Introduction to I nom1c sems Theory, Models, and Applications DavidG.Luenberger StanfordUniversity John Wiley & Sons New YorkChichesterBrisbaneToronto Copyright1979,byJohnWiley&Sons,Inc. AUrightsreserved.PublishedsimultaneouslyinCanada. Reproductionortranslationofanypartof thisworkbeyondthatpermittedbySections 107and108ofthe1976UnitedStatesCopyright Actwithoutthepermissionofthecopyright ownertsunlawful.Requestsforpermission or furtherinformationshouldbeaddressedto thePermassionsDepartment, JohnWiley&Sons. Library of Congressa ~ mPablkadoa Data Luenberger, David G .1937-Introduction to dynamic systems. Includes bibliographical references and index. 1.Systemanalysis.2.Differential equations. 3,Control theory.I. Title.II.Title:Dynamic Systems. QA402.L8400378-12366 ISBN0-471-02594-1 PrintedintheUnited States of America 10987654321 To my parents Preface ThisbookisanoutgrowthofacoursedevelopedatStanfordUniversityover thepastfiveyears.It issuitableasaself-containedtextbookforsecond-level undergraduatesorforfirst-levelgraduatestudentsinalmosteveryfieldthat employsquantitativemethods.Asprerequisites, itisassumedthatthestudent mayhavehad a firstcourseindifferential equations anda firstcourseinlinear algebraormatrixanalysis.Thesetwosubjects,however,arereviewedin Chapters2and3,insofarastheyarerequiredforlaterdevelopments. The objective of thebook,simplystated, istohelponedeveloptheability toanalyzerealdynamicphenomenaanddynamicsystems.Thisobjectiveis pursuedthroughthepresentationofthreeimportantaspectsofdynamic systems:(1)thetheory,whichexploresproperties ofmathematicalrepresenta-tions of dynamic systems,(2)examplemodels, which demonstrate how concrete situations can betranslated into appropriatemathematical representations, and (3)applications, whichillustratethekinds of questions that might beposed ina given situation, andhowtheory canhelpresolvethesequestions.Althoughthe highestpriorityis,appropriately,giventotheorderlypresentationofthe theory,significantsamplesofaUthreeoftheseessentialingredientsare containedinthebook. Theorganizationofthebookfollowstheoreticallines-asthechapter titles indicate. The particular theoretical approach, or style,however,isa blend ofthetraditionalapproach,asrepresentedbymanystandardtextbookson differentialequations,andthemodernstate-spaceapproach,nowcommonly usedasasetting forcontroltheory.Inpart,thisblendwasselectedsoasto vii viiiPreface broadenthescope-togettheadvantagesofbothapproaches;andinpartit wasdictatedbytherequirementsoftheapplicationspresented.Itisrecog-nized,however,that(asineverybranchofmathematics)therootideasof dynamicsystemstranscendanyparticularmathematicalframeworkusedto describe those ideas. Thus, although the theory in this book is presented within a certain framework,it isthe intent that whatistaught about dynamic systemsis richerandlessrestrictivethantheframeworkitself. Thecontent ofthebookis,of course,partlyareflection ofpersonaltaste, but inlarge portion it was selected todirectly relateto theprimary objective of developing theabilitytoanalyzereal systems,asstated earlier. Thetheoretical materialinChapters2through5isquitestandard,althoughinadditionto theorythesechaptersemphasizetherelationbetweentheoryandanalysis. Dominant . eigenvectoranalysisisusedasanextendedillustrationofthis relationship.Chapter6extendstheclassicalmaterialoflinearsystemstothe special andrichtopic of positivesystems.This chapter,perhaps morethanany other,demonstratestheintimaterelationbetweentheoryandintuition.The topic ofMarkov chains,inChapter 7, has traditionallybeen treated most often asa distinct subject.Nevertheless,althoughitdoeshavesomeuniquefeatures, a great dealofunityisachievedbyregarding this topic asabranch of dynamic systemtheory.Chapter8outlinestheconceptsofsystemcontrol-fromboth thetraditionaltransformapproachandthestate-spaceapproach.Chapters9 and10treatnonlinear systems,withtheLiapunov functionconceptservingto unifyboththetheoryandawideassortmentofapplications.Finally,Chapter 11surveystheexcitingtopicofoptimalcontrol-whichrepresentsanimpor-tant frameworkforproblemformulationinmanyareas.Throughoutallchap-tersthereisanassortmentofexamplesthatnotonlyillustratethetheorybut haveintrinsicvalueoftheirown.Althoughthesemodelsareabstractionsof reality,manyoftheseare"classic"modelsthathavestoodthetestoftime andhavehadgreatinfluenceonscientificdevelopment.Fordeveloping effectivenessinanalysis,thestudyoftheseexamplesisasvaluableasthe study of theory. Thebook contains enoughmaterial forafullacademic year course.There isroom,however,forsubstantialflexibilityindevelopingaplanofstudy.By omitting various sections,thebook has beenusedat Stanfordasthebasis for a six-monthcourse. Thechapterdependencychartshownbelowcanbeusedto plansuitableindividualprograms.Asafurtheraidtothisplanning,difficult sections ofthebook that are somewhat tangential tothemain development are designatedbyanasterisk*. Animportant component ofthebookistheset of problems attheend of thechapters.Someoftheseproblemsareexercises,whicharemoreorless straightforwardapplicationsofthetechniquesdiscussedinthechapter;some are extensionsofthetheory;and someintroducenewapplication areas.Afew ChapterDependency Chart(Achapterde-pendsonallchapters leadingtoitinthe chart.) Prefaceix ofeachtypeshouldbeattemptedfromeachchapter.Especiallydifficult problemsaremarkedwithanasterisk*. Thepreparationofthisbookhasbeenalongtaskthatcouldnothave beencompletedwithoutthehelpofmanyindividuals.Manyoftheproblems andexamplesinthebookweredevelopedjointlywithteachingassistantsand students.IwishtoacknowledgetheDepartmentofEngineering-Economic SystemsatStanfordwhichprovidedtheatmosphereandresourcestomake thisprojectpossible.Iwishtothankmyfamilyfortheirhelp,encour-agement,andendurance.IwishtothankLoisGoulartewhoefficientlytyped theseveraldraftsandhelpedorganizemanyaspectsoftheproject.Finally, Iwishto thank the scores of students, visitors, and colleagues who read primitive versionsofthemanuscriptandmademanyvaluableindividualsuggestions. DAVIDG.LUENBERGER Stanford,California January 1979 Contents 1INTRODUCTION 1.1DynamicPhenomena 1.2MultivariableSystems 1.3ACatalogofExamples 1.4TheStagesofDynamicSystemAnalysis 2DIFFERENCEANDDiffERENTIAlEQUATIONS 2.1DifferenceEquations 2.2ExistenceandUniquenessofSolutions 2.3AFirst-OrderEquation 2.4ChainLettersandAmortization 2.5TheCobwebModel 2.6LinearDifferenceEquations 2.7LinearEquationswithConstantCoefficients 2.8DifferentialEquations 2.9LinearDifferentialEquations 2 .. 10Harmonic MotionandBeats 2.11Problems NotesandReferences 3LINEARALGEBRA ALGEBRAICPROPERTIES 3.1Fundamentals 1 2 4 10 14 17 19 21 23 26 32 38 40 44 47 54 56 xi xiiContents 3.2Determinants 3.3InversesandtheFundamentalLemma

3.4VectorSpace 3.5Transformations 3.6Eigenvectors 3 .. 7DistinctEigenvalues 3.8RightandLeftEigenvectors 3.9MultipleEigenvalues 3.10Problems NotesandReferences 4LINEARSTATEEQUATIONS 62 66 69 73 77 80 83 84 86 89 4.1SystemsofFirst-OrderEquations90 4.2ConversiontoStateForm9 5 4.3DynamicDiagrams97 4.4HomogeneousDiscrete-TimeSystems99 4.5GeneralSolutiontoLinearDiscrete-TimeSystems108 4.6HomogeneousContinuous-TimeSystems113 4 .. 7GeneralSolutiontoLinearContinuous-TimeSystems118 *4.8EmbeddedStatics121 4.9Problems124 NotesandReferences130 5LINEARSYSTEMSWITHCONSTANTCOEFFICIENTS 5.1GeometricSequencesandExponentials 5.2SystemEigenvectors 5.3DiagonalizationofaSystem 5.4DynamicsofRightandLeftEigenvectors 5.5Example:ASimpleMigrationModel 5.6MultipleEigenvalues 5.7EquilibriumPoints 5.8Example:SurvivalinCulture 5.9Stability 5.10Oscillations 5.11Dominant Modes 5.12TheCohortPopulationModel *5.13TheSurprisingSolution totheNatchez Problem 5.14Problems NotesandReferences 133 135 136 142 144 148 150 152 154 160 165 170 174 179 186 Contentsxiii 6POSITIVELINEARSYSTEMS 6.1Introduction 188 6.2PositiveMatrices 190 6.3PositiveDiscrete-TimeSystems 195 6 .. 4QualityinaHierarchy-ThePeterPrinciple 199 6.5Continuous-TimePositiveSystems 204 6 .. 6Richardson'sTheoryofArmsRaces206 6.7ComparativeStaticsforPositiveSystems211 6.8Homans-SimonModelofGroupInteraction215 6.9Problems217 NotesandReferences222 1MARKOVCHAINS 7.1FiniteMarkovChains225 7.2RegularMarkovChainsandLimitingDistributions230 7.3ClassificationofStates235 7.4Transient State Analysis239 *7.5InfiniteMarkovChains245 7.6Problems248 NotesandReferences253 8CONCEPTSOf CONTROL 8.1Inputs,Outputs,andInterconnections254 TRANSFORMMETHODS 8.2z-Transforms 255 8 .. 3TransformSolutionofDifferenceEquations261 8.4StateEquationsandTransforms266 8 .. 5LaplaceTransforms 272 STATESPACEMETHODS 8.6Controllability 276 8.7Observability285 8.8CanonicalForms289 8.9Feedback296 8.10Observers300 8.11Problems309 NotesandReferences314 9ANALYSISOFNONLINEARSYSTEMS 9.1Introduction316 9.2EquilibriumPoints320 9.3Stability322 xivContents 9.4LinearizationandStability 9.5Example:ThePrincipleofCompetitiveExclusion 9.6LiapunovFunctions 9.7Examples *9.8InvariantSets 9.9ALinearLiapunovFunctionforPositiveSystems 9.10AnIntegralLiapunovFunction *9.11AQuadraticLiapunovFunctionforLinearSystems 9.12CombinedLiapunovFunctions 9.13GeneralSummarizingFunctions 9.14Problems NotesandReferences 10SOMEIMPORTANT-DYNAMICSYSTEMS 10.1EnergyinMechanics 10.2EntropyinThermodynamics 10.3InteractingPopulations 10.4Epidemics 10.5StabilityofCompetitiveEconomicEquilibria 10.6Genetics 10.7Problems NotesandReferences 11OPTIMAl CONTROL 11.1TheBasicOptimalControlProblem 11.2Examples 11.3ProblemswithTerminalConstraints 11.4FreeTerminalTimeProblems 11.5LinearSystemswithQuadraticCost 11.6Discrete-TimeProblems 11.7DynamicProgramming *11.8StabilityandOptimalControl 11.9Problems NotesandReferences REFERENCES INDEX 324 328 332 339 345 347 349 350 353 354 356 363 365 367 370 376 378 382 389 391 394 401 405 409 413 416 419 425 427 435 436 441 Introduction to Dynamic Systems chapterl Introduction 1.1DYNAMICPHENOMENA Thetermdynamicreferstophenomenathatproducetime-changing patterns,thecharacteristicsofthepatternatonetimebeinginterrelatedwith thoseatothertimes.Thetermisnearlysynonymouswithtime-evolutwnor pattern of change.It referstotheunfolding ofevents in a continuing evolution-ary process. Nearlyallobserved phenomenainour dailylivesor inscientificinvestiga-tionhaveimportantdynamicaspects.Specificexamplesmayarisein(a)a physical system, suchasatraveling space vehicle,ahomeheating system,or in theminingofamineraldeposit;(b)asocialsystem,suchasthemovement withinanorganizational hierarchy,theevolutionof atribal class system,or the behaviorofaneconomicstructure;or (c)alifesystem,suchasthatofgenetic transference, ecological decay,or population growth.But whiletheseexamples illustratethepervasivenessofdynamicsituationsandindicatethepotential valueofdevelopingthefacilityforrepresentingandanalyzingdynamicbe-havior,itmustbe emphasizedthat thegeneral concept ofdynamicstranscends theparticularoriginor settingoftheprocess. Manydynamicsystemscanbeunderstoodandanalyzedintuitively,with-outresorttomathematicsandwithoutdevelopmentofageneraltheoryof dynamics.Indeed,weoftendealquiteeffectivelywithmanysimpledynamic situations inour dailylives.However, inorder to approachunfamiliar complex situations efficiently,it isnecessarytoproceed systematically.Mathematics can providetherequiredeconomyoflanguageandconceptualframework. 