Gutman_Systems-&-Control-Letters_Robust-and-adaptive-control-fidelity-or-an-open-relationship

智能计算 和 自适应控制

Available online at

Systems&ControlLetters49(2003)9–

19

/locate/sysconle

Robustandadaptivecontrol:ÿdelityoranopenrelationship?

Per-OlofGutman

FacultyofCivilandEnvironmentalEngineering,Technion—IsraelInstituteofTechnology,Haifa32000,Israel

Abstract

Robustandadaptivecontrolareessentiallymeanttosolvethesamecontrolproblem.GivenanuncertainLTImodelsetwiththeassumptionthatthecontrolledplantslowlydriftsoroccasionallyjumpsintheallowedmodelset,ÿndacontrollerthatsatisÿesthegivenservoanddisturbancerejectionspeciÿcations.Speciÿcationsonthetransientresponsetoasuddenplantchangeor“plantjump”areeasilyincorporatedintotherobustcontrolproblem,andifasolutionisfound,therobustcontrolsystemdoesindeedexhibitsatisfactorytransientstoplantjumps.Thereasontouseadaptivecontrolisitsability,whentheplantdoesnotjump,tomaintainthegivenspeciÿcationswithalower-gaincontrolaction(ortoachievetighterspeciÿcations),andalsotosolvethecontrolproblemforalargeruncertaintysetthanarobustcontroller.Certainlyequivalence-basedadaptivecontrollers,however,oftenexhibitinsu cientrobustnessandunsatisfactorytransientstoplantjumps.Itisthereforesuggestedinthispaperthatadaptivecontrolalwaysbebuiltontopofarobustcontrollerinordertomarrytheadvantagesofrobustandadaptivecontrol.Theconceptiscalledadaptiverobustcontrol.Itmaybecomparedwithgainscheduling,two-timescaleadaptivecontrol,intermittentadaptivecontrol,repeatedauto-tuning,orswitchedadaptivecontrol,withtheimportantdi erencethatthecontrolisswitchedbetweenrobustcontrollersthatarebasedonplantuncertaintysetsthattakeintoaccountnotonlythecurrentlyestimatedplantmodelsetbutalsothepossiblejumpsanddriftsthatmayoccuruntiltheearliestnexttimethecontrollercanbeupdated.c2003ElsevierScienceB.V.Allrightsreserved.

Keywords:Adaptivecontrol;Robustcontrol;Identiÿcationforrobustcontrol;Plantuncertainty;QFT

1.Introduction

Plantuncertaintywasalwaysattheheartoffeed-backcontroltheory.Variousrobustcontrolmethodsweredevelopedtohandleextendedplantuncertainty.Basedonclassicalcontrol,quantitativefeedbackthe-ory(QFT)wasinvented,see[14,15].Basedonmod-erncontroltheory,H∞;H2,andthe -methodsweredeveloped,see[18].Robustpoleplacementisde-scribedin[1].Mostrobustmethodsarealsostrongly

Tel.:+972-4-829-2811;fax:+972-4-822-1529.

E-mailaddress:peo@tx.technion.ac.il(P.-O.Gutman).

supportedbyanddependentondesignsoftware,e.g.[11].

TherobustcontrolproblemisingeneralNP-hard.Still,theavailablecomputationaltoolshaveprovedtobeveryuseful.Ifsuccessful,everyrobustdesignresultsinaÿxedLTIcontrollerthatcontrolsany“frozen”plantintheplantuncertaintyset,satisfyingthegivenspeciÿcations.Itisnotpossibletotell,apri-ori,ifagivenrobustcontrolproblemhasasolutionwithoutperforming(apartof)thedesign.

