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Unstable跨越华盛Unstable跨越华盛顿州的塔科马峡谷的首座桥——开通于1940年7月日。只要有风，这座大桥就会晃灾难发灾难发生(a)(b)(c)(a)(b)(c)About(P145.Section9.1.1DefineontheStabilityofClosed-loopWhenthetransferfunctionofsystemAbout(P145.Section9.1.1DefineontheStabilityofClosed-loopWhenthetransferfunctionofsystemisФ(s),theoutputG(s)KMC(s)Φ(s)R(s)R(s)1G(s)Hn(ssir(t)nKMsC(s)R(s)n(ssii9.1Routh’sStabilitynC(t)L1[C(s)]nC(t)L1[C(s)]ResiThesystemissthene nThentotalC(t)ciesitThesystemisThesystemisstableThesystemisstableonlywhenalltheclosed-polesarelocatedintheleft-handhalfofthescomplexplane(LHP);Thesystembecomeunstableassoonasoneclosed-looppoleislocatedintheright-handhalfofthescomplexplane(RHP).s1,s2,s1,s2,LAnditscharacteristicroots9.2AlgebraStabilityOnthebasisoftherelationshipsbetweenrootsandcoefficientsoftheequation,weknowIfwerequestthattheserootss1-nhaveallnegativereal-part,those1a0ina2sOnthebasisoftherelationshipsbetweenrootsandcoefficientsoftheequation,weknowIfwerequestthattheserootss1-nhaveallnegativereal-part,those1a0ina2s,,Ljmustalli,jia3nsisska0i,j,kijOtherwisethereisonepositivereal-partrootatleast.Mn(1)nnai0stablestableOriginalCalculatedOriginalCalculatedetdynnem——ehsd (6RouthTable:(E.J.aLsa01nLaaaaaaa0246 Lsa1a4RouthTable:(E.J.aLsa01nLaaaaaaa0246 Lsa1a4a0Lbbbbb12342a1Lcccc1234aaa b3a2sd21b1a3e1b1OriginalanRouthCriterion(A)–Thesufficientandnecessaryconditionis:AllthedataofthefirstcolumnofRouthCriterion(A)–Thesufficientandnecessaryconditionis:AllthedataofthefirstcolumnofRouth’sarraymustbepositive.KKK64K30 K304KSP.Theopen-looptransferG(s)K'(ss2(TsK,τ,TwhichSP.Theopen-looptransferG(s)K'(ss2(TsK,τ,TwhichthesystemisThecharacteristicequations2(Ts1)K(s1)Ts3s2KsK0T1KKKKTT,0KKmmnfThecoefficientsofanyrowmaybemultipliedordividedbyapositivenumberwithoutchangingthesignsofthefirstcolumn.ThelaborofevaluatingthecoefficientsinRouth’sarraycanbereducedbymultiplyingordividinganyrowbyaconstant.Thismayresult,forexample,inreducingevaluationoftheremaining9.3TheoremsaboutRouth3s53s52s49s35s212s20WhenthefirstterminaWhenthefirstterminarowiszerobutnotallthetermsarezero,thefollowingmethodscanbeSubstitutes=1/xintheoriginalequation,thensolvetherootsxwithpositiverealparts.Thenumberrootsxwithpositiverealpartswillbethesameasthenumberofsrootswithpositiverealparts.Multiplytheoriginalpolynomialbythefactor(s+1),whichintroducesanadditionalnegativeroot.ThenformtheRouth’sarrayforthenewpolynomial.Substituteasmallvariableε(ε>0,andε→0)forthiszeroelement.m.AotneThereareTherearerootswithpositiverealparts.MethodQ1(s)Q(MethodQ1(s)Q(s)(s1)2s44s27s12234975Thesameresultisobtainedbybothmethod.Therearetwochangesofsigninthefirstcolumn,sotherearetwozerosofQ(s)withpositiverealparts.METHODQ(s)2s2METHODQ(s)2s22s11225255052 TheoremAzeroTheoremAzeroWhenallthecoefficientsofonerowarezero,procedureisasTheauxiliaryequationcanbeformedfromtheprecedingrow,asshownbelow.TheRouth’sarraycanbecompletedbyreplacingtheall-zerorowwiththecoefficientsobtainedbydifferentiatingtheauxiliaryequation.Therootsoftheauxiliaryequationarealsorootsoftheoriginalequation.Theserootsoccurinpairandarethenegativeofeachother.Therefore,theserootsmaybeimaginary(complexconjugates)orreal(onepositiveandonenegative),maylieinquadruplets(twopairsofcomplex-conjugateroots),etc.ConsiderthesystemthathasthecharacteristicQ(s)ConsiderthesystemthathasthecharacteristicQ(s)11s218s12129(Afterdividingby(Afterdividingby 920Thepresenceofazerorow(thes1row)indicatesthattherearerootsthatarethenegativeofeachother.Thenextstepistoformtheauxiliaryequationfromtheprecedingrow,whichisthes2row.Thehighestpowerofsiss2,andonlyevenpowerofsappear.Therefore,theauxiliaryequationis Example(Theorem3.Azero99s2Infact,alltherootsofclosed-loopsystems1 s2 s31 s41Example–CompleteSolutionQ(s)s418s1 19(Afterdividingby Example–CompleteSolutionQ(s)s418s1 19(Afterdividingby (Afterdividingby029009Theauxiliaryequation:s2d(s29)2sThesystemisAndalltherootsofclosed-loopsystemss s1 11234 ConceptsandRouthAcriterionforConceptsandRouthAcriterionfordeterminingstabilityofasystembyexaminingthecharacteristicequationofthetransferfunction.ThecriterionstatesthatthenumberofrootsofthecharacteristicequationwithpositiverealpartsisequaltothenumberofchangesofsignofthecoefficientsinthefirstcolumnoftheRouth’sarray.Aperformancemeasureofasystem.systemisstableifallthepolesofthetransferfunctionhavenegativerealparts.StableAdynamicsystemwithasystemresponsetoaboundedProblemProblemProblemProblem1n1nft–satfesfcnepmsanfenepr1G(s)H(s)G(s)H(s)C(s)1G(s)HmK(szjs j nf(KG(s)H(s)(spi9.4RootLocusMethod:MagnitudeandPhaseEquationAsecond-order2KKG(s) H(s)nAsecond-order2KKG(s) H(s)n2ss(s22snn1Locationofrootsforthe - - - - - - - - -1.0