Nonlinear Stabilization under Sampled and Delayed Measurements, and with Inputs Subject to Delay and Zero-Order Hold

Transcription

1 Nolea Sablao de Sapled ad Delayed Meaee, ad wh Ip Sbjec o Delay ad Zeo-Ode Hold Iao Kaafyll ad Molav Kc Abac Saplg ae laeoly wh p ad op delay ewoked cool ye. Whe he delay lef copeaed, he aplg peod geeally eqed o be ffcely all, he delay ffcely ho, ad, fo olea ye, oly eglobal paccal ably geeally acheved. Fo eaple, global ablao of c-feedfowad ye de apled eaee, apled-daa ablao of he oholooc cycle wh abaly pae aplg, ad apled-daa ablao of LI ye ove ewok wh log delay, ae ope poble. I h pape we pee wo geeal el ha adde hee eaple poble a pecal cae. F, we pee global aypoc able fo fowad coplee ye de abaly log p ad op delay, wh abaly log aplg peod, ad wh coo applcao of he cool p. Secod, we code ye wh apled eaee ad wh cool appled hogh a eo-ode hold, de he apo ha he ye ablable de apled-daa feedback fo oe aplg peod, ad he coc apled-daa feedback law ha acheve global aypoc ablao de abaly log p ad eaee delay. All he el eploy oal feedback law deged fo he coo-e ye he abece of delay, cobed wh pedcobaed copeao of delay ad he effec of aplg. Keywod: feedback ablao, e-delay ye, apled-daa ye, olea cool.. Iodco Movao. Saplg ae laeoly wh p ad op delay ay cool poble, o oably cool ove ewok. I he abece of delay, apled-daa cool of olea ye eglobal paccal ably geeally gaaeed [5,7,8,9], wh he deed ego of aaco acheved by ffcely fa aplg. Aleavely, global el ae acheved de ecve codo o he ce of he ye [4,7,,,4,]. Dep. of Evoeal Eg., echcal Uvey of Cee, 7, Chaa, Geece, eal: kaafyl@eveg.c.g. Dep. of Mechacal ad Aeopace Eg., Uvey of Calfoa, Sa Dego, La Jolla, CA 99-4, U.S.A., eal: kc@cd.ed.

2 O he ohe had, pely coo-e olea cool, p delay of abay legh ca be copeaed [5,9,] b o apled-daa eeo of ch el ae avalable. Slaeo codeao o aplg ad delay ehe phycal o aplg-dced gve he leae o cool of lea ad olea ye ove ewok [,,6,6,,,4,5,7], b all avalable el ely o delay-depede codo fo he eece of ablg feedback. Depe he eakable accoplhe he feld of apled-daa, ewoked, ad olea delay ye, he followg eaple poble ea ope: global ablao of c-feedfowad ye de apled eaee ad coo cool, apled-daa ablao of he oholooc cycle wh p appled va eo-ode hold ad de abaly pae aplg, ad apled-daa ablao of LI ye ove ewok wh log delay. I h pape we odce wo faewok fo olvg ch poble:. We pee global aypoc able fo fowad coplee ye de abaly log p ad op delay, wh abaly log aplg peod, ad wh coo applcao of he cool p.. We code ye wh apled eaee ad wh cool appled hogh a eoode hold, de he apo ha he ye ablable de apled-daa feedback fo oe aplg peod, ad he coc apled-daa feedback law ha acheve global aypoc ablao de abaly log p ad eaee delay. I boh faewok we eploy oal feedback law deged he abece of delay, cobed wh pedco-baed copeao of delay. Poble Saee. A [5,9,,,,,4,6,8], we code ye wh p delay, whee,..., R, R, f R R R f, ad & f,. : a locally Lpch appg wh a coa. I [5,9,,,8], he feedback deg poble fo ye. addeed by ag a feedback able k fo ye. wh o delay,.e.. wh, o & f,.

3 ad applyg a delay copeao pedco ehodology baed o he kowledge of he delay. I h pape, we copoae alo a codeao of eaee delay, aely, we adde he poble of ablao of. wh op y R. whee a coa,.e., we code delayed eaee. he ovao fo a laeo codeao of p ad eaee delay ha ay checal poce cool poble he eaee delay of coceao of checal pece ca be lage. We alo ae ha he op avalable a dcee e a he aplg e wh, whee > he aplg peod. Vey few pape have ded h > poble a ecepo [8] whee p ad eaee delay ae codeed fo lea ye b he eaee o apled. he poble of ablao of. wh op gve by. aely elaed o he ablao of ye. aloe. o ee h, oce ha he op y of.,. afe he followg ye of dffeeal eqao fo all : y& f y, Code he copao bewee wo poble decbed by he ae dffeeal eqao: he poble of ablao of. wh p delay > ad o eaee delay.e., & fo all ad he poble of ablao of.,. wh o p delay f, ad eaee delay >.e., y& f y, fo all. he wo poble ae o decal: he f ablao poble he appled p vale fo [, ] ae gve a al codo, whle he ecod ablao poble he appled p vale fo [, ] be coped baed o a abay al codo, [,] epecve of he ce vale of he ae. heefoe, eo echcal e coceg he eece of he olo fo [, ] ae fo he ecod ablao poble ee Reak.b below. Rel of he pape. We eablh wo geeal el:. A olo fo he ablao of. wh op gve by. de he apo ha ye. globally ablable ad fowad coplee ad he p ca be cooly adjed heoe.. he popoed dyac apled-daa coolle e vale of he op. a he dcee e a Z, whee > he aplg peod,

