Chapter9

StabilityintheFrequencyDomainIntroductionTheNyquistCriterionRelativeStabilityandtheNyquistCriterionTime-DomainPerformanceCriteriaintheFrequencyDomainStabilityintheFrequencyDomainUsingControlDesignSoftwareSequentialDesignExample:DiskDriveReadSystem9.1IntroductionVarioustoolstodeterminethestability(relativestability)Routh-HurwitzstabilitycriterionRootlocusmethodFrequencydomainmethod—NyquiststabilitycriterionItinvestigatesthestabilityofasystemintherealfrequencydomainItcanbeutilizedtoinvestigatetherelativestabilityofasystemwhenthesystemparametervalueshavenotbeendetermined.Itwouldbeusefulfordeterminingsuitableapproachestoadjustingtheparametersofasysteminordertoincreaseitsrelativestability.9.1IntroductionTodeterminetherelativestabilityofaclosed-loopsystem,wemustinvestigatethecharacteristicequationofthesystem:

(9.1)Toensurestability,wemustascertainthatallthezerosofF(s)lieintheleft-hands-plane.Nyquistthusproposedamappingoftheright-hands-planeintotheF(s)-plane.TouseandunderstandNyquist'scriterion,weshallfirstconsiderbrieflythemappingofcontoursortrajectoriesinthecomplexplaneintoanotherplanebyarelation

functionF(s).

9.2MappingContoursinthes-planeAcontourmap:s-plane—F(s)=u+jv-planeExample:F(s)=2s+1S-plane,s=σ+jω;F(s)-plane:F(s)=u+jv=2(σ+jω)+1 u=2σ+1,v=2ωUnitsquarecontourSquarecontourwiththecentershiftedbyoneunitandthemagnitudeofasidemultipliedby2.Conformalmappingtheanglesofthes-planecontourandF(s)-planecontouristhesametheclosedcontourinthes-planeresultsinaclosedcontourintheF(s)-planedirectionofcontoursassumeclockwisetraversalofacontourtobepositivetheareaenclosedwithinthecontourtobeontherightofthetraversalofthecontourExample:F(s)=s/(s+2)Anotherexample:F(s)=s/(s+1/2)Cauchy’stheorem-principleoftheargumentMappingofthefunctionwithafinitenumberofpolesandzeroswithinthecontourCauchy’stheorem:IfacontourΓs

inthes-planeencirclesZzerosandPpolesofF(s)anddoesnotpassthroughanypolesandzerosofF(s)andthetraversalisintheclockwisedirectionalongthecontour,thecorrespondingcontourΓF

intheF(s)-planeencirclestheoriginoftheF(s)-planeN=Z-Ptimesintheclockwisedirection.78OtherpatternsoftheuseofCauchy’stheoremOpenlooptransferfunctionClosed-looptransferfunction＋－10LetUsuallyareopenpolesThusZeorsofassistantfunctionequalstothepolesofclosed-loopsystemPolesofassistantfunctionequalstothepolesofopen-loopsystem119.3TheNyquistcriterionConsiderthecharacteristicequationF(s)=0Forasystemtobestable,allthezerosofF(s)mustlieintheleft-hands-plane,thatistheleftofthejω-axisinthes-planeInthes-plane,chooseaspecialcontourΓs

thatenclosestheentireright-hands-planeIntheF(s)-plane,examineN,i.e.,ΓF

’sencirclementnumberoftheorigin.AccordingtoN=Z-P,determinewhetheranyzerosofF(s)liewithinΓs.Wewillget

Z=N+PPisthenumberofpolesofF(s)intheright-hands-plane.Thus,ifP=0,asisusuallythecase,wefindthatthenumberofunstablerootsofthesystemisequaltoN,thenumberofencirclementsoftheoriginoftheF(s)-plane.IfZ>0,unstable.IfZ=0,stable.TheNyquistcontourthatenclosestheentireright-hands-planeisshownintheFigure.ThecontourΓspassesalongthejω-axisfrom–j∞to+j∞.Thecontouriscompletedbyasemicircularpathofradiusr,whererapproachesinfinitysothispartofthecontourtypicallymapstoapoint.ThiscontourΓFisknownastheNyquistdiagramorpolarplot.HowtoknowNaccordingtoGH(jω)?Mappingofthecharacteristicequation

F(s)=1+GH(s)FunctionGH(s)=F(s)-1ConstructΓGH(s)correspondingtotheselectedΓsjω:0→∞,G(jω),frequencyresponsecharacteristicjω:-∞→0,besymmetricaltotheaboveThecircle:originThenumberofclockwiseencirclementsoftheoriginoftheF(s)-planebecomesthenumberofclockwiseencirclementofthe(–1,j0)pointintheGH(s)-plane.Nyquiststabilitycriterion:Afeedbackcontrolsystemisstableifandonlyif,inthecontourΓGH,thenumberofcounterclockwiseencirclementsofthe(-1,j0)isequaltothenumberofpolesofGH(s)withpositiverealparts.Z=N+P=0SpecialconditionP=0AfeedbacksystemisstableifandonlyifthecontourΓGHintheGH(s)-planedoesnotencirclethe(-1,j0)pointwhenthenumberofpolesGH(s)intheright-hands-planeiszero.(P=0)Z=N=0Example9.1systemwithtworealpolesExample9.2SystemwithapoleattheoriginContourΓscannotpassthroughanypolesorzerosofF(s).ThusselecttheaboveΓsinthes-plane.ConstructthecontoursΓGH---4portions(1)Theoriginofthes-plane.Thesmallsemicirculardetouraroundthepoleattheorigincanberepresentedbysettings=εej?

