


013

013

1

013

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013

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013

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013

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013

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013

9

014

12141618

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

minus20

minus10

i a(A

)Zo

om o

fia

ia

Zoom

of

Wre

famp

Wm

(rpm

)

iaref

Tem

Tem

(Nmiddotm

)






013

013

1

013

2

013

3

013

4

013

5

013

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013

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013

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9

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010


013

013

1

013

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013

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013

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013

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013

7

013

8

013

9

014

101520

013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014


013

013

1

013

2

013

3

013

4

013

5

013

6

013

7

013

8

013

9

014

81216

013

013

1

013

2

013

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013

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013

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013

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013

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013

8

013

9

014


13

013

1

013

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013

3

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013

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013

6

013

7

013

8

013

9

014

1216

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

i a(A

)i a

(A)

i a(A

)

minus10

010

minus10

010

minus10

Tem

(Nmiddotm

)Tem

(Nmiddotm

)Tem

(Nmiddotm

)






Referencemodel (RM)

Adaptivemodel (AM)

Correctionmechanism

Output RM

Output AM

Inputs+

_








120579119903 = 1205872 minus tanminus1 (119890120573119890120572) (12)



119909119879 = [120579119903 120596119903 119894119904120572 119894119904120573] (13)



119889120579119903119889119905 = 120596119903 (14)

119889120596119898119889119905 = 1119869 (32 (119890120572119894120572 + 119890120573119894120573120596119898 ) minus 119879119871 minus 119861120596119898) (15)

119889119894119904120572119889119905 = minus119877119904119871 119904 119894119904120572 +1119871 119904 V119904120572 minus

1119871 119904 119890120572 (16)

119889119894119904120573119889119905 = minus119877119904119871 119904 119894119904120573 +1119871 119904 V119904120573 minus

1119871 119904 119890120573 (17)


= 119860119909 + 119861119906 + Φ (119909 119906) + 119863119891 (119910 119906) 119910 = 119862119909 (18)






1 = 119860111199091 + 119860121199092 + 1198611119906 + Φ1 (119909 119906) 2 = 119860211199091 + 119860221199092 + 1198612119906 + Φ2 (119909 119906) + 1198632119891 (119910 119906) 119910 = 11986221199092

(19)



119879 = [1198682 1198710 1198682]

119879minus1 = [1198682 minus1198710 1198682 ] (20)



2 = 119860211199111 + (11986022 minus 11986021119871) 1199112 + 1198612119906 + Φ2 (119879minus1119911 119906)+ 1198632119891 (119910 119906)

119910 = 11986221199112

(21)



1199112 = 119860211 + (11986022 minus 11986021119871) 2 + 1198612119906 + Φ2 (119879minus1 119906)+ 1198632119891 (119910 119906)

119910 = 11986222

(22)

where

V = 120588 sdot sign (119910 minus 11986222) (23)




(25)




Current controllers

+

Speed controller

+

+

+

pwm

pwm

pwm

Determination of


voltage


observer

PMBLmotor

Determinationof

three-phase

referenceharmonic

currents

harmonic

120596lowastm

minus

E1 E5 E7

minus

minus

minusTlowaste

m120579r

is120572 is120573 s120572 s120573

PWMinverter

ilowastas

ilowastbs

ilowastcs

+ Vdc minus



1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870 (27)


[11986011 + 11987111986021 0119862211986021 1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870] (28)


119878 = (1198901 119890119910) | 119890119910 = 0 (29)




0 005 01 0150

100200300400


0 005 01 0150

50010001500


0 005 01 015

05


Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

minus10

minus5

(rpm

)W

mamp

Wmh

at120579 r

(deg

)




0 005 01 0150

1020304050


012 0125 013 0135 014 0145 015101214161820

Time (Sec)

Time (Sec)

Teamp

Tlo

ad(N

middotm)

Zoom

(Nmiddotm

)


0 005 01 015

020

Current response

014

014

1

014

2

014

3

014

4

014

5

014

6

014

7

014

8

014

9

010

Zoom

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

0

Curr

ents

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

910111213

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

ref

minus20

minus10

10

minus10

iref

i qamp

i qre

f(A

)i a

ampI r

ef(A

)




