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Fall 2008 1
ME451: Control SystemsME451: Control Systems
Dr. JongeunDr.Jongeun ChoiChoi Department of Mechanical EngineeringDepartment of Mechanical Engineering Michigan State UniversityMichigan State University
Lecture 22 Lecture 22 Frequency responseFrequency response
Course roadmapCourse roadmap
Laplace transformLaplace transform
Transfer functionTransfer function
Models for systemsModels for systems
- • electricalelectrical
- • mechanicalmechanical
- • electromechanicalelectromechanical Block diagramsBlock diagrams LinearizationLinearization
ModelingModeling^ AnalysisAnalysis^ DesignDesign Time responseTime response
- • TransientTransient
- • Steady stateSteady state Frequency responseFrequency response
- • Bode plotBode plot StabilityStability
- • RouthRouth--HurwitzHurwitz
- • NyquistNyquist
Design specsDesign specs Root locusRoot locus Frequency domainFrequency domain PID & LeadPID & Lead--laglag Design examplesDesign examples
(Matlab (Matlab simulations &) laboratoriessimulations &) laboratories
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33
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What is frequency response?What is frequency response?
We would like to analyze a system property byWe would like to analyze a system property by applying aapplying a test sinusoidal inputtest sinusoidal input u(t) andu(t) and observing a response y(tobserving a responsey(t).).
Steady state response ySteady state responsey ssss (t(t) (after transient dies) (after transient dies out) of a system to sinusoidal inputs is calledout) of a system to sinusoidal inputs is called frequency response frequency response ..
SystemSystem
A simple exampleA simple example
RC circuitRC circuit
Input a sinusoidal voltage u(tInput a sinusoidal voltageu(t))
What is the output voltage y(tWhat is the output voltagey(t)?)?
RR
u(t)u(t) (^) CC y(t)y(t)
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Response to sinusoidal inputResponse to sinusoidal input
How is the steady state output of a linear systemHow is the steady state output of a linear system when the input is sinusoidal?when the input is sinusoidal?
Steady state outputSteady stateoutput Frequency is same as the input frequencyFrequencyis same as the input frequency Amplitude is that of input (A) multiplied byAmplitudeis that of input (A) multiplied by Phase shiftsPhaseshifts GainGain
G(sG(s))
y(ty(t))
Frequency response functionFrequency response function
For a stable system G(sFor a stable systemG(s),), G(jG(jωω) () (ωω is positive) isis positive) is calledcalled frequency response function (FRF)frequency response function (FRF) ..
FRF is a complex number, and thus, has anFRF is a complex number, and thus, has an amplitude and aamplitudeand a phasephase..
First order exampleFirst order example
Re Re
ImIm
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Another example of FRFAnother example of FRF
Second order systemSecond order system
Re Re
ImIm
First order example revisitedFirst order example revisited
FRFFRF
Two graphs representing FRFTwo graphs representing FRF Bode diagram (Bode plot) (Today)Bode diagram (Bode plot) (Today) Nyquist diagram (Nyquistdiagram (NyquistNyquist plot)plot)
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Exercises of sketching Bode plotExercises of sketching Bode plot
First order systemFirst order system
Remarks on Bode diagramRemarks on Bode diagram
Bode diagram shows amplification and phaseBode diagram shows amplification and phase shift of a system output for sinusoidal inputs withshift of a system output for sinusoidal inputs with various frequencies.various frequencies.
It is very useful and important in analysis andIt is very useful and important in analysis and design of control systems.design of control systems.
The shape of Bode plot contains information ofThe 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.)
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System identificationSystem identification
^ Sweep frequencies of sinusoidal signals andSweep frequencies of sinusoidal signals and obtain FRF data (i.e., gain and phase).obtain FRF data (i.e., gain and phase).
^ SelectSelect 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 Analyzer
UnknownUnknown systemsystem
Generate sin signalsGenerate sin signals Sweep frequenciesSweep frequencies
Collect FRF dataCollect FRF data Select G(sSelectG(s))
Summary and exercisesSummary and exercises
Frequency response is a steady state responseFrequency response is a steady state response of systems to a sinusoidal input.of systems to a sinusoidal input. For a linear system, sinusoidal input generatesFor a linear system, sinusoidal input generates sinusoidal output with same frequencysinusoidal output withsame frequency butbut different amplitude and phase.different amplitude and phase. Bode plot is a graphical representation ofBode 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 functions Exercise: Read Section 8.Exercise: Read Section 8.
