控制工程(周川)a

Ch8StabilityintheFrequencyDomain,Maincontent,Mappingcontoursinthes-planeTheNyquistCriterionRelativeStabilityandNyquistCriterionClosed-loopfrequencyresponseSystemBandwidthExamplesandSimulationSummary,Introduction,FeedbackControlSystem,Howtodeterminethestability?,RootsevaluationRouth-Hurwitzcriterion(absolute/relativestability)RootlocusNyquistcriterion(absolute/relativestability),8.1Mappingcontoursinthes-plane,ContourmapConformalmapping,s-planeF(s)-plane,Acontourmappingthatretainstheanglesonthes-planeontheF(s)-plane,Figure1mappingF(s)=2s+1,Figure2MappingforF(s)=s/(s+2),Figure3MappingforF(s)=s/(s+1/2),Principleoftheargument,Ifacontourinthes-planeencirclesZzerosandPpolesofF(s)anddoesnotpassthroughanypolesorzerosofF(s)andthetraversalisintheclockwisedirectionalongthecontour,thecorrespondingcontourinF(s)-planeencirclestheoriginofF(s)-planeN=Z-Ptimesintheclockwisedirection.,Therulesofcontour,Clockwisetraversalofacontourisassumedtobepositive.Encirclementisdefinedbytherule:“Clockwiseandeyesright”or“Counterclockwiseandeyesleft”,Figure4EvaluationofthenetangleofF(s)-plane2*Pi*N=2*Pi*Z-2*Pi*P,Figure5EvaluationofthenetangleofF(s)-planeN=Z-P=3-1=2,Figure6EvaluationofthenetangleofF(s)-planeN=Z-P=0-1=-1,8.2TheNyquistCriterion,Figure7Single-loopfeedbackcontrolsystem,Theclosed-loopcharacteristicequation:,Figure8Nyquistcontourins-plane,ThenumberofzerosofF(s)withinthecontouris:,Nyquiststabilitycriterion,AfeedbacksystemisstableifandonlyifthecontourintheL(s)-planedoesnotencirclethe(-1,0)pointwhenthenumberofpolesofL(s)intheright-hands-planeiszero(P=0)Afeedbacksystemisstableifandonlyif,forthecontour,thenumberofcounterclockwiseencirclementsofthe(-1,0)pointisequaltothenumberofpolesofL(s)intheright-hands-plane.,ApplicationofNyquistcriterion,Example1:systemwith2realpolesExample2:systemwithapoleatoriginExample3:systemwith3polesExample4:systemwith2polesatoriginExample5:systemwithapoleinRHPExample6:systemwithazeroinRHP,RefertoP506-515,Figure9NyquistcontourandmappingforGH(s),Example1,Example2,Figure10NyquistcontourandmappingforGH(s),Figure11NyquistcontourandmappingforGH(s),Example3,Figure12NyquistcontourforGH(s)when(a)K=1,(b)K=2,(c)K=3,Example4,Figure13NyquistcontourforGH(s),Example6,Figure10NyquistdiagramandmappingforGH(jw)/K,稳定性分析的补充举例,开环传递函数不含积分环节,例1：系统开环传递函数为试判定闭环系统的稳定性。,闭环系统稳定,例2：系统开环传递函数为试判定闭环系统的稳定性。,闭环系统稳定,例3：系统开环传递函数为试判定闭环系统的稳定性。,闭环系统不稳定,-1.59,开环传递函数含积分环节,需对开环幅相曲线作修正：从=0+处，逆时针补画90半径为无穷大的圆弧。,例4：系统开环传递函数为试判定闭环系统的稳定性。