Stability analysis of critical nonlinear systems using normal form techniques

136 Pages·1996·4.4 MB·English

by  Yan, Aiguo

STABILITYANALYSISOFCRITICALNONLINEARSYSTEMS
USINGNORMALFORMTECHNIQUES
By
AIGUOYAN
ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL
OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT
OFTHEREQUIREMENTSFORTHEDEGREEOF
DOCTOROFPHILOSOPHY
UNIVERSITYOFFLORIDA
1996
UNIVERSITYOFFLORIDALIBRARIES
ToMyWifeWeiXiongandMySonPongpong
&
ToMyParentsJingzhongandBaiyin
ACKNOWLEDGEMENTS
IwillalwaysbegratefulfortheopportunityIhavehadtoworkwithmyadvisorand
committeechairperson, Dr. CarlaA. Schwartz. I thankherforher academic as well
asnonacademicadviceandassistanceduringmystudyatboththeUniversityofFlorida
andtheUniversityofVermont. Manyideas inthis dissertation wereinspired through
discussionswithher.Thisdissertationwouldnothavebeenpossiblewithoutherdirection,
encouragementandsupport.
IwouldliketothankDr. HaniphA.Latchman,Dr. JacobHammer,Dr. OscarD.
CrisalleandDr. JohnK.Schuellerforservinginmysupervisorycommittee,advisingme
onvariousaspectsofthisdissertationandprovidingassistancewheneverandwithwhatever
Irequested.
Mythanksarealsoofferedtootherprofessors,staffmembersandfriendsinboththe
DepartmentofElectricalandComputerEngineeringattheUniversityofFloridaandthe
DepartmentofElectricalEngineeringandComputerScienceattheUniversityofVermont.
Specialthanksgotomyparentsandparents-in-law,whoneverhesitatetodelivertheir
love,encouragementandsupporttoenrichmylife. Finally,Iwouldliketodedicatethis
dissertationtomybelovedwife,WeiXiong,whosharedthehappinessandstruggleIexpe-
riencedduringtheworkofthisdissertation,andtomyson,Pongpong.
iii
TABLEOFCONTENTS
ACKNOWLEDGEMENTS
ffi
ABSTRACT vj
1 INTRODUCTION 1
1.1 ProblemFormulation 1
1.2 BriefLiteratureReview 8
1.3 OutlineoftheDissertation 10
2 INTRODUCTIONTOCENTERMANIFOLDTHEORY 13
3 INTRODUCTIONTONORMALFORMTHEORY 21
3.1 IntroductiontoNormalForms 21
3.2 ComputationofNormalForms 32
3.2.1 AGeneralAlgorithmforCoordinateTransformation 32
3.2.2 AProcedureforComputationofNormalForms 38
3.3 FurtherDiscussionaboutNormalForms 40
3.4 Summary 44
4 RELATIONSHIPBETWEENCENTERMANIFOLDANDNORMALFORMTHE-
ORIES 45
4.1 EquivalenceofCenterManifolds 45
4.2 NormalFormsofCriticalNonlinearSystems 49
4.3 Relationship 53
4.4 AExample 54
4.5 Summary 58
5 CONSTRUCTIONOFLYAPUNOVFUNCTIONS-I 59
5.1 GeneralProcedureForConstructingLyapunovFunctions 59
5.2 ConstructionofLyapunovFunctionsforNonlinearSystemsinTwoSpecial
Cases g3
5.2.1 CASE1:NonlinearSystemswithOneZeroEigenvalue 64
5.2.2 CASE2:NonlinearSystemswithaPairofPurelyImaginaryEigen-
values gg
5.3 LyapunovFunctionsintheOriginalCoordinates 69
iv
5.4 ExamplesandDiscussions 73
5.5 Summary 80
6 SIMPLIFICATIONOFLYAPUNOVFUNCTIONS 82
6.1 HeuristicExamples 82
6.2 MainResults: SimplificationofLyapunovFunctions 88
6.3 Examples 92
6.4 Summary 94
7 CONSTRUCTIONOFLYAPUNOVFUNCTIONS-II 96
7.1 AHeuristicExample 96
7.2 MainResults 98
7.3 SimplificationofLyapunovFunctions 102
7.4 ComputationofPartialNormalForms 105
7.5 Examples 109
7.6 Summary 112
8 APPLICATIONS 113
8.1 NonlinearSystemswithMultipleCriticalEigenvalues 113
