Automatica. Stabilization of linear strict-feedback systems with delayed integrators

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1 Auomaca 46 (1) Coes lss avalable a SceceDrec Auomaca joural homepage: wwwelsevercom/locae/auomaca Bref paper Sablzao of lear src-feedback sysems wh delayed egraors Nkolaos Bekars-Lbers, Mroslav Krsc Deparme of Mechacal ad Aerospace Egeerg, Uversy of Calfora, Sa Dego, La Jolla, CA , USA arcle fo absrac Arcle hsory: Receved 1 December 9 Receved revsed form 9 May 1 Acceped 8 July 1 Avalable ole 1 Augus 1 Keywords: Delay sysems Predcor Src-feedback sysems The problem of compesao of pu delays for usable lear sysems was solved he lae 197s Sysems wh smulaeous pu ad sae delay have remaed a challege, alhough epoeal sablzao has bee solved for sysems ha are o epoeally usable, such as chas of delayed egraors ad sysems he feedforward form We cosder a geeral sysem src-feedback form wh delayed egraors, whch s a eample of a parcularly challegg class of epoeally usable sysems wh smulaeous pu ad sae delays, ad desg a predcor feedback coroller for hs class of sysems Epoeal sably s prove wh he ad of a Lyapuov Krasovsk fucoal ha we cosruc usg he PDE backseppg approach 1 Elsever Ld All rghs reserved 1 Iroduco Sablzao of lear sysems wh pu delays coues o be a acve area of research Varous corol schemes for sysems wh pu delay have bee developed, wh he sarg po for may of hem beg he Smh predcor (Smh, 1959) The mos mpora eesos of he Smh predcor have bee desgs based o he fe specrum assgme framework (Arse, 198; Fagbedz & Pearso, 1986; Jakovc, 9a, 1; Kwo & Pearso, 198; Maus & Olbro, 1979; Mode & Mchels, 3; Olbro, 1978; Rchard, 3; Zhog, 6) I addo o hese desgs, adapve versos of predcorbased lear corollers are proposed Evesque, Aaswamy, Nculescu, ad Dowlg (3a), Lu ad Krsc (1), Nculescu ad Aaswamy (3b), Zhou, Wag, ad We (8), as adapve corollers for ukow delay have bee developed recely Bekars-Lbers ad Krsc (1), Bresch-Per ad Krsc (9a,b) ad Yldray, Aaswamy, Kolmaovsky, ad Yaakev (1) Moreover, varous corol desgs for olear sysems es (Jakovc, 1, 3, 9b; Karafylls, 6; Krsc, 8a, 1; Mazec & Blma, 4; Mazec, Mode, & Fracsco, 4) Despe he fac ha umerous papers deal wh lear sysems wh pu delay, he problem of coroller desg for sysems wh smulaeous pu ad sae delay has bee ackled oly a The maeral hs paper was o preseed a ay coferece Ths paper was recommeded for publcao revsed form by Assocae Edor Faryar Jabbar uder he dreco of Edor Robero Tempo Correspodg auhor Tel: ; fa: E-mal addresses: bekar@ucsdedu, kosbekars@gmalcom (N Bekars-Lbers), krsc@ucsdedu (M Krsc) few Refs Fagbedz ad Pearso (1986), Jakovc (9a), Jakovc (1), Loseau (), Maus ad Olbro (1979) ad Waaabe, Nobuyama, Kamor, ad Io (199) I hs work we cosder a specally chose possbly ope-loop usable lear sysem, wh he specal form of a src-feedback sysem wh delayed egraors Specfcally, we cosder he followg -dmesoal lear sysem X 1 () = ā 11 X1 () b 1 X ( D 1 ) (1) X () = ā 1 X1 () ā X () b X3 ( D ) () X () = ā 1 X1 () ā X () b Ū( D ), (3) X (), ā j, Ū() R, b =, ad D R For hs sysem we develop a predcor-based coroller (Seco ) To acheve hs we use ools from he boudary corol of frs order lear hyperbolc PDEs (Krsc & Smyshlyaev, 