System Identification and Trajectory Optimization for Guided Store Separation

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Title:
System Identification and Trajectory Optimization for Guided Store Separation
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1 online resource (275 p.)
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english
Creator:
Carter, Ryan Elliot
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University of Florida
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Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Aerospace Engineering, Mechanical and Aerospace Engineering
Committee Chair:
Lind, Richard C
Committee Members:
Ukeiley, Lawrence S.
Barooah, Prabir
Boginski, Vladimir L.

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Subjects / Keywords:
autopilot -- control -- identification -- neighboring -- optimal -- optimization -- separation -- store -- system -- trajectory
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
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Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract:
Combat aircraft utilize expendable stores such as missiles, bombs, flares, and external tanks to execute their missions.  Safe and acceptable separation of these stores from the parent aircraft is essential for meeting the mission objectives.  In many cases, the employed missile or bomb includes an onboard guidance and control system to enable precise engagement of the selected target.  Due to potential interference, the guidance and control system is usually not activated until the store is sufficiently far away from the aircraft.  This delay may result in large perturbations from the desired flight attitude caused by separation transients, significantly reducing the effectiveness of the store and jeopardizing mission objectives.  The purpose of this research is to investigate the use of a transitional control system to guide the store during separation.  The transitional control system, or "store separation autopilot", explicitly accounts for the nonuniform flow field through characterization of the spatially variant aerodynamics of the store during separation.  This approach can be used to mitigate aircraft-store interference and leverage aerodynamic interaction to improve separation characteristics.  This investigation proceeds in three phases.  First, system identification is used to determine a parametric model for the spatially variant aerodynamics.  Second, the store separation problem is recast into a trajectory optimization problem, and optimal control theory is used to establish a framework for designing a suitable reference trajectory with explicit dependence on the spatially variant aerodynamics.  Third, neighboring optimal control is used to construct a linear-optimal feedback controller for correcting deviations from the nominal reference trajectory due varying initial conditions, modeling errors, and flowfield perturbations.  An extended case study based on actual wind tunnel and flight test measurements is used throughout to illustrate the effectiveness of the approach and to highlight the anticipated benefits of guided store separation.
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In the series University of Florida Digital Collections.
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Includes vita.
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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 Ryan Elliot Carter.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Lind, Richard C.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-08-31

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UFE0044483:00001


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SYSTEMIDENTIFICATIONANDTRAJECTORYOPTIMIZATIONFOR GUIDEDSTORESEPARATION By RYANE.CARTER ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2012

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c r 2012RyanE.Carter 2

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DedicatedtomywifeMelanieandmyvechildrenEthan,Isaac ,Levi,Micah,andRose, fortheirunconditionallove,encouragement,andsupport. Tomychildren:Mayyoube blessedwiththeopportunitytofulllyourdreamsasIhavem ine. 3

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ACKNOWLEDGMENTS IfIhaveseenfurther,itisbystandingupontheshouldersof giants. —SirIsaacNewton, LettertoRobertHooke,February1676 Thoughindividualsoftenreceivethecreditforsignicant accomplishments,itis usuallyacollaborativeeffortwithmanycontributionsmad ebehindthescenes.Suchis thecaseforthiswork. IwouldliketothanktheAirForceSEEKEAGLEOfce(AFSEO)fo rsponsoring thisresearchandacademicendeavor.TheAFSEOhasprovided achallenging andsatisfyingworkenvironmentformanyyearsandIlookfor wardtomoretocome. Additionally,IamappreciativetheScience,Mathematics, andResearchforTransformation (SMART)ScholarshipProgramwhichprovidedthemajorityof thenancialresourcesto conductthisresearch.IwouldespeciallyliketothankDr.R ichardLindforhisguidance andadvicethroughoutmydoctoralstudies. Iwouldalsoliketothankmyfriendsandfamilyfortheirsupp ortandencouragement. MywifeMelanieandourvechildren(Ethan,Isaac,Levi,Mic ah,andRose)haveplayed asignicantroleinmyacademicachievementsbyprovidinga livelyandourishing homelife.Myfatherisoftenmysourceofmotivationandmyde eprespectforhim hasledtomanyachievementsthatIwouldhaveotherwiseneve rattempted.Most signicantly,IamthankfulformyrelationshipwithJesusC hrist,theauthorandnisher ofmyfaith. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 9 LISTOFFIGURES ..................................... 10 ABSTRACT ......................................... 16 CHAPTER 1BACKGROUNDANDINTRODUCTION ...................... 18 1.1Motivation .................................... 18 1.2ProblemDescription .............................. 19 1.3ResearchObjectives .............................. 20 1.4RepresentativeCaseStudy .......................... 23 1.5Contributions .................................. 24 2MATHEMATICALMODELINGOFSTORESEPARATION ............ 27 2.1Overview .................................... 27 2.2StoreSeparationEquationsofMotion .................... 28 2.2.1ReferenceFramesandCoordinateSystems ............. 29 2.2.2CoordinateFreeEquationsofMotion ................. 37 2.2.3StandardBody-AxisEquationsofMotion ............... 40 2.2.3.1Translationaldynamics ................... 40 2.2.3.2Translationalkinematics ................... 46 2.2.3.3Rotationaldynamics ..................... 47 2.2.3.4Rotationalkinematics .................... 48 2.2.3.5Collectedbody-axisequationsofmotion .......... 49 2.2.4Wind-AxisEquations .......................... 51 2.2.4.1Ancillaryequations ...................... 52 2.2.4.2Wind-axisequationsofmotion ............... 54 2.2.5PositionandVelocityoftheStoreRelativetotheAirc raft ...... 58 2.2.5.1Straightandlevelight ................... 60 2.2.5.2Steadyclimbordive ..................... 60 2.2.5.3Constantloadfactormaneuver ............... 61 2.3AerodynamicModeling ............................. 63 2.3.1AerodynamicCoefcients ....................... 63 2.3.2Delta-CoefcientMethodology ..................... 65 2.3.3RepresentativeCaseStudy ...................... 70 2.3.3.1Freestreamdata ....................... 70 2.3.3.2Griddata ........................... 72 2.4FlightTestValidation .............................. 75 5

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2.4.1FlightTestDataReduction ....................... 76 2.4.1.1Trajectoryreconstruction .................. 76 2.4.1.2Trajectoryanalysis ...................... 79 2.4.2FlightTestResults ........................... 80 2.5ChapterSummary ............................... 85 3SYSTEMIDENTIFICATION ............................. 86 3.1Overview .................................... 86 3.1.1SystemIdentication .......................... 86 3.1.2FlightVehicleSystemIdentication .................. 87 3.1.3StoreSeparationSystemIdentication ................ 90 3.2IdenticationMethods ............................. 91 3.2.1InputDesign ............................... 92 3.2.2ModelStructureDetermination .................... 95 3.2.3ParameterEstimation ......................... 101 3.2.4ModelPostulation ............................ 104 3.2.4.1Uniformowcontribution .................. 105 3.2.4.2Non-uniformowcontribution ................ 106 3.2.4.3Spatialvariation ....................... 106 3.3Example:StoreSeparation .......................... 111 3.3.1FreestreamSystemIdentication ................... 111 3.3.1.1Simulatedmaneuver ..................... 111 3.3.1.2Modelvalidation ....................... 114 3.3.2SpatiallyVariantSystemIdentication ................ 116 3.3.2.1Piecewise-continuousmaneuver .............. 116 3.3.2.2Modelvalidation ....................... 120 3.3.3FlightTestComparison ......................... 120 3.3.3.1Trajectorycomparison .................... 121 3.3.3.2Aerodynamiccomparison .................. 121 3.4Example:PlanarStoreSeparation ...................... 123 3.5ChapterSummary ............................... 128 4TRAJECTORYOPTIMIZATION ........................... 130 4.1Overview .................................... 130 4.2OptimalControl ................................. 132 4.2.1FirstOrderOptimalityConditions ................... 132 4.2.2InterpretationoftheCostate ...................... 137 4.2.3InterpretationoftheHamiltonian ................... 137 4.2.4LinearQuadraticRegulator ...................... 138 4.2.5NumericalMethods ........................... 140 4.3OptimalStoreSeparation ........................... 142 4.3.1PerformanceIndex ........................... 142 4.3.2FirstOrderOptimalityConditions ................... 144 4.3.3Example:PlanarStoreSeparation .................. 148 6

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4.3.3.1Modelequations ....................... 148 4.3.3.2Aerodynamicmodel ..................... 149 4.3.3.3Optimalityconditions .................... 151 4.3.3.4Results ............................ 151 4.4ChapterSummary ............................... 157 5NEIGHBORINGOPTIMALCONTROL ....................... 158 5.1Overview .................................... 158 5.2NeighboringOptimalControl ......................... 159 5.2.1SecondOrderOptimalityConditions ................. 160 5.2.2NeighboringExtremal ......................... 164 5.2.3NeighboringExtremalwithTerminalConstraints ........... 167 5.2.4NeighboringExtremalwithPath/ControlConstraints ........ 171 5.2.5NeighboringExtremalwithParameterVariations .......... 173 5.2.6SufcientConditionsforOptimality .................. 176 5.3StoreSeparationAutopilot ........................... 177 5.3.1FeedbackUsingNeighboringOptimalControl ............ 178 5.3.2InniteHorizonNeighboringOptimalControl ............. 180 5.4Example:PlanarStoreSeparation ...................... 182 5.4.1ModelEquations ............................ 182 5.4.2NeighboringOptimalControl ..................... 184 5.4.3NeighboringOptimalControlwithInequalityConstra ints ...... 190 5.4.4NeighboringOptimalControlwithTerminalCost ........... 196 5.4.5NeighboringOptimalControlwithTerminalConstrain ts ....... 198 5.4.6InniteHorizonNeighboringOptimalControl ............. 201 5.4.6.1Responsetoowelddisturbances ............ 205 5.4.6.2Responsetoparametervariations ............. 211 5.5ChapterSummary ............................... 214 6GUIDEDSTORESEPARATION .......................... 216 6.1Overview .................................... 216 6.2TrajectoryOptimization ............................. 217 6.2.1OptimalControl ............................. 218 6.2.1.1Problemstatement ...................... 218 6.2.1.2Optimaltrajectory ...................... 220 6.2.2FeedbackControl ............................ 226 6.2.2.1Problemstatement ...................... 228 6.2.2.2Neighboringoptimaltrajectory ............... 229 6.2.2.3Responsetovaryinginitialconditions ........... 232 6.2.2.4Responsetorandomdisturbances ............. 235 6.2.2.5Responsetoparametervariations ............. 237 6.3FlightTestComparison ............................ 240 6.3.1SubsonicFlightTest .......................... 241 6.3.2SupersonicFlightTest ......................... 245 7

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6.4ChapterSummary ............................... 250 7CONCLUSIONS ................................... 251 7.1Summary .................................... 251 7.2Contributions .................................. 255 7.3FutureWork ................................... 257 7.3.1SystemIdentication .......................... 258 7.3.2TrajectoryOptimization ......................... 259 7.3.3FeedbackControl ............................ 259 7.4ConcludingRemarks .............................. 260 REFERENCES ....................................... 262 BIOGRAPHICALSKETCH ................................ 275 8

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LISTOFTABLES Table page 3-1Parametersusedinplanarstoreseparationaerodynamic model. ........ 127 4-1Aerodynamicderivatives,controlderivatives,andspa tiallyvariantparameters usedinplanarstoreseparationaerodynamicmodel. ............... 150 6-1FactorsforParametricAnalysis ........................... 243 9

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LISTOFFIGURES Figure page 1-1Relationshipbetweensystemidentication,trajector yoptimization,andfeedback controlforguidedstoreseparation. ......................... 23 1-2F-16releaseofarepresentativeguidedmunition. ................. 23 2-1Denitionofaircraftaxiscoordinatesystem. .................... 32 2-2Denitionofstorebodyaxiscoordinatesystem. .................. 32 2-3Denitionofearth,inertial,andightaxes. ..................... 35 2-4Denitionofstorewindaxiscoordinatesystem. .................. 36 2-5Earth-to-inertialaxistransformation. ........................ 43 2-6Freestreamaerodynamiccoefcientsvs.angleofattack atxedsideslipangle forarepresentative 1 = 20 th scalemodelatMach0.8. ............... 71 2-7PitchingandYawingmomentcoefcientvs.angleofattac kforarepresentative 1 = 20 th scalemodelatMach0.8forfullrangeofangleofattackandsi deslip angle. ......................................... 72 2-8DualsupportmechanismforF-16storeseparationwindtu nneltest.Excerpt fromAEDC-TR-09-F-19[1]. ............................ 73 2-9Aerodynamicpitchingmomentandyawingmomentdeltacoe fcientsvs.vertical storepositionforvariouspitchandyawangles,forarepres entative 1 = 20 th scaledmodelatMach0.8. .............................. 74 2-10Aerodynamicpitchingmomentandyawingmomentdeltaco efcientsvs.vertical storepositionforvariouspitchandyawangles,forarepres entative 1 = 20 th scaledmodelatMach1.2. ............................. 75 2-11Measured6DOFtelemetrydataforF-16SeparationFligh tTest4535(Mach 1.2/600KCAS). ................................... 80 2-12Reconstructedighttesttrajectorycomparisonwithw indtunnelbasedsimulation forF-16SeparationFlightTest4535(Mach1.2/600KCAS). .......... 81 2-13Reconstructedighttesttrajectorycomparisonwithw indtunnelbasedsimulation forF-16SeparationFlightTest2265(Mach0.9/550KCAS). .......... 82 2-14Visualcomparisonofighttestandwindtunnelbasedsi mulationtrajectories forF-16SeparationFlightTest4535(Mach1.2/600KCAS). .......... 83 2-15Comparisonofighttestandwindtunnelaerodynamicco efcientsforF-16 SeparationFlightTest4535(Mach1.2/600KCAS). ............... 84 10

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2-16Comparisonofighttestandwindtunnelaerodynamicco efcientsforF-16 SeparationFlightTest2265(Mach0.9/550KCAS). ............... 84 3-1Multisineexcitationfortwoorthogonalinputsfrom0to 5Hzwithamplitude rangefrom+/-5. ................................... 96 3-2RegressormapforthemultisineinputsshowninFigure31. .......... 96 3-3Denitionofatrainingmaneuverwithmultisineinputsf or(top)air-incidence anglesand(bottom)angularratescomputedusingkinematic relationships. .. 112 3-4Aerodynamiccoefcientcomparisonbetweensimulation andsystemidentication resultsforastoreinfreestreamightconditionsatMach0. 9/550KCAS/ 4800ft. ........................................ 113 3-5Staticaerodynamiccoefcientcomparisonbetweeninte rpolatedwindtunnel data(solidlines)andsystemidenticationresults(dashe dlines)forastorein freestreamightconditionsatMach0.9/550KCAS/4800ft. .......... 115 3-6Validationresultsshowinganindependentmaneuverand aerodynamiccoefcient comparisonbetweeninterpolatedwindtunneldata(solidli nes)andsystem identicationresults(dashedlines)forfreestreamight conditionsatMach 0.9/550KCAS/4800ft. .............................. 115 3-7Comparisonofsimulation(solid)andsystemidenticat ion(dashed)results foraerodynamicpitchingmomentandnormalforcedeltacoef cientsat(a) z=0ft,(b)z=5ft,and(c)z=10ft. ........................... 117 3-8SpatialvariationofmodelparametersatMach0.9/550KC AS/4800ft. .... 119 3-9SpatialvariationofmodelparametersatMach0.9/550KC AS/4800ft. .... 120 3-10Validationresultsshowingpitchingmomentandnormal forcedeltacoefcient comparisonbetweensimulationandsystemidenticationfo ranindependent maneuver. ....................................... 121 3-11Trajectorycomparisonbetweenighttest,convention alsimulation,andsimulation withsystemidentication. .............................. 122 3-12Aerodynamiccomparisonbetweenighttest,conventio nalsimulation,and simulationwithsystemidentication. ........................ 122 3-13Angleofattack,pitchrate,andelevatorinputforplan arsystemidentication. 124 3-14Planaraerodynamiccoefcientcomparisonbetweensim ulationandsystem identicationforastoreinfreestreamightconditionsat Mach0.8/10kft. ... 125 3-15Planaraerodynamiccontroleffectcomparisonbetween simulationandsystem identicationforastoreinfreestreamightconditionsat Mach0.8/10kft. ... 126 11

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3-16Pitchingmomentdeltacoefcientforsimpliedparame tricmodelusingsystem identication. ..................................... 127 4-1Conceptualtrajectoriesdemonstratingsafetyandacce ptabilitycriteria. ..... 143 4-2Anextremaltrajectoryforplanarstoreseparationwith weightingfactorsselected tominimizeangleofattack. ............................. 152 4-3Anextremaltrajectoryforplanarstoreseparationwith weightingfactorsselected tominimizepitchrate. ................................ 154 4-4Seriesofneighboringextremaltrajectoriesforvaried initialpitchrates. .... 155 4-5Seriesofneighboringoptimaltrajectoriesforvariedi nitialangleofattack. ... 156 5-1NeighboringOptimalControlblockdiagram. ................... 166 5-2Optimaltrajectorywithneighboringoptimalfeedbackc ontrol. .......... 185 5-3SolutiontoRiccatidifferentialequation. ...................... 185 5-4Neighboringoptimalfeedbackgains. ....................... 186 5-5Optimalandneighboringoptimaltrajectoriesforvaryi nginitialpitchrate. ... 187 5-6Optimalandneighboringoptimaltrajectoriesforvaryi nginitialangleofattack. 187 5-7Optimalandneighboringoptimaltrajectoriesforvaryi nginitialpitchrateand initialangleofattack. ................................ 188 5-8Optimalandneighboringoptimaltrajectoriesforlarge perturbationsininitial angleofattack. ................................... 189 5-9Optimalandneighboringoptimaltrajectorieswithcons trainedelevatordeection. ............................................. 192 5-10Constraintmultiplierforoptimalandneighboringopt imaltrajectorieswithconstrained elevatordeection. ................................. 193 5-11Optimalandneighboringoptimaltrajectorieswithcon strainedelevatordeection forvaryinginitialpitchrate. ............................. 193 5-12Constraintmultiplierforoptimalandneighboringopt imaltrajectorieswithconstrained elevatordeectionforvaryinginitialpitchrate. .................. 194 5-13Optimalandneighboringoptimaltrajectorieswithcon strainedelevatordeection forvaryinginitialpitchrate. ............................. 195 5-14Optimalandneighboringoptimaltrajectorieswithcon strainedelevatordeection forextremelyadverseinitialconditions. ...................... 196 12

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5-15Optimalandneighboringoptimaltrajectorieswithter minalcost. ........ 197 5-16SolutiontoRiccatiequationandfeedbackgainsfornei ghboringoptimaltrajectories withterminalcost. .................................. 198 5-17Optimalandneighboringoptimaltrajectorieswithcum ulativeandterminal cost. ......................................... 199 5-18Optimalandneighboringoptimaltrajectorieswithter minalconstraints. .... 199 5-19Solutiontodifferentialequationsforneighboringop timaltrajectorieswithterminal constraints. ...................................... 200 5-20Optimalandneighboringoptimaltrajectoriesforvary inginitialpitchrate. ... 202 5-21SolutiontoRiccatiequationandfeedbackgainsfornei ghboringoptimaltrajectories. ............................................. 203 5-22Optimalandneighboringoptimaltrajectoriesforvary inginitialpitchrate,extended beyond t = t f .................................... 204 5-23Optimalandneighboringoptimaltrajectoriesforvary inginitialpitchrate,with additionalcoston q ( t ) ............................... 205 5-24Optimalandneighboringoptimaltrajectoriesforvary inginitialpitchrateand initialangleofattack. ................................. 206 5-25Aerodynamiccoefcientsestimatedfromighttestdat a. ............. 207 5-26Optimalandneighboringoptimaltrajectorieswithran domdisturbancesrepresentative ofaerodynamicturbulence. ............................. 208 5-27Optimalandneighboringoptimaltrajectorieswithamp liedrandomdisturbances representativeofaerodynamicturbulence. ..................... 209 5-28Non-stationarysignalrepresentativeofaturbulentw indgusteffectonpitching moment. ........................................ 210 5-29Optimalandneighboringoptimaltrajectorieswithamp liedturbulentwindgusts. 210 5-30Unguidedtrajectorieswithparametervariations. ................. 212 5-31Guidedtrajectorieswithparametervariations. ................... 213 6-1Relationshipbetweensystemidentication,trajector yoptimization,andfeedback controltoappliedforguidedstoreseparation. ................... 216 6-2Optimaltrajectoryforratecapture.Initialcondition sandightconditionsare basedonighttest2265(Mach0.9/550KCAS/4800ft). ............ 221 13

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6-3Optimaltrajectoryforangle-of-attackcapture.Initi alconditionsandightconditions arebasedonighttest2265(Mach0.9/550KCAS/4800ft). ......... 222 6-4Optimaltrajectoriesforvaryinginitialpitchrate.Fl ightconditionsarebased onighttest2265(Mach0.9/550KCAS/4800ft). ............... 223 6-5Optimaltrajectoriesforvaryinginitialyawrate.Flig htconditionsarebasedon ighttest2265(Mach0.9/550KCAS/4800ft). ................. 224 6-6Optimaltrajectoryforangle-of-attackcapture.Initi alconditionsandightconditions arebasedonFlightTestMission4535(Mach1.2/600KCAS/18k ft). ..... 225 6-7Optimaltrajectoriesforvaryinginitialpitchandyawr ate.Flightconditionsare basedonFlightTestMission4535(Mach1.2/600KCAS/18kft) ....... 226 6-8Comparisonofoptimal(guided)andighttest(unguided )trajectoriesforsubsonic andsupersonicightconditions. .......................... 227 6-9NeighboringOptimalControlblockdiagram. ................... 229 6-10Optimaltrajectoryandextendedneighboringoptimalt rajectoryformission 2265(Mach0.9/550KCAS/4800ft). ....................... 230 6-11TimevaryingfeedbackcontrolgainsandRiccatisoluti onforMission2265 (Mach0.9/550KCAS/4800ft). .......................... 231 6-12Optimaltrajectoryandextendedneighboringoptimalt rajectoryformission 4535(Mach1.2/600KCAS/18kft). ........................ 232 6-13TimevaryingfeedbackcontrolgainsandRiccatisoluti onformission4535 (Mach1.2/600KCAS/18kft). ........................... 233 6-14Optimalandneighboringoptimaltrajectorieswithvar yinginitialratesformission 2265(Mach0.9/550KCAS/4800ft). ....................... 234 6-15Optimalandneighboringoptimaltrajectorieswithvar yinginitialincidenceangles formission2265(Mach0.9/550KCAS/4800ft). ................ 235 6-16Optimalandneighboringoptimaltrajectorieswithvar yinginitialconditionsfor mission4535(Mach1.2/600KCAS/18kft). ................... 236 6-17Aerodynamiccoefcientsestimatedfromighttesttel emetrydataformission 2265(Mach0.9/550KCAS/4800ft). ....................... 237 6-18Optimaltrajectoryandneighboringoptimaltrajector yresponsetorandomdisturbances formission2265(Mach0.9/550KCAS/4800ft). ................ 238 6-19Optimaltrajectoryandneighboringoptimaltrajector yresponsetoamplied randomdisturbancesformission2265(Mach0.9/550KCAS/48 00ft). .... 238 14

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6-20Unguidedtrajectorieswithparametervariations. ................. 239 6-21Guidedtrajectorieswithparametervariations. ................... 240 6-22Comparisonofoptimaltrajectoryandneighboringopti malwindaxissimulations formission2265(Mach0.9/4800ft/550KCAS). ................ 242 6-23Comparisonofoptimaltrajectoryandneighboringopti malwindaxissimulations withvaryinginitialconditionsformission2265(Mach0.9/ 4800ft/550KCAS). 243 6-24Parametricanalysis(incidenceangles)forjettisona ndguidedstoreseparation correspondingtoighttest2265(Mach0.9/4800ft/550KCAS ). ....... 245 6-25Parametricanalysis(verticalvelocityandtranslati on)forjettisonandguided storeseparationcorrespondingtoighttest2265(Mach0.9 /4800ft/550 KCAS). ........................................ 246 6-26Parametricanalysis(pitch)forjettisonandguidedst oreseparationcorresponding toighttest2265(Mach0.9/4800ft/550KCAS). ................ 246 6-27Parametricanalysis(yaw)forjettisonandguidedstor eseparationcorresponding toighttest2265(Mach0.9/4800ft/550KCAS). ................ 247 6-28Comparisonofoptimaltrajectoryandneighboringopti malwindaxissimulations formission4535(Mach1.2/600KCAS/18kft). ................. 248 6-29Comparisonofoptimaltrajectoryandneighboringopti malwindaxissimulations withvaryinginitialconditionsformission4535(Mach1.2/ 600KCAS/18kft). 248 6-30Parametricanalysis(incidenceangles)forjettisona ndguidedstoreseparation correspondingtoighttest4535(Mach1.2/600KCAS/18kft) ........ 249 6-31Parametricanalysis(pitch)forjettisonandguidedst oreseparationcorresponding toighttest4535(Mach1.2/600KCAS/18kft). ................. 249 15

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy SYSTEMIDENTIFICATIONANDTRAJECTORYOPTIMIZATIONFOR GUIDEDSTORESEPARATION By RyanE.Carter August2012 Chair:RichardC.LindMajor:MechanicalandAerospaceEngineering Combataircraftutilizeexpendablestoressuchasmissiles ,bombs,ares,and externaltankstoexecutetheirmissions.Safeandacceptab leseparationofthesestores fromtheparentaircraftisessentialformeetingthemissio nobjectives.Inmanycases, theemployedmissileorbombincludesanonboardguidancean dcontrolsystemto enablepreciseengagementoftheselectedtarget.Duetopot entialinterference,the guidanceandcontrolsystemisusuallynotactivateduntilt hestoreissufcientlyfar awayfromtheaircraft.Thisdelaymayresultinlargepertur bationsfromthedesired ightattitudecausedbyseparationtransients,signican tlyreducingtheeffectiveness ofthestoreandjeopardizingmissionobjectives.Thepurpo seofthisresearchisto investigatetheuseofatransitionalcontrolsystemtoguid ethestoreduringseparation. Thetransitionalcontrolsystem,or“storeseparationauto pilot”,explicitlyaccountsfor thenonuniformoweldthroughcharacterizationofthespa tiallyvariantaerodynamics ofthestoreduringseparation.Thisapproachcanbeusedtom itigateaircraft-store interferenceandleverageaerodynamicinteractiontoimpr oveseparationcharacteristics. Thisinvestigationproceedsinthreephases.First,system identicationisusedto determineaparametricmodelforthespatiallyvariantaero dynamics.Second,thestore separationproblemisrecastintoatrajectoryoptimizatio nproblem,andoptimalcontrol theoryisusedtoestablishaframeworkfordesigningasuita blereferencetrajectorywith explicitdependenceonthespatiallyvariantaerodynamics .Third,neighboringoptimal 16

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controlisusedtoconstructalinear-optimalfeedbackcont rollerforcorrectingdeviations fromthenominalreferencetrajectoryduevaryinginitialc onditions,modelingerrors,and oweldperturbations.Anextendedcasestudybasedonactu alwindtunnelandight testmeasurementsisusedthroughouttoillustratetheeffe ctivenessoftheapproachand tohighlighttheanticipatedbenetsofguidedstoresepara tion. 17

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CHAPTER1 BACKGROUNDANDINTRODUCTION 1.1Motivation Tacticalghterandbomberaircrafthavebeenusedtocarrya nddeliverordinance sinceshortlyafterthedawnofaviation.Intheearlieststa gesofaircombat,separation ofstoresfromtheparentaircraftwasoflittleconcern.How ever,duringtheVietnamWar, theemploymentofheavystoresfromlargerjet-poweredairc raft,suchastheMcDonnell DouglasF-4Phantom,begantopresentdifcultiesforaircr aft-storecompatibility. Specically,scenariosinhigh-speedightwereencounter edwherethereleasedstore failedtoseparatecleanlyfromtheaircraftandinsteadbec ameaprojectilethreatening theaircraftandonoccasionre-contactingtheaircraftin ightcausingcatastrophic damageandlossoflife. Astorereleasedfromanaircraftinightmusttraverseanon uniformandunsteady oweldthatmayincludecomplexshockinteractions,large velocitygradients,regions oflocallyseparatedorreversedairow,andsevereowangu larityintheformof sidewashanddownwash.Storesreleasedfromaninternalwea ponsbaymayalso besubjectedtoawakedisturbancefromthespoiler,dynamic pressureandvelocity gradientsacrosstheshearlayer,highfrequencyvibration sduetoacousticnoise,and largeperturbationsinowpropertiesduetocavityoscilla tions. Althoughtheregionofnonuniformowneartheaircraftisex ceedinglysmall comparedtothefulllengthofthestoreballisticory-outt rajectory,theeffectsare signicant.Theoweldcharacteristicsmaycausethestor etoexhibitbehaviorthat compromisesthesafetyoftheairframeandcreworthatcompr omisestheeffectiveness ofthestoreitself.Predictionoftheightcharacteristic softhestoreinthevicinityof theaircraftisthereforevitallyimportantforensuringth esafetyandeffectivenessof therelease.ModelingandSimulationcapabilitiesalsopla yanintegralroleinthe cost-effectiveassessmentofseparationcharacteristics forarangeofaircraftandstore 18

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congurationsthroughouttheaircraftightenvelope.For challengingprograms,the correspondinggroundandighttestactivitiesrequireasi gnicantamountoftimeand resourcestocomplete.Thus,storeseparationengineering isanintegralpartofair combatsystemdevelopment. 1.2ProblemDescription Storeseparationengineering,asubsetofaircraft-storec ompatibility,isconcerned withtheightcharacteristicsofastoreinproximityofthe aircraftandotherstores. Groundtest,ighttest,simulation,andanalysisprocedur eshavebeendevelopedwhich largelyaddressthesafety-of-ightissuesrstencounter edintheVietnamera.Inmost cases,thestorecanbeejectedawayfromtheaircraftwithas ufcientverticalvelocity andnose-downpitchratetoensuresafeseparation.However ,withtheadventofsmart weapons,standoffcapabilities,andfocusedlethalitythe challengeinsuccessfulstore separationhasshiftedfromsafetytoacceptability[ 2 ].Whereasanunsafeseparation maythreatentheparentaircraft,anunacceptableseparati onmayresultinafailed missionorsignicantcollateraldamageduetoguidancepro blems,lossofcontrol,or damagetothestorecausedbytheseparationtransients. Modernsophisticated“smart”weaponsareequippedwithsen sitiveonboard electronicsincludinginertialmeasurementsystems,GPSu nits,sensors,seekers,and guidancecomputers.Standoffcapability(thedesirableab ilitytoreleaseamunition farawayfromtheintendedtarget)hasresultedincomplexae rodynamicshapeswith neutraldynamicstabilitymarginsdesignedformaximumgli deperformanceandminimal energyloss.Focusedlethality(thedesirableabilitytode stroyadesignatedtargetwhile minimizingcollateraldamage)hasresultedinmunitionsth ataresmallerandlighter andthereforemoredramaticallyaffectedbytheexigentow eldsurroundingthe aircraftinight.Thesetendencieshaveincreasedthesens itivitytoseparation-induced transients[ 2 ],potentiallyleadingtolargeangularratesandattitudes ,excessiveenergy loss,sensorsaturation,structurallimits,ordeparturef romstableightmodes[ 3 ].The 19

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challengeinstoreseparationisthustoensuresafetywhile alsomaintainingacceptability acrosstheightenvelope. Modernmunitionsaredesignedwithanonboardguidanceandc ontrolsystem toenablepreciseengagementoftheintendedtargets.Howev er,thecontrolsystem isnotusuallyactivateduntilthestoreissufcientlyfara wayfromtheaircrafttoavoid anypotentialinterference.Often,theseparation-induce dtransientsresultinlarge perturbationsfromthedesiredightattitudesthatrequir eadedicated“rate-capture” phaseforrecoverybeforethemunitioncanbeginthey-outt rajectory.Intherelatively fewcaseswheretheautopilotisengagedearlier(toprevent build-upofirrecoverable ratesandattitudes),themutualaerodynamicinterference betweenthestoreandaircraft isneglectedintheautopilotdesignleadingtoincreasedri skthroughreducedcondence insimulationcapabilitiesandpotentiallyunsafebehavio roftheautopilotreactingtoow eldperturbationswithoutconsiderationofthenearbyair craft. Thepurposeofthisresearchistoinvestigatethefeasibili tyofusingatransitional controlsystem,designedwiththeseparation-inducedtran sientsinmind,toguidethe storealongapreferredtrajectory. 1.3ResearchObjectives Theprimaryobjectiveofthisresearchistodevelopacompre hensiveapproachto improvetheseparationcharacteristicsofmodernejectorlaunchedguidedmunitions byutilizingatransitionalcontrolsystem,or“storesepar ationautopilot”,toguide thestorealonganoptimaltrajectory.Thisinvestigationi sintendedtoshowthe signicantincreaseinsafetyandacceptabilitythatcanbe achievedthroughguided storeseparationwithminimaladditionincostandcomplexi tyoftheguidanceandcontrol system. Thisdissertationisoutlinedasfollows.First,abriefove rviewofstoreseparation modeling&simulationmethodologyispresented.Emphasisi splacedonestablished windtunnelbasedpredictionmethodswithsomediscussiono fcomplementaryCFD 20

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basedmethods.Therigidbodyequationsofmotionarederive dandtheaerodynamic modelingapproachusedmostfrequentlyinstoreseparation analysisispresented. Anoverviewofighttestdatareductionandtrajectorymatc hingisdiscussedand demonstratedwithactualighttestresults.Thisbriefpre sentationofestablished storeseparationmodelingandsimulationcapabilitiespro videsthefoundationforthe remainingdevelopments. Next,SystemIdentication(SID)techniquesareusedtodev elopaparametric modelforthespatiallyvariantaerodynamiccharacteristi csofastoreduringseparation. Fornonlinearaerodynamicmodelingofaircraft,itiscommo ntouseamultivariate polynomialmodelstructurewithconstantcoefcientsdete rminedfromexperimental data[ 4 ].Thisapproachisextendedtothestoreseparationproblem bypostulatinga multivariatepolynomialmodelwithspatiallyvariantcoef cients.Thespatiallyvariant coefcientsareparametrizedasnonlinearfunctionsconsi stentwiththedominantow eldcharacteristicsandphysicallymeaningfulboundaryc onditions.Theresultofthis ventureintosystemidenticationisanonlinearparametri cmodelcapableofcapturing thesalientstoreseparationaerodynamicsinacompactmatr ixexpression.Thisreduced ordermodelisavaluableassetinitsownright,offeringins ightintophysicaldrivers ofstoreseparationandprovidingananalyticalframeworkf orcontrolsystemanalysis anddesign.Althoughtheparametricmodelinthisstudyisan analyticalrepresentation oftabulartime-averagedwindtunneldata,thetechniquesc ouldalsobeappliedto aerodynamicdataobtainedfromtime-accurateComputation alFluidDynamics(CFD) trajectories.Extensionofthesemethodstotime-variant owelddata(e.g.turbulence) isleftasafuturedevelopment,thoughsomepointersaregiv enforhowthismaybe achieved. Third,thestoreseparationproblem,withassociatedsafet yandacceptability objectives,isrestatedasatrajectoryoptimizationprobl emandapplicationofclassical optimalcontroltheoryisusedtodevelopasolutionmethodo logythatyieldsacandidate 21

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optimaltrajectorywithrespecttothechosencostsandcons traints.Indirectoptimal controlmethodsareemphasizedduetothedecreasedcomputa tionalburdencompared todirectmethods,aswellastheadditionalphysicalinsigh tofferedbydevelopmentof the1 st orderoptimalityconditions.Anexampleoftrajectoryopti mizationispresented usingtheplanarstoreseparationequationsofmotionandas impliedparametric aerodynamicmodel.Thesimpliedexampleisinstructivean dprovidesasolidstarting pointforamorein-depthcasestudyconsideredsubsequentl y. Fourth,withtheopenloopoptimaltrajectorysolutionasar eference,astore separationautopilotisdevelopedusingtheconceptofneig hboringoptimalcontrol (NOC).Theseparationautopilotisalinear-optimalfeedba ckcontrollerthatcorrects fordeviationsfromthenominaltrajectoryduetodisturban ces,modelinguncertainties, andvaryinginitialconditions.Theproposedinnitehoriz onneighboringoptimalcontrol (IHNOC)strategyaccountsforthespatiallyvariantaerody namicsneartheaircraftand convergestoalineartimeinvariantcontrollerinfareldc onditions.Theperformance oftheseparationautopilotisexaminedsubjecttoarangeof initialconditions,random perturbations,andparametervariations. Finally,theresultsaboveareextendedtothefullsixdegre e-of-freedom(6DOF), nonlinear,windtunnelbasedsimulationforselectaircraf tcongurationsandight conditions.Thesimulatedperformanceofthecontrolledst oreseparationapproachis comparedwithuncontrolledtrajectoriesforthesameiniti alconditions.Comparisons withighttestdataillustratethesignicantimprovement achievablewithguidedstore separation. Thecumulativeprocessofsystemidentication,trajector yoptimizationand feedbackcontrolrepresentsacomprehensiveapproachfora chievingguidedstore separationinarealisticenvironment.Agraphicaldepicti onofthisprocessisshownin Figure 6-1 .Acasestudyincludingarepresentativestorewithassocia tedwindtunnel 22

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andighttestdataispresentedthroughouttodemonstratet heanticipatedbenetsof guidedstoreseparation. System Identification Trajectory Optimization Feedback Control Guided Store Separation Figure1-1.Relationshipbetweensystemidentication,tr ajectoryoptimization,and feedbackcontrolforguidedstoreseparation. 1.4RepresentativeCaseStudy Thetechniquesdevelopedduringthecourseofthisresearch areapplicabletoa widerangeofaircraft/storecombinations.However,toill ustratethemethods,aparticular casestudywasselected.ThecasestudyincludestheF-16Fig htingFalconanda representativemid-size,verticallyejector-launchedgu idedmunition.Separationofa representativemunitionfromtheF-16isshownin 1-2 Figure1-2.F-16releaseofarepresentativeguidedmunitio n. TheF-16isamulti-rolesupersonicghteraircraftorigina llydevelopedbyGeneral Dynamics.TheF-16canbeconguredinanair-to-airorair-t o-groundcongurationand 23

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isequippedtocarryarangeofexternalstores.Theselected F-16casestudyprovides anidealexampleforpresentingestablishedstoreseparati onanalysistechniquesas wellasextendingtheexistingcapabilitytoincludesystem identicationandtrajectory optimization.Additionaldetailonthecasestudyisdiscus sedinlatersectionsand representativewindtunnelandighttestdataarepresente dthroughout. 1.5Contributions Theprimarycontributionofthisresearchisthedevelopmen tanddemonstration ofafeasiblestrategyforimplementingguidedstoresepara tion.Previousstudieshave highlightedtheuseofactivecontroltoimproveseparation characteristics,primarilyasa sidebenetofdemonstratingtheuseofanewCFDcapability[ 5 6 ].However,thisisthe rstworktoconsiderguidanceandcontrolspecicallyfors toreseparation.Guidance hereinreferstothedeterminationofthepreferredpathfro mreleasetoastabletrimmed ightconditionwithexplicitdependenceontheaerodynami cinteractionbetweenthe storeandaircraft.Controlhereinreferstothemanipulati onoftheaerodynamicforces usingcontrolsurfacedeectionstosteerthestorealongth epreferredtrajectoryin thepresenceofdisturbances.Inparticular,thisisthers tattempttodeterminean optimaltrajectoryforstoreseparation.Thisisalsother stattempttodesigna“store separationautopilot”withexplicitdependenceontheaero dynamicinteractionbetween thestoreandaircraft.Thedependenceoftheguidanceandco ntrolsystemonthe spatiallyvariantaerodynamicsreliesonaparametricmode ldevelopedusingsystem identication.Thismodelisalsotherstattempttodevelo paparametricmodelfor storeseparationaerodynamics.Thesecontributionsinter msofsystemidentication, trajectoryoptimization,andfeedbackcontrolaresummari zedbelow Applicationofsystemidenticationtoparametricmodelin gofstoreseparation aerodynamics.Freestreamaerodynamicsaremodeledusinga nonlinear multivariatepolynomialwithconstantcoefcients.Aircr aft/storeaerodynamic interferenceismodeledusinganonlinearmultivariatepol ynomialwithspatially variantcoefcients.Thefreestreamandinterferenceaero dynamicmodels areuniedinacompactmatrixrepresentationthatcaptures thedominant 24

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aerodynamiccharacteristicsofastoreduringseparationa ndsatisescertain physicallymeaningfulboundaryconditions. Applicationofoptimalcontroltheorytodetermineanoptim alstoreseparation trajectoryforaparticularightconditionandoweldofi nterest.Safetyand acceptabilityperformancemetricsarequantiedasaquadr aticcostfunction. The1 st ordernecessaryconditionsforoptimalstoreseparationwi thexplicit dependenceonspatiallyvariantaerodynamicsaredevelope d.Solutionstothe optimalcontrolproblemareapproximatedusingindirectnu mericalmethods tosolvethetwo-pointboundaryvalueproblem.Usingthisap proach,the aerodynamicinteractionbetweenthestoreandaircraftisl everagedtoimprove separationcharacteristics. Applicationofneighboringoptimalcontroltodevelopalin ear-optimalfeedback controllerthataccountsfordeviationsfromtheoptimalst oreseparationtrajectory. Anovelstatementoftheneighboringoptimalcontrolproble mwithjudicious selectionofcostandconstraintsleadstoaformulationref erredtoasInnite HorizonNeighboringOptimalControl(IHNOC).IHNOCisused toconstructastore separationautopilotthataccountsforthespatiallyvaria ntaerodynamicsnearthe aircraftandconvergestoalineartimeinvariantcontrolle rinfareldconditions. Performanceofthestoreseparationautopilotinthepresen ceofvaryinginitial conditions,randomdisturbances,andparametervariation sisconsidered. Thecontributionsstatedabovearespecictotheeldofsto reseparation engineering.Indeed,thefocusandintentofthisresearchh asbeenapplicationof identicationandcontroltheorytotheparticularchallen gesinstoreseparationwiththe visionofdevelopingtechnologywhichmayndreal-worldap plicationinthenearfuture. However,thisisnottoexcludetheacademiccontributionsa lsogainedbythisresearch. Storeseparationmaybeconsideredoneparticularrealizat ionofaclassofsystems withrapidlyvaryingparametersthatdependnonlinearlyon thestate,e.g.anonlinear parametervaryingsystem(NLPV).Generaltheoryforcontro lofaNLPVsystemdoes notexist.However,considerationofstoreseparationprov idesvaluableinsightintosuch asystemandthemethodsdevelopedhereinmaybeusefulforap plicationtosimilar systems. Theconceptofinnitehorizonneighboringoptimalcontrol (IHNOC)introduced hereisalsoavaluableacademiccontribution.TheIHNOCstr ategycanbeusedto improveperformanceforasystemthatmusttraverseanonlin earoperatingorstartup 25

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condition,followedbyoperationnearanequilibriumcondi tionforanindeterminate lengthoftime.Anad-hocapproachtocontrollingasystemth atexhibitsthistypeof behavioristoswitchbetweentwodisparatecontrollers,bu tthisisaninefcientmethod thatmayintroducediscontinuitiesinperformanceandexac erbatenonlinearities.The alternativeIHNOCapproachconsideredhereutilizesasing lecontinuousfull-state feedbackcontrollerthatconvergestoatime-invariantlin ear-optimalcontrollerasthe systemnonlinearitiesaredissipated.Thestoreseparatio nproblemcanbefurther identiedasaNLPVsystemthatconvergestoalineartime-in variant(LTI)systemunder suitablecontrol,andtheIHNOCprovidesanidealsolutionf orthisparticularsystem. TheapplicationofIHNOCtosystemsotherthanstoreseparat ionisbeyondthescopeof thisresearch,buttheextendedexampleconsideredherepro videsasolidfoundationfor furtherdevelopment. Finally,identicationandcontrolofightcharacteristi csinarapidlyvarying nonuniformoweldisnotlimitedtostoreseparation.Itis alsoimportanttoother aerospaceproblemsincludinglandingofanaircraftingrou ndeffectorvariablewinds, aircraftwake-vortexencounter,ightthroughamicroburs torwindshear,ightthrough severewindeldsinanurbanenvironment,employmentofhyp ersonicresearchvehicles fromhigh-altitudecarrierplanes,andightofmultipleai rcraftincloseproximitysuchas cooperativecongurationoraerialrefueling,tonameafew .Extensionofthemethods developedhereintotheseandsimilarproblemsisbeyondthe scopeofthepresentwork. Nevertheless,thetheoreticalinsightgainedbyapplicati onofidenticationandcontrolto aparticularexampleofproblemswithspatiallyvariantaer odynamicsisinstructive. 26

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CHAPTER2 MATHEMATICALMODELINGOFSTORESEPARATION 2.1Overview Mathematicalmodelingandsimulation(M&S)isusedtoreduc etherisk,cost,and scheduleofaircraft-storescompatibilityeffortsandpro videsavaluabletoolsetforthe practicingstoreseparationengineer.Inageneralstorese parationprogram,M&Swillbe usedtodevelopastreamlinedandcost-effectivewindtunne ltest,performsensitivityand uncertaintyanalysestodeterminebestandworst-casepred ictedoutcomes,downselect tocongurationsandightconditionsofinterestforight test,assesstheriskofeach individualighttestevent,andultimatelyprovideacerti cationrecommendationtothe authorizingagencyforoperationaluse. Themodelingandsimulationframeworkforstoreseparation typicallyincludes aerodynamicdataobtainedfromighttesting,computation aluiddynamics(CFD), andwindtunneltesting.CFDprovidesadirectandincreasin glyreliablemethod fordeterminingstoreseparationcharacteristicsandisof tenusedasanacceptable alternativetowindtunneltestdata.CFDcanbeusedtogener atetime-accurate dynamicstoreseparationtrajectoriestomatchighttest[ 7 ]orstatictime-averagedow eldsolutionsforuseinofinesimulations[ 8 9 ].Insomeanticipatedlow-riskcases, CFDhasbeenusedasthesolesourceofaerodynamicdatatodet ermineseparation characteristicspriortooreveninlieuofighttest.Howev er,inmanyapplicationsCFD isnotcapableofreplacingthewindtunnelentirely.Windtu nneltestinghaslongbeen thedominantapproachforcharacterizingstoreseparation aerodynamics.Although notwithoutlimitations,windtunneltestingremainstheme thodofchoiceforprojects withlimitedpriortesting,aggressivecapabilityrequire ments,orasubstantialnumber ofcongurationsandightconditions.Inthemostgeneralc ase,CFDisusedasa valuablecomplementtowindtunneltestdata,leveragingth estrengthofeachresource tocompletetheprogramwithasuitablebalancebetweencost s,schedule,andrisk. 27

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Emphasisforthepresentanalysishasbeenplacedonmodelin gwithwind tunneldata.Theprimaryreasonforthisselectionisthereq uirementtoperformrapid simulationsforidenticationandoptimization.However, giventheinterchangeableroles ofCFDandwindtunnelresourcesforstoreseparationanalys is,someefforthasbeen madetomaintainapplicabilityoftheresearchmethodologi estobothdatasources. Mathematicalmodelingforstoreseparationinvolvesstate mentoftherigidbody equationsofmotionanddevelopmentofmathematicalmodels foreachoftherelevant componentsthatinuencethestoreduringseparation.Thes emodelsmayinclude aircraftmaneuvering,ejectorperformance[ 10 11 ],constraintmechanismssuchas railsorpivots[ 12 ],aerodynamiceffects[ 13 ],congurationchangessuchasdeploying wingsorns[ 14 ],andcontrolsurfacedeections[ 15 ].Therigorousdevelopmentof eachofthesemodelsisbeyondthescopeofthepresentwork;i nstead,emphasis willbeplacedprimarilyonrigidbodydynamics,aerodynami cmodeling,andighttest validation.Thesethreeareasprovideasufcientfoundati oninstoreseparationto supportdevelopmentofsystemidenticationandtrajector yoptimizationtechniquesin Chapters 3 through 5 2.2StoreSeparationEquationsofMotion Mathematicalmodelingofstoreseparationisaccomplished bydividingthe separationeventintothreesequentialphasesbasedonthec haracteristicmotionof thestorerelativetotheaircraft.Duringtherstphase,th estoreisincontinuouscontact withtheaircraftandrigidlyattachedtotheaircraftsothe storeandaircraftactasa singlerigidbody.Duringthesecondphase,thestoreandair craftareincontinuous contactbutthestoreismovingrelativetotheaircraft.The contactforcesbetween theaircraftandstoreduetotheejectororconstrainthardw aredeterminetherelative motion.Finally,duringthethirdphase,thestoreisinfree ightandmovingrelativeto theaircraftundertheinuenceofthenonuniformoweld. 28

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Therstphase(asingleaircraft-storerigidbody)isvalua bleforanalysisofight testresults,buthaslittlebearingonthesimulationofast oreduringseparation.Indeed, thetrajectorytimeisinitiatedattheinstantofrelease,s oallpriormotionhasonlya secondaryeffectontheseparationthroughspecicationof theinitialconditionsof inertialandoweldproperties. Thesecondphaseishighlysituationdependent.Developmen tofasuitableejector orconstraintmodelcanbeasignicantundertakinginitsel f,oftenrequiringsubstantial groundtestdataandanalysistobemodeledadequately[ 11 ].Formostapplications, simulationofthecontactforcesappliedtothestoreduring separationisstraightforward whensufcientgroundtestdataareavailable[ 16 ].Thisresearchpresumesthatan adequateejectorand/orconstraintmodelcanbespecied.T heend-of-strokeconditions (measuredorpredicted)formtheinitialconditionsforthe free-ightphase. Thethirdphase,consistingofthestoreinfreeightrelati vetotheaircraft,is historicallythemostdifculttocharacterizeduetomutua laerodynamicinterferenceand itisalsotheregioninwhichsafetyandacceptabilityarede termined.Therefore,this regionisofparticularinterestforsystemidenticationa ndtrajectoryoptimization. Theequationsofmotionforastoreinfreeightaresimilart otheequationsof motionforanaircraftinfreeightsincebotharegovernedb ythesamephysical principles.However,acarefuldistinctionisnecessarydu etotherelativemotionof twoneighboringbodies.Ofparticularinterestisthestore motionrelativetotheaircraft. Forthisreason,theconventionalightdynamicsreference frames,coordinatesystems, motionvariablesandnomenclaturearemodiedtosuittheun iqueapplicationtostore separation.2.2.1ReferenceFramesandCoordinateSystems Analysisofastoreseparatingfromanaircraftrequiresthe developmentofavariety ofreferenceframesandcoordinatesystemstoparameterize themotionvariables. Inertialquantitiesareusedtopredictthetrajectoryofth estore,relativepositionand 29

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velocityareusedtorelatethemotionofthestoretoamaneuv eringaircraft,andthelocal velocityrelativetothewindisnecessarytoestimatetheae rodynamicloadsonthestore duringrelease. Referenceframesandcoordinatesystemsaredissimilarent ities.Referenceframes aremodelsofphysicalobjectsconsistingofmutuallyxedp oints,whereascoordinate systemsareabstractmathematicaldeviceswithnorealphys icalcounterpart[ 17 ].A referenceframe,denedasacollectionofatleastthreepoi ntsinthree-dimensional Euclideanspacesuchthatthedistancebetweenanytwopoint sinthecollectiondoes notchangewithtime[ 18 ],isrepresentativeofanidealizedrigidbody.Incontrast ,a coordinatesystemisamathematicalconstructusedtomeasu retheparametersof motionbetweenreferenceframes. Distinguishingbetweenreferenceframesandcoordinatesy stemsallowsderivation ofavectorformofthedifferentialequationsofmotionthat iscoordinate-freeand appropriateforimplementationinanysuitablecoordinate system.Thisprovidesa physicalbasisforunderstandingtheequationsofmotionbe forecoordinatesystems areintroducedwiththeassociatedalgebraiccomplexity.O ncethedifferentialequations havebeenexpressedinaparticularcoordinatesystem,the nalsteppriortomodeling andsimulationistodevelopamatrixrepresentationofthee quationsofmotion.The matrixrepresentationprovidesaconvenientformforprogr ammingtheequationsof motioninadigitalsimulation.Thisthree-partprocessofc oordinate-freemodeling, coordinatesystemimplementation,andmatrixrepresentat ionprovidesasystematic waytoapproachacomplicateddynamicsproblem.Severalnot edauthorsincluding KaneandLevinson[ 19 ],Rao[ 18 ],andZipfel[ 17 ]havehelpedtodevelopandrenethis process,addingsignicantlytothetheoreticalunderstan dingandpracticalapplication ofrigidbodydynamics.Theauthorhasfoundthisprocesstob eparticularlyvaluablefor storeseparationgiventhedistinctivenatureandrelative complexityoftheproblem. 30

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Thereferenceframesofinteresttostoreseparationinclud etheearthframe,inertial frame,ightframe,aircraftframe,storebodyframe,atmos phericwindframe,andstore windframe.Whileinprincipleitispossibletohavemultipl ecoordinatesystemsattached toeachframe(andorienteddifferently),thecoordinatesy stemsareuniquelyrelatedto thecorrespondingframe.Inparticular,wemakeuseoftheea rthaxis,inertialaxis,ight axis,aircraftaxis,storebodyaxis,atmosphericwindaxis ,andstorewindaxis.Although thenamesofthereferenceframesandcoordinateaxesoverla p,itshouldbenotedthat referenceframesandcoordinatesystemsaredifferententi tiesandusedindifferent ways.Thedistinctionshouldbecomeclearthroughtheconte xtofthesubsequent discussion.Forcompletion,thereferenceframesandcoord inateaxesaredenedhere. AircraftAxis A f a x a y a z g .TheAircraftAxis A f a x a y a z g isestablishedinthe AircraftReferenceFrame A .Theaircraftaxisiscloselyrelatedtotheconventional aircraftbodyaxisencounteredinightdynamicsliteratur e(thedirectionsarealigned; onlythepointsoforiginaredifferent).Theoriginoftheai rcraftaxis, O A ,iscoincident withthestoreCGatreleaseandxedwithrespecttotheaircr aft.Theaircraftreference framerotatestomaintainconstantorientationwithrespec ttotheaircraftatalltimes. The a x directionisparalleltotheaircraftbodyaxisandpositive intheforwarddirection asseenbythepilot.The a y directionisperpendicularto a x andpositiveoutofthe rightwingoftheaircraft(starboard)asseenbythepilot.T he a z directionisdenedby a z = a x a y andbyconsequenceispositivedownwardasseenbythepilot. Agraphical depictionoftheaircraftaxisisshowninFigure 2-1 StoreBodyAxis B f b x b y b z g .TheStoreBodyAxis B f b x b y b z g isestablished intheStoreBodyReferenceFrame B .Theoriginofthestorebodyaxis, O B ,is determinedbythestoreCG,whichmayinprinciplechangedur ingthetrajectory(due tochangingcongurationsorburningpropellant).The b x directionisalignedwiththe centerlineofthestoreandpositiveoutofthenose.The b z directionisperpendicular to b x andpositivedownward.The b y directionisdeterminedby b y = b z b x andby 31

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a x a z a y Figure2-1.Denitionofaircraftaxiscoordinatesystem. consequenceispositiveoutofthestarboardsideofthestor e.Thebodyaxisisoriented withrespecttotheaircraftaxisbytheinstalledincidence angles.Thestorebodyaxisis showngraphicallyinFigure 2-2 b x b y b z Figure2-2.Denitionofstorebodyaxiscoordinatesystem. EarthAxis E f e x e y e z g .TheEarthAxis E f e x e y e z g isestablishedwithin theEarthReferenceFrame E .Fortypicalightvehiclenavigationequations,the north-east-downdirectionswithreferencetothelatitude andlongitudeofthevehicle determinetheearthaxisdirections.However,forstoresep aration,alocalapproximation issufcient.Implicitinthisapproximationistheassumpt ionofaatearth.Theearth axisorigin O E isdenedasthestoreCGatthemomentofrelease.Theorigini sxedin 32

