ANALYSIS OF A CLASS OF ADAPTIVE ROBUSTIFIED PREDICTORS IN THE PRESENCE OF NOISE UNCERTAINTY

Transcription

1 I. K. Kvačevć dr. Aalza jede vrste adaptra rbusg predktra u prsutst epzatg šuma ISS Prt, ISS Ole DOI: /V AALYSIS OF A CLASS OF ADAPIVE ROBUSIFIED PREDICORS I HE PRESECE OF OISE UCERAIY Ivaa Kstć Kvačevć, Jelea Gavrlvć, Brak Kvačevć Orgal scetfc paper A ew class f adaptve rbust predctrs has bee csdered the paper. Frst a ptmal predctr s develped, based the mmzat f a geeralzed mea square predct errr crter. Startg frm the btaed result, a adaptve rbust predctr s sytheszed thrugh mmzat f a mdfed crter whch a sutably chse -lear fuct f the predct errr s trduced stead f the quadratc e. Ukw parameters f the predctr are estmated at each step by applyg a recursve algrthm f stchastc gradet type. he cvergece f the prpsed adaptve rbustfed predct algrthm s establshed theretcally usg the Martgale thery. It has bee shw that the prpsed adaptve rbust predct algrthm cverges t the ptmal systems utput predct. he feasblty f the prpsed apprach s demstrated by slvg a practcal prblem f desgg a rbust vers f adaptve mmum varace ctrller. Keywrds: estmat, -Gaussa se, parameter estmat, recursve stchastc algrthms, rbust adaptve predct Aalza jede vrste adaptra rbusg predktra u prsutst epzatg šuma Izvr zastve člaak U radu je razmatraa va klasa adaptrah rbush predktra. ajprje je kcpra ptmal predktr utemelje a mmzacj geeralzrae sredje kvadrate pgreške predkcje. me je dređea struktura rbusg adaptrag predktra, čja je steza zvršea a temelju mmzacje mdfcrag krterja u kme je umjest kvadrate uvedea przvlja eleara fukcja pgreške predkcje. epzat parametr predktra prcjejuju se u svakm kraku prmjem rekurzvg algrtma tpa sthastčkg gradjeta. Kvergecja predlžeg algrtma adaptrae rbuse predkcje se teretsk utvrđuje krsteć terju Martgala. me je pkaza da algrtam adaptrae rbuse predkcje kvergra ptmalm sustavu zlaze predkcje. Dbve terjsk rezultat prmjeje su a rješavaje praktčg zadatka steze rbusg adaptvg regulatra mmale varjace. Ključe rječ: prcjea, parametarska prcjea, rekurzv algrtam, rbus adaptra predktr Itrduct asks typcally related t the mder systems thery are ctrl, sgal prcessg flter desg ad predct. hese tasks have bee extesvely studed ctrl thery, cmmucat thery, sgal prcessg ad statstcs [ 6, 4, ]. A wde varety f techques have bee develped fr slvg prblems vlvg these tasks. he basc requremet f all such techques s, the e had, maxmum use f avalable a prr frmat abut the prpertes f the system. hs requremet vlves the adpt f a apprprate presetat f the system.e. ts mathematcal mdel. O the ther had, a mprtat practcal requremet s rbustess f the develped prcedures terms f sestvty t departures frm the assumpts frm whch they are derved umdeled dyamcs, absece f full kwledge f se statstcs, statarty, etc.. Mrever, practce t s ecessary t be ccered wth utlers arsg frm may reass, such as meter ad cmmucat errrs, sesr falures, cmplete measuremets, errrs mathematcal mdels, etc. [7, 8, 9]. hese have very detrmetal effects the statstcal estmat schemes based the Gaussa stchastc dsturbace mdels [0, ]. herefre, frm the practcal pt f vew, t s very mprtat t aalyse rbustess prpertes f adaptve estmat schemes the presece f utlers. Rbust alteratves abud the rbustess lterature [7, 8, 9]. Althugh there are may meags f the wrd "rbust", ts purely data reted vers s the wrd "resstat" [0]. amely, a estmate s called resstat f chagg