Introduction to the Nyquist criterion The Nyquist criterion relates the stability of a - PDF document

IntroductiontotheNyquistcriterionTheNyquistcriterionrelatesthestabilityofaclosedsystemtotheopen-loopfrequencyresponseandopenlooppolelocation.Mapping .Ifwetakeacomplexnumberonthes-planeandsubstituteintoafunctionF(s),an-othercomplexnumberresults.e.g.substitut-ings=4+j3intoF(s)=s2+2s+1yields16+j30.Contour .Consideracollectionofpoints,calledacontourA.ContourAcanbemappedintoContourB,asshowninthenextFigure. Figureabove;MappingcontourAthroughF(s)tocontourB.1 AssumingF(s)=(sz1)(sz2)


(sp1)(sp2)


IfweassumeaclockwisedirectionformappingthepointsoncontourA,thecontourBmapsinaclockwisedirectionifF(s)hasjustonezero.IfthezeroisenclosedbycontourA,thencontourBencloseorigin.Alternatively,themappingisinacounterclock-wisedirectionifF(s)hasjustonepole,andifthepoleisenclosedbycontourA,thencontourBencloseorigin.IfthereistheonepoleandonezeroisenclosedbycontourA,thencontourBdoesnotencloseorigin.2 Figureabove;Examplesofcontourmapping.3 ConsiderthesystemintheFigurebelow. Figureabove;closedloopcontrolsystemLettingG(s)=NG DG;H(s)=NH DH;WefoundT(s)=G(s) 1+G(s)H(s)=NGDH DGDH+NGNHNotethat1+G(s)H(s)=DGDH+NGNH DGDH4 Thepolesof1+G(s)H(s)arethesameasthepolesofG(s)H(s),theopen-loopedsystem,thatareknown.Thezerosof1+G(s)H(s)arethesameasthepolesofT(s),theclosed-loopedsystem,thatareunknown.BecausestablesystemshaveT(s)withpolesonlyinthelefthalf-plane,weapplytheconceptofcontourtousetheentirerighthalf-planeascontourA,asshownintheFigurebelow. Figureabove;Contourenclosingrighthalf-planetodeterminestability.5 WetrytoconstructcontourBviaF(s)=G(s)H(s)whichisthesameasthatof1+G(s)H(s),exceptthatitisshiftedtotherightby(1;j0).ThemappingiscalledtheNyquistdiagramofG(s)H(s).AssumingthatAstartsfromorigin,Aisapathtravelingupthej!axis,fromj0toj1,thenasemicirculararc,withradius!1,followedbyapathtravelingupthej!axis,fromj1toorigin.Sosubstitutings=j!,with!changingfrom0to1,weobtainpartofcontourB,whichisexactlythepolarplotofG(s)H(s).6 Eachzeroorpoleof1+G(s)H(s)thatisin-sidecontourA(therighthalf-plane),yieldsarotationaround(1;j0)(clockwiseforzeroandcounterclockwiseforpole)fortheresul-tantNyquistdiagram.Thetotalnumberofcounterclockwiserevolution,N,around(1;j0)isN=PZ,wherePisthenumberofopen-looppoles,andZisthenumberofclosedlooppoles.Thuswedeterminethatthatthenumberofclosedlooppoles,Z,intherighthalf-planeequalsthenumberofopen-looppoles,P,thatareintherighthalf-planeminusthenumberofcounterclockwiserevolution,N,around1ofthemapping,i.e.Z=PN.UseNyquistcriteriontodeterminestabilityIfP=0(openloopstablesystem),foraclosedsystemstobestable(i.e.Z=0),weshouldhaveN=0.Thatis,thecontourshouldnotenclose(1;j0).ThisisasshowninnextFig-ure(a).7 Ontheotherhand,anothersystemwithP=0(openloopstable)hasgeneratedtwoclock-wiseencirclementof(1;j0),(N=2),asshowninFigure(b)below.ThusZ=PN=2.Thesystemisunstablewithtwoclosed-looppolesintherighthandplane. Figureabove;Mappingexamples:(a)contourdoesnotencloseclosedlooppoles;(b)contourdoesencloseclosedlooppoles;8 Example:ApplytheNyquistcriteriontodeter-minethestabilityofthefollowingunit-feedbacksystemswith(i)G(s)=s+3 (s+2)(s2+2s+25):(ii)G(s)=s+20 (s+2)(s+7)(s+50):(iii)G(s)=500(s2) (s+2)(s+7)(s+50):Solution:For(i)and(ii),checkpolarplotsinthepreviouslecture.ForbothsystemswehaveP=0(openloopstablesystem).Thetwonyquistplotsdoesnotenclose(1;j0),(N=0)ThusZ=PN=0.Bothsystems(i)and(ii)arestablesincetherearenoclose-looppolesintherighthalfplane.9 For(iii),werunnumg=500*[1-2];;deng=conv([12],[17]);deng=conv(deng,[150]);G=tf(numg,deng);nyquist(G);gridon; -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 0 dB -20 dB -10 dB -6 dB -4 dB -2 dB 20 dB 10 dB 6 dB 4 dB 2 dB Nyquist DiagramReal AxisImaginary Axis Figureabove;ThepolarplotsforG(s)=500(s2) (s+2)(s+7)(s+50):WehaveP=0(openloopstablesystem),butN=1,soSystem(iii)isunstablewithoneclosedlooppoleintherighthalfplane.10