Seron CONICET Depto de Electr57524onica Fac de Cs Exactas Ing y Agrim Universidad Nacional de Rosario Riobamba 245bis 2000 Rosario Argentina Centre for Complex Dynamic Systems and Control The Univers ity of Newcastle Callaghan NSW 2308 Australia Abs ID: 25377
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ControldesignwithguaranteedultimateboundforfeedbacklinearizablesystemsErnestoKofman;1FernandoFontenlaHernanHaimovichMaraM.SeronCONICET;Depto.deElectronica,Fac.deCs.Exactas,Ing.yAgrim.,UniversidadNacionaldeRosario,Riobamba245bis,2000Rosario,Argentina.CentreforComplexDynamicSystemsandControl,TheUniversityofNewcastle,Callaghan,NSW2308,Australia. Abstract:Foraclassofperturbedfeedbacklinearizablenonlinearsystems,weconsiderthecomputationandassignmentofprescribedultimateboundsonthesystemstates.Weemployarecentlyproposedcomponentwiseboundcomputationprocedure,whichdirectlytakesintoaccountboththesystemandperturbationstructuresbyperformingcomponentwiseanalysis.We rstderivesucientconditionstoensurethatthetrajectoriesoriginatingfrominitialconditionsinanappropriatesetareultimatelybounded.Secondly,andmostimportantly,forstate-feedback-linearizablenonlinearsystemswithmatchedperturbations,weprovideasystematicdesignproceduretocomputeastatefeedbackcontrolthatensuresaprescribedultimateboundfortheclosed-loopsystemstates.Theprocedurecombinesnonlinearstate-feedback-linearizingcontrolwithastate-feedbackmatrixcomputedviaaneigenstructureassignmentmethodpreviouslyreportedbytheauthors.Asimulationexampleillustratesthesimplicityandsystematicityoftheproposeddesignmethod. 1.INTRODUCTIONThispaperconsidersthecomputationandassignmentviastatefeedbackofultimateboundsontheclosed-looptra-jectoriesofperturbedstate-feedback-linearizablenonlinearsystems.Perturbationsindynamicsystemsmayarisefromunknowndisturbancesignals,modeluncertainty,compo-nentageing,etc.Boundsontheperturbationvariablesaretypicallyknownandmaybeusedtoobtain,undercer-tainconditions,ultimateboundsontheperturbedsystemtrajectories.AstandardapproachforthecomputationofultimateboundsistheuseoflevelsetsofsuitableLyapunovfunc-tions[see,forexample,Section9.2ofKhalil(2002)].AnalternativeapproachwasproposedinKofman(2005),Kof-manetal.(2007a)andHaimovich(2006),wherecompo-nentwiseultimateboundformulaewerederivedexploitingthesystemgeometryinmodalcoordinatesaswellastheperturbationstructure,withoutrequiringthecomputationofaLyapunovfunctionforthesystem.Forsomesystemandperturbationstructures,thiscomponentwiseapproachwasshowntoprovideboundsthataremuchtighterthanthoseobtainedviastandardLyapunovanalysis.Forlinearsystemswithmatchedperturbations(thatis,perturbationsthatspanthesamespacespannedbythecontrolinput),anultimateboundontheclosed-looptrajectoriescanbearbitrarilyassignedbystatefeedbackcontrol(SchmitendorfandBarmish,1986).Severalcontroldesignmethods,basedonaLyapunovapproach,whichcanachieveanarbitrarilysmallultimateboundforlinearuncertainsystemshavebeenreportedintherobustcontrol 