a small fract f the data by large amuts f results a small chage t the estmate. hs requremet s e f sestvty t utlers. I addt, e may als sst that small chages mst f the data result ly small chages the estmates. hs requremet s e f sestvty t rudg, grupg ad quatzat errrs, r patchy utlers. Furthermre, the term rbustess als has a prbablstc meag, ad at last three dstct prbablstc tats f rbustess ca be perceved. he ldest ad mst accessble s that f effcecy rbustess. amely, a estmatr s sad t be effcecy rbust f t has hgh effcecy, say greater tha 90 %, at a mal Gaussa mdel, ad hgh effcecy at a varety f strategcally chse -Gaussa dstrbuts [0]. Ather t s that f m-max rbustess ver a famly f dstrbuts [7, ]. ypcally, the famly f dstrbuts s fte, ad asympttc varaces are used as quatfable perfrmace csts. Fally, the thrd frm f rbustess s qualtatve rbustess [8]. hs s, fact, a ctuty requremet whch s the prbablstc embdmet f the t that small chages the data shuld prduce ly small chages the estmates, where small chages vlve bth large chages a small fract f the data ad small chages all the data. A mprtat ccept f ths rbustess s that f the fluece curve, whch measures the perturbat f the estmate caused by a sgle addtal bservat, the s-called ctamat [8]. Ufrtuately, the hghly techcal character f m-max rbustess ad qualtatve rbustess makes them relatvely accessble t appled wrkers. Hwever, e ca make estmat prcedures havg readly apparet resstace prpertes, alg wth desrable effcecy rbustess. he rbust adaptve predct algrthm prpsed ths paper s based the last apprach, ehčk vjesk, 605,

2 Aalyss f a class f adaptve rbustfed predctrs the presece f se ucertaty vlvg resstat rbustess prperty, alg wth effcecy rbustess. he paper frst develps the ptmal structure f the predctr, based a geeralzed mea square predct errr []. he, ctuat, a rbust adaptve estep predctr s sytheszed based the mmzat f a crrespdg -lear crter. hs crter s derved frm the geeralzed mea square crter thrugh substtut f the quadratc fuct by a sutably chse -lear fuct. Startg frm the effcecy rbustess prperty ad practcal mprtace f achevg sestvty t utlers ctamatg the Gaussa dsturbaces, ths learty shuld lk lke the quadratc fuct fr small value f the argumet, whereas t has t grw mre slwly tha the quadratc fuct fr large values f the argumet. Furthermre, the resstat rbustess prperty requres that the crter fuct dervatve be buded ad ctuus, sce budedess prvdes fr sgle bservat t have a arbtrarly large fluece, whle ctuty prvdes fr patchy utlers t t have a majr effects. he ukw parameters f the prpsed adaptve rbust predctr are estmated each step by applyg a recursve algrthm f the stchastc gradet type [, 3]. he cvergece f the adaptve rbustfed predctr algrthm s establshed theretcally usg the Martgale thery [, 5, 6, 7]. Startg frm a lear sgleput/sgle-utput ARMAX system represetat, t has bee shw that the prpsed adaptve rbust e-step ahead predct cverges, the Cesar sese, t the ptmal predct f the system utput. he derved theretcal results are used t desg a rbust adaptve mmum varace ctrller. Prblem statemet he ptmzed e-step predctr mmzes the crter based the geeralzed mea square predct errr f the frm [] where the predct errr s y y y y ν, ν, ad ŷ s the e-step predct f the utput frm the system Q q f y. Plymals P q ad sutable prpertes are pre-selected. Specfcally, fr a dyamc system descrbed by the ARMAX mdel the relats betwee the systems utput, y, systems put, u, ad