8a), ogeher wh he classcal backseppg procedure (Krsc, Kaellakopoulos, & Kokoovc, 1995) Specfcally, a fe dmesoal backseppg rasformao s used, ogeher wh a corol law, o cover he sysem o a epoeally sable ( a cera sese) arge sysem Usg he boudess of he backseppg rasformao ad s verse, we he prove epoeal sably of he closed-loop sysem usg a suably weghed Lyapuov Krasovsk fucoal (Seco 3) The effecveess of he proposed coroller s llusraed by a smulao eample of a secod order usable sysem (Seco 4) Coroller desg We sar by redefg he saes of sysem (1) (3) such ha he coeffces fro of he delayed erms are uy Tha s, we 5-198/$ see fro maer 1 Elsever Ld All rghs reserved do:1116/jauomaca178

2 N Bekars-Lbers, M Krsc / Auomaca 46 (1) defe X 1 () = X1 () (4) X () = b 1 X () (5) X 3 () = b 1 b X3 () (6) X () = b 1 b b 1 X () (7) Moreover, for oaoal cossecy we defe U() = b 1 b b Ū() (8) āj, f = j a j = (9) b j b 1 ā j, f > j I he ew varables, sysem (1) (3) s rasformed o Ẋ 1 () = a 11 X 1 () X ( D 1 ) (1) Ẋ () = a 1 X 1 () a X () X 3 ( D ) (11) Ẋ () = a 1 X 1 () a X () U( D ) (1) We sae here our coroller ad Seco 3 we aalyze he sably properes of he closed-loop sysem The coroller for he sysem (1) (1) s gve by U() = u(d, ) = α (D, ) 1 = a 1 P 1 a P () c (P () α 1 (D 1 D, )) α 1 (D 1 D, ), (13) α (, ) = a 1 P 1 a P k= c P α 1 (D 1, ) k= α 1(D 1, ),, (14) k= α 1 (, ) = (a 11 c 1 )P 1,,, (15) ad he c, = 1, are arbrary posve cosas I he above corol scheme we use he P () sgals, he k= secods ahead predcors of he X () sae (hs fac becomes clear laer o) Tha s, holds ha P () = X ( k= ) These sgals are gve by P 1 () = X 1 () (a 11 P 1 (θ) P (θ))dθ (16) P () = X () P () = X () k= (a 1 P 1 (θ D 1 ) a P (θ) P 3 (θ))dθ (17) D k= 1 a 1 P 1 θ 1 a P θ a P (θ) U(θ) dθ, (18) wh al codos θ P 1 (θ) = X 1 () (a 11 P 1 (σ ) P (σ ))dσ (19) θ P (θ) = X () P (θ) = X () k= (a 1 P 1 (σ D 1 ) a P (σ ) P 3 (σ ))dσ () θ D a1 P1 1 σ 1 a P σ a P (σ ) U(σ ) dσ, (1) k= θ s defed each P (θ) as θ [ k=, ] Noe here ha he oao α 1(D 1,) correspods o α 1(,) =D 1 whch cludes he me dervaves of he sgals P 1 (),, P 1 () These dervaves are obaed from (1) (1) ad (16) (18) 3 Sably aalyss We frs sae a heorem descrbg our ma sably resul ad he we prove usg a seres of echcal lemmas Theorem 1 Sysem (1) (1) wh he coroller (13) s epoeally sable he sese ha here es cosas κ ad λ such ha Ω() κω()e λ, () Ω() = 1 ad 1 X () 1 D 1 X (θ)dθ = D 1 D U (θ)dθ = = D 1 X (θ)dθ D U (θ)dθ, (3) D ξ (, )d =ξ () (4) u (, )d =u() (5) ξ (, ) = X ( D 1 ), [, D 1 ] (6) u(, ) = U( D ), [, D ] (7) We frs gve ad prove he followg lemmas

3 194 N Bekars-Lbers, M Krsc / Auomaca 46 (1) Lemma 1 The sgals P () defed (16) (18) are, respecvely he k= secods ahead predcors of he X () saes Moreover a equvale represeao for (16) (18) s gve by p 1, p, k= p (D, ) = X () p (, ) = P = X 1 () (a 11 p 1 (y, ) p (y D 1, ))dy (8) = X () k= (a 1 p 1 (y, ) a p (y, ) p 3 (y D, ))dy (9) D (a 1 p 1 (y, ) a p (y, ) u(y, ))dy, (3),, k= (31) Proof Cosder he equvale represeao of sysem (1) (1) usg raspor PDEs for he delayed saes ad corol Ẋ 1 () = a 11 X 1 () ξ (, ) (3) ξ (, ) = ξ (, ) (33) ξ (D 1, ) = X () (34) Ẋ () = a 1 X 1 () a X () ξ 3 (, ) (35) ξ 3 (, ) = ξ 3 (, ) (36) ξ 3 (D, ) = X 3 () (37) Ẋ () = a 1 X 1 () a X () u(, ) (38) u (, ) = u (, ) (39) u(d, ) = U() (4) Cosder he followg ODEs (o become clear ha hese are ODEs, cosder he me o ac as a parameer raher as a rug varable), p 1 (, ) = a 11 p 1 (, ) p ( D 1, ) (41) p (, ) = a 1 p 1 (, ) a p (, ) p 3 ( D, ) (4) p (, ) = a 1 p 1 (, ) a p (, ) u(, ), (43), for each p (, ), vares [, k= ] The al codos for he above sysem of ODEs are gve by p (, ) = X (),, (44) ad p (θ, ) = X ( θ ), θ [ D 1, ], =,, (45) Sce sysem (41) (43) s drve by he pu u(, ), whch sasfes a raspor PDE, he same holds for all he p (, ) (see for eample Krsc (8a, 1)) Thus, p (, ) = p (, ),,, (46) k= k= By akg o accou (44) we have ha p (, ) = X ( ),,, (47) k= To see ha (46) holds s suffce o prove ha (47) s he uque soluo of he ODEs gve by (41) (43) wh he al codos (44) (45) Sce he, he p (, ) are fucos of oly oe varable, amely, ad cosequely (46) holds Thus, remas o prove ha (47) s he uque soluo of he al value problem (41) (45) Toward hs ed, by akg o accou (7) we po ou ha (47) sasfy he al value problem (41) (45) The, assumg ha X ( θ ), =,, are couous for all θ [ D 1, ], usg Theorem 1 from Hale ad Verduy Luel (1993) we ca coclude ha (47) s he uque soluo of he ODEs gve by (41) (43) wh he al codos (44) (45) Thus, (46) holds From relao (47) becomes clear ha he p (, ) are he secods ahead predcors of he saes By defg p, = P (),, (48) k= we ge (31) By egrag from o (41) (43) we ge p 1 (, ) = X 1 () p (, ) = X () p (, ) = X () (a 11 p 1 (y, ) p (y D 1, ))dy (49) (a 1 p 1 (y, ) a p (y, ) p 3 (y D, ))dy (5) (a 1 p 1 (y, ) a p (y, ) u(y, ))dy (51) By seg each p (, ), = k= ad usg (31) we ge (8) (3) I s mpora here o observe ha he oal delay from he pu o each sae X () s k= Ths eplas he fac ha our predcor ervals are dffere for each sae ad specfcally mus be k= secods for each sae X () Our coroller desg s based o a recursve procedure ha rasforms sysem (1) (1) o a arge sysem whch s epoeally sable wh he coroller (13) The, usg he verbly of hs rasformao, we prove epoeal sably of he orgal sysem We ow sae hs rasformao, alog wh s verse Lemma The sae rasformao defed by Z 1 () = X 1 () (5) Z 1 () = X 1 () α (D, ), = 1,,, 1, (53) alog wh he rasformao of he acuaor sae w(, ) = u(, ) α (, ), [, D ], (54) he α (, ) are defed as (14) (15), rasforms he sysem (1) (1) o he arge sysem wh he corol law gve by (13) The arge sysem s gve by Ż 1 () = c 1 Z 1 () Z ( D 1 ) (55) Ż () = c Z () Z 3 ( D ) (56) Ż () = c Z () W( D ), (57)

4 N Bekars-Lbers, M Krsc / Auomaca 46 (1) W(θ) =, θ (58) Proof Before we sar our recursve procedure we rewre he arge sysem usg raspor PDEs as Ż 1 () = c 1 Z 1 () ζ (, ) (59) ζ (, ) = ζ (, ) (6) ζ (D 1, ) = Z () (61) Ż () = c Z () ζ 3 (, ) (6) ζ 3 (, ) = ζ 3 (, ) (63) ζ 3 (D, ) = Z 3 () (64) Ż () = c Z () w(, ) (65) w (, ) = w (, ) (66) w(d, ) = (67) Noe ha ζ (, ) = Z ( D 1 ), [, D 1 ] (68) Sep 1 Followg he backseppg procedure we frs sablze X 1 () wh he vrual pu α 1 (D 1, ) We defe ζ (, ) = ξ (, ) α 1 (, ), (69) he usg (3) we ge Ẋ 1 () = a 11 X 1 () ζ (, ) α 1 (, ) (7) By choosg α 1 (, ) = (a 11 c 1 )p 1 (, ) (oe he equvale represeao of α 1 (, ) usg (31)) ad by usg (44) we ge Ż 1 () = c 1 Z 1 () ζ (, ) (71) From (69) wh = D 1 ad (61) follows ha Z () = X () α 