一类具有L_范数有界扰动的非线性系统鲁棒控制_英文_

1、第19卷第3期2002年6月控制理论与应用CONTROLTHEORYANDAPPLICATIONSVol.19No.3June2002ArticleID:1000-8152(2002)03-0369-07RobustControlofaClassofNonlinearSystems3withL-BoundedDisturbanceWUMin1,2,ZHANGLingboandLIUGuoping12,3(1.CollegeofInformationScience&Engineering,CentralSouthUniversityChangsha,410083,P.R.China;2.

2、SchoolofMechanical,Materials,ManufacturingEngineeringandManagement,UniversityofNottinghamNottingham,UK;3.InstituteofAutomation,AcademiaSinicaBeijing,100080,P.R.China)Abstract:InthesenseofLnorm,robuststabilizationandtrackingcontrolproblemsarefirstdefinedforuncertainnon2linearsystems.Usingthetechnique

3、offeedbacklinearizationandLyapunovapproach,therobustcontrollerscorrespondingtotherobustcontrolproblemsaredesigned.Finally,asimulationresultshowsthecorrectnessofthedesign.Keywords:robustcontrol;nonlinearsystem;Lnorm;feedbacklinearizationDocumentcode:A一类具有L吴敏1,2张凌波13(11中南大学信息科学与工程学院长沙,410083;21英国诺丁汉;3

4、1摘要:,L.然后利用反馈线性化技术和Lyapunov方法,.关键词:;反馈线性化1IntroductionInrecentyears,thecontrolproblemsofnonlinearsys2temshavereceivedmuchattention.Robustcontrolofnonlinearsystemsisoneofthemaintopicsinthearea14.andI/Olinearization,therobustcontrolproblemsofaclassofnonlinearsystemsarestudiedintheframeofanalysisandsyn

5、thesis6,7.BasedontheLyapunovtheo2ry,therobuststabilizationandrobusttrackingofnonlin2earsystemswithparameteruncertainty,satisfyingthemismatchedandmatchedconditions,arediscussedintheliterature8,9InthesenseofL2-gain,thenonlinearHcontrolthe2oryisfoundedbasedontheHamilton-Jaccobiinequali2ties.Bydifferent

6、ialgameanddissipativetheory,theHcontrolproblemofnonlinearsystemscanbetrans2formedequivalentlyintothesolvabilityproblemoftheHamilton-Jaccobiinequalitiesornonlinearmatrixin2equalities,andtherobustcontrollerscanalsobecon2structedviathesolutionofinequalities.Buttheyalldidnotinvolvethegeneralexternaldist

7、urbancebecausethenecessaryassumptionstotheresultsaretoorigorousforgeneraldisturbance.ThispaperdealswiththerobustcontrolproblemsofaclassofnonlinearsystemswithL-boundeddistur2bance.Inthispaper,consideringthefactthatasuffi2cientlysmalldeviationfromtheidealcontrolobjectiveisoftenadmissible,therobuststab

8、ilizationandrobusttrackingproblemsarerespectivelydefinedinthesenseofL-norm.Inthepasttwodecades,animportantachievementintheresearchonnonlinearcontrolsystemsisthefounda2tionanddevelopmentofthedifferentialgeometricalmethod5.Throughfeedbacklinearization,nonlinear2Problemformulationandpreliminaries2.1Pro

9、blemformulationsystemsaretransformedintotheequivalentformsoflin2earsystems.Basedonthestate-feedbacklinearizationConsiderasingle-inputsingle-output(SISO)affine3Foundationitem:supportedbytheDoctorSubjectFoundation(2000053303)andNationalNaturalScienceFoundationofChina(693740147).Receiveddate:2000-10-23

10、;Reviseddate:2001-12-24.© 1994-2009 China Academic Journal Electronic Publishing House. All rights reserved. 370CONTROLTHEORYANDAPPLICATIONSVol.19nonlinearsystemx=f(x)+g(x)(w+u), y=h(x),nJ=supz.(1a)(1b)w(3)ThenthenonlinearsystemhasL-performanceifJforallwsatisfyingtheassumptionA1).wherexRisthest

11、ate,uRistheinput,wRisthedisturbance,yRistheoutput.f(x)andg(x)aretheknownsmoothfunctionswithcorrespondingdi2mensions.Thedisturbancewexistsintheinputchannel,andissupposedtosatisfythefollowingassumption:2.2ExactlinearizationConsiderthenominalsystemofthenonlinearsystem(1),whichisdescribedby(4a) =f(x)+g(

