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Control-Oriented Analysis of Aerothermoelastic Effects for a Hypersonic Vehicle

Permanent Link: http://ufdc.ufl.edu/UFE0024107/00001

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Title: Control-Oriented Analysis of Aerothermoelastic Effects for a Hypersonic Vehicle
Physical Description: 1 online resource (60 p.)
Language: english
Creator: Bhat, Sanketh
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: aerothermoelasticity, control, hypersonic, parameter, varying
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Hypersonic flight is seen as a feasible solution to make space travel faster, safer and more affordable. The design of the Air-breathing hypersonic vehicle is such that there is coupling between the structure and the propulsion system. Therefore, the aerodynamic, propulsion and the structural effects must be accounted to effectively model the vehicle. The vibrations from the structure affect the performance of the vehicle. Hence, vibration attenuation is a critical requirement for hypersonic vehicles. The problems of vibration are compounded by variations in heating during flight. Structural variations resulting from the tremendous heating incurred during hypersonic fight is mitigated by a thermal protection system (TPS); however, such mitigation is accompanied by an increase in weight that can be prohibitive. The actual design of a thermal protection system can be chosen to vary the level of heating reduction, and associated weight, across the structure. Our study examined the design of a Linear Parameter Varying controller for an hypersonic vehicle and describes the process of control-oriented analysis to suggest a better 'Thermal Protection System' for the vehicle. A Linear Parameter Varying control architecture was used that damps any thermal effects for a range of temperature profiles. Various designs are considered for a representative model to show the large variation in flight dynamics. Simulation results indicate that the proposed methodology may constitute a feasible approach toward the development of a robust Linear Parameter Varying controller to satisfactorily address the issue of temperature effects on the dynamics of the vehicle. From the above closed-loop design analysis, important information regarding the open-loop dynamics can be obtained. We then considered how such designs and resulting thermal gradients influence the ability to achieve closed-loop performance. The resulting closed-loop performance is characterized as a function of the induced thermal gradients to indicate the optimality of the design. It is also shown that the introduction of control synthesis merely adds a linear dependency onto a nonlinear dependency which does not overly increase the computational challenge.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sanketh Bhat.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Lind, Richard C.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0024107:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024107/00001

Material Information

Title: Control-Oriented Analysis of Aerothermoelastic Effects for a Hypersonic Vehicle
Physical Description: 1 online resource (60 p.)
Language: english
Creator: Bhat, Sanketh
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: aerothermoelasticity, control, hypersonic, parameter, varying
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Hypersonic flight is seen as a feasible solution to make space travel faster, safer and more affordable. The design of the Air-breathing hypersonic vehicle is such that there is coupling between the structure and the propulsion system. Therefore, the aerodynamic, propulsion and the structural effects must be accounted to effectively model the vehicle. The vibrations from the structure affect the performance of the vehicle. Hence, vibration attenuation is a critical requirement for hypersonic vehicles. The problems of vibration are compounded by variations in heating during flight. Structural variations resulting from the tremendous heating incurred during hypersonic fight is mitigated by a thermal protection system (TPS); however, such mitigation is accompanied by an increase in weight that can be prohibitive. The actual design of a thermal protection system can be chosen to vary the level of heating reduction, and associated weight, across the structure. Our study examined the design of a Linear Parameter Varying controller for an hypersonic vehicle and describes the process of control-oriented analysis to suggest a better 'Thermal Protection System' for the vehicle. A Linear Parameter Varying control architecture was used that damps any thermal effects for a range of temperature profiles. Various designs are considered for a representative model to show the large variation in flight dynamics. Simulation results indicate that the proposed methodology may constitute a feasible approach toward the development of a robust Linear Parameter Varying controller to satisfactorily address the issue of temperature effects on the dynamics of the vehicle. From the above closed-loop design analysis, important information regarding the open-loop dynamics can be obtained. We then considered how such designs and resulting thermal gradients influence the ability to achieve closed-loop performance. The resulting closed-loop performance is characterized as a function of the induced thermal gradients to indicate the optimality of the design. It is also shown that the introduction of control synthesis merely adds a linear dependency onto a nonlinear dependency which does not overly increase the computational challenge.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sanketh Bhat.
Thesis: Thesis (M.S.)--University of Florida, 2008.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0024107:00001


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saranyetrayambakegaurinarayaninamostute" 3

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IwouldliketoacknowledgethemanagementatNASAbecausenoneofthisworkwouldhavebeenpossiblewithouttheirsupport.IwouldalsoliketoextendmysincerethankstoDr.WarrenDixonandDr.AnilRaoforagreeingtobeonmycommittee.HeartfeltthanksgoouttotheseniorresearchersattheFlightControlLab,includingDr.JoeKehoe,Dr.RyanCausey,Dr.MujahidAbdulrahim,Dr.AdamWatkins,andDr.SeanRegisfordfortheirvaluableguidanceandsupport.ThiseortwouldnothavebeenpossiblewithoutthehelpofmyfellowresearchersattheFlightControlLab,includingBrianRoberts,RobertLove,BaronJohnson,RyanHurley,DongTranandofcourse,mydeskbuddy,DanielTexGrant,forlightingupmanyadayswithhishumor.IwouldliketoexpressmysincereanddeepgratitudetoDr.RickLindforhisguidance,support,andmostimportantlyforgivingmethisopportunitytoprovemyself.MyparentsandmybrotherdeservemuchcreditformakingmewhatIamandshowingmetherightwayalltheseyears.Ithankallwhohelpedmegetthisfar,butwhosenamesIinadvertentlymissedout. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 1.1Motivation .................................... 12 1.2Overview .................................... 13 2CONTROLDESIGN ................................. 17 2.1Vehicle ...................................... 17 2.2Aerothermoelasticity .............................. 18 2.3LinearParameterVarying ........................... 20 2.3.1Framework ................................ 20 2.3.2Synthesis ................................. 22 3CONTROL-ORIENTEDDESIGN ......................... 24 3.1Closed-LoopDesignSpace ........................... 24 3.2Feasibility-BasedOptimization ......................... 25 4EXAMPLE ...................................... 28 4.1Objective .................................... 28 4.2TheLPVContolDesign ............................ 29 4.2.1ControlIssues .............................. 29 4.2.2ModelingThermalProle ........................ 30 4.2.3FlightDynamics ............................. 32 4.2.4ControlDesign .............................. 34 4.2.4.1Open-loopsynthesis ..................... 36 4.2.4.2Closed-loopmodeling ..................... 39 4.2.5Results .................................. 39 4.3Control-OrientedAnalysis ........................... 44 4.3.1DesignSpace ............................... 44 4.3.2Control-OrientedDesign ........................ 46 4.3.3Analysis ................................. 49 4.3.4Sensitivity ................................ 52 5CONCLUSION .................................... 54 5

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....................................... 55 BIOGRAPHICALSKETCH ................................ 60 6

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Table page 4-1Naturalfrequenciesforthelineartemperatureproles ............... 32 4-2H1normsforsystemwithH1andLPVcontroller ................ 40 4-3Temperaturegradients ................................ 45 4-4Coecientsoftheopen-loopdynamicswhichvarywithtemperature ....... 51 7

