Trust-region
Vl01.7.No.4EARTHQUAKEENGINEERINGANDENGINEERINGVIBRATIONDecember,2008
;EarthqEng&EngVib(2008)7:447—464
;Trust-regionbasedinstanta
;DoI:10.1007/s11803.008.1002—9
;neousoptimalsemi--activecontroloflong--
;spanspatiallyextendedstructureswithMRF-04Kdamper
;LinWei”.LiZhongxian1,2{andDingYang,{
;1.SchooloJCivilEngineering,TianjinUniversity,Tianjin300072China ;2.TianjinKeyLaboratoryofCivilEngineeringStructuresandNewMaterials,Tianjin300072,China
;Abstract:Inthefieldofcivilengineering,magneto!heologicalfluid(MR/:)damper-basedsemi—activecontrolsystems
;havereceivedconsiderableattentionforuseinprotectingstructuresfromnaturalhazardssuchasstrongearthquakesand
;highwinds.InthisPaDetheMRFdamper-basedsemi—activecontrolsystemisappliedtoal
ong—spanspatiallyextended
;structureanditsfeasibilityisdiscussed.Meanwhile.a_trust—regionmethodbasedinstant
aneouso_ptimalsemi—activecontro1
;algorithm(TIOC1isproposedtoimprovetheperformanceofthesemi-activecontroisysteminamultipledampersituation.
;TheproposedTIOCdescribesthecontrolprocessasaboundedconstraintoptimizationproblem,inwhichanoptimalsemi—
;activecontro1forcevectorissolvedbythetrust—regionmethodineverycontrolsteptoml
mmlzethestructuralresponses.A
;numericalexampleofarailwaystationroofstructureinstalledwimMRF一
04Kdampersispresented.First,amodifiedBouc—
;WenmodelisutilizedtodescribethebehavioroftheselectedMRF一
04Kdamper.Then,twosemi-activecontrolsystems,
;includingthewell—knownclipped—optimalcontrollerandtheproposedTIOCcontroller,areconsidered.Basedonthe
;characteristicsofthe1ong—spanspatiallyextendedstructure.theperformanceofthecontrolsystemisevaluatedunderuniform
;earthquakeexcitationandtravelling—waveexcitationwithdifferentapparentvelocities.Thesimulationresultsindicatethat
;theMRfluiddamper—basedsemiactivecontrolsystemshavethepotentialtomitigatetheresponsesoft_u11一scalelong—span
;spatiallyextendedstructuresunderearthquakehazards.ThesuperiorityoftheproposedTIOCcontrollerisdemonstratedby
;comparingitscontroleffectivenesswiththeclippedoptimalcontrollerforseveraldifferent
cases.
;Keywords:long—spanspatiallyextendedstructure;instantaneousoptimalcontrol;semi
—activecontrol;trust—region
;method;magnetorheological(MR)damper;boundedconstraintoptimization;modifiedB
Wenmodel;clipped—optimal ouc—
;control
;lIntr0ductiOn
;Earthquakeshavealwaysbeenamajorhazardto
;society.RecentearthquakesinChinahavebroughtgreat ;lossestohumanlifeandtotheeconomy.Theseismic ;performanceoflong—spanspatiallyextendedstructures. ;e.g.,gymnasiums,railwaystations,andnuclearpower ;plants,etc.,arehighlyrelevanttopublicsafety.Inorderto ;findaneffectivewaytoprotectthesetypesofstructures, ;theirbehaviorunderearthquakeexcitationshouldbe ;studied.Earthquakemotionsvaryalongthelengthof ;extendedstructures,thusarigorousseismicanalysis ;ofspatiallyextendedstructuresshouldaccountforthe ;spatialvariabilityofthegroundmotion(Harichandran ;C0rrespOndencet0:LiZhongxian,SchoolofCivilEngineering, ;TianjinUniversity,Tianjin300072,China
;Tel:86—22—27402397.Fax:86—22—27407177
;E—mail:zxli@tiu.edu.cn
;tphDstudent;*+Professor
;Supportedby:NationalScienceFundforDistinguishedYoung ;ScholarsofChinaUnderGrantNo.50425824andthe ;NationalNatura1ScienceFoundationofChinaUnderGrant ;No.50578109.90715034and90715032
;ReceivedAugust11.2008;AcceptedOctober15.2008 ;andVanmarcke.1986;LohandYeh,1988;Kiureghian, ;1996).Non—unifoIrirlseismicresponsesmaysignificantly ;deviatefromthosebasedonthesimpleassumptionthat ;thefree—fieldgroundmotionsarespatiallyuniforrn ;(Zhangeta1.2005).Forthesamereason,thespatial ;effectsshouldalsobeconsideredwhenapplyingcontro1 ;algorithmstolong—spanstructures,andnon—uniform
;excitationsshouldbeusedtoevaluatethecontrol ;efficiencyofacontroller.
