上交自控讲义第十讲.ppt

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1、Module10RulesforPlottingRootLocus(W.R.Evans,1948)RequirementsontheModuleGrasptherulesforplottingtherootlocusGrasptheconceptofparameterrootlocusandthemethodtotransformitintoageneralizedrootlocusequationformDefinitionoftherootlocus:RootLocusisanengineerin

2、gmethodofgraphicallysolvingthecharacteristicequation.Rootlocusisthetrackoftheclosed-looppolesonsplanevariedaccordingtothevariationofsystemparameter(s)GoalsoftherootlocusareTodeterminewhetherasystemisstableornot;Todeterminetheappropriaterangesofforward-p

3、athgainK(orsomealternativeparameter)forconditionallystablesystems2RootLocusmethodAdvantagesProvideastraightforwardmethodofgraphically“solving”thecharacteristicequationonthecomplexplaneProvidemuchmoreinsightintothetime-domainresponseofthesystemProvideapo

4、werfuldesigncapability,somethingnotobtainablefromadirectsolutionofthecharacteristicequationForatrialvalueofs,itmustsatisfybothequationsbelowsimultaneouslyStrategiesFindallvaluesofssatisfyingtheargumentequationFindparticularvaluesofssatisfyingthemagnitud

5、eequationReIm[s]

6、s

7、MagnitudeequationArgumentequation3AnexampletoexplainrootlocusmethodAsthevalueofKincreasesfromzerotowardinfinity,thecorrespondinglocationofs“moves”fromanopen-looppoletowardinfinityReIm[s]

8、s

9、

10、s+1

11、ReIm[s]

12、s

13、

14、s+1

15、4Anotherexampletoexplainr

16、ootlocusmethodReIm[s]

17、s+2

18、

19、s+1

20、

21、s+3

22、ReIm[s]-1-2-35Anexample:Question:drawtherootlocusforthesystemwithopen-looptransferfunctionReIm[s]-2-52j-12°12°-2j6.32jK=326RulesGeneralizedRootLocusEquationform:VariableparameterKinthenumeratorEvansformRule#1.Drawthec

23、omplexplaneandmarkthenopen-looppolesandmzeros.ThelocusstartsatapoleforK=0andfinishesatazeroorinfinitywhenK=∞.Thenumberofsegmentsgoingtoinfinityisthereforen-m.Rule#2.Segmentsoftherealaxistotheleftofanoddnumberofpolesorzerosaresegmentsoftherootlocus,remem

24、beringthatcomplexpolesorzeroshavenoeffects.7Rules:Rule#3.Thelociaresymmetricalabouttherealaxissincecomplexrootsarealwaysinconjugatepairs.Theanglebetweenadjacentsasymptotesis360°/(n-m),andtoobeythesymmetryrule,thenegativerealaxisi