1email:kofman@fceia.unr.edu.arliterature(see,forexample,Barmishetal.,1983;TrinhandAldeen,1996;CaoandSun,1998;Oucheriah,1999).TakingadeparturefromLyapunovanalysisandexploit-ingtheaforementionedcomponentwiseapproachforulti-mateboundcomputation,weproposedinKofmanetal.(2008,2007b)anewcontrollerdesignmethodforlin-ear,continous-timesystemsthatguaranteesaprespeci edultimateboundontheclosed-loopsystemtrajectories.Themethodtakesadvantageofthedependencyofthecomponentwiseultimateboundexpressionsonthesystemeigenstructure.Usingtechniquesofeigenvalueandeigen-vectorassignmentbystatefeedback,weshowedthatastatefeedbackgaincanbedesignedsuchthattheulti-mateboundexpressiondecreasestozeroasa\scaling"parameter,associatedwiththemagnitudeoftheclosed-loopeigenvalues,increases.Theproposeddesignprocedureissystematicinthesensethat,onceadesired\normalized"con gurationischosenfortheclosed-loopeigenvalues,itonlyrequirestoincreasethescalingparameteratmostonceforthedesiredultimateboundoneachcomponentofthestatetobeachieved.InthecurrentpaperweextendthedesignmethodofKofmanetal.(2008,2007b)toperturbedstate-feedback-linearizablenonlinearsystems(Isidori,1995).Thelatteraresystemswhich,undersuitablenonlinearfeedbackandcoordinatetransformation,canbeexpressedasalinearasymptoticallystablesystemwithnonlinearperturbationterms.Forthesesystems,wederivesucientconditionstoensurethatthetrajectoriesoriginatingfrominitialconditionsinanappropriatesetareultimatelybounded.Moreover,forstate-feedback-linearizablenonlinearsys-temswithmatchedperturbations,weprovideanalgorithm tosystematicallydesignastatefeedbackcontrolthatensuresaprescribedultimateboundfortheclosed-loopsystemstates.Theproposedalgorithmcombinesstandardnonlinearfeedbacklinearizingcontrol[see,forexample,Isidori(1995)]withastate-feedbackmatrixdesignedbasedonthemethodofKofmanetal.(2008,2007b).Anexam-pleofasynchronousgeneratorillustratesthesimplicityandsystematicityoftheapproach.Simulationresultsforthisexampleunderpersistentperturbationssatisfyingap-propriateboundsalsodemonstratethepotentialofthemethodtoobtainrelativelytightboundsontheclosed-looptrajectories.Theremainderofthepaperisorganizedasfollows.Section2presentstheclassofnonlinearsystemsun-derconsiderationandoutlinesthegoalsofthepaper.Section3reviewssomepreliminaryresultsneededtoachievethedesiredgoals.Section4presentsthemaincontributionsofthepaper,namely,asystematicmethodtocomputecomponentwiseultimateboundsforstate-feedback-linearizablesystemsandasystematiccontrolde-signmethodologysothatanydesiredultimateboundisachievedinthecaseofmatchedperturbations.Section5illustratestheresultswithanexampleofasynchronousgenerator.Finally,Section6concludesthepaper.Notation.Inthesequel,RandCdenotethesetsofrealandcomplexnumbers,respectively.jMjandRe(M)denotetheelementwisemagnitudeandrealpart,respec-tively,ofa(possiblycomplex)matrixorvectorM.Theexpressionxy(x