Research Article Nonlinear Current Control for Reluctance ...downloads.hindawi.com/journals/jcse/2014/150345.pdfResearch Article Nonlinear Current Control for Reluctance Actuator with

Research ArticleNonlinear Current Control for Reluctance Actuator withHysteresis Compensation

Yu-Ping Liu1 Kang-Zhi Liu1 and Xiaofeng Yang2

1 Department of Electrical and Electronic Engineering Chiba University Chiba 263-8522 Japan2Department of Microelectronics Fudan University Shanghai 200243 China

Correspondence should be addressed to Yu-Ping Liu ericliuyupinggmailcom

Received 15 August 2014 Accepted 4 October 2014 Published 21 October 2014

Academic Editor Shengwei Mei

Copyright copy 2014 Yu-Ping Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The next-generation fine stage of the wafer scanner needs a suitable actuator to meet the requirements of high speed highacceleration and high precisionThe voice coil actuator is no longer the best choice because of its large size and the heat dissipationis difficult to solve The reluctance actuator can provide a big force based on a unique property of small volume and low currentmaking it a very suitable candidate But the strong nonlinearity such as the hysteresis between the current and force limits thereluctance actuator applications in nanometer positioning This paper proposes a nonlinear current control configuration withhysteresis compensation using the adaptive multilayer neural network Simulation results show that the hysteresis compensator iseffective in overcoming the hysteresis and is promising in precision control applications

1 Introduction

In high-precision applications such as semiconductor lithog-raphy systems the force density and accuracy of waferscanner increase rapidly in recent years [1] Forwafer scannerit needs accurate positioning tomeet the requirement of high-precision actuation It is realized by a fine stage mounted ona coarse stage The fine stage has a millimeter range with anaccuracy of nanometers and the coarse stage has a range ofmeters with the accuracy of submicrometers

In conventional design of fine stage actuators the rela-tively low variation of the force is the main reason to choosefor voice coil actuator But in the development of new short-stroke actuator the production speed and the positioningaccuracy gain more importance [2] A higher throughput isachieved by increasing the acceleration of the fine actuatorIf voice coil actuator is still used to achieve greater force itssize will become very large and the heat dissipation problemwill be very difficult to solve [3] Therefore the voice coilactuator is no longer the best choice as the main drivingactuator of fine stage Because the reluctance actuator whoseforce is proportional to the square of the excitation currentcan produce greater force with a small volume and lowcurrent than the voice coil motor it provides a solution to

meet the driving requirements of the next-generation finestage

The mentioned advantage of the reluctance actuatorcomes at a sacrifice of parasitic effects such as the eddy cur-rent and hysteresis which deteriorate the force predictabil-ity and add additional nonlinearity to the actuator Theseeffects can be reduced by using magnetic cores made ofthin laminations of soft ferromagnetic materials From theanalysis in reference paper [4] the eddy current is not visiblebelow kHz range with thin laminations of soft ferromagneticmaterial and the main factor causing the force error isthe hysteresis When specification of the force predictabilitybecomes very strict the current air-gap force is no longer asingle-valued function and hysteresis in soft ferromagneticmaterials becomes dominant So the hysteresis has to bemodeled and compensated to obtain a predictable force tomeet required positioning accuracy in nanometer range

A variety of the models and compensation algorithmshave been developed for the hysteresis [5] The classicalapproach is to construct a hysteresis inverse and use it asa feed-forward compensator [6] together with a feedbackcontrol The hysteresis compensation method via Preisachmodel inversion has been proposed in paper [7] An inverse

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2014 Article ID 150345 7 pageshttpdxdoiorg1011552014150345

2 Journal of Control Science and Engineering

r(k)

k

(a)

y(k)

r(k)

(b)

Figure 1 Input 119903(119896) (a) and the corresponding output 119910(119896) map (b) trajectories of the parametric hysteresis operator

parametric hysteresis model [8] was used in current-modeoperated reluctance force actuator However the abovemeth-ods need precise hysteresis model which is usually complexand hard to obtain

Owing to the online self-learning and estimate ability ofthe neural network it provides a good solution for solvingnonlinear problems In particular the multilayer neuralnetwork (MNN) is effectively used in nonlinear discrete-time system identification and control [9] Although a lot ofresearches have been done on the neural network applicationto the hysteresis [10 11] the neural network compensation forthe current-driven reluctance actuator with hysteresis has yetto be studied

Themain contribution of this paper is a nonlinear currentcontroller with hysteresis compensation for current-drivenreluctance linear actuator using the adaptive MNN proposedfor the first time Concretely speaking based on the input-output feature of the hysteresis in reluctance force and thelearning and approximation ability of neural network a hys-teresis current compensator is proposed using the adaptiveMNN [9] whose weight is updated by the error betweenthe desired force and the actual force The main advantageof the promosed compensation method is that the inversehysteresis model is not required Do the simulations onthe reluctance actuator model with hysteresis and EI pairsstage and the simulation results show that the nonlinearcurrent control with hysteresis compensator is effective inovercoming the hysteresis and promising in high-precisionand high-acceleration control applications