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Figure page 1-1Air-breathinghypersonicvehicle ........................... 12 1-2TheX-43modelduringGVT ............................ 14 2-1Modeshapeswiththermalvariation ........................ 19 3-1Closed-loopblockdiagram .............................. 24 4-1Dierenttemperatureproles ............................ 31 4-2TransferFunctionfrompitchratetoelevatordeection .............. 32 4-3Modeshapesforthevehicle ............................. 33 4-4Variationinthecoecientsofthestatematrices .................. 34 4-5H1normforthedierenttemperatureproles .................. 35 4-6Open-Loopnormparameterizedaroundopen-loopdynamics ........... 36 4-7Synthesisblockdiagram ............................... 37 4-8Transferfunctionforthenominalmodelandtargetmodel ............ 38 4-9Closed-loopdesign .................................. 40 4-10Normsoftheclosed-loopsystems .......................... 41 4-11Comparisonofthetransferfunctionsforthedierentsystems .......... 42 4-12Timeresponsefortheopen-loopandclosed-loopsystemswiththepointandLPVcontrollers ................................... 43 4-13Pole-zeromapoftheclosed-loopsystemwithH1andLPVcontrollers ..... 43 4-14Thermalprolescomprisingthedesignspace .................... 44 4-15Open-loopstabilitycoecientasafunctionofthedesignspace ......... 45 4-16Open-loopcontrolcoecientasafunctionofthedesignspace .......... 46 4-17Optimalcontrollerfrompitchratetoelevatorandcanarddeection ....... 47 4-18Actualanddesiredtransferfunction ........................ 48 4-19Inputelevatordeection ............................... 48 4-20Actualanddesiredclosed-loopresponse ...................... 49 4-21Closed-loopnormparametrizedaroundthedesignspace ............. 50 8

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........... 51 4-23Thermalprolesassociatedwithsimilarly-valuedlocalminima .......... 52 4-24Closed-loopperformanceforeachthermalprole ................. 53 9

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Hypersonicightisseenasafeasiblesolutiontomakespacetravelfaster,saferandmoreaordable.ThedesignoftheAir-breathinghypersonicvehicleissuchthatthereiscouplingbetweenthestructureandthepropulsionsystem.Therefore,theaerodynamic,propulsionandthestructuraleectsmustbeaccountedtoeectivelymodelthevehicle.Thevibrationsfromthestructureaecttheperformanceofthevehicle.Hence,vibrationattenuationisacriticalrequirementforhypersonicvehicles.Theproblemsofvibrationarecompoundedbyvariationsinheatingduringight.Structuralvariationsresultingfromthetremendousheatingincurredduringhypersonicghtismitigatedbyathermalprotectionsystem(TPS);however,suchmitigationisaccompaniedbyanincreaseinweightthatcanbeprohibitive.Theactualdesignofathermalprotectionsystemcanbechosentovarythelevelofheatingreduction,andassociatedweight,acrossthestructure. OurstudyexaminedthedesignofaLinearParameterVaryingcontrollerforanhypersonicvehicleanddescribestheprocessofcontrol-orientedanalysistosuggestabetter'ThermalProtectionSystem'forthevehicle.ALinearParameterVaryingcontrolarchitecturewasusedthatdampsanythermaleectsforarangeoftemperatureproles.Variousdesignsareconsideredforarepresentativemodeltoshowthelargevariationinightdynamics.SimulationresultsindicatethattheproposedmethodologymayconstituteafeasibleapproachtowardthedevelopmentofarobustLinearParameterVaryingcontrollertosatisfactorilyaddresstheissueoftemperatureeectsonthedynamicsofthe 10

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1-1 ),hasatightlyintegratedairframeandSCRAMjetpropulsionsystem( 1 ). Figure1-1. Air-breathinghypersonicvehicle Thedesignofhypersonicvehiclesismaturingwithrespecttotheaeropropulsiveinteractionsofthefuselageandengine;however,theaerothermoelasticcharacteristicsmustalsobeaddressed.Vibrationattenuationisacriticalrequirementforthesevehicles 12

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Anovelapproachtocontrolthehypersonicvehicle,namelyamulti-looparchitectureisformulatedthatcontainscompensatorsforvibrationsuppression,maneuveringandenginecontrol.Thisarchitecturedirectlymatchesamodelingschemefortheopen-loopdynamicsthatcouplesaerodynamicsandstructuraldynamicswithenginedynamics.Theinner-loopcontrollerisusedtoactivelyaugmentdampingofthestructuralmodes.Theouter-loopcontrolleristhenusedtoachieverigid-bodyperformancespecications.Finally,anenginecontrolleroperatescontinuouslytoguaranteeproperpropulsiondespitevariationsintheightdynamics.Also,thearchitectureincludesbothgain-scheduledelementsandadaptiveelements.Thegain-scheduledelementsrepresentpre-ightdesignsusinghigh-delitymodelswhereastheadaptiveelementsareusedtocancelanyresidualerrors.Essentially,theadaptiveelementsonlyaectthesystemwhenaerothermoelasticdynamicsvarybeyondtheoreticalrangesandthegain-scheduledcontrollerisunabletoachievethedesiredperformanceofeithertheightpathorenginepropulsion. Thisstudy,howeverrestrictsconsiderationonlytotheinner-loopLPVcontroller.Thisanalysisalsointroducesacontrol-orienteddesignforhypersonicvehiclesthatdirectlyconsidersmissioncapability.Inthiscase,thedesignseekstochooseathermalprotectionsystem(TPS)andassociatedcontrollerthatmaximizevibrationattenuation. 13

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1-2 ),demonstratedthesuitabilityofaSCRAMjetengineinthiscongurationforhypersonicight. Figure1-2. TheX-43modelduringGVT Therehavebeenseveralpapersinliteraturethathavediscussedchallengesassociatedwiththedynamicsandcontrolofhypersonicvehicles.AdetailedanalyticalmodelofthelongitudinaldynamicswasundertakenbyChavezandSchmidt( 2 ).AslightlydierentapproachtodevelopthemodelwasundertakenbyBolenderandDoman( 1 ; 3 ; 4 )whichisfurtherdevelopedbythesameauthors( 5 ; 6 ).Anothermodelofthehypersonicvehiclewasdevelopedusingpistontheory( 7 ).Usingtheabovemodelsasaxeddesign,severalapproachesforcontrolhavebeenconsideredincludingH1( 8 ),synthesis( 9 )andLinearParameterVarying(LPV)control( 10 ).Additionalworkhasevenconsideredsensorplacement( 11 { 15 ).ACFDapproachtodevelopamodelofhypersonicvehicleispresentedin( 16 ).Variouscontrolstrategiesforthehypersonicvehiclelikeadaptivecontrol( 17 { 20 )andotherlinearcontroltheories( 21 { 23 )havebeendiscussedinliterature. Thedynamicsofair-breathinghypersonicvehiclesincludecouplingsbetweentheengineandightdynamics,inadditiontotheinteractionsbetweenexibleandrigid 14

