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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2017.2700384, IEEE Transactions on Automatic Control 0018-9286 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 1Event-TriggeredGlobalRobustOutputRegulationforaClassofNonlinearSystemsWeiLiu,Member,IEEEandJieHuang,Fellow,IEEEAbstract—Thisnotestudiestheglobalrobustoutputregulationproblemforaclassofnonlinearsystemsbyanoutput-basedevent-triggeredcontrollaw.First,weconverttheproblemintotheevent-triggeredglobalrobuststabilizationproblemofawelldeﬁnedaugmentedsystembasedontheinternalmodelapproach.Then,wedesignanoutput-basedevent-triggeredcontrollawtogetherwithanoutput-basedevent-triggeredmechanismtosolvethestabilizationproblem,whichinturnleadstothesolutiontotheoriginalproblembyanoutput-basedevent-triggeredcontrollaw.Finally,weillustrateourapproachbyanexample.IndexTerms—Event-triggeredcontrol,nonlinearsystems,out-putregulation,robustcontrol.I.INTRODUCTIONTherobustoutputregulationproblemhasbeenoneofthebasicandimportantcontrolproblemssince1970s.Theproblemaimstodesignafeedbackcontrollawforanun-certainsystemsothattheoutputoftheclosed-loopsystemasymptoticallytrackssomereferenceinputsinthepresenceofsomeexternaldisturbances,whereboththereferenceinputsandtheexternaldisturbancesaregeneratedbyanexosystem.Theproblemhasbeenstudiedforbothlinearuncertainsystemsin,say,[4],[7],[8],andnonlinearuncertainsystemsin,say,[3],[14],[15],[16].Inthisnote,wewillfurtherstudytheglobalrobustoutputregulationproblemfornonlinearsystemsinnormalformwithunityrelativedegreebytheoutput-basedevent-triggeredcontrollaw.Inpractice,acontrollawisoftenimplementedinadig-italplatform.Oneapproachisthetraditionalsampled-dataimplementation[2],[9],wherethecontrolactuationupdatesperiodicallyevenafterthesystemhasachievedthecontrolgoalwithsufﬁcientaccuracy.Incontrast,theevent-triggeredcontrolapproachgeneratesthesamplingsandcontrolactuationwhenthesystemstateoroutputdeviatesmorethanacertainthresholdfromanacceptablevalue[11].Thustheevent-triggeredcontrolapproachismoreefﬁcientinutilizingthelim-itedcontroltaskexecutioncapabilitiesandenergyresources.Animportantissuewiththeevent-triggeredcontrolistoguaranteetheexistenceofapositivedwellingtimetopreventtheso-calledZenobehavior,i.e.,theexecutiontimesbecomearbitrarilycloseandresultinanaccumulationpoint[25].TheThisworkhasbeensupportedbytheResearchGrantsCounciloftheHongKongSpecialAdministrationRegionundergrantNo.14200515.WeiLiuandJieHuangarewiththeDepartmentofMechanicalandAutoma-tionEngineering,TheChineseUniversityofHongKong,Shatin,NewTerrito-ries,HongKong.E-mail:wliu@mae.cuhk.edu.hk,jhuang@mae.cuhk.edu.hkCorrespondingauthor:JieHuang.Themainresultofthispaperwithoutanyproofwillbepresentedatthe20thWorldCongressoftheInternationalFederationofAutomaticControl,July9-14,2017,Toulouse,France.event-triggeredcontrolproblemshavebeenﬁrststudiedforlinearsystems[6],[11],[19],[26].Speciﬁcally,reference[11]gaveanintroductiontotheevent-triggeredcontrolandstudiedthestabilizationproblemforaclassoflinearsystemsbyastate-feedbackevent-triggeredcontrollaw.Reference[26]furtherstudiedthestabilizationproblemforaclassofLTIsystemsbyanoutput-basedevent-triggeredcontrollaw.In[6],anoutput-basedevent-triggeredcontrollawwasproposedtoguaranteetheclosed-loopstabilityandtheL1-performanceforaclassoflinearsystems.Reference[19]furtherstudiedtherobustpracticaloutputregulationproblemforaclassoflinearuncertainminimum-phasesystemsbyanoutput-basedevent-triggeredcontrollaw.Recently,theevent-triggeredcontrolproblemshavealsobeenstudiedforvariousnonlinearsys-tems.Forexample,undertheassumptionoftheinput-to-statestabilitywithrespecttothemeasurementerror,reference[25]proposedastate-basedevent-triggeredcontrollawtosolvethestabilizationproblemforaclassofnonlinearsystems.In[18],byusingthecyclicsmallgaintheorem,therobuststabilizationproblemforaclassofnonlinearsystemssubjecttoexternaldisturbanceswassolvedbyastate-basedevent-triggeredcontrollaw.Reference[27]studiedtheasymptotictrackingproblemforaclassofnonlinearsystemsbyastate-basedevent-triggeredcontrollawanditwasshownthatthetrackingerrorwasuniformlyultimatelybounded.Reference[5]analyzedthestabilityandLp-performanceforaclassofnonlinearsystemsbasedonanoutputfeedbackevent-triggeredcontrollaw.In[1],anoutput-basedeventtriggeredcontrollawwasproposedtosolvethestabilizationproblemfornonlinearsystems.Someothereffortscanbefoundin[10],[23],[28],[31].Sofar,theevent-triggeredcontrolproblemsfornonlinearsystemsaremainlylimitedtostabilizationproblem[1],[10],[18],[23],[25],ortrackingproblem[27],[28].Theevent-triggeredglobalrobustoutputregulationproblemfornonlinearsystemshasneverbeenstudiedbefore.Comparedwiththoseexistingresults,ourproblemismorechallenginginthefol-lowingthreeways.First,weneedtoaddressboththetrackingproblemanddisturbancerejectionproblemsimultaneously.Second,weneedtohandleuncertainparametersthatbelongtosomearbitrarilylargeprescribedcompactset.Third,ourcontrollawisadynamicoutputfeedbackcontrollaw.Toimplementsuchacontrollawinadigitalplatform,weneedtoseekaspeciﬁcformofcontrollawandsamplenotonlytheoutputoftheplantbutalsothestateofthedynamiccompensator.Thus,thestabilityanalysisoftheclosed-loopsystemismorechallengingthanthestaticstateorstaticoutputfeedbackcase.Toovercomethesechallenges,wecombinethe

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2017.2700384, IEEE Transactions on Automatic Control 0018-9286 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 2internalmodelapproachwithanoutput-basedevent-triggeredmechanism.Thisevent-triggeredmechanismcontainsadesignparameterwhichnotonlydictatestheultimateboundofthetrackingerroroftheclosed-loopsystem,butalsothefrequencyofthetriggeringevents.Itturnsoutthatthiscontrollawtogetherwiththeevent-triggeredmechanismleadstothepracticalsolutiontoourprobleminthesensethatthesteady-statetrackingerroroftheclosed-loopsystemcanbemadearbitrarilysmallwithoutincurringtheZenobehaviorintheevent-triggeredmechanism.Notation.Foranycolumnvectorsai,i=1,...,s,denotecol(a1,...,as)=[aT1,...,aTs]T.∥x∥denotestheEuclideannormofvectorx.∥A∥denotestheinducednormofmatrixAbytheEuclideannorm.Z+denotesthesetofallnonnegativeintegers.Thebaseofthenaturallogarithmisdenotedbye.λmax(A)andλmin(A)denotethemaximumeigenvalueandtheminimumeigenvalueofasymmetricrealmatrixA,respectively.II.PROBLEMFORMULATIONANDPRELIMINARIESConsideraclassofnonlinearsystemsasfollows˙z=f(z,y,v,w)˙y=g(z,y,v,w)+b(w)ue=y−q(v,w)(1)where(z,y)∈Rn×Risthestate,e∈Ristheerroroutput,u∈Ristheinput,w∈Rnwisanuncertainconstantvector,andv(t)∈Rnvisanexogenoussignalrepresentingbothreferenceinputandexternaldisturbance.Itisassumedthatv(t)isgeneratedbyalinearexosystemasfollows˙v=Sv,y0=q(v,w).(2)Allfunctionsin(1)and(2)areassumedtobesufﬁcientlysmooth,andsatisfyf(0,0,0,w)=0,g(0,0,0,w)=0,q(0,w)=0forallw∈Rnw.System(1)takesthesameformasthatin[29]andiscalledanonlinearsysteminnormalformwithunityrelativedegree.Ourcontrollawisofthefollowingform:u(t)=ˆf(e(tk))+Ψη(tk)˙η(t)=Mη(t)+Nu(t),t∈[tk,tk+1),k∈Z+(3)whereˆf(·)issomenonlinearfunction,η∈Rsforsomeintegers,Ψ,MandNaresomematricestobespeciﬁedlater,tkdenotesthetriggeringtimeinstantwitht0=0,andisgeneratedbythefollowingso-calledevent-triggeredmechanismtk+1=inf{t>tk|h(˜e(t),˜η(t),e(t))≥δ}(4)whereδ>0issomeconstant,h(·)issomenonlinearfunction,and˜e(t)=e(tk)−e(t)˜η(t)=η(tk)−η(t)(5)foranyt∈[tk,tk+1)withk∈Z+.Remark2.1:Denotethemaximallydeﬁnedintervalforthesolutionxc(t)=col(z(t),η(t),y(t))oftheclosed-loopsystemcomposedof(1)and(3)withthetriggeringmechanism(4)by[0,TM)with0tk.Tomakeoursystemwellbehaved,weneedtomakesurethatourcontrollawtogetherwiththetriggeringmechanismwillguaranteeTM=∞,thusrulingouttheZenobehavior.Remark2.2:Thecontrollaw(3)iscalledadynamicoutput-basedevent-triggeredcontrollawandtheη-subsystemof(3)istheso-calledinternalmodelwhichwillbeintroducedlaterbasedontheresultin[29].Itisnotedthatthiscontrollawtakessomespeciﬁcforminthattheﬁrstequationislinearinη(tk)andthesecondequationislinearinη(t)andu(t).Thisspeciﬁcformlendsitselftothefollowingpiecewiseconstantimplementation:u(t)=