4 ad he al e. h jfe he e apled-daa. No eco fo he vale of he delay, o he aplg peod > ae poed. I geeal, we how ha hee o eed fo coo eaee fo global aypoc ablao of ay ablable fowad coplee ye wh abay p ad op delay.. A olo fo he ablao of. wh op gve by. de he apo ha ye. globally ablable ad fowad coplee ad he cool aco pleeed wh eo ode hold heoe.. Aga, he popoed apled-daa coolle e vale of he op. a he dcee e a Z, whee > he, aplg peod ad he al e. I h cae, we ca olve he ablao poble fo ye wh boh delayed p ad eaee povded ha he e chooe he aplg peod a he ao of he p delay ad ay ege. O delay copeao ehodology gaaee ha ay coolle coo o apleddaa deged fo he delay-fee cae ca be ed fo he eglao of he delayed ye wh p/eaee delay ad apled eaee. Fo eaple, all apled-daa feedback deg popoed [4,5,,4,7,8,9,] whch gaaee global ablao ca be eploed fo he ablao of a delayed ye wh p/eaee delay, apled eaee ad p appled wh eo ode hold. he el ae appled o he Lea e Ivaa LI cae, whee f, A B, A R, B R. h cae ha bee ecely ded eevely he coe of lea Newoked Cool Sye, whee vao delay ae. Delay-depede ad/o aplg peod-depede ffce codo fo he ablao of Newoked Cool Sye have bee popoed he leae [,,6,6,,,4,5,7]. Hee, we popoe a lea delay copeao ha gaaee epoeal ably of he cloed-loop ye de he ld eco ha he e chooe he aplg peod a he ao of he p delay ad ay ege, wh o addoal eco fo he delay Coollay.4. he copeao deged baed o he kowledge of lea feedback able fo he delay-fee cae. c-feedfowad ye [8,,], whch ae ded Eaple.4 ad.8. he ablao of he oholooc egao &, & &.4, wh boh delayed p ad eaee. he poble wa ecely ded [7] he peece of delay ad [4,9] he peece of aplg. Hee, o popoed 4

5 dyac apled-daa coolle appled wh o eco fo he vale of he delay o he e of he aplg peod. he ablao poble olved fo he cae whee he p ca be cooly adjed Coollay 4., a well a fo he cae whee he p ae appled wh eo ode hold Popoo 4.. Ogaao of he pape. I Seco he a el coceg he cae of he cooly adjed p ae aed ad ay coe ad eplaao ae povded. I Seco he a el coceg he cae of p appled wh eo ode hold ae povded. Specal el ae povded fo he cae of lea aooo ye ad fo he cae of olea ye whch ae dffeoophcally eqvale o a cha of egao. Seco 4 devoed o he applcao of he obaed el o he ablao of a hee-wheeled vehcle wh wo depede ea ooed wheel he oholooc egao. Fally, Seco 5 we pee he cocldg eak of he pee wok. he Apped coa he poof of cea el. Noao. hogho he pape we adop he followg oao: Fo a veco a o. I A R we deoe by al Ecldea o, by apoe. Fo a eal R, A R deoe apoe ad A : p{ A ; R, } R deoe he dey a. dced R deoe he e of o-egave eal be. Z deoe he e of o-egave ege. Fo evey, []. deoe he ege pa of,.e., he lage ege beg le o eqal o Fo he defo of he cla of fco KL, ee [6]. By C j A A ; Ω C j, whee j a o-egave ege, we deoe he cla of fco akg vale Ω ha have coo devave of ode j o A. Le :[ a, b R wh > a b ad. By we deoe he hoy of fo o,.e., : ; [,], fo [ a, b. By we deoe he ope hoy of fo o,.e., : ; [,, fo [ a, b. Le R I : [, be a eval. By L I; U L I; U we deoe he pace of eaable ad locally boded fco defed o I ad akg vale U R. Noce ha we do o defy fco L I; U whch dffe o a eae eo e. Fo L [,]; R o loc L [,; R we defe : p o : p. Noce ha p [,] [, [,] o 5

6 he eeal pe b he acal pe ad ha why he qae p [,] ad p [, do o cocde geeal. We wll alo e he oao M U fo he pace of eaable ad locally boded fco : R U. We ay ha a ye of he fo. fowad coplee f fo evey R, M U he olo of. wh al codo R coepodg o p M U e fo all. hogho he pape we adop he coveo L [,]; R R ad C [,]; R R fo. Fally, fo eade coveece, we eo he followg fac, whch a dec coeqece of Lea. [] ad Lea. []. he fac ed eevely hogho he pape. FAC: Sppoe ha ye. fowad coplee. he fo evey R, L [, ; R he olo loc of. wh al codo R coepodg o p L [, ; R e fo all. Moeove, fo evey > hee e a fco loc a K ch ha fo evey R, L [, ; R he olo of. wh al codo R loc coepodg o p L [, ; R afe a p, fo all, ]. < loc. Dyac Sapled-Daa Feedback fo Cooly Adjed Ip We a by peeg he apo fo ye.. O f apo coceg ye. fowad copleee. Hypohe H: Sye. fowad coplee. Apo H gaaee ha ye. fowad coplee a well: fo evey R, L [, ; R he olo loc way: 6 of. wh al codo R coepodg o p L [, ; R e fo all. heefoe, we ae a poo o defe he loc pedco appg Φ : R [,; R R L fo all, wh > he followg

7 fo evey R, [,; R L he olo of. wh al codo coepodg o p [,; R L afe Φ, By ve of he Fac, we ca gaaee he eece of Φ a K ch ha, a, fo all, R [,; R L. Ug. ad he fac ha f : R R R a locally Lpch appg, we ca gaaee he eece of a o-deceag fco L : R R ch ha Φ, Φ y, v L y v y v, fo all, R L [,; R, y, v R [,; R We ae e ha. globally ablable. L. Hypohe H cooly adjed p: hee e k C R R ; R k, g, K g wh, fo all, R R. ch ha R Ufoly Globally Aypocally Sable fo ye. wh k,,.e., hee e a fco σ KL ch ha fo evey, R R he olo of. wh k, ad al codo R afe he followg eqaly: σ,.4, Code ye. de hypohee H, H fo ye.. O popoed dyac apled-daa feedback ha ae, R [,]; R,, R [,]; R each bye: L ad p y R ad fo L he ae ae coped by he ecoeco of wo A apled-daa bye ee [] wh p y, R [, R L ]; : & f,, [, Φ R y,, Z.5 whee Z, ae he aplg e ad > he aplg peod. We e ha he popoed apleddaa dyac coolle e oly vale of he op a, whee Z. y R a he dcee e 7

