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ClassicControlTheoryWhathaveyoulearnedinthiscourse?1.linearsystemcontroltheory:modelingtimedomainmethodrootlocusanalysisfrequencydomainmethodSeriescorrection2.*nonlinearsystemcontroltheory:harmonicwaveequilibriummethod3.*samplingcontrolsystemStabilitycriterion&SteadystateerrorDetailLinearsystemtheory1.ModelingClassiccontroltheoryuseT.Fasitsmodel.There’reusuallytwomethodstoobtainapracticalsystem’sT.F:1.1Mechanismanalysismethod:analysisthesystemandusesomecertainphysicalorotherprinciplesofsystemtoobtaindifferentialequationsofsystem,thenusingLaplacetransformtogetsystem’sT.F.Example:LRCseriescircuit(pic.)1.2Testingmethod:Giveacertaininputtosystem,usuallystepinputorimpulseinput,thenrecordsystem’soutputcurve,andanalysisthecurvetogetaappropriateT.Fmodel.Example:A.Singlecapacitytank:thefirstordermodeling(pic.)Accordingtheoutputcurve,it’safirstorderobject,thegaincanbecalculatedasK=......Andthetimeconstantisthetimevalueof63.2%’soutputvalue,that’sT=t......SotheT.Fis......B.Doubletank:thesecondorderobject’smodeling(pic.)ThecurveisapproximateS-shape,soit’sasecondorahigherorderobject.ThegainK=...Supposeit’sasecondorder,thenthegeneralformulacanbelikethisG(s)=......andtimeconstantT1,T2canalsobecalculatedaccordingtothecurve,whichhavecertainformulatoreference.Andifthemodelturnsoutnotaccurateenough,wecanchangethemodel’sorderanddoitagain,untiltheresultiscloseenoughtotheoutputcurve.1.3Togetdiscretetimesystem’smodelwecanusesocalledleastsquaremethod,whichareusuallyappliedbycomputersoftware,likeMATLAB.Theleastsquaremethodisasfollowing,thebestestimatemodelistominimizethesumoftheerrors....../*最小二乘法待补充*/1.4MasonruleIfwehaveasystem’sblockdiagram,wecandirectlywriteoutthesystem’sclosedloopT.F,accordingtoMasonrule:T=......forexample:...2.TimedomainmethodThere’rethreeaspectstoanalysisasystem’sperformanceintimedomain:transientresponse,stabilityandsteadystateerror.2.1transientresponseWemainlyconcernabouttwoparametersintransientresponse:settlingtime(ts)andovershoot(Mp%).Settlingtimepresentsthetimewhenoutputiswithinplusandminusfivepercentageortwopercentageoferror.Andovershootequalsto(peakvaluedividedbysteadystatevalue).Forasecondordersystem,thetransientofstepresponseequalstoc(t)=...ts=...Mp%=...andthecurvelikethis:(pic.)Wecanalsostudythedistributionofopenlooppolestoanalysisthetransientresponse.Forexample(pic.).Thisdistancerepresents.....Andthecloserpolestoimaginaryaxisare,theshortersettlingtimewillbe.Thebiggercos(x)is,thelargerovershootwillbe.2.2stabilityThere’remanymethodstodeterminethestabilityofasysteminclassiccontroltheory,suchasthepoledistributionmethod,Routh-Hurwitzstabilitycriterion,Nyquistcriterion,rootlocusanalysisandsoon.Poledistributionmethod:isthemostbasicmethod.Ifalloftheopenlooppolesaresettledinthelefthalfpartofthes-plane,thentheclosedloopsystemisstable,otherwisethesystemisunstable.(pic.)Routh-Hurwitzstabilitycriterion:isaalgebraicmethodtodeterminethesystem’sstability.FirstlyweshouldgetthecharacteristicequationfromT.F,thenlistouttheRouthTable,ifalloftheelementsofthefirstcolumnarepositive,theclosedsystemcanbedeterminedasstable.E.g.(pic.)2.3steady-stateerrorSteady-stateerror(Es)isusedtoevaluatethesystem’ssteadystateaccuracy.IfweknowthegiveninputandtheT.FofopenloopthenwecanworkoutEs=...(finalvaluetheorem)./*表达式*/3.RootlocusanalysisRootlocusanalysisisagraphicalmethodforexamininghowtheclosedlooprootsofasystemchangewithvariationofacertainsystemparameter,commonlythegainofafeedbacksystem.Ithasseveralrulestofollowwhenplottingarootlocus.E.g(pic.)Rootlocuscanalsobeusedtoanalysisthesystem’stransientperformanceandstability.3.1secondordersystemForasecondordersystem,ifrootsmovedclosertotheimaginaryaxis,thenthesettlingtimewillbesmaller,andifcos(x)getbigger,thenovershootwillalsobehigher.Andwhenalloftherootsaresettledinleftpartofthes-plane,thesystemaredeterminedasstable.3.2higherordersystemForahigherordersystem,there’resocalleddominantpoles,whichplayamajorroleinthetransientperformance.Dominantpolesarethosepoleswhichareclosesttotheimaginaryaxis,thereareusuallytwodominantpoles.Bystudyingthesetwodominantpolesinasimilarway,thehigherordersystemcanbeanalyzedlikeasecondordersystem.E.g.(pic.)3.3effectsofsystemzerosInahigherordersystem,ifweaddaopenloopzeroinrealaxis,itwillchangetheshapeoftherootlocus,thenchangethesystem’stransientperformance.Generallyspeaking,therootlocuswillshifttotheleft,thussystem’stransientperformancewillbeimproved.Andthecloserthezerototheimaginaryaxisis,themoresignificanttheeffectwillbe.What‘smore,ifazeroisplacedclosetoapole,theimpactofthepolewillbereduced.Inextremecases,ifazeroandapolecoincide,whichbecomeadipole,theeffectofthisdipolecanbeignoredintransientanalysis.Themethodiscommonlyusedtoeliminatethebadimpactofapole.4.FrequencydomainmethodInfrequencydomainanalysis,wechosesinesignalasthereferenceinput,andFouriertransformisappliedinT.F,wecallitfrequencycharacteristic,whichincludemagnitudefrequencycharacteristicandphasefrequencycharacteristic.G(jω)=G(s).4.1Nyquistplot&NyquiststabilitycriterionInCartesiancoordinates,therealpartofthetran