1 (D 1, ) (7) By seg ow ζ 3 (, ) = ξ 3 (, ) α (, ), (73) ad usg (69), (33) ad (35) we have Ż () = ζ (, ) =D1 = a 1 X 1 () a X () ζ 3 (, ) α (, ) α 1(D 1, ), (74) α 1(D 1,) correspods o α 1(,) =D1 ad we use he fac ha α 1 (, ) = α 1 (, ) (whch s a cosequece of relao (15)) Sep By choosg α (, ) = a 1 p 1 (, ) a p (, ) c (p (, ) α 1 (D 1, )) α 1( D 1, ) = a 1 p 1 (, ) a p (, ) c (p (, ) (a 11 c 1 )p 1 (D 1, )) (a 11 c 1 )(a 11 p 1 ( D 1, ) p (, )), (75) we ge from (74) (wh he help of (44)) ha Ż () = c Z () ζ 3 (, ) (76) By seg ow = D (73) ad usg (64) we ge Z 3 () = X 3 () α (D, ) (77) If we ow defe ζ 4 (, ) = ξ 4 (, ) α 3 (, ), (78) he wh he help of (36) we ge Ż 3 () = a 31 X 1 () a 3 X () a 33 X 3 () ζ 4 (, ) α 3 (, ) α (D, ) (79) Sep Assume ow ha Ż 1 () = c 1 Z 1 () ζ (, ), (8) ad defe ζ 1 (, ) as ζ 1 (, ) = ξ 1 (, ) α (, ) (81) The from (53) wh = D 1 we have ha Ż () = a 1 X 1 () a X () ζ 1 (, ) Hece, wh α (, ) α 1(D 1, ) (8) α (, ) = a 1 p 1 (, ) a p (, ) c (p (, ) α 1 (D 1, )) α 1(D 1, ), (83) we ge Ż () = c Z () ζ 1 (, ) (84) Sep I he las sep we choose he coroller U() Sce Ż () = a 1 X 1 () a X () u(, ) α 1 (D 1, ) (85) The usg (14) for = we have ha Ż () = c Z () w(, ), (86) ad usg (13) w (, ) = w (, ) (87) w(d, ) = (88) Assumg a al codo for (87) as w(, ) = w (), (89) ad by defg a ew varable W( ) as w () = W( D), [, D ], (9) we ge ha W( D), D D w(, ) = (91), D Defg θ = D oe ges (58) Noe here ha based o (54), w () s gve by w () = u(, ) α (, ), [, D ] (9) We ow defe he verse rasformao of (5) (54) Lemma 3 The verse rasformao of (5) (54) s defed as X 1 () = Z 1 () (93) X 1 () = Z 1 () β (D, ), = 1,,, 1 (94) u(, ) = w(, ) β (, ), [, D ], (95)

5 196 N Bekars-Lbers, M Krsc / Auomaca 46 (1) he β (, ) are ow gve by β 1 (, ) = (a 11 c 1 ) 1 (, ),, (96) β (, ) = a 1 1 (, ) a ( (, ) β 1 (D 1, )) a ( (, ) β 1 (D 1, )) c (, ) β 1(D 1, ),,, =,,, (97) k= ad he (, ) (he predcors of he rasformed saes) are gve by he followg relaos 1 (, ) = Z 1 () (, ) = Z () (, ) = Z () ( c 1 1 (y, ) (y D 1, ))dy (98) ( c (y, ) 3 (y D, ))dy (99) ( c (y, ) w(y, ))dy, (1) each (, ), vares [, k= ] Proof Applyg smlar argumes as Lemma we prove ha he verse rasformao of (5) (54) ad (14) (15) s gve by (93) (97) We ow prove he sably of he rasformed sysem Lemma 4 The arge sysem s epoeally sable he sese ha here es cosas M 1,m 1 ad m such ha Ξ() M 1(1 D ma ) m Ξ() = 1 1 Z () 1 Ξ()e m 1 M 1, (11) = D 1 Z (θ)dθ D W (θ)dθ (1) D ma = ma {D },, (13) ad D 1 Z (θ)dθ = D 1 D W (θ)dθ = D ζ (, )d =ζ () (14) w (, )d =w() (15) Proof We cosder he followg Lyapuov-lke fuco V() = 1 k Z () 1 D 1 λ (1 )ζ (, )d λ 1 D = (1 )w (, )d (16) Noe ha he above fucoal ca be cosdered as a Corol Lyapuov Fucoal he sese of Karafylls ad Jag ( press) Ths fac reforces he sregh of he prese resul: a Corol Lyapuov Fucoal s acually cosruced By akg he me dervave of he above fuco alog he soluos of he Z() sysem ad by eplog he fac ha ζ (, ) ad w(, ) sasfy raspor PDEs (based o (6), (63) ad (66)), follows ha V() = c k Z () 1 k Z ()ζ 1 (, ) k Z ()w(, ) 1 1 = λ ζ (, ) 1 = λ 1 w (, ) λ 1 λ (1 D 1 )Z () D 1 λ = D ζ (, )d w (, )d, (17) we used egrao by pars he above egrals By choosg he weghs as k = λ c (1 D 1 ), =,, k 1 = (18) λ = 4 λ 1(1 D ), = 3,, 1 c 1 λ = 1 c 1, (19) ad afer some mapulaos ha corporae compleo of squares we ge V() 1 Defg λ 1 c k Z () 1 D D 1 λ = ζ (, )d w (, )d (11) M 1 = ma {k,λ 1 }, = 1,,, (111) c k m 1 = m, λ 1, = 1,,,, (11) (1 D ) follows ha V() m 1 M 1 V() (113) If we ow defe k m = m, λ 1, = 1,,,, (114) he Ξ() V() m e m 1 M 1 M 1(1 D ma ) m Ξ()e m 1 M 1 (115) We gve ow he followg lemma whch we prove he Apped Lemma 5 There es cosas G such ha p (, ) G X() D = u (y, )dy D 1, ξ (y, )dy,, (116) k=

