Stability of Non-Neutral and Neutral Dynamic Switched Systems Subject to Internal Delays

Transcription

1 mea Joual of ppled Sees (): 48-49, 5 ISS See Publaos Sably of o-eual ad eual Dyam Swhed Sysems Sube o Ieal Delays M. De la Se, J.L. Malaa,. Gallego ad J.C. Soo Isue of Reseah ad Developme of Poesses Depame of Eley ad Eleos, Campus of Leoa. pdo. 644 de Blbao Depame of ppled Mahemas, Campus of Blbao-La Caslla Uvesy Basque Couy, Spa bsa: hs sudy deals wh he quada sably ad lea sae-feedba ad oupu-feedba sablzao of swhed delayed lea dyam sysems wh, geeal, a fe umbe of o ommesuae osa eal po delays. he esuls ae obaed based o Lyapuov s sably aalyss va appopae asovsy-lyapuov s fuoals ad he elaed sably sudy s pefomed o oba boh delay depede ad delay depede esuls. I s poved ha he sablzg swhg ule s abay f all he swhed subsysems ae quadaally sable ad ha exss a ( geeal, o-uque) sablzg swhg law whe he sysem s polyop, sable a some eo po of he polyope bu wh o-eessaly sable paameezaos a he vees defg he subsysems. ey wods: sympo sably, Quada sably, Ufom sably, Covexy poblems, Uommesuae me-delay sysems, eual me- delay sysems IRODUCIO Swhg sysems ae hybd dyamal sysems omposed of subsysems wh he ow paameezaos sube o a ule ohesag he swhg law bewee he vaous subsysems. I he las yeas, hee has bee easg ees sably ad sablzao fo swhed dyam sysems [-] whee appopae swhg laws dede hough me whh subsysem paameezes he sysem so ha sably s guaaeed. I paula, swhg ules ae appled [8-] amog seveal esmao shemes of a gve lea pla whh ae he used o oba dffee me-updaed paameezaos of he adapve oolle. he swhg law ohesag he vaous esmaos o oba he ave oe whh paameezes he adapve oolle s epeed as a hghe heahal deso level of he whole adapve sysem whle he bas adapao sheme s he lowes deso level. he swhg law s desged so ha he defao eo s mmzed eal me whle he losed-loop sysem s guaaeed o be sable. ey movao fo sudyg swhed sysems s ha may paal sysems ae heely mul-model he sese ha seveal dyam subsysems desbe he whole behavo depedg o mulple evomeal faos [5]. O he ohe had, me-delay sysems offe a easg ees se may eal lfe examples ae sube o delays, le, fo sae, populao gowh models, sgal asmsso, ele-opeao poblems, wea/peae models ad auao mooed poesses wh osy sesos e., []. Delays may be lassfed as po delays o dsbued delays aodg o he aue ad as exeal (.e. he pus o oupus) ad eals (.e., he sae) aodg o he sgals hey fluee. Po delays may be ommesuae f eah delay s a ege mulple of a base delay o, moe geeally, ommesuae f hey ae abay eal umbes [-4]. he pesee of eal delays leads o a lage omplexy he esulg sysem s dyams se he whole dyamal sysem beomes fe-dmesoal. hs fa eases, addo, he dffuly he sudy of bas popees, le fo sae oollably, obsevably, sably ad sablzao ad obusess, ompaed o he delay-fee ase se he asfe fuos oss of asede umeao ad deomao quas-polyomals [-38]. By hose easos, he desg of exa o appoxmae poleplaeme oolles owads he aheveme of a fe o fe losed-loop speum beomes of eased dffuly elaed o he delay-fee ase [9-8]. eual delay sysems, whh ae hose whee he delayed me-devave fluees he sysem s dyams [,8,9], pese eve a hghe aalyss ad desg dffuly. gea effo has bee devoed o he vesgao of he behavo of me-delay sysems sldg mode ad he use of suh a popey fo syheszg appopae oolles [3-3,36-38] ludg applaos o vbaos hea exhage ubes ad aaf dyams [35,36]. mpoa po s ha dffee ypes of delays appea a aual way whe modelg dsee sysems ad some lasses of hybd sysems [39-4]. he obeve of hs sudy s o vesgae he sably ad sablzao popees of lea swhed me-delay dyam sysems sube o, geeal, mulple ommesuae ow eal po delays. Coespodg uho: M. De la Se, Isue of Reseah ad Developme of Poesses, Depame of Eley ad Eleos. Campus of Leoa. pdo. 644 de Blbao 48

