Optimal Control and Design Using Genetic Algorithm Accelerated by Neural Networks

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Optimal Control and Design Using Genetic Algorithm Accelerated by Neural Networks
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Nambi,Sara
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Aerospace Engineering, Mechanical and Aerospace Engineering
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Rao, Anil
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Dixon, Warren E

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genetic -- neural -- optimization
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The main feature of this paper is the incorporation of Artificial Neural Networks (ANN) to accelerate the processing time of Genetic Algorithms (GA). In this paper, we use ANN for data mining and approximation of the initial range required GA such that its search efficiency increases. Deterministic/Gradient-based methods have proven to be difficult to produce optimal solutions to problems whose objective functions are discontinuous, non-differentiable, non-linear and stochastic. On the other hand, GA can produce an accurate search for an optimal solution for such problems. The hybridization of a GA with ANN though complex and expensive than the deterministic methods, is found to be more efficient in regard to actual operation time and near-optimal solutions for a given optimization problem. The method developed in this paper is a two-stage approach. The initial population required by GA is obtained using an ANN. Using this initial population, an optimal solution is obtained by GA. The approach is demonstrated in two types of problem - an optimal control problem and a fin design problem. It was found to be a successful method for generating optimal solutions in a constrained environment with minimal input from the user.
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by Sara Nambi.
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Thesis (M.S.)--University of Florida, 2011.
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OPTIMALCONTROLANDDESIGNUSINGGENETICALGORITHMSACCELERATEDBYNEURALNETWORKSBySARASWATHINAMBIATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEUNIVERSITYOFFLORIDA2011

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c2011SaraswathiNambi 2

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ToVadivazhaghiaNambiArasappanPillai 3

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ACKNOWLEDGMENTS IwouldliketoexpressmygratitudetomymentorProfessorAnilV.Raoforgivingmetheopportunitytoworkonthisresearchprogram.Withouthissupportandencouragement,thisprojectwouldhavenotbeenpossible.Iwouldalsoliketothankmycommitteemember,ProfessorWarrenE.Dixonforhisinputsandexpertguidancetowardsmyproject.ItwasthroughhimthatIgotintroducedtoNeuralNetworks.IamgratefultohimforteachingmealltheconceptsinvolvedinNon-LinearControlTheoryandforpatientlylisteningandclarifyingallmyinquisitivequestions.Hispresenceandencouragementthroughouttheprojectisinvaluabletome.IamindebtedtoRushikeshKamalapurkar,whohasbeenassistingmeinallmyexperimentsdespitehisvariousresearchcommitmentsandtightschedules.Additionally,Iwishtotakethisopportunitytothankallmyothercolleaguesfortheirvaluablesupportduringmyresearchcareer.FinallyIwishtoexpressmydeepestappreciationtomyfamilyandfriendsforstandingbymeinallaspectsofmyacademicsandpersonallife. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 10 2GENETICALGORITHMS .............................. 13 2.1Overview .................................... 13 2.2TransitionRules ................................ 13 2.2.1Selection ................................ 13 2.2.2Reproduction .............................. 14 2.2.2.1Elitism ............................. 14 2.2.2.2Crossover ........................... 14 2.2.3Mutation ................................. 17 2.2.4Population ................................ 18 2.2.5FitnessScaling ............................. 19 3ARTIFICIALNEURALNETWORKS ........................ 20 3.1Overview .................................... 20 4PROBLEMFORMULATION ............................. 23 4.1Overview .................................... 23 4.2EquationsofMotion .............................. 23 4.2.1Problem1 ................................ 23 4.2.2Problem2 ................................ 24 4.3BoundaryConditions .............................. 24 4.3.1Problem1 ................................ 24 4.3.2Problem2 ................................ 25 4.4ControlandDesignConstraints ........................ 26 4.4.1Problem1 ................................ 26 4.4.2Problem2 ................................ 26 4.5CostFunctional ................................. 26 4.5.1Problem1 ................................ 26 4.5.2Problem2 ................................ 27 5

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5AUGMENTATIONOFGENETICALGORITHMSANDARTIFICIALNEURALNETWORKS ..................................... 28 5.1Overview .................................... 28 5.2Augmentation .................................. 29 6RESULTS ....................................... 34 7CONCLUSION .................................... 41 REFERENCES ....................................... 43 BIOGRAPHICALSKETCH ................................ 45 6

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LISTOFTABLES Table page 6-1GAParametersforProblem1andProblem2 ................... 34 6-2GAParametersforProblem1andProblem2 ................... 35 6-3OptimalDesignParametersobtainedusingGAandNN ............. 38 7

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LISTOFFIGURES Figure page 5-1GeneticAlgorithmFlowchart ............................ 29 5-2NeuralNetworkFlowchart .............................. 30 5-3NeuralNetworkforevery3generation-ActualModel .............. 31 5-4NeuralNetworkforevery3generations-Condensedform ............ 32 5-5AugmentationofGAandANN ........................... 33 6-1Statesofall3methods ............................... 35 6-2Controlofall3methods ............................... 36 6-3States-usingGAalone ............................... 36 6-4States-usingGAandasingleNN ......................... 36 6-5States-usingGAandaNNforevery3generations ............... 37 6-6Control-usingGAalone .............................. 37 6-7Control-usingGAandasingleNN ........................ 37 6-8Control-usingGAandaNNforevery3generations ............... 38 6-9Control-usingGAandaNNforevery3generations ............... 38 6-10BEST-FITIndividuals ................................. 39 6-11BEST-FITIndividuals ................................. 39 8

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AbstractofthesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceOPTIMALCONTROLANDDESIGNUSINGGENETICALGORITHMSACCELERATEDBYNEURALNETWORKSBySaraswathiNambiAugust2011Chair:AnilV.RaoMajor:AerospaceEngineeringThemainfeatureofthispaperistheincorporationofArticialNeuralNetworks(ANN)toacceleratetheprocessingtimeofGeneticAlgorithms(GA).Inthispaper,weuseANNfordataminingandapproximationoftheinitialrangerequiredGAsuchthatitssearchefciencyincreases.Deterministic/Gradient-basedmethodshaveproventobedifculttoproduceoptimalsolutionstoproblemswhoseobjectivefunctionsarediscontinuous,non-differentiable,non-linearandstochastic.Ontheotherhand,GAcanproduceanaccuratesearchforanoptimalsolutionforsuchproblems.ThehybridizationofaGAwithANNthoughcomplexandexpensivethanthedeterministicmethods,isfoundtobemoreefcientinregardtoactualoperationtimeandnear-optimalsolutionsforagivenoptimizationproblem.Themethoddevelopedinthispaperisatwo-stageapproach.TheinitialpopulationrequiredbyGAisobtainedusinganANN.Usingthisinitialpopulation,anoptimalsolutionisobtainedbyGA.Theapproachisdemonstratedintwotypesofproblem-anoptimalcontrolproblemandandesignproblem.Itwasfoundtobeasuccessfulmethodforgeneratingoptimalsolutionsinaconstrainedenvironmentwithminimalinputfromtheuser. 9