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spacerelativetothesurfaceoftheearth.Theprimarydirec tionofinterestfortheearth axisisthelocalvertical e z ,whichdenestheorientationofthegravityvector.Thevec tor e y isperpendicularto e z andispositiveoutoftherightwingoftheaircraft(starboa rd) asseenbythepilot.Thevector e x isdenedbythecrossproduct e x = e y e z andby consequenceliesinthelocalhorizontalplanepositiveint heforwarddirectionasseen bythepilot. InertialAxis I f i x i y i z g .TheInertialAxis I f i x i y i z g isestablishedintheInertial ReferenceFrame I .Thechoiceofaninertialreferenceframeisofcrucialimpo rtancefor dynamicsproblemsingeneralandightmechanicsproblemsi nparticular.Aninertial referenceframeisadequatelydescribedasareferencefram eatrestwithrespecttothe distantstars.Inmanyightvehicleapplications,theeart hitselfcanserveasasuitable referenceframe.Instoreseparationanalysis,afurtherap proximationiswarranted. Theinertialreferenceframeisselectedasahypotheticalf ramemovingataconstant translationalvelocityrelativetotheearth,wherethevel ocityandorientationoftheframe aredeterminedbytheaircraftightaxisvelocityandorien tationattheinstantthestore isreleased.Thisprovidesanimportantadvantageofrefere ncingtheinertialmotionof thestoreandtheaircrafttothesameinertialreferencefra meandprovidesaconvenient waytodescribetherelativemotionbetweenthestoreandair craft.Inthelimitingcase ofstraightandlevelightduringrelease,theaircraftits elfbecomesasuitableinertial referenceframeandthestoremotionrelativetotheaircraf tisdirectlyobtained.For moregeneralaircraftmaneuvers,atransformationisstrai ghtforwardusingthecommon inertialreferenceframe.Theoriginoftheinertialaxis, O I ,iscoincidentwiththestore CGatthemomentofreleaseandtravelsinastraightlinewith aconstantvelocityequal tothevelocityoftheightaxisattheinstantofrelease.Th e i x directionisdenedby thevelocityvectoroftheaircraftightaxisatthemomento freleaseandremainsata constantorientationwithrespecttotheearth.The i y directionispositiveinthedirection oftherightwingoftheaircraft(starboard)asseenbythepi lot.The i z directionis 33

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determinedby i z = i x i y andconsequentlyispositiveinthedownwarddirection.Ift he aircraftightpathangleandbankangleat t =0 arebothidenticallyzero,whichisthe mostcommonscenarioofinterestforstoreseparation,then theinertialaxis I f i x i y i z g isalignedwiththeearthaxis E f e x e y e z g ,withtheoriginoftheinertialaxismovingata constantvelocityrelativetotheoriginoftheearthaxis. FlightAxis F f f x f y f z g .TheFlightAxis F f f x f y f z g isestablishedintheFlight ReferenceFrame F .Theightaxisisdenedbythedirectionoftheaircraftvel ocity vectorthroughoutthetrajectory.Theightaxisiscoincid entwiththeinertialaxisatthe instantofrelease,buttranslatesandrotatesrelativetot heinertialaxisastheaircraft velocityvectorchanges.Forsteadyightconditions,the ightaxisandinertialaxis remaincoincidentthroughoutthetrajectory.Theoriginof theightaxis, O F ,isdened asapointcoincidentwiththestoreCGatthemomentofreleas ebutisxedrelative totheaircraftandthustranslatesalongwiththeaircraftd uringthemaneuver.The f x directionisalignedwiththeaircraftvelocityvectorthro ughoutthemaneuver.The orientationofthe f x directionwithrespecttothe e x directionisdeterminedbytheaircraft ightpathangle.The f y directionisperpendicularto f x andpositiveoutoftherightwing (starboard)asseenbythepilot.Theorientationofthe f y directionwithrespecttothe e y directionisdeterminedbytheaircraftbankangle.Iftheai rcraftightpathangleand bankangleareidenticallyzerothroughoutthetrajectory, thentheightaxisisaligned withtheearthaxis(andconsequentlytheinertialaxis).Th eorientationoftheaircraft axiswithrespecttotheightaxisisdeterminedbytheaircr aftincidenceangles(angleof attackandangleofsideslip).Insteadyight,theincidenc eanglesareconstantandthe orientationoftheaircraftaxiswithrespecttotheightax isisconstant.Therelationship betweentheightaxis,inertialaxis,andearthaxisisshow ninFigure 2-3 1 1 Displacementoftheinertialaxisisexaggeratedforclarit y. 34

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i x i z b x b z s x a x f x a z f z e z e x AMomentafterRelease, t > 0 ez fx, ix e x f z i z a x b x a z b z BMomentofRelease, t =0 Figure2-3.Denitionofearth,inertial,andightaxes. AtmosphericWindAxis W f w x w y w z g .TheAtmosphericWindAxis W f w x w y w z g isestablishedintheAtmosphericWindReferenceFrame W .Forthepurposesofstore separation,theatmosphericwindssurroundingtheaircraf tatreleaseareassumed tobemonolithic(uniform)andmovingataconstantvelocity relativetothesurface oftheearth.Theparticularvelocityofthewindisrarelyim portantsinceonlythe motionofthestorerelativetothewindandrelativetotheai rcraftisofinterest.The monolithicassumptionallowsthewindtobemodeledasanide alizedrigidbodyand representedbyareferenceframe.Theassumedconstanttran slationalvelocityalso qualiestheatmosphericwindframeasasuitableinertialr eferenceframe,afactthat willbeusefulinderivingtheso-calledwindaxisequations ofmotion.Theatmospheric windaxis W f w x w y w z g isalignedwiththeearthaxis E f e x e y e z g withtheoriginof theatmosphericwindaxis O W movingataconstantvelocityrelativetotheoriginofthe earthaxis O E ,wherethemagnitudeofthevelocityisdenedbytheatmosph ericwinds. Iftheatmosphericwindsareassumedtobeidenticallyzero, thentheatmosphericwind axisandearthaxisarecoincident. 35

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StoreWindAxis S f s x s y s z g .TheStoreWindAxis S f s x s y s z g isestablished intheStoreWindReferenceFrame.Thestorewindaxisisquit edifferentfromthe previouslydenedatmosphericwindaxis.Theoriginofthes torewindaxis O S is coincidentwiththestoreCGthroughoutthetrajectory.The velocityofthestorerelative tothelocalwindreferenceframedeterminestheorientatio nofthestorewindaxis.The velocityandorientationofthestorewindaxiswithrespect totheatmosphericwindaxis isessentialfordeterminingtheaerodynamicforcesacting onthestoreduringight.The s x directionrotatestomaintainalignmentwiththestorevelo cityvector.The s z direction isperpendicularto s x andpositiveinthedownwarddirection.The s y directionisdened by s y = s z s x .ThestorewindaxisisshowngraphicallyinFigure 2-4 sx b x b y b z Figure2-4.Denitionofstorewindaxiscoordinatesystem. Sofar,sevenreferenceframesandassociatedcoordinatesy stemshavebeen introduced.Inpractice,severalmorecoordinatesystemsa reneededtomodelstore separationincludingthesuspension,carriage,pylon,ref erence,andgridaxissystems. Formissiles,theaeroballisticcoordinatesystemisusual lyintroduced.Andforwind tunneltestingandighttestdatareduction,amultitudeof additionalcoordinatesystems 36

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areneeded.However,forthepurposesofderivingthestores eparationequationsof motionrelativetoamaneuveringaircraft,thesesevenden itionswillsufce. 2.2.2CoordinateFreeEquationsofMotion Usingreferenceframesasaphysicalentityrepresentingan idealizedrigidbody, theequationsofmotionforaightvehiclecanbederivedina coordinatefreemanner withouttheassociatedalgebraiccomplexities.Oncetheco ordinatefreeequations ofmotionhavebeendeveloped,theycanbeimplementedinany suitablecoordinate system(s). Let t =0 bedenedattheinstantthestorebeginstomoverelativetot heaircraft (themomentofrelease).Also,letthepositionofthecenter ofmassofthestoreatthe momentofreleasebedenedasthepoint O B .Itisassumedthatthestoreisactedon byauniformgravitationaleldsothecenterofmass(CM)and thecenterofgravity(CG) arecoincident.ThepositionoftheStoreCGrelativetoanin ertiallyxedpointatany time t > 0 isdenedas r B .Notethat r B isavectorthatrepresentsaphysicalentity,the existenceofwhichisnotdependentonaparticularcoordina tesystem.Inpreparationfor theuseofEuler'sLaws,itisnecessarytodifferentiatethe position r B togetthevelocity ofthestoreCGasseenbyanobserverintheinertialreferenc eframe.Theexpressionis showninEquation( 2–1 )wheretheleftsuperscript I [] denotesaderivativeasseenby anobserverxedinaninertialreferenceframe. I v B = I d dt r B (2–1) Similarly,theaccelerationofthestoreCGisshowninEquat ion( 2–2 ). I a B = I d dt I v B (2–2) ApplyingEulers1 st Law(ageneralizationofNewtons1 st Lawtoarigidbody) providestherstvectorequationofmotionforthetranslat ionaldynamics,Equation( 2–3 ). F = m I a B (2–3) 37

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InEquation( 2–3 ), F isthetotalgravitational,contact,andaerodynamicforce actingon thevehicleand m isthemassofthestore,assumedtobeconstantforthedurati onof thetrajectory(thisassumptioncanberelaxedformissiles expellingpropellant,buthere itisassumedthechangeinmassisnegligibleovertherelati velyshorttimeperiodof interestduringstoreseparation). EulersLawisonlyvalidforthevelocityandaccelerationre lativetoaninertial referenceframe.However,indetermininganexpressionfor thevelocityandacceleration ofthestoreCG,themostconciseformthatisconsistentwith themeasurements obtainedfromabody-xedmeasurementsystemisdesirable. Whenthevelocityofthe storeCGisexpressedinanysuitablebody-xedcoordinates ystem,theaccelerationis moreconciselydeterminedusingtherotationaltimederiva tive[ 17 ]ortransporttheorem [ 18 ].Theangularvelocityofthestorebodyframerelativetoth einertialreferenceframe isdenedas I B .TheaccelerationcanbeexpressedasshowninEquation( 2–4 ). I a B = B d dt I v B + I B I v B (2–4) RewritingEulers1 st lawresultsinthenalcoordinate-freeformofthetranslat ional equationsofmotionforaightvehicleasshowninEquation( 2–5 ). F = m B d dt I v B + I B I v B (2–5) Consideringtherotationalequationsofmotion,themoment ofinertiatensor referencedtothestoreCGandexpressedinabody-xedcoord inatesystemis denedas I BB .Again,themomentofinertiatensorrepresentsaphysicalq uantity andisthereforecoordinatefree.Onceexpressedinapartic ularcoordinatesystem,the representationofthemomentofinertiatensorwilltakeona conventionalmatrixform. Themomentofinertiatensorcanbeusedtodenetheangularm omentumofthestore framerelativetotheinertialframeasshowninEquation( 2–6 ). I H B = I BB I B (2–6) 38

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ApplyingEulers2 nd Law,thefollowingcoordinate-freeresultisobtained. M = I d dt I H B = I d dt I BB I B (2–7) InEquation( 2–7 ), M isthetotalmomentactingonthestoreduetoaerodynamic andcontactforces(themomentduetothegravitationaleld isneglectedsincethe gravitationaleldisassumedtobeuniform). Again,itisdesirabletoexpressallquantitiesinacoordin atesystemxedtothe body.Thisimpliesthatthemostefcientwaytoevaluatethe timederivativewithrespect totheinertialframeistoapplythetransporttheorem,assh owninEquation( 2–8 ). M = I d dt I BB I B = B d dt I BB I B + I B I BB I B (2–8) Forthisderivation,itisassumedthatthemomentofinertia tensorisconstantwhen expressedinanybody-xedcoordinatesystem.Thisassumpt ionisnotnecessarily thecaseformanyapplicationsinstoreseparationduetodep loyingnsormoving controlsurfaces.Thedifcultyinmodelingsucheffectsis notprimarilythecomplexity oftheequationsofmotion,whicharetractable,butinmodel ingthecomplexityofthe aerodynamiceffectsofchangingstorecongurations.Atle astoneefforthasbeenmade totaketheaerodynamicandinertialchangesintoconsidera tion[ 14 ].Forsimplicity ofthecurrentdevelopment,theinertiapropertieswillbea ssumedconstant.Thisis consideredajustiedassumptiondueto(1)theshorttimedu rationofinterestforstore separation(approximately1sec)and(2)therelativelymin oreffectontheequationsof motionduetoinherentlysmallmassofthecontrolsurfacesc omparedtothestoreitself. Theassumptioncouldberelaxedwithoutaffectingthevalid ityofChapters 3 through 6 buttheadditionalcomplexityisnotwarrantedforthisstud y. 39

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Giventheconstancyofthemomentofinertiatensorandlinea rpropertiesofthe timederivative,Eulers2 nd lawcannallybeexpressedasshowninEquation( 2–9 ). M = I d dt I BB I B = I BB B d dt I B + I B I BB I B (2–9) Itisnotedthatthetimederivative B d I B dt inEquation( 2–9 )isefcientlyevaluated sincetheangularvelocityisexpressedinacoordinatesyst emxedinthebody referenceframe.2.2.3StandardBody-AxisEquationsofMotion Thecoordinate-freeequationsofmotionderivedinSection 2.2.2 areapplicable toawiderangeofdynamicsystems,includinganyrigidbodyi nfreefall.Inthis section,theapplicationbeginstogetspecictoightvehi clesasthecoordinate systemsofinterestareselected.Thederivationofthestan dardbodyaxisequations ofmotionisaccomplishedbyexpressingthecoordinate-fre eequationsinthebody axis B f b x b y b z g .Thebodyaxisequationsarethemostfrequentlyusedformof the equationsofmotioninstoreseparationanalysis.Thederiv ationcanbeseparatedinto translationaldynamics,rotationaldynamics,translatio nalkinematics,androtational kinematics.2.2.3.1Translationaldynamics ToapplyEulers1 st Law,itisnecessarytospecifyacoordinatedformofthevect or quantitiesdenedinSection 2.2.2 .Forsimplicity,generalizedcoordinatesforthe velocityandangularaccelerationareintroduced[ 20 ].Equation( 2–10 )providesthe velocityofthestoreCGasseenbyanobserverxedinaninert ialreferenceframeand measuredinacoordinatesystemxedtothebody.Thescalarc omponents I u B B I v B B and I w B B inEquation( 2–10 )representthegeneralizedvelocitycoordinatesofthebod y ( [] B )framerelativetotheinertial( I [] )frame,expressedinthebody( [] B )frame. I v B = I u B B b x + I v B B b y + I w B B b z (2–10) 40

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Equation( 2–11 )givesthetimederivativeofthevelocityasseenbyanobser ver inthebodyframe,wherethe“dot”notation(e.g. I u B B )isshorthandforthescalartime derivative. B d dt I v B = I u B B b x + I v B B b y + I w B B b z (2–11) Similarly,Equation( 2–12 )givestheangularvelocity,wherethescalarcomponents I p B B I q B B I r B B representthegeneralizedangularvelocitycoordinatesof thebody( [] B ) framerelativetotheinertial( I [] )frame,expressedinthebody( [] B )frame. I B = I p B B b x + I q B B b y + I r B B b z (2–12) Thetimederivativeoftheangularvelocityinthebodyframe immediatelyfollows. B d dt I B = I p B B b x + I q B B b y + I r B B b z (2–13) Thetranslationandangularvelocitiescanalsobewritteni nacompactmatrix notation,asshowninEquations( 2–14a )and( 2–14b ). I v B B = I u B B I v B B I w B B T (2–14a) I B B = I p B B I q B B I r B B T (2–14b) Thequantities I v B B and I B B arethematrixrepresentationofthevectors I v B and I B inthebodyaxis.Thematrixrepresentationisaconvenientf ormforprogramming theequationsandgivesrisetoaself-deningnamingconven tionthatisparticularly usefulindevelopingacomputersimulation[ 21 ]. Thetranslationalequationsofmotionincludethetotalfor ce F ,whichaccountsfor thegravitational,contact,andaerodynamicforcesapplie dtothestoreduringseparation, withtheresultingvectorequationshowninEquation( 2–15 ). F = F G + F C + F A (2–15) 41

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Equation( 2–15 )iscoordinatefree,butmustbeparameterizedinaparticul ar coordinatesystemtobeusedfurther.Thecontactandaerody namicforcesaresuitably expressedinabody-xedcoordinatesystem. F C = F C x b x + F C y b y + F C z b z (2–16) F A = F A x b x + F A y b y + F A z b z (2–17) Thegravitationalforceisexpressedinanearthxedcoordi natesystem,asshown inEquation( 2–18 ). F G = F G g = mg e z (2–18) Theorientationofthestorebodyaxiswithrespecttotheine rtialaxisisgivenbythe standard(yaw-pitch-roll)Eulerrotationsequenceandthe correspondingtransformation matrix,asshowninEquation( 2–19 ). [ T ] BI = T IB BY T I B YX T I B XI (2–19) Thequantities I B I B IB aretheyaw,pitch,androllanglesofthestorebodyframe withrespecttotheinertialframeandtheframes X and Y areintermediatereference frames.Therotationscorrespondtothreesimplerotations aboutthe z y ,and x axes(a so-called3-2-1rotation).Theexpandedtransformationma trixisshownin( 2–20 ),where thetrigonometricterms sin and cos havebeenabbreviatedas s and c ,respectively. [ T ] BI = 266664 1000cos IB sin IB 0 sin IB cos IB 377775 266664 cos I B 0 sin I B 010 sin I B 0cos I B 377775 266664 cos I B sin I B 0 sin I B cos I B 0 001 377775 = 266664 c I B c I B c I B s I B s I B c I B s IB s I B c IB s I B c IB c I B +s IB s I B s I B c I B s IB s IB s I B +c IB c I B s I B c IB s I B s I B c I B s IB c IB c I B 377775 (2–20) 42

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Inthemostgeneralcase,theinertialaxisisorientedwithr especttotheearthaxis throughtheightpathangleoftheaircraftat t =0 ,denedas r E I ,whichisalways constantthroughoutthesimulation,regardlessoftheairc raftmaneuver.Thebank angleoftheaircraftat t =0 ,denedas EI ,isalsousedtodenetheorientation oftheinertialaxisrelativetotheearthaxisandbecomesim portantindeningthe directionofgravity.Theheadingangleoftheaircraftat t =0 isimmaterialfroma storeseparationperspectiveduetotheatearthassumptio n.Agraphicaldepiction oftheearth-to-inertialtransformationisshowninFigure 2-5 .Theearth-to-inertial transformationmatrixisgiveninEquation( 2–21 ). g I E f I E f I E e x i x e z i z i y e y Figure2-5.Earth-to-inertialaxistransformation. [ T ] IE = 266664 1000cos EI sin EI 0 sin EI cos EI 377775 266664 cos r E I 0 sin r E I 010 sin r E I 0cos r E I 377775 = 266664 cos r E I 0 sin r E I sin r E I sin EI cos EI cos r E I sin EI cos EI sin r E I sin EI cos r E I cos EI 377775 (2–21) Usingthecombinedtransformations [ T ] BE = [ T ] BI [ T ] IE ,theorientationofthe gravityvectorinthebodyaxisisgivenasshowninEquations ( 2–22 )and( 2–23a ) through( 2–23c ). 43

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[ F G ] B = [ T ] BE [ F G ] E = [ T ] BI [ T ] IE [ F G ] E (2–22) F G b x = mg (cos r E I cos EI sin I B +cos I B cos I B sin r E I cos r E I cos I B sin EI sin I B ) (2–23a) F G b y = mg (sin r E I (cos IB sin I B cos I B sin IB sin I B ) +cos r E I sin EI (cos IB cos I B +sin IB sin I B sin I B ) +cos r E I cos EI cos I B sin IB ) (2–23b) F G b z = mg (sin r E I (sin IB sin I B +cos IB cos I B sin I B ) +cos r E I sin EI (cos I B sin IB cos IB sin I B sin I B ) cos r E I cos IB cos EI cos I B ) (2–23c) ReturningtoEulers1 st Lawappliedtoaightvehicle,Equation( 2–5 ),and combiningEquations( 2–16 ),( 2–17 ),( 2–18 ),and( 2–22 )resultsinthethreescalar translationalequationsofmotionshowninEquations( 2–24a )through( 2–24c ). Substitutionofthescalarproductsisstraightforwardusi ngEquations( 2–23a )through ( 2–23c ). F C x + F A x + F G b x = m I u B B + m I q B B I w B B I r B B I v B B (2–24a) F C y + F A y + F G b y = m I v B B m I p B B I w B B I r B B I u B B (2–24b) F C z + F A z + F G b z = m I w B B + m I p B B I v B B I q B B I u B B (2–24c) Equations( 2–24a )through( 2–24c )representthegeneralcasewhentheaircraft bankangleandightpathanglearenon-zeroatthemomentofr elease.Twosimplied casesarealsoofinterest.First,whentheaircraftisinwin gs-levelight( EI =0 ),the gravityvectorexpressedinthebodyaxisisgivenbyEquatio n( 2–25 ).Theresulting 44

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translationalequationsofmotionaregivenbyEquations( 2–26a )through( 2–26c ). [ F G ] B = mg 266664 (c r E I s I B +c I B c I B s r E I ) (s r E I (c IB s I B c I B s IB s I B )+c r E I c I B s IB ) (s r E I (s IB s I B +c IB c I B s I B ) c r E I c IB c I B ) 377775 (2–25) F C x + F A x mg (c r E I s I B +c I B c I B s r E I )= m ( I u B B + I q B B I w B B I r B B I v B B ) (2–26a) F C y + F A y + mg (s r E I (c IB s I B c I B s IB s I B )+c r E I c I B s IB )= m ( I v B B I p B B I w B B I r B B I u B B ) (2–26b) F C z + F A z mg (s r E I (s IB s I B +c IB c I B s I B ) c r E I c IB c I B )= m ( I w B B + I p B B I v B B I q B B I u B B ) (2–26c) Second,whentheinertialaxesarealignedwiththeearthaxe s(e.g.theaircraftis instraight,wings-levelightat t =0 ),theconstants r E I = EI =0 ,andthegravityforce expressedinbodyaxesreducestothematrixrepresentation givenbyEquation( 2–27 ). [ F G ] B = mg 266664 sin I B cos I B sin IB cos IB cos I B 377775 (2–27) Inthisspecialcase,thescalartranslationalequationsof motionreducetothe followingclassicalformforaightvehicle[ 22 ],asshowninEquations( 2–28a )through ( 2–28c ). F C x + F A x mg sin I B = m I u B B + m ( I q B B I w B B I r B B I v B B ) (2–28a) F C y + F A y + mg cos I B sin IB = m I v B B m ( I p B B I w B B I r B B I u B B ) (2–28b) F C z + F A z + mg cos IB cos I B = m I w B B + m ( I p B B I v B B I q B B I u B B ) (2–28c) 45

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NotethateventhoughtheformoftheEquations( 2–28a )through( 2–28c )is identicaltotheclassicalform,themeaningofthedynamicm otionvariablesisdifferent. Thetranslationalvelocitiesaredenedrelativetoamovin ginertialreferenceframe, sothevelocitiesareidenticallyzeroatthemomentofrelea seeventhoughthestore ismovingrelativetothesurroundingairmass(andthesurfa ceoftheearth).This distinctionbecomesimportantwhendeterminingtheaerody namicforcesandmoments. 2.2.3.2Translationalkinematics ThepositionvectorofthestoreCGwithrespecttotheorigin oftheinertialaxis systemisgivenby r B ,whichisconvenientlyexpressedininertialcoordinates. r B = I x I B i x + I y I B i y + I z I B i z (2–29) Thetimederivativeofthepositionvectorasseenbyanobser verxedinthe inertialreferenceframegivesthevelocityofthestorewit hrespecttotheinertialframe, expressedininertialaxes. I v B = I d r B dt = I x I B i x + I y I B i y + I z I B i z (2–30) Thequantity I v B wasdenedinEquation( 2–10 )tobe I v B = I u B B b x + I v B B b y + I w B B b z andsincebothexpressionsrepresentthesamephysicalquan tity,theexpressionsmust beequivalent. I v B = I x I B i x + I y I B i y + I z I B i z = I u B B b x + I v B B b y + I w B B b z (2–31) Usingthebody-to-inertialtransformationmatrixinEquat ion( 2–20 ),thetwo expressionscanbeevaluatedinthesamecoordinatesystem, showninvectorform in( 2–32 )andmatrixformin( 2–33 ). I v B I = T BI I v B B = [ T ] IB I v B B (2–32) 46

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266664 I x I B I y I B I z I B 377775 = [ T ] IB 266664 I u B B I v B B I w B 377775 (2–33) Equation( 2–33 )canbeexpressedasthreescalarkinematicdifferentialeq uations forthetranslationalequationsofmotion,whichapplyrega rdlessoftheaircraftmaneuver atrelease. I x I B = I w B B (sin IB sin I B +cos IB cos I B sin I B ) I v B B (cos IB sin I B cos I B sin IB sin I B )+ I u B B cos I B cos I B (2–34a) I y I B = I v B B (cos IB cos I B +sin IB sin I B sin I B ) I w B B (cos I B sin IB cos IB sin I B sin I B )+ I u B B cos I B sin I B (2–34b) I z I B = I w B B cos IB cos I B I u B B sin I B + I v B B cos I B sin IB (2–34c) Inordertodeterminetheproximityofthestorewithrespect totheaircraft,the translationalkinematicsmustalsoaccountfortheaircraf tmaneuverrelativetothe inertialframe.ThisderivationiscontinuedinSection 2.2.5 2.2.3.3Rotationaldynamics Euler's2 nd Lawforarigidbodyappliedtoaightvehiclehasbeenprevio uslygiven inEquation( 2–9 ).Inordertocarrytheaboveoperationsfurther,themoment ofinertia tensor I BB mustbeexpressedincoordinateformasshowninEquation( 2–35 ). I BB = I xx b x n b x + I yy b y n b y + I zz b z n b z + I xz b x n b z + I zx b z n b x (2–35) InEquation( 2–35 ), I xx I yy I zz arethemomentsofinertia, I xz = I zx istheproduct ofinertia.Theremainingproductsofinertiaarezero, I xy = I yx = I yz = I zy =0 ,dueto symmetryoftheightvehicleinthe x–z plane.Theexpressions b i n b j for i j = x y z representthedyadicproductbetweentobasisvectors b i and b j .Theassumptionof symmetryoftheightvehicleisnotstrictlynecessaryandt hederivationcouldbe continuedwithouttheassumption,albeitwithmorelengthy expressions.However, 47

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nearlyallstoresaresymmetricornearlysymmetricinthe x–z plane,sotheassumption ismadeforthesakeofsimplicityintheresultingequations ofmotion.Thechoiceofthe bodyaxisasthepreferredcoordinatesystemiscrucialsinc ethebodyframeistheonly frameinwhichthemomentofinertiatensorisconstant.Them omentofinertiatensor canalsoberepresentedasamatrixquantity,asshowninEqua tion( 2–36 ). I BB = 266664 I xx 0 I xz 0 I yy 0 I xz 0 I zz 377775 (2–36) GiventhedyadicrepresentationinEquation( 2–35 )withpriorresultsfor I B and B d I B dt inEquations( 2–13 )and( 2–13 ),Eulers2 nd Lawforaightvehiclecanbe writtenasthreescalardifferentialequationsofmotionsh owninEquation( 2–37a ) through( 2–37c ). M C x + M A x = I p B B I xx I r B B I xz +( I q B B )( I r B B ) ( I zz I yy ) ( I q B B )( I p B B ) I xz (2–37a) M C y + M Ay = I q B B I yy +( I p B B )( I r B B ) ( I xx I zz ) I p B B 2 I r B B 2 I xz (2–37b) M C z + M A z = I r B B I zz I p B B I xz +( I p B B )( I q B B ) ( I yy I xx ) +( I q B B )( I r B B ) I xz (2–37c) 2.2.3.4Rotationalkinematics Theangularvelocityofthestorerelativetotheinertialfr ameisexpressedinterms ofthegeneralizedcoordinates I p B B I q B B I r B B ,asshowninEquation( 2–38 ). I B = I p B B b x + I q B B b y + I r B B b z (2–38) Thebody-to-inertialtransformationmatrix,showninEqua tion( 2–39 )andexpanded inEquation( 2–40 ),givestheangularorientationofthestorerelativetothe inertialaxis asaseriesofthreeconsecutiverotations,whereXandYarei ntermediatereference frames. [ T ] BI = T IB BY T I B YX T I B XI (2–39) 48

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[ T ] BI = 266664 1000 c IB s IB 0 s IB c IB 377775 266664 c I B 0 s I B 010 s I B 0 c I B 377775 266664 c I B s I B 0 s I B c I B 0 001 377775 (2–40) Theangularvelocitycanalsobeexpressedasthecombinatio nofthreesimple rotationalvelocitiesaboutthecorrespondingaxes.Since I B representsaphysical quantity,theseexpressionsmustbeequivalent,asshownin vectorforminEquation ( 2–41 )andmatrixforminEquation( 2–42 ). I B = I B x z + I B y y + IB b x = I p B B b x + I q B B b y + I r B B b z (2–41) I B B = I B T IB BY T I B YX 266664 001 377775 + I B T IB BY 266664 010 377775 + IB 266664 100 377775 (2–42) Equatingthetwosetsofexpressionsresultsinthreescalar kinematicdifferential equations,giveninEquations( 2–43a )through( 2–43c ). I p B B = IB I B sin( I B ) (2–43a) I q B B = I B cos I B sin IB + I B cos( IB ) (2–43b) I r B B = I B cos I B cos IB I B sin( IB ) (2–43c) 2.2.3.5Collectedbody-axisequationsofmotion Thetranslationalandrotationalrelationshipsaboveresu ltintwelverstorder differentialequationsofmotioncorrespondingtotherigi dstoresixdegreesoffreedom. Theequationsofmotioncanbewritteninstate-spaceformsu itableforanalysisusing moderncontrolmethodsorintegrationwithanumericalordi narydifferentialequation (ODE)solver.Thetwelvedifferentialequationspreviousl yderivedcanbeexpressedin statespaceformbysolvingfortheindividualscalarderiva tives.Theresultsarecollected below.Theseresultsreectequationscorrespondingtoai ghtpathangleandbank angleidenticallyequaltozeroatrelease,renderingasimp liedexpressionforthe 49

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gravityvectorinbodycoordinates.Furthermore,itisunde rstoodthattheseequations arevalidinthebodyaxisandrepresentingmotionvariables relativetoaninertialaxis. Assuch,thesuperscriptnotationhasbeenomitted.Forthis specialcase,theresults areconsistentwiththeequationspresentedinthepredomin antightdynamicsliterature [ 17 22 23 ]. TranslationalKinematics x = w (s s +c c s ) v (c s c s s )+ u c c (2-44a) y = v (c c +s s s ) w (c s c s s )+ u c s (2-44b) z = w c c u s + v c s (2-44c) RotationalKinematics = p + ( q sin + r cos ) tan (2-45a) = q cos r sin (2-45b) = ( q sin + r cos ) sec (2-45c) TranslationalDynamics u = ( F x mg s mqw + mrv ) = m (2-46a) v = ( F y + mpw mru + mg c s ) = m (2-46b) w = ( F z mpv + mqu + mg c c ) = m (2-46c) RotationalDynamics p = I xz M z + I zz M x + I 1 qr + I 2 pq = I xx I zz I 2 xz (2-47a) I 1 := I yy I zz I 2 xz I 2 zz I 2 := ( I xx I xz I xz I yy + I xz I zz ) q = M y I xz p 2 + I xz r 2 I xx pr + I zz pr = I yy (2-47b) r = ( I xz M x + I xx M z + I 3 qr + I 4 pq ) = I xx I zz I 2 xz (2-47c) I 3 := ( I xz I yy I xx I xz I xz I zz ) I 4 := I 2 xx + I 2 xz I xx I yy 50

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2.2.4Wind-AxisEquations Thechoiceofasuitableinertialaxisisimportantforexpre ssingthephysicallaws thatgovernrigidbodymotionbuttheinertialaxisaloneisn otsufcientfordescribing theforcesactingonavehicleinight.Rather,theaerodyna micforcesandmoments aredeterminedbythevehiclemotionrelativetothesurroun dingairmass.Intraditional ightdynamicsliterature,thewind-axisoriginislocated attheCGoftheightvehicle (andthustranslateswiththevehicletracingoutthetrajec toryoftheCGovertime),but rotatesrelativetothevehicletomaintainconstantalignm entwiththevelocityvector. However,byconventioninstoreseparationanalysis,thewi ndreferenceframeisused todenotetheatmospheresurroundingtheaircraftatthetim eofrelease.Here,the symbol W denotesthereferenceframeafxedtotheatmosphericwindt hatisassumed tobeuniformandmovingataconstantvelocitywithrespectt otheearthframe.The symbol S willbeusedtodenotethestorewind-axis,whichisxedatth estoreCGand alignedwiththestorevelocityvector,inamannerconsiste ntwiththepredominantight dynamicsliterature. Thevelocityofthestorerelativetothewindreferencefram eisshowninEquation ( 2–48 ),where W V B isthemagnitudeofthevelocityandtheunitvector s x isdenedby thedirectionofthestorevelocity. W v B = W V B s x (2–48) Theorientationofthevelocityvectorwithrespecttothest orebodyaxisisgivenby thetwoCartesianincidenceangles, s and s .Todeterminethetransformationmatrix relatingthestorebodyaxisandthestorewindaxis,itiscon venienttointroducethe intermediatereferenceframeX(whichsupersedesanyprevi ouslydenedintermediate referenceframes).Therotationoftheintermediateframew ithrespecttothebodyframe isgivenbyEquation( 2–49 ),where s isthestoreangle-of-attack. 51

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[ T ] XB = 266664 cos s 0sin s 010 sin s 0cos s 377775 (2–49) Therotationofthestorewind-axiswithrespecttotheinter mediateframeisgivenby Equation( 2–50 ),where s isthestoreangle-of-sideslip. [ T ] SX = 266664 cos s sin s 0 sin s cos s 0 001 377775 (2–50) Thetransformationforthestorewind-axiswithrespecttot hebodyaxisisgivenby thecorrespondingcompoundtransformation,asshowninEqu ation( 2–51 ). [ T ] SB = [ T ] SX T BX = 266664 cos s cos s sin s cos s sin s cos s sin s cos s sin s sin s sin s 0cos s 377775 (2–51) Themagnitudeofthestorevelocityrelativetothewind, W V B ,andtheCartesian incidenceangles, s and s ,playanimportantroleincharacterizingtheaerodynamic forcesandmomentsforthestoreinfar-eld(freestream)co nditions.Ifthebody-axis equationsofmotionaresolved,thevelocityandincidencea nglescanbedetermined asancillaryequations.Alternatively,thetranslational equationsofmotioncanbe expressedinthewind-axisandsolveddirectly.Therstmet hodisthemostcommonin storeseparationanalysisbutthesecondmethodwillbecome particularlyvaluablefor trajectoryoptimization.Bothapproachesarebrieyconsi deredhere. 2.2.4.1Ancillaryequations Thegeneralizedvelocities, I u B B I v B B I w B B ,representthecomponentsofthevelocity ofthevehiclewithrespecttotheinertialreferenceframe, whichisitselfmovingata 52

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constantvelocitydeterminedbytheaircraftvelocityatth einstantofrelease.Itisalso valuabletodeneasetofgeneralizedvelocities W u B B W v B B W w B B ,whichrepresentthe velocityofthestoreframe( [] B )withrespecttotheatmosphericwindframe( W [] ), expressedinthestorebodyaxis( [] B ).Whenthevelocityofthestorewithrespectto theinertialframe, I v B = I u B B b x + I v B B b y + I w B B b z ,isknownfromthesolutionofthe body-axisdifferentialequations,andthevelocityofthei nertialframerelativetothewind, W v I = W u I I i x ,isknownfromtheinitialconditions,thevelocityofthest orerelativetothe windcanbereadilydetermined.ThevectorformisgivenbyEq uation( 2–52 ). W v B = I v B + W v I (2–52) Equation( 2–52 )canbewritteninmatrixformusingthetransformationmatr ix [ T ] BI W v B B = I v B B + [ T ] BI W v I I (2–53) Expandingthematrixequationresultsinthreescalarequat ions. 266664 W u B B W v B B W w B B 377775 = 266664 I u B B + W u I I cos I B cos I B I v B B W u I I (cos IB sin I B cos I B sin IB sin I B ) I w B B + W u I I (sin IB sin I B +cos IB cos I B sin I B ) 377775 (2–54) Finally,themagnitudeofthevelocity, W V B ,isgivenbytheEuclideannorm. W V B = q W u B B 2 + W v B B 2 + W w B B 2 (2–55) Todeterminetheincidenceanglesusingthegeneralizedvel ocities,expandthe matrixequation W v B S = [ T ] SB W v B B asshowninEquation( 2–56 ). 266664 W V B 00 377775 = 266664 cos s cos s sin s cos s sin s cos s sin s cos s sin s sin s sin s 0cos s 377775 266664 W u B B W v B B W w B B 377775 (2–56) 53

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Usingthesethreescalarequations,thecorrespondingexpr essionsfor s and s canbedetermined.Notethat W u B B =0 resultsin s =90 degforany W w B B 6 =0 ,whereas W V B =0 resultsin s beingundened. s =tan 1 W w B B = W u B B (2–57) s =sin 1 W v B B = W V B (2–58) 2.2.4.2Wind-axisequationsofmotion Thetranslationequationsofmotioncanalsobederivedusin gthetranslationaland angularvelocitiesexpressedinthestorewindaxis.Thevel ocityofthestorewithrespect totheatmosphericwindisgivenbyEquation( 2–59 ). W v B = W V B s x (2–59) Sincethewindisassumedtobemonolithic(uniform),thewin distreatedasan idealizedrigidbodyandcanberepresentedbyareferencefr ame.Furthermore,since thewindisassumedtobemovingataconstanttranslationalv elocitywithrespecttothe earth,thewindreferenceframeisasuitableinertialrefer enceframeforthepurposeof storeseparation.Assuch,Eulers1 st lawcanbeapplied.Applyingthetransporttheorem toachieveacompactsetofdifferentialequations,theacce lerationofthestorebodywith respecttothewindframeisgivenbyEquation( 2–60 ). W d dt W v B = S d dt W v B + W S W v B (2–60) Theangularvelocityofthestorewindframewithrespecttot hewindreference frameisgivenbythevectoradditionoftheintermediatebod yframe. W S = W B + B S (2–61) Sincethewindframeisaninertialreferenceframe(albeitm ovingwithadifferent translationalvelocitythantheprimaryinertialreferenc eframe),theangularvelocity 54

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ofthebodyrelativetothewindframeisequivalenttotheang ularvelocityofthebody relativetotheprimaryinertialframe(e.g.theangularvel ocityofthewindframewith respecttotheprimaryinertialframeiszero–indeed,ifthi swerenotthecasethewind framewouldnotqualifyasaninertialreferenceframe). W B = I B (2–62) Theangularvelocityofthestorewindaxisrelativetothest orebodyaxisisgivenby theadditionoftwosimpleangularvelocities,asshowninEq uation( 2–63 ). B S = x y + s z (2–63) ThevectorexpressioninEquation( 2–63 )canbewritteninmatrixformusingthe transformationmatrixbetweentheintermediatereference frameXandthestorewind axis. B S S = [ T ] SX 266664 010 377775 + 266664 001 377775 (2–64) Expressingthevectorinthestorewindcoordinatesystemgi vestheangularvelocity inaconsistentform,asshowninEquation( 2–65 ). B S = sin s x cos s y + s z (2–65) Usingthetransformationmatrixforthestorewindaxiswith respecttothestore bodyaxis,theangularvelocityofthestorewindaxisrelati vetotheatmosphericwind axiscanbewritteninmatrixformasEquation( 2–66 ). W S S = [ T ] SB I B B + B S S (2–66) 55

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Expandingthematrixequationgivesthefollowingthreesca lardifferentialequations. W S S = 266664 I q B B sin s s sin s + I p B B cos s cos s + I r B B cos s sin s I q B B cos s s cos s I p B B cos s sin s I r B B sin s sin s s + I r B B cos s I p B B sin s 377775 (2–67) Toexpresstheequationsofmotioninthestorewindaxis,the gravityvectormustbe properlycoordinatedusingthetransformationmatrices [ T ] SB and [ T ] BI F G = mg [ T ] SB [ T ] BI i z (2–68) Forthesakeofbrevity,ithasbeenassumedthattheaircraft ightpathangle atreleaseiszeroandasaconsequencetheinertialaxisisal ignedwiththeearth axis.Relaxingthisassumptionintroducesathirdtransfor mationmatrix, [ T ] IE ,which signicantlylengthenstheresultingequationswithoutfu ndamentallychangingthe derivation.Expandingthematrixequationabovegivestheg ravityvectorexpressedin thestorewindaxiscoordinatesystem. F G = mg 266664 s s c I B s IB c s c s s I B +c s c IB s s c I B c s c I B s IB +c s s s s I B c IB s s s s c I B s s s I B +c s c IB c I B 377775 (2–69) Ignoringanycontactforcesbetweentheaircraftandstorea ndtreatingthe atmosphericwindframeasaninertialreferenceframe,Eule rs1 st Lawcanbewrittenas showninEquation( 2–70 ),where F A istheaerodynamicforceexpressedinthestore windaxis. F A + F G = m S d dt W v B + W S W v B (2–70) Theaerodynamicforceistypicallyparameterizedintermso fLift( F L ),Drag( F D ), andSideforce( F Y ),asshowninEquation( 2–71 ). F A = F D W s x + F Y W s y F L s z (2–71) 56

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Thedragandsideforceinthestorewindaxisarerelatedtoth econventionally deneddragandsideforce(inthestabilityaxis)byEquatio ns( 2–72 )and( 2–73 ). F D W = F D cos s F Y sin s (2–72) F Y W = F Y cos s + F D sin s (2–73) Finally,thetranslationaldifferentialequationsofmoti onexpressedinthestorewind axisaregivenasEquations( 2–74a )through( 2–74c ). F D W = V W B m mg cos s cos s sin I B + mg sin s cos I B sin IB + mg cos s cos IB sin s cos I B (2–74a) F Y W = mV W B ( s + I r I B cos s I p I B sin s ) mg cos s cos I B sin IB mg cos s sin s sin I B + mg cos IB sin s sin s cos I B (2–74b) F L = mV W B (_ s cos s I q I B cos s + I p I B cos s sin s + I r I B sin s sin s )+ mg sin s sin I B + mg cos s cos IB cos I B (2–74c) Equations( 2–74a )through( 2–74c )canbeputintostate-spaceformbysolvingfor thetimederivatives, V W B s ,and s ,asshowninEquations( 2–75a )through( 2–75c ). V W B = F D W + g c IB c I B s s c s +s IB c I B s s s I B c s c s (2–75a) s = F L mV W B c s + I q B B tan s I p B B c s + I r B B s s + g V W B c s c IB c I B c s +s I B s s (2–75b) s = F Y W mV W B + I p B B s s I r B B c s + g V W B c s s IB c I B + s s V W B g c s s I B g s s c IB c I B (2–75c) Combinedwiththe12statespaceequationsinthebodyaxis,t hedifferential equationsarenow15innumber.However,only12oftheseequa tionsareindependent. 57

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Whennecessary,thetranslationalwind-axisequationsofm otionareusedinlieuofthe translationalbody-axisequationsofmotion.Thechoiceof onesystemoveranotheris drivenbytheparticularapplication. The15differentialequationspresentedthusfararesufci entfordeterminingthe storemotionrelativetotheinertialreferenceframeandat mosphericwindframe.For storeseparation,theprimaryinterestisinthestoremotio nrelativetotheaircraft,which isachievedthroughanalgebraicextensiontotheequations ofmotion. 2.2.5PositionandVelocityoftheStoreRelativetotheAirc raft Inthesimplestcase,theaircraftisassumedtobeyinginun iformunaccelerated ight(suchasaconstantclimbordiveangleorstraightandl evel)andistherefore alignedwiththeinertialaxis.Theequationsofmotionderi vedinSections 2.2.3 and 2.2.4 applyimmediately,e.g.theaircraft-relativetrajectory isidenticaltotheinertial trajectory.However,inthemoregeneralcase,theaircraft ismaneuveringduring therelease,resultinginmotionoftheaircraftrelativeto theinertialreferenceframe. Themotionmaybearbitraryinthesensethattheaircraftmov esalonganyallowable trajectory,orthemotionmaybesimpliedbyassumingaspec icmaneuver,such asapull-uporpush-over.Ifthemotionisindeedarbitrary, itisassumedthatthe aircrafttrajectoryisknownapriori.Inotherwords,simul ationoftheaircrafttrajectory isnotconsidered.Regardless,theequationsofmotionofth estorerelativetothe inertialreferenceframeareunmodied.Determiningthepo sitionandvelocityofthe storerelativetotheaircraft,anessentialsteppriortoae rodynamicmodelingofthe nonuniformoweld,ismoreinvolved. Theightaxisorientationisdeterminedbytheaircraftvel ocityvectorthroughout thetrajectory.Thepositionofthestorerelativetotheig htaxisisgivenbythevector relationshipshowninEquation( 2–76 ). F r B = I r B I r F (2–76) 58

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Similarly,thevelocityisgivenbyEquation( 2–77 ). F v B = I v B I v F (2–77) InEquations( 2–76 )and( 2–77 ), I r B = I x I B i x + I y I B i y + I z I B i z and, I v B = I x I B i x + I y I B i y + I z I B i z arethepositionandvelocityofthestoreCGrelativetothe inertialaxis(givenbysimulation).Similarly, I r F = I x I F i x + I y I F i y + I z I F i z and I v F = I x I F i x + I y I F i y + I z I F i z arethepositionandvelocityoftheightaxisoriginrelati vetothe inertialaxis(givenbytheproblemstatement).Theangular velocityofthestorerelative totheaircraftisgivenbythecoordinatefreeexpressionsh owninEquation( 2–78 ), where I B = I p B B b x + I q B B b y + I r B B b z isthestoreangularvelocityrelativetotheinertial frameand I F = I p F F f x + I q F F f y + I r F F f z istheightaxisangularvelocityrelativetothe inertialframe. F B = I B I F (2–78) Inmatrixform,theangularvelocityofthestorerelativeto theaircraftisshownin Equation( 2–79 ),where [ T ] BF isthetransformationmatrixthatrepresentstheorientati on ofthestorewithrespecttotheightaxis,asshowninEquati on( 2–80 ). F B B = I B B [ T ] BF I F F (2–79) [ T ] BF = [ T ] BI T FI (2–80) Thematrix [ T ] FI isdeterminedusinga3-2-1Eulerrotationinamannersimila rto [ T ] BI [ T ] FI = 266664 1000cos IF sin IF 0 sin IF cos IF 377775 266664 cos I F 0 sin I F 010 sin I F 0cos I F 377775 266664 cos I F sin I F 0 sin I F cos I F 0 001 377775 (2–81) Equations( 2–76 )through( 2–81 )areapplicableinanymaneuveringaircraft simulation,providedthattheaircraftmotionrelativetot hecommoninertialframeis 59

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knownorcanbedetermined.However,thegeneralityofthese relationshipsareseldom required.Nearlyallstoreseparationsimulationscanbemo deledusingsimplifying assumptions,including(1)straightandlevelight,(2)st eadyclimbordive,(3)steady pull-uporpush-over.Eachofthesemaneuversissteady(sot heaircraftvelocityand angleofattackareconstant)andconnedtothepitch-plane (sothelateraltranslational andangularvelocitiesarezero),resultinginsignicants implicationsofEquations ( 2–76 )through( 2–81 ). 2.2.5.1Straightandlevelight Whentheaircraftifyingstraightandlevelduringrelease ,theightaxisis coincidentwiththeinertialaxis,resultingintherelatio nshipsshowninEquations ( 2–82 )and( 2–83 ). I r F = I v F = I F =0 (2–82) [ T ] FI = 266664 100010001 377775 (2–83) Itisclearthatthesimulationresults I r B I v B I B ,and [ T ] BI areuseddirectlyto relatetheposition,velocity,andorientationofthestore relativetotheaircraft.Ineffect, theaircraftitselfisusedasaninertialreferenceframe.T hisisthesimplestformof relativemotionandthemostcommonlyusedapproachinstore separationtrajectory prediction.2.2.5.2Steadyclimbordive Theorientationoftheinertialreferenceframewithrespec ttotheearthisdetermined bytheightpathangleoftheaircraftattheinstantthestor eisreleased.Whenthe aircraftisinasteadyclimbordive,theightpathanglerem ainsconstant.Asaresult, theightaxisremainscoincidentwiththeinertialaxisthr oughoutthetrajectory.Once again,thesimulationresults I r B I v B I B ,and [ T ] BI areuseddirectlytorelatethe position,velocity,andorientationofthestorerelativet otheaircraft.Notethatthe 60

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simulationresultswillingeneralbedifferentfromthecas eofstraightandlevelight,due totheorientationofthegravityforceactingonthestore.2.2.5.3Constantloadfactormaneuver Awings-levelconstantloadfactorpitch-planemaneuverca nbeidealizedasa circulararcinaplanenormaltothelocalearthhorizontal[ 21 ].Theidealizedmotion oftheaircraftleadstoaclosed-formsolutionfortheight axismotionrelativetothe inertialaxis.Thestraightforwardderivationispresente dbyKeen[ 21 ]andtheresultsare includedinthissection. Sincetheaircraftmotionisassumedtobeacirculararcwith constantforward velocity,theaccelerationoftheightaxisrelativetothe inertialaxisisgivenbyEquation ( 2–84 ),where I az F F isthenormalacceleration, N Z istheloadfactor(fornormalstraight andlevelight N Z =1 )and g isthelocalaccelerationofgravity. I a F = I az F F f z = ( 1 N Z ) g f z (2–84) Theangularvelocityoftheightaxeswithrespecttotheine rtialaxesisgivenby Equation( 2–85 ),where I q F F istheightaxispitchrateand W u I I isthex-componentofthe velocityoftheinertialaxisrelativetothewindaxis(equi valenttotheaircraftairspeedat release). I F = I q F F f y = I az F F W u I I f y (2–85) Duetotheplanarmotion,theorientationoftheightaxesre lativetotheinertial axesisgivenbydirectintegrationofthepitchrate. I F = t Z 0 I q F F dt = I q F F t (2–86) 61

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Theorientationiseasilyrepresentedusingasimpletransf ormationmatrix. [ T ] FI = T I F FI = 266664 cos I F 0 sin I F 010 sin I F 0cos I F 377775 (2–87) Usingthenormalaccelerationandpitchrateoftheconstant loadfactormaneuver, theapparentradiusofcurvatureisasshowninEquation( 2–88 ). R = I az F F I q F F 2 (2–88) Theradiusofcurvaturecanbeusedtodeterminetheposition andvelocityofthe ightaxesrelativetotheinertialaxes,asshowninEquatio ns( 2–89 )and( 2–90 ). I r F = I x I F i x + I y I F i y + I z I F i z I x I F = W u I I t R sin I F I y I F =0 I z I F = R 1 cos I F (2–89) I v F = I u I F i x + I v I F i y + I w I F i z I u I F = W u I I 1 cos I F I w I F = W u I I sin I F I v I F =0 (2–90) Equations( 2–84 )through( 2–90 )provideafullspecicationoftheightaxes motionwithrespecttotheinertialaxesforaconstantloadf actormaneuver.Since thedifferentialequationsprovidethemotionofthestorer elativetotheinertialaxes, theposition,orientation,andvelocityofthestorerelati vetotheightaxisarereadily determined. 62

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2.3AerodynamicModeling Theequationsofmotionforstoreseparationareaformidabl esetofdifferential equationsdueprimarilytothealgebraiccomplexityandnon linearcoupling,butthe approximatesolutiontothissystemofequationsisreadily obtainedusingstandard numericalsolvers.Ontheotherhand,theaerodynamicmodel ingwhichisnecessary toinformtheequationsofmotionissubstantiallymoredif cult.Aerodynamicsofight vehiclesisamajorstudyinandofitself,andstoreseparati oncomplicatesthematter byconsideringmutuallyinterferingaerodynamicsbetween thestoreandaircraft. Thenonuniforminterferenceoweldrequiresthattheaero dynamicloadsonthe storetakeintoaccountthelocationandorientationofthes torerelativetotheaircraft. Aerodynamiccoefcientsareusedtomodeltheforcesandmom entsactingonthestore inightandthedelta-coefcientmethodologyisusedtoacc ountforthespatialvarying aerodynamicscausedbythenonuniforminterferenceowel d.Thesemethodsare wellestablishedintheightdynamicscommunityingeneral ,andthestoreseparation communityinparticular.2.3.1AerodynamicCoefcients Theaerodynamicforcesandmomentsarisefromairpressurea ctingonthe surfaceofthestore.Thepressuredistributionisafunctio noftheoweld(andvaries withlocationinanonuniformoweld)aswellasthemotiono fthestorethroughthe surroundingairmass.Inpractice,thepressuredistributi onismodeledasasingle force/momentsetactingatthestorecenterofgravity.Thes edimensionalforcesand momentsarerepresentedbytheterms F A i and M A i i = x y z ,inEquations( 2–34a ) through( 2–34c )and( 2–37a )through( 2–37c ). Theaerodynamicforcesandmomentsarenon-dimensionalize dbasedonthesize oftheightvehicleandthedynamicpressureoftheairow.I nawindtunnel,forces andmomentsareusuallymeasuredandanalyzedinthebodyaxi s,givingrisetothe so-called“bodyaxiscoefcients”: C X or C A C Y C Z or C N C l C m and C n .Additionally, 63

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theforcesandmomentsarephysicallyrelatedtothemagnitu deanddirectionofthe storevelocityrelativetotheair,givingrisetothe“winda xiscoefcients”: C D C Y C L Notethatthemomentcoefcientsarethesameforbothsystem s.Bothsystemsare usedprevalentlyinpractice.Bodyaxiscoefcientsareuse dexclusivelytoquantifythe aerodynamicinterferencebetweenthestoreandaircraft,b utwind-axiscoefcientsare usefulforbuildingphysicalintuitionandforautomaticco ntrolsystemdesign. Thenon-dimensionalforcecoefcientsarerelatedtothedi mensionalaerodynamic forcesandscaledbythedynamicpressure, q 1 ,andreferencearea, S ref C i = F i q 1 S ref for i = X Y Z L D (2–91) Similarly,thenon-dimensionalmomentcoefcientsaresca ledbythedynamic pressure,referencearea,andreferencelength, l ref ,asshowninEquation( 2–92 ).In general,thereferenceareaandreferencelengthmaybediff erentbetweencoefcients. However,moststoresareaxisymmetricornearlysoandthepa rametersarethesame forallforceandmomentcoefcients. C i = M i q 1 S ref l ref for i = l m n (2–92) Conversionbetweenthebodyaxisandwindaxisforcecoefci entsisdependent onthestoreangleofattackandreadilyobtainedfromthetra nsformationmatrixin Equation( 2–93 ). 266664 C D C Y C L 377775 = 266664 cos s 0sin s 010 sin s 0cos s 377775 266664 C X C Y C Z 377775 (2–93) Variousmethodsareusedtodeterminethevalueoftheaerody namiccoefcients foraparticularightconditionandtrajectoryofaightve hicle,includinganalytical, numerical,andempiricalmethods.Analyticalmethodsbase donrstprincipleswill provetobetoosimplisticformoststoreseparationproblem s.Numericalmethods, 64

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includingCFD,arewidelyusedtoestimateaerodynamiccoef cientsforstoreseparation, butrequiresufcientcomputationalresources.Empirical methods,basedprimarilyon windtunneldataandaccomplishedusingadelta-coefcient methodology,provide adirectsourceforrapidlygeneratingstoreseparationtra jectoriesforidentication andcontrolapplications.InChapter3,systemidenticati onwillbeusedtointroduce asemi-empiricalformulationoftheaerodynamiccoefcien ts.Thesemi-empirical classicationisduetoapostulatedanalyticalformofthee quationscompletedwith parametersestimatedfromempiricaldata.2.3.2Delta-CoefcientMethodology Theaerodynamicloadsactingonthestorearedueprimarilyt othreeidentiable physicalcontributions,including(1)thelocalowveloci tycombinedwiththeposition andorientationofthestore,(2)thetranslationalvelocit yofthestorethroughtheow eld,and(3)therotationalvelocityofthestorethroughth eoweld[ 21 ].Physically, thesecontributionsareinterrelatedandinseparable.Com putationalFluidDynamics hastheadvantageoftreatingtheseeffectsinaphysicallym eaningfulwaybysolving fortheoweldsurroundingthestoreduringthetrajectory andintegratingthepressure distributiondirectlytoobtaintheaerodynamicloads.Inc ontrast,whenwindtunnel andanalyticalmethodsareapplied,theeffectsarenecessa rilydeterminedseparately andsubsequentlycombinedtoestimatethetotalaerodynami cloads.Experiencehas shownthatthisbuild-upapproachissufcientlyaccuratef ormostapplicationsinstore separation. Ingeneral,theaerodynamicloadsonaightvehicleinunifo rmowdepend nonlinearlyonpresentandpastvaluesofairspeed,angleso fincidence,linearand angularaccelerations,controlsurfacedeections,unste adyowproperties,viscous forces,compressibilityeffects,atmosphericproperties ,andotherphysicalfactors [ 24 ].Forastoreinthevicinityoftheaircraft,eachoftheseef fectsiscomplicatedby thevariationoftheoweldinspaceandtime.Afulldescrip tionoftheaerodynamic 65