se r dsturbace, e, are expressed by [ 4,5, ]. k Aq y q Bq u+ Cq e, 3 I. K. Kvačevć et al. where q s the ut delay peratr, q y y, ad k s the prcess delay. Here A, B ad C are plymals defed by expresss A q aq a q m m B q b bq b q C q cq cq se e s a whte dscrete radm sequece f zer mea value ad varace σ. Fr the system defed by Eq. 3, the e-step * ptmzed predctr y, whch mmzes the perfrmace dex, s defed by the equats mre detals are Appedx A P q C q y q Q q B q u + q G q y where the plymal G satsfes the prespecfed equat P q Cq Q q Aq + q Gq 6 Here the plymal G s defed by , max, G q g gq g q 7 he geeralzed predct errr s defed by the relat P Q q q e 4 5 ν 8 ad the crter reaches the mmum { } * I p m I p Ε e σ 9 Relat 5, whch defes the ptmzed predctr, may be wrtte a mre cmpact frm as q B q u q G q C q y Q q y + P q Q q 0 If e detes the fltered quattes f the put, u, ad utput frm the system, y, as Q q y u u, y, P q Q q 466 echcal Gazette, 605,

3 I. K. Kvačevć dr. Aalza jede vrste adaptra rbusg predktra u prsutst epzatg šuma the the relat 0 reduces t q B q u + q G q y C q y Usg 4 ad 7, relat may be wrtte the lear regress frm as y * Z θ 3 where the regress vectr s Z y y u k 4... u m y* y* + ad θ s the predctr parameter vectr, whch ctas the ceffcets f the plymals Cq, Bq, Gq, respectvely, that s I practce, parameters θ f the ptmzed predctr 3, 4 are geerally ukw. he basc requremet s t defe a parameter estmat prcedure based avalable measured data the put t ad utput frm the system, arrvg at the adaptve frm f the predctr. 3 A ew rbust adaptve predctr Sce the crter weghts all predct errrs equally, e ca expect that t wll be susceptble t utlers ad hece be -rbust. Oe smple way t rbustfy estmat prcedure s t use a -lear crter { H, } H,, { } I θ Ε ν Ε ν θ, 5 stead f the quadratc e, where H s a rbust scre r lss fuct that has t suppress the fluece f utlers. Let us w shw why crter ca be geeralzed by relat 5. amely, the express fr the mea square crter gradet fllws frm relat, ad by equatg t t zer we derve the cdt fr the mmum f the crter q P q ν I p Ε ν 0, 6 Q q Gve that Eq. 8 apples t the ptmzed slut 6 fr θ θ*, whch crrespds t the crter mmum, ad sce e ad dv are ucrrelated radm quattes, the geeral case t s pssble t trduce a -lear fuct f the geeralzed predct errr, s that the ptmalty cdt 6 s stll satsfed. I rder t acheve ths requremet, ths fuct has t be equal t zer fr zer-value argumets ad features a eve prperty. Mrever, as meted befre, wth regard t the practcal mprtace f achevg sestvty t utlers ctamatg the Gaussa dsturbaces, a eve lss fuct H 5 shuld lk lke the quadratc e fr small values f the argumet, whereas t has t grw mre slwly tha the quadratc fuct fr large values f the argumet. I addt, a resstat rbustess prperty requres that the lss fuct dervatve ψ H ' be bded ad ctuus. hs crrespds, fr example, t the chce f the Huber s lss fuct [7, 9] H c +, f σ σ + c, f < σ σ, 7 where c, are apprprately defed cstats ad Δ s chse t gve the desred effcecy at the mal zermea Gaussa se sequece {e} 3, wth varace σ. herefre, the tug parameter Δ 7 has t be chse s as t prvde the desred effcecy at the mal Gaussa mdel. A cmm chce s Δ,5, kw as the,5-hber s rbust prcedure [7]. hs s, fact, a effcet rbustess requremet. O the ther had, the dervatve Ψ f the H-fuct 7, the scalled fluece fuct qualtatve rbustess [3, 4], s gve by x Ψ x H' x m max,, 8 σ σ σ ad s buded ad ctuus. hs s, tur, a resstat rbustess requremet. hus, the chce f the Huber s lss fuct 7 prvdes fr resstat rbustess, alg wth a effcet rbustess prperty. Applcat f the Rbbs-Mr apprach [, 3] results a stchastc gradet algrthm fr estmatg ukw vectrs f parameters θ f a adaptve rbust predctr, where the gradet f crter defed by Eq. 5 s gve by θ I θ dy Ε{, θ Ψ, ν, θ };, θ 9 I relat 9 the -lear fuct ψ H ', where H s gve by 5, 7. he frm f the stchastc gradet algrthm s [, 3, 7] θ θ +γ, θ Ψ, ν, θ 0 ehčk vjesk, 605,

