A Rule-Based Symbiotic MOdified Differential Evolution for Self-Organizing Neuro-Fuzzy Systems

ContentslistsavailableatScienceDirect

Applied

Soft

Computing

jo u r n al h om e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / a s o c

A

Rule-Based

Symbiotic

MOdified

Differential

Evolution

for

Self-Organizing

Neuro-Fuzzy

Systems

Miin-Tsair

Su

a

_,

_Cheng-Hung

_Chen

b

_,

_Cheng-Jian

_Lin

c,∗

_,

_Chin-Teng

_Lin

a

a_{DepartmentofElectricalEngineering,NationalChiao-TungUniversity,Hsinchu300,Taiwan,ROC} b_{DepartmentofElectricalEngineering,NationalFormosaUniversity,YunlinCounty632,Taiwan,ROC}

c_{DepartmentofComputerScienceandInformationEngineering,NationalChin-YiUniversityofTechnology,Taichung411,Taiwan,ROC}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received9March2009

Receivedinrevisedform24March2011 Accepted26June2011

Availableonline8July2011 Keywords: Neuro-fuzzysystems Symbioticevolution Differentialevolution Entropymeasure Control

a

b

s

t

r

a

c

t

This study proposes a Rule-Based Symbiotic MOdified Differential Evolution (RSMODE) for Self-OrganizingNeuro-FuzzySystems(SONFS).TheRSMODEadoptsamulti-subpopulationschemethatuses eachindividualrepresentsasinglefuzzyruleandeachindividualineachsubpopulationevolves sepa-rately.TheproposedRSMODElearningalgorithmconsistsofstructurelearningandparameterlearning fortheSONFSmodel.Thestructurelearningcandeterminewhetherornottogenerateanew rule-basedsubpopulationwhichsatisfiesthefuzzypartitionofinputvariablesusingtheentropymeasure. Theparameterlearningcombinestwostrategiesincludingasubpopulationsymbioticevolutionanda modifieddifferentialevolution.TheRSMODEcanautomaticallygenerateinitialsubpopulationandeach individualineachsubpopulationevolvesseparatelyusingamodifieddifferentialevolution.Finally,the proposedmethodisappliedinvarioussimulations.Resultsofthisstudydemonstratetheeffectiveness oftheproposedRSMODElearningalgorithm.

©2011ElsevierB.V.Allrightsreserved.

1. Introduction

Neuro-fuzzysystems(NFSs)[1–3]havebeendemonstratedto solvingmanyengineeringproblems.Theycombinethecapability ofneuralnetworkstolearnfromprocessesandthecapabilityof fuzzyreasoningunderlinguisticinformationpertainingto numer-icalvariables.Ontheotherhand,recentdevelopmentingenetic algorithms(GAs)hasprovidedamethodforneuro-fuzzysystem design.Geneticfuzzysystems(GFSs)[4–6]hybridizethe approx-imatereasoningoffuzzysystemswiththelearningcapabilityof geneticalgorithms.

GAsrepresenthighlyeffectivetechniquesforevaluating sys-temparametersandfindingglobalsolutionswhileoptimizingthe overallstructure.Thus,manyresearchershavedevelopedGAsto implement fuzzysystems and neuro-fuzzy systems in orderto automatethedeterminationofstructuresandparameters[7–16]. Carseetal. [7]presenteda GA-basedapproach toemploy vari-ablelengthrulesetsandsimultaneouslyevolvesfuzzymembership functionsandrelationscalledPittsburgh-stylefuzzyclassifier sys-tem.Herreraetal.[8]proposedageneticalgorithm-basedtuning approach for the parameters of membership functions usedto definefuzzyrules.Thisapproachreliedonasetofinput–output training data and minimized a squared-error function defined

∗ Correspondingauthor.

E-mailaddress:[email protected](C.-J.Lin).

intermsof thetrainingdata.HomaifarandMcCormick[9] pre-sented amethodthat simultaneouslyfoundtheconsequentsof fuzzyrulesandthecenterpointsoftriangularmembership func-tions in the antecedent using genetic algorithms. Velasco [10] described aMichigan approach whichgeneratesa specialplace whererulescanbetestedtoavoidtheuseofbadrulesforonline genetic learning. Ishibuchiet al. [11] applied a Michigan-style genetic fuzzy system to automatically generate fuzzy IF-THEN rulesfordesigningcompactfuzzyrule-basedclassificationsystems. The genetic learning process proposed is based on the itera-tiverulelearningapproachanditcanautomaticallydesignfuzzy rule-basedsystemsbyCordonetal.[12].AGA-basedlearning algo-rithmcalledstructurallearningalgorithminavagueenvironment (SLAVE)wasproposedin[13].SLAVEusedaniterativeapproachto includemoreinformationintheprocessoflearningoneindividual rule.

Moreover,averyinterestingalgorithmwasproposedbyRusso in [14] which attemptedtocombine allgood features of fuzzy systems,neuralnetworksandgeneticalgorithmforfuzzymodel derivationfrominput–outputdata.Chungetal.[15]adoptedboth neuralnetworksandGAstoautomaticallydeterminethe param-eters offuzzylogicsystems.Theyutilized afeedforwardneural networkforrealizingthebasicelementsandfunctionsofafuzzy controller. In [16], a hybrid of evolution strategies and simu-latedannealingalgorithmsisemployedtooptimizemembership functionparametersandrulenumberswhicharecombinedwith geneticparameters.