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CHAPTER1INTRODUCTIONOvertheyearsseveraltechniquesforoptimizationofaircraftandspacecrafttrajectorieshavebeendeveloped.OptimizationtheoriesarenotrestrictedtoAerospaceapplicationsalonebutarefoundinuseforIndustrialEngineering,BusinessManagement,Mechanicaldesignandotherstreamsaswell[ 1 ].Themethodshavebeenlargelyclassiedasgradient-basedapproachandheuristicapproach.Initiallytheoriesforoptimizationwereproposedusingmathematicalmodelsandlaterautomatedtoolsweredesigned,developedandmodiedtoproducesolutionsforalloptimizationproblems.Theautomatedoptimizationtoolsfornumericalmethodsinrecenttimeshavebeenknowntosolveunconstrained,constrained,non-linear,linear,quadratic,non-linearleastsquares,sparseandstructuredobjectiveandmulti-objectiveproblems.Extensiveresearchhasbeencarriedoutindesigninganddevelopingtoolsforoptimizationproblemswithapplicationsinaerospaceengineering.Ingeneral,optimizationproblemsdonothaveanyanalyticalsolutionsandaresolvedusingeithernumericalmethodsorheuristicmethods.Optimizationofatrajectoryusingglobalcollocation[ 2 ]whereniteandinnitehorizonproblemshavebeensolvedusingLegendre-Gauss-Radau(LGR)points.SpacetrajectorieshavebeenoptimizedusingParticleSwarmOptimizationwhichisastochasticmethodinspiredbybehaviorofbirdsandantswhilesearchingforfood[ 3 ].Similarlyamulti-objectiveproblemusingdeterministicCollaborativeRobustOptimizationhasbeendevelopedandisassistedbyanapproximationforuncertainintervals[ 4 ].Theseresearchersusedmethodsinvolvinggradient-basedanddeterministicapproachessincetheobjectivefunctionwascontinuousanddifferentiableoveraspeciedintervalwhichconstrainedthesolutionstobelocal.Problemariseswhentheobjectivefunctioninreal-timeapplicationsarediscontinuousandnon-differentiable.Inliterature,ahandfulofheuristictechniqueshavebeenpublished[ 5 ],[ 6 ],[ 7 ],[ 8 ]whichovercomethedisadvantageofgradient-basedmethods, 10

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discontinuousandnon-differentiableobjectives.Anear-optimalsolution[ 9 ]wasachievedforanEarth-MarsTrajectoryusingastochasticapproach(GeneticAlgorithm).TheliteraturepublishedwastoidentifywhetheraGAOptimizerwassuitableforarealisticmodel.Infact,asimilarapproachusingcollocationandnon-linearprogrammingwassolved[ 10 ].ThemainreasonbehindtheimplementationwasthatGAOptimizerwassimpleandrequirednoinitialguessesorpriorinformation.Similarly,GAwasdevelopedasapreprocessingalgorithmtoformulateaninitialguessofthesolutionforadirectcollocationwithnon-linearprogrammingmethod(semi-DCNLP).SeveralarticlesbasedonanaugmentedGAandANNhavebeenproposed,ofwhichallproblemspertaintotrainingNeuralNetworkforaparticularsetofdesigndata(uidandthermalscienceorstructures).Afterthetraininghasbeenaccomplished,theapproximateddataisusedastheobjectivetobeoptimizedbyGAOptimizer.Toaddresstheproblemdealingwithobjectivefunctionswhicharenotsmoothandhavediscontinuitiesorlargederivatives,anapproachisdevelopedinthispapersuchthatthesolutionisoptimalandconsistent,obtainedusingGeneticAlgorithms(GA)acceleratedbyArticialNeuralNetworks(ANN).GeneticAlgorithmsdependuponnaturalselectionandnaturalgenetics.Initiallyasetofsolutionsarecreatedbyrandomgenerationandthissetiscalledpopulation.Thesolutiontoaspecicproblemischosenfromthiscurrentpopulationdependingontheirbesttness.Newpopulationsarecreatedforeverygenerationbychoosingthebestindividualsasparentsfromcurrentpopulationandusingreproduction,crossoverandmutationoperatorstoproduceoffspring(individuals)fornextgeneration.Overevolvingsuccessivegenerations,thepopulationconvergestoanoptimalsolution.MoreoveraninitialguessofsolutionoranyotherinputprovidedbytheuserisnotrequiredforGeneticAlgorithms,asisthecasefornumericalmethods.GAusesaninitialrangeforpopulationandndsanoptimalsolutionfromthepopulationdependingupontheirbesttnessvalue.Ifaninitialrangeisreallyhugewhencomparedtotherangeinwhichtheoptimalsolution 11

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lies,GAtendstoleadaveryslowoptimizationprocesswhichmightprolongformanylonghoursandneverconvergetoasolution.Theotherpossibilityoftherangebeingtoosmallwillresultinprematureconvergenceofanoptimalsolution.SoitishighlyessentialthataproperinitialrangeischosensuchthatevaluationscarriedoutbyGAareinexpensive.LiteratureworksshowthatdeterminingoptimaltrajectoriesandoptimaldesignparametersofspacevehiclesinminimalcomputationaltimewithlessornoinputfromtheuserishighlyimportantforimprovingtheefciencyofspacecraftsandUnmannedAerialVehicles(UAVs).Inthispaper,wesolveforoptimalsolutionsusingmulti-stageapproachheuristically.Thesolutionsforoptimalcontrolanddesignproblemsareobtainedusingahybridsolver-GeneticAlgorithmsandArticialNeuralNetworks.ANNgeneratestheinitialrangeforGAfromwhichaninitialpopulationiscreatedrandomly.Usingthisinitialrange,GAsolvesforoptimalsolutionsofthecontrolanddesignproblem.Thesolutionsobtainedusingthisapproacharecomparedwithoptimalsolutionsobtainedusingnumericalmethods,hybridizationofGAandasingleANNandGAintheabsenceofANN. 12