4 Aalyss f a class f adaptve rbustfed predctrs the presece f se ucertaty Here γ s a sequece f egatve umbers whch ctrl the speed f parameter estmates cvergece, ad ψ H ' s gve by 8, 9. accelerate cvergece f algrthm 0 the eghburhd f the mmum f fuctal 5, ga γ ca be multpled by the pstvely deftve matrx, resultg a ewt Raphs type algrthm [3, 6] R, Ψ, ν, θ θ θ +γ θ Matrx R s Hessa ad s determed by recursve relat [3, 6, 7] +γ θ R, R R Ψν ', ν, θ, θ where ' ψ Ψ. ν Eq. fllws frm the fact that the secd dervatve f crter 5, defg the Hessa matrx, s gve by Iθ Ε, θ Ψ, ν, θ +, ν ',,, } + θ Ψ ν θ θ 3 Mrever, the vcty f ptmal slut θ θ, the Eq. 3 ca be apprxmated by the relat {, ν ',,, } Iθ E θ Ψ ν θ θ 4 sce the v, θ e ad Ψ e 0 mea value f e s zer., due t the fact that Fally, let us apprxmate the mathematcal expectat 4 by the crrespdg arthmetc mea, I θ R, l k θ k k ', Ψ k θ k l k, θ k v frm whch t fllws R R + l, θ, θ l, θ ' v Ψ γ. Let us assume further that algrthm, the γ ad let us trduce the matrx he last relat represets the Eq. wth scalar factr s I. K. Kvačevć et al. R R. Let us als te that the algrthm, was derved fr the geeral mdel. Sce ths research csders a ARMAX mdel 3, t s pssble t, θ mre determe the predct dervatve specfcally. Mrever, e fte resrts t apprxmat [, 3] y, θ Z 5 he apprxmat 5 reflects that the mplct depedecy f vectr Z predctr parameter θ s dsregarded. If e adpts the matrx trace symbls r trr, ad replaces the matrx ga factr R wth the scalar ga r, the deftve frm f the algrthm f stchastc apprxmat type fr adaptve rbust predctr parameter estmat s gve by θ θ + Z Ψ, ν, θ 6 r +Ψ ν ν θ 0 r r ',, Z Z, r u k m y y,, 7 Z y y u k 8 y Z θ ν θ y y 9 he Eq. 7 fllws frm after trducg γ ad replacg the matrx R wth the prevusly defed matrx R, as well as by takg the matrx trace perat the s btaed relat. he Eq. 8 ad 9 are lke befre derved equat, 3 ad 4 respectvely. Eqs. 6 9 defe the adaptve rbust predctr. hs algrthm represets a cmprmse betwee rate f cvergece ad cmputatal cmplexty. he ext task s t aalyse the cvergece f algrthm 6 9. It s mprtat because f the deft f strget cdts whch the algrthm s applcable. 4 Cvergece aalyss he cvergece prperty f the prpsed adaptve rbust predctr ca be vestgated usg the Martgale thery [, 6, 7, 8]. he basc cvergece result s the lemma f eveu [8]. he result s restated a umber f frms that sute better specfc theretcal aalyss. A ufed treatmet f a umber f almst sure cvergece therems, based fact that the prcesses vlved pssess a cmm "almst super-martgale" prpertes, has bee prpsed the lterature [9]. be precse, let { Ω, F, P} be a prbablty space ad F F a 468 echcal Gazette, 605,

