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Control-Oriented Design of Systems with Operating Ranges to Optimize Closed-Loop Performance

Permanent Link: http://ufdc.ufl.edu/UFE0042357/00001

Material Information

Title: Control-Oriented Design of Systems with Operating Ranges to Optimize Closed-Loop Performance
Physical Description: 1 online resource (165 p.)
Language: english
Creator: Bhat, Sanketh
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

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

Notes

Abstract: Traditionally, the design of structural systems and vehicles has focused first on the design of the structure and then the control and other aspects are considered. But systems of the future will be evaluated based on closed-loop metrics and will be designed directly for 'high-performance' applications. Such capability depends on a synergistic integration of different disciplines and sub-systems, which cannot be captured by the traditional sequential design approach. Hence, it is imperative that a formulation for design be introduced that inherently considers the interactions between different disciplines and brings in controls earlier in the design process for optimal closed-loop performance. But existing approaches to the simultaneous design and optimization of the dynamics and the controller have some issues for realistic systems. This dissertation introduces an approach that overcomes some of the drawbacks of the existing simultaneous design approaches. This research introduces a formulation for multi-disciplinary design optimization with an emphasis on control. This approach notes that since mission capability is evaluated using a closed-loop configuration; the design must consider closed-loop performance when choosing system parameters. The approach does not actually optimize both the open-loop plant and controller simultaneously; rather, the approach minimizes a closed-loop norm by optimizing over the open-loop plant while constrained to satisfying conditions for controller existence. In this way, the approach finds the optimal plant along with the open-loop variables for which a controller exists such that the closed-loop system is optimal. This approach is developed for H-infinity, H-2 and linear parameter varying (LPV) control synthesis techniques. Surrogate-based design optimization with multiple surrogates is used to find the optimal configuration of the design variables. The proposed approach is investigated for vibration attenuation of a hypersonic vehicle. Structural damage resulting from the tremendous heating incurred during hypersonic flight is mitigated by a thermal protection system; 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. This research considers how such designs and resulting thermal gradients influence the ability to achieve closed-loop performance. The proposed control-oriented design approach answers the question on how to design the thermal protection system for the optimal thermal profile for which a controller exists and optimizes closed-loop performance.
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 (Ph.D.)--University of Florida, 2010.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042357:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042357/00001

Material Information

Title: Control-Oriented Design of Systems with Operating Ranges to Optimize Closed-Loop Performance
Physical Description: 1 online resource (165 p.)
Language: english
Creator: Bhat, Sanketh
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

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

Notes

Abstract: Traditionally, the design of structural systems and vehicles has focused first on the design of the structure and then the control and other aspects are considered. But systems of the future will be evaluated based on closed-loop metrics and will be designed directly for 'high-performance' applications. Such capability depends on a synergistic integration of different disciplines and sub-systems, which cannot be captured by the traditional sequential design approach. Hence, it is imperative that a formulation for design be introduced that inherently considers the interactions between different disciplines and brings in controls earlier in the design process for optimal closed-loop performance. But existing approaches to the simultaneous design and optimization of the dynamics and the controller have some issues for realistic systems. This dissertation introduces an approach that overcomes some of the drawbacks of the existing simultaneous design approaches. This research introduces a formulation for multi-disciplinary design optimization with an emphasis on control. This approach notes that since mission capability is evaluated using a closed-loop configuration; the design must consider closed-loop performance when choosing system parameters. The approach does not actually optimize both the open-loop plant and controller simultaneously; rather, the approach minimizes a closed-loop norm by optimizing over the open-loop plant while constrained to satisfying conditions for controller existence. In this way, the approach finds the optimal plant along with the open-loop variables for which a controller exists such that the closed-loop system is optimal. This approach is developed for H-infinity, H-2 and linear parameter varying (LPV) control synthesis techniques. Surrogate-based design optimization with multiple surrogates is used to find the optimal configuration of the design variables. The proposed approach is investigated for vibration attenuation of a hypersonic vehicle. Structural damage resulting from the tremendous heating incurred during hypersonic flight is mitigated by a thermal protection system; 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. This research considers how such designs and resulting thermal gradients influence the ability to achieve closed-loop performance. The proposed control-oriented design approach answers the question on how to design the thermal protection system for the optimal thermal profile for which a controller exists and optimizes closed-loop performance.
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 (Ph.D.)--University of Florida, 2010.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042357:00001


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CONTROL-ORIENTED DESIGNOFSYSTEMSWITHOPERATINGRANGESTO OPTIMIZECLOSED-LOOPPERFORMANCE By SANKETHBHAT ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2010

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c r 2010Sank ethBhat 2

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Sarv amangalamangalyesivesarvardhasadhikesaranyetrayambakegaurinarayani namostute Dedicatedwithlovetomyparents,tomybrotherandtoallthechildrenwho didnotgettheopportunitytohaveadesktostudyon. 3

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ACKNO WLEDGMENTS IwouldliketoexpressmysincereanddeepgratitudetomyPhD.advisor,Dr.Rick Lindforhisguidance,support,encouragementandmostimportantlyforgivingmean opportunitytoprovemyselfonthatbrightmorninginearlyJanuary2007.ThankstoDr. PramodKhargonekar,Dr.WarrenDixonandDr.PrabirBarooahforagreeingtoserve onmycommitteeandfortheusefultechnicaldiscussionsIhavehadwiththem.Noneof thisworkwouldhavebeenpossiblewithoutthenancialsupportgivenbyNasa. SpecialthankstoDr.F.A.C.Vianna,Mr.VijayJagdaleandDr.ZachWilcoxforthe technicalhelpwithSurrogate-basedDesignOptimization,NSGA-IIcodeandnon-linear controlinmyproject.HeartfeltthanksgoouttotheseniorresearchersattheFlight ControlLab,includingDr.JoeKehoe,Dr.RyanCausey,Dr.MujahidAbdulrahim,Dr. AdamWatkins,andDr.SeanRegisfordfortheirvaluableguidanceandsupportearlier oninmyPhD.ThankstoAbePachikaraandDongTranwhohavebeenahugehelp especiallywithrestartingmylabPCwhenIwashavingissuesremotelyconnectingto itwhileIwasinCambridge,MAforthesummerwritingmydissertationandfortheir tremendoussupport.Thiseffortwouldnothavebeenpossiblewithoutthehelpofmy fellowresearchersattheFlightControlLab,includingBrianRoberts,RobertLove,Baron Johnson,RyanHurley,StephenSorleyandofcourse,mydeskbuddy,Daniel Tex Grant, forlightingupmanyadayswithhishumor. MyparentsandmybrotherdeservemuchcreditformakingmewhatIamand showingmetherightwayalltheseyears.Mybrotheralsodeservesmanythanksfor helpingmewithmyprofessionalpresentationsandfor`settingthebarsohigh'thatI hadtostretchtoreachtohisexpectations.Thankstoallthevolunteersof`Ashafor Education',particularlyatAshaUFlorida,myAmericanparentsDanaMoserand RickSwensonandmynon-academicmentor,Dr.MeeraSitharam,whohavemademy graduateschoollifeverymemorableandforhavinggivenmeanewpriorityinlife. Ithankallwhohelpedmegetthisfar,butwhosenamesIinadvertentlymissedout. 4

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TABLE OFCONTENTS page ACKNO WLEDGMENTS ..................................4 LISTOFTABLES ......................................9 LISTOFFIGURES .....................................10 ABSTRACT .........................................15 CHAPTER 1INTRODUCTION ...................................17 1.1Background ...................................17 1.1.1SequentialDesignApproach .....................17 1.1.2SimultaneousDesignApproach ....................17 1.1.3System-of-systemsDesignPhilosophy ................17 1.2Motivation ....................................20 1.3ProblemStatement ...............................23 1.4Objectives ....................................23 1.5LiteratureSurvey ................................24 1.6DissertationOutline ..............................25 1.7Contribution ...................................26 2CONTROLSYNTHESIS ...............................30 2.1Introduction ...................................30 2.2Closed-loopSystemRepresentation .....................30 2.3 H 1 Theory ...................................33 2.4 H 2 Theory ....................................35 2.5LinearParameterVarying(LPV)Theory ...................36 2.5.1OverviewofGainScheduling .....................36 2.5.2LinearParameterVarying .......................37 2.6Summary ....................................39 3OPTIMIZATIONALGORITHMS ...........................41 3.1Introduction ...................................41 3.2GlobalSearch .................................41 3.3One-dimensionalIteration ...........................41 3.4Surrogate-basedDesignOptimization ....................44 3.4.1SurrogateModeling ...........................44 3.4.2Optimization ...............................45 3.4.2.1Efcientglobaloptimization(EGO)algorithm .......46 3.4.2.2Multi-objectiveoptimization .................49 3.5Summary ....................................50 5

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4CONTR OL-ORIENTEDDESIGN ..........................51 4.1Backgroundon `Multi-disciplinaryDesignOptimization' (MDO) ......51 4.2Control-orientedDesign ............................51 4.2.1SystemDesign .............................51 4.2.2ProblemFormulation ..........................55 4.2.2.1 H 1 controlsynthesis ....................57 4.2.2.2 H 2 controlsynthesis .....................59 4.2.2.3LPVcontrolsynthesis ....................61 4.2.3SolutionMethodology .........................62 4.2.4AdvantagesofProposedApproach ..................62 4.3Applications ...................................63 4.4Summary ....................................63 5LPVCONTROLDESIGNFORVIBRATIONATTENUATIONOFAHYPERSONIC VEHICLE .......................................65 5.1Introduction ...................................65 5.2Backgroundon Aerothermoelasticity .....................66 5.3ControlIssues .................................70 5.4LiteratureSurvey ................................72 5.5Objective ....................................73 5.6ApproachtoIntroduceThermalProledependencyontheDynamics ...73 5.7Vehicle .....................................74 5.8SetsofThermalProles ............................75 5.8.1SetA ...................................75 5.8.2SetB ...................................76 5.9ResultsforSetA ................................76 5.9.1EffectofTemperatureontheModalProperties ...........76 5.9.2Open-loopDynamics ..........................77 5.9.3ControlSynthesis ............................81 5.9.4ResultsandDiscussion ........................85 5.10ResultsforSetB ................................87 5.10.1Open-loopDynamics ..........................87 5.10.2Performance ..............................88 5.10.2.1Norm .............................88 5.10.2.2Class6 ............................89 5.11Summary ....................................90 6CONTROL-ORIENTEDDESIGNOFHYPERSONICVEHICLEFOR H 2 AND H 1 PERFORMANCE ................................97 6.1Introduction ...................................97 6.2Objective ....................................98 6.3DesignSpace ..................................98 6.4ControlSynthesis ................................99 6

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6.5Optimization .................................99 6.5.1GlobalSearch ..............................99 6.5.2Surrogate-BasedDesignOptimization ................102 6.6Analysis .....................................105 6.7Summary ....................................107 7CONTROL-ORIENTEDDESIGNWITHMULTI-OBJECTIVEOPTIMIZATION ANDINCLUSIONOFWEIGHTINGFUNCTIONSINTHEDESIGNPROCESS FORTHEHYPERSONICVEHICLE ........................108 7.1Introduction ...................................108 7.2Objective ....................................109 7.3ControlSynthesisArchitecture ........................109 7.4OptimizationProblemFormulation ......................110 7.5ResultsandDiscussion ............................113 7.6AdvantagesofusingMultipleSurrogates ...................117 7.7Summary ....................................120 8CONTROL-ORIENTEDDESIGNOFOPERATINGRANGEFORLINEARPARAMETER VARYINGSYSTEMS ................................121 8.1Introduction ...................................121 8.2Example1:InvertedPendulumandCartSystem .............121 8.2.1Objective ................................121 8.2.2LinearizedDynamics ..........................123 8.2.3DesignSpace ..............................123 8.2.4ProblemFormulation ..........................124 8.2.5OptimizationResults ..........................124 8.2.5.1Globalsearch ........................124 8.2.5.2Surrogate-basedoptimization ................126 8.3Example2:HypersonicVehicle .......................128 8.3.1DesignSpace ..............................129 8.3.2Objective ................................130 8.3.3SolutionMethodology .........................131 8.4Summary ....................................133 9CONTROL-ORIENTEDDESIGN:PARAMETRICUNCERTAINTYANALYSIS FORAMASS-SPRING-DAMPERSYSTEM ....................134 9.1Introduction ...................................134 9.2IncorporatingUncertaintyintheSystem ...................134 9.3Example1:Minimize H 1 -norm ........................136 9.4Example2:Minimize L 2 -normfor LPV systems ..............137 9.5Example3:Multi-objectiveOptimization-maximizeperformancebyusing minimumcontrolactuationand H 1 controlsynthesis ............138 9.6Summary ....................................142 7

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10CONCLUSIONS ANDRECOMMENDATIONS ...................146 10.1DissertationSummaryandConclusions ...................146 10.2Contribution ...................................147 10.3FutureDirections ................................149 10.3.1Off-shootsofresearch .........................150 10.4ImplicationsofResearch ............................150 10.5Limitations ...................................152 REFERENCES .......................................153 BIOGRAPHICALSKETCH ................................165 8

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LISTOF TABLES Tab le page 1-1Motiv ationfor`simultaneousdesignofthedynamicsandthecontroller'from fewrelevantliterature ................................28 1-2Someissueswithexistingapproachestosimultaneousdesign .........29 2-1Summaryofdifferentcontrolsynthesistechniquesusedinthisresearch ....40 3-1Differentsurrogatesusedinthesurrogate-basedoptimization ..........46 3-2Optimizationalgorithmsusedinthisresearchalongwithmultiplesurrogates ..46 4-1Objectivesusedfordifferentapplicationsinthisdissertation ...........63 5-1Naturalfrequencies, i forthelinearthermalprolesinFigure5-6A .......77 5-2Closed-loop H 1 -normsforsystemswith H 1 andLPVcontrollerforthelinear proles ........................................86 5-3Thermalprolesandnaturalfrequenciesofthe 1 st bendingmode ........88 5-4Normofclosed-loopsystem .............................88 6-1Weightingfunctions .................................99 6-2Surrogate-baseddesignoptimizationusingmultiplesurrogatesfor H 1 synthesis103 6-3Surrogate-baseddesignoptimizationusingmultiplesurrogatesfor H 2 synthesis 103 7-1Designvariables. -operatingrange, -weightingfunction ..........111 7-2Informationaboutthesetofsurrogatesusedinthesurrogate-basedoptimization 114 7-3 %PRESS RMS (PRESS RMS = Range %) valuesfordifferentsurrogatemodels fortheinitialconguration. ..............................114 8-1DesignofExperiments(DOE)with15pointsintheinitialset ...........126 8-2DesignofExperiments(DOE) ............................131 9-1Parametervalues ...................................136 9-2Designvariables. -operatingrange, -weightingfunction ...........140 9-3Closed-loop H 1 -normforthe3setofpointsontheparetofront. ........142 9

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LISTOF FIGURES Figure page 1-1Illustr ationofsequentialdesignofopen-loopdynamicsandcontrol .......18 1-2Illustrationofsimultaneousdesignofopen-loopdynamicsandcontrol .....19 1-3Illustrationofintegrateddesignapproach .....................20 1-4Simpliedintegratedvehicledesignprocess ....................21 1-5Motivationforefcientdesigntechniquesforhypersonicvehicles ........23 1-6Problemstatement ..................................24 1-7Overviewofdissertation ...............................27 2-1Evolutionofcontrol ..................................31 2-2Featuresofacontrolsystem ............................31 2-3Overviewofcontrol ..................................32 2-4Closed-loopmodel ..................................33 2-5Synthesismodel, S ..................................33 2-6Physicalinterpretationof H 1 -normforaSISOsystem ..............34 2-7Stepsingainscheduling ...............................38 2-8IllustrationofthedifferencebetweengainschedulingandLPV .........40 3-1Standardoptimizationproblem ...........................42 3-2Objective, J (x ) ofasingledesignvariable, x ...................42 3-31-Diteration.Objectiveistominimize r ina2-Ddesignspace .........43 3-4Stepsinsurrogate-baseddesignoptimization. x 1 and x 2 aredesignvariables 47 3-5IllustrationofoptimizationprocesswithEGOalgorithm ..............48 3-6Singlecycleoftheefcientglobaloptimization(EGO)algorithmshowingcost functiondueto(a)krigingand(b)expectedimprovement ............48 3-7UpdatedcycleoftheEGOalgorithmshowingcostfunctiondueto(a)kriging and(b)expectedimprovement ...........................49 3-8Illustrationofamulti-objectiveoptimizationproblem ...............50 4-1MDOcategories ...................................52 10

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4-2Subsystems ofahypersonicvehicle ........................52 4-3Designspaceofthesubsystemsofahypersonicvehicle .............53 4-4Closed-loopmodelillustratingthedesignspace ..................53 4-5Systemlevelperspectiveofcontrol-orienteddesign ................57 4-6MDOapproach ....................................64 5-1Air-breathinghypersonicvehicle ..........................66 5-2Coupleddynamicsofthehypersonicvehicle ....................67 5-3Aerothermoelasticitytetrahedron ..........................68 5-4Variationinmodeshapeswiththermalprolesforarepresentativemodel ...69 5-5Fuselagesection ...................................73 5-6Differentsetsofthermalproles ..........................76 5-7Transferfunctionfromelevatordeection, e topitchrate, q forthedifferent thermalprolesshowninFigure5-6A .......................77 5-8Modeshapesforarepresentativethermalproleforthehypersonicvehicle ..78 5-9Variationinthecoefcientsofthestatematrices .................79 5-10Open-loop H 1 -normforthesystemswiththedifferentthermalproles .....80 5-11Open-loop H 1 -normparameterizedaroundtheopen-loopdynamicco-efcient, A(7,6) ........................................80 5-12Multi-loopcontrolarchitecture ............................82 5-13Synthesismodel, S usedtocreatetheLPVcontroller, K .............83 5-14Responsefromelevatordeection, e topitchrate, q forthenominalmodel ()andtargetmodel(-.-.-.) .............................83 5-15Pitchrate(q )timeresponsefor200sec. ......................84 5-16StepsinLPVcontrolsynthesis ...........................85 5-17Normsoftheclosed-loopsystems .........................86 5-18Frequencyresponsefromelevatordeection, delta e topitchrate, q fortheve linearthermalprole .................................91 5-19Pitchrate(q )timeresponsefortheopen-loopandclosed-loopsystemswith the H 1 andLPVcontrollers .............................92 11

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5-20P ole-zeromapoftheclosed-loopsystemwiththe H 1 andLPVcontrollersfor arepresentativemodel ................................93 5-21Open-loopstabilitycoefcientrepresentingtheinuenceofthevelocity( v )on thevelocityoftherstbendingmode( 1 ), A(7,1) asafunctionoftemperature 93 5-22Open-loopcontrolcoefcientrepresentingtheinuenceoftheelevatordeection ( e )onthevelocityoftherstbendingmode( i ), B (7,1) asafunctionoftemperature 94 5-23Actual(),desired(-.-.)andopen-loop( )-340()-340()]TJ /T1_0 11.955 Tf [()frequencyresponsefrom elevatordeection( e )topitchrate( q )forclass6 .................94 5-24Actual(),desired(-.-.)andopen-loop( )-222()-222()]TJ /T1_0 11.955 Tf [()timeresponseforclass6 ...95 5-25A)ElevatorcommandsandB)tailtemperature ..................95 5-26Actual(),desired(-.-.)andopen-loop( )-222()-222()]TJ /T1_0 11.955 Tf [()timeresponse .........96 6-1Thermalprolescomprisingthedesignspace ...................98 6-2Synthesismodel, S usedtocreatethecontroller, K ................100 6-3Closed-loopnorm,( r )for H 1 and H 2 performanceparameterizedaroundthe designspace .....................................101 6-4Magnitudeoftransferfunctionfromelevatortopitchrate( q )ofoptimaldesign ()anddesired( )-221()-223()]TJ /T1_0 11.955 Tf [()for H 1 and H 2 synthesis ................101 6-5Pitchrate( q )ofoptimaldesign()anddesired(\000)Tj /T1_0 11.955 Tf 221.9 0 Td [()for H 1 and H 2 synthesis102 6-6Elevatordeection( e )ofoptimaldesign()for H 1 and H 2 synthesis .....102 6-7Surrogatetsandexpectedimprovementmodels, E [I (x )] forkriging(KRG) andsurrogate2(Surr 2 )forcaseAatthe1stiteration ...............104 6-8Surrogatetsandexpectedimprovementmodels, E [I (x )] forkriging(KRG) andsurrogate2(Surr 2 )forCaseAafterthe2nditeration .............105 6-9Surrogatetsandexpectedimprovementmodels, E [I (x )] forkriging(KRG) andsurrogate2(Surr 2 )forCaseAafterthenaliteration ............106 6-10Closed-loopnormfor H 1 and H 2 asafunctionoftheopen-loopdynamiccoefcient, B (7,1) .........................................107 7-1Synthesismodel, S usedtocreatethecontroller, K () ..............110 7-2Open-loopdesignspace, .............................112 7-32-DObjectivespace( p RMSE and c MS arenormalizedbetween0and1using theinitialdatasetof700points) ..........................115 12

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7-4Ev olutionofparetofrontfordifferentiterationsusingmultiplesurrogates. -initialconguration(700points), 3-after 1 st iteration(880points), -after 2 nd iteration(1100points)( p RMSE and c MS arenormalizedbetween 0and1usingtheinitialdataset) ..........................116 7-5Paretofrontwithtimeanalysisperformedfor3points ...............118 7-6Timeresponsefor2pointsshownontheparetofront. )-261()-262()]TJ /T1_0 11.955 Tf 37.48 0 Td [(correspondsto 3 ,correspondsto ...............................118 7-7Timeresponsefor2pointsshownontheparetofront.correspondsto )Tj /T1_3 11.955 Tf (. )Tj /T1_3 11.955 Tf 11.95 0 Td (. correspondsto ? ...............................119 7-8Paretoplotsfortheoriginalsetandafterthenaliteration ............119 7-9ParetofrontjustforPRS( )andcombinationofPRSandKRG( 3) ......120 8-1Motivationforcontrol-orienteddesignforLPVsystems ..............122 8-2Invertedpendulumandcartsystem ........................122 8-3(a)Designspaceand(b)krigingtoftheclosed-loopnorm( r )parameterized aroundthedesignspace, =[M m ] .......................125 8-4Surrogatetsandexpectedimprovement E [I (x )] modelsforinitialsetforcase A.KRG-Kriging,Surr2-Surrogate2 .......................127 8-5Surrogatetsandexpectedimprovement E [I (x )] modelsafternaliteration forcaseA.KRG-Kriging,Surr2-Surrogate2 ..................128 8-6Boxplotofthedesignvariablesfora.)Initialandb.)FinalforcaseA.Design variable:1Mass cart ,2Mass pendulum .......................129 8-7Noseandtailtemperatureforthethermalprolesacrossthefuselageforthe hypersonicvehicle ..................................130 8-8(a)Pitchrateerrorand(b)elevatordeectiontimeresponse(logscale)for 200secforCaseA.istheresponsefortheupperbound, [T nose + T tail + ] and )-222()-222()]TJ /T1_0 11.955 Tf 36.53 0 Td [(isthelowerbound, [T nose )Tj /T1_7 11.955 Tf 11.95 0 Td ( T tail )Tj /T1_7 11.955 Tf 11.96 0 Td ( ],where =50 ........132 8-9(a)Pitchrateerrorand(b)elevatordeectiontimeresponse(logscale)for 200secforCaseB.istheresponsefortheupperbound, [T nose + T tail + ] and )-222()-222()]TJ /T1_0 11.955 Tf 36.53 0 Td [(isthelowerbound, [T nose )Tj /T1_7 11.955 Tf 11.95 0 Td ( T tail )Tj /T1_7 11.955 Tf 11.96 0 Td ( ],where =50 ........132 9-1Mass-spring-dampersystem ............................134 9-2Synthesismodelfordisturbancerejection .....................137 9-3Boxplotoftheclosed-loopnorm, r .1-nouncertainty,210% uncertainty,3 20% uncertainty ..................................138 13

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9-4Bo xplotoftheclosed-loopnorm, r withLPVcontrolsynthesis.1-nouncertainty, 210% uncertainty,320% uncertaintyin m ...................139 9-5Synthesismodelfortracking ............................141 9-6Paretofrontofobjectiveswithnouncertainty( ),with 10% uncertainty(3) andwith 20% uncertainty( ) ............................142 9-7Boxplotoftheclosed-loopnorm, r .1-nouncertainty,210% uncertainty,3 20% uncertainty ...................................143 9-8Paretofrontofobjectiveswithnouncertainty( ),with 10% uncertainty(3) andwith 20% uncertainty( )forT=200sec ...................143 9-9Analysisof3setofpointsontheparetofront ...................144 9-10Closed-loopfrequencyresponsefromcommandtoA)positionandB)velocity ofcartforset1 ....................................144 9-11Closed-loopfrequencyresponsefromA)commandtopositionandB)velocity ofcartforset2 ....................................145 9-12Closed-loopfrequencyresponsefromcommandtoA)positionandB)velocity ofcartforset3 ....................................145 10-1Summaryofthedissertation ............................147 10-2Importanttechnicalconceptsusedinthisresearch ................149 10-3Integrationofthecontrol-orienteddesignapproachinthecompletedesign framework .......................................151 14

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Abstract ofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy CONTROL-ORIENTEDDESIGNOFSYSTEMSWITHOPERATINGRANGESTO OPTIMIZECLOSED-LOOPPERFORMANCE By SankethBhat December2010 Chair:RichardC.LindJr. Major:AerospaceEngineering Traditionally,thedesignofstructuralsystemsandvehicleshasfocusedrston thedesignofthestructureandthenthecontrolandotheraspectsareconsidered. Butsystemsofthefuturewillbeevaluatedbasedonclosed-loopmetricsandwillbe designeddirectlyfor`high-performance'applications.Suchcapabilitydependson asynergisticintegrationofdifferentdisciplinesandsub-systems,whichcannotbe capturedbythetraditionalsequentialdesignapproach.Hence,itisimperativethata formulationfordesignbeintroducedthatinherentlyconsiderstheinteractionsbetween differentdisciplinesandbringsincontrolsearlierinthedesignprocessforoptimal closed-loopperformance.Butexistingapproachestothesimultaneousdesignand optimizationofthedynamicsandthecontrollerhavesomeissuesforrealisticsystems. Thisdissertationintroducesanapproachthatovercomessomeofthedrawbacksofthe existingsimultaneousdesignapproaches. Thisresearchintroducesaformulationformulti-disciplinarydesignoptimizationwith anemphasisoncontrol.Thisapproachnotesthatsincemissioncapabilityisevaluated usingaclosed-loopconguration;thedesignmustconsiderclosed-loopperformance whenchoosingsystemparameters.Theapproachdoesnotactuallyoptimizeboth theopen-loopplantandcontrollersimultaneously;rather,theapproachminimizesa closed-loopnormbyoptimizingovertheopen-loopplantwhileconstrainedtosatisfying conditionsforcontrollerexistence.Inthisway,theapproachndstheoptimalplant 15

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alongwith theopen-loopvariablesforwhichacontrollerexistssuchthattheclosed-loop systemisoptimal.Thisapproachisdevelopedfor H 1 H 2 andlinearparametervarying (LPV)controlsynthesistechniques.Surrogate-baseddesignoptimizationwithmultiple surrogatesisusedtondtheoptimalcongurationofthedesignvariables. Theproposedapproachisinvestigatedforvibrationattenuationofahypersonic vehicle.Structuraldamageresultingfromthetremendousheatingincurredduring hypersonicightismitigatedbyathermalprotectionsystem;however,suchmitigation isaccompaniedbyanincreaseinweightthatcanbeprohibitive.Theactualdesignof athermalprotectionsystemcanbechosentovarythelevelofheatingreduction,and associatedweight,acrossthestructure.Thisresearchconsidershowsuchdesignsand resultingthermalgradientsinuencetheabilitytoachieveclosed-loopperformance.The proposedcontrol-orienteddesignapproachanswersthequestiononhowtodesignthe thermalprotectionsystemfortheoptimalthermalproleforwhichacontrollerexistsand optimizesclosed-loopperformance. 16

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CHAPTER1 INTR ODUCTION Thischapterintroducestheproblembeingtackledinthisdissertation,relevantliterature reviewalongwiththeoverviewoftheremainderofthisdissertation. 1.1Background 1.1.1SequentialDesignApproach Traditionally,thedesignofaerospacevehiclesandstructuresgenerallyfollowsa twostepprocedure.Intherststep,thedetailedstructureandgeometryaredesigned andinthenextstepthecontroller,electronicsandotherfeaturesareintroduced[ 13]. Figure 1-1 illustratesthesequentialprocessforthedesignoftheopen-loopdynamics andthecontroller 1 .Theplanttobecontrolledisgivenaprioritothecontrolsengineer andhe/shehasnosayintheplantdesignprocess.Butquietoftensimplechangesin theopen-loopdesignvariablescandrasticallyimproverobustperformanceetc[ 4].In aircraftforexample,thissequentialdesignapproachoftenleadstocontrolsurfacewhich areusuallyover-sizedwhichcomesattheexpenseofcost,stealth,dragandweight[ 2]. 1.1.2SimultaneousDesignApproach Literaturehassuggestedintegrateddesign-optimizationofstructuralsystems toaccountfortheinteractionsbetweenthedifferentdisciplines[ 2 21 ],asshownin Figure 1-2.Theemphasisisonimprovementspossiblebytheintegrationofdifferent disciplinesandtheideaistobringcontrolsearlierinthedesignprocess. 1.1.3System-of-systemsDesignPhilosophy `System-of-systems'areacollectionofheterogeneoussystemswhichwhen consideredasawholeoffersmorefunctionalityandperformancethansimplythesumof 1 Theopen-loop dynamicswillinterchangeablybereferredtoasplant.Theopen-loop designvariablesistheoperatingrange. 17

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Figure1-1. Illustrationofsequentialdesignofopen-loopdynamicsandcontrol theindividualsystems,i.e.theconstituentsystemscouldbeverydifferentandoperate independently,buttheirinteractionstypicallyexhibitdifferentproperties[ 22, 23]. Features: Thisdesignphilosophygivesanintegratedandpanoramicviewofsystemdesign. Thescopeanddenitionofthesystemhasbeenenlargedandhencethedesign spaceofthesystemtobecontrolledislargerthansumoftheindividualdesign spaces. Traditionalsequentialapproachesconsiderthedesignofindividualcomponents andanalyzethemforindividualdiscipline-relatedmetrics.Systemsareevaluated byclosed-loopmetricsandkeepingthemissioncapabilityinmind[2, 3, 68]. Simultaneousdesignandoptimizationmaypotentiallygivebetterperformanceand functionalitycomparedtosequentialdesign.Forexample,Ettenetal.[ 24]showed thatintegrateddesigncanobtainagreaterlevelofaeroelasticdampingwithlesser controlleractivitycomparedtothesequentialdesignforaexibleaircraftwhich 18

