Research ArticleAn Adaptive Nonlinear Extended State Observer forthe Sensorless Speed Control of a PMSM

Minggang Gan12 and Chenyi Wang12

1School of Automation Beijing Institute of Technology Beijing 100081 China2Key Laboratory of Intelligent Control and Decision of Complex Systems Beijing 100081 China

Correspondence should be addressed to Chenyi Wang chimes298163com

Received 13 December 2014 Accepted 5 March 2015

Academic Editor Andrzej Swierniak

Copyright copy 2015 M Gan and C WangThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a sensorless speed control strategy for a permanent-magnet synchronous motor (PMSM) based on an adaptivenonlinear extended state observer (ANLESO) In this paper an extended state observer (ESO) which takes back-EMF (backelectromotive force) as an extended state is used to estimate the rotor position and the rotor speed because of its simpler structureand higher accuracy Both linear ESO (LESO) and nonlinear ESO (NLESO) are considered to estimate the back-EMF of PMSMand NLESO is finally implemented due to its obvious advantage in convergence The convergence characteristics of the estimationerror of the observer are analyzed by the Lyapunov theory In order to take both stability and steady-state error into considerationan adaptive NLESO is proposed which adaptively adjusts the parameters of NLESO to a compromised value The performance ofthe proposed method was demonstrated by simulations and experiments

1 Introduction

In recent years permanent-magnet synchronousmotors haveincreasingly gained lots of applications due to their highefficiency high dynamic response and high torque to currentratio [1 2] Compared to DC motors PMSMs are superiorin robustness and overload capability and they do not needmechanical commutation As the fast switch components anddigital signal processors become economical PMSMs arewidely used as an alternative for energy saving applicationsBesides some countries have kept using low-cost inductionmotors due to their simple structure and easiness in manu-facturing However induction motors are less efficient thanPMSMs in terms of performance [3 4]

PMSM drives require a position sensor with high reso-lution to achieve an efficient vector control such as a shaftencoder or a resolver Unfortunately these sensors are expen-sive and have a limited lifetime Besides the size of the motorassembly will become larger Therefore many sensorlesscontrol methods have been developed to eliminate thesemechanical sensors In particular in some extreme envi-ronments where position sensors may work abnormally

sensorless control methods are more robust than those usingposition sensors [4]

By means of injecting high frequency signal into motorthe impedance difference caused by magnetic saturationwhich contains the information about the rotor positioncan be calculated [5] However injecting high frequencysignal brings noise and torque ripples Because of goodstandstill performance the method is usually used withinlow-speed range [6 7]The extended Kalman filter (EKF) wasalso employed for rotor speed and position estimation dueto its good dynamic performance and strong antijammingcapability [8] Algorithm complexity and large calculatingquantity are disadvantages of EKF which limits its practicalapplicationNeural network technology has a good advantagein terms of parameter identification and it has been usedin sensorless control system by many scholars [9] In thesepapers the rotor position is identified online by neuralnetwork However the method needs a long computationtime and brings great burden on the digital signal processingwhich gives rise to choke points in applications Zhang andLi proposed the model reference adaptive scheme (MRAS)to implement a sensorless control system The idea of MRAS

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 807615 14 pageshttpdxdoiorg1011552015807615

2 Mathematical Problems in Engineering

is to build a reference model of the PMSM system withoutcontaining unknown parameters and to obtain the differencebetween the reference model outputs and actual systemoutputs Next an appropriate adaptive lawwill be designed toadjust parameters of referencemodel in real time according tothe difference [10] But the accuracy of this method dependson the designed reference model which may prohibit theconvergence of the system Moreover in [11 12] the fluxlinkage is used as state variable and is estimated directly Someauthors used a slidingmode observer (SMO) approach [6 13ndash16] The structure of SMO is simple and convenient to beused in practical applications Unfortunately the estimationaccuracy of SMO would be affected by chattering

The extended state observer (ESO) first proposed byHuang andHan in [17] is the key step of the active disturbancerejection control (ADRC) that is taking off as a technologyafter numerous successful applications in engineering ESOhas simple structure and it is able to estimate unmodeleddynamics accurately in many cases In terms of ADRC aclass of nonlinear ESOs is designed to estimate the sum ofboth the states and external disturbances [18] After thatGao proposed a class of linear ESOs (LESO) and providedguidance on how to choose the optimal parameters in thecontroller design In this way the number of parameters tobe configured is reduced [19] At present ESO is mainly usedin control system to estimate disturbances and to compensatethem via a feed-forward cancellation technique [20ndash22]Besides ESO can be extended to multi-inputndashmulti-outputsystems as well [22]

In this paper a sensorless speed control scheme for aPMSM based on an ANLESO is proposed ESO is appliedto estimate the rotor position and rotor speed of PMSM forthe first time The information of the rotor position and therotor speed of PMSM can be calculated by estimating theback-EMF For this reason the estimation accuracy of back-EMF affects the accuracy of the position and speed directlyESOusually treats all the external disturbances andor systemuncertainties as the ldquototal disturbancerdquo and extends it as anadditional state variable [16] Therefore system states andunknown parts of system can be estimated by ESO in themeantime without knowing the model of the unknown parts(the derivative of the unknown parts) In this scheme back-EMF is the unknown part of the system and it would betreated as an additional state variable to be estimated Back-EMF contains the information of the rotor position and speedthat cannot be obtainedwithout sensor For this reason back-EMF cannot be modeled directly As mentioned previouslyESOwould treat back-EMF as ldquodisturbancerdquo and estimate theback-EMF accurately without knowing its model By meansof Lyapunov stability analysis an adaptive control scheme isadopted to balance stability and steady-state error The adap-tive control scheme could adaptively adjust the parametersof ESO to a compromised value and the steady-state errorwill converge to a minimum on the premise of system stableCompared to other sensorless controlmethods ANLESOhassimpler structure smaller chattering and higher accuracy

The state equations of the stator current in a stator-fixedreference frame and the back-EMF for each phase in the fixedframe of the PMSM are discussed in Section 2 In Section 3

an adaptive nonlinear extended state observer for the sen-sorless control scheme is designed and discussed Section 4gives the simulation results and Experimental results Finallyconclusions and suggestions for future improvements aregiven in Section 5

2 Mathematical Model

The voltages in the three phases can be transformed into thesynchronous coordinates in the two phases for vector controlThe state equations of the stator current written in a stator-fixed reference frame (120572120573) take the following form [23]

119889119894120572

119889119905= minus

119877

119871119894120572minus

1

119871119890120572+

1

119871119881120572 (1)

119889119894120573

119889119905= minus

119877

119871119894120573minus

1

119871119890120573+

1

119871119881120573 (2)

where 119894120572120573

119890120572120573

and 119881120572120573

represent the current electromotiveforce and voltage for each phase respectively 119877 and 119871

represent the stator resistance and inductance respectivelyThe back EMF for each phase can be represented in the

fixed frame as119890120572= minus 119870

119864120596119903sin 120579

119890120573= 119870119864120596119903cos 120579

(3)

where 119870119864 120596119903 and 120579 represent the magnetic flux of the PM

the electric angular velocity and the rotor angle respectively

3 Proposed Adaptive Nonlinear ESO

In the sensorless control system the back-EMF 119890120572 119890120573are

usually figured out by the information of control signals 119881120572

119881120573and output currents 119894

120572 119894120573 In other words from the system

formulated by (1) and (2) 119890120572and 119890

120573can be treated as the

unknown parts of the system Thus the back-EMF wouldbe extended as additional state variables In this sectionlinear ESO and nonlinear ESO are described respectively ForLESO the bounded-input bounded-output (BIBO) stabilitycan be reached For NLESO a Lyapunov stability analysisis shown At last an adaptive control scheme is designed inorder to improve the estimated accuracy

The block diagram of the sensorless control system forPMSM is shown in Figure 1 The space vector pulse widthmodulation (SVPWM) algorithm is used as modulationstrategy to generate a sinusoidal current of the stator Tocontrol the PMSM the three-phase coordinates need tobe transformed into 119889119902 synchronous coordinates by Clarketransformation and Park transformation An ADC (analog todigital converter) module is used to detect phase current andbus voltage The proportional-integral (PI) control is usedin speed loop and current loop All the sensorless controlalgorithms are simulated by MatLab and executed on a DSPchip

31 Linear ESO Note that this system is symmetric and weonly need to design an observer to estimate 119890

120572(the structure

Mathematical Problems in Engineering 3

3-phaseinverter

PWMHW

PMSMmotor

PWM1AB

PWM2ABPWM3AB

SVGEN

Park Clake

PIPISpd

Phase voltage

ADCHW

BusVolt

Adaptive nonlinear

ESO

Estimatedposition

and speed

PIRef

RefFdb

FdbFdbRef

Speed

V120572

V120572

V120573

V120573

I120572I120573

120579

I_park

Vdc

Ia

IbIc

TaTbTc

Id

Iq

Ds

Qs

As

Bs

Figure 1 Block diagram of the sensorless vector control of PMSM

and parameters of the observer) In this way the order of theobserver can be reduced

Set 1199091

= 119894120572and 119909

2= minus119890120572as augmented states ℎ is the

derivative of 1199092 which is a variable that we do not need to

know The state equation (1) is written in the following form

1= minus

119877

1198711199091+

1

1198711199092+

1

1198711199061

2= ℎ

(4)

The output can be written as

119910 = 119872[1199091

1199092

] (5)

where119872 = [1 0]Now 119909

2= minus119890120572can be estimated using a linear extended

state observer as follows

[1

2

] = [

[

minus119877

119871

1

1198710 0

]

]

[1199111

1199112

] + [

[

1

1198710

]

]

119906 + 119866 (119910 minus 1199111) (6)

where 1199111and 1199112are used to estimate the value of 119909

1and 119909

2

respectively 119866 = [minus1205731

minus 1205732]119879 is the observer gain vector

which can be obtained using any knownmethod for examplethe pole placement technique

Let 119890 = [1199091minus1199111

1199092minus1199112] then the error equation can be written as

119890 = (119860119900minus 119866119872) 119890 + 119867 (7)

where

119860119900= [

[

minus119877

119871

1

1198710 0

]

]

119867 = [0 ℎ]119879

(8)

Obviously the LESO is bounded-input bounded-output(BIBO) stable if (119860

119900minus 119866119862) is stable and ℎ is bounded [19]

32 Nonlinear ESO For the sake of reducing the steady-stateerror a nonlinear extended state observer is proposed asfollows

[1

2

] = [

[

minus119877

119871

1

1198710 0

]

]

[1199111

1199112

] + [

[

1

1198710

]

]

119906

+ [minus1205731sdot 1198901

minus1205732sdot10038161003816100381610038161198901

100381610038161003816100381612

sdot sign (1198901)]

(9)

Let 119890 = [1199091minus1199111

1199092minus1199112] then the error equation can be written as

1198901= minus (

119877

119871+ 1205731) 1198901+

1

1198711198902

1198902= minus ℎ minus 120573

2

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1)

(10)

When the steady state is reached we can obtain

minus ℎ minus 1205732

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1)

= minus (119877

119871+ 1205731) 1198901+

1

1198711198902= 0

(11)

Hence the steady-state errors are

1198901= (

ℎ

1205732

)

2

(12)

1198902= (119877 + 119871120573

1) (

ℎ

1205732

)

2

(13)

Theorem 1 For system (4) the state error of the nonlinearextended state observer (9) converges asymptotically if thefollowing conditions are satisfied

(C1) (119877119871 + 1205731) gt 0

(C2) 1205732gt 0

4 Mathematical Problems in Engineering

Proof The goal is to prove that there exists a positive definiteenergy function whose derivative is always made negativeThe following energy function is chosen [24]

119881 (1198901 1198902)

= 11986010038161003816100381610038161198901

100381610038161003816100381632

minus 11986111989011198902+ 1198621198902

2

= 11986010038161003816100381610038161198901

1003816100381610038161003816minus12 [

[

(1198901minus

11986110038161003816100381610038161198901

100381610038161003816100381612

21198601198902)

2

minus1198612 10038161003816100381610038161198901

1003816100381610038161003816

411986021198902

2

]

]

+ 1198621198902

2

= 11986010038161003816100381610038161198901

1003816100381610038161003816minus12

(1198901minus

11986110038161003816100381610038161198901

100381610038161003816100381612

21198601198902)

2

+ (119862 minus1198612

10038161003816100381610038161198901100381610038161003816100381612

4119860) 1198902

2

(14)

where 119860 119861 and 119862 are the parameters to be determinedSuitable 119860 119861 and 119862 need to be chosen to let the energyfunction meet the Lyapunov stability theorem

In order to guarantee that 119881(1198901 1198902) is positively definite

the following rules need to be followed

(R1) 119860 gt 0

(R2) 119862 gt 1198612|1198901|124119860 gt 0

Since |1198901|12 is bounded (R2) always can be satisfied by

choosing a suitable 119862Calculate the partial derivative of 119881(119890

1 1198902) as follows

120597119881

1205971198901

=3

2119860

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1) minus 119861119890

2

120597119881

1205971198902

= minus 1198611198901+ 2119862119890

2

(15)

The derivative of the energy function is as follows

=120597119881

1205971198901

1198901+

120597119881

1205971198902

1198902 (16)

By substituting (15) into (16) can be given as

= (3

2119860

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1) minus 119861119890

2)(minus(

119877

119871+ 1205731) 1198901+

1

1198711198902)

+ (minus1198611198901+ 2119862119890

2) (minusℎ minus 120573

2

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1))

(17)

From (17) the following expression can be deduced

= (3

2

119860

119871minus 2119862120573

2+ 119861(

119877

119871+ 1205731)10038161003816100381610038161198901

100381610038161003816100381612

)

sdot10038161003816100381610038161198901

1003816100381610038161003816minus14 10038161003816100381610038161198901

100381610038161003816100381634 sign (119890

1) 1198902

minus119861

1198711198902

2minus (

3

2119860(

119877

119871+ 1205731) minus 119861120573

2)10038161003816100381610038161198901

10038161003816100381610038162(34)

+ (119861ℎ1198901minus 2119888ℎ119890

2)

(18)

Assume that

119883 =3

2119860(

119877

119871+ 1205731) minus 119861120573

2

119884 = (3

2

119860

119871minus 2119862120573

2+ 119861(

119877

119871+ 1205731)10038161003816100381610038161198901

100381610038161003816100381612

)10038161003816100381610038161198901

1003816100381610038161003816minus14

119885 =119861

119871

(19)

Equation (18) can be rewritten as

= minus 11988310038161003816100381610038161198901

10038161003816100381610038162(34)

+ 11988410038161003816100381610038161198901

100381610038161003816100381634 sign (119890

1) 1198902

minus 1198851198902

2minus (minus119861ℎ119890

1+ 2119862ℎ119890

2)

= 1198811minus 1198812

(20)

where1198811= minus119883|119890

1|2(34)+119884|119890

1|34 sign(119890

1)1198902minus11988511989022is a quadra-

tic function with regard to |1198901|34 sign(119890

1) and 119890

2 1198812

=

(minus119861ℎ1198901+ 2119862ℎ119890

2) is a plane function with regard to 119890

1and 1198902

A necessary and sufficient condition for the positively definiteproperty of the quadratic function 119881

1can be obtained

A 119883 gt 0

B 119885 gt 0

C 1198842 minus 4119883119885 lt 0

FromA we have

(R3) (32)119860(119877119871 + 1205731) gt 119861120573

2

FromB we have

(R4) 119861119871 gt 0 which means 119861 gt 0

Then fromC we can derive that

(3

2

119860

119871minus 2119862120573

2+ 119861(

119877

119871+ 1205731)10038161003816100381610038161198901

100381610038161003816100381612

)2

100381610038161003816100381611989011003816100381610038161003816minus12

lt 4119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2)

(21)

Then the following expression can be obtained

(3

2

119860

119871minus 2119862120573

2+ 119861(

119877

119871+ 1205731)10038161003816100381610038161198901

100381610038161003816100381612

)

lt 2radic119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2)10038161003816100381610038161198901

100381610038161003816100381614

(22)

Assume that

119886 = 119861(119877

119871+ 1205731)

119887 = radic119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2)

119888 =3

2

119860

119871minus 2119862120573

2

119909 =10038161003816100381610038161198901

100381610038161003816100381614

(23)

Mathematical Problems in Engineering 5

Equation (22) can be rewritten as

119888 + 1198861199092

lt 2119887119909 (24)

It can be verified that there exists some meaningful 119909

satisfying inequation (24) and meeting the rules (R1)sim(R4)if and only if the following conditions are satisfied

(119877

119871+ 1205731) gt 0 120573

2gt 0 (25)

Therefore 119886 = 119861(119877119871 + 1205731) gt 0 and inequality (24)

denotes a parabola going upwards Due to the existence ofsome meaningful 119909 satisfying inequality (24) the vertex ofthe parabola must be negative

1198872

minus 119886119888 gt 0 (26)

Hence119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2) minus 119861(

119877

119871+ 1205731)(

3

2

119860

119871minus 2119862120573

2) gt 0

1198611205732(2119862(

119877

119871+ 1205731) minus

119861

119871) gt 0

2119862(119877

119871+ 1205731) gt

119861

119871

(27)

Inequality (26) is established between the two real roots119909 = (119887plusmnradic1198872 minus 119886119888)119886 of the quadratic equation 1198861199092minus2119887119909+119888 =

0 In order to separate the two roots as far as possible andmake the smaller one very small a very large 119861 needs to beselected which makes 119861120573

2≫ 0 and the choice of parameter

119862 is required to ensure 2119862(119877119871 + 1205731) minus 119861119871 gt 0 Finally

choose appropriate 119860 to make parameters 119861 and 119862 satisfyrules (R1)sim(R4) In this way 119881

1will be a negative definite

function with variable 1198901in a very wide range and arbitrary

1198902 Meanwhile the area in which = 119881

1minus1198812does not satisfy

negative conditions (the area gt 0) is the part above theintersecting line of the quadratic function 119881

1and the plane

function 1198812

The value of 1198901in the intersecting line is the same order

as 1198901= (119861ℎ119883)

2 which is the root of the following equation

119861ℎ1198901= 119883

1003816100381610038161003816119890110038161003816100381610038162(34)

(28)

Since 119883 is the same order as 1198611205732 the value of 119890

1in the

intersecting line is the same order as (ℎ1205732)2 Therefore a

large enough 1205732can guarantee that 119890

1is stable in a very wide

rangeThe larger 1205732we set the smaller the unstable boundary

of 1198901 and then it will result in a better convergenceIn the same way the value of 119890

2in the intersecting line

is the same order as 1198902

= (2119862ℎ119885) which is the root of thefollowing equation

2119862ℎ1198902= 1198851198902

2 (29)

Consider that 119885 = 119861 the value of 1198901in the intersecting

line is the same order as (2119862ℎ119861) From inequality (21) theinequality 2119862(119877 + 119871120573

1)119861 gt 1 holds Therefore a large

enough (119877 + 1198711205731) by setting 120573

1 can guarantee the existence

of appropriate 119861 and 119862 to make |1198902| = |(2119862ℎ119861)| small

enough

Yes

Yes

No

No

IF |e2| gtmax(h)R + L1205731

1205731 = 1205731 minus ΔV lowast 120572

120573out = 12057311205731 = 120573out 1205731 = 1205731 lowast p

IF 1205731 minus ΔV lowast 120572 gt 0

Figure 2 Flow chart of the algorithm to calculate 1205731

Remark 1 Obviously the state error would not actually goto zero but enter and is contained within a neighborhoodof the origin by this method Therefore by means of settingappropriate 120573

1and 120573

2 the performance required in practical

engineering can be achieved

Remark 2 Using a sigmoid function as the nonlinear func-tion in ESO can reduce chatting in the system [15] Consider

[1

2

] = [

[

minus119877

119871

1

1198710 0

]

]

[1198851

1198852

] + [

[

1

1198710

]

]

119906

+ [minus1205731sdot 1198901

minus1205732sdot10038161003816100381610038161198901

1003816100381610038161003816 sdot sigmoid (1198901)]

(30)

where

sigmoid (1198901) = (

2

1 + exp (minus1198861198901)) minus 1 (31)

33 Proposed Adaptive Nonlinear ESO Although a large (119877+

1198711205731) will render 119890

2stable in a wider range the steady-state

error still has to be considered here From (13) we knowthat a large (119877 + 119871120573

1) leads to the increase of 119890

2in steady

state The improved method proposed in this paper is toadaptively adjust the parameters ofNLESO to a compromisedvalue which takes both stability and steady-state error intoconsideration

Rewrite inequality (27) as the following expression

2119862

119861gt

1

119877 + 1198711205731

(32)

Thus the roots of (29) hold the following equality

100381610038161003816100381611989021003816100381610038161003816 =

2119862 |ℎ|

119861gt

|ℎ|

119877 + 1198711205731

(33)

6 Mathematical Problems in Engineering

0 005 01 015 02 025 03 035 04

0

5

Time (s)

Back

-EM

F (V

)

Estimated e120572Estimated e120573

minus10

minus5

(a)

0 005 01 015 02 025 03

10

Time (s)

Estimated e120572Estimated e120573

0

5

Back

-EM

F (V

)

minus10

minus5

(b)

Figure 3 Simulation waveforms of estimated back EMF when speed is 500 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

05

0 005 01 015 02 025 03 035 04Time (s)

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

Curr

ent (

A)

minus5

05

minus5

(a)

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

0

5

minus5

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

0

5

minus5

(b)

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

0

5

minus5

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

0

5

minus5

(c)

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

05

minus5

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

05

minus5

(d)

Figure 4 Simulation waveforms of actual and estimated 119894120572 119894120573when speed is 500 rmin and the load torque is 1 Nm (a) Actual and estimated

119894120572obtained by the proposed ANLESO (b) Actual and estimated 119894

120573obtained by the proposed ANLESO (c) Actual and estimated 119894

120572obtained

by the conventional SMO (d) Actual and estimated 119894120573obtained by the conventional SMO

This means that the lower bound of 1198902is larger than

|ℎ|(119877 + 1198711205731) 1205731is a design parameter so the size of 119877 + 119871120573

1

can be changed by means of adjusting 1205731 The main idea of

this method is to decrease 1205731gradually to reduce steady-state

error and increase 1205731when the systemmay become unstable

By several iterations the value of 1205731would approximate the

compromised value gradually

Figure 2 illustrates the algorithm to calculate 1205731 where

0 lt 120572 lt 1 and 119901 gt 1 are adjustment factors Δ119881 is the errorof command velocity and measured velocity

34 Rotor Position and Speed Estimate Using the estimatedback-EMF obtained by the ANLESO the position and speed

Mathematical Problems in Engineering 7

0 005 01 015 02 025 03 035 040

200

400

600

Time (s)

Roto

r spe

ed (r

min

)

Actual speedEstimated speed

(a)

0 005 01 015 02 025 03

0

200

400

600

Time (s)

Roto

r spe

ed (r

min

)

Estimated speedActual speed

(b)

Figure 5 Simulation waveforms of actual and estimated speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by theproposed ANLESO and (b) obtained by the conventional SMO

0

5

0

5

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)

0 005 01 015 02 025 03 035 04

05

Time (s)

0 005 01 015 02 025 03 035 04Time (s)

0 005 01 015 02 025 03 035 04Time (s)

minus5

(a)

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)0

5

0

5

0 005 01 015 02 025 03

010

Time (s)

0 005 01 015 02 025 03Time (s)

0 005 01 015 02 025 03Time (s)

minus10

(b)

Figure 6 Simulation waveforms of actual rotor position estimated rotor position and estimated error when speed is 500 rmin and the loadtorque is 1 Nm (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO

signals can be calculated easily According to (3) the rotorspeed can be obtained as follows

119903=

radic(119890120572

2 + 119890120573

2)

119870119864

(34)

Similarly the rotor position can be deduced from (3) asfollows

120579 = minus arctan[119890120572

119890120573

] (35)

4 Simulation and Experimental Results

41 Simulation Results In order to show the high-speedperformance of the proposed ANLESO it is necessary tocompare it with the conventional SMO through the Mat-LabSimulink programming environmentThe parameters ofPMSM used in simulation are listed in Table 1

Figures 3ndash6 show the simulation waveforms when thereference speed is changed from zero to 500 rmin and

Table 1 The parameters of PMSM

Parameters ValuesRated power 15 KWRated speed 2500 rminInput voltage (DC) 310VCurrent 6AStator resistance 118Ω

Stator inductance 5326mHRotational inertia 133 times 10minus3 kgsdotm2

Flux of permanent magnet 00356WbPoles 8

the load torque is 1 Nm In these figures part (a) gives thesimulation waveforms obtained by the proposed ANLESOand part (b) gives the simulation waveforms obtained bythe conventional SMO Estimated back-EMF actual andestimated phase currents actual and estimated rotor speed

8 Mathematical Problems in Engineering

015 02 025 03

0

20

40

Time (s)

Back

-EM

F (V

)

minus20

minus40

Estimated e120572Estimated e120573

(a)

Time (s)015 02 025 03

0

20

40

Back

-EM

F (V

)

minus20

minus40

Estimated e120572Estimated e120573

(b)

Figure 7 Simulation waveforms of estimated back EMF when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

(a)

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

(b)

015 02 025 03Time (s)

015 02 025 03Time (s)

0

5

Curr

ent (

A)

minus5

0

5

Curr

ent (

A)

minus5

(c)

0

5

015 02 025 03Time (s)

015 02 025 03Time (s)

0

5

Curr

ent (

A)

Curr

ent (

A)

minus5

minus5

(d)

Figure 8 Simulation waveforms of actual and estimated 119894120572 119894120573when speed is 2000 rmin and the load torque is 1 Nm (a) Actual and estimated

119894120572obtained by the proposed ANLESO (b) Actual and estimated 119894

120573obtained by the proposed ANLESO (c) Actual and estimated 119894

120572obtained

by the conventional SMO (d) Actual and estimated 119894120573obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 3ndash6 the accuracy of esti-mation has been greatly improved with the help of proposedANLESO Different from the conventional SMO the back-EMF in Figure 3(a) almost has no chattering Comparedwith Figures 4(c) and 4(d) the estimated results of phase

currents in Figures 4(a) and 4(b) are better Due to the goodperformance of estimating back-EMF and currents thechattering phenomenon of the estimated rotor position andspeed is reduced which can be seen from Figures 5(a) 5(b)6(a) and 6(b) In part (a) of these figures the estimated speedis almost the same as actual speed however the estimatedspeed in part (b) of these figures exhibits the chattering

Mathematical Problems in Engineering 9

Time (s)0 005 01 015 02 025 03 035 04

0

1000

2000

Roto

r spe

ed (r

min

)

Estimated speedActual speed

(a)

Time (s)

Estimated speedActual speed

0 005 01 015 02 025 03

0

1000

2000

Roto

r spe

ed (r

min

)

(b)

Figure 9 Simulation waveforms of actual and estimated speeds when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by theproposed ANLESO and (b) obtained by the conventional SMO

0

5

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)

0

5

015 02 025 03

05

Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

minus5

(a)

5

Roto

r pos

ition

(rad

)0

5

Roto

r pos

ition

(rad

)

0

010

015 02 025 03Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

Roto

r pos

ition

(rad

)

minus10

(b)

Figure 10 Simulation waveforms of actual rotor position estimated rotor position and estimated error when speed is 2000 rmin and theload torque is 1 Nm (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO

Figure 11 Platform of 15 kw PMSM sensorless control system based on DSP

phenomenon Besides it can be seen that the estimated errorof the position in part (a) is smaller than that shown in part(b) of these figures

Figures 7ndash10 show the two sets of simulation waveformswhen the reference speed is changed from zero to 2000 rmin

and the load torque is 1 Nm In these figures part (a) gives thesimulation waveforms obtained by the proposed ANLESOand part (b) gives the simulation waveforms obtained bythe conventional SMO Estimated back-EMF actual andestimated phase currents actual and estimated rotor speed

10 Mathematical Problems in Engineering

(a) (b)

Figure 12 Operating waveforms of estimated back EMF in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained bythe proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 13 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 14 Operating waveforms of estimated rotor speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 7ndash10 when the rotor speedis raised to 2000 rmin advantages of ANLESO becomemoreapparent than conventional SMO Comparing the estimatedback-EMF in Figures 7(a) and 7(b) we can find that theformer has smaller chattering As a result the estimationof rotor speed becomes more accurate by ANLESO whichcan be seen in Figures 9(a) and 9(b) The waveform formspresented in Figures 10(a) and 10(b) show the estimatedrotor position The position estimation error is reduced andthe accuracy of rotor position estimation is improved inFigure 10(a)

42 Experimental Results To further verify the performanceof the new SMO for estimating rotor position and speed an

experimental systemhas been designed to control the 130SJT-M060D (15 kw) sinusoidal PMSMmotor made by GSK

Figure 11 shows a photograph of the sensorless controlsystemTheHighVoltage DigitalMotor Control Kit made byTI is utilized in the system which contains power modulesswitching devices current and voltage sensing circuit ADCmodules and PWM DAC (digital to analog converter)modules A floating-point TMS320F28335 DSP is employedas the digital controller The current control cycle is 100120583sand the velocity control cycle is 1ms A 2 120583s dead time ischosen for the PWM switching All of the control variablesare monitored in real time by an oscilloscope after theyare converted to analog signals through the PWM DACsmodules which use an external low pass filter to generatethe waveformsThemotor parameters used in experiment aregiven in Table 1

Mathematical Problems in Engineering 11

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 15 Operating waveforms of rotor position and estimate error when speed is 500 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

(a) (b)

Figure 16 Operating waveforms of estimated back EMF in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtainedby the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 17 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

Figures 12ndash14 show the operating waveforms when thereference speed is changed from zero to 500 rmin and theload torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO Estimated back-EMF estimated phase

currents and estimated rotor speed are shown in thesefigures In Figure 15 actual rotor position estimated rotorposition and their error are shown

Figures 12(a) and 13(a) show the experimental resultsof the proposed ANLESO Compared to the waveformsobtained by SMO (see part (b) in Figures 12 and 13)

12 Mathematical Problems in Engineering

(a) (b)

Figure 18 Operatingwaveforms of estimated rotor speedswhen speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 19 Operating waveforms of rotor position and estimate error when speed is 2000 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

thewaveforms of the estimated back-EMF and current shownin Figures 12(a) and 13(a) are more accurate The waveformspresented in Figures 14(a) and 14(b) show the differences ofrotor speed estimated by two kinds of observers The formerspeed waveform has less chattering In other words ANLESOhas higher estimated precision

Figure 15 shows the experimental results of positionresponseThe estimated error shown in Figure 15(c) is smallerthan that in Figure 15(d) which verifies the high precision ofANLESO

Figures 16ndash18 show the operating waveforms when thereference speed is changed from zero to 2000 rmin andthe load torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO In these figures estimated back-EMFestimated phase currents and estimated rotor speed areshown In Figure 19 actual rotor position estimated rotorposition and their error are shown

Figures 12ndash15 show the operating waveforms when thereference speed is changed from zero to 2000 rmin Figures12 and 13 provide the operating waveforms obtained by theproposed ANLESO Figures 14 and 15 show the simulationwaveforms obtained by the conventional SMO In thesefigures estimated back-EMF estimated phase currents esti-mated rotor speed and actual and estimated rotor positionare shown

Figures 16ndash18 show the experimental results of estimatedback-EMF current and rotor speed when the rotor speed israised to 2000 rmin It can be seen that the waveforms inpart (a) of these figures have smaller chattering and higheraccuracy In this way the advantages of ANLESO have beenverified

As shown in Figure 19 when the rotor speed is raisedto 2000 rmin the advantages of ANLESO to estimate rotorposition becomemore apparent than conventional SMOTheestimation error in Figure 19(c) is significantly less than thatin Figure 19(d) That is to say ANLESO has higher accuracy

Mathematical Problems in Engineering 13

and smaller chattering when being applied to estimate posi-tion than SMO

5 Conclusions

In this paper an adaptive nonlinear extended state observerhas been designed for the sensorless control of a PMSMTheconvergence of this observer has been proved by means of aLyapunov stability analysis An adaptive algorithm is adoptedto calculate the compromised parameter of ESO in order totake both stability and steady-state error into considerationThegoodperformance of the proposed sensorless control sys-tem was verified by several experimental results The resultsshow that ANLESO has more advantages than conventionalSMO Compared to conventional SMO ANLESO has smallerchattering higher accuracy and no phase delay

In future works we will explore the stability of the ESOunder parameter uncertainties in sensorless control system

Conflict of Interests

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

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China (Grant no 61304097) and Projects ofMajor International (Regional) Joint Research ProgramNSFC (Grant no 61120106010)

References

[1] S Lu X Tang andB Song ldquoAdaptive PIF control for permanentmagnet synchronousmotors based onGPCrdquo Sensors vol 13 no1 pp 175ndash192 2013

[2] A Gomez-Espinosa V M Hernandez-Guzman M Bandala-Sanchez et al ldquoA new adaptive self-tuning Fourier Coefficientsalgorithm for periodic torque ripple minimization in perma-nentmagnet synchronousmotors (PMSM)rdquo Sensors vol 13 no3 pp 3831ndash3847 2013

[3] Y Zhang C M Akujuobi W H Ali C L Tolliver and L-SShieh ldquoLoad disturbance resistance speed controller design forPMSMrdquo IEEE Transactions on Industrial Electronics vol 53 no4 pp 1198ndash1208 2006

[4] J C Gamazo-Real E Vazquez-Sanchez and J Gomez-GilldquoPosition and speed control of brushless DC motors usingsensorless techniques and application trendsrdquo Sensors vol 10no 7 pp 6901ndash6947 2010

[5] J-H Jang S-K Sul J-I Ha K Ide and M Sawamura ldquoSen-sorless drive of surface-mounted permanent-magnet motor byhigh-frequency signal injection based on magnetic saliencyrdquoIEEE Transactions on Industry Applications vol 39 no 4 pp1031ndash1039 2003

[6] A Piippo M Hinkkanen and J Luomi ldquoSensorless control ofPMSM drives using a combination of voltage model and HFsignal injectionrdquo in Proceedings of the 39th IAS Annual MeetingConference Record of the IEEE Industry Applications Conferencevol 2 pp 964ndash970 October 2004

[7] G Wang R Yang and D Xu ldquoDSP-based control of sensorlessIPMSM drives for wide-speed-range operationrdquo IEEE Transac-tions on Industrial Electronics vol 60 no 2 pp 720ndash727 2013

[8] Z Wang Y Zheng Z Zou and M Cheng ldquoPosition sensorlesscontrol of interleaved CSI fed PMSM drive with extendedKalman filterrdquo IEEE Transactions on Magnetics vol 48 no 11pp 3688ndash3691 2012

[9] T D Batzel and K Y Lee ldquoAn approach to sensorless operationof the permanent-magnet synchronous motor using diagonallyrecurrent neural networksrdquo IEEE Transactions on Energy Con-version vol 18 no 1 pp 100ndash106 2003

[10] B Zhang andY Li ldquoAPMSMslidingmode control systembasedon model reference adaptive controlrdquo in Proceedings of the 3rdInternational Power Electronics and Motion Control Conference(IPEMC rsquo00) vol 1 pp 336ndash341 IEEE Beijing China 2000

[11] S Shinnaka ldquoNew sensorless vector control using minimum-order flux state observer in a stationary reference frame forpermanent-magnet synchronousmotorsrdquo IEEETransactions onIndustrial Electronics vol 53 no 2 pp 388ndash398 2006

[12] A Khlaief M Bendjedia M Boussak and M Gossa ldquoA non-linear observer for high-performance sensorless speed controlof IPMSM driverdquo IEEE Transactions on Power Electronics vol27 no 6 pp 3028ndash3040 2012

[13] H Lee and J Lee ldquoDesign of iterative sliding mode observerfor sensorless PMSM controlrdquo IEEE Transactions on ControlSystems Technology vol 21 no 4 pp 1394ndash1399 2013

[14] Y-S Han J-S Choi and Y-S Kim ldquoSensorless PMSM drivewith a sliding mode control based adaptive speed and statorresistance estimatorrdquo IEEE Transactions on Magnetics vol 36no 5 pp 3588ndash3591 2000

[15] V C Ilioudis and N I Margaris ldquoPMSM sensorless speedestimation based on sliding mode observersrdquo in Proceedings ofthe IEEE Power Electronics Specialists Conference (PESC rsquo08) pp2838ndash2843 IEEE Rhodes Greece June 2008

[16] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[17] Y Huang and J Han ldquoAnalysis and design for the second ordernonlinear continuous extended states observerrdquoChinese ScienceBulletin vol 45 no 21 pp 1938ndash1944 2000

[18] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[19] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence vol 6 pp 4989ndash4996 IEEE June 2003

[20] Y Xia Z Zhu and M Fu ldquoBack-stepping sliding mode controlfor missile systems based on an extended state observerrdquo IETControl Theory and Applications vol 5 no 1 pp 93ndash102 2011

[21] X Li and S Li ldquoExtended state observer based adaptive controlscheme for PMSM systemrdquo in Proceedings of the 33rd IEEEChinese Control Conference (CCC rsquo14) 2014

[22] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

14 Mathematical Problems in Engineering

[23] G ZhuAKaddouri L ADessaint andOAkhrif ldquoAnonlinearstate observer for the sensorless control of a permanent-magnetAC machinerdquo IEEE Transactions on Industrial Electronics vol45 no 2 pp 291ndash296 1996

[24] J Han Active Disturbance Rejection Control TechniquemdashTheTechnique for Estimating and Compensating the UncertaintiesNational Defense Industry Press Beijing China 2013 (Chi-nese)

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

is to build a reference model of the PMSM system withoutcontaining unknown parameters and to obtain the differencebetween the reference model outputs and actual systemoutputs Next an appropriate adaptive lawwill be designed toadjust parameters of referencemodel in real time according tothe difference [10] But the accuracy of this method dependson the designed reference model which may prohibit theconvergence of the system Moreover in [11 12] the fluxlinkage is used as state variable and is estimated directly Someauthors used a slidingmode observer (SMO) approach [6 13ndash16] The structure of SMO is simple and convenient to beused in practical applications Unfortunately the estimationaccuracy of SMO would be affected by chattering

The extended state observer (ESO) first proposed byHuang andHan in [17] is the key step of the active disturbancerejection control (ADRC) that is taking off as a technologyafter numerous successful applications in engineering ESOhas simple structure and it is able to estimate unmodeleddynamics accurately in many cases In terms of ADRC aclass of nonlinear ESOs is designed to estimate the sum ofboth the states and external disturbances [18] After thatGao proposed a class of linear ESOs (LESO) and providedguidance on how to choose the optimal parameters in thecontroller design In this way the number of parameters tobe configured is reduced [19] At present ESO is mainly usedin control system to estimate disturbances and to compensatethem via a feed-forward cancellation technique [20ndash22]Besides ESO can be extended to multi-inputndashmulti-outputsystems as well [22]

In this paper a sensorless speed control scheme for aPMSM based on an ANLESO is proposed ESO is appliedto estimate the rotor position and rotor speed of PMSM forthe first time The information of the rotor position and therotor speed of PMSM can be calculated by estimating theback-EMF For this reason the estimation accuracy of back-EMF affects the accuracy of the position and speed directlyESOusually treats all the external disturbances andor systemuncertainties as the ldquototal disturbancerdquo and extends it as anadditional state variable [16] Therefore system states andunknown parts of system can be estimated by ESO in themeantime without knowing the model of the unknown parts(the derivative of the unknown parts) In this scheme back-EMF is the unknown part of the system and it would betreated as an additional state variable to be estimated Back-EMF contains the information of the rotor position and speedthat cannot be obtainedwithout sensor For this reason back-EMF cannot be modeled directly As mentioned previouslyESOwould treat back-EMF as ldquodisturbancerdquo and estimate theback-EMF accurately without knowing its model By meansof Lyapunov stability analysis an adaptive control scheme isadopted to balance stability and steady-state error The adap-tive control scheme could adaptively adjust the parametersof ESO to a compromised value and the steady-state errorwill converge to a minimum on the premise of system stableCompared to other sensorless controlmethods ANLESOhassimpler structure smaller chattering and higher accuracy

The state equations of the stator current in a stator-fixedreference frame and the back-EMF for each phase in the fixedframe of the PMSM are discussed in Section 2 In Section 3

an adaptive nonlinear extended state observer for the sen-sorless control scheme is designed and discussed Section 4gives the simulation results and Experimental results Finallyconclusions and suggestions for future improvements aregiven in Section 5

2 Mathematical Model

The voltages in the three phases can be transformed into thesynchronous coordinates in the two phases for vector controlThe state equations of the stator current written in a stator-fixed reference frame (120572120573) take the following form [23]

119889119894120572

119889119905= minus

119877

119871119894120572minus

1

119871119890120572+

1

119871119881120572 (1)

119889119894120573

119889119905= minus

119877

119871119894120573minus

1

119871119890120573+

1

119871119881120573 (2)

where 119894120572120573

119890120572120573

and 119881120572120573

represent the current electromotiveforce and voltage for each phase respectively 119877 and 119871

represent the stator resistance and inductance respectivelyThe back EMF for each phase can be represented in the

fixed frame as119890120572= minus 119870

119864120596119903sin 120579

119890120573= 119870119864120596119903cos 120579

(3)

where 119870119864 120596119903 and 120579 represent the magnetic flux of the PM

the electric angular velocity and the rotor angle respectively

3 Proposed Adaptive Nonlinear ESO

In the sensorless control system the back-EMF 119890120572 119890120573are

usually figured out by the information of control signals 119881120572

119881120573and output currents 119894

120572 119894120573 In other words from the system

formulated by (1) and (2) 119890120572and 119890

120573can be treated as the

unknown parts of the system Thus the back-EMF wouldbe extended as additional state variables In this sectionlinear ESO and nonlinear ESO are described respectively ForLESO the bounded-input bounded-output (BIBO) stabilitycan be reached For NLESO a Lyapunov stability analysisis shown At last an adaptive control scheme is designed inorder to improve the estimated accuracy

The block diagram of the sensorless control system forPMSM is shown in Figure 1 The space vector pulse widthmodulation (SVPWM) algorithm is used as modulationstrategy to generate a sinusoidal current of the stator Tocontrol the PMSM the three-phase coordinates need tobe transformed into 119889119902 synchronous coordinates by Clarketransformation and Park transformation An ADC (analog todigital converter) module is used to detect phase current andbus voltage The proportional-integral (PI) control is usedin speed loop and current loop All the sensorless controlalgorithms are simulated by MatLab and executed on a DSPchip

31 Linear ESO Note that this system is symmetric and weonly need to design an observer to estimate 119890

120572(the structure

Mathematical Problems in Engineering 3

3-phaseinverter

PWMHW

PMSMmotor

PWM1AB

PWM2ABPWM3AB

SVGEN

Park Clake

PIPISpd

Phase voltage

ADCHW

BusVolt

Adaptive nonlinear

ESO

Estimatedposition

and speed

PIRef

RefFdb

FdbFdbRef

Speed

V120572

V120572

V120573

V120573

I120572I120573

120579

I_park

Vdc

Ia

IbIc

TaTbTc

Id

Iq

Ds

Qs

As

Bs

Figure 1 Block diagram of the sensorless vector control of PMSM

and parameters of the observer) In this way the order of theobserver can be reduced

Set 1199091

= 119894120572and 119909

2= minus119890120572as augmented states ℎ is the

derivative of 1199092 which is a variable that we do not need to

know The state equation (1) is written in the following form

1= minus

119877

1198711199091+

1

1198711199092+

1

1198711199061

2= ℎ

(4)

The output can be written as

119910 = 119872[1199091

1199092

] (5)

where119872 = [1 0]Now 119909

2= minus119890120572can be estimated using a linear extended

state observer as follows

[1

2

] = [

[

minus119877

119871

1

1198710 0

]

]

[1199111

1199112

] + [

[

1

1198710

]

]

119906 + 119866 (119910 minus 1199111) (6)

where 1199111and 1199112are used to estimate the value of 119909

1and 119909

2

respectively 119866 = [minus1205731

minus 1205732]119879 is the observer gain vector

which can be obtained using any knownmethod for examplethe pole placement technique

Let 119890 = [1199091minus1199111

1199092minus1199112] then the error equation can be written as

119890 = (119860119900minus 119866119872) 119890 + 119867 (7)

where

119860119900= [

[

minus119877

119871

1

1198710 0

]

]

119867 = [0 ℎ]119879

(8)

Obviously the LESO is bounded-input bounded-output(BIBO) stable if (119860

119900minus 119866119862) is stable and ℎ is bounded [19]

32 Nonlinear ESO For the sake of reducing the steady-stateerror a nonlinear extended state observer is proposed asfollows

[1

2

] = [

[

minus119877

119871

1

1198710 0

]

]

[1199111

1199112

] + [

[

1

1198710

]

]

119906

+ [minus1205731sdot 1198901

minus1205732sdot10038161003816100381610038161198901

100381610038161003816100381612

sdot sign (1198901)]

(9)

Let 119890 = [1199091minus1199111

1199092minus1199112] then the error equation can be written as

1198901= minus (

119877

119871+ 1205731) 1198901+

1

1198711198902

1198902= minus ℎ minus 120573

2

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1)

(10)

When the steady state is reached we can obtain

minus ℎ minus 1205732

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1)

= minus (119877

119871+ 1205731) 1198901+

1

1198711198902= 0

(11)

Hence the steady-state errors are

1198901= (

ℎ

1205732

)

2

(12)

1198902= (119877 + 119871120573

1) (

ℎ

1205732

)

2

(13)

Theorem 1 For system (4) the state error of the nonlinearextended state observer (9) converges asymptotically if thefollowing conditions are satisfied

(C1) (119877119871 + 1205731) gt 0

(C2) 1205732gt 0

4 Mathematical Problems in Engineering

Proof The goal is to prove that there exists a positive definiteenergy function whose derivative is always made negativeThe following energy function is chosen [24]

119881 (1198901 1198902)

= 11986010038161003816100381610038161198901

100381610038161003816100381632

minus 11986111989011198902+ 1198621198902

2

= 11986010038161003816100381610038161198901

1003816100381610038161003816minus12 [

[

(1198901minus

11986110038161003816100381610038161198901

100381610038161003816100381612

21198601198902)

2

minus1198612 10038161003816100381610038161198901

1003816100381610038161003816

411986021198902

2

]

]

+ 1198621198902

2

= 11986010038161003816100381610038161198901

1003816100381610038161003816minus12

(1198901minus

11986110038161003816100381610038161198901

100381610038161003816100381612

21198601198902)

2

+ (119862 minus1198612

10038161003816100381610038161198901100381610038161003816100381612

4119860) 1198902

2

(14)

where 119860 119861 and 119862 are the parameters to be determinedSuitable 119860 119861 and 119862 need to be chosen to let the energyfunction meet the Lyapunov stability theorem

In order to guarantee that 119881(1198901 1198902) is positively definite

the following rules need to be followed

(R1) 119860 gt 0

(R2) 119862 gt 1198612|1198901|124119860 gt 0

Since |1198901|12 is bounded (R2) always can be satisfied by

choosing a suitable 119862Calculate the partial derivative of 119881(119890

1 1198902) as follows

120597119881

1205971198901

=3

2119860

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1) minus 119861119890

2

120597119881

1205971198902

= minus 1198611198901+ 2119862119890

2

(15)

The derivative of the energy function is as follows

=120597119881

1205971198901

1198901+

120597119881

1205971198902

1198902 (16)

By substituting (15) into (16) can be given as

= (3

2119860

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1) minus 119861119890

2)(minus(

119877

119871+ 1205731) 1198901+

1

1198711198902)

+ (minus1198611198901+ 2119862119890

2) (minusℎ minus 120573

2

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1))

(17)

From (17) the following expression can be deduced

= (3

2

119860

119871minus 2119862120573

2+ 119861(

119877

119871+ 1205731)10038161003816100381610038161198901

100381610038161003816100381612

)

sdot10038161003816100381610038161198901

1003816100381610038161003816minus14 10038161003816100381610038161198901

100381610038161003816100381634 sign (119890

1) 1198902

minus119861

1198711198902

2minus (

3

2119860(

119877

119871+ 1205731) minus 119861120573

2)10038161003816100381610038161198901

10038161003816100381610038162(34)

+ (119861ℎ1198901minus 2119888ℎ119890

2)

(18)

Assume that

119883 =3

2119860(

119877

119871+ 1205731) minus 119861120573

2

119884 = (3

2

119860

119871minus 2119862120573

2+ 119861(

119877

119871+ 1205731)10038161003816100381610038161198901

100381610038161003816100381612

)10038161003816100381610038161198901

1003816100381610038161003816minus14

119885 =119861

119871

(19)

Equation (18) can be rewritten as

= minus 11988310038161003816100381610038161198901

10038161003816100381610038162(34)

+ 11988410038161003816100381610038161198901

100381610038161003816100381634 sign (119890

1) 1198902

minus 1198851198902

2minus (minus119861ℎ119890

1+ 2119862ℎ119890

2)

= 1198811minus 1198812

(20)

where1198811= minus119883|119890

1|2(34)+119884|119890

1|34 sign(119890

1)1198902minus11988511989022is a quadra-

tic function with regard to |1198901|34 sign(119890

1) and 119890

2 1198812

=

(minus119861ℎ1198901+ 2119862ℎ119890

2) is a plane function with regard to 119890

1and 1198902

A necessary and sufficient condition for the positively definiteproperty of the quadratic function 119881

1can be obtained

A 119883 gt 0

B 119885 gt 0

C 1198842 minus 4119883119885 lt 0

FromA we have

(R3) (32)119860(119877119871 + 1205731) gt 119861120573

2

FromB we have

(R4) 119861119871 gt 0 which means 119861 gt 0

Then fromC we can derive that

(3

2

119860

119871minus 2119862120573

2+ 119861(

119877

119871+ 1205731)10038161003816100381610038161198901

100381610038161003816100381612

)2

100381610038161003816100381611989011003816100381610038161003816minus12

lt 4119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2)

(21)

Then the following expression can be obtained

(3

2

119860

119871minus 2119862120573

2+ 119861(

119877

119871+ 1205731)10038161003816100381610038161198901

100381610038161003816100381612

)

lt 2radic119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2)10038161003816100381610038161198901

100381610038161003816100381614

(22)

Assume that

119886 = 119861(119877

119871+ 1205731)

119887 = radic119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2)

119888 =3

2

119860

119871minus 2119862120573

2

119909 =10038161003816100381610038161198901

100381610038161003816100381614

(23)

Mathematical Problems in Engineering 5

Equation (22) can be rewritten as

119888 + 1198861199092

lt 2119887119909 (24)

It can be verified that there exists some meaningful 119909

satisfying inequation (24) and meeting the rules (R1)sim(R4)if and only if the following conditions are satisfied

(119877

119871+ 1205731) gt 0 120573

2gt 0 (25)

Therefore 119886 = 119861(119877119871 + 1205731) gt 0 and inequality (24)

denotes a parabola going upwards Due to the existence ofsome meaningful 119909 satisfying inequality (24) the vertex ofthe parabola must be negative

1198872

minus 119886119888 gt 0 (26)

Hence119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2) minus 119861(

119877

119871+ 1205731)(

3

2

119860

119871minus 2119862120573

2) gt 0

1198611205732(2119862(

119877

119871+ 1205731) minus

119861

119871) gt 0

2119862(119877

119871+ 1205731) gt

119861

119871

(27)

Inequality (26) is established between the two real roots119909 = (119887plusmnradic1198872 minus 119886119888)119886 of the quadratic equation 1198861199092minus2119887119909+119888 =

0 In order to separate the two roots as far as possible andmake the smaller one very small a very large 119861 needs to beselected which makes 119861120573

2≫ 0 and the choice of parameter

119862 is required to ensure 2119862(119877119871 + 1205731) minus 119861119871 gt 0 Finally

choose appropriate 119860 to make parameters 119861 and 119862 satisfyrules (R1)sim(R4) In this way 119881

1will be a negative definite

function with variable 1198901in a very wide range and arbitrary

1198902 Meanwhile the area in which = 119881

1minus1198812does not satisfy

negative conditions (the area gt 0) is the part above theintersecting line of the quadratic function 119881

1and the plane

function 1198812

The value of 1198901in the intersecting line is the same order

as 1198901= (119861ℎ119883)

2 which is the root of the following equation

119861ℎ1198901= 119883

1003816100381610038161003816119890110038161003816100381610038162(34)

(28)

Since 119883 is the same order as 1198611205732 the value of 119890

1in the

intersecting line is the same order as (ℎ1205732)2 Therefore a

large enough 1205732can guarantee that 119890

1is stable in a very wide

rangeThe larger 1205732we set the smaller the unstable boundary

of 1198901 and then it will result in a better convergenceIn the same way the value of 119890

2in the intersecting line

is the same order as 1198902

= (2119862ℎ119885) which is the root of thefollowing equation

2119862ℎ1198902= 1198851198902

2 (29)

Consider that 119885 = 119861 the value of 1198901in the intersecting

line is the same order as (2119862ℎ119861) From inequality (21) theinequality 2119862(119877 + 119871120573

1)119861 gt 1 holds Therefore a large

enough (119877 + 1198711205731) by setting 120573

1 can guarantee the existence

of appropriate 119861 and 119862 to make |1198902| = |(2119862ℎ119861)| small

enough

Yes

Yes

No

No

IF |e2| gtmax(h)R + L1205731

1205731 = 1205731 minus ΔV lowast 120572

120573out = 12057311205731 = 120573out 1205731 = 1205731 lowast p

IF 1205731 minus ΔV lowast 120572 gt 0

Figure 2 Flow chart of the algorithm to calculate 1205731

Remark 1 Obviously the state error would not actually goto zero but enter and is contained within a neighborhoodof the origin by this method Therefore by means of settingappropriate 120573

1and 120573

2 the performance required in practical

engineering can be achieved

Remark 2 Using a sigmoid function as the nonlinear func-tion in ESO can reduce chatting in the system [15] Consider

[1

2

] = [

[

minus119877

119871

1

1198710 0

]

]

[1198851

1198852

] + [

[

1

1198710

]

]

119906

+ [minus1205731sdot 1198901

minus1205732sdot10038161003816100381610038161198901

1003816100381610038161003816 sdot sigmoid (1198901)]

(30)

where

sigmoid (1198901) = (

2

1 + exp (minus1198861198901)) minus 1 (31)

33 Proposed Adaptive Nonlinear ESO Although a large (119877+

1198711205731) will render 119890

2stable in a wider range the steady-state

error still has to be considered here From (13) we knowthat a large (119877 + 119871120573

1) leads to the increase of 119890

2in steady

state The improved method proposed in this paper is toadaptively adjust the parameters ofNLESO to a compromisedvalue which takes both stability and steady-state error intoconsideration

Rewrite inequality (27) as the following expression

2119862

119861gt

1

119877 + 1198711205731

(32)

Thus the roots of (29) hold the following equality

100381610038161003816100381611989021003816100381610038161003816 =

2119862 |ℎ|

119861gt

|ℎ|

119877 + 1198711205731

(33)

6 Mathematical Problems in Engineering

0 005 01 015 02 025 03 035 04

0

5

Time (s)

Back

-EM

F (V

)

Estimated e120572Estimated e120573

minus10

minus5

(a)

0 005 01 015 02 025 03

10

Time (s)

Estimated e120572Estimated e120573

0

5

Back

-EM

F (V

)

minus10

minus5

(b)

Figure 3 Simulation waveforms of estimated back EMF when speed is 500 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

05

0 005 01 015 02 025 03 035 04Time (s)

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

Curr

ent (

A)

minus5

05

minus5

(a)

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

0

5

minus5

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

0

5

minus5

(b)

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

0

5

minus5

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

0

5

minus5

(c)

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

05

minus5

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

05

minus5

(d)

Figure 4 Simulation waveforms of actual and estimated 119894120572 119894120573when speed is 500 rmin and the load torque is 1 Nm (a) Actual and estimated

119894120572obtained by the proposed ANLESO (b) Actual and estimated 119894

120573obtained by the proposed ANLESO (c) Actual and estimated 119894

120572obtained

by the conventional SMO (d) Actual and estimated 119894120573obtained by the conventional SMO

This means that the lower bound of 1198902is larger than

|ℎ|(119877 + 1198711205731) 1205731is a design parameter so the size of 119877 + 119871120573

1

can be changed by means of adjusting 1205731 The main idea of

this method is to decrease 1205731gradually to reduce steady-state

error and increase 1205731when the systemmay become unstable

By several iterations the value of 1205731would approximate the

compromised value gradually

Figure 2 illustrates the algorithm to calculate 1205731 where

0 lt 120572 lt 1 and 119901 gt 1 are adjustment factors Δ119881 is the errorof command velocity and measured velocity

34 Rotor Position and Speed Estimate Using the estimatedback-EMF obtained by the ANLESO the position and speed

Mathematical Problems in Engineering 7

0 005 01 015 02 025 03 035 040

200

400

600

Time (s)

Roto

r spe

ed (r

min

)

Actual speedEstimated speed

(a)

0 005 01 015 02 025 03

0

200

400

600

Time (s)

Roto

r spe

ed (r

min

)

Estimated speedActual speed

(b)

Figure 5 Simulation waveforms of actual and estimated speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by theproposed ANLESO and (b) obtained by the conventional SMO

0

5

0

5

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)

0 005 01 015 02 025 03 035 04

05

Time (s)

0 005 01 015 02 025 03 035 04Time (s)

0 005 01 015 02 025 03 035 04Time (s)

minus5

(a)

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)0

5

0

5

0 005 01 015 02 025 03

010

Time (s)

0 005 01 015 02 025 03Time (s)

0 005 01 015 02 025 03Time (s)

minus10

(b)

Figure 6 Simulation waveforms of actual rotor position estimated rotor position and estimated error when speed is 500 rmin and the loadtorque is 1 Nm (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO

signals can be calculated easily According to (3) the rotorspeed can be obtained as follows

119903=

radic(119890120572

2 + 119890120573

2)

119870119864

(34)

Similarly the rotor position can be deduced from (3) asfollows

120579 = minus arctan[119890120572

119890120573

] (35)

4 Simulation and Experimental Results

41 Simulation Results In order to show the high-speedperformance of the proposed ANLESO it is necessary tocompare it with the conventional SMO through the Mat-LabSimulink programming environmentThe parameters ofPMSM used in simulation are listed in Table 1

Figures 3ndash6 show the simulation waveforms when thereference speed is changed from zero to 500 rmin and

Table 1 The parameters of PMSM

Parameters ValuesRated power 15 KWRated speed 2500 rminInput voltage (DC) 310VCurrent 6AStator resistance 118Ω

Stator inductance 5326mHRotational inertia 133 times 10minus3 kgsdotm2

Flux of permanent magnet 00356WbPoles 8

the load torque is 1 Nm In these figures part (a) gives thesimulation waveforms obtained by the proposed ANLESOand part (b) gives the simulation waveforms obtained bythe conventional SMO Estimated back-EMF actual andestimated phase currents actual and estimated rotor speed

8 Mathematical Problems in Engineering

015 02 025 03

0

20

40

Time (s)

Back

-EM

F (V

)

minus20

minus40

Estimated e120572Estimated e120573

(a)

Time (s)015 02 025 03

0

20

40

Back

-EM

F (V

)

minus20

minus40

Estimated e120572Estimated e120573

(b)

Figure 7 Simulation waveforms of estimated back EMF when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

(a)

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

(b)

015 02 025 03Time (s)

015 02 025 03Time (s)

0

5

Curr

ent (

A)

minus5

0

5

Curr

ent (

A)

minus5

(c)