Adaptivecontrolemergedasanalternativetohan-dleuncertainplants.Theideaistocombineanon-lineidentiÿcationalgorithmwithacontroldesignmethodtoyieldatime-varyingcontrollerthatfollowsthe

c2003ElsevierScienceB.V.Allrightsreserved.0167-6911/03/$-seefrontmatter

doi:10.1016/S0167-6911(02)00339-0

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10P.-O.Gutman/Systems&ControlLetters49(2003)9–19

changingceptualplant,see[3].Inspiteoftheirobviouscon-fort,theadaptiveappeal,controllers,andanimpressiveinparticulardevelopmentthoseef-ubiquitouscertaintyseemsanteedtobeinequivalencethatindustryclosed-loopasexpected.principle,havenotbasedbecomeonstabilityThecannotreasonbeforthisearonthesamelevelofconÿdenceaswithguar-lin-havecontrollers,tationunsatisfactorythatadaptivetransientcontrollersbehaviouroftenduringseemadap-tothethatadaptivetoaplantneltheydemandcontrollerchange(e.g.duringastart-upwhenhighlyisinitiallyskilledandwronglyeducatedtuned),person-andparisonThisfortuningpaperdesignbetweenisandnotmaintenance.

variousmeantrobusttobeaandreviewadaptiveoforcontrolacom-trolforthemsomefrommethods.ofarobustInstead,wetrytoviewadaptivecon-theirrespectivepointofview,shortcomingsandsuggestaremedybustnesstoeachotherinasuitableway.Webydiscussmarryingtorobustnessimproverelevanton-linesystemidentiÿcationidentiÿcation,inandideashowro-SISOusedLTIexamplesrequirementswithofadaptiveordercontrol.toSimplereduceicalconcepts,nature,forexpositorypurposes.parametricBecauseuncertaintyofitsgraph-aredesignwithoutQFTendorsingisusedasitaortoolanytootherillustrateparticularthebetweenSeveralmethod.

previouscontributionstreattherelationshipceptInplant[7ofrobustandadaptivecontrol.In[9]thecon-]itrobustisproposedcertaintytoequivalencewasintroduced.quencymodelwithunstructuredestimateuncertaintyon-lineainnominalthefre-designrobustindomainanadaptivetobeusedforrobustcontrolsysteme.g.adaptive[5,and20].adaptiveIn[8]itcontrolsetting.ispointedisThefurtherintegrationoutexpoundedbetweenin,signcontrolarecomplementary,andthatthatrobustonlyandde-viatedlimitationsdiscusseswiththehelpduetoofplantuncertaintycanbealle-controlThepapertheseadaptation.Thecurrentpaperveryproblemisorganizedpointsfurther.

isdeÿned.asSectionsfollows.In3andSection42thecontrol,briefpleplantisgivenrespectively.descriptionsofthetransientInofSectionrobustcontrolandadaptivecontainbehaviours5anillustratingafteraexam-respectively,changeofdisadvantages.witharobustsuddenTheaandadaptivecontrolsystem,argumentsummaryisofmadetheirclearadvantagesinSection

and6spective.whereadaptive7Robust,leadingTheupr controlisseenfromarobustper-tooletheofadaptationisdiscussedinSectioninControlinSectionsuggested8.AparadigmconclusionofAdaptiveisbibliography.Section9,followedbyanAcknowledgementfoundanda2.Problemdeÿnition

withForLTIplant

uncertainty.simplicity,weGivenconsideranuncertain,onlytheSISOstrictlyLTIpropercaseP(s)∈{Pi(s)}=n(s;p)e

pqs

(1+M(s))(1)withtheuncertainparametervectorp∈ ∈Rqthe,and|proper,M(jmultiplicative!)|6m(!unstructureduncertaintysatisfyingknown.andinThethe).M(s)isassumedtobestableandindexhigh-frequencyiin(gainsignof(1)isdisturbances,thesetandnotTheDenumeration.1)only),areassumedInput,denotestoactstate,membershiponandtheoutputi(sspeciÿcation,

closed-loopspeciÿcationsaregivenasaplant.servoa(!)6|Y(j!)=R(j!)|6b(!);

(2)

whereofY(j!)andR(j!sensitivitythecontrolledspeciÿcation,output)andarereference,theLaplacerespectively,transformsa|S(j!)|6x(!)

