控制工程（周川）ac-11

Ch8

StabilityintheFrequencyDomainMaincontentMappingcontoursinthes-planeTheNyquistCriterionRelativeStabilityandNyquistCriterionClosed-loopfrequencyresponseSystemBandwidthExamplesandSimulationSummaryIntroductionFeedbackControlSystemHowtodeterminethestability?

Rootsevaluation

Routh-Hurwitzcriterion(absolute/relativestability)

Rootlocus

Nyquistcriterion(absolute/relativestability)

8.1Mappingcontoursinthes-planeContourmapConformalmappings-plane

F(s)-planeAcontourmappingthatretainstheanglesonthes-planeontheF(s)-plane

Figure1mappingF(s)=2s+1Figure2MappingforF(s)=s/(s+2)Figure3MappingforF(s)=s/(s+1/2)PrincipleoftheargumentIfacontourinthes-planeencirclesZzerosandPpolesofF(s)anddoesnotpassthroughanypolesorzerosofF(s)andthetraversalisintheclockwisedirectionalongthecontour,thecorrespondingcontourinF(s)-planeencirclestheoriginofF(s)-planeN=Z-Ptimesintheclockwisedirection.TherulesofcontourClockwisetraversalofacontourisassumedtobepositive.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

8.2TheNyquistCriterionFigure7Single-loopfeedbackcontrolsystemTheclosed-loopcharacteristicequation:Figure8Nyquistcontourins-planeThenumberofzerosofF(s)withinthecontouris:

NyquiststabilitycriterionAfeedbacksystemisstableifandonlyifthecontourintheL(s)-planedoesnotencirclethe(-1,0)pointwhenthenumberofpolesofL(s)intheright-hands-planeiszero(P=0)Afeedbacksystemisstableifandonlyif,forthecontour,thenumberofcounterclockwiseencirclementsofthe(-1,0)pointisequaltothenumberofpolesofL(s)intheright-hands-plane.ApplicationofNyquistcriterionExample1:systemwith2realpolesExample2:systemwithapoleatoriginExample3:systemwith3polesExample4:systemwith2polesatoriginExample5:systemwithapoleinRHPExample6:systemwithazeroinRHP

RefertoP506-515Figure9NyquistcontourandmappingforGH(s)Example1Example2Figure10

NyquistcontourandmappingforGH(s)Figure11

NyquistcontourandmappingforGH(s)Example3Figure12

Nyquistcontourfor

GH(s)when(a)K=1,(b)K=2,(c)K=3Example4Figure13

Nyquistcontourfor

GH(s)Example6Figure10NyquistdiagramandmappingforGH(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-1.59-1开环传递函数含积分环节需对开环幅相曲线作修正：从ω=0+处，逆时针补画ν×90°半径为无穷大的圆弧。例4：系统开环传递函数为