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A Linear Input-Varying Framework for Modeling and Control of Morphing Aircraft

Permanent Link: http://ufdc.ufl.edu/UFE0042424/00001

Material Information

Title: A Linear Input-Varying Framework for Modeling and Control of Morphing Aircraft
Physical Description: 1 online resource (200 p.)
Language: english
Creator: GRANT,DANIEL THURMOND
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

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

Notes

Abstract: Morphing, which changes the shape and configuration of an aircraft, is being adopted to expand mission capabilities of aircraft. The introduction of biological-inspired morphing is particularly attractive in that highly-agile birds present examples of desired shapes and configurations. A previous study adopted such morphing by designing a multiple-joint wing that represented the shoulder and elbow joints of a bird. The resulting variable-gull aircraft could rotate the wing section vertically at these joints to alter the flight dynamics. This paper extends that multiple-joint concept to allow a variable-sweep wing with independent inboard and outboard sections. The aircraft is designed and analyzed to demonstrate the range of flight dynamics which result from the morphing. In particular, the vehicle is shown to have enhanced crosswind rejection which is a certainly critical metric for the urban environments in which these aircraft are anticipated to operate. Mission capability can be enabled by morphing an aircraft to optimize its aerodynamics and associated flight dynamics for each maneuver. Such optimization often consider the steady-state behavior of the configuration; however, the transient behavior must also be analyzed. In particular, the time-varying inertias have an effect on the flight dynamics that can adversely affect mission performance if not properly compensated. These inertia terms cause coupling between the longitudinal and lateral-directional dynamics even for maneuvers around trim. A simulation of a variable-sweep aircraft undergoing a symmetric morphing for an altitude change shows a noticeable lateral translation in the flight path because of the induced asymmetry. The flight dynamics of morphing aircraft must be analyzed to ensure shape-changing trajectories have the desired characteristics. The tools for describing flight dynamics of fixed-geometry aircraft are not valid for time-varying systems such as morphing aircraft. This paper introduces a method to relate the flight dynamics of morphing aircraft by interpreting a time-varying eigenvector in terms of flight modes. The time-varying eigenvector is actually defined through a decomposition of the state-transition matrix and thus describes an entire response through a morphing trajectory. A variable-sweep aircraft is analyzed to demonstrate the information that is obtained through this method and how the flight dynamics are altered by the time-varying morphing. Also, morphing vehicles have inherently time-varying dynamics due to the alteration of their configurations; consequently, the numerous techniques for analysis and control of time-invariant systems are inappropriate. Therefore, a control scheme is introduced that directly considers a concept of time-varying pole to command morphing. The resulting trajectory minimizing tracking error for either a state response or a pole response.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by DANIEL THURMOND GRANT.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042424:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042424/00001

Material Information

Title: A Linear Input-Varying Framework for Modeling and Control of Morphing Aircraft
Physical Description: 1 online resource (200 p.)
Language: english
Creator: GRANT,DANIEL THURMOND
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

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

Notes

Abstract: Morphing, which changes the shape and configuration of an aircraft, is being adopted to expand mission capabilities of aircraft. The introduction of biological-inspired morphing is particularly attractive in that highly-agile birds present examples of desired shapes and configurations. A previous study adopted such morphing by designing a multiple-joint wing that represented the shoulder and elbow joints of a bird. The resulting variable-gull aircraft could rotate the wing section vertically at these joints to alter the flight dynamics. This paper extends that multiple-joint concept to allow a variable-sweep wing with independent inboard and outboard sections. The aircraft is designed and analyzed to demonstrate the range of flight dynamics which result from the morphing. In particular, the vehicle is shown to have enhanced crosswind rejection which is a certainly critical metric for the urban environments in which these aircraft are anticipated to operate. Mission capability can be enabled by morphing an aircraft to optimize its aerodynamics and associated flight dynamics for each maneuver. Such optimization often consider the steady-state behavior of the configuration; however, the transient behavior must also be analyzed. In particular, the time-varying inertias have an effect on the flight dynamics that can adversely affect mission performance if not properly compensated. These inertia terms cause coupling between the longitudinal and lateral-directional dynamics even for maneuvers around trim. A simulation of a variable-sweep aircraft undergoing a symmetric morphing for an altitude change shows a noticeable lateral translation in the flight path because of the induced asymmetry. The flight dynamics of morphing aircraft must be analyzed to ensure shape-changing trajectories have the desired characteristics. The tools for describing flight dynamics of fixed-geometry aircraft are not valid for time-varying systems such as morphing aircraft. This paper introduces a method to relate the flight dynamics of morphing aircraft by interpreting a time-varying eigenvector in terms of flight modes. The time-varying eigenvector is actually defined through a decomposition of the state-transition matrix and thus describes an entire response through a morphing trajectory. A variable-sweep aircraft is analyzed to demonstrate the information that is obtained through this method and how the flight dynamics are altered by the time-varying morphing. Also, morphing vehicles have inherently time-varying dynamics due to the alteration of their configurations; consequently, the numerous techniques for analysis and control of time-invariant systems are inappropriate. Therefore, a control scheme is introduced that directly considers a concept of time-varying pole to command morphing. The resulting trajectory minimizing tracking error for either a state response or a pole response.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by DANIEL THURMOND GRANT.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042424:00001


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ALINEARINPUT-VARYINGFRAMEWORKFORMODELINGANDCONTROLOFMORPHINGAIRCRAFTByDANIELT.GRANTADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011 1

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c2011DanielT.Grant 2

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\IcandoallthingsthroughChristwhichstrengthensme" Philippians4:13 Dedicatedwithlovetomywifeandfamily 3

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ACKNOWLEDGMENTS IwouldrstliketoacknowledgetheUniversityofFloridaandUnitedStatesAirForceforsupportingmyambitionandgivingmetheopportunitytoconductsuchresearch.ThanksshouldbegiventoDr.WarrenDixon,Dr.PrabirBarooah,Dr.PeterIfjuandDr.OscarCrisalleforprovidingdirectionandservingasmycommitteemembers.IwouldalsoliketothankmyseniorlabfellowsDr.MujahidAbdulrahim,Dr.AdamWatkins,Dr.RyanCausey,Dr.JosephKehoeandDr.SeanRegisfordfortheirpatienceandguidancewhilementoringme.MuchthanksisgiventomycolleaguesSankethBhat,BrianRoberts,RobertLove,BaronJohnson,RyanHurley,AbePachikara,DongTranandStevenSorelyfortheirsupport,inspiration,andperseverance.IwouldliketoextendmysincerestthanksandgratitudetoDr.RickLindforhiseortsinsupportingmyeducation,guidanceinacademia,andprovidingmewithaninvaluableopportunitytoachievesuccess.Withoutthecontinuoussupportandunconditionalloveofmyfamilyandfriends,noneofthisworkwouldhavebeenpossible.Lastandmostimportant,IwouldliketothankmylovingwifeforbeingtheshininglightbehindallthatIdo. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 11 ABSTRACT ........................................ 17 CHAPTER 1INTRODUCTION .................................. 19 1.1Motivation .................................... 19 1.2ProblemDescription .............................. 20 1.3ProblemStatement ............................... 23 1.3.1Contributions .............................. 24 1.3.2Papers .................................. 25 1.4DissertationOverview ............................. 28 2EQUATIONSOFMOTION ............................. 29 2.1AircraftAxisSystem .............................. 29 2.1.1BodyAxisSystem ............................ 29 2.1.2StabilityAxisSystem .......................... 29 2.1.3EarthAxisSystem ........................... 30 2.2CoordinateTransformations .......................... 31 2.2.1EarthtoBodyFrame .......................... 31 2.2.2StabilitytoBodyFrame ........................ 33 2.3NonlinearEquationsofMotion ........................ 34 2.3.1DynamicEquations ........................... 34 2.3.1.1Forceequations ........................ 34 2.3.1.2Momentequations ...................... 39 2.3.2KinematicEquations .......................... 43 2.3.2.1Orientationequations .................... 43 2.3.2.2Positionequations ...................... 44 2.3.3TheEquationsCollected ........................ 45 2.4LinearizedEquationsofMotion ........................ 46 2.5Examples .................................... 49 2.5.1Linearization ............................... 50 2.5.2AsymmetricMorphing ......................... 53 2.5.3SymmetricConguration ........................ 54 2.6ATechnicalApproachtotheEquationsofMotion .............. 55 5

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3LINEARTIME-VARYINGEIGENSTRUCTURESANDTHEIRSTABILITY:ASURVEY ...................................... 57 3.1DenitionofanLTVSystem .......................... 57 3.2Kamen'sConceptofPolesofanLTVSystem ................. 57 3.2.1TwoStateSystem ............................ 57 3.2.2FourStateSystem ............................ 58 3.3ZhuandJohnson ................................ 60 3.3.1GeneralizedPD-Eigenvectors ...................... 62 3.3.2StabilityCriteria ............................ 63 3.4Wu ........................................ 64 3.4.1Formulation ............................... 64 3.4.2StabilityCriteria ............................ 66 3.5O'BrienandIglesias .............................. 67 3.5.1Formulation ............................... 67 3.5.2StabilityCriteria ............................ 69 3.5.3SpecialCases .............................. 71 3.6MethodologyComparisons ........................... 71 4LINEARTIME-VARYINGMODALANALYSIS ................. 74 4.1SystemDynamics ................................ 74 4.1.1LinearTime-InvariantSystems:Formulation ............. 74 4.1.2LinearTime-InvariantSystems:Response ............... 75 4.1.3LinearTime-VaryingSystems:Formulation .............. 76 4.1.4LinearTime-VaryingSystems:Response ............... 77 4.2ModalInterpretation:Kamen ......................... 78 4.2.1Damping ................................. 79 4.2.2Frequency ................................ 80 4.3ModalInterpretation:O'Brien ......................... 81 4.3.1DampingandNaturalFrequency .................... 81 4.3.2ModeShapes .............................. 82 4.4Example:SimpleMechanicalSystem ..................... 83 5CONTROLDESIGN ................................. 94 5.1InherentClosed-LoopNonlinearity ...................... 94 5.2Quasi-StaticApproach ............................. 95 5.2.1Synthesis ................................. 95 5.2.2Example:Gull-WingedAircraft .................... 96 5.3StabilizingControl:DisturbanceRejection .................. 97 5.3.1Synthesis ................................. 97 5.3.2Example:Chord-VaryingMorphing .................. 100 5.4FeedForwardOptimalControl ......................... 102 5.4.1State-ResponseTracking ........................ 102 5.4.2Pole-ResponseTracking ......................... 103 6

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5.4.3Example:Mass-SpringSystem .................... 104 5.4.3.1System ............................. 104 5.4.3.2MorphingBasis ........................ 104 5.4.3.3TrackingaSystemResponse:PolynomialMorphing .... 106 5.4.3.4TrackingaSystemResponse:Piecewise-PolynomialMorphing 107 5.4.3.5TrackingaPoleResponse:PolynomialMorphing ..... 108 5.4.3.6TrackingaPoleResponse:Piecewise-PolynomialMorphing 110 5.5H1FeedbackControl:WithoutInertialEects ............... 112 5.6FutureWorkandChallenges .......................... 117 6EXAMPLEOFVARIABLESWEEPAIRCRAFT ................. 120 6.1Design ...................................... 120 6.1.1BiologicalInspiration .......................... 120 6.1.2MechanicalDesign ............................ 121 6.1.3TechnicalSpecications ......................... 123 6.2Modeling ..................................... 125 6.2.1ComputationalTools .......................... 125 6.2.2SweepDetermination .......................... 127 6.3AerodynamicProperties ............................ 128 6.3.1SymmetricCongurations:Aerodynamics ............... 128 6.3.2SymmetricCongurations:FlightDynamics ............. 130 6.3.3AsymmetricCongurations ....................... 133 6.3.3.1Flightdynamics ........................ 135 6.3.3.2Modecharacterization .................... 136 6.3.3.3Crosswindrejection ...................... 138 6.4DynamicProperties ............................... 140 6.4.1MissionScenario ............................. 140 6.4.1.1Divemaneuver ........................ 141 6.4.1.2Turnmaneuver ........................ 141 6.4.2MassDistribution ............................ 142 6.4.3ManeuverAssumptions ......................... 143 6.4.4DiveManuever ............................. 144 6.4.4.1Modeling ........................... 144 6.4.4.2Altitudecontroller ...................... 145 6.4.4.3Time-varyingdynamics .................... 146 6.4.4.4Simulation ........................... 147 6.4.4.5Missionevaluation ...................... 149 6.4.4.6Eectsoftime-varyinginertia ................ 150 6.4.5CoordinatedTurnManeuver ...................... 152 6.4.5.1Modeling ........................... 152 6.4.5.2Turncontroller ........................ 153 6.4.5.3Time-varyingdynamics .................... 154 6.4.5.4Simulation ........................... 155 6.4.5.5Missionevaluation ...................... 159 7

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6.4.5.6Eectsoftime-varyinginertia ................ 160 6.5Time-VaryingModalAnalysis ......................... 161 6.5.1Kamen'sMethod ............................ 161 6.5.1.1Statespacemodel ...................... 161 6.5.1.2Morphingtrajectory ..................... 163 6.5.1.3Time-varyingpoles ...................... 163 6.5.1.4Modalinterpretation ..................... 167 6.5.2O'Brien'sMethod ............................ 169 6.5.2.1Statespacemodel ...................... 169 6.5.2.2Time-varyingpolesandstabilitymodes ........... 172 6.6FeedforwardControl .............................. 178 6.6.1MorphingBasis ............................. 178 6.6.2TrackingASystemResponse ...................... 182 6.6.2.1Polynomialmorphing ..................... 182 6.6.2.2Piecewise-polynomialmorphing ............... 183 6.6.3TrackingaPoleResponse ........................ 184 6.6.3.1Polynomialmorphing ..................... 184 6.6.3.2Piecewise-polynomialmorphing ............... 186 7CONCLUSION .................................... 188 REFERENCES ....................................... 190 BIOGRAPHICALSKETCH ................................ 200 8

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LISTOFTABLES Table page 1-1Publicationsresultingfromresearch ......................... 27 4-1Mass-spring-dampersystemparameters ....................... 84 5-1Gainsforchord-varyingaircraftdisturbancerejectioncontroller ......... 100 5-2Boundsoncoecientofmorphingbasis ...................... 106 5-3Optimalpolynomialmorphingtotrackdesiredpolynomialmorphing ...... 106 5-4Optimalpolynomialmorphingtotrackdesiredsinusoidalmorphing ....... 106 5-5Optimalpiecewise-polynomialmorphingtotrackdesiredsinusoidalmorphing .. 108 5-6Optimalpolynomialmorphingtotrackdesiredpolynomialmorphing ...... 109 5-7Optimalpolynomialmorphingtotrackdesiredsinusoidalmorphing ....... 109 5-8Optimalpolynomialmorphingtotrackdesiredsinusoidalmorphing ....... 111 6-1Empennagespecications .............................. 124 6-2Referenceparametersforsymmetricsweep ..................... 125 6-3Setofeigenvalues ................................... 137 6-4Timeconstantsofnon-oscillatorymodes ...................... 137 6-5Modeshapesofnon-oscillatorymodes ....................... 138 6-6Modalpropertiesofoscillatorymodes ........................ 138 6-7Modeshapesofoscillatorymodes .......................... 139 6-8Individualpointmasses ............................... 143 6-9Characteristicsofelementsgivenascentroidposition(in)andmomentsofinertia(gin2) ........................................ 143 6-10Boundsoncoecientofmorphingbasis ...................... 181 6-11Optimalpolynomialmorphingtotrackdesiredpolynomialmorphing ...... 182 6-12Optimalpolynomialmorphingtotrackdesiredsinusoidalmorphing ....... 182 6-13Optimalpiecewise-polynomialmorphingtotrackdesiredsinusoidalmorphing .. 184 6-14Optimalpolynomialmorphingtotrackdesiredpolynomialmorphing ...... 185 6-15Optimalpolynomialmorphingtotrackdesiredsinusoidalmorphing ....... 185 9

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6-16Optimalpiecewise-polynomialmorphingtotrackdesiredsinusoidalmorphing .. 187 10

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LISTOFFIGURES Figure page 1-1Surveillancemissionthroughanurbanenvironment ................ 19 1-2Vision-basedpathplanning ............................. 20 1-3Readinessformissioncapability ........................... 21 1-4MorphingMAVs:A)CapableofhorizontalmorphingB)Capableofverticalmorphing ....................................... 21 2-1Body-FixedCoordinateFrame ............................ 29 2-2StabilityCoordinateFrame ............................. 30 2-3Earth-FixedCoordinateFrame ........................... 30 2-4Rotationthrough .................................. 31 2-5Rotationthrough .................................. 32 2-6Rotationthrough .................................. 33 2-7AsymmetricCongurations ............................. 53 2-8SymmetricCongurations .............................. 54 4-1Mass-Spring-DamperSystem ............................ 84 4-2Time-VaryingResponses:A)Non-Inertal:State1(|),State2()-356()-356()]TJ /F1 11.955 Tf 36.42 0 Td[()B)Inertial:State1(|),State2()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[() ....................... 86 4-3Time-VaryingModes:A)Non-Inertal:Mode1(|),Mode2(\000)]TJ /F1 11.955 Tf 27.9 0 Td[()B)Inertial:Mode1(|),Mode2()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[() ........................... 87 4-4Time-VaryingPoles:A)Non-Inertal:Pole1(\000)]TJ /F1 11.955 Tf 27.9 0 Td[(),Pole2(\001)]TJ /F1 11.955 Tf 21.92 0 Td[(),RealportionofdiscreteLTIpoles(|),Poleaverage()B)Inertial:Pole1()-267()-267()]TJ /F1 11.955 Tf 34.29 0 Td[(),Pole2()-222()]TJ /F1 11.955 Tf 27.23 0 Td[(),RealportionofdiscreteLTIpoles(|),Poleaverage() ....... 88 4-5Non-InertialTime-VaryingEigenvectors:A)Eigenvector1:(1,1)(|),(2,1)(\000)]TJ /F1 11.955 Tf 9.3 0 Td[()B)Eigenvector2:(1,1)(|),(2,1)()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() .................. 89 4-6InertialTime-VaryingEigenvectors:A)Eigenvector1:(1,1)(|),(2,1)()-119()-118()]TJ /F1 11.955 Tf 30.74 0 Td[()B)Eigenvector2:(1,1)(|),(2,1)()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() .................... 90 4-7Non-InertialQcolumnvectorvalues:A)Q11(|),Q21()-278()-278()]TJ /F1 11.955 Tf 34.55 0 Td[()B)Q12(|),Q22()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() ..................................... 90 4-8Qcolumnperiodicity:Q11(|),Q21()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[() ................... 91 11

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4-9Qcolumnfrequency:ImaginarypartofdiscreteLTIpole(|),1Qcolumnperiodicity(\000)]TJ /F1 11.955 Tf 9.3 0 Td[() ........................................... 91 4-10InertialQcolumnvectorvalues:A)Q11(|),Q21(\000)]TJ /F1 11.955 Tf 27.9 0 Td[()B)Q12(|),Q22(\000)]TJ /F1 11.955 Tf 9.3 0 Td[() ........................................... 92 4-11Qcolumnperiodicity:Q11(|),Q21()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[() ................... 93 4-12Qcolumnfrequency:ImaginarypartofdiscreteLTIpole(|),LTVapproximatedfrequency()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() .................................. 93 5-1Model-FollowingSystem ............................... 97 5-2SimulationsoftheGull-MorphingAircraftModel ................. 97 5-3Chord-VaryingMAVModel(m) ........................... 100 5-4AngleofAttackResponseandAssociatedChordofAircraft ........... 101 5-5ResponsestoChord-VaryingMorphing ....................... 102 5-6ArchitectureforState-ResponseTracking ...................... 103 5-7ArchitectureforPole-ResponseTracking ...................... 104 5-8Mass-Spring-DamperSystem ............................ 104 5-9DesiredResponse(|)andOptimalResponse(\000)]TJ /F1 11.955 Tf 27.89 0 Td[()forA)First-OrderMorphingandB)Second-OrderMorphingandC)Third-OrderMorphing .......... 107 5-10DesiredResponse(|)andOptimalResponse()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()forSinusoidalMorphing 108 5-11DesiredResponse(|)andOptimalResponse()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()forSinusoidalMorphing 109 5-12DesiredResponse(|)andOptimalResponse(\000)]TJ /F1 11.955 Tf 27.89 0 Td[()forA)First-OrderMorphingandB)Second-OrderMorphingandC)Third-OrderMorphing .......... 110 5-13DesiredResponse(|)andOptimalResponse()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()forSinusoidalMorphing 111 5-14DesiredResponse(|)andOptimalResponse()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()forSinusoidalMorphing 111 6-1PicturesofSeagulls .................................. 120 6-2JointsonWing .................................... 121 6-3FloatingElbowJoint ................................. 122 6-4Feather-LikeElements ................................ 122 6-5Trackandrunnersystem ............................... 123 6-6Underwingsparstructure .............................. 124 12

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6-7Modelingoftheliftvectors ............................. 125 6-8Modelingofthetrailinglegvectors ......................... 126 6-9SweepCongurations ................................. 127 6-10SweepAngles ..................................... 128 6-11VariationofLiftwithAngleofAttackforSymmetricSweep ........... 129 6-12VariationofPitchMomentwithAngleofAttackforSymmetricSweep ..... 129 6-13VariationofRollMomentwithRollRateforSymmetricSweep .......... 130 6-14VariationofYawMomentwithAngleofSideslipforSymmetricSweep ..... 130 6-15Modeltostate-spaceowchart ........................... 131 6-16NumberofUnstablePolesofLongitudinalDynamicsforSymmetricSweep ... 131 6-17NumberofUnstablePolesofLateral-DirectionalDynamicsforSymmetricSweep 132 6-18NumberofOscillatoryPolesforLongitudinalDynamicswithSymmetricSweep 132 6-19NumberofOscillatoryPolesforLateral-DirectionalDynamicswithSymmetricSweep ......................................... 133 6-20VariationofLiftwithAngleofAttackforAsymmetricSweep ........... 134 6-21VariationofPitchMomentwithAngleofAttackforAsymmetricSweep ..... 134 6-22VariationofRollMomentwithRollRateforAsymmetricSweep ......... 134 6-23VariationofYawMomentwithAngleofSideslipforAsymmetricSweep ..... 135 6-24VariationofCoupledAerodynamicsforAsymmetricSweep ............ 135 6-25NumberofUnstablePolesforDynamicswithAsymmetricSweep ........ 136 6-26NumberofOscillatoryPolesforDynamicswithAsymmetricSweep ....... 136 6-27EectiveAnglesofSideslip .............................. 139 6-28MaximumAngleofSideslipatwhichAircraftcanTrim .............. 140 6-29PointMassLocations ................................. 143 6-30ChangeinVelocityBasedonSymmetricMorphing:0deg( {2{ ),5deg( {/{ ),10deg( {{ ),15deg( {{ ),20deg( {.{ ),25deg( {{ ),30deg( {4{ ) ...... 144 6-31Closed-LoopBlockDiagram ............................. 146 6-32PlantModelwithTrimLogic ............................ 147 13

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6-33MorphingCongurationforFastMorphing( | ),SlowMorphing( )-222()-222()]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.955 Tf 398.84 -11.96 Td[() .... 148 6-34AltitudeinResponsetoFastMorphing( | ),SlowMorphing( ::: ),FixedSwept( \000)]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.955 Tf 36.22 -50.31 Td[(),FixedStraight( )-221(\001 ) .............................. 149 6-35PitchAngle(left)andPitchRate(right)inResponsetoFastMorphing( | ),SlowMorphing( ::: ),FixedSwept( )-221()-223()]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.955 Tf 239.46 -88.66 Td[(),FixedStraight( )-222(\001 ) ........ 149 6-36AltitudeinResponsetoFastMorphing( | ),SlowMorphing( ::: ),FixedSwept( \000)]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.955 Tf 36.22 -127.02 Td[(),FixedStraight( )-221(\001 ) .............................. 150 6-37AltitudeinResponsetoFastMorphing( | ),SlowMorphing( ::: ),FixedSwept( \000)]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.955 Tf 36.22 -165.37 Td[(),FixedStraight( )-221(\001 ) .............................. 151 6-38AltitudeinResponsetoFastMorphing( | ),SlowMorphing( ::: ),FastMorphingWithoutInertia( )-222(\001 ),SlowMorphingWithoutInertia( )-222()-222()]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.955 Tf 363.15 -203.72 Td[() ........ 152 6-39ChangeinVelocityBasedonAsymmetricMorphing:0deg( {2{ ),5deg( {/{ ),10deg( {{ ),15deg( {{ ),20deg( {.{ ),25deg( {{ ),30deg( {4{ ) ...... 153 6-40Open-LoopBlockDiagram .............................. 154 6-41PlantModelwithTrimLogic ............................ 155 6-42MorphingCongurationforFastMorphing( | ),SlowMorphing( )-222()-222()]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.955 Tf 398.84 -313.8 Td[() .... 156 6-43TurninResponsetoFastMorphing( | ),SlowMorphing( ::: ),FixedSwept( \000)]TJ ET 0 0 1 RG 0 0 1 rg BT /F1 11.955 Tf 36.22 -352.15 Td[(),FixedStraight( )-169(\001 ):A)ThecompleteturnproleB)Theturnproleat270degreesC)Theturnproleat180degrees ................... 157 6-44RollAngle(left)andRollRate(right)inResponsetoFastMorphing(|),SlowMorphing()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() .................................. 157 6-45YawAngle(left)andYawRate(right)inResponsetoFastMorphing(|),SlowMorphing()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() .................................. 158 6-46PitchAngle(left)andPitchRate(right)inResponsetoFastMorphing(|),SlowMorphing()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() ............................... 158 6-47TurninResponsetoFastMorphing(|),SlowMorphing(:::),FixedSwept(\000)]TJ /F1 11.955 Tf 9.3 0 Td[(),FixedStraight()-221(\001) .............................. 159 6-48TurninResponsetoFastMorphing(|),SlowMorphing(:::),FastMorphingWithoutInertia()-222(\001),SlowMorphingWithoutInertia()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() ........ 160 6-49LongitudinalStatesduringMorphingfrom+30degto0degover1sec:ForwardVelocity(upperleft),VerticalVelocity(upperright),PitchRate(lowerleft),PitchAngle(lowerright) ............................... 164 14

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6-50LinearTime-VaryingPoles(|)andLinearTime-InvariantPoles()-20()-21()]TJ /F1 11.955 Tf 28.39 0 Td[()duringMorphingfrom+30degto0degover1sec:RealPart(upperleft)andImaginaryPart(lowerleft)ofp41andp42,RealPart(upperright)andImaginaryPart(lowerright)ofp43andp44 ................................. 165 6-51ModesAssociatedwithTime-VaryingPolesduringMorphingfrom+30degto0degover1sec:RealPart(upperleft)andImaginaryPart(lowerleft)of41and42,RealPart(upperright)andImaginaryPart(lowerright)of43and44 166 6-52NormalizedEigenvectorsAssociatedwithTime-VaryingModesduringMorphingfrom+30degto0degover1sec:Magnitude(upperleft)andPhase(lowerleft)ofv1andMagnitude(upperright)andPhase(lowerright)ofv2 ......... 167 6-53NaturalFrequencyAssociatedwithLinearTime-VaryingPoles(|)andLinearTime-InvariantPoles()-94()-95()]TJ /F1 11.955 Tf 30.16 0 Td[()duringMorphingfrom+30degto0degover1sec:Poles1and2(left)andPoles3and4(right) ................... 168 6-54EnvelopeAssociatedwithLinearTime-VaryingPoles(|)andLinearTime-InvariantPoles()-257()-257()]TJ /F1 11.955 Tf 34.04 0 Td[()duringMorphingfrom+30degto0degover1sec:Poles1and2(left)andPoles3and4(right) .......................... 169 6-55DampingRatioAssociatedwithLinearTime-VaryingPoles(|)andLinearTime-InvariantPoles()-257()-257()]TJ /F1 11.955 Tf 34.04 0 Td[()duringMorphingfrom+30degto0degover1sec:Poles1and2(left)andPoles3and4(right) .......................... 169 6-56Stabilitymodechangevstimechange:10degrees(|),20degrees()-258()-259()]TJ /F1 11.955 Tf 34.08 0 Td[(),30degrees(---)A)2secondsB)5secondsC)10secondsD)20seconds ..... 173 6-57Stabilitymodechangevsdegreechange:20seconds(|),10seconds()-216()-216()]TJ /F1 11.955 Tf 33.06 0 Td[(),5seconds(---),2seconds()A)10degreesB)20degreesC)30degrees .... 174 6-58Polechangevstimechange:10degrees(|),20degrees()-327()-328()]TJ /F1 11.955 Tf 35.73 0 Td[(),30degrees(---)A)2secondsB)5secondsC)10secondsD)20seconds .......... 175 6-59Polechangevsdegreechange:20seconds(|),10seconds()-281()-283()]TJ /F1 11.955 Tf 34.64 0 Td[(),5seconds(---),2seconds()A)10degreesB)20degreesC)30degrees ......... 176 6-60Longitudinalaircraft30degree,10secondmorphA)polesets:polesets1and2()-271()-271()]TJ /F1 11.955 Tf 34.38 0 Td[(),polesets3and4(---),time-invariantfrozentimeeigenvalues(|)B)stabilitymodes:mode1(|)mode2()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()mode3(---)mode4() .. 177 6-61LongitudinalaircraftQcolumnvectors1and2for30degree,10secondmorph:A)Q11(|)Q21()-252()-252()]TJ /F1 11.955 Tf 33.93 0 Td[()Q31(---)Q41()B)Q12(|)Q22()-252()-252()]TJ /F1 11.955 Tf 33.93 0 Td[()Q32(---)Q42() .................................... 178 6-62LongitudinalaircraftQcolumnvectors3and4for30degree,10secondmorph:A,B)Q13(|)Q23()-256()-256()]TJ /F1 11.955 Tf 34.03 0 Td[()Q33(---)Q43()C,D)Q14(|)Q24()-256()-256()]TJ /F1 11.955 Tf 34.02 0 Td[()Q34(---)Q44() ................................. 179 15

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6-63LongitudinalaircraftstateresponsesA)state1(u)B)state2(w)C)state3(q)D)state4() ................................... 180 6-64Longitudinalaircraftcolumneigenvectorsfor30degree,10secondmorph:A)V11(|)V21()-100()-101()]TJ /F1 11.955 Tf 30.3 0 Td[()V31(---)V41()B)V12(|)V22()-100()-101()]TJ /F1 11.955 Tf 30.3 0 Td[()V32(---)V42()C)V13(|)V23()-98()-98()]TJ /F1 11.955 Tf 30.24 0 Td[()V33(---)V43()D)V14(|)V24()-98()]TJ -391.46 -14.45 Td[()]TJ /F1 11.955 Tf 9.3 0 Td[()V34(---)V44() ............................... 181 6-65DesiredResponse(|)andOptimalResponse(\000)]TJ /F1 11.955 Tf 27.89 0 Td[()forA)First-OrderMorphingandB)Second-OrderMorphingandC)Third-OrderMorphing .......... 183 6-66DesiredResponse(|)andOptimalResponse()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()forSinusoidalMorphing 184 6-67DesiredResponse(|)andOptimalResponse()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()forSinusoidalMorphing 185 6-68A)ResponsetoMorphingTrajectory:Desired(|),Optimized()-281()-282()]TJ /F1 11.955 Tf 34.63 0 Td[()B)ResponsetoMorphingTrajectory:Desired(|),Optimized(\000)]TJ /F1 11.955 Tf 27.89 0 Td[()C)ResponsetoMorphingTrajectory:Desired(|),Optimized()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() ............ 186 6-69ResponsetoMorphingTrajectory:Desired(|)andOptimized()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[() .... 187 6-70ResponsetoMorphingTrajectory:Desired(|)andOptimized()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[() .... 187 16

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyALINEARINPUT-VARYINGFRAMEWORKFORMODELINGANDCONTROLOFMORPHINGAIRCRAFTByDanielT.GrantMay2011Chair:RickLindMajor:AerospaceEngineering Morphing,whichchangestheshapeandcongurationofanaircraft,isbeingadoptedtoexpandmissioncapabilitiesofaircraft.Theintroductionofbiological-inspiredmorphingisparticularlyattractiveinthathighly-agilebirdspresentexamplesofdesiredshapesandcongurations.Apreviousstudyadoptedsuchmorphingbydesigningamultiple-jointwingthatrepresentedtheshoulderandelbowjointsofabird.Theresultingvariable-gullaircraftcouldrotatethewingsectionverticallyatthesejointstoaltertheightdynamics.Thispaperextendsthatmultiple-jointconcepttoallowavariable-sweepwingwithindependentinboardandoutboardsections.Theaircraftisdesignedandanalyzedtodemonstratetherangeofightdynamicswhichresultfromthemorphing.Inparticular,thevehicleisshowntohaveenhancedcrosswindrejectionwhichisacertainlycriticalmetricfortheurbanenvironmentsinwhichtheseaircraftareanticipatedtooperate. Missioncapabilitycanbeenabledbymorphinganaircrafttooptimizeitsaerodynamicsandassociatedightdynamicsforeachmaneuver.Suchoptimizationoftenconsiderthesteady-statebehavioroftheconguration;however,thetransientbehaviormustalsobeanalyzed.Inparticular,thetime-varyinginertiashaveaneectontheightdynamicsthatcanadverselyaectmissionperformanceifnotproperlycompensated.Theseinertiatermscausecouplingbetweenthelongitudinalandlateral-directionaldynamicsevenformaneuversaroundtrim.Asimulationofavariable-sweepaircraftundergoingasymmetric 17

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morphingforanaltitudechangeshowsanoticeablelateraltranslationintheightpathbecauseoftheinducedasymmetry. Theightdynamicsofmorphingaircraftmustbeanalyzedtoensureshape-changingtrajectorieshavethedesiredcharacteristics.Thetoolsfordescribingightdynamicsofxed-geometryaircraftarenotvalidfortime-varyingsystemssuchasmorphingaircraft.Thispaperintroducesamethodtorelatetheightdynamicsofmorphingaircraftbyinterpretingatime-varyingeigenvectorintermsofightmodes.Thetime-varyingeigenvectorisactuallydenedthroughadecompositionofthestate-transitionmatrixandthusdescribesanentireresponsethroughamorphingtrajectory.Avariable-sweepaircraftisanalyzedtodemonstratetheinformationthatisobtainedthroughthismethodandhowtheightdynamicsarealteredbythetime-varyingmorphing. Also,morphingvehicleshaveinherentlytime-varyingdynamicsduetothealterationoftheircongurations;consequently,thenumeroustechniquesforanalysisandcontroloftime-invariantsystemsareinappropriate.Therefore,acontrolschemeisintroducedthatdirectlyconsidersaconceptoftime-varyingpoletocommandmorphing.Theresultingtrajectoryminimizingtrackingerrorforeitherastateresponseorapoleresponse. 18

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CHAPTER1INTRODUCTION 1.1Motivation Miniatureairvehiclesareresourceswhosecharacteristics,suchassizeandspeed,enablearangeofmissionproles.Thesevehiclesareideallysuitedtooperatewithinurbanenvironmentsataltitudesthatareinaccessibletolargeraircraftduetodenseobstacles.Tasksincludingsurveillanceandtrackingwillbegreatlyfacilitatedbyvehiclesthatcanyattreetoplevelandintobuildings.Anillustrationofapossiblesurveillancemission,astraversedthroughanurbanenvironment,maybeseeninFig. 1-1 Figure1-1. Surveillancemissionthroughanurbanenvironment Agilityisincreasinglyrequiredforthesevehiclesasthemissiontasksconsidertheightconditionsassociatedwithurbanenvironments.Theclosespacingofobstacleswillrequireavehiclethatcanturnsharplyinasmallradiusbutyetloiterandcruise.Thewindsaroundtheseobstaclessignicantlyvaryindirectionwhichwillrequirethevehicletoincurlargeanglesofsidesliptomaintainsensorpointing.Thedurationforwhichthesensorismaintainedonthetargetiscrucialtocompletingmissionobjectivessuchaslaser-basedswathmapping[ 108 ]andvision-basedpathplanning[ 64 ],asshowninFig 1-2 19

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Figure1-2. Vision-basedpathplanning Suchdisparaterequirementsplaceconstraintsonthedesignwithinwhichasinglevehiclecannotlie.Therefore,morphingisbeingincorporatedtoenablemulti-rolecapabilitiesofasinglevehicle.Essentially,thevehiclechangesshapebyalteringparameters,suchasspanorcamber,duringight.Theresultingrangeofcongurationswillhaveanassociatedrangeofightdynamicsand,consequently,maneuvering. 1.2ProblemDescription BattleeldenvironmentshavepreviouslyservedasaperformancestageformanyprovencommercialgradeUAVs.TheseUAVsaretypicallylargerinsize,anddesignedprimarilyforsurveillancefromhigheraltitudes,relativetoitssmallercounterpart,theMAVorMiniatureAerialVehicle.ThelargerUAVsmaylacktheadvantageofsizeandmaneuverabilityovertheMAV,butsignicantlymake-upforthefactwiththeirtechnologicalreadinessformissioncapability,asshowninFig. 1-3 Modernavionics,andtheirrespectivesub-systems,haverecentlymadelargeadvancesinthereductionoftheiroverallweightandsize.Asaresult,MAVsarebeginningtobeoutttedwithmoresophisticatedsensorpackagesandcontrolsystems.Itshouldbenotedhowever,thatevenwiththeseadvances,thelargerUAVisstillsuperiortotheMAVintermsofbeingmissioncapable.Duetothefactofthistechnologicalgap,ithasbeensuggestedthataviancharacteristics,andtheireectivebenets,bestudied. 20