(3)

and/oramongItisanyassumedotherdisturbancethattheplantrejectionP(speciÿcation.change{environment.ofPs)“slowly”driftsioperating(s)}.Suchpoint,adriftormayabecausedbywear,sionally”fromjumpsMoreover,within{itPis(sassumedchange)},i.e.suddenlythatinPthe(s)outside“occa-imaypartialbeoneLTIplantinstancetoanother.Suchchangesajumparmfailure,causedorbyunknownachangeloadingofplantequipment,oraonpicksupanunknownload),or(e.g.whenwhenswitchingarobotofsumedsetthe(control1)describessystemthewithoutknowingwhichmemberlowerthatthefrequencyplantoftheatthatmoment.Itisas-andthan,thatthanthethebandwidthofthejumpsclosed-loopisconsiderablysystemtheclosed-loope.g.thespeedspeedofsystem,ofthethedriftseestepisconsiderablylowere.g.response[3].Onetransientcouldsay

of

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that[troller6].Thethe“productcontroldesignofplantproblemchangeisandthentimeissmall”speciÿcations.thatmakestheclosed-loopsystemtoÿndsatisfyacon-the3.Robustcontrol

controllerTherobustcontrolrameters,G(s)andapreÿlterproblemFis(s)towithÿndafeedbacketc.,{aresatisÿedseeFig.1for,suchthatspeciÿcationsconstant(2),(pa-3),mayP(s)}.NotethattheactualeachcontrollermemberintheplantsetialsoIfdi erasolutionfromtheisfound,canonicalthenformtheshownimplementationinFig.1.assumptionsatisÿedtionsinforSectionslowplant2.If,drift,induespeciÿcationstothequasi-LTIaretoincludetherejectionofdisturbancesaddition,theequivalentspeciÿca-found,theenvisaged“occasionaljumps”,andasolutionisplantjumps.

thenthespeciÿcationsarealsosatisÿedduringtheSomebeintroduction.robustcontrolHeremethodswerementionedinthegiven,likereadersinceafewdetailsaboutQFTwillfamiliarQFTwillbeusedforillustration.ForproblemtopointwiththeH∞-method,wewoulduncertaintyforaoutthattheSISOrobustsensitivity(cf.andFig.2.17only,plantiniswith[18identicalunstructuredforHmultiplicative∞andforQFTeachtheplantuncertainty]).InsetQFT,{Pthespeciÿcationsi(s)}giverise,suchfrequencysatisÿedthatforG(jeach!)!∈,toacomplexvaluedsetBforG(j!)memberBG(j!) in{theP(sspeciÿcations)}.LetBareiL(j!

)=

Fig.1.Theclosed-loopcontrolsystemforrobust

control.

Fig.gether2.withThetheHorowitznominalboundsopenloop,@BL(j!L)forsomefrequencies,to-nom(j!),inaNicholschart.

BtraryG(j!speciÿcationsnominal)Pnom(j!plant.),whereThenPnomL(s)(j∈!{)P∈i(Bs)}isanarbi-nomL(j!)wherenalLaresatisÿedforeachmemberin{P thei(s)},nom(s)=Pnom(s)G(s)isdeÿnedboundcompensateddisplayed@Bopenloop.Ingeneral,theastheHorowitznomi-Lforin(j!a),Nicholsdeÿnedchart.astheWithboundaryHorowitzofBL(jbounds!),isuallyseveralFig.2loopshapedfrequencies.

inorderdisplayed,tosatisfyLnomthe(jbounds.!)isman-See4.Adaptivecontrol

[Theidealadaptivecontrolwouldbedualcontrolplant3],inwhichthecontrolsignalisoptimalcontrolestimationandcontrol.Unfortunately,forbothdualadaptiveiscapabilities:controlcomputationallytheestimateshouldthecurrenthaveprohibitive.theLTI-basedplantfollowingmodel,redesigndesired[all6].controller,Manytypesandofdecideadaptivewhencontrollerstoestimate/redesigndonotadaptivethesefeatures;instead,apracticaldeÿnitionhaveofacontrolcouldbe:“anadaptivecontrollerisnismcontrollerareforadjustment”withadjustable[3].Seeparameters,Fig.3.Theandamecha-tionsadjusted(2),(3),suchetc.,thataresatisÿed.

