<%BANNER%>

Identification and Analysis of Time-Varying Modal Parameters

Permanent Link: http://ufdc.ufl.edu/UFE0042601/00001

Material Information

Title: Identification and Analysis of Time-Varying Modal Parameters
Physical Description: 1 online resource (96 p.)
Language: english
Creator: Sorley, Stephen
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: aircraft, control, eigenvalue, identification, kamen, mode, morphing, pole, time, varying
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this thesis, methods for identifying and analyzing the modal parameters of a linear, time-varying system with no control input are proposed. These methods are derived from two related definitions for time-varying poles and eigenvectors found in the literature. The limitations of the analysis methods used for each time-varying eigenpair definition are explored. Practical requirements regarding the quantity and quality of the experimental data used in each identification method are also addressed. Finally, a morphing-wing aircraft model is analyzed using the proposed techniques, and the results are compared to traditional frozen-time analysis for a particular morphing trajectory.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Stephen Sorley.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Lind, Richard C.
Local: Co-adviser: Chakravarthy, Animesh.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042601:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042601/00001

Material Information

Title: Identification and Analysis of Time-Varying Modal Parameters
Physical Description: 1 online resource (96 p.)
Language: english
Creator: Sorley, Stephen
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: aircraft, control, eigenvalue, identification, kamen, mode, morphing, pole, time, varying
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this thesis, methods for identifying and analyzing the modal parameters of a linear, time-varying system with no control input are proposed. These methods are derived from two related definitions for time-varying poles and eigenvectors found in the literature. The limitations of the analysis methods used for each time-varying eigenpair definition are explored. Practical requirements regarding the quantity and quality of the experimental data used in each identification method are also addressed. Finally, a morphing-wing aircraft model is analyzed using the proposed techniques, and the results are compared to traditional frozen-time analysis for a particular morphing trajectory.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Stephen Sorley.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Lind, Richard C.
Local: Co-adviser: Chakravarthy, Animesh.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042601:00001


This item has the following downloads:


Full Text

PAGE 1

IDENTIFICATIONANDANALYSISOFTIME-VARYINGMODALPARAMETERSBySTEPHENL.SORLEYATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEUNIVERSITYOFFLORIDA2010

PAGE 2

c2010StephenL.Sorley 2

PAGE 3

IdedicatethisthesistomysaviorJesusChrist,withoutwhosehelpIwouldn'tbehere. 3

PAGE 4

ACKNOWLEDGMENTS Iwouldliketorstthankmyparents,CurtisandCynthiaSorley,fortheirconstantencouragementinthisendeavor.IwouldalsoliketothankmyadvisorRickLindandcoworkersDanGrantandAnimeshChakravarthyforthemanycontributionstheyhavemadetothiswork.Finally,Iwouldliketothankeveryoneinthelabforprovidingsuchanenjoyableworkingenvironment,boththroughtheircamaraderieandtheirabilitytolaugh. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 12 1.1Motivation .................................... 12 1.2Contributions .................................. 12 2BACKGROUND ................................... 14 2.1LTVSystemDenition ............................. 14 2.2Time-varyingSystemIdentication ...................... 15 2.3SurveyofAvailableLTVEigenpairDenitions ................ 16 3LTVANALYSISMETHODS ............................. 20 3.1InadequacyofFrozen-TimeEigenpairAnalysis ............... 20 3.2DenitionofLTVPolesandEigenvectors .................. 22 3.2.1LTVResponseEquation ........................ 22 3.2.2LTVEigenrelation ............................ 22 3.2.3RelationshipbetweenLTVEigenpairsandStability ......... 24 3.2.4RelationshipbetweenLTVEigenpairsandOscillation ........ 27 3.3ModeVectorAnalysis ............................. 28 3.3.1EigenpairsandLinearIndependence ................. 29 3.3.2TransformationsbetweenEquivalentEigenpairs ........... 30 3.3.3DenitionofModeVectors ....................... 35 3.4KamenAnalysis ................................ 39 3.4.1ADifferentPerspectiveonKamen'sMethod ............. 39 3.4.2SufciencyofKamen'sPolestoDetermineStability ......... 43 3.4.3KamenAnalysisviaEquivalentEigenpairTransformations ..... 45 3.4.4ProblemswithKamenAnalysis .................... 47 3.5Summary .................................... 56 4DIRECTIDENTIFICATIONALGORITHMS .................... 58 4.1DevelopmentofIdenticationAlgorithms ................... 58 4.1.1Completevs.IncrementalOptimization ................ 58 4.1.2UnnormalizedEigenvectorMethod .................. 61 4.1.3CanonicalNormalizedEigenvectorMethod ............. 62 4.2DemonstrationofPracticalUsageIssues .................. 65 5

PAGE 6

4.2.1EffectofNumberofDatasets ..................... 66 4.2.2EffectofLinearIndependenceofInitialStates ............ 74 4.2.3EffectofStateMeasurementNoise .................. 77 4.3Example:AircraftModelwithVariableWingSweep ............. 81 4.3.1LateralDynamics ............................ 82 4.3.2LongitudinalDynamics ......................... 87 5CONCLUSIONS ................................... 92 REFERENCES ....................................... 94 BIOGRAPHICALSKETCH ................................ 96 6

PAGE 7

LISTOFFIGURES Figure page 3-1Eigenvectors(left)andpoles(right)fromFTEanalysisofEquation 3 ..... 21 3-2StateResponseofEquation 3 forx(0)=[10](left)andx(0)=[01](right) 21 3-3Eigenvectors(left)andpoles(right)fromanLTVsolutionofEquation 3 ... 25 3-4Validitychecksforeigenpair1(left)andeigenpair2(right)fromanLTVsolutionofEquation 3 .................................... 25 3-5StatetrajectoriesfromEquation 2 versusthemodalresponseofanLTVsolutionofEquation 3 ,forx(0)=[10] ...................... 26 3-6Eigenvectors(left)andpoles(right)fromanLTVsolutionforEquation 3 withoscillationrepresentedbycomplexpoles ................... 27 3-7Eigenvectors(left)andpoles(right)fromanLTVsolutionforEquation 3 withoscillationrepresentedbyrealpoles ..................... 28 3-8Eigenvectors(left)andpoles(right)fromanLTVsolutionforEquation 3 withoscillationrepresentedbyeigenvectors .................... 29 3-9Eigenvectors1(left)and2(right)fromtheoriginalandtransformedsolutionsofEquation 3 .................................... 33 3-10PolesfromtheoriginalandtransformedsolutionsofEquation 3 ....... 34 3-11Validitychecksforeigenpair1(left)andeigenpair2(right)fromthetransformedsolutionofEquation 3 ............................... 34 3-12StatetrajectoriesfromEquation 2 versusthemodalresponseofthetransformedsolutionofEquation 3 ,forx(0)=[10](left)andx(0)=[01](right) ..... 35 3-13Modevectors1(left)and2(right)fromtheoriginalandtransformedsolutionsofEquation 3 .................................... 39 3-14Eigenvectors(left)andpoles(right)fromtheKamensolutiontoEquation 3 43 3-15Validitycheckofeigenpairs1and2(left)andstatetrajectoriesversusmodalresponse(right)fortheKamensolutiontoEquation 3 ............. 44 3-16Eigenvectors(left)andpoles(right)fromtheoriginalLTVsolutionofEquation 3 .......................................... 46 3-17Eigenvectors(left)andpoles(right)fromboththeKamensolutionandthetransformednon-KamensolutiontoEquation 3 ................. 47 3-18PolesfromKamen(left)andtransformed(right)solutionstoEquation 3 .. 48 7

PAGE 8

3-19Firstcomponentofbotheigenvectorsforsolution#1(left)andsolution#2(right)ofEquation 3 ................................... 49 3-20StatetrajectoriesobtainedfromEquation 3 withx(0)=[10](left)andx(0)=[01](right) .................................. 51 3-21PolesfromKamen(left)andtransformed(right)solutionstoEquation 3 .. 51 3-22Stateresponsetox(0)=[10]forEquation 3 andEquation 3 ...... 54 3-23Normofmodevector1fromsolutionstoEquation 3 andEquation 3 .. 54 3-24Stateresponsetox(0)=[10]forEquation 3 andEquation 3 ...... 55 3-25Non-canonicalOscillator,Originalvs.CanonicalMatrix:modevectornorms .. 56 3-26PolesfromKamensolutiontoEquation 3 .................... 57 3-27Modevector1(left)andnormofmodevector1(right)fromsolutiontoEquation 3 .......................................... 57 4-1PolesfromtheKamenandCNEsolutionsofRun#1(left),andvaliditycheckofeigenpair1fromtheCNEsolutionofRun#1(right) .............. 67 4-2StatetrajectoriesofEquation 4 versusthemodalresponseoftheCNEsolutionofRun#1,forx(0)=[57](left)andx(0)=[75](right) ............. 68 4-3PolesfromtheKamenandCNEsolutionsofRun#2(left),andvaliditycheckofeigenpair1fromtheCNEsolutionofRun#2(right) .............. 68 4-4StatetrajectoriesofEquation 4 versusthemodalresponseoftheCNEsolutionofRun#2,forx(0)=[57](left)andx(0)=[75](right) ............. 69 4-5PolesfromtheKamenandCNEsolutionsofRun#3(left),andvaliditycheckofeigenpair1fromtheCNEsolutionofRun#3(right) .............. 70 4-6StatetrajectoriesofEquation 4 versusthemodalresponseoftheCNEsolutionofRun#3,forx(0)=[57](left),x(0)=[75](center)andx(0)=[31](right) 70 4-7Normofmodevector1fromtheKamenandUEsolutionsofRun#1(left),andvaliditycheckofeigenpair1fromtheUEsolutionofRun#1(right) ..... 71 4-8StatetrajectoriesofEquation 4 versusthemodalresponseoftheUEsolutionofRun#1,forx(0)=[57](left)andx(0)=[75](right) ............. 71 4-9Normofmodevector1fromtheKamenandUEsolutionsofRun#2(left),andvaliditycheckofeigenpair1fromtheUEsolutionofRun#2(right) ..... 72 4-10StatetrajectoriesofEquation 4 versusthemodalresponseoftheUEsolutionofRun#2,forx(0)=[57](left)andx(0)=[75](right) ............. 73 8

PAGE 9

4-11Normofmodevector1fromtheKamenandUEsolutionsofRun#3(left),andvaliditycheckofeigenpair1fromtheUEsolutionofRun#3(right) ..... 73 4-12StatetrajectoriesofEquation 4 versusthemodalresponseoftheUEsolutionofRun#3,forx(0)=[57](left),x(0)=[75](center),andx(0)=[31](right) 74 4-13PolesfromKamenandCNEsolutionsofRun#4(left),Run#5(center),andRun#6(right) ..................................... 75 4-14StatetrajectoryofEquation 4 versusthemodalresponseoftheCNEsolutionofRun#4(left),Run#5(center),andRun#6(right),forx(0)=[57] ...... 76 4-15Normofmodevector1fromKamenandUEsolutionsofRun#4(left),Run#5(center),andRun#6(right) ........................... 77 4-16StatetrajectoryofEquation 4 versusthemodalresponseoftheUEsolutionofRun#4(left),Run#5(center),andRun#6(right),forx(0)=[57] ...... 77 4-17StatetrajectoriesofEquation 4 with(15%uniformerroradded),forx(0)=[10](left)andx(0)=[01](right) .......................... 78 4-18Pole1(left)andpole2(right)fromtheKamensolutionandthenoisyCNEsolutionofEquation 4 ............................... 79 4-19ModalresponseofthenoisyCNEsolutiontoEquation 4 ,forx(0)=[10](left)andx(0)=[01](right) ............................. 79 4-20Normofmodevector1fromtheKamensolutionandthenoisyUEsolutionofEquation 4 ..................................... 80 4-21ModalresponseofthenoisyUEsolutiontoEquation 4 ,forx(0)=[10](left)andx(0)=[01](right) ............................. 80 4-22Statetrajectoriesforlateral(left)andlongitudinal(right)dynamicsofaircraftmodel,forx(0)=[1000] .............................. 82 4-23Eigenvectors1(left)and2(right)fromUEsolutionoflateraldynamics ..... 83 4-24Eigenvectors3(left)and4(right)fromUEsolutionoflateraldynamics ..... 83 4-25PolesfromUEsolutionoflateraldynamics .................... 83 4-26Modevectors1(left)and2(right)fromUEsolutionoflateraldynamics ..... 84 4-27Modevectors3(left)and4(right)fromUEsolutionoflateraldynamics ..... 84 4-28Modevectornorms1(left)and2(right)fromUEsolutionoflateraldynamics 85 4-29Modevectornorms3(left)and4(right)fromUEsolutionoflateraldynamics 86 4-30Eigenvectors1(left)and2(right)fromUEsolutionoflongitudinaldynamics .. 87 9

PAGE 10

4-31Eigenvectors3(left)and4(right)fromUEsolutionoflongitudinaldynamics .. 88 4-32PolesfromUEsolutionoflongitudinaldynamics ................. 88 4-33Modevectors1(left)and2(right)fromUEsolutionoflongitudinaldynamics .. 89 4-34Modevectors3(left)and4(right)fromUEsolutionoflongitudinaldynamics .. 89 4-35Modevectornorms1(left)and2(right)fromUEsolutionoflongitudinaldynamics 90 4-36Modevectornorms3(left)and4(right)fromUEsolutionoflongitudinaldynamics 90 10

PAGE 11

AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceIDENTIFICATIONANDANALYSISOFTIME-VARYINGMODALPARAMETERSByStephenL.SorleyDecember2010Chair:RickLindMajor:MechanicalEngineeringInthisthesis,methodsforidentifyingandanalyzingthemodalparametersofalinear,time-varyingsystemwithnocontrolinputareproposed.Thesemethodsarederivedfromtworelateddenitionsfortime-varyingpolesandeigenvectorsfoundintheliterature.Thelimitationsoftheanalysismethodsusedforeachtime-varyingeigenpairdenitionareexplored.Practicalrequirementsregardingthequantityandqualityoftheexperimentaldatausedineachidenticationmethodarealsoaddressed.Finally,amorphing-wingaircraftmodelisanalyzedusingtheproposedtechniques,andtheresultsarecomparedtotraditionalfrozen-timeanalysisforaparticularmorphingtrajectory. 11

PAGE 12

CHAPTER1INTRODUCTION 1.1MotivationThemodalparametersofasystemareimportantevaluatorsofstabilityandperformance.Forlinearsystemswithconstantcoefcients(lineartime-invariantsystems),methodsforndingusefulmodalparametersarewell-establishedandmature.Thesamecannotbesaidforlinearsystemswhosecoefcientsarefunctionsoftime(lineartime-varyingsystems).Boththeidenticationoflineartime-varyingmodels[ 1 ]andthedenitionofusefulmodalparametersforsuchsystems[ 2 ]areopenresearchquestions.Lineartime-varying(LTV)systemshavemanypracticalapplications.Anaircraftcanleveragevariablegeometrytoactivelyenhanceperformanceduringamaneuver,oradapttoconictingmissionrequirements[ 3 ].Inspeechanalysis,someelementsofthesignalvarytoorapidlyfortime-invariantmethodstobeeffective[ 4 ].ActiveautomobilesuspensionshavebeenmodeledasLTVsystems[ 5 ].Themechanicsofbiologicalsystemsalsoprovidemanyapplications,suchastheproblemofdynamicanklejointstiffness[ 6 ].Theexistenceofsomanyusefulapplicationsmotivatesndingmodalparameterswhichcharacterizethebehaviorofthesesystems.Unfortunately,thesystemdynamicswillnotbeknownexplicitlyformanyreal-worldsituations.Oneapproachtodealwithsuchsystemsistouseidenticationmethodstoapproximatethestateequations,beforeperformingsomesortofmodalanalysisontheidentieddynamics(suchastheworkdonebyLiu[ 1 ]).Analternativeapproachistoidentifythemodalparametersdirectly. 1.2ContributionsTheLTVpolesandeigenvectorsproposedbyWu[ 7 ][ 8 ]andKamen[ 9 ]areevaluated.Wu'spoleandeigenvectordenitionisshowntobenon-unique,andatransformationbetweenlinearly-dependenteigenpairsunderWu'sdenitionisderived. 12

PAGE 13

Kamen'sdenitionisshowntobeaspecialcaseofWu'sdenition,reachablefromanysolutiontoWu'sdenitionviathederivedtransformation.Amathematicalquantitycalledthemodevector,whichremainsinvariantacrossthesetransformations,isthendened.Ananalysisprocedurebasedonthisquantity(modevectoranalysis)isdescribedandcomparedtotheanalysismethodoutlinedbyKamen.ThisanalysisprocedureisshowntooperatecorrectlyinsituationswhereKamenanalysisfailstoprovideagoodresult.TwoalgorithmsareproposedtoidentifyLTVpolesandeigenvectorsdirectlyfromstatemeasurements,withoutexplicitknowledgeofthetime-varyingdynamicsofthesystem.Thesealgorithmsareshowntoproducevalideigenpairsforavarietyofsystems.Finally,amorphing-wingaircraftmodelisexaminedusingmodevectoranalysis,andtheresultsareevaluatedinthecontextoftraditionallineartime-invariant(LTI)aircraftmodes. 13

PAGE 14

CHAPTER2BACKGROUND 2.1LTVSystemDenitionThisworkdealsprimarilywithsystemsdenedbycoupledlinearhomogeneousdifferentialequations,asshowninEquation 2 .InthisequationA:R+!Rnnisthecoefcientmatrixandx:R+!Rnisthestatevectorofthesystem,wherenisthenumberofindividualstates._x(t)referstotherstderivativeofthestatevectorx(t)withrespecttotime. _x(t)=A(t)x(t)(2)AsystemissaidtobeincanonicalformifthecoefcientmatrixA(t)hasthespecialstructureshowninEquation 2 ,wherea0:R+!Rthroughan)]TJ /F7 7.97 Tf 6.59 0 Td[(1:R+!Raretheonlytime-varyingcomponentsofthecoefcientmatrix. Acanon(t)=266666664010......001)]TJ /F3 11.955 Tf 9.3 0 Td[(a0(t))]TJ /F3 11.955 Tf 9.3 0 Td[(a1(t))]TJ /F3 11.955 Tf 35.2 0 Td[(an)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)377777775(2)Asaconsequenceofthisspecialstructure,thecomponentsofthestatevectorallbecomederivativesofthesamescalarquantity.ThisisshowninEquation 2 ,wherexs:R+!Rreferstothissinglestatefromwhichthestatevectorisderivedandx(n)sreferstothenthderivativeofxs(t)withrespecttotime. 266666664_xs(t)xs(t)...x(n)s(t)377777775=Acanon(t)266666664xs(t)_xs(t)...x(n)]TJ /F7 7.97 Tf 6.58 0 Td[(1)s(t)377777775(2) 14