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CHAPTER2GENETICALGORITHMS 2.1OverviewGeneticAlgorithmsareheuristicsearchalgorithmsbasedonnaturalgeneticstheory.ItwasdevelopedbyJohnHollandandhisstudentsatUniversityofMichiganin1975[ 11 ].Itisanarticialsystemthatfollowsthemechanismsofnaturalgeneticsandnaturalselection.Themainadvantageofthismethodhasbeenitsrobustnessandexibilityincomplexspaces.Itisknownforitssimplicitywhichinvolvescopying,crossingoverandmutatingthestrings.Theydonotworkwiththeparametersthataretobeoptimizedbutwithacodedformoftheparameterset.Initiallytheparametersarecodedtobinarystrings(individuals).Thecodingofparametersisnotrestrictedtoasinglestringbutalsobyapplyingtransitionrules(Selection 2.2.1 ,Reproduction 2.2.2 andMutation( 2.2.3 )togeneratenewstringsforeachtrial.Theseindividualstogetherformtheinitialpopulationofsize'n',wherenisthenumberoftrialsusedtogeneratenewstrings.Thisisfollowedbysuccessivegenerationofpopulations.Searchingforanoptimalsolutioninapopulationofpoints(parallelcomputing)reducestheprobabilityofndingafalsesolutioninsteadofmovinggingerlyfromasinglepointtoanotherusingcalculus-basedtransitionrulewhichultimatelyleadstolocationoffalsesolutionsinmultimodalsearchspaces.Takingintoconsiderationthedirectcodingofparameters,searchfromapopulationofstrings,andnoneedforauxiliaryinformationtogetheraccountfortherobustnessoftheGAoptimizer. 2.2TransitionRulesGeneticAlgorithmsusesthreemainoperatorstocreatesuccessivegenerationsfromcurrentpopulationare: 2.2.1SelectionTheseareruleswhichselectthebesttindividualsforthegivenobjective.Itfollowsthearticialversionofnaturalselection-Darwiniantheorywhereindividualswithmore 13

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thavebetterpotentialtosurviveandmostpopularlyknownbythephrase'survivalofthettest'.Thesebesttindividualsbecometheparentsofcurrentgenerationandwillreproduceagaintoformthenextgeneration.Therearedifferenttypesoffunctionfortheprocess-Stochasticuniform,Remainder,Uniform,RouletteandTournament.InStochasticuniformtypeoffunction,alineislaidandeachparentcorrespondstoasectionoflinewhichisproportionaltoitsscaledvalue.InRemainderfunction,parentsareassigneddeterministicallyfromanindividual'sscaledvalueandrouletteselectionisusedtofortheremainingfractionalpart.Aftertheparentsareassigned,theyarechosenstochastically.Theprobabilitythataparentisbeingchoseninthismethodisproportionaltoitsfractionalpartofitsscaledvalue.AsforUniformfunction,parentsarechosenaccordingtoexpectationsandnumberofparents.Thismethodisnotaneffectivesearchstrategybutcanbeusedfordebuggingandtesting.InRoulette,parentsarechosenusingasimulatedroulettewheel,wheretheareaofasectionofthewheel(ofanindividual)isproportionaltotheindividual'sexpectation.TheparentsinTournamentfunctionarechosenbychoosingarandomtournamentsizeandbestindividualsoutofthattournamentsetarechosenasparents. 2.2.2ReproductionThisprocessspecieshowthechildrenarebeingcreatedforthenextgeneration.Theyareoftwotypes 2.2.2.1ElitismItdenesthenumberofindividualsthatwillsurvivethenextgenerationandisuserspeciedandcanbedenotedasarealnumber. 2.2.2.2CrossoverAsetofruleswhichswapthecharactersortraitsbetweentwostringsorindividualswhichresultinpartialstringexchanges.Socrossingrandomlyselectedbest-t 14

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individualsresultinanewpopulationconstructedbyvaryingtheirtraits.TherearedifferentoptionsforcrossoverfunctionsuchasScattered,Singlepoint,TwoPoint,Intermediate,HeuristicandArithmetic.InScatteredoption,arandombinarystringiscreated,theonesinthebinarystringarereplacedbytheelementsofparent1,correspondingtoones'position.Inasimilarmannerthezerosofthestringarereplacedbytheelementsofparent2.Forexample,if v1=qwerty(2)and v2=123456(2)areparents,thenthecreatedbitstringbeing b=101100(2)thenthechildwouldbe c=q2er56(2)AsforSinglepointcrossover,arandomnumberfrom1tonischosen,wherenisthenumberofvariablesischosen.Thenthevectorelementslesserthanorequaltotherandomnumberischosenfromparent1,similarlyvectorelementsgreaterthanorequaltorandomnumberischosenfromparent2.Theseelementsareconcatenatedtoformachildvector.Forexample,if v1=qwerty(2)and v2=123456(2)Iftherandomnumberis4,thenthechildis c=qwer56(2) 15

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Two-pointcrossoverissimilartoSingle-pointcrossoverbuttworandomnumbersareselectedfrom1tonumberofvariables.Usingthesameexample,ifv1andv2areparents,then v1=qwerty(2)and v2=123456(2)Iftherandomnumberis2and4,thenthechildis c=qw34ty(2)InIntermediatecrossover,thechildiscreatedbyusingtheweightedaverageoftheparents.Theweightisuser-denedandisasingleparameterwhichcanbeascalaroravector.Thefollowingformulaisusedtocreatethechild. c1=v1+p(v2)]TJ /F5 11.955 Tf 10.95 0 Td[(v1)(2)wherec1isthechild,pistheweight,v2isparent2,andv1isparent1.Iftheweightliesbetweentherange[0;1],thechildrengeneratedliewithinthehypercubecreatedbyparentswhoareplacedinoppositevertices.Ifotherwise,thechildrenlieoutsidethehypercubegenerated.Iftheweightisascalar,thenthechildrenlieinthesamelineconnectingthevertices.AsforHeuristicmethod,thechildrenlieinalinecontainingthetwoparents,averysmalldistancefromthebetterttedparentandawayfromtheworstttedparent.Thedistancefromthebetterttedparentisuserdenedandthisfunctionfollowstheequation c1=v2+h(v1)]TJ /F5 11.955 Tf 10.95 0 Td[(v2)(2)wherec1isthechild,histhedistancefromparent1,v2isparent2(worst-tted),andv1isparent1(better-tted).IntheArithmeticoption,thechildrenarereproducedwhicharethe 16

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weightedmeanoftheactualparents.Thesechildrenarealwaysfeasibleevenwithlinearconstraintsandbounds. 2.2.3MutationMutationisasetofruleswhichrandomlyaltersthevalueinastring.Itisneededbecauseofthefactthatcrossoverandselectionoccasionallymaylosepotentiallyusefulinformationretainedinastring.ThedifferenttypesofmutationareGaussianandUniform.IntheGaussiantype,aGaussiandistributioniscreatedwithameanofzero.Arandomnumberischosenfromthisdistributionandaddedtotheentriesoftheparent.Thedistributioniscreatedusingtheinitialrangespeciedbytheuser.Soiftheinitialrangeisavectoroftworowsandcolumnsequaltothenumberofvariables,thenastandarddeviationisgeneratedusingtheformula SD=psd(vi;2)]TJ /F5 11.955 Tf 10.95 0 Td[(vi;1)(2)wherepsdisaparameterdenedbytheuserwhichcanbeanintegeranddeterminestheinitialvalueofstandarddeviationintherstgeneration,v2isparent2,v1isparent1andiistheco-ordinatecorrespondingtotheparentvectors.Anotherparametergsddeterminesorcontrolsthespreadofthestandarddeviation.Thestandarddeviationforkthisgivenby SDi;k=SDi;k)]TJ /F8 8.966 Tf 6.97 0 Td[(11)]TJ /F5 11.955 Tf 10.95 0 Td[(gsdk generations(2)whereiistheco-ordinatecorrespondingtotheparentvectors.IntheUniformMutationprocess,initiallyafractionofvectorentriesoftheparentsarechosenformutationandeachentryhasaprobabilityrateofbeingmutated.Aftertheentriesarechosen,theywillbereplacedbyarandomnumberselected(uniformselection)fromtherangeofthatentry. 17

