Chaoticinvasiveweedoptimizationalgorithmwithapplicationtoparameteresti - PDF document

1 Chaoticinvasiveweedoptimizationalgorithm
ChaoticinvasiveweedoptimizationalgorithmwithapplicationtoparameterestimationofchaoticsystemsMohamadrezaAhmadi,HamedMojallaliElectricalEngineeringDepartment,FacultyofEngineering,UniversityofGuilan,Rasht,IranarticleinfoArticlehistory:Received12October2010Accepted27May2012Availableonline20July2012 Correspondingauthor.Address:ElectricalEngineeringDepartment,FacultyofEngineering,UniversityofGuilan,P.O.Box3756,Rasht,Iran.Tel.:+981316690276-8;fax:+981316690271.E-mailaddresses:(M.Ahmadi),(H.Mojallali). Chaos,Solitons&Fractals45(2012)1108…1120 ContentslistsavailableatSciVerseScienceDirectChaos,Solitons&FractalsNonlinearScience,andNonequilibriumandComplexPhenomenajournalhomepage:www.elsevier.com/loc variablesisatthecoreofchaoticoptimization.AccordingtothenumericalresultsgiveninRef.Ref.,thechaoticsearchcanescapefromlocaloptimamoreeasilycomparedwithotherstochasticsearchoptimizationalgorithms,andexposessuperiorhill-climbingability.However,COAhasanumberofproblemsinpractice.Chaoticsequencesareextremelysensitiveoninitialcondi-tionsandslowinlocatingtheoptimalareaasthesearchspaceextends.Thus,appropriateinitialvaluesshouldbetunedcarefullybeforehand,andthesolutionspaceshouldbecon“ned.Debatecontinuesaboutalternativeoptimizationstrate-giestoovercomethedetrimentsofCOA.Oneavenuethatresearchershavefollowedintheirattemptto“ndmorerobustalgorithmsisthroughintegratingmeta-heuristicalgorithmswithCOACOA.Inn,authorssuggestedahybridoptimizationalgorithmincorporatingchaosandtabusearch.Zilongetal.al.combinedsomecharacteristicsofsimulatedannealingwiththoseofchaoticoptimization.Xiangetal.al.createdanimprovedPSOalgorithmusingthepiece-wiselinearchaoticmapbringingforwardthecha-oticPSO(CPSO)algorithm.Yongetal.al.combinedtheGAwithCOAinordertobeexploitedinneuralnetworks.Theobjectiveofthispaperistoproposeanoveloptimiza-tionmethodbyincorporatingtheCOAandtheIWOalgo-rithm.Differentchaoticmapsareappliedtothealgorithm,andthemostcomputationalef“cientoneisselectedaccord-ingly.Afterwards,thealgorithmisassessedtooptimizebenchmarkfunctions.ItisshownthattheproposedCIWOmethodoutperformsothermethodslikeIWOandPSO

2 whichrelyonlyonrandomdistribution.Furthe
whichrelyonlyonrandomdistribution.Furthermore,theCIWOalgorithmisutilizedforparametricidenti“cationofchaoticsystems.Inordertofurnishabetterinsightintothecapabil-ityoftheCIWOalgorithm,theresultsobtainedfromothermethods,e.g.IWO,PSO,CPSO,GAandCOA,arealsoincluded.ThecorrespondingresultsverifytheCIWOandthecorre-spondingidenti“cationschemesaccurateness.Thebalanceofthispaperproceedsasfollows.Thesec-ondsectionofthispaperbrie”yreviewsthetraditionalIWOmethod.SectionconsiderstheproposedCIWOalgorithm.ThenumericalresultsestablishedupontheproposedschemearegiveninSection.AparameterestimationstrategyusingtheCIWOalgorithmisproposedandtestedinSection.Thepaperendswithconclusionsin2.Invasiveweedoptimization2.1.KeytermsPriortodescribingtheIWOalgorithm,thekeytermsareexplainedasfollows::eachunitinthecolonywhichencompassesavalueforeachvariableintheoptimizationproblembefore“tnessevaluation.:anyseedthatisevaluatedgrowstoaweedorplant.:avaluecorrespondingtothegoodnessofeachunitafterbeingevaluated.:thesearch/solutionspace.Maximumweedpopulation:aparameterpresetrepre-sentingthemaximumnumberofpossibleweedsinthe“eld.2.2.DescriptionoftraditionalIWOmethodTheprocess”owoftheIWOalgorithmisoutlined1.Randomlydistributetheinitialseedsisthenumberofselectedvariables,overthesearchspace.Consequently,eachseedcontainsrandomvaluesforeachvariableinthesolutionspace.2.The“tnessofeachindividualseediscalculatedaccord-ingtotheoptimizationproblem,andtheseedsgrowtoweedsabletoproducenewunits.3.Eachindividualisrankedbasedonits“tnesswithrespecttootherweeds.Subsequently,eachweedpro-ducesnewseedsdependingonitsrankinthepopula-tion.Theweedswhichhaveacquiredmoreresourceshaveabetterchanceofproducingseeds,andthosewhicharelessadaptedtothe“eldareunlikelytorepro-ducetherebycreatinglessseeds.Thatis,thenumberofseedstobecreatedbyeachweedalterslinearlyfromwhichcanbecomputedusingtheequationgivenbelowNumberofseeds Inwhichisthe“tnessofthweed.,andnotethebestandtheworst“tnessintheweedpopula-tion.Thisstepensuresthateachweedtakepartinthereproductionprocess.4.Thegeneratedseedsarenormallydistributedoverthe“eldwithzeromeanandavaryingstan

3 darddeviationdescribedby arethemaximumnu
darddeviationdescribedby arethemaximumnumberofiterationcyclesassignedbytheuser,andthecurrentiterationnumberrespectively.representthepre-de“nedinitialand“nalstandarddeviations.calledthenonlinearmodulationindex.Thisisarela-tivelycriticalparameterwhichcanin”uencethecon-vergenceperformanceoftheIWOalgorithm.Throughasetofsimulations,ithasbeendiscernedthatthemostappropriatevalueformodulationindexis3(seeFig.1whichportraysthedecreaseinnormalizedstandarddeviationfordifferentnonlinearmodulationindicesasthenumberofiterationcyclesaugment).Havingselectedthissuitablevaluefornonlinearmodulationindex,thealgorithmstartswitharelativelyhighdistri-butionvariancetoguaranteeacompletescanofthesolutionspace.AstheiterationnumberincreasesandM.Ahmadi,H.Mojallali/Chaos,Solitons&Fractals45(2012)1108…1120 thedispersionvariancedwindle,thesearchwouldberestrictedtotheneighborhoodsaroundthereproducingplantwhichhasattainedanaptlevelof“tnesstherebyincreasingtheestimationaccuracy.5.The“tnessofeachseediscalculatedalongwiththeirparentsandthewholepopulationisranked.Thoseweedswithless“tnessareeliminatedthroughcompe-titionandonlyanumberofweedsremainwhichareequaltoMaximumweedpopulation.6.Theprocedureisrepeatedatstep2untilthemaximumnumberofiterationsallowedbytheuserisreached.2.3.AdvantagesanddisadvantagesofimplementingIWOOntheonehand,TheIWOalgorithmcerti“esthatallpossiblecandidateswouldparticipateinthereproductionprocess.Incontrast,mostmeta-heuristicalgorithmswouldnotallowtheless-“ttedindividualstoproduceoffspringsuchastheGA.Besides,theIWOalgorithmisstraightfor-wardanditincludeslessdealofcomputationalburdenun-likeothermethods.Asagoodillustration,onecanconsiderthePSOalgorithm.PSOneedstoupdateboththepositionandvelocityofindividualsineachiterationroundwhichrequiresomeextracalculationsto“ndthebestpositionintheneighborhoodofeachparticleaswellasthewholewhole.Ontheotherhand,incaseofproblemswithanoutsizedsearchspace,onehastoapplyagreaternumberofseedstoeachplantsothatthesearchspacecanbecompletelyin-spected.Thiswouldaugmentthecomputationtimedra-matically,nottomentionthefactthattheproblembecomesevenmoresevereas

4 thenumberofvariablesto Fig.1.Evolutionof
thenumberofvariablesto Fig.1.Evolutionofthestandarddeviationcorrespondingtodifferentvaluesofthemodulationindex. Fig.2.TheFeigenbaumdiagramforlogisticmap.M.Ahmadi,H.Mojallali/Chaos,Solitons&Fractals45(2012)1108…1120 betunedincreases.Thisisbecausethesearchshouldbeperformedinabulkiermulti-dimensionalspace.Inaddi-tion,thegradualreductionofstandardvariancewhichplaysakey-roleintheIWOmethodcanbringaboutimma-tureconvergence,andsometimesitisratherdif“culttomakeanadequatetrade-offbetweenapproximationpre-cisenessandavoidingconvergencetolocaloptima.ForamorecomprehensivecomparisonbetweenIWOandpreviouslysuggestedmeta-heuristicalgorithmstheinterestedreaderisreferredtoRefs.Refs..3.TheChaoticinvasiveweedoptimizationalgorithmInordertoovercometheshortcomingsofIWO,chaoticsearchisintegratedintheIWOalgorithm.Itisworthnot-ingthatchaoticsearchmethodshaveagreaterabilitytoescapefromthelocalminimaminima;therefore,theCIWOalgorithmhasalesserchanceofpre-matureconvergencecomparedtoIWO.Besides,duetogreaterscanningandsearchcapabilities,theimplementationofthechaoticse-quencesprecludestheneedforincreasingthenumberofseedsinthesolutionspace,therebyrelativelyreducingthecomputationalcostoftheoverallalgorithm.Inthissection,“rstlythechaoticmapsutilizedintheCIWOalgorithmareexplained.Afterwards,theproposedCIWOschemeisdescribed.Formoreinformationregardingchaoticdynamicsrefertoto.3.1.Chaoticmaps3.1.1.LogisticmapOneofthesimplestandwell-knownchaoticmapswhichhasbeenutilizedinseveralstudies[20,21,23…26]isthelogisticmapintroducedbySirRobertMayin1976.Animplicationofthisisthepossibilitythatasimpledeter-ministicdynamicsystemcanexposecomplexchaoticbehaviordevoidofanystochasticdisruptions.Thelogisticmapisgivenby[0,4]isthecontrolparameterand[0,1]standsforthechaoticvariable.Bymanipulatingthecontrol Fig.3.Thebifurcationdiagramforsinusoidaliterator. Fig.4.ThepseudocoderepresentingtheCIWOalgorithm.M.Ahmadi,H.Mojallali/Chaos,Solitons&Fractals45(2012)1108…1120 parameteronecandeterminewhetherthesystemisinthechaoticstate,orinthestablestate.Thebifurcationdiagram(orFeigenbaumdiagram)forlogisticmap,whichshowsthedistributi