PAGE 15

Notethatasystemofcoupleddifferentialequationsthatisincanonicalformcanbealternativelyexpressedasascalarnthorderhomogeneousdifferentialequation,asshowninEquation 2 x(n)s(t)+a0(t)xs(t)+a1(t)_xs(t)++an)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)x(n)]TJ /F7 7.97 Tf 6.59 0 Td[(1)s(t)=0(2) 2.2Time-varyingSystemIdenticationIdenticationofthecoefcientmatrixofanLTVsystemismorecomplicatedthaninLTIsystems.InLTIsystems,statevaluescollectedatanytimecanbeusedtondthecoefcientmatrix,sincethematrixisconstant.Asaresult,thecoefcientmatrixofanLTIsystemcanbefoundfromasinglestatetrajectoryofsufcientlength.InLTVsystems,sincethecoefcientmatrixvarieswithtime,thestatevaluescollectedatagiventimecanonlybeusedtoidentifythecoefcientmatrixatthatspecictime(unlesssomeknowledgeofhowthecoefcientmatrixdevelopswithtimeisalreadyknown).Thus,multiplestatetrajectoriesresultingfromperformingthesametime-variationmultipletimesfromdifferentinitialconditionsarerequiredtoidentifythecoefcientmatrixaccurately[ 6 ].StrategiesforidentifyingthecoefcientmatrixA(t)ofalineartime-varyingsystemcanbedividedintothreeprimarycategories[ 6 ][ 1 ].First,quasi-time-invariantmethodsallowforidenticationfromonlyonestatetrajectorywhenthecoefcientmatrixchangesslowlywithrespecttotime(i.e.,jjd dtA(t)jj0).Inthesemethods,thecoefcientmatrixisassumedtobeconstantovershortintervalsofasinglestatetrajectory.Overeachinterval,thestandardLTIsystemidenticationtechniquesareapplied.AnexampleofthismethodwasexploredbyKamen[ 10 ].Temporalexpansionmethodscanbeusedforsystemswhosecoefcientmatricesareknowntovaryperiodically.Sincethevaluesofthecoefcientmatrixrepeatthemselvesaftereveryperiod,asinglestatetrajectoryofsufcientlengthcanbeusedtoidentifythesystem.Whilethistypicallyrequiresamuchlongerstatetrajectory 15

PAGE 16

thanisnecessaryforLTIanalysis,anaccurateresultcanbeachieved.AnexampleofthismethodcanbefoundinworkbyVerhaegen[ 11 ].Finally,ensemblemethodsareusedfornon-periodicsystemswhosecoefcientmatricesvarytooquicklyforquasi-time-invariantmethodstobeaccurate.Inanensemblemethod,statemeasurementsarecollectedfrommultipleexperimentalrunsusingdifferentinitialconditions.Ineachexperimentalrun,thecoefcientmatrixmustperformthesametime-variationatthesametimepoints.Inpractice,thisisdifculttoachieveexactly;however,itistheonlymethodwhichisguaranteedtoproduceaccurateresultsforasystemaboutwhichnothingisknownbeforehand.ExamplesofensemblemethodscanbefoundinMacNeil[ 6 ]andLiu[ 1 ]. 2.3SurveyofAvailableLTVEigenpairDenitionsAnumberofdifferentdenitionsforLTVpolesandeigenvectorsexist.Zhu[ 12 ][ 13 ]denedLTVpolesandeigenvectorsusingvectorpolynomialdifferentialoperators.O'BrienandIglesias[ 14 ][ 2 ]createdamoregeneraldenitionbyperformingaQRdecompositionofthesystem'stransitionmatrix.Inboththesedenitions,theconceptofpolesandeigenvectorsaregeneralizedtoincludecross-termsintheresponseequation.Inthesegeneralizations,eachpolemaybeassociatedwithseveraldifferenteigenvectorsintheresponseequation.Inotherwords,thepolesarefoundbytransformingthestateequationintoanupper-triangularmatrixinsteadofadiagonalmatrix.Wu[ 7 ][ 8 ]gavethedenitionforLTVpolesandeigenvectorsshowninEquation 2 ,where"i:R+!CNaretheLTVeigenvectors,andpi:R+!CaretheLTVpoles. A(t)"(t)=p(t)"(t)+_"(t)(2)TheLTVpolesofthisdenitionwerethenshowntoremaininvariantunderanalgebraictransformation.ThisledtothedenitionforsimilarLTVmatricesshowninEquation 2 ,whereT:R+!Rnnisthealgebraictransformationappliedtothesystemand 16

PAGE 17

A:R+!Rnnistheresultingsimilarmatrix. A(t)=T)]TJ /F7 7.97 Tf 6.58 0 Td[(1(T)A(t)T(t))]TJ /F3 11.955 Tf 11.95 0 Td[(T)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)_T(t)(2)IfthealgebraictransformationT(t)wasdenedsuchthateachcolumnofT(t)wasalinearly-independenteigenvectorunderEquation 2 ,thesimilarmatrixresultingfromEquation 2 wasshowntoequaladiagonalmatrixcomposedofthepoles.Wuthenshowedthatthesepolesarearbitrary,inthatatransformationmatrixcanbefoundwhichchangesA(t)intoanydesireddiagonalmatrix[ 7 ].Sincethesepolesarearbitraryandtheeigenvectorsaretime-varying,boththepolesandtheeigenvectorsarerequiredtocharacterizethestabilityofthissystem.Stabilitywasthendenedusingtheresponsemodes1ofthesystem,asshowninEquation 2 (t)=expZtt0p()d"(t)(2)Wuusedresponsemodestodenestabilityintwoways: Thelineartime-varyingsystemgiveninEquation 2 isstableifandonlyifthenormofeveryresponsemodejji(t)jjofA(t)isbounded.ThisisexpressedinEquation 2 ,whereC2Cissomearbitraryniteconstant. jji(t)jj0,8i=1,...,n(2) Thissystemisasymptoticallystable,ifinadditiontobeingstable,thenormofeveryresponsemodeapproaches0astimegoestoinnity. jji(t)jj!0ast!1,8i=1,...,n(2)Kamen[ 9 ]denedLTVpolesandeigenvectorsforscalar,nthorderdifferentialequations(asshowninEquation 2 ),byfactoringscalarpolynomialdifferentialoperators(SPDO's).Usingtheseoperators,Equation 2 wasrewrittenasshown 1Wureferredtothesequantitiesasmodevectors,buttheyhavebeenrenamedinthisworktoavoidconfusionwithadifferentquantitydenedlaterthathasthesamename. 17

PAGE 18

inEquation 2 .Inthisequation,D:R!Risanoperatorwhichdenotesscalardifferentiationwithrespecttotime,andallotherquantitiesaredenedasforEquation 2 .Notethatannthorderdifferentialequationresultsinapolynomialofordernwithrespecttotheexponentofthedifferentialoperator.a(D,t)xs(t)=0a(D,t)=Dn+n)]TJ /F7 7.97 Tf 6.59 -.01 Td[(1Xi=0ai(t) (2)Kamen'spoleswerefoundbyfactoringtheSPDOa(D,t).Inotherwords,pi:R+!CisapoleofthesystemifanSPDOe(D,t)canbefoundforwhichEquation 2 istrue.SincetheSPDOisannthorderpolynomial,thereshouldbeatmostnpossiblefactorizations.Thus,Kamen'spolesareunique[ 9 ](unlikeWu'spoles). a(D,t)=e(D,t)[D)]TJ /F3 11.955 Tf 11.96 0 Td[(pi(t)],8i=[1,...,n](2)TheoperatorSpiwasdenedasinEquation 2 ,forsomedifferentiablefunctionf:R+!C. Spi(f(t))=pi(t)f(t)+_f(t)(2)Usingthisoperator,KamendenedtheeigenvectorsasshowninEquation 2 ,whereeacheigenvector"i:R+!CnformstheithcolumnofthegeneralizedVandermondematrix[ 9 ]. "i(t)=2666666666641pi(t)Spi(pi(t))...Sn)]TJ /F7 7.97 Tf 6.58 0 Td[(2pi(pi(t))377777777775(2)Themodes i:R+!CofthesystemwerethendenedbyKamenasinEquation 2 .Notethatthesemodesarescalarquantitieswhichdependonlyonthepoles, 18

PAGE 19

unliketheresponsemodesdenedbyWu. i(t)=expZtt0pi()d(2)ItwasthenshownthatallsolutionsofthedifferentialequationgiveninEquation 2 andEquation 2 werecomposedofalinearcombinationofthesemodes.ThisresultisshowninEquation 2 ,whereCi2Carearbitraryconstants. xs(t)=nXi=1Ci (t)(2)Fromthisequation,Kamenstatedthatthesystemwasasymptoticallystableifandonlyiftheabsolutevalueofeachmodeconvergestozero.IncontrasttoWu'sresults,underKamen'sdenitionthestabilityofthesystemisnotaffectedbytheeigenvectors. j i(t)j!0ast!1fori=[1,...,n](2) 19

PAGE 20

CHAPTER3LTVANALYSISMETHODS 3.1InadequacyofFrozen-TimeEigenpairAnalysisInlineartime-invariant(LTI)systems,thepolesandeigenvectorsofannthordersystemwithconstantcoefcientsaredenedbyEquation 3 .InthisequationA2Rnnisthecoefcientmatrix,p2Cisapole,and"2Cnisaneigenvector[ 15 ]. A")]TJ /F3 11.955 Tf 11.96 0 Td[(p"=0(3)Asarststeptowardsexamininglinear,time-varyingsystems(systemswithacoefcientmatrixthatvarieswithtime),itseemslogicaltoextendthisLTIeigenrelationdirectlytothetimedomainbysimplymakingthepolesandeigenvectorsinEquation 3 functionsoftime. A(t)"(t))]TJ /F3 11.955 Tf 11.96 0 Td[(p(t)"(t)=0(3)Practically,thismethodndspolesandeigenvectorsbysolvingtheLTIeigenrelationindependentlyateachdiscretetimestep.Thepolescanthenbeexaminedtodeterminethestabilityorotherperformancemetrics.ThisprocedureisknownasFrozen-TimeEigenpair(FTE)analysis[ 10 ].WhileFTEanalysisprovidesanintuitiveextensionofLTIanalysisandhasprovidedusefulresultsinsomecases,itcannotcorrectlycharacterizethestabilityofLTVsystemsingeneral.TodemonstratetheinsufciencyofFTEanalysis,anexamplegivenbyKhalil[ 16 ]isintroduced.Khalil'sexampleconcernsalineartime-varyingsystemasdenedinEquation 2 ,withacoefcientmatrixspeciedasinEquation 3 A(t)=264)]TJ /F4 11.955 Tf 9.29 0 Td[(1+1.5cos2(t)1)]TJ /F4 11.955 Tf 11.96 0 Td[(1.5sin(t)cos(t))]TJ /F4 11.955 Tf 9.3 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(1.5sin(t)cos(t))]TJ /F4 11.955 Tf 9.3 0 Td[(1+1.5sin2(t)375(3)First,theFTEpolesandeigenvectorsarefoundbysolvingEquation 3 ateachtimestep.ThesepolesandeigenvectorsareshowninFigure 3-1 .Theeigenvectorsarecomplex-conjugate,periodic,andboundedfortheentiretimerange.Notethat,inthe 20

PAGE 21

Figure3-1. Eigenvectors(left)andpoles(right)fromFTEanalysisofEquation 3 legend,eijdenotesthejthcomponentoftheitheigenvector).Thepolesareaconstantcomplex-conjugatepairwithanegativerealpart.Sincetheeigenvectorsareboundedandthepoleshavepositiverealparts,FTEanalysiswouldconcludethatthesystemisstable. Figure3-2. StateResponseofEquation 3 forx(0)=[10](left)andx(0)=[01](right) However,ifthesystem_x(t)=A(t)x(t)issolvednumericallytoobtainstatetrajectories,itbecomesobviousthatthesystemisunstableforcertaininitialconditions.InFigure 3-2 ,thestatetrajectoriesresultingfromtwodifferentsetsofinitialconditions([10]and[01])aredepicted.Whilethesecondinitialconditionvectorresultsinastabletrajectory,therstdiverges.Athoroughtreatmentofthisexamplehasbeendetailedby 21

PAGE 22

MarkusandYamabe[ 17 ].Anexampleoftheinsufciencyoffrozen-timemethodsformoregeneralsystemidenticationproblemsisdescribedbyMacNeil[ 6 ]. 3.2DenitionofLTVPolesandEigenvectorsFTEanalysisprovidessomeusefulresultsonoccasion,butitisnotapplicableingeneral.Thisproblemmotivatesadifferentdenitionforpolesandeigenvectorsthatcorrectlycharacterizethesystem.Onesuchdenitionisexaminedinthissection. 3.2.1LTVResponseEquationAresponseequationthatrelatestheLTVpolesandeigenvectorstothestatesx(t)mustrstbedened.TokeepthedifferencesbetweentheproposedLTVeigenpairdenitionandtraditionalLTIanalysistoaminimum,theresponseequationisdenedusingthemodedenitionsprovidedbyWu[ 8 ]andKamen[ 9 ].ThisresponseequationisgiveninEquation 3 ,wherefori=[1,...,n],Ci2Careconstants,i:R+!CNarelinearlyindependentsolutionstoEquation 2 ,"i:R+!CNaretheLTVeigenvectors,andpi:R+!CaretheLTVpoles.Theeigenvectors"i(t)areassumedtobedifferentiable.x(t)=NXi=1Cii(t)=NXi=1Ci"i(t)expZt0pi()d (3)Notethateachlinearly-independentsolutioni(t)canalsobethoughtofasamodeofthesystem[ 8 ].Wheni(t)isusedinthiscontext,itwillbereferredtoastheithresponsemodeofthesystem. 3.2.2LTVEigenrelationTherelationshipbetweentheeigenpairsandthestatesgiveninEquation 3 isusedtodenearelationshipbetweentheeigenpairsandthecoefcientmatrix.First,anexpressionfortheithresponsemode(oneparticularsolutiontoEquation 2 )is 22

PAGE 23

extractedfromEquation 3 i(t)="i(t)expZt0pi()d(3)ThederivativewithrespecttotimeofEquation 3 isthenobtainedinEquation 3 ._i(t)=_"i(t)expZt0pi()d+"i(t)pi(t)expZt0pi()d=[_"i(t)+"i(t)pi(t)]expZt0pi()d (3)Sincetheresponsemodesaresolutionstothesystem,Equation 3 andEquation 3 canbeinsertedintoEquation 2 ._i(t)=A(t)i(t)[_"i(t)+"i(t)pi(t)]expZt0pi()d=A(t)"i(t)expZt0pi()d (3)Theexponentialtermsareguaranteedtobenon-zeroifthepolecontainsnosingularpointsoverthetimeintervalinquestion,whichallowstheexponentialstobecanceledfrombothsidesofEquation 3 .ThiscancelationyieldsEquation 3 ,whichrelatestheLTVeigenpairstothecoefcientmatrix._"i(t)+"i(t)pi(t)=A(t)"i(t)_"i(t)=A(t)"i(t))]TJ /F12 11.955 Tf 11.95 0 Td[("i(t)pi(t) (3)Equation 3 willhereafterbereferredtoastheLTVeigenrelation.AnLTVpoleandeigenvectorthatsatisesthisequationwillbereferredtoasanLTVeigenpair.ThisdenitionisidenticaltotheonespeciedbyWu[ 7 ].NotethatEquation 3 isunder-dened.Forasystemwithnstates,thisequationhasn+1unknowns(neigenvectorcomponentsand1scalarpole),butonlynscalarequations.ThismeansthatthereareaninnitenumberofLTVeigenpairswithagiveninitialvaluethatsolveEquation 3 .ThisisalsotrueinLTIsystems,wheresolutions 23

PAGE 24

areonlyuniqueuptoascalarconstantmultipliedontotheeigenvector.Bymultiplyingdifferentconstantsontotheeigenvector,aninnitenumberofLTIeigenpairsthatsatisfyEquation 3 canbeobtained.InLTVsystems,eigenpairsolutionsareuniqueuptoascalar,differentiablefunctionoftime.Thesefunctionsaremultipliedontotheeigenvector,andmustmodifythepoleinaparticularwaytoresultinanotherLTVeigenpairthatsatisesEquation 3 .ThisideaisexploredmorethoroughlyinSection 3.3 3.2.3RelationshipbetweenLTVEigenpairsandStabilityThetransitionfromLTItoLTVanalysisintroducesseveraladditionalproblems.Sincetheeigenvectorsandpolesbothvarywithtime,thestabilityofthesystemisdeterminedbybothquantitiesinsteadofthepolesalone[ 8 ].Evenifthepolesarenegativeandbounded,thesystemcanstillbemadeunstablebyrapidlydivergingeigenvectors.Toillustratethispoint,anunstablesystemwiththecoefcientmatrixspeciedinEquation 3 isintroduced. A(t)=26401)]TJ /F4 11.955 Tf 9.3 0 Td[(0.2t)]TJ /F4 11.955 Tf 11.95 0 Td[(0.5375(3)AnLTVeigenpairsolutioniscomputedforthissystembychoosingapairofcomplex-conjugatepolevalueswithnegativerealparts,thensolvingtheLTVeigenrelation(giveninEquation 3 )onceforeachpolevaluetoobtaintheappropriateeigenvectorcomponents.Sincethissystemisunstable,choosingthepolevaluestohavenegativerealpartsforcestheinstabilityinthesystemtobereectedintheeigenvectors.ThetwoeigenpairsfoundbythisprocessareshowninFigure 3-3 .Notethatthissolutionisnotunique,sinceadifferentchoiceofpoleswouldhaveproducedadifferentsetofeigenpairs.ToverifythatthetwoeigenpairsshowninFigure 3-3 satisfytheLTVeigenrelation,thenumericalderivativeoftheeigenvectorcomponentscanbecomparedtothe 24

