NUMERICAL SCHEMES FOR NONLINEAR PREDICTOR FEEDBACK

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1 UMEIAL SHEMES FO OLIEA PEDIO FEEDBAK Io Kfyll * d Molv Kc ** * Dep. of Evoel E. eccl Uvey of ee 73 Geece el: fyl@eve.c. ** Dep. of Meccl d Aeopce E. Uvey of lfo S Deo L Joll A 993- U.S.A. el: c@cd.ed Ac Ipleeo coo pole w feedc lw w ded dely. ppe foce o pecfc pec of e pleeo pole fo pedco-ed feedc lw: e pole of e ppoo of e pedco pp. I ow e ecl ppoo of e pedco pp y e of ecl cee coco w yd feedc lw e pled eee c e ed fo e lol lo of ll fowd coplee ole ye e lolly ypoclly lle d loclly oelly lle e dely-fee ce. Specl el e povded fo e le e v ce. Eplc fole e povded fo e eo of e pee of e el yd cool cee. Keywod: ole ye dely ye feedc lo ecl eod.. Iodco Feedc lw w ded dely e we pedco-ed eodoloe e ppled o ye w p o eee dely. e lee fo pedco-ed feedc lw evewed [] d e ece wo [893] ve eeded e pedco-ed eodoloe o ole ye d e-vy ye. e ede c col [9] fo deled evew of le cool ye w ded dely. A coo pole w feedc lw w ded dely pleeo. e ecl ppoo of ded dely y dcee dely c led o ly: pole w f peeed [6] e coe of pedco-ed feedc. e pole w codeed po ope pole [3] d ee e y wo wc e devoed o e olo of e pleeo pole: ee [ ]. Mo of e el e foced o e le ce. ppe foce o dffee pec of e pleeo pole fo pedco-ed feedc lw: e pole of e ppoo of e pedco pp. pole vey coo fo ole ye: fo ole ye vey e e olo p

2 ow lyclly. e ece wo [5] w devoed o e ppoo of e pedco pp w cceve ppoo ppoc: e eod le fo lolly Lpc ye. e pole of ppoo of e pedco pp vey po pec of e pleeo pole fo pedco-ed feedc lw dffee fo e l pole of e ppoo of ded dely y dcee dely. e le pole wll o e ded e pee wo. e de of e ecl ppoo of e pedco pp y e of ecl cee fo olv ody dffeel eqo e lly pole eod fo olv e pole of ppoo of e pedco pp. Howeve ce oco e wc e o ecoeed dd ecl ly el. e f oco e eece of p: cool eoy cle ye w p cool ye wee dd el ecl ly e del w dycl ye ye wo p. Ecepo e wo [] ee lo efeece ee. A ecod pole e ccy of lc fole fo e ppoo eo wc cocde w e o-clled lol dceo eo ecl ly: o ce e ee of e ppoo eo e qlve ee [3]. I wo we ow e ecl ppoo of e pedco pp y e of ecl cee fo olv ody dffeel eqo deed oe eodoloy c e ed w cce fo ye wc e o lolly Lpc. Moe pecfclly we foc o e lc Ele cee. We lo dy e pl pole.e. e pole wee e eee o vlle o-le oly vlle dcee e e pl e. e pole olved y e of yd feedc lw d e el ve e. eoe.: ode e dely-fee ye: = f. wee f : cooly dffeele pp w f =. Ae : A Sye. fowd coplee. A ee e cooly dffeele fco ; w = c Glolly Aypoclly Sle d Loclly Epoelly Sle eql po of e cloed-loop ye. w =. e fo evey > > ee e loclly oded pp : {3...} co > d loclly Lpc o-dece fco : w = c fo evey po of w p fo evey d L [ ; U e olo { } = of e cloed-loop ye = f. w

