IEEE TRANSACTIONS ON AUTOMATIC CONTROL VOL - PDF document

IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.53,NO.6,JULY2008 Fig.1.Thethreecomponentsoftheangularvelocityerror versustime. Fig.2.Unitquaterniontrackingerror ;q ;q versustime.proofofTheorem1)sincetheequilibriumpointof(18),inviewofthefactthattendstozero,ischaracterizedby.Onceconvergetozero,theconvergenceoftozeroisguaranteedinviewofthesystemdynamics(1)andthestructureofthecontrollaw(13).Remark4:Intheregulationcase,ourcontrollaw(22)isapurequaternionfeedback(sincearethevectorpartsofunitquater-nion).Since,itisclearthatthecontroleffortisboundedasandhenceanaturalsaturation,intermsofthecontrolgains,isachievedandthedesignercansetthelimitsofthecontroleffortthroughthecontrolgains.Thisconclu-sioncannotbeachievedwiththeregulationcontrollerof[10]sincethetermsubstitutingtheangularvelocityisnotaunit-quaternionandisfrequencydependant.V.SIMULATIONESULTSInthissection,wepresentsomesimulationresultsshowingtheeffec-tivenessoftheproposedcontroller.Theinertiamatrixhasbeentakenas Fig.3.=(~ ;~q ;~q ;~q versustime. Fig.4.Controlinputversustime.=diag(2030).WeappliedthecontrollawofTheorem1,with=20=diag(3.Theinitialconditionshavebeentakenasfollows:(0)=(0(0)=(0Thereferencetrajectoryisgivenby(9)with(0)=(11sin(0)[1.ThesimulationwasperformedwithSimulinkforatimespanof50s.Fig.1showstheevolutionofthethreecomponentsoftheangularvelocitytrackingerrorwithrespecttotime.Fig.2,showstheevolutionoftheunit-quaterniontrackingerror,describingthede-viationbetweentheorientationofthebodyandthedesiredorientation,withrespecttotime.Fig.3,showsthetimeevolutionoftheunit-quater-nionerror,describingthedeviationbetween.Fig.4,showsthecontrolinputversustime. IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.53,NO.6,JULY2008,andlettheinputoftheauxiliarysystem(12)beThevectorsarethevectorpartsoftheunit-quater-,respectively.Then,areglobally,andlim)=lim)=lim)=0,where):=Proof:ThedynamicalequationfortheangularvelocitytrackingerrorisgivenbyAftersomealgebraicmanipulations,onecanshowthat dt 1 2~ TIf~ ,itisclearthatisaskewsymmetricmatrixandhence =0.There- dt 1 2~ TIf~ Using(11)and(12),onecanshowthat dt(Q1)=1 2Q?~Q+1 2~ = 1 2~qT(~ );1 2~q0(~  2~q(~) :=(ConsiderthefollowingLyapunovfunctioncandidate: +(~ +1 (qe)Tqee01)2 +1 )+2 whosetime-derivative,inviewof(11),(17)and(18)isgivenby dt 1 2~ TIf~ )+whichinviewof(13)and(14),leadstoTherefore,onecanconcludethataregloballybounded.Therefore,itisclearthatisbounded.Hence,invokingBarbalatLemma,onecanconcludethat)=0,whichimpliesTheglobalboundednesshereindicatesthatthestatesareboundedforany ~Q(0); (0))2   .Notethattheunit-quaternion areboundedbyde.Consequently,onecanshowthatboundedsinceisbounded,andhence)=0,whichinturns,from(18),impliesthat))=0.Since)=0,itisclear,from(14),that)=0Consequently,onecanconcludethat)=0.Usingthefactthatisboundedandthepreviousboundednessresults,onecanshowthatisbounded,andhence,onecanconclude)=0.Asgoestoinnity,from(15),wehave.Therefore,from(13),itisclearthat))=0,whichim-pliesthat)=0)=0.Finally,.Sincetendsto,whentgoestonity,itiscleargoestoandhence,tendstoConsequently,))=0 