afterconvergence,parametersspeciÿca-

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Fig.3.Theclosed-loopcontrolsystemforadaptivecontrol.

combinesThecertaintyequivalence(CE)adaptivecontrollersomeparameterestimation(RLS,LMS,etc.)withetc.).controltheThecontrollerdesignmethodparameters(poleareplacement,computedMRAS,asifconditionscurrentparameterestimateistrue.Underidealelling,ednessminimum-phase(nonoise,nodisturbances,noundermod-non-idealandconvergenceplant)[MRASgivesbound-zone,conditions,detuning3].TofromhandleCE,suchsomeasofdeadthesuggested -modiÿcation,handles[6].Afunctioningback-stepping,CEadaptiveetc.,havecontrollerbeensientveryafterslowtheplantoccasionaldriftveryplantwell.jumpTheis,adaptationtran-theAnunsatisfactory,auto-tuner[3]asisillustratedinthehowever,nextsection.oftenconvergencecontrollawanadaptivecontrollerwherehumanofisautomaticallyupdatedonlyafterthesion.trollerThus,operatortheparameterestimate,andonlyonthedemandandunderoperatorsupervi-neitherareavoided,disadvantagesbutanauto-tuneroftheCEisableadaptivetocon-tervention.

plantdriftnorplantjumpswithouthumanhandlein-pre-computedGainscheduling[3]ingseparatecondition.controllersisaschemewheredi erentTheareappliedforeachoperat-naltweensignalsidentiÿcationoperatingconditionisgiveninaorloopbasedonmeasuredexter-thebilitysystemthedi erentprocessvariables.Oftensofttransferbe-problems.isalmostcontrollersisimplemented.SinceGainLTIschedulingatalltimes,handlesthereareplantnodrift

sta-well,robust.ifforeachoperatingconditionthecontrollerisalsopriately.

handledPlantjumpsifthelocalwithincontrolleranoperatingisdesignedconditionappro-aredueThecontroltodi cultiestheinterferencewithCEadaptivecontrolseemtobecontrollersloopsunderhas[6been].Therefore,betweentheidentiÿcationandsuggestedaandnewtypeofadaptivetrol,namessuchastwotimescaleresearchedadaptivelately,tiveintermittentofcontrol,attemptingadaptivecontrol,orswitchedadap-con-andauto-tuningeterscontrolloopsandaregainseparatedscheduling.tocombinewiththeThetheadvantagescontrolidentiÿcationparam-identiÿcationbeingupdatedonlywhennecessaryandwhentheLTIityat(almost)hasallconverged.Hencetheclosedloopistrollersproblem.plantbehaveItis,timesandthereisnolocalstabil-duringhowever,notclearhowthesecon-paper.

jumps.Thisisoneslowoftheplantissuesdriftdiscussedandoccasionalinthis5.Robustvs.adaptive

lustrateInthissection,asimpleexampleisgiventoil-suddensystem,plantthedi erencechangeofofatransientrobustandbehavioursadaptivecontrolafteratainTheplantexamplerespectively.

istakenfrom[2].Considertheuncer-P(s)=

kwithk∈[1;4]andT∈[0:5;2]:

(4)

AcationQFTsultshavingdesignawasperformedwithaservospeciÿ-was

areshownbandwidthinFig.4.ofThe2radresulting=s.Simulationcontrollerre-G(s)=4×107

(s+0:25)(s+1:5)

×

1

andF(s)=2:89

1

:

(5)

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Fig.4.Simulationoftherobustcontrolsystem.Theupperandlowergraphsshowstepresponsesandthecontrolsignals,respectively.Ontheleft,theplantgainkchangesfrom1to4attimet=15swhileT=1.Ontheright,thetimeconstantTchangesfrom1to0.5attimet=15swhilek=parethetransientoscillationsofthestepresponsesatt=12–15sand32–35stoascertainthatthestepresponsesareslightlydi erentbeforeandaftertheplant

changes.