PAGE 25

Figure3-3. Eigenvectors(left)andpoles(right)fromanLTVsolutionofEquation 3 right-handsideofEquation 3 .Ifthetwovaluesareroughlyequal(thenumericalderivativeintroducessomeerror),thisindicatesthattheeigenpairsatisestheequation.Thischeckisperformedonbotheigenpairs,andtheresultsareshowninFigure 3-4 .NotethatinthelegendofFigure 3-4 ,deijreferstothenumericalderivativeofthejthcomponentoftheitheigenvector,andmdeijreferstothevalueofthesamecomponentpredictedbytheright-handsideoftheLTVeigenrelation.ThegureindicatesthatthetwoeigenpairssatisfytheLTVeigenrelation. Figure3-4. Validitychecksforeigenpair1(left)andeigenpair2(right)fromanLTVsolutionofEquation 3 Asanalcheck,thetwoeigenpairsareinsertedintoEquation 3 (LTVresponseequation).Theconstantsinthisequationarecomputedforaparticularinitialstate 25

PAGE 26

vectorx(0)byinsertingtheinitialvaluesoftheeigenpairsintotheresponseequation,thensolvingfortheconstants.ThestatesproducedasaresultofsolvingtheresponseequationarethencomparedtotheactualstatetrajectoriesofthesystemobtainedbysolvingEquation 2 withanumericalODEsolver.Figure 3-5 showsboththestatetrajectoriesandthemodalresponse(stateestimatescomputedfromtheresponseequation)forx1(t)andx2(t).Notethatthestateestimatescomputedfromthemodalresponsearedenotedinthelegendasxhat1andxhat2.ThisgureindicatesthatthetwoeigenpairssatisfyEquation 3 Figure3-5. StatetrajectoriesfromEquation 2 versusthemodalresponseofanLTVsolutionofEquation 3 ,forx(0)=[10] TheseguresshowthatthepolesalonearenotenoughtodeterminethestabilityofanLTVsystem.ItispossibletoobtainvalidLTVeigenpairsthatdividethestabilityinformationbetweenthepolesandtheeigenvectorssuchthatthepoleshavenegativerealparts,eventhoughthesystemisunstable.Thechoiceofpolessimplyforcedtheeigenvectorstodivergefasterthantheexponentsofthepolesweredecaying.Sincethesesetsofeigenpairsaren'tunique,itwouldbepossibletogenerateequallyvalideigenpairsforwhichtherealpartsofthepoleswerepositive.Thisnewsetofeigenpairswouldsimplypartitionthestabilityinformationinadifferentmanner. 26

PAGE 27

3.2.4RelationshipbetweenLTVEigenpairsandOscillationAnotherissueintroducedbythemovefromLTItoLTVanalysisinvolvesthesourcesofoscillationintheresponse.InLTIanalysis,thefrequenciesofanyoscillatorybehaviorwerecompletelycharacterizedbytheimaginaryvalueofthepole.InLTVanalysistheoscillatorybehaviormayarisefromthreesources:complexpoles,realpoles,andeigenvectors.Toillustrateeachofthesesources,asystemwiththecoefcientmatrixgiveninEquation 3 isintroduced: A(t)=26401)]TJ /F4 11.955 Tf 9.3 0 Td[(0.2sin(8t)375(3)Threedifferenteigenpairsolutionsarecomputedforthissystem,eachofwhichrepresentstheoscillationinadifferentpartoftheeigenpairs.EachsetofeigenpairssatisesEquation 3 (theLTVeigenrelation)andEquation 3 (theLTVresponseequation).InFigure 3-6 ,therstsetofeigenpairsdemonstrateshowtheoscillationinthissystemcanberepresentedbythecomplexvaluesofthepoles.ThisdivisionofinformationreectsthestandardbehaviorofLTIpolesandeigenvectors. Figure3-6. Eigenvectors(left)andpoles(right)fromanLTVsolutionforEquation 3 withoscillationrepresentedbycomplexpoles InFigure 3-7 ,asecondsetofeigenpairsdemonstrateshowthesamesystemcanhaveitsoscillationrepresentedbytherealpartsofthepoles.Asinewave(sin(8t)) 27

PAGE 28

ischosenforthevaluesofthepoles,withnoimaginarycomponent.Bychoosingthecorrectfrequencyitispossibletoverynearlyeliminatetheoscillationfromtheeigenvectors,leavingalmostallofthesystem'soscillatorybehaviorintherealpartsofthepoles. Figure3-7. Eigenvectors(left)andpoles(right)fromanLTVsolutionforEquation 3 withoscillationrepresentedbyrealpoles Finally,Figure 3-8 showsathirdsetofeigenpairsthatdemonstratesthatthissamesystemcanhaveitsoscillationrepresentedbythevaluesoftheeigenvectors.Inthisset,thepolesarechosentoberealandconstantsothattheycannotcontributeanythingtotheoscillatorybehaviorofthesystem.Instead,theoscillationisforcedtoshowupintheeigenvectors.Whilethesethreeeigenpairsolutionsarechosensuchthattheoscillatorybehaviorisarticiallylimitedtoonepartoftheeigenpair,eigenpairsingeneralwillrepresentoscillatoryinformationassomecombinationoftherealandimaginarypartsofboththepolesandtheeigenvectors.Theentireeigenpairmustbeanalyzedtodeterminetheoscillatorycharacteristicsoftheresponse. 3.3ModeVectorAnalysisGiventhedenitionforLTVeigenpairsdescribedinSection 3.2 ,neitherthestabilitynortheoscillatorycharacteristicsofanLTVsystemcanbedeterminedingeneralfromeithertheLTVpolesoreigenvectorsalone.Instead,eacheigenpairmustbeanalyzed 28

PAGE 29

Figure3-8. Eigenvectors(left)andpoles(right)fromanLTVsolutionforEquation 3 withoscillationrepresentedbyeigenvectors asawhole.SinceLTIanalysistechniquesarebasedontheassumptionthatthestabilityandoscillatorycharacteristicsareisolatedinthepoles,newanalysistechniquesareclearlynecessary.Thissectionproposesasuitablemethod:ModeVectorAnalysis. 3.3.1EigenpairsandLinearIndependenceRecallthatthestatetrajectoriesofanLTVsystemwithnstatescanberepresentedbyneigenpairs.EacheigenpairformsaresponsemodeofthesystemaccordingtoEquation 3 ,andalinearcombinationoflinearly-independentresponsemodesformsaparticularstatetrajectoryaccordingtoEquation 3 .Theconceptoflinearindependence(anddependence)canbeextendedtotime-varyingvectorsbysimplyapplyingthetime-invariantdenitionateachindividualtimestep[ 15 ].Thus,twotime-varyingvectorsarelinearlydependentifonevectorisequaltotheothervectormultipliedbyascalarfunctionoftime.Linearindependenceisdenedtobethefailuretomeetthiscondition.Underthisdenition,linearindependenceisthoughtofasactingbetweenonlytwovectorsatatime.Themoregeneraldenition,thatavectorislinearlyindependentfromagroupofothervectorsifitcannotbeexpressedasalinearcombinationofthoseothervectors,isomittedfromthisdiscussionforthesakeofsimplicity. 29

PAGE 30

Sincetheexponentialoftheintegralofapoleisascalarvalue,theresponsemodei(t)islinearlyindependentfromanotherresponsemodej(t)underthisdenitionifandonlyiftheeigenvector"i(t)islinearlyindependentfrom"j(t).Thisresultmeansthatforasystemwithnstates,neigenpairswithlinearly-independenteigenvectorsareenoughtocompletelycharacterizeallsolutionsofthesystem.Anyoftheseeigenpairscanbereplacedbyanothereigenpairwhoseeigenvectorislinearlydependentwiththeoriginal'seigenvector,andthesetofeigenpairswillstillcompletelycharacterizethesystem.Forthisreason,thetermequivalenteigenpairswillbeusedtodescribetwoeigenpairswhoseeigenvectorsarelinearly-dependent. 3.3.2TransformationsbetweenEquivalentEigenpairsTheeigenvectorportionofanequivalenteigenpaircanbefoundfromanexistingeigenpairbymultiplyingtheeigenvectorbysomescalarfunctionoftime.However,thepolevaluemayalsoneedtochangeinorderfortheneweigenpairtobeavalidsolutionofEquation 3 (theLTVeigenrelation).Atransformationequationmustbederivedthatdescribeshowthepoleneedstochangeforagiveneigenvectortransformation.Assumethatthereexistssomeknowneigenpairh"(t),p(t)ithatsatisesEquation 3 forthecoefcientmatrixA(t).LetT:R+!Cnnbeaninvertible,differentiabletransformationappliedtotheeigenvector,asshowninEquation 3 .Inthisequation, ":R+!Cndenotestheeigenvectorthatresultsafterthetransformationisapplied. "(t)=T)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)"(t)"(t)=T(t) "(t) (3)SubstitutingEquation 3 intoEquation 3 yieldsEquation 3 A(t)T(t) "(t))]TJ /F3 11.955 Tf 11.95 0 Td[(T(t)p(t) "(t)=_T(t) "(t)+T(t)_ "(t)(3) 30

PAGE 31

MultiplyingbothsidesoftheequationbyT)]TJ /F7 7.97 Tf 6.58 0 Td[(1(t)producesEquation 3 .T)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)A(t)T(t) "(t))]TJ /F3 11.955 Tf 11.95 0 Td[(p(t) "(t)=T)]TJ /F7 7.97 Tf 6.58 0 Td[(1(t)_T(t) "(t)+_ "(t)T)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)A(t)T(t) "(t))]TJ /F10 11.955 Tf 11.95 9.68 Td[(p(t)+T)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)_T(t) "(t)=_ "(t) (3)Thegoalistondanew,transformedeigenpairthatisvalidfortheoriginalA(t)matrix.Toaccomplishthisobjective,itisnecessarytorestrictT(t)suchthatthemultiplicationA(t)T(t)iscommutative.ThisrestrictionisshowninEquation 3 T)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)A(t)T(t)=A(t)(3)IfT(t)ischosentosatisfyEquation 3 ,Equation 3 reducestoEquation 3 A(t) "(t))]TJ /F10 11.955 Tf 11.95 9.68 Td[(p(t)+T)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)_T(t) "(t)=_ "(t)(3)Let p:R+!Cbethetransformedpole,asdenedinEquation 3 p(t)=p(t)+T)]TJ /F7 7.97 Tf 6.58 0 Td[(1(t)_T(t)(3)ThistransformedpoleisthensubstitutedintoEquation 3 A(t) "(t))]TJ ET q .478 w 222.74 -402.12 m 229.31 -402.12 l S Q BT /F3 11.955 Tf 222.74 -409.43 Td[(p(t) "(t)=_ "(t)(3)Equation 3 istheLTVeigenrelationgiveninEquation 3 .So,thepairh "(t), p(t)iformsavalideigenpairofthematrixA(t).Forpracticalpurposes p(t)shouldtoberestrictedtoascalarvalue,sinceaneigenpairinvolvingamatrix-valuedpolewouldbedifculttointerpretinanalysis.AsecondrestrictiononT(t)isthereforenecessary.ThisrestrictionisgiveninEquation 3 ,whereg:R+!Cissomescalarfunctionoftime.T)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)_T(t)=g(t)Inn (3) 31

PAGE 32

Inotherwords,T(t)shouldbechosensuchthatT)]TJ /F7 7.97 Tf 6.58 0 Td[(1(t)_T(t)evaluatestoascalartime-varyingmatrix.TomeetthetworestrictionsgiveninEquation 3 andEquation 3 ,T(t)willbechosenfromtheclassofscalarmatricesgiveninEquation 3 .Thischoiceissufcienttomeetthetworestrictions,butnotnecessary.Notethatthefunctionf(t)isrequiredtobedifferentiableandinvertible. T=T(t):T(t)=f(t)Inn,f:R+!C(3)TransformationschosenfromthisclassmeettherstrestrictiongiveninEquation 3 :T)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)A(t)T(t)=1 f(t)InnA(t)f(t)Inn=A(t) (3)TransformationsofthisformalsomeetthesecondrestrictiongiveninEquation 3 :T)]TJ /F7 7.97 Tf 6.58 0 Td[(1(t)_T(t)=1 f(t)Inn_f(t)Inn=_f(t) f(t)Inn(scalarmatrix) (3)Insummary,transformationsoftheformT(t)=f(t)InnappliedtoanLTVeigenpairh"(t),p(t)iofA(t)resultinnewequivalenteigenpairsh "(t), p(t)iofA(t)withthefollowingform: "(t)=1 f(t)Inn"(t) p(t)=p(t)+_f(t) f(t) (3)Todemonstratethatthesetransformationequationswork,theunstablesystemgiveninEquation 3 isagainexamined.Acompleteeigenpairsolutionforthissystemisobtainedbychoosingvaluesforeachofthetwopoles,thensolvingEquation 3 withanODEsolvertoobtaintheeigenvectorcomponents.Thetwoeigenpairsthatformthe 32

PAGE 33

solution,h"1(t),p1(t)iandh"2(t),p2(t)i,arethentransformedaccordingtoEquation 3 .Thefunctionf1(t)=2+sin(t)isusedtotransformthersteigenpair,asshowninEquation 3 "1(t)=1 f1(t)Inn"1(t)=1 2+sin(t)Inn"1(t) p1(t)=p1(t)+_f1(t) f1(t)=p1(t)+cos(t) 2+sin(t) (3)Thefunctionf2(t)=exp(2t)isusedtotransformthesecondeigenpair,asshowninEquation 3 "2(t)=1 f2(t)Inn"2(t)=exp()]TJ /F4 11.955 Tf 9.3 0 Td[(2t)Inn"2(t) p2(t)=p2(t)+_f2(t) f2(t)=p2(t)+2 (3)Figure 3-9 showstheeigenvectorsoftheoriginalsolution(UEinthelegend),plottedagainstthetransformedeigenvectors(TRinthelegend).Figure 3-10 showsthepolesoftheoriginalsolution,plottedagainstthosefromthetransformedeigenpair.Theseguresdemonstratethatthetransformedeigenpairsaremeaningfullydifferentfromtheoriginaleigenpairs. Figure3-9. Eigenvectors1(left)and2(right)fromtheoriginalandtransformedsolutionsofEquation 3 33

PAGE 34

Figure3-10. PolesfromtheoriginalandtransformedsolutionsofEquation 3 Thoughthetransformedeigenpairsaresignicantlydifferentthantheoriginalones,theystillsatisfyEquation 3 (LTVeigenrelation).ThisisdemonstratedbytwoeigenpairvaliditychecksinFigure 3-11 .NotethatinthelegendofFigure 3-11 ,deijreferstothenumericalderivativeofthejthcomponentoftheitheigenvector,andmdeijreferstothevalueofthesamecomponentpredictedbytheright-handsideofEquation 3 (LTVeigenrelation). Figure3-11. Validitychecksforeigenpair1(left)andeigenpair2(right)fromthetransformedsolutionofEquation 3 Finally,Figure 3-12 showsthatthetransformedeigenpairsatisesEquation 3 (theLTVresponseequation).Forbothstatetrajectories(x(0)=[10]ontheleft,x(0)=[01]ontheright),themodalresponsecomputedusingthetwotransformedeigenpairs 34

PAGE 35

matchesthestatetrajectoriescomputedbysolvingEquation 2 directly.TheseguresverifythatequivalenteigenpairsarevalidsolutionstotheLTVeigenrelationandLTVresponseequation. Figure3-12. StatetrajectoriesfromEquation 2 versusthemodalresponseofthetransformedsolutionofEquation 3 ,forx(0)=[10](left)andx(0)=[01](right) 3.3.3DenitionofModeVectorsInordertoperformanalysisusingtheeigenpairsofanLTVsystem,itisdesirabletoobtainsomequantitythatencapsulatesalltheinformationintheeigenpairandremainsinvarianttoequivalenteigenpairtransformations.Basingtheanalysisofasystemonsuchaquantitywouldpreventproblemscausedbythepartitioningoftime-varyinginformationinbetweenthepoleandtheeigenvector,asdescribedinSections 3.2.3 and 3.2.4 .Amathematicalentitythatmeetsthesegoalsisdenedbyadaptingthedenitionoftheresponsemodesofthesystem.ThestabilityofanLTVsystemhasbeensuccessfullycharacterizedthroughtheuseofthesystem'sresponsemodes,denedinEquation 2 (seeSection 2.3 ).Whiletheresponsemodescombinethepartsofeacheigenpairinawaythatpreservesthestabilityinformation,theydonotremaininvarianttoequivalenteigenpairtransformations.Amodicationoftheresponsemodeisthereforenecessary.Thisnewquantitywillbecalledamodevectorofthesystem.First,thecomponentsofeacheigenvectorare 35

PAGE 36

speciedusingthenotationgiveninEquation 3 ,where"i:R+!CnistheLTVeigenvectoroftheitheigenpair,eij:R+!Cisthejthcomponentofthateigenvector,andnisthenumberofstatesinthesystem."i(t)=266664ei1(t)...ein(t)377775 (3)ThemodevectorsarethendenedinEquation 3 ,wherei:R+!Cnisthemodevectoroftheitheigenvector,ij:R+!Cisthejthcomponentofthatmodevector,andpi:R+!CistheLTVpoleoftheitheigenpair.i(t)=266664i1(t)...in(t)377775ij=expZt0pi()+_eij() eij()d (3)Movingtheeigenvectorcomponentsoutsideoftheintegralyieldsasecond,equivalentdenitiongiveninEquation 3 .ij=eij(t) eij(0)expZt0pi()d (3)NotethatEquation 3 differsfromthedenitionoftheresponsemodesinEquation 3 byonlytheinitialvalueofeacheigenvectorcomponent,whichisaconstantfactor.Aslongaseverycomponentoftheeigenvectorsattimet=0isnon-zero,thetwostabilityresultsderivedbyWu[ 8 ]fortheresponsemodesshouldworkequallywellforthemodevectors. Thelineartime-varyingsystemgiveninEquation 2 isstableifandonlyifthenormofeverymodevectorjji(t)jjofA(t)isbounded.ThisisexpressedinEquation 3 ,whereC2Cissomearbitraryniteconstant. jji(t)jj0,8i=1,...,n(3) 36