5 onofagainstdifferentvaluesof,isde-picted
onofagainstdifferentvaluesof,isde-pictedinFig.2.Thechaoticbehaviorofthesequenceisen-suredwhen=4,providedthattheinitialvalueforthechaoticvariable()isintherangeof(0,1)exceptforpoints={0.25,0.5,0.75}.3.1.2.SinusoidalmapThesinusoidalmaporthesinusoiditeratorisde“nedbywhichensureschaoticbehaviorinthespanof(0,1).Fig.3showsthevariationsofthechaoticvariableversuschangesinthecontrolparameter.Asitcanbeseen,theperfor-manceofthesystembecomeschaoticwhen=2.3.ItisapparentformFigs.2and3thatthelogisticmapandthesinusoidalmapareanalogous.3.1.3.TentmapThetentmapexhibitschaoticdynamics.Thismappinggenerateschaoticsequencesindatarange(0,1).Thefol-lowingequationde“nesthetentmap:3.2.ChaoticinvasiveweedoptimizationThegoaloftheoptimizationalgorithmistominimize...subjectto...ThestepsoftheproposedCIWOalgorithmproceedsas Fig.5.Thequadraticfunctionwithtwovariableandaglobalminimumat()=(1,1,0). Table1ParameterattributesoftheCIWOalgorithmforsolvingthequadraticfunctionoptimizationproblem.Maximumweed300.010.000015125 Table2Thenumericalresultsofquadraticfunctionoptimization.MeanMaxMinMedianSDLogisticmap1.00011.00040.99991.00011.6728e41.9545eSinusoidalmap1.00011.00020.99971.00002.1542e41.9053eTentmap1.05651.14560.99641.01650.07110.0362PSO0.99981.00020.99961.00002.7431e42.1116eCPSO1.00031.02100.97521.00316.7839e42.5033eIWO1.01041.12080.98740.99390.09230.0639GA1.05201.23370.75080.79340.17370.1141M.Ahmadi,H.Mojallali/Chaos,Solitons&Fractals45(2012)1108…1120 Fig.6.TheRosenbroksfunctionwithtwovariablesandaglobalminimumat()=(0,0,0). Table3ParametervaluesoftheCIWOalgorithmforsolvingtheRosenbrokfunctionminimizationproblem.Maximumweedpopulation300.0250.000015120 Table4ThenumericalresultsofRosenbrokfunctionoptimization.MeanMaxMinMedianSDLogisticmap63.0175e51.8445e42.702eSinusoidalmap000000Tentmap5.1627e47.4979e43.1353e44.7165e41.6140e42.8507ePSO0.00080.00030.99960.00016.0127e42.7846eCPSO2.8428e45.2628e43.638243.4529eIWO0.00890.07010.00570.01090.08420.0649GA0.03860.15920.00310.06230.12050.1641 Fig.7.TheRastriginsfunctionwithtwovariablesandaglobalminimumat()=(0,0,0).M.Ahmadi,H.Mojallali/Chaos,Solitons&Fr

6 actals45(2012)1108…1120 (1)Initially,set
actals45(2012)1108…1120 (1)Initially,setthemaximumandminimumvalueforeachvariableexploitedintheoptimizationof“tnessfunction.Chaoticallydistributethepioneeringseedsoverthe“eldusingthechaoticmapsdescribedin.Itisworthnotingthatthevariablesshouldbenormalizedtotherangeof(0,1)beforeapplyingachaoticmap.Thenormalizationproce-dureisdescribednext:I.Transformvariablecon“nedinthedatarange(0,1): II.Applythechaoticsequencetothetransformedproducinganewvalue.III.Translateintotherange((2)Evaluateeachweed,andrankthemaccordingtotheir“tnessinthepopulation.(3)ProducenewseedswithrespecttoeachweedsrankinginthepopulationusingEq..Thenewlycreatedseedsaredispersedrandomlyonthe“eldwiththestandarddeviationcomputedbyEq.