8 A bye decbed by Fcoal Dffeece Eqao ee [] wh p R : k,, > [, R L ];.6 O f a el ow aed. heoe.: Le >,, wh > ad ppoe ha hypohee H, H hold fo ye.. he he cloed-loop ye.,..5,.6 Ufoly Globally Aypocally Sable, he ee ha hee e a fco ~ σ KL ch ha fo evey,,, C [,]; R R [,]; R L, he olo,, R R R of he cloed-loop ye.5,.6,.,. wh al codo R, L [, R, C [,]; R ]; : afe he followg eqaly fo all ~ σ.7, Soe eak fo he dyac apled-daa feedback gve by.5,.6 ae ode befoe we poceed o he poof of heoe.. Reak.: a he dyac apled-daa coolle.5,.6 e-vayg f k e-vayg. If k peodc he he dyac apled-daa coolle.5,.6 peodc oo. b Sce he op y gve by. of ye. afe he ye of dffeeal eqao y& f y,, fo all, whee he al e, we ca pcple apply he pedco-baed delay copeao appoach decbed [] eeded fo e-vayg feedback law, whch gve he ac feedback law k, Φ y,. h he pao fo he coco of he apled-daa dyac feedback.5,.6: fo all, whee Z, he vale of coped by.5,.6 eacly k, Φ y,. Howeve, hee ae echcal poble wh he applcao of he ac feedback law k, Φ y, : gve al codo C [,]; R, [,]; R eaable ad eeally boded k, Φ, L, we cao gaaee eece of a :[, ] R afyg he egal eqao fo all, ]. Fo he cae, oe ffce codo fo 8

9 he eece of a olo of he egal eqao ha he al codo C [,]; R, [,]; R L afe he eqao & f, fo all, ] : h cae he olo k, fo [, ], whee he olo of he al vale poble & f, k, wh. Ohe ecve ffce codo fo he eece of a olo of he egal eqao ca be obaed by g fed po heoy. he poof of heoe. how ha h e ca be copleely avoded fo he dyac apled-daa feedback.5,.6. c Fo evey al codo he vale of coped by.5,.6 eacly k, Φ y, fo all wh Z afyg, o o dyac apleddaa feedback baed o he pedco pcple. d Fo he pleeao of he coolle.5,.6, we kow he pedco appg Φ : R L [,; R R. h appg ca be eplcly coped fo Lea ye & A B, wh R, R. I h cae Coollay.4 below he pedco appg Φ : R [,; R R A ep Φ, : ep Aw B w dw. L gve by he eplc eqao Blea ye & A B C, wh R, R ad AC CA. I h cae he pedco appg Φ : R [,; R R Φ, : ep Nolea ye of he followg fo: L gve by he eplc eqao A epc d ep Aw & & M & a f a, a,,...,,..., R epc f,, R w f,,..., db w dw. whee all appg a, f,..., ae locally Lpch. I h cae he pedco appg [,; R R Φ : R L ca be coced dcvely. Fo eaple, fo he a d f w dw. w pedco appg gve by Φ, ep a d ep Eaple.4 below apple heoe. o a hee-deoal olea ye of he above cla. Moeove, he oholooc egao.4 belog o he above cla ad heoe. ca be appled ee Coollay 4.. 9

10 v Nolea ye & f,, fo whch hee e a global dffeooph Θ : R R ch ha he chage of coodae Θ afo he ye o oe of he above cae Coollay.7 below. Fo globally Lpch ye, oe ca le appoae pedco appg [,; R R Φ : R L a how [5] de addoal ad oe ecve hypohee. Poof of heoe.: We a wh he followg cla, whch we pove he Apped. Cla : hee e a fco G K ch ha fo evey ad,, C [,]; R R [,]; R L he olo,, R R R of he cloed-loop ye.5,.6,.,. wh al codo R, L [, R, C [,]; R G ]; e fo all, ] ad afe [, fo all, ].8 [ By ve of dco ad Cla, he followg cla hold. Cla : hee e a fco,, C [,]; R R G K ch ha fo evey, p Z, p ad [,]; R L he olo,, R R R of he cloed-loop ye.5,.6,.,. wh al codo R L [, R, C [,]; R p G ];, e fo all [, p ] ad afe, fo all [, p ].9 p whee G : G 4 K 4 o G o. p e We oce ha fo all wh Z afyg he olo,, R R R of he cloed-loop ye.5,.6,.,. afe:. Coeqely, fo all wh Z afyg hold ha: k,. Hypohe H cojco wh eqaly.4 ad eqao. ple ha he followg eqaly hold: σ,,.

11 Defe p. Ug.9,. ad., follow ha he followg eqaly hold: p G p, Defe ~ p p, : σ G, p g σ G, p ~ p p σ, : G g G fo all [, p σ, p. σ fo all p ad coclde ha.7 hold. he poof coplee. <. Ug.,.,. ad. we ca Reak.: he poof of heoe. how ha f k C R R ; R wh k, fo all, a o-fo e able fo ye ee [9] he we ca pove ha he cloedloop ye.5,.6,.,. o-foly e Globally Aypocally Sable,.e., hee e fco ~ σ KL ad a pove coo fco β : R, ch ha fo evey,,, C [,]; R R [,]; R L, he olo,, R R R of he cloed-loop ye.5,.6,.,. wh al codo L [, R, C [,]; R ]; : R, afe he followg eqaly fo all β ~ σ.4, I h cae, hee o eed o ae ha eqaly. hold. We e pee a eaple whch how how he obaed el ca be appled o feedfowad olea ye. Eaple.4 Cool of c-feedfowad ye wh abaly pae aplg: Code he followg eaple ake fo []: &,,, R whee &, R, &.5 Hee, we code he ablao poble fo.5 wh op gve by. avalable oly a he dcee e a he aplg e wh >, whee > he aplg peod. Hypohe H hold fo ye.5 ad he pedco appg ca be eplcly epeed by he eqao: Φ, : φ,, d q dq d, d.6

12 , φ w d dq q dq dw d q w dq d q.7 Moeove, hypohe H hold a well wh he ooh, e-depede feedback law: : k.8 I follow fo heoe. ha he dyac apled-daa coolle k wh,,,, R & & &, fo, [.9 ad Φ Z y,,. whee,; [ : R R R Φ L defed by.6,.7 ad R R : k defed by.8, gaaee global aypoc ably fo ye.5. he eade hold oce ha he dyac apled-daa coolle.9,.,. ca ll be ed eve f o delay ae pee b he ae avalable oly a he dcee e a he aplg e wh >, whee > he aplg peod. Hece, h eco we have povded, a a pecal cae, he f olo o he poble of global aypoc ablao of c feedfowad ye wh abaly pae e aplg of he ae ad wh coo cool. <. Sapled-Daa Feedback fo Ip Appled wh Zeo Ode Hold. Geeal Deg h eco devoed o he cae whee he p appled wh eo ode hold. I h eco we ae ha. globally ablable wh feedback appled wh eo ode hold. h vey dffee fo hypohe H he pevo eco. Hypohe H p appled wh eo ode hold: hee e k R R :, K g, > ch ha g k, fo all R.

13 ad ch ha R Ufoly Globally Aypocally Sable fo he apled-daa ye & f, k, [,, l R. he ee ha hee e a fco σ KL ch ha fo evey R he olo of. wh al codo R afe eqaly.4 wh fo all. Reak.: Hypohe H ee lke a ecve hypohe, becae dead global ablably by ea of apled-daa feedback wh pove aplg ae. Howeve, hypohe H ca be afed fo: Lea ablable ye, whee f, A B, A R B R, ee Coollay.4 ad Reak.5 below, Nolea ye of he fo, & f g, R R, whee he veco feld f : R R globally Lpch ad he veco feld g R R : locally Lpch ad boded, whch ca be abled by a globally Lpch feedback law k ee [7]. Nolea ye of he fo f, g, & f, g,, whee he df e f,,..., codo f L... L,..., & fo,..., ad afy he lea gowh fo cea coa L ad hee e coa b a > ch ha a g, b fo all,...,, R, R ee []. v Aypocally coollable hoogeeo ye wh pove al powe ad eo degee ee [4]. v Sye afyg he eachably hypohee of heoe. [4], o hypohee A, A, A Seco 4 of [], v Nolea ye & f,, fo whch hee e a global dffeooph Θ : R R ch ha he chage of coodae Θ afo he ye o oe of he above cae. Code ye. de hypohee H, H fo ye.. I h cae we popoe a feedback law ha ply a copoo of he feedback able ad he delay copeao: whee Z, Φ y,, [, k ae he aplg e ad Φ : R [,; R R. L he pedco appg volved.,.. he cool aco appled wh eo ode hold,.e.,

14 coa o, ; howeve he cool aco affecg ye. ea coa o he [ eval [,. O a el aed e. heoe.: Le >,, wh > ad ppoe ha hee e l Z ch ha l. Moeove, ppoe ha hypohee H, H hold fo ye.. he he cloed-loop ye. wh.,.e., he followg apled-daa ye & f, k Φ,,, [,, Z Ufoly Globally Aypocally Sable, he ee ha hee e a fco ha fo evey, C [,]; R [,; R, R.4 wh al codo L [, R, C [,]; R followg eqaly fo all :.4 ~ σ KL ch L, he olo R of ye ; afe he ~ σ,.5 Fally, f ye. afe he dead-bea popey of ode j, whee j Z pove,.e., fo all R he olo of. wh al codo R afe fo all j he ye.4 afe he dead-bea popey of ode j l, whee he ege pa of,.e., fo evey, C [,]; R L [,; R, he olo, R R of ye.4 wh al codo L [,; R [, R C ]; afe fo all j l., Reak.: a If we deoe o be he delay ecevg he eaed daa, he copao e fo he qay v k Φ,, R L [,; R, ad he e fo he daa o, whee each he acao, he oe hold oce ha ad. b I pacce, whe, ae gve, he cool pacoe hold look fo apled-daa able afyg hypohe H fo cea aplg peod > wh l fo oe l Z. heefoe, he vale of he aplg peod deeed afe he eao of he p 4

15 delay. If he oe ca chooe ay aplg peod > he codo l hold wh l. Howeve, f > he he aplg peod coaed o be le o eqal o >. h ay ee o be paccal fo he cae whee > all. Howeve, pacce, a delay ha alle ha a eaoable aplg peod wold be ypcally goed, o he cool egee ca dce a p delay of agde eqal o he aplg peod fo eaple, by delayg he ao of he cool aco. Poof of heoe.: Ug he Fac, we ca gaaee he eece of evey, C [,]; R [,; R L [,; R, C [,]; R b b K ch ha fo L he olo R of. wh al codo e fo all [, ] ad afe, fo all [, ].6 If > he he p ake eacly l vale o he eval [, wh p vale fo [,,,..., l he followg eae:. We alo e l l. Ug. ad., we oba a g ga a, fo q q,..., l Ug he above eqale, he val eqaly ad he eqale b fo,..., l whch ae dec coeqece of.6, we ca coc a fco h K ch ha h we ay coclde ha hee e h, fo,..., l.7 H K ch ha H Ieqaly.8 hold vally fo he cae., fo all [, ].8 We e coe wh he followg cla. I poof povded he Apped. Cla : hee e a fco e fo all [, ] ad afe G K ch ha he olo, R R of ye.4 G By ve of dco ad Cla, he followg cla hold., fo all [, ].9 5