y)denotesthesetofcomponent-wise(strict)inequalitiesbetweentheelementsoftherealvectors(ormatrices)xandy,andsimilarlyforxy(x y).R+andR+;0denotethepositiveandnonnegativerealnumbers,respectively.Forc2C, cdenotesitscomplexconjugate.2.PROBLEMSTATEMENTConsiderthefollowingcontinuous-timenonlinearsystemwithnstates,minputs,andkdisturbancevariables:_x=f(x)+mXi=1gi(x)ui+kXj=1hj(x)wj;(1)wherex(t)2Rn,f(0)=0,andsuchthatf;g1;:::;gm,h1;:::;hkaresmooth(C1)vector eldsde nedonanopensetUxRncontainingtheorigin.Thedisturbancevariablesw1;:::;wkareassumedtobeboundedasfollows:jwj(t)jj(x(t));forallt0;forj=1;:::;k;(2)wherej(x)0forallx2Ux.Wewrite(2)incondensedformasfollows:jw(t)j(x(t));forallt0;(3)withw=col(w1;:::;wk)and=col(1;:::;k).Also,wehereafterdenoteu=col(u1;:::;um),g=[g1j:::jgm]andh=[h1j:::jhk].Associatedwith(1)isthenominalsystem_x=f(x)+mXi=1gi(x)ui=f(x)+g(x)u:(4)Thenominalsystem(4)isassumedtobestate-feedback-linearizableinUx(Isidori,1995),thatis,thereexistacoordinatetransformation(di eomorphism)z=(x)andapairoffeedbackfunctions (x)and (x),allde nedonUx,sothat (x)isnonsingularforallx2Uxand@ @x(f(x)+g(x) (x))x= 1(z)=A0z;(5)@ @x(g(x) (x))x= 1(z)=B0;(6)whereA0=diag(A1;:::;Am);B0=diag(b1;:::;bm);(7)Ai2Rdidi,bi2Rdi1,Ai=266666664010:::0......1..................0............10:::::::::0377777775;bi=26666640......013777775;(8)fori=1;:::;m,andPmi=1di=n.Therefore,inthiscaseapplicationofthestate-feedbacklawu= (x)+ (x)vtosystem(1),wherev=col(v1;:::;vm)isanewinput,jointlywiththechangeofcoordinatesz=(x),yields_z=A0z+B0v+@ @xh(x)x= 1(z)w:(9)Applicationoftheadditionalstate-feedbacklawv=Kzto(9)or,equivalently,applicationofu= (x)+ (x)K(x)(10)to(1),yields_z=(A0+B0K)z+@ @xh(x)x= 1(z)w:(11)Aparticularimportanttypeofsystemsoftheform(1)isgivenbythecaseof\matchedperturbations",namelythecasewhenh(x)=g(x)\r(x),forsomematrix\r(x)ofsmoothfunctionsde nedonUx.Forthistypeofsystems,applicationof(10)yieldstheclosed-loopsystem_z=(A0+B0K)z+B0 1(x)\r(x)x= 1(z)w:(12)Thisworkhasthefollowinggoals:G1)toprovidesucientconditionstoensurethatthetrajectoriesofsystem(1),underafeedbacklawoftheform(10),areultimatelybounded.G2)toestimateanultimateboundforsuchclosed-loopsystem,andG3)todesignmatrixKin(10)sothatthesystemstatetrajectoriesareultimatelyboundedandsatisfyagivenultimateboundinthecaseofmatchedpertur-bations.Beforeproceedingwithsomepreliminaryresultsrequiredtoachievetheabovegoals,weobservethatthesetting(1)-(3)canaccommodateanycombinationofthefollowingtypesofuncertainty:Uncertaintyinthesystemevolutionfunction,where_x(t)=(f(x)+f(x))+g(x)u(t),andjf(x)j (x),8t0;inthiscase,wecantakeh(x)=Inin(1)and(x)= (x)in(3).Uncertaintyinthesysteminputfunction[assumingafeedbacku=(x)in(1)],where_x(t)=f(x)+(g(x)+g(x))(x),andjg(x)j\r(x),8t0;inthiscase, wecantakeh(x)=Inin(1)and(x)=\r(x)j(x)jin(3).Boundeddisturbances,wherejw(t)jw,8t0;inthiscase,wecantake(x)=win(3).3.PRELIMINARYRESULTSInthissection,westatepreviousresultsthatareneededtoachievethegoalsoutlinedabove.Section3.1presentsamethodforcomputingacomponentwiseultimateboundforalinearsystemwithanadditiveperturbationhavingaboundthatmaydependonthelinearsystemstate.Section3.2recallsaresultthatshowshowalinearstatefeedbackmaybedesignedtoassignclosed-loopeigenvaluesandeigenvectorsforalinearsystem.3.1UltimateBoundofaPerturbedLinearSystemThefollowingresultcomputesacomponentwiseultimateboundforaLTIsysteminpresenceofperturbationsthatarecomponentwiseboundedbyfunctionsofthesystemstate.Thisresultisamodi edversionofTheorem2ofKofmanetal.(2007a).Theorem1.Considerthesystem_x(t)=Ax(t)+Hv(t);(13)wherex(t)2Rn,v(t)2Rk,H2Rnk,andA2RnnisHurwitzwith(complex)Jordancanonicalform=V 1AV.Supposethatjv(t)j(x(t))8t0;(14)where:Rn!Rk+;0isacontinuousmapverifyingjx1jjx2j)(x1)(x2):(15)ConsiderthemapT:Rn!Rn+;0de nedbyT(x),jVjSH(x);(16)whereSH, [Re()] 1 V 1H :(17)Supposethatthereexistsxm2Rnsatisfying,T(xm) xm.Then,1)b,limr!1Tr(xm)existsandsatis es0b xm.2)IfjV 1x(0)jSH(xm),thena)jV 1x(t)jSH(xm)forallt0.b)Also,givenapositivevector2Rn+,a nitetimetfexistssothatforallttf,i)jV 1x(t)jSH(b)+.ii)jx(t)jb+jVj.TheproofofthistheoremisalmostidenticaltothatofTheorem2ofKofmanetal.(2007a),theonlydi erencebeingthepresenceofmatrixH.Inthesequel,wewillequivalentlyexpressTheorem1part2)b)ii)asfollows:IfjV 1x(0)jSH(xm),thenxisultimatelyboundedtotheregionfx:jxjbg.3.2EigenvalueandEigenvectorAssignmentbyFeedbackThefollowingresultispartofTheorem4.1ofKofmanetal.(2007b)andalsopartofTheorems4.2and4.3ofKofmanetal.(2008).Thisresultshowshowastate-feedbackmatrixcanbecomputedsothatdesiredeigenvaluesandeigenvectorsareassignedtotheclosed-looplinearsystem.Inaddition,theresultestablisheshowsomematriceschangeasthedesiredeigenvaluesarescaledaccordingtoascalingfactor.Theorem2.Take�0andselectaneigenvaluematrix=~=diag(~1;:::;~n),where~i2C,i=1;:::;n,satisfy~i=~jwheneveri=j,Re(~i)0,andif~i=2R,theneither~i 1= ~ior~i+1= ~i.Selectcomplexnumbersei;j2C,i=1;:::;m,j=1;:::;nsothatei;j+1= ei;jwhenever~j+1= ~jandsuchthatthematrixVde nedasV,"V1;1:::V1;n.........Vm;1:::Vm;n#;Vi;j,264ei;j(~j) (di 1)...ei;j(~j) 1ei;j375;(18)haslinearlyindependentcolumns,wheredi,i=1;:::;m,arethedimensionsofAiandbiin(7){(8).De neR,jVj [Re()] 1 V 1B0 ;(19)and,K=(BT0B0) 1BT0(VV 1 A0):(20)Then,i)TheentriesofthematrixRarenonincreasingfunctionsof.ii)A0+B0K=VV 