0

5

015 02 025 03Time (s)

015 02 025 03Time (s)

0

5

Curr

ent (

A)

Curr

ent (

A)

minus5

minus5

(d)

Figure 8 Simulation waveforms of actual and estimated 119894120572 119894120573when speed is 2000 rmin and the load torque is 1 Nm (a) Actual and estimated

119894120572obtained by the proposed ANLESO (b) Actual and estimated 119894

120573obtained by the proposed ANLESO (c) Actual and estimated 119894

120572obtained

by the conventional SMO (d) Actual and estimated 119894120573obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 3ndash6 the accuracy of esti-mation has been greatly improved with the help of proposedANLESO Different from the conventional SMO the back-EMF in Figure 3(a) almost has no chattering Comparedwith Figures 4(c) and 4(d) the estimated results of phase

currents in Figures 4(a) and 4(b) are better Due to the goodperformance of estimating back-EMF and currents thechattering phenomenon of the estimated rotor position andspeed is reduced which can be seen from Figures 5(a) 5(b)6(a) and 6(b) In part (a) of these figures the estimated speedis almost the same as actual speed however the estimatedspeed in part (b) of these figures exhibits the chattering

Mathematical Problems in Engineering 9

Time (s)0 005 01 015 02 025 03 035 04

0

1000

2000

Roto

r spe

ed (r

min

)

Estimated speedActual speed

(a)

Time (s)

Estimated speedActual speed

0 005 01 015 02 025 03

0

1000

2000

Roto

r spe

ed (r

min

)

(b)

Figure 9 Simulation waveforms of actual and estimated speeds when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by theproposed ANLESO and (b) obtained by the conventional SMO

0

5

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)

0

5

015 02 025 03

05

Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

minus5

(a)

5

Roto

r pos

ition

(rad

)0

5

Roto

r pos

ition

(rad

)

0

010

015 02 025 03Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

Roto

r pos

ition

(rad

)

minus10

(b)

Figure 10 Simulation waveforms of actual rotor position estimated rotor position and estimated error when speed is 2000 rmin and theload torque is 1 Nm (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO

Figure 11 Platform of 15 kw PMSM sensorless control system based on DSP

phenomenon Besides it can be seen that the estimated errorof the position in part (a) is smaller than that shown in part(b) of these figures

Figures 7ndash10 show the two sets of simulation waveformswhen the reference speed is changed from zero to 2000 rmin

and the load torque is 1 Nm In these figures part (a) gives thesimulation waveforms obtained by the proposed ANLESOand part (b) gives the simulation waveforms obtained bythe conventional SMO Estimated back-EMF actual andestimated phase currents actual and estimated rotor speed

10 Mathematical Problems in Engineering

(a) (b)

Figure 12 Operating waveforms of estimated back EMF in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained bythe proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 13 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 14 Operating waveforms of estimated rotor speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 7ndash10 when the rotor speedis raised to 2000 rmin advantages of ANLESO becomemoreapparent than conventional SMO Comparing the estimatedback-EMF in Figures 7(a) and 7(b) we can find that theformer has smaller chattering As a result the estimationof rotor speed becomes more accurate by ANLESO whichcan be seen in Figures 9(a) and 9(b) The waveform formspresented in Figures 10(a) and 10(b) show the estimatedrotor position The position estimation error is reduced andthe accuracy of rotor position estimation is improved inFigure 10(a)

42 Experimental Results To further verify the performanceof the new SMO for estimating rotor position and speed an

experimental systemhas been designed to control the 130SJT-M060D (15 kw) sinusoidal PMSMmotor made by GSK

Figure 11 shows a photograph of the sensorless controlsystemTheHighVoltage DigitalMotor Control Kit made byTI is utilized in the system which contains power modulesswitching devices current and voltage sensing circuit ADCmodules and PWM DAC (digital to analog converter)modules A floating-point TMS320F28335 DSP is employedas the digital controller The current control cycle is 100120583sand the velocity control cycle is 1ms A 2 120583s dead time ischosen for the PWM switching All of the control variablesare monitored in real time by an oscilloscope after theyare converted to analog signals through the PWM DACsmodules which use an external low pass filter to generatethe waveformsThemotor parameters used in experiment aregiven in Table 1

Mathematical Problems in Engineering 11

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 15 Operating waveforms of rotor position and estimate error when speed is 500 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

(a) (b)

Figure 16 Operating waveforms of estimated back EMF in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtainedby the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 17 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

Figures 12ndash14 show the operating waveforms when thereference speed is changed from zero to 500 rmin and theload torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO Estimated back-EMF estimated phase

currents and estimated rotor speed are shown in thesefigures In Figure 15 actual rotor position estimated rotorposition and their error are shown

Figures 12(a) and 13(a) show the experimental resultsof the proposed ANLESO Compared to the waveformsobtained by SMO (see part (b) in Figures 12 and 13)

12 Mathematical Problems in Engineering

(a) (b)

Figure 18 Operatingwaveforms of estimated rotor speedswhen speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 19 Operating waveforms of rotor position and estimate error when speed is 2000 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

thewaveforms of the estimated back-EMF and current shownin Figures 12(a) and 13(a) are more accurate The waveformspresented in Figures 14(a) and 14(b) show the differences ofrotor speed estimated by two kinds of observers The formerspeed waveform has less chattering In other words ANLESOhas higher estimated precision

Figure 15 shows the experimental results of positionresponseThe estimated error shown in Figure 15(c) is smallerthan that in Figure 15(d) which verifies the high precision ofANLESO

Figures 16ndash18 show the operating waveforms when thereference speed is changed from zero to 2000 rmin andthe load torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO In these figures estimated back-EMFestimated phase currents and estimated rotor speed areshown In Figure 19 actual rotor position estimated rotorposition and their error are shown

Figures 12ndash15 show the operating waveforms when thereference speed is changed from zero to 2000 rmin Figures12 and 13 provide the operating waveforms obtained by theproposed ANLESO Figures 14 and 15 show the simulationwaveforms obtained by the conventional SMO In thesefigures estimated back-EMF estimated phase currents esti-mated rotor speed and actual and estimated rotor positionare shown

Figures 16ndash18 show the experimental results of estimatedback-EMF current and rotor speed when the rotor speed israised to 2000 rmin It can be seen that the waveforms inpart (a) of these figures have smaller chattering and higheraccuracy In this way the advantages of ANLESO have beenverified

As shown in Figure 19 when the rotor speed is raisedto 2000 rmin the advantages of ANLESO to estimate rotorposition becomemore apparent than conventional SMOTheestimation error in Figure 19(c) is significantly less than thatin Figure 19(d) That is to say ANLESO has higher accuracy

Mathematical Problems in Engineering 13

and smaller chattering when being applied to estimate posi-tion than SMO

5 Conclusions

In this paper an adaptive nonlinear extended state observerhas been designed for the sensorless control of a PMSMTheconvergence of this observer has been proved by means of aLyapunov stability analysis An adaptive algorithm is adoptedto calculate the compromised parameter of ESO in order totake both stability and steady-state error into considerationThegoodperformance of the proposed sensorless control sys-tem was verified by several experimental results The resultsshow that ANLESO has more advantages than conventionalSMO Compared to conventional SMO ANLESO has smallerchattering higher accuracy and no phase delay

In future works we will explore the stability of the ESOunder parameter uncertainties in sensorless control system

Conflict of Interests

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

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China (Grant no 61304097) and Projects ofMajor International (Regional) Joint Research ProgramNSFC (Grant no 61120106010)

References

[1] S Lu X Tang andB Song ldquoAdaptive PIF control for permanentmagnet synchronousmotors based onGPCrdquo Sensors vol 13 no1 pp 175ndash192 2013

[2] A Gomez-Espinosa V M Hernandez-Guzman M Bandala-Sanchez et al ldquoA new adaptive self-tuning Fourier Coefficientsalgorithm for periodic torque ripple minimization in perma-nentmagnet synchronousmotors (PMSM)rdquo Sensors vol 13 no3 pp 3831ndash3847 2013

[3] Y Zhang C M Akujuobi W H Ali C L Tolliver and L-SShieh ldquoLoad disturbance resistance speed controller design forPMSMrdquo IEEE Transactions on Industrial Electronics vol 53 no4 pp 1198ndash1208 2006

[4] J C Gamazo-Real E Vazquez-Sanchez and J Gomez-GilldquoPosition and speed control of brushless DC motors usingsensorless techniques and application trendsrdquo Sensors vol 10no 7 pp 6901ndash6947 2010

[5] J-H Jang S-K Sul J-I Ha K Ide and M Sawamura ldquoSen-sorless drive of surface-mounted permanent-magnet motor byhigh-frequency signal injection based on magnetic saliencyrdquoIEEE Transactions on Industry Applications vol 39 no 4 pp1031ndash1039 2003

[6] A Piippo M Hinkkanen and J Luomi ldquoSensorless control ofPMSM drives using a combination of voltage model and HFsignal injectionrdquo in Proceedings of the 39th IAS Annual MeetingConference Record of the IEEE Industry Applications Conferencevol 2 pp 964ndash970 October 2004

[7] G Wang R Yang and D Xu ldquoDSP-based control of sensorlessIPMSM drives for wide-speed-range operationrdquo IEEE Transac-tions on Industrial Electronics vol 60 no 2 pp 720ndash727 2013

[8] Z Wang Y Zheng Z Zou and M Cheng ldquoPosition sensorlesscontrol of interleaved CSI fed PMSM drive with extendedKalman filterrdquo IEEE Transactions on Magnetics vol 48 no 11pp 3688ndash3691 2012

[9] T D Batzel and K Y Lee ldquoAn approach to sensorless operationof the permanent-magnet synchronous motor using diagonallyrecurrent neural networksrdquo IEEE Transactions on Energy Con-version vol 18 no 1 pp 100ndash106 2003

[10] B Zhang andY Li ldquoAPMSMslidingmode control systembasedon model reference adaptive controlrdquo in Proceedings of the 3rdInternational Power Electronics and Motion Control Conference(IPEMC rsquo00) vol 1 pp 336ndash341 IEEE Beijing China 2000

[11] S Shinnaka ldquoNew sensorless vector control using minimum-order flux state observer in a stationary reference frame forpermanent-magnet synchronousmotorsrdquo IEEETransactions onIndustrial Electronics vol 53 no 2 pp 388ndash398 2006

[12] A Khlaief M Bendjedia M Boussak and M Gossa ldquoA non-linear observer for high-performance sensorless speed controlof IPMSM driverdquo IEEE Transactions on Power Electronics vol27 no 6 pp 3028ndash3040 2012

[13] H Lee and J Lee ldquoDesign of iterative sliding mode observerfor sensorless PMSM controlrdquo IEEE Transactions on ControlSystems Technology vol 21 no 4 pp 1394ndash1399 2013

[14] Y-S Han J-S Choi and Y-S Kim ldquoSensorless PMSM drivewith a sliding mode control based adaptive speed and statorresistance estimatorrdquo IEEE Transactions on Magnetics vol 36no 5 pp 3588ndash3591 2000

[15] V C Ilioudis and N I Margaris ldquoPMSM sensorless speedestimation based on sliding mode observersrdquo in Proceedings ofthe IEEE Power Electronics Specialists Conference (PESC rsquo08) pp2838ndash2843 IEEE Rhodes Greece June 2008

[16] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[17] Y Huang and J Han ldquoAnalysis and design for the second ordernonlinear continuous extended states observerrdquoChinese ScienceBulletin vol 45 no 21 pp 1938ndash1944 2000

[18] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[19] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence vol 6 pp 4989ndash4996 IEEE June 2003

[20] Y Xia Z Zhu and M Fu ldquoBack-stepping sliding mode controlfor missile systems based on an extended state observerrdquo IETControl Theory and Applications vol 5 no 1 pp 93ndash102 2011

[21] X Li and S Li ldquoExtended state observer based adaptive controlscheme for PMSM systemrdquo in Proceedings of the 33rd IEEEChinese Control Conference (CCC rsquo14) 2014

[22] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

14 Mathematical Problems in Engineering

[23] G ZhuAKaddouri L ADessaint andOAkhrif ldquoAnonlinearstate observer for the sensorless control of a permanent-magnetAC machinerdquo IEEE Transactions on Industrial Electronics vol45 no 2 pp 291ndash296 1996

[24] J Han Active Disturbance Rejection Control TechniquemdashTheTechnique for Estimating and Compensating the UncertaintiesNational Defense Industry Press Beijing China 2013 (Chi-nese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

3-phaseinverter

PWMHW

PMSMmotor

PWM1AB

PWM2ABPWM3AB

SVGEN

Park Clake

PIPISpd

Phase voltage

ADCHW

BusVolt

Adaptive nonlinear

ESO

Estimatedposition

and speed

PIRef

RefFdb

FdbFdbRef

Speed

V120572

V120572

V120573

V120573

I120572I120573

120579

I_park

Vdc

Ia

IbIc

TaTbTc

Id

Iq

Ds

Qs

As

Bs

Figure 1 Block diagram of the sensorless vector control of PMSM

and parameters of the observer) In this way the order of theobserver can be reduced

Set 1199091

= 119894120572and 119909

2= minus119890120572as augmented states ℎ is the

derivative of 1199092 which is a variable that we do not need to

know The state equation (1) is written in the following form

1= minus

119877

1198711199091+

1

1198711199092+

1

1198711199061

2= ℎ

(4)

The output can be written as

119910 = 119872[1199091

1199092

] (5)

where119872 = [1 0]Now 119909

2= minus119890120572can be estimated using a linear extended

state observer as follows

[1

2

] = [

[

minus119877

119871

1

1198710 0

]

]

[1199111

1199112

] + [

[

1

1198710

]

]

119906 + 119866 (119910 minus 1199111) (6)

where 1199111and 1199112are used to estimate the value of 119909

1and 119909

2

respectively 119866 = [minus1205731

minus 1205732]119879 is the observer gain vector

which can be obtained using any knownmethod for examplethe pole placement technique

Let 119890 = [1199091minus1199111

1199092minus1199112] then the error equation can be written as

119890 = (119860119900minus 119866119872) 119890 + 119867 (7)

where

119860119900= [

[

minus119877

119871

1

1198710 0

]

]

119867 = [0 ℎ]119879

(8)

Obviously the LESO is bounded-input bounded-output(BIBO) stable if (119860

119900minus 119866119862) is stable and ℎ is bounded [19]

32 Nonlinear ESO For the sake of reducing the steady-stateerror a nonlinear extended state observer is proposed asfollows

[1

2

] = [

[

minus119877

119871

1

1198710 0

]

]

[1199111

1199112

] + [

[

1

1198710

]

]

119906

+ [minus1205731sdot 1198901

minus1205732sdot10038161003816100381610038161198901

100381610038161003816100381612

sdot sign (1198901)]

(9)

Let 119890 = [1199091minus1199111

1199092minus1199112] then the error equation can be written as

1198901= minus (

119877

119871+ 1205731) 1198901+

1

1198711198902

1198902= minus ℎ minus 120573

2

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1)

(10)

When the steady state is reached we can obtain

minus ℎ minus 1205732

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1)

= minus (119877

119871+ 1205731) 1198901+

1

1198711198902= 0

(11)

Hence the steady-state errors are

1198901= (

ℎ

1205732

)

2

(12)

1198902= (119877 + 119871120573

1) (

ℎ

1205732

)

2

(13)

Theorem 1 For system (4) the state error of the nonlinearextended state observer (9) converges asymptotically if thefollowing conditions are satisfied

(C1) (119877119871 + 1205731) gt 0

(C2) 1205732gt 0

4 Mathematical Problems in Engineering

Proof The goal is to prove that there exists a positive definiteenergy function whose derivative is always made negativeThe following energy function is chosen [24]

119881 (1198901 1198902)

= 11986010038161003816100381610038161198901

100381610038161003816100381632

minus 11986111989011198902+ 1198621198902

2

= 11986010038161003816100381610038161198901

1003816100381610038161003816minus12 [

[

(1198901minus

11986110038161003816100381610038161198901

100381610038161003816100381612

21198601198902)

2

minus1198612 10038161003816100381610038161198901

1003816100381610038161003816

411986021198902

2

]

]

+ 1198621198902

2

= 11986010038161003816100381610038161198901

1003816100381610038161003816minus12

(1198901minus

11986110038161003816100381610038161198901

100381610038161003816100381612

21198601198902)

2

+ (119862 minus1198612

10038161003816100381610038161198901100381610038161003816100381612

4119860) 1198902

2

(14)

where 119860 119861 and 119862 are the parameters to be determinedSuitable 119860 119861 and 119862 need to be chosen to let the energyfunction meet the Lyapunov stability theorem

In order to guarantee that 119881(1198901 1198902) is positively definite

the following rules need to be followed

(R1) 119860 gt 0

(R2) 119862 gt 1198612|1198901|124119860 gt 0

Since |1198901|12 is bounded (R2) always can be satisfied by

choosing a suitable 119862Calculate the partial derivative of 119881(119890

1 1198902) as follows

120597119881

1205971198901

=3

2119860

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1) minus 119861119890

2

120597119881

1205971198902

= minus 1198611198901+ 2119862119890

2

(15)

The derivative of the energy function is as follows

=120597119881

1205971198901

1198901+

120597119881

1205971198902

1198902 (16)

By substituting (15) into (16) can be given as

= (3

2119860

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1) minus 119861119890

2)(minus(

119877

119871+ 1205731) 1198901+

1

1198711198902)

+ (minus1198611198901+ 2119862119890

2) (minusℎ minus 120573

2

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1))

(17)

From (17) the following expression can be deduced

= (3

2

119860

119871minus 2119862120573

2+ 119861(

119877

119871+ 1205731)10038161003816100381610038161198901

100381610038161003816100381612

)

sdot10038161003816100381610038161198901

1003816100381610038161003816minus14 10038161003816100381610038161198901

100381610038161003816100381634 sign (119890

1) 1198902

minus119861

1198711198902

2minus (

3

2119860(

119877

119871+ 1205731) minus 119861120573

2)10038161003816100381610038161198901

10038161003816100381610038162(34)

+ (119861ℎ1198901minus 2119888ℎ119890

2)

(18)

Assume that

119883 =3

2119860(

119877

119871+ 1205731) minus 119861120573

2

119884 = (3

2

119860

119871minus 2119862120573

2+ 119861(

119877

119871+ 1205731)10038161003816100381610038161198901

100381610038161003816100381612

)10038161003816100381610038161198901

1003816100381610038161003816minus14

119885 =119861

119871

(19)

Equation (18) can be rewritten as

= minus 11988310038161003816100381610038161198901

10038161003816100381610038162(34)

+ 11988410038161003816100381610038161198901

100381610038161003816100381634 sign (119890

1) 1198902

minus 1198851198902

2minus (minus119861ℎ119890

1+ 2119862ℎ119890

2)

= 1198811minus 1198812

(20)

where1198811= minus119883|119890

1|2(34)+119884|119890

1|34 sign(119890

1)1198902minus11988511989022is a quadra-

tic function with regard to |1198901|34 sign(119890

1) and 119890

2 1198812

=

(minus119861ℎ1198901+ 2119862ℎ119890

2) is a plane function with regard to 119890

1and 1198902

A necessary and sufficient condition for the positively definiteproperty of the quadratic function 119881

1can be obtained

A 119883 gt 0

B 119885 gt 0

C 1198842 minus 4119883119885 lt 0

FromA we have

(R3) (32)119860(119877119871 + 1205731) gt 119861120573

2

FromB we have

(R4) 119861119871 gt 0 which means 119861 gt 0

Then fromC we can derive that

(3

2

119860

119871minus 2119862120573

2+ 119861(

119877

119871+ 1205731)10038161003816100381610038161198901

100381610038161003816100381612

)2

100381610038161003816100381611989011003816100381610038161003816minus12

lt 4119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2)

(21)

Then the following expression can be obtained

(3

2

119860

119871minus 2119862120573

2+ 119861(

119877

119871+ 1205731)10038161003816100381610038161198901

100381610038161003816100381612

)

lt 2radic119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2)10038161003816100381610038161198901

100381610038161003816100381614

(22)

Assume that

119886 = 119861(119877

119871+ 1205731)

119887 = radic119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2)

119888 =3

2

119860

119871minus 2119862120573

2

119909 =10038161003816100381610038161198901

100381610038161003816100381614

(23)

Mathematical Problems in Engineering 5

Equation (22) can be rewritten as

119888 + 1198861199092

lt 2119887119909 (24)

It can be verified that there exists some meaningful 119909

satisfying inequation (24) and meeting the rules (R1)sim(R4)if and only if the following conditions are satisfied

(119877

119871+ 1205731) gt 0 120573

2gt 0 (25)

Therefore 119886 = 119861(119877119871 + 1205731) gt 0 and inequality (24)

denotes a parabola going upwards Due to the existence ofsome meaningful 119909 satisfying inequality (24) the vertex ofthe parabola must be negative

1198872

minus 119886119888 gt 0 (26)

Hence119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2) minus 119861(

119877

119871+ 1205731)(

3

2

119860

119871minus 2119862120573

2) gt 0

1198611205732(2119862(

119877

119871+ 1205731) minus

119861

119871) gt 0

2119862(119877

119871+ 1205731) gt

119861

119871

(27)

Inequality (26) is established between the two real roots119909 = (119887plusmnradic1198872 minus 119886119888)119886 of the quadratic equation 1198861199092minus2119887119909+119888 =

0 In order to separate the two roots as far as possible andmake the smaller one very small a very large 119861 needs to beselected which makes 119861120573

2≫ 0 and the choice of parameter

119862 is required to ensure 2119862(119877119871 + 1205731) minus 119861119871 gt 0 Finally

choose appropriate 119860 to make parameters 119861 and 119862 satisfyrules (R1)sim(R4) In this way 119881

1will be a negative definite

function with variable 1198901in a very wide range and arbitrary

1198902 Meanwhile the area in which = 119881

1minus1198812does not satisfy

negative conditions (the area gt 0) is the part above theintersecting line of the quadratic function 119881

1and the plane

function 1198812

The value of 1198901in the intersecting line is the same order

as 1198901= (119861ℎ119883)

2 which is the root of the following equation

119861ℎ1198901= 119883

1003816100381610038161003816119890110038161003816100381610038162(34)

(28)

Since 119883 is the same order as 1198611205732 the value of 119890

1in the

intersecting line is the same order as (ℎ1205732)2 Therefore a

large enough 1205732can guarantee that 119890

1is stable in a very wide

rangeThe larger 1205732we set the smaller the unstable boundary

of 1198901 and then it will result in a better convergenceIn the same way the value of 119890

2in the intersecting line

is the same order as 1198902

= (2119862ℎ119885) which is the root of thefollowing equation

2119862ℎ1198902= 1198851198902

2 (29)

Consider that 119885 = 119861 the value of 1198901in the intersecting

line is the same order as (2119862ℎ119861) From inequality (21) theinequality 2119862(119877 + 119871120573

1)119861 gt 1 holds Therefore a large

enough (119877 + 1198711205731) by setting 120573

1 can guarantee the existence

of appropriate 119861 and 119862 to make |1198902| = |(2119862ℎ119861)| small

enough

Yes

Yes

No

No

IF |e2| gtmax(h)R + L1205731

1205731 = 1205731 minus ΔV lowast 120572

120573out = 12057311205731 = 120573out 1205731 = 1205731 lowast p

IF 1205731 minus ΔV lowast 120572 gt 0

Figure 2 Flow chart of the algorithm to calculate 1205731

Remark 1 Obviously the state error would not actually goto zero but enter and is contained within a neighborhoodof the origin by this method Therefore by means of settingappropriate 120573

1and 120573

2 the performance required in practical

engineering can be achieved

Remark 2 Using a sigmoid function as the nonlinear func-tion in ESO can reduce chatting in the system [15] Consider

[1

2

] = [

[

minus119877

119871

1

1198710 0

]

]

[1198851

1198852

] + [

[

1

1198710

]

]

119906

+ [minus1205731sdot 1198901

minus1205732sdot10038161003816100381610038161198901

1003816100381610038161003816 sdot sigmoid (1198901)]

(30)

where

sigmoid (1198901) = (

2

1 + exp (minus1198861198901)) minus 1 (31)

33 Proposed Adaptive Nonlinear ESO Although a large (119877+

1198711205731) will render 119890

2stable in a wider range the steady-state

error still has to be considered here From (13) we knowthat a large (119877 + 119871120573

1) leads to the increase of 119890

2in steady

state The improved method proposed in this paper is toadaptively adjust the parameters ofNLESO to a compromisedvalue which takes both stability and steady-state error intoconsideration

Rewrite inequality (27) as the following expression

2119862

119861gt

1

119877 + 1198711205731

(32)

Thus the roots of (29) hold the following equality

100381610038161003816100381611989021003816100381610038161003816 =

2119862 |ℎ|

119861gt

|ℎ|

119877 + 1198711205731

(33)

6 Mathematical Problems in Engineering

0 005 01 015 02 025 03 035 04

0

5

Time (s)

Back

-EM

F (V

)

Estimated e120572Estimated e120573

minus10

minus5

(a)

0 005 01 015 02 025 03

10

Time (s)

Estimated e120572Estimated e120573

0

5

Back

-EM

F (V

)

minus10

minus5

(b)

Figure 3 Simulation waveforms of estimated back EMF when speed is 500 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

05

0 005 01 015 02 025 03 035 04Time (s)

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

Curr

ent (

A)

minus5

05

minus5

(a)

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

0

5

minus5

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

0

5

minus5

(b)

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

0

5

minus5

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

0

5

minus5

(c)

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

05

minus5

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

05

minus5

(d)

Figure 4 Simulation waveforms of actual and estimated 119894120572 119894120573when speed is 500 rmin and the load torque is 1 Nm (a) Actual and estimated

119894120572obtained by the proposed ANLESO (b) Actual and estimated 119894

120573obtained by the proposed ANLESO (c) Actual and estimated 119894

120572obtained

by the conventional SMO (d) Actual and estimated 119894120573obtained by the conventional SMO

This means that the lower bound of 1198902is larger than

|ℎ|(119877 + 1198711205731) 1205731is a design parameter so the size of 119877 + 119871120573

1

can be changed by means of adjusting 1205731 The main idea of

this method is to decrease 1205731gradually to reduce steady-state

error and increase 1205731when the systemmay become unstable

By several iterations the value of 1205731would approximate the

compromised value gradually

Figure 2 illustrates the algorithm to calculate 1205731 where

0 lt 120572 lt 1 and 119901 gt 1 are adjustment factors Δ119881 is the errorof command velocity and measured velocity

34 Rotor Position and Speed Estimate Using the estimatedback-EMF obtained by the ANLESO the position and speed

Mathematical Problems in Engineering 7

0 005 01 015 02 025 03 035 040

200

400

600

Time (s)

Roto

r spe

ed (r

min

)

Actual speedEstimated speed

(a)

0 005 01 015 02 025 03

0

200

400

600

Time (s)

Roto

r spe

ed (r

min

)

Estimated speedActual speed

(b)

Figure 5 Simulation waveforms of actual and estimated speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by theproposed ANLESO and (b) obtained by the conventional SMO

0

5

0

5

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)

0 005 01 015 02 025 03 035 04

05

Time (s)

0 005 01 015 02 025 03 035 04Time (s)

0 005 01 015 02 025 03 035 04Time (s)

minus5

(a)

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)0

5

0

5

0 005 01 015 02 025 03

010

Time (s)

0 005 01 015 02 025 03Time (s)

0 005 01 015 02 025 03Time (s)

minus10

(b)

Figure 6 Simulation waveforms of actual rotor position estimated rotor position and estimated error when speed is 500 rmin and the loadtorque is 1 Nm (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO

signals can be calculated easily According to (3) the rotorspeed can be obtained as follows

119903=

radic(119890120572

2 + 119890120573

2)

119870119864

(34)

Similarly the rotor position can be deduced from (3) asfollows

120579 = minus arctan[119890120572

119890120573

] (35)

4 Simulation and Experimental Results

41 Simulation Results In order to show the high-speedperformance of the proposed ANLESO it is necessary tocompare it with the conventional SMO through the Mat-LabSimulink programming environmentThe parameters ofPMSM used in simulation are listed in Table 1

Figures 3ndash6 show the simulation waveforms when thereference speed is changed from zero to 500 rmin and

Table 1 The parameters of PMSM

Parameters ValuesRated power 15 KWRated speed 2500 rminInput voltage (DC) 310VCurrent 6AStator resistance 118Ω

Stator inductance 5326mHRotational inertia 133 times 10minus3 kgsdotm2

Flux of permanent magnet 00356WbPoles 8

the load torque is 1 Nm In these figures part (a) gives thesimulation waveforms obtained by the proposed ANLESOand part (b) gives the simulation waveforms obtained bythe conventional SMO Estimated back-EMF actual andestimated phase currents actual and estimated rotor speed

8 Mathematical Problems in Engineering

015 02 025 03

0

20

40

Time (s)

Back

-EM

F (V

)

minus20

minus40

Estimated e120572Estimated e120573

(a)

Time (s)015 02 025 03

0

20

40

Back

-EM

F (V

)

minus20

minus40

Estimated e120572Estimated e120573

(b)

Figure 7 Simulation waveforms of estimated back EMF when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

(a)

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

(b)

015 02 025 03Time (s)

015 02 025 03Time (s)

0

5

Curr

ent (

A)

minus5

0

5

Curr

ent (

A)

minus5

(c)

0

5

015 02 025 03Time (s)

015 02 025 03Time (s)

0

5

Curr

ent (

A)

Curr

ent (

A)

minus5

minus5

(d)

Figure 8 Simulation waveforms of actual and estimated 119894120572 119894120573when speed is 2000 rmin and the load torque is 1 Nm (a) Actual and estimated