3 d = f = fo.3 [ =. wee : = p = d = f d fo =... d =.5 3 d l codo = d = fo [ fe e follow eqly fo ll : p.6 p e oo of Glol Aypoc Sly d Locl Epoel ly eployed e ee of eoe. e e dd oo ed e lee ee []. e oo of fowd copleee fo. e dd oo ee eece of e olo of. fo ll e ll l codo d ll pole p ee []. oce.5 e pplco of e lc Ele ecl cee o e cool ye. w ep e =. Sce e e of e d po : = p fco of e e d e p cle dffee e ep e ed ec e ew eee ve. eoe. poved y e of coed Lypov d ll- eodoloy d poof cocve. I Seco 3 e cool pcoe wll fd lc fole fo e copo of : = p wc eqe e owlede of ppope Lypov fco fo e cloed-loop ye. w =. oce e fc e fco : volved.6 loclly Lpc w = ee e loe of locl oel ly fo coplced ye c e cloed-loop ye. w.3. d.5 ye w dely d yd fee ce e ee Ω p fo ll p old fo ce ppope co Ω > d fo l codo w p ffcely ll. eefoe o lol ypoc ly d locl oel ly e peeved depe e dely e pled eee d e ecl ppoo. e pole of ppoo of e pedco pp c e po eve le ye. Fo eple le cle ye dffcl o cope e oel well e covolo el volve e oel. Moeove we eee e o vlle cooly e o-dd cool ppoce e ed. Ideed we ow e pole olved y e of yd feedc lw d e el fo le ye ve e.

4 eoe.: ode e le ye: = A B.7 wee > A B. Le e c A B Hw. Fo evey > ee e ee c fo evey ee e co Q > w e follow popey: fo evey po { } = of w p e olo of e cloed-loop ye.7 w = A B = fo.8 [ = I A = I A B d.9 d y l codo = = fo [ fe e ee: p p Q. A oce.9 e op of e epeed pplco of e lc Ele ecl cee o cool ye.7.e. e pplco of e lo = I A B d fo =... w = d ep e =. eoe. poved y e of ll- eodoloy wc dffee fo e poof of eoe. d poof cocve. I old e eped eoe. co e oed y e peclo of eoe. o e le ce. Moeove Seco e cool pcoe c clly fd lc fole fo e copo of. e ce of e ppe follow: Seco povde oe el fo e ecl lc Ele cee fo cool ye wc e ecey fo e poof of e el. e el Seco e o vlle ecl ly eoo e poof e de e e wy w e coepod el fo ye wo p. Seco 3 devoed o e poof of eoe.. Seco devoed o e poof of eoe. d e peeo of ple eple. Seco 5 povde e cocld e of e pee wo. e Apped co e poof of ce ly el.

5 oo. oo e ppe we dop e follow oo: Fo veco we deoe y l Eclde o y poe. e o A of A defed y A = { A : = }. I deoe e. Fo d ε > we deoe y B ε e cloed ll o d ε > ceeed.e. Bε : = { y : y ε }. deoe e e of o-eve el e. Z deoe e e of o-eve ee. Fo evey [] deoe e ee p of.e. e le ee e le o eql o. A po { } π = = of ce eqece of e w = d. We y ce coo fco γ : of cl K f γ =. We y ce coo fco γ : of cl K f γ = d l γ =. By KL we deoe e e of ll coo fco = : w e popee: fo ec e pp of cl K ; fo ec e pp o-ce w l =. By A A ; Ω wee A Ω o-eve ee we deoe e cl of fco vle Ω ve coo devve of ode o A. Le :[ w > d. By we deoe e ope oy of fo o.e. θ : = θ ; θ [ fo [. Le I : = [ e evl. By L I; U we deoe e pce of ele d oded fco defed o I d vle U. oce we do o defy fco L I; U wc dffe o ee eo e. Fo L [ ; we defe : = p θ. oce p θ o e eel pe e cl θ [ θ [ ] pe d wy e qe p θ θ [ ] d p θ θ [ do o cocde eel. e f opeo δ p ec fco :[ U o e fco δ :[ U w δ = fo ll [. A fco f : A wee A pove defe f f = d f > fo ll. A fco f : dlly oded f e e { : f M } oded o epy fo evey M >.. ecl Appoo of e Solo of Fowd oplee Sye We code ye. de e follow po: H f : loclly Lpc veco feld w f = fe: y y f f y L fo ll y. L f fo ll. wee L : [ coo o-dece fco. 5

