1、The Stability of Linear FeedbackTheStabilityofLinearFeedbacksystemsy Astablesystemisadynamicsystemwithabounded response to a bounded inputboundedresponsetoaboundedinput.( ) Sbl(b) Nl( ) Ubl(a)Stable(b)Neutral(c)Unstable塔科马大桥倒塌视频塔科马大桥倒塌视频relativestability:inordertoenablethesystembehaviorwithreliablep
2、erformance,itnotonlyrequirethestabilityp,yqybutalsoenoughstabilitymarginsothatthesystembehavewellindistursbanceorchangingenvironment.relativestabilitystabilitymarginGainMarginPhaseMarginIntermsoflinearsystems,werecognizethatthestabilityrequirementmaybedefinedintermsofthelocationofthepolesofhld lffhl
3、d lftheclosedlooptransferfunction.TheclosedloopsystemtransferfunctioniswrittenasMizsK)(RmmmmQkkNiisssszsKsqspsT122211)(2)()()()()(whereq(s)=A(s)=0isthecharacteristicequationwhoserootsarethepolesofthld ltThttfilftimk11theclosedloopsystem.Theoutputresponseforanimpulsefunctioninput(whenN=0)isthenRmmtmQ
4、tkteBeAtymk)sin(1)(Toobtainaboundedresponse,thepolesoftheclosedloopsystemmustbeinthelefthandportionofthe5plane.Thus,anecessarymmmmmkky11)()(pp,yandsufficientconditionforafeedbacksystemtobestableisthatallthepolesofthesystemtransferfunctionhavenegativerealparts.Thth df t bilititiThemethodsofstabilityc
5、riterionThe routh hurwith stability criterionTherouthhurwith stabilitycriterionNyquist stability criterionNyquist stabilitycriterionLyapunov stabilitycriterionSTABILITYCRITERIONSTABILITYCRITERIONyapunov stability criterion.Therouthhurwith stabilitycriterion0)()(0111asasasasqsnnnncharacteristicequati
6、on:42naaas1)()(aaaaaa531421nnnnnnnaaaaaass3121132111)()(nnnnnnnnnnnaaaaaaaaaab53153132nnnnnnnncccbbbss514131nnnnnnaaaaab5310nnnhs3111nnnbbaabc10nhs311nnnbbbRouth table:four conditionsRouthtable:fourconditions Condition 1:none of the members of begin column is Condition1:noneofthemembersofbegincolumn
7、is0.di if hbf bi Condition2:1:someofthemembersofbegincolumnis0,butnotforall. Condition3:1:someofthemembersofbegincolumnis0,andforall.Condition4:Ithasmultiplerootsinimaginaryaxis.Example 1Example1AnalysisthestabilityofthefollowingsystemLiapunov stability analysisLiapunovs firstmethod(李雅普诺夫第一法)require
8、stheexplicitformsofsolutions ofthedifferentialequationscalled direct method of LiapunovcalleddirectmethodofLiapunovLiapunovs second method(李雅普诺夫第二法)Liapunov s secondmethod(李雅普诺夫第二法)notrequirethesolutions ofthedifferentialequationsmoreuseful(especiallyfornonlinearsystems,exactsolutionsmaynotbeunobtai
9、nable)ButrequiregenerationofaLiapunov function (energyfunction)called direct method of LiapunovcalleddirectmethodofLiapunovExample2:Example2:Aspringqualitydampersystemasfollows.Themotionofthesystem can be described as follows:systemcanbedescribedasfollows:0kxx fxm 1m0kf set1m0kxx fxsetxxxxxchoose st
10、ate variablexx 112xxxchoosestatevariableThenwegetstateequation:fk21xx 212fxkxxInanytime, thetotalenergyofthesystem2122212121),(kxxxxE00)(0Obviously，when,when0 x0)x(E0 x0)(0E221dd)(dfxxxxkxxExExxE22211221121dd),(dfxxxxkxtxtxxxEtOnlyat,x00/ddtEOtherwise0/ddtETheenergyofthesystemisdecresing,sothesystem