This paper is organized as follows In Section 2 paramet-ric hysteresis operator is reviewed In Section 2 reluctanceactuator models with and without hysteresis are reviewedA nonlinear current control with hysteresis compensatorusing the adaptive MNN is presented in Section 3 Finallysimulation results are illustrated in Section 4

2 Parametric Hysteresis Operator

Hysteresis is encountered over a wide range of applicationsthat usually involve magnetic ferroelectric mechanical oroptical systems It is a complex nonlinearity that displaysthe properties of bifurcation and nonlocal memory Thehysteresis can be defined as a loop in the input-outputmap In this paper the parametric hysteresis operator [4] is

reviewed which is defined in the discrete-time domain sothe implementation is more straightforward

Definition 1 Let 119903(119896) 119910(119896) be bounded 1205821 1205822gt 0 Then the

parametric hysteresis operator

119910 (119896) = 119867hys [119903 (119896) 1205821 1205822] (1)

is defined as followswhen 119903(119896 + 1) minus 119903(119896) ge 0

119910 (119896) = 119903 (119896) + 1199100minus 1199030+ 119898 minus

1

1205822

119882(1205822119898 sdot 1198901205822(119898+119903(119896)minus119903

0))

(2)

when 119903(119896 + 1) minus 119903(119896) lt 0

119910 (119896) = 119903 (119896) + 1199100minus 1199030minus 119899 +

1

1205822

119882(1205822119899 sdot 1198901205822(119899minus119903(119896)+119903

0))

(3)

where 119898 = 1205821+ 1199030minus 1199100and 119899 = 120582

1minus 1199030+ 1199100 1199030= 119903(119896lowast) and

1199100

= 119910(119896lowast) The indicator 119896

lowast denotes the last time instantbefore 119896 when the difference of 119903 changed sign that is anextremum occurred which corresponds to a corner point ofthe 119910-119903 curve in Figure 1 The parameter 120582

1represents the

amount of hysteresis around the straight line 119903(119896) = 119910(119896)

and defines the asymptotes while the parameter 1205822defines

the smoothness of the hysteresis loops that is the rate ofconvergence towards the left or right asymptote 119882 is theprincipal branch of the Lambert 119882 function [12]

3 Problem Statement

31 Reluctance Actuator Models with and without HysteresisReluctance linear actuator is a type of electric motor thatinduces nonpermanent magnetic poles on the ferromagneticrotor and produces temporary magnetic The reluctanceactuator is named as such because it usesmagnetic reluctanceto generate force which can be called reluctance force TheEI-core actuator is the most basic linear reluctance linearactuator [13] As shown in Figure 2 the actuator includes agenerally Cobalt-Iron ldquoErdquo shaped electromagnet and an ldquoIrdquoshaped targetThe electromagnet has an electrical coil woundaround the center section Current flowing through the coilgenerates a magnetic flux and this flux creates a reluctance

Journal of Control Science and Engineering 3

i

N lm

xg

E-core I-target

FF

B

Figure 2 Sketches of the EI actuator

B

H

Figure 3 Typical B-H hysteresis curve

force on the target The amount of current determinesthe amount of reluctance force The reluctance force actingon the ldquoIrdquo target is described [13] by

119865 = 119896

1198942

1199092

119892

(4)

where 119896 = 120583011987321198604 120583

0is the permeability of air 119873 is the

number of turns in coil on the center leg of the E-core and119860

is the area of the air gap It is a lumpedmodelwhich disregardsnonmodeled effects such as hysteresis leakage fringing andsaturation

However the E-core coil uses the soft magnetic materialwhich has magnetic hysteresis [14] between the magneticfield 119867 and the bulk magnetic flux density 119861 and the typical119861-119867 curve is illustrated in Figure 3 It is assumed that thehysteretic 119861-119867 curve can be modeled [4] by the parametrichysteresis operator (4) Then it can get a reluctance linearactuator model with hysteresis [8] as

119865 = 119896

[119867hys (119894 1205821 1205822)]2

1199092

119892

(5)

This model contains both the hysteresis and obvioussquare linearity between the input and output

32 Nonlinear Current Control for Reluctance Actuatorwithout Hysteresis Compensator A typical structure of thecurrent-driven reluctance linear actuator control is shown inFigure 4 Due to the unstable nature of the electromagnetic

i F

Setpoint Fd id

C Nonlinearcompensator

Poweramplifier

Reluctanceactuator Mass

Positionsensor

xg

minus

Figure 4 Typical current-driven reluctance actuator control loop

forces a feedback control loop is required to achieve stableperformance In Figure 4 ldquoCrdquo denotes a linear positioncontroller such as the proportional-integral-derivative (PID)controller which guarantees the stability of the closed loopThe position controller uses the position in the air gap mea-sured by a position sensor to generate the force command 119865

119889

The power amplifier supplies the current for the reluctanceactuator

From (4) a nonlinear current controller without hystere-sis compensator [15] can be obtained as

119894119889=

radic

119865119889sdot 1199092

119892

119896119889

(6)

where 119896119889is the estimate of real constant 119896

The above nonlinear current compensator does not con-sider the hysteresis in the reluctance force From the analysisof the hysteresis influence in the section of introduction it isnecessary to find a hysteresis compensation method for thereluctance linear actuator in the high-precision positioningIn the following a hysteresis compensator for the reluctancelinear actuator based on the adaptive MNNwill be proposed