5 I. K. Kvačevć dr. Aalza jede vrste adaptra rbusg predktra u prsutst epzatg šuma sequece f creasg sub-sgma-algebras f F. Fr each let z, β, ξad ζ be - egatve F - measurable radm varables such that the cdtal mathematcal expectat { z + F } z Ε +β +ξ z 30 he, the fllwg therem ca be prve [9]. herem : lm z exsts ad s fte, ad ζ < wth prbablty e w.p.. β <, ξ <. I addt, the fllwg prpsts sequeces are frequetly used establshg cvergece results [0]. Lemma. Let the sequeces cverge. he the sequece x y ad k y k x als cverges uder the cdt +. k k he results f herem ad Lemma ca be used t prve the cvergece f the prpsed adaptve rbust predctr 6 9. Hwever, frst we eed t prve the fllwg auxlary lemmas. Lemma. Csder the mdel 3 ad the algrthm 6 9. Let us assume further that the frst dervatve ', Ψ fuct satsfes Ψ ν f the Ψ ν ', 0, kp, where 0 k p Z Z sequeces cverges lm r r < <. he the { } r. he prf s gve Appedx B. Z Lemma 3. he sequece Z dverges { } lm r uder the assumpts f Lemma. he prf s gve Appedx C. Startg frm the results f herem ad Lemmas, ad 3 e ca prve the fllwg cvergece therem. herem : Csder the mdel 3 ad the algrthm 6 9 subject t the cdts: C: All zers f the plymal C q are sde the ut crcle. e s a sequece f buded, depedet ad C: { } detcally dstrbuted..d radm varables, such that the prbablty dstrbut fuct P s symmetrc, sup e < wp.., whle the cdtal ad mea ad varace Ε e F 0, Ε e F σ < { } { } C3: he fluece fuct Ψ s dd ad ctuus almst everywhere. C4: he fluece fuct Ψ ad the prbablty dstrbut fuct P have a cmm rasg pt,.e. Ψ z+ε >Ψ z ε, P z+ε > P z ε fr ε > 0. C5: he fuct Ψ s buded,.e. Ψ z < k, k 0, C6: he dervatve Ψ ν ', f the Ψ fuct satsfes Ψ ν ', 0, kp < k p <. C7: he plymal C q, where 0 3 s strctly pstve real,.e. the real part Re > 0 C q. C8: he system put ad utput sgals satsfy sup u < wp..., sup y < wp... he the adaptve rbust predct ŷ 9 cverges, the Cesar sese, t the ptmal e- step * ahead predctr y 3 wth prbablty e w.p.,.e. he prf s gve Appedx D. he cdts C ad C7 are cmmly used whe the stadard martgale results are appled fr cvergece aalyss [6, 7]. he assumpt C represets a stadard se cdt the rbust estmat []. he cdts C3 C6 defe a class f Ψ, r fluece fucts, that have t leartes cut ff the utlers. May Ψ fucts that are cmmly used rbust estmat, except the Huber s fluece fuct 8, such as Hampel s, ukey s r Adrew s learty, satsfy the abve assumpts [7 9]. Furthermre, t s farly bvus that sme cdt the put sequece must be trduced rder t secure a reasable result. Clearly, a put that s detcally zer wll t be able t yeld full frmat abut the system put- utput prpertes. he cdt C8 represets a reasable practcal assumpt that put- utput sequeces are dscrete-tme sgals wth fte eergy. Remark: he cvergece, Cesar sese, allws the umber f departures f adaptve rbust predct frm the ptmal e t be fte. herefre, a strger result s t prve almst surely cvergece, r cvergece wth prbablty e, fr whch ehčk vjesk, 605,

6 Aalyss f a class f adaptve rbustfed predctrs the presece f se ucertaty { * } P lm y y 0 5 Practcal example demstrate the feasblty f derved theretcal results, the prpsed apprach wll be appled t the prblem f desgg a rbust adaptve mmumvarace ctrller [,, ]. Let the dyamc plat uder csderat be represeted by ARMAX mdel 3 wth ut delay fr whch parameter k s equal t. he dyamc plat has t be ctrlled t make the behavur f the etre ctrl system wth a statary radm setpt apprach the desred behavur f the pre-specfed referece system [,, ]. I ther wrds, the system utput, y, shuld dffer as lttle as pssble, sme sese, frm the desred utput y, wth the gve set pt. he measure f ths dfferece ca be specfed by perfrmace dex wth P Q, where the msalgmet v s defed as v y y. Here the e-step predct f the utput, ŷ, s replaced by the desred utput, y. Wth the