Email arcakmrpiedu Visteon Corporation 36800 Plymouth Rd Livonia MI 48150 Tel 7346662714 Email mlarsenvisteoncom Department of Electrical and Computer Engineering University of California Santa Barbara CA 931069560 Email petareceucsbedu Abstract The ID: 26524
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CIRCLEANDPOPOVCRITERIAASTOOLSFORNONLINEARFEEDBACKDESIGNMuratArcakMichaelLarsenPetarKokotovi¶cDepartmentofElectrical,ComputerandSystemsEngineering,RensselaerPolytechnicInstitute,Troy,NY12180-3590.Email:arcakm@rpi.edu ResearchsupportedinpartbytheNationalScienceFoundationundergrantECS-9812346andtheAirForce positivereal(SPR),thusachievingglobalasymp-toticstability(GAS)fromthecirclecriterion.Inthispaperweidentifystructuralobstaclestothefeasibilityofthecirclecriteriondesign,anddevelopnewdesignprocedureswhichcircumventtheseobstacles.We¯rstgiveananalyticaltestforthefeasibilityofthecirclecriteriondesign,basedontherecentindirectpassivationconditionsde-rivedbyArcakandKokotovi¶c[2001].Thistestre-vealsthatfeasibilityisdeterminedbytherelativedegreeofthelinearblock,andtheunstablepartofitszerodynamics.Next,therelativedegreeob-stacleisremovedwithanextendeddesignwhichemploysderivativesofthenonlinearityinthefeed-backcontrollaw.ThezerodynamicsrestrictionsarerelaxedbyaPopovmultiplierdesign,whichalsoreducesthecomplexityofthecirclecriteriondesignwhenbothdesignsareapplicable.Thisisillustratedonajetenginesurgesubsystemexam-ple.ereviewthebasiccirclecriteriondesigninSec-tion2,andanalyzeitsfeasibilityinSection3.Anextendeddesign,presentedinSection4,removestherelativedegreeobstacleofthebasiccirclecriteriondesign.ThePopovmultiplierdesignispresentedinSection5.Theproofsareomittedduetospacelimitations.2.BASICCIRCLECRITERIONDESIGNWestartwithanintroductoryexampleofasingledegree-of-freedomactivemagneticbearingmodelduetoTsiotrasandVelenis[2000],_x1=x2_x2=²x3+x3jx3j(1)_x3=u;wherex1istherotorposition,x2isthevelocity,x3isthemagnetic°ux,andtheparameter²�0representsthebias°ux.Letustrytostabilizethesystematx=0usingalinearfeedbackandacopyofthenonlinearity;thatis,u=k1x1+k2x2+k3x3¡¯x3jx3j:(2)Theresultingclosed-loopsysteminFigure1isthefeedbackinterconnectionofalinearblockandthenonlinearity'(x3)=x3jx3j.Thefeasibilityofthecirclecriteriondesign(2)dependsonwhethertheparametersk1,k2,k3and¯canbefoundtorenderthelinearblockSPR.Ifso,thesectorpropertyx3'(x3)¸0ensuresGASoftheequi-libriumx=0fromthecirclecriterion.Afurtherquestioniswhethersuchadesignispossiblewith¯=0,thatis,withoutanonlinearterminthecontrollaw.Thiswouldeliminatetheneedforknowledgeaboutthenonlinearity,otherthanitssectorproperty.Asweshallsee,oneofourresultsimpliesthatthecirclecriteriondesignforsystem(1)isnotfeasiblewith¯=0,whichmeansthatthenonlinearterm¯x3jx3jiscrucial._x1=x2_x2=²x3¡w_x3=k1x1+k2x2+k3x3+¯wx3w¡'(x3)Fig.1.Theclosed-loopsystem(1)withthecirclecriteriondesign(2).Wenowformulatethecirclecriteriondesignforthesystem_x=Ax¡G'(z)+Bu(3)z=Hx;wherex2IRn,u2IRm,z2IRp,andthenonlinearity'(¢):IRp!IRpsatis¯esthesectorpropertyzT'(z)¸0,andiscontinuous,sothat'(0)=0.Weemploythecontrollawu=Kx¡¯'(z)(4)which,with¯6=0,requireseithertheknowledgeofthenonlinearity,ortheavailabilityofthesignalw:=¡'(z):(5)InviewofthePositiveRealLemma,theproblemofrenderingtheclosed-loopsystem_x=(A+BK)x+(G+B¯)w(6)SPRfromtheinputwtotheoutputz=HxisequivalenttotheexistenceofmatricesP=PT�0andQ=QT�0suchthat(A+BK)TP+P(A+BK)+Q·0(7)P(G+B¯)=HT:(8)ThisSPRpropertyguaranteesGASoftheequi-libriumx=0,becausetheLyapunovfunctionV(x)=xTPxsatis¯es_V=¡xTQx¡2zT'(z);(9)wheretheright-handsideisnegativede¯nitebecauseofthesectorpropertyzT'(z)¸0.Although(7)-(8)isnotanLMI,multiplying(7)frombothsides,and(8)fromtheleft,byX:=P¡1resultsin X(A+BK)T+(A+BK)X+~Q·0(10)(G+B¯)=XHT;(11)whichisanLMIinX=XT�0,~Q=~QT:=XTQX�0,XKTand¯.Thismeansthatwecanusethee±cientnumericaltoolsavailableforLMI'stodeterminewhetherthedesignisfeasibleand,ifso,tocomputeKand¯inthecontrollaw(4).3.FEASIBILITYCONDITIONSTocharacterizetheclassesofsystemstowhichthecirclecriteriondesignisapplicable,ourtaskistodeterminewhenthereexistK,¯,X=XT�0and~Q=~QTsatisfying(10)-(11).Thecasewhere¯isconstrainedtobezeroisofseparateinterestbecause,then,thecontrollaw(4)islinear,andtheexactknowledgeofthenonlinearity'(z)isnotrequiredforitsimplementation.Thefeasibilityconditionsaregivenforsystemswithasinglenonlinearityandasinglecontrolinput;thatis,u;z2IR.Formultivariablenonlinearities,analo-gousresultscanbeobtainedwithmorecumber-somecalculations.Whenthelinearpartof(3)iscontrollableandobservable,thatis,whenthetriple(H;A;B)isminimal,astateandfeedbacktransformationresultsinthenormalform_»i=Ai»i+Ei0y1+Gi0w;i=1;2;3;(12)_y1=y2+g1w_y2=y3+g2w.(13)_yr=u+grwz=y1;(14)whererdenotestherelativedegreefromtheinpututotheoutputz,andthespectraofAiinthezerodynamicssubsystem(12)are¾(A1)½jC+;¾(A2)½jC0;¾(A3)½jC¡:(15)Thefollowinglemma,provedinArcakandKoko-tovi¶c[2001],showsthattheobstaclestofeasibilityareprimarilyduetotheunstablepartofthezerodynamics:emma1.(¯=0)Considerthesystem(12)-(14),andthematricesU=UT,V=VTde¯nedbyA1U+UA1T=(E10¡G1)(E10¡G1)T(16)A1V+VA1T=(E10+G1)(E10+G1)T:(17)Astatefeedbackcontrollawu=Kxthatrenderstheclosed-loopsystemSPRfromwtoz=y1existsifandonlyifg1�0;g20;U¡V�2g1G1G1T;(18)and,foreveryeigenvectorpofA2T,p¤(E20¡G2)(E20¡G2)Tp�(19)p¤(E20+G2)(E20+G2)Tp:2Wenextgivethefeasibilityconditionsforthecase¯6=0:Lemma2.