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Linear and nonlinear control of unmanned rotorcraft

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Title:
Linear and nonlinear control of unmanned rotorcraft
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English
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Raptis, Ioannis
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Subjects / Keywords:
Helicopter Control
Aerial Robotics
Unmanned Aerial Vehicles
System Identification
Backstepping Control
Dissertations, Academic -- Electrical Engineering -- Doctoral -- USF   ( lcsh )
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non-fiction   ( marcgt )

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Abstract:
ABSTRACT: The main characteristic attribute of the rotorcraft is the use of rotary wings to produce the thrust force necessary for motion. Therefore, rotorcraft have an advantage relative to fixed wing aircraft because they do not require any relative velocity to produce aerodynamic forces. Rotorcraft have been used in a wide range of missions of civilian and military applications. Particular interest has been concentrated in applications related to search and rescue in environments that impose restrictions to human presence and interference. The main representative of the rotorcraft family is the helicopter. Small scale helicopters retain all the flight characteristics and physical principles of their full scale counterpart. In addition, they are naturally more agile and dexterous compared to full scale helicopters. Their flight capabilities, reduced size and cost have monopolized the attention of the Unmanned Aerial Vehicles research community for the development of low cost and efficient autonomous flight platforms. Helicopters are highly nonlinear systems with significant dynamic coupling. In general, they are considered to be much more unstable than fixed wing aircraft and constant control must be sustained at all times. The goal of this dissertation is to investigate the challenging design problem of autonomous flight controllers for small scale helicopters. A typical flight control system is composed of a mathematical algorithm that produces the appropriate command signals required to perform autonomous flight. Modern control techniques are model based, since the controller architecture depends on the dynamic description of the system to be controlled. This principle applies to the helicopter as well, therefore, the flight control problem is tightly connected with the helicopter modeling. The helicopter dynamics can be represented by both linear and nonlinear models of ordinary differential equations. Theoretically, the validity of the linear models is restricted in a certain region around a specific operating point. Contrary, nonlinear models provide a global description of the helicopter dynamics. This work proposes several detailed control designs based on both dynamic representations of small scale helicopters. The controller objective is for the helicopter to autonomously track predefined position (or velocity) and heading reference trajectories. The controllers performance is evaluated using X-Plane, a realistic and commercially available flight simulator.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2010.
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Includes bibliographical references.
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by Ioannis Raptis.
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LinearandNonlinearControlofUnmannedRotorcraft by IoannisA.Raptis Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofElectricalEngineering CollegeofEngineering UniversityofSouthFlorida Co-MajorProfessor:KimonP.Valavanis,Ph.D. Co-MajorProfessor:WilfridoA.Moreno,Ph.D. KennethBuckle,Ph.D. EliasStefanakos,Ph.D. AbrahamKandel,Ph.D. GeorgeVachtsevanos,Ph.D. DateofApproval: November30,2009 Keywords:HelicopterControl,AerialRobotics,UnmannedA erialVehicles,SystemIdentication, BacksteppingControl c r Copyright2010,IoannisA.Raptis

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InthelovingmemoryofEuthimiosP.Roussis,IoannisG.Rapt isandGeorgiosI.Raptis

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Acknowledgements FirstofallIwouldliketothankmyadvisor,KimonValavanis whogavemetheopportunity toworkinhislaboratory.Iamindebtedforhisguidance,sup portandfriendshipinalltheseyears thatwehaveworkedtogether.Mostimportantly,Iwouldlike tothankhimforgivingmethehelp andencouragementtoexploreanyideasandthoughtsassocia tedwithmyresearch.Iwouldalso liketothankmyco-advisorDr.WilfridoMorenoforhelpingm eandguidingmewiththelabyrinth oftheUniversity'sandtheEEdepartment'sprocedures. IwouldalsoliketothankmychildhoodfriendsNikosKalamar asandKostasVelivasakisfor theirfriendshipandencouragementinallthedifculttime sofmylife.Ihavenowordstoexpress mygratitudetomybeautifulgirlfriendDianaforherlovean dpatienceallthistime.Hersupport andencouragementhelpedmekeepgoing. IwouldliketothankmyunclePeriklisRoussisforhislovean dsupport.Ihavealwaysvalued hisopinionandappreciatedhishelpbyallmeansthroughout myacademicstudies. Mostimportantly,Iwouldliketoexpressmygratitudetomys isterMarthaandmyparents AnastasiosandAlexandra.WithouttheirsacricesIwouldh aveneverbeenabletochasemy dreamsandambitions. Thisresearchhasbeenbeenpartiallysupportedbygrants:A ROW911NF-06-1-0069,SPAWAR N00039-06-C-0062andNSFIIP-0856311(DenverUniversityg rantnumber36563)

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TableofContents ListofTables v ListofFigures vi ListofAcronyms xi Abstract xii Chapter1Introduction 1 1.1Motivation 1 1.2ProblemStatement 4 1.3MethodsofSolutionandContributions8 1.3.1HelicopterLinearControl 8 1.3.2HelicopterNonlinearControl10 1.4DissertationOutline 12 Chapter2LiteratureReview 13 2.1LinearControl 13 2.2NonlinearControl 18 Chapter3HelicopterBasicEquationsofMotion 21 3.1HelicopterEquationsofMotion 21 3.2PositionandOrientationoftheHelicopter26 3.2.1HelicopterPositionDynamics283.2.2HelicopterOrientationDynamics31 3.3CompleteHelicopterDynamics 33 3.4Remarks 34 Chapter4SimpliedRotorDynamics 35 4.1Introduction 35 4.2BladeMotion 37 4.3SwashplateMechanism 40 4.4FundamentalRotorAerodynamics 42 4.5FlappingEquationsofMotion 48 i

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4.6RotorTip-Path-PlaneEquation 51 4.7FirstOrderTip-Path-PlaneEquations534.8MainRotorForcesandMoments 54 4.9Remarks 56 Chapter5FrequencyDomainSystemIdentication 57 5.1MathematicalModeling 57 5.1.1FirstPrinciplesModeling 58 5.1.2SystemIdenticationModeling59 5.2FrequencyDomainSystemIdentication605.3AdvantagesoftheFrequencyDomainIdentication625.4HelicopterIdenticationChallenges 62 5.5FrequencyResponseandCoherenceFunction635.6The CIFER c r Package 66 5.7TimeHistoryDataandExcitationInputs685.8LinearizationoftheEquationsofMotion705.9StabilityandControlDerivatives 72 5.10ModelIdentication 73 5.10.1ExperimentalPlatform 74 5.10.2ParametrizedStateSpaceModel755.10.3IdenticationSetup 79 5.10.4TimeDomainValidation 89 5.11Remarks 90 Chapter6LinearTrackingControllerDesignforSmallScale UnmannedHelicopters91 6.1HelicopterLinearModel 91 6.2ControllerOutline 93 6.3DecomposingtheSystem 97 6.4VelocityandHeadingTrackingControl101 6.4.1Lateral-LongitudinalDynamics1016.4.2Yaw-HeaveDynamics 110 6.4.3StabilityoftheCompleteSystemErrorDynamics114 6.5PositionandHeadingTracking 116 6.6PIDControl 121 6.7ExperimentalResults 123 6.8Remarks 124 Chapter7NonlinearTrackingControllerDesignforUnmanne dHelicopters129 7.1Introduction 130 7.2HelicopterNonlinearModel 131 7.2.1RigidBodyDynamics 131 7.2.2ExternalWrenchModel 132 7.2.3CompleteRigidBodyDynamics136 7.3TranslationalErrorDynamics 136 7.4AttitudeErrorDynamics 142 ii

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7.4.1YawErrorDynamics 143 7.4.2OrientationErrorDynamics1447.4.3AngularVelocityErrorDynamics145 7.5StabilityoftheAttitudeErrorDynamics1457.6StabilityoftheTranslationalErrorDynamics1537.7NumericSimulationResults 162 7.8Remarks 167 Chapter8TimeDomainParameterIdenticationandAppliedD iscreteNonlinear ControlforSmallScaleUnmannedHelicopters1738.1Introduction 173 8.2DiscreteSystemDynamics 174 8.3DiscreteBacksteppingAlgorithm 177 8.3.1AngularVelocityDynamics1778.3.2TranslationalDynamics 177 8.3.3YawDynamics 180 8.4ParameterEstimationUsingRecursiveLeastSquares1838.5ParametricModel 184 8.6ExperimentalResults 185 8.6.1TimeHistoryDataandExcitationInputs1858.6.2Validation 186 8.6.3ControlDesign 187 8.7Remarks 189 Chapter9TimeDomainSystemIdenticationforSmallScaleU nmannedHelicopters UsingFuzzyModels 193 9.1Introduction 193 9.2Takagi-SugenoFuzzyModels 194 9.3ProposedTakagi-SugenoSystemforHelicopters1969.4ExperimentalResults 197 9.4.1TuningoftheMembershipFunctionsParameters1999.4.2Validation 199 Chapter10ComparisonStudies 202 10.1SummaryoftheControllerDesigns20210.2ExperimentalResults 203 10.3FirstManeuver:ForwardFlight 204 10.4SecondManeuver:AggressiveForwardFlight20510.5ThirdManeuver:8Shaped 206 10.6FourthManeuver:Pirouette 207 10.7Remarks 209 Chapter11ConclusionsandFutureWork 223 11.1SummaryofContributions 225 11.2ResultsandReal-LifeImplementation227 iii

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11.3FutureWork 227 ListofReferences 229 Appendices 239 AppendixA:BacksteppingControl 240 AbouttheAuthor EndPage iv

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ListofTables Table5.1Experimentalhelicoptermodelbasicspecicatio ns.75 Table5.2Frequencysweepsparameters. 80 Table5.3Selectedfrequencyresponsesandtheircorrespon dingfrequencyranges(in rad=sec ). 81 Table5.4Linearstatespacemodelidentiedparameters.84Table5.5Transferfunctionscostsforeachinput-outputpa ir.85 Table6.1Lineartrackingcontrollerfeedbackgains.125Table6.2PIDcontrollergains. 125 Table7.1Helicopterparameters. 164 Table7.2Dragandservoparameters. 164 Table7.3Controllergains. 164 Table7.4Controlleroutline. 166 Table8.1Identiedsystemparameters. 188 Table8.2Valuesofthediagonalgainmatrices. 188 Table9.1Gaussianmembershipfunctions. 196 Table9.2Gaussiancentersandspreads. 198 Table9.3MeanerroroftheTakagi-SugenoRLSincomparisonw ithRLSidenticationoverthevericationdata. 198 v

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ListofFigures Figure1.1Typicalhelicopterconguration. 2 Figure1.2Thisblockdiagramillustratesthehelicopterco ntroldesignproblem.7 Figure3.1Body-xedcoordinatesystem. 22 Figure3.2Helicopterorientation. 27 Figure3.3Interconnectionofthehelicopterdynamicsinth espace SE (3) .33 Figure4.1Representationoftherotor3DOF. 38 Figure4.2Connectionofthepitchhorntothepitchlink.41Figure4.3Basiccongurationoftheswashplatemechanism. 42 Figure4.4Directionsofthevelocitycomponentsseenbythe bladeelement.45 Figure4.5Illustrationofatwodimensionalbladeelement. 47 Figure4.6Aerodynamic,inertiaandcentrifugalforcesact ingonabladeelement.49 Figure4.7Effectofeachharmonicgivenby(4.23)totheTPP. 52 Figure5.1Blockdiagramoftheexperimentalplatform'scom municationinterface.76 Figure5.2On-axisfrequencyresponsesoftheightdata(so lidline)andfrequency responsespredictedbythestatespacemodel(dashedline). 86 Figure5.3Off-axisfrequencyresponsesoftheightdata(s olidline)andfrequency responsespredictedbythestatespacemodel(dashedline). 87 Figure5.4Timedomainvalidation. 88 vi

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Figure6.1Interconnectionofthetwohelicopterdynamicss ubsystems.99 Figure6.2Strict-feedbackinterconnectionofthelongitu dinal-lateralhelicopterdynamicssubsystem. 103 Figure6.3Interconnectionoftheyaw-heavehelicopterdyn amicssubsystem.111 Figure6.4Cascadeconnectionoftheclosedlooperrordynam icssubsystems.115 Figure6.5Cascadeconnectionoftheerrordynamicssubsyst emsrelatedwiththepositiontrackingproblem. 119 Figure6.6Referencetrajectory(solidgreenline),actual positiontrajectoryofthelinear (greendashedline)andPID(dashed-dottedredline)design s,expressedin inertialcoordinateswithrespecttotime. 125 Figure6.7Orientationanglesofthelinear(solidline)and PID(dashedline)designs.126 Figure6.8Referencepositiontrajectory(solidline)andt heactualtrajectoryofthe linear(dashedline)designwithrespecttotheinertiaaxis .126 Figure6.9Referencepositiontrajectory(solidline)andt heactualtrajectoryofthePID (dashedline)design,withrespecttotheinertiaaxis.127 Figure6.10Controlinputsofthelineardesign. 127 Figure6.11ControlinputsofthePIDdesign. 128 Figure7.1Thehelicopter'sbody-xedframe,theTip-PathPlaneanglesandthethrust vectorsofthemainandtailrotor. 132 Figure7.2Thisblockdiagramillustratestheconnectionof thegeneratedthrustsofthe mainandtailrotorwiththehelicopterdynamics.133 Figure7.3Thisblockdiagramillustratestheinterconnect ionoftheapproximatedhelicopter'sdynamics. 136 Figure7.4Thisblockdiagramillustratesthetranslationa lerrordynamicssubsystem.138 Figure7.5Resultingsystemdynamicsafterthechoiceof v I d d and T M .140 Figure7.6Thisgureillustratesthehelicopter'svertica lorientationvectors 3 d with respecttoinertiaframefor 3 ; 3 ; d; 3 > 0 .142 vii

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Figure7.7Thisgureillustratesthatonlytheexponential convergenceof e % cannot guaranteethat k % k < 1 forevery t t 0 148 Figure7.8Thisgureillustratestheexistenceofavalue C ? with C max C ? then R ( k % k ) < 0 150 Figure7.9Blockdiagramofthecompletehelicopterdynamic safterthetransformation ofthetranslationalerrorstates. 154 Figure7.10 Firstmaneuver :Referencepositiontrajectory(dashedline)andactualhe licoptertrajectory(solidline)expressedintheinertial coordinateswith respecttotime. 167 Figure7.11 Secondmaneuver :Referencepositiontrajectory(dashedline)andactual helicoptertrajectory(solidline)expressedintheinerti alcoordinateswith respecttotime. 168 Figure7.12 Firstmaneuver :Referencepositiontrajectory(solidline)andactualhel icoptertrajectory(dashedline)withrespecttotheinertia laxis.170 Figure7.13 Secondmaneuver :Referencepositiontrajectory(solidline)andactualhel icoptertrajectory(dashedline)withrespecttotheinertia laxis.170 Figure7.14 Firstmaneuver :Euler'sorientationangles.171 Figure7.15 Secondmaneuver :Euler'sorientationangles.171 Figure7.16 Firstmaneuver :Mainandtailrotorthrust T M ;T T andtheappingangles a;b 172 Figure7.17 Secondmaneuver :Mainandtailrotorthrust T M ;T T andtheappingangles a;b 172 Figure8.1Interconnectionofthehelicopterdynamicsusin g(8.23)-(8.27).179 Figure8.2Comparisonbetweentheactual(solidline)andes timated(dashedline) linearvelocitiesusingthevericationdata.190 Figure8.3Comparisonbetweentheactual(solidline)andes timated(dashedline) angularvelocitiesusingthevericationdata.190 Figure8.4Referencetrajectory(dashedline)andactualve locitytrajectory(solidline) ofthehelicopterexpressedininertialcoordinateswithre specttotime.191 Figure8.5Euler'sorientationangles. 191 viii

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Figure8.6Referencepositiontrajectory(solidline)andt heactualhelicopterposition (dashedline)withrespecttotheinertialaxis.192 Figure8.7Controlinputs. 192 Figure9.1Comparisonbetweentheactual(solidline)andes timated(dottedline)linear velocitiesusingthevericationdata. 200 Figure9.2Comparisonbetweentheactual(solidline)andes timated(dottedline)angularvelocitiesusingthevericationdata. 201 Figure10.1 Firstmaneuver(Forwardight) :Referencevelocitytrajectory(greendashed line)andactualvelocitytrajectoryofthelinear(solidbl ueline),PID(red dasheddottedline),nonlinear(dasheddottedblackline)c ontrollerdesigns, expressedininertialcoordinateswithrespecttotime.211 Figure10.2 Firstmaneuver(Forwardight) :Orientationanglesofthelinear(solid blueline),PID(dashedredline)andnonlinear(dasheddott edblackline) controllersdesigns. 211 Figure10.3 Firstmaneuver(Forwardight) :Controlinputsofthelinear(solidblue line),PID(dashedredline)andnonlinear(dasheddottedbl ackline)controllerdesigns. 212 Figure10.4 Firstmaneuver(Forwardight) :Referencepositiontrajectory(solidline) andactualtrajectoryofthecontrollerdesigns(dashedlin e)withrespectto theinertialaxis. 213 Figure10.5 Secondmaneuver(Aggressiveforwardight) :Referencevelocitytrajectory (greendashedline)andactualvelocitytrajectoryoftheli near(solidblue line),PID(reddasheddottedline),nonlinear(dasheddott edblackline) controllerdesigns,expressedininertialcoordinateswit hrespecttotime.214 Figure10.6 Secondmaneuver(Aggressiveforwardight) :Orientationanglesofthe linear(solidblueline),PID(dashedredline)andnonlinea r(dasheddotted blackline)controllersdesigns. 214 Figure10.7 Secondmaneuver(Aggressiveforwardight) :Controlinputsofthelinear (solidblueline),PID(dashedredline)andnonlinear(dash eddottedblack line)controllerdesigns. 215 Figure10.8 Secondmaneuver(Aggressiveforwardight) :Referencepositiontrajectory (solidline)andactualtrajectoryofthecontrollerdesign s(dashedline)with respecttotheinertialaxis. 216 ix

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Figure10.9 Thirdmaneuver(8shaped) :Referencepositiontrajectory(greendashed line)andactualpositiontrajectoryofthelinear(solidbl ueline),PID(red dasheddottedline),nonlinear(dasheddottedblackline)c ontrollerdesigns, expressedininertialcoordinateswithrespecttotime.217 Figure10.10 Thirdmaneuver(8shaped) :Orientationanglesofthelinear(solidblueline), PID(dashedredline)andnonlinear(dasheddottedblacklin e)controllers designs. 217 Figure10.11 Thirdmaneuver(8shaped) :Controlinputsofthelinear(solidblueline), PID(dashedredline)andnonlinear(dasheddottedblacklin e)controller designs. 218 Figure10.12 Thirdmaneuver(8shaped) :Referencepositiontrajectory(solidline)and actualtrajectoryofthecontrollerdesigns(dashedline)w ithrespecttothe inertialaxis. 219 Figure10.13 Fourthmaneuver(Pirouette) :Referencepositiontrajectory(greendashed line)andactualpositiontrajectoryofthelinear(solidbl ueline),PID(red dasheddottedline),nonlinear(dasheddottedblackline)c ontrollerdesigns, expressedininertialcoordinateswithrespecttotime.220 Figure10.14 Fourthmaneuver(Pirouette) :Orientationanglesofthelinear(solidblue line),PID(dashedredline)andnonlinear(dasheddottedbl ackline)controllersdesigns. 220 Figure10.15 Fourthmaneuver(Pirouette) :Controlinputsofthelinear(solidblueline), PID(dashedredline)andnonlinear(dasheddottedblacklin e)controller designs. 221 Figure10.16 Fourthmaneuver(Pirouette) :Referencepositiontrajectory(solidline)and actualtrajectoryofthecontrollerdesigns(dashedline)w ithrespecttothe inertialaxis. 222 x

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ListofAcronyms SISOSingle-InputSingle-OutputMIMOMultiple-InputMultiple-OutputDOFDegreesOfFreedomCGCenterofGravityRCRadioControlledFAAFederalAviationAdministrationUDPUserDatagramProtocolPIDProportionalIntegralDerivativePDProportionalDerivativeTPPTip-Path-PlaneSMDSpring-Mass-DamperGASGloballyAsymptoticallyStableUGASUniformlyGloballyAsymptoticallyStableUGBUniformlyGloballyBoundedLTILinearTimeInvariantRLSRecursiveLeastSquares xi

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LinearandNonlinearControlofUnmannedRotorcraft IoannisA.Raptis ABSTRACT Themaincharacteristicattributeoftherotorcraftistheu seofrotarywingstoproducethe thrustforcenecessaryformotion.Therefore,rotorcrafth aveanadvantagerelativetoxedwing aircraftbecausetheydonotrequireanyrelativevelocityt oproduceaerodynamicforces.Rotorcrafthavebeenusedinawiderangeofmissionsofcivilianan dmilitaryapplications.Particular interesthasbeenconcentratedinapplicationsrelatedtos earchandrescueinenvironmentsthat imposerestrictionstohumanpresenceandinterference. Themainrepresentativeoftherotorcraftfamilyistheheli copter.Smallscalehelicoptersretain alltheightcharacteristicsandphysicalprinciplesofth eirfullscalecounterpart.Inaddition,they arenaturallymoreagileanddexterouscomparedtofullscal ehelicopters.Theirightcapabilities, reducedsizeandcosthavemonopolizedtheattentionoftheU nmannedAerialVehiclesresearch communityforthedevelopmentoflowcostandefcientauton omousightplatforms. Helicoptersarehighlynonlinearsystemswithsignicantd ynamiccoupling.Ingeneral,they areconsideredtobemuchmoreunstablethanxedwingaircra ftandconstantcontrolmustbe sustainedatalltimes.Thegoalofthisdissertationistoin vestigatethechallengingdesignproblem ofautonomousightcontrollersforsmallscalehelicopter s.Atypicalightcontrolsystemis composedofamathematicalalgorithmthatproducestheappr opriatecommandsignalsrequired toperformautonomousight. Moderncontroltechniquesaremodelbased,sincethecontro llerarchitecturedependsonthe dynamicdescriptionofthesystemtobecontrolled.Thispri ncipleappliestothehelicopteraswell, xii

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therefore,theightcontrolproblemistightlyconnectedw iththehelicoptermodeling.Thehelicopterdynamicscanberepresentedbybothlinearandnonlin earmodelsofordinarydifferential equations.Theoretically,thevalidityofthelinearmodel sisrestrictedinacertainregionarounda specicoperatingpoint.Contrary,nonlinearmodelsprovi deaglobaldescriptionofthehelicopter dynamics. Thisworkproposesseveraldetailedcontroldesignsbasedo nbothdynamicrepresentations ofsmallscalehelicopters.Thecontrollerobjectiveisfor thehelicoptertoautonomouslytrack predenedposition(orvelocity)andheadingreferencetra jectories.Thecontrollersperformanceis evaluatedusing X-Plane ,arealisticandcommerciallyavailableightsimulator. xiii

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Chapter1:Introduction 1.1Motivation ThetermUnmannedAerialVehicles(UAVs)isusedtodescribe unpilotedyingvessels.This termreferstovehiclesthatareremotelypilotedorautonom ouslycontrolledfortheexecutionofa predenedighttask.Inbothcasesthekeyattributeofthes evehiclesistheabsenceofahuman pilotonboard[106].TheapplicabilityofUAVsispredomina ntintheexecutionofpotentially dangerousightmissionsorincaseswherethesmallsizeoft hevehiclerestrictsthepresenceof apilot[70]. PotentialusageofUAVscanbefoundinmilitaryandcivilian applications,althoughmilitary applicationsdominatethenon-militaryones.Civilianapp licationsinvolvepipelinesandpower linesinspection,surveillance,rescuemissions,borderp atrol,oilandnaturalgasresearch,re prevention,topography,agriculturalapplications[106] ,lmmaking[70],trafcmonitoring,ight inhazardousenvironments(i.e.inaradioactiveenvironme nt)[11]. UAVsarefurtherclassiedintotwomaincategories.Thers tcategoryarexed-wingUAVs (e.g.,unmannedairplanes)thatrequirerelativevelocity fortheproductionofaerodynamicforces andarunawayfortake-offandlanding[105].Thesecondcate goryaretherotorcraftUAVs.The advantagesoftherotorcraftuniqueightcapabilitieshav edrawnmuchattentionthroughtheyears. Theprimarycharacteristicattributeoftherotorcraftist heuseofrotarywingstoproducethethrust forcenecessaryformotion.Themainbenetofusingarotorc raftisitsabilitytomoveinalldirectionsoftheCartesianspacewhilepreservinganindependen theading.Therefore,rotorcrafthave anadvantagerelativetoxedwingaircraftbecausetheydon otrequireanyrelativevelocityto produceaerodynamicforces[40]andalsoduetotheirvertic alightcapability. 1

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Main rotor Tail rotor Figure1.1:Typicalhelicopterconguration.Thehelicopt ermotionisproducedbytwoengine drivenrotors:Themainandtailrotor. Themainrepresentativeoftherotorcraftfamilyistheheli copter.Thetypicalcongurationofa helicopterinvolvestwoenginedrivenrotors:Themainandt hetailrotor.Themainrotorproduces thethrustforcefortheverticalliftofthehelicopter.The tailrotorcompensatesthetorqueproducedbythemainrotorandcontrolstheheadingofthehelico pter.Thechangeofthehelicopter's fuselageattitudeanglesresultsinthetiltofthemainroto rand,therefore,theproductionofthe propulsiveforcesforthelongitudinal/lateralmotionoft hehelicopter. Smallscalehelicoptersretainalltheightcharacteristi csandphysicalprinciplesoftheirfull scalecounterpart.Inaddition,theyarenaturallymoreagi leanddexterouscomparedtofullscale helicopters.Theirightcapabilities,reducedsizeandco sthavemonopolizedtheattentionof theUAVresearchcommunityforthedevelopmentoflowcostan defcientautonomousight platforms. Thedesignofanautonomoussmallscalehelicopterightpla tformrequiresseveralexpertiseindiverseeldsofengineering.Someofthechallenges towardsthedevelopmentofanautonomouslyyinghelicopterinvolvesensorintegrationan dsensorfusiontoobtainaccuratemeasurements,ightcontrollerdesign,pathplanningandcomm unications.Advancesinsensortechnology,computationalefciencyandtheconstantlyreduce dsizeofprocessorsprovideasignicantboostinthedevelopmentofon-boardhardwarefortheUA Vs. Thegoalofthisdissertationistoexaminethechallengingd esignproblemofautonomous ightcontrollersforsmallscaleunmannedhelicopters.At ypicalightcontrollersystemiscomposedofamathematicalalgorithmthatproducestheappropr iatecommandsignalsrequiredto 2

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performanyautonomousight.Thecontrolalgorithmreceiv esthemeasurementsignalsfrom severalsensorsandtriggersasuitableoutputforoperatin gthehelicopter.Thecontroller'soutput isalsoreferredtoasthecontroller'sfeedbacksignal.Ani mportantrequirementofthecontroller designistoguaranteethestabilityofthehelicopterdurin gtheautonomousightoperation. Themostreliableapproachfordesigningthecontrolalgori thmandalsoexaminingthestabilitypropertiesoftheautonomousightsystem,isviamod erncontroltheory.Accordingtothis theoreticalframework,theightcontrollerdesignisbase donthehelicopterdynamicmodel.This modelisamathematicalsystemofordinarydifferentialequ ations.Thedynamicmodeldescribes thehelicopterresponsetoanygiveninput. Helicoptersarehighlynonlinearsystemswithsignicantd ynamiccoupling.Thedynamic couplingisattributedtotwomainsources.Therstoneisth ehelicopternonlinearequationsof motion.Thesecondoneisthedynamiccouplingbetweenthege neratedaerodynamicforcesand moments.Inaddition,thereisalsosignicantparameteran dmodeluncertaintyduetocomplicated aerodynamicnatureofthethrustgeneration.Furthermore, helicoptersareconsideredtobemuch moreunstablethanxedwingaircraftandconstantcontrola ctionmustbesustainedatalltimes. Theabovehelicoptercharacteristicsconstituteverychal lengingobstaclestothecontrollerdesign problem. Asinmostcontrolapplications,thehelicoptermodelthati susedforcontroldesignpurposes isjustanapproximationoftheactualnonlinearhelicopter dynamics.Tothisextent,inorderto developagenericightcontrolsystemwhichappliestomost standardsmallscalehelicopterplatforms,thedesignermustsuccessfullysolvethefollowingi ntermediatetasks: Derivethestructureandtheorderofaparametricdynamicmo delthatbestdescribesthe helicoptermotion.Theorderofthemodelshouldbekepttomi nimumsuchthattheparametricmodelincludesonlytheabsolutelynecessaryvariab lesthatarerequiredfortherepresentationofthehelicopterdynamics.Dynamicsystemsof highorderareveryimpractical sincetheysignicantlyincreasethecomplexityofthecont roldesign.Theparametricmodel shouldprovideaphysicallymeaningfuldynamicdescriptio nforalargefamilyofsmall scalehelicopters. 3

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Basedontheparametrichelicoptermodel,determineanomin alfeedbackcontrollawsuch thatthehelicoptertracksapredenedreferencetrajector y.Thedesignshouldguaranteethat thecontrolinputsremainboundedwhilethehelicoptertrac ksthereferencetrajectory. Finally,foraparticularhelicopter,determinewhichisth ebestmethodologyforaccurately extractingthevaluesoftheparametricmodel. Mostofthecurrentworkpublishedintheeldofhelicopterc ontrolrestrictitsanalysisonly inasubsetoftheabovedesignchallenges.Thisdissertatio nisoneofthefewresearchefforts thatencompassathoroughexaminationofalloftheabovedes ignissues.Thecharacteristicsof thehelicopterdynamics(highuncertainty,nonlinearcoup leddynamics)constitutethehelicopter controlproblemstimulatingforbothitstheoreticalandre al-lifeimplementationviewpoint.The objectiveofthisworkistoprovidemathematicallyconsist entmethodologiesthatcanbeapplied intoactualsmallscalehelicopterplatforms.1.2ProblemStatement Thehelicopterdynamicsareinherentlynonlinearwithsign icantdynamiccouplingamong thestatevariablesandcontrolinputs.Thedynamiccouplin gexpressesthefactthatanychangein acontrolinputaffectsmultiplestatevariablesoftheheli copter.Therefore,eachinputeffectsnot onlythestatevariablesofinterest,butalsoproducesunin tendedsecondaryresponses.Tosuppress theunwantedexcitationofsecondarystatevariablesasimu ltaneouscoordinationofallthecontrol inputsisrequiredatalltimeinstances.Thenonlinearnatu reandthecrosscouplingeffectofthe helicopterdynamicsplacesthemamongthemostcomplexaeri alvehicles. Thehelicopterhasfourcontrolinputs.Twocycliccommands thatmanipulatethelongitudinal/lateralmotion,onecollectivecommandthatcontrol stheverticalmotionandnallythe pedalcommandthatcontrolstheheadingmotionofthehelico pter.Sincethecontrolinputsare signicantlylessthanthemotionvariables,thehelicopte risfurtherclassiedasanunderactuated system. 4

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Theactualhelicopterdynamicsarerepresentedinmathemat icaltermsbythedifferentialequationsofthefollowingnonlinearsystem: x = f ( x;u c ) (1.1) where x 2 R n isthehelicopter'sstateand u c 2 R 4 isthecontrolinputvector.Controltechniques basedonmoderncontroltheoryaremodelbased,inthesenset hatthecontrollerarchitecturedependsonthedynamicdescriptionofthesystem.Therefore,k nowledgeofthehelicopter'sdynamic modelisrequiredforthedesignofautonomousightcontrol lers. However,theactualhelicopterdynamicsareunknownandasi nmostengineeringapplications, theyareapproximatedbyphysicallymeaningfulmathematic almodelsoflowerorder.Tothis point,itmustbestatedthattheapproximatedmodelisjusta n“abstraction"sinceitispractically impossibletoprovideacompleterepresentationoftheactu alhelicopterdynamics[81].However, thisdoesnotmeanthatitisimpossibletodevelopamodel,th atsufcientlyrepresentsthedynamicsofthehelicopterundercertainoperatingightconditi ons. Generally,therearetwowaystoapproximatetheactualheli copterdynamics.Therstisbya LinearTimeInvariant(LTI)model.Thesecondrepresentati onisviaamodelofnonlineardifferentialequations.Typically,thevalidityoftheLTImodeli srestrictedinthevicinityofaparticular operatingconditionofthehelicopter.Forthedescription ofawideportionoftheightenvelope, multiplelinearmodelsarerequiredfordifferentoperatin gconditions.TheLTImodelisrepresentedbyasetofrst-orderlineardifferentialequations ,writtenintheform: x l = Ax l + Bu c y = C l x l (1.2) y m = C m l x l where x l isthevectorofthehelicopter'slinearmodelstatevariabl es, y m isthevectorofthehelicoptersavailablemeasurementsand y isthevectorofthehelicopteroutputsthatneedtobecon5

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trolled.Thedimensionoftheoutputvectorcannotexceedth enumberofthecontrolinputs.The designproblemistondafeedbacklawofthemeasurementvec tor,i.e., u l = l ( y m ) ,suchthat when u c = u l ,thenthehelicopteroutputasymptoticallytracksarefere ncetrajectorydenotedby y r .Hence,theobjectiveis: lim t !1 k y ( t ) y r ( t ) k =0 (1.3) Byapplyingmoderncontroldesigntechniques,thearchitec tureofthefeedbacklaw u l willbe(in general)dependedonthestructureofthelinearsystemgive nby(1.2). Nonlinearmodelsareusedtoprovideaglobaldescriptionof thehelicopterdynamicsforthe completeightenvelope.Theyaremoreelaborateandcomple xcomparedtolinearmodels,however,onlyasinglemodelisrequiredforthedescriptionoft hehelicopterdynamics.Whenanonlineardynamicrepresentationischosen,thehelicopterdy namicscanbewrittenas: x n = ( x n ;u c ; ) y = C n x n (1.4) y m = C m n x n where denotestheparametervectorofthenonlinearmodel.Ofcour se,eveninthecaseofthe nonlinearrepresentation,theoutputandthemeasurementv ectorofthehelicopterareidentical withthelinearmodelcase.However,thedimensionsofthest atevectors x n and x l are(ingeneral) differentsincethetwomodelsmighthavedifferentorders. Similarlytothelinearcase,thecontrol objectiveisthedesignofafeedbacklaw u n = n ( y m ) suchthatwhen u c = u n ,thenthe asymptotictrackingof(1.3)isachieved.Since u n dependsonthestatespaceequationsof(1.4) then,inprincipal, u l and u n willbedifferent.Theblockdiagramofthehelicoptertrack ingcontrol problemisillustratedinFigure1.2.Ineithercasethedesi gnchallengesare: Thedeterminationoftheorderandstructureoftheparametr icmodel(1.2)or(1.4).These parametricmodelsshouldencapsulatethedynamicbehavior ofalargefamilyofsmallscale helicopters. 6

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Linear Model Nonlinear Model) (m n y ) (m l y c l lBu A x x ) , ( c n nu x x ) (cu x f x lunu Controller design based on the linear helicopter model Controller design based on the nonlinearhelicopter modelm y Figure1.2:Thisblockdiagramillustratesthehelicopterc ontroldesignproblem.Thehelicopter dynamicscanberepresentedbyalinearornonlinearsystemo fdifferentialequations.Ineither casethefeedbackcontrollawdependsonthemodelchoice. 7

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Thederivationofaconsistentmethodologyfordesigningth efeedbacklaws u l = l ( y m ) or u n = n ( y m ) whichguaranteethatthetrackingobjectivegivenin(1.3)i sachieved. Thecalculationofthematrices A B ortheparametervector suchthatthepredictedresponsefrom(1.2)and(1.4)isthesamewiththeactualhelico pterresponseobtainedbyight data.Theidentiedparametersarerequiredfortheimpleme ntationofthecontrollaws u l and u n ,respectively. 1.3MethodsofSolutionandContributions Thisresearchprovidesacompleteandconsistentsolutiont othehelicoptercontrollerdesign problem.Allintermediatechallengesassociatedwiththeh elicoptercontrollerdesignareaddressed forboththelinearandthenonlinearrepresentationsofthe helicopterdynamics.Theproposed solutionsincorporateanebalancebetweentheoreticalco ntrolchallengesandreal-lifeapplication issues.Theproposedcontrollersperformanceandapplicab ilityareevaluatedusingthecommerciallyavailableightsimulator X-Plane .Theexperimentalpartofthisresearchwasconducted inthe X-Plane environmentforasmallscale Raptor90SE RadioControlled(RC)helicopter. Dependingonthehelicoptermodelrepresentation,thecont rollerdesignsproposedinthiswork areclassiedaslinearandnonlinear.1.3.1HelicopterLinearControl TheproposedcontroldesignisbasedonalinearMultiple-In putMultiple-Output(MIMO) coupledhelicoptermodel.Typicaldesigntechniquesthatd ealwiththetrackingproblemoflinear systemsaretheinternalmodelapproachandtheintegralcon troldesign.Thedisadvantageofthe internalmodelapproachisitscomplexdesignwhiletheinte gralcontrolisrestrictedonlyincases wherethereferenceoutputisaconstantsignal.Thepropose ddesignguaranteestheasymptotic trackingofarbitrarycontinuousreferencetrajectoriesw iththeonlyrequirementthatthereference signalanditshigherderivativesarebounded. 8

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Themainnoveltyoftheproposedcontrollerisitsabilityto “pass”theintuitivenotionofhelicopterpilotingtothemathematicalcontrollerdesign.T hisisachievedbydecouplingtherotor dynamicsintotwoseparatesubsystems.Therstsubsystemi nvolvesthecoupleddynamicsofthe longitudinal/lateralmotionwhilethesecondsubsystemis composedbytheyaw/heavedynamics ofthehelicopter.Thisseparationprovidesamoredistinct effectofthehelicopterinputstothe statevariablesofthetwosubsystems.Theintuitiveoperat ionofthevehicledictatesthatthetwo cycliccommandsareusedforthegenerationoflongitudinal andlateralmotion.Thetwocollective commandsofthemainandtailrotoraremainlyusedforthepro ductionoftheverticalliftand regulatingthehelicopter'sheading. Thebasicideaofthecontrollerdesignistodetermineadesi redstatevectorforeachsubsystem suchthatwhenthehelicopterstatevariablesconvergetoth eirdesiredstatevaluesthenthetracking errortendsasymptoticallytozero.Thedesiredstatevecto rsforeachsubsystem,arecomposedby thecomponentsofthereferenceoutputsvectorsandtheirhi gherderivatives. Thesecondcontributionoftheproposeddesignisthedevelo pmentofarecursiveprocedure forthederivationoftheaforementioneddesiredstatevect orsforeachsubsystem.Therecursive procedureisbasedonthebacksteppingdesignofsystemsinp urefeedbackform.However,the linearhelicopterdynamicsarenotisfeedbackform.Thisfa ctisattributedtothecouplingbetween thehelicopter'sexternalforcesandmoments.Similarlyto [47],asimpliedhelicoptermodelthat neglectsthecouplingbetweenthehelicopterforcesandmom entsisinpurefeedbackform.This approximationisbasedontherationalassumptionthatthef orcesproducedbytheappingmotion ofthemainrotorbladesarenegligiblecomparedtotheforce sproducedbythetiltofthefuselage. Sincetheapproximatesystemisinpurefeedbackform,itisa lsofeedbacklinearizableanddifferentiallyat.Thederivationofthedesiredstatevectorsis basedonthedifferentialatnessproperty ofthetwosubsystems. Forthelinearmodelrepresentationofthehelicopterdynam icsthemodelstructureproposed in[70]isadopted.Thislinearmodelhasbeensuccessfullyu sedfortheparametricidentication ofseveralsmallscalehelicoptersofdifferentspecicati ons[8,10,27,28,89,90].Theproposed modelisalinercoupledsystemofthehelicoptermotionvari ablesandthemainrotorapping 9

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dynamics.Themodelvalidityisevaluatedbyperformingfre quencydomainsystemidentication usingighttestdataobtainedforthe Raptor90SE .Thefrequencydomainidenticationprocedureofthe Raptor90SE takesplacebyusingthe CIFER c r packagedevelopedbytheNASA RotorcraftDivision(AmesResearchCenter)[105].Theiden tiedmodelislaterusedtoevaluate thecontroller'sperformance. Finally,asecondcontrollerisintroducedwhichdoesnotre quiretheknowledgeofthehelicoptermodel.InmanypracticalcontrolapplicationstheMI MOdynamicmodelofthehelicopter isnotavailable.Afundamentalcontrollercomposedbyfour SISOProportionalIntegralDerivative (PID)feedbackloopsispresented.Thiscontrolschemeisve rycommonstartupdesignpointin real-lifeapplications,sinceitdoesnotrequireknowledg eofthehelicoptermodelandthecontrollergainscanbeempiricallytuned.1.3.2HelicopterNonlinearControl Theadoptednonlinearmodelofthehelicopterdynamicsisba sedon[47].Thehelicopter modelisrepresentedbytherigidbodynonlinearequationso fmotionenhancedbyasimplied modelofforceandtorquegeneration.Therstcontrollerde signisbasedonthebackstepping designprincipleforsystemsinfeedbackform.Theintermed iatebacksteppingcontrolsignals (a.k.a.pseudocontrols)foreachlevelofthefeedbacksyst emareappropriatelychosentostabilize theoverallhelicopterdynamics.Theresultingsystemerro rdynamicscanbeseparatedintwointerconnectedsubsystemsrepresentingtheerrorintransla tionalandattitudedynamics,respectively. Thisseparationreectstheinheritedtimescalingthatexi stsinthehelicopterdynamics.Theattitudedynamicsaresignicantlyfastercomparedtothedynam icsofthetranslationalmotion. Oneofthenoveltiesoftheproposedcontrolleristhattheth rustmagnitudeisusedtocompensatethetranslationalerrordynamicsinallCartesiandire ctionsandnotonlyfortheheavedynamics.Furthermore,apartfromstabilizingtheattitudedyna mics,thecontroldesigncanguarantee thatthehelicopterwillnotoverturnforeveryallowedrefe rencetrajectory.Inaddition,theuseof 10

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nestedsaturationsintheintermediatepseudocontrolsoft hetranslationaldynamicscanguarantee thatthephysicalconstraintsofthehelicoptermotionandp owerwillbepreserved. Theoretically,theproposedcontrollerisapplicableforb othfullscaleandsmallscalehelicopters.However,theadoptednonlinearmodelissignican tlysimpliedanddoesnotinclude higherorderdynamicssuchasengine,inowvelocityandmai nrotorlead-lagdynamicsthatare requiredforthemodelingoffullscalehelicopters. Althoughthiscontrollerhassignicanttheoreticalpoten tial,theextractionofthemodelparametersfromthecontinuoustimenonlinearmodelusingtim edomainidenticationiscomputationallyinefcient.Theidenticationprocedureissig nicantlysimpliedwhenthenonlinear dynamicmodelisdiscretized.Asecondcontrollerisintrod ucedthatappliesthebackstepping methodologyforthediscretetimesystem.Similarlytothec ontinuoustimecase,thediscretized modelhasacascadestructure.Themaincontributionofthed evelopedcontrolleristhedesign freedomintheconvergencerateforeachstatevariableofth ecascadestructure.Thisisofparticularinterestsincecontroloftheconvergencerateineachl evelofthecascadestructureprovides betterightresults.Furthermore,thestabilityoftheres ultingdynamicscanbesimplyinspected bytheeigenvaluesofalinearsystemwithoutthenecessityo fLyapunov'sfunctions.Thoseeigenvaluesaredeterminedbythedesigner. Fortheidenticationoftheparametersofthenonlineardis cretetimesystem,asimplerecursiveleastsquaresalgorithmisperformed.Theidentiedmo delandthecontrollerperformance wereevaluatedforthe Raptor90SE .Finally,theidenticationresultsofthepreviousmethod ologycanbesignicantlyimprovedifthediscretenonlinearh elicopterdynamicsarerepresentedby aTakagi-Sugenofuzzysystem.AfterthedevelopmentoftheT akagi-Sugenosystem,astandard RLSalgorithmisusedtoestimateitsparameters.Theresult ingfuzzysystemisaninterpolatorof nonlineardiscretesystems,whichdependsonthehelicopte r'sightcondition. 11

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1.4DissertationOutline Thisdissertationisorganizedasfollows.Chapter2presen tstheliteraturereviewrelatedtothe helicoptercontrolproblem.Thereviewincludesadescript ionofseveralightcontrolsystemsthat havebeenimplementedtoavarietyofhelicoptertypes. ThenexttwoChaptersprovidethenecessaryinformationfor theunderstandingofbothlinear andnonlinearhelicoptermodels.Inparticular,Chapter3p resentsananalyticalderivationofthe helicopter'skinematicequationofmotion,whenthehelico pteristreatedasarigidbody. ThegoalofChapter4istopresentasimpliedmodelofthemai nrotordynamicsthatencapsulatesthecouplingeffectsbetweenthefuselagemotionan dthemainrotor.Chapter4presents thesequenceofalltheintermediateeventsthattakeplacef romtheimplementationofthecyclic commandstothegenerationofthebladesappingmotion.The conceptsdescribedinthisChapter areimportantfortheunderstandingoftheexternalaerodyn amicforcesandmomentsmodels,used byboththelinearandnonlinearrepresentationsoftheheli copterdynamics. TheChapters5and6arerelatedtothelinearcontrollerdesi gnforhelicopters.Chapter5gives adescriptionofthefrequencydomainidenticationmethod whichisusedfortheextractionoflow orderlinearhelicoptermodels. Chapter6introducesatrackingcontrollerdesignbasedont helinearhelicopterdynamics. Chapter7providesabacksteppingtrackingcontrollerbase donthenonlinearhelicopterdynamics. Chapter8introducesadiscretetimeappliedbacksteppingc ontrollerandasimpletimedomain identicationmethodforthedeterminationofhelicopter' smodelunknownparameters. Chapter9showshowaTakagi-Sugenofuzzysystemcanimprove thetimedomainidenticationresults. Chapter10providesanextensivecomparisonandevaluation ofthecontrollerdesignsthathave beenpresentedinthepreviousChapters. ConcludingremarksandfutureworkfollowinChapter11.Fin ally,AppendixAprovides backgroundinformationaboutthebacksteppingcontrolmet hod. 12

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Chapter2:LiteratureReview ThisChapterpresentstheliteraturereviewofseveraligh tcontrollerdesignsforrotorcraft. Flightcontrolsystemshavebeentestedinawiderangeofrot orcrafttypesandcongurations. Thereviewincludesapplicationsforseveralrotorcraftty pessuchasfull-scale,small-scaleand experimentalplatforms,whicharegimbaledonaverticalst and.Theightcontrolsystemsthat existintheliteratureusetoolsfromalltheeldsofcontro ltheorybyincorporatingintothedesign classical,modernandintelligentcontroltechniques. Flightcontrolsystemsaremainlyclassiedaslinearandno nlinear.Typically,thisclassicationisbasedontherotorcraftmodelrepresentationthatis usedbythecontroller.Linearcontrol designsaremoreapplication-orientedandhavebeenimplem entedonthemajorityofrotorcraft autonomousplatforms.Theirpopularitystemsfromthesimp licityofthecontroldesign,which minimizesboththecomputationaleffortandthedesigntime .Onthecontrary,nonlinearcontrollersaremostlyvaluedfortheirtheoreticalcontribut iontotherotorcraftcontrolproblemand theirimplementationtoactualplatformsislimited.Inwha tfollowsbothlinearandnonlinear controldesignsarecoveredandcompared.2.1LinearControl Classicalcontroltechniquesdisregardthemultivariable natureoftherotorcraftdynamicsand thestrongcouplingthatexistsbetweentherotorcraftstat esandthecontrolinputs.Inthecontroller designsofthistype,eachcontrolinputisresponsiblefort heregulationofaparticularrotorcraft output.Theinteraxiscouplingsthatexistbetweentheroto rcraftoutputsaredisregarded,andeach controlinputisassociatedwithaSingle-InputSingle-Out put(SISO)feedbackloop.TheSISO 13

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feedbackloopsthatcorrespondtothecontrolinputsarecom pletelyindependentwitheachother. TheSISOfeedbackloopsaredesignedbasedontypicalloopsh apingtechniques.Thestability ofthefeedbackloopisdeterminedbythephaseandgainmargi nsofthelatter.Thesemargins dictatetheadmissibleamountofgainandphasethatcanbein jectedbythecontrollersuchthat thefeedbackloopdynamicsarestable.Thesemargins,howev er,caneasilyleadtoerroneous conclusionsinthecaseofmultivariablesystems[108]. In[89]aPIDcontrollercomposedbyfourindependentSISOlo opsisappliedtothe Kyosho Concept60Graphite smallscaleradiocontrolledhelicopteraspartofthe BerkeleyAeRobot (BEAR) project.Inordertoevaluatetheclosedloopcharacteristi csofthePIDschemeanelevenstate linearmodelwasidentiedbasedonthemodelstructureprop osedby[72].Themodelparameterswereidentiedbyusingthepredictionerrormethodth atisatimedomainidentication approach.ThePIDdesigndidnotmanagetosuppressthecoupl ingeffectbetweenthelateraland longitudinalmotionofthehelicopterandtheightcontrol lerwaslimitedonlytohoveright.The resultsindicatethatSISOtechniqueshavemoderateperfor manceandmultivariableapproaches arerequiredtoeliminatetheinherentcrosscouplingeffec tofthehelicopterdynamics.Asimilar multi-loopPIDdesignhasbeenimplementedin[44]fora YamahaR-50 smallscalehelicopter. Similarshortcomingsofthisclassicalcontrolapproachre strictedtheautonomousightofthe helicopteronlytohovermode. AsimpleclassicalcontroldesigncomposedofProportional Derivative(PD)SISOfeedback loopsisalsoinvestigatedin[70]forthe YamahaR-50 helicopter.Thehelicoptermodelisderived byperformingafrequencydomainidenticationmethod.The identiedhelicopterdynamics arerepresentedbyathirteenstatelinearmodelofthemotio nvariables,therotorandstabilizer barcharacteristics.Theidentiedlinearmodelisusedfor theoptimizationoftheightcontrol system.Inthisparticularcase,theuseofanotchlterissu ggestedforcompensatingtheeffect ofthestabilizerbaronthehelicopter'sattitudedynamics .Theparticularcasestudyindicatesthat althoughtheperformanceofightcontrolsystemsbasedonc lassicalcontroltechniquesislimited, accurateknowledgeofthehelicopter'smodelcansignican tlyimprovethedesignofthefeedback loops. 14

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Themajorityoflinearightcontrollersthathavebeenappl iedtoautonomoushelicopterplatforms,arebasedonthe H 1 feedbackcontrolapproach.The H 1 controlschemewasinitially introducedin[68].Themainadvantageofthe H 1 approach,isitsabilitytocopewithbothmodel uncertaintyanddisturbancerejection.The H 1 basedcontrollerdesigncanbeeasilyadjustedto classicalcontroltechniquesandatthesametimecompensat eforthemultivariableeffectsofthe helicopter.Theworkreportedin[80]providesverystronga rgumentsofwhythe H 1 approachis areasonableandsuitablecontrolsolutionforightvehicl es. Thetypicalstructureofan H 1 controlleriscomposedoftwoparts.Therstpartistheloop shapingportionoftheproblemwheretheinputchannelispre -compensatedandpost-compensated inasimilarwaythattakesplaceintheclassicalcontroltec hniquesofSISOsystems.TheprecompensatorincludesProportionalIntegral(PI)compensa torsforincreasingthelow-frequency gainofthesystem,disturbancerejectionandattenuatethe steadystateerror.Thepostcompensatoristypicallyusedfornoiseelimination,therefore,i tistypicallycomposedbylowpasslters. Thesecondportionofthecontroller,isthe H 1 synthesispart,whereastaticfeedbackgainis calculatedinordertostabilizethemultivariablesystemd ynamicsandatthesametimebeing optimalwithrespecttoaperformanceindex.Moreabout H 1 controlcanbefoundin[12,17,78, 92,113]. In[108]anobserverbasedmultivariablecontrollerwasdes igned,usingasingularvalueloop shapingmethodbasedonatwodegreeoffreedom H 1 optimization.Thecontrollerobjective wasthedevelopmentofanAttitude-CommandAttitude-Hold( ACAH)ightsystemforthefull scaleWestlandLynxhelicopter.Contrarytoautonomousig htapplications,theACAHight systemisintegratedtomannedightoperations.Thegoalof theACAHightcontrollerisfor thehelicoptertotrackanattitudeandheavevelocitycomma ndthatisgeneratedbythepilot'sstick input.Theprincipleofthecontrollerdesignistosuppress theinteraxiscouplingofthehelicopter dynamics,thusdecreasingthepilot'sworkload.Thepiloti sonlychargedwiththegenerationof thereferenceattitudeandheavevelocitycommandsthatare necessaryforthehelicopter'smotion. The H 1 controllerdesignwasbasedonaneightrigid-bodystatesan dfouractuatorstateslinear model.Themodelwasobtainedbylinearizingamoreelaborat enonlinearmodelinhovermode. 15