2 m. J. ppled S., (): 48-49, 5 oao: * he ses R (Z), R (Z ) ad R (Z ) deoe, espevely, he ses of eal (ege) umbes, posve eal (ege) membes ad o egave eal (ege) umbes. * I s sad ha a omplex max s sly Huwza, o a sably max, f all s egevalues have egave eal pas. * ufoed lea sysem wh fe ommesuae eal po delays h of sae equao x( ) = x( ) x( h ) ɺ has wo = assoae sysems whou delays, amely: zɺ ( ) = z ( ) Whh desbes he above = so-alled ue delay-fee sysem me-delay sysem whe h = ; =, ; ad zɺ ( ) = z ( ) whh s alled he omal delay-fee sysem whh desbes he above me-delay sysem whe =, o whe h ; =,. Boh sysems have o be sable ode ha he delay sysem s a sable depede of he delays. he sysem s sad o be g.a.s. f s globally asympoally Lyapuov sable ad g.u.a.s. If s globally ufomly asympoally sable: * he l -om of a max (o veo ) M s deoed / M = λ M M. I Euldea veos, as Max suh a om odes wh he Euldea om. posve defe (semdefe) max M s deoed as M > ( M ). egave defe (semdefe) max M s deoed as M < (M ). * he oao fo he subse {,,, } of Ζ s abbevaed as. SBILIY D SBILIZIO WIH RBIRRY SWICHIG LW whee, x( ) R m p, u( ) R, y( ) R ae he - sae, m- pu ad p-oupu, espevely, ( ), ; = {,,, }, ae eal squae - maes desbg, espevely, he delay-fee dyams ad he vaous delayed dyams ad B ( ) R ad p C ( ) R ae eal ool ad oupu maes. he al odo of () s ay absoluely ouous ϕ : h, R plus, eveually, a fuo of fuo [ ] zeo measue of solaed bouded dsoues defed o [ h, ], wh x( ) = ϕ ( ) = x, whee ( ) h = Max h, wh h, beg he delays. he me fuo :[, ) = {,,..., } s a swhg fuo amog he vaous subsysems Σ defed a me by ()-() fo ()= beg paameezed wh he oespodg maes, ;, B ad C ;, Σ Σ, Σ,..., Σ fo all.he followg se s odued by oveee: S ˆ ε h = M =,,..., suh ha he. hus, { } { } swhed ufoed sysem absssa ; ( ε ) Σ s g.u.a.s. wh sably ;.e. wh all s egevalues ( ) sasfyg Max Re λ (.) = ε < ε, whee ĥ = {,,..., } S ε ( ˆ ) h h h, ay. oe, paula, ha s he se of paameezaos of he ufoed (.a) suh ha he delay- fee sysem s all he se of maes Σ s g.u.a.s ;.e.,, ; ;, suh ha ε I s sly Huwza. oe = S ˆ S ˆ ε, ε. he also ha ε, [ ] ε followg esul s oeed wh abay swhg laws whle geealzg pevous esuls [] o mulple po delays. sympo sably depede of ad depede o he delays: Cosde he me- vayg swhed lea dyam sysem: Σ ( ) : ( ) ( ) ( ) ( ) xɺ ( ) = x ( ) x h B u ( ) () y( ) = = C ( ) x ( ) () 48 heoem : he followg ems hold:. he swhed ufoed sysem Σ ( ) s g.u.a.s. wh quada sably depede of he delays fo ay abay swhg law : [, ) ad (,,..., ) ( ˆ ε ) M = S h,, hˆ, ˆ ˆ f hee exs ε R ad eal maes P = P >, S = S >,, suh ha he followg se of max osas holds:

3 m. J. ppled S., (): 48-49, 5 a = Q = P P S P S P < ε I ; (3) If (3) holds wh ε he he swhed ufoed sysem Σ ( ) s g.a.s. Wh quada sably depede of he delays fo ay : [, ). = = α h I h β P M R M P < ε I ; (7.b) oe ha f heoem () holds he heoem () holds wh h = ( ). he, he sably depeds o he delays may be heed fom (7.b) o esablsh a age of maxmum allowable delays.. he swhed ufoed sysem Σ ( ) s g.u.a.s. wh quada sably fo ay abay swhg law (), ad ay delays h eal, h,, f hee exs ε R ad maes P = P >, S = S >, S = S >,,, suh ha he followg se of max osas holds: Q a = P P = = = = h S h P M R M P < ε I (4) Whee: (,,..., ) M = ; (5) (,,..., ) R = Blo Dag S S S m ( ) m ( ) ; R (6) If (4) holds wh ε he he swhed ufoed sysem Σ ( ) s g.a.s. wh quada sably depede of he delays fo ay : [, ) ad ay delays h, h,. Rema : oe ha f (3)-(4) hold fo some ses of ( ) maes S, S,, he hey also hold fo some eal salas α,. hus, heoem () holds f: > β wh α β, ( α β ) = P P I P S P < ε I; (7.a) P P = = sympo sablzao depede of ad depede o he delays: he esuls of Seo. may be appled o foed sablzable sysems f a sablzg egulao ool law s appled. he dsusso s lmed o sae ad oupu lea feedba. he fs defos ae fs gve. Defos : he swhed ufoed sysem Σ ( ) s sad o be globally ufomly asympoally sablzable (g.u.a.s.) [espevely, globally ufomly asympoally oupu sablzable (g.u.a.o.s.)] wh quada sably va a lea delay-fee ool law fo ay abay swhg law : [, ) f hee s a lea egulao feedba ool law x,,..., u ) = ( wh ( ) { } [espevely, a oupu egulao feedba ool law u = () C ( ) x ] fo some eal maes p R [espevely, R ] ; suh ha he losed-loop sysem: Σ ( ) : ( ( ) ( ) ( ) ) ( ) ( ) = xɺ ( ) = B C x( ) x h [espevely: x( ) = ( B C ) x( ) ( ) x( h ) = ɺ ] (8) s g.u.a.s. Wh quada sably. he followg esul whose poof s omed holds. heoem : he followg ems hold:. ssume ha he swhed ufoed sysem Σ ( ) s o g.u.a.s. fo all delays h, h some h >,, fo a abay swhg law : [, ) (. e. hee s a oempy se suh ha s o sly Huwza fo. he, a eessay ad suffe odo fo he foed Σ ( ) o be g.u.a.s.

4 (g.u.a.o.s.) fo all delays h >, h m. J. ppled S., (): 48-49, 5, h some, s ha he pa ( ) sablzable (espevely, he ple (, B, C ), B be be sablzable ad deeable) fo all. hose odos guaaee ha he swhed foed sysem Σ s g.u.a.s. (espevely, g.u.a.o.s.) ( ) depede of he delays povded ha, s suffely small. ;. ssume ha he swhed ufoed sysem Σ ( ) s g.u.a.s. fo abay swhg law : [, ) h, h fo all delays [ ] some fe h >,. he hee s a (ouque) lea egulao sae-feedba ool law u ( ) = () x ( ) wh,,..., fo some eal { } maes R [espevely, a (ouque) lea egulaed oupu-feedba ool law u ( ) = () C ( ) x ( ) wh p R ]; suh ha he esulg losed-loop sysem Σ ( ) s g.u.a.s. wh quada sably fo all delays h, h some fe h > h,, beg depede, s o he paameezao, f B ompleely oollable (espevely, he ple (, B, C ) s ompleely oollable ad Max ) fo all. obsevable ad ( m, p ) Defos may be geealzed a aual way fo lea sae ad oupu- sablzably va lea egulao delay- depede ool laws [-3,3,33]. Fo ha pupose, osde he followg ool laws: = ( ) ( ) u x x h (9) = = ( ) ( ) ( ) u C x C x h () = wh {,,..., } {,,..., },, ( ) {,,..., } { } ad ( ) ( ), ( ),..., ( ) 484 m ( ( ), ) fo eal maes R, m R m p, R m p ad R ;, spefyg he oolle gas. Defe x eal maes: Â = B ; Â = B C ; () he followg ehal esul, oeed wh he hoe of he oolle gas oespodg o he delayed dyams f he ool laws (9) o (), so ha he losed-loop delayed dyams s ahlaed o ealy ahlaed. Lemma :. If (, B ) s a ompleely oollable pa he always exss a R suh ha all he zeos of he polyomal p s = De s I B (o, equvalely, of s oeffes) ae loaed abay pefxed posos., B, C s a ompleely oollable. If ad obsevable ple he always exss a m p R suh ha all he zeos of he polyomal p s = De s I B C (o, equvalely, of s oeffes) ae loaed abaly lose o a gve se of pefxed posos.. If a ( B ) = m < he fo ay gve eal max * Â, exss a uque B * B ˆ B = gves he * mmum value o ˆ ˆ ( ). v. If a ( B ) = m he fo ay pesbed eal max suh ha R, exss a uque R * Â * Â = Â. v. If a ( B ) ( m, p ) = m ad a ( C ) = p wh Max he fo ay gve eal max * Â, exss a uque max: ( ˆ B B ) B C ( C C ) * = () suh ha gves he mmum value of ˆ * ˆ.

