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1、MassachusettsInstituteofTechnologyDepartmentofElectricalEngineeringandComputerScience6.243j(Fall2003):DYNAMICSOFNONLINEARSYSTEMSbyA.MegretskiLecture1:Input/OutputandState-SpaceModels1Thislecturepresentssomebasicdefinitionsandsimpleexamplesonnonlineardynamicalsystemsmodeling.1.1BehavioralMo
2、dels.Themostgeneral(thoughrarelythemostconvenient)waytodefineasystemisbyusingabehavioralinput/outputmodel.1.1.1Whatisasignal?kIntheselectures,asignalisalocallyintegrablefunctionz:R+≤�R,whereR+denotesthesetofallnon-negativerealnumbers.Thenotionof“localintegrability”comesfromtheLebesquemeasu
3、retheory,andmeanssimplythatthefunctioncanbesafelyandmeaningfullyintegratedoverfiniteintervals.Generalizedfunctions,suchasthedeltafunction�(t),arenotallowed.Theargumentt→R+ofasignalfunctionwillbereferredtoas“time”(whichitusuallyis).Example1.1Functionz=z(·)definedby�t−0.9sgn(cos(1/t))fort>0,z
4、(t)=0fort=01VersionofSeptember3,20032isavalidsignal,while�1/tfort>0,z(t)=0fort=0andz(t)=�˙(t)arenot.Thedefinitionaboveformallycoverstheso-calledcontinuoustime(CT)signals.Discretetime(DT)signalscanberepresentedwithinthisframeworkasspecialCTsignals.kMoreprecisely,asignalz:R+≤�RiscalledaDTsig
5、nalifitisconstantoneveryinterval[k,k+1)wherek=0,1,2,....1.1.2Whatisasystem?Systemsareobjectsproducingsignals(calledoutputsignals),usuallydependingonothersignals(inputs)andsomeotherparameters(initialconditions).Inmostapplications,mathematicalmodelsofsystemsaredefined(usuallyimplicitly)bybeh
6、aviorsets.Foranautonomoussystem(i.e.forasystemwithnoinputs),abehaviorsetisjustasetB={z}kconsistingofsomesignalsz:R+≤�R(kmustbethesameforallsignalsfromB).Forasystemwithinputvandoutputw,thebehaviorsetconsistsofallpossibleinput/outputpairsz=(v(·),w(·)).Thereisnorealdifferencebetweenthetwodefin
7、itions,sincethepairofsignalsz=(v(·),w(·))canbeinterpretedasasinglevectorsignalz(t)=[v(t);w(t)]containingbothinputandoutputstackedoneovertheother.Notethatinthisdefinitionafixedinputv(·)mayoccurinmanyorinnopairs(v,w)→B,whichmeansthatthebehaviorsetdoesnotnecessarilydefine