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Figure1-2. Illustrationofsimultaneousdesignofopen-loopdynamicsandcontrol achievesdesiredyingqualities,attenuatesstructuralvibrationsandisrobustto modelingerrorswithinthedesignightenvelope. Thisapproachmayincreasethecomplexityofthedesignproblem,butitwill potentiallyreducethedesigncycletimeandensurethatclosed-looprequirements andcontrolissueslikecontrollabilityandobservabilitywillbemetatthepreliminary designstageitself[3, 25 ].Italsocapturesthemulti-disciplinaryandmulti-objective natureofthedesignproblem. Traditionaldesignapproachesaregenerallyacompromisedesigntoachievea system'smissionrequirements.Specialistinadisciplinemaydesignaparticular subsystemconsideringthevariablesandconstraintsoftheirowndiscipline.This maynotalwaysbebenecialandotherdisciplinesneedtoabsorbthesideeffects ofthis,whichcouldadverselyaffecttheperformanceoftheoverallsystem.For example,highaspectratiowingaircraftdesigns[ 8 ]havegoodefciencyinterms oflifttodragratio,buthavestructuralandmaneuverabilityissues. Considertheexampleofanautomobiledesign.Traditionally,rstthegeometry isdesigned,thenthecontrol,electronicsandotheraspectsareconsideredasshown inFigure 1-3.Inintegrateddesignapproach,thedifferentaspectsaredesignedand optimizedsimultaneously.Theresultingpotentialbenetsbeing,areductionindesign 19

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cycletime andachievingmissionrequirements.Anobjectiveofintegratedautomobile designexamplecouldbe:Highendvehicleshapeoptimizationwhileimprovingcar safetyforxedperformancelevelandgivengeometricconstraints,[ 26]. Figure1-3. Illustrationofintegrateddesignapproach 1.2Motivation Missioncapabilityofavehicleisultimatelyevaluatedbyclosed-loopperformance. Suchcapabilitydependsonasynergisticintegrationofaerodynamics,structures, propulsion,andcontrolthatresultsinightdynamicswhichareoptimalforthe mission.Unfortunately,mostsystemsaretraditionallydesignedusingasequential seriesofopen-loopoptimizationsthatcannotaccount,noroptimize,foranysynergistic integrations.Aformulationfordesignthatinherentlyconsiderscontrolmustthereforebe developedtoenableoptimalclosed-loopperformance. Topresentthemotivationforsimultaneousdesignoftheopen-loopdynamicsand thecontroller,Table 1-1 showssnippetsfromsomerelevantliterature.Theunderlying themeinallthecitedliterature,isthatforfuturisticsystemsitisbenecialtouse integrateddesignapproachcomparedtothetraditionalsequentialdesignapproach. 20

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Figure1-4. Simpliedintegratedvehicledesignprocess Fuetal.[ 27]showedthatfornon-decomposableproblems 2 ,theintegrateddesign approachwillgivebetteroptimalclosed-loopperformancecomparedtothesequential approach.Fordecomposableproblems,theintegratedapproachwillgivesimilar performanceassequentialdesign.Soloway[ 25 ]highlightedthebenetsofusinga simultaneousapproachforaerospacevehicledesign,showninFigure 1-4.Therearea plethoraofapproachestalkingaboutsimultaneousdesigninliterature. Butthereareissueswiththeexistingapproachestosimultaneousdesign, someofwhicharehighlightedinTable 1-2.Thissimultaneousapproachoften leadstonon-convex,constrained,multi-objectiveoptimizationproblemswhichare computationallyintensiveandforwhichitisdifculttondaglobaloptimalsolutionfor realisticsystems.Itisdifculttoincorporaterobustnesswithrespecttoparametric uncertaintyintheframeworkandtheremayberestrictionsonthechoiceofthe optimizationobjectives. 2 Adecomposab ledesignproblemcanbedenedasaprobleminwhichthe integratedoptimizationproblemcanbemathematicallydecomposedbyoptimization theory. 21

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Thisdisser tationintroducesacontrol-orientedapproachfordesignthatavoidsthe commondifcultiesofsimultaneousopen-loopdynamics-controldesignwhichisknown tobenon-convex[ 21, 3032].Theapproachactuallyconsidersanexistencequestion thatnotesifacontrollerexistsforagivenstructurethatachievesadesiredlevelof performance.Theapproachdoesnotdesignboththedynamicsandcontroltooptimize aclosed-loopnorm;rather,itdesignsastructureforawhichacontrollerexiststhat optimizesaclosed-loopnorm.Formulationsusing H 1 H 2 and LPV controlsynthesisto minimizean H 1 -norm,an H 2 -norm,an L 2 -normmetricandmulti-objective2-normfor performanceandcontrolactuationarederivedusingappropriateexistenceconditions. Asimportantly,asolutionmethodologyisutilizedbasedonsurrogatemodelingtoavoid theiterationsandexpensivecomputationsassociatedwithtechniquesdoingdesignwith LMIexpressions.Thesurrogate-basedoptimizationisshowntobeefcientandeffective atexploringadesignspacetooptimizetheclosed-loopmetrics. Thisdissertationconsidersthesimultaneousdesignofthedynamicsandthe controllerforvibrationsuppressionofahypersonicvehicle.Hypersonicvehiclesare increasinglygainingsignicantattentionthroughouttheglobeasafeasiblesolution forvariousmissions 3 .Thereistremendousamountofheatingofthestructurein hypersonicightbecauseofshockwaveswhichcoulddamagethestructure.Hence, thevehicleneedsathermalprotectionsystem(TPS).ButtheTPSaddsweightto thestructurewhichisdetrimenttothehighperformancerequirementofthevehicle. SothereisaneedtodevelopefcientmethodstodesigntheTPStoprotectthe structurealongwithoptimizingtheclosed-loopperformanceforthevehicle,illustratedin Figure 1-5. 3 Chapter 5 describes thedetailsofthehypersonicvehicle. 22

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Figure1-5. Motivationforefcientdesigntechniquesforhypersonicvehicles [ http://www.dfrc.nasa.gov/Gallery/Photo/X-43A/HTML/ED97-43968-1.html reprintedwithpermission ] 1.3ProblemStatement Systemsofthefuturewillhavecoupled,time-varyingdynamicswhichrequires complexinteractionsamongthedynamics.Futureclassesofvehicleswillneedtobe designeddirectlyformissioncapabilityandwilloperateatoff-cruiseconditionsand utilizehighagility.Traditionalapproaches,whicharesatisfactoryfortraditionalvehicles, willnotbeabletomaximizemissioncapabilityfornextgenerationvehicles. Themainfocusofthisresearchisdevelopanintegrateddesignframeworkwhich accountsforthesecomplexinteractionsbetweensub-systemsandtosimultaneously designtheopen-loopdynamicsandthecontroller.Theproposedapproachattemptsto designtheopen-loopdynamicsalongwiththeoptimaloperatingrangeandcontrollerto optimizemissionandclosed-loopperformance,asillustratedinFigure 1-6. 1.4Objectives Theobjectiveofthisresearchistodevelopaframeworkforintegrateddesignwith thefollowingfeatures: Notcomputationallyexpensiveandcomplex Easytoincorporaterobustnessanalysis Generalenoughtoincorporatedifferentoptimizationobjectiveslikesingle-objective, multi-objective,mixtureoftimeandfrequencydomainmetrics 23

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Figure1-6. Problemstatement Hasanefcientsolutionmethodologywithrespecttotheoptimizationwhichcan accountfornon-convex,multi-modal,constrainedoptimizationproblems 1.5LiteratureSurvey Thissectionenlistssomeoftherelevantliteraturetothisresearch.Theissueof costfunctionisactuallyquitecriticaltotheinclusionofcontrolsynthesisfordesign optimization.Everydisciplinehasmetricsthatareuniquetotheirobjectivessoasingle costthatencompassesallthesemetricscanbechallengingtoformulate.Oneapproach thatconsidersvibrationcontrolusesnorms,bothforvibrationlevelandeffort,asa costinalinear-quadraticframework[ 1].Amixed-normapproachisformulatedthat considersboth H 2 and H 1 insummationtorepresentindependentmetricsofthe design[ 33].Also,apositive-realconditionacrossfrequencyisintroducedasacostthat hastime-domaininterpretationsfordesign[ 34 ]. Thedominantissueintheliteratureisactuallysolutionmethodologiestooptimize theresultingcostfunction.Sensitivityanalysisisusedinsomeforminmostapproaches suchasutilizationofnon-linearprogramming[ 13 ],adoptionofamulti-levelframework usinganupper-levelcoordinationformulation[ 17],amodalmodelthatdetermines stability[ 10 ].Iterativeapproachesareusedinseveralmethodsincludingonethat preserveseithertheclosed-loopsystemmatrixorthecovariancematrix[ 4 ]andanother 24

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thatsuccessiv elyredesignsusingaconstrainedgradientoptimization[ 5]andeven iterativeconstructionofasequenceofdesigns[ 18 ].Additionaltoolsforoptimization thatareappliedtomulti-disciplinarydesignincludephysicalprogramming[ 6],genetic algorithms[ 11 ],nestedoptimization[ 20],scaling[29],goalprogramming[ 35],simulated annealing[ 36 ],mixed-integerdynamicoptimization[ 37 ]andcooperativegame theory[ 38].Homotopymethodsarealsousedinseveralinstancesincludingan investigationusingactivesetalgorithmstotraceparametrizedoptima[ 39]and evaluationoffamiliesofdesign[ 40 ].Asetofapproachesaredevelopedspecically formulti-objectiveoptimizationusingParetofronts[ 19 41].Finally,theuseofsurrogate modelingisextensivelyutilizedfordesignoptimization[ 42]. Severalformulationsformulatecostfunctionsandsolutionmethodologiesfor designsthatincludelinearmatrixinequalities(LMI)associatedwith H 1 -normsynthesis. Onegeneratesanon-convexformulationandusesiterationstosolvetheassociated optimization[ 2 ].Anotherapproximatesfunctionsassociatedwithperturbedstate-space matricesasLMIstobesolvedusinganiterativeapproach[ 3].A2-stepprocedureis usedforanoptimizationcoupledwithanLMIsolverforthecontroller[ 43]asanMDO approach.Anotherapproachconsidersaniterativesequentialcontroldesignanda coupledredesignwitheachiterationinvolvingthesolutionofanLMI[ 28]. 1.6DissertationOutline ThisdissertationcanbesplitintothreecategoriesasshowninFigure 1-7.This chapterintroducesthetraditionalsequentialandsimultaneousdesignapproaches andtriestomotivatethenecessityoffollowingthesimultaneousdesignapproach forfuturisticsystemswith`high-performance'requirementandcouplingbetweenthe sub-systems. Backgroundtheory Chapter 2 discussesthethreepopularcontrolsynthesistechniquesusedin thisproject,namely, H 1 H 2 andlinearparametervarying(LPV)theory. 25

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InChapter 3 ,differentoptimizationalgorithmslikeglobalsearch,1-Diteration, surrogate-basedoptimizationalongwithefcientglobaloptimization(EGO) andnon-sortedgeneticalgorithms(NSGA-II)algorithmsarediscussed. Proposedapproach Chapter 4 introducesthecontrol-orienteddesignapproachwhichattemptsto simultaneouslydesigntheopen-loopdynamicsalongwiththeoperatingrange andcontrollertooptimizeclosed-loopperformance. Applicationsandcase-studies Chapter 5 introducestheapplicationofthehypersonicvehicleandmotivates theproblemtryingtobeaddressed.Aframeworkforlinear-parametervarying (LPV)controllerisalsodevelopedforthevibrationattenuationofthevehicle. Chapters 6 8 demonstratesthecontrol-orienteddesignframework,proposed inChapter 4,todesignthehypersonicvehiclealongwiththeoperating rangefor H 2 H 1 and LPV controllersanddifferentoptimizationmetrics.A case-studyofaninvertedpendulumandcartsystemisalsoinvestigated inChapter 8.Thecase-studiesinthesechaptersconsiderthereisno uncertaintyassociatedwiththeparametersandsignals. Chapter 9 performsuncertaintyanalysisandtriestoanalyzetheinuence ofuncertaintyontheclosed-loopperformancewithanapplicationtoa mass-spring-damperexample. Thenalchapterreviewsthekeyideasandcontributionsofthisproposal.Italso highlightsthepathwhichcanbepursuedinthefutureinthisareaanddiscussesthe implicationsandlimitationsofthisresearch. 1.7Contribution Thekeycontributionsofthisprojectare: Proposedacontrol-orienteddesignapproachforstructuralsystemsforapplications whereclosed-loopperformanceiscriticalandthereiscouplinginthedynamics. Standardcontroltechniquesareusedinthisformulationandthereisafreedomon thechoiceoftheobjectivestobeoptimized. Demonstratedthesimultaneousdesignapproachforthedesignoflinear parametervaryingsystemswhichdesignsthevehiclealongwiththeoperating rangeandcontrollerforoptimalperformance. Investigatedthefeasibilityofusingsurrogate-basedoptimizationwiththeefcient globaloptimization(EGO)algorithmandnon-sortedgeneticalgorithm(NSGA-II) 26

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Figure1-7. Overviewofdissertation fordesignoptimizationwithcontrols.Demonstratedthebenetsofusingmultiple surrogatesinthedesignoptimizationprocess. Proposedanapproachtondoptimalsolutionswhentheoptimizationobjectives arenon-normmetricsfornorm-basedcontrollersbyintroducingweightingfunction variablesinthedesignprocess. Theframeworkforcontrol-orienteddesignisanalyzedindetailforahypersonic vehicle. Introducedthelinear-parametervaryingframeworktocompensateforthe aerothermoelasticeffects.Themulti-loopcontrolarchitecturewiththeinner-loop beingtheLPVcontrollerisusefultocontroltheparametervaryingdynamics acrossthehypersonicightenvelope. Developtheframeworkforthedesignofathermalprotectionsystemfroma controlsperspective.Theactualdesignofathermalprotectionsystemcanbe chosentovarythelevelofheatingreduction,andassociatedweight,across thestructure.Thisresearchconsidershowsuchdesignsandresultingthermal gradientsinuencetheabilitytoachieveclosed-loopperformance. 27

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Tab le1-1.Motivationfor`simultaneousdesignofthedynamicsandthecontroller'from fewrelevantliterature Publication Motiv ation Grigor iadisetal.[ 28] Introduction :Simultaneousdesignofthestructure andthecontrollercanachievegreaterreductionsincontrol effortandimprovedperformancethanwhatcanbe achievedthroughsequentialdesignsynthesis. Venkayyaetal.[ 29] Introduction :Theinteractionbetweenthestructuresand controlsdesignershasbeenveryminimalinthepast. However,therehasbeenstrongindicationinrecentyears thatsignicantperformanceaswellascostimprovements canberealizedbyoptimizingthestructureandthe controlstogether. Lustetal.[ 18 ] Introduction :Thedesignofefcientstructuralsystemsisof fundamentalinteresttobothstructuralandcontrolengineers. However,forthemostpart,thesedesigntechniques, havebeenappliedindependentlywithintheoveralldesign process.Thedesignchallengepresentedbylargeexible spacestructureshasgeneratedinterestininterdisciplinary approachestothedesignproblem. Messacetal.[ 1] Introduction :Futuremilitaryandcommercialspacesystems willpresentincreasinglystringentperformancerequirements. Currentdesignpracticesarecharacterizedbyaschism betweenthestructuralandthecontroldesign processes.Thesetwodesignphasesareperformed separately,followingtwodisparatepaths,withlittleorno interaction.Asaresultofthislimitedinterdisciplinaryinteraction, thenalsystemdesignisseldomoptimalinany globalsense.Thisisparticularlytrueinthecaseofcomplex systemswhereexperienceorintuitionusuallyfailstoproduce optimal(ornear-optimal)designs. Hiramotoetal.[ 3 ] Introduction :Thislimitationofthe(sequential)design freedommaybecomeaseriousobstaclewhena highlytightspecicationisimposedontheclosed-loop systemandshouldberesolvedinsomesensefor thegenuineoptimalcontrolsystemsdesign.Theconceptof integrateddesignofstructuralandcontrolsystems,isa prescriptionfortheaboveprobleminthegeneraland currentframeworkofcontrolsystemdesign.Intheintegrated designschemedesignparametersbothintheplantandthe controlleraredealtwithequallyandoptimizedsimultaneously. Therefore,wecanexpecttoobtainbetterclosed-loop performancethanthatoftheconventionaltwostep design. 28

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Tab le1-2.Someissueswithexistingapproachestosimultaneousdesign.BMI-bilinear matrixinequality,NP-hard-non-deterministicpolynomial-timehard,LMIlinearmatrixinequality Publication Motivation Venka yyaetal.[ 29] Introduction :However,thereisalsoarealizationthat, foranumberofreasons,thiscombinedoptimization(of structureandcontroller)problemisnoteasilytractable. Hiramotoetal.[ 3 ] Introduction :However,unfortunatelyallintegrated designproblemsresultinakindofBMIproblemeven ifweassumethatthecoefcientmatricesoftheplant statespace(ordescriptor)formarelinearfunctions onstructuraldesignparameters.Itiswellknownthatthe BMIproblemisanNPhardproblem,i.e.,roughly speaking,therearenoefcientmethodsto obtaintheglobaloptimalsolution. Gilbertetal.[ 17 ] Introduction :Withthisapproach,thedesignproblem maybeofhighordersincethedesignmathematicalmodel mustincludenotonlythedynamicsofboththecontrol systemandthestructure,butalsothecombinedconstraint anddesignvariablesets. Messacetal.[ 1] Introduction :The(integrated)approachpresentedinthis paperishighlycomputationalinnature. 29

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CHAPTER2 CONTR OLSYNTHESIS Thischapterprovidesbackgroundonthethreecontrolsynthesistechniquesusedinthis dissertationresearch. 2.1Introduction Controltheoryisaninterdisciplinarysubjectthatdealswiththebehaviorof dynamicalsystems[ 44].Dr.D.BernsteinfromtheUniversityofMichigansummarized theevolutionofthetheoryoffeedbackcontroloverthepastcenturyinaseminaratthe DepartmentofMechanicalandAerospaceEngineering,UniversityofFloridainMarch 2009andisshowninFigure 2-1.Figure 2-2 andFigure 2-3 highlightthefeaturesofa typicalcontrolsystem.Controldesignbasicallyaddresseswhichcontroleffectorsneed tobevariedandbyhowmuchsoastoachieveclosed-loopobjectiveslikestabilityand robustperformancewhenwehaveknowledgeabouttheeffectorslikethelimitonthe actuatoroperatingrange. Thethreecontrolstrategiesusedinthisresearchare: H 1 H 2 LinearParameterVarying(LPV) 2.2Closed-loopSystemRepresentation Ageneralizeddescriptionoftheclosed-loopsystemisrepresentedinFigure 2-4 as afeedbackrelationshipbetweenoperators.Theopen-loopmodelgivenas P contains thenominalestimatesofthedynamics.Thefeedbackcontrollerisgivenas K and containstherelationshipbetweensensormeasurements, y andactuatorcommands, u Anotherelement, ,representstheuncertaintyassociatedwitherrorsandunknown 30

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Figure2-1. Evolutionofcontrol Figure2-2. Featuresofacontrolsystem[25] 31

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Figure2-3. Overviewofcontrol[45] valuesoftheopen-loopdynamics.Theremainingelements, W ,representsthe frequency-dependentweightingfunctionsthatnormalizetheerrors 1 Ageneralizedmodelisformulatedthatcapturestherelationshipbetweenthe open-loopdynamics,differentsignalsanddesiredpropertiesoftheclosed-loopsystem. Thismodel,calledassynthesismodel, S isshowninFigure 2-5 andthedynamics of S aregiveninEquation 2. S isusedtoformulatethefeedbackcontroller, K in Figure 2-4. 1 Itis muchmoreconvenienttoreectthesystemperformanceobjectivesby choosingappropriateweightingfunctions.Somecomponentsofasignalmaybe moreimportantthanothersandnotallthesignalsmaybemeasuredinthesame units.Weightingfunctionsessentially,canbeusedtomakesuchsignalcomponents comparable.Also,frequency-dependentweightshelptorejecterrorsinthedesired frequencyrange.TheweightingfunctionsinFigure 2-4 ischosentoreectthedesign objectivesandknowledgeofthedisturbanceandsensornoise.Forexample,the performanceweightingmaybeusedtoreectrequirementsontheshapeofclosed-loop transferfunctionsandtheactuatorweightingmaybeusedtoreectlimitsonthe actuatoroutputs[ 46].So,weightsareusedtoformulateperformanceobjectivesinto mathematicallytractableproblems.Ifthecontroldesignobjective(forascalarcase)isto ensurethatthenormofthetransferfunctionfrom d to e isinsomebounds,( kT de k ") itcanrephrasedas kWT de k 1 where kW k =1=". 32

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P K yu e d W Figure2-4. Closed-loopmodel.Sub-systems: P -open-loopdynamics, K -controller, W -weightingfunctions, -uncertaintyinsignalsandparameters.Signals: d vectorofexogenousinputsordisturbancesincludingreferencecommands, e -vectoroferrorstobeminimized, y -sensoroutputs, u -actuatorinputs P yu e d W Figure2-5. Synthesismodel, S S ( s )= 2 6 6 6 6 4 A B 1 B 2 C 1 0 D 12 C 2 D 21 0 3 7 7 7 7 5 2 6 6 6 6 4 x d u 3 7 7 7 7 5 (2) 2.3 H 1 Theory In thecontextofthisdissertation, H 1 isdenedasthespaceofmatrix-valued functionsthatareanalyticandboundedintheopenright-halfofthecomplexplane denedbyRe(s) > 0[ 46].The H 1 -norm, r isthemaximumsingularvalueofthe functionoverthatspace.Mathematically,itisgivenbyEquation 2 ,where S (s ) isa matrixoftransferfunctionsand isthe maximumsingularvalueofthematrix S (s ).This normcanbeinterpretedasamaximumgaininanydirectionandatanyfrequency.For 33

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Figure2-6. Physicalinterpretationof H 1 -normforaSISOsystem example,insingleinputsingleoutput(SISO)systems,thisiseffectivelythepeakofthe magnitudeofthefrequencyresponseasshowninFigure 2-6 2 kS k 1 =sup Re(s )>0 [ S (s ) ] =sup 2 R [ S (j ) ] (2) The H 1 controlsynthesis generallyfollowstheblockdiagramshowninFigure 2-4, asafeedbackrelationshipbetweentheopen-loopplant, P ,controller, K ,weights, W anduncertainty, [47]. H 1 techniquescanbeusedtominimizetheclosed-loopimpact ofaperturbation.Dependingontheproblemformulation,theimpactwilleitherbe measuredintermsofstabilizationorperformance[ 48 ]. H 1 -normcanbeseenasagain, sothecontrolobjectivecanbeformulatedas`minimizethegain( r )fromdisturbances (d )toerror( e )'.Thebasicdesignobjectiveistondacontroller, K ,tostabilizethe closed-loopsystemandtominimize kF l (S K )k 1 where F l (.,.) representsthelower linearfractionaltransformation(LFT)ofthesystem.Ithasbeenshowninliterature thatndinganoptimal H 1 controllerisnumericallyandtheoreticallycomplicated[ 49]. 2 Theph ysicalinterpretationofthe 1-normisthemaximumpeakofthesignalfora SISOsystem. 34

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Hence, asuboptimalcontrollerisdesignedsuchthatgivenascalar r> 0 ,nd K (s ) for whichEquation 2 isvalid[ 46, 47]. k F l ( S K )k 1 ( XY ) X > 0 Y > 0 (2) ThecontrolleristhengivenbyEquation 2. K (s )= 2 6 4 A +( r )Tj /T1_8 7.97 Tf 6.59 0 Td (2 B 1 B T 1 )Tj /T1_3 11.955 Tf 11.96 0 Td (B 2 B T 2 )X )Tj /T1_2 11.955 Tf 11.95 0 Td (( I )Tj /T1_1 11.955 Tf 11.96 0 Td (r )Tj /T1_8 7.97 Tf 6.59 0 Td (2 YX ) )Tj /T1_8 7.97 Tf (1 YC T 2 C 2 (I )Tj /T1_1 11.955 Tf 11.96 0 Td (r )Tj /T1_8 7.97 Tf 6.59 0 Td (2 YX ) )Tj /T1_8 7.97 Tf 6.59 0 Td (1 YC T 2 B T 2 X 0 3 7 5 (2) 2.4 H 2 Theory Similar tothe H 1 -norm,the H 2 -normisdenedinthecontextofthisdissertation,in thefrequencydomain,asinEquation 2.Itcanbeinterpretedastherootmeansquare valueoftheimpulseresponse. k S k 2 2 = tr 1 2 Z 1 S ( j ) S (j ) d (2) 35

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The H 2 controlsynthesis problemistondaproper,realandrationalcontroller, K that internallystabilizes S andminimizesthe H 2 -normofthetransfermatrix S ed from d to e [46]. ControllerExistenceConditions Inthe H 2 synthesisframework,anuniquefeedbackcontroller, K existswhich minimizesthe H 2 -normgiveninEquation 2 forasynthesismodel,ifandonlyifthere existssolutionsto X and Y inEquation 2.Here, X = X and Y = Y 0= X (A )Tj /T1_3 11.955 Tf 11.96 0 Td (B 2 R )Tj /T1_2 7.97 Tf 6.59 0 Td (1 1 D 12 C 1 ) +(A )Tj /T1_3 11.955 Tf 11.95 0 Td (B 2 R )Tj /T1_2 7.97 Tf 6.59 0 Td (1 1 D 12 C 1 ) X )Tj /T1_3 11.955 Tf 11.95 0 Td (XB 2 R )Tj /T1_2 7.97 Tf 6.59 0 Td (1 1 B 2 X +C 1 (I )Tj /T1_3 11.955 Tf 11.95 0 Td (D 12 R )Tj /T1_2 7.97 Tf (1 1 D 12 C 1 ) C 1 0= Y (A )Tj /T1_3 11.955 Tf 11.95 0 Td (B 1 D 21 R )Tj /T1_2 7.97 Tf 6.59 0 Td (1 2 C 2 ) +(A )Tj /T1_3 11.955 Tf 11.95 0 Td (B 1 D 21 R )Tj /T1_2 7.97 Tf 6.59 0 Td (1 2 C 2 )Y )Tj /T1_3 11.955 Tf 11.95 0 Td (YC 2 R )Tj /T1_2 7.97 Tf 6.59 0 Td (1 2 C 2 Y +B 1 (I )Tj /T1_3 11.955 Tf 11.95 0 Td (D 21 R )Tj /T1_2 7.97 Tf 6.59 0 Td (1 2 D 21 )B 1 (2) where, R 1 = D 12 D 12 > 0 R 2 = D 21 D 21 > 0 2.5LinearParameterVarying(LPV)Theory 2.5.1OverviewofGainScheduling Whenthedynamicsofthesystemchangeasafunctionoftheparametersoverthe operatingenvelope,itisdifculttomaintaindesiredperformancelevelthroughoutthe envelopeusinglineartimeinvariant(LTI)controllers.TheuseofLTIcontrollerseems somewhatlimitedintermsofstabilityandperformancethroughouttheoperatingrange. Forexample,themassofanairplanesignicantlychangesinightasthefuelisburnt, 36

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changingthe dynamics,butmostmaturecontroltechniquesaddressessinglepoint designforaparticularmass[ 2 ].Forsuchsystems,itmaybebenecialtodevelop controllersthatcanbelinkedtocertainpartofthedynamicsandthenschedulethemas afunctionoftheparametervariations. `GainScheduling' issaidtobetheregulationofasystemwhoseparameters arefunctionsoftheoperatingconditions.Itisbasicallya`divideandconquer'control strategywheretheoperatingconditionsarebrokendownintolinearsub-problems[ 50]. ItcanbebrokendownintothreestepsasshowninFigure 2-7 [51].Firstly,separatethe operatingrangeintosubspacesandcreateparameterizedmodelforeachsubspace. Then,createcontrollersforeachofthemodelsandthendevelopaschedulingscheme byinterpolatingbetweentheseregionalcontrollersforthelocalsubspaces.Thoughthis workswellformanyrealisticsystems,therearesomeissueswiththistechnique.For example,thereisnoguaranteeofstabilityandrobustnesswithrespecttouncertainties inthedynamicsandthereisalsoapossibilityofskippingbehaviorduringswitch betweencontrollers.Moreover,developingtheinterpolationschemetoswitchbetween controllercanbearigorousandtimeconsumingprocess.LPVthoughagainscheduling techniquetriestoaddresssomeoftheseissues. 2.5.2LinearParameterVarying LPVsystemsareemergingasanimportantclassofsystemswithmanyaerospace[52 58],mechanical[59, 60],electrical[61]andbiomedicalsystems,manufacturing andchemical[ 62 63 ]processesfallingintothiscategory.Therearemanypapersin literaturetryingtoaddressthecontrolissuesforthisclassofsystems[ 5266]. AtypicalcaseofaLPVplant, S (., ),whosedynamicalequationsdependon physicalcoefcientsthatvaryduringoperation,hastheformasgiveninEquation 2 37

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ASteps BIllustr ationofanexample Figure2-7.Stepsingainscheduling.Billustratesanexampleofaaircraftwhose dynamics, P andcontroller, K arefunctionofthealtitude( h ) where ( t ) isgivenbyequation 2. S (., )= 8 > > > > < > > > > : x = A()x + B 1 () d + B 2 ()u e = C 1 ()x + D 11 ()d + D 12 ()u y = C 2 ()x + D 21 ()d + D 22 ()u 9 > > > > = > > > > ; = 2 6 4 A((t )) B ((t )) C ((t )) D ((t )) 3 7 5 2 6 6 6 6 4 x d u 3 7 7 7 7 5 (2) ( t ) =( 1 (t ),..., n (t )), i i (t ) i (2) ( t ) isa timevaryingvectorofphysicalparameters,forexample,velocity,angleof attack,temperature; A, B 1 B 2 C 1 C 2 D 11 D 12 D 21 D 22 areafnefunctionsof (t ), x isthestatevector, y isthemeasuredoutput, e istheregulatedoutputorerrors, d istheexogenousdisturbances,and u istheregulatedinputshowninFigure 2-4. Iftheparametervector, (t ) takesvaluesin R n withcorners f i g N i =1 (N =2n ),the 38

