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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 12Lecture 12TimeTime--domain specificationsdomain specifications2Course 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 &) laboratories3What we do nextWhat we do nextWe learned stability. We learned stability. Definition in time domainDefinition in time domainCondition in sCondition in s--domaindomainRouthRouth--Hurwitz criterion to check the conditionHurwitz criterion to check the conditionStability is a necessary requirement, but not Stability is a necessary requirement, but not sufficient in most control problems. sufficient in most control problems. Specifications other than stabilitySpecifications other than stabilityHow to evaluate a system quantitatively in time domain?How to evaluate a system quantitatively in time domain?How to give specifications in time domain?How to give specifications in time domain?What are the corresponding conditions in sWhat are the corresponding conditions in s--domain?domain?4Time responseTime responseWe would like to analyze a system property by We would like to analyze a system property by applying a applying a test inputtest inputr(tr(t) and observing a time ) and observing a time response response y(ty(t).).Time response is divided asTime response is divided asSystemSystemTransient responseTransient responseSteadySteady--state responsestate response(after (after yyttdies out)dies out)5Example of transient &Example of transient &steadysteady--state responsesstate responsesTransient responseTransient responseSteadySteady--state resp.state resp.Step ResponseTime (sec)Amplitude0 2 4 6 8 10 1200.511.522.53Step responseStep responseTime (sec)Time (sec)6Usage of time responsesUsage of time responsesModelingModelingSome parameters in the system may be estimated by Some parameters in the system may be estimated by time responses.time responses.AnalysisAnalysisEvaluate transient and steadyEvaluate transient and steady--state responses state responses (Satisfactory or not?)(Satisfactory or not?)DesignDesignGiven design specs in terms of transient and steadyGiven design specs in terms of transient and steady--state responses, design controllers satisfying all the state responses, design controllers satisfying all the design specs.design specs.7Typical test inputsTypical test inputsStep functionStep function(Most popular)(Most popular)Ramp functionRamp functionParabolic Parabolic functionfunctionSinusoidal input Sinusoidal input will be dealt with will be dealt with later.later.8Steady state value for step test signalSteady state value for step test signalSuppose that Suppose that G(sG(s) is stable) is stable..By the final value theorem:By the final value theorem:Step response converges to some finite value, Step response converges to some finite value, calledcalledsteady state value steady state value ..G(sG(s))9Typical unit step responseTypical unit step response10SteadySteady--state error for reference state error for reference uuss(t(t))11Peak value, peak time, and Peak value, peak time, and percent overshootpercent overshoot12Delay, rise, and settling timesDelay, rise, and settling timesDelay timeDelay time: time to reach 0.5 : time to reach 0.5 yyssssRise timeRise time: time to rise from 0.1y: time to rise from 0.1yssssto 0.9yto 0.9yssssSettling timeSettling time: time to settle within 5% of : time to settle within 5% of yyssss13An example revisitedAn example revisitedFor the example in a previous slide,For the example in a previous slide,SteadySteady--state error : 2state error : 2Delay time around 1.5 secDelay time around 1.5 secRise time around 5 secRise time around 5 secSettling time around 6 secSettling time around 6 secStep ResponseTime (sec)Amplitude0 2 4 6 8 10 1200.511.522.53RemarkRemark: There is no peak in : There is no peak in this case, so peak value, peak this case, so peak value, peak time and percent overshoot time and percent overshoot cannot be defined.cannot be defined.14Remarks on timeRemarks on time--domain responsesdomain responsesSpeed of responseSpeed of responseis measured byis measured byRise time, delay time, and settling timeRise time, delay time, and settling timeRelative stabilityRelative stabilityis measured byis measured byPercent overshootPercent overshootIn general In general ……..Fast response Fast response ÆÆLarge percent overshootLarge percent overshootLarge percent overshoot Large percent overshoot ÆÆsmall stability marginsmall stability marginWe need to take tradeWe need to take trade--off between response off between response speed and stability.speed and stability.15Summary and ExercisesSummary and ExercisesTime response and time domain specificationsTime response and time domain specificationsTime response can be used for Time response can be used for ••Parameter estimationParameter estimation••Design specification of the feedback systemDesign specification of the feedback systemTime response is difficult to compute analytically, Time response is difficult to compute analytically, except 1st and 2nd order systems (weexcept 1st and 2nd order systems (we’’ll study later).ll study later).NextNextWhen does steady state error become zero?When does steady state error become zero?ExercisesExercisesRead about performance of feedback control Read about performance of feedback control


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MSU ME 451 - Control Systems

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