PAGE 37

Thissystemisasymptoticallystableif,inadditiontobeingstable,thenormofeverymodevectorapproaches0astimegoestoinnity.ThisisexpressedinEquation 3 jji(t)jj!0ast!1,8i=1,...,n(3)Likeresponsemodes,modevectorsencapsulatetheentireeigenpairinawaythatpreservesthestabilityinformation.Whatremainstobeseeniswhetherthemodevectorsremaininvarianttoequivalenteigenpairtransformations.Toprovethattheydoremaininvariant,themodevectorofaneigenpairafteranequivalenteigenpairtransformationiscomparedtothemodevectoroftheeigenpairbeforethetransformation.First,assumethataneigenpairh"i(t),pi(t)iofA(t)isgiven.Givensomescalardifferentiablefunctionf(t):R+!C,anequivalenteigenpairh "i(t), pi(t)iisobtainedviathetransformationsinEquation 3 ,asshowninEquation 3 pi(t)=pi(t)+_f(t) f(t) "i(t)=266664 ei1(t)... ein(t)377775=266664f)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)ei1(t)...f)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)ein(t)377775 (3)Theobjectiveistoshowthatthemodevectorofthetransformedeigenpairisequaltothemodevectoroftheoriginaleigenpair.ThisisdepictedinEquation 3 ,where i:R+!Cnisthemodevectorofthetransformedeigenpairh "i(t), pi(t)iandi:R+!Cnisthemodevectoroftheoriginaleigenpairh"i(t),pi(t)i. i(t)=i(t)(3)ThemodevectorsoneithersideofEquation 3 mustbeequalifthecorrespondingcomponentsofeachmodevectorareequal.UsingtherstdenitiongiveninEquation 37

PAGE 38

3 ,thetwomodevectorsarecomparedcomponent-to-componentinEquation 3 ij(t)=ij(t)expZt0 pi()+_ eij() eij()d=expZt0pi()+_eij() eij()d (3)IftheintegrandsinEquation 3 areequalforalltime,theexpressionsonbothsidesoftheequationmustbeequal.ThissimpliesEquation 3 toEquation 3 pi(t)+_ eij(t) eij(t)=pi(t)+_eij(t) eij(t) (3)Theleft-handsideofEquation 3 isthenexpandedbyinsertingthedenitionsfor pi(t)and "i(t)giveninEquation 3 pi(t)+_ eij(t) eij(t)=pi(t)+_f(t) f(t)+f(t) eij(t)d dteij(t) f(t)=pi(t)+_f(t) f(t)+f(t) eij(t)_eij(t) f(t))]TJ /F4 11.955 Tf 14.65 10.74 Td[(_f(t)eij(t) f(t)2=pi(t)+_f(t) f(t)+_eij(t) eij(t))]TJ /F4 11.955 Tf 14.64 10.75 Td[(_f(t) f(t)=pi(t)+_eij(t) eij(t)Proof. (3)Todemonstratethatthemodevectorsfromtwoequivalenteigenpairsarethesame,themodevectorsofthetwoLTVeigenpairsolutionsofEquation 3 coveredintheexampleinSection 3.3.2 arecalculated.ThemodevectorsfromtheoriginalsolutionaregraphedagainstthosefromthetransformedsolutioninFigure 3-13 .Thegureshowsthatthemodevectorsfrombothsolutionsarenearlyidentical.Itistrivialtoshowthatastherateatwhichtheeigenvectorschangewithtimedropstozero,thevaluesofeachcomponentofthemodevectorconvergetothetraditionaldenitionofthemodesusedinLTIandFTEanalysis,exphRt0pi()di.Thisresultcanbeseenbymakingthesubstitution_eij()=0intoEquation 3 38

PAGE 39

Figure3-13. Modevectors1(left)and2(right)fromtheoriginalandtransformedsolutionsofEquation 3 Insummary,themodevectorisamathematicalquantitythatcanbeanalyzedinplaceofthepolesandeigenvectorsofaparticularLTVsolution.Thisnewquantityhasthreepropertiesthatmakeitveryusefulforanalysis: Modevectorsdonotchangebetweenequivalenteigenpairs. Modevectorsincorporateinformationfromboththepoleandtheeigenvectorintoasinglequantity. ModevectorsreducetothemodaldenitionusedinLTIandFTEanalysis,whenthesystemisslowly-time-varyingorLTI. 3.4KamenAnalysisAsanalternativetomodevectoranalysis,amethodbasedonresearchdonebyKamen[ 9 ][ 18 ]isexaminedindetail.Whereasmodevectoranalysisproceedsthroughtheuseofamathematicalquantitythatisthesamenomatterhowthedataispartitionedbetweenthepoleandtheeigenvector,inKamenanalysisaspecicpartitioningischosenthatisguaranteedtoisolatethestabilityinformationinthepoles,givensomeconstraintonthecoefcientmatrixA(t). 3.4.1ADifferentPerspectiveonKamen'sMethodKamen'sapproachtoanalyzingLTVsystemshasalreadybeendescribedinSection 2.3 .Usingscalarpolynomialdifferentialoperators(SPDO's),Kamendevelopsthe 39

PAGE 40

equationforthepolesofasecond-orderLTVsystemoftheformshowninEquation 2 .Kamen'sresultisgiveninEquation 3 ,wherep:R+!CistheKamenpoleforthissystem.Solvingthisquadraticequationwillyieldtwosolutionsforthepole,eachofwhichformsaneigenpairofthesolution[ 9 ]. p2(t)+_p(t)+a1(t)p(t)+a0(t)=0(3)TheeigenvectorsoftheKamensolutionofasecond-ordersystemaregivenaccordingtoEquation 3 .Inthisequation,"i:R+!CnistheeigenvectoroftheithKameneigenpair,andpiisthepoleoftheithKameneigenpair(theithsolutiontoEquation 3 ). "i(t)=2641pi(t)375(3)InordertolinkKamen'spolestotheLTVpoleandeigenvectordenitionsgiveninSection 3.2 ,theresultgiveninEquation 3 isderivedfromEquation 3 (theLTVeigenrelation).Equation 3 isreexpressedforasecond-ordersysteminEquation 3 264_ei1(t)_ei2(t)375=264a11(t)a12(t)a21(t)a22(t)375264ei1(t)ei2(t)375)]TJ /F10 11.955 Tf 11.95 27.62 Td[(264pi(t)ei1(t)pi(t)ei2(t)375(3)Thisequationisunder-dened,inthattherearethreeunknowns(thetwoeigenvectorcomponents,andthepole)butonlytwoscalarequations.Togetaparticularsolutiontothisequation,anassumptionmustbemadetoeliminateoneofthevariables.Differentassumptionsproduceeigenpairswithpotentiallyverydifferentdivisionsofinformationbetweenthepolesandeigenvectors.ToderiveKamen'sequationtherstcomponentofeveryeigenvectorisassumedtobeequalto1foralltime,asdetailedinEquation 3 ei1(t):=1,8t0,8i(3) 40

PAGE 41

Equation 3 isthenincorporatedintoEquation 3 2640_ei2(t)375=264a11(t)a12(t)a21(t)a22(t)3752641ei2(t)375)]TJ /F10 11.955 Tf 11.95 27.61 Td[(264pi(t)pi(t)ei2(t)375(3)RecallthatKamen'sresultisderivedforasystemdescribedbyasecond-orderscalardifferentialequation.Assumingthesystemishomogeneous(nocontrolinput),thisdifferentialequationisexpressedinEquation 3 [ 9 ]. x(t)+a1(t)_x(t)+a0(t)=0(3)Thisequationcanbeexpressedasarst-ordermatrixequationoftheformgiveninEquation 2 ,wherethecoefcientmatrixA(t)isinthecanonicalformdescribedinEquation 2 .ToderivetheKamenmethod,itisthereforenecessarytoassumethatthecoefcientmatrixinEquation 3 isinthiscanonicalform. A(t)=26401)]TJ /F3 11.955 Tf 9.3 0 Td[(a0(t))]TJ /F3 11.955 Tf 9.3 0 Td[(a1(t)375(3)UpdatingEquation 3 withthisformforthecoefcientmatrixyieldsEquation 3 2640_ei2(t)375=26401)]TJ /F3 11.955 Tf 9.3 0 Td[(a0(t))]TJ /F3 11.955 Tf 9.3 0 Td[(a1(t)3752641ei2(t)375)]TJ /F10 11.955 Tf 11.96 27.62 Td[(264pi(t)pi(t)ei2(t)375(3)Equation 3 isthenexpressedastwoscalarequations.TherstscalarequationisgiveninEquation 3 .0=ei2(t))]TJ /F3 11.955 Tf 11.95 0 Td[(pi(t)ei2(t)=pi(t) (3) 41

PAGE 42

ThesecondscalarequationextractedfromEquation 3 isgiveninEquation 3 ._ei2(t)=)]TJ /F3 11.955 Tf 9.29 0 Td[(a0(t))]TJ /F3 11.955 Tf 11.95 0 Td[(a1(t)pi(t))]TJ /F3 11.955 Tf 11.95 0 Td[(pi(t)ei2(t)_ei2(t)+a0(t)+a1(t)pi(t)+pi(t)ei2(t)=0 (3)SubstitutingEquation 3 intoEquation 3 yieldsEquation 3 ,whichissimplyarestatementofKamen'sresultgiveninEquation 3 p2i(t)+_pi(t)+a1(t)pi(t)+a0(t)=0(3)ThisresultshowsthatKamen'spolescanbethoughtofasoneparticularsetofeigenpairschosenoutofallthepossibleeigenpairsthatsatisfytheLTVeigenrelation,forasystemwhosecoefcientmatrixisincanonicalform.TodemonstratethattheKameneigenpairsolutionissimplyoneofthesolutionstotheLTVeigenpairrelation,theunstablesystemrstintroducedinSection 3.2.3 asEquation 3 willagainbeexamined.Notethatthissystemisalreadyincanonicalform._x=A(t)x(t),A(t)=26401)]TJ /F4 11.955 Tf 9.29 0 Td[(0.2t)]TJ /F4 11.955 Tf 11.95 0 Td[(0.5375TheeigenvectorsandpolesoftheKamensolutiontothesystemgiveninEquation 3 areshowninFigure 3-14 .Boththeeigenvectors(leftside)andthepoles(rightside)arecomplexconjugatepairs.AspredictedbyEquation 3 ,therstcomponentofeacheigenvectorisequaltoonefortheentiretimeintervalandthesecondcomponentofeacheigenvectorisequaltothecorrespondingpole.TherealvaluesoftheKamenpolesstartoutslightlynegative,buttransitiontoastrongpositivevaluehalfwaythroughthetimeinterval.Thiscorrectlyindicatesthatthesystemisunstable,sincetheintegralofthepolebecomespositive.TheresultsofboththeeigenpairvalidityandthemodalresponsechecksareshowninFigure 3-15 .Theeigenpairvaliditycheck(leftside)containsthevaliditychecksfor 42

PAGE 43

Figure3-14. Eigenvectors(left)andpoles(right)fromtheKamensolutiontoEquation 3 botheigenpairsonthesamegraph.Asbefore,deijreferstothenumericalderivativeofthejthcomponentoftheitheigenvector,andmdeijreferstothevalueofthesamecomponentpredictedbytheright-handsideofEquation 3 (theLTVeigenrelation).Apartfromsomeerrorintroducedbycomputingthenumericalderivative,thisgraphshowsthatbotheigenpairsoftheKamensolutionsatisfytheLTVeigenrelation.Themodalresponsecheck(rightsideofgure)veriesthatKamen'seigenpairsolutionsatisesEquation 3 (theLTVresponseequation).ThestatetrajectoriesreproducedbytheresponseequationalmostexactlymatchthoseproducedbysolvingEquation 2 directly. 3.4.2SufciencyofKamen'sPolestoDetermineStabilityKamenshowedthatthepolesdenedbyEquation 3 aresufcienttodeterminethestabilityofanLTVsystem[ 9 ].Whilethisfacthasalreadybeenproveninotherworks,someadditionalinsightisgainedbyreexaminingthisresultfromtheviewpointoftheLTVeigenrelationderivationdescribedinSection 3.4.1 .InSection 3.2.3 itwasshownthatevenifthesystemisunstable,LTVeigenpairsthatsatisfytheLTVeigenrelationcanbefoundthathaverealpolevaluesthatindicatestability.IfKamen'smethodguaranteesthatthepolesaresufcienttodeterminethestabilityofthesystemwhiletheLTVeigenrelationdoesnot,thisguaranteemustbe 43

PAGE 44

Figure3-15. Validitycheckofeigenpairs1and2(left)andstatetrajectoriesversusmodalresponse(right)fortheKamensolutiontoEquation 3 aresultofthetwoassumptionsthatweremadetoextracttheKamenpolesfromthegeneralLTVeigenrelation.NotethatKamen'smethodrequiresthatthesystembeincanonicalform.Inotherwords,thesystem'sstatevectorisrequiredtobecomposedonlyofonestateanditsderivatives,asstatedinEquation 2 .Equation 3 isreexpressedwiththisspecialstatevectorforasecond-ordersysteminEquation 3 .x(t)=C1e11(t)expZt0p1()d+C2e21(t)expZt0p2()d (3)_x(t)=C1e12(t)expZt0p1()d+C2e22(t)expZt0p2()d (3)TheotherassumptionusedinderivingKamen'spolesfromtheLTVeigenrelationisthattherstcomponentoftheeigenvectorsshouldalwaysequalone,asstatedinEquation 3 .IncorporatingthisassumptionintoEquation 3 yieldsEquation 3 .FromEquation 3 itisobservedthattheresponseforx(t)dependssolelyonthepoles.So,ifx(t)isunstable,thiscanonlybereectedinthepoles.Iftheintegralsofeachpolearenegative,theresponseforx(t)mustbestable,regardlessofwhat 44

PAGE 45

happensto_x(t).x(t)=C1expZt0p1()d+C2expZt0p2()d (3)_x(t)=C1e12(t)expZt0p1()d+C2e22(t)expZt0p2()d (3) 3.4.3KamenAnalysisviaEquivalentEigenpairTransformationsInSection 3.4.1 ,theKameneigenpairsolutionwasshowntobeaspecialcaseoftheLTVeigenrelationappliedtosystemswithcanonicalcoefcientmatrices.SincetheKameneigenpairsolutionisjustoneofmanypossiblesolutionstotheLTVeigenrelation,itshouldbepossibletotransformsomeothereigenpairsolutionintotheKamensolutionthroughtheuseoftheequivalenteigenpairtransformationsdescribedinSection 3.3.2 .Section 3.3.1 denedequivalenteigenpairsaseigenpairswithlinearly-dependenteigenvectors(i.e.,oneeigenvectorcanbechangedintotheotherbymultiplyingitwithascalarfunctionoftime).Recallthatifthesystemtobeanalyzedisincanonicalform,theonlyadditionalassumptionmadetoobtaintheKameneigenpairsolutionisthattherstcomponentofeacheigenvectorisalwaysequaltoone.AnysolutiontotheLTVeigenrelationcanbetransformedtomeetthisassumptionbysimplydividingeacheigenvectorbyitsrstcomponent,thenupdatingthepolesappropriatelyaccordingtothetransformationoutlinedinEquation 3 .Assumethatasecond-ordersystemincanonicalformisprovided,andthattwolinearlyindependentLTVeigenpairsthatsatisfyEquation 3 (theLTVeigenrelation)havealreadybeenobtained.TotransformthiseigenpairsolutionintotheKamensolutionforthesystem,thetwoeigenpairswouldbetransformedasshowninEquation 3 andEquation 3 .Intheseequations,"i:R+!Cnreferstotheeigenvectoroftheithoriginaleigenpair, "i:R+!Cnreferstotheithtransformedeigenpair,pi:R+!Creferstothepoleoftheithoriginaleigenpair,and pi:R+!Creferstothepoleoftheithtransformedeigenpair.Theadditionofasecondsubscriptjindicatesthejth 45

PAGE 46

componentoftheassociatedvector.ThesetransformedeigenpairsshouldnowbeequaltotheKamensolution,iftheKamensolutionistrulyanequivalenteigenpairtotheothersolutionsoftheLTVeigenrelation. "1(t)=1 e11(t)Inn"1(t) p1(t)=p1(t)+_e11(t) e11(t) (3)Thesecondeigenpairwouldthenbetransformedinasimilarfashion. "2(t)=1 e21(t)Inn"2(t) p2(t)=p2(t)+_e21(t) e21(t) (3)TodemonstratethattheKamensolutioncanbereachedbytransformingadifferentsolutionoftheLTVeigenrelation,theunstablesystemgivenasEquation 3 inSection 3.4.1 isagainexamined.Recallthat,althoughthissystemisunstable,Section 3.2.3 detailedaneigenpairsolutionthatsatisedEquation 3 buthadpolesthatindicatedstability.Thissetofeigenpairs(referredtoasthenon-Kamensolutionforthisdemonstration)isshowninFigure 3-16 Figure3-16. Eigenvectors(left)andpoles(right)fromtheoriginalLTVsolutionofEquation 3 46

PAGE 47

Thenon-KamensolutioninFigure 3-16 istransformedusingthespecialequivalenteigenpairtransformationdetailedinEquation 3 andEquation 3 .TheresultsofthistransformationaregraphedagainsttheKamensolutioninFigure 3-17 .Takingintoaccounterrorsintroducedbytakingthenumericalderivativeduringthetransformationofthepoles,thetransformedeigenpairsmatchtheKameneigenpairs.ThisexampledemonstratesthataKamensolutioncanbeobtainedfromanexistingLTVsolutionwithoutsolvingEquation 3 directlyorevenusinganyinformationbeyondwhatiscontainedinthenon-Kameneigenpairs. Figure3-17. Eigenvectors(left)andpoles(right)fromboththeKamensolutionandthetransformednon-KamensolutiontoEquation 3 3.4.4ProblemswithKamenAnalysisKamenanalysishassomedeniteadvantagesovermodevectoranalysis.Specically,theKamensolutionisauniquesolution,anditguaranteesthatthestabilityisisolatedinthepoles.ThesefeaturesresultinananalysismethodthatisclosertoLTItechniquesthanmodevectoranalysis.Unfortunately,thetwoassumptionsintroducedinordertoderivetheKamensolutionfromtheLTVeigenrelation(seeSection 3.4.1 )revealproblemsthatcanlimittheapplicabilityofthismethod. Unboundedpolesduetoeigenvectornormalization.Forsomesystems,certaininitialpoleandeigenvectorvaluescausesingularpointstoappearinthepolesandeigenvectorsoftheKamensolution.Todemonstratethis,anunstablesystemwiththe 47

