CHARACTERIZATIONS OF THE NON-UNIFORM IN TIME ISS PROPERTY AND APPLICATIONS

CHARACTERIZATIONS OF THE NON-UNIFORM IN TIME ISS PROPERTY AND APPLICATIONS I. Karafyllis ad J. Tsiias Depare of Maheaics, Naioal Techical Uiversiy of Ahes, Zografou Capus 578, Ahes, Greece Eail: jsi@ceral.ua.gr. Iroducio Absrac For ie-varyig corol syses various characerizaios of he o-uifor i ie ipu-o-sae sabiliy propery (ISS are provided. These characerizaios eable us o derive sufficie codiios for ISS ad feedback sabilizaio for coposie syses. This paper cosiues a coiuaio of auhors works [5,6,7,8,9] o he cocep of ouifor Ipu-o-Sae Sabiliy (ISS ad is applicabiliy o sabiliy ad feedback sabilizaio of ie-varyig syses: = f (, R, u R f ( where is of class C R R R ; R ad saisfies f (,, =.,, beig ally Lipschiz wih respec o (. ( I Secio we firs provide he defiiios of he o-uifor i ie Robus Global Asypoic Sabiliy (RGAS ad Ipu-o-Sae Sabiliy (ISS as give i [6]. I should be poied ou ha he cocep of o-uifor i ie ISS as proposed i [6] eeds he ISS propery as described i [6] for he auooous case. I Secio 3 we give he o-uifor i ie eesio of he failiar oio of uifor i ie ISS as proposed by Soag i [] (see also [-4] cocerig auooous syses ad we provide i Proposiio 3. equivalece bewee his oio ad he cocep of ISS as suggesed i [6]. Furher equivale characerizaios of ISS are also give i Proposiio 3. ad liks bewee ISS ad he coceps of CICS (Covergig-Ipu-Covergig-Sae ad BIBS (Bouded-Ipu- Bouded-Sae are provided i Corollary 3.4. Aoher ipora cosequece of Proposiio 3. is Corollary 3.5 cocerig auooous syses: = f ( R, u R (. Paricularly, we prove ha, if (. is forward coplee ad saisfies he -GAS propery, aely, R is GAS wih respec o he uforced syse: = f ( (.3

he (. saisfies he (o-uifor i ie ISS propery. I should be ephasized here ha, as proved i [], syse (. uder he sae hypoheses fails i geeral o saisfy he uifor i ie ISS propery. I Secio 4 we provide sufficie codiios for he (o-uifor i ie ISS for coposie ie-varyig syses: = f (, y, (.4a y = g(, y, (.4b k R, y R, u R, where f (,,, =, g(,,, = ad we assue ha f ad g are C appigs beig ally Lipschiz wih respec o ( y,. I is kow (see for isace [3,4,5] ha for auooous syses he uifor ISS for (.4a wih ( y, as ipu ad for (.4b wih ( as ipu, respecively, leads o a siple characerizaio of a sufficie codiio uder which he overall syse saisfies he ISS propery fro he ipu u. For he ie-varyig case (.4 Theore 4. assers ha a se of addiioal codiios cocerig qualiaive behavior of he soluios of (.4a ad (.4b guaraees ISS for he overall syse (.4. For he paricular case of syses (.4 whe g( is idepede of, aely for he cascade iercoecio: = f (, y, (.5a y = g(, y, (.5b k R, y R, u R, he sufficie codiios of Theore 4. are cosiderably siplified. Paricularly, Corollary 4. provides sufficie codiios for ISS for he case (.5 ad geeralizes a well-kow resul cocerig he auooous case. Fially, by eploiig he oio of o-uifor i ie Ipu-o-Oupu Sabiliy (IOS, we sudy he proble of he propagaio of he o-uifor i ie ISS propery hrough oe iegraor. Specifically, cosider he syse: = f (, y (.6a y = g(, y d(, y u R, y R,, u R (.6b We assue ha he dyaics f, g, d are C, ally Lipschiz wih respec o ( y ad saisfy f (,, =, g(,, =. Theore 4.3 provides sufficie codiios for he eisece of a oupu saic feedback sabilizer: u = k(, y v (.7 ha reders he closed-loop syse (.6 wih (.7 (o-uiforly ISS wih v as ipu.

