Table Of ContentSpatial Filtering for the Control of Smart Structures
James E. Hubbard, Jr.
Spatial Filtering for the
Control of Smart Structures
An Introduction
123
Prof.JamesE.Hubbard,Jr.
UniversityofMaryland
100ExplorationWay
Hampton,VA23666-6147
USA
[email protected]
[email protected]
ISBN978-3-642-03803-7 e-ISBN978-3-642-03804-4
DOI10.1007/978-3-642-03804-4
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TotheYoungDragons–
James,DrewandJordan
and
TheFairPrincess–
Adrienne
and
TheRuleroftheKingdom–
TheAlphaandOmega
Iunderstoodthattogrowadream
YouneedmorethantheoneIwas.
YouneedtheBelieveofchildhood
TheDoofYouthand
theThinkofExperience.
-ATaleofWonder,WisdomandWishes
-SusanV.Bosak
An Historical Prologue and Preface
What follows is my personal perspective on early events that played a significant
roleintheformationofthefieldnowknownasSmartStructures.Itisbynomeans
meanttobeallinclusiveordefinitiveinanyway,butmerelyanaccountofpersonal
experiences that ultimately lead to the development of the material contained and
presentedherein.
On March 23, 1983 then President Ronald Reagan announced his intentions to
develop a new system to reduce the threat of nuclear attack and end the strategy
ofmutualdeterrenceinanaddresstothenationentitled,AddresstotheNationon
Defense and National Security. The system he proposed became known as “Star
Wars,”afterthepopularmovie,becauseitwasmeanttoprovideaprotectiveshield
overthenationfromspace.Hisspeechmobilizedtheentirenationonaresearchand
developmentpathtowardthisend.
Investigations were conducted into new areas such as space based radar, large
apertureantennaeandlargeflexiblemirrorconcepts.Theseproposedsystemsrep-
resentedanentirelynewclassofstructuresthatprovedtoprovidenewchallengesin
materials,structures,controlsystemsandmodeling.Forexampleantennaeneeded
tomonitorlargeareasofrealestateinthecontinentalUnitedStatesrequiredaper-
turesontheorderof100m.Thiscoupledwiththeheftycostoflaunchtospace,on
theorderof$10,000perpound,resultedinthedesignoflightweight,highlyflex-
ible, lightly damped structures. Analysis of such structures revealed some never
before seen characteristics such as very high modal densities, large numbers of
paired modes due to the symmetries associated with the designs, lightly damped
modesandconcomitantlargeordermodels.Itbecameclearthattheresearchcom-
munityandtheacademiccommunityinparticularneededtodevelopnewtoolsand
techniques to cope with the issues associated with these Large Space Structures
(LSS).
DuringthisperiodDr.TonyAmos,thenaProgramManagerwiththeAirForce
Office of Scientific Research (AFOSR) began holding a series of invitation only
workshops to discuss these systems, associated problems and potential solutions.
The list of invitee’s included members of government, academia and the private
sector who were all active in this area of research. Senior, mid-career and junior
researchers from diverse fields that encompassed structural vibrations, active con-
trol,fluiddynamics,appliedmathematicsandmorewereinattendance.Thegroup
vii
viii AnHistoricalPrologueandPreface
includedanumberofwell-known,distinguishedscholarssuchasProfessorLeonard
Meirovitchinthefieldofvibrations,ProfessorHoltAshleythefieldofaeroelastic
structuresandunsteadyaerodynamics,andProfessorMichaelAthansinthefieldof
control theory. This was during a period in which multidisciplinary collaborations
were rare and it quickly became apparent to the group that we lacked a common
nomenclature for discussion. A simple example involved terminology to describe
structural damping. Some spoke of loss factor, some of damping ratio and others
of “Q” factor. This of course led to numerous philosophical debates and vocifer-
ousdiscussionsonthefundamentalsrequiredforcharacterizationandperformance
assessmentforLSS.Thisgroupmetregularlyforseveralyearsandculminatedwith
the formation of an annual meeting at Virginia Polytechnic and State University
hostedbyProfessorMeirovitchentitledVPI&SU/AIAASymposiumonDynamics
andControlofLargeStructuresinBlacksburgh,Virginia.Itbecameclearovertime
that LSS were lightly damped structures that could have demanding performance
tolerancessuchastheshapeorprofilesrequiredforantennaeinspacebasedradar
applications. It also became clear that these systems could benefit from advanced
activecontroltechniquesfordampingaugmentationandperformanceenhancement.