;Inrecentyears,semi—activecontrolsystemshave
;receivedconsiderableattentionamongthevarious ;responsecontrolstrategies,becausetheynotonly ;offerthereliabilityofpassivecon订olsystems.butalso
;maintaintheversatilityandadaptabilityoffullyactive
;con仃olsystems.Amonganumberofcontroldevices ;developedforseismicprotection,semi—activedevices,
;suchasmagnetorheological(MR1damperswhich ;combinethebestfeaturesofactiveandpassivedevices, ;havethepotentialtoimprovetheseismicbehaviorof ;fu11一scalecivilengineeringstructures(Spencereta1. ;1997;Duaneta1..2005).WiththedevelopmentofMR ;dampingtechnology,characteristicsofMRdampers ;havebecomebetterunderstoodandanumberofmodels ;havebeenbuilttoreproducetheirhystereticbehavior ;448EARTHQUAKEENGINEERINGANDENGINEERINGVIBRATIONVl01.7 ;Spencerela1.f1997)proposedaphenomenological ;modelbasedontheBOBC—Wenhysteresismodelwhere
;14parametersneedtobedetermined.Changand ;Roschkef1999)proposedaforwardmodeljnterms ;ofamulti—layerperceptronneuralnetworktoemulate ;thedynamicbehavioroftheMRdamper.Schurterand ;Roschke(2000)presentedaneuro—fuzzymodelofan
;MRdamperbasedonanadaptiveneuro—fuzzyinference
;systemfANFIS).
;Avarietyofsemi.activecontrolalgorithms ;havebeendevelopedintheliterature.includingthe ;Lyapunovstabilitytheory—basedcontrol(Leitmann.
;1994).decentralizedbang—bangcontrol(McClamroch
;andGavin,1995),clippedoptimalcontrol(Dykeeta1.. ;1996),modulatedhomogeneousfrictioncontrol(Inaudi. ;1997)slidingmodecontro1.neuralnetworkbased ;control(Nieta1.,2002;Xueta1.,2003),adaptivecontrol ;(TerasawaandSano,2005)andfuzzycontrol(Bhardwai ;andDatta.2006).etc.Analyticalandexperimental ;studieshavedemons~atedthecontrole珩cienciesof
;semi—activecontrolsystemsandtheirsuperiorityover ;passivecontrolsystems(Dykeeta1.,l999;Yi口,..
;2001).Mostofthesecontrolalgorithms,suchasthe ;clipped..optimalandthedecentralizedbang..bangcontrol ;algorithms,areactivecontrol—basedalgorithms,which
;aredevelopedbasedonoptimalactivecontroltheory. ;Atypicalactivecontro1.basedsemi.activecontrol ;systemhasthebasicconfigurationshowninFigure1. ;Theoptimalactivecontrolforceisfirstcalculatedby ;anactivecontrolalgorithm,suchastheLOR,LOGand ;Halgorithms.Then.semi.activecontrol1awsareused ;todeterminethecommandvoltagestothesemi—active
;devicestoallowthedampingforcetobegeneratedto
;approachthetarget.