6 H Sye. fowd coplee. Apo H d H ve po coeqece fo ye.. e we po o wo coeqece wc wll e ed eco: ee e fco W : [ wc dlly oded co c > d fco p c K W f cw p fo ll.3 Fo evey > ee e fco K c e olo of. w y l codo = coepod o y ele d eelly oded p :[ fe fo ll [ ]. wee : = ep [ Moeove fo evey > ee e co M > c M fo ll []. = e eece of fco W : [ d e eece of fco K fy e eqee of po e dec coeqece of eoe oolly.3 [] d po H. Le P : e o-dece coo fco fe: P L { W ξ : ξ L } fo ll.5 Le Q : e o-dece coo fco fe: Defe fo ll : Q : W c c W y p fo ll.6 y c : L Q.7 A = : L Q L.8 B = ode e follow ecl cee wc eeo of e lc Ele eod o ye w p: we elec pove ee d defe fo =. = f d fo =

7 eoe.: ode ye. de po H H. Le > e pove co d le fco W : [ wc dlly oded co c > fco p K K e c po d old. Le P : Q : A : B : e coo fco fy Le y d y ele d eelly oded p :[. If P Q e e follow eqle old: c B A. A Q fo ll =.... wee e olo of. w l codo = coepod o p :[ e =. e.: Ieqly. ow f we ow e l codo = d e ppled p :[ e we c ee ll qe volved.. Moeove f we w e ppoo eo o e le ε > ffce o elec e pove ee o : B P Q A εa c oce e d-de of e ove eqly c e evled efoe we pply e cee.9. e eco poed ode o o e fo od povded y. d ecey fo e cool of e cee of e fco W ecly e e p ep e cool w ppled [7] fo e cool of e decee of e Lypov fco. e od povded y. efl fo e poof of eoe.. e poof of eoe. deped o ee eccl le wc e ed elow d e poved e Apped. Le.3: ode ye. de e po of eoe.. If > d cw P wee P : e fco volved.5 e W c W c p d. Le.: ode ye. de e po of eoe.. If e W P Q c c W c p d fo ll =....3 wee Q : e fco volved.6. 7

8 Le.5: ode ye. de e po of eoe.. Defe e : = {... } wee e olo of. w l codo = coepod o p :[ d ppoe P Q c A A. e e B fo ll {... }. wee e fco A B : e defed y.7.8. We e ow edy o povde e poof of eoe.. Poof of eoe.: All po of Le. d Le.5 old. oeqely eqle.3. old. Ieqly. follow fo e fc A A d defo = coco w. fo =. Moeove eqly.3 ple c W c W p. e pevo c eqly coco w.6 ple.. e poof coplee. eoe. llow o coc pp wc ppoe e olo of. e ed w eed cccy level. Ideed le ; e pove defe fco w l f > eqo:. Defe e pp Φ L [ ; : y e of e Φ : =.5 wee =... e defed y e ecl cee.9 w = d 8 : P Q = L A c fo d > fo =. = By ve of. e pp Φ L [ ;.6 : =.7 : fe.8 Φ Ieqle.. coco w.8 d. ple e follow eqly: Q Φ.9

9 oce e pp defed y.6 d.7 loclly oded. Ideed ee e co M > c = M M fo ll d fo evey ele d eelly oded p :[ w ffcely ll. eefoe coy of ll fco volved.6 coco w e fc l f > ple fo ll. eefoe we coclde: p oolly.6: ode ye. de e po of eoe.. Fo evey pove defe fco ; w l f > d fo evey > code e pp Φ : L [ ; defed y.5 fo ll L [ ; defed y e ecl cee.9 w wee e eqle.8.9 old fo ll L [ ; =... e = d : = defed y.6.7. wee deoe e olo of. w l codo = coepod o p :[ d : = ep. Moeove eqly. old fo ll. [ 3. Poof of eoe. eco devoed o e poof of eoe.. e poof of eoe. cocve d fole wll e ve e fo e loclly oded pp : {3...} volved e yd dyc feedc lw defed y.3. d.5. I ode o plfy e pocede of e poof we e e poof p o wo ep. F Sep: oco of feedc Secod Sep: e of poof e cool pcoe wo o eeed ed e del of e poof y ed oly e f ep of e poof. F Sep: oco of feedc e feedc lw eely ve y.3-.5 ecep fo e fco : {3...} woe coco we ve ee. We e e owlede of fco L : [ fco W : [ d fco K fy e eqee of po of Seco. A eed e pevo eco e eece of fco L : [ fco W : [ d fco K fy e eqee of po H e dec coeqece of eoe oolly.3 [] d e fc f : cooly dffeele pp w = f. Moeove we eed o e e owlede of Lypov fco fo e cloed-loop ye. w =. Moe pecfclly we e e eece of pove defe 9

