Fall 2008 1ME451: Control SystemsME451: Control SystemsDr. Dr. JongeunJongeunChoiChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 12Lecture 12TimeTime--domain specificationsdomain specificationsFall 2008 2Course 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 &) laboratoriesFall 2008 3What we did and what we do nextWhat we did and what we do nextWe have learned stability.We have learned stability.Definition in time domainDefinition in time domainCondition in sCondition in s--domaindomainRouthRouth--Hurwitz criterion to check the conditionHurwitz criterion to check the conditionStability 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 stabilityHow 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?Fall 2008 4Time responseTime responseWe 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)Fall 2008 5Example of transient &Example of transient &steadysteady--state responsesstate responsesTransient responseTransient responseSteadySteady--state resp.state resp.Step ResponseTime (sec)Amplitude0 2 4 6 8 10 1200.511.522.53Step responseStep responseTime (sec)Time (sec)Fall 2008 6Usage of time responsesUsage of time responsesModelingModelingSome parameters in the system may be estimated by Some parameters in the system may be estimated by time responses.time responses.AnalysisAnalysisEvaluate transient and steadyEvaluate transient and steady--state responses state responses (Satisfactory or not?)(Satisfactory or not?)DesignDesignGiven 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.Fall 2008 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.Fall 2008 8Steady state value for step test signalSteady state value for step test signalSuppose 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))Fall 2008 9Typical unit step responseTypical unit step responseFall 2008 10SteadySteady--state error for reference state error for reference uuss(t(t))Fall 2008 11Peak value, peak time, and Peak value, peak time, and percent overshootpercent overshootFall 2008 12Delay, 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 yyssssFall 2008 13An example revisitedAn example revisitedFor the example in a previous slide,For the example in a previous slide,SteadySteady--state error : 2state error : 2Delay time around 1.5 secDelay time around 1.5 secRise time around 5 secRise time around 5 secSettling 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.Fall 2008 14Remarks on timeRemarks on time--domain responsesdomain responsesSpeed of responseSpeed of responseis measured byis measured byRise time, delay time, and settling timeRise time, delay time, and settling timeRelative stabilityRelative stabilityis measured byis measured byPercent overshootPercent overshootIn general In general ……..Fast response Fast response ÆÆLarge percent overshootLarge percent overshootLarge percent overshoot Large percent overshoot ÆÆsmall stability marginsmall stability marginWe need to take tradeWe need to take trade--off between response off between response speed and stability.speed and stability.Fall 2008 15Summary and ExercisesSummary and ExercisesTime response and time domain specificationsTime response and time domain specificationsTime response can be used for Time response can be used for ••Parameter estimationParameter estimation••Design specification of the feedback systemDesign specification of the feedback systemTime 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).NextNextWhen does steady state error become zero?When does steady state error become zero?ExercisesExercisesRead
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