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loadsisnotpracticalandinpracticeisnotnecessary.Rath er,avarietyofsimplifying assumptionsareintroduced. 1. Theaerodynamiceffectsareassumedtobeseparable(uncoup led)allowing superpositiontobeusedtodeterminethetotalaerodynamic loads.Thisisa practicalassumptionthatisnotrigorouslyjustiedbutis usuallynecessaryto maketheproblemtractable. 2. Theowisquasi-steady,whichimpliesthattheoweldreac hesasteadystate instantaneouslyafterchangesintheboundaryconditions. Thisassumptionlimits theaerodynamicloadstodependencyonthepresentvaluesof theindependent variables,ignoringtheeffectofpastvaluesoftheindepen dentvariables.Some nonlineareffects,suchashysteresis,areneglectedunder thisassumption. 3. Theatmospheresurroundingtheaircraftisuniformandunal teredbythepresence oftheaircraftandstore.Thisimpliesthattheatmospheric properties(pressure, temperature,density)areconstantandthatlocalatmosphe ricwindsaremonolithic withaconstanttranslationalvelocity.Thisassumptionis reasonable,duetothe shortdurationofatypicalstoreseparationevent(approxi mately1second). 4. Theaircraftisinasteady-stateightconditionandthecor respondingowaround theaircraftissteady(e.g.nottime-variant).Thisassump tionallowsuseof wind-tunnelmeasurementsthataretime-averagedvalues,e quivalenttomean owcomponentswhenturbulentuctuationsarepresent.Thi sassumptionis justiedwhen(a)thelengthscalesoftheturbulencearesig nicantlysmaller thanthelengthscaleofthestore,and(b)thefrequenciesof theturbulence aresignicantlyhigherthanthefrequenciesofstoremotio n,asisthecasein mostcircumstances.Asanotableexception,large-scaleva riationsinoweld propertiescausedbycavityaerodynamicsforweaponsbaysh avebeensuspected tocausesignicantvariationsinstoretrajectories[ 25 26 ].However,currentstore separationwindtunneltestmethodologiesareinsufcient formodelingthese effects.CFDremainstheprimarymethodologyfordealingwi thunsteadyow effectsonstoreseparation.TheF-16casestudyusedinthis researchisa“clean” congurationwithastorereleasedfromawingstationinstr aightandlevelight, soturbulenceisnotexpectedtobeasignicantissue. Theseprimaryassumptionsallowaconsiderablereductioni nthecomplexityof theaerodynamicmodeling.Whentheassumptionsaboveareim posed,thestore aerodynamicloadsareafunctionoftheaircraftvelocityan dorientation,theposition andorientationofthestorerelativetotheaircraft,andth etranslationalandrotational velocityofthestorerelativetotheaircraft.Infunctiona lform,thenon-dimensional 66

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forceormomentcoefcientscanbewrittenasanonlinearfun ctionoftheindependent parameters. C i ( t )= f M A A ; r B = A ( t ), B = A ( t ), A v B ( t ), A B ( t ), ( t ) (2–94) InEquation( 2–94 ), C i representsanyoftheaerodynamiccoefcients, M isthe aircraftMachnumber,and A and A aretheaircraftincidenceangles(eachassumed constantthroughoutthetrajectoryconsistentwithastead ystateightcondition).The time-dependentparameters r B = A ( t ) B = A ( t ) A v B ( t ) A B ( t ) ,and ( t ) ,aretheposition, orientation,velocity,andangularvelocityofthestorere lativetotheaircraftandthe controlsurfacedeections,respectively. Evenwiththepriorsimplications,theabovefunctionalfo rmistoocomplicated tobeofpracticaluse.Givensufcientresources,currentq uasi-staticwindtunneltest methodologiesarecapableofcapturingtheaerodynamicdep endencieson r B = A ( t ) and B = A ( t ) .Thisisaccomplishedbyplacingthestoreatamatrixofpred etermined positionsandattitudesinthevicinityoftheaircraftandr ecordingthetime-averaged aerodynamicloads(theresultingdatasetisreferredtoasg riddata).However,the dependencieson A v B ( t ) and A B ( t ) wouldrequireadynamicrigcapableofperturbing thestore(inthevicinityoftheaircraft)atscaledvelocit iesrepresentativeofthose encounteredinight.Fortunately,thetranslationalandr otationalvelocitiesofthestore neartheaircraftarerelativelysmall,andtheaerodynamic dependenciesonthedynamic parameters A v B ( t ) and A B ( t ) canbeadequatelyapproximatedusingauniform oweld(e.gneglectingtheaircraftinterference).Thus, theaerodynamiceffectsare decomposedintocontributionsfromauniformandnonunifor moweld.Notethatthis limitationdoesnotapplytoCFDmethods,andthedynamicint erferenceeffectscouldin principlebeestimatedusingCFD. Theindependentparameters r B = A ( t ) and B = A ( t ) representsixquantities(namely, threepositions x B = A y B = A z B = A andthreerotations A B A B AB )thatmaytakeonarange 67

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ofvaluesduringanygiventrajectory.Toestablishawindtu nneldatabasewithsufcient coverageforallsixparametersisprohibitivelyexpensive .Forinstance,ifeachvariable wereevaluatedattendifferentvalues,thedatabasewouldr equire 10 6 distinctdata points.Thiscomplexityiscompoundedbythevariousaircra ftcongurationsandight conditions.Clearly,amoreefcientapproachisrequired. Typically,spatialvariationswith y B = A areneglectedtoreducetheparameterspace andconnetheaerodynamicinterferenceeffectstothe x–z planealignedwiththestore atcarriage.Variationswith x B = A and z B = A arecombinedintoasingleradialdirection r B = A projectingverticallydownwardandslightlyaftoftherele asepoint,withtheradial directionchosentoberepresentativeoftheafttranslatio nofthestoreduringseparation duetoaerodynamicdrag.Measurementsalongthisradialdir ectionareaccomplished usingaquasi-staticsweepthateconomicallycapturesthea erodynamicfeaturesover anemeshofdiscretevalues.Theyawandpitchattitudes, A B and A B ,aredecoupled fromtherollattitude, AB ,requiringfarfewercombinedorientations(andimplicitl y neglectinginteractionswiththerollangle,alimitationt hatispartiallycorrectedusing anofineapproximation).Thenetresultofthesepractical reductionsisasignicantly smallerparameterspace.However,evenwiththisreducedpa rameterspace,asufcient numberofdatapointsareprohibitivelyexpensivetoobtain inacostlywind-tunneltest. Typically,thenalpredeterminedgridvaluesselectedfor theangularorientations A B A B ,and AB ,aretoosparsetobeusedindependently,i.e.thegriddatac ollected areinsufcienttoadequatelydescribetheaerodynamicsof thestoreneartheaircraft. Rather,inamannersimilartothedynamicparameters,theae rodynamicdependencies onthestorepositionandorientationaredecomposedfurthe rintouniformand nonuniformcontributions.Theuniformowcontributionsa reindependentofthe aircraftandrapidlyachievedoveralargerrangeandnerin crementsofindependent parameters.Thenonuniformowcontributionsarethenadde dasanincremental correctionduetothepresenceoftheaircraft.Thisso-call eddelta-coefcientapproach 68

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(rstdevelopedbyBamber[ 27 ]andfurtherdocumentedbyMorgret[ 28 ])providesan efcientandeconomicalwaytomapthespatiallyvariantaer odynamiccharacteristics. Thedecompositionoftheaerodynamiccontributionsisassu medtobeindependent andsuperpositionisusedtodeterminethecumulativeaerod ynamicloads.Thus, thetotalaerodynamiccoefcientisgivenbyaninitialfree stream(FS)estimate,plus correctionsforthedynamiceffectsduetotranslational(T RANS)androtational(ROT) velocities,theaircraftinterferenceorgrid(GR)data,an dtheuniformowcontrol (C)effect.Theaerodynamicmodelisrepresentedbysuperpo sitionofindividual contributions,asshowninEquation( 2–95 ). C = C FS + C GR + C DD + C C (2–95) Thesecontributionsarefrequentlydeterminedusingsepar ateaerodynamictables obtainedfromwindtunneltesting,tabulatedasafunctiono fmultipleindependent parameters.Infunctionalform,thecoefcientisdetermin edbytherelationshipshown inEquation( 2–96 ),where I p B B I q B B ,and I r B B aretherotationalvelocitiesrelativetothe inertialwindframeand e a ,and r aretheelevator,aileron,andrudderdeections, respectively. C = C FS ( M S S ) + C G r B = A A B A B AB + C ROT I p B B I q B B I r B B + C C ( e a r ) (2–96) Notethatthetranslationalvelocitiesofthestorerelativ etothewind, W u B B W v B B ,and W w B B areimplicitinthecomputationof s and s ,usingtheancillarywindaxisequations developedintheprevioussection.Thus,theterm C TRANS isquitenaturallyaccounted forthrough C FS ( M s s ) Theprimaryrationaleforusingthedelta-coefcientmetho d,asopposedtodirect interpolationofmeasuredgriddata,isthatarelativelysm allfreestreamdatabase 69

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canbeleveragedefcientlytoextendtherangeofapplicabi lityofagriddatabase [ 28 ].Thefreestreamdataisoftenavailableatanerresolutio n,andsometimeswith increasedaccuracyduetoaseparatelarge-scalefreestrea mwindtunneltest,sothe delta-coefcientmethodprovidesasignicantimprovemen tinmodelingaccuracyand efciency.Thedelta-coefcientmethodhasbeenusedsucce ssfullyformanyyears withinthestoreseparationcommunityandexperiencehassh ownittobequitereliable whensufcientgriddataareavailableandowconditionsar enottoodissimilarfrom freestreamconditions[ 28 ]. 2.3.3RepresentativeCaseStudy ThediscussioninSection 2.3 onaerodynamicmodelingforstoreseparation isclariedbyconsiderationofaparticularrepresentativ eexample.Thecasestudy selectedhighlightsmanyofthemodelingprecedentsdiscus sedthusfarandprovides asolidframeworkforinvestigatingsystemidenticationa ndtrajectoryoptimization techniques.2.3.3.1Freestreamdata TherepresentativefreestreamdatawerecollectedattheAr noldEngineering DevelopmentCenter(AEDC)testfacilityinthe4Ttransonic windtunnel[ 29 ].The AerodynamicWindTunnel4Tisaclosed-loop,continuous-o w,variable-density tunnelwithaMachnumberrangefrom0.05to2.0.Thetestsect ionis4ftsquare and12.5ftlongwithperforatedwallstoreduceboundarylay ereffects.Thetestwas accomplishedusinga 1 = 20 th scalestoremodelandincludesMachnumbersranging from0.8to1.2andstoreincidenceangles( s and s )rangingfrom-40to+40degrees. Theaerodynamicforcesandmomentsweremeasuredwitha0.18 8-indiameter, sting-mounted,strain-gaugedmomentbalance.Thefreestr eamwindtunneltestwas accomplishedinJune2009andisdocumentedinAEDC-TR-09-F -19[ 1 ]. 70

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Thefreestreamdatawerecollectedinaseriesofangleofatt achsweeps(i.e.alpha sweeps)atxedsideslipangles(i.e.xedbeta).Figure 2-6 showsasnapshotofa seriesofalphasweepsatmultiplesideslipangles. -20 -10 0 10 20 -2 0 2 Yaw Moment Coeff, C nAngle of Attack, a s (deg) -20 -10 0 10 20 -1 0 1 Side Force Coeff, C YAngle of Attack, a s (deg) -20 -10 0 10 20 -2 0 2 Pitch Moment Coeff, C m b s = 10 (deg) b s = 5 b s = 0 b s = -5 b s = -10 -20 -10 0 10 20 -2 0 2 4 Normal Force Coeff, C N Figure2-6.Freestreamaerodynamiccoefcientsvs.angleo fattackatxedsideslip angleforarepresentative 1 = 20 th scalemodelatMach0.8. Therepresentativestoreisnearlyaxisymmetricandthesym metryisapparentin thecollectedfreestreamdata,especiallyinthelateralsi deforceandyawingmoment measurements, C Y and C n .Therelationshipbetween C N and s demonstratesan increasingnormalforceathigheranglesofattackconsiste ntwithphysicalintuition. Thelinearityofthe C N s curveoverarelativelylargerangeof s isaninteresting andbenecialfeatureforsubsequentapplicationofsystem identication.Thenegative slopeofthepitchingmomentcoefcientoverarangeof s ,especiallynear s =0 isindicativeofanegativepitchmomentderivative, C m < 0 ,aclassicalcondition necessary(butnotsufcient)forstaticlongitudinalstab ility. Itisclearthatthefreestreamdataarewellbehavedandcons istentovertherange ofindependentvariablespresentedthusfar.However,sepa ration-inducedtransients maydrivethestorewellbeyondthelinearrangeofaerodynam ics.Thereforethe freestreamdataarecollectedatawiderangeof s and s .Figure 2-7 showsthealpha 71

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sweepsextendedtothefull+/-40degrangewith2degincreme nts,withxedbeta valuesfrom-40to+40degat20degincrements.Thecoarsenes softhexedbeta valuesischosenforclarity;thedatabaseincludesbetaval uesovertheentire+/-40deg rangeat5degincrements. -40 -20 0 20 40 -5 0 5 Yaw Moment Coeff, C nAngle of Attack, a s (deg) -40 -20 0 20 40 -4 -2 0 2 Pitch Moment Coeff, C mAngle of Attack, a s (deg) b s = 40 (deg) b s = 20 b s = 0 b s = -20 b s = -40 Figure2-7.PitchingandYawingmomentcoefcientvs.angle ofattackfora representative 1 = 20 th scalemodelatMach0.8forfullrangeofangleof attackandsideslipangle. Itisclearthatthefullrangeof s and s includesignicantnonlinearitiesand simplisticanalyticalmodelsarenotfeasible.Rather,the freestreamdatasimilartothose inFigure 2-7 arecollectedintoafreestreamdatabasesuitableforfurth eranalysisand interpolationduringsimulation.2.3.3.2Griddata Thefreestreamdataareusedtodeterminethestoreaerodyna micsinuniform owapplicabletofar-eldconditions.Intheproximityoft heaircraft,theaerodynamics arespatiallyvariantduetothenonuniformoweld.Theaer odynamicinterferenceis modeledusingthedelta-coefcientsdescribedinSection 2.3.2 .Thedelta-coefcients aredeterminedonlineduringthewindtunneltestbymeasuri ngthetotalforcesand momentsactingonthestoreinproximitytotheaircraft,and thensubtractingoffthe interpolatedfreestreamcoefcientsforthesameightcon ditionsandincidenceangles. Measurementofthegriddatarequiresadual-supportstruct ureinthewindtunnel,to 72

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articulateboththeaircraftandthestoreindependently(t heaircraftremainsstationary duringtheverticalsweepofthestore).Theinvertedaircra ftwindtunnelmodeland metricstoreareshowninFigure 2-8 Figure2-8.DualsupportmechanismforF-16storeseparatio nwindtunneltest.Excerpt fromAEDC-TR-09-F-19[ 1 ]. Thegriddataarecollectedasafunctionofthestorepositio nandattituderelativeto theaircraft(3dimensions),aswellastheightconditions andaircraftangle-of-attack, foratotalof5independentvariablesforeachaircraftcon guration.Typically,the Machnumberandaircraftangle-of-attackareheldconstant throughoutthesimulated trajectory,soathree-dimensionalinterpolationissufc ientfordeterminingthe aerodynamiccoefcientsofinterest.Figure 2-9 showsasnapshotofrepresentative griddatadelta-coefcientsfortherepresentative 1 = 20 th scalemodel. ThegriddatapresentedinFigure 2-9 illustratethevariationinthepitchingand yawingmomentdeltacoefcientswithverticalposition.Fo rillustration,thepitching momentvariationisshownforvariousxedpitchanglesandt heyawingmoment variationisshownforvariousyawangles.Thecollecteddat aincludeeverycombination 73

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0 10 20 30 -3 -2.5 -2 -1.5 -1 -0.5 0 Pitch Moment Delta Coeff, D C mVertical Position, Z (ft) q = 10 (deg) q = 0 q = -10 q = -20 q = -30 0 10 20 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Yaw Moment Delta Coeff, D C nVertical Position, Z (ft) y = 10 (deg) y = 0 y = -10 Figure2-9.Aerodynamicpitchingmomentandyawingmomentd eltacoefcientsvs. verticalstorepositionforvariouspitchandyawangles,fo rarepresentative 1 = 20 th scaledmodelatMach0.8. ofthe5discretepitchanglesand3discreteyawangles,fora combinationof15distinct attitudes.Eachattitudeisheldconstantthroughouttheve rticalsweep.Thevertical sweepisconductedslowly,sothatthedataateachdiscretez -locationrepresent quasi-static,time-averageddata.Thecollecteddataareo rganizedbypositionand attitudeinadatabasesuitableforfurtheranalysisormult i-dimensionalinterpolation. ThemostprominentfeatureofthegriddatashowninFigure 2-9 istheconsistent decayofthedelta-coefcientswithincreasingdistancefr omtheaircraft.Forthis particularapplicationtheaircraftinuenceisnegligibl e( C G =0 )beyond30feet, indicatingauniformoweldinwhichthestoreaerodynamic scanbemodeledusing onlyfreestreamdata.Forcomparison,Figure 2-10 showssimilardatacollectedatMach 1.2. Again,theinuenceoftheaircraftoweldonthestoreaero dynamicsisnegligible beyond30ft.However,ratherthanaconsistentdecay,there isasignicantperturbation inthecoefcientsbetween10and20feetbelowtheaircraft. Thislevelofcomplexity intheoweldisnotunusualanddemonstratestheimportanc eofahigh-resolution databaseforstoreseparationmodelingandsimulation. 74

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0 10 20 30 0 0.2 0.4 0.6 0.8 1 Yaw Moment Delta Coeff, D C nVertical Position, Z (ft) y = 10 (deg) y = 0 y = -10 0 10 20 30 -2.5 -2 -1.5 -1 -0.5 0 0.5 Pitch Moment Delta Coeff, D C mVertical Position, Z (ft) q = 10 (deg) q = 0 q = -10 q = -20 q = -30 Figure2-10.Aerodynamicpitchingmomentandyawingmoment deltacoefcientsvs. verticalstorepositionforvariouspitchandyawangles,fo rarepresentative 1 = 20 th scaledmodelatMach1.2. Foralargeraircraft,theverticalpositionatwhichthespa tialvariationisnegligible willbedifferent,butthetrendswillbesimilar.Theuniver salityofthedelta-coefcient decayinfar-eldconditionsisanimportantfeatureuseful fordataqualitycheckinganda characteristicthatwillbeexploitedintheapplicationof systemidenticationmethodsto modelingspatiallyvariantaerodynamics. 2.4FlightTestValidation Inordertomakethestoreseparationproblemtractable,ava rietyofassumptions havebeenintroduceduptothispoint.Afewoftheseassumpti onsarerelatedtothe kinematicsanddynamics,butmostarerelatedtotheaerodyn amiccharacteristics. Someoftheseassumptionsarejustiedbyexaminingthenatu reoftheproblem(and haveaninsignicanteffectontheresult)andsomeareneces sarytomakethesolution accessibleusingpracticalmethods(andmayhaveasignica nteffectontheresultthat isdifculttoquantify).Giventhevarietyofassumptionst hathavebeenproposed,the questionfacedisoneofaccuracy:istheproposedmethodolo gysufcientlyaccurateto characterizethestoretrajectoryduringseparation? Themostdirectanswertothisquestioncomesfromcompariso ntoighttestdata. Flighttestingprovidesadependablewaytovalidatethesim ulation,andsimulation providesaneffectivewayofreducingighttesting.Assuch ,aniterativeapproach 75

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toighttestingandsimulationvalidationhasproventobea neffectivemethodfor accomplishingstoreseparationanalyses.2.4.1FlightTestDataReduction Thetwoprimarysourcesofdataforstoreseparationightte stvalidationare photogrammetricandtelemetrydata.Photogrammetricdata areobtainedfrom high-speedvideocamerasmountedontheaircraftinlocatio nsthatminimizeow elddisturbanceandprovidemultipleperspectivesforobs ervingthemotionofthestore relativetotheaircraft.Photogrammetricdataareusedini tiallyforqualitativeassessment ofthestoretrajectory,butspecializedtrackingsoftware isalsousedtodeterminea quantitativedescriptionofthestoretrajectory[ 30 ]. Telemetrydataareobtainedfromaninertialmeasurementun it(IMU)attached tothestoreandtransmittedtoanindependentgroundstatio nduringseparation. TheSpectrumSensors&Controls65210Asixdegree-of-freed om(6DOF)inertial measurementunitprovidestri-axialaccelerationandangu larratemeasurements.The self-containedIMUwithabuilt-inFMtransmitterprovides simultaneoussamplingatup to42,500samples/secwithrecongurableoutputrangeand lterfrequency.Thesedata canbeusedtocompletelyreconstructthestoreseparationt rajectoryandanalyzethe trajectoryrelativetotheaircraft[ 31 32 ]. 2.4.1.1Trajectoryreconstruction Thepurposeoftrajectoryreconstructionistoutilizethem easuredratesand accelerationstoestimatetheremainingstatevariables.R ecognizingthepresence ofmeasurementnoise,thetrajectoryreconstructionproce sscouldbeaccomplished usingnon-deterministicstateestimationmethods,suchas Kalmanlteringoroptimal smoothing.However,deterministicmethodsbasedonkinema ticrelationshipstypically providesufcientaccuracyovertheshorttimeintervalofi nterestforstoreseparation purposes. 76

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TheIMUattachedtothestoremeasuresthestoreangularrate sinthebodyaxis, whichareequivalenttothegeneralizedangularvelocities showninEquation( 2–97 ). I B = I p B B b x + I q B B b y + I r B B b z (2–97) Themeasuredbodyratescanbeusedtodeterminetheorientat ionofthestore relativetotheinertialreferenceframebysolvingfortheE uleranglesusingthekinematic differentialequationsinstatespaceform. 266664 IB I B I B 377775 = 266664 I p B B + I r B B cos IB tan I B + I q B B tan I B sin IB I q B B cos IB I r B B sin IB ( I r B B cos IB + I q B B sin IB )sec I B 377775 (2–98) Thesethreescalarnonlinearequationsarecoupledandmust besolvedsimultaneously usingadifferentialequationsolver,suchasa4thorderRun geKuttaalgorithm.Notethat thekinematicrelationshipshaveasingularityat I B = 90 deg ,whichcorrespondstoa straightclimbordive.Thissituationrarelyarisesinstor eseparation,butnonetheless quaternionrelationshipscanbeusedinsteadforamorerobu stmethodofdetermining theEulerangles[ 33 ]. Oncetheorientationofthestorebodyaxiswithrespecttoth einertialaxisis available,thetranslationalkinematicscanbeconsidered .Recallthatthepositionofthe storeCGrelativetotheoriginoftheinertialaxissystemis givenby r B .Similarly,letthe positionofthetelemetryIMUcenterofnavigationrelative totheinertialaxissystembe denotedby r T .ThenthepositionofthestoreCGrelativetotheIMUisdenot edby r B = T ThethreevectorsarerelatedbyEquation( 2–99 ). r B = r T + r B = T (2–99) 77

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Takingthederivativeoftheaboveexpression,asseenbyano bserverxedinthe inertialreferenceframegivesEquation( 2–100 ). I d r B dt = I d r T dt + I d r B = T dt (2–100) Theterm r B = T isxedinbodycoordinates(assumingtheCGisxed)andknow n frompre-ightmeasurements.Thederivativeismostefcie ntlyevaluatedusingthe transporttheorem. I d r B dt = I d r T dt + B d r B = T dt + I B r B = T (2–101) Notethattheterm B d r B = T dt =0 sincethepositionoftheCGrelativetotheIMUis constant.Alsonotethatthederivativeof r B isequivalenttothevelocityofthestoreCG, I d r B / dt = I v B .Withthesesimplications,Equation( 2–101 )canberewrittenasshown inEquation( 2–102 ). I v B = I v T + I B r B = T (2–102) IntroducingtheaccelerationofthestoreCGas I a B ,takingthetimederivativeof Equation( 2–102 )andsimultaneouslyapplyingthetransporttheoremgivest heresult showninEquation( 2–103 ),where I B istheangularaccelerationofthestorerelative totheinertialaxis.Theangularaccelerationisdetermine dbynumericaldifferentiation (withappropriatesmoothing)ofthemeasuredangularrates I a B = I a T + I B r B = T + I B I B r B = T (2–103) Equation( 2–103 )providesacoordinate-freeexpressionfortheaccelerati on ofthestoreCGwhenthetranslationalandrotationalaccele rationsareknownfrom measurements.Sincethemeasurementsarenecessarilymade inthebodyaxis, equation( 2–103 )istypicallyevaluatedinthebodyaxis.Theresultingacce lerations areeasilytransformedfromthebodyaxistotheinertialaxi susingacoordinate transformation. I a B I = [ T ] IB I a B B (2–104) 78

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Oncetheaccelerationsareexpressedintheinertialaxis,t heaccelerationdue togravitycanbeaccountedfor,dependingonthetypeofacce lerometersused (someaccelerometersmeasureabsoluteacceleration,othe rsmeasurechangesin acceleration).Thevelocityandpositionofthestoreinine rtialcoordinatescanbe determinedviadirectintegrationofthetransformedaccel erations. I v B ( t ) = t Z 0 I a B ( t ) dt (2–105) r B ( t ) = t Z 0 I v B ( t ) dt (2–106) Oncethetrajectoryreconstructionisaccomplished,acomp letedescriptionofthe storetrajectoryduringseparationisavailable,providin gavaluableresourceforfurther analysisincludingdeterminationofanaircraft-relative trajectoryandinvestigationofthe in-ightaerodynamiccharacteristics.2.4.1.2Trajectoryanalysis Theinertialtrajectorydeterminedfromtrajectoryrecons tructionismathematically equivalenttoaninertialtrajectorygeneratedviasimulat ion.Therefore,oncethe reconstructedtrajectoryisquantiedininertialcoordin ates,thestoremotionrelative totheaircraftcanbedeterminedusingthemathematicalrel ationshipsdescribedin Section 2.2.5 for(1)anarbitraryaircraftmaneuveror(2)asteady-state aircrafttrajectory including(a)straightandlevelight,(b)steadyclimbord ive,or(c)steadypull-upor push-over.Theresultingaircraft-relativetrajectoryis suitableforqualitativecomparison tophotogrammetricdataobtainedfromcamerasxedtotheai rcraft. Oncethetwelvestatevariableshavebeenestimatedusingth etrajectoryreconstruction process,the6DOFequationsofmotioncanbeusedtorelateth eobservedmotionto thecorrespondingaerodynamicforcesresponsibleforthem otion.Returningtothe translationalandrotationalequationsofmotiongivenbyE quations( 2–28 )and( 2–37 ),it isapparentthatwhenthestoremotionisknownapriori,dete rminingtheaerodynamic 79

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forcesandmomentscanbeaccomplishedalgebraically,apro cesssometimesreferred toasreversedynamics.Thus,ighttestdatacanbeuseddire ctlytoestimatethe aerodynamicforcesandmomentsactingonthestoreduringse paration.Thisprovidesa valuableresourceforvalidatingtheseparationcharacter isticsandaerodynamicmodels. Considerationofaparticularexamplewillprovideincreas edclaritytothetrajectory reconstructionandtrajectoryanalysisprocedures.2.4.2FlightTestResults Tworepresentativecaseswereselectedforcomparisonofsi mulationandight testdataincludingonereleaseatMach0.9/550KCASandanot heratMach1.2/600 KCAS.Bothighttestswereaccomplishedinawings-level,s teady,unacceleratedight condition.Theinert,unguided,instrumentedseparationt estvehiclewasequippedwith anhigh-qualityinertialmeasurementunit.Themeasured6D OFtelemetrydataforthe Mach1.2releaseisshowninFigure 2-11 3 4 5 6 7 8 -1 0 1 2 Long., a xAcceleration (g) 3 4 5 6 7 8 -4 -2 0 2 Lat., a y 3 4 5 6 7 8 -20 -10 0 10 Time (sec)Vertical, a z 3 4 5 6 7 8 -200 -100 0 100 Roll, pAngular Rate (deg/sec) 3 4 5 6 7 8 -200 0 200 Pitch, q 3 4 5 6 7 8 -100 0 100 Time (sec)Yaw, r Flight Test Figure2-11.Measured6DOFtelemetrydataforF-16Separati onFlightTest4535 (Mach1.2/600KCAS). Thetelemetrydatashownincludesthetranslationalaccele rations a x a y a z andangularvelocities, p q r ,correspondingtoscalarcomponentsoftheinertial vectors I a B B and I B B ,respectively.Thelargepeakinverticalacceleration( a z ) 80

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atapproximately3.5secondsistheejectionforceandrepre sentsthestartofthe separationevent(redenedas t =0 forsubsequentplots).Theinherentstabilityof thestoreisevidentfromthetelemetrydatainthatthesepar ation-inducedtransients diminishafterabout5secondsofighttime,drivingthesto retowardasteady-stateight condition.However,itisdifculttovisualizethenatureo fthestoretrajectorynearthe aircraftusingonlythetelemetrydata.Thetrajectoryreco nstructionprocessprovides amoredirectanalysis.Thereconstructedtrajectory,alon gwithawindtunnelbased simulation,isshowninFigure 2-12 0 0.2 0.4 0.6 0.8 1 -10 0 10 20 Time (sec)Yaw, y 0 0.2 0.4 0.6 0.8 1 -40 -20 0 20 Pitch, q 0 0.2 0.4 0.6 0.8 1 -60 -40 -20 0 20 Roll, fOrientation (deg) Simulation Flight Test 0 0.2 0.4 0.6 0.8 1 -20 0 20 40 60 Time (sec)Vert. Z 0 0.2 0.4 0.6 0.8 1 -2 0 2 4 Lat., Y 0 0.2 0.4 0.6 0.8 1 -30 -20 -10 0 10 Long., XTranslation (ft) Figure2-12.Reconstructedighttesttrajectorycomparis onwithwindtunnelbased simulationforF-16SeparationFlightTest4535(Mach1.2/6 00KCAS). Theighttesttrajectoryininertialcoordinates( X Y Z correspondingtothe scalarcomponentsof [ r B ] I and correspondingto IB I B I B )wasdetermined usingthetrajectoryreconstructionprocessdescribedinS ection 2.4.1.1 .Thesimulated trajectorywasdeterminedusingthewindtunnelbasedmetho dsdiscussedinSection 2.3 .Thecomparisonshowsexcellentagreementfortheprimaryd egreesoffreedom ofinterest(namely,pitchangleandverticaltranslation) .Anotabledeviationinrollis 81

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apparentfromthecomparison.Thisdeviationisnotunusual forstoreseparationand isduetoacombinationoflargeuncertaintiesinwindtunnel measurementsduetoan exceedinglysmallrollmomentmeasurementandalowroll-ax ismomentofinertiaofthe full-scalestore(thus,thetrajectoryissensitivetoavar iablethatisdifculttomeasure accurately).Forthenearlyaxisymmetricstoreusedinthis analysis,thedeviationinroll anglebetweenightandsimulationdoesnotpresentaproble m.Figure 2-13 showsa similartrendforighttestdataatMach0.9/550KCAS. 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 10 Time (sec)Yaw, y 0 0.2 0.4 0.6 0.8 1 -40 -20 0 20 Pitch, q 0 0.2 0.4 0.6 0.8 1 -60 -40 -20 0 20 Roll, fOrientation (deg) Simulation Flight Test 0 0.2 0.4 0.6 0.8 1 -20 0 20 40 60 Time (sec)Vert. Z 0 0.2 0.4 0.6 0.8 1 -1 0 1 2 Lat. Y 0 0.2 0.4 0.6 0.8 1 -15 -10 -5 0 5 Long. XTranslation (ft) Figure2-13.Reconstructedighttesttrajectorycomparis onwithwindtunnelbased simulationforF-16SeparationFlightTest2265(Mach0.9/5 50KCAS). Theighttestandsimulationtrajectorycomparisonforthe Mach0.9release indicatessimilarresultstotheMach1.2release.Bothtraj ectoriesarecharacterized byaslightnose-downpitchangleneartheaircraft–anear-i dealseparationtrajectory. Recoverytoasmallnose-uppitchangleisevidentinbothcas es,againindicatingthe inherentlongitudinalstabilityofthestore.Figure 2-14 showsavisualcomparisonofthe ightandsimulationtrajectoriesfortheMach1.2release. Thesimilaritybetweenthe ighttestandsimulatedtrajectoriesandthecharacterist icmotionofthestorenearthe aircraftisevidentfromthevisualization. 82

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Figure2-14.Visualcomparisonofighttestandwindtunnel basedsimulation trajectoriesforF-16SeparationFlightTest4535(Mach1.2 /600KCAS). Forbothcases,theagreementbetweentheighttestandsimu latedtrajectory isverystrong,supportingthewindtunnelbasedtrajectory predictionmethodology. Consideringthecomparisonofestimatedaerodynamiccoef cientsfromtheighttest trajectorieswithwindtunnelpredictionsforanequivalen tprescribedtrajectoryprovides furthervalidationoftheaerodynamicmodelingapproach.F igures 2-15 and 2-16 show theaerodynamiccomparisonforbothighttests. Theighttestaerodynamiccoefcientsweredeterminedasp artofthetrajectory reconstructionprocess.Itisimportanttonotethat(1)the estimatesarenoisydueto thenumericaldifferentiation,despitesignicantsmooth ingofthemeasureddataand (2)thattheejectorforcesareincludedintheaerodynamicc oefcientsfortherst50 msecofthetrajectory.Forcomparison,thewindtunnelbase daerodynamiccoefcients weredeterminedusingthedelta-coefcientmethodologyfo rtheprescribedight testtrajectory.Theagreementbetweentheighttestandwi ndtunnelaerodynamic coefcientsisstronginbothcases.Thedeviationintheaxi alforcecoefcient, C A ,is expectedduetothedifcultyofmeasuringaxialforceinthe windtunnelforasmall-scale 83

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0 0.2 0.4 0.6 0.8 1 -4 -2 0 2 4 Time (sec)Yaw, C n 0 0.2 0.4 0.6 0.8 1 -4 -2 0 2 4 Pitch, C m 0 0.2 0.4 0.6 0.8 1 -0.4 -0.2 0 0.2 Roll, C lMoment Simulation Flight Test 0 0.2 0.4 0.6 0.8 1 -15 -10 -5 0 5 Time (sec)Normal, C N 0 0.2 0.4 0.6 0.8 1 -2 0 2 Side, C Y 0 0.2 0.4 0.6 0.8 1 -1 0 1 2 Axial, C AForce Figure2-15.Comparisonofighttestandwindtunnelaerody namiccoefcientsforF-16 SeparationFlightTest4535(Mach1.2/600KCAS). 0 0.2 0.4 0.6 0.8 1 -4 -2 0 2 Time (sec)Yaw, C n 0 0.2 0.4 0.6 0.8 1 -4 -2 0 2 4 Pitch, C m 0 0.2 0.4 0.6 0.8 1 -0.4 -0.2 0 0.2 Roll, C lMoment Simulation Flight Test 0 0.2 0.4 0.6 0.8 1 -15 -10 -5 0 5 Time (sec)Normal, C N 0 0.2 0.4 0.6 0.8 1 -1 0 1 2 Side, C Y 0 0.2 0.4 0.6 0.8 1 -1 0 1 2 Axial, C AForce Figure2-16.Comparisonofighttestandwindtunnelaerody namiccoefcientsforF-16 SeparationFlightTest2265(Mach0.9/550KCAS). model.Fortunately,theaxialforcehasonlyaminuteeffect ontheoveralltrajectory.Itis unclearifthehighfrequencyperturbationsintheighttes tdataearlyinthetrajectory (lessthan0.25sec)areduetonoiseinthemeasurement(perh apsstructuralvibration 84

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fromtheejection)orturbulentaerodynamics.Regardless, thetime-averagednature ofthewindtunneldataisclearlyevidentwhencomparedtome asuredighttestdata. Despitethesedifferences,thesimulationresultsshowast rongcorrelationwithight test,furthervalidatingthetrajectorypredictionmethod ology,providingavaluable resourceforfurtheranalysisandcontrolsystemdesign. 2.5ChapterSummary Thepurposeofthischapteristosummarizemathematicalmod elingofstore separationinordertoprovideaframeworkfortheremaining developmentsinsystem identicationandtrajectoryoptimization.Storeseparat ionisaninter-disciplinaryeld leveragingconceptsfromaerospace,mechanicalandsystem sengineering.Rigidbody dynamics,aerodynamics,windtunneltestingandighttest validationwerepresentedto providethenecessarybackgroundforimplementingsystemi denticationandtrajectory optimizationforguidedstoreseparation.Thestoresepara tionequationsofmotion werederived,usingasetofreferenceframesandcoordinate systemsuniquetostore separation,andtheequationswereextendedtodeterminean aircraft-relativetrajectory formultipletypesofaircraftmaneuvers.Thewell-establi sheddelta-coefcientmethod waspresentedasanextensiontotraditionalaerodynamicmo delingforightdynamics. Representativewindtunneldatawereincludedtoclarifyth eaerodynamicmodeling approach.Finally,ighttesttrajectoryreconstructionf rominertialmeasurementswas consideredandrepresentativeighttestdatawerepresent ed.Comparisonsofthe ighttestdatatowindtunnelbasedmodelingandsimulation werepresentedtovalidate theproposedmethodology.Althoughtheagreementwithigh ttestisnotperfect,the modelpresentedhereissufcientlyrepresentativetousea safoundationforsystem identicationandtrajectoryoptimization. 85

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CHAPTER3 SYSTEMIDENTIFICATION 3.1Overview Thepurposeofsystemidenticationinthecontextofguided storeseparationisto developaparametricmodelthatdescribesthespatiallyvar iantaerodynamicsofastore duringseparation.Thismodelcanbeusedtoassessthemajor contributorsaffecting separationcharacteristics,determinethetrajectoryoft hestoregivenprescribedinitial conditions,performtrajectoryoptimization,anddesigna controlsystemusinglinear andnonlinearcontroltechniques.Thischapterprovidesab riefdescriptionofrelevant techniquesinsystemidentication,followedbyapplicati onofsystemidentication methodstodetermineaparametricmodelforthespatiallyva riantaerodynamicsofa storeduringseparation.3.1.1SystemIdentication Systemidentication(SID)istheprocessofdeterminingan adequatemathematical model,usuallycontainingdifferentialequations,withun knownparametersthathave tobedeterminedindirectlyfrommeasuredexperimentaldat a[ 34 ].Practically,SID encompassesawiderangeofactivitiesincludingmathemati calmodeling,designing suitableexperiments,collectingmeasurements,performi ngdiagnosticsandstatistical analysis,andvalidatingidentiedmodels.Systemidenti cationiscloselyrelatedto parameterestimation(orparameteridentication),butth etwodisciplinesarenotthe same.SIDinvolvesarangeofactivitiesnecessarytodeterm ineanadequatemodel foranexperimentalsystem,whereasparameterestimationi saclassofstatistical techniquesthatusemeasureddatatoestimatetheparameter swithinapostulated model.Thus,parameterestimationisasubsetofsystemiden tication. ThepurposeofSIDistouseexperimentalmeasurements,post ulatedphysical relationships,andstatisticalinferencestodetermineam athematicalmodelthat adequatelydescribesthephysicalsystemofinterestovera specieddomainof 86

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operation.SIDisabroaddisciplineandhasbeenappliedinm anyareasincluding biology,medicine,chemicalprocesses,economics,geolog y,materials,civiland mechanicalengineering,andightdynamics[ 34 ].Systemidentication,inthebroadest sense,isaninverseapproachtomodelingcause-and-effect dynamicsystemsusing observedinput-outputrelationships. Systemidenticationiswelldocumentedinthetechnicalli terature.Thesurvey paperbyAstromandEykhoff[ 35 ]andseveralnotabletextbooksincludingthoseby Eykhoff[ 36 ],GoodwinandPayne[ 37 ],Ljung[ 38 ],Schweppe[ 39 ],SageandMelsa[ 40 ], Hsia[ 41 ]andNorton[ 42 ]aregoodstartingpoints.Additionally,theproceedingsf rom theInternationalFederationonAutomaticControl(IFAC)S ymposiaonIdenticationand SystemParameterEstimationonthree-yearintervalsfrom1 967topresentprovidea richresourceforfurtherinvestigation.3.1.2FlightVehicleSystemIdentication Flightvehiclesystemidenticationconsistsofthespecia lizationofgeneralSID techniquestoavarietyofightvehiclesincludingaircraf t,helicopters,andmissiles. Theightcharacteristicsofthesevehiclesaredetermined bythedynamicrelationships describedbytheequationsofmotioncombinedwithaerodyna micandpropulsiveforces andmomentsactingonthevehicle.Therigidbodyaircrafteq uationsofmotionarewell knownandcloselyrelatedtothestoreseparationequations ofmotionderivedinSection 2.2.3.5 .Propulsiveforcesaregenerallycharacterizedinaground testenvironment withcorrectionsforin-ightperformance.Therefore,SID ofightvehiclesreducesto usingmeasureddatatodeterminethemodelstructureforthe aerodynamicforcesand momentsandtoestimateoftheunknownparameterscontained inthemodel[ 24 ]. Windtunneltestingandcomputationaluiddynamics(CFD)a recommonsources forquantifyingaerodynamiccharacteristicsoftheightv ehicleearlyinthedesign phaseandthroughoutthelifecycle.Inmostcases,windtunn eltestingandCFD provideamorecosteffectivewaytocharacterizetheaerody namicsincomparison 87

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toighttesting,buteachmethodhassignicantlimitation s(suchasscalingeffects anddynamiclimitationsforwindtunneltestingandcomputa tionalresourcesanderror propagationforCFD).SIDprovidesawaytocharacterizethe aerodynamicsusing in-ightmeasurementsfortheactualightvehicleinopera tionalconditions.System identication(1)providesanimportantindependentsourc eofvalidationforwindtunnel testingandCFD,(2)providesameansoftestingightcondit ionsandmaneuversthat arenoteasilyreproducedinagroundtestenvironment,(3)p rovidesawaytoexpand theightenvelopeforexistingightvehicles,(4)provide sameansofvericationfor specicationcompliance,and(5)providesareliablesourc eofaerodynamicmodeling forautomaticcontrolsystemdesign[ 24 ].Ideally,eachofthethreeprimaryresourcesfor aerodynamicmodeling:windtunneltesting,computational uiddynamics,andsystem identication,areusedinacoherentfashiontoproduceaco mprehensiveandreliable modelofightvehicleaerodynamics. AlthoughightvehicleSIDisappropriatelycategorizedas aspecializedapplication ofSID,theuseofin-ightmeasurementstodetermineaerody namicparameterslong precededtheadventofSIDasatechnicaldiscipline.Oneoft herstapproachesfor obtainingstaticanddynamicaerodynamicparametersfrom ightdatawasgivenby Milliken[ 43 ]in1947.Afewyearslater,Greenberg[ 44 ]andShinbrot[ 45 ]established moregeneralandrigorousmethodsfordeterminingaerodyna micparametersfrom transientmaneuvers. Withtheintroductionofdigitalcomputersinthe1960sand1 970s,ightvehicle SIDbecameatremendouseldofresearchwithmuchrapidadva ncement.Pioneering contributionsweremadebyindividualssuchasTaylorandIl iff[ 46 ],Mehra[ 47 ],Stepner andMehra[ 48 ],andGerlach[ 49 ].Theseearlycontributionswereprimarilyinthearea ofdevelopingvariousparameterandstateestimationtechn iques.Introductionofhighly maneuverableandunstableaircraftpresentedmanychallen gestothetheoryand practiceofsystemidentication,manyofwhichwereaddres sedbyKlein[ 50 ]andKlein 88

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andMurphy[ 51 ].Withagrowingrelianceonsystemidenticationforaircr aftcertication andcontrolsystemdesigncombinedwithadesiretoreduceth eextentofighttesting, researchinoptimalmaneuverdesignwaspursuedbyMulder[ 52 ],Mehra[ 53 54 ],and Morelli[ 55 56 ].Severalusefultechnicalreferencesforaircraftsystem identicationhave beenpublished,includinganextensivebibliographycompi ledbyIliffandMaine[ 57 ],and broadsurveypapersbyKlein[ 58 59 ],andHamelandJategaonkar[ 60 ].Padeld[ 61 ] andHamelandKaletka[ 62 ]haveprovidedsimilarreferencearticlesforrotorcrafts ystem identication. Withmultipleoptionsforinputdesignandparameterestima tiontechniqueswell inhand,currentresearchtrendsarefocusedonmodelstruct uredeterminationfor nonlinearandtime-variantaerodynamicphenomenon,inclu dinghigh-alphaight[ 63 ], unsteadyaerodynamics[ 64 ],wakevortexencounter[ 65 66 ],groundeffect[ 67 ]and othercongurationchanges[ 68 ].TheinterestinSIDforadaptivecontrolsystemshas alsoledthedevelopmentofvariousrecursiveparameterest imationtechniques[ 38 69 ], includingrecursiveleastsquaresandextendedKalmanlte ringwithstateaugmentation [ 34 ].TheapplicationofSIDtostoreseparationpursuedinthis researchisproperly understoodinthiscontexttobeanextensionoftheseburgeo ningtechniquestoa nonlinearand,inthiscase,spatiallyvariantaerodynamic modelingproblem. Theurgentdesiretoreduceighttesting,combinedwiththe increasingreliability andversatilityofcomputationaluiddynamicshasgivenri setoapromisingendeavor toapplysystemidenticationtechniquestoCFDmodeling.C FDcanbeuseddirectlyto estimatelinearandnonlinearaerodynamiccharacteristic sbuttheseapplicationshave traditionallybeenlimitedtostationaryightconditions .However,thecomputational powerneededtoperformdynamicmaneuversinavirtualenvir onmentisnowtechnically feasibleandincreasinglyavailable,closingthegapbetwe entraditionalightmechanics methodsandcomputationalmethods.Promisingresultshave beendocumentedby DeanandMorton[ 70 71 ]Green[ 72 ],Bodkin[ 73 ],andClifton[ 74 ].Itisrecognized 89

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thatwhereasconventionalSIDisappliedtotheactualight vehicle,CFD-basedSID providesamodelofamodel,e.g.aparametricmodelorreduce dordermodel.However, theCFD-basedparametricmodelhasmanybenecialusesthat maydramatically reducetheextentoffull-scaleighttests,resultinginco standschedulesavings [ 70 ].Furthermore,manyoftheinaccuraciesthatarisefromig httesting(suchas measurementerror,stateestimation,atmosphericturbule nce,controlsystemcoupling, collinearityamongstatevariables,etc)arenolongerlimi tationswithinacomputational environment.Thoughmodelinginaccuraciesareinherentin computationalmethods, thetrade-offbetweenmodelingandight-testlimitations levelstheplayingeldbetween thetwoapproaches,providingfurthercredibilitytoCFD-b asedSID.AsCFDowsolvers continuetoadvanceinsophisticationandaccuracywithsim ultaneousadvancementof largescalecomputingcapability,CFD-basedSIDwillbecom eincreasinglyrelevantfor ightdynamicsandcontrol. ApplicationofCFD-basedSIDtostoreseparationisofparti cularinterest.System identicationrequiresexecutionofdynamicmaneuverstha tarephysicallyimpossible oratleastimmenselyimpracticalforastoreinthevicinity ofanaircraft.However, suchmaneuversareeasilyaccomplishedinacomputationale nvironment,providing thecapabilityofparametricmodelingusingSIDinaoweld environmentthatis otherwisedifculttodescribeanalytically.Althoughthi sresearchemphasizeswind tunnelbasedSID,thetransferofthesemethodologiestoaCF Denvironmentshouldbe straightforward.3.1.3StoreSeparationSystemIdentication Theapplicationofsystemidenticationtoaerodynamicmod elingforstore separationleadstoaparametricmodelthatdescribesthesp atiallyvariantaerodynamics ofthestoreduringseparation.Ofinewindtunnelbasedtra jectorysimulationsprovide avirtualenvironmentforaccomplishingmaneuversnecessa rytoperformsystem identicationinanexploratoryfashionduetotherapidcom putationoftrajectorieson 90

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anymodernpersonalcomputer.StoreseparationSIDisaproc edurethatincludes(1) modelpostulation,(2)inputdesign,(3)modelstructurede termination,(4)parameter estimation,and(5)modelvalidation.Forthepurposeofthi sresearch,thepostulated modelisamultivariatepolynomialwithspatiallyvariantc oefcients.Themodelstructure isdependentontheinputandightconditionsandisdetermi nedusingmultivariate orthogonaldecomposition.Theinputisdenedusingamulti sinedynamicoptimization algorithm.Parameterestimationisaccomplishedusingthe equationerrormethodand modelvalidationisaccomplishedbycomparingpredictedre sultswithindependent simulations.TheseconceptsarediscussedindetailinSect ion 3.2 .Theapplicationof thesetechniquestodetermineaparametricmodelforstores eparationisoneofthe primarycontributionsofthisresearch. 3.2IdenticationMethods Thecoreactivitiesofsystemidenticationareprimarilym odelstructuredetermination andparameterestimation.Inpractice,ightvehicleSIDen compassesamuchbroader rangeofactivity.AccordingtoKleinandMorelli[ 24 ],ightvehicleSIDincludes modelpostulation,experimentdesign,datacompatibility analysis,modelstructure determination,parameterandstateestimation,collinear itydiagnostics,andmodel validation.Anin-depthdevelopmentofeachoftheseareasi sprovidedbyKleinand MorelliandsimilardevelopmentsareprovidedbyJategaonk ar[ 34 ].Whenapplying SIDtostoreseparationinavirtualenvironment,theproces sisslightlymodiedand insomewayssimpler.Experimentdesignrequiresconsidera tionofinstrumentation requirements,whicharenotrelevantforsimulation-based study.Rather,theexperiment designstepreducestoinputdesign,whichisanareaofparam ountimportancefor simulation-basedstudieswhereshorterdurationmaneuver sarehighlydesirable. Datacompatibilityanalysisisdrivenbytheneedtosynthes izemeasurementsfrom variousinstrumentationsystems(typicallytheaircrafta irdatasystemandinertial measurementsystem)andagainthisstepisomittedforsimul atedtrajectorysolutions. 91

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Stateestimationisanecessarytoolforighttestingsince manyofthestatevariables arenotmeasureddirectlyandeventhosethataremeasureddi rectlyarecorrupted bynoise.Stateestimationtechniques(usuallyaKalmanlt ervariant)provideaway toreconstructtheightpathwithoptimalestimatesofthet ruestategivenassumed measurementerrorcharacteristics.Inasimulationenviro nment,thestatesarespecied andrecordedwithnumericalprecision;allstatesareknown andmeasurementnoiseis notincurred.Finally,inighttesting,thestatevariable sareinherentlyrelatedthrough dynamicandkinematicrelationshipsandinsomecasestheva riablesarenearlylinearly dependent,causingsignicantproblemsforidentiabilit y. 1 Insuchcases,collinearity diagnosticsarenecessarytodeterminetheaccuracyofthep arameterestimationby consideringthecorrelationamongtheinputvariables.Ina simulationenvironment,the inputismorefreelyselectedandtheinputscanbechosentob emutuallyorthogonal, providingmaximuminformationcontentintheinputsignala ndeliminatingcollinearity amonginputvariables. Withtheabovesimplications,thestoreseparationSIDpro cedureisreducedto thefollowingsteps:modelpostulation,inputdesign,mode lstructuredetermination, parameterestimation,andmodelvalidation.Theseprocedu resarediscussedinmore detailinSections 3.2.1 through 3.2.4 3.2.1InputDesign Thetaskofinputdesignistodetermineasuitablereference trajectoryfromwhich parameterestimationcanbeaccomplished.Theobjectiveis todesignaninputthatwill excitethedynamicsystemsothatthedatacontainsufcient informationforaccurate 1 Theconceptofidentiabilityiscloselyrelatedtocontrol labilityandobservability. Identiability,asthenamesuggests,quantiestheabilit ytodeterminetheinuenceofa giveninputonaspeciedoutput.Whentwoaremoreinputsare perfectlycorrelated, theinuenceofeachinputontheobservedoutputbecomesimp ossibletoidentify mathematically. 92

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modeling.Itisalsodesirabletomaximizetheinformationi ntheinputsignalinorder toreducethedurationofthetrajectory.Severaltypesofin putcommonlyusedin aircraftsystemidenticationincludeanimpulse,step,sq uarewave,doublet,multistep, andfrequencysweep[ 24 ].Thesesignalsndwideapplicationinaircraftsystem identicationdueprimarilytotheeaseofimplementation. However,theseinputs generallyrequirelongdurationmaneuverstogeneratesuf cientmodelingresults. Alternatively,multisineinputsproduceafavorableinput signalforsimulation-based systemidenticationwithrichinformationcontentoverar elativelyshorttimeinterval. Themultisineinputisasumofsinusoidswithvariousfreque ncies,amplitudes,and phaseangles,optimizedtoprovidethemaximuminformation contentoveraspecied rangeoffrequencyandamplitude[ 24 ].TheSchroedersweepisonesuchmaneuver consistingofasummationofmultipleharmoniccosineterms [ 75 ].Previousworkhas shownthattheSchroedersweepprovidesaninputwithgoodfr equencycontentandlow peakfactor[ 76 ].Thepeakfactorisameasureoftheratioofmaximuminputam plitude toinputenergycontainedinthesignal.Inputswithlowpeak factorsareefcientinthe senseofprovidinggoodfrequencycontentwithoutlargeamp litudesinthetimedomain [ 76 ].ModicationsoftheSchroedersweephavebeeninvestigat edandappliedina varietyofsituations[ 77 78 ].Multisineinputsaredifcultforapilottocreatemanual ly, buthavefoundstrongsupportinapplicationsinvolvingsim ulationwheretheprimary interestisinaninformation-richinputsignalwithcompac tduration[ 79 80 ].Morelli developedanextensionoftheSchroedersweepinputdesignm ethodtoincludemultiple orthogonalinputswithoptimizedpeakfactors[ 76 ].Thecorrespondingtechnique wasimplementedintheMatlab R r toolboxSystemIdenticationProgramsforAircraft (SIDPAC R r ),alsodevelopedbyMorelli[ 81 ].Themultisineinputmethodcangenerate anarbitrarynumberofmultipleinputsthataremutuallyort hogonalandadheretoa uniformpowerspectrum,providinganexcellentframeworkf orsimulation-basedsystem identication.ThedevelopmentproposedbyMorelliissumm arizedhere. 93

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Eachmultisineinput u j iscomprisedofasetofsummedharmonicsinusoidswith individualphaselags K ,asshowninEquation( 3–1 ),where M isthetotalnumber ofharmonicfrequencies, T isthetimelengthoftheexcitation,and A K and K arethe amplitudeandphaseanglestobechosenforeachoftheharmon iccomponents. u j = M X k =1 A k cos 2 k t T + K (3–1) Thephaseanglesarechosenusingasimplexoptimizationalg orithmtoproducea lowpeakfactorPF,denedbyEquation( 3–2 ). PF ( u j ) = [ max( u j ) min( u j ) ]/ 2 q u Tj u j N (3–2) Whentheinputexcitation u j oscillatessymmetricallyaboutzero,thepeakfactorcan beexpressedintermsofthenorms k u j k 1 and k u j k 2 PF ( u j ) = [ max( u j ) min( u j ) ]/ 2 2rms( u j ) = k u j k 1 k u j k 2 (3–3) Theterm k u j k 1 k u j k 2 isreferredtoasthecrestfactorinsignalprocessingliter ature andisameasureofthepeaktoaverageratio.Asinglesinusoi dalcomponentfrom thesummationinEquation( 3–1 )has PF = p 2 .Therelativepeakfactor,denedby Equation( 3–4 ),isameasureofthepeakfactorrelativetoasinglesinusoi d. RPF ( u j ) = [ max( u j ) min( u j ) ] 2 p 2rms( u j ) = PF ( u j ) p 2 (3–4) Therelativepeakfactorisameasureofefciencyofaninput forparameter estimationpurposes,intermsoftheamplituderangeofthei nputsignaldividedby themeasureofthesignalenergy[ 76 ].Lowerrelativepeakfactorsaredesirablefor parameterestimationwheretheobjectiveistoexcitethesy stemwithoutdrivingit toofarawayfromthenominaloperatingpoint[ 76 ].Inmanycases,thepostulated modelislinear(orslightlynonlinear)andkeepingtheairc raftneartheoperating conditionisanessentialfeatureforvalidinputdesign.In thecaseofsimulation-based 94

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systemidentication,alowrelativepeakfactoralsoimpli esaninformation-richsignal, coveringthedesiredinputrangeinashorttimeperiod.Furt hermore,theorthogonal multisineinputallowsmultipleinputstobeexcitedsimult aneouslywhilepreservinggood identiabilitybetweentheoutputandinputsignalsduetot heorthogonalityoftheinputs. Thisiscomparedtoconventionalsquare-waveandfrequency -sweepinputs,whichare executedsequentiallytoavoidcorrelationintheinputvar iables,greatlyextendingthe durationofthemaneuver.Bycomparison,itisapparentthat (1)conventionalinputsare idealformanualimplementationinaircraftwithsufcient lylongmaneuverdurations,and (2)multisineinputsareidealforcomputersimulationwith compactmaneuversandgood energycontent. Figure 3-1 showsamultisineinputsignalfortwoorthogonalinputsgen eratedusing theSIDPAC R r function mkmsswp .Thefrequencyrangespeciedisfrom0to5Hz, thedurationis5seconds,andtheamplitudeis+/-5.Theinpu tsarerepresentativeof controlsurfacedeectionsduringaparticularmaneuver. Parameterestimationinvolvesdevelopingamathematicalr elationshipbetweenthe speciedinputandobservedoutput.Therefore,itisimport anttouseaninputsignalthat sufcientlycoverstheinputspaceofinterest.Figure 3-2 showsanalternativeviewof inthesametwoinputsignals,calledaregressormap,whichc anbeusedtovisualize thecoverageoftheinputspace.FromFigure 3-2 ,itisapparentthatthemultsineinput providesgoodcoverageoftheentirespacewithemphasisnea rtheboundariesand corners.Thisparticularfeatureisbenecialforcapturin gnonlinearitiesintheobserved dataandusefulforpredictingvalueswithintheinputspace 3.2.2ModelStructureDetermination ThepostulatedmodelforstoreseparationSIDincludesamul tivariatepolynomial withconstantcoefcientsforuniformowcontributionsco mbinedwithamultivariate polynomialwithspatiallyvariantcoefcientsforthenonu niformowcontribution.The postulatedmodelisgeneralinthesensethatitcouldbeappl iedtoanyaircraft/store 95