1568-4946/$–seefrontmatter©2011ElsevierB.V.Allrightsreserved. doi:10.1016/j.asoc.2011.06.015

searchability and thefastconvergenceability over GAs orany othertraditionaloptimizationapproach,especiallyforrealvalued problems[19].Inaddition,theDEalgorithmhasgraduallybecome morepopularandhasbeenusedinmanypracticalareas,mainly manyresearches[20–24]demonstratedthatDEisrobust,simple inimplementationanduse,easytounderstand,andrequiresonly afewcontrolparametersforparticleswarmoptimization(PSO), originalGAandsomemodifiedGAs.

This study proposes a RSMODE for a SONFS. The neuro-fuzzy system is based on our previous research [25], and combinesafuzzysystemwithafunctional linkneuralnetwork (FLNN)[26]. Theconsequentpartof thefuzzyrules that corre-spondstoanFLNNcomprises thefunctionalexpansionofinput variables.

TheproposedRSMODElearningalgorithmconsistsofstructure learningtogenerateinitialrule-basedsubpopulation,and parame-terlearningtoadjusttheSONFSparameters.Thestructurelearning candeterminewhetherornottogenerateanewrule-based sub-populationwhichsatisfiesthefuzzypartitionofinputvariables. Initially,thereisnotanysubpopulation.Therule-based subpop-ulationisautomaticallygeneratedfromtrainingdatabyentropy measure.Theparameterlearningcombinestwostrategies

includ-Section4presentstheresultsofsimulationsofvariousproblems. Finally,thelastsectiondrawsconclusions.

2. StructureofSONFS

Thissubsection describestheSONFS[25],which usesa non-linearcombinationofinputvariables(FLNN)[26].Eachfuzzyrule corresponds totheFLNN,comprising a functional expansionof inputvariables.TheSONFSmodelrealizesafuzzyif-thenrulein thefollowingform.

Rulej: IF ˆx1 is A1j and ˆx2 is A2j... and ˆxi isAij... and ˆxN is ANj

THEN ˆy_j=

M



k=1

wkjk=w1j1+w2j2+...+wMjM (1)

where ˆxiand ˆy_jaretheinputandlocaloutputvariables,

respec-tively;Aijisthelinguistictermofthepreconditionpart;Nisthe

numberofinputvariables;wkjisthelinkweightofthelocal

out-put;kisthebasistrigonometricfunctionofinputvariables;Mis

thenumberofbasisfunction,andRulejisthejthfuzzyrule.

ThestructureoftheSONFSmodelisshowninFig.1,inwhich linguisticlayerbyaGaussian-typemembershipfunction,Aij(ˆxi),

definedby Aij(ˆxi)=exp



−[ˆxi−mij] 2 2 ij



(2)

wheremijand␴ijarethemeanandvarianceoftheGaussian

mem-bership function, respectively, of the jthterm of the ithinput variablexi.

ThecollectionoffuzzysetsAj={A1j,...,ANj}pertainingtothe

preconditionpartofRulejformsafuzzyregionthatcanberegarded

asa multi-dimensionalfuzzysetwhosemembershipfunctionis determinedby Aj(ˆx)= N



i=1 Aij(ˆxi) (3)

Nodesinconsequentlayeronlyreceivethesignal,whicharethe outputfromrulelayerandafunctionallinkneuralnetwork.Finally, theoutputnodeintegratesalloftheactionsrecommendedbyrule layerandconsequentlayerandactsasadefuzzifierwith,

ˆy=



R j=1Aj(ˆx)ˆy  j



R j=1Aj(ˆx) =



R j=1Aj(ˆx)



_M



k=1 wkjk





R j=1Aj(ˆx) (4)

whereRisthenumberoffuzzyrules, ˆyistheoutputoftheSONFS model,wkjisthecorrespondinglinkweightoffunctionallinkneural

network,andkisthefunctionalexpansionofinputvariables[26].

Thefunctionalexpansionusesatrigonometric polynomialbasis function,givenby[ˆx1 sin(ˆx1) cos(ˆx1) ˆx2 sin(ˆx2) cos(ˆx2)]

fortwo-dimensionalinputvariables.Therefore,Misthenumberof basisfunctions,M=3×N,whereNisthenumberofinputvariables.

3. Arule-basedsymbioticmodifieddifferentialevolution fortheSONFSmodel

ThissectionrepresentstheproposedRSMODEfortheSONFS. TheRSMODEcomprisesstructurelearningandparameterlearning. Thestructurelearningusestheentropymeasurethatdetermines properinputspacepartitioningandfindsthemeanandvarianceof theGaussianmembershipfunctionandthenumberofrules.Next, theinitialrule-basedsubpopulationiscreatedaccordingtoarange ofthemeanandvarianceofthemembershipfunction.The param-eterlearningconsistsofasubpopulationsymbioticevolution(SSE) andamodifieddifferentialevolution(MODE).Eachindividualin eachsubpopulationevolvesseparatelyusingamodifieddifferential evolution.Butinordertoevaluateeachindividual,theindividual iscomposedafuzzysystemusingotherindividuals(rules)inother subpopulations.ThedetailedflowchartoftheproposedRSMODE learningalgorithmispresentedinFig.2.