8.2 NormalFormsForDiscrete-TimeNonlinearSystems 117
8.3 Summary 119
9 SUMMARYOFTHEDISSERTATION 120
REFERENCES 123
BIOGRAPHICALSKETCH 126
v
AbstractofDissertationPresentedtotheGraduateSchool
oftheUniversityofFloridainPartialFulfillmentofthe
RequirementsfortheDegreeofDoctorofPhilosophy
STABILITYANALYSISOFCRITICALNONLINEARSYSTEMS
USINGNORMALFORMTECHNIQUES
By
AiguoYan
August1996
Chairperson: Dr. CarlaA.Schwartz
MajorDepartment:ElectricalandComputerEngineering
Nonlineardynamicalmodelsarisefromboththeoreticalresearchandrealapplications,
includingrobotics,aircraftcontrol,adaptivecontrol,chemicalprocess,numericalalgorithm,
andpowersystems. Themostimportantanalysisfornonlinearsystemsisthedetermina-
tionofstability. Onlycriticalnonlinearsystemsneedfurtherinvestigation,anduptonow,
stabilityanalysishasbeenpredominantlyperformedusingcentermanifoldandbifurcation
theoriesin combinationwith Lyapunovtechniques. Whenadynamic systemislocally
asymptoticallystable(LASinshort),aLyapunovfunctionmustexist.Manysystemprop-
erties,suchasstabilityregion,convergencerate,androbustness,areassociatedwithand
canbeestimatedusingLyapunovfunctions. Theworkonthedeterminationofthelocal
vi
stabilityofcriticalnonlinearsystems,ortheconstructionofLyapunovfunctionswhenthey
areLAS,isfarfromcomplete.
Inthisdissertation,criticalnonlinearsystemsarestudiedusingnormalformtheory.
Onceacriticalnonlinearsystemistransformedintoanormalform,thedynamicsofthe
criticalstatesaredecoupledlocallyfromthoseofthestablestates.Similartoresultsderived
usingcentermanifoldtheory,thestabilityoftheentirenonlinearsystemisdeterminedby
that ofcritical dynamics. Therelationship between normalform and center manifold
theoriesisthenexplored. Ifthecriticaldynamicsis LAS, andaLyapunovfunctionis
available,aLyapunovfunctionfortheentiresystemcanbeeasilyconstructed.
The convergenceis not alwaysguaranteedfornormalforms. ConstructionofLya-
punovfunctionsforpurelycriticalnonlinearsystemsisonlyknowninafewspecialcases.
Therefore,criticalnonlinearsystemsintwospecialcasesarestudiedintensivelyusingap-
proximatenormalforms. Theexplicitstabilitycriteria,aswellasexplicitalgorithmsfor
constructionofLyapunovfunctions,aredeveloped. MethodsforsimplificationofLyapunov
functionsarealsoproposed.TheconstructionofLyapunovfunctionsrequiresderiving(ap-
proximate)normalformsandtheassociatedtransformations,andisverycomputationally
intensive.Theissueofreducingthecomplexityofcomputingnormalformsandtheassoci-
atedLyapunovfunctionsisalsostudied. Criticalnonlinearsystemswithmultiplecritical
eigenvaluesareexamined,andsomesufficientconditionsfordeterminingstabilityandcon-
structingLyapunovfunctionsareobtained.
Thisdissertationdealswithcontinuous-timesystems. Alloftheresultsareapplicable
todiscrete-timesystemswithslightmodifications.
vii
CHAPTER1
INTRODUCTION
Nonlineardynamicalmodelsarisefromboththeoreticalresearchandrealapplications,
includingrobotics,aircraftcontrol,adaptivecontrol,chemicalprocesses,numericalalgo-
rithms,andpowersystems.Themostimportantanalysisfornonlineardynamicalsystems
isthedeterminationofstability.Linearsystemtheoryhasbewell-developedinlastdecades.
