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IEEETRANSACTIONSONROBOTICS,VOL25,NO2,APRIL2009459CONTROLOFACLASSOFUNDERACTUATEDMECHANICALSYSTEMSUSINGSLIDINGMODESVSANKARANARAYANAN,MEMBER,IEEE,ANDARUNDMAHINDRAKAR,MEMBER,IEEEABSTRACTINTHISPAPER,WEPRESENTASLIDINGMODECONTROLALGORITHMTOROBUSTLYSTABILIZEACLASSOFUNDERACTUATEDMECHANICALSYSTEMSTHATARENOTLINEARLYCONTROLLABLEANDVIOLATEBROCKETT’SNECESSARYCONDITIONFORSMOOTHASYMPTOTICSTABILIZATIONOFTHEEQUILIBRIUM,WITHPARAMETRICUNCERTAINTIESINDEFININGTHECLASSOFSYSTEMS,AFEWSIMPLIFYINGASSUMPTIONSAREMADEONTHESTRUCTUREOFTHEDYNAMICS;INPARTICULAR,THEDAMPINGFORCESAREASSUMEDTOBELINEARINVELOCITIESWEFIRSTPROPOSEASWITCHINGSURFACEDESIGNFORTHISCLASSOFSYSTEMS,ANDSUBSEQUENTLY,ASWITCHEDALGORITHMTOREACHTHISSURFACEINFINITETIMEUSINGCONVENTIONALANDHIGHERORDERSLIDINGMODECONTROLLERSTHESTABILITYOFTHECLOSEDLOOPSYSTEMISINVESTIGATEDWITHANUNDEFINEDRELATIVEDEGREEOFTHESLIDINGFUNCTIONSTHECONTROLLERGAINSAREDESIGNEDSUCHTHATTHECONTROLLERSTABILIZESTHEACTUALSYSTEMWITHPARAMETRICUNCERTAINTYTHEPROPOSEDCONTROLALGORITHMISAPPLIEDTOTWOBENCHMARKPROBLEMSAMOBILEROBOTANDANUNDERACTUATEDUNDERWATERVEHICLESIMULATIONRESULTSAREPRESENTEDTOVALIDATETHEPROPOSEDSCHEMEINDEXTERMSMOBILEROBOT,NONHOLONOMIC,SLIDINGMODE,UNDERACTUATEDUNDERWATERVEHICLEUUVIINTRODUCTIONCONTROLOFUNDERACTUATEDMECHANICALSYSTEMS,SYSTEMSWITHFEWERNUMBEROFCONTROLINPUTSTHANTHEDEGREESOFFREEDOM,HASRECEIVEDMUCHATTENTIONOVERTHEPASTFEWYEARSTHISISBECAUSEOFTHETHEORETICALCHALLENGESASWELLASPRACTICALAPPLICABILITYAMONGTHEHOSTOFCONTROLMETHODSPROPOSEDFORTHEINDIVIDUALAPPLICATIONS,SOMEGENERALMETHODOLOGIESHAVEALSOBEENPROPOSEDTOSTABILIZETHESESYSTEMS1–3THESEMETHODOLOGIESUSEPASSIVITYTECHNIQUES,ENERGYSHAPING,ANDDISCONTINUOUSCONTROLMETHODSACHALLENGINGPROBLEMISTOSTABILIZEAPARTICULARCLASSOFSYSTEMSTHATARENOTLINEARLYCONTROLLABLEANDDISOBEYBROCKETT’SNECESSARYCONDITION4WHILETHEREEXISTSPIECEWISEANALYTICORTIMEPERIODICCONTINUOUSFEEDBACKLAWSTHATCANASYMPTOTICALLYSTABILIZETHEEQUILIBRIUM,THEPROBLEMOFROBUSTANDBOUNDEDCONTROLLERDESIGNARETWOIMPORTANTISSUESTHATARESTILLOPENINTHISFIELDINTHISPAPER,WECONSIDERTHEROBUSTSTABILIZATIONOFTHISPARTICULARCLASSOFSYSTEMSINPARTICULAR,WEPRESENTTHEDESIGNOFASLIDINGMODECONTROLLERFORTHEAFOREMENTIONEDCLASSOFSYSTEMSSLIDINGMODEISONEOFTHEROBUSTCONTROLLERDESIGNMETHODSANDHASBEENSUCCESSFULLYAPPLIEDTOUNDERACTUATEDANDNONHOLONOMICSYSTEMS5–8STABILIZATIONOFSEVERALUNDERACTUATEDSYSTEMSUNDERTHISPARTICULARCLASSHAVEBEENSOLVEDUSINGSLIDINGMODECONTROLTECHNIQUESSEE,FOREXAMPLE,THENONHOLONOMICINTEGRATORANDITSEXTENDEDVERSION9HOWEVER,TOOURKNOWLEDGE,THEREISNOCONSTRUCTIVEPROMANUSCRIPTRECEIVEDMARCH11,2008;REVISEDJULY1,2008ANDSEPTEMBER13,2008FIRSTPUBLISHEDFEBRUARY6,2009;CURRENTVERSIONPUBLISHEDAPRIL3,2009THISPAPERWASRECOMMENDEDFORPUBLICATIONBYASSOCIATEEDITOREPAPADOPOULOSANDEDITORHARAIUPONEVALUATIONOFTHEREVIEWERS’COMMENTSTHEWORKOFADMAHINDRAKARWASSUPPORTEDBYTHEDEPARTMENTOFSCIENCEANDTECHNOLOGY,INDIA,UNDERTHERESEARCHGRANTELE/0708/153/DSTX/ARUDVSANKARANARAYANANISWITHTHEDEPARTMENTOFELECTRICALANDELECTRONICSENGINEERING,NATIONALINSTITUTEOFTECHNOLOGY,TIRUCHIRAPALLI620015,INDIAEMAILVSANKARNITTEDUADMAHINDRAKARISWITHTHEDEPARTMENTOFELECTRICALENGINEERING,INDIANINSTITUTEOFTECHNOLOGYMADRAS,CHENNAI600036,INDIAEMAILARUN_DMIITMACINCOLORVERSIONSOFONEORMOREOFTHEFIGURESINTHISPAPERAREAVAILABLEONLINEATHTTP//IEEEXPLOREIEEEORGDIGITALOBJECTIDENTIFIER101109/TRO20082012338CEDURETODESIGNASWITCHINGSURFACEANDASLIDINGMODECONTROLLERFORTHISCLASSOFSYSTEMSFIRST,WEPRESENTANOUTLINETODESIGNTHESWITCHINGSURFACEBYESTABLISHINGACONNECTIONBETWEENTHEREGULARFORM10ANDTHEDEFINITIONOFTHESWITCHINGSURFACETHREECASESARECONSIDEREDTODESIGNTHESWITCHINGSURFACESUBSEQUENTLY,THESLIDINGMODECONTROLLERDESIGNISPRESENTEDTOREACHTHESWITCHINGSURFACEINFINITETIMEUSINGBOTHCONVENTIONALANDHIGHERORDERSLIDINGMODECONTROLLERSINADDITION,ASWITCHEDALGORITHMISALSOPROPOSEDTOREACHTHESWITCHINGSURFACEIFTHERELATIVEDEGREEOFTHESLIDINGFUNCTIONSARENOTWELLDEFINEDFINALLY,WEPRESENTTHEDESIGNOFTHECONTROLLERGAINSTORENDERTHESYSTEMSTABLEWITHPARAMETRICUNCERTAINTIESTHERESTOFTHEPAPERISORGANIZEDASFOLLOWSACLASSIFICATIONOFUNDERACTUATEDMECHANICALSYSTEMSINTHECONTEXTOFSLIDINGMODEISPRESENTEDINSECTIONIIINSECTIONIII,ADESIGNOUTLINEISPRESENTEDFORTHEGENERATION,FINITETIMEREACHABILITYWITHWELLDEFINEDANDUNDEFINEDRELATIVEDEGREESOFTHESLIDINGFUNCTIONSINSECTIONSIVANDV,THEPROPOSEDMETHODOLOGYISAPPLIEDTOTWOBENCHMARKEXAMPLESAMOBILEROBOTANDTHEUNDERACTUATEDUNDERWATERVEHICLEUUV,RESPECTIVELYSIMULATIONRESULTSAREPRESENTEDINSECTIONVI,FOLLOWEDBYCONCLUSIONSINSECTIONVIIWEBEGINWITHTHEDEFINITIONOFTHECLASSOFUNDERACTUATEDMECHANICALSYSTEMSCONSIDEREDINTHISPAPERIICLASSOFUNDERACTUATEDMECHANICALSYSTEMSTHEDYNAMICSOFMANYMECHANICALSYSTEMSCANBEEXPRESSEDASDQQCQ,˙QFU1WHEREQQ1,,QPISTHEPARAMETRIZATIONOFTHECONFIGURATIONSPACEQ,DQ∈IRPPISTHEINERTIAMATRIX,CQ,˙Q∈IRPCONSISTSOFDAMPING,CORIOLIS,STIFFNESS,ETC,U∈IRMISTHEVECTOROFEXTERNALINPUTS,ANDF∈IRPMISTHECORRESPONDINGINPUTMATRIXWECONSIDERACLASSOFUNDERACTUATEDMECHANICALSYSTEMSTHATISCHARACTERIZEDASFOLLOWS1THEINERTIAMATRIXDISDIAGONALANDCONSTANT2THEVECTORCIN1DEPENDSONLYON˙Q,ANDFURTHER,THEDAMPINGFORCESARELINEARINVELOCITIES3MPANDRANKFMEXAMPLESOFSYSTEMSSATISFYING1–3ARESPACEROBOTS,UNDERWATERROBOTICSYSTEMS,ANDMOBILEROBOTSMOVINGINAHORIZONTALPLANE,ANDTHEYCANALLBEREPRESENTEDINTHEFORMD˙VCVFU2WHEREVISTHEVELOCITYVECTOROFTHESYSTEMINBODYCOORDINATEEQUATION2ALONGWITHTHEKINEMATICEQUATIONSBECOMES˙QHQ,V3˙V−D−1CVGU4WHEREGTRIANGLED−1FITISEASYTOSEETHATTHEUNDERACTUATEDSYSTEMOFTHEFORM3–4HAVENOPOTENTIALFORCES,ANDHENCE,ARENOTLINEARLYCONTROLLABLEINTHENEXTSECTION,WEPRESENTWELLKNOWNCONCEPTSFROMTHESLIDINGMODETHEORYTOFORMULATETHEROBUSTSTABILIZATIONPROBLEMIIISLIDINGMODECONTROLCONSIDERANONLINEARAFFINECONTROLSYSTEMOFTHEFORM˙XFXMSUMMATIONDISPLAYJ1GJXUJ515523098/25002009IEEEAUTHORIZEDLICENSEDUSELIMITEDTONANCHANGUNIVERSITYDOWNLOADEDONJANUARY12,2010AT2012FROMIEEEXPLORERESTRICTIONSAPPLY460IEEETRANSACTIONSONROBOTICS,VOL25,NO2,APRIL2009WHEREX∈M⊆IRN,UJ∈IR,F,GJARESMOOTHVECTORFIELDS,FXE0,ANDWITHOUTLOSSOFGENERALITY,LETXE0BETHEEQUILIBRIUMOFINTERESTFURTHER,LETLFHXTRIANGLE∂H/∂XFXDENOTETHELIEDERIVATIVEOFASMOOTHREALVALUEDFUNCTIONHALONGASMOOTHVECTORFIELDFWEDENOTEBYSJXTHEJTHSLIDINGFUNCTIONSUCHTHATSJM−→IRANDTHECORRESPONDINGSLIDINGSURFACEISTHESET{X∈MSJX0}WEDEFINEO⊂M,TERMEDASSWITCHINGSURFACE,SUCHTHATITSATISFIESTHEFOLLOWINGPROPERTIES1OISCONNECTEDANDCONTAINSTHEEQUILIBRIUMPOINTXE2THECLOSEDLOOPSYSTEMCONFINEDTOTHISSETO,CALLEDEQUIVALENTDYNAMICS,ISSTABLEINSLIDINGMODECONTROL,THECONTROLOBJECTIVEISTOREACHASUBSETOINTHESTATESPACEINFINITETIMEANDMAINTAINTHETRAJECTORYINITTHEDESIGNPROCEDURECANTHEREFOREBESPLITINTOTWOIMPORTANTSTEPS111DESIGNOFAPROPERSWITCHINGSURFACESUCHTHATTHECLOSEDLOOPSYSTEMBEHAVESASDESIRED2REACHTHESWITCHINGSURFACEINFINITETIMEANDMAINTAINITTHEREPOSITIVEINVARIANCEASWITCHINGSURFACEDESIGNTODESIGNTHESWITCHINGSURFACE,WETRANSFORMSYSTEM5TOREGULARFORM,WHICHISDEFINEDASFOLLOWSDEFINITION31THEFOLLOWINGFORMOF5˙X1F1X˙X2F2XMSUMMATIONDISPLAY1GJXUJ6WHEREX1∈IRN−M,X2∈IRM,ANDGJISRELATEDTOGJASGJ01N−MGLATTICETOPJLATTICETOPISREFERREDTOASTHEREGULARFORMTHEEXISTENCEOFTHEREGULARFORMANDTHECORRESPONDINGTRANSFORMATIONSAREEXPLOREDIN11AND12,ANDITISEASYTOTRANSFORM3AND4INTOTHEREGULARFORM6FROMTHEVIEWOF6,THEAIMOFTHESLIDINGMODECONTROLISTORESTRICTTHEMOTIONOFTHESYSTEMTOX1DYNAMICS,ANDSUBSEQUENTLY,WELOOKFORASWITCHINGSURFACEOFTHEFORMOTRIANGLE{X∈MF1XLX1}SUCHTHATTHEEQUIVALENTDYNAMICS˙X1LX1POSSESSESSOMESTABILITYPROPERTYWENEXTCONSIDERTHREECASESTOOBTAINLX1THATHELPSINDEFININGTHESWITCHINGSURFACEOCASE1INTHISCASE,WEASSIGNALINEARDYNAMICSTOLX1OFTHEFORMLX1−KX1,WHEREK∈IRN−MN−M0THISLEADSTOANEQUIVALENTDYNAMICS˙X1−KX1,WHICHISEXPONENTIALLYSTABLEBUTTHISCHOICEOFLX1RESULTSINASWITCHINGSURFACETHATISTHEINTERSECTIONOFN−MSLIDINGSURFACES,ANDMOREOVER,IFTHECORRESPONDINGSLIDINGFUNCTIONSARELINEARLYINDEPENDENT,THENTHEYCANNOTBEREACHEDINFINITETIMEUSINGMCONTROLINPUTSCASE2TOOVERCOMETHECONSERVATIVEPROCEDUREOUTLINEDINCASE1,WEPRESENTALYAPUNOVFUNCTIONBASEDDESIGNOFLX1CONSIDERTHEFOLLOWINGCANDIDATELYAPUNOVFUNCTIONVIRN−M−→IRDEFINEDASVX11/2XLATTICETOP1X1,ANDFURTHERCONSTRUCTLX1SUCHTHAT˙VX1≤0ONO,ANDFURTHER,OISTHEINTERSECTIONOFMSLIDINGSURFACESCASE3THISCASEPERTAINSTOTHEDESIGNOFLX1WHEREINTHEEQUIVALENTDYNAMICSISNOTLYAPUNOVSTABLEWITHRESPECTTOVX1INSUCHACASE,CONSTRUCTLX1SUCHTHATTHEEQUIVALENTDYNAMICSPOSSESSESAWEAKNOTIONOFSTABILITYSUCHASCONVERGENCETOTHEORIGINBFINITETIMEREACHABILITYOFTHESWITCHINGSURFACEONCETHESWITCHINGSURFACEISDESIGNED,THECONTROLTASKISREDUCEDTOREACHINGTHESWITCHINGSURFACEINFINITETIMEWENEXTDEFINEFINITETIMEREACHABILITY13OFTHESWITCHINGSURFACEDEFINITION32THESWITCHINGSURFACEOISSAIDTOBEFINITETIMEREACHABLEIFFORANYX0∈N⊆IRN,THEREEXISTST∈0,∞ANDANADMISSIBLECONTROLU0,T−→IRMWITHTHEPROPERTYTHATXT∈OWEASSUMETHATTHEDESIGNPROCEDUREOUTLINEDINTHEEARLIERSECTIONYIELDSMFUNCTIONSSJXSUCHTHATOINTERSECTIONTEXTMJ1SJX0THEDYNAMICSOBTAINEDAFTERDIFFERENTIATINGEACHSJFORSIMPLICITY,WESUPPRESSTHEDEPENDENCEONX,RJNUMBEROFTIMES,ALONGTHETRAJECTORIESOF5YIELDSSR11SR22SRMMRXQXU7WHERERXLATTICETOPLR1FS1LR2FS2LRMFSMLATTICETOPANDQXLG1LR1−1FS1LGMLR1−1FS1LG1LR2−1FS2LGMLR2−1FS2LG1LRM−1FSMLGMLRM−1FSMBYTHEDEFINITIONOFWELLDEFINEDVECTORRELATIVEDEGREE14,QXISINVERTIBLEATXE,ANDTHUS,FORALLXINTHENEIGHBOURHOODOFXE,WEHAVEUQ−1X{PX−RX}8THECHOICEOFPX∈IRMDEPENDSONTHERELATIVEDEGREEOFTHESLIDINGFUNCTIONSFOREXAMPLE,IFTHEVECTORRELATIVEDEGREEIS{1,2},THENAFEWPOSSIBLECHOICESOFPXP1XP2XLATTICETOPAREP1XBRACELEFTBIGG−K1SIGNS1−SΛ1P2X−SIGNS2|S2|A−SIGN˙S2|˙S2|B−SIGN˙S2|˙S2|1/3−SIGNPARENLEFTBIGΧS2,˙S2PARENRIGHTBIGVEXTENDSINGLEVEXTENDSINGLEVEXTENDSINGLEΧS2,˙S2VEXTENDSINGLEVEXTENDSINGLEVEXTENDSINGLE1/5−Γ1SIGN˙S2Γ2|S2|1/2SIGNS2WHEREK10,Λ,B∈0,1,AB2−B,Γ1,Γ20,ANDΧS2,˙S2TRIANGLES235˙S5/32THEFINITETIMECONVERGENCEPROPERTIESOFP1X,P2XCANBEFOUNDIN15–17IFRJ1,THENTHESLIDINGSURFACESCANBEREACHEDINFINITETIMEUSINGHIGHERORDERSLIDINGMODECONTROLLERSFROM7,ITISCLEARTHATTHESLIDINGSURFACESSJ0,J1,,M,CORRESPONDINGTOTHESYSTEM5AREFINITETIMEREACHABLEITISTHENSTRAIGHTFORWARDTODESIGNTHECONTROLLERSUJSUCHTHATTHETRAJECTORYCANREACHTHEINDIVIDUALSLIDINGSURFACESSJ0INFINITETIMEONCETHEINDIVIDUALSURFACEISREACHED,THETRAJECTORIESARECONFINEDTOTHESWITCHINGSURFACEOSINCETHESWITCHINGSURFACEISCONNECTEDANDCONTAINSTHEORIGIN,ANDFURTHERTHECONTROLLERISSUCHTHATITRENDERSEACHOFTHESLIDINGSURFACESFINITETIMESTABLE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本文标题:外文翻译---使用滑动模式控制一类欠驱动机械系统 英文.pdf
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