Itisclearthatourcontrolschemeincludestheattituderegulationproblemasaparticularcase,i.e.,.Thevelocity-freeattituderegulationschemeisgiveninthefollowingCorollary.Corollary1:Considersystem(1)underthefollowingcontrollaw: isthevectorpartof =(0 .Then,aregloballybounded)=lim)=0Remark1:FromtheproofofTheorem1,itisclearthatfortheclosedloopsystem,atthefollowingfourequilibriumpoints=0),andawayfromtheseequilib-riumpoints.Notethatthesefourequilibriarepresentthesamephysicalequilibriumfortherigidbody().Ifinitially,theclosed-loopsystemisatoneofthesefourequilibria,itwillremainthereforallsubsequenttime.Inthecasewheretheclosed-loopsystemisnotatoneofthefourequilibria,itwillconvergetotheattractiveequi-libriumpoint=0)forwhichThethreeisolatedequilibriumpoints=0)=0)=0)notattractors,butrepellerequilibria[8].Remark2:Theintroductionoftheauxiliarysystem(12)allowstogenerateapassivemap[5].Infact,thiscanbeeasilyseenbysubstituting(13)in(20)toget dt)=(~.Therefore,theauxil-iarysysteminputcanbedesignedinastraightforwardmannerasin(14).Theresultingclosed-loopsystemisafeedbackinterconnectionofapassivesystemandaconstantgain.This,guaranteesglobalbound-ednessofandtheconvergenceoftozero.Finally,thankstothefactthatthelargestpositivelyinvariantsetissimplytheRemark3:Itisworthnotingthatthemainpurposesoftheauxiliarydynamicalsystem(12)are1)togenerateapassivemappingbetweenandthevectorpartoftheunitquaternionerror;2)toguaranteethattheequilibriumof(18),inviewofthefactthattendstozero,ischaracterizedby.Infact,underthecontrollaw(13)andforcingtheinputoftheauxiliarysystem(12)tobeproportionaltoweensureasymptoticconvergenceoftozero.Theconvergenceoftozerowillguaranteetheconvergenceoftozero(asshownintheTheglobalboundednesshere,meansforany   . IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.53,NO.6,JULY2008VI.CAnewquaternion-basedsolutiontotheattitudetrackingproblem,withoutvelocitymeasurement,hasbeenproposed.Ourapproachisbasedontheuseofaunit-quaternionauxiliarysystemwhoseinputisrelatedtothevectorpartoftheunitquaternionerrorviaapas-sivemap,underanappropriateunitquaternion-basedfeedback.Theproposedcontrolschemeincludestheattituderegulationproblemasaparticularcase,andguaranteesglobalasymptoticstabilityoftheequilibriumpoint().Intheregulationcase,ourcontrolschemeisapurequaternionfeedback,andconsequently,thedesignercanset,inastraightforwardmanner,theupperboundforthecontroleffortintermsofthecontrolgains.[1]M.R.Akella,J.T.Halbert,andG.R.Kotamraju,Rigidbodyatti-tudecontrolwithinclinometerandlow-costgyromeasurements,Contr.Lett.,vol.49,pp.151159,2003.[2]S.P.BhatandD.S.Bernstein,Atopologicalobstructiontocontin-uousglobalstabilizationofrotationalmotionandtheunwindingphe-Syst.Contr.Lett.,vol.39,pp.6370,2000.[3]F.CaccavaleandL.Villani,OutputfeedbackcontrolforattitudeSyst.Contr.Lett.,vol.38,no.2,pp.9198,1999.