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1、Stability, Convergence of Harmonious Particle Swarm Optimizer and Its ApplicationStability,ConvergenceofHarmoniousParticleSwarmOptimizerandItsApplicationPANFeng(潘峰),CHENJie(陈杰),CAITao(蔡涛),GANMinggang(甘明刚),WANGGuanghui(王光芒)(SchoolofInforrnationScienceandTechnology,BeijingInstituteofTechnology,Beijing
2、100081,China)Abstract:Particleswarmoptimizer(PSO),anewevolutionarycomputationalgorithm,exhibitsgoodperfortiaweightarededucedinthispaper.Thevalueoftheinertiaweightwisenhancedto(一1.1).FurthermoreanewadaptivePSOalgorithm-harmoniousPSO(HPSO)isproposedandprovedthatHPSOisaglobalofmodeluncertaintyandnoisef
3、romsensors,HPSOareappliedtooptimizetheparametersoffuzzyPIDcontroller.Theexperimentresultsdemonstratetheefficiencyofthemethods.Keywords:evolutionarycomputation;particleswarmoptimizer;asymptoticstability;globalconvergence;fuzzyPIDCLCnumber:TP181Documentcode:AArticleID:10040579(2021)01003506Particleswa
4、rlTloptimizationalgorithmisapopulationbasedparalleloptimizationtechnique,whichexhibitsgoodperformanceforoptimizationproblems.Therearesomeadjustableparameters,suchastheinertiaweight,accelerationfactor,scaledfactor,andsoon,whichgreatlyinfluencetheconvergenceperpers2-4aboutthestabilityanalysishavebeenp
5、ublishedwithmanyproposedmethods.Attimek+1,theupdateequationsofthei-thparticleinthedthdimensionsearchspaceofthestandardPSOalgorithmaredefinedasfollows:譬=wk+flrid(一z乞)+c2r(户一xS),z=zk+,(1)(2)whichsatisfiesIlVm,wherewistheinertiaweight;zisthecurrentpositionoftheparticle;kisthevelocity;isthepersonalbestp
6、ositionoftheparticle;istheswarmbestpositionamongallparticles;clandc2areaccelerationfactors,respectively;Yidandrgdarerandomnumbersintherange0,1.Asanupperboundofthevelocityvectorineveryepoch,llVcanbepresentedastheLipschitzconditionofparticlesdynamicsystemslz一z乞lV.(3)Inthesearchspace,theparticlesflytot
7、hetargetguidedbytheswarminformationanditsowninformation.Inthispaper,thestandardparticleswalTDoptimizer(PSO)algorithmisanalyzedasadiscretedystabilityarededuced.Onthebasisoftheanalysis,anewadaptivePSOalgorithm-harmoniousPSO(HPReceived20061117SponsoredbytheTeachingandResearchAwardProgramforOutstandingY
8、oungTeacherinHigherEducationInstituteofMOE(20010248)BeijingEducationCommitteeCoorperationBuildingFoundationBiographyPANFeng(1978一),lecturer,andropanfeng126.corft.一35JournalofBeijingInstituteofTechnology,2021,Vo1.so)isproposedandprovedtobeaglobalsearchalgorithm.Furthermore,HPSOareappliedtooptimizethe
9、parametersoffuzzyPIDcontrollerofalinearmotordrivenplatform.1StabilityAnalysisofPSOAdjustableparametersofPSOarealwaysadjustsimplifiedandanalyzedasaconstantcoefficientdvnamxcsystem.Inthissection,thesufficientasymptoticstabilityconditionswithoutLipschitzconditionconstrainsforaccelerationfactorandinerti
10、aweightarededuced.First,alemmaisintr0一duced51.Lemma1Atimevaryingdiscretedynamicsvstemsisdefinedasz(r/+1)=A()z().Thesufficientconditionforasymptoticstabilityofthediscrete-timesystemisthatthereexistM>2and志MsatisfyinglD(T(k,whereT(k,kM)=matnx,PisthespectralradiusandP(T(k,kM)=maxJi靠<1.Basedonlemma
11、1,theorem1canbededuoed.Theorem1Thesufficientconditionforasymptionfactorandinertiaweight硼satisfythefollowingconditions:r/w(一1,1),(4)UIuc(,k+1),(,u)一刀(1+ProofThepressedas1+一仰standardPSOalgorithm=(1+r/w1H一0十一JI1J1-(5)canbeex(6)whereaccelerationfactor=+1,ciri,andinputvectorisP=(P5+k)/.Thetransfermatrixo