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24 )discussesthedevelopmentofacontrol-orientedmodelinclosedformbyreplacingcomplexforceandmomentfunctionswithcurve-ttedapproximations,neglectingcertainweakcouplings,andneglectingslowerportionsofthesystemdynamics.Theadvantageofthisapproachisthatthelinearcontrolstrategiesneednotbeused,butthemodelingprocessistime-consuming. Forthecontroldesign,thereareseveralissuesthatmustbeaddressed.Thecontrollermustaccountforstronglycoupledaerodynamics-propulsiondynamicsandactivelysuppressmodalvibrations.Aerothermoelasticeectscannotbeignoredinahypersonicightandmustbecompensated.Thechoiceofcontrolarchitectureiscloselyrelatedtothepreviousissues.Generatingasinglestate-spacecontrollerthatprovidesstabilityandperformanceseemssomewhatlimitedbecauseitmaybeadvantageoustolinkcertainpartsofthecontrollertocertaindynamicsofthemodel.Also,thetheoriesinvolvingH1andsynthesisonlyconsideredasingleightcondition. Investigationsintoaerothermoelasticdesignarenotasmaturebecauseofthechallengesassociatedwithsimultaneousoptimizationofboththestructureandthecontroller.Manypreviouseortsintothegeneralproblemofstructure-controldesignhavenoteditsinherentnonlinearitiesthatcanbesolvedusingavarietyofformulationsincludinglinearmatrixinequalitiesandbi-linearmatrixinequalities( 25 { 28 ). Thecontrol-orienteddesignisoptimizedusingaparametrizedsolutiontoaRiccatiequation.Systemdesignisintractablewhentryingtooptimizebothopen-loopdynamicsandfeedbackcompensatorsimultaneously;alternatively,systemdesignisactuallymanageablewhentryingtooptimizetheopen-loopdynamicswithrespecttoafeasibilityconditionthatguaranteestheexistenceofafeedbackcompensator.Inthisway,theactualcontrollerdoesnotneedtobecomputedbutmerelyanopen-loopdesignforwhichacontrollerisguaranteedtoexistwillbedesigned. Thisconceptofcontrol-orienteddesignrepresentsasignicantadvancementtothestate-of-the-artforthecommunityandisparticularlyadvantageousfornext-generation 15

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5 )usedinourstudy,islimitedtolongitudinalmotionandisdevelopedwithelevenstates,fourinputsandelevenoutputs.Thestatesincludeverigidbodystates,velocity(V),angleofattack(),pitchrate(Q),altitude(h),pitchattitude()andsixelasticstatesfortherstthreefuselagebendingmodes(i;_i).Theinputsincludeelevatordeection(e),canarddeection(c),diuserarearatio(Ad)andfuelowratio().Themodelhasfullstatefeedbacki.e.alltheelevenstatesareusedasfeedbacktothecontroller.Aerodynamic,inertial,propulsive,andelasticforceswereusedtoderivetheequationsofmotionforthehypersonicvehicle( 1 ). Toincorporatestructuraldynamicsandaerothermoelasticeectsinthehypersonicvehicledynamicmodel,anassumedmodesmodelisconsideredforthelongitudinaldynamics( 5 )as, _V=Tcos()D mV+Q+g Vcos()_=Q_Q=M Iyyi=2&i!i_i!2ii+Ni;i=1;2;3m2Rdenotesthevehiclemass,Iyy2Risthemomentofinertia,g2Ristheaccelerationduetogravity,T(t)2Rdenotesthrust,D(t)2Rdenotesdrag,L(t)2Rislift,i;!i2Rarethedampingfactorandnaturalfrequencyoftheithexiblemode,respectively,andNi2Rdenotegeneralizedelasticforces.Theequationsthatdenetheaerodynamicandgeneralizedmomentsandforcesarelengthyandareomittedforbrevity.Detailsofthemomentsandforcesareprovidedin( 1 ).Becauseofaerothermolelastic 17

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29 { 37 ). Ithasbeenshowninliteraturethattheexactcomputationofthethermaleectsontheaerodynamicsofanaerospacecraftinthehypersonicregimeisdicult( 38 ).Hence,thisstudyconcentratesonlyontheeectsofaerothermoelasticity.Thewaythisisdoneisbynotingthevariationsinthestructuralpropertiesasafunctionoftemperature. Thehypersonicvehicleissubjectedtoextremetemperaturesandheatingduringthehypersonicightregime.Hence,thestructureneedstobeprotectedbyaThermalProtectionSystem(TPS).Tostudythetemperatureeects,varioustemperaturegradientsalongthefuselageofthevehicleareintroducedintothemodelsimulatingthetemperaturesattainedbythevehicleinight.KnowingthematerialpropertiessuchasYoung'smodulus(E)( 39 )asafunctionoftemperature,theeectsonthestructuralpropertieslikethemodeshapes,naturalfrequenciesanddampingareanalyzed.Forexample,thevariationsintherepresentativemodeshapesforabeamatdierenttemperaturescanbeanalyzedtostudytheeectoftemperatureonthedynamics(Figure 2-1 ).Inthiscase,thebendingmodeisextractedtoindicatechangessuchasnodelocation,anti-nodelocation,andmagnitudeofoscillation.Thisbehaviorisincorporatedintoightmodelsthroughaerothermoelasticdynamics. Ithasbeenimpressedbeforethattorepresentthedynamicsofthevehicleaccurately,themodelmustbeformulatedtoincludetheeectsofstructuralexibilityinaddition 18

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Modeshapeswiththermalvariation tothedynamicsoftherigid-body.Thissectiondescribestheformulationofastate-spacemodelthatincludesrigid-bodyandstructuraldynamics.Themodelstructuredevelopedhereisbasedontheworkof( 40 { 42 ).Thegeneralformofthestate-spacemodelis( 43 ):_x=Ax+Bu Withthestructuraleectsincluded,AandBareoftheform,A=266666666664RigidBodyTermsAeroelasticCouplingTermsRigidBodyCouplingTermsStructuralFlexibilityTerms377777777775B=264RigidBodyControlStructuralModeControl375 19