119894120572obtained by the proposed ANLESO (b) Actual and estimated 119894

120573obtained by the proposed ANLESO (c) Actual and estimated 119894

120572obtained

by the conventional SMO (d) Actual and estimated 119894120573obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 3ndash6 the accuracy of esti-mation has been greatly improved with the help of proposedANLESO Different from the conventional SMO the back-EMF in Figure 3(a) almost has no chattering Comparedwith Figures 4(c) and 4(d) the estimated results of phase

currents in Figures 4(a) and 4(b) are better Due to the goodperformance of estimating back-EMF and currents thechattering phenomenon of the estimated rotor position andspeed is reduced which can be seen from Figures 5(a) 5(b)6(a) and 6(b) In part (a) of these figures the estimated speedis almost the same as actual speed however the estimatedspeed in part (b) of these figures exhibits the chattering

Mathematical Problems in Engineering 9

Time (s)0 005 01 015 02 025 03 035 04

0

1000

2000

Roto

r spe

ed (r

min

)

Estimated speedActual speed

(a)

Time (s)

Estimated speedActual speed

0 005 01 015 02 025 03

0

1000

2000

Roto

r spe

ed (r

min

)

(b)

Figure 9 Simulation waveforms of actual and estimated speeds when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by theproposed ANLESO and (b) obtained by the conventional SMO

0

5

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)

0

5

015 02 025 03

05

Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

minus5

(a)

5

Roto

r pos

ition

(rad

)0

5

Roto

r pos

ition

(rad

)

0

010

015 02 025 03Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

Roto

r pos

ition

(rad

)

minus10

(b)

Figure 10 Simulation waveforms of actual rotor position estimated rotor position and estimated error when speed is 2000 rmin and theload torque is 1 Nm (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO

Figure 11 Platform of 15 kw PMSM sensorless control system based on DSP

phenomenon Besides it can be seen that the estimated errorof the position in part (a) is smaller than that shown in part(b) of these figures

Figures 7ndash10 show the two sets of simulation waveformswhen the reference speed is changed from zero to 2000 rmin

and the load torque is 1 Nm In these figures part (a) gives thesimulation waveforms obtained by the proposed ANLESOand part (b) gives the simulation waveforms obtained bythe conventional SMO Estimated back-EMF actual andestimated phase currents actual and estimated rotor speed

10 Mathematical Problems in Engineering

(a) (b)

Figure 12 Operating waveforms of estimated back EMF in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained bythe proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 13 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 14 Operating waveforms of estimated rotor speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 7ndash10 when the rotor speedis raised to 2000 rmin advantages of ANLESO becomemoreapparent than conventional SMO Comparing the estimatedback-EMF in Figures 7(a) and 7(b) we can find that theformer has smaller chattering As a result the estimationof rotor speed becomes more accurate by ANLESO whichcan be seen in Figures 9(a) and 9(b) The waveform formspresented in Figures 10(a) and 10(b) show the estimatedrotor position The position estimation error is reduced andthe accuracy of rotor position estimation is improved inFigure 10(a)

42 Experimental Results To further verify the performanceof the new SMO for estimating rotor position and speed an

experimental systemhas been designed to control the 130SJT-M060D (15 kw) sinusoidal PMSMmotor made by GSK

Figure 11 shows a photograph of the sensorless controlsystemTheHighVoltage DigitalMotor Control Kit made byTI is utilized in the system which contains power modulesswitching devices current and voltage sensing circuit ADCmodules and PWM DAC (digital to analog converter)modules A floating-point TMS320F28335 DSP is employedas the digital controller The current control cycle is 100120583sand the velocity control cycle is 1ms A 2 120583s dead time ischosen for the PWM switching All of the control variablesare monitored in real time by an oscilloscope after theyare converted to analog signals through the PWM DACsmodules which use an external low pass filter to generatethe waveformsThemotor parameters used in experiment aregiven in Table 1

Mathematical Problems in Engineering 11

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 15 Operating waveforms of rotor position and estimate error when speed is 500 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

(a) (b)

Figure 16 Operating waveforms of estimated back EMF in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtainedby the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 17 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

Figures 12ndash14 show the operating waveforms when thereference speed is changed from zero to 500 rmin and theload torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO Estimated back-EMF estimated phase

currents and estimated rotor speed are shown in thesefigures In Figure 15 actual rotor position estimated rotorposition and their error are shown

Figures 12(a) and 13(a) show the experimental resultsof the proposed ANLESO Compared to the waveformsobtained by SMO (see part (b) in Figures 12 and 13)

12 Mathematical Problems in Engineering

(a) (b)

Figure 18 Operatingwaveforms of estimated rotor speedswhen speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 19 Operating waveforms of rotor position and estimate error when speed is 2000 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

thewaveforms of the estimated back-EMF and current shownin Figures 12(a) and 13(a) are more accurate The waveformspresented in Figures 14(a) and 14(b) show the differences ofrotor speed estimated by two kinds of observers The formerspeed waveform has less chattering In other words ANLESOhas higher estimated precision

Figure 15 shows the experimental results of positionresponseThe estimated error shown in Figure 15(c) is smallerthan that in Figure 15(d) which verifies the high precision ofANLESO

Figures 16ndash18 show the operating waveforms when thereference speed is changed from zero to 2000 rmin andthe load torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO In these figures estimated back-EMFestimated phase currents and estimated rotor speed areshown In Figure 19 actual rotor position estimated rotorposition and their error are shown

Figures 12ndash15 show the operating waveforms when thereference speed is changed from zero to 2000 rmin Figures12 and 13 provide the operating waveforms obtained by theproposed ANLESO Figures 14 and 15 show the simulationwaveforms obtained by the conventional SMO In thesefigures estimated back-EMF estimated phase currents esti-mated rotor speed and actual and estimated rotor positionare shown

Figures 16ndash18 show the experimental results of estimatedback-EMF current and rotor speed when the rotor speed israised to 2000 rmin It can be seen that the waveforms inpart (a) of these figures have smaller chattering and higheraccuracy In this way the advantages of ANLESO have beenverified

As shown in Figure 19 when the rotor speed is raisedto 2000 rmin the advantages of ANLESO to estimate rotorposition becomemore apparent than conventional SMOTheestimation error in Figure 19(c) is significantly less than thatin Figure 19(d) That is to say ANLESO has higher accuracy

Mathematical Problems in Engineering 13

and smaller chattering when being applied to estimate posi-tion than SMO

5 Conclusions

In this paper an adaptive nonlinear extended state observerhas been designed for the sensorless control of a PMSMTheconvergence of this observer has been proved by means of aLyapunov stability analysis An adaptive algorithm is adoptedto calculate the compromised parameter of ESO in order totake both stability and steady-state error into considerationThegoodperformance of the proposed sensorless control sys-tem was verified by several experimental results The resultsshow that ANLESO has more advantages than conventionalSMO Compared to conventional SMO ANLESO has smallerchattering higher accuracy and no phase delay

In future works we will explore the stability of the ESOunder parameter uncertainties in sensorless control system

Conflict of Interests

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

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China (Grant no 61304097) and Projects ofMajor International (Regional) Joint Research ProgramNSFC (Grant no 61120106010)

References

[1] S Lu X Tang andB Song ldquoAdaptive PIF control for permanentmagnet synchronousmotors based onGPCrdquo Sensors vol 13 no1 pp 175ndash192 2013

[2] A Gomez-Espinosa V M Hernandez-Guzman M Bandala-Sanchez et al ldquoA new adaptive self-tuning Fourier Coefficientsalgorithm for periodic torque ripple minimization in perma-nentmagnet synchronousmotors (PMSM)rdquo Sensors vol 13 no3 pp 3831ndash3847 2013

[3] Y Zhang C M Akujuobi W H Ali C L Tolliver and L-SShieh ldquoLoad disturbance resistance speed controller design forPMSMrdquo IEEE Transactions on Industrial Electronics vol 53 no4 pp 1198ndash1208 2006

[4] J C Gamazo-Real E Vazquez-Sanchez and J Gomez-GilldquoPosition and speed control of brushless DC motors usingsensorless techniques and application trendsrdquo Sensors vol 10no 7 pp 6901ndash6947 2010

[5] J-H Jang S-K Sul J-I Ha K Ide and M Sawamura ldquoSen-sorless drive of surface-mounted permanent-magnet motor byhigh-frequency signal injection based on magnetic saliencyrdquoIEEE Transactions on Industry Applications vol 39 no 4 pp1031ndash1039 2003

[6] A Piippo M Hinkkanen and J Luomi ldquoSensorless control ofPMSM drives using a combination of voltage model and HFsignal injectionrdquo in Proceedings of the 39th IAS Annual MeetingConference Record of the IEEE Industry Applications Conferencevol 2 pp 964ndash970 October 2004

[7] G Wang R Yang and D Xu ldquoDSP-based control of sensorlessIPMSM drives for wide-speed-range operationrdquo IEEE Transac-tions on Industrial Electronics vol 60 no 2 pp 720ndash727 2013

[8] Z Wang Y Zheng Z Zou and M Cheng ldquoPosition sensorlesscontrol of interleaved CSI fed PMSM drive with extendedKalman filterrdquo IEEE Transactions on Magnetics vol 48 no 11pp 3688ndash3691 2012

[9] T D Batzel and K Y Lee ldquoAn approach to sensorless operationof the permanent-magnet synchronous motor using diagonallyrecurrent neural networksrdquo IEEE Transactions on Energy Con-version vol 18 no 1 pp 100ndash106 2003

[10] B Zhang andY Li ldquoAPMSMslidingmode control systembasedon model reference adaptive controlrdquo in Proceedings of the 3rdInternational Power Electronics and Motion Control Conference(IPEMC rsquo00) vol 1 pp 336ndash341 IEEE Beijing China 2000

[11] S Shinnaka ldquoNew sensorless vector control using minimum-order flux state observer in a stationary reference frame forpermanent-magnet synchronousmotorsrdquo IEEETransactions onIndustrial Electronics vol 53 no 2 pp 388ndash398 2006

[12] A Khlaief M Bendjedia M Boussak and M Gossa ldquoA non-linear observer for high-performance sensorless speed controlof IPMSM driverdquo IEEE Transactions on Power Electronics vol27 no 6 pp 3028ndash3040 2012

[13] H Lee and J Lee ldquoDesign of iterative sliding mode observerfor sensorless PMSM controlrdquo IEEE Transactions on ControlSystems Technology vol 21 no 4 pp 1394ndash1399 2013

[14] Y-S Han J-S Choi and Y-S Kim ldquoSensorless PMSM drivewith a sliding mode control based adaptive speed and statorresistance estimatorrdquo IEEE Transactions on Magnetics vol 36no 5 pp 3588ndash3591 2000

[15] V C Ilioudis and N I Margaris ldquoPMSM sensorless speedestimation based on sliding mode observersrdquo in Proceedings ofthe IEEE Power Electronics Specialists Conference (PESC rsquo08) pp2838ndash2843 IEEE Rhodes Greece June 2008

[16] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[17] Y Huang and J Han ldquoAnalysis and design for the second ordernonlinear continuous extended states observerrdquoChinese ScienceBulletin vol 45 no 21 pp 1938ndash1944 2000

[18] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[19] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence vol 6 pp 4989ndash4996 IEEE June 2003

[20] Y Xia Z Zhu and M Fu ldquoBack-stepping sliding mode controlfor missile systems based on an extended state observerrdquo IETControl Theory and Applications vol 5 no 1 pp 93ndash102 2011

[21] X Li and S Li ldquoExtended state observer based adaptive controlscheme for PMSM systemrdquo in Proceedings of the 33rd IEEEChinese Control Conference (CCC rsquo14) 2014

[22] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

14 Mathematical Problems in Engineering

[23] G ZhuAKaddouri L ADessaint andOAkhrif ldquoAnonlinearstate observer for the sensorless control of a permanent-magnetAC machinerdquo IEEE Transactions on Industrial Electronics vol45 no 2 pp 291ndash296 1996

[24] J Han Active Disturbance Rejection Control TechniquemdashTheTechnique for Estimating and Compensating the UncertaintiesNational Defense Industry Press Beijing China 2013 (Chi-nese)

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Proof The goal is to prove that there exists a positive definiteenergy function whose derivative is always made negativeThe following energy function is chosen [24]

119881 (1198901 1198902)

= 11986010038161003816100381610038161198901

100381610038161003816100381632

minus 11986111989011198902+ 1198621198902

2

= 11986010038161003816100381610038161198901

1003816100381610038161003816minus12 [

[

(1198901minus

11986110038161003816100381610038161198901

100381610038161003816100381612

21198601198902)

2

minus1198612 10038161003816100381610038161198901

1003816100381610038161003816

411986021198902

2

]

]

+ 1198621198902

2

= 11986010038161003816100381610038161198901

1003816100381610038161003816minus12

(1198901minus

11986110038161003816100381610038161198901

100381610038161003816100381612

21198601198902)

2

+ (119862 minus1198612

10038161003816100381610038161198901100381610038161003816100381612

4119860) 1198902

2

(14)

where 119860 119861 and 119862 are the parameters to be determinedSuitable 119860 119861 and 119862 need to be chosen to let the energyfunction meet the Lyapunov stability theorem

In order to guarantee that 119881(1198901 1198902) is positively definite

the following rules need to be followed

(R1) 119860 gt 0

(R2) 119862 gt 1198612|1198901|124119860 gt 0

Since |1198901|12 is bounded (R2) always can be satisfied by

choosing a suitable 119862Calculate the partial derivative of 119881(119890

1 1198902) as follows

120597119881

1205971198901

=3

2119860

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1) minus 119861119890

2

120597119881

1205971198902

= minus 1198611198901+ 2119862119890

2

(15)

The derivative of the energy function is as follows

=120597119881

1205971198901

1198901+

120597119881

1205971198902

1198902 (16)

By substituting (15) into (16) can be given as

= (3

2119860

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1) minus 119861119890

2)(minus(

119877

119871+ 1205731) 1198901+

1

1198711198902)

+ (minus1198611198901+ 2119862119890

2) (minusℎ minus 120573

2

10038161003816100381610038161198901100381610038161003816100381612 sign (119890

1))

(17)

From (17) the following expression can be deduced

= (3

2

119860

119871minus 2119862120573

2+ 119861(

119877

119871+ 1205731)10038161003816100381610038161198901

100381610038161003816100381612

)

sdot10038161003816100381610038161198901

1003816100381610038161003816minus14 10038161003816100381610038161198901

100381610038161003816100381634 sign (119890

1) 1198902

minus119861

1198711198902

2minus (

3

2119860(

119877

119871+ 1205731) minus 119861120573

2)10038161003816100381610038161198901

10038161003816100381610038162(34)

+ (119861ℎ1198901minus 2119888ℎ119890

2)

(18)

Assume that

119883 =3

2119860(

119877

119871+ 1205731) minus 119861120573

2

119884 = (3

2

119860

119871minus 2119862120573

2+ 119861(

119877

119871+ 1205731)10038161003816100381610038161198901

100381610038161003816100381612

)10038161003816100381610038161198901

1003816100381610038161003816minus14

119885 =119861

119871

(19)

Equation (18) can be rewritten as

= minus 11988310038161003816100381610038161198901

10038161003816100381610038162(34)

+ 11988410038161003816100381610038161198901

100381610038161003816100381634 sign (119890

1) 1198902

minus 1198851198902

2minus (minus119861ℎ119890

1+ 2119862ℎ119890

2)

= 1198811minus 1198812

(20)

where1198811= minus119883|119890

1|2(34)+119884|119890

1|34 sign(119890

1)1198902minus11988511989022is a quadra-

tic function with regard to |1198901|34 sign(119890

1) and 119890

2 1198812

=

(minus119861ℎ1198901+ 2119862ℎ119890

2) is a plane function with regard to 119890

1and 1198902

A necessary and sufficient condition for the positively definiteproperty of the quadratic function 119881

1can be obtained

A 119883 gt 0

B 119885 gt 0

C 1198842 minus 4119883119885 lt 0

FromA we have

(R3) (32)119860(119877119871 + 1205731) gt 119861120573

2

FromB we have

(R4) 119861119871 gt 0 which means 119861 gt 0

Then fromC we can derive that

(3

2

119860

119871minus 2119862120573

2+ 119861(

119877

119871+ 1205731)10038161003816100381610038161198901

100381610038161003816100381612

)2

100381610038161003816100381611989011003816100381610038161003816minus12

lt 4119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2)

(21)

Then the following expression can be obtained

(3

2

119860

119871minus 2119862120573

2+ 119861(

119877

119871+ 1205731)10038161003816100381610038161198901

100381610038161003816100381612

)

lt 2radic119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2)10038161003816100381610038161198901

100381610038161003816100381614

(22)

Assume that

119886 = 119861(119877

119871+ 1205731)

119887 = radic119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2)

119888 =3

2

119860

119871minus 2119862120573

2

119909 =10038161003816100381610038161198901

100381610038161003816100381614

(23)

Mathematical Problems in Engineering 5

Equation (22) can be rewritten as

119888 + 1198861199092

lt 2119887119909 (24)

It can be verified that there exists some meaningful 119909

satisfying inequation (24) and meeting the rules (R1)sim(R4)if and only if the following conditions are satisfied

(119877

119871+ 1205731) gt 0 120573

2gt 0 (25)

Therefore 119886 = 119861(119877119871 + 1205731) gt 0 and inequality (24)

denotes a parabola going upwards Due to the existence ofsome meaningful 119909 satisfying inequality (24) the vertex ofthe parabola must be negative

1198872

minus 119886119888 gt 0 (26)

Hence119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2) minus 119861(

119877

119871+ 1205731)(

3

2

119860

119871minus 2119862120573

2) gt 0

1198611205732(2119862(

119877

119871+ 1205731) minus

119861

119871) gt 0

2119862(119877

119871+ 1205731) gt

119861

119871

(27)

Inequality (26) is established between the two real roots119909 = (119887plusmnradic1198872 minus 119886119888)119886 of the quadratic equation 1198861199092minus2119887119909+119888 =

0 In order to separate the two roots as far as possible andmake the smaller one very small a very large 119861 needs to beselected which makes 119861120573

2≫ 0 and the choice of parameter

119862 is required to ensure 2119862(119877119871 + 1205731) minus 119861119871 gt 0 Finally

choose appropriate 119860 to make parameters 119861 and 119862 satisfyrules (R1)sim(R4) In this way 119881

1will be a negative definite

function with variable 1198901in a very wide range and arbitrary

1198902 Meanwhile the area in which = 119881

1minus1198812does not satisfy

negative conditions (the area gt 0) is the part above theintersecting line of the quadratic function 119881

1and the plane

function 1198812

The value of 1198901in the intersecting line is the same order

as 1198901= (119861ℎ119883)

2 which is the root of the following equation

119861ℎ1198901= 119883

1003816100381610038161003816119890110038161003816100381610038162(34)

(28)

Since 119883 is the same order as 1198611205732 the value of 119890

1in the

intersecting line is the same order as (ℎ1205732)2 Therefore a

large enough 1205732can guarantee that 119890

1is stable in a very wide

rangeThe larger 1205732we set the smaller the unstable boundary

of 1198901 and then it will result in a better convergenceIn the same way the value of 119890

2in the intersecting line

is the same order as 1198902

= (2119862ℎ119885) which is the root of thefollowing equation

2119862ℎ1198902= 1198851198902

2 (29)

Consider that 119885 = 119861 the value of 1198901in the intersecting

line is the same order as (2119862ℎ119861) From inequality (21) theinequality 2119862(119877 + 119871120573

1)119861 gt 1 holds Therefore a large

enough (119877 + 1198711205731) by setting 120573

1 can guarantee the existence

of appropriate 119861 and 119862 to make |1198902| = |(2119862ℎ119861)| small

enough

Yes

Yes

No

No

IF |e2| gtmax(h)R + L1205731

1205731 = 1205731 minus ΔV lowast 120572

120573out = 12057311205731 = 120573out 1205731 = 1205731 lowast p

IF 1205731 minus ΔV lowast 120572 gt 0

Figure 2 Flow chart of the algorithm to calculate 1205731

Remark 1 Obviously the state error would not actually goto zero but enter and is contained within a neighborhoodof the origin by this method Therefore by means of settingappropriate 120573

1and 120573

2 the performance required in practical

engineering can be achieved

Remark 2 Using a sigmoid function as the nonlinear func-tion in ESO can reduce chatting in the system [15] Consider

[1

2

] = [

[

minus119877

119871

1

1198710 0

]

]

[1198851

1198852

] + [

[

1

1198710

]

]

119906

+ [minus1205731sdot 1198901

minus1205732sdot10038161003816100381610038161198901

1003816100381610038161003816 sdot sigmoid (1198901)]

(30)

where

sigmoid (1198901) = (

2

1 + exp (minus1198861198901)) minus 1 (31)

33 Proposed Adaptive Nonlinear ESO Although a large (119877+

1198711205731) will render 119890

2stable in a wider range the steady-state

error still has to be considered here From (13) we knowthat a large (119877 + 119871120573

1) leads to the increase of 119890

2in steady

state The improved method proposed in this paper is toadaptively adjust the parameters ofNLESO to a compromisedvalue which takes both stability and steady-state error intoconsideration

Rewrite inequality (27) as the following expression

2119862

119861gt

1

119877 + 1198711205731

(32)

Thus the roots of (29) hold the following equality

100381610038161003816100381611989021003816100381610038161003816 =

2119862 |ℎ|

119861gt

|ℎ|

119877 + 1198711205731

(33)

6 Mathematical Problems in Engineering

0 005 01 015 02 025 03 035 04

0

5

Time (s)

Back

-EM

F (V

)

Estimated e120572Estimated e120573

minus10

minus5

(a)

0 005 01 015 02 025 03

10

Time (s)

Estimated e120572Estimated e120573

0

5

Back

-EM

F (V

)

minus10

minus5

(b)

Figure 3 Simulation waveforms of estimated back EMF when speed is 500 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

05

0 005 01 015 02 025 03 035 04Time (s)

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

Curr

ent (

A)

minus5

05

minus5

(a)

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

0

5

minus5

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

0

5

minus5

(b)

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

0

5

minus5

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

0

5

minus5

(c)

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

05

minus5

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

05

minus5

(d)

Figure 4 Simulation waveforms of actual and estimated 119894120572 119894120573when speed is 500 rmin and the load torque is 1 Nm (a) Actual and estimated

119894120572obtained by the proposed ANLESO (b) Actual and estimated 119894

120573obtained by the proposed ANLESO (c) Actual and estimated 119894

120572obtained

by the conventional SMO (d) Actual and estimated 119894120573obtained by the conventional SMO

This means that the lower bound of 1198902is larger than

|ℎ|(119877 + 1198711205731) 1205731is a design parameter so the size of 119877 + 119871120573

1

can be changed by means of adjusting 1205731 The main idea of

this method is to decrease 1205731gradually to reduce steady-state

error and increase 1205731when the systemmay become unstable

By several iterations the value of 1205731would approximate the

compromised value gradually

Figure 2 illustrates the algorithm to calculate 1205731 where

0 lt 120572 lt 1 and 119901 gt 1 are adjustment factors Δ119881 is the errorof command velocity and measured velocity

34 Rotor Position and Speed Estimate Using the estimatedback-EMF obtained by the ANLESO the position and speed

Mathematical Problems in Engineering 7

0 005 01 015 02 025 03 035 040

200

400

600

Time (s)

Roto

r spe

ed (r

min

)

Actual speedEstimated speed

(a)

0 005 01 015 02 025 03

0

200

400

600

Time (s)

Roto

r spe

ed (r

min

)

Estimated speedActual speed

(b)

Figure 5 Simulation waveforms of actual and estimated speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by theproposed ANLESO and (b) obtained by the conventional SMO

0

5

0

5

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)

0 005 01 015 02 025 03 035 04

05

Time (s)

0 005 01 015 02 025 03 035 04Time (s)

0 005 01 015 02 025 03 035 04Time (s)

minus5

(a)

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)0

5

0

5

0 005 01 015 02 025 03

010

Time (s)

0 005 01 015 02 025 03Time (s)

0 005 01 015 02 025 03Time (s)

minus10

(b)

Figure 6 Simulation waveforms of actual rotor position estimated rotor position and estimated error when speed is 500 rmin and the loadtorque is 1 Nm (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO

signals can be calculated easily According to (3) the rotorspeed can be obtained as follows

119903=

radic(119890120572

2 + 119890120573

2)

119870119864

(34)

Similarly the rotor position can be deduced from (3) asfollows

120579 = minus arctan[119890120572

119890120573

] (35)

4 Simulation and Experimental Results

41 Simulation Results In order to show the high-speedperformance of the proposed ANLESO it is necessary tocompare it with the conventional SMO through the Mat-LabSimulink programming environmentThe parameters ofPMSM used in simulation are listed in Table 1

Figures 3ndash6 show the simulation waveforms when thereference speed is changed from zero to 500 rmin and

Table 1 The parameters of PMSM

Parameters ValuesRated power 15 KWRated speed 2500 rminInput voltage (DC) 310VCurrent 6AStator resistance 118Ω

Stator inductance 5326mHRotational inertia 133 times 10minus3 kgsdotm2

Flux of permanent magnet 00356WbPoles 8

the load torque is 1 Nm In these figures part (a) gives thesimulation waveforms obtained by the proposed ANLESOand part (b) gives the simulation waveforms obtained bythe conventional SMO Estimated back-EMF actual andestimated phase currents actual and estimated rotor speed

8 Mathematical Problems in Engineering

015 02 025 03

0

20

40

Time (s)

Back

-EM

F (V

)

minus20

minus40

Estimated e120572Estimated e120573

(a)

Time (s)015 02 025 03

0

20

40

Back

-EM

F (V

)

minus20

minus40

Estimated e120572Estimated e120573

(b)

Figure 7 Simulation waveforms of estimated back EMF when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

(a)

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

(b)

015 02 025 03Time (s)

015 02 025 03Time (s)

0

5

Curr

ent (

A)

minus5

0

5

Curr

ent (

A)

minus5

(c)

0

5

015 02 025 03Time (s)

015 02 025 03Time (s)

0

5

Curr

ent (

A)

Curr

ent (

A)

minus5

minus5

(d)

Figure 8 Simulation waveforms of actual and estimated 119894120572 119894120573when speed is 2000 rmin and the load torque is 1 Nm (a) Actual and estimated

119894120572obtained by the proposed ANLESO (b) Actual and estimated 119894

120573obtained by the proposed ANLESO (c) Actual and estimated 119894

120572obtained

by the conventional SMO (d) Actual and estimated 119894120573obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 3ndash6 the accuracy of esti-mation has been greatly improved with the help of proposedANLESO Different from the conventional SMO the back-EMF in Figure 3(a) almost has no chattering Comparedwith Figures 4(c) and 4(d) the estimated results of phase

currents in Figures 4(a) and 4(b) are better Due to the goodperformance of estimating back-EMF and currents thechattering phenomenon of the estimated rotor position andspeed is reduced which can be seen from Figures 5(a) 5(b)6(a) and 6(b) In part (a) of these figures the estimated speedis almost the same as actual speed however the estimatedspeed in part (b) of these figures exhibits the chattering

Mathematical Problems in Engineering 9

Time (s)0 005 01 015 02 025 03 035 04

0

1000

2000

Roto

r spe

ed (r

min

)

Estimated speedActual speed

(a)

Time (s)

Estimated speedActual speed

0 005 01 015 02 025 03

0

1000

2000

Roto

r spe

ed (r

min

)

(b)

Figure 9 Simulation waveforms of actual and estimated speeds when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by theproposed ANLESO and (b) obtained by the conventional SMO

0

5

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)

0

5

015 02 025 03

05

Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

minus5

(a)

5

Roto

r pos

ition

(rad

)0

5

Roto

r pos

ition

(rad

)

0

010

015 02 025 03Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

Roto

r pos

ition

(rad

)

minus10

(b)

Figure 10 Simulation waveforms of actual rotor position estimated rotor position and estimated error when speed is 2000 rmin and theload torque is 1 Nm (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO

Figure 11 Platform of 15 kw PMSM sensorless control system based on DSP

phenomenon Besides it can be seen that the estimated errorof the position in part (a) is smaller than that shown in part(b) of these figures

Figures 7ndash10 show the two sets of simulation waveformswhen the reference speed is changed from zero to 2000 rmin

and the load torque is 1 Nm In these figures part (a) gives thesimulation waveforms obtained by the proposed ANLESOand part (b) gives the simulation waveforms obtained bythe conventional SMO Estimated back-EMF actual andestimated phase currents actual and estimated rotor speed

10 Mathematical Problems in Engineering

(a) (b)

Figure 12 Operating waveforms of estimated back EMF in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained bythe proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 13 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 14 Operating waveforms of estimated rotor speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 7ndash10 when the rotor speedis raised to 2000 rmin advantages of ANLESO becomemoreapparent than conventional SMO Comparing the estimatedback-EMF in Figures 7(a) and 7(b) we can find that theformer has smaller chattering As a result the estimationof rotor speed becomes more accurate by ANLESO whichcan be seen in Figures 9(a) and 9(b) The waveform formspresented in Figures 10(a) and 10(b) show the estimatedrotor position The position estimation error is reduced andthe accuracy of rotor position estimation is improved inFigure 10(a)

42 Experimental Results To further verify the performanceof the new SMO for estimating rotor position and speed an

experimental systemhas been designed to control the 130SJT-M060D (15 kw) sinusoidal PMSMmotor made by GSK

Figure 11 shows a photograph of the sensorless controlsystemTheHighVoltage DigitalMotor Control Kit made byTI is utilized in the system which contains power modulesswitching devices current and voltage sensing circuit ADCmodules and PWM DAC (digital to analog converter)modules A floating-point TMS320F28335 DSP is employedas the digital controller The current control cycle is 100120583sand the velocity control cycle is 1ms A 2 120583s dead time ischosen for the PWM switching All of the control variablesare monitored in real time by an oscilloscope after theyare converted to analog signals through the PWM DACsmodules which use an external low pass filter to generatethe waveformsThemotor parameters used in experiment aregiven in Table 1

