Stable neural-network-based adaptive control for sampled-data nonlinear systems

StableNeural-Network-BasedAdaptiveControl

forSampled-DataNonlinearSystems

FuchunSun,Member,IEEE,ZengqiSun,SeniorMember,IEEE,andPeng-YungWoo,Member,IEEE

Abstract—Foraclassofmultiinput–multioutput(MIMO)sampled-datanonlinearsystemswithunknowndynamicnonlinearities,astableneural-network(NN)-basedadaptivecontrolapproachwhichisanintegrationofanNNapproachandtheadaptiveimplementationofthevariablestructurecontrolwithasector,isdeveloped.Thesampled-datanonlinearsystemisassumedtobecontrollableanditsstatevectorisavailableformeasurement.Thevariablestrucurecontrolwithasectorservestwopurposes.OneistoforcethesystemstatetobewithinthestateregioninwhichtheNN’sareusedwhenthesystemgoesoutofneuralcontrol;andtheotheristoprovideanadditionalcontroluntilthesystemtrackingerrormetriciscontrolledinsidethesectorwithinthenetworkapproximationregion.TheproofofacompletestabilityandatrackingerrorconvergenceisgivenandthesettingofthesectorandtheNNparametersisdiscussed.Itisdemonstratedthattheasymptoticerrorofthesystemcanbemadedependentonlyoninherentnetworkapproximationerrorsandthefrequencyrangeofunmodeleddynamics.Simulationstudiesofatwo-linkmanipulatorshowtheeffectivenessoftheproposedcontrolapproach.

IndexTerms—Adaptivecontrol,neuralnetworks,sampled-datanonlinearsystems,variablestructurecontrol.

I.INTRODUCTION

HEinherentparallelismcombinedwiththeabilitytolearnmakesneuralnetworks(NN’s)apowerfulreal-timecontroltool.TheearlyapplicationofNN’sintheidentificationandcontrolofanonlinearsystemisusuallyrealizedthroughgradientmethodlearningbasedonstaticordynamicbackprop-agation[1],[2].Adrawbackofgradient-basedmethodsisthedifficultyencounteredinassuringthestability,robustness,andperformancepropertiesoftheoverallsystem,especiallywhenthecontrollerparametersareadaptedon-line[2],[16].Inanattempttoovercomesomeoftheseproblems,stableNN-basedon-lineadaptivecontrolbothincontinuousanddiscrete-timefornonlinearsystemshasbeenrecentlyinvestigatedbymanyresearches[3]–[24].DependingonthetypesofNN’susedinthecontrolscheme,theseresearchescanbefurtherdividedintotwocategories.OneisbasedonthemultilayerNN’s[3]–[15],andtheotherisbasedonthelinearlyparameterizedNN’s[19]–[24]whichincludealargeclassofNN’ssuchasradialbasisfunction(RBF)-likenetwork,thecerebellar

ManuscriptreceivedOctober10,1996;revisedAugust17,1997andJune10,1998.ThisworkwassupportedbytheNationalScienceFoundationofChinaunderGrant69685002.

F.C.SunandZ.Q.SunarewiththeDepartmentofComputerScienceandTechnology,StateKeyLaboratoryofIntelligentTechnologyandSystems,TsinghuaUniversity,Beijing100084,China.

P.Y.WooiswiththeDepartmentofElectricalEngineering,NorthernIllinoisUniversity,Dekalb,IL60115USA.

PublisherItemIdentifierS1045-9227(98)06828-3.