Toacertainextent,thelocalstabilityofanonlinearsystemcanbecompletelycharacterized
byitslinearizationattheequilibriumpointofinterest. Thelocalstabilityofanonlinear
systemcannotbedeterminedinthecasethatthelinearizationpossessesatleastonecriti-
caleigenvaluewhileallothereigenvaluesarestable.Thisdissertationwillstudynonlinear
dynamicalsystemswhoselinearizationsexhibitsuchaproperty. Thelocalstabilizability
ofnonlinearcontrolsystemspossessinglinearlyuncontrollablecriticaleigenvalueswillalso
bediscussed. TheemphasiswillbeontheconstructionofLyapunovfunctionsforcritical
nonlinearsystemswhichareknowntobelocallyasymptoticallystable(LAS).Thischapter
servesasanintroductiontothedissertation.InSection1.1,wepresentmuchoftheback-
groundmaterialwhichwillbeusedthroughoutthisdissertation. Section1.2givesabrief
literaturereview.Theorganization,alongwiththemajorcontributions,ofthisdissertation
ispresentedinSection1.3.
1.1 ProblemFormulation
Inthissection,wepresentmuchofthebackgroundmaterialthatwillbeusedthroughout
thisdissertation.
1
2
Amapg:ftC Rm-*RnissaidtobeamapofclassCk(or,simplyCk),denoted
asg€ Cfc,forsomenonnegativeintegerkifitspartialderivativesofuptoorderA;with
respecttoitsvariablesexistandarecontinuousinft. Ifwecanalwayschoosek—+00,
thenwesaythatgisamapofclassC°°(or,simplyC°°). Furthermore,gissaidtobe
analytic(denotedasC")ifitisC°°andforeachpointx0€ ftTaylorseriesexpansionof
gat£0convergestog(x)foralla;€ ft-
Thegoalofthisdissertationistostudythelocalasymptoticstability(LASYinshort)
ofnonlineardynamicalsystemsofform
*=/(*), xeRn (1.1)
where/:ft—Rnisasufficientlydifferentiate(orCkwithksufficientlylarge)mapfrom
adomainftCRnintoRn.
Anequilibriumsolution(orequilibriumpoint,orsimplyequilibrium)of(1.1)isapoint
xo€ ftsuchthat
/(x0)=0,
i.e.,asolutionwhichdoesnotchangeintime.
Forconvenience,westateallthedefinitionsandtheoremsforthecasewhentheequi-
libriumpointisattheoriginofRn;i.e.,xQ=0. Thereisnolossofgeneralityindoing
thissinceanyequilibriumpointcanbeshiftedtotheoriginviaachangeofvariables[37].
Occasionally, theterm"zerosolution" isused,instead, tosignifythattheoriginisthe
equilibriumpointofinterest.
3
Thelinearizationof(1.1)isdefinedas
x=Ax, (1.2)
whereA=Df(0)isthefirstorderderivative,ortheJacobianmatrixoff(x):Df(x)=
df(x)/dx,evaluatedattheorigin. Thedefinitionofeigenvaluesoreigenvectorsiswell-
known. ThespectrumofannxnmatrixAisthesetofeigenvaluesofA,anddenoted
asa(A)= {Ai,A2,---,An}whereA,needsnottobedifferentfromXj. AssumeAisan
eigenvalueofthenxnmatrixA.TheeigenvalueAissaidtobehyperbolicifitsrealpartis
nonzero(denotedasRe(X)^0),stableifitsrealpartisnegative(denotedasRe(X)<0),
unstableifitsrealpartispositive(denotedasRe(X)>0),andcriticalifitsrealpartis
zero(denotedasRe(X)=0). Anonlinearsystemofformof(1.1)issaidtobecriticalif
itslinearizationonlypossessesbothstableandcriticaleigenvalues, andpurelycriticalif
itslinearizationonlypossessescriticaleigenvalues. Theequilibriumpointx0=0issaid
tobehyperbolicifnoneoftheeigenvaluesofA=Df(0)havezerorealpart. Similarly,a
nonlinearcontrolsystem
x=f(x,u)
issaidtobecriticalifthelinearizationaboutx=0;u=0hasatleastonecriticaleigenvalue
thatisnotcontrollablewhileallothereigenvaluesarestabilizable.
Theterm"stability" hasdifferentdefinitionsindifferent areas. Inthisdissertation,
westudythestabilityofnonlinearsystemsinthesenseofLyapunov. Letx(t)denotethe
solution(trajectory)ofthesystem(1.1)whichattheinitialtimei0=0passesthroughthe
initialpointXo=x(0).
Definition1 ([37],[28])Theequilibriumpointx=0of(1.1)is