Table Of ContentModeling and Control of Magnetic Fluid
Deformable Mirrors for Adaptive Optics Systems
Zhizheng Wu • Azhar Iqbal (cid:129) Foued Ben Amara
Modeling and Control of
Magnetic Fluid Deformable
Mirrors for Adaptive Optics
Systems
123
ZhizhengWu AzharIqbal
ShanghaiUniversity UniversityofToronto
Shanghai TorontoOntario
China,People’sRepublic Canada
FouedBenAmara
UniversityofToronto
TorontoOntario
Canada
ISBN978-3-642-32228-0 ISBN978-3-642-32229-7(eBook)
DOI10.1007/978-3-642-32229-7
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012951037
©Springer-VerlagBerlinHeidelberg2013
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
with reviews or scholarly analysis or material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Preface
Adaptiveoptics(AO)systemsareusedinvariousapplicationareastoenhancethe
performanceof optical systems, such as imaging systems, laser systems, and free
space optical communication systems. Recent developments in optomechatronic
systems targeted at AO applications cover two main areas, namely, the devel-
opment of novel wavefront correctors and the development of advanced control
algorithms. Adaptive optics (AO) systems make use of active optical elements,
namely, wavefront correctors, to improve the resolution of imaging systems by
compensatingforcomplexopticalaberrations.Recently,magneticfluiddeformable
mirrors (MFDM) were proposed as a novel type of wavefront correctors that
offer cost and performance advantages over existing wavefront correctors. These
mirrors are developed by coating the free surface of a magnetic fluid with a thin
reflectivefilmofnanoparticles.Thereflectivesurfaceofthemirrorscanbedeformed
using a locally applied magnetic field and thus serves as a wavefront corrector.
MFDMs have been found particularly suitable for ophthalmic imaging systems,
wherethey can be used to compensatefor the complexaberrationsin the eye that
blur the images of the internal parts of the eye, or astronomical imaging systems
where an MFDM can be easily deployed and maintained. However, the practical
implementationofMFDMsinrealapplicationsishamperedbythelackofeffective
methodstocontroltheshapeoftheirdeformablesurface.
The recent developmentsreported in this book present solutions to the surface
shapecontrolproblemin anMFDM thatwill makeit possibleforsuch devicesto
becomeintegralcomponentsofadaptiveoptics(AO)systems.Thecontentsofthis
bookaddressthefollowingmainobjectives:
(cid:129) Model development. This first objective involves the development of a com-
prehensive model of the MFDM surface shape response to the applied control
magneticfield.Themodelshouldaccuratelycapturethedynamicsofthesurface
shape and shouldbe simple enoughto allow for the design of effectivesurface
shapecontrolalgorithms.
(cid:129) DesignanddevelopmentofanMFDMandAOsystem. Thisobjectiveconcerns
thedesignanddevelopmentofaprototypeMFDMtobeusedinthevalidationof
v
vi Preface
themirrormodelandintheevaluationofthedesignedcontrollers.Italsoincludes
thedesignandassemblyofanexperimentalAOsetuptotesttheperformanceof
theMFDM.TheAO setupcouldbefurtherextendedtoanophthalmicimaging
systemtocompensatefortheaberrationsinthehumaneye.
(cid:129) Controller design. The third objective involves the design of effective MFDM
surfaceshapecontrollerssuchthatthemirrorcouldbeusedinaclosed-loopAO
system to cancelthe opticalaberrationsofinterest. The designof the proposed
controllersisbasedonconsideringvariousclosed-loopAOsystemperformance
specifications.Astheaberrationsinthehumaneyeandinotherapplicationsare
dynamic in nature and are not known a priori, the control problem is cast as a
regulation problem where it is desired for the mirror surface shape to track a
time-varyingunknownreferenceshape.
(cid:129) Experimentalvalidationandevaluation.Thefinalobjectiveistoexperimentally
evaluatetheeffectivenessoftheclosed-loopAOsysteminvolvingtheprototype
MFDM and the developed control algorithms. The closed-loop AO system
performanceevaluationisbasedonanalyzingthemirrorsurfaceshape,tracking
accuracy for given desired mirror surface shapes that are needed to cancel the
opticalaberrations.
