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Reduced Order Models and the Approximation of Stokes Flow Control Problems


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REDUCEDORDERMODELSANDTHEAPPROXIMATION OFSTOKESFLOWCONTROLPROBLEMS By YUNFEIFENG ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2003

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ACKNOWLEDGMENTS Iwishtoexpressmydeepestgratitudetomyadvisor,Dr.Andr ewJ.Kurdila, whoseconstantsupportandpatientguidanceprovidedaclea rpathformyresearch. Ithankhimforhiseortinteachingandguidingmeinthisres earchfeld.Ihave learnedagreatdealfromhim.IwouldalsoliketothankDr.Da vidW.Mikolaitis andDr.RickLindfortheireortsinguidingmyresearch.Ith ankDr.WeiShyy, Dr.NormanG.Fitz-Coy,andDr.OscarD.Crisallefortheirhe lpfuldiscussionsand suggestions,andtheirtimeandeorttoserveonmycommitte e. Additionally,IwouldliketothankDr.YongshengLian,Jion gyangWu,and JellieJacksonfortheiradviceandtheireortinprovidin gCFDdataformyresearch work. IwouldliketothankNASAforfundingthiswork. Finally,Iwouldliketothankmywifeandmyparentsfortheir immeasurable advice,encouragement,andsupport. ii

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.............................ii LISTOFTABLES.................................vi LISTOFFIGURES................................vii NOMENCLATURE................................xi ABSTRACT....................................xiv CHAPTERS 1INTRODUCTIONANDLITERATUREREVIEW............1 1.1IntroductionandMotivation.....................1 1.1.1IntroductiontoFlowControl.................1 1.1.2Motivation...........................3 1.2Objectives...............................4 1.3ReducedOrderModelsofIncompressibleViscousFluidFl ows..5 1.3.1GoverningEquations.....................5 1.3.2ReducedOrderModeling...................6 1.3.3ConstructionofBasisFunctionsinROM..........7 1.4ControlStrategiesinthisThesis................... 9 1.4.1OptimalFlowControl.....................9 1.4.2ModelPredictiveControl(RecedingHorizonControl) ...12 1.4.3GainSchedulingandLinearParameter-varyingSystem s.13 2REDUCED-ORDERMODELINGOFSTOKESFLOW.........16 2.1GeneralStokesFlowProblem....................17 2.1.1StokesEquationandTimeScaleIssue............17 2.1.2ProblemFormulation.....................20 2.1.3OrderReductionFramework.................28 2.2DrivenCavityStokesFlowProblem.................36 2.2.1ControlProblemDescription.................36 2.2.2ReducedOrderModelofDrivenCavityProblem......38 2.2.3StationaryDrivenCavityFlow................41 2.3PODandUnsteadyDrivenCavityFlowProblem.........43 2.3.1ProperOrthogonalDecomposition(POD)..........43 2.3.2PODforUnsteadyDrivenCavityFlow...........45 iii

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2.3.3ROMbyPODforUnsteadyDrivenCavityStokesFlow..55 3OPTIMALCONTROLOFDRIVEN-CAVITYSTOKESFLOW....63 3.1VariationalOperator ........................63 3.2OptimalControlFramework.....................64 3.2.1FixedTerminalTimeProblems................66 3.3OptimalControlofDrivenCavityFlow...............6 9 3.3.1ControlProblemDescription.................69 3.3.2OptimalitySystemsandtheClassicalRiccatiEquatio n..71 3.3.3NumericalSimulations....................74 3.4DirectDiscretizationMethod.....................7 7 3.4.1MethodDescription......................77 3.4.2NumericalSimulations....................79 3.5RecedingHorizonControl(RHC)..................85 3.5.1RecedingHorizonFormulation................85 3.5.2NumericalSimulations....................86 4ROBUSTCONTROLOFDRIVENCAVITYSTOKESFLOW.....88 4.1 H 1 -basedModelingandControl..................88 4.1.1SingularValues, H 1 )Tj/T1_0 11.9552 Tf9.23997 0 Td[(norm..................89 4.1.2 H 1 OptimalControlSynthesis................91 4.2 H 1 OptimalControlofDrivenCavityFlow............92 4.2.1Open-loopModel.......................93 4.2.2ControlSynthesis.......................94 4.2.3Closed-loopResponse.....................96 4.3 H 1 RobustControl..........................108 4.3.1Uncertainties..........................108 4.3.2SmallGainTheorem.....................108 4.3.3LinearFractionalTransformation..............11 0 4.3.4 H 1 RobustControlSynthesis................111 4.3.5StructuredUncertainties...................112 4.3.6StructuredSingularValue: .................113 4.3.7ControlDesignwith -Synthesis...............114 4.4 H 1 RobustControlofDrivenCavityFlow.............115 4.4.1UncertaintyModel.......................115 4.4.2ControlObjectives.......................118 4.4.3ControllerSynthesis......................118 4.4.4RobustnessAnalysis......................120 4.4.5Closed-loopSimulation....................123 4.5LPVControl..............................129 4.5.1LPVSystem..........................129 4.5.2Parameter-dependentState-SpaceModels.......... 131 4.5.3LinearMatrixInequalities(LMIs)..............13 2 4.5.4GainScheduled H 1 Control.................133 iv

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4.6LPVControlofDrivenCavityFlow.................136 4.6.1ControlSynthesis.......................137 4.6.2Closed-LoopResponsesforFixed Values.........137 4.6.3Closed-LoopResponsesforTime-Varying .........141 5CONCLUSIONS...............................143 REFERENCES...................................145 BIOGRAPHICALSKETCH............................152 v

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LISTOFTABLES Table page 2.1Singularvaluesfor =0 : 1........................49 2.2Singularvaluesfor =1.........................49 2.3Singularvaluesfor =2 : 5........................50 4.1Controllerperformance..........................1 20 4.2Gainandphasemarginoftheclosed-looptransferfuncti ons......124 vi

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LISTOFFIGURES Figure page 1Geometry.................................xi 1.1Drivencavityrow.............................4 2.1Finiteelements..............................22 2.2Bilinearquadrilateralelementdomainandlocalnodeor dering....23 2.3Apressurequadrilateralanditsfourassociatedveloci tyquadrilaterals26 2.4Geometry.................................29 2.5Stokesdrivencavityrowproblem.................... 37 2.6Stationarydrivencavityrow....................... 41 2.7Comparisonofreducedordersolutionsofvelocitiestof ullordersolution42 2.8TogenerateensembletowhichPODistaken............. .47 2.9ContourplotsofPODeigenmodeswithstreamlinesat =0 : 1....47 2.10ContourplotsofPODeigenmodeswithstreamlinesat =1.....48 2.11ContourplotsofPODeigenmodeswithstreamlinesat =2 : 5....48 2.12Erroraveragedovertimestepsisboundedbythesingula rvalues m +1 51 2.13Errorofeachtimestepisboundedbythesingularvalues .......52 2.14Erroraveragedovertimestepsisboundedbythesingula rvalues m +1 52 2.15Errorofeachtimestepisboundedbythesingularvalues .......53 2.16Erroraveragedovertimestepsisboundedbythesingula rvalues m +1 53 2.17Errorofeachtimestepisboundedbythesingularvalues .......54 2.18Unsteadydrivencavityrow.......................5 5 2.19Contourplotofinitialvelocityfeld............... ....56 2.20ErroranalysisofROMapproximation................ ..57 vii

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2.21ErroranalysisofROMapproximationat =0 : 1............58 2.22ErroranalysisofROMapproximation................ ..59 2.23ErroranalysisofROMapproximationfor =1............60 2.24ErroranalysisofROMapproximationfor =2 : 5...........61 2.25ErroranalysisofROMapproximationfor =2 : 5...........62 3.1Statessolutionforthetrackingproblem............. ....75 3.2Controlhistoryforthetrackingproblem............. ...75 3.3Riccatisolutionforthetrackingproblem............ ....76 3.4Historyof ( t )forthetrackingproblem.................76 3.5Targetrowfor N t =31.........................80 3.6Controlledrowfor N t =11 ;N u =4..................81 3.7Controlledrowfor N t =11 ;N u =8..................82 3.8Controlledrowfor N t =31 ;N u =8..................83 3.9Solutionsof f ( t )vstime.........................84 3.10Optimalstatessolutionscomparingtothetargetstate s........87 3.11RHCcontrolhistory...........................87 4.1Closed-loopsystem............................91 4.2Synthesismodel..............................95 4.3Transferfunctionsat =1........................97 4.4Disturbancerejectioneectofthecontrollerdesineda t =1.....97 4.5Open-loopvelocityoutputathorizontalcenter-line.. ........98 4.6Closed-loopvelocityoutputathorizontalcenter-line ..........98 4.7Statesvariablesandcontrolinput.................. ..99 4.8Fullorderclosed-loopblockdiagramforpointcontroll er........99 4.9Transferfunctionsfortheclosed-loopsystem........ ......100 4.10Closed-loopvelocityoutputathorizontalcenter-lin e..........100 4.11Open-loopvelocityoutputathorizontalcenter-line. .........101 viii

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4.12Closed-loopvelocityoutputathorizontalcenter-lin e..........101 4.13Transferfunctionsat =10.......................102 4.14Disturbancerejectioneectofthecontrollerdesigne dat =10....103 4.15Open-loopvelocityoutputathorizontalcenter-line. .........103 4.16Closed-loopvelocityoutputathorizontalcenter-lin e..........104 4.17Statesvariablesandcontrolinput................. ...105 4.18Fullorderclosed-loopblockdiagramforpointcontrol ler........105 4.19Transferfunctionsfortheclosed-loopsystem....... .......106 4.20Closed-loopvelocityoutputathorizontalcenter-lin e..........106 4.21Open-loopvelocityoutputathorizontalcenter-line. .........107 4.22Closed-loopvelocityoutputathorizontalcenter-lin e..........107 4.23BlockdiagramfortheSmallGainTheorem............. ..109 4.24Linearfractionaltransformation F u ( G; )...............110 4.25Linearfractionaltransformation F ` ( G; )................110 4.26 H 1 robustcontrolproblem........................111 4.27 H 1 -synthesisconsideringrobustperformance............. 112 4.28LFTsystemforrobuststabilityanalysisusing ............114 4.29Synthesismodel..............................119 4.30Extractedsynthesismodel........................ 120 4.31Original H 1 controller..........................121 4.32Balancedtruncatedcontroller.................... ..122 4.33Closed-loopsystemwithsynthesismodel............ ....123 4.34Controllerperformance.......................... 123 4.35Closed-loopsystemfortimeresponse............... ...125 4.36Transferfunctionat5 th sensorlocation.................125 4.37Timeresponseat t =0 : 2.........................126 4.38Timeresponsefornominalmodel.................... 126 ix

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4.39Timeresponsefortruthmodel...................... 127 4.40Transferfunctionat5 th sensorlocation.................127 4.41Timeresponseat t =0 : 2.........................127 4.42Timeresponsefornominalmodel.................... 128 4.43Timeresponsefortruthmodel...................... 128 4.44Gainscheduled H 1 controlproblem...................135 4.45Transferfunctionsatconstant .....................138 4.46Flowvelocityatconstant ........................139 4.47Timeresponsefor =1.........................139 4.48Timeresponsefor =2 : 5........................140 4.49Timeresponsefor =5.........................140 4.50Timeresponsefor =7 : 5........................140 4.51Controlhistorywhen =1........................141 4.52Exampleofparametertrajectoryof ( t ).................141 4.53TimeresponseoftheLPVsystemalongparametertraject ory.....142 x

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NOMENCLATURE SymbolsforthePhysicalDomain Throughoutthisthesis,weusefollowingsymbols n:thephysicaldomain.@ n:theboundaryofthedomain u :thesetofindicesofnodalpointsassociatedwiththeinter iorofthe domain :thesetofindicesofnodalpointsassociatedwiththebound ary Frequently,itisnecessarytopartition intosubsets.Forexample,inthecaseas showningure1theboundaryispartionedinto c and k .Controlsacton c .We Figure1:Geometry write @ n= c [ k = c [ k where c and k aresetsofindicesofnodalpointsassociatedwith c and k respectively.SymbolstoExpressVelocities (1)Vectorsofvelocityelds xi

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Thefollowingsymbolsareusedtorepresentvelocityeldsw hicharefunctions of(x,y,t): ~ V ( x;y;t )||Velocityeldtobesolved ~ V 0 ( x;y )||Initialvelocityeld ~ ( x;y )||Velocityeldofcontrolbasis ~ ( x;y )||VelocityeldofROMbasis (2)Nodalvelocityvectors Boldfontsymbolsareusedtorepresentnodalvelocityvecto rs.Throughout thisthesis,tilde(~)isusedtoemphasizeanodalvelocityf romsingleprescribed basisorcombinationofprescribedbases.Therearetwotype ofprescribedbases,i.e. ~ ~ V 0 and ~ ~ Thefollowingconstitutesthemostfrequentlyusedsymbols : ~ V ||Unknownvelocityassociatedwith u ~ ~ V ||Nodalvelocityassociatedwithallallprescribedbases ~ ~ V 0 ||Nodalvelocityassociatedwithboth u and ~ ~ ||Nodalvelocityassociatedwithboth u and ~ ||NodalvelocityofROMbasisassociatedwith u [ ~ ]||Ensembleofcontrolbases:[ ~ ] 4 = [ ~ ~ 1 ~ ~ 2 ] []||EnsembleofROMbases:[] 4 = [ ~ 1 ~ 2 ] Thefollowinglistsummarizesfrequentlyencounteredcomp onentsofvariousvectors: ~ V k ||Nodalvelocity, k th nodeof ~ V ;k 2 u V k;i ||the i th componentofthenodalvelocity ~ V k ;k 2 u ;i =1 ; 2 ~ ~ V u ||Thesubsetof ~ ~ V associatedwith u ~ ~ V ||Thesubsetof ~ ~ V associatedwith ~ ~ V 0 ;u ||Thesubsetof ~ ~ V 0 associatedwith u ~ ~ V 0 ; ||Thesubsetof ~ ~ V 0 associatedwith xii

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~ ~ V 0 ;T ; ~ ~ V 0 ;B ; ~ ~ V 0 ;c ; ~ ~ V 0 ; 1 ;etc ||Thesubsetof ~ ~ V 0 ; associatedwithsome specicsubsetof ,like T ; B ; c ; 1 ;etc: ~ ~ V 0 ;u;k ||Nodalvelocityofthe k th nodeof ~ ~ V 0 ;u ;k 2 u ~ ~ V 0 ; ;n ||Nodalvelocityofthe n th nodeof ~ ~ V 0 ; ;n 2 ~ j ||the j th Controlbasis, j =1 ; 2 ; ;N c ~ ~ j;u ||Thesubsetof ~ ~ j associatedwith u only, j =1 ; 2 ; ;N c ~ ~ j; ||Thesubsetof ~ ~ j associatedwith j =1 ; 2 ; ;N c ~ ~ j;T ; ~ ~ j;B ; ~ ~ j;c ; ~ ~ j; 1 ;etc ||Thesubsetof ~ ~ j associatedwithsome specicindicessubsetof like T ; B ; c ; 1 ;etc: ~ ~ j;u;k ||Nodalvelocityofthe k th nodeof ~ ~ j ;j =1 ; 2 ; ;N c ;k 2 u ~ ~ j; ;n ||Nodalvelocityofthe n th nodeof ~ ~ j ;j =1 ; 2 ; ;N c ;n 2 ~ l ||the l th ROMbasis, l =1 ; 2 ; ;N s ~ l;k ||Nodalvelocityofthe k th nodeof ~ l ;l =1 ; 2 ; ;N s ;k 2 u Intheabovelist, N c isthenumberofcontrolbasesand N s isthenumberof ROMbases.

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy REDUCEDORDERMODELSANDTHEAPPROXIMATION OFSTOKESFLOWCONTROLPROBLEMS By YunfeiFeng December2003 Chair:AndrewJ.KurdilaMajorDepartment:MechanicalandAerospaceEngineering Numericalsimulationsofruidrowswhicharegovernedbythe Navier-Stokes equationsaretoocostlyforoptimizationandcontrol.Mode lsofunsteadyrows arenotamenabletooptimizationduetotheirhighdimension alityandnonlinearity. Thus,areduced-ordermodel(ROM)isproposedinthisthesis toprovidesuitable strategyforactivecontrolofruiddynamicalsystems.Stok esrowisstudiedindetail. Thisrowregimeisstudiedbecauseitgeneratesalinearmode l.Basedonthederived ROM,optimalcontrolstrategiesforthetrackingproblemar edevelopedandnumerical simulationsareprovided.Moreover,gain-schedulingcont rollersofanassociatedLPV systemareinvestigated.ALPV(linearparameter-varying) systemisaparameterdependentsystem.Itischaracterizedasalinearsystemtha tdependsontime-varying realparameters.Inthisthesis,theparameterisselectedt obetheReynoldsnumber. InthespecialcaseinwhichtheLPVplantisalinearfraction altransformation(LFT) dependentontheparameter,controllersynthesisisfullyc haracterizedasaconvex problemandissolvedeciently.Thisthesisinvestigatest hedevelopmentofaROM ofStokesrowforcontrolsynthesis.Numericalsimulations areprovided. xiv

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CHAPTER1 INTRODUCTIONANDLITERATUREREVIEW 1.1 Introduction and Motivation Flowcontrolinruiddynamicshaslongbeenasubjectofinter esttoengineersand scientists.Inpastyears,increasingattentionhasbeende votedtothedevelopment oftechniquescapableofenhancingourabilitytocontrolun steadyrowinawide varietyofenvironmentssuchasengineinletsandnozzles,c ombustors,automobiles, aircraft,andmarinevehicles.Controllingtherowinthese problemscanleadto greatlyimprovedeciencyandperformance.Duringthepast centurytremendous advanceshavebeenreportedforrowcontrolproblems. 1.1.1 Introduction to Flow Control ThescienceofrowcontroloriginatedwithPrandtlin1904,o neyearafterthe rstpoweredright.Prandtlintroducedtheconceptofthebo undarylayer[73]and proposedameanstocontrolitsattachmenttoasolidsurface .Sincethen,variousrow controltheoriesandapplicationsinruidmechanics,e.g.l aminarrowcontrol(LFC), separationcontrol,optimalcontrol,etc.,havebeenstudi ed.Reviewsofexperimental andnumericalworkhavebeenwrittenonthesubjectby,forex ample,Moinand Bewley(1994)[68],Joslin,ErlebacherandHussaini(1996) [60],Gad-el-Hak(1996) [38],LumleyandBlossey(1998)[67],GreenblattandWygnan ski(2000)[44],and Bewley(2001)[11].Articlesmainlyconcerningthemathema ticalaspectsofthe optimizationmethodsusedforrowcontrolcanbefoundinthe bookseditedbye.g. Gunzburger(1995)[46]andSritharan(1998)[80]. Flowcontrolcantakemanydierentformsandsomekindofcla ssicationof dierentstrategiesisuseful.Passivecontrolisusedwhen therowisaectedwithout 1

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2 requiringaninputofenergy.Forinstance,laminarrowcanb eobtainedpassivelyover theforwardpartofairplaneliftingsurfaces(wingsandtai ls)thathaveleading-edge sweepanglesoflessthanabout18degrees.Thisisachievedb ydesigningthesurface cross-sectionalcontoursothatthelocalpressureinitial lydecreasesoverthesurfacein thedirectionfromtheleadingedgetowardsthetrailingedg e[15].However,thisform ofcontrolisnotcapableofadjustingtoinstantaneousrowc onditions.Asopposed topassivecontrol,activecontrolrequiresanenergyinput whichhastraditionally beenassociatedwiththeinjectionorsuctionofruid,orthe motionofasurface adjacenttothestream.Anothermechanismofrowcontroluse soscillatoryaddition ofmomentumasopposedtotraditionalsteadyadditionofmom entum|oscillatory blowinginsteadofsteadyblowingandsuction.Activecontr olperformedaccordingto apredeterminedschemeinaopenloop,frommeasureddataina nopen,feedforward loop,orfromaclosed,feedbackloop.Anexampleofanactive ,openloop,approach torowcontrolisblowingahighspeedjetoveraderectedrapt oenabletheairplane tolandatalowerspeed. Flowcontrolproblemsofteninvolvepassiveoractivedevic estoeectabenecial changeinwall-boundedorfree-shearrows.Exampleofanext ernalwall-boundedrow canbetherowthatdevelopsontheexteriorsurfaceofanairc raftorasubmarine.In contrasttofree-shearrows,wall-boundedrowsaremoreimp ortant.Carefulstudy andmanipulationoftheserowscouldleadustopreventsepar ation,increaseliftforce, reducepressuredragandnoise,etc.Forexample,theskin-f rictiondragandrowinducednoisearereducedbymaintainingasmuchofaboundar ylayerinthelaminar stateaspossible.Thistypeofrowcontroliscalledlaminar rowcontrol(LFC).A reviewofLFCcanbefoundinJoslin[59].Dragreductioneec t,ofsteadylateral walloscillationsonturbulentboundarylayercharacteris ticswasexaminedbyTrujillo andotherauthors[82]byintroducinghotlmmeasurementso fmeanvelocityinthe viscoussublayertodeterminethedragreduction.Theexper imentsshowthatat

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3 least25%dragreductioncanbeachievedforaturbulentboun daryrowusingsteady oscillationofthesurface.OtherLFCstudiescanbefoundin [7,27,60,61,63,72]. Anothermostoftenstudiedrowcontrolproblemisrowsepara tioncontrol.Separationisgenerallyaccompaniedbylossesofsomekind,suc hasthelossoflift.By postponingseparation,dragcanbereduced,dynamicstalli sdelayed,liftisenhanced andpressurerecoveryisimproved.Fromapracticalstandpo int,separationcontrol canbeachievedbysteadyinjectionofruid[16,78]to,orsuc tionofaruid[9,47,73] from,aboundarylayer.Alternatively,themotionofthewal ltangenttothedirection ofstreamcanbemodied.Furthermore,Wygnanski(1997)[84 ]demonstratedby experimentsthatoscillatoryexcitationcanbeusedtodela yrowseparation.GreenblattandWygnanski(2000)[44]reviewthedevelopmentofco ntrolofrowseparation fromsolidsurfaceswithemphasisonperiodicalexcitation method. 1.1.2 Motivation Despitesignicantprogressoverthepastdecade,thecontr olofruidrowremains oneofthemostchallenging,unsolvedproblemsintheeld.C ontrolproblemsinvolvingpartialdierentialequationsareformidableproblems tosolveinrealtime.The mostsignicantdicultiesareperhapsthenonlinearityof theincompressibleNavierStokesequationsthatrepresentthedynamicsoftherow,and thehighdimensionality oftypicalapproximationoftheseequations.Forexample,t hedimensionofatypical ruiddynamicssystemcanbeintheorderof O (10 6 ) O (10 7 )degreesoffreedom. Ifoneweretosolvesuchproblemsusingstandardniteeleme ntmethod(FEM)or nitedierencemethod(FDM)[33],theresultingsystemisp rohibitivelyhuge.It isclearthatsignicantlydierentstrategiesarerequire dforrowcontrolproblems. LowdimensionalapproximationofNavier-Stokesequations isjustsuchanapproach thattriestodescribetheruidrowbyusinglowestpossibleo rdermodelthatcapturesthemajorcharacteristics.Thisstrategyhasrourish edduringthelastseveral

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4 years.Variousreduced-ordermodelshavebeenderivedanda ppliedtotherowcontrolproblem.Thisthesiswillfocusonreduced-ordermodel ingoftheroweldand theincorporationofthisapproximationinoptimalandrobu stcontrol. 1.2 Objectives Asstatedabove,thegoalsofthisthesisare(1)derivationo freduced-order modelsoftheincompressibleviscousrowand(2)theirappli cationinoptimaland robustcontrolmethods.Thereduced-ordermodel(ROM)inth isthesisisderived forthegeneralcaseandcanbeapplied,inprinciple,toanyg eometry.Aparticular case,drivencavityrowasshowninFigure1.1,willbeusedth roughoutthisthesisas aprototypicalexampletodemonstratethemethodology. Figure1.1:Drivencavityrow Thesegoalsareachievedinseveralsteps: 1.Reduced-ordermodeling.Bothstaticandunsteadyrowsar einvestigatedto demonstratethefeasibilityofobtainingaccuratereduced -ordermodels. 2.Studyofopen-loopoptimalcontrol.Asimpletrackingpro blemisintroduced andsolvedbyadirectdiscretiziationmethod,theclassica lRiccatisolution,and recedinghorizoncontrolmethod.

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5 3.Studyofclosed-looprobustcontrol.Gainschedulingcon trollersynthesistechniqueisdiscussedforthelinearparameter-varying(LPV)s ystems. (a)Pointcontrollersynthesizingforthereduced-ordermo delbythe H 1 controlapproach. (b)Fullordersimulationisusedtoassestheperformanceof thereduced-order modelinrobustcontrol. (c)GainschedulingfortheLPVsystemisconstructedandnum ericallytested. 1.3 Reduced Order Models of Incompressible Viscous Fluid Flows Systemspossessingalarge(potentiallyinnite)numberof degreesoffreedom oftenexhibitsimpledynamicalbehaviorinthesensethatth elong-termmotionis characterizedbyasmallphasespacedimension.Thismotiva teslow-dimensionalrepresentationsofsuchlargedynamicalsystems(typicallyof theorderof10 6 ODEsfor atwo-dimensionalproblem).Developmentoflow-dimension aldescriptionsofthese systemsincomputationalmethodsforcontrolofruidshasbe enthetopicofrecent studiesbymanyresearchers.Thebasicprincipleistoappro ximatethenonlineardynamicsoftheequationsofruidrowbyreducedordermodels(R OM),andthenderive adequatecontrolstrategiesfromthereducedordersystem. Thebriefdiscussionbelow willdescribeageneralapproachtosetupareducedordermod elforincompressible viscousruidrow. 1.3.1 Governing Equations 1.3.1.1 Navier-Stokes Flow Two-dimensionalunsteadyincompressibleviscousruidrow isgovernedbythe Navier-Stokesequaions.Innon-dimensionalformtheseequ ationsare @ ~ V @t 1 Re ~ V + ~ V r ~ V + r p = ~ f (1.1)

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6 Massconsevationisexpressedbythecontinuityequation r ~ V =0(1.2) where ~ V ( x;y;t )isthevelocityeldofruidrowand p isthepressure.TheReynolds numberisdenedtobe Re = UL where thedensity, theviscosity, L thelengthscale, U thevelocityscale.For instance,thecenterlineReynoldsnumber Re cl = U c h iscommonlyusedforlaminar channelrowwith U c thecenterlinevelocityand h thehalfheightofthechannel.The Reynoldsnumber Re = V T 0 L isusedforthedrivencavityrowmodelinthisthesis. Itisbasedonthetopslipvelocity V T 0 andwidthofthecavity L 1.3.1.2 Stokes Flow Inthisthesis,thetwo-dimensionalunsteadyStokesrowisg overnedbythenondimensionalequations @ ~ V @t ~ V + r p = ~ f (1.3) Theseequationsarelinearintherowvelocity ~ V ( x;t ).Thederivationoftheseequationsandthemeaningofparameter willbedescribedindetailinChapter2. 1.3.2 Reduced Order Modeling Reducedordermodelsofeitheroftheabovegoverningequati onsarebasedon theexpansion ~ V ( x;y;t )= ~ V 0 ( x;y;t )+ N c X j =1 j ~ j ( x;y )+ N s X l =1 l ~ ( x;y )(1.4)

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7 where ~ V 0 ( x;y;t )issomeprescribedvelocityeldsatisfyingtheboundaryc onditions, andwhere ~ j ( x;y )and ~ l ( x;y )arebasisfunctionsthatareassumedtobeindependentoftimet.Theunknownsnowbecome f l g N s l =1 and f j g N c j =1 .Ifanaccurate approximationofthevelocityeldcanbeachievedwithvery small N s and N c ,the reducedordermodelisdeemedtobeasuccessfulapproximati onofthefullorder dynamicalsystem. 1.3.3 Construction of Basis Functions in ROM Inthereducedbasisapproach,oneusesasabasisfunctionst hataresomehow physicallymotivatedbytherowunderconsideration.Thisi sincontrasttothenite elementGalerkinschemeswherethebasiselementsareessen tiallyunrelatedtothe physicalpropertiesofthesystemthattheyapproximate.Va riousapproachesbased onROMwithdierentconstructionofthebasisfunctionshav eappearedintheliteratureoverthepastfewdecades.Properorthogonaldecomp ostion(POD)wasrst introducedinruidmechanicsbyLumleytoidentifycoherent spatialstructures[66]. Relatedinvestigationscanbefoundin[5,17,25,52,53,64, 75,76].Essentially,PODextractsanoptimalorthonormalbasistorepresenttheensemb leofdataobtainedfrom numericalcomputationsorexperimentalmeasurement.Them ethodisoptimalin thesensethatitcapturesmoreenergythananyotherorthono rmalset.PODmodes serveasabasisforaGalerkinreductionofthegoverningequ ations.ThePODmodes typicallyarecreatedfromasetof\rowsnapshots"generate dfromexperimentsor numericalsimulations.Byconstruction,PODmodesformaco mpletesetofbasis functions(satisfyingtheboundaryconditionsandtheinco mpressibilityconstraint) foraxeddynamicalsystem.Forexample,theymaybecreated ataxedReynolds number,whichcouldbeaproblemwhenparametervariationsa reencountered.Systemswithdierentparametervaluesmayhavedierentdomin antmodes[62].One optimalPODsetmaynotalwayscapturemaximalenergyforadi erentrowregime. Thiscouldresultinreducedordermodelsrequiringhigherd imensiontoprovidea

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8 goodapproximationofthefulldynamicsystems.Althoughso meenhancedmethods likeweightedPOD(w-POD)andpredenedPOD(p-POD)werepro posedbyErik AdlerChristensen[22],PODmaybeinappropriateasapracti caltoolforproviding low-dimensionalmodelsofparameter-dependentproblems. Incontrast,anotherapproachistoderivethereducedorder methodfromphysicalconsiderations.ItoandRavindran[56,57]haveshownv iaempiricalnumerical evidencethatreducedorderNavier-Stokessimulationscan yieldaccuratereducedordermodels.NumericalresultsweregiveninItoandRavindra n[56]fordrivencavity andchannelrowcontrolproblemsinsteadyviscousincompre ssiblerowsgoverned bytheequations 1 Re ~ V + ~ V r ~ V + r p = ~ fin n(1.5) r ~ V =0 in n Themethodologyisessentiallyrestrictedtosomespecicb oundaryconditions.A seriesofbasiselements f u i g Mi =1 wereobtainedfromsolutionsofthesteadyNavierStokesequations(1.5)withdierentinhomogeneousbounda ryvalues.Subsequently, linearcombinationsofthe f u i g Mi =1 wereconstructedtogettestfunctions n ~ i o N s i =1 whichsatisfyhomogeneousboundaryconditions.Otherline arcombinationsofthe f u i g Mi =1 areusedtogenerate n ~ i o N c i =1 ,whichsatisfyhomogeneousboundaryconditionsexceptforcontrolboundary c .Thepaperdemonstratedrstthatthereduced ordermethodabovecanachieveveryniceapproximationofth efullordersolution. Thentwooptimalcontrolproblemsweresolvedsuccessfully basedonthereduced ordermodels.Thecrucialpointinthisapproachisthechoic eofthetestfunctions n ~ i o N s i =1 .Theymustbechosensuchthattheexpansioninequation(1.4 )describes thenonlineardynamicsoftherowoveralargerangeofrowreg imes.

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9 1.4 Control Strategies in this Thesis 1.4.1 Optimal Flow Control Optimalcontrolmethodologieshavebeenrecentlyappliedt oavarietyofrow controlproblemsinvolvingreductionofdragoreliminatio nofthetransitionfrom laminartoturbulentrow.Sophisticatedoptimalcontrolst rategiesinengineering applicationscoupledwithalargenumberofpartialdieren tialequationsarenow feasibleduetothedevelopmentofcomputationalruiddynam ics(CFD)algorithms whichhavereachedasucientlyhighlevelofmaturity,gene rality,andeciency. 1.4.1.1 Introduction Optimalcontroltheoryismathematicallyrigorousandisam enabletothederivationofthemathematicalformulationofabstractrowcontro lproblems.Overthepast severalyears,carefulandrigorousstudiesoftheanalytic alaspectsofcontrolforthe Navier-Stokesequationshaveappearedintheliterature.S ignicantadvanceshave beenmadeintheareaofcontrollability.Imanuvilov[54]st udiedthelocalexactcontrollabilityproblemfortheNavier-Stokesequationsthat describeanincompressible ruidrowinaboundeddomainwithcontroldistributedinanar bitraryxedsubdomain.Itisshownthatthereexistsalocallydistributedcon trolsuchthatatagiven momentoftimethesolutionoftheNavier-Stokesequationsc oincideswithaspecic stationarypointprovidedtheinitialconditionissucien tlyclosetoit[18,23,35,37]. Thecarefulmathematicalworkontheproofoftheexistenceo foptimalcontrolfor somenonconvexproblemswascarriedoutbyFattoriniandSri tharan(1992)[29].In asubsequentpaper[30],thePontryaginmaximumprinciplew asusedtoestablishthe necessaryconditionsfortheoptimalcontrolproblems.Des aiandIto[26]formulated twocontrolproblemsthatconsiderthedrivencavityandrow throughachannelwith suddenexpansion.Theyestablishedtheexistenceandrstorderoptimalityconditionfortheoptimalcontrol.Generalstudiesofvariousopt imalcontrolproblems, includingboundarycontrol,distributedcontrol,2Dprobl ems,3Dproblems,etc.,can

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10 befoundinCuvelier[24],Fattorini[28],FattoriniandSri tharan[32],Gunzburger andLee[45],HouandYan[50],HouandRavindran[49]. Dierentoptimalcontrolstrategieshavebeendevelopedov erthepastseveral years.AbergelandTemam(1990)[1]developedoptimalcontr oltheoryforsuppressingturbulenceinanumericallysimulated,two-dimen sionalNavier-Stokesrow. However,thecontrolrequiresmeasurementandfeedbackoff ullrow-eldinformation. Choiandotherauthors(1993)[21]developedamorepractica l,suboptimalcontrol strategyseekinganoptimalconditionoverashorttimeperi od.Oneadvantageof thismethodisthatonly\wall-measurements"areusedforfe edback.Thesuboptimal controlprocedurewassuccessfullyappliedtotheone-dime nsionalstochasticBurgers equations.BewleyandMoin(1994)[10]appliedthesuboptim alcontrolprocedure rsttoanumericallysimulatedturbulentchannelrowwith1 7%dragreductionreported.ItoandKang(1994)[55]describeanapproachtoobta insuboptimalsolutions totheHamilton-Jacobi-Bellmanformoftheclosedloopsolu tionforoptimalcontrol problemsgovernedbytheBurgersandtheNavier-Stokesequa tions.Laterapplicationsofsuboptimalcontroltheoryarereportedin[6,20,48 ,65]. 1.4.1.2 Framework of Optimal Control Problems Anoptimalcontrolproblemisdenedinsuchawaythatitmini mizesanobjectivefunctionwhichissubjecttotheevolutionequationsan dboundaryconditions. Forexample,Cuvelier[24]studiesadistributedoptimalco ntrolproblem,wherethe admissiblecontrolisselectedsuchthatthetemperaturedi stribution Te ( x;T ; v )at timet=Tisascloseaspossibletoadesireddistribution z d ( x ) ;x 2 n.Theobjective functionaltobeminimizedis J ( v )= 1 2 Z n j Te ( x;T ; v ) z d ( x ) j 2 dx + Z T 0 Z 1 j v ( x;t ) j 2 d dt (1.6) where 1 istheboundarywherecontrolisapplied.

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11 Thersttermofthefunctional J ( v )isintroducedbythemotivationtominimize thedistancebetweenthecontrolledroweldandthedesired roweld.Dierentfunctionals,e.g.disturbancerejection,vorticityminimizat ion,canbedenedindiverse waysfordierentproblemsdependingontheapplicationsin whichtherowsituation occurs.Somepopularapplicationsaredragreduction[36,6 5],andtransitiondelay, forexample. Thesecondterminthefunctional J ( v )isregardedaspenaltytoachievethe desiredtemperaturedistributionandisintroducedtoredu cetheenergyinput.This termisfrequentlyconsideredinstudiesofdierentoptima lproblemswithactive control. Foroptimalcontrolproblemsthatarisewithquadraticcost functionalsandlinearevolutionequations,thefeedbacksolutionisoftengiv enintermsofthematrix solutionoftheRiccatiequation.Thecontinuousalgebraic Riccatiequationtakesthe form: XRX + XA + A X + Q =0 where Q and R areHermitianmatrices,and A isthecomplexconjugateofthe transposeof A .Inparticular,continuousalgebraicRiccatiequationsar iseinthe problemndingsolutionswhichareoptimalinthesenseofmi nimizingsomeobjective functionalsthatarequadraticin x ( t )and u ( t ),andsubjecttothelinearevolution equations x ( t )= Ax ( t )+ Bu ( t ).ThegrowthofinterestinalgebraicRiccatiequations (ARE)hasbeenexplosiveoverthepastfewdecades.Primaril y,thishasbeendriven bytheimportantroleplayedbytheseequationsinoptimall terdesignandcontrol theory.Aprodigiousnumberofresearchpublicationshavea ppearedwhichhave steadilyincreasedourunderstandingoftheseequationsan dtheirsolutions.

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12 1.4.1.3 Nonconvex Control Oneexampleofamoregeneralformofpenaltytermis Z T 0 r ( u ( x;t )) dt where r ( )isafunctionofcontrol u ( x;t ).Mostoftheliteraturequotedearlierdeal withthespecialcasewith r ( )convexand U ad isaconvexset.Sincethefunction r ( )isdeterminedbyengineeringrequirements(andlimitatio ns)itisdesirableto developatheorywithouttheconvexityassumption.Fattori ni(1995)[31]studiesa classofrowcontrolproblemswheretheruidiscontrolledby adistributedforcing ataportionoftheboundaryandthecostfunctionalisnoncon vexwithrespectto thecontrolvariable.Thischoiceismotivatedbynonconvex rowcontrolproblems thatariseinpracticewherenonconvexfunctionalssuchasa veragelifttodragratio bydistributedboundarycontrol.Inthispaper,Fattorinif ormulatedandproved theexistenceofoptimalchatteringcontrolsfortherelaxe doptimalcontrolsystem. ThissystemisobtainedutilizingtheYoungmeasuresdened oninnitedimensional controlsets.Otherpracticalnonconvexproblemsinvolvin gsuppressionofKarman vortexshedding,obstaclesmovingincurvedpaths,etc.can befoundin[80,81]. 1.4.2 Model Predictive Control (Receding Horizon Control) Modelpredictivecontrol(MPC)referstoaclassofalgorith msthatcomputea sequenceofvariableadjustmentsinordertooptimizethefu turebehaviorofaplant. Thetechniqueisusedtoaddressthepracticalissuesassoci atedwiththecontrolof large,multivariableprocesseswherethereareconstraint sonthesystems.Originally developedtomeetthespecializedcontrolneedsofpowerpla ntsandpetroleumreneries,MPCmethodologiescannowbefoundinawidevariety ofareasincluding applicationsinthechemical,foodprocessing,automotive ,aerospace,metallurgy,and pulpandpaperindustries.