ˆf(e(tk))+Ψη(tk)η(tk+1)=eM(tk+1tk)η(tk)+N(ˆf(e(tk))+Ψη(tk))∫tk+1tkeM(tk+1)dτ(6)foranyt∈[tk,tk+1)withk∈S.Itisworthmentioningthat,asopposedtothetraditionalperiodicsamplingtechniquein,say,[2],in(6),thesamplingisaperiodicandthesequence{tk}k2Sisdeterminedbytheoutput-basedevent-triggeredmechanism(4).Itisalsonotedthat(3)ismorecomplexthanthestaticstate-basedevent-triggeredcontrollawsin[18],[21],[25],[27],sinceweneedtosamplenotonlytheerroroutputoftheplantbutalsothestateoftheinternalmodel.Similarly,ourevent-triggeredmechanism(4)isalsomorecomplexsinceitnotonlydependson˜e(t),e(t),butalso˜η(t).Asaresult,itismorechallengingtoanalyzethestabilityoftheclosed-loopsystemandpreventZenobehaviorfromhappening.Nowwedescribeourproblemasfollows.Problem2.1:Giventheplant(1),theexosystem(2),somecompactsubsetsV⊂RnvandW⊂Rnwwith0∈Wand0∈V,andanyϵ>0,designacontrollawoftheform(3)andanevent-triggeredmechanismoftheform(4)suchthattheresultingclosed-loopsystemhasthefollowingproperties:foranyv∈V,w∈W,andanyinitialstatesz(0),y(0),η(0),thetrajectoryoftheclosed-loopsystemexistsandisboundedforallt≥0;limt!1sup|e(t)|≤ϵ.Remark2.3:WecallProblem2.1astheevent-triggeredglobalrobustpracticaloutputregulationproblem.WecallacontrollawthatsolvesProblem2.1asapracticalsolutiontotheglobalrobustoutputregulationproblem.Problem2.1is

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2017.2700384, IEEE Transactions on Automatic Control 0018-9286 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 3morechallengingthantheproblemin[29]inthefollowingthreeaspects.First,weneedtodesignbothanevent-triggeredmechanismandapiecewiseconstantcontrollaw.Second,thestabilitybehaviorofourclosed-loopsystemismorecomplexsincetheclosed-loopsystemsisahybridsystem.Third,inordertopreventtheZenobehavior,weneedtoguaranteeTM=∞,i.e.,thesolutionoftheclosed-loopsystemexistsforallt∈[0,∞).Tosolveourproblem,asin[29],weneedtoﬁrstintroducetheinternalmodelfor(1)and(2),andformtheso-calledaugmentedsystem.Forthispurpose,weﬁrstintroducesomestandardassumptionswhichcanalsobefoundin[29],[30].Assumption2.1:Theexosystemisneutrallystable,i.e.,alltheeigenvaluesofSaresemi-simplewithzerorealparts.Assumption2.2:b(w)isacontinuousfunctionandsatisﬁesb(w)>0forallw∈Rnw.Remark2.4:UnderAssumption2.2,foranyknowncom-pactsubsetW⊆Rnw,therealwaysexistsomeknownpositivenumbersbmandbMsuchthat,bm≤b(w)≤bMforallw∈W.Sinceb(w)u=(−b(w))(−u)=ˆb(w)ˆuwithˆb(w)=−b(w)andˆu=−u.Ifb(w)<0forallw∈Rnw,bychangingutoˆu,thenˆb(w)willstillsatisfyAssumption2.2.Assumption2.3:Thereexistsagloballydeﬁnedsmoothfunctionz:Rnv×Rnw7→Rnwithz(0,w)=0suchthat∂z(v,w)∂vSv=f(z(v,w),q(v,w),v,w)(7)forall(v,w)∈Rnv×Rnw.Remark2.5:UnderAssumption2.3,lety(v,w)=q(v,w)u(v,w)=b1(w)(∂q(v,w)∂vSv−g(z(v,w),q(v,w),v,w)).Then,z(v,w),y(v,w)andu(v,w)arethesolutiontotheregulatorequationsassociatedwith(1)and(2)[3],[13],[29].Itisknownthatthesolvabilityoftheregulatorequationsisanecessaryconditionforthesolvabilityoftheoutputregulationproblem.Thesolutionoftheregulatorequationsprovidesthenecessaryfeedforwardcontrolinformationtothecontroller.Nevertheless,u(v,w)cannotbedirectlyusedfordesigningfeedbackcontrollawasitdependsontheexogenoussignalvandtheunknownparameterw.Weneedtodesignadynamiccompensatorcalledinternalmodelwhichcanasymptoticallyprovidetheinformationofu(v,w).Fortheexistenceofalinearinternalmodel,weneedonemoreassumption.Assumption2.4:Thefunctionu(v,w)isapolynomialinvwithcoefﬁcientsdependingonw.Remark2.6:Asremarkedin[29],underAssumptions2.1to2.4,thereexistsanintegersandarealcoefﬁcientpolynomialP(λ)=λs−ϱ1−ϱ2λ−···−ϱsλs1(8)whoserootsarealldistinctwithzerorealpart,suchthat,foralltrajectoriesv(t)oftheexosystem(2)andallw∈W,u(v,w)satisﬁesdsu(v,w)dts=ϱ1u(v,w)+ϱ2du(v,w)dt+···+ϱsds1u(v,w)dts1.(9)LetΦ=01···0............00···1ϱ1ϱ2···ϱs,Γ=10...0T.Since(Γ,Φ)isobservableandtheeigenvaluesofΦhavezerorealpart,foranycontrollablepair(M,N)withM∈RssaHurwitzmatrixandN∈Rs1acolumnvector,thereisauniquenonsingularmatrixTsatisfyingthefollowingSylvesterequation:TΦ−MT=NΓ.