10 dlly oded fco ; fo. co ε K > d fco ρ K c e follow old: f ρ 3. K B ε 3. K B ε 3.3 f B ε 3. e eece of Lypov fco fo e cloed-loop ye. w = fy dec coeqece of Popoo. [7]. Bed o e owlede of ll e fco d co deced ove we e poceed o e coco of ew fco. e f fco o defe e e coo o-dece fco P : Q : A : B : fy e we defe: fco K =... d co fy: 3 > fo ll 3.5 d 3 fo ll 3.6 : = : = = 3 : 3 : = fo ll [ ε ] 3.7 coo o-dece fco M : [ fe: f f M fo ll 3.8 e ede old oce e eece of fco M : [ K =... d co 3 > fy d 3.8 dec coeqece of e fc ; pove defe d dlly oded ee Le 3.5 [] of Le. [6] c of eqle d d of e fc f : d : e cooly dffeele pp w =. Moeove defe fo ll : = M L q : = 3.9 D : 3 wee K e fco volved. w eplced y > d L : [ e fco volved po H. e elec co δ > c : δ ε 3.

11 Hv eleced δ > we e poo o elec co γ > o : Defe: δ δ ρ γ 3. δ δ : = 3. 3M L δ δ L : = L 3.3 d oeove elec co > o : Flly defe: : = γ D q Q oce y ve of 3.7 d e fc Q fo ll follow fo defo : {3...} defed y.6.7 loclly oded d e pp Φ : L defed y.5 fe eqle.8.9 fo ll 3.5 l f = >. eefoe oolly.6 ee e pp [ ; L [ ; wee coepod o p deoe e olo of. w l codo = :[ d : = e p. [ Secod Sep: e of poof Hv copleed e de of e feedc lw y coc e fco : {3...}.6 we e ow edy o pove oe c el coce e cloed-loop ye. w.3..5 d.6. e follow cl ow pccl lo ceved. I poof povded e Apped. l : ee e KL c fo evey po { } = of w p fo evey codo d L [ ; = = e olo of..3. d.5 w l fe e follow eqly fo ll : ρ 3.6 γ wee ρ K e fco volved 3. d γ > e co volved 3. d 3.5.

12 e follow cl ow locl oel lo ceved. I poof povded e Apped. l : ee e co Q Q > c fo ec po { } = of w p fo ec d L [ ; e olo of..3. d.5 w l codo = = fe e follow eqle: Q p w w Q p w w fo ll 3.7 fo ll 3.8 wee e lle pl e fo wc old co volved 3. d 3.. δ wee δ > e e follow cl ee oded. I poof povded e Apped. l 3: ee e o-dece fco G : c fo ec po { } = of w p fo ec d L [ ; e olo of..3. d.5 w l codo = = fe e follow eqly fo ll : G 3.9 We e ow edy o pove eoe.. Le y po { } = of w p L [ ; d code e olo of..3. d.5 w y l codo = =. Ieqle 3.5 d. ply e lle pl e fo wc δ old = fo e ce δ. Moeove e fc ee e co M > c = M fo ll [] coco w eqle llow o coclde ee e co Ω > Ω fo ll 3. povded δ. M Popoo 7 [5] coco w d e fc p llow o ee e eece of o-dece fco : c e lle pl e fo wc δ fo ll old fe

