Design and implementation of FPGA-based LQ control of active magnetic bearings

Rafa Piotr Jastrz bski

DESIGN AND IMPLEMENTATION OF FPGA-BASED LQ CONTROL OF ACTIVE MAGNETIC BEARINGS

Thesis for the degree of Doctor of Science (Technology) to be presented with due per- mission for public examination and criticism in the Auditorium 1382 at Lappeenranta University of Technology, Lappeenranta, Finland on the 19 ^th of December, 2007, at noon.

Acta Universitatis Lappeenrantaensis 296

LAPPEENRANTA

UNIVERSITY OF TECHNOLOGY

Rafa Piotr Jastrz bski

DESIGN AND IMPLEMENTATION OF FPGA-BASED LQ CONTROL OF ACTIVE MAGNETIC BEARINGS

Thesis for the degree of Doctor of Science (Technology) to be presented with due per- mission for public examination and criticism in the Auditorium 1382 at Lappeenranta University of Technology, Lappeenranta, Finland on the 19 ^th of December, 2007, at noon.

Acta Universitatis Lappeenrantaensis 296

LAPPEENRANTA

UNIVERSITY OF TECHNOLOGY

Department of Electrical Engineering Lappeenranta University of Technology Lappeenranta, Finland

Professor Riku Pöllänen

Department of Electrical Engineering Lappeenranta University of Technology Lappeenranta, Finland

Reviewers Professor Jerzy T. Sawicki

Department of Mechanical Engineering Cleveland State University

Cleveland, USA Dr. Adam Pi at

Department of Automatics

AGH University of Science and Technology Kraków, Poland

Opponent Professor Josu Jugo

Electricity and Electronics department University of the Basque Country Leioa (Bizkaia), Spain

ISBN 978-952-214-508-6 ISBN 978-952-214-509-3 (PDF)

ISSN 1456-4491

Lappeenrannan teknillinen yliopisto Digipaino 2007

Department of Electrical Engineering Lappeenranta University of Technology Lappeenranta, Finland

Professor Riku Pöllänen

Department of Electrical Engineering Lappeenranta University of Technology Lappeenranta, Finland

Reviewers Professor Jerzy T. Sawicki

Department of Mechanical Engineering Cleveland State University

Cleveland, USA Dr. Adam Pi at

Department of Automatics

AGH University of Science and Technology Kraków, Poland

Opponent Professor Josu Jugo

Electricity and Electronics department University of the Basque Country Leioa (Bizkaia), Spain

ISBN 978-952-214-508-6 ISBN 978-952-214-509-3 (PDF)

ISSN 1456-4491

Lappeenrannan teknillinen yliopisto

Digipaino 2007

RafaªPiotr Jastrz¦bski

Designand implementationof FPGA-based LQ control ofactive

magneticbearings

Lappeenranta2007

159p.

ActaUniversitatisLappeenrantaensis296

Diss. LappeenrantaUniversityofTechnology

ISBN978-952-214-508-6,ISBN 978-952-214-509-3(PDF),ISSN1456-4491

The needfor high performance, high precision, and energy saving in rotating

machinerydemands analternativesolutiontotraditionalbearings. Becauseof

thecontactlessoperationprinciple,therotatingmachinesemployingactivemag-

neticbearings(AMBs)providemanyadvantagesoverthetraditionalones. The

advantages such ascontamination-freeoperation, lowmaintenancecosts, high

rotational speeds, low parasitic losses, programmable stiness and damping,

and vibrationinsulation come at expenseof high cost, and complex technical

solution. Allthese propertiesmaketheuse ofAMBsappropriateprimarilyfor

specic andhighlydemandingapplications.

High performance and high precisioncontrol requires model-based control

methods and accurate models of the exible rotor. In turn, complex models

leadto high-ordercontrollers and feature considerable computational burden.

Fortunately,inthelastfewyearstheadvancementsinsignalprocessingdevices

providenewperspectiveonthereal-timecontrolofAMBs. Thedesignandthe

real-timedigital implementation ofthe high-orderlinear-quadraticcontrollers,

which focusonfastexecutiontimes,arethesubjectsofthiswork.

In particular, the control designand implementation in theeld program-

mablegatearray(FPGA)circuitsareinvestigated. Theoptimaldesignisguided

by the physical constraints of the system for selecting the optimal weighting

matrices. Theplantmodeliscomplementedbyaugmentingappropriatedistur-

bancemodels. Thecompensationoftheforce-eldnonlinearitiesisproposedfor

decreasingthe uncertaintyof the actuator. A disturbance-observer-based un-

balancecompensationforcancelingthemagneticforce vibrationsorvibrations

in themeasuredpositions ispresented. Thetheoretical studiesare veried by

thepracticalexperimentsutilizingacustom-builtlaboratorytest rig. Thetest

rigusesaprototypingcontrolplatformdevelopedin thescopeofthiswork.

Tosumup,theworkmakesastepinthedirectionofanembeddedsingle-chip

FPGA-basedcontrollerofAMBs.

Keywords: Activemagneticbearings,linear-quadraticcontrol,unbalancecom-

pensation, eldprogrammablegatearrays

UDC681.587: 621.822: 621.318

ThisthesisistheresultoftheresearchworkIcarriedoutatLappeenrantaUni-

versity of Technology (LUT) during the years 2004-2007. These years I have

worked in the Laboratory of Control Engineering and DigitalSystems, in the

Department of ElectricalEngineering. It is impossible to thank all who con-

tributedtothiswork.Nevertheless,Iwouldliketoexpressmysinceregratitude

tosomepeopleto whomIamdeeplyindebted.

Firstof all, I wish to thank you, my supervisorsProfessor Olli Pyrhönenand

ProfessorRikuPöllänenfortheopportunityandpriviledgetobeyourstudent,

forscienticfreedom,countlessdiscussions,andyourinvaluablescienticguid-

ance.

IwouldexpressmythankstoProfessorAkiMikkola,whohasproofreadpartof

thethesis. IthankyouDr. JusiSopanen,Dr. AnttiKärkkäinen,andM.Sc. Aki

PenttinenforthefruitfulcollaborationontheAMBproject. Thanksto allmy

workingcolleaguesatLUTforcreatingtherealinspiringworkingenvironment.

I wouldliketo thanktheengineersfromtheElectronicsDesign Center forthe

trickycontrolelectronics and other laboratorypersonel. The practicalresults

wouldnotbepossiblewithoutyourcontribution.

Special thanks to Ph.D. Hanna Niemelä for her patient languageinspections

andnumerouscommentsonthelayoutimprovements.

Iamgratefultothepre-examinersProfessorJerzyT.SawickifromtheCleveland

StateUniversityandDr. AdamPiªatfromtheAGHUniversityofScienceand

Technologyforexaminingthemanuscriptandtheirvaluablecomments.

The project was co-nanced by the Finnish Funding Agency for Technology

andInnovationTEKES.Financialsupport bytheLappeenrannanteknillisen

yliopistontukisäätiöisgreatlyacknowledged.

Veryspecial thanksto myparentsfor theirlove, encouragement,and support

throughoutallmystudies.

LastbutnotleastmydearestOlathankyouforyourtruelove,patience,and

understandingduringthelengthypreparationofthiswork.

Lappeenranta,November2007

Rafaª Piotr Jastrz¦bski

I R. Jastrz¦bski,R. Pöllänen,O. Pyrhönen, Analysis of System Archi-

tectureofFPGA-basedEmbeddedControllerforMagneticallySuspended

Rotor,IEEEProceedingsoftheInternationalSymposium onSystem-

on-Chip,Tampere,Finland,pp. 128-132,2005.

II R.Jastrz¦bski,R.Pöllänen,O.Pyrhönen,Real-TimeEmulatorofMag-

netic Suspension System for FPGA-based Controller, IEEE Proceed-

ingsoftheInternationalConferenceonMechatronics,Budapest, Hun-

gary,July3-5,pp. 98-103,2006.

III R. Jastrz¦bski, R. Pöllänen, O. Pyrhönen, Real-Time Emulation of

Magnetic Suspension System for Flexible Rotor in FPGA, WSEAS

Trans. onCircuitsandSystems,vol. 5,no. 7,pp. 1081-1088,2006.