33 Nonlinear Current Control for Reluctance Actuator withHysteresis Compensator In control engineering artificial NNcan be used as a universal function approximator whichinspires using NN as an emulator to approximate unknownnonlinearity Many types of neural networks have beenapplied such as the high-order neural networks (HONN)radial basis function neural networks (RBF) and multilayerneural network (MNN) For MNN its universal approx-imation ability parallel distributed processing ability andlearning and adaptation ability make it one of the mostpopular tools in nonlinear control [9]

MNN is a static feed-forward network that consists of anumber of layers each layer having a number of McCulloch-Pitts neurons Due to its hidden layers the MNN has animportant character that it can approximate any continu-ous nonlinear function Once the hidden layers have beenselected only the adjustable weights have to be determinedto specify the networks completely Owing to the aboveadvantages of MNN a nonlinear current controller withhysteresis compensator for the current-driven reluctanceactuator force using the adaptive MNN can be obtained asshown in Figure 5 The feedback force 119865 can be estimatedfrom the relation 119865 = 119898119886 with 119898 the stage mass and 119886 theactual accelerationThe actual acceleration 119886 can bemeasuredby an accelerometer or computed from the measured gap


a

Fd id idd

+ic

i

V W

+

minus120576

MNN

m

Nonlinearcompensator

Poweramplifier


Adaptive laws

Hysteresis current compensator

AccelerometerF

Figure 5 Structure of hysteresis current compensator using adap-tive MNN

position 119909119892by a digital double differentiator The double

differentiation may introduce a noise However when usinga filter such as the least-square method and considering thenoise level it makes double differentiation acceptable as amethod to calculate the stage acceleration [16] In this paperthe acceleration 119886 is measured by the accelerometer Thedetails of the adaptive MNN are as follows

The nonlinear current controller with hysteresis compen-sator can be defined as

119894119889119889

(119896) = 119894119889(119896) + 119894

119888(119896) (7)

where the hysteresis current compensator in (7) is estimatedby the following MNN

119894119888(119896) = WT

(k) S (VT(k) x (k)) (8)

whereW(k) isin R119897 andV(k) isin R(119899+1)times119897 are weighting matrices119897 is the number of hidden-layer neurons x(119896) = [x119879(119896) 1]119879 isin

R119899+1 denotes the input vector of the neural network andS(V119879(119896)x(119896)) = [119904(V119879

1(119896)x(119896)) 119904(V119879

119897(119896)x(119896)) 1]119879 isin R119897

with 119904(119909) = 1(1 + 119890minus119909

)The force error is

120576 (119896 + 1) = 119865119889(119896) minus 119865 (119896) (9)

The adaptive update law of the neural network weights ischosen as

W (119896 + 1) = W (119896) minus Proj119882

[120574119908S (119896) 120576 (119896 + 1)] (10)

V (119896 + 1) = V (119896) minus ProjV [120574Vz119897W119879(119896) S1015840 (119905) 120576 (119896 + 1)]

(11)

where 120574119908and 120574V are the learning rates with positive constants

S(119896) = diag1199041(119896) 119904

119897(119896) with 119904

119894(119896) = 119904(V119879

119894x(119896)) and

z119897= [1radic119897 1radic119897]

119879

(z = 1) is a vector compatible with

V(119896) Since the output of the hidden layer neural network isdirectly related to the input of the outputweight so the outputweightW is introduced in (10) S1015840(119896) = diag1199041015840

1(119896) 119904

1015840

119897(119896)

is a diagonal matrix with 1199041015840

119894(119896) = 119904

1015840(V119879119894x(119896)) The projection

function Proj120579(lowast) is defined as Proj

120579(lowast) = Proj

120579(lowast119895119896)whose

element in the 119895th row and 119896th column is

Proj120579(lowast119895119896) =

minuslowast119895119896 if

120579119895119896

= 120588120579119895119896max lowast

119895119896lt 0

120579119895119896

= 120588120579119895119896min lowast

119895119896gt 0

lowast119895119896 otherwise

(12)

where lowast is either a vector or matrix with its element beinglowast119895119896 120588120579119895119896min and 120588

120579119895119896max are presumed as upper and lower

boundaries of 120579119894119895 and 120579 denotes W and V However due

to the fact that those bounds may not be known a prioricertain fictitious large enough bounds can be used [17] It isjust needed to limit the output amplitude of the output ofadaptive MNN equation (7) for example corresponding tothe power amplifier input limit plusmn10 v

The hysteresis compensation algorithm using adaptiveMNN can be summarized as follows

Step 1 Construct the input of adaptive neural network x(119896)and estimate 119894

119888(119896)

Step 2 Calculate the hysteresis compensation current 119894119888(119896)

using (8)

Step 3 Calculate the force error 120576(119896 + 1) using (9)

Step 4 Update the neural networksweight using (10) and (11)

Step 5 Let 119896 = 119896 + 1 return to Step 1

4 Simulation

Example 1 A simulation for the current-driven reluctanceactuator is conducted to verify the performance of theproposed hysteresis compensation configuration shown inFigure 7