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0 1 2 3 4 5 -6 -4 -2 0 2 4 6 Time (sec)Input, u(t) u 2 (t) u 1 (t) Figure3-1.Multisineexcitationfortwoorthogonalinputs from0to5Hzwithamplitude rangefrom+/-5. -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 u 1 (t)u 2 (t) Figure3-2.RegressormapforthemultisineinputsshowninF igure 3-1 congurationortoanindividualstore/aircraftinavariet yofightconditionsand congurations,buttobeofpracticalusethemodelmustbeco mpletelyspecied. Inotherwords,modelpostulationissuitableforagenericd escriptionofnonlinear 96

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aerodynamics,butmodelstructuredeterminationandparam eterestimationare necessarytonalizethemodelforaparticularapplication ofinterest. Ingeneral,thegoalofmodelstructuredeterminationisto ndacompactmodel thatstillhasadequatecomplexitytocapturethenonlinear functionaldependencies betweentheindependentanddependentvariables[ 82 ].Keepingthenumberoftermsin themodellowisanessentialfeaturethatimprovestheident iabilityoftheparameters inthemodel,resultinginamoreaccuratemodelwithgoodpre dictivecapability[ 82 ].In thisresearch,themodeltermsaredeterminedusingmultiva riateorthogonalpolynomial functionsbasedonaprocedureproposedbyMorelli[ 4 ]andimplementedintheMatlab R r toolboxSIDPAC R r Theproposedtechniqueusesmultivariateorthogonalmodel ingfunctionsgenerated directlyfromthemeasureddatatodetermineacompactnonli nearmodelstructure. Theorthogonalmodelsarethentransformedbackintotheori ginalregressorspaceto determineamultivariatepolynomialvalidformodelingnon linearaerodynamics.Thenal modelisanitemultivariatepowerseriesexpansionforthe dependentvariableinterms oftheindependentvariables. Determiningthemodelstructureinthetransformedorthogo nalcoordinatesprovides severaladvantagesovertraditionaliterativeoradhocmet hods(suchasstepwise regression).First,eachtransformedregressorismutuall yorthogonaltoallother regressors,eliminatinganydifcultiesthatarisefromco rrelatedinputs.Second,the estimateofeachparameterwithinthemodelisdecoupledfro mtheestimatesofall otherparameters.Thisdecouplingprovidesawaytoassesst hecontributionofeach individualmodeltermindependentlyofanyotherterminthe model,resultingina straightforwardmethodfordeterminingthemostsignican ttermsinthemodel.In general,themoretermspresentinthemodel,thebetterthem odelwilltthemeasured data.However,atsomepoint,theexplanatorypowerofthemo delisexhaustedand theremainingincreasesinaccuracyareduecompletelytoov er-parameterizingthe 97

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modeland“ttingthenoise”inthemeasureddata.Obviously ,atrade-offbetween goodness-of-tandthenumberofparametersinthemodelisn ecessary.Orthogonal functionsprovideameansforachievingthistrade-offinas tructuredfashionbyselecting themodelthatminimizesthepredictedsquarederror. Modelstructuredeterminationbeginswithavectorofmeasu reddependentvariable valuesconsolidatedasanN-dimensionalvector y y = [ y 1 y 2 ,..., y N ] T (3–5) Themeasuredvaluesaremodeledbyalinearcombinationofmu tuallyorthogonal modelingfunctions, p j j =1,2,..., n .Each p j isitselfanN-dimensionalvector,whichis ingeneralafunctionoftheindependentvariables x i i =1,2,..., m .Theindependent variablescanbeconsolidatedintoaregressormatrix. X = [ x 1 x 2 ,..., x m ] X 2 R Nxm (3–6) Similarly,thevaluesoforthogonalfunctionscanbewritte nasatransformed regressormatrix,asshowninEquation( 3–7 ),whereitisunderstoodthat P = P ( X ) P = [ p 1 p 2 ,..., p n ] P 2 R Nxn (3–7) Amethodfordetermining P ( X ) usingGraham-Schmidtorthogonalizationis presentedbyMorelli[ 24 ].Thedependentvariable y canbeexpressedasalinear combinationoftheorthogonalmodelingfunctions,asshown inEquation( 3–8 ),where a j ( j =1,2,..., n )areconstantsyettobedeterminedand isavectorofmodelingerrors. y = a 1 p 1 + a 2 p 2 +...+ a n p n + (3–8) Inmatrixform,equation( 3–8 )canbewrittenasEquation( 3–9 ),where a = [ a 1 a 2 ,..., a N ] T y = Pa + (3–9) 98

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Thegoalistodeterminethevaluesof a whichminimizetheleastsquarescost function. J = ( y Pa ) T ( y Pa ) (3–10) ItwillbeshownintheSection 3.2.3 thattheleastsquaresestimate ^a isgivenby Equation( 3–11 ). ^a = P T P 1 P T y (3–11) Thematrixinversion P T P 1 isthesourceofdifcultyforparameterestimation problemswithcorrelatedinputs.Fortheinversetoexist,t hevectors p j mustbe independent.Highlycorrelatedinputsthereforeresultin anill-conditionedmatrix, signicantlyweakeningtheaccuracyoftheparameterestim ates.However,forthecase ofmutuallyorthogonalmodelingfunctions,thefollowingi mportantpropertyexists. p Ti p j =0, i 6 = j i j =1,2,..., n (3–12) Asaresult,thematrix P T P isdiagonal,thematrixinversionistrivialized,andthe leastsquaresparameterestimatesaredecoupled.Followin gEquation( 3–11 ),theleast squaresparameterestimatesaredeterminedasasetofscala requations. a j = p Tj y p Tj p j j =1,2,..., n (3–13) Forthepurposesofmodelstructuredetermination,theresu ltshowninEquation ( 3–13 )issignicant.Inessence,thedeterminationofthe j th parameterestimate hasbeendecoupledfromallothertermsinthemodel.Onlythe dependentvariable measurementsandasingleorthogonalvectorarerequiredto determineeachparameter estimate.Thisallowseachparametertobedetermineduniqu elyregardlessofthe otherparametersunderconsideration.Furthermore,thees timatedcostfunctioncanbe writtenasEquation( 3–14 ). ^ J = y T y n X j =1 p Tj y 2 p Tj p j (3–14) 99

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Thesecondterminthisexpressionisalwayspositiveanditi sthereforeapparent thateachorthogonalfunctionwillnecessarilyreducethel eastsquarescostfunction. Moreimportantly,thequantitativereductioninthecostfu nctioncanbedetermined independentlyforeachorthogonalfunction,providingame ansofselectingthemost signicanttermsrstandeliminatingtheleastsignicant termsaltogether. Equation( 3–14 )indicatesthatthecostfunctionwillcontinuetobereduce das longastermsareaddedtothemodel.Intuitively,theremust bealimittothephysically meaningfultermsthatexplainthefunctionaldependencies inthedata.Assuch,the numberoforthogonalmodelingtermsarechosentominimizet hepredictedsquared error,denedby[ 83 ]andshowninEquation( 3–15 ). PSE = ^ J N + 2 o n N (3–15) ThemaximumestimateofthevarianceisgivenbyEquation( 3–16 ),where y is simplytheaveragevalueoftheindependentvariablemeasur ements. 2 o = 1 N N X i =1 ( y i y ) 2 (3–16) TherstterminthePSE(Equation( 3–15 ))representsthemeansquareterror, andwillalwaysbereducedasthenumberofthetermsinthemod elincreases.The secondtermisaproductofthemaximumvarianceinthedataan daratioofthenumber oftermsinthemodelwiththenumberofmeasurementsintheda ta.Therefore,the secondtermrepresentsapenaltyforover-ttingthedata.S incethersttermis guaranteedtodecreasewitheachadditionalterm,andthese condtermisalways positiveandguaranteedtoincreasewitheachadditionalte rm,thePSEisguaranteed tohaveatrueglobalminimum.Furtherdetailsonthestatist icalpropertiesofthe PSEmetric,includingjusticationforitsuseinmodelingp roblemsisdocumentedby Barron[ 83 ].Thenumberoftermsinthemodelisselectedtoachieveamin imumPSE, resultinginamodelthatissufcientlycomplextocapturet hefunctionaldependencies 100

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inthemeasureddataandsufcientlycompacttoprovidepara meteridentiability andpredictivecapability.Examplesusingmultivariateor thogonalfunctionswillbe consideredinSections 3.3 3.2.3ParameterEstimation Themostcommonlyappliedparameterestimationtechniques canbebroadly classiedintothreecategories:1)equationerror,2)outp uterror,and3)ltererror methods[ 60 ].Thechoiceofaparticularmethodisgenerallydictatedby themodel structureandbytheassumedcharacteristicsofthemeasure mentandprocessnoise inherentinthesystem. Theoutputerrormethodisanonlinearoptimizationmethodt hathasbeenmost widelyusedforaircraftparameterestimation[ 60 ].Thekeyfeatureoftheoutputerror methodisthecapabilitytoaccountforthemeasurementnois einherentinight-test datausingtheprincipleofmaximumlikelihood.Intheoutpu terrormethod,model parametersareadjustediterativelytominimizetheerrorb etweenthemeasuredoutput andthemodel-estimatedresponses[ 34 ],hencethename“outputerror”.Thus,the nonlinearequationsofmotionareevaluatedduringeachite rationresultinginanonlinear optimizationproblem.Acomprehensivedescriptionoftheo utputerrormethodwith severalexamplesisprovidedbyJategaonkar[ 34 ]andMorelli[ 24 ]. Theoutputerrormethodallowsformeasurementerrorinthed ependentvariables butassumesperfectmeasurementofthestatevariables.In ight-testapplications,the statemeasurementsarealsocorruptedbyerror.Particular lyinsituationswithsignicant atmosphericturbulence,theoutputerrormethodmayyieldp oorestimatesofthemodel parameters[ 34 ].Insuchcases,theltererrormethodismorereliable.The ltererror methodaccountsforbothmeasurementnoiseandprocessnois e(suchasturbulence) byconsideringastochasticsystem.Asaconsequence,asuit ablestateestimatoris requiredtopropagatethestates.Thestateestimationispe rformedusingaKalman lteroranExtendedKalmanFilter(EKF),dependingontheli nearityofthepostulated 101

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model.Filtererrormethodsrepresentthemostgeneralappr oachtoaircraftparameter estimation.Althoughtheltererrormethodissubstantial lymorecomplexthanthe outputerrormethod,themethodisapplicabletoawidervari etyofaircraftidentication problems,includingparameterestimationfornonlinearae rodynamicmodelsinthe presenceofturbulence.Theltererrormethodwasrstprop osedbyBalakrishnan[ 84 ], withpioneeringdevelopmentsbyMehra[ 85 ]andIliff[ 86 ]inthe1970s.Acomprehensive descriptionoftheltererrormethodwithseveralexamples isprovidedbyJategaonkar [ 34 ]andMorelli[ 24 ]. ThethirdparameterestimationtechniquecommonlyusedinS IDistheequation errormethod.Synthesisoftheaerodynamicforcesandmomen tsactingonaight vehiclethroughTaylorseriesexpansionleadstoamodeltha tislinearintheparameters (thoughitmaybenonlinearinthestatevariables).Theequa tionerrormethod,based onclassicalregressiontechniques,isideallysuitedforp arameterestimationof multivariatepolynomialmodels.Theequationerrormethod isthesimplestofthethree mainstreamparameterestimationmethods,butitmayleadto leastaccurateestimates inthepresenceofsignicantmeasurementnoise.Thuspract icalimplementationof theequationerrormethodreliesonsmoothight-testcondi tions,qualityight-test instrumentation,andpost-ighttrajectoryreconstructi on. Forsimulation-basedsystemidentication,theselimitat ionsarenotofconcern. Inasimulationenvironmentthestatesareknownwithnumeri calprecisionandthe measurementnoiseislimitedtothenoisepresentintheCFDo rwindtunneldatabase. Furthermore,theequationerrormethodprovidesagreatdea lofexibilityinmodeling structuresandlendsitselfnicelytomodelstructuredeter minationmethods,suchasthe multivariateorthogonalpolynomialpreviouslydescribed .Theequationerrormethod doesnotrelyonthetemporalarrangementoftheight-testd ata,allowingmultiple ight-testmaneuverstobeconcatenatedtogetherforasing leestimationproblemor alternativelyalargescalemaneuvertobedecomposedintos mallerlocallyvalidmodels 102

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(aprocessreferredtoasdatapartitioning)[ 34 ].Finally,evenfornonlinearequations ofmotionandnonlinearmultivariatepolynomials,theequa tionerrormethodremains alinearestimationproblemsolvedefcientlyinasinglepa ssusinglinearalgebraic methods. Theequationerrormethodisfundamentallybasedonthewell -knownleastsquares technique,whichisbrieyconsideredhere.Asaresultofth epostulatedaerodynamic modelforastoreinuniformow,theaerodynamiccoefcient srepresentedbythe responsevariable y ( t ) canbeexpressedasalinearcombinationofvariables,assho wn inEquation( 3–17 ),where 1 2 ,..., n areingeneralnonlinearcombinationsofthestate variables(e.g. 2 etc.). y ( t )= 0 + 1 1 ( t )+ 2 2 ( t )+...+ n n ( t ) (3–17) Despitethepotentialnonlinearrelationshipbetweenther esponsevariableandthe statevariables,themodelequationfor y ( t ) islinearintheparameters 0 1 ,..., n .The modelequationcanbewritteninmorecompactformasshownin Equation( 3–18 ). y = X (3–18) InEquation( 3–18 ), y= [ y (1) y (2)... y ( N ) ] T isan N 1 vector, = [ 0 1 ... n ] T isan n p 1 vectorwiththenumberofparameters n p = n +1 ,and X =[ 1 1 ... n ] T isan N n p matrix.Theresponsevariablesarenotknown precisely.Rather,themeasurementequationisgivenbyEqu ation( 3–19 ),where z =[ z (1) z (2)... z ( N ) ] T and = [ (1) (2)... ( N ) ] T areboth N 1 vectors and arethemeasurementerrors. z = X + (3–19) Thebestestimatoroftheparameters inaleast-squaressensecomesfrom minimizingthesumofsquareddifferencesbetweenthemeasu rementsandthemodel. 103

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TheassociatedcostfunctioncanbewrittenasEquation( 3–20 ). J ( ) = 1 2 ( z X ) T ( z X ) (3–20) Theparameterestimate ^ thatminimizesthecostfunctionmustsatisfyEquation ( 3–21 ),whichcanbesimpliedtoEquation( 3–22 ). @ J @ = X T z + X T X ^ = 0 (3–21) X T z X ^ = 0 (3–22) Providedthattherequiredinverseexists,Equation( 3–22 )canbesolvedtoyieldthe leastsquaresestimator. ^ = X T X 1 X T z (3–23) Thus,giventhemeasurements z andamatrixofpostulatedregressors X ,the parameters thatminimizetheresidualerror canbedetermineddirectly.The mathematicalsimplicityofEquation( 3–23 )istheprimaryadvantageoftheequation errormethodandtheprinciplefeaturethatleadstoavariet yofapplicationsand modelingexibility.Examplesusingtheequationerrormet hodappliedtostore separationsystemidenticationwillbepresentedinSecti on 3.3 3.2.4ModelPostulation Forstoreseparation,theestablisheddeltacoefcientmet hodology( 2.3.2 )provides asolidstartingpointforapplicationofsystemidenticat ionmethods.Usingthe superpositionapproachinherentinthedelta-coefcientm ethodology,thefunctional formoftheaerodynamiccoefcientsisshowninEquation( 3–24 ). C = C FS + C TRANS + C ROT + C G + C C (3–24) ThefourseparatetermsinEquation( 3–24 )resultfromfourdistincttypesofwind tunneltesting,providingalogicalframeworkforthefunct ionalexpression.Itisalso 104

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notedthatthefreestream,rotational,andcontrolterms: C FS C ROT ,and C C ,areall determinedinauniformowenvironment,whereasthegridda tacorrection, C G ,is determinedfromanonuniformow.Thisdichotomysuggestsa notherwaytoformulate theaerodynamiccoefcientsasshowninEquation( 3–25a ),where C U representsthe uniformowcontributiontothetotalaerodynamiccoefcie ntand C NU representsthe nonuniformcorrection.Itisintuitiveandconvenienttoap plysystemidenticationto eachofthesetermsseparatelyandrecombinetheexpression stodeterminethetotal aerodynamiccoefcient. C = C U + C NU (3–25a) C U = C FS ( M S S ) + C ROT W p B B W q B B W r B B + C C ( e a r ) (3–25b) C NU = C G r B = A A B A B AB (3–25c) 3.2.4.1Uniformowcontribution Duetotheassumptionofaquasi-steadyoweld,thefunctio nalformofthe uniformowcontributionisamenabletoaTaylorseriesexpa nsion.Theexpansion canberepresentedbyamultivariatepolynomialthatisline arintheparameters butpotentiallynonlinearinthestatevariables.Thegener alnonlinearformofthe Taylorseriesexpansionisconvenientlyexpressedusingma trixnotation,asshownin Equation( 3–26 ). C i U = X ( s s p q r ) (3–26) InEquation( 3–26 ), X 2 R 1 N isamatrixofregressorscomposedoflinearand nonlinearfunctionsoftheoriginalstatevariables,and 2 R N 1 isamatrixofconstant coefcients,composedinpartofconventionalaerodynamic derivatives.Forsimplicity, ithasalsobeenassumedthatthevariationinthecoefcient withrespecttoMach numberandvelocityisnegligibleovertheintervalofinter est(approximately1sec). 105

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GiventhepostulatedmodelinEquation( 3–26 ),severalmethodsexistthatmaybeused todetermineasuitablemodelstructure X andmodelcoefcients .Inthisresearch, theregressorsweredeterminedusingmultivariateorthogo nalpolynomialsandthe coefcientsweredeterminedusingordinaryleastsquares, asdiscussedin 3.2.3 .An exampleusingthisapproachispresentedinSection 3.3 3.2.4.2Non-uniformowcontribution Thenonuniformoweldischaracterizedinwindtunneltest ingbymeasuring aerodynamicforcesandmomentsactingonthestoreatmultip lediscretelocations withintheoweld.Atanyonelocation,thenonuniformcorr ectioncanbeformulatedas showninEquation( 3–27 ). C NU ( z = const ) = C NU ( ) (3–27) ATaylorseriesexpansioncanbeappliedappropriately,whe reitisrecognized thattheexpansioniscarriedoutataparticularstationary locationwithintheoweld. Intuitively,theexpansionsateachlocationwithintheow eldcanbeformulatedina nonlinearcontinuousfashion,resultinginamultivariate polynomialmodelwithspatially variantcoefcients.Thegeneralnonlinearformofthismod elcanbeconveniently expressedinmatrixnotation,asshowninEquation( 3–28 ). C NU ( z ) = E ( ) c 0 ( z ) c 1 ( z )... c n 1 ( z ) T (3–28) Themodelstructure E ( ) 2 R 1 n iscomposedoflinearandnonlinear combinationsoftheEuleranglesandcanbedeterminedusing multivariateorthogonal polynomials.Themodelcoefcients c 0 ( z ) c 1 ( z ) c n 1 ( z ) ,etc.representspatiallyvariant termsandrequirefurtherconsideration.3.2.4.3Spatialvariation Identicationofanonuniformdelta-coefcientmodelcons istsoftwointerrelated tasks,including(1)determinationofthemodelstructurei nthestatevariablesand 106

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(2)determinationofthespatiallyvariantcoefcients.In thisresearch,themodel structureisdeterminedusingmultivariateorthogonalpol ynomialsinamannersimilar tothefreestreammodel.Thefunctionalformofthespatiall yvariantcoefcientsis speciedbyconsideringthedominantcharacteristicsofdi minishingnonuniformow eldaerodynamicsandtheparameterswithinthefunctional formareestimatedusing nonlinearleastsquares.Theresultisacompactparametric modelthatcanbeusedto representthespatiallyvariantaerodynamicsofstoreduri ngseparationforavarietyof applications. Thediminishingnatureofthenonuniformdeltacoefcienta sthestoreseparates fromtheaircraftprovidesanimportantboundarycondition forthespatiallyvariant coefcientsinEquation( 3–28 ).Underthequasi-steadyanduniformatmosphere assumptionspositedinChapter2,therealwaysexistsadist ancebetweentheaircraft andstoreatwhichtheaircraftoweldeffectonthestoreis negligible.Thisphysical realizationcanbeexpressedmathematicallybyconsiderin gthelimitas z !1 ,i.e. onecanbeassuredthatthelimitingvalueofthenonuniformd eltacoefcientiszero,as showninEquation( 3–29 ). lim z !1 C NU =0 (3–29) GiventhemultivariatepolynomialmodelpostulatedinEqua tion( 3–28 ),itisevident thateachterminthemodelisindependent.Therefore,Equat ion( 3–29 )implies thateachcoefcientinthemodelwillapproachzeroindepen dently,asshownin Equation( 3–30 ),where n isthenumberoftermsinthemodel. lim z !1 c i ( z )=0 i =1,..., n (3–30) Thisguidingcriterionisbasedonthephysicalprincipleth atasthedistance betweentheaircraftandstoreincreaseswithoutbound,the effectoftheaircraftow eldonthestoreaerodynamicsbecomesnegligible.Inpract ice,itisreasonableto 107

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expecttheeffectoftheaircraftonthestoreaerodynamicst obecomesmallatavertical distanceequivalentto1-2lengthsoftheaircraft. Oneparticularmathematicalexpressionthatmeetstheboun darycondition expressedinEquation( 3–30 )istheexponentialfunction, e z wheretherateofdecay, ,needstobeestimatedusingmeasureddata.However,theexp onentialfunction ismonotonicallydecreasing.Forcomplexowelds,onecan beassuredthatthe aircrafteffectwilldiminishasthedistancebetweentheai rcraftandstorebecomes large,butthereisnoassurancethatthisdecreasewillbemo notonic.Forexample, onemightexpectthattheoweldeffectwouldbecomemoresi gnicantasastore passesthroughashockorexpansionwaveduringsupersonic ight.Apostulatedmodel thatsatisesEquation( 3–30 )andallowsanon-monotoniccurvatureisgivenbyan exponential-polynomialproduct,asshowninEquation( 3–31 ). c i ( z )= e z 0 + 1 z + 2 z 2 +...+ m z m (3–31) Equation( 3–31 )representsanonlinearequationforsinglemodelcoefcie ntwith unknownparameters 0 1 ... m thatmustbeestimatedfrommeasureddata. Thenonuniformdeltacoefcientcanbewrittenasamatrixeq uation,asshownin Equation( 3–32 ). C NU ( z ) = E ( ) c 0 ( z ) c 1 ( z )... c n 1 ( z ) T (3–32) Themodelstructureisamatrixcomposedoflinearandnonlin earregressors,as showninEquation( 3–33 ). E ( ) = 1 2 ... n E 2 R 1 n (3–33) Usingthepostulatedformofthespatiallyvariantmodelcoe fcientsgivenin Equation( 3–31 ),themodelcoefcientscanalsobewritteninageneralmatr ixform,as 108

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showninEquation( 3–34 )andfurthercondensedinEquation( 3–35 ). c i ( z )= e i z 0 1 ... m 1 z ... z m T (3–34) c i ( z )= e i z ( i Z i ) (3–35) InEquation( 3–35 ), Z i =[ 1 z ... z m ] T isa p 1 vectorand i =[ 0 1 ... m ] isa 1 p vector,where p = m +1 .Ifitisfurtherpresumedthatthemodelstructure withinthepolynomialportionofeachmodeltermisheldxed ,then Z = Z i 2 R p 1 for i =1,..., n .Consequently,thevectors i canbecombinedintoan n p matrix. = T1 T2 ... Tn T 2 R n p (3–36) ThematrixproductinEquation( 3–35 )resultsinan n 1 vector. Z 2 R n 1 (3–37) Furthermore,thescalarexponentialfunctionsof z canbewrittenasamatrix exponential,asshowninEquation( 3–38 ). e z = 266664 e 1 z 0 ... ... 0 e n z 377775 2 R n n (3–38) CombiningtheresultsfromEquation( 3–37 )and( 3–38 ),thematrixproductin Equation( 3–39 )correspondstotheoriginalcoefcientmatrix. e z Z= c 0 ( z ) c 1 ( z )... c n 1 ( z ) T 2 R n 1 (3–39) ConsidertheoriginalformofEquation( 3–32 ).UsingtheresultinEquation( 3–39 ), thenonuniformdeltacoefcientcanbewritteninthecompac tparametricformshownin Equation( 3–40 ). C NU ( z ) = E ( ) e z Z ( z ) (3–40) 109

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Equation( 3–40 )issignicant.GivensufcientwindtunnelorCFDdata,thi s compactparametricmodelisasuitablecandidateformodeli ngthespatiallyvariant aerodynamicsofastoreduringseparation.Theterms E ( ) and Z ( z ) arematrices composedoftheoriginalstatevariables.Thecomponentsof thematrix E ( ) are determinedusingmultivariateorthogonalpolynomialsand thecomponentsof Z ( z ) arechosenbytheanalystfortheparticularapplicationath and(acubicpolynomial including z z 2 and z 3 isgenerallysufcient).Theterms and areconstantmatrices determinedusingnonlinearleastsquaresestimation.Thus ,allfourtermsarereadily determinedandquantiedusingtheSIDtechniquesdiscusse dinSection 3.2 ReturningtoEquation( 3–26 ),theuniformowcontributiontotheaerodynamic coefcientcanalsobeexpressedinmatrixform,where X isamatrixofregressorsand isaconstantvector.Thegeneralmatrixformforthetotalae rodynamiccoefcientis givenbyEquation( 3–41 ). C = X ( S S p q r ) +E ( ) e z Z ( z ) (3–41) Ofparticularimportanceisthenatureoftheexponentialte rm, e z .Duringasafe separation,thevariablezisalwayspositiveandmonotonic allyincreasing.Therefore, providedthatthematrix ispositivedenite,theexponentialtermwillcontinually diminishas z increases,andtheaerodynamiccoefcientwillapproachth efreestream value,asshowninEquation( 3–42 ). lim z !1 X +E e z Z = X ) lim z !1 ( C U + C NU ) = C U (3–42) Equation( 3–41 )canbeconsideredasacandidatemodelforanystoreina continuousspatiallyvaryingoweldforwhichtime-avera gedwindtunnelorCFD dataareavailable.Theaccuracyofthemodelinreproducing thesourcedatawillbe dependentonthecomplexitiesoftheoweldandtheexperti seoftheanalyst.An 110

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extendedexamplewillprovidegreaterclaricationinthea pplicationandutilityofthis parametricmodel. 3.3Example:StoreSeparation ThesystemidenticationapproachdiscussedinSection 3.2 isrstappliedto determineaparametricfreestreammodel,validforastorei nfar-eld,uniformow conditions;seeSection 3.3.1 .Next,thenonuniformdeltacoefcientismodeled usingsystemidenticationtodetermineaspatiallyvarian tparametricmodel;see Section 3.3.2 .Thefreestreamanddeltacoefcientmodelsforthisexampl earebased completelyonwindtunneldata,butinprinciplecouldalsob edeterminedusingCFD. Finally,inSection 3.3.3 thetwomodelsarecombinedtoestimatethetotalaerodynami c forcesandmomentsandthecumulativemodeliscomparedtoi ght-testdatawith favorableresults.3.3.1FreestreamSystemIdentication Modelingtheaerodynamiccharacteristicsofastoreinfareld,uniform(freestream) ightconditionsisstraightforwardusingestablishedsys temidenticationtechniques. Thissectionpresentssimulatedresultsusedformodeliden tication,andprovides validationoftheparametricfreestreammodel.3.3.1.1Simulatedmaneuver InthetypicalapplicationofightvehicleSID,theinputsa respeciedandthe responseofthevehicleismeasuredinight.However,thisa pproachmayresultin correlationbetweenstatevariablesandlowinformationco ntentintheaircraftresponse, requiringlongdurationmaneuvers.Inasimulationenviron ment,directmanipulation ofthestateandcontrolinputsresultsinamoreefcientman euverwithimproved identiability[ 49 70 ].Multisineinputscanbeusedtogenerateorthogonalsigna lsforthe desiredstateandcontrolinputsandthekinematicequation sofmotioncanbeusedto maintainphysicalrelationshipsbetweenstatevariables. Forthecurrentapplication,the airincidenceangleswerespeciedusingmultisineinputs, theangularbodyrateswere 111

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computedusingkinematicrelationships,thevelocityofth estorerelativetotheairwas heldconstant,andthepositionofthestoreCGwasheldxed( i.e.thestoreispinnedat astationarylocationbutfreetorotate,conceptuallysimi lartoawindtunneltest). Thekinematicrelationshipsbetweenthetimederivativeof theEuleranglesandthe angularratescanbeextractedfromtheconventionalrigida ircraftequationsofmotion [ 24 ].ForthespecialcaseofapinnedstoreCG,therollorientat ionisspeciedsuchthat = s = s and :=0 .Asaresultthekinematicrelationshipreducestothe expressionsinEquation( 3–43 ). pqr T = s sin s s s cos s T (3–43) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1000 -500 0 500 1000 Time (sec)Angular Rates (deg/sec) Yaw, r Pitch, q Roll, p 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -40 -20 0 20 40 Time (sec)Air-Incidence Angles (deg) b s a s Figure3-3.Denitionofatrainingmaneuverwithmultisine inputsfor(top)air-incidence anglesand(bottom)angularratescomputedusingkinematic relationships. ThetopportionofFigure( 3-3 )showsthespeciedairincidenceanglesofthestore inuniformow,generatedasorthogonalmultisineinputswi thafrequencyrangeof 0-5Hzandanamplituderangeof 20 degfor s and 10 for s .Thebottomportion ofFigure( 3-3 )showstheangularratescomputedusingthekinematicrelat ionshipsin ( 3–43 ). 112

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Withalltwelvestatevariablesspecied,theaerodynamicc oefcientsforthe prescribedtrajectorycanbedeterminedwiththesameinter polationmethodsused foraconventional6DOFsimulation.Multivariateorthogon alpolynomialsareusedto determineasuitablemodelstructurethatminimizesthepre dictedsquarederrorandthe ordinaryleastsquaresapproachisusedtoestimatethemode lparameters.Theresults forthepitchingmomentcoefcient, C m ,andthenormalforcecoefcient, C N ,areshown inFigure( 3-4 ). 0 1 2 3 4 5 -3 -2 -1 0 1 2 3 Time (sec)Pitching Moment Coeff, C m Sys ID Simulation APitchingMoment 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2 3 4 Time (sec)Normal Force Coeff, C N Sys ID Simulation BNormalForce Figure3-4.Aerodynamiccoefcientcomparisonbetweensim ulationandsystem identicationresultsforastoreinfreestreamightcondi tionsatMach0.9/ 550KCAS/4800ft. Theresultsfromsimulationandthepredictedresponsefor C m and C N areingood agreementoverarangeofoutputvalues,withR-squaredvalu esof R 2 =98.5% and R 2 =99.7% ,respectively.Theidentiedmultivariatepolynomialmod els,a4thorder modelfor C m and3rdordermodelfor C N ,aregivenbyEquations( 3–44 )and( 3–45 ). C m = 4.94 74^ q 94.71 2 +2.44 2 +10.07 3 77.75 2 2 (3–44) C N =5.25 +45.656 2 +7.90 3 (3–45) Similarresultswereobtainedforsideforce( R 2 =99.1% )andyawmoment ( R 2 =97.3% )coefcients.Theaxialforce( R 2 =84.9% )androllmoment( R 2 =91.2% ) coefcientshavelowerpredictionaccuracydueinparttoag reatermeasurement 113

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uncertainty.Fortunately,thetrajectorypredictionisle sssensitivetothesecoefcients. Overall,themultivariatepolynomialswithconstantcoef cientscharacterizethe freestreamaerodynamicsquitewell.Itshouldbenotedthat thesemodelsarevalid fortherangeof 20 degfor s and 10 for s ,commensuratewiththeprescribed maneuver.Thoughthemodelsmaybeevaluatedoutsideofthis range,theaccuracyof thepredictionquicklydegrades.Ifalargerrangeisnecess ary,theinputmaneuvercan beredesigned,thoughmodelpredictiveaccuracymaydegrad eiftheinputdesignistoo large.Alternatively,thesystemidenticationprocessca nbeappliedoveraneighboring rangeofinputvalues(say, 30 10 degfor s )toidentifyacomplementarymodel.The complementarymodelscanthenbysynthesizedinasimulatio nenvironment. 3.3.1.2Modelvalidation Thepredictivequalityofthemodelcanbeevaluatedinmulti pleways;twoparticular comparisonsareexaminedhere.First,comparisonofthemod eltothestaticfreestream databasefromwhichitisderivedisconsidered.Second,com parisonwithasimilarbut independentmaneuverisconsidered. Astaticfreestreammodelcanberecoveredbysettingalldyn amictermsinthe modeltozero.Forthisapplication,thisimpliesthat ^ p =^ q =^ r =0 .Theresulting modelisreadilydeducedfromEquations( 3–44 )and( 3–45 ).Itshouldbenotedthat therecoveredstaticmodelisnotnecessarilyidenticaltot hemodelthatwouldhave beenestimatedif ^ p =^ q =^ r =0 fromthebeginning.However,itiscloseenoughto demonstratethevalidityofthedynamicmodelinthisrestri ctedsense.Theresultsfor thestaticpitchingmomentcoefcient, C m ,andthestaticnormalforcecoefcient, C N areshowninFigure( 3-5 ).Again,theresultsarefavorable. Thepredictivequalityofthemodelisfurtherjustiedbyco nsideringasimilarbut independentmaneuver.Figure( 3-6 )showsairincidenceanglesgeneratedusing amultisineinputwithafrequencyrangeof0-3Hzandanampli tuderangeof+/-20 degfor s and+/-10for s (top).Figure( 3-6 )showsacomparisonoftheresulting 114

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-20 -15 -10 -5 0 5 10 15 20 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Angle-of-Attack, a sPitching Moment Coeff. C m b s = 10 b s = 5 b s = 0 (deg) APitchingMoment -20 -10 0 10 20 -3 -2 -1 0 1 2 3 Angle-of-Attack, a sNormal Force Coeff. C N b s = 10 b s = 5 b s = 0 (deg) BNormalForce Figure3-5.Staticaerodynamiccoefcientcomparisonbetw eeninterpolatedwindtunnel data(solidlines)andsystemidenticationresults(dashe dlines)forastore infreestreamightconditionsatMach0.9/550KCAS/4800ft 0 0.5 1 1.5 2 2.5 3 -30 -20 -10 0 10 20 30 Time (sec)Air Incidence Angles (deg) b s a s 0 0.5 1 1.5 2 2.5 3 -4 -2 0 2 4 Time (sec)Aero. Coeff. Normal, C N Pitch, C m Figure3-6.Validationresultsshowinganindependentmane uverandaerodynamic coefcientcomparisonbetweeninterpolatedwindtunnelda ta(solidlines) andsystemidenticationresults(dashedlines)forfreest reamight conditionsatMach0.9/550KCAS/4800ft. pitchingmomentandnormalforcecoefcientsalongthetraj ectory(bottom).Thesystem identicationmodelisshowntoagreefavorablywiththesim ulationresults,indicating goodpredictivequalityforanindependenttrajectory. 115

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3.3.2SpatiallyVariantSystemIdentication Identicationofaconstantparametermodelforaerodynami csofastorein auniformoweldisastraightforward,asshowninSection 3.3.1 .Modelingthe aerodynamicsofastoreinanonuniformoweldismoreinvol vedduetothespatially varianttermsinthemodel.Thissectiondescribestheuseof apiecewise-continuous maneuvertodetermineaspatiallyvariantaerodynamicmode landvalidationofthe modelagainstanindependentsimulatedmaneuver.3.3.2.1Piecewise-continuousmaneuver Systemidenticationofthestoreinfreestreamconditions includedspecication ofamaneuverfollowedbymodelstructuredeterminationand parameterestimation usingmultivariateorthogonalpolynomials.Forthepiecew ise-continuousapproach tomodelingthespatiallyvariantaerodynamics,thefreest reammodelingapproachis simplyrepeatedatmultiplestationarylocationswithinth enonuniformoweldandthe resultsarecombinedtodetermineasuitablemultivariatep olynomialmodelwithspatially variantcoefcients. Thesamemaneuverusedforfreestreamsystemidentication ,showninFigure ( 3-3 ),isusedateachstationarylocationwithinthenonuniform oweld.Itshould benotedthatthefreestreamaerodynamiccoefcientsarepa rameterizedinterms oftheairincidenceangles s and s ,thenon-dimensionalangularrates ^ p ,^ q ,^ r ,and optionallythecontrolsurfacedeections a e r .Incontrast,thedeltacoefcientsare parameterizedintermsofthedistancealongtheverticaldi rectionfromtheaircraft, z andtheorientationofthestorerelativetotheaircraft, .Sinceeachmaneuveris conductedatastationarypositionwithinthenonuniformo weld, = s = s and =0 .Therefore,themaneuverspeciedinFigure( 3-3 )canbeuseddirectly. Allotherstatevariablesarespeciedusingtherelationsh ipsspeciedin 3.3.1.1 .A non-zerorollangleiscorrectedforusingacoordinatetran sformationduringsimulation. Thisapproachneglectstheinteractionbetweentheaircraf toweldandthestoreroll 116

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angle,anadequateapproximationforastorethatisnearlya xisymmetric.Aslightly moresophisticated“rolleddeltacoefcient”approachcan beusedwhenthestoreisnot axisymmetric[ 3 ],buttheadditionalcomplexityisunwarrantedforthisexa mple. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -2 -1 0 Time (sec)Deltea Coeff.Z = 10 ft 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -2 -1 0 Deltea Coeff.Z = 5 ft 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -2 -1 0 Deltea Coeff.Z = 0 ft Normal, D C N Pitch, D C m Figure3-7.Comparisonofsimulation(solid)andsystemide ntication(dashed)results foraerodynamicpitchingmomentandnormalforcedeltacoef cientsat(a) z=0ft,(b)z=5ft,and(c)z=10ft. Figure( 3-7 )comparessimulationandsystemidenticationresultsfor thepitching momentandnormalforcedeltacoefcientsatthreerepresen tativestationarypositions inthenonuniformoweld.TheordinateaxisscaleinFigure ( 3-7 )isheldxedto emphasizethediminishingvalueofthedeltacoefcientast heverticaldistance increases.Theresultsindicatethattheidentiedmodelma tchesthetrainingdata wellateachstationarypositionwithintheoweld. Theidentiedmodelequationsforthepitchingmomentdelta coefcientareshown inEquations( 3–46 )through( 3–48 ),asaretheR-squaredvaluesforeachposition. Themodelstructurewasdeterminedusingmultivariateorth ogonalpolynomialsforthe rstmaneuver( z =0 )andheldxedforallremainingpositions.Thecoefcients were 117

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determinedusingordinaryleastsquaresestimation. C m ( z =0)= 1.02+2.83 +16.24 2 +9.7 11.41 3 0.53 20.82 2 40.22 3 +131.00 2 2 2.31 2 R 2 ( z =0)=93.2% (3–46) C m ( z =5)= 0.47+1.16 +1.47 2 +3.85 3.64 3 0.42 8.56 2 6.86 3 +58.89 2 2 1.48 2 R 2 ( z =5)=95.4% (3–47) C m ( z =10)= 0.18+0.51 +0.95 2 +1.20 1.55 3 0.06 4.86 2 1.81 3 +26.06 2 2 0.53 2 R 2 ( z =5)=94.3% (3–48) Equations( 3–46 )through( 3–48 )arethreediscretemodelsthatmayberepresented byacontinuousmodelwithspatiallyvariantcoefcients,a sshowninEqn.( 3–49 ). C m ( z 0)= c 0 ( z )+ c 1 ( z ) + c 2 ( z ) 2 + c 3 ( z ) + c 4 ( z ) 3 + c 5 ( z ) + c 6 ( z ) 2 + c 7 ( z ) 3 + c 8 ( z ) 2 2 + c 9 ( z ) 2 R 2 =90.1% (3–49) EachofthespatiallyvariantcoefcientsinEquation( 3–50 )canfurtherbemodeled byanonlinearexponential-polynomialproduct,asdiscuss edinSection 3.2.4.3 and showninEquation( 3–50 ),where n isthenumberoftermsinthemodeland m is theorderofthepolynomialin z .Forthepitchingmomentdeltacoefcientmodelin Equation( 3–49 ), n =10 and m =3 c i ( z )= e z 0 + 1 z + 2 z 2 +...+ m z m i =1... n (3–50) Thepitchingmomentdeltacoefcientmodelnowconsistsofe levenequations includingEquation( 3–49 )andtennonlinearequationsrepresentedbyEquation( 3–50 ). Theseelevenequationsmaybewritteninaconcisematrixfor m,asshowninEquation ( 3–51 ),where C NU representsanyofthenonuniformowelddeltacoefcients .A similarmodelmaybeidentiedforthenormalforcedeltacoe fcient,aswellasthefour remainingdeltacoefcients. C NU ( z ) = E ( ) e z Z ( z ) (3–51) 118

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Thespatialvariationofthemodeltermsforthepitchingmom entandnormalforce deltacoefcientmodelsisshowninFigure( 3-8 ).Thesolidlinesrepresentnumerical results;thedashedlinesareparameterizedusingEquation ( 3–51 ).Thecurvesshownin Figure( 3-8 )havebeenscaledbythestandarderrorofeachparameter.As aresult,the magnitudeofeachparameterisindicativeoftherelativeim portancewithinthemodel. ThelegendentriesinFigure( 3-8 )listtheparametersinorderofdecreasingstatistical signicance. 0 10 20 30 -400 -300 -200 -100 0 100 200 Vert. Pos, z (ft)Model Coeff. 1 q y 2 y q q 3 y y 2 q y q 3 y 2 q 2 q 2 APitchingMoment 0 10 20 30 -150 -100 -50 0 50 100 150 200 250 Vert. Pos, z (ft)Model Coeff. q 1 y 2 q 2 y 2 y q y q 2 q 3 y q 2 BNormalForce Figure3-8.SpatialvariationofmodelparametersatMach0. 9/550KCAS/4800ft. Figure( 3-9 )showssimilarresultsforasupersonicightcondition.Ag ain,themodel structurewasdeterminedusingamaneuvernearcarriageand modelparameterswere estimatedusingordinaryleastsquaresateachdiscretepos itionintheoweld.The magnitudeofthescaledmodelcoefcientsisrepresentativ eoftheinuenceeachterm hasontheresult,andthelegendentriesarelistedindecrea singorderofstatistical signicance.ThesolidlinesinFigure( 3-9 )representthediscretenumericalresults andthedashedlinesaretheparameterizedresultintheform ofEquation( 3–51 ).The supersonicoweldismorecomplex,andasaresulttheparam eterizedmodelisnot asaccurateasthesubsoniccase.However,theparametricmo delstillcapturesthe salientfeaturesofthetrainingdata.Considerationofani ndependentmaneuverand comparisontoight-testdatawillfurtherjustifythepred ictivecapabilityofthemodel. 119

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0 10 20 30 -500 -400 -300 -200 -100 0 100 200 Vert. Pos, z (ft)Model Coeff. 1 q y 2 q 2 y y 2 q 2 y 3 q q 3 y q 3 y q APitchingMoment 0 10 20 30 -100 -50 0 50 100 150 200 250 Vert. Pos, z (ft)Model Coeff. 1 q y 2 q 2 y y 2 q 2 y 2 q q 3 BNormalForce Figure3-9.SpatialvariationofmodelparametersatMach0. 9/550KCAS/4800ft. 3.3.2.2Modelvalidation Modelvalidationisconsideredbyapplicationoftheidenti edmodeltoan independentsimulation.Figure( 3-10 )showsasimulatedtrajectorycreatedusing orthogonalmultisineinputsforthestoreorientationwith respecttotheaircraft,based onafrequencyrangeof0-3Hzandanamplituderangeof 20 degfor and 10 for .Theverticalpositionvariescontinuouslyfrom0to30ftli nearlythroughoutthe3sec trajectory.ThesolidlinesinthelowerportionofFigure( 3-10 )representthepitching momentandnormalforcedeltacoefcientsobtainedfromcon ventionalsimulationfor theprescribedtrajectory.Thedashedlinesrepresentthep arameterizedmodelderived frompiecewisecontinuoussystemidentication.Itisappa rentthattheparametric modelisanadequaterepresentationofthewindtunneldelta coefcients.Thesubsonic resultsareslightlymoreaccurate,asexpectedbasedonthe relativecomplexitiesofthe individualowelds.However,inbothcases,thesystemide nticationresultscapture thesalientfeaturesofthespatiallyvariantaerodynamics 3.3.3FlightTestComparison Thefreestreamandspatiallyvariantparametricmodelsint roducedinSection 3.3.1 and 3.3.2 havebeenshowntoadequatelyrepresentthewindtunnelsour cedata onwhichthemodelsarebased.Furthercondenceinthepredi ctivecapabilityofthe combinedaerodynamicmodelcanbegainedbycomparisontoac tualight-testdata. 120

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0 0.5 1 1.5 2 2.5 3 -40 -20 0 20 40 Orientation (deg) q y 0 5 10 15 20 25 30 -2 -1 0 1 Vertical Distance, ZDelta Coefficient D C N D C m AMach0.9/550KCAS/4800ft 0 5 10 15 20 25 30 -2 -1 0 1 Vertical Distance, ZDelta Coefficient D C N D C m 0 0.5 1 1.5 2 2.5 3 -40 -20 0 20 40 Orientation (deg) q y BMach1.2/600KCAS/1800ft Figure3-10.Validationresultsshowingpitchingmomentan dnormalforcedelta coefcientcomparisonbetweensimulationandsystemident icationforan independentmaneuver. Inthissection,comparisonbetweensimulation,parametri cmodeling,andight-test dataindicatetheparametricmodeladequatelycharacteriz esthestoreseparation aerodynamicsandmaybeusefulfortrajectoryprediction.O therapplications,including sensitivityanalysis,trajectoryoptimization,andcontr olsystemdesign,arealsopossible giventhevalidatedparametricmodel.3.3.3.1Trajectorycomparison Figure( 3-11 )showsacomparisonbetweenight-testdata,conventional 6DOF simulationusingawindtunneldatabase,and6DOFsimulatio nusingaparametric model.ResultsareshowninFigure( 3-11A )forighttest2265(Mach0.90/552KCAS /4820ft)andFigure( 3-11B )forighttest4535(Mach1.19/595KCAS/17900ft).In bothcases,thetrajectorypredictiondeterminedusingthe identiedparametricmodel closelymatchestheconventionalsimulationandight-tes tresults,especiallyinthe primarymotionvariables:verticaltranslation, z ,andpitchangle, 3.3.3.2Aerodynamiccomparison Thereconstructedtrajectorycanbeused,alongwiththeequ ationsofmotion,to estimatethefull-scaleaerodynamicforcesandmomentsact ingonthestoreduring 121

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0 0.5 1 -10 0 10 Time (sec)Yaw, y 0 0.5 1 -40 -20 0 20 Pitch, q 0 0.5 1 -100 -50 0 50 Roll, fOrientation (deg) Sim, SysID Sim, WT Flight Test 0 0.5 1 -50 0 50 Time (sec)Vert. Z 0 0.5 1 -1 0 1 2 Lat. Y 0 0.5 1 -20 -10 0 10 Long. XTranslation (ft) AMach0.90/552KCAS/4820ft 0 0.5 1 -10 0 10 20 Time (sec)Yaw, y 0 0.5 1 -40 -20 0 20 Pitch, q 0 0.5 1 -100 -50 0 50 Roll, fOrientation (deg) Sim, SysID Sim, WT Flight Test 0 0.5 1 -50 0 50 Time (sec)Vert. Z 0 0.5 1 -2 0 2 4 Lat. Y 0 0.5 1 -40 -20 0 20 Long. XTranslation (ft) BMach1.19/595KCAS/17900ft Figure3-11.Trajectorycomparisonbetweenighttest,con ventionalsimulation,and simulationwithsystemidentication. separation.Theseestimatescanbecomparedtowindtunnele stimatesevaluated alongtheight-testtrajectoryforamoredirectassessmen toftheaerodynamicmodel. Figure( 3-12 )showsacomparisonbetweentheestimatedfull-scaleaerod ynamic coefcientsand(1)simulatedcoefcientsusingconventio nalmethodsand(2)simulated coefcientsusingparametricmodeling.Again,bothtypeso fsimulationcloselyresemble theight-testdata.Inparticular,theparametricmodelis observedtomatchighttest withnearlythesamelevelofaccuracyasconventionalmetho ds. 0 0.5 1 -4 -2 0 2 Time (sec)Yaw, C n 0 0.5 1 -5 0 5 Pitch, C m 0 0.5 1 -0.5 0 0.5 Roll, C lMoment Sim, SysID Sim, WT Flight Test 0 0.5 1 -20 -10 0 10 Time (sec)Normal, C N 0 0.5 1 -1 0 1 2 Side, C Y 0 0.5 1 -1 0 1 2 Axial, C AForce AMach0.90/552KCAS/4820ft 0 0.5 1 -5 0 5 Time (sec)Yaw, C n 0 0.5 1 -5 0 5 Pitch, C m 0 0.5 1 -0.5 0 0.5 Roll, C lMoment Sim, SysID Sim, WT Flight Test 0 0.5 1 -20 -10 0 10 Time (sec)Normal, C N 0 0.5 1 -2 0 2 Side, C Y 0 0.5 1 -1 0 1 2 Axial, C AForce BMach1.19/595KCAS/17900ft Figure3-12.Aerodynamiccomparisonbetweenighttest,co nventionalsimulation,and simulationwithsystemidentication. 122

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3.4Example:PlanarStoreSeparation Storeseparationisoftendominatedbyverticaltranslatio nandpitchattitude.In mostcases,lateraltranslationandyawattitudearefairly benignandofsecondary interest.Forinstructivepurposes,considerationofasto reconnedtothevertical x– z planeduringseparationmaintainstheprimaryscopeofinte restandconsiderably reducesthecomplexityoftheparametricmodel. ThesystemidenticationapproachdescribedinSection 3.3 canalsobeusedto determineasimpliedparametricmodeldescribingthelong itudinalaerodynamicsof astoreconnedtotheverticalplane.InSection 3.3 theobjectivewastoconstructa parametricmodelthatcouldadequatelycapturethesalient featuresofsixdimensional windtunneldatabaseforapplicationtotrajectorypredict ionandighttestmatching. Asimpliedplanarmodelprovidesareducedordermodelthat retainstheessential featuresofthemorecomplexmodel,butinasimplecompactex pression.This simpliedparametricmodelwillbeusedextensivelyinChap ters 4 and 5 toillustrate theapplicationofoptimalcontroltheorytostoreseparati on.Thefullsixdimensional aerodynamicmodelwillbeconsideredinChapter 6 Forplanarstoreseparation,thedominanteffectofthenonu niformoweldis onthepitchingmoment.Assuch,considerthefollowingspat iallyvariantquasi-linear aerodynamicmodel. C A = C A 0 = constant (3–52) C N = C N + C N e e (3–53) C m = C m + C m q ^ q + C m e e + e ( z ) ( 0 + 1 z ) (3–54) Thevariable e istheelevatorcontrolsurfacedeection,theonlyinputof interest fordynamicslimitedtothe x–z plane.Thevariable ^ q isthenon-dimensionalpitch rateintroducedforunitconsistency.Thevariables C N C A 0 C m ,and C m q arethe classicalaerodynamicderivativesinthebodyaxis.Thevar iables C N e and C m e are 123

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theclassicalcontrolderivatives.Theaerodynamicandcon trolderivativescanbe estimatedusingfreestreamsystemidentication.Theexpo nential-polynomialform C m ( z )= e ( z ) ( 0 + 1 z ) isaspecialcaseofthemoregeneralparametricmodel presentedinSection 3.3 .Theconstants 0 ,and 1 canbeestimatedusingspatially variantsystemidentication.Thenalresultisasimplie dquasi-linearaerodynamic modelthatcanbeusedtoinvestigateguidanceandcontrolof astoreduringseparation. Thefollowingexampleisbasedonastoreaerodynamicdataba seatMach0.8 /10kft.ThelongitudinalinputisshowninFigure 3-13 .Theangleofattack s was speciedasamultisineinputusinganamplitudeof 5 deg,afrequencyrangeof 0 5 Hz,andadurationof t =5 seconds.Thepitchrate q wascomputedusingthekinematic relationshipforpitchrateandangleofattack.Forthecase ofplanarmotionwitha pinnedcenterofgravity, q =_ s .Finally,theelevatorcontrolsurfacedeection e was speciedusinganorthogonalmultisineinputwiththesamea mplitude,frequencyrange, anddurationastheangleofattack. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -10 0 10 Angle of Attack a s (deg) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -200 0 200 Pitch Rate q (deg/sec) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -10 0 10 Elevator InputTime (sec) d e (deg) Figure3-13.Angleofattack,pitchrate,andelevatorinput forplanarsystem identication. Thefreestreamsimulationresultsforthepitchingmomenta ndnormalforceare showninFigure 3-14 .Theaerodynamiccoefcientsweredeterminedusinginterp olation ofthefreestreamwindtunneldatabasealongthetrajectory showninFigure 3-13 .The 124

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systemidenticationresultsweredeterminedusingtheequ ationerrormethoddescribed inSection 3.2.4.1 .Somediscrepanciesbetweenthemodelandthedataareevide nt,as onemightexpectgiventhesimplelinearstructureoftheaer odynamicmodel.However, forthislimitedrangeofangleofattack, j s j 5 deg,thelinearmodeladequately capturesthefreestreamaerodynamiccharacteristics.The normalforceandpitching momentderivativeswereestimatedtobe C N =4.56 rad -1 and C m = 3.38 rad -1 respectively.Thepitchingmomentdampingderivativeiskn ownfromthewindtunneltest reporttobe C m q = 74 rad -1 .Thenegativevalueof C m forsmall s isindicativeofthe inherentlongitudinalstabilityofthestore. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec)Pitching Moment, C m System ID Simulation APitchingMoment 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Time (sec)Normal Force, C N System ID Simulation BNormalForce Figure3-14.Planaraerodynamiccoefcientcomparisonbet weensimulationandsystem identicationforastoreinfreestreamightconditionsat Mach0.8/10kft. Figure 3-15 showssimulationresultsforthepitchingmomentandnormal force controlincrement.Again,thesimulationresultsweredete rminedusinganaerodynamic databaseandthesystemidenticationresultsweredetermi nedusingtheequationerror 125

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method.Thenormalforceandpitchingmomentcontrolderiva tiveswereestimatedtobe C N e =2.09 rad -1 and C m e = 8.16 rad -1 ,respectively. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1 -0.5 0 0.5 1 1.5 Time (sec)Pitching Moment, D C m ( d e ) System ID Simulation APitchingMoment 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Time (sec)Normal Force, D C N ( d e ) System ID Simulation BNormalForce Figure3-15.Planaraerodynamiccontroleffectcomparison betweensimulationand systemidenticationforastoreinfreestreamightcondit ionsatMach0.8/ 10kft. Finally,Figure 3-16 showsthepitchingmomentdeltacoefcient C m ( z ) .Thegure includesdataextractedfromthewindtunneldatabaseforan ominalyawangleof A B = 0 .Thewindtunneldatashowthemeasureddeltacoefcientfor increasinglynose-down pitchattitudes.Thepitchattitudeisheldconstantthroug houttheverticalsweep.The deltacoefcientsconvergetozeroatabout 30 feetbelowtheaircraft,indicatingthe aircrafteffectisnolongersignicant.Thenonlinearexpo nential-polynomialproductis alsoshown. Theconstants 0 and 1 weredeterminedusingthespatiallyvariantsystem identicationtechniquedescribedinSection 3.2.4.2 .Thegureindicatesthatthe 126

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0 5 10 15 20 25 30 35 40 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 Time (sec)Pitching Moment, D C m (z) q =10 q =0 q =-10 q =-20 q =-30 D C m (z) = e m z ( h 0 + h 1 z) Figure3-16.Pitchingmomentdeltacoefcientforsimplie dparametricmodelusing systemidentication. parametricmodelcloselyresemblesthegriddataforapitch anglebetween 0 < A B < 10 deg.Thisresultisconsistentwiththetrainingmaneuver,w hichisbasedonapitchangle between 5 < A B < 5 deg.ThespatiallyvariantparametersaresummarizedinTab le 3-1 ,alongwiththefreestreamderivatives. Table3-1.Parametersusedinplanarstoreseparationaerod ynamicmodel. AerodynamicDerivativesControlDerivativesSpatiallyVa riantParameters C N =4.56 rad 1 C m e = 8.16 rad 1 =0.115 C M = 3.38 rad 1 C N e =2.09 rad 1 0 = 1.23 C M q = 74 rad 1 1 =0.042 ft 1 C A 0 =0.201 ThecompactaerodynamicmodelinEquations( 3–52 )through( 3–54 )canbeused inconjunctionwiththethreedegree-of-freedomequations ofmotiontosimulatethe separationofastoreconnedtothevertical x–z plane.Thisreducedordermodel 127