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13 Thefundamentalconceptsconcerningmodelpredictivecont rolcanbetraced backto1963andtheworkofPropoi[74]wherethemovinghoriz onapproachwas rstproposed.Yet,itwasnotuntilthelate1970sandthecon tributionofRichalet andcoauthors[77],thatthemethodwaswidelyaccepted.How ever,theapplicationofmodelpredictivecontrolisnotlimitedtolinearsys tems.Therehasbeena rapidlyincreasinginterestinusingmodelpredictivecont rolschemesofbothlinear andnonlinearsystems[58,69]. 1.4.3 Gain Scheduling and Linear Parameter-varying Systems Oneofthemostpopularmethodsforapplyinglineartime-inv ariant(LTI)control theorytotime-varyingand/ornonlinearsystemsisthemeth odofgainscheduling. Theclassicalapproachofgainschedulinginvolvesthecrea tionofaparameterized plant,developinglinearizationatmultipleequilibriump oints,designinganLTIcontrollaw(pointdesign)tosatisfylocalperformanceobject ivesforeachdesignpoint, andtheninterpolationofthecontrollergains(scheduling ).Thisapproachoften workswellinpractice.However,itdoesnottaketheparamet ervariationsintoaccountinthecontrolsynthesisandthuscannotprovideperfo rmanceandrobustness guarantees.Arobustapproach,bywhichasinglecontroller isdesignedsuchthatthe closed-loopsystemisrobusttoanyvalueoftheparameterwi thinsomeanticipated setofvalues,givesguaranteesofbothstabilityandperfor mance.Itmaybeoverly conservativetondsuchacontrollerwhenthevaluesofthep arametercanhave largevariations.Asasystematicmeansofsynthesizinggai n-scheduledcontrollers, linearparameter-varying(LPV)systemshaverecentlyrece ivedconsiderablyattention.LPVsystemsarecharacterizedaslinearsystemsthatd ependontime-varying realparameters.Theseparametersareassumedtobeexogeno ussignalsthatarenot knowninadvancebutareconstrainedaprioritolieinsomekn own,boundedset.It isassumedthattheycanbemeasuredinrealtime.Themeasure mentoftheseparametersprovidesreal-timeinformationonthevariationsof theplant'scharacteristics.

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14 Hence,itisdesirabletodesigncontrollersthatareschedu ledbasedonthisinformation.Thegeneralformofarepresentativesystemcanbewrit teninstate-spaceform withstatevectorx,exogenousinputsd,controlinputsu,re gulatedoutputse,and measurementsy 266664 x e y 377775 = 266664 A () B 1 () B 2 () C 1 () D 11 () D 12 () C 2 () D 21 () D 22 () 377775 266664 x d u 377775 (1.7) whereisatimevaryingparametervector.TheLPVsystemhas astructure, whichissimilartoalineartime-invariantstatespacesyst em,andcontroldesign methodswithsomesimilaritytolinearstatespacemethodsc anindeedbeused. Sincethestatematricesarewrittendirectlyasfunctionso ftheparametervector, suchasystemiscalledaparameter-dependentmodel.LPVsys temsaredierent fromtheirstandardlineartime-varyingcounterpartdueto thecausaldependence oftheircontrollergainsonthevariationsoftheplantdyna mics.Itispossibleto treatgainscheduledcontrollersasasingleentity,withth egain-schedulingachieved entirelybytheparameter-dependentcontroller. 1.4.3.1 Literature Review Recently,severalsynthesisalgorithmshavebeendesigned forthesystematic determinationofgain-schedulingcontrollersforLPVsyst ems. Aparameter-dependentcontrollersynthesistechniquebas edonsmallgaintheoremwasproposedbyPackard(1994)[70]fordiscrete-timesy stemsandbyApkarian andGahinet(1995)[2]forcontinuous-timesystems.Inboth papers,LPVplantsare assumedtobelinearfractionaltransformation(LFT)depen dentontheparameters. Theexistenceofsuchagain-scheduledcontrollerisfullyc haracterizedintermsoflinearmatrixinequalities(LMIs).Theunderlyingsynthesisp roblemisthereforeaconvexproblemandcanbesolvedusingstandardconvexoptimiza tionalgorithms[14,40].

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15 TheresultingLPVcontrolleristime-varyingandsmoothlys cheduledbythemeasurementsoftheparameter.Thisisasignicantbenetovertrad itionalgainscheduling controllersbecauseitguaranteesthestabilityofthecorr espondingclosed-loopsystemsforanyvalueofparametersandalsoforanytime-varytr ajectoryoftheparameters.Itwasalsodemonstratedthattheoriginalgain-sch edulingproblemcanbe re-formulatedasoneofrobustperformancewithspecialpla nt/uncertaintystructure andthecontrollercanbesynthesizedusinglinear H 1 controlapproaches[2]. Analternativeapproachtothegain-schedulingproblemwas recentlyproposed byWuandcoauthorsin[83].TheLPVsystemisallowedtohaveg eneralparameterdependence.Insteadofusingthescaledsmall-gaintheorem ,Wuet.al.consider aparameter-dependentLyapunovfunction.Thisapproachwa srstproposedfor theanalysisofgain-scheduledcontrolbyFromionandother authors[34].Themain motivationhereistotakeintoaccountaprioriboundsonthe rateofvariationofthe parameters.Unfortunately,contrarytothepreviousappro achforthegain-scheduling problem,theresultingconditionscanbecheckedonlyappro ximately,withahigh computationalburden.Moreprecisely,Wu'sconditionsreq uireagriddlingonthe setoftheparametersinconjunctionwiththeresolutionofa largenumberofLMI problems.Thisapproachhasoneofthedrawbacksofthetradi tionalgain-scheduling engineeringpractice:thereisnoguaranteeonthestabilit yoftheclosed-loopsystem.

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CHAPTER2 REDUCED-ORDERMODELINGOFSTOKESFLOW Undernormalconditions,mostcommonruidssuchaswater,ai r,andglycerin areNewtonianruidswhicharedenedasthoseforwhichshear stressisdirectly proportionaltorateofdeformation.Themainsubdivisiono fNewtonianruidrowis betweeninviscidandviscousrows.Viscousrowisofparamou ntimportanceinthe studyofruiddynamics,andcanbefurtherclassiedaslamin arroworturbulentrow accordingtotherowstructure.Laminarrowmovesinlaminae ,orlayers,smoothly whileturbulentrowmovesinarandom,three-dimensionalma nner.Reynoldsnumber,whichisdenedastheratioofinertialforcestoviscou sforceswithintheruid, isthecriterionthatdeterminesinwhichregimeistherowpr esent.Itisexpressedas Re = VD where V and D areruidcharacteristicvelocityanddistance, and aredensityand viscosityoftheruid.Forexample,forruidrowinginapipe, V couldbetheaverage ruidvelocity,and D wouldbethepipediameter.Typically,viscousstresseswit hina ruidtendtostabilizeandorganizetherow,whereasexcessi veruidinertiatendsto disruptorganizedrow,leadingtochaoticturbulentbehavi or.Thus,rowwithhigher Reynoldsnumberwillbemorelikelyturbulent.Forinstance ,weusuallysaythata piperowisalaminaroneforReynoldsnumberupto2000,andat urbulentoneif theReynoldsnumberisbeyond4000.Whilebetweenthesetwon umbers,transitional rowmaybedened.Thisthesisfocusesonstudyofnonturbule ntrows. ThemotionofincompressiblenonturbulentNewtonianrowis governedbythe Navier-Stokesequations.Theseequationsarenonlinear,p artialdierentialequations 16

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17 havinginnitedegreeoffreedom.Stokesrowisaparticular caseofNavier-Stokes row,obtainedinthelimitofvanishinglysmallReynoldsnum ber.Stokesrowis governedbyalineardynamicalsystem,whereastheNavier-S tokesequationsarenot. Theviscousforcesinthistypeofrowaremuchlargerthanthe inertialforces,and theconvectivetermintheNavier-Stokesequationscanbene glected.Stokesrow isalsocalledcreepingrow.Practicalcasesofcreepingrow aretheruidrowsin whichahighviscosityleadstoslowmotion.Thechemicaland materialprocessing industriesfrequentlydealwithveryviscoussubstances.I nothercases,theruidmay beofmodestviscositybutthelengthdimensionissmall. 2.1 General Stokes Flow Problem Inthissection,wewillbeconcernedwiththemathematicalt heoryofStokes row.Detailedanalysiswillbecarriedoutinthissectionto setupamathematical frameworkforreducedordermodelingoftheStokesrowprobl em.Theanalysis isbasedonthenon-dimensionalanalysisofNavier-Stokese quation.Itshouldbe declaredthatthemethodologyproposedinthissectionisan abstracttreatmentwhich isapplicabletothegeneralcaseofStokesproblem.Themean ingwillbesignicantly reinforcedbythefollowingsectionswheretheapproachwil lbeappliedtoadriven cavityproblem. 2.1.1 Stokes Equation and Time Scale Issue Twodimensionalincompressibleviscousrowisgovernedbyt heNavier-Stokes equationgivenby @ ~ V @t + ~ V r ~ V = r p + ~ V (2.1) withacharacteristicdimensionL,characteristicvelocit y V s .Asstatedbefore,Stokes rowisthecasewhentheReynoldsnumber, Re 0.Inthiscase,theconvective

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18 termcanbeneglected,andpressureforceshouldbalancethe viscousstresses P L ~ V L 2 Thus,theappropriatenon-dimensionalpressure P mustsatisfy P = P V s L Also,non-dimensionalvariables x = x L ;y = y L ; ~ V = ~ V V s ;t = t L=V s aredened. ThenthenondimensionalNavier-Stokesequationbecomes Re @ ~ V @t + Re ~ V r ~ V = r p + ~ V (2.2) wheretheReynoldsnumberisdenedasRe= V s L .Theconvectivetermwillbe neglectedaswellastheunsteadytermwhenStokesrowiscons idered,i.e.Re 1. TheaboveequationissimpliedtoStokesequation r p = ~ V (2.3) whichmeansthenetforcesonaruidparticlemustaddtozero. Flowsgovernedby Equation(2.3)aretermedStokesrows.Stokes(1851)wasthe rsttoproposethis simplicationwhenhestudiedtheproblemofStokesrowarou ndasphere. TherowsgovernedbytheEquation(2.3)arequasi-steadysin cetimedoesnot appearexplicitlyinthisequation.Itmeansthatanytime-d ependentmotionof amasslessruidarisingfromunsteadyboundaryconditionsi squasi-static.This conclusionrestsontheassumptionthatthetimescaleinthe substantialderivative termofEquation(2.2)is t = t L=V s .TogettheappropriateformofStokesequationfor dedicatedboundaryconditions,wemustconsideralsoanoth ertimescaleintroduced bythemotionofaboundarycondition. FortheincompressibleNavier-StokesrowgovernedbyEquat ion2.1subjectalso toboundaryconditionwithcharacteristicfrequency f ,thenewpotentialtimescale

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19 mustbe 1 f .If f istoobigcomparedtothecharacteristicfrequency V s L ,theexcitation ontheboundaryvariestoofastanditcannotpenetrateintot heinnereld.Onthe otherhand,itwillbequasi-steadyif f isverysmall,whichisconsistentwiththecase discussedabove.Inthisthesis, f isassumedtobecomparabletothecharacteristic frequency V s L suchthatthecontrolinputcanbeusedtochangethevelocity eld. Thus,wewillconsiderthetimescale = 1 f .Bykeepingothernon-dimensional denitions,theresultingnon-dimensionalNavier-Stokes equationis Re St @ ~ V @t + Re ~ V r ~ V = r p + ~ V (2.4) where St isStrouhalnumberdenedas St = Lf V s When( Re St ) O (1)andtheReynoldsnumber Re = V s L 1,thetermsonthe righthandsideoftheequation(2.4)willtrytobalancethe rsttermonthelefthand side.Thesecondtermwillbeneglected.Thustherowisappro ximatelyStokesrow. Tosimplifytheequation,anewparameter isintroduced,whichisdenedas theinverseoftheproductoftheReynoldsnumberandtheStro uhalnumber,i.e. = 1 = 1 Re St (2.5) Nowre-scalethepressure, ^ P = 1 P .Thenequation(2.4)becomes @ ~ V @t ~ V + r ^ p =0(2.6) Thisisthenon-dimensionalformofthegoverningequationf orStokesrow.For convenience,itiswrittenas @ ~ V @t ~ V + r p =0(2.7)

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20 2.1.2 Problem Formulation TheEquation(2.7)isapartialdierentialequation.Toapp lyamodelreduction technique,avariationalformulationandtheniteelement methodarerstusedto obtainitsapproximation.Thisprocessyieldsanordinaryd ierentialequaionform. 2.1.2.1 Function Spaces and Auxiliary Notations BeforeformulatefortheStokesproblem,somevariationaln otationsshouldbe introduced.Assumeatwo-dimensionalproblemthus N =2.Denotethebounded domainnwithcontinuousboundary @ n.Intheusualway, L 2 (n)isdenedtobe thesetofthosefunctionsthataresquareintegrableinthed omainnwithnorm k f k 2L 2 (n) = Z n f 2 ( x ) dx Let H 1 (n)betheHilbertspaceoffunctionswhoserstweakderivat iveisin L 2 (n). H 1 (n)= f 2 L 2 (n): @f @x i 2 L 2 (n) ;i =1 ; 2 anddenote H 1 0 (n)= f 2 H 1 (n): f j @ n =0 L 20 (n)= f 2 L 2 (n): Z n fdx =0 Theinnerproductdenedin L 2 (n)is ( ~u;~v ) [ L 2 (n)] 2 = 2 X i =1 Z n u i v i dx Introducebinlinearandtrilinearforms a ( ~u;~v ) = 2 X i =1 2 X j =1 Z n @u i @x k @v i @x k dx b ( ~u;~v;~w ) = 2 X i =1 2 X j =1 Z n u l @v k @x l w k dx

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21 thatdenethemapping\ ~u;~v a ( ~u;~v )"on H 1 (n) H 1 (n)and\ ~u;~v;~w; b ( ~u;~v;~w )"on H 1 (n) H 1 (n) H 1 (n). 2.1.2.2 Weak Form of Stokes Equation TheStokesproblemisgovernedbyEquation(2.7)andcontinu ityequation r ~ V =0(2.8) Tosolveit,theproblemisrstreformulatedinaweaksensea nditsvariational formulationisderived.Let( ~ V;p )beaclassicalsolutionoftheStokesproblem. MultiplyingEquation(2.7)witharbitrary ~ 2 H 1 0 (n)andintegratingovern,by Green'stheorem Z n ( ~ V ~ ) dx = Z @ n @ ~ V @ ^ n ~ ds + Z n 2 X i =1 2 X j =1 @V i @x j @ i @x j dx (2.9) Z n ( r p ~ ) dx = Z @ n ( p ~ ^ n ) ds Z n ( p r ~ ) dx (2.10) weobtain Z n ( @ ~ V @t ~ ) dx + Z n 2 X i =1 2 X j =1 @V i @x j @ i @x j dx Z n ( p r ~ ) dx Z @ n @ ~ V @ ^ n ~ ds + Z @ n ( p ~ ^ n ) ds =0 (2.11) Z n r r ~ V =0 (2.12) forall ~ 2 V 0 \ V h and r 2 P h .Since ~ 2 H 1 0 (n),wehave ~ j @ n =0.Theabove equationscanbefurthersimpliedas Z n 2 X i =1 V i i dx + Z n 2 X i =1 2 X j =1 @V i @x j @ i @x j dx Z n ( p r ~ ) dx =0(2.13) Z n r r ~ V =0 (2.14)

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22 TheseequationsaretheweakformofStokesequation(2.7)an dcontinuityequation (2.8). 2.1.2.3 Galerkin Formulation Weshallnowdescribeamethod,calledGalerkin'sMethod,fo robtainingapproximatesolutionstoStokesproblembasedupontheweakformul ation.Thebasicidea istoapproximatethesolution( ~ V;p )andfunction ~ byconvenient,nite-dimensional collectionsoffunctions.Thevariationalequationsareth ensolvedinthisnitedimensionalcontext.Therststepindevelopingthemethod istodiscretizethe physicaldomain.Letsviewourdomainnas\discretized"int oniteelementdomains,orsimplyelements,n e ; 1 e N ele ,where N ele isthenumberofelements inthedomain.Twotypeofsimpleelements|trianglesandqua drilaterals,asshown inFigure2.1|areusedmostfrequentlyforthetwodimension alphysicaldomain. Theverticesofelementsarecallednodalpoints,orsimplyn odes.Itshouldbedeclaredthatnodalpointsmayexistanywhereonthedomainbut usuallyarechosen toappearattheelementvertices.Eachnodeisnumberedanda llnodesareordered e e W = W Figure2.1:Finiteelements whichiscalledglobalorderingofnodalpoints.Denote u asthesetofunknown nodalpoints,andby thesetofboundarynodalpoints.Theapproximationofthe

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23 solution ~ V ( x;y;t )isassumedtohavetheform ~ V = 8><>: V 0 ; 1 V 0 ; 2 9>=>; + X k 2 u 8><>: V k; 1 V k; 2 9>=>; N k ( x;y ) Inmoredetail,itcanbewrittenas ~ V ( x;y;t )= X n 2 8><>: ~ V n; 1 ~ V n; 2 9>=>; N n ( x;y )+ X k 2 u 8><>: V k; 1 ( t ) V k; 2 ( t ) 9>=>; N k ( x;y )(2.15) where n ~ V n; 1 ~ V n; 2 o isaconstantvectorcorrespondingtotheboundaryconditio natnode n ,and n V k; 1 V k; 2 o istheunknownvelocitycorrespondingtotheinnerdomainno de k N n ( x;y )and N k ( x;y )intheequationabovearereferredtoasshape,basis,orint erpolationfunctions.Forrstorderinterpolation,shapefu nctionsareoftenchosento bepiecewisepolynomials.Inthisthesis,wechoosebilinea rquadrilateralelementsas showninFigure2.2. x h x y (-1,-1) (1,-1) (-1,1) (1,1) ) ( 1 1 e e y x ) ( 4 4 e e y x ) ( 3 3 e e y x ) ( 2 2 e e y x 1 2 3 4 1 2 43 Figure2.2:Bilinearquadrilateralelementdomainandloca lnodeordering Theunitdomainissometimescalledtheparentdomain.Point ( ; )inthe parentdomainisrelatedtothepoint( x;y )inthephysicaldomainnbyamapping

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24 oftheform x ( ; )= 4 X a =1 N a ( ; ) x ea (2.16) y ( ; )= 4 X a =1 N a ( ; ) y e a (2.17) Assumethatthebilinearshapefunctionshavetheform N a ( ; )= 1 4 (1+ a )(1+ a )(2.18) Bytheabovemappingrule,thefollowingconditionsissatis ed x ( a ; a )= x ea y ( a ; a )= y e a Astandardnumericalsubroutinecanbebuilttocalculateth erequiredintegralsover theparentdomainbyGuassianquadraturerule.Bythemappin grulesabove,the integralsoverthephysicaldomainintheweakformofthegov erningequationscan betransformedontotheparentdomain,andthestandardsubr outineswillgivethe requirednumericalresults. Todistinguishwithothercommonvelocitynotations,bolds ymbolsareusedin thisthesistoindicatenodalvelocityvalues.Forinstance ,thenodalvelocityvector associatedwithunknownsintheinteriorofthedomainwillb e n ~ V ( t ) o = 8>>>>>>>>>><>>>>>>>>>>: ~ V 1 ( t ) ~ V 2 ( t ) ... ~ V k ( t ) ... 9>>>>>>>>>>=>>>>>>>>>>; ; where ~ V k ( t )= 8><>: V k; 1 ( t ) V k; 2 ( t ) 9>=>; ;k 2 u

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25 whilethenodalvelocityvectorforaprescribedvelocitye ldwillbe n ~ ~ V ( t ) o = 8><>: ~ ~ V ( t ) ~ ~ V u ( t ) 9>=>; where ~ ~ V ( t )isthepartofthenodalvelocityvectorassociatedwithbou ndaries,and ~ ~ V u ( t )isthepartofthenodalvelocityvectorassociatedwiththe interiorofthe domain.Inmoredetail, n ~ ~ V ( t ) o = 8>>>>>>>>>><>>>>>>>>>>: ~ ~ V 1 ( t ) ~ ~ V 2 ( t ) ... ~ ~ V n ( t ) ... 9>>>>>>>>>>=>>>>>>>>>>; ; where ~ ~ V n ( t )= 8><>: ~ V n; 1 ( t ) ~ V n; 2 ( t ) 9>=>; ;n 2 n ~ ~ V u ( t ) o = 8>>>>>>>>>><>>>>>>>>>>: ~ ~ V 1 ( t ) ~ ~ V 2 ( t ) ... ~ ~ V k ( t ) ... 9>>>>>>>>>>=>>>>>>>>>>; ; where ~ ~ V k ( t )= 8><>: ~ V k; 1 ( t ) ~ V k; 2 ( t ) 9>=>; ;k 2 u Thus,Equation(2.15)canbeexpressedas V i ( x;y;t )= X n 2 ~ V n;i N n ( x;y )+ X k 2 u V k;i ( t ) N k ( x;y ) ;i =1 ; 2(2.19) Wechoose ~ ( x;y )= ~e s N m ( x;y ) ;m 2 u ;s =1 ; 2(2.20) ~e 1 = 8><>: e 11 e 12 9>=>; = 8><>: 10 9>=>; ; ~e 2 = 8><>: e 21 e 22 9>=>; = 8><>: 01 9>=>;

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26 Figure2.3:Apressurequadrilateralanditsfourassociate dvelocityquadrilaterals Alsopiecewiseconstantsarechosenasthepressurespacewi threspecttothelarger quadrilateralsofthesubdivision Q h of nasshowninFigure2.3,whichisdivided furtherintofoursmallerrectanglesbyjoiningtheopposit emidsides,thuscreating anothersubdivision Q h= 2 of nintorectangles.Thevelocitydegreesoffreedomare simplyfunctionvaluesattheinteriorverticesoftherecta nglesin Q h= 2 .Thereisone pressuredegreeoffreedomassociatedwitheachrectanglei n Q h .Thisisrequired tosatisfytheBabuska-BrezziConditionrequiredfornite elementmodelsofruid row[].Thuspressureeldcanbeexpandedinto p = N P X r =1 H r ( x;y ) p r (2.21) where H r ( x;y )isthepressureshapefunctionat r th pressureelement(thelarger quadrilateral).Substitutingequation(2.19),(2.20),(2 .21)intoeq(2.13),thefollowing equationisobtained 2 X i =1 Z n ( X n 2 ~ V n;i N n + X k 2 u V k;i N k )( e s;i N m ) d n + 2 X i =1 2 X j =1 Z n @ @x j ( X n 2 ~ V n;i N n + X k 2 u V k;i N k ) @ @x j ( e s;i N m ) d n Z n N P X r =1 H r p r @ @x 1 ( e s; 1 N m )+ @ @x 2 ( e s; 2 N m ) d n=0

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27 Collectingterms,itbecomes 2 X i =1 X k 2 u Z n e s;i N k N m d n V k;i N P X r =1 Z n H r @N m @x s d n p r + 2 X i =1 X k 2 u 2 X j =1 Z n e s;i @N k @x j @N m @x j d n # V k;i = 2 X i =1 X n 2 Z n e s;i N n N m d n ~ V n;i 2 X i =1 X n 2 2 X j =1 Z n e s;i @N n @x j @N m @x j d n # ~ V n;i s =1 ; 2 m 2 u (2.22) Substitutingequation(2.21),(2.23)intoequation(2.14) andsupposingthat r = H r ( x;y ) ;r =1 ; 2 ;:::;N p (2.23) theothergoverningequationis 2 X i =1 X k 2 u Z n H r @N k @x i d n V k;i = 2 X i =1 X n 2 Z n H r @N n @x i d n ~ V n;i r =1 ; 2 ; ;N P (2.24) Equation(2.22)and(2.24)arethenite-dimensionalappro ximationofweakform governingequations.Nowtheinnite-dimensionalPDEprob lemhasbeentransformedintoanite-dimensionalODEproblem,whichisfeasi bletodonumerical manipulationsandtoapplycontrolstrategies. Thegoverningequationsabovecanbewrittenintothematrix form 264 [ M u ][0] [0][0] 375 8><>: ~ V 0 9>=>; + 264 [ A u ][ B u ] [ B u ] T [0] 375 8><>: ~ V ~ P 9>=>; = 264 [ A ] [ B ] T 375 n ~ ~ V o 264 [ M ] [0] 375 ~ ~ V (2.25)

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28 where [ M u ] ( m;s )( i;k ) = Z n e s;i N k N m d n s;i =1 ; 2 m;k 2 u [ M ] ( m;s )( i;n ) = Z n e s;i N n N m d n s;i =1 ; 2 m 2 u n 2 [ A u ] ( m;s )( i;k ) = 2 X j =1 Z n e s;i @N k @x j @N m @x j d n s;i =1 ; 2 m;k 2 u [ A ] ( m;s )( i;n ) = 2 X j =1 Z n e s;i @N n @x j @N m @x j d n s;i =1 ; 2 m 2 u n 2 [ B u ] ( m;s )( r ) = Z n H r @N m @x s d n s =1 ; 2 r =1 ; 2 ; ;N P m 2 u [ B ] ( n;s )( r ) = Z n H r @N n @x s d n s =1 ; 2 r =1 ; 2 ; ;N P n 2 ItshouldbenoticedthatinGalerkinprocedure,theglobalo rderingandordering ofequationsinthematrixsystemmayormaynotcoincidewith eachother.The numberofequationsiscalculatedbysubtractingthenumber ofboundarynodesfrom thenumberoftotalglobalnodes.Inthegoverningequations above,eachelementin thecoecientmatricesinvolvesanintegral.Standardsubr outineswillcalculatethese integralsovertheparentdomaincorrespondingtoeachnit eelement.Subsequently, theassemblyprocedureisusedtoconstructtheglobalcoec ientmatrices.More detaileddiscussionoftheniteelementmethodcanbefound in[51]. 2.1.3 Order Reduction Framework ConsidernowhowtosolvetheStokesprobleminEquation(2.2 5)forlargesystems.Supposethereisatwodimensionalsystemwith10 6 interiordomainnodes, thushavingatotalof2 10 6 unknowns.Thematurityofdevelopmentofcomputationalruiddynamics(CFD)techniquesmakesthishugeprobl emsolvableandyields high-resolutionsolutionsatxedrowregimes.However,it maynotbeappropriate touseCFDdirectlyforcontroldesign.Thoughtherehavebee neortswhichattempttomodifyCFDintoausefultoolforcontroldesign,the resultingmodelshave roughlythesamenumberofstates(unknowns)astheoriginal CFDsimulation.This

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29 modelsizeistoolargeandcomputationallycumbersomeforc ontrolapplications.So reductiontechniqueshavebeenmotivatedtoreducethesela rgemodelsbyseveral ordersofmagnitudewithoutsignicantlychangingthedyna micresponse.Theresult isanaccurate,easytouse,low-ordermodelthattakeslesst imetogeneratethan thosegeneratedbytraditionalmeans. Figure2.4:Geometry Herethemathematicalframeworkofamodelreductiontechni queisdescribed. ConsidertheStokesproblemoverthephysicaldomainnassho wninFigure2.4.The boundaryisgivenby @ n= [ k k [ c Boundarysegments k ;k =1 ; 2 ;::: mightbeseparatedcurveswithdierentconditions.Forexample,foraparticularproblemwemighthave inrowandoutrow boundaries,andalsosomesolidboundaries. c isthecontrolboundarywherecontrol activitiesareapplied.Thepossiblecontrolactivitiesmi ghtbemovingwalls,blowing orsuckingjets,orsyntheticjets.Itisassumedthattheinp utvelocityimpartedby theactuatorcanberepresentedintermof N c functions ~ i ;i =1 :::N c .Eachofthe functions ~g i aredenedontheentiredomainn,butareassumedtoexhibits pecic propertiesontheboundary.Werequirethat ~ i j @ n ( x;y )= 8><>: ~ i;c ( x;y ) for ( x;y ) 2 c 0 for ( x;y ) 2 @ n n c

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30 Additionallyitisrequiredthat Z @ n ~ i ^ ndS =0 for i =1 :::N c .Thislastconditionisrequiredtoguaranteecompatibilit yoftherow eldwiththecontinuityequation.Theactuatorimpartsave locityalong c whose spatialdistributionisdenedintermsofknownfunctions ~ i;c Spatiallyvaryingfunctions ~ l l =1 :::N s arenowintroducedtoconstitutethe reducedbasiswhichisusedtorepresentthe N s statesinthecontrolmodel.There areseveralmethodstogenerate ~ l .Properorthogonaldecomposition(POD)has beenutilizedin[5][17],andbymanyotherauthors,toconst ructareducedbasis determinedviaanoptimalitycriteria.Incontrast,eecti vereducedbasescanbe derivedfromphysicalconsiderations.ItoandRavindranin [56]and[57]haveshown viaempiricalnumericalevidencethatreducedorderNavier -Stokessimulationscan yieldaccuratereducedordermodels.Itwillbediscussedin detaillaterinthischapter howtoobtainreducedbasisbyPOD. Assumethespatiallyvaryingfunctions ~ i ,for i =1 :::N c ,havelikewisebeen derivedfromeitherparticleimagevelocimetryorsimulati on.Thefunctions ~ i comprisetheinruencefunctionsthatdeterminethecontrolsac tingontheruidrow.Itis assumedthatthesefunctionssatisfythefollowingconditi ons,whichareconventional inmanyreducedbasisformulations: 8>>>>>>><>>>>>>>: r ~ l =0for l =1 :::N s r ~ i =0for i =1 :::N c ~ l j @ n =0for l =1 :::N s ~ i j @ n =0for x= 2 c 1 [ c 2 (2.26)

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31 Intermsofthesereducedbasisfunctions,thevelocityeld associtatedwithdomain nisassumedtotaketheform ~ V ( x;y;t )= ~ V 0 ( x;y;t )+ N C X i =1 ~ i ( x;y ) i ( t )+ N S X i =1 ~ l ( x;y ) l ( t )(2.27) Expressingvelocityeldsinnodalvelocityvectors,oneob tains ~ V 0 ( x;y;t )= X n 2 ~ ~ V 0 ; ;n ( t ) N n ( x;y )+ X k 2 u ~ ~ V 0 ;u;k N k ( x;y )(2.28) ~ i ( x;y )= X n 2 ~ ~ i; ;n N n ( x;y )+ X k 2 u ~ ~ i;u;k N k ( x;y ) ;i =1 ; 2 ;:::;N c (2.29) ~ l ( x;y )= X n 2 ~' l; ;n N n ( x;y )+ X k 2 u ~ l;k N k ( x;y ) ;l =1 ; 2 ;:::;N s (2.30) where ~ ~ V 0 ; ( t )and ~ ~ V 0 ;u ( t )arenodalvelocityvectorsassociatedwith and u respectively. ~ ~ i; and ~ ~ i;u arenodalvelocityvectorsassociatedwith and u respectively for i =1 ; 2 ;:::;N c .Thevectors ~' i; and ~ arenodalvelocityvectorsassociatedwith and u respectivelyfor l =1 ; 2 ;:::;N s .Denote ~ ~ V 0 = 8><>: ~ V 0 ; ~ V 0 ;u 9>=>; ~ ~ i = 8><>: ~ ~ i;c ~ ~ i;u 9>=>; ItisobviousfromEquation(2.26)that ~' l; ;n j n 2 = 8><>: 00 9>=>; (2.31)

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32 Thus,withthenodalvelocityvectors,Equation(2.27)canb erewrittenas ~ V ( x;y;t )= X n 2 ~ ~ V 0 ; ;n ( t ) N n ( x;y ) + N c X i =1 X n 2 c ~ ~ i;c;n N n ( x;y ) i ( t )+ X k 2 u ~ ~ V 0 ;u;k ( t ) N k ( x;y ) + N c X i =1 X k 2 u ~ ~ i;u;k N k ( x;y ) i ( t )+ N s X l =1 X k 2 u ~ l;k N k ( x;y ) l ( t ) (2.32) Thevelocity ~ V ( x;y;t )canbeseparatedintotwoparts|oneisassociatedwith u ,theothercorrespondsto .AccordingtoEquations(2.32),thenodalvelocity vectorfor u canbeexpressedas ~ V ( t )= ~ ~ V 0 ;u ( t )+ N c X i =1 ~ ~ i;u i ( t )+ N s X l =1 ~ l l ( t ) ~ ~ V 0 ;u ( t )+ h ~ u i ~ ( t )+[] ~ ( t )(2.33) where h ~ u i isthecollectionofcontrolfunctions ~ ~ i;u ,for i =1 ; 2 ;:::;N c ,[]isthe collectionofreducedbasis ~ l ,for l =1 ; 2 ;:::;N s .Denote h ~ u i = h ~ ~ 1 ;u ; ~ ~ 2 ;u ;:::; ~ ~ N c ;u i h ~ c i = h ~ ~ 1 ;c ; ~ ~ 2 ;c ;:::; ~ ~ N c ;c i h ~ i = 264 h ~ c i h ~ u i 375 [] = h ~ 1 ; ~ 2 ;:::; ~ N s i Substitutingeq(2.33)intotheupperpartofeq(2.25),oneo btains [ M u ] ~ ~ V 0 ;u ( t )+ h ~ u i ~ ( t )+[] ~ ( t ) +[ A u ] n ~ ~ V 0 ;u ( t )+ h ~ u i ~ ( t )+[] ~ ( t ) o +[ B u ] ~ P ( t )(2.34) = [ M ] ~ ~ V 0 ; ( t ) [ M c ] h ~ c i ~ ( t ) [ A ] ~ ~ V 0 ; h 0 ( t ) [ A c ] h ~ c i ~ ( t )

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33 Premultipliedby[] T ,Equation(2.34)isrewritteninthematrixform [] T [ M u ][][] T [ M u ] h ~ u i +[ M c ] h ~ c i 8><>: ~ ( t ) ~ ( t ) 9>=>; + [] T [ A u ][][] T [ A u ] h ~ u i +[ A c ] h ~ c i 8><>: ~ ( t ) ~ ( t ) 9>=>; = [] T [ M u ] ~ ~ V 0 ;u ( t )+[ M ] ~ ~ V 0 ; ( t ) [] T [ A u ] ~ ~ V 0 ;u ( t )+[ A ] ~ ~ V 0 ; ( t ) [] T [ B u ] ~ P ( t ) (2.35) Forthereducedbasis ~ l ( x;y ),nomatterifitwasderivedfromPODorthemethod introducedbyItoandRavindranin[56]and[57],itsassocia tednodalvelocityvector canalwaysbeinterpretedas 8><>: ~' l; ~ l 9>=>; = X m l;m ~ u l;m (2.36) whre ~ u l;m arethecoecientsoffullorderStokessolutions.Substitu tingEquation (2.23)intoweakformofcontinuityequation(2.14),oneobt ains 2 X s =1 Z n H r @ @x s X n 2 X m l;m ~ u ( l;m ) ;n N n + X k 2 u X m l;m ~ u ( l;m ) ;k N k d n=0(2.37) r =1 ; 2 ; ;N P Itcanbewrittenas X m l;m 2 X s =1 X k 2 u Z n H r @N k @x s d n ~ u ( l;m ) ;k + X m i;m 2 X s =1 X n 2 Z n H r @N n @x s d n ~ u ( l;m ) ;n =0 r =1 ; 2 ; ;N P (2.38)

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34 Recallingthat [ B u ] ( k;s )( r ) = Z n H r @N k @x s d n s =1 ; 2 r =1 ; 2 ; ;N P k 2 u [ B ] ( n;s )( r ) = Z n H r @N n @x s d n s =1 ; 2 r =1 ; 2 ; ;N P n 2 SubstitutingtheseexpressionsintoEquation(2.38) X m l;m [ B u ] T ~ u l;m j u + X m l;m [ B ] T ~ u l;m j =0(2.39) AccordingtoEquation(2.36), [ B u ] T ~ l +[ B ] T ~ ~ l =0 ByEquation(2.31) [ B u ] T ~ l =0 Thus,thisexpressioncanbewritteninmatrixform [ B u ] T []=0 Consequently,onecantakethetransposetoobtain [] T [ B u ]=0(2.40) So,nally,itispossibletowrite [] T [ B u ] ~ P =0(2.41)

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35 ThisequationshowsthatthepressuretermintheEquation(2 .35)vanishes.Thus, thereduced-ordermodel(ROM)ofgeneralStokesproblemis [] T [ M u ][][] T [ M u ] h ~ u i +[ M c ] h ~ c i 8><>: ~ ( t ) ~ ( t ) 9>=>; + [] T [ A u ][][] T [ A u ] h ~ u i +[ A c ] h ~ c i 8><>: ~ ( t ) ~ ( t ) 9>=>; = [] T [ M u ] ~ ~ V 0 ;u ( t )+[ M ] ~ ~ V 0 ; ( t ) [] T [ A u ] ~ ~ V 0 ;u ( t )+[ A ] ~ ~ V 0 ; ( t ) (2.42) Thegoverningequationfor theROMoftheStokesFlowControlProblem is 264 [] T [ M u ][][] T [ M u ] h ~ u i +[ M c ] h ~ c i ; I 375 8><>: ~ ( t ) ~ ( t ) 9>=>; + 264 [] T [ A u ][][] T [ A u ] h ~ u i +[ A c ] h ~ c i ;; 375 8><>: ~ ( t ) ~ ( t ) 9>=>; (2.43) = 8><>: [] T [ M u ] ~ ~ V 0 ;u ( t )+[ M ] ~ ~ V 0 ; ( t ) [] T [ A u ] ~ ~ V 0 ;u ( t )+[ A ] ~ ~ V 0 ; ( t ) ; 9>=>; + 8><>: ;I 9>=>; ~ ( t )

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36 where I istheidentitymatrix.Actually,byusinginnerproductand bilinearform denedatthebeginningofthissection,thisequationcanbe writtenas 264 [ a ( i ; l )][ a ( i ; j )] ; I 375 8><>: ~ ( t ) ~ ( t ) 9>=>; + 264 h ( i ; l ) [ L 2 (n)] 2 ih ( i ; j ) [ L 2 (n)] 2 i ;; 375 8><>: ~ ( t ) ~ ( t ) 9>=>; (2.44) = 8><>: a ( i ; ~ ~ V 0 ) i ; ~ ~ V 0 [ L 2 (n)] 2 ; 9>=>; + 8><>: ; I 9>=>; ~ ( t ) for i;l =1 ; 2 ;:::;N s and j =1 ; 2 ;:::;N c .Where[ a ( i ; l )]representmatrixwhose elementat i t h -rowand j t h -columnis a ( i ; l ).Similardenitionsholdfortheother matricesintheaboveequation. 2.2 Driven Cavity Stokes Flow Problem Controlofdrivencavityrowhasbecomeoneofthestandardex amples[56]in therowcontrolliterature.Inthissection,thisparticula rtypeofrowproblemis studiedusingthemathematicalframeworkdeducedinthelas tsection.Stokesrow isconsideredonly.Controlapplicationsaremostlyconcer nedforthereducedorder model.Thenextsectionwillshowwithnumericalexamplesho wwellthismodel performs. 2.2.1 Control Problem Description ConsiderthephysicaldomaininFigure2.5.Theboundaryoft hesquaredomain nisdenotedby @ n= B [ R [ T [ L