5 v. If a ( B ) = a ( C ) ( m, p ) m. J. ppled S., (): 48-49, 5 = wh Max he fo ay gve eal max * Â, exss a uque max = Blo Dag, wh * R * defed by: ( ˆ ˆ ) * * = B B C C (3) gves exa mahg ˆ = ˆ (pefxed abaly) * whee max paos [ B, B ] [ C, C ] C = ae used wh B = ad B, C beg squae eal -maes all. Lemma mgh be used ombed wh heoem he sese ha a ool law volvg delays allows o edue he om of  assoaed wh he delayed dyams afe feedba (defed by oe of he wo equaos () depedg o he use of sae o oupu feedba) elaed o ha of ude he vaous gve odos of oollably/ obsevably. hs allows o aomplsh wh he odos of losed-loop asympo sably depede of he delays o o ease he sze of he maxmum allowable delay guaaeeg losed-loop asympo sably va sae/oupu feedba. SYMPOIC SBILIY D SBILIZIO WIH SWICHIG LW MOG HE VERICES OF POLYOPIC SYSEM he ma esul of hs seo follows below. heoem 3: he followg wo ems hold:. ssume ha hee exs squae eal -maes P P = >, S = S ( ), ε R ad > a eal sala (, ) suh ha (3), λ fulfllg λ =, λ a wh a = Q = Q < Q defed. hus, fo ay al odo, hee s a (o-uque) swhg law : [, ) whh s peewse osa o [ α, ) fo some eal osa α, suh ha he swhed ufoed sysem Σ ( ) s g.u.a.s. wh quada sably depede of he delays. If Q fo he se of 485 ufoed sysems Σ,, he always exss a (o-uque) sablzg swhg law, whh s pee-wse osa o : [ ) [, ) α, some eal α, suh ha he swhed foed sysem Σ ( ) s g.u.a.s. wh quada sably depede of he delays povded ha ay of he odos below hold:, s sablzable fo all ad a. he pa ( B ) a ool law u ( ) x ( ) wh {,,..., } gas m R s appled. b. he ple (, B, C ) deeable fo all = s geeaed,, fo oolle s sablzable / ad a oupu-feedba ool law u ( ) C x ( ) = s geeaed, wh {,,..., } gas m p R., fo oolle Fuhemoe, f Eo! Booma o defed. fo he se of ufoed sysems Σ,, bu hee exss a se of maes salas (, ) R, ad a se of eal λ fulfllg λ = suh ha Q = λ Q a < fom, wh = Q a beg edefed Q a by eplag B ( ), he he swhed losed-loop sysem Σ ( ) fo he ool law u ( ) x ( ) = ad some sablzg swhg law : [, ) s g.u.a.s. wh quada sably depede of he delays (eve f λ, λ B, s o sablzable (, B ) ad he ( ) fo all ).. ssume ha hee exss squae symme posve defe maes P, S, S (, ) ad some eal salas λ (, ) fulfllg Q Q a wh a = = λ < λ =, suh ha Q defed (4)-(5),. hus, fo ay al odos, hee s a (o-uque) swhg law : [, ), whh s pee-wse osa o [ α, ) some eal α, suh ha he swhed ufoed sysem Σ ( ) s g.u.a.s. wh quada sably depede

6 of he delays,. If Σ, m. J. ppled S., (): 48-49, 5 h, h, some h >, all Q fo he se of ufoed sysems, he always exss a (o-uque) sablzg swhg law : [ ) pee-wse osa o [, ), whh s α some eal α, suh ha he swhed losed-loop sysem Σ ( ) s g.u.a.s. wh quada sably depede of he delays, h, h, some h >, all, povded ha smla odos as (a)-(b) Iem () hold. Fuhemoe, fo ay pefxed se h, always exs maes ( ) (, ) h ( ) γ (, ) ad a posve eal osa γ depede o small sysems suh ha fo suffely, he swhed Σ ( ) obaed ude lea sae feedba va a oolle of ga {,,..., } s g.u.a.s. wh quada sably depede of he delays, h, h fo some swhg law : [, ). Fuhemoe, f Q fo he se of ufoed Σ, P R, (, ), bu hee exss a se of maes ad a se of eal salas λ fulfllg λ = Q fom = λ Q a <, wh = a suh ha Q beg edefed Q a by eplag B C Σ ( ), he he swhed losed-loop sysem fo he ool law u ( ) C x ( ) = ad some sablzg swhg law : [, ) s g.u.a.s. wh quada sably depede of he delays (eve f, B C λ, λ B, λ C, s o ( ) ad he sablzable ad deeable fo all ). epeao of heoem 3 s as follows. he odos of heoem 3 () mply ha f a polyop sysem: x() ɺ = λ x() x ( h ) B u() ; = = λ (, ), λ = (4) = 486 s g.u.a.s. fo zeo ool pu guaaeed by he odo = λ Q < ;.e., s g.u.a.s. depede a of he delays a some eo po of he polyope beg he se defed by some ombao of vees defed by he maes Q ( ) he hee s a swhg law : [, ) suh ha he oespodg ufoed swhed Σ ( ) s g.u.a.s. he same dea mgh be exeded by swhg losed-loop sysem fo he sae o oupu lea feedba ude he oespodg modfaos gve odos as well as fo sably depede of he delays. ha meas, oughly speag, ha sably a a po sde he polyope mples sably a ay eo po of he polyope (fo some swhg law) eve f he sysem s o sable a ay veex. oe ha Q > Q a < Q > Q some a a < bu oly x ( ) Q x ( ) < x ( ) Q x ( ) < a all ozeo x () ad all ( [,4-6,8] fo a delay-fee sysem). I s ow eesg o vesgae quada sably of a swhed sysem omposed of wo subsysems wh a (uea) polyop-ype paameezao whh ae o eessaly sable. ssume ha he swhed ufoed sysem Σ Σ, Σ s defed fo all by oe of ( ) { } he wo subseque subsysems: Σ : xɺ = x x h ; =, (5) = a fo ssume ha Σ (=,) ae uea polyop sysems defed by: = µ ; () = µ ( ) (6) () wh salas (, ) µ ;, =, sube o µ ; =, ; ad eal squae -maes ad ;,, ; defg he delay-fee ad ( ) delayed dyams a he vees. Fo smply of exposo ad mahemaal poofs, s assumed he sequel whou loss of geealy ha he umbe of exeme pos = ; =,. I ode o mae he subseque dsusso oval, he followg assumpo s made. ssumpo : Boh Σ, =, ae o quadaally sable h, h, some h >, all =, ;.e.

7 m. J. ppled S., (): 48-49, 5 hee does o exs eal squae symme -maes P > ( =,) suh ha: ( ) ( ) ( ) ( ) P P < ; = =,, (7) he max equaly (7) holds whe fo eah () ( ) =,, hee a leas oe max fo = = o whh s o sly Huwza so ha he polyop sysem Σ s o quadaally sable fo =, a he oespodg veex fo zeo delays. By ouy of he haaes oos, hee s some eghbohood of values of delays aoud zeo suh ha he oespodg polyop sysem s o quadaally sable ;.e. hee exs h suh ha Σ (=,) s o quadaally sable fo all h, h. oe ha f ssumpo holds he heoem ao be appled beause of he sably o al sably a he vees. he followg esul, whose poof s omed, s elaed o he sablzao of (5) va swhg., ; maes S > λ = ; =, ad eal squae - > P = P >, S = S, = S (, ) suh ha : λ λ P ( P λ ) λ h S = λ h P M R M P ( ) ( ) ( λ ) < ;, =, (9) h P M R M P ε I SYMPOIC SBILIY OF CLSS OF UFORCED EURL SYSEMS heoem 4: he followg wo ems hold ude ssumpo :. he swhed sysem Σ ( ) s quadaally sable depede of he delays, va some o-uque =,, f hee λ swhg law : [, ) { } exs osa eal salas >, ; maes ha: ε, (, ) λ ;, λ = ; =, ad eal squae - P = P >, ( λ ) ( λ ) ( ) P P λ ( ) S = S > ( ) suh λ S P S P < ε I ( λ ) S P S P ;, =, (8). he swhed sysem Σ ( ) s quadaally sable fo all delays h, h, some h > ( ) va some swhg law : [, ), f hee exs osa eal salas > ε, (, ) sadad