008 009 01 011 012 013 014 015 016

0200


008 009 01 011 012 013 014 015 0160

200400


0 002 004 006 008 01 012 014 016 018 020

1020

Erro

r (de

g)

0 002 004 006 008 01 012 014 016 018 020

1000


008 009 01 011 012 013 014 015 016

05

Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

minus200

minus5

(rpm

)W

mamp

Wmh

ate a

(V)

120579 r(d

eg)





0 002 004 006 008 01 012 014 016 018 020

204060


008 009 01 011 012 013 014 015 01610

15

20

0 002 004 006 008 01 012 014 016 018 02

0

50Current response

008 009 01 011 012 013 014 015 016

010

Zoom

(A)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Zoom

(Nmiddotm

)

minus50

minus10

Teamp

Tlo

ad(N

middotm)

i aamp

I ref

(A)


6 Conclusion



Appendix


119860 =

11986011⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 10 00 00 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟11986021

11986012⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0 0

0 0minus 119877119871 119904 00 minus 119877119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

11986022

119861 =

1198611⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 01119871 119904 00 1119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198612

119862 =1198621⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[ 0 00 0

1198622⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 0

0 1 ]

119863119891 (119910 119906) =

1198631⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 0

minus01119877119904119871 119904 00 minus01119877119904119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198632

Φ (119909 119906) = [1119869 32119870119890 (1198752 )2 (119894119904120572 (sin 120579119903 + 025 sin 3120579119903


(A1)






(B1)

where

119875 fl 119875 [1198682 119871] 119860 fl [1198601111986021]

(B2)



(B3)

As a result

le 1198901198791 (119860119879119875119879 + 119875119860) 1198901 + 12057611989011987911198751198751198791198901+ 1120576 (119871Φ)2 1003817100381710038171003817119890110038171003817100381710038172

= 1198901198791 (119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868) 1198901(B4)


119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868 lt minus120572119875 (B5)


le minus12057211989011987911198751198901 = minus120572119881 (B6)


Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Referencemodel (RM)

Adaptivemodel (AM)

Correctionmechanism

Output RM

Output AM

Inputs+

_








120579119903 = 1205872 minus tanminus1 (119890120573119890120572) (12)



119909119879 = [120579119903 120596119903 119894119904120572 119894119904120573] (13)



119889120579119903119889119905 = 120596119903 (14)

119889120596119898119889119905 = 1119869 (32 (119890120572119894120572 + 119890120573119894120573120596119898 ) minus 119879119871 minus 119861120596119898) (15)

119889119894119904120572119889119905 = minus119877119904119871 119904 119894119904120572 +1119871 119904 V119904120572 minus

1119871 119904 119890120572 (16)

119889119894119904120573119889119905 = minus119877119904119871 119904 119894119904120573 +1119871 119904 V119904120573 minus

1119871 119904 119890120573 (17)


= 119860119909 + 119861119906 + Φ (119909 119906) + 119863119891 (119910 119906) 119910 = 119862119909 (18)






1 = 119860111199091 + 119860121199092 + 1198611119906 + Φ1 (119909 119906) 2 = 119860211199091 + 119860221199092 + 1198612119906 + Φ2 (119909 119906) + 1198632119891 (119910 119906) 119910 = 11986221199092

(19)



119879 = [1198682 1198710 1198682]

119879minus1 = [1198682 minus1198710 1198682 ] (20)



2 = 119860211199111 + (11986022 minus 11986021119871) 1199112 + 1198612119906 + Φ2 (119879minus1119911 119906)+ 1198632119891 (119910 119906)

119910 = 11986221199112

(21)



1199112 = 119860211 + (11986022 minus 11986021119871) 2 + 1198612119906 + Φ2 (119879minus1 119906)+ 1198632119891 (119910 119906)

119910 = 11986222

(22)

where

V = 120588 sdot sign (119910 minus 11986222) (23)




(25)