IV R.Jastrz¦bski,R.Pöllänen,O.Pyrhönen,A.KärkkäinenandJ.Sopa-

nen, Modeling and Implementationof ActiveMagnetic Bearing Rotor

System for FPGA-based Control, Proceedings of the tenth Interna-

tional Symposium on Magnetic Bearings, Martigny, Switzerland, Au-

gust21-23,CD proceedings,2006.

V R.Jastrz¦bski,R.Pöllänen,O.Pyrhönen,LinearizationofForceChar-

acteristicsofActiveMagneticBearingsfortheFPGA-basedLQ-control-

ler,IEEEInternationalConferenceonIndustrialTechnology,Mumbai,

India,December15-17,CDproceedings,2006.

VI A. Penttinen, R.Jastrz¦bski, Pöllänen, O. Pyrhönen, "Run-Time De-

bugging and Monitoringof FPGA CircuitsUsing EmbeddedMicropro-

cessor",IEEENinthWorkshoponDesignandDiagnosticsofElectronic

CircuitsandSystems,Prague,CzechRepublic,pp. 147-148,2006.

Abstract 3

Acknowledgments 5

Listof Appended Publications 7

Nomenclature 13

1 Introduction 21

1.1 Frominventionofthewheeltomagnetic

levitation . . . 21

1.2 Activemagneticbearingsinrotorsuspension . . . 23

1.3 Digitalcontrol forembeddedsystem . . . 26

1.4 Relatedwork . . . 28

1.5 Outlineofthethesis . . . 31

1.6 Scienticcontribution . . . 35

2 AMBsystem as a mechatronic product 37 2.1 Mechatronicproduct . . . 37

2.2 Mechanicalsubsystem . . . 38

2.3 Actuators . . . 39

2.3.1 Electromagnets . . . 39

2.3.2 Ampliers . . . 43

2.4 Sensors . . . 44

2.4.1 Currenttransducers . . . 44

2.4.2 Positionsensors. . . 47

2.4.3 Analogto digitalconverters . . . 48

2.5 Prototypingplatform. . . 49

2.5.1 Controlelectronics . . . 49

2.5.2 Processingunits . . . 51

2.6 Summary . . . 53

3 AMBsystem modeling 55 3.1 Rigidrotor . . . 55

3.1.1 Rigidrotor2-DOFmodel . . . 56

3.1.2 Rigidrotor4-DOFmodel . . . 57

3.2 Flexiblerotor . . . 59

3.2.1 Discretizationtechniques . . . 59

3.3 Modelingofactuators . . . 67

3.3.1 ElectromechanicalcharacteristicsofradialAMBs . . . 67

3.3.2 CharacteristicsofaxialAMBs. . . 69

3.3.3 Actuatordynamics . . . 69

3.4 Measurementsandltering . . . 77

3.5 Overallplantmodel . . . 78

3.6 Summary . . . 80

4 Magnetic bearing control 83 4.1 Controlprinciples. . . 83

4.1.1 Spilloverproblem . . . 83

4.1.2 Non-collocationoftheprototype . . . 84

4.1.3 Objectivesandcontrollayout . . . 85

4.2 Currentcontrolloop . . . 88

4.3 Force-eldlinearization . . . 90

4.3.1 Invertednonlinearforceeld . . . 90

4.3.2 Modelreferencemethod . . . 93

4.4 PID-basedpositioncontrol . . . 94

4.5 SISOcaseLQpositioncontrol. . . 95

4.5.1 Selectionofoptimalcontrollayout . . . 95

4.5.2 Selectionofoptimalweightingmatrices . . . 98

4.5.3 ComparisonofLQandPIDcontrollers. . . 103

4.6 MIMOcaseposition control . . . 104

4.6.1 Selectionofoptimalweightingmatrices . . . 106

4.6.2 Unbalanceforcerejectioncontrol . . . 108

4.6.3 ComparisonofLQandPIDcontrollers. . . 110

4.7 Discrete-timecontrol . . . 115

4.8 Numericalconditioning. . . 119

4.9 Adaptivecontrol . . . 121

4.10 Robustness . . . 121

4.11 Summary . . . 122

5 FPGA-based control platform 125 5.1 Toolsanddesignmethodology. . . 125

5.2 Digitalarithmetic functions . . . 126

5.2.1 Interpolation . . . 126

5.2.2 Matrixoperations . . . 127

5.2.3 Floating-pointarithmetics . . . 127

5.3 Magneticbearingcontroller . . . 128

5.3.1 Hardware-softwarepartitioning anddataow . . . 128

5.3.2 Integratedactuatorcontroller . . . 129

5.3.3 Uppercontrolalgorithmsandinterconnections . . . 129

5.3.4 Designtesting, verication,andmonitoring . . . 130

5.3.5 FPGA-basedemulation . . . 131

6.1 Summary . . . 133

6.1.1 Controldesign . . . 133

6.1.2 FPGA-based implementation . . . 134

6.2 Outlook . . . 135

A Derivations and algorithms 145 A.1 Electromagneticforce . . . 145

A.2 MagneticenergyinRNM . . . 146

A.3 Calibrationof positionsensors. . . 146

A.4 Describingfunction foractuatorsaturation . . . 150

A.5 Per-unitquantities . . . 151

A.6 Stabilityconditionsoftheleadcompensator . . . 153

A.7 Closed-looptransferfunctions . . . 154

B Detailsof the prototype platform 157 B.1 Prototypedimensions . . . 157

B.2 Specicationofcontrolelectronics . . . 158

Symbols

a

^coecient

A

^{statematrixin} state-spacerepresentation

b

^coecient

B

^{inputmatrixin} state-spacerepresentation

B, B

^{magneticuxdensity;alsocalledmagneticeld (scalar,vector)}

c

^closeduxpath

C

^{outputmatrixin}state-spacerepresentation

d

^{thickness,length}

d M

^{dampingin mechanicalmodel}

d r

^{rotordiameter}

D

feedthrough(orfeedforward)matrixin state-spacerepresentation

D M

^{dampingmatrixinmechanicalmodel}

D

^{electricuxdensity}

e

^{vectorofreferenceerrorsignals}

E, E

^{electriceld(scalar,vector)}

f, f

^{force(scalar,vector)}

F

^function

g

^{gravityconstant}

G

^{transferfunction}

G P

proportionalgain

G F

feedforwardgain

G

^{gyroscopicmatrixin mechanicalmodel}

G pl

^{transferfunctionmatrixofplant}

G c

^{transferfunctionmatrixofcontroller}

H, H

^{magneticeldstrength (scalar,vector)}

i

^current

i cc

compensatedcontrolcurrent

i cp

compensation current

i

^{vectorcontainingcurrents}

I

^unitymatrix

I

^{massmomentofinertia}

j

^{incrementvariableorindexofsummation, integer}

J, J

^{currentdensity(scalar,vector)}

J

^{Jacobeanmatrix}

J Q

^{costfunctionor}performanceindex

K

^gainmatrixin state-feedbacklaw

k

^{incrementvariableorindexofsummation, integer}

k i

^{currentstiness(actuatorgain)}

k x

^{positionstiness(negativeopen-loopstiness)}

k M

^{stiness inmechanicalmodel}

k u

velocity-inducedvoltagecoecient

K i

^{matrixofactuatorgains}

K M

^{stiness matrixinmechanicalmodel}

K x

^{open-loopstinessmatrix}

l, l

^{lengthofuxpath(scalar,vector)}

l 0

^{nominalair-gap}

L

^inductance

L a , L _a

^{angularmomentum (scalar,vector)}

L br

^{bearingcorelength}

L

proportionalgainmatrixin estimatordesign

m

^mass

M

^{numberofmodes,uppersummationindex}

M

^{massmatrixin mechanicalmodel}

n

^{densityofcarrierelectronsin conductor}

N

^{numberofcoilturns}

N

^{linkagematrix}

P

^{numberofnodes}

P p

^polynomial

r

^radius

R

^resistance

R

^{controlweightingmatrix}

R v

measurementnoiseintensitymatrixorcovariance

R w

^{processnoiseintensitymatrixorcovariance}

s

^{Laplacevariable}

S, S

^{area(scalar,vector)}

S m

^positionmeasurementsensorsmatrix

t

^time

T

^sampleperiod

T

transformationmatrixofstatevariables

T u

transformationmatrixofinputvariables

T y

transformationmatrixofoutputvariables

u

^voltage

u

^{vectorofinputsin} state-spacerepresentation

q

^{electroncharge}

Q

^{stateweightingmatrix}

v

^,

v

^{velocity(scalar,vector)}

V

^volume

w

^{vectoroftotalcontroleort(attheoutputofthe}controller)