16 Cla 4: hee e a fco G K ch ha fo evey p Z, p he olo, R R of ye.4 e fo all [, p ] ad afe p G p whee G : G 4 K 4 o G o. p e, fo all [, p ]. We oce ha fo all Z wh he olo, R R of ye.4 afe: k, [,. Hypohe H cojco wh eqaly.4 wh ad eqao. ple ha he followg eqaly hold fo all Z wh : Defe p [ / ] σ,,.. Ug. ad., he followg eqaly hold: σ p G, p Defe ~ p p σ, : σ G H, p g σ G H, p p ad ~ p p, : G H g G H, p, p. fo all σ fo all [. Ug.,.8,.,. ad. we ca coclde ha.5 hold. Noce ha f σ, fo all ad j.e., he dead-bea popey of ode j he. ple ha fo all j p l.e., he dead-bea popey of ode j p l. he poof coplee. < heoe. ca be appled o all fowad coplee ye afyg Reak.d ad Reak.. Hee we foc o wo pecal cae: he cae of ablable Lea e Ivaa LI ye ad he cae of ye whch ae Dffeoophcally Eqvale o a Cha of Iegao DECI. he lae cae clde he leaable c feedfowad ye ee [8].. Deg fo Sablable LI Sye Fo he Lea e Ivaa LI cae, hee ae ace A R, B R ch ha f, A B. I h cae he pedco appg gve by he eplc epeo Φ, ep A ep A B dw. he lea feedback law k K, whee K R, 6

17 afe hypohe H f ad oly f hee e > ch ha all he egevale of he a ep A I ep Aw dw BK ae cly de he ccle o he cople plae. he e of he apled-daa feedback law Ky, whee ad y y, doe o geeal gaaee global aypoc ably fo he delayed cae, whee >. Howeve, heoe. ca be ed fo he deg of a delay copeao, whch gaaee global aypoc ably fo he coepodg cloed-loop ye. he followg coollay a dec coeqece of heoe. ad poof oed. Coollay.4 Sablao of Lea Newoked Cool Sye wh Delay: Le >,, wh > ad ppoe ha hee e l Z ch ha l. Defe q : ad Aw dw BK ae cly de he ccle o he cople plae. he he cloed-loop LI ye ~ q. Moeove, ppoe ha all he egevale of he a ep A I ep & A B, [,, wh p appled wh eo ode hold gve by whee y ad y Q Q Z.4 l q K ep A y K Q B.5 p ep l q Ap ep A ep ~ p A l q ep A p d p d, p,..., l q.6 Globally Epoeally Sable, he ee ha hee e coa M, σ > ch ha fo evey, C [,]; R L [,; R, he olo, R R.5 wh al codo L [, R, C [,]; R followg eqaly fo all : ; of ye.4, σ M ep afe he.7 7

18 Reak.5: If he pa of ace A, B ablable he hee e K R ch ha he a A BK Hw, a yec pove defe a P R ad a coa μ > ch ha P A BK A BK P μ I <. Ug Coollay 4. [] wh V P, a : λ, abay λ,, ad he fac ha A, R : BK A k λ k, whee k : a P, k : P, oe ca how ha all he egevale of he a ep A I ep Aw dw BK ae cly de he ccle o he cople plae fo all > afyg k A BK BK < ad k PBK < μ BK Of coe, he eae fo he a allowable aplg peod povded by he above eqale coevave o cae. Ohe eae fo he a allowable aplg peod ca be fod []. Eaple.6: We code he cala cool ye &.8 whee R, R. he ye ca be epoeally abled by he lea feedback k wh k > appled wh eo ode hold,.e., whee he aplg peod > afy, [, k, Z.9 < l k. he e of he ae feedback law fo he cae whee eaee delay ae pee decbed by he eqao: &, [, k, Z. whee he eaee delay. Necal epee fo he cloed-loop ye. how ha fo each pa of k > ad > afyg., hee e > ch ha c f < c he ye. globally epoeally able, 8

19 f > c he ye. ad epoeally gowg olo. Fo he cae k, he vale of he ccal eaee delay afe.,.. Fge how he evolo of he ae fo ye. wh k,,. c ad al codo fo [,]. he ae covege epoeally o eo. Fge how he evolo of he ae fo ye. wh k,,. ad ae al codo fo [,]. I h cae, he ae gow epoeally, dcag ably.,8,6,4, -, -,4 -,6 -,8-5 5 Fge : he evolo of he ae fo ye. wh codo fo [,] k,,. ad al Fge : he evolo of he ae fo ye. wh codo fo [,] 9 k,,. ad al

20 I clea ha fo he cae he ccal eaee delay.,. > oe eed a delay copeao. Noce ha fo he cae k, c oly a all faco of he aplg peod. c he al pacce wold be o goe he delay ad h wold gve e o copleely accepable el. Coollay.4 how ha he feedback law:, [,. k ep k ep wll gaaee global epoeal ably fo he cloed-loop ye.8 wh. whe <. Ideed, Fge how he evolo of he ae fo he cloed-loop ye.8 wh., k,,. ae covege epoeally o eo. ad al codo fo [,], 4 fo [,. he Fge : he evolo of he ae fo he cloed-loop ye.8 wh.,. ad al codo, 4 fo [,] fo [, k,, h eaple deoae ha he delay-copeag pedco-baed feedback. eed he age of eaee delay fo whch ablao acheved fo gve k > ad > afyg.. <. Deg fo Coollable Sye Leaable by Coodae Chage he cla of ye ha ae Dffeoophcally Eqvale o a Cha of Iegao DECI he cla of all olea ye f, &, R R,, whee f : R R R locally Lpch wh f,, fo whch hee e a global dffeooph Θ : R R ch ha he

21 chage of coodae Θ afo he ye o he lea ye & A B B,...,,, A a,, j,..., } {, j wh a fo,..., ad a f,, j j., whee I h cae, fo evey > hee e K R ep, ch ha all he egevale of he a ep A I A w dw B K ae eo. Fo eaple, fo he veco K R defed by K,. If all egevale of he a ep A I ep A w dw BK ae eo he he apled-daa coolle wh eo ode hold appled o he lea ye elg cloed-loop ye,.e., K, [,, Z & A B wll gaaee he dead-bea popey of ode fo he, fo all ad fo all al codo R h, we ca coclde ha he apled-daa coolle wh eo ode hold K Θ, [,, Z appled o he olea ye & f, wll gaaee he dead-bea popey of ode fo he elg cloed-loop ye. heefoe, heoe. ad Coollay.4 lead o he followg coollay. Coollay.7 Pedco fo Leaable Coollable Sye: Le >,, wh > ad ppoe ha hee e l Z ch ha l. Defe q : ad ~ q. Code ye. wh ad ppoe ha hee e a global dffeooph Θ : R R ch ha Θ f, A Θ B, fo all R R D whee DΘ he Jacoba of Θ, B,...,,, A a,, j,..., } {, j,. wh a fo,...,, ad a f j. Le K R be ch ha all egeevale of he a, j ep ep A I A w dw B K ae cly de he ccle o he cople plae. he he cloed-loop ye. wh p appled wh eo ode hold gve by, [, Z.4,

