1ME451: Control SystemsME451: Control SystemsDr. Dr. JongeunJongeunChoiChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 20Lecture 20Root locus: Lead compensator designRoot locus: Lead 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). (How?)). (How?)G(sG(s))C(sC(s))PlantPlantControllerControllerCompensationCompensationFixed!Fixed!Designable!Designable!(for time domain specs)(for time domain specs)4General effect of addition of polesGeneral effect of addition of polesPulling root locus to the RIGHTPulling root locus to the RIGHTLess stableLess stableSlow down the settlingSlow down the settlingReReImImReReImImReReImImAdd a poleAdd a poleAdd a poleAdd a pole5General effect of addition of zerosGeneral effect of addition of zerosPulling root locus to the LEFTPulling root locus to the LEFTMore stableMore stableSpeed up the settlingSpeed up the settlingReReImImAdd a zeroAdd a zeroReReImImReReImImReReImIm6Some remarksSome remarksAdding only zero Adding only zero often problematic because such controller amplifies often problematic because such controller amplifies the the highhigh--frequency noisefrequency noise. . Adding only pole Adding only pole often problematic because such controller generates often problematic because such controller generates a a less stableless stablesystem (by moving the closedsystem (by moving the closed--loop poles loop poles to the right). to the right). These facts can be explained by using These facts can be explained by using frequency response analysisfrequency response analysis..Add both zero and pole!Add both zero and pole!7Lead and lag compensatorsLead and lag compensatorsLeadLeadcompensatorcompensatorG(sG(s))C(sC(s))PlantPlantControllerControllerReReImImLagLagcompensatorcompensatorReReImImWhy these are called Why these are called ““leadlead””and and ““laglag””??We will see that from frequency response in this class.We will see that from frequency response in this class.8Lead compensatorLead compensatorPositive angle contributionPositive angle contributionReReImImTest pointTest pointss--zz11--pp119Lag compensatorLag compensatorNegative angle contributionNegative angle contributionReReImImTest pointTest pointss--zz22--pp2210Roles of lead and lag compensatorsRoles of lead and lag compensatorsLead compensator (Today)Lead compensator (Today)Improve Improve transient responsetransient responseImprove Improve stabilitystabilityLag compensator (Next)Lag compensator (Next)Reduce Reduce steady state errorsteady state errorLeadLead--lag compensator (Next)lag compensator (Next)Take into account all the above issues.Take into account all the above issues.11Radar tracking systemRadar tracking system12Lead compensator designLead 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/sPerformance specificationPerformance specificationDamping ratio Damping ratio ζζ=0.5=0.5UndampedUndampednatural freq. natural freq. ωωnn=4 =4 rad/srad/sG(sG(s))C(sC(s))PlantPlantControllerControllerReReImImDesired poleDesired poleCL pole CL pole with with C(sC(s)=1)=113Angle and magnitude conditions Angle and magnitude conditions (review)(review)A point s to be on root locus A point s to be on root locus ÅÆÅÆit satisfiesit satisfiesAngle conditionAngle conditionFor a point on root locus, gain K is obtained byFor a point on root locus, gain K is obtained byMagnitude conditionMagnitude conditionOdd numberOdd number14Lead compensator design (contLead compensator design (cont’’d)d)Evaluate Evaluate G(sG(s) at the desired pole.) at the desired pole.ooIf If angle conditionangle conditionis satisfied, is satisfied, compute the corresponding K.compute the corresponding K.ooIn this example, In this example, Angle condition is not satisfied.Angle condition is not satisfied.ReReImImDesired poleDesired poleAngle deficiencyAngle deficiency15Lead compensator design (contLead compensator design (cont’’d)d)To compensate angle deficiency, To compensate angle deficiency, design a lead compensator design a lead compensator C(sC(s))satisfying satisfying ReReImImDesired poleDesired poleThere are many ways to design such There are many ways to design such C(sC(s)!)!16Lead compensatorLead compensatorPositive angle contributionPositive angle contributionTriangleTriangleReReImImTest pointTest pointss--zz11--pp1117How to select pole and zero?How to select pole and zero?Draw horizontal line PADraw horizontal line PADraw line PODraw line PODraw bisector PBDraw bisector PBDraw PC and PDDraw PC and PDPole and zero of Pole and zero of C(sC(s) are shown in the figure.) are shown in the figure.ReReImImDesired poleDesired polePPAAOOBBCCDD--z(=z(=--2.9)2.9)--p(=p(=--5.4)5.4)18Comparison of root locusComparison of root locusG(sG(s))ReReImImG(s)C(sG(s)C(s))ReReImImImproved stability!Improved stability!19How to design the gain K?How to design the gain K?Lead
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