第九章讲义——拉普拉斯变换

SignalsandSystems,Chapter9TheLaplaceTransformLiuKe,SchoolofAutomationEngineering,9.0Introduction,RepresentthesignalsandLTIsystems,Defect:cantbeusedinunstablesystem,Fouriertransform:,AnalysistheLTIsystemsinotherdomain,FouriertransformAnalysisthesysteminfrequencydomain,LaplacetransformAnalysisthesysteminSdomain,ConvolutionAnalysisthesystemintimedomain,Thecontentsofthechapter,TheLaplacetransformandInverseLaplacetransformTheROCforLaplacetransformGeometricevaluationoftheFouriertransformfromthePole-ZeroplotPropertiesoftheLaplacetransformAnalysisandcharacterizationofLTIsystemusingtheLaplacetransformSystemfunctionalgebraandblockdiagramrepresentationsTheunilateralLaplacetransform,9.1TheLaplacetransform,ConceptionLaplacetransformROCPole-ZeroplotTherelationshipbetweenLaplacetransformandFouriertransform,TherepresentationoftheLaplacetransform,Fouriertransform:,Laplacetransform:,TherelationshipbetweentheLaplacetransformandtheFouriertransform,If,FouriertransformisaparticularformofLaplacetransform,LaplacetransformisFouriertransformof,Example9.1,Example9.2,Regionofconvergence(ROC),ROCtherangeofvaluesofSforwhichintegralinconverges,Laplacetransformincludes:thealgebraicexpressionROC,TherepresentationofROCComplexplane(S-plane),Example9.3,Example9.4,Pole-Zeroplot,Laplacetransformmaybearatioofpolynomials,PolestherootsofD(S)ZerostherootsofN(S),TherepresentationofpolesandzerosPole-Zeroplot,AnotherrepresentationofLaplacetransformPole-ZeroplotandROC,Theorderofpoleorzero,Example9.5,Summarization,TherepresentationofLaplacetransformTherelationshipbetweenLaplacetransformandFouriertransformROCPole-Zeroplot,9.2TheROCforLaplacetransform,ThecharacteristicofROCTherelationshipbetweentheROCandthesignals,Property1theROCofX(S)consistsofstripsparalleltothejw-axisinthes-plane,Property2forrationalLaplacetransforms,theROCdoesnotcontainanypoles,Property3ifx(t)isoffinitedurationandisabsolutelyintegrable,thentheROCistheentires-plane,Example9.6,Property4ifx(t)isrightsided,andifthelineRes=isintheROC,thenallvaluesofSforwhichReswillalsobeintheROC,Property5ifx(t)isleftsided,andifthelineRes=isintheROC,thenallvaluesofSforwhichReswillalsobeintheROC,Property6ifx(t)istwosided,andifthelineRes=isintheROC,thentheROCwillconsistofastripinthes-planethatincludethelineRes=,Example9.7,Property7iftheLaplacetransformX(S)ofx(t)isrational,thenitsROCisboundedbypolesorextendstoinfinity.Inaddition,nopolesofX(S)arecontainedintheROC,Property8iftheLaplacetransformX(S)ofx(t)isrationalifx(t)isrightsided,theROCistheregioninthes-planetotherightoftherightmostpoleifx(t)isleftsided,theROCistheregioninthes-planetotheleftoftheleftmostpole,Example9.8,9.3TheinverseLaplacetransform,TherepresentationoftheinverseLaplacetransformTheusualmethodofdeterminetheinverserationalLaplacetransform,TherepresentationoftheinverseLaplacetransform,TheusualmethodofdeterminetheinverserationalLaplacetransform,Partial-fractionexpansion,Example9.9,Example9.10,Example9.11,9.4GeometricevaluationoftheFouriertransformfromthePole-Zeroplot