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ThereareotherGeneticoperatorsaswell;howevertheabovethreearethemainoperatorswhichhaveprovedtobeeffectiveandefcientinsolvingmanyoptimizationproblems.ThethreemainparameterswhichdecideupontheefciencyofaGAOptimizerarethepopulation,cross-overfractionandmutation.Thenominalratesforcross-overandmutationare0.8and0.035respectively.ItisaknownfactthatGAperformancerequiresthechoiceofahighcrossoverprobabilityandalowmutationpropertyandamoderatepopulationsize. 2.2.4PopulationThepopulationsizedeneshowmanyindividualsaregeneratedforeachgeneration.Thedeterminationofpopulationoccursfromapredenedinitialpopulationrange.Theinitialrangecanbeuserdenedanditspeciestherangeofthevectorsintheinitialpopulationthatistobegenerated.Itisofutmostimportancebecauseofthefactthesearchspaceisafunctionoftheinitialrange.Ifthegivensearchspaceislargeenough,thesolutionsofGAareknowntobesmoothandunimodalorifthesearchspaceisnotlargeenough,itwouldconductanexhaustivesearchandndasolution,butitwouldbealocaloptimumratherthanglobaloptimumandifthesearchspaceisexorbitantlylargetheGAwouldcontinuesolvingandeventuallywouldtakeinnitetimetoarriveatanoptimalsolution.Thetypesofpopulationfortheentriescanbegeneratedasbitstringsorvectors.Ifthepopulationisavector,multiplesubpopulationsaregeneratedwhichequalsthelengthofthevectordened.Insuchcase,Migrationspecieshowmanyindividualsarepassedontothenextgeneration.Theworstindividualsofaspecicpopulationarereplacedbythebestindividualsofanotherpopulation.Therearethreetypesofpossiblewaysinwhichmigrationcanbecarriedout-byDirection,IntervalandFraction.Direction-asthenameindicates,thedirectioninwhichthemigrationshouldtakeplaceisspecied.Itcaneitherbeforwardorboth.Ifmigrationistotakefromnthgeneration,theninforwardmigration,n+1thgenerationisreplacedbynthgeneration. 18

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Ifthemigrationtakesplaceinbothdirectionsthenthen+1thgenerationandn)]TJ /F10 11.955 Tf 11.36 0 Td[(1thgenerationarereplacedbynthgeneration.Interval-Themigrationtakesplaceafteraspecicinterval.Iftheintervalischosentobe10thenmigrationtakesplaceevery10generationsoftheoptimizationprocess.Fraction-Thenumberofindividualswhichcanmigratebetweenpopulationsisdenedbythefractionfunction. 2.2.5FitnessScalingThetnessfunctionvalueswhicharebeingreturnedaftereachgenerationcanbescaled.Rawtnessscoreswillbereturnedbythetnessfunctionandtheycanbescaleddowntovalueswhichwillbesuitablefortheselectionfunctionandthisiscalledtnessscaling.Thedifferenttypesoftnessscalingarerank,proportional,andshiftlinear.Inthetyperank,therawscorearescaledaccordingtheirrank(itspositiondependingonsortedscores).Thistypeofscalingremoveseffectofspreadofthescores.Inproportionalscalingthescaledvalueofanindividualisproportionaltoitsrawtnessscore.Asforshiftlinear,therawscoresarescaledsuchthattheexpectedttestindividualisequaltoaconstant(userdened)multipliedbyaveragetnessscore.TheabovesaidparametersarethedifferentoptionswithwhichaGeneticAlgorithmOptimizercanbedesigned.Parameterscanalsobeuserdenedandcustommadeaccordingtoaspecicproblem.Thediscussedoptionsarethemostcommonlydenedandessentialoptions.Theseclearlydenetherulesusingwhichanoptimizationprocesscanbecarriedoutwithoutanydiscrepancies.Theyimprovetheperformanceofthefunctiontowardsanoptimalpointi.e.anapproachwhichleadstoimprovementtowardsanoptimalpoint.Calculus-basedmethodshavealwaysfocusedonconvergenceandnotontheinterimperformance.Onthecontrary,GeneticAlgorithmsstrivetoimprovethetnessfunctionusingtheabovediscussedoptionstoattainanoptimalsolution. 19

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CHAPTER3ARTIFICIALNEURALNETWORKS 3.1OverviewArticialNeuralNetworksaremathematicalmodelsofhumannervoussystem.Theyarecomposedofarticialneuronsornodes.Theeffectofsynapticterminalsiscarriedoutbyweightfunctionswhichmodulatetheeffectofinputsandthenonlinearityofthenodesisrepresentedbyatransferfunction.Theinputsdenedhereinthepictureareequivalenttoinputsreceivedbythedendritesofahumanneuronandtheoutputtransmittedisequivalenttotheoutputtransmittedbytheaxonofahumanneuron.TheoutputofthearticialneuronisgivenbytheUniversalApproximationProperty yi=Ws(b;xi)+e(3)whereWistheweightvector,xistheinputvector,biasweightb,eistheapproximationerrorandsistheactivationfunction.Thetransmissionofsignalsiscalculatedasaweightedsumoftheinputsandistransformedbythetransferfunction.Byalteringthevaluesoftheweightsaneuralnetworkcanbetrainedtoperformaparticularfunction.Theycanbeessentiallytrainedtoperformdifferentfunctionssuchasttingafunction,recognizingapattern,clusteringdatasets,approximationofdataandcontrollingsystems.Thetransferfunctionsortheactivationfunctionsaremainlyclassiedbythreetypes.TheyareStepfunction,LinearfunctionandSigmoidfunction.TheStepfunctionreturnszeroiftheoutputisequaltozeroorlesserthanzeroandreturnsoneifitisgreaterthanzero.Itrepresentedbytheequation y=8><>:1ifx00ifx<0(3)whereyistheoutputfunctionandxistheinputfunction.TheLinearTransferFunctionusuallyspansbetween)]TJ /F4 11.955 Tf 9.29 0 Td[(to+anddenotedbystraightline.Itisafunctionofasinglevariableinput.Itoutputissameastheinputora 20