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Thecontrollerperformancewasevaluatedthroughightsim ulations.Althoughthecontroller wasdesignedforhoverandlowspeedoperations,thesimulat ionresultsindicatedsatisfactory performanceforspeedsupto 90 knots ThedesignofanACAHightsystembasedonastatic H 1 loopshapingapproach,isalso reportedin[83]fortheBell205fullscalehelicopter.This workaddressesthecommonproblem thatexistsinmultivariablemoderncontroltheory,accord ingtowhichthecontrollerorderisequal totheorderoftheplanttobecontrolled.Thisfactisofpart icularimportanceforthedesignof helicopterightcontrolsystems,sincetheorderofafulls calehelicoptermodelmayreachupto thirtystates!Theorderofthecontrollercanbereducedbym odelreductiontechniques,however, itispreferabletodesignfromthebeginningaightcontrol lerofminimumorderviatheuseof outputfeedback.Whenthecompletestatevectorofasystemi snotavailableforfeedbackpurposes,instead,onlyasubsetofthestatevariablescanbeus edbythecontroller;thenthecontrol lawisclassiedasanoutputfeedbackcontroller.Thisrese archdemonstratedthedesignofhigh performanceandloworder H 1 controllersbyapplyinglinearmatrixinequalityoptimiza tion techniques.Thehelicoptermodelwasderivedbylinearizin gathirtytwostatesnonlinearmodel athover.Thelinearizedmodelwasfurthertruncatedtotwel vestatesbyremovingthedynamics associatedwiththemainrotor.Theperformanceofthedevel opedACAHsystemwastestedina seriesofhelicoptermaneuverswithsatisfactoryresults. Analternate H 1 staticoutputfeedbackcontrollerdesignisproposedin[26 –28]forthestabilizationofanautonomoussmallscalehelicopterathover.T heoutputfeedbackapproachallows thedesignofmultivariablefeedbackloopsusingareduceds etofstateswhichresultsinminimizationoftheightcontroller'sorder.Thestructureofthepr oposedfeedbackloopsreectthephysicalightintuitionforhelicopterssuchthatthecontroll erdesigniswellsuitedfortheparticular application.Theloopshapingpartofthe H 1 controllerattenuatestheeffectsofhelicopterhigh frequencyunmodeleddynamics.Inmostcases,theoutputfee dbackcontrollerdesignproblem requiresthesolutionofthreenonlinearcoupledmatrixequ ations.Inthereportedwork,anovel iterativealgorithmisintroducedthatsolvesthe H 1 synthesispartofthecontrollerbysolving onlytwo-coupledmatrixequationsanddoesnotrequirethek nowledgeofaninitialstabilizing 16

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gain.Thecontrollerstructureiscomposedoftwomainloops .Therstloopisresponsibleforthe stabilizationoftheattitudedynamicswhilethesecondloo pisusedforpositiontracking.Thecontrollerdesignisbasedonathirteenstatelinearmodelofth ecoupledfuselageandrotordynamics. Themodelorderandstructureareobtainedby[70].Theident iedparametervalueshavebeen obtainedforthesmallscale Raptor90 radiocontrolledhelicopter.Thecontrollerperformancei s evaluatedbynumericsimulationsanditisrestrictedtohov erights. Promisingightresultsforanautonomoussmallscalehelic opterhavebeenobtainedinthe workreportedin[51,53–55].Inthisresearch,an H 1 loopshapingcontrollerwasimplemented ontheCarnegieMellonUniversity's YamahaR-50 .Thisapproachappliesablendingofmultivariablecontroltechniquesandsystemidenticationfort hedevelopmentoftheightcontrol system.Thehelicopternonlinearmodelisderivedbyusingt heMOdelingforFlightSimulation andControlAnalysis(MOSCA)modelingtechnique[52].MOSC Acombinesrstprinciplesand systemidenticationtechniquesforthederivationofboth linearandnonlinearhelicoptermodels. Athirtystatenonlinearmodelisderivedthatincludesthef uselage,mainrotor,stabilizerbarand inowdynamics.Thehelicopternonlineardynamicsarefurt herlinearizedinseverallinearmodels whichcorrespondtocertainoperatingconditionsofthehel icopter.Basedonthemultiplelinear modelsagainscheduled H 1 loop-shapingcontrollerisapplied. Gainscheduling isacontroltechniqueaccordingtowhichthegainsofthecon trollerarevaryingdependingoncertainvariables,whicharecalled schedulingvariables .Theschedulingvariablescouldbefunctionsofthesystem'sstatevariablesore xogenousvariablesthatdescribethe operatingconditionsofthesystem.Themaindesignideaist ocontrolanonlinearsystemusing afamilyoflinearcontrollers.Thenonlinearsystemdynami csarelinearizedoveranitenumber ofoperatingpoints.Theoperatingpointsareparametrized bytheschedulingvariables.Foreach linearizedmodelthatcorrespondstoaparticularoperatin gpoint,alinearcontrollerisdesigned. Theoverallcontrollawoperatesasaninterpolatorofthemu ltiplelinearcontrollerswhosegain parametersdependontheschedulingvariables.Moredetail saboutgainschedulingcanbefound in[43,87].Thegainschedulingapproachhasemergedfromav ionicscontrolapplications,where 17

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thelinearizationofthevehicle'snonlineardynamicsarou ndseveraloperatingpointsisacommon procedure. Aninterestingcomparativestudybetweenseveralcontroll erdesignsisgivenin[109,110]. Bothclassicalandmultivariablelinearcontrollersarein cludedinthestudy.Aneighteenstate linearmodel,whichrepresentsthehelicopterdynamicsath over,wasusedfortheightcontrollers design.Theightcontrollersweretestedinaradiocontrol ledhelicoptermountedonamechanicalstructurethatallowsthemotionofthehelicopterinal ldirectionsoftheCartesianspace.For hoveringthemultivariabletechniqueshadsuperiorperfor manceincomparisonwiththeclassical controldesigns.FromthemultivariabledesignsLQR, H 2 and H 1 designswereevaluated.The ightvalidationindicatesthatinthemultivariabledesig ncaseitispreferabletodesignmultiple feedbackloopswhichcorrespondtoindependentsubsystems ofthehelicopterdynamics,thus, decomposingtheproblem.Thisapproachispreferablefrome stablishingthecontrollerdesign directlytothecompletehelicopterdynamics.Theloworder subsystemsshouldappealtothe physicalightintuitionandshouldbeasdecoupledaspossi ble.Intheparticularcasetheinitial linearmodelwasdecomposedtoasubsystemrepresentingthe longitudinal/lateralmotionanda secondsubsystemoftheheaveandyawdynamics. Anexampleofalinearcontrollerdesignforahelicopterina verticalstandisalsogivenin [56].Thegimbaledlikedeviceonwhichthehelicopterwasco nnectedto,allowsonlyathree degreesoffreedommotionofthelatter.AdiscreteLinearQu adraticRegulatorisusedwithan augmentedKalmanlterforstateestimation.Theworkin[2] comparesasimpleeigenstructure assignmentwithfullstatefeedbackcontrollerversusatyp icalLQRdesign.Thehelicoptermodel underconsiderationdoesnotincludetheappingdynamicsa ndthevericationtakesplaceby numericalsimulations.Otherrobustdesignsofhelicopter controlarereportedin[6,50,82,97] 2.2NonlinearControl Ingeneral,mostcontroldesignsarebasedonlinearizedhel icopterdynamicsusingthewidely adoptedconceptofstabilityderivatives.However,inrece ntyearsthereisconsiderableresearch 18

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relatedtohelicopterightcontrolbasedonnonlineardyna micrepresentations.Thenonlinear controllerdesignsaremostlyvaluedfortheirtheoretical contributiontothehelicopterightcontrolproblem.Theirapplicabilityisstillanopenchalleng emainlyduetotheincreasedorderand nonlinearstructureofthecontroller.However,theircont ributiontotheunderstandingofthelimitationsandcapabilitiesofthehelicoptercontrolproblem issignicant. Detailedmodelsofhelicopternonlineardynamicscanbefou ndin[40,79,84].However,such modelsareofhighorderandimpracticalforcontrollerdesi gnpurposes.In[47,48]asimplied nonlinearmodelofthehelicopterdynamicsisintroduced.T hehelicoptermodelisrepresented bythenonlineardynamicequationsofmotionofthehelicopt erenhancedbyasimpliedmodel oftheaerodynamicforceandtorquegeneration.Theparticu larmodelhasbeenadoptedinmost workrelatedtothehelicopternonlinearcontrollerdesign .Thereportedworkindicatesthatexactinput-outputlinearizationfailstolinearizetheheli coptermodelresultinginunstablezero dynamics.Thisworkhasshownthattheuseofanapproximatem odelthatdisregardsthethrust forcesproducedbythemainrotorappingmotion,isfullsta telinearizable.Thisderivationisvery importantsinceifthesystemdynamicsarenotinput-output linearizablemostnonlinearcontrol techniqueswouldbeinapplicable.Afeedbacklinearizatio ncontrollerisproposedbasedontheapproximatedmodeldynamics.Itisproventhattheproposedco ntroller,basedontheapproximated model,achievesboundedtrackingofthepositionandyawref erencetrajectories. However,helicoptersarecharacterizedbysignicantpara metricandmodeluncertaintydue tothecomplicatedaerodynamicnatureofthethrustgenerat ion.Therefore,linearizationandnonlineartermscancellationtechniquesarepoorlysuited.It isimportantthatthecontrollerdesign exhibitssufcientrobustnesstowardspotentiallysigni cantuncertainty.Adesignthatguarantees boundedtrackinginthepresenceofparametricandmodelunc ertaintyisreportedincite[37].The proposedcontrollawincorporatesstabilizationtechniqu esforfeedforwardsystemswithinput saturationandadaptivenonlinearoutputregulationtechn iques. Theworkreportedin[66,67]addressesthedesignofanautop ilotforthehelicoptercapableof lettingitsvertical/lateralandlongitudinaldynamicsan dyawattitudedynamicstrackingarbitrary referenceswithonlysomeboundrequirementsonthehighero rdertimederivativesimposedby 19

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functionalcontrollability.Thisworkisanextensionof[3 7]byincludingthemainrotordynamics andallowingthetrackingofarbitrarytrajectories.Inadd ition,inthereportedworkthecontroller designisbasedonthepitch-roll-yawattitudeconventioni nsteadofquaternionswhichareuse in[37].Similarlyto[37],thenalcontrolstructureisami xoffeedforwardactionsandnested saturationcontrollaws.Theproposedcontrollerisableto enforceveryaggressivemaneuvers characterizedbylargeattitudeanglesandtocopewithposs iblelargeuncertaintiesaffectingthe physicalparameters. Aspreviouslymentioned,mostnonlineardesignsneglectth eeffectofthrustforcecomponents associatedwiththetiltofthemainrotordisk.Thisiscommo npracticesincethoseparasiticforces haveaminimaleffectontranslationaldynamics.Thissimpl icationresultsinasetofsystem equationshavingafeedbackform,whichisidealforbackste ppingcontroldesignestablishedin [49].Backsteppingcontrolimplementationforhelicopter sispresentedin[11,21,64,65]and similardesignsforaquadrotorin[32,33,42]. ApproachesofnonlinearcontrolthatuseNeuralNetworks(N N)andnonlinearinversionare reportedin[14,15,34,38,39,45].Inalltheaforementione dcases,thenonlinearinversionrequirementandtheaugmentationofaNNincreasessignicant lytheorderofthecontroller.To thisextentthederivationofthecontrollerusingthenonli nearequationofmotionofthehelicopter becomesimpractical.Thereforethesecaseshaveappliedth econtrollersbasedonthelinearized dynamicsofthehelicopteraroundhover.In[34,45]theanal ysisisevenmorerestrictedbyusinga simpliedmodelofonlythelongitudinalandheavemotionof thehelicopter.In[38,39]thecontrollerwasexperimentallyimplementedtoa YamahaR-50 helicopterforasimplestepcommand response. 20

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Chapter3:HelicopterBasicEquationsofMotion TheobjectiveofthisChapteristoprovidethebasicequatio nsofmotionofthehelicopter, whenthehelicopteristreatedasarigidbody.Theequations ofmotionarederivedbyimplementingNewton'ssecondlawthatdealswithvectorsummationsof allforcesandmomentsasapplied tothehelicopter,relativetoaninertialreferenceframe. However,forpracticalreasons,analysis maybesignicantlysimpliedifmotionisdescribedrelati vetoareferenceframerigidlyattached tothehelicopter.Whenthisisthecase,theequationsofmot ionarederivedrelativetothisnoninertialbody-xedframe.TheendresultofthisChapterist hecompletestatespacerepresentation ofthehelicopterequationsofmotionsinthecongurations pace. 3.1HelicopterEquationsofMotion Therstassumptiontowarddynamicmodelingofahelicopter istoconsideritasarigidbody withsixDegreesOfFreedom(DOF).TheDOFdictatetheminima lnumberofparametersthatare requiredtospecifyanobject'sconguration[95].Themoti onofarigidbodyisdenedrelativeto aCartesianinertialframe.Aframeiscomposedofapointins paceandthreeorthonormalvectors thatformabasis.Therefore,inordertoderivetheequation sofmotion,twoframesarerequired. Therstoneistheinertialframe(Earth-xedframe)dened as F I = f O I ; ~ i I ; ~ j I ; ~ k I g .Atypical conventionoftheEarth-xedframe,istheNorth-East-Down systemwhere ~ i I pointsNorth, ~ j I pointsEastand ~ k I pointsatthecenteroftheEarth.Thesecondframeisthebody -xedreference framedenedas F B = f O B ; ~ i B ; ~ j B ; ~ k B g wherethecenter O B islocatedattheCenterofGravity (CG)ofthehelicopter.Thevector ~ i B ispointingforwardthroughthehelicopternose, ~ j B ispointingattherightsideofthefuselageand ~ k B pointsdownwards,suchthat f ~ i B ; ~ j B ; ~ k B g constitutesa 21

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r N ,p L ,w Z ,u X q M ,v Y Bi Bj Bk B O Figure3.1:Body-xedcoordinatesystem.Thecomponentsof theexternalforcesandmoments actingonthefuselagearedenotedby f B =[ XYZ ] T and B =[ LMN ] T ,respectively. Thelinearandangularvelocitycomponentsaredenotedby v B =[ uvw ] T and B =[ pqr ] T respectively.right-handedCartesiancoordinateframe( ~ k B = ~ i B ~ j B ).Thedirectionsofthebody-xedframe orthonormalvectors f ~ i B ; ~ j B ; ~ k B g areshowninFigure3.1. Therearetwowaystorepresentfreevectorsinspace.Thers tisthroughthesyntheticapproach,wherethefreevectorsareconsideredasgeometrice ntities.Inthesecondapproach,the geometricentitiesarerepresentedbycoordinates.Thisis calledanalyticapproach[95].Inthe analyticapproach,thevectorrepresentationdependsonth ecoordinateframeofreference.The advantageoftheanalyticapproachisthattheoperationsbe tweenvectorsmaybetackledbyalgebraicmethods(equations).Forexample,avector ~w canberepresentedanalyticallybythecoordinatetriple w B =[ w B 1 w B 2 w B 3 ] T 2 R 3 ,withrespecttothebody-xedframe,orbythetriple w I =[ w I 1 w I 2 w I 3 ] T 2 R 3 ,withrespecttotheinertialframe.Ingeneral,thetriples w B and w I willbedifferent,however,theybothrepresentthesamegeo metricentity ~w .Inordertoprovidea clearunderstandingofthederivationofthehelicopter'se quationsofmotion,inthisChapterboth approacheswillbeadopted. Aninertialframemakestheanalysisimpracticalsincemome ntsandproductsofinertiavary withtime.Thisisnotthecasewhenabody-xedframeisconsi dered,wheremomentsandprod22

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uctsofinertiaareconstant.Therefore,theequationsofmo tionwillbederivedwithrespecttothe body-xedframe. ThelinearvelocityvectorofthefuselageCGisdenotedby ~v .Thecoordinatevectorofthe linearvelocityis v B =[ uvw ] T ,withrespecttothebody-xedframe.Similarly,theangula r velocity ~! ofthefuselage,isrepresentedinthebody-xedframeby B =[ pqr ] T Thesumofallexternalforcesactingonthefuselagearedeno tedby f B =[ XYZ ] T ,withrespecttothebody-xedframe.Similarly,thesumofallexter nalmoments(torques)aredenotedby B =[ LMN ] T ,asshowninFigure3.1.Positivedirectionoftheangularve locityandmoment componentsreferstotheright-handruleabouttherespecti veaxis. TheequationsofNewton'ssecondlawarevalidonlyinaniner tialreferenceframe.Therefore, Newton'ssecondlawforthetranslationalmotionoftheheli copterisgivenby: ~ f = m d~v dt I (3.1) where m denotesthetotalmassofthehelicopter.Theoperand d ( ) dt I denotesthetimederivative ofavectorinspaceasviewedbyanobserverintheinertialre ferenceframe.Frombasickinematic principles,whichcanbefoundin[31,111],thetimederivat iveof ~v withrespecttotheinertial referenceframe,isequalto: d~v dt I = d~v dt B + ~! ~v (3.2) Theoperator ( ) isthevectorcrossproduct.Theterm d~v dt B denotesthetimederivativeofthe velocityvector ~v withrespecttothebody-xedreferenceframe.Ingeneral, d ( ) dt B denotesthe derivativeofavectorfromtheviewpointofanobserverinth ebody-xedframe.Atthispointa commentshouldbemadeaboutvectordifferentiation:Asind icatedin[31],theoperands d ( ) dt I and d ( ) dt B whenperformedonafreevectorinspacewillprovideingener aladifferentresult.The rstoneisthetimerateofchangeofavectorasviewedbyanob serverfromtheinertialframe, whilethesecondoneisthetimerateofchangeviewedbyanobs erverofarotatingframe.The changeofthevector'sdirectionduetotheangularvelocity ofthebody-xedframe,isnotcon23

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ceivablebytheobserveronthebody-xedframe.Onthecontr ary,thischangeisdetectedbythe observeroftheinertialframe.Asimplecoordinateconvers ionwillnotprovideaccurateresults sincebothofthemareviewingadifferentchange. Since ~v = u ~ i B + v ~ j B + w ~ k B ,then d~v dt B =_ u ~ i B +_ v ~ j B +_ w ~ k B .Therefore,substituting(3.2)to (3.1),theanalyticexpressionofNewton'ssecondlawforth etranslationalmotionis: X=m =_ u + qw rv Y=m =_ v + ru pw (3.3) Z=m =_ w + pv qu Toconcludethederivationoftheequationsofmotion,Newto n'ssecondlawisappliedtoall momentsthatactontheCG.Thereferencepointforcalculati ngtheangularmomentumandthe externalmomentsisrigidlyattachedtothe CG ofthehelicopter.Furthermore,usingthebodyxedreferenceframefortheanalysisisadvantageoussince themomentsandtheproductsof inertiadonotvarywithtimegiventhatthemassdistributio nofthehelicopterdoesnotchange. Let ~ H denotethevectorofthehelicopterangularmomentumand H B =[ h x h y h z ] T its coordinateswithrespecttothebody-xedframe.From[31], theangularmomentumcomponents ofthebody-xedreferenceframearegivenby H B = I B ,where I denotestheinertiamatrix: I = 266664 I xx I xy I xz I yx + I yy I yz I zx I zy + I zz 377775 (3.4) Therespectivemomentsofinertiaare: I xx = X ( y 2 m + z 2 m ) dm I yy = X ( x 2m + z 2 m ) dm I zz = X ( x 2m + y 2 m ) dm 24

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Theproductsofinertiaare: I xy = I yx = X x m y m dm I xz = I zx = X x m z m dm I yz = I zy = X y m z m dm Theabovesumsapplytoallelementarymasses dm ofthehelicopter,and x m y m and z m arethe distancesofeachelementarymassfromtheCG.Itisassumedt hattheprincipalaxescoincidewith theaxesofthebody-xedframe,therefore,itfollowsthat I xy = I yx =0 I yz = I zy =0 and I zz = I zx =0 Newton'ssecondlawfortherotationalmotiondictatesthat theexternalmomentsactingon thehelicopterareequaltothetimerateofchangeoftheangu larmomentumwithrespecttothe inertialreferenceframe.Therefore: ~ = d ~ H dt I (3.5) Fromdifferentiationoffreevectors,onehas: d ~ H dt I = d ~ H dt B + ~! ~ H (3.6) Theterm d ~ H dt I isthetimerateofchangeoftheangularmomentumwithrespec ttotheinertial frame.Thetimederivativecomponentsoftheangularmoment um d ~ H dt B ,aregivenby: h x = I xx p h y = I yy q (3.7) h z = I zz r Substituting(3.6)and(3.7)to(3.5),theanalyticexpress ionofNewton'ssecondlawfortherotationalmotionofthehelicopteris: L = I xx p + qr ( I zz I yy ) 25

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M = I yy q + pr ( I xx I zz ) (3.8) N = I zz r + pq ( I yy I xx ) Therefore,thenalformoftheequationsofmotionwithresp ecttotheinertiaframe,butexpressedinthebody-xedframecoordinatecomponents,areg ivenby(3.3)forthetranslationaland by(3.8)fortherotationalmotionofthehelicopter.Acompa ctformofthehelicopterequationsof motionexpressedinthebody-xedframe,isthefollowing: 264 mI 3 0 0 I 375 264 v B B 375 + 264 B mv B B I B 375 = 264 f B B 375 (3.9) From[75],theaboveequationsarecalled Newton-Eulerequations inthebody-xedframe'scoordinates.3.2PositionandOrientationoftheHelicopter Themotionofthehelicopterisdenedbythepositionandori entationofthebody-xedframe relativetotheinertialframe.TheNewton-Eulerequations provideinformationaboutthetranslationalandangularvelocityofthehelicopter.However,nei therofthemgiveinformationaboutthe helicopter'scurrentpositionandorientation.Thehelico pterequationsofmotionarecompletedby determiningthepositionandorientationdynamicsofthela tter.Derivationfollows[20]butwith additionaldetailsforclaricationpurposes. Let F 1 = f O B ; ~ i 1 ; ~ j 1 ; ~ k 1 g deneanintermediateframethatisalignedwith F I andcentered ontheCGofthehelicopter.Thehelicopterorientationatan ytimeinstantmaybeobtainedby performingthreeconsecutiverotationsof F 1 untilitisalignedwith F B .Therotationsareperformedataspecicorder,theycannotbeconsideredasvecto rsandtheyarenotcommutative [111].Therefore,therotationorderisimportantforconsi stency,asfollows(seeFigure3.2): 26

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B O " IOIk 1i 1j Bj 3 2j j ,Bk 2 1k k ,3i iB ,3k 2i Ij Ii Figure3.2:Helicopterorientation. Arotationofanangle about ~ k 1 .Thisrotationmovesthehelicoptertothedirectiondenedby F 2 = f O B ; ~ i 2 ; ~ j 2 ; ~ k 2 g ,bringing ~ i 2 paralleltotheplanedenedby ~ i B and ~ k 1 Arotationofanangle about ~ j 2 .Thisrotationmovesthehelicoptertothedirectiondescribedby F 3 = f O B ; ~ i 3 ; ~ j 3 ; ~ k 3 g ,aligning ~ i 3 with ~ i B Arotationofanangle aboutaxis ~ i 3 bringing F 3 toitsnalorientation F B Intheaboveconvention,eachrotationisperformedaboutan axiswhoselocationdependsonthe precedingrotations[16].Theintermediateframesandeach rotationisshownindetailinFigure3.2.Theseangleswiththeparticularsequenceofrotati onsarealsoknownas Z-Y-XEuler angles .TheEuleranglesorientationvectorisdenotedby =[ ] T .Positivedirectionofeach anglereferstotheright-handruleabouttherespectiveaxi s.Anyarbitraryrotationofthebodyxedframerelativetotheinertiaframecanbeparametrized bythethreeEulerangles. 27

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3.2.1HelicopterPositionDynamics Expressingthehelicopterpositionrelativetothebody-x edframeismeaninglessandsuchan actioncannottakeplace.Therefore,thepositiondynamics arederivedwithrespecttotheinertial frame.Beforewepresentthepositiondynamics,weintroduc ethedescriptionthatrelatesthe coordinatevectorsofthebody-xedandinertialframes.Th isdescriptioniscalledthe rotation matrix anditprovidesasystematicwaytoexpresstherelativeorie ntationofthetwoframes. Denoteby v I =[ v I x v I y v I z ] T thelinearvelocity'scoordinatevectorwithrespecttothe inertial frame.Thelinearvelocityvectorofthehelicopter,relati veto F B and F I ,respectively,is: ~v = u ~ i B + v ~ j B + w ~ k B (3.10a) ~v = v I x ~ i 1 + v I y ~ j 1 + v I z ~ k 1 (3.10b) UsingthedenitionoftheEulerangles,theunitvectorsoft hebody-xedframe F B arewritten relativetotheframe F 3 as: 266664 ~ i B ~ j B ~ k B 377775 = 266664 1000cos sin 0 sin cos 377775 266664 ~ i 3 ~ j 3 ~ k 3 377775 = R T ( )[ ~ i 3 ~ j 3 ~ k 3 ] T (3.11) Similarly,theunitvectorsoftheframe F 3 areexpressedrelativetotheframe F 2 as: 266664 ~ i 3 ~ j 3 ~ k 3 377775 = 266664 cos 0 sin 010 sin 0cos 377775 266664 ~ i 2 ~ j 2 ~ k 2 377775 = R T ( )[ ~ i 2 ~ j 2 ~ k 2 ] T (3.12) 28

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Finally,theunitvectorsoftheframe F 2 relativeto F 1 areexpressedas: 266664 ~ i 2 ~ j 2 ~ k 2 377775 = 266664 cos sin 0 sin cos 0 001 377775 266664 ~ i 1 ~ j 1 ~ k 1 377775 (3.13) = R T ( )[ ~ i 1 ~ j 1 ~ k 1 ] T (3.14) Byconsecutivesubstitutionsof(3.11),(3.12)and(3.13)t o(3.10a),oneobtains: ~v =[ uvw ][ ~ i B ~ j B ~ k B ] T =[ uvw ] R T ( )[ ~ i 3 ~ j 3 ~ k 3 ] T =[ uvw ] R T ( ) R T ( )[ ~ i 2 ~ j 2 ~ k 2 ] T =[ uvw ] R T ( ) R T ( ) R T ( )[ ~ i 1 ~ j 1 ~ k 1 ] T (3.15) Denoteby R () theproduct: R ()= R ( ) R ( ) R ( ) (3.16) Equatingtherighthandsidesof(3.10b)and(3.15),onegets : 266664 v I x v I y v I z 377775 = R () 266664 u v w 377775 (3.17) where: R ()= 266664 cos cos sin sin cos cos sin cos sin cos +sin sin cos sin sin sin sin +cos cos cos sin sin sin cos sin sin cos cos cos 377775 (3.18) 29

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Thematrix R () iscalledtherotationmatrixanditisparametrizedwithres pecttothethreeEuler angles.Therotationmatrix R isusedtomapvectorsfromthebody-xedframe F B totheinertial frame F I .TherotationmatrixbelongstotheSpecialOrthogonalgrou poforder 3 denotedby SO (3) Property3.1. Therotationmatrixhasthefollowingproperties[95]: 1. RR T = R T R = I 2. det ( R )=1 3.Eachcolumn(andeachrow)of R isaunitvector 4.Eachcolumn(andeachrow)of R aremutuallyorthogonal WhentherotationmatrixisparametrizedbytheZ-Y-XEulera ngles,singularitiesoccurat = = 2 .Morespecically,when = = 2 ,then,theinverseproblemofretrievingtheEuler anglesfromtherotationmatrix,doesnothaveasolution[75 ].Suchsingularitiesoccurinany3-D representationof SO (3) Therotationmatrixfacilitatesthederivationoftheposit ionandtranslationalvelocitydynamics withrespecttotheinertialframe.Denoteby p I =[ p Ix p Iy p Iz ] T thepositionofthehelicopterCG. Then,thepositionandvelocitydynamicswithrespecttothe inertialframeare: p I = v I (3.19) v I = 1 m Rf B (3.20) Anyrigidmotionisdenedbytheorderedpair ( p I ;R ) where p I 2 R 3 and R 2 SO (3) .The group SE (3)= R 3 SO (3) isthecongurationspaceofthehelicopteranditisknownas the SpecialEuclideangroup 30

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3.2.2HelicopterOrientationDynamics Considerthatduringaninnitesimaltimeinterval dt thehelicopterissubjectedtothreeinnitesimalrotations d d and d resultinginapositiondenedbyangles + d + d and + d .Althoughniterotationscannotbetreatedasvectors,in nitesimalrotationsmaybetreated assuch,thus,accordingto[20],thevectorthatrepresents theaboverotationis: ^ n = d ~ i B + d ~ j 3 + d ~ k 2 (3.21) Then,theangularvelocitycanbeexpressedas: ~! = d ^ n dt = ~ i B + ~ j 2 + ~ k 1 (3.22a) and: ~! = p ~ i B + q ~ j B + r ~ k B (3.22b) Byusingtheexpressions(3.11)-(3.13)andequatingtherig hthandsidesof(3.22a)and(3.22b), onehas: 266664 p qr 377775 = 266664 00 377775 + R T ( ) 266664 0 0 377775 + R T ( ) R T ( ) 266664 00 377775 ) 266664 p qr 377775 = 266664 10 sin 0cos sin cos 0 sin cos cos 377775 266664 _ 377775 (3.23) 31

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Basedontheaboveequation,theorientationdynamicsofthe helicopteraregivenby: =() B (3.24) where: ()= 266664 1sin tan cos tan 0cos sin 0sin = cos cos = cos 377775 (3.25) Foranarbitrarymotion,thecomponentsoftherotationmatr ixaretimevarying.Thederivativeof therotationmatrixisgivenby: R = R ^ B (3.26) where ^ B denotestheskewsymmetricmatrixofthevector B .Foravector w =[ w 1 w 2 w 3 ] T the skewsymmetricmatrixisdenedas: ^ w = 266664 0 w 3 w 2 w 3 0 w 1 w 2 w 1 0 377775 Themultiplicationofthematrix ^ w withavector h ,producesthecoordinatesofthecrossproduct w h Proposition3.1. Fortwovectors g 1 and g 2 of R 3 ,theskewsymmetricmatrixhasthefollowing properties: 1. ^ g 1 g 1 =0 2. R (^ g 1 g 2 )= \ ( Rg 1 )( Rg 1 ) 3. ^ g 1 +^ g T 1 =0 4. R ^ g 1 R T = d Rg 1 32

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_ v I = 1 m Rf B p I = v I R = R ^ B I B = B ( I B )+ B =() B v I B R f B B Figure3.3:Interconnectionofthehelicopterdynamicsint hespace SE (3) Thederivationof(3.26)isnotpresentedherebecauseitiso utofthescopeofthisChapter. However,moredetailsmaybefoundin[75,95].Therotationm atrixdynamicsareveryimportant,sincetheyappearinthelinearvelocitydynamicsgive nin(3.20).Althoughtheorientation dynamicsarealsogivenin(3.25),workingwiththerotation matrixincontrolapplicationsismore preferableduetothespecialpropertiesoftherotationmat rix. 3.3CompleteHelicopterDynamics Havingdenedthepositionandorientationdynamics,theco mpletestatespacerepresentation ofthehelicopterequationsofmotioninthecongurationss pace SE (3) is: p I = v I (3.27) v I = 1 m Rf B (3.28) R = R ^ B (3.29) I B = B ( I B )+ B (3.30) where [ p I v I R! B ] 2 R 3 R 3 SO (3) R 3 .Integrationoftheaboveequationsprovides alltherequiredinformationfordeterminingthehelicopte rmotioninthecongurationspace.The interconnectionofthehelicopterdynamicsin SE (3) isillustratedinFigure3.3. 33

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Asmentionedearlier,theorientationofthehelicopterisp arametrizedbytheZ-Y-XEuler angles.Inthiscaseeachintermediaterotationtakesplace aboutanaxisofaframethatisproduced byaprecedingrotation.Inaviationapplicationsitispref erablethateachrotationtakesplaceabout theaxisofaxedframe.Exactlythesameequationsarederiv edifthenalorientationisproducedbya angleabouttheaxis ~ i I ,thenanangle about ~ j I andnallyanangle abouttheaxis ~ k I .Inthisconventiontheangles and arecalledpitch,rollandyawangles,respectively. Thehelicopterrigidbodydynamicsgivenin(3.27)-(3.30)a recompletedbydeningtheexternalbodyframeforce f B andtorque B 3.4Remarks ThisChapterhaspresentedananalyticalderivationoftheh elicopter'sbasicequationsofmotion.Thelinearandangularvelocitydynamicsareobtained fromNewton'ssecondlawfortranslationalandrotationalmotion.Theorientationofthehelico pterwithrespecttoastationaryinertial frameisdeterminedbythreeorientationangles.Therotati onmatrixisparametrizedbytheorientationanglesandconstitutesasystematictoolformapping vectorsfromtheinertialframetothe bodyxedframeandviseversa.Thepositionandorientation dynamicscompletethedescription ofthehelicopter'smotioninthecongurationsspace.The nalrequirementtowardsthederivation ofthehelicopter'smathematicalmodelisthedeterminatio noftheexternalforcesandmoments appliedtothehelicopter.Themainsourceofforceandtorqu egenerationofthehelicopterisproducedbythemainandtailrotor.Themainrotoritselfisadyn amicalsystem.Adetailedmodel oftheaerodynamicforcesandmomentsofthemainrotorwould beofhighorderandsignicant complexity.ThenextChapterpresentsasimpliedmodeloft hemainrotordynamicswhichis suitableforcontroldesignpurposes. 34

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Chapter4:SimpliedRotorDynamics Thehelicopter'smainsourceofpropulsionisprovidedbyth emainandtailrotor.Theaerodynamicforcesandmomentsarenonlinearfunctionsofmotionc haracteristicsandcontrols.Due tothecomplexityandtheuncertaintyassociatedwiththeae rodynamicphenomena,adetailed modeloftheforcesandmomentsproducedbythemainrotorwou ldbeofhighorderandcompletelyimpracticalforanycontrollerdesign.InthisChap ter,themodelingapproachpresentedin [47,56,70,72]isfollowed,whichprovidesasimpliedderi vationofthemainrotordynamicsand theproducedthrustforcevector,adequateforcontrollerd esignpurposes. 4.1Introduction Therearefourcontrolcommandsassociatedwithhelicopter piloting.Thecontrolinputvector isdenedas u c =[ u lon u lat u ped u col ] T ,where u col and u ped arethecollectivecontrolsofthe mainandtailrotor,correspondingly.Thecollectivecomma ndscontrolthemagnitudeofthemain andtailrotorthrustbyauniformchangeinthepitchangleso falltherotor'sblades.Theother twocontrolcommands, u lon and u lat ,arethecycliccontrolsofthehelicopter,whichcontrolth e inclinationoftheTip-Path-Plane(TPP)onthelongitudina landlateraldirection.TheTPPisthe planeonwhichthetipsofthebladeslieanditisusedtoprovi deasimpliedrepresentationofall therotorblades[70]. Forthemainrotorthrustgeneration,asimpliedapproachi ffollowedbasedon[47,70,72]. Accordingtothat,thethrustvectorproducedbytherotordi skisperpendiculartotheTPP.The mainrotorbladesapartfromrotatingabouttheshaftaxis,t heyalsoexhibitaappingmotion 35

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normaltotheplaneofrotation.Sincethethrustvectorisno rmaltotheTPP,bycontrollingthe TPPinclination,thepilotindirectlycontrolsthedirecti onofthepropulsionforces. TheTPPisitselfadynamicsystem.ThedynamicsoftheTPPrep resenttherotordynamics. Therotorisaffectedbyboththepilot'scontrolcommandsan dthehelicopter'smotion.Onthe otherhand,thehelicopter'smotionitselfiscontrolledby theappliedrotorforcesandmoments. Therefore,thereisanobviouscouplingbetweentherotoran dfuselagedynamics.Theworkpresentedin[70]and[104]providesasimpliedmodeloftherot ordynamicsthatisintegratedwith therigidbodymodel,inordertoarriveata“hybridmodel”of thehelicopterdynamics. ThegoalofthisChapteristopresentasimpliedmodelofthe rotordynamics,whichencapsulatesthecrosscouplingeffectbetweentherotorandthef uselage.Thesecondtaskistoderive apracticaldescriptionofthethrustforceandmomentcompo nents,producedbythemainrotor. Ingeneral,therotormathematicalmodelingisaverycomple xprocedure.Thecomplexityofthe model,withoutconsideringanysimplicationassumptions ,willsignicantlyincrease.Aspointed in[18],themodelcomplexitydependsontheapplicationthe modelisdesignedfor.Forcontrol applications,theproposedmodelprovidesapracticalandp hysicallymeaningfuldescriptionofthe rotordynamics.ThemainresultsofthisChapterassociated withtherotordynamicsarebasedon [70]. Inorderforthereadertounderstandthenalderivationoft hesimpliedrotordynamicsandto obtainafairinsightofthephysicalconceptsthateffectth erotorbehavior,aseriesofintermediate stepsarepresented.Therststepistointroducetheadditi onalDOFoftheblades.Thecontrol oftherotorismainlyproducedbythevariationoftheblades pitchangle.Bychangingthepitch angle,theaerodynamicloadsofthebladesarealsoaltered. Thisisawayofcontrollingthelift forcesappliedtoeachblade.Tothisextent,agenericdescr iptionofthebasicmechanicaldesign thatproducesthevariationofthepitchangleisgiven. Simpliedaerodynamicsconceptsarepresentednext,which resultinthederivationofthe aerodynamicforcesappliedtoeachblade.Bygivingadescri ptionoftheaerodynamicforcesand byconsideringtheadditionalinertiaforcesactingontheb lade,theblade'sequationsofmotion arederived.Theadoptionofsomephysicallymeaningfulsim plicationassumptionsleadsto 36

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thederivationofthesocalledTip-Path-Planedynamicequa tions,whichessentiallyarethemain rotordynamics.Finally,usingtheTip-Path-Planeequatio ns,theforceandmomentcomponents producedbythemainrotorarederived.4.2BladeMotion Themostcommonrotorcongurationconsistsoftwo(ormore) identicalbladesattachedto therotorhub[40].Therotorhubisconnectedtotherotorsha ft.Thebladesperformrotational motionaroundtherotorshaftwithaconstantangularveloci ty n Apartfromtherotationalmotionaroundtheshaft,theblade salsohavethreeadditionalDOF. TheseDOFareillustratedinFigure4.1.Morespecically: Flapping :ThisDOFproducesamotionofthebladethatisparalleltoth eplanethatincludes thebladeandtheshaft,anditisdenotedbytheappingangle .Theappingangleis denedtobepositivewhentheblademovesupwards. Lead-Lagging :ThisDOFproducesamotionofthebladethatisparalleltoth ehubplane. Thelaggingangleisdenotedby .Laggingispositivewhenthebladeopposesthedirection ofrotationproducedbytherotor. Feathering :ThisDOFproducesapitchingmotionofthebladeaboutthebl adespan.The featheringangleisdenotedby .Featheringangleisconsideredpositivefornoseupmotion oftheblade. Thenecessityforfreemotionofthebladewithrespecttothe seadditionalDOFwasapparentfromearlyhelicopterdesigns.Thefeatheringanglecon trolstheaerodynamicforcesthatare generatedontheblades.Thoseaerodynamicforcescontrolt hethrustforcethatisnecessaryfor themotionofthehelicopter.However,thegenerationofaer odynamicforceshasasaresultthe appearanceoflargemomentsontherootoftheblade.Thosemo mentsaretransmittedtothehub andthentotherestofthehelicopter'sbody.Arotorcongur ationthatallowstheappingmotion 37

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! #rotor shaft (a)The3DOFoftherotorbladeinspace.TheFigureisbasedon[40]. flapping feathering lead-lagging hub center (b)TopviewoftherotorhubwhereeachDOFoftherotorbladeisrepresentedbyabladehinge.TheFigureisbasedon[40]. Figure4.1:Representationoftherotor3DOF. 38

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ofthebladeisneededinordertoreliefthebladerootfromth osearisingmoments.Theimmediate resultoftheappingmotionisthegenerationofCoriolismo mentsonthebladeintheplaneof rotation[7].Asecondcongurationisneededtoallowthela ggingmotionofthebladesothose momentsarerelieved. Thereareseveralhubdesignsthatallowthemotionofthebla des.Thetraditionalapproach istheuseofmechanicalhingesatthebladerootfortheappi ngandlaggingmotion.Modern designshavesubstitutedtheuseofhingesbyexibleelemen tsintherootofthehubthatallow theappingandlaggingmotion.Inaddition,therearecong urationsthatusebothapproaches.A generalclassicationoftherotorhubdependingonthemech anicalcongurationthatisusedto facilitatetheappingandlaggingmotionaccordingto[40, 58]isthefollowing: Articulatedrotor :Thistypeofrotorhubprovidesaapandalaghingeforevery individualblade.Thereisalsoafeatheringbearingforthecontrol ofthebladepitch.Thisisthe mostclassicalmeanstoprovideblademotion.Thiscongura tionallowsthebladetomove independentlyfromtheothers. Teeteringrotor :Thistypeofrotoriscomposedoftwobladesthatareconnect edtogether, formingacontinuousstructurewithasingleaphinge.Thet wobladesareconnectedto theaphingeinsuchawaythatwhentheonebladeapsupwards theotherbladeaps downwards.Thistypeofrotordoesnotincludelaghinges. Hingelessrotor :Thehingelessrotorallowstheapandlagmotionbystructu ralbending intherootoftheblade.Thiscongurationdoesnotrequireh inges.Thestructuralbending attherootofthebladeismadebyanattachmenttothehubofac antileverrootrestraint. Afeatheringbearingorhingeisusedforchangesinthepitch angleofeachblade.This designprovidesarelativestiffrotorhubandasaresultthe hubandbladeloadsarehigher thanthoseofhingedcongurations. 39

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4.3SwashplateMechanism Helicopterightcontrolisachievedbyvaryingthepitchan gleoftheblades.Featheringis thepitchingmotionofthebladeaboutthespanoftheblades. Thefeatheringmotionchanges theblade'sangleofattack,providingawaytocontroltheth rustandtherotormomentsthatare appliedtotherotor.Thefeatheringangle(aswellastheap pingangle)aremeasuredrelativelyto areferenceplane.Thisreferenceplaneisperpendicularto therotorshaftanditisdenotedasthe hubplane .Thetotalpitchangleofeachbladeisgivenbytheequation: = 0 1 c cos b 1 s sin b (4.1) Theangle 0 iscalledcollectivepitchanditcontrolsthemagnitudeoft hethrustvector.Thetwo angles 1 c and 1 s arecalledcyclicpitchangles.Thetwocyclicpitchanglesc ontroltheorientationofthethrustvector.Morespecically, 1 c controlsthelateralorientationofthethrustvector while 1 s controlsthelongitudinalorientation.Theblade'spositi onisdescribedbytheazimuth angle b =n t .Theazimuthangleisconsideredzerowhenthebladeisalign edwiththetailfacing backwards. Thereareseveraltypesofmechanicaldesignsthatproducet hecollectiveandcyclicangles oftheblades.Agenericdescriptionofthemoststandardcon gurationisgivenin[40]anditis describedhere.Thiscongurationiscomposedoftwomainme chanicalparts.Therstpartis associatedwiththecreationoftheblade'sfeatheringangl eanditisillustratedinFigure4.2.The pitchmotionofthebladestakesplaceaboutapitchbearingo rahinge.Thisbearingisrigidly attachedtooneofthetipsofthepitchhorn.Theothertipoft hepitchhornisconnectedtothe pitchlink.Thepitchhornandthepitchlinkareconnectedin suchawaythattheverticalmotionof thepitchlinkproducestheblade'spitchmotion.Whatisnee dedisamechanicalarrangementthat providestheperiodicpitchangledescribedby(4.1).Themo ststandardmechanicalconguration forthistaskistheuseoftheswashplatemechanism. 40

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Figure4.2:Connectionofthepitchhorntothepitchlink.Th epitchlinkisalsoattachedtothe swashplate.Theblade's3DOFarerepresentedbythreeblade hinges.ThisFigureisbasedon [40]. Thereisawidevarietyofdesignsfortheswashplate.Here,w epresentthefundamentalprincipleoftheswashplate'sfunction.Thisdescriptionisbas edon[40].Aschematicofthebasic swasplate'scomponentsisillustratedinFigure4.3. Theswashplateiscomposedoftworingsthatareconcentricw iththeshaft.Oneoftherings hastheabilitytorotateabouttheshaftwhiletheotheronei sconstantlynonrotating.Bearingslie betweenthetworings.Thebladepitchlinksareattachedtot herotatingwingwhilethepilot's controlsareattachedtothenonrotatingring.Thetworings areattachedtotheshaftinsuchaway thattheswashplatesurfacecantakeanarbitraryorientati onrelativetotheshaft. Movingtheswashplateverticallytotheshaftresultsinaun iformchangeoftheblade'spitch independentlyofthepositionoftheblade.Therefore,thev erticalmotionoftheswashplateproducesthecollectivepitchangle 0 .Ontheotherhand,alongitudinalorlateraltiltoftheswas hplatecreatesasinusoidalvariationofthepitchangledepe ndingontheazimuthalpositionofthe 41

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Figure4.3:Basiccongurationoftheswashplatemechanism .ThisFigureisbasedon[40]. blade.Itisobviousthatthecontroloftheswashplatetiltp roducesthecycliccontrolangles 1 s and 1 c oftherotorblades. Therefore,thecycliccontrolanglescanbewrittenaslinea rfunctionsofthecontrolsinputsof thepilot'sstick.Hence: 1 c = B lat lat 1 s = A lon lon (4.2) 4.4FundamentalRotorAerodynamics TheobjectiveofthisSectionistoprovidearelativelysimp liedanalysisoftherotoraerodynamics.Themathematicalanalysiswillbekepttothemini mumrequiredinordertoreduce complexity,howeveritwillprovideinsighttothedominati ngbehavioroftherotor.Inorderto determinetheaerodynamicforcesthatareappliedtothebla detherststepistoanalyzethevelocitycomponentsofthebladerelativetotheair,overthec ompletebladespan.Thisanalysis,in general,isaverydifculttask.Thisisduetothecomplexit yassociatedwiththemodelingofthe inowvelocitythroughouttherotordisk. 42

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Asindicatedin[40]and[58]thebladeelementanalysiscons iderseachbladeelementasatwo dimensionalairfoil.Theaerodynamicbehaviorofneighbor ingbladeelementsisindependentof eachother.Aninducedinowvelocityoneachbladeelements houldbeaccounted,whichisa productoftherotorwake.Analyticalwaysofcalculatingth einducedvelocitymaybefoundusing momentumtheory,vortextheoryornonuniforminowcalcula tions[40].Ingeneralthecalculation oftheinowvelocityisaverychallengingtask,duetoitsno nuniformityacrossthebladespan, somathematicalsimplicationsshouldbeappliedinordert ominimizethecomplexityofthe analysis.Finally,afterdeterminingthevelocitycompone ntsofthebladeelement,wecalculate theaerodynamicforcesactingonthiselement.Thecomplete dynamicbehaviorofthebladeis obtainedbyintegratingtheappliedforcesoftheindividua lelementsthroughoutthebladespan. Inwhatfollows,thehubplaneisconsideredasthereference plane.Tofacilitatetheanalysis denoteby F h = f O h ; ~ i h ; ~ j h ; ~ k h g areferenceframeattachedtothemainrotorwhere ~ i h = ~ i B ~ j h = ~ j B and ~ k h = ~ k B .Thecenter O h islocatedatthecenteroftherotorhubsuchthat ~ i h is alignedwiththebladewhen b =0 Let V 1 denotethefreestreamvelocitywhichisthehelicopter'sfo rwardvelocitywithrespect totheair.Thefreestreamvelocity,illustratedinFigures 4.4(a)and4.4(c),isdirectedstraightto thefrontpartofthehelicopterwithanangle hb withrespecttothehubplane(positivewhenthe freestreamvelocityisfacingdownwardstothehub).Theref ore,thefreestreamvelocityhasa component V 1 cos hb ,whichliesintheplaneofthehub,andacomponent V 1 sin hb ,which isnormaltothehubplane.Usuallyintheliterature,theinp lanecomponentisdenedasthenon dimensionalquantitycalledrotoradvanceratiodenotedby thatistheinplanefreestreamcomponentnormalizedbytheblade'stipspeed.Therefore: = V 1 cos hb n R b (4.3) where R b denotestheblade'sradius.Therotorbladesperformthreet ypesofmotion.Therstone isoutofplaneappingmotiondescribedbytheappingangle .Thereisalsofeatheringmotion 43

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aboutthebladeaxiswithafeatheringangle measuredrelativetothehubplane.Last,theblade performsarotationalmotionabouttherotor'sshaftwithan gularvelocity n Thevelocityaccountedbyeachbladeelementisduetothehel icopterforwardmotion,the blade'sappingmotion,therotor'sinowvelocityandther otor'srotationabouttheshaft. Threevelocityvectorsarerequiredforthedescriptionoft hetotalairvelocity U asseenby thebladeelement.Thosevectorsaretwoinplanecomponents andoneoutofplanecomponent normaltothehubplane.Therstinplanecomponentisdenote dby U T .Itistangentialtothe bladeandparalleltothediskplane.Weconsiderthatthepos itivedirectionof U T isopposingthe rotationalblademotion. Thesecondinplanecomponentistheradialcomponentoftheb lade,denotedby U R thatlies onthehubplane,itisparalleltothebladeaxisandpositive directionisconsideredoutwards.Both ofthemcanbeseeninFigure4.4(a).Finallytheoutofplanec omponentisdenotedby U P andit isperpendiculartothehubplanewithpositivedirectionfa cingdownwardsasillustratedinFigures 4.4(a)and4.4(b). Thetangentialvelocity U T isaffectedbytherotorrotationandtheforwardvelocity.T hecomponentduetorotorrotationis n r (where r istheradialdistanceofthebladeelement),whilethe tangentialtothebladeforwardvelocitycomponentis ( V 1 cos hb )sin b .Therefore,thecompleteformof U T withrespecttotheazimuthalangle b andtheradialdistance r oftheblade elementisgivenby: U T ( r; b )=( V 1 cos hb )sin b +n r (4.4) Theradialcomponentofthebladeelementissolelyproduced bythefreestreamvelocity,therefore: U R ( b )=( V 1 cos hb )cos b (4.5) 44

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R UTU hbV cos hi hj (a)Topviewoftherotor. R U P Uhub planehk (b)Sideviewoftherotor. V hb i uhk hi (c)Directionofthefreestreamandinowvelocityrelativetothehubplane. Figure4.4:Directionsofthevelocitycomponentsseenbyth ebladeelement.ThisFigurealso illustratesthedirectionofthefreestreamandinowveloc ity. 45

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Ingeneraltheeffectoftheradialcomponenttowardsthecal culationoftheairvelocityoftheblade elementisneglected.However,thiscomponentshouldbecon sideredwhencalculatingexplicitly theeffectoftherotordrug[58]. Theoutofplanevelocityvectorconsistsoffourvelocityco mponents.Therstoneisthevelocityduetobladeappinggivenby r .Thesecondoneistheperpendiculartothebladeelement componentduetotheradialvelocity U R givenby U R sin .Thethirdistheeffectoftheforward velocitydescribedby ( V 1 sin hb )cos .Lastly,thereistheinuenceoftheinowvelocity u i whichisperpendiculartotherotorhubwithcomponent u i cos .Thecompleteoutofplanevelocityisgivenby: U P ( r; b )= r + U R sin +( V 1 sin hb )cos +( u i )cos (4.6) Byconsideringasmallappingangle ,thefollowingsimpliedequationisobtained: U P ( r; b )= r + U R +( V 1 sin hb )+ u i (4.7) Aschematicdescriptionofthevelocities,aerodynamicang lesandelementalforcesactingon abladeelementisgiveninFigure4.5.Themagnitudeoftheve locityseenbythebladeelementis givenby: U = q U 2 T + U 2 P (4.8) Therelativeinowangle(orinducedangleofattack)isgive nby: b =tan 1 U P U T (4.9) Theblade'sangleofattackisafunctionofthebladepitchan gle andtheproducedinowangle b .Thecompleteexpressionoftheangleofattackisgivenby: b = b (4.10) 46