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Figure1-3. Readinessformissioncapability TwoMAVsutilizingtheconceptofwingmorphing,orshapechanging,(acommoncharacteristicofavianight)canbeshowninFig. 1-4 A B Figure1-4. MorphingMAVs:A)CapableofhorizontalmorphingB)Capableofverticalmorphing TheightdynamicsoftheaircraftshowninFig. 1-4 aresomewhatuniqueanddierentfromthosedenedforsymmetricxed-wingight.Withtheintroductionofmorphing,afewpreviouslymadeassumptionsmustnowbereconsidered.Itshouldbenotedthatmorphingchangesthesystemfromtime-invarianttotime-varying,andasaresult,introducesnewinertialtermsintothedynamics. 21

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Themomentsofinertiaofabodyobviouslyhaveaprofoundinuenceonthedynamicsandassociatedmotionofthatbody.Certainlyaerospacesystems,withmultipledegreesoffreedomfortranslationandrotation,mustproperlyaccountforinertiatoahighlevelofaccuracyinordertomodelthedynamics.Thetime-varyingaspectoftheseinertiasmustbeconsideredwithsimilaraccuracytonoteitsinuenceonthesystem. Someextensiveandrigorousevaluationsoftraditionalcausesoftime-varyinginertia,suchasfuelexpenditureandmulti-bodyrotation,havebeenperformedforspacesystems.Theeectsoftranslatingmasswithinaspacestationarederivedunderanassumptionofharmonicmotionandusedtocomputelibrationalstability[ 114 ].Movingmasswasalsoincludedinthedynamicsofavehiclewithasolarsailthatcouldmoveforcontrolpurposes[ 124 ].Thedynamicsandassociatedtime-varyinginertiawasmodeledforatwo-vehicleformationinwhichaCoulombtethercontrolledtherelativedistanceandmassdistribution[ 82 ].Anotherstudyoptimizedadesignforatwo-vehicleformationwithaexibleappendagewhosemotionalteredtheinertiaproperties[ 89 ].Theinuenceofthrusters,whichexpendmassthroughactivationandthusvarytheinertia,wasinvestigatedusingaformulationoffeedbackandfeedforwardtocanceltheeects[ 118 ].Thetime-varyinginertiaduetothrusterswascoupledwitheectsofuidsloshinginanotherexaminationofspacecraftdynamics[ 51 ]. Thesetraditionalcauseshavealsobeenexaminedwithrespecttotheireectsonaircraftalthoughnotnecessarilytothesamedegreeasspacecraft.Fuelburnisoftenneglectedsinceitstimeconstantisslowerthantheightdynamicsofmanyaircraft;however,thateectsontime-varyinginertiawereshownforthecaseofaerialrefuelinginwhichmasswasrapidlytransferredfromthetankertotherecipient[ 120 ].Onasmallerscale,thedynamicsofaapping-wingmicroairvehiclewerestudiedbynotingtheeectofwingmotion[ 121 ]. Theintroductionofmorphing,orshape-changingactuation,toanaircraftwillaltertheshapeandmassdistributionofthevehicle,andasaresult,producetime-varying 22

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inertias.Manystudiesintomorphingaircrafthavefocusedonthesteady-statebenetsofalteringacongurationforissuessuchasfuelconsumption[ 14 ],rangeandendurance[ 43 ],costandlogistics[ 15 ],actuatorenergy[ 93 ],maneuverability[ 99 ]andairfoilrequirements[ 103 ].Additionally,aeroelasticeectshavebeenoftenstudiedrelativetomaximumrollrate[ 66 44 9 ]andactuatorloads[ 77 ]. Morphinghasalsobeenintroducedtomicroairvehiclesforthepurposeofmanueveringcontrol[ 1 2 ].Specically,anaircraftisdesignedthatusesindependentwing-sweep,asshowninFig. 1-4 ,ofinboardandoutboardsectionsonboththerightandleftwings[ 46 ].Thataircraftisshowntousethemorphingforalteringtheaerodynamicsandachieveperformancemetricsrelatedtosensorpointing.Thewingsareabletosweepontheorderofasecond;consequently,thetemporalnatureofthemorphingmustbeconsidered. 1.3ProblemStatement Performanceofamorphingaircraftiscriticaltothesuccessofanymission;therefore,methodstoachievethisperformancemustbeconsidered.Designinganeectivecontrolschemeformorphingaircraftisanon-trivialtask,butcanbemademoreapparentbyrstunderstandingthesystemdynamics.Thedynamicperformanceoflinearsystemshaveclassicallybeenstudiedusingtime-invariantmethodssuchaseigenvalueandeigenvectoranalysis[ 97 ].Thisanalysishelpsdeterminesystemstabilityaswellasdynamicightmodes. Morphingcanbemodeledasatime-invariantsystemwithdiscretechanges[ 46 ],butisconsideredmoreaccuratewhenmodeledasatime-varyingsystem.Itiswellknown[ 127 ]thattime-invarianteigenvaluesprovidenoinformationabouttime-varyingstability.Asaresult,conceptsanalogoustotime-invarianteigenvaluesandeigenvectorshavebeendevelopedfortime-varyingsystems[ 59 86 126 130 131 127 ].Theparticularmethodologyforwhichtheseconceptsarederived,varybaseduponthesystemrepresentation.Oneapproachaddressesthenotionofpolesetsderivedfromadierentialequationwithtime-varyingcoecients[ 59 131 ].Anotherapproachaddressesthenotionofpolesets 23

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derivedbytransformingthestateequationintoanupper-triangularstateequation,viaastabilitypreservingvariablechange[ 86 ]. Thechallengeariseswhenconsideringmorphingasaplausiblecontrolscheme.Typically,anaircraftiscontrolledaboutitsaxesbycommandingadeection,fromrelativetrim,inoneofitsclassicallydenedcontrolsurfaces.Itisseenhowever,thatnocontrolsurfacesarepresentinavianight,yetmaneuveringisstilldoneeciently.Thisobservationcomesasadirectresultofwingmorphing,andthus,servesasaeldofinterestincontroldesign.Themainthoughtbeing,isitpossibletoeectivelycontrolanaircraftwithmorphing,andifitis,howdoyoudescribetheresultingtime-varyingeects? 1.3.1Contributions TheclassofsystemsunderinvestigationcanberepresentedasinEquation 1{1 .Thisformdescribesasystemwithnxstatesalongwithnmorphingactuatorsandnutraditionalactuatorssuchthatx2Rnxisthestatevector,2Rnisthecommandtomorphing,andu2Rnuisthecommandtotraditionalcontrolsurfaces.ThecoecientsintheequationsofmotionarecombinedintoA2RnxnxandB2Rnxnuwhicharefunctionsofboththemorphingconguration,,andtheightcondition,. _x=A(;)x+B(;)u(1{1) Thisresearchhasresultedinthedevelopment,building,andtestingofinnovativeandtheoreticaladvancementsusedtoaddressthechallengesrelatedtotime-varyingmorphingaircraft.Asetofsolutionsareoutlinedinthedissertationtoaddressthisnovelapproachdevelopedforsystemsanalysisandcontrolsynthesis.Theresultingapproachhasresultedinthefollowingspeciccontributions. MorphingAircraft { Amorphingplatformwasdesignedsuchthatactuationofanindependentmulti-joint,asymmetricwingwasmadepossible. { Missionperformancewasevaluatedforthisaircraftthroughsimulationandighttesting. 24

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Time-VaryingInertias { Retainedallinertialtermsintheequationsofmotiontostudytheirtime-varyingeectswithinmorphingightdynamics. { Theresultingtime-varyinginertialeectswerecharacterizedandparamterizedbasedontheirinuencewithinthemodalstructureoftheightdynamics. Time-varyingFlightDynamics { Interpretationsweredevelopedforthetime-varyingpolesandmodalcharacteristicsofamorphingaircraft. { Theseinterpretationswerecompletedusingtwodierentmethodologiespreviouslyuntestedfortime-varyingightdynamics. MorphingControl { Aframeworkforlinearinput-varyingsystemswasformulatedtoaccountforavarietyofsystemswhosedynamicsaredependentinasetofcontroleectors. { Developedandsynthesizedoptimalfeedforwardtrackingfordesiredtime-varyingpolesandsignals. { Developedandsynthesizedoptimalfeedbackforaspeciccaseoftime-varyingsystem.Thissystemwasredenedinabilineartime-invariantformwherenoinputmatrixwasavailableforcontrolpurposes. 1.3.2Papers ThecontributionslistedinSection 1.3.1 haveresultedinmultiplepublicationsincludingjournalpapers,conferencepapers,andabookchapter,aswellas,televisiondocumentariesandaU.S.patent.Thesepublications,alongwiththedocumentariesandpatent,arelistedasfollows: BookChapter 1. D.T.Grant,S.Sorley,A.Chakravarthy,R.Lind,"FlightDynamicsofMorphingAircraftwithTime-VaryingInertias,"inMorphingVehiclesandStructures:AnAerospacePerspective,editedbyJ.Valasek,Wiley[Acceptedforpublication,scheduledforreleaseJanuary2011] JournalPapers 25

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2. D.T.Grant,M.AbdulrahimandR.Lind,"DesignandAnalysisofBiomimeticJointsforMorphingofMicroAirVehicles,"InternationalJournalofMicroAirVehicles,[Accepted]. 3. D.T.Grant,M.AbdulrahimandR.Lind,"FlightDynamicsofaMorphingAircraftUtilizingIndependentMultiple-JointWingSweep,"BioinspirationandBiomimetics,[Accepted]. 4. A.Chakravarthy,D.T.GrantandR.Lind,"Time-VaryingDynamicsofMicroAirVehicleWithVariable-SweepMorphing,"JournalofGuidance,Control,andDynamics,[InReview]. 5. D.T.GrantandR.Lind,"OptimalFeedbackControlforTime-VaryingMorphingAircraft,"[FutureWorkinProgress]. ConferencePapers 6. D.T.Grant,M.AbdulrahimandR.Lind,"FlightDynamicsofaMorphingAircraftUtilizingIndependentMultiple-JointWingSweep,"AIAAAtmosphericFlightMechanicsConference,Keystone,CO,August2006,AIAA-2006-6505.[BestStudentPaper] 7. D.T.Grant,M.AbdulrahimandR.Lind,"EnhancingMissionCapabilityforMicroAirVehiclesusingBiomimeticJointsandStructures,"AustralianInternationalAerospaceCongress,Melbourne,Australia,March2007,accepted. 8. D.T.Grant,M.AbdulrahimandR.Lind,"TheRoleofMorphingtoEnhanceMissionCapabilityofMicroAirVehicles,"WorldForumonSmartMaterialsandSmartStructuresTechnology,invitedpaper,ChongqingandNanjing,China,May2007. 9. D.T.Grant,M.AbdulrahimandR.Lind,"AnInvestigationofBiologically-InspiredJointstoEnableSensorPointingofMorphingMicroAirVehicles,"Interna-tionalForumonAeroelasticityandStructuralDynamics,Stockholm,Sweden,June2007,IF-069. 10. D.T.GrantandR.Lind,"EectsofTime-VaryingInertiasonFlightDynamicsofanAsymmetricVariable-SweepMorphingAircraft,"AIAAAtmosphericFlightMechanicsConference,HiltonHead,SC,August2007,AIAA-2007-6487. 11. D.T.Grant,A.ChakravarthyandR.Lind,"ModalInterpretationofTime-VaryingEigenvectorsofMorphingAircraft,"AIAAAtmosphericFlightMechanicsCon-ference,Chicago,IL,AIAA-2009-5848. 12. D.T.GrantandR.Lind,"OptimalTrackingofTime-VaryingModesForControlofMorphingAircraft,"AIAAGuidance,Navigation,andControlConference,Toronto,ON,Canada,August2010,AIAA-2010-8203. 26

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13. A.Chakravarthy,D.T.GrantandR.Lind,"Time-VaryingDynamicsofaMicroAirVehiclewithVariable-SweepMorphing,"AIAAGuidance,NavigationandControlConference,Chicago,IL,August2009,AIAA-2009-6304. TelevisionDocumentariesandFilms 14. "FlyingMosters3D":Sky3DTVDocumentary(3DImaxFeatureFilm)[AiredDecember2010] 15. "MorphingMAV":NationalGeographicChannel(5minutesegmentonMadLab)[FirstAired:Apr2008] 16. "BirdWingsforAircraft":VPRO-publictelevisioninTheNetherlands(25minutesegmentonCopyCats)[FirstAired:Dec2006] 17. "TheMagicofMotion":PBS(20minutesegmentonNatureTechDocumentary)[FirstAired:Nov2006] 18. "MorphingMAV":ScienceChannel(5minutesegmentonDiscoveriesThisWeek)[FirstAired:Jul2006] 19. "Biologically-InspiredMorphingMAV":DiscoveryChannel(5minutesegmentonBeyondTomorrow)[FirstAired:May2006] 20. "MorphingMAV":DiscoveryChannel(5minutesegmentonDailyPlanet)[FirstAired:Oct2005] Patent 21. MorphingAircraft:USPatentApplicationSerialNumberPCT/US09/54917 Thepublicationslistedabovearederivedfromcertainsectionsofthisdissertation.Relationshipsbetweenthesetopics,dissertationsections,andresultingpublicationscanbefoundinTable 1-1 Table1-1. Publicationsresultingfromresearch TopicDissertationSectionWork PlatformDesign 6.1 6.3 1. 2. 3. 6. 7. 8. 9. Time-VaryingInertias 2.1 2.6 6.4 10. Time-VaryingFlightDynamics 3.1 3.6 4.1 4.4 4. 11. 13. OptimalFeedforwardControl 5.1 5.4 12. OptimalFeedbackControl 5.1 5. 27

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1.4DissertationOverview Chapter1introducesthefollowingdocumentbystrategicallyaddressingthekeyissuesdirectlyrelatedtothedissertation'smaintopicofresearch.Theseissuesarewrittenintosectionsbasedupontheircontributiontotheoverallunderstandingoftheproblembeingconsidered.Sectiontopicsinclude:motivationforconductingresearch,abackgrounddescriptionoftheproblem,andastatementformallydeningtheactualproblem. Chapter2beginsbyrecallingthetraditionalderivationandlinearizationofthenonlinearaircraftequationsofmotion.Theseaircraftequationsofmotionarethenlaterrederived,andlinearized,basedupontheassumptionthatthesystemistime-varying(aresultofmorphing).Thischapterisconcludedbyexaminingtwoexampleswhereboththesymmetricandasymmetricmorphingequationsofmotionarereducedtothetraditionallydenedequationsofmotion,asdescribedpreviouslyinthechapter. Chapter3updatesthereaderwiththemostinuentialmethodsindeningthepolesandzerosforalineartime-varyingsystem.Thefourmost-commonapproachesandtheirmethodsaregiven,eachincludingabriefdescriptionofbackgroundtheoryanddenitions.Thestabilityofeachmethodisalsoconsidered,therefore,necessaryandsucientconditionspertainingtodierentlevelsofstabilityaredened. Chapter4beginsbyreviewingthefundamentalframeworksusedtointerpretbothtime-invariantandtime-varyingsystemcharacterisitcs.Theresultingtime-varyingframeworkisthenfurtherinvestigatedfortwoofthefourmethodologiesdescribedinChapter3.Chapter4closesbyprovidingatime-varyingexamplespecictotheframeworkpreviouslymention. Chapter5motivesthederivationofasystematicapproachtoclassifyingandcontrollingthetime-varyingsystemproducedbymorphing.Morphingcontrolchallangesandapproachesaredecribedandthenfollowedbyexamplesoutliningcertainapproaches.Bothstabilizingandoptimizingperformanceareconsideredthroughtheoreticalderivationandthenimplementation.Thechatpterconcludeswithafutureworksectionwhichexpandsthepreviousworkintopossibleavenuesforfutureworkandproposestheoreticalderivations. Chapter6presentsanextensiveexamplewhichinvestigatestheeectsofbothsymmetricandasymmetricmorphing.Generatedstate-spacesystemsareusedtosimulatebasiclineartime-invariant(quasi-static)controlledmaneuverssuchasdivingandturning,aswellas,theeectsoftime-varyingmorphingonthesystem'smodalcharacterisiticsandcontrollability. 28

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CHAPTER2EQUATIONSOFMOTION 2.1AircraftAxisSystem Threeaxissystemsarecommontodescribeaircraftmotion.Thesesystemsincludethebody-axissystem(xedtotheaircraft),theEarth-axissystem(assumedtobeaninertialaxissystemxedtotheEarth),andthestability-axissystem(denedwithrespecttothelocalwind). 2.1.1BodyAxisSystem Thebody-xedcoordinatesystemhasitsoriginlocatedattheaircraft'scenterofgravity.Theaxesareorientedsuchthat^xB,pointsdirectlyoutthenose,^yB,pointsdirectlyouttherightwing,and^zB,pointsdirectlyoutthebottom,asshowninFigure 2-1 Figure2-1. Body-FixedCoordinateFrame 2.1.2StabilityAxisSystem Thestabilityaxissystemsharesthesameoriginlocationasthebody-xedframe,butisrotatedrelativetothebody-xedaxistoalignwiththevelocityvector,asshowninFigure 2-2 .Theresulting^xSaxispointsinthedirectionoftheprojectionoftherelativewindontothexzplaneoftheaircraft.The^ySaxisisoutoftherightwingcoincidentwith^yB,whilethe^zSaxispointsdownwardinthedirectioncompletingthevectorsetdescribedbytheright-handruleandshowninFigure 2-2 29

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Figure2-2. StabilityCoordinateFrame 2.1.3EarthAxisSystem TheEarth-xedcoordinatesystemisxedtothesurfaceoftheEarthwithits^zEaxispointingtothecenteroftheEarth.The^xEand^yEaxesareorthogonalandlieinthelocalhorizontalplane,asshowninFigure 2-3 Figure2-3. Earth-FixedCoordinateFrame 30

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2.2CoordinateTransformations 2.2.1EarthtoBodyFrame AvectormaybetransformedfromtheEarth-xedframeintothebody-xedframebythreeconsectutiveconsecutiverotationsaboutthez-axis,y-axis,andx-axis,respectively.TraditionalightmechanicsdenetheanglesthroughwhichthesecoordinateframesarerelativelyrotatedastheEulerangles.TheEuleranglesareexpressedasyaw ,pitch,androll,anditisimportanttonotethataparticularsequenceofEuleranglerotationsisunique.Thefollowingillustratesthetransformationofavector,PE,intheEarth-xedframe,asdenedinEquation 2{1 ,intothebody-xedframe. PE=x^iE+y^jE+z^kE=266664XEYEZE377775(2{1) Therstrotationisthroughtheangleaboutthevector^kE,asshowninFigure 2-4 Figure2-4. Rotationthrough Therotationabout^kEbytheangle, ,isreferredtoasR3( ),whereR3( )istheshort-handnotationusedtodescribetherotationmatrixdenedinEquation 2{2 31

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266664X1Y1Z1377775=266664cos sin 0)]TJ /F1 11.955 Tf 11.3 0 Td[(sin cos 0001377775266664XEYEZE377775(2{2) Thesecondrotationisthroughtheangleaboutthevector^j1=^j2,asshowninFigure 2-5 Figure2-5. Rotationthrough Therotationabout^j1bytheangle,,isreferredtoasR2(),whereR2()istheshort-handnotationusedtodescribetherotationmatrixdenedinEquation 2{3 266664X2Y2Z2377775=266664cos0)]TJ /F1 11.955 Tf 11.29 0 Td[(sin010sin0cos377775266664X1Y1Z1377775(2{3) Thethirdandnalrotationisthroughtheangleaboutthevector^i2=^i,asshowninFigure 2-6 Therotationabout^i2bytheangle,,isreferredtoasR1(),whereR1()istheshort-handnotationusedtodescribetherotationmatrixdenedinEquation 2{4 32

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Figure2-6. Rotationthrough 266664XBYBZB377775=2666641000cossin0)]TJ /F1 11.955 Tf 11.3 0 Td[(sincos377775266664X2Y2Z2377775(2{4) Therefore,anyvectorintheEarth-xedframe,PE,canbetransformedintothebody-xedframe,PB,usingtherelationshipdenedinEquation 2{5 PB=R1()R2()R3( )PE(2{5) Forexample,Earth-xedgravityforcescanbeexpressedinthebody-xedcoordinatesystembyimplementingEquation 2{5 ontheEarth-xedweightvector FGE=26666400mg377775E=266664)]TJ /F5 11.955 Tf 9.3 0 Td[(mgsinmgsincosmgcossin377775B(2{6) 2.2.2StabilitytoBodyFrame Typically,aerodynamicforcesaredenedbythestabilityaxisforconvenienceyet,forcomputationalreasons,theyneedtobedenedbythebody-xedaxis.Recallingthatthistransformationcanbeaccomplishedbyrotatingthestabilityaxisthroughapositiveangleofattack,thetransformationfromthestabilitycoordinateframetothebody-xed 33

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coordinateframeissimplyarotationaboutthevector^yBthroughtheangle,asdenedinEquation 2{7 266664FAxFAyFAz377775B=266664cos0)]TJ /F1 11.955 Tf 11.29 0 Td[(sin010sin0cos377775266664)]TJ /F5 11.955 Tf 9.29 0 Td[(DFAy)]TJ /F5 11.955 Tf 9.29 0 Td[(L377775S(2{7) 2.3NonlinearEquationsofMotion 2.3.1DynamicEquations TherigidbodyequationsofmotionareobtainedformNewton'ssecondlaw,whichstatesthatthesummationofallexternalforcesactingonabodyisequaltothetimerateofchangeofthemomentumofthebody;andthesummationoftheexternalmomentsactingonthebodyisequaltothetimerateofchangeofthemomentofmomentum(angularmomentum).ThetimeratesofchangeoflinearandangularmomentumarerelativetoaninertialorNewtonianreferenceframe(Earth-xedframe).Newton'ssecondlawcanbeexpressedbythevectorsdenedinEquations 2{8 2{9 XF=d dt(mv)E(2{8) XM=d dtHE(2{9) 2.3.1.1Forceequations Afewadditionalassumptionsmustbemadeinordertodeveloptheright-handside,orresponseside,ofequation 2{8 .Theseassumptionsincludethattheaircraftbeconsideredarigidbody,andthatthemassoftheaircraftremainconstant.Asaresultofassumingthemasstobeconstant,themassterm,seeninEquation 2{8 ,canbemovedoutsideofthetimederivativeandredenedasEquation 2{10 34

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XF=md dtvE=maE(2{10) Theaccelerationofanaircraftisnormallymeasuredinthebody-xedframe,therefore,touseEquation 2{10 ,thebody-xedaccelerationmustbetransformedintotheEarth-xedframe.Itisnotedthatsincethetransformationinvolvesarotationofcoordinates,anyvectorinthebody-xedframecanbecalculatedintheEarth-xedframebyusingthetransporttheorem[ 95 ],asseeninEquation 2{11 .Itshouldalsobenotedthatforthefollowingderivations,allofthevectorswillbeexpressedinthebody-xedcoordinatesystem,denotedbythesubscript,B,unlessotherwisestated.Forexample,thevelocityvectorasseenbyanobserverintheEarth-xedframe,expressedinthebody-xedcoordinatesystem,willbedontedasvEB. db dtA=db dtB+A!Bb(2{11) Thevelocityvector,vE,representstherateofchangeofthepositionvector,r,asviewedbyanobserverintheEarthxedreferenceframe.Sincethepositionvector,r,isnormallymeasuredwithrespecttothebody-xedcoordinatesystem,itmustbersttransformedintotheEarth-xedframe,inordertocomputethedesiredEarth-denedvelocityvector,VEB.Thus,thetransporttheoremisusedtorelatethepositionvector,r,totheEarth-denedvelocityvector,vEB,asseeninasseeninEquation 2{12 vEB=dr dtE=dr dtB+E!Br(2{12) TheEarth-denedveloctyvectorcanbeconvenientlyrewrittenanddened,asseeninEquation 2{13 35

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vEB=u^i+v^j+w^k=266664UVW377775EB(2{13) ThesttermontherightsideofEquation 2{12 iscalledthebody-denedvelocity.Thebody-denedvelocityvector,vBB,issimplythetimederivativeofthepositionvector,r,asseeninEquation 2{14 .Notethatthisisonlythecasewhenthepositionvectorisdescribedbythebody-xedcoordinatesystem. dr dtB=_x^i+y^j+_z^k=vBB(2{14) ThesecondtermontherightsideofEquation 2{12 ,isdescribedbythecrossproductbetweentheEarth-denedvelocityvector,vEBandatermcalledtheangularvelocityvector.Theangularvelocityvector,E!B,describestheangularvelocityofreferenceframeB(body-xedframe)asviewedbyanobserverinreferenceframeE(Earthxedframe),representedinthebody-xedcoordinatesystem. Theangularveloctyvectorcanbeconvenientlyrewrittenanddened,asseeninEquation 2{15 B!E=p^i+q^j+r^k=266664_p_q_r377775EB(2{15) Itshouldbenoted,thattheangularvelocityvectors,E!BandB!E,arerelatedthroughasimplerelationship,asseeninEquation 2{16 E!B=)]TJ /F11 7.97 Tf 9.3 4.94 Td[(B!E(2{16) Itshouldalsobenoted,thatangularvelocityvectorsrelatingmultiplerotations,maybelinearlyaddedintooneangularvelocityvectorthatrelatestheentiretransformation. 36

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Thisadditionofangularvelocityvectorscanbedescribedbytheangularvelocityadditiontheorem[ 95 ],whichisshowninEquation 2{17 A1!An=A1!A2+A2!A3+:::+An)]TJ /F15 5.978 Tf 5.75 0 Td[(1!An(2{17) ThederivationoftheEarth-denedaccelerationvector,aEB,issimilartothepreviousdenitionoftheEarth-denedvelocityvector,vEB.Thatis,theEarth-denedaccelerationvector,aEB,representstherateofchangeoftheEarth-denedvelocityvector,vEB,asviewedbyanobserverintheEarth-xedreferenceframe.Therefore,tosolvefortheaccelerationvector,aEB,thetransporttheoremmustbeappliedtothevelocityvector,vEB,asseeninEquation 2{18 aEB=dvEB dtE=dvEB dtB+E!BvEB(2{18) TheEarth-denedaccelerationvectorcanbeconvenientlyrewrittenanddened,asseeninEquation 2{19 aEB=u^i+v^j+w^k=266664_u_v_w377775B(2{19) NoticethatifthesttermontherightsideofEquation 2{19 isdenedbythebody-xedcoordinatesystem,thenonlythetimederivativeneedstobetaken.Ifthistermisnotdenedbythebody-xedcoordinatesystem,thenthetransporttheoremmustbeappliedtocompensateforthecoordinatechange. TherighthandsideofEquation 2{19 cannowbesolvedforbyrstcomputingthecrossproductbetweentheEarth-denedangularvelocityvector,B!E,andtheEarth-denedvelocityvector,vEB.Thecrossproductisthenaddedtothebody-denedaccelerationvector,aBB.Itshouldnotedthatthebody-denedaccelerationisnotsimplythesecondderivativeoftheofthebody-denedposition.Thistermisdenedastherate 37

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ofchangeoftheEarth-denedvelocityvector,asseenbyanobserverinthebody-xedframe,andtherefore,canbecomputedbytakingthetimederivativeoftheEarth-denedvelocityvector. Theresultingterm,showninEquation 2{20 ,representstheaccelerationoftheaircraftasseenbyanobserverintheinertially-xedEarthframe,representedinthebody-xedcoordinatesystem. aB=266664_u+qw)]TJ /F5 11.955 Tf 11.96 0 Td[(rv_v+ru)]TJ /F5 11.955 Tf 11.95 0 Td[(pw_w+pv)]TJ /F5 11.955 Tf 11.95 0 Td[(qu377775B(2{20) Todenethelefthandside,orappliedforceside,ofEquation 2{8 ,itisrstassumedthatonlythemostsignicantforcesaectthemotionoftheaircraft.Theappliedforcescanthenbebrokendownintovectorcomponentsandarrangedinamannersuchthattheyaredenedbythevector,FEB,asseeninEquation 2{21 FEB=266664FxFyFz377775EB=266664FGx+FAxFGy+FAyFGz+FAz377775EB=XF(2{21) Aresultingsetoffull-order,nonlinearforceequations,asseeninEquation 2{22 ,canbederivedbyinsertingEquations 2{20 2{21 intoEquation 2{8 andrecallingthattheappliedforcesaregivenbyEquations 2{6 2{7 m(_u+qw)]TJ /F5 11.955 Tf 11.96 0 Td[(rv)=)]TJ /F5 11.955 Tf 9.3 0 Td[(mgsin+()]TJ /F5 11.955 Tf 9.3 0 Td[(Dcos+Lsin)m(_v+ru)]TJ /F5 11.955 Tf 11.96 0 Td[(pw)=mgsincos+FAym(_w+pv)]TJ /F5 11.955 Tf 11.96 0 Td[(qu)=mgcoscos+()]TJ /F5 11.955 Tf 9.29 0 Td[(Dsin)]TJ /F5 11.955 Tf 11.96 0 Td[(Lcos)(2{22) 38

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2.3.1.2Momentequations Itisshowninequation 2{2 thatNewton'ssecondlawissatisedbyequatingthesummationofmomentstothetotalrateofchangeofthemomentofmomentum(angularmomentum).ThesamerelationshipusedpreviouslytodenetheEarth-denedaccelerationcanbeusedtodenetheEarth-denedangularmomentum.Itshouldbeagainnotedthatforthefollowingderivations,allofthevectorswillbeexpressedinthebody-xedcoordinatesystem,denotedbythesubscript,B,unlessotherwisestated.Forexample,theangularmomentumvectorasseenbyanobserverintheEarth-xedframe,expressedinthebody-xedcoordinatesystem,willbedontedasHEB. Traditionally,theangularmomentumiscomputedfrommeasurementstakeninthebody-xedcoordinatesysytem.InordertoproperlydescribetheEarth-denedangularmomentumvector,itmustbersttransformedintotheinertially-xedEarthframe.Thistransformationcanbeaccomplishedbythetransporttheorem,asseeninEquation 2{23 dHEB dtE=dHEB dtB+E!BHEB(2{23) Thegeneralexpressionforangularmomentumcanbetakendirectlyfrombasicphysicsanddescribedasanobject'sinertialtensor,I,multipliedbythatobject'sangularratevector,!,asseeninEquation 2{24 H=I!(2{24) TheangularmomentumdenedinEquation 2{24 canbemaderelavanttoanaircraftbydeningtheinertialtensorintheaircraft'sbody-xedcoordinatesystem,asseeninEquation 2{25 ,andrecallingthattheEarth-denedangularvelocityvectorwaspreviouslydenedinEquation 2{15 39

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IB=266664Ixx)]TJ /F5 11.955 Tf 9.3 0 Td[(Ixy)]TJ /F5 11.955 Tf 9.3 0 Td[(Ixz)]TJ /F5 11.955 Tf 9.3 0 Td[(IyxIyy)]TJ /F5 11.955 Tf 9.3 0 Td[(Iyz)]TJ /F5 11.955 Tf 9.3 0 Td[(Ixz)]TJ /F5 11.955 Tf 9.3 0 Td[(IyzIzz377775B(2{25) Equations 2{15 and 2{25 canbeinsertedintoEquation 2{24 toproducetheEarth-denedangularmomentumvector,asseeninEquation 2{26 HEB=266664Ixx)]TJ /F5 11.955 Tf 9.3 0 Td[(Ixy)]TJ /F5 11.955 Tf 9.3 0 Td[(Ixz)]TJ /F5 11.955 Tf 9.3 0 Td[(IyxIyy)]TJ /F5 11.955 Tf 9.3 0 Td[(Iyz)]TJ /F5 11.955 Tf 9.3 0 Td[(Ixz)]TJ /F5 11.955 Tf 9.3 0 Td[(IyzIzz377775EB266664PQR377775EB(2{26) TheEarth-denedangularmomentumvector,HEB,canbeconvenientlywritteninvectornotation,asseeninEquation 2{27 HEB=Hx^i+Hy^j+Hz^k=266664HxHyHz377775EB(2{27) InsertingEquation 2{27 intothelefthandsideofEquation 2{26 andcarryingoutthematrixmultiplicationontherighthandside,resultsintheEarth-denedvectornotationoftheaircraft'sangularmomentum,asseeninEquation 2{28 HEB=266664pIx)]TJ /F5 11.955 Tf 11.95 0 Td[(qIxy)]TJ /F5 11.955 Tf 11.96 0 Td[(rIxzqIy)]TJ /F5 11.955 Tf 11.95 0 Td[(rIz)]TJ /F5 11.955 Tf 11.95 0 Td[(pIxyrIz)]TJ /F5 11.955 Tf 11.96 0 Td[(pIxz)]TJ /F5 11.955 Tf 11.95 0 Td[(qIyz377775(2{28) Themomentsofineritaaredescibedasindicatorstotheresistancetorotationaboutthataxis,asdenedinEquations 2{29 2{31 .Therefore,Ixindicatestheresistancetorotationaboutthex-axis(relativelydened).Theproductsofinertiaaredescribedasindicatorstothesymmetryoftheaircraft,asdenedinEquations 2{32 2{34 40

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Ix=Z(y2+z2)dm (2{29) Iy=Z(x2+z2)dm (2{30) Iz=Z(x2+y2)dm (2{31) Ixy=Z(xy)dm (2{32) Ixz=Z(xz)dm (2{33) Iyz=Z(yz)dm (2{34) Duetothefactthatthemasseswereassumedtobepointmasses,theintegralsinEquations 2{29 2{34 canbereducedto: Ix=m(y2+z2) (2{35) Iy=m(x2+z2) (2{36) Iz=m(x2+y2) (2{37) Ixy=m(xy) (2{38) Ixz=m(xz) (2{39) Iyz=m(yz) (2{40) ThecorrespondinginertialratescanbecalculatedbytakingthetimederivativeofEquations 2{35 2{40 ,asshowninEquationrefeqrates. 41

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_Ix=m[(2y)(_y)+(2z)(_z)]_Iy=m[(2x)(_x)+(2z)(_z)]_Iz=m[(2x)(_x)+(2y)(_y)]_Ixy=)]TJ /F5 11.955 Tf 9.29 0 Td[(m[(x)(_y)+(y)(_x)]_Ixz=)]TJ /F5 11.955 Tf 9.3 0 Td[(m[(x)(_z)+(z)(_x)]_Iyz=)]TJ /F5 11.955 Tf 9.3 0 Td[(m[(y)(_z)+(z)(_y)](2{41) Thersttermontherighthandsideofequation 2{23 isdescribedastherateofchangeoftheEarth-denedangularmomentumvectorasseenbyanoberverinthebody-xedcoordinatesystem,representedinthebody-xedcoordinatesystem.ThistermcanbefoundbysimplytakingthetimederivativeofEquation 2{28 ,asseeninEquation 2{42 dHEB dt=266664_pIx)]TJ /F1 11.955 Tf 14.12 0 Td[(_qIxy)]TJ /F1 11.955 Tf 13.78 0 Td[(_rIxz+p_Ix)]TJ /F5 11.955 Tf 11.96 0 Td[(q_Ixy)]TJ /F5 11.955 Tf 11.96 0 Td[(r_Ixz_qIy)]TJ /F1 11.955 Tf 13.78 0 Td[(_rIyz)]TJ /F1 11.955 Tf 14.24 0 Td[(_pIxy+q_Iy)]TJ /F5 11.955 Tf 11.95 0 Td[(r_Iz)]TJ /F5 11.955 Tf 11.96 0 Td[(p_Ixy_rIz)]TJ /F1 11.955 Tf 14.24 0 Td[(_pIxz)]TJ /F1 11.955 Tf 14.11 0 Td[(_qIyz+_rIz)]TJ /F5 11.955 Tf 11.96 0 Td[(p_Ixz)]TJ /F5 11.955 Tf 11.95 0 Td[(q_Iyz377775(2{42) ThesecondtermontherightsideofEquation 2{23 canbefoundbytakingthecrossproductbetweenthepreviouslydenedangularmomentumvector,HEB,asseeninEquation 2{28 ,andtheangularvelocityvector,!EB,asseeninEquation 2{19 .Thistermcanbethentemporarilydenedas,HT,andrewrittenintheformshownbyEquation 2{43 HT=266664qrIz)]TJ /F5 11.955 Tf 11.95 0 Td[(qpIxz)]TJ /F5 11.955 Tf 11.95 0 Td[(q2Iyz)]TJ /F5 11.955 Tf 11.96 0 Td[(qrIy+r2Iyz+rpIxyrpIx)]TJ /F5 11.955 Tf 11.95 0 Td[(qrIxy)]TJ /F5 11.955 Tf 11.95 0 Td[(r2Ixz)]TJ /F5 11.955 Tf 11.96 0 Td[(rpIz+p2Ixz+qpIyzpqIy)]TJ /F5 11.955 Tf 11.96 0 Td[(rpIyz)]TJ /F5 11.955 Tf 11.96 0 Td[(p2Ixy)]TJ /F5 11.955 Tf 11.95 0 Td[(pqIx+q2Ixy+rqIxz377775(2{43) TherightsideofEquation 2{23 cannowbesolvedforintermsoftheEarth-denedangularmomentumvector,HEB,byinsertingEquations 2{42 and 2{43 intoEquation 2{23 ,asseeninEquation 2{44 42