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plantsystem matrixgiveninEquation 2 rangesinamatrixpolytopewithvertices S ( i ) [ 51].Givenaconvexdecomposition,asshowninEquation 2 of ,overthe cornersoftheparameterregion,thesystemmatrixisgivenbyEquation 2,if i exists. ( t ) = 1 1 +...+ N N i 0, N P i =1 i =1 (2) S ( ) = 1 S ( 1 ) +...+ N S ( N ) (2) Thissuggestsseekingaparameterdependentcontroller,asgiveninEquation 2 with thecurrentparametervalue (t ) given. K ( ., ) 8 > < > : = A K ( ) + B K ( ) y u = C K ( ) + D K ( ) y (2) Thecontrollerstate-spacematrices, A K (), B K (), C K ( ) and D K () inEquation 2 at theoperatingpoint ( t ) areobtainedbyconvexinterpolationoftheLTIvertexcontrollers asgiveninEquation 2. K i := 0 B @ A K ( i ) B K ( i ) C K ( i ) D K ( i ) 1 C A (2) Thisyieldsasmoothschedulingschemeofthecontrollermatricesbytheparameter measurements (t ). Inthisway,LPVcontrolsynthesisoffsetssomeofthedisadvantagesofgain schedulingbycreatingonecontrollerovertheentireoperatingrangewhichguarantees performanceandstability.Thecontrollerisinherentlyscheduledasafunctionofthe parameter, sonoexplicitinterpolationschemeisneed.Figure 2-8 highlightsthe differencebetweenLPVandgainscheduling. 2.6Summary Table 2-1 summarizedthedifferentcontrolsynthesistechniquesdiscussedinthis chapter. 39

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Figure2-8. IllustrationofthedifferencebetweengainschedulingandLPV Table2-1.Summaryofdifferentcontrolsynthesistechniquesusedinthisresearch H 1 H 2 LPV System LTI L TI LPV RobustnessAnalysisYes Yes Yes ControllerDynamicDynamicDynamic Controlobjectivemin H 1 -normmin H 2 -normmin L 2 -norm 40

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CHAPTER3 OPTIMIZA TIONALGORITHMS Thischapterprovidesaconcisebackgroundontheoptimizationalgorithmsusedinthe examplesinChapters 6 8. 3.1Introduction Figure 3-1 showsatypical`constrainedoptimizationproblem'withobjective, J (x ),where x isthevectorofdesignvariables.Thisoptimizationproblemforrealistic systemscanpotentiallybemulti-objective,complex,computationallyintensiveorhave non-convexdependenciesin x .Avarietyofnumericalapproachescanbeappliedto theminimizationincludingbranchandbound,simulatedannealing,neuralnetworks, globalsearch,geneticalgorithmsanditsvariations,althoughnoneofthetechniques providesaguaranteethattheglobalminimumwillbefound,if J ( x ) isnon-convex andmulti-modalashighlightedinFigure 3-2.Sincethisresearchisproposingan approachtosimultaneouslydesigndifferentsub-systemsforrealisticsystems,asolution methodology(optimizationalgorithm)isrequiredwhichdoesnotmaketheproblem computationallyexpensiveandatthesametimegivesgoodresults. 3.2GlobalSearch Abasicapproachtotheoptimizationoftheobjective, J (x ) issimplytoperforma globalsearch.Essentially,theconceptrequires J (x ) tobeevaluatedateveryfeasible valueof x 2 [x l x u ],where x l isthelowerboundand x u istheupperboundonthestates andthensearchoverallvaluesof J tondthe x whichoptimizes J .Thisconceptofa globalsearchwillcertainlyndtheglobaloptimumbutsuchoptimizationcomesatthe expenseofahighcomputationalcost. 3.3One-dimensionalIteration AnotherapproachtosolvingtheoptimizationprobleminFigure 3-1 canutilize aniterativeseriesof 1 -dimensionaloptimizations.Thefundamentalconceptforan n-dimensionaldesignspacewouldrequireholding n )Tj /T1_3 11.955 Tf 13.1 0 Td (1 variablesatxedvalues 41

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Figure3-1. Standardoptimizationproblem AUnimodal (Convex) J ( x ) BMulti-modal (Non-convex) J ( x ) Figure3-2.Objective, J ( x ) ofasingledesignvariable, x 42

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whilending theoptimalvalueoftheremainingvariable.Thisconceptwoulditerateby cyclingovereachofthe n variablesinthedesignspace.Thenaldesignwouldresult bychoosingthesetofvariableswiththelowestobjective.Arepresentativeconceptis showninFigure 3-3 foradesignspaceconsistingof2variablesthatmayeachassume 3possiblevalues. Figure3-3. 1-Diteration.Objectiveistominimize r ina2-Ddesignspace Algorithm : Chooseonedesignvariablerandomly Searchtheminimum J forthedesignspacevariablechosen.Notethevalueofthe seconddesignvariablecorrespondingtothis J Nowfortheseconddesignvariable,searchfortheminimum J .Notethevalueof therstdesignvariablecorrespondingtothis J Ifthesetofdesignvariablesfromtherstandtheprevioussteparethesame,then theminimumisalocalminimum.Repeattheabovestepsfornewvalueoftherst designvariable Else,continuewiththeprocess,tillalllocalminimaarefoundandallvaluesofrst designvariablesaresearched 43

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From allthevaluesoflocalminimaof J ,ndtheglobalminimum Thistechniqueisrelativelystraightforwardtoapplyand,duetoconstrainingeach optimizationtoa 1 -dimensionaldesignspace,hasarelativelylowcomputationalcost. Suchadvantagesinoptimizingaresomewhatoffsetbyadditionaldifculties.Oneissue tonoteisthelackofanyguaranteethattheglobalminimumfor J willbefound.Another issueisthepotentialforinnitecyclinginthattheiterativemaysimplyswitchbetween identicallocalminimaasthefreevariableischangedinthedesignspace. 3.4Surrogate-basedDesignOptimization 3.4.1SurrogateModeling Typically,severaldesignvariablecongurationswillbeexploredinadesign optimizationprocesswhichpotentiallyrequiresalargenumberofexpensivesimulations and/orexperiments[ 8].Moreover,theoptimizationobjectivesandconstraintsmaynot beasmoothfunctionoronlyabunchofdiscretepointvaluescorruptedwithnoise. Methodsthatcreateapproximationmodels(alsoknownassurrogateormeta-models) provideawayofobtaininghigh-delitymodelinformationintheoptimizationprocess withoutthecomputationalexpenseofsimulationmodels[ 42 67 69 ].Theexpensive actualsimulationresponse, y (x ) isapproximatedbyacheapermodel y pred basedon (i)assumptionsonthenatureof y (x ) ,and(ii)ontheobservedvaluesof y (x ) ataset of p datapointscalledexperimentaldesign.MoreexplicitlyitisgiveninEquation 3, where x =[ x 1 ,..., x d ] T isareal d -dimensionalvectorand "(x ) representsboththe errorofapproximationandmeasurement(random)errors.Itisimportanttonotethat thesurrogate, y pred maybemoreaccuratesince,theactualdatamaybecorruptedwith noise. y ( x )= y pred + "(x ) (3) e RMS = v u u t 1 V Z V e 2 (x )dx = v u u t 1 V Z V [ y pred )Tj /T1_2 11.955 Tf 11.96 0 Td (y ( x ) ] 2 dx (3) 44

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Theaccur acyofasurrogateismeasuredbytherootmeansquareerror, e RMS shown inEquation 3 .Duetothecomputationalcostofestimating e RMS ,crossvalidationis oftenusedasanalternativeforbothassessingaccuracyandsurrogateselection.It isattractivebecauseitdoesnotdependonthestatisticalassumptionsofaparticular surrogatetechniqueanditdoesnotrequireextratestpoints[ 70, 71].Crossvalidationis aprocessofestimatingerrorsbyconstructingthesurrogatewithoutsomeofthepoints andcalculatingtheerrorsattheseomittedpoints.Thesurrogateisttoallpointsexcept one,anderroriscomputedforthatomittedpoint.Thisprocessisrepeatedforallpoints toproduceavectorofcross-validationerrors e XV calledas PRESS RMS ,where PRESS standsforpredictionsumofsquaresandisgiveninEquation 3. PRESS RMS = r 1 p e T XV e XV (3) Ifthe surrogatemodelsareinexpensivetoobtain,manymoredesignvariablecongurations canbeevaluatedwithoutworryingaboutthecomputationalburden.Thedifferent surrogatemodelingtechniquesexploredinthisresearcharekriging(KRG)[ 72, 73], supportvectorregression(SVR)[ 7476],radialbasisneutralnetwork(RBNN)[77, 78] andpolynomialresponsesurface(PRS)[ 79, 80]asshowninTable 3-1. 3.4.2Optimization Surrogate-basedoptimizationevolvesinacyclethatconsistsofanalyzinganumber ofdesigns,ttingasurrogate,performingoptimizationbasedonthesurrogate,and nallyperformingexactsimulationatthedesignobtainedbytheoptimizationasshown inFigure 3-4.Thisworktakesadvantageofmultiplesurrogateswhenperforming optimization.Severalsurrogatesarettothesimulationcodesinsteadofone.Then, theoptimizationissolvedseveraltimes,eachoneusingdifferentsurrogatesforthe objectives.Useofmultiplesurrogatesisbenecialbecausettingmanysurrogatesand repeatingoptimizationsischeapcomparedtocostofsimulationsandmostaccurate 45

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Tab le3-1.Differentsurrogatesusedinthesurrogate-basedoptimization Surrogates Details KRG Krigingmodel: Constant trendfunctionandGaussiancorrelation Secondandthirdorderfullmodels RBNN Radialbasisneuralnetwork: Goal =(0.05 y ) 2 and Spread =1 = 3 SVR Supportvectorregression: kernel exp-Exponential functions lin-Linear poly-Polynomial gau-Gaussian linespl-Linearspline anospl-AnovaBSpline PRS Polynomialresponsesurface: Secondandthirdorderfullmodels surrogatedoes notguaranteetheglobaloptimum[ 81 82, 85].Table 3-2 highlightsthe optimizationalgorithmsusedalongwithmultiplesurrogatesinthisresearch. Table3-2.Optimizationalgorithmsusedinthisresearchalongwithmultiplesurrogates OptimizationObjectiv e Algorithm Single EfcientGlobal Optimization(EGO) MultiNon-sortedGeneticAlgorithm(NSGA-II) 3.4.2.1Efcient globaloptimization(EGO)algorithm Recentsurrogate-basedtechniquesusepredictionalongwithsurrogateerror estimatesinthedesignprocess.ThisresearchutilizestheEfcientGlobalOptimization (EGO)algorithmtoguidethedesignoptimizationprocessforsingleobjectiveproblems[ 72, 73, 83, 84 ].EGOcombinesthepredictionvariancefromkrigingandthepredictionfrom anyothersurrogatemodelingtechnique(possibly,krigingitself)andseekstheregion wherethedesigncanbeexpectedtoimproveintheoptimizationprocess. Therststepinthesurrogate-baseddesignoptimization(SBDO)usingtheEGO algorithmistotasurrogatefromanumberofsimulations.Thealgorithmthen 46

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Figure3-4. Stepsinsurrogate-baseddesignoptimization. x 1 and x 2 aredesign variables iterativelyaddsdatapoints,byevaluatingthecostatadditionalpointsinthedesign space,totheinitialsetinanattempttoimproveuponthepresentbestsample, y best .At eachiterationthenextpointtobesampledistheonewhichmaximizestheexpected improvement E [I (x )] asgiveninEquation 3 where b(.) and (.) arethecumulative densityfunction(CDF)andprobabilitydensityfunction(pdf)ofanormaldistribution; and (x ) isthepredictionstandarddeviation(hereestimatedasthesquarerootof thepredictionvariance).Figure 3-5 illustratestheoptimizationprocessusingtheEGO algorithm. E [I (x )]= (x )[ u b( u )+ (x )] u =[y best )Tj /T1_1 11.955 Tf 11.96 0 Td (y pred (x )] = (x ) (3) 47

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Figure3-5. IllustrationofoptimizationprocesswithEGOalgorithm Thisprocesswillbeexplainedusingthekrigingtechnique;however,themethodology remainsthesameforothertechniques.Thisdemonstrationconsiderssomecost function, Y ,thatisafunctionofaone-dimensionaldesignspaceof X asshownin Figure 3-6.Thesurrogatemodelconsidersthecostatadiscretesetofdesignpoints toextrapolatethroughoutthedesignspacewhileeliminatingerroratthatdiscreteset. Alongwiththemodel,theassociatedfunctionofexpectedimprovementindicatesthe probabilityofreducingerrorinthesurrogatebyaddingthecosttotheoptimizationat eachpointinthedesign. AKr igingmodel BExpected Improvement Figure3-6.Singlecycleoftheefcientglobaloptimization(EGO)algorithmshowing costfunctiondueto(a)krigingand(b)expectedimprovement Theprocessiteratesbyaddingobjectivesfromnewpointsinthedesignspace andupdatingthesurrogatemodel.TheexamplefromFigure 3-6 isupdatedbyadding 48

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asingle point,correspondingto X =0.5 whichisexpectedtoprovidethebiggest improvement,toresultinthesurrogatemodelinFigure 3-7.Notethevaluesofexpected improvementaredramaticallysmallinFigure 3-7 ascomparedtoFigure 3-6 dueto theinclusionoftheadditionalpoint.Thisprocessisrepeatedatleastfor5iterations andaddingamaximumofnpoints,wherenisthenumberofpointsintheinitial conguration. AKr igingmodel BExpected Improvement Figure3-7.UpdatedcycleoftheEGOalgorithmshowingcostfunctiondueto(a)kriging and(b)expectedimprovement 3.4.2.2Multi-objectiveoptimization Manyrealisticdesignproblemsoftenneedtosimultaneouslyoptimizeseveral objectivesanditisexpectedthatnosingledesignwillsimultaneouslyperformbest forallmetrics.Manytimestheobjectivescompetewitheachother.Forexample, structuraloptimizationproblemsmayneedtomaximizestiffnessalongwithminimizing weightofthestructure,controlobjectivesmaybeformulatedas`maximizeperformance byusingminimumcontrolenergy'.Insuchcases,itisconvenienttorepresentthe objectivesinatrade-offcurve(Paretooptimalfront)asshowninFigure 3-8.Pointsin theobjectivespacewhichareclearlypoorinboththeobjectivesarecalleddominated andpointswhicharenotdominatedarecalledasnon-dominatedorParetooptimal. Allnon-dominatedpointsmakeuptheparetofront.Inthisresearch,themulti-objective 49

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optimizationprob lemishandledusingtheelitistnon-dominatedsortinggeneticalgorithm (NSGA-II)alongwithmultiplesurrogatestondtheoptimalparetofront[ 8688]. Figure3-8. Illustrationofamulti-objectiveoptimizationproblem.Theproblemistond thedesignvariables, [x 1 x 2 ] whichminimizesboth y 1 and y 2 3.5Summary Thischapterdiscussestheglobalsearch,one-dimensionaliterationandsurrogate-based optimization.Thisresearchusesthesurrogate-basedoptimizationalongwithefcient globaloptimization(EGO)algorithmandnon-sortedgeneticalgorithm(NSGA-II) extensively. 50

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CHAPTER4 CONTR OL-ORIENTEDDESIGN 4.1Backgroundon `Multi-disciplinaryDesignOptimization' (MDO) `Multidisciplinary/integratedoptimization'canbedescribedasamethodologywhere theinteractionofseveraldisciplinesmustbeconsideredforthedesignofsystems,and wherethedesignerisfreetosignicantlyaffectthesystemperformanceinmorethan onediscipline[ 9].MDOproblemcanbediscernedintothreecategoriesshowninFigure 4-1.Itisbasicallya`simultaneousdesignandoptimization'approachforthedesignof thedifferentsubsystemsandexaminesthecouplingbetweenthem,asdescribedin Section 1.1.TheMDOproblemcanbesummarizedas Howtodecidewhattochange, andtowhatextenttochangeit,wheneverythinginuenceseverythingelse [8, 89 ]. Forexample,Figure 4-2 showsthedifferentsubsystemswhichmustbedesigned foragenerichypersonicvehicle.Thedesignofthevehicleincludesthedesignofthe geometry,selectionofthematerialforthestructure,enginecomponents,designofthe controlleramongstothers. 4.2Control-orientedDesign Chapter 1 motivatedtheneedforsimultaneousdesignoftheopen-loopdynamics andthecontroller.Thisresearchinvestigatesaframeworkforsimultaneousdesign foroptimalclosed-loopperformance.Theapproachattemptstodesignthedynamics alongwiththeoperatingrangeandacontroller,sointhisway,thedesignproblemis multi-disciplinaryinnature. 4.2.1SystemDesign Systemsareevaluatedbasedonclosed-loopmetrics,sothedesignspaceneeds toincorporateallthevariableswhichaffectthisability.Theclosed-loopoperationof realisticsystemssuggestthedecompositionofthedesignspaceintodiscipline-related subspaces.ForthehypersonicvehicleshowninFigure 4-2,someofthedesign variableswhichwillaffecttheclosed-loopoperationisillustratedinFigure 4-3. 51

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Figure4-1. MDOcategories[9] Figure4-2. Subsystemsofahypersonicvehicle [ http://www.dfrc.nasa.gov/Gallery/Photo/X-43A/HTML/ED98-44824-1.html reprintedwithpermission], [ http://nix.ksc.nasa.gov/info?id=ED97-43968-1&orgid=7, reprintedwith permission], [ http://commons.wikimedia.org/wiki/File:Scramjet operation.png repr inted withpermission] 52

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Figure4-3. Designspaceofthesubsystemsofahypersonicvehicle [ http://www.dfrc.nasa.gov/Gallery/Photo/X-43A/HTML/ED98-44824-1.html reprintedwithpermission], [ http://nix.ksc.nasa.gov/info?id=ED97-43968-1&orgid=7 reprintedwith permission], [ http://commons.wikimedia.org/wiki/File:Scramjet operation.png repr inted withpermission] P ( ) K () yu e d W (! ) Figure4-4. Closed-loopmodelillustratingthedesignspace.Sub-systems: P open-loopdynamics, K -controller, W -weightingfunctions, -uncertainty insignalsandparameters.Signals: d -vectorofexogenousinputsor disturbancesincludingreferencecommands, e -vectoroferrorstobe minimized, y -sensoroutputs, u -actuatorinputs.Designvariables: open-loopdynamicvariablesandoperatingrange, -controllervariables, -weightingfunctionvariables 53

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Theclosed-loop operationofthesystemshowninFigure 2-4,suggeststhe decompositionofthedesignspaceasshowninFigure 4-4.Partofthevariablesthat composethedesignspacearerelatedtotheopen-loopdynamics.Thisspace,dened as P,canincludeawidevarietyofvariablesincludinggeometry,structure,materials andotheraspectsrelatedtovehicledesign.Aparticularcongurationoftheopen-loop dynamicsisthusrepresentedbythevector, 2 P,withinthedesignspace. Partofthevariablesthatcomposethedesignspacearerelatedtothecontroller elementsthatmaybevaried.Thisspace,denedas K,canincludeaspectsofthe feedbackcompensatorsuchasgainsorvariablesusedtoselectstate-spacematrices. Anycontrolleristhusformulatedusingthevector, 2 K,fromwithinthedesignspace. Theremainingvariablesconsiderstheweightingfunctionsthatmaybevaried. Thisspace,denedas W,canincludepolesandzerosoftransferfunctionsalongwith gains[ 46].Aspecicsetofweightingsthatnormalizestheerrorsisassociatedwitha vector, 2 W,withinthedesignspace. Thesetofclosed-loopsystemsthatarepossiblecandidatesfortheoptimal congurationcanberepresentedby T .Thissetnotesthattheopen-loopplant, P ( ), dependsonthedesignspaceof P,weightingfunction, W (! ),on W andthecontroller, K ( ) ,dependsonthedesignspaceof K.Finally,thesetofallclosed-loopsystems T canbedescribedasaLFTasgiveninEquation 4. T = fF u fW ( ) F l ( P ( ), K () ) ,g 2 P, 2 W, 2 Kg (4) Also,thesetof T canutilizeastandardreduced-ordermodeloftheopen-loop dynamics.Standardtoolscancomputestate-spacemodelsusinghigh-delity approachesfromcomputationaluiddynamicsorcomputationalstructuraldynamics. Abasicrepresentationofastate-spacemodel, P = fA B C D g ,isintroduced inEquation 4 ,althoughotherrepresentationscaneasilybesubstitutedintothe 54

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approach. T = f F u fW ( ) F l ( fA( ), B ( ), C D g K () ) g 2 P 2 W, 2 K g (4) 4.2.2ProblemFormulation Theapproachforcontrol-orienteddesignisexpressedasminimizingclosed-loop metrics,givenas f 2R m showninEquation 4,bysearchingoverthedesignspace, givenas P, W and K,whilemaintainingthefeasibilityconstraintsforcontrollerexistence, asinEquation 4,where m isthenumberofobjectivefunctions. min 2 P 2 W 2 K f ( ) (4) suchthat g ( ) 0 (4) Controlsynthesisisaprocessofchoosingacontroller, K suchthatcertainweighted signalsaremadesmall[ 46].Thisresearchtakesadvantageofthewell-established theoryoflinearrobustcontrolsynthesistechniqueslike H 1 H 2 andLPVtodenethe smallnessmetric.Forexample, H 1 -normfor H 1 controlsynthesis. Theobjective, f ( ) showninEquation 4 canbesingleormulti-objective.As mentionedearlier,theissueofcostfunctionisactuallyquitecriticaltotheinclusionof controlsynthesisfordesignoptimization.Everydisciplinehasmetricsthatareuniqueto theirobjectivessoasinglecostthatencompassesallthesemetricscanbechallenging toformulate.Theycanbeacombinationofnormandnon-normmetrics,bothinthe timeandfrequency-domain.Thisresearchwillexploredesignproblemsinvolvingboth, multipleandsingle-objectiveoptimization. 55

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Theconstr aints, g ( ) showninEquation 4 arethecontroller-existence conditions,whichifsatisedguaranteestheexistenceofafeedbackcontroller.The computationof H 1 and H 2 controlsynthesisusingmoderntechniquesactuallyfollowsa two-stepprocedure. Theinitialstepiteratesoverafeasibilitycheckthatindicatesifacontrollerexiststo achieveaparticularvalueofclosed-loopperformance. Thenalstepcomputesthegainforthefeedbackcompensatorthatachievesthe optimalclosed-loopperformance. Thistwo-stepprocedureisimplementedinaprofessionalsoftwaresuchas MATLAB,becauseasetoffeasibilityconditionsissignicantlylesscomputationally expensivethanasetofsynthesisconditions.Soifacontrollerdoesn'texistinstep 1,it willbeknownrelativelyquicklyandlotmoredesigncongurationscanbeexplored. Inthisway,theoptimaldesignactuallydoesnotneedtocomputeboththe open-loopplant, P andcontroller, K simultaneously;instead,thedesigncansimply ndtheopen-loopdynamicsforwhichacontrollerexiststhatachievesthelowest closed-loopmetrics( f ).Thisissueofexistence,orfeasibility,isthecentralissuethat enablesclosed-loopdesign.Thedesignproblemcanberephrasedsoastondthebest open-loopdesignvariables, andtheweightingfunctiondesignvariables, ,forwhicha controller, K existsi.e.therearesolutionstothecontrollerdesignvariables, forwhich theclosed-loopperformanceisoptimal. Figure 4-5 summarizestheprocessofcontrol-orienteddesign.Thecosttobe optimizedisdirectlyrelatedtotheperformance.Thesubsystemstobedesignedare theopen-loopdynamicsandthecontroller.Theconstraints, g ,actuallycapturethe couplingbetweenthetwosubsystems.Theremainderofthissectiondiscussesthe optimizationobjectives, f andtheconstraints, g forthedifferentcontrolsynthesis techniquesexploredinthisdissertation. 56

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Figure4-5. Systemlevelperspectiveofcontrol-orienteddesign 4.2.2.1 H 1 controlsynthesis Therearetwodifferentformulationsexploredwiththe H 1 controllers,oneinvolvinga normconditionandtheothernon-normmetricsformulti-objectiveoptimization. 1.Normconditionsforobjective( f ) Themetricfordesigncanbecastasan H 1 -normconditionontheclosed-loop system.Suchanormrelatesthegainfromdisturbances, d toerrors, e inFigure 4-4, suchthatanincreaseinmissionperformanceisdirectlyreectedbyadecreasein thisnorm.Assuch,thedesignseekstondtheoptimalvaluesforboththeopen-loop dynamicsandacontrollertominimizethe H 1 -normcondition. Theapproachforclosed-loopdesignisnowexpressedasminimizingtheclosed-loop norm, r ,withrespecttothedesignspacewhilemaintainingthefeasibilityconstraints. SuchanoptimizationisformulatedinEquation 4 asminimizingthevalueof r by searchingoverthespaceofopen-loopvariables, ,andthesolutions, X Y ,tothe Riccatiequationswhichcomprisetheexistenceconditionsforacontroller.Inthis 57

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for mulation,theweightingfunctionvariables, areusedtoreectrequirementsonthe performanceandactuationpenaltyandremainxed[ 90]. ThecontrollerexistenceconditionsareexpressedasthepairofRiccatiequations inEquation 4 andEquation 4 alongwiththeconstraintofspectralradius( )in Equation 4.Theopen-loopvariables( )areintroducedthroughthedynamicsofthe open-loopsynthesismodel, A( ), B i ( ) and C i ( ) where i =1,2 showninEquation 2. Theopen-loopsynthesismodelisthemodelwithoutthecontroller, K asshownin Figure 2-5.Theelements, of K = fX Y jX = X > 0, Y = Y > 0 g arethesolutions totheRiccatiequationsneededforcontrollerexistence.Theoptimizedsystemis determinedbytheargumentswhichminimize r inEquation 4. min 2 P X = X > 0 Y = Y > 0 r (4) subjectto 0= XA( )+ A ( ) X +X ( 1 r 2 B 1 ( )B 1 ( ) )Tj /T1_2 11.955 Tf 11.96 0 Td (B 2 ( )B 2 ( ) )X +C 1 ( ) C 1 ( ) (4) 0= A ( )Y + YA( ) +Y ( 1 r 2 C 1 ( ) C 1 ( ) )Tj /T1_2 11.955 Tf 11.96 0 Td (C 2 ( ) C 2 ( ))Y +B 1 ( )B 1 ( ) (4) r 2 > (XY ) (4) 2.Non-normconditionsforobjective( f ) Theactualmetricsthatrelateperformancecanoftenbecastasnormsonthe transferfunctionsfromcommandstoresponsesasshowninEquation 4;however, 58

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somemetr icsmustberepresentedbynon-normmetrics.Forexample,metricsfor handlingqualitiesofaircraftareamixofnormandnon-normmetrics[ 91]. Suchnon-normmetricsforperformancestillusenorm-basedcontrollerslike H 1 controllerswhichminimizetheclosed-loop H 1 -norm.Inordertoincorporatesuch non-normmetrics,thisapproachintroducesweightingfunctionvariables, tothedesign spacethatrepresentweightingsontheerrorsoftheclosed-looptransferfunction.These weights, W (! ) aredeterminedbydesiredcharacteristicsfortheclosed-looptransfer functionanditsassociatednorm;however,itisunclearhowtodeterminetheseweights fornon-normmetricsofthatclosed-looptransferfunction.Assuch,theintroductionof theweightsinthedesignprocessallowsanoptimal H 1 controllertobesynthesized thatminimizesanonH 1 metric.Theweightingfunctionvariables, enterthrough thedynamicsofthesynthesismodel, A( ), B i ( ) and C i ( ) where i =1,2 in Equation 2. Forexample,iftheobjectiveistominimizetheperformanceerroralongwiththe controlpowerusing H 1 controlsynthesis,thephysicalinterpretationisasfollows: Theobjectiveistominimizethenormofavectorwithcomponentsofcontrolpower andtrackingerrorforTseconds A H 1 controllerisdesignedthatminimizesthe H 1 -normforasynthesismodel Thedesignvariablesaretheoperatingrangeandweightingfunctions.The weightingfunctionsaffectthecontroldesignbutarenotdirectlypartoftheactual costfunction Chapter 7 describesthisprocessindetailforthedesignofarealisticsystem. 4.2.2.2 H 2 controlsynthesis Ametriccanalsobeexpressedasthe H 2 -normconditionontheclosed-loop system.Suchagainagainreectstheperformanceassizeoferrorsgivendisturbances soadecreaseinnormindicatesanincreaseinperformance.Theresultingoptimization isformulatedinEquation 4 asminimizingthenorm, r ,whilesearchingoverthe open-loopvariables, ,andthesolutions, X Y ,associatedwiththeexistenceconditions 59

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for acontroller.Thenorm r inthiscaseisactuallya2-normmetricfromdisturbanceto errors 1 min 2 P X = X > 0 Y = Y > 0 r (4) subjectto 0= X (A( ) )Tj /T1_5 11.955 Tf 11.95 0 Td (B 2 ( )R 1 ( ) )Tj /T1_8 7.97 Tf 6.59 0 Td (1 D 12 ( )C 1 ( )) +(A( ) )Tj /T1_5 11.955 Tf 11.95 0 Td (B 2 ( ) R 1 ( ) )Tj /T1_8 7.97 Tf 6.59 0 Td (1 D 12 ( ) C 1 ( )) X )Tj /T1_5 11.955 Tf 11.95 0 Td (XB 2 ( )R 1 ( ) )Tj /T1_8 7.97 Tf (1 B 2 ( )X (4) +C 1 ( ) (I )Tj /T1_5 11.955 Tf 11.96 0 Td (D 12 R )Tj /T1_8 7.97 Tf 6.59 0 Td (1 1 D 12 C 1 )C 1 ( ) 0= Y (A( ) )Tj /T1_5 11.955 Tf 11.95 0 Td (B 1 ( ) D 21 ( ) R 2 ( ) )Tj /T1_8 7.97 Tf (1 C 2 ( )) (4) +(A( ) )Tj /T1_5 11.955 Tf 11.95 0 Td (B 1 ( ) D 21 ( ) R 2 ( ) )Tj /T1_8 7.97 Tf (1 C 2 ( ))Y )Tj /T1_5 11.955 Tf 11.95 0 Td (YC 2 ( )R 2 ( ) )Tj /T1_8 7.97 Tf (1 C 2 ( )Y +B 1 ( )(I )Tj /T1_5 11.955 Tf 11.95 0 Td (D 21 R )Tj /T1_8 7.97 Tf (1 2 D 21 )B 1 ( ) (4) where, R 1 = D 12 D 12 > 0 R 2 = D 21 D 21 > 0 Non-normmetricscanalsobeoptimizedusingthenorm-based H 2 controllerbythe inclusionoftheweightingfunctionsinthedesignprocess.Butthisdissertationdoesnot demonstratethisapproachfor H 2 controllersynthesisusinganexample. 1 Refer Chapter 2 fordetailson H 2 -norm. 60