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T Ubc dr r bR U b b TU P U dDdLb hk hub plane Figure4.5:Illustrationofatwodimensionalbladeelement .Thegureillustratesthevelocity componentsofthebladeelement,theaerodynamicanglesand theelementalaerodynamicforces. Thisgureisbasedon[70].Theaerodynamicliftanddrugvectorsofthebladeelementar enormalandparallel,respectively, totheresultantvelocity U seenbythebladeelement. From[58]theincrementallift dL producedatthebladeelementis: dL = 1 2 a U 2 c b C l b dr (4.11) Intheaboveequation a istheairdensity, c b isthebladechordand C l istheairfoil'sliftcurve slope.Thedragcomponent,denoted dD ,oftheelementbladeisgivenby: dD = 1 2 a U 2 c b C d dr (4.12) where C d isadragconstantwhichdependsontheblade'sgeometry.The componentsoftheforces actingparallelandperpendiculartothehubplanearegiven by: dF x = dL sin b + dD cos b (4.13) 47

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dF z = dL cos b dD sin b (4.14) Thecompleteforcesareobtainedbyintegratingtheaboveeq uationsforallthebladeelements alongtheblade'slength.Theaboveequationsindicatethat thecyclicinputsandthehelicopter forwardmotionthroughtheair,produceperiodicaerodynam icforceswithafrequencyrelatedto n .Actually,asindicatedin[7,40,58,70],theperiodicaero dynamicloadsproducedbyfeathering haveafrequencyequalorclosedto n .Ananalyticaldescriptionoftheaerodynamicforcesis tocomplexandititoutofthescopeofthiswork.Theseperiod icforcesresulttotheperiodic appingmotionoftheblade.Theblade'sappingmotionisde scribedinthenextSection. 4.5FlappingEquationsofMotion ThisSectionpresentstherotorequationsofmotionassocia tedwiththeappingoftheblades. Flappingisassumedtotakeplaceaboutahingelocatedatthe intersectionoftheshaftwiththehub plane(nohingeoffset).Tocompletethemodeloftheapping hinge,alineartorsionalspringis addedatthehingewithstiffness K .Thismodelapproachisbasedon[7,79]anditisasuccessful waytorepresentuniformlyavarietyofhingedandhingeless rotors.Thismodelingapproachis alsoabletocapturetheeffectofthehingeoffset.Apartfro mtheappingmotion,thebladeis rotatingwithangularvelocity n abouttheshaft.Theeffectoftherotationalandtranslatio nal accelerationsofthefuselageontheblademotionisdisrega rded.Thisisatypicalsimplication assumption,however,detailsaboutthiseffectcanbefound in[79].Furthermore,massuniformity ofthebladeisassumed.Themassperunitlengthofthebladei sdenotedby m b .Themassofa bladeelementwithradialdistance r fromthebladerootis m b dr Therstthingtowardsthisanalysisisthedeterminationof theforcesactingonthebladeelement.Therstforcecomponentistheperiodicaerodynamic liftforce dF a ,actingontheblade element.Thisforcecomponentisperpendiculartotheblade elementfacingupwards.Inaddition,therearetwoinertiaforcesactingontheblade.Ther stoneistheinertiaforcecomponent opposingtheappingmotion.Theaccelerationofthebladee lementduetoappingis r ,there48

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cdF i dF K a dFdr mbhub planehk Figure4.6:Aerodynamic,inertiaandcentrifugalforcesac tingonabladeelement.Theapping angleofthebladeisdenotedby .Acenteredtorsionalspringofstiffness K isplacedattheroot oftheblade.Thisgureisbasedon[70].fore,theinertiaforceduetoapping dF i is m b dr r ,whichisperpendiculartothebladefacing downwards.Thesecondinertiaforceisthecentrifugalforc e dF c = m b dr n 2 r cos ,whichis paralleltothehubplanedirectedradiallyoutwards,dueto thecentripetalacceleration n 2 r cos TheinertiaforceduetoCoriolisacceleration(thisforcei sinthein-planedirection)andtheweight forceactingonthebladearedisregardedsincetheyproduce signicantsmallerforcesthanthe forcesproducedbyapping. Theappingequationofmotionisderivedbyequatingallmom entsthatactontheblade.The totalmomentisderivedbycalculatingtheelementarymomen tsactingonabladeelementandthen byintegratingalongthecompletebladelength.Sincethefo rcecomponentsthatarecollinearwith thethebladeaxisdonotproduceanymoments,themomentequa tiontakestheform: Z R b 0 m b n r 2 cos sin dr + Z R b 0 m b r 2 dr + K = Z R b 0 rdF a dr (4.15) Byassumingsmallangleapproximationfor ,theaboveequationstakestheform: +n 2 Z R b 0 m b r 2 dr + K = Z R b 0 rdF a dr (4.16) 49

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Theintegralofthersttermistheinertiaofthebladegiven by: I b = Z R b 0 m b r 2 dr (4.17) Equation(4.16)takesamoreintuitiveformiftheappingan gle isexpressedasafunction oftheazimuthalangle b oftheblade,insteadoftime.Theoperand ( 0 ) denotesthederivativeof withrespectto b .Therelationbetweentheazimuthalangleandtimeisgivenb y b =n t so regardingthederivativesof withrespectto b thefollowingequalitieshold: = @ @ b @ b @t =n 0 (4.18) = @ @ b @ b @t =n 2 00 (4.19) Considering(4.18)and(4.19),then(4.16)resultsin: 00 + 2 = 1 n 2 I b Z R b 0 rdF a dr (4.20) wheretheappingfrequencyratio [70,79]isgivenbytheexpression: 2 = K n 2 I b +1 (4.21) Thedynamicsof(4.20)resembletheequationofmotionofasi ngleDOFSpring-Mass-Damper (SMD)system.Thedescriptionofthelatterisgivenbytheeq uation m x + c x + kx = F where m denotesthemassoftheobject, c isthedampingcoefcient, k isthespringstiffnessand F isthe externalappliedforce.Forthissystem,thenaturalfreque ncyisgivenby n = p k=m anditis independentofthedampingcoefcient.For(4.20)itisobvi ousthatthenaturalfrequencyofblade appingisequaltotheappingfrequencyratio .Theaerodynamictermintherighthandside of(4.20)includesthedampingterm. 50

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4.6RotorTip-Path-PlaneEquation FromtheanalysisofthepreviousSection,itisapparenttha ttheappingmotiondepends ontheazimuthalangleoftheblade.Therefore,theappingm otionisaperiodicfunctionwith fundamentalfrequency n andperiod T b =2 = n .Everyperiodicfunctioncanbeexpressedasa Fourierseries,sotheappingmotioncanbeexpandedtothef ollowinginnitesum: ( b )= 0 1 Xn =1 ( b nc cos n b + b ns sin n b ) = 0 b 1 c cos b b 1 s sin b b 2 c cos2 b b 2 s sin2 b ::: (4.22) where 0 nc ; and ns denotetheFourierseriescoefcients.Practicalobservat ionshaveshown thatonlytherstharmonicsoftheinniteseriesaresufci enttoapproximatetheappingbehaviorofthebladesincethecontributionofhigherharmonicsc anbeconsiderednegligible.Inthis case,followingtheclassicalapproachof[13],theformoft heappingangle isrepresentedby therstharmonictermsof(4.22)withtimevaryingcoefcie nts,therefore: ( b )= 0 ( t ) 1 c ( t )cos b 1 s ( t )sin b (4.23) Theaboveequationindicatesthatthetipsofthebladecurve acircularlypath.Theplanethatthis circularpathlieson,isreferredtoas Tip-Path-Plane(TPP) or rotordisk .Inorderforthereaderto understandtheblademotiondescribedby(4.23)thefollowi nganalysisexaminesindividuallythe effectoftherst-harmoniccoefcientstotheTPP.Forsimp licity,thecoefcients 0 1 c ,and 1 s areconsideredconstantwithtime.Denoteby [ x h y h z h ] T thecoordinatesofthetipoftheblade withrespecttothehubframe F h Iftheappingangleiscomposedonlybythe 0 coefcient,thenthebladesformaconeas theyrotateandtheTPPisacircleparalleltothehubplaneas illustratedinFigure4.7(a). 51

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0 0 hk TPP (a)Coning c 1 hk hi TP P (b)LongitudinaltiltoftheTPP s 1 hk hj T P P (c)LateraltiltoftheTPP Figure4.7:Effectofeachharmonicgivenby(4.23)totheTPP Regardingthe 1 c term,ifsmallangleapproximationisusedandtheappingan gleisgivenby ( b )= 1 c cos b ,thenthecoordinateofthetipofthebladeonthe ~ k h axisis: z h = R b sin R b = R b 1 c cos b 1 c x h (4.24) InthiscasetheTPPliesonaplanethatistiltedaboutthe ~ j h axiswithanangle 1 c downwardsas illustratedinFigure4.7(b).Followingthesameanalysisf orthemotionof ( b )= 1 s sin b oneobtains: z h = R b sin R b = R b 1 s sin b 1 s y h (4.25) andtheTPPwillbeaplanetiltedaboutthe ~ i h axisdownwardshavinganangle 1 s withthereferenceplane.ThelateraltiltoftheTPPisillustratedinFigu re4.7(c).TheTPPequationdescribed by(4.23)resultsinalongitudinalandlateraltiltoftheco neproducedby 0 .Thetiltanglesofthe coneare 1 c and 1 s ,respectively. Thedynamicsoftherstharmonictermsof(4.23)providethe dynamicequationsoftheTPP. Thoseequationsarederivedbysubstituting(4.23)to(4.20 ),andequating,respectively,thenonperiodicterm,thetermsincluding cos b andthetermswith sin b .Adetailedanalysisofthis approach,providingathoroughmathematicalrepresentati onisgivenin[13].Let a =[ 0 ab ] T denotethestatevectoroftheTPP(followingthenotationgi venin[70])where a standsfor 1 c and b for 1 s .TheTPPdynamicequationsaregivenbythefollowingdiffer entialequationofthestate 52

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vector a : a + D a + K a = F (4.26) where D isthedampingmatrix, K isthestiffnessmatrixand F isthematrixoftheforcingfunction.Asmentionedearlier,thecompleteformulationonthe aboveequationcanbefoundin[13]. Thoseequationsarefurthersimpliedinordertoprovideap racticalmodeloftheTPPdynamics. Thosesimplicationsareintroducedin[70]andtheyarepre sentedinthenextSection. 4.7FirstOrderTip-Path-PlaneEquations Forthederivationofasimpliedmodeloftherotordynamics theworkin[70]hasadopted thedetaileddynamicequationsoftheTPPpresentedin[13]a lsoconsideringsomeadditional simplicationassumptions.Themodelproposedin[70]issu itableforsystemidenticationsince itincludesthenecessarycomponentsthatcapturethedynam icbehaviorthataffectthehelicopter withoutburdeningthemodelwithunnecessarycomplexity.T hesimplicationassumptionsarethe following: Theeffectoftheinowratioisdisregarded. Theconingangleisconsideredconstant,thereforeitsasso ciateddynamicsareomitted. Theeffectofthehingeoffsetisdisregarded. Thepitch-apcouplingratioiszero. Theeffectoftheforwardvelocityisdisregarded( =0 ). TheTPPmodelpresentedin[13]providesaveryextensivedes criptionoftheTPPdynamics.If wedonotconsidertheabovesimplicationassumptionsther esultingTPPmodelisgoingtobe verycomplexandcompletelyimpracticalforcontroldesign purposes.Thenbasedon[70],the 53

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simpliedappingdynamicsaregivenby: f a = a f q + A b b + A lon u lon (4.27a) f b = b f p + B a a + B lat u lat (4.27b) TheaboveequationsareanapproximationoftheTPPdynamics producedbythehelicoptermotionandcontrolinputs.Theterm f denotesthemainrotortimeconstantanditisgivenby: f = 16 r n (4.28) Therotor'stimeconstantdependsontheangularvelocity n andtheLocknumber r .TheLock numberisgivenby: r = a ac b R 4 b I b (4.29) Finally,themainrotorcrosscouplingterms A b and B a are: A b = B a = 8 r ( 2 1) (4.30) 4.8MainRotorForcesandMoments Thenalpartoftherotordescriptiondealswiththederivat ionofasimpliedmodelofthe forcesandmomentsproducedbythemainrotor.Thethrustvec torproducedbythemainrotor isconsideredperpendiculartotheTip-Path-Plane(TPP).S incethethrustvectorisnormaltothe TPP,bycontrollingtheTPPinclination,thepilotindirect lycontrolsthedirectionofthepropulsion forces. Let ~ T M denotethethrustvectorofthemainrotorand T M itsmagnitude.Thebody-xedframe coordinatevectorofthethrustisdenotedby T B M .Bysimplegeometrythefollowingequationsare 54

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derived: T B M = 266664 X M Y M Z M 377775 = 266664 sin a cos b cos a sin b cos a cos b 377775 T M 266664 a b 1 377775 T M (4.31) Theaboveequationsaresimpliedbyassumingsmallangleap proximation( cos( ) 1 and sin( ) ( ) )fortheappingangles.Thesmallangleassumptionisadopt edby[40,47,70]. Thegeneratedthrusttorqueistheresultoftheaboveforcea ndtherotor'sstiffnessmoments. Denoteby h BM =[ x m y m z m ] T thepositionofthemainrotorshaft.Let ~ denotethevectorof themainrotormomentsduetothehubstiffness K .Then,themainrotormomentvectorisgiven by ~ M = ~ h M ~ T M + ~ .Thecomponentsofthehubstiffnessmomentsvectorinthebo dy-xed framearegivenby: B = 266664 L M N 377775 = 266664 b a 0 377775 K (4.32) IntheidealcasethattheCGisalignedwiththeshaft,i.e. h BM =[00 l h ] thenthepitchandroll momentsofthemainrotoraregivenby: L M = ( l h ) Y M + L M M = l h X M + M Hence: L M =( l h T M + K ) b (4.33a) M M =( l h T M + K ) a (4.33b) Therefore,thepitchandrollmomentsabouttheCGdependont hemainrotorthrustmagnitude andthestiffnessofthehub.Theabovesimpliedcaseispres entedbecauseitprovidesinsightto thedevelopmentofthelinearhelicoptermodel.Inthecaset hatthenonlinearhelicopterdynamics 55

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areconsideredthemoreelaboratedescription ~ M = ~ h M ~ T M + ~ ,isusedfortherepresentation ofthemomentproducedbythemainrotor.4.9Remarks ThisChapterhaspresentedadescriptionoftheintermediat econceptsthatarerelatedwith theappingdynamicsoftheblades.Theappingmotionisini tiallytriggeredbyachangein thecyclicpitchoftheblades.Thepitchvariationaltersth eblade'sangleofattackresultingto thegenerationofperiodicaerodynamicforcesthatactupon theblade.Theappingmotionis producedbytheaerodynamic,centrifugal,inertialandhub stiffnessmomentsthatactontheblade. Theappingdynamicsequationsarebasedontheworkpresent edin[70].Inthereportedworkthe simpliedrotordynamics(appingdynamics)arederivedby signicantlysimplifyingthemore elaboratemodelpresentedin[13].Theparticularrotormod elisphysicallymeaningfulandhas beensuccessfullyappliedtosystemidenticationmodelin gofseveralhelicopters.Theapping dynamicsgivenin(4.27)aresuitableforsmallscalehelico pterssinceforfullscalehelicoptersan accuratemodelwouldalsorequiretheadditionoftheconing dynamicseffect.Therotormodel isaugmentedtotherigidbodydynamicstoproducethecomple tehelicoptermodel.Themain rotorthrustvectorisconsideredperpendiculartotheTPP. Thismodelingassumptionisadopted bybothlinearandnonlinearhelicoptermodels.Thetaskoft henextsectionistopresentareliable systemidenticationmethodologyfortheextractionoflin earhelicoptermodels.Thepresented methodologyisbasedintheworkreportedin[70,105]andita successfulapproachforthesystem identicationmodelingofsmallscalehelicopters. 56

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Chapter5:FrequencyDomainSystemIdentication Helicopterightcontrollerdesignrequiresknowledgeofa mathematicalmodelthataccurately describesthedynamicbehaviorofthehelicopter.Thismath ematicalmodelisrepresentedbyaset ofordinarydifferentialequations.Establishingsuchamo delinthecaseofhelicoptersisachallengingtask.ThisChapterprovidesathoroughdescription ofafrequencydomainidentication procedurefortheextractionoflinearmodelsthatcorrespo ndtocertainoperatingconditionsofthe helicopter.Thismethodologyhasbeenestablishedin[105] andhasbeensuccessfullyappliedfor asmallscalehelicopterintheworkreportedin[70].Thefre quencydomainidenticationprocedureisevaluatedforanexperimentalsmallscaleRadioCont rolled(RC) Raptor90SE helicopter throughthe X-Plane ightsimulator.The Raptor90SE helicopterisusedfortheevaluationand comparisonoftheseveralcontrollerdesignsandidentica tionmethodsthatarepresentedinthis research.5.1MathematicalModeling Helicopterdynamicsarenonlinearandofhighorder.Fortyp icalaircraftmodelsthereisa distinctseparationbetweenthedynamicsassociatedwitht helateralandlongitudinalmotion. Thisseparationcannottakeplaceinthecaseofahelicopter ,wherethereexistsastrongcoupling amongthesystemdynamics. Theprimecouplingeffectisencounteredbytheinteraction ofthefuselageandmainrotor dynamics.AsindicatedfromthepreviousChapter,therotor isadynamicalsystemitself,affected byboththeenvironment,throughtheairow(inow)passing throughtherotorblades,andthe fuselagemotion.Inmanycases,thefuselagerigidbodydyna micsrepresentationisnotadequate 57

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andtheadditionaleffectoftherotorshouldbeencountered [70].Anadditionalsourceofcomplexityisthedescriptionoftheaerodynamicforcesandmom entsactingonthehelicopter.Those forcesandmomentsarecomplicated,withsignicantchange sintheirbehavior,dependingonthe operatingconditionofthehelicopter. Twoapproachesmaybefollowedforthederivationofamathem aticalmodelrepresenting thehelicopterdynamics.Therstmodelingapproachisthed erivationofamathematicalmodel from rstprinciples modeling,whilethesecondisthrough systemidentication .Insomeparts thosetwomethodsarecomplementarytoeachotherandinmany casestheuseofbothofthemis mandatoryforincreasingtheaccuracyofthederivedmodel.5.1.1FirstPrinciplesModeling Whentherstprinciplesmodelingmethodisused,thesystem equationsarederivedbythe implementationofphysicslaws.Obviously,thisapproach, requiresanaprioriknowledgeofall theparametersthataffectthehelicoptermotionandaerody namics.Thetypicalendresultofrst principlesmodelingisasetofnonlineardifferentialequa tionsofhighorderthatcoverawide portionoftheightenvelope.Acommonuseoftherstprinci plesmodelingmethodisforthe developmentofsimulationmodels.Themaindisadvantageof thisapproachisthelargenumber ofparameterstobedetermined.Thoseparametersinvolvege ometricalcharacteristics,massand inertias,dragcoefcientsandaerodynamicparameters.Ma nyofthelatterparameterscanbeeasilyobtainedbysimpleexperimentaltests(suchasmassesan dinertias),howevertheirmajority requiresmoresophisticatedexperimentmethodssuchaswin dtunneltests[105].Thedifculty ofobtaininganaccurateestimateofmanyofthehelicopterp arametersrendertherstprinciples modelingmethodimpracticalformanyapplications. 58

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5.1.2SystemIdenticationModeling Systemidenticationistheprocedureofderivingamathema ticalmodelofthesystembased onexperimentaldataofthesystem'scontrolinputsandmeas uredoutputs.Twotypesofmodels canbederivedbythismethod.Thersttypeisthe nonparametricmodels andexamplesofsuch modelsaretheimpulseresponseandfrequencyresponses.Th enonparametricmodelsaredirectlyproducedbyexperimentaldataandprovideaninput-o utputdescriptionofthesystem.These modeltypesarejustcollectionsofdataanddonotrequirean yknowledgeofthesystemstructure. Thechallengeofthesystemidenticationprocedure,istod erivea parametricmodel ofthe system.Examplesofparametricmodelsarethetransferfunc tionsandthestatespacemodels. Therststeptowardstheextractionofaparametricmodel,i sthederivationofaparametrized model,whichwillserveasalogicalguessoftheactualsyste mmodel.Theuseofanoptimization algorithmdeterminestheparametersofthemodelthatminim ize(inaleast-squaresense)theerror betweentheactualsystemresponsesandthemodelresponses .Therstquestionthatarisesis whatisasuitableguessoftheinitialparametrizedmodelin termsofmodelorder,structureand theinitialvaluesoftheparameters.Estimatesofthosecha racteristicscanbeobtainedbyanalysis ofthenonparametricmodelcombinedwithinformationobtai nedbytherstprinciplesapproach. Thesystemidenticationprocedureisaniterativeprocess .Dependingontheidentication results,theparametrizedmodelcanberenedintermsoford erandstructureuntilasatisfactory identicationerrorisachieved.Whentheparametrizedmod elisknown,thesystemidentication methodreducestotheparameterestimationproblem.Theree xistmanysystemidentication methods,whicharewelldescribedin[61,62,93].Amajorcla ssicationamongstthesystem identicationmethodologiesdependsonwhetherthecompar edresponsesareconsideredinthe timedomainorthefrequencydomain.Frequencydomainsyste midenticationhasbeenprovena successfulapproachforextractingaccuratelinearmodels ofaircraftandhelicopters. 59

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5.2FrequencyDomainSystemIdentication Theinabilityoftherstprinciplesmodelingapproachtopr ovideaccurateandpracticalmodels forcontroldesign,leadtothedevelopmentofmoresuitable systemidenticationapproaches.In particular,frequencydomainidenticationhasbeenregar dedasanidealsolutionforextracting linearhelicoptermodelsofhighaccuracy.Oneofthemainad vantagesofthisapproachistheuse ofactualightdataforthederivationandvalidationofthe model.Additionally,thishasacoherent owofthedesignstepsstartingfromtheinput-outputchara cterizationofthehelicopter(nonparametricmodeling),continuingwiththeextractionofthesta tespacemodel(parametricmodeling) andnallyvalidatingthepredictedmodelinthetimedomain .Thismethodisclassiedasan output-errormethodwherethettingerrorisdenedbetwee ntheactualightdatafrequency responsesandthefrequencyresponsespredictedbythemode l. Theinitialstepoftheidenticationprocedureistheexcit ationofthehelicopterbyspecially designedinputsignalssuchasfrequencysweeps.Theintent ionofthetestdatainputsistoexcite thehelicopterdynamicsoveradesiredfrequencyrange.The choiceofthedesiredfrequencyrange (modelbandwidth)hasanimportantroleintheidenticatio nprocess.Themodelbandwidthhasto bewideenoughinordertoencapsulateallthedynamiceffect sofinterest(i.e.,fuselagedynamics androtordynamics). Aftersomepreprocessingtoeliminatethenoiseeffectsand othertypesofinconsistenciesin thetimedomainoutputdata,thesecondphaseisthecomputat ionoftheinput-outputfrequency responsesusingaFastFourierTransform.Thisphaseofthep rocessconstitutesthenonparametric modelofthehelicopter. Thenextstepisthedesignoftheparametrizedlinearstates pacemodel,usinginformation fromtherstprinciplesphysicallawsandthenonparametri cmodelingphase.Thelinearmodel hastheform: x ( t )= A () x ( t )+ B () u c ( t ) (5.1) y ( t )= Cx ( t )+ u c ( t ) (5.2) 60

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where x isthestatespacevector, y isthemeasurementvector, denotestheunknownmodel parametervectorand isthesystem'sdelay.Thematrices C D areusuallyknown,basedon standardkinematicequations.Theobjectiveofparametric modelingistheextractionofthemodel matrices A B (dependedon )andthetimedelay Thefrequencydomainidenticationmethodisonlysuitable forthederivationoflinearstate spacemodels.Althoughthehelicopterdynamicsarenonline ar,aroundcertaintrimmedight conditions,thenonlinearitiesfromtheequationsofmotio nandaerodynamicsarerelativelymild. Whenthisisthecase,alinearizedmodelisadequatetoaccur atelypredictthehelicopter'sresponse.Usually,thevalidityofthelinearizedmodelissat isfactoryinarelativelywideareaofthe ightenvelopearoundthetrimpoint.However,asingleline armodelinmostcasesisnotenough foraglobalrepresentationoftheightenvelope.Differen tmodelsarerequiredforeachoperating condition. Afterthedeterminationofthelinearizedmodel,anoptimiz ationalgorithmisusedtotune theidenticationparameters,suchthatagoodtisachieve dbetweentheparametrizedsystem's responsesandtheightdataresponses.Thefrequencyrespo nsemagnitudeandphaseerrorsare denotedbythevector ( !; ) forafrequency .Theobjectiveistheminimizationofacostfunction J () ,whichisthesumoftheweightedsquarederrors ( !; ) overanitenumberoffrequencies.Morespecically: J ()= n X j =1 ( i ; ) T W ( i ; ) (5.3) where W isaweightmatrix.Theaboveproceduresconstitutethepara metricmodelingpartofthe problem.Iftheparameteridenticationdoesnotprovideas atisfactoryresult,theparametrized modelisrevisitedintermsoforderandstructureuntilasat isfactoryminimizationofthecost functionisachieved. Thenalstepoftheidenticationprocedureisthevalidati onofthemodel.Thissteptakes placeinthetimedomain,withdifferentightdatafromthei denticationprocedure.Forthesame inputsequence,thehelicopterresponsesfromtheightdat aarecomparedwiththepredicted 61

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valuesofthemodel,obtainedbyintegrationofthestatespa cemodel.Again,ifthevalidation portionoftheproblemisnotsatisfactorythedesignershou ldmodifytheparametricmodeling setupandrepeattheprocedure.5.3AdvantagesoftheFrequencyDomainIdentication Basedon[70,105],someoftheadvantagesforusingfrequenc ydomainidenticationfor helicoptermodelingarethefollowing: Biasesandreferenceshiftsfromthetrimconditionareremo vedbytheidenticationprocess. Thefrequencyresponseestimatesareunbiasedfrommeasure mentnoise,giventhatthe latterisuncorrelatedwiththeexcitationsignals. Accurateidenticationoftimedelays. Thefrequencyrangeofeachfrequencyresponseisselectedi ndividually.Therefore,onlythe mostaccuratedataareinvolvedinthecalculations. Themodelstructureandorderselectionarefacilitatedbyt henonparametricmodel. Thefrequencydomainidenticationiscomputationallymor eefcientfromitstimedomain counterpart.Thetimedomainidenticationrequiresthein tegrationofthesystemstate spaceequationsforeachiterativestep.Integrationofthe systemequationdoesnottake placeinthefrequencydomainscheme.Inaddition,frequenc ydomainidenticationrequires lessdatapointsthanthetimedomainidentication. 5.4HelicopterIdenticationChallenges Theidenticationprocessencounterssomeparticulardif cultiesinthecaseofhelicopters. Basedon[70,105]thosedifcultiesarelistedbelow: 62

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Inmanycaseswherethehelicopterisoperatingatlowveloci ties(hover,lowspeedcruising) thecontrolinputhassimilarmagnitudewiththemeasuremen tnoise.Commonnoisesource couldbeproducedbystructuralvibrationscausedfromgear boxes,theengineaswellasthe rotor. ThehelicopterisaMIMOsystemwithsignicantdynamiccoup ling(or interaxiscoupling ). Foranyprimaryaxisresponse( on-axis response)causedbyoneoftheinputs,unintended secondaryaxisresponses( off-axis responses)result. Alinearmodelbasedsolelyontherigidbodydynamicswillno tbesufcienttoaccurately describethehelicopterresponses.Amodelofhigherorderi sneededincludingadditional subsystemssuchastherotordynamics.Furthermore,therot ordynamicsarenotindependentfromtherestofthemodelsoacoupledfuselage-rotormo delisrequired. Thehelicopterdynamicsareingeneralunstableorcritical lystable.Duringtheexecution oftheexcitationcontrolsignals,requiredfortheexperim entaldatacollection,additional feedbackisrequiredtosustainthevehicleinarangeofacer tainoperatingcondition.The presenceoffeedbackdeterioratestheidenticationresul ts. 5.5FrequencyResponseandCoherenceFunction ConsideraLinearTimeInvariant(LTI)systemwithinputand outputsignals x ( t ) and y ( t ) respectively.Denoteby h ( t ) ,theimpulseresponsethatcharacterizesthepreviousLTIs ystem.The timedomainrelationoftheoutput y ( t ) withrespecttotheinput x ( t ) ofthesystem,isgivenbythe convolutionintegral[23,77],namely: y ( t )= Z 1 1 h ( t ) x ( ) d (5.4) 63

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Thefrequencydomainrepresentationofthesignals x ( t ) y ( t ) and h ( t ) isgivenbythe Fourier transform .Morespecically: X ( j! )= Z 1 1 x ( t ) e j!t dt Y ( j! )= Z 1 1 y ( t ) e j!t dt (5.5) H ( j! )= Z 1 1 h ( t ) e j!t dt where istherealcontinuoustimeangularfrequencyvariableinra dians.Thesysteminput-output mappingiseasierrepresentedinthefrequencydomainby: Y ( j! )= H ( j! ) X ( j! ) (5.6) TheFouriertransform H ( j! ) oftheimpulseresponseiscalledfrequencyresponseofthes ystem. Itisacomplexvaluedfunctionwithrealandimaginaryparts H R ( j! ) and H I ( j! ) ,respectively. Thefrequencyresponsecanbeexpressedinpolarformas: H ( j! )= j H ( j! ) j e j \ H ( j! ) (5.7) where: j H ( j! ) j = q H 2 R ( j! )+ H 2 I ( j! ) and \ H ( j! )=tan 1 H I ( j! ) H R ( j! ) (5.8) Thefrequencydomaincanbealsoderivedbytheinputandoutp utspectraldensities.Thequantities S xx and S xy aretheautospectraldensityandcrossspectraldensity,re spectively.Theauto spectraldensityandcrossspectraldensityarefunctionsc ommonlyusedinstochasticprocesses [5,46].Thetwo-sidedautospectraldensity S xx ( j! ) andcrossspectraldensity S xy ( j! ) aregiven 64

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by: S xx ( j! )=2 Z 1 1 R xx ( ) e j! dS xy ( j! )=2 Z 1 1 R xy ( ) e j! d (5.9) where R xx ( ) and R xy ( ) denotestheautocorrelationandcrosscorrelation,respec tively,given by: R xx ( )=lim T !1 1 T Z 1 1 x ( t ) x ( t + ) dt R xy ( )=lim T !1 1 T Z 1 1 x ( t ) y ( t + ) dt (5.10) Theequalitythatrelatesthespectraldensitieswiththefr equencyresponseis: S xy ( j! )= H ( j! ) S xx ( j! )= ) H ( j! )= S xy ( j! ) S xx ( j! ) (5.11) Animportantquantity,particularlyusefulinthefrequenc ydomainidenticationofMIMO systemsisthecoherencefunction.Thelatterisdenedfort heSISOcaseas: r 2 xy ( j! )= j S xy ( j! ) j 2 j S xx ( j! ) jj S yy ( j! ) j (5.12) Thecoherencefunctionisanormalizedmetricwithitsvalue srangingforzerotounity.Itisan indicatorofthelinearitybetweentheinputandtheoutput[ 46].Avalueofthecoherencefunction closetounity,indicatesthattheoutputissignicantlyli nearlycorrelatedwiththeinputofthe system.Possiblecausesforalowvalueofthecoherencefunc tionare[46]: Presenceofnoise Theinput-outputmappingisnonlinear Theinputdoesnoteffecttheoutput InthecaseofMIMOsystemstheequivalentmetricisdenoteda spartialcoherence.AlowpartialcoherenceinaMIMOsystem,isusuallyanindicatorofth atthespecicinput-outputpair 65

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isuncorrelated,therefore,thecorrespondingfrequencyr esponseshouldnotbeincludedinthe identicationprocess.Moreaboutpartialcoherencecanbe foundin[105]. Alloftheabovefunctionswillbecalculatedinadigitalcom puter.Thediscretizationofthe continuoussignals x ( t ) and y ( t ) byasamplingperiod T s willleadtotheconceptofthe Discrete FourierTransform (DFT).Denote N thetotalnumberofsampleddata.TheDFTsforthe N samplesof x ( t ) and y ( t ) aregivenby[73,76]: X ( k n s )= N 1 X n =0 x ( t 0 + nT s ) e j 2 kn=N (5.13) Y ( k n s )= N 1 X n =0 y ( t 0 + nT s ) e j 2 kn=N (5.14) where n s isthefrequencyresolutionand t 0 istherstsamplingtimeinstant.Finallythediscrete estimatesoftheautospectralandcrossspectraldensity, ^ S xx and ^ S xy ,respectively,aregivenby [46,70]: ^ S xx ( k n s )= 2 NT s j X ( k n s ) j 2 (5.15) ^ S xy ( k n s )= 2 NT s X y ( k n s ) Y ( k n s ) (5.16)(5.17) wheretheupperscript y denotesthecomplexconjugatevalueofthevariable. 5.6The CIFER c r Package The CIFER c r packageisaneffectivetooltotackletheaircraftandrotor craftcompleteidenticationproblem. CIFER c r (ComprehensiveIdenticationfromFrEquencyResponses)[ 105]has beendevelopedasajointventureoftheArmy/NASARotorcraf tDivision(AmesResearchCenter).Theprogramiscomposedofsixutilitypackagesthatin teractwithasophisticateddatabase offrequencyresponses.Theimportanceofawellorganizeda ndexibledatabasesystemisvery 66

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crucialinalargescaleMIMOidenticationprocedureofana irvehicle.The CIFER c r packageis designedtocoveralltheintermediatestepsnecessaryfort hedevelopmentofanairvehicleparametricmodel.Thekeycharacteristicof CIFER c r isitsabilitytogenerateandanalyzehighquality frequencyresponsesforMIMOsystems,byusingsophisticat edDFTandwindowingalgorithms. Thesixutilitypackagesof CIFER c r are[70,105]: FRESPID :ThisutilitypackagecalculatestheSISOfrequencyrespon sesforeachinputoutputpair.ForthecalculationoftheFFTsachirp-zalgori thmisused.Theuserprovides totheutilitythetimedomainightrecordsoftheinputando utputmeasurements.Biases andshiftsareremovedbythetimedomaindata,andtheightr ecordsareconcatenatedinto asinglerecord.Thetimedomaindataareadditionallylter ed(toeliminatehighfrequency noise)andadditionallyprocessedbyoverlappingwindowin g.Thelateractionsarenecessarytoimprovethedelityandthespeedofthechirp-ztrans form.Finallythedatabasedis updatedwiththeestimatedfrequencyresponsesandcoheren cefunctions MISOSA :Thisutilitypackagereceivesthefrequencyresponsespre viouslycalculatedfrom FRESPIDandremovestheeffectofsecondaryinputswhichare possiblycorrelatedwiththe primaryinput(conditioning).MISOSAoutputstheconditio nedfrequencyresponsesand partialcoherence. COMPOSITE :Thismoduleoptimizesthefrequencyresponsesforeachspe ctralwindowappliedbyFRESPIDandMISOSA,toprovidethebestpossibleest imatedfrequencyresponse andhighestcoherencefunction,overthedesiredbandwidth NAVFIT :Thismodulebelongstotheparametricportionoftheidenti cationprocedure. NAVFITcalculatesthetransferfunctionmodelthatbestts theestimatedSISOfrequency response. DERIVID :ThisprogramestimatestheMIMOstatespacerepresentatio nwhosefrequency responseisthebesttfortheestimatedfrequencyresponse sobtainedbytheightdata. Theparametersofthemodelcanbeconsideredfreeorconstra inedbyadifferentparameter, 67

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duringtheidenticationprocess.Theunknownparametersa reextractedbytheapplication ofanonlineariterativesecantalgorithm. VERIFY :Thismodulesisthenalstepoftheidenticationprocedur es.VERIFYcompares thetimedomainresponseoftheidentiedmodelversustheex perimentaldata.Thedata usedbyVERIFYshouldbedissimilarwiththeightrecordsob tainedbytheidentication procedure. 5.7TimeHistoryDataandExcitationInputs Anissueofprimaryconcernisthedesignoftheexcitationin putsusedtocollectdataforthe identicationpart.Itisimportanttonotethatthebehavio roftheactualmodelthatisrequiredto beencapsulatedbytheidentiershouldbeincludedintheda tausedfortheidentication[105].In generalregardingsystemidentication,thedesignofthee xcitationsignalisanopensubjectwhich dependsonthemodeltobeidentied.Theexcitationsignalm ustbecapableofexcitingtheactual systemmodesthatareneededtoappearintheidentiedmodel Adescriptionofexcitationsignalsspeciallydesignedfor aircraftidenticationmaybefound in[46].Someofthosesignalsarefrequencysweeps,impulse multisinesanddoublets.Inthiswork frequencysweepsareused.Frequencysweepsaresinusoidal signalswithvariablefrequency.The frequencyofthesignalincreaseslogarithmicallyovertim e.Followingthisapproachtheexcitation signaliscapableofcoveringthedesiredfrequencyband.Fr equencysweepsarecommonlyused infrequencyidenticationtechniqueswherethemodelisid entiedoverapredenedfrequency range. Observationsregardingthefrequencysweepsarepresented in[46,105].Themostimportant featureisthattheyarenotrequiredtohaveconstantamplit ude.Variationsinthefrequencysweeps insteadofbeingavoidedarewelcomesincetheyenrichthefr equencycontentofthesignal.The symmetryofthosesignalsallowsthehelicoptertosustaini tspositionaroundacertainoperating condition. 68

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Whenthefrequencysweepisappliedtooneofthehelicopter' scontrolinputstherestshould beimplementedinsuchawaytoadjustthehelicopterinthene ighborhoodoftheoperatingpoint. Asindicatedin[105]therestofthecontrolinputsshouldbe uncorrelatedwiththemainexcitation signalandatthesametimesuppressanyunwantedightbehav ior.Duringthesystemidenticationprocedure,frequencysweepdatacollectedbysevera lmaneuverscanbeconcatenated,so itisveryimportantthatthedatastartandendatthetrimcon dition.A 3 sec periodintrimatthe beginningandattheendissuggested. Thedesignofthefrequencysweepsrequiresthatthefrequen cybandwidthisdeterminedapriori.Ingeneralagoodbandwidthforhelicopteridentic ationliesbetween0.3-12rad/sec[105]. Therecordedlengthofthedataforeachsweepfollowingarul eofthumbshouldbefourtove timestheperiodthatcorrespondstotheminimumfrequency. Let [ min max ] bethedesired frequencyintervalthattheexcitationsignalshouldconta in.Then,theperiodthatcorresponds tothesmallestfrequencywillbe T max =2 =! min .Thesuggestedrecordedlengthshouldbe T rec 4 T max .Theproposedexcitationsignalisgivenby u = A sin[ f ( t )] where A isthe amplitudeofthesignaland: f ( t )= Z T rec 0 v ( t ) dt (5.18) K ( t )= C 2 [ exp ( C 1 t=T rec ) 1] (5.19) v ( t )= min + K ( t )( max min ) (5.20) From[105],theproposedparametersof(5.19)are C 1 =4 : 0 and C 2 =0 : 0187 .Further,basedon [105]abriefsummaryofthemostimportantguidelinesthats houldbeaccountedinthefrequency sweepsignals,arethefollowing: Thesinusoidalshouldbeassymmetricaspossibletomaintai nthehelicopterattrim.The symmetricinputwillalsoassisttheFFTtoidentifyandremo vethetrimvalues. Thesweepsignalshouldprovidesatisfactoryexcitationov erthefrequencyrangeofinterest.Specialattentionshouldbegiventothelowfrequencye xcitation(0.3-1 rad=sec ).At 69

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leasttwoperiodsoftheminimumfrequencyofinterestshoul dbeincludedintheexcitation signal. Theamplitudedoesnothavetobeconstant. Theincreaseinfrequenciesisnotimportant.Furthermore, themaneuvershouldstartand endwitha3 sec operationattrim. Mostimportantly,thesecondarycontrolcommandsshouldbe asuncorrelatedaspossible withtheprimaryexcitation.Theuseoflowfrequencypulses isrecommendedtokeepthe off-axis responsesbounded.However,althoughthe off-axis responsesshouldnotdiverge fromthetrimcondition,theyshouldnotbesuppressedeithe r.Thoseeffectsareproducedby thecross-couplednatureofthehelicopterdynamicsandthi sinformationshouldbeincluded intheidenticationprocess. 5.8LinearizationoftheEquationsofMotion Equationsdescribingthehelicoptermotionarenonlineard ifferentialequations.Linearizing theseequations,underspecicassumptions,isacommonpra cticethatsimpliesgreatlycalculationsandatthesametimeprovidesanadequatedescriptio noftheactualbehaviorofthehelicopter.Derivationsfollowtheworkdescribedin[20]. Modellinearizationisbasedonsmalldisturbancetheory.A ccordingtothattheory,analysisis doneundersmallperturbationsofmotioncharacteristics( relatedtoforces,momentums,velocities, angularvelocities,etc.)fromasteadynon-acceleratingr eferenceight.Therationalebehind thisapproachisthefactthatexternalaerodynamicforcesa ndmomentsactingontheCGdepend mainlyonhelicopter'scontrolinputsandmotionvariables suchaslinearandangularvelocities. Whenthisisthecase,theperturbedaerodynamicforcesandm omentsmaybeconsideredaslinear functionsofthedisturbances[20]. Thehelicopterisassumedtoperformareferencetrimmedig htwhenthedisturbancesoccur. Inthisequilibriumoperation,thestatevariable x ofthehelicoptercanbeapproximatedby x = 70

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x 0 + x ,where x 0 isthetrimmedvalueofthestateand x theperturbationfromthereference ightcondition.Thesmallperturbationslogicappliesfor thecontrolinputsaswell.Sinceinthe identicationprocedurewearegoingtoconsideronlytheho verrepresentationofthehelicopter, theequilibriumstatevalueswillbe: u 0 = v 0 = w 0 = p 0 = q 0 = r 0 = 0 = 0 =0 Theperturbationquantitiesandtheirderivativeswillhav everysmallvalues;therefore,their productsarenegligible.Withoutlossofgenerality,itisa ssumedthatthetrigonometricquantities oftheperturbedvariables,forexample ,willbe cos =1 and sin = .Therefore: sin( 0 + )=sin 0 cos +cos 0 sin = (5.21) cos( 0 + )=cos 0 cos sin 0 sin =1 (5.22) Basedontheaboveassumptions,substitutionsinto(3.3)an d(3.8)resultinthefollowingperturbed equations: m u = mg + X 0 + X m v = mg + Y 0 + Y m w = mg + Z 0 + Z (5.23) I xx p = L 0 + L I yy q = M 0 + M I zz r = N 0 + N (5.24) = q = p = r (5.25) 71

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Intheaboveequations X Y Z denotetheperturbedvaluesoftheexternalaerodynamic forcesand L M N denotetheperturbedvaluesofthemomentsabouttheCG.When the helicopterisattrim,thetrimmedvaluesofthemomentsabou ttheCGwillbezero.Inaddition, onlythetrimmedforcecomponent Z 0 iscompensatingforthegravitationalforce.Hence,attrim : u = g + X=m v = g + Y=m w = Z=m (5.26) p = L= I xx q = M= I yy r = N= I zz (5.27) 5.9StabilityandControlDerivatives Thelaststeptowardsthelinearizationoftheinitialrigid bodyequationrelatestoexpressing theperturbedvaluesoftheexternalaerodynamicforcesand momentsinalinearway.Theanalysis oftheperturbedexternalaerodynamicforcesandmomentsfo llowstheassumptionthatthelatter arecontinuousfunctionsofthehelicopterdisturbedmotio nvariablesandthehelicoptercontrols [20,70,79].Thelinearizationofthoseperturbedvaluesis averycommonmethodwithverypracticalresultsalthoughitisnotbasedonaconsistentmathem aticalbackground,andtothisextent theremightbecasesthatthismodelingmethodwillnotprovi deadequateresults[20,79]. Duetotheassumptionthattheperturbedforcesandmomentsa refunctionsofthedisturbed valuesofthehelicopter'smotionandcontrols,itfollowst hattheformercanbeexpressedasa Taylorseries.Thelinearformofthosequantitiesfollowsb yneglectinghighorderterms.Notation wise,theexpansionoftheaerodynamicforce(ormoment)isn ormalizedbythemass(orcorrespondinginertia).Anexampleistheexpansionoftheaerody namicmoment L ,as: 72

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1 I xx L = 1 I xx @L @u u + ::: + 1 I xx @L @p p + ::: + 1 I xx @L @a a::: + 1 I xx @L @u i u i (5.28) where u i denotesahelicopter'scontrolvariable.Typically,thepr oductsofthepartialderivatives arenotatedi.eas: L u = 1 I xx @L @u (5.29) Theabovepartialderivatives,withrespecttothehelicopt er'sperturbedmotionvariablesandcontrolinputs,arecalledstabilityandcontrolderivatives, respectively.Thosederivativesarecalculatedunderthetrimightcondition.Thecalculationofthe stabilityderivativesisbeyondthe scopeofthiswork;however,detailsmaybefoundin[7,79,84 ,86].Ingeneralnotallstability derivativesarenecessaryforlinearizationoftheforceso rmoments.Asmentionedin[70]animportantpartofsystemidenticationistodecidewhichderi vativesareimportantinthecalculations oftheperturbedforcesandmoments.Everythingwilltakepl aceathover. 5.10ModelIdentication ThepreviousSectionsofthisChapterprovidedanoutlineof thefrequencydomainidenticationmethodforhelicoptermodeling.ThisSectionpresents theidenticationresultsobtainedby CIFER c r forasmallscalehelicopter,operatinginaightsimulator environment.Theighttests throughoutthisworkareconductedusingthe X-Plane ightsimulatorforaRC Raptor90SE helicopter.Atrst,thedescriptionoftheexperimentalpl atformisgiven.Theparametrizedmodel withtheassociatedstabilityderivativesisalsoprovided .Afterthepresentationoftheparametrized model,theset-upandnalresultsoftheidenticationproc edureobtainedby CIFER c r follow. Finallytheaccuracyoftheextractedmodelisvalidatedint hetimedomain.Theendresultofthis Sectionwillbealineardynamicsystemrepresentingthehel icopterresponseathover. 73

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5.10.1ExperimentalPlatform Thesystemidenticationaccuracyandtheperformanceofth econtrollerdesignsareevaluated byusingthecommercialightsimulator X-Plane .Thehelicoptermodelin X-Plane istreatedas the“blackbox”portionoftheproblem,sincenoa-priorikno wledgeofthemodelparametersis usedintheidenticationprocessorthecontroldesign. X-Plane isanawardedightsimulator certiedbytheFederalAviationAdministration(FAA). Apartfromtherealisticightsimulationcapabilities, X-Plane incorporatesaseriesofadditionalusefulfeatures,makingitanidealsolutionforexpe rimentationandvalidationofunmanned ight.Theuserhastheabilitytomodifyandcustomizethose modelsinordertoachievethedesiredightcharacteristics.Inaddition, X-Plane suppliesaplethoraofightdata,whicharerequiredforthemodelidenticationprocessandthecontrolf eedback.Themainadvantageof XPlane ,incomparisonwithothersimulatorssuchas Microsoft'sFlightSimulator and FlightGear istheabilitytoimportandexportreal-timedata.Thisisof particularimportance,sincethecontrol inputscanbeobtainedbyanexternalautopilot.Inaddition ,theautopilotrequiresthehelicopter's stateateverysamplinginstant,whichisavailablebytheex porteddataof X-Plane Thehelicopterusedforexperimentationin X-Plane ,isacustomized Raptor90SE RChelicopter,basedonthe Raptor70 ightmodel[19].Thebasicspecicationsofthismodelcanb e foundinTable5.1.The X-Plane helicoptermodel,hasbeenadditionallycalibratedbyanex periencedpilot,insuchawaythattheightbehaviorofthel atterwillaccuratelyresemblethe behavioroftheactualhelicopter.However,inthesoftware model,theyawrateexhibitssignicant sensitivitytothepedalinput.Thissensitivityintheyawr ateresultsfromtheabsenceofagyro feedbackmechanisminthesimulatormodel.Thegyroisatypi calfeatureofactualsmallscale helicoptersandinsertsadditionalfeedbackforcontrolli ngtheheading. Theexperimentalplatform,inwhichtheighttestingtookp lace,isbasedonacommunicationinterfacebetween MATLAB / SIMULINK and X-Plane .Thecodeofthecontrolalgorithmis developedandstoredin SIMULINK .Ateverysamplinginstant,thecontrolalgorithmreceives thestatemeasurementfrom X-Plane andoutputsthecontrolcommands.Theightsimulator 74

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Table5.1:Experimentalhelicoptermodelbasicspecicati ons. Fulllengthoffuselage 6.6(ft) Fullwidthoffuselage 1(ft) Totalheight 2.12(ft) Mainrotorradius 3(ft) Tailrotorradius 0.7(ft) Mainrotordesignedangularspeed 1250(RPM) Tailrotordesignedangularspeed 5000(RPM) Fullequippedweight 16(lb) receivesthecontrolcommandsandvisualizestheightresp onse.Thecommunicationbetween SIMULINK and X-Plane takesplacethroughaUserDatagramProtocol(UDP)connecti on.The blockdiagramofthecommunicationinterconnectionsisdep ictedinFigure5.1.Thecommunicationofthesoftwarepackagesisbasedontheworkpresen tedin[19].Thesamplingrateis slightlyvariablearoundanaveragevalue.Thisaverageval uecanbechosenbytheuserandithas amaximumvalueof 100 Hz .Mostoftheexperimentswerecontactedat 60 Hz 5.10.2ParametrizedStateSpaceModel Oneofthemostcriticalpartsinthefrequencydomainidenti cationmethodisthedeterminationoftheparametrizedmodel.AsindicatedinSection5.9, thekeychallengeistodecideabout whichstabilityderivativesshouldbeincludedinthedevel opmentoftheparametrizedmodel. Thelinearparametrizedmodelusedforparameteridentica tionofthe Raptor90SE isbasedon Mettler'smodelthatisdescribedin[70–72]fortheCarnegi eMellon's YamahaR-50 andMIT's X-Cell.60 75

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` ` Figure5.1:Blockdiagramoftheexperimentalplatform'sco mmunicationinterface. ThestructureoftheparametrizedmodelproposedbyMettler hasbeenalreadysuccessfully usedfortheparametricidenticationofseveralhelicopte rs,ofdifferentsizesandspecications [8,10,27,28,89,90].Theabilityofthismodelstructureto establishagenericsolutiontothe smallscalehelicopteridenticationproblemisbasedontw oimportantfactors:Therstfactor isthatMettler'sparametrizedmodelprovidesaphysically meaningfulrepresentationofthesystem dynamics.Allstabilityderivativesincludedinthismodel arerelatedtokinematicandaerodynamiceffectsofthefuselageandthemainrotor.Thesecondc omponentistheabilitytorepresent theseveralcrosscouplingeffectsthatdominatethehelico ptermotion.Thisabilitystemsfromthe integrationoftherotormodelwiththelinearizedequation sofmotion. Theadoptedparametrizedmodelinthisworkhastwomaindiff erenceswithrespecttoMettler'smodel.Therstdifferenceistheabsenceofthestabi lizerbardynamics.Thestabilizerbar providesadditionaldampingtothepitchandrollrates.Thi smechanismisnotincludedinthe XPlaneRaptor90SE helicoptermodel.Inaddition,asmentionedinSection5.10 .1,theRaptordoes notincludeagyrofeedback.Theabsenceofthegyroresultsi nveryhighyawrateresponseto 76

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thepedalinput.Thisfactwasanobstacleintheapplication ofthefrequencysweepsofthepedal command.Smallsinusoidaloscillationofthepedalresulte dinveryhighdeviationsoftheyaw rates.Totacklethisproblem,thepedalinputusedwas: u ped = r r + u ped (5.30) where r isapositivegain.Thiswasapracticalwaytoprovidesomead ditionalfeedbacktothe yawresponse,inordertoconducttheexperiments.Thefrequ encysweepexcitationisapplied throughtheinput u ped insteadofadirecttransmissionthrough u ped .Althoughtheexperiments associatedwiththepedalcommandwereconductedinclosedl oop,thisdidnotcreateaproblem intheidenticationprocedure.Theadditionalyawdamping fromthefeedbacktermin(5.30)is absorbedbythestabilityderivative N r .Inthiscase,itisimportanttoclarify,thattheparametri zed modelconsiders u ped asthepedalinputcommand. Theparametrizedmodelrepresentsthelinearizeddynamics oftheperturbedstatesandcontrolinputsofthehelicopterfromatrimmedreferenceight condition.Thetrimoperatingconditionconsideredisthehovermode.Althoughtheparametri zedmodelisassociatedwiththe perturbedvaluesofthestatesandinputs,fornotationsimp licity,the 'sdenedinSection5.8will bedropped.Thelinearstate-spaceparametrizedmodelisgi venby: x = Ax + Bu wherethestateandcontrolvectorsare,respectively: x =[ uvqpabwr ] T and u c =[ u lon u lat u col u lat ] T Thematrices A and B oftheparametrizedmodelarecomposedbythestabilityandc ontrolderivativesofthehelicopter.Thestatespacematricesofthepara metrizedlinearmodel,forthe Raptor 77