22 l q K ep A Θ y K Q B.5 whee y y ad he ace p p Q,..., l q p p p ae defed by.6 wh A place of A, Globally Aypocally Sable. Moeove, f all egeevale of he a ep ep A I A w dw B K ae eo he fo evey, C [,]; R L [,; R, he olo, R R of ye.,.4,.5 wh al codo L ; [, R, C [,]; R afe:, fo all l q.6 Eaple.8: Dead-bea cool wh a pedco ca be appled o ay delayed -deoal c feedfowad ye,.e., ay ye of he fo: whee & p, &.7 p : R R a ooh fco ad he eaee ae apled ad gve by.. he dffeooph gve by ee [8] Θ p w dw, afo ye.7 wh o a cha of wo egao. heefoe, he feedback law.8 p w dw.9 appled wh eo ode hold ad aplg peod > acheve global ablao of ye.7 wh whe o eaee delay ae pee. Moeove, he dead-bea popey of ode gaaeed fo he coepodg cloed-loop ye. Ieegly, he feedback law.9 alo globally aypocally ablg a a coo-e coolle, placg he cloed-loop pole he Θ -coodae a ± 4 7 fo ay >. We e code he cae whee we have eaee delay > afyg <. I h cae l q, ~ we apply Coollay.7 ad we ca coclde ha he feedback law.4 wh p w dw. gaaee he dead-bea popey of ode fo he coepodg cloed-loop ye. Sla fola o. ae obaed fo ohe cae, whee > o. <

23 4. Sablao of a Noholooc Moble Robo Ove a Log-Dace Cocao Newok wh Abaly Spae Saplg he edced-ode odel of a hee-wheeled vehcle wh wo depede ea ooed wheel ca be decbed by he followg ye of dffeeal eqao: v co, y& v, & ω & 4. whee, y ae he coodae of he cee of a of he vehcle ad he agle bewee he a of he vehcle ad he hooal a. he p v ad ω ae lea cobao of he agla veloce of he wo ea wheel. he coodae afoao: ad he p afoao: co y 4.a y co 4.b 4.c v ω, ω 4. bg ye 4. o he fo.4. May eeache have obaed el fo he ablao of he eqlb po R of ye.4. Hee, we ae ha he eaee y,, whee >,,,, ae avalable a dcee e a whch dffe by a coa >. Moeove, we ae ha hee a e delay bewee he coped cool aco ad he appled p cocao delay. I h cae he eqao of he vehcle ae: wh eaee v co, y& v, & ω & 4.4, y ad. he eade hold oce ha hypohee H, H hold fo ye 4.. Paclaly, hee e ooh e-peodc feedback able fo ye.4 ee [5,] ad coeqely we ca gaaee ha hypohe H hold. he pedco appg fo ye 4.4 gve by he followg eqao: Φ, y,, v, ω : v co ω p dpd, y v ω p dpd, ω d 4.5

24 4 Ug ay ablg feedback fo [5,] fo he oholooc egao.4, he coodae afoao 4. ad he p afoao 4. ad heoe., we ave a he followg coollay. Coollay 4.: Ae ha ; R R R C k wh, k fo all, a e peodc fo able fo.4,.e., he feedback law,,, k,,,, k foly, globally able R fo ye.4. he fo evey,, > he apled-daa dyac feedback co v v y ω & & &,, [ 4.6 co y d d dp p v y d dp p v ω ω ω 4.7 whee, Z ad, co, co, co, co, co, k k v y y y y y ω ω 4.8 acheve fo global ablao of R fo ye 4.4. A h po hold be ephaed ha f he ooh, e-vayg feedback popoed [9] fo he ablao of he oholooc egao wee ed 4.8 he he cloed-loop ye 4.4 wh 4.6, 4.7, 4.8 wold be o-foly e globally aypocally able ee Reak. above. Moeove, he eade hold copae he el of Coollay 4. wh he el [7]: o eco fo he agde of he delay ae poed he pee wok. If he cool aco pleeed wh eo ode hold, he a dffee pocede ha o be appled. he eade hold oce ha fo evey aplg peod > he dcoo feedback able

25 k g / k k k / { }, fo C ξ R : ξ ξ ξ ξ 4.9a :, fo C { ξ R : ξ, ξ ξ } k k 4.9b :, fo C : { ξ R : ξ ξ ξ } 4.9c afe hypohe H fo ye.4. o ee h oce ha eqaly. hold wh g :. Fheoe, by eplc copao of he olo oe ca how ha: f C he he olo of.4 wh k afe,.e., C C, f C he he olo of.4 wh k afe,.e., C, f C he he olo of.4 wh k afe. I follow ha he apled-daa pleeao of he feedback 4.9 wh aplg peod > gaaee he dead-bea popey of ode fo he coepodg cloed-loop ye. he eqaly p p p < < < whch hold fo he olo of.4 fo evey al codo ad fo evey appled p L [, ]; R, cojco wh eqaly. gaaee fo evey > fco he eece of a a K ch ha fo evey al codo he olo of.4 wh he apled-daa pleeao of he feedback 4.9 wh aplg peod > afe p a <. he dead-bea popey of ode cojco wh he pevo eae gaaee ha he dcoo feedback able defed by 4.9 afe hypohe H fo ye.4. he eade hold oce ha he feedback deg pocede decbed [4] ca be appled a well fo he oholooc egao.4 ce.4 a aypocally coollable hoogeeo ye wh pove al powe ad eo degee. 5