3.1. Structurelearning

Inthisstudy,wecanfinishthestructurelearningfromtraining datainthefirstgeneration.Thissubsection introducesthe pro-ductionofinitialrule-basedsubpopulation,covering thecoding andinitializationsteps.Thecodingstepinvolvesthemembership functionsandthefuzzyrulesofafuzzysystemthatrepresent indi-vidualsandaresuitableforsubpopulationsymbioticevolution.The initializationstepassignsthenumberofsubpopulationbeforethe evolutionprocessbegins.

3.1.1. Codingstep

ThefirststepinRSMODElearningalgorithmisthecodingofa fuzzyruleintoanindividual.Fig.3showsanexampleofafuzzyrule codedintoanindividualwhereiandjaretheithdimensionandthe jthrule.Fig.3describesafuzzyrulegivenbyEq.(1),wheremijand

␴ijarethemeanandvarianceofaGaussianmembershipfunction,

respectively,andwkjrepresentsthecorrespondinglinkweightof

theconsequentpartthatisconnectedtothejthrulenode.Inthis study,arealnumberrepresentsthepositionofeachindividual. 3.1.2. Initializationstep

Fortrainingdata,findingtheoptimalsolutionisdifficultbecause therangeoftrainingdataiswide.Therefore,thedatamustbe nor-malized.Lettrainingdatabetransformedtotheintervalof[0,1]: ˆxi=

ˆx_i− ˆxi min

ˆx_imax− ˆx_i _min (5)

where ˆxiisthevalueafternormalization; ˆxiisthevectoroftheith

dimensiontobenormalized; ˆx_{i min}istheminimumvalueofvector ˆx_i; ˆx_i _maxisthemaximumvalueofvector ˆx_i.

BeforetheRSMODEmethodisdesigned,theindividualsthat willconstituteRinitialsubpopulationmustbecreated.Thefirst stepinstructurelearningistocreatetheinitialfirstindividualin eachsubpopulationtosatisfythefuzzyrulepartitionofinput vari-ables.Thefuzzyrulepartitionstrategycandeterminewhethera newruleshouldbeextractedfromthetrainingdataanddetermine thenumberoffuzzyrulesintheuniversalofdiscourseofeachinput variable,sinceoneclusterintheinputspacecorrespondstoone potentialfuzzylogicrule.Foreachincomingdata ˆx_i,therulefiring strengthcanberegardedasthedegreetowhichtheincomingdata belongstothecorrespondingcluster.Entropymeasurebetween eachdatapointandeachmembershipfunctioniscalculatedbased onasimilaritymeasure.Adatapointofclosedmeanwillhaslower entropy.Therefore,theentropyvaluesbetweendatapointsand currentmembershipfunctionsarecalculatedtodeterminewhether ornottoaddanewruleintotheinitialfirstindividualandcreatea newrule-basedsubpopulationspace.Forcomputationalefficiency, theentropymeasurecanbecalculatedusingthefiringstrength fromAij(ˆxi)asfollows: EMj=− N



i=1 Dij log2 Dij (6) whereD_ij=exp(uAij(ˆxi) −1_{)and EM} j∈[0,1].AccordingtoEq.(6),

themeasureisusedtogenerateanewfuzzyruleandnew func-tionallinkbasesfornewincomingdataisdescribedasfollows.The maximumentropymeasure

EMmax=max

1≤j≤REMj (7)

isdetermined,whereRisthenumberofexistingrules.IfEMmax≤

EM,thenanewruleandanewrule-basedsubpopulationspaceare generated,whereEM∈[0,1]isaprespecifiedthreshold.

Onceanewrulehasbeengenerated,thenextstepistoassignthe initialfirstindividualinthenewrule-basedsubpopulationbythe initialmeanandvariancetothenewmembershipfunctionandthe correspondinglinkweight.Hence,themean,varianceandweight forthenewrulearesetasfollows:

mij= ˆxi (8)

ij=init (9)

wkj=random[−1,1] (10)

Fig.3. CodingafuzzyruleintoanindividualintheproposedRSMODEmethod.

Thesecondstepistocreateotherindividualsineach subpopula-tionaccordingtoarangeoftheinitialfirstindividual.Thefollowing formulationsshowtheproductionoftheotherindividuals. Mean: Individual [d]=mij+random [0,1]×ij,

where d=1,3,... ,2×N−1 (11)

Variance : Individual [d]₌2_×random [0,1]_×_ij,

where d=2,4,...,2×N (12)

Otherparameters: Individual [d]=random [−1,1],

where d>2×N (13)

where d is thesite of each individual and mij and ij are the

correspondingmeanandvariance,respectively,oftheinitialfirst individual.