13 [ ; L. o w e pevo eqly llow o coclde e eece of o-dece fco G : c e follow eqly old fo ll : G 3. oeqely 3. d 3. we coclde.6 old w > d : = wee : = w dw fo ll G l : = Ω l Ω Ω G fo ll [ l] : = fo ll > l l : = M δ e poof of eoe. coplee.. Le Sye We f eco y povd e fole d oe lo fo eoe.. I eco we pove e ee = e ee of eoe. c e eleced e lle ee fe e eqly A Aγ A B A A γ A. wee γ > co fo wc e ee M λ γ p λ. old fo e olo of = A B w y l codo = coepod o y ele d loclly eelly oded p :[ fo ce ppope co M λ >. e co γ > c e eleced o e y co w B PB γ > wee P yec pove defe d > co fe P A B A B P P d P I. 3

14 I cle e e I A.9 ppoo of A. Moeove e e I A B = = d.9 ppoo of A B dw = A B dw. eefoe ppoo of = A A B dw. e fol.9 l o e fol.7 o pe 77 of e oo [9]. Howeve ee dffeece ewee fol.9 d fol.7 o pe 77 of [9]: fol.9 e oel A d A e ppoed y I A d I A epecvely. I oe wod fol.9 doe o eqe e copo of e oel. fee c e po fo le-cle ye. Aoe dve of o ly e copo of lc od fo e ee =. I [9] ow e eo covee o eo we eoe.6 o pe 87 o lc od povded. Flly e oed e coolle.8.9 eqe eee dcee e e pl e d doe o eqe coo eee of e e veco. Moeove ee oe o peo of e pl cedle: ee. old fo ll po { } = of w p. Fo po o we pov eoe.. We f pove e follow eccl el. Le.: Fo evey ee = A B fe d L ; [ e olo of.3 wee = B. = I A B d =... = = p = = A d Poof: If we defe e = fo =... e follow e = I A e A d fo =.... eefoe we e e I A e A d fo =....

15 Fo ll [ ] old A w dw A B. e pevo eqly coco w e Gowll-Bell Le ple A B A fo ll [ ]. e pevo eqly coco w. d e fc I A A ve: e A e A A B A fo =....5 U.5 d e fc e = d =... we o: e A B A A Fo ll [ ] old A w dw B. e pevo eqly coco w e Gowll-Bell Le ple B A [ ]. oeqely e pevo ee coco w.6 ve:.6 fo ll e.7 wee = A d B =. Ieqly.7 coco w e fc d = ve e deed eqly.3. We e ow edy o ve e poof of eoe.. Poof of eoe.: e poof eled y e of ll- e. Le λ] e ffcely ll o γ.8 = wee γ λ > e e co volved. = A d B = eed y.. Le { } = =. e eece of ffcely ll λ] e y po of w p d code e olo of e cloed-loop ye.7 w.8.9 d y l codo = L [ ;. oce eqly. ple fo ll we ve: M γ p w w w w Sce = fo ll we e fo.9: M γ p w w w w =.9. 5

16 Fo ll [ old. Moeove eqly.3 ple p wee = d =. e ove eqle coco w e fc ply e follow eqly old fo ll [ p : oce ce = fo ll we e p p d p p ve fo ll Z w. fo ll Z w. oeqely ce we p p p fo ll Z w d [ p p p fo ll Z w d [ o e wo ove ce d. follow e follow ee old fo ll : [ p. e vl eqle p p p d p wc old fo ll [ coco w. ply e follow ee old fo ll : p p p.3 Fo ll [ ] old A w dw B. e pevo eqly coco w e Gowll-Bell Le d defo = A d = B ple fo ll [ ]. oeqely follow e follow ee old fo ll :. e pevo eqly coed w. d.3 ple 6

17 7 p p p. p p γ w w w M M w.5 o..5 d.8 we o fo ll : γ γ p M M.6 γ γ γ γ p M M M M.7 Sce follow fo.7 : γ γ γ γ p M M M M.8 e eece of co > Q fo wc. old dec coeqece of d e follow eqle: p p p p p p wc old fo ll. e poof coplee. e we pee ple eple wc ow e efle of e fole wee povded eco.

