Fall 2008 1ME451: Control SystemsME451: Control SystemsDr. Dr. JongeunJongeunChoiChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 2Lecture 2Laplace transformLaplace transformFall 2008 2Course roadmapCourse roadmapLaplace transformLaplace transformTransfer functionTransfer functionModels for systemsModels for systems••electricalelectrical••mechanicalmechanical••electromechanical electromechanical Block diagramsBlock diagramsLinearizationLinearizationModelingModelingAnalysisAnalysisDesignDesignTime responseTime response••TransientTransient••Steady stateSteady stateFrequency responseFrequency response••Bode plotBode plotStabilityStability••RouthRouth--HurwitzHurwitz••((NyquistNyquist))Design specsDesign specsRoot locusRoot locusFrequency domainFrequency domainPID & LeadPID & Lead--laglagDesign examplesDesign examples((MatlabMatlabsimulations &) laboratoriessimulations &) laboratoriesFall 2008 3Laplace transformLaplace transformOne of most important math tools in the course!One of most important math tools in the course!Definition: For a function Definition: For a function f(tf(t) () (f(tf(t)=0 for t<0),)=0 for t<0),We denote Laplace transform of We denote Laplace transform of f(tf(t) by ) by F(sF(s).).f(tf(t))tt00F(sF(s))(s: complex variable)Fall 2008 4Examples of Laplace transformExamples of Laplace transformUnit step functionUnit step functionUnit ramp functionUnit ramp functionf(tf(t))tt0011f(tf(t))tt00(Memorize this!)(Integration by parts)Fall 2008 5Integration by partsIntegration by partsEX. Fall 2008 6Examples of Laplace transform (contExamples of Laplace transform (cont’’d)d)Unit impulse functionUnit impulse functionExponential functionExponential functionf(tf(t))tt00f(tf(t))tt0011(Memorize this!)Width = 0 Width = 0 Height = Height = infinfArea = 1Area = 1Fall 2008 7Examples of Laplace transform (contExamples of Laplace transform (cont’’d)d)Sine functionSine functionCosine functionCosine function(Memorize these!)Remark:Remark:Instead of computing Laplace Instead of computing Laplace transform for each function, and/or transform for each function, and/or memorizing complicated Laplace transform, memorizing complicated Laplace transform, use the use the Laplace transform table Laplace transform table !!Fall 2008 8Laplace transform tableLaplace transform table(Table B.1 in (Table B.1 in AppedixAppedixB of the textbook)B of the textbook)Inverse Laplace TransformInverse Laplace TransformFall 2008 9Properties of Laplace transformProperties of Laplace transform1.1.LinearityLinearityEx.Ex.Proof.Proof.Fall 2008 10Properties of Laplace transformProperties of Laplace transform2.Time delay2.Time delayEx.Ex.Proof.Proof.f(tf(t))00TTf(tf(t--T)T)tt--domaindomainss--domaindomainFall 2008 11Properties of Laplace transformProperties of Laplace transform3.3.DifferentiationDifferentiationEx.Ex.Proof.Proof.tt--domaindomainss--domaindomainFall 2008 12Properties of Laplace transformProperties of Laplace transform4.4.IntegrationIntegrationProof.Proof.tt--domaindomainss--domaindomainFall 2008 13Properties of Laplace transformProperties of Laplace transform5.5.Final value theoremFinal value theoremEx.Ex.if all the poles of if all the poles of sF(ssF(s) are in ) are in the left half planethe left half plane(LHP)(LHP)Poles of Poles of sF(ssF(s) are in LHP) are in LHP, so final value , so final value thmthmapplies.applies.Ex.Ex.Some poles of Some poles of sF(ssF(s) are not in LHP) are not in LHP, so final value , so final value thmthmdoes does NOTNOTapply.apply.Fall 2008 14Properties of Laplace transformProperties of Laplace transform6.6.Initial value theoremInitial value theoremEx.Ex.Remark: In this theorem, it does not matter if Remark: In this theorem, it does not matter if pole location is in LHS or not. pole location is in LHS or not. if the limits exist.if the limits exist.Ex.Ex.Fall 2008 15Properties of Laplace transformProperties of Laplace transform7.7.ConvolutionConvolutionIMPORTANT REMARKIMPORTANT REMARKConvolutionConvolutionFall 2008 16Summary & ExercisesSummary & ExercisesLaplace transform (Important math tool!)Laplace transform (Important math tool!)DefinitionDefinitionLaplace transform tableLaplace transform tableProperties of Laplace transformProperties of Laplace transformNext Next Solution to Solution to ODEsODEsvia Laplace transformvia Laplace transformExercisesExercisesRead Appendix A, B.Read Appendix A, B.Solve Problems B.1 (a), (b); B.2 (a), (c), (d).Solve Problems B.1 (a), (b); B.2 (a), (c),
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