,ReviewtherelationshipbetweenFouriertransformandLaplacetransformAsimplemethodgeometricevaluation,ReviewtherelationshipbetweentheLaplacetransformandFouriertransform,Thebasicknowledgeofgeometricevaluation,9.5PropertiesoftheLaplacetransform,LinearityTimeshiftingShiftinginthes-domainTimescalingConjugationConvolutionpropertyDifferentiationinthetimedomainDifferentiationinthes-domainIntegrationinthetimedomainTheinitial-andfinal-valuetheorems,Example9.13,Linearity,Timeshifting,Shiftinginthes-domain,Timescaling,Conjugation,Convolutionproperty,Differentiationinthetimedomain,DifferentiationintheS-domain,Example9.14,Example9.15,Integrationinthetimedomain,Theinitial-andfinal-valuetheorems,Example9.16,9.7AnalysisandcharacterizationofLTIsystemusingtheLaplacetransform,TherelationshipbetweenthepropertiesofsystemandH(S)HowcangettheX(S)orH(S)orY(S),Severalimportantpropertiesofsystem,Causality,TheROCassociatedwiththesystemfunctionforacausalsystemisaright-halfplane,Forasystemwitharationalsystemfunction,causalityofthesystemisequivalenttotheROCbeingtheright-halfplanetotherightoftherightmostpole,Example9.17,Thesystemiscausal,sotheROCofH(S)isaright-halfplane,Example9.18,Thesystemisnotcausal,theROCofH(S)isnotaright-halfplane,Example9.19,ROCisaright-halfplane,butthesystemisnotcausal,unlessthesystemisrational,Stability,AnLTIsystemisstableifandonlyiftheROCofitssystemfunctionH(S)includestheentirejw-axisi.e.ReS=0,Example9.20,AcausalsystemwithrationalsystemfunctionH(S)isstableifandonlyifallofthepolesofH(S)liesintheleft-halfofthes-planei.e.,allofthepoleshavenegativerealparts,Example9.21,LTIsystemcharacterizedbylinearconstant-coefficientdifferentialequations,SomeexampleTherelationshipbetweensystembehaviorandthesystemfunction,Themethodofdetermineh(t)bydifferentialequations,Usepartialfractionexpansion,Example9.23,Example9.24,Example9.25,Example9.26,AnLTIsystemincludetheinformationasbelow:1.thesystemiscausal2.thesystemfunctionisrationalandhasonlytwopoles,ats=-2ands=43.Ifx(t)=1,theny(t)=04.thevalueoftheimpulseresponseatt=0+is4,Example9.27,Astableandcausalsystem,H(S)isrational,containsapoleats=-2,anddoesnothaveazeroattheorigin,pleasedeterminethestatementsbelow:,P723-14P728-27P729-34,9.8Systemfunctionalgebraandblockdiagramrepresentations,ReviewthesystemfunctionsforinterconnectionsofLTIsystemSignal-passplotTheexampleofblockdiagramrepresentations,Parallelinterconnection,Seriescombination,Feedbackinterconnection,Signal-passplot（信流图）,Someconcepts节点表示一个信号。只有信号输出的节点称为源点，只有信号输入的节点是阱点，既有信号输入、又有信号输出的节点称为混合节点。节点所代表的信号等于输入该节点的全部信号的和，与输出支路无关。支路节点之间的有向线段称为支路通路一条或几条同方向的支路组成通路。通路的转移函数等于各支路转移函数的乘积。不闭合的通路称为开路，闭合的通路称为环路。,信流图分析梅森规则（相关术语1）,接触没有公共节点的通路称为不接触通路切断支路切断某条支路意味着取消这条支路而仍然保存两端的节点（不移去节点）移去节点移去某节点意味着截断与该节点连接的全部支路（移去该节点和与该节点相连接的所有支路）移去通路移去某一条通路意味着移去该通路上的所有节点,信流图分析梅森规则（相关术语2）,信流图行列式,信流图分析梅森规则,梅森规则应用举例,Example9.28,Example9.29,Example9.30,Directform,Cascadeform,Parallelform,Example9.31,Directform,Cascadeform,Parallelform,P723-17,9.9TheunilateralLaplacetransform,RepresentationoftheunilateralLaplac