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HEB=266664_pIx)]TJ /F1 11.955 Tf 14.11 0 Td[(_qIxy)]TJ /F1 11.955 Tf 13.78 0 Td[(_rIxz+p_Ix)]TJ /F5 11.955 Tf 11.95 0 Td[(q_Ixy)]TJ /F5 11.955 Tf 11.95 0 Td[(r_Ixz_qIy)]TJ /F1 11.955 Tf 13.78 0 Td[(_rIyz)]TJ /F1 11.955 Tf 14.24 0 Td[(_pIxy+q_Iy)]TJ /F5 11.955 Tf 11.96 0 Td[(r_Iz)]TJ /F5 11.955 Tf 11.96 0 Td[(p_Ixy_rIz)]TJ /F1 11.955 Tf 14.24 0 Td[(_pIxz)]TJ /F1 11.955 Tf 14.11 0 Td[(_qIyz+_rIz)]TJ /F5 11.955 Tf 11.96 0 Td[(p_Ixz)]TJ /F5 11.955 Tf 11.96 0 Td[(q_Iyz377775+266664qrIz)]TJ /F5 11.955 Tf 11.96 0 Td[(qpIxz)]TJ /F5 11.955 Tf 11.96 0 Td[(q2Iyz)]TJ /F5 11.955 Tf 11.95 0 Td[(qrIy+r2Iyz+rpIxyrpIx)]TJ /F5 11.955 Tf 11.96 0 Td[(qrIxy)]TJ /F5 11.955 Tf 11.96 0 Td[(r2Ixz)]TJ /F5 11.955 Tf 11.95 0 Td[(rpIz+p2Ixz+qpIyzpqIy)]TJ /F5 11.955 Tf 11.95 0 Td[(rpIyz)]TJ /F5 11.955 Tf 11.96 0 Td[(p2Ixy)]TJ /F5 11.955 Tf 11.96 0 Td[(pqIx+q2Ixy+rqIxz377775(2{44) TheleftsideofEquation 2{44 canbeputintovectornotation,suchthatthevectorcomponentsoftheEarth-denedangularmomentumvector,HEB,aredenedbyindividualmomentterms,asshownbyEquation 2{45 .` dH dtEB=266664LMN377775EB(2{45) Afull-ordersetofnonlinearmomentequationscanbefoundbyrstinsertingEquation 2{45 intoEquation 2{44 andthenequatingsides,asseeninEquation 2{46 L=_pIx)]TJ /F5 11.955 Tf 11.96 0 Td[(qrIy+qrIz+(pr)]TJ /F1 11.955 Tf 14.11 0 Td[(_q)Ixy)]TJ /F1 11.955 Tf 11.95 0 Td[((pq+_r)Ixz+(r2)]TJ /F5 11.955 Tf 11.96 0 Td[(q2)Iyz+p_Ix)]TJ /F5 11.955 Tf 11.96 0 Td[(q_Ixy)]TJ /F5 11.955 Tf 11.96 0 Td[(r_IxzM=prIx+_qIy)]TJ /F5 11.955 Tf 11.96 0 Td[(prIz)]TJ /F1 11.955 Tf 11.95 0 Td[((qr)]TJ /F1 11.955 Tf 14.24 0 Td[(_p)Ixy+(p2)]TJ /F5 11.955 Tf 11.95 0 Td[(r2)Ixz+(pq)]TJ /F1 11.955 Tf 13.78 0 Td[(_r)Iyz+q_Iy)]TJ /F5 11.955 Tf 11.96 0 Td[(p_Ixy)]TJ /F5 11.955 Tf 11.96 0 Td[(r_IyzN=)]TJ /F5 11.955 Tf 9.3 0 Td[(pqIx+pqIy+_rIz+(q2)]TJ /F5 11.955 Tf 11.95 0 Td[(p2)Ixy+(qr)]TJ /F1 11.955 Tf 14.24 0 Td[(_p)Ixz)]TJ /F1 11.955 Tf 11.96 0 Td[((pr+_q)Iyz+r_Iz)]TJ /F5 11.955 Tf 11.95 0 Td[(p_Ixz)]TJ /F5 11.955 Tf 11.95 0 Td[(q_Iyz(2{46) 2.3.2KinematicEquations Thesixequationsofmotion,previouslydevelopedtorepresenttheforcesandmomentsactingontheaircraft,arenecessarybutnotsucient.Additionalequationsmustbeaddedinordertosolvetheoverallaircraftproblem.TheseadditionalequationsarenecessaryduetothefactthattheEuleranglesrepresentedintheforceequationscreatemorethansixunknowns. 2.3.2.1Orientationequations Threenewequationscanbeobtainedbyrelatingthethreebody-axissystemrates(p;q;r)tothethreeEulerrates(_ ;_;_).Itisnotedthatthisrelationshipcan 43

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beillustratedbyavectorequationwherethemagnitudeofthethreebodyratesequalsthethreeEulerratesand,asseeninEquation 2{47 !B=p^i+q^j+r^k=_ i+_j+_k(2{47) Toequatebothsidesofequation 2{24 ,itisnecessarythatbothvectorsarerepresentedinthesamecoordinateframe.Thus,bytransformingtheEuleranglesintothebody-xedcoordinatesystem,threenonlinearbodyrateequationscanbewritten,asseeninEquation 2{48 p=_)]TJ /F1 11.955 Tf 15.65 3.16 Td[(_ sinq=_cos+_ cossinr=_ coscos)]TJ /F1 11.955 Tf 14.2 3.15 Td[(_sin(2{48) ThesethreebodyrateequationcanalsobedenedintermsoftheEulerangles,asseeninEquation 2{49 _=p+q(sin+rcos)tan_=qcos)]TJ /F5 11.955 Tf 11.95 0 Td[(rsin_ =(qsin+rcos)sec(2{49) 2.3.2.2Positionequations Anadditionalthreeequationsarederivedfromtheightvelocitycomponentsrelativetotheinertiallydenedreferenceframe.Inordertoderivetheseequations,theinertiallydenedvelocitycomponents,representedinthebody-xedcoordinatesystem,mustrstbedened,asseeninEquation 2{50 266664xyz377775EB=266664dx/dtdy/dtdz/dt377775EB(2{50) 44

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AnEulertransformationmaybeusedtotransformthebody-denedvelocities,(u;v;w),intothedesiredEarth-denedvelocities.Thistransformationiscompletedbyapplyingequation 2{5 ,inreverseorder,tothebody-denedvelocities,asseeninEquation 2{51 266664dx/dtdy/dtdz/dt377775EB=266664coscos sinsincos )]TJ /F1 11.955 Tf 11.95 0 Td[(cossin cossincos +sinsin cossin sinsinsin )]TJ /F1 11.955 Tf 11.95 0 Td[(coscos cossinsin +sincos )]TJ /F1 11.955 Tf 11.29 0 Td[(sinsincoscoscos377775266664uvw377775B(2{51) Afull-ordersetofnon-linearvelocityequationscanthusbefoundbyrstcompletingthethematrixmultiplicationontherighthandsideofEquation 2{51 andthenequatingbothsides,asseeninEquation 2{52 _xEB=uBcoscos +vB(sinsincos )]TJ /F1 11.955 Tf 11.95 0 Td[(cossin )+wB(cossincos +sinsin )_yEB=uBcossin +vB(sinsinsin )]TJ /F1 11.955 Tf 11.95 0 Td[(coscos )+wB(cossinsin +sincos )_zEB=)]TJ /F5 11.955 Tf 9.3 0 Td[(uBsin+vB(sincos)+wBcoscos(2{52) IntegratingEquation 2{52 yieldstheairplane'spositionrelativetotheinertially-xedreferenceframe. 2.3.3TheEquationsCollected Thenonlinearaircraftequationsofmotioncanbecollectedintoaformalset,asshowninEquation 2{53 45

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m(_u+qw)]TJ /F5 11.955 Tf 11.95 0 Td[(rv)=)]TJ /F5 11.955 Tf 9.3 0 Td[(mgsin+()]TJ /F5 11.955 Tf 9.29 0 Td[(DcosA+LsinA)+TsinTm(_v+ru)]TJ /F5 11.955 Tf 11.95 0 Td[(pw)=mgsincos+FAy+FTym(_w+pv)]TJ /F5 11.955 Tf 11.96 0 Td[(qu)=FGz+FAz+FTzL=_pIx)]TJ /F5 11.955 Tf 11.96 0 Td[(qrIy+qrIz+(pr)]TJ /F1 11.955 Tf 14.11 0 Td[(_q)Ixy)]TJ /F1 11.955 Tf 11.95 0 Td[((pq+_r)Ixz+(r2)]TJ /F5 11.955 Tf 11.95 0 Td[(q2)Iyz+p_Ix)]TJ /F5 11.955 Tf 11.96 0 Td[(q_Ixy)]TJ /F5 11.955 Tf 11.96 0 Td[(r_IxzM=prIx+_qIy)]TJ /F5 11.955 Tf 11.96 0 Td[(prIz)]TJ /F1 11.955 Tf 11.95 0 Td[((qr)]TJ /F1 11.955 Tf 14.24 0 Td[(_p)Ixy+(p2)]TJ /F5 11.955 Tf 11.95 0 Td[(r2)Ixz+(pq)]TJ /F1 11.955 Tf 13.78 0 Td[(_r)Iyz+q_Iy)]TJ /F5 11.955 Tf 11.96 0 Td[(p_Ixy)]TJ /F5 11.955 Tf 11.96 0 Td[(r_IyzN=)]TJ /F5 11.955 Tf 9.3 0 Td[(pqIx+pqIy+_rIz+(q2)]TJ /F5 11.955 Tf 11.95 0 Td[(p2)Ixy+(qr)]TJ /F1 11.955 Tf 14.24 0 Td[(_p)Ixz)]TJ /F1 11.955 Tf 11.96 0 Td[((pr+_q)Iyz+r_Iz)]TJ /F5 11.955 Tf 11.95 0 Td[(p_Ixz)]TJ /F5 11.955 Tf 11.95 0 Td[(q_Iyz_=p+q(sin+rcos)tan_=qcos)]TJ /F5 11.955 Tf 11.96 0 Td[(rsin_ =(qsin+rcos)sec_xEB=uBcoscos +vB(sinsincos )]TJ /F1 11.955 Tf 11.95 0 Td[(cossin )+wB(cossincos +sinsin )_yEB=uBcossin +vB(sinsinsin )]TJ /F1 11.955 Tf 11.95 0 Td[(coscos )+wB(cossinsin +sincos )_zEB=)]TJ /F5 11.955 Tf 9.3 0 Td[(uBsin+vB(sincos)+wBcoscos(2{53) 2.4LinearizedEquationsofMotion Oftentimes,thenonlinearsetofmotionequationsislinearizedforuseinstabilityandcontrolanalysis.Thelinearizationiscarriedoutbymeansofthesmall-disturbancetheory.Whenusingthesmall-disturbancetheory,itisassumedthatthemotionoftheaircraftconsistsofsmalldeviationsfromareferenceconditionofsteadyight.Limitationsdoapplytothesmall-disturbancetheory,inthatproblemscontaininglargedisturbanceangles(i.e.:==2)cannotbelinearizedusingthismethod. Whenusingthesmall-disturbancetheory,allthevariablesintheequationsofmotionarereplacedbyareferencevalueplussomeperturbationasshowninEquation 2{54 46

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u=uo+up=p0+px=x0+xM=M0+Mv=v0+vq=q0+qy=y0+yN=N0+Nw=w0+wr=r0+rz=z0+zL=L0+L(2{54) Forconvenience,thereferenceightconditionisassumedtobesteadytrimmedightwithasymmetriccongurationandnoangularvelocity.TheseassumptionscanbephysicallyillustratedasshowninEquation 2{55 v0=p0=q0=r0=0=0=0(2{55) Furthermore,thex-axisisassumedtobeinitiallyalignedalongthedirectionoftheaircraft'svelocityvector,thus,w0=0.Asaresult,u0isequaltothereferenceightspeedand0tothereferenceangleofclimb.Forreasonsofsimplication,itisnotedthattheatrigonometricidentitymaybeapplied,asdenedinEquation 2{56 sin(0+)=sin0cos+cos0sin:=sin0+cos0cos(0+)=cos0cos)]TJ /F1 11.955 Tf 11.95 0 Td[(sin0sin:=cos0)]TJ /F1 11.955 Tf 11.96 0 Td[(sin0(2{56) Ageneralsetoflinearizedmotionequationsmaybeobtained,asshowninEquation 2{57 ,byapplyingthesmall-disturbancetheory,combinedwiththepreviouslymadeassumptions,tothenonlinearsetofmotionequations,givenbyequation 2{53 ,andretainingonlytherstorderterms. 47

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x0+x)]TJ /F5 11.955 Tf 11.95 0 Td[(mg(sin0+cos0)=m_uy0+y+mgcos0=m(_v+u0r)z0+z+mg(cos0)]TJ /F1 11.955 Tf 11.96 0 Td[(sin0)=m(_w)]TJ /F5 11.955 Tf 11.95 0 Td[(u0q)L0+L=Ix_p)]TJ /F5 11.955 Tf 11.95 0 Td[(Izx_rM0+M=Iy_qN0+N=)]TJ /F5 11.955 Tf 9.3 0 Td[(Izx_p+Iz_r_0+_=q_0_=p+rtan0_ 0_ =rsec0_xE0+_xE=(u0+u)cos0)]TJ /F5 11.955 Tf 11.95 0 Td[(u0sin0+wsin0_yE0+_yE=u0cos0+v_zE0+_zE=)]TJ /F1 11.955 Tf 9.3 0 Td[((u0+u)sin0)]TJ /F5 11.955 Tf 11.95 0 Td[(u0cos0+wcos0(2{57) IfallofthedisturbancesinEquation 2{57 aresetequaltozero,thentheresultingsetoflinearizedmotionequationsarerepresentativeofthosedenedforreferenceight. Iftheassumptionismadethattheaircraftisatitsreferenceightcondition,thedisturbancequantitiesareconsiderednegligibleandthereforesetequaltozero.Applyingthisassumptiontoequation 2{57 ,itisseenthatasetofequationsisdeveloped,asseeninEquation 2{58 ,whichcanbeusedtoeliminateallofthereferenceforcesandmomentsfoundinEquation 2{57 X0)]TJ /F5 11.955 Tf 11.95 0 Td[(mgsin0=0Y0=0Z0+mgcos0=0L0=M0+N0=0_xE0=u0cos0_yE0=0_zE0=)]TJ /F5 11.955 Tf 9.3 0 Td[(u0sin0(2{58) 48

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Equation 2{58 canbesustitutedbackintoEquation 2{57 ,suchthattheresultinglinearizedmotionequationscanberewrittenanddened,asseeninEquation 2{59 _u=x m)]TJ /F5 11.955 Tf 11.96 0 Td[(gcos0_v=y m+gcos0)]TJ /F5 11.955 Tf 11.95 0 Td[(u0r_w=z m)]TJ /F5 11.955 Tf 11.95 0 Td[(gsin0+u0qL=Ix_p)]TJ /F5 11.955 Tf 11.95 0 Td[(Izx_rM=Iy_qN=)]TJ /F5 11.955 Tf 9.3 0 Td[(Izx_q_=q_=p+rtan0_=rsec0_xE=ucos0)]TJ /F5 11.955 Tf 11.96 0 Td[(u0sin0+wsin0_yE=u0cos0+v_zE=)]TJ /F1 11.955 Tf 9.3 0 Td[(usin0)]TJ /F5 11.955 Tf 11.96 0 Td[(u0cos0+wcos0(2{59) TheperturbationtermsrepresentaerodynamicforcesandmomentsthatcanbeexpressedbymeansofaTaylorseriesexpansion.TheTaylorseriesexpansionmaycontainallofthemotionvariables,butisnormallyreducedtoonlythesignicanttermsrelevanttothatpaticularforceormoment.Forexample,theTaylorseriesexpansionforthechangeinrollmoment,L,maybeexpressedasafunctionofthemoments,forcesandcontrolsurfacedeections,asseeninEquation 2{60 L=@L @uu+@L @vv+@L @ww+@L @qq+@L @pp+@L @rr+@L @aa+@L @rr+@L @ee(2{60) 2.5Examples Toillustratethederivationandlinearizationprocessfurther,anexamplewillbegiven.Theexamplewillexamineanaircraftthatiscapableofmorphingboth 49

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symmetricallyandasymmetrically.Itisnoticedthattheonlyequationsthatvarywithsymmetryarethemomentequationsgivenbyequation 2{46 .Thenonlinearmomentequationswillrstbelinearizedandthenreducedaccordingtocongurationandaerodynamicassumptions. 2.5.1Linearization Startingwiththenonlinearmomentequations,asdenedinequation 2{46 ,thesmall-disturbancetheoryisapplied,thusresultingintheperturbationequationsshowninEquation 2{61 L0+L=_pIx+(r0q+q0r)(Iz)]TJ /F5 11.955 Tf 11.95 0 Td[(Iy)+(p0r+r0p)]TJ /F1 11.955 Tf 11.96 0 Td[(_q)Ixy)]TJ /F1 11.955 Tf 11.96 0 Td[((p0q+q0p+_r)Ixz+(2r0)]TJ /F1 11.955 Tf 11.96 0 Td[(2q0q)Iyz+p_Ix)]TJ /F1 11.955 Tf 11.96 0 Td[(q_Ixy)]TJ /F1 11.955 Tf 11.95 0 Td[(r_IxzM0+M=(r0p+p0r)(Ix)]TJ /F5 11.955 Tf 11.96 0 Td[(Iz)+_qIy)]TJ /F1 11.955 Tf 11.95 0 Td[((q0r+r0q+_p)Ixy+(2p0p)]TJ /F1 11.955 Tf 11.96 0 Td[(2r0r)Ixz+(q0p+p0q)]TJ /F1 11.955 Tf 11.95 0 Td[(_r)Iyz+q_Iy)]TJ /F1 11.955 Tf 11.95 0 Td[(p_Ixy)]TJ /F1 11.955 Tf 11.96 0 Td[(r_IyzN0+N=(p0q+q0p)(Iy)]TJ /F5 11.955 Tf 11.95 0 Td[(Ix)+_rIz)]TJ /F1 11.955 Tf 11.96 0 Td[((2q0q)]TJ /F1 11.955 Tf 11.96 0 Td[(2p0p)Ixy+(r0q)]TJ /F5 11.955 Tf 11.96 0 Td[(q0r)]TJ /F1 11.955 Tf 11.96 0 Td[(_p)Ixz)]TJ /F1 11.955 Tf 11.96 0 Td[((r0p+p0r)]TJ /F1 11.955 Tf 11.95 0 Td[(_q)Iyz+r_Iz)]TJ /F1 11.955 Tf 11.96 0 Td[(p_Ixz)]TJ /F1 11.955 Tf 11.95 0 Td[(q_Iyz(2{61) Equation 2{61 canberearrangedinsuchamannerthatitisexpressedasadierentialequation,asseeninEquation 2{62 Ix_p)]TJ /F5 11.955 Tf 11.96 0 Td[(Ixy_q)]TJ /F5 11.955 Tf 11.95 0 Td[(Ixz_r=L0+L+()]TJ /F5 11.955 Tf 9.3 0 Td[(r0Ixy+q0Ixz)]TJ /F1 11.955 Tf 14.68 3.02 Td[(_Ix)p+()]TJ /F5 11.955 Tf 9.3 0 Td[(r0(Iz)]TJ /F5 11.955 Tf 11.95 0 Td[(Iy)+p0Ixz+2q0Iyz+_Ixy)q+()]TJ /F5 11.955 Tf 9.3 0 Td[(q0(Iz)]TJ /F5 11.955 Tf 11.96 0 Td[(Iy))]TJ /F5 11.955 Tf 11.96 0 Td[(p0Ixy)]TJ /F1 11.955 Tf 11.96 0 Td[(2r0Iyz+_Ixz)r)]TJ /F5 11.955 Tf 9.29 0 Td[(Ixy_p)]TJ /F5 11.955 Tf 11.96 0 Td[(Iy_q)]TJ /F5 11.955 Tf 11.95 0 Td[(Iyz_r=M0+M+()]TJ /F5 11.955 Tf 9.3 0 Td[(r0(Ix)]TJ /F5 11.955 Tf 11.96 0 Td[(Iz))]TJ /F1 11.955 Tf 11.96 0 Td[(2p0Ixz)]TJ /F5 11.955 Tf 11.95 0 Td[(q0Iyz+_Ixy)p+(r0Ixy)]TJ /F5 11.955 Tf 11.95 0 Td[(p0Iyz)]TJ /F1 11.955 Tf 14.68 3.02 Td[(_Iy)q+()]TJ /F5 11.955 Tf 9.3 0 Td[(p0(Ix)]TJ /F5 11.955 Tf 11.96 0 Td[(Iz)+q0Ixy+2r0Ixz+_Iyz)r)]TJ /F5 11.955 Tf 9.29 0 Td[(Ixz_p)]TJ /F5 11.955 Tf 11.96 0 Td[(Iyz_q+Iz_r=N0+N+()]TJ /F5 11.955 Tf 9.3 0 Td[(q0(Iy)]TJ /F5 11.955 Tf 11.96 0 Td[(Ix)+2p0Ixy+r0Iyz+_Ixz)p+()]TJ /F5 11.955 Tf 9.3 0 Td[(p0(Iy)]TJ /F5 11.955 Tf 11.95 0 Td[(Ix))]TJ /F5 11.955 Tf 11.96 0 Td[(r0Ixz)]TJ /F1 11.955 Tf 11.96 0 Td[(2q0Ixy+_Iyz)q+()]TJ /F5 11.955 Tf 9.3 0 Td[(q0Ixz+p0Iyz)]TJ /F1 11.955 Tf 14.68 3.02 Td[(_Iz)r(2{62) 50

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Theaircraftisassumedtobeatstraightandleveltrimmed(reference)ight,therefore,thedisturbancevaluescanbesetequaltozeroandEquation 2{62 canbefurtherreduced,asshownbyEquation 2{63 Ix_p)]TJ /F5 11.955 Tf 11.95 0 Td[(Ixy_q)]TJ /F5 11.955 Tf 11.96 0 Td[(Ixz_r=L)]TJ /F1 11.955 Tf 14.68 3.02 Td[(_Ixp+_Ixyq+_Ixzr)]TJ /F5 11.955 Tf 9.3 0 Td[(Ixy_p+Iy_q)]TJ /F5 11.955 Tf 11.96 0 Td[(Iyz_r=M+_Ixyp)]TJ /F1 11.955 Tf 14.68 3.02 Td[(_Iyq+_Iyzr)]TJ /F5 11.955 Tf 9.3 0 Td[(Ixz_p)]TJ /F5 11.955 Tf 11.96 0 Td[(Iyz_q+Iz_r=N+_Ixzp+_Iyzq)]TJ /F1 11.955 Tf 14.69 3.03 Td[(_Izr(2{63) Notethattheinertialrates(_Iterms)areretainedinthelinearizedmomentequations.Theretentionofthesetermsisdoneinordertoaccountforthefactthattheaircraftiscapableofmorphing.Solvingequation 2{63 intermsof_p;_qand_rresultsinthethreeequationsshowninEquation 2{64 _p=Ppp+Pqq+Prr+PLL+PMM+PNN D_q=Qpp+Qqq+Qrr+QLL+QMM+QNN D_r=Rpp+Rqq+Rrr+RLL+RMM+RNN D(2{64) ThecoecientsoftheperturbationtermsinEquation 2{63 areexpressedasfunctionsoftheinertialmoments,productsandrates,asseeninEquation 2{65 51

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Pp=)]TJ /F5 11.955 Tf 9.3 0 Td[(I2yz_Ix+IzIy_Ix)]TJ /F5 11.955 Tf 11.96 0 Td[(IxyIz_Ixy)]TJ /F5 11.955 Tf 11.96 0 Td[(IxyIyz_Ixz)]TJ /F5 11.955 Tf 11.95 0 Td[(IxzIyz_Ixy)]TJ /F5 11.955 Tf 11.95 0 Td[(IxzIy_IxzPq=IxyIz_Iy)]TJ /F5 11.955 Tf 11.95 0 Td[(IxyIyz_Iyz+IxzIyz_Iy)]TJ /F5 11.955 Tf 11.95 0 Td[(IxzIy_Iyz)]TJ /F5 11.955 Tf 11.95 0 Td[(IzIy_Ixy+Iyz_IxyPr=)]TJ /F5 11.955 Tf 9.3 0 Td[(IxyIz_Iyz+IxyIyz_Iz+I2yz_Ixz+IxzIy_Iz)]TJ /F5 11.955 Tf 11.95 0 Td[(IxzIyz_Iyz)]TJ /F5 11.955 Tf 11.95 0 Td[(IyIz_IxzPL=I2yz)]TJ /F5 11.955 Tf 11.95 0 Td[(IzIyPM=)]TJ /F5 11.955 Tf 9.29 0 Td[(IxyIz)]TJ /F5 11.955 Tf 11.95 0 Td[(IxzIyzPN=)]TJ /F5 11.955 Tf 9.29 0 Td[(IxyIyz)]TJ /F5 11.955 Tf 11.95 0 Td[(IxzIyQp=)]TJ /F5 11.955 Tf 9.3 0 Td[(IxIz_Ixy)]TJ /F5 11.955 Tf 11.96 0 Td[(IxIyz_Ixz)]TJ /F5 11.955 Tf 11.96 0 Td[(IxzIxy_Ixz+IyzIxz_Ix+I2xz_Ixy+IxyIz_IxQq=IxIz_Iy)]TJ /F5 11.955 Tf 11.96 0 Td[(IxIyz_Iyz)]TJ /F5 11.955 Tf 11.95 0 Td[(IxzIxy_Iyz)]TJ /F5 11.955 Tf 11.96 0 Td[(IyzIxz_Ixy)]TJ /F5 11.955 Tf 11.95 0 Td[(I2xz_Iy)]TJ /F5 11.955 Tf 11.96 0 Td[(IxyIz_IxyQr=)]TJ /F5 11.955 Tf 9.3 0 Td[(IxIz_Iyz+IxIyz_Iz+IxzIxy_Iz)]TJ /F5 11.955 Tf 11.96 0 Td[(IyzIxz_Ixz+I2xz_Iyz)]TJ /F5 11.955 Tf 11.96 0 Td[(IxyIz_IxzQL=)]TJ /F5 11.955 Tf 9.3 0 Td[(IyzIxz)]TJ /F5 11.955 Tf 11.95 0 Td[(IxyIzQM=)]TJ /F5 11.955 Tf 9.3 0 Td[(IxIz+I2xzQN=)]TJ /F5 11.955 Tf 9.3 0 Td[(IxIyz)]TJ /F5 11.955 Tf 11.96 0 Td[(IxzIxyRp=I2xy_Ixz+IxzIy_Ix)]TJ /F5 11.955 Tf 11.96 0 Td[(IxIyz_Ixy+IyzIxy_Ix)]TJ /F5 11.955 Tf 11.95 0 Td[(IxzIxy_Ixy)]TJ /F5 11.955 Tf 11.96 0 Td[(IxIy_IxzRq=)]TJ /F5 11.955 Tf 9.29 0 Td[(IxzIy_Ixy+I2xy_Iyz+IxIyz_Iy)]TJ /F5 11.955 Tf 11.96 0 Td[(IyzIxy_Ixy+IxzIxy_Iy)]TJ /F5 11.955 Tf 11.96 0 Td[(IxIy_IyzRr=)]TJ /F5 11.955 Tf 9.29 0 Td[(IxzIy_Ixz)]TJ /F5 11.955 Tf 11.96 0 Td[(I2xy_Iz)]TJ /F5 11.955 Tf 11.95 0 Td[(IyzIxy_Ixz+IxIy_Iz)]TJ /F5 11.955 Tf 11.95 0 Td[(IxzIxy_Iyz)]TJ /F5 11.955 Tf 11.96 0 Td[(IxIyz_IyzRL=)]TJ /F5 11.955 Tf 9.3 0 Td[(IxzIy)]TJ /F5 11.955 Tf 11.95 0 Td[(IyzIxyRM=)]TJ /F5 11.955 Tf 9.3 0 Td[(IxzIxy)]TJ /F5 11.955 Tf 11.95 0 Td[(IxIyzRN=)]TJ /F5 11.955 Tf 9.3 0 Td[(IxIy+I2xy(2{65) Thecommondenominator,D,foundinEquation 2{64 isalsoexpressedasafunctionoftheinertialmoments,products,andrates,asseeninEquation 2{66 D=)]TJ /F5 11.955 Tf 9.29 0 Td[(IxIyIz+I2xzIy+IzI2xy+IxI2yz+2IyzIxyIxz(2{66) MotionvariablescanbeaccountedforinthemomentequationsbyexpressingthemasatermintheTaylorseriesexpansion,asseeninEquation 2{67 52

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L=@L @uu+@L @vv+@L @ww+@L @qq+@L @pp+@L @rr+@L @aa+@L @rr+@L @eeM=@M @uu+@M @vv+@M @ww+@M @qq+@M @pp+@M @rr+@M @aa+@M @rr+@M @eeN=@N @uu+@N @vv+@N @ww+@N @qq+@N @pp+@N @rr+@N @aa+@N @rr+@N @ee(2{67) ThefullyexpandedsetoflinearizedmomentequationscanbeobtainedbyinsertingEquations 2{63 and 2{67 intoequation 2{63 andmultiplyingouttheterms. 2.5.2AsymmetricMorphing Figure2-7. AsymmetricCongurations Theasymmetriccaseassumesthatthereisnosymmetrytakenwithrespecttotheaircraft'scenterofgravity.Itisalsoassumedthattheaircraftisactivelymorphing,andtherefore,theinertialratesareretained.Themomentequationsspecictotheasymmetricmorphingcase,havepreviouslybeendenedandareshowninEquations 2{63 2{67 .Ifitisdeterminedthattheaircraftnolongermorphs,theinertialratesgotozeroandequation 2{63 canbefurtherreduced,asseeninEquation 2{68 Ix_p)]TJ /F5 11.955 Tf 11.96 0 Td[(Ixy_q)]TJ /F5 11.955 Tf 11.95 0 Td[(Ixz_r=L)]TJ /F5 11.955 Tf 9.3 0 Td[(Ixy_p+Iy_q)]TJ /F5 11.955 Tf 11.95 0 Td[(Iyz_r=M)]TJ /F5 11.955 Tf 9.3 0 Td[(Ixz_p)]TJ /F5 11.955 Tf 11.95 0 Td[(Iyz_q+Iz_r=N(2{68) 53

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Figure2-8. SymmetricCongurations 2.5.3SymmetricConguration Thesymmetriccaseassumesthatthereissymmetryinthexy-andyz-planes.Asadirectresult,allofthetermsrelateddescribedbyIxyandIyzgotozero.Thesymmetricexampleissimilartoasymmetriccase,inthatitisalsoassumedtobeactivelymorphing.Therefore,theinertialratesareagainretainedandtheresultingmomentequationsforsymmetricmorphingcanbeshownbyEquation 2{69 Ix_p)]TJ /F5 11.955 Tf 11.96 0 Td[(Ixz_r=L)]TJ /F1 11.955 Tf 14.68 3.02 Td[(_Ixp+_Ixzr)]TJ /F5 11.955 Tf 9.3 0 Td[(Iy_q=M)]TJ /F1 11.955 Tf 14.68 3.03 Td[(_Iyq)]TJ /F5 11.955 Tf 9.3 0 Td[(Ixz_p+Iz_r=N+_Ixzp)]TJ /F1 11.955 Tf 14.69 3.02 Td[(_Izr(2{69) Ifitisdeterminedthattheaircraftnolongermorphs,theinertialratesagaingotozeroandequation 2{69 canbereducedtothemomentequations,asseeninEquations 2{59 and 2{70 Ix_p)]TJ /F5 11.955 Tf 11.96 0 Td[(Ixz_r=LIy_q=M)]TJ /F5 11.955 Tf 9.3 0 Td[(Ixz_p+Iz_r=N(2{70) 54

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2.6ATechnicalApproachtotheEquationsofMotion Theequationsofmotion,includingasymmetricandtime-varyingshapesofassociatedmorphingactuation,canbesimpliedintotheformgiveninEquation 2{71 .Thisformdescribesasystemwithnxstatesalongwithnmorphingactuatorsandnutraditionalactuatorssuchthatx2Rnxisthestatevector,2Rnisthecommandtomorphing,andu2Rnuisthecommandtotraditionalcontrolsurfaces.ThecoecientsintheequationsofmotionarecombinedintoA2RnxnxandB2Rnxnuwhicharefunctionsofboththemorphingconguration,,andtheightcondition,. _x=A(;)x+B(;)u(2{71) ThemodelinEquation 2{72a representstheightdynamicsforanaircraftwithaxedconguration.Inotherwords,thismodeldescribesanaircraftwithoutmorphing.Thismodelutilizesmatricesthatdependonanexogenous,butmeasurable,variablesuchasMachoraltitude.Thisformisactuallythelinearparameter-varying(LPV)dynamicswhichhavebeenextensivelystudied[ 42 ]. _x=A()x+B()uu=K()x(2{72a) ThemodelinEquation 2{72b representstheightdynamicsforanaircraftataxedightcondition.Theaircraftisallowedtomorph;however,theaerodynamicsdonotvarybecauseofMachoraltitudechanges.Theresearcherproposestodenotethisformaslinearinput-varyingdynamics(LIV)becauseofthedependencyonthecommandinputfordynamicswhicharelinearinthestates. _x=A()x+B()uu=K()x(2{72b) 55

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Clearlyamorphingaircrafthashighlystructureddynamics.Inparticular,notethattheightdynamicsreverttoalinearstate-spaceforminEquation 2{72c whenbothmorphingandightconditionarexed.Themodelthusrepresentsanimportantphysicalaspectofmorphing;specically,theaircraftwillhavetraditionaldynamicsformaneuveringaroundtrimwhenmorphedintoaxedconguration. _x=Ax+Buu=Kx(2{72c) 56

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CHAPTER3LINEARTIME-VARYINGEIGENSTRUCTURESANDTHEIRSTABILITY:ASURVEY 3.1DenitionofanLTVSystem Alinear-time-varying(LTV)systemcanbedenedbythecoupledlinearhomogeneousdierentialequationsshowninEquation 3{1 _x=A(t)x(t)+B(t)u(t)y(t)=C(t)x(t)+D(t)u(t)(3{1) Forthepurposesofstudyinthispaper,theLTVstateequationwillberestrictedtoformshowninEquation 3{2 _x=A(t)x(t)(3{2) FromEquation 3{2 ,itisnotedthatcoecientmatrix,A(t),iscomposedsuchthatA2R+!Rnxn,whilethestatevector,x(t),iscomposedsuchthatx2R+!Rn,andnisthenumberofstatesinthesystem.Usingthisdenitionofalineartime-varyingsystem,multipleconceptsforLTVeigenpairsandtheirstabilityhavebeendeveloped. 3.2Kamen'sConceptofPolesofanLTVSystem 3.2.1TwoStateSystem ConsiderasecondorderLTVsystemoftheform: x+a1(t)_x+a0(t)x(t)=0(3{3) whichcanbewrittenusingoperatornotationD=d dt,as (D2+a1(t)D+a0(t))x(t)=0(3{4) Ifthereexistfunctionsp1(t)andp2(t)suchthatonecanwrite (D2+a1(t)D+a0(t))x(t)=(D)]TJ /F5 11.955 Tf 11.95 0 Td[(p1(t))f(D)]TJ /F5 11.955 Tf 11.95 0 Td[(p2(t))x(t)g(3{5) 57

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andwedenea(noncommutative)polynomialmultiplicationosuchthat f(D)]TJ /F5 11.955 Tf 11.96 0 Td[(p1(t))o(D)]TJ /F5 11.955 Tf 11.95 0 Td[(p2(t))gx(t)=(D)]TJ /F5 11.955 Tf 11.96 0 Td[(p1(t))f(D)]TJ /F5 11.955 Tf 11.96 0 Td[(p2(t))x(t)g(3{6) thencomparing 3{5 and 3{6 ,onecandeneDop2(t)=p2(t)D+_p2(t),andnallyarriveatanequationforp2as p22(t)+_p2(t)+a1(t)p2(t)+a0(t)=0(3{7) fromwhichonecanthendeterminep1using p1(t)+p2(t)=)]TJ /F5 11.955 Tf 9.3 0 Td[(a1(t)(3{8)(p1(t);p2(t))thenformapoleset,andp2(t)iscalledarightpole.Notethatthisisanorderedpoleset,andevenwhenp1(t)andp2(t)arecomplex,theyneednot,ingeneral,becomplexconjugate[ 59 ]. Themodeassociatedwitharightpolep2(t)isdenedas p2(t;0)=eRt0p2()d(3{9) andonecanwrite x(t)=C121(t;0)+C222(t;0)(3{10) wherep21andp22representtworightpolesdeterminedbysolvingequation 3{7 fromtwodierentinitialconditions(whicharetypicallythetwotimeinvariantpolesatt=0,alsoreferredtoasthe"frozen-time"rootsof 3{3 att=0);andC1andC2areconstants.21(t;0)and22(t;0)giveinformationaboutthestabilityoftheLTVsystem[ 59 ]. 3.2.2FourStateSystem Thegeneralizedformofa4-statesystemisexpressedinEquation 3{11 .Inthiscase,thederivativesofthestate,w(t),aremultipliedbyrealcoecientsofA0(t);A1(t);A2(t);A3(t).Theexpressionisalteredusingoperatornotation,givenasD=d dtasinEquation 3{4 ,togenerateEquation 3{12 58

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0=d4w dt4+A3(t)d3w dt3+A2(t)d2w dt2+A1(t)dw dt+A0(t)w (3{11) =)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(D4+A3(t)D3+A2(t)D2+A1(t)D+A0(t)w(t) (3{12) Theconceptofanordersetofpoles,(p1(t);p2(t);p3(t)p4(t)),isintroducedandrelatetothedynamicsasinEquation 3{13 )]TJ /F5 11.955 Tf 5.48 -9.69 Td[(D4+A3(t)D3+A2(t)D2+A1(t)D+A0(t)w(t)=(D)]TJ /F5 11.955 Tf 9.3 0 Td[(p1(t))[(D)]TJ /F5 11.955 Tf 9.3 0 Td[(p2(t)f(D)]TJ /F5 11.955 Tf 9.3 0 Td[(p3(t))[D)]TJ /F5 11.955 Tf 9.3 0 Td[(p4(t)]g]w(3{13) Therightpole,whichistheonlypoleneededtofullycharacterizethesystemisgeneratedasasolutiontoEquation 3{14 .Notethatagaintherightpole,p4,actuallyhas4valuesofp4iresultingfromchoiceofinitialconditionsforthepoles. 0=d3p4i dt3+(4p4i(t)+A3(t))d2p4i dt2+(6p24i(t)+A2(t)+3A3(t)p4i(t))dp4i dt+3(dp4i dt)2+p44i(t)+A3(t)p34i(t)+A2(t)p24i(t)+A1(t)p4i(t)+A0(t) (3{14) Asetofeigenvectorsareagainassociatedwitheachpole.Eacheigenvector,vi,anditassociatedpole,p4i,mustsatisfyEquation 3{15 (A(t))]TJ /F5 11.955 Tf 11.95 0 Td[(p4i(t))vi(t)=_vi(t)(3{15) Thestatesofthesystemarecomputedasalinearcombinationoftheseeigenvectorsandthemodes,givenasi=exp(Rt0p4i(t)),associatedwitheachpole.Theresultingexpressionisgivenalongwiththescalarconstants,Ci,inEquation 3{16 x(t)=C1v1(t)41(t)+C2v2(t)42(t)+C3v3(t)43(t)+C4v4(t)44(t)(3{16) 59

