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••electromechanicalelectromechanicalLinearizationLinearizationModelingModelingAnalysisAnalysisDesignDesignTime 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.5Why no proof in textbooks?Why no proof in textbooks?““most undergraduate students are exposed to the most undergraduate students are exposed to the RouthRouth––Hurwitz criterion in their first introductory Hurwitz criterion in their first introductory controls course. This exposure, however, is at the controls course. This exposure, however, is at the purely algorithmic level in the sense that no attempt purely algorithmic level in the sense that no attempt is made whatsoever to explain why or how such an is made whatsoever to explain why or how such an algorithm works.algorithm works.””An Elementary Derivation of the Routh–Hurwitz CriterionMing-Tzu Ho, Aniruddha Datta, and S. P. BhattacharyyaIEEE Transactions on Automatic Controlvol. 43, no. 3, 1998, pp. 405-409.6Why no proof in textbooks? (contWhy no proof in textbooks? (cont’’d)d)““The principal reason for this is that the classical The principal reason for this is that the classical proof of the proof of the RouthRouth--Hurwitz criterion relies on the Hurwitz criterion relies on the notion of Cauchy indexes and Sturmnotion of Cauchy indexes and Sturm’’s theorem, s theorem, both of which are beyond the scope of both of which are beyond the scope of undergraduate students.undergraduate students.””““RouthRouth--Hurwitz criterion has become one of the few Hurwitz criterion has become one of the few results in control theory that most control engineers results in control theory that most control engineers are compelled to accept on faith.are compelled to accept on faith.””7Example 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)8Example 1: Example 1: K(sK(s)=K)=KCharacteristic equationCharacteristic equationRouthRoutharrayarray9Example 1: Example 1: K(sK(s)=K)=KPP+K+KII/s/sCharacteristic equationCharacteristic equationRouthRoutharrayarray10-1 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,11Example 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.12Example 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 response13Example 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.14Example 2 (contExample 2 (cont’’d)d)15Example 2 (contExample 2 (cont’’d)d)Characteristic equationCharacteristic equation16Example 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 equation17Summary 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 specificationsExercisesExercisesRead Chapter 6 again.Read Chapter 6 again.Redo Examples 1 and 2Redo Examples 1 and 2Do Problem 6.6Do Problem 6.6--(a) and 6.7(a) and 6.7--(b)(b)--Find the range of K for which the Find the range of K for which the system is stable.system is stable.18More 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).)).)19More 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.442220More example 3More example 3RouthRoutharrayarrayOne sign changesOne
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