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4.2.2.3LPV controlsynthesis IntheLPVcontrolsynthesisframework,theperformancemetricistheinduced L 2 norm, r denedinEquation 4 [65, 66].Theapproachforcontrol-orienteddesign isnowexpressedasminimizingclosed-loopmetrics,givenas r ,bysearchingover thedesignspace,givenas P and K,whilemaintainingthefeasibilityconstraintsfor controllerexistenceshowninEquation 4,Equation 4 andEquation 4 2 where i =1,..., N isfromEquation 2 N X and N Y arethenullspacesof (B 2 T D 12 T ) and ( C 2 D 21 ) respectively, r isaclosed-loopnorm, A( ), B 1 ( ), B 2 ( ), C 1 ( ), C 2 ( ), D 11 ( ), D 12 ( ), D 21 ( ) D 22 ( ) inEquation 2 ,Equation 4,Equation 4 and Equation 4 arethedynamicsofthesynthesismodel, S showninFigure 2-5 [ 92, 93]. Asmentionedearliertheweightingfunctionvariables, remainxedfornormmetrics. r =sup 2,k d k 2 6=0 ke k 2 kd k 2 (4) min 2 P 2 K r ( ) (4) suchthat, 2 6 4 N X 0 0 I 3 7 5 T 2 6 6 6 6 4 A i X + XA T i XC T 1 i B 1i C 1i X )Tj /T1_4 11.955 Tf (r I D 11 i B T 1i D T 11i )Tj /T1_4 11.955 Tf (r I 3 7 7 7 7 5 2 6 4 N X 0 0 I 3 7 5 < 0 (4) 2 Thee xplicitdependenceonthedesignvariableshasbeenomittedforbrevity. 61

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2 6 4 N Y 0 0 I 3 7 5 T 2 6 6 6 6 4 A T i Y + YA i YB 1 i C T 1i B T 1i Y )Tj /T1_6 11.955 Tf (r I D T 11 i C 1i D 11i )Tj /T1_6 11.955 Tf (r I 3 7 7 7 7 5 2 6 4 N Y 0 0 I 3 7 5 < 0 (4) 2 6 4 XI I Y 3 7 5 0 (4) 4.2.3SolutionMethodology Theconstrainedoptimizationproblempresentedinthischapterrequiresndinga minimumtoanonlinearfunction.Theoperatorsof X and Y ,iftheyexist,canbefound foranyxedvalueof and usingstandardalgorithms;however,theyarealmost certaintohavenon-convexdependencieswhenconsideringall 2 P and 2 W. ThisresearchusestheoptimizationalgorithmspresentedinChapter 3 tosolveforthe optimizationformulations. 4.2.4AdvantagesofProposedApproach Thissimultaneousdesignapproachwithfocusoncontroller-existenceattempts toovercomesomeofthedrawbacksofthesequentialandsimultaneousoptimization approachesinliterature. Advantagesoversequentialapproach[ 25]: Reductionindesigncycletime.Since`controls'isconsideredearlierinthedesign process,thisreducestheiterationsnecessarybetweenthedifferentdesignteams toaccommodatedifferentobjectivesasmentionedinChapter 1 Optimalchoiceofactuators,sensorplacementetc.Closed-loopsystemproperties likecontrollabilityandobservabilityareguaranteedtobemetatthepreliminary designstageitself. Advantagesoverexisting`simultaneousoptimization'approaches: Notcomputationallyexpensive.Thisapproachdoesnotmaketheproblem computationallyexpensiveassolvingfortheexistenceconditionsiscomputationally easiercomparedtothesynthesisconditions.Useofsurrogatetechniquesfor optimizationbottlesdownthecomplexityofthedesignoptimizationproblemas well[ 42]. 62

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Robustness analysis.Itiseasytoincorporaterobustnessanalysisinthis framework. Choiceofobjectiveforoptimization.Thisframeworkgivesthefreedomto choosetheobjectiveswhichcanbesingleormulti-objective,mixtureofnorm andnon-normmetrics,inthetimeandfrequencydomain. 4.3Applications ThisdesignformulationisimplementedfordifferentexamplesinChapters 6 9.The optimizationobjectiveforthedifferentapplicationsissummarizedinTable 4-1. Table4-1.Objectivesusedfordifferentapplicationsinthisdissertation Control Objective System Design Synthesis min f Variables H 1 H 1 -norm mass-spr ing-damper hypersonicvehicle performanceerror,controlpowermass-spring-damper hypersonicvehicle H 2 H 2 -norm hypersonicvehicle LPV L 2 -norm Inverted-pendulum andcart mass-spring-damper hypersonicvehicle 4.4Summar y Thischapterintroducedacontrol-orienteddesignapproachtodesigntheopen-loop dynamicsalongwiththecontrollerandtheoperatingrangeforoptimalperformance forstructuralsystems.Thisisacontrollerexistence-basedformulationwhichtries toovercomethedrawbacksofsomeoftheothersimultaneousdesignapproaches presentedinliterature.Figure 4-6 illustratestheprocess.Firstyouchooseasetof designvariables,runsimulationsandevaluatediscretevaluesoftheobjective,then theoptimizationalgorithmsdiscussedinChapter 3 isusedtondtheoptimalvalueof thedesignvariables.ThisresearchtakesadvantageofsomeMatlabin-builtfunctions 63

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Figure4-6. MDOapproach likehinfsyn(),h2syn()[ 94 ]andhinfgs()[92 93, 95]tosolveforthecontrollerexistence conditionsforthe H 1 H 2 andLPVcontrolsynthesistechniquesrespectively. 64

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CHAPTER5 LPV CONTROLDESIGNFORVIBRATIONATTENUATIONOFAHYPERSONIC VEHICLE Flightathypersonicspeedswillinducesignicantheatingonthevehicle.The resultingthermalgradientswillcausespatially-varyingtemperaturesthatrangeacross thefuselage.Thesethermalgradientswillaffectthemodalpropertiesofthestructural dynamicsand,duetoaerothermoelasticcoupling,affecttheightdynamics.This chapterintroducesthelinearparameter-varyingframeworktocompensatefortheeffects ofthethermalproles.Thedynamicsofahypersonicvehicleareexpressedwitha dependencyonthetemperatureatdifferentlocationsalongthefuselage.Acontroller isthensynthesizedthatisinherentlyscheduledonthesetemperaturevalues.The approachisdemonstratedtoachievevibrationattenuationofastructuralmodefora widerangeofthermalproles. 5.1Introduction Hypersonicightisbeingaggressivelypursuedasacapabilitytotraversethe worldinafewhours.Aclassofvehiclesunderconsiderationutilizeadesigninwhicha wedge-shapedfuselageprovidesliftandactsasaninletfortheSCRAMjetengineas showninFigure 5-1.Thiscongurationanditsassociatedaeropropulsivecharacteristics wassuccessfullydemonstratedontheX-43prototype. Aprimarycharacteristicofthisvehicleaffectingthedynamicsistheintegrated fuselageandpropulsionsystemhighlightedinFigure 5-2.Thefuselageisactually designedtobepartoftheenginesystembyusingtheforebodyasacompressor andtheaftbodyasanexternalnozzle.Thisdesignintroducesasignicantamountof couplingbetweentheaerodynamicsandpropulsiondynamics.Firstly,theairowacross theforebodyintroducesaliftforceandanose-uppitchingmomentwhiletheairow throughtheexternalnozzleintroducesaliftforceandanose-downpitchingmomentso variationsinpropulsionperformancealtertheaerodynamiccharacteristics.Conversely, anyvariationinangleofattackandsideslipaffectstheengineinletconditionssothe 65

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Figure5-1. Air-breathinghypersonicvehicle [ http://mix.msfc.nasa.gov/abstracts.php?p=2468, reprintedwithpermission ] propulsionperformanceisalteredbyvariationsinaerodynamiccharacteristics.Also, thebendingstructurechangestheowturnanglewhichchangestheinletconditions whichinturnchangesthethrustanglesothereisanespeciallystrongandfastcoupling betweenpitchandpropulsion. Acommonmaneuverissimplyaprescribedchangetoairspeedandaltitude; however,severaldifcultiesmustbecircumventedforthisbasicmaneuver.Since thepropulsionsystemistightlycoupledtothestructuraldynamicsofthefuselage, vibrationscancauselossofengineperformance.Thevibrationsarecompoundedbythe introductionofthermalgradientswhichresultfromthetremendousheatingacrossthe fuselagethroughoutight.Assuch,vibrationattenuationbecomesacriticalaspectof missionperformance. 5.2Backgroundon Aerothermoelasticity `Aerothermoelasticity'canbesaidtobetheresponseofelasticstructuresto aerodynamicheatingandloading.Atetrahedron,showninFigure 5-3,canbeusedto deneaerothermoelasticity[ 96, 97]. A, I E ,and H representtheaerodynamic,inertia, 66

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Figure5-2. Coupleddynamicsofthehypersonicvehicle elasticandthermalforcesrespectively.Thevarioussub-elementsofaerothermoelasticity areindicatedbytheedgesandfacesofthetetrahedron.Forexample, 1 representsthe naturalvibrationswhichareduetointeractionsbetweentheinertialandelasticterms, 11 representstheentiretetrahedron.Detailedexplanationsforeachofthesubelements canbefoundinthecitedliterature[ 96, 97]. Thevarioussubelementsarelistedbelow: 1-Vibration 2-Stability 3-Aerothermodynamics 4-Thermoelasticity 5-Aeroelasticity,static 6-Thermalmolecularprocesses 7-Aeroelasticity,dynamic 8-Stabilityandheat 67

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9Aerothermoelasticity,static 10-Vibrationandheat 11-Aerothermoelasticity,dynamic Figure5-3. Aerothermoelasticitytetrahedron Theaerothermoelasticeffectsduringatypicalightprolewerestudiedforthe NationalAerospacePlane(NASP)[ 98, 99].TheNASPhadasimilarshapetothe hypersonicvehicleshowninFigure 5-1.Thisstudynotedthatthesurfacetemperatures couldrangefrom 0 o F tonearly 5000 o F atcertainpointsandresultinlargesurface gradients.Consequently,thenaturalfrequenciesanddampingofthestructuralmodes canvarysignicantlybyupto30%.Theseeffectswillbeusedasrepresentativeeffects thatmaybeencounteredforthegeneralclassofvehiclesconsideredinthisstudy. Theaerothermoelasticeffectswerenotedtocauseadecreaseinnaturalfrequency anddampingofthestructuralmodes.Thiseffectisincorporatedbyformulatingthestate matrixasanafnefunctionoftemperature.Therangeoftemperaturesconsideredfor thismodelischosenas 2 (0 o F ,1000 o F ) tomatchtheoperatingrangeofTitanium(Ti). 68

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Hypersonic VehicleandAerothermoelasticity Inrecentyears,thefocusintheareaofhypersonicaeroelasticityand aerothermoelasticityhasbeenpredominantlyonthedevelopmentofcomputational aeroelasticandaerothermoelasticmethodscapableofstudyingcompletehypersonic vehicles[ 100108]. Ithasbeenshowninliteraturethattheexactcomputationofthethermaleffects ontheaerodynamicsofanaerospacecraftinthehypersonicregimeisdifcult[ 109]. Hence,thisstudyconcentratesonlyontheeffectsofaerothermoelasticity.Theway thisisdoneisbynotingthevariationsinthestructuralpropertiesasafunctionof temperature.Forexample,Figure 5-4 showsthevariationsintherepresentativemode shapesforabeamatdifferenttemperatureswhichcanbeanalyzedtostudytheeffect oftemperatureonthedynamics.Inthiscase,thebendingmodeisextractedtoindicate changessuchasnodelocation,anti-nodelocation,andmagnitudeofoscillation.This behaviorisincorporatedintoightmodelsthroughaerothermoelasticdynamics. Figure5-4. Variationinmodeshapeswiththermalprolesforarepresentativemodel Inordertorepresentthedynamicsofthevehicleaccurately,themodelmustbe formulatedtoincludetheeffectsofstructuralexibilityinadditiontothedynamicsof therigid-body.Aformulationforthestate-spacemodelisintroducedthatincludes rigid-bodyandstructuraldynamics.Themodeldevelopedhereisbasedonthecited literature[ 110112].Thegeneralformofthestate-spacemodelisgiveninEquation 5 69

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where, x is thevectorofstates, u isthevectorofcontrolinputs, A isthestabilitymatrix and B isthematrixofcontrolinputs[113]. x = Ax + Bu (5) Withthestructuraleffectsincluded,AandBareoftheform, A = 2 6 6 6 6 6 6 6 6 6 6 4 RigidBody Terms Aeroelastic CouplingTerms RigidBody CouplingTerms Structural FlexibilityTerms 3 7 7 7 7 7 7 7 7 7 7 5 B = 2 6 4 RigidBodyControl StructuralModeControl 3 7 5 wherethe`RigidBodyTerms'aretherigid-bodyportionsofthemodel,the`Structural FlexibilityTerms'arethedynamicsofthestructuralmodesincludedinthemodel,and the`AeroelasticCoupingTerms'and`RigidBodyCouplingTerms'arethecoupling betweentherigid-bodyandthestructuralexibilityofthevehicle. 5.3ControlIssues AtypicalmissionforthisvehiclewouldbetoputapayloadintolowEarthorbitwhich wouldrequirethevehicletooperateinmanyightregimessuchassubsonic,transonic, supersonic,hypersonicandorbital,asmentionedearlier.Eachregimeintroducescontrol problemsthatmustbealleviatedforasuccessfulmission.Forexample,thecontrol surfaceswillprobablybesmallsoastominimizeheatingduringhypersonicight,but thismaycreatedifcultiesforproperlycontrollingthevehicleatlowsupersonicspeeds. Anotherpotentialcontrolproblemmayarisefromtheshocksgeneratedbyunsteady aerodynamicsattransonicight.Also,theissueoforbittransfersforpayloaddelivery 70

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whilein spaceisacontrolproblemforthistypeofvehiclethatintroducesissuesnot usuallyaffectingatmosphericight.Thecontrolproblemsineveryightregimeare important;however,thisstudywilllimitconsiderationtothehypersonicregimewhile stillintheatmosphere.Onereasonforlimitingconsiderationtothisregimeissimplyto concentrateonasmallersetofproblemssothatausefulsolutioncanbeformulated inashorttime.Anothervalidreasontorestrictattentiontothislimitedightregimeis becauseitavoidstheissueofchoosingasingle-stageortwo-stagetoorbitvehicle. Everyair-breathingvehiclemustpassthroughtheatmospherichypersonicregime regardlessofwhetheraboosterwasusedinitiallyoradifferenton-boardpropulsion systemplacedthevehicleatlowhypersonicspeeds.Thus,thisprojectwillassumeitis feasibletoplacethevehicleathypersonicightconditionsandfocusonthedifcultiesin thisregime. Severalcontrolissueswereidentiedforinvestigatinghypersonicightthrough theatmospherethatmustbeinvestigated.Firstly,thecontrollermustactivelysuppress modalvibrations.Theaerothermoelasticeffectsmustbecompensated.Thisissueis generallynotconsideredfortraditionalaircraftbutisquiteimportantforhypersonicight. Thedegreeofheatingresultingfromhypersonicightathighdynamicpressurecanbe tremendousandresultinchangingmaterialpropertiessuchasstiffness.Thischange instiffnesscanhaveadramaticeffectonclosed-looppropertiesbecausethecontroller mustaccountforthelowfrequencyfuselagebendingmodeandalsothechangesto thosemodaldynamicsbecauseofaerothermoelasticeffects. Secondly,thechoiceofacontrolarchitectureiscloselyrelatedtotheprevious issue.Generatingasinglestate-spacecontrollerthatprovidesstabilityandperformance seemssomewhatlimitedbecauseitmaybeadvantageoustolinkcertainpartsof thecontrollertocertaindynamicsofthemodel.Inparticular,vibrationsuppression isanextremelydifculttasksoitseemslogicaltoseparatesomeelementsofmodal controlfromthemainightcontrollertolocalizesomeaspectsoftheaerothermoelastic 71

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dynamics. Thisstructuredapproachallowsthedesignertomakesmallchangesto onlyasmallpartofthesynthesismodeltoimproveaspecicclosed-loopperformance problem.Thecontrollermustincludegain-schedulingstrategiesthatareparticularly suitableforhypersonicight. 5.4LiteratureSurvey Controlofthemodelsrepresentingtheightdynamicsofhypersonicvehicles,which arecharacterizedbyaeropropulsiveandaeroelasticinteractions,haveconsideredmany approachestoaddressthenumerouschallenges[ 114, 115].Theseapproachesinclude classicaldesign[ 116]alongwith H 1 [117], [118]andalinearparameter-varying synthesis[ 119].Anadaptiveslide-modecontrol[120]wasconsideredaswasa modelreferenceadaptivecontrol[ 121]andnonlineardynamicinversion[ 122].Other approachesconsideredamodel-predictiveschemeofcontrolallocation[ 123]anda robustapproachbasedonservomechanismtheory[ 124 ].Theoveralllimitsofcontrol performancewerenotedforcontrolsynthesisresultingfromtheinteractionsofexible effectsonhypersonicvehicles[ 125127].Additionaldiscussionsofcontrolstrategies andsensorplacementhavebeenperformed[ 128, 129]. Theissueofheatingmustalsobeconsideredinthesynthesisofcontrollers forhypersonicight.Computationalanalysisindicatesthetemperaturemayrange signicantlyacrossthefuselageduringatypicalmission[ 98].Theassociatedstructural dynamicswillalsovarysignicantlyintermsofmodaldampingandnaturalfrequency.A linear-quadraticregulatorwasdesignedthatconsideredthedynamicsattheworst-case temperaturewiththelowestdampingofthestructuraldynamicsandassumedtobe sufcientforanytemperaturethatresultedinhigherdamping[ 99].Again-scheduled approachwasinvestigatedthatdesignedarobustcontrollerwhichvariedwithasingle valueoftemperaturethatwasassumedtobeuniformacrossthefuselage[ 130]. 72

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5.5Objective Flight athypersonicspeedswillinducesignicantheatingonthevehicle.The resultingthermalgradientswillcausetime-varyingtemperaturesthatrangeacross thefuselage.Thesethermalgradientswillaffectthemodalpropertiesofthestructural dynamicsand,duetoaerothermoelasticcoupling,affecttheightdynamics.The objectiveofthisexampleistointroduceacontrollerintheLPVframeworktocompensate fortheeffectsofthethermalproles.Amodel-followingapproachisused,wherethe closed-loopsystemisexpectedtoemulateareferencemodelwithdesiredclosed-loop properties. 5.6ApproachtoIntroduceThermalProledependencyontheDynamics Variousthermalgradientsalongthefuselageofthevehicleareintroducedinto themodelsimulatingthetemperaturesattainedbythevehicleinightandtostudy thethermaleffects.Structuralparameterslikenaturalfrequencies,modeshapes, dampingdependonthematerialpropertiesliketheYoung'smodulus(E)andthe Young'smodulusinturnaredependentonthetemperature[ 131].Bymodelingthe Young'smodulusasthefunctionofthethermalprolealongthevehicle,theeffectsof temperatureontheightdynamicsarecaptured.Thefuselagehasbeendividedinto nineequalsectionsasshowninFigure 5-5 fortheaerothermoelasticanalysiswitheach sectionhavingaconstanttemperaturevalueandhenceconstantmaterialproperties. Figure5-5. Fuselagesection 73

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5.7V ehicle Arepresentativemodelisusedinthisresearchthatdescribestheightdynamics forabaselineconguration[ 132137].Thisreduced-ordermodelhas5rigid-bodystates correspondingtotheforwardvelocity (v ),angleofattack ( ),altitude ( h ),pitchrate (q ) andpitchangle ( ) .Additionally,itcontains6exible-bodystatescorrespondingto3 bendingmodesforthefuselage ( i ,_ i ) .Thedynamicsincluderesponsesto4control effectorsgivenaselevatorangle ( e ),canardangle ( c ),diffuser-arearatio ( A d ) and fuel-owratio (b) Anassumedmodesmodelisconsideredforthelongitudinaldynamicstoincorporate structuraldynamicsandaerothermoelasticeffectsinthehypersonicvehicledynamic modelasgiveninEquation 5 where m 2 R denotesthevehiclemass, I yy 2 R isthe momentofinertia,g 2 R istheaccelerationduetogravity, T h ( t ) 2 R denotesthrust, D (t ) 2 R denotesdrag, L (t ) 2 R islift, i i 2 R arethedampingfactorandnatural frequencyofthe i th exiblemode,respectively,and N i 2 R denotegeneralizedelastic forces[ 135].Thecouplingbetweenthedifferentdisciplinesisintroducedthroughthe moments, M ,theaerodyanmicforces, L and T h andthegeneralizedforces, N i .Inthis research,onlytherst-bendingmodeisincludedsincetheeffectsofthehighermodes arenegligibleontheightdynamics. Alinearizedmodelisobtainedfromperturbationsaboutatrimconditionforthis model.Theequationsthatdenetheaerodynamicandgeneralizedmomentsand forcesarelengthyandareomittedforbrevity.Detailsofthemomentsandforcesare providedinBolenderandDomain[ 132].Errorsduetolinearizationofthedynamicsare accountedforintheouter-loopcontrollerforrigid-bodyperformance. 74

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_v = T h cos( ) )Tj /T1_0 11.955 Tf 11.96 0 Td (D m )Tj /T1_0 11.955 Tf (gsin( )Tj /T1_2 11.955 Tf ( ) h= vsin( )Tj /T1_2 11.955 Tf 9.29 0 Td ( ) = )Tj /T1_0 11.955 Tf 10.49 8.09 Td [(L+T h sin( ) mv +q+ g v cos( )Tj /T1_2 11.955 Tf ( ) =q q= M Iyy i =2 & i i i )Tj /T1_2 11.955 Tf 11.96 0 Td (! 2 i i + N i i =1,2,3 (5) 5.8SetsofThermalProles ThisexampleusestwodifferentsetsofthermalprolestocreatetheLPV controllersasshowninFigure 5-6.Thetemperaturealongthefuselagewillvary duringaight;however,thenosetemperatureshouldalwaysbeatleastashotasthe tailtemperature.Differentthermalgradientsalongthefuselageareintroducedintothe model.ThemodelassumesastiffnesscorrespondingtoTitanium( Ti )so 950 o F isthe maximumtemperaturethatcanbereachedbeforematerialchangesphase[ 134, 136]. Thetemperaturesarenotedaseffectivebecausethesurfacetemperaturesofthe vehiclewillbeexceedinglyhighbutthetemperatureofthefuselagestructurewillbe noticeablycoolerbecauseofthethermalprotectionsystem[ 98, 99].Inthiscase,a reduced-ordermodelofthefuselageisgeneratedasatitaniumbeamsotherange ofeffectivetemperaturesusedinFigure 5-6 arethuslimitedbymaterialpropertiesof titanium. 5.8.1SetA InSetA,fteenthermalprolesareintroducedintothemodelinitially,shownin Figure 5-6A.Therstveprolesarelineari.ethethermalgradientslinearlydecrease fromnosetotail,thenextvehavegradientslesserthanthelinearprolesandthelast vehavegradientsgreaterthanthelinearproles. 75

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ASet A BSet B Figure5-6.Differentsetsofthermalproles.IntheanalysispresentedlaterforSetA, thethermalprolesarenumberedasfollows:1-hottestlinearprolewith [T nose T tail ]=[900,500] ,5-linearprolewith [T nose T tail ]=[500,100] 5.8.2SetB AlargersetofthermalprolesdepictedinFigure 5-6B isusedtoperformtheLPV closed-loopanalysis.Asetofappropriatethermalproles,areusedtodescribepossible variations. 5.9ResultsforSetA 5.9.1EffectofTemperatureontheModalProperties Theregulationofthepitchrate, q isimportantasitdirectlyaffectstheintakeofair intotheengine,henceaffectingthepropulsiondynamics.Italsoaffectsthepressure distribution,henceinuencingtheaerodynamics,whichinturnalterstheperformance. Thevariationsinpitchrate, q becauseofthevariationsinheatingneedstobeminimized usingtheelevator.Theopen-looptransferfunctionfromelevatordeection, e topitch rate, q forthedifferentthermalprolesinFigure 5-6A areplottedinFigure 5-7.Note thepeaknear0.04 rad/s isassociatedwitharigid-bodyightmodewhilethepeak near22 rad/s isassociatedwiththestructuralmodethatshouldbeattenuated.Itis observedfromFigure 5-7 thatthereisavariationbothinthedampingandthenatural frequenciesofthestructure.Table 5-1 showsthevariationinnaturalfrequenciesfor 76

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the5 linearthermalproles.Thereisabout 7% variationinthenaturalfrequenciesfor thethreemodes.However,themodeshapes,shownforarepresentatvethermalprole inFigure 5-8,exhibitverylittlevariationwithtemperature.Theasymmetricnatureof themodeshapesisduetothedependenceofthestructuralpropertiesonthethermal proles. Figure5-7. Transferfunctionfromelevatordeection, e topitchrate, q forthedifferent thermalprolesshowninFigure 5-6A Table5-1.Naturalfrequencies, i forthelinearthermalprolesinFigure 5-6A Mode1( hottest ) 2345Reduction( %) 123.01 23.5023.9024.3124.736.96 249.8750.8951.7852.6253.546.85 398.90100.95102.7104.4106.216.88 5.9.2Open-loop Dynamics Theopen-loopmodelsoftheightdynamicsareparametrizedaroundthese thermalproles.Boththestatematrix, A() 1 andtheinputmatrix, B () ofthe 1 (t ) isthe parametervector.Inthisexample,itisthethermalprolealongthe fuselage. 77

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Figure5-8. Modeshapesforarepresentativethermalproleforthehypersonicvehicle linearizedmodelinEquation 5 areexpressedwithatemperaturedependency. Thesedependenciesareactuallyquitestronginthattheelementsofeachmatrixwill varydramaticallywithtemperature. x ( t )= A()x (t )+ B ()u (t ) (5) A(i j ) and B (i k ) representtheeffectofthe j th stateor k th controlinputontherateof changeofthe i th state.Thecoefcients A(3,2) A(7,6) and B (3,1) showconsiderable variationswithtemperature. A(3,2) representstheinuenceofangleofattack( )onthe pitchrate( q ), A(7,6) representstheinuenceoftherstbendingmodedisplacement ( 1 )ontherstbendingmodevelocity( 1 )and B (3,1) representstheinuenceofthe elevatordeection( e )onthepitchrate( q ).Itisobservedthat A(3,2) affectsonlythe unstablerigidbodymodewhereas A(7,6) and B (3,1) affectsonlytheexiblemodes. ThevariationofthecoefcientsforthedifferentplantmodelsisshowninFigure 5-9. A(7,6) decreaseswithadecreaseintemperatureasexpected. A(3,2) and B (3,1) have anonlinearnature,butshowasimilartrendwithrespecttothedifferentthermalproles. 78

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A A (3,2) B A (7,6) C B (3,1) Figure 5-9.Variationinthecoefcientsofthestatematrices.Theabscissarepresents thethermalprole,1representsthehottestlinearprolewith [T nose T tail ]=[900,500] and5representsthelinearprolewith [T nose T tail ]=[500,100] Figure 5-10 showsthe H 1 -normfortheopen-loopstablesystems 2 Itcanbe deducedthatbetterperformancecanbeachievedwhenthevehicleiscoolerandthe gradientofthethermalproledoessignicantlyaffecttheperformance.Thetrend shownbytheopen-loopnormseemstobesimilartothecoefcient A(7,6) .Inorder 2 Theunstab lerigidbodydynamicsisstabilizedusinganominal H 1 controller,N. ThestabilizerwillbediscussedmoreinSection 5.9.3. 79

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toe xplorethisrelationshipmorethe H 1 -normisplottedasafunctionoftheopen-loop dynamiccoefcient, A(7,6) inFigure 5-11.Thisnormseemstoshowamonotonic dependencyontheopen-loopdynamics. Figure5-10. Open-loop H 1 -normforthesystemswiththedifferentthermalproles.The abscissarepresentsthethermalprole.1,6,11-linear,lessthanlinear andmorethanlinearthermalproleswith [T nose T tail ]=[900,500] respectively. Figure5-11. Open-loop H 1 -normparameterizedaroundtheopen-loopdynamic co-efcient, A(7,6) 80

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5.9.3Contr olSynthesis Theeffectsofthermalprolesontheightdynamicsshouldbeminimized;however, therigid-bodymaneuveringshouldnotbealtered.Assuch,thecontrolobjective becomesvibrationattenuationofthestructuraldynamicsusingacompensatorthatdoes notsignicantlyuselow-frequencyenergyneartherigid-bodymodes.Thisobjective allowstheresultingcontrollertotwithinamulti-loopschemeinvolvingaseparate controllerforvibrationattenuationandanotheroneformaneuveringandguidance asshowninFigure 5-12 [109].Suchaloopdecompositionrecognizesthatthermal effectsarepredominantlylimitedtothestructuraldynamicsrelatedtovibration.The outer-loopcontrolleristhusdesignedwithoutconsiderationoftemperatureeffectssince theinner-loopcontrollerisassumedtoprovideadequatecompensation.Thisresearch concentratesonlyonthedesignoftheinner-loopcontroller. Astabilizer, N ,isinitiallydesignedfortheopen-loopdynamicspriortosynthesizing thevibration-attenuationcontroller.Thiselementsimplystabilizestherigid-body dynamicsbylow-frequencyactuation.Suchastabilizerisrequiredbecausethe synthesisforvibrationattenuationmuststabilizetheclosed-loopsystem;however, thevibrationattenuationcontrollershouldnotaffecttherigid-bodydynamics.Thisinitial stabilizerensuresthesynthesisforvibrationattenuationdoesnotneedtoutilizeany low-frequencyactuationtoaffecttherigid-bodymodes.Itusesmeasurementsofpitch rate( q )todeterminecommandsfortheelevatorandcanardtoachievethisstabilization. Theblockdiagramusedtosynthesizethecompensatorforvibrationattenuationis giveninFigure 5-13.Thefeedbacktothecontrolleris y q whichisanoisymeasurement oferrorinpitchrate( q )whilethecommandsfromthecontrollerare u e and u c as actuationfortheelevatorandcanard.Theouter-loopcontroller,whichgenerates commandsformaneuveringandguidance,providesthevaluesof f e c d f g asthe elevator,canard,diffuser-arearatio,andfuel-owratio. 81