18 Eple.: ode e cl ye = w.9 e eple codeed o pe 88 of e oo [9]. Hee p >. e co γ > c e eleced o e y co w = = d = p wee γ > y e p P = []. eefoe eqly. ow e ee = c e eleced o > f p = p e e p e p Fe ow e evo of f p w epec o p > fo =. e ed p =.93 w f p 6. 7 ow le 65 d po old e ed fo e Ele ppoo of e olo of e ope-loop ye ove e wdow of le w cccy ee e fo of.3. I follow fo evey ee e co Q > w e follow popey: Fo evey po w { } = d y l codo of w p e olo of e cloed-loop ye.9 = p = p = = fo. [ d. = = L [ ; fe ee fp p Fe : e evo of f p w epec o p > fo = 8

19 5. ocld e wo foced o ey pec of e pleeo pole fo pedco-ed feedc lw: e pole of e ppoo of e pedco pp. I w ow e ecl ppoo of e pedco pp y e of e lc Ele ecl cee coco w yd feedc lw e pled eee c e ed fo e lol lo of ll fowd coplee ole ye e lolly ypoclly lle d loclly oelly lle e dely-fee ce. Specl el wee povded fo e le e v ce. Bo fo e le d ole ce lc fole e povded fo e ee fo e de pee of e el yd cool cee. e pee ppe oe eyod e ppoo el [5] y eov e lol Lpc eco. Moe e o e doe fo e ppoo of e el volved e lc Ele cee y ely pleele fole. el fo le ye e ledy ve [ ]. Feoe oe co oe e poly of dffee ecl cee ecep e lc Ele cee; ee []: e e of plc ecl cee y eqe fewe d po e d po eeded fo e lc Ele cee. Flly ee e clle pole of ecl ppoo fo ce wee e eed op o ecely e e veco d ee eee dely ee [89]. efeece [] Ael D. d E.D. So Fowd opleee Uoded Oevly d e Lypov ceo Sye d ool Lee [] Güe L. d P. E. Kloede He Ode ecl Scee fo Affely oolled ole Sye ece Me [3] He E. S. P. oe d G. We Solv Ody Dffeel Eqo I off Pole d Ed. Spe-el Bel-Hedele 993. [] He E. d G. We Solv Ody Dffeel Eqo II Sff d Dffeel- Alec Pole d Edo Spe-el Bel-Hedele. [5] Kfyll I. Slo By Me of Appoe Pedco fo Sye w Delyed Ip SIAM Jol o ool d Opo [6] Kfyll I. d Z.-P. J Sly d Slo of ole Sye Spe- el Lodo See: oco d ool Eee. [7] Kfyll I. d L. Güe Feedc Slo Meod fo e ecl Solo of Sye of Ody Dffeel Eqo Dcee d oo Dycl Sye: See B [8] Kfyll I. d M. Kc ole Slo de Spled d Delyed Meee d w Ip Sec o Dely d Zeo-Ode Hold IEEE co o Aoc ool [9] Kfyll I. d M. Kc Pedco-Bed Op Feedc fo ole Dely Sye ed o Aoc. [] Kll H. K. ole Sye d Edo Pece-Hll 996. [] Kc M. Lypov ool fo pedco feedc fo dely ye: Ivee oply d oe o dely c Aoc