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90SE ,are: A = 26666666666666666666666666664 X u 0 g 000 X a 000 0 Y v 0 g 000 Y b 00 00001000000000010000 M u M v 0000 M a 000 L u L v 00000 L b 00 0000 10 1 = f A b 00 00000 1 B a 1 = f 00 000000 Z a Z b Z w Z r 0 N 000 N p 00 N w N r 37777777777777777777777777775 B = 26666666666666666666666666664 000000000000000000000000 A lon A lat 00 B lon B lat 00 00 Z col 0 00 N col N ped 37777777777777777777777777775 Tonalizethedescriptionoftheparametrizedmodel,weare goingtoprovidesomeadditional detailsforsomeofthekeystabilityandcontrolderivative softheabovematrices.Sincethetrim operatingconditionisthehovermode,itisassumedthatthe magnitudeofthemainrotorthrust willbeequaltotheweightofthehelicopter.Therefore T M = mg .Basedon(4.31)thelinear velocitystabilityderivativescanbeapproximatedby: X a = 1 m @X @a = 1 m @ ( T M a ) @a = g Y b = 1 m @Y @b = 1 m @ ( T M b ) @b = g Theaboveequationsimposeaconstrainttothevaluesof X a and Y b ,reducingthenumberofthe unknownparametersintheparameterestimationphase.Base don(4.33),thestabilityderivatives forthepitchandrollmoments,canbecalculatedby: M a = 1 I xx @M @a = 1 I xx @ [( l h T M + K )] a @a = l h mg + K I xx L b = 1 I yy @L @b = 1 I yy @ [( l h T M + K )] b @b = l h mg + K I yy 78

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Someadditionalstabilityderivativesthatrequirefurthe rclaricationarethefollowing: A lat B lon :Thesestabilityderivativesareaddedtotheappingdynam icstocapturepotentialunmodeledoff-axiseffects. M u M v and L u L v :Accordingto[70],thesespeedderivativesareincludedto capturethe effectofairspeedtotheangulardynamics.Intheory,thean gulardynamicsarenotaffected bytheairspeed.Itwouldmakemoresensetoincludetheminth erotordynamics.However, asindicatedin[70],theidenticationresultsaresignic antlybetterwhenthosemoments areincludedinthepitchandrollequations. Asmentionedearlier,theaboveparametrizedmodelprovide sanexcellentgenericdescriptionofthesmallscalehelicopterdynamics.Thedimensions oftheparametrizedmodelcanbe increasedbytheinclusionofthestabilizerbarandgyrofee dbackdynamics.Thechallengeis determinewhichofthoseparametersshouldbeincludedinth emodelandthedeterminationof theirarithmeticvalues.5.10.3IdenticationSetup Theidenticationprocedureforthe Raptor90SE startswiththecollectionoftheexperimental timedomainightdata.Forthecollectionofeachightdata record,thehelicopterissettohover andacomputerizedfrequencysweepexcitationsignalisapp liedtooneofthefourcontrolinputs. Whilethefrequencysweepisexecutedbytheprimaryinputof interest,therestofthecontrol commandsshouldmaintainthehelicopterinthevicinityoft hereferenceoperatingpoint.Inaddition,asindicatedinSection5.7,thesecondaryinputssh ouldbeasuncorrelatedaspossiblefrom themaininput.Foreachcontrolinput,vetosixightrecor dsarecollected.Thebandwidthof theexcitationsignalisrangingbetween 0 : 3 rad=sec 28 rad=sec .Thecomputerizedsweepsappliedarebasedon(5.18)-(5.20).Theminimumandmaximumfr equencyoftheexcitationsweeps aswellasthedurationoftheightrecords,foreachcontrol inputaregiveninTable5.2. 79

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Table5.2:Frequencysweepsparameters.Thoseparametersc orrespondto(5.18)-(5.20). min max T rec ( rad=sec )( rad=sec )( sec ) u lon 1287 T max u lat 0 : 8287 T max u col 0 : 3274 T max u ped 0 : 8257 T max Foreachightrecord,themaximumfrequency max ,ofthecorrespondingexcitationsignalis slightlyvariedfromthevaluegiveninTable5.2.Thisvaria tionwillproduceadifferentexcitation signalforeachightrecord.Identicalexcitationsdonotp rovideadditionalspectralinformation. Thesamplingrateoftheexperimentswassetat 60 Hz X-Plane providesavailabilitytoallthe helicopterstatesandcontrolinputs.Thecollectedmeasur ementsfortheidenticationprocess,are thefollowing: Eulerangles , Angularvelocities p q r Bodyframeaccelerations u v andlinearvelocity w Fortranslationalmotion,thebodyframeaccelerations u v werechoseninsteadofthevelocity measurements u and v ,respectively.Thebodyframeaccelerationmeasurementsf orthesedirectionsprovideamoresymmetricalresponsearoundthetrimva lue,facilitatingthecalculationsof therespectiveFFTs. Afterthecollectionofthetimedomainexperimentaldata, ightrecordsexcitedbythesame primarycontrolinput,areconcatenatedintoasinglerecor d.Theconcatenatedightrecordsare additionallylteredbyalowpasslterwithacutofffreque ncyof 13 Hz .Thetimedomainexperimentaldataareinsertedtothe CIFER c r software.Thethreemodules FRESPID MISOSA and COMPOSITE ,processthetimedomainexperimentaldatatoproduceahigh qualityMIMO 80

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Table5.3:Selectedfrequencyresponsesandtheircorrespo ndingfrequencyranges(in rad=sec ). Thedashedentriesindicatethatthespecicinput-outputp airwasnotincludedintheidentication process.Theboldentrieshighlightthe on-axis responses. u lon u lat u col u ped u 0.5-12.5 v 0.51-22 w 0.20942-27 0.51-27 0.5-18 p 0 : 5 18 0.51-27 q 0.5-18 0 : 51 27 r 0 : 51 271 10 1-10 frequencyresponsedatabase.Thisdatabaseiscomposedbyt he conditionedfrequencyresponses and partialcoherences foreachinput-outputpair. Afterthecalculationoftheightdatafrequencyresponses ,thenexttaskistheextractionof theparametricmodel. CIFER c r usesthe DERIVID moduletodeterminetheparametersofthe statespacemodel,suchthattheestimatedfrequencyrespon sesfromthelatter,arethebesttsto theightdatafrequencyresponses. Therstactionrequiredbytheparametricmodelingprocess isthedeterminationoftheight datafrequencyresponseinput-outputpairs,whichwillbei ncludedintheidenticationprocess. Fromthesefrequencyresponses,thefrequencyrangeofinte restshouldalsobedetermined.For the Raptor90SE ,theselectedfrequencyresponsesandtheircorresponding rangesaredepicted inTable5.3.Thecriterionforthefrequencyresponseselec tionisthecoherencefunction r 2 .Frequencyresponsesforwhichthecoherencefunctionhasvalue sgreaterthan 0 : 7 overthedesired frequencyrangeofthemodelwillbeincluded.Frequencyres ponseswith r 0 : 7 overtheirentire rangearedropped. 81

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Afterdeterminingthefrequencyresponsepairsthatwillbe includedintheidenticationprocess,wearereadytoproceedwiththeextractionofthestate spacemodel.Thispartinitiallyrequiresthedeterminationofthestructureandorderofthepa rametrizedstatespacemodel.The selectedparametrizedmodelisdescribedinSection5.10.2 .Thenextstepistodecideaboutlogicalinitialguessesforthevaluesofthemodelparameters. DERIVID usesanoptimizationalgorithmwhichcalculatestheparametervector ,suchthatthecostfunctiondenedin(5.3)foreach input-outputpair,isminimized.Theoptimizationalgorit hmisbasedonaniterativerobustsecant algorithmthatreducesthephaseandmagnitudeerrorbetwee nthestatespacemodelandtheight datafrequencyresponses.Theexecutionoftheoptimizatio nalgorithmcontinues,untiltheaverage oftheselectedfrequencyresponsescostfunctions J a ,isminimized. Theextractionoftheparametricmodelisaniterativeproce dure,whichcontinuesuntilthe mostsuitablestabilityandcontrolderivativesofthestat espacemodelareselected.Inorderto determinewhichstabilityorcontrolderivativesaregoing toparticipateinthestatespacemodel, apartfromthefrequencyresponsescostfunctions, DERIVID providestwoadditionalstatistical metrics.TherstoneisthepercentageoftheCramr-Rao(CR )boundforeachparameter.The CRboundgivesalowerboundofthestandarddeviationofthep arameter.AhighCRboundindicatesthattheparameterisunreliableandshouldbedisqu aliedfromthemodel,orxedtoa certainvalue.Thesecondstatisticalmetricisthepercent ageoftheinsensitivityofeachparameter withrespecttothecostfunction.Ahighinsensitiveparame terwillhaveaminimaloranyeffectto thecalculationofthecostfunction.Therefore,thisparam etershouldbedroppedfromthemodel. Asummaryoftheguidelinesfortheselectionofthestatespa cemodel'sderivativesbasedon[105] is: J a 100 CR % 20% Insensitivity % 10% Theidentiedstabilityandcontrolderivativesforthe Raptor90SE ,withtheirrespectiveCR boundandinsensitivitypercentage,canbeseeninTable5.4 .Theon-axisfrequencyresponses, 82

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obtainedbytheightdataandthosepredictedbythestatesp acemodelaregiveninFigure5.2. Thesamecomparisonfortheoff-axisresponsesisgiveninFi gure5.3.Theidenticationresults illustrateaverygoodtbetweenthefrequencyresponsesob tainedbytheightdataandthose predictedbythestatespacemodel.Thecostvalueforeachfr equencyresponseoftheinput-output pairsthatparticipatedintheidenticationprocess,isde pictedinTable5.5.Theaveragecost J a ,is wellbelowthesuggestedguidelinevalue.Thoseresultsind icatethattheidenticationprocedure hasaccuratelyextractedalinearstatespacemodelofthe Raptor90SE dynamics. Table5.4indicatesthatsomeoftheidentiedparametersex hibithighCRboundsandinsensitivities.Thelargervaluesareencounteredinthetra nslationalvelocitydampingderivatives X u and Y v .Thesameissuewiththespecicparameterswasalsoencount eredforthe Yamaha R50 modeldescribedin[70].Althoughthesignandthevalueofth isparametersmakessense, thestatisticalmetricsindicatethattheyarecompletelyu nreliable.Accordingto[70],thelarge uncertaintyofthespecicstabilityderivativesresulted fromthelackoflowfrequencyexcitation. Highstatisticalmetricsarealsoassociatedwiththespeed derivativesoftherollandpitchrates. Inparticular, M v and L u L v exhibitveryhighCRboundsandinsensitivities.Thosepara meters couldbedroppedfromthemodelwithoutsacricingtheaccur acyoftheidenticationresults. However,theywereintensionallypreservedtokeepthenal statespacedynamicsascloseas possibletotheparametrizedmodel. Finally,themismatchintheheaveresponsesdepictedinFig ure5.2,indicatethat X-Plane accountsforthemainrotorinowdynamics.Themostimportant parametersofthestatespacemodel arethemainrotorappingspringderivatives M a and L b .Thehighvalueofthosetwovariables indicatethethe Raptor90SE isasupermaneuverableandhighlyagilehelicopter.Thiswa san anticipatedresultsincesmallscalehelicoptersofthisty pehaveveryrigidblades.Apartfromthe excellenttoftheactualandpredictedfrequencyresponse s,theidenticationresultindicatethat theightsimulatormayduplicaterealightapplications. 83

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Table5.4:Linearstatespacemodelidentiedparameters.T hedashedentriesindicatethatthe specicderivativeswerenotincludedinthestatespacemod el. ValueCR%Insensitivity% ValueCR%Insensitivity% A matrix X u 0 : 03996118 : 758 : 24 B a 0 : 61689 : 0901 : 923 Y v 0 : 05989127 : 462 : 24 Z a M u 0 : 254212 : 254 : 195 Z b M v 0 : 0601328 : 957 : 091 Z w 2 : 0557 : 3512 : 546 M a 307 : 5716 : 8151 : 097 Z r L u 0 : 0244036 : 8110 : 63 N 2 : 9826 : 9911 : 908 L v 0 : 1173246 : 694 : 13 N p L b 1172 : 48175 : 7511 : 462 N w 0 : 707615 : 954 : 400 A b 0 : 77138 : 8961 : 860 N r 10 : 716 : 7291 : 233 g 9 : 3893 : 3310 : 9953 1 = f 30 : 717 : 4740 : 9838 B matrix A lon 4 : 0593 : 0050 : 9285 Z col 13 : 115 : 0261 : 688 A lat 0 : 0161014 : 663 : 356 N col 3 : 7497 : 1612 : 602 B lon 0 : 0101723 : 797 : 206 N ped 26 : 906 : 1891 : 825 B lat 4 : 0852 : 9000 : 8280 84

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Table5.5:Transferfunctionscostsforeachinput-outputp air. u=u lon 54 : 087 =u lon 56 : 108 p=u lon 48 : 502 q=u lon 60 : 196 v=u lat 29 : 704 =u lat 36 : 271 p=u lat 38 : 068 q=u lat 55 : 421 r=u lat 42 : 551 w=u col 89 : 496 r=u col 20 : 147 r= u ped 20 : 178 Average 45 : 894 85

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15 0 20 45 u=u lonMagnitude (dB) 450 350 250 150 Phase (deg) 10 1 10 0 10 1 10 2 0.7 0.8 0.9 1 CoherenceFrequency (rad/sec) 20 0 20 40 v=u lat 250 150 50 50 10 1 10 0 10 1 10 2 0.7 0.8 0.9 1 Frequency (rad/sec) 25 5 15 35 w=u col 400 300 200 100 10 1 10 0 10 1 10 2 0.7 0.8 0.9 1 Frequency (rad/sec) 20 0 20 40 q=u lonMagnitude (dB) 200 100 0 100 Phase (deg) 10 1 10 0 10 1 10 2 0.7 0.8 0.9 1 CoherenceFrequency (rad/sec) 15 5 25 45 p=u lat 200 100 0 100 10 1 10 0 10 1 10 2 0.7 0.8 0.9 1 Frequency (rad/sec) 25 5 15 35 r= u ped 200 100 0 100 10 1 10 0 10 1 10 2 0.7 0.8 0.9 1 CoherenceFrequency (rad/sec) Figure5.2:On-axisfrequencyresponsesoftheightdata(s olidline)andfrequencyresponses predictedbythestatespacemodel(dashedline). 86

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60 40 20 0 q=u latMagnitude (dB) 500 400 300 200 Phase (deg) 10 0 10 1 10 2 0.7 0.8 0.9 1 CoherenceFrequency (rad/sec) 60 40 20 0 p=u lon 450 350 250 150 10 1 10 0 10 1 10 2 0.7 0.8 0.9 1 Frequency (rad/sec) 60 25 10 45 r=u lat 400 300 200 100 10 1 10 0 10 1 10 2 0.7 0.8 0.9 1 Frequency (rad/sec) 30 10 10 30 =u lonMagnitude (dB) 300 200 100 0 Phase (deg) 10 1 10 0 10 1 10 2 0.7 0.8 0.9 1 CoherenceFrequency (rad/sec) 30 10 10 30 =u lat 300 200 100 0 10 1 10 0 10 1 10 2 0.7 0.8 0.9 1 Frequency (rad/sec) 18 11 4 3 r=u col 125 75 25 25 10 1 10 0 10 1 10 2 0.7 0.8 0.9 1 CoherenceFrequency (rad/sec) Figure5.3:Off-axisfrequencyresponsesoftheightdata( solidline)andfrequencyresponses predictedbythestatespacemodel(dashedline). 87

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0.05 0 0.05 u lon 6 0 6 u ( m=sec 2 ) 6 0 6 v ( m=sec 2 ) 6 0 6 w ( m=sec 2 ) 0.7 0 0.7 ( rad ) 0.7 0 0.7 ( rad ) 0.7 0 0.7 p ( rad=sec ) 0.7 0 0.7 q ( rad=sec ) 0 1 2 3 4 5 6 1.5 0 1.5 time (sec) r ( rad=sec ) 0.1 0 0.1 u lat 6 0 6 u ( m=sec 2 ) 6 0 6 v ( m=sec 2 ) 6 0 6 w ( m=sec 2 ) 0.7 0 0.7 ( rad ) 0.7 0 0.7 ( rad ) 0.7 0 0.7 p ( rad=sec ) 0.7 0 0.7 q ( rad=sec ) 0 1 2 3 4 5 6 1.5 0 1.5 time (sec) r ( rad=sec ) 0.1 0.2 0.5 u col 6 0 6 u ( m=sec 2 ) 6 0 6 v ( m=sec 2 ) 6 0 6 w ( m=sec 2 ) 0.7 0 0.7 ( rad ) 0.7 0 0.7 ( rad ) 0.7 0 0.7 p ( rad=sec ) 0.7 0 0.7 q ( rad=sec ) 0 1 2 3 4 5 6 1.5 0 1.5 time (sec) r ( rad=sec ) 0.3 0.15 0 u ped 6 0 6 u ( m=sec 2 ) 6 0 6 v ( m=sec 2 ) 6 0 6 w ( m=sec 2 ) 0.7 0 0.7 ( rad ) 0.7 0 0.7 ( rad ) 0.7 0 0.7 p ( rad=sec ) 0.7 0 0.7 q ( rad=sec ) 0 1 2 3 4 5 6 1.5 0 1.5 time (sec) r ( rad=sec ) Figure5.4:Timedomainvalidation. 88

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5.10.4TimeDomainValidation Thenalstepoftheidenticationprocedureisthevalidati onoftheextractedstatespacemodel inthetimedomain.Thetimedomainvalidationisimportantf orevaluatingthepredictiveaccuracyandlimitationsoftheidentiedmodel.Thetimedomain ightdatausedforthevalidation partareobtainedbyapplyingspecialcontrolinputswhicha redissimilarwiththeonesusedin theidenticationprocess.Theseinputsarestepsorroughl ysymmetricdoublets.Thesetypesof inputsareusedduetotheirrelativelargefrequencyconten t[70].Thetimedomainresponsesof theidentiedmodelobtainedbytheintegrationofthestate spaceequations,arecomparedwith thecorrespondingresponsesoftheightdata.Theinputsto thestatespacemodelusedforthe integrationprocessareidenticalwiththeonesobtainedby theightdata. Toobtainthevalidationightdata,fourindividualightr ecordsarecollected,eachcorrespondingtooneofthecontrolinputs.Ineveryindividuali ghtrecord,aroughlysymmetricdoubletisappliedbythecorrespondingprimaryinput,whileth erestofthecontrolcommandsretain theirtrimmedvalue.Thedoubletshouldbeappliedinsuchaw aythattheon-axisresponsesofthe correspondinginputaresufcientlydivergedfromthetrim medcondition.Alargedeviationfrom theoperatingpointwillrevealtheidentiedmodelpredict ivelimitations.Beforeeachdoubletis applied,thehelicopterissettohovermode.Thetimedomain validationcomparisonresultsare depictedinFigure5.4,inasimilarwaywith[70].Thetimedo mainresponsesforeachrecordare illustratedincolumns.Therstrowshowstheexecuteddoub letofeachprimarycontrolinput. Thevalidationcomparisonindicatesanexcellenttbetwee nthepredictedvaluesfromthelinear statespacemodelandtheightdata.Therefore,theidenti edmodelprovidesareliabledynamic representationofthehelicopteraroundthehoveringopera tingconditionanditisappropriatefor controldesign. 89

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5.11Remarks TheidenticationprocessdescribedinthisChapterconsid ershoveringasthereferenceight operatingpoint.Therefore,themodelislimitedtoanareao ftheightenvelopearoundthespecicoperatingcondition.Toderivealinearmodelforadiff erentightmode,thesameprocedures shouldberepeated.However,theexecutionofthefrequency sweepsforadifferentreferenceight conditionfromhoverisaverytediousprocess.Forexample, inthecaseofforwardight,the helicoptershouldcruiseinaconstanttranslationalveloc itywhenthesweepsareapplied.This experimentalprocedureintroducespracticallimitations .Firstly,itisverydifculttosustaina constanttranslationalvelocityinalltheightrecords.I naddition,theretainmentofthehelicopter aroundthedesiredoperatingpointwhenthesweepsareappli edisanadditionallimitingfactor. Thislimitationismoreapparentwhenthelowfrequencyport ionofthesweepisexecuted.Tothis extent,theexperimentaldataacquiredfromthecruisemode haveinferiorqualitycomparedwith thedatacollectedwhenthehelicopterisinhover.Therefor e,thesystemidenticationmodeling methodhaspotentialshortcomingsinthedevelopmentoflin earmodelswhichcorrespondtoight modesdifferentfromhover.Havingdecidedtheorderandthe structureofagenericparametric linearhelicoptermodelathover,thenextstepisthedevelo pmentofasystematicprocedurefor thedesignoflinearhelicopterightcontrollers.Thenext Chapterprovidesapositionandheading trackingcontrollerbasedonthelinearhelicoptermodel. Theindividualexperimentsarearrangedincolumnsforthed oubled-inputexperiments.The rstrowshowsthepiloteddoubletappliedtotherespective controlinputandtheremainingrows showtheresponsestothevehiclesstates. 90

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Chapter6:LinearTrackingControllerDesignforSmallScal eUnmannedHelicopters InthepreviousChapterweprovidedananalyticalmethodolo gyfortheextractionofalinear dynamicmodelforasmallscalehelicopterbasedon[70,105] .Moderncontroltechniquesare modelbased,inthesensethatthecontrollerarchitectured ependsonthedynamicdescriptionof thesystem.Therefore,theknowledgeofthehelicopterline ardynamicmodelisveryvaluable forthedesignofautonomousightcontrollers.ThisChapte rpresentsasystematicprocedurefor thedesignofaightcontrollerbasedonthelineardynamicr epresentationofthehelicopter.The controllerobjectiveisforthehelicoptertotrackpreden edreferencetrajectoriesoftheinertial positionandtheyawangle.6.1HelicopterLinearModel ThegoalofthisSectionistoderiveaightcontrollerbased onthehelicopter'slineardynamic model.Theproposedcontrollershouldalsobeapplicableto anysmallscalehelicopter.Thisclaim requirestheadoptionofanominallineardynamicmodelstru cture,whichiscapableofcapturing thedynamicbehaviorofawidefamilyofsmallscalehelicopt ers.AnidealsolutiontothisrequirementistheuseoftheparametrizedmodeldescribedinSectio n5.10.2asabasisforthecontroller design. Thespecicmodelrepresentsthedynamicresponseofthehel icopterperturbedstatevector fromthereferenceightcondition.Inthiscase,therefere nceoperatingconditionishover.At hover,thetrimvaluesofthelinearandangularvelocityare : v B o = B o =[000] T (6.1) 91

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Fromtheaboveequationsitisapparentthatwhenthehelicop teroperatesaroundhover,thehelicopter'sstateisequaltotheperturbedstatevectoraboutt hereferenceoperatingpoint.ThehelicopterlinearmodelisbasedonSection5.10.2anditisrepea tedhereforclaricationpurposes. Theadoptedstatespacemodelis: x = Ax + Bu c (6.2) wherethestateandcontrolvectorsare: x =[ uvqpabwr ] T and u c =[ u lon u lat u col u lat ] T (6.3) Thematrices A and B ofthestatespacemodelaregivenby: A = 266666666666666666666666666666664 X u 0 g 000 X a 0000 0 Y v 0 g 000 Y b 000 0000100000000000100000 M u M 0000 M a 0000 L u L v 00000 L b 000 0000 10 1 = f A b 000 00000 1 B a 1 = f 000 000000 Z a Z b Z w Z r 0 0 N 000 N p 00 N w N r 0 00000000010 377777777777777777777777777777775 B T = 266666664 000000 A lon B lon 000 000000 A lat B lat 000 00000000 Z col N col 0 000000000 N ped 0 377777775 92

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Theabovestatespacerepresentationisslightlydifferent fromtheparametrizedmodelofSection5.10.2,sinceitincludestheyawdynamicsgivenby = r .Theyawdynamicsareexcluded fromtheidenticationprocesssincetheydonotincludeany unknownstabilityderivativesand alsotheyawisdecoupledfromtherestofthestatevariables .However,thecontrollerdesignrequirestheinclusionoftheyawtothestatespacemodel.Theo veralldynamicsconstituteacoupled linearsystemofthehelicoptermotionvariablesandthemai nrotorappingdynamics. Theorderoftheabovemodelcanbeincreasedbyincludingthe dynamicsofthestabilizerbar andtheyawdampingsystem.Thesetwosubsystemsprovideadd itionaldampingtotheangular velocitydynamics.Sincetheyconstituteadditionalfeedb acksourcesoftheangulardynamics, theirpresenceinthestatespacesystemdoesnotinuenceth econtrollerdesign.Therefore,their effecthasbeenomittedfromthehelicoptermodel. Theproposedlinearmodel(usuallywiththeinclusionofthe yawgyrodynamics)hasbeen successfullyadoptedforcontrolapplicationsinalargenu mberofsmallscaleunmannedhelicopters[8,10,27,28,89,90].Tothisextent,thelinearmod elproposedby[70]providesageneralizedandphysicallymeaningfulsolutiontothedevelopme ntofpracticallinearmodelsforsmall scalehelicopters.Foranyparticularsmallscalehelicopt er,thenumericvaluesofthematrices A and B entriescanbeestimatedbyfollowingtheidenticationpro ceduredescribedintheprevious Chapter.6.2ControllerOutline Havingestablishedthehelicopterlineardynamicmodel,th enextstepisthedesignoftheautonomousightcontroller.Thecontroller'sultimateobje ctiveisforthehelicoptertoautonomously trackpredenedboundedpositionandheadingreferencetra jectories.Thelinearmodelgivenin (6.2)doesnotincludethehelicopterpositiondynamics.Th erefore,thecontrollerdesignstarts withthetrackingproblemofareferencetranslationalvelo cityandheadingprole.Theintegrationofthepositiontrackingtothecontrolproblemfollows .Theinitialoutputofinterestofthe 93

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helicopteris: y =[ uvw ] T = Cx (6.4) Therstdesigntaskisforthehelicoptertotracktherefere nceoutput y r =[ u r v r w r r ] T Thetrackingproblemrequiresthedeterminationofthecont rolsignal u c ( t ) asafunctionofthe statevariablesofthevector x ( t ) andthereferenceoutput y r ( t ) (withitshigherderivatives)such that: lim t !1 k y ( t ) y r ( t ) k =0 (6.5) whilethestateofthesystem x ( t ) and,thus,thecontrolinput u c ( t ) remainboundedforanybounded referenceoutput y r ( t ) .Anadditionaldifcultyofthetrackingcontrolproblemis theavailability ofthestatevariablesfrommeasurements.Notalloftheheli copterstatescanbemeasured,hence onlyasubsetofthestatevariablescanbeusedbythecontrol lerforfeedbackpurposes.Inreal lifeapplications,onlythehelicoptermotionstatevariab lescanbedirectlymeasured.Ontheother hand,theappinganglesaretypicallyabsentfromtheavail ablemeasurements.Itisassumedthat thereisavailabilityofthefollowingmeasurementvector: y m =[ uvwpqr ] T = C m x (6.6) Thecompletestatecanbereconstructedforcontrolpurpose sbyaKalmanlterorastateestimator[3,23,41].Bothofthesechoicesincreasethesystemdyn amicsorder.However,inmanned ightapplications,thepilotisabletooperatethehelicop terwithoutaccountingfortheapping angles.Therefore,wesetthesamerequirementfortheunman nedcaserestrictingthecontroller's feedbackinformationonlytothemeasuredvector y m .Thisproblemisclassiedasoutputfeedback.When y m = x ,thenwehavefullstatestatefeedback. Inthecaseoflinearsystems,thetrackingproblemwithoutp utfeedbackcanbetackledwith twodifferentapproaches.Trackingwithintegralcontrola ndtrackingviatheuseofaninternal model.Intheinternalmodelapproach,thereferenceoutput signalisgeneratedbyaxedreferencedynamicsystemdrivenbyaboundedinput.Thisreferenc esystemiscalled internalmodel 94

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Thestructureoftheinternalmodelisusedbythecontroller yieldingadynamicfeedbackscheme. Typicalapplicationofsuchcontroldesignismetwhenthere ferenceoutputisaconstantsignalor sinusoidalwithconstantfrequency[43].Theinternalmode lapproachhasveryimportantrobust andadaptiveproperties,howeverthedesignisrelativelyc omplex.InthecaseofMIMOsystems thegeneratedinternalmodelshouldconsidertherelatived egreevectorthatcorrespondstothe output(therelativedegreevectorcomponentsindicatesho wmanytimeeachoutputshouldbe differentiateduntiltheinputappears).Likewisewiththe integralcontrol,theuseoftheinternal modelbecomesrelativelycomplicatedwhenthedesiredoutp utisanarbitrarycontinuoussignalof time.Moredetailsabouttheinternalmodelapproachcanbyf oundin[9,36]. Theuseofintegralcontrolforthetrackingproblemresults inthedesignofadynamicfeedbackcontroller.Integralcontrolprovidesareliableandc onsistentsolutionwhenthedesiredoutput hasconstantvaluesovertime.However,inthecaseofatimev aryingoutputprole,theintegral controldesignrequiresthedeterminationofasteadystate response x ss ( t ) andasteadystatecontrolinput u ssc ( t ) ,suchthatwhen y ( t ) tendsto y r ( t ) ,thefollowingequalityholds: x ss = Ax ss + Bu ssc (6.7) Thedeterminationofthepair ( x ss ;u ssc ) isadifculttask,renderingtheintegralcontroldesign impracticalforthetrackingproblemofatimevaryingoutpu t.Moredetailsabouttheintegral controloflinearsystemscanbefoundin[23,43]. Insteadoffollowingtheabovestandardmethodologies,wea doptatrackingdesignwhichis simple,mathematicallyconsistentandwellsuitedtothesp ecicproblem.Therstpartofthe designinvolvesthedeterminationofadesiredstatevector x d whichiscomposedonlybythe componentsofthereferenceoutputvector y r andtheirhigherderivatives.Denote e = x x d theerrorbetweentheactualhelicopterstateanditsdesire dvalue.Thedesiredvector x d shouldbe choseninsuchawaythat,given: lim t !1 k e ( t ) k =0 then lim t !1 k y ( t ) y r ( t ) k =0 (6.8) 95

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Theproposedcontrollerdesignprovidesarecursivemethod ologyforthederivationofadesired statevector x d andadesiredcontrolinput u dc thatsatises(6.8)andalso: x d = Ax d + Bu dc (6.9) Theroleofthedesiredstatevector x d andthecontrolinput u dc isidenticalwiththesteadystate vector x ss andtheinputvector u ssc whichisrequiredbytheintegralcontrolmethodology.The contributionoftheproposeddesignisthedevelopmentofas implerecursiveprocedureforthe derivationofthepair ( x d ;u dc ) thatsatises(6.8)-(6.9). Thechoiceofthepair ( x d ;u dc ) isbasedonthebacksteppingdesignapproach.Detailsabout thebacksteppingdesignmethodologycanbefoundintheAppe ndixA.Intheparticularcasethe backsteppingdesignisnotusedforthestabilizationofthe trackingerrorbutitisrestrictedtothe determinationofthedesiredstateandcontrolinputvector s.Backsteppingprovidesasystematic methodologyfortheoutputtrackingproblemofsystemsinfe edbackform. Duetothepresenceofthestabilityderivatives X a and Y b in(6.2),thehelicoptermodelcan notbecategorizedinthisclassofsystems.Acommonsimpli cationpractice,followedin[37,47, 66],istoneglecttheeffectofthelateralandlongitudinal forcesproducedbytheTPPtilt.Those parasiticforceshaveaminimaleffectonthetranslational dynamicscomparedtothepropulsion forcesproducedbythestabilityderivatives X and Y (in(6.2)aredenotedby g and g ,respectively).Thisassumptionisphysicallymeaningfulandresu ltsintoalinearsystemoffeedback form. Systemsofstrict-feedbackformarefeedbacklinearizable andthereforedifferentiallyat. Thedifferentiallyatnesspropertyisthekeyattributeof theapproximatedsystemtowhichthe controllerdesignisbasedon.Asystemiscalleddifferenti allyatwhenthereexistsoutputfunctions(calledatoutputs)suchthatallthestateandinputv ectorscanbeexpressedintermsofthe atoutputsandtheirhigherderivatives[48].Detailsabou tthedifferentialatnesspropertyof nonlinearsystemsmaybefoundin[22,107].Theconceptofdi fferentialatnesshasbeenalso 96

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usedin[47,48]forthedevelopmentofanonlinearcontrolle rbasedonnonlinearinversionforthe helicoptertrackingproblem. Havingdenedthedesiredstate x d andcontrolvector u dc ,weintroducethestabilizingcontrollerofthesystem.Thecontrollersignalisconstructed bythefollowingsuperposition: u c = u dc + u fbc (6.10) Thentheerrordynamicstaketheform: e = Ae + Bu fbc (6.11) Theabovesystemisidenticalwiththesystemgivenin(6.2). Thedifferenceisthatthestatespace vectorissubstitutedbytheerrorvector.Thesecondcontro lcomponentcanbechosenfroma varietyofoutputfeedbacktechniques,suchthattheerror e isrenderedgloballyasymptotically stable(GAS).6.3DecomposingtheSystem Itisemphasizedthatthecontrollerdesignmustincorporat ethephysicallimitationsofhelicopteright.Acommonmistakeinthedevelopmentofightco ntrollersistheblindadoptionofa mathematicalcontrolschemewithoutconsideringthephysi calstructureofthehelicoptermodel.It istypicalthattheightcontrolproblemisforcedtosuitas peciccontrollerdesignratherthanthe controllerdesignbeingtailoredbasedontheproblem.Acha llengingandrigorousmathematical controlschemewillperformsignicantlypoorinareallife applicationifthefundamentalnotion ofhelicopterightisdisregardedbythedesigner. Thehelicopterpilotingfundamentalintuitiondictatesth atthecycliccommands u lon and u lat areusedtomanipulatethepitchandrollmomentswithultima teobjectivetheproductionoftranslationalmotion.Thecollectivecommand u col controlsthemagnitudeofthethrustofthemain rotorproducingthenecessaryliftingforce,whilethepeda lcommandcontrolstheheadingofthe 97

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helicopter.Tothisextenttheidealsolutionisforeachcon trolcommandtobeasmuchindependentaspossiblefromtheothers.Theidealsolutiontothepr oblemistoconstruct4independent SISOfeedbackloopsforeachcontrolinput.However,sincet hesystemisahighlycoupledlinear systemthisapproachcannotguaranteearigorousandmathem aticallyconsistentstabilityanalysis. Havingsaidthat,acloseinspectionofthemodelstructureg ivenin(6.2),indicatesthatthe helicopterdynamicscanbeseparatedintotwointerconnect edsubsystems.Therstsubsystem representsthehelicopterlongitudinalandlateralmotion .Thesecondsubsystemrepresentsthe coupledyawandheavedynamics.Inparticular,thelaterallongitudinalsubsystemisgivenby: x ll = A ll x ll + B ll u ll (6.12) where: x ll =[ uvqpab ] T and u ll =[ u lon u lat ] T (6.13) and: A ll = 26666666666666666666664 X u 0 g 000 X a 0 0 Y v 0 g 000 Y b 0000100000000100 M u M v 0000 M a 0 L u L v 00000 L b 0000 10 1 = f A b 00000 1 B a 1 = f 37777777777777777777775 B ll = 26666666666666666666664 000000000000 A lon A lat B lon B lat 37777777777777777777775 (6.14) Theyaw-heavedynamicssubsystemisgivenby: x yh = A yh x yh + B yh u yh + D yh x ll (6.15) 98

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_ x yh = A yh x yh + B yh u yh + D yh x ll Yaw-Heavesubsystem x ll = A ll x ll + B ll u ll Longitudinal-Lateralsubsystem x ll Figure6.1:Interconnectionofthetwohelicopterdynamics subsystems. where: x yh =[ wr ] T and u yh =[ u col u ped ] T (6.16) and: A yh = 266664 0010 Z w Z r 0 N w N r 377775 B yh = 266664 000 Z col N ped N col 377775 D yh = 266664 00000000000000 Z a Z b 00000100 377775 (6.17) TheinterconnectionofthetwosubsystemsisshowninFigure 6.1.Thecontrollerdesignrequires thatthefollowingassumptionsassociatedwiththehelicop terlinearmodelgivenin(6.2),should hold:Assumption6.1. Thematrixpairs ( A ll ;B ll ) and ( A yh ;B yh ) arecontrollable. Assumption6.2. Thematrix B 2 R 8 4 hasfourlinearlyindependentrows. Assumption6.3. Thestabilityderivatives g M a and L b arenonzero. Theaboveassumptionsaresubstantiallynecessaryconditi onsrequiredbythecontrollerdesign.Ifthelinearmodeldoesnotsatisfyalloftheabovecon ditionsthenmostlikelythemodeling 99

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identicationprocesshasleadtoerroneousresults.Theyr eectthefactthatthelinearmodelhas tobephysicallymeaningful.Intuitively,frommannedigh tapplications,thepilotcommandscan regulatethepositionandheadingofthehelicopterinallof thecongurationspace.Regarding Assumption6.1,lackofcontrollabilityindicatespooride nticationresults,wrongmodelstructure orahelicopterthatcannotyproperly!Inaddition,eachin putmusthaveadirecteffecttothe helicopter'smotion,therefore,Assumption6.3shouldhol daswell.Finally,if M a =0 or L b =0 itimpliesthatnomomentsaretransmittedtothehelicopter .Therefore,theaboveassumptions provideavaliditycheckofthehelicopterlinearmodel. Beforeweproceed,weintroduceapreliminarycontrolactio nfortheinputvectors u ll u yh that cancelsoutthecouplingeffectofthecontrolderivativesa ndnormalizesthe B ll and B yh matrices, respectively.Hence: u ll =( B n ll ) 1 v ll u yh =( B n yh ) 1 v yh (6.18) where: B n ll = 264 A lon A lat B lon B lat 375 B n yh = 264 0 Z col N ped N col 375 (6.19) BasedonAssumption6.3theaboveinversematricesarenonsi ngular.Singularityinanyofthem indicateserroneousparametervalues.Substitutingtheab ovepreliminarycontrolactionsthetwo subsystemsbecome: x ll = A ll x ll + B ll v ll (6.20) x yh = A yh x yh + B yh v yh + D yh x ll (6.21) where: B ll = 264 0 6 2 I 2 375 B yh = 264 0 2 1 I 2 375 (6.22) 100

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Fromtheaboveanalysis,theinitialsystemisnowviewedast wointerconnectedsubsystemsin cascadeform.Thebacksteppingdesignisperformedindepen dentlyforeachsubsystemresulting inthecascadederrordynamicsofthehelicopter.Stabiliza tionofnonlinearsystemsincascade hasbeenextensivelystudiedin[63,94,98].Contrarytothe nonlinearsystems,thecaseforthe LTIsystemsismuchmoreeasierintermsofanalysis.Iftheco ntrollerisdesignedsuchthatthe twoerrordynamicssubsystemsarerenderedGAS(byignoring theinterconnectioneffect),then thecompleteerrordynamicssystemisrenderedGAS,aswell. Thisapproachisbasedonthe separationprinciple ,whichemergesfromthe superpositionproperty ofLTIsystems.Thestability analysisofthecontrollerdesignisgivenindetailinthefo llowingSections. Atthispoint,thecontrollerstructurerequiresthedesign oftwoindependentfeedbackloops foreachsubsystem.Thisapproachresultsinamathematical lyconsistentandsystematicmethodology,whichreectstheintuitiveightnotion.Thelatera l/longitudinalmotionisregulatedindependentlyfromtheheadingandverticalmotionofthehelico pter.Thesamedecompositionofthe helicopterdynamicsisalsoreportedin[109].6.4VelocityandHeadingTrackingControl ThisSectionprovidesadetailedpresentationofthecontro llerdesignforthevelocityandheadingtrackingofthehelicopter.Thecontrolproblemisfocus edonthedesignoftwofeedbackloops foreachsubsystem.Aftertheintroductionofthetwofeedba ckloopsthestabilityanalysisofthe overallsystemdynamicsisgiven.6.4.1Lateral-LongitudinalDynamics Thelongitudinalandlateralmotionofthehelicopterareno tdirectlycontrolledthroughthe cyclicinputsbutratherviaasequenceofintermediateeven ts.Thecyclicinputsproducepitchand rollmomentstothehelicopterfuselage.Thosemomentsresu ltinachangeofthepitchandroll attitudeangles.Theattitudechangeresultsinthetiltoft hehelicoptermainrotordisc.Bytilting 101

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therotordiscthemainrotorthrustisalsotiltedtoproduce thenecessarypropulsionforcesfor lateralandlongitudinalmotion.Theeffectofthetranslat ionalforcesproducedbytheapping motionofthethemainrotorisparasiticandnegligiblecomp aredtothemainsourceofpropulsion, whichistherollandpitchtiltofthemainrotor. AsindicatedinSection6.2,neglectingtheeffectofthesta bilityderivatives X a and Y b isa commonpracticethatresultsinamorephysicallymeaningfu ldesign.Whenthelatterstability derivativesareomittedfromthehelicoptermodel,thelate ral-longitudinaldynamicshaveastrictfeedbackform. Thecompletedescriptionofthelongitudinal-lateralsubs ystemisgivenby: x ll = A fbll x ll + B ll v ll y ll = C ll x ll (6.23) y m ll = C m ll x ll where: x ll =[ uvqpab ] T v ll =[ v lon v lat ] T y ll =[ uv ] T y m ll =[ uvqp ] T Intheaboveequations y m ll isthemeasurementvectoravailableforfeedbackand y ll istheoutputof thesubsystem.Thereferenceoutputvectoris y r ll =[ u r v r ] T .Thematrix A fbll ,isidenticalto A ll withtheonlydifferencethatthestabilityderivatives X a and Y b areomitted.Theinterconnection oftheapproximatedlongitudinal-lateralsubsystemissho wninFigure6.2. FromSection6.2,therstgoalofthecontrollerdesignfort hissubsystemistodeterminea desiredstatevector x dll andadesiredcontrolinput v d ll ,withbothofthembeingfunctionsofthe y r ll 102

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_ a = q 1 f a + A b b + v lon b = p 1 f b + B a a + v lat q = M u u + M v v + M a a p = L u u + L v v + L b b = q = p u = X u u g v = Y v v + g a;b q;p ; u;v u v v lon v lat Figure6.2:Strict-feedbackinterconnectionofthelongit udinal-lateralhelicopterdynamics subsystem.Thetermsassociatedwiththe X a and Y b stabilityderivativesaredisregarded. componentsandtheirhigherderivatives,suchfortheerror e ll = x ll x dll giventhat: lim t !1 k e ll k =0 then lim t !1 k y ll ( t ) y r ll ( t ) k =0 (6.24) Todosothecontrollawofthissubsystemisobtainedbythefo llowingsuperposition: v ll = v d ll + v fb ll = 264 v d lon v d lat 375 + 264 v fb lon v fb lat 375 (6.25) Theinitialtaskistoselectthepair ( x dll ;v d ll ) suchthattheysatisfytherequirmentof(6.24)andalso: x dll = A fbll x dll + B ll v d ll (6.26) Ifthepair ( x dll ;v d ll ) satisestheaboveequationthentheerrordynamicsbecome: e ll = A fbll e ll + B ll v fb ll (6.27) Thenalstepistheselectionofanoutputfeedbackcontroll aw v fb ll whichstabilizes e ll suchthat thetrackingobjectiveof(6.24)isachieved. Forthederivationofthedesiredstatevector x dll andcontrolinput v d ll wearegoingtoapply arecursiveprocedurebasedonthebacksteppingmethodolog y.Thebacksteppingapproachis idealforthecontroldesignofsystemsinfeedbackform.Int hiscase,however,thebackstepping procedureisnotusedforthestabilizationofthesystembut itisonlyrestrictedtothederivation ofthepair ( x dll ;v d ll ) suchthat(6.24)and(6.26)aresatised.Theapplicability ofthisapproach 103

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isbasedonthefactthatthelongitudinal-lateralsubsyste misinstrict-feedbackformthereforeit isdifferentiallyat.Therefore,thederivationofthedes iredstateandthenominaldesiredinput basedonthereferenceoutputisfeasible. Thederivationoftheerrordynamicsandtheselectionofthe desiredstatesandinputsisgoing totakeplacesimultaneously.Thebasicideaoftherecursiv eprocedureistostartfromthetopstate equationsofthesubsystemandgraduallyderivethedesired statevariablesandtheerrordynamics ofeachlevelbymovingdownwardsineachstep,untilthebott omsetofstateequationsisreached. Ineachstepthedesiredvaluesofthestatevariablesoflowe rlevelsischoseninsuchawaythat theycanceloutthedesiredvaluesofstatevariablesofhigh erlevels. Theprocedurebeginsbyderivingtheerrordynamicsofthetr anslationalvelocityvariables. Therefore,onehas: e u =_ u u d = u d + X u ( e u + u d ) | {z } u g ( e + d ) | {z } = u d + X u u d g d + X u e u ge (6.28) e v =_ v v d = v d + Y v ( e v + v d ) | {z } v + g ( e + d ) | {z } = v d + Y v v d + g d + X u e v + ge (6.29) Thedesiredpitchandandrollanglesarechosensuchthatthe ycanceloutthevalues u d u d and v d v d ,respectively.Moreprecisely: d = 1 g [_ u d X u u d ] d = 1 g [_ v d Y v v d ] (6.30) Thechoiceofthedesiredtranslationalvelocitycomponent sis u d = u r and v d = v r suchthat when: lim t !1 rr [ e u e v ] T rr =0 then lim t !1 k y ll ( t ) y r ll ( t ) k =0 (6.31) 104

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Itisapparentthatthedesiredanglesof(6.30)arefunction sofonlythe y r ll vectorcomponentsand theirrstderivatives(i.e. d := w (_ u r ;u r ) and d := w (_ v r ;v r ) ).Theparticularchoiceof (6.30)isalsophysicallymeaningfulsinceitindicatestha tthedesiredattitudeisproportionaltothe referenceaccelerationandvelocity.Withtheabovechoice ofthedesiredrollandpitchangles,the translationalvelocityerrordynamicsbecome: e u = X u e u ge (6.32) e v = Y v e v + ge (6.33) Theattitudeangleserrordynamicsare: e = _ d = d +( e q + q d ) | {z } q = d + q d + e q (6.34) e = _ d = d +( e p + p d ) | {z } p = d + p d + e p (6.35) Thedesiredvaluesofthepitchandrollangularvelocitiesa rechosensuchthatthecanceloutthe effectof d and d .Therefore: q d = d p d = d (6.36) Therollandpitchattitudeerrordynamicsbecome: e = e q (6.37) e = e p (6.38) 105

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Similarly,theangularvelocityerrordynamicsare: e q =_ q q d = q d + M u ( e u + u d ) | {z } u + M v ( e v + v d ) | {z } v + M a ( e a + a d ) | {z } a = q d + M u u d + M v v d + M a a d + M u e u + M v e v + M a e a (6.39) e p =_ p p d = p d + L u ( e u + u d ) | {z } u + L v ( e v + v d ) | {z } v + L b ( e b + b d ) | {z } b = p d + L u u d + L v v d + L b b d + L u e u + L v e v + L b e b (6.40) Thevaluesofthedesiredappingangles a d and b d arechosenas: a d = 1 M a [_ q d M u u d M v v d ] b d = 1 L b [_ p d L u u d L v v d ] (6.41) Hence,theangularerrorvelocitydynamics,become: e q = M u e u + M v e v + M a e a (6.42) e p = L u e u + L v e v + L b e b (6.43) Finally,theappingangleserrordynamics,are: e a =_ a a d = a d ( e q + q d ) | {z } q 1 f ( e a + a d ) | {z } a + A b ( e b + b d ) | {z } b + v lon = a d q d 1 f a d + A b b d e q 1 f e a + A b e b + v d lon + v fb lon (6.44) e b = b b d = b d ( e p + p d ) | {z } p 1 f ( e b + b d ) | {z } b + B a ( e a + a d ) | {z } a + v lat = b d p d 1 f b d + B a a d e p 1 f e b + B a e a + v d lat + v fb lat (6.45) Thecomponentsofthecontrolvector v d ll arechosensuchthattheycanceloutthetermsofall thedesiredstatevaluesandonlytheerrorstatevariablesr emaintotheappingerrordynamic 106

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equations.Thus: v ds lon =_ a d + q d + 1 f a d A b b d v ds lat = b d + p d + 1 f b d B a a d (6.46) Itiseasytoverifythatthederivedpair ( x dll ;v d ll ) satisesthedifferentialequationof(6.26).The componentsof x dll and v d ll arecomposedbythereferencevalues u r and v r andtheirhigherderivativesuptothefourthorder.Thereforethecomponentsof y r ll shouldbelongto C 4 .Thenalform ofthelongitudinal-lateralsubsystemerrordynamicsis: e ll = A fbll e ll + B ll v fb ll Y ll = e ll (6.47) Y m ll = C m ll e ll where: e ll =[ e u e v e e e q e p e a e b ] T Y m ll =[ e u e v e e e q e p ] T Theinitialtrackingproblemofthelongitudinalandlatera ldynamicshasbeenconvertedto thestabilizationproblemoftheerrorvector e ll .Themeasurementvector Y m ll doeshaveavailable allthestatevariablesofthesystem(6.47)sincetheappin gangles a and b cannotbemeasured. Whenthecompletestatevectorofasystemisnotavailablefo rfeedbackpurposesandonlya subsetofthestatevariablescanbeusedbythecontroller,t henthecontrollawisclassiedasan outputfeedbackcontroller.Inparticular,insteadofinte gratingintheinitialsystemthedynamics ofastateestimator,werequireastaticfeedbackcontrolla woftheform: v ll = K ll Y m ll (6.48) 107

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suchthatfortheclosedloopsystem: e ll =( A fbll B ll K ll C m ll ) e ll (6.49) theclosedloopmatrix A clll = A fbll B ll K ll C m ll isHurwitz.AsquarematrixiscalledHurwitzifall ofitseigenvalueshavestrictlynegativerealparts. Averygoodstudyoftheoutputfeedbackproblemisgivenin[9 9]and[100].Stabilization viaoutputfeedbackcanbeachievedbytwoways:Eigenvaluep lacementandinthecontextofthe LinearQuadraticRegulator(LQR).Theeigenvalueplacemen tapproach,typicallyrequiresthe solutionofverycomplicatedheuristicalgorithmsforthec alculationoftheoutputfeedbackgain. ForthisreasonweadopttheLQRapproach.Inthiscase,theob jectiveistochose K ll of(6.48) suchthat A clll isHurwitzand,inaddition,thegainselectionminimizesth efollowingquadratic performanceindex: J ll = Z 1 t 0 e Tll Q ll e ll + v fb ll T R ll v fb ll dt (6.50) where Q ll 0 (positivesemi-denite)and R ll > 0 (positivedenite)arediagonalmatrices.The Q ll and R ll matricesarethedesignparametersoftheLQRcontroller.Th eprincipleoftheoptimalityproblemistoregulatethestateerrorvectortozero,wit htheleastpossiblestatedeviationand controlenergy.Thetradeoffbetweencontrolenergyandsta tedeviationisspeciedbytherelative valuesof Q ll and R ll .Foralargerweightingmatrix R ll ,thecontrolinputisforcedtobesmaller inmagnituderelativetothestatenorm.Contrary,alarger Q ll matrix,requiresthattheerrorstate vectordeviateslessfromzerobyinjectingmorecontrolene rgytothesystem. TheLQRcontrollerdesignforLTIsystemswithoutputfeedba ckwasinitiallyintroducedin [59].Thenecessaryconditionforthesolutionoftheaboveo ptimalityproblemistheexistence ofthreematricesnamely, K ll S ll and P ll ,whicharesolutionstothefollowingcoupledequations [59,74]: 0= A clll T S ll + S ll A clll + Q ll +( C m ll ) T K T ll R ll K ll C m ll (6.51) 0= P ll A clll T + A clll P ll + I 8 (6.52) 108

