北京理工大学810自动控制原理考研课件9

1Chapter7StabilityintheFrequencyDomain7.1Introduction7.2MappingContoursinthes-plane7.3NyquistStabilityCriterion7.4StabilityMarginofSystem7.5Dynamicsperformanceofclosed-loopfromopen-loopfrequencycharacteristic7.6Summary27.1Introduction•DevelopedbyH.Nyquistin1932.•BasedonCauchy’stheorem.3•Thefrequencyresponsecanbeobtainedexperimentally.•Itcanbeutilizedtoinvestigatetherelativestability.理硕教育—专注北理工考研辅导•本资料由理硕教育整理，理硕教育是全国唯一专注于北理工考研辅导的学校，相对于其它机构理硕教育有得天独厚的优势。丰富的理工内部资料资源与人力资源确保每个学员都受益匪浅，确保理硕教育的学员初试通过率89%以上，复试通过率接近100%，理硕教育现开设初试专业课VIP一对一，初试专业课网络小班，假期集训营，复试VIP一对一辅导，复试网络小班，考前专业课网络小班，满足学员不同的需求。因为专一所以专业，理硕教育助您圆北理之梦。详情请查阅理硕教育官网451()1()()()cFsLsGsGsHsG(s)R(s)Y(s)Fig7.1Aclosed-loopsystemGc(s)+H(s)-•WhereL(s)isarationalfunctionofs.•Toensurestability,itmustbeascertainedthatallzerosofF(s)lieintheleft-hands-plane.•Proposeamappingoftheright-hands-planeinF(s)-plane.67.2MappingContoursinthes-plane•AcontourmapisacontourinoneplanemappedintoanotherplanebyarelationF(s).Example:j01-1j-jABCDs-22)(sssFj01-1j-jABCDF(s)7Cauchy’stheorem:IfacontourΓsinthes-planeencirclesZzerosandPpolesofF(s)anddoesnotpassthroughanypolesandzerosofF(s)andthetraversalisintheclockwisedirectionalongthecontour,thecorrespondingcontourΓFintheF(s)-planeencirclestheoriginoftheF(s)-planeN=Z-Ptimesintheclockwisedirection.8Anotherexample:()12sFssj01-1j-jABCDs-2s1abjv01ABCDF(s)u911()1()1()()()()()()niiMkkNsFsLsDsKszDsNsDssp•ThepolesofF(s)arethepolesofL(s).•ThezerosofF(s)arethecharacteristicrootsofthesystem.7.3NyquistStabilityCriterion10j0r=∞•Forasystemtobestable,allthezerosofF(s)mustlieintheleft-hands-plane.•ChooseacontourΓsinthes-planethatenclosestheentireright-hands-plane,thenumberofencirclementsoftheoriginoftheF(s)-planeisN=Z-P.Z:zerosinRHPP:polesinRHP•SothenumberofunstablepolesofthesystemisZ=N+P11•ThecontourΓFisknownastheNyquistdiagramorploarplotofF(s).•AsL(s)=F(s)-1,thenumberofencirclementsoftheorigininF(s)-planebecomesthenumberofencirclementsof-1pointinL(s)-plane.•L(s)istheopen-looptransferfunction.12Nyquiststabilitycriterion1.AfeedbacksystemisstableifandonlyifthecontourΓLintheL(s)-planedoesnetencirclethe(-1,0)pointwhenthenumberofpolesofL(s)intheright-hands-planeiszero(P=0).2.Afeedbacksystemisstableifandonlyif,forthecontourΓL,thenumberofcounter-clockwiseencirclementsofthe(-1,0)pointisequaltothenumberofpolesofL(s)withpositiverealparts.1312()()()(1)(1)KLsGsHsTsTsExample7.10ReIm=0=∞K=1/T-1N=Z=0,sothesystemisstable.14Example7.2Assumingopenlooptransferfunctionis()(101)(21)(0.21)KLssssdeterminethestabilityofthesystematK=20andK=100.15001startlim()lim0LjKK.：2endlim()0270Lj＋.：＝Weneedtofindthecross-overpointandcompareitwith-1!3circledirection()clockwise.：，163234Thecross-overpoint()(412.2)22.41Thecross-overpointisat412.203.05,where(3.05(122.43.05)68.32KLjjKKLj.，＝)＝－SothesystemisstableatK=20andunstableatK=100.17－120－0.3K=2018K=2019100K=10020－1－1.48K=10021K=10022()()()(1)KLsGsHssTsExample7.3j0r=∞0ReIm=0+=+∞=-∞=0-r=∞23(a)Theoriginofthes-plane,0:90090jse000lim()limlimjjKKLseej0r=∞0ReIm=0+=+∞=-∞=0-r=∞24=++(b)Theportionfrom=0toisthepolarplotofL(s).=(c)Theportionfrom=+toismappedintotheoriginoftheL(s)-plane.=0(d)Theportionfrom=toissymmetricaltothepolarplot.2511()lim()lim0()()whenRe,,thecontourwillbemappedintozerooraconstant.Sowecanonlyconsiderthe-axis,i.e.()():mjjnssiijkszLsmnspsRjsjLsLj,,Note:1.26L-+Theplotofcontourfortherange-0willbethecomplexconjugateoftheplotfortherange0,andthepolarplotof()willbesymmetricalinthe()-planeabouttherealaxis.SoitisLsLsL+sufficienttoconstructthecontourfor(0),andtheconclusionchangestothatforthecontourinthe()-plane,thenumberofcounterclockwiseencirclementsofthe(1,j0)pointisequalLs＋toP/2.2.273.Theconclusioncanbeexpandedtothesystemincludingdelayunit.4.IfthecontourL(j)overpassthe(-1,j0)point,thatisoneclose-looppoleonthej-axis,thesystemiscriticallystable.285.Systemwithvpolesattheorigin•Thesupplementcurvemustbedraw.•Thesmallsemicirculardetouraroundthepoleattheorigincanberepresentedbysetting:90090)jse（0,000then,lim()limlimi.e.thecontourencirclestheorigin/2timesintheclockwisedirection,andtheradiusis.jvvjvvKKLseev296.Ifthenumberofcounter-clockwiseencirclementsisN≠P,thentheclosed-loopsystemisunstablewithZunstablepoles,whereZ=P-N.30正负穿越正穿越：相角增加负穿越：相角减少极坐标图穿越点（-1，0）左边实轴的正负穿越次数之差等于极坐标图逆时针方向包围点（-1，0）的周数。Nyquist判据：极坐标图穿越点（-1，0）左边实轴的正负穿越次数之差应等于P/2。P：开环传递函数正实部极点数。31Example7.412()(1)(1)KLssTsTs0-1ReIm0-1ReImKissmallKislargeItispossibletoencirclethe-1point.32atrealaxis321212121()()()KLjjjTTTTTTAtrealaxis,12atrealaxis12()KTTLjTTSothesystemisstablewhen121212121,orKTTTTKTTTT33Example7.52(1)()(1)KsLssTs2200(1)()lim()lim(1)KjKLjLjjT(),Andif0,(),LjTLjT340-1ReImT0-1ReImTSothesystemisstablewhenT,andisunstablewhenT.35Example7.6non-minimumphasesystem6(0.331)()(1)sLsss01.startlim()1809002702.endlim()0180