In the ensuing years a number of test platforms were constructed to allow
researcherstogainfirsthandexperimentalknowledgeofthisnewclassofstructures.
ProminentamongthesewastheNASAHoopColumnAntenna;aLangleyResearch
Centerconceiveddesignforincreasedsensitivitytogroundorspace-basedsignals.
Theantennaconsistedofadeployablecentralcolumnanda15-mhoop,stiffened
bycableintoastructurewithahightolerancesurfaceandoffsetfeedlocation.The
surfacewasbeenconfiguredtohavefouroffsetparabolicapertures,eachabout6m
indiameter,andmadeofgoldplatedmolybdenumwiremesh.Vibrationanalysisof
thisstructureyieldedmodaldensitiesthathadnotroutinelybeenencounteredbefore
revealing 70 significant modes over a bandwidth of just 4.1–6.2 Hz. Conventional
control synthesis techniques often lead to large order state space models and con-
trollers, ill-conditioned matrices that required inversion and sometimes needed to
accountforspatiallydistributednonlinearities.Traditionallyengineersmodeledand
designedsuchsystemsusinglinear,timeinvariant,lumpedparametermethodsand
techniques for control. This yielded a system of low order ordinary differential
equations(typicallywithconstantcoefficients)thatcouldbereadilyanalyzedusing
moderncomputationalandlinearalgebratools.Whenappliedtolargeordersystems
newissuesofmodelreduction,stability,physicalrealizabilityetc.begantosurface.
This suggested to me at the time that these structures represented true continu-
umsthatdisplayedbothtemporalandspatialdynamicsandshouldbemodeledand
controlledusingtechniquesappropriateforDistributedParameterSystems(DPS).
DistributedParameterSystemsmayinvolveparametersthataretimevaryingand
distributedovercertainspatialdomains.Thedynamicbehaviorofthesesystemsis
governedbypartialdifferentialequations,integral,orintegro-differential,andocca-
sionallybymoregeneralfunctionalequations.Becauseofthefundamentalnatureof
suchsystems(allphysicalsystemshavespatialextent),andtheimportanceofappli-
cationsareas,thestudyofDPShashadtheattentionofmathematiciansandcontrol
theoristformanyyears.Thedichotomywasthatwhilemuchhadbeendevelopedin
AnHistoricalPrologueandPreface ix
thisareabywayofmathematics,verylittleifanycontrolsystemswerebuiltbecause
theresultingdesignscouldnotbephysicallyrealized.
ItwasatthistimethatIbecameawareoftheimportantworkthatwasbeingdone
byR.L.ForwardandC.J.Swigertinso-called“electronicdamping”duringthelate
seventies and early eighties. These were two Air Force researchers that had been
experimentingwithusingleadzirconiumtitanate(PZT)todampopticalstructures
with good results. Their approach of using such exotic materials as actuators for
activevibrationcontrolwasquiteunconventionalandnovel.Thesematerialswere
lightweightandusedverylittlepowerandhenceappearedtobeidealforapplication
tolargespacestructures.Unlikeconventionalactuationdeviceswhichappliedcon-
trolauthorityatasinglepointinspace;itappearedthatthesematerialsrepresented
actuatorsthatappliedcontrolforceswhichweredistributedoverspaceandtimeand
hencewerecharacteristicofDPS.ThemergingofthisclassofactuatorswithDPS
“plants” for control seemed a natural undertaking. A single actuator could cover
anentirestructureandprovidearelativelylowordercontrollerthatwasphysically
realizable.In1985ThomasBailey,aMastersThesisstudentofmineatMIT,built
andtestedsuchasystem.Thestructureconsistedofasteelcantileveredbeamwith
tip mass, covered by a thin sheet of polyvinylidene fluoride (PVDF) as an active
damper for vibration control. Both the structure and the actuator were modeled as
DPSandenergybasedcontroltechniques(Lyapunaov’s2ndmethod)wereusedto
synthesizean“electronicdamper”.Theresultwasademonstrationofatrulyadap-
tive structure which could significantly increase its damping when subjected to an
outsidedisturbance.