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Figure5-12. Multi-loopcontrolarchitecture Atargetmodel, T ,isformulatedthatrepresentsanacceptablesetofdynamics. Therigid-bodydynamicsareidenticaltothehypersonicmodelbutthestructuralmode hashigherdampingthanthehypersonicmodel.Thisincreaseindampingresultsina considerabledifference,asshowninFigure 5-14,betweenthetransferfunctionsfrom elevatortopitchrate. Amodel-followingapproachisadoptedsuchthatanerror, e q ,isformulatedasthe differencebetweenthemeasuredpitchrate( q )andthedesiredpitchrate( q d ).This errorisnormalizedtoreecttheperformancegoalsacrossfrequencyusingaweighting, W q = s +100 s +5 onthedifferenceinpitchrate.Thisweightingindicatesthedifference inpitchrateshouldbelessthan0.05 rad/s atlowfrequenciesbutmaybeashighas 82

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e e e c W k F 6 6 6 6 e e e e P q e W q e q 6 6 N T q d ? ? e W n n y q u e u c e c d f Figure5-13. Synthesismodel, S usedtocreatetheLPVcontroller, K AOpen-loop frequencyresponse BF requencyresponsewithstabilizer Figure5-14.Responsefromelevatordeection, e topitchrate, q forthenominalmodel ()andtargetmodel(-.-.-.) 1 rad/s athighfrequencies.Figure 5-15 showsthetimeresponseofthepitchratefor 200sec. Additionalerrorsaregeneratedaspenaltiesontheactuation.Theerrorof e e is denedforthepenaltyonelevatorand e c isdenedforthepenaltyoncanard.These errorsarenormalizedbyscalingthecontrollercommandsthroughthediagonalmatrix 83

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APitch rate( q q d )-478()-479()]TJ /T1_0 9.963 Tf 32.78 0 Td [()timeresponsewith stabilizer, N BTime responseoftheerrorbetween q and q d Figure5-15.Pitchrate(q )timeresponsefor200sec. of W k = n s +10 s +100 s + 10 s +100 o Thisweightingallowsactuationupto10 deg atlow frequencybutonly1 deg athighfrequencyforeithertheelevatororcanard. Noise, n ,isassociatedwiththesensormeasurementofpitchrate.Thissignalis weightedthrough W n =0.01 tolimittherelativesizeofthisnoiseincomparisontothe pitchrate. Also,alter, F = 1000 s +1000 is includedtoeliminatetheeffectivedependenceof theinputmatrixontime[ 93 ].Suchalterdoesnotaltertheresultsinanyappreciable fashion;however,itsatisesassumptionsrequiredbythesynthesisalgorithms[ 95 ]. Thedynamicsoftheopen-loopsynthesismodel, S isusedtocreatetheLPV controller, K (),usingtheLMI ControlToolbox [95].ThestepsintheLPVcontrol synthesisarereiteratedinFigure 5-16. H 1 controllersarecreatedforeachofthe thermalprolestocomparetheperformanceoftheLPVcontroller 3 3 The H 1 controllerswill alsobereferredtoaspointdesigncontrollerorsimplypoint controllers. 84

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Figure5-16. StepsinLPVcontrolsynthesis 5.9.4ResultsandDiscussion Thefrequency-domainandtime-domainresponsesarecomparedfortheopen-loop stabilizedsystem 4 ,theclosed-loopsystemwiththe H 1 controllerandtheclosed-loop systemwiththeLPVcontroller.Onemetricforcomparisonbetweentheclosed-loop systemswiththe H 1 controllerandtheLPVcontrolleristheclosed-loop H 1 -norm.This H 1 -normreectstheupperboundonthemaximumsingularvalueoftheclosed-loop synthesismodelandthesynthesismodelisnormalizedsuchthatagainofunityisthe optimalupperbound. TheresultingnormsshowninTable 5-2 andFigure 5-17 ,arenotabletoguarantee alltheperformancegoalsforallthethermalproleswiththeLPVcontrollersincethe normisgreaterthan1.Itmustbenotedthatthisupperboundisaconservativebound forrobustperformancewithrespecttouncertainties.Thismetricevaluatesthecontroller 4 Thesystem isstabilizedwiththestabilizer, N inFigure 5-13. 85

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perfor manceifthevehiclecaninstantaneouslytransitionfromhottocoldtohotas determinedbyanyproleinFigure 5-6A.Actually,itconsidersthetailandnosetobe independentbutsimplynorm-boundedsoreversedproleswhichwouldhavethetail hotterthanthenoseareconsidered.Assuch,theguaranteeprovidedbythismetricis importantbutnotnecessarilyrequiredforrealisticperformance. Table5-2.Closed-loop H 1 -normsforsystemswith H 1 andLPVcontrollerforthelinear proles Model H 1 LPV 10.38 5.05 20.282.60 30.270.88 40.384.98 50.516.88 Figure5-17. Normsoftheclosed-loopsystems Thetransferfunctionsfortheopenloopsystem,closed-loopsystemwiththe H 1 controllerandwiththeLPVcontrollerforthevelineartemperatureprolesareplotted inFigure 5-18.ItcanbeseenthattheLPVcontrollerachievestheobjectiveofvibration attenuationbydampingouttherstbendingmode. 86

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Figure 5-19 shows thetimeresponseoftheopenloop,closed-loopsystems withthe H 1 controllerandwiththeLPVcontrollerareplotted.Theopen-looptime responseshowsoscillationswhichisduetothelackofstructuraldampingandshould beeliminatedbythecontroller.ItcanseenthattheLPVandthe H 1 controllersadd dampingtothesystemandtheoscillationsareeliminated. Thepole-zeromap,asshowninFigure 5-20,oftheclosed-loopsystemwith H 1 andLPVcontrollerforanominalmodelwereanalyzedtounderstandthesystembetter. Thetargetmodel, T hasanundershootinthetimeresponseasitisdesignedonthe basisoftheopen-loopsystemwhichisanon-minimumphaseandunstablesystem. Since,thisisa`model-matching'approachanundershootshouldbeexpectedinthe closed-looptimeresponse.Robustperformanceforguidanceormaneuveringwillbe guaranteedbytheouter-loopcontroller. 5.10ResultsforSetB TheprolesinSetB,showninFigure 5-6B ,areseparatedintoclassessuch thateachprolewithinaclasshasthesametemperatureatthenose( T nose ).The temperaturevariationswithintheseclassesalongwiththeassociatedvariationsinthe naturalfrequencyoftherstbendingmodeofthefuselagerst-bendingmodearenoted inTable 5-3.Foreachclass,thelowestvalueofnaturalfrequencycorrelatestothe hottestvalueoftailtemperature( T tail ). 5.10.1Open-loopDynamics Theopen-loopmodelsoftheightdynamicsareparametrizedaroundthese thermalproles.Boththestatematrix, A andtheinputmatrix, B ofthestate-space modelareexpressedwithatemperaturedependency.Thesedependenciesare actuallyquitestronginthattheelementsofeachmatrixwillvarydramaticallywiththe temperature.Asetofvariablesthatarerepresentativeofthisparametrizationarenoted inFigure 5-21 forthestate-matrixelementdescribingtheinuenceofbending-mode 87

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Tab le5-3.Thermalprolesandnaturalfrequenciesofthe 1 st bendingmode Class T nose (F) T tail (F) Natural F requency ( rad/s ) 1900 800-10023.47-24.75 2850750-10023.85-24.85 3800700-10023.82-24.89 4750650-10023.90-25.17 5700600-10024.23-25.14 6650550-10024.53-25.27 7600500-10024.47-25.22 8550450-10024.99-25.67 9500400-10025.09-25.41 10450350-10025.44-25.57 displacement( 1 ) onthevelocity( v )andinFigure 5-22 fortheinput-matrixelement describingtheinuenceoftheelevator( e )onbending-modevelocity( 1 ). 5.10.2Performance 5.10.2.1Norm Theinitialmetricforperformanceresultsinthissetisthenormoftheclosed-loop synthesismodel.Thisnormreectstheworst-casegainfromexogenousinputs( d )to regulatederrors( e )inFigure 2-4 asdeterminedbytheratioofthe2-normvalues.The synthesisisnormalizedsuchthatagainofunityisoptimal.AscanbeseeninTable 5-4 theresultingnormsarenotabletoguaranteealltheperformancegoalsforanyclassof thermalproles. Table5-4.Normofclosed-loopsystem ClassNor m 16.51 2 4.50 35.38 421.14 56.76 611.21 75.28 86.02 95.42 104.22 88

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This2-nor mmetricisactuallyindicativeoftheworst-casegaindespiteany time-varyingchangesinthethermalprole.AsmentionedinSetA,theguarantee providedbythe2-normmetricisimportantbutnotnecessarilyrequiredforrealistic performance. 5.10.2.2Class6 Thepropertiesoftheclosed-loopmodelsforthermalprolesinclass6are consideredindetailtoevaluatethecontrollerperformance.Thesemodelshavea relativelyhighnorm,asgiveninTable 5-4,andactuallyprovideagoodrepresentationof theclosed-looppropertiesforeachthermalproleinFigure 5-6B. Thefrequency-domaintransferfunctionsarealsoappropriateforevaluationofthe controllers.Theclosed-looptransferfunctionsareshowninFigure 5-23 forthetarget modelalongwiththehypersonicvehicleassociatedwitheachproleintheclass6 alongwiththeopen-loopmodel.Thesetransferfunctionsindicatethatthecontroller isindeedabletomatchthetargetmodelandthehypersonicmodel.Thematchshows somedeviationsatlowfrequencieswhereperformanceisnotcriticalbutshowsexcellent performancenearthestructuralmodelwhereperformanceisindeedcritical. Asetoftime-domainresponsesaregeneratedusingasingleimpulsetothe elevator.TheseresponsesasshowninFigure 5-24 ,indicatethattheclosed-loop responsesareveryclosetothetargetmodelandaresignicantlymoredampedthan theopen-loopmodel. Finally,atime-varyingresponseisgeneratedthatconsidersthepitchratewhile thethermalproleischanging.Inthiscase,asetofelevatordoubletsarecommanded showninFigure 5-25A whilethenoseremainsconstantat 650 o butthetailisrandomly changingittemperatureasinFigure 5-25B. Theresponsetothistemperature-varyingsimulationshowninFigure 5-26, demonstratesthegain-scheduledcontrollerachievesvibrationattenuationthroughout 89

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ther ange.Specically,theclosed-loopresponseissimilartothedesiredtargetdespite thevariationsinthetemperatureatthetailresultingfromthermalgradients. Theperformanceisonlyshownusingthethermalprolesinclass6;however,these resultsareindicativeoftheremainingproles.Thecontrollerisabletocloselymatchthe closed-looptransferfunctiontothetargetmodelforeveryproleinFigure 5-6. 5.11Summary ThischapterconsideredtheapplicationofLPVtheorytohypersonicvehicles. Flightathypersonicspeedswillinducesignicantheatingonthevehicle.Theresulting thermalgradientswillcausetime-varyingtemperaturesthatrangeacrossthefuselage. Thesethermalgradientswillaffectthemodalpropertiesofthestructuraldynamics and,duetoaerothermoelasticcoupling,affecttheightdynamics.Thisexample introducedthelinearparameter-varyingframeworktocompensatefortheeffects ofthethermalproles.Thedynamicsofahypersonicvehicleareexpressedwitha dependencyonthetemperatureatdifferentlocationsalongthefuselage.Acontroller isthensynthesizedthatisinherentlyscheduledonthesetemperaturevalues.The approachisdemonstratedtoachievevibrationattenuationofastructuralmodefora widerangeofthermalproles. 90

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A B C D E Figure5-18. Frequencyresponsefromelevatordeection, delta e topitchrate, q forthe velinearthermalprole 91

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AOpen-loop B H 1 CLPV Figure 5-19.Pitchrate(q )timeresponsefortheopen-loopandclosed-loopsystems withthe H 1 andLPVcontrollers.Thedesiredpitchrate, q d isdepictedby )-222()-222()]TJ /T1_0 11.955 Tf (. 92

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A H 1 point BLPV Figure 5-20.Pole-zeromapoftheclosed-loopsystemwiththe H 1 andLPVcontrollers forarepresentativemodel Figure5-21. Open-loopstabilitycoefcientrepresentingtheinuenceofthevelocity(v ) onthevelocityoftherstbendingmode( 1 ), A(7,1) asafunctionof temperature 93

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Figure5-22. Open-loopcontrolcoefcientrepresentingtheinuenceoftheelevator deection( e )onthevelocityoftherstbendingmode( i ), B (7,1) asa functionoftemperature Figure5-23. Actual(),desired(-.-.)andopen-loop( )-221()-223()]TJ /T1_0 11.955 Tf 33.2 0 Td [()frequencyresponsefrom elevatordeection( e )topitchrate( q )forclass6 94

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Figure5-24. Actual(),desired(-.-.)andopen-loop()-222()-222()]TJ /T1_0 11.955 Tf [()timeresponseforclass6 A e B T tail Figure5-25. A)ElevatorcommandsandB)tailtemperature 95

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Figure5-26. Actual(),desired(-.-.)andopen-loop( )-221()-223()]TJ /T1_0 11.955 Tf 33.2 0 Td [()timeresponse 96

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CHAPTER6 CONTR OL-ORIENTEDDESIGNOFHYPERSONICVEHICLEFOR H 2 AND H 1 PERFORMANCE 6.1Introduction Anapproachformulti-disciplinarydesignoptimizationisformulatedinChapter 4 thatminimizesaclosed-loopnormasthecostmetric.Theconceptsearchesoverthe designspacethatincludesopen-loopcongurationsandsolutionstoRiccatiequations thatguaranteecontrollerexistence.Inthisway,theapproachdesignsanopen-loop modelforwhichacontrollerexiststhatoptimizesaclosed-loopnorm.Thisconcept ofdesignusingexistenceconditionsisdemonstratedfor H 1 -normperformanceand H 2 -normperformance.Asurrogatemodelisshowntoapproximatetheobjectiveasa functionofthedesignspaceandthusoptimizethedesignwithlimitedcomputations. Acontrol-orienteddesignisoptimizedforahypersonicvehicletominimize aerothermoelasticeffectsdiscussedinSection 5.2 .Sucheffectsareinitiallycaused bycouplingbetweentheaerodynamics,thepropulsiondynamicsandthestructural dynamicsofthefuselage.Thechallengeiscompoundedbytheintroductionofthermal gradientswhichresultfromthetremendousheatingacrossthefuselagethroughout ight.Assuch,vibrationattenuationbecomesacriticalaspectofmissionperformance. Structuraldamageresultingfromthetremendousheatingincurredduring hypersonicightismitigatedbyathermalprotectionsystem;however,suchmitigation isaccompaniedbyanincreaseinweightthatcanbeprohibitive.Theactualdesign ofathermalprotectionsystemcanbechosenintelligentlytovarythelevelofheating reduction,andassociatedweight,acrossthestructure.Thisexampleconsidershow suchdesignsandresultingthermalgradientsinuencetheabilitytoachieveclosed-loop performance. 97

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6.2Objective The objectiveofthisexampleistondtheoptimaloperatingrange, 1 forwhich H 2 and H 1 controllerexistsandminimizesclosed-loopmetrics,inordertodesignthe thermalprotectionsystemfromacontrol'sperspective. 6.3DesignSpace Thedesignspacefortheopen-loopdynamicsconsistsofa2-dimensionalset, P, relatedtoeffectivetemperatures, [T nose T tail ],ofthefuselagestructureatthenoseand tail.Thedesignisactuallychoosingtheamountofthermalprotectionsystemonthe structure;however,theeffectivetemperatureareadirectresultofthatthermalprotection systemandthusrepresentthedesignvariables.Thesetofeffectivetemperaturesinthe designspacearelimitedtocongurationswithmonotonicdecreasefromthenosetothe tailasshowninFigure 6-1. Figure6-1. Thermalprolescomprisingthedesignspace 1 Refer Figure 4-4 98

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Theopen-loop dynamicsareparametrizedasfunctionoftheseeffectivetemperatures. Elementsofthelinearizedstate-spacemodelsindicatethenonlineardependencyonthe designspace.Asetofvariablesthatarerepresentativeofthisdependencyarenoted inFigure 5-21 fortheinuenceofairspeedonthebending-modevelocityusedinthe statematrixandinFigure 5-22 fortheinuenceofelevatoronthebending-mode velocityusedinthecontrol-effectivenessmatrix.Thesevaluesindicatethatthe aeroservoelasticcouplingshowninFigure 5-21 islargeandthecontroleffectiveness showninFigure 5-22 forcongurationswithhightemperaturesatbothnoseandtail. Suchdependencymakesdesignchallengingsinceanidealcongurationshouldhave smallcouplingbutlargeeffectiveness. 6.4ControlSynthesis Thesynthesismodel, S discussedinSection 5.9.3,isusedinthisexampleandis illustratedagaininFigure 6-2,withthedependenceoftheopen-loopdesignvariables, onthedynamics, P highlighted.Theseerrorsarespecicallychosensuchthattheirsize isdirectlyinversetotheclosed-loopperformanceforvibrationattenuationi.e.the H 1 and H 2 -norms.ThesetofweightingfunctionsusedinthisanalysisisgiveninTable 6-1 andisselectedtoreectdesiredperformanceusingthedifferentnorms. Table6-1.Weightingfunctions Var iables H 1 -normperformance H 2 -normperformance W q s +100 s +5 s + 150 s +140 W k s + 10 s +100 s + 70 s +100 6.5Optimization 6.5.1 GlobalSearch Thetechniqueofglobalsearchisutilizedtondtheoptimaldesign.Inthiscase, acontrolleriscomputedusingthearchitectureofFigure 6-2 foreachofthepotential prolesinFigure 6-1.Thus,atotalof105controllerswerecomputedtogenerate 99

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e e e c W k W k 6 6 6 6 e e e e P ( ) q e W q e q 6 6 N T q d ? ? e W n n y q u e u c e c d f Figure6-2. Synthesismodel, S usedtocreatethecontroller, K .Thedependenceofthe designvariable, onthedynamics, P ishighlighted.Thedynamicsofthe synthesismodelarethe A( ), B i ( ) and C i ( ), i =1,2 in Equation 4 ,Equation 4,Equation 4 andEquation 4 theassociatedclosed-loopnormassociatedwithvibrationattenuationasshownin Figure 6-3 for H 1 -normperformanceand H 2 -normperformance. ThevariationsacrossthedesignspaceinFigure 6-3 showsomeinteresting differencesbetweenthetwonorms.Certainlybothhaveageneralnonlinearrelationship thatpeaksatacongurationassociatedwiththehottesttemperaturesfornoseand tail;however,otherimportanttrendsaredissimilar.Themagnitudeofvariationisquite disparateinthattheuseof H 1 asthenormshowsonlya2.2%variationacrossthe designspacewhiletheuseof H 2 asthenormshowsalarge29.1%variation.Also,the closed-loopnormassociatedwiththelargestthermalgradientsareroughlydecreasing intermsof H 1 asthegradientgetslargerbutvarysporadicallyintermsof H 2 Theresultingcongurationsthatareoptimalare = [ 900,100 ] withrespectto theclosed-loop H 1 normand = [ 600,100 ] withrespecttotheclosed-loop H 2 norm. Theassociatednormsare0.22asmeasuredby H 1 and0.69asmeasuredby H 2 Theclosed-looptransferfunctionsfromexogenouselevatorcommandtopitchrateare 100

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A H 1 B H 2 Figure6-3. Closed-loopnorm,(r )for H 1 and H 2 performanceparameterizedaround thedesignspace showntocloselymatchthedesiredmodelasshowninFigure 6-4.Similarly,thetime responseofpitchrateinFigure 6-5 andelevatordeectioninFigure 6-6 inresponseto anouter-loopcommandindicatetheclosed-loopsystemfollowsthetargetmodel. A H 1 B H 2 Figure6-4. Magnitudeoftransferfunctionfromelevatortopitchrate(q )ofoptimal design()anddesired( )-222()-222()]TJ /T1_0 11.955 Tf [()for H 1 and H 2 synthesis 101

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A H 1 B H 2 Figure6-5. Pitchrate(q )ofoptimaldesign()anddesired( )-221()-223()]TJ /T1_0 11.955 Tf 33.2 0 Td [()for H 1 and H 2 synthesis A H 1 B H 2 Figure6-6. Elevatordeection( e )ofoptimaldesign()for H 1 and H 2 synthesis 6.5.2Surrogate-BasedDesignOptimization Theuseofsurrogatemodelsisevaluatedtooptimizethedesignwithlowcomputational cost.Inthiscase,arandomsetof15congurationsarechosenfromthe105possibilities inFigure 6-1.Surrogatemodelsarecomputedusingkrigingalongwithradialbasis neuralnetworks(RBNN)andsupportvectorregression(SVR)withvarioustypesof radialbasisfunctions(RBF)asgiveninTable 3-1.TheDACEtoolboxofLophavenet al.[ 138],thenativeneuralnetworksMATLABtoolbox[139]andthecodedevelopedby 102

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Gunn[ 140 ]wereusedtogeneratethekriging,radialbasisneuralnetworkandsupport vectorregressionmodelsrespectively.TheSURROGATEStoolboxofViana[ 141]was usedtorunEGOalgorithmandfortheeasymanipulationofthesurrogates. TheEGOapproachusesminimalcomputationstooptimizethedesignstobethe samecongurationsasarefoundbythecomputationally-expensiveglobalsearch.The implementationalwayschoosesasurrogatefromkrigingalongwiththesurrogatefrom Table 3-1 thatgeneratesthelowestestimateofcross-validationserrors,whichestimates accuracyofthesurrogate,knownasthe PRESS RMS value[ 83].Thedesignisinitiated with3differentsetsof15randomly-chosencongurationswhoseoptimalclosed-loop normisgiveninTable 6-2 for H 1 (representedasCaseA,BandC)andinTable 6-3 for H 2 (representedasCaseD,EandF).Also,thetableslistthesecondsurrogatethathad thelowest PRESS RMS valueandthenalcongurationoftheoptimaldesign. Table6-2.Surrogate-baseddesignoptimizationusingmultiplesurrogatesfor H 1 synthesis.SVR-SupportVectorRegression,RBF-Radialbasisfunction. CaseSurrogate 1Surrogate2 InitialBest Solution [T nose T tail ] FinalBest Solution [T nose T tail ] AKr igingSVRgaussianRBF[850,150] [900,100] BKrigingSVRpolynomial[850,150] [900,100] CKrigingSVRpolynomial[750,150] [900,100] Tab le6-3.Surrogate-baseddesignoptimizationusingmultiplesurrogatesfor H 2 synthesis.SVR-SupportVectorRegression,RBF-Radialbasisfunction. CaseSurrogate 1Surrogate2 InitialBest Solution [T nose T tail ] FinalBest Solution [T nose T tail ] DKr igingSVRpolynomial[900,100] [600,100] EKrigingSVRLinearSpline[800,150] [600,100] FKrigingSVRLinearSpline[900,100] [600,100] Thematur ationofthesurrogatemodelusingEGOisdemonstratedusingCaseA forthe H 1 norm.Thesurrogatesandassociatedexpectedimprovementresulting fromthe15initialcongurationsareshowninFigure 6-7.Theactualclosed-loop designsaregeneratedfor3additionalcongurationscorrespondingtothelocationsof 103

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highe xpectedimprovementfromFigure 6-7 andthennewsurrogatewithassociated expectedimprovementaregeneratedasinFigure 6-8.Atotalof5iterationsareusedto resultinthenalsurrogateandassociatedexpectedimprovementofFigure 6-9.Inthis case,theiterationsareallowedtostopsincetheexpectedimprovementisrelativelyat forthekrigingmodeltoindicatethatfurtheradditionstothedesignspacemaynotyield muchimprovementinaccuracy. A KRG B Surr 2 CExpected Improvement(KRG) DExpected Improvement(Surr 2 ) Figure6-7.Surrogatetsandexpectedimprovementmodels, E [I (x )] forkriging(KRG) andsurrogate2(Surr 2 )forcaseAatthe1stiteration Thevalueofmodelingwithmultiplesurrogatesisevidentinthisexample.The natureofkrigingisabasicinterpolationsothecostfunctioninFigure 6-7 doesnotseem 104

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A KRG B Surr 2 CExpected Improvement(KRG) DExpected Improvement(Surr 2 ) Figure6-8.Surrogatetsandexpectedimprovementmodels, E [I (x )] forkriging(KRG) andsurrogate2(Surr 2 )forCaseAafterthe2nditeration anidealapproximationtoFigure 6-3;however,thestatisticsprovidedbykrigingenable thesupportvectorregressiontogenerateamodelwithhighaccuracy. 6.6Analysis Therelationshipbetweenthedesignspaceandtheclosed-loopperformancecanbe explored.Inparticular,thecomplexitybetweenopen-loopdesignandclosed-loopdesign shouldbeevaluatedtodeterminetheadditionalcostinducedbytheadditionofcontrol synthesistotheprocedure. Thedifcultyofoptimizinganopen-loopdesignareunderstood.Certainlythe open-loopdynamics,asevidencedinFigure 5-21 andFigure 5-22,haveanonlinear 105

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A KRG B Surr 2 CExpected Improvement(KRG) DExpected Improvement(Surr 2 ) Figure6-9.Surrogatetsandexpectedimprovementmodels, E [I (x )] forkriging(KRG) andsurrogate2(Surr 2 )forCaseAafterthenaliteration parameterizationaroundthedesignspace.Afunctionalbasedonthisnonlinear parametrizationwouldthushavetobeminimizedtoobtainoptimalityinanyopen-loop design. Theclosed-loopnormisparameterizedasafunctionoftheopen-loopdynamics insteadofthedesignspace.Theclosed-loopnormandassociatedperformancefor trackingisactuallydirectlyrelatedtothevariablesoftheopen-loopstate-spacemodel. Thisresultiscertainlyexpected;however,theindependenceofthatrelationshipfrom temperatureasshowninFigure 6-10 isnotcompletelyanticipated. 106

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A H 1 B H 2 Figure6-10. Closed-loopnormfor H 1 and H 2 asafunctionoftheopen-loopdynamic coefcient, B (7,1).Eachlinerepresentsthenormanddynamics correspondingtoproleswiththesamenosetemperaturebutwithdifferent tailtemperaturesinFigure 6-1. Acontrol-orienteddesignisthusdemonstratedtobesimilarindifcultytoopen-loop design.Theintroductionofcontrolsynthesismerelyaddsamonotonicdependencyonto anonlineardependencywhichdoesnotoverlyincreasethecomputationalchallenge. 6.7Summary Simultaneousdesignofanopen-loopplantandcontrollerisexceedinglychallenging. Acontrol-orienteddesignisintroducedthatdoesnotactuallycomputeboththeplant andcontroller;rather,theplantisdeterminedforwhichacontrollerexiststhatminimizes aclosed-loopnorm.Theconceptusesexistenceconditionsforthecontrollerwhichcan beparametrizedaroundadesignspace.Theactualoptimizationresultsfromefcient explorationofthatdesignspaceusingsurrogatemodeling.Arepresentativemodelofa hypersonicvehicleisusedtodemonstratethisapproachcanindeedgenerateanoptimal design. 107

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CHAPTER7 CONTR OL-ORIENTEDDESIGNWITHMULTI-OBJECTIVEOPTIMIZATIONAND INCLUSIONOFWEIGHTINGFUNCTIONSINTHEDESIGNPROCESSFORTHE HYPERSONICVEHICLE 7.1Introduction Thischapterdiscussesa`open-loopdynamics/control'simultaneousdesign approachwiththeintroductionoffrequency-dependentweightingfunctionsinthe H 1 -controlsynthesisframeworkasdesignvariablesandformulti-objectiveoptimization. Thiscontrol-orienteddesignapproachminimizesclosed-loopmetricsforoptimal performancebysearchingoverthedesignspaceencompassingtheopen-loopdynamic, weightingfunctionsandcontrollervariables,whilemaintainingconstraintsforcontroller existence.Thismethodologyhasbeenimplementedforvibration-attenuationin hypersonicvehicles.Thiscase-studyusesmultiplesurrogatesalongwithNSGA-II foroptimization. WeightingFunctionsintheDesignProcess Typically,designoptimizationproblemscanbeformulatedasmulti-objective problemssuchas`maximizeperformancebyusingminimumcontrolactuation'. Performancecanbeevaluatedintermsoftime-domainmetricsliketheerrorinthetime responseforaspeciccombinationofinput-output.Minimizingthe H 1 -normasinthe case-studyinChapter 6,maynotnecessarilyguaranteemaximumperformanceinthe requiredtransferfunctions.Thisisduetothefactthatthe H 1 -norm 1 triestominimize thenormoverallcombinationsoftransferfunctionsfora`multi-input/multi-output' system.Forsuchdesignoptimizationproblems,itmaybebenecialtoincludethe weightingfunctionsofthesynthesismodel,showninFigure 5-13,inthedesignprocess itself. 1 Thisdiscussion holdsfor H 2 -normaswell. 108