PAGE 48

coefcientmatrixspeciedinEquation 3 isintroduced. A(t)=26401)]TJ /F4 11.955 Tf 9.3 0 Td[(0.2t)]TJ /F4 11.955 Tf 11.95 0 Td[(3.0375(3)ThepolesresultingfromaKamenanalysisofthissystemareshowninFigure 3-18 .ThepolesresultingfromsolvingfortheKamensolutionstraightfromEquation 3 areshownontheleft.Thesepolesdivergeatabout0.933secondsanddonotrecover.TherightsideofFigure 3-18 showsthepolesachievedbyrstcalculatinganon-Kameneigenpairsolutionthatstaysbounded,thentransformingitaccordingtoEquation 3 andEquation 3 .Sincetheoriginalnon-Kamensolutioncontainednosingularpoints,andsincethetransformationequationsoperateindependentlyateachpointintime,thebehaviorofthepoleafterthesingularpointcanbeseen.NotethatthetransformedsolutionhasasingularpointatthesamepointintimeastheKamensolution. Figure3-18. PolesfromKamen(left)andtransformed(right)solutionstoEquation 3 ThisproblemcanbeexplainedbytheequationsthatdescribetheequivalenteigenpairtransformationsusedtoreachtheKamensolutionfromthenon-Kamensolution(Equation 3 andEquation 3 ).Notethatbothofthesetransformationsdivideeacheigenvectorbythersteigenvectorcomponent.Ifthersteigenvectorcomponentofanyeigenvectorcrosseszero,asingularpointisintroducedintoboth 48

PAGE 49

thepoleandtheeigenvector,causingtheobserveddivergence.NotealsothattheKamenpolesobtaineddirectlyfromKamen'sresult(Equation 3 )containthissamedivergence.Thisindicatesthattheproblemisnotwiththetransformationequations.Instead,thetransformationequationshavesimplyexpressedinadifferentwayaproblemthatalreadyexistswithKamen'ssolution.Twoadditional,non-KameneigenpairsolutionstothesystemdescribedbyEquation 3 areshowninFigure 3-19 .Thisgureplotstherstcomponentofbotheigenvectorsineachsolutionagainstalinepassingthroughzero.Theleftandrightsidesareplotsfromtwodifferentnon-Kamensolutions(Solution#1andSolution#2).Forbothsolutions,therstcomponentofthesecondeigenvectorcrosseszeroat0.933seconds.Thiscorrespondstothedivergenceofthesecondpoleat0.933secondsinFigure 3-18 .Notethattherstcomponentofeigenvector2ontheleftsideofFigure 3-19 isnotasingularpoint,butissimplyalarge-magnitudeoscillation.Thegraph'srangewasreducedtoshowthepointwherethecomponentcrossedzero,causingthefullplotofthecomponenttobecutoff. Figure3-19. Firstcomponentofbotheigenvectorsforsolution#1(left)andsolution#2(right)ofEquation 3 ThesesingularpointscanbetroublesomeinKamenanalysis,ifthesystembeinganalyzedhaseigenvectorsthatoftencrosszero.Inthecontextofthiswork,theeffectsofthesesingularpointsaredifculttointerpret.Eventhoughtheuseofthe 49

PAGE 50

transformationequationsallowsasolutiontobeobtainedfortheentiretimeinterval,thecorrectvaluefortheintegralofthepoleisveryuncertainafterthesingularpoint.Thisisduetothefactthatallnumericalcomputationsoncomputersystemsarenecessarilydiscrete,andthesingularpointwillalmostneverbeperfectlycenteredbetweentwotimepoints.Thetimepointnearesttothesingularpointwillhavealargermagnitudethanthetimepointontheotherside.Thus,theplacementofthesediscretetimepointsmaysometimesmodifywhetherornottheintegralofthepoleevaluatestoapositiveornegativenumber,makinganystabilitydeterminationbasedontheintegralofthatparticularpolecompletelyarbitrary.Interpretationofsingularpointsisalsomadedifcultbythefactthatthepresenceofsingularpointsinatimeintervaldoesnotimplythatthesystemisunstableoverthattimeinterval.Toillustratethispoint,acanonicalsystemwiththecoefcientmatrixspeciedinEquation 3 isintroduced.A(t)=26401a0(t)a1(t)375a0(t)=1489.6t3)]TJ /F4 11.955 Tf 11.96 0 Td[(4202.6t2+1530.8t)]TJ /F4 11.955 Tf 11.96 0 Td[(104.1a1(t)=7.6677t4)]TJ /F4 11.955 Tf 11.96 0 Td[(30.0238t3+40.1319t2)]TJ /F4 11.955 Tf 11.95 0 Td[(23.0603t)]TJ /F4 11.955 Tf 11.95 0 Td[(34.8975 (3)Thissystemisstableovertheintervalt=[0,1]seconds.ThisisconrmedbothbytheconditionspeciedbyEquation 3 andbyFigure 3-20 .Thisgureshowsthestatetrajectoriesresultingfromtwodifferentinitialconditions(x(0)=[10]andx(0)=[01]).Thoughthesystemisstable,theKamenpolescontainasingularpointoverthissametimeinterval.ThissingularpointisshowninFigure 3-21 .TheKamenpolesresultingfromsolvingEquation 3 directlyareshownontheleft,andthosethatresultfromtransforminganon-Kameneigenpairsolutionareshownontheright.Bothsetsofpolesreachasingularpointbefore0.8seconds,eventhoughFigure 3-20 showsthat 50

PAGE 51

Figure3-20. StatetrajectoriesobtainedfromEquation 3 withx(0)=[10](left)andx(0)=[01](right) thesystemresponsecontinuestodecaytowardszeroupthroughtheendofthetimeinterval. Figure3-21. PolesfromKamen(left)andtransformed(right)solutionstoEquation 3 Asaresultoftherstassumption(Equation 3 )usedinderivingKamen'spolesfromtheLTVeigenrelation,someKamensolutionscontainsingularpoints.Thoughthesolutionfortheentiretimeintervalcanbeobtainedindirectlybytransforminganon-Kameneigenpairsolution,theresultsachievedmaynotprovideusefulinformationaboutthestabilityofthesystem.Thisisduetouncertaintiesfromnumericalerror,andthefactthatsingularpointscanoccurintheKamensolutionsofbothstableandunstablesystems. 51

PAGE 52

Effectsoftransformingsystemtocanonicalform.ThesecondassumptionmadeinderivingKamen'spolesfromtheLTVeigenrelationisthatthesystemisgivenincanonicalform.Usefulsystemsareoftennotinthisformnaturally,sinceitrequiresthateverystateofthesystembeasuccessivelyhigherderivativeoftherststate.Togetasystem'scoefcientmatrixincanonicalform,A(t)isusuallytransformedtoakinematicallysimilarcanonicalmatrix.Therstissuethisraisesinvolvestheuseofidentication.TherequirementthatthesystembetransformedrstbeforethepolesandeigenvectorsarecomputedmeansthatKamen'spolesandeigenvectorscannoteasilybeidentiedfromstatedatawithoutrstlearninganLTVmodel.Thisproblemisduetothefactthatdirectidenticationmethodsoperateonstatemeasurements.Ifthesystemisnotnaturallyincanonicalform,themeasuredstatevectorsneedtobetransformedsothattheymeetthespecialformoutlinedinEquation 2 beforetheyareinputintoaKamenidenticationroutine.ThesecondissuearisesfromthefactthattransformationsofthecoefcientmatrixbehavedifferentlyinLTVsystems,sincetheyaretime-varying.Thesetransformationscansometimeschangetheoscillatorypropertiesofthesystem.InLTIanalysis,thecanonicalformisreachedbyapplyingaLyapunovtransformationtothecoefcientmatrix.Thisresultsinakinematicallysimilarmatrix,As=T)]TJ /F7 7.97 Tf 6.59 0 Td[(1AT,withthesamepolesastheoriginalmatrix.Theeigenvectorsofthenewcoefcientmatrixaredifferent,butthisdoesnotmatterbecausethestabilityandoscillatorycharacteristicsofanLTIsystemaredeterminedbythepolesalone.InLTVsystems,thissituationbecomesmorecomplicated.Wu[ 7 ]denedsimilarityinthetime-varyingcaseasfollows: As(t)=T)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)A(t)T(t))]TJ /F3 11.955 Tf 11.96 0 Td[(T)]TJ /F7 7.97 Tf 6.59 0 Td[(1(t)_T(t)(3)Wuthenprovedthatthissimilaritytransformationdoesnotchangethepolesofthesystem.However,asinLTIsystemsthetransformationdoesactupontheeigenvectors. 52

PAGE 53

Themultiplicationofatime-varyingtransformationmatrixontotheeigenvectorsisaproblem,becausetheeigenvectorsofatime-varyingsystemcontributetoboththestabilityandtheoscillatorybehavioroftheresponse(asdiscussedinSections 3.2.3 and 3.2.4 .IfthetransformationT(t)satisestheLyapunovconditions,itwillnotchangethestabilityinformationpresentintheeigenvectors.Inotherwords,iftheeigenvectorisboundedoriginally,itwillstayboundedaftertheapplicationoftheLyapunovtransformation[ 19 ].However,aLyapunovtransformationmakesnoguaranteesabouttheoscillatoryinformationpresentintheeigenvectors.Sincethetransformationistime-varying,evenaLyapunovtransformationcouldeasilychangetheoscillatorycharacteristicsoftheeigenvectors.Thiswouldresultinamodicationoftheoscillatorycharacteristicsoftheentireresponse,sincethepolesareguaranteedtoremaininvarianttoanalgebraictransformationofthecoefcientmatrix[ 7 ].TodemonstratethataLyapunovtransformationcanchangeoscillatorybehaviorintheresponse,thestableoscillatingsystemdenedbyEquation 3 isagainused._x=A(t)x(t),A(t)=26401)]TJ /F4 11.955 Tf 9.29 0 Td[(0.2sin(8t)375Atime-varyingLyapunovtransformationmatrixT1(t)isintroducedinEquation 3 T1(t)=2642+sin(5.86t)002+sin(3t)375(3)AkinematicallysimilarmatrixisthendenedthroughthetransformationequationgiveninEquation 3 As(t)=T)]TJ /F7 7.97 Tf 6.59 0 Td[(11(t)A(t)T1(t))]TJ /F3 11.955 Tf 11.96 0 Td[(T)]TJ /F7 7.97 Tf 6.59 0 Td[(11(t)_T1(t)(3)ThestatetrajectoriesforthesystemsdenedbyboththeoriginalcoefcientmatrixandthekinematicallysimilaroneareplottedinFigure 3-22 .Inthisgurestate1isgraphedontheleftside,andstate2isgraphedontheright.Thisgureshows 53

PAGE 54

Figure3-22. Stateresponsetox(0)=[10]forEquation 3 andEquation 3 signicantdifferencesintheoscillatorybehaviorofthestatesbetweentheoriginalandthekinematicallysimilarsystem.Thesedifferencesshowupevenmoreclearlyinaplotofthe2-normoftherstmodevectorofeachsystem(themodevectorsineachofthesesystemsarecomplex-conjugatepairs,sothesecondmodevectorisredundant).ThisplotisshownasFigure 3-23 Figure3-23. Normofmodevector1fromsolutionstoEquation 3 andEquation 3 Transformationsingeneralareseentohavethecapacitytochangetheoscillatorybehaviorofthesystem.Suchchangesarestillpossiblewhenthetransformationinquestionisrestrictedtomatricesthattransformagivencoefcientmatrixintocanonicalform.Toillustratethispoint,astablenon-canonicalsystemwithacoefcientmatrixas 54

PAGE 55

speciedinEquation 3 isintroduced. A(t)=264)]TJ /F4 11.955 Tf 9.3 0 Td[(3+sin(t)2+cos(3t)4+sin(5t))]TJ /F4 11.955 Tf 9.3 0 Td[(5+sin(t)375(3)AmatrixT2(t)isdenedthattransformsEquation 3 intoasimilarcanonicalmatrix(asspeciedbyEquation 2 ). T2(t)=264103)]TJ /F7 7.97 Tf 6.58 0 Td[(sin(t) 2+cos(3t)1 2+cos(3t)375(3)ThisnewcanonicalmatrixisobtainedbyapplyingWu'stransformationequation,asshowninEquation 3 As(t)=T)]TJ /F7 7.97 Tf 6.59 0 Td[(12(t)A(t)T2(t))]TJ /F3 11.955 Tf 11.96 0 Td[(T)]TJ /F7 7.97 Tf 6.59 0 Td[(12(t)_T2(t)(3)ThestateresponsesoftheoriginalsystemversusthoseofthecanonicalsystemareplottedinFigure 3-24 .Amarkeddifferenceintheoscillatorybehaviorisseenthatisnotlimitedtochangesinmagnitudeandphase. Figure3-24. Stateresponsetox(0)=[10]forEquation 3 andEquation 3 Thedifferencesbetweenthetwosystemsarealsoreectedbythe2-normsofeachmodevector,asshowninFigure 3-25 .Bothmodevectornormsareshowninthisgure,sincethesystem(Equation 3 )hasreal-valuedmodevectors.Whilethesecond 55

PAGE 56

modevectornormdecaystooquicklytobeofuse,therstdisplaysadifferenceintheoscillatorybehaviorbetweenthetwosystems. Figure3-25. Non-canonicalOscillator,Originalvs.CanonicalMatrix:modevectornorms Whileakinematically-similarcanonicalmatrixcanbefoundforagiventime-varyingmatrix,thetransformationnecessarytoreachthiscanonicalmatrixcanchangetheoscillatorybehaviorofthesystem.Thiscanoccurevenwhenthestabilityinformationisnotchanged.Forsomeapplicationsthisdifferencemightbeminor,butforothersitcouldhaveasignicantimpactontheanalysisofoscillatorybehavior. 3.5SummaryThischapterhasshownthatclassicfrozen-timeeigenpairanalysisisinsufcientforgeneralLTVsystems,andanalternativedenitiongivenbyWuhasbeenexaminedindetail.Itwasshownthatinformationregardingthestabilityandtheoscillatorybehaviorofthesystemisdividedbetweenthepolesandtheeigenvectors,andthatforeverysystemmultiplesolutionsexistinwhichthisinformationispartitioneddifferently.Theconceptsofequivalenteigenpairsandmodevectoranalysiswereproposedandexploredindetail.Finally,Kamen'sanalysismethodswerelinkedtothesenewconcepts,whichhelpedtoclarifyandexplaintheshortcomingsofKamen'smethod.RecallthatKhalil'sexample(Equation 3 )isanLTVsystemforwhichthefrozen-timeeigenpairsindicatestability,buttheactualstateresponseofthesystem 56

PAGE 57

isunstable.Figure 3-26 showsthepolesreturnedbyaKamenanalysisofthissystem.TheresultsoftheKamenanalysisareinconclusive,duetothehighnumberofsingularitiespresent.Incontrast,modevectoranalysisisabletosuccessfullycharacterizethisproblematicsystem.Figure 3-27 showstherstmodevectorontheleft,andthe2-normofthatsamemodevectorontheright.Thesecondmodevectorisnotshownbecauseitismerelythecomplexconjugateoftherst.Theoscillationpresentinthesystemisreectedbythemodevectorontheleft,whiletheinstabilityofthesystemisshowninthemodevectornormontheright. Figure3-26. PolesfromKamensolutiontoEquation 3 Figure3-27. Modevector1(left)andnormofmodevector1(right)fromsolutiontoEquation 3 57

PAGE 58

CHAPTER4DIRECTIDENTIFICATIONALGORITHMS 4.1DevelopmentofIdenticationAlgorithmsThemethodsdescribedinChapter 3 arederivedfromtheLTVeigenrelationgiveninEquation 3 ,whichrelatestheLTVpolesandeigenvectorstothecoefcientmatrixA(t).TheidenticationalgorithmsproposedinthischapterarederivedfromtheLTVresponseequationgiveninEquation 3 ,whichrelatesthemeasuredstatestothepolesandeigenvectors.ThisisrestatedasEquation 4 ,where^x:R+!Rnisavectorofmeasuredstates,^pi:R+!Caretheunknowntime-varyingpoles,^":R+!Cnaretheunknowntime-varyingeigenvectors,andCi2Careconstants. ^x(t)=NXi=1Ci^"i(t)expZt0^pi()d(4)Existingworkintime-varyingsystemidenticationindicatesthatthecollectionofstatetrajectoriesfrommultipleexperimentalrunsmaybenecessaryforsuccessfulidentication,asdiscussedinSection 2.2 .Eachexperimentalrunmuststartatadifferentinitialstatevector,butthesystemparametersmustrepeatthesamevariationintimeforeachrun.AllowingdatafrommultipleexperimentsmodiesEquation 4 asshowninEquation 4 .Inthisequation,^xk:R+!RnisavectorofmeasuredstatesfromthekthexperimentandCi,k2Cistheconstantfortheitheigenpair,computedusingtheinitialstatevectorfromthekthexperiment.AllothervariablesaredenedasinEquation 4 .Notethatthepolesandeigenvectorsstaythesameforallexperimentalruns. ^xk(t)=NXi=1Ci,k^"i(t)expZt0^pi()d(4) 4.1.1Completevs.IncrementalOptimizationThevaluesofthepolesandeigenvectorsateachtimestepshouldbeconsideredasseparatevariablesintheoptimization,sinceanydependenciesbetweenadjacent 58

PAGE 59

timesarenotknownapriori.Twostrategiestondthesepolesandeigenvectorspresentthemselves: Makeeachpoleandeigenvectorcomponentateachdiscretetimepointanoptimizationvariable,thensolveforeveryoneofthesevariablesatonceinasingle,largeoptimizationproblem.Thisstrategyisreferredtoasthecompleteoptimizationalgorithm. Solveaseparateoptimizationproblemateachdiscretetimestep,usingonlythepoleandeigenvectorcomponentsatthatparticulartimestepasvariablesintheoptimization.Thisstrategyisreferredtoastheincrementaloptimizationalgorithm.ThecompleteoptimizationalgorithmisgivenasAlgorithm 4.1 .Theunderlyingsystemisassumedtohavenstates.Itisfurtherassumedthatvectorsofmeasuredstatesfrommexperimentalrunsareavailable.Timeistreatedasdiscrete,andisrestrictedtosomeniteinterval[t0,...,tend].Theoptimizationmethodusedinthisalgorithmisassumedtobelocal(someformofgradientdescent),sothechoiceofinitialguessheavilyinuencestheaccuracyoftheresultfornon-convexsolutionspaces.Theoptimizationprobleminthisalgorithmrequiresaninitialguessforeverypoleandeigenvectorcomponent,ateverytimestep. Algorithm4.1CompleteOptimization 1. Let^p1(t),...,^pn(t)and^"1(t),...,^"n(t)bethecurrentestimateoftheeigenpairs(polesandeigenvectors)atalltimest2[t0,...,tend]. 2. Initializetheeigenpairestimatestosomechoseninitialguess(~pi(t)and~"i(t))atalltimest.^pi(t)=~pi(t),^"i(t)=~"i(t)8t,8i2[1,...,n] 3. OptimizesomecostfunctionJ(t)tondthecorrectvaluesoftheeigenpairs.p1(t),...,pn(t),"1(t),...,"n(t)=argmin^p1(t),...,^"1(t),...J(^x1(t),...,^xm(t),^p1(t),...,^pn(t),^"1(t),...,^"n(t)) TheincrementaloptimizationalgorithmisgivenasAlgorithm 4.2 .Again,theunderlyingsystemisassumedtohavenstates,mexperimentalrunshavebeenperformed,andtimeisdiscreteandrestrictedtoaniteinterval.Thisalgorithmonly 59