20 [] Kc M. Dely opeo fo ole Adpve d PDE Sye Bäe Boo 9. [3] Kc M. Ip Dely opeo fo Fowd oplee d Sc-Feedfowd ole Sye IEEE co o Aoc ool [] Kc M. Lypov Sly of Le Pedco Feedc fo e-y Ip Dely IEEE co o Aoc ool [5] Mcel W. S. Mode d D. ooe ecey d ffce codo fo fe pleeo of ded dely cool Poceed of e S-SF Woop: Advce e-dely Sye P Fce [6] M L. Ae ded-dely cool lw clly ppole? Poceed of e IFA Woop o e-dely Sye DS 3 ocqeco Fce 3. [7] M L. "O e ppoo of ded-dely cool lw" Sye & ool Lee [8] Mode S.. Loo d J. olldo ee poce-odel cool fo le ye w dely Poceed of e IEEE ofeece o Deco & ool Oldo Flod USA [9] Mode S. M. De O. So Appoo of cool lw w ded dely: ecey codo fo ly Poceed of e IFA Sypo o Sye Sce d ool Pe ec eplc. [] Mode S. d W. Mcel Fe pec e of le e-dely ye w fe pleeo IEEE co o Aoc ool [] clec S.I. Dely Effec o Sly A o ool Appoc Lece oe ool d Ifoo Scece Hedele Gey Spe-el. [] Po J.. d P. M. Ml ol ppoo of ded-dely coolle Ieol Jol of ool [3] cd J.-P. e-dely ye: A ovevew of oe ece dvce d ope pole Aoc [] So O. d S. Mode ool lw volv ded e dely: oe of e pleeo Poceed of e Aec ool ofeece [5] So E.D. oe o Iel of ISS Sye d ool Lee [6] Ace. M. De J.F. Lfy d J.P. cd Soe pole e pleeo of ded-dely cool lw Poceed of e 38 IEEE ofeece o Deco & ool Poe Ao USA [7] Ace. M. De J.F. Lfy d J.P. cd Ipleeo of ded cool lw fo cl of ye w dely Poceed of e 3d IFA Woop o e-dely Sye USA [8] Zo Q.-. d L. M ool of el pocee w ded e P : Qve ly IEE Poceed of ool eoy & Applco [9] Zo Q.-. o ool of e-dely Sye Spe-el Lodo.

21 Apped Poof of Le.3: Defe e fco: fo [] λ W λ A. = λ. e follow eqle old fo ll λ [] : d d λ = W λ d λ = dλ λ W λ A. Moeove oce y ve of. d.9 old L pevo eqly coco w.5 d A. ve:. e d λ P dλ A.3 wee P : e fco volved.5. Feoe eqly.3 coco w.9 d A. ve: d d = W λ o A. A.3 d A. we e: f d cw p d W = c W p d P A. A.5 Ieqly A.5 coco w e follow eqly c W p d P c W c p d wc old fo ll cw P ply. old. e poof coplee. Poof of Le.: We wll f pove f ee e {... } c d > =... dco. P Q c e.3 old fo ll =.... e poof y F oce.3 old fo =. Sppoe old fo oe {... }. lely c eqly.3 ple c W p. e pevo eqly W c Q coco w.6 ple. oeqely e fc P : o-

22 dece d W ply cw P Le.3 ow : W P Q c cw P c W c p. Sce > d e ove eqly coco w.3 ow.3 old fo eplced y. e ce ee e {... } w c e eed e follow wy. = =... Le {... } e e lle ee w. ple = d = = =... d. Sce f =.9 ple = fo ll =.... oeqely.3 old fo ll =.... e poof coplee. Poof of Le.5: oce y ve of.9 e follow eqo old fo ll {... } : e = e f f d A.6 Ieqly. ple e follow eqly fo ll {... } d [ ] f f L : A.7 U e defo e : = d eqle.. we e fo ll {... } d [ ] : e e e L L A.8 oce ll ypoee of Le. old. eefoe eqly.3 old fo ll =.... c lely eqly.3 ple c W p. e pevo eqly W c Q... coco w.6 ple fo ll =. Eplo e fc Q fo ll =... d A.6 A.7 A.8 we o fo ll {... } : e e L Q L L Q e A.9 Defo.7.8 coco w eqly A.9 ow e follow ecve elo old fo ll {... }

23 e A e B A. U e fc e = coco w elo A. ve e deed eqly.. e poof coplee. Poof of l : F we ow fo ec po { } = of w p fo ec d L [ ; = = e olo of..3. d.5 w l codo qe d e fo ll. e olo of..3. d.5 deeed y e follow poce: Il Sep: Gve = = we deee e olo of. fo [ ]. oce e olo qe. Ieqly. ple e follow ee: fo ll [ ] A. - Sep: Gve fo [ ] d fo [ we deee fo [ ] d fo [. e olo of.3 fo [ w l codo = qe y ve of e fc f d e loclly Lpc pp. Ieqly 3. ple: fo ll A. [ We deee fo [ y e of e eqo =. oce eqle coco w A. ply e follow eqly fo ll : [ = A.3 Flly we deee e olo of. fo [ ]. oce e olo qe. e fc coco w eqly. w eplced y > d eqly A.3 ple e ee: fo ll [ ] A. e we evle e dffeece fo. Eplo. we e: [ = L f f d U e eqly 3.5 eqle A. A.3 A. coco w e ove eqly we o: d 3