(¯6=0)Whenr=1,acontrollawu=Kx+¯wthatrenders(12)-(14)SPRfromwtoz=y1existsifandonlyifU�V,and(19)holdsforeveryeigenvectorofA2T.Whenr=2,g1�0andU¡V�2g1G1G1Tarerequiredinaddition.Whenr¸3,alltheconditionsofLemma1arerequired.2AnimportantimplicationofLemma2isthat,whentherelativedegreeisr=1orr=2,thenonlinearterm¯'(z)inthecontrollaw(4)rendersthefeasibilityconditionslessrestrictive.However,whenr¸3,theconditionsofLemma1andLemma2arethesame;thatis,ifthedesignisfeasible,itisalsofeasiblewiththelinearcontrollawu=Kx.Example1.Themagneticbearingsystem(1)hasr=1,andwiththenewvariablesy1:=x3,and»2:=(x1;x2)T,itappearsintheform(12)-(14),whereA2=·0100¸;E20=·0²¸;G2:=·0¡1¸(20)andw:=¡x3jx3j.Thebasiccirclecriteriondesign(2)isfeasiblewith¯6=0becausetheeigenvectorofA2T0isp=[01]T,whichsatis¯es(19)forevery²�0.AsolutiontotheLMI(10)-(11)for²=1yieldsthecontrollawu=¡0:6777x1¡2:1724x2¡1:3706x3¡2:8833x3jx3jwhichachievesGASoftheequilibriumx=0.However,thelineardesignwith¯=0isnotfeasiblebecauseg1=0.2Example2.Thesurgesubsystemofanaxialcom-pressormodelhasbeenusedtoillustrateseveralnonlineardesigns,includingamodi¯ed(\lean")versionofbackstepping;Krsti¶cetal.[1995].Herewepresentabasiccirclecriteriondesignforthesamesurgemodel:_Á=¡Ã¡32Á2¡12Á3(21)_Ã=u;(22) whereÁandÃarethedeviationsofthemass°owandthepressurerisefromtheirsetpoints,andthecontrolinputuisthe°owthroughthethrottlewithapreliminarylinearfeedback.Becausethequadraticterminthenonlinearity32Á2+12Á3violatesthesectorproperty,weaddandsubtractthelinearterm98Áin(21),andobtain_Á=¡Ã+98Á¡'(Á)(23)_Ã=u;(24)where'(Á):=98Á+32Á2+12Á3(25)satis¯esthesectorconditionÁ'(Á)¸0.Thismodelhasrelativedegreer=2,andwithy1:=Áandy2:=¡Ã+98Á,itsnormalformis(12)-(14)withnozerodynamics.Thebasiccirclecriteriondesignu=k1Á+k2á¯'(Á)(26)isfeasiblewith¯6=0becauseg1�0.However,thelinearcontrollawwith¯=0isnotfeasiblebecauseg2�0.Togainfurtherinsightintothefeasibilityconditions,werepresent(23),(24),(26)asthefeedbackinterconnectionofthetransferfunctionG(s)=s¡k2¡¯s2¡(k2+98)s+(k1+98k2)(27)andthesectornonlinearity'(¢).AsecondordertransferfunctionisSPRifandonlyifitspolesarestable,anditszeroislocatedintheinterval(¾;0),where¾representsthesumofthepoles.Forthetransferfunction(27),wehave¾=k2+98and,hence,itszerok2+¯mustsatisfyk2+98k2+¯0:(28)Thismeansthatthecirclecriteriondesignisnotfeasiblewith¯=0.However,with¯�98,anychoiceofk1,k2satisfyingk2+¯0andk1+98k2�0,rendersG(s)SPRand,thus,ensuresGASoftheequilibrium(Á;Ã)=0.24.EXTENDEDCIRCLECRITERIONDESIGNInthissectionweextendtheapplicabilityofthebasiccirclecriteriondesigntosystemswhichviolatetheconditionsg1�0andg20ofLemma2.We¯rstconsidertherelativedegreetwocase,thatis,thesystem_»=A0»+E0y1¡G0'(y1)_y1=y2¡g1'(y1)(29)_y2=u¡g2'(y1):BecausethebasiccirclecriteriondesignofSection2isnotfeasiblewheng10,welet~g1�0,andde¯ne~y2:=y2¡(g1¡~g1)'(y1);(30)whichresultsinthenewequations_»=A0»+E0y1¡G0'(y1)_y1=~y2¡~g1'(y1)(31)_~y2=u¡g2'(y1)¡(g1¡~g1)_';where_'isavailableasafunctionofy1andy2:_'=@'@y1(y2¡g1'(y1)):(32)Withthefeedbacktransformationu=~u¡(~g1¡g1)_'(y1;y2);(33)system(31)becomes_»=A0»+E0y1¡G0'(y1)_y1=~y2¡~g1'(y1)(34)_~y2=~u¡g2'(y1)where~g1�0.Thus,thenewvariable~y2andthefeedbacktransformation(33)eliminatedtherestrictiong1�0fromLemma2.Withthedesign~u=K~x¡~¯'(y1),where~x=[»Ty1~y2]T,andKand¯areobtainedfromtheLMI(10)-(11)for(34),the¯nalformofourcontrollawisu=Kx¡¯'(y1)¡(~g1¡g1)_'(y1;y2);(35)wherex=[»Ty1y2]T;thatis,thenonlineartermin~y2isincorporatedin¯'(y1).Whentherelativedegreeisthreeormore,are-peatedapplicationoftheprocedureaboveelim-inatesbothg1�0andg20fromLemma2.Indeed,withthechangeofvariables(30)and~yi=yi¡(g1¡~g1)'(i¡2)(y1;¢¢¢;yi¡1)(36)¡(g2¡~g2)'(i¡3)(y1;¢¢¢;yi¡2);i=3;¢¢¢;r;where'(k)denotesthek-thtimederivativeof'(y1),whichisavailableasafunctionofy1;¢¢¢;yk+1;and,withthepreliminaryfeedbacku=~u+(g1¡~g1)'(r¡1)(y1;¢¢¢;yr)(37)+(g2¡~g2)'(r¡2)(y1;¢¢¢;yr¡1);system(12)-(14)becomes _»=A0»+E0y1¡G0'(y1)(38)_y1=~y2¡~g1'(y1)_~y2=~y3¡~g2'(y1)_~y3=~y4¡g3'(y1)(39)._~yr=~u¡gr'(y1):Choosing~g1and~g2tosatisfy(18),andapplyingthestateandinputtransformations(36)-(37),weobtainacontrollawoftheformu=Kx¡¯'(y1)¡¯1_'(y1;y2)¡¢¢¢(40)¡¯r¡1'(r¡1)(y1;¢¢¢;yr):Thefollowingtheoremsummarizesourderiva-tions:orem1.(Feasibilityoftheextendedcirclecri-teriondesign)Considerthesystem(3),repre-sentedasin(12)-(14),wherew=¡'(z),ristherelativedegreefromthecontrolinpututotheoutputz;andletUandVbede¯nedby(16)and(17),respectively.Acontrollawoftheform(40)renderstheclosed-loopsystemSPRfromtheinputwtotheoutputzifandonlyifU�V,and(19)holdsforeveryeigenvectorofA2T.If,inaddition,(18)holds,thenthelinearcontrollawu=Kxisalsofeasible.2Itisofinteresttointerprettheconditionsintheabovetheoremasstructuralcausesforfeasibil-ityorinfeasibilityoftheextendedcirclecrite-riondesign.Therelativedegreerdeterminestheformofthecontrollawandtherestrictionsunderwhichthelinearcontrollawisalsofeasible.Ascanbeexpected,forahigherrelativedegree,thecomplexityofthecontrollawisalsohigher.An-otherimportantobservationisthattheobstaclestofeasibilityareprimarilyduetothesystem'sunstablezeros(eigenvaluesofA1),becausethecorrespondingmatricesUandVde¯