(4)Thenewlygeneratedseedsarechaoticallydistrib-utedintheneighborhoodofthe”oweringweedusingoneofthechaoticmapsoutlinedinSection.Ifthecurrentchaoticallydistributedseedhasabetterestimationthanthepreviousseed,keepthenewone.Otherwise,thechaoticsequenceiscontin-ued.Bytakingadvantagesofthelocalsearchsuperi-oritiesofchaoticsearch,thealgorithmisguaranteedtoconvergemuchfaster.(5)Theseedsarerankedagain,andthosewithlower“t-nessareeliminatedtoreachthemaximumnumberofweedsallowedwhichispresetbytheuser.(6)Thealgorithmcontinuesatstep3untilmaximumnumberofiterationsisreachedorapredeterminedprecisenesscriterionissatis“ed.ThepseudocodeoftheoverallCIWOalgorithmisgivenFig.44.Numericalresults4.1.QuadraticfunctionThe“rst“tnessfunctionsuggestedisthesimplequa-draticfunctionwhichisquiteprevalentinnonlinearopti-mizationproblemsencounteredinengineering.Thequadraticfunctionisdescribedaswherethevariablesarerestrictedto(10,10).Itisobviousthattheminimaliesin0as=(1,1,,1).Fig.5illustratesa3-Dquadraticfunction.Forthepurposeofoptimization,theparametersoftheCIWOalgorithmaresetasillustratedinTable1.NumericalresultsbasedontheproposedCIWOmethodareprovidedinTable24.2.RosenbrokfunctionThe3-DRosenbrokfunction(alsoknownastheRosen-brokvalleyorbanana)isdepictedinFig.6.Thisnon-convexfunctionhasaspecialinteractionamongitsvariableswhichiswidelyusedforevaluatingoptimizationalgorithms.TheRosenbrokfunctionisgivenasfollowswh

7 erethevariablescanalterintherange(4,4).P
erethevariablescanalterintherange(4,4).Parame-terselectionoftheCIWOalgorithmisaddressedinTable3OptimizationresultsusingCIWOarepresentedinTable44.3.RastriginfunctionRastriginsfunctionisatypicalexampleofmulti-modalfunctionswhichisusedtotestthecapabilitiesofoptimiza-tionmethods.Fig.7showsthe3-DRastriginfunction.EquationbelowexpressestheRastriginfunctionasusedinthisstudy Table5ParameterattributesoftheCIWOalgorithmintheRastriginfunctionoptimizationproblem.Maximumweedpopulation300.010.0000013110 Table6ThesimulationresultsofRastriginfunctionminimization.MeanMaxMinMedianSDLogisticmap52.4195e52.2130e44.8126eSinusoidalmap000000Tentmap0.00120.005400.00338.9421e30.00202PSO0.00390.00810.00190.00560.05420.0112CPSO1.3327e51.5375e43.0027e49.3492eIWO0.03730.04940.00120.02180.09870.0555GA0.08080.01380.00560.03490.0110.0739M.Ahmadi,H.Mojallali/Chaos,Solitons&Fractals45(2012)1108…1120 10cosTheproblemvariablesarecon“nedto5,5).CIWOparametersaregiveninTable5,andthesimulationresultsareaddressedinTable64.4.GriewangkfunctionGriewangkfunctionisacontinuousnon-convexmulti-modalquadratictestfunctionwhichisportrayedinFig.8.Thefunctionisgivenbytheequationgivenbelow x2i4000Y10i¼1cos