1ME451: Control SystemsME451: Control SystemsDr. Dr. JongeunJongeunChoiChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 11Lecture 11RouthRouth--Hurwitz criterion: Control examplesHurwitz criterion: Control 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 &) laboratories3Stability summary (review)Stability summary (review)(BIBO, asymptotically) stable(BIBO, asymptotically) stableififRe(sRe(sii)<0 for all i.)<0 for all i.marginally stablemarginally stableififRe(sRe(sii)<=0 for all i, and)<=0 for all i, andsimple root for simple root for Re(sRe(sii)=0)=0unstableunstableififit is neither stable nor it is neither stable nor marginally stable.marginally stable.Let Let ssiibe be polespolesof of rational G. Then, G is rational G. Then, G is ……4RouthRouth--Hurwitz criterion (review)Hurwitz criterion (review)The number of roots The number of roots in the right halfin the right half--plane plane is equal to is equal to the number of sign changesthe number of sign changesin the in the first columnfirst columnof of RouthRoutharray.array.5Example 1Example 1Design Design K(sK(s) that stabilizes the closed) that stabilizes the closed--loop loop system for the following cases.system for the following cases.K(sK(s) = K (constant)) = K (constant)K(sK(s) = K) = KPP+K+KII/s (PI (Proportional/s (PI (Proportional--Integral) controller)Integral) controller)6Example 1: Example 1: K(sK(s)=K)=KCharacteristic equationCharacteristic equationRouthRoutharrayarray7Example 1: Example 1: K(sK(s)=K)=KPP+K+KII/s/sCharacteristic equationCharacteristic equationRouthRoutharrayarray8-1 0 1 2 3 4 5 6 7 8 900.511.522.533.5Example 1: Range of (KExample 1: Range of (KPP,K,KII) ) From From RouthRoutharray,array,9Example 1: Example 1: K(sK(s)=K)=KPP+K+KII/s (cont/s (cont’’d)d)Select KSelect KPP=3 (<9)=3 (<9)RouthRoutharray (contarray (cont’’d)d)If we select different KIf we select different KPP, the range of K, the range of KI I changes.changes.10Example 1: What happens if KExample 1: What happens if KPP=K=KII=3=3Auxiliary equationAuxiliary equationOscillation frequencyOscillation frequencyPeriodPeriod0 2 4 6 8 10 12 14 16 18 2000.20.40.60.811.21.41.61.82Unit step responseUnit step response11Example 2Example 2Determine the range of K and a that stabilize the Determine the range of K and a that stabilize the closedclosed--loop system.loop system.12Example 2 (contExample 2 (cont’’d)d)13Example 2 (contExample 2 (cont’’d)d)Characteristic equationCharacteristic equation14Example 2 (contExample 2 (cont’’d)d)RouthRoutharrayarrayIf K=35, oscillation frequency is obtained by the If K=35, oscillation frequency is obtained by the auxiliary equationauxiliary equation15Summary and ExercisesSummary and ExercisesControl examples for Control examples for RouthRouth--Hurwitz criterionHurwitz criterionP controller gain range for stabilityP controller gain range for stabilityPI controller gain range for stabilityPI controller gain range for stabilityOscillation frequencyOscillation frequencyCharacteristic equationCharacteristic equationNextNextTime domain specificationsTime domain specificationsExercisesExercises16More example 1More example 1RouthRoutharrayarrayNo sign changesNo sign changesin the first columnin the first columnNo root in OPEN(!) RHPNo root in OPEN(!) RHP22Derivative of auxiliary poly.Derivative of auxiliary poly.(Auxiliary poly. is a factor of (Auxiliary poly. is a factor of Q(sQ(s).)).)17More example 2More example 2RouthRoutharrayarrayNo sign changesNo sign changesin the first columnin the first columnNo root in OPEN(!) RHPNo root in OPEN(!) RHP44Derivative of auxiliary poly.Derivative of auxiliary poly.442218More example 3More example 3RouthRoutharrayarrayOne sign changesOne sign changesin the first columnin the first columnOne root in OPEN(!) RHPOne root in OPEN(!) RHPDerivative of auxiliary poly.Derivative of auxiliary
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