6 N Bekars-Lbers, M Krsc / Auomaca 46 (1) X() = X (), (117) ad he boud (116) s depede of Lemma 6 There ess a cosa M such ha Ξ() MΩ() (118) Proof From (5) (54) follows ha Z () (X () α 1 (D, )), =,, (119) ζ (, ) (ξ (, ) α 1 (, )), [, D 1 ], =,, (1) w (, ) (u (, ) α (, )), [, D ] (11) Moreover, from relaos (14) (15) oe ca see ha he α (, ) are lear fucos of he predcors p 1 (, ),, p (, ), hece holds ha α (, ) b p (, ), k, (1) for some cosas b By employg he boud of Lemma 5, he lemma s prove k= Lemma 7 There es cosas F such ha D 1 (, ) F Z() ζ (y, )dy D = w (y, )dy,, (13) Proof Immedaely oe ha he relao for he (, ) s smlar o he relao for p (, ) Noe here ha hs case he dervao of he eplc boud s easer due o he specal form of he (, ) (98) (1) Lemma 8 There ess a cosa M such ha MΩ() Ξ() (14) Proof Usg relaos (93) (95) we ge X () (Z () β 1 (D, )), =,, (15) ξ (, ) (ζ (, ) β 1 (, )), [, D 1 ], =,, (16) u (, ) (w (, ) β (, )), [, D ] (17) By observg ha β (, ) are learly depede o 1 (, ),, (, ) we coclude ha here es cosas d such ha β (, ) d (, ), k, (18) Usg Lemma 7 he lemma s prove Proof of Theorem 1 Combg Lemmas 6 ad 8 we have ha MΩ() Ξ() MΩ() (19) k= k= Hece, Fg 1 Sysem s respose for he smulao eample Ω() Ξ() M, (13) ad by Lemma 4 we ge Ω() MM 1(1 D ma ) Mm Thus Theorem 1 s prove wh Ω()e m 1 M 1 (131) κ = MM 1(1 D ma ) Mm (13) λ = m 1 M 1 (133) 4 Smulaos We llusrae here our coroller wh a secod order eample wh parameers a 11 = a 1 = a =, D 1 = 4, D = 8 ad c 1 = c = The al codos for he coroller are gve by (19) (1) ad for he sysem are X 1 () = X () = 1 ad X (θ) = 1,θ [ D 1, ] Ths sysem s usable (o see hs oe ca use Olgac ad Spah ()) I he prese case he coroller wll have he form U() = u(d, ) = α (D, ) = a 1 p 1 (D, ) a p (D, ) c (p (D, ) (a 11 c 1 )p 1 (D 1 D, )) (a 11 c 1 )(a 11 p 1 (D D 1, ) p (D, )) = a 1 P 1 ( D 1 ) a P () c (P () (a 11 c 1 )P 1 ()) (a 11 c 1 )(a 11 P 1 () P ()), (134) P 1 () ad P () are calculaed usg he egral represeao (16) (18) Noe also ha hese egrals are compued usg he rapezodal rule Fg 1 shows ha he predcor coroller epoeally sablzes he sysem The corol sgal reaches frs X (), sce he delay from he pu o X () s 4, whch s smaller ha he oal delay from he pu o X 1 () Afer 1 s, whch s he oal delay from he pu o X 1 (), he coroller sars sablzg X 1 () The boh X 1 () ad X () coverge epoeally o zero (see Fg )

