1ME451: Control SystemsME451: Control SystemsDr. Dr. JongeunJongeunChoiChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 4Lecture 4Modeling of electrical systemsModeling of electrical systems2Course 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 &) laboratories3Controller 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 circuits4Representation 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 systemsystemModelModelModelingModelingInputInputOutputOutput5Modeling 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 models6Transfer 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)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::SystemSystem8Models 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 transformtransform9ImpedanceImpedanceGeneralized 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!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 =0.11KirchhoffKirchhoff’’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.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)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)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)OutputOutputCC15Modeling example (contModeling example (cont’’d)d)Transfer functionTransfer functionvv11(t)(t)i(ti(t))RR22InputInputRR11vv22(t)(t)OutputOutputCC(first(first--order system)order system)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))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)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=0CC1119More 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. . 20Summary & ExercisesSummary & ExercisesModelingModelingModeling is an important task!Modeling is an important task!Mathematical modelMathematical modelTransfer functionTransfer functionModeling of electrical systemsModeling of electrical systemsNext, modeling of mechanical systemsNext, modeling of mechanical systemsExercisesExercisesRead Sections 2.2, 2.3 Read Sections 2.2, 2.3 Solve problems
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