It is known as Nyquist stability criterion It is based on the complex analysis result known as Cauchys principle of ar gument Note that the system transfer function is complex function By applying Cauchys principle of ar gument to the openloop syste ID: 26075
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NyquistStabilityCriterionAstabilitytestfortimeinvariantlinearsystemscanalsobederivedinthefrequencydomain.ItisknownasNyquiststabilitycriterion.ItisbasedonthecomplexanalysisresultknownasCauchy'sprincipleofargument.Notethatthesystemtransferfunctionisacomplexfunction.ByapplyingCauchy'sprincipleofargumenttotheopen-loopsystemtransferfunction,wewillgetinformationaboutstabilityoftheclosed-loopsystemtransferfunctionandarriveattheNyquiststabilitycriterion(Nyquist,1932).TheimportanceofNyquiststabilityliesinthefactthatitcanalsobeusedtodeterminetherelativedegreeofsystemstabilitybyproducingtheso-calledphaseandgainstabilitymargins.Thesestabilitymarginsareneededforfrequencydomaincontrollerdesigntechniques. WepresentonlytheessenceoftheNyquiststabilitycriterionandde nethephaseandgainstabilitymargins.TheNyquistmethodisusedforstudyingthestabilityoflinearsystemswithpuretimedelay.ForaSISOfeedbacksystemtheclosed-looptransferfunctionisgivenbywhererepresentsthesystemandisthefeedbackelement.Sincethesystempolesaredeterminedasthosevaluesatwhichitstransferfunctionbecomesin nity,itfollowsthattheclosed-loopsystempolesareobtainedbysolvingthefollowingequationwhich,infact,representsthesystemcharacteristicequation. Inthefollowingweconsiderthecomplexfunctionwhosezerosaretheclosed-looppolesofthetransferfunction.Inaddition,itiseasytoseethatthepolesofarethezerosof.Atthesametimethepolesofaretheopen-loopcontrolsystempolessincetheyarecontributedbythepolesof,whichcanbeconsideredastheopen-loopcontrolsystemtransferfunctionÐobtainedwhenthefeedbackloopisopenatsomepoint.TheNyquiststabilitytestisobtainedbyapplyingtheCauchyprincipleofargumenttothecomplexfunction.First,westateCauchy'sprincipleofargument. Cauchy'sPrincipleofArgumentLetbeananalyticfunctioninaclosedregionofthecomplexplanegiveninFigure4.6exceptata nitenumberofpoints(namely,thepolesof).Itisalsoassumedthatisanalyticateverypointonthecontour.Then,astravelsaroundthecontourinthe-planeintheclockwisedirection,thefunctionencirclestheorigininthe-planeinthesamedirectiontimes(seeFigure4.6),withgivenbywhereandstandforthenumberofzerosandpoles(includingtheirmultiplicities)ofthefunctioninsidethecontour. Theaboveresultcanbealsowrittenaswhichjusti estheterminologyused,ªtheprincipleofargumentº.++++++Im{s}Re {s}Z=3P=6N=-3s-planeIm{F(s)}Re{F(s)}F(s)-planeFigure4.6:Cauchy'sprincipleofargument NyquistPlotTheNyquistplotisapolarplotofthefunctionwhentravelsaroundthecontourgiveninFigure4.7.+++s-planeR r0Im{s}Re{s}Figure4.7:Contourinthe-planeThecontourinthis gurecoversthewholeunstablehalfplaneofthecomplexplane,.Sincethefunction,accordingtoCauchy'sprincipleofargument,mustbeanalyticateverypointonthe contour,thepolesofontheimaginaryaxismustbeencircledbyin nitesimallysmallsemicircles.NyquistStabilityCriterionItstatesthatthenumberofunstableclosed-looppolesisequaltothenumberofunstableopen-looppolesplusthenumberofencirclementsoftheoriginoftheNyquistplotofthecomplexfunction.Thiscanbeeasilyjusti edbyapplyingCauchy'sprincipleofargumenttothefunctionwiththe-planecontourgiveninFigure4.7.Notethatandrepresentthenumbersofzerosandpoles,respectively,ofintheunstablepartofthecomplexplane.Atthesametime,thezerosofaretheclosed-loopsystempoles,andthepolesofaretheopen-loopsystempoles(closed-loopzeros). Theabovecriterioncanbeslightlysimpli edifinsteadofplottingthefunction,weplotonlythefunctionandcountencirclementoftheNyquistplotofaroundthepoint,sothatthemodi