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isusefulfordesignandanalysisofpotentialcontrolstrat egies,aswillbeshownin Chapters 4 and 5 3.5ChapterSummary Thepurposeofthischapteristoshowthattheexistingaerod ynamicdatacan berepresentedasaparametricmodelusingsystemidentica tiontechniques. Theparametricmodelofferstheadvantageofrepresentingt hestoreseparation aerodynamicsinanalyticalform,suitablefortrajectoryp rediction,sensitivityanalysis, trajectoryoptimization,andcontrolsystemdesign. Theparametricmodelisexpectedtobenonlinearandspatial lyvariantdueto thecomplexityofthenonuniformoweldsurroundingtheai rcraftduringight.The modelingtaskisdecomposedintotwocomplementarymodels: afreestreammodel fortheuniformowcontributionandadelta-coefcientmod elforthenonuniformow contribution.Thefreestreamaerodynamicmodelisbasedon amultivariatepolynomial withconstantcoefcients,themodelstructureisidentie dusingorthogonalpolynomials, andthecoefcientsareestimatedusingordinaryleastsqua res.Thedeltacoefcient modelisbasedonamultivariatepolynomialwithspatiallyv ariantcoefcients.The spatiallyvariantcoefcientsaregivenaparticularexpon ential-polynomialformbasedon physicalconsiderationsandknownboundaryconditions.Th eunknownparametersare estimatedusingnonlinearleastsquares.Theresultisacom pactnonlinearparametric modelthatisasuitablecandidateforanystoreseparationp roblemwheretime-averaged aerodynamicdataareavailable. Theproposedparametricmodelingapproachisappliedtoare presentative example.Theexampleincludessmall-scalewindtunneldata andfull-scaleight-test dataforarepresentativestoreseparatingfromtheF-16air craftatbothsubsonicand supersonicconditions.Inallcasesconsidered,theparame tricmodeladequately representedtheunderlyingwindtunneldataandaccurately reproducedtheight-test results.Assuch,thevalidatedmodelisconsideredsuitabl eforfurtheranalysis, 128

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includingtrajectoryoptimizationandcontrolsystemdesi gn.Specically,developmentof astoreseparationguidanceandcontrolsystemusingthemod elsdiscussedhereinis consideredinChapter 6 129

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CHAPTER4 TRAJECTORYOPTIMIZATION 4.1Overview Trajectoryoptimizationistheprocessofdeterminingcont rolandstatehistories foradynamicsysteminordertominimize(ormaximize)ameas ureofperformance whilesatisfyingprescribedboundaryconditionsand/orpa thconstraints[ 87 ].Trajectory optimizationiscloselyrelatedtoOptimalControlTheory, andindeed,thetermsareoften usedinterchangeably.Thedynamicsystemisgenerallymode ledinthetimedomain usingastatespacerepresentation.Themeasureofperforma ncerepresentsametric orcombinationofmetrics(e.g.time,energy,controleffor t,deviationfromadesired operatingcondition,etc.)thatquantifythedesiredperfo rmanceofthesystem.The boundaryconditionsincludelimitationsontheinitialand /ornalstateofthedynamic system,aswellaslimitsonthecontrol(e.g.actuatorlimit s,controlsaturation).Path(or state)constraintsareusedtoexcludetrajectoriesthatvi olateapredeterminedrange ortypeofundesirablemotion.Eachofthesecomponentsares tatedwithmathematical precisionandcombinedtocreateanoptimalcontrolproblem Fromanhistoricalpointofview,theadventofoptimalcontr oltheoryarosefrom thestringentrequirementsonaerospacesystems[ 88 ].Suchsystemsareinherently nonlinear,theyaresubjecttovariousconstraints(fuel,t hrust,acceleration,etc.),and theyaregreatlybenetedbyoptimizationduetohighoperat ionalcosts.Prominent applicationsofoptimalcontroltoaerospacesystemsinclu deorbittransferand rendezvous[ 89 ],optimalguidanceandcontroltodirectthepoweredighto fa missile[ 90 ],controlofare-entryvehicle[ 91 92 ],maximumrangeofamissile[ 89 ], andautomaticlandingofanaircraft[ 93 ].Aprominentcontributionofoptimalcontrol theorycamefromastudypublishedbyBryson[ 94 ].Inthisgroundbreakingwork, Brysonpresentedtheoptimalightpathforasupersonicgh tertoclimbtoadesired altitudeinminimumtimeandacompanionsolutionforaghte rtoachievemaximum 130

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altitude,acapabilityofintenseinterestduringthecoldwarerainwhichtheresultswere obtained.Thesolutionswerepairedwithactualighttestr esultsthatcloselymatched thepredictedtrajectories.Thenon-intuitivebutrealist icallydemonstratedresultshelped optimalcontroltheorygaintractionamongpractitioners. Thisclassicproblemisstill usedasaclearillustrationofthecapabilityofoptimalcon trol[ 95 ]. Morerecentlyoptimalcontroltechniqueshavebeenusedtos olvespacecraft orientationandnavigationproblemswithminimumtimeand/ orfuelconsumption[ 96 97 ],ightcontrolofareusablereentryvehicle[ 91 92 ],multiplespacecraftcooperative formation[ 98 99 ],andtime-optimalUAVightthroughanurbanenvironment[ 100 101 ]. Alsoofparticularrelevancearethenumerousstudiesconsi deringightpathtrajectory optimizationthroughnon-uniformwindelds,suchaswindcorrectedightpathplanning formicro-airvehicles[ 102 ],minimum-timeightpathsthroughhigh-altitudeatmosph eric winds[ 103 104 ],optimalrecoveryfrommicroburstwindshear[ 105 106 ],andthe maximumrangetrajectoryofaglideringroundeffectandwin dshear[ 107 ].Optimal controlhasalsobeensuccessfullyusedinvariousotherapp lications,includingsuch diverseapplicationsasmotionplanningforautonomousgro undvehicles[ 108 ],quantum mechanics[ 109 ],andenvironmentalandeconomicprocesses[ 110 ],tonamejusta few.Severalexcellenttextbookshavealsobeenwrittenont hetopicofoptimalcontrol, includingthosebyBryson[ 87 111 – 113 ],Kirk[ 114 ],AthansandFalb[ 115 ]andStengel [ 116 ]. Theclassicalapproachtosolvinganoptimalcontrolproble minvolvestheuse ofPontryaginsminimumprinciple[ 117 ].Pontryaginsminimumprinciple(PMP)is fundamentallyanextensionandapplicationofthecalculus ofvariationstotheoptimal controlproblem[ 88 ].Thisapproachcanbeusedtodevelopasetofdifferential equationsthatprovidenecessary(butnotsufcient)condi tionsforlocaloptimality. Thesenecessaryconditionscanbeusedtondanalyticalsol utionsforanarrow rangeofoptimalcontrolproblems,butinpracticenumerica lmethodsarenecessary 131

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toapproximateasolution.Inmorerecentliterature,numer icalmethodsareusedto approximateasolutionmoredirectly,butPMPstillplaysak eyroleinvalidatingthe necessaryoptimalityconditionsfortheproposedsolution [ 118 119 ].Inthisresearch, theclassicalindirectapproachtooptimalcontrolisconsi deredexclusively,dueprimarily tothegreatercomputationalefciencyandadditionalinsi ghtgainedbyanalytical formulationoftheoptimalityconditions.Moredetailabou tthenumericalmethodsused inthecourseofthisstudyarepresentedinSection 4.2.5 4.2OptimalControl Theobjectiveofanoptimalcontrolproblemistodeterminea nadmissiblecontrol inputthatminimizes(ormaximizes)thedesiredperformanc eindexsubjecttothe speciedboundaryconditionsanddynamicconstraints.The solutionusinganindirect methodisbasedonthecalculusofvariations.Ordinarycalc ulusispredominantly concernedwiththecalculusoffunctions,characterizedby thedifferentialoperator. Comparatively,thecalculusofvariationsisconcernedwit hthecalculusoffunctionals, characterizedbythevariationaloperator.4.2.1FirstOrderOptimalityConditions Theobjectiveofanoptimalcontrolproblemistominimizeth eBolzaformofthecost functional J = x ( t 0 ), t 0 x ( t f ), t f + t f Z t 0 L x ( t ), u ( t ) dt (4–1) subjecttothedynamicconstraints _x ( t )= f x ( t ), u ( t ) (4–2) theinequalitypathandcontrolconstraints C x ( t ), u ( t ) 0 (4–3) 132

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andtheboundaryconditions x ( t 0 ), t 0 x ( t f ), t f =0 (4–4) where x ( t ) 2 R n isthestate, u ( t ) 2 R m isthecontrol, f : R n R m R n isasystem ofordinarydifferentialequations, C : R n R m R s arethe(optional)pathandcontrol constraints, : R n R R n R R q aretheboundaryconditions,and t istime.The Mayercostisgivenby : R n R R n R R andtheLagrangianis L : R n R m R Theaugmentedcostfunctionalisformedbymultiplyingthec onstraintswith associatedLagrangemultipliersandjoiningtheproductwi ththecostfunctional. J a = x ( t 0 ), t 0 x ( t f ), t f T x ( t 0 ), t 0 x ( t f ), t f + t f Z t 0 L x ( t ), u ( t ) T ( t ) _x ( t ) f x ( t ), u ( t ) + T ( t ) C x ( t ), u ( t ) dt (4–5) where T 2 R q T ( t ) 2 R n ,and T ( t ) 2 R s aretheLagrangemultipliers.The Hamiltonianisintroducedintermsofthestate x ( t ) andthecostate T ( t ) asa combinationoftheLagrangianandaugmenteddynamicconstr aints. H x ( t ), u ( t ), ( t ), ( t ) := L x ( t ), u ( t ) + T ( t ) f x ( t ), u ( t ) + T ( t ) C x ( t ), u ( t ) (4–6) UsingtheHamiltonian,theaugmentedcostfunctionalcanbe writtenasfollows. J a = x ( t 0 ), t 0 x ( t f ), t f T x ( t 0 ), t 0 x ( t f ), t f + t f Z t 0 H x ( t ), u ( t ), ( t ), ( t ) T ( t ) _x ( t ) dt (4–7) The1 st ordernecessaryconditionsforanextremaltrajectoryared eterminedby takingtherstvariationoftheaugmentedcostfunctionalw ithrespecttoeachfree variable. 133

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J a = @ @ x ( t 0 ) x 0 + @ @ t 0 t 0 + @ @ x ( t f ) x f + @ @ t f t f T T @ @ x ( t 0 ) x 0 + @ @ t 0 t 0 + @ @ x ( t f ) x f + @ @ t f t f + H T _x t = t f t f H T _x t = t 0 t 0 + t f Z t 0 @ H @ x x + @ H @ u u T ( _x f ) T _x + T C dt (4–8) ThetermwithintheintegralofEquation( 4–8 )containing _x requiresspecial treatment.Integratingbyparts,theintegralcanbeexpand edasfollows. t f Z t 0 T _x dt = T ( t f ) x ( t f )+ T ( t 0 ) x ( t 0 )+ t f Z t 0 T x dt (4–9) Furthermore,theterms x ( t 0 ) and x ( t f ) canbeexpandedtorstorderusingthe denitionofavariationinamanneranalogoustoaTaylorser iesexpansion. x 0 = x ( t 0 )+ _x ( t 0 ) t 0 (4–10) x f = x ( t f )+ _x ( t f ) t f (4–11) Substitutingtheseexpressionsintoequation( 4–8 ),cancellingtermsinvolving _x ( t 0 ) and _x ( t f ) ,andfactoringthevariationofeachfreevariable,theaugm entedcost functionalisgivenbyEquation( 4–12 ). J a = @ @ x ( t 0 ) T @ @ x ( t 0 ) + T ( t 0 ) x 0 + @ @ x ( t f ) T @ @ x ( t f ) T ( t f ) x f + @ @ t 0 T @ @ t 0 H ( t 0 ) t 0 + @ @ t f T @ @ t f + H ( t f ) t f T + t f Z t 0 @ H @ x + T x + @ H @ u u T ( _x f ) + T C dt (4–12) 134

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Inamanneranalogoustoordinarycalculus,theextremalcon ditionisfoundby requiring J a =0 .However,eachvariationisindependent,soforanextremal solutionto existeachterminequation( 4–12 )mustbezeroindividually.Asaresult,thefollowing1 st orderoptimalityconditionsareobtained. _x T = @ H @ (4–13) T = @ H @ x (4–14) 0 = @ H @ u (4–15) T ( t 0 )= @ @ x ( t 0 ) + T @ @ x ( t 0 ) (4–16) T ( t f )= @ @ x ( t f ) T @ @ x ( t f ) (4–17) H ( t 0 )= @ @ t 0 T @ @ t 0 (4–18) H ( t f )= @ @ t f + T @ @ t f (4–19) Equations( 4–13 )and( 4–14 )representasetof 2 n coupledordinarydifferential equations.Theboundaryconditionsaregivenbyequations( 4–16 )through( 4–19 ). Dependingonthespecicationsoftheproblem,oneormoreof theseboundary conditionsmaynotapply.However,theboundaryconditions usuallyincludespecications attheinitialandterminalconditions.Thus,equations( 4–13 )and( 4–14 )canusuallybe classiedasasetof 2 n nonlinear,coupledordinarydifferentialequationswiths plit boundaryconditions,referredtoasaHamiltonianboundary valueproblem(HBVP). Finally,notethatequation( 4–15 )canoftenbeusedtodetermineanexplicitformforthe optimalcontrolintermsofthestateandcostate. Usingthecomplementaryslacknesscondition,theLagrange multiplier ( t ) is relatedtothedynamicconstraint C ( x ( t ), u ( t ) ) 0 ,asshowninEquation( 4–20 ). 135

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Lagrangemultiplier ( t ) : 8>><>>: i ( t )=0 if C i ( x ( t ), u ( t ) ) < 0; i =1,..., s i ( t ) > 0 if C i ( x ( t ), u ( t ) ) =0; i =1,..., s (4–20) Thepositivityof i when C i =0 isinterpretedsuchthatimprovingthecostmayonly comefromviolatingtheconstraint[ 113 ].Furthermore, i ( t )=0 when C i < 0 statesthat thisconstraintisinactiveandcanbeignored.Inthiscase, theHamiltonianreducesto theclassicalformgiveninequation( 4–21 ). H = L ( x ( t ), u ( t ) ) + T ( t ) f ( x ( t ), u ( t ) ) (4–21) Theconstraint C ( x ( t ), u ( t ) ) 0 canbeusedtolimitthecomponentsofthestate orthecontroltopredeterminedhardlimits.Statedanother way,anytrajectorythatdoes notsatisfytheseconstraintsisinadmissibleandwillnots atisfythenecessaryconditions. Frequently,thehardlimitscanbereplacedby“softlimits” imposedbypenalizingthe componentsofthestateandcontrolinthecostfunctional.W henthisisthecase,the additionalconditionsrequiredbythedynamicstateandcon trolinequalityconstraints maybeomitted,andEquations( 4–13 )through( 4–19 )providethenecessaryconditions foranextremalorstationarysolution. Theseconditionsalonearenotsufcienttoconcludethatth eextremaltrajectory isindeedalocaloptimaltrajectory.Inamanneranalogoust oordinaryfunction minimization,considerationofthesecondvariationprovi desasufcientcondition todemonstratethatthestationarysolutionisalsoalocalm inimumsolution.Further discussionofthenecessaryandsufcientconditionsforop timalitywillbepresentedin Chapter5,anditwillbeshownthatinorderforaneighboring optimalcontroltoexist, thestationarysolutionmustbealocalminimum(i.e.thesec ondvariationispositive deniteovertheentirepath).Fornow,asolutionthatsatis esequations( 4–13 )through 136

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( 4–19 )maybesaidtobeanextremalorstationarytrajectory,whic hmayormaynotbe anoptimaltrajectory.4.2.2InterpretationoftheCostate Thenecessaryconditionspresentedinequations( 4–13 )through( 4–19 )introduce anadditionalsetofdifferentialequationsreferredtoast hecostate.Thecostateis aLagrangemultiplierthatwasintroducedpurelyasamathem aticaldevicetoallow formulationoftheminimizationproblemsubjecttothedyna micconstraints.However, thecostatecanberelatedtothecostfunctionalalongastat ionarypath,asshownby BrysonandHo[ 113 ]. T = @ L min @ f (4–22) Hence,thecostatecanbeinterpretedasthesensitivityoft heLagrangecost functionaltochangesinthesystemdynamicsevaluatedalon gtheextremaltrajectory. Thisrelationshipprovidesmeaningfulinsightintotheopt imalperformanceofthesystem, asdiscussedinSection 4.3.3.4 4.2.3InterpretationoftheHamiltonian ConsidertheformoftheHamiltonianwithoutinequalitycon straintsgivenby equation( 4–21 ),whichcanbewrittenasfunctionalforminequation( 4–23 ). H = H x ( t ), u x ( t ), T ( t ) T ( t ), t = H x ( t ), T ( t ), t (4–23) Takingthetotalderivativewithrespecttotimeyieldsequa tion( 4–24 ). dH dt = @ H @ x _x + @ H @ + @ H @ t (4–24) NotingthatthederivativesoftheHamiltonianwithrespect tothestateandcostate aregivenbythe1 st orderoptimalityconditionsinequations( 4–13 )and( 4–14 ),thetotal derivativecanbestatedasequation( 4–25 )andreducedtoequation( 4–26 ). dH dt = T _x + _x T + @ H @ t (4–25) 137

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dH dt = @ H @ t (4–26) Inmanycases,theHamiltonianisnotanexplicitfunctionof time,soequation ( 4–26 )canbetakenfurthertoimplythattheHamiltonianisaconst ant. dH dt = @ H @ t =0 ) H = const (4–27) ThesignicanceofEquation( 4–27 )isnoteworthy.Equation( 4–27 )impliesthat whentheHamiltonianisnotanexplicitfunctionoftime,itm ustbeconstantalongan extremalsolution.Analytically,thisresultcanfrequent lybeusedasanadditionalknown valueinsolvingoptimalcontrolproblems.Numerically,th econstancyoftheHamiltonian canbeusedasaqualitychecktoensurethatanextremalsolut ionhasindeedbeen found.4.2.4LinearQuadraticRegulator Recallthatthecostfunctionalisgiveningenericformbyeq uation( 4–1 ),restated here. J = ( x ( t 0 ), t 0 x ( t f ), t f ) + t f Z t 0 L ( x ( t ), u ( t ) ) dt (4–28) InEquation( 4–28 ), ( x ( t 0 ), t 0 x ( t f ), t f ) istheendpointorMayercost,and L ( x ( t ), u ( t ) ) istheaccumulatedorLagrangecost.Inmanyapplications,a quadratic costfunctionalisapplied,asinequation( 4–29 ). J = x T ( t f ) P x ( t f )+ 1 2 t f Z t 0 x T Q x + u T R u dt (4–29) Thematrix Q 2 R n n isapositivesemi-denitematrix( Q 0 )and R 2 R m m is apositivedenitematrix( R > 0 )chosenbythecontroldesignertoachievefavorable trajectorycharacteristicswithinsuitablestateandcont rollimits.Thequadraticcost functionalisquiteexibleandhasanintuitivephysicalsi gnicance.Thevaluesof Q determinewhichstatevariableswillberegulatedtozero.T hevaluesof P penalizethe 138

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deviationfromadesiredend-state.Thevaluesof R determinetheamountofcontrol effortavailabletoachievetheobjective. Thequadraticcostfunctionalisespeciallyusefulfordyna micsystemsrepresented byalineartimevariantstatespacerelationship. _x = A ( t ) x ( t )+ B ( t ) u ( t ) (4–30) Usingequation( 4–30 ),theHamiltoniancanbewrittenasfollows. H = 1 2 x T Q x + 1 2 u T R u + T ( t ) A ( t ) x ( t )+ T ( t ) B ( t ) u ( t ) (4–31) Thecostatedifferentialequationsandoptimalcontrolcon ditionarereadilydetermined, asshowninequations( 4–32 )and( 4–33 ). = @ H @ x T = Q x ( t ) A T ( t ) ( t ) (4–32) 0 = @ H @ u = R u + B T ( t ) ( t ) (4–33) Equation( 4–33 )canbesolvedtodeterminetheoptimalcontrolintermsofth e systemparametersandthecostate,asshownin( 4–34 ). u = R 1 B T ( t ) ( t ) (4–34) Suppose,duetothelinearityofthesystemdynamics,thatth ecostatecanbe writtenasalinearcombinationofthestatevariables.This isshowninequation( 4–35 ), where K ( t ) isanunknownmatrix.Thederivativeof( 4–35 )isgivenby( 4–36 ). ( t )= K ( t ) x ( t ) (4–35) ( t )= K ( t ) x ( t )+ K ( t ) _x ( t ) (4–36) Substitutingintheexpressionsfor _x and andsolvingfor K providesthefollowing matrixdifferentialRiccatiequation. 139

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_ K ( t )= Q A T ( t ) K ( t ) K ( t ) A ( t )+ R 1 B T ( t ) K ( t ) (4–37) TheRiccatiequationisstableinbackwardtimeandcanbesol vedusingstandard numericaldifferentialequationsolvers,suchasaRungeKu ttamethod.Theresultisa setoftimevaryinggains K ( t ) .Asaresult,theoptimalcontrollawcanbewrittenasa combinationofthestatevariablesasinEquation( 4–38 ). u = R 1 B T ( t ) K ( t ) x ( t ) (4–38) Noticethatthecostateequationsareeliminatedfromtheso lutionprocess andtheboundaryvalueproblemisdecoupled.TheRiccatiequ ationcanbesolved independently,allowingthecontrolgainstobetabulateda ndstoredofine.Theoptimal trajectorycanbeimplementedreal-timeusingthestoredga insandfull-statefeedback. Thisresultsharesmanycommonalitieswithneighboringopt imalcontrol,andwillbe furtherconsideredinChapter 5 4.2.5NumericalMethods Numericalmethodsforoptimalcontrolcanbeclassiedasdi rectorindirect methods.Followingtheindirectmethod,thenecessarycond itionsarederivedfrom thecalculusofvariationsandPontryagin'sminimumprinci ple(PMP),resultingina Hamiltonianboundary-valueproblem(HBVP).Thedifferent ialequationsposedbythe HBVPnecessarilyincludeasubsetofvariablesthatareunst able,makingnumerical solutionsdifculttoobtainoverlongdurations.Examples ofindirectmethodsinclude shooting,multipleshooting[ 120 ],nitedifference[ 121 ],andcollocation[ 122 ].The numericaldifcultieswithindirectmethods,combinedwit htherequirementtoderive theoptimalityconditionsinanalyticalform,haveledtoth edevelopmentofalternative numericalmethodsthatattempttondanapproximatesoluti ondirectly. Inadirectmethod,theoptimalcontrolproblemistranscrib edintoanite-dimensional nonlinearprogrammingproblem(NLP),whichcanbesolvedus ingavarietyofexisting 140

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softwarepackages.Examplesofdirectmethodsincludedire ctshootingmethods[ 123 ] anddirectcollocationmethods[ 124 ].Asubsetofdirectcollocationmethodsreferredto as p-methods or pseudospectralmethods [ 125 ]havegainedimmensepopularityinthe optimalcontrolcommunityduetothediversityofproblemst hatcanbesolved,favorable convergenceproperties,computationalefciency,andeas eofproblemstatementwhen littleisknownabouttheoptimalperformanceofthesystem. Full-featuredsoftware packagesimplementingpseudospectralmethodsarenowavai lableincommercial[ 119 ] andopensourcepackages[ 125 ]. Collocation,avaluablenumericalmethodforsolvingbound aryvalueproblems, involvesrepresentingthesolutiontothedifferentialequ ationasalow-orderpolynomial overameshofspeciedsub-intervals.Asaresult,thediffe rentialequationisrepresented byasystemofnonlinearalgebraicequationsthatinherentl ysatisfytheendpoint constraintsandmaybesolvedusingasuitablecomputationa lalgorithm.Whereas conventionalnumericalintegrationmethods(e.g.Euler,R unge-Kutta,etc.)require propagationofthedifferentialequationsintime,colloca tionarrivesatasolutionoverthe entireintervalsimultaneously.Thispropertylargelymit igatesthenumericaldifculties associatedwithunstablemodesofaHBVP.Assuch,collocati onprovidesanefcient waytondasolutiontoanoptimalcontrolproblemfollowing eithertheindirectordirect approach. Theindirectapproachrequiresanalyticalstatementofthe 1 st orderoptimality conditions,butthisalsooffersanadvantageofadditional insightintotheoptimal performanceofthedynamicsystem.Additionally,theindir ectapproachusingcollocation generallyresultsinamoreaccuratesolutionobtainedwith lesscomputationalresources [ 95 ].Finally,theindirectmethodisassuredtosatisfythers torderoptimalityconditions andsecondorderconditionsmaybeusedtoverifythesolutio nasaminimum(as opposedtoamaximum,saddle,orsingularsolution). 141

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Forthecurrentapplicationtostoreseparationtrajectory optimization,theshort durationofastoreseparationevent(approximately1sec)m akestheproblemamenable toindirectmethodsandthephysicalinsightgainedfromthe 1 st orderoptimality conditionsisavaluableassetforstoreseparationtraject oryanalysis.Forthesereasons, theemphasisinthisresearchisonindirectmethods.Thenum ericalsolutionspresented hereinarebasedontheMatlab R r program bvp4c ,whichimplementsathree-stage LobattoIllaformula.The bvp4c algorithmprovidesanapproximatesolutionthatis fourth-orderaccurateanddifferentiableoverthespecie dinterval[ 126 ].Theoptimal controlsolutionspresentedinSection 4.3.3 aresolvedexclusivelyusingthismethod. 4.3OptimalStoreSeparation 4.3.1PerformanceIndex Successfulstoreseparationisabalancebetweentwocompet ingobjectives.First, asuccessfulstoreseparationtrajectorymustbesafeandno texhibitanythreatening motiontowardtheaircraft.Insomecases,lateralmotionis theprimaryconcerndueto tighttolerancesbetweenthestoreandadjacentaircraftco mponentsoradditionalstores. However,inmostcases,safeseparationisdominatedbythev erticaltranslationofthe store.Ifthestoreescapestheaircraftoweldwithamonot onicallyincreasingvertical velocity,thenthetrajectoryisconsideredsafe.Ifthesto rehesitatesorbeginstoyback totheaircraft,thetrajectoryisconsideredunsafe.Dueto uncertaintiesinseparation predictionmethodsandvariationsinstoreandaircraftpro perties,ight-testingofunsafe trajectoriesisusuallyavoidedaltogether. Inmostcases,thestoreislaunchedfromanejectorprovidin ganinitialvertical velocity.Inordertoyback,thestoremustgenerateenough aerodynamiclifttorst arresttheverticalvelocityandthenbegintranslationina nupwarddirection.Thus, ybackisalwaysprecededbyasignicantdurationataposit iveangleofattack.For moststores,limitingtheangleofattackcanensureasafese paration.Thesafetymargin isincreasedwhentheangleofattackisnegativethroughout muchofthetrajectory, 142

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generatingaerodynamicforcesinthedirectionoftranslat ionandacceleratingthestore awayfromtheaircraft. Asecondcriterionforasuccessfulseparationisthatthetr ajectorymustbe acceptable,i.e.thetransitoryeffectsoftheseparationm ustnotcompromisethe abilityofthestoretoachieveaspeciedmission.Anunsafe trajectorycannotbe acceptable,butasafetrajectorymaybeunacceptable.Ther efore,safetyisasubset ofacceptability.Incomparisontosafety,itisgenerallym oredifculttoquantifyand ensureacceptability.However,forthepurposeofthisinve stigation,acceptabilitycanbe adequatelyaddressedbythefollowingfournecessarycondi tions. 1. Thestoretotalaerodynamicangleofattackshouldnotexcee dthespeciedrange forwhichthestoreautopilothasbeendesignedtofunctionp roperly. 2. Theangularratesandaccelerationsshouldnotexceedthesp eciedrangefor whichtheonboardinstrumentationissufcienttomeasure. 3. Thecontrolinputsshouldnotexceedthespeciedcapabilit yofthecontrol actuators. 4. Thetotalaerodynamicloadsshouldnotexceedthesafetymar ginsforthe structuralintegrityofthestoreandempennage. Aconceptualcomparisonofsafeandacceptabletrajectoryc haracteristicsisshown inFigure 4-1 .InFigure 4-1A ,thebenigntrajectoryissafeandacceptable.InFigure 4-1B ,thetrajectoryisunsafeduetoybackandthereforeunacce ptable.InFigure 4-1C thetrajectoryissafe,butunacceptableduetoover-rotati on. ASafe/Acceptable BUnsafe/Unacceptable CSafe/Unacceptable Figure4-1.Conceptualtrajectoriesdemonstratingsafety andacceptabilitycriteria. 143

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Precisestatementoftheacceptabilityconditionsrequire sconsiderationofaspecic system.Ingeneralterms,acceptabilitycanbeachievedbyk eepingthetotalangleof attackandangularrateslowandbylimitingcontroleffort. Anarrowbutusefulsufcient conditionforacceptability,especiallyincontrolsystem design,istorequirethestateand inputbemaintainedwithinacertainpredenedoperatingra ngeoverwhichthecontrol systemhasbeendesignedtofunctionproperly. Finally,itisrecognizedthataseparationautopilotisatr ansitionalcontrolsystem, intendedtoguidethestorethroughthenonuniformowelda ndtransferthecontrolto themissionautopilot.Assuch,theobjectiveofaseparatio nautopilotistosafelydrive thestoretoanear-equilibriumstateatorbeforethetransi tiontothemissionautopilot. Therefore,itisdesirablenotonlyforcertaincomponentso fthestatetobenearzero,but alsoforcertaincomponentsofthederivativeofthestateto benearzero. Thefollowingdiscussioncanbesummarizedasthefollowing sufcientconditionfor safeandacceptablestoreseparation:minimizethetotalae rodynamicangleofattack anddeviationfromanear-equilibriumconditionattheendstatesubjecttopredened limitsonthestateandcontrolinputs.Thisobjectivecanre adilybeachievedusinga quadraticcostfunctionalandcontrolinequalityconstrai ntswhennecessary. 4.3.2FirstOrderOptimalityConditions Thegeneral1 st orderoptimalityconditionspresentedinEquations( 4–13 )through ( 4–19 )canreadilybeappliedtostoreseparation.Thestatespace representationofthe storeseparationequationsofmotionaresimilartotheclas sicalaircraftightdynamics equationsprevalentintheliterature[ 17 22 23 ],andaredescribedindetailinChapter 2 .Ingeneralform,thenonlineardifferentialequationsare writtenasshowninequation ( 4–39 ). _x ( t )= f ( x ( t ), u ( t ) ) (4–39) Here, x ( t ) 2 R n isthestatewith n =12 and u ( t ) 2 R m isthecontrolwith m =3 for afullsixdegree-of-freedom(6DOF)model.Theequationsof motionaredependenton 144

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theaerodynamicforcesandmoments,giveninequations( 4–40 )and( 4–41 ).Ingeneral, C F 2 R N F and C M 2 R N M .Inthe6DOFcase, N F = N M =3 C TF = [ C D C Y C L ] T = 1 q 1 S [ D Y L ] T (4–40) C TM = [ C l C m C n ] T = 1 q 1 S c [ M X M Y M Z ] T (4–41) Theaerodynamicforcesandmomentsarethemselvesfunction softhestateand controlinputs.Inaverygeneralsense,theequationsofmot ioncanbecastinthe followingfunctionalform. _x ( t )= f x ( t ), u ( t ), C TF x ( t ), u ( t ) C TM x ( t ), u ( t ) (4–42) Intheabsenceofinequalityconstraints,theHamiltonianc anbeformedasequation ( 4–43 ). H = L + T f x ( t ), u ( t ), C TF x ( t ), u ( t ) C TM x ( t ), u ( t ) (4–43) ApplyingtheoptimalityconditionsfromSection 4.2.1 givesthecostatedifferential equationandthenecessaryconditionforunconstrainedopt imalcontrol. = @ H @ x = @ L @ x + T @ f ( x u ) @ x + @ f ( x u ) @ C F @ C F @ x + @ f ( x u ) @ C M @ C M @ x (4–44) @ H @ u = @ L @ u + T @ f ( x u ) @ u + @ f ( x u ) @ C F @ C F @ u + @ f ( x u ) @ C M @ C M @ u = 0 (4–45) SeveralofthederivativesinEquations( 4–44 )and( 4–45 )canbeevaluated analyticallyusingonlytheequationsofmotionwithoutref erencetotheparticular functionalformoftheaerodynamiccoefcients.Therstte rminsidetheparenthesesin eachofequations( 4–44 )and( 4–45 )areJacobianmatricessimilartothecanonicalform ofthestatespacerepresentationforlinearcontroltheory ^ A := @ f ( x u ) @ x 2 R n n (4–46) ^ B := @ f ( x u ) @ u 2 R n m (4–47) 145

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Similarly,thederivativesof f ( x u ) withrespecttotheforceandmomentcoefcients canbeanalyticallydeterminedasJacobianmatricesfromth eequationsofmotion. C F := @ f ( x u ) @ C F 2 R n N F (4–48) C M := @ f ( x u ) @ C M 2 R n N M (4–49) TheremainingfourderivativeswithintheparenthesesofEq uations( 4–44 )and ( 4–45 )canberecognizedastheaerodynamicderivatives(Equatio ns( 4–50 )and( 4–51 )) andcontrolderivatives(Equations( 4–52 )and( 4–53 )),respectively. C F x := @ C F @ x 2 R N F n (4–50) C M x := @ C M @ x 2 R N M n (4–51) C F u := @ C F @ u 2 R N F m (4–52) C M u := @ C M @ u 2 R N M m (4–53) Thesematricesmaybedeterminedanalyticallywhenthefunc tionalformofthe aerodynamicdependenciesonthestateandcontrolareknown ,usuallyasaresult ofsystemidentication.Alternatively,thesematricesma ybeevaluatednumerically fromtabulatedvaluesusingnitedifferencingoranaltern ativenumericaldifferentiation technique. Usingthenotationabove,thedifferential-algebraicequa tionsdescribingthe1 st ordernecessaryconditionsforanoptimalstoreseparation trajectoryaregivenas follows. _x ( t )= f x ( t ), u ( t ), C F ( x ( t ), u ( t ) ) C M ( x ( t ), u ( t ) ) (4–54) ( t )= h L x + T ^ A ( t )+ C F ( t ) C F x ( t )+ C M ( t ) C M x ( t ) i (4–55) @ H @ u = h L u + T ^ B ( t )+ C F ( t ) C F u ( t )+ C M ( t ) C M u ( t ) i = 0 (4–56) 146

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ThequadraticcostfunctionalintroducesinSection 4.2.4 providesaexibleand intuitivestartingpointformanyoptimalcontrolproblems .Thematrices P Q ,and R canbeselectedtodrivethesystemtoaspeciedend-statewi thoutexpendingtoo muchcontroleffort.Forthepurposeofstoreseparation,it isdesirabletominimize certaincomponentsoftheend-state(forexample, ( t f ) and q ( t f ) )aswellascertain componentsofthederivativeoftheend-state(forexample, ( t f ) and q ( t f ) ).The quadraticcostfunctioncanbeusedtoachievethisthroughc arefulselectionofthe stateandcontrolweightingfactors.Inthisresearch,thes calarweightingfactors correspondingtodiagonalentriesof P Q ,and R aredenoted Q P q R e ,etc. Off-diagonalscalarweightingfactorsarenotusedherein. J = x T ( t f ) P x ( t f )+ 1 2 t f Z t 0 x T Q x + u T R u dt (4–57) Usingthequadraticcostfunctional,the1 st orderoptimalityconditionscanbe simpliedevenfurtherandtheoptimalcontrolcanbedeterm inedexplicitly.Notethatthe matrix R 1 isguaranteedtoexistsince R ispositivedenite. _x ( t )= f x ( t ), u ( t ), C F ( x ( t ), u ( t ) ) C M ( x ( t ), u ( t ) ) (4–58) ( t )= h Q x + T ^ A ( t )+ C F ( t ) C F x ( t )+ C M ( t ) C M x ( t ) i (4–59) u ( t )= R 1 T ^ B ( t )+ C F ( t ) C F u ( t )+ C M ( t ) C M u ( t ) (4–60) Theinitialconditionsarespeciedby x ( t 0 )= x 0 .The 2 n boundaryconditions arecompletedusingthecostateterminalvaluesforthe1 st orderoptimalityconditions givenbyequation( 4–17 ).Intheabsenceofterminalconstraints,thecostatetermi nal conditionsreducetoEquation( 4–61 ). ( t f )= P x ( t f ) (4–61) 147

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Equations( 4–58 )through( 4–61 )representasetof 2 n nonlinear,coupled, algebraic-differentialequationswithsplitboundarycon ditions.Providedasolution canbefound,thesolutiontothissetofequationsprovidesa nextremaltrajectorysubject tothespeciedconstraintsandcostfunctional.Theextrem altrajectoryisacandidate optimaltrajectory,andmaybeshowntobeatleastlocallyop timalusingthesecond variationofthecostfunctional,developedinChapter5.4.3.3Example:PlanarStoreSeparation Storeseparationismostoftendominatedbyverticaltransl ationandpitchattitude. Inmostcases,lateraltranslationandyawattitudearefair lybenignandofsecondary interest.Forinstructivepurposes,considerationofasto reconnedtothevertical x z planeduringseparationmaintainstheprimaryscopeofinte restandconsiderably reducesthecomplexityoftheoptimalcontrolproblem.4.3.3.1Modelequations Forthissimpliedproblemstatement,thestatespaceequat ionsofmotion,inmixed windandbodyaxesaregivenasequation( 4–62 ). 266666666664 V q z 377777777775 = 266666666664 D / m g sin r q L / mV + g / V cos r M = I yy q V sin r 377777777775 (4–62) Thecomponentsofthestate,V(t), ( t ) ,q(t), ( t ) andz(t),aretheair-relative velocity,angleofattack,pitchrate,pitchangle,andvert icalposition,respectively.The ightpathangleisgivenby r ( t )= ( t ) ( t ) andthelocalaccelerationofgravity isdenotedby g .Thevariables L D ,and M representthedimensionallift,drag,and pitchingmoment,respectively.Finally, I yy isthepitch-axismomentofinertiaand m isthe 148

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massofthestore.TheanalyticalJacobianmatricesareeval uateddirectly,resultingin equations( 4–63 )through( 4–66 ). @ f ( x u ) @ x := ^ A = 266666666664 0 g cos r 0 g cos r 0 q 1 SC L = mV 2 g cos r= V 2 g sin r= V 1 g sin r= V 0 0000000100 sin r V cos r 0 V cos r 0 377777777775 (4–63) @ f ( x u ) @ u := ^ B = 0 (4–64) @ f ( x u ) @ C F := C F = 266666666664 0 Sq 1 = m Sq 1 = mV 0 000000 377777777775 (4–65) @ f ( x u ) @ C M := C M = 266666666664 00 q 1 S c = I yy 00 377777777775 (4–66) 4.3.3.2Aerodynamicmodel Toproceedfurther,itisnecessarytospecifyanexplicitae rodynamicmodelforthe storeduringseparation;asimplisticspatiallyvariantmo delbasedonwindtunneldata ischosenforillustration.Amorecomplexmultivariatenon linearmodelcouldalsobe usedwithintheframeworkpresentedhere,attheexpenseofs ignicantlymorecomplex mathematicalexpressions.Forplanarstoreseparation,th edominanteffectofthe 149

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nonuniformoweldisonthepitchingmoment.Assuch,consi derthefollowingspatially variantquasi-linearaerodynamicmodel. C L = C L + C L e e (4–67) C D = C D 0 + KC 2 L + C D e e (4–68) C m = C m + C m q ^ q + C m e e + e ( z ) ( 0 + 1 z ) (4–69) Thevariable e representstheelevatorcontrolsurfacedeection,theonl yinputof interestfordynamicslimitedtothe x z plane.Thevariable ^ q isthenon-dimensional pitchrateintroducedforunitconsistency.Thevariables C L C D 0 C m ,and C m q representtheclassicalaerodynamicderivatives.Thevari ables C L e C D e ,and C m e representtheclassicalcontrolderivatives.Forthecurre ntexample,theaerodynamic andcontrolderivativesareconstantandestimatedusingfr eestreamwindtunneldata. Theterm e ( z ) ( 0 + 1 z ) istheonlyspatiallyvariantcontributiontotheaerodynam ic model.Theexponential-polynomialformisseenasaspecial caseofthemoregeneral parametricmodelpresentedinChapter 3 .Theconstants 0 ,and 1 wereestimated usinganonlinearleastsquarescurvetofrepresentativew indtunneldataatanominal pitchattitude.Forthisparticularexample,thevaluesoft heconstantsareprovidedin Table( 4-1 ). Table4-1.Aerodynamicderivatives,controlderivatives, andspatiallyvariantparameters usedinplanarstoreseparationaerodynamicmodel. AerodynamicDerivativesControlDerivativesSpatiallyVa riantParameters C L =3.81 rad 1 C M e = 8.24 rad 1 =0.158 C M = 2.11 rad 1 C L e =2.21 rad 1 0 = 0.953 C M q = 74 rad 1 C D e = 0.22 rad 1 1 =0.064 ft 1 C D 0 =0.201 K =0.1856 150

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4.3.3.3Optimalityconditions UsingthesimpliedaerodynamicmodelinEquations( 4–67 )through( 4–69 ),the Jacobianmatricesfortheaerodynamicandcontrolderivati vescanbedetermined. C F x = 264 0 C L 000 02 KC 2 L ( t )000 375 (4–70) C M x = 0 C m C m q 0 e ( z ) ( ( 0 + 1 z ) + 1 ) (4–71) C F u = 264 C L e C D e 375 (4–72) C M u = C m e (4–73) Asaresult,theoptimalcontrolisgivenintermsofthecosta tesandcontrol derivatives,asshowninEquation( 4–74 ). u ( t )= e ( t )= R 1 q ( C m e q 1 S c ) = Iyy ( C L e q 1 S ) = mV V ( C D e q 1 S ) = m (4–74) Equation( 4–74 )canbeseenasaparticularcaseoftheoptimalcontrolforal inear timevariantsystem: u ( t )= e ( t )= R 1 B ( t ) T ( t ) .However,thenonlineardynamics inthepresentcaseleadstoanonlinearrelationshipbetwee nthestatesandcostates, soanoptimalfeedbackcontrollawisnotforthcoming.Itisa ssumedthattheinitial conditionsareknown, x ( t 0 )= x 0 .Forplanarstoreseparation,itisdesirabletominimize thetotalangleofattack ( t ) andtheterminalconditions ( t f ) and q ( t f ) .Thisisreadily accomplishedusingthescalarvalues P P q Q ,and R e 4.3.3.4Results Figure 4-2A showsanextremaltrajectoryforaparticularsetofinitial conditions. Theweightingfactorswerechosentobe P = P q = Q =1 and R e =10 withallother factorssettozero.Theinitialconditionsaregiveninequa tion( 4–75 ),correspondingtoa 151

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standarddayreleaseatMachnumberof0.8andaltitudeof10k ft. x ( t 0 ):= V =861 ft / sec =5 degq = 50 deg / sec =0 degz =0 ft T (4–75) 0 0.2 0.4 0.6 0.8 1 860 860.5 861 861.5 862 time (sec)Vel (ft/sec) 0 0.2 0.4 0.6 0.8 1 -15 -10 -5 0 5 time (sec)angle (deg) 0 0.2 0.4 0.6 0.8 1 -60 -40 -20 0 20 time (sec)Pitch Rate (deg/sec) 0 0.2 0.4 0.6 0.8 1 -4 -2 0 2 time (sec)Elevator Defl (deg) 0 0.2 0.4 0.6 0.8 1 0 50 100 time (sec)Z (ft) 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 time (sec)Aero Coeff Vel g q a q d e Z Pos Cm Cl Cd AOptimalTrajectory 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.1 -0.05 0 0.05 0.1 time (sec) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -2 -1 0 1 x 10 -3 time (sec) l q l q l a Hamiltonian l z l v BHamiltonianandCostate Figure4-2.Anextremaltrajectoryforplanarstoreseparat ionwithweightingfactors selectedtominimizeangleofattack. TheresultsshowninFigure 4-2A arerepresentativeofanidealstoreseparation trajectory.Theinitialpitchrateisnose-down,consisten twithadesirablepitchrate imposedbytheejectionforces.Theinitialangleofattack, (0) ,is5deg,consistentwith alargedownwardvelocityandlevelattitudeimposedbythee jection.Theoptimalcontrol rstdrivesthestorepitchratemorenegative,simultaneou slybringing ( t ) negative,and thengentlybeginsaslightlyunderdampedpitchrateoscill ation,driving ( t ) and q ( t ) to zeroatthe t f =1 .Thelevelslopeofthe ( t ) and q ( t ) curvesfurtherindicatethatthe derivatives ( t f ) and q ( t f ) arealsonearzero,consistentwiththedesiredperformance Thestoreexhibitsbenignmotionthroughoutthetrajectory andisinanear-equilibrium stateatthetransitionalnaltime, t f =1 TheHamiltonianandcostatevariablesareshowninFigure 4-2B .TheHamiltonian isconstantasexpectedforanextremaltrajectory.Thecost atevariables ( t ) and q ( t ) 152

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showthemostsignicantvariation.Thisvariationisdirec tlyrelatedtothedependence oftheaerodynamicmodelon ( t ) and q ( t ) ,aswellastheinclusionof ( t ) and q ( t ) in thecostfunctional. Considerthecostatevariables z ( t ) and V ( t ) .Thenorm k V k =3.416 E 5 is nearlyzeroandessentiallyconstantthroughoutthetrajec tory,indicatinganegligible sensitivityofthecostfunctionaltochangesinvelocity.T hisisprimarilyduetothe functionalformoftheaerodynamicmodel,whichdoesnotinc ludeanyexplicitdependence on V ( t ) (thedependenceisbuilt-intothewindtunneldatausedines timatingthe aerodynamiccoefcients).Thecostate V ( t ) rapidlyapproacheszeroandisessentially negligibleafter200msec.Thisindicatesthatthecostfunc tionalisonlysensitiveto theverticallocation z ( t ) earlyinthetrajectory,nearesttheaircraft.Thisconclus ionis consistentwithphysicalintuitionandtheexponentialdec ayoftheaerodynamicmodel. Thepreviousextremaltrajectorywasdeterminedwiththeac cumulated(Lagragian) costontheangleofattacksettoone, Q =1 ,withzeropenaltyforthepitchrate, Q q =0 .Asanalternative,considerthecasewhere Q =0 and Q q =1 ,placing theemphasisonminimizingthepitchratethroughoutthetra jectory.Theterminalcost P = P q =0 remainunchanged.Theresultingextremaltrajectoryissho wninFigure 4-3A andtheHamiltonianandcostatevariablesareshowninFigur e 4-3B IncomparisonwiththetrajectoryshowninFigure 4-2A ,itisclearthatthepitch rateisreducedrapidlyandconvergestoaconstantvaluefor thedurationbetween 300msecand900msec,strikingabalancebetweentheminimum pitchrateandthe minimumcontrolinput.Therapiddecreaseinthepitchratec omesastheexpenseof controleffort,marginallyexceedingthelinearoperating rangeofthecontrolmodel.Also notethattheangleofattackremainsslightlypositivethro ughoutthetrajectory,another adversecharacteristic.Assuch,thepreferreddesignchoi ceforthisparticularexample istoomitthepenaltyonpitchrateandplacetheemphasisont heangleofattackand controleffort.Theresultingnegativeincreaseinpitchra te(seeFigure 4-2A )isactually 153

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0 0.2 0.4 0.6 0.8 1 -0.5 0 0.5 1 1.5 time (sec)Aero Coeff Cm Cl Cd 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 time (sec)Z (ft) Z Pos 0 0.2 0.4 0.6 0.8 1 -20 -15 -10 -5 0 time (sec)Elevator Defl (deg) d e 0 0.2 0.4 0.6 0.8 1 -50 -40 -30 -20 -10 0 time (sec)Pitch Rate (deg/sec) q 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 time (sec)angle (deg) g q a 0 0.2 0.4 0.6 0.8 1 860 860.5 861 861.5 862 time (sec)Vel (ft/sec) Vel AOptimalTrajectory 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -5 -4 -3 -2 -1 0 1 x 10 -3 time (sec) Hamiltonian l z l v 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.2 -0.1 0 0.1 0.2 time (sec) l q l q l a BHamiltonianandCostate Figure4-3.Anextremaltrajectoryforplanarstoreseparat ionwithweightingfactors selectedtominimizepitchrate. advantageousfromasafeseparationperspective,andwellb eneaththeangularrate limitsrequiredforanacceptableseparation. EachsolutiontotheHamiltonianBVPisdependentonthespec iedinitial conditions, x ( t 0 )= x 0 .Assuch,itisworthwhiletoconsiderhowchangesininitial conditionsaffectchangesintheextremaltrajectory.Figu re 4-4 showsaseriesof neighboringtrajectoriesforarangeofinitialpitchrates from0deg/secto-150deg/sec in50deg/secincrements.Theremaininginitialconditions areheldconstantatthe valuesprescribedinequation( 4–75 ).Theweightingfactorsarerestoredtotheoriginal values, P = P q = Q =1 and R e =10 withallotherfactorssettozero. AllofthetrajectoriesinFigure 4-4 exhibitasimilartrend.Fromastoreseparation perspective,alargernose-downpitchrateisdesirablesin ceitdrivestheangleof attackmorenegative,resultinginincreasedsafetywithad equateacceptabilitymargins. However,anoverlyaggressivenegativeinitialpitchrate( q < 150deg = sec )resultsin anunacceptabletrajectory,duetolargeelevatorcontrold eection( e = 10deg )and excessiveangleofattack( < 15deg ),exceedingthelinearrangeoftheaerodynamic 154

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0 0.5 1 858 860 862 Vel (ft/sec) 0 0.5 1 -30 -20 -10 0 10 Angle (deg) 0 0.5 1 -200 -100 0 100 Pitch Rate (deg/sec) 0 0.5 1 -10 -5 0 5 Elevator Defl (deg) 0 0.5 1 0 50 100 150 Time (sec)Z (ft) 0 0.5 1 -3 -2 -1 0 1 Time (sec)Aero Coeff Vel a q q d e Z Pos Cd Cl Cm Figure4-4.Seriesofneighboringextremaltrajectoriesfo rvariedinitialpitchrates. model.Apitchrateintheneighborhoodof 50 > q > 100deg = sec isthepreferred rangeofinitialpitchrateforthisparticularexample. Changesintheinitialangleofattackalsohaveasignicant effectontheextremal trajectory.Aseriesofneighboringextremaltrajectories areshowninFigure 4-5 ,using arangeofvaluesfortheinitialangleofattack, 0 (0) 5deg .Theremaininginitial conditionsarethesameasequation( 4–75 ). Notethattheinitialpitchangleiszeroinallcases, (0)=0deg .Beringinmind thattheinitialpitchangleisxedbytheaircrafthardware atcarriage,theinitialangle ofattackisnotmanipulatedbyreorientingthestore,butby ejectingthestorewith increasingvaluesofdownwardvelocity.Aninitial (0)=0deg isequivalenttoa gravity-release,i.e.zeroejectionvelocity.Aninitial (0)=5deg isequivalenttoalarge ejectionvelocityof 75 ft = sec (fortheselectedightconditionwith V =861 ft = sec ).Note thatanegativeinitial (0) < 0 for (0)=0deg isimpossiblesinceitwouldphysically requireaninitialverticalvelocityintheupwarddirectio n,interferingwiththeaircraft 155

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0 0.5 1 856 858 860 862 Vel (ft/sec) 0 0.5 1 -15 -10 -5 0 5 Angle (deg) 0 0.5 1 -100 -50 0 50 Pitch Rate (deg/sec) 0 0.5 1 -10 -5 0 5 Elevator Defl (deg) 0 0.5 1 0 50 100 Time (sec)Z (ft) 0 0.5 1 -2 -1 0 1 Time (sec)Aero Coeff Vel a q q d e Z Pos Cd Cl Cm Figure4-5.Seriesofneighboringoptimaltrajectoriesfor variedinitialangleofattack. hardware.Asaresultofthevarying (0) ,theverticaltranslationcurvesinFigure 4-5 showawiderangeofvalues.Fromasafetyperspective,itisc learlydesirabletomove thestoreawayfromtheaircraftasquicklyaspossible,indi catingthatalargeinitial (0) ispreferable(equivalenttoalargeverticalejectionvelo city).However,thedramatic increaseinverticaltranslationforlarge (0) ,alsoresultsinasignicantpotentialenergy loss.Forastorewithalong-rangemission,thismaybeunacc eptable.Thepreferred initialangleofattackforthisproblemisintherangeof 1 > (0) > 2deg ,equivalenttoa quiterealisticejectionvelocityof 15 30 ft = sec Theprecedingdiscussiondemonstratesthatthereexistsar angeofinitialconditions overwhichacandidateoptimaltrajectorycanbefoundthats atisessafetyand acceptabilitycriteria.Knowledgeofthepreferredrangeo finitialconditionscanbe usedtospecifyejectorperformancecriteria.However,the exactinitialconditions cannotbedeterminedaprioriandtheoptimalcontrolclearl yrequiresadjustment basedontheinitialconditions.Thisdifcultyleadstothe importanceofconsideringa 156

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feedbackcontrollertoaccountforthevariationsintheini tialconditionsaswellasmodel inaccuraciesandunmodeleddisturbances.Neighboringopt imalcontrolprovidesthe frameworkforimplementingsuchacontroller.Thisdiscuss ionwillbecontinuedinthe Chapter 5 withtheintroductionofneighboringoptimalcontrol. 4.4ChapterSummary Thepurposeofthisresearchistoinvestigateastoresepara tionguidanceand controlsystemwhichexplicitlyaccountsforthespatially variantaerodynamicsofthe storeduringseparationandleveragestheaerodynamicinte ractionbetweenthestore andaircrafttoimproveseparationcharacteristics.Thiso bjectiveisaccomplished,in part,usingoptimalcontroltodeterminea“bestcase”traje ctory. Acandidateoptimaltrajectoryforstoreseparationisdete rminedbysolving aHamiltonianboundaryvalueproblem,asspeciedbythe1 st orderoptimality conditionsinoptimalcontrol.Thegeneraloptimalitycond itionsareappliedtoa storeduringseparationandthe1 st ordernecessaryconditionsforoptimalstore separationarederived.Asimpliedexample,namelyplanar storeseparationwith aspatially-variantaerodynamicmodel,isconsideredtoil lustratetheuseofthese methods.The“open-loop”candidateoptimaltrajectoryisa suitablestartingpoint forconsiderationof“closed-loop”feedbackusingneighbo ringoptimalcontrol.This discussionistakenupinChapter 5 157

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CHAPTER5 NEIGHBORINGOPTIMALCONTROL 5.1Overview Previousstudieshavehighlightedtheuseofactivecontrol toimproveseparation characteristics,primarilyasasidebenetofdemonstrati ngtheuseofanewCFD capability.Themostsignicantdocumentedcontributiont ocontrolledstoreseparation comesfromR.H.NicholsandA.G.Denny(ArnoldEngineeringD evelopmentCenter) [ 5 ].Theauthorspresentacasestudyforthenumericalsimulat ionofanon-thrusted ejectorlaunchedAIM-120CAdvancedMediumRangeAir-to-Ai rMissile(AMRAAM)in controlledseparationfromawingstationoftheF-15EStrik eEagle.Theresultsindicate asignicantimprovementoftheseparationcharacteristic swithanactiveautopilotduring separation. C.A.Atwood(NASAAmesResearchCenter)madeanothernotabl econtribution tocontrolledstoreseparation[ 6 ].AtwooddemonstratedaCFDsimulationofa canard-controlledstorereleasedfromanopenowrectangu larcavityatsupersonic freestreamconditions.Again,thesimulationshowedsigni cantimprovementsinthe separationcharacteristicsusingabasicpitchattitudeco ntroller,incomparisontothe canard-xedstore. Inbothofthesestudies,theemphasiswasondemonstratingc ontrolledseparation withahighdelityCFDsimulation.Assuch,theinvestigati onoftheautopilotitselfwas limited.Remarkably,literatureoninvestigationofacont rolsystemdesignedspecically forthenonuniformoweldencounteredbyastoreduringsep arationisvoid. Thisresearchisfocusedonthedevelopmentofatransitiona lguidanceandcontrol system,designedwiththeseparation-inducedtransientsi nmind,toachievethebest casetrajectoryforavarietyofightconditionsandcongu rations.Inparticular,the objectiveistoimprovetheseparationcharacteristicsofe jector-launchedguided munitionsusingaseparationautopilottoachieveanoptima ltrajectorywithrespect 158