W 1

disturbanceattenuationfactor

W 2

^{additiveplant}discrepancybound

W 3

multiplicativeplantdiscrepancybound

W ce

electromagneticeld co-energy

W fe

electromagneticeld energy

W P

dissipationenergy

W T

^{kineticenergy}

W V

^{potentialenergy}

W

^{weightingmatrix}

x

displacementin x axis

x

^space coordinates, vectorof displacements or statevariables in state- spacerepresentation

y

displacementin y axis

y

^{vector of outputs in} state-space representation or vector of measure- ments

y j

^{deection ofapointoftheelasticbody}

z

displacementin z axis

z 0

^{locationofthecenterofgravityonz axis}

α

distributionofsensorsonstatorcircumference,measuredindegrees

β

^{tiltingangleoftherotor}

δ

^{anglebetweenidealsensinglineandactualsensingline}

ζ

^dampingratio

η

^{modalamplitude}

η

^{vectorofmodalamplitudes ormodal}coordinates

θ

^{gainandosetin measuredposition}

ϑ

^{parameterinthePincerprocedure}

µ 0

permeabilityofvacuum

µ

^relativepermeability

ξ

^timedelay

ρ

^{vectorofinput}disturbancesignals

σ

^{realpartofcomplexpole}

¯

σ

^{thegreatestsingularvalue}

ς

^phaseangle

τ

^timeconstant

υ

^{vectorofimput}disturbances

φ

^phase

φ

^{modeshapevector}

χ

^{forceactingangle}

ψ

^{magneticuxlinkage}

ω

^{angularfrequency}

Γ

^{describingfunction}

Θ

^torque

Λ

determinant

Σ

^summation

Φ

^{modeshapematrix}

Φ m

^magneticux

Φ _m

^{vectorofmagneticuxes}

Φ _ml

^{vectorofmagneticloopuxes}

Ψ

^{vectorofmagneticuxlinkages}

Ω

^{rotationalspeed}

Ω p

^{angularvelocityofprecession}

<

^reluctance

<

reluctancematrix

∇×

^curloperator

×

^{vectorproduct}

Subscripts

A

^{referringto theend-A oftherotorwithbearingA}

a

^actuator

air

^air

b

^bearingcoordinates

B

^{referringto theend-BoftherotorwithbearingB}

base

^basevalue

bias

^bias

BW

^bandwidth

c

^control

cl

^closedloop

d

^damped

DC

^{directcurrent}

del

^delay

dist

disturbance

dyn

^dynamic

f

^lter

fb

^feedback

ff

feedforward

Fe

^iron

flex

^exible

H

^Hall

in

^input

I

^integral

ld

^lead

L

^Lorentz

LL

^{lineto line}

m

^measured

max

^maximal

md

^modulation

ol

^open-loop

out

^output

p

^primary

P

proportional

pi

proportionalintegral

pl

^plant

r

^rotor

ref

^reference

rigid

^rigid

rise

^rise

s

^settling

sd

^secondary

sat

^saturation

tri

^triangle

x

^{x axis}

y

^{y axis}

z

^{z axis}

Superscripts

g

^{global completemodelin physical}coordinatesin FEM orglobalco- ordinates

m

^modal

pu

^per-unit

r

^residual

T

^{matrixtranspose}

Other symbols

¯

^{estimatedsignal}

∠

^angle

Abbreviations

ADC analog-to-digitalconverter

ALU arithmeticlogicunit

AMB activemagneticbearing

ASIC application-specicintegratedcircuit

AU arithmeticunit

CAD computer-aideddesign

CAM computer-aidedmanufacturing

DAC digital-to-analogconverter

DFT discreteFouriertransform

DSP digitalsignalprocessor

EDA electronic designautomation

EMC electromagneticcompatibility

FE niteelement

FEM niteelementmethod

FPGA eld programmablegatearray

HDL hardwaredescriptionlanguage

IDE integrateddesignenvironment

INL inversenonlinearities

I/O input-output

IP intellectualproperty

LTI lineartime-invariant

LQ linear-quadratic

LQR linear-quadraticregulator

MAGLEVmagneticallylevitatedvehicles

MDOF multi-degreeoffreedom

MIMO multi-input, multi-output

mmf magnetomotiveforce

MAC multiply-accumulate

NoC Network-on-a-chip

RAM randomaccessmemory

RDS rotor delevitation system, also known as retainer bearings, auxiliary

bearings,touch-downbearingsandsafetybearings

RISC reducedinstructionset computer

RNM reluctancenetwork method

SDRAM synchronousdynamicrandomaccessmemory

SISO single-inputsinle-output

SoC System-on-a-chip

TTL transistor-transistorlogic

UFRC unbalanceforcerejection control

ZOH zero-orderhold

VHDL veryhighspeedintegratedcircuitshardwaredescriptionlanguage

Introduction

1.1 From invention of the wheel to magnetic

levitation

Theinventionofthewheelisregardedasoneoftheoldestandmostfamoushu-

maninventions. ItoriginatedinthegeographicalareawateredbytheEuphrates

andTigrisRiversreferredtoas theCradleofCivilization inthefth millen-

nium BC, which falls in the late Neolithic (early village communities) before

theBronzeAge. IntheancientMesopotamia(modernIraq),therstfunction

ofthe inventionwasapotter'swheel, which wasfollowedby awheel-and-axle

concept. However,thewheelappearedalsointheNeolithicEurope. According

toMaªecki(1996),itissupposedthattheearliestdepictionofthefunctionasa

four-wheel,two-axlesvehicle(wagon)is onaclaypot, ca. 4000BC,excavated

inBronocicevillagein centralEurope.

It is notable that when taking into account themacroscopic nature, there

areno spinning wheels in plantsand animals. Therefore, theinventionof the

wheel canbe regarded as oneof the breakthrough concepts in the historyof

engineering. AccordingtothenotablelaureateoftheNationalMedalofScience 1

TheodorevonKármán: Scientistsstudytheworldasitis;engineerscreatethe

world that hasnever been. Whetherweagree ornotthat theconcept of the

wheelisacreationofhumaningenuity,itcertainlygainedsignicantpublicity.

Consideringthat thereis notmuch useofawheelwithoutanaxleandsocket,

which are predecessors of asleevebearing, it isregrettable that the invention

ofthebearingisbelittled.

Since theancienttimes, thedevelopmentof bearingshasbeendrivenby a

demand for higher speeds and extreme working conditions. An early type of

woodenballbearingswasfoundinthewreckofaRomanshipdatedto40BC.

TheideaofthecagedballbearingswasdescribedbyanItalianphysicistGalileo

Galilei in the year 1600. Ocially,the cagedball bearingswere inventedand

appliedbyanEnglish clockmakerJohnHarrisonin his marinechronometerto

successfullydeterminelongitudeat sea,in themid

18 ^th

^{century. Attheendof}

the

18 ^th

^{centuryand at thebeginning of the}

19 ^th

^{century, ball bearings were}

employed in an increasing number of applications. The modern self-aligning

ballbearingwaspatentedin1907byaSwedishengineerSvenWingquist, who

1

foundedtheSKF,today'slargestbearingmanufacturerintheworld. Generally,

ballbearingsarenotexpensivebuttheyarelimitedbyahighfriction,lowload-

carryingcapacity,and short lifefor high-speedoperation. Theperformance of

ball bearings can be evaluated using a top rotational speed and bearing size,

namely aDNnumber 2

. Modernhybridball bearingsare capableof achieving

theDN

≈ 1.5 · 10 ⁶

^{(Popoli,2000). The}limitationsofmechanicalbearingscanbe overcomebytheuseofnon-contactbearings. Forthenon-contactbearings,the

DNnumberislimitedbytherotorstrengthtocentrifugalforces. Toexemplify,

assumingtheachievableperipheralspeed

v =

^{200m/s,theDN=}

200 · 60 · 10 ³ /π = 3.820 · 10 ⁶ .