T

modelarticulationcontroller(CMAC),thebasis(B)-splinenetwork,thewaveletnetwork,andacertainclassoffuzzylogicnetworks.Insomecircumstances,multilayerNN’scanalsoberegardedaslinearlyparameterizedNN’siftheNNweightsarenearthedesiredweights[16].Especially,iftheNNweightsandactivationfunctionsareassumedtobefixedexceptforthoseintheoutputlayerbeinglinearfunctions,multilayerNN’sbecamethelinearlyparameterizedNN’s[3].ThebasiclimitationinapplyingmultilayerNN’stononlin-earsystemcontrolisthattheunknownparametersgothroughnonlinearactivationfunctionssuchthattheadjustableweightsappearnonaffinelywithrespecttothenonlinearitiesoftheNNstructure.TherearenotmanyanalyticalresultsconcerningthestabilitypropertiesofsuchnetworksinproblemsdealingwithlearningindynamicenvironmentuntilrecentworkbyLewisetal.[3]–[4],[12],Chenetal.[5]–[8],Rovithakis[13]andSadegh[14]–[15],especiallytheworkbyJagannathanonmultilayerdiscrete-timeNNcontrollerwithguaranteedperformance[9]–[11].

ThestableNN-basedadaptivecontrolusinglinearlypa-rameterizedNN’swasoriginallyformulatedbyPolycarpouetal.[16]andSanneretal.[19]forcontinuoustimenonlinearsystems.Theirworkinstigatesfurtherresearchesinthisarea[17]–[18],[20]–[23].ByusinglinearlyparameterizedNN’s,linearityoftheparametersholds.ThereforetherigorousresultsofadaptivecontrolbecomeapplicabletotheNNweighttuning,andstableclosed-loopsystemareeventuallyachieved.Usually,Lyapunovstabilitytheory[16]–[23]orpassivetheory[3]isemployedtodesignaclosed-loopcontrolsystemoftheglobalstability.Atypicaleffortinthisrespectistocombinedirect,indirectadaptivemethodswithvariablestructuretoobtainimprovedperformance[19]–[23].However,mostoftheaccomplishmentsinthisareaareforcontinuous-timesystems,andthevariablestructurecontrolemployedintheirNN-basedadaptivecontrolschemesisstatic,i.e.,thevariablestructurecontrolisonlyaswitching-typeoneorasaturation-typeonerelatedtothesystemtrackingerrormetricorasisoftentermed,aswitchingfunctioninvariablestructurecontrol(VSC)theory.ThispaperisconcernedwiththestableNN-basedadaptivecontrolusingtheRBFnetworksforaclassofunknownsampled-datanonlinearsystems.Ofcourse,theresultsob-tainedinthispaperaredirectlyapplicabletoothertypesofthelinearlyparameterizedNN’s.ThecontrolschemeconsideredhereistheintegrationofanNNapproachandtheadaptiveimplementationofthevariablestructurecontrolwithasector[24].Thesector,definedbythestatetrackingerrorsofthesystemandNNbasisfunctions,isareachableregion

1045–9227/98$10.00©1998IEEE

SUNetal.:STABLENEURAL-NETWORK-BASEDADAPTIVECONTROLofattractionaroundtheswitchinghyperplane.ThevariablestructurecontrolwithasectorisconstructedasthemodulationofNNbasisfunctionsandstatetrackingerrorsofthesystemsuchthatitsmagnitudewilldecreasedynamicallyalongwiththeapproachofthesystemtrackingerrormetrictothesectoruntilzerowhenthetrackingerrormetricofthesystemisdriveninsidethesector.Thatis,thevariablestructurecontrolwithasectorisessentiallyadynamicone,whichdistinguishesthepaperfromotherworkbasedonthestaticvariablestructurecontrolwithonlyaswitching-typeoneorasaturation-typeone.Assuch,thevariablestructurecontrolwithasectorservestwopurposes:oneistoprovidetheglobalstabilityoftheclosed-loopsystemwhenthesystemgoesoutoftheneuralcontrol,andtheotheristoimprovethetrackingperformanceofthesystemwithintheNNapproximationregion.Whilethesystemtrackingerrormetriciscontrolledinsidethesector,theNN-basedadaptivecontrolisassuredtobestable.Ontheotherhand,forfeedbacklinearization,howtouseNN’stoapproximatethegeneralcontrollerstructure

isalsoaproblemconsideredbyChen

[5]–[8]andJagannathan[9]–[11],etc.Tosolvetheproblem,differenttechniquesexistintheliteraturethatassurelocalandglobalstabilitywithanadditionalknowledge.UnlikeChenandJagannathan,theapproachusedhereistoestimateinsteadof,whichavoidsmanycomplications.ThispaperoffersrigorousproofsofperformanceintermsoftrackingerrorstabilityandboundedNNweights.ThesettingofthesectorandNNparametersisdiscussed.Simulationstudiesofatwo-linkmanipulatorareusedtoillustratethefeaturesoftheproposedcontrolapproach.