2Introduction Withthisview,thetermdynamicssoontakesonsomewhatofadual meamng.Itis,first.asstatedearlier,atermforthetime-evolutmnary phenomenaintheworldaboutus,and,second,itisatermforthatpartof mathematicalsciencethatisusedfortherepresentationandanalysisofsuch phenomena.In themost profound sensethetermrefers simultaneously toboth aspects:thereal,theabstract,andtheinterplaybetweenthem. Althoughthereareendlessexamplesofinterestingdynamicsituations arisinginaspectrum of areas,thenumber of correspondinggeneralformsfor mathematicalrepresentationisrelativelysmalLMostcommonly,dynamic systems are represented mathematically interms of either differential or differ-enceequations.Indeed,thisissomuchthecasethat,intermsofpure mathematicalcontent,atleasttheelementarystudyofdynamicsisalmost synonymouswiththetheory of differentialand differenceequations.It isthese equationsthatprovidethestructureforrepresentingtimelinkagesamong variables. Theuseofeither differentialor differenceequations to represent dynamic behaviorcorresponds,respectively,towhetherthebehaviorisviewedas occurringincontinuousor discretetime.Continuoustimecorrespondstoour usualconception,wheretimeisregardedasacontinuousvariableandisoften viewedasflowmgsmoothlypast us.In mathematical terms, continuous timeof thissortisquantifiedintermsofthecontinuumof realnumbers.Anarbitrary valueofthiscontinuoustimeisusuallydenotedbythelettert.Dynamic behaviorviewedincontinuoustimeisusuallydescribedbydifferentialequa-tions,whichrelatethederivativesofadynamicvariabletoitscurrentvalue. Discretetimeconsistsofanorderedsequenceofpointsratherthana continuum.Intermsofapplications,itisconvenienttointroducethtskindof ttmewheneventsandconsequenceseither occurorareaccountedforonlyat discretetimeperiods,suchasdaily,monthly,oryearly.Whendevelopinga populationmodel,forexample,itmaybeconvenienttoworkwithyearly populationchangesratherthancontinuoustimechanges.Discretetimeis usuallylabeled bysimply indexing,inorder, the discrete timepoints, startingat a convenientreferencepoint.Thustimecorresponds tointegers 0,1,2,and so forth,andanarbitrarytimepointisusuallydenotedbytheletterk.Accord-ingly,dynamicbehaviorviewedindiscretetimeisusuallydescribedbyequa-tions relating thevalue of a variableat one timetothe values at adjacenttimes. Suchequationsarecalleddifferenceequations. 1.2MULTIVARIABLESYSTEMS The termsystem,asappliedtogeneral analysis,wasoriginated asa recognition thatmeaningfulinvestigationofaparticularphenomenoncanoftenonlybe 1.2MultivariableSystems3 achievedbyexplicitlyaccountingforitsenvironment.Theparticularvariables ofinterestarelikelytorepresentstmplyonecomponentofacomplex, consistingofperhapsseveralothercomponents.Meamngfulanalysismust consider theentire systemand the relations among its components. Accordingly, mathematicalmodelsofsystemsarelikelytoinvolvealargenumberof interrelatedvariables-andthisisemphasizedbydescribingsuchsituationsas multivariable systems.Some examplesillustratingthepervasiveness andimpor-tanceofmultivariablephenomenaariseinconsiderationof(a)themigration patternsofpopulationbetweenvariousgeographicalareas,(b)thesimultane-ousinteractionofvariousindividualsinaneconomic system,or (c)thevarious agegroupsinagrowingpopulatwn. Theabilitytodealeffectivelywithlargenumbersofinterrelatedvanables isone of themostimportant characteristics ofmathematical systemanalysis.It isnecessarythereforetodevelopfacilitywithtechniquesthathelponeclearly thinkaboutandsystematicallymampulatelargenumbersofsimultaneous relations.For one's ownthinkingpurposes, inorder tounderstand theessential elementsofthesituation,onemustlearn,first,toviewthewholesetof relationsasaunit,suppressingthedetails;and,second,toseetheimportant detailedinterrelationswhenrequired.Forpurposesofmanipulation,withthe primaryobjective of computationrather thanfurtheringinsight,onereqmresa systematicandefficientrepresentatiOn. Therearetwomainmethodsforrepresentingsetsofinterrelations.The firstisvectornotation,whichprovidesanefficientrepresentationbothfor computationandfortheoreticaldevelopment.Byitsverynature,vector notationsuppressesdetailbutallowsforitsretrievalwhenrequued.Itis thereforeaconvenient,effective,andpracticallanguage.rv-toreover,oncea situationiscastinthisform,theentirearrayoftheoreticalresultsfromlinear algebraisavailable forapplication.Thus,thislanguageisalsowellmatchedto mathematicaltheory. Thesecondtechniqueforrepresentinginterrelationsbetweenvariablesis byuseofdiagrams.Inthisapproachthevariouscomponentsofasystemare representedbypointsorblocks,wtthconnectinglinesrepresentingrelations betweenthecorrespondingcomponents.Thisrepresentationisexceedingly helpfulforvisualizationofessentialstructureinmanycomplexsituations; however,itlacksthefullanalyticalpowerofthevectormethod.Itisforth1s reasonthat,althoughbothmethodsaredevelopedinthisbook,primary emphasisisplacedonthevectorapproach. Mostsituationsthatweinvestigatearebothdynamicandmult1variable. Theyare,accordingly,characterizedbyseveralvariables,eachchangingwith timeand each linkedthroughtimeto other variables.Indeed,this combination of multivariableand time-evolutionary structure characterizes the setting of the moderntheoryofdynamicsystems. 4Introduction Thatmostdynamicsystemsarebothtime-evolutionaryandmultivariable implies something about thenature ofthemathematicsthat formsthebasisfor the1ranalysis.The mathematical toolsare essentially a combination of differen-ual (or difference)equations and vector algebra. The differential (or difference) equations providethe element of dynamics,and the vector algebra providesthe notationformultivariablerepresentation.Thecombinationandinterplaybe-tweenthesetwobranches ofmathematicsprovidesthebasic foundationforall analysisinthisbook.Itisforthisreasonthatthtsintroductorychapteris followedfirstbya chapter ondifferentialand differenceequationsand thenby achapteronmatrixalgebra. 1.3ACATALOGOFEXAMPLES As inallareas of problem formulationand analysis,the process of passing from a"realworld"dynamicsituationtoasuitableabstractionintermsofa mathematicalmodelrequiresanexpertisethatisrefinedonlythroughexperi-ence.Inanygivenapplicationthereisgenerallynosingle"correct''model; rather,thedegreeofdetail,theemphasis,andthechoiceofmodelformare subjecttothediscretionarychoiceoftheanalyst.Thereare,however,a number of models that are considered "classic" in that they are well-known and generallyaccepted.Theseclassicmodelsserveanimportantrole,notonlyas models of the situation that they wereoriginally intended torepresent,butalso asexamplesofthedegreeofclarityandrealityone should strivetoachievein newsituations.Aproficientanalystusuallypossessesalargementalcatalog of theseclassicmodelsthatserveasvaluablereferencepoints-aswell-founded pointsofdeparture. Theexamplesinthissectionareinthissenseallclassic,andassuchcan formthebeginningsofacatalogforthereader.Thecatalogexpandsasone workshtswaythroughsucceedingchapters,andthisgrowthofwell-founded exampleswithknownpropertiesshouldbeoneofthemostimportantobjec-tivesofone's study.Adiversecatalog enrichestheprocessofmodeldevelop-ment. Thefirstfourexamplesareformulatedindiscretetimeandare,accord-mgly,definedbydifferenceequations.Thelasttwoaredefinedincontinuous timeandthusresultindifferentialequations.It willbeapparentfromastudy of the examples that the choiceto develop a continuous-time or a discrete-time modelofaspecificphenomenonissomewhatarbitrary.Thechoiceisusually resolvedonthebasisofdataavailability,analyticaltractability,established conventionintheapplicationarea,or simplypersonalpreference. Example1(GeometricGrowth).Asimplegrowthlaw,usefulinawide assortmentofsituations(suchasdescribingtheincreaseinhumanorother 1.3ACatalogofExampies5 5 4 " 3 2 1 01234567k Figure1.1.Geometricgrowth. populations,thegrowthofvegetation,accumulatedpublicationsinascientific field,consumptionofrawmaterials,theaccumulationofinterestonaloan, etc.),isthelinearlawdescribedbythedifferenceequation x(k + 1) = ax(k) Thevaluex(k)representsthemagnitudeofthevariable(e.g.,population)at timeinstantk.Theparameteraisaconstantthatdeterminestherateof growth.Forpositivegrowth,thevalueofamustbegreaterthanunity-then eachsuccessivemagnitudeisafixedfactorlargerthanitspredecessor. If aninitialmagnitudeisgiven,sayx(O)= 1,thesuccessivevaluescanbe foundrecursively.Inparticular, it iseasyto seethat x(1) =a, x(2) = a2,and,in general,x(k) = ~ c fork =0, 1, 2, ....Atypicalpatternofgrowthresulting fromthismodelisshowninFig.1. 1. The growthpattern resultingfromthis simple linear model isreferred toas geometricgrowthsincethevaluesgrowastheterms ofageometric series.This formofgrowthpatternhasbeenfoundtoagreecloselywithempirical datain manysituations,andthereisoftenstrongaccompanyingtheoreticaljustifica-tionforthemodel,atleastoverarangeofvalues. Example2(CohortPopulationModel) ..Formanypurposes(particularlyin populationswherethelevelofreproductiveactivityisnonuniformovera normallifetime)thesimplegrowthmodelgivenaboveisinadequatefor comprehensiveanalysisofpopulationchange.Moresatisfactorymodelstake accountoftheagedistributionwithinthepopulation.Theclassicalmodelof thistypeisreferredtoasacohortpopulationmodel. Thepopulationisdividedintoagegroups(orcohorts)ofequalagespan, sayfiveyears.Thatis,thefirstgroupconsistsofallthosemembersofthe populationbetween the ages of zero and fiveyears, the second consists of those betweenfiveandtenyears,andsoforth.Thecohortmodelitselfisa discrete-timedynamicsystemwiththedurationofa singletimeperiod corres pondingtothebasic cohort span(fiveyearsinour example).Byassumingthat themaleandfemalepopulationsareidenticalindistribution,itispossibleto 6Introduction simplifythemodelbyconsideringonlythefemalepopulation.Let xi(k)bethe (female)populationoftheithagegroupattimeperiodk.Thegroupsare indexedsequentiallyfrom0throughn,with0representingthelowestage groupandnthelargest.Todescribesystembehavior,itisonlynecessaryto describehowthesenumberschangeduringonetimeperiod. First,asidefromthepossibilityofdeath,whichwillbeconsideredina moment,it isdear that during one time period thecohorts intheith agegroup simplymoveuptothe{i + 1)thagegroup.Toaccountforthedeathrateof individuals withina givenagegroup,thisupward progressionisattenuated bya survivalfactor.The net progression canbe described bythe simpleequations i = 0, 1, ... , n -1(1-1) where(3,isthe survivalrate of theith age group during one period. The factors {3;canbedeterminedstatisticallyfromactuarialtables. The onlyagegroup not determined bythe equationabove isx0{k + 1),the group ofindividualsborn duringthelasttimeperiod.Theyareoffspring of the populationthatexistedintheprevioustimeperiod.Thenumberinthisgroup depends on thebirthrate ofeach ofthe other cohort groups,and on howlarge eachofthesegroupswasduringthepreviousperiod.Specifically, x0(k + 1) = a0x0(k) + a1x1(k)+ a2x2(k)+ + anxn(k)(1-2) wherea.isthebirthrate oftheithagegroup (expressedinnumber of female offspring per time period per member of agegroupi). The factoraialsocan be usuallydeterminedfromstatisticalrecords. TogetherEqs.(1-1)and(1-2)definethesystemequations,determining howxi(k + 1)'sarefoundfromxi(k)'s.Thisisanexcellentexampleofthe combinationofdynamicsandmultivariablesystemstructure.Thepopulation systemismostnaturallyvisualizedintermsofthevariablesrepresentingthe populationlevelsofthevariouscohortgroups,andthusitisamultivariable system. Thesevariablesarelinkeddynamicallybysimpledifferenceequations, andthusthewholecanberegardedasacomposite of differenceequationsand rnultivariablestructure. Example 3(National Economics).There are several simplemodels of national economicdynamics.*Wepresentoneformulatedindiscretetime,wherethe timebetweenperiodsisusuallytakenasquartersoffullyears.Ateachtime periodtherearefourvariablesthatdefinethemodel.Theyare Y(k) =National Incomeor NationalProduct C(k) =Consumption I(k) =Investment G(k) =Government Expenditure *SeethenotesandreferencesforSect.4.8,attheendofChapter4. 