PAGE 60

requiresaninitialguessforthersttimestep,sincetheinitialguessforeachsubsequenttimestepissettothevaluefoundintheprevioustimestep.Ifthetimestepsareclosetogetherandtheeigenpairsolutionisconstrainedtobesmoothintherstderivative,theseguessesarelikelytobeclosetothecorrectvalue. Algorithm4.2IncrementalOptimization 1. Assumethattheeigenpairsattimet=t0(givenasp1(t0),...,pn(t0),"1(t0),...,"n(t0))areknown. 2. Letj=1,wheretjisthejthdiscretetimestepfromthesequence[t0,t1,...,tend].Let^p1(tj),...,^pn(tj)and^"1(tj),...,^"n(tj)bethecurrentestimateoftheeigenpairsattimet=tj. 3. Initializeeigenpairestimatesfortimet=tjtothevaluesfromtheprevioustimestep.^pi(tj)=pi(tj)]TJ /F7 7.97 Tf 6.59 0 Td[(1),8i2[1,...,n]^"i(tj)="i(tj)]TJ /F7 7.97 Tf 6.58 0 Td[(1),8i2[1,...,n] 4. OptimizesomecostfunctionJtondthecorrecteigenpairsfortimet=tj.p1(tj),...,pn(tj),"1(tj),...,"n(tj)=argmin^p1(tj),...,^"1(tj),...J(^x1(tj),...,^xm(tj),^p1(tj),...,^pn(tj),^"1(tj),...,^"n(tj)) 5. Incrementjby1. 6. Iftjtend,GOTO 3 .Otherwise,STOP. Theincrementalmethodissuperiortothecompletealgorithminseveralways.Thesingle,high-dimensionaloptimizationprobleminthecompletealgorithmisfarmorecomputationallyexpensivethanthemanylow-dimensionalproblemsthataresolvedintheincrementalalgorithm.Thehigh-dimensionaloptimizationproblemismorepronetobecomingstrandedinlocaloptimawhicharefarfromfromthecorrectsolution.Finally,thecompletealgorithmrequiresagoodinitialguesstobeknownbeforehand,foreverytimestep.Findingsuchaninitialguessrequiresknowledgeofthetime-varyingdynamicsthroughoutthetimeintervalofinterest.Theincrementalalgorithmonlyrequiresaguessattheinitialtimet0.Ifthesystemisinitiallytime-invariant,noknowledgeofthetime-varyingdynamicsisrequired. 60

PAGE 61

Forthesereasons,theincrementalalgorithmischosenfortheidenticationmethodsproposedinthiswork.Twomethodsthatimplementtheincrementalalgorithmareintroduced:theUnnormalizedEigenvector(UE)method,andtheCanonicalNormalizedEigenvector(CNE)method.BothmethodsfollowAlgorithm 4.2 ,buteachusesadifferentcostfunctionJ.ThecostfunctionofeachmethodistailoredtobeusedwithoneoftheanalysismethodsdescribedinChapter 3 4.1.2UnnormalizedEigenvectorMethodGivenstatevectors^xk(t)frommexperimentalruns,acostfunctionmaybeformulatedbysubtractingtherightsideofEquation 4 fromtheleft,squaringthe2-normoftheresult,thensummingoverallexperimentalruns.ThiscostfunctionisgivenasEquation 4 .Inthisequation,Jueisthetotalcostofthecurrenteigenpairestimates,^xk:R+!Rnisthestatetrajectoryproducedbythekthexperimentalrun,Ci,k2Cistheconstantcalculatedfromtheitheigenpairandtheinitialstatesfromthekthexperimentalrun,^"i:R+!CNistheeigenvectorestimatefortheitheigenpair,and^pi:R+!Cisthepoleestimatefortheitheigenpair. Jue(^x1(t),...,^p1(t),...,^"1(t),...)=mXk=1^xk(t))]TJ /F8 7.97 Tf 18.31 14.95 Td[(nXi=1Ci,k^"i(t)expZt0^pi()d2(4)Thiscostmaybeinterpretedasthedifferencebetweenthemeasuredstatesandthemodalresponseproducedbythecurrentsetofeigenpairs.Minimizingthiscostoverallpossiblevaluesofthepolesandeigenvectorsshouldproduceasetofeigenpairsthatcanaccuratelyreproducethemeasuredstates.Theeigenpairestimatesthatmakethecostfunctionequalzeroreproducethestatetrajectories^xk(t)exactly.However,successinmatchingthetrainingdatadoesnotnecessarilyimplythattheeigenpairscanbeusedtoaccuratelyreproduceanyotherstatetrajectory.Annthordersystemrequirestrainingagainstnstatetrajectorieswithlinearly-independentinitialstatevectorstoobtainasetofeigenpairsthatcanreproduce 61

PAGE 62

anystatetrajectory,asdescribedbyVerhaegen[ 11 ]andLiu[ 1 ].Aneigenpairestimatewhichcansuccessfullyreproduceanyofasystem'sstatetrajectoriesisavalideigenpairsolutionofthesystem.TheUEmethodisperformedbyfollowingtheincrementalalgorithmgiveninAlgorithm 4.2 withEquation 4 chosenasthecostfunction.Thiscostfunctionisunder-denedwith)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(n2+nunknownsandonlyn2scalarequations,sotherearemultiplesetsofeigenpairvaluesthatwillsatisfyit.ThissamebehaviorwasseeninSection 3.2.2 ,whereitwasnotedthattheLTVeigenrelationEquation 3 hadanadditionalunknownvariableineacheigenpair.Sincethecostfunctionhasanon-uniquesolution,anoptimizationmethodthatusesthiscostfunctionisnotguaranteedtoproduceanyparticularpartitioningofinformationbetweenthepolesandtheeigenvectors.ThismeansthatthepolesofaUEsolutioncannotbeexaminedinisolationtodeterminethestabilityoroscillatorycharacteristicsofasystem,asdemonstratedinSections 3.2.2 through 3.2.4 .Setsofeigenpairsthatsatisfythiscostfunctionarethereforebestanalyzedthroughtheuseofmodevectors,whicharedescribedinSection 3.3 4.1.3CanonicalNormalizedEigenvectorMethodUEsolutionscanbeusedtoperformKamenanalysisviaequivalenteigenpairtransformationsiftheunderlyingsystemisincanonicalform,asdescribedinSection 3.4.3 .ThisprocedureforKamenanalysisisunfortunatelynotideal,sincenoiseisinsertedintothesystembythenumericalderivativecomputationnecessarytoperformthetransformation.ToavoidusinganeigenpairtransformationtodoKamenanalysis,thecostfunctioncanbemodiedsuchthatthealgorithmoptimizesfortheKamensolutiondirectly.RecallfromSection 3.4.1 thetwoassumptionsusedtoderiveKamen'sresultfromtheLTVeigenrelation: Therstcomponentofeveryeigenvectorisassumedtobeexactlyequalto1foralltime. 62

PAGE 63

Thesystemisassumedtobegivenincanonicalform,asdescribedinEquation 2 andEquation 2 .ToreproducetheKamenresultsfromtheincrementalupdatealgorithm,thecostfunctioninEquation 4 mustbemodiedtoincludetheseassumptions.Tomeettherstassumption,theeigenvectorestimates^"i(t)willberestrictedsuchthattherstcomponentofeacheigenvectorisxedat1.Eigenvectorsthathavebeenrestrictedinthisfashionwillbereferredtoasnormalizedeigenvectors.Thecomponentsofthenormalizedeigenvectors "i(t)aredenotedasshowninEquation 4 "i(t)=2666666641 ei2(t)... ein(t)377777775,8i2[1,...,n](4)Thesecondassumption,thatthecoefcientmatrixisincanonicalform,canthenbeusedtoeliminatethepoleestimateasanoptimizationvariable.Recallthatifthecoefcientmatrixisincanonicalform,thestatevectoriscomposedonlyofderivativesofasinglestate.ThisformforthestatevectorisshowninEquation 4 ,where^xk:R+!Rnisthestatemeasurementvectorfromthekthexperimentalrun,and^xs,k:R+!Rissomescalardifferentiablefunction. ^xk(t)=266666664^xs,k(t)_^xs,k(t)...^xs,k(n)(t)377777775(4)TheequationsforthersttwocomponentsofthestatevectorinEquation 4 (theLTVresponseequation)arerewrittenassumingthattheeigenvectorsarenormalizedgiveninEquation 4 .TheresultisgivenasEquation 4 .Inthisequation,^xkj:R+!Rdenotesthejthcomponentofthestatemeasurementvectorfromthekthexperimental 63

PAGE 64

run,Ci,k2Cistheconstantcalculatedfromtheitheigenpairandtheinitialstatesfromthekthexperimentalrun, "i:R+!CNisthenormalizedeigenvectorestimatefortheitheigenpair,and^pi:R+!Cisthepoleestimatefortheitheigenpair.^xk1(t)=^xs,k(t)=NXi=1Ci,kexpZt0^pi()d^xk2(t)=_^xs,k(t)=NXi=1Ci,k ei2(t)expZt0^pi()d (4)Usingthefactthat^xk2(t)issimplythederivativeof^xk1(t)withrespecttotime,analternativeequationfor^xk2(t)isshowninEquation 4 .^xk2(t)=d dt NXi=1Ci,kexpZt0pi()d!=NXi=1Ci,kpi(t)expZt0pi()d (4)BycomparingEquation 4 withEquation 4 ,itisobservedthat^pi(t)= ei2(t),8i2[1,...,n].Makingthisreplacementfor^pi(t),thenalformforthecostfunctionforCNEoptimizationisreachedinEquation 4 .Inthisequation,Jcneisthetotalcostofthecurrentsetofeigenpairvalues,^xk:R+!Rnisthestatetrajectoryproducedbythekthexperimentalrun,Ci,k2Cistheconstantcalculatedfromtheitheigenpairandthekthexperimentalrun,and "i(t):R+!CNisthenormalizedeigenvectorestimateoftheitheigenpair,asdenedinEquation 4 .Thequantity ei2:R+!Cissimplythesecondcomponentofthenormalizedeigenvectorestimatefromtheitheigenpair. Jcne(^x1(t),...,^"1(t),...)=nXk=1^xk(t))]TJ /F8 7.97 Tf 18.31 14.94 Td[(nXi=1Ci,k "i(t)expZt0 ei2()d2(4)ToperformCNEoptimization,thecostfunctiongivenasEquation 4 issimplyinsertedintoAlgorithm 4.2 .Notethatthiscostfunctionisfully-dened,withnvariablesandnunknowns.Solutionsthatareproducedbythiscostfunctionwillhaveaspecicpartitioningofinformationbetweenthepolesandtheeigenvectors.Becauseofthe 64

PAGE 65

assumptionschosentomaketheproblemfully-dened,theresultingsolutionwillequaltheKamensolutionofthesystem.ThissolutioncanthenbeinterpretedaccordingtoKamenanalysismethods,asdescribedinSection 3.4 4.2DemonstrationofPracticalUsageIssuesInthissection,thebehavioroftheunnormalizedeigenvectorandcanonicalnormalizedeigenvectoroptimizationmethodsareexamined.Unlessotherwisestated,thedemonstrationsallusethestable,canonicalsecond-ordersystemwithcomplex-conjugatepolesdenedinEquation 4 .A(t)=26401)]TJ /F4 11.955 Tf 9.3 0 Td[(0.20.5)]TJ /F3 11.955 Tf 11.96 0 Td[(t375 (4)Intheexamplesfoundinthissection,thestatevectorsusedasinputintoeachidenticationmethodarefoundbysolvingthesystemwithanumericalODEsolver.Differentexperimentalrunsaresimulatedbysolvingthesystemwithdifferentsetsofinitialconditions.Thestatevectorsdonothaveadditionalnoiseadded,exceptinSection 4.2.3 .Severaltypesofchecksareperformedineachexampletoshowtheaccuracy(orinaccuracy)oftheachievedsolution.Notethatthesechecksarenotrequiredinpracticewhenperforminganidenticationusingrealexperimentaldata.Instead,thesechecksaresimplyprovidedforconcreteevidencethatthealgorithmsworkcorrectly.First,theidenticationsolutionsarecomparedtoanexactsolutionfoundbyusingKamen'sequationforatwo-statesystem(Equation 3 )withtheknowncoefcientmatrixA(t).TheCNEpolesshouldbeidenticaltotheKamenpoles,sotheyareplottedagainstoneanothertocheckforcorrectness.ThemodevectornormsoftheUEsolutionandthoseoftheKamensolutionshouldalsobeidenticalwhenplottedagainstoneanother,asdescribedinSection 3.4.3 65

PAGE 66

Second,aneigenpairvaliditycheckisperformed.ThischeckshowswhetherornotagivensolutionsatisesEquation 3 (theLTVeigenrelation),repeatedhereforconvenience.A(t)pi(t))]TJ /F12 11.955 Tf 11.95 0 Td[("i(t)pi(t)=_"i(t),8i2[1,...,n]TheeigenpairvaliditycheckproceedsbyrstinsertingtheestimatedpolesandeigenvectorsintotheleftsideofEquation 3 .Thisevaluatestoapredictedvalueforthederivativeoftheeigenvector.IfthegivenpoleandeigenvectorestimatessatisfytheLTVeigenrelation,thispredictedvalueshouldmatchthenumericalderivativeoftheeigenvectorcomponentswhenplottedagainstoneanother.Thenoiseintroducedbythenumericalderivativemustbetakenintoaccountwheninterpretingthischeck.Third,themodalresponsesoftheeigenpairestimatesarecomputedbyinsertingtheestimatesintoEquation 3 .TheactualstatetrajectoriesofthesystemarecomputedbyusinganODEsolvertosolveEquation 2 .Theactualstatetrajectoriesshouldmatchthemodalresponsewhentheyareplottedagainsteachother.Itisimportanttonotethatcorrectlymatchingthestatetrajectoriesusedasinputintotheidenticationalgorithmdoesnotensurethatthesolutioniscorrect. 4.2.1EffectofNumberofDatasetsSection 4.1.2 statedthattoobtainasetofeigenpairsthatcanreproduceeverypossiblestatetrajectoryofageneraln-ordersystem,theoptimizationmethodshouldusestatetrajectoriesfromndifferentexperimentalruns.Todemonstratethisbehavior,bothoptimizationmethodsareperformedonthreedifferentsetsoftrainingdata. Run#1:Trainingsetcontainsonestatetrajectory,whichbeginsatx(0)=[57]. Run#2:Trainingsetcontainsonestatetrajectory,whichbeginsatx(0)=[75]. Run#3:Trainingsetcontainstwostatetrajectories,onestartingatx(0)=[57]andtheotherstartingatx(0)=[75]. CNEmethod.Theseoptimizationrunsarerstconductedusingthecanonicalnormalizedeigenvectormethod.IftheeigenpairsproducedbytheCNEmethodare 66

PAGE 67

correct,thepolesineacheigenpairshouldmatchtheKamenpolesforthesystemfairlyclosely.ValidLTVeigenpairsmustsatisfytheLTVeigenrelationgiveninEquation 3 ,andthemodalresponsesproducedfromtheeigenpairsviaEquation 3 mustmatchthestatetrajectoriesofthesystem.Thisshouldbetrueevenforstatetrajectoriesthatwerenotincludedinthetrainingset. Figure4-1. PolesfromtheKamenandCNEsolutionsofRun#1(left),andvaliditycheckofeigenpair1fromtheCNEsolutionofRun#1(right) CNE-Run#1:TheCNEpolesareplottedagainsttheKamenpolesontheleftsideofFigure 4-1 .TheCNEpolesindicateanunstableresultovertheentiretimeinterval,whereastheKamenpolesindicateastableresult.TheresultsofaneigenpairvaliditycheckarethenshownontherightsideofFigure 4-1 (onlythersteigenpairischecked,sincethesecondisjustthecomplexconjugateoftherst).Notethatdeijreferstothenumericalderivativeofthejthcomponentoftheitheigenvector,andmdeijreferstothevalueofthesamecomponentpredictedbytheright-handsideoftheLTVeigenrelation.ThefactthatthesetwovaluesarenotthesameindicatesthattheCNEsolutiondoesnotsatisfytheLTVeigenrelation(Equation 3 ).Figure 4-2 showsthestatetrajectoriesplottedagainstthemodalresponseoftheCNEsolution,fortwodifferentinitialstatevectors.ThemodalresponseforRun#1(indicatedinthelegendbyxhat1andxhat2)matchesthestatetrajectoryexactlyforx(0)=[57],sincethisstatetrajectorywasusedassolesourceoftrainingdatafor 67

PAGE 68

Figure4-2. StatetrajectoriesofEquation 4 versusthemodalresponseoftheCNEsolutionofRun#1,forx(0)=[57](left)andx(0)=[75](right) thisrun.Themodalresponseforx(0)=[75]doesnotreproducethecorrespondingstatetrajectory,indicatingthattheCNEsolutionforthisrunisnotabletoreproducestatetrajectoriesoutsideitsowntrainingset. Figure4-3. PolesfromtheKamenandCNEsolutionsofRun#2(left),andvaliditycheckofeigenpair1fromtheCNEsolutionofRun#2(right) CNE-Run#2:TheresultsforRun#2(singledatasetwithstatetrajectorystartingatx(0)=[75])areverysimilartothosefromRun#1.TheleftsideofFigure 4-3 comparestheKamenpolesforthesystemagainsttheCNEresultsfromRun#2.Thetwosetsofpolesareagainquitedifferent,andindicatedifferentstabilityresults.TherightsideofFigure 4-3 displaystheresultsofaneigenpairvaliditycheckforthersteigenpairofthe 68