Mathematical Problems in Engineering 11

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 15 Operating waveforms of rotor position and estimate error when speed is 500 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

(a) (b)

Figure 16 Operating waveforms of estimated back EMF in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtainedby the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 17 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

Figures 12ndash14 show the operating waveforms when thereference speed is changed from zero to 500 rmin and theload torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO Estimated back-EMF estimated phase

currents and estimated rotor speed are shown in thesefigures In Figure 15 actual rotor position estimated rotorposition and their error are shown

Figures 12(a) and 13(a) show the experimental resultsof the proposed ANLESO Compared to the waveformsobtained by SMO (see part (b) in Figures 12 and 13)

12 Mathematical Problems in Engineering

(a) (b)

Figure 18 Operatingwaveforms of estimated rotor speedswhen speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 19 Operating waveforms of rotor position and estimate error when speed is 2000 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

thewaveforms of the estimated back-EMF and current shownin Figures 12(a) and 13(a) are more accurate The waveformspresented in Figures 14(a) and 14(b) show the differences ofrotor speed estimated by two kinds of observers The formerspeed waveform has less chattering In other words ANLESOhas higher estimated precision

Figure 15 shows the experimental results of positionresponseThe estimated error shown in Figure 15(c) is smallerthan that in Figure 15(d) which verifies the high precision ofANLESO

Figures 16ndash18 show the operating waveforms when thereference speed is changed from zero to 2000 rmin andthe load torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO In these figures estimated back-EMFestimated phase currents and estimated rotor speed areshown In Figure 19 actual rotor position estimated rotorposition and their error are shown

Figures 12ndash15 show the operating waveforms when thereference speed is changed from zero to 2000 rmin Figures12 and 13 provide the operating waveforms obtained by theproposed ANLESO Figures 14 and 15 show the simulationwaveforms obtained by the conventional SMO In thesefigures estimated back-EMF estimated phase currents esti-mated rotor speed and actual and estimated rotor positionare shown

Figures 16ndash18 show the experimental results of estimatedback-EMF current and rotor speed when the rotor speed israised to 2000 rmin It can be seen that the waveforms inpart (a) of these figures have smaller chattering and higheraccuracy In this way the advantages of ANLESO have beenverified

As shown in Figure 19 when the rotor speed is raisedto 2000 rmin the advantages of ANLESO to estimate rotorposition becomemore apparent than conventional SMOTheestimation error in Figure 19(c) is significantly less than thatin Figure 19(d) That is to say ANLESO has higher accuracy

Mathematical Problems in Engineering 13

and smaller chattering when being applied to estimate posi-tion than SMO

5 Conclusions

In this paper an adaptive nonlinear extended state observerhas been designed for the sensorless control of a PMSMTheconvergence of this observer has been proved by means of aLyapunov stability analysis An adaptive algorithm is adoptedto calculate the compromised parameter of ESO in order totake both stability and steady-state error into considerationThegoodperformance of the proposed sensorless control sys-tem was verified by several experimental results The resultsshow that ANLESO has more advantages than conventionalSMO Compared to conventional SMO ANLESO has smallerchattering higher accuracy and no phase delay

In future works we will explore the stability of the ESOunder parameter uncertainties in sensorless control system

Conflict of Interests

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

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China (Grant no 61304097) and Projects ofMajor International (Regional) Joint Research ProgramNSFC (Grant no 61120106010)

References

[1] S Lu X Tang andB Song ldquoAdaptive PIF control for permanentmagnet synchronousmotors based onGPCrdquo Sensors vol 13 no1 pp 175ndash192 2013

[2] A Gomez-Espinosa V M Hernandez-Guzman M Bandala-Sanchez et al ldquoA new adaptive self-tuning Fourier Coefficientsalgorithm for periodic torque ripple minimization in perma-nentmagnet synchronousmotors (PMSM)rdquo Sensors vol 13 no3 pp 3831ndash3847 2013

[3] Y Zhang C M Akujuobi W H Ali C L Tolliver and L-SShieh ldquoLoad disturbance resistance speed controller design forPMSMrdquo IEEE Transactions on Industrial Electronics vol 53 no4 pp 1198ndash1208 2006

[4] J C Gamazo-Real E Vazquez-Sanchez and J Gomez-GilldquoPosition and speed control of brushless DC motors usingsensorless techniques and application trendsrdquo Sensors vol 10no 7 pp 6901ndash6947 2010

[5] J-H Jang S-K Sul J-I Ha K Ide and M Sawamura ldquoSen-sorless drive of surface-mounted permanent-magnet motor byhigh-frequency signal injection based on magnetic saliencyrdquoIEEE Transactions on Industry Applications vol 39 no 4 pp1031ndash1039 2003

[6] A Piippo M Hinkkanen and J Luomi ldquoSensorless control ofPMSM drives using a combination of voltage model and HFsignal injectionrdquo in Proceedings of the 39th IAS Annual MeetingConference Record of the IEEE Industry Applications Conferencevol 2 pp 964ndash970 October 2004

[7] G Wang R Yang and D Xu ldquoDSP-based control of sensorlessIPMSM drives for wide-speed-range operationrdquo IEEE Transac-tions on Industrial Electronics vol 60 no 2 pp 720ndash727 2013

[8] Z Wang Y Zheng Z Zou and M Cheng ldquoPosition sensorlesscontrol of interleaved CSI fed PMSM drive with extendedKalman filterrdquo IEEE Transactions on Magnetics vol 48 no 11pp 3688ndash3691 2012

[9] T D Batzel and K Y Lee ldquoAn approach to sensorless operationof the permanent-magnet synchronous motor using diagonallyrecurrent neural networksrdquo IEEE Transactions on Energy Con-version vol 18 no 1 pp 100ndash106 2003

[10] B Zhang andY Li ldquoAPMSMslidingmode control systembasedon model reference adaptive controlrdquo in Proceedings of the 3rdInternational Power Electronics and Motion Control Conference(IPEMC rsquo00) vol 1 pp 336ndash341 IEEE Beijing China 2000

[11] S Shinnaka ldquoNew sensorless vector control using minimum-order flux state observer in a stationary reference frame forpermanent-magnet synchronousmotorsrdquo IEEETransactions onIndustrial Electronics vol 53 no 2 pp 388ndash398 2006

[12] A Khlaief M Bendjedia M Boussak and M Gossa ldquoA non-linear observer for high-performance sensorless speed controlof IPMSM driverdquo IEEE Transactions on Power Electronics vol27 no 6 pp 3028ndash3040 2012

[13] H Lee and J Lee ldquoDesign of iterative sliding mode observerfor sensorless PMSM controlrdquo IEEE Transactions on ControlSystems Technology vol 21 no 4 pp 1394ndash1399 2013

[14] Y-S Han J-S Choi and Y-S Kim ldquoSensorless PMSM drivewith a sliding mode control based adaptive speed and statorresistance estimatorrdquo IEEE Transactions on Magnetics vol 36no 5 pp 3588ndash3591 2000

[15] V C Ilioudis and N I Margaris ldquoPMSM sensorless speedestimation based on sliding mode observersrdquo in Proceedings ofthe IEEE Power Electronics Specialists Conference (PESC rsquo08) pp2838ndash2843 IEEE Rhodes Greece June 2008

[16] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[17] Y Huang and J Han ldquoAnalysis and design for the second ordernonlinear continuous extended states observerrdquoChinese ScienceBulletin vol 45 no 21 pp 1938ndash1944 2000

[18] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[19] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence vol 6 pp 4989ndash4996 IEEE June 2003

[20] Y Xia Z Zhu and M Fu ldquoBack-stepping sliding mode controlfor missile systems based on an extended state observerrdquo IETControl Theory and Applications vol 5 no 1 pp 93ndash102 2011

[21] X Li and S Li ldquoExtended state observer based adaptive controlscheme for PMSM systemrdquo in Proceedings of the 33rd IEEEChinese Control Conference (CCC rsquo14) 2014

[22] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

14 Mathematical Problems in Engineering

[23] G ZhuAKaddouri L ADessaint andOAkhrif ldquoAnonlinearstate observer for the sensorless control of a permanent-magnetAC machinerdquo IEEE Transactions on Industrial Electronics vol45 no 2 pp 291ndash296 1996

[24] J Han Active Disturbance Rejection Control TechniquemdashTheTechnique for Estimating and Compensating the UncertaintiesNational Defense Industry Press Beijing China 2013 (Chi-nese)

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Mathematical Problems in Engineering

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Mathematical Problems in Engineering 5

Equation (22) can be rewritten as

119888 + 1198861199092

lt 2119887119909 (24)

It can be verified that there exists some meaningful 119909

satisfying inequation (24) and meeting the rules (R1)sim(R4)if and only if the following conditions are satisfied

(119877

119871+ 1205731) gt 0 120573

2gt 0 (25)

Therefore 119886 = 119861(119877119871 + 1205731) gt 0 and inequality (24)

denotes a parabola going upwards Due to the existence ofsome meaningful 119909 satisfying inequality (24) the vertex ofthe parabola must be negative

1198872

minus 119886119888 gt 0 (26)

Hence119861

119871(3

2119860(

119877

119871+ 1205731) minus 119861120573

2) minus 119861(

119877

119871+ 1205731)(

3

2

119860

119871minus 2119862120573

2) gt 0

1198611205732(2119862(

119877

119871+ 1205731) minus

119861

119871) gt 0

2119862(119877

119871+ 1205731) gt

119861

119871

(27)

Inequality (26) is established between the two real roots119909 = (119887plusmnradic1198872 minus 119886119888)119886 of the quadratic equation 1198861199092minus2119887119909+119888 =

0 In order to separate the two roots as far as possible andmake the smaller one very small a very large 119861 needs to beselected which makes 119861120573

2≫ 0 and the choice of parameter

119862 is required to ensure 2119862(119877119871 + 1205731) minus 119861119871 gt 0 Finally

choose appropriate 119860 to make parameters 119861 and 119862 satisfyrules (R1)sim(R4) In this way 119881

1will be a negative definite

function with variable 1198901in a very wide range and arbitrary

1198902 Meanwhile the area in which = 119881

1minus1198812does not satisfy

negative conditions (the area gt 0) is the part above theintersecting line of the quadratic function 119881

1and the plane

function 1198812

The value of 1198901in the intersecting line is the same order

as 1198901= (119861ℎ119883)

2 which is the root of the following equation

119861ℎ1198901= 119883

1003816100381610038161003816119890110038161003816100381610038162(34)

(28)

Since 119883 is the same order as 1198611205732 the value of 119890

1in the

intersecting line is the same order as (ℎ1205732)2 Therefore a

large enough 1205732can guarantee that 119890

1is stable in a very wide

rangeThe larger 1205732we set the smaller the unstable boundary

of 1198901 and then it will result in a better convergenceIn the same way the value of 119890

2in the intersecting line

is the same order as 1198902

= (2119862ℎ119885) which is the root of thefollowing equation

2119862ℎ1198902= 1198851198902

2 (29)

Consider that 119885 = 119861 the value of 1198901in the intersecting

line is the same order as (2119862ℎ119861) From inequality (21) theinequality 2119862(119877 + 119871120573

1)119861 gt 1 holds Therefore a large

enough (119877 + 1198711205731) by setting 120573

1 can guarantee the existence

of appropriate 119861 and 119862 to make |1198902| = |(2119862ℎ119861)| small

enough

Yes

Yes

No

No

IF |e2| gtmax(h)R + L1205731

1205731 = 1205731 minus ΔV lowast 120572

120573out = 12057311205731 = 120573out 1205731 = 1205731 lowast p

IF 1205731 minus ΔV lowast 120572 gt 0

Figure 2 Flow chart of the algorithm to calculate 1205731

Remark 1 Obviously the state error would not actually goto zero but enter and is contained within a neighborhoodof the origin by this method Therefore by means of settingappropriate 120573

1and 120573

2 the performance required in practical

engineering can be achieved

Remark 2 Using a sigmoid function as the nonlinear func-tion in ESO can reduce chatting in the system [15] Consider

[1

2

] = [

[

minus119877

119871

1

1198710 0

]

]

[1198851

1198852

] + [

[

1

1198710

]

]

119906

+ [minus1205731sdot 1198901

minus1205732sdot10038161003816100381610038161198901

1003816100381610038161003816 sdot sigmoid (1198901)]

(30)

where

sigmoid (1198901) = (

2

1 + exp (minus1198861198901)) minus 1 (31)

33 Proposed Adaptive Nonlinear ESO Although a large (119877+

1198711205731) will render 119890

2stable in a wider range the steady-state

error still has to be considered here From (13) we knowthat a large (119877 + 119871120573

1) leads to the increase of 119890

2in steady

state The improved method proposed in this paper is toadaptively adjust the parameters ofNLESO to a compromisedvalue which takes both stability and steady-state error intoconsideration

Rewrite inequality (27) as the following expression

2119862

119861gt

1

119877 + 1198711205731

(32)

Thus the roots of (29) hold the following equality

100381610038161003816100381611989021003816100381610038161003816 =

2119862 |ℎ|

119861gt

|ℎ|

119877 + 1198711205731

(33)

6 Mathematical Problems in Engineering

0 005 01 015 02 025 03 035 04

0

5

Time (s)

Back

-EM

F (V

)

Estimated e120572Estimated e120573

minus10

minus5

(a)

0 005 01 015 02 025 03

10

Time (s)

Estimated e120572Estimated e120573

0

5

Back

-EM

F (V

)

minus10

minus5

(b)

Figure 3 Simulation waveforms of estimated back EMF when speed is 500 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

05

0 005 01 015 02 025 03 035 04Time (s)

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

Curr

ent (

A)

minus5

05

minus5

(a)

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

0

5

minus5

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

0

5

minus5

(b)

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

0

5

minus5

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

0

5

minus5

(c)

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

05

minus5

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

05

minus5

(d)

Figure 4 Simulation waveforms of actual and estimated 119894120572 119894120573when speed is 500 rmin and the load torque is 1 Nm (a) Actual and estimated

119894120572obtained by the proposed ANLESO (b) Actual and estimated 119894

120573obtained by the proposed ANLESO (c) Actual and estimated 119894

120572obtained

by the conventional SMO (d) Actual and estimated 119894120573obtained by the conventional SMO

This means that the lower bound of 1198902is larger than

|ℎ|(119877 + 1198711205731) 1205731is a design parameter so the size of 119877 + 119871120573

1

can be changed by means of adjusting 1205731 The main idea of

this method is to decrease 1205731gradually to reduce steady-state

error and increase 1205731when the systemmay become unstable

By several iterations the value of 1205731would approximate the

compromised value gradually

Figure 2 illustrates the algorithm to calculate 1205731 where

0 lt 120572 lt 1 and 119901 gt 1 are adjustment factors Δ119881 is the errorof command velocity and measured velocity

34 Rotor Position and Speed Estimate Using the estimatedback-EMF obtained by the ANLESO the position and speed

Mathematical Problems in Engineering 7

0 005 01 015 02 025 03 035 040

200

400

600

Time (s)

Roto

r spe

ed (r

min

)

Actual speedEstimated speed

(a)

0 005 01 015 02 025 03

0

200

400

600

Time (s)

Roto

r spe

ed (r

min

)

Estimated speedActual speed

(b)

Figure 5 Simulation waveforms of actual and estimated speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by theproposed ANLESO and (b) obtained by the conventional SMO

0

5

0

5

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)

0 005 01 015 02 025 03 035 04

05

Time (s)

0 005 01 015 02 025 03 035 04Time (s)

0 005 01 015 02 025 03 035 04Time (s)

minus5

(a)

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)0

5

0

5

0 005 01 015 02 025 03

010

Time (s)

0 005 01 015 02 025 03Time (s)

0 005 01 015 02 025 03Time (s)

minus10

(b)

Figure 6 Simulation waveforms of actual rotor position estimated rotor position and estimated error when speed is 500 rmin and the loadtorque is 1 Nm (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO

signals can be calculated easily According to (3) the rotorspeed can be obtained as follows

119903=

radic(119890120572

2 + 119890120573

2)

119870119864

(34)

Similarly the rotor position can be deduced from (3) asfollows

120579 = minus arctan[119890120572

119890120573

] (35)

4 Simulation and Experimental Results

41 Simulation Results In order to show the high-speedperformance of the proposed ANLESO it is necessary tocompare it with the conventional SMO through the Mat-LabSimulink programming environmentThe parameters ofPMSM used in simulation are listed in Table 1

Figures 3ndash6 show the simulation waveforms when thereference speed is changed from zero to 500 rmin and

Table 1 The parameters of PMSM

Parameters ValuesRated power 15 KWRated speed 2500 rminInput voltage (DC) 310VCurrent 6AStator resistance 118Ω

Stator inductance 5326mHRotational inertia 133 times 10minus3 kgsdotm2

Flux of permanent magnet 00356WbPoles 8

the load torque is 1 Nm In these figures part (a) gives thesimulation waveforms obtained by the proposed ANLESOand part (b) gives the simulation waveforms obtained bythe conventional SMO Estimated back-EMF actual andestimated phase currents actual and estimated rotor speed

8 Mathematical Problems in Engineering

015 02 025 03

0

20

40

Time (s)

Back

-EM

F (V

)

minus20

minus40

Estimated e120572Estimated e120573

(a)

Time (s)015 02 025 03

0

20

40

Back

-EM

F (V

)

minus20

minus40

Estimated e120572Estimated e120573

(b)

Figure 7 Simulation waveforms of estimated back EMF when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

(a)

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

(b)

015 02 025 03Time (s)

015 02 025 03Time (s)

0

5

Curr

ent (

A)

minus5

0

5

Curr

ent (

A)

minus5

(c)

0

5

015 02 025 03Time (s)

015 02 025 03Time (s)

0

5

Curr

ent (

A)

Curr

ent (

A)

minus5

minus5

(d)

Figure 8 Simulation waveforms of actual and estimated 119894120572 119894120573when speed is 2000 rmin and the load torque is 1 Nm (a) Actual and estimated

119894120572obtained by the proposed ANLESO (b) Actual and estimated 119894

120573obtained by the proposed ANLESO (c) Actual and estimated 119894

120572obtained

by the conventional SMO (d) Actual and estimated 119894120573obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 3ndash6 the accuracy of esti-mation has been greatly improved with the help of proposedANLESO Different from the conventional SMO the back-EMF in Figure 3(a) almost has no chattering Comparedwith Figures 4(c) and 4(d) the estimated results of phase

currents in Figures 4(a) and 4(b) are better Due to the goodperformance of estimating back-EMF and currents thechattering phenomenon of the estimated rotor position andspeed is reduced which can be seen from Figures 5(a) 5(b)6(a) and 6(b) In part (a) of these figures the estimated speedis almost the same as actual speed however the estimatedspeed in part (b) of these figures exhibits the chattering

Mathematical Problems in Engineering 9

Time (s)0 005 01 015 02 025 03 035 04

0

1000

2000

Roto

r spe

ed (r

min

)

Estimated speedActual speed

(a)

Time (s)

Estimated speedActual speed

0 005 01 015 02 025 03

0

1000

2000

Roto

r spe

ed (r

min

)

(b)

Figure 9 Simulation waveforms of actual and estimated speeds when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by theproposed ANLESO and (b) obtained by the conventional SMO

0

5

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)

0

5

015 02 025 03

05

Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

minus5

(a)

5

Roto

r pos

ition

(rad

)0

5

Roto

r pos

ition

(rad

)

0

010

015 02 025 03Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

Roto

r pos

ition

(rad

)

minus10

(b)

Figure 10 Simulation waveforms of actual rotor position estimated rotor position and estimated error when speed is 2000 rmin and theload torque is 1 Nm (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO

Figure 11 Platform of 15 kw PMSM sensorless control system based on DSP

phenomenon Besides it can be seen that the estimated errorof the position in part (a) is smaller than that shown in part(b) of these figures

Figures 7ndash10 show the two sets of simulation waveformswhen the reference speed is changed from zero to 2000 rmin

and the load torque is 1 Nm In these figures part (a) gives thesimulation waveforms obtained by the proposed ANLESOand part (b) gives the simulation waveforms obtained bythe conventional SMO Estimated back-EMF actual andestimated phase currents actual and estimated rotor speed

10 Mathematical Problems in Engineering

(a) (b)

Figure 12 Operating waveforms of estimated back EMF in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained bythe proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 13 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 14 Operating waveforms of estimated rotor speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 7ndash10 when the rotor speedis raised to 2000 rmin advantages of ANLESO becomemoreapparent than conventional SMO Comparing the estimatedback-EMF in Figures 7(a) and 7(b) we can find that theformer has smaller chattering As a result the estimationof rotor speed becomes more accurate by ANLESO whichcan be seen in Figures 9(a) and 9(b) The waveform formspresented in Figures 10(a) and 10(b) show the estimatedrotor position The position estimation error is reduced andthe accuracy of rotor position estimation is improved inFigure 10(a)

42 Experimental Results To further verify the performanceof the new SMO for estimating rotor position and speed an

experimental systemhas been designed to control the 130SJT-M060D (15 kw) sinusoidal PMSMmotor made by GSK

Figure 11 shows a photograph of the sensorless controlsystemTheHighVoltage DigitalMotor Control Kit made byTI is utilized in the system which contains power modulesswitching devices current and voltage sensing circuit ADCmodules and PWM DAC (digital to analog converter)modules A floating-point TMS320F28335 DSP is employedas the digital controller The current control cycle is 100120583sand the velocity control cycle is 1ms A 2 120583s dead time ischosen for the PWM switching All of the control variablesare monitored in real time by an oscilloscope after theyare converted to analog signals through the PWM DACsmodules which use an external low pass filter to generatethe waveformsThemotor parameters used in experiment aregiven in Table 1

Mathematical Problems in Engineering 11

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 15 Operating waveforms of rotor position and estimate error when speed is 500 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

(a) (b)

Figure 16 Operating waveforms of estimated back EMF in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtainedby the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 17 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

Figures 12ndash14 show the operating waveforms when thereference speed is changed from zero to 500 rmin and theload torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO Estimated back-EMF estimated phase

currents and estimated rotor speed are shown in thesefigures In Figure 15 actual rotor position estimated rotorposition and their error are shown

Figures 12(a) and 13(a) show the experimental resultsof the proposed ANLESO Compared to the waveformsobtained by SMO (see part (b) in Figures 12 and 13)

12 Mathematical Problems in Engineering

(a) (b)

Figure 18 Operatingwaveforms of estimated rotor speedswhen speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 19 Operating waveforms of rotor position and estimate error when speed is 2000 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

thewaveforms of the estimated back-EMF and current shownin Figures 12(a) and 13(a) are more accurate The waveformspresented in Figures 14(a) and 14(b) show the differences ofrotor speed estimated by two kinds of observers The formerspeed waveform has less chattering In other words ANLESOhas higher estimated precision

Figure 15 shows the experimental results of positionresponseThe estimated error shown in Figure 15(c) is smallerthan that in Figure 15(d) which verifies the high precision ofANLESO

Figures 16ndash18 show the operating waveforms when thereference speed is changed from zero to 2000 rmin andthe load torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO In these figures estimated back-EMFestimated phase currents and estimated rotor speed areshown In Figure 19 actual rotor position estimated rotorposition and their error are shown

Figures 12ndash15 show the operating waveforms when thereference speed is changed from zero to 2000 rmin Figures12 and 13 provide the operating waveforms obtained by theproposed ANLESO Figures 14 and 15 show the simulationwaveforms obtained by the conventional SMO In thesefigures estimated back-EMF estimated phase currents esti-mated rotor speed and actual and estimated rotor positionare shown

Figures 16ndash18 show the experimental results of estimatedback-EMF current and rotor speed when the rotor speed israised to 2000 rmin It can be seen that the waveforms inpart (a) of these figures have smaller chattering and higheraccuracy In this way the advantages of ANLESO have beenverified

As shown in Figure 19 when the rotor speed is raisedto 2000 rmin the advantages of ANLESO to estimate rotorposition becomemore apparent than conventional SMOTheestimation error in Figure 19(c) is significantly less than thatin Figure 19(d) That is to say ANLESO has higher accuracy

Mathematical Problems in Engineering 13

and smaller chattering when being applied to estimate posi-tion than SMO

5 Conclusions

In this paper an adaptive nonlinear extended state observerhas been designed for the sensorless control of a PMSMTheconvergence of this observer has been proved by means of aLyapunov stability analysis An adaptive algorithm is adoptedto calculate the compromised parameter of ESO in order totake both stability and steady-state error into considerationThegoodperformance of the proposed sensorless control sys-tem was verified by several experimental results The resultsshow that ANLESO has more advantages than conventionalSMO Compared to conventional SMO ANLESO has smallerchattering higher accuracy and no phase delay

In future works we will explore the stability of the ESOunder parameter uncertainties in sensorless control system

Conflict of Interests

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

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China (Grant no 61304097) and Projects ofMajor International (Regional) Joint Research ProgramNSFC (Grant no 61120106010)

References

[1] S Lu X Tang andB Song ldquoAdaptive PIF control for permanentmagnet synchronousmotors based onGPCrdquo Sensors vol 13 no1 pp 175ndash192 2013

[2] A Gomez-Espinosa V M Hernandez-Guzman M Bandala-Sanchez et al ldquoA new adaptive self-tuning Fourier Coefficientsalgorithm for periodic torque ripple minimization in perma-nentmagnet synchronousmotors (PMSM)rdquo Sensors vol 13 no3 pp 3831ndash3847 2013

[3] Y Zhang C M Akujuobi W H Ali C L Tolliver and L-SShieh ldquoLoad disturbance resistance speed controller design forPMSMrdquo IEEE Transactions on Industrial Electronics vol 53 no4 pp 1198ndash1208 2006

[4] J C Gamazo-Real E Vazquez-Sanchez and J Gomez-GilldquoPosition and speed control of brushless DC motors usingsensorless techniques and application trendsrdquo Sensors vol 10no 7 pp 6901ndash6947 2010

[5] J-H Jang S-K Sul J-I Ha K Ide and M Sawamura ldquoSen-sorless drive of surface-mounted permanent-magnet motor byhigh-frequency signal injection based on magnetic saliencyrdquoIEEE Transactions on Industry Applications vol 39 no 4 pp1031ndash1039 2003

[6] A Piippo M Hinkkanen and J Luomi ldquoSensorless control ofPMSM drives using a combination of voltage model and HFsignal injectionrdquo in Proceedings of the 39th IAS Annual MeetingConference Record of the IEEE Industry Applications Conferencevol 2 pp 964ndash970 October 2004

[7] G Wang R Yang and D Xu ldquoDSP-based control of sensorlessIPMSM drives for wide-speed-range operationrdquo IEEE Transac-tions on Industrial Electronics vol 60 no 2 pp 720ndash727 2013

[8] Z Wang Y Zheng Z Zou and M Cheng ldquoPosition sensorlesscontrol of interleaved CSI fed PMSM drive with extendedKalman filterrdquo IEEE Transactions on Magnetics vol 48 no 11pp 3688ndash3691 2012

[9] T D Batzel and K Y Lee ldquoAn approach to sensorless operationof the permanent-magnet synchronous motor using diagonallyrecurrent neural networksrdquo IEEE Transactions on Energy Con-version vol 18 no 1 pp 100ndash106 2003

[10] B Zhang andY Li ldquoAPMSMslidingmode control systembasedon model reference adaptive controlrdquo in Proceedings of the 3rdInternational Power Electronics and Motion Control Conference(IPEMC rsquo00) vol 1 pp 336ndash341 IEEE Beijing China 2000

[11] S Shinnaka ldquoNew sensorless vector control using minimum-order flux state observer in a stationary reference frame forpermanent-magnet synchronousmotorsrdquo IEEETransactions onIndustrial Electronics vol 53 no 2 pp 388ndash398 2006

[12] A Khlaief M Bendjedia M Boussak and M Gossa ldquoA non-linear observer for high-performance sensorless speed controlof IPMSM driverdquo IEEE Transactions on Power Electronics vol27 no 6 pp 3028ndash3040 2012

[13] H Lee and J Lee ldquoDesign of iterative sliding mode observerfor sensorless PMSM controlrdquo IEEE Transactions on ControlSystems Technology vol 21 no 4 pp 1394ndash1399 2013

[14] Y-S Han J-S Choi and Y-S Kim ldquoSensorless PMSM drivewith a sliding mode control based adaptive speed and statorresistance estimatorrdquo IEEE Transactions on Magnetics vol 36no 5 pp 3588ndash3591 2000

[15] V C Ilioudis and N I Margaris ldquoPMSM sensorless speedestimation based on sliding mode observersrdquo in Proceedings ofthe IEEE Power Electronics Specialists Conference (PESC rsquo08) pp2838ndash2843 IEEE Rhodes Greece June 2008

[16] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[17] Y Huang and J Han ldquoAnalysis and design for the second ordernonlinear continuous extended states observerrdquoChinese ScienceBulletin vol 45 no 21 pp 1938ndash1944 2000

[18] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[19] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence vol 6 pp 4989ndash4996 IEEE June 2003

[20] Y Xia Z Zhu and M Fu ldquoBack-stepping sliding mode controlfor missile systems based on an extended state observerrdquo IETControl Theory and Applications vol 5 no 1 pp 93ndash102 2011

[21] X Li and S Li ldquoExtended state observer based adaptive controlscheme for PMSM systemrdquo in Proceedings of the 33rd IEEEChinese Control Conference (CCC rsquo14) 2014

[22] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

14 Mathematical Problems in Engineering

[23] G ZhuAKaddouri L ADessaint andOAkhrif ldquoAnonlinearstate observer for the sensorless control of a permanent-magnetAC machinerdquo IEEE Transactions on Industrial Electronics vol45 no 2 pp 291ndash296 1996

[24] J Han Active Disturbance Rejection Control TechniquemdashTheTechnique for Estimating and Compensating the UncertaintiesNational Defense Industry Press Beijing China 2013 (Chi-nese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 005 01 015 02 025 03 035 04

