1ME451: Control SystemsME451: Control SystemsDr. Jongeun ChoiDr. Jongeun ChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 13Lecture 13SteadySteady--state errorstate error2Course 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(Matlab simulations &) laboratories(Matlab simulations &) 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.(Today(Today’’s lecture)s lecture)(From next lecture)(From next lecture)4SteadySteady--state error: state error: unity feedbackunity feedbackSuppose that we want output y(t) to track r(t).Suppose that we want output y(t) to track r(t).Error Error SteadySteady--state errorstate errorFinal value theorem Final value theorem (Suppose CL system is stable!!!)(Suppose CL system is stable!!!)Unity feedback!Unity feedback!We assume that the We assume that the CL system is stable!CL system is stable!5Error constantsError constantsStepStep--error (positionerror (position--error) constanterror) constantRampRamp--error (velocityerror (velocity--error) constanterror) constantParabolicParabolic--error (accelerationerror (acceleration--error) constanterror) constantKp, Kv, Ka :Kp, Kv, Ka :ability to reduce steadyability to reduce steady--state errorstate error6SteadySteady--state error for step r(t)state error for step r(t)KpKp7SteadySteady--state error for ramp r(t)state error for ramp r(t)KvKv8SteadySteady--state error for parabolic r(t)state error for parabolic r(t)KaKa9System typeSystem typeSystem type of GSystem type of Gis defined as the order is defined as the order (number) of poles of G(s) at s=0.(number) of poles of G(s) at s=0.ExamplesExamplestype 1type 1type 2type 2type 3type 310Zero steadyZero steady--state error state error If error constant is infinite, we can achieve zero If error constant is infinite, we can achieve zero steadysteady--state error. (Accurate tracking)state error. (Accurate tracking)For step r(t)For step r(t)For ramp r(t)For ramp r(t)For parabolic r(t)For parabolic r(t)11Example 1Example 1G(s) of type 2G(s) of type 2Characteristic equationCharacteristic equationCL system is NOT stable for any K.CL system is NOT stable for any K.e(t) goes to infinity. (Done(t) goes to infinity. (Don’’t use todayt use today’’s results if s results if CL system is not stable!!!)CL system is not stable!!!)G(s)G(s)12Example 2Example 2G(s) of type 1G(s) of type 1By RouthBy Routh--Hurwitz criterion, CL is stable iffHurwitz criterion, CL is stable iffStep r(t)Step r(t)Ramp r(t)Ramp r(t)Parabolic r(t)Parabolic r(t)G(s)G(s)13Example 3Example 3G(s) of type 2G(s) of type 2By RouthBy Routh--Hurwitz criterion, we can show that CL Hurwitz criterion, we can show that CL system is stable.system is stable.Step r(t)Step r(t)Ramp r(t)Ramp r(t)Parabolic r(t)Parabolic r(t)G(s)G(s)14A control exampleA control exampleClosedClosed--loop stable?loop stable?Compute error constantsCompute error constantsCompute steady state errorsCompute steady state errors15Summary and ExercisesSummary and ExercisesSteadySteady--state errorstate errorFor For unity feedbackunity feedback(STABLE!) systems, the system (STABLE!) systems, the system type of the forwardtype of the forward--path system determines if the path system determines if the steadysteady--state error is zero.state error is zero.The key tool is the The key tool is the final value theoremfinal value theorem!!Next, time response of 1stNext, time response of 1st--order systemsorder systemsExercisesExercisesGo over the examples in this lecture.Go over the examples in this
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