PAGE 69

CNEsolution.TheeigenpairvaliditycheckshowsthattheCNEsolutionforRun#2doesnotsatisfytheLTVeigenrelation,Equation 3 .Figure 4-4 showsthesametwostatetrajectoriesasbefore,plottedagainstthemodalresponsefromtheCNEsolutionofRun#2.Thestatetrajectorythatwasusedastrainingdata(x(0)=[75],rightsideofgure)isreproducedalmostexactly.Theotherstatetrajectory(x(0)=[57],leftsideofgure)isnotcorrectlyreproduced. Figure4-4. StatetrajectoriesofEquation 4 versusthemodalresponseoftheCNEsolutionofRun#2,forx(0)=[57](left)andx(0)=[75](right) CNE-Run#3:TheresultsfromRun#3(twodatasets,eachstartingatadifferentpoint)areverydifferentfromtheearlierruns,eventhoughthetrainingdataconsistsofthesametwodatasetsthatwereusedseparatelyinRuns#1and#2.TheleftsideofFigure 4-5 showsthattheCNEpolesforRun#3arenearlyidenticaltotheKamenpoles.TheeigenpairvaliditycheckshownontherightsideofFigure 4-5 indicatesthattheCNEsolutionsatisestheLTVeigenrelation,Equation 3 .Figure 4-6 showsthatthesolutionproducedbyRun#3oftheCNEmethodcorrectlyreproducesbothofthestatetrajectoriesusedfortraining.Acomparisonbetweenthemodalresponseandthestatetrajectoryforx(0)=[57]isshownontheleftside,andthesamecomparisonismadefortheinitialstatevectorx(0)=[75]inthecenterplot.Toverifythatthissolutioncanreproducestatetrajectoriesdifferentthanthetrainingdata,athirdcomparisonthatusesx(0)=[31]isshownontherightsideofthegure. 69

PAGE 70

Figure4-5. PolesfromtheKamenandCNEsolutionsofRun#3(left),andvaliditycheckofeigenpair1fromtheCNEsolutionofRun#3(right) Figure4-6. StatetrajectoriesofEquation 4 versusthemodalresponseoftheCNEsolutionofRun#3,forx(0)=[57](left),x(0)=[75](center)andx(0)=[31](right) UEmethod.Theunnormalizedeigenvectormethodisnotguaranteedtoproduceanyparticularpartitioningbetweenthepolesandtheeigenvectors(Section 4.1.2 ),soitspolescannotbedirectlycomparedtotheKamenpoles.Recallthatthemodevectorsoftwovalideigenpairsolutionstothesamesystemshouldbeequal,regardlessofhoweachsolutionpartitionedthedatainsidetheeigenpairs(Section 3.3.3 ).IftheeigenpairsproducedbytheUEmethodarecorrect,theirmodevectorsshouldthereforebeidenticaltothemodevectorsoftheKameneigenpairs. UE-Run#1:ThenormofthemodevectorsoftheUEandKamensolutionsforRun#1(onestatetrajectory,startingatx(0)=[57])arecomparedtoeachotherontheleft 70

PAGE 71

Figure4-7. Normofmodevector1fromtheKamenandUEsolutionsofRun#1(left),andvaliditycheckofeigenpair1fromtheUEsolutionofRun#1(right) sideofFigure 4-7 .ThemodevectornormoftheKamensolutiondecaystowardszero,signifyingstability.ThemodevectornormoftheUEsolutioncontradictsthisresult.TheresultsofaneigenpairvaliditycheckareshownontherightsideofFigure 4-7 .Notethatdeijreferstothenumericalderivativeofthejthcomponentoftheitheigenvector,andmdeijreferstothevalueofthesamecomponentpredictedbytheright-handsideofEquation 3 .ThefactthatthesetwovaluesaresubstantiallydifferentintheplotindicatesthatthiseigenpairsolutiondoesnotsatisfytheLTVeigenrelation. Figure4-8. StatetrajectoriesofEquation 4 versusthemodalresponseoftheUEsolutionofRun#1,forx(0)=[57](left)andx(0)=[75](right) 71

PAGE 72

Figure 4-8 showsthestatetrajectoriesfortwodifferentinitialstatevectors,plottedagainstthemodalresponsefromtheUEsolutionforthesameinitialstates.Asexpected,themodalresponseforRun#1matchesthestatetrajectoryforx(0)=[57](thestatetrajectoryusedastrainingdataforRun#1).Themodalresponseforx(0)=[75]doesnotmatchthestatetrajectory,indicatingthattheUEsolutionforthisrundoesnotsatisfytheLTVresponseequation,giveninEquation 3 Figure4-9. Normofmodevector1fromtheKamenandUEsolutionsofRun#2(left),andvaliditycheckofeigenpair1fromtheUEsolutionofRun#2(right) UE-Run#2:TheresultsforRun#2(singlestatetrajectory,startingatx(0)=[75])mirrorthosefromRun#1.TheleftsideofFigure 4-9 comparestherstmodevectornormsoftheKamenandUEsolutionsforthisrun.TheKamenmodevectornormsconvergedownwards(indicatingstability),whilethemodevectornormsfromtheUEsolutiondiverge.TherightsideofFigure 4-9 showstheresultsofaneigenpairvaliditycheckforthersteigenpairoftheUEsolution.ThisplotshowsthattheUEsolutionforRun#2doesnotsatisfytheLTVeigenrelationgiveninEquation 3 .Figure 4-10 showsthesametwostatetrajectories,plottedagainstthemodalresponsefromtheUEsolutionofRun#2.Thestatetrajectorythatwasusedfortraining(x(0)=[75],rightsideofgure)ismatchedalmostexactly,whiletheotherstatetrajectory(x(0)=[57],leftsideofgure)isnotcorrectlyreproduced.TheseresultsmirrorthoseobtainedfromRun#1. 72

PAGE 73

Figure4-10. StatetrajectoriesofEquation 4 versusthemodalresponseoftheUEsolutionofRun#2,forx(0)=[57](left)andx(0)=[75](right) UE-Run#3:AsintheCNEresults,thesolutionproducedfromRun#3(twodatasets,eachstartingatadifferentpoint)isfundamentallydifferentfromtheearlierruns.ThisoccursinspiteofthefactthatthetrainingdataconsistsofthesametwodatasetsthatwereusedseparatelyinRuns#1and#2.TheleftsideofFigure 4-11 showsthatthenormoftherstUEmodevectorisnearlyidenticaltotheKamenmodevectornorm.TheeigenpairvaliditycheckshownontherightsideofFigure 4-11 indicatesthattheUEsolutionsatisestheLTVeigenrelationinEquation 3 Figure4-11. Normofmodevector1fromtheKamenandUEsolutionsofRun#3(left),andvaliditycheckofeigenpair1fromtheUEsolutionofRun#3(right) 73

PAGE 74

Figure 4-6 showsthatthesolutionproducedbyRun#3oftheUEmethodcorrectlyreproducesbothofthestatetrajectoriesusedfortraining.Acomparisonbetweenthemodalresponseandthestatetrajectoryforx(0)=[57]isshownontheleftside,andthesamecomparisonismadefortheinitialstatevectorx(0)=[75]inthecenterplot.Toverifythatthissolutioncanreproducestatetrajectoriesdifferentthanthetrainingdata,athirdcomparisonthatusesx(0)=[31]isshownontherightsideofthegure.Inallthreecases,theUEsolutionisabletoreproducethestatetrajectoriesnearlyperfectly. Figure4-12. StatetrajectoriesofEquation 4 versusthemodalresponseoftheUEsolutionofRun#3,forx(0)=[57](left),x(0)=[75](center),andx(0)=[31](right) 4.2.2EffectofLinearIndependenceofInitialStatesSection 4.1.2 alsostatedthattoobtainasetofeigenpairsthatcanreproduceeverypossiblestatetrajectoryforannthordersystem,theinitialconditionsofthenstatetrajectoriesusedmustbelinearlyindependentfromoneanother.Thissectiondemonstrateshowtheoptimizationsolutionsdegradewhentheinitialvaluesofthetrajectoriesusedastrainingdataapproachlineardependence.Bothoptimizationmethodsarerunonthreedifferenttrainingsets,eachcomposedoftwostatetrajectories. Run#4:Trainedusingtwodatasets,x(0)=[12.00]andx(0)=[24.30]. Run#5:Trainedusingtwodatasets,x(0)=[12.00]andx(0)=[24.03]. Run#6:Trainedusingtwodatasets,x(0)=[12.00]andx(0)=[24.00]. 74

PAGE 75

ThesestatetrajectoriesaregeneratedbysolvingthesystemgiveninEquation 4 numerically,usingtheappropriateinitialstatevectorx(0)ineachcase.Thersttwopairsofinitialstatevectors(Run#4andRun#5)arechosentobeclosetolineardependence,withthesecondpair(Run#5)beingcloserthantherst.Thethirdpair(Run#6)ischosentobeexactlylinearlydependent. CNEmethod.TheresultsfromtherunsusingtheCNEmethodarepresentedrst.InFigure 4-13 ,theKamenpolesareplottedalongsidethepolesgeneratedfromtheCNEoptimization.AcorrectCNEsolutionshouldproducepolesthatareveryclosetotheKamenpoles.Thenoiseinthepolesolutionisseentoincreasefromtheleftplottothecenterplot,astheinitialstatevectorsofthetrainingsetapproachlineardependence.Whentheinitialstatevectorsreachlinearindependenceintherightplot,theCNEpolesstopfollowingtheKamensolution.AcomparisonofthisplotandFigure 4-1 showsthatthepolesfromthethirdrunareverysimilartothoseproducedfromatrainingsetcontainingonlyonestatetrajectory. Figure4-13. PolesfromKamenandCNEsolutionsofRun#4(left),Run#5(center),andRun#6(right) Figure 4-14 plotsacomparisonbetweenastatetrajectoryproducedfromtheinitialstatevectorx(0)=[57]andthecorrespondingmodalresponseoftheCNEsolution,foreachrun.ForRun#4(left),themodalresponsematchesthestatetrajectoryveryclosely.ForRun#5(center),themodalresponsestillmatchesthestatetrajectoryfairlyclosely.Aslightamountoferrorisvisibleuponcloseexamination.Finally,theCNE 75

PAGE 76

solutionforRun#6(right)isunabletoreproducethestatetrajectory.TheresultsforRun#6showninbothFigure 4-13 andFigure 4-14 indicatethattheCNEalgorithmcouldnotproduceavalidsolutionwhenthetwostatetrajectoriesinitstrainingsethadlinearlydependentinitialvalues. Figure4-14. StatetrajectoryofEquation 4 versusthemodalresponseoftheCNEsolutionofRun#4(left),Run#5(center),andRun#6(right),forx(0)=[57] UEmethod.TheresultsfortheUEmethodarepresentedinasimilarfashion.Figure 4-15 plotsthenormsoftherstmodevectoroftheKamensolutionandtheUEsolutionforeachrun.Becausetheeigenpairsofthissystemarecomplexconjugate,thesecondmodevectornormisredundantandisnotpicturedhere.IftheUEsolutioniscorrect,itsmodevectorsshouldbeequaltothosefromtheKamenmethod.Theleftandcenterplotsshowthatthenoiseincreasesdrasticallyastheinitialstatevectorsapproachlineardependence(Run#4andRun#5).Whentheinitialstatevectorsreachexactlineardependence(Run#6,rightmostplot),themodevectornormsfromtheKamenandtheUEsolutionsarecompletelydifferent.Notethatthisparticularplotisverysimilartotheresultachievedwhenusingonlyonestatetrajectoryinthetrainingset(seeFigure 4-7 ).Figure 4-16 plotsacomparisonbetweenastatetrajectoryproducedfromtheinitialstatevectorx(0)=[57]andthecorrespondingmodalresponseoftheUEsolution,foreachrun.TheleftmostplotshowsthattheUEsolutionforRun#4successfullyreproducesthetrajectory.ThecenterplotshowsthatthesolutionforRun#5isalso 76

PAGE 77

Figure4-15. Normofmodevector1fromKamenandUEsolutionsofRun#4(left),Run#5(center),andRun#6(right) abletoreproducethetrajectory,thoughnotwithoutintroducingsomenoise.However,theUEsolutionforRun#6(right)isunabletoreproducethestatetrajectory.TheseresultsforRun#6mirrorthoseobtainedfortheCNEsolution.TheUEalgorithmisalsounabletoproduceavalidsolutionwhenthetrainingsetcontainstrajectorieswithlinearly-dependentinitialstatevectors.Inaddition,proximitytolineardependencewasenoughtointroducenoiseintothesolution,forboththeCNEandtheUEalgorithms. Figure4-16. StatetrajectoryofEquation 4 versusthemodalresponseoftheUEsolutionofRun#4(left),Run#5(center),andRun#6(right),forx(0)=[57] 4.2.3EffectofStateMeasurementNoiseThissectioninvestigatestheeffectsofaddingnoisetothestatetrajectoriesusedasinputintotheoptimizationmethods.Thisismeanttosimulatetheeffectsofreal-worldmeasurementerror.Forthisdemonstration,bothoptimizationmethodsarerunonatrainingsetconsistingoftwostatetrajectories,bothgeneratedfromthesystemgiven 77

PAGE 78

inEquation 4 .Theinitialstatevectorx(0)=[10]wasusedforthersttrajectory,andx(0)=[01]wasusedforthesecond.Zero-meannoiseisthenaddedtothestatetrajectoriesateverytimestep(notincludingtheinitialvalues).Thestatevaluesoftheresultingnoisysignalareuniformlydistributedinsuchawaythattheyliewithin15%ofthetruevalue,oneitherside.Figure 4-17 showsthesetwonoisystatetrajectories,withx(0)=[10]ontheleftandx(0)=[01]ontheright. Figure4-17. StatetrajectoriesofEquation 4 with(15%uniformerroradded),forx(0)=[10](left)andx(0)=[01](right) CNEmethod.TheresultsfromtheCNEmethodaredescribedrst.ThenoisystatetrajectoriesareinputdirectlyintotheCNEoptimizationmethod,withoutperforminglteringofanykind.Figure 4-18 comparestheKamenpolesproduceddirectlyfromthecoefcientmatrixA(t)withthoseproducedbytheCNEmethodfromthenoisystatetrajectories.WhiletheCNEpolesarenoisy,thisgureshowsthatbothpolesfollowthegeneralpathoftheKamenpoles.ThemodalresponsesoftheCNEmethodforbothinitialstatevectorsareshowninFigure 4-19 .Forstate2,theCNEmethodexactlyreproducesthenoisystatetrajectory.However,theCNEmethodproducesacleanmodalresponsesignalforstate1.Themethodappearstobeimplicitlylteringtheinputstatetrajectorysignalforthisstate.Thereasonsforthisbehaviorarecurrentlyunknown,buttheymayberelatedtothe 78

PAGE 79

Figure4-18. Pole1(left)andpole2(right)fromtheKamensolutionandthenoisyCNEsolutionofEquation 4 factthattheCNEcostfunctiongiveninEquation 4 isover-dened(thereisonemoreequationthanunknownforeacheigenpair). Figure4-19. ModalresponseofthenoisyCNEsolutiontoEquation 4 ,forx(0)=[10](left)andx(0)=[01](right) UEmethod.TheresultsfortheUEmethodarepresentedhere.Asbefore,thenoisystatetrajectoriesareinputintotheoptimizationmethodwithoutanyltering.Figure 4-20 comparesthenormoftherstmodevectoroftheKamensolutionwiththenormoftherstmodevectoroftheUEsolution.Becausethetwoeigenpairsinthissolutionarecomplexconjugate,thesecondmodevectornormisidenticaltotherst. 79

PAGE 80

ThegureshowsthatwhiletheUEmodevectornormisnoisy,itfollowsthesamepathastheKamenmodevectornorm. Figure4-20. Normofmodevector1fromtheKamensolutionandthenoisyUEsolutionofEquation 4 Figure4-21. ModalresponseofthenoisyUEsolutiontoEquation 4 ,forx(0)=[10](left)andx(0)=[01](right) ThemodalresponsesoftheUEmethodforbothinitialstatevectorsareshowninFigure 4-21 .Forbothstates,theUEmethodexactlyreproducesthenoisystatetrajectorysignals.NoimplicitlteringeffectsarenotedfortheUEmethod.ThislimitsthepossiblecausesofthisimplicitlteringtothedifferencesbetweentheUEandCNEalgorithms.Thesetwoalgorithmshaveonlytwofundamentaldifferences.First,theCNEmethod'scostfunctionisover-denedandtheUEmethod'scostfunctionis 80

PAGE 81

under-dened.Second,theCNEmethodassumesthattheeigenvectorsarenormalizedsuchthattheirrstcomponentsareequaltooneforalltime.ThismeansthatintheCNEmethod,onlythepolescontributetotheresponseoftherststate.IntheUEmethod,boththepolesandtheeigenvectorscontributetotheresponseofallthestates.Itisalmostcertainthattheimplicitlteringoftherststateiscausedbyoneorbothofthesedifferences,butnoconclusionisdrawninthepresentwork. 4.3Example:AircraftModelwithVariableWingSweepTheunnormalizedeigenvector(UE)optimizationalgorithmisusedtoanalyzethedynamicsofamorphing-wingaircraftsimulationtakenfromGrant[ 20 ].Inthissimulation,thewingsaresweptfrom0degrees(perpendiculartobody)to30degreesrearward,linearlyintwosecondsofmorphing.Nocontrolinputisappliedtothesystem.Thelateralandlongitudinaldynamicsareassumedtobedecoupled,andaretreatedasseparate4x4systems.TheanalysisisperformedbyrstsolvingthedynamicsequationsusinganODEsolver,toobtainsimulatedstatemeasurementsforalltime.Theunnormalizedeigenvectoroptimizationmethodisthenperformedusingthesesimulatedstatemeasurementstogeneratepoleandeigenvectorestimates.TheinitialvaluesoftheeigenpairsarechosentobeequaltotheLTIpolesandeigenvectorsoftheaircraftmodelinitsinitial(unmorphed)conguration.Theresultingeigenpairs,modevectors,andmodevectornormsarethenpresentedforboththelateralandlongitudinaldynamics.Anexampleofthemodel'sstateresponseforthedescribedmorphingtrajectoryisprovidedinFigure 4-22 .Thesetwographsshowthestatetrajectoriesforbothlateralandlongitudinaldynamics,withx(0)=[1000]Tastheinitialconditionusedinboth.Inthesimulation,threeadditionalsetsofmeasurementswerecollectedforbothlateralandlongitudinaldynamics.Eachadditionalstatetrajectorywasgeneratedusingadifferent,linearlyindependentinitialstatevector.TheseadditionalinitialstatevectorswerechosentobetheotherstandardbasisvectorsforR4. 81