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0= R ll K ll C m ll P ll ( C m ll ) T B T ll S ll P ll ( C m ll ) T (6.53) Generally,optimalcontrolwithoutputfeedback,resultsi nsuchcouplednonlinearmatrix equations[60].Thereareseveraliterativealgorithmsfor thesolutionoftheaboveproblem.However,themostpracticalconvergentalgorithmthatresults inalocalminimumsolutionisgivenin [60]basedon[74].Theiterativealgorithmisthefollowing : Step1:Initializetheiterationprocedurebysetting n =0 .Determineaninitialgain K ll; 0 suchthatthe A clll; 0 = A fbll B ll K ll; 0 C m ll isHurwitz. Step2( n -thiteration):Set A clll;n = A fbll B ll K ll;n C m ll .Solvefor S n and P n thefollowing Lyapunovequations: 0= A clll;n T S n + S n A clll;n + Q +( C m ll ) T K T ll;n R ll K ll;n C n ll 0= P n A clll;n T + A clll;n P T n + I Set J ll;n = tr ( S n ) andevaluatethegainupdatedirection: K = R 1 ll B T ll S n P n ( C m ll ) T C m ll P n ( C m ll ) T 1 K n Updatethefeedbackgainby: K ll;n +1 = K ll;n + K Intheaboveequationchose 2 (01] suchthattheclosedloopmatrix A clll;n isHurwitzand: J ll = k J ll;n +1 J ll;n k = k tr ( S n +1 ) tr ( S n ) k where isaverysmallnumber.If J ll proceedtoStep3,elseset n = n +1 andrepeat Step2. 109

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Step3:Terminatethealgorithmbysetting K ll = K ll;n +1 and J ll = J ll;n +1 Thedisadvantageofthespecicnumericalalgorithm,isthe requirementtoguessaninitialstabilizinggain K ll; 0 ,attherststepofthealgorithm.Apracticalsolutiontoth isproblemistoinitially calculatethestatefeedbackgainbyaregulareigenvaluepl acementalgorithm.Then,omitthe entriesthatcorrespondtotheunmeasuredstates,anduseth erestofthegaincomponentsthat correspondtothemeasurestatesastheinitialoutputfeedb ackgain K ll; 0 .Theabovealgorithmwas presentedbecausestandardsoftwarepackagessuchas MATLAB donotincludebuilt-inroutines forthecalculationoftheoutputfeedbackgain.Contrary, MATLAB providesacompletesetof algorithmsforthesolutionofgeneralizedLyapunovequati onsandtheextractionoffullstate feedbackgainsviaeigenvalueplacementorperformanceind exoptimization. 6.4.2Yaw-HeaveDynamics ThegoalofthisSectionisthedesignofthesecondcontrolla w,responsiblefortheheading andverticalvelocitytracking.Theyaw-heavedynamicssub system,issummarizedbythefollowingequations: x yh = A yh x yh + B yh v yh + D yh x ll y yh = C yh x yh (6.54) y m yh = x yh where: x yh =[ rw ] T v yh =[ v ped v col ] T y yh =[ w ] T Intheaboveequations, y yh istheoutputvectorand y m yh isthemeasurementvector.Thereference 110

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_ r = N r r + N w w + N v v + N p p + v ped w = Z w w + Z r r + Z a a + Z b b + v col = r w r x ll v ped v col Figure6.3:Interconnectionoftheyaw-heavehelicopterdy namicssubsystem.Theyaw-heave dynamicsareadditionallyperturbedbythelongitudinal-l ateraldynamicsstatevector x ll outputisdenotedby y r yh =[ r w r ] T .Theyaw-heavesubsystemisincascadeconnectionwith thelongitudinal-lateralsubsystemviathematrix D yh .Theinterconnectionoftheyaw-heavesubsystemdynamicsisshowninFigure6.3.Thedesignprocedure issimilarwiththeonepresentedin Section6.4.1.Thecontrollerdesignrequiresthedetermin ationofadesiredstatevector x dyh anda desirednominalcontrolinput v d yh ,suchthatwhentheerror e yh = x yh x dyh isregulatedtozero, thentheoutput y yh oftheyawheavesubsystemasymptoticallytrackstherefere nceoutputvector y r yh .Theobviouschoiceofthedesiredyawandheavevelocityis d = r and w d = w r .Thus, when: lim t !1 rr [ e e w ] T rr =0 then lim t !1 rr y yh ( t ) y r yh ( t ) rr =0 (6.55) Thecontrollawfortheyaw-heavesubsystem,isobtainedast hefollowingsuperposition: v yh = v d yh + v fb yh = 264 v d ped v d col 375 + 264 v fb ped v fb col 375 (6.56) Thechoiceofthecontrollercomponent v d yh andthedesiredstatevector x dyh shouldsatisfy: x dyh = A yh x dyh + B yh v d yh + D yh x dll (6.57) wherethestatevector x dll isdenedinSection6.4.1.Theinput v d yh andthedesiredstate x dyh ,are derivedbyusingasimilarrecursivebacksteppingprocedur ewiththeonedescribedinSection6.4.1. 111

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Thechoiceof v d yh and x dyh componentsemergefromtheinspectionoftheerrorvector e yh = x yh x dyh dynamics.Theerrordynamicsoftheyaw-heavesubsystemare givenby: e = _ d = d +( e r + r d ) | {z } r = d + r d + e r (6.58) e r =_ r r d = r d + N ( e v + v d ) | {z } v + N p ( e p + p d ) | {z } p + N w ( e w + w d ) | {z } w + N r ( e r + r d ) | {z } r + v ped = r d + Nv d + N p p d + N w w d + N r r d + Ne v + N p e p + N w e w + N r e r + v d ped + v fb ped (6.59) e w =_ w w d = w d + Z a ( e a + a d ) | {z } a + Z b ( e b + b d ) | {z } b + Z r ( e r + r d ) | {z } r + Z w ( e w + w d ) | {z } w + v col = w d + Z a a d + Z b b d + Z r r d + Z w w d + Z a e a + Z b e b + Z r e r + Z w e w + v ds col + v fb col (6.60) Thedesiredangularvelocity r d ,andthecomponentsof v d yh ,arechosensuchthattheycancelout allthetermsassociatedwiththerestdesiredstatevariabl esandonlytheerrortermsremaintothe yaw-heavesubsystemerrordynamics.Thus: r d = d (6.61) v ds ped =_ r d Nv d N p p d N w w d N p p d (6.62) v ds col =_ w d Z a a d Z b b d Z r r d Z w w d (6.63) Basedontheabovechoice,itiseasytoverifythat(6.57)iss atised.Thedesiredstatevector x dyh andthecontrolinput v d yh arefunctionsofthecomponentsofthe y r yh y r ll vectorsandtheir higherderivatives.Moreover, r and w r shouldbelongto C 2 and C 1 ,respectively.Thedependenceof v d yh tothecomponentsof y r ll stemsfromtheinterconnectionofthetwosubsystemsthroug h thematrix D yh .Usingtheequationsgivenin(6.61)-(6.63),theerrordyna micsoftheyaw-heave 112

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subsystembecome: e yh = A yh e yh + B yh v fb yh + D yh e ll Y yh = e yh (6.64) Y m yh = e yh where: e yh =[ e e r e v ] T Intheaboveequations Y m yh denotesthevectorofavailablemeasurements.Similarlywi ththelongitudinallateralsubsystem,thetrackingproblemof y r yh isconvertedtotheregulationof e yh tozero.However,intheparticularcase,thefullstatevectorofthesys temin(6.64)isavailableforfeedback. Thedesignobjectiveistodetermineastaticfeedbacklaw v fb yh oftheform: v fb yh = K yh e yh (6.65) suchthattheclosedloopstabilitymatrix A clyh = A yh B yh K yh oftheyaw-heaveerrorsubsystem isHurwitz.Asitwillbeillustratedlaterifthiscondition issatised,thesolutionofthecomplete errordynamicsisGASgiventhat A clll isHurwitzaswell. Sincefullstatefeedbackisavailable,thereisavarietyof optionsfordeterminingthefeedback gain K yh .Therstchoiceforcalculating K yh isviatheLQRmethod.Similarlywiththeoutput feedbackcase, K yh iscalculatedsuchthat A clyh isHurwitz,andthegainselectionminimizesthe followingperformanceindex: J yh = Z 1 t 0 e Tyh Q yh e yh + v fb yh T R yh v fb yh dt (6.66) Intheaboveequality, Q yh 0 and R yh > 0 arediagonalmatricesofappropriatedimensions. Likewiseto Q ll and R ll ,thematrices Q yh and R yh arechosenbythedesignersuchthatane 113

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balancebetweenthesystemresponseandthecontrolefforti sachieved.Inthecaseoffullstate feedback,theparticularoptimizationproblemismucheasi erthanitsoutputfeedbackcounterpart. Thecontrollerstatefeedbackgainisgivenby: K yh = R 1 yh B T yh P yh (6.67) wherethematrix P yh isthesolutionofthe algebraicRiccatiequation : 0= P yh B yh R 1 yh B T yh P yh Q yh P yh A yh A Tyh P yh (6.68) ThesolutionofthealgebraicRiccatiequation,isprovided by MATLAB byusingthe care.m built-inroutine.Adifferentapproachistodeterminethef eedbackgain K yh bydirecteigenvalue placement.Theadvantageofthismethodisthattheeigenval uepositionprovidesaquantitative perceptionofthesystem'sresponse. MATLAB providesthe place.m built-inroutine,foraccurateeigenvalueplacementwithfullstatefeedbackforMIMO systems. 6.4.3StabilityoftheCompleteSystemErrorDynamics InSections6.4.1and6.4.2,wehavegivenadetailedpresent ationofhowtodenethefeedbackgainmatrices K ll and K yh ,suchthatthethecloseloopmatrices A clll = A fbll B ll K ll C m ll and A clyh = A yh B yh K yh areHurwitz.Byapplyingthecontrollaws v fb ll and v fb yh ,thecompleteerror systemdynamicstaketheform: 264 e yh e ll 375 = 264 ( A yh B yh K yh ) D yh 0 8 3 ( A fbll B ll K ll C m ll ) 375 264 e yh e ll 375 (6.69) Thecascadeconnectionoftheclosedlooperrordynamicsiss howninFigure6.4.Thestabilityof thecompleteerrordynamicssystemgivenin(6.69),isspeci edbythefollowingTheorem: 114

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_ e yh = A clyh e yh + D yh e ll Yaw-Heavesubsystem e ll = A clll e ll Longitudinal-Lateralsubsystem e ll Figure6.4:Cascadeconnectionoftheclosedlooperrordyna micssubsystems. Theorem6.1. Giventhatthefeedbackgains K ll and K yh areselectedsuchthatthematrices A clll = A fbll B ll K ll C m ll and A clyh = A yh B yh K yh areHurwitz,thenthesolution e ( t )= [ e yh ( t ) e ll ( t )] ofthecompleteerrordynamicssystemof (6.69) isGAS. Proof. TheproofoftheTheorembeginswithastandardresultfromli nearalgebra.If A2 R n n B2 R m m aresquarematrices,and C2 R n m ,thenthefollowingpropertyholds: det 0B@ 264 AC 0 m n B 375 1CA = det ( A ) det ( B ) where det ( ) denotesthedeterminantofamatrix.Denoteby theeigenvaluesofthecomposite errordynamicssystemof(6.69).Bydenition,theeigenval uesof(6.69)satisfythefollowing equalities: det 0B@ 264 A clyh I 3 3 D yh 0 A clll I 8 8 375 1CA = det A clyh I 3 3 det A clll I 8 8 =0 Thereforetheeigenvaluesofthecompositeerrorsystem,ar etheunionoftheeigenvaluesof A clyh and A clll .SincebothofthosematricesareHurwitz,thenalltheeigen valuesof(6.69)havestrictly negativerealparts.Thereforethecompleteerrordynamics systemof(6.69)isGAS. 115

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6.5PositionandHeadingTracking Theultimategoalofthecontrollerdesignisforthehelicop tertotrackapredenedposition trajectoryoftheinertialframeexpressedbythereference vector p Ir =[ p Ir;x p Ir;y p Ir;z ] T .The helicopterpositionexpressedinthebody-xedframe,isde notedbythecoordinatevector p B = p Bx p By p Bz T .Thepositionerrorexpressedinthebody-xedframeisgive nby e Bp = p B p Br Thepositionerrordynamicsarederivedbyusingthepropert iesoftherotationmatrix R ,described inChapter3.Therotationmatrixisusedformappingcoordin atevectorsfromthebody-xed frametotheinertialframe.Forthepositionerrorexpresse dinthebody-xedframethefollowing equalitieshold: e Bp = p B p Br = R T p I R T p Ir (6.70) UsingtheanalysisofChapter3,thepositionerrordynamics aregivenby: e Bp = R T (_ p I p Ir )+ R T ( p I p Ir ) = R T ( v I v I d )+( R ^ B ) T ( p I p Ir ) = v B v B d +(^ B ) T ( p B p Ir ) = e Bv ^ B e Bp = e Bv +^ e Bp B (6.71) Forderivingthepositionerrordynamicswehaveusedthefol lowing: v I d =_ p Ir v I =_ p I R = R ^ B ^ B e Bp = ^ e Bp B (6.72) Thepositionerrordynamicsarenotlinearsincetheyinclud ethenonlinearterm ^ e Bp B .The lattertermexpressesthecontributionoftheangularveloc itytothepositionerrordynamics. Thechoiceofalinearmodelfortherepresentationofthehel icopterdynamicsislimitedto acertainrangeofaparticularoperatingcondition.Inthis case,theoperatingconditionofinter116

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estisthehoverightmode.Sincethelinearmodelof(6.2)is restrictedtoacertainrangeofthe hovermode,thetrackingproblemofarbitrarypositionandv elocitytrajectoriesbecomesdubious.However,experimentalresultsofreallifeapplicatio nsindicatethattheaccuracyoflinear dynamicmodelsissatisfactoryenoughforarelativewidera ngeoftheightenvelopearound thereferenceoperatingcondition.Therefore,itisassume dthattheadoptedlinearmodelof(6.2) providesaquasi-globaldescriptionofthehelicopterdyna mics.Linearizationisalsoappliedtothe nonlinearpositionerrordynamics,assumingthat e Bp istheperturbedvalueofthepositionerror fromthereferencesteadystatevector e Bp;o =[000] T .Similarly, B isconsideredastheangular velocity'sperturbedvaluefromthetrimvector B o =[000] T .Inthiscase,theterm ^ e Bp B canbe disregardedsinceitisconsideredasaproductoftwopertur bedvalues 1 .Thisapproximationadds uptoallsimplicationassumptionsthattakeplaceinorder toobtainthelineardynamicmodelof thehelicoptergivenin(6.2).Therefore,theapproximated positionerrordynamicsaregivenby: e Bp = e Bv (6.73) Thecompositeerrorsystemisadditionallyenhancedbythei ntegralofthepositionandyaw errordynamics.Thepresenceofintegraltermsinthecontro llawisverybenecialintermsof robustnessperformance.Thefeedbackintegralcomponents attenuatethesteadystatetracking errorcausedbypotentialparametricandmodeluncertainty .Denoteby B p =[ B x B y B z ] T and theintegralofthepositionandyawerror.Thus: B p = e Bp and = e (6.74) Thestructureofthecontrollawsforthepositiontrackingp roblemwillbeidenticaltothe velocitytrackingcase.Thecompositeerrordynamicsarest illseparatedintotwosubsystemscorrespondingtothelateral-longitudinalandyaw-heavemoti on.Havingsaidthat,thelongitudinal1 MoredetailsaboutlinearizationmaybefoundinSection5.8 117

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lateraldynamicsaregivenby: ll = A ll ll + B ll v fb ll (6.75) Y m ll = C m ll ll where: ll = B x B y e Bx e By e u e v e e e q e p e a e b T Y m ll = B x B y e Bx e By e u e v e e e q e p T and: A ll = 264 0 4 2 I 4 4 0 4 6 0 8 2 0 8 2 A fbll 375 B ll = 264 0 4 2 B ll 375 (6.76) Theyaw-heaveerrordynamicsaregivenby: yh = A yh yh + B yh v fb yh + D yh ll (6.77) Y m yh = yh where: yh =[ B z e Bz e e w e r ] T and: A yh = 264 0 3 2 I 3 3 0 3 1 0 3 2 0 3 1 A yh 375 B yh = 264 0 3 2 B yh 375 D yh = 264 0 3 3 O 3 8 0 3 4 D yh 375 (6.78) Theinterconnectionofthenewcompleteerrordynamicssubs ystemsisillustratedinFigure6.5. Similarlytothevelocitytrackingcase,thecontroldesign isreducedtothecalculationoftwo 118

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_ yh = A yh yh + B yh v fb yh + D yh ll Yaw-Heavesubsystem ll = A ll ll + B ll v fb ll Longitudinal-Lateralsubsystem ll Figure6.5:Cascadeconnectionoftheerrordynamicssubsys temsrelatedwiththeposition trackingproblem.feedbackgainmatrices K ll and K yh ,suchthatbyapplyingthefollowingfeedbackcontrollaws: v fb ll = K ll Y m ll (6.79) v fb yh = K yh Y m yh (6.80) theclosedloopmatrices A clll = A ll B ll K ll C m ll and A clyh = A yh B yh K yh areHurwitz.The feedbackgainscanbecalculatedbyperformingthemethodol ogiesdescribedinSections6.4.1and 6.4.2.Forexample,followingtheLQRmethodthegainsarese lectedsuchthattheyminimizethe followingquadraticperformanceindexes: J ll = Z 1 t 0 Tll Q ll ll + v fb ll T R ll v fb ll dt (6.81) J yh = Z 1 t 0 Tyh Q yh yh + v fb yh T R yh v fb yh dt (6.82) However,inordertofollowtheLQRoreigenvalueplacementm ethodologies,thepairs ( A ll ; B ll ) and ( A yh ; B yh ) mustbecontrollable.Thenecessaryconditionforcontroll abilityofthepairs ( A ll ; B ll ) and ( A yh ; B yh ) isestablishedbythefollowingTheorem: Theorem6.2. GiventhatAssumptions6.1,6.2and6.3hold,thenthepairs ( A ll ; B ll ) and ( A yh ; B yh ) arecontrollable. 119

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Proof. BasedonAssumptions6.1and6.2,thepair A fbll ; B yh iscontrollable.Let T ( s )= h sI 8 A fbll j B ll i where s 2 R .FromthePopov-Belevitch-Hautus(PBH)test,forevery s 2 R wehave rank ( T ( s ))=8 .Weneedtoshowthat rank ( T ( s ))=12 forevery s 2 R ,where T ( s )=[ sI 12 A ll jB ll ] For s 6 =0 onehas: rank ( T ( s ))= rank 0BBBB@ 266664 sI 2 I 2 0 2 2 0 2 6 0 4 2 0 2 2 sI 2 I 2 0 2 6 0 2 2 0 2 2 A fbll B ll 377775 1CCCCA Since s 6 =0 ,therstfourrowsarelinearlyindependent.Therefore: rank ( T ( s ))=4+ rank h A fbll j B ll i =4+8=12 For s =0 onehas: rank ( T (0))= 0BBBB@ 266664 0 2 2 I 2 0 2 2 0 2 6 0 4 2 0 2 2 0 2 2 I 2 0 2 6 0 2 2 0 2 2 A fbll B ll 377775 1CCCCA Thersttworowsarelinearlyindependent.Therefore: rank ( T (0))=2+ 0B@ 264 I 2 0 2 6 0 2 2 A fbll B ll 375 1CA Thematrixoftherighthandsideoftheaboveequation,issqu areandlowertriangularwith nonzeroelementsinitsmaindiagonal(thisfactisguarante edbyAssumption6.3).Hence, therankofthismatrixis10and rank ( T (0))=12 Wehaveprovedthatforevery s 2 R ,wehave rank ( T ( s ))=12 .Thereforegiventhatthe pair ( A fbll ; B ll ) iscontrollable,thenthepair ( A ll ; B ll ) iscontrollableaswell.Theproofforthe 120

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controllabilityof ( A yh ; B yh ) basedonthecontrollabilityofthepair ( A yh ; B yh ) isderivedsimilarly totheaboveanalysis. Byapplyingthecontrollaws v fb ll = K ll Y m ll and v fb yh = K yh Y m yh ,thecompleteerrorsystem dynamicstaketheform: = A cl" (6.83) where: = 264 yh ll 375 A = 264 ( A yh B yh K yh ) D yh 0 8 3 ( A ll B ll K ll C m ll ) 375 (6.84) Thestabilityofthecompleteerrordynamicssystemof(6.83 )isestablishedbythefollowingTheorem:Theorem6.3. Giventhatthefeedbackgains K ll and K yh areselectedsuchthatthematrices A clll = A ll BK ll C m ll and A clyh = A yh B yh K yh areHurwitz,thenthesolution ( t )=[ yh ( t ) ll ( t )] ofthecompleteerrordynamicssystemin (6.83) ,isGAS. Proof. TheproofisderivedsimilarlytoTheorem6.1.Theeigenvalu esof(6.83)havestrictlynegativerealpartsbasedonthedeterminantpropertyofsquare matricesinblocktriangularform. 6.6PIDControl InmanypracticalcontrolapplicationstheMIMOdynamicmod elofthehelicopterisnotavailable.InthisSectionwepresentafundamentalcontrollerco mposedbyfourSISOProportional IntegralDerivative(PID)feedbackloops.Thiscontrolsch emeisaverycommonstartupdesign pointinreallifeapplications,sinceitdoesnotrequireth eknowledgeofthehelicoptermodeland thecontrollergainscanbeempiricallytuned. Thedesignofthecyclicfeedbackloopsisbasedonthesimple factthatthelongitudinaland lateralvelocityofthehelicopterisproducedfromthepitc handrolltiltofthefuselage.Therefore, thehelicoptervelocityisconsideredproportionaltotheh elicopterattitude[70].Thestructureof thefeedbacklawiscomposedbytwomainloops:The innerloop andthe outerloop .Theinner 121

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loopregulatesthehelicopterattitudetothedesiredangle s des and des .Thefeedbacksignalof theinnerloopisproportionaltotheattitudeerror.Theout erloopgeneratesthedesiredattitude angles.Thedesiredpitchandrollanglesareproportionalt othepositionandvelocityerrorinthe longitudinalandlateraldirections,respectively.Thecy cliccommandsaregivenby: u lon = K ( des )= K ( K ;x B x K x e Bx K u e u ) (6.85) and: u lat = K ( + des )= K ( + K ;y B y + K y e By + K v e v ) (6.86) Inorderfortheabovefeedbacklawtoperformwell,theattit udeerrorshouldberegulatedtozero fasterthanthetranslationalerror.Todoso,thecontrolla wgainsshouldbechosenappropriately suchthatadistincttimescalingisachievedbetweentheatt itudedynamicsandthetranslational dynamics.Thepedalandcollectivefeedbackloopsaremored irectthanthecyclicloops.Each ofthemiscomposedsolelyfromtheyawandheaveerrorandthe ircorrespondingvelocityerror. Thereforethepedalandthecollectiveinputaregivenby: u ped = K ; K e K r e r (6.87) and: u col = K ;z B z K y e Bx K v e v (6.88) ThePIDcontroldesigndoesnottakeintoconsiderationthec rosscouplingeffectthatusuallyexistsinthehelicopterdynamics.Therefore,thefourclosed loopsarecompletelyindependentwith eachother.Thegainsofthecontrolfeedbacklooparetunedb ysimpletrialanderror.Thegain tunningprocedurecanbesignicantlyimprovedbytheknowl edgeofasimplenon-parametric 122

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modelofthehelicopter.Thenon-parametricmodelcanbeext ractedwiththemethodologiesdescribedinChapter5.6.7ExperimentalResults Theperformanceoftheproposedlineartrackingcontroller andthePIDdesignisevaluated usingthe Raptor90SE RChelicopterinthe X-Plane simulator.DetailsabouttheRaptormodel and X-Plane canbefoundinSection5.10.1.Thestabilityandcontrolder ivativesoftheRaptor's linearmodelaregiveninTable5.4.Bothcontrollerperform ancewastestedbytheexecutionofa velocitytrackingmaneuver.Thedesiredmaneuverisatrape zoidalvelocityproleinthelateral andlongitudinaldirectionsoftheinertialspace.Through outthemaneuverthedesiredheading remainsconstantwiththevalue d =0 .Thelineartrackingcontroller'sgainsof(6.79)-(6.80) areshowninTable6.1.ThePIDgainsaregiveninTable6.2.Th econtrollerresponsesversusthe desiredtrajectoryareillustratedinFigure6.6.Thepitch ,rollandyaworientationanglesforthe twocontrollersaredepictedinFigure6.7.Thepositionoft hehelicopterintheinertialcoordinates isgiveninFigures6.8and6.9.Finallythecontrolinputsfo rthetwodesignsaregiveninFigures 6.10and6.11. Basedontheresults,theperformanceofbothcontrollerdes ignswassatisfactory.Although thereferencetrajectoryrequiresthatthehelicopterexec utesacruisingmaneuver(longitudinal velocityupto 17 m=sec andlateralvelocityupto 3 m=sec )asinglelinearcontrollerbasedonly onthehoverlinearmodel,wasadequate.Tothisextent,thei denticationofmultiplemodelsfor differentoperatingconditionswasredundant.Itwasexpec tedthatthePIDperformancewould beinferiortothelineardesign,howevertheightresultsi ndicatethatboththedesignsprovided equallysuccessfulresults.ThesuccessofthePIDcontroll erisattributedtotheattenuatedcross couplingeffectamongsttheRaptordynamics.Thisfactissu pportedbytheoff-axisresponsesof thehelicopterillustratedinFigure5.3.Themagnitudeoft he q=u lat and p=u lon responseslieinthe zoneof 20 to 40 dB .Thisisanindicatorofnegligiblecrosscouplingbetweent hehelicopter dynamics. 123

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6.8Remarks ThisChapterhaspresentedaposition(orvelocity)andhead ingtrackingcontrollerforsmall scalehelicopters.Theanalysisisrestrictedtothisclass ofrotorcraftbecausetheadoptedgeneric linearmodel,towhichthecontrollerisbasedon,maybeinad equateforfullscalehelicopters. Modelsforfullscalehelicoptersareinprincipleofhigher orderbyincludingadditionaldynamics suchasconing,enginedynamicsandotheraerodynamiceffec tsliketheinowvelocity'sdynamics.Thelineardesignisbasedonthelinearizedhelicopter dynamicsaroundhover.Thedesigncan beexpandedsuchthattheoverallcontrollawcanbeaninterp olatorofmultiplecontrollerswhere eachofthemcorrespondstoalinearmodelofadifferentoper atingconditionofthehelicopter. Itisimportanthoweverthatallofthelinearizedmodelshav ethesamestructureandorderwith thebasehovermodelandonlytheirparametersmayvary.Inad dition,itisimportantthatforall thelinearmodels,itisphysicallymeaningfultobeapproxi matedbyasystemofstrict-feedback formsuchthattheprincipleofdifferentialatnessholds. Theoutputfeedbackcontrollers v fb ll and v fb yh arenotrestrictedonlytotheproposeddesignsofthisChapt erbuttheycouldbechosenfrom awidevarietyoflinearcontrollerdesignsthatexistinthe literature.Tothisextent,thepopular methodof H 1 maybealsoapplied.Thesuggestedoutputfeedbackcontroll awsofthisChapter areonlyindicatorsforastraightforwarddesign. Toeliminatethenecessityofmultiplelinearmodelsasingl enonlinearmodelshouldbeused leadingtoanonlinearcontrollerdesign.Thisisthegoalof thenextChapterwhereanonlinear backsteppingcontrollerisproposedbasedonthenonlinear helicopterdynamics.Thehelicopter dynamicsarebasedonthecompletenonlinearequationsofmo tionsenhancedbyasimplied modelofthemainandtailrotorforcesandmomentsgeneratio n. 124

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Table6.1:Lineartrackingcontrollerfeedbackgains. K ll = 24 1 : 91870 : 4710 4 : 37111 : 0374 3 : 13530 : 68829 : 80541 : 90410 : 56620 : 2395 0 : 12420 : 6031 0 : 27341 : 3663 0 : 18470 : 96820 : 50382 : 96870 : 0632 0 : 5391 35 K yh = 24 0042010 : 94510 0006001 35 Table6.2:PIDcontrollergains. K 0.7566 K y 0.3252 K ;x 0 K v 0.2493 K x 0.3256 K ; 0 K u 0.1628 K 3 K 0.4569 K r 0.35 K ;y 0 K ;z 0 K w 0.6060 K z 1.6018 0 10 20 30 40 50 60 70 10 0 10 20 vx I (m/sec) 0 10 20 30 40 50 60 70 2 0 2 4 vy I (m/sec) 0 10 20 30 40 50 60 70 5 0 5 time (sec)vz I (m/sec) Figure6.6:Referencetrajectory(solidgreenline),actua lpositiontrajectoryofthelinear(green dashedline)andPID(dashed-dottedredline)designs,expr essedininertialcoordinateswith respecttotime. 125

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0 10 20 30 40 50 60 70 0.5 0 0.5 q (rad) 0 10 20 30 40 50 60 70 0.2 0.1 0 0.1 f (rad) 0 10 20 30 40 50 60 70 1 0 1 2 y (rad)time (sec) Figure6.7:Orientationanglesofthelinear(solidline)an dPID(dashedline)designs. Figure6.8:Referencepositiontrajectory(solidline)and theactualtrajectoryofthelinear(dashed line)designwithrespecttotheinertiaaxis. 126

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Figure6.9:Referencepositiontrajectory(solidline)and theactualtrajectoryofthePID(dashed line)design,withrespecttotheinertiaaxis. 0 10 20 30 40 50 60 70 0.2 0 0.2 ulon 0 10 20 30 40 50 60 70 0.1 0 0.1 ulat 0 10 20 30 40 50 60 70 1 0 1 uped 0 10 20 30 40 50 60 70 1 0 1 ucoltime (sec) Figure6.10:Controlinputsofthelineardesign. 127

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0 10 20 30 40 50 60 70 0.4 0.2 0 u lon 0 10 20 30 40 50 60 70 0.2 0 0.2 u lat 0 10 20 30 40 50 60 70 1 0.5 0 u ped 0 10 20 30 40 50 60 70 1 0 1 u coltime (sec) Figure6.11:ControlinputsofthePIDdesign. 128

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Chapter7:NonlinearTrackingControllerDesignforUnmann edHelicopters ThepreviousChapterpresentedatrackingcontrollerofthe positionandheadingofahelicopterbasedonthelinearizedhelicopterdynamics.Theado ptedparametriclinearmodel,towhich theightcontrollerisbasedon,representedthequasistea dystatebehaviorofthehelicopterdynamicsathover. Reallifecasestudiesindicatethatthevalidityoflinearm odelsisrestrictedonlytoightoperationaroundthetrimpointofreference.Awiderdescript ionoftheightenveloperequiresthe identicationofmultiplelinearmodelswhereeachofthemc orrespondstoadifferentoperating conditionofthehelicopter.Therefore,multiplecontroll ersshouldbedesignedwhereeachofthem isbasedonthelinearmodelofaparticularoperatingcondit ion.Theoutputofoverallcontrollaw isproducedbyaschedulingprocessofthesemultiplecontro llersdependingonthehelicopter's operatingcondition. However,asindicatedinChapter5theexperimentalprocedu refortheextractionoflinear modelsparameters,foroperatingconditionsotherthanhov er,isatediousandinmanycasesunreliableprocess.Theidealsolutiontothisproblemwouldbet hedesignofasinglecontrollerbased onamodelthatprovidesaglobaldescriptionofthehelicopt erdynamics.ThegoalofthisChapter isthedesignofapositionandheadingcontrollawbasedonth enonlinearhelicopterdynamics. Theresultingcontrollaw,fromatheoreticalviewpoint,is validforthecompleteightenvelope andisapplicabletobothfullscaleandsmallscalehelicopt ers. 129

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7.1Introduction Ingeneral,mostcontrollerdesignsarebasedonthelineari zedhelicopterdynamicsusingthe widelyadoptedconceptofstabilityderivatives[25,28,54 –56,89].However,inrecentyearsthere isconsiderableresearchrelatedtohelicopterightcontr olbasedonnonlineardynamicrepresentations[24,30,47,88,91]. ThisChapterpresentsanonlineartrackingcontrollerdesi gnforhelicopters.Themainobjectiveisforthehelicoptertotrackapredened,possiblyagg ressive,positionandyawreference trajectorieswithcertainboundsthatreectthehelicopte r'sphysicallimitations.Thehelicopter modelisrepresentedbytherigidbodyequationsofmotionen hancedbyasimpliedmodelof forceandtorquegeneration.Thehelicopternonlinearmode lisbasedontheworkreportedin[47]. Thecontrollerisbasedonthebacksteppingdesignprincipl eforsystemsinfeedbackform.The intermediatebacksteppingcontrolsignals(a.k.a.pseudo controls)foreachlevelofthefeedback systemareappropriatelychosentostabilizetheoverallhe licopterdynamics.Theresultingsystemerrordynamicscanbeseparatedintwointerconnectedsu bsystemsrepresentingtheerrorin translationalandattitudedynamics,respectively.Thedi stinctionofthetwosubsystemsindicate thetimescalingseparationthatexistsinactualhelicopte rswherethepositiondynamicsaresignicantlyslowerthantheattitudedynamics. Theincorporationofnestedsaturationfeedbackfunctions inthebacksteppingdesignpreserves thehelicopter'smotionandpowerphysicalconstraints.Th eintermediatecontrolsignalsrelatedto theattitudedynamicsexploitthestructuralpropertiesof therotationmatrixandareenhancedwith termsthatguaranteethatthehelicopterwillnotoverturnw hiletrackingthedesiredpositiontrajectory.Theattitudedynamicsarerenderedexponentiallysta blewhilethetranslationaldynamicsare globallyasymptoticallystable.Numericalsimulationsil lustratetheapplicabilityoftheproposed design. 130

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7.2HelicopterNonlinearModel Beforeweproceedwiththehelicopternonlinearmodelweint roducesomemathematicalnotationthatisrequiredforthefollowinganalysis.Theabbrev iations C t and S t with t 2 R represent thetrigonometricfunctions cos( t ) and sin( t ) ,respectively.Theoperands k ( ) k j ( ) j denotethe Euclideannormandthe k ( ) k 1 normofavector,respectively. ThehelicoptermodelconsideredinthisSectioniscomposed bythenonlinearequationsof motionaccompaniedbyasimpliedmodeloftheforcesandmom entsthatareproducedbythe mainandtailrotor.Theseaerodynamicforcesandmomentsar ecomplexnonlinearfunctionsof themotioncharacteristicsandcontrolswhicharedominate dbyhighuncertainty.Detailedmodels ofthehelicopternonlineardynamicscanbefoundin[7,40,8 4].However,suchmodelsareof highorderandimpracticalforthedevelopmentofightcont rollers.InthisSection,thederivation oftheexternalforcesandmomentsthatactonthehelicopter arebasedonthesimpliedmodelof thegeneratedmainrotorthrustthatiscoveredChapter4.7.2.1RigidBodyDynamics Thehelicopterrigidbodynonlinearequationsofmotionhav ebeenalreadyderivedinChapter 3andarebrieyrepeatedhereforclaricationpurposes.Le t p I =[ p Ix p Iy p Iz ] T denotetheposition vectoroftheCGofthehelicopterwithrespecttotheinertia lcoordinates,and v I =[ v I x v I y v I z ] T denotethelinearvelocityvectorininertialcoordinates. Theangularvelocitywithrespecttothe bodyframeis B =[ pqr ] T .BasedonChapter3,thecompleterigidbodydynamicequatio nsof thehelicopterinthecongurationspace SE (3)= R 3 SO (3) are: p I = v I (7.1) v I = 1 m Rf B (7.2) R = R ^ B (7.3) I B = B ( I B )+ B (7.4) 131

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Figure7.1:Thehelicopter'sbody-xedframe,theTip-Path -Planeanglesandthethrustvectorsof themainandtailrotor. Therotationmatrix R isparametrizedwithrespecttothethreeEuleranglesroll( ),pitch ( )andyaw( )andmapsvectorsfromthebodyxedframe F B totheinertiaframe F I .The controllerdesignofthisChaptermakesextensiveuseofthe rotationmatrixsoitscomponentsare repeatedhere: R = 266664 C C S C + C S S S S + C S C S C C C + S S S C S + S S C S C S C C 377775 Theorientationvectorisgivenby =[ ] T andtheassociatedorientationdynamicsare governedby: =() B (7.5) Thecomponentsof () matrixaregivenin(3.25).Thehelicopter'srigidbodydyna micsgiven in(7.1)-(7.4)arecompletedbydeningtheexternalbodyxedframeforce f B andtorque B Thevector F B =[ f B B ] T iscalledtheexternalwrenchthatactsonthehelicopter[75 ]. 7.2.2ExternalWrenchModel ThisChapterfollowsthemodelingapproachof[47,56,70,72 ],whichprovidesasimplied externalwrenchmodeladequateforcontrollerdesignpurpo ses.Mostoftheconseptsassociated 132

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Thrustgeneration ~ T M ; ~ T T ~ f = ~ T M + ~ T T + ~ W B ~ = ~ Q + ~ h M ~ T M + ~ h T ~ T T v I = 1 m Rf B R = R ^ B I B = B ( I B )+ B T M T T a b ~ T M ; ~ T T R f B B Figure7.2:Thisblockdiagramillustratestheconnectiono fthegeneratedthrustsofthemainand tailrotorwiththehelicopterdynamics.Thevector ~ W B representstheweightforceexpressedin thebodyxedframe.withthederivationofthesimpliedexternalwrenchmodelh avebeenalreadycoveredinChapter 4.Themainassumptionisthatthethrustvectorproducedbyt hemainrotorisconsideredperpendiculartotheTPP. Therearefourcontrolinputsassociatedwithhelicopterpi loting.Thecontrolinputvectorin thisChapterisdenedas u c =[ abT M T T ] T .Thecomponents T M and T T arethemagnitudes ofthegeneratedthrustsbythemainandtailrotor,respecti vely.Themagnitudeofthemainand tailrotorthrustisproducedbyauniformchangeinthepitch anglesofthemainandtailrotor blades.Theappingangles a b representthetiltoftheTPPatthelongitudinalandlateral axis, respectively.Thevectorsofthebody-xedframe,theappi nganglesandthethrustvectorsare depictedinFigure7.1. FromSection4.8thecomponentsofthemainrotorthrustvect or ~ T M ,expressedinthebodyxedframe,aregivenby: T B M = 266664 X M Y M Z M 377775 = 266664 S a C b C a S b C a C b 377775 T M 266664 a b 1 377775 T M (7.6) AsindicatedfromSection4.8,theaboveequationissimpli edbyassumingsmallangleapproximation( cos( ) 1 and sin( ) ( ) )fortheappingangles.Thesmallangleassumptionis 133

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adoptedby[40,47,70].Forthebody-xedcomponentsofthet ailrotorthrustvector,onehas: T B T = 266664 0 Y T 0 377775 = 266664 0 1 0 377775 T T (7.7) Therefore,byincludingthehelicopter'sweightthecomple teforcevectoris: f B = 266664 X M Y M + Y T Z M 377775 + R T 266664 00 mg 377775 (7.8) Acommonsimplicationpracticefollowedin[37,47,66]ist oneglecttheeffectofthelateral andlongitudinalforcesproducedbytheTPPtiltandtheeffe ctofthetailrotorthrust.Thoseparasiticforceshaveaminimaleffectonthetranslationaldyna micscomparedtothe Z M component 1 Inthiscase,theonlytwoforcesappliedtothehelicopterar ethemainrotor'sthrustvectoratthe directionof ~ k B ofthebodyframeandtheweightforce.Therefore,(7.8)beco mes: f B = 266664 00 T M 377775 + R T 266664 00 mg 377775 (7.9) Thegeneratedtorquesaretheresultoftheaboveforcesandt herotorsmoments.Denote h BM = [ x m y m z m ] T and h BT =[ x t y t z t ] T asthepositionvectorsofthemainandtailrotorshafts,res pectively(expressedinthebody-xedcoordinateframe).Let ~ M = ~ h M ~ T M and ~ T = ~ h T ~ T T be 1 Theoverrideofthe f B componentsinthe ~ i B and ~ j B directionsofthebody-xedframeachievesthedecouplingo f thehelicopterexternalforceandmomentmodel.Theworkrep ortedin[47]indicatesthatifthecompletedescriptionof theforcevectorgivenin(7.8)isused,thenthestatespaced ynamicsofthenonlinearhelicoptermodelcannotbeinputoutputlinearizableandthezero-dynamicsofthesystemwil lbeunstable.Ifthesystemdynamicsarenotinput-output linearizablemostofthestandardcontrolmethodologieswi llbeinapplicable.Iftheproposedapproximationtakespla ce, thehelicopternonlinearmodelbecomesfullstatelineariz ablebyconsideringthepositionandtheyawasoutputs.To theauthorsknowledgethereisnotanycontrollerdesignint heliteraturethatisbasedontheexactmodelandinallcase studiesthisapproximationisperformed.Theuseoftheappr oximatedmodelalsotookplaceinChapter6indicatingthat forthehelicoptercontrolproblemonlypracticalstabilit ycanbeachievedbasedontheapproximatedmodel. 134

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thetorquesgeneratedby ~ T M and ~ T T ,respectively.Thecompletetorquevectorwillbe: B = B Q + 266664 y m Z M z m Y M z t Y T z m X M x m Z M x m Y M y m X M + x t Y T 377775 (7.10) with B Q =[ R M M M N M ] T .The ~ Q isproducedbythemainrotormomentvector ~ duetothe hubstiffnessandthemainrotoranti-torquedenotedby Q M .Thecomponentsof B M =[ R M M M N M ] T are: R M = K b Q M S a C b M M = K a + Q M S b C a N M = Q M C a C b Q M = C M j T M j 1 : 5 + D M Thepositiveconstants C M and D M areassociatedwiththegenerationofthereactiontorque Q M Adetaileddescriptionof ~ Q canbefoundin[30,47].Figure7.2depictstheassociationo fthe generatedthrustswiththehelicopter'srigidbodydynamic s.Substituting(7.6),(7.7)to(7.10)a morecompactformofthetorquecanbegivenas: B = A ( T M ) v c + B ( T M ) (7.11) where: v c =( abT T ) T (7.12) with A ( T M ) 2 R 3 3 beinganinvertiblematrixforbounded T M and B ( T M ) 2 R 3 1 135

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I B = B ( I B ) + A ( T M ) v c + B ( T M ) Re 3 = R ^ B e 3 v I = 1 m 3 T M + ge 3 p I = v I =() B B 3 v I B T M T T a b TranslationalDynamics AttitudeDynamics Figure7.3:Thisblockdiagramillustratestheinterconnec tionoftheapproximatedhelicopter's dynamics.7.2.3CompleteRigidBodyDynamics Usingtheforcesimplicationassumptiongivenin(7.9)and theappliedtorquegivenby(7.11), thetranslationalandangularvelocityhelicopterdynamic sareexpressedas: v I = 1 m Re 3 T M + ge 3 (7.13) I B = B ( I B )+ A ( T M ) v c + B ( T M ) (7.14) where e 3 =[001] T .Theinterconnectionofthehelicopterdynamicsisshownin Figure7.3.The helicopterdynamicscanbefurtherseparatedintwointerco nnectedsubsystemsrepresentingthe attitudeandthetranslationaldynamics,respectively.7.3TranslationalErrorDynamics Considerahelicopterdescribedbythedynamicequations(7 .1),(7.3)and(7.13),(7.14).The objectiveistodesignacontrollerregulatingposition p I andtheyawangle tothereferencevalues p Ir =[ p Ir;x p Ir;y p Ir;z ] T and r ,respectively.Theproposedcontrollerdesignrequiresth atthe componentsof p Ir andtheirhighertimederivativesarebounded.Thisisanexp ectedrestriction, whichreectsthehelicopter'sphysicalconstraints.Furt hermore,thecontrollerdesignassumes availabilityofallhelicopter'sstatevariablesofthetra nslationalandattitudedynamics.Thecon136

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trollerdesignisbasedonthebacksteppingprocedureforsy stemsinfeedbackform.Adescription ofthebacksteppingmethodologycanbefoundinAppendixA. Let R =[ 1 2 3 ] where i with i =1 ; 2 ; 3 arethecolumnvectorsoftherotationmatrix. Denote i;j theelementofthe j th rowand i th columnoftherotationmatrix.Let e denotethe orientationerrorbetweentheactualdirectionofthethrus tvector 3 ,minusadesireddirection denotedby d =[ d; 1 d; 2 d; 3 ] T .Followingstandardprocedureofthebacksteppingdesign, the translationalerrordynamicsofthehelicoptercanbewritt enas: e p =_ p I p Ir = p Ir + v I d + e v (7.15) e v =_ v I v I d = ge 3 v I d 1 m d T M 1 m e T M (7.16) Theelementsoftheunitaryvector 3 expresstheinertiacoordinatesofthebody'sframevector ~ k B .Theterm 3 T M representsthehelicopter'sthrustforce.Obviously, 3 dictatesthedirection ofthethrustvectorwhile T M itsmagnitude.AsillustratedinFigure7.3,thethrustmagn itude T M isadirectcontrolcommandwhilethedirectionvector 3 isindirectlymanipulatedbytheattitude dynamics.Thetranslationalerrordynamicssubsystemissh owninFigure7.4. Themaindesignideaofthisstepistochoosethedesiredvelo citydynamics v I d ,thedesired directionandmagnitudeofthethrustvector( d and T M ,respectively)insuchawaysothatthe translationalerrordynamicswillbegloballyasymptotica llystable(GAS)bydisregardinginitially theeffectof e .Theresultingtranslationalerrordynamicssubsystemcan beviewedasGASnominalsystemperturbedbytheorientationerror e .Asitwillbeillustrated,theproposedchoiceof v I d ; d ;T M followedbytheexponentialstabilityoftheorientationer ror e ,willguaranteethatthe completetranslationalerrordynamicswillbeuniformlygl oballyasymptoticallystable(UGAS) foranyinitialconditionofthepositionandtranslational velocity. Thefollowingdesiredvalueswillbechosen: v I d =_ p Ir (7.17) 137

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_ e v = ge 3 v I d 1 m d T M 1 m e T M e p = p Ir + v I d + e v v I d T M d 3 e e v Figure7.4:Thisblockdiagramillustratesthetranslation alerrordynamicssubsystem. d = p Ir + ge 3 + 2 e v + 1 W ( e v + e p ) rrr p Ir + ge 3 + 2 e v + 1 W ( e v + e p ) rrr (7.18) T M = m rrr p Ir + ge 3 + 2 e v + 1 W ( e v + e p ) rrr (7.19) where W = diag ( w 1 ;w 2 ;w 3 ) with w i > 0 for i =1 ; 2 ; 3 and: S ( e p ;e v )= 2 e v + 1 W ( e v + e p ) = 266664 2 ; 1 e v;x + 1 ; 1 w 1 ( e v;x + e p;x ) 2 ; 2 e v;y + 1 ; 2 w 2 ( e v;y + e p;y ) 2 ; 3 e v;z + 1 ; 3 w 3 ( e v;z + e p;z ) 377775 (7.20) Thefunction denotesasaturationfunction,whichisdenedasfollows: Denition7.1. Thefunction : R R isacontinuous,twicedifferentiable,nondecreasing functionforwhichgiventwopositivenumbers L M with L M thefollowingpropertieshold: P.1. ( s )= s when j s j L ; P.2. j ( s ) j M forevery s 2 R ; P.3. s ( s ) > 0 forevery s 6 =0 ; P.4. j ( s ) jj s j forevery s 2 R ; 138

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P.5. ( s ) isgloballyLipschitzin s ,withLipschitzconstant L .Hence: 8 s 1 ;s 2 2 R j ( s 1 ) ( s 2 ) j L j s 1 s 2 j Theabovedenitionofthelinearsaturationfunctionissim ilartothedenitiongivenin[102]. Twoadditionalpropertiesareadded.Thetwicedifferentia bilityandthegloballyLipschitzproperty(P.5)thatarenecessaryforthebacksteppingdesign. Thechoiceofthedesiredthrustvector d T M givenin(7.18),(7.19)istwofold.Firstly,by (7.18)itisobviousthat d ischosentobeaunitaryvector.Secondly,duetotheuseofth enested saturationfeedback,giventhatthedesiredacceleration p Ir isboundedby(7.19)thethrustmagnitude T M willbeboundedaswell.Thisfactisofparticularimportanc esinceduetothethephysicalconstraintsofthehelicopteractuation,stabilitysh ouldbeachievedwithlimitedcontrolresources. Thehelicopterduringtheightoperationisrequirednotto overturnwhiletrackingthereferencemaneuver.Morespecicallyitisrequiredthat j ( t ) j <= 2 and j ( t ) j <= 2 forevery t t 0 .Apartfromthephysicalhelicopterightlimitations,thi sconditionisnecessarytoavoid singularitiesintherotationmatrixrepresentationbythe Eulerangles.Similarconstraintsapplyby theuseofquaternionsfortheattituderepresentation[4,3 7].Since 3 ; 3 = C C thehelicopterwill notoverturniftheinequality 3 ; 3 ( t ) > 0 ispreservedforevery t t 0 .Whenthehelicopteris trackingitsdesiredorientation,dictatedbythedirectio nalvector d ,thesamelimitationshould apply.Inotherwords, j d ( t ) j <= 2 and j d ( t ) j <= 2 forevery t t 0 .From(7.18)an additionalconstraintisimposedonthechoiceofthesatura tionvector S ( e p ;e v ) andthedesired positiontrajectory.Thisconstraintissufcienttoguara nteethat d; 3 = C d C d > 0 forevery t t 0 Property7.1. Ifforevery t t 0 thesaturationlevel M 2 ; 3 ofthefunction 2 ; 3 andthepredened valueof p Ir;z satisfytheinequality: g M 2 ; 3 > max t t 0 p Ir;z ( t ) 139

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AttitudeDynamics TranslationalErrorDynamics e p = e v e v = S ( e p ;e v ) e U ( t;e p ;e v ) d ( p Ir ;e p ;e v ) e 3 Figure7.5:Resultingsystemdynamicsafterthechoiceof v I d d and T M then d; 3 ( t ) > 0 andconsequently j d ( t ) j ; j d ( t ) j <= 2 forevery t t 0 Theabovepropertycanbeeasilyveriedbythefollowingser iesofinequalities: d; 3 ( t ) > 0 ) p Ir;z ( t )+ g + 2 ; 3 e v;z + 1 ; 3 w 3 ( e v;z + e p;z ) > 0 ) g M 2 ; 3 > max t t 0 p Ir;z ( t ) Substitutionofthedesiredvaluesgivenin(7.17)-(7.19)w illresultinthefollowingrepresentation ofthetranslationalerrordynamics: e p = e v (7.21) e v = S ( e p ;e v ) ( 3 () d ( t )) | {z } e U ( t;e p ;e v ) (7.22) where: U ( t;e p ;e v )= rrr p Ir + ge 3 + 2 e v + 1 W ( e v + e p ) rrr (7.23) Regarding U ( ) thefollowingpropertywillhold: Property7.2. Giventhat d; 3 ( t ) > 0 forevery t t 0 ,thenthefollowinginequalitieswillhold: U min U ( t;e p ;e v ) U max 140

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with: U min = g M 2 ; 3 max t t 0 p Ir;z ( t ) > 0 U max =max t t 0 k p Ir ( t ) k + g + p 3( M 2 ; 1 + M 2 ; 2 + M 2 ; 3 ) Theresultingsystemdynamics,uptothispoint,canbeseeni nFigure7.5.Thetranslational errordynamicssubsystemcanbeconsideredasaGASnominals ystemofasingleintegratorcontrolledbyanestedsaturationfeedbacklaw.Chainsofinteg ratorscontrolledbylinearsaturation functionshavebeenextensivelyinvestigatedin[102].The nominalsystemisperturbedbyabounded termoftheorientationerror e .Thestabilityanalysisoftheresultingtranslationalerr ordynamics willbeinvestigatedindetailinSection7.6,afterweestab lishsomeusefulstabilityresultsassociatedwiththeattitudeerrordynamicssubsystem. Beforeweproceedwiththeanalysisoftheattitudedynamics subsystem,thefollowingobservationismentioned.Since 3 and d areunitaryvectorsthereisanadditionalconstraintexpre ssed bytheequality 3 ; 3 = q 1 23 ; 1 23 ; 2 giventhat 3 ; 3 0 .Duetothisconstraintitisshownthat onlyexponentialdecayofthevector e % = % % d with % =[ 3 ; 1 3 ; 2 ] T and % d =[ d; 1 d; 2 ] T is required.Thevectors % and % d lieinthe x y planeoftheinertiaframe.Giventhatthecontroller designguaranteesthatthehelicopterwillnotoverturn( 3 ; 3 ( t ) > 0 forevery t>t 0 )theexponentialconvergenceof 3 ; 3 to d; 3 follows.Arepresentationoftheorthonormalvectors 3 ; d canbe seeninFigure7.6.Denition7.2. Denotetheopenandconnectedsets: 1. P =(01] 2.Thetwodimensionalset Q = v 2 R 2 : k v k < 1 3.Thetwodimensionalset E =( 22) ( 22) Aconsequenceoftheanglebounds j j ; j j <= 2 and j d j ; j d j <= 2 arethestatementsof thefollowingProposition: 141