system parameters 3 kw, what we have s the prblem f desgg a ptmal ctrller, mmzg the adpted crter. he ptmal ctrl s gve by 5, wth P Q ad k, that s B q u C q y + G q y 3 where the plymal G s defed by 7, ad satsfes the equat 6,.e. Cq Aq + q Gq 3 akg t accut 8 ad 9, e ccludes that fr the ptmal ctrl the msalgmet,ν, s equal t the whte se, e. Mrever, the mmal value f the crter s equal t the se varace. he ctrller equat 3 may be represeted a explctly recursve frm, whch s mre cveet adaptve systems. amely, by trducg the tat f the ctrller parameters vectr q0 q 0 qq b g 0 bbm c c c l 33 ad the bservat vectr Z 4, e ca rewrte 3 as + 0, u y Z y b 0 where frm 4 fllws Z0, s y y g ru u u m s s + l I. K. Kvačevć et al. Here the parameter r s ether 0 r, ad Z 4 crrespds t the value f r equal t e. Hwever, wth a ukw system parameters 3, the eed fr a adaptve system arses. hs system emplys a adapt algrthm whch chages ctrller parameters 33 t make the etre ctrl system meet the requremets. Adaptve ctrl system ca be btaed several ways [,]. Oe pssblty s t determe drectly the ukw ctrller parameters 33 []. hs ca be de by rbust algrthms predctg the desred referece value y, ad mmzg the fuctal f the predct errr, ν, 5. he slut f the predct prblem reles e step ahead predct, ŷ, f y, ad the ctrller equat 34. hs leads t the recursve algrthm 6 9 wth Z beg equal t Z y, 35, ad ν y y. It shuld be ted that the relat 34 ca be rewrtte the frm f equat 3, that s Z θ y ; Z Z y ; 36 hus, the mmum varace strategy s btaed by predctg rbustly e-step ahead the utput, y, wth 9 ad the chsg a ctrl, u, that makes the predct, ŷ, equal t the desred utput, y, as s shw 36. he perfrmaces f ths algrthm, cmpared wth the cveet -rbust mmum varace type adaptve ctrller, are aalysed by smulats the lterature []. 6 Cclus A ew adaptve rbust e-step ahead predctr was sytheszed by mmzg a sutable chse -lear predct errr crter. Gve the mprtace f ccurrece f pulse se, r utlers, wth the Gaussa samples f the measuremet se ppulat, ths learty shuld lk lke a quadratc fuct fr small values f the argumet, whereas t has t grw mre slwly tha the quadratc e fr large value f the argumet. I addt, the -lear lss fuct dervatve, amed the fluece fuct, has resstat rbustess prperty, alg wth the effcecy rbustess. he ukw parameters f the prpsed adaptve rbust predctr are estmated each step by applyg a recursve algrthm f the stchastc gradet type. he cvergece f the adaptve rbust predct algrthm, the Cesar sese, s establshed theretcally usg stadard Martgale thery. It has bee shw that the prpsed adaptve rbust predct cverges t the ptmal systems utput predcts. he btaed theretcal results are used t slve the prblem f desgg a rbust vers f a adaptve mmum varace type ctrller. Further prblems the rbust predct ctext, that are f practcal terest, clude a mult-step predctr that plays a sgfcat rle prcesses vlvg delay. It s well-kw frm egeerg practce that the delay pheme reders the geerat f adequate ctrl act rather dffcult. It wuld als be f terest, because 470 echcal Gazette, 605,