12、x)u,x(4b)y=h(x).Thestatespaceexactlinearizationproblemofthenonlin2earsystem(4a)and(4b)is:givenapointx0,findaneighborhoodUofx0,afeedbacku=(x)+(x)v(5)isdefinedonU,andatransformation=<(x)isalsodefinedonU,suchthatinthecoordinates=<(x)thecorrespondingclosedloopsystemxf(x)+x)g(x)(x)vis(6)(7a)(7b)A1)

13、ThedisturbancewisL-norm-bounded,i.e.wl,forL-norm.(2)wherelisaknownpositivevalue,andstandsTherobuststabilizationproblemisaddressedasfol2lows.Definition1Givenascalar>0,therobuststa2bilizationproblemofthenonlinearsystem(1)istode2signarobustcontrollawusuchthat:a)Whenthereisnodisturbancew,thenonlinear

14、systemisasymptoticallystableatthepointx0;+Bv,y=Cislinearandcontrollable,i.e.b)Whentherethew,thenonlinearsystemtothesetx|x-x0.()fx+g(x)(x)x()()gxxxx=<-1x=<-1(),=A()=B,Moreover,therobusttrackingproblemisaddressedbelow.h(x)x=<-1()=CforsomesuitablematrixARsatisfyingtheconditionn×nDefinitio

15、n2Givenascalar>0andareferenceinputyR(t),therobusttrackingproblemofthenonlin2earsystem(1)istodesignarobustcontrollawusuchthat:)Whenthereisnodisturbancew,theoutputyofathenonlinearsystemconvergesasymptoticallytothepre2scribedoutputyR(t)astimetendstoinfinity;)Whenthereexiststhedisturbancesatisfyingth

16、ebLnormcondition(2),thedifferencebetweentheout2putyofthenonlinearsystemandthereferenceinputyR(t)convergestothesete|e.Inordertosolvetheaboveproblems,anappropriatepenaltyoutputtoreflecttheinfluenceofdisturbanceisnecessary.Customarily,zisusedtodenoteit.Thedefi2nitionofL-performanceisintroducedasfollows

17、.andvectorBRnrankBABAn-1B=n.Lemma15Thestatespaceexactlinearizationproblemofthenonlinearsystem(4a)canbesolvedifandonlyifthereexistsasmoothfunction(x)suchthatthesystemx=f(x)+g(x)u,y=(x)hasrelativedegreen,i.e.ka)LgLf(x)=0forallxinaneighborhoodUofx0andall0k<n-1;r-1b)LgLf(x0)0.WhereLf(x)denotestheLied

18、erivativeof(x)alongf(x).kDefinition3Considerthenonlinearsystem(1).Foragivenpositivenumber,letLemma25Thestate-spaceexactlinearizationproblemofthenonlinearsystem(4a)canbesolvednear(x)functionforapoint,i.e.thereexistsan“output”whichthesystemhasrelativedegreenatx0,ifandonly© 1994-2009 China Academi

19、c Journal Electronic Publishing House. All rights reserved. No.3RobustControlofaClassofNonlinearSystemswithL-BoundedDisturbance371ifthefollowingconditionsaresatisfied:-2a)Thematrixg(x0)adfg(x0)adng(x0)fv=Kcontrolleru=(x)+(x)K<(x)(1).(13)stabilizesthesystem(7).Moreover,thecorresponding(14)adfadfn-

20、1g(x0)hasrankn;g(x)isinvolutivenearx0.b)ThedistributionD=spang(x)adfg(x)n-15Lemma3Ifnonlinearsystem(4)hasrelativestabilizesthenominalsystem(4)ofthenonlinearsystemdegreenatthepointx0,thenthereexistsacoordinatetransformationTakingthedisturbancewintoaccount,inthecoordi2nates=<(x),theclosed-loopnonli

21、nearsystem(1)withcontroller(14)canbeexactlylinearizedas=<(x)suchthatthesystemcanbetransformedintotheform: i=i+1,i=1,n-1,)+b()u, n=a(y=1,whereh(x)Lfh(x)(8a)(8b)(8c)holds: =(A+BK)+Bw,y=C(15a)(15b)w.Thenthefollowingexpressionwherew=b(wkl.Constructadynamicmodelofthe<(x)=1Ln-h(x)fnLf,(9a) =(A),(16a