In this simulation since the linear power amplifier hasa high bandwidth and rapid response it is ignored A forcecommand 119865

119889is imposed without a counter balance force the

I-target would be pulled to the E-core For this reason it isassumed that the I-target is fixed so that we may investigatethe force tracing performanceThe reluctance linear actuatorwith hysteresis is modeled by (5) whose parameters arechosen as in the patent [15] the maximum force is about200N the maximum gap 119909

119892between the E-core and I-target

is 04mm and the constant 119896 = 773 times 10minus6 To reproduce

a reluctance actuator model with nonsmooth hysteresis theparameter 120582

2of the hysteretic operator would have to be

infinitely large Here the hysteresis operator parameters arechosen as 120582

1= 002 and 120582

2= 10

The control objective is to make the system output 119865

follow the reference force 119865119889 A nontrivial reference force is

depicted in Figure 6 This profile is used to test the propo-sed nonlinear controller with hysteresis compensation with


0 2 4 6 8 10 12 14 160

50

100

150

200

Refe

renc

e for

ce (N

)

t (s)

Figure 6 Reference force

0 50 100 150 200

0

05

1

15

2

Reference force

Forc

e err

or (N

)

Without hysteresis compensationWith hysteresis compensation

minus2

minus15

minus1

minus05

Figure 7 The maps between the reference force and the forcetracking error

respect to higher order reversal curvesThe parameters of theadaptive MNN are determined as follows the input vector ofthe neural network is chosen as x(119896) = [119865

119889(119896+1) 119865

119889(119896) 119865(119896+

1) 119865(119896)] the number of hidden-layer neurons is 119897 = 40 theinitial neural network weighting matrices are w(0) = 0 andv(0) = 0 and the learning rate is selected as 120574

119908= 00006 and

120574V = 001 respectivelyThe maps between the reference force 119865

119889and the force

tracking error with nonlinear current control without hys-teresis compensation and with nonlinear current controlwith hysteresis compensation are shown in Figure 7 Thelargest force tracking error without hysteresis compensationis about plusmn2N With hysteresis compensation it is aboutplusmn01 N Furthermore Figure 8 shows that the ratio betweenthe output force 119865 and its reference command 119865

119889with and

without hysteresis compensator and the ratio with hysteresiscompensation is very close to 1 From the simulation resultsit is clear that the force with hysteresis compensation hasa much smaller force tracking error than without hysteresiscompensation

Example 2 In order to verify if the proposed hysteresis com-pensation method can be applied in high-precision systems

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

160

180

200

Reference force

Out

put f

orce


80 85 9070

80

90

100

Figure 8 Comparison of the force hysteresis loop between forcegenerations without hysteresis compensation and with hysteresiscompensation

E-core 2 I-target 2

i2F2

xg2 xg1

i1F1

Stage

x

E-core 1I-target 1

Figure 9 EI-core type actuator pairs

we do the following simulation using the reluctance actuatorto drive a one degree of freedom stage

Due to the nature of the reluctance force an EI coreactuator can only generate a unidirectional attractive forceTo generate an active force in the opposite direction asecond actuator needs to be placed on the opposite side ofthe I-target Figure 9 illustrates a simplified actuator pairsincluding electromagnetic actuators E-core 1 and E-core 2 aswell as a stage [15]The gap 119909

1198921separates the E-core 1 and the

I-target 1 and the gap1199091198922separates E-core 2 and I-target 2The

stage mass is 10 kg the position sensor resolution is 1 nm thesampling time is 05ms and the bias force is119865

0= 15Nwhich

is applied on both EI actuators The E-core electromagneticconstant is 119896 = 773times10

minus6 The gaps 1199091198921and 119909

1198922between the

E-cores 1 2 and the I-targets 1 2 are in the range of 0 120583m to400 120583m and the initial gap is 119909

1198920= 400 120583m The parameters

of hysteresis operator are set as in Example 1The stage control system uses the proportional-integral

(PI) controller to guarantee the stability of the closed-loopposition system while the nonlinear current controller with


0

02

04

Posit

ion

(mm

)

0

5

10

Velo

city

(mm

s)

0 001 002 003 004 005 006 007 008t (s)

Figure 10 Reference position and velocity

0 001 002 003 004 005 006 007 008

0

10

20

30


StopConstantvelocity

minus10

minus20

minus30

Trac

king

erro

r (120583

m)

t (s)

Figure 11 Comparison of the position tracking

hysteresis compensator is employed to reduce the influenceof the hysteresis and improve the performance of the closed-loop position system The stage control system is formedby a combination of Figures 6 and 7 The adaptive MNNparameters for the two EI actuators are the same as inExample 1

The control objective is to make the stage position 119909

follow the reference position Use the 3rd trajectory [18] asthe reference position profile The position and velocity areshown in Figure 10 The largest displacement is 350120583m themaximum velocity is 10mms the maximum acceleration is1ms2 and the jerk is 200ms3