3.2. Parameterlearning

Theparameter learningcombines two strategies includinga subpopulationsymbioticevolution(SSE)andamodified differen-tialevolution(MODE).Eachsubpopulationallowstheindividual (rule)itselftoevolvebyevaluatingthecomposedfuzzysystem. Fig.4showsthestructureoftheindividualintheRSMODE.The parameterlearningprocessisdescribedstep-by-stepbelow.

Step1:GeneratetheinitialbestFuzzysystem

Inthisstep,weorderlyselectthefirstindividualfromeach sub-population,andcomposeafuzzysystemastheinitialbestfuzzy system.

Step2:UpdateeachindividualineachsubpopulationusingMODE Inordertoupdateeachindividualineachsubpopulation,weuse amodifieddifferentialevolutiontoselectthebetterindividualto thenextstep.Fig.5givesanexampleoftheMODEprocess.Hence, thisstepcomprisesofthreecomponents–parentchoicephase, offspringgenerationphase,andsurvivorselectionphase.

Fig.5. IllustrationoftheMODEprocessfor8-dimensionalvector.

Step2.1:Parentchoicephase

Eachindividualinthecurrentgenerationisallowedtobreed throughmatingwithotherrandomlyselectedindividualsfromthe subpopulation.Specifically,foreachcurrentindividualxk,g,k=1,

2,...,PS,wheregdenotesthecurrentgenerationandPSdenotes thepopulationsize,threeotherrandomindividualsxr1,g,xr2,gand

xr3,g areselectedfromthesubpopulationsuchthatr1,r2,andr3

∈



1,2,...PS



andk /= r1 /= r2 /= r3.Thisway,aparentpoolof

fourindividualsisformedtobreedanoffspring. Step2.2:Offspringgenerationphase

Afterchoosingtheparents,MODEappliesadifferential oper-ation togenerate a mutatedindividual

v

_k,g+1,according tothe followingequation:

v

k,g+1=xr1,g+(1−F)·(xr2,g−xr3,g)+F·(xbest−xr1,g) (14)

whereF,commonlyknownasscalingfactor,isdefinedasg/Gto controltherateatwhichthesubpopulationevolves,gdenotesthe currentgeneration,Gisthemaximumnumberofgenerations,and xbestisthecorrespondingparameterofthecurrentbestfuzzy

sys-tem.Tocomplementthedifferentialoperationsearchstrategy,then usesacrossoveroperation,oftenreferredtoasdiscrete recombi-nation,inwhichthemutatedindividual

v

_k,g+1ismatedwithxk,g

andgeneratestheoffspringuk,g+1.Theelementoftrialindividual

uk,g+1areinheritedfromxk,gand

v

k,g+1,determinedbyaparameter

calledcrossoverprobability(CR∈[0,1]),asfollows: =

v

kd,g+1, if Rand(d)≤CR

xkd,g, if Rand(d)>CR

(15) whered=1,2...,Ddenotesthedthelementofindividualvectors. Rand(d)∈[0,1]isthedthevaluationofarandomnumbergenerator. Forsearchinginnonseparableandmultimodallandscapes,CR=0.9 isagoodchoice[19]inthisstudy.

Step2.3:Survivorselectionphase

MODEappliesselectionpressureonlywhenselectingsurvivors. First,thecurrentcomposedfuzzysystemembedsthecurrent indi-vidualxk,gintothebestfuzzysystemandthetrialcomposedfuzzy

Fig.6. AmutationoperationintheRSMODE.

systemembedsthetrialindividualuk,g+1intothebestfuzzy

sys-tem.Second,aknockoutcompetitionisplayedbetweenthecurrent composedfuzzysystemandthetrialcomposedfuzzysystem,and thecorrespondingindividualofthewinnerisselected determinis-ticallybasedonobjectivefunctionvaluesandpromotedtothenext phase.Inthisstudy,weadoptafitnessfunction(i.e.,objective func-tion)toevaluatetheperformanceofthesecomposedfuzzysystems. Thefitnessfunctionisdefinedasfollows.

F= 1

1+

1/Nt



N_l=1t (yl− ¯yl)2

(16)

whereylrepresentsthemodeloutputofthelthdata; ¯ylrepresents

thedesiredoutputofthelthdata,andNtrepresentsthenumberof

thetrainingdata.

Step3:Updatethebestfuzzysystem

Comparethefitnessvalueofthecurrentcomposedfuzzy sys-tem,thetrialcomposedfuzzysystemandthebestfuzzysystem.If thefitnessvalueofthecurrentcomposedfuzzysystemexceeds those of the best fuzzy system, then the best fuzzy system is replacedwiththecurrentcomposedfuzzysystem.Ifthefitness valueofthetrialcomposedfuzzysystemexceedsthoseofthebest fuzzysystem,andthenthebestfuzzysystemisreplacedwiththe trialcomposedfuzzysystem.