Fig.5.Simulationoftheadaptivecontrolsystem.Theupperandlowergraphsshowstepresponsesandthecontrolsignals,respectively.Ontheleft,theplantgainkchangesfrom1to4attimet=15swhileT=1.Ontheright,thetimeconstantTchangesfrom1to0.5attimet=15swhilek=paretheovershootsofthestepresponsesatt=12and42stoascertainthatthestepresponsesarethesameforthetwoplantsaftertheconvergenceoftheadaptivecontroller.

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14P.-O.Gutman/Systems&ControlLetters49(2003)9–19

Astiveaparameterpolecomparison,placementancontroller,explicitsecond-orderadap-plingestimatorwasimplementedwithRLSwithaassam-thehaddampingaintervalnaturalfrequency=0:3s.The=1:5requiredrad=sandpolethelocationrelativedesigns=0:707.Thus,thespeciÿcationsofthetwocontrolsystemweresimilar.arefoundSimulationsinFig.oftheadaptivecontrollerItisclearfromthesimulations5.

thattheadaptiveadaptation,exhibitsHowever,whiletheunsatisfactoryrobustcontrollertransientsworksduringÿne.systemforhasafterthesameconvergence,referencethestepadaptiveresponse,controltemdi erentwithingivesslightlyplants,whereastherobustcontrolevensys-Thespeciÿcations,di erentforthestepdi erentresponses,plantbutcases.stillin[3].

sameconclusionisdrawnfromExample10.1theRobustdrift.plantrestrictiveACEduringcontroladaptiveoccasionalisstablecontrollerplantandsatisfactorilyjumpscontrolsisstableandonlyslowplantingduringplantsolutionslowjumps,assumptions,ornobutdrift.controlsexhibitsuglytransientsunderdur-Infact,theplantanmoreuniformlyjumps)maybefound(withtheexceptionadaptiveofcontrolplantiÿcations,forlengewhenalargeruncertaintysetorfortighterspec-gettheisbesthowoftobothmarrynorobustworlds.

robustsolutionandadaptiveexists.Thecontrolchal-to6.AdaptivecontrolfromarobustperspectivebackInthissection,thefundamentaltrade-o inandhelpclosed-loopcontrolbetweenspeciÿcationsfeedbackgain,feed-isplantuncertainty,tationofsimpleexamples,anditisillustrateddiscussedhowwiththeIfFig.may6showsshiftthethetrade-o .

adap-trade-o in(withintheplantuncertaintyincreases,allmorefeedbackfeedbackdesign.gainquirement)theperformanceislimitneededsettobetheclosed-loopstabilityre-ÿcationsthewillplantuncertaintygettighter,speciÿcations.maintainIfthetheperformancesameclosed-loopspeci-remainsmorefeedbackunchanged.gainisneedediftoplantFig.bewith1illustrated,letnoPdynamics,(s)=withk∈a[andkscalarexample.TheReferringtrade-o min;Gkmax(s)]=bega¿scalar0a

scalar

gainFig.stability6.Theandtrade-o otherfundamental

infeedbacklimitations.

design,disregardingclosed-loopFig.of7.Theremainingclosed-loopuncertainty uncertaintythefeedbackkgaingfortwodi erentvaluesofastheaopen-loopfunctionmax=kmin.

compensator.kLettheplantuncertaintybedeÿnedbythemaxclosed-loop=kmin.ReferringuncertaintytoEq.(2 ),beletgiventhespeciÿcationby

for =

maxk|S

k|kmax1+kmingbkk=minmax6;(6)whereisb¿a¿0aregiven,and k=PkG=(1+PkG)plant.thecomplementaryFig.Aplotof assensitivityS

afunctionfunctionofgisforshownthekththetrade-o 7fortwomentioneddi erentabovevaluesholds.ofkinmaxWith=kminQFT,.Clearlythe