1.4.MAINRESULTSInthissection,wepresentthemaincontributionofthepa-per.Namely,weprovideasystematicmethodtocomputecomponentwiseultimateboundsforthestate-feedback-linearizablesystem(1),(3)andasystematiccontrolde-signmethodologysothatanydesiredultimateboundisachievedinthecaseofmatchedperturbations,thatis,foraclosed-loopsystemoftheform(12).4.1ComponentwiseUltimateBoundAnalysisThefollowingtheorempresentssucientconditionstoensuretheultimateboundednessofthestatetrajectoriesofaperturbedsystemunderstate-feedbacklinearization.Thistheoremalsoshowshowtocomputeanultimatebound.Theorem3.Considersystem(1),wheretheperturbationw(t)satis es(3).Supposethattheassociatednominalsystem(4)isstate-feedback-linearizable,andletz=(x)bethecoordinatetransformation,and (x)and (x)bethefeedbackfunctions,sothat(1)istransformedinto(11)underapplicationofthefeedbacklaw(10).LetKbechosensothatA,A0+B0KisHurwitz,andlet=V 1AVbethe(complex)JordancanonicalformofA.ConsiderSI,de nedin(17)withH=I,andletR,jVjSI.De ne(z),sup:jjjzj @ @xh(x) (x)x= 1()(21)andletT(z),R(z).LetUz,(Ux)andsupposethatT(zm) zmforsomezm2Uz.Inaddition,letBm,fz:jV 1zjSI(zm)gandsupposethatBmUz.Then, i)bz,limr!1Tr(zm)existsandifx(0)2 1(Bm)thetrajectoriesoftheclosed-loopsystem(1),(10)areultimatelyboundedtotheregionjxjb,supz:jV 1zjSI(bz)j 1(z)j:(22)ii)If,inaddition,h(x)=g(x)\r(x)[matchedperturba-tions,recall(12)],theni)abovealsoholdsifwereplaceSIabovewithSB0[de nedasin(17)withH=B0andB0asin(6)]andabovebym,de nedasm(z),sup:jjjzjh 1(x)\r(x) (x)ix= 1():(23)Proof.De ningv(t),@ @xh(x)x= 1(z(t))w(t);(24)system(11)canberewrittenas_z(t)=Az(t)+v(t)(25)From(3)and(21),itfollowsthatjv(t)j(z(t))where(z)veri es(15).If,inaddition,h(x)=g(x)\r(x),thenv(t)=B0s(t)[see(12)],wheres(t)= 1(x)\r(x)x= 1(z(t))w(t)(26)satis esjs(t)jm(z(t))[see(23)]andmalsoveri- es(15).Then,Theorem1,part1)establishestheex-istenceofbz.Next,applyingTheorem1,part2)a)tosystem(25)weconcludethatanytrajectorystartingfromaninitialconditionz(0)2BmdoesnotleavetheregionBmUz.Moreover,itfollowsfromTheorem1,part2)b)i)thatz(t)isultimatelyboundedtotheregionBz=fz:jV 1zjSI(bz)gBmUz.Asimilarargumentap-pliedtothecaseofmatchedperturbationsyieldsthatz(t)isultimatelyboundedtotheregionBz=fz:jV 1zjSB0m(bz)gBmUz.Takingintoaccountthatx= 1(z)(with 1beingadi eomorphismde nedinUz),andthatz(t)cannotabandonUz,itfollowsthatx(t)isultimatelyboundedtotheregionBx= 1(Bz).Then,takingbasthesupremumofjxjinBxweobtaintheultimateboundofEq.(22).24.2RobustcontroldesignThenexttheoremestablishesdesignconditionstoachieveadesiredultimateboundinaperturbedsystemwithstate-feedback-linearizationunderthehypothesisofmatchedperturbations.Theresultisthentranslatedintoadesignalgorithmthatconstitutesthecoreoftheproposedsys-tematiccontroldesignmethodology.Theorem4.Considersystem(1),wheretheperturbationw(t)satis es(3).Supposethattheassociatednominalsystem(4)isstate-feedback-linearizable,andletz=(x)bethecoordinatetransformation,and (x)and (x)bethefeedbackfunctions,sothat(5){(8)hold.Assumealsothath(x)=g(x)\r(x)(matchedperturbations).SelectV;accordingtoTheorem2,calculateRfrom(19),andconsidermasde nedin(23)andT(z),Rm(z)(27)Letzc2Uz,zc 0,besuchthatfz:jzjzcgUz.Then,i)ThecomponentsofthevectorT(zc)arenon-increasingfunctionsoffor�0.ii)Let�0besuchthatT(zc) zc.Then,thefeedbacklaw(10)withK=Kgivenby(20)ensuresthattheclosed-loopsystem(12)isultimatelyboundedtotheregionjxjb,supz:jV 1zjSB0m(bz)j 1(z)j(28)whereSB0andmareasde nedinTheorem3ii)andbz=limr!1Tr(zc) zc.Proof.i):theprooffollowsstraightforwardlyfromTheo-rem2i)andthefactthatm(z)doesnotdependon.ii):De ningv(z;w), 1( 1(z))\r( 1(z))w,theclosed-loopsystem(11)undermatchedperturbationsandwithK=Ktakestheform_z(t)=(A0+B0K)z(t)+B0v(z(t);w(t)):(29)AccordingtoTheorem2ii),A,A0+B0K=VV 1,which,since=diag(~1;:::;~n)andRe(~i)0ensuresthatAisHurwitz.From(3)and(23),itfollowsthatjv(z;w)jm(z).Also,itcanbestraightforwardlyveri edthatmsatis es(15).Then,takingH=B0andapplyingTheorem1ii)tosystem(29)weconcludethatanytrajectorysuchthatjV 1z(0)jSB0(zc)cannotleavetheregionBc=fz:jV 1zjSB0(zc)g.NoticethatBcisentirelycontainedintheregionfz:jzjzcgUzandthenBcUz.Then,applicationofTheorem3yieldsthatxisultimatelyboundedtotheregion(28).2Theorem4i)statesthatthecomponentsofthevectorT(zc)decrease(atleastinverse-linearly)with.There-fore,theconditionT(zc) zccanalwaysbeachievedthroughthechoiceofasucientlylargevalueof.The-orem4ii)canbeappliedinordertodesignmatrixKin(10)sothattheclosed-loopsystemexhibitsanarbitraryultimatebound.Thefollowingalgorithmimplementsthisidea.Algorithm1.Givenadesiredcomponentwiseultimateboundb2Rn+;0forthestatex,(i)Findthechangeofcoordinatesz=(x),jointlywiththefeedbackfunctions (x)and (x),sothat(5){(8)aresatis ed.(ii)Takezc 0sothatjzjzc)z2Uz,and,supz:jzjzcj 1(z)jb(30)(iii)Calculatem(z)accordingto(23).(iv)Selectanarbitrary&#x-5.1;䘘0,andandVasspeci edinTheorem2.(v)ComputeRfrom(19).(vi)EvaluateT(zc)accordingto(27).IfT(zc) zc,gotostep(viii).(vii)Compute2maxi((T(zc))i=zci)andsetthenewequaltothisvalue.ReevaluateVaccordingto(18).(viii)ComputeK=Kaccordingto(20).Algorithm1,accordingtoTheorem4,. ndsthefeedbackmatrixKofthefeedbacklaw(10)thatguaranteesanultimateboundbgivenby(28).Thisboundbcanbeproventobelessthanorequaltob(thedesiredultimatebound)asfollows. 2(T(zc))idenotesthei-thcomponentofT(zc).Similarlyforzci. First,noticethattheconditionjV 1zjSB0m(bz)in(28)impliesthatjzj zcsincejzj=jVV 1zjjVjjV 1zjjVjSB0m(bz)=T(bz)=bz zc.Thus,thesupremumofj 1()jonz:jzjzcisgreaterthanorequaltothesupremumonz:jV 1zjSB0m(bz).Then,comparing(28)and(30)itfollowsthatbb.Remark1.Thedesignprocedurebeginsbyfeedbacklin-earizingthenonlinearplant.Then,eigenvalue/eigenvectorassignmentisperformed,similarlytoKofmanetal.