Based on the aforementioned objectives, several research developments have
been made with respect to the design, modeling, and control of magnetic fluid
deformablemirrors.Thefirstmajorcontributionofthisresearchisthedevelopment
of an accurate analytical model of the dynamics of the mirror surface shape.
The model is developed by analytically solving the coupled system of fluid-
electromagneticequationsthatgovernthedynamicsofthesurfaceshape.Themodel
is presented in state-space form and can be readily used in the development of
surface shape control algorithms. The second major contribution of the research
work is a novel and innovative design of the MFDM. The design change was
promptedby the findings of the analytical work undertakento develop the model
mentioned above and is aimed at linearizing the response of the mirror surface.
Theproposeddesignalsoallowsformirrorsurfacedeflectionsthataremuchlarger
than those provided by the conventional MFDM designs. A third contribution of
this book involves the development of control algorithms that allowed the first-
ever use of an MFDM in a closed-loop adaptive optics system. A decentralized
proportional-plus-integral(PI)controlalgorithmdevelopedbasedontheDCmodel
of the wavefront corrector is presented to deal mostly with static or slowly time-
varying aberrations. To improve the stability robustness of the closed-loop AO
system,adecoupleduncertainmodeloftheplantisconsideredandadecentralized
robust proportional-integral-derivative (PID) controller is developed. To improve
the performanceof the closed-loop AO system in dealing with complex dynamic
aberrations,centralizedcontrolmethodsaredevelopedandincludeamultivariable
PIDcontrolapproachwhereafixedstructurecontrollerisdesignedbasedonsolving
linear matrix inequalities (LMIs) and an optimal control approach based on the
mixed-sensitivityH1 design method.The proposedmodelprototypeMFDM and
control algorithms are experimentally tested and evaluated. The aforementioned
developmentsareorganizedintoninechaptersassummarizedinthefollowing.
Preface vii
Chapter 1 presents an introduction to the topics discussed in this book. The
advantagesofmagneticfluiddeformablemirrorsovertheconventionalsolidmirrors
are presented, and the major contributions of this book are summarized. Some
preliminaries that help the readers better understand the contents in this book are
alsobrieflyintroduced.
In Chap. 2, a brief review of the history on adaptive optics systems and their
applications is presented. The basic concept of a wavefront and the wavefront-
related optical aberrations is introduced. The basic setup of a typical adaptive
optics system is illustrated, and a brief exposition of the ophthalmic technologies
whichhavebenefitedfromadaptiveopticssystemsisintroduced.Themotivationfor
consideringAOsystemsbasedonmagneticfluiddeformablemirrorsisdiscussed.
In Chap. 3, the briefhistory of the developmentof MFDMs is introduced.The
workingprincipleofMFDMsincludingthedescriptionoftheirmainstructureand
compositionispresented.TheadvantagesoftheMFDMsandsignificantchallenges
arediscussed.Ananalysisoftheproblemsimpedingthepotentialutilizationofthese
mirrorsinpracticalapplicationsispresented.
Chapter4introducesananalyticalmodeloftheMFDMsurfaceshapedynamics.
Themodeldescribesthedynamicsofthemirrorsurfaceshapeintermsofdeflections
ofthemirrorsurfacedrivenbyanarrayofelectromagneticcoilsandisdevelopedin
boththeCartesianandthecylindricalcoordinatesystems.Aninnovativemethodof
linearizingtheequationsgoverningthedynamicsofthemirrorsurfaceisintroduced.
Thelinearizationmethodisalsoutilizedtoproposeanimportantmodificationinthe
MFDMdesign.
ThedesignoftheprototypeMFDMandofanexperimentalAOsetupispresented
inChap.5.Thenovelconceptualdesignispromptedbythefindingsoftheanalytical
workpresentedinChap.4.ThedetailsofthelayoutoftheAOsetupforexperimental
validationandthe correspondingcomponentsusedinthesetup aredescribed.The
prototype mirror and the experimental setup are used to validate the developed
analytical model of the mirror, including the static and dynamic response of the
prototype MFDM, as well as validate the obtained new features of the MFDM
comparedwiththeconventionalMFDMs,suchasthelinearizationoftheresponse,
thebidirectionaldisplacementsofthemirrorsurface,andtheamplifiedmagnitude
ofthesurfacedeflections.