^{Theexamplesof}non-contactbearingsareuidbearingsandmagnetic bearings.

Theuidbearingssupporttheloadonathinlayerofliquidorgas. Theycan

providemuchlowerfrictionthanmechanicalbearings,theyareabletosupport

heavier loads, and do not require frequent maintenance. As an example, the

rst Kingsbury/Michell uid bearing (named after its inventors) in the USA

wasinstalledinHoltwoodHydroelectricPowerPlant,andithasbeeninservice

without maintenance since 1912. It supports a water turbine and generator

with arotating mass of about 165tonnes. The up-to-day technologies bring

new extreme applications, which require contamination-free solutions, higher

rotational speeds, lower vibrations, contactless support, operation in dicult

environments(e.g. process gases, corrosiveuids, and high temperatures). In

these applications, the conventional mechanical oruid bearings have proven

insucient. From thepoint ofviewof such extreme orspecial conditions,the

sucientconceptismagneticlevitation.

The idea of using magnetic levitation proved to be anything but trivial.

The instability of theferromagnetic body kept in free hovering in the six de-

greesoffreedombyxedmagnetsandelectricchargeswasshownbyEarnshaw

(1842)andisknownasErnshaw'stheorem. Thistheoremappliestotheclassical

Maxwell'selectromagnetism,which neglectsquantummechanics;itstatesthat

astatic magnetic suspensionof the ferromagneticbody cannot reach astable

equilibrium when the actingforces are inverselyproportionalto the square of

thedistances. However,themagneticlevitationispossiblebyviolatingthethe-

orem'sassumptions,suchastheuseofdiamagneticandsuperconductingobjects

oranoscillatingmagneticeldthatinducesanalternatingcurrentinaconduc-

tor,thusgeneratingthelevitatingforce. Anotherwaytoovercomethetheorem

is an active feedback. It took roughly a century to nd out that stable free

hoveringandlarge forcescanbeachievedbyferromagnetsand acontinuously

adjustedmagneticeld.Theeectsofthisevolutionmaybetracedthroughthe

patentsissuedin twoareas: anewmeansoftransportationthemagnetically

levitated vehicles (MAGLEVs) and activemagnetic bearings (AMBs) applied

torotatingbodies.

As the MAGLEVs' predecessor,wemayconsider Kemper'spatent in 1937

(Kemper, 1937), and his practical experiments. Kemper described a system

consisting of a load-carrying electromagnet, valve ampliers and sensors. In

fact,thepropertiesofsuchaload-carryingmagnethaveanamethatremindsof

thefamousinventionmentionedearlier,amagneticwheel. Themagneticwheel

has not won the market, owing to the high costs of the tracks. The rst in

2

DNnumberequalstothe product ofabearingdiameterin[mm]andmaximalspeedin

theworldcommercialhigh-speedconventionalMAGLEVrailwayconnectedthe

Pudong internationalairport and Shanghaiin 2004 (inaugurated in 2002). In

spiteof thetragicMAGLEV accident caused byahumanerroron September

22 ^nd

^{,2006,atthetesttrackinLathen,Germany,thetechnologyisregardedas}

thesafest,fastestandmostreliablegroundtransportationmethodeverinvented.

Thesuccessfuluseofthemagneticlevitationingroundtransportationisleading

tootherinterestingapplications;forexampleFoster-Millerisbuildingasystem,

calledmaglifter,tolaunchrocketsintoouterspace(Foster-Miller,2007).

Whenconsideringthesupportofrotors,in1937,Beams andHolmesat the

Universityof Virginia used anelectromagnetic suspensionfor testingmaterial

strength. One of theirexperiments concerned the electromagnetic suspension

ofasteel ball. Theballwas0.8 mmin diameter andit reachedtherotational

speed of 18 million rpm,in vacuum. At the University of Virginia, this work

continuedwithrelationto ultracentifugesused forpuricationofisotopesdur-

ingtheWorldWarII.TheearlyAMB patentconcerningthisworkisassigned

toaforementionedBeamsandHolmes(1941). However,atthatpoint,theAMB

technologywasnotyetmatured. Thetechnologystartedtomaturewiththead-

vancesofsolid-stateelectronics,digitalcontrol,andwiththepioneering works

ofSchweitzerandLange(1976)andHabermannandLiard(1979). Thenewso-

lutionsin thiseldhavebeenreportedinthebiannualInternationalSymposia

on Magnetic Bearings, which havebeen organized since 1988. The advances

of the last two decades in such elds ashigh-speed microprocessors for com-

plex control algorithms, precise sensor technology, materials and coatings for

high-stress andhigh-temperatureoperations,safetybearings technology, rotor

dynamics,andmodeling havemadetheAMBsamorecompetitivesolutionfor

rotatingmachinery. Despite these advances, the overall complexity and price

arestilllimitingfactorsin widespreadindustrialuse.

1.2 Active magnetic bearings in rotor suspension

Inthissection,theprinciplesofoperation,advantages,disadvantagesandappli-

cationsofthemagneticlevitationarediscussed. Themagneticlevitationcanbe

implementedusing dierenttechnologiesanddierentmaterials. Forinstance,

theclassicationofdierentmagneticbearingsandtypesofmagneticlevitation

ispresentedbySchweitzeret al. (2003). Inaddition, possibleapplicationsare

numerous. Therefore, wenarrowthescopeto ferromagneticbodiessupported

by electromagnets. In particular, classical active magnetic bearings in rotor

suspensionareconsidered.

Theactivemagneticlevitationisbasedonanattractivemagneticforcepro-

duced byan electromagneton aferromagneticbody. Theforces acting in the

opposite directionsrequireapairofelectromagnetspositionedontheopposite

sidesof the suspendedbody. Theprinciple ofoperationfor asingledegreeof

freedom,usingonehorseshoeelectromagnetisdepictedinFig.1.1. Thedevia-

tionofthebody

x m

^{fromthecentralpositionismeasuredbyaproximityprobe}

and is used by the control unit as afeedback signal. The control unit reacts

accordingtoaspeciccontrollawwiththecontrolcurrent

i c

^intheelectromag- net,whichproducesmagneticforce

f

^{insuchawaythatthebody withamass}

m

^{remainsin free hovering. This principleextends in thecase of therotorto}

f E l e c t r o m a g n e t

m

x m N

R o t o r

S a i r

F m

i c

C o n t r o l U n i t A m p l i f i e r

P o s i t i o n S e n s o r

x m

Figure1.1: Principleofelectromagneticlevitationfor1-DOFmagneticbearing

system, where

Φ m , S air , N,

^and

m

^{are themagnetic ux crossingthe air-gap,}

thesmallestcross-sectionareaoftheelectromagnet(i.e.,usuallyastatortooth),

numberof coilturns,andmass.

R a d i a l

s e n s o r A R a d i a l

A M B A R a d i a l

s e n s o r B R a d i a l

A M B B A x i a l A M B

A x i a l s e n s o r

S a f e t y b e a r i n g s

Figure1.2: Principleofelectromagneticlevitationfor5-DOFmagneticbearing

system

bearingrotorsystem, tenelectromagnetsandveposition measurementscon-

trol ve degreesof freedom of a rotatingrotor with respect to the stationary

partnamelythestator.

Ingeneral,theAMBsystemcomprisesthree distinctparts: magneticbear-

ings, acontrol unit, auxiliary bearings or a rotordelevitation system (RDS).

Thebearings comein two congurations. Therst oneisaradial bearing, in

which typically four electromagnetsare symmetricallydistributed around the

rotor. That is, they generatetheforces acting in two dimensionsalongx and

y axes. In horizontal applications, the magnets are positioned in such away

thatthegravityforceiscompensatedbytwoelectromagnets. Thisincreasesthe

loadcapabilityofthebearings. Thesecondcongurationisanaxialbearing. It

operatesliketheradialbearingbut inonedimensiononly,that isalongz axis.