22、)(16b)<(x0),thenitisobviously()x=<-1(isequaltothestatexofthenonlinearsystem(1)intheidealcasewithoutdisturbancew.Choose-1)z=x-<()=a(h(x),n-1)Lb(Letu=h(.-1LgLnh(fx)-Lfnh(x)+v.(10)(17)asthepenaltyoutputfortherobuststabilizationproblembecauseiteffectivelyreflectsthedeviationofthestatexfromtheid

23、ealcase,resultingfromthedisturbancew.Thentheclosed-loopsystembecomesalinearcontrol2lablesystemoftheform(7),wherethematricesA,BandCareBasedontheLyapunovtheory,thefollowingproposi2100100A=0000000000T10tioncanbeobtained.,B=,C=.100010(11)Proposition1Givenpositivenumber>0,forthenonlinearsystem(1)satis

24、fyingtheassumptionsA1)-1)ofthecoor2andA2),supposetheinversemap<(dinationtransform=<(x)satisfiestheLipschizscon2ditionBesidestheassumptionsA1),thefollowingassumptionissupposedtoholdthroughoutthepaper:)k,wherekisapositiveconstant.A2)b(A3)<-1(x)-<-1(y)Lx-y.Thenthecontrolleru2=TT,-kl,BPe0and

25、eTBPe3Mainresults3.1Robuststabilization0,otherwise(18)Accordingto5,thereexiststhefollowinglemma.5Lemma4IfthereexistsamatrixKsuchthatthefollowinginequalityTP(A+BK)+(A+BK)P<0=(12)2L2,e=<(x)-,maxholds,wherePisapositivedefinedmatrix,thenthecontrollersuchthatthesystems(1a),(16a)and(17)havetheL-perf

26、ormance.© 1994-2009 China Academic Journal Electronic Publishing House. All rights reserved. 372CONTROLTHEORYANDAPPLICATIONST-22ePe=Lmin,areincludedintheset2Vol.19InordertoproveProposition1,weintroduceaprop2ertyoftheEuclidnorm.nnLemma5SupposethevectorxR,yRandtheanglebetweenthevectorxandyis.Then

27、x-xyxTy=xycosy.e|e.2LmaxAccordingto(20),foralleinthesete|e22Lmax(19)4Definition4Givenaset=<(t,x,w)|wW,tR+,whereWistheadmissibledisturbanceset.isinvariantforthesystem(1)ifforallwWandx,<,(e)0.ThismeansthatattheedgepointsofthereisVT-22Te|ePeLmin,thevalueofV(e)=ePe(t,x,w).willdecrease.Thus,forthec

28、losed-loopsystem(1a),(16a)and(17)withthecontroller(18),eTPe-22Lzminalwaysholds.Thenthesetz|BasedonLemma5,Proposition1canbeprovedasfollows.ProofofProposition1Obviously,toprovetheProposition1isequivalenttoprovingthat,withthecon2trollawu,thesetz|zisaninvariantsetforallw,eand.isaninvariantsetfortheclose

29、d-loopsystem.There2fore,thepropositionisproved.BasedonLemma3and1,thefollowingbeByusingLemma5,wehavezzL<()L22minForsystem(1)anda,ifthesystemsatisfiestheassump2),A2)andA3),therobuststabilizationprob2T2e2minlemofthedefinition1canbesolvedbythecontrolleru=u1+u2=(x)+(x)K<(x)+u2,(21)whereePe).Sothesu

30、fficientconditionthatthesetz|zisaninvariantsetisthat,withthecontroller(18),T-22ePeLmin =(A+BK),T(22a)holdsforallw,eand.NowconsiderthetimederivativeofthequadraticfunctionV(e)=eTPe,i.e.(e)=eP(A+BK)+(A+BK)Pe+VTT,-kl,BTPe0andeTBPeu20,otherwise.(22b)2ePB(w+u2).By(12),thereiseTP(A+BK)+(A+BK)TPe0.FromLemma

31、5,itcanbeobtainedthatePBwePBw=ePBkl.Takingtheformofthecontroller(18)intoaccount,wecanconcludethatforallesatisfyinge,ePBu2-klTTTTTxdenotesthestateofcertainnonlinearsystem(1),and(0)=<(x0),=2,2Lmaxe=<(x)-.ProofCorrespondingtothedefinition,theproofisdividedintotwoparts:a)Thefirstistoconsiderthecas