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TheaerothermoelasticeectsduringatypicalightprolewerestudiedfortheNationalAerospacePlane(NASP)( 44 ).Thisstudynotedthatthesurfacetemperaturescouldrangefrom0oFtonearly5000oFatcertainpointsandresultinlargesurfacegradients.Consequently,thenaturalfrequenciesanddampingofthestructuralmodescanvarysignicantlybyupto30%.Theseeectswillbeusedasrepresentativeeectsthatmaybeencounteredforthegeneralclassofvehiclesconsideredinthisstudy. Theaerothermoelasticeectswerenotedtocauseadecreaseinnaturalfrequencyanddampingofthestructuralmodes.Thiseectisincorporatedbyformulatingthestatematrixasananefunctionoftemperature.Therangeoftemperaturesconsideredforthismodelischosenas2(0oF;1000oF)tomatchtheoperatingrangeofTitanium(Ti). 2.3.1Framework 45 ).ThethreemainclassesofaerospacesystemsusingtheconceptofgainschedulingareLinearTimeInvariant(LTI),LinearTimeVarying(LTV)andLinearParameterVarying(LPV). GainSchedulingcanbebrokendownintothreesteps( 46 ).Firstly,separatetheoperatingrangeintosubspacesandcreateparameterizedmodelforeachsubspace.Then,createcontrollersforeachofthemodelsandthendevelopaschedulingschemebylinearlyinterpolatingbetweentheseregionalcontrollersforthelocalsubspaces.However,thereisnoguaranteeofstabilityandrobustnesswithrespecttouncertaintiesinthedynamicsandthereisapossibilityofskippingbehaviorduringswitchbetweencontrollers.Also,developingthelinearinterpolationlawcanberigorousandtimeconsuming. 20

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RobustidenticationtechniquesforspecialclassesofLPVsystemsarepresentedin( 47 ; 48 ).Amethodofidentifyingmulti-variableLPVstatespacesystemsthatarebasedonlocalparameterizationandgradientsearchintheresultingparameterspaceispresentedin( 49 ).Twoidenticationmethodswereproposedin( 50 )foraclassofmulti-inputmulti-outputdiscretetimelinearparametervaryingsystems.Bothmethodsarebasedonthesubspacestatespacemethod,whichwassuggestedby( 51 ).( 52 )suggeststwomethodsformodelingaircraftdynamics,namely,theboundingboxandsmallhullapproach.AnotherapproachtosolvetheLPVsystemswhicharecharacterizedbyasetoflinearmatrixinequalities(LMI)ispresentedin( 53 ). AtypicalcaseofalinearparametervaryingplantP(:;),whosedynamicalequationsdependonphysicalcoecientsthatvaryduringoperation,hastheform where isatimevaryingvectorofphysicalparameters,forexample,velocity,angleofattack,temperature;A,B,C,Dareanefunctionsof(t),xisthestatevector,yisthemeasuredoutput,eistheregulatedoutputorerrors,distheexogenousdisturbances,anduistheregulatedinput. 21

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46 ) rangesinamatrixpolytopewithverticesS(i).Givenaconvexdecomposition, ofoverthecornersoftheparameterregion,thesystemmatrixisgivenby Thissuggestsseekingaparameterdependentcontrollerswithequations andwithavertexpropertywhereagivenconvexdecomposition(t)=nPi=Niiofthecurrentparametervalue(t).Thecontrollerstate-spacematricesattheoperatingpoint(t)areobtainedbyconvexinterpolationoftheLTIvertexcontrollers Thisyieldsasmoothschedulingschemeofthecontrollermatricesbytheparametermeasurements(t). TherearemanytechniquestomaketheLPVcontrolleroncethesystemhasbeenputintheLPVframework.Thethreemainsynthesistechniquesaresynthesisdesign 22

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54 ),LinearQuadraticGaussian(LQG)design( 55 ),andH1( 56 ).Inthisstudy,theH1techniquehasbeenusedandthecontrolproblemcanbeformulatedas'linearmatrixinequalities'(LMI),whichasshownin( 57 )isaconvexoptimizationproblem.( 58 )hasanexampleofcreatingaconvexoptimizationproblemwithLMIexpressionsfortheuseofndinganLPVcontrollerfortheattitudecontrolofanX-33. 23

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3-1 ),asafeedbackrelationshipbetweentheopen-loopplantandacontroller.Inthediagram,disthevectorofexogenousinputsordisturbancesincludingreferencecommandsandeisthevectoroferrorstobeminimized.Therehavebeenon-goingstudiestodevelopalgorithmsforoptimizingthedesignspacefor K ed Closed-loopblockdiagram aerospacesystems.Mostoptimizationdesignsstartwiththexeddesignspace.But,sincethefeasibleregioninthexedspaceisverysmallandtheprobabilityofndingapropersolutionislow,( 60 )proposesaprobabilisticapproachforthefeasibilityimprovementofthedesignspace.( 61 )presentsanovelhybridoptimizationmethodtoecientlyndtheglobaloptimalofcomplex,highlymultimodalsystems.( 62 )proposesamethodologyfortheanalysisanddesignofsystemssubjecttoparametricuncertaintyinwhichdesignrequirementsarespeciedviahardinequalityconstraints.Hardconstraintsarethosethatmustbesatisedforallparameterrealizationswithinagivenuncertaintymodel. Adesignspaceisformulatedusingtheparametersthataecttheopen-loopdynamics.Thisspace,denedasP,canincludeawidevarietyofparametersincludinggeometry, 24

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Anotherdesignspaceisformulatedthatcontainsthecompensatorelementsthatmaybevaried.Thisspace,denedasK,canincludeaspectsofthefeedbackcompensatorsuchasgains,lags,bandwidth,andadaptionratesalongwithsensorsandactuators.Anycontrolleristhusformulatedusingthevector,2K,fromwithinthedesignspace. Thesetofclosed-loopsystemsthatarepossiblecandidatesfortheoptimalcongurationcanberepresentedbyT.Thissetnotesthattheopen-loopplant,P(),dependsonthedesignspaceofPandthecompensator,K(),dependsonthedesignspaceofK.Finally,thesetofallclosed-loopsystemsTcanbedescribedasaLinearFractionalTransformation(LFT),T=fFl(P();K())2P;2Kg 25

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ThemetricfordesigncanbecastasanH1-normconditionontheclosed-loopsystem.Assuch,thedesignseekstondtheoptimalvaluesforboththeopen-loopdynamicsandanH1-normcontroller.Thecomputationforthatcontrollerisactuallysomewhatmatureusingastate-spacesolutionalthoughthejointcomputationofbothdynamicsandcontrollerisintractable. Theoptimaldesignactuallydoesnotneedtocomputeboththeopen-loopdynamicsandcontrollersimultaneously;instead,thedesigncansimplyndtheopen-loopdynamicsforwhichacontrollerexiststhatachievesthelowestH1-normvalue. Thesynthesisofcontrollersusingmoderntechniquesactuallyfollowsatwo-stepprocedure.Theinitialstepiteratesoverafeasibilitycheckthatindicatesifacontrollerexiststoachieveaparticularvalueofclosed-loopperformance.Thenalstepcomputesthegainforthefeedbackcompensatorthatachievestheoptimalclosed-loopperformance.Thistwo-stepprocedureisimplementedinprofessionalsoftwaresuchasMATLAB,becauseasetoffeasibilityconditionsissignicantlylesscomputationallyexpensivethanasetofsynthesisconditions. Theapproachforcontrol-orienteddesignisnowexpressedasminimizingtheclosed-loopnorm()withrespecttothedesignspacewhilemaintainingthefeasibilityconstraints,min2PX=X>0Y=Y>0