11、isstable.gyyg,yLiapunov stability analysisStabilityinthesenseofLiapunovA spherical region of radius r about an equilibrium state x aspyyAsphericalregion ofradiusr aboutanequilibriumstatexeasWhere x xe iscalledtheEuclideannorm anddefinedasrexxLetS()consistofallpointss.t.1/22221122()()()eennexxxxxxexx
12、xxLetS()consistofallpointss.t.exx000( ;, ),ttfor all ttexxIfthereisanS()correspondingtoeachS()s.t.trajectoriesstartinginS() donotleaveS() astincreasesindefinitely,anequilibriumstate xeissaidtobestableinthesenseofLiapunoveTherealnumber dependsonandalsodependsont0If doesnotdependont0,theequilibriumsta
13、texeissaidtobeuniformlystable12Liapunov stability analysisAsymptoticStabilityAn equilibrium state x is said to be asymptotically stablepyyAnequilibriumstatexeissaidtobeasymptoticallystableifitisstableinthesenseofLiapunovifeverysolutionstartingwithinS() convergestoxe,withoutygS()ge,leavingS() astincr
14、easesindefinitelyInfactitismoreimportantthanmerestabilityItisalocal concept,sosimplytoestablishasymptoticstabilitydoesnotnecessarilymeanthatthesystemwilloperateproperly13Liapunov stability analysisAsymptoticStabilitypyydomainofattraction:thesizeofthelargestregion ofasymptotically stabilityasymptotic
15、allystabilityEvery trajectory originating in the domain of attraction isEverytrajectoryoriginatinginthedomainofattraction isasymptoticallystableItisthestatespace inwhichasymptoticallystablepy pytrajectoriesoriginate14Liapunov stability analysisAsymptoticStabilityinthelargepyyIfasymptoticstability ho
16、ldsforallstates(allpointsinthestatespace)fromwhichtrajectoriesoriginate,theequilibriumstatexeissaidtobejg,qeasymptoticallystableinthelargeThatis,everysolution ofthestateequationconvergestoxeastincreases indefinitelyincreasesindefinitelyAnecessarycondition forasymptoticstableinthelarge isthattherelil
17、ib it t i thh lt tare onlyoneequilibriumstate inthewholestatespaceInpracticeitisadesirablefeaturetodetermine aregionofibililh hdibilld iasymptoticstabilitylargeenough thatnodisturbancewillexceedit15Liapunov stability analysisInstabilitypyyAnequilibriumstatexeissaidtobeunstableifforsomerealnumber0and
18、anyrealnumber 0,nomatterhowsmall,thereis alwaysastatex0inS() s.t.thetrajectorystartingatthisstate leavesS()16Liapunov stability analysisGraphicalrepresentationThe region S() bounds the initial state xpyyTheregionS()bounds theinitialstatex0TheregionS()bounds thetrajectory startingfromanyinitialstatex
19、0intheregionS()17Liapunov stability analysisPositivedefiniteness （正定性） ofscalarfunctionspyyDefn. AscalarfunctionV(x) issaidtobepositivedefineinaregion (whichincludestheoriginofthestatespace)ifV(x) fllhdf(0)0 forallnonzerostatesx intheregion andifV(0) =0Defn. AtimevaryingfunctionV(x,t) issaidtobeposi
20、tivedefine iny g( , )pfaregion (whichincludestheoriginofthestatespace)ifitisbounded frombelowbyatimeinvariantpositivedefinitefhf hd ff( )function,thatis,ifthereexistsapositivedefinitefunctionV(x)s.t.00( , )( ),( , )0,VtVfor all ttVtfor all ttxx0180( , )0,Vtfor all tt0Liapunov stability analysisNegat
21、ivedefiniteness ofscalarfunctionspyyDefn.AscalarfunctionV(x) issaidtobenegativedefineifV(x) ispositivedefinitePositivesemidefiniteness ofscalarfunctionsDefn.AscalarfunctionV(x) issaidtobepositivesemidefinite ifit is positive at all states in the region except at the originitispositiveatallstatesinth