(10)WefurtherletΨ=ΓT1andθ(v,w)=Tcol(u(v,w),˙u(v,w),···,u(s1)(v,w)).Thenitcanbeveriﬁedthatθ(v,w)hasthefollowingproper-ties:u(v,w)=Ψθ(v,w)∂θ(v,w)∂vSv=(M+NΨ)θ(v,w).(11)Moreover,deﬁneadynamiccompensatorasfollows[22]:˙η=Mη+Nu.(12)Intheliterature[3],[13],[22],thedynamiccompensator(12)iscalledalinearinternalmodelof(1),whichisusedtorepro-ducethefunctionθ(v,w)asymptotically,thusasymptoticallyprovidingtheinformationofthefunctionu(v,w).Remark2.7:Assumptions2.1-2.4havebeenusedin[29],[30]andotherliteraturefordealingwiththeglobalrobustoutputregulationproblemforthesamesystemas(1).Itmayworthfurtherdiscussingtherationalityoftheseassumptionsandthepossibilityofrelaxingthem:UnderAssumption2.1,foranyv(0)∈V0withV0beingsomeknowncompactset,thereexistsanothercompactsetVsuchthatv(t)∈Vforallt≥0.Itispossibletorelaxthisassumptionbyconsideringthenonlinearexosystemaswedidin[20].Assumption2.2assumesthatthesignofthecontrolgainb(w)isknown.Itisalsopossibletofurtherconsiderthecasewherethesignofthecontrolgainisunknownbyusingtheso-calledNussbaumgaintechniqueaswedidin[29].Assumption2.3isanecessaryconditionforthesolvabil-ityoftheregulatorequationsandthusitisanecessaryconditionforthesolvabilityoftheoutputregulationproblemforthesystem(1).AsnotedinRemark2.6,Assumption2.4guaranteestheexistenceofthelinearinternalmodel(12).Itispossibletofurtherrelaxthisassumptionbyconsideringthenonlinearinternalmodelaswedidin[20].Thecompositionoftheoriginalplant(1)andtheinternalmodel(12)iscalledtheaugmentedsystem.Next,wewillshowthattheoutputregulationproblemof(1)canbeconvertedtothestabilizationproblemoftheaugmentedsystem.For

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2017.2700384, IEEE Transactions on Automatic Control 0018-9286 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 4thispurpose,performingthefollowingcoordinateandinputtransformationontheaugmentedsystem¯z=z−z(v,w),¯η=η−θ(v,w)−Nb1ee=y−q(v,w),¯u=u−Ψη(tk),t∈[tk,tk+1),k∈S(13)gives˙¯z=¯f(¯z,e,µ)˙¯η=M¯η+MNb1e−Nb1¯g(¯z,e,µ)˙e=¯g(¯z,e,µ)+bΨ¯η+ΨNe+b¯u+bΨ˜η(14)whereµ=(v,w),and¯f(¯z,e,µ)=f(¯z+z,e+q,v,w)−f(z,q,v,w)¯g(¯z,e,µ)=g(¯z+z,e+q,v,w)−g(z,q,v,w).Clearly,foranyµ∈Rnv×Rnw,wehave¯f(0,0,µ)=0and¯g(0,0,µ)=0.Considerapiecewiseconstantcontrollawasfollows:¯u(t)=ˆf(e(tk)),t∈[tk,tk+1),k∈S(15)whereˆf(·)isagloballydeﬁnedsufﬁcientlysmoothfunctionvanishingattheorigin.Letthestateoftheclosed-loopsystemcomposedoftheaugmentedsystem(14)andthecontrollaw(15)bedenotedby¯xc=col(¯z,¯η,e).Thenwegivethefollowingproposition.Proposition2.1:UnderAssumptions2.1-2.4,foranyϵ>0,anyknowncompactsetsV∈RnvandW∈Rw,ifacontrollawoftheform(15)issuchthat,forall¯xc(0),allv(t)∈Vandallw∈W,¯xc(t)existsandisboundedforallt∈[0,∞),andsatisﬁeslimt!1sup∥¯xc(t)∥≤ϵ,(16)thentheglobalrobustpracticaloutputregulationproblemforthesystem(1)issolvablebythefollowingpiecewiseconstantoutputfeedbackcontrollawu(t)=ˆf(e(tk))+Ψη(tk)˙η(t)=Mη(t)+Nu(t),t∈[tk,tk+1),k∈S.(17)Proof:First,notethat(16)impliesthatlimt!1sup|e(t)|≤limt!1sup∥¯xc(t)∥≤ϵ.(18)Thatistosay,thesecondpropertyinProblem2.1issatisﬁed.Next,weprovethatthestatexc(t)=col(z(t),η(t),y(t))oftheclosed-loopsystemcomposedof(1)and(17)existsandisboundedforallt∈[0,∞).Accordingtothecoordinatetransformation(13),wehavexc(t)=col(z(v(t),w),θ(v(t),w)+Nb1(w)e,q(v(t),w))+¯xc(t).(19)Sincez(v(t),w),b(w),θ(v(t),w)andq(v(t),w)arealls-moothfunctions,andtheboundariesofthecompactsetsVandWareknown,z(v(t),w),b(w),θ(v(t),w)andq(v(t),w)areallboundedforallt∈[0,∞).Since¯xc(t)existsandisboundedforallt∈[0,∞),e(t)alsoexistsandisboundedforallt∈[0,∞).Thus,by(19),xc(t)existsandisboundedforallt∈[0,∞),thatistosay,theﬁrstpropertyinProblem2.1isalsosatisﬁed.Wecalltheproblemofdesigningacontrollawoftheform(15)toachieve(16)astheglobalrobustpracticalstabilizationproblemfortheaugmentedsystem(14).Remark2.8:Notethat,toobtainapiecewiseconstantcontrollaw,thetransformation(13)isdifferentfromthosein[29],[30],sincewereplacethetermη(t)byη(tk).Thuswegetamorecomplexaugmentedsystem(14)thanthosein[29],[30].Moreover,sinceourcontrollawisahybridcontrollaw,theclosed-loopsystemisalsoahybridsystem.Thus,aswillbeseeninnextsectionthat,thestabilityanalysisoftheclosed-loopsystemwillalsobemorecomplexthanthosein[29],[30].III.MAINRESULTInthissection,wefocusontheglobalrobustpracticalstabilizationfortheaugmentedsystem(14).Forthispurpose,weneedonemoreassumption.Assumption3.1:ForanycompactsubsetΩ⊂Rnv×Rnw,thereexistsaC1functionV1(¯z)suchthat,forany¯z,andanye,α1(∥¯z∥)≤V1(¯z)≤¯α1(∥¯z∥)(20)∂V1(¯z)∂¯z¯f(¯z,e,µ)≤−α1(∥¯z∥)+γ1(e)(21)whereα1(·)and¯α1(·)aresomeclassK1functions,γ1(·)isaknownsmoothpositivedeﬁnitefunction,andα1(·)isaknownclassK1functionsatisfyinglims!0+sup(s2/α1(s))<∞.Remark3.1:Assumption3.1isastandardassumptionandhasalsobeenusedin[29],[30].Thisassumptionguaranteesthat,foranyµ∈Ω,thesubsystem˙¯z=¯f(¯z,e,µ)isinput-to-statestable(ISS)witheastheinput[24].Beforegivingourmainresult,weintroducesomenotation.Deﬁneϑ(t)=−ρ(e(t))e(t)˜ϑ(t)=ϑ(tk)−ϑ(t),t∈[tk,tk+1),k∈S(22)whereρ(·)isasufﬁcientlysmoothpositivefunctiontobespeciﬁedlater.Thenweconsiderthefollowingoutputfeed-backcontrollaw¯u(t)=ϑ(tk),t∈[tk,tk+1),k∈S(23)togetherwiththefollowingoutput-basedevent-triggeredmechanismtk+1=inf{t>tk||˜ϑ(t)+Ψ˜η(t)|−σ|ϑ(t)|≥δ},(24)whereσ>0andδ>0aresomeconstantstobedeterminedlater.Fromtheevent-triggeredmechanism(24),foranyt∈[tk,tk+1)withk∈S,wehave|˜ϑ(t)+Ψ˜η(t)|≤σ|ϑ(t)|+δ.(25)¿From(22),wehaveϑ(tk)=˜ϑ(t)+ϑ(t),t∈[tk,tk+1),k∈S.(26)

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2017.2700384, IEEE Transactions on Automatic Control 0018-9286 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 5Thus,foranyt∈[0,TM),theclosed-loopsystemcomposedof(14)and(23)canbeputtothefollowingform:˙Z(t)=F(Z(t),e(t),µ)˙e(t)=˜g(Z(t),e(t),µ)+bϑ(t)+b˜ϑ(t)+bΨ˜η(t)(27)whereZ(t)=col(¯z(t),¯η(t)),F(Z(t),e(t),µ)=col(¯f(¯z(t),e(t),µ),M¯η(t)+MNb1e(t)−Nb1¯g(¯z(t),e(t),µ))and˜g(Z(t),e(t),µ)=¯g(¯z(t),e(t),µ)+bΨ¯η(t)+ΨNe(t).Tofacilitateouranalysis,wefurtherput(27)intothefollowingcompactform:˙¯xc(t)=fc(¯xc(t),µ)(28)where¯xc(t)isthesameasthatdeﬁnedbefore,andfc(¯xc(t),µ)=col(F(Z(t),e(t),µ),˜g(Z(t),e(t),µ)+bϑ(t)+b˜ϑ(t)+bΨ˜η(t)).Forsimplicity,wewilluse¯xc,Z,¯z,¯η,˜η,ϑ,˜ϑ,e,insteadof¯xc(t),Z(t),¯z(t),¯η(t),˜η(t),ϑ(t),˜ϑ(t),e(t)whennoambiguitywilloccur.Lemma3.1:UnderAssumptions2.1-2.4and3.1,thereexistsasmoothpositivefunctionρ(·),aC1functionU(¯xc)andtwoclassK1functionsβ(·)and¯β(·)suchthat,foranyµ∈Ω,andany¯xc,β(∥¯xc∥)≤U(¯xc)≤¯β(∥¯xc∥)(29)∂U(¯xc)∂¯xcfc(¯xc,µ)≤−∥¯xc∥2,∀∥¯xc∥≥δ.(30)Asaresult,forany¯xc(0),thesolutionof(28)isboundedover[0,TM).Proof:First,notethattheZ=col(¯z,¯η)subsystemof(14)isinthesameformasequation(17)of[30]withz1=¯z,z2=¯η,u=e,φ1=¯f(¯z,e,µ),φ2=M¯η+MNb1e−Nb1¯g(¯z,e,µ)andA=MwhichisHurwitz.Then,underAssumption3.1,byapplyingLemma3.1of[30],thereexistsaC1functionV2(Z)suchthat,foranyµ∈Ω,anyZ,andanye,α2(∥Z∥)≤V2(Z)≤¯α2(∥Z∥)(31)∂V2(Z)∂ZF(Z,e,µ)≤−∥Z∥2+γ2(e)(32)whereα2(·)and¯α2(·)aretwoclassK1functions,andγ2(·)isaknownsmoothpositivedeﬁnitefunction.Then,byapplyingthechangingsupplypairtechnique[24],foranygivensmoothfunction∆(Z)>0,thereexistsaC1functionV3(Z)suchthat,foranyµ∈Ω,anyZ,andanye,α3(∥Z∥)≤V3(Z)≤¯α3(∥Z∥)(33)∂V3(Z)∂ZF(Z,e,µ)≤−∆(Z)∥Z∥2+π(e)e2(34)whereα3(·)and¯α3(·)aresomeclassK1functions,andπ(·)isaknownsmoothpositivefunction.Since˜g(Z,e,µ)issmoothand˜g(0,0,µ)=0forallµ∈Ω,byLemma7.8of[13],thereexistsomesmoothfunctionsF1(Z)andF2(e)satisfyingF1(0)=0andF2(0)=0,suchthat,forallZ∈Rn+s,alle∈Randallµ∈Ω,|˜g(Z,e,µ)|≤F1(Z)+F2(e)(35)whichimpliesthat|˜g(Z,e,µ)|2≤F21(Z)+2F1(Z)F2(e)+F22(e)≤2F21(Z)+2F22(e)≤φ(Z)∥Z∥2+χ(e)e2(36)forsomesmoothpositivefunctionsφ(Z)andχ(e).Notethattheexistenceofφ(Z)andχ(e)satisfyingthethirdinequalityof(36)isguaranteedbythefactthatF1(0)=0andF2(0)=0.LetV4(e)=12e2.Then,accordingto(25),(27)and(36),foranyµ∈Ω,anyZ,andanye,∂V4(e)∂e(˜g(Z,e,µ)+bϑ+b˜ϑ+bΨ˜η)=e(˜g(Z,e,µ)+bϑ+b˜ϑ+bΨ˜η)=e˜g(Z,e,µ)−bρ(e)e2+be(˜ϑ+Ψ˜η)≤14e2+|˜g(Z,e,µ)|2−bmρ(e)e2+bM(σ|e||ϑ|+δ|e|)≤−(bmρ(e)−14)e2+φ(Z)∥Z∥2+χ(e)e2+bMσρ(e)e2+b2M4e2+δ2=−((bm−bMσ)ρ(e)−1+b2M4−χ(e))e2+φ(Z)∥Z∥2+δ2.(37)LetU(¯xc)=V3(Z)+V4(e).Clearly,thereexisttwoclassK1functionsβ(·)and¯β(·)suchthat(29)issatisﬁed.Also,accordingto(34)and(37),foranyµ∈Ω,andany¯xc,wehave∂U(¯xc)∂¯xcfc(¯xc,µ)≤−((bm−bMσ)ρ(e)−1+b2M4−χ(e)−π(e))e2−(∆(Z)−φ(Z))∥Z∥2+δ2.(38)Chooseσc.IfU(¯xc(t))>cforallt∈[0,TM)whichimplies∥¯xc(t)∥≥δforallt∈[0,TM),then,from(30),wehave@U(xc)@xcfc(¯xc(t),µ)≤0,whichtogetherwith(29)implies∥¯xc(t)∥≤β1(U(¯xc(t)))≤β1(U(¯xc(0)))≤β1(¯β(∥¯xc(0)∥))forallt∈[0,TM).Otherwise,thereexists0<τcforallt∈[0,τ)andU(¯xc(τ))=c.Then,from(30)again,