(2008).Westressthattheprocedureisnotjustastraightforwardapplicationofthelatterresultssince,evenafterfeedbacklinearization,theperturbationsexhibitnonlineardepen-denceonthestatevariables.TheultimateboundishenceestimatedviathetechniquedevelopedinKofmanetal.(2007a),wheretheultimateboundestimateisgivenbythe xedpointofanonlinearmapinsteadofthesimpleclosed-formexpressionofKofmanetal.(2008).Inaddition,thenonlinearcoordinatetransformationrequiredposesanad-ditionalproblem,becauserelationshipsbetweenboundsontheoriginalandthelinearcoordinatesmustbecomputed.Thecontributionofthisworkliespreciselyinprovidingasolutiontoalltheseproblems.5.EXAMPLESThedynamicsofasynchronousgeneratoronanin nitebuscanbeexpressedbyEq.(1),withf(x)="x2 p[(1+x3)sin(x1+d) sind] qx2 rx3+s[cos(x1+d) cosd]#;g(x)=g1(x)=h(x)=h1(x)=[001]T;wherep,q,r,s,darerealparametersandwhereweconsideraboundedperturbationtermjw(t)jwm.Theassociatednominalplant(4)isstate-feedback-linearizable.Themapz=(x),where(x)="x1x2 p[(1+x3)sin(x1+d) sind] qx2#;jointlywiththefeedbackfunctions (x)= px2(1+x3)cos(x1+d) pqsind+q2x2 psin(x1+d)++q(1+x3)+rx3 s[cos(x1+d) cosd])and (x)= 1 psin(x1+d);issuchthat(5){(8)aresatis edwithA0=A1="010001000#;B0=b1="001#;forallx2Ux=fx:0x1+dg.Also,Uz=(Ux)=fz:0z1+dg.ThegoalistodesignthefeedbackmatrixKin(10)sothattheclosed-loopsystem(1),(10)isultimatelyboundedtotheregionjxjb=[0:10:050:001]T.Weconsiderthesetofparametersp=136:0544;q=4;r=0:4091;s=0:2576;d==4,andtheperturbationboundwm=0:001.WenextfollowAlgorithm1.Step(i)ofAlgorithm1wasperformedabove.AtStep(ii),zccanbechoseninmanyways.Apossibleselectioniszc=[0:05000:00255:0098]T:(31)Notethatjzj zc)z2Uz.Step(iii)requiresthecalculationofm(z),accordingto(23),where\r(x)=1and(x)=wm=0:001.Thiscomputationyieldsm(z)=psin(jz1j+d)wmif(jz1j+d)=2;pwmotherwise.AtStep(iv)wetake=1,andproposetheeigenvaluecon guration=diag( 1; 5; 20).ForV,wetakee1;j=1forj=1;2;3in(18).ThischoiceyieldsV="1:00000:04000:0025 1:0000 0:2000 0:05001:00001:00001:0000#:(32)Step(v)yieldsR=[0:01670:03330:1667]TandforStep(vi)wecalculateT(zc)=Rm(zc)=[0:0016820:0033630:01682]T.SincetheconditionT(zc) zcisnotsatis ed,weproceedwithStep(vii),computingmaxi((T(zc))i=zci)=1:3452,setting=1:3452andrecalculatingVforthisnewvalueof.Finally,Step(viii)givesthefeedbackmatrixK=[ 243:4 226:2 34:98].Theorem4ensuresthattheultimateboundoftheclosed-loopsystem(1),(10)withK=Kisatleastastightasb.Wecanalsoemploy(28)toestimatea(possibly)tighterbound.The xedpointbzofmapTiteratedfromzcisbz=[0:65911:773311:93]T10 3,andthen,using(28)weconcludethatjx(t)jisboundedtobx=[0:65911:77330:7101]T10 3.Figures1and2showtheregionde nedinthestatespacebytheboundbxandsimulationresultsfortheclosed-looptrajectorieswithzeroinitialconditionsandfordi erentperturbationssatisfyingEq.