The position tracking errors of control without hysteresiscompensation and with hysteresis compensation are shownin Figure 11 We define the constant velocity settling time andthe stop time as the time instants after which the positiontracking error is less than 1 120583m From the enlarged view ofthe constant velocity segment as shown in Figure 12 it can beseen that the constant velocity settling timewithout hysteresiscompensation (00198 s) is about 132 times longer than withhysteresis compensation (0015 s) and from the enlarged viewof the stop segment as shown in Figure 13 the stop time

0015 002 0025 003 0035

0015 s 00198 s

0

1

2

3

4

minus2

minus3

minus4

minus1

Trac

king

erro

r (120583

m)

t (s)


Figure 12 Comparison of the position tracking error during theconstant velocity segment

005 0055 006 0065 007 0075 008

0516 s

05945 s

0

1

2

3

4

minus2

minus3

minus4

minus1

Trac

king

erro

r (120583

m)

t (s)


Figure 13 Comparison of the position tracking error in the stopsegment

without hysteresis compensation (05945 s) is about 11521times longer than with hysteresis compensation (0516 s)

From the simulation results of the one degree EI stageit is verified that the adaptive MNN hysteresis compensatorprovides a feasible solution to reduce the influence of reluc-tance actuator with hysteresis

5 Conclusion

This paper has proposed a nonlinear current control con-figuration with hysteresis compensator for the reluctanceactuator using the discrete-time adaptiveMNNwhich is usedas a hysteresis learning machine The proposed hysteresiscompensator compensates the current based on the desired


force and the actual force using the adaptive MNN Thishysteresis compensator does not need the inverse hysteresismodel Due to the simplicity of the compensation configura-tion it may have a wider range of applications Simulationresults show that the proposed methods can reduce thehysteresis influence effectively and are promising in high-precision control applications

Conflict of Interests

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

Acknowledgment

This work is supported by the National Natural ScienceFoundation (NNSF) of China under Grant 61274109

References

[1] C Grant ldquoHigh-resolution patterning a view of the futurerdquo inProceedings of the Conference of Advanced Lithography San JoseCalif USA 2012

[2] H Butler ldquoPosition control in lithographic equipmentrdquo IEEEControl Systems Magazine vol 31 no 5 pp 28ndash47 2011

[3] N H Vrijsen J W Jansen and E A Lomonova ldquoComparisonof linear voice coil and reluctance actuators for high-precisionapplicationsrdquo in Proceedings of the 14th International PowerElectronics andMotion Control Conference (EPE-PEMC rsquo10) ppS329ndashS336 Ohrid Republic of Macedonia September 2010

[4] A Katalenic J De Boeij H Butler and P P J Van DenBosch ldquoLinearization of a current-driven reluctance actuatorwith hysteresis compensationrdquo Mechatronics vol 23 no 2 pp163ndash171 2013

[5] N H Vrijsen J W Jansen and E A Lomonova ldquoForceprediction including hysteresis effects in a short-stroke reluc-tance actuator using a3d-FEM and the preisach modelrdquoAppliedMechanics and Materials vol 416-417 pp 187ndash194 2013

[6] R V Iyer and X Tan ldquoControl of hysteretic systems throughinverse compensationrdquo IEEE Control Systems Magazine vol 29no 1 pp 83ndash99 2009

[7] S Mittal and C-H Menq ldquoHysteresis compensation in elec-tromagnetic actuators through Preisach model inversionrdquoIEEEASMETransactions onMechatronics vol 5 no 4 pp 394ndash409 2000

[8] A Katalenic and J de Boeij ldquoLinearization of the reluctanceforce actuator based on the parametric hysteresis inverse and a2D splinerdquo in Proceedings of the 8th International Symposium onLinear Drives for Industry Applications (LDIA rsquo11) EindhovenThe Netherlands July 2011

[9] S S Ge J Zhang and T H Lee ldquoDirect MNN control ofcontinuous stirred tank reactor based on input-output modelrdquoin Proceedings of the 41st SICE Annual Conference (SICE rsquo02)vol 5 pp 2770ndash2775 Osaka Japan August 2002

[10] F-J Lin H-J Shieh and P-K Huang ldquoAdaptive waveletneural network control with hysteresis estimation for piezo-positioning mechanismrdquo IEEE Transactions on Neural Net-works vol 17 no 2 pp 432ndash444 2006

[11] A A Adly and S K Abd-El-Hafiz ldquoUsing neural networksin the identification of preisach-type hysteresis modelsrdquo IEEETransactions on Magnetics vol 34 no 3 pp 629ndash635 1998

[12] R M Corless G H Gonnet D E Hare D J Jeffrey and D EKnuth ldquoOn the Lambert 119882 functionrdquo Advances in Computa-tional Mathematics vol 5 no 1 pp 329ndash359 1996

[13] E P Furlani Permanent Magnet and Electromechanical DevicesMaterials Analysis and Applications Academic Press 2001

[14] G Bertotti Hysteresis in Magnetism For Physicists MaterialsScientists Academic Press 1998

[15] T C Teng and B Yuan US Patent No 6069417 US Patentand Trademark Office Washington DC USA 2000

[16] H Butler ldquoAdaptive feedforward for a wafer stage in a litho-graphic toolrdquo IEEE Transactions on Control Systems Technologyvol 21 no 3 pp 875ndash881 2013