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~ i I ~ j I ~ i I ~ k I 3 ; 1 3 ; 2 d; 1 d; 2 % % d 3 ; 3 d; 3 e ; 3 e ; 1 e ; 2 Topview Sideview k 3 ; 3 k = q 1 k % k 2 k d; 3 k = q 1 k % d k 2 O B O B % r Figure7.6:Thisgureillustratesthehelicopter'svertic alorientationvectors 3 d withrespectto inertiaframefor 3 ; 3 ; d; 3 > 0 Proposition7.1. When 3 ; 3 ; d; 3 2P then: 1. j j ; j d j ; j j ; j d j <= 2 2. %;% d 2Q 3. e % 2E ThisSectionhasintroducedtheappliedpseudocontrolsass ociatedwiththetranslationalerror dynamics.Additionalcommentsandconditionswerepresent edrelatedtotheorientationrestrictionsofthehelicopterduringtheightmaneuver,thataren ecessaryfortheanalysisoftheattitude dynamics.Thedetailedstabilityanalysisofthetranslati onalerrordynamicssubsystemisgiven inSection7.6,aftersomeusefulresultsassociatedwithth estabilityoftheattitudedynamicsare establishedinSections7.4and7.5.7.4AttitudeErrorDynamics ThisSectionpresentstheattitudeerrordynamicssubsyste m.Furthermore,theproposedpseudo controlsandtheinputvector v c forthestabilizationoftheattitudeerrorareprovided.Ap artfrom 142

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thestabilizationpart,additionalgoalforthecontrollaw istokeep j ( t ) j ; j ( t ) j <= 2 forevery t t 0 foranyinitialconditionoftheattitudedynamicsforwhich thehelicopterisnotoverturned. 7.4.1YawErrorDynamics Theyawdynamicsareobtainedbytheequation: = 3 () B (7.24) where 3 () isthethirdrowofthematrix () denedin(3.25).Let e = r betheerror oftheyawangle,thentheerrordynamicswillbe: e = r + 3 () B = r + S C q + C C r (7.25) Usingtheyawangularvelocity r aspseudocontrol,theerrordynamicsfortheyawanglecanbe writtenas: e = r + S C q + C C r d + ( ; ) e (7.26) where e = B B d ,with e =[ e !;x e !;y e !;z ] B d =[ p d q d r d ] T and ( ; )= h 00 C C i .The valueof r d willbechoseninsuchawaytocanceloutthenonlineartermsa ndstabilizetheyaw errordynamics.Thechoiceis: r d = C C r S C q e (7.27) where isapositivegain.Theyawdynamicsbecome: e = e + ( ; ) e (7.28) 143

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7.4.2OrientationErrorDynamics Asmentionedearlierduetotheconstraintoforthonormalit yofthevector 3 theorientation analysiscanberestrictedtothevector % 2E .Asitwillbeshown,exponentialstabilizationof theerrordynamics e % = % % d willguaranteetheexponentialstabilizationof e .Thereduced orientationerrordynamicsare: e % = % d + Z () 264 p d q d 375 + Z () 264 e !;x e !;y 375 (7.29) where: Z ()= 264 2 ; 1 1 ; 1 2 ; 2 1 ; 2 375 with 2 Z 1 ()= 1 3 ; 3 264 1 ; 2 1 ; 1 2 ; 2 2 ; 1 375 (7.30) Thechoiceoftheangularvelocitypseudocontrolsis: 264 p d q d 375 = Z 1 () % d 1 e % k 3 ; 3 e % (7.31) where 1 = diag ( 1 ; 1 ; 1 ; 2 ) with 1 ;i ;k> 0 for i =1 ; 2 .Thereducedorientationerrordynamics taketheform: e % = 1 e % k 3 ; 3 e % + Z () 264 e !;x e !;y 375 = 1 e % k 3 ; 3 e % + Z 0 () e (7.32) with Z 0 ()=[ Z ()0 2 1 ] .Itcanbeeasilyveriedthat k Z () k = k Z 0 () k =1 2 Notethat 3 ; 3 = 1 ; 1 2 ; 2 1 ; 2 2 ; 1 144

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7.4.3AngularVelocityErrorDynamics Theangularvelocityerrordynamics e basedon(7.14)havethefollowingform: I e = I (_ B B d ) = I B d ^ B I B + A ( T M ) v c + B ( T M ) = I B d ^ e I B ^ B d I B + A ( T M ) v c + B ( T M ) (7.33) Theinitialobjectiveof v c istoremovetheeffectof A ( T M ) and B ( T M ) .Thereforetheinitial choiceof v c is: v c = A 1 ( T M )[ B ( T M )+~ v ] (7.34) Thevector ~ v isanadditionalstabilizingtermofthefollowingform: ~ v = I B d +^ B d I B e ( ; ) T 2 e (7.35) where 2 2 R 3 3 isadiagonalmatrixofpositivegains. 7.5StabilityoftheAttitudeErrorDynamics Applyingthecontrol v c of(7.34),(7.35)andthepseudocontrolsgivenin(7.27),(7 .31),the errorattitudedynamicsbecome: e % = 1 e % k 3 ; 3 e % + Z 0 () e e = e + ( ; ) e (7.36) I e = ^ e I B e ( ; ) T 2 e 145

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Thecompleteerrorvectoroftheattitudedynamicsisgivenb ythestatevector [ e e % e ] T 2Z where Z = R E R 3 .Preconditionforthecontinuityoftherighthandsideof(7 .36)isfor 3 ; 3 tobelongtotheset P Theorem7.1. Giventhat 3 ; 3 ( t ) andthedesiredvalueof d; 3 ( t ) belongto P forevery t t 0 ,and thechoiceofgains: 1 ; 1 = 1 + 2 1 1 ; 2 = 2 + 2 1 2 ;min = + 2 2 + 2 2 where 2 ;min istheminimumentryofthegainmatrix 2 and 1 ; 2 ; 1 ; 2 ;> 0 with 1 2 1 = 2 1 2 1 = 2 ,thentheerrordynamicsofthesystemdescribedbyequation s (7.36) areexponentiallystableforanyinitialcondition [ e ( t 0 ) e % ( t 0 ) e ( t 0 )] 2Z Proof. Thestabilityanalysisoftheattitudedynamicsbeginsbeco nsideringthebelowLyapunov quadraticfunctionoftheassociatedattitudevariables: V ( e ;e % ;e )= 1 2 e 2 + 1 2 e T% e % + 1 2 e T! I e Thederivativeof V ( e ;e % ;e ) alongthetrajectoriesoftheattitudedynamics,forevery [ e e % e ] 2 Q and 3 ; 3 2P willbe: V ( e ;e % ;e )= e e + e T% e % + e T! I e = e 2 e T% 1 e % k 3 ; 3 e T% e % e T! 2 e + e T% Z 0 () e k e k 2 k 3 ; 3 e T% e % 1 ; 1 rr e ; 1 rr 2 1 ; 2 rr e ; 2 rr 2 2 ;min k e k 2 + e ; 1 [10] Z 0 () e + e ; 2 [01] Z 0 () e k e k 2 1 ; 1 rr e ; 1 rr 2 1 ; 2 rr e ; 2 rr 2 2 ;min k e k 2 + 1 rr e ; 1 rr 2 k e k 2 + 1 rr e ; 1 rr 2 k e k 2 + rr e ; 1 rr k e k + rr e ; 2 rr k e k 146

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k e k 2 1 ; 1 2 1 rr e ; 1 rr 2 1 ; 2 2 1 rr e ; 2 rr 2 (2 1 2 1) rr e ; 1 rr k e k (2 1 2 1) rr e ; 2 rr k e k 2 ;min 2 2 2 2 k e k 2 k e k 2 1 rr e ; 1 rr 2 2 rr e ; 2 rr 2 k e k 2 Thisprovesthetheorem. Theexponentialdecayofthevector e % fromTheorem7.1resultsinthefollowinginequalities: k e ; 1 kk e ; 1 ( t 0 ) k e 1 ( t t 0 ) and k e ; 2 kk e ; 2 ( t 0 ) k e 2 ( t t 0 ) ; 8 t t 0 (7.37) Theorem7.2. Forthesystemin (7.36) ,givenadesiredorientationvector d ( t ) withthevector component d; 3 ( t ) > 0 forevery t t 0 ,thehelicopterwillnotoverturn,satisfying 3 ; 3 ( t ) > 0 for every t t 0 .Thelatterinequalityofthevectorcomponent 3 ; 3 holdsforeveryinitialstateofthe angularvelocityandtheorientationofthethrustvector,g iventhat 3 ; 3 ( t 0 ) > 0 Proof. Thenecessaryconditionforthehelicopternottooverturni s 3 ; 3 ( t ) > 0 forevery t t 0 Thisconditionrequiresthat k % k < 1 forevery t t 0 IfProperty7.1holds,then d; 3 ( t ) > 0 forevery t t 0 .Let min t t 0 d; 3 ( t )= c min > 0 .Dene thepositiveconstant C max givenby max t t 0 2d; 1 ( t )+ 2d; 2 ( t ) = C 2 max .Since: min t t 0 2d; 3 ( t )=1 max t t 0 2d; 1 ( t )+ 2d; 2 ( t ) ) c 2min =1 C 2 max itfollowsthat 0 C max < 1 .FromTheorem7.1,theerrorvariables e %; 1 and e %; 2 areexponentiallystablein E .Theexponentialstabilityof e % itselfcannotguaranteethat 3 ; 3 ( t ) > 0 8 ;t t 0 .Consideringonlytheexponentialstabilityof e % onegets: k e %;i ( t 0 ) k e i ( t t 0 ) + d;i 3 ;i k e %;i ( t 0 ) k e i ( t t 0 ) + d;i (7.38) 147

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~ i I ~ j I 3 ; 1 3 ; 2 d; 1 d; 2 t t % d % ( t ? ) O B 1 C max k e %; 1 ( t 0 ) k k e %; 2 ( t 0 ) k k e %; 1 k k e %; 2 k Figure7.7:Thisgureillustratesthatonlytheexponentia lconvergenceof e % cannotguarantee that k % k < 1 forevery t t 0 .Inthedepictedcasealthoughtheinequalities(7.38)hold there mightexistatime t ? forwhich k % ( t ? ) k =1 for i =1 ; 2 .Theaboveinequalityindicatesthattheremightexistinit ialconditions e % ( t 0 ) ,adesiredvector % d andatime t ? suchthat k % ( t ? ) k =1 .ThiscaseisdepictedinFigure7.7.Therefore, thequestionthatarisesiswhathappenswhen k % k! 1 .Ofcoursethegoalisforevery t t 0 to hold k % k < 1 From(7.32)theratesofchangeofthevector 3 ( t ) inthe x and y directionoftheinertiaframe aregivenby: % =_ % d 1 e % k 3 ; 3 e % + Z 0 () e (7.39) Considerthequadraticfunction R ( k % k )=(1 = 2) k % k 2 of k % k .Theobjectiveistoprovethateach time k % k tendstothevicinityof 1 ,then R ( k % k ) 0 .Thederivativeof R ( k % k ) is: R ( k % k )= % T % = % T % d % T 1 e % k % T e % 3 ; 3 + % T Z 0 () e % T % d % T 1 e % + k % kk Z 0 () kk e k k % T e % 3 ; 3 148

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_ R ( k % d k )+ e T% % d % T 1 e % + k e ( t 0 ) k e ( t t 0 ) k % T e % 3 ; 3 R ( k % d k )+( k % d k + k % k ) k e % k + k e ( t 0 ) k e ( t t 0 ) k % T e % 3 ; 3 R ( k % d k )+ k e % ( t 0 ) k ( k % d k + ) e ( t t 0 ) + k e ( t 0 ) k e ( t t 0 ) k %% T % T % d q 1 k % k 2 R ( k % d k )+2( k % d k + ) e ( t t 0 ) + k e ( t 0 ) k e ( t t 0 ) k k % k ( k % kk % d k ) q 1 k % k 2 k ( t;% d ; % d ; k e ( t 0 ) k ) k $ ( k % k ) q 1 k % k 2 = R k ( ) k ; k % k where =min( 1 ; 2 ) =max( 1 ; 1 ; 1 ; 2 ) and: ( )= R ( k % d k )+2( k % d k + ) e ( t t 0 ) + k e ( t 0 ) k e ( t t 0 ) $ ( )= k k % k ( k % k C max ) When k % k liesinsidetheset C max =( C max 1) itisobviousthat $ ( k % k ) > 0 .Bysolving R ( k k ; k % k ) < 0 ,withrespectto k % k when k % k2C max ,aftersomealgebraiccalculationsitis easytoshowthatthereexistsa C ? ( k ( ) k ) ,with C max C ? then R ( k % k ) < 0 .Thevalueof C ? isgivenby: If C max > 0 then: C ? ( r 1 )= C max + r 1 p r 2 1 +1 C 2 max 1+ r 2 1 where: r 1 ( k ( ) k )= k ( t;% d ; % d ; k e ( t 0 ) k ) k kC max 149

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~ i I ~ j I 3 ; 1 3 ; 2 d; 1 d; 2 % d % R ( k % k ) < 0 e ; 1 e ; 2 O B 1 C ? C max Figure7.8:Thisgureillustratestheexistenceofavalue C ? with C max C ? then R ( k % k ) < 0 .Thedenitionof R ( k % k ) isgivenintheproofofTheorem7.2. If C max =0 then k % d k = k % d k =0 forevery t 0 ,andthevalueof C ? isgivenby: C ? ( r 2 )= s r 2 p r 2 2 +4 r 2 2 2 where: r 2 ( k ( ) k )= k ( t; 0 ; 0 ; k e ( t 0 ) k ) k k Since R ( k % k ) isapositivedenitefunctionof k % k and R ( k % k ) < 0 forevery k % k >C ? with C ? < 1 ,then k % k isdecreasingintheinterval ( C ? 1) andneverreaches1,sothehelicopterwill neveroverturn.Thisprovesthetheorem.Agraphicrepresen tationclarifyingthendingsofthis proofcanbeseeninFigure7.8. Duetothefactthat 3 ; 3 = C C ,Theorem7.2impliesthat j ( t ) j ; j ( t ) j <= 2 forevery t t 0 giventhat j ( t 0 ) j ; j ( t 0 ) j <= 2 150

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Lemma7.1. GiventhattheconditionsofTheorem7.1aremetforthesyste min (7.36) ,thedynamicsof e ; 3 willexponentiallydecaytozero,withthebound: k e ; 3 k 2 p 2 c min k e % ( t 0 ) k e ( t t 0 ) where =min( 1 ; 2 ) Proof. FromTheorem7.2ithasbeenprovedthat 3 ; 3 > 0 and d; 3 c min forevery t t 0 .Thus: 3 ; 3 + d; 3 c min ) 1 3 ; 3 + d; 3 1 c min Regarding e ; 3 onehas: e ; 3 = 3 ; 3 d; 3 = 23 ; 3 2d; 3 3 ; 3 + d; 3 = 23 ; 1 23 ; 2 + 2d; 1 + 2d; 2 3 ; 3 + d; 3 = ( 3 ; 1 + d; 1 )( 3 ; 1 d; 1 ) ( 3 ; 2 + d; 2 )( 3 ; 2 d; 2 ) 3 ; 3 + d; 3 = e %; 1 ( 3 ; 1 + d; 1 ) e %; 2 ( 3 ; 2 + d; 2 ) 3 ; 3 + d; 3 Thenormof e ; 3 willbe: k e ; 3 k rrrr 3 ; 1 + d; 1 3 ; 3 + d; 3 rrrr k e %; 1 k + rrrr 3 ; 2 + d; 2 3 ; 3 + d; 3 rrrr k e %; 2 k 2 p 2 c min k e % k 2 p 2 c min k e % ( t 0 ) k e ( t t 0 ) AnimmediateconsequenceofTheorem7.1andLemma7.1isthef ollowingproperty,which summarizestheboundsofthenorm k e k .Thoseboundsareusefulintheanalysisofthetranslationalerrordynamics.Property7.3. GiventhatTheorem7.1andLemma7.1hold, k e k willhavethefollowingbounds: 7.3.1. k e k 2 151

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7.3.2.Forthecomponentsoftheerrorvector e : k e ;i k i k e % ( t 0 ) k e ( t t 0 ) where i =1 for i =1 ; 2 and 3 =2 p 2 =c min 7.3.3.Thevector e isexponentiallystableforevery e ( t 0 ) 2EP withtheexponentially decayingbound: k e k c min +2 p 2 c min k e ( t 0 ) k e ( t t 0 ) Proof. Duetoorthonormality k 3 k ; k d k =1 .Consequently,Property7.3.1isderivedby: k e k = q ( 3 d ) T ( 3 d )= q T3 3 + Td d 2 T3 d = q 2 2 T3 d 2 Property7.3.2canbeeasilyderivedbyTheorem7.1andLemma 7.1.Fortheexponentialboundof Property7.3.3thefollowingwillhold: k e kk e % k + k e ; 3 k k e % ( t 0 ) k e ( t t 0 ) + 2 p 2 c min k e % ( t 0 ) k e ( t t 0 ) c min +2 p 2 c min k e % ( t 0 ) k e ( t t 0 ) c min +2 p 2 c min k e ( t 0 ) k e ( t t 0 ) Lemma7.1andProperty7.3.3provideaveryconservativebou ndon k e ; 3 k and k e k .However,theusefulattributeofthoseistheexponentialdecay of e ; 3 and e ,whichisnecessaryforthe stabilityanalysisofthetranslationalerrordynamics. 152

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InthisSection,Theorem7.1establishestheexponentialst abilityoftheattitudeerror [ e e % e ] T InadditionTheorem7.2guaranteesthatthehelicopterwill notoverturninitsefforttotrackthe referencetrajectory,achievingtheboundingcondition j j ; j j <= 2 forevery t t 0 .Basedon thosetworesults,fromProperty7.3.3,theexponentialdec ayoftheorientationerror e follows. 7.6StabilityoftheTranslationalErrorDynamics ThisSectionexaminesthestabilityofthetranslationaler rordynamics.Therststeptowards thestabilityanalysisistoperformthefollowinglinearst atetransformation: y = 264 y 1 y 2 375 = 264 I 3 3 I 3 3 0 I 3 3 375 264 e p e v 375 (7.40) Thestatetransformationabovewillfacilitatethestabili tyanalysisofthisSection.Theresulting formofthetranslationaldynamicsis: y = f ( y )+ g ( t;y ) e = G ( t;y;e ) (7.41) where: f ( y )= 264 y 2 2 y 2 + 1 ( Wy 1 ) 2 y 2 + 1 ( Wy 1 ) 375 g ( t;y )= 264 I 3 3 I 3 3 375 U ( t;y ) (7.42) Thefollowingpropertiesarerequiredtoproveglobalasymp toticstabilityofthesystemin(7.41). Property7.4. Forthenominalsystem: y = f ( y ) (7.43) with f ( y ) denedin (7.42) y =0 isanequilibriumpoint.Giventhat,forthesaturationleve lsof thevector S (denedin (7.20) ),thefollowinginequalitieshold: 1. L 2 ;i M 2 ;i and L 1 ;i M 1 ;i for i =1 ; 2 ; 3 2. M 1 ;i < 1 3 L 2 ;i for i =1 ; 2 ; 3 153

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_ y = f ( y )+ g ( t;y ) e TranslationalErrorDynamics AttitudeErrorDynamics e Figure7.9:Blockdiagramofthecompletehelicopterdynami csafterthetransformationofthe translationalerrorstates.Then,basedonthendingsof[102],thenominalsystemof (7.43) isGAS. Theresultinghelicopterdynamicsafterthestatetransfor mationcanbeseeninFigure7.9.The translationaldynamicssubsystemcanbeviewedasaperturb edUGASnominalsystemwhere theperturbationtermisdrivenby e .Thenalformofthecompletehelicopterdynamicsisa nonlinearcascadedtime-varyingsystem.Thestabilitypro pertiesforthisclassofsystemshas beeninvestigatedin[63].Accordingto[63],inorderforth esolutionsofthesystemin(7.41)to beUGAS,thefollowingsufcientconditionsshouldholdsim ultaneously: C.1:Thenominalsystemof(7.43)isUGAS C.2:Theintegralcurvesof e areUGAS C.3:Thesolutionsofthesystemin(7.41)areuniformlyglob allybounded(UGB). ConditionsC.1andC.2areguaranteedbyProperties7.4and7 .3.3,respectively.Thesystemin (7.41)isnotInputtoStateStable(ISS).TheISSpropertywo uldsignicantlyfacilitatetheproof ofconditionC.3.Consequently,adifferentapproachisfol lowed,whichexploitstheLipschitz propertiesof G ( t;y;e ) withrespectto y andtheboundsof e providedbyProperty7.3. Property7.5. Thefunction f ( y ) denedin (7.42) ,isgloballyLipschitzin y ,withLipschitzconstant: D f = p 6 1+2 L 2 +2 w max L 1 L 2 154

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where w max =max( w 1 ;w 2 ;w 3 ) and L 1 L 2 positiveconstantssuchthat: 8 s 1 s 2 2 R 3 j i ( s 1 ) i ( s 2 ) j L i j s 1 s 2 j for i =1 ; 2 : Proof. Forthefunction f : R 6 R 6 denedin(7.42),forany y;z 2 R 6 thefollowinginequalitieswillhold: k f ( y ) f ( z ) k = = rrrrrrr 264 y 2 z 2 2 y 2 + 1 ( Wy 1 ) + 2 z 2 + 1 ( Wz 1 ) 2 y 2 + 1 ( Wy 1 ) + 2 z 2 + 1 ( Wz 1 ) 375 rrrrrrr y 2 z 2 2 y 2 + 1 ( Wy 1 ) + 2 z 2 + 1 ( Wz 1 ) + 2 y 2 + 1 ( Wy 1 ) + 2 z 2 + 1 ( Wz 1 ) j y 2 z 2 j +2 2 y 2 + 1 ( Wy 1 ) + 2 z 2 + 1 ( Wz 1 ) j y 2 z 2 j +2 L 2 j y 2 1 ( Wy 1 ) z 2 + 1 ( Wz 1 ) j (1+2 L 2 ) j y 2 z 2 j +2 w max L 1 L 2 j y 1 z 1 j (1+2 L 2 +2 w max L 1 L 2 ) j y 1 z 1 j + j y 2 z 2 j (1+2 L 2 +2 w max L 1 L 2 ) p 6 k y z k Therefore f ( y ) isgloballyLipschitzin y Theexistenceof L 1 L 2 isguaranteedbypropertyP.5ofDenition7.1. Property7.6. Foranyvectorfunction d ( t ) 2 R 3 thatisuniformcontinuouswithrespectto t and k d ( t ) k 0 forevery t t 0 with 0 apositiveconstant,thefunction g ( t;y ) d ( t ):=( t;y ) is globallyLipschitzin y withLipschitzconstant: D g ( 0 )= 0 L 2 + w max L 1 L 2 p 12 U max U min 155

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Proof. Let a ( t )= p Ir + ge 3 .Forthefunction ( t;y )= g ( t;y ) d ( t ) with :[0 1 ] R 6 R 6 forany y;z 2 R 6 thefollowinginequalitieswillhold: k ( t;y ) ( t;z ) k p 2 k d ( t ) U ( t;y ) d ( t ) U ( t;z ) k 0 p 2 k U ( t;y ) U ( t;z ) k 0 p 2 rrrr U 2 ( t;y ) U 2 ( t;z ) U ( t;y )+ U ( t;z ) rrrr 0 p 2 2 U min rr 2 a T ( t )( S ( y ) S ( z )) +( S ( y )+ S ( z )) T ( S ( y ) S ( z )) rr 0 p 2 U min 2 k a ( t ) k + k S ( y )+ S ( z ) k k S ( y ) S ( z ) k 0 p 2 U max U min 2 y 2 + 1 ( Wy 1 ) 2 z 2 + 1 ( Wz 1 ) 0 p 2 U max U min ( L 2 j y 2 z 2 j + w max L 1 L 2 j y 1 z 1 j ) 0 ( L 2 + w max L 1 L 2 ) p 12 U max U min k y z k Theexistenceof U min ;U max isguaranteedfromProperty7.2giventhatProperty7.1issa tisedandthesecondderivativesof p Ir ( t ) coordinatesarebounded.Theaboveinequalityimplies thattherealwaysexistsaLipschitzconstantforeveryappr opriatechoiceof p Ir ( t ) andforevery bounded d ( t ) 2 R 3 .Therefore g ( t;y ) d ( t ) isgloballyLipschitzin y ThefollowinglemmaisanimmediateconsequenceofProperti es7.5and7.6. Lemma7.2. Foranyvector d ( t ) denedinProperty7.6,theperturbedsystem: y = f ( y )+ g ( t;y ) d ( t ):=( t;y ) (7.44) isgloballyLipschitzin y withLipschitzconstant: D 0 ( 0 )= D f + D g ( 0 ) 156

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Therefore,thesolutionsof (7.44) exist,areuniqueanddonothaveaniteescapetimeforany arbitrarilylargetimeinterval. Theerrorvector e iscontinuousandfromProperty7.3.1 k e k 2 forevery e % ( t 0 ) 2E Therefore:Lemma7.3. BasedonLemma7.2,duetothecontinuityandboundednessoft hevector e ,the systemin (7.41) isgloballyLipschitzin y ,withLipschitzconstant D = D 0 (2) ,thereforethe solutionsof (7.41) exist,areuniqueanddonothaveaniteescapetimeforanyar bitrarilylarge timeinterval. Lemma7.3isofparticularinterestfortheproofofthefollo wingtheorem,whichguarantees theglobaluniformboundednessofthesolutionsofthesyste min(7.41). Theorem7.3. GiventhatTheorems7.1and7.2hold,thesolutionsofthesys temgivenby (7.41) areUGBforeverytime t t 0 Proof. Thenominalsystem z = f ( z ) (7.45) of(7.43),basedon[102]isgloballyasymptoticallystable (GAS).Sinceitisanautonomoussystem,itwillbeuniformlygloballybounded(UGB)aswell.The reforeforany > 0 (arbitrarily large)thereexists > 0 whichmaydependon suchthat: k z ( t 0 ) k )k z ( t ) k ( ) 8 t t 0 Fortheperturbedtermofthesystemin(7.41),forany y 2 R 6 usingProperty7.3.1thefollowing boundwillhold: k g ( t;y ) e k p 2 k U ( t;y ) e k 2 p 2 U max = E 157

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ApplyingtheGronwall-Bellmaninequalitytotheintegralc urvesofthenominal(7.45)andperturbedsystem(7.41),with z ( t 0 )= y ( t 0 ) foranynitetimeintervalwith t t 0 oneobtains: k y ( t ) kk z ( t ) kk y ( t ) z ( t ) k E D h e D ( t t 0 ) 1 i )k y ( t ) k ( )+ E D h e D ( t t 0 ) 1 i = B ( ;t t 0 ) (7.46) with D denedinLemma7.3.Let y 1 ;i ;y 2 ;i and e ;i with i =1 ; 2 ; 3 denotethe i th component ofthevectors y 1 ;y 2 and e correspondingly.Thedynamicsofthe i th componentoftheperturbed system(7.41)willbe: y 1 ;i = y 2 ;i 2 ;i y 2 ;i + 1 ;i ( w i y 1 ;i ) r i ( t;y;e ;i ) y 2 ;i = 2 ;i y 2 ;i + 1 ;i ( w i y 1 ;i ) r i ( t;y;e ;i ) where r i ( t;y;e ;i )= U ( t;y ) e ;i .UsingProperty7.3.2onehas: k r i ( t;y;e ;i ) k = k U ( t;y ) e ;i k U max k e ;i k U max i k e %;i ( t 0 ) k e ( t t 0 ) 2 U max i e ( t t 0 ) Toproveuniformboundednessof y itissufcienttoshowuniformboundednessof y 1 ;i ;y 2 ;i for i =1 ; 2 ; 3 .Fromthispointforwardofthisproof,thesubscript i willbeomittedtoeasethe notation. Fromtheexponentialdecayingboundof r ( ) therealwaysexistsanitetime T = t 0 + t with t 0 suchthat: 2 U max e t ? L 1 4 158

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ConsidertheLyapunovfunction V 2 = 1 2 y 2 2 .Fromtheaboveinequalityandusing t 0 = T t ,the derivativeof V 2 alongthetrajectoriesoftheperturbedsystemwillbe: V 2 = y 2 2 y 2 + 1 ( wy 1 ) y 2 U ( t;y ) e y 2 2 y 2 + 1 ( wy 1 ) + j y 2 j U max k e % ( t 0 ) k e ( t t 0 ) y 2 2 y 2 + 1 ( wy 1 ) + j y 2 j 2 U max e t ? e ( t T ? ) y 2 2 y 2 + 1 ( wy 1 ) + j y 2 j L 1 4 e ( t T ? ) Forevery k y 2 k M 1 + L 1 2 = 2 andforevery t T ? onewillget: V 2 y 2 2 y 2 + 1 ( wy 1 ) + L 1 4 j y 2 j L 1 2 j y 2 j + L 1 4 j y 2 j L 1 4 j y 2 j Thenfrom[43,Theorem4.18]forevery j y 2 ( T ) j 2 andforevery t T thereexistsa KL function 2 andanitetime t 1 0 dependentof y 2 ( T ) and 2 suchthattheintegralcurveof y 2 ( t ) satises: k y 2 ( t ) k 2 ( k y 2 ( T ) k ;t T ) 8 T t T 1 k y 2 ( t ) k 2 8 t T 1 where T 1 = T + t 1 .Clearly,if j y 2 ( T ) j 2 then j y 2 ( t ) j 2 forevery t T rendering t 1 =0 and T 1 = T .Thosefactsindicatethattherealwaysexistanitetime T 1 T afterwhich theintegralcurveof y 2 ( t ) willremainboundedintheset 2 = f y 2 : j y 2 j 2 g foranyinitial condition y 2 ( t 0 ) 2 R .Moreover,theasymptoticconvergence(ortheconnementw hen t 1 =0 )of y 2 ( t ) totheboundedset 2 beginsatthenitetime T .Lemma7.3guaranteesthatthetrajectory of y 2 ( t ) doesnothaveaniteescapetimeintheinterval [ t 0 T ] andremainsbounded. 159

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From(7.46),giventhat k y 2 ( t 0 ) k thetrajectoryof y 2 ( t ) for t 2 [ t 0 T ] willbeboundedby k y 2 ( t ) k B ( ;t )= B 2 ( ) .Hence,forevery > 0 with k y 2 ( t 0 ) k : k y 2 ( t ) k max B 2 ( ) ; 2 ( B 2 ( ) ; 0) ; 2 = R 2 ( ) 8 t t 0 Obviouslythebound R 2 ( ) > 0 isindependentfrom t 0 .Therefore,thesolution y 2 ( t ) isUGB. Afterthetimethreshold T 1 theargumentofthesaturationfunction 2 willbeboundedby: j y 2 + 1 ( wy 1 ) jj y 2 j + j 1 ( wy 1 ) j 2 M 1 + L 1 2 5 6 L 2 (7.47) Tothisextent,when t T 1 ,thesaturationfunction 2 ( ) operatesinitslinearregion.Continuing theaboveprocedure,considertheLyapunovfunction V 1 = 1 2 y 2 1 .Thederivativeof V 1 forevery t T 1 willbe: V 1 = y 1 1 ( wy 1 ) U ( t;y ) e ;i y 1 1 ( wy 1 )+ L 1 4 j y 1 j Consequently,forevery j y 1 j L 1 =w = 1 and t T 1 willyield, V 1 3 4 L 1 j y 1 j .Once morethereexistsa KL function 1 andanitetime t 2 dependedof y 1 ( T 1 ) and 1 suchthatwhen j y 1 ( T 1 ) j 1 ,theintegralcurveof y 1 ( t ) satises: k y 1 ( t ) k 1 ( k y 1 ( T 1 ) k ;t T 1 ) 8 T 1 t T 2 k y 1 ( t ) k 1 8 t T 2 where T 2 = T 1 + t 2 .If j y 1 ( T 1 ) j 1 then y 1 ( t ) remainsboundedintheset 1 = f y 1 : j y 1 j 1 g forevery t T 1 rendering t 2 =0 .Ineithercaseforanyinitialcondition y 1 ( t 0 ) 2 R there existsanitetime T 2 T 1 afterwhichthetrajectory y 1 ( t ) remainsboundedintheset 1 .The convergence(ortheconnementwhen t 2 =0 )of y 1 ( t ) to 1 startswhen t T 1 .Theexistenceof y 1 ( t ) inthetimeinterval [ t 0 T 1 ] isguaranteedbyLemma7.3. 160

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From(7.46),giventhat k y 1 ( t 0 ) k thetrajectoryof y 1 ( t ) for t 2 [ t 0 T 1 ] willbeboundedby k y 1 ( t ) k B ( ;t + t 1 )= B 1 ( ;t 1 ) .Hence,forevery > 0 and t t 0 with k y 1 ( t 0 ) k : k y 1 ( t ) k max B 1 ( ;t 1 ) ; 1 ( B 1 ( ;t 1 ) ; 0) ; 1 = R 1 ( ;t 1 ) Thetime t 1 isdependentonthevalue y 2 ( T ? ) and 2 .Bothofthemareindependentof t 0 .Tothis extent R 1 ( ;t 1 ) doesnotdependontheinitialtime t 0 whichprovestheuniformglobalboundednessofthetrajectory y 1 ( t ) Since y 1 ;i ( t ) ;y 2 ;i ( t ) areUGBfor i =1 ; 2 ; 3 thensameholdsforthecompletestates y 1 ( t ) y 2 ( t ) ofthesystemin(7.41). Theorem7.3satisestheremainingconditionC.3whichisre quiredtoguaranteethatthe solutionsof(7.41)areUGAS.Basedontheworkof[63,94,103 ]thestabilityofthehelicopter translationalerrordynamicsisformallystatedinthefoll owingtheorem: Theorem7.4 ([63,103]) Giventhatthenominalsystemin (7.43) isUGAS(Property7.4),the orientationerror e isexponentiallyconvergentandbounded(Property7.3),an dthesolutionsof (7.41) areUGB(Theorem7.3),thenthesolutionsoftheperturbedsy stemin (7.41) areUGAS. Theorems7.1,7.2and7.4guaranteethatthecontrollerdesi gnobjectivesaremet.Morespecic,foranydesiredpositionreferencetrajectory p Ir withboundedhigherderivativessatisfyingthe requirementsofProperty7.1andforeverydesiredyawheadi ng r : lim t !1 k p I p Ir k =0lim t !1 k r k =0 and j ( t ) j ; j ( t ) j <= 2 8 t t 0 foranyinitialcondition [ p I ( t 0 ) v I ( t 0 ) B ( t 0 ) ( t 0 )] T 2 R 10 giventhatthehelicopterisnot initiallyoverturned( j ( t 0 ) j ; j ( t 0 ) j <= 2 ). 161

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7.7NumericSimulationResults ThisSectionpresentsthenumericsimulationresultsofthe controlalgorithm.Forthehelicoptermodel,thecompleterepresentationofthethrustvec torisusedgivenin(7.8),whichincludestheparasiticelements X M ;Y M and Y T .However,thecontrollerdesignwasbasedonthe simpliedforcevectorrepresentationof(7.9).Furthermo re,thetotalbodyforceandmomentvectorsof(7.8)and(7.10)areadditionallyperturbedbytheto taldragforceandmomentvectors f B d and B d ,respectively.Thedragforcesandmomentsareproducedbyt heeffectoftherelativewind velocityandairpressure,tothesurfacesofthehelicopter 'sfuselage,verticalnandhorizontal stabilizer.Torepresentthecompletedragforceandmoment vectorswehaveadoptedthemodel givenin[66],whichisasimpliedversionofthemoreelabor atedescriptionpresentedin[29]. Thosevectorsare: f B d = 266664 d fx v B a;x V 1 d fy v B a;y V 1 d vfy j v vf j v vf d fz v B a;z + u i V 1 + d hsz j v hs j v hs 377775 B d = 266664 z t d vfy j v vf j v vf x hs d hsz j v hs j v hs x t d vfy j v vf j v vf 377775 (7.48) where d fx ;d fy ;d fz ;d vfy ;d hsz areconstantparametersthatdependontheairdensityaswel lasthe geometryofthefuselage,theverticalnandhorizontalsta bilizer.Theconstant u i denotesthe mainrotor'sinducedvelocitywhile x hs isthecoordinateofthehorizontalstabilizerinthe ~ i B directionofthebodyframe.Therelativewindvelocityvect or v B a =[ v B a;x v B a;y v B a;z ] T isgiven by v B a = v B v B w ,where v B w denotesthewindvelocityinthebodyframecoordinates.The restof thevelocitycomponentsinvolvedinthedragforceandmomen tmodel,are: v vf = v B a;y + x t rv hs = v B a;z x hs q (7.49) V 1 = q v B a;x 2 + v B a;y 2 + v B a;z + u i 2 (7.50) Inadditiontothewindeffects,thenumericsimulatorinclu destheservodynamicswhichare typicallyrepresentedbyarstorderlter[30].Therefore ,theservooutputs T M T T ofthemain 162

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andtailrotoraregivenby: s T M = T M + T M s T T = T T + T T (7.51) where s istherotorstimeconstant.Theappliedappingangles a b areproducedbytheapping dynamicsmodelestablishedin[30,70],namely: f a = f y a + a f b = f x b + b (7.52) where f isthemainrotor'sdynamicstimeconstant.Theappingangl es a b arealsosaturated to 0 : 25 rad ,complyingwithrealisticlimitationsofactualrotorcon gurations.Thenominal helicoptermodelparameters,usedbythecontroller,areob tainedby[29]fortheMIT'ssmallscale helicopterX-Cell : 60 andpresentedinTable7.1.Theparametersrelatedtothedra gforcesand momentsaswellastheservostimeconstantsaregiveninTabl e7.2.Theactualhelicoptermodel ofthesimulator,includesparametricuncertaintythatrea chadifferenceofupto 30% withrespect tothenominalvaluesusedbythecontroller.Alloftheabove uncertaintyinjectionisnecessary forinvestigatingtherobustcapabilitiesofthecontrolle rundermodelandparametricuncertainty whichoccursinreallifeapplications. Theproposedcontrolschemecanbeeasilymodiedinorderto includeintegralcomponents thatwillattenuatethesteadystatetrackingerror,caused bytheparametricandmodeluncertainty. Inparticular,thenestedsaturationvector S andthedesiredangularvelocitycomponent r d (denedin(7.20)and(7.27),repsectively),canbeenhancedwi ththepositionandyawintegralerror, asfollows: S ( p ;y 1 ;y 2 )= 3 y 2 + 2 W 2 y 1 + 1 W 1 ( p + y 1 ) (7.53) r d = C C r S C q e (7.54) 163

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Table7.1:Helicopterparameters. I = diag (0 : 18 ; 0 : 34 ; 0 : 28) kg m 2 ;m =8 : 2 kg;g =9 : 81 m=sec 2 x t = 0 : 91 m;z t = 0 : 08 m;z m = 0 : 235 m;x m = y m = y t =0 K =52 N m=rad;C M =0 : 004452 m= p N;D M =0 : 6304 N m Table7.2:Dragandservoparameters. d fx =0 : 06 ;d fy =0 : 132 ;d fz =0 : 09 ;d vfy =0 : 0072 ;d hsz =0 : 006 kg=m; x hs = 0 : 71 m;u i =4 : 2 m=sec; s =0 : 1 sec; f =0 : 1 sec Table7.3:Controllergains. M 3 ;i 22 1 diag(3.1,3.1) L 3 ;i 21.5 2 diag(6,6,3) M 2 ;i 7 W 1 diag(8,8,8) L 2 ;i 6.5 W 2 diag(0.1,0.1,0.1) M 1 ;i 2 2 L 1 ;i 1.5 2 for i =1 ; 2 ; 3 k 0.1 where p = e p = e > 0 and W 1 W 2 arediagonalmatricesofpositivegains.Inthiscase, therequirementsofProperty7.4become, L i;j M i;j for i;j =1 ; 2 ; 3 while M j;i
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fortherstmaneuveris: p Ir ( t )= 0BBBB@ 20 20 e 0 : 25 t 30+30 e 0 : 25 t 10+10 e 0 : 45 t 1CCCCA r ( t )=0 Thesecondmaneuveriscomposedoftwoparts.Intherstpart thehelicopterliftsvertically for 7 seconds.Thenitperformsan“ 8 shaped”curvedpathwhileitcontinuestolift.Throughout thewholemaneuvertheverticalvelocityisexponentiallyd ecreasingwhiletheheadingremains constant.Forthesecondmaneuver,thedesiredpositionand headingare: p Ir ( t )=(00 7(1 e 0 : 3 t )) T for t 7 p Ir ( t )= 0BBBB@ 20 1 cos 2 23 ( t 7) 10sin 4 23 ( t 7) 7(1 e 0 : 3 t ) 1CCCCA for t> 7 r =0 Duringtheexecutionofbothofthemaneuvers,thecomponent softhewindspeedintheinertia coordinatesare(in m=sec ): v I w ( t )=2sin( t ) v I w ( t )=2cos(0 : 75 t + = 2) v I w ( t )=0 Thecontrollergainsassociatedwiththeattitudedynamics aretunedbasedonthegainrequirementsofTheorem7.1.Theyaresufcientlyhighinorderfort hehelicoptertorapidlyobtainits desiredorientation.Thesaturationgainsaretunedbasedo nthegainrequirementsofProperty7.4. Inaddition, p Ir;z and M 3 ; 3 complywithProperty7.1.Tocompensatetheeffectoftheant i-torque Q M andthemodeluncertainty,asteadystatevalueoftheappin ganglesisrequired.Thissteady statevalue,throughtheparasiticforces X M ;Y M and Y T causesanoffsetinthetranslationalpositionerror.Thissteadystateoffsetisminimizedbyincre asingthegainsofthediagonalmatrices 165

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Table7.4:Controlleroutline. v I d =_ p Ir d = p Ir + ge 3 + S ( e p ;e v ) k p Ir + ge 3 + S ( e p ;e v ) k T M = m k p Ir + ge 3 + S ( e p ;e v ) k p d q d = Z 1 () % d 1 e % k 3 ; 3 e % r d = C C h r S C q e i ~ v = I B d +^ B d I B e ( ; ) T 2 e v c = A 1 ( T M )[ B ( T M )+~ v ] W 1 ;W 2 .ThecontrollergainsusedforthesimulationareshowninTa ble7.3.Thechoiceofthe linearsaturationfunctionsatisfyingtherequirementsof Denition7.1isthefollowing: ( s )= 8>>>>>><>>>>>>: s j s j L sgn ( s ) h sin j s j L 2( M L ) M L + 1 2 ( j s j L )+ L i L< j s j 2 M L sgn ( s ) M j s j > 2 M L Thepositionresponseintheinertiacoordinates,versusth edesiredtrajectorieswithrespectto time,areillustratedinFigure7.10andFigure7.11forthet womaneuvers.Thehelicopterposition ininertiacoordinatesisillustratedinFigure7.12andFig ure7.13.Theorientationangles,forthe twocontrolschemes,aredepictedinFigure7.14andFigure7 .15.Finally,therotorsthrustsand theappinganglescanbeseeninFigure7.16andFigure7.17. Thenumericalresultsillustratethe controller'ssuccessfultrackingperformance.Eventhoug h,theproposeddesignisamodelbased controller,itexhibitssignicantrobustnessattributes towardsconsiderableparametricandmodel uncertainty.Figures7.14and7.15indicatethattherollan dpitchboundwhichguaranteethatthe helicopterwillnotoverturn,ismetevenintheaggressivep artofthemaneuvers. 166

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0 5 10 15 20 25 30 0 7 14 21 x (m)time (sec) 0 5 10 15 20 25 30 32 24 16 8 0 y (m)time (sec) 0 5 10 15 20 25 30 12 8 4 0 z (m)time (sec) Figure7.10: Firstmaneuver :Referencepositiontrajectory(dashedline)andactualhe licopter trajectory(solidline)expressedintheinertialcoordina teswithrespecttotime. 7.8Remarks ThisChapterhaspresentedabacksteppingpositionandhead ingtrackingcontrollerforhelicopters.Thehelicoptermodelisrepresentedbytherigidbo dyequationsofmotionenhancedbya simpliedmodelofforceandtorquegeneration.Thecontrol lerassumesfullavailabilityofallthe helicopter'sstatevariablesofthetranslationalandatti tudedynamics.Thedesignoutlinefollows atypicalbacksteppingdesignforfeedbacksystems.Thecho iceofthepseudocontrolsistaken withcautionavoidingunnecessarytermscancellations.Th isresultsinacontrollerthatincludes aminimalamountoftermsrequiredtostabilizetheoveralls ystem.Asummaryofthecontroller inputsandpseudocontrolsisgiveninTable7.4. Themainideaofthedesignistheuseofthedirectionandmagn itudeofthethrustvectorto stabilizethepositionerrordynamics.Thechoiceofthebac ksteppingpseudocontrolsresultsin twointerconnectedsubsystemsrepresentingthetranslati onalandattitudedynamicserrorscorrespondingly. 167

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0 5 10 15 20 25 30 2 9 20 31 42 x (m)time (sec) 0 5 10 15 20 25 30 12 6 0 6 12 y (m)time (sec) 0 5 10 15 20 25 30 10 5 0 z (m)time (sec) Figure7.11: Secondmaneuver :Referencepositiontrajectory(dashedline)andactualhe licopter trajectory(solidline)expressedintheinertialcoordina teswithrespecttotime. Thetranslationalerrordynamicsarecontrolledbyanested saturationfeedbacktermandatthe sametimeareperturbedbyaboundedfunctionofthedirectio nalerror.Theattitudecontroldesign isbasedonthestructuralpropertiesoftherotationmatrix anditisenhancedwithspecialterms thatcanguaranteethatthehelicopterwillnotoverturnini tsefforttotrackthepredenedposition referencetrajectory.Theattitudeerrordynamicswillber enderedexponentiallystabledrivingthe translationalerrordynamicsgloballyuniformlyasymptot icallystable. Thephilosophyofthisworkdictatesthatforeachcontrolle rdesignastandardidentication procedureisproposedthatwillprovidethemodelparameter softhehelicopterbasedonexperimentalightdata.Theapplicabilityofthecontrollerisl imitedifthedesignerdoesnothavea practicalmethodtoextractthemodelparametersoftheheli copter.Theparametricidentication ofnonlinearcontinuousdynamicsystemscanonlytakeplace inthetimedomain.However,time domainparametricidenticationmethodsforightsystems arecomputationallyinefcientand lesseffectivecomparedtofrequencydomainidentication methods[105].Inthetimedomain approacheachiterationoftheidenticationalgorithmreq uirestheintegrationofthenonlinear differentialequationsofthesystemforthecalculationof thecostfunctionvalue.Thisprocedure 168

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signicantlyincreasesthecomputationalload.Inadditio n,inreallifeapplicationsthecontroller algorithmisexecutedinamicroprocessoronboardthehelic opter.Theprocessingofthealgorithmstakesplaceindiscretetimeandthesamplingeffects houldbetakenintoaccount. Althoughtheproposedcontrollerexhibitssignicantrobu stnesstoparametricuncertainty,still afairknowledgeofthemodelparametersisnecessary.Dueto thelackofanefcientidentication methodthetestingoftheproposedalgorithmisrestrictedo nlytonumericsimulationsbasedonthe MITs X-Cell.60 smallscalehelicopterparameters. ThegoalofthenextChapteristopresentabacksteppingalgo rithmbasedonthediscretenonlinearhelicopterdynamics.Thediscretizationoftheheli copterdynamicsfacilitatestheidenticationproceduresinceasimplerecursiveleastsquarealgori thmcanbeusedforthedeterminationof themodelparametersbasedontheightdata.Duetothediscr etizationofthehelicopterdynamics thenewdesignisnotequivalentwiththebacksteppingcontr ollerdescribedinthisChapter.The proposedcontrollerofthenextChapterprovidesapractica lsolutionwhichcanbedirectlyapplied toreallifeapplications.Theperformanceofthecontrolle risevaluatedusingthe X-Plane simulator. 169

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Figure7.12: Firstmaneuver :Referencepositiontrajectory(solidline)andactualhel icopter trajectory(dashedline)withrespecttotheinertialaxis. Figure7.13: Secondmaneuver :Referencepositiontrajectory(solidline)andactualhel icopter trajectory(dashedline)withrespecttotheinertialaxis. 170

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0 5 10 15 20 25 30 0.7 0.35 0 0.35 0.7 f (rad)time (sec) 0 5 10 15 20 25 30 0.5 0.25 0 0.25 0.5 q (rad)time (sec) 0 5 10 15 20 25 30 0.4 0.2 0 0.2 0.4 y (rad)time (sec) Figure7.14: Firstmaneuver :Euler'sorientationangles. 0 5 10 15 20 25 30 0.8 0.4 0 0.4 0.8 f (rad)time (sec) 0 5 10 15 20 25 30 0.6 0.3 0 0.3 0.6 q (rad)time (sec) 0 5 10 15 20 25 30 0.3 0 0.3 y (rad)time (sec) Figure7.15: Secondmaneuver :Euler'sorientationangles. 171

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0 10 20 30 100 150 200 TM (N)time (sec) 0 10 20 30 5 10 15 TT (N)time (sec) 0 10 20 30 0.25 0.1 0 0.1 0.25 a (rad)time (sec) 0 10 20 30 0.25 0.1 0 0.1 0.25 b (rad)time (sec) Figure7.16: Firstmaneuver :Mainandtailrotorthrust T M ;T T andtheappingangles a;b 0 10 20 30 80 100 120 140 TM (N)time (sec) 0 10 20 30 4 6 8 TT (N)time (sec) 0 10 20 30 0.25 0.1 0 0.1 0.25 a (rad)time (sec) 0 10 20 30 0.25 0.1 0 0.1 0.25 b (rad)time (sec) Figure7.17: Secondmaneuver :Mainandtailrotorthrust T M ;T T andtheappingangles a;b 172

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Chapter8:TimeDomainParameterIdenticationandApplied DiscreteNonlinearControl forSmallScaleUnmannedHelicopters ThisChapterdealswiththedualproblemofparametricident icationandnonlinearcontrol ofhelicopters.ThegoalofthisChapteristhedevelopmento fpracticalidenticationandcontrol solutionfordirectapplicationtoanautonomoushelicopte rightsystem.Althoughmostcontrollerdesignsareincontinuoustime,thischapterconsid ersthediscretetimedynamicsofthehelicopter.Theshiftoftheinitialhelicoptercontrolprobl emtothediscretetimeistwofold:Control algorithmsareexecutedbymicroprocessors.Thediscretiz ationeffectofthehelicopterdynamics shouldbeaccountedbythecontroller.Inaddition,timedom ainparametricidenticationismuch simplerandcomputationallymoreefcientwhenthesysteme quationsarediscretized. AsimpleRecursiveLeastSquare(RLS)algorithmisusedfort heparameteridenticationin thetimedomain,theobjectivebeingthederivationofsyste mdynamicsthatarebothminimalin complexityandaccurateforcontroldesignindiscretetime .Thecontrollerisdesignedbasedona discretetimebacksteppingtechnique,forthetrackingofp redenedpositionandyawtrajectories. Thedevelopedcontrollerprovidesdesignfreedominthecon vergencerateforeachstatevariable ofthecascadestructure.Thisisofparticularinterestsin cecontroloftheconvergencerateineach levelofthecascadestructureprovidesbetterightresult s.Boththeidenticationpartandcontrol performanceareevaluatedusing X-Plane 8.1Introduction Theconceptofbacksteppingcontrolforcontinuoustimesys temsinacascadeformhasbeen wellstudiedandanalyzed[43]includingadaptivemodicat ions[49]tocopewithsystemsinclud173

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ingparameteruncertainties.Inthecaseofthediscretetim esystemstherehasbeensignicant lessworktothespeciceld.Themostdistinctiveworkisfr om[112]dealingwiththeadaptive backsteppingcontrolfordiscretetimesystems. TherstobjectiveofthisChapteristhedesignofanonlinea rcontrollerfortrackingofpredinedpositionandyawtrajectories.Adiscretetimebackst eppingcontrollerbasedonthenonlineardiscretizedequationsofthehelicopterisproposed .Thecontrollerprovidesmoredesign freedomcomparedtothecontinuousbacksteppingcounterpa rtalgorithmproposedin[11,21], sincetheconvergencerateofeachstatevariableofthecasc adestructurecanbemanipulated. Furthermore,thestabilityoftheresultingdynamicscanbe simplyinspectedbytheeigenvaluesof alinearsystemwithoutthenecessityofLyapunov'sfunctio ns.Thoseeigenvaluesaredetermined bythedesigner. ThesecondtaskofthisChapteristoexamineastandardRecur siveLeastSquare(RLS)algorithmforparameterestimationofthenonlineardiscreteti medynamicsofthehelicopter.Boththe identicationandthecontrolresultswheresuccessfullyt estedin X-Plane forthe Raptor90SE RC helicopter.8.2DiscreteSystemDynamics Thediscretenonlinearmodelofthehelicopterdynamicsisd erivedbydirectdiscretization ofthecontinuoustimemodelpresentedinthepreviousChapt er.TheTPPdynamicsareassumed tobeveryfastincomparisonwiththerigidbodydynamicsand onlytheirsteadystateeffectwill beregarded.Thisisatypicalassumptionthattakesplacein thenonlinearcontrollerdesignsthat existsintheliterature.Thedynamicsoftheappingmotion aretreatedasunmodeleduncertainty whichiscompensatedbytherobustnessofthecontrolalgori thm.Therefore,regardingtheTPP anglesthefollowinghold: a = K a u lon (8.1) b = K b u lat (8.2) 174