Therestofthispaperisorganizedasfollows.InSectionII,ageneralformulationoftheproblemofdesigninganadaptivecontrollerusingNN’sforunknownmultiinput/multiputput(MIMO)sampled-datanonlinearsystemsisgiven.Asta-bleNN-basedadaptivecontrolstrategyforunknownMIMOsampled-datanonlinearsystemsisformulatedinSectionIII,whereacompletecontrolstructureandthelearningalgorithmforthefreeNNparametersarepresented,andthecompleteproofofstabilityandtrackingerrorconvergenceisofferedinAppendixA.InSectionIV,thesettingofthesectorandNNparametersisgiven.SectionVisdevotedtoanapplicationexample.Finally,SectionVIconcludesthepaper.

II.SYSTEMDESCRIPTION

ConsiderthefollowingMIMOcontinuous-timenonlinearsystem:

(1)

where

independent

coordinates.isthecontrolgainmatrix.

Bothof

andarenonlinearfunctionsofthesystem

statevector.

957

(2)

where

and

(3)

with

where

denotestheLiederivativeof

alongthevectorfield

and[26].

Indiscrete-timesystems,thetrackingerrormetricisdefined

as

958IEEETRANSACTIONSONNEURALNETWORKS,VOL.9,NO.5,SEPTEMBER1998

wherethsubsystemcorrespondingtotheindependent

coordinate

thsubsystem.Bycomparing(7)with(5)and(6),amatchingrelationisobtainedas

(8)

From(3)and(5),itfollowsthat

matrixand

matrixwithzerodiagonalelements.Substituting(10)into(9)gives

(12a)

and

(12b)

Undertheassumptionthatthesystemnonlinearfunctions

(14)

where

thsubsystemis

(15)

where

,wherecouldbeassmallaspossibleby

carefullychoosingNNparameters.

SUNetal.:STABLENEURAL-NETWORK-BASEDADAPTIVECONTROL959

Fig.1.Neural-adaptivecontrolstructure.

whichactasafeedforwardcontroller,areusedtoapproximatetheunknownnonlinearfunctions

withastheknown

upper-boundonthemodelreconstructionerror,whichisusedtoenhancesystemrobustnessagainsttheNNapproximationerror.Themagnitudeoftheslidingcontroleffortistheboundlimitvalueoftheapproximationerror,whichcanbedesignedassmallaspossiblebycarefullychoosingtheNNstructure.

inthefigureandInaddition,thevariable

(21)

where

.Thisoperationisusedinthe“time

delaycontrol”conceptofYoucef-Toumi[29]forsuchdynamicsystemas(1)andin[25]fortheperturbationestimation.Itisshownthatifallthecomponentsinthedynamicsshowslowervariationswithrespecttothesamplinginverval,thisoperationisvalid.Inaddition,anapproximationerrorcausedbythisoperationwillbeincludedinthemodelreconstructionerrorandwillalsobecompensatedbythevariablestructurecontrolwithasector.

Substituting(17)into(11)yields

(22)

Define

(23)

Equation(23)canbewritteninadecoupledformas

(24)

960IEEETRANSACTIONSONNEURALNETWORKS,VOL.9,NO.5,SEPTEMBER1998

where

andthen

(25b)

(26)

wherethefirstthreetermsarethevariablestracturecontrol

components,thelasttermistheslidingcontrol.

isthesignfunction,and

andareswitchingtypecoefficientswhichwill

bedeterminedby(30).