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functionofadditionandmultiplicationoftheinput.Theequationwhichrepresentsthefunctionisgivenby y=mx+c(3)wheremandcarerealconstants,ytheoutputandxtheinput.AsfortheSigmoidfunction,isaprogressivefunctionwhichspreadsfromsmallvaluesandwithingivenperiodoftimeacceleratestoreachamaximumvalue.Theequationwhichrepresentsthefunctionisgivenby y(x)=1 1+e)]TJ /F16 8.966 Tf 6.97 0 Td[(x(3)where,ytheoutputandxtheinput.Abasicneuralnetworkhasthreelayersnamelyinputlayer,hiddenlayerandoutputlayer.Theinputlayercanbeascalarinputorinputofvectors.Herethehiddenlayerisfunctionoftheweightswandbiasb.AnANNistobedesignedsuchthattheinputsproduceadesiredoutputandthiswhollydependsupontheweightsofthenetwork.Variousmethodsexisttostrengthentheconnection;oneofthemistosettheweightsexplicitlyandanotheristhatbytrainingtheneuralnetworksandusingalearningrule.InthisparticularpaperweareusingRadialBasisNeuralNetworkstocarryoutapproximationofinitialrangefromthegenerationdataobtainedfromGA.Theapproximationofthegenerationdataresultsingivesanestimatesetofvaluesforthenextgeneration,fromwhichaninitialrangeforthenextgenerationcanbededuced.TheRadialBasisNeuralNetworkhasthreelayersnamelyinput,hiddenradialbasislayerandoutputlinearlayer.TheRadialBasisFunctionisaGaussiandistributionwhichismainlyknownforitsapproximationtechniqueinstatistics.TheNeuralNetworkusedinthisliteratureisforthesameapproximationofclustereddata.Theentireworkingoftheneuralnetisexplainedinthischapter.Thenetinputtothetransferfunctionisthatthevectordistancebetweenclustercentercandinputvectorxmultipliedbybiasb.Thistypeofsettingweightsiscalled 21

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clusteringtechnique.Theconceptisthatpatternspresentintheinputvectorformclusters.ThecenteroftheclustersisknownandtheEuclideandistancebetweentheinputvectorandtheclustercenterismeasured.Astheinputmovesawayfromthecenter,theactivationvaluereduces.Thetransferfunctionisgivenas s(b;x)=e)]TJ /F16 8.966 Tf 6.97 0 Td[(n(3)wheren=kc)]TJ /F5 11.955 Tf 10.95 0 Td[(xkbisthecoefcientofthetransferfunctionofthehiddenlayer,cisthecenterofthecluster,xistheinputvector,bisthespreadofthetransferfunction.Theradialbasisfunctionhasamaximumof1whentheinputis0,i.e.itproduces1whentheinputxisequaltoitsweightw.Thebiasvectorballowsthesensitivityoftheradialbasisneurontobeadjusted.Thefactorn=kc)]TJ /F5 11.955 Tf 10.95 0 Td[(xkbistheweightofthelayerwhichisawidthparameterthatcontrolsthespreadofthecurve.Thehiddenlayertransformstheinputstoanonlinearfunctionfromtheinputspacetothehiddenspace,whereastheoutputlayerappliesalinearspacefromhiddentooutputspace.ThehiddenlayerusestheRadialBasisFunctionwhiletheoutputusesLinearFunction.TheuseofthisapproximationtoapproximatethevaluesobtainedaftertheinitialrunoftheGeneticAlgorithmsSolverfordeterminingtherangeofinitialpopulationwillbediscussedindetailinthecomingchapter. 22

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CHAPTER4PROBLEMFORMULATION 4.1OverviewTherearearetwokindofproblemssolvedinthispapertotesttheworkingoftheproposedalgorithm,developedusingMATLAB.Theobjectiveofoneproblemistondtheoptimaltrajectoryofamovingvehicleusingthecontrolangleandtheotherproblemdealswithndinganoptimaltrajectoryofamodelrocketusingthendesign.Theproblemformulationforthedesignofoptimaltrajectoryisasfollows. InSection( 4.2 ),theequationsofmotionareformulated. InSection( 4.3 ),theboundaryconditionsaredeveloped. InSection( 4.4 ),thecontrolconstraintsareformulated. Section( 4.5 ),describestheobjectivefunctionthatistobeminimized. 4.2EquationsofMotion 4.2.1Problem1TheequationsofmotionforthevehiclemovingatavelocityaoveratEartharegivenasfollows x=u(4) y=v(4) u=acos(b)(4) v=asin(b)(4)wherex(t)andy(t)arethehorizontalandverticalcomponentsofposition,uandvarethehorizontalandverticalcomponentsofvelocitya.bisthecontrolangle. 23

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4.2.2Problem2Theequationsofmotionforthemodelrocketwithpropellantmassmandtotalmass,M,withthrustTactingonit,withadragofD,andmovingwithanaccelerationaisgivenby y=T(t))]TJ /F5 11.955 Tf 10.95 0 Td[(m(t;a)g)]TJ /F5 11.955 Tf 10.95 0 Td[(D(t) m(t;a)(4)wheretheDragequationisgivenby D=1 2rv2SiCdi(4)theThrustisgivenby T=mve(4)themassowrateofthepropellantisgivenby m=m(a)]TJ /F5 11.955 Tf 10.95 0 Td[(g))]TJ /F5 11.955 Tf 10.95 0 Td[(Dve2 ve(4)wherey(t)istheverticalcomponentofposition,visthevelocity,aistheaccelerationoftherocket.veistheexitvelocityofthepropellantwithrespecttotherocket,SiisthesurfaceareaoftherocketpartsandCDistheco-efcientofdrag,Dandgistheaccelerationduetogravity. 4.3BoundaryConditions 4.3.1Problem1Theboundaryconditionsforthevehiclearegivenasfollows x(t0)=0(4) y(t0)=0(4) u(t0)=0(4) v(t0)=0(4) 24

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wheret0istheinitialtime.Next,theterminalconditionsofeachconditionsofeachaircraftaregivenas x)]TJ /F5 11.955 Tf 5.18 -9.69 Td[(tf=xf(4) y)]TJ /F5 11.955 Tf 5.18 -9.69 Td[(tf=yf=0(4) u)]TJ /F5 11.955 Tf 5.18 -9.69 Td[(tf=uf(4) v)]TJ /F5 11.955 Tf 5.18 -9.69 Td[(tf=vf=0(4)wheretf=10isthenaltimeofthemovingvehicle. 4.3.2Problem2Theboundaryconditionsfortherocketaregivenasfollows y(t0)=0(4) v(t0)=0(4) a(t0)=0(4)wheret0istheinitialtime.Next,theterminalconditionsofeachconditionsofeachaircraftaregivenas y)]TJ /F5 11.955 Tf 5.18 -9.69 Td[(tf=yf=0(4) v)]TJ /F5 11.955 Tf 5.18 -9.69 Td[(tf=vf(4) a)]TJ /F5 11.955 Tf 5.18 -9.69 Td[(tf=af=0(4)wheretf=13secondsisthenaltimeofthemovingvehicle. 25