0

5

Time (s)

Back

-EM

F (V

)

Estimated e120572Estimated e120573

minus10

minus5

(a)

0 005 01 015 02 025 03

10

Time (s)

Estimated e120572Estimated e120573

0

5

Back

-EM

F (V

)

minus10

minus5

(b)

Figure 3 Simulation waveforms of estimated back EMF when speed is 500 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

05

0 005 01 015 02 025 03 035 04Time (s)

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

Curr

ent (

A)

minus5

05

minus5

(a)

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

0

5

minus5

0 005 01 015 02 025 03 035 04Time (s)

Curr

ent (

A)

0

5

minus5

(b)

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

0

5

minus5

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

0

5

minus5

(c)

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

05

minus5

0 005 01 015 02 025 03Time (s)

Curr

ent (

A)

05

minus5

(d)

Figure 4 Simulation waveforms of actual and estimated 119894120572 119894120573when speed is 500 rmin and the load torque is 1 Nm (a) Actual and estimated

119894120572obtained by the proposed ANLESO (b) Actual and estimated 119894

120573obtained by the proposed ANLESO (c) Actual and estimated 119894

120572obtained

by the conventional SMO (d) Actual and estimated 119894120573obtained by the conventional SMO

This means that the lower bound of 1198902is larger than

|ℎ|(119877 + 1198711205731) 1205731is a design parameter so the size of 119877 + 119871120573

1

can be changed by means of adjusting 1205731 The main idea of

this method is to decrease 1205731gradually to reduce steady-state

error and increase 1205731when the systemmay become unstable

By several iterations the value of 1205731would approximate the

compromised value gradually

Figure 2 illustrates the algorithm to calculate 1205731 where

0 lt 120572 lt 1 and 119901 gt 1 are adjustment factors Δ119881 is the errorof command velocity and measured velocity

34 Rotor Position and Speed Estimate Using the estimatedback-EMF obtained by the ANLESO the position and speed

Mathematical Problems in Engineering 7

0 005 01 015 02 025 03 035 040

200

400

600

Time (s)

Roto

r spe

ed (r

min

)

Actual speedEstimated speed

(a)

0 005 01 015 02 025 03

0

200

400

600

Time (s)

Roto

r spe

ed (r

min

)

Estimated speedActual speed

(b)

Figure 5 Simulation waveforms of actual and estimated speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by theproposed ANLESO and (b) obtained by the conventional SMO

0

5

0

5

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)

0 005 01 015 02 025 03 035 04

05

Time (s)

0 005 01 015 02 025 03 035 04Time (s)

0 005 01 015 02 025 03 035 04Time (s)

minus5

(a)

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)0

5

0

5

0 005 01 015 02 025 03

010

Time (s)

0 005 01 015 02 025 03Time (s)

0 005 01 015 02 025 03Time (s)

minus10

(b)

Figure 6 Simulation waveforms of actual rotor position estimated rotor position and estimated error when speed is 500 rmin and the loadtorque is 1 Nm (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO

signals can be calculated easily According to (3) the rotorspeed can be obtained as follows

119903=

radic(119890120572

2 + 119890120573

2)

119870119864

(34)

Similarly the rotor position can be deduced from (3) asfollows

120579 = minus arctan[119890120572

119890120573

] (35)

4 Simulation and Experimental Results

41 Simulation Results In order to show the high-speedperformance of the proposed ANLESO it is necessary tocompare it with the conventional SMO through the Mat-LabSimulink programming environmentThe parameters ofPMSM used in simulation are listed in Table 1

Figures 3ndash6 show the simulation waveforms when thereference speed is changed from zero to 500 rmin and

Table 1 The parameters of PMSM

Parameters ValuesRated power 15 KWRated speed 2500 rminInput voltage (DC) 310VCurrent 6AStator resistance 118Ω

Stator inductance 5326mHRotational inertia 133 times 10minus3 kgsdotm2

Flux of permanent magnet 00356WbPoles 8

the load torque is 1 Nm In these figures part (a) gives thesimulation waveforms obtained by the proposed ANLESOand part (b) gives the simulation waveforms obtained bythe conventional SMO Estimated back-EMF actual andestimated phase currents actual and estimated rotor speed

8 Mathematical Problems in Engineering

015 02 025 03

0

20

40

Time (s)

Back

-EM

F (V

)

minus20

minus40

Estimated e120572Estimated e120573

(a)

Time (s)015 02 025 03

0

20

40

Back

-EM

F (V

)

minus20

minus40

Estimated e120572Estimated e120573

(b)

Figure 7 Simulation waveforms of estimated back EMF when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

(a)

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

(b)

015 02 025 03Time (s)

015 02 025 03Time (s)

0

5

Curr

ent (

A)

minus5

0

5

Curr

ent (

A)

minus5

(c)

0

5

015 02 025 03Time (s)

015 02 025 03Time (s)

0

5

Curr

ent (

A)

Curr

ent (

A)

minus5

minus5

(d)

Figure 8 Simulation waveforms of actual and estimated 119894120572 119894120573when speed is 2000 rmin and the load torque is 1 Nm (a) Actual and estimated

119894120572obtained by the proposed ANLESO (b) Actual and estimated 119894

120573obtained by the proposed ANLESO (c) Actual and estimated 119894

120572obtained

by the conventional SMO (d) Actual and estimated 119894120573obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 3ndash6 the accuracy of esti-mation has been greatly improved with the help of proposedANLESO Different from the conventional SMO the back-EMF in Figure 3(a) almost has no chattering Comparedwith Figures 4(c) and 4(d) the estimated results of phase

currents in Figures 4(a) and 4(b) are better Due to the goodperformance of estimating back-EMF and currents thechattering phenomenon of the estimated rotor position andspeed is reduced which can be seen from Figures 5(a) 5(b)6(a) and 6(b) In part (a) of these figures the estimated speedis almost the same as actual speed however the estimatedspeed in part (b) of these figures exhibits the chattering

Mathematical Problems in Engineering 9

Time (s)0 005 01 015 02 025 03 035 04

0

1000

2000

Roto

r spe

ed (r

min

)

Estimated speedActual speed

(a)

Time (s)

Estimated speedActual speed

0 005 01 015 02 025 03

0

1000

2000

Roto

r spe

ed (r

min

)

(b)

Figure 9 Simulation waveforms of actual and estimated speeds when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by theproposed ANLESO and (b) obtained by the conventional SMO

0

5

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)

0

5

015 02 025 03

05

Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

minus5

(a)

5

Roto

r pos

ition

(rad

)0

5

Roto

r pos

ition

(rad

)

0

010

015 02 025 03Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

Roto

r pos

ition

(rad

)

minus10

(b)

Figure 10 Simulation waveforms of actual rotor position estimated rotor position and estimated error when speed is 2000 rmin and theload torque is 1 Nm (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO

Figure 11 Platform of 15 kw PMSM sensorless control system based on DSP

phenomenon Besides it can be seen that the estimated errorof the position in part (a) is smaller than that shown in part(b) of these figures

Figures 7ndash10 show the two sets of simulation waveformswhen the reference speed is changed from zero to 2000 rmin

and the load torque is 1 Nm In these figures part (a) gives thesimulation waveforms obtained by the proposed ANLESOand part (b) gives the simulation waveforms obtained bythe conventional SMO Estimated back-EMF actual andestimated phase currents actual and estimated rotor speed

10 Mathematical Problems in Engineering

(a) (b)

Figure 12 Operating waveforms of estimated back EMF in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained bythe proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 13 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 14 Operating waveforms of estimated rotor speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 7ndash10 when the rotor speedis raised to 2000 rmin advantages of ANLESO becomemoreapparent than conventional SMO Comparing the estimatedback-EMF in Figures 7(a) and 7(b) we can find that theformer has smaller chattering As a result the estimationof rotor speed becomes more accurate by ANLESO whichcan be seen in Figures 9(a) and 9(b) The waveform formspresented in Figures 10(a) and 10(b) show the estimatedrotor position The position estimation error is reduced andthe accuracy of rotor position estimation is improved inFigure 10(a)

42 Experimental Results To further verify the performanceof the new SMO for estimating rotor position and speed an

experimental systemhas been designed to control the 130SJT-M060D (15 kw) sinusoidal PMSMmotor made by GSK

Figure 11 shows a photograph of the sensorless controlsystemTheHighVoltage DigitalMotor Control Kit made byTI is utilized in the system which contains power modulesswitching devices current and voltage sensing circuit ADCmodules and PWM DAC (digital to analog converter)modules A floating-point TMS320F28335 DSP is employedas the digital controller The current control cycle is 100120583sand the velocity control cycle is 1ms A 2 120583s dead time ischosen for the PWM switching All of the control variablesare monitored in real time by an oscilloscope after theyare converted to analog signals through the PWM DACsmodules which use an external low pass filter to generatethe waveformsThemotor parameters used in experiment aregiven in Table 1

Mathematical Problems in Engineering 11

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 15 Operating waveforms of rotor position and estimate error when speed is 500 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

(a) (b)

Figure 16 Operating waveforms of estimated back EMF in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtainedby the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 17 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

Figures 12ndash14 show the operating waveforms when thereference speed is changed from zero to 500 rmin and theload torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO Estimated back-EMF estimated phase

currents and estimated rotor speed are shown in thesefigures In Figure 15 actual rotor position estimated rotorposition and their error are shown

Figures 12(a) and 13(a) show the experimental resultsof the proposed ANLESO Compared to the waveformsobtained by SMO (see part (b) in Figures 12 and 13)

12 Mathematical Problems in Engineering

(a) (b)

Figure 18 Operatingwaveforms of estimated rotor speedswhen speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 19 Operating waveforms of rotor position and estimate error when speed is 2000 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

thewaveforms of the estimated back-EMF and current shownin Figures 12(a) and 13(a) are more accurate The waveformspresented in Figures 14(a) and 14(b) show the differences ofrotor speed estimated by two kinds of observers The formerspeed waveform has less chattering In other words ANLESOhas higher estimated precision

Figure 15 shows the experimental results of positionresponseThe estimated error shown in Figure 15(c) is smallerthan that in Figure 15(d) which verifies the high precision ofANLESO

Figures 16ndash18 show the operating waveforms when thereference speed is changed from zero to 2000 rmin andthe load torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO In these figures estimated back-EMFestimated phase currents and estimated rotor speed areshown In Figure 19 actual rotor position estimated rotorposition and their error are shown

Figures 12ndash15 show the operating waveforms when thereference speed is changed from zero to 2000 rmin Figures12 and 13 provide the operating waveforms obtained by theproposed ANLESO Figures 14 and 15 show the simulationwaveforms obtained by the conventional SMO In thesefigures estimated back-EMF estimated phase currents esti-mated rotor speed and actual and estimated rotor positionare shown

Figures 16ndash18 show the experimental results of estimatedback-EMF current and rotor speed when the rotor speed israised to 2000 rmin It can be seen that the waveforms inpart (a) of these figures have smaller chattering and higheraccuracy In this way the advantages of ANLESO have beenverified

As shown in Figure 19 when the rotor speed is raisedto 2000 rmin the advantages of ANLESO to estimate rotorposition becomemore apparent than conventional SMOTheestimation error in Figure 19(c) is significantly less than thatin Figure 19(d) That is to say ANLESO has higher accuracy

Mathematical Problems in Engineering 13

and smaller chattering when being applied to estimate posi-tion than SMO

5 Conclusions

In this paper an adaptive nonlinear extended state observerhas been designed for the sensorless control of a PMSMTheconvergence of this observer has been proved by means of aLyapunov stability analysis An adaptive algorithm is adoptedto calculate the compromised parameter of ESO in order totake both stability and steady-state error into considerationThegoodperformance of the proposed sensorless control sys-tem was verified by several experimental results The resultsshow that ANLESO has more advantages than conventionalSMO Compared to conventional SMO ANLESO has smallerchattering higher accuracy and no phase delay

In future works we will explore the stability of the ESOunder parameter uncertainties in sensorless control system

Conflict of Interests

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

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China (Grant no 61304097) and Projects ofMajor International (Regional) Joint Research ProgramNSFC (Grant no 61120106010)

References

[1] S Lu X Tang andB Song ldquoAdaptive PIF control for permanentmagnet synchronousmotors based onGPCrdquo Sensors vol 13 no1 pp 175ndash192 2013

[2] A Gomez-Espinosa V M Hernandez-Guzman M Bandala-Sanchez et al ldquoA new adaptive self-tuning Fourier Coefficientsalgorithm for periodic torque ripple minimization in perma-nentmagnet synchronousmotors (PMSM)rdquo Sensors vol 13 no3 pp 3831ndash3847 2013

[3] Y Zhang C M Akujuobi W H Ali C L Tolliver and L-SShieh ldquoLoad disturbance resistance speed controller design forPMSMrdquo IEEE Transactions on Industrial Electronics vol 53 no4 pp 1198ndash1208 2006

[4] J C Gamazo-Real E Vazquez-Sanchez and J Gomez-GilldquoPosition and speed control of brushless DC motors usingsensorless techniques and application trendsrdquo Sensors vol 10no 7 pp 6901ndash6947 2010

[5] J-H Jang S-K Sul J-I Ha K Ide and M Sawamura ldquoSen-sorless drive of surface-mounted permanent-magnet motor byhigh-frequency signal injection based on magnetic saliencyrdquoIEEE Transactions on Industry Applications vol 39 no 4 pp1031ndash1039 2003

[6] A Piippo M Hinkkanen and J Luomi ldquoSensorless control ofPMSM drives using a combination of voltage model and HFsignal injectionrdquo in Proceedings of the 39th IAS Annual MeetingConference Record of the IEEE Industry Applications Conferencevol 2 pp 964ndash970 October 2004

[7] G Wang R Yang and D Xu ldquoDSP-based control of sensorlessIPMSM drives for wide-speed-range operationrdquo IEEE Transac-tions on Industrial Electronics vol 60 no 2 pp 720ndash727 2013

[8] Z Wang Y Zheng Z Zou and M Cheng ldquoPosition sensorlesscontrol of interleaved CSI fed PMSM drive with extendedKalman filterrdquo IEEE Transactions on Magnetics vol 48 no 11pp 3688ndash3691 2012

[9] T D Batzel and K Y Lee ldquoAn approach to sensorless operationof the permanent-magnet synchronous motor using diagonallyrecurrent neural networksrdquo IEEE Transactions on Energy Con-version vol 18 no 1 pp 100ndash106 2003

[10] B Zhang andY Li ldquoAPMSMslidingmode control systembasedon model reference adaptive controlrdquo in Proceedings of the 3rdInternational Power Electronics and Motion Control Conference(IPEMC rsquo00) vol 1 pp 336ndash341 IEEE Beijing China 2000

[11] S Shinnaka ldquoNew sensorless vector control using minimum-order flux state observer in a stationary reference frame forpermanent-magnet synchronousmotorsrdquo IEEETransactions onIndustrial Electronics vol 53 no 2 pp 388ndash398 2006

[12] A Khlaief M Bendjedia M Boussak and M Gossa ldquoA non-linear observer for high-performance sensorless speed controlof IPMSM driverdquo IEEE Transactions on Power Electronics vol27 no 6 pp 3028ndash3040 2012

[13] H Lee and J Lee ldquoDesign of iterative sliding mode observerfor sensorless PMSM controlrdquo IEEE Transactions on ControlSystems Technology vol 21 no 4 pp 1394ndash1399 2013

[14] Y-S Han J-S Choi and Y-S Kim ldquoSensorless PMSM drivewith a sliding mode control based adaptive speed and statorresistance estimatorrdquo IEEE Transactions on Magnetics vol 36no 5 pp 3588ndash3591 2000

[15] V C Ilioudis and N I Margaris ldquoPMSM sensorless speedestimation based on sliding mode observersrdquo in Proceedings ofthe IEEE Power Electronics Specialists Conference (PESC rsquo08) pp2838ndash2843 IEEE Rhodes Greece June 2008

[16] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[17] Y Huang and J Han ldquoAnalysis and design for the second ordernonlinear continuous extended states observerrdquoChinese ScienceBulletin vol 45 no 21 pp 1938ndash1944 2000

[18] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[19] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence vol 6 pp 4989ndash4996 IEEE June 2003

[20] Y Xia Z Zhu and M Fu ldquoBack-stepping sliding mode controlfor missile systems based on an extended state observerrdquo IETControl Theory and Applications vol 5 no 1 pp 93ndash102 2011

[21] X Li and S Li ldquoExtended state observer based adaptive controlscheme for PMSM systemrdquo in Proceedings of the 33rd IEEEChinese Control Conference (CCC rsquo14) 2014

[22] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

14 Mathematical Problems in Engineering

[23] G ZhuAKaddouri L ADessaint andOAkhrif ldquoAnonlinearstate observer for the sensorless control of a permanent-magnetAC machinerdquo IEEE Transactions on Industrial Electronics vol45 no 2 pp 291ndash296 1996

[24] J Han Active Disturbance Rejection Control TechniquemdashTheTechnique for Estimating and Compensating the UncertaintiesNational Defense Industry Press Beijing China 2013 (Chi-nese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 005 01 015 02 025 03 035 040

200

400

600

Time (s)

Roto

r spe

ed (r

min

)

Actual speedEstimated speed

(a)

0 005 01 015 02 025 03

0

200

400

600

Time (s)

Roto

r spe

ed (r

min

)

Estimated speedActual speed

(b)

Figure 5 Simulation waveforms of actual and estimated speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by theproposed ANLESO and (b) obtained by the conventional SMO

0

5

0

5

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)

0 005 01 015 02 025 03 035 04

05

Time (s)

0 005 01 015 02 025 03 035 04Time (s)

0 005 01 015 02 025 03 035 04Time (s)

minus5

(a)

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)0

5

0

5

0 005 01 015 02 025 03

010

Time (s)

0 005 01 015 02 025 03Time (s)

0 005 01 015 02 025 03Time (s)

minus10

(b)

Figure 6 Simulation waveforms of actual rotor position estimated rotor position and estimated error when speed is 500 rmin and the loadtorque is 1 Nm (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO

signals can be calculated easily According to (3) the rotorspeed can be obtained as follows

119903=

radic(119890120572

2 + 119890120573

2)

119870119864

(34)

Similarly the rotor position can be deduced from (3) asfollows

120579 = minus arctan[119890120572

119890120573

] (35)

4 Simulation and Experimental Results

41 Simulation Results In order to show the high-speedperformance of the proposed ANLESO it is necessary tocompare it with the conventional SMO through the Mat-LabSimulink programming environmentThe parameters ofPMSM used in simulation are listed in Table 1

Figures 3ndash6 show the simulation waveforms when thereference speed is changed from zero to 500 rmin and

Table 1 The parameters of PMSM

Parameters ValuesRated power 15 KWRated speed 2500 rminInput voltage (DC) 310VCurrent 6AStator resistance 118Ω

Stator inductance 5326mHRotational inertia 133 times 10minus3 kgsdotm2

Flux of permanent magnet 00356WbPoles 8

the load torque is 1 Nm In these figures part (a) gives thesimulation waveforms obtained by the proposed ANLESOand part (b) gives the simulation waveforms obtained bythe conventional SMO Estimated back-EMF actual andestimated phase currents actual and estimated rotor speed

8 Mathematical Problems in Engineering

015 02 025 03

0

20

40

Time (s)

Back

-EM

F (V

)

minus20

minus40

Estimated e120572Estimated e120573

(a)

Time (s)015 02 025 03

0

20

40

Back

-EM

F (V

)

minus20

minus40

Estimated e120572Estimated e120573

(b)

Figure 7 Simulation waveforms of estimated back EMF when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

(a)

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

(b)

015 02 025 03Time (s)

015 02 025 03Time (s)

0

5

Curr

ent (

A)

minus5

0

5

Curr

ent (

A)

minus5

(c)

0

5

015 02 025 03Time (s)

015 02 025 03Time (s)

0

5

Curr

ent (

A)

Curr

ent (

A)

minus5

minus5

(d)

Figure 8 Simulation waveforms of actual and estimated 119894120572 119894120573when speed is 2000 rmin and the load torque is 1 Nm (a) Actual and estimated

119894120572obtained by the proposed ANLESO (b) Actual and estimated 119894

120573obtained by the proposed ANLESO (c) Actual and estimated 119894

120572obtained

by the conventional SMO (d) Actual and estimated 119894120573obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 3ndash6 the accuracy of esti-mation has been greatly improved with the help of proposedANLESO Different from the conventional SMO the back-EMF in Figure 3(a) almost has no chattering Comparedwith Figures 4(c) and 4(d) the estimated results of phase

currents in Figures 4(a) and 4(b) are better Due to the goodperformance of estimating back-EMF and currents thechattering phenomenon of the estimated rotor position andspeed is reduced which can be seen from Figures 5(a) 5(b)6(a) and 6(b) In part (a) of these figures the estimated speedis almost the same as actual speed however the estimatedspeed in part (b) of these figures exhibits the chattering

Mathematical Problems in Engineering 9

Time (s)0 005 01 015 02 025 03 035 04

0

1000

2000

Roto

r spe

ed (r

min

)

Estimated speedActual speed

(a)

Time (s)

Estimated speedActual speed

0 005 01 015 02 025 03

0

1000

2000

Roto

r spe

ed (r

min

)

(b)

Figure 9 Simulation waveforms of actual and estimated speeds when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by theproposed ANLESO and (b) obtained by the conventional SMO

0

5

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)

0

5

015 02 025 03

05

Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

minus5

(a)

5

Roto

r pos

ition

(rad

)0

5

Roto

r pos

ition

(rad

)

0

010

015 02 025 03Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

Roto

r pos

ition

(rad

)

minus10

(b)

Figure 10 Simulation waveforms of actual rotor position estimated rotor position and estimated error when speed is 2000 rmin and theload torque is 1 Nm (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO

Figure 11 Platform of 15 kw PMSM sensorless control system based on DSP

phenomenon Besides it can be seen that the estimated errorof the position in part (a) is smaller than that shown in part(b) of these figures

Figures 7ndash10 show the two sets of simulation waveformswhen the reference speed is changed from zero to 2000 rmin

and the load torque is 1 Nm In these figures part (a) gives thesimulation waveforms obtained by the proposed ANLESOand part (b) gives the simulation waveforms obtained bythe conventional SMO Estimated back-EMF actual andestimated phase currents actual and estimated rotor speed

10 Mathematical Problems in Engineering

(a) (b)

Figure 12 Operating waveforms of estimated back EMF in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained bythe proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 13 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 14 Operating waveforms of estimated rotor speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 7ndash10 when the rotor speedis raised to 2000 rmin advantages of ANLESO becomemoreapparent than conventional SMO Comparing the estimatedback-EMF in Figures 7(a) and 7(b) we can find that theformer has smaller chattering As a result the estimationof rotor speed becomes more accurate by ANLESO whichcan be seen in Figures 9(a) and 9(b) The waveform formspresented in Figures 10(a) and 10(b) show the estimatedrotor position The position estimation error is reduced andthe accuracy of rotor position estimation is improved inFigure 10(a)

42 Experimental Results To further verify the performanceof the new SMO for estimating rotor position and speed an

experimental systemhas been designed to control the 130SJT-M060D (15 kw) sinusoidal PMSMmotor made by GSK

Figure 11 shows a photograph of the sensorless controlsystemTheHighVoltage DigitalMotor Control Kit made byTI is utilized in the system which contains power modulesswitching devices current and voltage sensing circuit ADCmodules and PWM DAC (digital to analog converter)modules A floating-point TMS320F28335 DSP is employedas the digital controller The current control cycle is 100120583sand the velocity control cycle is 1ms A 2 120583s dead time ischosen for the PWM switching All of the control variablesare monitored in real time by an oscilloscope after theyare converted to analog signals through the PWM DACsmodules which use an external low pass filter to generatethe waveformsThemotor parameters used in experiment aregiven in Table 1

Mathematical Problems in Engineering 11

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 15 Operating waveforms of rotor position and estimate error when speed is 500 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

(a) (b)

Figure 16 Operating waveforms of estimated back EMF in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtainedby the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 17 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

Figures 12ndash14 show the operating waveforms when thereference speed is changed from zero to 500 rmin and theload torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO Estimated back-EMF estimated phase

currents and estimated rotor speed are shown in thesefigures In Figure 15 actual rotor position estimated rotorposition and their error are shown

Figures 12(a) and 13(a) show the experimental resultsof the proposed ANLESO Compared to the waveformsobtained by SMO (see part (b) in Figures 12 and 13)

12 Mathematical Problems in Engineering

(a) (b)

Figure 18 Operatingwaveforms of estimated rotor speedswhen speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 19 Operating waveforms of rotor position and estimate error when speed is 2000 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

thewaveforms of the estimated back-EMF and current shownin Figures 12(a) and 13(a) are more accurate The waveformspresented in Figures 14(a) and 14(b) show the differences ofrotor speed estimated by two kinds of observers The formerspeed waveform has less chattering In other words ANLESOhas higher estimated precision

Figure 15 shows the experimental results of positionresponseThe estimated error shown in Figure 15(c) is smallerthan that in Figure 15(d) which verifies the high precision ofANLESO

Figures 16ndash18 show the operating waveforms when thereference speed is changed from zero to 2000 rmin andthe load torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO In these figures estimated back-EMFestimated phase currents and estimated rotor speed areshown In Figure 19 actual rotor position estimated rotorposition and their error are shown

Figures 12ndash15 show the operating waveforms when thereference speed is changed from zero to 2000 rmin Figures12 and 13 provide the operating waveforms obtained by theproposed ANLESO Figures 14 and 15 show the simulationwaveforms obtained by the conventional SMO In thesefigures estimated back-EMF estimated phase currents esti-mated rotor speed and actual and estimated rotor positionare shown

Figures 16ndash18 show the experimental results of estimatedback-EMF current and rotor speed when the rotor speed israised to 2000 rmin It can be seen that the waveforms inpart (a) of these figures have smaller chattering and higheraccuracy In this way the advantages of ANLESO have beenverified

As shown in Figure 19 when the rotor speed is raisedto 2000 rmin the advantages of ANLESO to estimate rotorposition becomemore apparent than conventional SMOTheestimation error in Figure 19(c) is significantly less than thatin Figure 19(d) That is to say ANLESO has higher accuracy

Mathematical Problems in Engineering 13

and smaller chattering when being applied to estimate posi-tion than SMO

5 Conclusions

In this paper an adaptive nonlinear extended state observerhas been designed for the sensorless control of a PMSMTheconvergence of this observer has been proved by means of aLyapunov stability analysis An adaptive algorithm is adoptedto calculate the compromised parameter of ESO in order totake both stability and steady-state error into considerationThegoodperformance of the proposed sensorless control sys-tem was verified by several experimental results The resultsshow that ANLESO has more advantages than conventionalSMO Compared to conventional SMO ANLESO has smallerchattering higher accuracy and no phase delay

In future works we will explore the stability of the ESOunder parameter uncertainties in sensorless control system

Conflict of Interests

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

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China (Grant no 61304097) and Projects ofMajor International (Regional) Joint Research ProgramNSFC (Grant no 61120106010)

References

[1] S Lu X Tang andB Song ldquoAdaptive PIF control for permanentmagnet synchronousmotors based onGPCrdquo Sensors vol 13 no1 pp 175ndash192 2013

[2] A Gomez-Espinosa V M Hernandez-Guzman M Bandala-Sanchez et al ldquoA new adaptive self-tuning Fourier Coefficientsalgorithm for periodic torque ripple minimization in perma-nentmagnet synchronousmotors (PMSM)rdquo Sensors vol 13 no3 pp 3831ndash3847 2013

[3] Y Zhang C M Akujuobi W H Ali C L Tolliver and L-SShieh ldquoLoad disturbance resistance speed controller design forPMSMrdquo IEEE Transactions on Industrial Electronics vol 53 no4 pp 1198ndash1208 2006

[4] J C Gamazo-Real E Vazquez-Sanchez and J Gomez-GilldquoPosition and speed control of brushless DC motors usingsensorless techniques and application trendsrdquo Sensors vol 10no 7 pp 6901ndash6947 2010

[5] J-H Jang S-K Sul J-I Ha K Ide and M Sawamura ldquoSen-sorless drive of surface-mounted permanent-magnet motor byhigh-frequency signal injection based on magnetic saliencyrdquoIEEE Transactions on Industry Applications vol 39 no 4 pp1031ndash1039 2003

[6] A Piippo M Hinkkanen and J Luomi ldquoSensorless control ofPMSM drives using a combination of voltage model and HFsignal injectionrdquo in Proceedings of the 39th IAS Annual MeetingConference Record of the IEEE Industry Applications Conferencevol 2 pp 964ndash970 October 2004

[7] G Wang R Yang and D Xu ldquoDSP-based control of sensorlessIPMSM drives for wide-speed-range operationrdquo IEEE Transac-tions on Industrial Electronics vol 60 no 2 pp 720ndash727 2013

[8] Z Wang Y Zheng Z Zou and M Cheng ldquoPosition sensorlesscontrol of interleaved CSI fed PMSM drive with extendedKalman filterrdquo IEEE Transactions on Magnetics vol 48 no 11pp 3688ndash3691 2012

[9] T D Batzel and K Y Lee ldquoAn approach to sensorless operationof the permanent-magnet synchronous motor using diagonallyrecurrent neural networksrdquo IEEE Transactions on Energy Con-version vol 18 no 1 pp 100ndash106 2003

[10] B Zhang andY Li ldquoAPMSMslidingmode control systembasedon model reference adaptive controlrdquo in Proceedings of the 3rdInternational Power Electronics and Motion Control Conference(IPEMC rsquo00) vol 1 pp 336ndash341 IEEE Beijing China 2000

[11] S Shinnaka ldquoNew sensorless vector control using minimum-order flux state observer in a stationary reference frame forpermanent-magnet synchronousmotorsrdquo IEEETransactions onIndustrial Electronics vol 53 no 2 pp 388ndash398 2006