inwhichtheboundis600,600).Asitisobvious,theminimumofthefunctionoccursat.TheCIWOalgorithmistunedaccordingtotheparametersgivenTable7,andoptimizationresultsareshowninTable84.5.OptimizationproblemswithnonlinearconstraintsInordertofurtherillustratetheusefulnessofthepro-posedCIWOmethod,theperformanceoftheCIWOalgo-rithmistestedagainsttwooptimizationproblemswithnonlinearconstraintsfromRef.Ref..Notethatthesenon-linearconstrainedproblemscanbereadilydealtwithbyaddingastepwhichcheckswhethertheconstraintisfea-sibleforapossiblesolution.TheparameterselectionoftheCIWOalgorithmforbothproblemsprovidedinthissectionissimilartothosegiveninTable1The“rstproblemistominimizeÞ¼ð Subjectto=10and0(=1,2,,10).Theglobalmin-imumislocatedat0005.TheresultsofapplyingCIWOalongwithIWO,PSO,CPSO,andGAaregiveninTable9ThesecondproblemderivedfromfromistominimizeÞ¼ðSubjecttowherein00(=1,2,3)and=3,4,5.Thefeasi-bleregionofsearchspaceconsistsof4disjo

8 intspheres.TheoptimumsolutionisTable10sh
intspheres.TheoptimumsolutionisTable10showstheresultsobtainedutilizingtheCIWO Fig.8.TheGriewangksfunctionwithtwovariablesandaglobalminimumat()=(0,0,0)(a)withoutenlargement(b)mediumenlargement(c)highenlargement. Table7ParametervaluesoftheCIWOalgorithmintheGriewangkfunctionminimizationproblem.Maximumweedpopulation300.0010.000018210M.Ahmadi,H.Mojallali/Chaos,Solitons&Fractals45(2012)1108…1120 Havingassessedthesimulationresultsgiveninthissec-tion,weadoptthesinusoidalmapasthechaoticsequencegeneratorinCIWOmethodology.5.Parameteridenti“cationofchaoticsystemsusingCIWOalgorithm5.1.Theidenti“cationalgorithmTheparameteridenti“cationofchaoticsystemsisanactiveresearchsubjectsubject.Thusfar,severalinvalu-ableinvestigationshavebeenconductedbyresearcherssuggestingtheutilizationofevolutionaryalgorithms,suchasPSO,GA,COA,andetc.,foridenti“cationofchaoticsys-sys-.Considerachaoticdynamicalsystemde-“nedbydenotesthestatevector,representstheun-knownparametervector,and()isthederivativeoperator.Theidenti“cationmethodbasedonCIWOalgorithmisdis-cussednext.5.1.1.PreliminariesEachweedincludesastringofdifferentvaluesofunknownparameters.Thesuggested“tnessfunctionisgivenbelowþþðInwhichistheestimatedthstateattimestepthenumberofsamplesfromthesystemtobeimplementedforidenti“cation.denotesthenumberofstatevariables.Notethatincaseofdiscrete-timechaoticsystemsthe“tnessfunctionconvertstoþþðOncetheinformationaboutthestatesofachaoticsystemisavailable,theidenti“cationalgorithmcanbereadilyapplied.Inthisstudy,thesystemparametersaresetinadvance,andapre-de“nedlevelofnoiseisaddedtothe Table8ThesimulationresultsofGriewangkfunctionminimization.MeanMaxMinMedianSDLogisticmap62.4260e51.7569e62.1494e42.8834eSinusoidalmap000000Tentmap3.5468e40.001044.7771e46.8058e46.8313ePSO0.00010.00040.99980.00021.9984e43.1398e64.9250e53.2636e63.7428e44.8258eIWO0.12680.15440.08430.09970.08990.1392GA0.18350.33360.07310.05420.19810.4325 Table9Thenumericalresultsregardingtheoptimizationproblemde“nedbyEqs.(12)and(13)MeanMaxMinMedianSDLogisticmap0.31720.31850.30290.31630.0332Sinusoidalmap0.31620.31710.31580.31670.0002Tentmap0

9 .31830.32110.29870.30990.0013PSO0.31600.