32、eofw=0.Bythe)ofDefinition2isobvious;lemma,thepartaTTBPeePBBPeTTb)Thesecondistoconsiderthecaseofw0andsatisfiestheassumptionA1).Accordingtotheabovelemmas,asymptoticallyconvergestotheorigin,which(20)convergestotheorigin.ByProposi2impliesthat<-1()tion1,xisalwayskeptinthesetx|x-<-1()ofDefi2).Sothep

33、artb,whosecenteris<-1(-klePB0.Then,forallesatisfyinge,(e)0.VBecauseeTPe22maxe,theedgepointsofT-22e|ePeLmin,i.e.thepointssatisfying© 1994-2009 China Academic Journal Electronic Publishing House. All rights reserved. No.3RobustControlofaClassofNonlinearSystemswithL-BoundedDisturbance373nition2

34、holds.where(0)=h(x0)h(1)(x0)h(n-1)(x0),andThenTheorem1isproved.3.2RobusttrackingA,B,Caretakenasin(11).Clearly,becauseofthefeedbackequivalence,thedynamicbehaviorofmodel(27)isequivalenttothedynamicbehaviorofthenomi2nalsystem,i.e.theidealcasewithoutdisturbancew.Sotheerrorbetweenthepracticaloutputofthes

35、ystemandtheoutputoftheartificialnominalsystem,canberepresentedbythedynamicmodel,(28) =(A+BK)e+Bwe,wheree(0)=0.Choosez=easthepenaltyoutput.Proposition2Givenapositivenumber>0,thecontroller(23a)(23b)T,-kl,BPe0andeTBPeu2TGivenareferenceinputyR(t),fortheI/Olinearizedsystem(7)ofthenominalsystem(4),intr

36、oducethetrajectoryerror(t)asthedifferencebetweentherealoutputandthereferenceoutputyR(t),i.e.1(t)(t)=2(t)n(t)thenthesystem(7)becomes=1-yR(t)(1)2-yR(t)(n-1)(t)n-yR+Bv, =Ay=1.Sohereistheresult.0,otherwise.(29)22max5Lemma6ForagivenreferenceinputyR(t),itassumes:a)Thereexiststhe(n-1)-thordertimederivative

37、yR(t);thesystemb)Theoutputy(t)inn.Then,thesolvedbythecon2trolleru1=)b(e=(A+BK)e+Bw +Bu2,z=e,(30a)(30b)wheree(0)=0,satisfiestheL-performancez(-a()+yR(n)-K),(24)whereK=k0k1kn-1satisfiesthatallrootsoftheequationsn+kn-1sn-1+k1s+k0=0aretakenasinLemma3.(25)andb()lieintheleft-halfcomplexplane,anda(.HereKis

38、takenasinLemma6.ProofTheproofissimilartotheproofofProposi2tion1,andthereforeisomittedhere.Hereistheresults.Theorem2Foragivenvalue>0andareferenceinputyR(t),iftheassumptionsinLemma6aresatis2fied,thentherobusttrackingproblemcanbesolvedbythecontrollerThroughI/Olinearization,thenonlinearsystemwiththec

39、ontroller(24)hasthefollowingform:+B =A,y=Cholds:(n)yR+K(-(1)yRyR(2)yR(n-1)yR)+Bw,(26a)(26b)Tu=u1+u2=n)(-Lfnh(x)+y(+u2),(31)-KRn-1LgLfh(x)where)w.Thenthefollowingexpressionwherew=b(=-yRy(R1)y(R2)y(Rn-1)T,=h(x)h(1)(x)h(n-1)(x),+By(Rn)+K(-yRy(R1)y(R2)y(Rn-1)T), =A(32a)e=-,Twkl,where(0)=h(x0)h(1)(x0)h(n

40、-1)(x0),andA,B,Caretakenasin(11).Constructadynamicmodelasfollows:+By(Rn)+K(-yRy(R1)y(R2)y(Rn-1)T), =A(27a)(32b)T,-kl,BPe0andeTBPeu20,otherwise.(32c),yn=C(27b)© 1994-2009 China Academic Journal Electronic Publishing House. All rights reserved. 374CONTROLTHEORYANDAPPLICATIONSVol.19(0)=h(x0)h(1)(x

41、0)h(n-1)(x0),=22whereu2(x)=,17.962331.460023.8069<(x)-22Lmax.max-2exp(2)xdenotesthestateofuncertainnonlinearsystem(1),KistakenasinProposition2.if<(x)-=2×0.3×0.1=0.06ProofTheproofissimpleandisomitted.4ExampleConsidertheuncertainnonlinearsystem(1)withtheformx3f(x)=2x1+x2+x3,g2(x)=2and1