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ThedecompositionofthestatesintotheformofEquation 3{16 indicatesthetime-varyingparametersessentiallydiagonalizethesystem.ConsiderthatamatrixdenedasV(t)=[v1(t)jv2(t)jv3(t)jv4(t)]willdiagonalizethesystemmatrixaslongasV(t)isinvertibleandbounded. ThestabilityofthesystemisdeterminedbytherelationshipinEquation 3{16 .Essentially,thesystemhasasymptoticstabilityforwhichstateswilltendtoequilibriumifandonlyifthemagnitudeofthemodegoestozeroastimegoesincreases.Thiscondition,whichisexpressedasj4ij!0ast!1foreachi=1;2;3;4,isequivalenttoaconditionontherealpartofthepolebeingR10pR()d<0. 3.3ZhuandJohnson Annth-order,scalar,lineartime-varying(LTV)system y(n)=Xnk=1k(t)y(k)]TJ /F7 7.97 Tf 6.59 0 Td[(1)(3{17) canbeconvenientlyrepresentedasDfyg=0usingthescalarpolynomialdierentialoperator(SPDO) D=n+Xnk=1k(t)k)]TJ /F7 7.97 Tf 6.59 0 Td[(1(3{18) where=d=dtisthederivativeoperatordenedonthedierentialring(D-ring),K. UsingatechniquedevelopedbyFloquet[ 39 ],theSPDOcanbefactoredinto D=()]TJ /F5 11.955 Tf 11.96 0 Td[(n(t))()]TJ /F5 11.955 Tf 11.96 0 Td[(2(t))()]TJ /F5 11.955 Tf 11.96 0 Td[(1(t))(3{19) IfDisdenedasanSPDOoperatorwithtime-varyingcoecients,k2K,k=1;2;:::;n,thenthescalarfunctionsk2K,k=1;2;:::;ngivenbythefactorizationofequation 3{19 arecalledSeriesD-eigenvalues(SD-eigenvalues)ofD.Moreover,ifthereexistssome(t)suchthat(t)=1(t),then(t)iscalledaParallelD-eigenvalue(PD-eigenvalues)ofD. 60

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Amulti-setofSD-eigenvalues,)]TJ /F11 7.97 Tf 165.63 -1.79 Td[(=fk(t)gnk=1,canbedenedasaSeriesD-spectrum(SD-spectrum)forDifk(t)satisesequation 3{19 ;whereasamulti-setofPD-eigenvalues=fk(t)gnk=1canbedenedasaParallelD-spectrum(PD-spectrum)forDifk(t)arePD-eigenvaluesforDandfyk=exp(Rk(t)dt)gnk=1constitutesafundamentalsetofsolutionstoDfyg=0. Assumingthatfk(t)gnk=1isaSD-spectrumforannth-orderSPDOD,thenthePD-spectrumfk(t)gnk=1forDcanbedirectlyobtainedby k(t)=1(t)+_qk(t)q)]TJ /F7 7.97 Tf 6.58 0 Td[(1k(t);k=2;3;:::;n(3{20) where, qk(t)=Z21(t)Z32(t)ZZk;k)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)dk)]TJ /F7 7.97 Tf 6.59 0 Td[(1t(3{21) and ij(t)=eR(i(t))]TJ /F11 7.97 Tf 6.58 0 Td[(j(t))dt(3{22) Thefactthatascalarequationcanalwaysbetransformedintoanequivalentcompanioncanonicalvectorequation,_x=Ac(t)x,leadsonetodenethecompanionmatrixassociatedwithD,Ac(t)as Ac(t)=2666666666640100...........................0001)]TJ /F5 11.955 Tf 9.3 0 Td[(1)]TJ /F5 11.955 Tf 9.3 0 Td[(2)]TJ /F5 11.955 Tf 61.1 0 Td[(n377777777775(3{23) whichdirectlyresultsinthematrices 61

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\(t)=2666666666641(t)10002(t).....................0............100n(t)377777777775(3{24) and (t)=diag1(t)2(t)n(t)(3{25) denedasaSeriesSpectralcanonicalform(SScanonicalform)andaParallelSpectralcanonicalform(PS-canonicalform),repectively.ItisimportanttonotethatbothmatricesaredenedoverthebasisdescribedbyDandAc(t),andthatforanytime-invariantD,therealwaysexistsanSD-spectrumoftime-invariantisatisfyingequation 3{19 Welldened(freeofnite-timesingularities)SD-andPD-spectracanbeusedtoderiveanalyticalsolutionsandstabilitycriteriaforLTVsystemsinawaysimilartothoseforLTIsystems.Inparticular,iffk(t)gnk=1isawell-denedPD-spectrumforD,thenthegeneralsolutiony(t)toDfyg=0isgivenby y(t)=nXi=1CieRi(t)dt whereCiareconstantsofintegrationdeterminedbyinitialconditions. 3.3.1GeneralizedPD-Eigenvectors AssumingthatDisannth-orderSPDOwithaPD-spectrumfk(t)gnk=1andthatfyi(t)gnk=1isafundamentalsetofsolutionsforDfyg=0,suchthatyi(t)=eR(i(t)dt),thenthematrix,D,canbedenotedbythediagonalmatrix D=diag[y1y2:::yn](3{26) 62

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Then WD)]TJ /F7 7.97 Tf 6.58 0 Td[(1=V(12:::n)=266666666664111D1f1gD2f1gDnf1gD21f1gD22f1gD2nf1g............Dn)]TJ /F7 7.97 Tf 6.58 0 Td[(11f1gDn)]TJ /F7 7.97 Tf 6.59 0 Td[(1nf1g377777777775(3{27) and detV=detWnk=1y)]TJ /F7 7.97 Tf 6.59 0 Td[(1k(3{28) whereD1=(+i),Dki=DiDk)]TJ /F7 7.97 Tf 6.58 0 Td[(1i,andW=W(y1y2:::yn)istheWronskianmatrixassociatedwithfyigni=1. ThecanonicalcoordinatetransformationmatrixV(t)iscalledthemodalcanonicalmatrixforDassociatedwiththePD-spectrumfigni=1andthedeterminantinequation 3{28 iscalledtheassociatedmodaldeterminant.Itisnotedthatthecolumnvectorsvi(t)ofV(t)satisfy Ac(t)vi(t))]TJ /F5 11.955 Tf 11.96 0 Td[(i(t)vi(t)=_vi(t)(3{29) andtherowvectorsuTi(t)ofV(t)satisfy UTi(t)Ac(t))]TJ /F5 11.955 Tf 11.96 0 Td[(i(t)UTi(t)=_UTi(t)(3{30) Thus,vi(t)anduTi(t)havebeencalledcolumnPD-eigenvectorsandrowPD-eigenvectors,respectively,ofDassociatedwithit. 3.3.2StabilityCriteria AssumingthatDisawell-denednth-orderSPDOwithawell-denedPD-spectrumfk(t)gnk=1inI=[T0;1),suchthatvk(t)anduTi(t)arecolumnPD-eigenvectorsandrow 63

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PD-eigenvectorsassociatedwithk(t),respectively.ThenthenullsolutiontoDfyg=0isuniformlyasymptoticallystableforallt0T0ifandonlyif thereexistsa00and0
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issaidtobethemode-vectorofA(t)associatedwiththex-eigenpairfi(t);ui(t)gandisusedtodeterminethestabilityofthelineartime-varyingsystem. IfA(t)hasndistincteigenvalues,thenA(t)canalwaysbediagonalizedbyanon-singularmatrixformedbytheeigenvectorsofA(t).Thus,thex-eigenvectorsofA(t)arefoundbyusingtheeigenvectorsofA(t)andthederivedmatricesobtainedfromA(t). Toobtainthex-eigenpairsofA(t),itisrstassumedthatj=1andAj(t)=A(t).TheeigenpairsofA(t),fji(t);uji(t)gi=1;2;:::;narethencomputedtocheckfordistinctnessoftheeigenvalues,fji(t)gni. Iftheeigenvaluesaredistinctthen Sj(t)=[u1j(t)u2j(t):::unj(t)](3{34) where j(t)=diag[1j(t)2j(t):::nj(t)](3{35) butiftheeigenvalues,fji(t)gni,arenotdistinct,thenanon-singularmatrixSj(t)ischosen.OnceSj(t)andj(t)hasbeendetermined,thedierentialequation Ej(t)=_Sj(t)S)]TJ /F7 7.97 Tf 6.59 0 Td[(1j(t)+Aj(t))]TJ /F9 11.955 Tf 11.95 0 Td[(A(t)(3{36) issolved.ForcaseswhereEj(t)0,thenSj(t)=S(t)andj(t)=(t).Thecolumns,ui(t)ofS(t)givethex-eigenvectorsofA(t)andtheelementsof(t),fi(t);ui(t)g,givethex-eigenvaluesofA(t).ForcaseswhereEj(t)6=0,thenj=j+1andAj(t)=Aj)]TJ /F7 7.97 Tf 6.58 0 Td[(1(t))]TJ /F9 11.955 Tf 11.45 0 Td[(Ej)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t).TheprocedureisthencontinuedbyrecalculatingtheeigenpairsofA(t),fji(t);uji(t)gi=1;2;:::;nandcheckingfortheirdistinctness.TheiterationprocessesendswhenEj(t)0. Thecalulatedx-eigenpaircanthenbeusedtoproduceasystemresponsesuchthat 65

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x(t)=C1eR1(t)dte1(t)+C2eR2(t)dte2(t) whereC1andC2arecontantsofintegrationdeterminedbyinitialconditions. 3.4.2StabilityCriteria Anecessaryconditionforasymptoticstabilityandthesucientconditionfortheinstabilityofthesystem,equation 3{31 ,canbeexplicitlyanddirectlycheckedformthesystemmatrixA(t). Thenecessaryconditionforthesystem,equation 3{31 ,tobeasymptoticallystable,isthatforanyt>t0,tRt0trA()d!t!1.WheretristhetraceofA(t)andisrepresentedasA(t)=nPi=1aii(t). Thesucientconditionforinstabilityofthesystem,equation 3{31 ,isthatforanyt>t0,tRt0trA()d!1t!1. Necessaryandsucientconditionsforasymptoticstabilityandinstability,respectively,canalsobedescribedforthelineartime-varyingsystem y(n)+a1(t)y(n)]TJ /F7 7.97 Tf 6.59 0 Td[(1)+:::+an(t)y=0(3{37) Asucientconditionforinstabilityofthesystem,equation 3{37 ,isdescribedby tZtoa1()d!t!1(3{38) whereanecessaryconditionforasymptoticstabilityisdescribedby tZtoa1()d!1t!1(3{39) Thenecessaryandsucientconditionspresentedthusfaronlyprovidepartialanswerstothestabilityoflineartime-varyingsystems.Therefore,necessaryandsucientconditionsforthestabilityoflineartime-varyingsystems,intermsofmode-vectors, 66

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aredeveloped.Recallthatthemode-vectorsarenon-zerodierentiablevectorsofA(t),associatedwiththex-eigenpairfi(t);ui(t)g. Thelineartime-varyingsystem, 3{31 ,isstableifandonlyifeverymode-vectormi(t)ofA(t)isboundedsuchthat kmi(t)k<1t>t0;i=1;2;:::;n(3{40) andasymptoticallystableifandonlyif,inadditiontoequation 3{39 kmi(t)k!0t!1;i=1;2;:::;n(3{41) 3.5O'BrienandIglesias 3.5.1Formulation Consideralineartime-varyingsystemasdescribedinEquation 3{42 forsomestatevector,x(t)2Rn,andassociatedmatrix,A(t)2Rnn,forallt2R. _x(t)=A(t)x(t)(3{42) Introduceanewvectorofstates,z(t)2Rn,whichrelatestox(t)throughatransformation,S(t)2Rnn,forallt2R.Therelationshipresultsasz(t)=S)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)x(t).TheexpressioninEquation 3{42 forx(t)isthuswrittenintermsofz(t)inEquation 3{43 _S(t)z(t)+S(t)_z(t)=A(t)S(t)z(t) (3{43) S(t)_z(t)=A(t)S(t)z(t))]TJ /F1 11.955 Tf 15.25 3.02 Td[(_S(t)z(t) (3{44) _z(t)=S)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)A(t)S(t))]TJ /F5 11.955 Tf 11.96 0 Td[(S)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)_S(t)z(t) (3{45) Denethematrix,P(t)=S)]TJ /F7 7.97 Tf 6.58 0 Td[(1(t)A(t)S(t))]TJ /F5 11.955 Tf 12.42 0 Td[(S)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)_S(t),suchthatEquation 3{45 isexpressedasEquation 3{46 67

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_z(t)=P(t)z(t)(3{46) ThestabilityofthesysteminEquation 3{42 isequivalenttothestabilityofthesysteminEquation 3{46 ifthematrix,S(t)isaLyapunovtransformation.Suchequivalenceisparticularlyadvantageousinthatcertainconditionscanbeimposedsuchthatthestatematrix,P(t),inEquation 3{46 isuppertriangularforallt2R.Assuch,thepolesandassociatedstabilityofP(t)arerelativelyeasytoextract. ApolesetofP(t)isdenedbasedonitsnatureasuppertriangular.Thisset,asgiveninDenition 3.5.1 ,utilizestransitionmatricesforbothP(t)andS(t). Denition3.5.1. ThesetoffunctionsP:=fp1;:::;pnginL1(C)isapolesetofAifthereexistfunctionskij2CL1,i=1;:::;n;j>iandaninvertiblenxnmatrixS0suchthat S(t)=A(t;0)S0)]TJ /F7 7.97 Tf 6.59 0 Td[(1P(t;0);t2R+(3{47) isaLyapunovtransformationwhereP=fpijgand pij=8>>>><>>>>:pi:i=jkij:ij(3{48) AsuitablechoiceofS(t)hasbeenshowntobetheuniquesolutionofthematrixdierentialequation,giveninEquation 3{49 ,whichresultsinaLyapunovtransformation[ 126 127 ].NotethisexpressionagreeswiththepreviousdenitionforP(t)fromEquation 3{45 _S(t)=A(t)S(t))]TJ /F9 11.955 Tf 11.96 0 Td[(S(t)P(t);S(0)=S0(3{49) ThepolesetasgiveninDenition 3.5.1 dependsonthestate-transitionmatrix,A(t;0),associatedwiththesysteminEquation 3{42 .Thismatrixcanbedirectly 68

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computedusingknowledgeofA(t)soitrepresentsaknownquantity.Furthermore,thismatrixcanbeseparatedintoelementsusingastandardQR-decompositionusinganorthogonalmatrixofQ2RnnwithQQ1=IandanuppertriangularmatrixofR2RnnasinEquation 3{50 A(t;0)=Q(t)R(t)(3{50) TheelementsinEquation 3{50 arecomparedtotherelationshipinEquation 3{47 whichcanbeexpressedasA(t;0)=S(t)P(t;0)S)]TJ /F7 7.97 Tf 6.59 0 Td[(10.Inthisway,thecomputationofS(t)isdirectlycomputedastheorthogonalmatrixoftheq-rdecompositionwhilethetransitionmatrixassociatedwiththestabilitymatrixofP(t)iscomputedasthetriangularmatrixusingtheequivalanceinEquation 3{51 A(t;0)=[Q(t)][R(t)]=[S(t)]P(t;0)S)]TJ /F7 7.97 Tf 6.59 0 Td[(10(3{51) Inthisway,astraightforwardq-rdecompositionisusedtodetermineboththepolesofP(t)andthemodeshapesofS(t).Suchanapproachishighlyadvantageouscomputationallysincetheelementsareeasilyfoundusingmaturealgorithms. Anissueofnoteisthenon-uniquenatureofthisdecomposition.Essentially,thematricesassociatedwithaq-rdecompositioncanbescaledbyanyunitarymatrix.Theinformationrelatingtostabilityisretained;however,theneedtointerpretthepolesissomewhathighlightedbythisresult.Considerthatthisnon-uniquenatureimpliesthatthesignofanyrealvaluealongthediagonalofR(t)bearbitrarilychangedsotheconceptofamodemustreecttheabilitytovarythesignofP(t;0)usingtherelationshipfromEquation 3{51 3.5.2StabilityCriteria Anotionofstabilitymustbedenedforthetime-varyingsystemgiveninEquation 3{42 thatrelatesusefulproperties.Inthiscase,stabilityreferstothestateremainingbounded 69

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byanexponentialdecay.ThisnotionisuniformexponentialstabilityandisdescribedinDenition 3.5.2 Denition3.5.2. Thestateequation,_x(t)=A(t)x(t)withx(0)=xo,isuniformlyexponentiallystableifthereexistnite,positiveconstantsof;2RsuchthatkA(t;)ke)]TJ /F11 7.97 Tf 6.59 0 Td[((t)]TJ /F11 7.97 Tf 6.59 0 Td[() Asimilarnotionofstabilityisassociatedwiththepolesandtheirstabilitymodes.ThisnotionalsorequiresthetransitionmatrixtoremainboundedbyanexponentialdecayasnotedinDenition 3.5.3 [ 86 ]. Denition3.5.3. Thestabilitymodesassociatedwithp2CL1(C)isuniformlyexponentiallystableifthereexistsnite,positiveconstantsof;2Rsuchthatjp(t;t0)je)]TJ /F11 7.97 Tf 6.59 0 Td[((t)]TJ /F11 7.97 Tf 6.59 0 Td[(t0);8(tt00) Arelaxednotionofstabilityisalsodenedinwhichthetransitionremainsboundedandconvergestotheorigin.ThisnotionofasymptoticalstabilityisdescribedinDenition 3.5.4 [ 86 ]. Denition3.5.4. Thestabilitymodeassociatedwithp2CL1(C)isasymptoticallystableif,foranyto2R+,thereexistsanite,positiveconstantsuchthatjp(t;t0)j6;tt0andjp(t;t0)j!0;t!1. AnotionofsimplyremainingboundedisalsousedtodescribestabilityasuniformlystableinDenition 3.5.5 [ 86 ]. Denition3.5.5. Thestabilitymodeassociatedwithp2CL1(C)isuniformlystableifthereexistsanite,positiveconstantsuchthatjp(t;t0)j6;8(tt00). Finally,ageneralizedconceptofboundednessisgivenasnonexponentialinDenition 3.5.6 Denition3.5.6. Thestabilitymodesassociatedwithp2CL1(C)isnon-exponentialifthereexistsnite,positiveconstants1;2suchthat16jp(t;t0)j62;8(tt00). ThefundamentaltheoremisstatedinTheorem 3.5.7 whichnoteshowconditionsonthestabilitymodescanguaranteestabilityofthesystem.Theproofisnotpresentedhere;however,suchaproofisgivenintheliterature[ 86 ]. 70

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Theorem3.5.7. SupposethatA(t)hasapolesetofP.ThenthesystemofEquation 3{42 isuniformlyexponentiallystableifandonlyifpicorrespondstoanexponentiallystablemodefori2[1;n]. 3.5.3SpecialCases SpecialcasesdoexistwherethematrixfunctionA(t)isneitherconstantwithtimenorcontinuouslyvaryingwithtime.Infact,A(t)canfollowthenatureofaT-periodicsystemoronewithanitetime-variation. Recallingthefactthataperiodicstateequationcanequalatime-invariantstateequationbymeansofaFloquetdecomposition[ 39 ];Itcanbeshownthatthetime-invarianteigenvaluesdeterminethestabilityoftheequivalentperiodicstateequationandarecalledcharacteristicexponents[ 25 ].Thissetofcharacteristicexponentscanthenbesaidtoformapolesetoftheperiodicstateequation.Moreprecisely,supposethatAisT-periodic;thatis,thereexistsaniteT0suchthatA(t+T)=A(t)forallt2R.Thenitcanbeshown[ 86 ]thatthecharacteristicexponentsofAareapolesetofA. Whenthematrixfunction,A(t),hasanitetime-variation;thatisA(t)isconstantoutsideaniteinterval,itcanbesaidthatthefrozen-timeeigenvaluesformapolesetofA.Asmentionedearlier,thisisnotthecasewhenA(t)iscontinuouslyvaryingwithtime.Itisdirectlyduetothefactthatthefrozen-timeeigenvaluesmayormaynotcontaininformationabouttheoriginalstateequation,equation 3{42 .Moreprecisely,supposethatthereexistsaniteT>0suchthatA(t)=ATforalltT.Thenitcanbeshown[ 86 ]thesetoffrozen-timeeigenvaluesfi(t)gni=1satisfyingdet(A(t))]TJ /F5 11.955 Tf 12.53 0 Td[(i(t)I)=0ateacht2R+isapolesetofA. 3.6MethodologyComparisons Theeigenstructureforalineartime-varyingsystemcanbefoundbyvariousmethodologies,yeteachisspecictoitsown.Individualmethodsaredevelopedfromabasicsetofassumptionsspecictothemathematicalapproachitisbuiltupon,andthereforepossessesitsowncomparativeadvantages. 71

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Kamen'smethodinvolvescomputinganth-orderinput-outputdierenceequationwithtimevaryingcoecients.Thisdierenceequationisbasedonthesystem'slineartime-varyingdynamics,andfromit,anorderedpairofrightandlefteigenvaluescanbecalculated.Therighteigenvaluesarethenusedtodescribethesystem'stransientperformanceandstability.AfewbenetsofKamen'smethodinclude Arightpoleset,p21(t);p22(t),canbelinkedtothetime-invariantsystemviainitialconditions(frozentimeeigenvaluesatt=tdesired) TheVandermondematrix,V(t),diagonalizesthesystemmatrixwhenV(t)isinvertibleandbounded ThecalculatedLTVpolesareuniqueforalmostallinitialconditions Integratingtherightpolesetcaninuencethestabilityofthesystem whereas,thedisadvantagesinclude LTVpolescansometimeshaveanite-timesingularity,resultinginp21(t);p22(t)! Higherordersystemsaremorediculttoconvertintoaninput-outputordinarydierentialequation ZhuandJohnson'smethodalsoutilizesthenth-orderinput-outputdierenceequation,butinadditionaddsastatespaceapproachbyndingasimilaritytransformation,L(t),withdet(L(t))=constant,suchthat_z=L)]TJ /F7 7.97 Tf 6.59 0 Td[(1(AL)]TJ /F1 11.955 Tf 15.65 3.02 Td[(_L)z.InsteadofdeningLTVpolesbyrightandleftsets,ZhuandJohnsonuseorderedpolessetscalledSeriesD-spectra(SD-spectra)andParallelD-spectra(PD-spectra).AdeningbenetofZhuandJohnson'smethod,isthattheLTVpolesetscanbederivedaswell-dened(freeofnite-timesingularities). O'BrienandIglesias'smethodintroducesastate-spaceonlyapproachtosolvingforthetime-varyingeigenstructure.ItisshownthatbyapplyingaQRdecompositiontothecalculatedtransitionmatirx,eigenvectorsandeigenvaluescanbecomputed.BenetstoO'BrienandIglesias'smethodinclude Polesgiveinformationonstabilitythroughassociatedstabilitymodes 72

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Stabilitymodesarealwayscalculatedbeforepoles,thereforeassuringthatthepolesarealwaysbounded Moresuitableforhigherordersystems StabilityispreservedthroughLyapunovtransformations whereas,thedisadvantagesinclude TheLTVpolesblowuparoundii=0,wherepii=_ii.ii Thismethoddoesnotalwaysdiagonalizethesystem TheLTVpolescannotbelinkedtothetime-invariantsystem TheLTVpolesarenon-uniqueduetothenon-uniquenessoftheQRdecomposition Wu'smethodintroducesaslightydierentapproachtosolvingforlineartime-varyingsystemsinthestate-spaceform.X-eigenpairs,fi(t);ui(t)g,arefoundbyevaluatingthematrixdierential, 3{36 ,withanonsingularmatrix,S(t),developedfromtheeigenvectorsofA(t).BenetsofWu'smethodinclude Thesystemmatrixcanalwaysbediagonalized Thex-eigenpairinuencesstability whereas,thedisadvantagesinclude Thex-eigenpairisnotnecessarilylinkedtotheinitialLTIsystem Thex-eigenpairisnotunique 73

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CHAPTER4LINEARTIME-VARYINGMODALANALYSIS 4.1SystemDynamics Theconceptofaneigenvalue/eigenvectoranditsrelationshiptomodaldynamicsiswelldenedfortime-invariantsystems[ 88 ];however,thissamerelationshipissomewhatunderdevelopedfortime-varyingsystems.Thepurposeofthischapteristointroducenewconceptsusedfortime-varyingmodalanalysis.Tobegin,areviewofsomebasictime-invariantconceptswillbegiven. 4.1.1LinearTime-InvariantSystems:Formulation AsystemisconsideredtobeLinearTime-Invariant(LTI)ifitexhibitsbothlinearityandtimeinvariance. Linearitycanbedenedasanyrelationshipbetweenasystem'sinputandoutput,suchthatitpreservestheoperationsofsuperpositionandscalarmultiplication.Thepropertiesofscalarmultiplicationandsuperpositionofalinearsystem,y=f(x),areshowninEquation 4{1 ,respectively. f(x)=f(x)f(x1+x2)=f(x1)+f(x2)(4{1) Time-invariancealsoreferstoshift-invariance,whichstatesthatifaninputtoasystemisshiftedbysometime,t,thentheoutputofthatsystemwillalsobetimeshiftedbyt,asshowninEquation 4{2 y(t)=f(t)y(t)]TJ /F1 11.955 Tf 11.96 0 Td[(t)=f(t)]TJ /F1 11.955 Tf 11.96 0 Td[(t)(4{2) Inotherwords,atime-invariantsystemisonewhosebehavior(itsresponsetoinputs)doesnotchangewithtime. ThefundamentalresultinLTIsystemtheoryisthatanyLTIsystemcanbecharacterizedentirelybyasinglefunctioncalledthesystem'simpulseresponse.The 74

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outputofthesystemissimplytheconvolutionoftheinputtothesystemwiththesystem'simpulseresponse.Thismethodofanalysisisoftencalledthetimedomainpoint-of-view.Thesameresultistrueofdiscrete-timelinearshift-invariantsystemsinwhichsignalsarediscrete-timesamples,andconvolutionisdenedonsequences Equivalently,anyLTIsystemcanbecharacterizedinthefrequencydomainbythesystem'stransferfunction,whichistheLaplacetransformofthesystem'simpulseresponse(orZtransforminthecaseofdiscrete-timesystems).Asaresultofthepropertiesofthesetransforms,theoutputofthesysteminthefrequencydomainistheproductofthetransferfunctionandthetransformoftheinput.Consequently,convolutioninthetimedomainisequivalenttomultiplicationinthefrequencydomain MostLTIsystemconceptsaresimilarbetweenthecontinuous-timeanddiscrete-time(linearshift-invariant)cases. 4.1.2LinearTime-InvariantSystems:Response ConsideranLTIsystemdenedbyadierentialequation,asrepresentedinEquation 4{3 oritsequivalentstate-spacerepresentation,asrepresentedinEquation 4{4 xn+a1xn)]TJ /F7 7.97 Tf 6.59 0 Td[(1+:::+an)]TJ /F7 7.97 Tf 6.58 0 Td[(1_x+anx=p(t)(4{3) _x=Ax+Bu(4{4) Thecompletesolution,x(t),ofEquation 4{3 isderivedfromtwoparts,theparticularsolution(forcedresponse)andthecomplimentarysolution(naturalresponse).Theparticularsolutiondependsonthefunctionalformoftheinput,p(t),whilethecomplimentarysolutionisfoundbysettingtheinputfunctiontozeroandsolvingtheassociatedhomogeneousdierentialequation.Forthepurposesofthispaper,onlythecomplimentaryornaturalresponsewillbeconsidered. Foradynamicsystem,boththenaturalandforcedresponseconsistsofatransientandsteady-stateportion.Thetransientresponseisdenedastheresponsegeneratedby 75

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thesystem'schangefrominitialconditiontonalstate,whereasthesteady-stateresponseisdenedasthebehaviorofthesystem'soutputastimegoestoinnity. TheunforcedornaturalresponseforeachstatecanthenbedescribedasthesolutiontohomogeneoussolutiontoEquation 4{4 ,asseeninEquation 4{5 x(t)=nXi=1viexpit(4{5) TheeigenvectorisshowninEquation 4{5 asvi,whiletheeigenvaluesarerepresentedbyi.TakingthederivativeonbothsidesofEquation 4{5 andsubstitutingintheunforcedversionofEquation 4{4 for_x,resultsinthewellknowneigenvector-eigenvaluerelationshowninEquation 4{6 Ax=x(4{6) 4.1.3LinearTime-VaryingSystems:Formulation Lineartime-Varying(LTV)systemssharethesamelinearpropertiesasthepreviouslymentionedLTIsystem,yettheLTVsystemisnottime-invariantorshift-invariant.Thisfactsuggeststhatifaninputtoasystemisshiftedbysometime,t,thentheoutputofthatsystemwillnotbetimeshiftedbythesamet,asshowninEquation 4{7 y(t)]TJ /F1 11.955 Tf 11.96 0 Td[(t)6=f(t)]TJ /F1 11.955 Tf 11.96 0 Td[(t)(4{7) Assumingthatthelineartime-varyingstateequationisdenotedinstate-spaceform,asshowninEqutaion 4{8 ,andthatAisacontinuous,uniformlyboundedfunction(A2CL1),thenthestatespacerealizationcanbeshowntohavetheformdescribedbyEquation 4{9 _x(t)=A(t)x(t);x(0)=x0(4{8) 76

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G:=264A B C D375=8><>:_x(t)=A(t)x(t)+B(t)u(t)y(t)=C(t)x(t)+D(t)u(t)9>=>;(4{9) ItwillalsobeassumedthateachcoecientmatrixgeneratedbyG,isagainanelementofthecontinuous,uniformlyboundedspace(CL1). Forthecaseofmultivariablelinear,time-varyingsystems,aneigenstructureequation,reminiscentoftheclassicallydenedeigenvector-eigenvalueequationforconstantmatrices(Equation 4{6 ),canbedenedandshownasEquation 4{10 (A(t))]TJ /F5 11.955 Tf 11.96 0 Td[(pi(t)I)x(i)(t)=_x(i)(t);i=1;:::;n(4{10) InEquation 4{10 ,thetime-varyingeigenvalueisdenedaspi(t),wheretheassociatedtime-varyingeigenvectorisdenedasx(i)(t).Thestabilityofthestateequation,Equation 4{8 ,hasbeenshowntohavearelationdirectlylinkedtothebehaviorofthetermsinEquation 4{11 [ 126 ];[ 127 ]. Zt0pi()dx(i)(t);i=1;:::;n(4{11) Thestabilityofthestateequationhasalsobeenshowntohavearelationdirectlyrelatedtothemodesasscoiatedwitheachpolepi(t)[ 86 ];[ 130 ];[ 131 ],asshowninEquation 4{12 Zt0pi()d;i=1;:::;n(4{12) 4.1.4LinearTime-VaryingSystems:Response First,considerthetime-varyingresponseequationdenedusingthemodedenitionsprovidedbyWu[ 126 ]andKamen[ 59 ],asshowninEqaution 4{13 x(t)=nPi=1Cii(t)exphRt0pi()dix(t)=nPi=1Cii(t)(4{13) 77

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InEquation 4{13 ,thetime-varyingpoles,pi,arecomposedsuchthatpi:R+!C,whilethetime-varyingeigenvectors,i,arecomposedsuchthati:R+!CN.Itshouldbenotedthatthetime-varyingeigenvectorsareassumedtobedierentiableforalltime,tandthateacheachlinearly-independentsolution,i(t),canbedenedasamodeofthesystem[ 126 ]. Asecondtime-varyingresponseequationcanbedenedusingthemodedenitionsprovidedbyO'Brien[ 86 ],asshowninEqaution 4{14 x(t)=A(t;0)x0(4{14) InEquation 4{14 ,A(t;0)isrepresentativeofthetime-varyingstatetransitionmatrixandcanberelatedtothetime-varyingpolesandeigenvectorsviaEquation 3{51 BothEquation 4{13 and 4{13 mayberearrangedsuchthatarelationshipcanbedevelopedbetweenthetime-varyingpoles/eigenvectorsandthecoeeicentmatrix,A(t).ThisrelationshipisdeemedanLTVEigenrelation[ 112 ],anddependingonthetime-varyingalgorithmused,canbeshownaseitherEquation 3{29 3{32 ,or 3{49 .Anysetoftime-varyingpolesandeigenvectorsthatsatisfytheseequationswillbereferredtoasanLTVEigenpair[ 112 ]. GiventhisdenitionofanLTVeigenpair,itcanbeshownthatneitherthestabilitynorthefrequencycharacteristicsofanLTVsystemcanbedeterminedfromeithertheLTVpolesoreigenvectorsalone.Instead,eacheigenpairmustbeanalyzedtogetherasawhole.SinceLTIanalysistechniquesarebasedontheassumptionthatstabilityandoscillatorycharacteristicsareisolatedwithinthepoles,newanalysistechniquesareclearlynecessary. 4.2ModalInterpretation:Kamen Considertheoscillatoryresponseofa2-statesystem,asoriginallygiveninEquation 3{10 ,withthecoecientsnormalizedtoeasepresentationasinEquation 4{15 .SubstitutethedenitionofmodeintermsofpolesfromEquation 3{9 togenerateEquation 4{16 .Also, 78

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assumethegeneralizedpolestobecomplexconjugatessuchthatp21=p22=pR+|pIasinEquation 4{17 .ThecomplexexponentialscanthenbeexpressedintermsofsinesandcosinesasinEquation 4{18 andcombinedtogenerateEquation 4{19 x(t)=1(t)+2(t) (4{15) =eRt0p21()d+eRt0p22()d (4{16) =eRt0(pR()+|pI())d+eRt0(pR()+|pI())d (4{17) =eRt0pR()dcosZt0pI()d+|eRt0pR()dsinZt0pI()d+eRt0pR()dcosZt0pI()d)]TJ /F5 11.955 Tf 11.96 0 Td[(|eRt0pR()dsinZt0pI()d (4{18) =2eRt0pR()dcosZt0pI()d (4{19) AnoscillatoryresponsewithdecayisdemonstratedinEquation 4{19 tobegeneratedbyapairofpoleswhicharecomplexconjugates.Thenatureoftheoscillationsandthedecayarebothdeterminedbytheintegralsofrealandimaginarypartsofthesepoles.Also,theequal-but-oppositenatureoftheimaginarypartsofthesepolesmeanstheresponseinEquation 4{19 issimplydoubletherealpartofthemodeasnotedinEquation 4{18 4.2.1Damping Thedecayingnatureoftheresponse,whichissimilartothedampingratioofalineartime-invariantsystem,isdeterminedbythevaryingmagnitudeoftheexponentialinEquation 4{19 .TheresultingenvelopeisgiveninEquation 4{20 usingtherealpartofthepoleandequivalentlyinEquation 4{21 byaddingthecomplex-conjugatepoles. envelope(x(t))=eRt0pR()d (4{20) =eRt0p21()+p22() 2d (4{21) 79

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4.2.2Frequency Theoscillationsoccurwithafrequencyrelatedtotheimaginarypartofthetime-varyingpole.Atime-varyingequivalenttonaturalfrequency,!(t),isgeneratedbycomparingthecosinetermofEquation 4{19 toastandardtermindynamicsofcos(!t).Thetime-varyingequivalenttoanaturalfrequencyresultsfromthiscomparisonandisgiveninEquation 4{22 .Notethatanystablesystemwillhave!tendto0astimeincreases. !(t)=Rt0pI()d t(4{22) TheperiodicityoftheoscillationsresultsdirectlyfrominvertingthefrequencyofEquation 4{22 andisgiveninEquation 4{23 .Alternatively,suchperiodicitycanresultsimplybynotingwhentheangleinthecosinetermrepeatsitselfasgiveninEquation 4{24 T(t)=2t Rt0pI()d (4{23) =maxT>02R2 2T:Zt0pI()d=Zt+T0pI()d=0 (4{24) Also,atime-varyingequivalenttodampingratioiscomputedbyrelatingtheenvelopeofEquation 4{20 andthefrequencyinEquation 4{22 .Essentially,theenvelopeisequivalentto)]TJ /F5 11.955 Tf 9.3 0 Td[(!nandthefrequencyisrelatedto!np 1)]TJ /F5 11.955 Tf 11.95 0 Td[(2.theresultingdampingratioisgiveninEquation 4{25 (t)=vuut 1 1+Rt0pI()d Rt0pR()d2(4{25) 80