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tospeciedsafetyandacceptabilityperformancemetrics. Theopen-loopoptimal trajectoryisdeterminedusingclassicaloptimalcontrolt heory,asshowninChapter 4 Closed-loopfeedbackcontrolisaccomplishedusingneighb oringoptimalcontrol,as developedhere. 5.2NeighboringOptimalControl TherstorderoptimalityconditionsdiscussedinSection 4.2.1 providethe necessaryconditionsforanextremaltrajectory.Thecontr olisdeterminedbasedon solutionofthetwopointHamiltonianboundaryvalueproble mandimplicitlyassumes perfectknowledgeofthesystemoperatinginadisturbancefreeenvironment.However, deterministicdisturbancesorvariationsintheinitialco nditions,terminalconditions,and systemparametersaltertheoptimalstateandcontrolhisto ry,requiringcomputationof uniquesolutionforeachvariation.Statedanotherway,the optimalcontrolstrategy is“open-loop”,meaningthecontrolisspeciedaprioriand xedregardlessof perturbationsthatmayaffectthesystemduringoperation. Incontrast,a“closed-loop” controllawismoredesirableasitaccountsforvariationsi ninitialconditionsand disturbancesalongtheoptimalpath.Neighboringoptimalc ontrol(NOC)provides apowerfulapproachforimplementingfeedbackcontrolalon ganoptimalpathby consideringlinearperturbationsalongtheextremalsolut ion.NOCreliesonalocally linearizeddynamicmodelinconjunctionwithaquadraticco stfunctionalderivedfrom thesecondvariationoftheoriginalcostfunctional.Thene ighboringoptimalsolutionis thenapproximatedasthesumoftheoriginaloptimaltraject oryplusthelinear-optimal solution[ 116 ]. Neighboringoptimalcontrolwasrstintroducedintheearl y1960'sbyKelly [ 127 ]andBreakwell,SpeyerandBryson[ 128 ].Theseearlycontributionswerestated informallyasanextensiontotheaccessoryminimumproblem (AMP)andimmediately gainedtractionasaconvenientapproachforimplementingo ptimalcontrolforareal worldsystem[ 129 ]aswellasaviablenumericalmethodforsolvingopen-loopo ptimal 159

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controlproblems[ 128 130 ].SignicantcontributionstoNOCinthepresenceofpath andequalityconstraintsweremadebyJacobson[ 131 ],Lee[ 132 ],Pesch[ 133 134 ], Hymas[ 135 ]andFisher[ 136 ].SimilarcontributionsforNOCinthepresenceof parametervariationswereprovidedbyLee[ 137 ],D'Souza[ 138 ],andHull[ 139 140 ]. ApplicationsofNOCintheaerospacecommunityincludeadva ncedlaunchsystems [ 141 ],spaceshuttleguidance[ 134 ],hypersonicvehicledescent[ 142 ],ightvehicle guidance[ 143 ],andmissileguidanceagainstamaneuveringtarget[ 144 ],toname afew.Conventionalnonlineartrajectoryoptimizationpro blemsrequiresubstantial computationalresourcestondanextremalsolution,which typicallyprohibitsareal-time implementationofoptimalcontrol.However,NOCiseasilyi mplementedandmay beusedtoprovidereal-timeoptimalcontrolinthepresence ofsmalldisturbances [ 145 – 147 ].Severalapplicationsforaircraftightthroughnonline arspatially-varyingow eldsdevelopedbyJardinandBrysonarealsoofparticulari nterestduetoconceptual similaritytoguidanceandcontrolofastoreduringseparat ion[ 148 – 150 ]. CurrentresearchtrendsinNOCincludeimprovementsinwork ingwithpathand controlconstraints[ 145 151 ],robustanalysisforcontrolofsystemswithuncertaintie s [ 152 – 154 ],implementationofreal-timeoptimalcontrol[ 146 ],andapplicationto increasinglycomplexrealworldsystemsincludingairtran sportationmanagement [ 103 104 147 155 ],reusablelaunchvehicles[ 152 154 ],andightthroughadverse oweldconditions[ 156 ].Inthiscontext,storeseparationguidanceandcontrolis a challengingapplicationofestablishedprinciplesinNOC. Additionally,theconceptof innitehorizonneighboringoptimalcontrol(IHNOC)isint roducedheretodemonstrate howNOCcanbeconstructedtoprovidecontinuationoftheopt imalcontrolbeyondthe originalopen-loopnite-horizonsolution.5.2.1SecondOrderOptimalityConditions ConsidertheBolzaproblemintroducedinSection 4.2.1 withacostfunctional givenbyEquation( 5–1 ),dynamicconstraintsgivenbyEquation( 5–2 ),pathandcontrol 160

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inequalityconstraintsgivenby( 5–3 ),andterminalconstraintsgivenbyEquation( 5–4 ), whereitisassumedthattheinitialconditions, x ( t 0 ) ,andnaltime, t f ,arespecied. J = x ( t f ) + t f Z t 0 L x ( t ), u ( t ) dt (5–1) _x ( t )= f x ( t ), u ( t ) (5–2) C x ( t ), u ( t ) 0 (5–3) x ( t f ) =0 (5–4) Thecorresponding1 st ordernecessaryconditionsaredeterminedbytakingthers t variationoftheaugmentedcostfunctionalwithrespecttoe achfreevariable,resulting intheEuler-Lagrangeequationsandtransversalitycondit ions.TheEuler-Lagrange equationsaregivenbyEquations( 5–5 )and( 5–6 ),andthetransversalityconditionis givenbyEquation( 5–7 ),where H = L + T f istheHamiltonian. T = @ H @ x (5–5) 0 = @ H @ u (5–6) T ( t f )= @ @ x ( t f ) T @ @ x ( t f ) (5–7) Thestate x ( t ) andcostate ( t ) formaHamiltoniansystemwithsplitboundary conditionsgivenby x ( t 0 ) and ( t f ) ,resultinginatwo-pointHamiltonianboundary valueproblem(HBVP).ThesolutiontotheHBVP,denoted ( x ( t ), u ( t ) ) ,representsan extremaltrajectorywithopen-loopcontrolofthetypeenco unteredinChapter 4 Now,considerasmallvariationintheinitialconditionsgi venby x ( t 0 ) .Itisintuitive toexpectthattheextremaltrajectorywillalsovaryslight lybytheamount x ( t ) .With theperturbedtrajectory,thecostfunctionalcanbeapprox imatedtosecondorderby Equation( 5–8 ).Itisnotedthatinordertosatisfytherstordernecessar yconditions,the rstvariationinthecostfunctionaliszeroalongtheoptim alpath, J =0 .Thistermis 161

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consequentlyommittedfromEquation( 5–8 ). J [ x ( t )+ x ( t ) ] = J [ x ( t ) ] + 2 J [ x ( t ) ] (5–8) FromEquation( 5–8 )itisapparentthatinordertominimizetheoriginalcostfu nction alongtheperturbedtrajectory,itisnecessarytominimize 2 J .Thesecondvariationof theaugmentedcostfunctionalisobtainedbyexpandingtheo riginalcostfunctionalto secondorderandtheconstraintstorstorder,eliminating termswhicharenecessarily zeroalongtheextremaltrajectory[ 113 ].TheresultisgivenbyEquation( 5–9 ),wherethe subscriptnotationindicatespartialdifferentiation. 2 J a = 1 2 x T xx + T x x x t = t f + 1 2 t f Z t 0 x T T 264 H xx H xu H ux H uu 375 264 x 375 dt (5–9) Thus,theobjectiveistondacontrolvariation u ( t ) ,correspondingtovariations inthestate x ( t ) andcostate ( t ) ,whichminimizesthesecondvariationofthecost functional.Thisminimizationproblemcanbefurtherconst ructedbylinearizingthe1 st ordernecessaryconditionsalongtheextremaltrajectory, asshowninEquation( 5–10 ) through( 5–14 ). _x ( t )= f x x + f u u (5–10) ( t )= H xx x f T x H xu u (5–11) 0= H ux x + f T u + H uu u (5–12) T ( t f )= xx + T x x x + Tx t = t f (5–13) = [ x x ] t = t f (5–14) Thevariations x u ,and aredenedasperturbationsalongtheoptimal trajectory.Inparticular, x = x ( t ) x ( t ) u = u ( t ) u ( t ) ,and = ( t ) ( t ) are thevariationsinthestate,control,andcostaterespectiv ely. 162

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Thus,fortheneighboringoptimalcontrolproblem,theobje ctiveistominimize thecostfunctionalgivenbyEq.( 5–9 )subjecttotheconstraintsgiveninEquations ( 5–10 )through( 5–14 ).Uponcarefulinspection,itbecomesapparentthatthepro blem statementissynonymouswiththelinearquadraticproblem( LQP)consideredinSection 4.2.4 andtheestablishedsolutionmethodologycanbeapplieddir ectlytotheNOC problem. Theoptimalcontrolproblemstatementandcorrespondingne cessaryconditions describedinEquations( 5–1 )through( 5–7 )includebothpath/controlinequality constraintsandterminalconstraints.Tobeconsideredana llowabletrajectory,theNOC extremaltrajectorymustalsosatisfytheseconstraints.H owever,thepresenceofthese constraintsaddssignicantcomplexitytothemathematica lexplanation.Therefore,the NOCproblemwithoutpath,control,orterminalconstraints willbeconsideredrstin Section 5.2.2 .TerminalconstraintsareconsideredinSection 5.2.3 andpath/control inequalityconstraintsareconsideredinSection 5.2.4 ItisnotedthattheNOCproblemdiscussedthusfarconsiders onlyperturbations ininitialconditions.Sinceanypointalongtheextremalpa thisavalidstartingpoint, thisdevelopmentisdirectlyapplicabletodisturbancesal ongtheextremalpath. However,variationsinthesystemparametersalsoresultin perturbationsoftheextremal trajectory.ThesubjectofNOCinthepresenceofparameterv ariationsisaddressedin Section 5.2.5 Finally,itisimportanttonotethattherstordernecessar yconditionsresultinan extremaltrajectorywhichmayormaynotbeoptimal.Theseco ndvariationofthecost functionalisanalogoustothesecondderivativeincalculu sandcanbeusedtoestablish sufcientconditionsforoptimality.Thiswillbediscusse dinSection 5.2.6 Thetheoreticaldevelopmentherehasmanypracticalapplic ations.Application ofNOCtostoreseparationisconsideredinSection 5.3 withanextendedexamplein Section 5.4 .ApplicationofNOCtoarealisticcasestudyistakenupinCh apter 6 163

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5.2.2NeighboringExtremal The1 st ordernecessaryconditionsforaneighboringextremalwith outterminal conditionsaregivenbyEquations( 5–15 )through( 5–18 ). _x ( t )= f x x + f u u (5–15) ( t )= H xx x f T x H xu u (5–16) 0= H ux x + f T u + H uu u (5–17) T ( t f )= [ xx x ] t = t f (5–18) Itshouldbenotedthattheendpointcost ( x ( t f )) isstillpresentinthesimplied problemsoitisstillpossibletoachieveanarbitrarilypre ciseterminalconditionwhen desired.Thefunction ( x ( t f )) isoftenreferredtoasa“soft”constraint,whereas ( x ( t f )) isa“hard”constraint. Equation( 5–17 )canbeusedtosolveforthecontrolvariationexplicitlypr ovidedthat thematrixinverse H 1 uu exists.ThecontrolvariationisgivenbyEquation( 5–19 ). u = H 1 uu H ux x + f T u (5–19) NotethatwhentheHamiltonianislinearinthecontrol,them atrix H uu =0 andthe inversedoesnotexist.Inthiscase,thecontrolmustbedete rminedusingtheminimum principle,oftenresultinginmaximumcontroleffortforth edurationofthetrajectory,a strategyreferredtoas“bang-bang”control[ 111 114 ]. Substituting( 5–19 )intothedifferentialequations( 5–15 )and( 5–16 )resultsinthe linearsystemofequations( 5–20 )and( 5–21 ),wherethetime-varyingmatrices A ( t ) B ( t ) ,and C ( t ) ,givenbyEquations( 5–22 )through( 5–24 ),areevaluatedalongthe extremalpath. _x = A ( t ) x B ( t ) (5–20) = C ( t ) x A T ( t ) (5–21) 164

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A ( t )= f x f u H 1 uu H ux (5–22) B ( t )= f u H 1 uu f T u (5–23) C ( t )= H xx H xu H 1 uu H ux (5–24) Theboundarycondition ( t f )= xx x from( 5–19 )suggestsaparticularformof perturbedcostateasalinearcombinationoftheperturbeds tate,asshowninEquation ( 5–25 ),withthecorrespondingboundaryconditiongivenby( 5–26 ). ( t )= S ( t ) x ( t ) (5–25) S ( t f )= xx (5–26) Differentiationof( 5–25 )resultsinEquation( 5–27 ).Substitutionof( 5–20 )and( 5–21 ) into( 5–27 )resultsinthedifferentialRiccatiEquation( 5–28 ). ( t )= S ( t ) x ( t )+ S ( t ) _x ( t ) (5–27) S ( t )= SA A T S + SBC C (5–28) TheRiccatiequation,subjecttotheterminalcondition( 5–26 ),isstableinbackward timeandindependentofthevariations x ( t ) ( t ) ,and u ( t ) .Asaresult,oncean extremaltrajectoryhasbeendeterminedusingthe1 st ordernecessaryconditions,the Riccatiequationcanbeusedtocomputeandstore S ( t ) alongtheextremalpath. Equation( 5–25 )canbeusedwith S ( t ) toeliminatethecostatefromtheproblem. Substitutionof( 5–25 )into( 5–19 )providesthecontrolvariation u ( t ) u = H 1 uu H ux + f T u S x (5–29) Recallingthedenitionoftheperturbedstateandcontrol, x = x ( t ) x ( t ) and u = u ( t ) u ( t ) ,where x ( t ) and u ( t ) representthenominalextremaltrajectory, Equation( 5–29 )canbeexpressedas( 5–30 )and( 5–31 ). 165

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u ( t )= u ( t ) K ( t ) ( x ( t ) x ( t ) ) (5–30) K ( t )= H 1 uu H ux + f T u S (5–31) Equations( 5–30 )and( 5–31 )representaNeighboringOptimalFeedbackLawthat canbeusedtocorrectforvaryinginitialconditionsordist urbancesalongtheextremal path.Theneighboringoptimalcontrolstructureisshowngr aphicallyinFigure 5-1 Nominal Opt Trajectory x 0 ( t ), u 0 ( t ) NOC Gains Dynamic System x ( t ), u ( t ) K ( t ) u 0 ( t ) x 0 ( t ) D x ( t ) x ( t ) D u ( t ) u ( t ) + + + Figure5-1.NeighboringOptimalControlblockdiagram. Althoughthemathisarduous,thenalresultiscompactande asilyimplemented. Theneighboringoptimalcontrolprocessissummarizedasfo llows. 1. Usingthe1 st orderoptimalityconditions,computeandstorethenonline ar, open-loop,optimaltrajectoryforthegivenproblemstatem ent, x ( t ) and u ( t ) 2. UsingthematrixRiccatiequation,computeandstorethefee dbackgains, K ( t ) alongtheoptimalpath.Ifcollocationisused,theRiccatie quationcanbesolved commensuratewiththeopen-loopoptimaltrajectoryinstep 1. 3. Usingthestoredmatrix K ( t ) ,thestorednominaloptimaltrajectory x ( t ) and u ( t ) andtheestimatesoftheactualstate x ( t ) ,determinethecontrol u ( t )= u ( t )+ u duringoperation. Withtheabovesynopsis,neighboringoptimalcontrolissee ntobeacompact andefcientapproachtoimplementingfeedbackcontrolalo nganoptimalpathwhen deviationsfromtheoptimalpathareexpectedtobesmall.Ap plicationofNOCtostore separationisstraightforward,resultinginafeedbackcon trolsystemwhichmaybeused toaccountforvariationsininitialconditionsandoweld perturbations.Extensionofthe 166

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neighboringoptimalfeedbacklawforprescribedterminalc onditions,constrainedcontrol inputs,andparametervariationsarediscussedsubsequent ly. 5.2.3NeighboringExtremalwithTerminalConstraints The1 st ordernecessaryconditionsforanextremaltrajectorywith terminal constraints ( x ( t f ))= 0 aregiveninEquations( 5–10 )through( 5–14 ).Theinclusion ofterminalconstraintsaffectsthecomputationofaneighb oringextremalintwoprimary waysincluding(1)modicationoftheboundaryconditionsf ortheRiccatiequationand (2)modicationofthefeedbackstructuretoincludeexplic itdependenceontheterminal conditions. Theboundaryconditionsonthevariationofthecostateandt hevariationofthe terminalconditionsaregiveninEquations( 5–32 )and( 5–33 ).Iftheoriginalterminal constraint ( x ( t f ))= 0 issatisedbytheneighboringextremal,then = 0 as aconsequence.However,insomecasesitmaybedesirabletoa djusttheterminal constraintsduringoperation.Therefore, maybeconsideredauserspecied parameterandweseekafeedbackcontrollawwithexplicitde pendenceontheoriginal terminalconstraintandanyspeciedperturbation T ( t f )= xx + T x x x + Tx t = t f (5–32) = [ x x ] t = t f (5–33) TheboundaryconditionsinEquations( 5–32 )and( 5–33 )suggestaparticular formforthevariationinthecostate ( t ) andterminalconditions ( t ) asalinear combinationofthevariationinthestate x ( t ) andLagrangemultiplier ,asshownin Equations( 5–34 )and( 5–35 ).Thecorrespondingmatrixboundaryconditionsgivenin ( 5–36 )through( 5–38 ). ( t )= S ( t ) x ( t )+ R ( t ) (5–34) ( t )= R ( t ) T x ( t )+ Q ( t ) (5–35) 167

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S ( t f )= xx + T x x t = t f (5–36) R ( t f )= Tx t = t f (5–37) Q ( t f )=0 (5–38) DifferentiationofEquation( 5–34 )andsubstitutionofEquations( 5–20 )and( 5–21 ) resultsinthesamedifferentialRiccatiequationasbefore ,Equation( 5–28 ),withthe newboundaryconditionsspeciedinEquation( 5–36 ).Takingnotethat isaconstant vectoranddifferentiatingEquation( 5–35 )resultsinthedifferentialequation( 5–39 ). 0= R T ( t ) x ( t )+ R T ( t ) _x ( t )+ Q ( t ) (5–39) SubstitutionofEquation( 5–20 )for _x ( t ) withsubsequentfactoringofthecoefcients givesEquation( 5–40 ). 0= R T + R T ( A BS ) x + Q R T BR (5–40) Sincethevariations x and areindependent,bothcoefcientsmustbezero independently,givingtwoadditionalmatrixdifferential equations. R = A T SB R (5–41) Q = R T BR (5–42) Thecompletesetofneighboringextremaldifferentialequa tionsandcorresponding boundaryconditionsaresummarizedinEquations( 5–43 )through( 5–45 ). S ( t )= SA A T S + SBC C S ( t f )= xx + T x x t = t f (5–43) R = A T SB R R ( t f )= Tx t = t f (5–44) Q = R T BR Q ( t f )=0 (5–45) Equations( 5–43 )through( 5–45 )areasetofmatrixdifferentialequationswithall necessaryboundaryconditionsspeciedatthenaltime t f .Theequationsarestablein 168

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backwardtimeandindependentofthevariations x ( t ) ( t ) ,and u ( t ) .Asaresult,the equationscanbesolvednumericallyalonganextremalpathy ieldingthetimevarying matrices S ( t ) R ( t ) ,and Q ( t ) RecallthatthecontrolvariationisgivenbyEquation( 5–19 ),restatedherefor convenience. u = H 1 uu H ux x + f T u (5–46) Substitutionof = S x + R fromEquation( 5–34 )givestheresultshownin Equation( 5–47 ). u = H 1 uu H ux x H 1 uu f T u ( S x + R ) (5–47) Equation( 5–35 )canbeusedtoeliminatethedependenceofthecontrolonthe Lagragemultiplier ,asshowninEquation( 5–48 ). = R T x + Q ) = Q 1 R T x (5–48) Substitutionofthisexpressionfor intoEquation( 5–47 )andsubsequentfactoring ofthecoefcientsgivesEquation( 5–49 ). u = H 1 uu H ux + f T u S f T u RQ 1 R T x H 1 uu f T u RQ 1 (5–49) Equation( 5–49 )canbewrittenconciselyasafeedbacklawwithtime-varyin ggains 1 ( t ) and 2 ( t ) u = 1 ( t ) x 2 ( t ) (5–50) 1 ( t )= H 1 uu H ux + f T u S f T u RQ 1 R T (5–51) 2 ( t )= H 1 uu f T u RQ 1 (5–52) ThefeedbackcontrollawinEquation( 5–50 )showsanexplicitdependenceonthe variationinthestate(arisingfromperturbedinitialcond itionsordisturbancesalong theextremaltrajectory)andthevariationintheterminalc onditions(arisingfroma 169

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user-speciedchangeinterminalconditionsfromthenomin alextremalsolution).Some simplicationresultsif =0 u = H 1 uu H ux + f T u S f T u RQ 1 R T x (5–53) Recallthattheneighboringoptimalcontrollawintheabsen ceofterminalconstraints isgivenbyEquation( 5–29 ),restatedhereas ^u ( t ) ^u ( t )= H 1 uu H ux + f T u S x (5–54) ThefeedbacklawwithterminalconstraintsfromEquation( 5–53 )canbeexpressed asacombinationofthecontrolvariationwithoutterminalc onstraints, ^u ( t ) ,andan additionaltermduetoterminalconstraints,asshowninEqu ation( 5–55 ). u = ^u ( t )+ H 1 uu f T u RQ 1 R T x (5–55) Inorderfor u toexistandbenite,thematrixinverse Q 1 ( t ) mustalsoexistfor t 2 [ t 0 t f ] .However,theboundaryconditionsspeciedinEquation( 5–45 )requirethat Q ( t f )=0 ,whichimplies Q 1 ( t f )= 1 .Asaresult, u ( t ) !1 as t t f Whatatrstseemslikeaninconvenientmathematicalartifa ctactuallyalludesto anintuitivephysicalprinciple.Namely,toguaranteethat thesystemarrivesatanexact terminalcondition ( x ( t f ))=0 inthepresenceofdisturbances x ( t ) requiresaninnite amountofcontrol.Statedanotherway,astheterminalcondi tionisapproached,an inniteamountofcontrolisrequiredoveraninntelyshort periodoftimetoreachthe speciedendstate. Thisproblemisalsomanifestbyexaminingthetime-varying gainsinEquations ( 5–51 )and( 5–52 ),whichapproachinnityneartheterminaltime. lim t t f 1 ( t )= 2 ( t )= 1 (5–56) 170

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Thedivergenceofthetime-varyinggainsnear t = t f introducesadifcultyin usingNOCwithterminalconstraintsforrealisticsystems. Thisproblemiswell-known intheliteratureandasimpleworkaroundistoboundthegain susingasaturation function[ 128 ].Asaresult,thegainsareheldconstantoverthenaldurat ionofthe trajectory.Anotheralternativeistospecifyboundsonthe control,intheformofan inequalityconstraint.Thesystemwillthennaturallyempl oymaximumcontroleffort neartheterminalconditionsinordertocomeasclosetothet erminalconditionas possible.ConsiderationofNOCinthepresenceofinequalit yconstraintsonthecontrol isdiscussednext.5.2.4NeighboringExtremalwithPath/ControlConstraints Theoptimalperformanceofmanysystemsinvolvesoperation atornearaphysical boundary.Examplesincludemaximumcontroleffortina“ban g-bang”controlsystem andmaximumaccelerationtoreachatargetinminimumtime.I nteriorconstraints canbeclassiedas(1)pathconstraints,(2)controlconstr aints,or(3)pathand controlconstraints.Theconstraintsmaybefurtherclassi edasequalityorinequality constraints,foratotalofsixpossibleproblemtypes,each withasimilarbutdistinct solutionapproach[ 111 ].Foramoreindepthtreatment,thereaderisreferredto literaturethataddressesneighboringoptimalcontrolwit hpathandcontrolconstraints indetail[ 131 – 133 136 151 ].Thegeneralsolutionapproach,brieyconsideredhere, consistsofjoiningtheconstraintstotheHamiltonianusin gcorrespondingLagrange multipliers. Considerthepath/controlinequalityconstraintintroduc edinEquation( 5–3 ). C x ( t ), u ( t ) 0 (5–57) AsdiscussedinChapter4,thepath/controlconstraintcanb ejoinedtothe HamiltonianusingtheLagrangemultiplier ( t ) H ( x u ) = L ( x u ) + T f ( x u ) + T C ( x u ) (5–58) 171

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Usingthecomplementaryslacknesscondition,theLagrange multiplier ( t ) is relatedtothevalueofthedynamicconstraint C ( x ( t ), u ( t ) ) 0 ,asshowninEquation ( 5–59 ),where s isthenumberofconstraintsandthedimensionofthevector C x u Lagrangemultiplier ( t ) : 8>><>>: i ( t )=0 if C i ( x ( t ), u ( t ) ) < 0; i =1,..., s i ( t ) > 0 if C i ( x ( t ), u ( t ) ) =0; i =1,..., s (5–59) Thepositivevalueof i when C i =0 isinterpretedsuchthatimprovingthecostmay onlycomefromviolatingtheconstraint[ 113 ].Furthermore, i ( t )=0 when C i < 0 states thatthisconstraintisinactiveandcanbeignored.Inthisc ase,theHamiltonianreduces totheunconstrainedformgiveninequation( 5–60 ). H ( x u ) = L ( x u ) + T f ( x u ) (5–60) Similarly,theEuler-Lagrangeequationsaredependentont heactivityofthe constraintinequality. T = H x = 8>><>>: L x T f x C x C =0 L x T f x C < 0 (5–61) Thenecessaryconditiontodeterminetheoptimalcontrolis givenbyEquation ( 5–62 ). H u = L u + T f u + C u =0 (5–62) Whentheconstraintisinactive, ( t )=0 and u ( t ) isdeterminedfrom L u + T f u =0 intheusualunconstrainedmanner.Otherwise,whenthecons traintisactive, u ( t ) is determinedfrom C ( x u )=0 and ( t ) isdeterminedfromEquation( 5–62 ).Asimple checktoensure ( t ) 0 issufcienttoverifythestationarityoftheextremalalon gthe constrainedarc.Whentheinequalityconstraintdoesnotin cludethecontrolexplicitly, e.g. C = C ( x ) ,additionaleffortisneededtointroducethecontrolthrou ghdifferentiation of C ( x ) andsubstitutionof _x ( t ) untilthecontrolappearsexplicitlyintheconstraint 172

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equation.Thisadditionalcomplexityisnotnecessaryfort hecurrentapplicationandonly constraintsoftheform C ( x u ) willbeconsideredherein. ThisargumentcanbeextendedtoNOCasfollows.Considerthe variationofthe function H u ( x u )=0 ,asshowninEquation( 5–63 ). H u ( x u )= H ux x + f T u + H uu u + C Tu =0 (5–63) When C < 0 ,thevariation =0 andthecontrolcanbedeterminedinamanner identicaltotheunconstrainedneighboringextremal.Inth ecasethat C =0 ,the constrainedcontrol u c canbedeterminedfromtheconstraintand u = u c u isalso known.Assuch,thevariation canbedeterminedfromEquation( 5–63 ).Thetwo possibilitiesaresummarizedinEquation( 5–64 ). When: 8>><>>: C < 0, =0, u = H 1 uu H ux + f T u S x C =0, u = u c u = C T u H ux + f T u S x + H uu u (5–64) Thevariation carriesthesamesignicanceastheoriginalmultiplier andthe complementaryslacknessconditioninEquation( 5–59 ).Namely,apositive implies thatthecostcanonlybeimprovedbyviolatingtheconstrain ts.Consequently,asimple checkof 0 issufcienttoensurethestationarityoftheneighboringe xtremal alongtheconstrainedarc.Thesevericationsareimplicit tothenumericalexamples presentedsubsequentlyandwhenaconstrainedarcisencoun tered,itcanbeassumed bythereaderthatthesequalicationshavebeenmet.5.2.5NeighboringExtremalwithParameterVariations Considerasystemthatisdependentonasetofconstantparam eters ,asshown inEquation( 5–65 ). _x ( t )= f ( x u t ) (5–65) Itisassumedthattheinitialvalueof isknownandusedinthesolutionofthe nominalHBVP.Thegoalistondaneighboringoptimalfeedba cklawthatminimizes 173

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theoriginalcostfunctionaltosecondorderinthepresence ofdisturbances x ( t 0 ) and parametervariations .ThedevelopmentissimilartothediscussioninSection 5.2.3 Forthepresentdiscussion,itwillbeassumedthatnotermin alconstraintshavebeen specied,althoughextensiontothecasewithterminalcons traintsisstraightforward.For amoreindepthtreatment,thereaderisreferredtoliteratu rethataddressesneighboring optimalcontrolwithparametersindetail[ 137 138 140 ]. Thevariationinthestateandcostatedifferentialequatio nswithparameter variationsaregivenbyEquations( 5–66 )and( 5–67 ). _x ( t )= f x x + f u u + f (5–66) ( t )= H xx x H xu u H x f T x (5–67) Thenecessaryconditionforoptimalcontrol H u =0 followedbytheresulting expressionforthecontrolvariation u isgivenbyEquation( 5–68 ). 0= H ux x + H uu u + H u + f T u ) u = H 1 uu H ux x + H u + f T u (5–68) Substitutionof u fromEquation( 5–68 )intothedifferentialequations( 5–66 )and ( 5–67 )resultsinthefollowinglinearsystemofequationsinterm sof x ,and _x ( t )= A ( t ) x B ( t ) + U ( t ) (5–69) ( t )= C ( t ) x A T ( t ) + V ( t ) (5–70) Thematrices A ( t ) B ( t ) ,and C ( t ) aregivenbyEquations( 5–22 )through( 5–24 )in Section 5.2.2 .Thematrices U ( t ) and V ( t ) aredeterminedinasimilarmanner. U ( t )= f f u H 1 uu H u (5–71) V ( t )= H xu H 1 uu H u H x (5–72) 174

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Theterminalcondition ( t f )= xx x ( t f ) suggestsaparticularformforthecostate asalinearcombinationofthevariationofthestate x ( t ) andtheconstantparameter vector ,asshowninEquation( 5–73 ). ( t )= S ( t ) x + W ( t ) (5–73) DifferentiationofEquation( 5–73 )followedbysubstitutionof _x and resultsinthe followingmatrixdifferentialequationsfor S ( t ) and W ( t ) .Thecorrespondingboundary conditionsaredeterminedbycomparisonwith ( t f )= xx x ( t f ) S ( t )= SA A T S + SBS C S ( t f )= xx (5–74) W ( t )= V A T W S ( U BW ), W ( t f )=0 (5–75) Equations( 5–74 )and( 5–75 )areasetofmatrixdifferentialequationswithall necessaryboundaryconditionsspeciedatthenaltime t f .Theequationsarestable inbackwardtimeandindependentofthevariations x u ,and .Asaresult,the equationscanbesolvednumericallyalonganextremalpathy ieldingthetimevarying matrices S ( t ) and W ( t ) Recallthattheneighboringoptimalcontrollawintheabsen ceofparameter variationsorterminalconstraintsisgivenbyEquation( 5–29 ),restatedhereas ^u ( t ) ^u ( t )= H 1 uu H ux + f T u S x (5–76) Thefeedbacklawwithparametervariations,whichresultsf romthesubstitutionof ( 5–73 )into( 5–68 ),canbeexpressedasacombinationofthecontrolvariation without parametervariations, ^u ( t ) ,andanadditionaltermduetoparametervariations,as showninEquation( 5–77 ). u = ^u ( t ) H 1 uu H u + f T u W ( t ) (5–77) 175

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Equation( 5–77 )representsaneighboringoptimalfeedbackcontrollawtha t minimizestheoriginalcostfunctiontosecondorderinthep resenceofdisturbances andparametervariations. Whentheparametervariation isknownormaybeestimatedduringoperation, NOCmaybeusedtoimplementafeedbackcontrollawthatisexp licitlydependent ontheperturbedparameters.However,when isunknown,thecontrolvariation isnecessarilysub-optimal.Althoughthefeedbackcontrol canstillaccommodate disturbancesthatresultfromapplyingthenominalopen-lo op(feedforward)controlto thesystemwithperturbedparameters,thefeedbackwillnot beoptimaltosecondorder andsomeadditionalcostwillbeincurred.Therefore,theus eofNOCforasystemwith unknownparametervariationscanbedescribedas“nearopti mal”,providedthatthe parametervariationsremainsmall.5.2.6SufcientConditionsforOptimality The1 st orderoptimalityconditionsspecifythenecessaryconditi onsforanextremal trajectory.However,anextremaltrajectorythatsatises thenecessaryconditionsmayor maynotbelocallyoptimal.Inordertodemonstratethatthee xtremaltrajectoryisindeed alocallyoptimaltrajectory,itisnecessarytoconsider2 nd ordersufcientconditions. Whereasthe1 st ordernecessaryconditionsrequire J a =0 ,the2 nd ordersufcient conditionsforlocaloptimalityrequire 2 J a > 0 [ 113 ].The2 nd variationoftheaugmented costfunctionalisgivenbyEquation( 5–78 ),whichisthesamecostfunctionalusedfor solvingtheneighboringoptimalcontrolproblem. 2 J a = 1 2 x T xx + T x x x t = t f + 1 2 t f Z t 0 x T T 264 H xx H xu H ux H uu 375 264 x 375 dt (5–78) 176

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Thequadraticformofthecostfunctionalensuresthatifane ighboringextremalcan befound,then 2 J a > 0 .Thus,simplystated,theexistenceofaneighboringoptima l solutionissufcientforensuringtheextremalpathisinde edlocallyoptimal. Thesufcientconditionsfortheexistenceofaneighboring extremal,whichare synonymouswiththesufcientconditionsforlocaloptimal ity,canbededucedfrom themathematicaldevelopmentintheprecedingsections.Ri gorousdevelopmentsare documentedelsewhere[ 87 113 116 ].Consideringthemoregeneralcasewithterminal constraints,thesufcientconditionsaresummarizedinEq uations( 5–79 )through ( 5–81 ). ConvexityCondition: H uu ( t ) > 0 for t 0 t t f (5–79) NormalityCondition: Q ( t ) < 0 for t 0 t < t f (5–80) JacobiCondition: S ( t ) R ( t ) Q 1 ( t ) R T ( t ) nitefor t 0 t < t f (5–81) Forthesimpliedcasewithoutterminalconstraints,thesu fcientconditionsreduce to H uu > 0 and S ( t ) niteovertheentireextremaltrajectory.Thesecondition scanbe easilyveriedalongtheextremalpathusingtheformulatio nsdevelopedinprevious sections. 5.3StoreSeparationAutopilot Flightvehicles,suchasaircraftandguidedstores,useig htmanagementsystems (FMS)toachieveguidanceandcontrolthroughouttheightp role.ThepilotorFMSwill frequentlyswitchbetweenautopilotsthatperformdiffere ntfunctions,suchasaltitude hold,climb/descent,bank-to-turn,etc.Inthiscontext,a storeseparationautopilotis atransitionalcontrolsystem,designedtoeffectivelytra nsferthestorefromrelease toastabletrimmedightcondition.Thespatiallyvarianta erodynamiccharacteristics areaccountedforthroughthenominaloptimaltrajectoryus ingthemethodsdiscussed inChapter 4 .Responsetovaryinginitialconditionsandowelddistur bancesare accountedforusingfull-statefeedbackbasedonneighbori ngoptimalcontroltechniques. 177

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5.3.1FeedbackUsingNeighboringOptimalControl Applicationofneighboringoptimalcontroltostoresepara tionisstraightforward. Aquadraticcostfunctional,givenbyEquation( 5–82 )issufcientforthisinvestigation, where Q isaconstantpositivesemi-denitematrix Q 0 and R isaconstantpositive denitematrix R > 0 .Theweightingmatrices Q and R arechosenbytheuserto inuencethemagnitudeofthestateandcontrolvector,resp ectively.Thematrix S f 0 isspeciedbytheusertoachievesatisfactoryterminalcon ditions. J = 1 2 x ( t f ) T S f x ( t f )+ 1 2 t f Z t 0 x T Q x + u T R u dt (5–82) Usingthequadraticcostfunctional,the1 st orderoptimalityconditionswithout terminalconstraintsarestatedinEquations( 5–83 )through( 5–85 ). _x ( t )= f ( x ( t ), u ( t ) ) x ( t 0 ) specied(5–83) ( t )= Q x ( t ) f T x ( t ) ( t ), ( t f )= S f x ( t f ) (5–84) u ( t )= R 1 f T u ( t ) ( t ) (5–85) Duetothedifcultyofmodelingstoreseparationaerodynam ics,itisdesirable toisolatetheaerodynamictermsappearinginthestateequa tions _x ( t )= f ( x u ) Thisallowstheoptimalityequationsdevelopedhereintobe usedwithavarietyof aerodynamicmodels.Recognizingthattheaerodynamicterm sarealsofunctionsof thestateandcontrol,thestateequationscanbewritteninf unctionalformasshownin Equation( 5–86 ),where C F ( x u ) and C M ( x u ) aretheaerodynamicforceandmoment coefcients,respectively. _x ( t )= f ( x C F ( x u ), C M ( x u ) ) (5–86) 178

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UsingthenotationinEquation( 5–86 ),theJacobianmatricesinEquations( 5–84 ) and( 5–85 )canbeexpandedasfollows. f x @ f ( x C F C M ) @ x = @ f 0 @ x + @ f @ C F @ C F @ x + @ f @ C M @ C M @ x (5–87) f u @ f ( x C F C M ) @ u = @ f 0 @ u + @ f @ C F @ C F @ u + @ f @ C M @ C M @ u (5–88) Equations( 5–89 )and( 5–90 )canbewrittenmoreconciselyusingsubscriptnotation torepresentpartialdifferentiation,wherethenotation f x 0 and f u 0 impliesthederivativeis takenwhileholdingtheaerodynamiccoefcientsconstant. f x = f x 0 + f C F C F X + f C M C M X (5–89) f u = f u 0 + f C F C F U + f C M C M U (5–90) Thematrices C F X and C M X representtheaerodynamicstabilityderivatives,andthe matrices C F U and C M U representtheaerodynamiccontrolderivatives.Thesematr ices maybedeterminedanalyticallywhenaparametricformofthe aerodynamicmodelis available,oftenasaresultofsystemidentication.Alter natively,theycanbeestimated numericallyusingnitedifferencingoranalternativenum ericalrecipe. BeginningwiththenecessaryconditionsinEquations( 5–83 )through( 5–85 ),the lineardifferentialequationsforaneighboringextremala riseimmediatelyfromtheresults inSection 5.2.2 andaresummarizedinEquations( 5–91 )through( 5–93 ). _x ( t )= f x x + f u u (5–91) u ( t )= R 1 f T u S x (5–92) S ( t )= S f x f T x S + S f u R 1 f T u S Q S ( t f )= S f (5–93) Equations( 5–91 )through( 5–93 )areacompactsetofdifferentialequationsthat canbeusedtoimplementaStoreSeparationAutopilotthatmi nimizestheoriginal costfunctiontosecondorderinthepresenceofdisturbance salongapredetermined 179

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optimaltrajectory.ThematrixRiccatiequation( 5–93 )isevaluatedalongtheoptimal trajectorytodeterminethefeedbackgains K ( t )= R 1 f T u S andtheresultsarestored alongwiththenominalstateandcontrol, x ( t ) and u ( t ) .Theneighboringoptimal controlinputcanbedeterminedreal-timeusingfeedforwar dofthenominalcontrol plusfeedbackproportionaltothedeviationofthemeasured statefromthereference trajectory, u ( t )= u ( t ) K ( t ) x ( t ) 5.3.2InniteHorizonNeighboringOptimalControl Theaerodynamiccharacteristicsofthestoreinthevicinit yoftheaircraftare inherentlynonlinear.Aerodynamicnonlinearitiesappear throughlargeoweld gradientsneartheaircraftaswellasdecayoftheaircrafte ffectsinfareldconditions. Thus,thestoretransitionsthroughatime(orspatially)va riantnonlinearregimeand rapidlyapproachesatrimmedfreestreamightconditionth atcanbeadequately approximatedbytimeinvariantlinearbehavior.Oneapproa chtocontrollingthestore inthesetwodisparateightregimesistoswitchbetweenano nlineartimevariant controllerandalineartimeinvariantcontroller.Another approachistodesignasingle controlsystemthataccountsforthenonlinearightregime andconvergestoalinear timeinvariantcontrollerinfareldconditions.Thelatte rapproachisadoptedhereina processhereinreferredtoasInniteHorizonNeighboringO ptimalControl. Theneighboringoptimalfeedbackgains K ( t )= R 1 f T u S aredeterminedinpartby thesolutiontothematrixdifferentialRiccatiequation( 5–94 ). S ( t )= S f x f T x S + S f u R 1 f T u S Q S ( t f )= S f (5–94) TheJacobianmatrices f x and f u areingeneraltime-varying.Forstoreseparation, thesematricesresultfromlinearizationalongapredeterm inedtrajectoryandvarywith timeand/ordistancefromtheaircraftduetothenonlineara erodynamiccharacteristics. However,asthedistancebetweenthestoreandaircraftbeco meslarge,theeffectofthe aircraftoweldbecomesnegligibleandtheJacobianmatri cesconvergetoconstant 180

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freestreamquantities,denotedhereas F and G lim t t f f x ( t ) F (5–95) lim t t f f u ( t ) G (5–96) Inthislimitingcase,thematrixdifferentialRiccatiequa tion(DRE)approachesa constantsolution,resultinginanalgebraicRiccatiequat ion(ARE)withmaybesolved numericallytoyield S f 0= S f F F T S f + S f GR 1 G T S f Q (5–97) ThesolutiontotheAREcanbeusedtodeterminetheconstantf eedbackgains K f = R 1 G T S f .Theresultinglineartimeinvariantcontrolsystemismath ematically equivalenttoaLinearQuadraticRegulator(LQR). Returningtotheoriginalquadraticcostfunctional,resta tedhereasEquation ( 5–98 ),itisapparent,withobviousforethought,thatthematrix S f isusedtodenotea user-speciedweightingmatrixthatdeterminestheendpoi nt(Mayer)cost. J = 1 2 x ( t f ) T S f x ( t f )+ 1 2 t f Z t 0 x T Q x + u T R u dt (5–98) ChoosingtheMayercosttobeconsistentwiththesolutionto theAREresultsin timevaryinggainmatrixthatapproachesaconstantquantit yasthesystemconverges toatimeinvariantsystem.Thetimeinvariantgainscanbeus edtomaintainthesystem nearthedesiredoperatingconditionindenitely. lim t t f K ( t ) K f (5–99) Insummary,InniteHorizonNeighboringOptimalControl(I HNOC)consistsof threesequentialsteps.First,optimalcontroltheoryisus edtodetermineanominal referencetrajectorythatoptimizesadesiredperformance indexforadynamicsystem 181

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withtransitorynonlinearcharacteristics.Next,neighbo ringoptimalcontrolisusedto implementafeedbackcontrolsystemthatoptimizestheorig inalperformanceindexto secondorderinthepresenceofdisturbancesalongtheoptim alpath.Finally,asthe systemapproachesanoperatingconditionthatisadequatel yrepresentedbyalinear systemmodel,thefeedbackcontrollerconvergestoalinear timeinvariantregulatorthat maybeusedtokeepthesystemnearthedesiredoperatingcond itionindenitely. IHNOCprovidesaframeworkfordesigningastoreseparation autopilotthatguides thestoreawayfromtheaircraftalonganoptimaltrajectory ,respondsinanoptimal mannertodisturbancesalongthenominalpath,andconverge stoatimeinvariant linearfeedbackcontrollerinfareldconditions.Anexten dedexampledemonstrating theefcacyofthisapproachispresentedinSection 5.4 andarealisticcasestudyis examinedinChapter 6 5.4Example:PlanarStoreSeparation Storeseparationisoftendominatedbyverticaltranslatio nandpitchattitude.In mostcases,lateraltranslationandyawattitudearefairly benignandofsecondary interest.Forinstructivepurposes,considerationofasto reconnedtothevertical x z planeduringseparationmaintainstheprimaryscopeofinte restandconsiderably reducesthecomplexityoftheoptimalcontrolproblem.Thee xampleconsideredhereis acontinuationoftheresultsdiscussedinChapter 4 .Extensionoftheapproachtoafull nonlinearsixdegreeoffreedomcasestudyistakenupinChap ter 6 5.4.1ModelEquations Themodelequationsforplanarstoreseparationweregiveni nSection 4.3.3.1 Thestatespaceequationsofmotioninmixedwindandbodyaxe saregivenin Equation( 5–100 ). 182

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266666666664 V q z 377777777775 = 266666666664 D / m g sin r q L / mV + g / V cos r M = I yy q V sin r 377777777775 (5–100) Thecomponentsofthestate,V(t), ( t ) ,q(t), ( t ) andz(t),aretheair-relative velocity,angleofattack,pitchrate,pitchangle,andvert icalposition,respectively.The ightpathangleisgivenby r ( t )= ( t ) ( t ) andthelocalaccelerationofgravity isdenotedby g .Thevariables L D ,and M representthedimensionallift,drag,and pitchingmoment,respectively.Finally, I yy isthepitch-axismomentofinertiaand m is themassofthestore.TheJacobianmatrices f x 0 f u 0 f C F ,and f C M canbeevaluated analyticallyfromtheequationsofmotionandareprovidedi nSection 4.3.3.1 Asimplisticspatiallyvariantnonlinearaerodynamicmode lbasedonwindtunnel datahasbeenselectedtodemonstratetheapplicationofNOC tostoreseparation.The modelisthesamemodelusedinSection 4.3.3.2 C L = C L + C L e e (5–101) C D = C D 0 + KC 2 L + C D e e (5–102) C m = C m + C m q ^ q + C m e e + e ( z ) ( 0 + 1 z ) (5–103) Thevariable e representstheelevatorcontrolsurfacedeectionand ^ q isthe non-dimensionalpitchrateintroducedforunitconsistenc y.Theconstants C L C D 0 C m ,and C m q representtheclassicalaerodynamicderivatives.Thecons tants C L e C D e ,and C m e representtheclassicalcontrolderivatives.Forthecurre ntexample,the aerodynamicandcontrolderivativesareconstantandestim atedusingfreestreamwind tunneldataandthesystemidenticationmethodsdiscussed inChapter 3 .Theterm e ( z ) ( 0 + 1 z ) istheonlyspatiallyvariantcontributiontotheaerodynam icmodel.The 183

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exponential-polynomialformisseenasaspecialcaseofthe moregeneralparametric modelpresentedinChapter 3 .Theconstants 0 ,and 1 wereestimatedusinga nonlinearleastsquarescurvetofrepresentativewindtun neldataatanominalpitch attitudeandaretabulatedinTable( 4-1 ).TheJacobianmatrices C F X C M X C F U ,and C M U arereadilydeterminedfromtheaerodynamicmodelandprovi dedinSection 4.3.3.1 5.4.2NeighboringOptimalControl Figure 5-2 showsanopenloopoptimaltrajectoryandaneighboringopti mal trajectoryforthesameinitialconditions.Theopenlooptr ajectorywassolvedas anHBVPusingtheplanarnonlinearequationsofmotionandsi mpliednonlinear aerodynamicmodeldescribedinSection 5.4.1 .Theneighboringoptimaltrajectorywas determinedusingtheequationsderivedinSection 5.4 withaquadraticcostfunctional. Thecostwasselectedsuchthat Q =10 and R e =10 withallotherweighting parameterssettozero,including S f =0 .Theresultsagreeverycloselyasexpected, indicatingtheNOCformulationisworkingwellintheabsenc eofanydisturbancesor parametervariations. Thesolution S ( t ) totheRiccatidifferentialequationalongtheoptimaltraj ectoryis showninFigure 5-3 .NotethatalloftheRiccatigainsconvergetozeroat t = t f .Thisis consistentwiththeboundaryconditionresultingfromthew eightingmatrix S f =0 Thecorrespondingcontrolgains, K ( t )= R 1 f T u S ( t ) ,areshowninFigure 5-4 Thevariouscontrolgainsdifferbythreeordersofmagnitud eandthesubplotsinFigure 5-4 areorganizedbyorderofmagnitudetoenhanceclarity.Thel argestgainsarethose associatedwiththestatevariables ( t ) ( t ) ,and q ( t ) ,asonemightexpectbasedon thespeciedcostparameters.Notethatthecontrolgainsin Figure 5-4 alsoconvergeto zeroat t = t f ,commensuratewith S f =0 TheneighboringextremalinFigure 5-2 agreeswellwiththeHBVPoptimaltrajectory withthesameinitialconditions.Fortheopenlooptrajecto ry,theHBVPmustbesolved againforeachnewsetofinitialconditions.However,NOCca nbeusedtoestimate 184

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0 0.2 0.4 0.6 0.8 1 859.5 860 860.5 861 861.5 862 time (sec)Vel (ft/sec) 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 time (sec)angle (deg) a q 0 0.2 0.4 0.6 0.8 1 -60 -40 -20 0 20 time (sec)Pitch Rate (deg/sec) 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 time (sec)Elevator Defl (deg) d e 0 0.2 0.4 0.6 0.8 1 0 50 100 time (sec)Z (ft) Z Pos 0 0.2 0.4 0.6 0.8 1 -0.5 0 0.5 1 time (sec)Aero Coeff Normal Force, CN Pitching Moment, Clm Optimal Neighboring Optimal Velocity qbbi Figure5-2.Optimaltrajectorywithneighboringoptimalfe edbackcontrol. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 TimeRiccati Solution, S(t) Figure5-3.SolutiontoRiccatidifferentialequation. 185

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -2 -1 0 1 Control Gains, K(t)Order 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.5 0 0.5 Order 10 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.02 -0.01 0 0.01 Order 10 -2Time (sec) Figure5-4.Neighboringoptimalfeedbackgains. anewoptimaltrajectoryforeachsetofdisturbancesusingo nlyasinglereference trajectoryandstoredgainmatrix.Figure 5-5 showsaseriesofoptimaltrajectories computedforvariousinitialpitchrates, q ( t 0 )= 50 50 deg/sec.Thesolutionswere obtainedwith Q =1 and R e =10 andallotherweightingparameterssetto0.The reducedweight Q wasselectedtoavoidsaturationofthecontrolinputs. Theoptimalsolutions(solidlines)inFigure 5-5 weresolvedindependentlyfor eachsetofinitialconditions.Therstoptimalsolution,c orrespondingto q ( t 0 )= 50 deg/sec,wasusedasareferencetrajectoryfortheremainin gfourNOCsolutions.Note thateachofthefourNOCsolutionsagreeswellwiththetrueo ptimalsolutionforthe speciedinitialconditions.Thus,theNOCformulationisa dequatelyapproximatingthe trueoptimalperformanceofthesysteminthepresenceofvar yinginitialconditions. Figure 5-6 showsasimilarresultforperturbationsin ( t 0 )=5 2 deg. Again,theneighboringoptimaltrajectoriesareadequatea pproximationsofthetrue optimaltrajectories,withsomeminordegradationattheex tremevalues.Despitethe degradation,theperformanceoftheneighboringoptimalco ntrollerisexcellent. 186

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0 0.2 0.4 0.6 0.8 1 -15 -10 -5 0 5 10 Angle of Attack (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -120 -100 -80 -60 -40 -20 0 20 40 60 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 4 Elevator Defl (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -1.5 -1 -0.5 0 0.5 1 1.5 Pitching Momenttime (sec) Figure5-5.Optimalandneighboringoptimaltrajectoriesf orvaryinginitialpitchrate. 0 0.2 0.4 0.6 0.8 1 -15 -10 -5 0 5 10 Angle of Attack (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -80 -60 -40 -20 0 20 40 60 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.2 0.4 0.6 0.8 1 -5 -4 -3 -2 -1 0 1 2 Elevator Defl (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -1.5 -1 -0.5 0 0.5 1 Pitching Momenttime (sec) Figure5-6.Optimalandneighboringoptimaltrajectoriesf orvaryinginitialangleof attack. 187

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Figure 5-7 showssimilarresultsforperturbationsinboth q ( t 0 )= 50 50 deg/secand ( t 0 )=5 2 deg.Theresultsfurtherdemonstratetheperformanceofthe neighboringoptimalcontrollerinthepresenceoflargeini tialdisturbances. 0 0.2 0.4 0.6 0.8 1 -20 -15 -10 -5 0 5 10 Angle of Attack (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -150 -100 -50 0 50 100 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 4 Elevator Defl (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 Pitching Momenttime (sec) Figure5-7.Optimalandneighboringoptimaltrajectoriesf orvaryinginitialpitchrateand initialangleofattack. TheconsistencyoftheresultspresentedinFigures 5-5 through 5-7 isindeed promisingasastartingpoint.Itisimportanttonotethatth isexampleisbasedon anaerodynamicmodelthatislinearbyconstruction,withth eexceptionofasingle nonlinearspatiallyvariantterm.Afullscalesixdegreeof freedomaerodynamicmodel willnecessarilybemorecomplicatedandonewouldexpectth atasthenonlinearities becomestrongertheeffectivenessoftheneighboringoptim alcontrollerwilldecrease. Thelimitationoftheneighboringoptimalcontrollercanbe exposedbyconsidering increasinglylargerperturbationsininitialconditions. Forexample,Figure 5-8 shows threetrajectoriesforinitialangleofattack ( t 0 )=5 5 deg.Theneighboringtrajectories werecomputedusingonlythenominaloptimaltrajectoryasa reference.Theresults 188

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indicatethateventhoughtheNOCperformsadequately,then eighboringtrajectories differsubstantiallyfromthetrueoptimaltrajectories,e speciallyat ( t 0 )=0 deg. 0 0.2 0.4 0.6 0.8 1 -20 -15 -10 -5 0 5 10 15 Angle of Attack (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -80 -60 -40 -20 0 20 40 60 80 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.2 0.4 0.6 0.8 1 -10 -8 -6 -4 -2 0 2 Elevator Defl (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -2 -1.5 -1 -0.5 0 0.5 1 Pitching Momenttime (sec) Figure5-8.Optimalandneighboringoptimaltrajectoriesf orlargeperturbationsininitial angleofattack. Incomparisonoftheoptimalandneighboringoptimalsoluti onsinFigure 5-8 ,two thingsareimportanttonote.First,therangeininitial ( t 0 ) isextensive.Theinitialpitch angle ( t 0 ) hasbeenheldconstant ( t 0 )=0 foralloftheprecedingcomparisons. Thus,thechangeinangleofattackcorrespondsphysicallyt oadifferentinitialvertical velocity.Aninitial ( t 0 )=0 correspondstozeroverticalvelocity,e.g.agravityrelea se. Thenominal ( t 0 )=5 degcorrespondstoanejectionvelocityof 75 ft = sec ,orabout 46 g whichissubstantial.Themaximuminitialvalueof ( t 0 )=10 deg corresponds toa 150 deg = sec ejectionvelocity,orabout 93 g ,whichfarexceedsanyrealisticinitial condition. Thesecondimportantnoteisthateveninthemostextremecon ditions,the neighboringoptimalcontrollerperformswell,drivingthe storetoabenignightcondition inthedesiredtimeinterval.Thesignicantdecreaseinver ticalvelocityfor ( t 0 )=0 189

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hastheeffectofimmersingthestorewithintheaircraftow eldforalongerduration, makingthetrajectorymoresensitivetothenonlineartermi ntheaerodynamicmodel. Asaresult,thestoreexperiencesastrongernosedownpitch ingmomentthanthe controllerwasdesignedforandthecontrollerundercompen satesforthepitching moment,allowingthestoretoreachalargernosedownangleo fattack.Thereduced controlcorrespondingtothereducedverticalvelocityisa ctuallyadvantageous,inthatit increasesthesafetymarginbyallowingtheaerodynamiceff ecttoacceleratethestore awayfromtheaircraft,albeitatahighercostthanthetrueo ptimalsolution. Theabovediscussioncanbesummarizedasfollows.Forthesi mplequasi-linear aerodynamicmodelconsideredinthiscase,theinitialcond itionsmustbevaried beyondreasonablelimitsinordertondasituationinwhich theneighboringoptimal controllerfailstoadequatelyapproximatethetrueoptima lsolution.Eveninthemost extremecases,theperformanceoftheneighboringoptimalc ontrollerisstableand convergentandasignicantimprovementovertheequivalen tuncontrolledtrajectory. However,thecomparisondoessuggestthattheperformanceo ftheneighboring optimalcontrollerislimitedinthepresenceoflargedevia tionsawayfromthenominal solution.ThisisconsistentwiththeNOCtheorydevelopedi nSection 5.2 ,which assumedsmallvariationsinthestate x ( t ) andcontrol u ( t ) .Inthepresenceof sufcientlylargeperturbationsininitialconditions,th eneighboringoptimalcontrol performanceismarginal.Furtherinvestigationoftheperf ormanceandlimitationsofthe NOCformulationwillbeconsideredinChapter 6 5.4.3NeighboringOptimalControlwithInequalityConstra ints TheequationsforapplyingNOCtoaproblemwithinequalityc onstraintswere discussedinSection 5.2.4 .Theconstraintsareadjoinedtotheaugmentedcost functionalusingtheLagrangemultiplier ( t ) .Theresultisasetoftwoequations: onesetisvalidalonganunconstrainedarc,andtheotherisv alidalongtheconstrained arc.Thedifcultyarisesatthejunctionpoints.Historica lly,thishasbeentreatedas 190