Current controllers

+

Speed controller

+

+

+

pwm

pwm

pwm

Determination of


voltage


observer

PMBLmotor

Determinationof

three-phase

referenceharmonic

currents

harmonic

120596lowastm

minus

E1 E5 E7

minus

minus

minusTlowaste

m120579r

is120572 is120573 s120572 s120573

PWMinverter

ilowastas

ilowastbs

ilowastcs

+ Vdc minus



1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870 (27)


[11986011 + 11987111986021 0119862211986021 1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870] (28)


119878 = (1198901 119890119910) | 119890119910 = 0 (29)




0 005 01 0150

100200300400


0 005 01 0150

50010001500


0 005 01 015

05


Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

minus10

minus5

(rpm

)W

mamp

Wmh

at120579 r

(deg

)




0 005 01 0150

1020304050


012 0125 013 0135 014 0145 015101214161820

Time (Sec)

Time (Sec)

Teamp

Tlo

ad(N

middotm)

Zoom

(Nmiddotm

)


0 005 01 015

020

Current response

014

014

1

014

2

014

3

014

4

014

5

014

6

014

7

014

8

014

9

010

Zoom

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

0

Curr

ents

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

910111213

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

ref

minus20

minus10

10

minus10

iref

i qamp

i qre

f(A

)i a

ampI r

ef(A

)




008 009 01 011 012 013 014 015 016

0200


008 009 01 011 012 013 014 015 0160

200400


0 002 004 006 008 01 012 014 016 018 020

1020

Erro

r (de

g)

0 002 004 006 008 01 012 014 016 018 020

1000


008 009 01 011 012 013 014 015 016

05

Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

minus200

minus5

(rpm

)W

mamp

Wmh

ate a

(V)

120579 r(d

eg)





0 002 004 006 008 01 012 014 016 018 020

204060


008 009 01 011 012 013 014 015 01610

15

20

0 002 004 006 008 01 012 014 016 018 02

0

50Current response

008 009 01 011 012 013 014 015 016

010

Zoom

(A)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Zoom

(Nmiddotm

)

minus50

minus10

Teamp

Tlo

ad(N

middotm)

i aamp

I ref

(A)


6 Conclusion



Appendix


119860 =

11986011⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 10 00 00 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟11986021

11986012⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0 0

0 0minus 119877119871 119904 00 minus 119877119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

11986022

119861 =

1198611⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 01119871 119904 00 1119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198612

119862 =1198621⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[ 0 00 0

1198622⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 0

0 1 ]

119863119891 (119910 119906) =

1198631⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 0

minus01119877119904119871 119904 00 minus01119877119904119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198632

Φ (119909 119906) = [1119869 32119870119890 (1198752 )2 (119894119904120572 (sin 120579119903 + 025 sin 3120579119903


(A1)






(B1)

where

119875 fl 119875 [1198682 119871] 119860 fl [1198601111986021]

(B2)



(B3)

As a result

le 1198901198791 (119860119879119875119879 + 119875119860) 1198901 + 12057611989011987911198751198751198791198901+ 1120576 (119871Φ)2 1003817100381710038171003817119890110038171003817100381710038172

= 1198901198791 (119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868) 1198901(B4)


119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868 lt minus120572119875 (B5)


le minus12057211989011987911198751198901 = minus120572119881 (B6)


Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119889120579119903119889119905 = 120596119903 (14)

119889120596119898119889119905 = 1119869 (32 (119890120572119894120572 + 119890120573119894120573120596119898 ) minus 119879119871 minus 119861120596119898) (15)

119889119894119904120572119889119905 = minus119877119904119871 119904 119894119904120572 +1119871 119904 V119904120572 minus

1119871 119904 119890120572 (16)

119889119894119904120573119889119905 = minus119877119904119871 119904 119894119904120573 +1119871 119904 V119904120573 minus

1119871 119904 119890120573 (17)


= 119860119909 + 119861119906 + Φ (119909 119906) + 119863119891 (119910 119906) 119910 = 119862119909 (18)






1 = 119860111199091 + 119860121199092 + 1198611119906 + Φ1 (119909 119906) 2 = 119860211199091 + 119860221199092 + 1198612119906 + Φ2 (119909 119906) + 1198632119891 (119910 119906) 119910 = 11986221199092