Step4:Mutation

Aftertheaboveprocessyieldedoffspring,nonewinformationis introducedtotheeachsubpopulationatthesiteofanindividual.As asourceofnewsites,mutationshouldbeusedsparinglybecauseit isarandomsearchoperator.Inthefollowingsimulations,a muta-tionratewassetto1/(2*N+M),meaningthat,onaverage,onlyone trialparameterismutated,whereNisthenumberofinput vari-ables,MisthenumberofbasisfunctionofSOFNS,and2*N+Mis thelengthof eachindividual.Mutationisanoperator that ran-domlyaltersthealleleofanelement.The mutationadoptedin MODEtoyielddiversity.Theindividualsuffersfromamutationto avoidfallinginalocaloptimalsolutionandtoensurethesearching capacityofapproximateglobaloptimalsolution.Fig.6showsthe mutationofanindividual.Themutationvalueisgenerated accord-ingtoEqs.(11)–(13),wheremijandijarethecorrespondingmean

andvariance,respectively,ofthecurrentindividual.Followingthe mutationstep,anewindividualcanbeintroducedintotheeach subpopulation.

4. Simulationresults

ThisstudyevaluatedtheperformanceoftheproposedRSMODE fora SONFS tocontrol nonlinearsystems.Thissectionpresents several examples and compares the performance with that of other methods. In the nonlinear system control problems,

Table1

Initialparametersbeforelearning.

Parameter Value

Populationsize 50

Maximumnumberofgeneration 2000

Crossoverrate 0.9

Mutationrate 1/(2×N+M)

Codingtype Realnumber

SONFS–RSMODEisadoptedtodesigncontrollersinthree

simu-lations–controlofwaterbathtemperaturesystem[28],controlof

theballandbeamsystem[29],andcontrolofbackingupthetruck [30].Table1presentstheinitialparametersbeforelearningused inthethreecomputersimulations.

Example1. Controlofwaterbathtemperaturesystem

Thegoalofthissectionistoelucidatethecontrolofthe temper-atureofawaterbathsystemaccordingto,

dy(t) dt = u(t) C + Y0−y(t) TRC (17) wherey(t)istheoutputtemperatureofthesystemin◦C;u(t)is theheatflowingintothesystem;Y0isroomtemperature;Cisthe

equivalentthermalcapacityofthesystem,andTR isthe

equiva-lentthermalresistancebetweenthebordersofthesystemandthe surroundings.

TRandCareassumedtobeessentiallyconstant,andthesystem

inEq.(17)isrewrittenindiscrete-timeformtosomereasonable approximation.Thesystem

y(k₊1)₌e−˛Tsy(k)₊(ı/˛)(1−e−˛Ts)

1+e0.5y(k)−40 u(k)+[1−e−˛Ts]y0 (18)

isobtained, where␣and ␦aresomeconstantvalues ofTR and

C.Thesystemparametersusedinthisexampleare˛=1.0015e−4, ı=8.67973e−3andY0=25.0(◦C),whichwereobtainedfromareal

waterbathplantconsideredelsewhere[28].Theinputu(k)is lim-itedto0,and5Vrepresentvoltageunit.Thesamplingperiodis Ts=30.

Theconventionalonlinetrainingschemeisadoptedforonline training.Fig.7presentsablockdiagramfortheconventionalonline trainingscheme.Thisschemehastwophases–thetrainingphase andthecontrolphase.Inthetrainingphase,theswitchesS1and S2areconnectedtonodes1and2,respectively,toforma train-ingloop.Inthisloop,trainingdatawithinputvectorI(k)=[yp(k+1)

yp(k)]anddesiredoutputu(k)canbedefined,wheretheinput

vec-toroftheSONFScontrolleristhesameasthatusedinthegeneral inversemodeling[31]trainingscheme.Inthecontrolphase,the switchesS1andS2areconnectedtonodes3and4,respectively, formingacontrolloop.Inthisloop,thecontrolsignal ˆu(k)is gen-eratedaccordingtotheinputvectorI’(k)=[yref(k+1)yp(k)],where

ypistheplantoutputandyrefisthereferencemodeloutput.

A sequence of random inputsignals urd(k)limited to 0 and

5VisinjecteddirectlyintothesimulatedsystemdescribedinEq.

Fig.8.LearningcurvesofbestperformanceoftheSONFS–REMODE,SONFS–RSDE, SONFS–DEandSONFS–GAinExample1.

(18),using theonline training scheme for theSONFS–RSMODE controller. The 120training patternsare selected basedon the input–outputscharacteristicstocovertheentirereferenceoutput. Thetemperatureofthewaterisinitially25◦C,andrises progres-sivelywhenrandominputsignalsareinjected.

In initialization phase, four subpopulations are generated. This dissertation compares the SONFS–RSMODE controller to the SONFS–RSDE controller, the SONFS–DE controller and the SONFS–GAcontroller. Eachofthesecontrollersisappliedtothe water bathtemperaturecontrol system.Theperformance mea-suresincludetheset-pointsregulation,theinfluenceofimpulse noise,andalargeparametervariationinthesystem,andthe track-ingcapabilityofthecontrollers.Fig.8plotsthelearningcurvesof thebestperformanceoftheSONFS–RSMODEcontrollerforthe fit-nessvalue,theSONFS–RSDEcontroller,theSONFS–DEcontroller andtheSONFS–GAcontroller,afterthelearningprocessof2000 generations.