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4.4ControlandDesignConstraints 4.4.1Problem1Thepolynomialusedforparameterizingthecontrolanglebisgivenby bNi=0cifi(t)(4)wheref(t)=1+t+t2+::::andciissomeconstant. 4.4.2Problem2Thedesignconstraintsfordesigningthenofthemodelrocketisgivenbythecenterofgravityandcenterofpressure.Thecenterofpressureisthepointontherocketwherealloftheaerodynamicforcesact.Thiswasdeterminedbysimplifyingthetraditionalcalculations.Thecenterofpressuretimesthetotalprojectedarea,A,isequaltothesumofthecenterofpressureofeachcomponent,d,timestheprojectedareaofeachcomponent,a CpA=anosednose+afuselagedfuselage+afinsdfins(4)Thecenterofgravitywascomputedusingthebasicequationsfordeterminingthecenterofmassofanobjectwithmultiplecomponents. CG=midi M(4)wherethetotalmassoftheobject,M,isequaltothesumoftheproductofeachindividualcomponentsmass,m,withitscorrespondingdistance,d,fromareferenceline. 4.5CostFunctional 4.5.1Problem1Thecostfunctionalforthemovingvehicleistomaximizeitsnalhorizontalposition,whichisgivenby J=)]TJ /F5 11.955 Tf 9.29 0 Td[(xtf(4) 26

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4.5.2Problem2Thecostfunctionalfortherocketisacombinationofthestaticmargin,SMandheightofapogee,givenby J=s 1500)]TJ /F5 11.955 Tf 10.95 0 Td[(y 15002+2)]TJ /F5 11.955 Tf 10.95 0 Td[(SM SM2(4)Usingthesecostfunctionalsthetrajectoriesofboththeproblemsaredeterminedsubjecttothedynamicconstraints(Section( 4.2 )),boundaryconstraints(Section( 4.3 )),controlanddesignconstraints(Section( 4.4 )).TheoptimalproblemsaresolvedusingMATLABandSimulinkversionofGeneticAlgorithmandNeuralNetworksusingdefaultfeasibilityandoptimalitytolerances. 27

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CHAPTER5AUGMENTATIONOFGENETICALGORITHMSANDARTIFICIALNEURALNETWORKS 5.1OverviewOvertheyears,manyworksinliteraturehaveamalgamatedNeuralNetworkswithGeneticAlgorithms.MostoftheseliteraryworkshavebeenusedintheeldsofAerospaceEngineering[ 14 ],[ 20 ],[ 12 ],ComputationalFluidDynamics[ 13 ],[ 15 ],SolidMechanics[ 19 ]orotherDesignproblems[ 17 ],[ 18 ].ThegeneraloutlinesoftheseproblemshavebeenthatanoptimaldesignhasbeenachievedbyapproximatingtheresultsobtainedfromtheGeneticAlgorithmusingNeuralNetworks.TheseworkswhichhavealsobeenoptimizedbyNeuralNetworksaloneorthedataobtainedusingGeneticAlgorithmhavebeenusedfortrainingtheNeuralNetworksandthencreatinganoff-linemethodtoobtainoptimalsolutions.Asfarasthisresearchisconcerned,theoptimizationiscarriedoutbyGeneticAlgorithmsandtheapproximationisdonebyNeuralNetworks,butanovelwayisbeingimplemented.Theproblemaddressedhereisa)processingtimeandb)consistencyofproducingnear-optimalsolutionsofGeneticAlgorithms.Thoughthereareseveralpaperswhichhavepreviouslyaddressedthesameproblem[ 16 ]butthesolutionsprovidedaredifferent.In[ 16 ]theobjectivefunctionvaluesfromGeneticAlgorithmsarestoredinadatabaseandthenareapproximatedtofunctionorsetofvaluesandthenstringsareselectedfromthisparticularapproximation.Themainfeaturetobenotedisthatobjectivevaluesobtainedfromapproximationareinsertedinthepopulationsetofeverygenerationandthevaluesareagainobtainedforeverymodiedgeneration.Throughthissolutionisfeasibleandmakesperfectsense,itseemstohighlyexpensiveandstillaccountstotakeupmuchoftheprocessingtime.Secondly,theideaofspoilingtheentirepopulationishinderingthemainideaofnaturalselectionandreproductionofgenetics.ThemaindesignofthispaperisthatthedataobtainedfromtheGAOptimizerisapproximatedbyaNeuralNetworktodetermine 28

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theinitialrangeofthepopulationofaGA,therebynothinderingthenaturaltheoryofgeneticsandbuildinganinexpensivealgorithmtocarryouttheprocess. 5.2AugmentationFirstlyaGeneticAlgorithmOptimizerisbuiltaccordingtotheuser'sdenition.Toimplementasearchforoptimalsolutionallparametersoftheproblemareinitiallycodedintoachromosomeorstringwhereeachparameterisapartofthestring.The Figure5-1. GeneticAlgorithmFlowchart algorithmforgeneratingaGAOptimizerisasfollows: Step1:Asdiscussedbeforearandominitialpopulationofstringsisgenerated. Step2:Thetnessofeachstring/chromosomeiscalculated. Step3:Thestringsarecheckedforend-conditions.Iftheyhavebeenmet,thestringsareselectedas'best-t'individualsfornalpopulation;elsetheyareselectedtogenerateanewpopulation. 29

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Step4:Apopulationisgeneratedusingselection,crossover,andmutationoperators. Step5:Twostringsareselectedasparentsbasedontheirtnessvaluetogeneratenewsetofoffspring. Step6:Acrossoverprobabilityisusedtocarryoutcrossoverfunctionbetweenthetwoparentstogenerateanewoffspring. Step7:Amutationprobabilityisusedtomutatethenewoffspringgenerated. Step8:Close-theloop;gotoStep2.Afterthebest-tindividualsareobtainedtheyaredecodedtotheiroriginalvalues.ThefactorswhichweretakenintoconsiderationfordevelopingaGAOptimizerwereSelection,Reproduction,MutationandPopulation.InitiallytheGAisallowedtocomputefromainitialrangewhichcaneitherbeexorbitantlylargeorinnitesimallysmall.Theinitialpopulation,generation1isformed Figure5-2. NeuralNetworkFlowchart 30