. Review of he Noio of RGAS ad he No-Uifor i Tie ISS Propery I his secio we provide he defiiios of o-uifor i ie Robus Global Asypoic Sabiliy (RGAS ad ISS as precisely give i [6]. We cosider ie-varyig syses: = f (, d R,, d D (. where D R is a copac se, f R : R D R is a C ap beig ally Lipschiz wih respec o R ad saisfies f (,, d = for all he se of all easurable fucios fro o. R D (, d R D. By M we deoe D Defiiio. We say ha zero R is (o-uiforly i ie Robusly Globally Asypoically Sable (RGAS for (., if for every, d M D ad R he correspodig soluio ( of (. eiss for all ad saisfies he followig properies: P (Sabiliy. For every ε >, T, i holds ha ad here eiss a δ : = δ ( ε, T > such ha { ( : d M,,, [, ] } < sup D ε T (.a δ, [, T ] ( ε,, d M D (.b P (Araciviy. For every ε >, T ad R, here eiss a τ : = τ ( ε, T, R, such ha R, [, T ] ( ε, τ, d M D (.c Defiiio. We say ha syse (. saisfies he -GAS propery, if R is GAS for (. wih D ={}, or equivalely, for he uforced syse = f (,. Defiiio. Cosider syse (. ad le γ ( (, s : R be a C fucio, ally Lipschiz i s, such ha for each fied he appig γ (, is posiive defiie. We say ha (. saisfies he weak (o-uifor i ie Ipu-o-Sae Sabiliy propery (wiss wih gai γ (, if each soluio ( = (,, ; of (. eiss for all ad saisfies Properies P ad P of Defiiio., provided ha R γ (, (, a.e. for (.3 3

If i addiio for each he fucio γ (, is of class K, he we say ha (. saisfies he (o-uifor i ie Ipu-o-Sae Sabiliy propery (ISS wih gai γ (. The followig proposiio suarizes soe useful equivale descripios of he ISS propery. Proposiio.4 ([6] Le γ ( (, s : R be a C fucio, ally Lipschiz i s, such ha for each fied he appig γ (, is a posiive defiie fucio. The he followig saees are equivale: R (i Syse (. saisfies he wiss propery wih gai γ (. (ii R is o-uiforly i ie RGAS for he syse: = f (, γ (, d R, d B[,] R, (.4 where B[,] deoes he closed sphere of radius aroud R. (iii There eis a fucio σ of class KL ad a fucio β : R R of class he followig propery holds for all : τ γ ( τ, ( τ a.e. for [, ] τ ( ( β (, K such ha σ (.5 (iv There eis a C fucio V R R R, fucios a, : ha he followig hold for all (, R R R : a ad β such u a ( V (, a ( β ( (.6a γ (, V (, V (, (.6b (. Reark.5: The precise descripio of propery (iii, which is give i [6], is as follows: (v There eis fucios a, a of class K ad a fucio ha he followig propery holds for all : ~ : R R β of class K such ~ τ γ ( τ, ( τ a.e. for τ [, ] a ( ep( ( β ( a ( (.7 I ca be easily esablished ha (.5 ad (.7 are equivale. ( The followig proposiio eeds a well-kow resul cocerig he auooous case (see for isace [4,7]. 4

