1ME451: Control SystemsME451: Control SystemsDr. Dr. JongeunJongeunChoiChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 16Lecture 16Root locusRoot locus2Course 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 &) laboratories3Lecture planLecture planL16: Root locus, sketching algorithmL16: Root locus, sketching algorithmL17: Root locus, examplesL17: Root locus, examplesL18: Root locus, proofsL18: Root locus, proofsL19: Root locus, control examplesL19: Root locus, control examplesL20: Root locus, influence of zero and poleL20: Root locus, influence of zero and poleL21: Root locus, lead lag controller designL21: Root locus, lead lag controller design4What is Root Locus?What is Root Locus?W. R. Evans developed in 1948.W. R. Evans developed in 1948.Pole locationPole locationof the feedback system of the feedback system characterizes characterizes stabilitystabilityand and transient propertiestransient properties..Consider a feedback system that has one Consider a feedback system that has one parameter (gain) K>0 to be designed.parameter (gain) K>0 to be designed.Root locusRoot locusgraphically shows how poles of CL graphically shows how poles of CL system varies as K varies from 0 to infinity.system varies as K varies from 0 to infinity.L(sL(s))KKL(sL(s): open): open--loop TFloop TF5A simple exampleA simple exampleCharacteristic eq.Characteristic eq.K=0: s=0,K=0: s=0,--22K=1: s=K=1: s=--1,1,--11K>1: complex numbersK>1: complex numbersL(sL(s))KKClosedClosed--loop polesloop polesReReImIm6A more complicated exampleA more complicated exampleCharacteristic eq.Characteristic eq.It is hard to solve this analytically for each K.It is hard to solve this analytically for each K.Is there some way to Is there some way to sketch roughlysketch roughlyroot locus root locus by hand? (In by hand? (In MatlabMatlab, use command , use command ““rlocus.mrlocus.m””..))L(sL(s))KK7Root locus: Step 0Root locus: Step 0Root locus is symmetric Root locus is symmetric w.r.tw.r.t. the real axis.. the real axis.The number of branches = order of The number of branches = order of L(sL(s))Mark poles of L with Mark poles of L with ““xx””and zeros of L with and zeros of L with ““oo””..ReReImIm8Root locus: Step 1Root locus: Step 1RL includes all points on real axis to the left of an RL includes all points on real axis to the left of an odd number of real poles/zeros.odd number of real poles/zeros.RL originates from the poles of L and terminates RL originates from the poles of L and terminates at the zeros of L, including infinity zeros.at the zeros of L, including infinity zeros.ReReImImIndicate the direction Indicate the direction with an arrowhead.with an arrowhead.9Root locus: Step 2 (Asymptotes)Root locus: Step 2 (Asymptotes)Number of asymptotes = relative degree (r) of L:Number of asymptotes = relative degree (r) of L:Angles of asymptotes areAngles of asymptotes are10Root locus: Step 2 (Asymptotes)Root locus: Step 2 (Asymptotes)Intersections of asymptotesIntersections of asymptotesAsymptotes Asymptotes (Not root locus)(Not root locus)ReReImIm11Root locus: Step 3Root locus: Step 3Breakaway points are among roots ofBreakaway points are among roots ofFor each candidate s, check the positivity ofFor each candidate s, check the positivity ofPoints where two or more branches meet and break away.Points where two or more branches meet and break away.12Quotient ruleQuotient rule13Root locus: Step 3Root locus: Step 3ReReImImBreakaway pointBreakaway point14MatlabMatlabcommand command ““rlocus.mrlocus.m””Root LocusReal AxisImaginary Axis-3 -2.5 -2 -1.5 -1 -0.5 0-8-6-4-20246815A simple example: revisitedA simple example: revisitedAsymptotesAsymptotesRelative degree 2Relative degree 2IntersectionIntersectionBreakaway pointBreakaway pointL(sL(s))KKReReImIm16Summary and exercisesSummary and exercisesRoot locusRoot locusWhat is root locusWhat is root locusHow to roughly sketch root locusHow to roughly sketch root locusSketching root locus relies heavily on experience.Sketching root locus relies heavily on experience.PRACTICE!PRACTICE!To accurately draw root locus, use To accurately draw root locus, use MatlabMatlab..Next, more examplesNext, more examplesExercisesExercisesRead Chapter 7.Read Chapter
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