.31830.32110.29870.30990.0013PSO0.31600.34370.29680.30030.0089CPSO0.31580.31990.30030.31350.0314IWO0.33260.35280.26430.19970.0979GA0.42110.57310.23740.21140.1109 Table10Thenumericalresultsregardingtheoptimizationproblemde“nedbyEqs.(14)and(15)MeanMaxMinMedianSDLogisticmap4.99815.02464.97774.99730.0012Sinusoidalmap4.99995.00234.98974.99860.0002Tentmap4.98735.06824.89854.99690.0231PSO4.93935.09994.89934.89760.0013CPSO4.99725.01844.91934.99850.0019IWO4.80015.10664.81854.95870.0837GA4.72735.38274.52924.95730.1042M.Ahmadi,H.Mojallali/Chaos,Solitons&Fractals45(2012)1108…1120 system.Then,thesystemissimulatedandstatevaluesarecalculatedforeachtimestep.5.1.2.ThealgorithmI.Tobegin,asetofinitialvaluesareassignedtoeachparameter.Afterwards,thesevaluesareencodedintoweeds.Thiscanbedoneasitisdescribedinthe“rststepoftheCIWOalgorithm.II.FollowingtheCIWOsteps,theranking,thechaoticsearchandtheexecutionstagesareperformedaccordingly.III.Whenadesiredlevelofprecision;i.e.,isverysmallnumberclosetozero,orthemaximumnumberofiterationcyclesisreachedthealgorithmIV.Theoptimumvaluesofparametersareextractedfromthebestweedwhichhasmostaptlyminimizedthe“tnessfunction.5.2.Examples5.2.1.Rösslerssystem(Rösslerattractor)In1976,OttoE.Rösslerintroducedasimplecontinuous-timedynamicalsystemwhichdisplayedchaoticperfor-mance.Thesystemisreportedlyveryhelpfulinmodelingtheequilibriumofchemicalprocessesprocesses.RösslerssystemofdifferentialequationsisTheparameters,anddeterminethesystemsevolu-tion.Originally,Rösslerstudiedthecaseinwhich=0.2,=0.2,and=5.7;however,inthisinvestigation=0.1,=0.1,and=14arechosen,sincethissetofvaluesaremorefrequentlyused.Thenoiselevelischosentobe0.01whichisde“nedbythenoisestandarddeviationratiodividedbystandarddeviationofthenoise-freesystem.ThephasediagramsofthesimulatedRösslersystemaregivenFig.9.Theidenti“cationalgorithmasdiscussedinsec-.isapplied.ThealgorithmsettingsandestimatedparametersareprovidedinTables11and12,respectively.5.2.2.Lorenzssystem(Lorenzattractor)TheLorenzssystemwasproposedbyEdwardLorenzin1963ashewasundertakingresearchonweatherpredic-tion.Thedifferentialequati

10 ons,actually,designatesamathematicalmode
ons,actually,designatesamathematicalmodelforthermalconvection.Themodel Fig.9.TrajectoriesofRosslerssystemin-plane.(b)Rosslersattractorin-plane.(c)Rosslerssystemin Table12Parameteridenti“cationresultsofRosslerssystem.abcJCIWO0.10020.099913.99983.7413eIWO0.11480.101514.108718.9478CPSO0.10720.104514.02163.2738ePSO0.10000.100313.99938.1234eCOA0.10880.100713.754241.3926GA0.39360.240112.995989.4827 Table11ParameterselectionoftheCIWOalgorithmforparameterestimationofRosslerssystem.Maximumweed500.010.0000015120M.Ahmadi,H.Mojallali/Chaos,Solitons&Fractals45(2012)1108…1120 involvesdescriptionsofheatdistribution,themotionofviscous”uids(atmosphere),andthedrivingforceofther-malconvectionconvection.TheLorenzssystemisexpressedasThevaluesforsystemparametersarelistedbelowwhichensureachaoticbehaviorBysettingthesystemparametersasgivenaboveandaddingthesamenoiselevelasinthepreviousexample,thesystemperformanceissimulated.Thesystemtrajecto-riesaregiveninFig.10TheCIWOalgorithmasdiscussedisapplied.Tables13and14providetheidenti“cationresultsandtheCIWOalgorithmsparameterselection.