1.3ACatalogofExamples7 ThevariableYisdefinedtobetheNationalIncome:thetotalamountearned duringaperiodbyallmdiv1dualsintheeconomy.Alternatively,buteqmval-ently,YcanbedefinedastheNationalProduct:thetotalvalueofgoodsand servicesproducedintheeconomyduringtheperiod.ConsumptionC1sthe totalamount spent byindividualsforgoodsand services.It isthetotalofevery individual'sexpenditure.TheInvestmentIisthetotalamountinvestedinthe period.Finally,Gisthetotalamount spentbygovernmentduringthepenod, whichisequaltothegovernment'scurrentrevenue.Thebasicnatwnal accountingequationis Y(k) = C(k) + I(k) + G(k){1-3) Fromanincomeviewpoint,theequationstatesthattotalindividualincome mustbedividedamongconsumptionofgoodsandservices,investment,or payments tothe government.Alternatively, fromanationalproduct viewpoint, thetotalaggregateofgoodsandservicesproducedmustbedividedamong individualconsumption,investment,orgovernmentconsumption. Inadditiontothisbasicdefinitionalequation,tworelationshipsareintro-ducedthatrepresentassumptionsonthebehavioroftheeconomy.First,itis assumedthatconsumptionisafixedfractionofnationalincome.Thus, C(k)= mY(k)(1-4) forsomem.Thenumberm,whichisrestrictedtothevalues0 < m < 1,is referredtoasthemarginalpropensitytoconsume.Thisequationassumesthat ontheaverageindividualstendtoconsumeafixedportionoftheirincome. Thesecondassumptionconcerninghowtheeconomybehavesrelatesto theinfluenceof investment.Thegeneraleffectof investmentistoincreasethe productivecapacityofthenation.Thus,presentinvestmentwillincrease nationalincome {ornationalproduct)infutureyears.Specifically,itisassumed that theincreaseinnationalincomeisproportionaltothelevelof investment. Or, Y(k + 1)- Y(k) =ri(k)(1-5) Theconstantristhegrowthfactor,andit isassumedthatr > 0. Thesetofequations(1-3),(1-4),and(1-5)definestheoperationofthe economy.Of thethreeequations,onlythelastisdynamic.The firsttwo,(1-3) and(1-4),arestatic,expressingrelationshipsamongthevariablesthatholdat everyk.These two static equations can beused toeliminatetwovariables from themodel.Startingwith Y(k) = C(k) + I(k) + G(k) substitutionof(1-4)produces Y(k) =m Y(k) + I(k) + G(k) 8Introduction Substitutionof(1-5)thenproduces Y(k) ==m Y(k) + Y(k + l)- Y{k) + G(k) r Rearrangementleadstothefinalresult: Y(k + 1) =[1 + r(l- m)]Y(k)- rG(k)(1-6) The quantityG(k)appearsasaninputtothesystem.If G(k)wereheld equal tozero,themodelwouldbeidenticaltothefirst-order(geometric)growth modeldiscussedearlier. Example4(Exponential Growth)..Thecontinuous-timeversionofthesimple first-ordergrowthmodel(theanalogofgeometricgrowth)isdefinedbythe differentialequation dx(t) =rx(t) dt Thegrowthparameterr canbeanyrealvalue,butfor(increasing)growthit mustbegreater than zero. The solutionto theequation isfoundbywriting itin theform 1dx(t) - =r x(t)dt Bothsidescanthenbeintegratedwithrespecttottoproduce log x(t) = rt+ log c =log ert +log c wherecisanarbitraryconstant.Takingtheantilogyields x(t) = cert Finally,bysettingt = 0, itis seen that x(O) =c,sothe solution canbewritten x(t) = x(O)ert This istheequation ofexponential growth.The solutionissketched forvarious valuesofrinFig.1.2. Thepatternofsolutionsissimilartothatofgeometricgrowthshownin Fig.1.1inSect.1.6.Indeed,aseriesofvaluesfromthecontinuous-time solutionatequallyspacedtimepointsmakeupageometricgrowthpattern. Example 5(Newton's Laws)..Awealth of dynamic system examples isfoundin mechanicalsystemsgovernedbyNewton7Slaws.Infact,manyofthegeneral techniquesfordynamicsystemanalysiswereoriginallymotivatedbysuch applications.Asasimpleexample,considermotioninasingledimension-of, say,a streetcar or cablecar of mass Mmovingalong astraighttrack.Suppose 1.3ACatalogofExamples9 X figure1.2.Exponentialgrowth. thepositionofthecaralongthetrackattimetisdenotedbyy(t),andthe forceappliedtothestreetcar,paralleltothetrack,isdenotedbyu(t). Newton'ssecondlawsaysthatforceisequaltomasstimesacceleration,or, mathematically, Therefore,themotionisdefinedbyasecond-orderdifferentialequation. Amoredetailedmodelwould,ofcourse,havemanyothervariablesand equationstoaccountforspringaction,rocking,andbouncingmotion,andto accountforthefactthatforcesareappliedonlyindirectlytothemainbulk throughtorqueon thewheelsor fromafrictiongrip on acable.The degreeof detail constructed intothemodelwoulddepend on theusetowhichthemodel weretobeput. Figure1.3.Cable car. 10Introduction Example6(GGatsandWolves).Imagineanislandpopulatedprimarilyby goats and wolves. The goats survive byeating theisland vegetation. The wolves survivebyeatingthegoats. The modeling of thiskind of population system,referred to asapredator-preysystem,goesbacktoVolterrainresponsetotheobservationthat populationsofspeciesoftenoscillated.Inourexample,goatswouldfirstbe plentifulbutwolvesrare,andthenwolveswouldbeplentifulbutgoatsrare. Volterradescribedthesituationinthefollowingway. Let N1(t) =number of goatsattimet N2(t) =number of wolves at timet Theproposedmodelisthen dN1 (t) =aN1(t)- bN1(t)N2(t) dt dN2(t) = -cN2(t) + dN1(t)N2(t) dt wheretheconstantsa,b,c,anddareallpositive. Thismodel,whichisthearchetypeofpredator-preymodels,hasasimple biOlogicalinterpretation.Intheabsence of wolves[N2(t) =0],thegoat popula-tionisgovernedbysimpleexponential growth,withgrowthfactora.The goats thriveontheisland vegetation.In theabsence of goats [N1 (t) = 0],on the other hand, the wolf population isgoverned by exponential decline, declining at a rate -c. Thisinterpretation accounts forthefirstterms ontheright-hand sideof the differentialequations. When both goats and wolvesare present on the island,there are encounters betweenthetwogroups.Underanassumptionofrandommovement,the frequencyofencountersisproportionaltotheproductofthenumbersinthe twopopulations.Eachencounterdecreasesthegoatpopulationandincreases thewolfpopulation.Theeffectoftheseencountersisaccountedforbythe secondtermsmthedifferentialequations. 1.4THESTAGESOFDYNAMICSYSTEMANALYSIS Theprincipal objectives ofan analysisofa dynamic systemare asvariedasthe rangeofpossibleapplicationareas.Nevertheless,itishelpfultodistinguish four(oftenoverlapping)stagesofdynamicanalysis:representationof phenomena,generationofsolutions,explorationofstructuralrelations,and controlormodification.Mostanalysesemphasizeoneortwoofthesestages, 1.4TheStagesofDynamicSystemAnalysis11 withtheothershavingbeencompleted previously or lyingbeyond the reachof currenttechnique. Arecognitionofthesefourstageshelpsmotivatetheassortmentof theoreticalprinciplesassociatedwiththemathematicsofdynamicsystems,for thereis,naturally,greatinterplaybetweengeneraltheoryandtheanalysisof givensituations.Ontheonehand,theobjectivesforananalysisarestrongly influencedbyavailabletheory,and,ontheotherhand,developmentofnew theoryisoftenmotivatedbythedesiretoconductdeeperanalyses. Representation Oneoftheprimaryobjectivesoftheuseofmathematicsincomplexdynamic systems istoobtain amathematical representation ofthe system,and thisIsthe firststageofanalysis.Theprocessofobtainingtherepresentationisoften referredtoasmodeling,andthefinalproductamodel.Thisstageisclosely related tothesciences, forthe development ofa suitablemodelamountstothe employmentordevelopmentof scientifictheory.Thetheoryemployedinany givenmodelmaybewell-foundedandgenerallyaccepted,oritmaybebased onlyononeanalyst'shypothesizedrelationships.Acomplexmodelwilloften havebothstrongand weakcomponents.Butinanycasethemodeldescnption isanencapsulationofascientifictheory. Development of ameaningfulrepresentation ofa complex system requires morethanjustscientificknowledge.Theendproductislikelytobemost meaningfulifoneunderstandsthetheoryofdynamicsystemsaswellasthe relevantscientifictheory.Only thenisit possibletoassess,atleastmqualita-tiveterms,thedynamic significance of various assumptions,and thereby build a modelthatbehavesinamannerconsistentwithintuitiveexpectations. GenerationofSolutions The mostdirectuseofa dynamicmodelisthegenerationofaspecific solution toits describingequations. The resultingtimepattern of thevariablesthencan bestudiedforvariouspurposes. Aspecificsolutioncansometimesbefoundinanalyticalform,butmore oftenitisnecessarytogeneratespecificsolutionsnumericallybyuseofa calculator or digitalcomputer-aprocesscommonlyreferredtoassimulation. Asanexample of thisdirectuseofamodel,alarge cohort model of anation's populationgrowthcanbesolvednumericallytogeneratepredictionsoffuture population levels,catalogued byagegroup, sex,andrace.The results ofsuch a simulation mightbe usefulforvariousplanning problems.Likewise,a model of thenationaleconomycanforecastfutureeconomictrends,therebypossibly suggesting theappropriatenessof various corrective policies.Or,inthecontext ofanysituation,simulationmightbeusedtotestthereasonablenessofanew 12Introduction modelbyverifyingthataparticularsolutionhasthepropertiesusuallyas-sociatedwiththeunderlyingphenomena. It isofcourserarethatasinglesolutionofamodelisadequatefora meaningfulanalysis.Everymodelreallyrepresentsacollectionofsolutions, each determinedby different controlled inputs,differentparameter values,and different starting conditions.In the population system,forexample, the specific futurepopulationlevelisdependentonnationalimmigrationpolicy,onthe birthratesinfutureyears,andontheassumedlevelofcurrent population.Onemaythereforefindthatitisnecessarytogeneratesolutions correspondingtovariouscombinationsofassumptionsinordertoconducta meaningfulanalysisofprobablefuturepopulation. As a generalrule,thenumber of required solutions grows quickly withthe numberof differentparametersandinputsthatmustbevariedindependently. Thus,althoughdirectsimulationisaflexibleconcept applicabletoquitelarge andcomplexsystemswhereanalysisisdifficult,itissomewhatlimitedinits capabilitytoexploreallrangesofinputandparametervalues. ExplorationofStructuralRelations Muchofthetheoryofdynamicsystemsismotivatedbyadesiretogobeyond thestageofsimplycomputingparticularsolutionsofamodeltothepointof establishingvariousstructuralrelationsas,say,betweenacertainparameter anditsinfluenceonthesolution.Suchrelationsareoftenobtainedindirectly throughtheuseofauxiliaryconceptsofanalysis. Thepayoffofthistypeofstructuralexplorationmanifestsitselfintwo 1mportantandcomplementaryways.First,itdevelopsintuitiveinsightinto systembehavior.Withthisinsight,oneisoftenabletodeterminetherough outlinesofthesolutiontoacomplexsystemalmostbyinspection,and,more importantly,toforeseethenatureoftheeffectsofpossiblesystemmodifica-tions.Butitisimportanttostressthatthevalueofthisinsightgoeswell beyondthemereapproximationofasolution.Insightintosystembehavioris reflectedback,asan essential part of the creativeprocess,to refinementofthe formulationoftheoriginalmodel.Amodelwillbefinallyacceptedonlywhen oneisassured of itsreasonableness-both interms ofits structure and interms ofthebehaviorpatternsit generates. Thesecbndpayoff of structuralexplorationisthatitoftenenablesoneto explicitlycalculaterelationsthat otherwise couldbe deduced onlyafterexami-nation of numerous particular solutions. For example, as isshown in Chapter 5, thenaturalrateofgrowthofacohortpopulationmodelcanbedetermined directlyfromitsvariousbirthrateandsurvivalratecoefficients,without generatingevenasinglespecificgrowthpattern.Thisleads,forexample,toa specificrelationshipbetweenchangesinbirthratesandchangesincomposite 1.4TheStagesofDynamicSystemAnalysis13 populationgrowth.Inasimilarfashion,thestabilityofacomplexeconomic processofpriceadjustmentcanoftenbeinferredfromhsstructuralform, withoutgeneratingsolutions. Mostofthetheoreticaldevelopmentinthisbookisaimedatrevealing relationshipsofthiskindbetweenstructureandbehavior.Bylearningthis theorywebecomemorethanjustequationwritersandequationsolvers.Our analysisisnotlimitedinitsapplicationtoaparticular problemwithparticular numericalconstants,butinsteadisapplicabletowholeclassesofmodels;and resultsfromonesituationcanbereadilytransferredtoanother. ControlorModification Although study of aparticular dynamic situation issometimes motivated bythe simplephilosophicdesiretounderstandtheworldanditsphenomena,many analyseshavetheexplicitmotivationof devisingeffectivemeansforchanging asystem sothatitsbehavior patternisinsomewayimproved.Themeansfor affectingbehaviorcanbedescribedasbeingeithersystemmodificationor control.Modificationreferstoachangeinthesystem,andhenceinits describingequation.Thismightbeachangein various parameter values or the introductionofnewinterconnectivemechanisms.Examplesofmodification are:achangeinthebirthratesofapopulationsystem,achangeofmarriage rulesinaclasssociety,achangeofforecastingprocedureinaneconomic system,achangeofpromotionrateinanorganizationalhierarchy,andso forth.Control,ontheotherhand,generallyimpliesacontinuingactivity executedthroughouttheoperationofthesystem.TheFederalReserveBoard controlsthegenerationofnewmoneyintheeconomyonacontinuingbasis,a farmer controls the development of hisherd ofcattle bycontrolling theamount ofgrainthey arefed,apilotcontrolsthebehavior ofhisaircraftcontinuously, andsoforth. Determinationofasuitablemodificationorcontrolstrategyforasystem representsthe fourthstage of analysis,andgenerallymarkstheconclusionofa completeanalysiscycle.However,atthecompletionofthebestanalyses,the mainoutlinesofthesolutionshouldbefairlyintuitive-duringthecourseof analysistheintuitionshouldbeheightenedtoalevelsufficienttoacceptthe conclusions.Mathematicsservesasalanguagefororganizedthought,and thoughtdevelopment,notasamachineforgeneratingcomplexity.The mathematicsofdynamicsystemsisdevelopedtoexpediteourrequestsfor detailwhenrequired,andto enhanceour insightintothebehavior of dynamic phenomena weencounterintheworld. chapter2. Difference and Differential Equations Ordinarydifferenceanddifferentialequationsareversatiletoolsofanalysis. Theyareexcellentrepresentationsofmanydynamicsituations,andtheir associatedtheoryisrichenoughtoprovidesubstancetoone'sunderstanding. Theseequationsaredefinedintermsofasingledynamicvariable(thatis,a stnglefunctionoftime)andthereforerepresentonlyaspecialcaseofmore generaldynamicmodels.However,ordinarydifferenceanddifferentialequa-tionsarequiteadequateforthestudyofmanyproblems,andtheassociated theoryprovidesgoodbackgroundformoregeneralmultivariabletheory.In otherwords,bothwithrespecttoproblemformulationandtheoreticalde-velopment,differenceanddifferentialequations of a singlevariableprovidean importantfirststepindevelopingtechniquesforthemathematicalanalysisof dynamicphenomena. 