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where K a K b areconstantparameters.Themagnitudeofthemainandtailr otorthrustwillbe consideredproportionaltothecollectivecontrolcommand s,therefore: T M = K M u col (8.3) T T = K T u ped (8.4) where T M T T arethemagnitudeoftheforcesofthemainandtailrotorresp ectivelywhile K M K T areconstantparameters. Using(8.1)-(8.4)andbyignoringtheeffectoftheanti-tor que Q M to(7.10)forsimplication purposes,acompactformoftheexternaltorqueappliedtoth ehelicopteris: B = ~ Av c + ~ Bu col (8.5) where v c =( u lat u col u lon u col u ped ) T (8.6) with ~ A 2 R 3 3 and ~ B 2 R 3 1 beingparametermatrices. From(7.1),(7.13),(7.3),(7.14),(7.5)byusingEuler'sim plicitmethodfortheapproximation ofthecontinuousderivatives,thefollowingequationsare obtained: p Ik +1 = p Ik + T s v I k (8.7) v I k +1 = v I k + 1 R k e 3 u col; k + 2 e 3 (8.8) B k +1 = B k +( B k ) I ( I ;T s )+ A 0 v c; k + B 0 u col; k (8.9) k +1 = k + T s ( k ) B k (8.10) R k +1 = R k + T s R k ^ B k (8.11) where e 3 =[001] T and T s denotesthesamplingperiod.In(8.9) ( B k ) isamatrixof R 3 p composedonlybynonlinearfunctionsoftheangularvelocit ieswhile I ( I ;T s ) isavectorof R p 1 175

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composedbyinertiatermsandmultipliedbythesamplingper iod T s .Bothofthemsatisfy: ( B k ) I ( I ;T s )= T s I 1 [ I B k B k ] (8.12) Regardingtherestofthetermsin(8.8),(8.9)thefollowing holds: 1 = T s K M m (8.13) 2 = T s g (8.14) A 0 = T s I 1 ~ A (8.15) B 0 = T s I 1 ~ B (8.16) Animportantobservationshouldbegivenregardingthedisc reteapproximationof(8.11).Integrationoftranslationalandrotationdynamicsofarigid body'smotionunderapotentialrequires specialattention.From[57]Runge-Kuttamethodsdonotpre servetheLiegroupstructureofthe congurationspace.Mostimportantlythequantity R k +1 R T k +1 driftsfromtheidentitymatrixas thesimulationtimeincreases.Amoreaccurateintegration of(7.3)couldtakeplacebytheuseof discretevariationalintegrators[35,57],whichpreserve thegeometricpropertiesoftheLiegroup. Thedisadvantageofthisapproachisthattheproposedstruc tureofthediscreteequations-although providingmoreaccuratenumericalsolutions-isverycompl icatedforcontroldesign.Tothisextent animportantconditionfor(8.7)-(8.11)isthatthesamplin gfrequencyissmallenoughthat(8.11) canbeconsideredasaperturbationvalueoftherotationmat rix.Theexperimentalresultshave illustratedthatafrequencyof 50 Hz isadequateenoughfor(8.11)toprovideaccurateresultsev en uptoahorizonoftwotimestepsgiventhecurrentvalueofthe congurationmatrixandcanbe usedforcontroldesign. 176

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8.3DiscreteBacksteppingAlgorithm Considerahelicopterdescribedbythedifferenceequation s(8.7)-(8.11).Theobjectiveistodesignanonlinearcontrollerstabilizingtheposition p Ik andtheyawangle k totherefrencevalues p Ir; k and r; k ,respectively. 8.3.1AngularVelocityDynamics Considering(8.9)anobviouscontrolchoiceforcancelingo utthenonlineartermsoftheangularvelocitydynamicsis: v c; k = A 0 1 B k ( B k ) I ( I ;T s ) B 0 u col; k +~ v k (8.17) where ~ v k =[~ v 1 ; k ~ v 2 ; k ~ v 3 ; k ] T .Theangulardynamicsbecome: B k +1 =~ v k (8.18) while: 266664 u lat; k u lon; k u ped; k 377775 = 266664 u col; k 00 0 u col; k 0 001 377775 1 v c; k (8.19) Theexistenceoftheinverseoftheleftmatrixontherightha ndsideof(8.19)isguaranteedby thefactthatthecollectivecontrol u col; k shouldbeatalltimesdifferentthanzerosinceinight operationsomethrustisneededtocompensatefortheweight force. 8.3.2TranslationalDynamics Theequationoftranslationalvelocityisgivenby(8.8).Us ingthenotationofChapter7,let R k =[ 1 ; k 2 ; k 3 ; k ] where i; k with i =1 ; 2 ; 3 arethecolumnvectorsoftherotationmatrix.Then 177

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thedifferenceequationofthetranslationalvelocitycanb ewrittenas: v I k +1 = v I k + 1 3 ; k u col; k + 2 e 3 (8.20) Thecolumnvector 3 ; k isaunitvectorwithchangingdirectiondependingontheEul erangles.TheideasimilartoChapter7and[21]istochangethedi rectionof 3 ; k andatthesametime adjustthemagnitudeof u col; k toadesiredvectorwhichwillcontrolthetranslationalvel ocity dynamics.Thereforethedynamicsof 3 ; k u col; k arethefunctionwhichshouldbeforwardedin timetodevelopthebacksteppingscheme.Let u col; k +1 = k ,andbyconsidering(8.11)andalso ^ B k e 3 = ^ e 3 B k then: 3 ; k +1 u col; k +1 = R k +1 e 3 k = R k e 3 k T s R k ^ e 3 B k k = R k ( e 3 T s ^ e 3 B k ) k (8.21) Let k +1 = k thenbyforwardingintimetheaboveequationbecomes: 3 ; k +2 u col; k +2 = R k +1 e 3 T s ^ e 3 B k +1 k +1 = R k +1 ( e 3 T s ^ e 3 ~ v k ) k = R k +1 266664 T s ~ v 2 ; k k T s ~ v 1 ; k k k 377775 = X k (8.22) where X k isavectorasdenedbelow.From(8.22)thefollowingequali tieshold: k = e T3 R T k +1 X k (8.23) 264 ~ v 1 ; k ~ v 2 ; k 375 = 264 T s k 0 0 T s k 375 1 264 T2 ; k +1 X k T1 ; k +1 X k 375 (8.24) 178

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Z k +2 = X k z 2 v I k +1 = v I k + 1 Z k + 2 e 3 z 1 p Ik +1 = p Ik + T s v I k z 1 z 2 Z k v I k u col v 1 ; k v 2 ; k k R k +1 p Ik Figure8.1:Interconnectionofthehelicopterdynamicsusi ng(8.23)-(8.27).Theterm z 1 denotes aunittimedelay.Since k = u col; k +2 theexistenceoftheinvertibleoftheleftmatrixontherigh thandsideof(8.24) isguaranteedbythefactthatthecollectivecontrol u col; k shouldbedifferentfromzerosincein ightoperationsomethrustisneededtocompensateforthew eightforce. Let Z k + i = 3 ; k + i u col; k + i with i 2 N .Theassociatedequationsrelatedwiththetranslational dynamicsuptonoware: p Ik +1 = p Ik + T s v I k (8.25) v I k +1 = v I k + 1 Z k + 2 e 3 (8.26) Z k +2 = X k (8.27) Theerrordynamicsofthe p I v I and Z statevariablesare: e p; k +1 = p Ik +1 p Ir; k +1 = p Ir; k +1 + p Ik + T s v I d; k + T s e v; k (8.28) e v; k +1 = v I k +1 v I d; k +1 = v I d; k +1 + v I k + 1 Z d; k + 2 e 3 + 1 e Z d;k (8.29) e Z ;k +2 = Z k +2 Z d; k +2 = Z d; k +2 + X k (8.30) Choosethedesiredvalues: v I d; k = 1 T s p Ir; k +1 p Ik + K 1 e p; k (8.31) Z d; k = 1 1 v I d; k +1 v I k + K 2 e v; k 2 e 3 (8.32) X k = Z d; k +2 + 1 e Z ;k +1 + 2 e Z ;k (8.33) 179

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where K 1 ;K 2 ; 1 ; 2 arediagonalgainmatrices.Afterapplyingthedesiredvalu esof(8.31)(8.33)tothetranslationaldynamicsdescribed(8.28)-(8. 30)oneobtains: 266666664 e p; k +1 e v; k +1 e Z ;k +2 e Z ;k +1 377777775 266666664 K 1 T s 00 0 K 2 0 1 00 1 2 0010 377777775 266666664 e p; k e v; k e Z ;k +1 e Z ;k 377777775 (8.34) Theeigenvaluesoftheaboveequalityaredeterminedbytheg ains K 1 ;K 2 andthepolynomial z 2 1 z 2 .Providedthattheeigenvaluesoftheabovesystemlieinsid etheunitcirclethe translationaldynamicswillbegloballyasymptoticallyst able.Thisresultisveryimportantsince theconvergencerateoftheerrorvariablescanbedetermine dbythedesigner.Bytuningthegains ofthediagonalmatricesappropriately,smoothnessinthe ightbehaviorcanbeachieved.Real ightimplicationsofthisdesignaresignicant.Duetothe factthatsmallscalehelicoptersare verysensitivetocontrolinputs,regulatingtheconvergen cerateimprovestheightbehavior. 8.3.3YawDynamics TheyawdynamicsareobtainedbyEquation(8.10)andmorespe cically: k +1 = k + T s 3 ( k ) B k (8.35) where 3 ( k ) hasbeendenedin(7.24).Let e ; k = k r; k betheerrorintheyaw,thenthe yawerrordynamicswillbe: e ; k +1 = r; k +1 + k + T s 3 ( k ) B k (8.36) 180

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Theaboveequationwillbeshiftedforwardintimeinorderfo rthecontrolcommandstoappear. Thisleadsto: e ; k +2 = r; k +2 + k +1 + T s 3 ( k +1 ) B k +1 = r; k +2 + k +1 + T s 3 ( k +1 )~ v k = r; k +2 + k +1 + T s S k +1 C k +1 ~ v 2 ; k + C k +1 C k +1 ~ v 3 ; k (8.37) Anobviouschoicefortheselectionofthevalueof ~ v 3 ; k whichwillcanceloutthenonlinearterms andstabilizetheyawerrordynamicsis: ~ v 3 ; k = C k +1 C k +1 S k +1 C k +1 ~ v 2 ; k + 1 T s ( r; k +2 k +1 + Me ; k +1 ) (8.38) where M isadiagonalmatrixofgainswheretheabsolutevalueofeach diagonalentryissmaller thanunity.Applyingtheabovevaluefor ~ v 3 ; k theyawerrordynamicsbecome e ; k +2 = Me ; k +1 whichimpliestheasymptoticconvergenceof e ; k tozero.Thecontroldesignissummarizedby thefollowingalgorithm: Initialization:Attheinitialstep,whenthealgorithmise xecutedforrsttimeset u col (0) equaltoaverysmallquantityclosetozero.Thiswillguaran teetheexistenceoftheinvertiblematrixin(8.19). Executionattimestep k :Atanygiventimestep k thefullstatevectorisconsideredavailable.Tocalculatethedesiredcontrolcommandsobtainedby thebacksteppingalgorithmthe followingstepsshouldbefollowed. – Step1:Calculate (i) R k +1 from(8.11). (ii) v I k +1 from(8.8). (iii) v I k +2 from: v I k +2 = v I k +1 + 1 R k +1 e 3 k + 2 e 3 181

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– Step2:Calculatesequentiallythefollowingequations: p Ik +1+ i = p Ik + i + T s v I k + i for i =0 ; 1 ; 2 – Step3:Calculatesequentiallythefollowingequations: v I d; k + i = 1 T s p Ik + i p Ir; k +1+ i + K 1 p Ik + i p Ir; k + i for i =0 ; 1 ; 2 ; 3 – Step4:Calculatesequentiallythefollowingequations: Z d; k + i = 1 1 v I k + i v I d; k +1+ i + K 2 v I k + i v I d; k + i 2 e 3 for i =0 ; 1 ; 2 – Step5:Calculate X k from(8.33). – Step6:Calculate k from(8.23)and ~ v 1 ; k ; ~ v 2 ; k from(8.24). – Step9:Calculate (i) k from(8.10). (ii) ~ v 3 ; k from(8.38). – Step10:Calculate v c; k from(8.17). – Step11:Calculatethecontrolcommands u lat; k ;u lon; k and u ped; k from(8.19). – Step12:Setthefollowingvalues: u col; k = k k = k 182

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8.4ParameterEstimationUsingRecursiveLeastSquares Animportantpartofthedesignbeforetheimplementationof theightcontrolalgorithmisthe parameterestimationofthedifferenceequations(8.8),(8 .9).Suggestionsforonlinealgorithms [81]areRLSorGradientDescentmethods.InthisChapterast andardRLSalgorithmisused.The formoftheRLSalgorithmcanbefoundinmosttextbooksrelat edwithparameteridentication [69].Let y k bethemeasurementvectorwhere y k 2 R n and k 2 R N istheparametersvector whichisgoingtobeestimated.Then,themeasurementvector canbemodeledas: y k +1 = h k ^ k (8.39) where h k 2 R n N ,whilethemeasurementwillbeconsideredclearfromnoise. Theestimatesof theparametervectorareprovidedbytheiterativeexecutio nofthefollowingalgorithmeachtimea newmeasurementbecomesavailable: K k +1 = P k h Tk [ h k P k h Tk + I n n ] 1 (8.40) P k +1 =[ I N N K k +1 h k ] P k (8.41) ^ k +1 = ^ k + K k +1 [ y k +1 h k ^ k ] (8.42) TheseriesofcalculationsfortheaboveRLSalgorithmasind icatedby[69]is P k K k +1 P k +1 ^ k +1 .Theinitializationofthealgorithmissuggestedtobe P 0 = I N N where isa verylargenumberandforthe ^ 0 agoodinitialguessoftheparametersorjustazerovector. Forthedifferenceequations(8.8),(8.9)describingthetr anslationalandangularvelocitiesof thehelicoptertheaboveRLSalgorithmcanbemodiedinthef ollowingway: y k +1 = 264 v I k +1 v I k B k +1 B k 375 (8.43) 183

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h k = 264 R k e 3 u col; k e 3 00 00( B k ) k 375 (8.44) T k =[ 1 2 I T r T ] (8.45) where k :=( u lon; k ;u lat; k ;u ped; k ;u col; k ) isanmatrixbelongingto R 3 s composedonlybythe controlcommandswhilethevector r 2 R s aretheparametersassociatedwiththetorquevectorin suchamannerthat k r = B 8.5ParametricModel Theidenticationprocedureisaniterativeprocesswhichr equiresbackandforthtestingbetweenmodelingandverifying[70,85].Basedonthesystemeq uationsdescribedin(8.8)and(8.9) theproposedsystemdynamicsaredevelopedwiththedualobj ectiveofminimalcomplexityand satisfactoryresults.Thekeyfeatureistoinserttheterms thathaveadominanteffectinthehelicopterdynamicsandatthesametimeexcludethosethatdeter iorateordonoteffecttheidentier. Thosekeydynamicsareobtainedfromthehelicopterdynamic equationforlinearandangular velocitybysubstitutingtheforceandtorquegenerationde scribedin(7.8)and(7.10)respectively. Afterworkingbackandforthbetweenthesystemequationsan dthevericationoftheexperimentalresultsasimpliedparametricmodelwasconcludedwhic hhasphysicalrational. Thetranslationvelocitydynamicsarestraightforwardand easilyidentiedbyequation(8.8). Theactualinterestandcomplicationsisassociatedwithth eidenticationoftheangularvelocity dynamics.Forstarterssymmetrytotheprincipalaxesisass umed.Thisassumptionsimplies signicantlytheangularvelocitydynamics.Therefore ( B k )= diag ( qr;pr;pq ) and I ( I ;T s )= ( I 1 I 2 I 3 ) .Thesecondsimplicationassumesthatthepositionvector s ~ h M and ~ h T arealignedwith theunitaryvectors ~ j B and ~ k B respectively.Therefore, h BM =[00 z m ] T and h BT =[ x t 00] T Thentheparametersassociatedwiththecontrolcommandsar egivenby r =( r 1 r 2 r 3 ) .The effectofthecommandcontrolstotheangularvelocitydynam icsisgivenbythematrix k = diag ( u lat; k ;u lon; k ;u ped; k ) .Tofacilitatethecontroldesigntheeffectofthecollecti vecontrolcom184

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mandiscompletelydisregardedintheangularvelocitydyna mics.Itisassumedthatthecollective commandtakesthetrimvalue u col = mg=K M .If u col takessmallvalues,thentheinversematrix in(8.19)mayleadtoexcessivecyclicandpedalcommands.Th eexperimentalresultsindicate thatthisadditionalsimplicationassumptiondoesnothav easignicantimpactneithertothe parametricidenticationnortotheperformanceofthecont rolalgorithm.Then,theparametric modeloftheangularvelocitydynamicsisgivenby: p k +1 = p k + I 1 q k r k + r 1 u lat; k q k +1 = q k + I 2 p k r k + r 2 u lon; k (8.46) r k +1 = r k + I 3 q k p k + r 3 u ped; k 8.6ExperimentalResults Theparameterestimationalgorithmandthecontrollerdesi gnweretestedonthe Raptor90SE modelinstalledin X-Plane .Theuseof X-Plane providesagoodindicationoftheapplicabilityof theapproachtorealightapplications.Thelackofanyapri oriknowledgeofthesystemdynamics,makesitamorerealisticvalidationofthedesign.8.6.1TimeHistoryDataandExcitationInputs Animportantpartoftheparameterestimationproceduredes cribedinthisChapter,isthecollectionoftheexperimentalighttestdatawhicharerequir edfortheidenticationofthemodel. Theightdataoftheparametricidenticationprocedurear egeneratedbytheexecutionofspecial excitationinputstothehelicopter.Similarlytothefrequ encyidenticationcase,frequencysweeps werealsousedfortheexcitationofthehelicopter.Thedeta iledguidelinesofthefrequencysweeps inputsignalsaregiveninSection5.7.Foreachightrecord acomputerizedfrequencysweepis appliedtooneoftheinputswhiletherestremainasuncorrel atedaspossiblefromtheprimary 185

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inputofinterest.Duringtheexecutionofthefrequencyswe epitisimportantthethehelicopter doesnotdivergesignicantlyfromtheoperatingpoint. Apartfromthepedalcontrol u ped theamplitudeoftheexcitationsisadjustedinsuchamanner thatthehelicopterwillnotdriftawaysignicantlyfromth ehovertrimmedoperation.Sincethe Raptor modelinstalledin X-Plane doesnotincludeayawdamperoragyro,thebehaviorofthe helicopter'sheadingwasmuchmoresensitivethantheoneac countedinactualsmallscalehelicopters.Thedesignoftheexcitationsignalwasmuchmorech allengingthantherestofthecontrolssinceforthelongperiodofthesweeptheyawvelocityi ncreasessignicantly.Theexcitation signalappliedwasbasedonthefrequencysweepsandatthebe ginningofeachsinusoidalwaiving theamplitudewasdeterminedtopreservetheyawvelocitybe tweensomebounds. Theindividualightrecordsproducedbytheimplementatio nofthefrequencysweepsare concatenatedtoasinglerecord.Theconcatenatedrecordis processedbytheRLSalgorithmfor theestimationofthehelicopter'smodelparameters.Thesa mplingrateforthecollectionofthe ightdatawassetto 50 Hz 8.6.2Validation Inordertovalidatethemodeltheactualhelicopterissetto hovermodeanddoublets(symmetricalpulses)areappliedbythecontrolcommands.After eachdoubletthehelicopterreturnsto thehoveringmodeuntilanotherexcitationoccurs.Thoseex citationstakeplaceforallthecontrol inputs. Thecomparisonbetweentheactualandestimatedtranslatio nalandrotationalvelocitiescanbe seeninFigure8.2andFigure8.3,respectively.Basedonthe dataitcanbeseenthatthemodelalso providessufcientestimatesforlargevariationsintheli nearvelocities.Theidentiedparameters areshowninTable8.1.Thevericationresultsillustratet hepredictivecapabilityoftheidentied modelforthehorizonofonetimestep.Eachestimatedpointi nFigure8.2andFigure8.3isgeneratedbysubstitutingtheactualvalueofthehelicopter'sst ateandinputtotherighthandsideofthe differenceequations(8.8)and(8.9). 186

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8.6.3ControlDesign Allofthecontrolcommandsweresaturatedinordertolieint heinterval [ 11] since X-Plane does notacceptvaluesoutofthisscope.However,(8.19)require sthat u col 6 =0 foreverytimestep. Therefore,fortheexecutionofthecontrolalgorithmasimp lelineartransformationmodiedthe valuesofthecollectivecommandsuchthat u col 2 (01] .Forthepresentationofthecontroller resultsthecollectivesignalwasagainrevertedtotheinte rval [ 11] .Themodelingsimplication involvingthematrix resultedintheequality v c =( u lat u lon u ped ) T .Insteadofthepedalcontrolinputdescribedby(8.17)and(8.38)amoresimplePDcon trollerwithbiaswasappliedwith sufcientresults.Theproposedpedalcontrolcommandused was u ped; k = 0 : 5 e ; k 0 : 08 z; k 0 : 18 (8.47) Asecondmodicationthattookplacewasthechangeoftheide ntiedvalues r 1 ;r 2 .Thebacksteppingalgorithmisdesignbasedontheassumptionofperf ectknowledgeofthehelicopterdynamics.However,althoughtheidenticationresultswerea dequatethereisstillsomeuncertainty associatedwiththemodelsparametersespeciallywiththea ngularvelocitydynamicsdescribedby (8.9).Incasesofparameteruncertaintyexactdynamicscan cellationisnotagoodpractice.Since theinverseofthosevaluesisrequiredforthecalculationo fthecorrespondingcontrolcommand, thesmallerthevaluethehigherthecontrolcommandwillbe. Tothisextentthosevalueswere modiedtoregulatethecycliccontrolcommandstoachievet hedesiredtrackingperformance. Theparametersweresignicantlyincreasedwiththenewval uesbeing r 1 =20 ;r 2 =10 Ingeneral,thetimedomainparametricidenticationwaspr oventobesignicantlylesseffectivethanthefrequencydomainidenticationproceduredes cribedinChapter5.ThemaindifcultyoftheRLSalgorithmwasencounteredintheestimation oftheparametersassociatedwith theangularvelocitydynamics.Althoughthevericationre sultsweresatisfactory,theestimated parametersexhibitincreasedinsensitivityoftheangular velocitywithrespecttothecontrolinputs. 187

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Table8.1:Identiedsystemparameters. 1 2 I 1 I 2 I 3 r 1 r 2 r 3 -0.4857 0.0944 0.0256 0.0046 0.0452 0.7854 0.4994 0.1784 Table8.2:Valuesofthediagonalgainmatrices. K 1 0.92 0.92 0.93 K 2 0.93 0.93 0.94 1 0 0 0 2 0.9 0.9 0.95 Thepoorperformanceofthetimedomainidenticationcanbe signicantlyimprovedifsimple nonparametricmodelsofthefrequencydomainareusedasind icators. Thereferencemaneuverisatrapezoidalvelocityproleint helateralandlongitudinaldirectionsidenticaltotheonedescribedinSection6.7.Thro ughoutthemaneuverthereference headingremainsconstantwiththevalue r =0 .Thegainsofthediagonalmatricesusedfor thebacksteppingcontrollercanbeseeninTable8.2.Thetun ningofthecontrollergainsisavery straightforwardprocess.Theconvergentrateforeacherro rstatevariablein(8.34)shouldbefaster fromtheconvergentrateoferrorvariablesthatlieinhighe rlevelsofthesystem.Thisrequirement reectsthenaturaltimescalingbetweenthehelicopterdyn amics.Thetranslationaldynamics aresignicantlyslowerthantheattitudedynamics.Thehel icoptervelocityresponsesversusthe referencetrajectoryareillustratedinFigure8.4.TheEul eranglesofthehelicopteraredepictedin Figure8.5.Thepositionofthehelicopterintheinertialco ordinatesisgiveninFigure8.6.Finally thecontrolinputsareshowninFigure8.7.Theperformanceo fthenonlinearcontrollerwasexcellent.Thechangeinthevaluesof r 1 r 2 parametersresolvedtheshortcomingsofthetimedomain parameterestimationandresultedtoacontrollerdesignof hightrackingperformance. 188

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8.7Remarks ThisChapterhaspresentedatimedomainparameterestimati onschemeandanonlineardiscretetimecontrolalgorithmforhelicopters.AsimpleRLSa lgorithmisusedfortheparameter estimationprocedure.Theexcitationsignals,usedtoprod ucetheidenticationdata,werefrequencysweepsforeachofthecontrolcommands.Thesecondta skoftheChapteristhedesign ofanonlinearcontrollerbasedonthediscretetimediffere nceequationsofthehelicopter.Dueto thecascadeformofthesystemadiscretetimebacksteppingm ethodisproposed.Themaincontributionofthisdesignisthefactthattheconvergencerat eofthecascadesystem'sstatevariables totheirdesiredvalues,canbedeterminedbythedesigner.T unningthosegainsappropriately,resultsinsignicantimprovementoftheightbehavior.Thea bovecontroldesignconsidersperfect knowledgeofthehelicopterdynamics.Howeverasillustrat edbytheidenticationresultsthereis aparametricerrorassociatedwiththeangularvelocitydyn amics.The X-Plane simulatorisitself asourceofuncertaintyduetosmalluctuationinthesampli ngrate.Theexperimentalresultshave illustratedthateveninthatcasethecontrollerisrobuste noughtodealwithboththeendogenous andexogenousuncertainty. ThegoalofthenextChapteristhedevelopmentofanimproved timedomainsystemidenticationmethod.Thediscretehelicopterdynamicsarerep resentedbyaTakagi-Sugenofuzzy model.Insteadofusingasinglenonlinearmodelfortherepr esentationofthehelicopterdynamics, theTakagi-Sugenofuzzysystemisaninterpolatorofmultip lenonlinearmodelswhichdepend onthehelicopter'soperatingcondition.Theparametersof theTakagi-Sugenofuzzysystemare estimatedbythesimpleRLSalgorithmdescribedinthisChap ter.Theidenticationresultsofthe fuzzysystemindicatedsignicantimprovementrelativeto theparameterestimationapproachof thisChapter. 189

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20 40 60 80 100 120 2 0 2 vx I (m/sec)time (sec) 20 40 60 80 100 120 2 0 2 vy I (m/sec)time (sec) 20 40 60 80 100 120 0.5 0 0.5 1 vz I (m/sec)time (sec) Figure8.2:Comparisonbetweentheactual(solidline)ande stimated(dashedline)linear velocitiesusingthevericationdata. 20 40 60 80 100 120 2 0 2 p (rad/sec)time (sec) 20 40 60 80 100 120 2 0 2 q (rad/sec)time (sec) 20 40 60 80 100 120 2 0 2 r (rad/sec)time (sec) Figure8.3:Comparisonbetweentheactual(solidline)ande stimated(dashedline)angular velocitiesusingthevericationdata. 190

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0 10 20 30 40 50 60 70 10 0 10 20 vx I (m/sec) 0 10 20 30 40 50 60 70 2 0 2 4 vy I (m/sec) 0 10 20 30 40 50 60 70 4 2 0 2 time (sec)vz I (m/sec) Figure8.4:Referencetrajectory(dashedline)andactualv elocitytrajectory(solidline)ofthe helicopterexpressedininertialcoordinateswithrespect totime. 0 10 20 30 40 50 60 70 0.5 0 0.5 q (rad) 0 10 20 30 40 50 60 70 0.2 0 0.2 f (rad) 0 10 20 30 40 50 60 70 1 0 1 2 y (rad)time (sec) Figure8.5:Euler'sorientationangles. 191

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Figure8.6:Referencepositiontrajectory(solidline)and theactualhelicopterposition(dashed line)withrespecttotheinertialaxis. 0 10 20 30 40 50 60 70 1 0.5 0 0.5 u lon 0 10 20 30 40 50 60 70 0.2 0 0.2 u lat 0 10 20 30 40 50 60 70 1 0 1 u ped 0 10 20 30 40 50 60 70 1 0 1 u coltime (sec) Figure8.7:Controlinputs. 192

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Chapter9:TimeDomainSystemIdenticationforSmallScale UnmannedHelicopters UsingFuzzyModels TheobjectiveofthisChapteristopresentasystemidentic ationmethodsuitableforhelicopter.TheproposedmodeltobeidentiedisaTakagi-Sugen ofuzzysystem,representingthe translationalandrotationalvelocitydynamicsoftheheli copter.Fortheparameterestimationof theTakagi-SugenosystemaclassicalRLSalgorithmisused, whichallowstheidenticationto takeplaceon-linesinceparameterupdatesareproducedwhe neveranewmeasurementbecomes available.Thevalidityofthisapproachistestedusing X-Plane 9.1Introduction TheobjectiveofthisChapteristoexamineastandardtechni queoffuzzysystemidentication anditsapplicabilitytohelicopters.TheChapterillustra tesatimedomainidenticationapproach thatcanbeimplementedon-lineinthesensethatestimatesc anbemadeeachtimeanewstate measurementisavailable.Resultsillustratethatthismet hodissuccessfulofproducinganonlinear discretemodelofrelativelylowcomplexityandhighaccura cy.Theresultingmodelissuitablefor thedesignofmodelbasednonlinearfuzzycontrollers. Morespecically,aTakagi-Sugenofuzzysystemisdevelope dbasedonthediscretizeddynamicsoftranslationalandangularvelocityderivedinCha pter8.Afterthedevelopmentofthe Takagi-Sugenosystem,astandardRLSalgorithmisusedtoes timateitsparameters.Theresulting fuzzysystemisaninterpolatorofnonlineardiscretesyste mswhichdependsonthehelicopter's ightcondition. 193

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9.2Takagi-SugenoFuzzyModels ThisSectionillustrateshowRLScanbeusedtoidentifythep arametersofaTakagi-Sugeno fuzzymodel[101]usedtorepresentthediscretedynamicsof asinglestatemodel.Thisapproach willbemodiedtoidentifythecompleterotorcraftdynamic s.TheidenticationoftheTakagiSugenosystemproposedinthispaperisbasedonthemethodde scribedin[81]. TheTakagi-Sugenofuzzysystemsarecharacterizedas“func tionalfuzzysystems"[81]since theiroutputisafunctionratherthanamembershipfunction center.Thefuzzysystemisastatic nonlinearmappingbetweentheinputsandtheoutputsandthe yarecomposedbyRrulesofthe form If-Then .ItwillbeillustratedhowtheTakagi-Sugenosystemcanbeu sedtoadjustitsparametersinordertoprovidethebestestimate ^ y ( k +1) ofthestate y ( k ) giventheinputstothefuzzy system ( x 1 ;x 2 ;:::;x n ) 2 R n ,thestatevector Y ( k )=[ y ( k ) ;y ( k 1) ;:::;y ( k m )] 2 R m and theinputsoftheplant U ( k )=[ u 1 ( k ) ;u 2 ( k ) ;:::;u p ( k )] 2 R p .Followingsimilarnotationof[96] the i th ruleoftherulebasecanbewrittenas: If ( F j x 1 and F w x 2 and...and F l x n ) Then ^ y i ( k +1)= i; 1 1 ( Y ( k ) ;U ( k ))+ + i;d d ( Y ( k ) ;U ( k )) where ^ y i ( k +1) isthetheestimateof y ( k +1) givenbythe i th rule.Moreover, F b a isafuzzyset denedas: F b a := f a; F b a ( a ): a 2 R and F b a ( a ) 2 [01] g (9.1) Asmentionedin[81,96]themembershipfunction F b a ( a ) describesthecertaintythatthevalueof a representedbythelinguisticvariable ~ a canbedescribedbythelinguisticvalue ~ F b a .Themembershipfunctionsconsideredinthispaperarebelledshape dGaussianswithorwithoutasaturation portion.TheirformcanbeseeninTable9.1.Thefunctions s ( Y ( k ) ;U ( k )): R m + p R with s =1 ; 2 ;:::;d areusedtoindicatethattheparameteridenticationcanbe usedfornonlinear dynamicsystemswhicharelinearintheparameters.Theinfe rencemechanismusedtocalculate thepremiseofeachruleforthispaperwillbethedotproduct .Therefore,themembershipfunction 194

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representingthepremiseoftheabove i th rulewillbe: i ( x 1 ;x 2 ;:::;x n )= F j x 1 ( x 1 ) F w x 2 ( x 2 ) F l x n ( x n ) (9.2) After-centeraveragedefuzzicationtheestimatedoutput oftheidentierwillbe: ^ y ( k +1)= P Ri =1 ^ y i ( k +1) i P Ri =1 i (9.3) where i denotesthepremiseof i th rule i ( x 1 ;x 2 ;:::;x n ) forconvenience.Let: i = i P Ri =1 i (9.4) and: T ( k )=[ 1 ( k ) 1 1 ( k ) R d ( k ) 1 d ( k ) R ] (9.5) T =[ 1 ; 1 R; 1 1 ;d R;d ] (9.6) where ( k ) and arevectorsof R Rd .Fromtheabovetheestimatedstatecanbewrittenas: ^ y ( k +1)= T ( k ) (9.7) Theidenticationoftheparametervector takesplacewiththeRLSalgorithmdescribedinSection8.4.TheestimatesoftheparametervectorusingRLSare providedbythefollowingalgorithm: K ( k +1)= P ( k ) ( k )[ T ( k ) P ( k ) ( k )+1] 1 (9.8) P ( k +1)=[ I dR dR K ( k +1) T ( k )] P ( k ) (9.9) ^ ( k +1)= ^ ( k )+ K ( k +1)[ y ( k +1) T ( k ) ^ ( k )] (9.10) 195

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Table9.1:Gaussianmembershipfunctions. Left l ( x )= 8<: 1 if x c l exp 1 2 x c l l 2 otherwise Centers ( x )=exp 1 2 x c 2 Right r ( x )= ( 1 if x c r exp 1 2 x c r r 2 otherwise TheseriesofcalculationsfortheaboveRLSalgorithmasind icatedby[69]is P k K k +1 P k +1 ^ k +1 .Theinitializationofthealgorithmissuggestedtobe P (0)= I dR dR where isa verylargenumberandforthe ^ (0) agoodinitialguessoftheparametersorjustazerovector. Atthispointitshouldbementionedthattheinputstothefuz zysystem ( x 1 ;x 2 ;:::;x n ) could beasubsetofthestatevector.Ingeneralthechoiceofthein putstothefuzzysystemshouldbe descriptivevaluesoftheoperationalconditionofthesyst emtobeidentied. 9.3ProposedTakagi-SugenoSystemforHelicopters Aspreviouslystated,themainobjectiveofthispaperistoi dentifyaTakagi-Sugenofuzzy systemthatbestdescribesthediscretedynamicbehaviorof theactualhelicopter.Basedonthe systemequationspresentedin(8.8)and(8.9)aTakagi-Suge nosystemwillbedevelopedwiththe dualobjectiveofminimalcomplexityandsatisfactoryresu lts.TheTakagi-Suegnomodelisbased onthesimplicationassumptionsofSection8.5. Asindicatedby(8.8)thevelocitydynamicsdependontheori entationofthehelicopterand theforcevector.TheproposedTakagi-Sugenosystemrepres entingthetranslationaldynamicswill haveasinputthetranslationalvelocityvector v I ( k ) .Letthesystembecomposedby R 1 fuzzy rulesthenthe i th willbe: 196

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If ( F j v I x and F w v I y and F v I z ) Then ^ v I x ( k +1) i = v I x ( k )+ a i1 [sin ( k )sin ( k )+cos ( k )sin ( k )cos ( k )] u col ( k ) ^ v I y ( k +1) i = v I y ( k )+ a i1 [sin ( k )cos ( k ) cos ( k )sin ( k )sin ( k )] u col ( k ) ^ v I z ( k +1) i = v I z ( k )+ a i1 [cos ( k )cos ( k )] u col ( k )+ a i2 (9.11) where F j v I x F w v I y and F v I z arefuzzysetsrepresentingthelinguisticvaluesofthelin guisticvariables ~ v I x ~ v I y and ~ v I z .Fortheangularvelocities,let'sassumethatthefuzzysys temiscomposedby R 2 ruleswiththe i th rulebeing: If ( F e p and F g q and F c r ) Then p ( k +1) i = p ( k )+ b i1 q ( k ) r ( k )+ r i 1 u lat ( k ) u col ( k ) q ( k +1) i = q ( k )+ b i2 p ( k ) r B ( k )+ r i 2 u lon ( k ) u col ( k ) r ( k +1) i = r ( k )+ b i3 q ( k ) p ( k )+ r i 3 u ped ( k ) (9.12) where F e p F g q and F c r arefuzzysetsrepresentingthelinguisticvaluesofthelin guisticvariables ~ p ~ q and ~ r respectively.Theparametersofthefuzzysystemareunknow n.TheRLSalgorithmcan beusedsotheaboveequationinordertoprovideanestimateo ftheTakagi-Sugenoparametersat eachtimestepthatanewmeasurementisavailable.9.4ExperimentalResults SimilartoChapter8,thevalidationofthemodeltookplacef orthe Raptor90SE inthe XPlane simulator.Thesamplingratewassetto 50 Hz .Forthecollectionoftheidenticationdata thesameexcitationinputswereusedwiththeonesdescribed inSection8.6.1. 197

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Table9.2:Gaussiancentersandspreads. Output Linguistic Left Centers Right Variables c l l c c r r ~ v I x -0.5 0.01 0 1 0.5 0.01 ^ v I ~ v I y -1 0.03 0 3 1 0.03 ~ v I z -1 0.3 0 0.3 1 0.3 ~ q -1.5 0.01 0 6 1.5 0.01 ^ q ~ r -4 0.01 0 8 4 0.01 ~ p -0.5 1 0.5 1 ~ q ^ r ~ r -0.5 0.01 0 8 0.5 0.01 ~ p -1.5 0.03 0 6 1.5 0.03 ~ q -2 0.03 0 6 2 0.03 ^ p ~ r -0.5 0.01 0 8 0.5 0.01 ~ p Table9.3:MeanerroroftheTakagi-SugenoRLSincomparison withRLSidenticationoverthe vericationdata. StateEstimate Meanerror Improvement Fuzzy RLS RLS ~ v I x m=sec 0.0456 0.0457 0.2% ~ v I y m=sec 0.0049 0.0052 5.7% ~ v I z m=sec 0.0253 0.0255 0.7% ~ qdeg=sec 1.0432 1.2050 13.4% ~ rdeg=sec 2.2671 4.0852 43.7% ~ pdeg=sec 1.5554 1.8629 16.5% 198

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9.4.1TuningoftheMembershipFunctionsParameters ThecentersandthespreadsoftheGaussianmembershipfunct ionsoftherotorcraft'sTakagiSugenanofuzzysystem,describedby(9.11)-(9.12),aregiv eninTable9.2.The ( ) symbolindicatesthatthespeciclinguisticvariabledoesnotparti cipateintherulebase.Thechoiceof theseparametershasbeenbasedonintuitivecriteriarathe rthananoptimizingmethodoverthe trainingset.Themainideaisthatthelinguisticvaluescor respondingtohoveroperationshould haveawidespreadinordertodominateoverthelinguisticva riablesthatcorrespondtootheright operations.Theleftandrightmembershipfunctionsareuse dassupportivemeanstodescribe thebehaviorofthesystemwhentherotorcraftoperatesouts idetheboundsofthehovermode. Insteadofthisintuitiveapproachtherearemanyoptimizin gmethodstodeterminethemembership functionparametersoverthetrainingset.Agradientdesce nttuningmethodfordeterminingthe membershipfunctionparameters,isgivenin[81],howeverg radientdescentshouldbeusedtotune thefuzzymodelparametersaswell.Moreadvancemethodsfor updatingtherulebaseandthe parametersofthefuzzysystem,bysupervisedandunsupervi sedlearning,ispresentedin[1]. 9.4.2Validation Inordertovalidatethemodel,the Raptor90SE issettohovermode.Theappliedcontrol commandsareperiodicallyperturbingtherotorcrafttoane whoverstateuntilanewexcitation occurs.Thoseexcitationstakeplaceforallthecontrolinp uts. Thecomparisonbetweentheactualandestimatedtranslatio nalandrotationalvelocitiesis showninFigure9.1andFigure9.2correspondingly.Themean errorovertheidenticationdata isillustratedinTable9.3.ThesameTablepresentsthemean erroroftheRLSidenticationprocedureusingthestraightforwardmodelof(8.8),(8.9)inst eadofaTakagi-Sugenofuzzymodel. Thefuzzymodelhasasignicantimprovementintheangularv elocitydynamics,whicharethe biggestidenticationchallenge.Thevericationresults showthesuccessoftheapproachsincethe 199

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20 40 60 80 100 120 2 0 2 v x I (m/sec)time (sec) 20 40 60 80 100 120 2 0 2 v y I (m/sec)time (sec) 20 40 60 80 100 120 0.5 0 0.5 1 v z I (m/sec)time (sec) Figure9.1:Comparisonbetweentheactual(solidline)ande stimated(dottedline)linearvelocities usingthevericationdata.associatederroraresmallandboundedeveninthecaseofhig hexcitations.Basedonthedatait canbeseenthatthemodelalsoprovidessufcientestimates forlargevariationsinthevelocities. 200

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20 40 60 80 100 120 2 0 2 p (rad/sec)time (sec) 20 40 60 80 100 120 2 0 2 q (rad/sec)time (sec) 20 40 60 80 100 120 3 0 3 r (rad/sec)time (sec) Figure9.2:Comparisonbetweentheactual(solidline)ande stimated(dottedline)angular velocitiesusingthevericationdata. 201

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Chapter10:ComparisonStudies ThisChapterprovidesanextensiveevaluationandcomparis onofthecontrollerdesignsthat havebeenintroducedinthisresearch.Evaluationoftheig htcontrolsystemstakesisafunction theexecutionofseveralightmaneuversthataimtotestthe controllerdesignsintermsofstability andtrackingaccuracy.Thetestmaneuversareproducedbyre ferenceposition(orvelocity)and yawreferencetrajectories.Thereferencetrajectoriesar especiallydesignedinordertoexamine theperformanceofthecontrollerdesignsinmultipleopera tingconditionsthatcoverawideportionoftheightenvelope.Someofthereferencetrajectori esareparticularlyaggressiveinvestigatingthephysicallimitationsofthehelicopter.Thecont rollerswheretestedforthe Raptor90SE RChelicopterwhichoperatesinthe X-Plane ightsimulatorenvironment.Detailsregardingthe experimentalplatformtowhichtheexperimentswherecondu ctedaregivenininSection5.10.1. 10.1SummaryoftheControllerDesigns Thecomparisonstudyinvolvestheevaluationofthreecontr ollerdesignsthathavebeeninvestigatedthroughoutthisdissertation.ThisSectionprovid esabriefsummaryofthesedesigns.Two ofthedesignsarepresentedinChapter6.Thethirdcontroll erisdescribedinChapter8. Therstdesignisatrackingcontrollerbasedonthelineari zedhelicopterdynamics.Thecontrollawisseparatedintotwostaticfeedbackloops.Thers tisresponsiblefortheregulationof thelongitudinal/lateraldynamicsandthesecondisrespon siblefortheregulationoftheyaw/heave motion.Thecontrollerdesignisbasedonthestructureofap arametriclinearmodelproposedin [70].Theparametriclinearmodelisgivenin(6.2)andrepre sentsthehelicopterdynamicsathover. Thecontrollerisadditionallyenhancedwiththeintegralo fthepositionerror.Theinclusionofthe 202

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integratordynamicsachievestheattenuationofsteadysta teerrorsduetoparametricandmodeling uncertainty.The Raptor90SE linearmodelidentiedparametersaregiveninTable5.4.Th egain valuesforthetwofeedbackloopsofthecontrollawaregiven inTable6.1. ThesecondcontrollerdesignisbasedonfourindependentSI SOfeedbackloops.Thecontrol lawcompletelydisregardsthecrosscouplingbetweenthehe licopterdynamicsandassignsaPID controllerineachinputofthehelicopter.Themainadvanta geofthisapproachisitssimplicity sincetheparticulardesigndoesnotrequireanyknowledgeo fthehelicoptermodelandthefeedbackgainscanbeempiricallytuned.ThegainsforeachPIDfe edbacklooparegiveninTable6.2. Thethirddesignisadiscretetimenonlinearbacksteppingc ontroller.Theightcontrolsystem isbasedonthenonlinearhelicoptermodelcomposedafullde scriptionoftheequationsofmotion. Theattitudedynamicsandthecollectivecommandareusedto manipulatetheorientationandthe magnitudeofthethrustvectorthatisresponsibleforthege nerationofthehelicopterpropulsive forces.ThevaluesoftheRaptor'snonlinearmodelparamete rsaregiveninTable8.1.ThecontrollergainsaregiveninTable8.2.10.2ExperimentalResults Theperformanceofthecontrollersintermsoftrackingaccu racyanddexterityisexamined bytheexecutionoffourdifferentmaneuvers.Twoofthemane uversinvolvevelocitytracking whiletherestofthemrequirepositiontracking.Mostofthe maneuversrequireaggressiveight operationwhichistranslatedbyincreasedattitudeangles andthrustmagnitude.Themaneuvers arespeciallydesignedsuchthatthehelicoptertransition stomultipleoperatingightmodes.The executionofthemaneuversforcesthehelicoptertocoveraw ideareaoftheightenvelopeandin somecasestoreachitsphysicallimits. 203

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10.3FirstManeuver:ForwardFlight Therstmaneuverunderinvestigationrequiresthecruisin gofthehelicopterbytrackinga simpleforwardightroutine.Thereferencetrajectoryisa trapezoidalvelocityprole.Theheadingofthehelicopterremainsconstantthroughouttheexecu tionofthemaneuverwith r =0 .The forwardightmaneuveriscomposedbyveparts.Intherstp artthehelicopterissettohover byliftingverticallyfromitsstartingpointfromthegroun d.Inthesecondpartofthemaneuver, thehelicopteracceleratesforward.Afterreachingacerta invelocitythehelicopteriscruising withconstantspeed.Inthefourthpartofthemaneuverthehe licopterdeceleratesuntilitsvelocity reacheszero.Then,issettohoveragain.Thereferencevelo cityproleisgivenby: v I r ( t )=0 for t 18 v I r ( t )= 0022sin 30 t 18 T for 18 68 Thereferencevelocityandtheresponseofhelicopterveloc ityresponseproducedbythethree controllersisdepictedinFigure10.1.Thepitch,rollandy awanglesacquiredduringtheexecution ofthemaneuversforthethreedesignsaredepictedinFigure 10.2.Thecontrolinputsgenerated bytheightcontrolsystemsareshowninFigure10.3.Thepos itionandtheorientationofthe helicopterduringtheexecutionofthemaneuversisshownin Figure10.4. Duringtheexecutionofthemaneuverthehelicopterreaches amaximumvelocityof 22 m=sec Basedonextremeighttests,themaximumpossibleforwardv elocitythattheRaptorcanreachis 25 m=sec .ThisisthepickvelocitythattheRCmodelcanacquiredueto thepowerlimitationsof themainrotor.FromFigure10.2itisapparentthattheforwa rdvelocityandaccelerationofthe 204

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helicopterismanipulatedbythepitchangle .Allthecontrollerdesignssuccessfullytrackedthe referencevelocitytrajectory.10.4SecondManeuver:AggressiveForwardFlight Thesecondmaneuverisanaggressiveversionoftheprevious one.Theighttaskinvolves asimilarforwardightprole,however,inthiscasethehel icopterisexpectedtoacquirehigher acceleration.Thus,thehelicoptershouldreachitsmaximu mvelocityinashortertimeinterval. Sincethelongitudinal/lateralaccelerationofthehelico pterhasbeenproventobeproportionalto thepitch/rollangles,ahighertiltingofthefuselageisex pectedduringtheexecution.Theinterest ofthismaneuverfocusontheaccelerationphase.Again,the referenceheadingremainsconstant with r =0 .Thereferencevelocitytrajectoryproleisgivenby: v I r ( t )=0 for t 18 v I r ( t )= 0022sin 14 t 18 T for 18 60 Thereferencevelocitytrajectoryandthevelocityrespons eofthethreedesignsisdepictedin Figure10.5.Thepitch,rollandyawanglesduringtheexecut ionofthemaneuverareillustrated inFigure10.6.Thegeneratedcontrolinputsforthethreede signsareshowninFigure10.7.The positionandorientationofthehelicoptertotheCartesian spaceisshowninFigure10.8. Figure10.6indicatesthatduetotheaggressiveaccelerati onofthehelicopterthepitchangle takesasignicantlyhighervaluecomparedtothepreviousc asestudy.Forthenonlinearbacksteppingdesignthepitchanglemayreachavalueofupto 60 .Inaddition,duringtheaccelerationphase,thecollectivecommand u col issaturatedtoitsmaximumvalue.Thesimultaneous 205

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tiltingofthefuselageandtheincreasetothethrustmagnit udeproducethepropulsiveforcethat isnecessaryfortheaggressiveportionofthemaneuver.Fro mthethreedesigns,thePIDandthe nonlinearcontrollerexhibithigherpitchanglescompared tothelineardesign.Duringthisphase, sincethehelicopterisalreadyoperatingwithitsmaximuma vailablethrustpower,thehightiltof thefuselagedecreasestheverticalcomponentofthethrust vector.Thedecreaseofthethrust's verticalcomponentmakestheweightofthehelicopterthedo minantforceintheverticaldirection.Thisfactresultstothedivingmotionofthehelicopte rwhichisapparentinFigure10.8(b) andFigure10.8(c).SpeciallyinthecaseofthePIDcontroll er,thehelicopteralmosttouchesthe ground.Thedivingmotion,continuousuntilthehelicopter accumulatessufcientmomentum inthelongitudinaldirection,andtheabsolutevalueofthe pitchangleisdecreased.Thiseffect ispurelyrelatedwiththegainselectionofthecontrollers .InthePIDandnonlineardesignthe gainchoiceimposesignicantlyfasterconvergencetothel ongitudinal/lateralmotioncomparedto theheavedynamics.Thereforethecontrollersprioritizet hesedynamicsovertheverticalmotion. Thedivingmotionwouldbenegligibleintheidealcasethatt hecontrollerhadunlimitedpower resourcesandthemagnitudeofthethrustforcecouldcompen sateanydecreasetothevertical componentofthemainrotorthrustcausedbythetiltingofth efuselage. 10.5ThirdManeuver:8Shaped Forthethirdmaneuverthehelicopterisrequiredtoexecute an“8shaped”curvedpath.The headingofthehelicopterremainsconstantthroughoutthee xecutionofthemaneuver.Thismaneuverisapositiontrackingchallenge.Themaneuveriscompos edbythreeparts.Intherstphase thehelicopterliftsverticallyfromthestartingpointand itissettohovermode.Inthesecondpart ofthemaneuverthehelicopterisexpectedtocurvean“8shap ed”pathinthelongitudinaland lateraldirectionwhileitsaltituderemainsconstant.Att heendofthepaththehelicopterissetto hoveragain.Thereferencepositiontrajectoryisgivenby: p Ir ( t )= 00 5 T for t 15 206

