Delay-Depede Robus Asypocally Sable for Lear e Vara Syses D. Behard, Y. Ordoha, S. Sedagha ABSRAC I hs paper, he proble of delay depede robus asypocally sable for ucera lear e-vara syse wh ulple delays s vesgaed. A ew delay-depede sably suffce codo s gve by usg he Lyapuov ehod, lear arx equaly (LMI, paraeerzed frs-order odel rasforao echque ad rasforao of he erval uceray o he or bouded uceray. A uercal exaple s preseed o llusrae our prese sably crero allows a upper boud whch s bgger o he sze of he delay coparso wh hose he leraure. KEYORDS Lyapuov-Krasovs fucoal, Lear arx equaly, Paraeerzed frs-order odel rasforao, e-delay syses.. INRODUCION e delay s frequely a source of sably ad s ofe ecouered varous areas of corol syses, such as ecoocal syses, bology [, ], egeerg, eural ewor [], raspor pheoea ad populao dyacs [3, 7]. A syse s sad o be sable depede of delay (delaydepede sable f s sable whe delay paraeer assues all oegave values. he sably of a syse s delay-depede f s sable soe doa of delays [9]. Delay depede or delay-depede sably ca be easly derved by a approprae choce of he ers volved he Lyapuov-Krasovs fucoal. May crera for checg he sably of e delay syses have bee gve so far. I hs paper, he lear arx equaly (LMI ehod wh he paraeerzed frs order odel rasforao echque s eployed o derve a ew delay-depede robus asypocally sable codo for he lear evara syses wh ulple delays.. NOAIONS he followg sadard oao wll be used hroughou he paper. Le R deoes he se of all real arces, A be he raspose of arx, τ >, ad A < B (resp., A B eas ha he arx B A s posve defe (resp., posve se-defe for ay wo syerc arces alogwh Le N [ ], M [ ] ad, e defe he se arces [ N, M ] { A [ a ] : a }. I addo PC (resp., PC deoes he space of all uforly bouded pecewse couous (resp., pecewse couous dffereable real arx-valued fucos defed o [,. he Baach space C ([ τ, ] C([ τ, ], of couous vecor fucos appg he delay erval o wh ufor covergece opology, where τ > wh sadard supreu or, φ Sup θ [ τ,] φθ for gve φ C ([ τ, ] ad. refers o eher he Eucldea vecor or or he duced arx -or. 3. SABILIY CRIERIA FOR IME-VARIAN SYSEMS IH MULIPLE DELAYS Le us cosder he lear e-vara syse wh ulple delays as follows, ( ( A( x( A ( x(- τ D. Behard, Mahs Dep., Alzahra Uv., vaa, hera, Ira, (e-al: behard@alzahra.ac.r or dbeh@yahoo.co Y. Ordoha, Mahs Dep., Alzahra Uv., vaa, hera, Ira, (e-al: ordoha@alzahra.ac.r S. Sedagha, Mahs Dep., Alzahra Uv., vaa, hera, Ira,, (e-al: saaz_84@yahoo.co Arabr / MISC / Vol. 4 / No. / Sprg 9 35
wh al codos of e sa x ( θ x ( θ φ( θ, θ [ τ,] ( where x ( ( x(,..., x ( s sae vecor a e usual sese. A ( PC, are sae arxes such ha her copoes are o ow precsely bu sasfyg A ( [ N, M ], for,,..., where N [ ], M ( [ (] PC wh ( ( for all [, ]. he vecor fuco φ ( s a elee of Baach space C ([ τ, ] ad τ τ τ <,,,..., are ucera e-vara delays where τ ax{ τ :,,..., }. he e-vara syse assocaed wh syse ( s of he for ( A x ( (3 A x( - τ where A [ N, M ],,,...,. Now defe B ( ( N M / b ], H ( ( M N / h ] where,,...,. ad E ( [ E,..., E ] such ha each E, l l,... s a array wh ery e h l for l ad 36 e for l where,,.... Also defe F [ F,..., F ] such ha each F l, l,... s a array wh ery f h l for l ad e for l where,,.... I s easy o verfy ha (4 E ( E ( dag ( h,..., h ad. (5 F ( F ( dag ( h,..., h Le [ I, I ],, where I s dey arx. I s obvous ha dag(,...,,...,,..., ε ε ε ε, such ha ε,,,..., ad furherore I,,,.... Le N [ N (, M (]{ A ( B ( E ( F (: [- I, I ]}, he we have he followg lea, [4, ]. Lea 3.. [4] For,,..., he equales [N (, M (] N[N (, M (] always hold. hs lea shows ha he lear e-vara erval syse ( s equvale o he followg lear syse subec o or bouded srucured uceraes descrbed by he equao ( ( B ( E ( F ( x ( (6 ( B E F x ( - τ. Correspodgly, assocaed wh syse (3 we have ( ( B E F x ( ( B E F x (-, τ (7 where [ I, I ],,. herefore, whe oe s loog for sably codo whch depeds o delay, he sadard sep s o replace he orgal syses ( ad (3 by he syses (6 ad (7, [6, 5, 9]. Also, he followg leas are esseal for he proof of he a heore, []. b ( Lea 3.. [] Le ω(. f ( s ds he he a θ followg s sasfed, d ( ( b a f ( ( b d ω & a f ( sds b (8 ( b& a& f ( s ds a Sce x ( s couously dffereable for, by usg he Lebz-Newo forula we have x ( τ x τ x & ( θ x ( τ Ax( θ Ad x( θτ whch s used referece [9] o rasfer he syse ( Ax ( Ad x ( - θ, (9 o he dsrbued delay syse, ( A C x ( A C x( τ d τ Ax ( θ Ad x ( θτ, ( where C s a paraerc arx whch derves he sably less resrcve o soe degree. Sce hs process oly oe egrao over oe delay erval s used, he process s called paraeerzed frs-order odel rasforao. he sably of ( ples he sably of he syse (9 for all τ [, τ ], see [9] ad refereces here. By applyg he above odel rasforao o he syse (6, we have Arabr / MISC / Vol. 4 / No. / Sprg 9
( [ B ( E ( F ( C ] x( [ B ( E ( F ( C ] x ( τ C [( B ( θ E ( θ F ( θ x ( θ τ ( B ( θ E ( θ F ( θ x ( θ τ. ] ( he sably of hs syse ples he sably of he syse (6. herefore we focus o sably of he las oe. V ( x x Px b x ( θ S ( θ x ( θ τ f x ( θ S( θ x ( θ τ (5 L x S x d d, τ θ θ θ θ τ τ where b, f, L for,,..., are arbrary cosa coeffces. he e-dervave of hs fucoal alog wh he posve half raecores of he syses (, ca be expressed as follows: V& ( x heore 3.3. he erval syse (6 s robus x ( PB B P ( x asypocally sable for ay τ [, τ ],,..., wh x PE F x ( A ( [ N, M ],,,...,, f here exs x ( ( PB ( x ( - τ posve cosa scalars, α, posve defe arces P P >, R R >, Q Q >, ad cosa x ( ( PE ( F ( x ( - τ arx, for,,... such ha x B( x( τ Ω PB ( B ( P F ( F ( x E( F( x ( τ bs ( f S ( τ x B ( x( τ τ τ Q τ LS ( L x E ( F ( x ( τ τ E E α L τ x (( bs ( x ( (6 ( [( PB ( R ( PB ( b b x ( τ S ( τ x( τ τ PE ( E ( P] Q f f τ x S x f x S x d τ E PE P< f τ α τ for,,... where L x ( S ( x ( τ τ S ( B ( Q B ( α F ( F ( ( S B ( Q B ( F τ τ τ α ( τ F ( τ Lx ( S ( x (. (3 Usg he followg equales for ay posve real uber β > ad ay posve defe arx D, S R F ( τ F ( τ, (4 -u b, b, b are ay posve cosa,d v u v βu D - u β - v Dv, s ay real cosa ad he correspodg odel rasforao arces ( s gve by C P. Proof. Cosder he Lyapuov-Krasovs fucoal defed as follows Arabr / MISC / Vol. 4 / No. / Sprg 9 37
where uv, R, [8, ]. e have x ( ( PB ( x ( - τ b (( ( ( ( x PB R PB x( (7 b x (- R τ x( - τ. x ( PE ( F ( x ( - τ ( ( b x PE E ( Px( (8 (- bx τ F ( F ( x ( - τ. x B( x ( τ τ f x Q x (9 f x B ( QB( x ( τ x E( F( x( τ x E E x ( f α τ α f x F ( F( x(. τ x B ( x( τ τ τ x ( Q x ( L L x B τ ( Q B ( x ( τ. τ ( x E ( F ( x ( τ τ x ( E E x ( d α L τ α L τ x τ F ( F ( x( τ. ( Subsug (7- o (6, we ge, Ω PB ( B ( F ( F ( bs ( f S ( τ τ Q τ LS ( L E E α L τ ( [( PB ( R ( PB ( b τ PE ( E ( P] Q f E E f τ α ( (. PE E P Sce Ω <, s easy o show ha V& ( x ( < f x ( ad V& ( x (, f ad oly f x(. herefore by Lyapuov-Krasovs sably heore, he org of he syse ( s robus asypocally sable for A ( [ N, M ],,,...,, ad τ [, τ ], cosequely he org of he syse (6 s robus asypocally sable whch coplees he proof. I he above heore, f we le,, S τ Q, S τ Q, R R, τ τ, α α, followg resul s edae. Corollary 3.4. Syse ( wh s robus α τα he he asypocally sable for ay A ( [ N, M ],, f here exs cosa scalers >, α >, syerc ad posve defe arces P P >, R R >, S S > such ha Ω PB ( B ( P ( f α F F ( b L α F ( τ F ( τ for, ad cosa arx br L B ( τ S B ( τ f B S B S S τ ( PE ( E ( P L f V & ( x x Ω x, where 38 Arabr / MISC / Vol. 4 / No. / Sprg 9
