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MSU ME 451 - Control Systems

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Spring 2010 1ME451: Control SystemsME451: Control SystemsProf. Clark Radcliffe,Prof. Clark Radcliffe,Prof. Prof. Jongeun ChoiJongeun ChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 2Lecture 2Laplace transformLaplace transformSpring 2010 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 &) laboratoriesSpring 2010 3Laplace transformLaplace transformOne of most important math tools in the course!One of most important math tools in the course!Definition: For a function f(t) (f(t)=0 for t<0),Definition: For a function f(t) (f(t)=0 for t<0),We denote Laplace transform of f(t) by F(s).We denote Laplace transform of f(t) by F(s).f(tf(t))tt00F(sF(s))(s: complex variable)Spring 2010 4Example of Laplace transformExample of Laplace transformStep functionStep function00f(t)f(t)tt55Remember L(u(t)) = 1/sf (t) = 5u(t) =5 t ≥ 00 t < 0⎧⎨⎪⎩⎪F(s) = f (t)e−stdt0∞∫= 5e−stdt0∞∫= 5 e−stdt0∞∫= 5 −1se−st⎡⎣⎤⎦0∞⎡⎣⎢⎤⎦⎥= 51s⎡⎣⎢⎤⎦⎥=5sSpring 2010 5Integration is HardIntegration is HardTables are EasierTables are EasierSpring 2010 6Laplace Laplace transform tabletransform table(Table B.1 in Appendix B of the textbook)(Table B.1 in Appendix B of the textbook)Inverse Laplace TransformInverse Laplace TransformSpring 2010 7Properties of Laplace transformProperties of Laplace transformLinearityLinearityEx.Ex.Proof.Proof.Spring 2010 8Properties of Laplace transformProperties of Laplace transformDifferentiationDifferentiationEx.Ex.Proof.Proof.tt--domaindomainss--domaindomainSpring 2010 9Properties of Laplace transformProperties of Laplace transformIntegrationIntegrationProof.Proof.tt--domaindomainss--domaindomainSpring 2010 10Properties of Laplace transformProperties of Laplace transformFinal value theoremFinal value theoremEx.Ex.if all the poles of if all the poles of sFsF(s) are in (s) 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.Spring 2010 11Properties of Laplace transformProperties of Laplace transformInitial 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 LHP or not. pole location is in LHP or not. if the limits exist.if the limits exist.Ex.Ex.Spring 2010 12Properties of Laplace transformProperties of Laplace transformConvolutionConvolutionIMPORTANT REMARKIMPORTANT REMARKConvolutionConvolutionL−1F1(s)F2(s)()≠ f1(t) f2(t)Spring 2010 13An advantage of An advantage of Laplace Laplace transformtransformWe can transform an ordinary differential We can transform an ordinary differential equation (ODE) into an algebraic equation (AE).equation (ODE) into an algebraic equation (AE).ODEODEAEAEPartial fraction Partial fraction expansionexpansionSolution to ODESolution to ODEtt--domaindomainss--domaindomain112233Spring 2010 14Example 1Example 11st Order ODE with input and Initial Condition1st Order ODE with input and Initial ConditionTake Take Laplace Laplace TransformTransformSolve for Solve for Y(s)Y(s) 5&y(t) +10y(t) = 3u(t)y(0) = 15 sY (s) − y(0)[]+10 Y(s)[]= 31s⎡⎣⎢⎤⎦⎥5s +10()Y(s) = 5y(0) + 31s⎡⎣⎢⎤⎦⎥Y(s) =55s +10()+3s 5s +10()=1s + 2()+0.6ss+ 2()(Initial Condition) + (Input)Spring 2010 15Example 1 (cont)Example 1 (cont)Use table to Invert Use table to Invert Y(s)Y(s)term by term to find term by term to find y(t)y(t)From the Table:From the Table:So thatSo thatY(s) =1s + 2()+0.6ss+ 2()L1s + a⎛⎝⎜⎞⎠⎟= e−at⇒ L−11s + 2⎛⎝⎜⎞⎠⎟= e−2tLass+ a()⎛⎝⎜⎞⎠⎟= 1− e−at()⇒ L−10.3()2ss+ 2()⎛⎝⎜⎞⎠⎟= 0.3 1− e−2t()y(t) = e−2t+ 0.3 1− e−2t()(Initial Condition) + (Input)Spring 2010 16Properties of Laplace transformProperties of Laplace transformDifferentiation (review)Differentiation (review)tt--domaindomainss--domaindomainSpring 2010 17Example 2Example 2s2Y(s) − s −1− Y(s) =1s2s2−1()Y(s) = (s +1) +1s2Y(s) =(s+1)s2−1()+1s2s2−1()Y(s) =As −1()+Bs +1()+Cs2(Find A, B and C)Y(s) =3/2()s −1()+−1/2()s +1()+(−1)s2y(t) = 3/2()et+−1/2()e−t+−1()tHow do we do that???Spring 2010 18Partial fraction expansionPartial fraction expansionMultiply both sides by (sMultiply both sides by (s--1) & let s 11) & let s 1Similarly,Similarly,unknownsunknownsY(s) =(s + 1)s2−1()+1s2s2−1()=As −1()+Bs +1()+Cs2s −1()Y(s)s→1= s − 1()As −1()+Bs +1()+Cs2⎡⎣⎢⎤⎦⎥= Aso A = s − 1()(s + 1)s2−1()+1s2s2−1()⎧⎨⎪⎩⎪⎫⎬⎪⎭⎪s→1= 1+1s2s +1()⎧⎨⎩⎫⎬⎭= 1+12=32B = s +1()Y(s )s→−1=−12C = s2()Y(s)s→0=−1Example 2 (contExample 2 (cont’’d)d)Spring 2010 19Example 3Example 3ODE with initial conditions (ICs)ODE with initial conditions (ICs)Laplace Laplace transformtransform(This also isn’t in the table…)Spring 2010 20Inverse Inverse Laplace Laplace transformtransformIf we are interested in only the final value of y(t), apply If we are interested in only the final value of y(t), apply Final Value Theorem:Final Value Theorem:Example 3 (contExample 3 (cont’’d)d)Spring 2010 21Example: NewtonExample: Newton’’s laws lawWe want to know the trajectory of x(t). By We want to know the trajectory of x(t). By Laplace Laplace transform,transform,MM(Total response)(Total response)= = (Forced response)(Forced response)+ + (Initial condition response)(Initial condition response)Spring 2010 22EX. Air bag and accelerometerEX. Air bag and accelerometerTiny MEMS


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MSU ME 451 - Control Systems

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