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37 Figure2.5:Stokesdrivencavityrowproblem Assumestaticsolidboundaryconditionsontheleftandrigh tsides, L and R .The ruidinthecavityisdrivenbytheimposedrowonthetop T ofthedomain ~ V = 8><>: u =1( x ) h 0 ( t ) x 2 T v =0 x 2 T Therowiscontrolledbymodifyingtherowonthebottomofthe domain B ~ V = 8><>: u =1( x ) ( t ) x 2 B v =0 x 2 B Movingwallisusedasthephysicalcontrol. ForincompressibleStokesrow,thegoverningequationsare theequation(2.7) andthecontinuityequation(2.8).Nowthe controlproblemforthedriven cavityrow isdescribedas Tondthevelocity ~ V andpressure p thatminimize\someperformancefunctional"andalsosatisfythegoverningequations @ ~ V @t 4 ~ V + r p =0 r ~ V =0

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38 subjectto ~ V j L =0 8 t 2 [0 ;T ] ~ V j R =0 8 t 2 [0 ;T ] ~ V j T = 1( x;y T ) h 0 ( t ) 0 FixedBC ~ V j B = 1( x;y B ) ( t ) 0 control Theperformancefunctionalwillbediscussedinmoredetail inthesectionsthatfollow. 2.2.2 Reduced Order Model of Driven Cavity Problem Becauseafullreneddiscretizationoftherowcanbycomput ationallyprohibitive,itismotivatedtoseekareducedordermodelforth erow.Assumethe velocityeldisapproximatedby ~ V ( x;y;t )= ~ V 0 ( x;y ) h 0 ( t )+ ~ ( x;y ) ( t )+ N s X l =1 ~ l ( x;y ) l ( t )(2.45) Inthisequation, ~ V 0 isthesolutionofthefull,steadyStokessolverfortheboun dary conditions ~ V 0 j T = 10 and ~ V 0 j B S R S L = 00 (2.46) ~ V 0 ( x;y )and h 0 ( t )togetherdescribethecavityeldwhenthetimevaryingdis turbance h 0 ( t )appliedonthetop,whileallothersideshavezerovelociti es.Comparingthis situationwithEquation(2.27),herethemeanvelocityeld ~ V 0 ( x;y;t )isseparated intothespatiallydistributedfunction ~ V 0 ( x;y )andfunction h 0 ( t ).Ingeneral,asinthe nonlinearNavier-Stokesrows,thiscannotbetrue.However ,itworksforStokesrow becausetherowislinearandanychangeintheboundarywilli nduceaproportional changeinthevelocityeldinthedomain.Duetothisreason, itisonlypossible

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39 tohaveonecontrolbasisthatsatisestheaboveboundaryco nditionsforthexed cavitygeometry,ifthecontrolbasis ~ ( x;y )ischosentobethefull,steadyStokes equationsfortheboundaryconditions ~ j B = 10 and ~ j T S R S L = 00 (2.47) Thus,thereisonlyonecontrolvariable ( t ).Itisrequiredthatthereducedbasis associatedwiththestatessatiseszeroboundaryconditio nsovertheboundary @ n. ~ l j T S B S R S L = 00 ;forl =1 ; 2 ;:::;N s (2.48) Togeneratethereducedbasis ~ l ,possiblemethodsincludePOD,orthemethod describedbyItoandRavindran[56].Itwillbediscussedlat erhowtousethesetwo methodstotestthevalidityofthereducedordermodel. Byniteelementmethod,theabovevelocityeldscanbeexpr essedusingnodal velocityvectors ~ V 0 ( x;y )= X n 2 T ~ ~ V 0 ;T;n N n ( x;y )+ X k 2 u ~ ~ V 0 ;u;k N k ( x;y )(2.49) ~ ( x;y )= X n 2 B ~ ~ B;n N n ( x;y )+ X k 2 u ~ ~ u;k N k ( x;y )(2.50) ~ l ( x;y )= X k 2 u ~ l;k N k ( x;y ) ;forl =1 ; 2 ;:::;N s (2.51) Fromconditions(2.46)and(2.47),itcanbeobtainedthat ~ ~ V 0 ;T;n = f 10 g and ~ ~ B;n = f 10 g .NowtheROMforthegeneralStokesproblemcanbewrittenint oaspecic

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40 formforthedrivencavitycontrolproblemusingtheaboveno tation 264 [] T [ M u ][][] T [ M u ] ~ ~ u +[ M B ] ~ ~ B 01 375 8><>: ~ ( t ) ( t ) 9>=>; + 264 [] T [ A u ][][] T [ A u ] ~ ~ u +[ A B ] ~ ~ B 00 375 8><>: ~ ( t ) ( t ) 9>=>; = 8><>: [] T [ M u ] ~ ~ V 0 ;u +[ M T ] ~ ~ V 0 ;T h 0 ( t ) [] T [ A u ] ~ ~ V 0 ;u +[ A T ] ~ ~ V 0 ;T h 0 ( t ) 0 9>=>; + 8><>: 01 9>=>; ( t ) (2.52) Inthisequation,thefollowingmatricesaredened: [ M u ] ( m;s )( i;k ) = Z n e s;i N k N m d n s;i =1 ; 2 m;k 2 u [ M T ] ( m;s )( i;n ) = Z n e s;i N n N m d n s;i =1 ; 2 m 2 u n 2 T [ M B ] ( m;s )( i;n ) = Z n e s;i N n N m d n s;i =1 ; 2 m 2 u n 2 B [ A u ] ( m;s )( i;k ) = 2 X j =1 Z n e s;i @N k @x j @N m @x j d n s;i =1 ; 2 m;k 2 u [ A T ] ( m;s )( i;n ) = 2 X j =1 Z n e s;i @N n @x j @N m @x j d n s;i =1 ; 2 m 2 u n 2 T [ A B ] ( m;s )( i;n ) = 2 X j =1 Z n e s;i @N n @x j @N m @x j d n s;i =1 ; 2 m 2 u n 2 B [ B u ] ( m;s )( r ) = Z n H r @N m @x s d n s =1 ; 2 r =1 ; 2 ; ;N P m 2 u [ B T ] ( n;s )( r ) = Z n H r @N n @x s d n s =1 ; 2 r =1 ; 2 ; ;N P n 2 T [ B B ] ( n;s )( r ) = Z n H r @N n @x s d n s =1 ; 2 r =1 ; 2 ; ;N P n 2 B

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41 2.2.3 Stationary Driven Cavity Flow Thevalidityofreducedordermodelforthedrivencavityrow isconcernednow. Asimplycase|stationarydrivencavityrow|willbetested rst.Morecomplicated rowswillbedealtwithinthenextsection.Stationarydrive ncavityrowcorresponds tothecasewhenthecavityissubjecttoasteadydrivenrowon thetopboundary andnocontrolisapplied.Thus,theboundaryconditionsare zeroexceptonthetop side, T .Weassumeaunithorizontalvelocityonthetopofthecavity .Graphically, Figure2.6:Stationarydrivencavityrow thestationarydrivencavityrowmustsatisfyboundarycond itionsasshowninFigure 2.6.Inthiscase,thelowdimensionalapproximationofthev elocityeldissimplied ~ V ( x;y;t )= ~ V 0 ( x;y ) h 0 ( t )+ N s X l =1 ~ l ( x;y ) l ( t )(2.53) andtheROM(2.52)willbewrittenas [] T [ M u ][] ~ ( t )=[] T [ A u ][] ~ ( t ) [] T [ M u ] ~ ~ V 0 ;u +[ M T ] ~ ~ V 0 ;T h 0 ( t ) [] T [ A u ] ~ ~ V 0 ;u +[ A T ] ~ ~ V 0 ;T h 0 ( t ) (2.54) Nowtheprocedure(referto[56])togeneratethemeanveloci ty ~ V 0 andreduced basis ~ l isintroduced.First,acollectionbasiselements f ~u i g Mi =1 areobtainedfromfull, steadyNavier-StokessolverforMdierentReynoldsnumber s.Eachbasiselementis

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42 subjecttotheboundaryconditions ~u =(1 ; 0)onthetopboundary,andeverywhere elseontheboundary @ nnoslipboundaryconditionsareassumed.Then,areduced ordersolutionisformedbysetting ~ V 0 = ~u M and ~ l = ~u i +1 ~u i ;i =1 ; 2 ; ;M 1. Incurrenttest,M=5.Thebasiselements f ~u i g 5i =1 forthereducedordermodel areobtainedfromCFDsolutionstothedrivencavityrowatRe ynoldsnumber100, 300,500,700,900.Followingtheaboveprocedure, ~ V 0 = ~u 5 ~ l = ~u i +1 ~u i ;i = 1 ; 2 ; ; 4.Fourcasestestedherecorrespondto N s =1 ; 2 ; 3 ; 4forequation(2.53). Aftersolvingequation(2.54)foreachcase,oneobtainssol utionsforROMstates f i g 4l =1 .Subsequently,itispossibletoreconstructvelocityeld fromEquation(2.53). Figure2.7comparesthex-velocityalongtheverticalcente rlineofthecavity.The ROMsolutionsarecomparedwiththefullordersolution.Its howsthattheROM solutionisclosetofullordersolutions.Itcanbeconclude dthattheROMcanbe usedinstationarycasetorepresentfullvelocityeld.Als othegureshowsthatthe fullordersolutioncanbebetterapproximatedbythereduce dordersolutionwhen thenumberofbasisfuunctionsincreases. Figure2.7:Comparisonofreducedordersolutionsofveloci tiestofullordersolution

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43 2.3 POD and Unsteady Driven Cavity Flow Problem Itwasshowninlastsectionthatthemeanvelocityeld ~ V 0 andthecontrolbasis ~ canbeobtainedfromthefull,steadyStokessolver.Thecons tructionofthereduced basis ~ wasdiscussedalsoforstationarydrivencavityproblem.Th issectioncontinues thedevelopmentofROMfromPOD.ThetheorysupportingPOD,i ncludingthebasic numericalprocedureandusefulconclusions,willbediscus sedrst.Then,itisshown thatPODcanbeusedtoconstructreducedordermodelsforuns teadydrivencavity row. 2.3.1 Proper Orthogonal Decomposition (POD) Theproperorthogonaldecomposition(POD)isasystematict echniquetoobtain lowdimensionalmodels.Ithasbeensuccessfullyappliedfo rruidrows.Theaimis togenerateanoptimalbasistorepresentanensembleofcomp utationaldata.The basisconstructedbyPODisoptimalinthesensethattheener gycontainedinitis greaterthananyotherbasishavingthesamedimension. 2.3.1.1 POD Procedure Thegoalistondbasisfunctions f ~ ( x;y ) g Nj =1 whichcouldelegantlyrepresent thegivenensembledata f ~u k g Nk =1 .Thiscollectionofdataissometimescalledsnapshots ofthesolutionatsomexedtimeinstants,where ~u 2 L 2 and L 2 isHilbertSpace. Mathematically ~ shouldbechosentomaximizetheaverageprojectionof ~u onto ~ Ourmaximizationproblemis max ~ 2 L 2 ([0 ; 1]) hj ( ~u; ~ ) j 2 i k ~ k 2 subjectto k ~ k =1(2.55) where( )and kk denoteinnerproductandnormon L 2 respectively.Thebrackets hi denotesatimeaverageoperation.Tosolvethisproblem,wei ntroducethe Lagrangian L h ~ i = hj ( ~u; ~ ) j 2 i k ~ k 2 1 (2.56)

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44 where istheLagrangianmultiplier.Anecessaryconditionforthe calculationof extremaofthisfunctionalisthatitsGateauxderivativesv anishforallvariations @ @ L h ~ + ~ i j =0 =0 Carryingoutthiscalculationyields Z 1 0 h ~u ( x;y ) ~u ( x 0 ;y 0 ) dxdy i ~ ( x 0 ;y 0 ) dx 0 dy 0 = ~ ( x;y ) Denotetheoperator R = R h ~u~u i dxdy .Theaboveequationiswrittenas R ~ = ~ (2.57) whichisaneigenvalueproblem.Anorthonormalbasis f ~ j ( x;y ) g Nj =1 canbeextracted fromtheeigenvectors,i.e. ~u ( x;y;t )= N X j =1 a j ( t ) ~ j ( x;y ) 2.3.1.2 Optimality Properties of the POD Assumethatthevelocityeld ~u ( x;y;t )in L 2 isstationaryrandomandthat n ~ i ; i j i =1 ;:::; 1 ; i i 1 > 0 o istheassociatedsetofPODbasisvectorsand associatedeigenvalues.Let ~u ( x;y;t )= X j =1 a j ( t ) ~ j ( x;y ) bethedecompositionwithrespecttothePODbasisand ~u ( x;y;t )= N X j =1 b j ( t ) ~ j ( x;y ) thedecompositionusinganyotherarbitraryorthonormalse t.Thenthefollowingcan beshowntohold. (i) h a i ( t ) a j ( t ) i = ij ij .ThePODprojectioncoecientsareuncorrelated.

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45 (ii)Foreverynwehave n X i =1 n a i ( t ) a j ( t ) = n X i =1 i n X i =1 n b i ( t ) b j ( t ) (2.58) ThisequationstatesthatPODisoptimalinthesensethatthe rstnPODbasis functionscapturemoreenergythenanyotherbasisnbasisve ctors. (iii)IftheensembleiswritteninmatrixformasAwitheachc olumnrepresenting asinglesnapshotattime t i ,then k +1 =min rank ( B ) k k A B k 2 ; 2 = k A A k k 2 ; 2 (2.59) whereA k = k P i =1 i u i v T i ;U =[ u 1 ;:::; u m ] ;V =[ v 1 ;:::; v n ] ;andA = USV T Thispropertyshowsthat k ~u i p k ( ~u i ) k 2 k +1 wheretheoperator p k ( ~u i )isusedtodenetheprojectionofthearbitraryvector ~u i ontothesubspaceformedbyusingknumberofPODmodes.Itsta testhat,intheory, theerrorbetweenagivensnapshotataparticulartimestepa ndthereconstructed approximation(usingkmodes)shouldinfactbeboundedbyth esingularvalue, k +1 oftheensemblecontainedinA. 2.3.1.3 Energy in the Eigenfunctions Thekineticenergyintheextractedoptimaleigenmodesis E = Z n u ( x;t ) u ( x;t ) dx + = X ij a i ( t ) a j ( t ) + = X ij ij ij = X k k (2.60) Thismeansthattheenergyineacheigenfunctionisjustthec orrespondingeigenvalue. 2.3.2 POD for Unsteady Driven Cavity Flow Nowlowdimensionalapproximationinequation(2.45)isuse dfordescriptionof unsteadydrivencaivtyrow.Thegenerationofthemeanveloc ityeld ~ V 0 andcontrol

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46 basis ~ havebeendescribedinsomedetail.Theyareconstructedbyt hefull,steady Stokessolutionswhichsatisfy(2.46)and(2.47).Itwassho wnthatPODcanbeused toobtainaoptimalorthonormalbasissetgivenanensembleo fcomputationalor experimentaldata.NowPODmethodwillbeinvestigatedtoge neratereducedbasis forunsteadydrivencavityrowbyPOD. First,anensembleofdata f ~ V D ( t k ) g Nk =1 isobtainedfromsolutionsofthefull, unsteadyStokesequationwhere ~ V D ( t k )isthefullordersteadysolution,attime instance t k ,correspondingtothetime-varyingboundaryconditions ~ V D j T = cos(2 ft ) 0 ; ~ V D j B = sin(2 ft ) 0 ; ~ V D j R S L = 00 for t 2 [0 ;T ].Then,forthesametimeset f t 1 ;t 2 ;:::;t N g ,insteadofunsteady solutions,aseriesofstaticfullordersolutions f ~ V ST;k g Nk =1 isobtained,whereeach solution ~ V ST;k satisesboundaryconditions ~ V ST;k j T = cos(2 ft k ) 0 ; ~ V ST;k j B S R S L = 00 for k =1 ; 2 ;:::;N .Itshouldbekeptinmindthatthisisnotanensembleof dynamicssolutions.Itisjustasetofstaticsolutionswith eachvectorcorresponding todierentboundaryconditions.Similarly,anotherstati censembleoffullorder solutions, f ~ V SB;k g Nk =1 ,isobtained,whereeachvectorsatisesboundaryconditio ns ~ V ST;k j B = sin(2 ft k ) 0 ; ~ V ST;k j T S R S L = 00 Rememberthereducedbasis f ~ l g N s l =1 isrequiredtosatisfyboundaryconditionsdescribedinEquation(2.48).Sotheabovedataensembleshave tobeprocessedwegot beforePODisusedtogeneratethereducedbasis.Equation(2 .48)impliesthatthe ensembleofdatatobeusedinPODmusthavezeroboundarycond itionsoverallof

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47 @ n.Anewensembleofdataisgeneratedbysubtracting ~ V ST;k and ~ V SB;k from ~ V D ( t k ) foreachtimeinstance t k ~ V k = ~ V D ( t k ) ~ V ST;k ~ V SB;k Graphically,thisprocessisshowninFigure2.8.NowPODmet hodisusedtogenerate ) 2 cos( k ft p ) 2 sin( k ft p 0 0 0 0 0 0) ( k D t V r k STV,r k SBV,r ) 2 cos(kft p ) 2 sin( k ft p 0 0Figure2.8:TogenerateensembletowhichPODistaken thereducedbasis f ~ i g N s i =1 basedfromthenewensemble f ~ V k g Nk =1 .TheresultingPOD orthonormalbasiswillhavezeroboundaryconditions.Fort heparticularcases = 0 : 1 ; =1,and =2 : 5,Figures2.9 2.11showthersttwoeigenmodesrespectively. Tables2.1-2.3listtherstsixlargestsingularvaluesres pectively.Itisclearfrom thesisguresandtablesthatthersttwoeigenmodeswillbe dominantforeachcase. Lateron,thesePODeigenmodeswillbeusedformodelreducti on.Sincethe Figure2.9:ContourplotsofPODeigenmodeswithstreamline sat =0 : 1 reducedordermodel(ROM)isbasedonthenon-dimensionalde scriptionofEquation

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48 Figure2.10:ContourplotsofPODeigenmodeswithstreamlin esat =1 Figure2.11:ContourplotsofPODeigenmodeswithstreamlin esat =2 : 5 (2.7),whichisvalidfor O (1),itisnecessarytodiscusstherangeof forwhich theROMisvalid.Toassessthevalidity,numericalsimulati onsarecarriedoutfor threecases: =0 : 1 ; =1,and =2 : 5. NowPODbasis f ~ i g Ni =1 isgenerated.BeforeconsideringusingthePODbasis forreducedordermodels,erroranalysisisnecessaryinord ertodetermineifthe methodofPODisappropriate.Theerrorhereisdenedasthed ierencebetween thereferencesolutionanditsprojectiononasuitablenumb erofPODeigenmodes.

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49 Table2.1:Singularvaluesfor =0 : 1 1 3709.6 2 344.65 3 3.4099e-8 4 4.2987e-10 5 2.0972e-10 6 3.5968e-11 Table2.2:Singularvaluesfor =1 1 5053.3 2 607.91 3 1.4597e-8 4 1.1099e-11 5 1.0314e-11 6 1.0111e-11 Inthiscase,itisthedierencebetweentheactualfullorde rsolutions f ~ V k g Nk =1 and thePODreconstructedvelocityelds. Introduce ` 2 -normofavector ~x 2 R n k ~x k 2 = n X i =1 j x i j 2 1 2 Denotevector ~e ( m ) k astheerror,fortimeinstance t k ,between ~ V k andPODreconstructionwithrst m eigenmodes ~e ( m ) k = ~ V k m X i =1 ~ i ~ V k ~ i (2.61) Theoretically,thePODreconstructedvelocityeldsshoul dbecomeclosertothe referencesolutionsasthenumberofeigenmodesincreases. Ifnodalvectorsareused intheequation(2.61)insteadof ~ V k and ~ i ,anodalerrorvector ~e ( m ) k 2 R 2 N nd is introduced,where N nd isthenumberofnodesintheunknowndomainn n @ n.The accuracyofapproximationisdenedbythe ` 2 -normofthiserrorvector,namely e ( m ) ` 2 ;k = k ~e ( m ) k k 2 .However,fortheunsteadyproblemabove,wemightactually care abouttheaverageerrorovertheentiretimeperiod.Inthisc ase,theerrorisdened

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50 Table2.3:Singularvaluesfor =2 : 5 1 5144.2 2 616.66 3 1.1779e-8 4 1.1955e-11 5 1.0716e-11 6 1.0014e-11 as e ( m ) ` 2 = 1 T N X k =1 e ( m ) L 2 ;k = 1 T N X k =1 k ~e mk k 2 = 1 T N X k =1 2 N nd X j =1 e mk;j 1 2 (2.62) NowthefeasibilityofapplicationofthePODmethodforther educedordermodel ofunsteadyStokesproblemcanbeinvestigated.Threeexamp leswillbetestedfor =0 : 1, =1and =2 : 5. (i) =0 : 1 Figure2.12plotstheaverageerrorversus m ,thenumberofmodesusedin thereconstruction.Itshowsthattheerrordecreasesasthe numberofeigenmodes increases.Theerrordropsveryfastaftertherstfewmodes .When m 5,the errorgoesbelow10 6 .Thistellsusthatwecanuseasfewas5PODeigenmodes toreconstructvelocityeldswithanaccuracyoftheordero f10 6 .Theresultis excellentbecauseinareducedordermodelwewanttouseasfe wmodesaspossible. ThisresulttellusthatthePODmightbeanappropriateande cientmethodto constructreducedbasisformodelreduction.Thiswillbedi scussedindetaillater. Ithasbeendiscussedbeforethattheerrorbetweenthefullo rdersolutionand PODreconstructedapproximationwith m eigenmodesshouldbeboundedbythe singularvalue, m +1 ,oftheoriginalensemble.Thetheorycanbeconrmedby inspectioninaboveFigure2.12andFigure2.13,aswell.

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51 Figure2.12:Erroraveragedovertimestepsisboundedbythe singularvalues m +1 (ii) =1 Figure2.14showssimilarlythatusingaverysmallnumberof PODeigenmodes canachieveaccurateapproximationofthefullordersoluti ons.Onceagain,theerror isboundedbythesingularvalue, m +1 ,fromtheFigure2.14and2.15. (iii) =2 : 5 Figure2.14similarlyillustratesthatPODeigenmodescanb everyecientin reconstructingvelocityelds.Figure2.16and2.17illust rate,again,theerrorbound.

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52 Figure2.13:Errorofeachtimestepisboundedbythesingula rvalues Figure2.14:Erroraveragedovertimestepsisboundedbythe singularvalues m +1

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53 Figure2.15:Errorofeachtimestepisboundedbythesingula rvalues Figure2.16:Erroraveragedovertimestepsisboundedbythe singularvalues m +1

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54 Figure2.17:Errorofeachtimestepisboundedbythesingula rvalues

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55 2.3.3 ROM by POD for Unsteady Driven Cavity Stokes Flow Nowtheapproximationofreducedordermodelwillbestudied fortheunsteady StokesrowinadrivencavityasshownintheFigure2.18.Assu methatonthetop sideofthecavity,thedrivenrowhasavelocity h 0 ( t )=cos(2 ft ),andonthebottom y x Figure2.18:Unsteadydrivencavityrow sideofthecavity,therowissubjecttotheboundaryconditi on ( t )=sin(2 ft ). ThePODeigenmodesobtainedinlastsectionareusedasourre ducedbasis. Combiningthesemodeswiththemeanvelocityeld ~ V 0 h 0 ( t )andcontrolbasis ~ ,now itispossibletoobtainacompletereducedorderapproximat ionofvelocity,using Equation(2.27),oftheunsteadyrow.Theinitialcondition ~ V ROM ( t =0),ischosen tobethesolutionofthestaticStokesrowmodelwithuniform horizontalvelocityon thetopandzeroboundaryconditionsonothersidesofthecav ity. ~ V j T = 10 and ~ V j B S R S L = 00 UsingthefullordersteadyStokessolver,theinitialrowe ldcanbeobtainedas showninFigure2.19.Nowthereducedordermodelcanbeusedt oapproximate theunsteadysolutionwhichisgovernedbyEquation(2.52). Discardthetransition responseandsavethesteadysolutionforonecycle.Boththe fullorderandthe reducedordersolutionsareobtainedatsametimeinstances t k ;k =1 ; 2 ;:::;N

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56 Figure2.19:Contourplotofinitialvelocityeld Thenerroriscalculatedateachtimestep.Theerrorisdene dasthe ` 2 -normofthe relativenodalerrorbetweenthefullorderandreducedorde rsolutions e mL 2 ;k = k ~e mk k 2 = k ~ V ROM ( t k ) ~ V Full ( t k ) k 2 k ~ V Full ( t k ) k 2 (2.63) Thesamecasesasbeforearetestedwith =0 : 1, =1,and =2 : 5. (i) =0 : 1 SinceStokesrowisunderconsideration,averyviscousglyc erineisanappropriate ruidexample.Assumea99.42%glycerineisinsidea1 cm 1 cm cavity(i.e., L =1 cm ) withthetopboundaryx-velocity u =1 cm=s .Thedensityandabsoluteviscosityat 20 C are =1 : 25958 g=cm 3 and =1259 : 2 mPa Sec respectively 1 .Thusthe Reynoldsnumberassociatedwiththisruidis Re = uL =0 : 1.ItisaStokesrow.If thetopboundaryischangingwithafrequency f =1 kHz ,thentheStrouhalnumber St =100.Thus,thecavityrowischaracterizedbytheparameter = 1 Re St =0 : 1. Figure2.20plotstheerroratdierenttimeinstances.Then umberofPOD eigenmodesusedforthereducedbasisvariesfrom1to6.Ther elativeerrorislarge. 1 http://www.dow.com/glycerine/resources

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57 Evenaddingothereigenmodestothereducedbasisdoesnothe lpmuch.Figure2.21 Figure2.20:ErroranalysisofROMapproximation showsthecomparisonbetweenthefullordersolutionandred ucedorderapproximation.Theguresdepictthex-velocitiesalongthevertical centerline.Itshowsinthe guresthatthenodalvelocitiesdierconsiderablyandthe rowstructuresarenot same.Allthesefactstellusthatthereducedordermodelisn otagoodapproximation ofthecavityrow.(ii) =1 Thesameglycerineruidandcavitysizeareconsideredasinc ase(i).Sothe Reynoldsnumberissame,i.e., Re =0 : 1.However,inthiscase,adierentboundary frequencyisconsidered.Forthechoiceof f =100 Hz ,theparameter =1. Figure2.22plotstheerroratdierenttimeinstances.Then umberofPODeigenmodesusedforthereducedbasisvariesfrom1to6.Nowtherel ativeerrorissmall. Also,thegureshowsthattheerrordecreasesasthenumbero feigenmodesincreases. Thisobservationisreasonableandexpected.Whenthenumbe rincreasesbeyond2, theerrordoeschangesignicantly.Thisisbecausetherst twoeigenmodesdominate

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58 0 0.2 0.4 0.6 0.8 1 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 t = 9.2 sec yV x (x=0.5,y) Full Order SolROM: 1 modeROM: 2~6 modes 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 t = 9.4 sec yV x (x=0.5,y) Full Order SolROM: 1 modeROM: 2~6 modes 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 t = 9.6 sec yV x (x=0.5,y) Full Order SolROM: 1 modeROM: 2~6 modes 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 t = 9.8 sec yV x (x=0.5,y) Full Order SolROM: 1 modeROM: 2~6 modes Figure2.21:ErroranalysisofROMapproximationat =0 : 1 andcapturenearlyalltheenergy|recallthattheenergyoft heensembleprojected onaneigenmodeisthecorrespondingeigenvalue.Thus,beyo ndthisnumber,adding othereigenmodestothereducedbasisdoesnothelpmuch.Fig ure2.23showsthe comparisonbetweenthefullordersolutionandreducedorde rapproximation.Itonly plotsthex-velocitiesalongtheverticalcenterline.The guresshowthatthereduced ordersolutionsareverygoodapproximationsofthefullord ersolutions.Theerror analysisandtheabovecomparisonillustratethatthereduc edordermodelisagood approximationofthecavityrowinthecase =1. (iii) =2 : 5

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59 Figure2.22:ErroranalysisofROMapproximation Considerthesamecavitysizeandglycerineruidasinthepre viousexample. TheReynoldsnumberis Re =0 : 1.However,inthiscase,boundaryfrequencyis modied.Choose f =40 Hz ,Thentheparametertakesthevalue =2 : 5. Figure2.24plotstheerroratdierenttimeinstances.Then umberofPOD eigenmodesusedforthereducedbasisvariesfrom1to6.Thee rrorissmallerasthe numberofeigenmodesincreases,andthereducedorderbasis isecientinrepresentingtherowstructure.TheROMinthiscaseisecient.Onceag ain,asinthecase when =1,theerrordoesnotchangewhenthenumberofbasisvectors exceeds 2.InFigure2.25,acomparisonattheverticalcenterlineof thecavity.Thegure veriesagainthatthereducedordermodelisaccuratefor =2 : 5. Comparingtheabovethreecases,when =1,theROMisaccurate.ThisisexpectedbecausetheROMandnon-dimensionalanalysisarebas edontheassumption O (1).If isfarfrom1,theROMmaynotbeaccurate.Thisfactisrerecte d bythepoorresultsobtainedinthecase =0 : 1.However,goodresultsareachieved inthecase =2 : 5.RememberthatEquation(2.7)wasobtainedwhenthecharac teristicfrequency, f ,ontheboundaryiscomparabletothecharateristicfrequen cy,

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60 0 0.2 0.4 0.6 0.8 1 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 t = 9.2 sec yVx(x=0.5,y) Full Order SolROM: 1 modeROM: 2~6 modes 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 t = 9.4 sec yVx(x=0.5,y) Full Order SolROM: 1 modeROM: 2~6 modes 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 t = 9.6 sec yVx(x=0.5,y) Full Order SolROM: 1 modeROM: 2~6 modes 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 t = 9.8 sec yVx(x=0.5,y) Full Order SolROM: 1 modeROM: 2~6 modes Figure2.23:ErroranalysisofROMapproximationfor =1 V s L ,associatedwiththecavity.Inthecase =0 : 1,itsosmallthatthersttermin theequationwouldbetheonlylargeterm|thepressurehasbe enrescaled|there isnothingtobalanceit.Aphysicalexplanationofthisfact isthattheboundary conditionschangesofast( f =1 Khz )thatthereexistsalayeralongthetopsurface whereunsteadyeectsareconned.Settheoriginofacoordi natesystematthe lowerleftcornerofthecavity.Thelocalyvariablewillbes caled,thatis =( y 1) = where ischosensothatthereisabalancebetweenthetimederivati vetermandthe Laplacianoperator.Thisanalysisleadstothechoice = p = r fL 2

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61 Figure2.24:ErroranalysisofROMapproximationfor =2 : 5 sothatthosetwotermsarebothlarge.Theresultingasympto ticlayerequationis @ ~ V @t @ 2 ~ V @ 2 Theboundaryconditionsonthisequationare ~ V ( x; =0 ;t )=cos(2 ft ) ~ V 0as !1 Performsamescalinglocallyaround y =0.TheproblemhereisessentiallyStokes 2 nd problem.Ignorethetransientresponseandconsidertheste ady-stateoscillation inthislayerthattakesplaceonafastertimescale.Outside thislayertheleading ordersolutionissteadyandhasnovelocity.Foradierentc onditionontheupper surfacethathadanon-zeromeanvelocity,thentherowoutsi dethislayertoleading orderwouldbedeterminedbythemeanvelocityoftheuppersu rface.Thepointis thattherowoscillationsare,forthemostpart,restricted toathinregionnearthe oscillatingsurface.Nowconsiderthecontrolproblemwher etheuppersurfaceisbeing actuatedaswellasthelowersurface.Iftheuppersurfaceis oscillatingquicklythere islittlethatcanbedonetoremovethoseoscillationsinduc edbythelowersurface.

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62 0 0.2 0.4 0.6 0.8 1 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 t = 9.2 sec yVx(x=0.5,y) Full Order SolROM: 1 modeROM: 2~6 modes 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 t = 9.4 sec yVx(x=0.5,y) Full Order SolROM: 1 modeROM: 2~6 modes 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 t = 9.6 sec yVx(x=0.5,y) Full Order SolROM: 1 modeROM: 2~6 modes 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 t = 9.8 sec yVx(x=0.5,y) Full Order SolROM: 1 modeROM: 2~6 modes Figure2.25:ErroranalysisofROMapproximationfor =2 : 5 Theregionoftherowinruencedbytherapidoscillationsoft helowersurfaceistoo thintoreachallthewayacrossthedomain. Numericalresultsforother valuesshowthatanexampleregionatwhichthe reducedordermodelworkswellis > 0 : 5.This regionwillbeusedinlateron numericalsimulations.Fortheglycerine-lledcavityrow ,thefrequencyfor =0 : 5 iscalculatedtobe f =200 Hz .Inthiscase,thereducedordermodelisappropriate for f< 200 Hz

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CHAPTER3 OPTIMALCONTROLOFDRIVEN-CAVITYSTOKESFLOW Thelastchapterhasdiscussedhowtousemodelreductiontec hniquestoobtain loworderdescriptionsofruidrowmodelshavinghighdimens ion.Ithasbeenshown innumericalexamplesthatthereducedordermodelfordrive ncavityrowcancapture theessentialstructuresofruidrowusingveryfewstates.H owever,thisdissertation investigatesifthereducedordermodelisappropriateforc ontrolapplications.Inthis chapter,thefeasibilityofthereducedordermodelfordie rentcontrolstrategiesis investigated. Optimalcontrolforruidrowswillbetherstconcern.Manyr owcontrol problemsarecastusingoptimalcontroltheory.Themathema ticalstructuresof theseproblemsareamenabletoabstractcontrolformulatio ns.Awidelyaccepted approachistousethemethodologiesofthecalculusofvaria tionstondthegoverning equations.Themotivationforthisapproachisthatasingle mathematicalframework canbeusedformanydierentpracticalproblems. 3.1 Variational Operator Theoperator ( ),calledvariationaloperator,iswidelyusedinengineeri ngtexts. Itplaysafundamentalroleinthederivationsthatfollow.I thasproperties: (P1) ( )actslikedierential d ( ) (P2) t =0fortheindependentvariablet (P3) ( ) j boundaries =0 Forexample,forthefunctional J = Z b a f ( x;y ( x ) ;y 0 ( x )) dx 63

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64 therehave J = Z b a f ( x;y ( x ) ;y 0 ( x )) dx = Z b a ( f ( x;y ( x ) ;y 0 ( x )) dx = Z b a @f @x x + @f @y y + @f @y 0 y 0 dx Since x =0accordingtoproperty(P2),therstterminthebracketis zero.Integratedbypartstothethirdterm,thethirdtermcanberewrit ten J = Z b a @f @y d dx @f @y 0 ydx + @f @y 0 y ba Byproperty(P3),thelasttermiszero.Ifthevirtualvariat ion J mustbezerofor allchoicesof y ,theintegrandmustvanish.Thegoverningequationsarethe refore @f @y d dx @f @y 0 =0 3.2 Optimal Control Framework Optimalcontroltheoryinmanyapplicationsconsidersafam ilyofoptimization problemsthathaveasurprisinglysimilarstructure.Itsee kstond u ,orcontrolinput,thatminimizesacostfunctional J ,whilesimultaneouslysatisfyingaset ofequationsofmotionandconstraints.Therstchapterint roducedmathematicalframeworkforadistributedcontrolproblemstatedbyCu velier[24].Thecost functional, J ,forthatproblemisaquadraticfunctionwhichissubjectto theNavierStokesequation.Ingeneral,theequationofmotionandthec ostfunctionalcanbeof anyform.Forinstance,inacommoncollectionofproblems,t heevolutionofphysical states x ( t )satisesanonlinearordinarydierentialequationandin itialcondition x ( t )= f ( x ( t ) ;u ( t ) ;t ) x (0)= x 0 (3.1)

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65 Inthisequation,thevariablessatisfy x ( t ) 2 R N u ( t ) 2 R M f : R N R M R R N Forthecontrolofruiddynamics,thephysicalstates x ( t )maybevelocities,pressures, temperaturesorsomecombinationoftheseobservables.The controls u ( t )might representimposedvoltages,currents,forceloadings,orr owratesthatalterthenature oftheresponseembodiedinthestatevariables x ( t ). u ( t )mighthavetosatisfy somephysicalconstraintsthushastobeacceptable.Eachco ntrolinput\drives"the evolutionofthestatesandgenerates x ( u ( t )).Therankingofcompetingcontrolsis accomplishedviatheintroductionofthecost(error,orobj ective)functional J ( u ). Thecostfunctionalmayrepresenttheerrorthatacertainco ntrolproducesina targettrackingproblem,orthefuelcostincurredinaspace craftmission,forexample. Typicalcostfunctionalshavetheform J ( u ) 4 = ( x ( T ) ;T )+ Z T 0 L ( x ( ) ;u ( ) ; ) d (3.2) Inthisexpression, ( ; )isameasureofnalstatecost,while L ( ; ; )represents accumulatederrorsorcostsincurredovertheentiretrajec tory.Thesefunctionsdene mappings : R N R R L : R N R M R R Theoptimalcontrolproblemcanthenbestatedasfollows:se ek u 2 U suchthat J ( u ) J ( v ) 8 v 2 U (3.3) where J ( )isdenedinEquation3.2andthetrajectory x ( ) ; 2 [0 ;T ]appearingin J satisesEquation3.1.