Thefirsttaskistocontrolthesimulatedsystemtofollowthree set-points. yref(k)=

⎧

⎪

⎨

⎪

⎩

35 ◦C, for k≤40 55 ◦C for 40<k≤80 75 ◦C, for 80<k≤120. (19)

Fig. 9(a) presents the regulation performance of the SONFS–RSMODE controller. The regulation performance was alsotestedusingtheSONFS–RSDEcontroller,theSONFS–DE con-trollerandtheSONFS–GAcontroller.Fig.9(b)plotstheerrorcurves of the SONFS–RSMODE controller, the SONFS–RSDE controller, theSONFS–DE controller, and theSONFS–GAcontroller. In this figure,theSONFS–RSMODEcontrollerobtainssmallererrorsthan

Fig.9.(a)FinalregulationperformanceofSONFS–RSMODEcontrollerinwaterbath system.(b)ErrorcurvesoftheSONFS–RSMODEcontroller,SONFS–RSDEcontroller, theSONFS–DEcontroller,andSONFS–GAcontroller.

theotherthreecontrollers.Totesttheirregulationperformance,a performanceindex,thesumofabsoluteerror(SAE),isdefinedby SAE=



k



yref(k)−y(k)



(20)

where yref(k)and y(k) are thereference output and theactual

outputof thesimulated system,respectively.TheSAEvalues of theSONFS–RSMODEcontroller, theSONFS–RSDE controller,the SONFS–DEcontroller, and theSONFS–GAcontroller are 352.66, 352.81,352.91,and 372.85, which values are given inthe sec-ondrowofTable2.TheproposedSONFS–RSMODEcontrollerhas amuchbetterSAEvalueofregulationperformancethantheother controllers.

Thesecond setof simulations isperformed toelucidate the noise-rejectionabilityofthefivecontrollerswhensomeunknown impulse noise is imposed on the process. One impulse noise value−5◦_{C is added tothe plantoutput at the60thsampling}

instant.Aset-pointof50◦Cisadoptedinthissetofsimulations. For the SONFS–RSMODE controller, the same training scheme, trainingdataand learning parametersaswereused inthefirst setof simulations. Fig.10(a) and (b) presents thebehaviors of

Fig.10.(a)BehaviorofSONFS–RSMODEcontrollerunderimpulsenoiseinwater bathsystem.(b)ErrorcurvesofSONFS–RSMODEcontroller,SONFS–RSDEcontroller, theSONFS–DEcontrollerandSONFS–GAcontroller.

theSONFS–RSMODE controller,the SONFS–RSDEcontroller,the SONFS–DEcontrollerandtheSONFS–GAcontrollerunderthe influ-enceofimpulsenoise,andthecorrespondingerrors,respectively. TheSAEvaluesoftheSONFS–RSMODEcontroller,theSONFS–RSDE controller,theSONFS–DEcontroller,andtheSONFS–GAcontroller are270.46,270.76,270.65,and282.21,whichareshowninthe thirdrowofTable2.TheSONFS–RSMODEcontrollerperformsquite well.Itrecoversveryquicklyandsteadilyaftertheoccurrenceof theimpulsenoise.

One common characteristic of many industrial-control pro-cessesisthattheirparameterstendtochangeinanunpredictable way. The value of 0.7×u(k−2) is added to the plant input after the 60th sample in the third set of simulations to test the robustness of the five controllers. A set-point of 50◦C is adopted in this set of simulations. Fig. 11(a) presents the behaviors of the SONFS–RSMODE controller when in theplant dynamicschange.Fig.11(b)presentsthecorrespondingerrorsof theSONFS–RSMODE controller,the SONFS–RSDEcontroller,the SONFS–DEcontrollerandtheSONFS–GAcontroller.TheSAE val-uesoftheSONFS–RSMODEcontroller,theSONFS–RSDEcontroller,

Table2

Comparisonofperformanceofvariouscontrollerstocontrolofwaterbathtemperaturesystem.

SAE=

120



k=1

|yref(k)−y(k)| SONFS–RSMODEcontroller SONFS–RSDEcontroller SONFS–DEcontroller SONFS–GAcontroller

Regulationperformance 352.66 352.81 352.91 372.85

Influenceofimpulsenoise 270.46 270.76 270.65 282.21

Effectofchangeinplantdynamics 262.63 263.21 263.25 270.66

Fig.11. (a)BehaviorofSONFS–RSMODEcontrollerwhenachangeoccursinthe waterbathsystem.(b)ErrorcurvesofSONFS–RSMODEcontroller,SONFS–RSDE controller,theSONFS–DEcontroller,andSONFS–GAcontroller.

theSONFS–DEcontroller,andtheSONFS–GAcontrollerare262.63,

263.21,263.25,and270.66,whichvaluesareshowninthefourth

row of Table 2. The results present the favorable control and disturbancerejectioncapabilitiesofthetrainedSONFS–RSMODE controllerinthewaterbathsystem.