Substituting(26)into(24)yields

(28)

thenthetrackingerrormetricofthesystemwillenterthesectordefinedasbelow

(30)

where

(31)

holds.Where

withasmallvalueof

(32)

thentheNN-basedadaptivecontrolinsidethesectorisstable.

SUNetal.:STABLENEURAL-NETWORK-BASEDADAPTIVECONTROLProof:SeeAppendixA.Remark3:

1)NotefromTheorem1thatthesystemtrackingperfor-manceiscloselyrelatedtothesizeofthesecor.ForanNNwithmorehiddenunits,theboundingconstant

andtheparamters

in(12a),(31)becomes

(34)

whichmeansthatiftheincrement

(35)

with

5)IntheproofofTheorem1(seeAppendixA),

isrequiredtobeaverysmallquantity,

whichimpliestherequirementoftheboundontheNNapproximationerrorcanbealleviatedif

961

thentheconclusionof

Theorem1stillholds.

Proof:If

duringtheNNadaptivelearning.If

theparameters

(37)

where

representtheminimalandmaximalboundsonuncertaintyofparameters,respectively.Whenthesystemtrackingerror

metricisdrivedtoapproachthesector,

isusuallyinasmallmagnitudesuchthatitcanbeassumedthat

withasmallvalueof.Referingto[24],

asimilarsufficientconditionfortheexistenceofthesector,whenthetrackingerrormetricofthesystemapproachesthesector,isgivenfrom(36),(27),and(30)as

[28],and

parameter

962IEEETRANSACTIONSONNEURALNETWORKS,VOL.9,NO.5,SEPTEMBER1998

Thederivationofformula(38)isinAppendixB.Besides,

andareusuallychosenas

andmustsatisfy

-dimensionallatticewiththecenterofabasisfunctionoccuringateverypointonthelattice.Underthisassumption,thenumberofbasisfunctionsdependsexponentiallyoninputdimensions.Thereforeitisnotsuitableforhighdimension.Inpracticalapplicationsthreemethodsareusuallyusedtoselectthestructureparametersofbasisfunctions.Thefirstapproachusesunsupervisedcompetitiveclusteringalgorithmforpositioningthecenters[30],[31].Thesecondapproachsimplytakesthecentresofthebasisfunctionsasnonlinearparametersandtrainsthemusingthenonlineargradientdescentmethod[32],[33].Inthethirdmethod,theinputdataaresetasthecentersofthebasisfunctions.Asmallsubsetofallthepossiblebasisfunctionsisselectedthroughaniterativeorthogonalleast-squaresalgorithm[34].Inthispaper,theunsupervisedcompetitiveclusteringalgorithmand

andothercentersbelongingto

itstopologicalneighborhood

denotestheintegerpartoftheargument,and

3)Updatewidithsby

closestcentersofthecenter

(kg

SUNetal.:STABLENEURAL-NETWORK-BASEDADAPTIVECONTROL963

(a)(b)

Fig.2.Jointangletrackingerrorsintheneural-network-basedadaptivecontrol.(a)Withoutasector.(b)Withasector.

(a)

Fig.3.Controlmoments.(a)Withoutasector.(b)Withasector.

(b)

m.With16levelsforeachinput

variable,thereare256divisionstopositointhecentersofbasisfunctionsaccordingtotheorthogonaldesign.Thus256basisfunctionsarerequiredtoapproximatethenonlinearfunction

Theinitiallearningratesforfurtherupdatingthecentersofthebasisfunctionsarechosenas

-nearestneighborheuristics

Fig.2presentstheangletrackingerrorsduringthefirst40secondsofoperation.Fig.2(a)isobtainedbytheNN-basedadaptivecontrolwithoutasector,whileFig.2(b)withasector.Itcanbeseenthattheintroductionofthevariablestructurecontrolwithasectorimprovestheconvergenceofthetrackingerror.Fig.3(a)and3(b)showsthecorrespondingcontroltorquesforthejointsintheNN-basedadaptivecontrolwithoutasector,andwithasector,respectively.Fig.4depictsthevariablestructurecontrolterm,thatisdefinedin(26),duringthefirst40sofoperation.ItisseenthatastheNN’slearn,themagnitudeofthevariablestructurecontrolterm

9IEEETRANSACTIONSONNEURALNETWORKS,VOL.9,NO.5,SEPTEMBER1998

Fig.4.Thevariablestructurecontroltermduringthefirst40s.