Proposiio.6 Syse (. saisfies he wiss propery fro he ipu u, if ad oly if i saisfies he -GAS propery. The proof of Proposiio.6 is a iediae cosequece of he followig lea, which is a direc eesio of Lea IV. i [] ad cosiues a powerful ool for he resuls of e secio. Lea.7 Cosider he syse (. ad suppose ha i saisfies he -GAS propery. The, for every fucio µ ( of class K, here eiss a C ap V : R R R, fucios a i ( ( i =,...,4 of class a K ad p (, κ ( of class K, such ha V (, a ( p(, (, R R (.8 ( V V (, (, f (, V (, ep( a3 a4 ( κ ( u µ ( (, R R R (.9 Siilarly wih ISS we ay eed he oio of uifor i ie Ipu-o-Oupu Sabiliy (IOS (as proposed i [] o he o-uifor case. Defiiio.8 Cosider he syse (. wih oupu y = h(,, where he ap h( is of class C ( R R ; R k ad saisfies h(, =. We say ha (. is (o-uiforly i ie Robusly Globally Asypoically Oupu Sable (RGAOS, if for every, d M ad D R he correspodig soluio ( of (. eiss for all ad saisfies he followig properies: P (Oupu Sabiliy. For every ε >, T, i holds ha { (, ( : d M,,, [, ] } < sup h D ε T (.a ad here eiss a δ : = δ ( ε, T > such ha δ, [, T ] h(, ( ε,, d M D (.b P (Oupu Araciviy. For every ε >, T ad R, here eiss a τ : = τ ( ε, T, R, such ha R, [, T ] h(, ( ε, τ, d M D (.c Defiiio.9 Cosider syse (. wih oupu y = h(,, where h ( h( C k wih (, = ad le γ (, s : R be a fucio, ally Lipschiz i s, such ha for each fied he appig γ (, is of class K. We say ha (. saisfies he R C ( R R ; R 5

(o-uifor i ie Ipu-o-Oupu Sabiliy propery (IOS wih gai γ (, if each soluio ( = (,, ; of (. eiss for all ad saisfies Properies P ad P of Defiiio.8, provided ha (.3 holds. 3. Characerizaios of he No-Uifor i Tie ISS Propery The followig proposiio provides equivale characerizaios of he (o-uifor i ie ISS propery. Proposiio 3. The followig saees are equivale: (i Syse (. saisfies he ISS propery. (ii There eis fucios ρ(, φ(, β ( ad σ ( L such ha ρ ( φ( ( a.e. for ( ( σ β (,, (3. ρ (, φ(, β ( ad ( R, R ad for every ipu u ( ( = (iii There eis fucios = of class L ([, soluio of (. wih eiss for all ad saisfies: σ ( L such ha, for every, he correspodig ( a σ ( β ( σ ( β ( τ ρ( φ τ τ τ,, sup (, (3. τ (iv There eis fucios ζ (, β (, δ ( ad σ ( L such ha, for every R (, R ad for every ipu u L [,, he correspodig soluio ( of (. wih ( = eiss for all ad saisfies: = of class ( ( a σ ( β (,, sup ζ ( δ ( τ τ (3.3 τ (v There eis a fucio δ ( such ha R θ (, beig ally Lipschiz o R, ad a fucio is RGAS for he syse: θ ( = f (, d, d ( M B[, ] (3.4 δ ( (vi Syse (. saisfies he -GAS Propery ad here eis fucios σ (, ζ ( δ (, µ (, β ( L ([, ad saisfies: such ha for every, R R ad u = of class he correspodig soluio ( ( of (. wih ( = eiss for all 6