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66 3.2.1 Fixed Terminal Time Problems Thissectionwilldiscusshowthecalculusofvariationscan beemployedtoderive necessaryoptimalityconditionsforsomeoptimalcontrolp roblems.Itonlyconsiders thoseproblemssubjecttotheconstraintthatthetrajector y x ( ) ; 2 [0 ;T ]appearing inthecostfunctionalmustsatisfythegoverningequations ofphysics. Theproblemofndinganoptimalcontrolforaxedterminalt imeisprecisely thecontrolproblemsummarizedinEquations(3.1)-(3.3).I tseeks u 2 U suchthat J ( u ) J ( v ) 8 v 2 U (3.4) where J ( v ) 4 = J ( x ( v ) ;v )= ( x ( T ) ;T )+ Z T 0 L ( x ( ) ;v ( ) ; ) d andwherethetrajectory x ( ) ; 0 T issubjectto x ( t )= f ( x ( t ) ;u ( t ) ;t ) x (0)= x 0 (3.5) Conventionalapproachestothisoptimalcontrolproblemde neanaugmentedcost functionalintermsofLagrangemultiplier p ( t ) 2 R N as J ( x;u;p )= J ( x;u )+ Z T 0 p T ( ) f ( x ( ) ;u ( ) ; ) x ( ) d (3.6) andsubsequentlytreattheproblemasanunconstrainedopti mizationproblem.In Equation(3.6),Ithasusedthenotation N X i =1 p i f i = p T f (3.7)

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67 tosimplifyanexpressionoftheinnerproductoftwovectors .Thesymbol f denotes acolumnvectorand p T denotesarowvector.Similarly,if h : R s R t ,denoteby @h @x @h i @x j i =1 ; 2 ;:::;t j =1 ; 2 ;:::;s (3.8) Thus, @ L @u isamatrixhaving1rowandMcolumns,forexample,since u 2 R M inthis section.Again,itisconventionaltointegrateEquation(3 .6)byparts J ( x;u;p )= J ( x;u )+ Z T 0 p T ( ) f ( x ( ) ;u ( ) ; ) d p T ( ) x ( t ) T0 + Z T 0 p T ( ) x ( ) d (3.9) anddenetheHamiltonian H ( x;u;p; ) H ( x;u;p; )= L ( x;u; )+ p T f ( x;u; )(3.10) Hence,thecostfunctionalis J ( x;u;p )= ( x ( T ) ;T )+ Z T 0 H ( x ( ) ;u ( ) ;p ( ) ; ) d + Z T 0 p T ( ) x ( ) d p T ( ) x ( ) T0 (3.11) Now,viaaformalapplicationofthecalculusofvariations J ( x;u;p )= @ J @x x + @ J @u u + @ J @p p (3.12) Fromequation(3.6),aformalapplicationofthevariationa loperator yields @ J @p p = Z T 0 p T ( ) f ( x ( ) ;u ( ) ; ) x ( ) d = Z T 0 f ( x ( ) ;u ( ) ; ) x ( ) T p ( ) d (3.13) Similarly,from(3.11)

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68 @ J @u u = Z T 0 @ H @u ( x ( ) ;u ( ) ;p ( ) ; ) u ( ) d (3.14) and @ J @x x = Z T 0 p T ( )+ @ H @x ( x ( ) ;u ( ) ;p ( ) ; ) x ( ) d + @ @x ( x ( T ) ;T ) x ( T )+ @ @x ( x ( T ) ;T ) x (0) p T ( T ) x ( T )+ p T (0) x (0) (3.15) Since x (0)= x 0 isknowninthisproblem, x (0)=0byhypothesis.Viaaformal applicationofthecalculusofvariations,eachcoeciento fthevariations p u and x mustvanishandthen x ( t )= @ H T @p ( x ( t ) ;u ( t ) ;t )(3.16) x (0)= x 0 (3.17) p ( t )= @ H T @x ( x ( ) ;u ( ) ;p ( ) ; )(3.18) p ( T )= @ T @x ( x ( T ) ;T )(3.19) 0= @ H @u (3.20) IthasbeenchosentoexpresseachoftheEquations(3.16)thr ough(3.20)ascolumn vectorsinthisconvention.Theseequationshavetheformty picalofmanyoptimal controlproblems.Equations(3.16)and(3.17)comprisethe originalsystemequations, andtheyevolveforwardintime.Equations(3.16)and(3.17) arereferredtoasthecostate,oradjoint,equations.Incontrasttothestateequat ions,theadjointequations evolvebackwardintimefromthenalconditionin(3.19).To gether,theEquations (3.16)through(3.20)compriseatwopointboundaryvaluepr oblemfortheoptimal controltrajectory u ( ) ; 2 [0 ;T ].

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69 3.3 Optimal Control of Driven Cavity Flow Inthissection,theoptimalcontrolofunsteadydrivencavi tyrowwillbediscussed.Themathematicalframeworkoflastsectionwillbea ppliedtoatracking problem.Detailalgebraicworkwillshowthattheresulting solutionsresorttoclassicalRiccatisolution. 3.3.1 Control Problem Description ConsiderthecontrolproblemdescribedbyFigure2.5andequ ation(2.52)with someinitialcondition X 0 andxedtimeterminal t 2 [0 ;T ].Deneanewsetofstates thatincludeboththeoriginalreducedstatevariables ( t ),andtheamplitudeofthe control ( t ),sothatthenewstateis X ( t )= 8><>: ~ ( t ) ( t ) 9>=>; Thenewsetofcontrolsisdenedtobethetimederivativeoft heoriginalcontrols u ( t )= n ( t ) o Withthesedenitionsofthestate X ( t )andcontrols u ( t ),itispossibletowritethe weakformofthegoverningequationsas [ ~ M ] X ( t )=[ ~ A ] X ( t )+ ~ Bu ( t )+ ~ F ( t ) Itisimportanttonotethatthematrix[ ~ M ]willbediagonalifthereducedbasis vectorsarederivedfromaproperorthogonaldecomposition .However,ingeneral, thereducedbasisfunctionswillnotbeorthogonal.Thenal formofthereduced stateequationscanbeobtainedbypremultiplyingbytheinv erseofthematrix[ ~ M ] X ( t )=[ ~ M ] 1 [ ~ A ] X ( t )+[ ~ M ] 1 ~ Bu ( t )+[ ~ M ] 1 ~ F ( t )

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70 Thisequationcanberewritteninthenalformwithinitialc onditionas X ( t )=[ A ] X ( t )+ Bu ( t )+ F ( t ) X (0)= X 0 (3.21) Nowthataloworder,nitedimensionalapproximationofthe governingpartial dierentialequationshasbeenachieved,asuitablecostfu nctionalmustbedenedto completethedenitionoftheoptimalcontrolproblem.Cons ideratrackingproblem. Forconvenience,letvector( t )bethesolutionofequation(3.21),forsomearbitrarily chosen,continuousfunction r ( t ).Thatis ( t )=[ A ]( t )+ Br ( t )+ F ( t ) (0)= 0 (3.22) Introducearelativelycommonquadraticcostfunctional J ( u ) 4 = J X ( u ) ;u 4 = 1 2 Z T 0 ( X ( ) ( )) T [ Q ]( X ( ) ( )) d + 1 2 Z T 0 Ru 2 ( ) d (3.23) where[ Q ]issymmetric,positivesemideniteandthescalarRisposi tive.Recallthat amatrix[ M ]ispositivesemideniteprovidedtheproduct T [ M ] 0(3.24) forall .Thematrix[ M ]ispositivedeniteprovidedthattheproductinequation (3.24)isequaltozeroonlywhen 0.Someintuitiveobservationsaboutequation (3.23)showwhyitisapopularchoiceinapplications.Theov erallnitedimensional optimalrowcontrolproblemisthenthefollowing:Findthec ontrol u thatminimizes J ( u ) J ( u )= J X ( u ) ;u (3.25)

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71 overallchoicesofadmissiblecontrol u ,where X ( u )isconstrainedtobethesolution oftheevolutionEquation(3.21).Since[ Q ]ispositivesemidenite,and R ispositive, thefunction J ( u )isnon-negative.Tominimize J ( u ),ittendstomakeeachofthenonnegativetermsin(3.23)smaller.Theresultisthatittends toreducethemagnitude ofthestates( X )andcontrol u appearinginthefunctional(3.23),i.e.,ittriesto keepstates X trackingthetargetstateswithacceptablecontrolcost.T herelative magnitudesofthestates( X )andcontrols u canbemodulatedbychangingthe valuesoftheweightmatrix[ Q ]andscalar R Insummary,then,theoptimalproblemcanbestatedasfollow s: Find u 2 C ad [0 ;T ]; U thesetofadmissiblecontrolfunctions thatminimizes J ( u ) 4 = J X ( u ) ;u 4 = 1 2 Z T 0 ( X ( ) ( )) T [ Q ]( X ( ) ( )) d + 1 2 Z T 0 Ru 2 ( ) d subjecttotheconstraint X ( t )=[ A ] X ( t )+ Bu ( t )+ F ( t ) X (0)= X 0 3.3.2 Optimality Systems and the Classical Riccati Equation Nowderivetheoptimalityconditionsforthiscontrolprobl embyintroducingthe Hamiltonian H 1 2 X ( t ) ( t ) T [ Q ] X ( t ) ( t ) + 1 2 Ru 2 ( t )+ p T ( t ) [ A ] X ( t )+ Bu ( t )+ F ( t )

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72 Theoptimalitycondition @ H @u =0yieldscontrollaw u ( t )= R 1 B T p ( t )(3.26) Thecondition X ( t )= @ H @p yieldsthestateequation X ( t )=[ A ] X ( t ) BR 1 B T p ( t )+ F ( t ) X (0)= X 0 Thecondition_ p ( t )= @ H T @X yieldstheco-stateequation p ( t )= [ A ] T p ( t ) [ Q ] X ( t ) ( t ) p ( T )=0 Itturnsoutthatthestates X ( t )andtheco-state p ( t )canberelatedbytheequation (referto[4]) p ( t ) 4 =[ S ( t )] X ( t )+ ( t )(3.27) SubstitutingEquation(3.27)intothestateequationgives X ( t )= [ A ] BR 1 B T [ S ( t )] X ( t ) BR 1 B T ( t )+ F ( t ) X (0)= X 0 (3.28) Combiningequations(3.27),(3.28),andco-stateequation gives X ( t )+[ S ( t )][ A ] X ( t ) [ S ( t )] BR 1 B T ([ S ( t )] X ( t )+ ( t ))+[ S ( t )] F ( t )+ ( t ) = [ A ] T ([ S ( t )] X ( t )+ ( t )) X ( t )+( t ) Pre-multiplytheaboveequationby X T ( t ).Collecttermsandsetterms X T ( t )( ) X ( t )= 0and X T ( t )( )=0.Itresultsin [ S ( t )]= [ S ( t )][ A ] [ A ] T [ S ( t )]+[ S ( t )] BR 1 B T [ S ( t )] [ Q ](3.29)

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73 ( t )= [ A ] BR 1 B T [ S ( t )] T ( t ) [ S ( t )] F ( t )+[ Q ]( t )(3.30) Equation(3.29)iscalleddierentialRiccatiequation(DR E).Itssolutionisnamed gainmatrix.Sincetheonlyconditionisat t = T ,ithastobesolvedbackwardin time.Itcanbeveriedthatthegainmatrixisareal,symmetr ic,andpositivedenite squarematrix.Noticethatthesolution[ S ( t )]isindependentofthedesiredoutput ( t ).Thismeansthatthegainmatrixiscompletelyspeciedonc ethesystem,the cost,andtheterminal T arespecied.Nowthecontrollawis u ( t )= R 1 B ([ S ( t )] X ( t )+ ( t ))(3.31) Thegainmatrix[ S ( t )]inthecontrolproblemhasthepropertythatas T !1 [ S ( t )]ofthedierentialRiccatiequationtendstotheconstant positivedenitematrix [ ^ S ].Thematrix[ ^ S ]isthesolutionofthealgebraicRiccatiequation(ARE) [ ^ S ][ A ] [ A ] T [ ^ S ]+[ ^ S ] BR 1 B T [ ^ S ] Q =0(3.32) Anusefulinterpretationofthisisthatas T !1 ,thesolutionofdierentialRiccati equation,[ S ( t )]withinitialcondition[ S ( T )]=0,goesbackwardintimetothe steady-statesolution[ ^ S ].Thecontrollawisapproximatedbythelinearcombination oftheAREsolutionandvector ( t ) u ( t )= R 1 B ([ ^ S ] X ( t )+ ( t ))(3.33) Noticethatthecoecientmatrixof X ( t )terminequation(3.30)isthenegative transposeofthecoecientmatrixof ( t )inequation(3.28).Theeigenvaluesofthe coecientmatrixforonesystemarethenegativeoftheother .Thisindicatesthatif thesystemequation(stateequation)isstablethenthesolu tionfor ( t )isunstableif botharesolvedforwardintime.Sincetheinitialcondition forequation(3.30)isnot

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74 known,butratherthenalcondition,itcannotbesolveddir ectlywiththecontrol law.Insteadtheequationwillbeinvestigatedtointegrate backwardintime 3.3.3 Numerical Simulations Forthisexample,thetargetstates( t )arechosentobethereducedorder solutionoftheStokesdrivencavityrowsasshowninFigure2 .5withinitialcondition ( t =0)=0,boundarycondition h 0 ( t )=sin(2 t ),anddrivingfunction r ( t )= 2 cos(2 t )inEquation(3.22).Thetrackingproblemistoseekoptimal control u ( t ) suchthatthereducedorderstatessolution X ( t )getsascloseaspossibletothetarget solution( t ).Ithasbeenshownthatthistypeoftrackingproblemcanbes olved usingequations(3.29),(3.30),(3.31),andstateequation Inthisexample,thesystemisdiscretizedwithsamplingtim einterval T s =0 : 01 s Bydoingthis, ( t )anddierentialRiccatiequationscanbesolvedbackwardi n timeeasilyandthedataarestoredeasilyaswell.Thecontro llawiscalculated basedonthestoreddata.Theoptimalstatetrajectoryissol vedforwardintime. T =5 s issimulated.Figure3.1showstheoptimaltrajectoryof X ( t )with[ Q ]= diag (1000 ; 1000 ; 1000)and R =1.Itcanbeobservedfromthegurethattheoptimal solution X ( t )nearlycoincideswiththetargetsolution( t ).Thestatessolution X ( t ) isfollowingthetargetandthetrackingtaskisthussuccess ful.Figure3.2shows thecontroltrajectoryapplied.Theguresillustratestha tthecontrolamplitudeis approximatelyafactorof6timesthedisturbanceamplitude ThedierentialRiccatisolution[ S ( t )]ingure3.3.Thedottedlinesarethe correspondingalgebraicRiccatisolutions.Theguretell susthatwhen t T ,[ S ( t )] behaveslikeaconstantsteady-statesolution.Thisisnume ricalevidencethatas T !1 ,gainmatrix[ S ( t )]tendstotheconstantalgebraicRiccatiequationsolutio n [ ^ S ].Figure3.4istheplotofthetimehistoryofthevector ( t ).

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75 Figure3.1:Statessolutionforthetrackingproblem Figure3.2:Controlhistoryforthetrackingproblem

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76 Figure3.3:Riccatisolutionforthetrackingproblem Figure3.4:Historyof ( t )forthetrackingproblem

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77 3.4 Direct Discretization Method Theoptimalcontrolproblemofdrivencavityrowcanbesolve dviamethodology ofcalculusofvariations.Thismethodinvolvesastandardm athematicalframework. Itiswidelyappliedbecausetheframeworkcanworkformanyd ierentpractical problems.Thissectionwillintroduceasimpler,directmet hodtorealizethesolution ofsameoptimaltrackingcavityproblem. 3.4.1 Method Description Considerthecavitytrackingproblemasdescribedinlastse ction.Recallthe cavityhastheboundaryconditionatthetopof h 0 ( t ).Itseekstondthecontrol u ( t )tominimize J ( u ) 1 2 Z T 0 rr X ( t ) ( t ) rr 2R N dt + 1 2 Z T 0 u ( t ) 2 dt subjecttotheconstraint X ( t )=[ A ] X ( t )+ Bu ( t )+ F ( t ) X (0)= X 0 Thetrackingconstraintembodiedin J ( u )iscastintermsofthetargetsolution( t ) oftheequation ( t )=[ A ]( t )+ Br ( t )+ F ( t )(3.34) (0)= 0 where r ( t )issomegivenfunctionoftimeandtheinitialcondition 0 isgiven.Introduceapartition f t 0 ;t 1 ;:::;t N 1 g

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78 ofthetimeinterval[0 ;T ]forthesolutionofEquation(3.34).Forsimplicityinthe discussion,assumethatthediscretizationisuniformwith t = t i +1 t i i =0 :::N 2 Usinganystandardintegratorforordinarydierentialequ ations,itisatrivialmatter toobtaintheapproximation f k g N 1 k =0 f ( t k ) g N 1 k =0 oftheordinarydierentialequation(3.34). Similarly,itispossibletoobtainadiscretizedapproxima tionofthecostfunction J ( u ).Introduceanapproximationofthecontrolfunction u ( t )= N u 1 X k =0 N k ( t ) u k (3.35) wherethefunctions f N k ( t ) g N u 1 k =0 areknownshapefunctions(b-splines,niteelements, ::: ,etc)andthecoecients f u k g N u 1 k =0 areunknown.Anapproximate,discretizedcostfunctioncanbewrittenas J ( f u k g ) 1 2 t N 1 X k =0 k X ( t k ; f u k g ) ( t k ) k 2R N + 1 2 t N 1 X k =0 u 2k inthisequation, f u k g N u 1 k =0 isdenotedas f u k g fornotationalsimplicity. Nowtheproblemistond f u k g thatminimizes J ( f u k g ).Thisisasimple optimizationproblem.Theremanychoicesofalgorithmsfor searchingfortheminimumof J ( f u k g )andtheoptimalcoecients f u k g .Someexamplesofthesealgorithmsarethesimplexsearchmethod,trustregionmethod,p reconditionedconjugate gradientsmethod,etc.ThesehavebeenincorporatedintheM atlabfunctionslike fminsearch;fminunc etc.Oncetheoptimalcoecients f u k g areobtained,the optimalcontrol u ( t )canbeapproximatedbyEquation(3.35).

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79 3.4.2 Numerical Simulations TheresultsofaspecicexamplearedepictedinFigures3.5t hrough3.8.Inall ofthesesimulations,thefollowingselectionshold r ( t ) Acos (n t ) A =2 n=2 Thetopboundaryconditionis h 0 ( t )=sin(2 t )forboththetargetrowandthe controlledcavityrow.Thetargetrowinthetrackingfunct ion J ( u )isdepicted inFigures3.5.Inthetargetrowapproximation, N =31wasselectedinthetime integrationof(3.34)with t = 1 120 and T =0 : 25.Figure3.6depictsthecontrolled rowthatisobtainedwith N =11and N u =4.Figure3.7depictsthecontrolled rowatthesameinstancesintimeforthechoice N =11and N u =8.Figure3.8 depictsthecontrolledrowwith N =31and N u =8.Clearly,anerpartitionof timeperiodwillgivebettertrackingresults.Thecontroll edvelocityeldsarecloser tothetargetrowelds.Similarresultscanbeobservedinth econtroldiscretization. Theresultofanertimegridandcontroldiscretizationcan beseenmoreclearlyin Figure3.9

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80 t =0 : 025 t =0 : 2 t =0 : 1 t =0 : 25 Figure3.5:Targetrowfor N =31

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81 t =0 : 025 t =0 : 2 t =0 : 1 t =0 : 25 Figure3.6:Controlledrowfor N =11 ;N u =4

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82 t =0 : 025 t =0 : 2 t =0 : 1 t =0 : 25 Figure3.7:Controlledrowfor N =11 ;N u =8

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83 t =0 : 025 t =0 : 2 t =0 : 1 t =0 : 25 Figure3.8:Controlledrowfor N =31 ;N u =8

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84 Figure3.9:Solutionsof ( t )vstime

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85 3.5 Receding Horizon Control (RHC) Recedinghorizoncontrol(RHC),alsoknownasmodelpredict ivecontrol,isa techniquetoobtain,foreachtimestep,theoptimalcontrol lawwithinashorthorizoncomparedtowholetimeperiod.Itarisesbecausemanytra ditionalmethods cannotsolveproblemswithnonlinearplantsandconstraint sonstateandcontrol variables.Also,thesolutionofthetwopointboundaryvalu eproblem(TPBVP)in section3.3coststoomuchtimeifterminaltime T islarge.Basically,RHCdivides thewholetimeperiodintosmallintervals.FortheTPBVP,in steadofsolvingfor entiretimerange[0 ;T ],itsolvesaseriesofTPBVP'sinmuchsmallertimeinterval s [0 ;t 1 ] ; [ t 1 ;t 2 ] ;:::; [ t n 1 ;T ]where t 1 T;t 2 t 1 T ,etc.Thiscanbeafasterprocedure.Inaddition,thiscanyieldafeedbackcontrolforasu cientlynepartitionof timewhiletheprocedureintheearliersectionsyieldssolu tionsforopen-loopoptimal controlproblems. 3.5.1 Receding Horizon Formulation Considerthesametrackingproblemasdescribedinlastsect ionwhichissolved viathedirectdiscretizationmethod.Thetargetsequence f k g N 1 k =0 isgiven.Introducethesameapproximationofcontrolfunction u ( t )= P N u 1 k =0 N k ( t ) u k .TheTPBVP canbedescribedas Findvector ( X ( t j ; f u k g ) ; f u k g N 1 k =0 ) thatminimizes J ( f u k g N u 1 k =0 )= 1 2 t N X j =1 k X ( t j ; f u k g ) j k 2R N + 1 2 t N u 1 X k =0 u 2k subjectto X ( t )= AX ( t )+ Bu ( t )+ F ( t ) X (0)=0 Arecedinghorizonimplementationistypicallyformulated byintroducingthe followingoptimizationproblem.First,dividethewholeti meperiod[0 ;T ]intomuch

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86 smallerintervals[0 ;t p 1 ] ; [ t 1 ;t p ] ;:::; [ t N p ;T ]where t = t 1 and p denotesthe lengthofthepredictionhorizon.Denote m asthelengthofthecontrolhorizon. Thusin i th ( i =1 ; 2 ;:::;N p +1)timeinterval,asub-optimalcontrolproblem canbedescribedas Findvector ( X ( i ) t ( i ) j ; f u ( i ) k g ; f u ( i ) k g m 1 k =0 ) thatminimizes J ( i ) ( p;m ) = J ( i 1) ( p;m ) + 1 2 t p X j =1 k X ( i ) t ( i ) j ; f u ( i ) k g ( i ) j k 2R N + 1 2 t m 1 X j =0 u ( i ) j 2 subjectto X ( i ) ( t )= AX ( i ) ( t )+ Bu ( i ) ( t )+ F ( i ) ( t ) X ( i ) (0)= X ( i 1) ( t ) Thislocalsub-optimalcontrolproblemcanbesolvedbythed irectdiscretizationmethoddescribedbefore.Let u ( i ) j ;j =0 ;:::;m 1betheminimizingcontrolsequenceforthelocalsub-optimalproblemin i th timeinterval,and X ( i ) j ;j = 0 ;:::;p 1bethelocaloptimalstatesolutions.Thentheglobaloptim alstatesolution X willbeupdatedbythelocaloptimalsolutionsas X i 1+ j = X ( i ) j ;j =0 ;:::;p 1. Thepartoftheglobaloptimalcontrolsequence u thatfallsintothe i th timeinterval willbeupdatedbythelocalminimizingcontrolsequence u ( i ) j ;j =0 ;:::;m 1. 3.5.2 Numerical Simulations Inthissimulation,theentiretimeperiod[0 ; 5]isdiscretizedinto641discretetime steps,andthecontrolsequenceisdiscretizedinto201step s.Thesimulationsetsthe lengthofthepredictionhorizon p =21andthelengthofthecontrolhorizon m =7. Figure3.10showstheoptimalstatessolutionscomparedtot hetargetstates.Since thesolidlines(targetsolutions)anddottedlines(RHCopt imalstatessolutions)are soclosetoeachother,theoptimalRHCstatessolutionsareg oodapproximationsto thetargetsolutions.Figure3.11showstheresultantcontr olhistory.Withcontrol magnitudeofabout0.6,goodtrackingperformanceisachiev ed.

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87 0 1 2 3 4 5 -1 -0.5 0 0.5 1 1.5 TimeSolutions a 1,Target a 2,Target b Target a 1,RHC a 2,RHC b RHC Figure3.10:Optimalstatessolutionscomparingtothetarg etstates 0 1 2 3 4 5 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 TimeControl Figure3.11:RHCcontrolhistory

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CHAPTER4 ROBUSTCONTROLOFDRIVENCAVITYSTOKESFLOW Inchapter3,areducedordermodelforoptimalcontrolisstu died.Theobjective isachievedforatrackingproblemviadierentmethodologi es.However,thedynamic modelispreciseandxed,i.e.,everyparameterandsignala reideal,withoutuncertainties.Thiscannotbetrueinactualphysicalcontroldes ign.Theplantmodel mighthavevariationsduringoperation.Forcreepingrowco nsideredwithinadriven cavity,theparameter mightchangeinreal-timeduetothechangeofexcitation frequencyorthechangeofReynoldsnumber.ThechangeofRey noldsnumbercould becausedby,forexample,adensityandviscositychangewit htemperature.Besides, therearemanyotherissuestoconsiderforarealisticcontr ollerdesign.Forinstance, theactuationmightnotfollowexactlythecontrolcommand. Thuscommanderrors mustbetakenintoaccount.Theaccuracyofmeasurementandn oiseintroduced bysensorsmightbeconsideredalso.Thus,arobustcontroll erhastobesynthesizedforsuchphysicalmodel.Thecontrollershouldgiveex pectedperformancewhile ittoleratesuncertainties. H 1 and synthesiscanbeusedtoachievesuchrobust performance.Inthischapter,theapplicationofrobustcon troltothereducedorder modelofcavityrowisconsidered.Thedesignedcontrollers arerobusttosensornoise, actuationerror,andparametervariationof .Forthisentirechapter,adisturbance rejectionproblemisstudied. 4.1 H 1 -based Modeling and Control Therealprobleminrobustmultivariablefeedbackcontrols ystemdesignisto synthesizeacontrollawwhichmaintainssystemresponsean derrorsignalstowithin prespeciedtolerancesdespitetheeectsofuncertaintyo nthesystem.Therecently 88

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89 developed H 1 theoryprovidesadirect,reliableprocedureforsynthesiz ingacontroller whichoptimallysatisessingularvalueloopshapingspeci cations. 4.1.1 Singular Values, H 1 norm Since1970s,thesingularvalueBodePlothasemergedasause fulindicator ofmultivariablefeedbacksystemperformance.Thesingula rvaluethusjoinssuch previouslyusedmeasuresofmultivariablefeedbacksystem performanceasdominant polelocations,transitionzerosandRMS(rootmeansquare) errorofcontrolsignals. The singularvalues ofarankrmatrix A 2 C m n ,denoted i arethenon-negative square-rootsoftheeigenvaluesof A T A orderedsuchthat 1 2 n If r
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90 thenthebiggestgainformatrix A isgivenby = max u k y k 2 k u k 2 = max u k Au k 2 k u k 2 Wecanextendthesingularvaluetofrequency-varyingmodel s y = P ( s ) u The maximummodulustheory 1 statesthatifthetransferfunction P ( s )isstable (boundedandanalyticforall s intherighthalfplane),thenthemaximumgain occursforsome s = j .Thus,dene 1 asthebiggestgainfor P atanyfrequency 1 = max u k y k 2 k u k 2 = max max u k P ( j ) u k 2 k u k 2 = max ( P ( j )) Thisvalueisactuallythedenationofthe H 1 norm k P k 1 = max ( P ( j ))(4.1) Moreprecisely,considerafrequencydomainsignal, f ( ) 2 L 2 .Itsenergyis measuredby k f k 2 k f k 2 = 1 2 Z 1 1 f ( j ) f ( j ) d! 1 2 where L 2 isthespaceofsquareintegrablesignals, L 2 = f f ( j! ): k f k 2 < 1g .Ifthe systemgainisintroducedsuchthat y = Puu 2 L 2 ;y 2 L 2 1 Thetheorysaysthatif f ( s )isdenedandcontinuousonaclosedset S ,and analyticontheinteriorof S whichisanopenset,thenmaximumof j f ( s ) j isonthe boundaryof S

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91 Figure4.1:Closed-loopsystem thenthemaximumvalueoftheratioofsignalsisgivenby k P k 1 = sup ( P ( j )) Thespace H 1 oftransferfunctionmatricesisdenedas H 1 = f P :Pisanalyticandrationaland k P k 1 < 1g 4.1.2 H 1 Optimal Control Synthesis H 1 controlcanbeusedtodesignafeedbackcontrollawsuchthat expectedperformancecanbeachieved.Basicallyitisaminimizationpro blem,i.e.,tominimize thegainfrominputstooutputsandthesize( H 1 norm)ofoutputs.Forexample,a trackingproblemcandescribedtominimizethedierencebe tweenacommand(input)andameasurement(output).Adisturbancerejectionpr oblemcanbedescribed tominimizethemotionofaparameter(output)inresponseto disturbance(input). AstandardcongurationisshowninFigure4.1.Inthissyste m, G istheopen-loop dynamics, K isthecontrollertobedesigned, w isthevectorofexogenousinputs likedisturbanceandnoise, z isthevectoroferrorsorperformance, y isthevectorof measurementsusedforfeedbackintothecontroller, u isthevectorofcommandsignal fromthecontroller.Denotetheclosed-loopsystemas k F ( P;K ) k 1 .Theoptimal H 1 controlseekstominimize k F ( P;K ) k 1 overallstabilizingLTIcontrollers K ( s ).From thedenition,the H 1 normofaSISO(singleinputsingleoutput)systemcanbe foundbysearchingforthemaximumgainovertheentirebandw idth.ForMIMO (multipleinputmultipleoutput)systems,itismorecompli cated.Thecalculationof the H 1 normcanbecomputationallyexpensive.Alternatively,ama ximumvalue

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92 r isspeciedfortheclosed-loopRMSgainanda suboptimal H 1 controlproblem is statedas: Findastabilizingcontroller K ( s ) suchthat k F ( P;K ) k 1
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93 isobviouslyanidealizationbutthisexampledoesserveasa testbedtodemonstrate themethodology. Theobjectiveofrowcontrolistorejecttheeectsoftheexo genousdisturbance atthetopofthedrivencavity.Physically,therowcontrols eekstominimizethe horizontalcomponentoftheruidatasetofsensorlocations .Thesesensorsare locatedat19pointsevenlydistributedalongthehorizonta lcenterlineofthecavity. 4.2.1 Open-loop Model ThereducedordermodelinEquation(2.52)canberewrittena s 8><>: _ 9>=>; = 264 [ A 1 ( )][ A 2 ( )] 00 375 8><>: 9>=>; + 264 [ E 1 ( )][ E 2 ] 00 375 8><>: h h 9>=>; + 264 [ B 1 ] 1 375 (4.2) Considerthetime-domainequationfor = A 1 ( ) ( t )+ A 2 ( ) ( t )+ B 1 ( t )+ E 1 ( ) h ( t )+ E 2 h ( t ) Formulatetheequivalentfrequency-domainexpression s ( s )= A 1 ( ) ( s )+ A 2 ( ) ( s )+ sB 1 ( s )+ E 1 ( ) h ( s )+ sE 2 h ( s )

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94 Solvefor usingstate-spacemodels ( s )= sE 2 + E 1 ( ) sI A 1 ( ) h 0 ( s )+ sB 1 + A 2 ( ) sI A 1 ( ) ( s ) = 264 A 1 ( ) E 1 ( )+ A 1 ( ) E 2 I E 2 375 h ( s ) + 264 A 1 ( ) A 2 ( )+ A 1 ( ) B 1 I B 1 375 ( s ) = 264 A 1 ( ) E 1 ( ) A 2 ( ) I E 2 B 1 375 8><>: h ( s ) ( s ) 9>=>; = P ( s ) 8><>: h ( s ) ( s ) 9>=>; (4.3) Thisplantcanbeusedfor H 1 = control.Itisdepictedschematicallybelow. 4.2.2 Control Synthesis Controllersaredesignedfordisturbancerejectionusingt hemodelshownin Figure4.2.Thismodelcontainstheopen-loopdynamicsasde scribedbytheelementsof f P;V M ; ; g andtheweightingfunctionsusedforloopshapinggivenas f W P ;W n ;W K g .Themathematicalobjectiveofthecontroldesignistochoo sea K suchthattheclosed-looptransferfunctionfromdisturban cestoerrorshasaninduced normlessthanunity. Thelter W P servestonormalizethemeasurementofrowvelocityobserve d bythesensors.Thislterisessentiallyaloop-shapinglt erthatdenesthedesign specicationsinthefrequencydomain.Thevalueof W P ischosentorerect(the inverse)ofacceptablevelocitiesinresponsetounitydist urbances.Assuch, W P

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95 Figure4.2:Synthesismodel attemptsloopshapingsuchthatroughlyspeaking,theveloc itiesatlowfrequencies are\scaleddown"lessthan1 = 2000andathighfrequenciesare\scaleddown"less than1 = 20.Theactuallterisrealizedas W P = 20( s +1000) s +10 Additionalltersareusedtoconstrainthecontroldesign. Thelter W K provides aweightingontheactuationpenalty.Itindicatestheamoun tofcontrolactuation thatispossible.Inthecurrenttest,itischosenas W k = s +200 s +2000 .Thus,thecontrol surfacerateis,roughlyspeaking,scaledbyafactorof10at lowfrequencyand1 athighfrequency.Thelter W n indicatesthelevelofnoisethatisexpectedinthe measurements.Inthiscase, W n =0 : 0004.Sincethemagnitudeofhorizontalvelocity inaclosed-looptestisaround0.002,thiscorrespondstoad dabout2%orlessnoise tothemeasurement. ThecontrollerissynthesizedusingtheLMIControlToolbox inMatlab.The resultingclosed-loopnormvalueis1.06.Thisvalueindica tesallthedesignobjectives aresatised.

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96 4.2.3 Closed-loop Response Theeectivenessofthecontrollerisdemonstratedbyinves tigatingtheclosedlooppropertiesofthedrivencavity.Theinitialtestofthe sepropertiesisperformed byconsideringthesystematxedvaluesoftheparameter .Inthiscase,theopenloopdynamicsandthecontrollersarecomputedfortheparti cularvaluesof f 1 : 0 ; 10 g for 4.2.3.1 =1 Thecontrollerwasdesignedtominimizetheinducednormoft hesynthesismodel withthisxedparametervalue.Figure4.3showstheclosedlooptransferfunctions fromexogenousinput, h ,toperformanceerrors, e 1 ,forthesynthesismodelinFigure4.2.Thetransferfunctionsinthegureindicatethatth econtrollerisindeed providingdisturbancerejection.Thevelocitiesathighfr equencyarerelativelyunaectedbythecontrollerbutthevelocitiesarelowfrequen ciesareattenuatedbya factorof1200. Thetime-domainpropertiesoftheclosed-loopsystemarein vestigatedusinga nonlinearsimulation.Themeasurementsofhorizontalvelo cityatthesensorlocations afterthenon-dimensionaltimeis0.2areshowninFigure4.4 .Clearlytheclosed-loop velocitiesaresignicantlylessthantheopen-loopveloci ties.Thecontrollerachieves disturbancerejection. Horizontalvelocityisfunctionoftimeduetotimechanging disturbance.Since thisvelocityismeasuredat19sensorlocationssimultaneo usly,theplotsofvelocities shouldbe3-dimensional.Figure4.5and4.6showthetimehis toryofnon-dimensional velocitiesversusthex-positionsofsensorsalongthehori zontalcenterlineforopenloopandclosed-loopsystems,respectively.Theseplotsgi veclearvisualizationofthe eectivenessofthedesignedcontrollerindisturbancerej ection. Oneconsiderationforcontrolsynthesisisthatthecontrol actuationshouldbe withinphysicallymeaningfullimits.Figure4.7showsthat thecontrolsurfacerate

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97 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 -5 10 0 Log MagnitudeFrequency (radians/sec) 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 -200 -100 0 100 200 Phase (degrees)Frequency (radians/sec) Open-loopClosed-loop Figure4.3:Transferfunctionsat =1 0 0.2 0.4 0.6 0.8 1 -0.2 -0.15 -0.1 -0.05 0 0.05 PositionV x open-loopclosed-loop Figure4.4:Disturbancerejectioneectofthecontrollerd esinedat =1 ( t )islessthan1andachieves1200timesattenuationofveloci tywhenthedisturbanceisasinusoidalfunctionwithamplitudeof1.Thisperf ormanceiswithinthe designspecicationsatlowfrequency.

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98 0 0.5 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time x-Position Velocity Vx Figure4.5:Open-loopvelocityoutputathorizontalcenter -line 0 0.5 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time x-Position Velocity Vx Figure4.6:Closed-loopvelocityoutputathorizontalcent er-line Thefeasibilityofthedesignedcontrollercouldbeveried bytestingtheclosedloopsystemwiththefullorderplantmodel.Thesimulationw asdoneasshownin Figure4.18inwhich PF istheplantrepresentingthefullorderStokesrowandisK isthedesignedROMcontrollerinFigure4.2.Figure4.9show sthetransferfunctions fortheclosed-loopdepictedabove.Figure4.10showstheme asurementsofhorizontal velocityatthesensorlocationsafterthenon-dimensional timeis0.2inthesimulation.Severalthree-dimensionalplotsofnon-dimensional velocitiesversustimeand x-positionsofsensorsaredepictedinFigure4.11-4.12.Cl early,theROMcontroller workswellwiththefullorderplantmodelforachievingdist urbancerejection.

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99 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -10 0 10 a1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5 0 5 a2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 0 1 x 10 -3 a3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 0 1 Timeb Figure4.7:Statesvariablesandcontrolinput Figure4.8:Fullorderclosed-loopblockdiagramforpointc ontroller

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100 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 -4 10 -3 10 -2 10 -1 10 0 Log MagnitudeFrequency (radians/sec) 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 -600 -400 -200 0 200 Phase (degrees)Frequency (radians/sec) Open-loopClosed-loop Figure4.9:Transferfunctionsfortheclosed-loopsystem 0 0.2 0.4 0.6 0.8 1 -0.2 -0.15 -0.1 -0.05 0 0.05 PositionV x open-loopclosed-loop Figure4.10:Closed-loopvelocityoutputathorizontalcen ter-line

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101 0 0.5 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time x-Position Velocity Vx Figure4.11:Open-loopvelocityoutputathorizontalcente r-line 0 0.5 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time x-Position Velocity Vx Figure4.12:Closed-loopvelocityoutputathorizontalcen ter-line

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102 4.2.3.2 =10 Thecontrollerwasdesignedtominimizetheinducednormoft hesynthesismodel withxed =10.Figure4.13showstheclosed-looptransferfunctionsf romexogenousinput, h ,toperformanceerrors, e 1 ,forthesynthesismodelinFigure4.2.The transferfunctionsinthegureindicatethatthecontrolle risindeedprovidingdisturbancerejection.Thevelocitiesathighfrequencyarere lativelyunaectedbythe controller,butthevelocitiesarelowfrequenciesareatte nuatedbyafactorof1440. 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 -5 10 0 Log MagnitudeFrequency (radians/sec) 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 -200 -100 0 100 200 Phase (degrees)Frequency (radians/sec) Open-loopClosed-loop Figure4.13:Transferfunctionsat =10 Asinthecaseof =1,time-domainpropertiesoftheclosed-loopsystemare evaluatedusinganonlinearsimulation.Themeasurementso fhorizontalvelocity atthesensorlocationsafterthenon-dimensionaltimeiseq ualto0.2areshownin Figure4.14.Clearly,theclosed-loopvelocitiesaresigni cantlylessthantheopenloopvelocities.Thecontrollerachievesdisturbancereje ction.

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103 0 0.2 0.4 0.6 0.8 1 -0.2 -0.15 -0.1 -0.05 0 0.05 PositionV x open-loopclosed-loop Figure4.14:Disturbancerejectioneectofthecontroller designedat =10 Inthesameway,aplotofthenon-dimensionalvelocityhisto ryof3-dimensionsis giveninFigures4.15and4.16.Theseplotsgiveclearvisual izationoftheeectiveness ofthedesignedcontrollerfordisturbancerejection. 0 0.5 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time x-Position Velocity Vx Figure4.15:Open-loopvelocityoutputathorizontalcente r-line Itisnecessarytomakesurethatthecontrolactuationshoul dbewithinphysically meaningfullimit.Figure4.17showsthatcontrolsurfacera te ( t )islessthan1while obtaining1440timesattenuationofvelocitywhenthedistu rbanceisasinusoidal functionwithamplitudeof1.Thisiswithindesignspecica tionsatlowfrequency. ThedesignedROMcontrollerwasusedinthesystemasshownin Figure4.18to testfeasibility.Inthegure, PF istheplantrepresentingthefullorderStokesrow andKisthedesignedROMcontrollerinFigure4.2.Figure4.1 9showsthetransfer

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104 0 0.5 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time x-Position Velocity Vx Figure4.16:Closed-loopvelocityoutputathorizontalcen ter-line functionsfortheclosed-loopdepictedabove.Figure4.20s howsthemeasurementsof horizontalvelocityatthesensorlocationsafterthenon-d imensionaltimeequals0.2 duringthesimulation.Somethree-dimensionalplotsofnon -dimensionalvelocities versustimeandx-positionsofsensorsaredepictedinFigur e4.21-4.22.Clearly,the ROMcontrollerworkswellwiththefullorderplantmodelfor achievingdisturbance rejection.

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105 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -10 0 10 a1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5 0 5 a2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 0 1 x 10 -3 a3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 0 1 Timeb Figure4.17:Statesvariablesandcontrolinput Figure4.18:Fullorderclosed-loopblockdiagramforpoint controller

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106 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 -5 10 0 Log MagnitudeFrequency (radians/sec) 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 -600 -400 -200 0 200 Phase (degrees)Frequency (radians/sec) Open-loopClosed-loop Figure4.19:Transferfunctionsfortheclosed-loopsystem 0 0.2 0.4 0.6 0.8 1 -0.2 -0.15 -0.1 -0.05 0 0.05 PositionV x open-loopclosed-loop Figure4.20:Closed-loopvelocityoutputathorizontalcen ter-line

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107 0 0.5 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time x-Position Velocity Vx Figure4.21:Open-loopvelocityoutputathorizontalcente r-line 0 0.5 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time x-Position Velocity Vx Figure4.22:Closed-loopvelocityoutputathorizontalcen ter-line

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108 4.3 H 1 Robust Control H 1 optimalcontrolwastestedinlastsectionforthenominalpl antmodelat =1 ; 10.Inthesecases,itisassumedthattheplantmodelisperfe ctlyaccurateand allsignalsareideal.However,arealisticmodelcannotbes operfect.Forinstance, thevalueof mightchangeduringoperation,whichmaybecausedbytemper ature change.Thiseectwillfurtherinduceachangeinthedensit yandviscosityand consequentlyachangeintheReynoldsnumber.Thus,thecont rollerdesignshould consider variation.Moreover,theaccuracyoftheactuatorsandsens orsmustbe alsoconsideredinarealcontrolproblem.Uncertaintyisin troducedtodescribethose variationsinthedynamicmodelsandsignals. 4.3.1 Uncertainties Basically,therearetwotypesofuncertainties: parametricuncertainty stemsfromtheinaccurateknowledgeofphysicalparametersorfromvariationsoftheirvaluesduringoperation. Itisoftenrealand constantandassociatedwithaparametricequationofmotio n.Forexample, the valueinthedrivencavitymodelmightchangeduetoachangeo fworking conditionssuchastemperature,excitationfrequency,etc dynamicuncertainty duetoneglectedplantdynamics(Forinstance,high-freque ncy rexiblemodes,etc).Itisassociatedwithasignalandhasfr equencycontent andmagnitudephase. 4.3.2 Small Gain Theorem TheSmallGainTheoremisasimplebutgeneraltoolforcertif yingstability androbustnessofuncertainsystems,linearandnonlinear. TheSmallGainTheorem statesthataclosed-loopfeedbacksystemofisinternallys tableiftheloopgainof thoseoperatorsisstableandboundedbyunity.

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109 ConsiderthefeedbackinterconnectionfromFigure4.23,wh ere G andarearbitrarycausalsystemswith G; 2 H 1 : L 2 L 2 andthefeedbackinterconnection iswell-posed.Theorem4.3.1(SmallGainTheorem) If k G k 1 < 1 ,thefeedbackinterconnectionisstable. Figure4.23:BlockdiagramfortheSmallGainTheorem Robuststabilityofaplantmodelcanbedirectlyanalyzedby thesmallgain theoreminthepresenceofasetofperturbationswhich,basi cally,areusedtodescribe uncertaintyofdynamics.Usuallytheexactvalueisnotknow n,butisnormbounded. Denetheset ofnormboundedoperatorsdescribingtheseperturbationsw hich aecttheplant G throughafeedbackrelationship. = f : k k 1 g Bytheinequality k G k 1 k G k 1 k k 1 aconditionforrobuststabilityoftheclosed-loopsystemc anbestatedasfollows Lemma4.1 Theplant G isrobustlystabletothesetofuncertaintyperturbations whichenterthesystemasinFigure4.23with k k 1 forall 2 if k G k 1 < 1 Thispresentsasucient,butnotnecessary,conditionforr obuststability.

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110 4.3.3 Linear Fractional Transformation Intherobustcontrolliterature,itiscommontorepresents ystemswithuncertaintyasalinearfractionaltransformation(LFT)ofan ominalsystemanda structuredoperatorthatbelongstoasetthatdescribesthe natureandsizeofthe uncertainty.Theselinearfractionalrepresentationscan beregardedasgeneralizationsofstate-spacerealizationsandprovideauniedfram eworktomodelnotonly systemswithuncertaintybutlineartime-varyingsystems, linearparametervarying systems,spatiallyinvariantsystemsandmultidimensiona lsystems. Considerablockpartitionedmatrix G = 264 G 11 G 12 G 21 G 22 375 Foranoperator,theupperandlowerLFTof G aredenedas F u ( G; )= G 22 + G 21 ( I G 11 ) 1 G 12 F ` ( G; )= G 11 + G 12 ( I G 22 ) 1 G 21 respectively(Figure4.24-4.25) Figure4.24:Linearfractionaltransformation F u ( G; ) Figure4.25:Linearfractionaltransformation F ` ( G; )

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111 Figure4.26: H 1 robustcontrolproblem TheLFTisausefulframeworkforanalysisofcomplicatedsys temswithmany feedbackandseriesinterconnectionsofoperators.Themai npropertyofLFT'sthat willbeutilizedinthischapteristhat feedbackandseriesinterconnectionsofLFT's maybeformulatedasasingleLFT .Thus,controlsynthesisofanycomplicatedsystem canbesimpliedintoasingleLFTcontrolproblem. 4.3.4 H 1 Robust Control Synthesis StabilityanalysisbasedontheSmallGainTheoremgiveninT heorem4.3.1can beusedtoguaranteethattheLFTisstableandwell-posed.Th us,therobustcontrol problemwithuncertaintyissolvedifwecanndasolutionto thefollowing H 1 robustcontrolproblem [19]: Givenatransferfunctionmatrix P ( s ) F u ( G ( s ) ; ) asinFigure4.26 P ( s ):= 266664 A B 1 B 2 C 1 D 11 D 12 C 2 D 21 D 22 377775 ndastabilizingcontroller K ( s ) suchthattheclosed-looptransferfunctionfrom d to e F ` ( P;K ) ,isinternallystableandits H 1 -normislessthanorequaltoone: k F ` ( P;K ) k 1 1

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112 Figure4.27: H 1 -synthesisconsideringrobustperformance where F ` ( P;K )isgivenbythelinearfractionaltransformation F ` ( P;K )= P 11 ( s )+ P 12 ( s )( I K ( s )) 1 K ( s ) P 21 ( s ) Therearethreeissuesassociatedwith H 1 robustcontroldesign: Norminalperformance(NP): H 1 -synthesiswithnouncertainty,i.e.,designthe H 1 controllerbyignoringsignals z and w Robuststability(RS): H 1 -analysiswithonlyuncertainty,i.e.,ignorethesignal d e andthusignoreperformance. Robustperformance(RP): H 1 -synthesiswithuncertainty,i.e.,controldesign consideringperformanceandrobuststabilityinthepresen ceofuncertainty. Thewaytogetrobustperformanceistomakeanarticialoper ator p thatrelates thesignal e to d asshowninFigure4.27.Nowdesignacontroller K thatminimizes thegainfrom( w d )to( z e ).Thus,attempttomaximizerobuststabilitybymaking z small,andtomaximizethenominalperformancebymaking e small.Consequently, robustperformanceisachievedbymakingboth( z e )small. 4.3.5 Structured Uncertainties Inthe H 1 robustcontrolproblemabove,theuncertaintyisageneral ,full matrixwherealltheinputs( z )canaectalltheoutputs( w ).Theuncertaintyand theassociatedsignals( z and w )canberealorcomplexinthesmallgaintheorem.The

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113 smallgaintheoremdoesnotdistinguishbetweenrealandcom plexuncertainty.This makes H 1 controlsynthesistooconservative.Forthephysicalplant model,thesignal anduncertaintiesareallreal,andtheuncertaintymatrix mightbeinsomesimpler andspecicformat.Thus,eachoutputmightbeinruencedbyo nlypartofinput signals.AnotherreasonthatmakestheSmallGainTheoremco nservativeisthatthe conditiontoguaranteestabilizationisasucientconditi on.Theconservative H 1 theorylimitstheachievableperformancebecausethedesig nwillsacriceperformance togetbetterrobustness.Italsolimitstheoperationalenv elopeofsystemsbecause itisnecessarytoavoidconditionsatwhichdynamicsmightc hange. Tomakeimprovement,\structureduncertainty"isconsider ed.Thematrix willbeablockdiagrammatrixinwhichonlysomeinputsarere latedtosomeoutputs. Adiagonalmatrixisanexampleofthistypeofuncertainty.I nmathematicalform, asetofuncertaintyisdenedas = f = diag ( 1 I r 1 ; 2 I r 1 ;:::; s I r s ; 1 ; 2 ;:::; F ): i 2 C ; j 2 C m j m j ; j i j 1 ; k j k 1 ;i 2 [1 ;s ] ;j 2 [1 ;F ] ; k k 1 g (4.4) where s isascalar, i I r i 2 C r i r i arescalarblockswhosedimensionaregivenby r i F isthenumberoffullblocks,and 2 C m j m j havedimensiongivenby m j 4.3.6 Structured Singular Value: TheSmallGainTheoremguaranteesstabilityof P withrespecttotheset if k P k 1 < 1.Thisrobustnessconditionmaybeoverlyconservative,as notedearlier. Fortheuncertaintysetdenedabove,thestructuredsingul arvalue, ,isdenedas analternativemeasureofrobustness.Forthegiventransfe rfunctionmatrix P and associatednormboundedsetofuncertaintyoperators, isdenedas ( P )= 1 min 2 f (): det ( I M )=0 (4.5)

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114 Figure4.28:LFTsystemforrobuststabilityanalysisusing Theallowablesizeoftheuncertaintymatricesforwhichthe plantisrobustly stableisdeterminedbythevalueof giveninthefollowingtheorem[71] Theorem4.3.2 GiventhesyteminFigure4.28, P isrobustlystablewithrespectto theset whichisnormboundedbyrealscalarthat k k 1 forall 2 ifand onlyif ( ( P ) < 1 Usually P isinternallyweightedsuchthat k k 1 1forall 2 .Thus,the robuststabilityconditionbecomes k k 1 1forall 2 ifandonlyif ( P ) < 1. Obviously, dependsontheblockstructureoftheset.Itisameasureof uncertaintysize,notanorm.Asaconsequence synthesisislessconservativethan the H 1 controlsynthesis.Unfortunately, isveryhardtocompute.Instead,the upperandlowerboundsarecalculated. 4.3.7 Control Design with -Synthesis The -synthesisapproachdesignsanoptimalcontroller, K ,tominimizethe structuredsingularvalue, .Thereareactuallytwoproblemsinvolvedinthisprocess. Theseproblemsarethecomputationofthecontrollertostab ilizetheclosed-loop systemandminimizethe H 1 -norm,andcomputationofthe boundsforthedesigned controller.Thougheachofthesetwoproblemsisaconvexpro blem,theoverall -synthesisproblemisnon-convex.Theredoesnotexistagen eralformulationto computetheoptimal synthesiscontroller. TheD-Kiterationprocedurein -Toolsgivesanapproximationtothe -synthesis controldesign.Itgeneratesasequenceofminimizations, rstoverthecontroller

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115 variableK(holdingDwhichisassociatedwiththescaledupp erboundxed),and thenovertheDvariable(holdingthecontrollerKxed).The D-Kiterationprocedure isnotguaranteedtoconvergetotheminimumvalue,butoften workswellinpractice. 4.4 H 1 Robust Control of Driven Cavity Flow Nowthetheorydiscussedinlastsectionwillbeusedtoderiv ean H 1 = robust controllerforadisturbancerejectionproblemfordrivenc avityrow.Thisstudy willincludetheparametricuncertainty inthemodel.Dynamicaluncertainties associatedwithactuationandsensormeasurementsarelike wiseconsidered. 4.4.1 Uncertainty Model ThediscussioninChapter2suggeststhatthereduced-order modelisafairly accuraterepresentationoftheopen-loopdynamicsforStok esrowinthedrivencavity with O (1).Theaccuracysignicantlydecreasesas changesfromunitybecause ofunmodelednonlinearities.Theparametricuncertaintya ssociatedwith should fallwithinthevalidrangeofthereducedordermodel.Itiss howninChapter2that when 2 [0 : 5 ; 1 ],thereducedordermodelisagoodapproximationofthecree ping rowinthedrivencavity.Inthecurrenttest,asmallrangeof variationsof ,say [0 : 5 ; 1 : 5]isconsidered.

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116 Taketheplantmodelat =1 0 asthenominalplantmodelfor synthesis whichisgivenby P 0 264 A 10 E 10 A 20 C 1 E 2 B 1 375 = 2666666666666664 63 : 82142 : 269 0 : 0056534 0 : 31981 0 : 31981 42 : 273 212 : 940 : 019249 0 : 201730 : 20173 0 : 00732330 : 03136 9515 : 5 0 : 000735260 : 0011442 100 2 : 69022 : 6902 010 1 : 6952 1 : 6952 001 0 : 004159 0 : 0035298 3777777777777775 Thisequationdenesalinearmodelwithinput h 0 ; 2 R ,andoutput 2 R 3 1 Thenumberofstatesis N s =3inthismodel.Consider assubjecttoparametric uncertainty,i.e., = 0 + W 2 [0 : 5 ; 1 : 5].Thismeansthat50%variationin parameter isallowed.Choose W =0 : 5,andthus j j 1.Previousmathematical modelingindicatesthatthematrices A 1 ( ) ;A 2 ( ) ;E 1 ( )inthetruthmodel P are linearlydependenton ,i.e. A 1 ( )=( 0 + W ) A 10 A 2 ( )=( 0 + W ) A 20 E 1 ( )=( 0 + W ) E 10 Nowthestateequationsbecome ( t )= 0 A 10 + 0 E 10 h ( t )+ 0 A 20 ( t )+ W A 10 + W E 10 h ( t )+ W A 20 ( t )

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117 Denote z = W A 10 + W E 10 h ( t )+ W A 20 ( t ) w = z = 266664 00 0 0 00 377775 z where z 2 R 3 1 ;w 2 R 3 1 .Theuncertaintyplantmodelis 8><>: z 9>=>; = ^ P 8>>>><>>>>: w h 9>>>>=>>>>; with ^ P = 266664 A 10 I 3 3 E 10 A 20 W A 10 ; W E 10 W A 20 C 1 ; E 2 B 1 377775 = 26666666666666666666666664 63 : 82142 : 269 0 : 0056534 1000 : 31981 0 : 31981 42 : 273 212 : 940 : 019249 010 0 : 201730 : 20173 0 : 00732330 : 03136 9515 : 5 0010 : 000735260 : 0011442 31 : 91121 : 134 0 : 0028267 0000 : 15991 0 : 15991 21 : 137 106 : 470 : 0096244 000 0 : 100870 : 10087 0 : 00366170 : 01568 4757 : 7 0000 : 000367630 : 00057209 100 000 2 : 69022 : 6902 010 0001 : 6952 1 : 6952 001 000 0 : 004159 0 : 0035298 37777777777777777777777775 Besides variation,physicalaccuracyofcontrolactionmustbecons idered.Assumethatthereexist 5%errorsintheactuation.Also,inreality,sensoroutputi s notexactlyidenticaltothetruevelocity.Areasonableerr orinthesensormeasurementsmightbe 2%.Also,noisewillbeintroducedinmeasurements.Sinceth ere

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118 areasmanyas19sensors,itisnumericallycostlytoconside rallsensorerrorsinthe controldesign.Instead,noisewithsamemagnitudelevelwi llbeintroducedtoaccountforthesesensorerrors.Previousopen-loopsimulati onshowsthatthemaximum magnitudeofnondimensionalvelocitiesalongthehorizont alcenterlineisabout0.2. Considernoiseassociatedwitheachsensorthathasamagnit udeof2% 0 : 2=0 : 004. 4.4.2 Control Objectives Insummary,theobjectistodesignacontrollertoachieveth efollowinggoals: Rejecttheeectsoftheexogenousdisturbanceatthetopoft hedrivencavity. Physically,attempttominimizethehorizontalcomponento ftheruidatthe sensors. Limittheactuationamplitudeandfrequency.Aphysicalact uatorusuallyprovideslargecommand|controlsurfacerate ( t )incurrenttestcase|atlow frequencyandsmallcommandathighfrequency.Accordingto thediscussion inChapter2,itisreasonabletoconsidertheactuatorworki ngatfrequenct below200 Hz .Thus { maximumcontrolsurfacerateischosentobe10persecondatl owfrequency( < 200 Hz ) { maximumcontrolsurfacerateischosentobe1persecondathi ghfrequency( > 2000 Hz ) Accountfornoiseanddisturbance { errorsintheactuationof 5% { noiseinmeasurementof0.004 4.4.3 Controller Synthesis 4.4.3.1 Synthesis Model ThesynthesismodelforcontrollerdesignisshowninFigure 4.29and4.30. Thismodelincludestheopen-loopdynamicsasdeterminedby f P;V 0 ; ; g andthe

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119 weightingfunctionsusedforloopshaping f W P ;W n ;W K g .Theuncertaintieshaving Figure4.29:Synthesismodel thestructure 264 ; ; a 375 ,where j a j 1,accountsfortheuncertaintyassociatedwith actuation.Theweightsfunctionsaresetasfollows: Theperformanceweight W P = 10( s +2000) s +200 hasagainof100atlowfrequencyand againof10athighfrequency.Sincesignal e 1 isnormalizedto1,onegoalisto minimize y 0 tobeboundedby0.01atlowfrequencyand0.1athighfrequenc y. Betterrejectionisexpectedatlowfrequency. Theactuationpenalty W k = s +200 s +2000 hasagainof0.1atlowfrequencyanda gainof1athighfrequency.Sincesignal e 2 isnormalizedto1,thisweightlimits thecontrolsurfacerate u ( t )to10atlowfrequencyandto1athighfrequency. Thenoiseweightis W n =0 : 004.Thissuggeststhattheactualnoiseaddedto themeasurementsignalsshouldbelessthan0.004.

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120 Figure4.30:Extractedsynthesismodel 4.4.3.2 Controller Realization An H 1 controller K isdesignedforthesynthesismodelabove.Balancedtruncationisusedtoreducethestatesof K from23statesto6states.Thismethod truncatestheoriginalstate-spacemodeltoretainallHank elsingularvaluesgreater thansometolerancevalue.Thetransferfunctionsoftheori ginalcontrollerandthe reducedcontrollersareshowninFigure4.31and4.32. 4.4.4 Robustness Analysis Theclosed-loopsystemusingthebalancedtruncatedcontro llerisshowninFigure4.33. Figure4.34depictstherobuststability(RS),nominalperf ormance(NP),and robustperformance(RP)oftheclosed-loopsystem.Theresu ltsaresummarizedin table4.1 Table4.1:Controllerperformance RSNPRP 0.50.9650351.01323 H 1 -norm0.50.9821891.4152

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121 10-4 10-3 10-2 10-1 100 101 102 103 104 100 101 102 103 Log MagnitudeFrequency (radians/sec) 10-4 10-3 10-2 10-1 100 101 102 103 104 -60 -50 -40 -30 -20 -10 0 Phase (degrees)Frequency (radians/sec) Figure4.31:Original H 1 controller Both valueandthe H 1 -normofrobuststabilityare0.5,whichtellsusthatthe closed-loopsystemwiththedesignedKisrobustlystableto a50%variationinthe parameter ,5%variationintheactuation,andapproximately2%sensor error/noise. The valueand H 1 -normofnominalperformancearelessthan1.Thismeansthat thecontrollerissucienttoachieveexpectedperformance whentheuncertainties areignored.The valueofrobustperformanceisverycloseto1whichindicate s thatthecontrollercanhelptoachievedesiredperformance fortheplantmodelwith uncertainties.Noticethatthe valueofisless H 1 -norm.Thisisavericationthat analysiswithscalingmakesimprovementinindicatingsyst emperformancewith compareto H 1 analysis.Thereasonisbecauseweconsiderthestructureof the uncertaintiesin analysis,i.e.= 264 I 3 3 ; ; a 375 ,insteadfull4 4blockin H 1 analysis.

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122 10-4 10-3 10-2 10-1 100 101 102 103 104 100 101 102 103 Log MagnitudeFrequency (radians/sec) 10-4 10-3 10-2 10-1 100 101 102 103 104 -60 -50 -40 -30 -20 -10 0 Phase (degrees)Frequency (radians/sec) Figure4.32:Balancedtruncatedcontroller Noticethat plotsareallrat( 0 : 5)atlowfrequency.Sorobustnessand performancearewellbalancedatlowfrequency.Thepeakval uesaretakennear 2000Hz,thisbecauseouractuationratehasaupperlimitof1 0at f< 200 Hz buta upperlimitof1at f> 2000 Hz Sincetheplotsareratandbalanced,D-Kiterationwillnoth elpheretoimprove performance.Thesimulationresultsshowtheproofofthisb ecausepeakvaluefor eachof3iterationsis1.014.Nevertheless,wewilldiscuss -synthesisresultsbyD-K iterationinthefollowing. Thegainmarginsandphasemarginsfor H i; 1 ;i =1 ; 2 ;:::; 19areshownintable 4.2where H i; 1 meansthetransferfunctionfromrstinput, h ,to i th output, e 1 ( i ). Theleastgainmarginis6.0206dBandallphasemarginare 1 .Soourclosed-loop systemusingdesignedcontrollerisstable.

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123 Figure4.33:Closed-loopsystemwithsynthesismodel 10-3 10-2 10-1 100 101 102 103 104 0 0.5 1 1.5 2 2.5 3 3.5 4 Frequency (rad/sec)Mu nominal performancerobust stabilityrobust performance Figure4.34:Controllerperformance 4.4.5 Closed-loop Simulation Nowwetestourcontrollerbytimeresponsefortheclosed-lo opsystemasshown inFigure4.35.TheplantmodelGcouldbenominalplantmodel ortruthmodel. Weconsidertwocases:(1) =0 : 5;(2) =1 : 5.Inbothcases,actuation variationis 5%,andthedisturbance h ( t )= sin (2 t ). 4.4.5.1 Closed-loop Simulation for =0 : 5 Weusenominalplantat 0 =1withuncertainty = 1.Thenactual = 0 + W =0 : 5.Theresultscomparedwiththetruthmodelat =0 : 5.Figure4.36

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124 Table4.2:Gainandphasemarginoftheclosed-looptransfer functions igainmarginphasemargin 16.0206Inf2319.36Inf3128.5Inf4236.16Inf5179.43Inf6162.04Inf7170.15Inf8156.5Inf9166.28Inf 10153.75Inf11164.26Inf12152.35Inf13163.36Inf14151.92Inf15163.36Inf16152.35Inf17164.26Inf18153.75Inf19166.28Inf showsthatthetransferfunctionswithnominalplanttested at5 th sensorlocation almostcoincidewiththetransferfunctionswiththetruthm odel.Figure4.37plots thehorizontalcomponentofvelocitiesatall19sensorloca tion(thex-axisofthe guregivesx-positionsofeachsensor)attimeinstance t =0 : 2.Itshowsthatthe designedcontrollerisecientinminimizingthevelocitie sforbothnominalplantand truthplant.Andtherejectioneectsarenearlysameforbot hplants. Sameconclusionscanbemadewhenweplotouttimeresponsesf orentiresimulatedtimeperiodandforallmeasurementlocations(Figur e4.38-4.39).Numerical resultsshowthatthedesignedcontrollercandecreasethev elocitiesfrom0.18(openloop)to0.001(closed-loop)fornominalplantwithuncerta intieswhilefrom0.18(openloop)to0.00096(closed-loop)fortruthplantmodel.

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125 Figure4.35:Closed-loopsystemfortimeresponse 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 FrequencyTransfer Function |H5| Norminal model, Open-loopNorminal model, Closed-loopTruth model, Open-loopTruth model, Closed-loop Figure4.36:Transferfunctionat5 th sensorlocation 4.4.5.2 Closed-loop Simulation for =1 : 5 Weusenominalplantat 0 =1withuncertainty =1.Then = 0 + W =1 : 5.Theresultscomparedwiththetruthmodelat =1 : 5.Figure4.40 showsthatthetransferfunctionswithnominalplanttested at5 th sensorlocationare almostcoincidentwiththetruthmodel.Figure4.41plotthe horizontalcomponent ofvelocitiesatall19sensorlocation(thex-axisofthegu regivesx-positionsofeach sensor)attimeinstance t =0 : 2.Itshowsthatthedesignedcontrollerisecientin

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126 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 PositionVi,x Norminal model, Open-loopNorminal model, Closed-loopTruth model, Open-loopTruth model, Closed-loop Figure4.37:Timeresponseat t =0 : 2 Open-loop 0 0.2 0.4 0.6 0.8 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time (s) Position Velocity Vx Closed-loop 0 0.2 0.4 0.6 0.8 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time (s) Position Velocity Vx Figure4.38:Timeresponsefornominalmodel minimizingthevelocitiesforbothnominalplantandtruthp lant.Andtherejection eectsarenearlysameforbothplants. Sameconclusionscanbemadewhenweplotouttimeresponsesf orentire simulatedtimeperiodandforallmeasurementlocations(Fi gure4.42-4.43).Numericalresultsshowthatthedesignedcontrollercandecre asethevelocitiesfrom 0.175(open-loop)to0.00102(closed-loop)fornominalpla ntwithuncertaintieswhile from0.18(open-loop)to0.00097(closed-loop)fortruthpl antmodel.

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127 Open-loop 0 0.2 0.4 0.6 0.8 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time (s) Position Velocity Vx Closed-loop 0 0.2 0.4 0.6 0.8 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time (s) Position Velocity Vx Figure4.39:Timeresponsefortruthmodel 10-3 10-2 10-1 100 101 102 103 10-5 10-4 10-3 10-2 10-1 100 101 FrequencyTransfer Function |H5| Norminal model, Open-loopNorminal model, Closed-loopTruth model, Open-loopTruth model, Closed-loop Figure4.40:Transferfunctionat5 th sensorlocation 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 PositionV i,x Norminal model, Open-loopNorminal model, Closed-loopTruth model, Open-loopTruth model, Closed-loop Figure4.41:Timeresponseat t =0 : 2

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128 Open-loop 0 0.2 0.4 0.6 0.8 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time (s) Position Velocity Vx Closed-loop 0 0.2 0.4 0.6 0.8 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time (s) Position Velocity Vx Figure4.42:Timeresponsefornominalmodel Open-loop 0 0.2 0.4 0.6 0.8 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time (s) Position Velocity Vx Closed-loop 0 0.2 0.4 0.6 0.8 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time (s) Position Velocity Vx Figure4.43:Timeresponsefortruthmodel

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129 4.5 LPV Control synthesistechniquecangivesatisfactoryresultsinmanya pplicationsinpresenceofuncertainties.However,sinceasingleLTIcontroll erissynthesizedbythis technique,itmaynotbefeasiblefortheslowlybutwidelypa rameter-varyingdynamicalmodels.Forthiskindofuncertainlineartime-varying (LTV)plants,available techniquesarerelativelyimmature.Gain-schedulingisaw idelyusedtechniquefor controllingcertainclassesofnonlinearorlineartime-va ryingsystems.Ratherthan seekingasinglerobustLTIcontrollerfortheentireoperat ingrange,classicalgain schedulingapproachconsistsindesigninganLTIcontrolle rforeachoperatingpoint andinterpolationofthecontrollergainsinbetweenthepoi nts.Althoughitseems toworkwellinpractice,thisheuristicdesignproceduredo esnottaketheparameter variationsintoaccount,soanyperformanceandrobustness guaranteedintheindividualoperatingregionsmightbelostinthetransitionreg ion.Alternately,linear parameter-varying(LPV)control,rstintroducedbyShamm ain[79]isasystematic approachtosolvethisproblem. 4.5.1 LPV System ConsidertheLPVplantswhosetimedependenceassumesthefo rm x ( t )= A ( ( t )) x ( t )+ B ( ( t )) u ( t )(4.6) y ( t )= C ( ( t )) x ( t )+ D ( ( t )) u ( t ) where ( t )isavectoroftime-varyingplantparameters(velocity,an gleofattack, stiness, ::: )whichcanbemeasuredinreal-timeand A ( ) ;B ( ) ;C ( ) ;D ( )arexed functionsof ( t ).SuchplantswillbereferredtoasLPVsystems.Thisisasim ple modelofsystemswhosedynamicalequationsdependonphysic alcoecientsthatvary

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130 duringoperation.Whenthesecoecientsundergolargevari ation,itisoftenimpossibletoachievehighperformanceovertheentireoperating rangewithasinglerobust LTIcontroller.LPVapproach,ontheotherhand,explicitly takesintoaccountthe relationshipbetweenreal-timeparametervariationsandp erformance.Thisenables controllerstobedesignedforwholerangesofoperationcon ditionswiththeoretical guaranteesofperformanceandrobustnessthroughoutthere gion. Parameter-dependentsystemtheoryimplicatesthecausald ependenceofthecontrollergainsonthevariationsoftheplantdynamics.Apart icularfamilyofthisLPV controlproblemiswhentheparameterdependencyinbothpla ntandcontrolleris linearfractional.Usingscaledsmall-gaintheory,asyste maticgainschedulingcontrol designtechniquehasbeendevelopedin[2].Theauthorsargu edthattheexistence ofsuchagain-scheduledcontrollerisfullycharacterized intermsoflinearmatrix inequalities(LMIs).Theunderlyingsynthesisproblemist hereforeaconvexproblem forwhichecientoptimizationtechniquesareavailable.A drawbackoftheLFT formulationisthatthevariationsof areallowedtobecomplex,thusintroducing someconservatismwhenparametersareknowntobereal.Sign icantimprovements canbeobtainedbyusinginsteadthenotionofquadratic H 1 performance[8].It seeksasinglequadraticLyapunovfunctiontoensure H 1 likeperformanceforall possibletrajectoriesoftheLPVplant.Inthisframework,t heparameteristreatedas realandshouldenterthestate-spacematricesoftheLPVpla ntinananefashion. Theimprovementessentiallycomesfromtheabilityofthequ adratic H 1 performance formalismtohandlerealparameters.However,thisapproac hremainsconservativein thefaceofslowlyvaryingparameters,sincequadraticLyap unovtechniquesallowfor arbitrarilyfastparametervariations.Inthischapter,ba sictheoriesandtechniques necessaryforLPVcontrolwillbeintroducedrst,thentheL PVapproachbased onquadratic H 1 performancewillappliedtoadisturbancerejectionproble mofthe DrivenCavity.

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131 4.5.2 Parameter-dependent State-Space Models Physicalmodelofasystemwithtime-varyingparameterscan alwaysberepresentedinstate-spaceform. E x = Ax + Bu y = Cx + Du wherethestate-spacematrices A;B;C;D;E dependontime-varyingparameters. Therearetwoparticulartypeofparameter-dependentstate -spacemodelsinwhich statematricesareinaneorpolytopicfunctionoftheparam eters. 4.5.2.1 Polytopic Models An polytopicsystem isdenedasalineartime-varyingsystem E ( t )_ x = A ( t ) x + B ( t ) u y = C ( t ) x + D ( t ) u whenthesystemmatrix S ( t )= 264 A ( t )+ jE ( t ) B ( t ) C ( t ) D ( t ) 375 isvaryingintheconvexhullofthesystemmatrices S 1 ;:::;S n (LTIsystems) S k = 264 A k + jE k B k C k D k 375 k =1 ; 2 ;:::;n withinaxedpolytopefashion,i.e., S ( t ) 2 Co f S 1 ;:::;S n g := ( n X k =1 k S k : k 0 ; n X k =1 k =1 ) Thenonegativenumbers 1 ;:::; n arecalledthepolytopiccoordinatesof S [40].

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132 4.5.2.2 Ane Parameter-Dependent Models Anane parameter-dependentsystem (PDS)isalinearPDSoftheform E ( p )_ x = A ( p ) x + B ( p ) u y = C ( p ) x + D ( p ) u where A ( ) ;B ( ) ;C ( ) ;D ( ) ;E ( )areanefunctionsofsomeparametervector p = ( p 1 ; ;p n ),i.e., A ( p )= A 0 + p 1 A 1 + + p n A n ;B ( p )= B 0 + p 1 B 1 + + p n B n andsoon.Aneparameter-dependentmodelsarewll-suitedf orLyapunov-based analysisandsynthesisandareeasilyconvertedtolinear-f ractionaluncertaintymodels forsmall-gain-baseddesign. Withthenotation S ( p )= 264 A ( p )+ jE ( p ) B ( p ) C ( p ) D ( p ) 375 ;S i = 264 A i + jE ( p ) B i C i D i 375 theanedependenceon p iswrittenmorecompactlyinsystemmatrixtermsas S ( p )= S 0 + p 1 S 1 + + p n S n 4.5.3 Linear Matrix Inequalities (LMIs) Manycontrolproblemscanbecastasoptimizationproblems. Onceformulatedin termsofLMIs,aproblemcanbesolvedexactlybyecientconv exoptimizationalgorithms(the"LMIsolvers").LMIsallowtorecastmanycontro landanalysisproblems inaunifyingwayandtocombineseveralcontrolobjectives( multi-objectivedesign). Whilemostproblemswithmultipleconstraintsorobjective slackanalyticalsolutions intermsofmatrixequations,theyoftenremaintractablein theLMIframework.This makesLMI-baseddesignavaluablealternativetoclassical "analytical"methods.