In thefinal setofsimulations, thetrackingcapability ofthe SONFS–RSMODEcontrollerwithrespecttoramp-referencesignals isstudied.Define yref(k)=

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

34 ◦C for k≤30 (34+0.5(k−30)) ◦C for 30<k≤50 (44+0.8(k−50)) ◦C for 50<k≤70 (60+0.5(k−70)) ◦C for 70<k≤90 70 ◦C for 90<k≤120 (21)

Fig. 12(a) presents the tracking performance of the SONFS–RSMODEcontroller.Fig.12(b)presentsthecorresponding errors of theSONFS–RSMODE controller, theSONFS–RSDE con-troller,theSONFS–DE controller,and the SONFS–GAcontroller. TheSAEvaluesoftheSONFS–RSMODEcontroller,theSONFS–RSDE controller,theSONFS–DEcontroller,andtheSONFS–GAcontroller are41.73,42.56,42.92,and62.02,whichareshownin thefifth row of Table 2. The results present the favorable control and disturbancerejectioncapabilitiesofthetrainedSONFS–RSMODE controllerinthewaterbathsystem.Theaforementioned simula-tionresults,presentedinTable2,demonstratethattheproposed SONFS–RSMODEcontrolleroutperformsothercontrollers.

Example2. ControloftheBallandBeamSystem

Fig.12.(a)TrackingofSONFS–RSMODEcontrollerwhenachangeoccursinthe waterbathsystem.(b)ErrorcurvesofSONFS–RSMODEcontroller,SONFS–RSDE controller,theSONFS–DEcontroller,andSONFS–GAcontroller.

Fig.13showstheballandbeamsystem[29].Thebeamismade torotateintheverticalplanebyapplyingatorqueatthecenterof rotationandtheballisfreetorollalongthebeam.Theballmust remainincontactwiththebeam.

Ballandbeamsystemcanbewritteninstatespaceformas

⎡

⎢

⎢

⎢

⎣

˙x1 ˙x2 ˙x3 ˙x4

⎤

⎥

⎥

⎥

⎦

=

⎡

⎢

⎢

⎢

⎣

x2 B(x1x24−G sin x3) x4 0

⎤

⎥

⎥

⎥

⎦

+

⎡

⎢

⎢

⎢

⎣

0 0 0 1

⎤

⎥

⎥

⎥

⎦

u, y=x1 (22) wherex=(x1,x2,x3,x4)T≡(r, ˙r,, ˙) T

isthestateofthesystem andy=x1≡ristheoutputofthesystem.Thecontroluistheangular

acceleration( ¨)andtheparametersB=0.7143andG=9.81areset inthissystem.Thepurposeofcontrolistodetermineu(x)such thattheclosed-loopsystemoutputywillconvergetozerofrom differentinitialconditions.

Accordingtotheinput/output-linearizationalgorithm[29],the control lawu(x) is determinedas follows:for statex, compute

v

(x)=−˛34(x)−˛23(x)−˛12(x)−˛01(x), where 1(x)=x1,

Fig.14.LearningcurvesofbestperformanceoftheSONFS–RSMODE,SONFS–RSDE, SONFS–DE,andSONFS–GAinExample2.

2(x)=x2, 3(x)=−BG sin x3, 4(x)=−BGx4 cos x3, and ˛i are

chosensothats4_+˛

3s3+˛2s2+˛1s+˛0 isaHurwitzpolynomial.

Computea(x)=−BGcosx3andb(x)=BGx24 sin x3;thenu(x)=

[

v

(x)−b(x)]/a(x).

In the simulation herein, the differential equations are solved using the second/third-order Runge–Kutta method. The SONFS is trained to approximate the aforementioned conven-tional controller of a ball and beam system. u(x)=[

v

(x)− b(x)]/a(x)is used togenerate the input/output train pairwith x obtained by randomly sampling 200 points in the region U=[−5,5]×[−3,3]×[−1,1]×[−2,2].Ininitializationphase,14 sub-populationsaregenerated.Thisexamplewassimulated30times. Fig.14 plotsthelearningcurvesofthebestperformanceofthe SONFS–RSMODEcontrollerforthefitnessvalue,theSONFS–RSDE controller,theSONFS–DEcontrollerandtheSONFS–GAcontroller, afterthelearningprocessof2000generations.TheSONFS–RSMODE controllerafterlearningwastestedunderthefollowingfour ini-tialconditions;x(0)=[2.4,−0.1,0.6,0.1]T_{,[1.6,0.05,−0.5,−0.05]}T_,

[−1.6,−0.05,0.5,0.05]T_{and[−2.4,0.1,−0.6,−0.1]}T_.Fig.15plots

theoutputresponsesoftheclosed-loopballandbeamsystem con-trolledby the SONFS–RSMODE controller and theSONFS–RSDE controller.These responsesapproximatethose of thecontroller underthefourinitialconditions.Inthisfigure,thecurvesofthe SONFS–RSMODEcontrollertendquicklytostabilize.Fig.16also showsthebehaviorofthefourstatesoftheballandbeamsystem, startingfortheinitialcondition[−2.4,0.1,−0.6,−0.1]T_.Inthis

fig-ure,thefourstatesofthesystemdecaygraduallytozero.Theresults

Fig.15. ResponsesofballandbeamsystemcontrolledbySONFS–RSMODE con-troller(solidcurves)andSONFS–RSDEcontroller(dottedcurves)underfourinitial conditions.

Fig.16.Responsesoffourstatesoftheballandbeamsystemunderthecontrolof theSONFS–RSMODEcontroller.

