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1、2IntroductiontoChaosTheoryandElectricDriveSystemsAsthisbookisahappymarriagebetweentwodisciplineschaostheoryandelectricdrivesystemsthecoverageofknowledgeissobroadthatitisdesirabletobrieflyintroducebasicchaostheoryandthefundamentalsofelectricdrivesystems.Inthischapter,thenecessarybackgroundkn
2、owledgeofthisbook,namelythebasictheoryofchaosandsomefundamentalsofelectricdrivesystems,arediscussed.2.1BasicChaosTheoryThissectiondescribessomeofthebasicprinciplesofchaos,includingtheconceptofdynamicalsystems,discretemaps,limitsets,attractors,stability,andmanifolds.Thecriteriaofchaosarethe
3、nillustratedusingtheLyapunovexponents,fractaldimensions,andentropy.Hence,bifurcationsandroutestochaoswillbothbediscussed.Finally,variousmethodsforchaosanalysisareintroduced,whichincludewaveforms,phaseportraits,thePoincaremap,bifurcationdiagrams,time-seriesreconstruction,andthecalculations
4、ofLyapunovexponents,embeddedunstableperiodicorbits,powerspectra,fractaldimensions,andentropy.2.1.1BasicPrinciples2.1.1.1DynamicalSystemsAutonomousDynamicalSystemsAnnth-orderautonomousdynamicalsystemisdefinedbyx_¼fðxÞ,xðt0Þ¼x0.Thecorrespondingsolutionisftðx0Þ,andftðx0Þiscalledasaflow.Forauton
5、omouscontinuoussystems,thevectorfieldfdoesnotdependontimet.NonautonomousDynamicalSystemsAnnth-ordernonautonomousdynamicalsystemisdefinedbyx_¼fðx;tÞ,xðt0Þ¼x0.Thecorres-pondingunderlyingflowisftðx;t0Þ.Fornonautonomouscontinuoussystems,thevectorfieldfdependsnotonlyonthestatevariablexbutalsoonthet
6、imet.ChaosinElectricDriveSystems:Analysis,ControlandApplication,FirstEdition.K.T.ChauandZhengWang.©2011JohnWiley&Sons(Asia)PteLtd.Published2011byJohnWiley&Sons(Asia)PteLtd.ISBN:978-0-470-82633-124ChaosinElectricDriveSystems2.1.1.2DiscreteMapsOrbitnnAdiscretesystemcanbedefinedbyamapP:R!Rwith
7、thestateequationxkþ1¼PðxkÞ,wherenxk2Rarethestatesatthekthiterativetime,andPmapsthestatexktothenextstatexkþ1.Startingfrom1aninitialconditionx0,repeatedapplicationofPgeneratesasequenceofpointsfxkgk¼0whichisknownasanorbit(ParkerandChua,1999).PoincareMapAclassica