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133 Alinearmatrixinequality(LMI)isaninequality A ( x ) < 0 where A isananefunctionmappinganitedimensionalvectorspace toaHermitian matrixset H n n ,i.e., A ( x ):= A 0 + x 1 A 1 + + x N A N < 0(4.7) where x = x ( x 1 ;:::;x N )isavectorofunknownscalars(thedecisionofoptimizatio n variables); A 0 ;:::;A N aregivensymmetricmatrices;\ < 0 00 standsfor\negative denite". TheLMI4.7isaconvexconstrainton x since A ( y ) < 0and A ( z ) < 0implythat A y + z 2 < 0.Asaresult,itssolutionset,calledthefeasibleset,isa convexsubset of R N ;andndingasolution x to4.7isaconvexoptimizationproblem. Convexityguaranteesinndingasolutionto4.7numericall ywhenoneexists though4.7hasnoanalyticalsolutioningeneral. 4.5.4 Gain Scheduled H 1 Control RatherthanseekingasinglerobustLTIcontrollerfortheen tireoperatingrange, gainschedulingconsistsindesigninganLTIcontrollerfor eachoperatingpointand inswitchingcontrollerwhentheoperatingconditionschan ge.ForaLPVsystemwith ananeparameter-dependentplant x = A ( p ) x + B 1 ( p ) w + B 2 u (4.8) z = C 1 ( p ) x + D 11 ( p ) w + D 12 u y = C 2 x + D 21 w + D 22 u where p ( t )=( p 1 ( t ) ;:::;p n ( t ))

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134 and A ( ) ;B 1 ( ) ;C 1 ( ) ;D 11 ( )areanefunctionsof p ( t ).Denotethesystemmatrix S ( p ):= 266664 A ( p ) B 1 ( p ) B 2 C 1 ( p ) D 11 ( p ) D 12 C 2 D 21 D 22 377775 given p ( t )= 1 1 + + N N ; i 0 ; N X i =1 i =1 thesystemmatrix S ( p )= 1 S ( 1 )+ + N S ( N ) Theparameter-dependentnatureoftheplantimpliesaparam eter-varyingnature ofthefeedbackgains.Providedthattheparametervaluesar emeasureinrealtime, thissuggestsseekingaparameter-dependentcontroller x K = A K ( p ) x K + B K ( p ) y u = C K ( p ) x K + D K ( p ) y whichincorporatessuchmeasurementstoadjusttothecurre ntoperatingconditions. Suchcontrollersaresaidtobescheduledbytheparameterme asurements.Andthis typeofsynthesiscanbeconsideredasanautomaticgainsche dulingdesign. Denotesystemmatrixofthecontroller S K ( p ):= 264 A K ( p ) B K ( p ) C K ( p ) D K ( p ) 375 then S K ( p )= N X i =1 i S K ( i )

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135 Figure4.44:Gainscheduled H 1 controlproblem whichisaconvexinterpolationoftheLTIvertexcontroller s S K ( i ).Thisyieldsa smoothschedulingofthecontrollermatricesbytheparamet ermeasurementsof p ( t ). The GainScheduled H 1 ControlProblem canbedescribedlike[40]: Designagainscheduledcontroller K ( p ) withvertexpropertysuchthatforall admissibleparametertrajectories p ( t ) ,theclosed-loopsysteminFigure4.44isstable; andtheworstcaseclosed-loopRMSgainfrom w to z doesnotexceedsomelevel r> 0 TheLPVsystem x = A ( p ) x + B ( p ) y u = C ( p ) x + D ( p ) y issaidtohave quadratic H 1 performance r ifandonlyifthereexistasinglematrix X> 0suchthat 0BBBB@ A T X + XAXBC T B T X rID T CD rI 1CCCCA < 0 and A isstableandthesystemgain k H k 1
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136 Findtwosymmetricmatrices R and S suchthat 0B@ N 12 0 0 I 1CA T 0BBBB@ A i R + RA Ti RC T 1 i B 1 i C 1 i R rID 11 i B T 1 D T 11 i rI 1CCCCA 0B@ N 12 0 0 I 1CA < 0 ;i =1 ;:::;N 0B@ N 21 0 0 I 1CA T 0BBBB@ A Ti S + SA i SB 1 i C T 1 i B T 1 i S rID 11 i T C 1 i D 11 i rI 1CCCCA 0B@ N 21 0 0 I 1CA < 0 ;i =1 ;:::;N 0B@ RI IS 1CA 0 where 0B@ A i B 1 i C 1 i D 11 i 1CA := S ( i ) and N 12 and N 21 arebasesofthenullspacesof ( B T 2 ;D T 12 ) and ( C 2 ;D 21 ) respectively. Oncetheadequate R and S havebeencomputed,aLyapunovmatrixcommon toallinequalitiesabovecanbeobtainedandinconsequence thevertexcontroller S K ( i )canbeobtained. Thecommands hinfgs andothersintheLMIToolboxofMatlabcanbeusedto solvethatabovecontrolproblemeciently. 4.6 LPV Control of Driven Cavity Flow Inthissection,wewillconsideramuchwiderrangeof thanwedidin synthesis.PrevioustestinginChapter2indicatedthatfor 2 [0 : 5 ; 10]themodelis reasonablyaccuratesothecontrollerswillonlybedesigne dandtestedforthisrange. Theobjectiveofrowcontrolforthistestistorejecttheee ctsoftheexogenous disturbanceatthetopofthedrivencavity.Physically,the rowcontrolseeksto minimizethehorizontalcomponentoftheruidatasetofsens orlocations.These

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137 sensorsarelocatedat19pointsevenlydistributedalongth ehorizontalcenterlineof thecavity. 4.6.1 Control Synthesis Controllersaredesignedfordisturbancerejectionusingt hemodelshowninFigure4.2.Thelter W P attemptsloopshapingsuchthatroughlyspeaking,theveloc itiesatlowfrequenciesarereducedtolessthan1 = 2000andathighfrequenciesare lessthan1 = 20.Theactuallterisrealizedas W P = 20( s +1000) s +10 .Theactuationpenalty lterischosenas W k = s +200 s +2000 .Thusthecontrolsurfacerateis,roughlyspeaking, scaledbyafactorof10atlowfrequencyandof1athighfreque ncy.Thenoiselter ischosenas W n =0 : 004.Sincethemagnitudeofhorizontalvelocityinopen-loo ptest isaround0.2,weareaddingabout2%ormorenoisetothemeasu rement. ThecontrollerhasbeensynthesizedusingtheLMIControlTo olboxinMatlab. Theresultingclosed-loopnormhasavalue10.35.Thisvalue indicatesthatnotall thedesignobjectivesaresatised.Forthissystem,itisdi culttoachievethedesired levelsofdisturbancerejectioninhighfrequency.Thisdi cultydoesn'taectthegoal ofcontrolbecausethemodelisdenedtobevalidatlowfrequ encyandthecontrol applicationismainlyconcernedwiththelowfrequencyregi me. Thesynthesizedparameter-dependentcontrollerisapolyt opicsystemconsisting ofthevertexcontrollerswithsystemmatrices K i = 264 A K ( i ) B K ( i ) C K ( i ) D K ( i ) 375 .Givenany xed-valueoftheparameter ( t ),thecorrespondingstate-spaceparameters K ( )= 264 A ( ) B ( ) C ( ) D ( ) 375 ofthegain-scheduledcontrollercanbecalculated. 4.6.2 Closed-Loop Responses for Fixed Values Theeectivenessofthecontrollerisdemonstratedbyinves tigatingtheclosedloopresponseofthedrivencavity.Theinitialtestofthese propertiesisperformedby

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138 consideringthesystematxedvaluesoftheparameter .Inthiscase,theopen-loop dynamicsandthecontrollerarecomputedforthespecicval uesof f 1 : 0 ; 2 : 5 ; 5 : 0 ; 7 : 5 g for Figure4.45showstheclosed-looptransferfunctionsfrome xogenousinput, h toperformanceerrors, e 1 ,forthesynthesismodelinFigure4.2.Thecontrollerwas designedtominimizetheinducednormofthismodelfortimevaryingparameter changessotheperformancewithxedparametersshouldalso bechecked.Thetransferfunctionsinthegureindicatethatthecontrollerisin deedprovidingdisturbance rejection.Thevelocitiesathighfrequencyarerelatively unaectedbythecontroller butthevelocitiesarelowfrequenciesareattenuatedbyafa ctorof160. 10-2 100 102 104 106 10-4 10-2 100 MagnitudeFrequency 10-2 100 102 104 106 -200 -150 -100 -50 Phase (degrees)Frequency Open-loop, q =1 Open-loop, q =2.5 Open-loop, q =5 Open-loop, q =7.5 Closed-loop, q =1 Closed-loop, q =2.5 Closed-loop, q =5 Closed-loop, q =7.5 Figure4.45:Transferfunctionsatconstant Thetime-domainpropertiesoftheclosed-loopsystemarein vestigatedusinga nonlinearsimulation.Themeasurementsofhorizontalvelo cityatthesensorlocations whent=0.2areshowninFigure4.46.Allvariablesarenon-di mensional.Clearly

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139 theclosed-loopvelocitiesaresignicantlylessthantheo pen-loopvelocities.The controllerisachievingdisturbancerejection. 0 0.2 0.4 0.6 0.8 1 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 PositionV i,x Open-loop, q =1,2.5,5,7.5 Closed-loop, q =1,2.5,5,7.5 Figure4.46:Flowvelocityatconstant Horizontalvelocityoftherowisafunctionoftimeduetosin usoidaldisturbance. Sincewemeasurethisvelocityat19sensorlocationssimult aneously,theplotsof velocitiesshouldbe3-dimensional.Figure4.47-4.50show thetimehistoryofvelocities versusthex-positionsofsensorsalongthehorizontalcent erline.Allvariablesarenondimensional.Theseplotsgiveclearvisualizationofthee ectivenessofthedesigned controllerfordisturbancerejection. Open-loop 0 0.5 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time x-Position Velocity Vx Closed-loop 0 0.5 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time x-Position Velocity Vx Figure4.47:Timeresponsefor =1 Oneconsiderationforcontrolsynthesisisthatthecontrol actuationshouldbe limitedtoaphysicallymeaningfullevel.Figure4.51shows thatcontrolsurfacerate

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140 Open-loop 0 0.5 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time x-Position Velocity Vx Closed-loop 0 0.5 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time x-Position Velocity Vx Figure4.48:Timeresponsefor =2 : 5 Open-loop 0 0.5 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time x-Position Velocity Vx Closed-loop 0 0.5 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time x-Position Velocity Vx Figure4.49:Timeresponsefor =5 Open-loop 0 0.5 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time x-Position Velocity Vx Closed-loop 0 0.5 1 0 0.5 1 -0.2 -0.1 0 0.1 0.2 Time x-Position Velocity Vx Figure4.50:Timeresponsefor =7 : 5 islessthan1toachieve160timesdegradationofvelocitywh enthedisturbanceisa sinusoidalfunctionwithamplitudeof1.Allvariableshere arenon-dimensional.This iswithinourlimitationof10atlowfrequency.

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141 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 TimeControl Figure4.51:Controlhistorywhen =1 4.6.3 Closed-Loop Responses for Time-Varying Thenonlinearsimulationisalsousedtoinvestigatetheper formancefortimevaryingparameter .Thisparameterisallowedtovaryacrossthefullrangeof interest, 2 [ : 5 ; 10],asshowninFigure4.52. 0 0.5 1 1.5 2 0 2 4 6 8 10 Timeq (t) Figure4.52:Exampleofparametertrajectoryof ( t ) The2-normmeasureofthehorizontalvelocitiesateachsens orlocationareshown inFigure4.53.Theopen-loopresponseschangeinmagnitude acrosstimeindicating theeectoftheparametervariation.Conversely,theclose d-loopresponseremains

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142 relativelyconstantwithlowmagnitudethroughouttheenti resimulation.Thus,the LPVcontrollerisabletoachievedisturbancerejectionfor thetime-varyingparameter. 0 0.5 1 1.5 2 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 TimeV 5,x (t) closed-loop open-loop Figure4.53:TimeresponseoftheLPVsystemalongparameter trajectory

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CHAPTER5 CONCLUSIONS Modelreductiontechniqueshavebeenusedtocharacterizer uidroweldsin ordertoreducecomputationcostandsetupfeasiblemodelsf orcontrolapplications. Thisthesispresentsanapproachtobuildreducedordermode lsbyGalerkinprojection andproperorthogonaldecomposition.Theresultedreduced ordermodelsareusedto modeldierentrowcontrolproblems.Thendierentcontrol theories,suchasoptimal control,recedinghorizoncontrol, H 1 = control,andLPVcontrol,areappliedto solvetheseproblems.AdrivencavityStokesrowisinvestig atedwithnumerical simulationstotestifytheapproach. ThisthesisstudiestransientStokesrows.Non-dimensiona lanalysisshowsthat thedeterministicparameterofthistypeofrowisreverseof theproductofReynolds numberandStrouhalnumber,denotedas .Theoretically,thereducedordermodels arepropertodescribetransientStokesrowonlywhen 1.Detailedanalysisand numericalsimulationsshowthatthereducedordermodelsar eaccurateevenwhen isarbitrarilylargerthansomevaluenotmuchsmallerthano ne.Herewendpoor resultsfor =0 : 1butexcellentresultsfor =1andlarger.Thiscanbeexplained physically.Forinstance,ifthechangeof iscausedbythevariationofboundary excitationfrequency,whichisrelatedtoStrouhalnumber, thenthischangecannot betoofast.Otherwisethereexistsalayeralongtheexcitat ionboundarywherethe eectsoftheexcitationareconned.Ontheotherhand,thee xcitationfrequency couldbeveryslow,whichiscorrespondingtoverylarge value.Inthiscase,the rowisquasi-static,andthereducedordermodelsarestillv alid. 143

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144 Atrackingproblemisinvestigatedfortheapplicationofre ducedmodelsinoptimalcontrol.Mathematicalanalysisandnumericalsimulati onsshowthattheproblem canbesolvedbydierentapproaches.Bothdirectdiscretiz ationmethodandrecedinghorizoncontrolapproacharesuccessfulinsolvingit.T hemethodassociatedwith thecalculusofvariationisalsoapplicable.Theresultoft hismethodtellsusthat algebraicRiccatisolutionscanbeusedtosolvethetrackin gproblemeciently. Thereducedordermodelsarealsoinvestigatedtotheapplic ationsofrobustand linear-parametervarying(LPV)control.Controllersared esignedforthereduced ordermodelwithvariationofparameter .Forthesmallvariation, H 1 = control areprovedbynumericalsimulationstobeeectiveandthede signedcontrollerscan toleratethisvariation.Forthemodelisscheduledtoworka crossofabigrangeof ,controllersaredesignedusingthelinear-parametervary ingframework.AdisturbancerejectionproblemisinvestigatedinbothrobustandL PVcontrol.Numerical simulationsshowthattheapplicationof H 1 = controlandLPVtechniqueleads tosubstantiallyimprovedperformance,decreasingthevel ocitybyalargefactorand usingreasonableactuationlevels. Analysisandsimulationsinthisthesisshowthatreducedor dermodelprovidesa goodchoicetomodelruidrowandrealizerowcontrolapplica tions.Theinvestigation focusonStokesrowwhilethemethodologyisapplicabletoNa vier-Stokesrowwith anygeometry. Finally,asacontinuationofthecurrentstudywesuggestth efollowing: ExaminationofreducedordermodelsintransientNavier-St okesrow. Furthertestingothergeometries,e.g.back-wardfacingst ep.

PAGE 159

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BIOGRAPHICALSKETCH YunfeiFengwasborninChifeng,NeiMongol,China,in1975.H ereceivedthe B.S.andtheM.S.degreesfromDepartmentofPrecisionInstr umentsatTsinghua University,Beijing,China,in1997and1999,respectively .In1999,hejoinedthe DynamicandControl(D&C)ResearchGroupattheDepartmento fMechanicaland AerospaceEngineering,UniversityofFloridaforhisPh.D. degree.Hereceivedoneof hisdoubledegrees,M.S.,fromDepartmentofElectricaland ComputerEngineering attheUniversityofFlorida,inMay2003.Hisresearchinter estsincluderuiddynamic modeling,andcontroltheories. 152


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REDUCED ORDER MODELS AND THE APPROXIMATION
OF STOKES FLOW CONTROL PROBLEMS















By

YUNFEI FENG


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2003















ACKNOWLEDGMENTS


I wish to express my deepest gratitude to my advisor, Dr. Andrew J. Kurdila,

whose constant support and patient guidance provided a clear path for my research.

I thank him for his effort in teaching and guiding me in this research field. I have

learned a great deal from him. I would also like to thank Dr. David W. Mikolaitis

and Dr. Rick Lind for their efforts in guiding my research. I thank Dr. Wei Shyy,

Dr. Norman G. Fitz-Coy, and Dr. Oscar D. Crisalle for their helpful discussions and

-i--.- -I i. -, and their time and effort to serve on my committee.

Additionally, I would like to thank Dr. Yongsheng Lian, Jiongyang Wu, and

Jelliffe Jackson for their advice and their effort in providing CFD data for my research

work.

I would like to thank NASA for funding this work.

Finally, I would like to thank my wife and my parents for their immeasurable

advice, encouragement, and support.















TABLE OF CONTENTS


page


ACKNOWLEDGMENTS ................

LIST OF TABLES .....................


LIST OF FIGURES ................................


NOMENCLATURE ....................

ABSTRACT ........................

CHAPTERS

1 INTRODUCTION AND LITERATURE REVIEW


1.1 Introduction and Motivation .. .......
1.1.1 Introduction to Flow Control .....
1.1.2 Motivation .............
1.2 O objectives . . . . .
1.3 Reduced Order Models of Incompressible Vis
1.3.1 Governing Equations .........
1.3.2 Reduced Order Modeling .......
1.3.3 Construction of Basis Functions in R(
1.4 Control Strategies in this Thesis .......
1.4.1 Optimal Flow Control .........


1.4.2
1.4.3


;cous Fluid Flows


M . .


Model Predictive Control (Receding Horizon Control)
Gain Scheduling and Linear Parameter-varying Systems


2 REDUCED-ORDER MODELING OF STOKES FLOW .......

2.1 General Stokes Flow Problem .. ...............
2.1.1 Stokes Equation and Time Scale Issue .. ........
2.1.2 Problem Formulation .. ...............
2.1.3 Order Reduction Framework .. ............
2.2 Driven Cavity Stokes Flow Problem .. ............
2.2.1 Control Problem Description .. ............
2.2.2 Reduced Order Model of Driven Cavity Problem .
2.2.3 Stationary Driven Cavity Flow .. ............
2.3 POD and Unsteady Driven Cavity Flow Problem .. .....
2.3.1 Proper Orthogonal Decomposition (POD) ........
2.3.2 POD for Unsteady Driven Cavity Flow .. .......









2.3.3 ROM by POD for Unsteady Driven Cavity Stokes Flow

3 OPTIMAL CONTROL OF DRIVEN-CAVITY STOKES FLOW .

3.1 Variational Operator 6.....................
3.2 Optimal Control Framework ..................
3.2.1 Fixed Terminal Time Problems .. ............
3.3 Optimal Control of Driven Cavity Flow .. ...........
3.3.1 Control Problem Description .. ............
3.3.2 Optimality Systems and the Classical Riccati Equation
3.3.3 Numerical Simulations .. ..............
3.4 Direct Discretization Method .. ................
3.4.1 M ethod Description .. ................
3.4.2 Numerical Simulations .. ..............
3.5 Receding Horizon Control (RHC) .. .............
3.5.1 Receding Horizon Formulation .. ...........
3.5.2 Numerical Simulations .. ..............

4 ROBUST CONTROL OF DRIVEN CAVITY STOKES FLOW .

4.1 X-based Modeling and Control .. .............
4.1.1 Singular Values, H ,I -norm .. .............
4.1.2 H, Optimal Control Synthesis .. ............
4.2 H, Optimal Control of Driven Cavity Flow .. ........
4.2.1 Open-loop M odel .. .................
4.2.2 Control Synthesis .. .................
4.2.3 Closed-loop Response .. ................
4.3 H Robust Control .. ....................
4.3.1 Uncertainties . . . . . .
4.3.2 Small Gain Theorem .. ...............
4.3.3 Linear Fractional Transformation .. .........
4.3.4 -,Q Robust Control Synthesis .. ...........
4.3.5 Structured Uncertainties .. ..............
4.3.6 Structured Singular Value: p .. ............
4.3.7 Control Design with p-Synthesis .. ...........
4.4 H, Robust Control of Driven Cavity Flow .. .........
4.4.1 Uncertainty M odel .. .................
4.4.2 Control Objectives .. .................
4.4.3 Controller Synthesis .. ................
4.4.4 Robustness Analysis .. .................
4.4.5 Closed-loop Simulation .. ...............
4.5 LPV Control . . . . . . .


4.5.1 LPV System ..


4.5.2
4.5.3
4.5.4


Parameter-dependent State-Space Models
Linear Matrix Inequalities (LMIs) .
Gain Scheduled XC, Control .......


...............


.









4.6 LPV Control of Driven Cavity Flow . . 136
4.6.1 Control Synthesis . . ......... .. 137
4.6.2 Closed-Loop Responses for Fixed 0 Values ........ 137
4.6.3 Closed-Loop Responses for Time-Varying 0 . ... 141

5 CONCLUSIONS .................. ............ .. 143

REFERENCES .................. ................ .. 145

BIOGRAPHICAL SKETCH ............. . . .. 152















LIST OF TABLES

Table page

2.1 Singular values for 0 = 0.1 .................. .... .. 49

2.2 Singular values for 0 = 1 .................. ..... .49

2.3 Singular values for 0 = 2.5 .................. .... .. 50

4.1 Controller performance .................. ....... 120

4.2 Gain and phase margin of the closed-loop transfer functions ..... ..124















LIST OF FIGURES


Figure page

1 Geometry ...................... ......... xi

1.1 Driven cavity flow .................. ... 4

2.1 Finite elements .................. ........... .. 22

2.2 Bilinear quadrilateral element domain and local node ordering .. 23

2.3 A pressure quadrilateral and its four associated velocity quadrilaterals 26

2.4 Geometry .................. .............. .. 29

2.5 Stokes driven cavity flow problem ................ 37

2.6 Stationary driven cavity flow .................. .. 41

2.7 Comparison of reduced order solutions of velocities to full order solution 42

2.8 To generate ensemble to which POD is taken . . ..... 47

2.9 Contour plots of POD eigenmodes with streamlines at 0 = 0.1 47

2.10 Contour plots of POD eigenmodes with streamlines at 0 = 1 . 48

2.11 Contour plots of POD eigenmodes with streamlines at 0 = 2.5 .. 48

2.12 Error averaged over timesteps is bounded by the singular values ac,+ 51

2.13 Error of each timestep is bounded by the singular values ...... ..52

2.14 Error averaged over timesteps is bounded by the singular values crm,+ 52

2.15 Error of each timestep is bounded by the singular values . 53

2.16 Error averaged over timesteps is bounded by the singular values crm,+ 53

2.17 Error of each time step is bounded by the singular values ...... ..54

2.18 Unsteady driven cavity flow .................. .. 55

2.19 Contour plot of initial velocity field .................. .. 56

2.20 Error analysis of ROM approximation ................. .. 57









2.21 Error analysis of ROM approximation at 0


2.22

2.23

2.24

2.25

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11


Error analysis of ROM approximation ......

Error analysis of ROM approximation for 0 = 1

Error analysis of ROM approximation for 0 2.

Error analysis of ROM approximation for 0 2.

States solution for the tracking problem .....

Control history for the tracking problem .

Riccati solution for the tracking problem .

History of A(t) for the tracking problem .....

Target flow for N = 31 ......

Controlled flow for N = 11, N 4 ......

Controlled flow for N 11, N 8 ......

Controlled flow for N = 31, N 8 ......

Solutions of 3(t) vs time .. ...........

Optimal states solutions comparing to the target

RHC control history .. ............

Closed-loop system .. .............

Synthesis model .. ...............

Transfer functions at 0 1 ......


Disturbance rejection effect of the controller designed at 0

Open-loop velocity output at horizontal center-line .

Closed-loop velocity output at horizontal center-line .

States variables and control input .. ...........

Full order closed-loop block diagram for point controller .

Transfer functions for the closed-loop system .......

Closed-loop velocity output at horizontal center-line .

Open-loop velocity output at horizontal center-line .


States


1 .









4.12 Closed-loop velocity output at horizontal center-line . ... 101

4.13 Transfer functions at 0 = 10 ................ .... 102

4.14 Disturbance rejection effect of the controller designed at 0 = 10 . 103

4.15 Open-loop velocity output at horizontal center-line . ... 103

4.16 Closed-loop velocity output at horizontal center-line . ... 104

4.17 States variables and control input ................ . 105

4.18 Full order closed-loop block diagram for point controller . ... 105

4.19 Transfer functions for the closed-loop system . . 106

4.20 Closed-loop velocity output at horizontal center-line . ... 106

4.21 Open-loop velocity output at horizontal center-line . ... 107

4.22 Closed-loop velocity output at horizontal center-line . ... 107

4.23 Block diagram for the Small Gain Theorem . . 109

4.24 Linear fractional transformation Ju(G, A) .............. .110

4.25 Linear fractional transformation Te(G, A) ... . . 110

4.26 -X robust control problem ............... ... 111

4.27 XH,-synthesis considering robust performance . . 112

4.28 LFT system for robust stability analysis using . . ... 114

4.29 Synthesis model ............... .......... .. 119

4.30 Extracted synthesis model ................ ..... 120

4.31 Original HX, controller .................. .. .... .. .. 121

4.32 Balanced truncated controller .................. ..... 122

4.33 Closed-loop system with synthesis model . . ..... 123

4.34 Controller performance .................. .. ..... 123

4.35 Closed-loop system for time response ................. 125

4.36 Transfer function at 5th sensor location ................ ..125

4.37 Time response at t = 0.2 ............. ... ..... 126

4.38 Time response for nominal model ................ . 126










4.39 Time response for truth model .......... . 127

4.40 Transfer function at 5th sensor location ................ 127

4.41 Time response at t = 0.2 .................. .... 127

4.42 Time response for nominal model ................... ... 128

4.43 Time response for truth model ............ .. 128

4.44 Gain scheduled 'o,, control problem .................. 135

4.45 Transfer functions at constant 0 .................. .. 138

4.46 Flow velocity at constant 0 .................. .. 139

4.47 Time response for 8 = 1 .................. .... .. 139

4.48 Time response for 0 = 2.5 .................. ..... 140

4.49 Time response for = 5 .................. .... .. 140

4.50 Time response for 0 = 7.5 .................. ..... 140

4.51 Control history when 0 = 1 .................. ... .. 141

4.52 Example of parameter trajectory of (t) . . ..... 141

4.53 Time response of the LPV system along parameter trajectory ..... 142















NOMENCLATURE


Slmiil'.. for the PI;,<. ,l Domain

Throughout this thesis, we use following symbols

f : the physical domain.

9O : the boundary of the domain

A, : the set of indices of nodal points associated with the interior of the

domain

Ar : the set of indices of nodal points associated with the boundary

Frequently, it is necessary to partition Ar into subsets. For example, in the case as

shown in figure 1 the boundary is partioned into F, and Fk. Controls act on Fc. We


Fk(









Figure 1: Geometry


write


f = F U Fk A A u Ak


where A, and Ak are sets of indices of nodal points associated with F, and Fk respec-

tively.

Slmii.'l, to Express Velocities

(1) Vectors of velocity fields









The following symbols are used to represent velocity fields which are functions

of (x,y,t):

V(x,y, t) Velocity field to be solved

Vo(x, y)- Initial velocity field

(x, y) Velocity field of control basis

((x, y)- Velocity field of ROM basis
(2) Nodal velocity vectors

Bold font symbols are used to represent nodal velocity vectors. Throughout

this thesis, tilde ( ) is used to emphasize a nodal velocity from single prescribed

basis or combination of prescribed bases. There are two type of prescribed bases, i.e.

Vo and 'I.

The following constitutes the most frequently used symbols:

Unknown velocity associated with A,

V- Nodal velocity associated with all all prescribed bases

Vo- Nodal velocity associated with both A1 and Ar

V Nodal velocity associated with both A, and Ar

K- Nodal velocity of ROM basis associated with A,

[]- Ensemble of control bases: [1] A [I '2 .]

[ ]- Ensemble of ROM bases: [1] A [+i 2 ..

The following list summarizes frequently encountered components of various vec-

tors:

Vk- Nodal velocity, kth node of V, k E A,

Vk,i- the ith component of the nodal velocity Vk, k A,, i = 1, 2
V, The subset of V associated with A1

Vr- The subset of V associated with Ar

Vo,u The subset of Vo associated with AZ

Vo,r- The subset of Vo associated with Ar









VO,T, Vo,B, V,, Vo,1, etc- The subset of Vo,r associated with some
specific subset of Ar, like AT, AB, Ac, A1 etc.

Vo,u,k- Nodal velocity of the kth node of Vo,,, k E A,

Vo,r,n- Nodal velocity of the nth node of Vo,r, n Ar
j- the jth Control basis, j = 1, 2, N

,j, j- The subset of Tj associated with A, only, j 1, 2, .. N

Tj,r- The subset of Tj associated with Ar j = 1, 2, N

1J,T, Jj,B, j,c, j,1, etc- The subset of jr associated with some
specific indices subset of Ar like AT, AB, AC, A etc.

4j,u,k- Nodal velocity of the kth node of j, j = 1, 2, .. No, k E

lj,r,n- Nodal velocity of the Vth node of Tj, j 1, 2,... N, n
1l the Ith ROM basis, I 1,2, ,N,

)l,k Nodal velocity of the kth node of 4 1, = 1,2,. N k E I
In the above list, Nc is the number of control bases and Ns is the number

ROM bases.


A,

Ar




;r of















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

REDUCED ORDER MODELS AND THE APPROXIMATION
OF STOKES FLOW CONTROL PROBLEMS

By

Yunfei Feng

December 2003

C'!I ii: Andrew J. Kurdila
M i r Department: Mechanical and Aerospace Engineering

Numerical simulations of fluid flows which are governed by the N ,i., i-Stokes

equations are too costly for optimization and control. Models of unsteady flows

are not amenable to optimization due to their high dimensionality and nonlinearity.

Thus, a reduced-order model (ROM) is proposed in this thesis to provide suitable

strategy for active control of fluid dynamical systems. Stokes flow is studied in detail.

This flow regime is studied because it generates a linear model. Based on the derived

ROM, optimal control strategies for the tracking problem are developed and numerical

simulations are provided. Moreover, gain-scheduling controllers of an associated LPV

system are investigated. A LPV (linear parameter-varying) system is a parameter-

dependent system. It is characterized as a linear system that depends on time-varying

real parameters. In this thesis, the parameter is selected to be the Reynolds number.

In the special case in which the LPV plant is a linear fractional transformation (LFT)

dependent on the parameter, controller synthesis is fully characterized as a convex

problem and is solved efficiently. This thesis investigates the development of a ROM

of Stokes flow for control synthesis. Numerical simulations are provided.














CHAPTER 1
INTRODUCTION AND LITERATURE REVIEW


1.1 Introduction and Motivation

Flow control in fluid dynamics has long been a subject of interest to engineers and

scientists. In past years, increasing attention has been devoted to the development

of techniques capable of enhancing our ability to control unsteady flow in a wide

variety of environments such as engine inlets and nozzles, combustors, automobiles,

aircraft, and marine vehicles. Controlling the flow in these problems can lead to

greatly improved efficiency and performance. During the past century tremendous

advances have been reported for flow control problems.


1.1.1 Introduction to Flow Control

The science of flow control originated with Prandtl in 1904, one year after the

first powered flight. Prandtl introduced the concept of the boundary l-iv-r [73] and

proposed a means to control its attachment to a solid surface. Since then, various flow

control theories and applications in fluid mechanics, e.g. laminar flow control (LFC),

separation control, optimal control, etc., have been studied. Reviews of experimental

and numerical work have been written on the subject by, for example, Moin and

Bewley(1994) [68], Joslin, Erlebacher and Hussaini (1996) [60], Gad-el-Hak (1996)

[38], Lumley and Blossey (1998) [67], Greenblatt and Wygnanski (2000) [44], and

Bewley (2001) [11]. Articles mainly concerning the mathematical aspects of the

optimization methods used for flow control can be found in the books edited by e.g.

Gunzburger (1995) [46] and Sritharan (1998) [80].

Flow control can take many different forms and some kind of classification of

different strategies is useful. Passive control is used when the flow is affected without









requiring an input of energy. For instance, laminar flow can be obtained passively over

the forward part of airplane lifting surfaces (wings and tails) that have leading-edge

sweep angles of less than about 18 degrees. This is achieved by designing the surface

cross-sectional contour so that the local pressure initially decreases over the surface in

the direction from the leading edge towards the trailing edge [15]. However, this form

of control is not capable of adjusting to instantaneous flow conditions. As opposed

to passive control, active control requires an energy input which has traditionally

been associated with the injection or suction of fluid, or the motion of a surface

.,l.i i,:ent to the stream. Another mechanism of flow control uses oscillatory addition

of momentum as opposed to traditional steady addition of momentum oscillatory

blowing instead of steady blowing and suction. Active control performed according to

a predetermined scheme in a open loop, from measured data in an open, feedforward

loop, or from a closed, feedback loop. An example of an active, open loop, approach

to flow control is blowing a high speed jet over a deflected flap to enable the airplane

to land at a lower speed.

Flow control problems often involve passive or active devices to effect a beneficial

change in wall-bounded or free-shear flows. Example of an external wall-bounded flow

can be the flow that develops on the exterior surface of an aircraft or a submarine. In

contrast to free-shear flows, wall-bounded flows are more important. Careful study

and manipulation of these flows could lead us to prevent separation, increase lift force,

reduce pressure drag and noise, etc. For example, the skin-friction drag and flow-

induced noise are reduced by maintaining as much of a boundary l1v-,r in the laminar

state as possible. This type of flow control is called laminar flow control (LFC). A

review of LFC can be found in Joslin [59]. Drag reduction effect, of steady lateral

wall oscillations on turbulent boundary 1livr characteristics was examined by Trujillo

and other authors [82] by introducing hotfilm measurements of mean velocity in the

viscous sublayer to determine the drag reduction. The experiments show that at









least 25' drag reduction can be achieved for a turbulent boundary flow using steady

oscillation of the surface. Other LFC studies can be found in [7,27,60,61,63,72].

Another most often studied flow control problem is flow separation control. Sep-

aration is generally accompanied by losses of some kind, such as the loss of lift. By

postponing separation, drag can be reduced, dynamic stall is d. 1 i-, .1 lift is enhanced

and pressure recovery is improved. From a practical standpoint, separation control

can be achieved by steady injection of fluid [16, 78] to, or suction of a fluid [9,47, 73]

from, a boundary l1 .-r. Alternatively, the motion of the wall tangent to the direction

of stream can be modified. Furthermore, Wygnanski (1997) [84] demonstrated by

experiments that oscillatory excitation can be used to delay flow separation. Green-

blatt and Wygnanski (2000) [44] review the development of control of flow separation

from solid surfaces with emphasis on periodical excitation method.


1.1.2 Motivation

Despite significant progress over the past decade, the control of fluid flow remains

one of the most (1! 11i l:. i-. unsolved problems in the field. Control problems involv-

ing partial differential equations are formidable problems to solve in real time. The

most significant difficulties are perhaps the nonlinearity of the incompressible Navier-

Stokes equations that represent the dynamics of the flow, and the high dimensionality

of typical approximation of these equations. For example, the dimension of a typical

fluid dynamics system can be in the order of 0(106) O(107) degrees of freedom.

If one were to solve such problems using standard finite element method (FEM) or

finite difference method (FDM) [33], the resulting system is prohibitively huge. It

is clear that significantly different strategies are required for flow control problems.

Low dimensional approximation of N ,i1. i -Stokes equations is just such an approach

that tries to describe the fluid flow by using lowest possible order model that cap-

tures the 1n ii r characteristics. This strategy has flourished during the last several









years. Various reduced-order models have been derived and applied to the flow con-

trol problem. This thesis will focus on reduced-order modeling of the flow field and

the incorporation of this approximation in optimal and robust control.


1.2 Objectives

As stated above, the goals of this thesis are (1) derivation of reduced-order

models of the incompressible viscous flow and (2) their application in optimal and

robust control methods. The reduced-order model (ROM) in this thesis is derived

for the general case and can be applied, in principle, to any geometry. A particular

case, driven cavity flow as shown in Figurel.l, will be used throughout this thesis as

a prototypical example to demonstrate the methodology.


input



FT
T, T /
// Fr /
/ L R

FB



Control

Figure 1.1: Driven cavity flow


These goals are achieved in several steps:

1. Reduced-order modeling. Both static and unsteady flows are investigated to

demonstrate the feasibility of obtaining accurate reduced-order models.

2. Study of open-loop optimal control. A simple tracking problem is introduced

and solved by a direct discretiziation method, the classical Riccati solution, and

receding horizon control method.









3. Study of closed-loop robust control. Gain scheduling controller synthesis tech-

nique is discussed for the linear parameter-varying (LPV) systems.

(a) Point controller synthesizing for the reduced-order model by the CX con-

trol approach.

(b) Full order simulation is used to asses the performance of the reduced-order

model in robust control.

(c) Gain scheduling for the LPV system is constructed and numerically tested.


1.3 Reduced Order Models of Incompressible Viscous Fluid Flows

Systems possessing a large (potentially infinite) number of degrees of freedom

often exhibit simple dynamical behavior in the sense that the long-term motion is

characterized by a small phase space dimension. This motivates low-dimensional rep-

resentations of such large dynamical systems (typically of the order of 106 ODEs for

a two-dimensional problem). Development of low-dimensional descriptions of these

systems in computational methods for control of fluids has been the topic of recent

studies by many researchers. The basic principle is to approximate the nonlinear dy-

namics of the equations of fluid flow by reduced order models (ROM), and then derive

adequate control strategies from the reduced order system. The brief discussion below

will describe a general approach to set up a reduced order model for incompressible

viscous fluid flow.


1.3.1 Governing Equations

1.3.1.1 N ,v1i. -Stokes Flow

Two-dimensional unsteady incompressible viscous fluid flow is governed by the

N ,.l ir-Stokes equaions. In non-dimensional form these equations are

at Re + Vp f (.)
8t Re









Mass conservation is expressed by the continuity equation


V.- 0 (1.2)


where V(x, y, t) is the velocity field of fluid flow and p is the pressure. The Reynolds

number is defined to be


Re -UL
Re = p-


where p the density, v the viscosity, L the length scale, U the velocity scale. For

instance, the centerline Reynolds number Rel = p- is commonly used for laminar

channel flow with Uc the centerline velocity and h the half height of the channel. The

Reynolds number Re = p To is used for the driven cavity flow model in this thesis.

It is based on the top slip velocity VT0 and width of the cavity L.

1.3.1.2 Stokes Flow

In this thesis, the two-dimensional unsteady Stokes flow is governed by the non-

dimensional equations

a eov + vp=f (1.3)
at

These equations are linear in the flow velocity V(x, t). The derivation of these equa-

tions and the meaning of parameter 0 will be described in detail in C'!i lpter 2.


1.3.2 Reduced Order Modeling

Reduced order models of either of the above governing equations are based on

the expansion

Nc N,
V(x, y,t) = Vo(x, y, t) + (xy) + .'(x,) (1.4)
j=1 1=1









where Vo(x, y, t) is some prescribed velocity field satisfying the boundary conditions,

and where 'j(x, y) and ((x, y) are basis functions that are assumed to be indepen-

dent of time t. The unknowns now become {Jl}7, and {3j} ,. If an accurate

approximation of the velocity field can be achieved with very small N, and N~, the

reduced order model is deemed to be a successful approximation of the full order

dynamical system.


1.3.3 Construction of Basis Functions in ROM

In the reduced basis approach, one uses as a basis functions that are somehow

physically motivated by the flow under consideration. This is in contrast to the finite

element Galerkin schemes where the basis elements are essentially unrelated to the

physical properties of the system that they approximate. Various approaches based

on ROM with different construction of the basis functions have appeared in the lit-

erature over the past few decades. Proper orthogonal decompostion (POD) was first

introduced in fluid mechanics by Lumley to identify coherent spatial structures [66].

Related investigations can be found in [5,17,25,52,53,64,75,76]. Essentially, POD ex-

tracts an optimal orthonormal basis to represent the ensemble of data obtained from

numerical computations or experimental measurement. The method is optimal in

the sense that it captures more energy than any other orthonormal set. POD modes

serve as a basis for a Galerkin reduction of the governing equations. The POD modes

typically are created from a set of "flow -~ il-lini' generated from experiments or

numerical simulations. By construction, POD modes form a complete set of basis

functions (satisfying the boundary conditions and the incompressibility constraint)

for a fixed dynamical system. For example, they may be created at a fixed Reynolds

number, which could be a problem when parameter variations are encountered. Sys-

tems with different parameter values may have different dominant modes [62]. One

optimal POD set may not ahv--, capture maximal energy for a different flow regime.

This could result in reduced order models requiring higher dimension to provide a









good approximation of the full dynamic systems. Although some enhanced methods

like weighted POD (w-POD) and predefined POD (p-POD) were proposed by Erik

Adler ('!!i -I. I,- i [22], POD may be inappropriate as a practical tool for providing

low-dimensional models of parameter-dependent problems.

In contrast, another approach is to derive the reduced order method from phys-

ical considerations. Ito and Ravindran [56,57] have shown via empirical numerical

evidence that reduced order Navier-Stokes simulations can yield accurate reduced or-

der models. Numerical results were given in Ito and Ravindran [56] for driven cavity

and channel flow control problems in steady viscous incompressible flows governed

by the equations


V+ ( VV V + Vp= f in R (1.5)

V = 0 in Q

The methodology is essentially restricted to some specific boundary conditions. A

series of basis elements {ui} were obtained from solutions of the steady Navier-

Stokes equations(1.5) with different inhomogeneous boundary values. Subsequently,

linear combinations of the {u}i were constructed to get test functions (i

which satisfy homogeneous boundary conditions. Other linear combinations of the

{ui},1 are used to generate I', which satisfy homogeneous boundary condi-
S i= 1
tions except for control boundary FP. The paper demonstrated first that the reduced

order method above can achieve very nice approximation of the full order solution.

Then two optimal control problems were solved successfully based on the reduced

order models. The crucial point in this approach is the choice of the test functions
r N,
{i 1 They must be chosen such that the expansion in equation(1.4) describes
the nonlinear dynamics of the flow over a large range of flow regimes.