[4]B.T.Costic,D.M.Dawson,M.S.d.Queiroz,andV.Kapila,quaternion-basedadaptiveattitudetrackingcontrollerwithoutvelocityAIAAJ.Guidance,Contr.Dynam.,vol.24,no.6,2001.[5]C.A.DesoerandM.Vidyasagar,FeedbackSystems:Input-OutputProperties,ser.ElectricalScience.NewYork:Academic,1975.[6]O.EgelandandJ.M.Godhavn,Passivity-basedadaptiveattitudecon-trolofarigidspacecraft,IEEETrans.Autom.Control,vol.39,pp.846,1994.[7]P.C.Hughes,SpacecraftAttitudeDynamics.NewYork:Wiley,1986.[8]S.M.Joshi,A.G.Kelkar,andJ.T.-Y.Wen,Robustattitudestabiliza-tionofspacecraftusingnonlinearquaternionfeedback,IEEETrans.Autom.Control,vol.40,pp.18001803,1995.[9]D.E.Koditschek,ApplicationofanewLyapunovfunctiontoglobaladaptiveattitudetracking,Proc.27thIEEECDC,Austin,TX,1988,pp.63[10]F.LizarraldeandJ.T.Wen,Attitudecontrolwithoutangularvelocitymeasurement:Apassivityapproach,IEEETrans.Autom.Control,vol.41,pp.468472,1996.[11]R.M.Murray,Z.Li,andS.Satry,AMathematicalIntroductiontoRoboticManipulation.BatonRaton,FL:CRC,1994.[12]S.Salcudean,Agloballyconvergentangularvelocityobserverforrigidbodymotion,IEEETrans.Autom.Control,vol.36,pp.1497,1991.[13]M.D.Shuster,Asurveyofattituderepresentations,J.Astronaut.Sci.vol.41,no.4,pp.439517,1993.[14]W.Stanley,Quaternionfromrotationmatrix,J.GuidanceControlvol.1,no.3,pp.223224,1978.[15]A.TayebiandS.McGilvray,Attitudestabilizationofaquadrotorair-IEEETrans.ControlSyst.Technol.,vol.14,pp.562571,2006.[16]A.Tayebi,Unitquaternionobserverbasedattitudestabilizationofarigidspacecraftwithoutvelocitymeasurement,Inproc.ofthe45thIEEEConferenceonDecisionandControl,SanDiego,CA,2006,pp.[17]P.Tsiotras,Furtherresultsontheattitudecontrolproblem,Trans.Autom.Control,vol.43,pp.15971600,1998.[18]J.T.-Y.WenandK.Kreutz-Delgado,Theattitudecontrolproblem,IEEETrans.Autom.Control,vol.36,pp.11481162,1991.[19]B.Wie,H.Weiss,andA.Arapostathis,Quaternionfeedbackregulatorforspacecrafteigenaxisrotations,AIAAJ.GuidanceControl,vol.12,no.3,pp.375380,1989. RobustControlofNonlinearJumpParameterSystemsGovernedbyUncertainChainsJasonJ.FordandValeryA.UgrinovskiiWeconsideraninÞnite-horizonminimaxoptimalcontrolproblemforstochasticuncertainsystemsgovernedbyadiscrete-stateuncertaincontinuous-timechain.Usingexistingrisk-sensitivecontrolre-sults,arobustsuboptimalabsolutelystabilizingguaranteedcostcontrollerisconstructed.Conditionsarepresentedunderwhichthissuboptimalcontrollerisminimaxoptimal.Wethenpresentanumericalgorithmforcalculatingarobust(sub)optimalcontrollerusingaMarkovchainapproximationtechnique.IndexTermsMarkovchainapproximations,Markovjumpparametersystems,robustcontrol,stochasticcontrol.I.INTRODUCTIONMinimaxrobustcontrolofuncertainstochasticsystems,inwhichperturbationsarerestrictedtosatisfyaconstraintonprobabilitylawsassociatedwithdisturbances,hasbeenactivelydevelopedinthepastdecade[1][3].ThistheorycoversproblemsofrobustLQGcontrolandltering,andalsononlinearcontrolsystems[1],[3],controllability,ob-servability,andperformanceaspectsofrobustcontrollersandlters[4],[5].Thetheoryishoweverlimitedinthatitonlyappliestosystemssub-jecttoGaussiandisturbances.Inthispaper,weexpandtheboundariesofthistheorytoincludenonlinearhybridstochasticsystemsgovernedbyadiscrete-stateuncertainmodeprocess.Inaddition,dynamicsofeachmodeofthesystemaresubjecttodisturbances.Theprobleminthefocusofthispaperisthatofnonlinearrobustswitchingcontroldesignviaoptimizationoftheworst-caseperfor-manceofanuncertainstochasticsystemdrivenbyanuncertainnoiseandsubjecttoabruptchangesofsystemparameters.Wewishtondastate-feedbackswitchingcontrolsolutiontotheworst-caseperfor-manceoptimizationproblemsup u;Q u;Q):=limsup T T0EQ dt:isthestateprocessanddescribesadiscrete-eventrandommechanismofmodechanges.Bothprocessesevolveunderanuncer-tainprobabilitymeasure,andhaveuncertainprobabilitydistribu-tionssubjecttotheconstraintisagivenset.WerefertoSectionIIforrigorousdenitions.Acontrollersoughtisallowedtoac-cessbothThemajornoveltyofthispaperisthehybriduncertaintymodelwhichcombinestheuncertaintiesinthediscrete-eventandcontinuous-statecomponentsofthesystem.Indeed,inahybridsystem,plantmod-elingerrorsmaydependonthestateofthemodeprocess.Also,proba-bilitiesofswitchingfromoneoperationmodetoanothermodemayde-ManuscriptreceivedJune28,2005;revisedJune5,2007.CurrentversionpublishedAugust29,2008.RecommendedbyAssociateEditorH.Hjalmarsson.ThisworkwassupportedbytheAustralianResearchCouncil.PartofthisworkwasdoneduringthesecondauthorsvisittotheAustralianNationalUniversity.J.J.FordiswiththeSchoolofEngineeringSystems,QueenslandUniversityofTechnology,Brisbane,Australia(e-mail:j2.ford@qut.edu.au).V.A.UgrinovskiiiswiththeSchoolofInformationTechnologyandElectricalEngineering,theAustralianDefenceForceAcademy,CanberraACT2600,Australia(e-mail:v.ougrinovski@adfa.edu.au).DigitalObjectIdentier10.1109/TAC.2008.9289110018-9286/$25.00©2008IEEE isasymmetricpositivedeÞniteconstantinertiamatrixofthebodywithrespecttothe0018-9286/$25.00©2008IEEE IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.53,NO.6,JULY2008UnitQuaternion-BasedOutputFeedbackfortheAttitudeTrackingProblemAbdelhamidTayebi,SeniorMember,IEEEInthisnote,weproposeaquaternion-baseddynamicoutputfeedbackfortheattitudetrackingproblemofarigidbodywithoutvelocitymeasurement.Ourapproachconsistsofintroducinganauxiliarydynam-icalsystemwhoseoutput(whichisalsoaunitquaternion)isusedinthecontrollawtogetherwiththeunitquaternionrepresentingtheattitudetrackingerror.Roughlyspeaking,thenecessarydampingthatwouldhavebeenachievedbythedirectuseoftheangularvelocitycanbeachieved,inourapproach,bythevectorpartoftheerrorsignalbetweentheoutputoftheauxiliarysystemandtheunitquaterniontrackingerror.Theresultingvelocity-freecontrolschemeguaranteesasymptoticstability   ,where isthesetofunit-quaternion.Furthermore,theclosed-loopsystemhasfourequilibriumpoints(threerepellerequilibriaandoneattractor)thataremathe-maticallydifferentbutrepresentthesamephysicalattitudeoftherigidbody.Alltrajectoriesstartingin   