7 198 N Bekars-Lbers, M Krsc / Auomaca 46 (1) Apped Here we gve he proof of Lemma 5 Proof of Lemma 5 By solvg (41) (43), ad by akg o accou ha hs ODE sysem s src-feedback form, we ge 5 Coclusos Fg Corol effor for he smulao eample We prese a backseppg desg for a epoeally usable sysem wh smulaeous pu ad sae delay Our desg s predcor-based sce uses he predced values of he saes o gve ervals Usg he boudess of he backseppg rasformao ad s verse, we prove epoeal sably of he closed-loop sysem usg a properly weghed Lyapuov Krasovsk fucoal A backseppg-lke desg for lear sysems wh oly sae delay s he oe cosdered Jakovc (9a) The major dfferece wh he desg Jakovc (9a) ad he oe cosdered here, s ha Jakovc (9a) delays are o allowed he vrual pus (whch s he dffcul case cosdered here) The prese procedure ca be modfed o corporae sae delays ha are oher posos oher ha he vrual pus I he case of a sysem wh oly pu delay (rrespecve of he form of he sysem, e, eher f he sysem s a cha of egraors wh pu delay, eg Mazec, Mode, ad Nculescu (3), or a sysem feedforward form, eg Jakovc (1), ec) he resulg corol law s he predcor-based/fe specrum assgme coroller from Arse (198), Fagbedz ad Pearso (1986), Krsc ad Smyshlyaev (8a) ad Maus ad Olbro (1979) wh he ga K beg desged usg he classcal backseppg procedure for lear sysems from Krsc e al (1995) I he case of a sysem wh smulaeous pu ad sae delays a backseppg-lke desg comparable wh he oe cosdered here s he oe Jakovc (1) for he specal case of a cha of delayed egraors ad pu delay I hs specal case he resulg corol law from he prese work urs ou o be he same wh he oe Jakovc (1) The prese resuls ca be also appled he case here are delays oher saes oo, ad o jus he vrual pus Thus, he class of sysems such ha he prese mehod ca be appled s o lmed Cosderg he problem he delays or he coeffces a j are ukow, s a compleely dffere ad very challegg problem Followg he fe dmesoal backseppg echque, hs problem has bee solved for he case here s oly ukow pu delay (Bresch-Per & Krsc, 9a) ad eeded o he case of ukow pu delay ad pla parameers Bresch-Per ad Krsc (9b) I Bekars- Lbers ad Krsc (1) a problem wh ukow pu ad sae delays s solved for a class of lear feedforward sysems Applcao of he desg mehods from Bekars-Lbers ad Krsc (1), Bresch-Per ad Krsc (9a,b) he prese case seems promsg ad ca be pursued as a forhcomg research opc p 1 (, ) = p (, ) = p (, ) = D1 v 11 ( y D 1 )p (y, )dy D 1 v 11 ()X 1 (),, D v ( y D )p 1 (y, )dy D v ()X (),, 1 D D k= v ( y D )p 1 (y, )dy v ()X () v ( y)u(y, )dy, (135) (136) [, D ], (137) v 11 () v 1 () v () e A = (138) v 1 () v () v () a 11 a 1 a A = (139) a 1 a a By applyg Youg s ad Cauchy Schwarz s equales o Eqs (135) (137) we ge D1 p (, ) 1 A 1 X () 1 p (y, )dy (14) p (, ) A p (, ) A D 1 X 1 () X () X () 1 u (y, )dy D D D D p 1 (y, )dy p 1 (y, )dy (141), (14), each of he above bouds, [, k= ], respecvely Also A = ma sup v (),, 1, k= sup, k= v(),

8 sup, k= sup, k= D1 D 1 D D v 1 ( y D 1 ) dy,, N Bekars-Lbers, M Krsc / Auomaca 46 (1) v ( y D ) dy (143) If we ake ow o accou ha p ( D 1, ) = X ( D 1 ) = ξ (, ) we ca rewre (14) (14) as p 1 (, ) A 1 p (, ) A p (, ) A X 1 () ξ () X () D X 1 D D1 ξ 1 () p (y, 1 )dy p (y, )dy (144) (145) () 1 ξ 1 () p (y, 1 )dy u (y, )dy, (146) each of he above relaos, [, k= ], respecvely From he above equaos, recursvely, we ca ake he upper boud of he lemma To see hs, we sar from relao (144) ad observe ha he boudess of p 1 (, ) depeds oly o he boudess of X 1 () ad ξ (, ) (has s, p 1 (, ) remas bouded for all [, ]), f for all [, k= ], p (, ) s upper bouded We proceed ow by provg ha he boudess of p (, ) depeds oly o he