12、fEq.(6)isT(k,志一2):IH一一L10JJ1+一仰一r/wL10_JFromlemma1,iftheconditionlD(T(k,k一2)=maxIAiI<1issatisfied,thesystemdescrjbedbyEq.(6)isasymptoticallystable.IET(k,k一2)l=LkL+rlwr/wL一L+r/w(7)whereL:(1+r/w一).ThecharacterjsticequationofEq.(7)isdescribedasD()+2础一Lk(1+)BasedonJurycriterion,ofP(?)<1is+Lkr/+(Vw
13、).(8)thesufficientcondition()<1,【l2础一Lk(1+)+Lk仰七+J<1+Bycalculation,(一1/7,I/v)Eq.(9)andalso1r/w十一仰,2(二()1+一仰().(9)satisfyingAttimek+1,theaccelerationfactor仰andinertiaweightwshouldmeetEq.(4),Eq.(5)toan-sureasymptoticstability.Iftherangeof仰satisfiesUI,k+1,theparticlesys-systemwilldiverge.Infact7i
14、saconstrictioncoefficient,whichmultipliestheinertiaweightandacce1eratlonfactortolimittheirvalues.Generally,itisadoptedas1.2DesignandGlobalConvergenceofHarmoniousPSoSimilartoGAandotherevolutionarycomputing一36一reSnarehS<一)惫)(MAMPANFeng(峰)eta1./Stability,ConvergenceofHarmoniousParticleSwarmOptimizer
15、andItsApplicationtechniques,PSOisfacingadilemmabetweenrapidtheanalysisresultoftheorem1,akindofadaptivePSOisproposedinthispaper.Ateveryepoch,accordingtothecurrentvalueofaccelerationfactor,canbeadjusteddynamically,soastoswaYln,whichexpandsandshrinkscontinuously,cansearchthesolutionspacerepeatedly.Thel
16、ogicalflowofHPS0isCreateandinitializeaPSOswarm;Evaluateeachparticleintheswarm;Iftheavailabletimehasexpired,orreachthetermination,returnthebestsolution.Ifnot,goto;IftheconsecutivefailuretimesexceedCmxorD<DmIn,updateparticlesbasedonEqs.(4)(5),toexpandtheswarm,elseifD>Dm,shrinkthesWarm:Updatepart
17、iclesgivenbyEqs.(4)(5)to.Intheaboveflow,Dsstandsforthediversityoftheswarm,thetermCmxistheupperboundofconsecutivefailuretimes,wherethefailuremeansthecurrentfitnessisworsethanthepreviousbestfitnessoftheswarm.D=Dev(Swarm)isdefinedasthesumofswarmvariance,andDmxdenotesthevarianceofsearchingspace.SOIisand
18、Wets6Drovidedsomeconditionsandresultsfortheglobalconvergenceofrandomsearchalgorithmsasfollows.(H1)f(D(z,)f(z)andifS,f(D(x,)f(),whereisgeneratedfromsamplespace(R,B,);z=D(x,),D:SR一S:SisasubsetofR;Bisthe一algebraofthesubsetofR;istheprobabilitymeaSUre.(H2)Forany(Bore1)subsetAofSwithu(A)>0,thereexists1
19、I1一(A)=0,where=0uisanonnegativemeasuredefinedonB,generallyisLebesguremeasure.Lemma2(Globalsearch)Supposethatfisameasurablefunction,SisameasurablesubsetofR,(H1)and(H2)aresatisfied.Letz0bealimPzR.M=1,oowherePzRe.Mistheprobabilityatstepkandthepointzisgeneratedbythealgorithmintheoptimalityregion.Ithasbe
20、enprovedthatPSOdoesntsatisfy(H2)andlemma2inRef.3.Inthispaper,aconchitzconstraincondition,Eqs.(1)(2)ofthestandardPSOcanberepresentedas=硼乞+十1(Pz),(10)z=zk+1,(11)wherePisahypercubewhosevertexesare户and户.Fromtheabove,wehavetheorem2.Theorem2HPSOisaglobalsearchalgorithm.ProofDfl1nctionofHPSOisdefinedbelow:
21、D(pd,xS)=户(Pkg,d)f(z)z,f(户)>,(z).ItisclearthatEq.(12)satisfys(H1).Attimek,thesupportofthei-thparticleisdefinedas=z+,乞+(户一XS),whereisahyperspherewhosecenteriszkandradiusisP=+(P一k).ForHPSO,searchingisaprocessoftheshrink-expand-shrinkcourse.Moreover,accordingtolemma1,thereisnorestrictiononthevalue,e
22、specially,iftheswarmhasnon1Pt叩S,S1(Pz乞)(13)thenIDkllS,Sill2,whereSandSiaretheupperandlowerboundofsearchingspaceS,respectively.If仰meetsEq.(13),thens,furtherm.reSu:1,u(u:1nS)=u(s).SoVAcS,II1一(A)=oand(H2)=0aresatisfied.一37Accordingtolemma2,HPSOisaglobalsearchalgorithm.3ExperimentsInthissection,itisfocu
23、sedonaservoplatformdirectlydrivenbylinearmotorontheverticaldirection(seeFig.1).Withthisconfiguration,allmechanicaltransmissions.suchasballorleadscrews,rackandpinions,beltsorpulleys,andgearboxesareanceandotherproblemsassociatedwiththesemechanicaltransmissions.Thenominalmodeloftheverticalplatformiside