Table3

Comparisonofperformanceofvariouscontrollerstocontrolofballandbeam system.

Method SONFS–RSMODE SONFS–RSDE SONFS–DE SONFS–GA Fitnessvalue (Avg) 0.9041 0.8737 0.8516 0.8287 Fitnessvalue (Best) 0.9653 0.9447 0.9441 0.9131

showtheperfectcontrolcapabilityofthetrainedSONFS–RSMODE

controller.TheperformanceoftheSONFS–RSMODEcontrolleris

comparedwiththatoftheSONFS–RSDEcontroller,theSONFS–DE

controller and the SONFS–GA controller. Table 3 presents the

comparison results. The resultsdemonstrate that the proposed SONFS–RSMODEcontrolleroutperformsothercontrollers.

Example3. Controlofbackingupthetruck

Backingatruckintoaloadingdockisdifficult.Itisa nonlin-earcontrolproblemforwhichnotraditionalcontrolmethodexists [30].Fig.17showsthesimulatedtruckandloadingzone.Thetruck positionisexactlydeterminedbythreestatevariables,xandy, whereistheanglebetweenthetruckandthehorizontal,andthe coordinatepair(x,y)specifiesthepositionofthecenteroftherear ofthetruckintheplane.Thesteeringangleofthetruckisthe con-trolledvariable.Positivevaluesofrepresentclockwiserotations ofthesteeringwheelandnegativevaluesrepresent counterclock-wiserotations.Thetruckisplacedatsomeinitialpositionandis backedupwhilebeingsteeredbythecontroller.Theobjectiveof thiscontrolproblemistousebackwardonlymotionsofthetruckto makethetruckarriveinthedesiredloadingdock(xdesired,ydesired)

atarightangle(desired=90◦).Thetruckmovesbackwardasthe

Fig.18.LearningcurvesofbestperformanceoftheSONFS–RSMODE,SONFS–RSDE, SONFS–DEandSONFS–GAinExample3.

steeringwheelmovesthroughafixeddistance(df)ineachstep.

Theloadingregionislimitedtotheplane[0,100]×[0,100]. TheinputandoutputvariablesoftheSONFS–RSMODEcontroller mustbespecified.Thecontrollerhastwoinputs,truckangleand crosspositionx.Whentheclearancebetweenthetruckandthe loadingdockisassumedtobesufficient,theycoordinateisnot consideredasaninputvariable.Theoutputofthecontrolleristhe steeringangle.Therangesofthevariablesx,andareasfollows.

0≤x≤100 (23)

−90◦_≤≤270◦ ₍₂₄₎

−30◦≤_≤30◦ (25)

Fig.19.Themovingtrajectoriesofthetruckwherethesolidcurvesrepresentthesix setsoftrainingtrajectoriesandthedottedcurvesrepresentthemovingtrajectories ofthetruckundertheSONFS–RSMODEcontroller.

Theequationsofbackwardmotionofthetruckare, x(k+1)=x(k)+df cos (k)+cos (k)

y(k+1)=y(k)+df cos (k)+sin (k)

(k+1)=tan−1



l sin (k)+dfcos (k)sin (k)

l cos (k)−dfsin (k)sin (k)



(26)

wherelisthelengthofthetruck.Eq.(26)yieldsthenextstatefrom thepresentstate.

Learninginvolvesseveralattempts,eachstartingfroman ini-tialstateandterminatingwhenthedesiredstateisreached;the

SONFSisthustrained.Ininitializationphase,7subpopulationsare

generated.Thisexamplewassimulated30times.Thefitnessvalue

oftheSONFS–RSMODEisapproximately0.9746andthelearning

curveofSONFS–RSMODEiscomparedwiththoseobtainedusing

theSONFS–RSDE,SONFS–DE,andSONFS–GA,asshowninFig.18.In

Fig.19,thesolidcurvesarethetrainingpathsandthedottedcurves arethepathsthatthetuckrunsunderthecontroloftheproposed controller.Asthisfigureshown,theSONFS–RSMODEcontrollercan smooththetrainingpaths.Fig.20(a)–(d)plotsthetrajectoriesofthe movingtruckcontrolledbytheSONFS–RSMODEcontroller, start-ingatinitialpositions(x,y,)=(a)(40,20,−30◦_{),(b)(10,20,150}◦_),

(c)(70,20,−30◦_{)and(d)(80,20,150}◦_{),afterthetrainingprocess}

hasbeenterminated.Theconsideredperformanceindicesinclude thebestfitnessandtheaveragefitnessvalue.Table4comparesthe results.Accordingtotheseresults,theproposedSONFS–RSMODE controlleroutperformsvariousexistingmethods.

5. Conclusion

This study proposes a RSMODE for a SONFS. The proposed RSMODElearningalgorithmconsistsofstructurelearningto gen-erateinitialrule-basedsubpopulation,andparameterlearningto adjusttheSONFSparameters.TheproposedRSMODElearning algo-rithmallowsthateachindividualineachsubpopulationevolves separatelyusingamodifieddifferentialevolution.The experimen-talresultsdemonstratethattheproposedRSMODEcanobtaina betterperformancethanotherexistingmethodsundersome cir-cumstances.

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