1.4 Control Strategies in this Thesis

1.4.1 Optimal Flow Control

Optimal control methodologies have been recently applied to a variety of flow

control problems involving reduction of drag or elimination of the transition from

laminar to turbulent flow. Sophisticated optimal control strategies in engineering

applications coupled with a large number of partial differential equations are now

feasible due to the development of computational fluid dynamics (CFD) algorithms

which have reached a sufficiently high level of maturity, generality, and efficiency.

1.4.1.1 Introduction

Optimal control theory is mathematically rigorous and is amenable to the deriva-

tion of the mathematical formulation of abstract flow control problems. Over the past

several years, careful and rigorous studies of the analytical aspects of control for the

N ,'i.. ;-Stokes equations have appeared in the literature. Significant advances have

been made in the area of controllability. Imanuvilov [54] studied the local exact con-

trollability problem for the N ',. ;i -Stokes equations that describe an incompressible

fluid flow in a bounded domain with control distributed in an arbitrary fixed subdo-

main. It is shown that there exists a locally distributed control such that at a given

moment of time the solution of the N ,i. ;, -Stokes equations coincides with a specific

stationary point provided the initial condition is sufficiently close to it [18, 23, 35,37].

The careful mathematical work on the proof of the existence of optimal control for

some nonconvex problems was carried out by Fattorini and Sritharan(1992) [29]. In

a subsequent paper [30], the Pontryagin maximum principle was used to establish the

necessary conditions for the optimal control problems. Desai and Ito [26] formulated

two control problems that consider the driven cavity and flow through a channel with

sudden expansion. They established the existence and first-order optimality condi-

tion for the optimal control. General studies of various optimal control problems,

including boundary control, distributed control, 2D problems, 3D problems, etc., can









be found in Cuvelier [24], Fattorini [28], Fattorini and Sritharan [32], Gunzburger

and Lee [45], Hou and Yan [50], Hou and Ravindran [49].

Different optimal control strategies have been developed over the past several

years. Abergel and Temam (1990) [1] developed optimal control theory for sup-

pressing turbulence in a numerically simulated, two-dimensional N ,i1. -Stokes flow.

However, the control requires measurement and feedback of full flow-field information.

Choi and other authors (1993) [21] developed a more practical, suboptimal control

strategy seeking an optimal condition over a short time period. One advantage of

this method is that only I-- !!-measuim ii, ii are used for feedback. The suboptimal

control procedure was successfully applied to the one-dimensional stochastic Burgers

equations. Bewley and Moin (1994) [10] applied the suboptimal control procedure

first to a numerically simulated turbulent channel flow with 17'. drag reduction re-

ported. Ito and Kang (1994) [55] describe an approach to obtain suboptimal solutions

to the Hamilton-Jacobi-Bellman form of the closed loop solution for optimal control

problems governed by the Burgers and the Navier-Stokes equations. Later applica-

tions of suboptimal control theory are reported in [6, 20, 48, 65].

1.4.1.2 Framework of Optimal Control Problems

An optimal control problem is defined in such a way that it minimizes an objec-

tive function which is subject to the evolution equations and boundary conditions.

For example, Cuvelier [24] studies a distributed optimal control problem, where the

admissible control is selected such that the temperature distribution Te(x, T; v) at

time t=T is as close as possible to a desired distribution zd(x), x GE The objective

functional to be minimized is

J(v) Te(x,T; v)- zd(x)12d +a j vx,t)2ddt (1.6)

where F1 is the boundary where control is applied.









The first term of the functional J(v) is introduced by the motivation to minimize

the distance between the controlled flow field and the desired flow field. Different func-

tionals, e.g. disturbance rejection, vorticity minimization, can be defined in diverse

v-i-; for different problems depending on the applications in which the flow situation

occurs. Some popular applications are drag reduction [36,65], and transition delay,

for example.

The second term in the functional J(v) is regarded as penalty to achieve the

desired temperature distribution and is introduced to reduce the energy input. This

term is frequently considered in studies of different optimal problems with active

control.

For optimal control problems that arise with quadratic cost functionals and lin-

ear evolution equations, the feedback solution is often given in terms of the matrix

solution of the Riccati equation. The continuous algebraic Riccati equation takes the

form:


XRX + XA + A*X + Q = 0


where Q and R are Hermitian matrices, and A* is the complex conjugate of the

transpose of A. In particular, continuous algebraic Riccati equations arise in the

problem finding solutions which are optimal in the sense of minimizing some objective

functionals that are quadratic in x(t) and u(t), and subject to the linear evolution

equations x(t) = Ax(t) + Bu(t). The growth of interest in algebraic Riccati equations

(ARE) has been explosive over the past few decades. Primarily, this has been driven

by the important role p1 i, d by these equations in optimal filter design and control

theory. A prodigious number of research publications have appeared which have

steadily increased our understanding of these equations and their solutions.









1.4.1.3 Nonconvex Control

One example of a more general form of penalty term is

0T
/ -j(u(xt))dt


where 7( ) is a function of control u(x, t). Most of the literature quoted earlier deal

with the special case with 7( ) convex and Uad is a convex set. Since the function

7( ) is determined by engineering requirements (and limitations) it is desirable to

develop a theory without the convexity assumption. Fattorini (1995) [31] studies a

class of flow control problems where the fluid is controlled by a distributed forcing

at a portion of the boundary and the cost functional is nonconvex with respect to

the control variable. This choice is motivated by nonconvex flow control problems

that arise in practice where nonconvex functionals such as average lift to drag ratio

by distributed boundary control. In this paper, Fattorini formulated and proved

the existence of optimal chattering controls for the relaxed optimal control system.

This system is obtained utilizing the Young measures defined on infinite dimensional

control sets. Other practical nonconvex problems involving suppression of Karman

vortex shedding, obstacles moving in curved paths, etc. can be found in [80,81].


1.4.2 Model Predictive Control (Receding Horizon Control)

Model predictive control (\!PC) refers to a class of algorithms that compute a

sequence of variable adjustments in order to optimize the future behavior of a plant.

The technique is used to address the practical issues associated with the control of

large, multivariable processes where there are constraints on the systems. Originally

developed to meet the specialized control needs of power plants and petroleum re-

fineries, MPC methodologies can now be found in a wide variety of areas including

applications in the chemical, food processing, automotive, aerospace, metallurgy, and

pulp and paper industries.









The fundamental concepts concerning model predictive control can be traced

back to 1963 and the work of Propoi [74] where the moving horizon approach was

first proposed. Yet, it was not until the late 1970s and the contribution of Richalet

and coauthors [77], that the method was widely accepted. However, the applica-

tion of model predictive control is not limited to linear systems. There has been a

rapidly increasing interest in using model predictive control schemes of both linear

and nonlinear systems [58,69].


1.4.3 Gain Scheduling and Linear Parameter-varying Systems

One of the most popular methods for applying linear time-invariant (LTI) control

theory to time-varying and/or nonlinear systems is the method of gain scheduling.

The classical approach of gain scheduling involves the creation of a parameterized

plant, developing linearization at multiple equilibrium points, designing an LTI con-

trol law (point design) to satisfy local performance objectives for each design point,

and then interpolation of the controller gains (scheduling). This approach often

works well in practice. However, it does not take the parameter variations into ac-

count in the control synthesis and thus cannot provide performance and robustness

guarantees. A robust approach, by which a single controller is designed such that the

closed-loop system is robust to any value of the parameter within some anticipated

set of values, gives guarantees of both stability and performance. It may be overly

conservative to find such a controller when the values of the parameter can have

large variations. As a systematic means of synthesizing gain-scheduled controllers,

linear parameter-varying (LPV) systems have recently received considerably atten-

tion. LPV systems are characterized as linear systems that depend on time-varying

real parameters. These parameters are assumed to be exogenous signals that are not

known in advance but are constrained a priori to lie in some known, bounded set. It

is assumed that they can be measured in real time. The measurement of these para-

meters provides real-time information on the variations of the plant's characteristics.









Hence, it is desirable to design controllers that are scheduled based on this informa-

tion. The general form of a representative system can be written in state-space form

with state vector x, exogenous inputs d, control inputs u, regulated outputs e, and

measurements y

x A(0) B,(O) B2(O) x

e C (e) Di1(O) D12(0) d (1.7)

y C2(O) D21(0) D22(O) u

where S is a time varying parameter vector. The LPV system has a structure,

which is similar to a linear time-invariant state space system, and control design

methods with some similarity to linear state space methods can indeed be used.

Since the state matrices are written directly as functions of the parameter vector,

such a system is called a parameter-dependent model. LPV systems are different

from their standard linear time-varying counterpart due to the causal dependence

of their controller gains on the variations of the plant dynamics. It is possible to

treat gain scheduled controllers as a single entity, with the gain-scheduling achieved

entirely by the parameter-dependent controller.

1.4.3.1 Literature Review

Recently, several synthesis algorithms have been designed for the systematic

determination of gain-scheduling controllers for LPV systems.

A parameter-dependent controller synthesis technique based on small gain theo-

rem was proposed by Packard (1994) [70] for discrete-time systems and by Apkarian

and Gahinet (1995) [2] for continuous-time systems. In both papers, LPV plants are

assumed to be linear fractional transformation (LFT) dependent on the parameters.

The existence of such a gain-scheduled controller is fully characterized in terms of lin-

ear matrix inequalities (LMIs). The underlying synthesis problem is therefore a con-

vex problem and can be solved using standard convex optimization algorithms [14,40].









The resulting LPV controller is time-varying and smoothly scheduled by the measure-

ments of the parameter. This is a significant benefit over traditional gain scheduling

controllers because it guarantees the stability of the corresponding closed-loop sys-

tems for any value of parameters and also for any time-vary trajectory of the para-

meters. It was also demonstrated that the original gain-scheduling problem can be

re-formulated as one of robust performance with special plant/uncertainty structure

and the controller can be synthesized using linear HC,, control approaches [2].

An alternative approach to the gain-scheduling problem was recently proposed

by Wu and coauthors in [83]. The LPV system is allowed to have general parameter-

dependence. Instead of using the scaled small-gain theorem, Wu et. al. consider

a parameter-dependent Lyapunov function. This approach was first proposed for

the analysis of gain-scheduled control by Fromion and other authors [34]. The main

motivation here is to take into account a priori bounds on the rate of variation of the

parameters. Unfortunately, contrary to the previous approach for the gain-scheduling

problem, the resulting conditions can be checked only approximately, with a high

computational burden. More precisely, Wu's conditions require a griddling on the

set of the parameters in conjunction with the resolution of a large number of LMI

problems. This approach has one of the drawbacks of the traditional gain-scheduling

engineering practice: there is no guarantee on the stability of the closed-loop system.















CHAPTER 2
REDUCED-ORDER MODELING OF STOKES FLOW


Under normal conditions, most common fluids such as water, air, and glycerin

are Newtonian fluids which are defined as those for which shear stress is directly

proportional to rate of deformation. The main subdivision of Newtonian fluid flow is

between inviscid and viscous flows. Viscous flow is of paramount importance in the

study of fluid dynamics, and can be further classified as laminar flow or turbulent flow

according to the flow structure. Laminar flow moves in laminae, or l-v-. rs, smoothly

while turbulent flow moves in a random, three-dimensional manner. Reynolds num-

ber, which is defined as the ratio of inertial forces to viscous forces within the fluid,

is the criterion that determines in which regime is the flow present. It is expressed as

Re pVD


where V and D are fluid characteristic velocity and distance, p and p are density and

viscosity of the fluid. For example, for fluid flowing in a pipe, V could be the average

fluid velocity, and D would be the pipe diameter. Typically, viscous stresses within a

fluid tend to stabilize and organize the flow, whereas excessive fluid inertia tends to

disrupt organized flow, leading to chaotic turbulent behavior. Thus, flow with higher

Reynolds number will be more likely turbulent. For instance, we usually -v- that a

pipe flow is a laminar one for Reynolds number up to 2000, and a turbulent one if

the Reynolds number is beyond 4000. While between these two numbers, transitional

flow may be defined. This thesis focuses on study of nonturbulent flows.

The motion of incompressible nonturbulent Newtonian flow is governed by the

Navier-Stokes equations. These equations are nonlinear, partial differential equations









having infinite degree of freedom. Stokes flow is a particular case of N ',.-. r-Stokes

flow, obtained in the limit of vanishingly small Reynolds number. Stokes flow is

governed by a linear dynamical system, whereas the Navier-Stokes equations are not.

The viscous forces in this type of flow are much larger than the inertial forces, and

the convective term in the Navier-Stokes equations can be neglected. Stokes flow

is also called creeping flow. Practical cases of creeping flow are the fluid flows in

which a high viscosity leads to slow motion. The chemical and material processing

industries frequently deal with very viscous substances. In other cases, the fluid may

be of modest viscosity but the length dimension is small.


2.1 General Stokes Flow Problem

In this section, we will be concerned with the mathematical theory of Stokes

flow. Detailed analysis will be carried out in this section to set up a mathematical

framework for reduced order modeling of the Stokes flow problem. The analysis

is based on the non-dimensional analysis of N ,i-,. --Stokes equation. It should be

declared that the methodology proposed in this section is an abstract treatment which

is applicable to the general case of Stokes problem. The meaning will be significantly

reinforced by the following sections where the approach will be applied to a driven

cavity problem.


2.1.1 Stokes Equation and Time Scale Issue

Two dimensional incompressible viscous flow is governed by the Navier-Stokes

equation given by


S+ -Vp + AV (2.1)


with a characteristic dimension L, characteristic velocity Vs. As stated before, Stokes

flow is the case when the Reynolds number, Re -> 0. In this case, the convective









term can be neglected, and pressure force should balance the viscous stresses

P V
L L2

Thus, the appropriate non-dimensional pressure P* must satisfy


P = P*
L

Also, non-dimensional variables x* = y* = V* = t* = are defined.
L Y L' L/V,
Then the nondimensional Navier-Stokes equation becomes


Re-V + Re V* V *) -v*p* + A*V* (2.2)
at*

where the Reynolds number is defined as Re = pV. The convective term will be

neglected as well as the unsteady term when Stokes flow is considered, i.e. Re < 1.

The above equation is simplified to Stokes equation


*p* = A*V* (2.3)


which means the net forces on a fluid particle must add to zero. Flows governed by

Equation (2.3) are termed Stokes flows. Stokes (1851) was the first to propose this

simplification when he studied the problem of Stokes flow around a sphere.

The flows governed by the Equation (2.3) are quasi-steady since time does not

appear explicitly in this equation. It means that any time-dependent motion of

a massless fluid arising from unsteady boundary conditions is quasi-static. This

conclusion rests on the assumption that the time scale in the substantial derivative

term of Equation (2.2) is t* = t- To get the appropriate form of Stokes equation for

dedicated boundary conditions, we must consider also another time scale introduced

by the motion of a boundary condition.

For the incompressible N ,',i, r-Stokes flow governed by Equation 2.1 subject also

to boundary condition with characteristic frequency f, the new potential time scale









must be 4. If f is too big compared to the characteristic frequency !, the excitation

on the boundary varies too fast and it can not penetrate into the inner field. On the

other hand, it will be quasi-steady if f is very small, which is consistent with the case

discussed above. In this thesis, f is assumed to be comparable to the characteristic

frequency such that the control input can be used to change the velocity field.
-L-

Thus, we will consider the time scale 6 By keeping other non-dimensional

definitions, the resulting non-dimensional Navier-Stokes equation is


Re St7 + Re (* V *) -V*p* + A*V* (2.4)
at*

where St is Strouhal number defined as

Lf
St =f
V,

When (Re St) ~ 0(1) and the Reynolds number Re = L < 1, the terms on the

right hand side of the equation (2.4) will try to balance the first term on the left hand

side. The second term will be neglected. Thus the flow is approximately Stokes flow.

To simplify the equation, a new parameter 0 is introduced, which is defined as

the inverse of the product of the Reynolds number and the Strouhal number, i.e.

1 1
0 1 (2.5)
SA Re St

Now re-scale the pressure, P = P*. Then equation (2.4) becomes

a OAV* + Vp= 0 (2.6)
at*

This is the non-dimensional form of the governing equation for Stokes flow. For

convenience, it is written as

a eAv + Vp 0 (2.7)
at









2.1.2 Problem Formulation

The Equation (2.7) is a partial differential equation. To apply a model reduction

technique, a variational formulation and the finite element method are first used to

obtain its approximation. This process yields an ordinary differential equaion form.

2.1.2.1 Function Spaces and Auxiliary Notations

Before formulate for the Stokes problem, some variational notations should be

introduced. Assume a two-dimensional problem thus N = 2. Denote the bounded

domain f with continuous boundary 9O. In the usual way, L2( 2) is defined to be

the set of those functions that are square integrable in the domain f with norm


If L2(Q) f2(x)dx

Let H1(Q) be the Hilbert space of functions whose first weak derivative is in L2(Q).


H1() = f eL2 (2): E L2 (Q), i 1,2

and denote


H0(Q) {f e H1() : fl a 0}

Lo(2) {= feL2()L : ffdx= 0

The inner product defined in L2( ) is
2
(, 0 [L2(Q)]2 .1
i= 1

Introduce binlinear and trilinear forms

a( Z) 2 2 Oui 2 Ovi
2 2
-v -> /t -- -'- ] 'dx

b(uV,w) = iI U "1 .1r
i=1 j= 1 x








that define the mapping "u, i -- a(u, i)" on H1(Q) x H1(Q) and "u, ,w,-
b(u, v, wJ)" on H1(Q) x H1(Q) x H1(Q).
2.1.2.2 Weak Form of Stokes Equation
The Stokes problem is governed by Equation (2.7) and continuity equation


V- = 0


(2.8)


To solve it, the problem is first reformulated in a weak sense and its variational
formulation is derived. Let ( V,p) be a classical solution of the Stokes problem.
Multiplying Equation (2.7) with arbitrary E Ho(Q) and integrating over Q, by
Green's theorem


-0 (AV ()dx

I (Vp )dx


f. 12 2
~ / --^ds+ 0 i >i 1 j i xj

_(p_-)ds I (pV )d0x
Jan J1


2 2
(f adx+ + 0 (p. Y. 0
Oa i- i iji O x j

o --ds + (po*)ds = 0
Ja Oh an


1 7V V = 0

for all Vo n Vh and 7 G Ph. Since G Ho (Q), we have an
equations can be further simplified as


(2.12)

0. The above


S f + o 2 2 (pv. )dx 0
Si= i=1 j 1


Sy7V = 0


(2.13)


(2.14)


we obtain


(2.9)

(2.10)


(2.11)









These equations are the weak form of Stokes equation (2.7) and continuity equation

(2.8).

2.1.2.3 Galerkin Formulation

We shall now describe a method, called Galerkin's Method, for obtaining approx-

imate solutions to Stokes problem based upon the weak formulation. The basic idea

is to approximate the solution (V, p) and function by convenient, finite-dimensional

collections of functions. The variational equations are then solved in this finite-

dimensional context. The first step in developing the method is to discretize the

physical domain. Lets view our domain Q as "discrelI. into finite element do-

mains, or simply elements, Qe, 1 < e < NAe, where N]Ae is the number of elements

in the domain. Two type of simple elements-triangles and quadrilaterals, as shown

in Figure 2.1-are used most frequently for the two dimensional physical domain.

The vertices of elements are called nodal points, or simply nodes. It should be de-

clared that nodal points may exist anywhere on the domain but usually are chosen

to appear at the element vertices. Each node is numbered and all nodes are ordered

n = une











Figure 2.1: Finite elements


which is called global ordering of nodal points. Denote A, as the set of unknown

nodal points, and by Ar the set of boundary nodal points. The approximation of the








solution V(x, y, t) is assumed to have the form

= Vo, I Nk (x,y)
Vo,2 J kA Vk,2

In more detail, it can be written as

V., VkI (t)
V(x, y,t)= Ni(x, y) + l Nk(x,y) (2.15)
neAr Vn,2 keA I Vk,2(t)

where j is a constant vector corresponding to the boundary condition at node
n, and {V } is the unknown velocity corresponding to the inner domain node k.
N1(x, y) and Nk(x, y) in the equation above are referred to as shape, basis, or inter-
polation functions. For first order interpolation, shape functions are often chosen to
be piecewise polynomials. In this thesis, we choose bilinear quadrilateral elements as
shown in Figure 2.2.



(x3,y )


1,1) (1,1 (xf,yl) (x ,2)





Figure 2.2: Bilinear quadrilateral element domain and local node ordering

The unit domain is sometimes called the parent domain. Point (, rl) in the
parent domain is related to the point (x, y) in the physical domain Q by a mapping









of the form
4
x(, ) N ) 'x (2.16)
a=1
4
y(,'I) = Na('I) y (2.17)
al1



Assume that the bilinear shape functions have the form


Na(,) (1 + a)(1 + T1) (2.18)
4

By the above mapping rule, the following conditions is satisfied


X(a,Tra) = '

y(a, r/a) = Ya

A standard numerical subroutine can be built to calculate the required integrals over

the parent domain by Guassian quadrature rule. By the mapping rules above, the

integrals over the physical domain in the weak form of the governing equations can

be transformed onto the parent domain, and the standard subroutines will give the

required numerical results.

To distinguish with other common velocity notations, bold symbols are used in

this thesis to indicate nodal velocity values. For instance, the nodal velocity vector

associated with unknowns in the interior of the domain will be

V1(t) ]

V2(t) k, (t

V(t) < ( where k(t) k (t) k A,

V ( ) l ( )








while the nodal velocity vector for a prescribed velocity field will be


,V (t) ( >(t)


where Vr(t) is the part of the nodal velocity vector associated with boundaries, and

V,(t) is the part of the nodal velocity vector associated with the interior of the
domain. In more detail,

V1(t)
V2(t)
{Vr(t) < >, where V,(t) ne Ar

V,2(tt
MVn,2



V (t)
V2(t) A
V(t) < where Vk(t) ke A,

VV, (t()


Thus, Equation (2.15) can be expressed as

V(x, y, t) V N(X, y) + Vk,i (t)Nk(x, y), i 1, 2 (2.19)
neAr kceA
We choose


(x, y) =t Nm(x, y) m A,, s 1,2 (2.20)


el 1 2 4C21 0
e12 0o e22J {
















Figure 2.3: A pressure quadrilateral and its four associated velocity quadrilaterals

Also piecewise constants are chosen as the pressure space with respect to the larger

quadrilaterals of the subdivision Qh of Q as shown in Figure 2.3, which is divided

further into four smaller rectangles by joining the opposite midsides, thus creating

another subdivision Qh/2 of Q into rectangles. The velocity degrees of freedom are

simply function values at the interior vertices of the rectangles in Qh/2. There is one

pressure degree of freedom associated with each rectangle in Qh. This is required

to satisfy the Babuska-Brezzi Condition required for finite element models of fluid

flow [].Thus pressure field can be expanded into

Np
p = H,(x,y)pr (2.21)
r=1

where Hr(x,y) is the pressure shape function at rth pressure element (the larger

quadrilateral). Substituting equation(2.19),(2.20),(2.21) into eq(2.13), the following

equation is obtained
2
zf, (Z ^ + S i (k,,eiN)
i=1 neAr kceA

+a( Y (xj A iN~ + Vk,iNk) a (es,,iN)d1
i=1 j=1 neAr kEA,

HrPr (es,iNm) + (es,2Nm) d2 = 0
T=l








Collecting terms, it becomes
Sc Np H ON, dG Pr



+ 0 j 1 OXj OXj
5 / sNkNd V,,kdQ Vkxi
1 0 k 2 2 87 n d \AA1

2
[i jsfi j k n ]i

E YIj CsiOnNmd]
i1 nCeA,


i-1 nAr, j 1
s 1,2 nm A, (2.22)

Substituting equation(2.21),(2.23) into equation(2.14) and supposing that

7= H (x,y), r 1,2,... ,N (2.23)

the other governing equation is

2 H dQl Vk,i 2 He dQ Vi
i= 1 k Axi i= 1 nArxi
r 1, 2,... Np (2.24)

Equation (2.22) and (2.24) are the finite-dimensional approximation of weak form
governing equations. Now the infinite-dimensional PDE problem has been trans-
formed into a finite-dimensional ODE problem, which is feasible to do numerical
manipulations and to apply control strategies.
The governing equations above can be written into the matrix form

'V ] [0] V [A,] [B,]
[0] [0] 0 [B ]T [0] 1 2 2
(2.25)
[Ar] { [Mr]
[Br ] [ [0]








where

],, .)(ik) ,iNkNmdf s, i 1,2 m, k E A

[Mr](,s)(i,) es,iNnNmdQ s, i 1, 2 m c A, n E Ar

f Nk 8N
[A](msk) 2 d si = 1,2 n, k c A,

[Ar](ms(i,) es2 xs,i 1,2 m- A2 n Ar
ONd Is1,2 r 1,2 TeN, nGA1

[Bu](ms)(r) 8 d,2 r ,2,. p m A

[Br](n,s)(r) Hr dQ s= 1, 2 r 1,2,. -,Np n Ar

It should be noticed that in Galerkin procedure, the global ordering and ordering
of equations in the matrix system may or may not coincide with each other. The
number of equations is calculated by subtracting the number of boundary nodes from
the number of total global nodes. In the governing equations above, each element in
the coefficient matrices involves an integral. Standard subroutines will calculate these
integrals over the parent domain corresponding to each finite element. Subsequently,
the assembly procedure is used to construct the global coefficient matrices. More
detailed discussion of the finite element method can be found in [51].

2.1.3 Order Reduction Framework

Consider now how to solve the Stokes problem in Equation (2.25) for large sys-
tems. Suppose there is a two dimensional system with 106 interior domain nodes,
thus having a total of 2 x 106 unknowns. The maturity of development of computa-
tional fluid dynamics (CFD) techniques makes this huge problem solvable and yields
high-resolution solutions at fixed flow regimes. However, it may not be appropriate
to use CFD directly for control design. Though there have been efforts which at-
tempt to modify CFD into a useful tool for control design, the resulting models have
roughly the same number of states (unknowns) as the original CFD simulation. This









model size is too large and computationally cumbersome for control applications. So

reduction techniques have been motivated to reduce these large models by several

orders of magnitude without significantly changing the dynamic response. The result

is an accurate, easy to use, low-order model that takes less time to generate than

those generated by traditional means.

Fk(





r r



Figure 2.4: Geometry


Here the mathematical framework of a model reduction technique is described.

Consider the Stokes problem over the physical domain f as shown in Figure 2.4. The

boundary is given by


f9 = UFk U Fc
k

Boundary segments Fk, k = 1, 2,... might be separated curves with different con-

ditions. For example, for a particular problem we might have inflow and outflow

boundaries, and also some solid boundaries. Fc is the control boundary where control

activities are applied. The possible control activities might be moving walls, blowing

or sucking jets, or synthetic jets. It is assumed that the input velocity imparted by

the actuator can be represented in term of Nc functions i 1... Nc. Each of the

functions gi are defined on the entire domain f, but are assumed to exhibit specific

properties on the boundary. We require that


x,y) for (x,y) E F
o for (x, y) \ F,









Additionally it is required that


/ idS 0
an

for i = 1... Nc. This last condition is required to guarantee compatibility of the flow

field with the continuity equation. The actuator imparts a velocity along FP whose

spatial distribution is defined in terms of known functions ia,c.

Spatially varying functions Q', I 1 ... N, are now introduced to constitute the

reduced basis which is used to represent the Ns states in the control model. There

are several methods to generate (. Proper orthogonal decomposition (POD) has

been utilized in [5] [17], and by many other authors, to construct a reduced basis

determined via an optimality criteria. In contrast, effective reduced bases can be

derived from physical considerations. Ito and Ravindran in [56] and [57] have shown

via empirical numerical evidence that reduced order N ',;i. -Stokes simulations can

yield accurate reduced order models. It will be discussed in detail later in this chapter

how to obtain reduced basis by POD.

Assume the spatially varying functions for i 1... Nc have likewise been

derived from either particle image velocimetry or simulation. The functions com-

prise the influence functions that determine the controls acting on the fluid flow. It is

assumed that these functions satisfy the following conditions, which are conventional

in many reduced basis formulations:

V- = 0 for I-= ... N,

V ,* = 0 for i = 1... N,
(2.26)
Ian = 0 for =i 1...N (2

S,. for x f P, U F,









In terms of these reduced basis functions, the velocity field associated with domain
Q is assumed to take the form
Nc Ns
V(x, y, t) = Vo(x, y, t) + x, + y) + f1(x, y)al(t) (2.27)
i= 1 i= 1
Expressing velocity fields in nodal velocity vectors, one obtains


Vo(x, y, t) = V0,r,(t)N.(x, y) + S VO,u,kNk(x, y) (2.28)
neAr kCeA


(x, y) = 4 ,,nNn(x, y) + C~i^ kNk (x, y), 1= 2,... ,N (2.29)
neAr kCeA


(x, y) Y, ,rn(x, y) + Y 1,kNk(x, y), I = 1, 2,... N (2.30)
neAr kCeA

where Vo,r(t) and Vo,u(t) are nodal velocity vectors associated with Ar and A, respec-
tively. 4i,r and Ti,, are nodal velocity vectors associated with Ar and A, respectively
for i 1, 2,... Nc. The vectors (ir and 1 are nodal velocity vectors associated with
Ar and A, respectively for 1 1, 2,... N,. Denote


Vo(r i-
Vo0,. i,.

It is obvious from Equation(2.26) that


l,rF,nnA { } (2.31)
0








Thus, with the nodal velocity vectors, Equation (2.27) can be rewritten as


neAr
Nc,
+ S S 4i,c,nNn(x, y) 30(t) + V0o,.,k(t)Nk(x, Y) (2.32)
i=1 neAc kceA
Nc N,
+ 5 Y Y i,u,kNk(X, Y)3i(t) + Y ,1,kNk(X, y)al(t)
i-1 kceA 11 kceA
The velocity l(x, y, t) can be separated into two parts-one is associated with
A,, the other corresponds to Ar. According to Equations (2.32), the nodal velocity
vector for A, can be expressed as
Nc Ns
V(t) Vo,n(t) + i pPA(t) + 5 11ai(t) Vo,.(t) + [] j3(t) + [4] (t) (2.33)
i-i l= 1
where L] is the collection of control functions f,,u, for i = 1, 2,. Nc, [4] is the
collection of reduced basis <:, for I 1, 2,... Ns. Denote

L 1 1 ,, q2,u, qNzN,u]
[ c j1,l,, 2,c, Ncc

[1.1 ^ 'M
E-H-
] A 1, 2, N]... ,]
Substituting eq(2.33) into the upper part of eq(2.25), one obtains

[ I{ V0Mt) [ + jN +t) [at)]

+[Au] {Vo,L(t) + [u] /(t) + [1] a(t)} + [B,]P(t) (2.34)

= -[Mr]Vor(t) [MC] c (t) [Ar]Vorho(t) [Ac,] (1)








Premultiplied by []T Equation(2.34) is rewritten in the matrix form


[ [q ]T[A.][dQ] [4 ]T ( [A.] [ 4P.] + [A,] [t]) ) }


+( []T[A][] []([A, +[A][]) (2.35)


= [lT ]- ]Vo,11(t) + [Mr]Vo,r(t))

[]T ([A,]Vo,.(t) + [Ar]Vo,r(t) -[i]T [B ](t)

For the reduced basis j(x, y), no matter if it was derived from POD or the method
introduced by Ito and Ravindran in [56] and [57], its associated nodal velocity vector
can alv-- be interpreted as


= l,ml,m (2.36)
Im

where dil,m are the coefficients of full order Stokes solutions. Substituting Equation
(2.23) into weak form of continuity equation (2.14), one obtains
2
H, Hr, z l,m(m),nNn + lm(1m),kNk dQ = 0 (2.37)
s=1 nGA, kceA mn
r= 1,2,.- ,Np

It can be written as

y ,2 HNk dQ) d(l,m),k
2
n s=1 keA

+ i, (f an 8 d (m),n (2.38)
m s=l neAr


r 1,2,... ,Np









Recalling that


f aNkd
H dox,


s 1,2 r 1,2,.- ,Np k A,



s 1,2 r 1,2,..- ,Np nE A


Substituting these expressions into Equation (2.38)


S l, m[Bu Tlm A, + .lm[B] d, UA
m, m


According to Equation (2.36),


[B,]T i + [Br] 1


By Equation (2.31)

[B,] i = 0

Thus, this expression can be written in matrix form


[B]T[,]= 0

Consequently, one can take the transpose to obtain


[]T [B,]


So, finally, it is possible to write


[4]T[B,] = 0


(2.41)


[BR](k,s)(r)



[Br](n,s)(r),


(2.39)


(2.40)







This equation shows that the pressure term in the Equation (2.35) vanishes. Thus,
the reduced-order model (ROM) of general Stokes problem is




+ []T[A.][
= -[T ]Vo,0(t) + [Mr] Vor(t)
[4]T ([A,]Vo,.(t) + [Ar]Vo,r(t)
The governing equation for the ROM of the Stokes Flow Control Problem is

[s ll[ ][4)] ([-V] [v])+ [X] [^c]) fa(t1


[I []T[A][I ] [)]T ([A,] [,,] + [A,] []) (t) 2.43)
0 0 j()
_-[4]T () ]Vo,_(t) + [Mr]Vor(t [@]" [A,]Vo,.(t)+ [Ar]Vo,r(t))


+ N(O)
I








where I is the identity matrix. Actually, by using inner product and bilinear form
defined at the beginning of this section, this equation can be written as



0 NI (t
S[a( )] [a(, )]] {()( })


0 0 NOt J

a -a(j,Vo) -(1i, &o) 0 L(t).

0 I '

for i, 1 1, 2,... Ns and j 1, 2,... Nc. Where [a(+i, 4I)] represent matrix whose
element at ith-row and jth-column is a((i, I1). Similar definitions hold for the other
matrices in the above equation.

2.2 Driven Cavity Stokes Flow Problem
Control of driven cavity flow has become one of the standard examples [56] in
the flow control literature. In this section, this particular type of flow problem is
studied using the mathematical framework deduced in the last section. Stokes flow
is considered only. Control applications are mostly concerned for the reduced order
model. The next section will show with numerical examples how well this model
performs.

2.2.1 Control Problem Description
Consider the physical domain in Figure 2.5. The boundary of the square domain
Q is denoted by


89 = FB U FR U FT U FL









\u= l(x)h/(t)
fixed input: uv = o


T1V
rT


TB


control: u=1(x)P(t)
[ v=0

Figure 2.5: Stokes driven cavity flow problem


Assume static solid boundary conditions on the left and right sides, FL and PR. The

fluid in the cavity is driven by the imposed flow on the top FT of the domain

u = 1(x)ho(t) x E FT

v 0 x FT

The flow is controlled by modifying the flow on the bottom of the domain FB

Ju 1 (4W 3(t) x cB

v 0 x E FB

Moving wall is used as the physical control.

For incompressible Stokes flow, the governing equations are the equation (2.7)

and the continuity equation (2.8). Now the control problem for the driven

cavity flow is described as

To find the ;. ... :1,/ V and pressure p that minimize -.i,,, performance func-

tional" and also <,li.fy the governing equations

-OAV + Vp 0
at
V- 0









subject to

VrL 0 Vt [0, T]

VF 0 Vt [0, T]


-0 -0
t (X, T)/3ho(t)


V FB = 1(, y 0(t) control
0

The performance functional will be discussed in more detail in the sections that follow.

2.2.2 Reduced Order Model of Driven Cavity Problem

Because a full refined discretization of the flow can by computationally pro-

hibitive, it is motivated to seek a reduced order model for the flow. Assume the

velocity field is approximated by
Ns
V(x, y, t) Vo(x, y)ho(t) + (x, y)03(t) + f(x, y)ai (t) (2.45)
l=1

In this equation, V0 is the solution of the full, steady Stokes solver for the boundary

conditions


Vor and VoIBr U r UrL (2.46)


Vo(x, y) and ho(t) together describe the cavity field when the time varying disturbance
ho(t) applied on the top, while all other sides have zero velocities. Comparing this

situation with Equation (2.27), here the mean velocity field o(x, y, t) is separated

into the spatially distributed function Vo(x, y) and function ho(t). In general, as in the

nonlinear Navier-Stokes flows, this can not be true. However, it works for Stokes flow

because the flow is linear and any change in the boundary will induce a proportional

change in the velocity field in the domain. Due to this reason, it is only possible








to have one control basis that satisfies the above boundary conditions for the fixed
cavity geometry, if the control basis (zx, y) is chosen to be the full, steady Stokes
equations for the boundary conditions

10}
r =B {} and |IPTUPRUPL { (2.47)


Thus, there is only one control variable 3(t). It is required that the reduced basis
associated with the states satisfies zero boundary conditions over the boundary 91.


rrTUrBUrRUrL {}, for l 1,2,... ,N, (2.48)
{00

To generate the reduced basis Q, possible methods include POD, or the method
described by Ito and Ravindran [56]. It will be discussed later how to use these two
methods to test the validity of the reduced order model.
By finite element method, the above velocity fields can be expressed using nodal
velocity vectors

V0(x, y) = Vo,T,nNn (x, y) + V0o,,kNk(X, y) (2.49)
neAT kCeA


((x, y) >= B,nNn(X, y) + > .,kNk((X, y) (2.50)
neAB kceA


01(x,yY) > ,,kNk(x,y), for 1 1,2,... ,N, (2.51)


From conditions (2.46) and (2.47), it can be obtained that Vor,T { ( } and W'B,n

{ 1 }. Now the ROM for the general Stokes problem can be written into a specific









form for the driven cavity control problem using the above notation

[4 ]Tl[3 ][4[] [,]T C zo a+[Ms] (t)

[ 0 1 (t)

S [A. ] [] ([A,]T + [ABI) ] (t) }
0 0 N(t)


[ -4 (I ]Vo,. + [Mr]Vo,r) ho(t)

00


+ f(t)
1


[4]T [AI.Vo,, + [ArlV,r ho(t)


(2.52)


In this equation, the following matrices are defined:


Se,iNkNmd Q

j es,,N,,NdQ


= E es,iNNmdGl
2 1 N k ON ,
j=1


s,i 1,2 m, kE A,

s,i 1,2 me A. n AT

s,i 1,2 m A. nE AB


s,i =1,2 m, keA


2 8 NT N,
S 13 esi -dQ"


2v N,/ O ,
#=1xj 9x d


L N

J 8 9x,
H, OxN,
L aNO
Hn odQ


s,i=1,2 meA, n AT


s,i 1,2 meA~ nE AB


s 1,2


s 1,2 r


1,2,- ,Np mE A,


1,2,. Np nE AB


'If 1 -, )(i,k)



[M-B](ms)(i,n)


[Ar](m,s)(i,n)


[A]B (m,s)(i,n)


[Bu](m,s)(r) =

[BT](n,s)(r)

[BB](n,s)(r)


s 1,2 r =1,2,... Np n AT









2.2.3 Stationary Driven Cavity Flow

The validity of reduced order model for the driven cavity flow is concerned now.