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4.3ModalInterpretation:O'Brien 4.3.1DampingandNaturalFrequency Considerthesame2-stateoscillatorysystemdescribedbyEquation 3{10 ,butnowrepresentedasthestate-spacesystemshowninEquation 4{4 Theresponseofthesystem'sdynamicswillvaryinmagnitudeaccordingtotheassociatedstability.Theparameterofdampingisusedtodescribetherateofvariationinthisresponsewithpositivedampingusedforstablesystemsthatdecaytozeroandnegativedampingusedforunstablesystemsthatgrowtoinnity. Therateofmagnitudevariationcanbeextractedfromthepairofpolesassociatedwiththemode.Considerthatthetime-invariantdynamicshavecomplex-conjugatepoleswith1=2=R+|IsoR=1+2 2.Asimilarrelationholdsforthepolesofthetime-varyingsystemasshowninEquation 4{26 envelope(x(t))=expp1(t)+p2(t) 2(4{26) Simulationresultssuggeststhatthetime-varyingfrequencyoftheoscillatorybehaviorcanbeapproximatedbythedominantzero-crossingperiodicityoftheeigenvectors.Forsimplicty,anapproximationisformulatedbyassumingthesystemresponsemaintainsasinusoidalnature.Inthiscase,thematricesderivedbydecomposingthestate-transitionmatrixwillalsohaveasinusoidalnature.Abasicpropertyofanysinusoidgivenasx(t)=sin(!t)hasx(t)=)]TJ /F5 11.955 Tf 9.3 0 Td[(!2sin(!t)sothefrequencyofoscillationisgeneratedby!=q )]TJ /F7 7.97 Tf 10.99 4.71 Td[(x x.Therefore,theresultingfrequencyofthetime-varyingsystemmaybeapproximatedbyEquation 4{27 !i(t)2r )]TJ /F5 11.955 Tf 10.49 8.09 Td[(qii qii(4{27) 81

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4.3.2ModeShapes Systemdynamicsoftenutilizemodeshapestodescribetheactualmotionofavehicle.Essentially,amodeshapepresentstherelativemagnitudebetweenstateswhichisindependentofanydecayoroscillations.Thesemodeshapesarenormallyfounddirectlyfromthetime-invariant,scalereigenvectorsderivedinEquation 4{6 .Unfortunately,thetime-varyingeigenvectorsfoundinthispaperdocontainfrequencyinformationandthereforearemorediculttointerpret. AmodeshapeisgeneratedbynotingtherelationshipbetweentheQandRmatrices;specically,considerthescalarvalues,rij(t),oftheupper-triangularmatrixgivenasR(t)andthecolumns,qi(t),oftheunitarymatrixgivenasQ(t). x(t)=nXj=1jXi=1qjrijxj()(4{28) Amodeshapecanbeimmediatelyrealizedfromtherstcolumn,q1(t)oftheQ(t)matrix.Thismodeshapeisidentiedbyconsideringthemotionthatwouldresultfromaninitialconditioninwhichtheonlynon-zerocomponentistherststate.Theresultingresponseissimplyascalar,r11(t)x1(),multipliedbythevectorofq1(t)asshowninEquation 4{29 .Themodeshapewhichdescribestherelativevariationsbetweeneachstateisthusgivenasv1(t)=q1(t). x(t)=q1(t)r11(t)x1()ifx()=[x1()0:::0] (4{29) =v1(t)r11(t)x1() (4{30) AnothermodeshapeisthendeterminedasacombinationoftherstandsecondcolumnsoftheQmatrix.Considertheresultingmotiongivenaninitialconditioninwhichonlythesecondstatehasanon-zerocomponent.ThemodeshapeinresponsetothismotionisactuallytheparentheticalexpressioninEquation 4{32 whichismultipliedbythescalarsofr22(t)andx2()togeneratetheresponse. 82

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x(t)=(q1r12+q2r22)x2()ifx()=[0x2()0:::0] (4{31) =q1(t)r12(t) r22(t)+q2(t)r22(t)x2() (4{32) =v2(t)r22(t)x2() (4{33) AgeneralizedexpressionforamodeshapeiscomputedbyextendingEquation 4{30 andEquation 4{33 tonth-ordersystems.SuchanexpressionisgiveninEquation 4{34 vi(t)=iXj=1qj(t)rji(t) rii(t)(4{34) Theresponseofthemorphingaircraftiswrittenintermsofthegeneralizedmodeshape,vi(t),fromEquation 4{34 .ThisresponseisalinearcombinationofthemodeshapesasshowninEquation 4{35 x(t)=nXi=1vi(t)rii(t)xi()(4{35) 4.4Example:SimpleMechanicalSystem Tohelpillustratetheconceptofdeterminingthelineartime-varyingpolesandstabilitymodes,usingaQRdecomposition[ 86 ],asimplesecondordersystemwillrstbeanalyzed.Inthisanalysis,correlationstolineartime-invariantsystemswillbemadeforthepurposeofcharacterizinglineartime-variantdynamicmodalproperties.Thisexampleiscomposedofamass-spring-dampersystem,asshowninFigure 4-1 ,andwillonlyconsiderthehomogeneousorunforcedsolution.Inaddition,anassumptionwillbemadethatthewheelsattachedtothecartarefrictionless. TheparametersforthisexamplearelistedinTable 4-1 ,whereitisseenthatonlythedampingcoecientchangeswithtime. Themassanddampingcoecientarechangedwithtime,asshowninEquation 4{36 ,insuchamannerthatthesystemexperiencesanincreaseinmassandspansallthree 83

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Figure4-1. Mass-Spring-DamperSystem Table4-1. Mass-spring-dampersystemparameters Parameterst=0st=20st=30s M(kg)279.5K(N/m)888C(Ns/m)0812011.5 dampingcharacteristics,simultaneously.Astimeprogresses,thesystemtransitionsfrominitiallybeingunderdampedandlighttocriticallydampedandmoderate,andnallytooverdampedandheavy.Thevaluesforthesetime-varyingmasses/dampingratiosandtheircorrelatedtimesareshowninTable 4-1 C(t)=0:4(t)M(t)=2+0:25(t)(4{36) StartingfrombasicprincipalsandNewton'ssecondlaw,asseeninEquation 4{37 ,theordinarydierentialequationdescribingthesystem'sdynamicscanbederived,asshowninEquation 4{40 .ItcanbeseenfromEquations 4{39 and 4{40 thatthetimerateofchangeinmass,_M(t),isrepresentativeofinertialeectsonthesystemanddirectlymodiesthecontributionofthetime-varyingdampingseeninthea1(t)coecient. 84

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F=d(Mv) dt (4{37) =Mdv dt+vdM dt (4{38) =Mx+_x_M (4{39) x+C(t)+_M(t) M(t)_x+K M(t)x=x+a1(t)_x+a0(t)x=0 (4{40) Fromobservation,itisdeterminedthatanequivalentstate-spacecanonicalrepresentationofEquation 4{40 canbeformedandshownasEquation 4{41 Ax+Bu=26401)]TJ /F5 11.955 Tf 9.3 0 Td[(a0(t))]TJ /F5 11.955 Tf 9.3 0 Td[(a1(t)375264x1x2375+264b1b23750(4{41) Thetwostatesofthesystem,representedinEquation 4{41 ,arethedisplacementposition,x,andtherateofchangeindisplacementposition,_x.Itisalsonotedthatthecoecientsfoundinthestate-spacestatematrix,a0(t)anda1(t),canberepresentedasfunctionsshowninEquations 4{42 and 4{43 a0(t)=8>>>><>>>>:0:t08 2+0:25t:0>>><>>>>:0:t00:2t+0:25t 2+0:25t:0
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Whenassumingthatinertialeectscanbeneglected,thenEquation 4{40 reducestotheEquation 4{44 andthestate-spacecoecient,a1(t),canberewrittenasshowninEquations 4{45 x+C(t) M(t)_x+K M(t)x=x+a1(t)_x+a0(t)x=0 (4{44) a1(t)=8>>>><>>>>:0:t00:2t 2+0:25t:0
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Thesystemresponseforeachcase,asshowninFigure 4-2 ,showthatthestatesinitallyoscillatebeforeeventuallyconverginguponasteadystateequilibriumvalue.Thisconvergenceoccursroughlyaroundtensecondsfortheinertialcaseandaround20secondsforthenon-inertialcase. A B Figure4-3. Time-VaryingModes:A)Non-Inertal:Mode1(|),Mode2()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[()B)Inertial:Mode1(|),Mode2()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() ReferencingFigure 4-3 ,itiscleartoseethatthemodalresponses,forbothinertialandnon-inertialcases,alsoexhibitaperiodofdecayingoscillationsbeforenallyconverginguponzero.Notethattheconvergencewindowineachcaseisthesameasthatpreviouslyseeninthesystemresponses.Itshouldalsobenoticedthatbothmodesforeachcaseareboundedbysomeconstant,,andconvergetozeroastimegoestoinnity,thereforesuggestingthatthesystemmodesareasymptoticallystable[ 86 ].Animportantobservationtonoteisthatwhenthemodesapproachzero,thecorrespondingpolesbecomeincreasinglyill-conditioned.Theill-conditioningisbroughtaboutbythederivationofanLTVpole,asshowninSection 3.5 ,whichisdenedbydividingamode'stimerateofchangebythemodeitself.Therefore,whenthevalueofeachmodecrossesbelowsomepre-determinedthreshold,thepolevalueisceasedtobecalculated,asshowninFigures 4-4A and 4-4B 87

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A B Figure4-4. Time-VaryingPoles:A)Non-Inertal:Pole1()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[(),Pole2()-221()]TJ /F1 11.955 Tf 27.23 0 Td[(),RealportionofdiscreteLTIpoles(|),Poleaverage()B)Inertial:Pole1()-219()-219()]TJ /F1 11.955 Tf 33.14 0 Td[(),Pole2()-218()]TJ /F1 11.955 Tf 27.15 0 Td[(),RealportionofdiscreteLTIpoles(|),Poleaverage() Theassociatedpolesforthemass-spring-dampersystem,showninFigure 4-4A ,arefoundtooscillate,inanequal-but-oppositefashion,aboutalinedrawnthroughtheaverageofthetwoLTVpoles,(pole1+pole2)=2.Thispoleaverageisfoundtomatch,atdiscretepointsintime,therealpartofthecomplex-conjugateLTIpoles,asshowninFigure 4-4A .AnotableobservationisthatboththeLTIandLTVcomplex-conjugatepolessharethesamesimilarity,suchthattheaverageofthetwocomplex-conjugatepolesequaltherealpartsofeachLTIpole.ThisobservationsuggeststhatthemodaldampingforanLTVsystem,excludinginertialeects,isdirectlyrelatedtotheaverageofitsoscillatingpolepairsandcanbeapproximatedastheLTIdamping,asshownbyEquation 4{46 !LTV=(pole1+pole2)=2=Re(pLTI)(4{46) Thecorrespondingnon-inertialeigenvectors,asderivedinSection 4.3.2 ,canbeshowninFigure 4-5 Conversely,fromFigure 4-4B itisseenthatwheninertialeectsareincluded,theLTVpoleaveragedoesnotcorrespondwiththerealpartofthecomplex-conjugateLTI 88

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A B Figure4-5. Non-InertialTime-VaryingEigenvectors:A)Eigenvector1:(1,1)(|),(2,1)()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()B)Eigenvector2:(1,1)(|),(2,1)()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() poles.Instead,itisobservedthattheLTVpoleaveragediersfromtherealpartoftheLTIpolebyafactorroughlyequivalenttothemassrateofchange,_M.ThisfactsuggeststhatthedampingoftheLTVsystem,experiencingmassdisplacment,cannotaccuratelyberepresentedastherealpartsofeachLTIpole.Rather,thedampingoftheinertialLTVsystemcanberelatedtothediscreteLTIdampingbythesummationoftheLTVpoleaverageandthemassrateofchange,asshowninEquation 4{47 .ThecorrespondinginertialeigenvectorscanbeshowninFigure 4-6 !LTV=(pole1+pole2)=2=Re(pLTI))]TJ /F1 11.955 Tf 17.59 3.02 Td[(_M(4{47) Althoughsimilarintheory,theillustrativerepresentationofsuchacomplex-conjugatepolevariessignicantlyfromtheclassicallydenedLTIsystem[ 97 ]totherelativelynewdenedLTVsystem[ 86 ];[ 126 ].Thecomplex-conjugaterepresentationfortheaforementionedLTVsystemisshowninFigure 4-4A asanoppositephase,oscillatingpolepair.Frequencyinformation,however,isnotastrivialtondfromanoppositephase,oscillatingLTVpolepairasisthedamping.AsmentionedinSection 4.1.4 ,thefrequencyinformationissplitbetweenthepoleandeigenvector;therefore,neitherthe 89

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A B Figure4-6. InertialTime-VaryingEigenvectors:A)Eigenvector1:(1,1)(|),(2,1)()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()B)Eigenvector2:(1,1)(|),(2,1)()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() polenortheeigenvector,alone,canprovideacompleterepresentationofthetime-varyingfrequency.However,simulationresultssuggestthatasucientapproximationofthemodalfrequenciesmaybefoundbyreferringtotheperiodicityofthecolumnvectorsoftheorthonormalmatirx,Q,asshowninFigures 4-7 4-12 .AssumingthatQissinusoidalinnature,thenthisresultcanbeshownasthesamefrequencyapproximationderivedpreviouslyinEquation 4{24 A B Figure4-7. Non-InertialQcolumnvectorvalues:A)Q11(|),Q21()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[()B)Q12(|),Q22()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() 90

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Figures 4-7A 4-7B representthecolumnsofthenon-inertial,time-varyingQmatrixwhichcorrespondtotheLTVpolesshowninFigure 4-4A .FromFigures 4-7A 4-7B ,itisseenthatthetermsQ11andQ22representthesameresponse,whilethetermQ12representsthesameresponseasQ21,justoutofphaseby180degreesorradians.Asaresult,theperiodicityforeachoftheseresponsesisidenticalandthereforesuggeststhatthemodalfrequencycanbefoundbyinvertingthemeasuredQcolumnperiodicityandmultiplyingby2,asshowninFigures 4-8 and 4-9 Figure4-8. Qcolumnperiodicity:Q11(|),Q21()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() Figure4-9. Qcolumnfrequency:ImaginarypartofdiscreteLTIpole(|),1Qcolumnperiodicity()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() 91

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Figures 4-10A 4-10B representtheinertialLTVQcolumnswhichcorrespondtotheLTVpolesshowninFigure 4-4B .ThesameconclusionscanbedrwanfromFigures 4-10A 4-10B ,asthatpreviouslydeterminedfromFigures 4-7A 4-7B .However,itshouldbenotedforthiscasethattheQtermsbecomeincreasinglyill-conditionedasthecorrespondingmodes,showninFigure 4-3B ,approachzero,asshowninFigures 4-6 and 4-10 .Thisconditionisadirectresultofthealgorithmusedtoanalyzethetime-varyingsystem. A B Figure4-10. InertialQcolumnvectorvalues:A)Q11(|),Q21()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()B)Q12(|),Q22()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() Aspreviouslyshown,themodalfrequencycanbefoundbyinvertingthemeasuredQcolumnperiodicityandmultiplyingby2.TheresultingQcolumnperiodictyandmodalfrequencyfortheinertialcasecanbeshowninFigures 4-11 and 4-12 ItcanbeseenfromFigures 4-9 and 4-12 thatthetime-varyingfrequencies(forboththeinertialandnon-inertialcases)aresimilartotheimaginarypartsofthetime-invariantpoles.Asaresult,itmaybeinferredforthisexample,thattheinclusionofinerialeectsdonotsignicantlyalterthefrequecyinformation;therefore,theimaginarypartsofthetime-invariantpolesmaybeusedtoapproximatethetime-varyingfrequency. 92

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Figure4-11. Qcolumnperiodicity:Q11(|),Q21()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() Figure4-12. Qcolumnfrequency:ImaginarypartofdiscreteLTIpole(|),LTVapproximatedfrequency()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() 93

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CHAPTER5CONTROLDESIGN 5.1InherentClosed-LoopNonlinearity Anymorphingsystem,regardlessofopen-looplinearity,willhaveaclosed-loopnonlinearityinthepresenceofstatefeedback[ 28 ].Theintroductionofthisnonlinearityhasbeenespeciallyproblematicforthecommunityinvestigatingactivesystems.Considerthatmorphingaircrafthavebeenextensivelystudiedbythecommunitybuthavefocusedpredominatelyonaerodynamicperformance[ 122 ]andmaterials[ 125 ].Inparticular,theseprogramshavedemonstratedclearbenetsofmorphingasmeasuredbyaerodynamicperformancerelatedtoliftanddrag[ 58 ].Aircrafthavebeenoptimized[ 18 ]usingmorphingtoimproveliftandminimizedragalongwithfuelconsumption[ 14 ],rangeandendurance[ 43 ],costandlogistics[ 15 ],actuatorenergy[ 93 ],andmaneuverability[ 99 ].Additionally,aeroelasticeectshavebeenoftenstudiedrelativetomaximumrollrate[ 66 44 9 ]andactuatorloads[ 77 ].Someinvestigationshaveconsideredcontrolofmorphingsystems;however,theydonotfullyaddressmanueveringight.Astudyusingpiezoelectricmaterialsdesignedcontrolonlytoachieveroll.[ 70 ]Avarietyofotherstudieshaveinvestigatedcontrolbutonlyinthecontextofacutationenergy[ 57 ]andcontrolauthority[ 19 ]forsimplechangesinightcondition.Otherstudieshaveconsideredmorphingforcontrolsurfacesbutnotassociatedcontrolsynthesis.[ 101 102 ]Also,astudydesignednonlinearcontrollersbutitsmorphingmodelusedsimplyadistributedsetofcontroleectorsratherthanashapewithfullytime-varyingdynamics.[ 117 ] Toillustratethenotionofthisinherentclosed-loopnonlinearity,rstconsidertheLIVgeneralizedmodel,asshowninEquation 5{1 .Thecorrespondingstatevectorisgivenasx2Rnwhilethecontroleectorisgivenas2R. _x=A()x+B()u(5{1) 94

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Assumingthereisnocontrolmatrix,B,andthatthestatedynamicshaveananedependence,usingA0;A1;A22Rnn,thenEquation 5{1 mayberewritteninthebilinearform[ 17 81 ],asshowninEquation 5{2 .Itshouldbenotedthatthemorphingrate,_,asseeninEquation 5{2 ,isgeneratedfromtheinclusionoftheinertialrates,_I,asdescribedinSection 2.3.1.2 _x=(A0+A1+A2_)x(5{2) Forthepurposeofsimplicity,themorphingrateterm,asshowninEquation 5{2 ,isassumedtobezero.Therefore,toachievegeneralizedfeedbackinthesystem,agenericcontrollaw,asdenedinEquation 5{3 ,isappliedtothenewlytruncatedformofEquation 5{2 =Kx(5{3) Asaresult,thegeneralizednonlinear(inthestates)expressioncanbedevelopedwhichdescribestheclosed-loopsystem,asseeninEquation 5{4 _x=(A0x+A1Kx2)(5{4) 5.2Quasi-StaticApproach 5.2.1Synthesis Quasi-staticvariationsareconsideredwhenthetimescalesofmorphingaremuchlessthanthetimescalesofmaneuvering.Suchquasi-staticmorphingisactuallythemostprevalentoftheschemesbeingadoptedforUAVandpilotedvehicles.ThevariablesweepontheF-14isanexampleofsuchaschemealongwiththevariablespanbeingproposedbyDARPA[ 123 ].Thequasi-staticmorphingisessentiallybeingusedtochangeightconguration,suchascruiseordive,relatedtosegmentsofthemissionprole.Theresultingdynamicsareslowlytime-varyingwithrespecttothemorphingduetothe 95

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disparity,whichmaybeordersofmagnitude,betweentimescalesofcongurationandmaneuvering. Thecontrollersneededtohandlethisquasi-staticmorphingwouldhavetobedesignedsuchthattheychosetheoptimalvalueofmorphingalongwithobtainingdesiredperformancemetricsformaneuvering.Ifitwereassumedthatanaircrafthadasetofslowactuatorstoalterightconguration,aswellas,asetoffastactuatorsformaneuvering,thenthefollowingactuationchoiceswouldensue.Thechoiceofmorphingactuationwouldbe,inessence,choosinganoptimaldesignfromwithinarangeadmittedbythemorphingscheme,whereasthechoiceofmaneuveringactuationwouldbeastandarddesignoftheautopilot. Asynthesismethodforthisapplicationwouldmostlikelybeformulatedasastandardmodel-followingapproachwhichallowedhandlingqualitiestobedirectlyutilized.Thecontrollerwouldseektocommandthemorphingsystem,P()=fA();B();C;Dg,torespondsimilarlytoadesiredsystem,X=f^A;^B;C;Dg,andtrackmaneuveringcommandsthroughouttheightenvelope.Therststepinthedesignprocesswouldneedtobethechoiceofmorphingvalueswhichminimizesthedierencebetweenthemorphingdynamicsandthedesireddynamics(forthemissionsegment),asinEquation 5{5 .Thenextstepwouldneedacontrollertocommandtheeectorsusedformaneuvering.ThiscontrollercouldbederivedfromtraditionalapproachesforrobustornonlinearsystemsasinEquation 5{6 =argmin2RW1kA())]TJ /F1 11.955 Tf 15.04 3.03 Td[(^Ak+W2kB())]TJ /F1 11.955 Tf 14.75 3.03 Td[(^Bk(5{5) K=argminK2SkX)]TJ /F5 11.955 Tf 11.96 0 Td[(Fl(P;K)k(5{6) 5.2.2Example:Gull-WingedAircraft Aninitialcontrollerhasbeensynthesizedforavariablegull-wingedaircraft[ 2 ].Asetofdesireddynamicswereformulatedforcruise-loiterandaggressive-maneuver 96

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r H() G P() 6 X() ? -e Figure5-1. Model-FollowingSystem congurationsbyvarying,forexample,frequencyoftheshort-periodmode.Thevalueofgull-wingmorphingisdeterminedusingEquation 5{5 toapproximatethesedesiredcharacteristics.AfeedbackcontrollerwasdesignedfromEquation 5{6 usinganinner-loopH1,G,andouter-loopPID,H,withgainschedulingacrossconguration,,asshowninFigure 5-1 .Apredenedmissionprolefromcruisetoaggressiveandbacktocruise,showninFigure 5-2 ,indicatesthatthemorphingvariedwithdesiredmodalpropertieswhiletheresponsestopitchdoubletstrackedthedesiredmaneuvers. Figure5-2. SimulationsoftheGull-MorphingAircraftModel Thisexampledemonstratestheeectivenessoftheapproach;however,severalissuesarenotyetrigorouslyaddressed.TherstissuewouldbedeningcomputationalstrategiesforEquation 5{5 whenvariousordersofpolynomials(A())wereusedwithmultipletypesofmorphing,.Asecondissuewouldinvolveconsiderationofndingastandardizedformulation,suchasLPV,wherethemaneuveringcontrollercouldinherentlyincludegainschedulingandrobustness. 5.3StabilizingControl:DisturbanceRejection 5.3.1Synthesis Aclassofsuboptimalcontrollersaresynthesizedfordisturbancerejection.Theabilitytorejectdisturbances,suchaswindgusts,isobviouslyanimportantfeatureofanyight 97

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controller.Theresultingclosed-loopsystemwillbeinherentlynonlinearduetotheeectsofstatefeedback;consequently,thecontrollerswillensuretheorigin,ortrimcondition,isgloballyasymptoticallystableforthesenonlineardynamics.[ 5 49 54 56 65 76 109 ] Anapproachfordisturbancerejectionhasbeenformulatedthatusesaconstant-gainmatrixforfeedback.ThisapproachguaranteesthedesiredstabilitypropertiesasstatedinLemma1.Essentially,theargumentresultsfromshowingthatalltrajectoriesoftheclosed-loopsystemareniteandconvergetozero. Lemma5.3.1. Giventhesystem,_x=(A0+A1)x,thentheoriginisgloballyasymptoticallystableusingthecontrollaw=Kxif 1. A0<0 2. A1K)]TJ /F5 11.955 Tf 16.14 0 Td[(A0<0 Proof:Theclosed-loopsystemisformulatedas_x=A0x+A1Kx2whichisobviouslynonlinear.Thesolutiontothisdierentialequationcanbeexpressedasx(t)=)]TJ /F11 7.97 Tf 6.58 0 Td[(A0 A1K)]TJ /F11 7.97 Tf 6.59 0 Td[(A0e)]TJ /F12 5.978 Tf 5.76 0 Td[(A0t.Inthiscase,thesteady-statevalueshowslimt!1x(t)=0ifA0<0soalltrajectoriesconvergetotheoriginifA0isnegativedenite.Asingularitywillalsoarisesuchthatlimt!1x(t)=1ifA1K)]TJ /F5 11.955 Tf 12.38 0 Td[(Aoe)]TJ /F11 7.97 Tf 6.59 0 Td[(A0t=0.Thissingularitycorrespondstot=)]TJ /F7 7.97 Tf 6.58 0 Td[(1 A0lnA1K A0whichhasasolutiont2Rwitht>0ifA0<0andK
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2. A1K1<0 Proof: Theclosed-loopsystemcanbeexpressedas_x=(A0+A1K0)x+(A1K2)x3.DeneaLyapunovfunctionasV(x)=1 2x2sothefollowingconditionsaresatised. 1. V(0)=0 2. V(x)0 3. _V(x)=(A0+A1K0)x2+(A1K2)x4 NowV(x)isavalidLyapunovfunction,andconsequentlytheoriginisgloballyasymptoticallystable,ifA0+A1K0<0andA1K2<0. 2 Animportantconsiderationistheformulationoftheopen-loopmodel.Specically,thesecontrollersassumetheLIVdynamicscanbeexpressedasarst-orderfunctionofthemorphinginput.Suchanassumptionwillrestrictthesystemswhichcanbeconsideredforcontrolsynthesis;however,therestrictionisnotnecessarilyunreasonable.Theexamplesinthispapercanbesatisfactorilymodeledasrst-orderfunctionsand,ifsecond-orderfunctionsareneeded,theLemmascanbeextendedtoconsiderhigherorders. Anotherconsiderationistheuseofcontroleectors.Thesedesignsassumethatshape,representedby,istheonlyeectorusedtorejectdisturbances.Theinclusionoftraditionaleectorscanbeconsideredbuttheinitialresultsassumemorphingistheonlyeectoravailableforcontrol. Also,thesecontrollersdonotguaranteeanypropertiesrelatedtoperformance.TheLemmasmerelyindicatethestabilityoftheoriginforthenonlinearclosed-loopsystem.Noinformationaboutthespeedatwhichresponsesconvergetotheorigincanbeprovided.Theseconditionsactuallyapplydirectlytoresponsesfrominititalconditionsandarenotevenformulatedtoconsiderotherperformancemetricssuchastrackingorrobustness.Assuch,theseresultsarelimitedinnaturetosimplestabilityforgustrejection. 99

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5.3.2Example:Chord-VaryingMorphing AnothermodeloftheMAVisgeneratedbyassumingapurevariationinchord.Thismodelindicatessomepropertiesoftheightdynamicsofachord-varyingmorphingwhilethespanandcamberarekeptconstant.Thechordforthismodelischosentorangefrom11.43cmto22.86cmwhilethespanremainsat60.96cm.TherepresentativemodelinTORNADO[ 80 ]resultedincongurationssuchasthoseshowninFig. 5-3 Figure5-3. Chord-VaryingMAVModel(m) TheLIVframeworkisagainusedtorepresenttheightdynamics.ThemodelinEq. 5{7 indicatesarst-orderttothestate-spacematricesfortheightdynamicsatasetofchordvalues.Theperturbationtochord,,rangesbetween0:05715m. _x=0BBBBBBB@266666664)]TJ /F1 11.955 Tf 9.3 0 Td[(0:075)]TJ /F1 11.955 Tf 9.3 0 Td[(0:1240)]TJ /F1 11.955 Tf 9.3 0 Td[(9:81)]TJ /F1 11.955 Tf 9.3 0 Td[(1:538)]TJ /F1 11.955 Tf 9.3 0 Td[(7:45612:0002:686)]TJ /F1 11.955 Tf 9.3 0 Td[(5:848)]TJ /F1 11.955 Tf 9.3 0 Td[(32:9340001:000377777775+266666664)]TJ /F1 11.955 Tf 9.3 0 Td[(0:330)]TJ /F1 11.955 Tf 9.3 0 Td[(2:09300)]TJ /F1 11.955 Tf 9.3 0 Td[(3:957)]TJ /F1 11.955 Tf 9.29 0 Td[(15:3310025:32495:187)]TJ /F1 11.955 Tf 9.3 0 Td[(160:16000003777777751CCCCCCCAx(5{7) ApairofcontrollersaredesignedusingLemma 5.3.1 andLemma 5.3.2 forthemodelinEq. 5{7 .ThismodelsatisestheconditionsoftheLemmasincludingstabilityfor=0.TheresultinggainsforthesecontrollersaregiveninTable 5-1 Table5-1. Gainsforchord-varyingaircraftdisturbancerejectioncontroller controlleralgorithm Lemma1=0)]TJ /F1 11.955 Tf 9.3 0 Td[(100:1xLemma2=[0]+0)]TJ /F1 11.955 Tf 9.3 0 Td[(10)]TJ /F1 11.955 Tf 9.3 0 Td[(0:5x2 100

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Responsestoagustfortheopen-loopsystemandclosed-loopsystemsareshowninFig. 5-4 .Theopen-loopsystemreturnstotrimtodemonstratetheexpectedstabilitywhenthechordisnotvariedfromthenominalcondition.Theclosed-loopsystems,usingLemma 5.3.1 andLemma 5.3.2 ,arealsostableandabletorejecttheinitialdisturbance. TheresponsesinFig. 5-4 indicatethatchord-varyingmorphingcanindeedbebenecialtorejectinggustssinceeachcontrollerreturnsthevehicletotrimfasterthantheopen-loopsystem.Theincreaseinrejectionisperhapsnotoverlysignicantbutatleastisclearlynoticeable.Thecontrollerchangesthechordwhichactuallyhastheeectofreducingthemomentofinertiaand,subsequently,increasingthestabilityderivatives.Astheactuationplotshows,thecontrollerreducesthechordtorejectthisdisturbancebutthencanquicklyincreasethechordtoreturntoaerodynamiceciency. Figure5-4. AngleofAttackResponseandAssociatedChordofAircraft TheabilityofthecontrollertoreturnthevehicletotrimisalsoevidentintheresponsesofallstatesasshowninFig. 5-5 .Eachstatereturnstoequilibriumalthoughtheratesofdecayintheoscillationsaftertheinitialdisturbanceclearlyvarybetweenresponses. Thebehaviorofthecontrollers,andconsequentlytheclosed-loopresponses,canbedeterminedfromFig. 5-5 inassociationwithTable 5-1 .Inparticular,thisassociationcanexplainthedierencesbetweenactuationresponsesinFig. 5-4 .ThechordvariationsusingthecontrollerofLemma 5.3.1 showanoscillatorynaturewhereasthechordvariations 101

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Figure5-5. ResponsestoChord-VaryingMorphing fromthecontrollerofLemma 5.3.2 settlestoaconstantsteady-statevalue.Thedierentbehaviorsiscausedbythelinearandquadraticnatureofthecontrollers.Essentially,asmallchordiscommandedbyscalingsmallvaluesofthestatesbutanevensmallerchordiscommandedbyscalingquadraticvaluesofthosesmallstates. 5.4FeedForwardOptimalControl 5.4.1State-ResponseTracking Morphingcanbeusedtotrackadesiredsignal,yd(t),forthesystemresponse.Theactualresponseisexpressedasy(t)=P((t))wheretheplantmodelisexplicitywrittenashavingtime-varyingdynamicsduetothemorphingcommand.Inthiscase,morphingisrestrictedtobetheonlycontroleector. Anoptimizationcomputesthemorphingcommandthataddressestracking.Thisoptimizationsearchesoverthemorphingtrajectorytominimizethetrackingerrorateveryinstanceintime,asshowninEquation 5{8 min(t)s Zt0kyd(t))]TJ /F5 11.955 Tf 11.95 0 Td[(P((t))k2dt(5{8) 102

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SuchanoptimizationiseectivelyafeedforwardschemeasdepictedinFigure 5-6 .Thecompensatoractuallygeneratesanoptimaltrajectoryforthemorphingbasedonano-lineminimization yd (t) P((t)) -y Figure5-6. ArchitectureforState-ResponseTracking Suchafeedforwardelementrepresentsacriticalaspectofmanynonlinearcontrolschemes.Themorphingdynamicscertainlyhaveanonlineardependencyontheinputcommandsoamulti-elementcontrollerislikelynecessary.Someaspectoffeedbackmaybeutilizedfornoiseandrobustness;however,designingthisfeedforwardcontrollerisaninitialsteptowardsamulti-elementcontroller. 5.4.2Pole-ResponseTracking Amorphingcommandcanalsobegeneratedfortrackingofapoleresponse.Denepd(t)asthedesiredlocationfortime-varyingpoles.Suchdesiredpolescandirectlyrelatetotraditionalparametersoffrequencyanddamping.Assuch,thetrackingofapoleresponseisanalogoustotrackingofmodalparameters. Thepole,p(t),isnotedasbeingrelatedtothediagonalelementsofthedecompositionofthestate-transitionmatrix.Specically,theithtermisgivenaspi(t)=r)]TJ /F7 7.97 Tf 6.58 0 Td[(1ii((t))d/dt(rii((t)))whereriiisthediagonaltermofR(t)asconstrainedbyQ(t)R(t)=A(t;0).Note,thecompletepoleformulationmaybeseeninSection 3.5 TheoptimizationtocomputethemorphingcommandisgiveninEquation 5{9 min(t)s Zt0kpd(t))]TJ /F5 11.955 Tf 11.95 0 Td[(rii((t))_rii((t))k2dt(5{9) Suchoptimizationisagainafeedforwardscheme.Theblockdiagramforthisscheme,asshowninFigure 5-7 ,diersfromFigure 5-6 byincludingseveralelementsrelatedtothepoles.Specically,ablockofQRisincludedtorepresenttheQRdecompositionandablockofisincludedtorepresentthecomputationofthestate-transitionmatrix. 103

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pd (t) (P((t))) QR s ? -r)]TJ /F7 7.97 Tf 6.59 0 Td[(1ii_rii Figure5-7. ArchitectureforPole-ResponseTracking 5.4.3Example:Mass-SpringSystem 5.4.3.1System Amorphingmass-springsystemiscontrolledtotrackdesiredstateresponsesandpolesignals.Thissystem,asshowninFigure 5-8 ,isasinglemassattachedtoawallthroughaspringanddamper. Figure5-8. Mass-Spring-DamperSystem Controlisactuatedthroughthedamperofthissystem.Essentially,thedampingisassumedtobeadjustableinordertoalterthesystemdynamics.TheequationofmotionisthusgiveninEquation 5{10 .Thesedynamicsareobviouslydependentonthedampingsocommandingadampingtrajectorywillrequireanalysisusingtime-varyingpoles. mx=kx)]TJ /F5 11.955 Tf 11.95 0 Td[((t)_x(5{10) Themassis8kgandthespringconstantis2N/m. 5.4.3.2MorphingBasis Theoptimalmorphing,thatischosentogenerateasystemresponseorpoleresponsethattracksadesiredresponse,isrestrictedtobeingathird-orderpolynomialoftime.AnymorphingtrajectoryisthusrestrictedtoliewithinthesetofallpolynomialswithrealcoecientsasnotedinEquation 5{11 104

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23Xi=0itii2R(5{11) Also,thecoecientsofthemorphingtrajectoryareboundedtopreventthesystemfromimmediatelybecomingoverdamped.Thehigher-ordertermsinthepolynomialhavethepotentialtodominateastimeincreases;consequently,thesetermsarerestrictedmorethanthelower-orderterms.Theexactvaluesoftheseboundsaredeterminedsuchthatanymorphingtrajectoryallowsthesystemtovarybetweenunderdampedandoverdamped.ThevalueofthedampingratiogiveninEquation 5{12 representstheboundarybetweenunderdampedandoverdamped. =(t) 2p km(5{12) Boundsonthecoecientsareinitiallychosentolimitthetimeatwhichthesystembecomescriticallydampedto7.5s.Suchatimeisarbitrarilychosenforthisexamplebutmayeasilybealteredtoconsiderdierenttypesofresponses.Giventhistimeconstraint,thecoecientsofEquation 5{11 mustsatisfytheexpressionsinEquation 5{13 .Theprocedureactuallyndsthecoecientssequentiallybyconsideringonlyonecoecienttobeanon-zerovalueatatime.Indongso,thesolutiondeterminesthemaximumcontributionneededbyeachmorphingcoecienttoachievethedesireddampingtransient. iti 2p km=Pi(7:5)i 8=1i=1;2;3(5{13) Also,aconstraintisintroducedsuchthat(t)>0forallvaluesoftime.Thisconstraintnotesthatthedesiredresponseisassumedtobeboundedandthusthesystemmusthavepositivedamping.Thecoecientsofeachpolynomialareadditionallylimitedtomaintainthispositivecondition. TheresultingboundsonthecoecientsaregiveninTable 5-2 105

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Table5-2. Boundsoncoecientofmorphingbasis CoecientUpperBoundLowerBound 04-411.07-1.0720.14-0.1430.02-0.02 5.4.3.3TrackingaSystemResponse:PolynomialMorphing Asetofdesiredresponsesaregeneratedusingpolynomialmorphing.Thesepolynomialsrangeasarst-order,second-orderandthird-orderfunctions.TheoptimizedvaluesofthemorphingcommandsaregeneratedforfeedforwardcontrolandshowninTable 5-3 .TheoptimizedmorphingisessentiallyidenticaltothedesiredmorphingwhichisanticipatedsincethedesiredmorphingcanbeexactlyduplicatedbythemorphingbasisofEquation 5{11 Table5-3. Optimalpolynomialmorphingtotrackdesiredpolynomialmorphing desiredactual 0:5+t0:5+t0:5t+0:076t20:5t+0:076t2)]TJ /F1 11.955 Tf 9.3 0 Td[(0:75t+0:1t2+0:02t3)]TJ /F1 11.955 Tf 9.3 0 Td[(0:75t+0:1t2+0:02t3 Theoptimizedstateresponses,muchliketheoptimizedmorphingtrajectories,areessentiallyidenticaltothedesiredstateresponses.TheseresponsesareshowninFigure 5-9 Anotherdesiredresponseisattemptedtobetrackedusingthepolynomialmorphing;however,thisdesiredresponseisassociatedwithamorphingcomposedofasinusoidaladdedtoapolynomial.ThisdesiredmorphingcannotbeduplicatedusingtherestrictedbasisofEquation 5{11 sosomeerrormustresultinthetracking.TheoptimizedpolynomialformorphingisgiveninTable 5-4 alongwiththisdesiredmorphing. Table5-4. Optimalpolynomialmorphingtotrackdesiredsinusoidalmorphing desiredactual 0:94t+sin(t)0:5343+1:0700t)]TJ /F1 11.955 Tf 11.96 0 Td[(0:0812t2+0:0094t3 106