Itcomprisesaferromagneticdisk,attachedtotherotorandelectromagnetslo-

catedat either side ofthedisk. Themoretechnicaltreatmentof thebearings

TheAMBs havespecicproperties,whichdierentiatethem frommechan-

icaland uid bearings. First, theyare able to provide completely contactless

anduidlesssupport. Thisleadstothefollowingadvantages:

•

contamination-freeandseal-freesolutions

•

environmentallyfriendly solutions(nolubrication oilandnowasteparts becauseof periodicalmaintenance)

•

^lowmaintenancecosts

•

^{high rotational speeds up to the rotor strength to} centrifugal forces as explainedearlier

•

^{low breaking torques for high speeds} (parasitic, hysteresis, and eddy- currentlosses),seee.g. (Schweitzeretal.,2003)

•

^{in general, low power losscompared with the uid lm bearings (Chen}

and Gunter,2005)becauseoflowparasiticlosses(powerfromdrivetrain

minuspowerconvertedtousefulwork)

•

^{operationindicult}environments;e.g.processgases,corrosiveuidsand hightemperatures

•

^{vibrationinsulation(lownoise)}

Second, they require active control owing to the system instability, which is

usuallyviewedasthemaindiculty. However,thepresenceofthecontrolunit

andsensorsprovidesthefollowingadvantages:

•

^adjustableparameters,i.e.,stinessanddamping,controllablesystemdy- namics

•

^{widespeedoperationrange}

•

^{activecontrolofbendingmodes}

•

^{automaticbalancing}possibility

•

diagnosticsandmonitoringbuiltinthecontrolunit

Third, the main AMB disadvantages pointed out in the literature, see e.g.

(Wassermannetal.,2003),(ChenandGunter,2005)are:

•

^{highcostswhencomparedwith}conventional bearings

•

^{breakdownoftheloadmaybecausedbyafailureofanysinglecomponent}

•

^{backup bearingsmustbeprovided;theyalsorequireextraspace}

•

^{limited load carrying capacity in regard to required space (i.e. bearing}

load to pole face area ratio); the loadthat canbecarried is lower than

that ofuid(oil-lm)bearings

For the time being, all these properties make the use of AMBs appropriate

•

^turbocompressors,e.g.innaturalgas,hydrogen, coolingand airprocess applications

•

^{rotating equipment operatingin vacuum without} contamination, e.g.in biotechnologyprocesses,cleanrooms,semiconductorindustry,andastur-

bomolecularpumps

•

^{medicalequipment,e.g.bloodpumps}

•

^{energygenerationproductsandenergystorage,e.g.ywheels,plantgen-}

erators

•

^{researchandtest}applications,e.g.testingnovelforceandvibrationcontrol methods, measurementsystems,testingrotordynamics

•

^{precisemachinetools,e.g.milling,grindingandhigh-speedspindles}

•

^high temperature bearings for aerospace and defense markets, e.g. gas turbine engines

TheexamplesofearlyindustrialapplicationsofAMBsincontactlesssupportro-

torsaregivenbyBrunet(1988)andDussaux(1990). Withvaryingapplications,

alsothesizes ofmagnetic bearingsand theirloadsvary considerably,from the

relativelysmall applicationssuchasan articialheart (Katohand Masuzawa,

2006),tothegas-turbineinanuclearpowerplant(Suyuanet al.,2006).

Asfarastheauthorknows,therstcompanytocommerciallymarketAMBs

was the French S2M(Wagner, 1988), formerly Société de Mécanique Magné-

tique, founded in 1976. Since then, the number of companies producing the

AMBshasbeenincreasing. SKF,whichhaditsownAMBsolutions,purchased

all the shares of S2M in summer 2007 (Brunet, 2007). Some other industrial

playersare: Foster-MillerTechnologies, High Speed Tech Oy Ltd, LEViTEC,

Levitronix,MECOSTraxlerAG,SynchronyInc.,andWaukesha Bearings.

1.3 Digital control for embedded system

This section addresses the background and nomenclature of the digital con-

trol. The propertiesof theembedded real-timecontrol systemsarestudied by

theexampleof AMB rotorsystems. Thepossiblestrengthsand weaknessesof

programmablelogiccircuitsin anAMB controlareintroduced.

The modern computer scienceis often associated with the English mathe-

matician,cryptographerandlogicianAlanTuring. Hisconceptsoflogicaldesign

andauniversalmachinehavecontributedtomoderndigitalcomputers(Turing,

1936). TheTuringmachineandvonNeumannarchitecturemarkthebeginnings

ofthedigitalrevolution,whichisaprerequisiteforadigitalcontrol.

The rst controllers of dynamicalsystems (e.g. AMB rotorsystems) were

implementedusinganalogelectronics. Themainadvantageoftheanalogcontrol

was its short input-output delay compared with its early digital equivalents.

However,mostofthepresent-daycontrolsystemsareimplementedusingdigital

computers,suchas microprocessors,microcontrollers,andlogiccircuits.

To introduce the control, the physical processes are usually rst approx-

cannotintegrate,andtheyoperateonsampledandquantizedsignalsrepresented

withbinarynumbers. Therefore,thedierentialequationshaveto beapproxi-

matedagainbyreductiontosumsandproductsonly. Thesereduced,algebraic,

recursiveequationsoperateondiscretevariablesandarecalleddierenceequa-

tions. The process of transferring continuous-time models into time-discrete

ones iscalled discretization. What ismore, theanalogsignalsarechangedby

the analog-to-digital converter (ADC) into digital ones, which can be repre-

sentedbybinarynumbers. Thisprocessissometimesreferredtoasdigitization.

Theconversionsoccurperiodicallywithperiodsoftimeequal

T

^{seconds(called}

sampleperiod)andwiththeaccuracylimitedbytheADC capabilitiesandthe

resultingnumberformat. Thediscretesignalscanbechangedbacktotheanalog

onesbyusingtheDACandthezero-order-hold(ZOH)circuitry,whichholdsthe

samevalue,usuallyvoltage,onitsoutput. Toputitbriey: Truthismuchtoo

complicatedto allowanythingbut approximationsthequotationattributed

toJohnvonNeumann.

TogetbacktotheAMBrotorsystem,themeasuredanalogsignalsrepresent

therotor'sdisplacementsfromthecentralpositionandcurrentsintheactuators,

whichproducethemagneticforces. Intheactuators,switchedpowerampliers

are used, which do not require any additional DACs. The sampling period

hasto beselectedaccordingto thesystem bandwidthto keep theerrorsfrom

the approximation small. In general, the smaller the physical dimensions of

therotor, thefaster the system'sdynamics and the widerthebandwidth. As

a result, the shorter the sampling period, the tougher the time and number

formatrequirementsforthe real-time digitalcontroller. Tooshort asampling

period canleadtonumericalproblemswhenintegrating andtostabilityissues

afterdiscretizationofthecontinuoustimecontrollers.

TheAMBcontrollerisanembeddedsystem,whichhasspecialpurposesand

specic requirementsunlikeageneral-purposecomputer. Thecontrollerhasto

performonlyafewpredened tasks:

•

^{collectingandprocessingthesystem's}measurementsdisplacements,cur- rents,rotationalspeed,andperformingacoordinatetransformationofthe

displacements

•

^{executing control algorithm, i.e., control of rotor position and rotordy-}

namics, adaptationof parametersaccordingtochangeablespeed,control

ofcurrentsin actuators,biasingofampliercurrentsignals

•

controlling system in safety critical situations, e.g. in the case of rotor touch-down

•

^{monitoringand}diagnostics

•

^{handlingvarious}non-time-criticaltasks,e.g.userinterface,calibrationof sensors,clearancechecking

Furthermore, theAMB controllers, aswell asmost of theembedded systems,

have to be compliant with special requirements, which are for instance small

size, low cost, very high reliability, or they must withstand severe conditions

such aselectromagneticinterferenceanddisturbance,vibration,radiation,and

Building and validation of the real-time embedded control for the active

magneticsupportofhigh-speed,exiblerotorsisaverydemandingengineering

task. High performance androbustcontrol systemsoftenemploy model-based

andnonlinearcontrolmethods. However,applyingaccurate,high-ordermodels

(for controller synthesis and its validation) require higher overall complexity,

modelingoffaster dynamics,andaboveall,powerfulprocessingdevices.

