1ME451: Control SystemsME451: Control SystemsDr. Dr. JongeunJongeunChoiChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 17Lecture 17Root locus: ExamplesRoot locus: Examples2Course 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 &) laboratories3What is Root Locus? (Review)What is Root Locus? (Review)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 the graphically shows how poles of the closedclosed--loop system varies as K varies from 0 to loop system varies as K varies from 0 to infinity.infinity.L(sL(s))KKL(sL(s): open): open--loop TFloop TF4Root locus: Step 0 (Mark pole/zero)Root locus: Step 0 (Mark pole/zero)Root 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””..ReReImIm5Root locus: Step 1 (Real axis)Root locus: Step 1 (Real axis)RL 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.6Root 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 are7Root locus: Step 2 (Asymptotes)Root locus: Step 2 (Asymptotes)Intersections of asymptotesIntersections of asymptotesAsymptote Asymptote (Not root locus)(Not root locus)ReReImIm8Root locus: Step 3 (Breakaway)Root locus: Step 3 (Breakaway)Breakaway points are among roots ofBreakaway points are among roots ofFor each candidate s, check the positivity ofFor each candidate s, check the positivity of9What is this value? What is this value? For what K?For what K?ReReImImRoot locus: Step 3 (Breakaway)Root locus: Step 3 (Breakaway)Breakaway pointBreakaway pointRouthRouth--Hurwitz!Hurwitz!10Finding K for critical stabilityFinding K for critical stabilityCharacteristic equationCharacteristic equationRouthRoutharrayarrayWhen K=30When K=30Stability conditionStability condition11ReReImImRoot locusRoot locusBreakaway pointBreakaway point12Example with complex polesExample with complex polesReReImImBreakaway pointBreakaway pointAfter Steps 0,1,2,3, we obtainAfter Steps 0,1,2,3, we obtainHow to compute How to compute angle of departure?angle of departure?13Root locus: Step 4Root locus: Step 4Angle of departureAngle of departureAngle condition: Angle condition: For s to be on RL,For s to be on RL,ReReImImIf s is close to pIf s is close to p1114Root locusRoot locusBreakaway pointBreakaway pointReReImIm15Summary and exercisesSummary and exercisesExamples for root locus.Examples for root locus.Gain computation for marginal stability, by using Gain computation for marginal stability, by using RouthRouth--Hurwitz criterionHurwitz criterionAngle of departure (Angle of arrival can be obtained Angle of departure (Angle of arrival can be obtained by a similar argument.)by a similar argument.)Next, sketch of proofs for root locus algorithmNext, sketch of proofs for root locus algorithmExercisesExercisesDraw root locus for K>0 (no need to consider K<0) for Draw root locus for K>0 (no need to consider K<0) for openopen--loop transfer functions inloop transfer functions in••Problems 7.5 and 7.7.Problems 7.5 and 7.7.16Exercises 1 Exercises 117Exercises 2Exercises 218Exercises 3Exercises 319Exercises 4Exercises
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