42、7.962331.460023.8069<(x)-0,otherwiseu2(2)=0.exp(x2)exp(x2).(33)x1-x2Thesimulationshowsthatthestabilizingcontrollerofnominalsystemisunabletostabilizetheuncertainsys2teminFig.1,andFig.2showsthattherobustcontroller(41),designedbyTheorem1,isvalid.ItverifiesthecorrectnessofTheorem1.Theorem2canalsobepr

43、ovedbysimulationinasimilarway.Supposethattheinitialstateis123andthedistur2banceisw=sin(t).gin000.(34)Thecontrolobjectiveistostabilizethesystemattheori2Forthenominalsystem(4)of(1),thecoordinatex3=<()=ItiseasytogetbycalculationL=0.5,x1-x2-x1-x.(35)(36)(37)a(x)=-x1-x2-2x23,b(x)=-2exp(x2).Thenthesyst

44、em(4)islinearizedasthesystem(7).Forthesystem(7),astabilizingcontrollerisv=-1-3-3andthepositivedefinedmatrix(38)83.1578P=60.5567136.36631.460017.31.460023.(39)60.556717.96235ConclusionsInpractice,duetotheinfluenceoftheuncertainfac2torssuchasdisturbance,asmalldeviationtotheidealstateisoftenpermissible

45、.Correspondingly,inthestabi2lizationproblem,thestateisoftenpermittedtoswinginasmallneighborhoodoftheexpectedequilibriumpoint.Similarly,inthetrackingproblem,asmalltrackinger2rorispermitted.Basedontheseviews,therobuststabi2lizationproblemandrobusttrackingproblemaredefinedrespectivelyintheframeworkofLn

46、orm.satisfiestheinequality(12).ByTheorem1,supposethepermissibleL-perfor2manceis0.1,thenarobuststabilizingcontrolleris010-301,-(40) =u(x)=-1b(x)-a(x)+-1-3-3<(x)+u2(x),(41)© 1994-2009 China Academic Journal Electronic Publishing House. All rights reserved. No.3RobustControlofaClassofNonlinearS

47、ystemswithL-BoundedDisturbance375Onthebasisofthetechniqueofexactlinearization,thispaperhasdiscussedtherobustcontrolofaclassofSISOaffinenonlinearsystemswithL-boundeddistur2bances,andcorrespondingrobustcontrollershavebeenobtained.Inthefinalpartofthepaper,asimulationhasbeencarriedbyMATLABandSIMLINK,whi

48、chillus2tratesthecorrectnessoftheresults.Clearly,theresultsinthepapercanfurtherbegeneralizedtothecaseofMI2MOnonlinearsystems,andthecaseswithmoreuncer2tainties,butfurtherresearchisneeded.References1VanDerSchaftAJ.L2-gainanalysisofnonlinearsystemsandnon2linearstatefeedbackHcontrolJ.IEEETrans.Automatic

49、Con2trol,1992,37(6):770-7842ShenTLandTamuraK.RobustHcontrolofuncertainnonlinearsystemsviastatefeedbackJ.IEEETrans.AutomaticControl,nonlinearsystemsJ.IEEETrans.AutomaticControl,1997,42(12):1662-16685IsdoriA.NonlinearControlSystemsM.2ndedition.Berlin:Springer-Verlag,19896WuM,PengZH,TangZH,etal.analysi

50、sandsynthesisfornonlinearrobustcontrolsystemsJ.J.ofCentralSouthUniversityofTechnology,1998,5(1):68-737PengZH,WuMandCaiZX.I/Olinearizationbasedonsynthe2sisfornonlinearrobustcontrolsystemsA.Proceedingof14thIFACConferenceC,Beijing,China,1999,117-1228LiuYJandQinHS.Globalstabilizationviadynamicoutputfeed

51、2backforaclassofuncertainnonlinearsystemJ.ControlTheoryandApplications,1999,16(6):830-8359HashimotoY,WuHSandMizukamiK.RobustoutputtrackingofnonlinearsystemswithmismatcheduncertaintiesJ.Int.J.Con2trol,1999,72(5):411-417本文作者简介吴敏见本刊2002年第2期第.张凌波2.,.1989年赴英国曼彻斯特大,并先后在英国约克大学、谢非尔德大学,现为英国诺丁汉大学高级讲师.研究领域为:先进控制理论及应用,过程控制,鲁棒控制和网络控制等.1995