(2).Forthissimulation,sev-eralsinusoidalandpulsetrainperturbationsofdi erentfrequencieswereapplied. -8 -6 -4 -2 0 2 4 6 8x 10-4 -2 -1 0 1 2x 10-3 -8 -6 -4 -2 0 2 4 6 8x 10-4 x1x3x2bxFig.1.Ultimateboundregion(dashedbox)andclosed-looptrajectoriesfordi erentperturbations.Weobservefromthe guresthattheproposeddesignprocedureachievesrelativelytightboundsunderalargerangeofpersistentperturbations.Alsonotefromthestepscarriedoutinthisexample,thesystematicityoftheapproachandthesimplicityofthecomputationsinvolved. 0 2 4 6 8x 10 4 2 1.5 1 0.5 0 0.5 1 1.5 2x 10 3 0 2 4 6 8x 10 4 8 6 4 2 0 2 4 6 8x 10 4 5 5 0 0.5 1 1.5 2x 10 3 8 6 4 2 0 2 4 6 8x 10 4 x1x1 x2x2x3x3Fig.2.Ultimateboundregion(dashedbox)andclosed-looptrajectoriesfordi erentperturbations.6.CONCLUSIONSForaclassofperturbedfeedback-linearizablenonlinearsystems,wehavepresentedasystematicdesignproceduretocomputeastatefeedbackcontrolthatensuresapre-scribedultimateboundfortheclosed-loopsystemstates.Theprocedureutilizesacomponentwiseboundcomputa-tionmethodpreviouslyintroducedbytheauthors,andcombinesnonlinearstate-feedback-linearizingcontrolwithlinearstatefeedbackcomputedviaeigenstructureassign-ment.Theproposedprocedurewasillustratedonanex-ampleofasynchronousgenerator.Futureworkwillconsiderthemorepracticaloutputfeed-backcase,bothforlinearandnonlinearsystemsREFERENCESB.R.Barmish,I.R.Petersen,andA.Feuer.Linearultimateboundednesscontrolofuncertaindynamicalsystems.Automatica,19(5):523{532,1983.Y.-Y.CaoandY.-X.Sun.Robuststabilizationofuncer-tainsystemswithtime-varyingmultistatedelay.IEEETrans.onAutomaticControl,43(10):1484{1488,1998.H.Haimovich.Quantisationissuesinfeedbackcontrol.PhDthesis,SchoolofElectricalEngi-neeringandComputerScience,TheUniversityofNewcastle,Australia,March2006.URLhttp://usuarios.fceia.unr.edu.ar/haimo.AlbertoIsidori.NonlinearControlSystems.Springer,London,3rdedition,1995.H.Khalil.NonlinearSystems.Prentice-Hall,NewJersey,3rdedition,2002.E.Kofman.NonconservativeultimateboundestimationinLTIperturbedsystems.Automatica,41(10):1835{1838,2005.E.Kofman,H.Haimovich,andM.M.Seron.Asystem-aticmethodtoobtainultimateboundsforperturbedsystems.InternationalJournalofControl,80(2):167{178,2007a.E.Kofman,M.M.Seron,andH.Haimovich.Robustcon-troldesignwithguaranteedstateultimatebound.In3rdInternationalConferenceonIntegratedModelingandAnalysisinAppliedControlandAutomationIMAACA,BuenosAires,Argentina,2007b.E.Kofman,M.M.Seron,andH.Haimovich.Controldesignwithguaranteedultimateboundforperturbedsystems.Automatica,2008.InPress.S.Oucheriah.Robusttrackingandmodelfollowingofuncertaindynamicdelaysystemsbymemorilesslinearcontrollers.IEEETrans.onAutomaticControl,44(7):1473{1477,1999.W.E.SchmitendorfandB.R.Barmish.Robustasymp-totictrackingforlinearsystemswithunknownparame-ters.Automatica,22:355{360,1986.H.TrinhandM.Aldeen.Outputtrackingforlinearuncertaintime-delaysystems.IEEProceedings,ControlTheoryandApplications,143(6):481{488,1996.