boudess of X 1(), X (), ξ (, ) ad ξ 3 (, ) (has s, p (, ) remas bouded for all [, k= ]), f for all [, k=3 ], p 3 (, ) s upper bouded From relao (145) (ad by og D ha 1 p (y, )dy p (y, )dy for all for whch hs equao holds, e, [, k= ]) by usg he comparso prcple ad by eplog he fac ha e A e A k=, [, k= ], we ge ha p (y, )dy A e A k= ξ 1 () D k k= y D X () p 3 (r, )drdy (147) Pluggg he above boud o relao (145) we ge a boud of p (, ) ha depeds o p 3 (, ) Moreover, usg he relao p (, ) A 3 X () ξ 1 () D 3 ad he prevous boud, we ge p 1 (y, )dy, (148) p 3 (, ) A 3 A 3 A e A 3 A e 3 A k= X () k= A D y k k= ξ 1 () X () D3 A 3 p (, ), 4, p 3 (r, )drdy A 3 ξ 1 () p 3 (, )dy (149) Noe ha he delayed erms he egral for p 3 (, ) ca be removed sce ow [, k=3 ] whch s he doma of defo for p 3 (, ) (ad of course hs egral s larger ha he delayed oe) By chagg he order of egrao he double egral of he prevous relao, we ca rewre y k=3 p (r, 3 )drdy = ( y)p 3 (y, )dy (15) p 3 (y, )dy, By observg ha ( y)p(y, 3 )dy k=3 k=3, ad applyg aga he comparso prcple for p(y, 3 )dy we ca boud p(, ) 3 from p 4 (, ) ad cosequely also p (, ) Repeag hs process ul p (, ) (he boudess of whch depeds oly o he boudess of u(, ) ), we derve he boud of he lemma Refereces Arse, Z (198) Lear sysems wh delayed corols: a reduco IEEE Trasacos o Auomac Corol, 7, Bekars-Lbers, N, & Krsc, M (1) Delay-adapve feedback for lear feedforward sysems Sysems ad Corol Leers, 59, Bresch-Per, D, & Krsc, Mroslav (9a) Delay-adapve full-sae predcor feedback for sysems wh ukow log acuaor delay Amerca Corol Coferece, Bresch-Per, D, & Krsc, Mroslav (9b) Adapve rajecory rackg despe ukow pu delay ad pla parameers Auomaca, 45, Evesque, S, Aaswamy, A M, Nculescu, S, & Dowlg, A P (3) Adapve corol of a class of me-delay sysems ASME Trasacos o Dyamcs, Sysems, Measureme, ad Corol, 15, Fagbedz, Y A, & Pearso, A E (1986) Feedback sablzao of lear auoomous me lag sysems IEEE Trasacos o Auomac Corol, 31, Hale, J K, & Verduy Luel, S M (1993) Iroduco o fucoal dffereal equaos New York: Sprger-Verlag Jakovc, M (1) Corol Lyapuov Razumkh fucos ad robus sablzao of me delay sysems IEEE Trasacos o Auomac Corol, 46, Jakovc, M (3) Corol of olear sysems wh me delay IEEE Coferece o Decso ad Corol, Jakovc, M (9a) Forwardg, backseppg, ad fe specrum assgme for me delay sysems Auomaca, 45(1), 9 Jakovc, M (9b) Cross-erm forwardg for sysems wh me delay IEEE Trasacos o Auomac Corol, 54(3), Jakovc, M (1) Recursve predcor desg for sae ad oupu feedback corollers for lear me delay sysems Auomaca, 46(3), Karafylls, I (6) Fe-me global sablzao by meas of me-varyg dsrbued delay feedback SIAM Joural o Corol ad Opmzao, 45(1), 3 34 Karafylls, I, & Jag, Z P (8) Necessary ad suffce Lyapuov-lke codos for robus olear sablzao ESAIM Corol, Opmzao ad Calculus of Varaos, press, (do:1151/cocv/99), avalable a: hp://wwwesam-cocvorg/ Krsc, M (8a) O compesag log acuaor delays olear corol IEEE Trasacos o Auomac Corol, 53, Krsc, M (1) Ipu delay compesao for forward complee ad feedforward olear sysems IEEE Trasacos o Auomac Corol, 55, Krsc, M, Kaellakopoulos, I, & Kokoovc, P V (1995) Nolear ad adapve corol desg Wley Krsc, M, & Smyshlyaev, A (8a) Backseppg boudary corol for frs-order hyperbolc PDEs ad applcao o sysems wh acuaor ad sesor delays Sysems ad Corol Leers, 57, Kwo, W H, & Pearso, A E (198) Feedback sablzao of lear sysems wh delayed corol IEEE Trasacos o Auomac Corol, 5, 66 69

