外文翻译---使用滑动模式控制一类欠驱动机械系统 英文.pdf

IEEETRANSACTIONSONROBOTICS,VOL.25,NO.2,APRIL2009459ControlofaClassofUnderactuatedMechanicalSystemsUsingSlidingModesV.Sankaranarayanan,Member,IEEE,andArunD.Mahindrakar,Member,IEEEAbstractInthispaper,wepresentaslidingmodecontrolalgorithmtorobustlystabilizeaclassofunderactuatedmechanicalsystemsthatarenotlinearlycontrollableandviolateBrockettsnecessaryconditionforsmoothasymptoticstabilizationoftheequilibrium,withparametricuncertainties.Indefiningtheclassofsystems,afewsimplifyingassumptionsaremadeonthestructureofthedynamics；inparticular,thedampingforcesareassumedtobelinearinvelocities.Wefirstproposeaswitchingsurfacedesignforthisclassofsystems,andsubsequently,aswitchedalgorithmtoreachthissurfaceinfinitetimeusingconventionalandhigherorderslidingmodecontrollers.Thestabilityoftheclosed-loopsystemisinvestigatedwithanundefinedrelativedegreeoftheslidingfunctions.Thecontrollergainsaredesignedsuchthatthecontrollerstabilizestheactualsystemwithparametricuncertainty.Theproposedcontrolalgorithmisappliedtotwobenchmarkproblems:amobilerobotandanunderactuatedunderwatervehicle.Simulationresultsarepresentedtovalidatetheproposedscheme.IndexTermsMobilerobot,nonholonomic,slidingmode,underactu-atedunderwatervehicle(UUV).I.INTRODUCTIONControlofunderactuatedmechanicalsystems,systemswithfewernumberofcontrolinputsthanthedegreesoffreedom,hasreceivedmuchattentionoverthepastfewyears.Thisisbecauseofthetheo-reticalchallengesaswellaspracticalapplicability.Amongthehostofcontrolmethodsproposedfortheindividualapplications,somegen-eralmethodologieshavealsobeenproposedtostabilizethesesys-tems13.Thesemethodologiesusepassivitytechniques,energyshaping,anddiscontinuouscontrolmethods.Achallengingproblemistostabilizeaparticularclassofsystemsthatarenotlinearlycon-trollableanddisobeyBrockettsnecessarycondition4.Whilethereexistspiecewiseanalyticortime-periodiccontinuousfeedbacklawsthatcanasymptoticallystabilizetheequilibrium,theproblemofrobustandboundedcontrollerdesignaretwoimportantissuesthatarestillopeninthisfield.Inthispaper,weconsidertherobuststabilizationofthisparticularclassofsystems.Inparticular,wepresentthedesignofaslidingmodecontrollerfortheaforementionedclassofsystems.Slidingmodeisoneoftherobustcontrollerdesignmethodsandhasbeensuccessfullyappliedtounderactuatedandnonholonomicsys-tems58.Stabilizationofseveralunderactuatedsystemsunderthisparticularclasshavebeensolvedusingslidingmodecontroltech-niques(see,forexample,thenonholonomicintegratoranditsextendedversion9).However,toourknowledge,thereisnoconstructivepro-ManuscriptreceivedMarch11,2008；revisedJuly1,2008andSeptember13,2008.FirstpublishedFebruary6,2009；currentversionpublishedApril3,2009.ThispaperwasrecommendedforpublicationbyAssociateEditorE.PapadopoulosandEditorH.Araiuponevaluationofthereviewerscom-ments.TheworkofA.D.MahindrakarwassupportedbytheDepart-mentofScienceandTechnology,India,undertheresearchGrantELE/07-08/153/DSTX/ARUD.V.