[17] J Q Gong and B Yao ldquoAdaptive robust control withoutknowing bounds of parameter variationsrdquo in Proceedings of the38th IEEE Conference on Decision and Control pp 3334ndash3339December 1999

[18] D Roover Motion Control of a Wafer Stage Delft UniversityPress Delft The Netherlands 1997

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r(k)

k

(a)

y(k)

r(k)

(b)

Figure 1 Input 119903(119896) (a) and the corresponding output 119910(119896) map (b) trajectories of the parametric hysteresis operator

parametric hysteresis model [8] was used in current-modeoperated reluctance force actuator However the abovemeth-ods need precise hysteresis model which is usually complexand hard to obtain

Owing to the online self-learning and estimate ability ofthe neural network it provides a good solution for solvingnonlinear problems In particular the multilayer neuralnetwork (MNN) is effectively used in nonlinear discrete-time system identification and control [9] Although a lot ofresearches have been done on the neural network applicationto the hysteresis [10 11] the neural network compensation forthe current-driven reluctance actuator with hysteresis has yetto be studied

Themain contribution of this paper is a nonlinear currentcontroller with hysteresis compensation for current-drivenreluctance linear actuator using the adaptive MNN proposedfor the first time Concretely speaking based on the input-output feature of the hysteresis in reluctance force and thelearning and approximation ability of neural network a hys-teresis current compensator is proposed using the adaptiveMNN [9] whose weight is updated by the error betweenthe desired force and the actual force The main advantageof the promosed compensation method is that the inversehysteresis model is not required Do the simulations onthe reluctance actuator model with hysteresis and EI pairsstage and the simulation results show that the nonlinearcurrent control with hysteresis compensator is effective inovercoming the hysteresis and promising in high-precisionand high-acceleration control applications

This paper is organized as follows In Section 2 paramet-ric hysteresis operator is reviewed In Section 2 reluctanceactuator models with and without hysteresis are reviewedA nonlinear current control with hysteresis compensatorusing the adaptive MNN is presented in Section 3 Finallysimulation results are illustrated in Section 4

2 Parametric Hysteresis Operator

Hysteresis is encountered over a wide range of applicationsthat usually involve magnetic ferroelectric mechanical oroptical systems It is a complex nonlinearity that displaysthe properties of bifurcation and nonlocal memory Thehysteresis can be defined as a loop in the input-outputmap In this paper the parametric hysteresis operator [4] is

reviewed which is defined in the discrete-time domain sothe implementation is more straightforward

Definition 1 Let 119903(119896) 119910(119896) be bounded 1205821 1205822gt 0 Then the

parametric hysteresis operator

119910 (119896) = 119867hys [119903 (119896) 1205821 1205822] (1)

is defined as followswhen 119903(119896 + 1) minus 119903(119896) ge 0

119910 (119896) = 119903 (119896) + 1199100minus 1199030+ 119898 minus

1

1205822

119882(1205822119898 sdot 1198901205822(119898+119903(119896)minus119903

0))

(2)

when 119903(119896 + 1) minus 119903(119896) lt 0

119910 (119896) = 119903 (119896) + 1199100minus 1199030minus 119899 +

1

1205822

119882(1205822119899 sdot 1198901205822(119899minus119903(119896)+119903

0))

(3)

where 119898 = 1205821+ 1199030minus 1199100and 119899 = 120582

1minus 1199030+ 1199100 1199030= 119903(119896lowast) and

1199100

= 119910(119896lowast) The indicator 119896

lowast denotes the last time instantbefore 119896 when the difference of 119903 changed sign that is anextremum occurred which corresponds to a corner point ofthe 119910-119903 curve in Figure 1 The parameter 120582

1represents the

amount of hysteresis around the straight line 119903(119896) = 119910(119896)

and defines the asymptotes while the parameter 1205822defines

the smoothness of the hysteresis loops that is the rate ofconvergence towards the left or right asymptote 119882 is theprincipal branch of the Lambert 119882 function [12]

3 Problem Statement

31 Reluctance Actuator Models with and without HysteresisReluctance linear actuator is a type of electric motor thatinduces nonpermanent magnetic poles on the ferromagneticrotor and produces temporary magnetic The reluctanceactuator is named as such because it usesmagnetic reluctanceto generate force which can be called reluctance force TheEI-core actuator is the most basic linear reluctance linearactuator [13] As shown in Figure 2 the actuator includes agenerally Cobalt-Iron ldquoErdquo shaped electromagnet and an ldquoIrdquoshaped targetThe electromagnet has an electrical coil woundaround the center section Current flowing through the coilgenerates a magnetic flux and this flux creates a reluctance


i

N lm

xg

E-core I-target

FF

B


B

H



119865 = 119896

1198942

1199092

119892

(4)

where 119896 = 120583011987321198604 120583





119865 = 119896

[119867hys (119894 1205821 1205822)]2

1199092

119892

(5)



i F

Setpoint Fd id


Poweramplifier


Positionsensor

xg

minus



119889



119894119889=

radic

119865119889sdot 1199092

119892

119896119889

(6)






a

Fd id idd

+ic

i

V W

+

minus120576

MNN

m


Poweramplifier


Adaptive laws


AccelerometerF





119894119889119889

(119896) = 119894119889(119896) + 119894

119888(119896) (7)