24 d L Defe = : ϕ. U e Gowll-Bell le e ove eqly d e fc we e fo ll [ : q L ϕ A.5 e we evle e qy f fo [. U eqly 3. we e: ρ f f f e follow ee follow fo d e ove eqly: 3 M f ρ U e ove eqly coco w eqly 3.5 eqle A. A. d defo : q = = : ϕ we e: 3 q M q f ϕ ϕ ρ A.6 o eqle A.5 A.6 d defo 3.9 we o fo ll [ : q D f ρ A.7 : 3 L M D = = : Q q D γ Sce = ecll. follow fo.8.9 d.. e follow eqle old fo ll... = : A.8 Q A.9 Sce ecll. we o fo A.7 A.8 A.9 d defo 3.5 fo ll [ : γ ρ d d A. U A. d Le. pe 8 [6] we o fo ll : γ ρ A.

25 fo ce fco KL. o 3.5 A. d A. we o eqly 3.6 w : = fo ll > d : = fo ll [ ]. e poof coplee. Poof of l : Le y po } = of w p [ { L ; d code e olo of..3. d.5 w y l codo = =. Ieqle 3. d 3.6 ee ee e qe c δ. lle pl e Moeove eqle A. 3. d 3.5 llow o coclde δ d δ U A.8 defo d A. we o fo ll fo ll A. : γ δ δ A.3 U A. 3.5 d A.3 we e fo ll : A. δ e we evle e dffeece fo. Eplo. d eqle A. A. d defo 3.3 we e fo ll d : [ = L d f f U e Gowll-Bell le e ove eqly d e fc d : [ L d ply fo ll A.5 e we evle e qy f fo [. U eqle A A. d A.5 d defo 3. we e fo ll d : [ f A.6 U A. d A.6 we e fo ll d : [ A.7 5

26 6 wee =. U A d A.7 we e fo ll d [ : A.8 Le e pove co ffcely ll o d A.9 e eece of fy A.9 eed y 3.. U A.8 d e fc p we o fo ll d [ : p p A.3 e dffeel eqly A.3 llow o coclde e follow dffeel eqly old fo.e.: p p A.3 Ie A.3 d ce we o fo ll : p p A.3 U A. d e fc we o fo A.3 fo ll : p p A.33

27 7 U A. A.8 d A. we o fo ll d [ : = A.3 Ieqly A.3 coco w e fc p ple: p eefoe we e fo e ove eqly fo ll : p p A.35 D e ce p p = d p p = we o fo A.35 fo ll : p p A.36 o A.33 d A.36 we e fo ll : p p

28 8 D e ce p p = p p = d e ove eqly we o fo ll : p p A A A A.37 wee = A. Ieqle A.36 A.37 ply ee e co > Q Q c old. e poof coplee. Poof of l 3: Le y po = } { of w p L ; [ d code e olo of..3. d.5 w y l codo = =. Defe: : = fo ll A.38 d oce K. Moeove oce defo A.38 d 3.5 ply fo ll A.39 Feoe defo A.38 d eqly A.3 ply e follow eqly fo ll Z d [ : A. Ieqle ply e eece of o-dece fco : c : fo ll A. By ve of A.8 A.39 d A. we e fo ll Z : p p p p

29 9 e ove eqly coco w A. ve fo ll Z : p p p p Feoe A. d e ove eqly we o fo ll Z : p p A. wee : = fo ll o-dece fco. Defe e eqece: p : F = A.3 oce defo A.3 d e fc = p p p coco w A. ply e follow eqly fo ll Z : F F A. Ieqly A. coco w e fc : F = llow o pove y dco e follow eqly old fo ll Z : F A.5 Ieqly A. coco w eqly A.5 d defo A.3 ply ee 3.9 old w G : = fo ll. e poof coplee.

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