SankaranarayananiswiththeDepartmentofElectricalandElectronicsEngineering,NationalInstituteofTechnology,Tiruchirapalli-620015,India(e-mail:).A.D.MahindrakariswiththeDepartmentofElectricalEngineering,IndianInstituteofTechnologyMadras,Chennai-600036,India(e-mail:arun_dmiitm.ac.in).Colorversionsofoneormoreofthefiguresinthispaperareavailableonlineat.DigitalObjectIdentifier10.1109/TRO.2008.2012338ceduretodesignaswitchingsurfaceandaslidingmodecontrollerforthisclassofsystems.First,wepresentanoutlinetodesigntheswitchingsurfacebyestab-lishingaconnectionbetweentheregularform10andthedefinitionoftheswitchingsurface.Threecasesareconsideredtodesigntheswitchingsurface.Subsequently,theslidingmodecontrollerdesignispresentedtoreachtheswitchingsurfaceinfinitetimeusingbothconventionalandhigherorderslidingmodecontrollers.Inaddition,aswitchedalgorithmisalsoproposedtoreachtheswitchingsurfaceiftherelativedegreeoftheslidingfunctionsarenotwelldefined.Finally,wepresentthedesignofthecontrollergainstorenderthesystemstablewithparametricuncertainties.Therestofthepaperisorganizedasfollows.AclassificationofunderactuatedmechanicalsystemsinthecontextofslidingmodeispresentedinSectionII.InSectionIII,adesignoutlineispre-sentedforthegeneration,finite-timereachabilitywithwell-definedandundefinedrelativedegreesoftheslidingfunctions.InSectionsIVandV,theproposedmethodologyisappliedtotwobenchmarkexam-ples:amobilerobotandtheunderactuatedunderwatervehicle(UUV),respectively.SimulationresultsarepresentedinSectionVI,followedbyconclusionsinSectionVII.Webeginwiththedefinitionoftheclassofunderactuatedmechanicalsystemsconsideredinthispaper.II.CLASSOFUNDERACTUATEDMECHANICALSYSTEMSThedynamicsofmanymechanicalsystemscanbeexpressedasD(q)q+C(q,q)=Fu(1)whereq=(q1,.,qp)istheparametrizationoftheconfigurationspaceQ,D(q)IRppistheinertiamatrix,C(q,q)IRpconsistsofdamping,Coriolis,stiffness,etc.,uIRmisthevectorofexternalinputs,andFIRpmisthecorrespondinginputmatrix.Weconsideraclassofunderactuatedmechanicalsystemsthatischaracterizedasfollows.1)TheinertiamatrixDisdiagonalandconstant.2)ThevectorCin(1)dependsonlyonq,andfurther,thedampingforcesarelinearinvelocities.3)m0.Thisleadstoanequivalentdynamicsx1=Kx1,whichisexponentiallystable.Butthischoiceofl(x1)resultsinaswitchingsurfacethatistheintersectionof(nm)slidingsurfaces,andmoreover,ifthecorrespondingslidingfunctionsarelinearlyindependent,thentheycannotbereachedinfinitetimeusingmcontrolinputs.Case2:Toovercometheconservativeprocedureoutlinedincase1,wepresentaLyapunov-function-baseddesignofl(x1).ConsiderthefollowingcandidateLyapunovfunctionV:IRnmIRdefinedasV(x1)=(1/2)xlatticetop1x1,andfurtherconstructl(x1)suchthatV(x1)0onO,andfurther,Oistheintersectionofmslidingsurfaces.Case3:Thiscasepertainstothedesignofl(x1)whereintheequiv-alentdynamicsisnotLyapunovstablewithrespecttoV(x1).Insuchacase,constructl(x1)suchthattheequivalentdynamicspossessesaweaknotionofstabilitysuchasconvergencetotheorigin.B.Finite-TimeReachabilityoftheSwitchingSurfaceOncetheswitchingsurfaceisdesigned,thecontroltaskisreducedtoreachingtheswitchingsurfaceinfinitetime.Wenextdefinefinite-timereachability13oftheswitchingsurface.Definition3.2:TheswitchingsurfaceOissaidtobefinite-timereachableifforanyx(0)NIRn,thereexistsT0,)andanadmissiblecontrolu:0,TIRmwiththepropertythatx(T)O.Weassumethatthedesignprocedureoutlinedintheearliersec-tionyieldsmfunctionsSj(x)suchthatO=intersectiontextmj=1(Sj(x)=0).ThedynamicsobtainedafterdifferentiatingeachSj(forsimplicity,wesup-pressthedependenceonx),rjnumberoftimes,alongthetrajectoriesof(5)yieldsS(r1)1S(r2)2.S(rm)m=R(x)+Q(x)u(7)whereR(x)latticetop=Lr1fS1Lr2fS2LrmfSmlatticetopandQ(x)=Lg1Lr11fS1.LgmLr11fS1Lg1Lr21fS2.LgmLr21fS2.Lg1Lrm1fSm.LgmLrm1fSm.Bythedefinitionofwell-definedvectorrelativedegree14,Q(x)isinvertibleatxe,andthus,forallxintheneighbourhoodofxe,wehaveu=Q1(x)P(x)R(x).(8)ThechoiceofP(x)IRmdependsontherelativedegreeoftheslidingfunctions.Forexample,ifthevectorrelativedegreeis1,2,thenafewpossiblechoicesofP(x)=P1(x)P2(x)latticetopareP1(x)=braceleftbiggK1sign(S1)S1P2(x)=sign(S2)|S2|asign(S2)|S2|bsign(S2)|S2|1/3signparenleftBig(S2,S2)parenrightBigvextendsinglevextendsinglevextendsingle(S2,S2)vextendsinglevextendsinglevextendsingle1/51sign(S2+2|S2|1/2sign(S2)whereK10,b(0,1),ab2b,1,20,and(S2,S2)triangle=(S2+35S5/32).Thefinite-timeconvergencepropertiesofP1(x),P2(x)canbefoundin1517.Ifrj1,thentheslidingsurfacescanbereachedinfinitetimeusinghigherorderslidingmodecontrollers.From(7),itisclearthattheslidingsurfacesSj=0,j=1,.,m,correspondingtothesystem(5)arefinite-timereachable.ItisthenstraightforwardtodesignthecontrollersujsuchthatthetrajectorycanreachtheindividualslidingsurfacesSj=0infinitetime.Oncetheindividualsurfaceisreached,thetrajectoriesareconfinedtotheswitchingsurfaceO.Sincetheswitchingsurfaceisconnectedandcontainstheorigin,andfurtherthecontrollerissuchthatitrenderseachoftheslidingsurfacesfinite-timestable,thetrajectoriesstaywithintheswitchingsurfaceO(positivelyinvariant).C.StabilizationWithUndefinedRelativeDegreeWenotethatfortheclassofsystemsconsideredinthispaper,thereexistsatleastonerjthatisundefinedatxe,i.e.,LgjLrj1fSj(xe)=0forsomej.ThisisreflectedinthelossofinvertibilitypropertyofthematrixQonasubsetcontainingtheequilibriumpointxe.Letj=xM:LgjLrj1fSj(x)=0anddefine=mj=1j,andAuthorizedlicenseduselimitedto:NanchangUniversity.DownloadedonJanuary12,2010at20:12fromIEEEXplore.Restrictionsapply.IEEETRANSACTIONSONROBOTICS,VOL.25,NO.2,APRIL2009461Fig.1.Illustrationofrelativestability.furtherletBbeasmallopenballofradius0aroundxe.Now,ifforanyarbitrarilysmallBnegationslash=(9)then,theconventionalnotionofstability,suchasLyapunovstability,doesnothold.Thismotivatesustoaddressthestabilizationproblemusingthenotionofrelativestability18.Definition3.3:ConsideranopensetDasshowninFig.1,andfurthertheequilibriumpointxe=0issaidtoberelativelystablewithrespecttothesetO,ifgivenanyepsilon10,thereexists0