119894119888(119896) = WT

(k) S (VT(k) x (k)) (8)



1(119896)x(119896)) 119904(V119879

119897(119896)x(119896)) 1]119879 isin R119897

with 119904(119909) = 1(1 + 119890minus119909

)The force error is

120576 (119896 + 1) = 119865119889(119896) minus 119865 (119896) (9)


W (119896 + 1) = W (119896) minus Proj119882

[120574119908S (119896) 120576 (119896 + 1)] (10)


(11)


S(119896) = diag1199041(119896) 119904

119897(119896) with 119904

119894(119896) = 119904(V119879

119894x(119896)) and

z119897= [1radic119897 1radic119897]

119879



1(119896) 119904

1015840

119897(119896)


119894(119896) = 119904




120579(lowast119895119896)whose


Proj120579(lowast119895119896) =


120579119895119896

= 120588120579119895119896max lowast

119895119896lt 0

120579119895119896

= 120588120579119895119896min lowast

119895119896gt 0


(12)







119888(119896)


using (8)




4 Simulation










1= 002 and 120582

2= 10





0 2 4 6 8 10 12 14 160

50

100

150

200

Refe

renc

e for

ce (N

)

t (s)


0 50 100 150 200

0

05

1

15

2

Reference force

Forc

e err

or (N

)


minus2

minus15

minus1

minus05



119889(119896+1) 119865

119889(119896) 119865(119896+


119908= 00006 and


119889and the force


119889with and



0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

160

180

200

Reference force

Out

put f

orce


80 85 9070

80

90

100


E-core 2 I-target 2

i2F2

xg2 xg1

i1F1

Stage

x

E-core 1I-target 1







0= 15Nwhich



1198922between the






0

02

04

Posit

ion

(mm

)

0

5

10

Velo

city

(mm

s)

0 001 002 003 004 005 006 007 008t (s)


0 001 002 003 004 005 006 007 008

0

10

20

30



minus10

minus20

minus30

Trac

king

erro

r (120583

m)

t (s)






0015 002 0025 003 0035

0015 s 00198 s

0

1

2

3

4

minus2

minus3

minus4

minus1

Trac

king

erro

r (120583

m)

t (s)



005 0055 006 0065 007 0075 008

0516 s

05945 s

0

1

2

3

4

minus2

minus3

minus4

minus1

Trac

king

erro

r (120583

m)

t (s)





5 Conclusion






Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










i

N lm

xg

E-core I-target

FF

B


B

H



119865 = 119896

1198942

1199092

119892

(4)

where 119896 = 120583011987321198604 120583





119865 = 119896

[119867hys (119894 1205821 1205822)]2

1199092

119892

(5)



i F

Setpoint Fd id


Poweramplifier


Positionsensor

xg

minus



119889



119894119889=

radic

119865119889sdot 1199092

119892

119896119889

(6)






a

Fd id idd

+ic

i

V W

+

minus120576

MNN

m


Poweramplifier


Adaptive laws


AccelerometerF





119894119889119889

(119896) = 119894119889(119896) + 119894

119888(119896) (7)


119894119888(119896) = WT

(k) S (VT(k) x (k)) (8)



1(119896)x(119896)) 119904(V119879

119897(119896)x(119896)) 1]119879 isin R119897

with 119904(119909) = 1(1 + 119890minus119909

)The force error is

120576 (119896 + 1) = 119865119889(119896) minus 119865 (119896) (9)


W (119896 + 1) = W (119896) minus Proj119882

[120574119908S (119896) 120576 (119896 + 1)] (10)


(11)


S(119896) = diag1199041(119896) 119904

119897(119896) with 119904

119894(119896) = 119904(V119879

119894x(119896)) and

z119897= [1radic119897 1radic119897]

119879



1(119896) 119904

1015840

119897(119896)


119894(119896) = 119904




120579(lowast119895119896)whose


Proj120579(lowast119895119896) =


120579119895119896

= 120588120579119895119896max lowast

119895119896lt 0

120579119895119896

= 120588120579119895119896min lowast

119895119896gt 0


(12)







119888(119896)


using (8)




4 Simulation










1= 002 and 120582

2= 10





0 2 4 6 8 10 12 14 160

50

100

150

200

Refe

renc

e for

ce (N

)

t (s)


0 50 100 150 200

0

05

1

15

2

Reference force

Forc

e err

or (N

)


minus2

minus15

minus1

minus05



119889(119896+1) 119865

119889(119896) 119865(119896+


119908= 00006 and


119889and the force


119889with and



0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

160

180

200

Reference force

Out

put f

orce


80 85 9070

80

90

100


E-core 2 I-target 2

i2F2

xg2 xg1

i1F1

Stage

x

E-core 1I-target 1







0= 15Nwhich



1198922between the






0

02

04

Posit

ion

(mm

)

0

5

10

Velo

city

(mm

s)

0 001 002 003 004 005 006 007 008t (s)


0 001 002 003 004 005 006 007 008

0

10

20

30



minus10

minus20

minus30

Trac

king

erro

r (120583

m)

t (s)