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whereisthespectralradius.Thisconstrainedoptimizationrequiresndingaminimumtoanonlinearfunction.TheoperatorsofXandY,iftheyexist,canbefoundforanyxedvalueofusingstandardalgorithms;however,theyarealmostcertaintohavenon-convexdependencieswhenconsideringall2P.Avarietyofnumericalapproachescanbeappliedtotheminimizationincludingbranchandbound,simulatedannealing,neuralnetworks,andsoon. Finally,theactualcontrollerthatachievestheoptimalclosed-loopsystemiscomputedusingthesolutions,XandY,totheseRiccatiequations.ThestandardsynthesisforH1-normcontrolisused. 27

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Thisexamplerepresentsasingleelementwithinalargermulti-looparchitecture( 38 ).Thefundamentalconceptusesapairofloopssuchthattheinner-loopcontrollerprovidesvibrationattenuationwhiletheouter-loopcontrollerprovidesmaneuvering.Suchaloopdecompositionrecognizesthatthermaleectsarepredominantlylimitedtothestructuraldynamicsrelatedtovibration.Theouter-loopcontrolleristhusdesignedwithoutconsiderationoftemperatureeectssincetheinner-loopcontrollerisassumedtoprovideadequatecompensation. AbaselinevehicleisadoptedfromanextensiveprogrambytheU.S.AirForceforareduced-ordermodel( 1 ; 3 { 7 ).Thismodelincludesvestatesfortherigid-bodyightdynamicsandanadditionalsixstatesassociatedwiththreeexible-bodystructuraldynamics.Themodelisparticularlyattractiveinthatitcontainsarigorousderivationoftheaerothermoelasticcouplingthatexplicitlyhighlightstheeectsofvibrationsonmissionperformance. Thisstudycanbeputintotwocategories.Therstpartinvolvesthedesignoftheinner-loopcontrollerusingLinearParameterVaryingtheory.Thisanalysisisdonebasicallyasa`proof-of-concept'.Inthesecondpart,control-orientedanalysisisperformedtogiveadesignwhichoptimizestheperformanceofthevehicle. 28

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4.2.1ControlIssues Severalcontrolissueswereidentiedforinvestigatinghypersonicightthroughtheatmospherethatmustbeinvestigated.Firstly,thecontrollermustactivelysuppressmodalvibrations.Theaerothermoelasticeectsmustbecompensated.Thisissueisgenerallynotconsideredfortraditionalaircraftbutisquiteimportantforhypersonicight.Thedegreeofheatingresultingfromhypersonicightathighdynamicpressurecanbetremendousandresultinchangingmaterialpropertiessuchasstiness.Thischangeisstinesscanhaveadramaticeectonclosed-looppropertiesbecausethecontrollermustaccountforthelowfrequencyfuselagebendingmodeandalsothechangestothosemodaldynamicsbecauseofaerothermoelasticeects. Secondly,thechoiceofacontrolarchitectureiscloselyrelatedtothepreviousissue.Generatingasinglestate-spacecontrollerthatprovidesstabilityandperformanceseemssomewhatlimitedbecauseitmaybeadvantageoustolinkcertainpartsofthecontrollertocertaindynamicsofthemodel.Inparticular,vibrationsuppressionisanextremely 29

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Alinearparameter-varying(LPV)synthesiscanbeusedtoformulatethecontroller.Theresultingcontrollerwillbeautomaticallygainscheduledovertemperaturesuchthatthegainsarealteredtoaccountforthethermalvariationsinnaturalfrequencyanddampingofthestructuralmodes.Also,theclosed-loopsystemcanbeguaranteedtosatisfystabilityandperformancemetricsassociatedwiththeuncertaintyoperators. Themodelusedtodesign,theinner-loopcontrollershouldcharacterizeperformancebyincreasingthedampingofthestructuralmodesoftheplants.Therefore,amodel-followingapproachisanacceptablesynthesismethod.Thisapproachwouldattempttomaketheinner-loopsystemsimilartoadesiredinner-loopsystemthathasacceptablemodaldamping.Weightingfunctionscanbeusedtoshapetheresultingcontrollersuchthatthereislittlegainsatlowandhighfrequencytoensuretheinner-loopcontrollerdoesnotadverselyinteractwiththeouter-loopcontroller. 30

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4 ; 6 ). Initially,fteentemperatureprolesintroducedintothemodel(Figure 4-1 ).Therstveprolesarelineari.ethetemperaturegradientlinearlydecreasesfromnosetotail,thenextvehavegradientslesserthanthelinearprolesandthelastvehavegradientsgreaterthanthelinearproles.Thefuselagehasbeendividedintonineequalsections. Figure4-1. Dierenttemperatureproles Thetransferfunctionsfrompitchratetoelevatordeectionforthedierenttemperatureprolesareplotted(Figure 4-2 ).Therstpeakrepresentstheunstablerigidbodymode.Itisobservedthatthereisavariationinboth,thedampingandthenaturalfrequenciesofthestructure.Itisexpectedthatthenaturalfrequencyshould 31

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4-1 ).However,themodeshapesshowverylittlechangewithtemperature(Figure 4-3 ).Theasymmetricnatureofthemodeshapeshowsthedependenceofthestructuralpropertiesonthetemperaturegradients. Figure4-2. TransferFunctionfrompitchratetoelevatordeection Table4-1. Naturalfrequenciesforthelineartemperatureproles Mode1(hottest)2345Reduction(%) 123.0123.5023.9024.3124.736.96249.8750.8951.7852.6253.546.85398.90100.95102.7104.4106.216.88 32

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Modeshapesforthevehicle _x(t)=A()x(t)+B()u(t)(4{1) showastrongdependenceontemperature. 4-4 ).ItcanbeseenA(7;6)decreaseswithadecreaseintemperature,butitisdiculttondastructurein 33

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Figure4-4. Variationinthecoecientsofthestatematrices AnominalH1controlleriscreatedtostabilizethevehicleforallthetemperatureproles,sothatthestructuraldynamicscontrollerwillnottrytoaltertherigid-body 34

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Togetaquickglimpseoftheopen-loopsystempropertieslikecontrollability,thetemperatureprolesarerelatedwiththedierentH1normsfortheopen-loopstablesystems(Figure 4-5 ).Itcanbeexpectedthatbetterperformanceisachievedwhenthevehicleiscoolerandthegradientofthetemperatureproledoeseecttheperformance.Thetrendshownbytheopen-loopnormseemstobesimilartothecoecientA(7;6).Toexploretherelationshipbetweentheopen-loopnormandA(7;6)abitmore,thenormisplottedasafunctionoftheopen-loopdynamiccoecient(Figure 4-6 ).Suchrelationshipsbetweentheperformancemetricandtheopen-loopdynamicswillbeexploredmoreinfutureanalysis. Figure4-5. 35

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Open-Loopnormparameterizedaroundopen-loopdynamics 4-7 ).Theopen-loopsystemhasanunstablerigid-bodymodeandformostcasesisnon-minimumphase. Asynthesismodel(Figure 4-7 )wasformulatedthatrelatestheopen-loopdynamicstoasetoferrorsanddisturbances.Theseerrorsarespecicallychosensuchthattheirsizeisdirectlyinversetotheclosed-loopperformanceforvibrationattenuation. Amodel-matchingapproachischosentospecifyadesiredlevelofvibrationattenuation.Assuch,atargetmodelisgivenasTinthesynthesismodelthatrepresentsdynamicswithappropriatedampingonthestructuralmode.Thetransferfunctionsareshownforboththenominalopen-loopdynamicsandthetargetdynamics(Figure 4-8 ). 36

