Friction generated limit cycles

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1、Proceedingsofthe1996IEEEInternationalConferenceonControlApplicationsWM031:OODearborn,MISeptember15-18,1996itCycles1HenrikOlssolnandKarlJ.AstromDepartmentofAutomaticControlLundInstituteofTechnologyBox118,S-22100Lund,Swedenemail:henrik@control.lth.se,kja@control.lth.selimitcycle

2、s.Acomparisonismadein[5]withamoreelaboratefrictionmodel[3].ThispaperpresentstechniquesthatadmitanexactFrictiongeneratedlimitcyclescanbedividedanalysisoflimitcyclescausedbyfriction.Necessaryintotwocategories,namelylimitcycleswithandconditionsforlimitcycleoscillationsaregivenasw

3、ellwithoutsticking.Forlimitcycleswithstickingtheasconditionsforlocalstabilityofthelimitcycles.velocityatthefrictioninterfaceiszeroduringaTherearetwotypesoflimitcycles,onewheremotiontimeinterval.Forlimitcycleswithoutstickingthesticksforatimeintervalandanotherwherethevelocityisz

4、eroonlyatisolatedtimeinstants.Invelocityiszeroonlyatsingletimeinstants.Thethiscaselimitcyclescanbeexplainedintermsofresultsareappliedtoanexamplewitharesonantrelayoscillations[2].Thevariablethatdeterminesifservo.Comparisonswithdescribingfunctionanalysisstickingoccursisthemagnit

5、udeoftheexternalforcearealsomade.Uatthefrictioninterfacewhenthevelocitybecomeszero.IfthemagnitudeoftheexternalforceislargerthanthestictionforceFsthenthelimitcyclewill1.Introductionbeapurerelayoscillation.Ifnot,thevelocitywillbezerountilsomemechanismhasmadetheexternalFrictionaf

6、fectscontrolsystemsindifferentways.forcelargerthanthestaticfriction.Itmaycauselargesteady-statecontrolerrorsbutThedifferencebetweenthetwotypesoflimitcy-alsooscillations.Inthispaperwestudylimitcyclesclescanalsobeillustratedbyconsideringaswitch-causedbyfriction.Weconsiderasingle

7、frictionin-ingplaneinthephasespacegivenbyU=0.ForterfacewithanassociatedrelativevelocityUandrel-purerelayoscillationsthesolutionpassesthroughativepositionx.Themotionatthefrictioninterfacetheswitchingplanemomentarilywhereasforlimithasonedegreeoffreedomandisgivenbycycleswithstick

8、ingthesolutionisconstrainedtolieintheswitchingplaneduringthep