0015 002 0025 003 0035

0015 s 00198 s

0

1

2

3

4

minus2

minus3

minus4

minus1

Trac

king

erro

r (120583

m)

t (s)



005 0055 006 0065 007 0075 008

0516 s

05945 s

0

1

2

3

4

minus2

minus3

minus4

minus1

Trac

king

erro

r (120583

m)

t (s)





5 Conclusion






Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










a

Fd id idd

+ic

i

V W

+

minus120576

MNN

m


Poweramplifier


Adaptive laws


AccelerometerF





119894119889119889

(119896) = 119894119889(119896) + 119894

119888(119896) (7)


119894119888(119896) = WT

(k) S (VT(k) x (k)) (8)



1(119896)x(119896)) 119904(V119879

119897(119896)x(119896)) 1]119879 isin R119897

with 119904(119909) = 1(1 + 119890minus119909

)The force error is

120576 (119896 + 1) = 119865119889(119896) minus 119865 (119896) (9)


W (119896 + 1) = W (119896) minus Proj119882

[120574119908S (119896) 120576 (119896 + 1)] (10)


(11)


S(119896) = diag1199041(119896) 119904

119897(119896) with 119904

119894(119896) = 119904(V119879

119894x(119896)) and

z119897= [1radic119897 1radic119897]

119879



1(119896) 119904

1015840

119897(119896)


119894(119896) = 119904




120579(lowast119895119896)whose


Proj120579(lowast119895119896) =


120579119895119896

= 120588120579119895119896max lowast

119895119896lt 0

120579119895119896

= 120588120579119895119896min lowast

119895119896gt 0


(12)







119888(119896)


using (8)




4 Simulation










1= 002 and 120582

2= 10





0 2 4 6 8 10 12 14 160

50

100

150

200

Refe

renc

e for

ce (N

)

t (s)


0 50 100 150 200

0

05

1

15

2

Reference force

Forc

e err

or (N

)


minus2

minus15

minus1

minus05



119889(119896+1) 119865

119889(119896) 119865(119896+


119908= 00006 and


119889and the force


119889with and



0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

160

180

200

Reference force

Out

put f

orce


80 85 9070

80

90

100


E-core 2 I-target 2

i2F2

xg2 xg1

i1F1

Stage

x

E-core 1I-target 1







0= 15Nwhich



1198922between the






0

02

04

Posit

ion

(mm

)

0

5

10

Velo

city

(mm

s)

0 001 002 003 004 005 006 007 008t (s)


0 001 002 003 004 005 006 007 008

0

10

20

30



minus10

minus20

minus30

Trac

king

erro

r (120583

m)

t (s)






0015 002 0025 003 0035

0015 s 00198 s

0

1

2

3

4

minus2

minus3

minus4

minus1

Trac

king

erro

r (120583

m)

t (s)



005 0055 006 0065 007 0075 008

0516 s

05945 s

0

1

2

3

4

minus2

minus3

minus4

minus1

Trac

king

erro

r (120583

m)

t (s)





5 Conclusion






Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 10 12 14 160

50

100

150

200

Refe

renc

e for

ce (N

)

t (s)


0 50 100 150 200

0

05

1

15

2

Reference force

Forc

e err

or (N

)


minus2

minus15

minus1

minus05



119889(119896+1) 119865

119889(119896) 119865(119896+


119908= 00006 and


119889and the force


119889with and



0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

160

180

200

Reference force

Out

put f

orce


80 85 9070

80

90

100


E-core 2 I-target 2

i2F2

xg2 xg1

i1F1

Stage

x

E-core 1I-target 1







0= 15Nwhich



1198922between the






0

02

04

Posit

ion

(mm

)

0

5

10

Velo

city

(mm

s)

0 001 002 003 004 005 006 007 008t (s)


0 001 002 003 004 005 006 007 008

0

10

20

30



minus10

minus20

minus30

Trac

king

erro

r (120583

m)

t (s)






0015 002 0025 003 0035

0015 s 00198 s

0

1

2

3

4

minus2

minus3

minus4

minus1

Trac

king

erro

r (120583

m)

t (s)



005 0055 006 0065 007 0075 008

0516 s

05945 s

0

1

2

3

4

minus2

minus3

minus4

minus1

Trac

king

erro

r (120583

m)

t (s)





5 Conclusion






Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

02

04

Posit

ion

(mm

)

0

5

10

Velo

city

(mm

s)

0 001 002 003 004 005 006 007 008t (s)


0 001 002 003 004 005 006 007 008

0

10

20

30



minus10

minus20

minus30

Trac

king

erro

r (120583

m)

t (s)






0015 002 0025 003 0035

0015 s 00198 s

0

1

2

3

4

minus2

minus3

minus4

minus1

Trac

king

erro

r (120583

m)

t (s)



005 0055 006 0065 007 0075 008

0516 s

05945 s

0

1

2

3

4

minus2

minus3

minus4

minus1

Trac

king

erro

r (120583

m)

t (s)





5 Conclusion






Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Nonlinear Current Control for Reluctance ...downloads.hindawi.com/journals/jcse/2014/150345.pdfResearch Article Nonlinear Current Control for Reluctance Actuator with