Fig.5.Thevariablestructurecontroltermduringthelast10s.

decreasesgraduallywhilethesystemtrackingerrormetricapproachestothesector,whichmeansthatthefeedforwardcompensationcontrolterm

(A.2)

SUNetal.:STABLENEURAL-NETWORK-BASEDADAPTIVECONTROL965

(a)(b)

Fig.6.Jointangletrackingerrorsintheneural-network-basedadaptivecontrol.(a)Withoutasector.(b)Withasector.

First,theoutsideofthesectorisconsidered,because

(A.3)

and(27)gives

966IEEETRANSACTIONSONNEURALNETWORKS,VOL.9,NO.5,SEPTEMBER1998

Ifitisassumedthat

(A.11)

andfrom(27),wehave

(A.13)

Duetothefactthattheonlynonzeroeigenvalueoftherecursivefunctionis

hasasmallvalueenoughtoassure

thenegativedefintenessof

SUNetal.:STABLENEURAL-NETWORK-BASEDADAPTIVECONTROLItisnotedfrom(18)and(25b)that

respectively.Itfollowsfrom(30)that

967

(A.22)

Bysubstituting(A.22)into(A.21),weobtain

968IEEETRANSACTIONSONNEURALNETWORKS,VOL.9,NO.5,SEPTEMBER1998

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FuchunSun(S’94–M’98)wasborninJiansuProvince,China,in19.HereceivedtheB.S.andM.S.degreesfromNavalAeronauticalEngineeringAcademy,Yantai,P.R.China,in1986and19,respectively,andthePh.D.degreefromtheDepartmentofComputerScienceandTechnology,TsinghuaUniversity,Beijing,China,in1998.

HeiscurrentlyengagedinpostdoctoralresearchintheDepartmentofAutomation,TsinghuaUniversity,Beijing.Hisresearchinterestsincludeintelligentcontrol,neuralnetworks,fuzzysystems,

variablestructurecontrol,nonlinearsystems,androbotics.

ZengqiSun(SM’93)receivedtheB.S.degreefromtheDepartmentofAutomaticControl,TsinghuUni-versity,China,in1966andthePh.D.degreeincontrolengineeringfromtheChalmasUniversityofTechnology,Sweden,in1981.

HeiscurrentlyProfessorandtheViceChairoftheDepartmentofComputerScienceandTechnol-ogy,TsinghuaUniversity,China.Heistheauthororcoauthorofmorethan100papersandeightbooksoncontrolandrobotics.Hiscurrentresearchinterestsincludeintelligentcontrol,robotics,fuzzy

systems,neuralnetworks,andevolutioncomputing.

Dr.SunisalsoacouncilmemberoftheChineseAssociationofAutomation,anexecutivememberofIEEEBeijingSection,andthevicechairoftheBeijingChapter,IEEEControlSystemSociety.

Peng-YungWoo(M’)wasborninShing-hai,China.HereceivedtheB.S.degreeinphysics/electricalengineeringfromFudanUni-versity,Shanghai,China,in1982andtheM.S.degreeinelectricalengineeringfromDrexelUniversity,Philadelphia,PA,in1993.In1988,hereceivedthePh.D.degreeinsystemengineeringfromtheUniversityofPennsylvania,Philadelphia,forresearchoncoordinationamongroboticmanipulators.

HeiscurrentlyatenuredAssociateProfessorin

theDepartmentofElectricalEngineeringofNorthernIllinoisUniversity.Hisresearchinterestsincluderobotics,intelligentcontrol,digitalsignalprocessing,fuzzysystems,neuralsystems,andotherrelatedfields.