andallowing?

tovaryfrom-90°atω=0-to+90°atω=0+.Becauseεapproacheszero,themappingforGH(s)istheangleofthecontourintheL(s)-planechangesfrom90°atω=0-to-90°atω=0+,passingthrough0°atω=0.TheradiusofthecontourintheL(s)-planeforthisportionofthecontourisinfinite.(2)Theportionfromω=0+toω=+∞Themagnitudeapproachesto0atanangleof-180°(3)Theportionfromω=+∞toω=-∞

Thecontourmovesfromanangleof-180°atω=+∞toanangleof+180°atω=-∞.Themagnitudeofthecontourwhenrisinfiniteisalwayszerooraconstant.(4)Theportionfromω=-∞toω=0-

StabilityP=0:Numberofpolesintheright-hands-planeiszero.N=0:ContourΓGHdonotencirclethe(-1,j0)pointintheGH-plane.Z=N+P=0.Thesystemisstable.Thegeneralconclusionfromthisexample:Thecontourfortherange-∞<ω<0—willbethecomplexconjugateoftheplotfromtherange0+<ω<+∞.ThereforeitissufficienttoconstructthecontourΓGHfrom0+<ω<+∞.ThemagnitudeofGH(s)ass=rejΦandr→∞willnormallyapproachzerooraconstant.Example9.3

SystemwiththreepolesThepointpassingthroughtheimaginarypartis,GH(s)=u+jv,v=0.,and ,therealpartThenthesystemisstablewhenwhenExample9.4SystemwithtwopolesattheoriginZ=2,UnstablesystemTherealfrequencypolarplotisobtainedwhens=jωtheangleofL(jω)isalways-180°orless,andthelocusofL(jω)isabovetheω-axisforallvaluesofω.As

ω

approaches0+,As

ωapproaches+∞,Atthesmallsemicirculardetourattheoriginofthes-planewheres=εej?Example9.5Systemwithapoleintheright-hands-planeP=1,N=1,Z=2,unstablesystemExample9.6Example9.7Systemwithazerointheright-hands-plane9.4RelativestabilityandtheNyquistcriterionMeasuretherelativestabilityRelativesettlingtimeofeachrootorpairofroots.Shortersettlingtimemeansmorerelativelystable.(chapter6)TheNyquiststabilitycriterionisdefinedintermsofthe(-1,0)pointonthepolarplotorthe0dB,-180°pointontheBodediagramorlog-magnitude-phasediagram.

StabilityMargins

DefinitionsofStabilityMarginsCalculationsofStabilityMargins31

StabilityMarginsTimedomain（t）ThedynamicalperformanceStableboundaryFrequencydomain(w)HowstablethesystemisImaginaryaxisDampedratio

xThedistanceto(-1,j0)(-1,j0)Stabilitymargins(Open-loopfrequencyindices)Howstableitis32

StabilityMargins§

TheDefinitionofStabilityMargins

ThegeometrymeaningofCutofffrequency

wcPhasemarginGainmarginThestabilitydepthontheGainPhaseGenerallyThephysicsmeaningofPhasecrossoverfrequency33DemonstrationexampleMarginalsituation:GainmarginandphasemargininBodeplot:Clearly,thefeedbacksystemL2(jω)isrelativelylessstablethanthesystemL1(jω).

§StabilityMargins§CalculationsofStabilityMargins

SolutionI：ObtaingandhbytheNyquistplot，obtain(1)LetT.&E.37

§StabilityMarginsLetWehave38

§StabilityMarginsRewriteG(jw)

intotherealpartplustheimaginarypartLetWehaveSubstitutetotheRP39

§StabilityMarginsFromL(w):WehaveSolutionII：DeterminebyBodediagram40

§StabilityMarginsSolution.Determineby

L(w)SolutionI:，DetermineSolutionII:41

ObtainwgRewrittenasWehave42

SummaryConcept(Open-loopfrequencyindex)Definitions

CalculationsCutofffrequencywc

PhasemargingPhasecrossoverfrequencywg

Amplitudemargins

hMeaningsThegeometrymeaningofThephysicalmeaningof43

SystemAnalysisbyFrequencyResponseCharacteristicsofOpen-LoopSystems

CorrelationbetweenL(ω)LowFrequencySectionandSteadyStateErrorsCorrelationbetweenL(ω)MidFrequencySectionandDynamicPerformancesImpactsofL(ω)HighFrequencySectiononSystemPerformances44

AnalysisbyFrequencyResponseofO.L.SystemsTri-BandinFrequency

1.Lowerfrequencybandof

L(w)