Digital signal processors (DSPs) are especially dedicated microprocessors

designed for real-time embedded control systems. They comprise the ADCs,

DACs andecient arithmetic logic units (ALUs). The rstsingle chip DSPs

wereproducedbyBellLabsandNEC in1978 and1980,respectively. In1983,

TexasInstruments,presenteditsrst DSPTMS32010 TM

. It workedon16-bit

data,hadaHarvardarchitecture,couldperformmultiplyaccumulateoperation

in 390 ns (Giang et al., 1988) and was very successful. Texas Instruments

is now the number one producer of DSPs. The modern DSPs provide some

features in order to increase parallelism in data processing, such as parallel

accumulator and multiplier, parallel memory architecture. Despite that, they

couldstillbeclassiedasserialprocessingdeviceswhencomparedwithmodern

eldprogrammablegatearrays(FPGAs).

Modern FPGAs oerevenmore computational powerand more exibility

thanmicroprocessors.TheFPGAswereinventedin1984byengineerRossFree-

man(Buelletal.,2007),whowasalsotheco-founderofXilinx(www.xilinx.com,

1994-2007)theworld'slargestmanufacturerofrecongurablehardwarechips.

TheFPGAsarebuiltofprogrammablecomponents,such aslogiccells,memo-

ries,andarithmetic blocks,whichareinterconnectedbyamatrixofwiresand

programmableswitches. Suchanarchitectureoersparallelprocessingcapabil-

itiesandplentyofinput-outputs(I/Os)forinterfacingwith theoutsideworld.

TheFPGAs employed in theAMB controlsystems bringnew possibilities for

implementingnovelcontrolstrategiesandextendtheareaofpossibleindustrial

applications.

Unfortunately, the designwork,which involveslargeFPGAs employed for

complex controllers, faces many diculties such as long and complex design

ow,lackofsuperiortools,awkwardtesting, andthe absenceofcompleteand

veriedlibrariesforusein controlapplications.

1.4 Related work

SincethebeginningsoftheAMBtechnology,thecomplexityofthesolutionsas

wellas constantlygrowingnumber of possibleapplications have resulted in a

signicantamountof scientic publicationsin the eld. Fromthe perspective

ofthisdissertation,theimportantreferencesconcern: themodel-basedcentral-

izedAMBcontrol,actuatorlinearization,detailsoftheimplementation,andin

particular,FPGA-based realizations.

Speakingaboutthepositioncontrol ofthelevitatedbody,in therstyears

oftheAMBsdevelopment,theresearcherswerelimitedbythecomputingpower

ofearlymicroprocessors,andthereforefocusedonthecontrollerswithlowcom-

putational burden. In 1984, Bleuler (1984) presented amethod for designing

the decentralized PD controllers for rigid rotors. For a predened structure,

betterresultsthanPD and PIDcontrol weregiven byLarsonneur'sdirect de-

the optimal low-order controllers, withouttaking into account the gyroscopic

eect. ThismethodclosedthegapbetweenthePIDandlinear-quadratic(LQ)

control methods. In regard to the low computational cost and direct use of

statefeedback,thePID-basedcontrolandotherdecentralizedsolutionsarestill

attractiveforvarious applications,(e.g. Polajzeret al.,2006). Nevertheless,in

somecases, neglectinga gyroscopic eect may cause even instability (Ahrens

etal.,1996).

Themodel-based,coupledcontrolstructures,whichtakeintoaccountagy-

roscopic eect, proved to be superior, in terms of unbalance response (while

still withoutspecialunbalance compensation), to the decentralized controllers

asstatedbyZhuravlyovet al.(1994),Ahrens etal.(1996),Zhuravlyov(2000),

Piªat(2002), and Grega and Piªat(2005). These controllers were based on a

LQ control of rigid rotors. Additionally, Piªat (2002) studied an LQ control

appliedtogetherwithafeedbacklinearization. Zhuravlyov(2000)presentedan

LQcontrolwithaswitchingcontroller(asinthatworkcopperlosseswereopti-

mal). Ingeneral,theLQcontrolissuitableforapplicationsthatholdlinearized

models, where theplant model can beaccurately determined, which in many

applicationsisthecaseforAMBs.

In AMBs, the model of a control plant consists of the actuators, ltered

measurements and exible rotor models. The accurate rotor model can be

derived using nite element modeling as given by Chen and Gunter (2005),

YamamotoandIshida(2001),andGentaetal.(1993). Themodelcanthenbe

furthercorrectedbytheexperimentalmodalanalysis,andthereductionofthe

numberof degreesoffreedomcan becarriedout. Finally, itisformedinto the

state-spacerepresentationandtransformedinasuitableformforthecontroller

development. Theformulationandscalingoftheexiblestate-spacerotormodel

usedinthisworkisbasedontheworksofLantto(1999)andGenta(2005). The

latteremploysthecomplexcoordinatesinformulationofthesystemdynamics.

All applications exhibit certain model uncertainties, which may beassoci-

atedeitherwithmodelingerrorsorwithchangingplant'sdynamics(e.g.dueto

aconsiderablechangeinworkingconditions). ForsuchuncertainAMBsystems,

adaptiveand robustcontrol couldbeapplied. Forinstance, the adaptivecon-

trollerbasedontherecursivepredictionerrormethodwasusedbyWurmsdobler

(1997), therobust control like

µ

^{-synthesiswasemployedby Lösch (2002)and}

H ^∞

^{wasutilizedbyRenet al.(2006). Acollectionofdierentcontrolschemes,}

basedon

H ^∞

^{waspresentedby}Aeschlimann(2002). Renetal.(2006)alsoused gainschedulingaccordingtotherotationalspeed. Therobustandadaptivecon-

trolcanalleviatemodeluncertaintiesuptosomepoint,yetthecontrolofAMBs

stillrequiresexpertknowledgeandapplication-specicdesign. Therefore,some

authors have tried to automate the controller design (reduce modeling eort,

minimizemanual systemidenticationand controller tuning); seefor example

Lösch (2002). Lösch's work is agood example how the model-based control,

which took into accounttheexible andgyroscopiceects, increasedthecom-

plexityofthecontroller. AccordingtoLösch(2002),theaugmentedmodelthat

includedtwo exible modes per planewasof order 44, and the resultingcon-

troller was up to order of 70. Such models were considered not practicalfor

DSP realizationso theywere reduced to about 24states and thendiscretized

before implementation. The controllers similar in size were used in the work

ofWurmsdobler (1997). Lösch's automatedcontroller wasableto identify the

overcomethisinitialknowledge,anautomaticinitiallevitationwithAMBswas

proposedbyGlaserandSandagol(2006),wherealmostnoinitialparametershad

tobeknown. Thecomparisonsofdierentpositioncontrol methodsappliedin

AMB rotor systemscan be found in the works of Knospeand Collins (1996),

Lösch(2002),andGregaandPiªat(2005). Lastbutnotleast,therobustdesign

methodsresultinhigh-order,robust,but oftenrelativelyslowcontrollers.

Asforthemodel-basedpositioncontrol,thestartingpointforthisworkisan

optimalstate-spacecontrollerwithadditionalintegrativefeedbacksugestedby

Wurmsdobler (1997)and Zhuravlyov(2000). Mostcomponentsof thestudied

AMB rotorsystemcanbeeectivelylinearized,andthereforetheycomprisea

plantsuitableforLQcontrolmethod. Furthermore,thecontrolloopsareorga-

nizedbythesametokenasthosegivenby(Larsonneur,1990)andWurmsdobler

(1997),thatis,thepositioncontrolservesasanoutercontrollooptotheinner

currentcontrolloops. Thecascadedcontrol structuremakestheinner current

control loopto look likethe linearoneto theposition control. Thefastinner

high-gainfeedback eectivelylinearizesthe nonlinearinductanceofthe coilof

theelectromagnet.