In Chap. 6, a concise review of the differentcontrolalgorithmsthat have been
usedfor the shapecontrolof deformablemirrorsin AO systemsis presented.The
control problem for magnetic fluid deformable mirrors is then discussed, and the
appropriateconfigurationoftheproposedMFDMcontrolsystemispresented.
In Chap. 7, two decentralized PID control algorithms are proposed to control
the surface shape of an MFDM in the closed-loopadaptiveoptics system. First, a
classical decentralized proportional-plus-integral(PI) controller is designed based
on statically decoupling the plant model using the plant DC gain. To overcome
stability robustness issues and minimize the effect of the model uncertainties,
especiallyinthehighfrequencyrange,adecentralizedrobustproportional-integral-
derivative(PID)controlleristhenproposedandverifiedexperimentally.
viii Preface
Toimprovethecontrolperformancewithrespecttodynamicaberrationsignals,
two centralizedoptimalcontrolalgorithmsare proposedin Chap.8 to addressthe
MFDM surface shape control problem. The first control algorithm is associated
with a multivariable PID controller designed based on properly formulated linear
matrix inequalities. Controller designs that yield closed-loop systems with H1
and H performance specifications are investigated. A robust mixed-sensitivity
2
H1 controller design approach is also developed to provide the desired tracking
performance, limit the input currents, and enhance the robustness of the closed-
loop system. The performances of both control algorithms are experimentally
evaluatedusingtheclosed-loopadaptiveopticssystem,showingsuccessfultracking
ofdynamicwavefrontshapesthathaveafrequencycontentconsistentwiththatof
thehumaneyeaberrations.
In Chap. 9, the concluding remarks and recommendationsfor future work are
given.ImportantaspectsofMFDMsthathavetobeaddressedbeforethelattercan
befinallyusedpracticallyinclinicalorastronomicalapplicationsarediscussed.
This book is primarily intended for researchers and engineers in the adap-
tive optics systems community. The contents of this book are presented in a
self-contained manner. The nine chapters contained in this book exploit several
independentyetrelatedtopicsindetail.Insteadofonlygivingthecontrollerdesign
procedure,someconceptsbehindthecontrolapproachesandcorrespondingdetails
of the derivationsare also presented selectively in this book to outline the design
insightsforthereader.
Weareveryfortunatetohavehadmanyhelpfulsuggestionsfromourcolleagues,
friends, and coworkers through many stimulating discussions in relation to the
developments presented in this book. First of all, we would like to express our
sincere gratitude to numerousvaluable suggestionsand commentsfrom Professor
ErmannoBorraatLavalUniversity.Wewouldalsoliketoexpressoursincereappre-
ciation to a numberof graduate and undergraduatestudents who have contributed
tothecollaborativestudiesofadaptiveopticssystems.TheyareMaurizioFicocelli,
Jason Li, Devina Dukhu, Geoff Fung, Steve Bristo, and Ryan Li. Special thanks
go to Becky Zhao and Li Shen, Associate Editors of Springer-Verlag, for their
consistentandreliablesupportandcommentsintheprocessofpublishingthisbook.
TheresearchworkpresentedinthisbookhasbeensupportedinpartbytheNatural
SciencesandEngineeringResearchCouncilofCanada,theCanadaFoundationfor
Innovation, the Provincial Government of Ontario, the National Natural Science
FoundationofChina,andShanghaiPujiangProgram.
We are greatly indebted to our families for their encouragementand sacrifices
overtheyearsittooktocompletetheresearchworkpresentedinthisbook.Itisour
pleasuretodedicatethisbooktothemfortheirinvaluablesupport.