ÐexceptthethreerepellerequilibriaÑwillconvergetotheuniqueattractorequilibrium.Formoredetailsaboutthetopologicalobstructiontocontinuousglobalstabilizationofrotationalmotion,thereaderisreferredto[2]. IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.53,NO.6,JULY2008whoseoriginisatthecenterofmass.Thevectoristhetorqueappliedtherigidbody,consideredastheinputvector.Thematrix( )isaskew-symmetricmatrixsuchthatforanyvector ,wheredenotesthevectorcross-product.Ourobjectiveistodesignafeedbackcontroller,withoutve-locitymeasurement,forthestabilizationoftheequilibriumpoint),whereisthedesiredorienta-tionandisthedesiredangularvelocity.III.UUATERNIONTheorientationofarigidbodywithrespecttotheinertialframecanbedescribedbyafour-parametersrepresentation,namelyunit-quater-nion[11].Aquaternioniscomposedofascalarcompo- andavector .Thesetofquaternion isafour-di-mensionalvectorspaceoverthereals,whichformsagroupwiththequaternionmultiplicationdenotedby.Thequaternionmultiplica-tionisdistributiveandassociativebutnotcommutative[11].Themul-tiplicationoftwoquaternionisdeas[11],[13]p;qandhasthequaternionastheidentityelement.Notethat,foragivenquaternion,wehave=(1Thesetofunit-quaternion isasubsetof suchthat u= Q02  3jq20+qTq Notethatinthecasewhere ,theunit-quaternioninverseisgivenbyArotationmatrixbyanangleabouttheaxisdescribedbytheunitvector ,canbedescribedbyaunit-quaternion suchthat 2 =cos 2 TherotationmatrixisrelatedtothequaternionthroughtheRo-driguezformula[7],[13])+2Algorithmsallowingtheextractionoffromarotationmatrix,canbefoundin[13],[14].Inthisnote,insteadofusingtherotationmatrixtodescribetheorientationoftherigidbody,wewillusetheunit-quaternion.Thedy-namicequation(2)canbereplacedbythefollowingdynamicequationintermsoftheunit-quaternion[7],[13]: 2(7)whereQ02 =(0 .Inthesequel,wewillusetodenotethequaternion.Wealsodenetheunit-quaternionerror,whichdescribesthediscrepancybetweentwounit-quaternion,asfollows:=(Notethattheunit-quaternioncoincideif=(1Itisalsoimportanttomentionthattheequilibriumpoint =0)for(1)and(2)isequivalenttotheequilibriumpoint =0)for(1)and(7).Sincecorrespondstocorrespondsto,itisclearthatspondtothesamephysicalpoint.Hence,thetwoequilibriumpoints =0)areinrealityauniquephysicalequilibriumpointcorrespondingto =0)IV.MESULTSAssumethatthedesiredorientationtobetrackedisgivenby 2Qd isthedesiredangularvelocity,whichisassumedtobeboundedaswellasitsrstandsecondtime-derivatives.Letusdenetheunit-quaterniontrackingerror,whichdescribesthediscrepancybetweentheactualunit-quaternionandthedesired,asfollows::=(.There-fore,wehaveDifferentiatingbothsidesoftheaboveequationwithrespecttotime,wehaveUsing(7)and(9),theerrorquaterniondynamicsisgivenby 2Q +1 2Qe=1 2Qe?(Qe)1 +1 Usingthefactthat =Q ,withisobtainedfrom(6)bysubstituting,wehave 2Qe = 1 2(qe)T~ ;1 2(qe0I+S(qe))~ :=(_Letusintroducethefollowingauxiliarysystem: (0)=( =(0 ,wheretheof(12)willbedesignedlater.Wedenetheunit-quaternion=(~ describingthediscrepancybetweentheunit-quaterniontrackingerrorandtheauxiliaryunit-quaternionNow,wecanstatethefollowingtheorem.Theorem1:Considersystem(1)underthefollowingcontrollaw