In the inner current control loop, the dierential driving mode is used to

linearize the force-current-displacement characteristics of the electromagnets.

However, in order to reduce losses, areduced premagnetization current (bias

current)isappliedresultinginthestillnonlinearcharacteristics-outoftheop-

eratingpoint. ThisessentialnonlinearityoftheAMBsshouldbecompensated,

ifhighperformanceisrequired. Thenonlinearperformancecharacteristicsofthe

radialAMBcanbesucientlydeterminedbynumericalcomputationmethods

studiedbyNergetal.(2005)andPolajzeretal.(2004). Theseobtainednonlin-

earmodels, afterverication,canbeused forbuilding theforcecompensation

andvalidationofcontrol.

Dierentmethodscanbeappliedto alleviatetheproblemofcompensating

the actuator nonlinearities. Oneof the most popular methods in control en-

gineering for compensating dierent nonlinear actuators is that of an inverse

nonlinearity (INL) control as explained by Franklin et al. (1998). It assumes

that nonlinearity isinvertibleandcanbeundone. Some variationsofthe INL

methodforAMBs werestudied byHomannetal.(1998),andbySkrickaand

Markert (2002),where apolynomialformulationwasused in the implementa-

tion;andalsobyOberbeckandUlbrich(2002),whereananalyticalmethodwas

employed. The alternativesolutionsuch asanextended Kalmanlterapplied

foraccurate position measurementsin the collocatedsystemwaspresentedby

Schuhmannetal.(2006).

In the real-time control systems, the minimization of an input-to-output

controllerdelayis ofmajorimportance. Therefore,functionalandsoftwarein-

tegration of the actuator compensation should not introduce extra delay into

the system. Homann et al. (1998) sugested theintegration of the INL com-

pensation,intothedigitalcontroller. Theimprovementinthisintegrationwere

givenbySkrickaandMarkert(2002).

Anotherintrinsicproblemin AMBapplicationsistheunbalancecompensa-

tion. Forexample,Bleuler etal.(1994)brieysummarizesdierentunbalance

compensation methods. Inthis work,anobserver-basedunbalance compensa-

tionis proposed. The proposed method extends and appliesto AMBs abasic

techniqueforestimationofsinusoidaldisturbances(Franklinetal.,1998).

namicsandnonlinearities,requires fastsignalprocessingandexiblehardware

platforms. Literature reports many examples of digital magnetic suspension

controllers realized with DSPs (Bleuler et al., 1994), (Knospe et al., 1997),

(Schroderet al.,1998),and(Krachet al.,2003). Additionally,onehasto bear

in mind that the implementation of such an embedded control system needs

careful prototyping. Regarding the ecient control prototyping, a PC-based

platformthatutilizesDSPfortheAMBcontrol,ispresentedbySchroderetal.

(1998)andthe controlimplementationin aRT-Linux andPC systemisgiven

byHumphrey et al.(1999). Considerationsonageneralsoftware architecture

oftheAMB controller,whenusingmicroprocessors,arepresentedbyBetschon

(2000)andSchweitzeret al.(2003). Additionally, Betschon(2000)studiesdif-

ferent approachesto reduce costand size ofthe AMB system. However,most

researchactivities focus on control and modeling. Usually researchteams use

industrialtestrigsandcontrol platformsin prototyping,forinstancedSPACE,

automated DSP code generation byMatlab

R and Simulink

R. The possibil- ity of improving performance of the AMB controller, by replacing DSP with

FPGAiasuggested by Krach et al. (2003),but onlysimple controlstructures

areconsidered.

Regardingthe useofFPGAsin the controldesign forpowerelectronic ap-

plications,such asinvertertypewelding machine andfrequencyconverter,the

comprehensivestudywascarriedbyRauma(2006). Onthecontrary,therewere

nosimilarstudiesabouttheuseofFPGAsorapplication-specicintegratedcir-

cuits(ASICs)inAMB control,which wouldinclude similarindetailaspectsof

realization.

Inthe controlof theAMBs,thelow levelcurrentcontrol algorithmsand a

PWM are typically realized with the help of analogdevices orspecial digital

logicdevices, that is, integratedcircuitsorprogrammablelogic. Inparticular,

notonlyDSPs but alsoASICsand FPGAsare commonlyused for thedigital

control of currents. This tendencyis similar in all applications that involvea

fast control of power electronics, for example fast inverter control algorithms

(Jastrzebskietal.,2003)inelectricmotors.

Inthe earlystage oftheauthor's doctoralstudies, hehasbeenworking on

the implementation of an induction motor control, motor emulation, and the

possibility ofintegrating amotor controlwith an inverter control(Jastrzebski

et al., 2004a)and (2004b), in asingle FPGA.These studies contribute to the

idea of a single chip AMB controller that utilizes an FPGA-based embedded

controlplatform.

ThebeginningsofAMB technology inFinland datebackto1985. In1988,

the Conference on High Speed Technology washeld in Lappeenranta, where

specialattentionwaspaidtomagneticbearings(Larjola,1988). Theelectrome-

chanicalpropertiesof theradialAMB were studied byAntila(1998), and the

robustcontrolofAMBsinsubcriticalmachineswasresearchedbyLantto(1999).

1.5 Outline of the thesis

Therst objectiveof this work is to examineand describe theuse of FPGAs

inthereal-timecontrolofhighperformanceAMBrotorsystems. Thecomplete

workingembeddedFPGA-basedcontrolplatformisbuilt,testedandvalidated

AMBcontroller,whichexplorestheadvantagesofthedesignedcontrolplatform,

takes into accountthe bending modes of the rotor (they controllability),and

nonlinearitiesoftheactuator. Therequirementsforthecontrollerincludeahigh

performance and unbalance compensation, at variable rotational speeds. For

theobtainedcontrollerarobustanalysisisconducted. Furthermore,thethesis

focusesnotonlyoncontrolandsystemleveldesignissues,butalsoitaddresses

the hardware description language (HDL) design and testing problems. The

specic designissuesare considered: singlechip controlsolution, emulation of

AMBsystem,built-intesting,designportabilityandreusability,customtailored

digitalarithmetic,on-chipinterconnections,andtheuseofembeddedprocessors.

Theavailableliteraturedoesnotprovideenoughinformationonalltheseissues.

Last but notleast, theresearch was apartof awider work that aimed at

building a completely custom AMB laboratory test rig with AMB actuators

available forexperimental testingof novelcontrol algorithmsand examination

ofrotordynamics atLappeenranta UniversityofTechnology. Theprojectwas

co-nancedbyTEKES 3

.

The dissertation is presented as an introduction to the collection of the

original scientic publications. In addition to that, the thesis includes some

original results, which have not been presented to the wider audience up to

now.

The highlights of the appended publications and the contribution of the

authorsarereportedbelow.

Publication I addressesadesignandimplementationofanapplicationspecic

architecture (of AMB controller), in an FPGA circuit. The publication

presents a prototyping platform that eases the development of control,

testing andintegrationoftestedalgorithmsinto theFPGA. Itshowsthe

FPGA implementation of an AMB currentcontroller of a set of ten H-

bridge switching ampliers. Dierent architectures and data ow of the

AMB controller are examined. The paper shows the opportunities and

threadsofaexibleHDLimplementationofanembeddedcontrolsystem.

Publication II analyzesthe benets of an FPGA-based real-time emulation

formechatronicapplications. Asacasestudy,AMBsforcontactlesssup-

portofrigidrotors,areselected. Thispublicationshowsthatforageneral

class ofstate-spacemodels withseparablenonlinearities,itis possibleto

build an optimized hardware implementation of the plant emulator or

stateobserver(abasicpartofastate-spacecontroller). Theemulatorcan

beusedforreal-timevalidationofthecontrolsystemanditscomponents.

PublicationIIusesthe4-DOFrotormodelfromsubsection 3.1.2.