22、eregion exceptattheoriginandatcertainotherstates,whereitiszero19Liapunov stability analysisNegativesemidefiniteness ofscalarfunctionspyyDefn.AscalarfunctionV(x) issaidtobenegativesemidefiniteif-V(x) ispositivesemidefiniteIndefiniteness ofscalarfunctionsDefn.AscalarfunctionV(x) issaidtobeindefinite i
23、fintheregion it assumes both positive and negative values noregion itassumesbothpositiveandnegativevalues,nomatterhowsmalltheregion is20Liapunov stability analysisLiapunovfunctions:pyyAscalar functionandpositivedefinite (p.d.)functionItiscontinuous togetherwith itsfirstpartialderivatives (withrespec
24、ttogfp(pitsarguments)intheregionabouttheoriginThetimederivative takenalongthetrajectory isnegativedefinite (ornegative semidefinite)negative semidefinite)isactuallythetotalderivative ofV(x,t)withrespecttot alonga( , )V x tsolutionofthesystemimplies: V(x,t)isadecreasingfunction oft( , )0V x t 2122122
25、1.( ). .V xxxp d222122.( ). .1xV xxp dx221213.( ).xV xxxp sd12221123.( ).4.( ). .V xxxp sdV xxxxnd 11221224.( ). .5.( )V xxxxndV xx xxindefinite122( )fTheorem：setlineartimeinvariantcontinuoussystem：Thenequilibriumstateis AsymptoticStabilityinthelarge（1）AllcharacteristicrootofAhavenegativerealparts.P
26、roposition1 AllcharacteristicrootofA()havenegative real parts equils to existing an real symmetric matrix P，nnRAnegativerealpartsequilstoexistinganrealsymmetricmatrixP，0 PAPATLet：AnyrealsymmetricmatrixQ0，ifitexistsrealsymmetric（2）。matrixP0，satisfyQPAPAT（2）PxxxVT)(AsliapunovfunctionTheorem1:setthesta
27、teequationofthesystem)( xfx Intheneiborhoodofequilibriumstate0exhascontinuousfirstorder)(xVpartialderivativeandsatisfys:。）。（）（ definite negative is)(2definite positive is)(1xVxVThenisuniformlyasymptoticallystable.If,thenisuniformlyAsymptoticStabilityinthelarge.0exx)(xV)(xVTh2t tht ttif tht)(fTheorem
28、2:setthestateequationofthesystemIntheneiborhoodofequilibriumstate0exhascontinuousfirstorderpartialderivativeandsatisfys:)(xV)( xfx py。）。（）（ tesemidefini negative is)(2definite positive is)(1xVxV(3)Except thatthere is points that makesbut not for all0 x0)(VThenisuniformlyasymptoticallystable.If,theni
29、suniformlyStabilityinthelarge.0exx)(xV)(xV(3)Exceptthat,thereispointsthatmakes,butnotforall.0ex0)(xVTheorem3:setthestateequationofthesystemIntheneiborhoodofequilibriumstate0exhascontinuousfirstorder)(xV)( xfx partialderivativeandsatisfys:。）。（）（ tesemidefini negative is)(2definite positive is)(1xVxVT
30、henisuniformlystable.0exIf,thenisuniformlyStabilityinthelarge.x)(xV)(xVTheorem4:setthestateequationofthesystem)( xfx Intheneiborhoodofequilibriumstate0exhascontinuousfirstorderpartialderivativeandsatisfys:)(xV。）。（）（ tesemidefini negativeor definite positive is)(2definite positive is)(1xVxVThenisunst
31、able.0 x0exyThestateequationsofsystemsasPleaseanalysisthestabilityofequilibriumpointWeobtain:Bl iS ltitifthdofmatrixP,wefindP isp.d.ByapplyingSylvesterscriterionforthep.d.ofmatrixP,wefindP isp.d.S thilib ii t i At ti St bilit i thlSothe equilibriumpointis AsymptoticStabilityinthelargeorwecan get it as follows:cangetitasfollows:cangetitasfollows:Weobtain:isp.d.)(xVWeobtain:isn.d.)(xV谢谢大家！谢谢大家！谢谢大家C