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A B C Figure5-9. DesiredResponse(|)andOptimalResponse()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()forA)First-OrderMorphingandB)Second-OrderMorphingandC)Third-OrderMorphing TheresultingstateresponseisshowninFigure 5-10 asreasonablyclosetothedesireresponse.Someerrorispresent;however,thedesiredmorphingisnotexcessivelyfaroutsidetherangeofthemorphingbasissothefeedforwardcontrolisnearlyabletoprovidethecorrecttracking. 5.4.3.4TrackingaSystemResponse:Piecewise-PolynomialMorphing Apiecewise-polynomialmorphingisconsideredtoexpandtherangeofthemorphingbasisseeninEquation 5{11 .Inthiscase,themorphingbasisisallowedtobearst-orderpolynomialwhosecoecientscanvaryatdiscretepointsintime.Anadditionalconstraint 107

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Figure5-10. DesiredResponse(|)andOptimalResponse()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[()forSinusoidalMorphing ofcontinuityisimposedonthispiecewise-polynomialmorphingtoensurephysicalactuatorscouldpotentiallyimplementsuchacommand. Anoptimalvalueofpiecewise-polynomialmorphingisgeneratedtotrackadesiredresponsethatcontainsasinusoid.TheresultingcommandisgiveninTable 5-5 alongwiththemorphingassociatedwiththedesiredresponse. Table5-5. Optimalpiecewise-polynomialmorphingtotrackdesiredsinusoidalmorphing desiredactual 0:94t+sin(t)0:068+1:734t:0t1:521:959+0:399t:1:52
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Figure5-11. DesiredResponse(|)andOptimalResponse()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[()forSinusoidalMorphing variousorders.Theoptimalmorphing,asgiveninTable 5-6 ,isabletoeectivelymatchthemorphingassociatedwiththedesiredpoles. Table5-6. Optimalpolynomialmorphingtotrackdesiredpolynomialmorphing desiredactual 1:07t1:07t1+0:75t+0:024t21+0:75t+0:024t2)]TJ /F1 11.955 Tf 9.3 0 Td[(0:2t+0:13t2+0:039t3)]TJ /F1 11.955 Tf 9.3 0 Td[(0:2t+0:13t2+0:039t3 Theoptimizedpolesignals,muchliketheiroptimizedmorphingtrajectories,areessentiallyidenticaltothedesiredpolesignals.Formodalreasons,thepolesignalswillbetruncatedat10seconds.ThesesignalsareshowninFigure 5-12 Anotherpoleresponseisattemptedtobetrackedusingthepolynomialmorphing.Thisdesiredresponseisagainassociatedwithamorphingcomposedofasinusoidaladdedtoapolynomial.Duetodampingconstraints,itremainsthesamedesiredsinusoidalmorphingasthatusedforthestateresponse.TheoptimizedpolynomialformorphingisgiveninTable 5-7 alongwiththisdesiredmorphing. Table5-7. Optimalpolynomialmorphingtotrackdesiredsinusoidalmorphing desiredactual 0:94t+sin(t)0:8532+0:7648t)]TJ /F1 11.955 Tf 11.96 0 Td[(0:1248t2+0:0200t3 109

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A B C Figure5-12. DesiredResponse(|)andOptimalResponse()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[()forA)First-OrderMorphingandB)Second-OrderMorphingandC)Third-OrderMorphing Althoughusingthesamedesiredmorphing,theoptimizedpolynomialinTable 5-7 isdierentfromthatinTable 5-4 .Thisresultissimplyduetotheoptimizationofadierentcostfunction. TheresultingpolesignalisshowninFigure 5-13 .Someerrorispresent;however,thefeedforwardcontrolisagainnearlyabletoprovidethecorrecttracking. 5.4.3.6TrackingaPoleResponse:Piecewise-PolynomialMorphing Thepiecewise-polynomialmorphingisusedtotrackapoleresponseassociatedwithasinusoidalmorphing.TheresultingmorphingandthedesiredtrajectoryaregiveninTable 5-8 110

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Figure5-13. DesiredResponse(|)andOptimalResponse()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[()forSinusoidalMorphing Table5-8. Optimalpolynomialmorphingtotrackdesiredsinusoidalmorphing desiredactual 0:94t+sin(t)0:104+1:673t:0t1:522:182+0:269t:1:52
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5.5H1FeedbackControl:WithoutInertialEects Feedforwardcontrol,asshowninSection 5.4 ,isonemethodwhichcanbeusedforoptimaltracking,whileanotherisH1feedbackcontrol[ 21 23 55 61 100 129 ].H1controltypicallyachievesoptimalityfromderivingacontroller(robustorstabilizing)whichsolvessomemathematicaloptimizationproblem[ 29 45 48 ].Inthecaseoftracking,arobustcontrollerisusedandisdesignedsuchthatsomedesiredlevelofperformanceisobtained.Thisrobustcontrollermustbecapableofmaintainingperformancedespitetheinclusionofexogeneousinputsintothesystem,suchasnoise,disturbance,andreferencesignals.Whenallexogeneoussignalsareavailabletothesystem,theproblemisthenconsidera"fullinformation"H1problem[ 24 71 ],asshowninEquations 5{14 5{16 _x(t)=Ax(t)+B2w(t)+B1u(t) (5{14) z(t)=C1x(t)+D11w(t)+D12u(t) (5{15) y(t)=C2x(t)+D21w(t)+D22u(t) (5{16) Itshouldbenotedthatxrepresentsthesystemstatevariables,yisthesystemmeasuredoutput,zisthesystemcontroloutput,uisthesystemcontrol,andallstatematriceshaveproperdimensions.Itshouldbenotedhowever,thatthefullinformationassumptionrequiresthattheirbefullstate,disturbance,andcontrolfeedback,asdescribedbyEquation 5{17 C2=[I00]TD21=[0I0]TD22=[00I]T(5{17) TheexogeneousinputsinEquation 5{17 maybedescribedbyEquation 5{18 w(t)=[drn]T(5{18) 112

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Forthepurposesofthisstudy,the"fullinformation"standardwillbeappliedtothelineartime-varyingproblem.However,itisobviouswhentransformedintothebilinearform,Equation 5{1 cannotaccuratelybedescribedbythetime-invariantsystemdenedinEquations 5{14 5{16 .Therefore,Equations 5{14 5{16 mayberedenedforthebilineartime-invariantsystem[ 105 ],asshowninEquations 5{19 5{21 _x(t)=(A0+A1u(t))x(t)+B2w(t)+(B0+B1x(t))u(t) (5{19) z(t)=C1x(t)+D11w(t)+D12u(t) (5{20) y(t)=C2x(t)+D21w(t)+D22u(t) (5{21) Thisnewbilineartime-invariantsystemisbaseduponEquation 5{2 withouttheinertialcontribution,asshowninEquation 5{22 _x=A0x+A1xu(5{22) Itshouldbenotedthatthelineartime-invariantsystem,showninEquation 5{1 ,wasassumedtohaveananedependencewithinthestates.Fromthisresult,itwasfurtherassumedthatthelineartime-varyingsystemhadananedependencewithintheinputcommand,u(t),thusgivingtheformofabilinearsystem.Inmakingthisassumption,itisseenthatthetime-varyingdependencehasmovedoutsideofthestatematrixintothecontrolinput,u,andstatevariable,x.Thisfactsuggeststhatabilineartime-invariantcontrollermaybeusedtosucientlycontroltheunderlyingtime-varyingsystem[ 4 68 ]. InordertosimplifytheH1bilinearproblemandprovideareasonablesolution,thesimplifyingassumptions,showninEquation 5{23 ,needtobemade. 113

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DT12C1=0DT12D12=IB0DT12=0D21DT21=ID11sucientlysmall(5{23) FromEquation 5{23 ,itismentionedthatD11mustbesucientlysmall.ThissuciencysuggeststhatthemaximalsingularvalueofD11mustbesmallerthanthecorrespondingH1normoftheclosed-loopcontrolsystem.ItisalsoseenfromEquation 5{23 thatA0andB0mustbestabilizable;However,itisrecalledthatforthetime-varyingprobleminquestion,thereexistsnoexternalinputindependentofthestatecommand.Asaresult,ithasbeenshown[ 106 ]thattheoptimalcontrolofabilinearsystemcanbeachievedifandonlyifthereexistsascalar,r,suchthattherealpartsoftheeigenvaluesof(A0)]TJ /F5 11.955 Tf 12.77 0 Td[(r(A1+B1))arenegativeandthecostfunction,asseeninEquation 5{24 ,isminimized. J=1 2Z10fzT(t)z(t))]TJ /F5 11.955 Tf 11.95 0 Td[(2wT(t)w(t)gdt(5{24) BycombiningEquation 5{24 withthepartialderivativeoftheclosed-loopLyapunovfunction,,theHamiltonianfunctioncanbedenedandwrittenasEquation 5{25 H=1 2zTz)]TJ /F5 11.955 Tf 13.15 8.09 Td[(2 2wTw+T_x(5{25) SubstitutingEquations 5{19 5{21 intoEquation 5{25 resultsinEquation 5{26 ,fromwhichthemaximalexogeneousinputvectorandminimalcontrolcommandmaybefound. 114

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H=1 2(xTCT1C1x+xTCT1D11w+xTCT1D12(u+r)+(u+r)TDT12D11w+wTDT11D11w+(u+r)TDT12D12(u+r))-222(2wTw)+T(Ax+(A1+B1)x(u+r)+B2w) (5{26) Themaximalexogeneousinputvector,w,andminimalcontrolcommand,u,arefoundbytakingthepartialderivativeofEquation 5{26 ,respectively,andsettingitequaltozero.TheresultingoptimaltermscanbedescribedbyEquations 5{27 5{28 u=)]TJ /F5 11.955 Tf 9.3 0 Td[(r)]TJ /F1 11.955 Tf 15.95 8.09 Td[(1 2(D12D11BT2)]TJ /F5 11.955 Tf 11.95 0 Td[(xT(A1+B1)T)+xT(A1+B1)T(5{27) w=)]TJ /F1 11.955 Tf 13.3 8.09 Td[(1 2(DT11D12xT(A1+B1)T)]TJ /F5 11.955 Tf 11.96 0 Td[(BT2)(5{28) UsingthesimplifyingassumptionsfoundinEquation 5{23 ,theclosed-loopformoftheHamiltonianmaybefoundbysubstitutingEquations 5{27 5{28 backintoEquation 5{25 .Theresultingclosed-loopformoftheHamiltoniancanbeshownbyEquation 5{29 HCL=1 2(xTCT1C1x+(A0)]TJ /F5 11.955 Tf 11.95 0 Td[(r(A1+B1))+(A0)]TJ /F5 11.955 Tf 11.95 0 Td[(r(A1+B1))TT)-222((A1+B1)xxT(A1+B1)TT+1 2((A1+B1)xxT(A1+B1)TT+B2BT2T)]TJ /F1 11.955 Tf 12.44 3.16 Td[((A1+B1)xDT12D11BT2)]TJ /F1 11.955 Tf 12.45 3.16 Td[(B2DT11D12xT(A1+B1))) (5{29) Theclosed-loopLyapunovfunction,,forbilinearsystemshasbeenshowntobeaquadraticfunctionofthestatevector[ 10 62 69 79 32 ]andthereforecantaketheformshowninEquation 5{30 =1 2xTPx(5{30) 115

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ThepartialderivativeofEquation 5{30 canbetakenwithrespecttoxtond,asshowninEquation 5{31 =@ @x=xTP(5{31) SubstitutingEquation 5{31 intoEquation 5{29 givesthenalversionoftheclosed-loopHamiltonian,asshownbyEquation 5{32 HCL=1 2xT(CT1C1+P(A0)]TJ /F5 11.955 Tf 11.95 0 Td[(r(A1+B1))+(A0)]TJ /F5 11.955 Tf 11.96 0 Td[(r(A1+B1))TP)-222(P(A1+B1)xxT(A1+B1)P+1 2(P(A1+B1)xxT(A1+B1)P+PB2BT2P)]TJ /F5 11.955 Tf 11.96 0 Td[(P(A1+B1)xDT12D11BT2P)]TJ /F5 11.955 Tf 11.95 0 Td[(PB2DT11D12xT(A1+B1)TP))x (5{32) IthasbeenshownforbilinearsystemsthatthealgebraicRiccatiequationisderivedfromthelinearpartsoftheclosed-loopHamiltonianfunction[ 106 ].Therefore,thealgebraicRiccatiequationrelatedtoEquation 5{32 canbeshownbyEquation 5{33 [ 92 ]. P(A0)]TJ /F5 11.955 Tf 11.95 0 Td[(r(A1+B1))+(A0)]TJ /F5 11.955 Tf 11.95 0 Td[(r(A1+B1))TP+CT1C1+1 2PB2BT2P=0(5{33) ForthesingleinputbilinearsystemdescribedbyEquation 5{22 ,itcanbesaidthattheclosed-loopnonlinearsystemsfromtheH1statefeedbackcontrolaregloballyasymptoticallystableifandonlyifthefollowingconditionsaremet: (A0;B2)stabilizable (A0;C1)observable eig(A0)]TJ /F5 11.955 Tf 11.95 0 Td[(r(A1+B1))<0 Itshouldbenotedthattheconstantvalue,r,canbefoundusingmethodssuchastheRouthcriterion. 116

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5.6FutureWorkandChallenges Multipleavenueshavebeenestablishedtocontinuethecontrolworkpresentedthroughoutthispaper.Twoareasofparticularinterestwouldbethethoroughinvestigationandderivationofadynamiccontrolapproach,aswellas,theinclusionofaninertialtermwithinboththefeedforwardandH1feedbackoptimaizations. Fromthesetwotopics,theareaclosesttorealizationatthistimewouldbetheinclusionoftheinertialterm,_,asshowninEquation 5{2 .Therefore,themainfocusofthissectionwillprimarilycenteraroundtheconceptsandchallangessurroundingtheintroductionofthisterm. ItisclearlyseenfromSection 2.3.1.2 thatinertiaplaysacrucialroleinthedynamicoutcomeofthedescribedtime-varyingsystem.Typically,inertialeectsareneglectedduetotheirsmallinuenceonthedynamics;However,theeectsofinertiaintime-varyingsystemsarestillyettobefullyunderstood,althoughintialtestingsuggeststhatslightchangesininertiacouldbesignicantifunaccountedfor[ 47 ]. Anobviouschallengeregardingtheintroductionoftheinertialterm,_,intotheoptimalcontrolarchitectureisthedecisionofwhichtermtakesprecedence,themorphingangleorthemorphingrate?Thesetwotermsarelinkedthrougharstderivativerelationshipandthereforecannotbecontrolledindependently.Asaresult,theoptimizationproblembecomeseitherndingthecorrectratewhichproducesthedesiredmorphingtrajectoryorviceversa.Thekeytothisconceptisnotjustndingthecorrectterm,butratherndingthecorrecttermwhichminimizestheeectoftheotherandmaintainsthedesiredobjective. Beforethisproblemcanbeaddressed,theabilitytoeectlystabilizethesystemshouldrstbeconsidered.ItwasshowninSection 5.3.1 thatatime-varyingsystem,withoutinertia,couldbesuboptimallycontrolledwhileguaranteeinggloballyasymptoticstability.Anaddendumtothisillustrationwouldbetoaddtheinertialterm,asshowninEquation 5{2 ,andrederiveasubotpimalcontroller. 117

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StartingwithEquation 5{2 ,itisseenbyredeningthemorphingrate,_,astheproductoftwopartialderivatives,thatEquation 5{2 canberedenedintermsof_x,asshowninEquation 5{34 _x=A0x+A1x+A2x@ @x@x @t=A0x+A1x+A2x@ @x_x(5{34) ItshouldbenotedthatsinceisscalarandEquation 5{34 representsaSISOsystem,thenEquation 5{34 maybewrittenassowithoutalgebraicissues. Afterrearrangingterms,Equation 5{34 canberewrittenintheformshowninEquation 5{35 _x=(I)]TJ /F5 11.955 Tf 15.86 0 Td[(A2x@ @x))]TJ /F7 7.97 Tf 6.59 0 Td[(1(A0x+A1x)=A)]TJ /F7 7.97 Tf 6.59 0 Td[(1~A(5{35) NoticingtherightsideofEquation 5{35 ,itisrecalledthat~Ahasalreadybeenshowntoachievegloballyasymptoticstabilityusingsuboptimalcontrollers.Therefore,Awouldneedtobegloballyasymptoticallystablefortheenitiresystemtoachievegloballyasymptoticstabilityoratleastgloballystableforthesystemtobestable. AnotherwaytolookattheproblemistoseethatEquation 5{35 mayberearrangedsuchthatittakestheformshowninEquation 5{36 _x=(I)]TJ /F5 11.955 Tf 15.85 0 Td[(A2x@ @x))]TJ /F7 7.97 Tf 6.59 0 Td[(1(A0+A1)x=A)]TJ /F7 7.97 Tf 6.59 0 Td[(1^Ax(5{36) Now,thequadraticLyapunovfunction,asshowninEquation 5{30 ,canbeusedtochecktheoverallstabilityofthenewlydenedtime-varyingsystem,asdescribedbyEquation 5{36 .SubstitutionofEquation 5{36 intothepartialderivativeofEquation 5{30 resultsinthestabilitymetricshowninEquation 5{37 @ @x=xT(I)]TJ /F5 11.955 Tf 15.86 0 Td[(A2x@ @x))]TJ /F7 7.97 Tf 6.58 0 Td[(1(A0+A1)x=xTAx(5{37) 118

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ItiscleartoseefromEquation 5{37 thatinorderforthetime-varyingsystem(withinertia)toachievestability,thenAwouldneedtobenegativedenite. 119

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CHAPTER6EXAMPLEOFVARIABLESWEEPAIRCRAFT 6.1Design 6.1.1BiologicalInspiration Theseagullisalogicalchoicefromwhichtoderivebiologicalinspirationsinceitissoadeptatagileyinginwindyconditions.Suchbirdsareroutinelyseentrackingboats,divingtocatchprey,andlandingonbuoysdespiteheavywindsandstronggustsfromdierentdirections.Themissionsenvisionedforaminiatureairvehiclerequireasimilarsetofabilities;therefore,abiomimeticapproachiswarranted. Theskeletalstructureoftheseagullisacriticalcomponentthatenablesightcapability.Inparticular,thejointsattheshoulderandelbowareusedtorotatethewingsandconsequentlyaltertheightdynamics.Suchrotation,asseeninFig. 6-1 ,causesdisplacementinbothverticalandhorizontaldirectionswhichcorrelatestowingdihedralandwingsweep. Figure6-1. PicturesofSeagulls Thewings,asshowninFig. 6-1 ,willusuallyvarythesweepbetweentheinboardandoutboard.Thevariationresultsfromtheindependentactuationabouttheshoulderandelbowtovarythehorizontalrotation.Thismorphingprovidesavarietyofchangesintheightcharacteristicssuchasstability,divespeed,andturnradius. Also,thewingsareshowninFig. 6-1 tovarythesweepbetweenrightandleftwingsalongwiththeinboardandoutboard.Thisvariationutilizes4degreesoffreedomresulting 120

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fromindependentactuationofshoulderandelbowoneachwing.Thismorphingenablesseveralmaneuversrelatedtohoming,rolling,andrejectingcrosswinds. Emphasisisplacedontherelationshipbetweenwingsweepandmaneuvers.Thesweepisalreadyadesignparameterwhoseeectsonaerodynamicshavebeenstudiedfortraditionalaircraft;however,thestudyofbirdsprovidesadditionalinsightintotheperformancethatmaybeachievableusingindependentmulti-jointsweep.Inthiscase,thecorrelationsbetweensweepanddiveareaugmentedwithcorrelationsbetweensweepandagilityforbothturningandtrimming. 6.1.2MechanicalDesign Avehiclewhichfeaturestheindependentmulti-jointcapabilityisdesignedbyretrottinganexistingaircraft[ 3 ].Thebasicconstructionusesskeletalmembersofaprepregnated,bi-directionalcarbonberweavealongwithrip-stopnylon.Thefuselageandwingsareentirelyconstructedoftheweavewhilethetailfeaturescarbonsparscoveredwithnylon.Theresultingstructureisdurablebutlightweight. Thewingsactuallyconsistofseparatesectionswhichareconnectedtothefuselageandeachotherthroughasystemofsparsandjoints.Thesejoints,asshowninFig. 6-2 ,arerepresentativeofashoulderandelbowwhichservetovarythesweepofinboardandoutboard.Therangeofhorizontalmotionadmittedbythesejointsisapproximately30deg. Figure6-2. JointsonWing 121

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Itisnotedthatconventionalaileroncontrolsurfacesareomittedfromtheaircraft'snaldesign.Thisfeatureisadirectresultofspan-wiseinconsistenciescreatedbythedynamicrangeofmorphingcongurations.Therefore,theelbowjointsaredesignedinsuchamannerthattheyallowbothhorizontalsweepandrollingtwist.Thismotionisaccomplishedbycreatingaoatingjointthatcloselymimicsthevariousrangesofmotionattainablebyanautomobile'suniversaljoint,asshownifFig. 6-3 Figure6-3. FloatingElbowJoint Thewingsurfacemustbekeptcontinuousforanycongurationofsweepingbecauseofaerodynamicconcerns.Thisvehicleensuressuchcontinuitybylayeringfeather-likestructures,asshowninFig. 6-4 ,withinthejoint. Figure6-4. Feather-LikeElements Thesestructuresretractontoeachotherunderthewingwhenboththeinboardandoutboardaresweptback.Conversely,theycreateafan-likecoveracrosstheensuinggap 122

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whentheinboardissweptbackandtheoutboardissweptforward.Thecontractionandexpansionofthesurfaceareacreatedbythesestructuresissmoothlymaintainedbyatractandrunnersystemimplementedontheouterregionsofeachmember,asseeninFigure 6-5 Figure6-5. Trackandrunnersystem Spars,formedfromhollowshaftsofcarbonber,areplacedalongtheleadingedgeofeachwing.Thesesparsactasbotharigidsourcetomaintaintheleading-edgecurvatureandaconnectionofeachindependentwingjoint.Theinboardsparistranslatedhorizontallybyaservo-drivenlinearactuatorlocatedinsidethefuselage.Theinboardsparisthenconnectedtotheinboardwingsectionattheshoulderjointlocatedontheoutsideofthefuselage.Theinboardsparthenconnectsattheelbowjointtooutboardsparatroughlythequarter-spanpoint.Theoutboardwingregionisactivatedindependentlyoftheinboardregionbymeansofaservoattachedattheelbow.Anillustrationofthesparconguration,withcorrespondingattachmentpoints,canbeseeninFig 6-6 6.1.3TechnicalSpecications Thevehicleisxed-wingdesignthatincludesafuselageandempennageforatotalweightof596g.Thefuselagehasatotallengthof48cmandiscomposedofacylindicalbaywithlargestdiameterof7cmthatstretchesfor30cmandaboomwithdiamterof1cmthatstretchesfor18cm.Thewingsaremountedonthetopofthecylindrical 123

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Figure6-6. Underwingsparstructure baywhiletheavionicsaremountedinsidethecylindricalbay.Thecomponentsoftheempennage,includingtheelevatorandrudderwhichactascontrolsurfaces,aredescribedinTable 6-1 Table6-1. Empennagespecications SectionReferenceReferenceReferenceSpan(cm)Chord(cm)Area(cm2) HorizontalTail30.487.62232.26VerticalTail15.247.62116.13Rudder30.484.57139.29Elevator15.243.0446.33 Referenceparametersforthemorphingwingvarybasedonsweepconguration.ArepresentativesetoftheseparametersaregiveninTable 6-2 foralimitedsetofsymmetriccongurationsinwhichtheleftandrightwingshaveidenticalsweep. Theavionicsconsistsofactuatorsandamotor.AsetofeightHitechHS-65MGmetalgearservosprovideactuationandaredistributedasthreeservostocontroltheinboardsweep,outboardsweepandwingtwistofeachwingalongwithtwoservostocontrolthe 124

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Table6-2. Referenceparametersforsymmetricsweep InboardOutboardReferenceReferenceReference(deg)(deg)Span(cm)Chord(cm)Area(cm2) -15-3066.1714.681028.11-10-2073.9713.121003.45-5-1078.8112.38976.250080.3911.84947.1151078.6111.62916.68102073.6111.69885.66153065.7212.13854.76 elevatorandrudder.ThemotorisanE-ightsix-seriesbrushlesselectricmotorpoweredbyathree-cellThunderPowerLi-polymer2100mAhbattery. 6.2Modeling 6.2.1ComputationalTools Aerodynamicsolutionsforthree-dimensionalwingsofanyshapeorsizecanbecalculatedbyusingavortex-latticemodel.Assumingtheowtobeincompressibleandinviscid,thewingismodeledasasetofliftingpanelswitheachcontainingasinglehorse-shoevortex.Bothspan-wiseandchord-wisevariationinliftcanbemodeledasasetofstepchangesfromonepaneltothenext,asshowninFig 6-7 Figure6-7. Modelingoftheliftvectors Locatedatthepanelquarter-chordpositionisaboundvortex,whichshedstwotrailingvortexlines.Therequiredstrengthoftheboundvortexoneachpanelwillneedtobecalculatedbyapplyingasurface-owboundarycondition.Thisboundaryconditionstatesthereiszeroownormaltothesurfaceofthewing.Foreachpanelthiscondition 125

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isappliedatthethree-quarter-chordpositionalongthecenterlineofthepanel.Thenormalvelocityismadeupofafreestreamcomponentandaninducedowcomponent.Thisinducedcomponentisafunctionofstrengthsofallvortexpanelsonthewing.Thus,foreachpanelanequationcanbesetupwhichisalinearcombinationoftheeectivestrengthsproducedfromallpanel.Bysolvingtheseequations,onecanproduceamodelthateectivelydescribestheaerodynamicqualitiesandcontrollabilityofanaircraft. AVL[ 30 ]isavortex-latticemodelthatisbestsuitedforaerodynamiccongurationswhichconsistmainlyofthinliftingsurfacesatsmallanglesofattackandsideslip.Thesesurfacesandtheirtrailingwakesarerepresentedassingle-layervortexsheets,discreditedintohorseshoevortexlaments,whosetrailinglegsareassumedtobeparalleltothelongitudinalx-axis,asshowninFig 6-8 Figure6-8. Modelingofthetrailinglegvectors AVLprovidesthecapabilitytoalsomodelslenderbodiessuchasfuselagesandnacellesviasource-doubletlaments.Theresultingforceandmomentpredictionsareconsistentwithslender-bodytheory,buttheaerodynamicsaregenerallychallengingtocompute,thereforethemodelingofbodiesshouldbedonewithcaution.Ifafuselageisexpectedtohavelittleinuenceontheaerodynamicloads,itshouldbeleftoutoftheAVLmodelentirely.Thisexclusionofthebodyisprescribedtoavoidpotentialinaccuraciesfromenteringtheoverallmodel. 126

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AVLassumesquasi-steadyow,whichallowsunsteadyvorticitysheddingtobeneglected.Moreprecisely,itassumesthelimitofsmallreducedfrequency,whichmeansthatanyoscillatorymotion(e.g.inpitch)mustbeslowenoughsothattheperiodofoscillationismuchlongerthanthetimeittakestheowtotraverseanairfoilchord.Thisassumptionisvalidforvirtuallyanyexpectedightmaneuver.Also,theroll,pitch,andyawratesusedinthecomputationsmustbeslowenoughsothattheresultingrelativeowanglesaresmall,asjudgedbythedimensionlessrotationrateparameters. 6.2.2SweepDetermination Thisvehicleisabletoachieveawiderangeofsweeporientationsinbothsymmetricandasymmetriccongurations.SomerepresentativecongurationsareshowninFig. 6-9 todemonstratetherange. Figure6-9. SweepCongurations Acoordinatesystemisdenedtofacilitatetheproperdescriptionofeachconguration.Sweepanglesassociatedwiththeinboardsectionsaredenotedas1fortherightwingand3fortheleftwing,whileoutboardsectionsuse2fortherightwingand4fortheleftwing.Theseangles,asshowninFig. 6-10 ,aredenedsuchthatpositivevaluesindicateabackwardsweep.Also,eachangleisdescribedrelativetotheright-sidereferencelinethatisperpendiculartothefuselagereference. 127

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6nose -rightside 1 2 3 4 Figure6-10. SweepAngles 6.3AerodynamicProperties 6.3.1SymmetricCongurations:Aerodynamics Theaerodynamicsareevaluatedforthesymmetriccongurationsinwhichthesweepoftherightwingisequivalenttothesweepoftheleftwing.Inthiscase,theonlydegreesoffreedomaretheinboardandoutboardangleswhicharesharedbyeachwing.Theaerodynamicsarecomputedusingavortex-latticemethodthatisdesignedtoconsiderthinairfoils[ 30 ].Asetofrepresentativedataispresentedthatisparticularlyinformativewithrespecttothemaneuversanticipatedforthisclassofvehicle. ThevariationofliftwithrespecttoangleofattackisshowninFig. 6-11 forarangeofsweepcongurations.ThedatashowthattheaircraftobtainsitshighestCLalongaridgelinecorrelatingtoequalbutoppositesweepofinboardandoutboard.Conversely,thisderivativedecreasessignicantlyforcongurationsofinboardandoutboardbeingbothsweptbackorbothsweptforward.Assuch,theliftismoredependentonangleofattackbyutilizingtheadditionaldegreeprovidedbytheelbowtoopposethesweepoftheshoulder. Anotherlongitudinalparameter,Cm,isshowninFig. 6-12 forthesweepcongurations.Thisparameterisdirectlyindicativeofthestaticstability;consequently,thepositivevaluesindicatetheaircraftbecomesmorestaticallyunstableasthewingsaregraduallysweptforward.Thisinstabilityisdemonstrativeofacenterofgravitywhichliesaftofthe 128

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Figure6-11. VariationofLiftwithAngleofAttackforSymmetricSweep neutralpoint.Thecenterofgravityisrelativetotheplacementofmasseswithin,oron,theaircraft,andthereforeshiftsaccordingtoeachmorphingconguration. Figure6-12. VariationofPitchMomentwithAngleofAttackforSymmetricSweep Thedamping-in-rollderivative,towhichClpiscommonlyreferred,isshowninFig. 6-13 forthecongurationspace.Arollratecausesvariationsinangleofattackalongthespanofthewingwhichcreatesarollingmoment.Thisderivativeisnegativeforallsweepcongurations,withthelargestvaluedmagnitudesoccurringinregionscorrespondingtocongurationswithequalbutoppositesweepoftheinboardandoutboardsections.Themagnitudedecreasesforcongurationswithinboardandoutboardbeingbothforwardsweptorbothbackwardsweptwhichsuggestsapotentialitytoauto-rotateorspin. ThevehiclehasdirectionalstaticstabilityasevidencedbythedatainFig. 6-14 forallsymmetriccongurations.Thisdatarelatesthederivativeofyawmomentwithangle 129

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Figure6-13. VariationofRollMomentwithRollRateforSymmetricSweep ofsideslipwhosepositivitydemonstratesthestabilitycondition.Thestabilityisincreasedasthebackwardsweepincreasesbecausethestabilizingcontributionsofthefuselageandverticaltaildominateasthewingloseseectiveness. Figure6-14. VariationofYawMomentwithAngleofSideslipforSymmetricSweep 6.3.2SymmetricCongurations:FlightDynamics Linearizedmodelsoftheightdynamicsarecomputedbyrelatingtheaerodynamiccoecientstothestandardequationsofmotionforight[ 33 ],asgiveninFig. 6-15 .Theselinearizedmodelshavedecoupledstatesthatallowseparateanalysisoflongitudinaldynamicsandlateral-directionaldynamics.Modelsarecomputedforeverysymmetriccongurationintherangeofsweepanglestoindicatethevariedstabilityproperties. Thelongitudinaldynamicsarestable,asshowninFig. 6-16 ,forthemajorityofobtainablecongurations.Largevaluesofforwardsweepfortheinboardrequirealargevalueofbackwardsweepfortheoutboardtomaintainstability.Thesweepof 130

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Figure6-15. Modeltostate-spaceowchart theoutboardsectionisallowedtodecreaseastheinboarddecreasesitsforwardsweep.Eventually,thevehiclecanremainstabledespiteasmallvalueofforwardsweepfortheoutboardaslongastheinboardhasalargevalueofbackwardsweep. Figure6-16. NumberofUnstablePolesofLongitudinalDynamicsforSymmetricSweep Stabilityofthelateral-directionaldynamics,asshowninFig. 6-17 ,isachievedforasmallsetofcongurations.Theonlyregionofstabilitycorrespondstocongurationswithlargevaluesofbackwardsweepofbothinboardandoutboard.Theoneunstablepole,showninFig. 6-17 ,correspondstoaclassicallydenedspiralmodethatiscommonlyfoundtobeunstablewithalargetimeconstant. SomemodalpropertiesofthelongitudinaldynamicsarepresentedinFig. 6-18 toindicatethenumberofcomplexpoles.Eachpairofpolesrelatestoanoscillatorymodeso 131

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Figure6-17. NumberofUnstablePolesofLateral-DirectionalDynamicsforSymmetricSweep responsecharacteristicscanbedirectlyinferred.Inthiscase,thevehicledemonstratesaclassicalsetofphugoidandshort-periodmodesforthemajorityofcongurationsincludingallthosewithbackwardsweepoftheoutboardsections.Thephugoidmodeislostastheoutboardsectionsincreaseinforwardsweepuntileventuallyeventheshort-periodmodeislostforlargevaluesofforwardsweepfortheoutboard.Itcanbesaidthattheintroductionofunstablepoles,asshowninFig. 6-16 ,isdirectlyrelatedtothelossofbothoscillatorymodes,asshowninFig. 6-18 ,orviceversa. Figure6-18. NumberofOscillatoryPolesforLongitudinalDynamicswithSymmetricSweep ThenumberofoscillatorypolesisshowninFig. 6-19 forthelateral-directionaldynamics.Itisseenthatvehicleretainstwo-oscillatorypolesregardlessofthesweepconguration.Therefore,itcanbeinferredthatthevehiclehasaclassicdutchrollmode 132

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forallcongurations.Itcanbesaidthattheintroductionofunstablepoles,asshowninFig. 6-17 ,isnotcausedbyachangeinmodenature. Figure6-19. NumberofOscillatoryPolesforLateral-DirectionalDynamicswithSymmetricSweep 6.3.3AsymmetricCongurations Theaerodynamicsarealsocomputedforasetofasymmetriccongurationsinwhichtherightwingandleftwinghavedierentsweep.Theindependenceofinboardandoutboardoneachwingpresentsasetofcongurationswith4degreesoffreedom;consequently,thedatamustberestrictedtofacilitatepresentation.Theaerodynamicsarepresentedhereforcongurationsinwhichtherightwingisxedwith1=2=0andtheleftwingismorphedfrom-30degto30deginboththeinboardandoutboard. Asetofstandardparameterscanbecomputedtodirectlycomparetheaerodynamicsofsymmetricandasymmetriccongurations.Thevariationwithangleofattackforlift,showninFig. 6-20 ,andmoment,showninFig. 6-21 ,canbecomparedwithFig. 6-11 andFig. 6-12 ,respectively.Theclearsimilaritybetweenthesymmetricandasymmetricvaluesindicatessomerelationshipbetweenthecongurationscanbeinferred.Inparticular,thevariationcausedbysweepingbackasinglewingaresimilarinnaturetothevariationcausedbysweepingbackbothwings.Themagnitudeissmallerwhensweepingbackthesinglewingsosomelossofeciencyissuggested;however,thestabilityderivativesdisplaythesameshapeforeachsituation. 133

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Figure6-20. VariationofLiftwithAngleofAttackforAsymmetricSweep Figure6-21. VariationofPitchMomentwithAngleofAttackforAsymmetricSweep ThevariationofrollmomentwithrollratecanbecomparedinFig. 6-22 withFig. 6-13 alongwithvariationofyawmomentwithangleofsideslipshowninFig. 6-23 andFig. 6-14 forasymmetricandsymmetriccongurations.Again,thevariationsaresimilarinnatureforeachsetofcongurationssuggestingasimilarityinowphysicsbutalossofeciencyintheeect. Figure6-22. VariationofRollMomentwithRollRateforAsymmetricSweep 134

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Figure6-23. VariationofYawMomentwithAngleofSideslipforAsymmetricSweep Anadditionalsetofaerodynamicparametersarecomputedfortheasymmetriccongurationsthatarenullforthesymmetriccongurations.Theseparameters,whichareshowninFig. 6-24 ,representthecouplingbetweenlongitudinaldynamicsandlateral-directionaldynamics.Thedatashowsthatsweepingtheleftwingcausesadramaticincreaseinmagnitudeoftheseparameters.Sucharesultisexpectedsincetheseparametersreecttheasymmetrythatincreaseswithsweep. Figure6-24. VariationofCoupledAerodynamicsforAsymmetricSweep 6.3.3.1Flightdynamics Asetofmodelsthatrepresenttheightdynamicsarealsocomputedfortheasymmetriccongurations.Theselinearizedmodelsdonothavelongitudinalparametersdecoupledfromlateral-directionalparameters;consequently,theanalysismustconsiderasinglecoupledsystem. Thedynamicsareunstable,asshowninFig. 6-25 ,foranycongurationofasymmetricsweep.Thesystemisshowntohaveoneunstablepoleforthemajorityofcongurations 135