PE ( E P ( PB ( R ( PB ( b τ ( f α τ E E ( L α E( E ( <. he correspodg odel rasforao arx s gve by C P. 4. EXAMPLE Cosder he sae erval syse as gve [], x & ( A ( ( ( ( -, x A x τ where A ( Λ ε s I, A( Λ ε cos I, such ha Λ, Λ are ow arces, ε, ε are ucera bu bouded as ε, ε. I s easy o see ha, A( [ N (, M (],, where N ( Λ -s I M ( Λ s I N ( Λ -cos I M ( Λ cos I. Hece, by assug B ( Λ, B ( Λ, H ( s I, H( cos I, ad E ( E ( F ( F ( s I E ( E ( F ( F ( cos I e have Ω P Λ Λ P b R s fα I ( b L α cos I L Λ S Λ f Λ S Λ τ ( S L S f P s I cos P P I P ( PΛ R ( PΛ b τ ( s. I cos. I. f α L α Also f we assue bb,ff,ll, α α α,, he we have Ω P Λ Λ P br ( f α I (b Lα I LΛ S Λ f Λ S Λ τ ( S L S f P τ ( PΛ R ( PΛ (. b α f L For exaple le b /6, f L / ad assue.5 Λ, Λ.9..5 he oe feasble soluo for assocaed lear arx equaly (LMI s p dag(7.344, 59.9684 R dag(37.5,.6 S dag(8.85, 8.8647 S dag(34.9834, 3.8433 dag( 83.77, 79.833 4.6665, α 56.8986 herefore, by subsug o he rgh had sde of equaly, Ω we ge -87.757 5.499 τ 7.46458 Ω. < 7.46458 -.8578 97.854τ herefore τ <.389767 ad τ <.99849. Cosequely he syse s robus asypocally sable for τ [,.99849] whch s a larger doa for delay wh respec o exaple of referece []. 5. CONCLUSION I hs paper, we have vesgaed he robus asypocal sably ssue of lear erval e vara syses wh ucera delays. A ew delay depede sably codo s derved by usg he Lyapuov ehod, (LMI, paraeerzed frs-order odel rasforao echque ad roducg geously real cosas. Based o a prese crero, a ew upper boud o he sze of delays s preseed. A uercal exaple s also provded o deosrae he effecveess of he ew resul. 6. REFERENCES [] Capbell,S. A. ad Belar, J., 99, Mulple-delayed dffereal equaos as odel for bologcal corol syses. I Proceedg orld Cogress of Nolear Aalyss 9, 3-37,apa. [] K, J.-H.,, Delay ad e-dervave Depede Robus Sably of e-delay Lear Syses wh Uceray. IEEE ras. Auo. cor., 46,(5, 789-79. [3] Kuag, Y., 993, Delay Dffereal Equaos wh Applcaos Populao Dyacs. Acadec Press, Boso. 9 [4] L, C. D. ad Lao, X. f., 6, A global expoeal robus sably crero for NN wh varable delays. Neurocopuag 69, 8-89. [5] L, X. ad de Souza, C. E., 995 LMI approach o delay - depede robus sably of ucera lear syses. Proc. of he 34h CDC, New Orleas, 364-369. [6] L, X. ad de Souza, C. E., 997, Delay depede robus sably ad sablzao of ucera lear delay syse: A lear Marx Iequaly Approach. IEEE ras. o Auoac Corol, 4, 44-48. Arabr / MISC / Vol. 4 / No. / Sprg 9 39
[7] Macdooald, N., 989, Bologcal Delay Syses: Lear Sably heory, Cabrdge Uversy Press, Cabrdge. [8] Nculescu, S.-I., Do, J.-M., Dugard, L., ad L, H., 997, Sably of lear syses wh several delays: A L.M.I. approach. JESA, specal ssue o Aalyss ad corol of edelay syses 3, 955-97. [9] Nculescu, S.-I.,, Delay effecs o sably: A robus approach. Sprger, Berl. [] Sepa, G., 998, Rearded dyacal syse sably ad characersc fuco. Research Noes Maheacs Seres, Joh ley, New Yor, P:. [] Su, J.H., 994, Furher resuls o he robus sably of lear syses wh a sgle delay. Syses ad Corol Leers, 3, 375-379. [] Zhag, Z., Lao, ad Ch. L, X., 6, Delay-depede robus sably aalyss for erval lear e-vara syse wh delays ad applcao o delayed eural ewors. Neurocopuag, do:.6/.euco.6.9.,. 4 Arabr / MISC / Vol. 4 / No. / Sprg 9