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amultipointboundaryvalueproblem,whichmaybesolvedusi ngamultipleshooting algorithmthatiteratestondthejunctionpoints.Thisfea tureintroducessignicant complexitytothenumericalsolutionusingsweepmethods[ 131 – 133 136 ]. Collocationmethodsresultinanentirelydifferentapproa ch.Withcollocation, theequationsarenotsweptforwardorbackward,butsolvedo verthedurationofthe trajectorysimultaneously.Theboundaryconditionsareac countedforexplicitly.As aresult,asimplesaturationfunctionisallthatisnecessa rytolimitcontrolauthority. TheequationsinSection 5.2.4 canbeusedtoverifythattheresultingtrajectoryisan extremal. Intheprevioussection,thecostfunctionalwasspeciedin suchawaytoavoid saturationoftheinputs.However,thestoreseparationtra jectorycanbeimprovedifthe systemisallowedtousefullcontrolauthoritywhennecessa ry.Forthecurrentexample, theelevatordeectionwillbelimitedto j e j 10 deg .TheequationsfromSection 5.2.4 willbeusedtoverifythestationarityoftheresultingtraj ectory,restatedinEquation ( 5–104 ). When: 8>><>>: C < 0, =0, u = H 1 uu H ux + f T u S x C =0, u = u c u = C T u H ux + f T u S x + H uu u (5–104) Specically,thecontrolisrstcomputedusing u = H 1 uu H ux + f T u S x .Ifthe inequalityconstraintisactive(e.g. j u j = j u + u j > 10 deg ),thecontrolislimitedto 10 deg andtheLagrangemultiplieris = C T u H ux + f T u S x + H uu u .Asimple checktoverify 0 issufcienttoensurestationarity. Figure 5-9 showsoptimalandneighboringoptimaltrajectorieswithth eelevator deectionconstrainedto j e j 10 deg .Theinitialconditionwasspeciedas ( t 0 )= 1 deg q ( t 0 )= 50 deg = sec .Thecostwasspeciedas Q =10 and R e =1 ,putting moreemphasisonminimizingtheangleofattack ( t ) attheexpenseofmorecontrol effort. 191

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0 0.2 0.4 0.6 0.8 1 856 858 860 862 time (sec)Vel (ft/sec) 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 time (sec)angle (deg) a q 0 0.2 0.4 0.6 0.8 1 -50 0 50 time (sec)Pitch Rate (deg/sec) 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 time (sec)Elevator Defl (deg) d e 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 time (sec)Z (ft) Z Pos 0 0.2 0.4 0.6 0.8 1 -1 0 1 2 time (sec)Aero Coeff Normal Force, CN Pitching Moment, Clm Optimal Neighboring Optimal Velocity qbbi Figure5-9.Optimalandneighboringoptimaltrajectoriesw ithconstrainedelevator deection. Theconstraintmultiplier ( t ) forboththeoptimalandneighboringoptimal trajectoriesisshowninFigure 5-10 .Duringtheconstrainedarc,theconstraint multiplierispositive,indicatingthatareductionincost couldonlybeaccomplished byviolatingtheconstraint.During,theunconstrainedarc ,theconstraintmultiplieriszero tonumericalprecision,indicatingtheconstraintisinact ive.Notethattheoptimaland neighboringoptimalconstraintmultipliersagreequitewe llforthesameinitialconditions, asexpected. Figure 5-11 showsaseriesofoptimalandneighboringoptimaltrajector iesfor varyinginitialpitchrate.Thereferencetrajectorywasco mputedusinganinitialpitch rateof q ( t 0 )= 50 deg = sec .Theremainingtwooptimaltrajectorieswerecomputedfor increasinglynegativevaluesofinitialpitchrate.Thenei ghboringoptimaltrajectories, computingusingthersttrajectoryasareference,agreequ itewellwiththetrueoptimal trajectories. 192

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0 0.2 0.4 0.6 0.8 1 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Time (sec)Constraint Multiplier, m (t) Optimal Neighboring Optimal Figure5-10.Constraintmultiplierforoptimalandneighbo ringoptimaltrajectorieswith constrainedelevatordeection. 0 0.2 0.4 0.6 0.8 1 -15 -10 -5 0 5 Angle of Attack (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -100 -50 0 50 100 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 10 Elevator Defl (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -1.5 -1 -0.5 0 0.5 1 1.5 2 Pitching Momenttime (sec) Figure5-11.Optimalandneighboringoptimaltrajectories withconstrainedelevator deectionforvaryinginitialpitchrate. Figure 5-12 showstheconstraintmultipliersfortheextremaltrajecto riesshownin Figure 5-11 .Thepositivevalueofthemultiplierduringtheconstraine darcindicates stationarity. 193

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0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (sec)Constraint Multiplier, m (t) Optimal Neighboring Optimal Figure5-12.Constraintmultiplierforoptimalandneighbo ringoptimaltrajectorieswith constrainedelevatordeectionforvaryinginitialpitchr ate. Theresultsaboveindicatethattheneighboringoptimalcon trolleradequately representstheoptimalperformanceofthesystemeveninthe presenceofconstraints. However,constraintsareinherentlynonlineartheresults arenotalwayssoexemplary. Figure 5-13 showsaseriesofoptimalandneighboringoptimaltrajector ieswithvarying initialpitchrate q ( t 0 ) ,withincreasinglysmallerinitialvalues. Thereferenceoptimaltrajectoryfor q ( t 0 )= 50 deg = sec isidenticaltothetrajectory shownpreviouslyinFigure 5-10 .However,theagreementbetweenthesubsequent optimalandneighboringoptimaltrajectoriesissubstanti allydifferent.Theneighboring optimalcontrollerisdesignedtominimizetheangleofatta ck ( t ) byusingmaximum controlefforttocounteractasignicantnosedownpitchra tenearcarriage.Whenthe initialpitchrateissubstantiallyreduced,theoptimaltr ajectorychangesdramatically resultinginlargevariationsthatdegradetheaccuracyoft heneighboringoptimal controller.Fortunately,theneighboringoptimalcontrol lerisstillstableandconvergent andagainbiasedtowardanincreasedsafetymarginbydrivin gthestorenosedown, 194

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0 0.2 0.4 0.6 0.8 1 -5 -4 -3 -2 -1 0 1 2 Angle of Attack (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -60 -40 -20 0 20 40 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 10 Elevator Defl (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -3 -2 -1 0 1 2 Pitching Momenttime (sec) Figure5-13.Optimalandneighboringoptimaltrajectories withconstrainedelevator deectionforvaryinginitialpitchrate. acceleratingthestoreawayfromtheaircraft,albeitatahi ghercostthanthetrueoptimal solution. Finally,Figure 5-14 showsanexamplewithextremelyadverseinitialconditions Thereferenceoptimaltrajectoryisthesameaspreviousexa mplesfor ( t 0 )=1 deg and q ( t 0 )= 50 deg = sec .Theperturbedinitialconditionsare ( t 0 )=0 deg and q ( t 0 )=50 deg = sec .Physically,theseinitialconditionsareequivalenttoag ravityrelease (zeroejectionvelocity)withaninitialnoseuppitchrate, apotentiallydangeroussituation thatmayleadtothestoreyingbacktowardtheaircraft.The refore,thisexampleisa “stresscase”fortheneighboringoptimalcontroller. Althoughthereisasubstantialdifferencebetweentheopti malandneighboring optimalsolutionsfortheperturbedinitialconditions,th eneighboringoptimalcontroller performsquitewellundertheseadverseconditions.Thenos e-uppitchrateisimmediately arrestedusingmaximumcontroleffort.Theneighboringext remalplungesmuchfurther 195

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0 0.2 0.4 0.6 0.8 1 -6 -4 -2 0 2 Angle of Attack (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -50 0 50 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 10 Elevator Defl (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -3 -2 -1 0 1 2 Pitching Momenttime (sec) Figure5-14.Optimalandneighboringoptimaltrajectories withconstrainedelevator deectionforextremelyadverseinitialconditions. nosedownthannecessary,butstillachievesanincreasings afetymarginincomparison withtheoptimaltrajectory. Theabovediscussioncanbesummarizedasfollows.Forthesi mplequasi-linear aerodynamicmodelconsideredinthiscase,neighboringopt imalcontrolinthepresence ofconstraintsadequatelyrepresentsthetrueoptimalperf ormancewhentheconstraints donotcauselargedisparitybetweentheoptimalandneighbo ringoptimalsolutions. However,eveninthemostadverseconditionsconsidered,th eneighboringoptimal controllerperformedwell,resultinginasafeandacceptab letrajectory. 5.4.4NeighboringOptimalControlwithTerminalCost Inallexamplesconsideredtothispoint,theterminalcosth asbeenneglected, S f =0 .Theterminalcostcanbeusedtoachieveadesiredterminalc ondition,focusing thecontroleffortneartheendofthetrajectory.Figure 5-15 showsareferenceoptimal trajectoryandaseriesofneighboringoptimaltrajectorie sforvaryinginitialpitchrate. TheLagrangecostwasspeciedas Q =0 and R e =10 ,whichlimitscontroleffort 196

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duringthetrajectory.Theendpointcost,orMayercost,was speciedas S f =10 to emphasizeaminimalangleofattackat t = t f 0 0.2 0.4 0.6 0.8 1 -30 -20 -10 0 10 20 Angle of Attack (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -150 -100 -50 0 50 100 150 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 10 Elevator Defl (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -3 -2 -1 0 1 2 Pitching Momenttime (sec) Figure5-15.Optimalandneighboringoptimaltrajectories withterminalcost. Thesolution S ( t ) totheRiccatiequationandthecorrespondingfeedbackgain s K ( t )= R 1 f T u S ( t ) areshowninFigure 5-16 .TheRiccatigainsarereducedto 1 = 100 th oftheinitialvaluesduringthemidsectionofthetrajector y,commensuratewith Q =0 .Thegainsthataffect ( t f ) areincreasedneartheendofthetrajectoryinorderto minimizetheendpointcost. ReferringbacktoFigure 5-15 ,notetheterminalcondition ( t f ) 0 comesatthe expenseofalargevariationintheterminalpitchrate 150 < q ( t f ) < 150 deg = sec .As aresult,thestoreisfarfromatrimmedightconditionat t = t f andadditionalcontrol wouldbenecessarytocapturethepitchrate.Fromthispersp ective,anon-zerocoston ( t ) ,suchas 1 Q 10 ,ispreferable. 197

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -40 -20 0 20 40 Riccati Solution, S(t) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 -2 0 2 4 Time (sec)Control Gains, K(t) Figure5-16.SolutiontoRiccatiequationandfeedbackgain sforneighboringoptimal trajectorieswithterminalcost. Finally,Figure 5-20 showsaseriesofneighboringtrajectoriesforvaryinginit ialpitch ratewithbothLagrangecost, Q =1 and R e =10 ,andMayercost, S f = S f q =1 Thetrajectoriesterminatenearatrimmedightconditionw ith ( t f ) and q ( t f ) nearzero withoutexcessivecontroleffort.Ifmoreemphasison ( t ) isdesired,ahighweighton Q canbeselected.Inthiscase,theterminalconditionsareve rynearzeroandtheend pointcostcanbeomitted.5.4.5NeighboringOptimalControlwithTerminalConstrain ts TheresultswithterminalcostfromSection 5.4.4 canbeextendedtothecasewith terminalconstraints.Thesolutionismoreinvolved,given therequirementofsolvingtwo additionalmatrixdifferentialequations. Figure 5-18 showsanoptimaltrajectoryandseveralneighboringtrajec toriesfor varyinginitialpitchrate.Inthiscase,theendpointconst raintwasselectedsuchthat ( t f )=0 and q ( t f )=0 198

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0 0.2 0.4 0.6 0.8 1 -15 -10 -5 0 5 Angle of Attack (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -100 -80 -60 -40 -20 0 20 40 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.2 0.4 0.6 0.8 1 -10 -8 -6 -4 -2 0 2 4 Elevator Defl (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 Pitching Momenttime (sec) Figure5-17.Optimalandneighboringoptimaltrajectories withcumulativeandterminal cost. 0 0.2 0.4 0.6 0.8 1 -15 -10 -5 0 5 Angle of Attack (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -100 -80 -60 -40 -20 0 20 40 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.2 0.4 0.6 0.8 1 -10 -8 -6 -4 -2 0 2 4 Elevator Defl (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 1.5 Pitching Momenttime (sec) Figure5-18.Optimalandneighboringoptimaltrajectories withterminalconstraints. 199

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Itisclearthatthetrajectoriesconvergeto ( t f )=0 and q ( t f )=0 asdesired. However,terminalconstraintsresultininnitegainsnear theendpoint,assuggestedin Figure 5-18 bythesharpdropinelevatordeectionat t = t f Thesolutiontothematrixdifferentialequationsareshown inFigure 5-19 .The matrix Q ( t ) convergestozeroat t = t f ,givingrisetotheinnitegainsthroughthe inverse Q 1 ( t f ) .Thisisclearlyanundesirablecharacteristicandoflimit edutilityfor storeseparation. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -10 0 10 S(t) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -10 0 10 R(t) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -100 -50 0 Q(t)Time (sec) Figure5-19.Solutiontodifferentialequationsforneighb oringoptimaltrajectorieswith terminalconstraints. Terminalconstraintsmaybepracticallyusefulorabsolute lynecessaryforsome systems.Forstoreseparation,itisdesirabletobenearatr immedightconditionat t = t f ,butitisnotrequiredtobeatanexactightcondition.Give ntheadditional complexityofinnitegainsarisingfromterminalconstrai nts,theendpointcostis preferredovertheendpointconstraintapproach. 200

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5.4.6InniteHorizonNeighboringOptimalControl Theprevioussectionsconsideredneighboringoptimalcont rolwithcumulativecost, endpointcost,inequalityconstraints,andendpointconst raints.Ineachcase,theHBVP isthesame;thecostsandconstraintsaffectprimarilytheb oundaryconditionsandthe neighboringoptimaldifferentialequations.Inallcases, theneighboringoptimalfeedback gainsaretimevarying,butthegainschangedramaticallyba sedontheboundary conditions. Inthissection,theperformanceofthecontrollerisextend edbeyondthenaltime t = t f usedtosolvetheHBVP.Inparticular,theopenloopoptimalc ontrolproblem issolvedovertheinterval t 2 [ t 0 t f ]=[0,1] ,buttheneighboringoptimalfeedback controllerisusedtocontrolthesystemwellbeyondtheopen loophorizon.Byjudicious specicationoftheterminalcost,theboundaryconditions arespeciedsuchthat theneighboringoptimalcontrollercontinuestorespondto disturbancesinanoptimal mannerindenitely,i.e.overaninnitehorizon. Thisapproach,hereinreferredtoasinnitehorizonneighb oringoptimalcontrol (IHNOC),hassignicantpracticalvaluefortransitionals ystemsingeneralandstore separationinparticular.IHNOCisusedtoguidethestoreth roughanonlinearight regimetoatrimmedightcondition.Asthestoreapproaches atrimmedightcondition, thecontrollerconvergestoalineartime-invariantcontro ller.Thetime-invariantcontroller, mathematicallyequivalenttoalinearquadraticregulator ,canbeusedtoholdthestore inatrimmedightconditionuntiltransitiontothemission autopilotiscomplete.This isclearlyadesirablecharacteristicformanysystemsthat exhibitnonlinearities.To theauthor'sknowledge,thisnovelapplicationofNOCtoasy stemwithtransitional nonlinearitieshasnotbeendocumentedelsewhere. Figure 5-20 showsaseriesofoptimalandneighboringoptimaltrajector iesfor varyinginitialpitchrate q ( t 0 )=0 50 deg = sec .Theneighboringoptimaltrajectoriesare determinedusingareferencetrajectorywith q ( t 0 )=0 deg = sec .Theagreementbetween 201

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theoptimalandneighboringoptimaltrajectoriesisgood,d espitethelargevariationin q ( t 0 ) andthecontrolinequalityconstraints. 0 0.2 0.4 0.6 0.8 1 -6 -4 -2 0 2 4 Angle of Attack (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -50 0 50 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.2 0.4 0.6 0.8 1 -10 -8 -6 -4 -2 0 2 4 Elevator Defl (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -2 -1.5 -1 -0.5 0 0.5 1 Pitching Momenttime (sec) Figure5-20.Optimalandneighboringoptimaltrajectories forvaryinginitialpitchrate. TheextremaltrajectoriesinFigure 5-20 aredeterminedusingaLagrangecostof Q = R e =10 .TheMayercostwasspeciedasthepositivedenitesolutio nofthe algebraicRiccatiequation( 5–105 ),withtheconstantmatrices F and G determinedby Equation( 5–106 ).TheresultingRiccatisolutionandcontrolgainsareshow ninFigure 5-21 0= S f F F T S f + S f GR 1 G T S f Q (5–105) F =lim t t f f x ( t ), G =lim t t f f u ( t ) (5–106) Thetime-varyingstructureofthecontrolgainsisevidentf romFigure 5-21 .The gainsbeginatamaximumvalueattheinitialtimewhenthesto reisnearestthe 202

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -20 -10 0 10 20 Riccati Solution, S(t) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1.5 -1 -0.5 0 0.5 Time (sec)Control Gains, K(t) Figure5-21.SolutiontoRiccatiequationandfeedbackgain sforneighboringoptimal trajectories. aircraftandconvergerapidlytoconstantvaluesastheaero dynamicnonlinearities becomeinsignicant.TheRiccatisolution S ( t ) andcontrolgains K ( t ) areconstantafter 0.9 sec ,indicatingthestoreissufcientlyfarawayfromtheaircr aftthattheaerodynamic interferenceisnegligible. ExtensionofIHNOCbeyond t f =1 istrivial.Thereferencestate x ,reference control u andfeedbackgains K ( t ) areheldconstantattheterminalvaluesforaslong asthecontrollerisactive.Figure 5-22 showstheresultingseriesoftrajectories.Note thattheoptimaltrajectoryisterminatedat t = t f ,buttheextremaltrajectoriescontinue smoothlyto t =2 .Allvesimulationsconvergetothesametrimmedightcond ition, despitethelargevariationsininitialconditions. Forcomparison,Figure 5-23 showsaseriesofneighboringtrajectories.The extremaltrajectoriesaredeterminedusingaLagrangecost of Q = Q q =1 and R e =10 ,puttingmoreemphasisonminimizingthepitchrate q ( t ) .Theresultsare 203

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0 0.5 1 1.5 2 -6 -4 -2 0 2 4 Angle of Attack (deg)time (sec) 0 0.5 1 1.5 2 -50 0 50 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.5 1 1.5 2 -10 -8 -6 -4 -2 0 2 4 Elevator Defl (deg)time (sec) 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 Pitching Momenttime (sec) Figure5-22.Optimalandneighboringoptimaltrajectories forvaryinginitialpitchrate, extendedbeyond t = t f consistentwiththepreviousexample.However,duetothead ditionalcostplacedon q ( t ) ,theangleofattack ( t ) ismoredispersedat t = t f anddoesconvergeuntil t =2 sec .Theresultsshowhowinnitehorizonneighboringoptimalc ontrolcontinuesto operateinanoptimalmannerbeyondtheoriginalnitehoriz on. Finally,Figure 5-24 showsaseriesofneighboringtrajectorieswith Q = R e =10 and Q q =0 .Theinitialconditionsincludevariationinpitchrate q ( t 0 )= 50 50 deg = sec andangleofattack ( t 0 )=2 2 deg .Theneighboringextremalsaredeterminedusing theoptimaltrajectorywith ( t 0 )=2 deg and q ( t 0 )= 50 deg = sec asareference trajectory.Theresultsareagainfavorable,demonstratin gconvergencetoatrimmed ightconditionforawiderangeofinitialconditions. 204

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0 0.5 1 1.5 2 -6 -4 -2 0 2 4 Angle of Attack (deg)time (sec) 0 0.5 1 1.5 2 -50 0 50 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.5 1 1.5 2 -10 -5 0 5 10 Elevator Defl (deg)time (sec) 0 0.5 1 1.5 2 -3 -2 -1 0 1 Pitching Momenttime (sec) Figure5-23.Optimalandneighboringoptimaltrajectories forvaryinginitialpitchrate, withadditionalcoston q ( t ) 5.4.6.1Responsetoowelddisturbances Neighboringoptimalcontrolisdesignedtocorrectforpert urbationsintheinitial conditions x ( t 0 ) .However,sinceanytimeisavalid“initial”time,NOCcanal sobeused tocorrectfordisturbancesthathappenalongtheoptimalpa th.Forstoreseparation, turbulentairneartheaircraftisofparticularconcern. Thestudyofaerodynamicturbulenceisnosmallmatterandat horoughanalysisof turbulenteffectsonstoreseparationisbeyondthescopeof thisinvestigation.Rather, amoredirectapproachwillbeconsideredtodemonstratethe performanceofthe controllerinthepresenceofrandomdisturbances. Thepresenceofrandomdisturbancesalsochangesthenature ofthedynamic systemfromadeterministicsystemtoastochasticsystem.M ethodsforaddressing optimalcontrolinthepresenceofrandomdisturbancesaret hesubjectofmuch 205

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0 0.5 1 1.5 2 -20 -15 -10 -5 0 5 Angle of Attack (deg)time (sec) 0 0.5 1 1.5 2 -100 -50 0 50 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.5 1 1.5 2 -10 -5 0 5 Elevator Defl (deg)time (sec) 0 0.5 1 1.5 2 -2 -1 0 1 2 Pitching Momenttime (sec) Figure5-24.Optimalandneighboringoptimaltrajectories forvaryinginitialpitchrate andinitialangleofattack. research[ 111 113 116 ].Foralineartime-varyingstochasticsystem,alinearqua dratic gaussian(LQG)controllermaybeutilized[ 113 ].TheLQGessentiallycombinesa KalmanlterforstateestimationandaLQRforcontrol.Alth oughsuchanapproach wouldbeanaturalextensiontothisinvestigation,thisisb eyondthescopeofthepresent work.Thisremainsanareaforcontinuedresearchinstorese paration. Figure 5-25 showsasnapshotofthelongitudinalaerodynamiccoefcien tsderived fromstoreseparationighttest2265(Mach0.9/450KCAS).T heleftmostsubgures showtheestimatedaerodynamiccoefcientbasedoninertia lmeasurements,aswell asa“smoothed”estimate.Thesmoothedestimatewasdetermi nedusinga51-point movingaverage.Theresidualsforthethreeaerodynamiccoe fcientsareshownon therightsideofFigure 5-25 .Theseresidualsareassumedtoincludeeffectsfrom aerodynamicturbulence,sensornoise,andstructuralvibr ation.Noattemptwillbemade 206

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toseparatetheeffectofeachsource.Rather,theresiduals willbeusedtointroduce disturbancesintothesystem,representativeofthedistur bancesencounteredinight. 0 0.5 1 1.5 2 -2 -1 0 1 2 Axial Force, CA 0 0.5 1 1.5 2 -0.5 0 0.5 1 Residuals 0 0.5 1 1.5 2 -15 -10 -5 0 5 Normal Force, CN 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 Residuals 0 0.5 1 1.5 2 -4 -2 0 2 4 Pitching Moment, CmTime (sec) 0 0.5 1 1.5 2 -2 -1 0 1 2 ResidualsTime (sec) Figure5-25.Aerodynamiccoefcientsestimatedfromight testdata. Figure 5-26 showsanoptimaltrajectoryandneighboringoptimalcontro linthe presenceofrandomdisturbances.Theeffectoftherandomdi sturbanceisevidentinthe normalforceandpitchingmomentcoefcients.Theinertiao fthesystemactsasalow passltertomitigatetheturbulenteffectsandthecontrol systemperformswellamidst therandomdisturbances. Figure 5-27 showsaseriesofneighboringtrajectorieswithincreasing valuesof turbulence.Thetrajectoriesweredeterminedwiththesame turbulencesignalamplied 207

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0 0.5 1 1.5 2 854 856 858 860 862 time (sec)Vel (ft/sec) 0 0.5 1 1.5 2 -10 -5 0 5 time (sec)angle (deg) a q 0 0.5 1 1.5 2 -60 -40 -20 0 20 time (sec)Pitch Rate (deg/sec) 0 0.5 1 1.5 2 -10 -5 0 5 time (sec)Elevator Defl (deg) d e 0 0.5 1 1.5 2 0 50 100 150 time (sec)Z (ft) Z Pos 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 time (sec)Aero Coeff Normal Force, CN Pitching Moment, Clm Optimal Neighboring Optimal Velocity qbbi Figure5-26.Optimalandneighboringoptimaltrajectories withrandomdisturbances representativeofaerodynamicturbulence. byone,three,andvetimestheoriginalmagnitude.Again,t heturbulenceis“averaged out”bytheinertiaofthestoreandthecontrollerperformsw ellinallthreecases. TherandomdisturbancesintroducedinFigure 5-25 arenotcompletelystationary signals.Itisclearfromcarefulinspectionoftheplotthat themagnitudeoftherandom disturbancesislargernearthebeginningofthesignal,per hapsduetoejection-induced structuralvibrations.Evenso,thehighfrequencyofthedi sturbancesresultsinthe effectbeingaveragedoutbytheinertiaofthestore.Tofurt herstressthecontroller,it isnecessarytoconsideranon-stationaryrandomdisturban cewithalowfrequency component. Considerthenon-stationarysignalshowninFigure 5-28 .Thepitchingmoment incrementisacompositionoftheoriginalrandomturbulenc esignalplusadeterministic 208

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0 0.5 1 1.5 2 -6 -4 -2 0 2 Angle of Attack (deg)time (sec) 0 0.5 1 1.5 2 -60 -40 -20 0 20 40 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.5 1 1.5 2 -10 -5 0 5 Elevator Defl (deg)time (sec) 0 0.5 1 1.5 2 -6 -4 -2 0 2 4 Pitching Momenttime (sec) Figure5-27.Optimalandneighboringoptimaltrajectories withampliedrandom disturbancesrepresentativeofaerodynamicturbulence. effectduetoahypotheticalverticalwindgustgivenby f ( t )= Ae 3 t sin(2 t ) with A =1 Figure 5-29 showstheneighboringoptimalcontrollerperformanceinth epresence ofnon-stationaryrandomdisturbances,representativeof turbulentwindgusts.The trajectoriesweredeterminedusing f ( t )= Ae 3 t sin(2 t ) with A = 2, 1,1,2 ,to representarangeofsevereverticalgusts.Itshouldbenote dthatthedeterministic disturbanceisanalogoustoanunknownchangeinsystempara metersandthe neighboringoptimaltrajectorynolongerminimizestheori ginalcostfunction.Even so,thenear-optimalcontrollerperformssatisfactorilyi nthepresenceofnon-stationary randomdisturbances. Considerationofthemagnitudeofthedisturbanceiswarran tedtofurtherappreciate theresults.Thepitchingmomentsensitivitytoangleofatt ackforthisexampleisgivenin Table( 4-1 )as C m = 4.05 rad 1 .Thus,aunitchangeinpitchingmoment C m = 1 correspondstoachangeinangleofattack = 14 deg .Forthecurrentexample,the freestreamvelocityis 450 KCAS .Achangeinangleofattackby = 14 deg would 209

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0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 Time (sec)Pitching Moment Increment, D Cm Non-Stationary Turbulence f(t)=Ae -3t sin(2 p t) Figure5-28.Non-stationarysignalrepresentativeofatur bulentwindgusteffecton pitchingmoment. 0 0.5 1 1.5 2 -15 -10 -5 0 5 Angle of Attack (deg)time (sec) 0 0.5 1 1.5 2 -60 -40 -20 0 20 40 60 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.5 1 1.5 2 -10 -5 0 5 Elevator Defl (deg)time (sec) 0 0.5 1 1.5 2 -1 -0.5 0 0.5 1 1.5 2 Pitching Momenttime (sec) Figure5-29.Optimalandneighboringoptimaltrajectories withampliedturbulentwind gusts. 210

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requireaverticalwindgustofapproximately 110 KCAS .Consideringtheserough estimates,themagnitudeofaverticalgusttoproduce C m = 1 issubstantial.The maximumelevatordeectionof e = 10 deg representsapitchingmomentchangeof C m = 1.5 ,sotheverticalgustconsideredhereisonparwiththefullc ontrolauthority ofthestore.Evenso,thecontrolleradequatelycapturesth estoremotionandsafely guidesthestoreawayfromtheaircraft. Theabovediscussionisnotintendedtobeformalorcomprehe nsiveinnature. Rather,itisa“quicklook”attheperformancethatcanbeexp ectedbyNOCinthe presenceofnon-stationaryrandomdisturbances,suchasae rodynamicturbulence.The readeriscautionedthatthisisadramaticallysimpliedin vestigationandconclusions beyondthescopeofthisexamplearenotwarranted.5.4.6.2Responsetoparametervariations InSection 5.2.5 itwasshownthatNOCcanbeusedtoprovideoptimalfeedback controlinthepresenceofsmallconstantparametervariati ons,providedtheparameter variationisknownorcanbeestimated.However,whenthepar ametervariationis unknown,thecontrolisnecessarilysub-optimal. Theperformanceofthenear-optimalcontrollerfortheongo ingexampleis consideredhere.Theparameterthattypicallyhasthemosts ignicanteffectonthestore separationtrajectoryisthelongitudinalcenterofgravit y, x CG .Themasspropertiesof everyindividualstoreareunique,subjecttovariationswi thinmanufacturingtolerances. Therefore,itisimportanttoconsidervariationsinmasspr operties,especiallyin x CG ,to determineiftherangeofpossiblestoretrajectoriesissaf eandacceptable. Thelocationofthestorelongitudinalcenterofgravityrel ativetotheaerodynamic centerisknowntodeterminetheaerodynamicstability.Inf act,ifthe x CG istoofaraft, thestorewillbecomeinherentlyunstable.Fortheexamples tore,thisoccurswhenthe x CG isshiftedbyapproximatelyteninches, x CG =10 inches .Figure 5-30 showsaseries ofunguided(jettison)trajectoriesforthecenterofgravi ty 10 x CG 10 at 5 inch 211

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increments.Theuncontrolledtrajectorywith x CG = 10 inches isclearlyunstableand quicklydepartsfromstableight,asevidencedbytheexces siveangleofattack ( t ) andpitchrate q ( t ) 0 0.2 0.4 0.6 0.8 1 -200 -150 -100 -50 0 50 Angle of Attack (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -300 -200 -100 0 100 200 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 Elevator Defl (deg)time (sec) 0 0.2 0.4 0.6 0.8 1 -5 0 5 10 15 Pitching Momenttime (sec) Figure5-30.Unguidedtrajectorieswithparametervariati ons. Figure 5-31 showstrajectoriesusingthesamerangeofmasspropertiesw ithNOC, basedonthenominaltrajectorywith x CG =0 .Theresultsaresubstantiallydifferent. Evenforaninherentlyunstablestorewith x CG = 10 inches ,thenear-optimalcontroller drivesthestoretoatrimmedightconditionandsafelyguid esthestoreawayfromthe aircraft.Allveofthetrajectoriesconvergetoasimilart rimmedightconditionand exhibitsafeandacceptableseparationcharacteristics.T heseveguidedtrajectoriesare amarkedimprovementovereventhebestcaseunguidedtrajec tories. Furtherimprovementoftheseparationcharacteristics(e. g.areductioninthe totalcost)couldbeattainediftheconstantparameterisme asuredaprioriorifthe 212

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0 0.5 1 1.5 2 -15 -10 -5 0 5 Angle of Attack (deg)time (sec) 0 0.5 1 1.5 2 -60 -40 -20 0 20 40 Pitch Rate (deg/sec)time (sec) Optimal Neighboring Optimal 0 0.5 1 1.5 2 -10 -8 -6 -4 -2 0 2 Elevator Defl (deg)time (sec) 0 0.5 1 1.5 2 -0.5 0 0.5 1 1.5 2 Pitching Momenttime (sec) Figure5-31.Guidedtrajectorieswithparametervariation s. parameterisestimatedduringight.Thisisavalidapproac hforstoreseparation. However,thenumberofunknownparametersforarealisticsy stemmaybeexceedingly largeanddifculttoestimateovertheshortdurationofint erest.Furthermore,manyof theidentiedparametersmaybetime-varying,furthercomp licatingtheidentication procedure.Asaresult,theneighboringoptimalcontroller isconsidered“nearoptimal”in thepresenceofsmallunknownparametervariations.Howeve r,theaboveexample indicatesthatevenforlargeparametervariationswithoff -nominalperformance, theneighboringoptimalcontrollermayperformadequately ,resultinginstableand convergentbehaviorevenforaninherentlyunstablesystem .Thiscursorylookatthe performanceofNOCinthepresenceofsignicantparameterv ariationsispromising, butagainthereaderiscautionedthatthisisadramatically simpliedinvestigationand conclusionsbeyondthescopeofthisexamplearenotwarrant ed.Furtherconsideration forarealisticcasestudywithparametervariationswillbe takenupinChapter6. 213

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5.5ChapterSummary Theapplicationofoptimalcontroltheorytostoreseparati onprovidesaframework fordeterminingabestcaseguidedtrajectoryforastoretra versinganonlinearspatially variantoweld.However,theextremalsolutionisdepende ntonthespecied parametersandinitialconditions,andachangeinparamete rsorinitialconditions requiresthecomputationofanewextremalsolution.Thecom putationalburden prohibitstheimplementationofareal-timeoptimalcontro ller,renderinganopen-loop controlstrategy.Inordertorespondtotheinevitablevari ationsininitialconditionsand systemparameters,aswellasdisturbancesalongtheoptima lpath,feedbackcontrolis necessary.Closed-loopfullstatefeedbackcontrolcanbea ccomplishedreadilyusing neighboringoptimalcontrol. Neighboringoptimalcontrolconsidersadynamicmodelline arizedalongthe optimalpathandcanbeusedtominimizetheoriginalcostfun ctionaltosecondorder. TheNOCproblemisinitiallyconstructedtoaccommodateper turbationsininitial conditions.However,sinceanytimealongtheextremaltraj ectoryisavalidstarttime, theneighboringoptimalcontrollernaturallyaccountsfor disturbancesalongtheoptimal path.NOCcanalsobeextendedtoprovidelinear-optimalcon trolforsystemswith perturbedconstantparameters,providedtheparametersar eknownaprioriorcan beestimatedduringoperation.Forsystemswithunknownpar ametervariations,NOC affordsnear-optimalcontrol,providedtheparametervari ationsaresmall.NOCcan alsobeextendedtoincludeproblemswithpath/controlcons traintsandterminalcostor constraints.Finally,NOCiscommensuratewiththesolutio ntotheaccessoryminimum problem(AMP),andtheexistenceofaneighboringoptimalso lutionprovidesasufcient conditionfordemonstratingtheoptimalityofacandidatee xtremalsolution.Thus,NOC isacomprehensiveframeworkfordeterminingalineartimev ariantfeedbackcontrollaw whenanominalextremaltrajectoryisknown. 214

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Neighboringoptimalcontrolisespeciallywellsuitedfors toreseparation.The transitionalnonlinearitiesduetothespatiallyvariant oweldsurroundingthe aircraftbecomenegligibleasthedistancebetweenthestor eandaircraftbecomes large.Judiciousselectionoftheboundaryconditionsandc ostfunctionalleadsto alinear-optimalcontrollerthatconvergestoatimeinvari antcontrollerinfareld conditions,referredtoasinnitehorizonneighboringopt imalcontrol(IHNOC).IHNOCis ideallysuitedforstoreseparationandprovidesalinearti mevariantfeedbackcontroller thatguidesthestoreawayfromtheaircraftandconvergesto atimeinvariantlinear quadraticregulator.IHNOCactsasatransitionalautopilo tandmaybeusedtohold thestoreinatrimmedightconditionindenitely,e.g.ove raninnitehorizon.The resultisacompacteasily-implementedcontrollerthatsig nicantlyimprovesseparation characteristicsincomparisontoanunguidedtrajectory. Therobustnessofthecontrollerwasinformallydemonstrat edbyapplicationto avarietyofchallengingscenariosincludingextremeiniti alconditions,non-stationary randomperturbationsrepresentativeofaerodynamicturbu lence,andlargeparameter variations.Forlargedeviationsfromthenominaltrajecto ry,theIHNOCtrajectoryis substantiallydifferentfromthetrueoptimaltrajectory. However,theneighboringoptimal controllerperformedadequatelyundereventhemostextrem ecasesconsidered, includingstabilizationofanotherwiseunstablestore.Th isperformanceanalysiswasfar fromcomprehensive,buttheresultsareneverthelesspromi singandfurtherinvestigation iswarranted. Alloftheexamplesconsideredinthischapterwererestrict edtoastoreconned totheverticalplaneduringseparation.Asaresult,theequ ationsofmotionand aerodynamicmodelwereconsiderablysimplied.However,t hetheorydeveloped hereinisequallyapplicabletoafullnonlinearsixdegreeo ffreedomproblemwithan arbitrarilycomplexaerodynamicmodel,providedtheaerod ynamicandcontrolgradients canbeestimated. 215

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CHAPTER6 GUIDEDSTORESEPARATION 6.1Overview Theprimaryobjectiveofthisstudyistodevelopacomprehen siveapproachto improvetheseparationcharacteristicsofmodernejectorlaunchedguidedmunitions byutilizingaseparationautopilottoguidethestorealong apreferredtrajectory.This investigationisintendedtoshowthesignicantincreasei nsafetyandacceptability thatcanbeachievedthroughguidedstoreseparationwithmi nimaladditionincostand complexityoftheguidanceandcontrolsystem. Developmentofaguidanceandcontrolsystemforstoresepar ationincludes (1)identicationofaparametricmodelforthespatiallyva riantaerodynamics,(2) determinationofa“bestcase”trajectorythatmeetssafety andacceptabilitycriteria, and(3)designofaneffectivefeedbackcontrollertoaccoun tformodeluncertaintiesand operatingdisturbances.Inthisresearch,theparametricm odelisidentiedusingsystem identication,the“bestcase”trajectoryisdeterminedus ingoptimalcontroltheory,and thefeedbackcontrolsystemisdesignedusingneighboringo ptimalcontrol;seeFigure 6-1 .Theresultisacompactstoreseparationautopilotthatexp licitlytakesintoaccount thespatiallyvariantaerodynamicsandleveragestheaerod ynamicinteractionbetween theaircraftandstoretodramaticallyimproveseparationc haracteristics. System Identification Trajectory Optimization Feedback Control Guided Store Separation Figure6-1.Relationshipbetweensystemidentication,tr ajectoryoptimization,and feedbackcontroltoappliedforguidedstoreseparation. Systemidentication,trajectoryoptimizationandfeedba ckcontrolhavebeen examinedindetailinChapters3,4,and5.Ineachchapter,th etheorywasdeveloped 216

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ingeneralandappliedtostoreseparationinparticular,fo llowedbynumericalexamples toclearlyillustratetheapplication.Forinstructivecla rity,theoptimalcontrolexamples werelimitedtoconsiderationofstoremotionconnedtoave rticalplane,resultingin asimpliedthreedegree-of-freedom(3DOF)system.Inthis chapter,thesametheory isappliedtoafullnonlinearsixdegree-of-freedom(6DOF) storeseparationanalysis. Inthesamewaythatearlierchapterswereprimarilyfocused ontheory,thischapteris primarilyfocusedonapplication.Asaresult,mathematica ldevelopmentissparseand quantitativeresultsareprevalent. Theintentofthischapteristoshowthegeneralapplicabili tyandreadinessofthe theorydevelopedinpreviouschapters.Section 6.2.1 illustratestheapplicationofoptimal controltheorytodetermineapreferredseparationtraject orythatoptimizessafety andacceptabilitycriteriausingaquadraticcostfunction .Section 6.2.2 illustratesthe applicationofNeighboringOptimalControl(NOC)todesign alinear-optimalfeedback controllerforstoreseparation.Inparticular,InniteHo rizonNeighboringOptimal Control(IHNOC)isusedtodesignacontrolsystemthataccou ntsforthespatially variantaerodynamicsneartheaircraftandconvergestoati me-invariantlinearquadratic regulatorinfareldconditions.Usingthisapproach,thes toreseparationautopilotmay beusedtosafelyandeffectivelytransferthestorefromrel easetostabletrimmedight inanoptimalmanner.Theseresultsrelycompletelyonthepa rametricmodeldeveloped inChapter 3 ;seeSection 3.3 .InSection 6.3 ,theIHNOCcontrollerdeterminedusing theparametricmodelisappliedtoafullnonlinearconventi onal6DOFsimulation representativeofighttest.Thestoreseparationautopil otistestedforawiderangeof initialconditionsandphysicalparameterswithpromising results. 6.2TrajectoryOptimization Trajectoryoptimizationistheprocessofdeterminingcont rolandstatehistoriesfor adynamicsysteminordertominimize(ormaximize)acostfun ction(ormeasureof performance)whilesatisfyingprescribedboundarycondit ionsand/orpathconstraints 217

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[ 87 ].Thedynamicsystemisgenerallymodeledinthetimedomain usingastatespace representation.Themeasureofperformancerepresentsame tricorcombinationof metrics(e.g.time,energy,controleffort,deviationfrom adesiredoperatingcondition, etc.)thatquantifythedesiredperformanceofthesystem.T heboundaryconditions includelimitationsontheinitialand/ornalstateofthed ynamicsystem,aswellaslimits onthecontrol(e.g.actuatorlimits,controlsaturation). Path(orstate)constraintsare usedtoexcludetrajectoriesthatviolateapredeterminedr angeortypeofundesirable motion.Eachofthesecomponentsarestatedwithmathematic alprecisionand combinedtocreateanoptimalcontrolproblem.6.2.1OptimalControl Theobjectiveofanoptimalcontrolproblemistodeterminea nadmissiblecontrol inputthatminimizes(ormaximizes)thedesiredcostfuncti on(orperformanceindex) subjecttothespeciedboundaryconditionsanddynamiccon straints.Thesolution usinganindirectmethodisbasedonthecalculusofvariatio ns.Ordinarycalculusis predominantlyconcernedwiththecalculusoffunctions,ch aracterizedbythedifferential operator.Comparatively,thecalculusofvariationsiscon cernedwiththecalculusof functionals,characterizedbythevariationaloperator.T hecalculusofvariationscanbe usedtoderivetheclassical1 st orderoptimalityconditions,asshowninSection 4.2.1 Theoptimalcontrolproblemstatementandoptimalitycondi tionsareincludedherefor completeness.6.2.1.1Problemstatement TheobjectiveofanoptimalcontrolprobleminBolzaformist ominimizeacost functionalgivenbyEquation( 6–1 ),subjecttodynamicconstraintsgivenbyEquation ( 6–2 ),pathandcontrolinequalityconstraintsgivenby( 6–3 ),andterminalconstraints givenbyEquation( 6–4 ).Forbrevity,itisassumedthattheinitialconditions, x ( t 0 ) ,and naltime, t f ,arespecied. 218

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J = x ( t f ) + t f Z t 0 L x ( t ), u ( t ) dt (6–1) _x ( t )= f x ( t ), u ( t ) (6–2) C x ( t ), u ( t ) 0 (6–3) x ( t f ) =0 (6–4) Applicationofoptimalcontroltostoreseparationisstrai ghtforward.Aquadratic costfunctional,givenbyEquation( 6–5 )issufcientforthisinvestigation,where Q isa constantpositivesemi-denitematrix Q 0 and R isaconstantpositivedenitematrix R > 0 J = 1 2 x ( t f ) T S f x ( t f )+ 1 2 t f Z t 0 x T Q x + u T R u dt (6–5) Theweightingmatrices Q and R arechosenbytheusertoinuencethemagnitude ofthestateandcontrolvector,respectively.Thescalarpa rameters Q Q Q q ,etc. aretheelementsof Q alongthediagonalandmaybeusedtoinuencetheparticular statevariableofinterest.Thesameistrueforthediagonal componentsof R : R a R e R r ,whichinuencetheaileron,elevator,andrudderinputsre spectively.Non-diagnonal termsfor Q and R arenotconsideredinthisstudy.Finally,thematrix S f 0 isspecied bytheusertoachievesatisfactoryterminalconditions. Usingthequadraticcostfunctional,the1 st orderoptimalityconditionswithout terminalconstraintsarestatedinEquations( 6–6 )through( 6–8 ),wherethesubscript notationimpliespartialdifferentiation(e.g. f x = @ f =@ x ). x ( t )= f ( x ( t ), u ( t ) ) x ( t 0 ) specied(6–6) ( t )= Q x ( t ) f T x ( t ) ( t ), ( t f )= S f x ( t f ) (6–7) u ( t )= R 1 f T u ( t ) ( t ) (6–8) 219

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Theseequationsrepresentaboundaryvalueproblemthatmus tbesolved numerically,providedthesystem x = f ( x u ) canbedescribedanalytically.Forstore separation,theequationsofmotionaredocumentedinChapt er 2 andtheaerodynamic modelisbasedontheparametricmodeldevelopedinChapter 3 .Explicitderivationof the1 st orderoptimalityequationsforthefull6DOFstoreseparati onprobleminvolves alotofalgebraicmanipulation,andisbestaccomplishedus ingcomputersoftware, suchastheMatlabSymbolicMathToolbox R r .Theresultingequationsarelengthy, buteasilybuiltintoasubroutineforusewithanumericalBV Psolver.Thenumerical solutionspresentedherearebasedontheMatlab R r program bvp4c ,whichimplements athree-stageLobattoIllacollocationformula.Furtherdi scussionofthenumerical methodsusedinthisstudyareprovidedinSection 4.2.5 6.2.1.2Optimaltrajectory Figure 6-2 showsanoptimaltrajectoryatasubsonicightcondition.T height conditionsarespeciedtobeconsistentwithstoreseparat ionighttest2265(Mach 0.9/550KCAS/4800ft).Theinitialconditions, x ( t =0) ,aretakenfromtheight testinertialmeasurementsatend-of-strokeandaretheref oreconsistentwithanactual ighttestevent.Theuserspeciedweightingmatriceswere chosentominimizethe magnitudeoftheangularrates.Inparticular, Q = Q =0 Q p = Q q = Q r =10 and R a = R e = R r =50 .Emphasisisonreducingtheangularrates p q and r without unduecontroleffort.Theterminalcostiszero, S f =0 TheresultsshowninFigure 6-2 indicatethattheoptimalcontrolquicklycaptures theangularratesanddrivesthestoretoastationaryattitu de.Notethatthespecic valuesoftheEulerangles arenotofparticularconcern;onlytherates p q r are includedinthecostfunction.Alsonotethattheangleofatt ackcontinuestoincrease steadily.Sincethepitchattitude isheldconstantasaresultofminimizing j q j ,the angleofattackiscontinuallyincreasedasthestorevertic alvelocityincreasesduetothe accelerationofgravity. 220

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0 0.5 1 -10 0 10 20 30 Angular Velocity (deg/sec)Time (sec) r(t) q(t) p(t) 0 0.5 1 -10 -5 0 5 10 Control DeflectionsTime (sec) d r d e d a 0 0.5 1 -4 -2 0 2 Euler Angles yaw(t) pitch(t) roll(t) 0 0.5 1 -2 -1.5 -1 -0.5 0 Incidence Angles b s (t) a s (t) Figure6-2.Optimaltrajectoryforratecapture.Initialco nditionsandightconditionsare basedonighttest2265(Mach0.9/550KCAS/4800ft). Figure 6-3 showsanoptimaltrajectorywithanemphasisonangleofatta ck andtheangleofsideslip .Theightconditionsandinitialconditionsareidentical tothoseshowninFigure 6-2 .Inthiscase,thecostwaschosentominimize and withoutexcessivecontroleffort.Inparticular, Q = Q =100 Q q = Q r =0 and R a = R e = R r =50 .Theterminalcostisspeciedas S f = S f r =10 to achieveadesirableendpoint.Thepitchandyawratesaredam pedduetothekinematic correlationwith and ,buttherollratecouplingisminimalforthisaxisymmetric store andtherollratemustbedampedexplicitly.Choosingacostw ith Q p =10 provides effectiverollratedamping. TheresultsinFigure 6-3 showanimprovementintheangleofattackandangleof sideslipprolewhilestillprovidingexcellentratecaptu reperformance.Thespecied costrequiresthattheincidenceangles and arekeptneartheorigin,which necessarilyimplieslowangularrates.Thus,theangleofat tackcaptureexplicitly 221

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0 0.2 0.4 0.6 0.8 1 -10 0 10 20 30 40 Angular Velocity (deg/sec)Time (sec) r(t) q(t) p(t) 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 10 Control DeflectionsTime (sec) d r d e d a 0 0.2 0.4 0.6 0.8 1 -3 -2 -1 0 1 2 Euler Angles yaw(t) pitch(t) roll(t) 0 0.2 0.4 0.6 0.8 1 -2 -1.5 -1 -0.5 0 0.5 Incidence Angles b s (t) a s (t) Figure6-3.Optimaltrajectoryforangle-of-attackcaptur e.Initialconditionsandight conditionsarebasedonighttest2265(Mach0.9/550KCAS/4 800ft). attenuatestheincidenceanglesandimplicitlyprovidesra tecaptureaswell.Sinceangle ofattackhasasignicantroleinassessmentofsafetyandac ceptability,theangleof attackcaptureispreferredoverpureratecapture.Alsonot ethatsincetheangleof attackisheldconstant,thepitchanglemustcontinuallyde creasetocompensatefor increasingverticalvelocity.Assuch,thesteadystatepit chrateisslightlynegativeand thestorecontinuestogently“noseover”intothewind. SolutiontothetwopointHBVPrequiresspecicationofthei nitialconditionsforall twelvestatevariables.Intheguresabove,theinitialcon ditionsweredeterminedusing ighttesttelemetrydata.However,foranactualighttest theinitialconditionsarenot knowninadvanceandmaybevariedfromonemissiontothenext .Determinationofthe optimaltrajectoryrequiresanewsolutionforeachsetofin itialconditions,alimitation thatisaddressedusingneighboringoptimalcontrolinthen extsection.Figure 6-4 showsaseriesofoptimaltrajectoriesforarangeofinitial pitchrates.Thepitchrateis 222

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variedbetween q (0)= 7 50 deg = sec .Theuserspeciedcostfunctionisselectedto minimizethetotalangleofattackusingthesameweightingv aluesasFigure 6-3 0 0.2 0.4 0.6 0.8 1 -5 0 5 10 Yaw Rate, R BI (deg/sec)Time (sec) 0 0.2 0.4 0.6 0.8 1 -60 -40 -20 0 20 40 60 Pitch Rate, Q BI (deg/sec)Time (sec) 0 0.2 0.4 0.6 0.8 1 -0.6 -0.4 -0.2 0 0.2 Angle of Sideslip, b s (deg) Optimal Trajectory 0 0.2 0.4 0.6 0.8 1 -6 -4 -2 0 2 Angle of Attack, a s (deg) Figure6-4.Optimaltrajectoriesforvaryinginitialpitch rate.Flightconditionsarebased onighttest2265(Mach0.9/550KCAS/4800ft). TheresultsinFigure 6-4 indicatethattheoptimalcontroladequatelyaccountsfor thevariationininitialpitchrateanddrivesthestoretoas tabletrimmedightcondition inlessthan1second.Notethatthevariabilityintheangleo fsideslip andtheyawrate r areminimal,indicatingonlyaweakaerodynamicandkinemat iccouplingbetweenthe longitudinalandlateralvariables.Thisistobeexpectedg iventheaxisymmetricnature ofthestore. Figure 6-5 showsasimilarseriesofoptimaltrajectoriesforvaryingi nitialyawrates, r (0)=8 50 deg = sec .Theoptimalcontrolproducesafamilyoftrajectoriesthat converge rapidlytoabenignightconditionusingacceptablelevels ofcontrol.Again,thelimited variabilityoftheangleofattack andpitchrate q isanindicatorofweakcoupling.It shouldbenotedthatthisdecouplingeffectisaconvenientr esultbutnotanecessary 223

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assumption.Thesameapproachusingoptimalcontrolwillal soworkforastorewith strongaerodynamicandkinematiccoupling. 0 0.2 0.4 0.6 0.8 1 -60 -40 -20 0 20 40 60 Yaw Rate, R BI (deg/sec)Time (sec) 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 4 Pitch Rate, Q BI (deg/sec)Time (sec) 0 0.2 0.4 0.6 0.8 1 -3 -2 -1 0 1 2 Angle of Sideslip, b s (deg) Optimal Trajectory 0 0.2 0.4 0.6 0.8 1 -2 -1.5 -1 -0.5 0 Angle of Attack, a s (deg) Figure6-5.Optimaltrajectoriesforvaryinginitialyawra te.Flightconditionsarebased onighttest2265(Mach0.9/550KCAS/4800ft). Alloftheprecedingplotsinthissectionarebasedonsubson icightconditions consistentwithstoreseparationighttest2265(Mach0.9/ 550KCAS/4800ft).Figure 6-6 showsanoptimaltrajectoryforighttest4535(Mach1.2/60 0KCAS/18kft). Thesupersonicoweldresultsinamuchlargernose-downae rodynamicpitching momentnearcarriage.Theoptimalcontrolusesmaximumcont rolauthority e = 10 degtoarrestthepitchrateandangleofattack.Thestronger oweldresultsinhigher deviationsinpitchratethroughoutthetrajectory.Evenin theseadverseconditionsthe optimalcontrolsuccessfullybringsthestoretoastabletr immedightconditionwithina 1secondtimeinterval.Figure 6-7 showsaseriesofoptimaltrajectoriesforarangeof initialpitchrates q (0)= 22 50 deg/secandinitialyawrates r (0)=21 50 deg/sec. 224

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Theresultsprovidefurtherevidencefortheeffectiveness oftheoptimalcontrolprogram overarangeofinitialconditions. 0 0.2 0.4 0.6 0.8 1 -40 -20 0 20 40 Angular Velocity (deg/sec)Time (sec) r(t) q(t) p(t) 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 10 Control DeflectionsTime (sec) d r d e d a 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 Euler Angles yaw(t) pitch(t) roll(t) 0 0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 2 Incidence Angles b s (t) a s (t) Figure6-6.Optimaltrajectoryforangle-of-attackcaptur e.Initialconditionsandight conditionsarebasedonFlightTestMission4535(Mach1.2/6 00KCAS/ 18kft). TheresultsshowninFigures 6-2 through 6-7 representasignicantimprovement overtheactualghttestseparationcharacteristics.Forc omparison,Figure 6-8A showsthereconstructedighttesttrajectoryformission2 265(Mach0.9/550KCAS/ 4800ft)withtheoptimaltrajectorydeterminedusingthesa meightconditions,initial conditions,andmassproperties.Figure 6-8B showsasimilarcomparisonformission 4535(Mach1.2/600KCAS/18kft).Inbothcases,thecontroll edseparationisa dramaticimprovementovertheighttesttrajectoryinterm sofacceptabilitymargins. Formission4535,themaximumangleofattackisreducedfrom max = 20 degto max = 5 deg.Themaximumpitchrateisreducedfrom q max =130 deg/secto q max =28 deg/sec.Comparisonoftheoptimaltrajectoriesbetweeni ghtconditions isalsovaluable.Whereastheighttesttrajectoriesaredr amaticallydifferentbetween 225

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0 0.2 0.4 0.6 0.8 1 -40 -20 0 20 40 60 80 Yaw Rate, R BI (deg/sec)Time (sec) 0 0.2 0.4 0.6 0.8 1 -100 -50 0 50 100 Pitch Rate, Q BI (deg/sec)Time (sec) 0 0.2 0.4 0.6 0.8 1 -5 -4 -3 -2 -1 0 1 Angle of Sideslip, b s (deg) Optimal Trajectory 0 0.2 0.4 0.6 0.8 1 -15 -10 -5 0 5 Angle of Attack, a s (deg) Figure6-7.Optimaltrajectoriesforvaryinginitialpitch andyawrate.Flightconditions arebasedonFlightTestMission4535(Mach1.2/600KCAS/18k ft). subsonicandsupersonicightconditions,theoptimaltraj ectoriesareverysimilar.The optimalcontrolprogramnotonlyprovidesameasurableimpr ovementinsafetyand acceptability,butitalsoreducesthevariabilityintraje ctorycharacteristicsbetweenight conditions.Theuniformitybetweenightconditionsisana dvantageforensuringsafe andacceptableemploymentacrosstheightenvelope.6.2.2FeedbackControl TherstorderconditionsdiscussedinSection 4.2.1 providethenecessary conditionsforanoptimaltrajectory.Thecontrolisdeterm inedbasedonsolutionof thetwopointHamiltonianboundaryvalueproblemandimplic itlyassumesperfect knowledgeofthesystemoperatinginadisturbance-freeenv ironment.However, deterministicdisturbancesorvariationsintheinitialco nditions,terminalconditions,and systemparametersaltertheoptimalstateandcontrolhisto ry,requiringcomputationof uniquesolutionforeachvariation.Statedanotherway,the optimalcontrolstrategy 226