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p Ir ( t )= 0BBBBB@ 20 1 cos 20 ( t 15) 14sin 10 ( t 15) 5 1CCCCCA for 15 55 Thereferencepositiontrajectoryversusthepositionresp onsesofthethreecontrollersare illustratedinFigure10.9.Theorientationanglesofthehe licopterduringtheexecutionofthe maneuversforthethreecontrollersdesignsaredepictedin Figure10.10.Thecontrolinputsfor thethreedesignsareshowninFigure10.11. Thetrackingperformanceofthecontrollerdesignswassati sfactory.Allofthecontrollers accuratelysucceedthetrackingtaskofthismoreinvolvedc oordinatemotion.Ingeneral,tracking controllersrequirethatthereferencetrajectoriesaresm ooth(thereferencefunctionsandtheir higherderivativesarecontinuous).Acloseinspectiontot heparticularcontinuoustrajectoryindicatesthatitsrstderivativeisapiecewisecontinuousf unction.Thepointsofdiscontinuityare locatedintheendandthestartpointsofthe8shapedcurveex ecutionwhenthehelicopterinitiates andnalizestohover.Thediscontinuitiesintherstderiv ativeofthereferencetrajectoryresults ininstantaneoustransientjumpsinthecontrolinputs.Toa voidthesetransientsitispreferableto usedifferentiablefunctionsasreferences.Ifthegenerat ionofsuchtrajectoriesisnotpracticalor limitingandsuchtransientsarehazardousfortheoperatio nofthehelicopter,itissuggestedthat thereferencetrajectoriesareprocessedbyanappropriate lowpasslterthatattenuatesthethe highfrequencycomponentsofthesignal.10.6FourthManeuver:Pirouette Thenalmaneuverunderinvestigationisthemostchallengi ngsinceitinvolvesthesimultaneousandsynchronizedhelicoptermotioninalldirectionsof thecongurationspace.Similarlywith theprevioustrajectoriesthehelicopterisinitiallysett ohover.Inthemainpartofthemaneuver, 207

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thehelicopterisrequiredtoexecuteacircularmotioninth elongitudinalandlateraldirections. Duringtheexecutionofthecircularmotionthehelicopteri ssimultaneouslyascendingvertically withexponentiallydecreasingvelocity.Thisresultstoas piralmotionofthehelicopterarounda ctionalcylinder.Attheexecutionofthefthspiralacorr ectionmaneuversetsthehelicopterat thesenderofthecylinder.Thereferencetrajectoryisgive nby: p Ir ( t )=(00 3) T for t 15 p Ir ( t )= 0BBBBB@ 5 1 cos 5 ( t 15) 5sin 5 ( t 15) 23+20 e 0 : 06( t 15) 1CCCCCA for 15 70 Thereferencetrajectoryandthehelicopterpositionrespo nsesforthethreecontrollerdesignsare illustratedinFigure10.13.Theorientationanglesaredep ictedinFigure10.14.Thecontrolinputs generatedbythecontrollersaredepictedinFigure10.15.F inally,thepositionandorientation ofthehelicopterforeachcontrollerdesignduringtheexec utionofthemaneuverisillustratedin Figure10.16. Thelastmaneuverwaspossiblythemostchallenging.Itisar elativeaggressivetrajectory sinceincertaintimeinstancestherollangleofthehelicop terreachesavaluecloseto 60 .Obviously,theperformanceofallthecontrollersissatisfacto ryevenforthisdemandingmaneuver. 208

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10.7Remarks Theextensivecomparisonandighttestingpresentedinthi sChapter,providessomeveryusefulobservationsrelatedwiththeproposeddesignsandtheh elicoptercontrolproblemingeneral. Allthecontrollerdesignswhichwereunderinvestigationi nthiscomparativestudy,exhibitrobustnessandhighaccuracytrackingcapabilitiesevenforrefer encetrajectoriesthatexpectcomposite andaggressivehelicoptermotion. Therstremarkisassociatedwiththelinearcontrollerdes ign.Theoretically,theidentied linearmodelofthe Raptor90SE providesaquasi-steadydynamicdescriptionwhichislimit edto mildightoperation(hover,cruisingwithlowspeed).Howe ver,theexecutedmaneuversrequired theoperationofthehelicopterinseveraloperatingcondit ions.Incertaincasesthereferencetrajectoriesimposedtheoperationofthehelicopterinaggressiv eandhighagilemaneuversthatrequired attitudeanglesofupto 60 .Insuchoperationseventhelinearityassumptionsofthemo delare violated.Asinglecontroller,basedonlyontheidentiedh overmodelwasadequate. Thesuccessofthelineardesignisattributedtothreekeych aracteristics.Thefrequencydomainidenticationmethodproducesmodelsofhighdelitya ndaccuracy.Theprocedureitself, providessignicantunderstandingofthehelicopterdynam ics.Thisinsightisevaluatedandexploitedbythecontrollerdesign.Furthermore,althoughth eoretically,themodelislimitedonlytoa neighborhoodofacertainoperatingcondition,inrealityi tcoversarelativewideareaoftheight envelope.Thesecondcharacteristicisthedecompositiono fthecontrollerdesigntotwofeedback laws,eachofthemresponsibleforadifferentsubsystemoft hehelicopterdynamics.Thisidea passesthephysicalightintuitiontothemathematicaldev elopmentofthecontroller. Asecondremarkworthmentioning,istheperformanceoftheP IDdesign.AsimilarcommentaboutthisissuehasbeenalreadymadeinSection6.7.It wasexpectedthatthePIDperformancewouldbesignicantlyinferiorcomparedtotherestde signs.However,theightresults indicatethatthePIDcontrollerexhibitssatisfactorybeh avior.ThesuccessofthePIDcontrolleris attributedtotheattenuatedcrosscouplingeffectamongst theRaptordynamics.Thisfactissupportedbytheoff-axisresponsesofthehelicopterillustra tedinFigure5.3.ThisFigureillustrates 209

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thatthemagnitudeofthe q=u lat and p=u lon responseslieinthezoneof 20 to 40 dB .Thisis anindicatorofnegligiblecrosscouplingbetweenthehelic opterdynamics. Finally,themostinterestingremarkisthefollowingobser vation:Themotionandcontrol responsesofallthecontrollerdesignsaresimilargiventh atthetrackingobjectiveisachieved. Thisfactindicatesthatduringtheexecutionofareference maneuverthehelicoptermotionand nominalinputsareconstrained.Theconstrainedmotiondep endsonthereferencetrajectoryitself. Foranymethodthatachievesasymptoticconvergenceoftheh elicopteroutputstotheirreference values,aftertheoccurrenceofsomeinitialtransients,th ehelicopterstateandcontrolinputswill asymptoticallyreachamanifold,whichisdictatedbythefu nctionalcontrollabilityofthesystem equations[66].Thesimplestapproximatedescriptionofth ismanifoldisgivenbythedesiredstate vector x d presentedinSection6.2.Forexample,basedon(6.30)thede siredpitchandrollangles aregivenby: d = 1 g [_ u r X u u r ] d = 1 g [_ v r Y v v r ] Theaboveequationindicatesthatthepitchandrollanglesa tasteady-stateconditionareproportionaltothereferencelateral/longitudinalacceleratio nandvelocityofthehelicopter.Anydiscontinuitiestothereferencevelocityandaccelerationwilla ppeartotheattitudeanglesaswell.The abilityoftheapproximatedlinearmodeltoprovidethedesc riptionofthissteady-statemanifoldis attributedtothedifferentialatnessproperty[47].Thek nowledgeofthissteady-statevectorcan beexploitedinthedevelopmentoftrajectorygenerators.F orinstance,fromtheaboveequation, thedesignerwillknowwhatattitudeanglesareexpecteddur ingtheexecutionofapredened referencevelocityprole. 210

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0 10 20 30 40 50 60 70 80 10 0 10 20 30 vx I (m/sec) 0 10 20 30 40 50 60 70 80 1 0 1 vy I (m/sec) 0 10 20 30 40 50 60 70 80 5 0 5 vz I (m/sec)time (sec) Figure10.1: Firstmaneuver(Forwardight) :Referencevelocitytrajectory(greendashedline) andactualvelocitytrajectoryofthelinear(solidbluelin e),PID(reddasheddottedline),nonlinear (dasheddottedblackline)controllerdesigns,expressedi ninertialcoordinateswithrespecttotime. 0 10 20 30 40 50 60 70 80 0.5 0 0.5 q (rad) 0 10 20 30 40 50 60 70 80 0.5 0 0.5 f (rad) 0 10 20 30 40 50 60 70 80 1 0 1 2 y (rad)time (sec) Figure10.2: Firstmaneuver(Forwardight) :Orientationanglesofthelinear(solidblueline), PID(dashedredline)andnonlinear(dasheddottedblacklin e)controllersdesigns. 211

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0 10 20 30 40 50 60 70 80 1 0 1 u lon 0 10 20 30 40 50 60 70 80 1 0 1 u lat 0 10 20 30 40 50 60 70 80 1 0 1 u ped 0 10 20 30 40 50 60 70 80 1 0 1 u coltime (sec) Figure10.3: Firstmaneuver(Forwardight) :Controlinputsofthelinear(solidblueline),PID (dashedredline)andnonlinear(dasheddottedblackline)c ontrollerdesigns. 212

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(a)Linearcontroller. (b)PIDcontroller. (c)Nonlinearcontroller. Figure10.4: Firstmaneuver(Forwardight) :Referencepositiontrajectory(solidline)andactual trajectoryofthecontrollerdesigns(dashedline)withres pecttotheinertialaxis. 213

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0 10 20 30 40 50 60 70 10 0 10 20 30 v x I (m/sec) 0 10 20 30 40 50 60 70 1 0 1 v y I (m/sec) 0 10 20 30 40 50 60 70 5 0 5 v z I (m/sec)time (sec) Figure10.5: Secondmaneuver(Aggressiveforwardight) :Referencevelocitytrajectory(green dashedline)andactualvelocitytrajectoryofthelinear(s olidblueline),PID(reddasheddotted line),nonlinear(dasheddottedblackline)controllerdes igns,expressedininertialcoordinateswith respecttotime. 0 10 20 30 40 50 60 70 1 0.5 0 0.5 q (rad) 0 10 20 30 40 50 60 70 0.5 0 0.5 f (rad) 0 10 20 30 40 50 60 70 1 0 1 2 y (rad)time (sec) Figure10.6: Secondmaneuver(Aggressiveforwardight) :Orientationanglesofthelinear(solid blueline),PID(dashedredline)andnonlinear(dasheddott edblackline)controllersdesigns. 214

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0 10 20 30 40 50 60 70 1 0 1 u lon 0 10 20 30 40 50 60 70 1 0 1 u lat 0 10 20 30 40 50 60 70 1 0 1 u ped 0 10 20 30 40 50 60 70 1 0 1 u coltime (sec) Figure10.7: Secondmaneuver(Aggressiveforwardight) :Controlinputsofthelinear(solidblue line),PID(dashedredline)andnonlinear(dasheddottedbl ackline)controllerdesigns. 215

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(a)Linearcontroller. (b)PIDcontroller. (c)Nonlinearcontroller. Figure10.8: Secondmaneuver(Aggressiveforwardight) :Referencepositiontrajectory(solid line)andactualtrajectoryofthecontrollerdesigns(dash edline)withrespecttotheinertialaxis. 216

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0 10 20 30 40 50 60 70 0 20 40 p x I (m) 0 10 20 30 40 50 60 70 10 0 10 p y I (m) 0 10 20 30 40 50 60 70 8 6 4 2 0 time (sec) p z I (m) Figure10.9: Thirdmaneuver(8shaped) :Referencepositiontrajectory(greendashedline)and actualpositiontrajectoryofthelinear(solidblueline), PID(reddasheddottedline),nonlinear (dasheddottedblackline)controllerdesigns,expressedi ninertialcoordinateswithrespecttotime. 0 10 20 30 40 50 60 70 0.2 0 0.2 q (rad) 0 10 20 30 40 50 60 70 2 0 2 f (rad) 0 10 20 30 40 50 60 70 1 0 1 2 y (rad)time (sec) Figure10.10: Thirdmaneuver(8shaped) :Orientationanglesofthelinear(solidblueline),PID (dashedredline)andnonlinear(dasheddottedblackline)c ontrollersdesigns. 217

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0 10 20 30 40 50 60 70 1 0 1 u lon 0 10 20 30 40 50 60 70 1 0 1 u lat 0 10 20 30 40 50 60 70 1 0 1 u ped 0 10 20 30 40 50 60 70 1 0 1 u coltime (sec) Figure10.11: Thirdmaneuver(8shaped) :Controlinputsofthelinear(solidblueline),PID (dashedredline)andnonlinear(dasheddottedblackline)c ontrollerdesigns. 218

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(a)Linearcontroller. (b)PIDcontroller. (c)Nonlinearcontroller. Figure10.12: Thirdmaneuver(8shaped) :Referencepositiontrajectory(solidline)andactual trajectoryofthecontrollerdesigns(dashedline)withres pecttotheinertialaxis. 219

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0 10 20 30 40 50 60 70 80 5 0 5 10 15 px I (m) 0 10 20 30 40 50 60 70 80 5 0 5 px I (m) 0 10 20 30 40 50 60 70 80 20 10 0 pz I (m)time (sec) Figure10.13: Fourthmaneuver(Pirouette) :Referencepositiontrajectory(greendashedline)and actualpositiontrajectoryofthelinear(solidblueline), PID(reddasheddottedline),nonlinear (dasheddottedblackline)controllerdesigns,expressedi ninertialcoordinateswithrespecttotime. 0 10 20 30 40 50 60 70 80 0.5 0 0.5 q (rad) 0 10 20 30 40 50 60 70 80 1 0 1 f (rad) 0 10 20 30 40 50 60 70 80 1 0 1 2 y (rad)time (sec) Figure10.14: Fourthmaneuver(Pirouette) :Orientationanglesofthelinear(solidblueline),PID (dashedredline)andnonlinear(dasheddottedblackline)c ontrollersdesigns. 220

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0 10 20 30 40 50 60 70 80 1 0 1 u lon 0 10 20 30 40 50 60 70 80 1 0 1 u lat 0 10 20 30 40 50 60 70 80 1 0 1 u ped 0 10 20 30 40 50 60 70 80 1 0 1 u coltime (sec) Figure10.15: Fourthmaneuver(Pirouette) :Controlinputsofthelinear(solidblueline),PID (dashedredline)andnonlinear(dasheddottedblackline)c ontrollerdesigns. 221

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(a)Linearcontroller. (b)PIDcontroller. (c)Nonlinearcontroller. Figure10.16: Fourthmaneuver(Pirouette) :Referencepositiontrajectory(solidline)andactual trajectoryofthecontrollerdesigns(dashedline)withres pecttotheinertialaxis. 222

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Chapter11:ConclusionsandFutureWork Helicoptersarehighlynonlinearsystemswithsignicantd ynamiccoupling.Ingeneral,they areconsideredtobemuchmoreunstablethanxedwingaircra ft.Thegoalofthisdissertationhas beentoexaminethedesignproblemofautonomousightcontr ollersforsmallscalehelicopters. Moderncontroltechniquesaremodelbased,inthesensethat thecontrollerarchitecturedependsonthedynamicdescriptionofthesystemtobecontroll ed.Thisprincipleappliestohelicopteraswell,therefore,theightcontrolproblemistigh tlyconnectedwiththehelicoptermodelingchallenge. Thehelicopterdynamicscanberepresentedbybothlinearan dnonlinearmodelsofordinary differentialequations.Themodeldescriptionshouldaccu ratelypredictthehelicopterresponsefor anygiveninput.Theorderandthestructureofeachmodelisp ostulatedbasedonstandardlaws ofphysicsandaerodynamicsaccompaniedbycertainsimpli cationassumptionsthatreduceas muchaspossiblethecomplexityofthedescription.Thepara metricmodelsshouldencapsulate thedynamicbehaviorofalargefamilyofsmallscalehelicop ters.Linearizedhelicoptermodels havealimitedrangeofvaliditywhichislimitedtoaightop erationinthevicinityofacertain operatingpoint.Ontheotherhand,nonlinearmodelprovide arelativeglobaldescriptionofthe ightenvelope.Itisimportantthatthemathematicalmodel isaccurateyetmanageableenoughfor thedesignofacontrolsystem. Inthisresearchthelinearandnonlinearmodelsstructurea ndorderareadoptedbywidely acknowledgedworksintheareaofthehelicoptercontroland identication.Thelinearmodelis adoptedby[70]anditconsistsofacoupledsystemoftheheli coptermotionvariablesandthe mainrotorappingdynamics.Inthecaseofthenonlinearrep resentationstructure,thiswork 223

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adoptsthemodelproposedin[47].Thismodelconsistsofthe helicopternonlineardynamicequationsofmotionenhancedbyasimpliedmodelofforceandtor quegeneration. Basedontheaboveparametricmodelrepresentations,thisw orkintroducesseveralcontroller designs.Theobjectiveofeachightcontrolsystemisforth ehelicoptertotrackapredened position(orvelocity)andyawreferencetrajectories.All theproposedcontrollerdesignsneglect thecouplingbetweenthehelicopterforcesandmoments.Inp articular,wedisregardtheproduced forcesfromthemainrotorappingmotionandthetailrotori nthelongitudinalandlateraldirectionsofthebody-xedframe.Thisisatypicalassumptionth attakesplaceinmostcontrollersfor helicopterthatexistintheliterature.Theseparasiticfo rceshaveaminimaleffectonthetranslationaldynamicscomparedtothetothepropulsiveforcespro ducedbytheattitudechangeofthe helicopter.Therefore,thisassumptionhasphysicalsense .Asindicatedin[47]theapproximate modelisfeedbacklinearizableand,therefore,infeedback form.Inthiswork,bothlinearand nonlinearproposedcontrollersuseconceptsfromthebacks teppingrecursivedesignmethodology whichissuitableforsystemsofthisform. Afterestablishingamathematicalcontrolframeworkbased onagenericparametrichelicopter model,thenalstepfortheimplementationofthecontrolle ristheextractionofthenumericvaluesofthemodelparameters.Themodelparametersshouldbec hosensuchthatthepredicted responsesofthemodelmatchtheactualightdataoftheheli copter.Theprocessofextracting thenumericvaluesofthemodelparametersbasedonexperime ntalightdatalieintheeldof systemidentication.Thesystemidenticationprocedure sarefurtherclassiedtofrequency domainandtimedomain.Thefrequencydomainidentication ismuchmoresuperiorintermsof calculationcomplexityandaccuracycomparedtothetimedo mainapproaches.However,themain disadvantageofthefrequencydomainidenticationisthat itisrestrictedonlytolinearmodels. Atthispointweneedtomakeclearthatthemainfocusofthisw orkliesinthetheoretical developmentoftheightcontrollers.Eachderivedcontrol lerisattachedwiththemostsuitable systemidenticationapproachinordertoexperimentallyv alidatetheapplicabilityofthedesign. Inareal-lifeapplicationthetheoreticalcontrolframewo rkisworthlessifthehelicoptermodel 224

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parametersareunavailable.Theexaminationofseveralide nticationschemesindicatewhichare themostsuitablepracticesfortheextractionofthehelico pterparameters. 11.1SummaryofContributions Asummaryofthemaincontributionspresentedinthisworkis : Amultivariabletrackingcontrollerbasedonthelinearhel icopterdynamics.Theproposed proposeddesignhassignicantadvantagesrelativetothei nternalmodelandintegralcontrol approach.Themaincontributionofthisdesignisitsabilit ytopasstheintuitivenotionof helicoptermannedpilotingtothemathematicaldevelopmen toftheautonomouscontroller. Thisisachievedbyseparatingthehelicopterdynamicsinto twointerconnectedsubsystems representingthelongitudinal/lateralandyaw/heavemoti on,respectively.Bydisregarding theeffectoftheforcesproducedbytheappingmotionofthe mainrotor,theapproximated subsystemsareinfeedbackformand,therefore,differenti allyat.Duetothedifferential atnessofthesystemdynamics,adesiredstatestateandinp utcanbedetermined,composedbythecomponentsofthereferenceoutputandtheirhig herderivatives.Thedesired statecanbeeasilyandsystematicallydeterminedbythebac ksteppingapproach.Whenthe helicopterstateisregulatedtothisdesiredstate,thetra ckingerrortendsasymptoticallyto zero.Similarlyto[47],thedesiredstatevectorcanbeused forthedesignofmeaningful trajectories.Theoverallcontrollawisasuperpositionof thedesiredinputandanoutput feedbackcomponent.Theoutputfeedbackcomponentcanbech osenbyanydesignthat existsintheliterature.Thedesignalsoallowstheschedul ingofmultiplesimilarcontrollers basedonlinearmodelsofthesamestructure. Atrackingcontroldesignbasedonthehelicopternonlinear dynamicmodeladoptedby[47]. Thisdesignadoptsthebacksteppingdesignprincipleforno nlinearsystemsinfeedback form.Thepseudocontrolsforeachlevelofthefeedbacksyst emareappropriatelychosen tostabilizetheoverallhelicopterdynamics.Thepseudoco ntrolscombinenestedsaturation 225

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feedbacklawsandanovelcontrolstrategyforthestabiliza tionoftheattitudedynamics. Oneofthenoveltiesoftheproposedcontrollerisitsminima listicdesign.Byusingadvance stabilityanalysisconceptsonlythenecessarypseudocont roltermsareincludedforthe stabilizationofthesystem,whicharesignicantlylessth anexistingbacksteppingdesigns. Furthermore,apartfromstabilizingtheattitudedynamics ,thecontroldesigncanguarantee thatthehelicopterwillnotoverturnforeveryallowedrefe rencetrajectory.Theintense theoreticalanalysisthatisusedforthederivationofthec ontroldesignemergesimportant conceptsthatshouldbeaccountedinthehelicopterightco ntrollers.Suchconceptsinvolve theexpectedrangeofthepitchandrollanglesforaggressiv ereferencemaneuversandthe effectsoftheactuatorssaturationlimitsinthehelicopte rperformance. Atrackingcontrollerbasedonthediscretizednonlinearhe licopterdynamics.Thecontrol problemissettothediscretetimesincetimedomainsystemi denticationismuchsimpler andcomputationallyefcient.Inaddition,thecontrolalg orithmsareexecutedbymicroprocessors,therefore,thediscretizationeffectshouldb eaccountedbythecontroller.The maincontributionofthedevelopedcontrolleristhedesign freedomtotheconvergencerate foreachstatevariableofthecascadestructureofthefeedb acksystem.Thisisofparticular interestsincecontroloftheconvergencerateineachlevel ofthecascadestructureprovides betterightresults.Thestabilityoftheresultingdynami cscanbesimplyinspectedbythe eigenvaluesofalinearerrorwithoutthenecessityofLyapu nov'sfunctions.ThetimedomainidenticationtakesplacewithasimpleRLSalgorithm. Finallythethetime-domainidenticationresultscanbefu rtherimprovedifthediscrete nonlineardynamicsarerepresentedbyaTakagi-Sugenofuzz ysystem.AfterthedevelopmentoftheTakagi-Sugenosystem,astandardRLSalgorithmi susedtoestimateitsparameters.Theresultingfuzzysystemisaninterpolatorofn onlineardiscretesystemswhich dependsonthehelicopterightcondition. 226

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11.2ResultsandReal-LifeImplementation Thelineartracking,thediscretebacksteppingandthePID( introducedinChapter6)controller designsweresuccessfullytestedin X-Plane ightsimulatortoaRaptor90SERChelicopter. Anextensivecomparisontookplacewhereeachightcontrol lerwasexpectedtotrackseveral aggressiveanddexterousmaneuvers.Althoughthelinearhe licoptermodelistheoreticallylimited onlyinaneighborhoodaroundhover,asinglecontrollerbas edonlyontheidentiedhovermodel wasadequate.ThesatisfactoryperformanceofthePIDdesig nisattributedtotheattenuatedcross couplingeffectsamongstthe Raptor90SE dynamics. Forareal-lifeapplicationitiscommonengineeringintuit iontostartwiththelesscomplex approach.ThereforetherstchoiceshouldbethePIDcontro llerwiththefourSISOloops.Ifthe crosscouplingeffectamongthesystemdynamicsissignica ntthentheMIMOlineartracking controllershouldbeadopted.Finally,ifthelinearcontro llerfailstoachievetrackinginawide rangeoftheightenvelopethenthenonlinearschemeshould beapplied. 11.3FutureWork Additionalfeaturescanbeincorporatedtotheproposedcon trollerdesignsfortheirreliable implementationtoactualsmallscalehelicopterplatforms .Futureworkinvolves: Thehelicopterdynamicsarecharacterizedbysignicantpa rametricandmodeluncertainty. Theproposedcontrollersareproventobesignicantlyrobu st.Inallthedesignsthecertaintyequivalenceprinciplewasadopted.Accordingtotha ttheidentiedmodelisconsideredbythecontrolengineerastheactualhelicoptermod el.Atheoreticalframework thatexaminestheuncertaintyeffectstothecontrollerper formancewouldbeanimportant contributiontotheightcontroldesignproblem. Mostcontrollerdesignsneglectthecouplingbetweenforce sandmoments.Therefore,only practicalstabilityofthehelicoptercanbeachievedbased ontheapproximatedmodels. 227

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Aninterestingresearchavenuewouldbetotheoreticallyst udytheboundednessanderror marginsintroducedbytheapproximatemodels. Inreal-lifeapplicationsthemeasuredsensorsignalsexhi bitsignicantnoiselevelswhich arefurtherdeterioratedbythehelicopter'senginevibrat ions.Theconsequencesofnoise andtheimplementationeffectsofKalmanlteringtothecon trollerdesignshouldbefurther analyzed. 228

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ListofReferences [1]P.P.AngelovandD.P.Filev.Anapproachtoonlineidenti cationofTakagi-Sugenofuzzy models. IEEETransactionsonSystems,Man,andCybernetics,PartB: Cybernetics ,34:484– 498,2004. [2]N.Antequera,M.Santos,andJ.M.DelaCruz.Ahelicopter controlbasedoneigenstructure assignment.In IEEEConferenceEmergingTechnologiesandFactoryAutomat ion ,2006. [3]J.S.Bay. LinearStateSpaceSystems .McGraw-Hill,1999. [4]M.Bejar,A.Isidori,L.Marconi,andR.Naldi.Robustver tical/lateral/longitudinalcontrolof ahelicopterwithconstantyaw-attitude.In 44thIEEEConferenceonDecisionandControl, and2005EuropeanControlConference.CDC-ECC ,2005. [5]J.S.BendatandA.J.Piersol. RandomData:Analysis&MeasurementProcedures .Willey InterScience,1971. [6]P.BendottiandJ.C.Morris.Robusthovercontrolforamo delhelicopter.In Proceedingsof theAmericanControlConference ,1995. [7]A.R.S.Bramwell,G.Done,andD.Balmford. Bramwell'sHelicopterDynamics .Butterworth Heinemann,2001. [8]A.BudiyonoaandS.S.Wibowob.Optimaltrackingcontrol lerdesignforasmallscale helicopter. JournalofBionicEngineering ,4(4):271–280,2007. [9]C.I.Byrnes,F.D.Priscoli,andA.Isidori,editors. Outputregulationofuncertainnonlinear systems .Birkhauser,1997. 229

PAGE 246

[10]G.Cai,B.M.Chen,K.Peng,M.Dong,andT.H.Lee.Modelin gandcontrolsystemdesign foraUAVhelicopter.In 14thMediterraneanConferenceonControlandAutomation ,2006. [11]P.Castillo,R.Lozano,andA.E.Dzul. ModellingandControlofMini-FlyingMachines Springer-Verlag,2005. [12]P.C.Chandrasekharan. RobustControlofLinearDynamicalSystems .AcademicPress,1996. [13]R.Chen.Effectsofprimaryrotorparametersonapping dynamics.TechnicalReportTP1431,NASA,1980. [14]A.J.Prasad,J.V.R.Corban,J.E.Calise.Implementati onofadaptivenonlinearcontrolfor ighttestonanunmannedhelicopter.In Proceedingsofthe37thIEEEConferenceon DecisionandControl ,volume4,pages3641–3646,December1998. [15]J.E.Corban,A.J.Calise,J.V.R.Prasad,J.Hur,andN.K im.Flightevaluationofadaptive high-bandwidthcontrolmethodsforunmannedhelicopters. In AIAAGuidance,Navigation, andControlConference ,2003. [16]J.J.Craig. IntroductiontoRobotics:MechanicsandControl .PrenticeHall,2004. [17]J.C.Doyle,B.A.Francis,andA.Tannenbaum. FeedbackControlTheory .Macmillan,1992. [18]M.E.Dreier. IntroductiontoHelicopterandTiltrotorFlightSimulatio n .AIAAEducation Series,2007. [19]D.Ernst,K.Valavanis,andJ.Craighead.Automatedpro cessforunmannedaerialsystems controllerimplementationusingMATLAB.In 14thMediterraneanConferenceonControl andAutomation,2006.MED'06 ,2006. [20]B.Etkin. DynamicsofFlight:StabilityandControl .JohnWiley&Sons,1959. [21]I.FantoniandR.Lozano. NonlinearControlforUnderactuatedMechanicalSystems Springer-VerlagNewYork,Inc,2001. [22]M.Fliess,J.Levine,P.Martin,andP.Rouchon.Flatnes sanddefectofnonlinearsystems: Introductorytheoryandapplications. InternationalJournalofControl ,61:1327–1361,1995. 230

PAGE 247

[23]G.F.Franklin,J.D.Powell,andA.Emami-Naeini. FeedbackControlofDynamicSystems PrenticeHall,2002. [24]E.Frazzoli,M.A.Dahleh,andE.Feron.Trajectorytrac kingcontroldesignforautonomous helicoptersusingabacksteppingalgorithm.In ProceedingsoftheAmericanControl Conference ,volume6,pages4102–4107,2000. [25]D.Fujiwara,J.Shin,K.Hazawa,andK.Nonami. H 1 hoveringandguidancecontrolfor anautonomoussmall-scaleunmannedhelicopter.In InternationalConferenceonIntelligent RobotsandSystems ,2004. [26]J.Gadewadikar,F.Lewis,K.Subbarao,andB.Chen. H 1 staticoutput-feedbackcontrolfor rotorcraft. JournalofIntelligentandRoboticSystems ,54:629 U646,2008. [27]J.Gadewadikar,F.Lewis,K.Subbarao,andB.Chen.Stru ctured H 1 commandandcontrolloopdesignforunmannedhelicopters. JournalofGuidance,Control,andDynamics 31:1093–1102,2008. [28]J.Gadewadikar,F.L.Lewis,K.Subbarao,K.Peng,andB. M.Chen. H 1 staticoutputfeedbackcontrolforrotorcraft.In AIAAGuidance,Navigation,andControlConference andExhibit ,2006. [29]V.Gavrilets,B.Mettler,andE.Feron.Dynamicalmodel foraminiatureaerobatichelicopter. Technicalreport,MassachussettsInstituteofTechnology ,2001. [30]V.Gavrilets,B.Mettler,andE.Feron.Nonlinearmodel forasmall-sizeacrobatichelicopter. In AIAAGuidance,Navigation,andControlConferenceandExhi bit ,2001. [31]D.T.Greenwood. PrinciplesofDynamics .Prentice-Hall,1965. [32]N.Guenard,T.Hamel,andV.Moreau.Dynamicmodelingan dintuitivecontrolstrategyfor an"X4-yer".In InternationalConferenceonControlandAutomation ,pages141–146, 2005. [33]T.Hamel,R.Mahony,R.Lozano,andJ.Ostrowski.Dynami cmodelingandconguration stabilizationforanX4-yer.In 15thTriennialWorldCongressofIFAC ,2002. 231

PAGE 248

[34]N.Hovakimyan,N.Kim,A.J.Calise,J.V.R.Prasad,andN .Corban.Adaptiveoutput feedbackforhigh-bandwidthcontrolofanunmannedhelicop ter.In AIAAGuidance, Navigation,andControlConference ,2001. [35]I.I.Hussein,M.Leok,A.K.Sanyal,andA.M.Bloch.Adis cretevariationalintegratorfor optimalcontrolproblemsonSO(3).In 45thIEEEConferenceonDecisionandControl 2006. [36]A.Isidori,L.Marconi,andA.Serrani. RobustAutonomousGuidance .Springer-Verlang, 2003. [37]A.Isidori,L.Marconi,andA.Serrani.Robustnonlinea rmotioncontrolofahelicopter. IEEE TransactionsonAutomaticControl ,48:413–426,2003. [38]E.N.JohnsonandS.K.Kannan.Adaptiveightcontrolfo ranautonomousunmanned helicopter.In AIAAGuidance,NavigationandControlConference ,2002. [39]E.N.JohnsonandS.K.Kannan.Adaptivetrajectorycont rolforautonomoushelicopters. JournalofGuidance,Control,andDynamics ,28:524–538,2005. [40]W.Johnson. HelicopterTheory .PrincetonUniversityPress,1980. [41]R.E.KalmanandR.S.Bucy.Newresultsinlinearlterin gandpredictiontheory. Journalof BasicEngineering ,83:95–108,1961. [42]F.Kendoul,D.Lara,I.Fantoni-Coichot,andR.Lozano. Real-timenonlinearembedded controlforanautonomousquadrotorhelicopter. JournalofGuidance,Controland Dynamics ,30:1049–1061,2007. [43]H.K.Khalil. NonlinearSystems .PrenticeHall,2002. [44]H.J.KimandD.H.Shim.Aightcontrolsystemforaerial robots:algorithmsand experiments. ControlEngineeringPractice ,11:1389–1400,2003. [45]N.Kim,A.J.Calise,N.Hovakimyan,J.V.R.Prasad,andE .Corban.Adaptiveoutput feedbackforhigh-bandwidthightcontrol. JournalofGuidance,ControlandDynamics 25:993–1002,2002. 232

PAGE 249

[46]V.KleinandE.A.Moreli. AircraftSystemIdenticationTheoryandPractice .AIAA EducationSeries,2006. [47]T.J.KooandS.Sastry.Outputtrackingcontroldesigno fahelicoptermodelbasedon approximatelinearization.In Proceedingsofthe37thIEEEConferenceonDecisionand Control ,volume4,pages3635–3640,1998. [48]T.J.KooandS.Sastry.Differentialatnessbasedfull authorityhelicoptercontroldesign.In Proceedingsofthe38thIEEEConferenceonDecisionandCont rol ,1999. [49]M.Krstic,I.Kanellakopoulos,andP.V.Kokotovic. NonlinearandAdaptiveControlDesign Wiley-Interscience,1995. [50]R.Kureemun,D.J.Walker,B.Manimala,andMVoskuijl.H elicopterightcontrollaw designusing H 1 techniques.In IEEEConferenceOnDecicionandControl ,2005. [51]M.LaCivita. Integratedmodelingandrobustcontrolforfullenvelopei ghtofrobotic helicopters .PhDthesis,CarnegieMellonUniversity,2002. [52]M.LaCivita,W.C.Messner,andT.Kanade.Modelingofsm all-scalehelicopterswith integratedrst-principlesandsystemidenticationtech niques.In Proceedingsofthe58th ForumoftheAmericanHelicopterSociety ,volume2,pages2505–2516,2002. [53]M.LaCivita,G.Papageorgiou,W.Messner,andT.Kanade .Designandighttestingofa high-bandwidth H 1 loopshapingcontrollerforarobotichelicopter.In AIAAGuidance, Navigation,andControlConferenceandExhibit ,2002. [54]M.LaCivita,G.Papageorgiou,W.C.Messner,andT.Kana de.Integratedmodelingand robustcontrolforfull-envelopeightofrobotichelicopt ers.In ProceedingsofIEEE InternationalConferenceonRoboticsandAutomation ,pages552–557,2003. [55]M.LaCivita,G.Papageorgiou,W.C.Messner,andT.Kana de.Designandighttestingofan H 1 controllerforarobotichelicopter. JournalofGuidance,Control,andDynamics ,pages 485–494,2006. [56]E.H.Lee,H.Shim,L.Park,andK.Lee.Designofhovering attitudecontrollerforamodel helicopter.In ProceedingsofSocietyofInstrumentandControlEngineers ,pages1385– 1390,1993. 233

PAGE 250

[57]T.Lee,N.H.McClamroch,andM.Leok.Optimalcontrolof arigidbodyusinggeometrically exactcomputationsonSE(3).In 45thIEEEConferenceonDecisionandControl ,2006. [58]J.G.Leishman. PrinciplesofHelicopterAerodynamics .CambridgeUniversityPress,2000. [59]W.LevineandM.Athans.Onthedeterminationoftheopti malconstantoutputfeedback gainsforlinearmultivariablesystems. TransactionsonAutomaticControl ,15:44–48,1970. [60]F.L.LewisandV.L.Syrmos. OptimalControl .Wiley-InterScience,1995. [61]L.Ljung. SystemIdentication .Prentice-Hall,1987. [62]L.Ljung. SystemIdentication:TheoryfortheUser .PrenticeHall,1999. [63]A.LoriaandE.Panteley. AdvancedTopicsinControlSystemsTheory:LectureNotesfr om FAP2004 ,chapter2,pages23–64.Springer-Verlag,2005. [64]R.MahonyandT.Hamel.Robusttrajectorytrackingfora scalemodelautonomous helicopter. InternationalJournalofRobustandNonlinearControl ,14(12):1035–1059, 2004. [65]R.Mahony,T.Hamel,andA.Dzul.HovercontrolviaLyapu novcontrolforanautonomous modelhelicopter.In Proceedingsofthe38thIEEEConferenceonDecisionandCont rol volume4,pages3490–3495,1999. [66]L.MarconiandR.Naldi.Robustfulldegree-of-freedom trackingcontrolofahelicopter. Automatica ,43:1909–1920,2007. [67]L.MarconiandR.Naldi.Aggressivecontrolofhelicopt ersinpresenceofparametricand dynamicaluncertainties. Mechatronics ,1:381–389,2008. [68]D.McFarlaneandK.Glover.Aloop-shapingdesignproce dureusing H 1 synthesis. IEEE TransactionsonAutomaticControl ,37:759–769,1992. [69]J.M.Mendel. Lessonsinestimationtheoryforsignalprocessing,commun ications,and control .PrenticeHallPTR,1995. 234

PAGE 251

[70]B.Mettler. IdenticationModelingandCharacteristicsofMiniatureR otorcraft .Kluwer AcademicPublishers,2003. [71]B.Mettler,T.Kanade,andM.B.Tischler.Systemidenti cationmodelingofamodel-scale helicopter.Technicalreport,CarnegieMellonUniversity ,2000. [72]B.Mettler,M.B.Tischler,andT.Kanade.Systemidenti cationofsmall-sizeunmanned helicopterdynamics.In PresentedattheAmericanHelicopterSociety55thForum ,May 1999. [73]Mitra. DigitalSignalProcessing:AComputer-BasedApproach .McGraw-Hill,2006. [74]A.Moerder,D.Calise.Convergenceofanumericalalgor ithmforcalculatingoptimaloutput feedbackgains. IEEETransactionsonAutomaticControl ,30(9):900–903,1985. [75]R.M.Murray,L.Zexiang,andS.Sastry. AMathematicalIntroductiontoRoboticManipulation .CRCPress,1994. [76]A.V.Oppenheim,R.W.Shafer,andJ.R.Buck. Discrete-TimeSignalProcessing .Prentice Hall,1999. [77]A.VOppenheim,A.S.Willsky,andI.T.Young. SignalsandSystems .PrenticeHall,1983. [78]H.Ozbay. IntroductiontoFeedbackControlTheory .CRCPress,1999. [79]G.D.Padeld. HelicopterFlightDynamics:TheTheoryandApplicationofF lyingQualities andSimulationModeling .AIAAEducationSeries,1996. [80]G.PapageorgiouandK.Glover. H 1 loop-shaping:Whyisitasensibleprocedurefor designingrobustightcontrollers.In AIAAConferenceonGuidance,Navigationand Control ,1999. [81]K.M.PassinoandS.Yurkovich. FuzzyControl .PrenticeHall,1998. [82]S.Pieper,J.K.sndBaillieandK.R.Goheen.Linear-qua draticoptimalmodel-following controlofahelicopterinhover.In AmericanControlConference ,1994. 235

PAGE 252

[83]E.PrempainandI.Postlethwaite.Static H 1 loopshapingcontrolofay-by-wirehelicopter. Automatica ,41:1517–1528,2005. [84]R.W.Prouty. HelicopterPerformance,StabilityandControl .KriegerPublishingCompany, 1995. [85]I.A.Raptis,K.P.Valavanis,A.Kandel,andW.A.Moreno .Systemidenticationfora miniaturehelicopterathoverusingfuzzymodels.Submitte dinthe47thIEEEConference onDecisionandControl,2008. [86]E.Seckel. StabilityandControlofAirplanesandHelicopters .AcademicPress,1964. [87]J.S.ShammaandM.Athans.Analysisofgainscheduledco ntrolfornonlinearplants. IEEE TransactionsonAutomaticControl ,35:898–907,1990. [88]H.Shim,T.J.Koo,F.Hoffmann,andS.Sastry.Acomprehe nsivestudyofcontroldesignfor anautonomoushelicopter.In Proceedingsofthe37thIEEEConferenceonDecisionand Control ,volume4,pages3653–3658,1998. [89]H.D.Shim,H.J.Kim,andS.Sastry.Controlsystemdesig nforrotorcraft-basedunmanned aerialvehiclesusingtime-domainsystemidentication.I n Proceedingsofthe2000IEEE InternationalConferenceonControlApplications ,pages808–813,2000. [90]J.Shin,K.Nonami,D.Fujiwara,andK.Hazawa.Model-ba sedoptimalattitudeand positioningcontrolofsmall-scaleunmannedhelicopter. Robotica ,23:51–63,2005. [91]M.Sira-Ramirez,H.andZribiandS.Ahmad.Dynamicalsl idingmodecontrolapproachfor verticalightregulationinhelicopters.In ControlTheoryandApplications ,volume141, 1994. [92]S.SkogestadandI.Postlethwaite. MultivariableFeedbackControl .Wiley,1996. [93]T.SoderstromandP.Stoica. SystemIdentication .PrenticeHall,1989. [94]E.D.Sontag.Remarksonstabilizationandinput-to-st atestability.In Proceedingsofthe28th IEEEConferenceonDecisionandControl ,volume2,pages1376–1378,1989. 236

PAGE 253

[95]M.W.Spong,S.Hutchinson,andM.Vidyasagar. RobotModelingandControl .Wiley,2005. [96]J.T.SpoonerandK.M.Passino.Stableadaptivecontrol usingfuzzysystemsandneural networks. IEEETransactionsonFuzzySystems ,4:339–359,1996. [97]X.D.SunandT.Clarke.Applicationofhybrid / H 1 controltomodernhelicopters.In InternationalConferenceonControl ,1994. [98]H.J.SussmannandP.V.Kokotovic.Thepeakingphenomen onandtheglobalstabilizationof nonlinearsystems. IEEETransactionsonAutomaticControl ,36:424–440,1991. [99]V.L.Syrmos,C.Abdallah,andP.Dorato.Staticoutputf eedback:Asurvey.In 33rd ConferenceonDecisionandControl ,1994. [100]V.L.Syrmos,C.Abdallah,P.Dorato,andK.Grigoriadi s.Staticoutputfeedback:Asurvey. Technicalreport,UniversityofNewMexico,1995. [101]T.TakagiandM.Sugeno.Fuzzyidenticationofsystem sanditsapplicationstomodeling andcontrol. IEEETransactionsonSystems,Man,andCybernetics ,15:116–132,1985. [102]A.R.Teel.Globalstabilizationandrestrictedtrack ingformultipleintegratorswithbounded controls. Systems&ContrlolLetters ,18:165–171,1992. [103]A.R.Teel.Usingsaturationtostabilizeaclassofsin gle-inputpartiallylinearcomposite systems.In IFACNOLCOS'92Symposium ,pages379–384,1992. [104]M.B.TischlerandM.G.Cauffman.Frequency-response methodforrotorcraftsystem identication:FlightapplicationstoBO-105coupledfuse lage/rotordynamics. Journalof theAmericanHelicopterSociety ,3:3–17,1992. [105]M.B.TischlerandR.K.Remple. AircraftandRotorcraftSystemIdentication .AIAA EducationSeries,2006. [106]K.P.Valavanis,editor. AdvancesinUnmannedAerialVehiclesStateoftheArtandthe RoadtoAutonomy ,volume33of IntelligentSystems,ControlandAutomation:Scienceand Engineering .Springer,2007. 237

PAGE 254

[107]M.J.VanNieuwstadt. TrajectoryGenerationforNonlinearControlSystems .PhDthesis, CaliforniaInstituteofTechnology,1997. [108]D.J.WalkerandI.Postlethwaite.Advancedhelicopte rightcontrolusingtwo-degree-offreedom H 1 optimization. JournalofGuidance,ControlandDynamics ,19:461–468,1996. [109]M.F.Weilenmann,U.Christen,andH.P.Geering.Robus thelicopterpositioncontrolat hover.In AmericanControlConference ,1999. [110]M.F.WeilenmannandP.Hans.Atestbenchforrotorcraf thovercontrol.In AIAAGuidance, NavigationandControlConference ,pages1371–1382,1993. [111]J.H.Williams. FundamentalsofAppliedDynamics .JohnWiley&Sons,1996. [112]J.ZhaoandI.Kanellakopoulos.Adaptivecontrolofdi screte-timestrict-feedbacknonlinear systems. ProceedingsoftheAmericanControlConference ,1:828–832,1997. [113]K.Zhou,J.C.Doyle,andK.Glover. RobustandOptimalControl .PrenticeHall,1996. 238

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Appendices 239

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AppendixA:BacksteppingControl ThisAppendixprovidesamathematicalbackgroundoftherec ursivebacksteppingcontrol method.Thepresentedmaterialisasummaryofmoredetailed descriptionsthatcanbefoundin [43,49].Lyapunov-basedcontrollerdesigncanbesystemat icallytackledbyarecursivedesign procedurecalledbackstepping.Backsteppingissuitablef orstrict-feedbacksystemswhicharealso knownas“lowertriangular”.Anexampleofastrict-feedbac ksystemsis: 1 = f 1 ( 1 )+ g 1 ( 1 ) 2 2 = f 2 ( 1 ; 2 )+ g 2 ( 1 ; 2 ) 3 ... (A.1) r 1 = f r 1 ( 1 ; 2 ;::: r 1 )+ g r 1 ( 1 ; 2 ;::: r 1 ) r r = f r ( 1 ; 2 ;::: r )+ g r ( 1 ; 2 ;::: r ) u where 1 ;:::; r 2 R and u 2 R isthecontrolinput.Atypicalfeedbacklinearizationappr oach inmostcasesleadstocancellationofusefulnonlinearitie s.Backsteppingdesignexhibitmore exibilitycomparedtofeedbacklinearizationsincetheyd onotrequirethattheresultinginputoutputdynamicstobelinear.Cancellationofpotentiallyu sefulnonlinearitiescanbeavoided resultingtolesscomplexcontrollers. Themainideaistousesomeofthestatestatevariablesof(A. 1)as“virtualcontrols”or“pseudo controls”,anddependingonthedynamicsofeachstatedesig nintermediatecontrollaws.The backsteppingdesignisarecursiveprocedurewhereaLyapun ovfunctionisdevelopedforthe entiresystem.TheLyapunovfunctioncanguaranteethatthe overalldynamicsareuniformlygloballystable.Therecursiveprocedurecanbeeasilyexpanded fromthenominalcaseofasystem augmentedbyanintegrator.Thiscasestudyisalsoreferred toasintegratorbackstepping.Based onthedesignprinciplesoftheintegratorbackstepping,th econtroldesigncanbeeasilyexpanded 240

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AppendixA:(continued)forthecaseofstrict-feedbacksystemsgivenby(A.1).More particular,considerthesystem: = f ( )+ g ( ) (A.2) = u (A.3) where [ ] T 2 R n +1 isthestatevectorand u 2 R isthecontrolinput.Theobjectiveisthe designofastatefeedbackcontrollawsuchthat ; 0 as t !1 .Itisassumedthatboth f and g areknown.Thissystemcanbeviewedasacascadeconnectiono ftwocomponents.Therst componentis(A.2)with asinputandthesecondcomponentistheintegrator(A.3).Th emain designideaistotreat asavirtualcontrolinputforthestabilizationof .Assumethatthereexist asmoothstatefeedbackcontrollaw = ( ) ,with (0)=0 ;suchthattheoriginof: = f ( )+ g ( ) ( ) (A.4) isasymptoticallystable.Assumethatforthechoiceof ( ) weknowaLyapunovfunction V ( ) suchthat: @V @ [ f ( )+ g ( ) ( )] W ( ) ; 8 2 R n (A.5) where W ( ) ispositivedenite.Byaddingandsubtracting g ( ) ( ) ontherighthandsideof (A.2),onehas: = f ( )+ g ( )[ ( )] (A.6) = u (A.7) Denoteby e theerrorbetweenthestate andthepseudocontrol ( ) ,thatis: e = ( ) (A.8) 241

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AppendixA:(continued)Writingtheinitialsysteminthe ( ;e ) coordinates,onehas: =[ f ( )+ g ( ) ( )]+ g ( ) e (A.9) e = u ( ) (A.10) Since f g and areknown,oneoftheadvantagesofthebacksteppingdesigni sthatitdoesnot requireadiffrentiator.Inparticular,thederivative canbecomputedbyusingtheexpression: = @ @ [ f ( )+ g ( ) ] (A.11) Setting u = v + ,where v 2 R isanominalcontrolinput,thetransformedsystemtakesthe form: =[ f ( )+ g ( ) ( )]+ g ( ) e (A.12) e = v (A.13) whichissimilartotheinitialsystem,exceptthatnowther stcomponenthasanasymptotically stableoriginwhentheinputiszero.Usingthisprocedureth epseudocontrol ( ) hasbeen“back stepped”throughtheintegratorfrom u = v + ( ) .Theknowledgeof V ( ) isexploitedinthe designof v forthestabilizationoftheoverallsystem.Using: V c ( ; )= V ( )+ 1 2 e 2 (A.14) asaLyapunovfunctioncandidate,weobtain: V c = @V @ [ f ( )+ g ( ) ( )]+ @V @ g ( ) e + e v W ( )+ @V @ g ( ) e + e v (A.15) 242

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AppendixA:(continued)Thecontrolinput v ischosenas: v = @V @ g ( ) ke ;k> 0 (A.16) Substitutingtheabovechoiceof v to(A.15),onehas: V c W ( ) ke 2 (A.17) whichshowsthattheorigin ( =0 ;e =0) isasymptoticallystable.Since (0)=0 ,and e 0 as t !1 ;thentheorigin ( =0 ; =0) isasymptoticallystableaswell.Substitutingfor v e and ,thenalformofthecontrollawis: u = @ @ [ f ( )+ g ( ) ] @V @ g ( ) k [ ( )] (A.18) 243

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AbouttheAuthor IoannisA.RaptiswasborninAthens,Greecein1979.Herecei vedhisDipl-Ing.inElectrical andComputerEngineeringfromtheAristotleUniversityofT hessaloniki,GreeceandhisMaster ofScienceinElectricalandComputerEngineeringfromtheO hioStateUniversityin2003and 2006,respectively.From2005until2006heconductedresea rchattheLocomotionandBiomechanicsLaboratoryoftheOhioStateUniversity.In2006hej oinedtheUnmannedSystemsLaboratoryattheUniversityofSouthFlorida.Since2006heispu rsuinghisPh.D.degreeinthedepartmentofElectricalEngineeringattheUniversityofSouthFl orida.Hisresearchinterestsinclude nonlinearsystemscontroltheory,nonlinearcontrolofele ctromechanical/roboticsystemsand rotorcraft/aircraftsystemidenticationandcontrol.


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Linear and nonlinear control of unmanned rotorcraft
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ABSTRACT: The main characteristic attribute of the rotorcraft is the use of rotary wings to produce the thrust force necessary for motion. Therefore, rotorcraft have an advantage relative to fixed wing aircraft because they do not require any relative velocity to produce aerodynamic forces. Rotorcraft have been used in a wide range of missions of civilian and military applications. Particular interest has been concentrated in applications related to search and rescue in environments that impose restrictions to human presence and interference. The main representative of the rotorcraft family is the helicopter. Small scale helicopters retain all the flight characteristics and physical principles of their full scale counterpart. In addition, they are naturally more agile and dexterous compared to full scale helicopters. Their flight capabilities, reduced size and cost have monopolized the attention of the Unmanned Aerial Vehicles research community for the development of low cost and efficient autonomous flight platforms. Helicopters are highly nonlinear systems with significant dynamic coupling. In general, they are considered to be much more unstable than fixed wing aircraft and constant control must be sustained at all times. The goal of this dissertation is to investigate the challenging design problem of autonomous flight controllers for small scale helicopters. A typical flight control system is composed of a mathematical algorithm that produces the appropriate command signals required to perform autonomous flight. Modern control techniques are model based, since the controller architecture depends on the dynamic description of the system to be controlled. This principle applies to the helicopter as well, therefore, the flight control problem is tightly connected with the helicopter modeling. The helicopter dynamics can be represented by both linear and nonlinear models of ordinary differential equations. Theoretically, the validity of the linear models is restricted in a certain region around a specific operating point. Contrary, nonlinear models provide a global description of the helicopter dynamics. This work proposes several detailed control designs based on both dynamic representations of small scale helicopters. The controller objective is for the helicopter to autonomously track predefined position (or velocity) and heading reference trajectories. The controllers performance is evaluated using X-Plane, a realistic and commercially available flight simulator.
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