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andthreeunstablepolesforasmallregion.Thesmallregionisindicatedbyalargeforwardsweepoftheinboardsectionandaequaldisplacementofsweeparoundtheoutboardneutralposition. Figure6-25. NumberofUnstablePolesforDynamicswithAsymmetricSweep ThenumberofoscillatorymodesispresentedinFig. 6-26 todemonstratesomepropertiesofthevehiclemotion.Inthiscase,thevehiclehastheclassical3oscillatorymodesformostcongurationswhentheoutboardhasneutralorbackwardsweep.Astheoutboardissweptforwardoftheneutralposition,aregionofmodeswappingiscreated.Thisregionofforwardsweepfortheoutboardsectionindicatesthatamodeisgainedorlossstrictlydependingonthesweepoftheinboardsection. Figure6-26. NumberofOscillatoryPolesforDynamicswithAsymmetricSweep 6.3.3.2Modecharacterization Themodesofarepresentativecongurationischaracterizedtodemonstratethecoupledmotionwhichresultsfromanasymmmetricsweepofthewings.Theconguration 136

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ischosentohaveastraightwingontherightwithnosweepso1=2=0andastraightwingontheleftwithbackwardssweepso3=4=15deg.Theeigenvaluesgeneratedfromthisconguration,asshowninTable 6-3 ,indicatesevenstablepolesandoneunstablepolewhichisadistributionexpectedfromFig. 6-25 Table6-3. Setofeigenvalues Eigenvalues -17.59426.401i-37.202-2.78413.394i-0.1920.685i0.0610 FromTable 6-3 ,itcanbeseenthatthedynamicshavetwonon-oscillatorymodes.Thetimeconstantsofthesemodes,asshowninTable 6-4 ,indicatestheonemodehasastableconvergenceandtheotherhasanunstabledivergence.Thestableconvergenceisatleasttwoordersofmagnitudefasterthantheunstabledivergence. Table6-4. Timeconstantsofnon-oscillatorymodes ModeEigenvalueTimeConstant 1-37.200.026920.061-16.393 Theightmotionassociatedwitheachofthesemodesisdeterminedbythemodeshapes.SuchshapesaregiveninTable 6-5 todescribetherelativevalueofeachstateduringtheresponse.Theconvergenttermischaracterizedbymostlyrollratewithminorcontributionsfromangleofattack,rollangleandpitchrate.Thismodeissimilarinnaturetotheclassicallydenedrollmode.Thedivergenttermischaracterizedbyfullycoupledmotioninwhichtherollangleisvaryingalongwithprimarilytheyawrate,butalsotheforwardvelocity.Thismodeissimilarinnaturetotheclassicallydenedunstablespiralmode. 137

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Table6-5. Modeshapesofnon-oscillatorymodes statemode1mode2 forwardvelocity-0.0007-0.1324angleofattack-0.03980.0163pitchrate-0.05380.0003pitchangle-0.00140.0053angleofsideslip-0.00030.0100rollrate0.99710.0557yawrate-0.02310.3811rollangle-0.02680.9131 Table 6-3 alsoindicatesthattheightdynamicshavethreeoscillatorymodes.ThevaluesofnaturalfrequencyanddampingaregiveninTable 6-6 foreachmode.Themodewiththelowestnaturalfrequencyisunstablewhiletheothermodesarestable. Table6-6. Modalpropertiesofoscillatorymodes ModeEigenvalueFrequency(rad/s)Damping 3-17.59426.401i31.730.5464-2.78413.394i13.680.2035-0.1920.685i0.710.270 Theeigenvectorsassociatedwiththesemodesarealsocomplexsothemagnitudeandphaseofeachmodeshapeisusedtoanalyzetherelationshipbetweenstates.Theresultingdata,giveninTable 6-7 ,showsthatmodes3and5areprimarilydominatedbylongitudinalmotionwithonlyasmallcouplingtothelateral-directionalmotion,whilemode4isthedirectopposite.Suchmotionisnotentirelyunexpectedsinceeventhesymmetriccongurationshadoscillatorymodesaectingboththelongitudinalandlateral-directionaldynamics.Assuch,mode3haspropertieswithsomesimilaritytoashort-periodmode,whilemode4haspropertiessimilartoadutch-rollmodeandmode5similartoaphugoidmode. 6.3.3.3Crosswindrejection Sensorpointinginurbanenvironmentsisaprimemissionforwhichmicroairvehiclesarebeingdeveloped.Crosswinds,bothsteady-statewindandtime-varyinggusts,presentasignicantchallengetomaintainingsensorpointingduringight.Thecommonapproach 138

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Table6-7. Modeshapesofoscillatorymodes Mode3Mode4Mode5statemagnitudephase(deg)magnitudephase(deg)magnitudephase(deg) forwardvelocity0.02-42.60.0195.30.990.0angleofattack0.49-102.60.0536.10.08-179.1pitchrate0.620.00.0787.70.050.2pitchangle0.02-123.70.005-14.00.07-105.5angleofsideslip0.0002-61.20.0781.30.0002-20.3rollrate0.61-28.50.24-66.50.0051.5yawrate0.02-129.10.970.00.003-82.8rollangle0.02-152.20.02-168.30.007-104.2 tosensorpointingdespitecrosswindsisturningintothewindandcrabbingdownrangetoperiodicallypointthesensor;however,suchanapproachiscertainlynotoptimalduetothelackofcontinuouscoveragebythesensoralongthedesiredlineofsight. Asymmetricwing-sweepcanenhancetheabilitytoperformsensorpointinginthepresenceofsuchcrosswinds.Inparticular,onewingcanbesweptdownwindwhileonewingissweptupwind.Theaircrafthas,inasense,rotatedthewingsintothewindwhilethefuselageremainspointedinitsoriginaldirection.Thischangeintheeectivesideslipanglecaneasilybeillustrated,asseeninFig. 6-27 Figure6-27. EectiveAnglesofSideslip Theangleofsideslipatwhichtheaircraftcantrimisanindicatoroftheamountofcrosswindinwhichtheaircraftcanmaintainsensorpointing.Arepresentativedemonstration,showninFig. 6-28 ,presentsthemaximumpositivevaluesforangleof 139

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sideslipatwhichtheaircraftcantrim.Thewingsareconstrainedinthisdemonstrationsuchthatinboardandoutboardanglesareidenticalwhichlimitsthedegreesoffreedomandfacilitatespresentation.Also,eachconditioncorrespondstothelargestangleofsideslipatwhichtheaircraftcantrimgivendeectionlimitsof15degfortherudderandelevatoralongwithaileron. Figure6-28. MaximumAngleofSideslipatwhichAircraftcanTrim ThedatainFig. 6-28 demonstratesthatwingsweepisbenecialforsensorpointing.Specically,aforward)]TJ /F1 11.955 Tf 9.3 0 Td[(30degsweepoftheleftwingandabackward30degsweepoftherightwingallowsanangleofsideslipof44degtobemaintained.Thismaximumangledecreasesastheleftwingdecreasesitsforwardsweepandtherightwingdecreasesitsbackwardsweep.Thevehicleiseventuallyunabletotrimatanypositiveangleofsideslipwhenthebothwingsaresweptbackward. 6.4DynamicProperties 6.4.1MissionScenario Thevariablewing-sweepvehicleisdesignedforsurveillanceinurbanoperations.Inparticular,itisdesignedtoallowightinconstrainedareaswithlimitedairspace.Arepresentativemissionisplacingasensorintoawindowonabuildingwhichisclosetootherbuildings. 140

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6.4.1.1Divemaneuver Themissionproleforsuchamissioninvolvesseveralsegments.Theinitialightisstandardstraight-and-leveloperationatanaltitudeabovethebuildings.Uponreachingtheopeningbetweenthosebuildings,thevehicleentersasteepdivetorapidlydecreasealtitudewithoutincurringmuchforwardorsidetranslation.Theaircraftceasesthediveandreturnstostraight-and-levelightasquicklyaspossiblewhenthealtitudereachesthedesiredvalueassociatedwiththewindow. Themorphingvariestoincreaseperformanceofthemetricsassociatedwitheachmissionsegment.Theinitialightusesacruisecongurationwiththewingshavingnosweep.Thedivethenutilizesafullyswept-backconguration.Finally,thevehiclereturnstoanominalcongurationforenteringthewindow.Ineachcase,theaircraftusesasymmetricsweepofthewingstoavoidanycouplingassociatedwithasymmetriccongurations[ 46 ]. 6.4.1.2Turnmaneuver Themissionproleforsuchamissionagaininvolvesseveralsegments.Theinitialightisstandardstraight-and-leveloperationatanaltitudewithinthebuildings.Uponreachingadesiredopeningbetweenbuildings,thevehicleentersacoordinatedturntorapidlychangeheadingwithoutincurringmuchchangeinaltitude.Theaircraftceasestheturnandreturnstostraight-and-levelightasquicklyaspossiblewhentheheadingreachesthedesiredvalueassociatedwiththewindow. Themorphingagainvariestoincreaseperformanceofthemetricsassociatedwitheachmissionsegment.Theinitialightusesacruisecongurationwiththewingshavingnosweep.Theturnthenutilizesanequal-but-oppositecongurationarrangement.Anexampleofthisequal-but-oppositemorphingwouldbetheleftwingfullyforwardandtherightwingfullyswept.Finally,thevehiclereturnstoanominalcongurationforenteringthewindow.Itshouldbenotedthattheasymmetricmorphingcausesthedynamicstoexperiencecross-coupling. 141

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6.4.2MassDistribution Anelementalbreakdownoftheaircraft'smassdistributionallowsformoreaccurateinertialmomentsandratestobecomputed.Thesemassesarerepresentedaspointmassesandarelocatedatsomedistancefromtheaircraft'scenterofgravity.Thecenterofgravityisafunctionofwingmorph;therefore,ittranslatesalongthethree-dimensionalbody-axisaccordingly.Themagnitudeofthistranslationalchangeisfoundtobenegligiblethroughoutthedesiredmorphingrange,sinceverylittlemassisactivelymovedwhensweepingthewings,andthereforeassumedstationaryforthismodel. Adirectresultofhavingthisxedcenterofgravityisthattheaircraft'snon-dynamicpointmassesprovideforaconstantinertialmoment,whereonlythedynamicpointmasses,suchastheleftandrightwingsections,createanon-constantmoment.Theoverallmagnitudeofthisnon-constantmomentisdirectlydependentonhowtheaircraftisbeingmorphed.Iftheaircraft'swingsaresymmetricalwithrespecttothecommonlyassignedxz-plane,thenonlythexzinertialproducttermappears,whereas,ifthewingsareasymmetricalwithrespecttothexz-plane,thenallthreeinertialproducttermswillappear. Thewingisdividedintoseparateinboardandoutboardsections,whereacentroidlocationisfoundforeach.Basedonhowthewingismorphed,athreedimensionalaverageistakenbetweentheinboardandoutboardcentroids.Thisaverageistakenwithrespecttothecenterofgravityandisrepresentativeofanoverallcentroidlocationfromwhichthatwing'spointmassislocated. Asimplediagramillustratingtheaircraft'ssectionaldistributionandcenterofgravitylocationisshowninFigure 6-29 .NotethatinFigure 6-29 ,thecoordinatesystemisuniquetothemodelingprogramandisorientedoppositetotheearlierdenedaircraftbody-axis.ThisgureisaccompaniedbyTable 6-8 ,whichliststhemassesforeachsection.ThesemassesarethenusedtocalculatetheindividualinertialmomentslistedinTable 6-9 142

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Figure6-29. PointMassLocations Table6-8. Individualpointmasses MB(g)MF(g)MM(g)MTB(g)MVT(g)MHT(g)MRW(g)MLW(g) 1302955015884545 Table6-9. Characteristicsofelementsgivenascentroidposition(in)andmomentsofinertia(gin2) elementSymbolXYZIxxIyyIzzIxyIxzIyz batteryB-2.50.0-1.753981211812.50.0-5960.0fuselageF0.00.0-1.759039030.00.00.00.0motorM4.50.01.5113112510130.0-3380.0tailboomTB8.50.0-3.0135121910840.03830.0verticaltailVT13.50.00.00.0145814580.00.00.0horizontaltailHT13.50.0-3.072153014580.03240.0 6.4.3ManeuverAssumptions Asetofplantmodelsaregeneratedtorepresenttheightdynamicsateachconguration.Essentially,theforcesandmomentsaectingthevehiclearecomputedusinganassumptionofsteady-stateconditions.Theresultingmodelsdonotproperlyrelatetheunsteadyaerodynamicsbutwillincludethedominantsteady-stateaerodynamics.Thetime-varyinginertiasarethenintroducedtoaccountforthemorphingusingtheexpressionsderivedinSection 2.3.1.2 143

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6.4.4DiveManuever 6.4.4.1Modeling Thevehicleisconstrainedinthismaneuvertosimplifythecongurationspace.Thephysicalvehiclehas4degrees-of-freedominthateachinboardandoutboardcansweepindependentlyontheleftandrightsides;however,thissimulationconstrainstheleftandrightwingstosymmetricsweepwiththeoutboardhavingtwiceasmuchsweepastheinboard.Thisconstraintlimitsthesystemtoasingledegree-of-freedomwhichisappropriateforthelongitudinalnatureofthealtitudechangeassociatedwiththemission. Theightdynamicsforeachcongurationaretrimmedforstraightandlevelight.Actually,thethrustisheldconstantforeachofthesecongurationstonotethepropulsionisnotaectedbythemorphing.Suchanapproachisparticularlyusefulforrelatingmodelsthatmaybetrimmedatvariousairspeeds.Inthiscase,thethrustanddragwereheldconstantsothatthetrimroutinefoundthecorrectairspeed,asshowninFigure 6-30 ,torelateeachmodel. Figure6-30. ChangeinVelocityBasedonSymmetricMorphing:0deg( {2{ ),5deg( {/{ ),10deg( {{ ),15deg( {{ ),20deg( {.{ ),25deg( {{ ),30deg( {4{ ) Figure 6-30 relatestheoveralldragtovelocitywhereeachtrendline(designatedbylinetype),representsamorphingconguration.AconstantthrustoftwoNewtonsischosenand,thusalineisdrawntorepresentthisvalue.Itisshownthatasthewingsaremorphedfurtherback,theintersectionatwhichthetrendlinescrosstheconstant 144

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thrustlinesteadilyincreases.ThispointofintersectionrepresentsthevelocitynecessarytomaintaintwoNewtonsofdragatthatconguration.Overall,Figure 6-30 suggeststhatbysweepingthewingsback,highervelocitiescanbeattained,whilemaintainingaconstantdrag(equaltothrustintrimmedcases).Thistrendisasensibleresultduetothefactthatlesssurfaceareaisexposedtotheoncomingairowwhilethewingsaremorphedbackward. Thevelocitiesfoundateachintersectionarethenusedascorrespondingtrimvelocitieswhenperformingthedivemaneuver. 6.4.4.2Altitudecontroller Acontrollerisformulatedforthisaircrafttoenableamaneuverenvisionedinitsmissionprole;namely,acompensatorwillcommandthevehicletotrackadesiredaltitudeprole.Amulti-looparchitectureisusedtorepresentthecontrollerwhichwillbederivedinamodularapproach. Apairofcontrollersareactuallycomputedforthisaircraftsuchthatonecontrollerisappropriateforthenominalcongurationwhiletheotherisappropriatefortheswept-backconguration.Essentially,thecontrollersarederivedusinganassumptionofinstantaneousmorphing.Suchanapproachsimplyswitchesbetweenfeedbackgainstomatchtheassumedswitchbetweennominaldynamicsandswept-backdynamics.Obviouslythemorphingisnotinstantaneous;however,thisassumptionisnotoverlyunreasonablegiventhefastrateofmorphingontheaircraft. Aninner-loopcontrollerisderivedtotrackcommandstopitchrate.Thiscompensatoriscomputedusingalinear-quadraticregulator[ 87 ]usingafeedbackelement,K,andafeedforwardelement,k,alongwithanintegrator.Actually,thedesignisbasedonshort-perioddynamicstoavoidthepolesandzeroesassociatedwiththephugoiddynamics.Theguaranteeofnosteady-stateerrorinresponsetoastepcommandisonlyassociatedwiththeshort-periodmodelbutthefull-orderdynamicsusedinthesimulationstillshowanacceptableresponse. 145

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Anouter-loopcontrollerisderivedtoaecterrorinaltitude.Thedierencebetweencommandedaltitudeandmeasuredaltitudeisprocessedthrougharst-orderlter,F,toshapethephaseproperties[ 115 ].Theresultingsignalissomewhatindicativeofapitchcommandsoitsderivativeisusedasapitch-ratecommandfortheinner-loopcontroller. Theresultingclosed-loopsystem,asshowninFigure 6-31 ,incorporatesthevariouselementsusingappropriatefeedbackinthemulti-looparchitecture.Anadditionalfeedforwardelement,Z,isincludedinthedesigntoaectthepitchduringtheresponse. a F s 1 s k P K 6 6 ? Z ?huwq Figure6-31. Closed-LoopBlockDiagram 6.4.4.3Time-varyingdynamics Thesimulationofthemorphingaircraftmustproperlyaccountforthetime-varyingdynamics.Essentially,themorphingiscommandedtochangethroughoutthesimulationsotheightdynamicsmustchangeaccordingly.Theplant,P,inFigure 6-31 ,isthussomewhatcomplicatedinitsimplementation. Adiscretizedtypeofmorphingisutilizedinthesimulationtorepresentthesymmetricwing-sweepvariations.ThephysicalaircraftshowninFigure 6-9 hasasweepthatvariesasacontinuousfunctionoftime;however,adiscretizedversionofthismorphingsimpliesthesimulationwithoutlossofgenerality.Inthiscase,thedynamicsassumea5-degreesweepcanbeaccomplishedinstantaneouslybuteach5-degreeincrementmustbeseparatedbysomeminimumtime. 146

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Astandardstate-spaceformulationisusedtorepresenttheplantasshowninFigure 6-32 .Theplantdynamics,whichvarywithmorphingpositionandrate,aregivenbythequadrupleoffA;B;C;Dg. u U B 1 s -y X A 6 D ? Figure6-32. PlantModelwithTrimLogic TheelementofXisusedtoaccountforthechangeintrimconditionsforeachconguration.Considerthatatanypointintime,t,thetotalstatevalue,x(t),isactuallytheadditionofatrimvalue,xo(t),andperturbation,x(t),suchthatx(t)=xo(t)+x(t).Thistotalstatemustremaincontinuousdespitethemorphingeventhoughthetrimvalue,xo(t),willchange.Sincetheplantmodelusesstateperturbationsasafeedback,thenthelogicofXissuchthatx(t+t)=xo(t)+x(t))]TJ /F5 11.955 Tf 11.2 0 Td[(xo(t+t)forsomeintegrationstep-sizeoft. TheelementofUissimilarinnaturetoX.Inthiscase,theelementisusedtoreectthevariationsinelevatorpositionassociatedwithtrim. 6.4.4.4Simulation Thewingcongurationaltersduringthemissiontoreectthedesiredperformance.Therstcongurationhasnosweeptoreectacruiseprole.Thewingsarethensweptbacktothemaximumvaluewhenthediveisinitiated.Uponreachingthedesiredaltitude,thevalueofthesweepisreducedtoreturntoaproleforenteringthewindow.Theangleofthesweep,asshowninFigure 6-33 ,isvariedforsimulationsbasedonfastmorphingorslowmorphing. 147

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Figure6-33. MorphingCongurationforFastMorphing( | ),SlowMorphing( )-222()-222()]TJ ET 0 G 0 g BT /F1 11.955 Tf 438.74 -186.8 Td[() Thealtitudevariation,showninFigure 6-34 ,isthefundamentalmetricusedtoevaluateperformance.Inthiscase,themorphingvehiclesareabletochangetherequiredaltitudeandreturntostraight-and-levelightwithin2.2s.Thefastandslowmorphingaresimilarfortherstsecondbutthenbegintodivergesomewhat.Thefastmorphinghasreachedafullyswept-backcongurationatthistimesoitsresponseahassomewhatlargerovershootthanthatseenwiththeslowmorphing.Thisobservationisadirectresultofthefastmorphingcaseacquiringahighervelocitysoonerandretainingitforalongerperiodoftime.Twoothercasesareconsiderwheremorphingisnotutilized.Thesecasesincludedivemaneuversperformedwiththenominalstraight-wingandfullysweptcongurations.ItisseeninFig. 6-34 thatthenominalcongurationreachesthethedesiredaltitudetheslowestbuthasthesmallestovershoot,whilethefullysweptcongurationreachesthedesiredaltitudemuchfaster,yethasthelargestovershoot. ThestatesassociatedwithpitchareshowninFigure 6-35 inresponsetothealtitudecommandwhilemorphing.Theslowmorphing,incomparisontothefastmorphing,incursagreaterpitchangleduringtheinitialresponsebutthenincursasmallerpitchangleasthevehiclereachesitsnalaltitude.ThisbehaviorcorrelateswiththealtituderesponseofFigure 6-34 148

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Figure6-34. AltitudeinResponsetoFastMorphing( | ),SlowMorphing( ::: ),FixedSwept( )-222()-222()]TJ ET 0 G 0 g BT /F1 11.955 Tf 139.37 -170.84 Td[(),FixedStraight( )-221(\001 ) Figure6-35. PitchAngle(left)andPitchRate(right)inResponsetoFastMorphing( | ),SlowMorphing( ::: ),FixedSwept( )-222()-222()]TJ ET 0 G 0 g BT /F1 11.955 Tf 279.35 -357.97 Td[(),FixedStraight( )-222(\001 ) Thelateral-directionalstatesarefoundnottovarywithalongitudinaldivemaneuver.Thisresultissomewhatexpected,duetothefactthattheaircraftissymmetricallymorphingandthusthecross-couplinginertiasarecancelledoutduetosymmetry. Finally,theelevatorangleisshowninFigure 6-36 todemonstratetheresponsedoesnotincurexcessiveactuation.Themorphingcertainlyinuencestheelevatorinthatthecontroleectivenessdecreasesasthewingsaresweptback.Assuch,theslowmorphinghastheslowestspeedbutalsousesthesmallestrotationofelevator. 6.4.4.5Missionevaluation Theclosed-loopsystemisnotabletosuccessfullycompletethemissionusingonlythenominalcontrollerdesignedforthenominalwingconguration.Thevehicleisabletodivebetweenthebuildingsandsuccessfullychangealtitudewithininanitetimeasshownin 149

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Figure6-36. AltitudeinResponsetoFastMorphing( | ),SlowMorphing( ::: ),FixedSwept( )-222()-222()]TJ ET 0 G 0 g BT /F1 11.955 Tf 139.37 -165.43 Td[(),FixedStraight( )-221(\001 ) Figure 6-34 ;however,ifconstrainedtoacertainspaceandtime,theresultingightpathsassociatedwithmorphingmissthewindowandintersectthesideofthebuilding,asseeninFigure 6-37 .Thisfailuredirectlyresultsfromtheinabilitytoaccountfortime-varyingeectsinthecontrollerdesign.NotethatinFigure 6-37 thecoordinatesarebasedonanEarth-xedframewithaxesdenedbyX,Y,andH. 6.4.4.6Eectsoftime-varyinginertia Theightcontrollerneedstobealteredtocompensateforthetime-varyingparametersassociatedwithmorphing.Althoughmorphingintroducesinertialratesintothedynamics;theseratesarefoundtohavelittleeectontheoverallplantdynamicsandthereforealtertheightpathslightly,asseeninFig. 6-38 .Itshouldbenotedthattheinertialrates,asshowninEquationrefeq41,arecalculatedfromwingscontaininglessthanfteenpercentoftheaircraft'soverallmass.Whenthemassesofthewingsegmentsareincreased(orthetimetomorphisdecreased,asshownbythefastmorphinginFig. 6-38 ),thenthecontributionsfromtheinertialratesbecomemoreprominentintheoveralldynamics.Theresultingchangeindynamicswouldaltertheightpathfurther,thusillustratingtheimportanceofinertialrates. Dierentapproachesmaybetakentoaltertheightcontroller,buteachhasit'sownchallenges.Acompensatortoregulatethepitchisdicultbecausethegainsmustbeoptimizedtothedynamicsbutthosedynamicsarerapidlychangingduringthemorphing. 150

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Figure6-37. AltitudeinResponsetoFastMorphing( | ),SlowMorphing( ::: ),FixedSwept( )-222()-222()]TJ ET 0 G 0 g BT /F1 11.955 Tf 139.37 -468.51 Td[(),FixedStraight( )-221(\001 ) Alternatively,atrackingcontrollercanbeusedafterthemaneuverwhenthedynamicsareknownandxedbutthevehiclemustre-hometowardstheoriginalwaypointorre-locatethewindowusingvisionfeedbackandthencomputeanewtrajectory. 151

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Figure6-38. AltitudeinResponsetoFastMorphing( | ),SlowMorphing( ::: ),FastMorphingWithoutInertia( )-221(\001 ),SlowMorphingWithoutInertia( )-222()-222()]TJ ET 0 G 0 g BT /F1 11.955 Tf 456.7 -396.51 Td[() 6.4.5CoordinatedTurnManeuver 6.4.5.1Modeling Thevehicleisconstrainedinthismanuevertoagainsimplifythecongurationspace.Thismaneuverconstrainstheleftandrightwingstoequal-but-oppositeasymmetricsweepwiththeoutboardhavingnorelativesweepcomparedtotheinboard.Thisconstraint,likethedivemaneuver,limitsthesystemtoasingledegree-of-freedomwhichisappropriateforthelateral-directionalnatureofthechangeinturnperformanceassociatedwiththemission. 152

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Theightdynamicsforeachcongurationaretrimmedforsteadybankedight,andagainthethrustisheldconstantforeachofthesecongurations.Therefore,thetrimroutinefoundthecorrectairspeed,asshowninFigure 6-39 ,torelateeachmodel. Figure6-39. ChangeinVelocityBasedonAsymmetricMorphing:0deg( {2{ ),5deg( {/{ ),10deg( {{ ),15deg( {{ ),20deg( {.{ ),25deg( {{ ),30deg( {4{ ) Fig. 6-39 relatestheoveralldragtovelocitywhereeachtrendlinerepresentsamorphingconguration.AconstantthrustoftwoNewtonsisagainchosenasthedesiredvalue.Itisshownthatasthewingsareasymmetricallymorphed(equalbutopposite,wheretheangleismeasuredwithrespecttotherightwing),theintersectionatwhichthetrendlinescrosstheconstantthrustlinesteadilyincreases.ThispointofintersectionrepresentsthevelocitynecessarytomaintaintwoNewtonsofdragatthatconguration.Overall,Fig. 6-39 suggeststhatbyasymmetricallysweepingthewingsinanequalbutoppositemanner,slowervelocities(inabankedturn)areattainedwhilemaintainingaconstantdrag(equaltothrustintrimmedcases). Thevelocitiesfoundateachintersectionarethenusedascorrespondingtrimvelocitieswhenperformingthecoordinatedturnmaneuver. 6.4.5.2Turncontroller Thecoordinatedturnmaneuverusesabasicclosed-loopapproachforcontrollingthesystem,althoughanopen-loopdesigncouldbeused.Anopen-loopcontrollerwould 153

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besucientbecauseeachplantmodelispreviouslytrimmedforabankedturn.Themodelingtool,AVL,incorporatesaninternalcommandusedfortrimmingmodelsatauserdenedbankangle.Therefore,ifnopilotcommandsaregivenandallexternaldisturbancesareneglected,theaircraftwillremaintrimmedandcompletethemaneuver. Itisnotedthatthemodelsusedtosimulatethismaneuverareperturbationmodelsandthereforeproducestateperturbations.Asaresult,abasicfeedbackloopisincorporatedtomeasurethedierencebetweenthebankangleperturbationandzero.Azerocommandisgiventorepresentthedesiredperturbationfromtrim.Themeasureddierenceisthenmultipledbyaproportionalgainandfedintotheplantasanaileroncontrolcommand.Thisfeedbackloopisincorporatedtoguaranteecontinuitythroughoutmorphing,asdescribedfurtherinSection 6.4.5.3 .Theresultingclosed-loopsystemcanbeseeninFigure 6-40 u k P ?uwqpr Figure6-40. Open-LoopBlockDiagram 6.4.5.3Time-varyingdynamics Likeinthedivemaneuver,themorphingisagaincommandedtochangethroughoutthesimulationandthereforetheightdynamicschangeaccordingly.Althoughtheplant,P,inFigure 6-40 ,isslightlydierentfromtheplantmodelderivedforthedivemaneuverinFigure 6-31 ,itisstillsomewhatcomplicatedinitsimplementation. 154

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Adiscretizedtypeofmorphingisagainutilizedinthesimulationtorepresenttheasymmetricwing-sweepvariations.Theturnsimulationisidenticaltothedivesimulationinthatthephysicalaircrafthasasweepthatvariesasacontinuousfunctionoftimeandadiscretizedversionofthismorphingisusedinthesimulation. Astandardstate-spaceformulationisusedtorepresenttheplantasshowninFigure 6-41 .Theplantdynamics,whichvarywithmorphingpositionandrate,aregivenbythequadrupleoffA;B;C;Dg. u E A R B 1 s -y X A 6 D ? Figure6-41. PlantModelwithTrimLogic TheelementofXisagainusedtoaccountforthechangeintrimconditionsforeachconguration.Thesameconsideration(asthatforthedivesimulation)ismadetoaccountforthecontinuityofthetotalstatedespitethemorphing,aswellasthefeedbackofperturbations. TheelementsA,EandRaresimilarinnaturetoU,asseeninFigure 6-32 .Inthiscase,theelementsareusedtoreectthevariationsinaileron,elevator,andrudderposition,respectively,associatedwithtrim. 6.4.5.4Simulation Amissionischosenfortheturnmaneuver,similartothedivemaneuver,suchthatitwillagaintryytointoaonemeterwindow.Thewingcongurationaltersduringthemissiontoreectthedesiredperformance.Therstcongurationhasnosweepto 155

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reectacruiseprole.Thewingsarethenasymmetricallyswept(equalbutopposite)toachieveamaximummorphedcongurationconsistingoftheleftwinghavingafullforwardsweepwhiletherightwingisequalbutopposite.Themaximummorphedcongurationismaintaineduntiltheendofthemaneuverwhenthevalueofthesweepisreducedtoreturntoaproleforenteringthewindow.Theangleofthesweep,asshowninFigure 6-42 ,isvariedforeachsimulationbasedonfastmorphingorslowmorphing. Figure6-42. MorphingCongurationforFastMorphing( | ),SlowMorphing( )-222()-222()]TJ ET 0 G 0 g BT /F1 11.955 Tf 438.74 -334.56 Td[() Theturnvariation,showninFigure 6-43 ,isthefundamentalmetricusedtoevaluateperformance.Inthiscase,themorphingvehiclesareabletochangetherequiredturningradiusandreturntostraight-and-levelightata270degreeheadingchange.Thefastandslowmorphingaresimilarduringtherstandlastsectionsofthemaneuverbutvarysomewhatthroughouttheactualturn.Theresponseoffastmorphingmorecloselyresemblesthatofthenon-morpedsweptresponseduetothefactthefastmorphingreachesthefullysweptcongurationatanearliertime.Therefore,thefastmorphingresponsecontainsalargerportionproducedfromthesweptwingcongurationthanthatoftheslowmorphingresponsewhichtakeslongertoachievethesameconguration. ThelateralperturbationstatesareshowninFigure 6-44 astherollangleandrollrate.Itisseenthatthefastermorphinginboththerollangleandrateperturbations 156

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A B C Figure6-43. TurninResponsetoFastMorphing( | ),SlowMorphing( ::: ),FixedSwept( )-222()-222()]TJ ET 0 G 0 g BT /F1 11.955 Tf 139.37 -355.98 Td[(),FixedStraight( )-221(\001 ):A)ThecompleteturnproleB)Theturnproleat270degreesC)Theturnproleat180degrees achievehighermagnitudes.Therollangledisplaysthistrendthroughouttheentiremaneuver,whiletherollrateonlyhaslargermagnitudesatthetransientregions. Figure6-44. RollAngle(left)andRollRate(right)inResponsetoFastMorphing(|),SlowMorphing()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() 157

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ThedirectionalperturbationstatesinFigure 6-45 indicatetheyawrateandsideslipangle.Itisnoticedthatthefastermorphingincurslargerperturbationsinbothdirectionalstatesthroughouttheentiremaneuver. Figure6-45. YawAngle(left)andYawRate(right)inResponsetoFastMorphing(|),SlowMorphing()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() Itisnotedthattheturnmaneuverrequirestheaircrafttomorphasymmetrically,andthereforeresultsincoupleddynamics.ThestateperturbationsassociatedwithpitchareshowninFigure 6-46 inresponsetothesimplecontrolsurfacecommandwhilemorphing.Thefastmorphing,incomparisontotheslowmorphing,incursagreaterpitchrateperturbationatbothtransientregions,yetbehavesverysimilarduringthesteadystateregion.Incontrast,thepitchangleperturbationassociatedwithfastmorphinghasahighermagnitudethroughouttheentiremaneuver. Figure6-46. PitchAngle(left)andPitchRate(right)inResponsetoFastMorphing(|),SlowMorphing()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() 158

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6.4.5.5Missionevaluation Theclosed-loopsystemisnotabletosuccessfullycompletethemission.Thevehicleisabletoturnbetweenthebuildingsandsuccessfullychangeheadingwithinthedesiredtimeandairspace.However,whentheeectsofmorphingareintroduced,theresultingightpathmissesthewindowandintersectsthesideofthebuilding,asseeninFigure 6-47 .Thisfailuredirectlyresultsagainfromtheinabilitytoaccountfortime-varyingeectsinthecontrollerdesign.NotethatinFigure 6-37 ,thecoordinatesarebasedonanEarth-xedframewithaxesdenedbyX,Y,andH. Figure6-47. TurninResponsetoFastMorphing(|),SlowMorphing(:::),FixedSwept()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[(),FixedStraight()-221(\001) 159

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TheeectsofmorphingonthetrajectoryoftheightpathcanbetterbeseeninFigs. 6.4.5.4 6.4.5.4 6.4.5.6Eectsoftime-varyinginertia Itwasseeninthepreviousmissionthatinertialrateswereintroducedasaresultofmorphing.Duetothesymmetryoftheaircraftinthatmaneuver,mostinertialmomentsandthereforerates,werecancelledout.Inthecaseoftheturnmaneuver,alltheinertialmomentsandratesarekept.Thepresenceofthesetermshasamorepronouncedeectontheturnradius,asseeninFigure 6-48 ,thanthatseenfordiveinthepreviousmission,asseeninFigure 6-38 Figure6-48. TurninResponsetoFastMorphing(|),SlowMorphing(:::),FastMorphingWithoutInertia()-221(\001),SlowMorphingWithoutInertia()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() 160

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Theightcontrollerforthismissionalsoneedstobealteredtocompensateforthetime-varyingparametersassociatedwithmorphing.Simplyintroducingacompensatortoregulatethelateral-directionalstatesischallengingbecausethegainsmustbeoptimizedtothedynamicsbutthosedynamicsarerapidlychangingduringthemorphing.Itispointedoutthatthedynamicsarenowcoupled(duetoasymmetricmorphing)andthereforethecontrollermustalsocompensateforthelongitudinalstates.Atrackingcontrollermayalsobeusedbutthesamedicultiesarisethatwerepointedoutwiththedivemaneuver. 6.5Time-VaryingModalAnalysis 6.5.1Kamen'sMethod 6.5.1.1Statespacemodel Theintroductionofsymmetricmorphingvariestheaerodynamicsandassociatedvaluesofstabilityderivatives.Thesevariationsresultingfrommorphingthesweepsymmetricallyareprimarilyrestrictedtosomederivativesofverticalforceandallthederivativesofpitchmoment.Assuch,thestandardstate-spacerepresentationofthelinearizeddynamicsismodiedinEquation 6{1 toreectthedependencyofthesederivativesonthemorphinggivenas.Ithasbeenshown[ 20 ]thatastheorderofthedynamicsystemincreases,sodoesthediucltytorepresentthem(usingKamen'smethod).Therefore,themodalanalysisutilizingKamen'smethodwillbelimitedtothefourthordersystemdescribedbyEquation 6{1 .Thestatesofthissystemaregivenasuforforwardvelocity,wforverticalvelocity,qforpitchrate,andforpitchanglewhilethestabilityderivativesincludeXiforderivativeoflongitudinalforcewithrespecttotheistate,Ziforderivativeofverticalforcewithrespecttotheistate,andMiforderivativeofpitchmomentwithrespecttotheistate. 161

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266666664_u_w_q_377777775=266666664XuXwXq)]TJ /F5 11.955 Tf 9.3 0 Td[(gcos0ZuZw()u0)]TJ /F5 11.955 Tf 9.3 0 Td[(gsin0Mu()Mw()Mq()00010377777775266666664uwq377777775(6{1) Theanalysisoftime-varyingdynamicsusingthepolesdenedinEquation 3{14 requiresthestate-spacesysteminEquation 6{1 tobeformulatedasadierentialequation.Inthiscase,theequationinvolvingtheverticalvelocity,w,isconsidered.Theequationsinvolvingtheotherstatescanalsobederived;however,thepolesassociatedwiththeverticalvelocityaresucienttocharacterizetheentiresystemsinceanypolesetofobtainedbyatransformationofthesepoles. Thegeneralformofthefourth-orderexpressionforverticalvelocityisgiveninEquation 6{2 byincludingtime-varyingcoecients,A1;A2;A32R,whicharelengthyandthusdenedintheAppendix. d4w dt4+A3(t)d3w dt3+A2(t)d2w dt2+A1(t)dw dt+A0(t)w=0(6{2) ThecoecientsinEquation 6{2 havesignicantlydierentdependenciesonthemorphinginthateachdependsonthestabilitiesderivativesofZw;Mw;Mq;muwhichvarywithmorphingbutnoteachdependsontheratesofchangesofthesestabilityderivatives.ThevalueofA0dependsonboththerstderivativeandsecondderivativewithrespecttotimeforall4ofthesestabilityderivatives;conversely,A3dependsonlyontherstderivativeandsecondderivativewithrespecttotimeforMqandMu.TheeectofmorphingonA3canevenbefurthermitigatedbynotingthatA3dependsonthemorphingvaluebutnotthemorphingrateif_Mu _Mq=Mu Mq=)]TJ /F11 7.97 Tf 10.5 4.71 Td[(Zu uo.Assuch,thetimevariationsofcertainstabilityderivativesonlyaectcertaincoecientsdependingontherateofchangeofthemorphingtrajectory. 162

