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MSU ME 451 - Control Systems

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1ME451: Control SystemsME451: Control SystemsDr. Dr. JongeunJongeunChoiChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 14Lecture 14Time response of 1stTime response of 1st--order systemsorder 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 &) laboratories3Performance measures (review)Performance measures (review)Transient responseTransient responsePeak valuePeak valuePeak timePeak timePercent overshootPercent overshootDelay timeDelay timeRise timeRise timeSettling timeSettling timeSteady state responseSteady state responseSteady state errorSteady state errorNext, we will connect Next, we will connect these measures these measures with swith s--domain.domain.(Done)(Done)(Today(Today’’s lecture)s lecture)4FirstFirst--order systemorder systemA A standard formstandard formof the firstof the first--order system:order system:DC motor exampleDC motor example5DC motor example (contDC motor example (cont’’d)d)If La<<Ra, we can obtain a 1stIf La<<Ra, we can obtain a 1st--order systemorder systemTF from motor input voltage toTF from motor input voltage tomotor motor speedspeedis 1stis 1st--oder oder motor motor positionpositionis 2ndis 2nd--order order 6Step response for 1stStep response for 1st--order systemorder systemInput a Input a unit step functionunit step functionto a firstto a first--order system. order system. Then, what is the output?Then, what is the output?0011u(tu(t))y(ty(t))00(Partial fraction expansion)(Partial fraction expansion)7How to eliminate steadyHow to eliminate steady--state errorstate errorMake a feedback system with a controller having Make a feedback system with a controller having an integrator (an integrator (copy of Laplace transform of a unit copy of Laplace transform of a unit step functionstep function):):00u(tu(t))ControllerControllerOne has to select controller parameters One has to select controller parameters to stabilize the feedback system. to stabilize the feedback system. Suppose K=T=1, and obtain such parameters!Suppose K=T=1, and obtain such parameters!118Meaning of K and TMeaning of K and TK : K : GainGainFinal (steadyFinal (steady--state) valuestate) valueT : T : Time constantTime constantTime when response Time when response rises 63% of final value rises 63% of final value Indication of Indication of speedspeedof of response (convergence)response (convergence)Response is faster as T Response is faster as T becomes smaller.becomes smaller.0 1 2 3 4 5 600.20.40.60.81K=1,T=1TimeAmplitude9DC gain for a general systemDC gain for a general systemDC gain : DC gain : Final valueFinal valueof a unit step responseof a unit step responseFor firstFor first--order systems, DC gain is K.order systems, DC gain is K.For a For a general stable system Ggeneral stable system G, DC gain is G(0)., DC gain is G(0).ExamplesExamplesFinal value theoremFinal value theorem10Settling time of 1stSettling time of 1st--order systemsorder systemsRelation between time and exponential decayRelation between time and exponential decay5% settling time is about 3T!5% settling time is about 3T!2% settling time is about 4T!2% settling time is about 4T!11Step response for some K & T Step response for some K & T 0 5 10012K=1,T=1TimeAmplitude0 5 10012K=1,T=2TimeAmplitude0 5 10012K=2,T=1TimeAmplitude0 5 10012K=2,T=2TimeAmplitude12System identificationSystem identificationSuppose that we have a Suppose that we have a ““blackblack--boxbox””systemsystemObtain step responseObtain step responseCan you obtain a transfer function? How?Can you obtain a transfer function? How?UnknownUnknown13Ramp response for 1stRamp response for 1st--order systemorder systemInput a Input a unit ramp functionunit ramp functionto a 1stto a 1st--order system. order system. Then, what is the output?Then, what is the output?00u(tu(t)=t)=ty(ty(t))00(Partial fraction expansion)(Partial fraction expansion)140 1 2 3 4 5012345K=1,T=1TimeAmplitudeu(tu(t)=t)=ty(ty(t))Ramp response for 1stRamp response for 1st--order systemorder systemSteady state response Steady state response We may want to modify the system We may want to modify the system s.ts.t..TimeTimeK=1,T=1K=1,T=1AmplitudeAmplitudeslopeslope15How to eliminate steadyHow to eliminate steady--state errorstate errorMake a feedback system with a controller having Make a feedback system with a controller having a double integrator (a double integrator (copy of Laplace transform of copy of Laplace transform of ramp functionramp function):):00u(tu(t)=t)=tControllerControllerOne has to select controller parameters One has to select controller parameters to stabilize the feedback system. to stabilize the feedback system. Suppose K=T=1, and obtain such parameters!Suppose K=T=1, and obtain such parameters!16Summary and exercisesSummary and exercisesTime response for 1stTime response for 1st--order systemsorder systemsStep and ramp responsesStep and ramp responsesTime constant and DC gainTime constant and DC gainSystem identificationSystem identificationNext, time response for 2ndNext, time response for 2nd--order systemsorder systemsExercisesExercisesReview examples in this lecture.Review examples in this


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MSU ME 451 - Control Systems

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