2.1DIFFERENCEEQUATIONS Supposethereisdefinedasequenceofpoints,perhapsrepresentmgdiscrete equallyspacedtimepoints,indexedbyk.Supposealsothatthereisavalue y(k) (arealnumber)associated with each of thesepoints.Adifferenceequation isanequationrelatingthevaluey(k),atpointk,tovaluesatother(usually netghboring)points.Asimpleexampleistheequation y(k + 1) =ay(k)k= 0, 1, 2, ...(2-1) Differenceequations may,however,bemuchmorecomplicatedthanthis.For 2.1DifferenceEquations15 example, ky(k + 2)y(k + 1) ==!.J y(k)y(k -1)k ::::::0,1, 2, ...(2-2) Adifferenceequationisreallyaset of equations;one equation foreachofthe indexpointsk.Therefore,partofthespecificationofadifferenceequationis the set of integerskforwhich it istohold.In general,thisset of integersmust be asequence of successivevalues,of either finiteor infiniteduration,such as k= 0,1,2,3, ... ,Nork = 0,1,2,3, .... Often,if thesequenceisnotex-plicitly stated,itistobeunderstoodthateitheritisarbitraryorthatitisthe mostfrequentlyusedsequencek= 0, 1, 2, 3, .... Inmanycasesthepractical context ofthe equationmakestheappropriaterangedear.In any event,once the sequence isdefined,thecorrespondingvaluesof kcaneachbe substituted into the difference equation to obtainan explicit equation relating variousy's. Asanexample,thesimpledifferenceequation(2-1)Iseqmvalenttothe following(infinite)setofequations: y(l) = ay(O) y(2) = ay(l) y{3) =ay(2) (2-3) Differenceequations,justasanysetofequations,canbeviewedmtwo ways.Ifthevaluesy(k)areknown,ordefinedthroughsomealternate description,thedifferenceequationrepresentsarelationamongthedifferent values.If, onthe other hand,the values arenotknown,thedifference equation isviewedasanequationthatcanbesolvedfortheunknowny(k)values.In either interpretation. it isoften usefultoregardy(k) as afunctiononthe index set.Thedifferenceequationthendefinesarelationshipsatisfiedbythefunc-tion. Thetermdifferenceequatwnisusedinordertoreflectthefactthatthe varioustimepointsintheequationslidealongwiththeindexk.Thatis,the termsinvolvetheunknownsy(k),y(k + 1),y(k +2),y(k -1),y(k- 2),andso forth,ratherthanamixtureoffixedandslidingindices,suchas,say,y ( k ), y(k -1),y(l),andy(8).Indeed,sinceallindices slidealongwithk,itis possibleby suitable(but generallytedious)manipulationtoexpress a difference equationintermsofdifferencesA'ofvariousorders,definedbyA0(k) = y(k), A 1(k) =A 0(k + 1)- A 0(k), A 2(k) =A 1(k + 1)- A 1(k), and soforth.This difference formulationarisesnaturallywhenadifferenceequationisdefinedasan approximationtoadifferentialequation,butinmostcasesthemoredirect formisbothmorenaturalandeasiertoworkwith. Theorderofa, differenceequationistheditlerencebetweenthehighest 16DifferenceandDifferentialEquations andlowestindicesthatappearintheequation.Thus(2-1)isfirst-order,and (2-2)isthird-order. Adifferenceequationissaidtobelinearifithastheform a" (k )y(k + n)+ an_1(k)y(k + n -1) + + a1(k)y(k + 1) + a0{k)y(k) = g(k) (2-4) forsomegivenfunctionsg(k)anda1(k),i = 0, 1, 2, ... , n.Theunknown functionyappears linearly in theequation. Thetlt(k)'s intheseequationsare referredtoascoefficientsofthelinearequation.If thesecoefficientsdonot dependonk,theequationissaidtohaveconstantcoefficientsortobe time-invariant.Thefunctiong(k)isvariouslycalledtheforcingterm,the drivingt e n n ~ or simplytheright-handside. Solutions Asolutionofa differenceequation isa functiony(k) that reducestheequation toanidentity.For example,corresponding to thefirst-orderequation y(k + 1) =ay(k) thefunctiony(k) =atereducestheequationtoanidentity,sincey(k + 1) = a"+1 = aarc=ay(k). Asolutiontoadifferenceequationcanalternativelybeviewedasa sequenceofnumbers.Thus,fortheequationabovewitha = 1/2asolutionis representedbythesequence1,l/2,1/4,1/8, ....Thesolutioniseasily eXpressed inthis caseas(1/2)k.In general,however,there maynot be a simple representation,anditisthereforeoftenpreferable,inordertosimplify conceptualization,toviewasolutionasasequence-steppingalongwiththe timeindexk.The twoviewpointsof asolutionassome(perhapscomplicated) functionofkandasasequenceofnumbersare,ofcourse,equivalent. Example1.Considerthelineardifferenceequation (k + 1)y(k + 1)- ky(k) = 1 fork =1, 2, ...Asolutionis y(k)=1-1/k Tocheckthiswenotethaty(k) = (k -1)/k,y(k + 1) = kl(k + 1),andthus. (k + 1)y(k + 1)- ky(k) =k- (k -1) = 1. Thereareother solutionsaswell.Indeed,itiseasilyseenthat y(k)= 1-A/k isasolutionforanyconstantA. 2.2ExistenceandUniquenessofSolutions17 Example 2.Anonlinear differenceequationthat arisesingenetics (seeChapter 10)is It hasthesolution y(k) y(k + 1) = 1 + y(k)' k=0, 1, 2, ... A y(k)= l+Ak whereAisanarbitraryconstant. Example3.Considerthenonlineardifferenceequation y(k+lf+y(k)2=-1 Sincey(k)isdefinedasareal-valuedfunction,theleft-handsidecanneverbe lessthanzero;hencenosolutioncanexist. 2.2EXISTENCEANDUNIQUENESSOFSOLUTIONS As withanyset of equations, a differenceequation need notnecessarily possess a solution,andifit does havea solution, thesolution maynot be unique. These factsareillustratedbytheexamplesinSect.2.1.Wenowturntoageneral examinationoftheexistenceanduniquenessquestions. InitialConditions One characteristicandessentialfeatureofadifferenceequationisthat, over a finiteintervaloftime,asindexedbyk,therearemoreunknownsthan equations.Forexample,thefirst-orderdifferenceequationy(k + 1) = 2y(k) whenenumeratedfortwotimeperiodsk = 0, 1becomes y(l) = 2y(O) y(2) = 2y(l) whichisasystemoftwoequationsandthreeunknowns.Therefore,fromthe elementarytheoryofequations,weexpectthat itmaybenecessarytoassigna valuetooneoftheunknownvariablesinorder tospecifyauniquesolution.If thedifferenceequationswereappliedtoalongersequenceofindexvalues, eachnewequationwouldaddbothonenewequationandonenewunknown. Therefore,nomatter howlongthesequence,therewouldalwaysbeonemore unknownthanequations. Inthemoregeneral situationwherethedifferenceequationforeachfixed kinvolvesthevalueofy(k)atn + 1successivepoints,therearenmore 18Difference andDifferentialEquations unknownsthanequationsinanyfiniteset.This canbeseenfromthefactthat thefirstequationinvolvesn + 1unknownvariables,andagaineachadditional equationaddsbothonemoreunknownandonemoreequation-keepingthe surplusconstantatn.Thissu'rplusallowsthevaluesofnvar-iablestobe specifiedarbitrarily,andaccordingly,therearendegreesoffreedominthe solutionofadifferenceequation.Thesedegreesoffreedomshowupinthe formofarbitraryconstantsintheexpressionforthegeneralsolutionofthe equation. In principle, thenarbitrarycomponents of the solution canbe specifiedin variousways.However,itismostcommon,particularlyinthecontextof dynamicsystems evolvingforwardintime,tospecifythefirstn values ofy(k); thatis,thevaluesy(O),y{l), ... , y(n -1). Thecorrespondingspecifiedvalues are referred toasinitial conditions.For many difference equations, specification ofaset of valuesforinitialconditions leads directlytoacorresponding unique solutionoftheequation. Example1 ..Thefirst-orderdifferenceequation y(k + 1) = ay(k) correspondingtogeometricgrowth,hasthegeneralsolutiony(k) =Cak.Sub-stitutingk= 0,weseethaty{O) =C,and the solution canbe written interms of theinitialconditionasy(k) = y(O)a". Example2.Considerthesecond-orderdifferenceequation y(k + 2) = y(k) Thisequationcanberegardedasapplyingseparatelytotheevenandtheodd indicesk.Oncey(O)isspecified,theequationimpliesthesamevalueofy(k) forallevenk's,butthesinglevalueofy(k)foralloddk'sremainsarbitrary. Oncey(l)isalsospecified,theentire sequenceisdetermined.Thus,specifica-tiOnofy(O)andy(l)determineaunique solution.The solution canbe written as ExistenceandUniqueness Theorem Although,ingeneralydifferenceequationsmaynotpossesssolutions,most differenceequations encountered inapplications do.Moreover, it isusuallynot necessarytoexhibitasolutioninordertobeassuredofitsexistence,forthe verystructure of themost commondifferenceequations impliesthat asolution exists. Asindicated above,even if existenceisguaranteed,wedonotexpectthat 2.3AFirst-OrderEquation19 thesolutiontoadifferenceequatiOnwillbeunique.Thesolutionmustbe restrictedfurtherbyspecifymgasetofinitmlconditions.Thetheoremproved belowisaformalstatementofthisfact.Theassumptionofsuitablestructure, togetherwithappropriatelyspecifiedinitialconditions,guaranteesexistenceof auniquesolution. Theessentialideaofthetheoremisquitesimple.Itimposesarather modestassumptionthatallowsthesolutionofadifferenceequationtobe computedforwardrecursively,startingwiththegivensetofmitialconditions and successivelydetermining thevalues of theother unknowns.Stated another way,thetheoremimposesassumptionsguaranteeingthatthedifferenceequa-tionrepresentsatrulydynamicsystem,whichevolvesforwardintime. ExistenceandUniquenessTheorem ..Letadifferenceequationof theform y(k + n) + f[y(k + n -1), y(k + n- 2), ... , y(k),k] =0(2-5) wherefisanarbitraryreal-valuedfunction,bedefinedoverafimteor infinitesequence of consecutive integer values of k (k = k0,k0 + 1, k0 + 2, ...) . .Theequationhasoneandonlyonesolutioncorrespondingtoeacharbitrary specificationof theninitialvaluesy(k0),y(k0+ 1), ... , y(k0+ n -1). Proof.Supposethevaluesy{k0), y(k0 + 1), ... , y(k0 + n -1) are specified. Then thedifferenceequation(2-5),withk= k0,canbe solved uniquelyfory(k0+ n) simplybyevaluatingthefunctionf.Then,oncey(k0 + n)isknown,the differenceequation(2-5)withk = k0 + 1canbesolved fory(k0 + n + 1),and so forthforallconsecutivevaluesofk.I Itshouldbenotedthatnorestrictionsareplacedonthereal-valued function f.The functioncan behighly nonlinear. The essentialingredient of the resultisthattheyofleadingindexvaluecanbedeterminedfromprevious values,andthisleadingindexincreasesstepwise.Aspecialclassofdifference equationswhichsatisfiesthetheorem'srequirementsisthenth-orderlinear differenceequation y(k + n) + tln-1(k)y(k + n -1) + + a0(k) y(k) =g(k) This equation conforms to(2-5), withthefunction f being just a sumof terms. 2.3AFIRST .. QRDEREQUATION Thefirst-orderdifferenceequation y(k + 1) = ay(k) + b(2-6) arisesinmanyimportantapplications,anditsanalysismotivatesmuchofthe generaltheoryofdifferenceequations.Theequationislinear,hasaconstant coefficienta,andaconstantforcingtermb. 20Difference andDifferentialEquations Thegeneralsolutiontothisequationiseasilydeduced.Themost straightforwardsolutionprocedureistodeterminesuccessivevaluesrecur-sively,as outlined intheprevious section. Thus. wearbitrarily specifythevalue ofyataninitialpointk0,sayk0 = 0,andspecifyy(O) = C.Thisleads immediatelytothefollowingsuccessivevalues: Thegeneraltermis y(O) = C y(l) = ay(O) + b =aC+ b y(2)= ay(1)+b= a2C+ab+ b y(3) = a3C+ a2b+ ab+ b y(k) = akC +(a it-t+ ak-2 +a+ 1)b Fora =1,theexpressionreducessimplyto y(k) =C+ kb (2-7) For a#- 1,the expression can be somewhat simplifiedbycollapsing thegeomet-- . ncsenes,usmg Therefore,thedesiredsolutioninclosed-formis {C+ kb,a= 1 y(k)=- 1-ak a kC + 1 b,a#- 1 -a (2-8) Thissolutioncanbecheckedbysubstitutingitintotheoriginaldifference equation(2-:6). Whena P-1anotherwayofdisplayingthegeneralsolution(2-8)issome-timesmoreconvenient: b y(k) = Dak +--1-a whereDisanarbitrary constant.ClearlythisnewconstantDisrelatedtothe earlierconstantCbyD = C- [b/(1- a)].Inthisform,itisapparentthatthe solution functionisthe sum oftwoelementaryfunctions:theconstant function b/(1- a)andthegeometricsequenceDak.. lnadditiontoacquiringfamiliaritywiththeanalyticsolutionst ~ simple differenceequations,itisdesirablethatonebeabletoinferthesesolutions intuitively.Tobegindevelopingthisability,considerthespecialcasecorres-pondingtoa= 1 in(2-6).For thiscase,the equationstates thatthenewvalue 2.4ChainLettersandAmortization21 ofyequalstheoldvalueplustheconstantb.Therefore,successivey'smerely accumulatesuccessiveadditionsoftheconstantb.Thegeneralsolutionis clearlyy(k) =C + kb,whereCistheinitialvalue,y(O). Ifa ::1- 1thedifferenceequationmultipliestheoldvaluebythefactora eachperiodandaddstheconstantb.Itislikestoringupvalueandeither payinginterest(ifa> 1)ordeductingatax(if0 0satisfiestheseconditiOns.