(19)



119879 = [1198682 1198710 1198682]

119879minus1 = [1198682 minus1198710 1198682 ] (20)



2 = 119860211199111 + (11986022 minus 11986021119871) 1199112 + 1198612119906 + Φ2 (119879minus1119911 119906)+ 1198632119891 (119910 119906)

119910 = 11986221199112

(21)



1199112 = 119860211 + (11986022 minus 11986021119871) 2 + 1198612119906 + Φ2 (119879minus1 119906)+ 1198632119891 (119910 119906)

119910 = 11986222

(22)

where

V = 120588 sdot sign (119910 minus 11986222) (23)




(25)




Current controllers

+

Speed controller

+

+

+

pwm

pwm

pwm

Determination of


voltage


observer

PMBLmotor

Determinationof

three-phase

referenceharmonic

currents

harmonic

120596lowastm

minus

E1 E5 E7

minus

minus

minusTlowaste

m120579r

is120572 is120573 s120572 s120573

PWMinverter

ilowastas

ilowastbs

ilowastcs

+ Vdc minus



1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870 (27)


[11986011 + 11987111986021 0119862211986021 1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870] (28)


119878 = (1198901 119890119910) | 119890119910 = 0 (29)




0 005 01 0150

100200300400


0 005 01 0150

50010001500


0 005 01 015

05


Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

minus10

minus5

(rpm

)W

mamp

Wmh

at120579 r

(deg

)




0 005 01 0150

1020304050


012 0125 013 0135 014 0145 015101214161820

Time (Sec)

Time (Sec)

Teamp

Tlo

ad(N

middotm)

Zoom

(Nmiddotm

)


0 005 01 015

020

Current response

014

014

1

014

2

014

3

014

4

014

5

014

6

014

7

014

8

014

9

010

Zoom

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

0

Curr

ents

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

910111213

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

ref

minus20

minus10

10

minus10

iref

i qamp

i qre

f(A

)i a

ampI r

ef(A

)




008 009 01 011 012 013 014 015 016

0200


008 009 01 011 012 013 014 015 0160

200400


0 002 004 006 008 01 012 014 016 018 020

1020

Erro

r (de

g)

0 002 004 006 008 01 012 014 016 018 020

1000


008 009 01 011 012 013 014 015 016

05

Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

minus200

minus5

(rpm

)W

mamp

Wmh

ate a

(V)

120579 r(d

eg)





0 002 004 006 008 01 012 014 016 018 020

204060


008 009 01 011 012 013 014 015 01610

15

20

0 002 004 006 008 01 012 014 016 018 02

0

50Current response

008 009 01 011 012 013 014 015 016

010

Zoom

(A)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Zoom

(Nmiddotm

)

minus50

minus10

Teamp

Tlo

ad(N

middotm)

i aamp

I ref

(A)


6 Conclusion



Appendix


119860 =

11986011⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 10 00 00 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟11986021

11986012⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0 0

0 0minus 119877119871 119904 00 minus 119877119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

11986022

119861 =

1198611⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 01119871 119904 00 1119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198612

119862 =1198621⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[ 0 00 0

1198622⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 0

0 1 ]

119863119891 (119910 119906) =

1198631⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 0

minus01119877119904119871 119904 00 minus01119877119904119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198632

Φ (119909 119906) = [1119869 32119870119890 (1198752 )2 (119894119904120572 (sin 120579119903 + 025 sin 3120579119903


(A1)






(B1)

where

119875 fl 119875 [1198682 119871] 119860 fl [1198601111986021]

(B2)



(B3)

As a result

le 1198901198791 (119860119879119875119879 + 119875119860) 1198901 + 12057611989011987911198751198751198791198901+ 1120576 (119871Φ)2 1003817100381710038171003817119890110038171003817100381710038172

= 1198901198791 (119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868) 1198901(B4)


119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868 lt minus120572119875 (B5)


le minus12057211989011987911198751198901 = minus120572119881 (B6)


Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Current controllers

+

Speed controller

+

+

+

pwm

pwm

pwm

Determination of


voltage


observer

PMBLmotor

Determinationof

three-phase

referenceharmonic

currents

harmonic

120596lowastm

minus

E1 E5 E7

minus

minus

minusTlowaste

m120579r

is120572 is120573 s120572 s120573

PWMinverter

ilowastas

ilowastbs

ilowastcs

+ Vdc minus



1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870 (27)


[11986011 + 11987111986021 0119862211986021 1198622 (11986022 minus 11986021119871)119862minus12 + 1198622119870] (28)


119878 = (1198901 119890119910) | 119890119910 = 0 (29)




0 005 01 0150

100200300400


0 005 01 0150

50010001500


0 005 01 015

05


Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

minus10

minus5

(rpm

)W

mamp

Wmh

at120579 r

(deg

)




0 005 01 0150

1020304050


012 0125 013 0135 014 0145 015101214161820

Time (Sec)

Time (Sec)

Teamp

Tlo

ad(N

middotm)

Zoom

(Nmiddotm

)


0 005 01 015

020

Current response

014

014

1

014

2

014

3

014

4

014

5

014

6

014

7

014

8

014

9

010

Zoom

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

0

Curr

ents

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

910111213

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

ref

minus20

minus10

10

minus10

iref

i qamp

i qre

f(A

)i a

ampI r

ef(A

)




008 009 01 011 012 013 014 015 016

0200


008 009 01 011 012 013 014 015 0160

200400


0 002 004 006 008 01 012 014 016 018 020

1020

Erro

r (de

g)

0 002 004 006 008 01 012 014 016 018 020

1000


008 009 01 011 012 013 014 015 016

05

Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

minus200

minus5

(rpm

)W

mamp

Wmh

ate a

(V)

120579 r(d

eg)





0 002 004 006 008 01 012 014 016 018 020

204060


008 009 01 011 012 013 014 015 01610

15

20

0 002 004 006 008 01 012 014 016 018 02

0

50Current response

008 009 01 011 012 013 014 015 016

010

Zoom

(A)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Zoom

(Nmiddotm

)

minus50

minus10

Teamp

Tlo

ad(N

middotm)

i aamp

I ref

(A)


6 Conclusion



Appendix


119860 =

11986011⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 10 00 00 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟11986021

11986012⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0 0

0 0minus 119877119871 119904 00 minus 119877119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

11986022

119861 =

1198611⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 01119871 119904 00 1119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198612

119862 =1198621⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[ 0 00 0

1198622⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 0

0 1 ]

119863119891 (119910 119906) =

1198631⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 0

minus01119877119904119871 119904 00 minus01119877119904119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198632

Φ (119909 119906) = [1119869 32119870119890 (1198752 )2 (119894119904120572 (sin 120579119903 + 025 sin 3120579119903


(A1)






(B1)

where

119875 fl 119875 [1198682 119871] 119860 fl [1198601111986021]

(B2)



(B3)

As a result

le 1198901198791 (119860119879119875119879 + 119875119860) 1198901 + 12057611989011987911198751198751198791198901+ 1120576 (119871Φ)2 1003817100381710038171003817119890110038171003817100381710038172

= 1198901198791 (119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868) 1198901(B4)


119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868 lt minus120572119875 (B5)


le minus12057211989011987911198751198901 = minus120572119881 (B6)


Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 005 01 0150

1020304050


012 0125 013 0135 014 0145 015101214161820

Time (Sec)

Time (Sec)

Teamp

Tlo

ad(N

middotm)

Zoom

(Nmiddotm

)


0 005 01 015

020

Current response

014

014

1

014

2

014

3

014

4

014

5

014

6

014

7

014

8

014

9

010

Zoom

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

0

Curr

ents

(A)

013

013

2

013

4

013

6

013

8

014

014

2

014

4

014

6

014

8

015

910111213

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

ref

minus20

minus10

10

minus10

iref

i qamp

i qre

f(A

)i a

ampI r

ef(A

)




008 009 01 011 012 013 014 015 016

0200


008 009 01 011 012 013 014 015 0160

200400


0 002 004 006 008 01 012 014 016 018 020

1020

Erro

r (de

g)