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Synthesisblockdiagram Notethepeaknear0.04rad/sisassociatedwitharigid-bodyightmodewhilethepeaknear22rad/sisassociatedwiththestructuralmodethatshouldbeattenuated.Sotheobjectiveoftheinner-loopcontrolleristogettheactualresponseascloseaspossibletothetargetresponse,especiallyneartherstbendingmodewithoutalteringthelowfrequencydynamics. Thesystemhassixinputvectorsandveoutputvectors.Theinputvector,n2Risrandomnoisewhichaectsthesensormeasurements.Theinputvector2R4correspondstothefourinputstothesystemi.eelevatordeection,canarddeection,diuserarearatioandthefuelowratio.Theinputvector,u2R2isthecontrolcommandaectingtheactuators.Theoutputvector,ep2Ristheweightedmeasurementsofthepitchrateasmeasuredbythesensors.Theoutputvectormeasurementsarethesensormeasurementsusedasfeedbacktothecontroller. 37

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Transferfunctionforthenominalmodelandtargetmodel AfeedbackcompensatorisgivenasXinthesynthesismodel.Thiscompensatorisonlyincludedtostabilizetheopen-loopdynamics.Essentially,thecontrollerisbeingdesignedonlytoaugmentdampingofthestructuralmodewithoutintroducinganyvariationstothelow-frequencybehavior.TheH1-normsynthesisisrequiredtostabilizetheclosed-loopsystemsoXisincludedtoensuretheresultingcontrollerdoesnotaecttherigid-bodymodesthroughstabilization.Thenalmulti-looparchitecturewillintroduceanouter-loopcontrollertoreplaceXandprovidebothstabilityandperformancefortherigid-bodymaneuvering. Anerrorsignal,ep2R,isdenedtorepresentthetrackingperformance.Thissignalisaweighteddierencebetweentheactualpitchrateandthedesiredpitchrate.Theweighting,WP=s+100 38

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Resultsoftheopen-loopsynthesisisusedtocreatetheLPVcontroller,K(),usingtheLMIControlToolbox( 59 ).TodeterminehowwelltheLPVcontrollerwillwork,H1pointcontrollersaredevelopedforplantscorrespondingtothedierenttemperatureprolesandtheresultsarecomparedforthetwocontrollers. 4-9 ). Theclosed-loopsystemhasfourinputsandelevenoutputscorrespondingtotheelevenstates.Kisthecontrollerusedtodampouttheundesiredstructuraldynamics. 4-2 ),(Figure 4-10 ).Thesmallgaintheoremneedstheclosed-loopH1normsofthesystembelessthanorequaltooneforcontrolapplicationstoguaranteerobustperformancewithrespecttotheobjective.AsthenormfromtheH1controllerislessthanone,itensuresbetterperformancethantheLPVcontroller.Oneofthelimitations 39

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Closed-loopdesign ofthemathematicalformulationintheLPVframeworkisthatitassumesthatthesystemcanchangeextremelyfastfromonetemperatureproletoanotherwhichcannothappeninreality,thevariationsinheatingisatimedependentprocess.AnotherlimitationoftheLPVformulationisthateverysectionofthefuselagecanattainanytemperature,butitisexpectedthatthetemperaturedecreasesfromthenosetothetailofthefuselage.Sincethetrendshownbytheopen-loopnormisnotsimilartothetrendshownbytheclosed-loopnorm,itcanbeconcludedthatthebestopen-loopsystemneednotnecessarilygivethebestclosed-loopperformance,atleastforthissystem. Table4-2. ModelH1LPV 10.38245.049220.27752.599830.26940.876940.37964.979050.51416.8835 40

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Normsoftheclosed-loopsystems Thetransferfunctionsfortheopenloopsystem,closed-loopsystemwiththeH1controllerandwiththeLPVcontrollerforthevelineartemperatureprolesareplotted(Figure 4-11 ).ItcanbeseenthattheLPVcontrollerachievestheobjectiveofdampingouttherstbendingmode. Thetimeresponseoftheopenloop,closed-loopsystemswiththeH1controllerandwiththeLPVcontrollerareplotted(Figure 4-12 ).Theopen-looptimeresponseshowsoscillationswhichisduetothelackofstructuraldampingandshouldbeeliminatedbythecontroller.ItcanseenthattheLPVandtheH1controllersadddampingtothesystemandtheoscillationsareeliminated. Inordertounderstandthesystemabitbetter,apole-zeroanalysisisperformed.Thepole-zeromapoftheclosed-loopsystemwithH1andwiththeLPVcontrollerforthenominalplantmodelwereanalyzed(Figure 4-13 ).Thetargetmodelhasanundershootinthetimeresponseasitisdesignedonthebasisoftheopen-loopsystemwhichisa 41

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Comparisonofthetransferfunctionsforthedierentsystems non-minimumphaseandunstablesystem.Since,thisisa`model-matching'approachanundershootshouldbeexpectedintheclosed-looptimeresponse.Robustperformanceforguidanceormaneuveringwillbeguaranteedbytheouter-loopcontroller. 42

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Timeresponsefortheopen-loopandclosed-loopsystemswiththepointandLPVcontrollers Figure4-13. Pole-zeromapoftheclosed-loopsystemwithH1andLPVcontrollers 43

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4-14 ),considersvariationsinboththetailtemperatureandnosetemperature(Table 4-3 ). Figure4-14. Thermalprolescomprisingthedesignspace 44

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Temperaturegradients SetTnoseRangeofTtail Theopen-loopdynamicsareparametrizedasafunctionoftheseeectivetemperaturestoreectvariationsintheYoung'smodulusatthenoseandtailwhichresultfromthestructuralelementsandthermalprotectionsystem.Asetofvariablesthatarerepresentativeoftheparametrizationaroundthedesignspacearenoted,(Figure 4-15 )fortheinuenceofbending-modedisplacementonthevelocityand,(Figure 4-16 ),fortheinuenceofelevatoronthebending-modevelocity. Figure4-15. Open-loopstabilitycoecientasafunctionofthedesignspace 45

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Open-loopcontrolcoecientasafunctionofthedesignspace ThedesignspaceislimitedtoPwhichcontainsparametersforthefuselagestructureandthethermalprotectionsystemalongwithKwhichcontainsparametersforafeedbackcontroller.Suchlimitationsnotethatthegeometryisrelativelyxedduetoaerodynamicissueswhilethethermalissuesandstructuraldynamicshaveconsiderablefreedomintheirdesign.Inthiscase,thedesignspaceisappropriatesincethethermalprotectionsystemandstructureinteracttodeterminethevibrationcharacteristicsofthefuselagealongwithassociatedheatingeects. 46