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fromtheinitialrangespeciedinthesolverandcanbeconsideredtoberangeofgeneration0.Similarly,GeneticAlgorithmsisallowedtocomputegeneration2fromgeneration1followingthetransitionrules.Using, x=generation0astheinputvectorand(5) t=generation1asthetargetvector(5)aNeuralNetworkisbuiltwithahiddenlayerusingthetransferfunction s(V;x)=e)]TJ /F16 8.966 Tf 6.97 0 Td[(n(5)wheren=kc)]TJ /F5 11.955 Tf 10.95 0 Td[(xkVisthecoefcientofthetransferfunctionofthehiddenlayer,cisthecenterofthecluster,xistheinputvector,Vistheweightofthetransferfunction. Figure5-3. NeuralNetworkforevery3generation-ActualModel Nowthesizeofthetransferfunctionisdetermined.Letnrandncbetherespectivevaluesfornumberofrowsandnumberofcolumns.Usingthisinformation,theweightsofthenetworkaredeterminedusing t=We264s(V;x)ones(1;nc)375(5) 31

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Usingthecalculatedweights,theformedNeuralNetwork,andgeneration2asanewinput-itotheNeuralNetwork,anoutputyisformed y=Ws(V;x)+e(5)whereeistheapproximationerror. Figure5-4. NeuralNetworkforevery3generations-Condensedform Themean-squareerror,(MSE)betweenthelinearlayeroutput,yandthetargetvector,tiscalculatedsuchthatweightWandtheapproximationewouldbeadjustedsuchthattheMSEconvergestozero.ANeuralNetworkisbuiltandistrainedforevery3generationsuntiltheoptimizationbyGeneticAlgorithmsleadsanear-optimalsolutionusingtheinitialrangeprovidedbythedevelopedneuralnetworks.TheowchartwhichdepictstheworkingoftheaugmentedsolverisshowninFigure( 5-5 )usingthecondensedformofNeuralNetworksFigure( 5-4 ). 32

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Figure5-5. AugmentationofGAandANN 33

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CHAPTER6RESULTSTheproblemsweresolvedusingtheMATLABversionofGeneticAlgorithmscombinedwithanencodedNeuralNetworkusingMATLAB.Thefollowingtableprovidestheoptionsandinputparametersused Table6-1. GAParametersforProblem1andProblem2 Options/ParametersValues InitialPopulationRangeTBDbyNeuralNetworkPopulationSize20EliteCount2CrossoverFraction0.75Generations100TimeLimit+FitnessLimit-StallGenerations50StallTimeLimit+ToleranceLimit10)]TJ /F8 8.966 Tf 6.97 0 Td[(6CreationFunctionUniformFitnessScalingRankSelectionFunctionStochasticUniformCrossoverFunctionScatteredMutationFunctionGaussianPlotFunctionBest-FitUpperBound-Problem1100LowerBound-Problem1-100UpperBound-Problem210LowerBound-Problem20 Thefollowingguresaretheplotsforproblemwhichdepictstates,control,comparisonofmethodsandbest-t,using a)onlyGA, b)GAandasingleNNforallgenerationscreatedand c)GAwithNNforevery3generationsrespectively.Figures( 6-1 )-( 6-2 )comparethe3differentmethodsfortheoptimalsolutionwhichalmostmatchesresultsobtainedusingaNLPsolver.Figures( 6-3 )-( 6-8 )depictthestateandcontrolplotforthethreedifferentmethods.Fromthecontrolplotsitisclear 34

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Table6-2. GAParametersforProblem1andProblem2 Options/ParametersValues MinimumRocketLength50.62inRocketOuterDiameter2.3inMasspropellant1.2lbsBurntime1.7sDensityofair1.225kg/m3Densityofbalsawood(nmaterial)160.0kg/m3Numberofns3.0Finspan4.6inFinthickness0.25inCd,nose0.02Cd,fuselage0.05Cd,interference0.02Cd,n0.005Ctip0.0inCroot4.6inUpperBound-Problem210.0LowerBound-Problem20.0 Figure6-1. Statesofall3methods thatthemethodofusingNNforevery3generationsofGAyieldsaconsistentandstableresult.TheencircledareasofFigure 6-3 andFigure 6-4 clearlydepictthatthecurvesarenotconsistentforconsecutive5runsofthesolverusingsameparametricconditions. 35

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Figure6-2. Controlofall3methods Figure6-3. States-usingGAalone Figure6-4. States-usingGAandasingleNN 36

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Figure6-5. States-usingGAandaNNforevery3generations InFigure 6-5 wecanobservethatthestatecurveisconsistentandstablefor5consecutiverunsoftheaugmentedsolver.SimilarresultswhereobservedinthecontrolplotsandaredemonstratedinFigures 6-6 6-8 Figure6-6. Control-usingGAalone Figure6-7. Control-usingGAandasingleNN 37

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Figure6-8. Control-usingGAandaNNforevery3generations SimilarresultswereobservedforProblem2andthenalresultsofthestates,andoptimalnparametersaredepictedinFigures 6-9 Table 6 respectively. Figure6-9. Control-usingGAandaNNforevery3generations Table6-3. OptimalDesignParametersobtainedusingGAandNN ParametersValues FinRootChord2.25inFinTipChord1.5inFinSpan2.25inRocketLength74.375in Thefollowingplotsshowthebest-tindividualsfor3methodsusedinthisresearch(Problem1)andthebest-tforProblem2usingGAandNNinevery3generations.FromtheplotsanddiscussionitisevidentthatGAcanbeacceleratedbyNN.Itisalsonotedthatitproducesaglobalresult.NNhelpsGAinavoidingpoorconvergence 38

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ABest-tusingGAalone BBest-tusingGAandasingleNNFigure6-10. BEST-FITIndividuals ABest-tusingGAandNNevery3generations-Problem1 BBest-tusingGAandNNevery3generations-Problem2Figure6-11. BEST-FITIndividuals ornoconvergenceforirregularinitialpopulation.NNapproximatedtheinitialrangetoalmostperfectrangesuchthatsearchspaceisneithertoosmallnortoolarge.Theaugmentedsolverwasobservedtorunforabout126.36secondsforProblem1and 39

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134.61secondsforProblem2,lessthanusingaGeneticAlgorithmswhichtakesaboutseveralhoursbeforeitconvergestoasolution.FromtheseresultsitisevidentthattheaugmentationofGeneticAlgorithmsandNeuralNetworksyieldsaconsistentnear-optimalsolutionusingverylittlecomputationaltime. 40

