1ME451: Control SystemsME451: Control SystemsDr. Dr. JongeunJongeunChoiChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 22Lecture 22Frequency responseFrequency response2Course 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 &) laboratories3333112211333What is frequency response?What is frequency response?We would like to analyze a system property by We would like to analyze a system property by applying a applying a test sinusoidal inputtest sinusoidal inputu(tu(t) and ) and observing a response observing a response y(ty(t).).Steady state response Steady state response yyssss(t(t) (after transient dies ) (after transient dies out) of a system to sinusoidal inputs is called out) of a system to sinusoidal inputs is called frequency responsefrequency response..SystemSystem4A simple exampleA simple exampleRC circuitRC circuitInput a sinusoidal voltage Input a sinusoidal voltage u(tu(t))What is the output voltage What is the output voltage y(ty(t)?)?RRCCu(tu(t))y(ty(t))5An example (contAn example (cont’’d)d)TF (R=C=1)TF (R=C=1)u(tu(t)=)=sin(tsin(t))0 5 10 15 20 25 30 35 40 45 50-1-0.8-0.6-0.4-0.200.20.40.60.81 ryAt steadyAt steady--state, state, u(tu(t) and ) and y(ty(t) has same frequency, ) has same frequency, but different amplitude and phase!but different amplitude and phase!6An example (contAn example (cont’’d)d)Derivation of Derivation of y(ty(t))Inverse LaplaceInverse Laplace0 as t goes to infinity.0 as t goes to infinity.Partial fraction expansionPartial fraction expansion(Derivation for general (Derivation for general G(sG(s) is given at the end of lecture slide.)) is given at the end of lecture slide.)7Response to sinusoidal inputResponse to sinusoidal inputHow is the steady state output of a linear system How is the steady state output of a linear system when the input is sinusoidal?when the input is sinusoidal?Steady stateSteady stateoutput output FrequencyFrequencyis same as the input frequencyis same as the input frequencyAmplitudeAmplitudeis that of input (A) multiplied byis that of input (A) multiplied byPhasePhaseshifts shifts GainGainG(sG(s))y(ty(t))8Frequency response functionFrequency response functionFor a stable system For a stable system G(sG(s), ), G(jG(jωω) () (ωωis positive) is is positive) is called called frequency response function (FRF)frequency response function (FRF)..FRF is a complex number, and thus, has an FRF is a complex number, and thus, has an amplitudeamplitudeand a and a phasephase..First order exampleFirst order exampleReReImIm9Another example of FRFAnother example of FRFSecond order systemSecond order systemReReImIm10First order example revisitedFirst order example revisitedFRFFRFTwo graphs representing FRFTwo graphs representing FRFBode diagram (Bode plot) (Today)Bode diagram (Bode plot) (Today)NyquistNyquistdiagram (diagram (NyquistNyquistplot) plot) 11Bode diagram (Bode plot) of Bode diagram (Bode plot) of G(jG(jωω))Bode diagram consists of Bode diagram consists of gain plotgain plot& & phase plotphase plotLogLog--scalescale12Bode plot of a 1st order systemBode plot of a 1st order systemTF TF 10-210-1100101102-50-40-30-20-10010-210-1100101102-100-80-60-40-200Corner frequencyCorner frequency13Exercises of sketching Bode plotExercises of sketching Bode plotFirst order systemFirst order system14Remarks on Bode diagramRemarks on Bode diagramBode diagram shows amplification and phase Bode diagram shows amplification and phase shift of a system output for sinusoidal inputs with shift of a system output for sinusoidal inputs with various frequencies.various frequencies.It is very useful and important in analysis and It is very useful and important in analysis and design of control systems.design of control systems.The shape of Bode plot contains information of The shape of Bode plot contains information of stability, time responses, and much more!stability, time responses, and much more!It can also be used for system identification. It can also be used for system identification. (Given FRF experimental data, obtain a transfer (Given FRF experimental data, obtain a transfer function that matches the data.)function that matches the data.)15System identificationSystem identificationSweep frequencies of sinusoidal signals and Sweep frequencies of sinusoidal signals and obtain FRF data (i.e., gain and phase).obtain FRF data (i.e., gain and phase).Select Select G(sG(s) so that ) so that G(jG(jωω) fits the FRF data.) fits the FRF data.Agilent Technologies: FFT Dynamic Signal AnalyzerAgilent Technologies: FFT Dynamic Signal AnalyzerUnknownUnknownsystemsystemGenerate sin signalsGenerate sin signalsSweep frequenciesSweep frequenciesCollect FRF dataCollect FRF dataSelect Select G(sG(s))16Summary and exercisesSummary and exercisesFrequency response is a steady state response Frequency response is a steady state response of systems to a sinusoidal input.of systems to a sinusoidal input.For a linear system, sinusoidal input generates For a linear system, sinusoidal input generates sinusoidal output with sinusoidal output with same frequencysame frequencybut but different amplitude and phasedifferent amplitude and phase..Bode plot is a graphical representation of Bode plot is a graphical representation of frequency response function. (frequency response function. (““bode.mbode.m””))Next, Bode diagram of simple transfer functionsNext, Bode diagram of simple transfer functionsExercise: Read Section 8.Exercise: Read Section 8.17Derivation of frequency responseDerivation of frequency
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