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4-17 ).Assuch,thevalueofindicatesthelowestclosed-loopnormisachievedifthethermalprotectionsystemischosentohaveanosetemperatureof750oFandatailtemperatureof450oF. Figure4-17. Optimalcontrollerfrompitchratetoelevatorandcanarddeection Thetransferfunctionoftheclosed-loopsystem(Figure 4-18 )issimilartothetransferfunctionofthetargetmodel.Therelationshipbetweenpitchrateandelevatorarecloseatallfrequenciesbutparticularlyclosenearthenaturalfrequencyofthebendingmode.Assuch,theobjectiveofhigh-frequencyvibrationattenuationwithoutalteringthelow-frequencydynamicsisessentiallyachieved.Thetimeresponseoftheinputelevatordeectionfortheprolegivingoptimalperformance(Figure 4-19 )andtheresultingvibrationattenuationandassociatedactuationareplotted(Figure 4-20 ). Theoptimalityofthesystemcanbeveriedbycomparingtheperformancemetricsforthecontrol-orienteddesigntoacompletedesignoversysteminthedesignspace.This 47

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Actualanddesiredtransferfunction Figure4-19. Inputelevatordeection 48

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Actualanddesiredclosed-loopresponse comparisonisrelativelyeasytodoforthissystem;however,itwouldbeprohibitivetocomputeclosed-loopdesignsforeachcongurationwithahigh-dimensionaldesignspace.Inthiscase,thecontrol-orienteddesignisabletoachieveaclosed-loopnormof0.22. Thedicultyofoptimizinganopen-loopdesignareunderstood.Certainlytheopen-loopdynamics,(Figure 4-15 ),haveahighlynonlinearparameterizationaroundthedesignspace.Afunctionalbasedonthisnonlinearparametrizationwouldthushavetobeminimizedtoobtainoptimalityinanyopen-loopdesign. Theclosed-loopnormcansimilarlybeparametrizedaroundthedesignspace.Inthiscase,asetofcontrollersaregeneratedforeachthermalprole(Figure 4-14 )and 49

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4-15 ).Theresultingclosed-loopnorm,(Figure 4-21 ),showsaremarkablysimilarparametrizationastheopen-loopdynamics. Figure4-21. Closed-loopnormparametrizedaroundthedesignspace ItwasnoticedthatquietafewcoecientsoftheAandBmatrixoftheopen-loopdynamicsvarywithtemperature.Onfurtherinvestigation,itcanbenotedthatthecoecientswhichvarywithtemperaturecanbeputintotwogroups.Therstgroupconsistsofcoecientswhichshowthesametrendasintheclosed-loopnormandthesecondgroupconsistsofcoecientswhichshowtheoppositetrendtotheclosedloop-norm(Table 4-4 ). Thereasonforthesimilaritybetweenparameterizationsofopen-loopdynamicsandclosed-loopdynamicsisfoundbyinvestigatingadierentrelationship;namely,theclosed-loopnormshouldbeparameterizedasafunctionoftheopen-loopdynamicsinsteadofthedesignspace.Theclosed-loopnormandassociatedperformancefortrackingisactuallydirectlyrelatedtotheparametersoftheopen-loopstate-spacemodel.Thisresult 50

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Coecientsoftheopen-loopdynamicswhichvarywithtemperature GroupIGroupII A(1,6)A(3,2)A(3,6)A(7,4)A(7,1)B(1,1)B(1,3)B(1,2)B(7,1)B(1,4)A(7,3)B(3,1)B(3,2)B(3,3)B(3,4)B(7,2)B(7,2) iscertainlyexpected;however,theindependenceofthatrelationshipfromtemperature(Figure 4-22 )isnotcompletelyanticipated. Figure4-22. Closed-loopnormparametrizedaroundopen-loopdynamics 51

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4-15 ),arestronglynonlinearacrossthedesignspacesotheoptimizationisalmostcertaintoreachonlyalocalminimum.Suchlocalminimaarenotnecessarilyaccompaniedbypoorperformancesinceseveralsuchlocalminimahaveclosed-loopnormswithin5%oftheglobalminimum.Thedata(Figure 4-23 )showsthatseveralthermalprolesassociatedwithlocalminimaandtheresultingperformance(Figure 4-24 )cancomparefavorablywiththeglobalminimaanditsresultingperformance. Figure4-23. Thermalprolesassociatedwithsimilarly-valuedlocalminima 52

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Closed-loopperformanceforeachthermalprole Thissensitivitypresentsaninterestingfeatureofthehypersonicvehicle;namely,similarlevelsofclosed-loopperformancecanbeachievedforseveralchoicesofthermalprolesiftheyaredesignedproperly.Theproles(Figure 4-23 )allowforsimilarclosed-loopperformancesotheassociatedthermalprotectionsystemscanbefurtherevaluatedforissuessuchasweightandcosttooptimizethedesignforadditionalmetrics. 53

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Vibrationattenuationisacriticalrequirementformaintaininghypersonicightforacoupledfuselage-engineconguration.Suchattenuationcanbefacilitatedbydesigningbothathermalprotectionsystem(TPS)andafeedbackcontrollerthatcancompensateforthevariationsinthestructuraldynamicsduetodierenttemperatureproleswhichvaryintimeduringahypersonicight.Therstpartofthisstudyconsideredthecontrolofthestructuraldynamicsofanair-breathinghypersonicvehicleusingaLinearParameterVarying(LPV)controller.Theeectoftemperaturevariationsontheopen-loopdynamicsofthesystemwasanalyzed.ThenaLinearParameterVaryingcontrollerisformulatedtodampouttheundesireddynamics.Thistypeofcontrollerischosenbecausethechangeinthedynamicscanbemodeledasananefunctionoftemperature.Thiscontrolleristhencomparedwiththepointcontrollersatvarioustemperatureproles.Theclosed-loopH1normsshowedthatthepointcontrollerguaranteesperformanceandtheLPVcontrollerdoesnotguaranteeperformanceforalltrajectories.However,keepinginmindthemathematicalrestrictionsimposedbythewaytheLinearParameterVaryingsystemisformulated,thesimulationresultsshowthattheLPVcontrollerperformssatisfactorily.Hence,theapproachofhavinganinner-loopLPVcontrollertodamptheundesiredstructuraldynamicsseemstobeafeasiblesolution.Inthesecondpartofthisstudy,acontrol-orienteddesignisintroducedbywhichtheopen-loopsystemisdesignedtoachievethemaximumlevelofperformanceforwhichacontrollerexists.Arepresentativemodelofahypersonicvehicleisusedtodemonstratethisapproachcanindeedgenerateadesign.Itisalsoshownthatthereareseveraltemperatureproleswhichgivesimilarlevelofclosed-loopoptimalperformance. 54