0 002 004 006 008 01 012 014 016 018 020

1000


008 009 01 011 012 013 014 015 016

05

Erro

r (rp

m)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

minus200

minus5

(rpm

)W

mamp

Wmh

ate a

(V)

120579 r(d

eg)





0 002 004 006 008 01 012 014 016 018 020

204060


008 009 01 011 012 013 014 015 01610

15

20

0 002 004 006 008 01 012 014 016 018 02

0

50Current response

008 009 01 011 012 013 014 015 016

010

Zoom

(A)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Zoom

(Nmiddotm

)

minus50

minus10

Teamp

Tlo

ad(N

middotm)

i aamp

I ref

(A)


6 Conclusion



Appendix


119860 =

11986011⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 10 00 00 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟11986021

11986012⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0 0

0 0minus 119877119871 119904 00 minus 119877119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

11986022

119861 =

1198611⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 01119871 119904 00 1119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198612

119862 =1198621⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[ 0 00 0

1198622⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 0

0 1 ]

119863119891 (119910 119906) =

1198631⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 0

minus01119877119904119871 119904 00 minus01119877119904119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198632

Φ (119909 119906) = [1119869 32119870119890 (1198752 )2 (119894119904120572 (sin 120579119903 + 025 sin 3120579119903


(A1)






(B1)

where

119875 fl 119875 [1198682 119871] 119860 fl [1198601111986021]

(B2)



(B3)

As a result

le 1198901198791 (119860119879119875119879 + 119875119860) 1198901 + 12057611989011987911198751198751198791198901+ 1120576 (119871Φ)2 1003817100381710038171003817119890110038171003817100381710038172

= 1198901198791 (119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868) 1198901(B4)


119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868 lt minus120572119875 (B5)


le minus12057211989011987911198751198901 = minus120572119881 (B6)


Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 002 004 006 008 01 012 014 016 018 020

204060


008 009 01 011 012 013 014 015 01610

15

20

0 002 004 006 008 01 012 014 016 018 02

0

50Current response

008 009 01 011 012 013 014 015 016

010

Zoom

(A)

Time (Sec)

Time (Sec)

Time (Sec)

Time (Sec)

Zoom

(Nmiddotm

)

minus50

minus10

Teamp

Tlo

ad(N

middotm)

i aamp

I ref

(A)


6 Conclusion



Appendix


119860 =

11986011⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 10 00 00 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟11986021

11986012⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0 0

0 0minus 119877119871 119904 00 minus 119877119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

11986022

119861 =

1198611⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 01119871 119904 00 1119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198612

119862 =1198621⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[ 0 00 0

1198622⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 0

0 1 ]

119863119891 (119910 119906) =

1198631⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

[[[[[[[[[

0 00 0

minus01119877119904119871 119904 00 minus01119877119904119871 119904

]]]]]]]]]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1198632

Φ (119909 119906) = [1119869 32119870119890 (1198752 )2 (119894119904120572 (sin 120579119903 + 025 sin 3120579119903


(A1)






(B1)

where

119875 fl 119875 [1198682 119871] 119860 fl [1198601111986021]

(B2)



(B3)

As a result

le 1198901198791 (119860119879119875119879 + 119875119860) 1198901 + 12057611989011987911198751198751198791198901+ 1120576 (119871Φ)2 1003817100381710038171003817119890110038171003817100381710038172

= 1198901198791 (119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868) 1198901(B4)


119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868 lt minus120572119875 (B5)


le minus12057211989011987911198751198901 = minus120572119881 (B6)


Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














(B1)

where

119875 fl 119875 [1198682 119871] 119860 fl [1198601111986021]

(B2)



(B3)

As a result

le 1198901198791 (119860119879119875119879 + 119875119860) 1198901 + 12057611989011987911198751198751198791198901+ 1120576 (119871Φ)2 1003817100381710038171003817119890110038171003817100381710038172

= 1198901198791 (119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868) 1198901(B4)


119860119879119875119879 + 119875119860 + 120576119875119875119879 + 1120576 (119871Φ)2 119868 lt minus120572119875 (B5)


le minus12057211989011987911198751198901 = minus120572119881 (B6)


Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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