suchthatx(0)OB=x(t)Bepsilon1t0,),whereBandBepsilon1areopenballsaroundtheoriginofradiiandepsilon1,respectively.Further,ifx(t)0ast,thenxe=0issaidtoberelativelyasymptoticallystablewithrespecttothesetO.Fortheundefinedrelativedegreeoftheslidingfunctionswithcon-dition(9),theclosed-loopsystempossessesonlyrelativestability.Forexample,wehaveOtriangle=M.Thus,inviewoftheundefinedrelativedegreeoncertainsets,thecontrolstrategyhasthefollowingswitchingmechanism:ui=braceleftbigPiRisummationtextmj=1inegationslash=jLgjLri1fSiujbracerightbigLgiLri1fSi,ifx(t)/i0,otherwise.(10)D.StabilizationWithParametricUncertaintyNext,weanalyzetherobustnessoftheproposedcontrollaw(8)withparametricuncertainties.Forsimplicity,werestrictourattentiontosystemswithrelativedegree1foreachoftheslidingfunctions,andfurtherwithoutlossofgenerality,weassumethatQ(x)islowertriangular.Wedenotebyfandgjthevectorfieldscontainingthenominalvaluesoftheparameters.Thecontrolobjectiveistostabilize(5)aroundtheequilibriumpointxewiththeknowledgeofthenominalsystemf(x),g(x).Inotherwords,weseekfeedbackcontrollawoftheformu=(f,gi,Sj),i,j=1,.,m,suchthattheclosed-loopsystemx=f(x)+g(x)(f,gi,Sj)hasthedesiredstabilityproperty.Considertherobustcontrollawobtainedbyexpressing(8)intermsofthenominalvaluesofthesystemparametersu=Q1(x)braceleftbigP(x)R(x)bracerightbig(11)Fig.2.Mobilerobot.whereP=K1sign(S1)Kmsign(Sm)latticetopandKj0arefree.ThedynamicsofStriangle=S1S2SmlatticetopIRmsubjectedtothecontrollaw(11)isgivenbyS=R(x)+Q(x)Q1(x)P(x)Q(x)Q1(x)R(x)andthederivativeof(1/2)S2j,j=1,.,m,withrespecttotime,alongtheclosed-loopsystemtrajectories(5)isgivenbySjSj=j(x)(Kjsign(Sj)+pj(x)Sj(12)wherejtriangle=LgjSjLgjSjiswelldefinedprovidedLgjSj(x)negationslash=0.Thisfur-therimpliesthatthepjsp1triangle=Lg1S1Lg1S1LfS1LfS1.pmtriangle=LgmSmLgmSm(LfSm+Lg1Smu1+LgmSmum1)(LfSm+Lg1Smu1+LgmSmum1)arewelldefined.Henceforth,weassumethefollowing.Assumption3.1:Thefunctionsjsatisfyj(x)0,j=1,.,m.Thisassumption,ingeneral,quantifiestheperturbationofthepa-rametersinthecontrolvectorfieldsabouttheirnominalvalues.Assumption3.2:ThereexistsclosedsetsU0pjandconstantsj0,suchthatifx(0)U0pj=x(t)Upjtriangle=x:|pj(x)|j,j=1,.,m.Further,letU0ptriangle=mj=1U0pj.Ifx(0)U0p,then(12)reducestoSjSj=j(Kj|Sj|+pj(x)Sj),andusingtheboundsonpj,wehaveSjSjj(Kj+j)|Sj|(13)andwithgainsKjjsoselected,theclosed-looptrajectoriesreachtheslidingsurfacesinfinitetime.Inmanysituations,itisnoteasytoanalyticallycharacterizethesetUpj.Insuchasituation,wefixKj0tobelarge,andinviewofassumption3.1,thereexistsasmalldomainUpjsuchthat(13)holds.Inthefollowingsection,weapplytheproposedslidingmodecontrolstrategytotwobenchmarkexamples:themobilerobotandtheUUVwithparametricuncertainty.IV.EXAMPLE1:MOBILEROBOTInthissection,weapplytheproposedmethodtorobustlystabilizethemobilerobot.WeconsideramobilerobotasshowninFig.2.TheAuthorizedlicenseduselimitedto:NanchangUniversity.DownloadedonJanuary12,2010at20:12fromIEEEXplore.Restrictionsapply.462IEEETRANSACTIONSONROBOTICS,VOL.25,NO.2,APRIL2009kinematicmodelofthismobilerobotcanbeexpressedasx=vcosy=vsin=(14)wherethetriple(x,y,)denotesthepositionandtheorientationofthevehiclewithrespecttotheinertialframe,andvandarethelinearandangularvelocitiesofthemobilerobot.Thedynamicmodelisobtainedbyusingthefollowingrelations:Mv=FI=(15)whereMisthemassofthevehicle,Iisthemomentofinertia,=(L/r)(12),andF=(1/r)(1+2),withLbeingthedis-tancebetweenthecenterofmassandthewheel,1and2,theleftandrightwheelmotortorques,andristheradiusoftherearwheel.Twoassumptionsaremadetosimplifythemodel:thecen-terofmassandtherear-axiscentercoincide,andthewheelsdonotslide.Thesecondassumptionleadstoavelocity-levelnonholo-nomicconstraintxsinycos=0.Withthefollowingchoiceofstatevector(x1=,x2=xcos+ysin,x3=(y2x)sin+(2y+x)cos,x4=,x5=v(xsinycos),(14)and(15)canbecombinedtoyieldaregularformonthemanifoldS1IR4asx1=f1(x)x2=f2(x)+g1(x1)+g2F(16)wherex1=(x1,x2,x3),x2=(x4,x5),f1(x)=(x4,x5,x1x5x2x4),f2(x)=(0,x24x2),g1(x1)=(k1,k1(x1x2x3)2),g2=(0,k2),k1triangle=1/I,k2triangle=1/M.Notethatthechangeofcoordinatesfrom(,x,y,v,)mapsto(x1,x2,x3,x4,x5)isaglobaldiffeomorphism.Thestabilizationofthemobilerobothasbeensolvedusingadis-continuouscontroller1922usingthesigmaprocesswhiletime-varyingcontrollershavebeenusedin2325,buttherobustnessissuesarenotdiscussedintheseworks.In26,theauthorshavead-dressedthetrajectoryandforcetrackingproblemforthemobilerobotusingthedynamicmodel.A.ControllerDesignThecontrolobjectiveistomovethemobilerobotfromanyinitialpo-sitionandorientation(alsowithinitialvelocities)toadesiredpositionandorientation(withzerofinalvelocities).Withoutlossofgenerality,thepoint-to-pointcontrolobjectiveisreducedtothestabilizationof(16)totheorigin.WenowdesigntheswitchingsurfaceusingthedesignoutlineproposedinSectionIII.WeonceagainwritetheswitchingsurfaceOtriangle=xM:f1(x)=l(x1).Itcanbeverifiedthat,aspercase1inSectionIII-A,ifweassignalineardynamicsKx1tol(x1)in(16),theresultingswitchingsurfaceistheintersectionofthreeslid-ingsurfaces.Further,thecorrespondingslidingfunctionsarelinearlyindependent,andhence,theswitchingfunctionscannotbereachedinfinitetimeusingthetwocontrolinputs.Then,wetrytheLyapunov-function-baseddesignaspresentedincase2,whichagainfailsforthisparticularexample.Weareleftwiththeprocedureoutlinedincase3,andconstructl(x1)asl(x1)=parenleftBiggax1ax2bx3x1bx3parenrightBigg,x1negationslash=0(17)wherea,b0arefree.TheresultingswitchingsurfaceisOtriangle=xMS1IR4:S1=S2=0,whereS1triangle=ax1+x4,S2triangle=x1x5x2x4+bx3,whichsatisfiesthebasicpropertiesoftheswitch-ingsurfacewithexponentialconvergenceasthestabilitypropertyoftheequivalentdynamics.Thistypeofstabilityismostcommonfornonholonomicsystems.Interestingly,theequivalentdynamicsofthissystemissimilartotheclosed-loopsystemofthenonholonomicinte-gratorpresentedin27.Next,weshowthattheclosed-looptrajectoriesonOconvergeexponentiallytotheorigininO.Lemma4.1:Thetrajectoriesof(16)originatinginOreachitsorigin(x1=x2=x3