24、ntifiedfirst,asEq.(14),sothatsomecontrolalgorithmscouldbedesignedoff?linebasedontherood?plication.G(s)=21400s+228.5s+19830(14)However,thereareSomenonlinearfactorsintheservosystems.Fistofall,duringthemovementoflinearmotor,duetothemachinestructural,theperturbationofparameterscausethemodeluncertainty.S
25、econdly,itisthemeasurementnoisecomefromthegyro.Takeintoaccountbothofthem.0.5%sinuSoidalvariationofmodelparametersand0.5%randomdisturbanceofgyrooutputareaddedtothenominamodelexpressedbyEq.(14).Inthissection,HPSOareappliedtooptimizethefuzzyPIDcontrollerforthenominalmodel,firstplant,consideringthosenon
26、linearfactors.ThefuzzyPIDcontrolsystem,combinedwithHPSO,isplottedasFig.2.Fig?2HPSO+fuzzyPIDcontrolsystemAndtheparametersofPIDcontrolleraretunedbytheoutputoffuzzyinferencesystem,asbelow:Kp:Km+AKp,KI:K10+AKI,KD:KD0+AKD.ErrorEanderrorrateEcareselectedastheinputvariable.Thefuzzyrules,membershipfunctionp
27、arametersandtheoutputgainofMamdanifuzzyinguisticvariablesE,EcandoutputlinguisticvariablesAKP,AKI,AKDarealldefinedasNs,Zo,PswhosemembershipfunctionsareGaussianfunctionfinedas一3,3forE,Ec,AKI,andAKD,respectivelyandas一0.06,0.06forAKp.Theparticlevectorisconstructedasz=Km,KIo,K瑚,PE,PEC,PP,PI,PD,wherePE,PE
28、C,PP,PI,PDisthevariablesetsofmembershipfunc33一dimensionvector.Itisconcernedtheovershoot,thetransienttimeandsteadystateerrorofthestepresponsetoevaluatesignedfortheHPSOcanbepresentedasEq.(15).J=f(k)+k=0e(k)l+0.1u(k),(15)r10000t100愚,k=1false,true,(16)wheree(k)isthesystemerror;u(k)isthecon一一38PANFeng(潘峰
29、)eta1./Stability,ConvergenceofHarmoniousParticleSwarmOptimizerandItsApplicationtrolleroutput;f(k),definedasEq.(16),isthenotreachthesteadystatewithinthecertainsimulatrariwise,fisequaltothetransienttime.TheparametervaluesofHPSOaregiveninTab.1.TheparametersoffuzzyPIDestimatedbyHPSOareX=0.1321,2.0648,0.
30、0540,一1.1255,2.1959,0.7671,0.1671,0.5965,0.9883,一0.2542,1.0189,一1.3990,一0.9151,0.9190,一2.2127,0.2234,一0.1607,0.1336,0.1191,一0.1034,0.2819,一0.0100,一0.0207,0.0212.一0.0073,一0.0169,一0.0064,0.4564,1.4293,0.0516,一0.7961,ferencesystem.ko(3)Thestepresponseofthenominalmode(Fig.4)showsgoodperformanceofthecont
31、rollerswithshorttransienttime,withoutovershoot.Consideringtheperturbationofparametersandmeasurementnoise,althoughthereisovershootforthesystem,showninFig.5,thecontrollercanreducetheovershootquicklywithashorttransienttime.-fm/一.|f_lI.t/scontrolledbyfuzzyPIDcontrolledbyfuzzyPIDgoodperformanceoftheoptim
32、izedcontrollerthatcanproduceaccuratecurvetrackingandtherobustnessthatcanachievethedesiredcontrolperformanceevenconsideringthenonlinearfactors.一39一controlledbyfuzzyPID4ConclusionsInthispaper,thestabilitywithoutLipschitzconstraintofPSOparametershasbeenexplored,thesufficientconditionsofasymptoticstabil
33、ityhavebeenobtainedtheoretically,themathematicalrelationbetween砌andisexplored,theinertiaweightWvalueisenhancedto(一1,1).Furthermore,theHPSOhasbeenproposed,whichcanadaptivelyadjustparametersanditsglobalconvergencehasbeenproved.Intheexperimentsection,HPSOisappliedtoducingthosenonlinearfactorsandadoptin
34、grandomsignalsasinput,responsecurvesshowthegoodperformanceoftheoptimizedcontrollersanddemonstratetheeffectivenessofHPS0.References:1KennedyJ,EberhartRC.C/ProceedingofIEEEPartideswaITnoptimizationInternationalConferenceon?-40?-NeuralNetworks.Piscataway,NJ:IEEEServiceCenter,1995:19421948.ClercM,Kenned
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