TheexperimentreceivedmuchinterestfromtheDPScommunityandProfessor
H.T.BanksofBrownUniversitywhowaswellversedintheissuesofDPScontrol
subsequentlycontactedmetodiscusstheimplicationsofmyexperiment.Thisbegan
an odyssey of lectures and seminars around the country, encouraged by Professor
Banks,toinitiatecollaborationsinthisarea.TheseincludedavisittotheInstitutefor
ComputerApplicationsinScienceandEngineering(ICASE),theAirForceRocket
Propulsion Laboratory (RPL) and presentations to the International Federation of
Automatic control (IFAC). These lectures and seminars help shaped my ideas and
thematerialpresentedinthistextbook.IamparticularlygratefultoProfessorsJohn
Brown, Gary Rosen and Steve Gibson for the insights that they provided into the
natureofcontinuumsystems.
Astimeprogressedseveralnotableexperimentaltestplatformsbecameavailable.
Dr. Alok Das for example established the RPL Experiment, which was a flexible
satellitetestbedlocatedinmylaboratoryattheCharlesStarkDraperLaboratory.
Dr.Jer-nanJuangestablishedacantileveredbeamtestbedattheNASALangley
ResearchCenterandhisworkandcontributionstherewerehighlyregarded.These
platforms and others help move the development of hardware implementation
rapidlyalong.Dr.FrancisMoonandhisthenstudentC.K.Leepioneeredtheappli-
cationofPVDFsensorstoboth1-Dand2-Dstructures.Dr.EdwardCrawleyandhis
studentsproducedseminalpapersontheuseofpiezocrystalsforthecontrolofLSS.
Professor Andrew Von Flowtow developed a novel means of resistor shunting of
piezoelectriccrystalstoproduceanelegantsolutiontotheactivedampingofflexible
x AnHistoricalPrologueandPreface
beams. Professor Amr Baz pioneered the use of active materials to develop active
constrainedlayer(ACLAD)dampingsolutionsandProfessorAlisonFlataubroad-
enedtheapplicationofelectrostrictivematerialstothisclassofproblems.Professor
ChrisFullerextendedtheapplicationofactivematerialstocontrolsoundradiation
fromvibratingstructures.Dr.DanInmandevelopedtechniquesforsensingandactu-
ationusingasingletransducer.Thereweremanystrongcontributionsbyothersto
thestateofapplication;toomanytoduejusticehere,butthoselistedhadapersonal
andsignificantimpactonmythinkingandworkpresentedhere.In2001Drs.Alok
Das and Ben K. Wada chronicled these contributions in an SPIE Milestone Series
ofSelectedPapersonSmartStructuresforSpacecraft.Thesepapersformthebasis
forwhathasnowbecomesimplythefieldofSmartStructures.
This was also a time of much activity in the development of Modern Robust
Control Theory and major developments were taking place in the design, synthe-
sisandrealizationoftemporalfiltersforthecontrolofLSS.Littleworkwasbeing
donehowevertowardastructureddesignoftheassociatedspatialfiltersneededfor
thecontrolofsuchplants.Issuessuchassensorandactuatorplacementwerebeing
addressed on an ad hoc basis, treated separately from overall control system syn-
thesis.Inactual practiceasignificant amount ofinformationisneeded todescribe
large scale systems. Traditional State Space approaches lead to the need for large
numbersofsensorsandactuatorstoidentifyandcontrolsuchstructures.
Thespatiallydistributed/continuumnatureofvibratorystructuresmakesitdiffi-
cult to apply modern lumped parameter control philosophy and techniques. While
there exists a substantial amount of technical literature on the control of DPS,
there are still relatively few applications or practical implementations of the the-
ory. Modal representations are commonly employed to succinctly approximate a
structuresbehavior.Thisrepresentationisofcoursecompletewhenalltermsofthe
expansionareincluded.Dr.MarkBalasinaseriesofseminalpapersdemonstrated
the practical limitations of the then current techniques and the unique challenge
thatsuchstructuresposedtothecontrolsengineer.Oftenforpracticalimplementa-
tion when one must truncate the modal expansion, it is then difficult to determine
the number of modes required to accurately model the structure, and to reconcile
thelocationofsensorsandactuatorsandtoaddressoverallsystemstabilityissues.