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15

Fig.with8.templatek=Originalwhich2;a=template,di ers3; =from0:6{;P(2j)(7!},from(7);thenominalplant)nin=that4; a=∈0[2:05;:5;3].

andthereducedtrade-o tratedisapparentateachfrequency.certainwithplantthefollowingexample.ConsiderThisistheillus-un-P(s)=k

s+a

ne sn

;k∈[2;5];a∈[1;3]; ∈[0:3;0:6];

∈[0;0:05]:

(7)

Thevalueuncertainty{set[18],orattemplateeachfrequency,[13],{!P,isdeÿnedasthei(j!)}.InFig.8,ofP(2j)}isillustrated.Noticethatthemaximumgaini.e.{ferringtheP(2j)plant}isgain27dBuncertaintyandtheminimumat2radgainis13dB,cationtranspiresintoFig.(2),=sis14dB.Re-9assumehasthattheoriginalservospeciÿ-taintyat2thatrad=thesmustremainingtobesatisÿed.Fromtheÿgure,itsatisfyclosed-loopgainuncer- (2)=

maxk|S

k(2j)|mink|S

k(2j)|63:73dB;(8)

where functionS

kbound,for(j!the)plementaryTheresultingsensitivitythesumegain@BoftheboundisshowndependsinFig.on10the.NoticeHorowitzL(2j),phase.thatAs-whichnowHorowitz (2)that62a:13tighterdB.Thenspeciÿcationitfollowsisthatdesiredtheforthanbefore,boundseeFig.for102rad,implying=sisaboutthatthe

5dBnewfeedback

higherFig.side9.arrows.

inThe(8)originalisgivenclosed-loopbythedi erence,servospeciÿcation.3:73dB,betweenTheright-handthetwo

Fig.and10.TheHorowitzbounds@BL(2j)fortighterthewithtighterspeciÿcationsoriginalspeciÿcationspeciÿcations(“...tighter(“original”);(unmarked).

specs”);andthetheoriginaloriginaltemplate

thereducedtemplatetemplateandcontrollerever,zeroatthegainmustincreasebyabout5dB.If,how-canislesscurrentuncertain,operatingsuchthatcondition,a∈[2:5e.g.;theplantalgorithm,bedetectedcanthenthebyantemplateson-line3],andthisandparameterHorowitzestimationboundstemplateberecomputed.templatefor2rad=sInisFig.displayed.8thereducedWeuncertaintytheoriginalisfatconsiderablyandtallone.thinnerTherefore,andnoticeshorterthatthetheensuing

than

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Fig.11.Thetrade-o infeedbackdesignwithadaptation.

HorowitzlowerboundforthetighterspeciÿcationoriginalgainwithHorowitz(Fig.10bound)andalmostbecomeequalwilltohavetheÿcation.thevalidfortheoriginalplanttighterknowledgespeciÿcationHenceoriginaltheuncertaintytrade-o andtheoriginalspeci-canbeo -setinFig.by6increasedisapparent.plantAplantSo,Thisuncertaintywhy.

notincreaseortighterthefeedbackgainwhenlargerthatever,iscantheofbasiscourseofberobustdonespeciÿcationscontrol.toacertainThereextent,requireare,how-andit?[closed-loop19],somefundamentalofthemlimitationsconnectedtointhefeedbackrequirementcontrolbackstability,thatprohibitexcessivefeed-fornon-minimumgain.Anyatphase,real-lifeorhasplantlargeincludesphasedelayuncertaintyorismenthighfrequencies.Thenthephase-marginrequire-imposestogetheronabandwidthwithBode’sgain–phaserelationshipnoisetheseewantedFig.isallowedlimitation,andhencealimitampliÿedfeedbackgain.Moreover,thesensor1.High-feedbackattheplantinputby G=(1+PG),causeofactuatoratthesensorsaturationnoiseandfrequencies,compensatorwear,orrequiresincegaintheitismaynotusetheByamoredecreasingexpensive,lownoisesensor.

thecations.trade-o fundamentalinfavourfeedbackplantuncertainty,ofgainlimitation,adaptationandÿghtsshiftsFig.11.

