1ME451: Control SystemsME451: Control SystemsDr. Dr. JongeunJongeunChoiChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 21Lecture 21Root locus: Lag compensator & Root locus: Lag compensator & LeadLead--lag compensator designlag compensator design2Course 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 &) laboratories3ClosedClosed--loop design by root locusloop design by root locusPlace closedPlace closed--loop poles at desired locationloop poles at desired locationby tuning the gain by tuning the gain C(sC(s)=K.)=K.If root locus does not pass the desired location, If root locus does not pass the desired location, then reshape the root locusthen reshape the root locusby adding poles/zeros to by adding poles/zeros to C(sC(s).).G(sG(s))C(sC(s))PlantPlantControllerControllerCompensationCompensationFixed!Fixed!Designable!Designable!4Lead and lag compensators (review)Lead and lag compensators (review)LeadLeadcompensatorcompensatorG(sG(s))C(sC(s))PlantPlantControllerControllerReReImImLagLagcompensatorcompensatorReReImImThe reason why these are called The reason why these are called ““leadlead””and and ““laglag””will be explained will be explained in frequency response approach (later in this course).in frequency response approach (later in this course).5Compensator realizationCompensator realizationOne example, using operational amplifiersOne example, using operational amplifiersRR22RR11CC22--++CC11--++RR33RR44vvii(t(t))vvoo(t(t))6Compensator realization (contCompensator realization (cont’’d)d)Transfer functionTransfer functionLeadLeadcompensatorcompensatorReReImImLagLagcompensatorcompensatorReReImIm7Roles of lead and lag compensatorsRoles of lead and lag compensatorsLead compensator (Done)Lead compensator (Done)Improve transient responseImprove transient responseImprove stabilityImprove stabilityLag compensator (Today)Lag compensator (Today)Reduce steady state errorReduce steady state errorLeadLead--lag compensator (Today)lag compensator (Today)Take into account all the above issues.Take into account all the above issues.8Radar tracking systemRadar tracking system9LeadLead--lag compensator designlag compensator designConsider a systemConsider a systemAnalysis of CL system for Analysis of CL system for C(sC(s)=1)=1Damping ratio Damping ratio ζζ=0.5=0.5UndampedUndampednatural freq. natural freq. ωωnn=2 =2 rad/srad/sRampRamp--error constant error constant KvKv=2=2Performance specificationPerformance specificationDamping ratio Damping ratio ζζ=0.5=0.5UndampedUndampednatural freq. natural freq. ωωnn=4 =4 rad/srad/sRampRamp--error constant error constant KvKv=50=50G(sG(s))C(sC(s))PlantPlantControllerControllerReReImImDesired poleDesired pole100 1 2 3 4 500.20.40.60.811.21.4Comparison of step responses Comparison of step responses (after lead compensation)(after lead compensation)Uncompensated system (Uncompensated system (C(sC(s)=1))=1)Compensated systemCompensated systemLead compensatorLead compensatorgives gives ••faster transient responsefaster transient response(shorter rise and settling time)(shorter rise and settling time)••improved stabilityimproved stability110 1 2 3 4 5012345Error constants Error constants (after lead compensation)(after lead compensation)StepStep--error constanterror constantRampRamp--error constanterror constantLag compensator can reduce steadyLag compensator can reduce steady--state error. state error. Unit ramp inputUnit ramp inputRamp responseRamp responseNOT SATISFACTORY!NOT SATISFACTORY!12How to design lag compensatorHow to design lag compensatorLag compensatorLag compensatorWe want to increase rampWe want to increase ramp--error constanterror constantTake, for example, z=10p.Take, for example, z=10p.We do not want to change CL pole location sWe do not want to change CL pole location s11so so much (already satisfactory transient).much (already satisfactory transient).13Guidelines to choose z and pGuidelines to choose z and pThe zero and the pole of a lag compensator The zero and the pole of a lag compensator should be should be close to each otherclose to each other, for , for The pole of a lag compensator should be The pole of a lag compensator should be close close to the originto the origin, to have a large ratio , to have a large ratio z/pz/p, leading to , leading to a large rampa large ramp--error constant error constant KvKv..However, the pole of a lag compensator too However, the pole of a lag compensator too close to the origin may be problematic:close to the origin may be problematic:Difficult to realize (recall opDifficult to realize (recall op--amp realization)amp realization)Slow settling (due to closedSlow settling (due to closed--loop pole near the origin)loop pole near the origin)14Root locus with lag compensatorRoot locus with lag compensatorWithout compensatorWithout compensatorss11With compensatorWith compensatorss1115How to design lag compensatorHow to design lag compensatorFor the desired CL poleFor the desired CL poleTake a small p (by trialTake a small p (by trial--andand--error!)error!)LeadLead--lag controllerlag controller16Root locusRoot locus-6 -4 -2 0-15-10-5051015-6 -4 -2 0-15-10-5051015Root LocusReal AxisImaginary AxisRoot LocusReal AxisImaginary AxisWith lead compensatorWith lead compensatorWith leadWith lead--lag compensatorlag compensatorDesired poleDesired pole17Comparison of step responsesComparison of step responses0 1 2 3 4 500.20.40.60.811.21.4UncompensatedUncompensatedWith lead compensatorWith lead
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