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CHAPTER7CONCLUSIONFromtheplotsanddiscussionitisclearthatanimplementationofapproximationbyNeuralNetworktoaccelerateGeneticAlgorithmsisfeasible.Apartfromthatitproducesaglobaloptimalresultandisinexpensiveastheothermethodsdenedinliterature.ThevaluesobtainedforProblem2wereusedtodesignthenfortherocketandtherockethadasuccessfulightandreachedamaximumheightof1507feet.TheGeneticAlgorithmacceleratedbyNeuralNetworksdeducedthemaximumheighttobe1543feet.AsfortheProblem1,ithadalocaloptimalsolutiondeducedusingDirectShootingMethodandwhenGeneticAlgorithmswithNNwasusedaglobaloptimalsolutionwasobtained.ItisclearfromtheresultsthatGeneticAlgorithmsiscapableofaglobalsolutionandtoavoiditspoorconvergenceornoconvergenceforirregularinitialpopulationrangestheaugmentationofNeuralNetworksprovedtobesuccessful.TheNeuralNetworkshasapproximatedtheinitialrangetoalmostperfectrangesuchthatthesearchspaceisneithertoolargenortoosmall;itistailor-madeforboththeproblemsdiscussedinthispaper.InChapter2,GeneticAlgorithmswerediscussedindetailalongwiththeoptionsusedinsolvingtheoptimizationproblems.Chapter3gaveanoverviewofArticialNeuralNetworksandthespecicNNusedforapproximationpurposes.InChapter4,theoptimalcontrolandoptimaldesignproblemswereformulatedanddescribed.Chapter5discussedthemethodofhybridizationandtheresultsoftheproblemsofChapter4.Inoneproblemwedevelopedasolutionforoptimalcontrolandintheotherwedevelopedasolutionforoptimaldesignbuttheobjectiveforboththeproblemsweresuchthatmaximumdistancehadtobecovered.Inordertoimprovethefeasibilityoftheoptimizer,betteruser-denedappletscanbedesigned.Sotoimprovetheaccessibility,theaugmentationcanbeusedasabaselinetodevelopaJavaAppletforcontrolengineerswhowishtooptimizetrajectoriesordesignobjectsordesignabusinessproject.SincetheGeneticAlgorithmandNeural 41

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Networkaresimpleinstructureandeasiertounderstand,anappletdesignisnotadifculttask.Modelinganddesigningsuchanappletwouldbethenextgoalinthislineofresearch. 42

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REFERENCES [1] D.Kirk,Optimalcontroltheory:anintroduction,.DoverPublications,2004. [2] P.M.F.C.Garg,D.,Directtrajectoryoptimizationandcostateestimationofnite-horizonandinnite-horizonoptimalcontrolproblemsusingaradaupseudospectralmethod,ComputationalOptimizationandApplications,vol.49,pp.124,2009. [3] M.PontaniandB.CONWAY,Particleswarmoptimizationappliedtospacetrajectories,JournalofGuidance,Control,andDynamics,vol.Vol.33,No.5,,pp.pp.1429.,2010. [4] M.EmmerichandB.Naujoks,Metamodel-assistedmultiobjectiveoptimisationstrategiesandtheirapplicationinairfoildesign,AdaptiveComputinginDesignandManufacture,vol.6,p.pp.249260.,2004. [5] K.HorieandB.Conway,Geneticalgorithmpreprocessingfornumericalsolutionofdifferentialgamesproblems,JournalofGuidance,ControlandDynamics,vol.Vol.27,No.6,p.pp.1075.,2004. [6] H.Y.Peng,F.andA.Ng,Testsofinatablestructureshapecontrolusinggeneticalgorithmandneuralnetwork,Proc.AIAA/ASME/ASCE/AHS/ASCStruc.Struct.Dyn.Mat.Conf,vol.Vol.5,p.pp.31283136.,2005. [7] P.R.WHITAKER,K.W.andR.MARKIN,Specifyingexhaustnozzlecontoursinreal-timeusinggeneticalgorithmtrainedneuralnetworks,AIAAJournal,vol.Vol.31,Issue2,pp.pp273,1993. [8] L.H.K.J.Chen,H.P.,Delaminationdetectionproblemsusingacombinedgeneticalgorithmandneuralnetworktechnique,in10thAIAA/ISSMOMultidisciplinaryAnalysisandOptimizationConference,2004. [9] B.WallandB.Conway,Near-optimallow-thrustearth-marstrajectoriesviaageneticalgorithm,JournalofGuidanceControlandDynamics,vol.Vol.28,No.5,p.pp.1027.,2005. [10] S.TangandB.Conway,Optimizationoflow-thrustinterplanetarytrajectoriesusingcollocationandnonlinearprogramming,JournalofGuidance,Control,andDynamics,vol.Vol.18,No.3,pp.pp.599.,1995. [11] D.Goldberg,Geneticalgorithmsinsearch,optimization,andmachinelearning,.Addison-wesley,1989. [12] H.L.S.Z.Q.K.A..N.K.Raque,A.F.,Multidisciplinarydesignandoptimizationofanairlaunchedsatellitelaunchvehicleusingahybridheuristicsearchalgorithm,EngineeringOptimization,vol.Vol.43,pp.pp.305,2011. 43

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[13] A.Abraham,Metalearningevolutionaryarticialneuralnetworks,Neurocomputing,vol.Vol.56,pp.pp.1.,2004. [14] L.A.N.L.Canavan,G.H.,U.S.D.ofEnergy.OfceofScientic,andT.Information,Optimaldetectionofnear-earthobjects,LosAlamosNationalLaboratory;distributedbytheOfceofScienticandTechnicalInformation,U.S.Dept.ofEnergy,LosAlamos,N.M.;OakRidge,Tenn.,,p.pp.10.,1997. [15] G.CoupeandT.H.Delft,Onthedesignofnear-optimumcontrolprocedureswiththeaidofthelyapunovstabilitytheory,Delft,p.pp.247.,1975. [16] R.DuvigneauandM.Visonneau,Hybridgeneticalgorithmsandarticialneuralnetworksforcomplexdesignoptimizationincfd,InternationalJournalforNumericalMethodsinFluids,,vol.Vol.44,No.11,pp.pp.1257.,2004. [17] T.KobayashiandD.Simon,Hybridneural-networkgenetic-algorithmtechniqueforaircraftengineperformancediagnostics,JournalofPropulsionandPower,vol.Vol.21,No.4,pp.pp.751,2005. [18] E.LaBudde,Adesignprocedureformaximizingaltitudeperformance,ResearchandDevelopmentProjectSubmittedatNARAM,,pp.pp.1,1999. [19] A.Mannelquist,near-eldscanningoptimalmicroscopyandfractalcharacterizationwithatomicforcemicroscopyandothermethods,,Ph.D.dissertation,LuleUniversityofTechnology,,2000. [20] M.W.K.K.Patre,P.,asymptotictrackingforuncertaindynamicsystemsviaamultilayernnfeedforwardandrisefeedbackcontrolstructure,,IEEE,pp.pp.5989.,2007. 44

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BIOGRAPHICALSKETCH SaraswathiNambiisaGraduateStudentinAerospaceEngineeringatUniversityofFlorida(UF).Herresearchinterestsencompassvariousledsliketrajectoryoptimization,neuralnetworks,geneticalgorithmsandrobotdesign.Currently,sheisinvolvedinoptimizingneuronalnetworksofhumanbrains,signalprocessingandoptimalcontrolusingdifferentheuristicandcalculus-basedmethods.SheaimstoworkasaControlsEngineerfororganizationsinareassuchasaerospace,robotics,bioengineeringandautomationinthefuture. 45