A simply case-stationary driven cavity flow-will be tested first. More complicated

flows will be dealt with in the next section. Stationary driven cavity flow corresponds

to the case when the cavity is subject to a steady driven flow on the top boundary

and no control is applied. Thus, the boundary conditions are zero except on the top

side, FT. We assume a unit horizontal velocity on the top of the cavity. Graphically,


1



0 0


0


Figure 2.6: Stationary driven cavity flow


the stationary driven cavity flow must satisfy boundary conditions as shown in Figure

2.6. In this case, the low dimensional approximation of the velocity field is simplified
N.
V(x, y, t) = Vo(x, y)ho(t) + > pi(x, y)ai(t) (2.53)
l=1

and the ROM (2.52) will be written as



[]T (r. ]Vo0. + [Mr]0,T) ho(t) (2.54)

[] ([A]Vo,.] + [Ar]TVor ho(t)

Now the procedure (refer to [56]) to generate the mean velocity Vo and reduced

basis fi is introduced. First, a collection basis elements {uli}f, are obtained from full,

steady Navier-Stokes solver for M different Reynolds numbers. Each basis element is









2.3 POD and Unsteady Driven Cavity Flow Problem

It was shown in last section that the mean velocity field Vo and the control basis

i) can be obtained from the full, steady Stokes solver. The construction of the reduced

basis Q was discussed also for stationary driven cavity problem. This section continues

the development of ROM from POD. The theory supporting POD, including the basic

numerical procedure and useful conclusions, will be discussed first. Then, it is shown

that POD can be used to construct reduced order models for unsteady driven cavity

flow.

2.3.1 Proper Orthogonal Decomposition (POD)

The proper orthogonal decomposition (POD) is a systematic technique to obtain

low dimensional models. It has been successfully applied for fluid flows. The aim is

to generate an optimal basis to represent an ensemble of computational data. The

basis constructed by POD is optimal in the sense that the energy contained in it is

greater than any other basis having the same dimension.

2.3.1.1 POD Procedure

The goal is to find basis functions {(x, y) }1 which could elegantly represent

the given ensemble data {uk}I This collection of data is sometimes called snapshots

of the solution at some fixed time instants, where u G L2 and L2 is Hilbert Space.

Mathematically ( should be chosen to maximize the average projection of u onto .

Our maximization problem is

max ( ) subject to I|| 1 (2.55)
A (L2([0o,1l) 11 112

where (.," ) and || || denote inner product and norm on L2 respectively. The brack-

ets (-) denotes a time average operation. To solve this problem, we introduce the

Lagrangian


L[] =( (it, 2) A( ,2 ) (2.56)









where A is the Lagrangian multiplier. A necessary condition for the calculation of

extrema of this functional is that its Gateaux derivatives vanish for all variations

6 L [+ ] 0o = 0


Carrying out this calculation yields





Denote the operator R = fJ(u'),'.1,/~ The above equation is written as


R A= (2.57)


which is an eigenvalue problem. An orthonormal basis {(j(x, y)}) can be extracted

from the eigenvectors, i.e.

N

j=1


2.3.1.2 Optimality Properties of the POD

Assume that the velocity field u(x,y,t) in L2 is stationary random and that

4i, A |i = 1,... oo, c; Ai > A1 > 0 is the associated set of POD basis vectors and

associated eigenvalues. Let


~x, y, t) aj(t)j (x, y)
j=1

be the decomposition with respect to the POD basis and

N

j=1

the decomposition using any other arbitrary orthonormal set. Then the following can

be shown to hold.

(i) (ai(t)a*(t)) = 6ijij. The POD projection coefficients are uncorrelated.









(ii) For every n we have


0i(t)a* ()) > > f ( ) (2.58)
i= 1 i 1 i 1
This equation states that POD is optimal in the sense that the first n POD basis

functions capture more energy then any other basis n basis vectors.

(iii) If the ensemble is written in matrix form as A with each column representing

a single snapshot at time ti, then


ck+1 min IA B 2,2 = A Ak I2, (2.59)
rank(B) k
where Ak >1i i'uf, U [,... ,Um], V [,... v, and A USVT
i=

This property shows that


|I7 Pk(Ui) \2 <
where the operator pk(ui) is used to define the projection of the arbitrary vector ui

onto the subspace formed by using k number of POD modes. It states that, in theory,

the error between a given snapshot at a particular time step and the reconstructed

approximation (using k modes) should in fact be bounded by the singular value, 7k+1,

of the ensemble contained in A.

2.3.1.3 Energy in the Eigenfunctions

The kinetic energy in the extracted optimal eigenmodes is


E- (i u(x,t)u("x,t)dx) a iat)a*t) i) JAj -= k (2.60)
\ij ) i k

This means that the energy in each eigenfunction is just the corresponding eigenvalue.


2.3.2 POD for Unsteady Driven Cavity Flow

Now low dimensional approximation in equation (2.45) is used for description of

unsteady driven caivty flow. The generation of the mean velocity field V0 and control









basis i have been described in some detail. They are constructed by the full, steady
Stokes solutions which satisfy (2.46) and (2.47). It was shown that POD can be used

to obtain a optimal orthonormal basis set given an ensemble of computational or
experimental data. Now POD method will be investigated to generate reduced basis

for unsteady driven cavity flow by POD.

First, an ensemble of data {VD(tk)}1 is obtained from solutions of the full,
unsteady Stokes equation where V (tk) is the full order steady solution, at time

instance tk, corresponding to the time-varying boundary conditions


r {cos(27rft) } sin(27rft) D L { rU}


for t E [0,T]. Then, for the same time set {t,2, ... ,tN}, instead of unsteady

solutions, a series of static full order solutions {VsT,k} =1 is obtained, where each
solution VST,k satisfies boundary conditions



0 0

for k 1, 2,... N. It should be kept in mind that this is not an ensemble of

dynamics solutions. It is just a set of static solutions with each vector corresponding
to different boundary conditions. Similarly, another static ensemble of full order

solutions, {VsB,k}N =1, is obtained, where each vector satisfies boundary conditions

sin(27 ftk) 0
0 0
VST,k FB { sin (2ftk) VST,k rTUUrRL { 2}


Remember the reduced basis { }7 is required to satisfy boundary conditions de-

scribed in Equation (2.48). So the above data ensembles have to be processed we got
before POD is used to generate the reduced basis. Equation (2.48) implies that the

ensemble of data to be used in POD must have zero boundary conditions over all of







47

90. A new ensemble of data is generated by subtracting VsT,k and VSB,k from D (tk)

for each time instance tk.


Vk VD (tk) VST,k -VSB,k


Graphically, this process is shown in Figure 2.8. Now POD method is used to generate

cos(2nftk) cos(2nftk )
0


o vD(tk) O 0


sin(27nftk


Tk 0 0

0


Figure 2.8: To generate ensemble to which POD is taken


the reduced basis {J(},i based from the new ensemble {Vk47 The resulting POD

orthonormal basis will have zero boundary conditions. For the particular cases 0

0.1, 0 1, and 0 = 2.5, Figures 2.9-2.11 show the first two eigenmodes respectively.

Tables 2.1-2.3 list the first six largest singular values respectively. It is clear from

thesis figures and tables that the first two eigenmodes will be dominant for each case.

Later on, these POD eigenmodes will be used for model reduction. Since the


















Figure 2.9: Contour plots of POD eigenmodes with streamlines at 0 = 0.1


reduced order model (ROM) is based on the non-dimensional description of Equation


VSB.k 0


sin(27ftk














II I IL ,, t i'

II A











Figure 2.10: Contour plots of POD eigenmodes with streamlines at 0 1


















Figure 2.11: Contour plots of POD eigenmodes with streamlines at 0 = 2.5


(2.7), which is valid for 0 ~ 0(1), it is necessary to discuss the range of 0 for which

the ROM is valid. To assess the validity, numerical simulations are carried out for

three cases: 0 0.1, 0 1, and 0 = 2.5.

Now POD basis {}i 1 is generated. Before considering using the POD basis

for reduced order models, error analysis is necessary in order to determine if the

method of POD is appropriate. The error here is defined as the difference between

the reference solution and its projection on a suitable number of POD eigenmodes.


I









Table 2.1: Singular values for 0 = 0.1

a1 3709.6
a2 344.65
a3 3.4099e-8
a4 4.2987e-10
as 2.0972e-10
a6 3.5968e-11

Table 2.2: Singular values for 0 = 1

a1 5053.3
a2 607.91
a3 1.4597e-8
a4 1.1099e-11
a5 1.0314e-11
a6 1.0111e-11


In this case, it is the difference between the actual full order solutions {V1}7 1and

the POD reconstructed velocity fields.

Introduce 2-norm of a vector x E IR


12 (Ai= 1


Denote vector ekm) as the error, for time instance tk, between Vk and POD recon-

struction with first m eigenmodes


e = V E V (2.61)
i=l
Theoretically, the POD reconstructed velocity fields should become closer to the

reference solutions as the number of eigenmodes increases. If nodal vectors are used

in the equation (2.61) instead of Vk and et, a nodal error vector Ce- -.) is

introduced, where Nd is the number of nodes in the unknown domain Q\0Q. The

accuracy of approximation is defined by the f2-norm of this error vector, namely

eC,' ek) I2. However, for the unsteady problem above, we might actually care
about the average error over the entire time period. In this case, the error is defined









Table 2.3: Singular values for 0 = 2.5

a1 5144.2
o2 616.66
a3 1.1779e-8
a4 1.1955e-ll
as 1.0716e-11
a6 1.0014e-11


as

( N N tN '2Nd \ 2
) () Yf C (2.62)
T l L2k T .' 114 I11: T J
k=1 k=1 k=l j=l

Now the feasibility of application of the POD method for the reduced order model

of unsteady Stokes problem can be investigated. Three examples will be tested for

S0.1, 0 1 and 0 2.5.



(i) 0 0.1

Figure 2.12 plots the average error versus m, the number of modes used in

the reconstruction. It shows that the error decreases as the number of eigenmodes

increases. The error drops very fast after the first few modes. When m > 5, the

error goes below 10-6. This tells us that we can use as few as 5 POD eigenmodes

to reconstruct velocity fields with an accuracy of the order of 10-6. The result is

excellent because in a reduced order model we want to use as few modes as possible.

This result tell us that the POD might be an appropriate and efficient method to

construct reduced basis for model reduction. This will be discussed in detail later.

It has been discussed before that the error between the full order solution and

POD reconstructed approximation with m eigenmodes should be bounded by the

singular value, am+l, of the original ensemble. The theory can be confirmed by

inspection in above Figure 2.12 and Figure 2.13, as well.









2.3.3 ROM by POD for Unsteady Driven Cavity Stokes Flow

Now the approximation of reduced order model will be studied for the unsteady

Stokes flow in a driven cavity as shown in the Figure 2.18. Assume that on the top

side of the cavity, the driven flow has a velocity ho(t) = cos(27rft), and on the bottom

uu =l(x)ho(t)
v=o

TF


rF
fu=o {u=o
--- --I --- --- ---
v=0 Lv=0



U = l(x)U( (t)
L 1 v=0
x

Figure 2.18: Unsteady driven cavity flow


side of the cavity, the flow is subject to the boundary condition P(t) = sin(27rft).

The POD eigenmodes obtained in last section are used as our reduced basis.

Combining these modes with the mean velocity field Voho(t) and control basis i, now

it is possible to obtain a complete reduced order approximation of velocity, using

Equation (2.27), of the unsteady flow. The initial condition, VROM(t = 0), is chosen

to be the solution of the static Stokes flow model with uniform horizontal velocity on

the top and zero boundary conditions on other sides of the cavity.


Vr, -{} and VBUrnUr L Ur


Using the full order steady Stokes solver, the initial flow field can be obtained as

shown in Figure 2.19. Now the reduced order model can be used to approximate

the unsteady solution which is governed by Equation (2.52). Discard the transition

response and save the steady solution for one cycle. Both the full order and the

reduced order solutions are obtained at same time instances tk, k = 1, 2,... N.























Figure 2.19: Contour plot of initial velocity field

Then error is calculated at each time step. The error is defined as the f2-norm of the

relative nodal error between the full order and reduced order solutions

m ROM(tk) -Full(tk) 112 (2.63)
eL2,k i (2.63)
||VFu (tk)11 2

The same cases as before are tested with 0 = 0.1, 0 1, and 0 = 2.5.



(i) 0 0.1

Since Stokes flow is under consideration, a very viscous glycerine is an appropriate

fluid example. Assume a 99.,!-'-. glycerine is inside a Icmx 1cm cavity (i.e., L = 1cm)

with the top boundary x-velocity u = 1cm/s. The density and absolute viscosity at

20C are p = 1.25958g/cm3 and p = 1259.2mPa Sec respectively' Thus the

Reynolds number associated with this fluid is Re = P" 0.1. It is a Stokes flow. If

the top boundary is changing with a frequency f = 1kHz, then the Strouhal number

St = 100. Thus, the cavity flow is characterized by the parameter 0 = 1- 0.1.

Figure 2.20 plots the error at different time instances. The number of POD

eigenmodes used for the reduced basis varies from 1 to 6. The relative error is large.


1 http://www.dow.com/glycerine/resources











t = 9.2 sec

- Full Order Sol
.... ROM: 1 mode
- ROM: 2-6 modes


II,
z


0.8 1


II

z














6
lr,



















I I
z


lr,
6
I I
z


0.2 0.4 0.6 0.8 1


t = 9.4 sec

- Full Order Sol
.... ROM: 1 mode
- ROM: 2-6 modes


-0.5 F


0 0.2 0.4 0.6 0.8

y
t = 9.8 sec


0.4

0.2

0

-0.2

-0.4

-0.6
-0.8
-1
0


Full Order Sol
.... ROM: 1 mode
ROM: 2-6 modes


0.2 0.4 0.6 0.8 1


Figure 2.21: Error analysis of ROM approximation at 0 = 0.1


and capture nearly all the energy-recall that the energy of the ensemble projected

on an eigenmode is the corresponding eigenvalue. Thus, beyond this number, adding

other eigenmodes to the reduced basis does not help much. Figure 2.23 shows the

comparison between the full order solution and reduced order approximation. It only

plots the x-velocities along the vertical centerline. The figures show that the reduced

order solutions are very good approximations of the full order solutions. The error

analysis and the above comparison illustrate that the reduced order model is a good

approximation of the cavity flow in the case 0 =1.


(iii) 0 = 2.5


y
t = 9.6 sec


0.4

0.2
0

-0.2

-0.4

-0.6
-0.8
-1
0


















II,


1


0.5


II o

-0.5


-1


t = 9.2 sec
I
Full Order Sol
3 .... ROM: 1 mode
ROM: 2-6 modes









0 0.2 0.4 0.6 0.8 1

y
t = 9.6 sec








Full Order Sol
3 ROM: 1 mode
S ROM: 2-6 modes

0 0.2 0.4 0.6 0.8 1


0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1
0


t = 9.4 sec

- Full Order Sol
.... ROM: 1 mode
- ROM: 2-6 modes


0 0.2 0.4 0.6 0.8 1


y
t = 9.8 sec







Full Order Sol
.. ROM: 1 mode
ROM: 2-6 modes


0.2 0.4 0.6 0.8 1


Figure 2.23: Error analysis of ROM approximation for 0 1


V, associated with the cavity. In the case 0 0.1, it so small that the first term in


the equation would be the only large term-the pressure has been rescaled there

is nothing to balance it. A physical explanation of this fact is that the boundary

conditions change so fast (f = 1Khz) that there exists a l-v -r along the top surface

where unsteady effects are confined. Set the origin of a coordinate system at the

lower left corner of the cavity. The local y variable will be scaled, that is



r=(Y -l)/6


where 6 is chosen so that there is a balance between the time derivative term and the

Laplacian operator. This analysis leads to the choice



SpfL2


Ir,
O
I I
X


lr,
6
I I
X


-












t = 9.2 sec

- Full Order Sol
.... ROM: 1 mode
- ROM: 2-6 modes


II


'O
I I
' X


0 0.2 0.4 0.6 0.8 1


0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1
0


t = 9.4 sec

- Full Order Sol
.... ROM: 1 mode
- ROM: 2-6 modes


0 0.2 0.4 0.6 0.8 1


y
t = 9.8 sec






Full Order Sol
.. ROM: 1 mode
ROM: 2-6 modes


0.2 0.4 0.6 0.8 1


Figure 2.25: Error analysis of ROM approximation for 0 = 2.5


The region of the flow influenced by the rapid oscillations of the lower surface is too

thin to reach all the way across the domain.

Numerical results for other 0 values show that an example region at which the

reduced order model works well is 0 > 0.5. This 0 region will be used in later on

numerical simulations. For the glycerine-filled cavity flow, the frequency for 0 = 0.5

is calculated to be f = 200Hz. In this case, the reduced order model is appropriate

for f < 200Hz.


II


0 0.2 0.4 0.6 0.8 1

t = 9.6 sec
t = 9.6 sec


I I
X
>


-0.5 [


-














CHAPTER 3
OPTIMAL CONTROL OF DRIVEN-CAVITY STOKES FLOW

The last chapter has discussed how to use model reduction techniques to obtain

low order descriptions of fluid flow models having high dimension. It has been shown

in numerical examples that the reduced order model for driven cavity flow can capture

the essential structures of fluid flow using very few states. However, this dissertation

investigates if the reduced order model is appropriate for control applications. In this

chapter, the feasibility of the reduced order model for different control strategies is

investigated.

Optimal control for fluid flows will be the first concern. Many flow control

problems are cast using optimal control theory. The mathematical structures of

these problems are amenable to abstract control formulations. A widely accepted

approach is to use the methodologies of the calculus of variations to find the governing

equations. The motivation for this approach is that a single mathematical framework

can be used for many different practical problems.

3.1 Variational Operator 6

The operator 6(.), called variational operator, is widely used in engineering texts.

It pliv a fundamental role in the derivations that follow. It has properties:
(P1) 6(.) acts like differential d(-)

(P2) t = 0 for the independent variable t

(P3) 6()l boundaries 0
For example, for the functional


J f (x:, y (:), y'(x)) dx
GJ/b








there have

6J 6 f (x, y (x), y(x)) dx
/b
= ~ (f(x, y(x), yt(x)) dx
db
b Qf f Of f 6Y
+= +6x +}6y + by'dx
jb{ +x iy ay'
Since 6x = 0 according to property (P2), the first term in the bracket is zero. Inte-
grated by parts to the third term, the third term can be rewritten

b Of d f f b
6J = o\ 7 +
I l9y dx \ y' O \ y' a
By property (P3), the last term is zero. If the virtual variation 6J must be zero for
all choices of 6y, the integrand must vanish. The governing equations are therefore

+f d + f
ay dx \9y' -

3.2 Optimal Control Framework
Optimal control theory in many applications considers a family of optimization
problems that have a surprisingly similar structure. It seeks to find u*, or con-
trol input, that minimizes a cost functional J, while simultaneously satisfying a set
of equations of motion and constraints. The first chapter introduced mathemati-
cal framework for a distributed control problem stated by Cuvelier [24]. The cost
functional, J, for that problem is a quadratic function which is subject to the Navier-
Stokes equation. In general, the equation of motion and the cost functional can be of
any form. For instance, in a common collection of problems, the evolution of physical
states x(t) satisfies a nonlinear ordinary differential equation and initial condition

x(t) =f(x(t), u(t), t)
(3.1)
x(0) = zo









In this equation, the variables satisfy

x(t) G RN

u(t) e RM

f: R N x R x R -> RN

For the control of fluid dynamics, the physical states x(t) may be velocities, pressures,

temperatures or some combination of these observables. The controls u(t) might

represent imposed voltages, currents, force loadings, or flow rates that alter the nature

of the response embodied in the state variables x(t). u(t) might have to satisfy

some physical constraints thus has to be acceptable. Each control input "d i,' the

evolution of the states and generates x(u(t)). The ranking of competing controls is

accomplished via the introduction of the cost (error, or objective) functional J(u).

The cost functional may represent the error that a certain control produces in a

target tracking problem, or the fuel cost incurred in a spacecraft mission, for example.

Typical cost functionals have the form


J(u) Q(x(T), T)+ L (x(r-), u(r), T)d (3.2)

In this expression, (., -) is a measure of final state cost, while L(-,.,-) represents

accumulated errors or costs incurred over the entire trajectory. These functions define

mapping

0: RN x R--- R

L: RN x RM x -- R

The optimal control problem can then be stated as follows: seek u* E U such that


J(u*) < J(v) Vv E U (3.3)


where J(.) is defined in Equation 3.2 and the trajectory x(r),r [0, T] appearing in

J satisfies Equation 3.1.








3.2.1 Fixed Terminal Time Problems
This section will discuss how the calculus of variations can be employ, ,1 to derive
necessary optimality conditions for some optimal control problems. It only considers
those problems subject to the constraint that the trajectory x(r), r E [0, T] appearing
in the cost functional must satisfy the governing equations of physics.
The problem of finding an optimal control for a fixed terminal time is precisely
the control problem summarized in Equations (3.1)-(3.3). It seeks u* E U such that

J(u*) < J(v) Vv E U (3.4)

where

J(v) (x(v), v) = (x(T), T) + L(x(r-), v(-), 7)d
Jo

and where the trajectory x(r), 0 < < T is subject to

x(t) f(x(t),u(t),t)
(3.5)
x(0) = zo

Conventional approaches to this optimal control problem define an augmented cost
functional in terms of Lagrange multiplier p(t) E RN as



d*(x,u, p) = (x,) +J pT (7){f (x(r), u(r), 7 (r) }dr (3.6)

and subsequently treat the problem as an unconstrained optimization problem. In
Equation (3.6), It has used the notation
N
YSpf p f (3.7)
i=1









to simplify an expression of the inner product of two vectors. The symbol f denotes
a column vector and pT denotes a row vector. Similarly, if h: R' -- Rt, denote by


hx- x jhi,... (3.8)

Thus, 2 is a matrix having 1 row and M columns, for example, since u G RM in this
section. Again, it is conventional to integrate Equation (3.6) by parts
*T
a*(x, u,p) = a(x, u)+ j pT(7)f(x(7), u(7), 7)dT
o (3.9)
pT (T)x(t) + jpT (T)x(T)dr
0 Jo
and define the Hamiltonian (x, u,p, pr)

X(x, u, p, T) = L(x, u, T) + pTf(x, u,T) (3.10)

Hence, the cost functional is

*(x, u, p) = (x(T), T)+ j (x(T), u(T),p(T), T)dT
o (3.11)
+ j ] (r)x(r)dT_- p(7)X(7)

Now, via a formal application of the calculus of variations



6 u*(x,u,p) 6x + u + p (3.12)
ox ou op

From equation (3.6), a formal application of the variational operator 6 yields

p6= 6pT (r){ f (x(), u(T), T) j(T) } dr
Op 7 }(3.13)

{ f (x(rT), u(rT), T) (T) } T6p(r)d
0J


Similarly, from (3.11)











_--Ju J %H(xc(T), u(r), p(r), Tr)6u ()dr (3.14)
u Jo Ou

and

0*6x= \ {p(T)+ (xr),u r),p r),7)}6x(r)dv
X j Oax

+ (x(T), T) x(T) + 0(x(T), T)6x(0) (3.15)

pT(T)6x(T) + pT(0)6x(0)

Since x(0) = xo is known in this problem, 6x(0) = 0 by hypothesis. Via a formal
application of the calculus of variations, each coefficient of the variations 6p, 6u and
6x must vanish and then

x(t) (x(t),u(t),t) (3.16)
op
x(0) =xo (3.17)
aru-T
p(t) (x(T),u(),p(r),r) (3.18)
OXT
p(T) x= (x(T), T) (3.19)

0 O= (3.20)

It has been chosen to express each of the Equations (3.16) through (3.20) as column
vectors in this convention. These equations have the form typical of many optimal
control problems. Equations (3.16) and (3.17) comprise the original system equations,
and they evolve forward in time. Equations (3.16) and (3.17) are referred to as the co-
state, or adjoint, equations. In contrast to the state equations, the adjoint equations
evolve backward in time from the final condition in (3.19). Together, the Equations
(3.16) through (3.20) comprise a two point boundary value problem for the optimal
control trajectory u(r), T [0, T].








3.3 Optimal Control of Driven Cavity Flow
In this section, the optimal control of unsteady driven cavity flow will be dis-
cussed. The mathematical framework of last section will be applied to a tracking
problem. Detail algebraic work will show that the resulting solutions resort to clas-
sical Riccati solution.

3.3.1 Control Problem Description
Consider the control problem described by Figure 2.5 and equation (2.52) with
some initial condition Xo and fixed time terminal t E [0, T]. Define a new set of states
that include both the original reduced state variables ac(t), and the amplitude of the
control /(t), so that the new state is


X(t) a{(t)}
X(t = (t


The new set of controls is defined to be the time derivative of the original controls

u(t) {=[(t}I

With these definitions of the state X(t) and controls u(t), it is possible to write the
weak form of the governing equations as

[M]X(t) = [A]X(t) + Bu(t) + F(t)

It is important to note that the matrix [Mf] will be diagonal if the reduced basis
vectors are derived from a proper orthogonal decomposition. However, in general,
the reduced basis functions will not be orthogonal. The final form of the reduced
state equations can be obtained by premultiplying by the inverse of the matrix [Mf]


X(t) [M]-l[A]X(t) + [M]-1Bu(t) + [M]-1F(t)









This equation can be rewritten in the final form with initial condition as

X(t) = [A]X(t) + Bu(t) + F(t)
(3.21)
X(0) = X0

Now that a low order, finite dimensional approximation of the governing partial
differential equations has been achieved, a suitable cost functional must be defined to

complete the definition of the optimal control problem. Consider a tracking problem.
For convenience, let vector T(t) be the solution of equation (3.21), for some arbitrarily

chosen, continuous function 7(t). That is

P(t) = [A]h(t) + By(t) + F(t)
(3.22)
,(0) -= o

Introduce a relatively common quadratic cost functional

J(u) a8(X(u),u)
S(X(r) (r7)) [Q] (X(r) (r)) dr + Ru2(rdr
2 Jo 2 Jo
(3.23)

where [Q] is symmetric, positive semidefinite and the scalar R is positive. Recall that
a matrix [M] is positive semidefinite provided the product


T[M]> > 0 (3.24)

for all The matrix [M] is positive definite provided that the product in equation

(3.24) is equal to zero only when -= 0. Some intuitive observations about equation
(3.23) show why it is a popular choice in applications. The overall finite dimensional

optimal flow control problem is then the following: Find the control u* that minimizes


J(u*) < J(u) (X (u), u)


(3.25)









over all choices of admissible control u, where X(u) is constrained to be the solution
of the evolution Equation (3.21). Since [Q] is positive semidefinite, and R is positive,

the function J(u) is non-negative. To minimize J(u), it tends to make each of the non-
negative terms in (3.23) smaller. The result is that it tends to reduce the magnitude

of the states (X 9) and control u appearing in the functional (3.23), i.e., it tries to

keep states X tracking the target states T with acceptable control cost. The relative
magnitudes of the states (X 9) and controls u can be modulated by changing the

values of the weight matrix [Q] and scalar R.
In summary, then, the optimal problem can be stated as follows:

Find

u* G Cad([0, T]; U) the set of admissible control functions

that minimizes

J(u) a (X (u), u)
A 1 2T
L I t (X(r) q(r)) [Q] (X(r) (r)) dr + Ru dr
2 Jo 2 Jo

subject to the constraint

X(t) = [A]X(t) + Bu(t) + F(t)

X(0) = X0


3.3.2 Optimality Systems and the Classical Riccati Equation

Now derive the optimality conditions for this control problem by introducing the
Hamiltonian

1 1()) + + Bu(
f-- (X2(t) q(t))T[Q](X(t) (t)) + Ru2 (t) +pT(t)([A]X(t) + Bu(t) + F(t))
2 2








The optimality condition 0 yields control law

u(t) = -R-BTp(t) (3.26)

The condition X(t) = yields the state equation

X(t) = [A]X(t) BR-1BTp(t) + F(t)

X(0) = Xo

The condition p(t) = yields the co-state equation

p(t) -[A]Tp(t)- [Q](X(t) I(t))

p(T)= 0

It turns out that the states X(t) and the co-state p(t) can be related by the equation
(refer to [4])

p(t) [S(t)]X(t) + A(t) (3.27)

Substituting Equation (3.27) into the state equation gives

X(t) = ([A] BR- BT[S(t)])X(t) BR- BTA(t) + F(t)
(3.28)
X(0) = Xo

Combining equations (3.27), (3.28), and co-state equation gives

X(t) + [S(t)][A]X(t) [S(t)]BR-1BT([S(t)]X(t) + A(t)) + [S(t)]F(t) + A(t)

= -[A]([S(t)]X(t) + A(t)) X(t) + I(t)

Pre-multiply the above equation by XT(t). Collect terms and set terms XT(t)( )X(t)
0 and XT(t)( ) = 0. It results in


[S(t)] = -[S(t)][A]- [A]T[S(t)] + [S(t)]BR- BT[S(t)]- [Q]


(3.29)









A(t) = -([A] BR-1BT[S(t)])A(t) [S(t)]F(t) + [Q]W(t) (3.30)

Equation (3.29) is called differential Riccati equation (DRE). Its solution is named
gain matrix. Since the only condition is at t = T, it has to be solved backward in
time. It can be verified that the gain matrix is a real, symmetric, and positive definite

square matrix. Notice that the solution [S(t)] is independent of the desired output
I(t). This means that the gain matrix is completely specified once the system, the

cost, and the terminal T are specified. Now the control law is

u(t) -R-1B([S(t)]X(t) + A(t)) (3.31)

The gain matrix [S(t)] in the control problem has the property that as T -- o,

[S(t)] of the differential Riccati equation tends to the constant positive definite matrix
[S]. The matrix [S] is the solution of the algebraic Riccati equation (ARE)

-[S][A] [A]T[S] + [S]BR-1BT[S] Q 0 (3.32)

An useful interpretation of this is that as T -- o, the solution of differential Riccati

equation, [S(t)] with initial condition [S(T)] = 0, goes backward in time to the
steady-state solution [S]. The control law is approximated by the linear combination

of the ARE solution and vector A(t)

u(t) =-R-'B([S]X(t) + A(t)) (3.33)

Notice that the coefficient matrix of X(t) term in equation (3.30) is the negative
transpose of the coefficient matrix of A(t) in equation (3.28). The eigenvalues of the

coefficient matrix for one system are the negative of the other. This indicates that if

the system equation(state equation) is stable then the solution for A(t) is unstable if

both are solved forward in time. Since the initial condition for equation (3.30) is not









known, but rather the final condition, it can not be solved directly with the control

law. Instead the equation will be investigated to integrate backward in time

3.3.3 Numerical Simulations

For this example, the target states I(t) are chosen to be the reduced order

solution of the Stokes driven cavity flows as shown in Figure 2.5 with initial condition

I(t = 0) = 0, boundary condition ho(t) = sin(27t), and driving function 7(t) =

27 cos(27t) in Equation (3.22). The tracking problem is to seek optimal control u(t)

such that the reduced order states solution X(t) gets as close as possible to the target

solution I(t). It has been shown that this type of tracking problem can be solved

using equations (3.29), (3.30), (3.31), and state equation.

In this example, the system is discretized with sampling time interval T, = 0.01s.

By doing this, A(t) and differential Riccati equations can be solved backward in

time easily and the data are stored easily as well. The control law is calculated

based on the stored data. The optimal state trajectory is solved forward in time.

T = 5s is simulated. Figure 3.1 shows the optimal trajectory of X(t) with [Q] =

1.:,.i(1000, 1000, 1000) and R = 1. It can be observed from the figure that the optimal

solution X(t) nearly coincides with the target solution I(t). The states solution X(t)

is following the target and the tracking task is thus successful. Figure 3.2 shows

the control trajectory applied. The figures illustrates that the control amplitude is

approximately a factor of 6 times the disturbance amplitude.

The differential Riccati solution [S(t)] in figure 3.3. The dotted lines are the

corresponding algebraic Riccati solutions. The figure tells us that when t < T, [S(t)]

behaves like a constant steady-state solution. This is numerical evidence that as

T oo, gain matrix [S(t)] tends to the constant algebraic Riccati equation solution

[S]. Figure 3.4 is the plot of the time history of the vector A(t).









3.4 Direct Discretization Method

The optimal control problem of driven cavity flow can be solved via methodology

of calculus of variations. This method involves a standard mathematical framework.

It is widely applied because the framework can work for many different practical

problems. This section will introduce a simpler, direct method to realize the solution

of same optimal tracking cavity problem.


3.4.1 Method Description

Consider the cavity tracking problem as described in last section. Recall the

cavity has the boundary condition at the top of ho(t). It seeks to find the control

u(t) to minimize


J(u) A 2 1 d
Jwu) rXa) Wt) Rdt + 2 u(t) 2

subject to the constraint

X(t) = [A]X(t) + Bu(t) + F(t)

X(O) = Xo

The tracking constraint embodied in J(u) is cast in terms of the target solution I(t)

of the equation


I(t) = [A],(t) + By(t) + F(t) (3.34)

S(0) = To

where 7(t) is some given function of time and the initial condition To is given. In-

troduce a partition


{to,tl, ,tN ,-1}









of the time interval [0, T] for the solution of Equation (3.34). For simplicity in the

discussion, assume that the discretization is uniform with

At ti+l ti i = 0 ... NP 2

Using any standard integrator for ordinary differential equations, it is a trivial matter

to obtain the approximation


ic }o- 1 A -(t)k"N 1

of the ordinary differential equation (3.34).

Similarly, it is possible to obtain a discretized approximation of the cost function

J(u). Introduce an approximation of the control function


u(t) = Nk(t )k (3.35)
k-0

where the functions {Nk(t)} N1 are known shape functions (b-splines, finite ele-

ments, ... etc) and the coefficients {uk}N-0 are unknown. An approximate, dis-

cretized cost function can be written as
1 N- N -1
J (uk}) 2 At I\X(t(; {Uk}) (tk) IR + 2 At k u
k=0 k=0

in this equation, {UkJ}N0 is denoted as {uk} for notational simplicity.

Now the problem is to find {uk} that minimizes J({uk}). This is a simple

optimization problem. There many choices of algorithms for searching for the min-

imum of J ({uk}) and the optimal coefficients {uk}. Some examples of these algo-

rithms are the simplex search method, trust region method, preconditioned conjugate

gradients method, etc. These have been incorporated in the Matlab functions like

fminsearch, fminunc etc. Once the optimal coefficients {uk} are obtained, the

optimal control u(t) can be approximated by Equation (3.35).









3.4.2 Numerical Simulations

The results of a specific example are depicted in Figures 3.5 through 3.8. In all

of these simulations, the following selections hold


7(t) A Acos(Qt)

A = 2

2 27


The top boundary condition is ho(t) =sin(27t) for both the target flow and the

controlled cavity flow. The target flow T in the tracking function J(u) is depicted

in Figures 3.5. In the target flow approximation, N1 = 31 was selected in the time

integration of (3.34) with At = and T = 0.25. Figure 3.6 depicts the controlled

flow that is obtained with NV =11 and N, = 4. Figure 3.7 depicts the controlled

flow at the same instances in time for the choice NV =11 and N,, = 8. Figure 3.8

depicts the controlled flow with N =31 and N, = 8. Clearly, a finer partition of

time period will give better tracking results. The controlled velocity fields are closer

to the target flow fields. Similar results can be observed in the control discretization.

The result of a finer time grid and control discretization can be seen more clearly in

Figure 3.9

















t = 0.025


t =0.2


t =0.25


Figure 3.5: Target flow for N, =31


t = 0.1

















t =0.025


t =0.2


t =0.25


Figure 3.6: Controlled flow for N~,


t = 0.1


11, N,= 4

















t =0.025


t =0.2


t =0.25


Figure 3.7: Controlled flow for N~1


t = 0.1


11, N,,= 8
















t 0.025 t 0.1


t =0.2 t 0.25


Figure 3.8: Controlled flow for N~


31, N,= 8









3.5 Receding Horizon Control (RHC)

Receding horizon control (RHC), also known as model predictive control, is a

technique to obtain, for each time step, the optimal control law within a short hori-

zon compared to whole time period. It arises because many traditional methods

can not solve problems with nonlinear plants and constraints on state and control

variables. Also, the solution of the two point boundary value problem (TPBVP) in

section 3.3 costs too much time if terminal time T is large. Basically, RHC divides

the whole time period into small intervals. For the TPBVP, instead of solving for

entire time range [0, T], it solves a series of TPBVP's in much smaller time intervals

[0, tl], [ti, t2], [tn-l, T] where t1 < T, t2 t1 < T, etc. This can be a faster proce-
dure. In addition, this can yield a feedback control for a sufficiently fine partition of

time while the procedure in the earlier sections yields solutions for open-loop optimal

control problems.

3.5.1 Receding Horizon Formulation

Consider the same tracking problem as described in last section which is solved

via the direct discretization method. The target sequence {TkjNk is given. Intro-

duce the same approximation of control function u(t) = -O Nk (t)uk. The TPBVP

can be described as

Find vector (X (ti; {uk}) {Uk}-1) that minimizes


J({uk} -1)= At ||X (tj; {uk) / + At
j=1 k=0

subject to

X(t) = AX(t) + Bu(t) + F(t)

X(0) 0

A receding horizon implementation is typically formulated by introducing the

following optimization problem. First, divide the whole time period [0, T] into much









smaller intervals [0, tp1], [ti,tp],... ,[tN,-p, T] where At = tl and p denotes the
length of the prediction horizon. Denote m as the length of the control horizon.

Thus in th (i = 1, 2,... NX -p + 1) time interval, a sub-optimal control problem

can be described as

Find vector (X t (ti; {ui} {u}m-}o) that minimizes


f i) f -1) + / ) } V 21 AtY (2))2
p m-1
+W 7^+lAt llx^ (t^i). kfi 11,,N -+-At U
(p2m) (pm) \j k -
j=1 j=0

subject to

X (t) = AX() (t) + Bu) (t) + F) (t)

X(i) (0) = X(i-1)* (A t)

This local sub-optimal control problem can be solved by the direct discretiza-

tion method described before. Let u i), j 0, ... m 1 be the minimizing con-

trol sequence for the local sub-optimal problem in ith time interval, and X', j

0,... p- be the local optimal state solutions. Then the global optimal state solution

X* will be updated by the local optimal solutions as Xi_+ = Xi), j = 0,... ,p 1.

The part of the global optimal control sequence u* that falls into the ith time interval

will be updated by the local minimizing control sequence u), j 0,... m 1.


3.5.2 Numerical Simulations

In this simulation, the entire time period [0, 5] is discretized into 641 discrete time

steps, and the control sequence is discretized into 201 steps. The simulation sets the

length of the prediction horizon p = 21 and the length of the control horizon m = 7.

Figure 3.10 shows the optimal states solutions compared to the target states. Since

the solid lines (target solutions) and dotted lines (RHC optimal states solutions) are

so close to each other, the optimal RHC states solutions are good approximations to

the target solutions. Figure 3.11 shows the resultant control history. With control

magnitude of about 0.6, good tracking performance is achieved.