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Chapter2MathematicalModelsofSystems2-1IntroductiontoSystemModeling2-2LaplaceTransform2-3DifferentialEquationsofPhysicalSystems2-4TheTransferFunctionofLinearSystems2-5BlockDiagramModels2-6Signal-FlowGraphModels2-7EquivalenceamongModelsandSummary
Lecture4:TheTransferFunctionandBlockDiagramModels
(2-4,2-5,2hours)2-4TheTransferFunctionofLinearSystems2-5BlockDiagramModels
2-3TransferFunctions1.DefinitionandPropertiesThegeneralformofthelineardifferentialequationofaLTICsystemissupposedtobeWherey(t)istheoutput,r(t)istheinput.Withallinitialconditionsassumedtobezero,wecanhavethefollowingratiobyLaplacetransform:
ThetransferfunctionofaLTICsystem:theratiooftheLaplacetransformoftheoutputvariabletotheLaplacetransformoftheinputvariable,withallinitialconditionsassumedtobezero.ItisusuallydenotedasG(s).
Thisguaranteesthecausalityofphysicalsystems.Ifmn,thesystemcan’tberealizedphysically.Properties(1)ThetransferfunctionsareonlyapplicabletoLTICsystemswithallinitialconditionsassumedtobezero.functionsofsystemparameters(2)Transferfunctionsareusuallyrationalfractionfunctionsofcomplexvariables.Thatmeans,andthecoefficientsareallrealconstants.
(3)Transferfunctionsareonlytheinput-outputdescriptionofthebehaviorofsystems,anddonotincludeanyinformationconcerningtheinternalstructureofthesystemsandtheirbehavior.Differentphysicalsystemsmaysharethesametransferfunction(analogoussystems),andagivensystemmayhavedifferenttransferfunctionswhendifferentphysicalvariablesarechosenasinputoroutput.Suitableforblack-boxsystemsimulation.
(4)Transferfunctionfocusesontheabilityofsignaltransferringoflinearstationarysystemitself,andhasnot
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