k-tkk-t-1 Figure2.8.Alternatemethod. 17.Findthesolutionstothefollowingdifferenceequations,fork= 0, 1, 2, (a)(k +2)2y(k + 1)- (k + 1)2y(k) = 2k + 3 y(O) =0. (b)y(k+2)-5y(k+1)+6y(k)=O y(O) =y(1) = 1. (c)y(k+2)+y(k+1)+y(k)=O y(O) = 0, y(1) =1. (d)y(k+2)-2y(k+1)-4y(k)=O y(O) = 1, y(l) =0. (e)y(k+2)-3y(k+1)+2y(k)=1 y(O) = 2,y(l) = 2. 2.11Problems53 18.RadioactiVe Dating.Normai carbonbasanatomic weightof12. The radioisotope C1\WithanatomiCwetgbt of14, isproducedcontinuously bycosmicradiatiOnand isdistributedthroughouttheearth'satmosphereinaformofcarbondioxide. Carbondioxideisabsorbedbyplants,theseplantsareeatenbyammals,and, consequently,alllivingmattercontainsradioactivecarbon.TheIsotopeC14 is unstable-byemittmganeiectron,iteventuallydisintegratestonitrogen.Sinceat deaththe carboninplant andanimal tissue isno longer replenished,the percentage of C14 insuchtissue begins to decrease.It decreases exponentially wttba half-life of 5685years(thatis,after5685yearsonehalfoftheC14 atomswillhavedismte-grated). Supposecharcoalfromanancientruinproducedacountof1 disintegration/min/gonageigercounterwhilelivingwoodgaveacountof7. Estimatetheageoftheruins. 19.NewtonCooling.AccordingtoNewton'slawofcooling,anobjectofh1gher temperaturethanitsenvironmentcoolsataratethatisproportionaltothe differenceintemperature. (a)Athermometer reading 70F,whichbas beenins&deabousefora longtime,is taken outside. After one minute the thermometer reads 60F;aftertwommutes itreads53F.Whatistheoutsidetemperature? (b)Suppose youare servedahot cupofcoffeeanda small pitcher of cream (wh1ch iscold).Youwanttodrinkthecoffeeonlyafteritcooistoyourfavorue temperature.If youwishtogetthe coffeetopropertemperatureasquicklyas possible,shouldyouaddthecreamimmediatelyor shouldyouwa1tawhile? 20.The equation isanexampleofaneqw-dimenswnaldifferentialequation.Findasetoflinearly independentsolutions.[Hint:Tryy(t) = tP.] 21.An ElementarySeismograph.Aseismographisanmstrument thatrecordssudden groundmovements.Thesimplestkindofseismograph,measuringhonzontaidis-placement,consistsofamassattachedtotheinstrumentframebyaspring.The framemoveswhenhitbya seismicwave,whereasthe mass,Isolatedbythespnng, initiallytendstoremamstill.Arecordingpen,attachedtothemass,tracesa displacement ina direction opposite to the displacement of the frame. The masswill ofcourse soonbeginto oscillate.Inorder to beableto faithfullyrecordadditiOnal seismicwaves,it istherefore des1rable tosuppressthe oscillationofthe massbythe additionofadamper(oftenconsistingofaplungerinav1scousfluid).Tobemost effectivetheseismographmusthaveapropercombinationofmass,spring,and damper. (See Fig.2.9.) If the forceexerted bythe sprmg on themass is proportiOnal to displacementxandinanopposite direction;the forceexertedbythe damper 1s proportional to the velocitydx/dtand inanopposite direction;and the total force1s 54DifferenceandDifferentialEquations ~~kl8 c ~ ~~--E--Seasmograph frame Figure 2.9.Elementary seismograph. equaitomasst1mesacceleration,theequationthatdescribesthemotiOn1s d2xdx m-+c-+lcx=O dt2 dt (a)FindtherootsofthecharacteristicequatiOnmtermsofc,m,andk. (b)Distinguishthreecases,overdampmg,underdampmg,andcnticaldamping basedonthereiationshlpamongc,m,andkasImpliedbythesolut1onofthe charactenst1cequatiOn.Findthegeneralsolutionsforx(t)forallthreecases. (c)Wh1chcase1sbestforaseismographasdescribedmthisproblem?Why? 22.Prove Theorems1,2,and3ofSect.2.9. NOTESANDREFERENCES General.Theelementarytheonesofdifferenceanddifferentialequationsareso s1milarthatmasteryof one essentiallyimpliesmasteryofthe other.However,because therearemanymoretextsondifferentialequationsthandifferenceequatiOns,the readermterestedmsupplementalmaterialmayfind1tmostconvenienttostudy differentialequations.SomeexcellentpopulargeneraltextsareRamvilleandBedient [Rl], Coddington [C5],Braun[Bll], andMartmandRe1ssner (M2].Anexcellenttext ondifferenceequatiOns,wh1chmcludesmanyexamples,isGoldberg. [G8].Seealso Miller[MS]. Section2.5.Thecobwebmodelisanimportantclassicmodel.Forfurtherdiscussion seeHendersonandQuant [H2]. Section2.7.TheGambler'srumproblem(Example5)1streatedextensively inFeller [Fl].ItisalsodiscussedfurthermChapter7ofth1sbook. Section 2 .. 11.Informationtheory,as discussedbriefly in Problem 9, isdue to Shannon. SeeShannonandWeaver [S4]. chapter 3. Linear Algebra Linearalgebraisanearlyindispensabletoolformodernanalysis.It prov1des bothastreamlinednotationforproblemswithmanyvariablesandapowerful formatfortherichtheoryoflinearanalysis.Thischapterisanintroductory accountofthatportionoflinearalgebrathatisneededforabasicstudyof dynamicsystems.Inparticular,thefirstthreesectionsofthechapterare essentialprerequisitesforthenextchapter,andtheremainingsectionsare prerequisitesforlaterchapters.Otherresultsfromlinearalgebrathatare importantintheanalysisofdynamicsystemsarediscussedinmdividual sectionsinlaterportionsofthetext. Insomerespectsthischaptercanberegardedasakindofappendixon linearalgebra.As suchitissuggestedthatthereader mayWishtoskimmuch ofthematerial,bneflyreviewingthatpartwhich1Sfamiliar,andspendingat leastsomepreliminaryeffortonthepartsthatareunfamiliar.Manyofthe conceptspresentedherestrictlyfromtheviewpoint oflinearalgebra,partlcu-lariythoserelatedtoeigenvectors,arereintroducedandelaboratedonwith appHcationsinChapter5inthecontextofdynamicsystems.Accordingly, mun!readers willfinditadvantageousto studythiSmaterialbyrefemng back andforthbetweenthetwochapters. Linear Algebra ALGEBRAICPROPERTIES 3.1FUNDAMENTALS Muchoflinearalgebraismotivatedbyconsiderationofthegeneralsystemof mlinearalgebraicequationsinnunknowns: auXt + a12x2 + + atnXn= Yt a21 Xt + a22X2 + + a2nXn=Y2 am 1Xt + (k, 0) =Akk 2::0 Thesepowerscan, of course,befoundnumericallybybruteforce,butinthis caseitisrelativelyeasytofindananalyticexpressionforthem.Wewntethe matrixAasA= I+ B.where 0 0 0 0 Wemaythenusethebinomialexpansiontowrite A k=(I+ B)k = Jk + ++ ... + Bk denotes the binomial coefficientk!!(k-i)!(i!). (The binomial expan-sionisvalidinthismatrixcasebecauseIandBcommute-thatis,because IB =Bl.)Theexpressionsimplifiesinthisparticular casebecause 000 000 000 000 108LinearStateEquations but B3 (andeveryhigherpower)iszero.Thus, Ak=(I+B)k =I+ kB+ k(k2-1) B2 or,explicitly, 1000 k100 Ak= k(k -1) 2 k10 -k(k+ 1) -k01 2 OnceA kisknown,itispossibletoobtainanexpressionfortheclass populationsatanygenerationkintermsoftheinitialpopulations.Thus, x(k)= x1(0) kx1 (0) + x2(0) !k(k -1)x1(0) + kx2(0) + x3(0) -!k(k +1)x1(0)- kx2(0) + X4(0) From thisanalytical solution onecan determinethebehavior ofthesocial system.First,it canbe directly verifiedthatthetotalpopulation of thesociety isconstantfromgenerationtogeneration.Thisfollowsfromtheearlier assumption(c)andcanbeverifiedbysummingthecomponentsofx(k). Next,itcanbeseenthatunlessx1(0) =x2(0) = 0,thereisnosteady distributionofpopulationamongtheclasses.If,however,x1(0) =x2(0) = 0, corresponding tono Suns or Nobles initially,there willbenoSuns or Nobles in anysuccessivegeneration .. Inthissituation,theHonoredandStinkard populationbehaveaccordingtox3(k+l)=x3(k),x4(k+1)=x4(k),andthere-foretheirpopulationsremainfixedattheirinitialvalues. If eithertheSunor Nobleclassisinitiallypopulated,thenthenumber of Stinkardswilldecreasewithk,andultimatelytherewillnotbeenough Stinkards tomarry allthemembers of theruling class.At thispoint the social system,asdefinedbythegivenmarriagerules,breaksdown. 4.5GENERALSOLUTIONTOLINEARDISCRETE-TIMESYSTEMS Wetumnowtoconsiderationoftheforcedsystem x(k+ 1) =A(k)x(k)+ B(k)u(k)(4-11) 4.5GeneralSolutiontoLinearDiscrete-timeSystems109 Asbeforex.(k)isann-dimensionalstatevector,A(k)isann x nsystem matrix,B(k)isann x mdistributionmatrix,andn(k)isanm-dimensional input vector. The general solutionto this systemcan be expressedquite simply intermsofthestate-transitionmatrixdefinedinSect.4.4.Thesolutioncan beestablishedeasilybyalgebraicmanipulation,andweshalldothisfirst. Interpretationofthesolutionis,however,justasimportantasthealgebraic verification,andamajorpartofthissectionisdevotedtoexpositionofthat interpretation. Proposition.Thesolutionof thesystem(4-11),intermsof theimtialstatex(O) andtheinputs,is k-1 x.(k) = tl>(k, O)x.(O) + L tl>(k,l + 1)B(Z)n(l)(4-12) i=O Proof.Toverifythattheproposedexpression(4-12)doesrepresentthesolu-tion,itisonly necessarytoverify that it satisfiesthebasicrecursion(4-11)and theinitialcondition.An importantrelationforthispurposeis tl>(k + 1,l + 1) =A(k)tl>(k, l + 1)(4-13) fromthebasicdefinitionof the state-transitionmatrix.Wenotefirstthatthe proposedsolutioniscorrectfork = 0,sinceitreducestox.(O) = x.(O).The verificationcanthereforeproceedbyinductionfromk =0. Theproposedsolution(4-12)whenwrittenwithk +1replacingkis k x.( k +1) = (k + 1, O)x.(O) +L tl>( k + 1, l + 1)B( l)n(l) The lastterm inthe summation (corresponding tol = k) can be separated from thesummationsigntoproduce k-1 x.(k + 1) =-tl>(k + 1, O)x.(O)+L tl>(k + 1, l + 1)B(l)n(l) + B(k)n(k) Usingrelation(4-13)thisbecomes k-1 x.(k + 1) =A(k)tl>(k, O)x.