nedby(16)and(17)arerequiredtosatisfyU�V.Thezerosontheimaginaryaxis(eigenvaluesofA2)arealsoanobstacleduetotherequirementthateveryeigenvectorofA2mustsatisfy(19).When(12)-(14)isminimumphase,thatiswhenA0isHurwitz,thecontrollaw(40)isalwaysfeasible.5.THEPOPOVMULTIPLIERDESIGNTheSPRrequirementimposedbythecirclecrite-riononthelinearblockG(s)hasbeenrelaxedbyvarious\multipliers"duetoPopov[1960],ZamesandFalb[1968],andotherauthors,whoexploitadditionalpropertiesofthesectornonlinearity'(¢)toestablishpassivityofthefeedbackpathinFigure2frominputu2tooutputy2.Thus,theSPRrequirementisimposedonthefeedforwardpathG(s)M(s),ratherthanonG(s).¡G(s)M(s)M¡1(s)'(¢)y2u2u1y1Fig.2.FeedbackinterconnectionofG(s)andnon-linearity'(¢).ThemultiplierM(s)isintro-ducedtorelaxtheSPRrestrictiononG(s).WenowproceedwithaPopovmultiplierdesign,whenM(s)=1+´s.Intheclosed-loopsystem(3)-(4),wedenoteG(s)=H(sI¡A¡BK)¡1(G+B¯);(41)and,asastate-spacerealizationof(1+´s)G(s),weuseA=A+BKB=G+B¯(42)C=H[I+´(A+BK)]D=´H(G+B¯):FromthePositiveRealLemmatheSPRpropertyof(1+´s)G(s)means·ATP+PA+QPB¡CTBTP¡C¡D¡DT¸·0:(43)SupposethatK,´and¯satisfythisSPRcondi-tionforsomeP=PT�0,Q=QT�0,andthat'(¢)isavectornonlinearitysatisfyingzT'(z)¸0.Then,GASofx=0followsfromtheLyapunovfunctionV(x)=xTPx+2zZ0'T(¾)d¾(44)whosederivativefortheclosed-loopsystemisnegativede¯nite:_V·¡xTQx¡2´zT'(z):(45)Adi±cultyinadesignbasedon(43)isduetothepresenceoftheadditionalparameter´.Theattempttoconvert(43)intoanLMIusingX:=P¡1,~Q=XQX,yields·XAT+AX+~QB¡XCTBT¡CX¡D¡DT¸·0(46)whichisnotanLMIjointlyinK,¯and´,becauseitisbilinearduetotheproducts´Xand´¯. Ratherthansolving(46)usingabilinearmatrixinequality,suggestedbySafonovetal.[1994],amoredirectapplicationoftheresultsinthispaperistotreat(46)asaone-parameterfamilyofLMI's.Startingwith´=0,whichisinfeasible,aone-parametersearchforincreasingvaluesof´allowsustorelaxthezerodynamicsconditionsofTheorem1:Theorem2.(FeasibilityofthePopovmultiplierdesign)Considerthesystem(3),representedasin(12)-(14),wherew=¡'(z),andtherelativedegreefromthecontrolinpututotheoutputzisr¸2.Let¹Ei0=(I+´Ai)¡1Ei0;i=1;2;(47)andletUandVbede¯nedby(16)-(17),withE10replacedwith¹E10.Acontrollawoftheform(40)isfeasibleforthePopovmultiplierdesignifandonlyifthereexists´�0suchthatU�Vand(19)holdsforeveryeigenvectorofA2T,withE20replacedwith¹E20.If,inaddition,g1�0,thenthelinearcontrollawu=Kxisalsofeasible.2When´=0,(47)implies¹Ei0=Ei0;thatis,werecoverthezerodynamicsconditions(19)andU�VofTheorem1.TheseconditionsarerelaxedinthePopovmultiplierdesign,becausetheyonlyneedtoholdforsome´¸0in(47),ratherthanfor´=0asrequiredintheextendedcirclecriteriondesign.Theorem2restrictstherelativedegreebyr¸2because,ifr=1,theny1inFigure2containsathroughputtermfromu1,andthePopovmultiplierdesignisnotapplicable.Thefollowingexampleshowsthat,evenwhenacirclecriteriondesignisfeasible,thePopovmultiplierdesignmayleadtoasimplercontrollaw:Example3.ConsidertheaxialcompressorsurgesubsysteminExample2,wherealinearcontrollawwasnotfeasibleforacirclecriteriondesign.ItisfeasibleforaPopovmultiplierdesignbecauseg1�0asinTheorem2.Indeed,withthechoiceoflinearfeedbackgainsk1=1+µ98+1´¶2k2=¡2´¡98;(48)itiseasytoverifythat(1+´s)G(s)=(1+´s)(s¡k2)s2¡(k2+98)s+(k1+98k2)isSPRforall´�0,and,hence,thelinearcontrollawu=k1Á+k2ÃachievesGASof(21)-(22).26.CONCLUSIONWehavestudiedfeasibilityofthebasiccirclecriteriondesign,andrevealedstructuralobstaclesarisingfromtherelativedegreeandtheunstablepartofthezerodynamics.Therelativedegreeob-staclehasbeenremovedwithanextendedschemewhichemploysderivativesofthenonlinearityinthefeedbackcontrollaw.ThePopovmultiplierhasrelaxedtheconditionsonthezerodynam-ics.Tofurtherimprovethedesign,apromisingresearchdirectionistoemployothermultipliers,suchastheoneduetoZamesandFalb[1968].ReferencesArcak,M.andP.V.Kokotovi¶c(2001).Feasibilityconditionsforcirclecriteriondesigns.SystemsandControlLetters42(5),405{412.Bernussou,J.,J.C.GeromelandM.C.deOliveira(1999).Onstrictpositiverealsystemsdesign:guaranteedcostandrobustnessissues.SystemsandControlLetters36,135{141.Boyd,S.,L.ElGhaoui,E.FeronandV.Balakrish-nan(1994).LinearMatrixInequalitiesinSys-temandControlTheory.Vol.15ofSIAMStud-iesinAppliedMathematics.SIAM.Philadel-phia,PA.Jankovi¶c,M.,M.LarsenandP.V.Kokotovi¶c(1999).Master-slavepassivitydesignforsta-bilizationofnonlinearsystems.In:Proceedingsofthe18thAmericanControlConference.SanDiego,CA.pp.769{773.Kokotovi¶c,P.V.andM.Arcak(2001).Construc-tivenonlinearcontrol:ahistoricalperspective.Automatica37(5),637{662.Krsti¶c,M.,I.KanellakopoulosandP.Kokotovi¶c(1995).NonlinearandAdaptiveControlDesign.JohnWiley&Sons,Inc..NewYork.Popov,V.M.(1960).Criterionofqualityfornon-linearcontrolledsystems.In:PreprintsoftheFirstIFACWorldCongress.Butterworths.Moscow.pp.173{176.Safonov,M.G.,K.C.GohandJ.H.Ly(1994).Controlsystemsynthesisviabilinearmatrixin-equalities.In:Proceedingsofthe1994AmericanControlConference.Baltimore,MD.pp.45{49.Tsiotras,P.andE.Velenis(2000).Low-biascon-trolofAMB'ssubjecttosaturationconstraints.In:Proceedingsofthe2000IEEEInternationalConferenceonControlApplications.Anchor-age,Alaska.Zames,G.andP.L.Falb(1968).Stabilitycon-ditionsforsystemswithmonotoneandslope-restrictednonlinearities.SIAMJournalofCon-trolandOptimization6,89{108.