lass of ufoed eual sysems volvg a sgle po delay s ow foused o [,8]. he exesos he ases of mulple po delays ad egulag pus ae de by usg de exesos wh he ools of hs lass of sysems. hey ae omed by he sae of smply. Cosde he eual sysem: xɺ x Σ : = ( ) ( ) ( ) ( ) ɺ ( ) x h D x h = ɶ x ɶ x h D ( ) ( ) ( ) ɺ xɺ h W x τ d τ h (.a) (.b) whee, he fuo of al odos s ay ϕ : h, R plus, absoluely ouous fuo [ ] eveually, a fuo of zeo measue of solaed h,, wh bouded dsoues defed o [ ] x( ) =φ ( ) = x ad W ( ) s hose so ha ɶ = ( ) W ( ) s sly Huwza fo all ( ) ad ~ ( ) = ( ) W ( ) wh : [, ) beg he swhg law. ll he maes of paamees () ae squae eal of ode wh: λ ;, ; 487 {,,..., }

8 ,,..., { } { D, D,..., D } ( ) { } D ; W W, W,..., W () fo all. Se ɶ ( ) s sly Huwza fo all, ( W ) s sly Huwza fo all. he followg esul s elaed o he sably of () va swhg: heoem 5: he eual sysem Σ ( ) s g.u.a.s. fo all delays [, h ] h ad some h > fo ay abay swhg law : [, ), f D < ad W s sly Huwza ( ), povded ha hee exs squae eal posve defe symme - maes P, S, R ad suh ha: ( ) Π = Blo Max Π ;, =,3 < () ( W ) P P ( W ) Π = R S h h P W W P P ( S h ) D Π = Π = P W S h ; Π 3 = Π 3 = all (3) m. J. ppled S., (): 48-49, 5 he poof of he subseque esul follows mmedaely fom heoem 5, va Shus ompleme of he las blo max of Π = Blo Max Π ;, =,3 <, he equvale ( ) es fo egave defe [43]. Coollay : heoem 5 holds f: Π = Blo Max ( Π ;, =,3 ) < ( ) ( ) ( ) - ( M, M 3 ) ( Π ) ( M, M 3 ) < W P P W R S h all (4) ow, he polyop suues of heoems 4-5 ae exeded fo he gve lass of swhg laws whh odoally sablze he swhed sysem ude ovexy-ype osas. he exesos of all he esuls hs seo o he ase of foed sysems ae dely obaed by usg lea sae/oupu feedba laws ude ehe oollably/obsevably o sablzably/deeably assumpos of he appopae paameezaos of he subsysems. Some sably esuls fo he eual sysem of hs seo based o he popees of s subsysems ae summazed he followg esul. 488 heoem 6: he followg ems hold:. he eual sysem Σ ( ) s g.u.a.s. fo all delays h, h ad some h > fo some (ouque) swhg law : [, ), f D < ad ( W ) ( ) s sly Huwza, povded ha hee exs squae eal symme -maes P, S, R ad ad some se of eal salas λ sasfyg λ =, suh ha = λ, wh he maes = Π = Π < Π ( ) beg defed ()-(3).. ssume ha Σ ( ) { Σ, Σ } fo ay swhg law :[, ) whee Σ = = {, } ad all me, beg defed by () whe (), ae o quadaally sable wh ɶ = W s sly Huwza, ɶ = W ad D < fo =,. ssume also ha Σ (=,) ae uea polyop sysems defed by: = µ ; () = µ ( ) () () (5.a) W = µ W ; () D = µ D ( )(5.b) va eal salas µ (, ) ;, =, sube o µ ; =,. he, hee s a o-uque swhg law : [, ) suh ha Σ ( ) s g.u.a.s. fo all delays h, h ad some h > povded ha he fou subseque lea max osas hold fo some squae eal symme - maes P, S, R ad ad some se of eal salas λ > sasfyg ( ) λ = ; =, : Π, = λ Π λ Π < ;, =,, (6) ( ) ( ) whee Π s defed smlaly as Π ()-() wh he followg eplaemes elaed o heoem 5: (, W,, D ) (, W,, D ) ; =,