(O) + A(k)L tl>(k,l + 1)B(l)n(l) + B(k)n(k) This,intum, withtheproposedformforx(k),becomes x(k + 1) =A(k)x(k)+ B(k)n(k) showingthattheproposedsolutioninfactsatisfiesthedefiningdifference equation. I Superposition The linearity of the system(4-11)implies that the solution canbe computed by theprinciple of superposition.Namely,thetotalresponseduetoseveralinputs 110Linear StateEquations isthesumoftheirindividualresponses,plusaninitialconditionterm.This leadsto.ausefulinterpretationofthesolutionformula(4-12). Letusinvestigateeachterminthegeneralsolution(4-12).The firstterm ~ ( k , O)x(O)isthe contributiontox(k)duetotheinitialcondition x(O).It isthe responseof the systemasifit werefree.Whennonzeroinputs are present this termisnoteliminated,othertermsaresimplyaddedtoit. Thesecondterm,whichisthefirstofthetermsrepresentedbythe summationsign,isthatassociatedwiththefirstinput.Thetermis ~ ( k , l)B(O)u(O). To seehowthis term arises, let us look again at theunderlying systemequation x(k + 1) = A(k)x(k) + B(k)u(k) Atk = 0thisbecomes x(1) =A(O)x(O)+ B(O)u(O) If weassumeforthemomentthatx(O)=0[whichwemightaswellassume, sincewehavealreadydiscussedthecontributionduetox(O)],thenwehave x(1) = B(O)u(O) Thismeansthattheshort-termeffectof theinputu(O)istosetthestatex(1) equaltothevector B(O)u(O).Eveniftherewerenofurtherinputs,thesystem wouldcontinuetorespondtothisvalueofx(1)inamannersimilartoits responsetoaninitialcondition.Indeed,thevectorx(l)actsexactlylikean initialcondition,butatk = 1ratherthank ==0.Fromourknowledgeofthe behavior of freesystems wecanthereforeeasily deduce that the corresponding response,fork > 1,is x(k) =( k , l)x(l) Intermsofo(O),whichproducedthisx(1),theresponseis x(k) = W(k,1)B(O)u(O) whichispreciselythetermintheexpressionforthegeneralsolutioncorres-pondingtou(O). Foraninputatanothertime,sayattimeZ,theanalysisisvirtually identical.Intheabsence of initialconditionsor otherinputs,theeffectofthe inputu(lf istotransfer. thestatefromzeroattimeltoB(l}u(l)attimel + 1. Fromthispointtheresponseatk > l + 1isdeterminedbythefreesystem, leadingto x(k) = W(k,l + l)B(l)u(Z) astheresponseduetou(l). Thetotalresponseofthesystemisthesuperpositionoftheseparate 4.5GeneralSolutiontoLinearDiscrete-timeSystems111 responsesconsideredabove;theresponsetoeachindividualinputbeing calculated asafreeresponsetotheinstantaneouschangeitproduces.Wesee, therefore,intermsofthisinterpretation,thatthetotalsolution(4-12)tothe system can be regarded as a sum of freeresponses initiated at different times. Timelnvariant Systems(ImpulseResponse) If the system(4-11) istime-invariant, the general solution and its interpretation canbeslightlysimplified.Thisleadstotheformalconceptoftheimpulse responseof alinear time-invariant systemthat isconsidered in greater detailin Chapter8. Correspondingtothelineartime-invariantsystem x(k + 1) = Ax(k)+ Bu(k)(4-14) thestate-transitionmatrixtakesthesimpleform ti.J(k,l + 1) =A e - l - t(4-15) Therefore,thegeneralsolutioncorrespondingto(4-14)is k-1 x(k) = A"x(O)+L Ak-t-1Bu(l)(4-16) i=O Everythingsaidaboutthemoregeneraltime-varyingsolutioncertainly appliestothisspecialcase.To obtainfurtherinsightinthiscase,however,let uslookmorecloselyattheresponseduetoasingleinput.Forsimplicity assumethattheinputisscalar-valued(i.e.,one-dimensional).Inthatcasewe writethedistributionmatrixBasbtoindicatethatitisinfactann-vector. Theresponseduetoaninputu(O)attimek =0is x(k) = Ak-1bu(O) If u(O) = 1,corresponding toaunitinput attimek =0,theresponsetakesthe form x(k) =Ak-1b This responseistermed the impulse response of the system.It isdefined asthe responseduetoaunitinputattimek = 0. Theimportanceoftheimpulseresponseisthatforlineartime-invariant systemsitcanbeusedtodeterminetheresponsetolaterinputsaswell.For example,letuscalculatetheresponsetoaninputu(l).Becausethesystemis time-invariant, the response due to an input at timelisidentical tothat dueto oneof equalmagnitudeattimezero,exceptthatitisshiftedbyltimeunits. Thustheresponseis x(k) =A"-1-1bu(l)fork 2: l + 1 Of course,theresponsefork < liszero. 112Linear StateEquations The response of a linear time-invariant systemtoan arbitrary sequence of inputs ismadeup fromthe basic response pattern of theimpulse response. This basicresponsepatternisinitiatedat varioustimeswithvariousmagnitudesby inputsatthosetimes;themagnitudeofaninputdirectlydeterminingthe proportionatemagnitudeofthecorrespondingresponsepattern.Thetotal response,whichmayappearhighlycomplex,isjust thesumoftheindividual (shifted)responsepatterns. Example(First-OrderSystem).Considerthesystem x(k + 1) = ax(k) + u(k) where0 (t, T)(4-18) (t, T)B(T)D(T) dT+ B(t)u(t) =A(t)x(t) + B(t)u(t) whichshowsthattheproposedsolutionsatisfiesthesystemequation. I Superposition Theprincipleofsuperpositionappliestolinearcontinuous-timesystemsthe sameasitdoestodiscrete-timesystems.Theoveralleffectduetoseveral 120LinearStateEquations u(t) Figure 4.5.Decompositionofinput. differentinputsisthesumoftheeffectsthatwouldbeproducedbythe individualinputs(if theinitialconditionwerezero).Thisideacanbeusedto interprettheformulaforthegeneralsolution. Thefirstterm on theright-handsideof the solution(4-32)representsthe responseduetotheinitialconditionx(O).Thisresponseisdetermineddirectly bythestate-transitionmatrixdevelopedforthefreesystem,anditisa component ofeverysolution. Tointerpretthesecondterm,imaginetheinputfunctionu(t)asbeing brokenupfinelyintoa sequence ofindividualpulses of width.6.,asillustrated in Fig.4.5.At timeTthe pulse willhavean (approximate) height of u('r). If the pulseatTweretheonlyinput,andiftheinitialstatewerezero,thenthe immediateeffectofthispulsewouldbetotransferthestatefromzero,just prior tothepulse,to.6.B( T )u( T)just after it.This isbecausethe resulting value of thestateistheintegralofthepulse. Afterthestatehasbeentransferredfromzeroto.6.B( T )u( T ),thelonger-termresponse,intheabsenceoffurtherinputs,isdeterminedbythefree system.Therefore,fort > TtheresponseduetothepulseatTwouldbe .6.4J(t, T)B(T)U(T).Thetotaleffectduetothewholesequenceofpulsesisthe sum of the individual responses,asrepresented in the limit by the integral term ontheright-handsideof thesolutionformula(4-32). Example(F:arst-OrderDetay).Considerafirst-ordersystemgovernedbythe equation x(t) = -rx(t)+ u(t)(4-33) wherer > 0. This isreferred to as a decay system, since inthe absence of inputs thesolutionis (4-34) whichdecaystozeroexponentially. Supposethe system isinitially at rest,attimet = 0,and an inputuof unit magnitudeisappliedstartingat timet = 0.Letuscalculatetheresultingtime 4.8EmbeddedStatics121 x(t) ~ - - - - - - - - - - - - - - - - - - - - ~ ~ t0 Figure4.6.Responseof decay system. response.Thestate-transitionmatrix(whichis1 x 1inthiscase)ts c,f)(t)=e-rt The solutionwithzeroinitialconditionandunityinputts,therefore, ThisresponseisillustratedinFig.4.6. *4.8EMBEDDEDSTATICS Informulatingequationstodescribeadynamicsituation,theequationsone writes may not initially be in the standard state variable form. It is, however, often most convenient to transform the equations to the standard form. This procedure IS usuallynotdifficult;indeed,in manyinstancesitissostraightforward that one performs thenecessary operations without hesitation. Nevertheless,it isworth-whiletorecognizethatthistransformationisinfactanecessarystep. Ageneral formthat islikelytoarise(arbitrarily expressedindiscrete-time just for -specificity)is Ex(k + 1) =Ax(k) + Bu(k)(4-35) whereEandAaren x nmatricesandBisann x mmatrix.Thesematrices mayingeneral depend onkwithoutaffecting theessence of our discussion.If Eisnonsingular,itissimpletotransformtheequations bymultiplyingbythe inverseof E.Thisyieldsthestandard statevectorform x(k + 1) = E-1 Ax(k) + E-1Bu(k)(4-36) If Eisnotinvertiblethesituationismoreinteresting.Thesystemthen consists of a mixture of static and dynamic equations; the staticequations being in somesense embedded withinthedynamicframework.Under rather general 122LinearStateEquations conditions(seeProblem21)suchasystemwithembeddedstaticscanbe transformedtoastatevectordynamicsystemhavinganorderlessthanthe dimensionoftheoriginalsystemofequations.Thefollowingexamplesillus-tratethtspoint. Example1.Considerthesystemdefinedby x1(k + 1)+ x2(k + 1) =x1(k)+ 2x2(k)+ u(k) 0 =2x1(k)+ xz(k) + u(k) Th1shastheformof(4-35)with E=~ ~ ](4-37) (4-38) which issingular. To obtain the reduced formforthisparticular system, weadd thetwoequationstoproduce x1(k + 1) +x2(k + 1) =3[x1(k) + x2(k)]+ 2u(k) Thisshowsthatthevariable (4-39) (4-40) can serve asa state variable forthe system. The dynamic portion of the system takestheform z(k+ 1) =3z(k)+2u(k)(4-41) Theoriginalvariablesx1 andx2 canbeexpressedintermsofzanduby solving(4-38)and(4-40)simultaneously. This leadsto x1(k) = -z(k)- u(k) x2(k)= 2z(k)+ u(k) (4-42) (4-43) Example2(NationalEconomics-TheHarrod-TypeModel).Adynamic model of thenational economy was proposed inSect.1.3. In terms of variables thathaveaspecific economic meaning,the basis forthemodel isthefollowing threeequations: Y(k) =C(k)+ I(k) + G(k) C(k)= mY(k) Y(k + 1)- Y(k) =rl(k) Intheseequations only thevariableG(k)isanmputvariable.Theothersare derivedvariablesthat,atleastinsomemeasure,describetheconditionofthe 4.8EmbeddedStatics123 system.Inavector-matrix formatthedefiningequationstaketheform

001Y(k + 1)r01Y(k)0 In this formitisclear that the original equations can beregarded asa dynamic systemwithembeddedstatics.Thisparticularsystemiseasytoreducetoa first-ordersystembyaseriesof substitutions,ascarriedout inSect.1.3.This leadstothefirst-orderdynamicsystem Y(k + 1) = [1 + m(1- r)] Y(k)- rG(k) The other variables can berecoveredby expressing them intermsofY(k) and G ( k).In particular, C(k)= mY(k) I(k) =(1-m) Y(k)- G(k) Example3(NationalEconomic=s-AnotherVersion).Thedynamicmodelof thenationaleconomypresentedabovecanberegardedasbeingbut oneofa whole familyofpossible(andplausible)models.Other formsthat arebased on slightlydifferenthypothesescanresultindistinctdynamicstructures.The relationshipsbetweenthesedifferentmodelsismostclearlyperceivedinthe nonreducedform;that isintheformthat containsembeddedstatics. Samuelsonproposedamodelofthenationaleconomybasedonthe followingassumptions.NationalincomeY(k)isequaltothesumof consump-tionC(k), investment I(k), and government expenditureG(k). Consumption is proportionaltothenationalincomeoftheprecedingyear;andinvestmentis proportionaltotheincreaseinconsumerspendingofthatyearoverthe precedingyear. Inequationform,theSamuelsonmodelis Y(k) =C(k) + I(k) + G(k) C(k+ 1)= mY(k) I(k + 1) = IL[ C(k + 1)- C(k)] Inourgeneralizedmatrixform,thesystembecomes