Thisisillustratedtighterintheclosedconceptloopsdiagramspeciÿ-intoThemoreadaptlandmarktheparameterspaper[21of]presentsarobustancontrolleralgorithmhowstratesplantknowledgebecomesavailable,anddemon-whentrolleristhedescribedbeneÿts.inGain[12,adaptation22].ofarobustcon-7.Ther oleofadaptation

weInviewofthesentedassertfyingbythattheobservationsr oleofadaptationinthepreviousforplantssection,repre-thattheplantlinearvaluemodelssetsshould{Pbeequivalenttoidenti-i(j!)quenciesconstrainfailed,atwhichthedesign,Li.e.inQFT}atthoseparlance,frequenciesthefre-nomtionisofL(j!)∈@BL(j!)or,ifthedesignplantnom(juncertainty!)∈BL(j!in).aTospeciÿclocalizefrequencytheidentiÿca-itselfsuggestedisnotundertakenalsoin[7,5rangein],althoughtheidentiÿcationforWewiththebelievedesignthatpurposetheidentiÿcationthefrequencytosatisfytheshoulddomain.speciÿcationsbehelpfule.g.leastpossiblefeedbackgain,andcouldtherefore,tersbeselectivewithrespecttowhatplantparame-oftoidentify.InFig.8,theimprovedidentiÿcationtemplateoneparameteronlywassu cienttodecreasethedecreasedsizethereby5sodB.thatFig.the12feedbackillustratesgaintherequirementcasewhensatisÿedisationtightlysensitivityforsomespeciÿcation,frequency.|S|6The6dBidentiÿca-thatisonlyofasmallertemplate(shadedarea)ishelpfultotheif6itdBincreasessensitivitythelocus.distanceThefromimplicationthetemplateis

that

Fig.12.Helpfulandunhelpfulidentiÿcationofasmallertemplate.

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on-linethethatplantsystemmodel,identiÿcationbutonlygetneednotexactlyidentifytuning.

isrelevanttothesubsequenttothecontrollermodelinformationdesignortoMoreover,itredesignortheretuneadaptivetherobustcontrollershouldbeableanother,isbased,andswitchfromonecontrollerrobustcontrolleronwhichtoandquasi-LTI.

inordertokeeptheclosed-loopsystemrobust8.Adaptiverobustcontrol

re-tuningItis,however,identiÿedoftherobustnotsu cientcontrollertoonbaseonlythetheredesignhelpfullyorisplantuncertaintyset.Certaintyequivalencemustnotratesbesuitable.basedonTheaplantcurrentlyuncertaintyappliedrobustcontrollerduringplantcasestowhichtheplantmaysetthatdriftincorpo-orcalldiagramthistheparadigmperiodtoadaptivethenextrobustcontrollercontrolupdate.jump.AblockWewithReferringisfoundinFig.13.

theconstructiontoFig.14of,aletusillustratetheconcepttoBeforetemplatepossibletakingquenciesmayplantintobedriftaccountcompositetemplatetemplate.

extensionduebasedandonplantprobingjumps,attheselectedidentiÿedfre-pointestimate,[4]ore.g.mayfromevensomebeafrequencycommon

recursive

functionFig.13.Theblockdiagramforadaptiverobust

control.

Fig.forstabilityperformance14.Theidentiÿeddesigntemplate(dashed),(solid),andthetheworstprobabilistictemplatepossibledesignplantdriftonlyand(shaded),plantjumps.

beforetemplatecaseextensiontemplateduefortoestimationthediagram)pointestimatealgorithmand[3].itsAprobabilistictemplate,e.g.high-performancecouldconstitute1 ellipse(intheNyquistservodesignisthetemplateonwhichaperformancespeciÿcation(2).Insuchbased,away,e.g.satisfyinghighservoa≈identiÿed60%ofthewillplantbecases.achievedAworst-caseforthemosttemplate,commone.g.serveasthewithtemplatesetmembershipformethods[10,17]couldofFinally,andthedesignthetrade-o modeluncertainty,betweenstabilityspeciÿcations,designonly.