?ess2.Middlefrequencybandof

L(w)?(s,ts)3.Higherfrequencybandof

L(w)?Theabilityofanti-high-frequencynoise

Thecorrelationbetweentheslopeof

L(w)and

j(w)ofminimumphasesystemsIdeally,L(w)crossesthe0dBlinewithaslopeof-20dB/decandretainsabroadband45

Bodediagramforthesystem46

Determinetheshapeof

j(w)Thecorrelationbetweenj(w)andL(w)(k=1)ofminimumphasesystem.§AnalysisbyFrequencyResponseofO.L.Systems47

Thecorrelationbetweenj(w)andL(w).(k=1)ofminimumphasesystems§AnalysisbyFrequencyResponseofO.L.Systems48

(1)Second-ordersystem

§AnalysisbyFrequencyResponseofO.L.Systems49

§AnalysisbyFrequencyResponseofO.L.Systems50

Considerthesystemshowninthefigure.Obtainthe

wc,s

and

ts.Solution.Sketch

L(w)RefertoFigTimedomainmethod:§AnalysisbyFrequencyResponseofO.L.Systems51

(2)Higher-ordersystem§5AnalysisbyFrequencyResponseofO.L.Systems52

Solution.Sketch

L(w)RefertoFig.

Obtain

wc,g,s

and

ts

fortheunityfeedbacksystem.§AnalysisbyFrequencyResponseofO.L.Systems53

Estimatethedynamicperformanceofhigher-ordersystembyfrequencyresponsemethod.§AnalysisbyFrequencyResponseofO.L.Systems54

TheminimumphasesystemL(w)isshowninthefigure,(1)Obtaintheopen-looptransferfunctionG(s)(2)determinethestabilitybyg(3)DeterminetheeffectofshiftingL(w)totherightby1dec.Solution.(1)(3)AftershiftingL(w)totherightby1dec(2)wc

increased→ts

decreasedAftershiftingnochange→s

nochangeStable55

MiddlebandTri-BandinFrequencyHigherbandLowerbandCorrespondingperformanceExpectation

L(w)Theabilityofanti-high-frequencynoiseOpen-loopgainKSystemtypevessCutofffrequencywc

PhasemargingDynamicPerformanceSteep，HighModerate,WideLow，SteepFrequencyBandTri-Banddoesnotgivethestepstodesignsystems,butitshowthewaytoadjustthesystemstructureforbetterperformance.§AnalysisbyFrequencyResponseofO.L.Systems56§NicholsChart§SystemAnalysisbyFrequencyResponseCharacteristicsofClosed-LoopSystems

57

§NicholsChartWhyClosed-Loopfrequencyresponses(1)Indicesofclosed-loopfrequencycharacteristicsareusedwidelyinpractice;(2)Theclosed-loopfrequencycharacteristicssystemareeasilyobtainedbyexperimentalmethod;(3)Thesystemperformancespecificationscanbeestimatedbytheclosed-loopfrequencyindices.58

VectorCorrelationbetweenOLandCLFrequencyResponses59

ConstantM/NcirclesConstantMcircle—AlocuscorrespondingtoaconstantLet

Rewrite:—ConstantMcirclesequation60

Let

Rewrite:—ConstantNcirclesequationConstantNcircle—Alocuscorrespondingtoaconstant61

ConstantM/Ncircles→Nicholschart62§5.7閉環(huán)頻率特性曲線的繪制（6）

Determine63Example9.7 StabilityusingtheNicholschartφωB

=-142°Mpω=+2.5dBωr=0.8φωr=-72°ωB

=1.339.6SystemBandwidthThebandwidthoftheclosed-loopsystemisanexcellentmeasurementoftherangeoffidelityofthesystem.ThespeedoftheresponsetoastepinputwillberoughlyproportionaltoωB,thusweseekalargebandwidthconsistentwithreasonablesystemcomponents.ThesystemsThesystemsBothsystemshaveζ=0.5.Thenaturalfrequencyis10and30forsystemsT3andT4,respectively.Thebandwidthis12.7and38.1forsystemsT3andT4,respectively.Bothsystemshavea16%overshoot,butT4hasapeaktimeof0.12secondcomparedto0.36forT3.Also,notethatthesettlingtimeforT4is0.27second,whilethesettlingtimeforT3is0.8second.Thesystemwithalargerbandwidthprovidesafasterresponse.9.8Designexample:RemotelycontrolledreconnaissancevehicleTheremotecontrolledvehicleThedesigngoal:goodoverallcontrolwithlowsteady-stateerrorandlow-overshootresponsetostepinputcommand,R(s)First,lowsteady-stateerrorWemayselectK=20,thusthetransferfunctionofopen-loopsystemis9.8Designexample:RemotelycontrolledreconnaissancevehicleK=20,20logMpω=12dBandMpω=3.98.Thephasemarginis15°.Wepredictanexcessiveovershootofapproximate61%.Toreducetheovershoottoastepinput,reducethegain.Tolimittheovershoot25%,ζ=0.4,thusrequireMpω=1.35and20lo