i)] =1-,_,.0Y(k + 1)0,-,_,. -l][I(k)][1] mC(k)+0G(k) 0Y(k)0 Thissystemcanbereducedtoasecond-order systeminstandardform. 124LinearStateEquations 4.9PROBLEMS 1.MovingAverage.Therearemanys1tuationswhererawdata1Ssubjectedtoan averaging processbeforeit isdisplayedor used fordecisionmaking.This smoothes thedatasequence,andoftenhighlightsthetrendswhilesuppressingindividual devtations. Supposea sequence of rawdata isdenotedu(k). As1mplefour-point averager producesacorresponding sequencey(k) suchthateachy(k)istheaverageofthe datapointsu(k),u(k-1), u(k-2), u(k-3). Find arepresentatwn forthe averager oftheform x(k + 1) =Ax(k)+ bu(k) y(k)=:[x1(k)+ u(k)] wherex(k)tsthree-dimensional,Aisa3 x 3matnx:,andbisa3 x 1(column) vector. 2.CohortModel.Supposethatthemputu(k)ofnewmachmesintheexamplem Sect.4.1ISchosentoexactlyequalthenumberofmachmesgomgoutofserv1ce that year.Writethecorrespondingstatespacemodelandshowthatittsaspecial caseofthegeneral cohortpopulatiOnmodel describedmChapter1.Repeat under the assumptiOnthat inadditionto replacements there arenewpurchases amounting toypercent ofthetotalnumberofmachinesinservice. 3.Considerthelineardifferenceequation y(k + n)+ Cln-tY(k + n -1)+ + aoy(k) =b,._1u(k+n-l)+b,._2u(k+n-2)+, +bou(k) ShowthatthiSequationcanbeputinstate spaceform bydefimng x(k + 1)=Ax(k)+bu(k) Xt(k) = -a0y(k -1) + b0u(k -1) x2(k) =-a0y(k- 2) + b0u(k- 2) -a1y(k -1) + btu(k -1) x,._1(k) = -a0y(k- n + 1) + b0u(k- n + 1) -a1y(k- n +2) + b1u(k- n +2) -Cin-2Y(k -1) + bn-2u(k -1) x..(k) = y(k) I i ) I~,' l '" ' -, .-- ,-:.") 4.9Problems125 4.NonlinearSystems.Considerthenonlineardifferenceequationof the form y(k + n) =F[y(k + n -1), ... , y(k), u(k + n -1), ... , u(k), k] (a)Findastatespacerepresentationofthedifferenceequation.(Hint:Therep-resentationwillbemorethann-dimensional.) (b)Find ann-dimensional representation inthe casewhere Fhas the special form n. F= Ifi(y(k+n-i), u(k+n-i)) t .. l 5.Labor-ManagementNegotiations.Considerawagedisputebetweenlaborand management. At each stage of thenegotiations, labor representatives submtt a wage demandtomanagement that,intum, presentsacounter offer.Sincethe wage offer willbeusuallylessthanthewagedemand,furthernegotiationsarerequired.One can formulatethis situat10n asa dynamtc system, where at eachperiod management "updates"itspreviousofferbytheadditionofsomefractionaofthedifference between lastperiod's demand and offer.Laboralso"updates" its prevtous demand bythesubtractionofsomefraction(3ofthedifferencebetweenthedemandand offerofthelastperiod.Let x1 equalthemanagementofferandx2 equalthelabor demand.Wntethedynamicstateequations(inmatrixform)forthesttuatton describedabove. 6.ConsiderthetwosoctaisystemswhosemarriagerulesaresummanzedmFig.4.7. Ineach systemtherearefour sociaiclasses,andeverychildborntoacertain class combinationbecomesamemberoftheclassdesignatedinthetable.Theassump-tlOns(a),(b),and(c)oftheNatchez Indianexamplehold aswell.For each system: (a)writethestateequattonforthesocmlsystem;and(b)computethesolutlonto the stateequations. 7.ASimplePuzzle.Wehavefourtimepieceswhoseperformanceisdescribedas follows:Thewallclocklosestwominutesinanhour.Thetableclockgetstwo .... Clol.t::. -cq u. 1 2 3 41 Mother 234 4 4 4 234 Figure 4.7.Socialsystems. .... Clol.t::. ... m u. l 2 3 4 l 2 Motner 2 l 3 34 2 3 44 126LinearStateEquations mmutesaheadofthewallclockinanhour.Thealarmclockfallstwominutes behmdthetableclockin anhour.The lastpiece,thewristwatch,getstwominutes ahead of the alarm clockin an hour.At noonallfourtunep1eceswere set correctly. Let x1 equal the wall clock reading,x2 bethetable clock reading,x3 the alarm clock reading,andx4 the wristwatchreading,and consider noonasthe starting time (i.e., k =0). (a)Writethedynamic equat10nscorrespondingtothefourgiven statementsabout performance.Directlytranslateintotheform Ex(k + 1) =Cx(k) + c (b)Convertthesystemtothestandardform x(k+1)=Ax(k)+b [Hint:(I- Br1 =I+ B + B2 + + Bk + wheneverBissuchthattheseries converges.] (c)Findthestate-transttionmatrixcf(k). (d)Findasimplegeneralformulaforx(k).Whattunewillthewristwatchshowat 7:00p.m.(i.e.,atk = 7)? 8.AClass1cPuzzle.Repeat Problem 7aboveforthe alternatedescnption:The wall dock losestwominutesinanhour. The tableclockgetstwominutesaheadofthe wallclockforeveryhourreg1steredonthewallclock.Thealarmclockfallstwo minutesbehmdthetableclockforeveryhourregisteredonthetableclock.The wristwatchgetstwominutesaheadofthealarmclockforeveryhour regiSteredon thealarmclock. 9.PropertiesofState- TransitzonMatrix.Letcf(t,r)bethestate-transitionmatrix correspondingtothelinear system i(t) =A(t)x(t) Show: (a)cf(t2,to)= cf(t2,ft)cf(t"t0) (b)cfJ{t,T)-1 =cf(T,t) (c)ddTO. (a)Definestatevanablesx1:::::yandx2 = yandfindarepresentationoftheform i(t) = A(t)x(t) + bu(t) (b)Findtwolinearlymdependentsolutionstothehomogeneousscalarequation. (Hint:Tryy =t".) (c)ConstructthematnxX(t),afundamentalmatnxofsolutions,basedonthe resultsofpart(b).. (d)Findthestate-transinonmatrix {3> 0,thegrowthfactorof rural imbalanceisalwayslessthanthatofpopulation,soeventuallytherelative imbalance(thatis,theratioofimbalancetototalpopulation)tendstodisap-pear.If {3>atthena- /3< 0andmigrationoscillates,beingfromruralto urbanoneyearandfromurbantoruralthenext.If {3coprovidedthattherealpartof..\isnegative.Inconclusion, therefore,thecompanionto Theorem1isstatedbelow.(AlsoseeFig.5. 7 .) Theorem2.Anecessaryandsufficzentconditionforanequilibriumpomt of the contmuous-time system(5-51)tobeasymptoticallystableisthat theeigen-valuesof Aallhavenegativerealpart(thatis,theetgenvaluesmustliein the left half of thecomplexplane).If atleast one eigenvalue has positivereal part,thepointisunstable. Asinthediscrete-timecase,stabilityofanequilibriumpointofalinear continuous-time system depends only on the structure of thematrix Aandnot explicitly on the equilibrium point itself. Thus, we say "the system isasymptoti-cally stable" or "the matrixAisasymptotically stable" when,inthecontext of a continuous-time linear system, the eigenvalues of Aare all insidethe left-half plane. MarginalStability For discrete- and continuous-time systems,thereisanintermediate casethat is notcoveredbytheabovestabilitytheorems.Fordiscretetimethiscaseis wherenoeigenvaluesare outsidetheunitcirclebut one or more isexactly on the boundary of thecircle.For continuoustime,itiswherenoeigenvaluesare intherighthalf ofthecomplexplane,but one or morehaverealpart exactly equaltozero.(SeeFigs.5.6and5.7 .) 5.9Stability159 Intheseintermediate situationsfurtheranalysisisrequiredtocharacterize thelong-termbehaviorofsolutions.If theboundaryeigenvaluesallhavea complete setofassociatedeigenvectors,thentheonlytermstheyproduceare oftheformA kindiscretetime,ore""tincontinuoustime.Thetermshave constant absolute value. Therefore, the state vector neither tends toinfimtynor tozero.The systemisneitherunstable,norasymptotically stable.Thisspecial intermediatesituationisreferredtoasmarginalstability.Itisanimportant specialcase,arisinginseveralapplications. Ex:ample.Somecommontwo-dimensionalsystemshavetheform i(t) = Ax(t) wheretheAmatrixhastheoff-diagonalform A= [0c1] c2 0 Forinstance,themassandspringmodelhasthisformwithc1 = -w2,c2 = 1. TheLanchestermodelof warfarehasthisformwithc1 andc2 bothnegative. ThecharacteristicequationassociatedwiththisAmatrixts Therelationshipoftheparametersc1 andc2 tostabilitycanbededuced directlyfromthisequation.Therearereallyonlythreecases. CAsE1.Theparametersc1 andc2 arenonzeroandhavethesamesign. Inthecasec 1 c2 ispositive,andthereforethetwoeigenvaluesareA 1 = ~2andA2 =-...rc:c2Thefirstof theseisitselfalwayspositive andtherefore thesystemisunstable. CAsE2.Theparametersc1 andc2 arenonzeroandhaveoppositesigns. Inthiscase(ofwhichthemassandspringisanexample)c1c2 isnegative and thereforethetwoeigenvalues are imaginary (lying ontheaxisbetweenthe leftandrighthalvesofthecomplexplane).Thisimpliesoscillatorybehavior andasystemwhichismarginally stable. CAsE3.Eitherc1 orc2 iszero. If both c1 andc2 are zero the system ismarginally stable. If only one of the twoparametersc1 andc2 iszero,thesystemhaszeroasarepeatedrootina chain of length two.Some components willgrow linearly withk and,therefore, thesystemisunstable. Inviewofthesethreecases,itisclearthatanoff-diagonalsystemcan neverbe stable(exceptmarginally). 160LinearSystems withConstantCoefficients 5.10OSCillATIONS Agood dealof qualitativeinformationabout solutionscanbeinferreddirectly fromtheeigenvalues of asystem,evenwithoutcalculationofthecorrespond-ingeigenvectors.Forinstance,asdiscussedinSect.5.9,stability propertiesare determinedentirelybytheeigenvalues.Othergeneralcharacteristicsofthe solution canbe deduced by considering the placement of the eigenvaluesinthe complex plane.Each eigenvalue defines both acharacteristic growth rate and a characteristicfrequencyofoscillation.Theserelationsareexaminedinthis section. It issufficienttoconsider only homogeneoussystems,sincetheir solutions underlie thegeneralresponse of linear systems.Eveninthiscase, however,the complete pattern of solutioncanbequitecomplex,forit depends onthe initial statevector andonthetimepatternsofeachoftheeigenvectorsthatin conjunctioncomprisethesolution.Todecomposethesolutionintocompo-nents, 'itisnaturalto consider thebehavior associated with asmgleeigenvalue, or acomplexconjugatepatr ofeigenvalues.Eachoftheseactsseparatelyand hasadefinitecharacteristicpattern. Continuous Time Let A be an eigenvalue of acontinuous-time system.It isconvenientto express AintheformA =JJ- + iw.Thecharacteristicresponseassociatedwiththis eigenvalueise"' =eJ.f.leiW'.Thecoefficientthatmultipliestheassociatedeigen-vectorvariesaccordingtothischaracteristicpattern. If Aisreal,thenA= p,andw =0.The coefficiente"risthenalwaysof the samesign.Nooscillationsarederivedfromaneigenvalueofthistype. If Aiscomplex,thenitscomplexconjugateX =p,- iwmustalsobean eigenvalue.Thesolutionitselfisalwaysreal,so,overall,theimaginarynum-berscancelout.ThecontributionduetoA andA inanycomponent therefore containstermsoftheform(A sin wt+B cos wt)ej.L'.Suchtermsoscillatewitha frequencywandhaveamagnitudethateithergrowsordecaysexponentially accordingtoe .......Insummary,forcontinuous-timesystemsthefrequencyof oscillation{inradiansperunittime)duetoaneigenvalueisequaltoits imaginarypart.Therateofexponentialgrowth(ordecay)tsequaltoitsreal part.{SeeFig.5 .8a.) Discrete Time Let A bean eigenvalue of a discrete-time system.In thiscase it isconvenient to expressAintheformA= rei8 = r(cos 8 + i sin8).Thecharacteristicresponse duetothiseigenvalueisAk= rkeike= rkcosk8+ irksinkO.Thecoefficientthat multipliestheassociatedeigenvectorvariesaccordingtothtscharacteristic pattern. p.=Rateof growth w = Frequency of osci II a t1 on p. ( a ~X t I lw I I l I I lw t I I X 5.10Oscillations161 r== Rate of growth 8=Frequency of oscillatton (b) Figure 5.8.Relation of eigenvalue location to oscillation. (a)Continuous time.