speedofthesizeidentiÿcationsizeofplantinandcontrollerjumps,andplantdriftupdatingthespeedofon-linetaintyFig.speciÿcations(represented15,whilethesizeofthedesignismodeldepicteduncer-asFig.aresubjectintheÿguretotheassametemplates)trade-o andastheintoplantbe11enlarged;thesizewithoftherespectdesigntomodeluncertaintyhasupdates.jumpsifControllerduringthepossibleplantdriftorupdatestimeperiodwillbetweencontrollerothertheaccuratehand,identiÿcationalongtakesalongoccurtime,while,moreseldomonthesigniÿcantlyplantestimateidentiÿcationiftheplanttimedoesmaygivenotchangeamoreClearly,methodmustajudiciousduringtheidentiÿcationdatacollection.bemade.

choiceoftheon-lineidentiÿcation

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Fig.tiÿcation15.Thestilljumps.

requiretimetrade-o largemaydesigngiveinadaptiverobustcontrol.Alongiden-templatesaccuratelydueidentiÿedtopossibletemplates,plantdriftbutwilland9.Conclusions

manyThewillinstances,mainideasofthispaperareasfollows.Whileinspeciÿcationssatisfactorilyoneareforthesatisfyÿxedparameterrobustcontrollerfullplantthespeciÿedclosed-loopfeedbackothercasesibilitycontrolwhenpreventthefundamentaluncertaintyset,therethisduelimitationsofuncertainty.betweenlution:Inthistightcase,speciÿcationstoadaptationandtheincompat-maylargeplantmayfulÿllsbebyreduced,on-lineandidentiÿcation,a“local”theplantuncertaintyyieldaso-Ittheisimportantthespeciÿcations,thatthemayberobustdesignedcontrollerortuned.thatrelevantpurposebyciesreducingfortothereducetheon-lineplantuncertaintyidentiÿcationinservesawaytherobustplantvaluecontrolsetsdesignforcriticalortuning,frequen-e.g.results,inaproposenothelpfulevenway,accuratenotingones,thatmaynotallidentiÿcationintermittentlythatadaptationtrolinordertoofpreservethecontrollerberelevant.takeplaceWeredesignsystem.theoftheMoreover,robustcontrollerweproposeathatquasi-LTItheon-linecon-onmayancurrentlyextendedidentiÿedplantuncertaintybebasedset,notbutonlyalsoonnextdriftorjumpplantduringuncertaintythetimesettoperiodwhichtheuntilplantandadaptivecontrollercontrolupdate.isThiscalledmarriageadaptivebetweenrobustcontrol.

robusttheregardTheideasandprinciplesdiscussedinthepaper,thatanderalsystemtherelationshipidentiÿcationbetweenareapplicablerobustness,adaptivityplants,framework,suchasnon-linearandinaverygen-nottaintyonlyandforverygeneraluncertaintystructures,multi-variableandchoseninforremainsherethetheSISOquasi-LTIplantswithuncer-forformdidacticoffrequencypurposes.functionvaluesetsinon-lineplace,toalsobedoneforthebeforesimplestadaptiveAlotofresearchofstructures.robustcontrolisfurtherselective,toidentiÿcationroutineshavetobedevelopedCurrentwhereway.identifyOneplantusefuluncertaintyideaintherequired,templateprobingadaptationidentiÿcation.atselectedmaybefoundin[4],Fewfrequenciesalgorithmsissuggestedforswitchedfor[should21]beingofrobustcontrollershavebeenpublished,ofmentionalandmark[16]exception.Nevertheless,onepaper.

theideasofselectiveasaadaptationverysuccessfulpresentedapplicationinthismarriageSo,isittobeÿdelityously,controladaptivebetweencontrolrobustorshouldandanadaptiveopenrelationshipinthesticktoÿdelity.control?Obvi-fattemplates.mayhoweverÿddlearound,butnotwithRobusttooAcknowledgements

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