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1、IntroductiontoModernControlTheoryWhendifferentialequationsareencountered,theyarelinearizedandsubjectedtowhateverconstraintsarenecessarytoestablishusefulinput-outputrelationships.Arecognitionoftheapplicabilityofwell-knownmethodsinotherfieldsofknowledge.Optimalcontrol
2、theoryoftendictatesthatnonlineartimevaryingcontrollawbeused,evenifthebasicsystemislinearandtime-invariant.Whennonlinearitiesandtimevariationarepresent,theverybasisforclassicaltechniquesisremoved.Somesuccessfultechniquessuchasphase-plane,describingfunction,andadhocme
3、thods,havebeendevelopedtoalleviatethisshortcoming.Withanadvancingtechnologicalsociety,thereisatrendtowardsmoreambitiousgoals.Thisalsomeansdealingwithcomplexsystemwithalargernumberofinterestingcomponents.Theneedforgreateraccuracyandefficiencyhaschangedtheemphasisonco
4、ntrolsystemperformance.Theclassicalspecificationsintermsofpercentovershoot,settlingtime,bandwidth,etc.,haveinmanycasesgivenwaytooptimalcriteriasuchasminimumenergy,minimumcost,andminimumtimeoperation.Optimizationofthesecriteriamakesitevenmoredifficulttoavoiddealingwi
5、thunpleasantnonlinearities.Optimalcontroltheoryoftendictatesthatnonlineartimevaryingcontrollawbeused,evenifthebasicsystemislinearandtime-invariantTheconceptofstateoccupiesacentralpositioninmoderncontroltheory.However,itappearinmanyothertechnicalandnon-technicalconte
6、xtaswell.Inthermodynamicstheequationsofstateareprominentlyused.Binarysequentialnetworksarenormallyanalyzedintermoftheirstate.Ineverydaylife,monthlyfinancialstatementsarecommonplace.ThePresident’stateoftheUnionmeesageisanotherfamiliarexample.Inalloftheseexamplestheco
7、nceptofstateisessentiallythesame.Itisacompletesummaryofthestatusofthesystemataparticularpointintime.Knowledgeofthestateatsomeinitialtimet0plusknowledgeofthesysteminputsaftert0,allowsthedeterminationofthestateatalatertimet1.Asfarasthestateatt1isconcerned,itmakesnodif
8、ferencehowtheinitialstatewasattained.Thusthestateatt0constitutesacompletehistoryofthesystembehaviorpriortot0,insofarasthathistoryaffectsfu
