一階線性系統(tǒng)的自適應(yīng)控制

一階線性系統(tǒng)的自適應(yīng)控制資料來(lái)源：程代展，《應(yīng)用非線性控制》，P222，例8.3例8.3設(shè)計(jì)自適應(yīng)控制器控制不穩(wěn)定系統(tǒng)：V=v+3u假定對(duì)象參數(shù)ap=-1，=3對(duì)于自適應(yīng)控制器是未知的。參考模型選擇為：x =-4x+4r即"m—4，bm—4。取自適應(yīng)增益"—2。兩個(gè)控制器參數(shù)的初值均取為0.仿真中使用了兩種不同的參考信號(hào)：r(t)=4。從圖8.9中知跟蹤誤差收斂到零，但參數(shù)誤差不收斂。r(t)=4sin(3t)從圖8.10中知跟蹤誤差和參數(shù)誤差都收斂到零。8.2AdaptiveControlofFirst-OrderSystemsLetusnowdiscusstheadaptivecomro!offirst-orderplantsusingiheMRACmdhod,asaniElusirationofhawtodesignandanalyzeanAdaptivecontrolsystem.Thecan&1網(wǎng)havepracticalvalueuiilself,beeatiseanumberofsimplesystemsofengineeringinterestmayberepnesaiileclbyaHrsl-urJermodel.Forexample,(hebrakingofanautcmobik,(liedischargeofanelectronicGash,ortheflowoffluii^romatankmaybeapprosimaielyrepresentedbyafirsiorderdifferentialequauon" (&削^herey誼theplantoutput,uisitsinput,anda,andbpareconstantptancparatneters.PROBLEMSPECIFICATIONIntheadaptivecontrolproblem,theplantparametersapandbpareassumedtobeunknown.Letth?desiredperfomuanwoftheadaptiveeonfrolsyslcmbespecifiedbyafirst-orderreferencem?del命=-做為+如，電) MSerb8.2 AdaptiveControlofFi性f~OrtierSystems 327finadiipfiUionlaw,such[hatlheresultingmcKjelfollowingerrory(￡}—j炯asyntplotieailyconvergestozero.InordertoaccompEisht捻知wehavetoassumethesignoftheparameterbtobeknown.Thisisaquitemildcondilion,whichi$oftensatisfiedimpractice.Fortxample,forthebrakingofacar,thisassumpiifinamounttolhesimplepiiysicafknowledgethatbrakingsfowsdownthecar.CHOICEOFCONTROLl^AWAsthefirsrstepint陡adaptivecontrollerdesign,letuschoosethewnfrollawtobea- +?y(0vwhcfft&? %at1&finablefeed痢clgai略一Withfhi$corttrulla虬th*，cJossd-loopdynamicsareV=- &站))十倉(cāng)站X。(8.22)(8,23)1'hereasonforthechoiceofcontrollawin(8.22}isckar:hatkwsthepossibilityofperfecttnodelmatching,liidecd,iftheplantparameterswereknown*thefollowingvaluesofcontrolparametersh)n *叩」% ’hPwoWdJcadioihcclosed-loopdynamics(8-24)y=-知v+Mwhichisidenticaltothereferencemodeldynamics,andyieldstrackingerror.Inthiscase,thefirsttermie(8-22)wgldresultintherightdLc.gain,whilethesecondterminlhecontrollaw(8.22}wouldachievethedualobjectivesofcancelingthelermngure8.8:AMRACsystmnforthefirst-orderplantcontroiferparameterprovidedbyLheadaptationlawandtheidealparainelers,L已,(S.25)Thedynamicsoftraekmgerrorcanbefoundbysubtracting(8.23)from(8.21k&=_板s粉+&「》十站勾》+%&-W=-<7mf+^(2,.r+fiyy) (3.26)ThiscanIkconveaienctyrepresentedasTOC\o"1-5"\h\zbn }e=—<<i,r+況V)=— r4-石狎) (K27)Pgys* ,

TRACKINGCONVERGENCEANALYSISWiihTheconirollawandadaptaiionlawchosenabove,wecannowanalyzethesysienVsstabilityandconvergencebehaviorusingLyapunovHzry,orequivalentlyLemma8,J.Specifically,lheLyapunovfunctionCandida招柄陌中)=擴(kuò)+京％眼+過(guò)｝ (S.29)canbeeasilyshowniohavethefol!owingderivativealongsystemtrajectories-V2Thus,theadaptivecmntrtjlsysiemisgloballysMbl%(hesignalse,arandarebounded.Furihermore,theglobalasymptoticcotivcrgcncx;ofthetrackingerror4(r)isguaranteedbyBarbalat'slemm^hbecausetheboundedftessofe,司.andavimplieslheboundednessafe(accordingto(8J6))andthereforelheuniformcontinuityofKIthinteresiingtowonderwhylheadaptationlaw(S.2K)leadstotruckingerrorcotivergence.Toundent削id出is.k(ussecintuitivelyhowthecontrolparametersshouldbechanged.Consider,withoutlos.sofgeneralhynlhecaseofapositivesgji(站人Assumethatataparticularinstanttthetrackingerrorwisnegative,indicatingIh機(jī)Theplarnoutputis(oosmallFrom(&2。