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 responsePeak valuePeak valuePeak timePeak timePercent overshootPercent overshootDelay timeDelay timeRise timeRise timeSettling timeSettling timeSteady state responseSteady state responseSteady 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 systemA 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 systemTF from motor input voltage toTF from motor input voltage tomotor motor speedspeedis 1stis 1st--oder oder motor motor positionpositionis 2ndis 2nd--order order 6Step response for 1stStep response for 1st--order systemorder systemInput 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 errorMake 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 TK : K : GainGainFinal (steadyFinal (steady--state) valuestate) valueT : T : Time constantTime constantTime 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 systemDC gain : DC gain : Final valueFinal valueof a unit step responseof a unit step responseFor 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 systemsRelation 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 identificationSuppose that we have a Suppose that we have a ““blackblack--boxbox””systemsystemObtain step responseObtain step responseCan you obtain a transfer function? How?Can you obtain a transfer function? How?UnknownUnknown13Ramp response for 1stRamp response for 1st--order systemorder systemInput 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 systemSteady 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 errorMake 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 exercisesTime response for 1stTime response for 1st--order systemsorder systemsStep and ramp responsesStep and ramp responsesTime constant and DC gainTime constant and DC gainSystem identificationSystem identificationNext, time response for 2ndNext, time response for 2nd--order systemsorder systemsExercisesExercisesRead Section 4.Read Section 4.Solve Problems 4.1 and 4.2.Solve Problems 4.1 and
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