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0 0.2 0.4 0.6 0.8 1 -20 -15 -10 -5 0 5 10 15 Incidence Angles 0 0.2 0.4 0.6 0.8 1 -50 -40 -30 -20 -10 0 10 Euler Angles roll(t) pitch(t) yaw(t) 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 Control DeflectionsTime (sec) d a d e d r 0 0.2 0.4 0.6 0.8 1 -150 -100 -50 0 50 100 150 Angular Velocity (deg/sec)Time (sec) p(t) q(t) r(t) a s (t) b s (t) Optimal Trajectory Flight Test AMach0.90/550KCAS/4800ft 0 0.2 0.4 0.6 0.8 1 -30 -20 -10 0 10 20 Incidence Angles 0 0.2 0.4 0.6 0.8 1 -30 -20 -10 0 10 20 Euler Angles roll(t) pitch(t) yaw(t) 0 0.2 0.4 0.6 0.8 1 -10 -5 0 5 10 Control DeflectionsTime (sec) d a d e d r 0 0.2 0.4 0.6 0.8 1 -150 -100 -50 0 50 100 150 Angular Velocity (deg/sec)Time (sec) p(t) q(t) r(t) a s (t) b s (t) Optimal Trajectory Flight Test BMach1.20/600KCAS/18000ft Figure6-8.Comparisonofoptimal(guided)andighttest(u nguided)trajectoriesfor subsonicandsupersonicightconditions. 227

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is“open-loop”,meaningthecontrolisspeciedaprioriand xedregardlessof perturbationsthatmayaffectthesystemduringoperation. Incontrast,a“closed-loop” controllawismoredesirableasitaccountsforvariationsi ninitialconditionsand disturbancesalongtheoptimalpath.Neighboringoptimalc ontrol(NOC)provides apowerfulapproachforimplementingfeedbackcontrolalon ganoptimalpathby consideringlinearperturbationsalongtheoptimalsoluti on.NOCreliesonalocally linearizeddynamicmodelincombinationwithaquadraticco stfunctionalderivedfrom thesecondvariationoftheoriginalcostfunctional.Thene ighboringoptimalsolutionis thenapproximatedasthesumoftheoriginaloptimaltraject oryplusthelinear-optimal solution[ 116 ]. 6.2.2.1Problemstatement The1 st ordernecessaryconditionsforaneighboringextremalwith outterminal constraintsaregivenbyEquations( 6–9 )through( 6–12 ),where H = L + T f isthe Hamiltonian. _x ( t )= f x x + f u u (6–9) ( t )= H xx x f T x H xu u (6–10) 0= H ux x + f T u + H uu u (6–11) T ( t f )= [ xx x ] t = t f (6–12) Thevariations x u ,and aredenedasperturbationsalongtheoptimal trajectory, ( x ( t ), ( t ), u ( t ) ) .Inparticular, x = x ( t ) x ( t ) u = u ( t ) u ( t ) ,and = ( t ) ( t ) arethevariationsinthestate,control,andcostaterespec tively. Equation( 6–11 )canbeusedtosolveforthecontrolvariationexplicitlypr ovided thatthematrixinverse H 1 uu exists.Thecostate canbeeliminatedfromtheproblem usingadifferentialRiccatiequation.TheresultisaNeigh boringOptimalFeedback Lawthatcanbeusedtocorrectforvaryinginitialcondition sordisturbancesalong theextremalpath.Thefeedbacklawwithtimevaryingcontro lgains K ( t ) isgivenby 228

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Equations( 6–13 )and( 6–14 ),where S ( t ) isthenumericalsolutiontothedifferential Riccatiequation.Theneighboringoptimalcontrolstructu reisshowngraphicallyin Figure 6-9 u ( t )= u ( t ) K ( t ) ( x ( t ) x ( t ) ) (6–13) K ( t )= H 1 uu H ux + f T u S (6–14) Nominal Opt Trajectory x 0 ( t ), u 0 ( t ) NOC Gains Dynamic System x ( t ), u ( t ) K ( t ) u 0 ( t ) x 0 ( t ) D x ( t ) x ( t ) D u ( t ) u ( t ) + + + Figure6-9.NeighboringOptimalControlblockdiagram. Theneighboringoptimalcontrollawcanbeusedtominimizet heoriginalcost functiontosecondorderinthepresenceofvaryinginitialc onditionsanddisturbances alongtheoptimalpath.TheNOCcontrollerprovidesnear-op timalcorrectionsfor unknownvariationsinsystemparameters,providedthechan geinparametersissmall. Therefore,NOCprovidesacompactandviablesolutionforim plementingreal-time optimalcontrolinarealisticenvironment. Neighboringoptimalcontrolisespeciallywellsuitedfors toreseparation.Innite HorizonNOC(IHNOC)providesacontrolstructurethataccou ntsforthespatially variantaerodynamicsneartheaircraftandconvergessmoot hlytoatimeinvariant controllerinfareldconditions.ApplicationofIHNOCtos toreseparationforvarying initialconditions,randomdisturbances,andparameterva riationsarediscussedin Sections 6.2.2.2 through 6.2.2.5 6.2.2.2Neighboringoptimaltrajectory Figure 6-10 showsoptimalandandneighboringoptimaltrajectoriesfor ight conditionsconsistentwithmission2265(Mach0.9/550KCAS /4800ft).Thecostwas 229

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chosentominimizetheincidenceangleswithoutexcessivec ontroleffort.Inparticular, Q = Q =100 Q q = Q r =0 Q p =10 and R a = R e = R r =50 .Also,theend pointcostfortheHBVPwasspeciedas S f = S f r = S f p =10 tokeeptheendpoint nearatrimmedightcondition.TheendpointoftheHBVPbeco mesthesetpointforthe neighboringoptimalcontroller.TheendpointcostfortheN OCproblemwasspeciedas thesolutiontothealgebraicRiccatiequation,asdescribe dinSection 5.3.2 .Asaresult, thetimevaryingneighboringoptimalcontrollerconverges smoothlytoatimeinvariant nonzerosetpointregulator.TheIHNOCpolicywillcontinue toholdthestoreneara trimmedightconditionuntiltransitiontoamissionautop ilotiscomplete. 0 0.5 1 1.5 2 -10 0 10 20 30 40 Angular Velocity (deg/sec)Time (sec) r(t) q(t) p(t) 0 0.5 1 1.5 2 -10 -5 0 5 10 Control DeflectionsTime (sec) d r d e d a 0 0.5 1 1.5 2 -6 -4 -2 0 2 Euler Angles yaw(t) pitch(t) roll(t) 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 Incidence Angles b s (t) a s (t) Optimal Neighboring Optimal Figure6-10.Optimaltrajectoryandextendedneighboringo ptimaltrajectoryformission 2265(Mach0.9/550KCAS/4800ft). Figure 6-11 showstheRiccatisolution S ( t ) andcontrolgainsfortheneighboring optimalcontroller K ( t )= R 1 f T u S ( t ) .Thetimevariationofthegainsfor t < 0.8 are aresultofthespatiallyvariantaerodynamicsofthestoret raversingthenonuniformow eld.Thegainsrapidlyconvergetoconstantvaluesbeyond t =0.8 (about23ftbelow theaircraft),indicatingtheaircraftoweldeffectisno longersignicant. 230

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -200 -100 0 100 200 Riccati Solution, S(t) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2 -1 0 1 2 Time (sec)Control Gains, K(t) Figure6-11.TimevaryingfeedbackcontrolgainsandRiccat isolutionforMission2265 (Mach0.9/550KCAS/4800ft). Figure 6-12 showssimilartrajectoriesforsupersonicightcondition sconsistentwith ighttest4535(Mach1.2/600KCAS/18kft).Again,thecostw aschosentominimize thetotalaerodynamicangleofattack,whichindirectlypro videsaratecaptureeffectas well.Inparticular, Q = Q =200 ;allotherweightingfactorsareunchangedfromthe previousexample.Thesupersonicoweldsurroundingthea ircraftinducesastrong nose-downpitchingmomentonthestore,resultinginanegat iveangleofattackand pitchangle,despitethemaximumelevatordeectionfor t < 0.20 sec.However,the controlquicklyovercomestheadverseoweldconditionan ddrivesthestoretoa trimmedightconditionwithina1sectimeinterval.Thenei ghboringoptimalcontroller emulatestheoptimaltrajectoryandmaybeusedtomaintaint hestorenearthesetpoint indenitely. TheRiccatisolution S ( t ) andcontrolgains K ( t ) formission4535areshownin Figure 6-13 .Again,thetimevariationofthegainsnearcarriageisevid ent,followed byconvergencetostationaryvalues.However,incontrastt oFigure 6-11 ,thegains 231

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0 0.5 1 1.5 2 -40 -20 0 20 40 Angular Velocity (deg/sec)Time (sec) r(t) q(t) p(t) 0 0.5 1 1.5 2 -10 -5 0 5 10 Control DeflectionsTime (sec) d r d e d a 0 0.5 1 1.5 2 -8 -6 -4 -2 0 2 Euler Angles yaw(t) pitch(t) roll(t) 0 0.5 1 1.5 2 -8 -6 -4 -2 0 2 Incidence Angles b s (t) a s (t) Optimal Neighboring Optimal Figure6-12.Optimaltrajectoryandextendedneighboringo ptimaltrajectoryformission 4535(Mach1.2/600KCAS/18kft). K ( t ) arenotmonotonicallydecreasing.Thesupersonicoweldi sconsiderablymore complexthanthesubsonicequivalent,resultinginamoredy namiccontrolschedule. Thegainsconvergetoapproximatelyconstantvaluesatabou t t =0.9 sec,whichfor thisexamplecorrespondstoabout30ftbelowtheaircraft.T heIHNOCstrategyadapts welltothemorerigorousoweldandprovidesaframeworkth atworkssimilarlyatboth adverseandbenignightconditions.6.2.2.3Responsetovaryinginitialconditions InordertosolvethenonlinearHBVPdiscussedinSection 6.2.1.2 ,theinitial conditionsmustbespeciedapriori.Thisasignicantlimi tationforimplementing optimalcontrolforstoreseparation,sincetheinitialcon ditionswillingeneralbedifferent foreachrelease.However,iftheHBVPissolvedforanominal setofinitialconditions, neighboringoptimalcontrolcanbeusedtoprovidelinear-o ptimalcorrectionforvarying initialconditions,providedtheperturbedinitialcondit ionsarenottoofarfromthe 232

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -2000 -1000 0 1000 2000 Riccati Solution, S(t) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -6 -4 -2 0 2 4 Time (sec)Control Gains, K(t) Figure6-13.TimevaryingfeedbackcontrolgainsandRiccat isolutionformission4535 (Mach1.2/600KCAS/18kft). nominalvalues.Forthesimulationsconsideredinthisstud y,NOCperformsquitewell evenwhentheperturbationsarelarge;seeSection 5.4.2 Figure 6-14 showsanoptimaltrajectoryandfamilyofneighboringoptim al trajectorieswithvaryinginitialconditionsforsubsonic ightconditionsconsistent withmission2265(Mach0.9/550KCAS/4800ft).Theneighbor ingoptimalsolutions weredeterminedusingafeedbackcontrolsimulationwithco ntrolgainsshownin 6-11 Theinitialconditionsincludevariationsinpitchrate q (0)= 7 50 deg/secandyaw rate r (0)=8 50 deg/sec.Thenominalinitialconditionsarebasedonrecons tructed ighttestdatafrommission2265.Thecostfunctionisthesa meastheexampleshown inSection 6.2.2.2 TheresultsshowninFigure 6-14 indicatethattheneighboringoptimalcontroller adequatelycapturestheangularratesandincidenceangles ,despitethelargevariation ininitialpitchandyawrates.Allvetrajectoriesconverg etothesametrimmedight 233

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0 0.5 1 1.5 -6 -4 -2 0 2 Angle of Attack, a s (deg) 0 0.5 1 1.5 -3 -2 -1 0 1 2 Angle of Sideslip, b s (deg) Optimal Trajectory Neighboring Optimal 0 0.5 1 1.5 -100 -50 0 50 Pitch Rate, Q BI (deg/sec)Time (sec) 0 0.5 1 1.5 -50 0 50 100 Yaw Rate, R BI (deg/sec)Time (sec) Figure6-14.Optimalandneighboringoptimaltrajectories withvaryinginitialratesfor mission2265(Mach0.9/550KCAS/4800ft). condition,afeaturethatisclearlyanadvantageforensuri ngsafeandacceptable separationinthepresenceofinitialperturbations. Figure 6-15 showsasimilarseriesoftrajectoriesforvaryinginitiali ncidenceangles with (0)= 1.5 2 degand (0)= 0.2 2 deg.Theneighboringoptimalcontroller continuestoperformwellinthepresenceofvaryinginitial conditions. Figure 6-16 showsafamilyofneighboringoptimaltrajectorieswithvar yinginitial conditionsforasupersonicightconditionconsistentwit hmission4535(Mach1.2 /600KCAS/18kft).Theinitialangularratesarevaried 50 deg/secandtheinitial incidenceanglesarevaried 2 deg.Inallvecasestheneighboringoptimalcontroller performswellanddrivesthestoretoatrimmedightconditi ondespitethelargeinitial perturbations. 234

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0 0.5 1 1.5 -4 -3 -2 -1 0 1 Angle of Attack, a s (deg) 0 0.5 1 1.5 -3 -2 -1 0 1 2 Angle of Sideslip, b s (deg) Optimal Trajectory Neighboring Optimal 0 0.5 1 1.5 -10 -5 0 5 10 15 Pitch Rate, Q BI (deg/sec)Time (sec) 0 0.5 1 1.5 -20 -10 0 10 20 Yaw Rate, R BI (deg/sec)Time (sec) Figure6-15.Optimalandneighboringoptimaltrajectories withvaryinginitialincidence anglesformission2265(Mach0.9/550KCAS/4800ft). 6.2.2.4Responsetorandomdisturbances Neighboringoptimalcontrolisdesignedtocorrectforpert urbationsintheinitial conditions.Sinceanytimeisavalid“initial”time,NOCcan alsobeusedtocorrectfor disturbancesalongtheoptimalpath.Forstoreseparation, turbulentairneartheaircraft isofparticularconcern. Thestudyofaerodynamicturbulenceisnosmallmatterandat horoughanalysisof turbulenteffectsonstoreseparationisbeyondthescopeof thisinvestigation.Rather, amoredirectapproachwillbeconsideredtodemonstratethe performanceofthe controllerinthepresenceofrandomdisturbances. Figure 6-17 showstheaerodynamiccoefcientsestimatedfromighttes ttelemetry dataformission2265(Mach0.9/550KCAS/4800ft).Thesmoot hedestimatewas determinedusinga51-pointmovingaverage.Theresidualsb etweentheestimatedand smoothedsignalsareassumedtoincludeeffectsfromaerody namicturbulence,sensor 235

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0 0.5 1 1.5 -15 -10 -5 0 5 Angle of Attack, a s (deg) 0 0.5 1 1.5 -4 -2 0 2 Angle of Sideslip, b s (deg) Optimal Trajectory Neighboring Optimal 0 0.5 1 1.5 -100 -50 0 50 100 Pitch Rate, Q BI (deg/sec)Time (sec) 0 0.5 1 1.5 -50 0 50 100 Yaw Rate, R BI (deg/sec)Time (sec) Figure6-16.Optimalandneighboringoptimaltrajectories withvaryinginitialconditions formission4535(Mach1.2/600KCAS/18kft). noise,andstructuralvibration.Theseresidualswillbeus edtointroducedisturbances intothesystem,representativeofthedisturbancesencoun teredinight. Figure 6-18 showsoptimalandneighboringoptimaltrajectoriesinthep resenceof randomdisturbancesrepresentativeofaerodynamicturbul ence.Itisapparentthatthe turbulencehasaminoreffectonthetrajectory.Theinertia ofthestoreactsasaphysical ltertoreducetheeffectofthehighfrequencydisturbance sandthecontrollerperforms adequately.Figure 6-19 showstheresponseoftheneighboringoptimalcontrollerfo r increasingvaluesofrandomdisturbances.Theturbulences ignalisampliedone,three, andvetimeswithanincreasinglydramaticeffectonthetra jectory.However,inall vecasestheoptimalcontrollerperformsadequatelyandco ntinuallymaintainsanear trimmedightconditioninthepresenceoflargerandomdist urbances. 236

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0 0.5 1 -2 0 2 Axial, CAForce Coefficients 0 0.5 1 -1 0 1 2 Side, CY 0 0.5 1 -20 -10 0 10 Normal, CNTime (sec) Flight Test Smoothed 0 0.5 1 -0.5 0 0.5 Roll, CLLMoment Coefficients 0 0.5 1 -5 0 5 Pitch, CLM 0 0.5 1 -4 -2 0 2 Yaw, CLNTime (sec) Figure6-17.Aerodynamiccoefcientsestimatedfromight testtelemetrydatafor mission2265(Mach0.9/550KCAS/4800ft). Furtheranalysisisneededtoassesstheperformanceofthen eighboringoptimal controllerforatrulystochasticsystem.However,thiscur sorylookattheeffectof turbulenceispromising,indicatingfurtheranalysisiswa rranted. 6.2.2.5Responsetoparametervariations NOCcanbeusedtoprovideoptimalfeedbackcontrolinthepre senceofsmall constantparametervariations,providedtheparametervar iationisknownorcan beestimated.Whentheparametervariationisunknown,thec ontrolisnecessarily sub-optimalbutmaystillperformadequately. Forstoreseparation,theparametervariationsintheformo fmodeluncertainties andmasspropertiesaresignicant.Modeluncertaintiesar elargelyduetoaerodynamic effects,andmaybeconsideredasubsetoftherandomdisturb ancesconsideredinthe previoussection.Trajectorycharacteristicsareoftense nsitivetomassproperties,such ascenterofgravity(CG)location.Oneparameterthattypic allyhasasignicanteffect 237

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0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 Incidence Angles a s (t) b s (t) 0 0.5 1 1.5 2 -6 -4 -2 0 2 4 Euler Angles roll(t) pitch(t) yaw(t) 0 0.5 1 1.5 2 -10 -5 0 5 10 Control DeflectionsTime (sec) d a d e d r 0 0.5 1 1.5 2 -10 0 10 20 30 40 Angular Velocity (deg/sec)Time (sec) p(t) q(t) r(t) Figure6-18.Optimaltrajectoryandneighboringoptimaltr ajectoryresponsetorandom disturbancesformission2265(Mach0.9/550KCAS/4800ft). 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 Angle of Attack, a s (deg) 0 0.5 1 1.5 2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 Angle of Sideslip, b s (deg) Optimal Trajectory Neighboring Optimal 0 0.5 1 1.5 2 -10 -5 0 5 10 15 Pitch Rate, Q BI (deg/sec)Time (sec) 0 0.5 1 1.5 2 -10 -5 0 5 10 15 20 Yaw Rate, R BI (deg/sec)Time (sec) Figure6-19.Optimaltrajectoryandneighboringoptimaltr ajectoryresponsetoamplied randomdisturbancesformission2265(Mach0.9/550KCAS/48 00ft). 238

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onthestoreseparationtrajectoryisthelongitudinalcent erofgravity, x CG .Thelocation ofthestorelongitudinalcenterofgravityrelativetothea erodynamiccenterisknown todeterminetheaerodynamicstability.Infact,ifthe x CG istoofaraft,thestorewill becomeinherentlyunstable.Fortheexamplestore,thisocc urswhenthe x CG isshifted byapproximatelyteninches, x CG =10 inches.Figure 6-20 showsaseriesofunguided (jettison)trajectoriesforthecenterofgravity 10 x CG 10 at 5 inchincrements.The trajectoriesinFigure 6-20 weredeterminedbysettingallcostparameterstozero,e.g. Q = Q = Q p = Q q = Q r =0 and R a = R e = R r =0 .Theuncontrolledtrajectorywith x CG = 10 inchesisunstableandquicklydepartsfromstableight. Figure 6-21 showsaseriesoftrajectoriesforthesamerangeofparamete rs.In thiscase,theneighboringoptimalcontrolstabilizesanot herwiseunstablestore.The controlsystemprovidesstabilityaugmentationanddrives thestoretoastabletrimmed ightconditionforalargerangeofparametervariations.T hus,theneighboringoptimal controllerclearlyprovidesanadvantageovertheuncontro lledjettison. 0 0.2 0.4 0.6 0.8 -100 -50 0 50 100 Angle of Attack, a s (deg) 0 0.2 0.4 0.6 0.8 -100 -50 0 50 100 Angle of Sideslip, b s (deg) Optimal Trajectory Neighboring Optimal 0 0.2 0.4 0.6 0.8 -500 0 500 Pitch Rate, Q BI (deg/sec)Time (sec) 0 0.2 0.4 0.6 0.8 -500 0 500 Yaw Rate, R BI (deg/sec)Time (sec) Figure6-20.Unguidedtrajectorieswithparametervariati ons. 239

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0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 Angle of Attack, a s (deg) 0 0.5 1 1.5 2 -0.6 -0.4 -0.2 0 0.2 0.4 Angle of Sideslip, b s (deg) Optimal Trajectory Neighboring Optimal 0 0.5 1 1.5 2 -10 -5 0 5 10 Pitch Rate, Q BI (deg/sec)Time (sec) 0 0.5 1 1.5 2 -5 0 5 10 Yaw Rate, R BI (deg/sec)Time (sec) Figure6-21.Guidedtrajectorieswithparametervariation s. 6.3FlightTestComparison TheexamplesconsideredinSections 6.2.2.2 through 6.2.2.5 arebasedonthe parametricmodelconstructedusingsystemidenticationm ethodsinChapter 3 .The HBVPissolvedusingtheparametricmodel,theNOCgainsared erivedusingthe parametricmodel,andthewindaxissimulationisexecutedu singtheparametricmodel. Thisapproachensuresconsistencybetweentheresults;ava luablecharacteristicfor preliminaryresearch.Theresultscanbefurtherexaminedb yconsideringablendof parametricmodelingandmoreconventionalmethods.Inpart icular,theparametric modelcanbeusedtosolvetheHBVPanddeterminetheNOCgains ,andthenthe closedloopcontrolsystemcanbeappliedtoa6DOFsimulatio nusingconventional multi-dimensionalinterpolationofanaerodynamicdataba se.Thisapproachprovides someadditionalassurancethatthecontrolsystemwilloper ateeffectivelyinthe intendedenvironment.UsingtheNOCfull-statefeedbackwi thtimevaryinggains, theimplementationofastoreseparationautopilotisstrai ghtforward.Theresultsare 240

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consideredhereincomparisonwithighttestdataforsubso nicandsupersonicight conditions.6.3.1SubsonicFlightTest Figure 6-22 showsacomparisonofanoptimaltrajectorydeterminedusin g parametricmodelingandawindaxissimulationusingtheNOC gainsandconventional windtunneldatabaseinterpolation.TheNOCgainsweredete rminedusingthe parametricmodelevaluatedalongtheoptimaltrajectory.T hewindaxissimulationis dependentontheparametricmodelonlythroughtheclosedlo opcontrolgainsderived fromtheHBVPandNOCsolution.Thenominaloptimaltrajecto ryandneighboring optimalcontrolgainsarecomputedandstoredexternaltoth ewindaxissimulation. Duringsimulation,thereferencetrajectory x ( t ) ,referencecontrol u ( t ) ,andcontrol gains K ( t ) areinterpolatedateachtimesteptodeterminethedesiredc ontrolinput u ( t )= u ( t ) K ( t ) ( x ( t ) x ( t ) ) .Theresultsindicatethatthecontrollerperformswell whenappliedtoarealisticightsimulation. Figure 6-23 showsthesamenominaloptimaltrajectorywithaseriesofne ighboring optimaltrajectorieswithvaryinginitialconditions.The neighboringoptimaltrajectories aredeterminedusingawindaxissimulationwithinterpolat ionofanaerodynamic databaseinconjunctionwithclosedloopfeedbackgainsdet erminedusingparametric modeling.Theresultssubstantiatetheperformanceofthec ontrollerforarealisticight simulationwitharangeofinitialconditions. Storeseparationtrajectorycharacteristicsareaffected bymanyvariables,including theinitialconditions,ightconditions,storemassprope rties,andejectionforces.Due tothevariabilityoftheseeffects,notwostoreseparation ighttesttrajectoriesare identical.Toensuresafeandacceptableseparationacross theightenvelope,itis importanttoconsiderarangeofmotionforeachintendedrel easecondition.Repetitive ighttestingisprohibitivelyexpensiveandtherangeofmo tionistypicallyquantied usingmodelingandsimulation.Arangeofvariablesincludi nginitialconditions,mass 241

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0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 Incidence Angles a s (t) b s (t) 0 0.5 1 1.5 2 -6 -4 -2 0 2 4 Euler Angles roll(t) pitch(t) yaw(t) 0 0.5 1 1.5 2 -10 -5 0 5 10 Control DeflectionsTime (sec) d a d e d r 0 0.5 1 1.5 2 -10 -5 0 5 10 15 Angular Velocity (deg/sec)Time (sec) p(t) q(t) r(t) Figure6-22.Comparisonofoptimaltrajectoryandneighbor ingoptimalwindaxis simulationsformission2265(Mach0.9/4800ft/550KCAS). properties,ejectionforces,etc.canbespecied,andamul titudeofsimulationscanbe conductedtodeterminetheexpectedrangeofmotion.Theinp utparameterscanbe variedrandomlyordeterministically,oracombinationofb oth,toachievethedesired rangeoftrajectories.Simulationswithrandomlyvaryingi nputsarereferredtoasMonte Carlosimulations.Thosewithdeterministicinputsareref erredtoasaparametric analysisoradesignedexperiment. Tofurtherexaminetheperformanceofthestoreseparationa utopilotinthe presenceofdeterministicparametervariations,amultitu deofighttestsimulations wereconductedwithandwithoutguidanceandcontrol.Thein tentofthiscomparison istoshowthedramaticimprovementinsafetyandacceptabil itythatcanbeachieved withguidedstoreseparation.Theparametricanalysisincl udedvariationsinrelease 242

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0 0.5 1 -10 -8 -6 -4 -2 0 2 Angle of Attack, a s (deg) 0 0.5 1 -3 -2 -1 0 1 2 Angle of Sideslip, b s (deg) Optimal Trajectory Neighboring Optimal 0 0.5 1 -60 -40 -20 0 20 40 60 Pitch Rate, Q BI (deg/sec)Time (sec) 0 0.5 1 -60 -40 -20 0 20 40 60 Yaw Rate, R BI (deg/sec)Time (sec) Figure6-23.Comparisonofoptimaltrajectoryandneighbor ingoptimalwindaxis simulationswithvaryinginitialconditionsformission22 65(Mach0.9/4800 ft/550KCAS). conditions,storemassproperties,ejectorproperties,an daerodynamicdamping derivatives.Thenominalandtoleranceofeachspecicpara meterisshowninTable 6-1 Table6-1.FactorsforParametricAnalysis ParameterNominalTolerance FlightConditionsMach0.9 0.05 AircraftAoA1deg 1deg StorePropertiesLong. x CG 9.6in 2in Lat. y CG 0in 0.5in EjectorPropertiesFwdForce3000lbPeakForce 10% AftForce8000lbPeakForce 10% TotalForce11000lbPeakForce 20% DampingDerivativesRollDamping c l p -2.5 50% PitchDamping c m q -74 50% YawDamping c n r -74 50% 243

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Nineofthetenfactors(excludingTotalForce)inTable 6-1 werevariedbetween nominal,high,andlow(threelevelseach)usinga 1 = 2 fractioncentralcomposite designedexperiment,foratotalof275runs. 1 Notethat9factorsat3levelseach have 3 9 =19683 possiblecombinations,andthecentralcompositedesignis avery economicalwayofsimulatingtherangeofparameters. Figure 6-24 showstheangleofattack s andangleofsideslip s fortheparametric analysis.Thegureincludes275jettison(unguided)traje ctories,275guidedtrajectories usingNOC,thenominalguidedtrajectory,andtheighttest telemetrydataforthis subsonicightcondition.Notethattheguidedtrajectorie sbeginat t =0.05 sec,which istheendoftheejectionstroke.Itisclearthatthestorese parationautopilotsignicantly reducesthevariabilityinthewindaxisincidenceangles.T heentirerangeofcontrolled trajectoriesarewithintherangeofmotiondemonstratedby asingleuncontrolledight test,whereastheuncontrolledjettisontrajectoriesexhi bitlargeexcursionsinangleof attack( 40 < s < 40 deg)andangleofsideslip( 40 < s < 30 deg). Figure 6-25 showstheverticalpositionandverticalvelocityforthesi mulated ighttesttrajectories.Theguidedtrajectoriesexhibita tightgroupingincomparisonto thejettisontrajectories.Notethatthenominalguidedtra jectorymanifestsaconstant slopeequalto 32.2 ft/sec 2 correspondingthetheaccelerationofgravity.Sincethe storeisaxisymmetricandmaintainednearzeroangleofatta ck,theaerodynamicliftis essentiallyzeroandthestoreacceleratesevenlyunderthe inuenceofgravity. Figures 6-26 and 6-27 showtheangularratesandorientationofthestoreduring separation.Thestoreseparationautopilotperformswellf ortheentirerangeof parametervariationsandconsiderablyreducestherangeof motionincomparison tothejettison(unguided)simulations. 1 TheTotalForceparameterwasnotvariedindependently.Rat heritisaproductof theFwdandAftforcevariations,leaving9independentpara meters. 244

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -50 0 50 AoA, a s (deg) Jettison Guided (NOC) Guided (nominal) Flight Test 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -40 -20 0 20 40 Time (sec)AoS, b s (deg) Figure6-24.Parametricanalysis(incidenceangles)forje ttisonandguidedstore separationcorrespondingtoighttest2265(Mach0.9/4800 ft/550 KCAS). 6.3.2SupersonicFlightTest Section 6.3.1 focusedonvalidationresultsforasubsonicightconditio n.A supersonicightconditioncorrespondingtomission4535( Mach1.2/600KCAS/ 18kft)providesamorestrenuoustestcaseduetothestronge rnonuniformoweld effects.Despitethestrongeroweldinuence,thesepara tionautopilotcontinuesto performwell. Figure 6-28 showsoptimalandneighboringoptimaltrajectoriesforcom parison.The optimaltrajectorywascomputedusingtheparametricmodel constructedwithsystem identication.Theoptimaltrajectoryandcontrolweresto redandusedtodeterminethe controlinputsfortheneighboringoptimaltrajectory.The neighboringoptimaltrajectory wascomputedusingawindaxissimulationwithinterpolatio nintoanaerodynamic database.Theonlydependencyofthewindaxissimulationon theparametricmodel isthroughthetimevaryingfeedbackcontrolgains.Figure 6-29 showsaseriesof 245

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -20 0 20 40 60 80 Vertical Position, Z BI (ft) Jettison Guided (NOC) Guided (nominal) Flight Test 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -20 0 20 40 60 80 100 Time (sec)Vertical Velocity, dZ/dt (ft/sec) Figure6-25.Parametricanalysis(verticalvelocityandtr anslation)forjettisonandguided storeseparationcorrespondingtoighttest2265(Mach0.9 /4800ft/550 KCAS). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -300 -200 -100 0 100 200 Pitch Rate, Q BI (deg/sec) Jettison Guided (NOC) Guided (nominal) Flight Test 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -60 -40 -20 0 20 40 Time (sec)Pitch Angle, q BI (deg) Figure6-26.Parametricanalysis(pitch)forjettisonandg uidedstoreseparation correspondingtoighttest2265(Mach0.9/4800ft/550KCAS ). 246

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -300 -200 -100 0 100 200 300 Yaw Rate, R BI (deg/sec) Jettison Guided (NOC) Guided (nominal) Flight Test 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -40 -20 0 20 40 60 Time (sec)Yaw Angle, y BI (deg) Figure6-27.Parametricanalysis(yaw)forjettisonandgui dedstoreseparation correspondingtoighttest2265(Mach0.9/4800ft/550KCAS ). neighboringoptimaltrajectorieswithvaryinginitialcon ditions.Theresultsfromboth guresindicatethattheneighboringoptimalcontrollerwo rkswellwhenappliedtoa realisticighttestsimulation. Finally,Figures 6-30 and 6-31 showtheresultsofaparametricanalysisusingthe samesetofparametersshowninTable 6-1 .Theresultsaresimilar,providingfurther validationofthecontrollerinthepresenceofastrongnonu niformoweld. 247

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0 0.5 1 1.5 2 -8 -6 -4 -2 0 2 Incidence Angles 0 0.5 1 1.5 2 -8 -6 -4 -2 0 2 Euler Angles roll(t) pitch(t) yaw(t) 0 0.5 1 1.5 2 -10 -5 0 5 10 Control DeflectionsTime (sec) d a d e d r 0 0.5 1 1.5 2 -40 -20 0 20 40 Angular Velocity (deg/sec)Time (sec) p(t) q(t) r(t) a s (t) b s (t) Optimal Neighboring Optimal Figure6-28.Comparisonofoptimaltrajectoryandneighbor ingoptimalwindaxis simulationsformission4535(Mach1.2/600KCAS/18kft). 0 0.5 1 -15 -10 -5 0 5 Angle of Attack, a s (deg) 0 0.5 1 -6 -4 -2 0 2 Angle of Sideslip, b s (deg) Optimal Trajectory Neighboring Optimal 0 0.5 1 -100 -50 0 50 100 Pitch Rate, Q BI (deg/sec)Time (sec) 0 0.5 1 -50 0 50 100 Yaw Rate, R BI (deg/sec)Time (sec) Figure6-29.Comparisonofoptimaltrajectoryandneighbor ingoptimalwindaxis simulationswithvaryinginitialconditionsformission45 35(Mach1.2/600 KCAS/18kft). 248

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -50 0 50 AoA, a s (deg) Jettison Guided (NOC) Guided (nominal) Flight Test 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -60 -40 -20 0 20 40 Time (sec)AoS, b s (deg) Figure6-30.Parametricanalysis(incidenceangles)forje ttisonandguidedstore separationcorrespondingtoighttest4535(Mach1.2/600K CAS/18kft). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -300 -200 -100 0 100 200 300 Pitch Rate, Q BI (deg/sec) Jettison Guided (NOC) Guided (nominal) Flight Test 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -50 0 50 Time (sec)Pitch Angle, q BI (deg) Figure6-31.Parametricanalysis(pitch)forjettisonandg uidedstoreseparation correspondingtoighttest4535(Mach1.2/600KCAS/18kft) 249

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6.4ChapterSummary Guidedstoreseparationinvolvesparametricmodelingofth espatiallyvariant aerodynamics,determinationofapreferredtrajectorytha tsatisessafetyand acceptabilityobjectives,anddesignofafeedbackcontrol lerthataccountsformodel uncertainties,varyinginitialconditions,andoweldpe rturbations.Theseobjectives havebeenaccomplishedindetailusingsystemidenticatio n,trajectoryoptimization, andneighboringoptimalcontrol.Chapters 3 through 5 presentedthenecessary theoryandexamplesusingasimpliedplanarstoreseparati onproblem.Thischapter demonstratestherigorousapplicationofthistheorytoafu llsix-dimensionalstore separationighttestanalysis.Theoptimaltrajectoriesa ndneighboringoptimalfeedback gainswerecomputedusingaparametricmodelandawindaxiss imulationusing conventionaldatabaseinterpolationmethodswasusedtova lidatethecontrollerina realisticighttestsimulationenvironment.Theresultsi ndicatethatthestoreseparation autopilotperformswellunderavarietyofconditions,demo nstratingthesignicant improvementinseparationcharacteristicsthatmaybeobta inedusingguidedstore separation.Whilethisanalysisisnotexhaustive,theresu ltsarepromisingandindicate thatfurtherresearchandapplicationiswarranted. 250

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CHAPTER7 CONCLUSIONS 7.1Summary Combataircraftutilizeexpendablestoressuchasmissiles ,bombs,ares,and externaltankstoexecutetheirmissions.Safeandacceptab leseparationofthesestores fromtheparentaircraftisessentialformeetingthemissio nobjectives. Astorereleasedfromanaircraftinightmusttraverseanon uniformandunsteady oweldthatmayincludecomplexshockinteractions,large velocitygradients,regions oflocallyseparatedorreversedairow,andsevereowangu larityintheformof sidewashanddownwash.Storesreleasedfromaninternalwea ponsbaymayalso besubjectedtoawakedisturbancefromthespoiler,dynamic pressureandvelocity gradientsacrosstheshearlayer,highfrequencyvibration sduetoacousticnoise,and largeperturbationsinowpropertiesduetocavityoscilla tions. Althoughtheregionofnonuniformowneartheaircraftisex ceedinglysmall comparedtothefulllengthofthestoreballisticory-outt rajectory,theeffectsare signicant.Theoweldcharacteristicsmaycausethestor etoexhibitbehaviorthat compromisesthesafetyoftheairframeandcreworthatcompr omisestheeffectiveness ofthestoreitself.Modernguidedmunitionsaredesignedwi thanonboardguidance andcontrolsystemtoenablepreciseengagementoftheinten dedtargets.However, thecontrolsystemisnotusuallyactivateduntilthestorei ssufcientlyfarawayfrom theaircrafttoavoidanypotentialinterference.Often,th eseparation-inducedtransients resultinlargeperturbationsfromthedesiredightattitu desthatrequireadedicated “rate-capture”phaseforrecoverybeforethemunitioncanb eginthey-outtrajectory. Intherelativelyfewcaseswheretheautopilotisengagedea rlier(topreventbuild-up ofirrecoverableratesandattitudes),themutualaerodyna micinterferencebetweenthe storeandaircraftisneglectedintheautopilotdesignlead ingtoincreasedriskthrough 251

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reducedcondenceinsimulationcapabilitiesandpotentia llyunsafebehaviorofthe autopilotreactingtooweldperturbationswithoutconsi derationofthenearbyaircraft. Analternativeapproachistodesignaguidanceandcontrols ystemthatleverages theaerodynamicinteractionbetweenthestoreandaircraft toachievedesirable trajectorycharacteristics.Thedevelopmentofastoresep arationguidanceandcontrol system,or“storeseparationautopilot”istheprimaryfocu softhisresearch. Thisresearchanddissertationproceedsinseveralphases. Thestoreseparation equationsofmotionandaerodynamicmodelingapproachared evelopedinChapter 2 Relevanttechniquesinsystemidenticationaredescribed inChapter 3 andappliedto storeseparationtodevelopaparametricmodelforthespati allyvariantaerodynamics ofastoreduringseparation.Trajectoryoptimizationisex ploredinChapter 4 andthe 1 st orderconditionsforoptimalstoreseparationaredevelope dusingclassicaloptimal controltheory.Chapter 5 focusesonfeedbackcontrolusingneighboringoptimal controltoaccountforvariationsininitialconditions,sy stemparameters,andoweld perturbations.Finally,Chapter 6 providesanextendedapplicationtodemonstratethe efcacyofguidedstoreseparationforarepresentativesto reseparatingfromanF-16 aircraft.Adetailedoutlineforeachchapterissummarized below. Chapter1 .Storeseparationengineeringisdescribedandthemotivat ionfor guidedstoreseparationisconsidered.Thethree-partproc essofsystemidentication, trajectoryoptimization,andfeedbackcontrolisintroduc edandabriefsketchofthe researchapproachandcontributionsforeachareaispresen ted.Finally,acasestudy consistingofarepresentativemid-sizedguidedmunitionr eleasedfromtheF-16tactical aircraftisintroduced. Chapter2 .Acomprehensiveframeworkforstoreseparationmodelinga nd simulationispresented.Theequationsofmotionarederive dfromrstprinciples withconsiderationofspeciccoordinatesystemsandaircr aftmaneuvers.Theequations ofmotionarederivedinbothbodyandwindaxesandconsolida tedinstatespaceform. 252

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Storeseparationaerodynamicmodelingusingwindtunnelte stdataisdescribedand somerepresentativedataarepresented.Thestoreseparati onequationsofmotion areextendedforthepurposeofighttestdatareductionand trajectoryreconstruction. Actualighttestdataarepresentedtovalidatethemodelin gapproachandighttest dataarecomparedtosimulatedtrajectorieswithfavorable results.Thisbriefoverviewof establishedstoreseparationmethodologyprovidesasolid foundationfortheremaining researchdevelopments. Chapter3 .Anoverviewofsystemidenticationasappliedtoightveh iclesis presentedandapplicationofSIDtostoreseparationiscons idered.Amultisinesignal isselectedtogenerateanarbitrarynumberofinputsthatar emutuallyorthogonal andadheretoauniformpowerspectrum,providinganexcelle ntframeworkfor simulation-basedsystemidentication.Modelstructured eterminationisaccomplished usingmultivariateorthogonalpolynomialsandparametere stimationisaccomplished usingtheequationerrormethod.Freestreamsystemidenti cationisconsideredrst, resultinginanonlinearmultivariatepolynomialmodelwit hconstantcoefcients.The sameapproachisextendedtospatiallyvariantsystemident icationresultingina nonlinearmultivariatepolynomialmodelwithspatiallyva riantcoefcients.Thesemodels arerestatedinacompactmatrixform,providingasinglecom pactparametricmodel determineddirectlyfromwindtunneltestdata.Theparamet ricmodeliscomparedto windtunnelandighttestdatawithfavorableresults. Chapter4 .Trajectoryoptimizationisintroducedandthestoresepar ationproblem isrestatedasanoptimalcontrolproblem.Storeseparation isparticularlywellsuitedfor trajectoryoptimizationusingsafetyandacceptabilityst andardsasaperformancemetric. Theequationsofmotion,parametricmodel,anddesirabletr ajectorycharacteristicsare combinedtoestablishawellformedoptimalcontrolproblem andtheclassicalcalculus ofvariationapproachisusedtoderivetherstordernecess aryconditionsforoptimal storeseparation.TheresultsformatwopointHamiltonianb oundaryvalueproblemthat 253

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mustbesolvednumericallytodetermineafeasibleopen-loo pextremaltrajectory.The HBVPisreadilysolvedusingcollocationtechniquesandpro videssignicantinsight intostoreseparationdynamicsandcontrol.Thenalresult isanonlinear,open-loop, extremaltrajectorythatsatisesthedynamicconstraints andminimizesasafetyand acceptabilityperformanceindex.Anexampleusingasimpli edaerodynamicmodeland motionconnedtotheverticalplaneisconsideredindetail Chapter5 .Theopen-loopextremaltrajectoryconsideredinChapter4 isan excitingdevelopmentbutposesasignicantlimitation.Ea chsetofinitialconditions requiresanindependentsolutionoftheHBVP.Thenumerical solutioniscomputationally intensiveandnotguaranteedtoconverge.Furthermore,the HBVPimplicitlyassumes perfectknowledgeofthemodelandoperationinanoisefreee nvironment.Forthese reasons,theopen-loopcontrolisnotsuitableforimplemen tation.Rather,neighboring optimalcontrolisusedtoconstructalinear-optimalfeedb ackcontrollerthataccounts forvariationsininitialconditions,modeluncertainties ,anddisturbancesalongthe optimalpath.Theneighboringoptimalcontrollerisbasedo nfeedforwardofthenominal extremalsolutionandfeedbackcontrolthatminimizestheo riginalcostfunctionto secondorderinthepresenceofdisturbancesalongtheoptim alpath.NOCisextended toconsidercaseswithterminalcosts,terminalconstraint s,path/controlinequality constraints,andparametervariations.InniteHorizonNe ighboringOptimalControlis introducedtoprovideacontrollerthataccountsforthespa tiallyvariantaerodynamics throughtimevaryinggainsneartheaircraftandconvergest oalineartimeinvariant controllerinfareldconditions.TheIHNOCstrategyiside allysuitedforstoreseparation andsimulationresultsshowfavorableperformanceevenint hepresenceofsignicant parametervariationsandturbulentwindgusts. Chapter6 .Finally,storeseparationmodelingandsimulation,syste midentication, trajectoryoptimization,andfeedbackcontrolareuniedi nasingleextendedcasestudy. Thecasestudyconsidersseparationofarepresentativegui dedmunitionfromthe 254

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F-16aircraft.Trajectoryoptimizationisappliedtotheno nlinearsix-degree-of-freedom equationsofmotionincorporatingtheparametricmodeldev elopedinChapter3.Innite HorizonNeighboringOptimalControlisusedtodetermineti me-varyinggainsthat explicitlydependonthespatiallyvariantaerodynamics.T heresultingstoreseparation autopilotisvalidatedusingarealisticighttestsimulat ionwithvariationsininitial conditions,systemparameters,andrandomdisturbances.T heanalysisisfarfrom exhaustivebuttheresultsarepromising,indicatingthatf urtherresearchiswarranted andrealworldapplicationmaybefeasibleinthenearfuture 7.2Contributions Theprimarycontributionofthisresearchisdevelopmentof acomprehensive frameworkforguidedstoreseparationthatincludesparame tricmodeling,trajectory optimization,andfeedbackcontrol.Thecontributionsdem onstratedinthisworkalong withpotentialapplicationsarediscussedbelow. SystemIdentication .Thisresearchistherstattempttoconstructaparametric modelforstoreseparationaerodynamicsusingsystemident ication.Thisparametric modelisusefulinitsownrightandprovidesinsightintothe aerodynamicinteraction betweenthestoreandaircraft.Thisapproachmayalsoprove tobeusefulforight throughanonuniformoweld,suchasanaircraftingrounde ffectoranaircraftying throughawakevortex,microburst,orwindshear.Thekeyfea tureofspatiallyvariant systemidenticationistheavailabilityofanaccurateaer odynamicdatabaseandthe techniquesdemonstratedhereinmayalsobeusefulformappi ngothernonlineareffects, suchashigh-alphaightorstalleffects.Otherapplicatio nsoutsideoftheaerospace industrymaybenetfromthedemonstrationofestablishedt echniquesinthecontextof simulation-basedsystemidentication. TrajectoryOptimization .Thisresearchistherstattempttoapplyoptimalcontrol theorytodeterminea“bestcase”separationtrajectory.Th emoreprevalenttrend instoreseparationistoemphasizethe“worstcase”traject orytoensuresafetyand 255

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acceptabilityineventhemostadverseconditions.Traject oryoptimizationprovidesa frameworkforthestoreseparationengineertoactuallydes ignapreferredtrajectory thatinherentlysatisestheconstraintsofthesystemunde rconsideration.Rather thanrelyingonpassivecontrolthroughaircraftlimitsand ejectorsettings,trajectory optimizationgivesthestoreseparationengineeranotherd egreeoffreedomfor maximizingthecapabilityofaspecicaircraft/storecomb inationwhileprovidinga directwaytoaddresssafetyandacceptability.Theapplica tionoftrajectoryoptimization forguidanceofastorethroughanonuniformoweldbearsso meresemblancetoother aerospacechallenges,includinglandingofanaircrafting roundeffectorvariable winds,aircraftwake-vortexencounter,ightthroughamic roburstorwindshear, ightthroughseverewindeldsinanurbanenvironment,emp loymentofhypersonic researchvehiclesfromhigh-altitudecarrierplanes,and ightofmultipleaircraftinclose proximitysuchascooperativecongurationoraerialrefue ling.Therstorderoptimality equationsdevelopedinChapter 4 aregeneralenoughtodescribemanyothertrajectory optimizationproblems,providedthattheaerodynamicscan bemodeledappropriately. FeedbackControl .Thisresearchistherstattempttodevelopafeedback controlsystemthatexplicitlyaccountsforthespatiallyv ariantaerodynamicsofastore duringseparation.Neighboringoptimalcontrolisanelega ntandversatileapproach forimplementingreal-timeoptimalcontrolanditispartic ularlywellsuitedforstore separation.Ratherthanreacttoadverseowconditionsdur ingseparation,NOCcan beusedtoaccountforthedeterministicoweldfeaturesap rioriandactuallyleverage theaerodynamicinteractionbetweenthestoreandaircraft toimproveseparation characteristics.Theheavycomputationalburdenisplaced upfrontandthenalresult isasimplelineartimevariantcontrollerthancanbereadil yimplementedwithexisting hardwareandrmwarecommontoanymodernguidedmunition.T hus,NOCprovides awaytoaccomplishdramaticimprovementinseparationchar acteristicswithvery littleadditionalcomplexityofthestoreguidanceandcont rolsystem.Theadditional 256

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featuresincorporatedbyInniteHorizonNOCprovideacont rollerthataccountsforthe spatiallyvariantaerodynamicsneartheaircraftandrapid lyconvergestoatimeinvariant regulatorinfareldconditions.Insomecases,thetimeinv ariantcontrollermayprove sufcientfortheremainingyouttrajectory,allowingcon structionofasinglecontinuous autopilotfortheentiremissionprole.TheIHNOCstrategy mayalsondapplication outsideofstoreseparation.Inparticular,IHNOCmaybeuse fulforanysystemthat musttraverseanonlinearoperatingorstartupcondition,f ollowedbyoperationnearan equilibriumconditionforanindeterminatelengthoftime. Regardless,theNOCapproach demonstratedhereprovidesasafeandeffectivemethodfori mplementingguidedstore separationinarealisticenvironment. 7.3FutureWork Theresultspresentedinthisresearchareinherentlydepen dentonthecase studyconsidered.Thecasestudywasselectedinpartduetot hebenignseparation characteristicsandadequatecontrolauthority.Theresul tsdemonstratedherein maynotberepresentativeofamorechallengingsystemorast orewithinsufcient controlauthority.However,everyefforthasbeenmadetode velopaframeworkthatis extensibleandapplicabletoavarietyofaircraft/storeco mbinations.Furtheranalysisof storesthatareinherentlysensitivetooweldconditions orstoresthatmusttraverse extremelychallengingoweldconditionswouldbeavaluab lenextstep.Inparticular, applicationofastoreseparationautopilotforastorerele asedfromaninternalweapons baywouldoffertremendousinsightintotheextensibilityo fthisapproach. Storeseparationfromaninternalweaponsbaywouldalsoint roduceanother challengingelement,namelyaturbulentandnon-stationar yoweld.Althoughsome effortwasmadeinthisstudytodemonstratetheeffectivene ssofastoreseparation autopilotinaturbulentoweld,amoreappropriateanalys iswouldinvolvedesignof anautopilotwithexplicitconsiderationoftheoweldcha racteristics.Theintroduction ofarandomlyvaryingdisturbancetothesystemdescription placestheprobleminthe 257

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domainofstochasticoptimalcontrol.Althoughtheapplica tionofstochasticoptimal controlisbeyondthescopeofthisresearch,theresultsdoc umentedhereprovidea solidframeworkforcontinuedresearch.Extensiontostoch asticcontrolaswellasother extensionsareexploredinmoredetailbelow.7.3.1SystemIdentication Theparametricmodelconstructedusingsystemidenticati oninthisstudyis basedentirelyonwindtunneltestdata.Thisresourcewasse lectedprimarilyduetothe easeofimplementationandrapiditerationnecessaryforap reliminaryinvestigation. Withimprovedexpectationsbasedontheseresults,avaluab lenextstepwouldbeto applysystemidenticationtoCFDtoconstructaparametric orreducedordermodel. Thebiggestchallengeinaccomplishingthistaskistoreduc etheamountof“training” datarequiredtoconstructthemodel.Systemidentication techniquesthathavebeen developedforusewithadaptivecontrol,suchasrecursivel eastsquares[ 24 34 ],may beusefulforidentifyinganaerodynamicmodelforastoreim mersedinanonuniform oweld. Theparameteridenticationmethodusedinthisstudyisthe equationerrormethod. However,thisapproachmaynotbevalidforCFD-basedsystem identication.Many viscousowsolversincludeaturbulencemodelthatisneces sarytoemulatethe owphysicsandintroducesrandomperturbationsintothepr edictedaerodynamic coefcients.Theequationerrormethodisknowntoperformp oorlyinthepresence ofturbulence[ 24 34 ].Amoresuitablechoicewouldbetheltererrormethod,whi ch reliesonaKalmanltervarianttopropagatethestates.Sev eralauthors,including Jategaonkar[ 34 ],Greenwell[ 64 ],Young[ 69 ],Klein[ 24 ]andMorelli[ 76 ],haveshown thattheltererrormethodcanbeusedwithstateaugmentati ontoestimatetime-varying parameters.ApplicationofthisapproachtoaCFD-basedsys temidenticationis promisingandmeritsfurtherinvestigation. 258

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7.3.2TrajectoryOptimization Theclassicalindirectapproachtooptimalcontrolusedint hisresearchisvery effectiveandprovidesatremendousamountofinsightintot hedynamicsandcontrolof storeseparation.However,itisalsodifculttoimplement correctlyandreliesonafairly complexaerodynamicmodel.Twomethodsforimprovementare suggested. First,itispossibletoreplacetheparametricmodelwithap urelyempiricalmodel. Inthelattercase,theaerodynamiccoefcientsaredetermi nedusinginterpolation ofawindtunnelorCFDdatabase,andtheaerodynamicgradien tsaredetermined bylocalnitedifferenceorsomeothernumericaldifferent iationmethod.Withthis approach,analgorithmcouldbedevelopedonceandappliedt oarangeofaircraft/ storecombinations.Apurelynumericalapproachwasconsid eredinthisresearch, withsomesuccess,butfoundtobetoocomputationallyexpen siveforapreliminary investigation. Second,trajectoryoptimizationusingdirectmethodsmayb eavaluablenext step.Optimalcontrolresearchisdominatedbyinvestigati onsintodirectnumerical optimizationtechniques,suchastheGaussPseudospectral method[ 95 108 118 119 125 157 ].Thesemethodsdonotrequireananalyticalrepresentatio noftheproblem andavoidchallengesassociatedwiththecostateand1 st orderoptimalityconditions. Theconveniencecomesatthepriceoflessinsightintothedy namicsandcontrol,but thismaybeanacceptabletradeoffforlongtermapplication ofoptimalcontroltostore separation.7.3.3FeedbackControl Theresultsoftheneighboringoptimalcontrolapproachuse dinthisresearchis promising.However,furtherdevelopmentisrequiredbefor ethisapproachissuitablefor real-worldimplementation.Inparticular,twolimitation sareimportanttonote. First,carefulinspectionoftheprecedingresultsindicat ethatthecontrolleris abletorespondinstantlytochangesinthetrajectory,whic hisofcourseanunrealistic 259

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approximation.Inreality,thecontrolsurfacedeections arelimitedbytheactuator response,whichisitselfadynamicsystem.Thecontrolactu atorsneedtobemodeled appropriatelytoassesstheimpactontheperformanceofthe controller.Thisfeature wasnotincludedinthepresentresearchbecause(1)anactua tormodelishighly specicandrequiresdetailedknowledgeofthesystemthati sunavailabletotheauthor and(2)anypotentialcontrolstrategyfacesthesameactuat orlimitation,sothislimitation issomewhatindependentoftheparticularcontrolstrategy employed. Second,thecontrolstrategyinthisresearchisbasedonful lstatefeedbackand itwastacitlyassumedthatallstatesareavailableforfeed back.Inpractice,notall ofthestatescanbemeasureddirectlyandeventhosethatare canbeaffectedby measurementnoise.Thus,fullstatefeedbackmustincorpor ateastateestimator toberealizedinpractice.Duetothenonlinearandspatiall yvaryingaerodynamics, developmentofasuitablestateestimationtechniqueisnot trivialandmaydegrade theperformanceofthecontroller.Thislimitationisbeyon dthescopeofthepresent research,butsomethingthatneedstobeinvestigatedtoimp rovethereadinessofthe technology. 7.4ConcludingRemarks Theintentofthisresearchistodemonstrateanimprovement insafetyand acceptabilitybyapplyingsystemidenticationandtrajec toryoptimizationtoachieve guidedstoreseparation.Inmanyways,theresultsexceeded theauthor'sexpectations. Theapplicationofsystemidenticationtostoreseparatio nrequiredathorough investigation,butresultedinacompactparametricmodelw ithseveraldesirable features.Themathematicsbehindtheclassicalapproachto optimalcontrolare formidable,butprovidedacomprehensiveandrobustapproa chforndingapreferred referencetrajectory.Theneighboringoptimalcontrolstr ategyrequiredatremendous amountofproblemsolvingandnumericalmethods,butresult edinaneffectiveand eleganttimevaryingcontrollerthatperformedwellinever ysituationexaminedsofar. 260

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Together,systemidentication,trajectoryoptimization ,andfeedbackcontrolprovide aviablesolutiontotheoriginalresearchobjectivewithpr omisingpotentialforfuture real-worldapplication. 261

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BIOGRAPHICALSKETCH RyanE.CarterisanaerospaceengineerfortheUnitedStates AirForce.Ryan earnedadegreeinAerospaceEngineeringfromAuburnUniver sityin2003anda MasterofSciencedegreeinengineeringfromtheUniversity ofFloridain2005.Hehas workedasaStoreSeparationEngineerfortheAirForceSEEKE AGLEOfceformore than8yearsandhashands-onexperiencewithavarietyofair craftincludingtheF-16, B-1BandF-22.In2009,Ryanwasawardedthehighlycompetiti veSMARTScholarship, acollaborativeeffortoftheAmericanSocietyforEngineer ingEducation(ASEE)and theNavalPostgraduateSchool(NPS).UndertheSMARTSchola rship,Ryanisworking full-timeonadoctorateattheUniversityofFlorida,witha nexpectedgraduationdateof August2012. Ryanisalsotheproudfatherofvebeautifulandlivelychil dren,includingfourboys andonegirlbetweentheagesof4and9.Ryanandhisfamilyliv einCrestview,FLand makethemostofeveryopportunitytoenjoyfamilyadventure sandoutdooractivities. Theirfavoriteactivitiesincludetraveling,camping,hik ing,andmountainbiking.After graduation,RyanplanstoresumehiscareerwiththeUSAFash econtinuestoinvestin thediscipleshipofhisfamily.Perhapstheywilleventakea triptoDisneyWorld. 275