氣世econtrolinputhshouMbe.increasedinordertoincreasetheplantouipuLFrom(8-22Xanincre強(qiáng)seofthecontrolinpul耕canbeachievedbyrnercasing句，(assumingttiair(r)ispositive).Thus,theadaptationhwHwiththevariationrateofarde|)ei—ffeemireferertcesignalsartu^edinthe^imultCKMu,r(O-4.—ffeemireferertcesignalsartu^edinthe^imultCKMu,r(O-4.Ii詁smifromFigurethaithetracking,eirorconvergeslozerobut由巳pararnetererrofdoesnoi.f(F)=^n(3r),his旋知fromFigure810lhaiboihthsUAckingerrorandparameterenurconvergetozero. 口time(sec) cinrie(&ec)Figiirc8,9:TrekingpcrfnrmanocandparametercstimadcHi.,r(J)-4Thebehavioroflheadapiivecontrollerisdemonstratedinthefollowinglimniationeiampfe.time(sec) cinrie(&ec)Figiirc8,9:TrekingpcrfnrmanocandparametercstimadcHi.,r(J)-4PARAMETERCONVKKCSliNCBANALYSISlaui<ierlugaininsighteabomUkbehavior<jtadaptivecontrolsysletn,letuswnderskrndtheconvergenceofe5timm*dparam^lers.FromdiesimulacionresultsofExample8.3,onenoresthattheestimatedparameterscoipivftrgeto(heexactparametervaluesforonereferencesignalbutnotfortheother.ThispromptsustospecifHarearelzidojibcf^eenthe企a【wc$vfdielefcrencesignalsandpammeieri；(MLVc：崔enrc：」"、rhetgtimatedparameterswillncrtconvergetothe.id偵IccnilroHerparametersunlesstherstefenre^nal?(rjccTfaiTScondiuons.IiKteect,suchai^ktiQUb宣地ventlie修alu傳sofreferencesignalandconveigeticeof稅由商tedparangierscanbeintyidvelyundeisLood,InMRACsystems.theo-bjecriveofiheadaptationmechanismistofindouitheparatneierswhichdrivethetrackingerrory-j晶tozero.Ifthereference.signalr(i)isverysimipie,￡uchasa^erooracansCant,itispossibleftxrraa^iyveckirsofoanQxjIlerparameterbesidestheidealparamefe!veclc￥T(oiejd：o[rackingenrorconvergence,Tlieibtlie□dapialionlawwillnotbothertofindoutdieidealparanietersPLee￡1denotetinesetcomposedofalltheparanierervectorswhicheanguarame^trackingerrorconvergenceforapanictilarreferenoesignalhssloryr(f).Tli€iisdependingontheimtia!conditions,thevectorofestimatedparametersmayconvenetoanypoimtinthesetorwonderaroundinthesetinsteadofconvergingtoihetruepatauieter^However^iftherefeteiKt或即M『(“is知complextharonlyrhetreeparametervectors*=[j/canleadtoirackingerrorconv^rgejicekthenweshallhaveparanieicrconvergeficc.Letusflowfindout山白txactcondition?forparameterconvergence.Weshallusesimplifiedargumenlsto"36tedi^uSidetails.Nmciliatrheompi眥oflhestablefillerin(8.27)convergesiozeroandthatitsinpuliseasilyshowntobeuniformlycoatintiDLi^.Tbits,補(bǔ)+碼？mustconvergetozero.Fromtheadapiarionlaw(8.28)andthe(rackingerrorqGnvergwc己-Uu?rateof<heparametefestii'nMfrsconiv^rgtstovectorMOW。]丁shouldsatisfyinorderforttieequationtohaveauniquezerosolution.Ifr(i}isaconsianc,thenforlarger,網(wǎng)=為=ywithabeingthed.c,gainofthereferencemcdeLThus,kyl=[l&]壽Equation(830)bacon瞄5Z+對(duì)*-0Clearlytthisimplies(hattheestimatedparameters,insteadofconvergingtozero,convergetoastraightlinemparameterspace.ForExample8.3,with?-LtheaboveequationimpliesrfiaEthesteadystareerrorsofihctwop^nameterssbotildbeofequMmagnitudesbutoppositesigns.ThisisobviouslyconfirmedinFigureK9.Hcwever*whenr(f)issuchthatthecorrespondingsignalvectorv(<)satisfiesthesocalled"bpersj5ten(excitaiion'1copdition,wecanshowthat(8.28)willguaranteeparameter