[12] A Khlaief M Bendjedia M Boussak and M Gossa ldquoA non-linear observer for high-performance sensorless speed controlof IPMSM driverdquo IEEE Transactions on Power Electronics vol27 no 6 pp 3028ndash3040 2012

[13] H Lee and J Lee ldquoDesign of iterative sliding mode observerfor sensorless PMSM controlrdquo IEEE Transactions on ControlSystems Technology vol 21 no 4 pp 1394ndash1399 2013

[14] Y-S Han J-S Choi and Y-S Kim ldquoSensorless PMSM drivewith a sliding mode control based adaptive speed and statorresistance estimatorrdquo IEEE Transactions on Magnetics vol 36no 5 pp 3588ndash3591 2000

[15] V C Ilioudis and N I Margaris ldquoPMSM sensorless speedestimation based on sliding mode observersrdquo in Proceedings ofthe IEEE Power Electronics Specialists Conference (PESC rsquo08) pp2838ndash2843 IEEE Rhodes Greece June 2008

[16] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[17] Y Huang and J Han ldquoAnalysis and design for the second ordernonlinear continuous extended states observerrdquoChinese ScienceBulletin vol 45 no 21 pp 1938ndash1944 2000

[18] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[19] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence vol 6 pp 4989ndash4996 IEEE June 2003

[20] Y Xia Z Zhu and M Fu ldquoBack-stepping sliding mode controlfor missile systems based on an extended state observerrdquo IETControl Theory and Applications vol 5 no 1 pp 93ndash102 2011

[21] X Li and S Li ldquoExtended state observer based adaptive controlscheme for PMSM systemrdquo in Proceedings of the 33rd IEEEChinese Control Conference (CCC rsquo14) 2014

[22] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

14 Mathematical Problems in Engineering

[23] G ZhuAKaddouri L ADessaint andOAkhrif ldquoAnonlinearstate observer for the sensorless control of a permanent-magnetAC machinerdquo IEEE Transactions on Industrial Electronics vol45 no 2 pp 291ndash296 1996

[24] J Han Active Disturbance Rejection Control TechniquemdashTheTechnique for Estimating and Compensating the UncertaintiesNational Defense Industry Press Beijing China 2013 (Chi-nese)

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

015 02 025 03

0

20

40

Time (s)

Back

-EM

F (V

)

minus20

minus40

Estimated e120572Estimated e120573

(a)

Time (s)015 02 025 03

0

20

40

Back

-EM

F (V

)

minus20

minus40

Estimated e120572Estimated e120573

(b)

Figure 7 Simulation waveforms of estimated back EMF when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

(a)

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

015 02 025 03

0

5

Time (s)

Curr

ent (

A)

minus5

(b)

015 02 025 03Time (s)

015 02 025 03Time (s)

0

5

Curr

ent (

A)

minus5

0

5

Curr

ent (

A)

minus5

(c)

0

5

015 02 025 03Time (s)

015 02 025 03Time (s)

0

5

Curr

ent (

A)

Curr

ent (

A)

minus5

minus5

(d)

Figure 8 Simulation waveforms of actual and estimated 119894120572 119894120573when speed is 2000 rmin and the load torque is 1 Nm (a) Actual and estimated

119894120572obtained by the proposed ANLESO (b) Actual and estimated 119894

120573obtained by the proposed ANLESO (c) Actual and estimated 119894

120572obtained

by the conventional SMO (d) Actual and estimated 119894120573obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 3ndash6 the accuracy of esti-mation has been greatly improved with the help of proposedANLESO Different from the conventional SMO the back-EMF in Figure 3(a) almost has no chattering Comparedwith Figures 4(c) and 4(d) the estimated results of phase

currents in Figures 4(a) and 4(b) are better Due to the goodperformance of estimating back-EMF and currents thechattering phenomenon of the estimated rotor position andspeed is reduced which can be seen from Figures 5(a) 5(b)6(a) and 6(b) In part (a) of these figures the estimated speedis almost the same as actual speed however the estimatedspeed in part (b) of these figures exhibits the chattering

Mathematical Problems in Engineering 9

Time (s)0 005 01 015 02 025 03 035 04

0

1000

2000

Roto

r spe

ed (r

min

)

Estimated speedActual speed

(a)

Time (s)

Estimated speedActual speed

0 005 01 015 02 025 03

0

1000

2000

Roto

r spe

ed (r

min

)

(b)

Figure 9 Simulation waveforms of actual and estimated speeds when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by theproposed ANLESO and (b) obtained by the conventional SMO

0

5

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)

0

5

015 02 025 03

05

Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

minus5

(a)

5

Roto

r pos

ition

(rad

)0

5

Roto

r pos

ition

(rad

)

0

010

015 02 025 03Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

Roto

r pos

ition

(rad

)

minus10

(b)

Figure 10 Simulation waveforms of actual rotor position estimated rotor position and estimated error when speed is 2000 rmin and theload torque is 1 Nm (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO

Figure 11 Platform of 15 kw PMSM sensorless control system based on DSP

phenomenon Besides it can be seen that the estimated errorof the position in part (a) is smaller than that shown in part(b) of these figures

Figures 7ndash10 show the two sets of simulation waveformswhen the reference speed is changed from zero to 2000 rmin

and the load torque is 1 Nm In these figures part (a) gives thesimulation waveforms obtained by the proposed ANLESOand part (b) gives the simulation waveforms obtained bythe conventional SMO Estimated back-EMF actual andestimated phase currents actual and estimated rotor speed

10 Mathematical Problems in Engineering

(a) (b)

Figure 12 Operating waveforms of estimated back EMF in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained bythe proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 13 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 14 Operating waveforms of estimated rotor speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 7ndash10 when the rotor speedis raised to 2000 rmin advantages of ANLESO becomemoreapparent than conventional SMO Comparing the estimatedback-EMF in Figures 7(a) and 7(b) we can find that theformer has smaller chattering As a result the estimationof rotor speed becomes more accurate by ANLESO whichcan be seen in Figures 9(a) and 9(b) The waveform formspresented in Figures 10(a) and 10(b) show the estimatedrotor position The position estimation error is reduced andthe accuracy of rotor position estimation is improved inFigure 10(a)

42 Experimental Results To further verify the performanceof the new SMO for estimating rotor position and speed an

experimental systemhas been designed to control the 130SJT-M060D (15 kw) sinusoidal PMSMmotor made by GSK

Figure 11 shows a photograph of the sensorless controlsystemTheHighVoltage DigitalMotor Control Kit made byTI is utilized in the system which contains power modulesswitching devices current and voltage sensing circuit ADCmodules and PWM DAC (digital to analog converter)modules A floating-point TMS320F28335 DSP is employedas the digital controller The current control cycle is 100120583sand the velocity control cycle is 1ms A 2 120583s dead time ischosen for the PWM switching All of the control variablesare monitored in real time by an oscilloscope after theyare converted to analog signals through the PWM DACsmodules which use an external low pass filter to generatethe waveformsThemotor parameters used in experiment aregiven in Table 1

Mathematical Problems in Engineering 11

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 15 Operating waveforms of rotor position and estimate error when speed is 500 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

(a) (b)

Figure 16 Operating waveforms of estimated back EMF in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtainedby the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 17 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

Figures 12ndash14 show the operating waveforms when thereference speed is changed from zero to 500 rmin and theload torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO Estimated back-EMF estimated phase

currents and estimated rotor speed are shown in thesefigures In Figure 15 actual rotor position estimated rotorposition and their error are shown

Figures 12(a) and 13(a) show the experimental resultsof the proposed ANLESO Compared to the waveformsobtained by SMO (see part (b) in Figures 12 and 13)

12 Mathematical Problems in Engineering

(a) (b)

Figure 18 Operatingwaveforms of estimated rotor speedswhen speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 19 Operating waveforms of rotor position and estimate error when speed is 2000 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

thewaveforms of the estimated back-EMF and current shownin Figures 12(a) and 13(a) are more accurate The waveformspresented in Figures 14(a) and 14(b) show the differences ofrotor speed estimated by two kinds of observers The formerspeed waveform has less chattering In other words ANLESOhas higher estimated precision

Figure 15 shows the experimental results of positionresponseThe estimated error shown in Figure 15(c) is smallerthan that in Figure 15(d) which verifies the high precision ofANLESO

Figures 16ndash18 show the operating waveforms when thereference speed is changed from zero to 2000 rmin andthe load torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO In these figures estimated back-EMFestimated phase currents and estimated rotor speed areshown In Figure 19 actual rotor position estimated rotorposition and their error are shown

Figures 12ndash15 show the operating waveforms when thereference speed is changed from zero to 2000 rmin Figures12 and 13 provide the operating waveforms obtained by theproposed ANLESO Figures 14 and 15 show the simulationwaveforms obtained by the conventional SMO In thesefigures estimated back-EMF estimated phase currents esti-mated rotor speed and actual and estimated rotor positionare shown

Figures 16ndash18 show the experimental results of estimatedback-EMF current and rotor speed when the rotor speed israised to 2000 rmin It can be seen that the waveforms inpart (a) of these figures have smaller chattering and higheraccuracy In this way the advantages of ANLESO have beenverified

As shown in Figure 19 when the rotor speed is raisedto 2000 rmin the advantages of ANLESO to estimate rotorposition becomemore apparent than conventional SMOTheestimation error in Figure 19(c) is significantly less than thatin Figure 19(d) That is to say ANLESO has higher accuracy

Mathematical Problems in Engineering 13

and smaller chattering when being applied to estimate posi-tion than SMO

5 Conclusions

In this paper an adaptive nonlinear extended state observerhas been designed for the sensorless control of a PMSMTheconvergence of this observer has been proved by means of aLyapunov stability analysis An adaptive algorithm is adoptedto calculate the compromised parameter of ESO in order totake both stability and steady-state error into considerationThegoodperformance of the proposed sensorless control sys-tem was verified by several experimental results The resultsshow that ANLESO has more advantages than conventionalSMO Compared to conventional SMO ANLESO has smallerchattering higher accuracy and no phase delay

In future works we will explore the stability of the ESOunder parameter uncertainties in sensorless control system

Conflict of Interests

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

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China (Grant no 61304097) and Projects ofMajor International (Regional) Joint Research ProgramNSFC (Grant no 61120106010)

References

[1] S Lu X Tang andB Song ldquoAdaptive PIF control for permanentmagnet synchronousmotors based onGPCrdquo Sensors vol 13 no1 pp 175ndash192 2013

[2] A Gomez-Espinosa V M Hernandez-Guzman M Bandala-Sanchez et al ldquoA new adaptive self-tuning Fourier Coefficientsalgorithm for periodic torque ripple minimization in perma-nentmagnet synchronousmotors (PMSM)rdquo Sensors vol 13 no3 pp 3831ndash3847 2013

[3] Y Zhang C M Akujuobi W H Ali C L Tolliver and L-SShieh ldquoLoad disturbance resistance speed controller design forPMSMrdquo IEEE Transactions on Industrial Electronics vol 53 no4 pp 1198ndash1208 2006

[4] J C Gamazo-Real E Vazquez-Sanchez and J Gomez-GilldquoPosition and speed control of brushless DC motors usingsensorless techniques and application trendsrdquo Sensors vol 10no 7 pp 6901ndash6947 2010

[5] J-H Jang S-K Sul J-I Ha K Ide and M Sawamura ldquoSen-sorless drive of surface-mounted permanent-magnet motor byhigh-frequency signal injection based on magnetic saliencyrdquoIEEE Transactions on Industry Applications vol 39 no 4 pp1031ndash1039 2003

[6] A Piippo M Hinkkanen and J Luomi ldquoSensorless control ofPMSM drives using a combination of voltage model and HFsignal injectionrdquo in Proceedings of the 39th IAS Annual MeetingConference Record of the IEEE Industry Applications Conferencevol 2 pp 964ndash970 October 2004

[7] G Wang R Yang and D Xu ldquoDSP-based control of sensorlessIPMSM drives for wide-speed-range operationrdquo IEEE Transac-tions on Industrial Electronics vol 60 no 2 pp 720ndash727 2013

[8] Z Wang Y Zheng Z Zou and M Cheng ldquoPosition sensorlesscontrol of interleaved CSI fed PMSM drive with extendedKalman filterrdquo IEEE Transactions on Magnetics vol 48 no 11pp 3688ndash3691 2012

[9] T D Batzel and K Y Lee ldquoAn approach to sensorless operationof the permanent-magnet synchronous motor using diagonallyrecurrent neural networksrdquo IEEE Transactions on Energy Con-version vol 18 no 1 pp 100ndash106 2003

[10] B Zhang andY Li ldquoAPMSMslidingmode control systembasedon model reference adaptive controlrdquo in Proceedings of the 3rdInternational Power Electronics and Motion Control Conference(IPEMC rsquo00) vol 1 pp 336ndash341 IEEE Beijing China 2000

[11] S Shinnaka ldquoNew sensorless vector control using minimum-order flux state observer in a stationary reference frame forpermanent-magnet synchronousmotorsrdquo IEEETransactions onIndustrial Electronics vol 53 no 2 pp 388ndash398 2006

[12] A Khlaief M Bendjedia M Boussak and M Gossa ldquoA non-linear observer for high-performance sensorless speed controlof IPMSM driverdquo IEEE Transactions on Power Electronics vol27 no 6 pp 3028ndash3040 2012

[13] H Lee and J Lee ldquoDesign of iterative sliding mode observerfor sensorless PMSM controlrdquo IEEE Transactions on ControlSystems Technology vol 21 no 4 pp 1394ndash1399 2013

[14] Y-S Han J-S Choi and Y-S Kim ldquoSensorless PMSM drivewith a sliding mode control based adaptive speed and statorresistance estimatorrdquo IEEE Transactions on Magnetics vol 36no 5 pp 3588ndash3591 2000

[15] V C Ilioudis and N I Margaris ldquoPMSM sensorless speedestimation based on sliding mode observersrdquo in Proceedings ofthe IEEE Power Electronics Specialists Conference (PESC rsquo08) pp2838ndash2843 IEEE Rhodes Greece June 2008

[16] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[17] Y Huang and J Han ldquoAnalysis and design for the second ordernonlinear continuous extended states observerrdquoChinese ScienceBulletin vol 45 no 21 pp 1938ndash1944 2000

[18] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[19] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence vol 6 pp 4989ndash4996 IEEE June 2003

[20] Y Xia Z Zhu and M Fu ldquoBack-stepping sliding mode controlfor missile systems based on an extended state observerrdquo IETControl Theory and Applications vol 5 no 1 pp 93ndash102 2011

[21] X Li and S Li ldquoExtended state observer based adaptive controlscheme for PMSM systemrdquo in Proceedings of the 33rd IEEEChinese Control Conference (CCC rsquo14) 2014

[22] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

14 Mathematical Problems in Engineering

[23] G ZhuAKaddouri L ADessaint andOAkhrif ldquoAnonlinearstate observer for the sensorless control of a permanent-magnetAC machinerdquo IEEE Transactions on Industrial Electronics vol45 no 2 pp 291ndash296 1996

[24] J Han Active Disturbance Rejection Control TechniquemdashTheTechnique for Estimating and Compensating the UncertaintiesNational Defense Industry Press Beijing China 2013 (Chi-nese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Time (s)0 005 01 015 02 025 03 035 04

0

1000

2000

Roto

r spe

ed (r

min

)

Estimated speedActual speed

(a)

Time (s)

Estimated speedActual speed

0 005 01 015 02 025 03

0

1000

2000

Roto

r spe

ed (r

min

)

(b)

Figure 9 Simulation waveforms of actual and estimated speeds when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by theproposed ANLESO and (b) obtained by the conventional SMO

0

5

Roto

r pos

ition

(rad

)Ro

tor p

ositi

on(r

ad)

Roto

r pos

ition

(rad

)

0

5

015 02 025 03

05

Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

minus5

(a)

5

Roto

r pos

ition

(rad

)0

5

Roto

r pos

ition

(rad

)

0

010

015 02 025 03Time (s)

015 02 025 03Time (s)

015 02 025 03Time (s)

Roto

r pos

ition

(rad

)

minus10

(b)

Figure 10 Simulation waveforms of actual rotor position estimated rotor position and estimated error when speed is 2000 rmin and theload torque is 1 Nm (a) obtained by the proposed ANLESO and (b) obtained by the conventional SMO

Figure 11 Platform of 15 kw PMSM sensorless control system based on DSP

phenomenon Besides it can be seen that the estimated errorof the position in part (a) is smaller than that shown in part(b) of these figures

Figures 7ndash10 show the two sets of simulation waveformswhen the reference speed is changed from zero to 2000 rmin

and the load torque is 1 Nm In these figures part (a) gives thesimulation waveforms obtained by the proposed ANLESOand part (b) gives the simulation waveforms obtained bythe conventional SMO Estimated back-EMF actual andestimated phase currents actual and estimated rotor speed

10 Mathematical Problems in Engineering

(a) (b)

Figure 12 Operating waveforms of estimated back EMF in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained bythe proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 13 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 14 Operating waveforms of estimated rotor speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 7ndash10 when the rotor speedis raised to 2000 rmin advantages of ANLESO becomemoreapparent than conventional SMO Comparing the estimatedback-EMF in Figures 7(a) and 7(b) we can find that theformer has smaller chattering As a result the estimationof rotor speed becomes more accurate by ANLESO whichcan be seen in Figures 9(a) and 9(b) The waveform formspresented in Figures 10(a) and 10(b) show the estimatedrotor position The position estimation error is reduced andthe accuracy of rotor position estimation is improved inFigure 10(a)

42 Experimental Results To further verify the performanceof the new SMO for estimating rotor position and speed an

experimental systemhas been designed to control the 130SJT-M060D (15 kw) sinusoidal PMSMmotor made by GSK

Figure 11 shows a photograph of the sensorless controlsystemTheHighVoltage DigitalMotor Control Kit made byTI is utilized in the system which contains power modulesswitching devices current and voltage sensing circuit ADCmodules and PWM DAC (digital to analog converter)modules A floating-point TMS320F28335 DSP is employedas the digital controller The current control cycle is 100120583sand the velocity control cycle is 1ms A 2 120583s dead time ischosen for the PWM switching All of the control variablesare monitored in real time by an oscilloscope after theyare converted to analog signals through the PWM DACsmodules which use an external low pass filter to generatethe waveformsThemotor parameters used in experiment aregiven in Table 1

Mathematical Problems in Engineering 11

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 15 Operating waveforms of rotor position and estimate error when speed is 500 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

(a) (b)

Figure 16 Operating waveforms of estimated back EMF in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtainedby the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 17 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

Figures 12ndash14 show the operating waveforms when thereference speed is changed from zero to 500 rmin and theload torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO Estimated back-EMF estimated phase

currents and estimated rotor speed are shown in thesefigures In Figure 15 actual rotor position estimated rotorposition and their error are shown

Figures 12(a) and 13(a) show the experimental resultsof the proposed ANLESO Compared to the waveformsobtained by SMO (see part (b) in Figures 12 and 13)

12 Mathematical Problems in Engineering

(a) (b)

Figure 18 Operatingwaveforms of estimated rotor speedswhen speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 19 Operating waveforms of rotor position and estimate error when speed is 2000 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

thewaveforms of the estimated back-EMF and current shownin Figures 12(a) and 13(a) are more accurate The waveformspresented in Figures 14(a) and 14(b) show the differences ofrotor speed estimated by two kinds of observers The formerspeed waveform has less chattering In other words ANLESOhas higher estimated precision

Figure 15 shows the experimental results of positionresponseThe estimated error shown in Figure 15(c) is smallerthan that in Figure 15(d) which verifies the high precision ofANLESO

Figures 16ndash18 show the operating waveforms when thereference speed is changed from zero to 2000 rmin andthe load torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO In these figures estimated back-EMFestimated phase currents and estimated rotor speed areshown In Figure 19 actual rotor position estimated rotorposition and their error are shown

Figures 12ndash15 show the operating waveforms when thereference speed is changed from zero to 2000 rmin Figures12 and 13 provide the operating waveforms obtained by theproposed ANLESO Figures 14 and 15 show the simulationwaveforms obtained by the conventional SMO In thesefigures estimated back-EMF estimated phase currents esti-mated rotor speed and actual and estimated rotor positionare shown

Figures 16ndash18 show the experimental results of estimatedback-EMF current and rotor speed when the rotor speed israised to 2000 rmin It can be seen that the waveforms inpart (a) of these figures have smaller chattering and higheraccuracy In this way the advantages of ANLESO have beenverified

As shown in Figure 19 when the rotor speed is raisedto 2000 rmin the advantages of ANLESO to estimate rotorposition becomemore apparent than conventional SMOTheestimation error in Figure 19(c) is significantly less than thatin Figure 19(d) That is to say ANLESO has higher accuracy

Mathematical Problems in Engineering 13

and smaller chattering when being applied to estimate posi-tion than SMO

5 Conclusions

In this paper an adaptive nonlinear extended state observerhas been designed for the sensorless control of a PMSMTheconvergence of this observer has been proved by means of aLyapunov stability analysis An adaptive algorithm is adoptedto calculate the compromised parameter of ESO in order totake both stability and steady-state error into considerationThegoodperformance of the proposed sensorless control sys-tem was verified by several experimental results The resultsshow that ANLESO has more advantages than conventionalSMO Compared to conventional SMO ANLESO has smallerchattering higher accuracy and no phase delay

In future works we will explore the stability of the ESOunder parameter uncertainties in sensorless control system

Conflict of Interests

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

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China (Grant no 61304097) and Projects ofMajor International (Regional) Joint Research ProgramNSFC (Grant no 61120106010)

References

[1] S Lu X Tang andB Song ldquoAdaptive PIF control for permanentmagnet synchronousmotors based onGPCrdquo Sensors vol 13 no1 pp 175ndash192 2013

[2] A Gomez-Espinosa V M Hernandez-Guzman M Bandala-Sanchez et al ldquoA new adaptive self-tuning Fourier Coefficientsalgorithm for periodic torque ripple minimization in perma-nentmagnet synchronousmotors (PMSM)rdquo Sensors vol 13 no3 pp 3831ndash3847 2013

[3] Y Zhang C M Akujuobi W H Ali C L Tolliver and L-SShieh ldquoLoad disturbance resistance speed controller design forPMSMrdquo IEEE Transactions on Industrial Electronics vol 53 no4 pp 1198ndash1208 2006

[4] J C Gamazo-Real E Vazquez-Sanchez and J Gomez-GilldquoPosition and speed control of brushless DC motors usingsensorless techniques and application trendsrdquo Sensors vol 10no 7 pp 6901ndash6947 2010

[5] J-H Jang S-K Sul J-I Ha K Ide and M Sawamura ldquoSen-sorless drive of surface-mounted permanent-magnet motor byhigh-frequency signal injection based on magnetic saliencyrdquoIEEE Transactions on Industry Applications vol 39 no 4 pp1031ndash1039 2003

[6] A Piippo M Hinkkanen and J Luomi ldquoSensorless control ofPMSM drives using a combination of voltage model and HFsignal injectionrdquo in Proceedings of the 39th IAS Annual MeetingConference Record of the IEEE Industry Applications Conferencevol 2 pp 964ndash970 October 2004

[7] G Wang R Yang and D Xu ldquoDSP-based control of sensorlessIPMSM drives for wide-speed-range operationrdquo IEEE Transac-tions on Industrial Electronics vol 60 no 2 pp 720ndash727 2013

[8] Z Wang Y Zheng Z Zou and M Cheng ldquoPosition sensorlesscontrol of interleaved CSI fed PMSM drive with extendedKalman filterrdquo IEEE Transactions on Magnetics vol 48 no 11pp 3688ndash3691 2012

[9] T D Batzel and K Y Lee ldquoAn approach to sensorless operationof the permanent-magnet synchronous motor using diagonallyrecurrent neural networksrdquo IEEE Transactions on Energy Con-version vol 18 no 1 pp 100ndash106 2003

[10] B Zhang andY Li ldquoAPMSMslidingmode control systembasedon model reference adaptive controlrdquo in Proceedings of the 3rdInternational Power Electronics and Motion Control Conference(IPEMC rsquo00) vol 1 pp 336ndash341 IEEE Beijing China 2000

[11] S Shinnaka ldquoNew sensorless vector control using minimum-order flux state observer in a stationary reference frame forpermanent-magnet synchronousmotorsrdquo IEEETransactions onIndustrial Electronics vol 53 no 2 pp 388ndash398 2006

[12] A Khlaief M Bendjedia M Boussak and M Gossa ldquoA non-linear observer for high-performance sensorless speed controlof IPMSM driverdquo IEEE Transactions on Power Electronics vol27 no 6 pp 3028ndash3040 2012

[13] H Lee and J Lee ldquoDesign of iterative sliding mode observerfor sensorless PMSM controlrdquo IEEE Transactions on ControlSystems Technology vol 21 no 4 pp 1394ndash1399 2013

[14] Y-S Han J-S Choi and Y-S Kim ldquoSensorless PMSM drivewith a sliding mode control based adaptive speed and statorresistance estimatorrdquo IEEE Transactions on Magnetics vol 36no 5 pp 3588ndash3591 2000

[15] V C Ilioudis and N I Margaris ldquoPMSM sensorless speedestimation based on sliding mode observersrdquo in Proceedings ofthe IEEE Power Electronics Specialists Conference (PESC rsquo08) pp2838ndash2843 IEEE Rhodes Greece June 2008

[16] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[17] Y Huang and J Han ldquoAnalysis and design for the second ordernonlinear continuous extended states observerrdquoChinese ScienceBulletin vol 45 no 21 pp 1938ndash1944 2000

[18] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[19] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence vol 6 pp 4989ndash4996 IEEE June 2003

[20] Y Xia Z Zhu and M Fu ldquoBack-stepping sliding mode controlfor missile systems based on an extended state observerrdquo IETControl Theory and Applications vol 5 no 1 pp 93ndash102 2011

[21] X Li and S Li ldquoExtended state observer based adaptive controlscheme for PMSM systemrdquo in Proceedings of the 33rd IEEEChinese Control Conference (CCC rsquo14) 2014

[22] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

14 Mathematical Problems in Engineering

[23] G ZhuAKaddouri L ADessaint andOAkhrif ldquoAnonlinearstate observer for the sensorless control of a permanent-magnetAC machinerdquo IEEE Transactions on Industrial Electronics vol45 no 2 pp 291ndash296 1996

[24] J Han Active Disturbance Rejection Control TechniquemdashTheTechnique for Estimating and Compensating the UncertaintiesNational Defense Industry Press Beijing China 2013 (Chi-nese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

(a) (b)

Figure 12 Operating waveforms of estimated back EMF in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained bythe proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 13 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 500 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 14 Operating waveforms of estimated rotor speeds when speed is 500 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

and actual and estimated rotor position are also shown inthese figures

As can be seen from Figures 7ndash10 when the rotor speedis raised to 2000 rmin advantages of ANLESO becomemoreapparent than conventional SMO Comparing the estimatedback-EMF in Figures 7(a) and 7(b) we can find that theformer has smaller chattering As a result the estimationof rotor speed becomes more accurate by ANLESO whichcan be seen in Figures 9(a) and 9(b) The waveform formspresented in Figures 10(a) and 10(b) show the estimatedrotor position The position estimation error is reduced andthe accuracy of rotor position estimation is improved inFigure 10(a)

42 Experimental Results To further verify the performanceof the new SMO for estimating rotor position and speed an

experimental systemhas been designed to control the 130SJT-M060D (15 kw) sinusoidal PMSMmotor made by GSK

Figure 11 shows a photograph of the sensorless controlsystemTheHighVoltage DigitalMotor Control Kit made byTI is utilized in the system which contains power modulesswitching devices current and voltage sensing circuit ADCmodules and PWM DAC (digital to analog converter)modules A floating-point TMS320F28335 DSP is employedas the digital controller The current control cycle is 100120583sand the velocity control cycle is 1ms A 2 120583s dead time ischosen for the PWM switching All of the control variablesare monitored in real time by an oscilloscope after theyare converted to analog signals through the PWM DACsmodules which use an external low pass filter to generatethe waveformsThemotor parameters used in experiment aregiven in Table 1

Mathematical Problems in Engineering 11

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 15 Operating waveforms of rotor position and estimate error when speed is 500 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

(a) (b)

Figure 16 Operating waveforms of estimated back EMF in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtainedby the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 17 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

Figures 12ndash14 show the operating waveforms when thereference speed is changed from zero to 500 rmin and theload torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO Estimated back-EMF estimated phase

currents and estimated rotor speed are shown in thesefigures In Figure 15 actual rotor position estimated rotorposition and their error are shown

Figures 12(a) and 13(a) show the experimental resultsof the proposed ANLESO Compared to the waveformsobtained by SMO (see part (b) in Figures 12 and 13)

12 Mathematical Problems in Engineering

(a) (b)

Figure 18 Operatingwaveforms of estimated rotor speedswhen speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 19 Operating waveforms of rotor position and estimate error when speed is 2000 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

thewaveforms of the estimated back-EMF and current shownin Figures 12(a) and 13(a) are more accurate The waveformspresented in Figures 14(a) and 14(b) show the differences ofrotor speed estimated by two kinds of observers The formerspeed waveform has less chattering In other words ANLESOhas higher estimated precision

Figure 15 shows the experimental results of positionresponseThe estimated error shown in Figure 15(c) is smallerthan that in Figure 15(d) which verifies the high precision ofANLESO

Figures 16ndash18 show the operating waveforms when thereference speed is changed from zero to 2000 rmin andthe load torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO In these figures estimated back-EMFestimated phase currents and estimated rotor speed areshown In Figure 19 actual rotor position estimated rotorposition and their error are shown