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7.2Objective The objectiveofthiscasestudyistondtheoperatingrange, 2 P (open-loop variables)todesignthethermalprotectionsystemalongwiththeoptimalweightings, 2 W 2 (weightingfunctionvariables)forwhichacontrollerexists,i.e.thereexists solutions, = fX Y g2 K (controllervariables)toEquation 4,Equation 4 and Equation 4 thatminimizesthevibrationsonthevehicle. 7.3ControlSynthesisArchitecture Theblockdiagramusedtosynthesizethecontrollerforvibrationattenuationissimilarto Section 5.9.3 andisshownagaininFigure 7-1 withthedesignvariableshighlighted. Amodel-followingapproachisadoptedsuchthatanerror, e q ,isformulatedasthe differencebetweenthemeasuredpitchrate( q )andthedesiredpitchrate( q d ).This errorisnormalizedtoreecttheperformancegoalsacrossfrequencyusingaweighting, W q = K 1 s + a 1 s + a 2 ,on thedifferenceinpitchrate.Theperformanceobjectiveisgenerallyto havegoodtrackingatlowfrequenciesandnoiseattenuationathighfrequencies.So W q needstonormalizetheerrors,suchthatthetrackingerrorsaresmallatlowfrequencies, butshouldnottracknoiseathighfrequencies.Hence,thegain( K 1 ),pole(a 2 )andzero (a 1 )ischosensoastomake W q alow-passlter. 2 Itis muchmoreconvenienttoreectthesystemperformanceobjectivesby choosingappropriateweightingfunctions.Somecomponentsofasignalmaybe moreimportantthanothersandnotallthesignalsmaybemeasuredinthesame units.Weightingfunctionsessentially,canbeusedtomakesuchsignalcomponents comparable.Also,frequency-dependentweightshelptorejecterrorsinthedesired frequencyrange.TheweightingfunctionsinFigure 2-4 ischosentoreectthedesign objectivesandknowledgeofthedisturbanceandsensornoise.Forexample,the performanceweightingmaybeusedtoreectrequirementsontheshapeofclosed-loop transferfunctionsandtheactuatorweightingmaybeusedtoreectlimitsonthe actuatoroutputs[ 46].So,weightsareusedtoformulateperformanceobjectivesinto mathematicallytractableproblems.Ifthecontroldesignobjective(forascalarcase)isto ensurethatthenormofthetransferfunctionfrom d to e isinsomebounds,( kT de k ") itcanrephrasedas kWT de k 1 where kW k =1=". 109

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e e e c W k (! ) F 6 6 6 6 e e e e P ( ) q e W q (! ) e q 6 6 S T q d ? ? e W n n y q u e u c e c d f Figure7-1. Synthesismodel, S usedtocreatethecontroller, K ( ) .Notethe dependenceofthedynamics, P andtheweightingfunctions, W q and W k on and respectively. Additionalerrorsaregeneratedaspenaltiesontheactuation.Theerrorof e e is denedforthepenaltyonelevatorand e c isdenedforthepenaltyoncanard.These errorsarenormalizedbyscalingthecontrollercommandsthroughthediagonalmatrix of W k = K 2 s + b 1 s + b 2 .This weightingallowshighactuationatlowfrequencies(forgood tracking)butlowactuationathighfrequencies(toavoidtrackinghighfrequencynoise) foreithertheelevatororcanard.Hence, W k isahigh-passlter. 7.4OptimizationProblemFormulation Thedesignspaceconsistsof P whichcontainsvariablesforthefuselagestructure andthethermalprotectionsystem, W fortheweightingfunctionvariablesalongwith K whichcontainsvariablesforafeedbackcontroller.Itisimportanttonotethatthe geometryisrelativelyxedduetoaerodynamicissueswhilethethermalissuesand structuralcomponentshaveconsiderablefreedomintheirdesign.Thedesignvariables arelistedinTable 7-1.Inthiscase,thedesignspaceisappropriatesincethethermal protectionsystemandstructureinteracttodeterminethevibrationcharacteristicsofthe fuselagealongwithassociatedheatingeffects. 110

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Tab le7-1.Designvariables. -operatingrange, -weightingfunction No. DesignVariableTypeBoundsConstraintsComments 1 T nose [450,900] Nose -hottestpart 2 T tail [100,800] T nose T tail +100 Tail-coldestpart 3 K 1 [0.1,1] K 1 > 0 4 K 2 [0.75,1.25] K 2 > 0 5 a 1 [75,125] 6 a 2 [4,6] a 1 > a 2 W q -low-passler 7 b 1 [8,12] 8 b 2 [80,120] b 1 < b 2 W k -high-passlter Thedesign spacefortheopen-loopdynamicsconsistsofa2-dimensionalset, P, relatedtoeffectivetemperature, =[ T nose T tail ].Inthiscase,asetofthermalproles arechosenthathaveconstantgradientfromthenosetotail.Thisset,asshownin Figure 7-2,considersvariationsinboththetailtemperatureandnosetemperaturewith therestrictionthatthenosemustbeatleast 100 o F hotterthanthetail.Thisrestriction isconsistentwithcomputationaluiddynamicsandcomputationalstructuraldynamics analysisonthehypersonicvehiclethatthenoseishottestpartofthestructurebecause ofshockwaves.Theopen-loopdynamics, P ( ) areparametrizedasfunctionofthese effectivetemperaturestoreectvariationsintheYoung'smodulusatthenoseandtail whichresultfromthestructuralelementsandthermalprotectionsystem. Thedesignspacefortheweightingfunctionsconsistsofa6-dimensionalset, W toreecttheperformanceandcontrolleractuationweightingelements, = [K 1 K 2 a 1 a 2 b 1 b 2 ].Theelementsbasicallyreecttheregionwhereperformance iscriticalandactuationenergyneedstobeused. Thegoaloffeedbackcontroldesignistodevelopacontrollertotrackareference outputi.e.minimizetheerrorbetweenthetargetandactualtrajectorybyutilizingthe minimumcontrolactuationpossible.Theobjectivecanberephrasedsoastooptimize performancebyminimizingsomeerror( e p )betweentheactualanddesiredresponse byutilizingtheminimumactuation( c )possiblealongwithmaintainingtheclosed-loop H 1 -norm, r tobe 1 toguaranteerobustnesswithrespecttouncertaintiesalong 111

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Figure7-2. Open-loopdesignspace, withtheconstraintsforcontrollerexistence.Thisexamplecanbeextendedtoinclude uncertainties,i.efor 6=0 butheretheconstraintguaranteesperformance.The constrainedoptimizationcanbestatedasinEquation 7 where e p istheerrormetric forperformance, c isthecontrolpowerutilizedand r istheclosed-loop H 1 -norm.Since amodel-matchingapproachisfollowedtominimizetheerrorbetweentheactualpitch rate( q )andthedesiredpitchrate( q d )usingtheelevatorasthecontrolsurface,the performancemetricchosenistherootmeansquareerror( e RMS )between q and q d Thecontrollerpowermetricisthemeansquare(MS)oftheactuationused,i.e.elevator deection( e )timehistory. min 2 W 2 P 0 B @ e p c 1 C A suchthat r 1 (7) 112

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Theobjectiv eschosenareshowninEquation 7 where e p and c arethe normalizedvaluesof p RMSE and c MS respectively,showninEquation 7.The timeresponseisobservedtohavesteady-stateerrorsin q ,sotheperformanceobjective consideredhasbothstead-stateandtransientcomponentsandisconsideredfor T =200 sec. p RMSE = v u u u t 1 T T Z 0 (q )Tj /T1_1 11.955 Tf 11.96 0 Td (q d ) 2 dt c MS = 1 T T Z 0 2 e dt (7) e p =(( p RMSE ) actual )Tj /T1_3 11.955 Tf 11.96 0 Td (( p RMSE ) min )=((p RMSE ) max )Tj /T1_3 11.955 Tf 11.96 0 Td ((p RMSE ) min ) c =(( c MS ) actual )Tj /T1_3 11.955 Tf 11.96 0 Td ((c MS ) min )=(( c MS ) max )Tj /T1_3 11.955 Tf 11.95 0 Td ((c MS ) min ) (7) 7.5ResultsandDiscussion Theprocessisstartedbysamplingthedesignspacewith700points(thisisenough tohaveapproximatelythreepointsperorthant) 3 usinglatinhypercubesampling[142]. Thisinitialsetofdatafortheobjectives,[ e p c ]andtheconstraint, r isusedtotfour differentsurrogatemodelingtechniques,showninTable 7-2.TheDACEtoolboxof Lophavenetal.[ 138]andthenativeneuralnetworksMATLABtoolbox[ 139]wereused togeneratethekrigingandradialbasisneuralnetworkmodels.TheSURROGATES toolboxofViana[ 141]wasusedtorunthepolynomialresponsesurfacealgorithmandit wasalsousedforeasymanipulationofthesurrogates. 3 Orthant istheanalogueofthen-dimensionalEucledianspaceoraquadrantinthe planeortheoctantinthreedimensions. 113

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Tab le7-2.Informationaboutthesetofsurrogatesusedinthesurrogate-based optimization Surrogates Details KRG Krigingmodel: Constant trendfunctionandGaussiancorrelation PRS2andPRS3 Polynomialresponsesurface: Secondandthirdorderfullmodels RBNN Radialbasisneuralnetwork: Goal =(0.05 y ) 2 and Spread =1=3 Theestimated accuracyofthesetofsurrogatesisrststudied.Table 7-3 gives %PRESS RMS forthesetofsurrogates.Asthe %PRESS RMS for c and r surrogatets islessthan 10 ,thesurrogatetsareagoodapproximationtotheactualdataset. Figure 7-3 showstheobjectivespaceandtheparetofrontfor700points.Figure 7-3B showsthatthetwoobjectivesareactuallycompetingwitheachother.Allclosed-loop systemsontheparetofrontsatisfytheconstraint,i.e. r 1. Table7-3. %PRESS RMS (PRESS RMS = Range %) valuesfordifferentsurrogatemodelsfor theinitialconguration.Range: e p =[0,1], c =[0,1]and r =[0,45](e p and c arethenormalisedvaluesof p RMSE and c MS inEquation 7) Metric KRG PRS 2 PRS 3 RBNN e p 1313 1316 c 108813 r 4444 Thetw osurrogatesgivingthelowest PRESS RMS valuesi.e KRG and PRS forthe objective, [e p c ] andconstraint, r isgiventotheNSGA-IIalgorithm.Figure 7-4 shows theevolutionoftheparetofrontwithmultiplesurrogates.Therearenewpointsbeing addedtotheparetofrontateveryiterationi.e.someofthepointswhichwereoriginally ontheparetofrontarenolongeronthenalparetofront.Thereareatotalof28points onthenalparetofrontoutof1100points.Also,thevaluesgoslightlynegativeinthe controlMSaxis,whichshowsthatoptimizationdoesindeedresultinbetterparetofront. Section 7.6 illustratestheoptimizationprocessusingonlypolynomialsurfacetsand 114

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AObjectiv espace BP aretofront Figure7-3.2-DObjectivespace(p RMSE and c MS arenormalizedbetween0and1using theinitialdatasetof700points) withmultiplesurrogates.Thebenetofusingmultiplesurrogatesisevidentfromthe analysisshown. The 3 ontheparetofrontinFigure 7-5 correspondstothemaximumcontrol energyutilizedalongwiththelowesterrorinperformance, ? correspondstothe 115

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AP aretofront BMagnied Figure 7-4.Evolutionofparetofrontfordifferentiterationsusingmultiplesurrogates. -initialconguration(700points), 3 -after 1 st iteration(880points), -after 2 nd iteration(1100points)(p RMSE and c MS arenormalizedbetween 0and1usingtheinitialdataset) 116

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minimu menergyutilizedandtheworstperformanceachievedand corresponds toacompromisebetweentheenergyutilizedandperformancecommanded.The correspondsonlytoamuchloweractuationusedcomparedto 3 alongwithonlya slightlyhigherperformance.Italsogivesmuchbetterperformancecomparedto ? byusinglittlemoreactuation.Itisinterestingtonotethatthepitchrateerrorismore populatedthanthecontrolpower.Also,forthehypersonicvehiclea 1 o e maybe signicant.Hence,thisstudywilltrytoavoidpointswhichhas e > 1 .Theextreme pointontheparetofrontontheControlMSNormaxisisonesuchpoint.Figure 7-6A andFigure 7-6B showsthetimeresponsesfor 3 and .Thepitchrate(q )for shows worsetransientresponsecomparedto 3.Also,Figure 7-7A andFigure 7-7B shows thatthecontrolisutilizedmorefor thanfor ? .Soinessence, isatrade-offbetween performanceandcontrolactuationused.Basedonthedesignrequirements,theoptimal designcanbechosenfromthisparetofront.Nowthatthethermalprotectionsystem isdesignedfromacontrol'sperspectiveitcanbefurtherevaluatedforissuessuchas weightandcosttooptimizethedesignforadditionalmetrics. 7.6AdvantagesofusingMultipleSurrogates Initially,optimizationisperformedusingtheNSGA-IIalgorithmandonlythebest amongthetwoPRStsfortheobjectives, [e p c ] andtheconstraint, r areused.This optimizationprocessisrepeatedforatotalofthreeiterationsadding100pointsatevery iteration.Figure 7-8 showsthetheparetofrontfortheinitialdatasetof700pointsand afterthenaliterationforatotalof1000points.Ascanbeseen,theparetofrontgoesin thenegativeregionofthenormalizedcontrolpower,implyingthattheoptimizerisindeed abletogetbetterobjectivevalues.Also,morepointsarepopulatedatregionswhere lowercontrolpowerisutilizedandwheretheperformanceisbetween0.8and1.5. Optimizationisthenperformedusingthebesttwosurrogatesgivingthelowest PRESS RMS valuesinTable 7-3 i.eKRGandPRS.Attherstiteration,90points areaddedusingKRGtand90pointsareaddedusing 2 nd -orderPRSt. P K and 117

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Figure7-5. Paretofrontwithtimeanalysisperformedfor3points APitch rateerrorfor 3 and BEle vatordeectionfor 3 and Figure7-6.Timeresponsefor2pointsshownontheparetofront. )-222()-222()]TJ /T1_0 11.955 Tf 36.53 0 Td [(correspondsto 3,correspondsto simulationresultsareevaluatedforthe180pointsandtheoptimizationisrepeated foraseconditeration.Figure 7-9 showstheparetoplotusingonlyPRS( )after3 iterationsandforatotalof1000pointsandcombinationofKRGandPRS( 3 )after2 118

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APitch rateerrorfor and ? BEle vatordeectionfor and ? Figure7-7.Timeresponsefor2pointsshownontheparetofront.correspondsto )Tj /T1_4 11.955 Tf (. )Tj /T1_4 11.955 Tf 11.95 0 Td (. correspondsto ? AInitial DataSet(700points) B 3 rd Iteration (1000points) Figure7-8.Paretoplotsfortheoriginalsetandafterthenaliteration iterationsandatotalof1100points.Theobjectiveoftheoptimizationistogetasclose aspossibletotheoriginonboththeaxisi.e.trytogetthelowest2-normintheobjective space.ThecombinationofKRGandPRSseemstobeaddingmorepointscloserto theorigin.Evidently,itisbenecialtousemultiplesurrogatesinoptimizationforfaster convergence. 119

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Figure7-9. ParetofrontjustforPRS( )andcombinationofPRSandKRG(3) 7.7Summary Thiscase-studydemonstratesthecontrol-orienteddesignapproachwhenthe frequency-dependentweightingfunctionsneedstobeintroducedinthe H 1 -control synthesisframeworkasdesignvariables.Theobjectiveofthiscase-studyistond theoperatingrangetodesignthethermalprotectionsystemalongwiththeoptimal weightingsforwhichacontrollerexiststhatminimizevibrationsonthehypersonic vehiclewith8designvariables.Amulti-objectiveconstrainedoptimizationproblem isformulatedtoachievethisandsurrogate-basedoptimizationusingtheNSGA-IIis usedfortheoptimization.Thiscasestudyalsohighlightsthebenetsofusingmultiple surrogatesascomparedtosinglesurrogates. 120

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CHAPTER8 CONTR OL-ORIENTEDDESIGNOFOPERATINGRANGEFORLINEARPARAMETER VARYINGSYSTEMS 8.1Introduction Anapproachformulti-disciplinarydesignoptimizationisformulatedforlinear-parameter varying(LPV)systemsthatoptimizesclosed-loopperformance,asintroducedin Chapter 4 .Thisconceptsearchesoverthedesignspacethatincludesopen-loop congurationsandsolutionstoasetoflinearmatrixinequalities(LMI's)thatguarantee controllerexistenceforLPVsystems.Inthisway,thisapproachdesignsanopen-loop model, P ( ) alongwiththeoperatingrange, andacontroller, K () thatoptimizes closed-loopperformance, r giveninEquation 8.Surrogate-baseddesignoptimization isusedtondtheoptimalcongurationofthedesignvariables.Thisapproachis demonstratedforthedesignofaninvertedpendulumandcartsystemandimplemented tominimizetheundesirablethermaleffectsonthestructuraldynamicsofahypersonic vehicle. r =sup 2,k d k 2 6=0 ke k 2 kd k 2 (8) Figure 8-1 illustrates themotivationfortheneedforcontrol-orienteddesignoflinear parametervaryingsystems.ItismentionedinChapter 2 thatthedynamicsofanaircraft dependsonthemassandthemassvariesasthefuelisburnt.Buthowtodeterminethe initialamountoffuelforoptimalclosed-loopperformance?Thischapterdemonstrates thecontrol-orienteddesignapproachtoaddressthisquestion. 8.2Example1:InvertedPendulumandCartSystem AninvertedpendulumandcartsystemshowninFigure 8-2 isusedtostudythe proposedmethodology[ 143]. 8.2.1Objective Theobjectiveofthisexampleistodesignthecartandthependulumtokeepthe pendulumintheuprightposition,giventhatthesystemstoresfuelandthemassofthe 121

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Figure8-1. Motivationforcontrol-orienteddesignforLPVsystems Figure8-2. Invertedpendulumandcartsystem.M-massofthecart,m-massofthe pendulum,l-lengthofthependulum,I-inertiaofthependulum,F-force input,x-positionofthecart, -angleofthependulum. 122

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cart (M)andthemassofthependulum(m) 1 varyasthefuelisburnt,buttheinitial valuesoffuelforoptimalclosed-loopperformanceisnotknown. 8.2.2LinearizedDynamics Thedynamicsarelinearizedabouttheuprightequilibriumpositionusingthesmall anglesassumptionandisgiveninEquation 8. b isthefrictioncoefcientofthe cart.Thecomponentsofthestatearetheposition( x )andvelocity( x )ofthecart,the angularposition( )andangularvelocity( )ofthependulum.Themeasurementsignals availablearethepositionofthecart( x )andtheangleofthependulum( )asmeasured fromtheverticalposition.Thecontrolinputistheforce( F )giventothecart. 2 6 6 6 6 6 6 6 4 x x 3 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 4 01 00 0 )Tj /T1_7 7.97 Tf 6.59 0 Td (( I + ml 2 )b I (M + m )+ Mml 2 m 2 gl 2 I (M + m )+ Mml 2 0 00 01 0 )Tj /T1_8 7.97 Tf 6.59 0 Td (mlb I (M + m )+ Mml 2 mgl (M + m ) I (M + m )+ Mml 2 0 3 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 4 x x 3 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 4 0 I +ml 2 I (M + m )+ Mml 2 0 ml I (M + m )+ Mml 2 3 7 7 7 7 7 7 7 5 F y = 2 6 4 10 00 0010 3 7 5 2 6 6 6 6 6 6 6 4 x x 3 7 7 7 7 7 7 7 5 + 2 6 4 0 0 3 7 5 F (8) 8.2.3DesignSpace Theopen-loopdesignvariables(operatingrange), inthisexamplearethemass (M)ofthecartandthemass(m)ofthependulum.Mvariesby 0.05 kgandmvaries by 0.005 kgduringoperation.Assuchthisvariationcouldmeanthatamaximumof 1 Mis interchangeablyreferredtoas Mass cart andmisreferredtoas Mass pendulum in thissection. 123

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0.1kg and0.01kgfuelisburntfromthecartandpendulumrespectively.Forexample, iftheinitialvaluesofMandmare0.4kgand0.2kgrespectivelyandnalvaluesare 0.3kgand0.19kg,thedesignvariablevaluesistakentobethemean,i.e. [M m ] = [0.35,0.195] 8.2.4ProblemFormulation TointroducethedynamicsintheLPVframework,thedynamics,P(M,m)are evaluatedatthelowerandupperboundi.e [M m ],thenitisputintotheLPV frameworkasshowninEquation 2 andEquation 2 where 1 =[M )Tj /T1_5 11.955 Tf 12.39 0 Td ( m )Tj /T1_5 11.955 Tf 12.39 0 Td ( ] and 2 =[ M + m + ].TheoptimizationproblemtobesolvedisEquation 4 tond theoperatingrange, andcontroller, K whichminimizes r showninEquation 8. 8.2.5OptimizationResults 8.2.5.1Globalsearch Itiscomputationallycheaptoperformacompletedesignover forthissystem. Aglobalextensivesearchisinvestigatedtondtheoptimalsolutionofthedesign variableswithminimizes r .Adesignofexperiment(DOE) 2 of1000pointsischosen bylatinhypercubesampling(LHS)[ 142]showninFigure 8-3A andLPVcontrollersare evaluatedateachofthesepoints.Figure 8-3B showstheclosed-loopperformance metric, r asafunctionoftheopen-loopdesignspace, .Itcanbeexpectedfrom Figure 8-3B thatbetterperformanceisachievedifthemassofthecart(M)isatthe lowerlimitof0.2kg.Also,performancedoesnotchangemuchbychanging m for agiven M i.e.theperformanceoftheclosed-loopsystemismoresensitivetothe variationsin M 2 DOEis theprocessofchoosingthesamplingpointsinthedesignspace 124

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A = fM m g B r asa functionof Figure8-3.(a)Designspaceand(b)krigingtoftheclosed-loopnorm(r ) parameterizedaroundthedesignspace, =[ M m ] 125

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8.2.5.2Surr ogate-basedoptimization Aftergettingimportantinsightsintothedesignproblemusingtheglobalsearch, surrogate-baseddesignoptimizationisusedtondthevaluesofthedesignvariables, =[ M 0.05, m 0.005] whichminimizes r .Theoptimizationprocessisstartedby samplingthedesignspacewith15pointsusingLHS.Thisinitialsetofdataisusedto teightdifferentsurrogatemodels,showninTable 3-1.The 2 nd surrogateused 3 along withkrigingintheEGOalgorithmismentionedinTable 8-1 forthe6caseswithdifferent initialcongurations.Table 8-1 highlightsthattheoptimizationseemstoconvergeto M =0.2 and m between [0.34,0.36] asexpectedfromFigure 8-3B. Table8-1.DesignofExperiments(DOE)with15pointsintheinitialset.SVR-Support VectorRegression, r -closed-loopperformancemetric, -designvariables, IBS-initialbestsolution,PBS-presentbestsolution,KRG-kriging, poly-polynomial,linspl-linearspline CaseSurrogate 1Surrogate2 IBS PBS AKRG SVR poly =[0.25 0.28], r =2.02 =[0.200.34], r =1.99 BKRG SVR poly =[0.210.38], r =2.00 =[0.200.35], r =1.99 CKRG SVR poly =[0.200.33], r =1.99 =[0.200.35], r =1.99 DKRG SVR poly =[0.200.21], r =2.00 =[0.200.36], r =1.99 EKRG SVR linspl =[0.210.28], r =2.00 =[0.200.34], r =1.99 FKRG SVR poly =[0.220.19], r =2.00 =[0.200.35], r =1.99 Theoptimization processisillustratedinFigure 8-4 andFigure 8-5 forcaseAin Table 8-1.Figure 8-4A andFigure 8-4B isthesurrogatetforkrigingandsurrogate 2fortheinitialset.TheexpectedimprovementmodelsareshowninFigure 8-4C and Figure 8-4D.Figure 8-5 showsthesurrogatetsandtheexpectedimprovementmodels afterthenaliteration.Theexpectedimprovement, E [I (x )] decreasesinthenal iterationasshowninFigure 8-5C andFigure 8-5D ascomparedtoFigure 8-4C and Figure 8-4D. 3 Basedon thelowest PRESS RMS valuesasdiscussedinSection 3.4 126

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AKRG BSurr 2 CExpected Improvement(KRG) DExpected Improvement(Surr2) Figure8-4.Surrogatetsandexpectedimprovement E [I (x )] modelsforinitialsetfor caseA.KRG-Kriging,Surr2-Surrogate2 Figure 8-5A andFigure 8-5B showsthatthereisclusteringofpointsaround M =0.2kg .Toexplorethisclusteringmore,aboxplotofthedesignvariablesinplotted asshowninFigure 8-6 4 .InFigure 8-6B thedesignvariable1hasmoretighterstatistics 4 Ina boxplot,theboxisdenedbylinesatthelowerquartile(25 %),median(50%), andupperquartile(75 %)values.Linesextendfromeachendoftheboxandoutiliers showthecoverageoftherestofthedata.Linesareplottedatadistanceof1.5timesthe inter-quartilerangeineachdirectionorthelimitofthedata,ifthelimitofthedatafalls within1.5timestheinterquartilerange.Outliersaredatawithvaluesbeyondtheendsof thelinesbyplacinga+signforeachpoint. 127

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AKRG BSurr 2 CExpected Improvement(KRG) DExpected Improvement(Surr2) Figure8-5.Surrogatetsandexpectedimprovement E [I (x )] modelsafternaliteration forcaseA.KRG-Kriging,Surr2-Surrogate2 comparedtoFigure 8-6A.Thisbasicallyreiteratesthatoptimalclosed-loopperformance isachievedbychoosingtheinitialnominal M tobe 0.2kg Thisanalysisbasicallyconrmsthatsurrogate-basedoptimizationconvergestothe globaloptimalsolution.Thisoptimizationalgorithmisabletogivesatisfactoryresultsin thisdesignframeworkwithoutthecomputationalburdenofothernumericaloptimization techniques. 8.3Example2:HypersonicVehicle Theconceptofcontrol-orienteddesignisdemonstratedforahypersonicvehicle describedinChapter 5.ThesynthesismodelisformulatedsimilartoFigure 6-2.The 128

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AInitial BFinal Figure 8-6.Boxplotofthedesignvariablesfora.)Initialandb.)FinalforcaseA.Design variable:1Mass cart ,2Mass pendulum resultingcontrollerissynthesizedtominimizethegain, r fromdisturbancestoerrors,as giveninEquation 4 where e = fe q e e e c g and d = f e c d f n g.Inthissense,the weightingfunctionssuchas W q and W k servetonormalizethesignalsandreectthe desiredperformanceasafunctionoffrequency.Highperformanceisdirectlycorrelated tolowvaluesof r whichensuresthattheclosed-loopsystemwiththeLPVcontroller tracksthedesiredpitchrate, q d byminimizingtheratiooferror, e q todisturbances, d whileconstrainingtheactuationenergy e e e c utilizedtobewithinlimits.Essentially,the normreectstheerrorcausedbyexcessiveactuationanderrorcausedbyaclosed-loop systemthatdoesnotmatchthetargetmodel. 8.3.1DesignSpace Thedesignspacefortheopen-loopdynamicsconsistsofa2-dimensionalset, P,relatedtoeffectivetemperature, =[ T nose T tail ].Inthiscase,asetofthermal prolesarechosenthathaveconstantgradientfromthenosetotail.Thisset,as showninFigure 8-7,considersvariationsinboththetailtemperatureandnose temperaturewiththerestrictionthatthenosemustbehotterthanthetail.Theopen-loop dynamics, P ( ) areparametrizedasafunctionoftheseeffectivetemperaturestoreect 129

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var iationsintheYoung'smodulusatthenoseandtailwhichresultfromthestructural elementsandthermalprotectionsystem.Inthiscase,thedesignspaceisappropriate sincethethermalprotectionsystemandstructureinteracttodeterminethevibration characteristicsofthefuselagealongwithassociatedheatingeffects. Figure8-7. Noseandtailtemperatureforthethermalprolesacrossthefuselageforthe hypersonicvehicle 8.3.2Objective Theobjectiveofthisexampleistodesignthevehiclealongwiththeoperatingrange andaLPVcontrollersoastominimize r i.e.theobjectiveistondtheopen-loopdesign variables( =[ T nose T tail ])withintherange [ + )Tj /T1_2 11.955 Tf 12.55 0 Td ( ] whichoptimizes r soasto minimizethevibrationsonthehypersonicvehiclewith =50.Assuch,thedesign seekstondthebestthermalprolegiventhatitwillchangeby 100 o F duringight duetoaerodynamicheating.Ifthetemperatureofthenose, T nose atthebeginningof ightis 600 o anditheatsupduringhypersonicighttoamaximumof 700 o because ofshockwaves,thedesignvalueof T nose istakentobe 650 o .Since,theopen-loop 130

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dynamics, P ( ) dependonthethermalproles,thesynthesismodel, S ( ) willvarywith temperatureandsowilltheresultingLPVcontroller, K ( ) andtheclosed-loopsystem. 8.3.3SolutionMethodology Since,itisaverytimeconsumingtoperformaglobalsearchforthisexample, surrogate-basedoptimizationisattempted.Twodifferentsetsofinitialcongurations areconsidered,onehaving15pointsandthesecondhaving25points.Thestatistical informationforthedifferentdesignofexperiments(DOE)arepresentedinTable 8-2 5 alongwiththeresultsoftheoptimizationusingtheEGOalgorithm.Table 8-2 presents aninterestingfeatureofhypersonicvehicles,namelysimilarlevelsofclosed-loop performancecanbeachievedfordifferentsetsofthermalprolesiftheyaredesigned properly.Theoptimalelementsofthedesignspacearechosenas =[750,630] for CaseAand =[795,600] forCaseB.Assuchthevalueof indicatesthelowest closed-loopnormisachievedifthethermalprotectionsystemischosentohaveanose temperatureof 750 o andatailtemperatureof 630 o forCaseAand 795 o and 600 o for CaseB 6 Table8-2.DesignofExperiments(DOE).CaseA-15points,CaseB-25points.IBSinitialbestsolution,PBS-presentbestsolution,KRG-kriging,SVRSupportVectorRegression,poly-polynomial CaseSurrogate 1Surrogate2 IBS PBS AKRG SVR poly =[744, 607], r =21.12 =[750,630] r =21.08 BKRG SVR poly =[765,515], r =21.46 =[795,600] r =21.09 Figure 8-8 andFigure 8-9 sho wstheerrorinpitchrate, e q andtheelevator deection, e fortheoptimalsolutionsforthetwocases.Sincethereisnotmuch differenceinthevibrationattenuationandthecontrolactuationbetweenthetwocases, 5 Refer Table 3-1 6 Notethatthethermalprolewith T nose and T tail isthenominalproleandthe thermalprotectionsystemneedstobedesignedsuchthatthetemperatureofthe structureisintherangeof =[T nose T tail ] inightforoptimalperformance. 131