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Also,thedynamicshaveapotentialtoexperienceabifurcationthatreducestheorderofthedynamics.ThecoecientsinEquation 6{2 eachcontainafractionwiththesamedenominatorwhichiszeroforcertainvaluesofmorphing.ThelossoforderisequivalentlyviewedbymultiplyingEquation 6{2 bythisdenominatorsothatthedependenceond4w dt4isscaledbyzero. 6.5.1.2Morphingtrajectory TheightdynamicsofthevehicleshowninFigure 6-9 areanalyzedduringsymmetricmorphingfromabackwardsweeptohavingnosweep.Specically,thesweepvariedfrom+30degto0degin1secandthenremainsatasweepangleof0degforeachwing.Thismorphingwouldbevaluablewhentransitioningfromadivetostraight-and-levelightsimilarinamannertobiologicalsystemslikegullsandhawks.Thistransition,especiallywhenoperatingimmersedindenseobstaclessuchasurbanenvironments,maystillrequirerapidmaneuveringforpositioningalongwithgustrejectionsotheightdynamicsduringthemorphinremainofcriticalimportance. TheresponseofthestatesareshowninFigure 6-49 asaresultofthemorphing.Theaircraftisalineartime-varyingsystemsfortheinitial1sec;however,theresponsestillresemblesthetraditionalmodesforalineartime-invariantsystem.Thepitchrateandverticalvelocityshowahigh-frequencyresponsethatisheavilydampedtoresembletheshort-periodmode;conversely,theairspeedandpitchanglearedominatedbyalow-frequencyresponsethatislightlydampedtoresembleaphugoidmode. 6.5.1.3Time-varyingpoles Thetime-varyingpolesassociatedwithFigure 6-49 arecomputedtosatisfyEquation 3{14 andshowninFigure 6-50 alongwiththetime-invariantpolesthatignorethetime-varyingeectsofmorphing.Theseresultindicateseveralcharacteristicsoftime-varyingpoles.Considerthatthetime-invariantpolesassociatedwiththeshort-periodmoderemainsimilarforanyvalueofmorphing;however,thetime-varyingpolesofp41andp42,whichareinitiallyclosestinvaluetotheshort-periodpoles,decaytozeroatarate 163

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Figure6-49. LongitudinalStatesduringMorphingfrom+30degto0degover1sec:ForwardVelocity(upperleft),VerticalVelocity(upperright),PitchRate(lowerleft),PitchAngle(lowerright) similartodecayofthestateresponseduetodamping.Also,notethatthepolesofp43andp44showthelowmagnitudeandslowdecayassociatedwithaphugiodmode;however,thesepolesvaryduringtheinitialresponsewhichisdominatedbytheashort-periodresponse.Assuch,thetime-varyingpolesdierinnaturefromtime-invariantpolesinthattheshort-periodmodeandphugoidaresomewhat,butnotcompletely,distinctandthemagnitudeofthepolesdecaysastheresponsedecays. ThemodesdenedinEquation 3{9 associatedwitheachpoleinFigure 6-50 areshowninFigure 6-51 .Thedecompositionoftheresponseactuallydependsontheeigenvectorsandthesemodes,asopposedtothetime-varyingpoles,sotheymustbeconsideredwhenevaluatingtheightdynamics.Theindistinctseparationbetweentheshort-periodmodeandthephugiodmodethatisevidentinthepolesisalsoevidentinthemodes.Themodesof41and42showtheinitialvariationthatwouldindicateashort-periodresponse;however,thesemodesremainsignicantafterthedecaydueto 164

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Figure6-50. LinearTime-VaryingPoles(|)andLinearTime-InvariantPoles()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()duringMorphingfrom+30degto0degover1sec:RealPart(upperleft)andImaginaryPart(lowerleft)ofp41andp42,RealPart(upperright)andImaginaryPart(lowerright)ofp43andp44 short-perioddampingandthenshowvariationmoreconsistentwiththephugoidmode.Themodesfor43and44showsomeinitialinconsistencyaround1secbutthenarequiteconsistentwiththephugiodmode.Indeed,therealpartsof43and44areremarkablysimilartotheresponseofforwardvelocity,whichispredominatelyduetothephuoidmode,inFigure 6-49 Thenatureofthemodesagreeswiththemathematicalpropertiesthatrelatethemtoboththeresponseandthepoles.TheresponsesofFigure 6-49 areoscillatoryandindeedthepolesofFigure 6-50 arecomplexconjugatessothemodesofFigure 6-51 arealsocomplexconjugates.TherealandimaginarypartsofthemodesarenotedinEquation 4{18 tobe90ooutofphaseandindeedthisphasedierenceisseenforthealltimesofthephugoidresponseof43andtheinitialtimesofshort-periodresponseof41until0:2secwhenthedampinghascausedtheresponsetodecay.Also,thestateresponseshouldbeproportionaltotherealpartofthemodeasnotedinEquation 4{19 whichis 165

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Figure6-51. ModesAssociatedwithTime-VaryingPolesduringMorphingfrom+30degto0degover1sec:RealPart(upperleft)andImaginaryPart(lowerleft)of41and42,RealPart(upperright)andImaginaryPart(lowerright)of43and44 demonstratedbytheverticalvelocityinFigure 6-49 matchingtherealpartof1andtheforwardvelocityinFigure 6-49 matchingtherealpartof3. TheissueofstabilityisdirectlyindicatedbythemodesofFigure 6-51 .Thesemodesdemonstratethatthesystemduringthismorphingtrajectoryhasasymptoticstabilitysincethemagnitudeofeachmodedecaystozeroastimeincreases.ThisresultcorrelateswiththeresponsesshowninFigure 6-49 thatobviouslyreturntoequilibrium.Notethatoneguaranteeforasymptoticstabilityishavingnegativereal-partforthetime-varyingpole.TherealpartofthepolesinFigure 6-50 predominatelyassociatedwiththephugoidmodeareindeedalwaysnegative;however,therealpartofthepolespredominatelyassociatedwiththeshort-periodmodearesometimespositivesothemodemustbecomputedtoascertainstability. Finally,theeigenvectorsassociatedwitheachmodeofFigure 6-51 aregraphedinFigure 6-52 toshowtherelativeresponseofeachvehiclestateasnormalizedbythe 166

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verticalvelocity.Theseeigenvectors,similarlyasthethepolesandmodes,showbothshort-periodcharacteristicsandphugiodcharacteristicsbuteachisclearlydominatedbyonetypeofdynamic.Theeigenvectorofv1initiallyshowsshort-periodmotion,withlittlevariationinforwardvelocityandaphasedierenceof90obetweenpitchrateandverticalvelocity,untilthedampingdecaysthatmotionandthephugoidresponseisevident.Theeigenvectorofv2steadilytransitionstothephugiodresponsewhichisprimarilymotioninforwardvelocityandpitchanglewhichare90ooutofphase.Also,notethattheseeigenvectorsnearlyconvergetosimilarmagnitudesandphasesexceptfora90odierenceinphaseofthepitchanglebetweenv1andv2. Figure6-52. NormalizedEigenvectorsAssociatedwithTime-VaryingModesduringMorphingfrom+30degto0degover1sec:Magnitude(upperleft)andPhase(lowerleft)ofv1andMagnitude(upperright)andPhase(lowerright)ofv2 6.5.1.4Modalinterpretation AmodalinterpretationofthepolesinFigure 6-50 isconductedtorelatethesemathematicalconstructstostandardparametersassociatedwithightdynamics.Theparametersaredirectlycomputedfromthetime-invariantpoleswhiletheircounterparts 167

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fromthetime-varyingpolesresultfrominterpretationsthatrelatecharacteristicsoftheresponses. ThenaturalfrequenciesasshowninFigure 6-53 havesomecommonalitiesbutalsosomecleardierenceswhencomparingthetime-varyingpolesandthetime-invariantpoles.Thevaluesarereasonablyclosefortheentiretrajectorywhenconsideringthepolesassociatedwiththephugoidmode;however,thevaluesareonlycloseforashorttimewhenconsideringtheshort-periodmode.Thedierenceinnaturalfrequenciesfortheshort-periodmoderesultsfromtherelationshipofthetime-varyingpolestothestates.Essentially,theshort-periodpoleisinitiallyrelatingtheoscillatorybehavioroftheresponsebutthesignicantdecreaseinresponsemagnitudeduetodampingisactuallyreectedbythetime-varyingpoledecayingtozero. Figure6-53. NaturalFrequencyAssociatedwithLinearTime-VaryingPoles(|)andLinearTime-InvariantPoles()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()duringMorphingfrom+30degto0degover1sec:Poles1and2(left)andPoles3and4(right) TheenvelopethatboundstheresponsesareshowninFigure 6-54 .Theparametersaresimilarforthephugoidmodebutagain,aswiththenaturalfrequency,theparametersdierforthetime-varyingpolesandthetime-invariantpolesfortheshort-periodmode.Inthiscase,theenvelopeboundstheresponseofthepitchrateandverticalvelocitywhichdominatetheshort-periodresponsebutthen,afterthatmodehasdampedout,theenvelopereecttheboundonthepitchangle. 168

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Figure6-54. EnvelopeAssociatedwithLinearTime-VaryingPoles(|)andLinearTime-InvariantPoles()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()duringMorphingfrom+30degto0degover1sec:Poles1and2(left)andPoles3and4(right) ThedampingratioisshowninFigure 6-55 forthetime-varyingpolesandtime-invariantpoles.Thesevaluesaresmallforbothtypesofpolesandthusarereasonablyclose. Figure6-55. DampingRatioAssociatedwithLinearTime-VaryingPoles(|)andLinearTime-InvariantPoles()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()duringMorphingfrom+30degto0degover1sec:Poles1and2(left)andPoles3and4(right) 6.5.2O'Brien'sMethod 6.5.2.1Statespacemodel AsopposedtoKamen'smethod,itwasshowninSection 3.5 ,thatO'Brien'smethodcaneasilybeimplementedonhigherordersystems.Therefore,individual,eightstate,quasi-staticplantmodelsaretakenatperiodiccongurationsovertheentiremorphing 169

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range.Thesemodelsareshowntobe8x8realstatematrices.Eachperiodiccongurationrepresentesadiscretepointintime.Plantmatricesarethenbrokendownbyelements,whereeachelementisplacedintocorrespondingvectors.Theelementvectorsaretheneachanalyticallyttoauniquepolynomial.Byplacingthesefunctionsbackintotheiroriginalpositionswithintheplantmatrix,anequivalenttime-varyingmatrix(ormodel)canbeformed,asshowninEquation 6{3 A(t)=2666666666666666666664f11(t)f12(t)f13(t)f14(t)f21(t)f22(t)f23(t)f24(t)f31(t)f32(t)f33(t)f34(t)f41(t)f42(t)f43(t)f44(t)00f55(t)f56(t)f57(t)f58(t)f65(t)f66(t)f67(t)f68(t)f75(t)f76(t)f77(t)f78(t)f85(t)f86(t)f87(t)f88(t)3777777777777777777775(6{3) ThefunctionsdenedinEquation 6{3 arefoundtohaveapolynomialstructure,asshowninEquation 6{4 170

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f11(t)=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:09t)]TJ /F1 11.955 Tf 11.96 0 Td[(0:33f55(t)=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:03t2+0:07t)]TJ /F1 11.955 Tf 11.95 0 Td[(0:77f12(t)=0:13t2)]TJ /F1 11.955 Tf 11.96 0 Td[(0:67t)]TJ /F1 11.955 Tf 11.96 0 Td[(0:29f56(t)=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:0013t2)]TJ /F1 11.955 Tf 11.95 0 Td[(0:002t)]TJ /F1 11.955 Tf 11.96 0 Td[(0:0017f13(t)=)]TJ /F1 11.955 Tf 9.29 0 Td[(0:03t2)]TJ /F1 11.955 Tf 11.96 0 Td[(0:08t)]TJ /F1 11.955 Tf 11.96 0 Td[(0:15f57(t)=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:0006t)]TJ /F1 11.955 Tf 11.95 0 Td[(0:99f14(t)=0:03t)]TJ /F1 11.955 Tf 11.96 0 Td[(9:79f58(t)=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:001t+0:41f21(t)=0:16t2)]TJ /F1 11.955 Tf 11.96 0 Td[(0:66t)]TJ /F1 11.955 Tf 11.96 0 Td[(1:49f65(t)=)]TJ /F1 11.955 Tf 9.3 0 Td[(31:3t2+643t)]TJ /F1 11.955 Tf 11.95 0 Td[(1013f22(t)=0:79t2)]TJ /F1 11.955 Tf 11.96 0 Td[(3:45t)]TJ /F1 11.955 Tf 11.96 0 Td[(8:95f66(t)=5:94t2)]TJ /F1 11.955 Tf 11.96 0 Td[(20:73t)]TJ /F1 11.955 Tf 11.95 0 Td[(25:23f23(t)=0:19t2)]TJ /F1 11.955 Tf 11.96 0 Td[(0:23t+22:41f67(t)=0:6t3)]TJ /F1 11.955 Tf 11.96 0 Td[(1:17t2)]TJ /F1 11.955 Tf 11.96 0 Td[(3:92t+14:76f24(t)=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:12t2+0:51t+0:62f68(t)=0f31(t)=)]TJ /F1 11.955 Tf 9.29 0 Td[(0:63t3+5:96t2)]TJ /F1 11.955 Tf 11.95 0 Td[(5:12t)]TJ /F1 11.955 Tf 11.95 0 Td[(9:33)]TJ /F1 11.955 Tf 11.96 0 Td[(0:33f75(t)=)]TJ /F1 11.955 Tf 9.3 0 Td[(2:82t3+25:38t2)]TJ /F1 11.955 Tf 11.96 0 Td[(70:61t+279f32(t)=)]TJ /F1 11.955 Tf 9.3 0 Td[(3:27t3+31:99t2)]TJ /F1 11.955 Tf 11.95 0 Td[(2:76t)]TJ /F1 11.955 Tf 11.95 0 Td[(81:14f76(t)=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:29t2+0:87t+0:05f33(t)=)]TJ /F1 11.955 Tf 9.29 0 Td[(1:06t3+4:47t2)]TJ /F1 11.955 Tf 11.95 0 Td[(3:19t)]TJ /F1 11.955 Tf 11.95 0 Td[(30:32f77(t)=)]TJ /F1 11.955 Tf 9.3 0 Td[(0:22t2+1:03t)]TJ /F1 11.955 Tf 11.95 0 Td[(6:57f34(t)=0f78(t)=0f41(t)=0f85(t)=0f42(t)=0f86(t)=1f43(t)=1f87(t)=0f44(t)=0f88(t)=0(6{4) Altogether,thetwelvestatevectorincludesthethreeorientationangles(,, ),thethreeangularrates(p,q,r),thepositionvector([xyz]),andthevelocityvector([uvw]).Forthepurposesofconvenience,thestatevectorwillbereducedtoeightstatesandaugmentedtoinclude.Thereduced,augmentedstatevectorcanbeshowninequation 6{5 171

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x(t)=2666666666666666666664u(t)w(t)q(t)(t)(t)p(t)r(t)(t)3777777777777777777775(6{5) 6.5.2.2Time-varyingpolesandstabilitymodes Applyingthetime-varyingalgorithmdenedinSection 3.5 tothetime-varyingmodelinequation 6{3 ,ateacht2R+,producesanLTVpoleset,stabilitymodes,andcorrespondingeigenvectors.Duetotheextensiveamountofdataproducedforeachmorphingconguration,aswellasthepotentialenvelopeforwhichtheaircraftcanmorph,onlycertainmorphingtrajectoriesandcongurationsarechosentostudythetime-varyingdynamics.Inadditiontolimitingtheamountofcongurations,onlythelongitudinalsystemdynamicsareconsidered.Thecongurationsarechosensuchthattheyincludedsymmetricsweepfromstraight(0degrees)tosweptpositionsof10,20,and30degrees.Eachofthesemorphingtrajectoriesareperformedatratesof2,5,10,and20seconds.Increasingtherateofmorphingalsoincreasestheoveralleectthetime-varyinginertiacontributesintothesystem'sdynamics. Thesetime-varyingeectsareevident,asshowninFigures 6-56 6-59 .Clearly,thestabilitymodesinFigures 6-56 6-57 showtwomodeswhichconvergeuponzeroquickly(modesAandB),whiletwomodesconvergeuponzeroaftersomelengthyperiodoftime(modesCandD).ObservationsshowthatmodesAandBshowlittlechangefromvaryingthemorphingrateorangle.ThisresultissimplyduetothefactmodesAandBhavealreadyconvergeduponzerobeforeanyinertialeectshavebeenintroduced. 172

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A B C D Figure6-56. Stabilitymodechangevstimechange:10degrees(|),20degrees()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[(),30degrees(---)A)2secondsB)5secondsC)10secondsD)20seconds Conversely,modesCandDdoexperiencetheeectsofinertiaandthereforeshowlargedierencesastimeprogresses.ObservationsofFigures 6-56 6-57 suggestthatastheallowablemorphingtimeisreduced,thedeviationincoincidentmodesshapesincrease.Thisobservationleadstothefollowingintuitiveresult;asthemorphingtimeisreduced,theoveralleectsofinertiaincrease,andthereforeproducelargersystemerrorswhicharepreviouslyunaccountedfor. ThepolesetscorrespondingtotheaforementionedstabilitymodesareshowninFigures 6-58 6-59 .Likethetrendshownwiththestabilitymodes,thepolesetsalsodeviateintimebaseduponthemoprhingrateorangle.Thisobservationisintuitivedue 173

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A B C Figure6-57. Stabilitymodechangevsdegreechange:20seconds(|),10seconds()-203()-204()]TJ /F1 11.955 Tf 32.77 0 Td[(),5seconds(---),2seconds()A)10degreesB)20degreesC)30degrees tothefactthatthepolesetsaredirectlycomputedfromthestabilitymodes.ItshouldbenotedthatinFigures 6-59 A6-59 B,thatsomepolesetsseemtoconvergeuponaconstantvalue.Thisconvergingactionisduetothefactthattherespectivepolesethasreachedtheendofitsmorphingmaneuverandhasindeedbecometime-invariant.Thesystem'stime-invariancedemonstratesthatthestateequationisT-periodicandthereforeavalidrepresentationofthetime-varyingpolesetcanbegivenbythesystem'sfrozentimeeigenvalue[ 86 ].AnotherimportantobservationderivedfromFigures 6-58 6-59 isthattwopolesetscreateahighfrequency,oppositephase,oscillatingpair,whiletheothertwopolesetscreatealowfrequency,oppositephase,oscillatingpair.Recallfromthe 174

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previouslyshownmechanicalsystemexamplethatanoppositephase,oscillatingpolepairisrepresentativeofdiscretecomplex-conjugatepolepair.Usingthisfact,itcanstatedthatthepolesetsgeneratedforthelongitudinalaircraftexamplearedirectlycomparabletotheclassicallydenedtimeinvariantphugoidandshortperioddynamicmodes. A B C D Figure6-58. Polechangevstimechange:10degrees(|),20degrees()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[(),30degrees(---)A)2secondsB)5secondsC)10secondsD)20seconds Togainabetterunderstandingforthedampingandfrequencyelementsoftheselongitudinalaircraftmodes,letusconsiderasinglecasewheretheaircraftmorphs30degreesin10seconds.Thecorrespondingpolesets,stabilitymodes,andorthonormalvectormatrixareshowninFigures 6-60 6-62 175

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A B C Figure6-59. Polechangevsdegreechange:20seconds(|),10seconds()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[(),5seconds(---),2seconds()A)10degreesB)20degreesC)30degrees ThepolesetsfoundinFigure 6-60 Aoscillatedirectlyaboutlineswhicharecontinuousrepresentationsofthefrozentimeeigenvalues.Aconclusion,similartothatfoundwiththemechanicalsystemexample,canbedrawnsuchthattheaverageoftheoppositephase,oscillatingpolesetpairscanbeusedtondthedampingofthedynamicmodalresponses.Figure 6-60 Aalsoillustratesaninitialtransientregionwheretwopolesetsswap.Thisregion,alongwiththeill-conditionedregionfortheshort-period-likepolesetsaredisregarded.ItcanbeseenfromFigure 6-60 A6-60 Bthattheill-conditionedregionfortheshort-period-likepolesetsdirectlycorrelatestothetime 176

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A B Figure6-60. Longitudinalaircraft30degree,10secondmorphA)polesets:polesets1and2()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[(),polesets3and4(---),time-invariantfrozentimeeigenvalues(|)B)stabilitymodes:mode1(|)mode2()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()mode3(---)mode4() thatthecorrespondingstabilitymodesconvergeuponzero.Thisisthesameintuitiveresultthatwasshownearlierinthemechanicalsystemexample. Animportantobservationtakenfromthepreviousmechanicalsystemexample,inSection 4.4 ,wasthatthedynamicmodalfrequenciescouldbestudiedbylookingattheperiodicityofthecolumnvectorsinQ.Thissameobservationcanbeappliedtothelongitudinalaircraftexample.Fromthisexample,itcanbeseenthatcolumnvectors1and2,asseeninFigure 6-61 ,describeresponseswithagreaterperiodicitythanthatfoundincolumnvectors3and4,asshowninFigure 6-62 Therelativedierencesinperiodicityonceagainsuggeststhatthereexistscomparabletraitsbetweenthecalculatedtime-varyingdynamicmodesandtheclassicallydenedtime-invariantphugoidandshort-perioddynamicmodes.Figures 6-62 A6-62 Drepresentthecolumnvectorsthatdescribethehighlydamped,higherfrequencytime-varyingpolesets,andlikethepolesets,thecolumnvectorsbecomeill-conditionedasthecorrespondingstabilitymodesconvergeuponzero.ThisresultisduetothefactthatthemultiplicationofQandRmustlinearlycombinetocreatethesimulatedstateresponses,asshowninFigure 6-63 177

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A B Figure6-61. LongitudinalaircraftQcolumnvectors1and2for30degree,10secondmorph:A)Q11(|)Q21()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()Q31(---)Q41()B)Q12(|)Q22()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()Q32(---)Q42() Thestatesforthisexampleareu(thebodyrelatedx-axisvelocity(m/s)),w(thebodyrelatedz-axisvelocity(m/s)),q(themeasuredpitchrate(deg/s)),and(themeasuredpitchangle(deg)),wheretheirrelativeresponsesareshowninFigures 6-63 A6-63 D.RecallthatthestateresponsesareunforcedresponseswiththeinitialconditionvectordescribedbyEquation 6{6 266666664u0w0q00377777775=26666666415000377777775(6{6) 6.6FeedforwardControl 6.6.1MorphingBasis Thepolynomialbasisusedtoconstrainthemass-springexample,showninSection 5.4.3 ,willalsobeusedasthebasisconstraintforthisexample.Recall,thatanymorphingtrajectoryisrestrictedtoathird-orderpolynomialoftimeandmustliewithinthesetofallpolynomialswithrealcoecients,asseeninEquation 5{11 178

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A B C D Figure6-62. LongitudinalaircraftQcolumnvectors3and4for30degree,10secondmorph:A,B)Q13(|)Q23()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()Q33(---)Q43()C,D)Q14(|)Q24()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()Q34(---)Q44() Thecoecientsofthemorphingtrajectoryareboundedsuchthattheaircraftneversweepsitswingspast30deg.Thehigher-ordertermsinthepolynomialhavethepotentialtodominateastimeincreases;consequently,thesetermsarerestrictedmorethanthelower-orderterms. Theprocedureofdeterminingtheinitialboundsactuallyndsthecoecientssequentiallybyconsideringonlyonecoecienttobeanon-zerovalueatatime.Indongso,thesolutiondeterminesthemaximumcontributionneededbyeachmorphingcoecienttoremainwithin30deg.ThisprocedurecanbeshowninEquation 6{7 179

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A B C D Figure6-63. LongitudinalaircraftstateresponsesA)state1(u)B)state2(w)C)state3(q)D)state4() iti=30 (max(t))ii=0;:::;3(6{7) Also,aconstraintisintroducedsuchthatj(t)j30forallvaluesoftime.Thisconstraintnotesthatthedesiredsignalisassumedtobeboundedandthusthesystemmustexperiencerealisticgeometrychange.Thecoecientsofeachpolynomialareadditionallylimitedtomaintainthisboundedcondition. TheresultingboundsonthecoecientsaregiveninTable 6-10 180

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A B C D Figure6-64. Longitudinalaircraftcolumneigenvectorsfor30degree,10secondmorph:A)V11(|)V21()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()V31(---)V41()B)V12(|)V22()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()V32(---)V42()C)V13(|)V23()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()V33(---)V43()D)V14(|)V24()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[()V34(---)V44() Table6-10. Boundsoncoecientofmorphingbasis CoecientUpperBoundLowerBound 030-3012-220.133-0.13330.009-0.009 181

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6.6.2TrackingASystemResponse 6.6.2.1Polynomialmorphing Asetofdesiredresponsesaregeneratedusingpolynomialmorphing.Thesepolynomialsrangeasarst-order,second-orderandthird-orderfunctions.TheoptimizedvaluesofthemorphingcommandsaregeneratedforfeedforwardcontrolandshowninTable 6-11 .TheoptimizedmorphingisessentiallyidenticaltothedesiredmorphingwhichisanticipatedsincethedesiredmorphingcanbeexactlyduplicatedbythemorphingbasisofEquation 5{11 Table6-11. Optimalpolynomialmorphingtotrackdesiredpolynomialmorphing desiredactual 5+1:5t5+1:5t1:8t)]TJ /F1 11.955 Tf 11.96 0 Td[(0:08t21:8t)]TJ /F1 11.955 Tf 11.96 0 Td[(0:08t27)]TJ /F1 11.955 Tf 11.95 0 Td[(0:25t)]TJ /F1 11.955 Tf 11.95 0 Td[(0:1t2+0:008t37)]TJ /F1 11.955 Tf 11.95 0 Td[(0:25t)]TJ /F1 11.955 Tf 11.95 0 Td[(0:1t2+0:008t3 Theoptimizedstateresponses,muchliketheoptimizedmorphingtrajectories,areessentiallyidenticaltothedesiredstateresponses.TheseresponsesareshowninFigure 6-65 Anotherdesiredresponseisattemptedtobetrackedusingthepolynomialmorphing;however,thisdesiredresponseisassociatedwithamorphingcomposedofasinusoidaladdedtoapolynomial.ThisdesiredmorphingcannotbeduplicatedusingtherestrictedbasisofEquation 5{11 sosomeerrormustresultinthetracking.TheoptimizedpolynomialformorphingisgiveninTable 6-12 alongwiththisdesiredmorphing. Table6-12. Optimalpolynomialmorphingtotrackdesiredsinusoidalmorphing desiredactual 0:75t+cos(t)0:7874+0:153t)]TJ /F1 11.955 Tf 11.95 0 Td[(0:0819t2+0:0025t3 TheresultingstateresponseisshowninFigure 6-66 asreasonablyclosetothedesireresponse.Someerrorispresentaroundthepeaks;however,thedesiredmorphingliescloseenoughtothetherangeofthemorphingbasissuchthatthefeedforwardcontrolisnearlyabletoprovidethecorrecttracking. 182

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A B C Figure6-65. DesiredResponse(|)andOptimalResponse()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[()forA)First-OrderMorphingandB)Second-OrderMorphingandC)Third-OrderMorphing 6.6.2.2Piecewise-polynomialmorphing Apiecewise-polynomialmorphingisagainconsideredtoexpandtherangeofthemorphingbasisseeninEquation 5{11 .Thesameconstraintsplacedonthepiecewise-polynomialtechniqueseeninthemass-springexampleapplyhere.Recall,themorphingbasisisallowedtobearst-orderpolynomialwhosecoecientscanvaryatdiscretepointsintime;whereitisalsorequiredthattheoverallmorphingbecontinuousthroughouttheentireresponse. Applyingthepiecewise-polynomialtechniquetothesinusoidalmorphingresponse,resultsintheoptimizedvaluesgiveninTable 6-13 183

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Figure6-66. DesiredResponse(|)andOptimalResponse()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[()forSinusoidalMorphing Table6-13. Optimalpiecewise-polynomialmorphingtotrackdesiredsinusoidalmorphing desiredactual 0:75t+cos(t)1:245)]TJ /F1 11.955 Tf 11.96 0 Td[(0:103t:0t2:973:693+0:852t:2:97
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Figure6-67. DesiredResponse(|)andOptimalResponse()-221()-223()]TJ /F1 11.955 Tf 33.2 0 Td[()forSinusoidalMorphing showninTable 6-14 .Itisagainseenthattheoptimizedmorphingisessentiallyidenticaltothedesiredmorphing. Table6-14. Optimalpolynomialmorphingtotrackdesiredpolynomialmorphing desiredactual 8+0:75t8+0:75t)]TJ /F1 11.955 Tf 9.3 0 Td[(3+1:25t+0:02t2)]TJ /F1 11.955 Tf 9.3 0 Td[(3+1:25t+0:02t2)]TJ /F1 11.955 Tf 9.3 0 Td[(5+2t)]TJ /F1 11.955 Tf 11.95 0 Td[(0:1t2+0:008t3)]TJ /F1 11.955 Tf 9.3 0 Td[(5+2t)]TJ /F1 11.955 Tf 11.96 0 Td[(0:1t2+0:008t3 Theoptimizedpolesignalsareessentiallyidenticaltothedesiredpolesignals.ThesesignalsareshowninFigure 6-68 Anotherdesiredsignalisattemptedtobetrackedusingthepolynomialmorphing.Thisdesiredsignalisagainassociatedwithamorphingcomposedofasinusoidaladdedtoapolynomial.Forsimplicity,itremainsthesamedesiredsinusoidalmorphingasthatusedforthestateresponse.TheoptimizedpolynomialformorphingisgiveninTable 6-15 alongwiththisdesiredmorphing. Table6-15. Optimalpolynomialmorphingtotrackdesiredsinusoidalmorphing desiredactual 0:75t+cos(t)1:007+0:078t+0:0736t2)]TJ /F1 11.955 Tf 11.96 0 Td[(0:0089t3 185

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A B C Figure6-68. A)ResponsetoMorphingTrajectory:Desired(|),Optimized()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()B)ResponsetoMorphingTrajectory:Desired(|),Optimized()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[()C)ResponsetoMorphingTrajectory:Desired(|),Optimized()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() Althoughusingthesamedesiredmorphing,theoptimizedpolynomialinTable 6-15 isdierentfromthatinTable 5-4 .Thisresult,likethatseeninthemass-springexample,issimplyduetotheoptimizationofadierentcostfunction. TheresultingpolesignalisshowninFigure 6-69 .Someerrorispresent;however,thefeedforwardcontrolisagainnearlyabletoprovidethecorrecttracking. 6.6.3.2Piecewise-polynomialmorphing Applyingthepiecewise-polynomialtechniquetothesinusoidalmorphingresponse,resultsintheoptimizedvaluesgiveninTable 6-16 186

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Figure6-69. ResponsetoMorphingTrajectory:Desired(|)andOptimized()-222()-222()]TJ /F1 11.955 Tf 33.21 0 Td[() Table6-16. Optimalpiecewise-polynomialmorphingtotrackdesiredsinusoidalmorphing desiredactual 0:94t+sin(t)1:218)]TJ /F1 11.955 Tf 11.96 0 Td[(0:047t:0t2:979:197)]TJ /F1 11.955 Tf 11.96 0 Td[(0:660t:2:97
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CHAPTER7CONCLUSION Thispaperinvestigatestheeectsofwingsweepontheightcharacteristicsforaminiatureairvehicle.Inparticular,thevariationsadmittedbyamulti-jointmechanismarestudied.Suchamechanismallowsindependentchoiceofinboardandoutboardforeachwing.Thisvehicleisactuallyincorporatingabiologically-inspiredapproachthatnotestherotationsofshouldersandelbowsbygulls.Theassociatedaerodynamicsvaryinbothforceandmomentandthewingsweeprangesacrosssymmetricandasymmetriccongurations.Theresultingightdynamicsdemonstratetheasymmetricsweepisbenecialtomaintainingsensorpointingdespitecrosswinds;consequently,thebiomimeticdesignhasenhancedmissioneectivenessforthisclassofvehicles. Theequationsofmorphingarederivedforamorphingaircraftandshowntoincludetermsassociatedwithasymmetriesandtime-varyinginertias.Acriticaleectoftheseelementstermsisacouplingbetweenthelongitudinalandlateral-directionaldynamics.Essentially,thesetermsindicatethatthetransientbehaviorofmorphingcanbesignicantlydierentthanitssteady-statebehavior.Simulationsoftheclosed-loopbehaviorsforavariablewing-sweepaircraft(bothindivingandturning)indicatetheadverseinuencethatsuchtransientbehaviorcanpotentiallyhaveonmissionperformance.Inbothcases,theresultinginertiasproducedduringthetransientresponseresultedinrespectivelongitudinalandlateraltranslationsthatcausedthevehicletomissitstarget. Thispaperalsopresentsndingsbasedonobservationstakenfromparametercomputations[ 86 ]fortwodierentlineartime-varyingsystems.Comparisonsbetweenthetwotime-varyingexamplesaremade,fromwhichconclusionsaredrawnonhowtosolveforanddescribeasystem'stime-varyingdynamicmodes.Specically,itisshownthatforaparticularmorphingaircraftmaneuver,thelongitudinaltime-varyingparametersresembletheclassicallydenedtime-invariantphugoidandshort-periodcharacteristics. 188

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Thedampingandfrequencyrelatedtothetime-varyingdynamicmodesarefoundtoberelatedtotheaverageoftwooppositephase,oscillatingpolesetsandcolumnvectorperiodicityoftheorthonormalQmatrix,respectively.Itshouldbenotedthatduetothenatureofthedynamicmodesderivedinthispaper,onlyoscillating,complex-conjugatepolesandmodeswerestudied. Lastly,thispaperpresentsndingsbasedonamorphingcontroloftwodierentlineartime-varyingsystems.Comparisonsbetweenthetwotime-varyingexamplesaremade,fromwhichconclusionsaredrawnforthetrackingcapablilityofafeedforwarddesign.Specically,itisshownforparticularaircraftmorphingtrajectories,thatfeedforwardtrackingcanbeachievedondesiredstateresponseandpolevaluesignals.Itisalsoshownthattrackingmaybeperformedusingpiecewise-polynomialtechniques.Currently,feedforwardtrackingispreformedo-lineusingpredenedsignalsasdesiredinput. Futureworkwillconsistofstudyingthelateral-directionalaircraftmodel,fromwhichdenitionsforbothpurelyrealandcomplex-conjugatepolesandmodeswillbediscussed.Also,futureworkwillconsistofimplementingthepreviouslydenedfeedforwardcontrollerinsomeformofmulti-elementfeedbackcontroller.Feedbackwillbeutilizedforpurposesofsignalgeneration,noisecompensationandrobustness. 189

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BIOGRAPHICALSKETCH DanielThurmondGrantwasborninGainesville,Floridain1983.Asachild,heoftenfoundhimselfintriguedbythemechanicaloperationofremote-controlledtoys.Whennotlostinapileofnutsandbolts,hefoundenjoymentinbuildingmodelairplaneswithhisfather.Thecombinationofthesetwohobbieseventuallyledhimtodevelopastrongpassionforaeronauticalengineering.DanielgraduatedfromSantaFeHighSchoolin2002,afterwhichheenrolledattheUniversityofFlorida.WhileanundergraduateattheUniversityofFlorida,DanieljoinedtheMicro-AerialVehicle(MAV)Laboratory,wherehelearnedhowtobuildandysmallUnmannedAerialVehicles(UAVs).Thishands-onexperienceallowedDanieltoco-leadaDesign,Build,Fly(DBF)teamthatplacedninthataninternationalAIAAstudentcompetitionheldinWichita,Kansas.Asasenior,DanieljoinedtheFlightControlLabattheUniversityofFlorida,wherehedesigned,builtandewamulti-jointed,wing-sweepingMAV.Upongraduatinginthefallof2006,DanielenrolledintograduateschoolattheUniversityofFlorida.Danielreceivedhismaster'sdegreeinthespringof2009andthenhisdoctorateinthespringof2011.BothdegreesDanielearnedwereinaerospaceengineeringwithaprimaryfocusonadvancedightcontrolsanddynamics.Throughouthiscollegiatecareer,Danielexperiencedamultitudeofhighlightingevents.Theseeventsincludedbutnotlimitedto: WinningthetheAtmosphericFlightMechanicsstudentbestpaperawardatthe2006Guidance,Navigation,andControlConferenceinKeystone,Colorado TravelingtoWashingtonD.C.topresenthisworkforCongressduringtheeveningofaPresidentialStateoftheUnionAddress ObtainingaU.S.patentforhisworkonmorphingandurbancamouage ReceivingtheopportunitytoincludehisworkinandauthorabookchapterpublishedfortheAIAAeducationalseries FeaturinghisworkinnumerousdocumenturiesincludingNationalGeogrpahic,theScienceChannel,andSirDavidAttenborough'sim,"FlyingMonsters3D" 200