(b) Discretetime. If A isrealandposztive,theresponsepatternisthegeometnc sequencer\ whichincreasesifr > 1 anddecreasesif r < 1.Nooscillationsare derived from apositiveefgenvalue.If A isrealandnegatwe,theresponseisanalternating geometric sequence. IfAiscomplex,itwillappearwithitscomplexconjugate.Thereai responseduetobotheigenvaluesisoftheformrk(A sin kO + BcoskO).If 0 =;!:.0,theexpressionwithintheparentheseswillchangesignaskvanes. However,theexact pattern of variationmay not be perfectly regular.Infact,if 0isanirrationalmultipleof277',theexpressiondoesnothaveafinitecycle length sinceeachvalueofkproducesadifferentvalueintheexpression.This isillustratedinFig.5.9a.Because of thisphenomenon,itisusefulto supenm-poseaquasi-continuoussolutiononthediscretesolutionbyallowingktobe continuous.Theperiodofoscillationthencanbemeasuredinthestandard way,and often willbe somenonintegralmultiple ofk.This isillustratedinFig. 5.9b.Insummary,fordiscrete-timesystemsthefrequencyofoscillation(in ,-, ,-\ ,-, I ~I ' II\ f \ I\ I\ ~ I\I\ ' k \ k \ \ ~ I ,_, ( a ~ (M Figure5.9.Adiscrete-timepatternanditssuperimposedquasicontinuous pattern. 162LinearSystemswithConstantCoefficients radiansper unittime)duetoaneigenvalue isequaltottsangleasmeasured in thecomplexplane.Therateofgeometricgrowth(ordecay)isequaltothe magnitudeoftheeigenvalue.{SeeFig.5.8b.) Example(fhe Hog Cycle).For nearlyacentury ithasbeenobservedthatthe productionofhogsischaracterizedby astrong,nearlycycleofabout four years' duration. Production alternately rises and falls, forcing prices tocycle correspondinglyintheoppositedirection.(SeeFig.5.10).Theeconoinic hardshiptofarmerscausedby thesecycleshasmotivatedanumber of govern-mentpoliciesattemptingtosmooththem,andhasfocusedattentiononthis rathercuriousphenomenon. Onecommonexplanationofthecycleisbasedonthecobwebtheoryof supply and demandinteraction,thebasic argumentbeingthatthecyclesarea result of the farmers'useof current prices inproduction decisions.By respond-ingquicklytocurrentpriceconditions,hogproducersintroducethecharac-teristiccobweb oscillations.However,asshownbelow,whenthepure cobweb theory is appliedtohog production,it predictsthatthe hog cyclewould have a periodofonlytwoyears.Thus,thissimpleviewofthebehaviorofhog producers isnot consistent withthe observed cycle length.It seems appropriate thereforetorevisetheassumptionsandlookforanalternativeexplanation. Onepossibleapproachisdiscussedinthesecondpartofthisexample. Hogpnces (Dollarsper100 19001910192019301940195019601970 Figure5.10.TheU.S.hogcycle. 5.10Oscillations163 The CobwebModel Asomewhat simplifieddescriptionofthehog production processtsasfollows: Therearetwohogproductionseasonsayear correspondingtospringandfall breeding.Itisaboutsixmonthsfrombreedingtoweaning,andfiveorsix monthsofgrowthfromweaningtoslaughter.Veryfewsowsarerequiredfor thebreeding stock, sofarmersgenerally slaughterandmarket essentiallythetr whole stock of mature hogs.Translated into analytical terms, itisreasonableto developadiscrete-timemodelwithabasicperiodlengthofsixmonths. Productioncanbeinitiatedatthebeginningofeachperiod,buttakestwo periodsto complete.Inthisdiscrete-time frameworkwedenotethenumber of maturehogsproducedatthebeginningofperiodkbyh(k),andthecorres-pondingunitpricebyp(k). Asisstandard,weassumethatdemandforhogsinanyperiodisdeter-minedbypriceaccordingtoalineardemandcurve.Inparticular,demandat periodkisgivenby d ( k) =d0- ap ( k)(5-53) Inasimilarway,weassumethatthereisasupply curveoftheform (5-54) The quantityh represents the levelof seasonal productionthat afarmer wouid maintainifhisestimate of futurehogpriceswerep. Inthecobwebmodel,itisassumedthatatthebeginningofpenodka fannerdecideshowmanyhogstobreedonthebasisofcurrentprice.Thts leadstotheequations h(k + 2) =s0 + bp(k) d(k) = d0- ap(k) (5-55) (5-56) whereh(k +2)correspondstothebreedingdecisionatpenodk,which specifiestheultimatenumber of maturehogsat periodk + 2.Equating suppiy anddemand[i.e.,settingd(k) = h(k)] andeliminatingp{k)leadsto h(k + 2) =-ph(k) + pd0 + s0 (5-57) wherep = b/ a.Thecharacteristicpolynomialofthisdifferenceequattonts (5-58) whichhasthetwoimaginaryrootsA= t.Ji,. Thisresultisslightlydifferentinformthanthatobtainedmtheusuai cobwebmodel since there are twoimaginary roots rather thanasinglenegative root.Thisisbecausetheproductionintervalisnowdividedintotwopenods. 164LinearSystemswithConstantCoefficients However,thesquaresoftheseimaginaryrootsareequaltotherootcorres-pondingtoafullproductionduration. Theinterpretationoftheresultsofthismodelisthathogcycleswould haveaperiodoffourtimeintervals{sincetherootsareimaginary)or, equivalently,oftwoyears.Furthermore,hogproductionisaside-lineopera-tiononmanyfarmsthatiseasilyexpandedorcontracted,dependingon economicincentives.Thissuggeststhatp > 1andhencethecobwebmodel predictsanunstablesituation. SmoothedPricePredictionModel AsanaiternatiVemodel,weassumethatfarmersareawareoftheinherent mstabilityandaccompanyingoscillatorynatureofthehogmarket.Accord-ingly,they actmoreconservatively thanimpliedbythepure cobwebmodelby basmg thetr estimate of futureprice on anaverageof past prices.For example. sincetheoscillationisroughlyfouryearsinduratwn,ittsreasonableto average over atimespanof at leasthalfthislong.Auniformaverage over two yearswouldhavetheform p(k) = Usingthisestimatedpriceinthesupplyequatton(5-54)andequatingsupply anddemandleadsto h(k +2)- ~ [h(k) + h(k -1)+ h(k -2)+ h(k- 3)+ h(k -4)] + pdo+so (5-60) This sixth-orderdifferenceequationhasthecharacteristicequation {5-61) It isof course difficulttocomputetheroots of thissixth-order polynomial. However,assumingp =2.07,itcanbeverifiedthattherootsoflargest magnitudeareA= ..ffi2 i..ffi2.Theserootshaveamagnitudeofoneand correspondtoacycleperiodofexactlyfouryears.Theresponseduetothese eigenvaluesisthusanoscillationofaperiodoffouryearsthatpersists indefinitely,neitherincreasingor decreasinginmagnitude. Onecanargueofcoursethatdifferentpredictorswouldleadtodifferent roots,andhence,todifferentcyclelengths.However,wearguethatinthe aggregate,producersaverageinsuchawaysoastojustmamtain(marginal) stability.Ashort-term averagewouldleadtoshortercyclesbut instability.An excessivelylong-termaveragewouldleadtolongcyclelengths,but asluggish systeminwhichfarmersdonotrespondtothestrongeconomicincentivesto 5.11DominantModes165 adjusttopricetrends.Asagrouptheytendtowalkthefinelinebetween stabilityandinstability.If p:::::: 2,thentoreachthisbalancepoint,thepnce estimateusedmustbecloseinformtotheonepresentedby(5-59),and consequentlythecyclelengthmustbeclosetofouryears. 5.11DOMINANTMODES Thelong-termbehaviorofalineartime-invariant systemoftenisdetermined largelybyonlyoneor twoofitseigenvaluesandcorrespondingeigenvectors. Thesedominanteigenvaluesandeigenvectorsarethereforeof spectalmterest totheanalyst. Discrete-TimeSystem Considerthediscrete-timesystem x{k + 1) = Ax(k)(5-62) Suppose thematnx Ahas eigenvaluesA1,A2,. ,AnwithIAtl > IA2I;;;;: IA3l,> jA,JTheeigenvalueA1 of greatestmagnitudetsthedommantetgenvalue.For simplicity,letusassumethatthereisacompletesetofeigenvectors,andthat theretsonlyoneeigenvalueofgreatestmagnitude.If thereweretwogreatest of equalmagnitude(astherewouldbeif theeigenvalueswerecomplex),then bothwouldbeconsidereddominant. Any initialcondition vector canbe expressed asa linear combination of all eigenvectorsintheform x{O) =a1e1 + a2e2 + +an en Correspondingly,thesolutionto(5-62)atanarbitrarytimek > 0ts x(k) =a1A+

+ +(5-63) (5-64) SincegrowsfasterthanA 7 fori =2,3, ... , n,itfollowsthatforlargek fori=2,3, ... ,n aslongasa1 0.Inotherwords,forlargevaluesofk,thecoefficientofthe firsteigenvectorintheexpansion{5-64)islargerelativetotheothercoeffi-cients.Hence,forlargevaluesofk,thestate vector x{k)isessentially aligned withtheeigenvector e1 If a1 = 0,thefirstcoefficientin{5-64)iszeroforallvaluesofk,andthe state vector willnot line upwith e1Theoretically, in this casethe eigenvalue of nextgreatestmagnitudewoulddeterminethebehavior ofthesystem forlarge values ofk.In practice, however,thedominant eigenvaluealmost alwaystakes hold-atleastultimately-foraslightperturbationatanystepintroducesa smalla1 0thatgrowsfasterthananyother term. 166LinearSystemswithConstantCoefficients Fromtheabovediscussion,itcanbededucedthatifthereisasingle dominant eigenvalue,thestatevectortendstoalignitself wtththe correspond-ingeigenvector.If there isnotasingledominanteigenvalue,but twothatare complexconjugates,itcanbesimilarlyinferredthatthestatevectortends towardthetwo-dimensionalspacedefinedbythetwocorresponding eigenvec-tors.Typically,inthiscase,oscillationsaregenerated,characteristicofthe complexdominateeigenvalues. Asimilaranalysisappliestononhomogeneoussystemssuchas x(k + 1) =Ax(k) +b Ifthereisanasymptoticallystableequilibriumpointi,thenthestate convergestoi. Therateatwhichitconvergesisgovernedessentiallybythe eigenvalue of greatestmagnitude(whichis,however,lessthan one). The error vector x(k) -i willbe closelyalignedwtththe corresponding eigenvector,asit tendstozero. Continuous.. TimeSystems Considerthesystem i(t) =Ax(t)(5-65) Suppose the matrixAhas eigenvaluesA11 A2, ,A"ordered now according to Re(A1)> Re(A2)>Re(A3)>Re(An).TheeigenvalueA1 withgreatestreal partisthedominant eigenvalueinthiscase. Asbeforeletussupposethereisasingledominanteigenvalueandthat thereisacompletesetofeigenvectors.Parallelingthepreviousanalysis,any initialstatevectorx(O)canbeexpressedintermsoftheeigenvectorsinthe form x(O).= a1e1 +a2e2 + + a"e" The correspondingsolutionto(5-65)is X( f)= al eAt tel+

+ + aneA.,.ten Writmgeacheigenvalueintermsofitsrealandimaginarypartsas =IL1c+ iWJ.; (5-66) (5-67) itiseasytoseethat= Thus,sincetherealpartdeterminesthe rateofgrowthof theexponentialit isclearthatas i:::::::2, 3, . ., n providedthata 1 t60.Therefore,thefirsttermin(5-67)dominatesallthe othersforlargevaluesoft,andhence,forlargevaluesoftthevectorx(t)is closelyalignedwiththedominanteigenvectore1 5.11DominantModes167 Subdominant Modes Thelong-termbehaviorofasystemisdeterminedmostdirectlybythe dominante