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[1] Bolender,M.,andDoman,D.,\ANon-linearModelfortheLongitudinalDynamicsofaHypersonicAir-BreathingVehicle,"AIAAGuidance,NavigationandControlConferenceandExhibit,August2005,AIAA-2005-6255. [2] Chavez,F.,andSchmidt,D.,\AnalyticalAeropropulsive/AeroelasticHypersonic-VehicleModelwithDynamicAnalysis,"JournalofGuidance,ControlandDynamics,Vol.17,No.6,Nov-Dec1994,pp.1308-1319. [3] Bolender,M.,andDoman,D.,\NonlinearLongitudinalDynamicalModelofanAir-BreathingHypersonicVehicle,"JournalofSpacecraftandRockets,Vol.44,No.2,March-April2007,pp.374-387. [4] Bolender,M.,andDoman,D.,\ModelingUnsteadyHeatingEectsontheStructuralDynamicsofaHypersonicVehicle,"AIAAAtmosphericFlightMechanicsConferenceandExhibit,August2006,AIAA-2006-6646. [5] Williams,T.,Bolender,M.,Doman,D.,andMorataya,O.,\AnAerothermalFlexibleModelAnalysisofaHypersonicVehicle,"AIAAAtmosphericFlightMechanicsConferenceandExhibit,August2006,AIAA-2006-6647. [6] Culler,A.,Williams,T.,andBolender,M.,\AerothermalModelingandDynamicAnalysisofaHypersonicVehicle,"AIAAAtmosphericFlightMechanicsConferenceandExhibit,August2007,AIAA-2007-6395. [7] Oppenheimer,M.,Skijins,T.,Bolender,M.,andDoman,D.,\AFlexibleHypersonicVehicleModelDevelopedWithPistonTheory,"AIAAAtmosphericFlightMechanicsConferenceandExhibit,August2007,AIAA-2007-6396 [8] Buschek,H.,andCalise,A.,\FixedOrderRobustControlDesignforHypersonicVehicles,"AIAAGuidance,NavigationandControlConference,AIAA-94-3662,1994. [9] Buschek,H.,andCalise,A.,\RobustControlofHypersonicVehiclesConsideringPropulsiveandAeroelasticEects,"AIAAGuidance,NavigationandControlConference,1993,AIAA-93-3762. [10] Lind,R.,\LinearParameter-VaryingModelingandControlofStructuralDynamicswithAeroelasticEects,"JournalofGuidance,ControlandDynamics,Vol.25,No.4,2001,pp.733-739. [11] Jankovsky,P.,Sigthorsson,D.,Serrani,A.,Yurkovich,S.,Bolender,M.,andDoman,D.,\OutputFeedbackControlandSensorPlacementforaHypersonicVehicleModel,"AIAAGuidance,Navigation,andControlConferenceandExhibit,August2007,AIAA-2007-6327. [12] Baruth,H.,andChoe,K.,\SensorPlacementinStructuralControl"JournalofGuidance,Control,andDynamics,Vol.13,No.3,1990,pp.524533. 55

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Xu,K.,Warnitchai,P.,andIgusa,T.,\OptimalPlacementandGainsofSensorsandActuatorsforFeedbackControl,"JournalofGuidance,Control,andDynamics,Vol.17,No.5,1994,pp.929934. [14] Lim,K.,\MethodforOptimalActuatorandSensorPlacementforLargeFlexibleStructures,"JournalofGuidance,Control,andDynamics,Vol.15,No.1,1992,pp.4957. [15] Tongco,E.,andMeldrum,D.,\OptimalSensorPlacementofLargeFlexibleSpaceStructures,"JournalofGuidance,Control,andDynamics,Vol.19,No.4,1996,pp.961963. [16] Mirmirani,M.,Wu,C.,Clark,A.,Choi,S.,andColgren,R.,\ModelingforControlofaGenericAirbreathingHypersonicVehicle,"AIAAGuidance,NavigationandControlConference,AIAA-2005-6256. [17] Fiorentini,L.,Serrani,A.,Bolender,M.,andDoman,D.,\NonlinearRobust/AdaptiveControllerDesignforanAir-breathingHypersonicVehicleModel,"AIAAGuidance,NavigationandControlConferenceandExhibit,August2007,AIAA-2007-6329. [18] Sigthorsson,O.,D,Jankovsky,P.,Serrani,A.,Yurkovich,S.,Bolender,A.,M.,andDoman,D.,\RobustLinearOutputFeedbackControlofanAirbreathingHypersonicVehicle,",JournalofGuidance,ControlandDynamics,Vol.31,No.4,July-August2008,pp.1052-1066. [19] Kuipers,M.,Mirmirani,M.,Ioannou,P.,andHuo,Y.,\AdaptiveControlofanAeroelasticAirbreathingHypersonicCruiseVehicle,"August2007,AIAAPaper2007-6326. [20] Sigthorsson,D.,O.,Serrani,A.,Yurkovich,S.,Bolender,M.,A.,andDoman,D.,B.,\TrackingControlforanOveractuatedHypersonicAirbreathingVehiclewithSteadyStateConstraints,"August2006,AIAAPaper2006-6558. [21] Huo,Y.,Mirmirani,M.,Ioannou,P.,andKuipers,M.,\AltitudeandVelocityTrackingControlforanAirbreathingHypersonicCruiseVehicle,"August2006,AIAAPaper2006-6695. [22] Groves,K.,P.,Sigthorsson,D.,O.,Serrani,A.,Yurkovich,S.,Bolender,M.,A.,andDoman,D.,B.,\ReferenceCommandTrackingforaLinearizedModelofanAirbreathingHypersonicVehicle,"August2005,AIAAPaper2005-6144. [23] Groves,K.,P.,Serrani,A.,Yurkovich,S.,Bolender,M.,A.,andDoman,D.,B.,\Anti-WindupControlforanAirbreathingHypersonicVehicleModel,"August2006,AIAAPaper2006-6557,. [24] Parker,T.,J.,Serrani,A.,Yurkovich,S.,Bolender,A.,M.,andDoman,B.,D.,\Control-OrientedModelingofanAir-BreathingHypersonicVehicle,"JournalofGuidance,Control,andDynamics,Vol.30,No.3,2007,pp.856-869. 56

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SankethBhatwasborninMumbai,Indiain1984.HedidhisschoolingatO.L.P.S.highschoolandattendedjuniorcollegeatK.J.SomaiyaCollegeofScience.HethenattendedV.J.T.I.EngineeringcollegealiatedwithMumbaiUniversityandgraduatedwithaBachelorofEngineering(B.E.)degreeinJune2006.HedidhissummerprojectatCASDE,DepartmentofAerospaceEngineering,IndianInstituteofTechnology(IIT),Bombay,andworkedonthepropulsionsystemofminiaerialvehicles.HealsoworkedasaCFDresearchengineeratZeusNumerixPrivateLimited,Mumbai,India.Sankethiscurrentlyasecond-yeargraduatestudentintheDepartmentofMechanicalandAerospaceEngineeringattheUniversityofFlorida.HeisstudyingunderDr.RickLind.Hisresearchinvolvescontrol-orientedanalysisofhypersonicvehicles.HeplanstostayattheUniversityofFloridatopursueadoctoraldegreeinaerospaceengineeringwithafocusonstructuraldynamicsandcontrol. 60