7 I. K. Kvačevć dr. Aalza jede vrste adaptra rbusg predktra u prsutst epzatg šuma f cvergece speed, t csder a rbust vers f the parameter estmat algrthm where the scalar ga factr s replaced wth a sutable matrx. hs, tur, creases the cmputg cmplexty f the parameter estmat algrthm. Appedx A: Dervat f Eqs. 5 9 Algrthm 5 9 s a mdfcat f the results preseted []. amely, t fllws frm Eq. 6 that ν + P q C q Q q A q y q G q y Q q A q y q G q y A Csderg Eq. 3, ad replacg the term Ay frm 3 t A, equat A acqures the frm ν + q Q q B q u q G q y P q C q Q q C q e + + Q q A q y q G q y A If the express square brackets A s equated t zer, the predctr ptmalty equat becmes Q q A q y + + q G q y q Q q B q u + + q G q y A3 Fally, by expressg QA 0 frm the adpted equat 6 ad substtutg ths result t A3, e btas * P q C q y q Q q B q u + q G q y represetg the Eq. 5. Eq. 8 fllws frm A ad A3, that s P Q q q e A4 ν A5 whle the Eq. 9 fllws drectly frm ad 8, thus cmpletg the prf. Appedx B: Prf f Lemma Startg frm the assumpt f Lemma, e f Ψ ', δ> 0. herefre, there exsts a ccludes ν fte pstve cstat k such that k Ψ ', ν, θ. akg t accut 7, ν e ca wrte k r r 0 + ', Ψν k θ k Z k Z k, B r 0 ad r kψν ', θ Z Z Z Z + Ψν ', θ k Ψ ', θ k k Z k Z k Z Z ν Ψν ' k, θ k Z k Z k k k < B he last equalty B results frm Abel-D s therem [9], whch ca be appled sce the sequece Z Z Ψν ', ν, θ dverges, whch cmpletes the prf. Appedx C: Prf f Lemma 3 Smlarly as the prf f Lemma, e ccludes that there exsts a fte pstve cstat k, such that k Ψ ', ν, θ. hus, e ca wrte further ν usg B k r Ψν ', θ Z Z + Ψν k θ k Z k Z k Z Z ', k C he rght had sde f relat C s a csequece f the Abel-D s therem [9], whch cmpletes the prf. Appedx D: Prf f herem Let us trduce Lyapuv s stchastc fuct, V θ θ θ θ D where θ s the predctr parameter vectr estmate, geerated by 6, whle θ s the true ukw predctr ehčk vjesk, 605,

8 Aalyss f a class f adaptve rbustfed predctrs the presece f se ucertaty parameter vectr t be estmated, ad θ s the estmated errr the -th step. Symbl detes the Eucldea rm. Fllwg the methdlgy preseted [], e btas fr the predct errr the relat Z ν q + e D C q akg t accut D, the relat 6 ca be rewrtte q q Z r Z Ψ q e C q frm whch t fllws q q q q q r Z Z Ψq e + C q r Z Z Z Ψ q e C q Addtally, let us defe the fucts Z Fq C q Т Z Е Ψq e F C q Z F q C q Т Z Е Ψ q e F C q D3 D4 D5 D6 Uder the hypthess C ad C5 f herem, e ccludes Z Φ q k, k 0, D7 C q I. K. Kvačevć et al. I addt, the hypthess C4 f herem assumes the fuct Ψ D5 t be mte creasg ad pstve egatve fr pstve egatve argumets. Mrever, uder the hypthess C3, C4 ad C7 f the herem, e btas Z q Z Φq > 0 fr C q θ Z 0 D8 Usg the relats D4, D7 ad D8, e ca wrte { } r Z Z E q F q kz r Z q q F + D9 C q I rder t apply the results f herem the ' relat D9, wth z cdts have t be fulflled: he radm quattes Z ξ k r Z, θ ad β 0, the fllwg q Z ζ q Z Φ r C q have t be -egatve F measurable radm varables. kz Z Z Z k < w.p.. r r he cdt s fulflled bvusly, whle the cdt s satsfed due t the Lemma. hus, by applyg the herem the relat D9, e ccludes lm θ θ w.p.. D0 q r Z Z q Φ < w.p.. D C q Furthermre, let us aalyse the relat D mre detals. akg t accut B ad 7, e ccludes 47 echcal Gazette, 605,

9 I. K. Kvačevć dr. Aalza jede vrste adaptra rbusg predktra u prsutst epzatg šuma Z r ψ ', ν, θ ν Mrever, usg D ad D, e ca wrte r q Z C q Z Ψ ',, ν ν q r r Z q Φ < D D3 Bearg md hyptheses C4 ad C5 f herem, ad by applyg Lemma, e btas Ψ ν ', ν, q r Ψν ', ν, q Z q Z Φ r C q D4 Z k < Furthermre frm D ad Lemma, whch k k, ad Z q x, r Z Ψν ', q q Z y Φ r C q e ccludes θ r Z < I addt, let us defe the quatty D5 p ν e D6 Smlarly as [], e ca shw q C q p Z D7 Bearg md the hypthess C f the herem ad the relats D5 ad D7, e further ccludes p < D8 r w, ext step s t shw that lm 0 r p k D9 k herefre, let us defe a quatty p k r D0 k frm whch t fllws p k + p + r r r k r k r ', mψν θ r p k p k k r r p + r k Z Z p + r where m ] expectat { F } D k ε,, ε> 0. By applyg the cdtal Ε D, e ccludes, uder the relat D8, that s a dscrete super-martgale, satsfyg herem wth p z, β 0, ξ, r ', kmψν θ Z Z z r hs, tur, results lm * D kmψν ', θ Z Z < D3 r Hwever, usg C6 f herem ad Lemma 3, t fllws frm D ad D3 that *0, s that the relat D9 s prve. Fllwg the methdlgy expsed [], e ca als shw frm D8 that lm 0 k ad sce p k D4 ehčk vjesk, 605,

