IEEE_WorkShop_Slides_Lavretsky_Adaptive_Control

AdaptiveControl:Introduction,Overview,andApplicationsRobustandAdaptiveControlWorkshopAdaptiveControl:Introduction,Overview,andApplicationsEugeneLavretsky,Ph.D.E-mail:[e-mail]:714-235-7736E.LavretskyE.Lavretsky2AdaptiveControl:Introduction,Overview,andApplicationsRobustandAdaptiveControlWorkshopCourseOverview•MotivatingExample•ReviewofLyapunovStabilityTheory–Nonlinearsystemsandequilibriumpoints–Linearization–Lyapunov’sdirectmethod–Barbalat’sLemma,Lyapunov-likeLemma,BoundedStability•ModelReferenceAdaptiveControl–Basicconcepts–1stordersystems–nthordersystems–RobustnesstoParametric/Non-ParametricUncertainties•NeuralNetworks,(NN)–Architectures–Usingsigmoids–UsingRadialBasisFunctions,(RBF)•AdaptiveNeuroControl•DesignExample:AdaptiveReconfigurableFlightControlusingRBFNN-sE.Lavretsky3AdaptiveControl:Introduction,Overview,andApplicationsRobustandAdaptiveControlWorkshopReferences•J-J.E.SlotineandW.Li,AppliedNonlinearControl,Prentice-Hall,NewJersey,1991•S.Haykin,NeuralNetworks:AComprehensiveFoundation,2ndedition,Prentice-Hall,NewJersey,1999•H.K.,Khalil,NonlinearSystems,2ndedition,Prentice-Hall,NewJersey,2002•RecentJournal/ConferencePublications,(availableuponrequest)E.Lavretsky4AdaptiveControl:Introduction,Overview,andApplicationsRobustandAdaptiveControlWorkshopMotivatingExample:RollDynamics(ModelReferenceAdaptiveControl)•UncertainRolldynamics:–pisrollrate,–isaileronposition–areunknowndamping,aileroneffectiveness•FlyingQualitiesModel:–aredesireddamping,controleffectiveness–isareferenceinput,(pilotstick,guidancecommand)–rollratetrackingerror:•AdaptiveRollControl:ailpailpLpLδδ=+()mmmpmpLpLtδδ=+()()()()0pmetptpt=−→ˆˆailpKpKδδδ=+ailδ(),ailpLLδ(),mmpLLδparameteradaptationlaws()()()()ˆ,,0ˆailailppmpmKpppKtppδδδγγγγδ⎧=−−⎪⎨=−−⎪⎩()tδE.Lavretsky5AdaptiveControl:Introduction,Overview,andApplicationsRobustandAdaptiveControlWorkshopMotivatingExample:RollDynamics(Block-Diagram)mmpLsLδ+ailpLsLδ+ˆpKˆKδ()tδmpp0pe→parameteradaptationloopdesiredflyingqualitiesmodelrolltrackingerrorunknownplant•AdaptivecontrolprovidesLyapunovstability•DesignisbasedonLyapunovTheorem(2ndmethod)•Yieldsclosed-loopasymptotictrackingwithallremainingsignalsboundedinthepresenceofsystemuncertaintiesAdaptiveControl:Introduction,Overview,andApplicationsRobustandAdaptiveControlWorkshopLyapunovStabilityTheoryE.Lavretsky7AdaptiveControl:Introduction,Overview,andApplicationsRobustandAdaptiveControlWorkshopAlexanderMichailovichLyapunov1857-1918•RussianmathematicianandengineerwholaidoutthefoundationoftheStabilityTheory•Resultspublishedin1892,Russia•TranslatedintoFrench,1907•ReprintedbyPrincetonUniversity,1947•AmericanControlEngineeringCommunityInterest,1960’sE.Lavretsky8AdaptiveControl:Introduction,Overview,andApplicationsRobustandAdaptiveControlWorkshopNonlinearDynamicSystemsandEquilibriumPoints•Anonlineardynamicsystemcanusuallyberepresentedbyasetofndifferentialequationsintheform:–xisthestateofthesystem–tistime•Iffdoesnotdependexplicitlyontimethenthesystemissaidtobeautonomous:•Astatexeisanequilibriumifoncex(t)=xe,itremainsequaltoxeforallfuturetimes:(),,with,nxfxtxRtR=∈∈()xfx=()0fx=E.Lavretsky9AdaptiveControl:Introduction,Overview,andApplicationsRobustandAdaptiveControlWorkshopExample:EquilibriumPointsofaPendulum•Systemdynamics:•Statespacerepresentation,•Equilibriumpoints:()2sin0MRbMgRθθθ++=()122212sinxxbgxxxMRR==−−()12,xxθθ==()221200sinxbgxxMRR==−−()210,sin0xx==(),0,1,2,0ekxkπ⎛⎞==±±⎜⎟⎝⎠…θMR1x2xE.Lavretsky10AdaptiveControl:Introduction,Overview,andApplicationsRobustandAdaptiveControlWorkshopExample:LinearTime-Invariant(LTI)Systems•LTIsystemdynamics:–hasasingleequilibriumpoint(theorigin0)ifAisnonsingular–hasaninfinityofequilibriumpointsinthenull-spaceofA:•LTIsystemtrajectories:•IfAhasallitseigenvaluesinthelefthalfplanethenthesystemtrajectoriesconvergetotheoriginexponentiallyfastxAx=0eAx=()()()()00expxtAttxt=−E.Lavretsky11AdaptiveControl:Introduction,Overview,andApplicationsRobustandAdaptiveControlWorkshopStateTransformation•Supposethatxeisanequilibriumpoint•Introduceanewvariable:y=x-xe•Substitutingforx=y+xeinto•Newsystemdynamics:•Newequilibrium:y=0,(sincef(xe)=0)•Conclusion:studythebehaviorofthenewsystemintheneighborhoodoftheorigin()xfx=()eyfyx=+E.Lavretsky12AdaptiveControl:Introduction,Overview,andApplicationsRobustandAdaptiveControlWorkshopNominalMotion•Letx*(t)bethesolutionof–thenominalmotiontrajectorycorrespondingtoinitialconditionsx*(0)=x0•Perturbtheinitialcondition•Studythestabilityofthemotionerror:•Theerrordynamics:–non-autonomous!•Conclusion:Insteadofstudyingstabilityofthenominalmotion,studystabilityoftheerrordynamicsw.r.t.theorigin()xfx=()000xxxδ=+()()()()()()()0,0efxtetfxtgetexδ∗∗=+−==()()()etxtxt∗=−E.Lavretsky13AdaptiveControl:Introduction,Overview,andApplicationsRobustandAdaptiveControlWorkshopLyapunovStability•Definition:Theequilibriumstatex=0ofautonomousnonlineardynamicsystemissaidtobestableif:•LyapunovStabilitymeansthatthesystemtrajectorycanbekeptarbitraryclosetotheoriginbystartingsufficientlyclosetoit(){}(){}0,0,00,RrxrtxtR∀∃⇒∀≥