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Nyquist Stability Criterion stability test for time invariant linear systems can also be derived in the frequency domain

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

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Nyquist Stability Criterion stability test for time invariant linear systems can also be derived in the frequency domain






Presentation on theme: "Nyquist Stability Criterion stability test for time invariant linear systems can also be derived in the frequency domain"— Presentation transcript:

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-planeRr0Im{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)Onthissemicirclethecomplexvariableisrepresentedinthepolarformbywith,sothatwehaveSincechangesfromatpointAtoatpointB, 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'().10.501086420246810Real AxisImag Axis10.5010.80.60.40.200.20.40.60.81Real 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 1.510.500.51086420246810Real AxisImag Axis10.500.20.150.10.0500.050.10.150.2Real AxisImag AxisFigure4.12:MATLABNyquistplotforExample4.24Notethattheverticalasymptoteisgivenby.Thus,wehave,andsothattheclosed-loopsystemisstable.TheMATLABfunctionmarginproduces-./.0