10 Aalyss f a class f adaptve rbustfed predctrs the presece f se ucertaty * p y y e y y e cmpletes the prf. 7 Refereces D5 [] Astrm, K. J.; Wttemark, B. Adaptve Ctrl. // Adds Wesley, ew Yrk, 989. [] Gdw, G. C.; S, K. C. Adaptve Flterg, Predct ad Ctrl. // Pretce Hall, ew Jersey, 984. [3] Bsch va der, P. P. J.; Klauw va der, A. C. Mdellg, Idetfcat ad Smulat f Dyamcal Systems. // CRC Press, Bca Rat, Flrda, 994. [4] Verhacger, M.; Verdult, V. Flterg ad System Idetfcat. // Cambrdge Uversty Press, Cambrdge, 009. [5] Mtra, S. W. Dgtal Sgal Prcessg: A Cmputer based Apprach. // McGraw-Hll, ew Yrk, 00. [6] Smth, S. W. Dgtal Sgal Prcessg: A Practcal Gude fr Egeers ad Scetsts. // ewess: a mprt f Elsever, ew Yrk, 006. [7] Huber, P. J.; Rchett, E. M. Rbust Statstcs. // Jh Wley, ew Yrk, 009. DOI: 0.00/ [8] Hampel, F. R.; Rchett, E. M.; Russeew, P. J.; Stahel, W. A. Rbust Statstcs: he Apprach based Ifluece Fuct. // Jh Wley, ew Yrk, 986. [9] Veables, W..; Rply, B. D. Mder Appled Statstcs wth S. // Sprger, Berl, 00. DOI: 0.007/ [0] Baret, V.; Lews,. Outlers Statstcal Data. // Jh Wley, ew Yrk 978. [] sypk, Ya. Z. Fudats f Ifrmatal hery f Idetfcat. // auka, Mscw, 984. [] Gdw, C. G.; Ramadge, P. J.; Caes, P. E. A glbally cverget adaptve predctr. // Autmatca. 7, 98, pp DOI: 0.06/ [3] Ljug, L.; Sderstrm,. hery ad Practce f Recursve Idetfcat. // MI Press, Cambrdge, 983. [4] Cady, J. V. Mdel-based Sgal Prcessg. // Jh Wley, ew Yrk, 006. [5] Kvačevć, B.; Đurvć, Ž. Fudametals f Stchastc Sgals, Systems ad Estmat hery. // Sprger- Verlag, Berl, 008. DOI: 0.007/ [6] Kvačevć, I.; Kvačevć, B.; Đurvć, Ž. O strg csstecy f a class f rbust stchastc gradet type systems detfcat algrthms. // WSEAS rasact Crcuts ad Systems, WSEAS Press, Athes. 5, 8006, pp [7] Kvačevć, I.; Kvačevć, B.; Đurvć, Ž. O strg csstecy f a class f recursve stchastc ewt- Raphs type algrthms wth applcats t rbust lear dyamc system Idetfcat. // Facta Uverstes Seres: Electrcs ad Eergetcs, Uversty Press, š., 008, pp. -. [8] eveu, J. Dscrete Parameter Martgals. // rth-hllad, Amsterdam, 975. [9] Rustag, J. S. Optmzg Methds Statstc. // Academc Press, ew Yrk, 97. [0] Hardy, G. H.; Plya, G.; Lttlewd, J. E. Iequaltes.// Cambrdge Uversty Press, Cambrdge, 934. [] Flpvć, V., Kvačevć, B., O rbustfed adaptve mmum-varace ctrller. // It. J. Ctrl. 65, 996, pp DOI: 0.080/ [] Kvačevć, B.; Bajac, Z.; Mlsavljevć, M. Adaptve Dgtal Flters. // Sprger-Verlag, Berl, 03. DOI: 0.007/ Authrs addresses I. K. Kvačevć et al. Ivaa Kstć Kvačevć, Ph. D., Asscate prfessr Faculty f Ifrmatcs ad Cmputg, Sgduum Uversty Dajelva 3, 000 Belgrade, Serba E-mal: kkvacevc@sgduum.ac.rs Jelea Gavrlvć, M.Sc., eachg Assstat Faculty f Ifrmatcs ad Cmputg, Sgduum Uversty Dajelva 3, 000 Belgrade, Serba E-mal: jgavrlvc@sgduum.ac.rs Brak Kvačevć, Ph. D., Full prfessr Schl f Electrcal Egeerg, Uversty Belgrade Bulevar kralja Aleksadra 73, 000 Belgrade, Serba E-mal: dekaat@etf.bg.ac.rs 474 echcal Gazette, 605,