;Theactivecontrol—basedsemi—activecontrol
;algorithmconsidersthesemi—activecontrolasa
;constrainedoptimalcontrolsystemandpostulates ;thatalinearfeedbackformismaintained,thusthe ;bestapproachingsemi—activecontrolforcemightbe
;theoptimalforceforthesemi—activecontrolsystem.
;However,literatureonoptimalsemi.activecontrol ;hasshownthattheclipped.optimallawcannotalways ;beoptimal(TsengandHedrick1994;Giorgettietal一
;2006).Forcomplexstructuressuchasspatiallyextended ;structureswithmultipledampers,theclippedsemi. ;activecontro1forcevectorobtainedbytracingeach ;elementoftheoptimalactivecontrolforcevectormay ;failtoproducesimilarcontro1ef-fortsasactivecontro1. ;InthisPrdpebasedonthecharacteristicsofanMR ;damper,anovelsemi—activecontrolalgorithmnamed
active ;thetrust—regionbasedinstantaneousoptimalsemi—
;controller(TIOC)isproposed.First.theone.step.ahead ;semi-activeperformanceindexissetupbyapplying ;thepreciseintegrationmethod.Thecontrolprocedure ;isthendescribedasatypicalbound.constrained ;optimizationproblem.Here.amodifiedBOHC.Wen ;modelisadoptedtopredicttheminimumandmaximum ;controlforcesthatcanbegeneratedbytheMRdampers ;andprovideboundedconstraintinformationateachtime ;steD.Finally,atrust—regionmethodisappliedtosolve
;theoptimizationproblemandtheoptimalsemi—active
;controlforcevectorisobtained.Thefeasibilityofthe ;applicationofMRdampersinalong.spanspatially ;extendedstructureisstudiedthroughanumerica1 ;example.Thenumericalsimulationisconductedon ;aseismicallyexcitedrailwaystationroofstructure ;installedwithMRF.04Kdampers.Theeffectiveness ;oftheMRdampersforstructura】responsereduction
;isproventhroughnumericalsimulationemploying ;differentsemi—activecontrolalgorithmsunderbOth ;multi—dimensionaluniformexcitationsandexcitations ;withwave—passageeffect.Theefficiencyoftheproposed ;semi—activecontrolalgorithmisalsodemonstratedby ;comparingittotheclipped—optimalcontrolalgorithm.
;2Systemmodeling
;2.1Equationofmotionunderspatiallyvarying ;groundmotion
;Consideralong—span
;thefinite.elementmethod.
;structurediscretizedby
;Geometricalormaterial
;nonlinearityisnotconsidered.Thetotaldegreesof ;freedom(DOFs)oftheinternalnodesandsupportsare ;dividedintounconstrainedDOFs(UDOFs)andsupport ;DOFs(SDOFs).Thestiffnessmatrix,massmatrixand ;dampingmatrix,denotedbyK,MandCrespectively, ;Fig.1Activecontrol-basedcontrolalgorithm ;No.4LinWeieta1.:TIOCalgorithmonlong—spanspatiallyextendedstructureswithMRF一04Kdamper449
;aredividedintothecomponentscorrespondingto ;UDOFsandSDOFsasindicatedbythesubscriptssand ;b.Theequationofmotionisthenwrittenas(Ohsaki, ;2001)
;MubJ[2u
;l+
;J
;CssCsb
;+
;KbbXb=
;A:diag[&,32….,】
;=
;[COSG~,cosflf,cos7f](8)
;inwhich.【,andyaretheanti—clockwiseangles
;betweentheaxesofthedampersandthecoordinateaxes ;x,Y,and三,respectively.