edNyquistcriterionhasthefollowingform.Thenumberofunstableclosed-looppoles(Z)isequaltothenumberofunstableopen-looppoles(P)plusthenumberofencirclements(N)ofthepointoftheNyquistplotof,thatis PhaseandGainStabilityMarginsTwoimportantnotionscanbederivedfromtheNyquistdiagram:phaseandgainstabilitymargins.ThephaseandgainstabilitymarginsarepresentedinFigure4.8.Im{H(s)G(s)}Re {H(s)G(s)}(\n0,-j)(-\n1,j0)1(\n1,j0)(\n0,j)PmGm w cgw cpFigure4.8:Phaseandgainstabilitymargins Theygivethedegreeofrelativestability;inotherwords,theytellhowfarthegivensystemisfromtheinstabilityregion.Theirformalde nitionsaregivenby\rwhereandstandfor,respectively,thegainandphasecrossoverfrequencies,whichfromFigure4.8areobtainedasand\r Example4.23:ConsideracontrolsystemrepresentedbySincethissystemhasapoleattheorigin,thecontourinthe-planeshouldencircleitwithasemicircleofanin nitesimallysmallradius.Thiscontourhasthreeparts(a),(b),and(c).Mappingsforeachofthemareconsideredbelow.(a)Onthissemicirclethecomplexvariableisrepresentedinthepolarformbywith.Substitutinginto,weeasilyseethat.Thus,thehugesemicirclefromthe-planemapsintotheorigininthe-plane(seeFigure4.9). +Im{s}Im{G(s)H(s)}Re{s}Re{G(s)H(s)}(a)\n(c)\n(c)\n(b)\nAB(a)\n(c)\n(c)\nw =0+w=+-8w =0--1(b)\nABFigure4.9:NyquistplotforExample4.23(b)Onthissemicirclethecomplexvariableisrepresentedinthepolarformby with ,sothatwehave Sincechangesfrom atpointAto atpointB, willchangefrom !to !.Weconcludethatthein nitesimallysmallsemicircleattheorigininthe-planeismappedintoasemicircleofin niteradiusinthe-plane.(c)Onthispartofthecontourtakespureimaginaryvalues,i.e.withchangingfromto.Duetosymmetry,itissuf cienttostudyonlymappingalong".Wecan ndtherealandimaginarypartsofthefunction,whicharegivenby!! Fromtheseexpressionsweseethatneithertherealnortheimaginarypartscanbemadezero,andhencetheNyquistplothasnopointsofintersectionwiththecoordinateaxis.For#weareatpointBandsincetheplotatwillendupattheorigin,theNyquistdiagramcorrespondingtopart(c)hastheformasshowninFigure4.9.NotethattheverticalasymptoteoftheNyquistplotinFigure4.9isgivenby$$sinceatthosepoints$$.FromtheNyquistdiagramweseethatandsincetherearenoopen-looppolesinthelefthalfofthecomplexplane,i.e.,wehavesothatthecorrespondingclosed-loopsystemhasnounstablepoles. TheNyquistplotisdrawnbyusingtheMATLABfunctionnyquistnum=1;den=[110];nyquist(num,den);axis([-1.50.5Ð1010]);axis([-1.20.211]);TheMATLABNyquistplotispresentedinFigure4.10.ItcanbeseenfromFigures4.8and4.9that,whichimpliesthat.Also,fromthesame guresitfollowsthat%&.Inorderto ndthephasemarginandthecorrespondinggaincrossoverfrequencyweusetheMATLABfunctionmarginasfollows[Gm,Pm,wcp,wcg]=margin(num,den) producing,respectively,gainmargin,phasemargin,phasecrossoverfre-quency,andgaincrossoverfrequency.Therequiredphasemarginandgaincrossoverfrequencyareobtainedas'(). 1 0.50 10 8 6 4 20246810Real AxisImag Axis 1 0.50 1 0.8 0.6 0.4 0.200.20.40.60.81Real AxisImag AxisFigure4.10:MATLABNyquistplotforExample4.23 Example4.24:Considernowthefollowingsystem,obtainedfromtheoneinthepreviousexamplebyaddingapole,thatisThecontourinthe-planeisthesameasinthepreviousexample.Forcases(a)and(b)wehavethesameanalysesandconclusions.Itremainstoexaminecase(c).Ifwe ndtherealandimaginarypartsof,weget******* Itcanbeseenthatanintersectionwiththerealaxishappensatatthepoint+.TheNyquistplotisgiveninFigure4.11.ThecorrespondingNyquistplotobtainedbyusingMATLABisgiveninFigure4.12.Im{G(s)H(s)}Re{G(s)H(s)}(a)\n(c)\n(c)\nw =0+w=+-8w =0--1(b)\nAB,-34-16Figure4.11:NyquistplotforExample4.24 1.5 1 0.500.5 10 8 6 4 20246810Real AxisImag Axis 1 0.50 0.2 0.15 0.1 0.0500.050.10.150.2Real AxisImag AxisFigure4.12:MATLABNyquistplotforExample4.24Notethattheverticalasymptoteisgivenby.Thus,wehave,andsothattheclosed-loopsystemisstable.TheMATLABfunctionmarginproduces-./.0