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thether malprotectionsystemcanbedesignedforeithercase.Nowthattheassociated thermalprotectionsystemisdesignedfromacontrol'sperspective,itcanbefurther evaluatedforadditionalissueslikeweightandcosttooptimizethedesignforadditional metrics. A e q = q )Tj /T1_1 9.963 Tf 9.97 0 Td (q d B e Figure8-8. (a)Pitchrateerrorand(b)elevatordeectiontimeresponse(logscale)for 200secforCaseA.istheresponsefortheupperbound, [T nose + T tail + ] and )-222()-222()]TJ /T1_0 11.955 Tf 36.53 0 Td [(isthelowerbound, [T nose )Tj /T1_6 11.955 Tf 11.96 0 Td ( T tail )Tj /T1_6 11.955 Tf 11.95 0 Td ( ],where =50 A e q = q )Tj /T1_1 9.963 Tf 9.97 0 Td (q d B e Figure8-9. (a)Pitchrateerrorand(b)elevatordeectiontimeresponse(logscale)for 200secforCaseB.istheresponsefortheupperbound, [T nose + T tail + ] and )-222()-222()]TJ /T1_0 11.955 Tf 36.53 0 Td [(isthelowerbound, [T nose )Tj /T1_6 11.955 Tf 11.96 0 Td ( T tail )Tj /T1_6 11.955 Tf 11.95 0 Td ( ],where =50 132

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8.4Summar y Therearenotmanyapproachessuggestedinliteratureforthesimultaneous designofthedynamicsandthecontrollerforlinearparametervaryingsystems.A control-orienteddesignapproachisintroducedforlinearparameter-varying(LPV) systemsthatdoesnotactuallycomputeboththeopen-loopdynamicsandcontroller; rather,theplantisdeterminedforwhichacontrollerexiststhatminimizesaclosed-loop norm.TheconceptusesexistenceconditionsfortheLPVcontrollerwhichcanbe parametrizedaroundadesignspace.Theactualoptimizationresultsfromefcient explorationofthatdesignspaceusingsurrogatemodeling.Representativecase studiesofaninvertedpendulumandcartsystemandahypersonicvehicleisusedto demonstratethatthisapproachcanindeedgenerateanoptimaldesign. 133

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CHAPTER9 CONTR OL-ORIENTEDDESIGN:PARAMETRICUNCERTAINTYANALYSISFORA MASS-SPRING-DAMPERSYSTEM 9.1Introduction Chapters 6 9 investigatedthecontrol-orienteddesignapproachtodifferent applications.Thischapterinvestigatesthe effectofparametricuncertaintieson theclosed-loopperformance.Asimplesystemofamass-spring-dampershownin Figure 9-1 isusedtoperformtherobustnessanalysis. Figure9-1. Mass-spring-dampersystem Thedifferentclosed-loopmetricsusedinthisstudyare: Minimizetheclosed-loop H 1 -norm, r aspresentedinEquation 4 Minimizetheclosed-loop L 2 -norm, r showninEquation 4 foraLPVsystem Minimizetheerrorbetweentheactualanddesiredperformancebyusingthe minimumcontrolpowerasexplainedinChapter 7 9.2IncorporatingUncertaintyintheSystem ThedynamicsofthesystemshowninFigure 9-1 aregiveninEquation 9 ,where m k and B arethemassoftheblock,stiffnessofthespringanddampingofthedamper respectivelyand F isthecontrolforce(input)fordisturbanceattenuationortracking. m x + B x + kx = F (9) 134

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Quietoften theexactvaluesof m k and B arenotknown,i.e.thereisuncertaintyin theparameters.Theuncertaintyisincorporatedbyreplacing m k and B inEquation 9 withEquation 9,where m o isthenominalvalueofthemass, W m istheweightingfor theuncertainty, m andsoon. m = m o (1+ W m m ) k = k o (1+ W k k ) B = B o (1+ W B B ) (9) UsingEquation 9 andEquation 9,thedynamicscanberephrasedasshownin Equation 9. m o x + B o x + k o x = F )Tj /T1_3 11.955 Tf 11.95 0 Td ((m o W m m x + B o W B B x + k o W k k x ) (9) Thetermwiththeuncertaintyisrenamedasthesignal`w 'intermsofanother signal,` z ',asshowninEquation 9 w 'srepresenttheinputand z 'srepresentthe outputoftheplant.Usingposition, x andvelocity, x asthecomponentofthestates,the state-spacemodelisgivenasinEquation 9. w m = z m m w B = z B B w k = z k k (9) where z m = m o W m x z B = B o W B x z k = k o W k x 135

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2 6 4 x x 3 7 5 = 2 6 4 01 )Tj /T1_2 11.955 Tf (k o = m o )Tj /T1_2 11.955 Tf (b o = m o 3 7 5 2 6 4 x x 3 7 5 + 2 6 4 0000 )Tj /T1_1 11.955 Tf (1 =m o )Tj /T1_1 11.955 Tf (1=m o )Tj /T1_1 11.955 Tf (1=m o )Tj /T1_1 11.955 Tf 9.29 0 Td (1=m o 3 7 5 2 6 6 6 6 6 6 6 4 w m w B w k F 3 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 4 z m z B z k y 1 y 2 3 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 4 )Tj /T1_2 11.955 Tf (k o )Tj /T1_2 11.955 Tf (b o 0 b o k o 0 10 0 1 3 7 7 7 7 7 7 7 7 7 7 5 2 6 4 x x 3 7 5 + 2 6 6 6 6 6 6 6 6 6 6 4 )Tj /T1_1 11.955 Tf (1 )Tj /T1_1 11.955 Tf (1 )Tj /T1_1 11.955 Tf (1 )Tj /T1_1 11.955 Tf (1 0000 0000 0000 0000 3 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 4 w m w B w k F 3 7 7 7 7 7 7 7 5 (9) 9.3Example1:Minimize H 1 -norm Theobjectiveofthisexampleistochoosethebestspringanddampertoreject anyexternaldisturbanceandtooptimizetheclosed-loopperformancemetric, r Theopen-loopdesignvariables( )arethespringstiffness, k o andthedamping, B o Table 9-1 summarizestherangeoftheparametervalues.Intuitively,itisexpectedthat higherthedampingandthestiffness,betterthedisturbancerejection. Table9-1.Parametervalues Par ameterValue(units) Mass(m) 5 Stiffness (k o )[0.1 )Tj /T1_1 11.955 Tf 11.95 0 Td (2] Damping (B o )[0.1 )Tj /T1_1 11.955 Tf 11.95 0 Td (2] Thecontrol synthesisarchitectureusedinthisexampleisshowninFigure 9-2. Theperformanceweightingusedis W x = s +2 s +1 and thecontrolactuationweightingis W u = s +1 s +5 represents theuncertaintyinthemass( m ),spring( k )anddamping(B ). Thedesignspaceof =[ k o B o ] ispopulatedwithauniformgridof10000points andthesimulationsarerunfor3cases;( 1)nouncertaintyintheparameters,( 2 ) 10% uncertaintyand( 3) 20% uncertainty.Figure 9-3 showstheboxplotofthe r valuesfor 136

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Figure9-2. Synthesismodelfordisturbancerejection.Sub-systems: P ( ) -open-loop dynamicsasafunctionofthedesignvariables; Wx -performanceweighting; Wu -controlleractuationweighting; Wn -weightingonthesensornoise. Signals: d exogenousdisturbance; r -referencecommand; n -sensornoise; y -measurementsignal; u -controlinput; e p -weightederrorbetween referencesignal, r andoutputof P ; e u -weightedcontrollerinput; w i and z i aretheoutputandinputfromtheuncertaintyblock, the3cases 1 .Asexpected,thecasewithnouncertaintygivesthebestclosed-loop performance.Also,theperformancedecreaseswithanincreaseinuncertainty. Moreover,theoptimalsolutionisattheupperboundof k o =2 and B o =2 unitsfor allthreecases. 9.4Example2:Minimize L 2 -normfor LPV systems Theobjectiveofthisexampleistodesignthesystemalongwiththeoperating parameters( =[ k o B o ])andcontroller,soastooptimizeclosed-loopperformance metric, r giventhat variesduetooperationalwearandtearby 0.1.Assuchthis variation,meansthatthereisamaximumchangeinthevaluesof k o and B o by 0.2 units 1 Ina boxplot,theboxisdenedbylinesatthelowerquartile(25 %),median(50%), andupperquartile(75 %)values.Linesextendfromeachendoftheboxandoutiliers showthecoverageoftherestofthedata.Linesareplottedatadistanceof1.5timesthe inter-quartilerangeineachdirectionorthelimitofthedata,ifthelimitofthedatafalls within1.5timestheinterquartilerange.Outliersaredatawithvaluesbeyondtheendsof thelinesbyplacinga+signforeachpoint. 137

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Figure9-3. Boxplotoftheclosed-loopnorm, r .1-nouncertainty,210% uncertainty,3 20% uncertainty inoperation.Soiftheinitialvalueof k o inoperationis1unitandthereisachangeinthe valueby0.2units,thedesignvalueof k o istakenas0.9units.Therangeofthedesign variablesare [0.5,5] unitsforboththevariables.ThesystemisputintheLPVframework asdescribedinSection 8.2.4.ThesamesynthesismodelshowninFigure 9-2 isused otherthanalter, F describedinSection 5.9.3.Simulationresultsarerunwithno, 10% and 20% uncertaintyin m .Figure 9-4 showstheboxplotforthe3cases.Similartothe previousexample,theperformanceseemstodegradewithanincreaseinuncertainty,as expected.Theoptimalsolutioninthiscasesistheupperboundfor k o and B o 9.5Example3:Multi-objectiveOptimization-maximizeperformancebyusing minimumcontrolactuationand H 1 controlsynthesis Section 4.2.2.1 andChapter 7 motivatedtheneedtoconsiderweightingfunction variablesinthedesignprocesswhentheoptimizationobjectivesarenonH 1 metricsin the H 1 framework.Thisexampleconsidershowsuchdesignformulationsareaffected bythepresenceofparametricuncertainties. 138

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Figure9-4. Boxplotoftheclosed-loopnorm, r withLPVcontrolsynthesis.1-no uncertainty,210% uncertainty,320% uncertaintyin m Theobjectiveistodesignthesystem,showninFigure 9-1,wherethecontrol objectiveisforthemass, m totrackagivenstepresponsebyusingtheminimumcontrol actuation.Sotheoptimizationobjectiveistominimizetheperformanceerroralong withminimizingcontrolpower,showninEquation 9,similartoSection 7.4,where p RMSE istherootmeansquare(RMS)oftheerrorbetweentheactualpositionof m and thedesiredstepresponse, c MS isthecontrolpowerandclosed-loop H 1 -norm, r< 1 ensuresgoodrobustperformance.TheobjectivesaredenedinEquation 9 ,where T = 20 sec, r a istheactualoutputoftheplant.Thedesignvariablesaresummarized inTable: 9-2,where a i and b i arethepolesandzerosoftheweightingfunctiontransfer functionsinthecontrolsynthesismodel 2 2 Thesettling timeisbetween5to8sec. 139

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min 2 W 2 P 0 B @ p RMSE c MS 1 C A suchthat r 1 (9) p RMSE = v u u u t 1 T T Z 0 (r )Tj /T1_3 11.955 Tf 11.95 0 Td (r a ) 2 dt c MS = 1 T T Z 0 F 2 dt (9) Tab le9-2.Designvariables. -operatingrange, -weightingfunction No. DesignVariableTypeBoundsConstraints 1 k o [0.1,2] 2 B o [0.1, 2] 3 a 1 [1.5,10] 4 a 2 [0.1,2] a 1 > a 2 5 b 1 [0.1,2] 6 b 2 [2,10] b 1 < b 2 Thecontrol synthesismodelforthisexampleisshowninFigure 9-5.Here,the weightingfunctionsaregivenas W x = s + a 1 s + a 2 and W u = s + b 1 s + b 2 Thedesign pointsarechosenbypopulatingthe6-dimensionaldesignspacewith 5000pointsusinglatinhypercubesampling.Simulationresultsarerunfor3cases similartoexample1inSection 9.3.Figure 9-6 showstheparetofrontfortheobjectives forthe3casesandFigure 9-7 istheboxplotoftheclosed-loop H 1 -norm.Inthis case,thenormshowssimilarstatisticsforthe3cases,buttheparetofrontseemsto suggesthighertheuncertainty,betteristheparetofront 3 .Thisseemstosuggestthat 3 Itis toberememberedthatalltheseresultsarewithoutanyoptimization.Inthe optimizationprocess,theconstraint r< 1 willbeanactiveconstraint. 140

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Figure9-5. Synthesismodelfortracking.Sub-systems: P ( ) -open-loopdynamicsasa functionofthedesignvariables; Wx -performanceweighting; Wu -controller actuationweighting; Wn -weightingonthesensornoise.Signals: d exogenousdisturbance; r -referencecommand; n -sensornoise; y measurementsignal; u -controlinput; e p -weightederrorbetweenreference signal, r andoutputof P ; e u -weightedcontrollerinput; w i and z i arethe outputandinputfromtheuncertaintyblock, youcangetbetterperformanceforthesamelevelofcontroleffortbyaccountingfor uncertaintyintheprocess.Figure 9-8 showstheparetofrontwithT=200sec,which againhighlightsthesametrend,namely,highertheuncertainty,bettertheparetofront. Inordertounderstandthisbetter,Figure 9-9 performsanalysisfor3setofpoints ontheparetofront.3pointsarechosenfromtheparetofrontwith 20% uncertainty representedas inFigure 9-8 andthecorrespondingpointsforthesystemswith nouncertaintyand 10% uncertaintyarechosen.Table 9-3 showsthe H 1 -normfor the3setofpoints.Thenormincreaseswithincreaseinuncertaintyforthe3cases asinthepreviousexamples.Figure 9-11 ,Figure 9-10 andFigure 9-12 showthe transferfunctionsfromcommandtopositionofthemassforthe3sets.Thereisno discernabledifferencebetweenthefrequencyresponseswithandwithoutuncertainty. Suchanalysiswillbeperformedinthefuturetounderstandtheimpactofuncertaintyon theclosed-loopsystem.Italsohighlightstheimportanceofconsideringtheweighting functionsinthedesignprocessforsuchmulti-objectiveoptimizationproblems. 141

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Figure9-6. Paretofrontofobjectiveswithnouncertainty( ),with 10% uncertainty( 3) andwith 20% uncertainty( ) Table9-3.Closed-loop H 1 -normforthe3setofpointsontheparetofront. Set 12 3 Nouncer tainty7.2585.0568.556 10% uncertainty7.2765.0609.066 20% uncertainty7.2885.0649.462 SymbolinFigure 9-9 3 9.6Summar y Thischaptertriedtounderstandtheimpactofparametricuncertaintyonthe closed-loopsystemfordifferentdesignformulations.Whentheoptimizationobjective istominimizetheclosed-loopnorm, r ,itisseenthathighertheuncertainty,pooreris theperformance.Butforthemulti-objectiveoptimization,uncertaintyisshowntobe benecialfortheclosed-loopperformance. 142

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Figure9-7. Boxplotoftheclosed-loopnorm, r .1-nouncertainty,210% uncertainty,3 20% uncertainty Figure9-8. Paretofrontofobjectiveswithnouncertainty( ),with 10% uncertainty( 3) andwith 20% uncertainty( )forT=200sec 143

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Figure9-9. Analysisof3setofpointsontheparetofront.Set1;Set2;Set33 Blue-nouncertainty;Red10% uncertainty;Blue20% uncertainty Ar tox Br to x Figure9-10.Closed-loopfrequencyresponsefromcommandtoA)positionandB) velocityofcartforset1.:nouncertainty,-.-.-.10% uncertainty, )-222()-222(\000 20% uncertainty 144

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Ar tox Br to x Figure9-11.Closed-loopfrequencyresponsefromA)commandtopositionandB) velocityofcartforset2.:nouncertainty,-.-.-.10% uncertainty, )-222()-222(\000 20% uncertainty Ar tox Br to x Figure9-12.Closed-loopfrequencyresponsefromcommandtoA)positionandB) velocityofcartforset3.:nouncertainty,-.-.-.10% uncertainty, )-222()-222(\000 20% uncertainty 145

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CHAPTER10 CONCLUSIONS ANDRECOMMENDATIONS Thisdissertationproposesanapproachforthedesignoftheopen-loopdynamics alongwiththeoperatingrangeandthecontrollertooptimizeclosed-loopperformance. Theapproachisdemonstratedfordifferentapplicationswiththerealisticsystemofa hypersonicvehiclebeingconsidered. 10.1DissertationSummaryandConclusions Theintegratedsystemdesignproblemarisesfromtheneedtodevelopefcient systemsforoptimalclosed-loopperformance.Chapter 1 motivatestheneedtoadopt simultaneousdesignphilosophywithrelevantsurveyofexistingliterature.Thisresearch considersthedesignofspecicaspectsofahypersonicvehicleindepth. Chapter 2 givesabriefoverviewoftheevolutionofcontroloverthepastcentury. Threeimportantcontrolsynthesistechniquesof H 2 H 1 andlinearparametervarying (LPV)arediscussed.Chapter 3 discussesthedifferentoptimizationalgorithmsusedin thisresearchi.e.globalsearch,surrogate-baseddesignoptimizationand1-Diteration. InChapter 4,acontrol-orienteddesignapproachisproposedforsimultaneous designoftheopen-loopdynamicsandthecontroller.Theemphasisison`controller existence'thatnotesifacontrollerexistsforagivenopen-loopdynamicsgivingsome levelofperformance.Theoptimizationproblemisformulatedfordifferentobjectives and`controller-existence'conditionsforthedifferentsynthesistechniquesdiscussedin Chapter 2 .Asolutionmethodologyusingsurrogate-basedoptimization,discussedin Chapter 3 ,isusedtoefcientlysolvetheformulatedoptimizationproblem.Thebenets ofthisapproacharesummarizedinSection 4.2.4. Chapter 5 introducesthehypersonicvehicleexampleusedextensivelytodemonstrate thecontrol-orienteddesignapproachproposedinChapter 4.Thevehiclehasdynamics coupledintermsofaerodynamics,propulsiondynamics,structuraldynamicsand control.Thesignicantheatinginthevehicle,becauseofshockwaves,affectsthe 146

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dynamicsleading toavariationinclosed-loopperformance.Moreover,thereare variationsintheheatinginight.ALPVcontrollerisdevelopedtomitigatetheundesired thermaleffectsonthevehicle. Chapters 6 9 demonstratetheapproachfordifferentapplications,optimization objectivesandcontrolsynthesistechniques.InChapter 6 ,theapproachisapplied tominimizeaclosed-loopnormusing H 1 and H 2 synthesisusingsurrogate-based optimizationandglobalsearch.InChapter 7,amulti-objectiveoptimizationisformulated withcompetingobjectivesofmaximizingperformancebyusingminimumcontrol actuationforthehypersonicvehicle.Adesignapproachforlinear-parametervarying systemsisdemonstratedinChapter 8 .Chapter 9 analyzestheeffectsonparametric uncertaintyonclosed-loopperformanceintheproposedframework.Thedissertationis brieysummarizedinFigure 10-1 Figure10-1. Summaryofthedissertation 10.2Contribution Thekeycontributionsofthisprojectisrepeatedhereagainforbetterperspective. Proposedacontrol-orienteddesignapproachforstructuralsystemsforapplications whereclosed-loopperformanceiscriticalandthereiscouplinginthedynamics. Standardcontroltechniquesareusedinthisformulationandthereisafreedomon thechoiceoftheobjectivestobeoptimized. 147

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Demonstrated thesimultaneousdesignapproachforthedesignoflinear parametervaryingsystemswhichdesignsthevehiclealongwiththeoperating rangeandcontrollerforoptimalperformance. Investigatedthefeasibilityofusingsurrogate-basedoptimizationwiththeEGO andNSGA-IIalgorithmsfordesignoptimizationwithcontrols.Demonstratedthe benetsofusingmultiplesurrogatesinthedesignoptimizationprocess. Proposedanapproachtondoptimalsolutionswhentheoptimizationobjectives arenon-normmetricsfornorm-basedcontrollersbyintroducingweightingfunction variablesinthedesignprocess. Theframeworkforcontrol-orienteddesignisanalyzedindetailforahypersonic vehicle. Introducedthelinear-parametervaryingframeworktocompensateforthe aerothermoelasticeffects.Themulti-loopcontrolarchitecturewiththeinner-loop beingtheLPVcontrollerisusefultocontroltheparametervaryingdynamics acrossthehypersonicightenvelope. Developtheframeworkforthedesignofathermalprotectionsystemfroma controlsperspective.Theactualdesignofathermalprotectionsystemcanbe chosentovarythelevelofheatingreduction,andassociatedweight,across thestructure.Thisresearchconsidershowsuchdesignsandresultingthermal gradientsinuencetheabilitytoachieveclosed-loopperformance. Thetechnicalconceptsusedinthisdissertationwithasimplisticcontribution overviewishighlightedinFigure 10-2.Thebroadconceptsusedinthisresearchare: Controlsynthesis H 1 synthesis H 2 synthesis LPVsynthesis Optimizationalgorithms GlobalSearch Surrogate-basedoptimizationwithmultiplesurrogates,EGOandNSGA-II algorithm Dynamic,robustnessandsensitivityanalysis 148

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Systemmodeling Figure10-2. Importanttechnicalconceptsusedinthisresearch 10.3FutureDirections Thisapproachcanbeextendedtothesimultaneousdesignofthestructureand thecontrollerquieteasily.Forexample,inthehypersonicvehiclecasestudy,the optimizationproblemcanbeformulatedasminimizingthethermalstressesalong withtheclosed-loopnorm.Suchextensionscanbeexploredinthefuture,when thestudyofaerothermoelasticinteractionsismoremature. Thisapproachcanbeevaluatedforothernon-linearcontrolsynthesistechniques aswell.Wilcoxetal.[144]istherststepinthisattempt. Theapproachcantakeadvantageofthestoppingcriteriarecentlyproposedby Queipoetal.[ 145]forsurrogate-basedoptimization.Thiswillbeadvantageousfor systemswhereinthesimulationsarecomputationallyexpensive,forexample,LPV controllerdesigndemonstratedinChapter 5 andChapter 8. 149

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Anoptimal blendingortuningalgorithmcanbedevelopedtointegratethedifferent LPVcontrollersdevelopedinChapter 5. Theeffectofparametricuncertaintyontheclosed-loopperformanceneedstobe analyzedmoreindetail. 10.3.1Off-shootsofresearch Theexperienceoftheauthorwiththecontrol-orienteddesignapproachand surrogatemodels,canpotentiallyleadtothebelowresearchpaths: Onlinetuningandcalibrationofcontrollergainsforoptimalperformance: KillingworthandKristic[146]proposedanapproachtothetuningofaPID controllerusingextremumseeking.Inmanyrealisticsystems,itisnotpossible tooptimallychoosethecontrollergainsoff-lineonaccountoftheeffectof disturbances,variationsinthesystemparameters,unmodeleddynamicsetc. Insuchcases,itmaybeimperativetodevelopaself-tuningoronline-tuning algorithmtochoosethegainsafteraproductisinstalledorcommissioned.The authorbelievesthatsurrogate-basedoptimizationcanbeexploredasapractical wayofoptimallychoosingthegainsonlinenotonlyforPIDcontrol,butother synthesisconditionsandestimationarchitecturesaswell. Control-orientedmodelingusingsurrogatemodels:Therehavebeenmany papersinliteraturedevelopingmodelsforthehypersonicvehiclewhicharemore amenableforcontroldesign[ 125127].Inadditiontobeingcomputationallycheap, surrogate-modelingtechniqueslikekriging,polynomialresponsesurface,radial basisneuralnetworksandsupportvectormachinesgiveaccuracymetricsofthe surrogatets.Thesetechniquescanbeusedtodevelopaccurateandefcient control-orientedmodels. Pathplanning:Thissimultaneousdesignapproachusingsurrogate-based optimizationiscurrentlybeingexploredbyresearchersattheUniversityofFlorida foroptimalpathplanninginaerospaceapplications[147]. 10.4ImplicationsofResearch Theapproachproposedinthisdissertationcanbeincorporatedinthecomplete designframeworkasillustratedinFigure 10-3 [ 8 ].Thedesignvariableswhichisthe outputofthecontrol-orienteddesignblockcanthenbefedintotheniteelement analysis(FEA)/computationaluiddynamics(CFD)codeandhelpinthedesignofthe differentsubsystems. 150

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Figure10-3. Integrationofthecontrol-orienteddesignapproachinthecompletedesign framework Theresearchpresentedinthisproposalcanimpactmanyareasoftheaerospace designcommunity.Theprimarybeneciarieswouldbetheaerospacevehicledesign community.Inintegratedsystemdesign,thescopeofthedesignspacehasbeen increasedallowingthedesigndecisionstobemadefromaholisticframeworkto producemoreefcientandoptimaldesigns.Ithelpstooptimallychoosesensors, actuatorsetc.Theresearchondesignoflinearparametervaryingsystemsisalso applicabletoaerospacecontroldesigncommunity.Generatingasinglestate-space controllerovertheentireightenvelopeseemssomewhatlimited.Thisresearch providesinsightintothedesignofthevehiclealongwiththeoperatingrangeand controllerforoptimalperformance.Thecontrol-orienteddesignmethodologycan potentiallyimpactthesystemdesigncommunityaswell.Thisapproachgivesadditional degreesoffreedomtotweakthestructurealongwiththecontrollertooptimize performanceandmeetsystemrequirements. 151

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10.5Limitations The limitationsoftheintegratedsystemdesignproblemarisesfromtwoareas: computationalrequirementsandmodeldelity.Thisresearchfocusesontryingtoapply whattheauthorknowsbesttorealworldproblems.Thisresearchdemonstratesthe benetsofconsideringalargersystemdesignspace;buttoutilizetheseconceptsthe modelsneedtobeconsideredingreaterdetailandthecomputationalcapacityboth inspaceandtimeneedtoimprovised.Forexample,thecomputationaltimetoputa systemintheLPVframeworkandcreatingtheLPVcontrollerisatimeconsuming process.Butthiscomputationwillbedoneoff-lineandstoredinlook-uptables,sothe computationaltimeduringightislow. Inthehypersonicvehicleexampleconsidered,themodeldelityneedstoveried. Duringhypersonicight,thetemperatureofthestructureisexpectedtoreachupto 5000 o F ,buttoconsidermodelsatsuchhightemperaturesthematerialpropertiesneed tobespecied.Presently,suchknowledgeisrestricted.Duetothehighlynon-linear natureofthemodel,thethermaleffectscanbeonlyobservedbutthereasonforthe variationsinthedynamicscannotbeascertained.Thedynamicsneedtobeanalyzedin depthtounderstandthesystembetter. 152

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REFERENCES [1]Messac, A.andMalek,K.,Control-StructureIntegratedDesign, AIAAJournal, Vol.30,No.8,1992,pp.2124-2131. [2]Niewoehner,R.J.,andKaminer,I.I.,IntegratedAircraft-ControllerDesignUsing LinearMatrixInequalities, JournalofGuidance,ControlandDynamics, Vol.16, No.13,1996,pp.523-533. [3]Hiramoto,K.andGrigoriadis,K.M.,IntegratedDesignofStructuralandControl SystemswithaHomotopylikeIterativeMethod, InternationalJournalofControl, Vol.79,No.9,2006,pp.1062-1073. [4]Grigoriadis,K.M.,Zhu,G.,andSkelton,R.E.,OptimalRedesignofLinear Systems,, JournalofDynamicSystems,MeasurementandControl, Vol.118, No.3,1996,pp.598-605. [5]Messac,A.andTurner,J.D.,DualStructural-ControlOptimizationofLargeSpace Structures, AIAAStructures,StructuralDynamicsandMaterialConference,1984, AIAAPaper1984-1042. [6]Messac,A.,Control-StructureIntegratedDesignwithClosed-FormDesign MetricsUsingPhysicalProgramming, AIAAJournal, Vol.36,No.5,May1998, pp.855-864. [7]Tekin,O.A.,Babuska,R.,Tomiyama,T.,andSchutter,B.D.,Towardaexible controldesignframeworktoautomaticallygeneratecontrolcodeformechatronic systems, IEEEAmericanControlConference, 2009,pp.4933-4938. [8]OliverdeWeckandKarenWillcox,coursematerialsfor16.888/ESD.77 MultidisciplinarySystemDesignOptimization,Spring2004.MIT OpenCourseWare(http://ocw.mit.edu),MassachusettsInstituteofTechnology. Downloadedon 5 th March2009. [9]Sobieski,J.S.,andHaftka,R.T.,MultidisciplinaryAerospaceDesignOptimization: SurveyofRecentDevelopments, AIAA34thAerospaceSciencesMeetingand Exhibit, AIAAPaper1996-0711,1996. [10]Kajiwara,I.,Tsujioka,K.,andNagamatsu,A.,ApproachforSimultaneous OptimizationofaStructureandControlSystem,AIAAJournal,Vol.32,No.4., 1994,pp.866-873. [11]Tsujioka,K.,Kajiwara,I.,andNagamatsu,A.,IntegratedOptimumDesign ofStructureand H 1 ControlSystem,AIAAJournal,Vol.34,No.1,1996, pp.159-165. [12]Kajiwara,I.,andHaftka,R.T.,SimultaneousOptimumDesignofShapeand ControlSystemforMicroAirVehicles, ProceedingsofAIAA 40 th SDMConference, 1999,pp.1612-1621. 153