ZhizhengWu
AzharIqbal
FouedBenAmara
Contents
1 IntroductionandPreliminaries............................................. 1
1.1 Introduction ............................................................. 1
1.2 MathematicalPreliminaries ............................................ 6
1.2.1 BasicsofFluidMechanicsandElectromagneticsModeling . 6
1.2.2 ControlSystemsTheory........................................ 17
1.3 Summary ................................................................ 39
References..................................................................... 40
2 AdaptiveOpticsSystems.................................................... 43
2.1 IntroductiontoAdaptiveOpticsSystems.............................. 43
2.1.1 TheBasicConceptofaWavefront............................. 45
2.1.2 RepresentationofAberrationswithBasisFunctions.......... 47
2.1.3 OpticalMetricsofAberrations................................. 52
2.1.4 WavefrontAberrationCorrection.............................. 57
2.2 OperatingPrincipleofanAdaptiveOpticsSystem.................... 59
2.2.1 WavefrontSensors.............................................. 60
2.2.2 WavefrontCorrectors........................................... 62
2.2.3 Controllers ...................................................... 66
2.3 RetinalImagingAdaptiveOpticsSystems............................. 68
2.3.1 CorrectionoftheAberrationsintheEyeUsingAOSystems 70
2.3.2 HistoryofOphthalmicAdaptiveOpticsSystems............. 72
2.3.3 ChallengestoOphthalmicAdaptiveOpticsSystems ......... 73
2.4 Summary ................................................................ 74
References..................................................................... 75
3 MagneticFluidDeformableMirrors ...................................... 81
3.1 BriefHistoryoftheDevelopmentofMFDMs......................... 81
3.2 PrincipleofOperation .................................................. 83
3.2.1 MagneticFluids................................................. 85
3.2.2 MELLFs......................................................... 90
3.3 CurrentStateofResearchandChallenges............................. 91
ix
x Contents
3.4 PerformanceRequirementsforMFDMsinOphthalmic
AOSystems ............................................................. 94
3.5 Summary ................................................................ 95
References..................................................................... 96
4 AnalyticalModelofaMagneticFluidDeformableMirror ............. 99
4.1 AnalyticalModelinCartesianGeometry.............................. 99
4.1.1 GoverningEquations ........................................... 100
4.1.2 SimplificationoftheGoverningEquations.................... 101
4.1.3 DerivationoftheSurfaceResponse............................ 104
4.2 AnalyticalModelinCircularGeometry ............................... 109
4.2.1 SimplificationoftheGoverningEquations.................... 110
4.2.2 DerivationoftheSurfaceResponse............................ 111
4.3 Current–PotentialRelationship......................................... 118
4.4 SimulationoftheMFDMModel....................................... 120
4.4.1 ModelParameters............................................... 120
4.4.2 StaticResponse ................................................. 121
4.4.3 DynamicResponse ............................................. 121
4.5 Summary ................................................................ 123
References..................................................................... 125
5 Design of a Magnetic Fluid Deformable Mirror
andExperimentalModelValidation....................................... 127
5.1 DesignofMagneticFluidDeformableMirror......................... 127
5.1.1 ConceptualDesign.............................................. 128
5.1.2 DetailedDesign................................................. 130
5.1.3 DescriptionofthePrototypeMFDM .......................... 133
5.2 ExperimentalSetup ..................................................... 137
5.2.1 LayoutoftheSystem........................................... 137
5.2.2 DescriptionoftheMainComponents.......................... 138
5.2.3 SystemAssembly............................................... 140
5.2.4 AlignmentProcedure........................................... 142
5.2.5 Miscellanea ..................................................... 145
5.3 ExperimentalEvaluation................................................ 146
5.3.1 Preliminaries.................................................... 146
5.3.2 ModelValidation................................................ 151
5.3.3 LinearityoftheMFDMResponse............................. 161
5.4 Summary ................................................................ 163
References..................................................................... 164
6 ControlSystemDesign ...................................................... 165
6.1 ReviewoftheControlMethods........................................ 165
6.1.1 ClassicalDecentralizedAOControlMethods................. 166
6.1.2 CentralizedOptimalControlMethods......................... 172
6.1.3 DistributedControlMethods................................... 177
6.1.4 AdaptiveControlMethodsandOthers ........................ 179