PAGE 82

Figure4-22. Statetrajectoriesforlateral(left)andlongitudinal(right)dynamicsofaircraftmodel,forx(0)=[1000] Forthelateraldynamics,x1(t)isthesidewaysvelocity,x2(t)istherollrate,x3(t)istheyawrate,andx4(t)istherollangle.Forthelongitudinaldynamics,x1(t)istheforwardvelocity,x2(t)isthedownwardvelocity,x3(t)isthepitchrate,andx4(t)isthepitchangle. 4.3.1LateralDynamics LTVeigenpairs.TheeigenvectorsfoundbytheUEmethodforthelateraldynamicsareshowninFigures 4-23 and 4-24 .ThepolesareshowninFigure 4-25 .TheUEmethodseeksrsttomeetitsobjectivesbymodifyingthevaluesoftheeigenvectors.Inthisparticularexample,itsucceedssowellthatthepolesremainvirtuallyunchangedfromtheirinitialvalues.However,recallthattheUEmethoddoesnotensureaparticularpartitioningofinformationbetweenthepolesandeigenvectors.ThefactthatthepartitioningchosenbytheUEmethodonthisrunledtopolesthatlookLTIcannotbeusedtodrawconclusionsaboutthesystem.ItisobservedontheleftsideofFigure 4-24 that,afterabrieftransitionalperiod,thederivativeofeachofeigenvector3'scomponentsdropstozero.Thebehaviorofthecorrespondingmodeshouldthereforeapproachthemodalbehaviorobservedwhenthetime-varyingcharacteristicsofthesystemarenottakenintoaccount. 82

PAGE 83

Figure4-23. Eigenvectors1(left)and2(right)fromUEsolutionoflateraldynamics Figure4-24. Eigenvectors3(left)and4(right)fromUEsolutionoflateraldynamics Figure4-25. PolesfromUEsolutionoflateraldynamics 83

PAGE 84

Modevectors.ThemodevectorsaregraphedinFigures 4-26 and 4-27 .Notethattheoscillatorybehaviorintheimaginarypartsofmodevectors1and2(showninFigure 4-26 )differsamongcomponentsofthesamemodevector.ThisillustratesanimportantdifferencebetweenLTIandLTVaircraftdynamics.Inlineartime-invariantanalysis,modesarescalarquantities.InLTVsystems,thetime-varyingnatureoftheeigenvectorsrequiresthattheybeincludedinthemode(seeSections 3.2 and 3.3.3 ).Sinceeigenvectorsarenotscalarquantities,themodesmustalsobedenedasavectorquantity. Figure4-26. Modevectors1(left)and2(right)fromUEsolutionoflateraldynamics Figure4-27. Modevectors3(left)and4(right)fromUEsolutionoflateraldynamics Thefrequencycharacteristicsofeachcomponentofthemodemaydifferfromtheothercomponents,sinceLTVmodesarevectorquantities.Thesedifferencesare 84

PAGE 85

notsimplyshiftsinphaseamongcomponents.IntheimaginarypartoftherightsideofFigure 4-26 ,theperiodofthefourthcomponentofthemodevectorissignicantlydifferentthanthatofanyothercomponent.Thismeansthatthefrequencyofanyoscillatorybehaviormayinfactvaryamongcomponentsofthesamemodevector.Themannerinwhichthisisbestinterpretedforanaircraftmodelisanopenquestion. Modevectornorms.Recallthatifasystemdoesnotvarystronglywithtime,itcanbeanalyzedusingthetraditionaleigenpairequation(A")]TJ /F3 11.955 Tf 12.7 0 Td[(p"=0)evaluatedateachdiscretetimestep(seeSection 3.1 ).TheeigenpairsthatresultareknownasFrozen-TimeEigenpairs(FTE's).Themodesoftheseeigenpairscanthenbecomputedateachtimestepas((t)=exp[p(t)t]).ThemodevectornormreducestotheabsolutevalueoftheFTEmodeifthesystem'seigenvectorsdonotchangewithtime(seeEquation 3 ).IfthemodevectornormsarethesameastheFTEmodesforsomesystem,thatsystemdoesnotrequireuseoftheLTVanalysismethodsdescribedinthiswork.Time-varyinganalysisisnecessarytogetanaccuratepictureofthemodalbehaviorwhenthemodevectornormsdiffersignicantlyfromtheabsolutevalueoftheFTEmodes. Figure4-28. Modevectornorms1(left)and2(right)fromUEsolutionoflateraldynamics Themodevectornormsoftheaircraftmodelaregraphedalongsidetheabsolutevalueofthefrozen-timeeigenpairmodesinFigures 4-28 and 4-29 .Therstthree 85

PAGE 86

Figure4-29. Modevectornorms3(left)and4(right)fromUEsolutionoflateraldynamics modevectornormsappeartoconvergetowardszeroovertheperiodofthemorphing,indicatingthatthismorphingtrajectoryissomewhatstable.ThismatchesthebehavioroftherstthreeFTEmodes.Incontrast,thefourthFTEmodeslowlydecayswhilethefourthmodevectornormgrowsoverthecourseofthemorphing.ThisindicatesthattheFTEmodesarenotsufcientforanalyzingthesystem,sincetheFTEmodeshowsstabilitywhilethemodevectornormindicatessomelevelofinstabilityduringthemorphing. Comparisontotraditionalanalysis.Somecorrelationwiththemodesdescribedintraditionalaircraftanalysisisdesirabletointerpretthemeaningofthesemodevectors.ThelateralmodesforatypicalaircraftintraditionalLTIanalysisaredescribedas[ 21 ]: DutchRoll-complex(oscillatory)response,weakdamping RollSubsidence-real(non-oscillatory)response,strongdamping Spiral-real(non-oscillatory)response,weakdampingThemodevectorsproducedbytheLTVanalysisarecomparabletothesemodesfromLTIaircraftanalysis.Modevectors1and2areoscillatory(Figure 4-26 )andonlymoderatelydamped(Figure 4-28 ).Thiscomplex-conjugatepairofmodevectorscorrespondstotheDutchrollmode.Modevectors3and4arenon-oscillatory(Figure 86

PAGE 87

4-27 ).TheleftsideofFigure 4-29 showsthatmodevector3isheavilydamped,whichcorrespondstotherollsubsidencemode.TherightsideofFigure 4-29 showsthatmodevector4isdivergentinmodevectoranalysis,butverylightlydampedinFTEanalysis.TheFTEanalysisidentiesmodevector4asthespiralmode.TheFTEanalysiscannotaccuratelycharacterizethemodalcharacteristicsofthisparticularmorphingaircraftmodel,thoughitdoeshelpidentifytheLTIaircraftmoderelatedtoeachLTVmodevector.ThemodevectoranalysisshowsthattheDutchrollandrollsubsidencemodesarestablewhilethespiralmodeisunstableduringmorphing. 4.3.2LongitudinalDynamics LTVeigenpairs.TheeigenvectorsfoundbytheUEmethodforthelongitudinaldynamicsareshowninFigures 4-30 and 4-31 .ThepolesareshowninFigure 4-32 .NotethatthepathtakenbytheUEalgorithmledtothepolesvaryingsignicantlywithtime,unlikeinthelateralcase. Figure4-30. Eigenvectors1(left)and2(right)fromUEsolutionoflongitudinaldynamics ThepolesandeigenvectorsforboththelateralandlongitudinaldynamicshighlightthedangerofdrawingconclusionsfromthepolesandeigenvectorsofaUEsolutioninsteadofthemodevectors.TheUEmethodmakesnoguaranteesthatthepolesoreigenvectorswillreecttheleveltowhichthesystemisdependentontime.Therefore,drawingconclusionsaboutasystembasedonhowtheUEmethodchosetodivideitseffortsbetweenthepolesandeigenvectorsisunwise. 87

PAGE 88

Figure4-31. Eigenvectors3(left)and4(right)fromUEsolutionoflongitudinaldynamics Figure4-32. PolesfromUEsolutionoflongitudinaldynamics Modevectors.ThemodevectorsforthelongitudinaldynamicsareshowninFigures 4-33 and 4-34 .Unlikethelateraldynamics,thelongitudinaldynamicshaveonlytwodistinctmodevectors(twocomplexconjugatepairs). Modevectornorms.Forthelongitudinaldynamics,themodevectornormsaregraphedalongsidetheabsolutevalueofthefrozen-timeeigenpair(FTE)modesinFigures 4-35 and 4-36 .BothconjugatepairsareverydifferentfromtheFTEmodes.Fortherstconjugatepair(Figure 4-35 ),theFTEmodesquicklydecaytozeroduringthemorphing.ThemodevectornormsinitiallyfollowtheFTEmodes,buttheybegintogrowafterabout0.2seconds.SowhiletheFTEmodesindicateaverystableresponse,themodevectoranalysispredictsanunstableresponseovertheperiodofthemorphing. 88

PAGE 89

Figure4-33. Modevectors1(left)and2(right)fromUEsolutionoflongitudinaldynamics Figure4-34. Modevectors3(left)and4(right)fromUEsolutionoflongitudinaldynamics Thisdifferenceisalsoapparentinthesecondconjugatepair(Figure 4-36 ).TheFTEmodesdecay,thoughveryslowly(aswouldbeexpectedforaphugoidmode).Thetime-varyingmodevectornormsgrowrapidlysoonaftermorphingbegins,andonlystartlevelingoffneartheendofthemorphingtrajectory.Again,theFTEmodalanalysisofthelongitudinaldynamicsfailstocaptureimportantbehaviorsthatoccurduringtheaircraft'smorphingphase. Comparisontotraditionalanalysis.InLTIaircraftanalysis,thelongitudinalmodesforatypicalaircraftaredescribedas[ 21 ]: Short-period-complex(oscillatory)response,strongdamping Phugoid-complex(oscillatory)response,weakdamping 89

PAGE 90

Figure4-35. Modevectornorms1(left)and2(right)fromUEsolutionoflongitudinaldynamics Figure4-36. Modevectornorms3(left)and4(right)fromUEsolutionoflongitudinaldynamics Theresultsofthemodevectoranalysisarenowcomparedtothesetraditionallongitudinalmodes.TheFTEmodes(and,tosomeextent,themodevectors)matchthegeneralcharacteristicsofeachoftheLTImodes.Therstconjugatepairofmodesisahigher-frequencyoscillation(Figure 4-33 )thatisquicklydampedintheFTEmodes(Figure 4-35 ),correspondingtotheshort-periodmode.Thesecondconjugatepairisanoscillationwithalongerperiod(Figure 4-34 )andweakdampingoftheFTEmodes(Figure 4-36 ),similartoaphugoidmode.NotethatthenormsofthemodevectorscontradictthestabilityresultindicatedbytheFTEmodes.Themodevectoranalysis 90

PAGE 91

showsthatboththephugoidandtheshort-periodmodesareunstableovertheperiodofthemorphing,withthephugoidmodeshowingthegreatestinstability. 91

PAGE 92

CHAPTER5CONCLUSIONSThisthesispresentedthefollowingcontributions:Wu'sdenitionwasshowntobenon-unique,andexamplesweregiventhatshowedthatthepolesobtainedbysolvingWu'sLTVeigenrelationcannotbeusedtodeterminethestabilityofthesystemwithoutalsoexaminingtheeigenvectors.Itwasfurthershownthattheoscillatorycharacteristicsoftheresponsemayberepresentedineithertheeigenvectors,therealpoleparts,ortheimaginarypolepartsofWu'seigenpairs.AtransformationequationwhichrelatedequivalentsolutionsofWu'sdenitionwasderived.Aconceptcalledthemodevector,whichremainsinvarianttothesetransformations,wasthendened.ThesemodevectorswereshowntoreducetotheLTImodeswhenthesystemwasLTI,andconditionsforstabilityandasymptoticstabilityweredenedusingthesevectors.Kamen'spoleswereshowntobeaspecialcaseofWu'sdenition,reachablefromagivensolutiontoWu'sdenitionviaequivalencetransformations.ThederivationofKamen'sresultfromWu'sdenitionrevealedthatKamen'spolessometimesbecomeunboundedbecauseoftheimplicitassumptionthattherstcomponentofeacheigenvectorisalwaysequalto1.TwoensembleidenticationalgorithmswhichproduceLTVeigenpairsdirectlyfrommeasuredstatetrajectoriesweredescribed.CanonicalnormalizedeigenvectoroptimizationfoundeigenpairswhichsatisedKamen'sdenition,whileunnormalizedeigenvectoroptimizationfoundeigenpairswhichwereonlyguaranteedtosatisfyWu'smoregeneraldenition.Theseidenticationalgorithmswereshowntorequirensetsofstatetrajectories,eachproducedbyrunningthesametime-variationfromdifferentlinearly-independentinitialstatevectors.Iftoolittledatawasprovided,thesolutionsproducedwerenotvalidLTVeigenpairs. 92

PAGE 93

Modevectoranalysisandunnormalizedeigenvectoroptimizationwereusedtoanalyzeamorphing-wingaircraftmodel,foramorphingtrajectorywherethewingsweresweptfromperpendiculartothebodyto30backwardsover2seconds.ConclusionsabouttheaircraftbehaviorweredrawnbycomparingthemodevectorstothetraditionalaircraftmodesusedinLTIanalysis.Forthelateraldynamics,theDutchrollandrollsubsidencemodesremainedstable,whilethespiralmodewasunstable.TheseresultsmatchedthestabilityconclusionspredictedbycomputingtheLTIpolesofthecoefcientmatrixateachtimestep.Forthelongitudinaldynamics,thephugoidandshort-periodmodeswerebothunstable,eventhoughtheLTIpolesateachtimestepindicatedstability. 93

PAGE 94

REFERENCES [1] K.Liu,Identicationoflineartime-varyingsystems,J.SoundVibrat.,vol.206,no.4,pp.487,1997. [2] R.O'BrienandP.Iglesias,Onthepolesandzerosoflinear,time-varyingsystems,IEEETrans.CircuitsSyst.I,Fundam.TheoryAppl.,vol.48,pp.565,May2001. [3] M.Abdulrahim,H.Garcia,andR.Lind,Flightcharacteristicsofshapingthemembranewingofamicroairvehicle,J.Aircr.,vol.42,p.131,Jan.2005. [4] K.Nathan,Y.-T.Lee,andH.Silverman,Atime-varyinganalysismethodforrapidtransitionsinspeech,IEEETrans.SignalProcess.,vol.39,pp.815,Apr.1991. [5] S.HuangandH.Chen,Adaptiveslidingcontrollerwithself-tuningfuzzycompensationforvehiclesuspensioncontrol,Mechatronics,vol.16,no.10,pp.607,2006. [6] J.MacNeil,R.Kearney,andI.Hunter,Identicationoftime-varyingbiologicalsystemsfromensembledata(jointdynamicsapplication),IEEETrans.Biomed.Eng.,vol.39,pp.1213,Dec.1992. [7] M.Wu,Anewconceptofeigenvaluesandeigenvectorsanditsapplications,IEEETrans.Autom.Control,vol.25,pp.824,Aug.1980. [8] M.Wu,Onstabilityoflineartime-varyingsystems,inProc.21stIEEECDC,OrlandoFL,pp.1211,Dec.1982. [9] E.Kamen,Thepolesandzerosofalineartime-varyingsystem,LinearAlgebraAppl.,vol.98,pp.263,1988. [10] E.Kamen,P.Khargonekar,andA.Tannenbaum,Controlofslowly-varyinglinearsystems,IEEETrans.Autom.Control,vol.34,pp.1283,Dec.1989. [11] M.VerhaegenandX.Yu,Aclassofsubspacemodelidenticationalgorithmstoidentifyperiodicallyandarbitrarilytime-varyingsystems,Automatica,vol.31,no.2,pp.201,1995. [12] J.Zhu,Auniedspectraltheoryforlineartime-varyingsystems-progressandchallenges,inProc.34thIEEECDC,NewOrleansLA,vol.3,pp.2540,Dec.1995. [13] J.Zhu,Pd-spectraltheoryformultivariablelineartime-varyingsystems,inProc.36thIEEECDC,SanDiegoCA,vol.4,pp.3908,Dec.1997. [14] R.O'BrienandP.Iglesias,Polesandzerosfortime-varyingsystems,inProc.ACC97,AlbuquerqueNM,vol.5,pp.2672,Jun.1997. 94

PAGE 95

[15] W.BoyceandR.DiPrima,ElementaryDifferentialEquationsandBoundaryValueProblems.JohnWileyandSons,eighthed.,2005. [16] H.Khalil,NonlinearSystems.PrenticeHall,thirded.,2002. [17] L.MarkusandH.Yamabe,Globalstabilitycriteriafordifferentialsystems,OsakaJ.Math.,vol.12,pp.305,1960. [18] E.Kamen,Ontheinnerandouterpolesandzerosofalineartime-varyingsystem,inProc.27thIEEECDC,AustinTX,vol.2,pp.910,Dec.1988. [19] J.Allwright,Orthogonallyapunovtransformationswithsomeapplicationstostabilitystudies,IEEProc.ControlTheoryAppl.,vol.146,pp.333,Jul.1999. [20] D.Grant,M.Abdulrahim,andR.Lind,FlightDynamicsofaMorphingAircraftUtilizingIndependentMultiple-JointWingSweep,inAIAAAtmosphericFlightMechanicsConferenceandExhibit,pp.21,2006. [21] B.EtkinandL.Reid,DynamicsofFlight:StabilityandControl.JohnWileyandSons,thirded.,1996. 95

PAGE 96

BIOGRAPHICALSKETCH StephenLeeSorleywasborninEustis,FLin1986.HegrewupinAstatula,FLandgraduatedwithahighschooldiplomafromChristianHomeandBibleSchoolandanAssociateinArtsdegreefromLake-SumterCommunityCollegeconcurrentlyin2004.HethenwentontoobtainBachelorofSciencedegreesinbothmechanicalengineeringandcomputersciencefromDukeUniversityin2008.AfterasummerinternshipatTheJohnsHopkinsUniversityAppliedPhysicsLab,hejoinedtheFlightControlsLabattheUniversityofFloridaasagraduatestudentinthefallof2008.StephenplanstopursueadoctorateincontrolsattheUniversityofFlorida. 96