Publication III includes the description of the real-time emulation of mag-

neticsuspensionsystem,withaexiblerotor,in FPGA.Thepublication

comparestherealizationsoftherigidandexiblerotormodels,whichcan

beusedforemulationorfortheimplementationofthecontroller. Itintro-

ducestheconceptsof: numericalconditioningof thexed-pointdiscrete-

time model, discreterealizationofdierentialequations,veryhighspeed

integrated circuits hardware description language (VHDL) implementa-

tion of the non-linearactuator model and state-space rotormodel. The

3

emulatorandtheLQcontrollerwithintegralfeedbackaretestedinVHDL

Mentor Graphics

R ModelSim

R simulations. PublicationIII utilizesthe exiblerotormodelfromsection3.2.

Publication IV presents FEM of the rotorand dierent variantsof theLQ

position control (considered in regard to the rotor model). In the pub-

lication, the rotor model is obtained using the FEM, modal reduction

techniqueandexperimental analysis. PublicationIVincludestheVHDL

implementation of the obtained exible rotor model with the nonlinear

actuator,butitfocusesonthedierentvariantsofthepositioncontroller,

theirHDLimplementations,andonanactivedampingofexiblemodes.

Publication V describes an LQ control and nonlinearities compensation of

a rotorsuspended by AMBs. TheLQ control is based ona state-space

controllerwithanadditionalintegrativefeedback,anddisturbanceestima-

tor. Aneectivecompensationofperformancevariationsindynamicforce

characteristicsof radial AMBs is performed using two nonlinear princi-

ples. Thepublicationstudiesthewell-knowninversenonlinearitymethod.

Next,itpresentsanovelmodel reference-basedmethod. It isshownthat

theproposedcompensationprovidescertainadvantagesoverconventional

solutions. Theoverallcontrol istested withtheexible rotormodeland

the nonlinear AMBs. Finally, the paper presents an integration of the

proposedcompensationinto thedigitalcontroller. Accuratemodelingof

the AMB nonlinearities in an FPGA isachievedby using multi-variable

interpolationandlook-uptable.

Publication VI illustratesthetechniqueofanin-circuitvericationwiththe

custommadeembeddedlogicanalyzer. Theanalyzerprovidesthemeans

to monitor internal signals, state of buses, and registers of the embed-

dedcontroller. Theheart ofthetoolcomprisesofthehardcorePowerPC

processor,in theVirtex

TM

-IIProFPGA-circuit. Inaddition tomonitor-

ing, thedescribed toolis afeasiblealternativeto thesimilarcommercial

solutions.

Looking at the major contributions to theappended publications, the author

developedallthe programsinVHDL andMatlab, carriedoutthesimulations,

andwasthemainauthorofallthepublicationsbutPublicationVI.Inaddition,

he was involved in the testing, designing of the software part, and commis-

sioningof the FPGA-based AMB rotorprototypingplatform. The text of all

theappended publications, with the exceptionof Publication VI and sections

of PublicationIV, was written bythe author. Theco-author R.Pöllänende-

veloped the reluctance network model used for obtaining the accurate force

eld of radialAMB. Theco-authorsA. Kärkkäinenand J.Sopanenwrote the

FEMcode,performedthemechanicalmodeling,obtainedtheexiblemodesand

eigenfrequenciesofthetestrotor(the naltestrotor,describedinthethesis,is

dierentthough). TheyalsowrotethesectionsofPublicationIV,whichrequired

expertisein mechanicalengineering. Theremainingsectionsof PublicationIV

waswritten bythe author. Theco-authorA. Penttinen wasthe main author

of PublicationVI. Hedeveloped theC++ code for thePowerPC processorin

ofthecommunicationbetweentheFPGA,PowerPC,andPCconsole. InPub-

lication VI,theAMB controllerand theconnection betweenthelogicand the

processorwere realizedand tested bytheauthor. Theco-authorsR.Pöllänen

and O. Pyrhönen supervised and inspired the appended publications and the

wholeAMB project.

The main body of the dissertation provides the reader with introductory

information,themaintopictreatedin theorganizedandconsistentmanner,as

well asthe key results and conclusions. Experimental and simulation results

are notorganizedin separate chapters,however, theyfollow throughallparts

ofthedissertation. Themainbodyisorganizedinto sixchapters.

Chapter 1 introducesthehistorical background,properties, applicationsand

principlesofoperationofthemagneticbearings. BasiccontroloftheAMB

rotorsystemisdiscussedin thecontextofdigitalandembeddedsystems.

A briefintroductiontomodern signalprocessingdevicesandtheirusein

embeddedcontrol applicationsisgiven. Therelatedwork andimportant

references in the studied eld are listed. The outline, main goals and

objectivesofthisdissertationarestated.

Chapter 2 considers therotor, magnetic bearings, and control system asan

mechatronic product. The hardware parts and control hardware of the

studiedprototypesystemareintroduced;physicalphenomenabehindtheir

principles of operationare shown. Thenecessarybackgroundforthe ac-

curatesystemmodelingisprepared.

Chapter 3 discusses modeling of the AMB rotor systemfor thecontrol syn-

thesisand itsvalidation. Thereaderis introduced tothe rotormodeling

and main system nonlinearities. The construction of the overall plant

model is presented. The viability of models with dierent complexities

is considered. In particular, a FEM rotormodel is developed and cor-

rected,accordingtoamodalanalysis,tomatchtheprototypesuciently

to designreliablemodel-basedcontrollers. Theeect ofthereducedpre-

magnetization current on the force characteristics of the radial AMB is

discussed.

Chapter 4 studies the design of the controller, its inner and outer control

loops, theirrequirementsand realizations. Theclassicaldecoupled PID-

basedpositioncontrolandLQmodel-basedcontrolmethodsarediscussed.

ThemainconcernareLQcontrolbasedonphysicalquantitiesanddierent

methods of the force-eldlinearization. Special attentionis paid to the

automated LQcontrollerdesignand selectionof theweightingfunctions

fortheLQperformanceindex. Furthermore,theunbalancecompensation,

robust stability analysis, and the controller evaluation accordingto the

selectedcriteriaarepresented.

Chapter 5 focusesontheissuesassociatedwithFPGA-basedcontrolplatform,

suchasthedesignmethodology,hardwaredescriptionlanguagetools,use

ofembeddedprocessorcores,onchipinterconnections,andFPGAemula-

tion. Thesoftwarearchitectureandrealizationofcontrollersub-functions

in VHDL aredescribed. Themeansforecientutilizationof theFPGA

Chapter 6 concludes themain results, summarizes the content and suggests

future researchwork.

1.6 Scientic contribution

Inthisdoctoralthesis(andappendedpublications),thefollowingmainscientic

contributions to control design and embedded control implementation can be

highlighted:

1. Utilization of the FPGA-based control platform in the outer and inner

controlloopsoftheAMBrotorcontrolsystem. TheemploymentofFPGA

forthisapplicationenablesadesignofthesinglechipcontrollersolution.

2. BuildingofthecustomblocksofdigitalarithmeticintheFPGAs,whichal-

leviatetheAMBcontrollerimplementation,inVHDL.Thewrittencustom

VHDLintellectualproperties(IPs)suchasstate-variable-formrealization

and nonlinear force eld (as apiece-wise interpolation) are general,and

canbeusedin manyotherembedded controlsystems.

3. Handlingofcomputationallyintensivetasksinareal-timeembeddedcon-

trolsystem,isanalyzed,both,inrespectofthenumericalconditioningof

themodel-basedcontrolmodels(theirdiscretization,digitization),aswell

asoftheirimplementationinFPGA.

4. Applicationofthedescribingfunctionanalysisforpredictingtheactuator

bandwidth. Thesaturationfrequencyof theactuator, andbandwidthof

theinnercurrentcontrolloop,fordierentsignalamplitudes,arestudied

(withdierentmethods).

5. Compensation of the actuator force-eld nonlinearity by the use of the

modelreferencemethod andtheinversenonlinearityprinciple.

6. Formulationofthemodel-basedLQcontrol,basedonphysicalquantities.

Theactivecontroloftheselectedexiblemodes,whentakingintoaccount

theircontrollability,isachieved.

7. Applicationoftheobserver-basedunbalancecompensationcontrollerwith

theoptimalgains. Onevariantofthecontrollercancancelthemagnetic

forcevibrations,andtheothervariantmaycancelthemeasureddisplace-

mentvibrations.