9 191 N Bekars-Lbers, M Krsc / Auomaca 46 (1) Lu, W-J, & Krsc, M (1) Adapve corol of Burgers equao wh ukow vscosy Ieraoal Joural of Adapve Corol ad Sgal Processg, 15, Loseau, J J () Algebrac ools for he corol ad sablzao of me-delay sysems Aual Revews Corol, 4, Maus, A Z, & Olbro, A W (1979) Fe specrum assgme for sysems wh delays IEEE Trasacos o Auomac Corol, 4, Mazec, F, & Blma, P-A (4) Backseppg desg for me-delay olear sysems IEEE Trasacos o Auomac Corol, 51, Mazec, F, Mode, S, & Fracsco, R (4) Global asympoc sablzao of feedforward sysems wh delay a he pu IEEE Trasacos o Auomac Corol, 49, Mazec, F, Mode, S, & Nculescu, S I (3) Global asympoc sablzao for chas of egraors wh a delay he pu IEEE Trasacos o Auomac Corol, 48(1), Mode, S, & Mchels, W (3) Fe specrum assgme of usable medelay sysems wh a safe mplemeao IEEE Trasacos o Auomac Corol, 48, 7 1 Nculescu, S-I, & Aaswamy, A M (3) A adapve Smh-coroller for medelay sysems wh relave degree Sysems ad Corol Leers, 49, Olbro, A W (1978) Sablzably, deecably, ad specrum assgme for lear auoomous sysems wh geeral me delays IEEE Trasacos o Auomac Corol, 3, Olgac, N, & Spah, R () A eac mehod for he sably aalyss of medelayed lear me-vara (LTI) sysems IEEE Trasacos o Auomac Corol, 47, Rchard, J-P (3) Tme-delay sysems: a overvew of some rece advaces ad ope problems Auomaca, 39, Smh, O J M (1959) A coroller o overcome dead me ISA Trasacos, 6, 8 33 Waaabe, K, Nobuyama, E, Kamor, T, & Io, M (199) A ew algorhm for fe specrum assgme of sgle-pu sysems wh me delay IEEE Trasacos o Auomac Corol, 37, Yldray, Y, Aaswamy, A, Kolmaovsky, I V, & Yaakev, D (1) Adapve poscas coroller for me-delay sysems wh relave degree Auomaca, 46(), Zhog, Q-C (6) Robus corol of me-delay sysems Sprger Zhou, J, Wag, W, & We, C (8) Adapve backseppg corol of ucera sysems wh ukow pu me delay I FAC World cogress Nkolaos Bekars-Lbers receved hs BS degree Elecrcal ad Compuer Egeerg from he Naoal Techcal Uversy of Ahes 7 He s ow workg oward he PhD degree he Deparme of Mechacal ad Aerospace Egeerg a Uversy of Calfora, Sa Dego Hs research eress clude corol of delay sysems, corol of dsrbued parameer sysems ad olear corol Mroslav Krsc s he Dael L Alspach Professor ad he foudg Drecor of he Cymer Ceer for Corol Sysems ad Dyamcs (CCSD) a UC Sa Dego He receved hs PhD 1994 from UC Saa Barbara ad was Asssa Professor a Uversy of Marylad ul 1997 He s a coauhor of egh books: Nolear ad Adapve Corol Desg (Wley, 1995), Sablzao of Nolear Ucera Sysems (Sprger, 1998), Flow Corol by Feedback (Sprger, ), Real-me Opmzao by Eremum Seekg Corol (Wley, 3), Corol of Turbule ad Mageohydrodyamc Chael Flows (Brkhauser, 7), Boudary Corol of PDEs: A Course o Backseppg Desgs (SIAM, 8), Delay Compesao for Nolear, Adapve, ad PDE Sysems (Brkhauser, 9), ad Adapve Corol of Parabolc PDEs (Prceo, 1) Krsc s a Fellow of IEEE ad IFAC ad has receved he Aelby ad Schuck paper przes, NSF Career, ONR Youg Ivesgaor, ad PECASE award He has held he appome of Sprger Dsgushed Vsg Professor of Mechacal Egeerg a UC Berkeley