9 oe ha f all he paameezaos Σ ae o sable he λ (,) fo all heoem () se Π = Π < s mpossble f λ =, λ = fo some. Fo heoem 6() he above osa has o hold as well by he same easos ha hose poed ou elaed o heoem 4. he exeso of heoem 6 () o geeal paameezaos defed by (5) wh 3 ; =, s de by usg moe osa (6) by volvg he oespodg eessay osas he same way as heoem 4 s exedable o hs suao. he exeso s omed fo he sae of smply. COCLUSIO hs sudy has bee devoed o vesgae he sably ad sablzao popees of lea swhed me-delay dyam sysems beg sube o, geeal, mulple ommesuae ow eal po delays. Fsly, he ufom asympo quada Lyapuov sably (boh depede of ad depede o he delays) fo ufoed sysems has bee vesgaed ude abay swhg laws opolyop sysems paameezed by a fe se of sable subsysems. he esuls have bee exeded o pove he exsee of sablzg swhg laws polyop sysems ude esable ovexy-ype odos fo he vees. Fuhe sably esuls have bee deved fom foed sysems fo lea sae/oupu feedba ool laws ude ea oollably ad obsevably/sablzably ad deeably odos. he sably esuls have bee also exeded o a lass of swhed polyop sysems ad wo swhed sysems ossg of a se of polyop subsysems, sube o muual swhgs hough me, whh fulfll a ovexy-ype odo by eah ombao of ses of vees, oe oespodg o eah polyope. umeal smulaed examples have ooboaed some of he obaed esuls. COWLEDGMES he auhos ae vey gaeful o MEC ad UPV/EHU by he paal suppo of hs wo hough Gas DPI 3-64 ad 9/UPV.I6.I6-563/3, espevely. hey ae also gaeful o M. Ibeas fo hs eesg ommes abou he sube. REFERECES. aeda,.s. ad V. Balahsa, 994. ommo Lyapuov fuo fo sable LI sysems wh ommug -maes. IEEE as. uoma. Co., 39: m. J. ppled S., (): 48-49, Ws, M., P. Pelees ad R. DeCalo, 998. Swhed oolle syhess fo he quada sablzao of a pa of usable lea sysems. Eu. J. Cool, 4: Xu, X. ad P.J. sals, 4. Sablzao of seod-ode LI swhed sysems. Il. J. Cool, 73: Su, Z. ad S.S. Ge, 3. Dyam oupu feedba sablzao of a lass of swhed lea sysems. IEEE as. O Cus ad Sysems I- Fudameal heoy ad pplao, 5: Zha, G., H. L ad P. sals, 3. Quada sablzably of swhed lea sysems wh polyop ueaes. Il. J. Cool, Mah 4, 76: Su, Z., 4. obus sablzg law fo swhed lea sysems. I. J. Cool, 77: Wag, F. ad V. Balasha,. Impoved sably aalyss ad ga-sheduled oolle syhess fo paamee-depede sysems. IEEE as. uomae. Co., 47: Bao,., 4. Sablzao by meas of sae spae depedg swhg ules. Sysems ad Cool Le., 53: loso-quesada, S. ad M.D.L. Se, 4. Robus adapve ool wh mulple esmao models fo sablzao of a lass of o-vesely sable me-vayg plas. sa J. Cool, 6: Ibeas,., M. De la Se ad S. loso- Quesada, 4. Sable mul-esmao model fo sglepu sgle-oupu dsee adapve ool sysems. I. J. Sysems S., 35: ulesu, S.I.,. Delay Effes o Sably. Robus Cool ppoah. Leue oes Cool ad Ifomao, sees o. 69 (M. homa ad M. Moa Eds.), Spge-Velag, Bel.. Xe, G. ad L. Wag, 4. Sably ad sablzao of swhed lea sysems wh sae delay: ouous- me ase. 6h Il. Symp. I Mahemaal heoy of ewos ad Sysems, aholee Uvesa Leuve, Belgum, MS 4, Leuve, July 5-9, Sesso Swhed Lea Sysems I, M. 3. Jugo, J. ad M. de la Se,. Ipu-oupu based pole-plaeme a lass of me-delay sysems. IEE Po. Cool heoy ad ppl., 49: De la Se, M., 4. O pole-plaeme oolles fo lea me-delay sysems wh ommesuae delays, Mahemaal Poblems Egeeg ( pess, 4). 5. De la Se, M.,. lloao of poles of delayed sysems edued o hose assoaed wh he udelayed ouepas. Ele. Le., 36: De la Se, M., 3. O he asympo hypesably of dyam sysems wh po delays. IEEE as. C. Sys. I- Fudameal heoy pplaos, 5:

10 m. J. ppled S., (): 48-49, 5 7. Galows,.,. Sae-spae ealzaos of lea -D sysems wh exesos o he geeal D ( > ) ase. Leue oes Cool ad Ifomao Sees, Vol. 63,. 8. Rhad, J.P., 3. me-delay sysems: a ovevew of some ee advaes ad ope poblems. uoma, 38: Yu, Y., 983. O sablzg uea lea delay sysems. J. Opmzao heoy ad pplaos, 4: Mahmoud, M.S. ad.f. lmuha, 994. Quada sablzao of ouous-me sysems wh sae-delay ad om-bouded me-vayg ueaes. IEEE as. uomae. Co., 39: Cho, H.H. ad M.J. Chug, 995. Memoyless sablzao of uea dyam- sysems wh me-vayg delayed saes ad ools. uoma, 3: L, X. ad C.E. desouza, 997. Cea fo obus sably ad sablzao of uea lea sysems wh sae delay. uoma, 33: Luo,.S., M. De la Se ad J. Rodella, 997. Robus sablzao of a lass of uea me delay sysems sldg mode. I. J. Robus ad olea Cool, 76: El-hazal, R., 998. Vaable suue obus ool of uea me-delay sysems. uoma, 34: Rhad, J.P., F. Gouasbau ad W. Peuque,. Sldg mode ool he pesee of delay, ybeea. 37: Olov, Y., W. Peuque ad J.P. Rhad, 3. Sldg mode ool syhess of uea medelay sysems. sa J. Cool, 5: Ya, J.J., 3. Sldg mode ool desg fo uea me-delay sysems subeed o a lass of olea pus. Il. J. olea ad Robus Cool, 3: Che, J.J., 4. ew sably ea fo a lass of eual sysems wh dsee ad dsbued medelays: a LMI appoah. ppled Mahemas ad Compuao, 5: u, Y., J. Lam ad X. Wag, 4. Sldg-mode ool fo uea eual delay sysems. IEE Po. Cool heoy ad pp., 5: u, Y., J. Lam, X. Wag ad D.W.C. Ho, 4. Obseve-based sldg mode ool of olea sae-delayed sysems. Il. J. Sysems S., 35: Hsu,.C., 999. Deealzed sldg mode ool of lage-sale me-delayed sysems wh sees oleaes. J. Dyam Sysems, Measueme ad Cool- as. Of SME, : Ouheah, S., 3. Expoeal sablzao of lea delayed sysems usg sldg- mode oolles. IEEE as. uomae. Co., 5: De la Se, M., 4. Suffey-ype sably ad sablzao ea fo lea me-vaa sysems wh osa po delay. a ppladae Mahemaae, 83: De la Se, M. ad.s. Luo, 4. O he ufom expoeal sably of a wde lass of lea medelay sysems. J. Mahemaal alyss ad pplao, 89: De Bedou, J.M., M.. Fahe ad.. Baa, 999. Robus ool of hao vbaos fo mpag hea exhage ubes oss-flow. J. Soud ad Vbao, 7: Jafaov, E.M. ad R. asal, 3. Robus sldg- mode ool fo he uea MIMO aaf model F-8. IEEE as. eospae ad Eleo Sysems, 36: 7-4 (). IEEE as. uomae. Co., 5: Zheg, F. ad P.M. Fa,. Fedmesoal vaable suue ool desg fo sysems wh a sgle me delay. IEEE as. eospae ad Eleo Sysems, 36: Zheg, F. ad P.M. Fa,. Fedmesoal vaable suue ool desg fo dsbued delay sysems. Il. J. Cool, 74: De la Se, M., 997. O he hypesably of a lass of hybd sysems. I. J. Sysems S., 8: Baea, R., I. Sagasabea ad M. De la Se,. Impovg he sably popees of he zeos of sampled daa sysems wh faoal ode hold. Po. IEEE Cool heoy ad ppl., 47: Lag, S., M. Ishob ad Q. Zhu, 3. Impoveme of sably of zeos dsee-me mulvaable sysems usg faoal ode hold. I. J. Cool, 76: edle, E. ad. Jameso, 97. Codos fo oegaveess of paoed maes. IEEE as. uomae. Co., 7: Zha, X.,. Max Iequales, Leue oes Mahemas Sees, o. 79, Spge-Velag, Bel. 49