5.2.3.HenonsSystemTheHenonSystemisoneofthemostwellstudiedchaoticdiscrete-timedynamicsystemssystems.ThedynamicsofthesystemaregovernedbyHenonmappossesschaoticbehavioras=1.4and=0.3.ThesystemissimulatedandthephasespacediagramofHenonmapisdepictedinFig.11.Theidenti“cationresultsandexperimentattributesarelistedinTables15and166.ConclusionsInthispaper,anewstraightforwardhybridoptimiza-tionalgorithmisproposedcombiningtheCOAandtheIWO.Thecapabilitiesoftheoptimizationalgorithmareveri“edthroughsolvingaseriesofoptimizationproblemsusingmulti-dimensionalbenchmarkfunctions.Asigni“-cantapplicationputforwardinthisstudyisthatthepro-posedCIWOalgorithmcanbeimplementedforthepurposeofparameteridenti“cationofchaoticsystems.TheresultsbasedontheproposedschemearecomparedwiththoseoftheIWO,COA,PSO,CPSO,andGA,whereinallcasestheCIWOstrategycontributestosuperiorestima-tionperformance.Asuggestedtopicforfurtherresearchis Fig.10.TheplotoftrajectoryLorenzssystem-plane.(b)TheplotoftrajectoryLorenzssystem-plane.(c)TheplotoftrajectoryLore

11 nzs Table13ParameterselectionoftheCIWOa
nzs Table13ParameterselectionoftheCIWOalgorithmforparameterestimationofLorenzssystem.Maximumweed700.010.0000018135 Table14ParameterapproximationresultsofLorenzssystem.abcJCIWO10.000028.00022.66663.846eIWO10.488427.68262.878211.5623CPSO10.002028.01012.66763.3359ePSO10.000127.99822.66677.641eCOA11.413326.30163.1582284.6153GA9.596829.54272.0152344.1973M.Ahmadi,H.Mojallali/Chaos,Solitons&Fractals45(2012)1108…1120 toexploittheCIWOalgorithmtocopewiththechallengingoptimizationproblemsariseinengineeringapplications.[1]KulkarniRV,VenayagamoorthyGK.Bio-inspiredalgorithmsforautonomousdeploymentandlocalizationofsensornodes.IEEETransSystManCybernetC2010;99:1…13.[2]JatmikoW,SekiyamaK,FukudaT.APSO-basedmobilerobotforodorsourcelocalizationindynamicadvection…diffusionwithobstaclesenvironment:theory,simulation,andmeasurement.IEEEComputIntellMag2007;2(2):37…51.[3]LiBB,WangL,LiuB.AneffectivePSO-basedhybridalgorithmformultiobjectivepermutation”owshopscheduling.IEEETransSystManCybernetA2008;38(4):818…31.[4]DoringoM,BirattariM,StutzleT.Antcolonyoptimization.IEEEComputIntellMag2006;1(4):28…39.[5]MehrabianAR,LucasC.Anovelnumericaloptimizationalgorithminspiredfromweedcolonization.EcolInform2006;1:355…66.[6]KarimkashiS,KishkAA.Invasiveweedoptimizationanditsfeaturesinelectromagnetics.IEEETransAntennasPropag2010;58(4):1269…78.[7]MehrabianAR,Youse“-KomaA.Anoveltechniqueforoptimalplacementofpiezoelectricactuatorsonsmartstructures.JFranklinInst2009. [8]ZhangX,WangY,CuiG,NiuY,XuJ.ApplicationofanovelIWOtothedesignofencodingsequencesforDNAcomputing.ComputMathAppl2009;57:2001…8.[9]LorenzEN.Deterministicnonperiodic”ow.JAtmosSci[10]MuraliK,LakshmananM.Observationofmanybifurcationsequencesinadrivenpiecewise-linearcircuit.PhysLettA[11]MatsumotoT,ChuaLO,KomoroM.Birthanddeathofthedoublescroll.PhysicaD1987;24(1-3):97…124.[12]HaghighiHS,MarkaziAHD.ChaospredictionandcontrolinMEMSresonators.CommunNonlinearSciNumerSimul2010;15:[13]ZilongG,SunanW,JianZ.Anovelimmuneevolutionaryalgorithmincorporatingchaosoptimization.PatternRecognLett2006;27:[14]LuHJ,ZhangHM,MaLH.Anewoptimizationalgorithmbasedonchaos.J

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