PAGE 154

[13]Kajiwar a,I.,andNagamatsu,A.,IntegratedDesignofStructureandControl SystemconsideringPerformanceandStability, ProceedingsoftheIEEEInternationalConferenceonControlApplications, 1999,pp.86-91. [14]Xianyu,W.,Xu,L.,Liang,J.,Shibin,L.,Xiaoqian,C.,andZhenguo,W.,TheMDO EnvironmentforHypersonicVehicleSystemDesignandOptimization, AIAAJoint PropulsionConferenceandExhibit,2006,AIAAPaper2006-5191. [15]Jing,C.,andShuo,T.,IntegratedOptimizationDesignofHypersonicVehicle, AIAAAerospaceSciencesMeetingandExhibit, 2008,AIAAPaper2008-142. [16]Fonseca,I.M.,Bainum,P.M.,LargeSpaceStructureIntegratedStructuraland ControlOptimization,UsingAnalyticalSensitivityAnalysis,JournalofGuidance, ControlandDynamics,Vol.24,No.5,2001,pp.978-982. [17]Gilbert,G.G.,andSchmidt,D.K.,IntegratedStructure/ControlLawDesignby MultilevelOptimization, JournalofGuidance,ControlandDynamics, Vol.14, No.5,pp.1001-1007,1989. [18]Lust,R.V.andSchmit,L.,ControlAugmentedStructuralSynthesis, AIAA Journal, Vol.26,No.1,1988,pp.86-95. [19]Caneld,R.A.,andMeirovitch,L.,IntegratedStructuralDesignandVibration SuppressionUsingIndependentModalSpaceControl, AIAAJournal, Vol.32, No.10,1994,pp.2124-2131. [20]Onada,J.andHaftka,R.T.,AnApproachtoStructure/ControlSimultaneous OptimizationforLargeFlexibleAircraft, AIAAJournal, Vol.25,No.8,1987, pp.1133-1138. [21]Shi,G.,andSkelton,E.R.,AnAlgorithmforIntegratedStructureandControl DesignwithVarianceBounds, IEEEConferenceonDecisionandControl,1996, pp.167-172. [22]Taylor,C.,IntegratedTransportationSystemDesignOptimization,PhD.Thesis, http://dspace.mit.edu/handle/1721.1/38644?show =full,Downloadedon 1 st March 2009. [23]Crossley,W.,Mane,M.,andNusawardhana,Variableresourceallocationusing multidisciplinaryoptimization:Initialinvestigationsforsystemofsystems AIAAISSMOMultidisciplinaryAnalysisandOptimizationConference, 2004,AIAAPaper 2004-4605. [24]Etten,C.V.,Balas,G.J.,andBennani,S.,Linearparametricallyvaryingintegrated ightandstructuralmodecontrolforaexibleaircraft, AIAAGuidance,Navigation,andControlConferenceandExhibit, 1999,AIAAPaper1999-4217. [25]http://ti.arc.nasa.gov/m/pub/1395h/1395%20(Soloway).pdf,Accessedon 24/02/2009. 154

PAGE 155

[26]G.Lombardi, A.Vicere,H.Paap,G.Manacorda,OptimizedAerodynamic DesignforHighPerformanceCars, AIAAMAOConference, 1998,AIAAPaper 1998-4789. [27]Fu,K.,Sun,D.,andMills,J.K.,SimultaneousMechanicalStrucutureandControl SystemDesign:OptimizationandConvexApproaches, Proceedingofthe2002 IEEEInternationalSymposiumonIntelligentControl, 2002,pp.746-751. [28]Grigoriadis,K.M.andSkelton,R.E.,IntegratedStructuralandControlDesign forVectorSecond-OrderSystemsviaLMIs, Proceedingsofthe1998American ControlConference ,1998,pp.1625-1629. [29]Venkayya,V.B.andTischler,V.A.,FrequencyControlanditsEffectonthe DynamicResponseofFlexibleStructures, AIAAJournal ,Vol.23,No.11,1985, pp.1768-1774. [30]Safonov,G.M.,Goh,C.K.,andLy,H.J.,ControlSystemSynthesisviaBilinear MatrixInequalities, IEEEAmericanControlConference,1994,pp.45-49. [31]Tanaka,T.,andSugie,T.,GeneralFrameworkandBMIFormulaefor SimultaneousDesignofStructureandControlSystems, IEEEConferenceon DecisionandControl,1997,pp.773-778. [32]Tuan,D.H.,andApkarian,P.,LowNonconvexity-RankBilinearMatrixInequalities: AlgorithmsandApplicationsinRobustControllerandStructureDesign, IEEE TransactionsonAutomaticControl,Vol.45,No.11,November2000,pp. 2111-2117. [33]Lu,J.andSkelton,R.E.,IntegratingStructureandControlDesigntoAchieve Mixed H 2 = H 1 Performance, InternationalJournalofControl,Vol.73,2000, pp.1449-1462. [34]Iwasaki,T.,Hara,S.andYamauchi,H.,Structure/ControlDesignIntegrationwith FiniteFrequencyPositiveRealProperty, ProceedingsoftheAmericanControl Conference ,2000,pp.549-533. [35]Suzuki,S.andYonezawa,S.,SimultaneousStructure/ControlDesign OptimizationofaWingStructurewithaGustLoadAlleviationSystem, JournalofAircraft ,Vol.30,No.2,1993,pp.268-274. [36]Hiramoto,K.andDoki,H.,SimultaneousOptimalDesignofStructuraland ControlSystemsforCantileveredPipesforConveyingFluid, JournalofSound andVibration ,Vol.274,No.3-5,July2004,pp.685-699. [37]Flores-Tlacuahuac,A.andBiegler,L.T.,ARobustandEfcientMixed-Integer Non-LinearDynamicOptimizationApproachforSimultaneousDesignand Control, ComputersandChemicalEngineering,Vol.31,2007,pp.588-600. 155

PAGE 156

[38]Rao, S.S.,Venkayya,V.B.andKhot,N.S.,GameTheoryApproachforthe IntegratedDesignofStructureandControls, AIAAJournal,Vol.26,No.4,1988, pp.463-469. [39]Rakowska,J.,Haftka,R.T.andWatson,L.T.,Multi-ObjectiveControl-Structure OptimizationviaHomotopyMethods, SIAMJournalonOptimization,Vol.3,1993, pp.654-667. [40]Milman,M.,Salama,M.,Schied,R.E.,Bruno,R.andGibson,J.S.,Combined Control-StructuralOptimization, JournalofComputationalMechanics ,Vol.8, No.1,January1991,pp.1-18. [41]Cheng,F.Y.andLi,D.,MultiobjectiveOptimizationofStructureswithandwithout Control, JournalofGuidance,ControlandDynamics ,Vol.19,No.2,1996, pp.392-397. [42]ForresterA.I.J.andKeaneA.J.,Recentadvancesinsurrogate-based optimization, ProgressinAerospaceSciences, Vol.45,No.1-3,2009,pp.50-79. [43]Kajiwara,I.andHaftka,R.T.,IntegratedDesignofAerodynamicsandControl SystemforMicroAerialVehicles, JSMEInternationalJournal:SeriesC ,Vol.43, No.3,2000,pp.684-690. [44]Controltheory.Wikipedia,TheFreeEncyclopedia, 16Feb2009,17:46UTC.25Feb2009, http://en.wikipedia.org/w/index.php?title Control theory&oldid 271154240. [45] KhargonekarPramod,coursematerialsforEEL6935AdvancedTopicsinControls andSystemsEngineering,Fall2009,UniversityofFlorida.Downloadedon 10 th August2010. [46]Zhou,K.,EssentialsofRobustControl, PrenticeHall,UpperSaddleRiver,New Jersey,1998,pp.50-350. [47]JonathanHow,coursematerialsfor16.31FeedbackControlSystems,Fall2007. MITOpenCourseWare(http://ocw.mit.edu),MassachusettsInstituteofTechnology. Downloadedon 19 th February2009. [48]H-innitymethodsincontroltheory.Wikipedia,TheFreeEncyclopedia.10Feb 2009,09:31UTC.25Feb2009,http://en.wikipedia.org/w/index.php?title=Hinnity methods in control theory&oldid 269738068. [49] Glover,K.,andDoyle,J.C.,Astatespaceapproachto H 1 optimalcontrol, in ThreeDecadesofMathematicalSystemsTheory:ACollectionofSurveys attheOccasionofthe50thBirthdayofJanC.Willems, H.NijmeijerandJ.M. Schumacher(Eds.),Springer-Verlag,LectureNotesinControlandInformation Sciences,Vol.135,1989. 156

PAGE 157

[50]Leith,D .,andLeithead,W.,SurveyofGain-SchedulingAnalysis&Design, InternationalJournalofControl,Vol.73,No.11,2000,pp.1011-1025. [51]Fitzpatrick,K.L.,Applicationsoflinearparameter-varyingcontrolforaerospace systems, M.S.Thesis,UniversityofFlorida,2003. [52]Sparks,A.G.,Linearparametervaryingcontrolforataillessaircraft, AIAA Guidance,Navigation,andControlConference, 1997,AIAAPaper1997-3636. [53]Lee,L.H.,Spillman,M.,Robust,reduced-order,linearparameter-varyingight controlforanF-16, AIAAGuidance,Navigation,andControlConference, 1997, AIAAPaper1997-3637. [54]Protz,J.,andSparks,A.,AnLPVcontrollerforataillessghteraircraft simulation, AIAAGuidance,Navigation,andControlConferenceandExhibit, 1998,AIAAPaper1998-4298. [55]Balas,G.J.,Mueller,J.B.,andBarker,J.,Applicationofgain-scheduled, multivariablecontroltechniquestotheF/A-18systemresearchaircraft, AIAA Guidance,Navigation,andControlConferenceandExhibit, 1999,AIAAPaper 1999-4206. [56]Mueller,J.B.,andBalas,G.J.,ImplementationandtestingofLPVcontrollersfor theNASAF/A-18SystemsResearchAircraft, AIAAGuidance,Navigation,and ControlConferenceandExhibit, 2000,AIAAPaper2000-4446. [57]Spillman,M.S.,RobustLongitudinalFlightControlDesignUsingLinear Parameter-VaryingFeedback, JournalOFGuidance,Control,andDynamics, Vol.23,No.1,2000,pp.101-108. [58]Marcos,A.,andBalas,G.,LinearparametervaryingmodelingoftheBoeing 747-100/200longitudinalmotion, AIAAGuidance,Navigation,andControl ConferenceandExhibit, 2001,AIAAPaper2001-4347. [59]Fialho,I.andBalas,G.J.,RoadAdaptiveActiveSuspensionDesignUsingLinear Parameter-VaryingGain-Scheduling, IEEETRANSACTIONSONCONTROL SYSTEMSTECHNOLOGY, Vol.10,No.1,Jan2002,pp.43-54. [60]Balas,G.J.,Linear,parameter-varyingcontrolanditsapplicationtoaturbofan engine, InternationalJournalofRobustandNonlinearControl, Vol.12,No.9, 2002,pp.763-796. [61]Butcher,M.andKarimi,A.,LinearParameter-VaryingIterativeLearningControl WithApplicationtoaLinearMotorSystem, IEEE/ASMETransactionsonMechatronics, Vol.15,No.3,2010,pp.412-420. [62]Wu,F.andGrigoriadis,K.M,LPVSystemswithparameter-varyingtimedelays: analysisandcontrol, Automatica, Vol.37,No.2,2001,pp.221-229. 157

PAGE 158

[63]Kothare ,M.V.,Mettler,B.,Morari,M.,Bendotti,P.,andFalinover,C.N.,Linear parametervaryingmodelpredictivecontrolforsteamgeneratorlevelcontrol, SupplementtoComputersandChemicalEngineering,6thInternationalSymposium onProcessSystemsEngineeringand30thEuropeanSymposiumonComputer, Vol21,Supplement1,1997,pp.S861-S866. [64]Marcos,A.andBennani,S.,LPVModeling,AnalysisandDesigninSpace Systems:Rationale,ObjectivesandLimitations, AIAAGuidance,Navigation,and ControlConference, 2009,AIAAPaper2009-5633. [65]Shin,J.Y.,andBalas,G.J.,OptimalBlendinginLinearParameterVaryingControl SynthesisforF-16Aircraft, AmericanControlConference, 2002,pp.41-46. [66]Wu,F.,ControlofLinearParameterVaryingSystems,PhD.Thesis, http://www.mae.ncsu.edu/homepages/wu/publications.html,Downloadedon 24 th March2009. [67]SacksJ,WelchW.J.,MitchellT.J.,andWynnH.P.,DesignandAnalysisof ComputerExperiments, StatisticalScience, Vol.4,No.4,1989pp.409-435. [68]Queipo,N.V.,Haftka,R.T.,ShyyW.,GoelT.,VaidyanathanR.,andTucker,P.K., Surrogate-basedanalysisandoptimization, ProgressinAerospaceSciences, Vol.41,2005,pp.1-28. [69]Simpson,T.W.,Toropov,V.,Balabanov,V.,andViana,F.A.C.,Designand AnalysisofComputerExperimentsinMultidisciplinaryDesignOptimization:a ReviewofHowFarWeHaveCome-orNot,in:12th AIAA/ISSMOMultidisciplinaryAnalysisandOptimizationConference ,2008,AIAAPaper2008-5802. [70]Meckesheimer,M.,Booker,A.J.,Barton,R.R.,andSimpson,T.W., Computationallyinexpensivemetamodelassessmentstrategies, AIAAJournal, Vol.40,No.10,2002,pp.2053-2060. [71]Viana,F.A.C.,Haftka,R.T.,andSteffen,Jr.V.,Multiplesurrogates:how cross-validationerrorscanhelpustoobtainthebestpredictor, Structuraland MultidisciplinaryOptimization, Vol.39,No.4,2009,pp.439-457. [72]Stein,M.L.,InterpolationofSpatialData:SomeTheoryforKriging,Springer, 1999. [73]KleijnenJPC,Krigingmetamodelinginsimulation:Areview, EuropeanJournalof OperationalResearch, Vol.192,No.3,2009,pp.707-716. [74]Muller,K.R.,Mika,S.,Ratsch,G.,Tsuda,K.andScholkopf,B.,Anintroductionto kernel-basedlearningalgorithms, IEEEtransactionsonneuralnetworks, Vol.12, No.2,2001,pp.181201. [75]Scholkopf,B.andSmola,A.J.,Learningwithkernels,TheMITPress,2002. 158

PAGE 159

[76]Smola,A.J .andScholkopf,B.,Atutorialonsupportvectorregression, Statistics andComputing, Vol.14,No.3,2004,pp.199222. [77]Park,J.,andSandberg,I.W.,UniversalApproximationUsing Radial-Basis-FunctionNetworks, NeuralComputation, Vol.3,No.2,1991, pp.246-257. [78]Cheng,B.,andTitterington,D.M.,Neuralnetworks:areviewfromastatistical perspective, StatisticalScience, Vol.9,No.1,1994,pp.2-54. [79]Box,G.E.P.,Hunter,J.S.,andHunter,W.G.,Statisticsforexperimenters:an introductiontodesign,dataanalysis,andmodelbuilding,JohnWiley & Sons,Inc, NewYork,USA,1978. [80]Myers,R.H.,andMontgomery,D.C.,ResponseSurfaceMethodology:Process andProductOptimizationUsingDesignedExperiments,JohnWiley & Sons,Inc, NewYork,USA,1995. [81]Glaz,B.,Goel,T.,Liu,L.,Friedmann,P.,andHaftka,R.T.,AMultipleSurrogate ApproachtoHelicopterRotorBladeVibrationReduction, AIAAJournal, Vol.47, No.1,2009,pp.271-282. [82]Samad,A.,Kim,K.,Goel,T.,Haftka,R.T.,andShyy.W.,MultipleSurrogate ModelingforAxialCompressorBladeShapeOptimization, JournalofPropulsion andPower, Vol.24,No.2,2008,pp.302-310. [83]Jones,D.,Schonlau,M.,andWelch,W,Efcientglobaloptimizationofexpensive black-boxfunctions,JournalofGlobalOptimization,Vol.13,No.4,1998, pp.455-492. [84]Martin,J.D.,andSimpson,T.W.,UseofKrigingmodelstoapproximate deterministiccomputermodels,AIAAJournal,Vol.43,No.4,2005,pp.853-863. [85]Viana,F.A.C.,Haftka,R.T.,andWatson,L.T.,Whynotrunthe efcientglobaloptimizationalgorithmwithmultiplesurrogates?, 51th AIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamics,andMaterials Conference, 2010,AIAAPaper2010-3090. [86]Deb,K.,Pratap,A.,Agarwal,S.,andMeyarivan,T.,AFastandElitist MultiobjectiveGeneticAlgorithm:NSGA-II, IEEETransactionsonEvolutionaryComputations, Vol.6,No.2,2002,pp.182-197. [87]Deb,K.,Agrawal,S.,Pratap,A.,andMeyarivan,T.,Afastelitistnon-dominated sortinggeneticalgorithmformulti-objectiveoptimization:NSGA-II,LectureNotes inComputerScience,2000,pp.849-858. [88]Zitzler,E.,Laumanns,M.,andBleuler,S.,ATutorialonEvolutionary Multi-ObjectiveOptimization, http://www.cs.cinvestav.mx/emooworkgroup/, Downloadedon 22 nd June2010. 159

PAGE 160

[89]AIAAMDO websitehttp://endo.sandia.gov/AIAA MDOTC/main.html. Accessedon 5 th March2009. [90]Carbonnel,C.,HinfControllerDesignandMu-Analysis:PowerfulToolsfor FlexibleSatelliteAttitudeControl, AIAAGuidance,NavigationControl, 2010,AIAA Paper2010-7907. [91]Balas,G.andHodgkinson,J.,ControlDesignMethodsforGoodFlyingQualities, AIAAGuidance,NavigationControl, 2009,AIAAPaper2009-6319. [92]Apkarian,P.,Gahinet,P.,andBecker,G.,Self-Scheduled H 1 ControlofLinear ParameterVaryingSystems, IEEEConferenceonDecisionandControl ,Vol.3, 1994,pp.2026-2031. [93]Apkarian,P.,andGahinet,P.,AConvexCharacterizationofGain-Scheduled H 1 Controllers, IEEETransactionsonAutomaticControl ,Vol.40,No.5,1995, pp.853-864. [94]Balas,G.,Chian,R.,Packard,A.,andSafonovM.,RobustControlToolbox3, User'sGuide,March2009. [95]Gahinet,P.,Nemirovski,A.,Laub,A.,andChilali,M., LMIControlToolbox-Users Guide,TheMathworks,Inc.Natick.,MA,1995. [96]Garrick,I.E.,andCunningham,H.J.,ProblemsandDevelopmentsin Aerothermoelasticity, ProceedingofSymposiumonAerothermoelasticity, ASD-TR-61-6435,Feb1962. [97]Doggett,V.R.,Ricketts,H.R.,Noll,E.T.,andMalone,B.J.,NASP AeroservothermoelasticityStudies,NASATM104058,April,1991. [98]Heeg,J.,Zeiler,A.T.,Pototzky,S.A,Spain,V.C.,andEngelund,C.W., AerothermoelasticAnalysisofaNASPDemonstratorModel, AIAAStructures, StructuralDynamics,andMaterialsConference ,1993,AIAAPaper1993-1366. [99]J.Heeg,M.G.GilbertandA.S.Pototzky,AerothermoelasticEffectsfora ConceptualHypersonicAircraft, JournalofAircraft, Vol.30,No.4,1993, pp.453-458. [100]McNamara,J.J.,andFriedmann,P.P.,AeroelasticandAerothermoelasticAnalysis ofHypersonicVehicles:CurrentStatusandFutureTrends,AIAA Structures, StructuralDynamics,andMaterialsConference ,2007,AIAAPaper2007-2013. [101]McNamara,J.,Friedmann,P.,Powell,K.,andThuruthimattam,B.,Aeroelastic andAerothermoelasticVehicleBehaviorinHypersonicFlow,AIAA/CIRCA 13th InternationalSpacePlanesandHypersonicSystemsandTechnologies,2005, AIAAPaper2005-3305. 160

PAGE 161

[102]Bisplinghoff, R.L.andDugundji,J.,InuenceofAerodynamicHeatingon AeroelasticPhenomenainHighTemperatureEffectsinAircraftStructures, Agardograph No.28,EditedbyN.J.Hoff,PergamonPress,1958,pp.288-312. [103]Garrick,I.E.,ASurveyofAerothermoelasticity, AerospaceEngineering,January, 1963,pp.140-147. [104]Ricketts,R.,Noll,T.,Whitlow,W.,andHuttsell,L.,AnOverviewof AeroelasticityStudiesfortheNationalAerospacePlane,Proc.34th AIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamicsandMaterials Conference,1993,pp.152-162. [105]Thuruthimattam,B.J.,Friedmann,P.P.,McNamara,J.J.,andPowell, K.G.,AeroelasticityofaGenericHypersonicVehicle, Proc.43rd AIAA/ASME/ASCE/AHSStructures,StructuralDynamicsandMaterialsConference ,2002,AIAAPaper2002-1209. [106]Thuruthimattam,B.J.,Friedmann,P.P.,McNamara,J.J.,andPowell,K.G., ModelingApproachestoHypersonicAerothermoelasticitywithApplicationto ReusableLaunchVehicles, Proc.44thAIAA/ASME/ASCE/AHSStructures, StructuralDynamicsandMaterialsConference ,2003,AIAAPaper2003-1967. [107]Rogers,M.,Aerothermoelasticity, AeroSpaceEngineering,October1958, pp.34-43. [108]Bisplinghoff,R.L.,Ashley,H.,andHalfman,R.L., Aeroelasticity,Addison-Wesley, 1955. [109]Lind,R.,Bufngton,J.,andSparks,A.,Multi-LoopAeroservoelasticControlofa HypersonicVehicle,1999,AIAAPaper1999-4123. [110]Waszak,M.R.,Buttrill,C.S.,andSchmidt,D.K.,ModelingandModel SimplicationofAeroelasticVehicles:AnOverview,NASATM-107691,1992. [111]Meirovitch,L.,andTuzcu,I.,IntegratedApproachtotheDynamicsandControlof ManeuveringFlexibleAircraft,NASACR-2003-211748,2003. [112]McLean,D.,AutomaticFlightControlSystems,Prentice-HallInternational, Hertfordshire,UK,1990. [113]Theodore,R.C.,Ivler,M.C.,Tischler,M.B.,Field,J.E.,Neville,L.R.,andRoss, P.H.,SystemIdenticationofLargeFlexibleTransportAircraft, AIAAAtmospheric FlightMechanicsConferenceandExhibit,AIAAPaper2008-6894,2008. [114]Chavez,F.R.andSchmidt,D.K.,AnalyticalAeropropulsive/Aeroelastic Hypersonic-VehicleModelwithDynamicAnalysis, JournalofGuidance,Control,andDynamics,Vol.17,No.6,1994,pp.1308-1319. 161

PAGE 162

[115]Schmidt,D .K.,IntegratedControlofHypersonicVehicles-ANecessityNotJusta Possibility,1993,AIAAPaper1993-3761. [116]Schmidt,D.K.,IntegratedControlofHypersonicVehicles,1993,AIAAPaper 1993-5091. [117]Gregory,I.,McMinn,J.,Shaughnessy,J.andChowdry,R.,HypersonicVehicle ControlLawDevelopmentUsing H 1 and -Synthesis,1992,AIAAPaper 1992-5010. [118]Buschek,H.andCalise,A.J.,UncertaintyModelingandFixed-OrderController DesignforaHypersonicVehicleModel, JournalofGuidance,Navigation,and Control,Vol.20,No.1,1997,pp.42-48. [119]Heller,M.,Sachs,G.,Gunnarsson,K.,Frank,H.andRylander,D.,Flight DynamicsandRobustControlofaHypersonicTestVehiclewithRamjet Propulsion,1998,AIAAPaper1998-1521. [120]Xu,H.,Mirmirani,D.andIoannou,P.A.,AdaptiveSlideModeControlDesignfor aHypersonicFlightVehicle, JournalofGuidance,ControlandDynamics,Vol.27, No.5,2004,pp.829-938. [121]Mooij,E.,NumericalInvestigationofModelReferenceAdaptiveControlfor HypersonicAircraft, JournalofGuidance,ControlandDynamics ,Vol.24,No.2, 2001,pp.315-323. [122]Wang,Q.andStengel,R.F.,RobustNonlinearControlofaHypersonicAircraft, JournalofGuidance,ControlandDynamics,Vol.23,No.4,2000,pp.577-585. [123]Luo,Y.,Serrani,A.,Yurkovich,S.,Oppenheimer,M.W.andDoman,D.B., Model-PredictiveDynamicControlAllocationSchemeforReentryVehicles, JournalofGuidance,ControlandDynamics,Vol.30,No.1,2007,pp.100-113. [124]Sigthorsson,D.O.,Jankovsky,P.,Serrani,A.,Yurkovich,S.,Bolender,M.A. andDoman,D.B.,RobustLinearOutput-FeedbackControlofanAirbreathing HypersonicVehicle, JournalofGuidance,ControlandDynamics ,Vol.31,No.4, 2008,pp.1052-1066. [125]Parker,J.T.,Serrani,A.,Yurkovich,S.,Bolender,M.A.andDoman,D.B., Control-OrientedModelingofanAir-BreathingHypersonicVehicle, Journal ofGuidance,ControlandDynamics,Vol.30,No.3,2007,pp.856-869. [126]Sigthorsson,D.,andSerrani,A.,DevelopmentofLinearParameter-Varying ModelsofHypersonicAir-BreathingVehicles, AIAAGuidance,Navigation,and ControlConference, 2009,AIAAPaper2009-6282. [127]Sigthorsson,D.,Serrani,A.,andBolender,M.,andDoman,D.,LPVControl DesignforOver-ActuatedHypersonicVehiclesModels, AIAAGuidance,Navigation,andControlConference, 2009,AIAAPaper2009-6280. 162

PAGE 163

[128]Jank ovsky,P.,Sigthorsson,D.,Serrani,A.,Yurkovich,S.,Bolender,M.and Doman,D.,OutputFeedbackControlandSensorPlacementforaHypersonic VehicleModel,2007,AIAAPaper2007-6327. [129]Fiorentini,L.,Serrani,A.,Bolender,M.andDoman,D.,Nonlinear Robust/AdaptiveControllerDesignforanAir-breathingHypersonicVehicle Model,2007,AIAAPaper2007-6329. [130]Lind,R.,LinearParameter-VaryingModelingandControlofStructuralDynamics withAeroelasticEffects, JournalofGuidance,ControlandDynamics,Vol.25, No.4,2001,pp.733-739. [131]Vosteen,L.F.,EffectofTemperatureonDynamicModulusofElasticityofSome StructuralAlloys,NACA,TN4348,August1958. [132]Bolender,M.,andDoman,D.,ANon-linearModelfortheLongitudinalDynamics ofaHypersonicAir-BreathingVehicle, AIAAGuidance,NavigationandControl ConferenceandExhibit,2005,AIAAPaper2005-6255. [133]Bolender,M.,andDoman,D.,NonlinearLongitudinalDynamicalModelofan Air-BreathingHypersonicVehicle, JournalofSpacecraftandRockets,Vol.44, No.2,March-April2007,pp.374-387. [134]Bolender,M.,andDoman,D.,ModelingUnsteadyHeatingEffectsonthe StructuralDynamicsofaHypersonicVehicle, AIAAAtmosphericFlightMechanics ConferenceandExhibit,2006,AIAAPaper2006-6646. [135]Williams,T.,Bolender,M.,Doman,D.,andMorataya,O.,AnAerothermalFlexible ModelAnalysisofaHypersonicVehicle, AIAAAtmosphericFlightMechanics ConferenceandExhibit,2006,AIAAPaper2006-6647. [136]Culler,A.,Williams,T.,andBolender,M.,AerothermalModelingandDynamic AnalysisofaHypersonicVehicle, AIAAAtmosphericFlightMechanicsConferenceandExhibit,2007,AIAAPaper2007-6395. [137]Oppenheimer,M.,Skijins,T.,Bolender,M.,andDoman,D.,AFlexibleHypersonic VehicleModelDevelopedWithPistonTheory,AIAA AtmosphericFlightMechanicsConferenceandExhibit,2007,AIAAPaper2007-6396. [138]LophavenSN,NielsenHB,andSondergaardJ,DACE-AMATLABKriging Toolbox,TechnicalReportIMM-TR-2002-12,InformaticsandMathematical Modeling, TechnicalUniversityofDenmark, 2002. [139]Mathworkscontributors,MATLABThelanguageoftechnicalcomputing, Version7.0Release14,TheMathWorksInc,2004. [140]S.R.Gunn,Supportvectormachinesforclassicationandregression,Technical report,ImageSpeechandIntelligentSystemsResearchGroup,Universityof Southampton,UK,1997. 163

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[141]VianaF AC,SURROGATESToolBoxUser'sGuide, http://fchegury.googlepages.com,2009. [142]MckayMD,BeckmanRJ,andConoverWJ,AComparisonofThreeMethods forSelectingValuesofInputVariablesfromaComputerCode, Technometrics, Vol.21,1979,pp.239-245. [143]http://www.engin.umich.edu/group/ctm/examples/pend/invpen.html. [144]Wilcox,Z.D,Bhat,S.,Lind,R.andDixon,W.E.,ControlPerformanceVariation duetoNonlinearAerothermoelasticityinaHypersonicVehicle:Insightsfor StructuralDesign, AIAAGNCConference,2009,AIAAPaper2009-6184. [145]Queipo,N.,Maracaibo,Z,Verde,A.,andNava,E.,SettingTargetsfor Surrogate-BasedOptimization, 51stAIAA/ASME/ASCE/AHS/ASCStructures, StructuralDynamics,andMaterialsConference, 2010,AIAAPaper2010-3087. [146]Killingsworth,N.,andKrstic,M.,PIDtuningusingextremumseeking,Control SystemsMagazine,Vol.26,February2006,pp.70-79. [147]Johnson,B.andLind,R.,ImprovingTree-BasedTrajectoriesThroughOrder Reduction/ExpansionandSurrogateModels, AIAAGNCConference, 2010,AIAA Paper2010-8020. 164

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BIOGRAPHICALSKETCH Sank ethBhatwasborninMumbai,Indiain1984.HedidhisschoolingatO.L.P.S. highschoolandattendedjuniorcollegeatK.J.SomaiyaCollegeofScienceboth inthesuburbsofMumbai.HethenattendedV.J.T.I.Engineeringcollegeafliated withthe`MumbaiUniversity'andgraduatedwithaBachelorofEngineering(B.E.) degreeinJune2006.HestartedgraduateschoolintheDepartmentofMechanical andAerospaceEngineering,UniversityofFlorida,Gainesville,FLandwentonto receivehisM.S.inAerospaceEngineeringinDecember2008undertheadvisement ofDr.RickLindandcontinuedtoworkonhisPhD.attheFlightControlLab.His researchinvolvessimultaneousdesignoftheopen-loopdynamicsandthecontrollerto optimizeclosed-loopperformanceandhisresearchinterestsareintheeldofcontrols, optimization,dynamicanalysisandstructuraldynamics.Sankethpursuedaninternship inthemechatronicsgroupatMistubishiElectricResearchLabs,Cambridge,MAin Summer2010andiscurrentlyaninternatHoneybeeRobotics,NewYork,NY.Hehopes tocontinueworkingasasystemsengineerinaresearchlabafterhisdissertation. 165