26 heefoe, we ae a poo o olve he ablao poble fo 4.4 fo he cae of p appled wh eo ode hold, wh eaee, y ad avalable a dcee e a. Ideed, f we elec a aplg peod > whch afe l fo oe l Z. he we ca apply heoe. fo he dcoo feedback defed by 4.9 ad oba he followg el. Popoo 4.: Le,, > ad ae ha l fo oe l Z. he R foly globally aypocally able fo he cloed-loop ye 4.4 wh ω k X co Θ Y Θ, X Θ Y co Θ, Θ, fo, v k X co Θ Y Θ, X Θ Y co Θ, Θ whee, Z, : R R k defed by 4.9 ad 4. [ X Θ Y co Θ ω, fo, 4. [ X Y y Θ v co v ω d ω p dpd ω p dpd 4. Moeove, fo evey al codo he olo of he cloed-loop ye 4.4 wh 4., 4., 4., whee, Z, : R R k defed by 4.9 afe y fo all l 4, whee he ege pa of. I hold be oced ha he cool aco coped by 4., 4., 4. appled wh eo ode hold,.e., v ad ω ae coa o each eval,, Z [ pecewe coa ove he egao eval, ] 4.. [, ad hece hey ae 5. Cocldg Reak Sablao ded fo olea ye wh p ad eaee delay, ad wh eaee avalable oly a dcee e a aplg e. wo dffee cae ae codeed: he cae whee he p ca be cooly adjed ad he cae whee he p appled wh eo ode hold. Ude he apo of fowad copleee ad cea 6

27 addoal ablably apo, how ha apled-daa feedback law wh a pedco-baed delay copeao ca gaaee global aypoc ably fo he cloed-loop ye wh o eco fo he agde of he delay. Addoally, whe he cool appled cooly ad oly he eaee ae apled, he aplg e ca be abaly log. Applcao o he ablao of lea ewoked cool ye, c feedfowad ye ad a oholooc oble obo ove a log-dace cocao ewok ae peeed. Fe wok wll adde he e of obe of he popoed feedback law wh epec o acao ad eaee eo, a well a he eeo of he obaed el o he cae whee he delayed ad apled eaed op doe o ecealy cocde wh he ae veco. Ackowledge: he aho wold lke o hak Pofeo Coa Kava fo bgg o he aeo he fac ha he pedco appg ca be eplcly coped fo blea ye &, wh R, R ad AC CA ee Reak.d. A B C Refeece [] Agel, D. ad E.D. Soag, Fowad Copleee, Uboded Obevably ad he Lyapov Chaaceao, Sye ad Cool Lee, 84-5, 999, 9-7. [] Fda, E., A. See ad J.-P. Rchad, Rob Sapled-Daa Sablao of Lea Sye: A Ip Delay Appoach, Aoaca, 4, 4, [] Gao, H.,. Che ad J. La, A New Sye Appoach o Newok-Baed Cool, Aoaca, 44, 8, 9-5. [4] Güe, L., Hoogeeo Sae Feedback Sablao of Hoogeo Sye, SIAM Joal o Cool ad Opao, 84,, [5] Güe, L. ad D. Nešć, Opao Baed Sablao of Sapled-Daa Nolea Sye va he Appoae Dcee-e Model, SIAM Joal o Cool ad Opao, 4,, 98-. [6] Heeel, M., A.R. eel, N. va de Wow ad D. Nešć, Newoked Cool Sye wh Cocao Coa: adeoff bewee ao Ieval, Delay ad Pefoace, o appea IEEE aaco o Aoac Cool. 7

28 [7] Hea, G., S.K. Spgeo ad C. Edwad, Dceao of Sldg Mode Baed Cool Schee, Poceedg of he 8 h Cofeece o Deco ad Cool, Phoe, Aoa, U.S.A., 999, [8] Jakovc, M., Recve Pedco Deg fo Sae ad Op Feedback Coolle fo Lea e Delay Sye, Aoaca, 46,, [9] Kaafyll, I. ad J. a, Global Sablao ad Aypoc ackg fo a Cla of Nolea Sye by Mea of e-vayg Feedback, Ieaoal Joal of Rob ad Nolea Cool, 6,, [] Kaafyll, I., A Sye-heoec Faewok fo a Wde Cla of Sye I: Applcao o Necal Aaly, Joal of Maheacal Aaly ad Applcao, 8, 7, [] Kaafyll, I. ad Z.-P. Jag, A Sall-Ga heoe fo a Wde Cla of Feedback Sye wh Cool Applcao, SIAM Joal Cool ad Opao, 464, 7, [] Kaafyll, I., ad C. Kava, Global Sably Rel fo Sye de Sapled-Daa Cool, Ieaoal Joal of Rob ad Nolea Cool, 9, 9, 5-8. [] Kaafyll, I., P. Pepe ad Z.-P. Jag, Sably Rel fo Sye Decbed by Copled Readed Fcoal Dffeeal Eqao ad Fcoal Dffeece Eqao, Nolea Aaly, heoy, Mehod ad Applcao, 77-8, 9, 9-6. [4] Kaafyll, I. ad C. Kava, Rob Global Sablably by Mea of Sapled-Daa Cool wh Pove Saplg Rae, Ieaoal Joal of Cool, 84, 9, [5] Kaafyll, I., Sablao By Mea of Appoae Pedco fo Sye wh Delayed Ip, bed o SIAM Joal o Cool ad Opao. [6] Khall, H. K., Nolea Sye, d Edo, Pece-Hall, 996. [7] Koja, K.,. Ogch, A. Alvae-Age ad H. Njeje, Pedco-Baed ackg Cool of a Moble Robo Wh e-delay, Poceedg of NOLCOS, Bologa, Ialy,, [8] Kc, M., Feedback Leaably ad Eplc Iegao Fowadg Coolle fo Clae of Feedfowad Sye, IEEE aaco o Aoac Cool, 49, 4, [9] Kc, M., Delay copeao fo olea, adapve, ad PDE ye. Sye & Cool: Fodao & Applcao. Bkhäe Boo, Ic., Boo, MA, 9. [] Kc, M., Ip Delay Copeao fo Fowad Coplee ad Feedfowad Nolea Sye, IEEE aaco o Aoac Cool, 55,, 87-. [] Kc, M. Lyapov ably of lea pedco feedback fo e-vayg p delay, IEEE aaco o Aoac Cool, 55,,

