Fall 2008 1ME451: Control SystemsME451: Control SystemsDr. Dr. JongeunJongeunChoiChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 4Lecture 4Modeling of electrical systemsModeling of electrical systemsFall 2008 2Course roadmapCourse roadmapLaplace transformLaplace transformTransfer functionTransfer functionModels for systemsModels for systems••electricalelectrical••mechanicalmechanical••electromechanicalelectromechanicalBlock diagramsBlock diagramsLinearizationLinearizationModelingModelingAnalysisAnalysisDesignDesignTime responseTime response••TransientTransient••Steady stateSteady stateFrequency responseFrequency response••Bode plotBode plotStabilityStability••RouthRouth--HurwitzHurwitz••NyquistNyquistDesign specsDesign specsRoot locusRoot locusFrequency domainFrequency domainPID & LeadPID & Lead--laglagDesign examplesDesign examples((MatlabMatlabsimulations &) laboratoriessimulations &) laboratoriesFall 2008 3Controller design procedure (review)Controller design procedure (review)plantplantInputInputOutputOutputRef.Ref.SensorSensorActuatorActuatorControllerControllerDisturbanceDisturbance1. ModelingMathematical modelMathematical model2. AnalysisControllerController3. Design4. ImplemenationWhat is the What is the ““mathematical modelmathematical model””??Transfer functionTransfer functionModeling of electrical circuitsModeling of electrical circuitsFall 2008 4Representation of the inputRepresentation of the input--output (signal) output (signal) relation of a physical systemrelation of a physical systemA model is used for the A model is used for the analysisanalysisand and designdesignof of control systems.control systems.Mathematical modelMathematical modelPhysical Physical systemsystemModelModelModelingModelingInputInputOutputOutputFall 2008 5Modeling is the Modeling is the most important and difficult taskmost important and difficult taskin control system design.in control system design.No mathematical model exactly represents a No mathematical model exactly represents a physical system.physical system.Do not confuse Do not confuse modelsmodelswith with physical systemsphysical systems!!In this course, we may use the term In this course, we may use the term ““systemsystem””to to mean a mathematical model.mean a mathematical model.Important remarks on modelsImportant remarks on modelsFall 2008 6Transfer functionTransfer functionA transfer function is defined byA transfer function is defined byA system is assumed to be at rest. (Zero initial A system is assumed to be at rest. (Zero initial condition)condition)Laplace transform of system outputLaplace transform of system outputLaplace transform of system inputLaplace transform of system inputFall 2008 7Impulse responseImpulse responseSuppose that Suppose that u(tu(t) is the unit impulse function ) is the unit impulse function and system is at rest.and system is at rest.The output The output g(tg(t) for the unit impulse input is called ) for the unit impulse input is called impulse responseimpulse response..Since Since U(sU(s)=1, the transfer function can also be )=1, the transfer function can also be defined as the defined as the Laplace transform of impulse Laplace transform of impulse responseresponse::SystemSystemFall 2008 8Models of electrical elements:Models of electrical elements:(constitutive equations)(constitutive equations)v(tv(t))i(ti(t))RRResistanceResistanceCapacitanceCapacitanceInductanceInductancev(tv(t))i(ti(t))LLv(tv(t))i(ti(t))CCLaplace Laplace transformtransformFall 2008 9ImpedanceImpedanceGeneralized resistance to a sinusoidal Generalized resistance to a sinusoidal alternating current (AC) alternating current (AC) I(sI(s))Z(sZ(s): ): V(sV(s)=)=Z(s)I(sZ(s)I(s))V(sV(s))I(sI(s))Z(sZ(s))Time domain Impedance Z(s)ResistanceResistanceCapacitanceCapacitanceInductanceInductanceElementMemorize!Memorize!Fall 2008 10KirchhoffKirchhoff’’s Voltage Law (KVL)s Voltage Law (KVL)The algebraic sum of voltage drops around any The algebraic sum of voltage drops around any loop is =0.loop is =0.Fall 2008 11KirchhoffKirchhoff’’s Current Law (KCL)s Current Law (KCL)The algebraic sum of currents into any junction The algebraic sum of currents into any junction is zero.is zero.Fall 2008 12Impedance computationImpedance computationSeries connectionSeries connectionProof (OhmProof (Ohm’’s law)s law)V(sV(s))I(sI(s))ZZ11(s)(s)ZZ22(s)(s)VV11(s)(s)VV22(s)(s)Fall 2008 13Impedance computationImpedance computationParallel connectionParallel connectionProof (OhmProof (Ohm’’s law)s law)KCLKCLV(sV(s))I(sI(s))ZZ11(s)(s)ZZ22(s)(s)II11(s)(s)II22(s)(s)Fall 2008 14Modeling exampleModeling exampleKirchhoff voltage lawKirchhoff voltage law(with zero initial conditions)(with zero initial conditions)By By Laplace transformLaplace transform,,vv11(t)(t)i(ti(t))RR22InputInputRR11vv22(t)(t)OutputOutputCCFall 2008 15Modeling example (contModeling example (cont’’d)d)Transfer functionTransfer functionvv11(t)(t)i(ti(t))RR22InputInputRR11vv22(t)(t)OutputOutputCC(first(first--order system)order system)Fall 2008 16vvddExample: Modeling of op ampExample: Modeling of op ampImpedanceImpedanceZ(sZ(s): ): V(sV(s)=)=Z(s)I(sZ(s)I(s))Transfer functionTransfer functionof the above op amp:of the above op amp:VVii(s(s))I(sI(s))InputInputZZii(s(s))VVoo(s(s))OutputOutput--++ii--Rule2: Rule2: vvdd=0=0Rule1: Rule1: ii--=0=0ZZff(s(s))IIff(s(s))Fall 2008 17Modeling example: op ampModeling example: op ampBy the formula in previous two pages,By the formula in previous two pages,vvii(t(t))i(ti(t))RR22InputInputRR11vvoo(t(t))OutputOutputCC--++ii--vvddVVdd=0=0ii--=0=0(first(first--order system)order system)Fall 2008 18vvddModeling exercise: op ampModeling exercise: op ampFind the transfer function!Find the transfer function!vvii(t(t))RR22InputInputRR11vvoo(t(t))OutputOutputCC22--++ii--VVdd=0=0ii--=0=0CC11Fall 2008 19More exercises in the textbookMore exercises in the textbookFind a transfer function from vFind a transfer function from v11to vto v22..Find a transfer function from vFind a transfer function from viito vto voo. . Fall 2008 20Summary & ExercisesSummary & ExercisesModelingModelingModeling is an important task!Modeling is an
View Full Document