Model Uncertainty and Robust Control

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1、ModelUncertaintyandRobustControlK.J.ÅströmDepartmentofAutomaticControlLundUniversity,Lund,SwedenEmail:kja@control.lth.se,Fax:+46461381181IntroductionAkeyreasonforusingfeedbackistoreducetheeffectsofuncertaintywhichmayappearindifferentformsasdisturbancesorasotherimperfectionsinthemodelsusedtodesignthe

2、feedbacklaw.Modeluncertaintyandrobustnesshavebeenacentralthemeinthedevelopmentofthefieldofautomaticcontrol.Thispapergivesanelementarypresentationofthekeyresults.Acentralproblemintheearlydevelopmentofautomaticcontrolwastoconstructfeedbackamplifierswhosepropertiesremainconstantinspiteofvariationsinsuppl

3、yvoltageandcomponentvariations.Thisproblemwasthekeyforthetelephoneindustrythatemergedinthe1920s.Theproblemwassolvedby[9].Wequotefromhispaper:“..bybuildinganamplifierwhosegainisdeliberatelymadesay40deci-belshigherthannecessary(10000foldexcessonenergybasis)andthenfeedingtheoutputbackontheinputinsuchawa

4、yastothrowawayexcessgain,ithasbeenfoundpossibletoeffectextraordinarilyimprovementinconstancyofamplificationandfreedomfromnonlinearity.”Blacksinventionhadatremendousimpactanditinspiredmuchtheoreticalwork.Thiswasrequiredbothforunderstandingandfordevelopmentofdesignmethod.Anovelapproachtostabilitywasdev

5、elopedin[36],fundamentallimitationswereexploredby[10]whoalsodevelopedmethodsfordesigningfeedbackamplifiers,see[11].AsystematicapproachtodesigncontrollersthatwererobusttogainvariationswerealsodevelopedbyBode.Theworkonfeedbackamplifiersbecameacentralpartofthetheoryofservomech-anismsthatappearedinthe1940

6、s,see[22],[27].Systemswerethendescribedusing6364ModelUncertaintyandRobustControltransferfunctionsorfrequencyresponses.Itwasverynaturaltocaptureuncertaintyintermsofdeviationsofthefrequencyresponses.Anumberofmeasuressuchasam-plitudeandphasemarginsandmaximumsensitivitieswerealsointroducedtodescriberobu

7、stness.DesigntoolssuchastheBodediagramintroducedtodesignfeedbackam-plifiersalsofoundgooduseindesignofservomechanisms.Bode'sworkonrobustdesignwasgeneralizedtodealwitharbitraryvariationsintheprocessbyHor