;Byintroducingz=r,Eq.(6)istransformed ;intoastandardstatespaceequationas
;inwhichFrepresentstheexcitati.nf0rcesatall=z+以+EF(9)
;supports.In0rdertosolVeEq?(1),isecomposedinoin ;which
;twoparts
;X:XqX
;whereXqandXdare,respectively,thequasi—static
;anddynamicdisplacementvectors,whichsatisfythe ;equations
;=一
;bb
;SubstitutingXqintoEq.(1)yields
;+c.+..=[M./CK一]+
;[K幽一](4)
;Usually,thesecondtermintherightsideofEq.(4)is
;relativelysmallandisneglected.Then,thedynamic ;displacementvectorX~issolvedfromtheequation ;M爻+Css文+KX[.一]藏
;(5)
;wherebisobtainedfromtheaccelerationtime—history
;ofeachsupport.
;2.2Statespaceequationofcontrolledspatial ;structure
;Assumethatxyzisaright-handCO0rdinatesystem ;inwhichboththex.axisandy-axislieonthehorizontal ;plane.Onlyhorizontalearthquakeexcitationsare ;consideredinthispaper.ConsiderMRdampersare ;installedinaspatialstructuretomitigatethedynamic ;responses.ThecontrolforcesprovidedbytheMR ;dampersaredecomposedalongtheCO0rdinateaxes,and ;theequationofmotionbecomes
;M爻+C文+KX=EB爻b十EDA?U,)
Msbl,Uisar×1control ;whereEB=lKsb—
;forcevector,Aisablock一&agonalmatrix,which
;denotesthedirectionsofMRdampers.
;=
;『一Icfss一0.],=[.0一E.],lIll
;E:]【_0J
;3Controlalgorithms
;(101
;3.1Clipped-optimalcontrolalgorithmforspatial ;structure
;Intheclipped—optimalcontroller,alinearoptimal ;controller)isdesignedforeachdamper,which ;providesthedesiredcontrolforce,=basedonthe ;measuredresponsesYandthecurrentcontrolforcef ;(Dykeeta1.,1996)
;彻)}
;whereL{.}istheLaplacetransform,andK()is ;obtainedfromtheLORstrategy.
;Inmispaper,theLORoptimalactivecontrolleris ;builttothespatialstructurecontrolproblemandthe ;optimalcontrolforceissetasthedesiredeontrolforce ;fortheclipped—optimalc987)andhasbeenfurther ;jnvestigatedthroughanumberofjmpressivenumerical ;andexperimentalstudies(WongandYang.2003; ;Tarantinoeta1.2004).Thetime—dependentperformance
;indexofIOCisexpressedas
;.oc(f)=z()Qz(f)+u(f)R()
;whereDisa2nx2npositivesemi—definiteweighting
;matrixandRisarxrpositivedefiniteweightingmatrix. ;Notethatthecontrolforcef1cannotchangethecurrent ;stateZ(t1immediately,sotominimizetheperformance ;index,Eq.(16)followsf1=0.Thus,aone—step.ahead
;optimalperformanceindexisusedinstead.andtheaim ;ofthecontrolistoreducethefuturesystemresponses. ;Jloc(t):z(f+At)Qz(t+?)+u(f)(f)
;Thesecondtermoftheperformanceindexjsthe ;penaltyofcontrolforces.Forasemi.activecontrol ;device.thereisalwaysanupperandalowerboundon ;thecontrolforcethattheMRdampercanproduceina ;givenstate.Hence,thesecondtermoftheperformance ;indexEq.(171isreplacedbyarestraintconditionand ;thentheperformanceindexoftheinstantaneousoptimal ;semi-activecontrolisexpressedas
;I..i(f)=z(f+At)QZ(t+At)
;l(f)?[u(f),()]
;wheretheminimumandmaximumdamperforces, ;Vm(f)andVm(f),aredependentonthepistonvelocity ;oftheMRdamperandareobtainedbysettingthe ;voltagetobeminimumandmaximum,respectively. ;Then.thesemi—activecontroloptimizationprocedureis ;formulatedas
;MinimizeJsi(f)=z(+At)QZ(t+at)
;Subjecttoz=AZ+B(AU1+(20)
;(f)?[U(),u()]
;(t)=f[2d(f),0],(t)=f[ka(f),](22)
;where()isthevelocityoftheMRdamperpiston, ;Visvoltage.whichcanbeconvertedtoproportional ;DCcurrentinputtotheMRdamper,thusVisthe ;maximumvoltagefortheMRdamper.