Figures 12ndash15 show the operating waveforms when thereference speed is changed from zero to 2000 rmin Figures12 and 13 provide the operating waveforms obtained by theproposed ANLESO Figures 14 and 15 show the simulationwaveforms obtained by the conventional SMO In thesefigures estimated back-EMF estimated phase currents esti-mated rotor speed and actual and estimated rotor positionare shown

Figures 16ndash18 show the experimental results of estimatedback-EMF current and rotor speed when the rotor speed israised to 2000 rmin It can be seen that the waveforms inpart (a) of these figures have smaller chattering and higheraccuracy In this way the advantages of ANLESO have beenverified

As shown in Figure 19 when the rotor speed is raisedto 2000 rmin the advantages of ANLESO to estimate rotorposition becomemore apparent than conventional SMOTheestimation error in Figure 19(c) is significantly less than thatin Figure 19(d) That is to say ANLESO has higher accuracy

Mathematical Problems in Engineering 13

and smaller chattering when being applied to estimate posi-tion than SMO

5 Conclusions

In this paper an adaptive nonlinear extended state observerhas been designed for the sensorless control of a PMSMTheconvergence of this observer has been proved by means of aLyapunov stability analysis An adaptive algorithm is adoptedto calculate the compromised parameter of ESO in order totake both stability and steady-state error into considerationThegoodperformance of the proposed sensorless control sys-tem was verified by several experimental results The resultsshow that ANLESO has more advantages than conventionalSMO Compared to conventional SMO ANLESO has smallerchattering higher accuracy and no phase delay

In future works we will explore the stability of the ESOunder parameter uncertainties in sensorless control system

Conflict of Interests

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

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China (Grant no 61304097) and Projects ofMajor International (Regional) Joint Research ProgramNSFC (Grant no 61120106010)

References

[1] S Lu X Tang andB Song ldquoAdaptive PIF control for permanentmagnet synchronousmotors based onGPCrdquo Sensors vol 13 no1 pp 175ndash192 2013

[2] A Gomez-Espinosa V M Hernandez-Guzman M Bandala-Sanchez et al ldquoA new adaptive self-tuning Fourier Coefficientsalgorithm for periodic torque ripple minimization in perma-nentmagnet synchronousmotors (PMSM)rdquo Sensors vol 13 no3 pp 3831ndash3847 2013

[3] Y Zhang C M Akujuobi W H Ali C L Tolliver and L-SShieh ldquoLoad disturbance resistance speed controller design forPMSMrdquo IEEE Transactions on Industrial Electronics vol 53 no4 pp 1198ndash1208 2006

[4] J C Gamazo-Real E Vazquez-Sanchez and J Gomez-GilldquoPosition and speed control of brushless DC motors usingsensorless techniques and application trendsrdquo Sensors vol 10no 7 pp 6901ndash6947 2010

[5] J-H Jang S-K Sul J-I Ha K Ide and M Sawamura ldquoSen-sorless drive of surface-mounted permanent-magnet motor byhigh-frequency signal injection based on magnetic saliencyrdquoIEEE Transactions on Industry Applications vol 39 no 4 pp1031ndash1039 2003

[6] A Piippo M Hinkkanen and J Luomi ldquoSensorless control ofPMSM drives using a combination of voltage model and HFsignal injectionrdquo in Proceedings of the 39th IAS Annual MeetingConference Record of the IEEE Industry Applications Conferencevol 2 pp 964ndash970 October 2004

[7] G Wang R Yang and D Xu ldquoDSP-based control of sensorlessIPMSM drives for wide-speed-range operationrdquo IEEE Transac-tions on Industrial Electronics vol 60 no 2 pp 720ndash727 2013

[8] Z Wang Y Zheng Z Zou and M Cheng ldquoPosition sensorlesscontrol of interleaved CSI fed PMSM drive with extendedKalman filterrdquo IEEE Transactions on Magnetics vol 48 no 11pp 3688ndash3691 2012

[9] T D Batzel and K Y Lee ldquoAn approach to sensorless operationof the permanent-magnet synchronous motor using diagonallyrecurrent neural networksrdquo IEEE Transactions on Energy Con-version vol 18 no 1 pp 100ndash106 2003

[10] B Zhang andY Li ldquoAPMSMslidingmode control systembasedon model reference adaptive controlrdquo in Proceedings of the 3rdInternational Power Electronics and Motion Control Conference(IPEMC rsquo00) vol 1 pp 336ndash341 IEEE Beijing China 2000

[11] S Shinnaka ldquoNew sensorless vector control using minimum-order flux state observer in a stationary reference frame forpermanent-magnet synchronousmotorsrdquo IEEETransactions onIndustrial Electronics vol 53 no 2 pp 388ndash398 2006

[12] A Khlaief M Bendjedia M Boussak and M Gossa ldquoA non-linear observer for high-performance sensorless speed controlof IPMSM driverdquo IEEE Transactions on Power Electronics vol27 no 6 pp 3028ndash3040 2012

[13] H Lee and J Lee ldquoDesign of iterative sliding mode observerfor sensorless PMSM controlrdquo IEEE Transactions on ControlSystems Technology vol 21 no 4 pp 1394ndash1399 2013

[14] Y-S Han J-S Choi and Y-S Kim ldquoSensorless PMSM drivewith a sliding mode control based adaptive speed and statorresistance estimatorrdquo IEEE Transactions on Magnetics vol 36no 5 pp 3588ndash3591 2000

[15] V C Ilioudis and N I Margaris ldquoPMSM sensorless speedestimation based on sliding mode observersrdquo in Proceedings ofthe IEEE Power Electronics Specialists Conference (PESC rsquo08) pp2838ndash2843 IEEE Rhodes Greece June 2008

[16] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[17] Y Huang and J Han ldquoAnalysis and design for the second ordernonlinear continuous extended states observerrdquoChinese ScienceBulletin vol 45 no 21 pp 1938ndash1944 2000

[18] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[19] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence vol 6 pp 4989ndash4996 IEEE June 2003

[20] Y Xia Z Zhu and M Fu ldquoBack-stepping sliding mode controlfor missile systems based on an extended state observerrdquo IETControl Theory and Applications vol 5 no 1 pp 93ndash102 2011

[21] X Li and S Li ldquoExtended state observer based adaptive controlscheme for PMSM systemrdquo in Proceedings of the 33rd IEEEChinese Control Conference (CCC rsquo14) 2014

[22] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

14 Mathematical Problems in Engineering

[23] G ZhuAKaddouri L ADessaint andOAkhrif ldquoAnonlinearstate observer for the sensorless control of a permanent-magnetAC machinerdquo IEEE Transactions on Industrial Electronics vol45 no 2 pp 291ndash296 1996

[24] J Han Active Disturbance Rejection Control TechniquemdashTheTechnique for Estimating and Compensating the UncertaintiesNational Defense Industry Press Beijing China 2013 (Chi-nese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 15 Operating waveforms of rotor position and estimate error when speed is 500 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

(a) (b)

Figure 16 Operating waveforms of estimated back EMF in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtainedby the proposed ANLESO and (b) obtained by the conventional SMO

(a) (b)

Figure 17 Operating waveforms of estimated 119894120572and 119894120573in steady state when speed is 2000 rmin and the load torque is 1 Nm (a) obtained by

the proposed ANLESO and (b) obtained by the conventional SMO

Figures 12ndash14 show the operating waveforms when thereference speed is changed from zero to 500 rmin and theload torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO Estimated back-EMF estimated phase

currents and estimated rotor speed are shown in thesefigures In Figure 15 actual rotor position estimated rotorposition and their error are shown

Figures 12(a) and 13(a) show the experimental resultsof the proposed ANLESO Compared to the waveformsobtained by SMO (see part (b) in Figures 12 and 13)

12 Mathematical Problems in Engineering

(a) (b)

Figure 18 Operatingwaveforms of estimated rotor speedswhen speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 19 Operating waveforms of rotor position and estimate error when speed is 2000 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

thewaveforms of the estimated back-EMF and current shownin Figures 12(a) and 13(a) are more accurate The waveformspresented in Figures 14(a) and 14(b) show the differences ofrotor speed estimated by two kinds of observers The formerspeed waveform has less chattering In other words ANLESOhas higher estimated precision

Figure 15 shows the experimental results of positionresponseThe estimated error shown in Figure 15(c) is smallerthan that in Figure 15(d) which verifies the high precision ofANLESO

Figures 16ndash18 show the operating waveforms when thereference speed is changed from zero to 2000 rmin andthe load torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO In these figures estimated back-EMFestimated phase currents and estimated rotor speed areshown In Figure 19 actual rotor position estimated rotorposition and their error are shown

Figures 12ndash15 show the operating waveforms when thereference speed is changed from zero to 2000 rmin Figures12 and 13 provide the operating waveforms obtained by theproposed ANLESO Figures 14 and 15 show the simulationwaveforms obtained by the conventional SMO In thesefigures estimated back-EMF estimated phase currents esti-mated rotor speed and actual and estimated rotor positionare shown

Figures 16ndash18 show the experimental results of estimatedback-EMF current and rotor speed when the rotor speed israised to 2000 rmin It can be seen that the waveforms inpart (a) of these figures have smaller chattering and higheraccuracy In this way the advantages of ANLESO have beenverified

As shown in Figure 19 when the rotor speed is raisedto 2000 rmin the advantages of ANLESO to estimate rotorposition becomemore apparent than conventional SMOTheestimation error in Figure 19(c) is significantly less than thatin Figure 19(d) That is to say ANLESO has higher accuracy

Mathematical Problems in Engineering 13

and smaller chattering when being applied to estimate posi-tion than SMO

5 Conclusions

In this paper an adaptive nonlinear extended state observerhas been designed for the sensorless control of a PMSMTheconvergence of this observer has been proved by means of aLyapunov stability analysis An adaptive algorithm is adoptedto calculate the compromised parameter of ESO in order totake both stability and steady-state error into considerationThegoodperformance of the proposed sensorless control sys-tem was verified by several experimental results The resultsshow that ANLESO has more advantages than conventionalSMO Compared to conventional SMO ANLESO has smallerchattering higher accuracy and no phase delay

In future works we will explore the stability of the ESOunder parameter uncertainties in sensorless control system

Conflict of Interests

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

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China (Grant no 61304097) and Projects ofMajor International (Regional) Joint Research ProgramNSFC (Grant no 61120106010)

References

[1] S Lu X Tang andB Song ldquoAdaptive PIF control for permanentmagnet synchronousmotors based onGPCrdquo Sensors vol 13 no1 pp 175ndash192 2013

[2] A Gomez-Espinosa V M Hernandez-Guzman M Bandala-Sanchez et al ldquoA new adaptive self-tuning Fourier Coefficientsalgorithm for periodic torque ripple minimization in perma-nentmagnet synchronousmotors (PMSM)rdquo Sensors vol 13 no3 pp 3831ndash3847 2013

[3] Y Zhang C M Akujuobi W H Ali C L Tolliver and L-SShieh ldquoLoad disturbance resistance speed controller design forPMSMrdquo IEEE Transactions on Industrial Electronics vol 53 no4 pp 1198ndash1208 2006

[4] J C Gamazo-Real E Vazquez-Sanchez and J Gomez-GilldquoPosition and speed control of brushless DC motors usingsensorless techniques and application trendsrdquo Sensors vol 10no 7 pp 6901ndash6947 2010

[5] J-H Jang S-K Sul J-I Ha K Ide and M Sawamura ldquoSen-sorless drive of surface-mounted permanent-magnet motor byhigh-frequency signal injection based on magnetic saliencyrdquoIEEE Transactions on Industry Applications vol 39 no 4 pp1031ndash1039 2003

[6] A Piippo M Hinkkanen and J Luomi ldquoSensorless control ofPMSM drives using a combination of voltage model and HFsignal injectionrdquo in Proceedings of the 39th IAS Annual MeetingConference Record of the IEEE Industry Applications Conferencevol 2 pp 964ndash970 October 2004

[7] G Wang R Yang and D Xu ldquoDSP-based control of sensorlessIPMSM drives for wide-speed-range operationrdquo IEEE Transac-tions on Industrial Electronics vol 60 no 2 pp 720ndash727 2013

[8] Z Wang Y Zheng Z Zou and M Cheng ldquoPosition sensorlesscontrol of interleaved CSI fed PMSM drive with extendedKalman filterrdquo IEEE Transactions on Magnetics vol 48 no 11pp 3688ndash3691 2012

[9] T D Batzel and K Y Lee ldquoAn approach to sensorless operationof the permanent-magnet synchronous motor using diagonallyrecurrent neural networksrdquo IEEE Transactions on Energy Con-version vol 18 no 1 pp 100ndash106 2003

[10] B Zhang andY Li ldquoAPMSMslidingmode control systembasedon model reference adaptive controlrdquo in Proceedings of the 3rdInternational Power Electronics and Motion Control Conference(IPEMC rsquo00) vol 1 pp 336ndash341 IEEE Beijing China 2000

[11] S Shinnaka ldquoNew sensorless vector control using minimum-order flux state observer in a stationary reference frame forpermanent-magnet synchronousmotorsrdquo IEEETransactions onIndustrial Electronics vol 53 no 2 pp 388ndash398 2006

[12] A Khlaief M Bendjedia M Boussak and M Gossa ldquoA non-linear observer for high-performance sensorless speed controlof IPMSM driverdquo IEEE Transactions on Power Electronics vol27 no 6 pp 3028ndash3040 2012

[13] H Lee and J Lee ldquoDesign of iterative sliding mode observerfor sensorless PMSM controlrdquo IEEE Transactions on ControlSystems Technology vol 21 no 4 pp 1394ndash1399 2013

[14] Y-S Han J-S Choi and Y-S Kim ldquoSensorless PMSM drivewith a sliding mode control based adaptive speed and statorresistance estimatorrdquo IEEE Transactions on Magnetics vol 36no 5 pp 3588ndash3591 2000

[15] V C Ilioudis and N I Margaris ldquoPMSM sensorless speedestimation based on sliding mode observersrdquo in Proceedings ofthe IEEE Power Electronics Specialists Conference (PESC rsquo08) pp2838ndash2843 IEEE Rhodes Greece June 2008

[16] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[17] Y Huang and J Han ldquoAnalysis and design for the second ordernonlinear continuous extended states observerrdquoChinese ScienceBulletin vol 45 no 21 pp 1938ndash1944 2000

[18] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[19] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence vol 6 pp 4989ndash4996 IEEE June 2003

[20] Y Xia Z Zhu and M Fu ldquoBack-stepping sliding mode controlfor missile systems based on an extended state observerrdquo IETControl Theory and Applications vol 5 no 1 pp 93ndash102 2011

[21] X Li and S Li ldquoExtended state observer based adaptive controlscheme for PMSM systemrdquo in Proceedings of the 33rd IEEEChinese Control Conference (CCC rsquo14) 2014

[22] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

14 Mathematical Problems in Engineering

[23] G ZhuAKaddouri L ADessaint andOAkhrif ldquoAnonlinearstate observer for the sensorless control of a permanent-magnetAC machinerdquo IEEE Transactions on Industrial Electronics vol45 no 2 pp 291ndash296 1996

[24] J Han Active Disturbance Rejection Control TechniquemdashTheTechnique for Estimating and Compensating the UncertaintiesNational Defense Industry Press Beijing China 2013 (Chi-nese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

(a) (b)

Figure 18 Operatingwaveforms of estimated rotor speedswhen speed is 2000 rmin and the load torque is 1 Nm (a) obtained by the proposedANLESO and (b) obtained by the conventional SMO

Estimated by ANLESO

Actual position

(a)

Estimated by SMO

Actual position

(b)

(c) (d)

Figure 19 Operating waveforms of rotor position and estimate error when speed is 2000 rmin and the load torque is 1 Nm (a) Actual andestimated rotor position obtained by the proposed ANLESO (b) Actual and estimated rotor position obtained by the conventional SMO (c)Estimated error obtained by the proposed ANLESO (d) Estimated error obtained by the conventional SMO

thewaveforms of the estimated back-EMF and current shownin Figures 12(a) and 13(a) are more accurate The waveformspresented in Figures 14(a) and 14(b) show the differences ofrotor speed estimated by two kinds of observers The formerspeed waveform has less chattering In other words ANLESOhas higher estimated precision

Figure 15 shows the experimental results of positionresponseThe estimated error shown in Figure 15(c) is smallerthan that in Figure 15(d) which verifies the high precision ofANLESO

Figures 16ndash18 show the operating waveforms when thereference speed is changed from zero to 2000 rmin andthe load torque is 1 Nm In these figures part (a) gives theoperating waveforms obtained by the proposed ANLESOand part (b) gives the operating waveforms obtained by theconventional SMO In these figures estimated back-EMFestimated phase currents and estimated rotor speed areshown In Figure 19 actual rotor position estimated rotorposition and their error are shown

Figures 12ndash15 show the operating waveforms when thereference speed is changed from zero to 2000 rmin Figures12 and 13 provide the operating waveforms obtained by theproposed ANLESO Figures 14 and 15 show the simulationwaveforms obtained by the conventional SMO In thesefigures estimated back-EMF estimated phase currents esti-mated rotor speed and actual and estimated rotor positionare shown

Figures 16ndash18 show the experimental results of estimatedback-EMF current and rotor speed when the rotor speed israised to 2000 rmin It can be seen that the waveforms inpart (a) of these figures have smaller chattering and higheraccuracy In this way the advantages of ANLESO have beenverified

As shown in Figure 19 when the rotor speed is raisedto 2000 rmin the advantages of ANLESO to estimate rotorposition becomemore apparent than conventional SMOTheestimation error in Figure 19(c) is significantly less than thatin Figure 19(d) That is to say ANLESO has higher accuracy

Mathematical Problems in Engineering 13

and smaller chattering when being applied to estimate posi-tion than SMO

5 Conclusions

In this paper an adaptive nonlinear extended state observerhas been designed for the sensorless control of a PMSMTheconvergence of this observer has been proved by means of aLyapunov stability analysis An adaptive algorithm is adoptedto calculate the compromised parameter of ESO in order totake both stability and steady-state error into considerationThegoodperformance of the proposed sensorless control sys-tem was verified by several experimental results The resultsshow that ANLESO has more advantages than conventionalSMO Compared to conventional SMO ANLESO has smallerchattering higher accuracy and no phase delay

In future works we will explore the stability of the ESOunder parameter uncertainties in sensorless control system

Conflict of Interests

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

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China (Grant no 61304097) and Projects ofMajor International (Regional) Joint Research ProgramNSFC (Grant no 61120106010)

References

[1] S Lu X Tang andB Song ldquoAdaptive PIF control for permanentmagnet synchronousmotors based onGPCrdquo Sensors vol 13 no1 pp 175ndash192 2013

[2] A Gomez-Espinosa V M Hernandez-Guzman M Bandala-Sanchez et al ldquoA new adaptive self-tuning Fourier Coefficientsalgorithm for periodic torque ripple minimization in perma-nentmagnet synchronousmotors (PMSM)rdquo Sensors vol 13 no3 pp 3831ndash3847 2013

[3] Y Zhang C M Akujuobi W H Ali C L Tolliver and L-SShieh ldquoLoad disturbance resistance speed controller design forPMSMrdquo IEEE Transactions on Industrial Electronics vol 53 no4 pp 1198ndash1208 2006

[4] J C Gamazo-Real E Vazquez-Sanchez and J Gomez-GilldquoPosition and speed control of brushless DC motors usingsensorless techniques and application trendsrdquo Sensors vol 10no 7 pp 6901ndash6947 2010

[5] J-H Jang S-K Sul J-I Ha K Ide and M Sawamura ldquoSen-sorless drive of surface-mounted permanent-magnet motor byhigh-frequency signal injection based on magnetic saliencyrdquoIEEE Transactions on Industry Applications vol 39 no 4 pp1031ndash1039 2003

[6] A Piippo M Hinkkanen and J Luomi ldquoSensorless control ofPMSM drives using a combination of voltage model and HFsignal injectionrdquo in Proceedings of the 39th IAS Annual MeetingConference Record of the IEEE Industry Applications Conferencevol 2 pp 964ndash970 October 2004

[7] G Wang R Yang and D Xu ldquoDSP-based control of sensorlessIPMSM drives for wide-speed-range operationrdquo IEEE Transac-tions on Industrial Electronics vol 60 no 2 pp 720ndash727 2013

[8] Z Wang Y Zheng Z Zou and M Cheng ldquoPosition sensorlesscontrol of interleaved CSI fed PMSM drive with extendedKalman filterrdquo IEEE Transactions on Magnetics vol 48 no 11pp 3688ndash3691 2012

[9] T D Batzel and K Y Lee ldquoAn approach to sensorless operationof the permanent-magnet synchronous motor using diagonallyrecurrent neural networksrdquo IEEE Transactions on Energy Con-version vol 18 no 1 pp 100ndash106 2003

[10] B Zhang andY Li ldquoAPMSMslidingmode control systembasedon model reference adaptive controlrdquo in Proceedings of the 3rdInternational Power Electronics and Motion Control Conference(IPEMC rsquo00) vol 1 pp 336ndash341 IEEE Beijing China 2000

[11] S Shinnaka ldquoNew sensorless vector control using minimum-order flux state observer in a stationary reference frame forpermanent-magnet synchronousmotorsrdquo IEEETransactions onIndustrial Electronics vol 53 no 2 pp 388ndash398 2006

[12] A Khlaief M Bendjedia M Boussak and M Gossa ldquoA non-linear observer for high-performance sensorless speed controlof IPMSM driverdquo IEEE Transactions on Power Electronics vol27 no 6 pp 3028ndash3040 2012

[13] H Lee and J Lee ldquoDesign of iterative sliding mode observerfor sensorless PMSM controlrdquo IEEE Transactions on ControlSystems Technology vol 21 no 4 pp 1394ndash1399 2013

[14] Y-S Han J-S Choi and Y-S Kim ldquoSensorless PMSM drivewith a sliding mode control based adaptive speed and statorresistance estimatorrdquo IEEE Transactions on Magnetics vol 36no 5 pp 3588ndash3591 2000

[15] V C Ilioudis and N I Margaris ldquoPMSM sensorless speedestimation based on sliding mode observersrdquo in Proceedings ofthe IEEE Power Electronics Specialists Conference (PESC rsquo08) pp2838ndash2843 IEEE Rhodes Greece June 2008

[16] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[17] Y Huang and J Han ldquoAnalysis and design for the second ordernonlinear continuous extended states observerrdquoChinese ScienceBulletin vol 45 no 21 pp 1938ndash1944 2000

[18] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[19] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence vol 6 pp 4989ndash4996 IEEE June 2003

[20] Y Xia Z Zhu and M Fu ldquoBack-stepping sliding mode controlfor missile systems based on an extended state observerrdquo IETControl Theory and Applications vol 5 no 1 pp 93ndash102 2011

[21] X Li and S Li ldquoExtended state observer based adaptive controlscheme for PMSM systemrdquo in Proceedings of the 33rd IEEEChinese Control Conference (CCC rsquo14) 2014

[22] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

14 Mathematical Problems in Engineering

[23] G ZhuAKaddouri L ADessaint andOAkhrif ldquoAnonlinearstate observer for the sensorless control of a permanent-magnetAC machinerdquo IEEE Transactions on Industrial Electronics vol45 no 2 pp 291ndash296 1996

[24] J Han Active Disturbance Rejection Control TechniquemdashTheTechnique for Estimating and Compensating the UncertaintiesNational Defense Industry Press Beijing China 2013 (Chi-nese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

and smaller chattering when being applied to estimate posi-tion than SMO

5 Conclusions

In this paper an adaptive nonlinear extended state observerhas been designed for the sensorless control of a PMSMTheconvergence of this observer has been proved by means of aLyapunov stability analysis An adaptive algorithm is adoptedto calculate the compromised parameter of ESO in order totake both stability and steady-state error into considerationThegoodperformance of the proposed sensorless control sys-tem was verified by several experimental results The resultsshow that ANLESO has more advantages than conventionalSMO Compared to conventional SMO ANLESO has smallerchattering higher accuracy and no phase delay

In future works we will explore the stability of the ESOunder parameter uncertainties in sensorless control system

Conflict of Interests

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

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China (Grant no 61304097) and Projects ofMajor International (Regional) Joint Research ProgramNSFC (Grant no 61120106010)

References

[1] S Lu X Tang andB Song ldquoAdaptive PIF control for permanentmagnet synchronousmotors based onGPCrdquo Sensors vol 13 no1 pp 175ndash192 2013

[2] A Gomez-Espinosa V M Hernandez-Guzman M Bandala-Sanchez et al ldquoA new adaptive self-tuning Fourier Coefficientsalgorithm for periodic torque ripple minimization in perma-nentmagnet synchronousmotors (PMSM)rdquo Sensors vol 13 no3 pp 3831ndash3847 2013

[3] Y Zhang C M Akujuobi W H Ali C L Tolliver and L-SShieh ldquoLoad disturbance resistance speed controller design forPMSMrdquo IEEE Transactions on Industrial Electronics vol 53 no4 pp 1198ndash1208 2006

[4] J C Gamazo-Real E Vazquez-Sanchez and J Gomez-GilldquoPosition and speed control of brushless DC motors usingsensorless techniques and application trendsrdquo Sensors vol 10no 7 pp 6901ndash6947 2010

[5] J-H Jang S-K Sul J-I Ha K Ide and M Sawamura ldquoSen-sorless drive of surface-mounted permanent-magnet motor byhigh-frequency signal injection based on magnetic saliencyrdquoIEEE Transactions on Industry Applications vol 39 no 4 pp1031ndash1039 2003

[6] A Piippo M Hinkkanen and J Luomi ldquoSensorless control ofPMSM drives using a combination of voltage model and HFsignal injectionrdquo in Proceedings of the 39th IAS Annual MeetingConference Record of the IEEE Industry Applications Conferencevol 2 pp 964ndash970 October 2004

[7] G Wang R Yang and D Xu ldquoDSP-based control of sensorlessIPMSM drives for wide-speed-range operationrdquo IEEE Transac-tions on Industrial Electronics vol 60 no 2 pp 720ndash727 2013

[8] Z Wang Y Zheng Z Zou and M Cheng ldquoPosition sensorlesscontrol of interleaved CSI fed PMSM drive with extendedKalman filterrdquo IEEE Transactions on Magnetics vol 48 no 11pp 3688ndash3691 2012

[9] T D Batzel and K Y Lee ldquoAn approach to sensorless operationof the permanent-magnet synchronous motor using diagonallyrecurrent neural networksrdquo IEEE Transactions on Energy Con-version vol 18 no 1 pp 100ndash106 2003

[10] B Zhang andY Li ldquoAPMSMslidingmode control systembasedon model reference adaptive controlrdquo in Proceedings of the 3rdInternational Power Electronics and Motion Control Conference(IPEMC rsquo00) vol 1 pp 336ndash341 IEEE Beijing China 2000

[11] S Shinnaka ldquoNew sensorless vector control using minimum-order flux state observer in a stationary reference frame forpermanent-magnet synchronousmotorsrdquo IEEETransactions onIndustrial Electronics vol 53 no 2 pp 388ndash398 2006

[12] A Khlaief M Bendjedia M Boussak and M Gossa ldquoA non-linear observer for high-performance sensorless speed controlof IPMSM driverdquo IEEE Transactions on Power Electronics vol27 no 6 pp 3028ndash3040 2012

[13] H Lee and J Lee ldquoDesign of iterative sliding mode observerfor sensorless PMSM controlrdquo IEEE Transactions on ControlSystems Technology vol 21 no 4 pp 1394ndash1399 2013

[14] Y-S Han J-S Choi and Y-S Kim ldquoSensorless PMSM drivewith a sliding mode control based adaptive speed and statorresistance estimatorrdquo IEEE Transactions on Magnetics vol 36no 5 pp 3588ndash3591 2000

[15] V C Ilioudis and N I Margaris ldquoPMSM sensorless speedestimation based on sliding mode observersrdquo in Proceedings ofthe IEEE Power Electronics Specialists Conference (PESC rsquo08) pp2838ndash2843 IEEE Rhodes Greece June 2008

[16] Z Qiao T Shi YWang Y Yan C Xia and X He ldquoNew sliding-mode observer for position sensorless control of permanent-magnet synchronous motorrdquo IEEE Transactions on IndustrialElectronics vol 60 no 2 pp 710ndash719 2013

[17] Y Huang and J Han ldquoAnalysis and design for the second ordernonlinear continuous extended states observerrdquoChinese ScienceBulletin vol 45 no 21 pp 1938ndash1944 2000

[18] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[19] Z Gao ldquoScaling and bandwidth-parameterization based con-troller tuningrdquo in Proceedings of the American Control Confer-ence vol 6 pp 4989ndash4996 IEEE June 2003

[20] Y Xia Z Zhu and M Fu ldquoBack-stepping sliding mode controlfor missile systems based on an extended state observerrdquo IETControl Theory and Applications vol 5 no 1 pp 93ndash102 2011

[21] X Li and S Li ldquoExtended state observer based adaptive controlscheme for PMSM systemrdquo in Proceedings of the 33rd IEEEChinese Control Conference (CCC rsquo14) 2014

[22] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012

14 Mathematical Problems in Engineering

[23] G ZhuAKaddouri L ADessaint andOAkhrif ldquoAnonlinearstate observer for the sensorless control of a permanent-magnetAC machinerdquo IEEE Transactions on Industrial Electronics vol45 no 2 pp 291ndash296 1996

[24] J Han Active Disturbance Rejection Control TechniquemdashTheTechnique for Estimating and Compensating the UncertaintiesNational Defense Industry Press Beijing China 2013 (Chi-nese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

[23] G ZhuAKaddouri L ADessaint andOAkhrif ldquoAnonlinearstate observer for the sensorless control of a permanent-magnetAC machinerdquo IEEE Transactions on Industrial Electronics vol45 no 2 pp 291ndash296 1996

[24] J Han Active Disturbance Rejection Control TechniquemdashTheTechnique for Estimating and Compensating the UncertaintiesNational Defense Industry Press Beijing China 2013 (Chi-nese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of