29 [] Maec, F., S. Mode ad R. Facco, Global Aypoc Sablao of Feedfowad Sye wh Delay a he Ip, IEEE aaco o Aoac Cool, 49, 4, [] Maec, F. ad P.-A. Bla, Backeppg Deg fo e-delay Nolea Sye, IEEE aaco o Aoac Cool, 5, 6, [4] Maec, F., M. Maloff ad Z. L, Fhe Rel o Ip-o-Sae Sably fo Nolea Sye wh Delayed Feedback, Aoaca, 44, 8, [5] Mo, P., J.-B. Poe ad C. Sao, Deg of Hoogeeo e-vayg Sablg Cool Law fo Dfle Coollable Sye Va Ocllaoy Appoao of Le Backe Cloed Loop, SIAM Joal o Cool ad Opao, 8, 999, -49. [6] Naghhab, P., J. Hepaha ad A. R. eel, Sably of Delay Iplve Sye Wh Applcao o Newoked Cool Sye, Poceedg of he 6 h Aeca Cool Cofeece, New Yok, U.S.A., 7. [7] Nešć, D., A.R. eel ad P.V. Kokoovc, Sffce Codo fo Sablao of Sapled-Daa Nolea Sye va Dcee-e Appoao, Sye ad Cool Lee, 84-5, 999, [8] Nešć, D. ad A.R. eel, Sapled-Daa Cool of Nolea Sye: A Ovevew of Rece Rel, Pepecve o Rob Cool, R.S.O. Mohea Ed., Spge-Velag: New Yok,, -9. [9] Nešć, D. ad A. eel, A Faewok fo Sablao of Nolea Sapled-Daa Sye Baed o he Appoae Dcee-e Model, IEEE aaco o Aoac Cool, 497, 4, -. [] Nešć, D. ad D. Lbeo, A fed faewok fo deg ad aaly of ewoked ad qaed cool ye, IEEE aaco o Aoac Cool, 544, 9, [] Nešć, D., A. R. eel ad D. Caevale, Eplc copao of he aplg peod elao of coolle fo olea apled-daa ye, IEEE aaco o Aoac Cool, 54, 9, [] Poe, J.-P., Eplc Deg of e-vayg Sablg Cool Law fo a Cla of Coollable Sye Who Df", Sye ad Cool Lee, 8, 99, [] Seplche, R., M. Jakovc, ad P. Kokoovc, Cocve Nolea Cool. New Yok: Spge, 997. [4] abbaa, M., D. Nešć ad A. R. eel, Newoked cool ye: elao baed deg, Newoked Cool Sye Ed. D. L ad F.-Y. Wag See Iellge Cool ad Iellge Aoao, Wold Scefc, 7. 9

30 [5] abada, P., Eve-ggeed Real-e Schedlg of Sablg Cool ak, IEEE aaco o Aoac Cool, 59, 7, [6] eel, A.R., Coeco bewee Rakh-ype heoe ad he ISS Nolea Sall Ga heoe, IEEE aaco o Aoac Cool, 47, 998, [7] Y, M., L. Wag,. Ch ad F. Hao, Sablao of Newoked Cool Sye wh Daa Packe Dopo ad ao Delay: Coo-e Cae, Eopea Joal of Cool,, 5, [8] Zhag, X., H. Gao ad C. Zhag, Global Aypoc Sablao of Feedfowad Nolea Sye wh a Delay he Ip, Ieaoal Joal of Sye Scece, 7, 6, Apped Poof of Cla he poof of heoe.: Le,, C [,]; R R [,]; R L be abay ad code he olo,, R R R of he cloed-loop ye.5,.6,.,. wh al codo R, [,]; R C [,]; R. I ccal o L, oce ha he olo of & f, k, wh R afe ξ, whee ξ he olo of ξ f ξ, k, ξ & wh ξ R. Ieqaly.4 ple ha he olo e fo all, ad ha he followg eqaly hold R [ σ,,, A [ I follow ha he olo of.6 e fo all,. Coy of k C R R ; R [ ad eqale., A ply ha he appg coo o, ad boded wh k, l ad k,, whee l. Noce l ha he l l e by ve of fo coy of he appg o [,. By ve of eqale. ad A we oba he eqaly σ, p g < A Ug hypohe H we ay coclde ha he olo R of. e fo all [, ]. Ideed, by ve of he Fac, we ca gaaee he eece of K ζ ch ha

31 p < ζ, fo all, ] he above eqaly cojco wh A ad he val eqaly p gve: g σ, ζ, [, ] A Fally, we defe k,, whee Φ,. Aga, g.,., A ad A we oba: ad g σ, ζ g σ, a a g σ, ζ g σ, g Ug A, A, A ad he above eqale we ae a poo o coc a fco G K ch ha eqaly.8 hold. he poof coplee. < Poof of Cla he poof of heoe.: Code f he cae >. Sce l fo oe l Z, follow ha. By ve of he Fac, we ca gaaee he eece of b K ch ha b Moeove, g he eqao k Φ,, [, we oba a g, fo all [, ] A, eqale. ad.,, fo all [, A4 Fally, g he val eqale p < p cojco wh A, A4, we ca coclde ha.9 hold wh G : b g a. ad Ne code he cae. Ug he eqao k Φ,, [, ad., we oba A4 wh. By ve of he Fac ad he apo b K ch ha, eqale., hee e b p, fo all, ] A5 < Cobg A4 ad A5 we oba: b g a, fo all [, ] A6

32 Aga, g he val eqale p < ad p cojco wh A6 ad A4 wh, we ca coclde ha.9 hold wh : a g a g b G. he poof coplee. <