;Inordertosolvetheoptimizationproblem.thefirst ;orderrestraintconditionneedstobetransfolrmedinto ;adiscretesystem.Foraspatiallyextendedstructure ;withalargedegreeof-freedom(D0F1,inorderto ;preventerrorenrolledbythisprocedure,theprecise ;integrationmethod(Zhong,2004),whichisbelieved ;tohavehighprecisionandefficiency,isappliedinthis ;paper.Moreover,theprecisionofthesolutionsprovided ;bythepreciseintegrationmethodvarieslittlewhen ;differenttimestepsarechosen.whichmakesthecontrol ;algorithmsuitablefordifferentcontrolproblems. ;Byusingthepreciseintegrationmethod.the
;derivativetransferfunctionEq.f91canbeconverted ;into
;where
;z(t十At):z1(t+At)+z2(f)(23)
;ZI(A
;6
;t{eASt/2[】+
;aav(t)+EF(t)}(24)
+EF(f一?f)]+ ;(f)=eaA’Z(『)+[一
;[]}f251
;Further,substitutingEq.(23)intoEq.(19),thesemi—
;activecontrolproblembecomes
;Minimize
;(f)=g(V(t—at),F(t一?),(f),F),At)(26)
;Subjectto(f)?[u(f),u(f)](27)
;(t)=f[xd(f),0],U.(t)=_厂[(f),](28)
;Theaboveequationsdescribeatypicalbound. ;constrainedminimizationproblemwhichcanbesolved ;bytheinterior-reflectiveNewtonmethod.Inthispaper, ;thesubspacetrust—regionmethod(Brancheta1..1999; ;ColemanandLi.1994;Rodriguezeta1.,20001isused ;tosolvetheoptimalsemi—activecontrolforcevectorat
;everytimestep.
;Inthetrustregionmethod,aninitialvalueof ;No.4LinWeieta1.:TIOCalgorithmonlong—spanspatiallyextendedstructureswithMRF一04Kdamper45l
;thevariableisprovided.And.asecond.order ;approximation,季(),oftheobjective,g(?,is
;successivelyminimizedwiththetrustregionregulating ;thelengthofthestepsineachiteration.Theglobal ;convergencepropertiesofthesemethodsrelyprimarily ;ontheassumptionthattheapproximationg(u)andthe ;actualfunctiong(x)matchuptothefirstorder.Ateach ;optimizationiteration(),
;I季(.):g(U’)
;lVg(V)=()(29)
;Then,atrustregionradiusA()isupdatedanda ;neighborhoodNaroundthepoint’isdecidedatthe
;iteration).Thisneighborhoodiscalledthetrustregion. ;And,thetrustregionsub—problemisexpressedas
;min季(‘)U’?N
;Theinitialpointatiteration+11issettobe ;‘)=(,(‘+(,,andthetrialstepcomputationis
;repeatedwithnewtrustregionradiusA’untilthe
;solutionreachesconvergence.
;Mathematically,theaboverulesforupdatingthetrust ;regionradiusmaybedescribedbychoosingconstantsto ;definetherangesoftheratiovalueforwhichreduction ;orenlargementisnecessary.Thepositiveconstants ;R,<<1,c,<1andc>1arechosensothatthetrust101’
;regionradiusisupdatedas
;HereareliabilityindexP(,whichmonitorshowwell ;thecurrentapproximationrepresentstheactualdesign ;space,is?