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2017.2700384, IEEE Transactions on Automatic Control 0018-9286 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 6wehave@U(xc)@xcfc(¯xc(t),µ)≤0forallt∈[0,τ),whichtogetherwith(29)implies∥¯xc(t)∥≤β1(¯β(∥¯xc(0)∥))forallt∈[0,τ).Moreover,sinceU(¯xc(τ))=c,wehave∥¯xc(t)∥≤δforallt∈[τ,TM).Inconclusion,forany¯xc(0),wehave∥¯xc(t)∥≤max{δ,β1(¯β(∥¯xc(0)∥))},∀t∈[0,TM).(40)Thustheproofiscompleted.Theorem3.1:UnderAssumptions2.1-2.4and3.1,foranyϵ>0,Problem2.1forthesystem(1)issolvablebythefollowingoutputfeedbackcontrollawu(t)=−ρ(e(tk))e(tk)+Ψη(tk)˙η(t)=Mη(t)+Nu(t),∀t∈[tk,tk+1)(41)togetherwiththeoutput-basedevent-triggeredmechanism(24)withδ=¯β1(β(ϵ)).Proof:Notethat,foranydifferentiablefunctionF:[0,∞)→R,d|F(t)|dt=d((F(t))2)12dt=((F(t))2)12F(t)˙F(t)≤|˙F(t)|.Then,foranyt∈[tk,tk+1)withk∈S,d|˜ϑ(t)+Ψ˜η(t)|dt≤|˙˜ϑ(t)+Ψ˙˜η(t)|=|−˙ϑ(t)−Ψ˙η(t)|=|(dρ(e)dee+ρ(e))˙e(t)−Ψ˙η(t)|=(dρ(e)dee+ρ(e))(˜g(Z,e,µ)−bρ(e(tk))e(tk)+bΨ(η(tk)−η(t)))−Ψ(Mη(t)+N(−ρ(e(tk))e(tk)+Ψη(tk))).(42)ByLemma3.1,¯xc(t)isboundedforallt∈[0,TM),whichimpliesthat¯z(t),¯η(t)ande(t)areboundedforallt∈[0,TM).Also,underAssumption2.1,v(t)isboundedforallt∈[0,∞).Thusθ(v(t),w)isboundedforallt∈[0,∞).Sinceη(t)=¯η(t)+θ(v(t),w)+Nb1e(t)from(13),η(t)isalsoboundedforallt∈[0,TM).Therefore,thereexistsapositivenumberc0dependingonδand¯xc(0)suchthatd|˜ϑ(t)+Ψ˜η(t)|dt≤c0(43)forallt∈[0,TM).Ontheotherhand,foranyk∈S,fromthesecondequationof(5)andthesecondequationof(22),wehave˜η(tk)=η(tk)−η(tk)=0˜ϑ(tk)=ϑ(tk)−ϑ(tk)=0(44)and,from(24),wehavelimt!tk+1|˜ϑ(t)+Ψ˜η(t)|≥limt!tk+1(σ|ϑ(t)|+δ)≥δ.(45)Combining(43),(44)and(45),weconcludethat,tk+1−tk≥c0foranyk∈S.Thus,lettingτd=c0givesinfk2S{tk+1−tk}≥τd>0(46)whichimpliesthattheZenobehaviorcannothappen,i.e.,Case1)isimpossible.Otherwise,wewouldhaveinfk2S{tk+1−tk}=0,whichcontradictswith(46).IfCase3)mentionedinRemark2.1happens,then,thereexistsaﬁnitetimetksuchthat,fort>tk,thereisnotriggering.Asaresult,thecontrollawandhencetheclosed-loopsystemwillreducetoacontinuous-timesystemforallt>tk.Since,by(40),¯xc(t)isboundedforallt∈[tk,TM),itmustholdthatTM=∞.Thatistosay,¯xc(t)isdeﬁnedforallt∈[0,∞).Sincethesolution¯xc(t)existsforallt∈[0,∞),byTheorem4.18of[17]andLemma3.1here,thesolution¯xc(t)oftheclosed-loopsystem(28)isgloballyultimatelyboundedwiththeultimateboundd(δ)=β1(¯β(δ)).Sinced(·)isaninvertibleclassK1function,foranyϵ>0,lettingδ=d1(ϵ)=¯β1(β(ϵ))giveslimt!1sup∥¯xc(t)∥≤ϵ.Thatistosay,thecontrollaw(23)togetherwiththeevent-triggeredmechanism(24)solvestheglobalrobustpracticalstabilizationproblemfortheaugmentedsystem(14).TheproofisthuscompleteduponusingProposition2.1.Remark3.2:Itisinterestingtonotethat,sincetheZenobehaviormayhappenintheclosed-loopsystem,wecannotassumethatTM=∞inLemma3.1.WehavetoproveTM=∞inTheorem3.1underourspeciﬁccontrollawandspeciﬁcevent-triggeredmechanism.Ifweletδ=0inthetriggeringmechanism(24),weobtainaspecialcaseofTheorem3.1wherelimt!1|e(t)|=limt!1∥¯xc(t)∥=0.Nevertheless,inthiscase,asindicatedin(45),ourprooftechniquecannotguaranteeτd>0.Thatiswhywehaveintroducedtheparameterδintheevent-triggeredmechanism(24).Moreover,sincealargerδusuallyresultsinalargerτd,andhenceincurslessfrequenttriggeringnumberbutleadstolargersteady-statetrackingerror.Thisfactwillbeillustratedbytheexampleinnextsection.Remark3.3:Itisofinteresttocompareourresultwithsomeexistingresultsontheevent-triggeredcontrolproblemsfornonlinearsystems.First,comparedwiththosestate-basedevent-triggeredcontrolproblemsin[10],[18],[21],[23],[25],[27],ourcontrollawhereisoutput-basedanditinvolvesthedesignofalineardynamiccompensatorcalledinternalmodel.Tofacilitatetheimplementationofourcontrollawinadigitalplatform,weneedtosamplenotonlytheerroroutputoftheplantbutalsothestateoftheinternalmodel.Second,comparedwiththoseoutput-basedevent-triggeredcontrolproblemsin[1],[5],[31],weneedtodealwithnotonlytheasymptotictrackingofaclassofreferenceinputs,butalsotheasymptoticrejectionofaclassofexternaldisturbancesinthepresenceoftheparameteruncertaintythatisallowedtovaryinanarbitrarilylargeprescribedcompactset.Incontrast,references[1],[31]donotdealwithexternaldisturbancesorplantuncertainties,andreference[5]doesnotdealwithparameteruncertainties.Third,asshowninRemark2.2,thespeciﬁcformofourcontrollawisamenabletoadirectimplementationinadigitalplatformwhilethedynamiccompensatorsin[1],[5],[31]donotpossessthisproperty.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2017.2700384, IEEE Transactions on Automatic Control 0018-9286 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 7IV.ANEXAMPLEConsideraclassofLorenzsystemstakenfrom[29]asfollows˙z1=a1z1−a1y˙z2=a2z2+z1y˙y=a3z1−y−z1z2+bue=y−v1(47)wherea,col(a1,a2,a3,b)isaconstantparametervectorsatisfyinga1<0,a2<0andb>0.Theexosystemsystemtakestheformof(2)withS=[01−10].Theuncertainparameteraisexpressedasa=¯a+w,where¯a=col(¯a1,¯a2,¯a3,¯b)=col(−4,−5,2,3)representsthenominalvalueofa,andw=col(w1,w2,w3,w4)representstheuncertaintyofa.Weassumethatw∈W={w|w∈R4,|wi|≤1,i=1,2,3,4},andv∈V={v|v∈R2,|vi|≤1,i=1,2}.Clearly,Assumptions2.1and2.2aresatisﬁed.Likein[29],wehavey(v,w)=v1,z1(v,w)=r11(w)v1+r12(w)v2z2(v,w)=r21(w)v21+r22(w)v22+r23(w)v1v2u(v,w)=r31(w)v1+r32(w)v2+r33(w)v31+r34(w)v32+r35(w)v21v2+r36(w)v1v22wherethecoefﬁcientscanbefoundin[29].Clearly,Assump-tions2.3and2.4aresatisﬁed.Also,wecanfurtherverifythatd4u(v,w)dt4+10d2u(v,w)dt2+9u(v,w)=0.ThuswehaveΦ=010000100001−90−100,Γ=1000T.Choosethecontrollablepair(M,N)asfollows,M=010000100001−6−17−17−7,N=0001.BysolvingtheSylvesterequation(10),wehaveΨ=ΓT1=[−3,17,7,7].Thenweperformthecoordinatetransformation(13)andgetthefollowingaugmentedsystem,˙¯z=¯f(¯z,e,µ)˙¯η=M¯η+MNb1e−Nb1¯g(¯z,e,µ)˙e=¯g(¯z,e,µ)+bΨ¯η+ΨNe+b¯u+bΨ˜η(48)where¯z=col(¯z1,¯z2),¯f(¯z,e,µ)=col(a1¯z1−a1e,a2¯z2+(¯z1+z1)(e+v1)−z1v1)and¯g(¯z,e,µ)=a3¯z1−e−¯z1¯z2−z1¯z2−¯z1z2.Forthe¯z-subsystem,choosetheLyapunovfunctioncandi-dateV1(¯z)=~2¯z21+~4¯z41+12¯z22forsomesufﬁcientlylarge~>0.Itispossibletoshowthat,foranyµ∈Ω,any¯zandanye,@V1(z)@z¯f(¯z,e,µ)≤−ℓ1¯z21−ℓ2¯z41−ℓ3¯z22+ℓ4e2+ℓ5e4forsomeconstantsℓs>0,s=1,···,5.Thatistosay,Assumption3.1isalsosatisﬁed.051015202530−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.100.10.2Time e202530−505x 10−3Fig.1:Trackingerrorforδ=0.02.01234500.050.10.150.20.250.30.35Time |˜ϑ(t)+Ψ˜η(t)|0.05|ϑ(t)|+δFig.2:Event-triggeredconditionforδ=0.02.051015202530−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.100.1Time e202530−0.0100.01Fig.3:Trackingerrorforδ=0.04.01234500.050.10.150.20.250.30.35Time |˜ϑ(t)+Ψ˜η(t)|0.05|ϑ(t)|+δFig.4:Event-triggeredconditionforδ=0.04.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2017.2700384, IEEE Transactions on Automatic Control 0018-9286 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 8TABLEI:Event-triggerednumbers.Thus,byTheorem3.1,wecandesignanoutputfeedbackcontrollawoftheform(41)withρ(e(tk))=6(e6(tk)+1),andanoutput-basedevent-triggeredmechanismoftheform(24)withσ=0.05,andδ=0.02orδ=0.04.Simulationisperformedwithw=[0.5,−0.6,0.7,−0.3]Tandtheinitialconditions[z1(0),z2(0),y(0)]=[−1.61,0.45,−1.68],v(0)=[−0.94,0.10]Tandη(0)=[0.49,−0.31,0.13,0.20]T.TableIshowstheevent-triggerednumbersforbothδ=0.02andδ=0.04.Figures1and3showthetrackingerrorsforδ=0.02andδ=0.04,respectively.Figures2and4showtheevent-triggeredconditionsforδ=0.02andδ=0.04,respectively.Wecanﬁndthatlimt!1sup|e(t)|≤5×103forδ=0.02,andlimt!1sup|e(t)|≤0.01forδ=0.04.Also,thetriggeringnumberforδ=0.02isgreaterthanthatforδ=0.04.Thusthesesimulationresultsillustratetheviewpointthatalargerδleadstolesstriggeringnumberbutleadstolargersteady-statetrackingerrorasnotedinRemark3.2.Moreover,theZenobehaviordoesnothappenforbothδ=0.02andδ=0.04.V.CONCLUSIONInthispaper,wehaveformulatedtheevent-triggeredglobalrobustoutputregulationproblemforaclassofnonlinearsystems,andpresentedapracticalsolutiontotheproblembyanoutput-basedevent-triggeredcontrollawtogetherwithanoutput-basedZeno-freeevent-triggeredmechanism.Itwouldbeinterestingtofurtherconsiderthepossibilityofﬁndingotherevent-triggeredmechanismsandoutput-basedevent-triggeredcontrollawstoobtainanexactsolutiontoourproblem.ACKNOWLEDGMENTTheauthorswishtoacknowledgetheassociateeditorandallreviewersfortheirvaluablecommentsandconstructivesuggestions.REFERENCES[1]M.Abdelrahim,R.Postoyan,J.Daafouz,andD.Nesic,“Stabilizationofnonlinearsystemsusingevent-triggeredoutputfeedbackcontrollers”,IEEETransactionsonAutomaticControl,vol.61,no.9,pp.2682-2687,2016.[2]K.J.Astrom,B.Wittenmark,Computercontrolledsystems.PrenticeHall,UpperSaddleRiver,1977.[3]C.I.Byrnes,F.Priscoli,A.IsidoriandW.Kang,“Structurallystableoutputregulationofnonlinearsystems,”Automatica,vol.33,no.3,pp.369–385,1997.[4]E.J.Davison,“Therobustcontrolofaservomechanismproblemforlineartime-invariantmultivariablesystems,”IEEETransactionsonAutomaticControl,vol.21,no.1,pp.25–34,Feb.1976.[5]V.S.Dolk,D.P.BorgersandW.P.M.H.Heemels,“Output-basedanddecentralizeddynamicevent-triggeredcontrolwithguaranteedLp-gainperformanceandZeno-freeness,”IEEETransactionsonAutomaticControl,vol.62,no.1,pp.34-49,2017.[6]M.C.F.Donkers,andW.P.M.H.Heemels,“Output-basedevent-triggeredcontrolwithguaranteedL1-gainandimprovedanddecentral-izedevent-triggering,”IEEETransactionsonAutomaticControl,vol.57,no.6,pp.1362-1376,2012.[7]B.A.Francis,“Thelinearmultivariableregulatorproblem,”SIAMJournalonControlandOptimization,vol.15,no.3,pp.486–505,1977.[8]B.A.FrancisandW.M.Wonham,“Theinternalmodelpricipleofcontroltheory,”Automatica,vol.12,no.5,pp.457–465,1976.[9]G.F.Franklin,J.D.Powel,A.Emami-Naeini,Feedbackcontrolofdynamicalsystems.PrenticeHall,UpperSaddleRiver,2010.[10]A.Girard,“Dynamictriggeringmechanismsforevent-triggeredcontrol,”IEEETransactionsonAutomaticControl,vol.60,no.7,pp.1992-1997,2015.[11]W.P.M.H.Heemels,K.H.Johansson,andP.Tabuada,“Anintroductiontoevent-triggeredandself-triggeredcontrol,”The51stIEEEConferenceonDecisionandControl,Maul,Hawaii,USA,2012,pp.3270-3285.[12]R.A.HornandC.R.Johnson,TopicsinMatrixAnalysis,NewYork:CambridgeUniversityPress,1991.[13]J.Huang,Nonlinearoutputregulation:theoryandapplications,Phildel-phia,PA:SIAM,2004.[14]J.HuangandZ.Chen,“Ageneralframeworkfortacklingtheoutputregulationproblem,”IEEETransactionsonAutomaticControl,vol.49,no.12,pp.2203-2218,2004.[15]J.HuangandC-F.Lin,“Onarobustnonlinearservomechanismprob-lem,”IEEETransactionsonAutomaticControl,vol.39,no.7,pp.1510–1513,1994.[16]H.K.Khalil,“Robustservomechanismoutputfeedbackcontrollersforfeedbacklinearizablesystems,”Automatica,vol.30,no.10,pp.1587–1599,1994.[17]H.K.Khalil,NonlinearSystems-thirdedition,PrenticeHall,2002.[18]T.LiuandZ.P.Jiang,“Asmall-gainapproachtorobustevent-triggeredcontrolofnonlinearsystems,”IEEETransactionsonAutomaticControl,vol.60,no.8,pp.2072-2085,2015.[19]W.LiuandJ.Huang,“Robustpracticaloutputregulationforaclassofuncertainlinearminimum-phasesystemsbyoutput-basedevent-triggeredcontrol,”InternationalJournalofRobustandNonlinearControl,DOI:10.1002/rnc.3815,2017.[20]M.LuandJ.Huang,“Aclassofnonlinearinternalmodelsforglobalrobustoutputregulationproblem,”InternationalJournalofRobustandNonlinearControl,vol.25,no.12,pp.1831-1843,2015.[21]N.Marchand,S.Durand,andJ.F.G.Castellanos,“Ageneralformulaforevent-basedstabilizationofnonlinearsystems,”IEEETransactionsonAutomaticControl,vol.58,no.5,pp.1332-1337,2013.[22]V.O.Nikiforov,“Adaptivenon-lineartrackingwithcompletecompen-sationofunknowndisturbances,”EuropeanJournalofControl,vol.4,no.2,pp.132-139,1998.[23]R.Postoyan,P.Tabuada,D.Nesic,andA.Anta,“Aframeworkfortheevent-triggeredstabilizationofnonlinearsystems,”IEEETransactionsonAutomaticControl,vol.60,no.4,pp.982-996,2015.[24]E.D.SontagandA.R.Teel,“Changingsupplyfunctionsininput/statestablesystems.”IEEETransactionsonAutomaticControl,vol.40,no.8,pp.1476-1478,1995.[25]P.Tabuada,“Event-triggeredreal-timeschedulingofstabilizin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