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1、IntroductiontoModernControlTheoryWhendifferentialequationsareencountered,theyarelinearizedandsubjectedtowhateverconstraintsarenecessarytoestablishusefulinput-outputrelationships.Arecognitionoftheapplicabilityofwell-knownmethodsinotherfieldsofknowledge
2、.Optimalcontroltheoryoftendictatesthatnonlineartimevaryingcontrollawbeused,evenifthebasicsystemislinearandtime-invariant.Whennonlinearitiesandtimevariationarepresent,theverybasisforclassicaltechniquesisremoved.Somesuccessfultechniquessuchasphase-plane
3、,describingfunction,andadhocmethods,havebeendevelopedtoalleviatethisshortcoming.Withanadvancingtechnologicalsociety,thereisatrendtowardsmoreambitiousgoals.Thisalsomeansdealingwithcomplexsystemwithalargernumberofinterestingcomponents.Theneedforgreatera
4、ccuracyandefficiencyhaschangedtheemphasisoncontrolsystemperformance.Theclassicalspecificationsintermsofpercentovershoot,settlingtime,bandwidth,etc.,haveinmanycasesgivenwaytooptimalcriteriasuchasminimumenergy,minimumcost,andminimumtimeoperation.Optimiz
5、ationofthesecriteriamakesitevenmoredifficulttoavoiddealingwithunpleasantnonlinearities.Optimalcontroltheoryoftendictatesthatnonlineartimevaryingcontrollawbeused,evenifthebasicsystemislinearandtime-invariantTheconceptofstateoccupiesacentralpositioninmo
6、derncontroltheory.However,itappearinmanyothertechnicalandnon-technicalcontextaswell.Inthermodynamicstheequationsofstateareprominentlyused.Binarysequentialnetworksarenormallyanalyzedintermoftheirstate.Ineverydaylife,monthlyfinancialstatementsarecommonp
7、lace.ThePresident’stateoftheUnionmeesageisanotherfamiliarexample.Inalloftheseexamplestheconceptofstateisessentiallythesame.Itisacompletesummaryofthestatusofthesystemataparticularpointintime.Knowledgeofthestateatsomeinitialtimet0plusknowledgeofthesyste
8、minputsaftert0,allowsthedeterminationofthestateatalatertimet1.Asfarasthestateatt1isconcerned,itmakesnodifferencehowtheinitialstatewasattained.Thusthestateatt0constitutesacompletehistoryofthesystembehaviorpriortot0,insofarasthathistoryaffectsfu