Computational limitations can also necessitate the need for truncation and model
orderreduction.Reduced-ordermodelshavebeenshowntosufferfromcontroland
observation spillover effects. Control and observation spillover can cause closed
loopinstabilityforevenasimpleflexiblebeamproblemthatisotherwiseopenloop
stable.
The work presented in this textbook addresses the issue of the design and
implementation of distributed parameter control schemes which exploit both
spatially distributed sensing and actuation through the use of modern smart
material technology. The merging of DPS with distributed parameter transduc-
ers leads to simple, realizable control system designs. It is hoped that this text
will provide a significant reference for practicing professionals, students and
researchers in the area of transducer design using smart materials for smart
structures.
AnHistoricalPrologueandPreface xi
FinallyIwouldberemissifIdidn’tacknowledgethecontributionsofmymany
studentsovertheyearsthathavecontributedtothedevelopment ofthetechniques
presented in this book. Thomas Bailey developed the basic tenants for using spa-
tially distributed actuation and energy based strategies for structural control. John
PlumpappliedthesetechniquessuccessfullytotheRPLstructureandlaterdefined
the concept of active constrained layer damping. Shawn Burke developed a uni-
fied approach to structural control using all of my previous students’ works and
his extensive background in the field of acoustics and control. Jeannie Sullivan
extendedourknowledgeoftheuseofspatiallydistributedactivematerialsforcon-
trolwithapplicationstotwodimensionalstructures.Thereareofcoursemanymore
to numerous to list here including my most recent students in the University of
Maryland’s Morpheus Laboratory who helped with the editing and problems sets
giveninthistext.Muchoftheclerical,administrativeandeditorialworkdonehereis
dueinlargeparttothededicationofLauriePostlewait,MollieBuechelandCarolyn
Sager. Finally I dedicate this book in its entirety to my lifelong role models Lillie
EcholsandJamesEdwardHubbardSr.Mysincereappreciationalsogoesouttomy
mentorsandtechnicaladvisorsovertheyearsfromM.I.T.,Drs.StephenH.Crandall,
DonaldC.Fraser,WesleyL.Harris,HankPaynterandHerbertH.Richardson.
The graphic support of the National Institute of Aerospace and Mr. Rene H.
Penziaisgratefullyacknowledged.
Contents
1 SmartStructureSystems . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 SmartStructureArchitectureandPerformance . . . . . . . . . . . 1
1.3 SmartMaterialTransducerConsiderations . . . . . . . . . . . . . 5
1.4 ContinuumRepresentationofSmartStructures . . . . . . . . . . 9
1.5 TimeDomainRepresentationofSmartStructureModels . . . . . 15
1.6 OrganizationoftheBook . . . . . . . . . . . . . . . . . . . . . . 18
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 SpatialShadingofDistributedTransducers . . . . . . . . . . . . . . 25
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 SpatialShadingofDistributedTransducers. . . . . . . . . . . . . 26
2.2.1 DesignbyExample:ACenterofPressureSensor . . . . . 26
2.2.2 ApproximatingShadedApertures. . . . . . . . . . . . . . 29
2.3 Analytical Modeling of Spatial Shading Functions
forDistributedTransducers . . . . . . . . . . . . . . . . . . . . . 34
2.3.1 A Compact Analytical Representation of
DistributedTransducers . . . . . . . . . . . . . . . . . . . 35
2.3.2 TwoDimensionalRepresentationofDistributed
TransducerswithNearlyArbitrarySpatialShading . . . . 45
2.4 ApplicationtoTwo-DimensionalShadingUsingSkewAngle . . . 51
2.4.1 Applications Including Finite Skew Angle of
MaterialAxes . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 ActiveVibrationControlwithSpatiallyShadedDistributed
Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 ControlSystemSynthesisBasedontheLyapunovDirectMethod . 70
3.3 ControlSystemSynthesisforBeams . . . . . . . . . . . . . . . . 71
3.3.1 CollocatedDistributedTransducersandLyapunovControl 74
3.3.2 PerformanceLimitationsofControlDesignswith
ShadedDistributions . . . . . . . . . . . . . . . . . . . . 76
xiii