( µ ( σ ( β ( sup ζ ( δ ( τ τ τ (3.5 (vii There eis a C fucio V : R R R, fucios a, a (, a ( of class ad fucios p (, q( of class a K such ha for all (, ( 3 R R R we have: V (, a ( p( (3.6a ( K V V (, (, f (, V (, a3 ( q( u (3.6b Reark 3.: Whe he fucios φ ( ad β ( are bouded, he (3. is equivale o he uifor i ie ISS propery, as give i [6,7]. Likewise, whe β ( ad δ ( are bouded, he (3.3 is equivale o ISS propery, as origially proposed i [] by Soag. The equivalece bewee (3. ad (3.3 geeralizes he well kow fac ha for he auooous case ad whe γ ( is idepede of, aely, γ ( is of class K, he uifor i ie ISS propery as give by Soag is equivale o he correspodig characerizaio give i [6,7]. Fially, we oe ha, whe p ( ad q( are bouded, he (3.6a,b coicides wih he Lyapuov characerizaio i [] for he uifor i ie ISS propery. Reark 3.3: By eploiig he resul of Proposiio 3. ad paricularly he equivalece bewee (i ad (iv, i ca be esablished ha he oio of o-uifor i ie ISS reais ivaria uder: τ S Scalig of ie = a( s ds, where a ( C ( R wih a( s >, for all s ad a( s ds =. S Trasforaios = Φ(, z of he sae, where Φ ( C ( R R ; R, wih Φ Φ(, =, Φ (, R = R ad de (, z for all z here eiss a pair of fucios a i ( ( =, (, z R R. (, z R R i wih ( z Φ(, z a ( z a ad such ha for all S3 Ipu rasforaios u = q(, v, where q( C ( R R ; R saisfyig q(, = for all, ad i such a way ha here eiss a fucio q ( C ( R R ; R wih u = q(, q (, respec o v R. for all (, R R ad such ha f (, q(, v is ally Lipschiz wih 7

I should be ephasized here ha he uifor i ie ISS, does o i geeral reai ivaria uder S or S3. A iediae cosequece of Proposiio 3. is he followig corollary, which provides liks bewee o-uifor i ie ISS ad he coceps of BIBS ad CICS. Corollary 3.4 Suppose ha syse (. saisfies he ISS propery, ad i paricular assue ha (3.3 holds for cerai fucios ζ, σ L, β, δ. Le u = be a ipu of ( class L R such ha: δ ( is bouded over R The for every (, R R, he soluio (,, ; is bouded over R as well. If i addiio li δ ( = he for every R, R, i holds ha li (,, ; =. ( Saee (vii of Proposiio 3. shows ha, uder a special ype of forward copleeess, he -GAS Propery for (. is equivale o (o-uifor i ie ISS for (.. For he auooous case we esablish below ha -GAS plus forward copleeess is equivale o ISS, which i geeral is o rue for he uifor i ie ISS (see []. Corollary 3.5 Cosider he auooous syse (., where respec o ( ad saisfies f (, =. Suppose ha: f is ally Lipschiz wih (i R is GAS for he uforced syse (.3 (-GAS Propery. (ii Syse (. is forward coplee wih u as ipu. The he soluios of (. saisfy (3.5 (saee (vii of Proposiio 3., ad herefore (. fulfills he ISS propery. Proof Sice R is GAS for he syse (.3, he by Lea IV. i [] ad Theore 3 i [4], here eiss a sooh fucio V : R R, fucios a, a, λ, δ of class K, such ha for all ( R R we have: ( V ( a ( a (3.7a V ( f ( V ( λ( δ (u (3.7b 8

Clearly, iequaliies (3.7a,b give he followig esiae for he soluio ( of (. iiiaed fro R a ie ad correspodig o soe ipu u = of class L ([, : ( ( ep( ( a ( ep( ( λ( ( τ δ ( τ a τ dτ, (3.7c Furherore, sice (. is forward coplee, Corollary. i [] guaraees he eisece of a sooh ad proper fucio W : R R, fucios a, σ of class ad a cosa R > such ha for all ( R R i holds: ( W a ( R 3, a4 a ( (3.8a 3 4 K W ( f ( W ( σ (u (3.8b I follows fro (3.8a,b ha he soluio ( of (. saisfies a Defie s : = λ a ( 3 ( ( ep( ( a4 ( R ep( σ ( τ τ τ d, (3.9 ~ λ ( ( 3 s. I he follows fro (3.7c ad (3.9 ha for all i holds: a ( ( ep( ( a ( s ( u s ds λ ep( τ σ ( δ ( τ ~ λ ~ ( ep( τ ( a ( R δ ( τ 4 τ dτ dτ (3. Corollary ad Reark i [4] guaraee he eisece of a fucio q ( ha ( rs δ ( rs λ ( rs q( r q( s, such ~ σ, r, s (3. ad le φ be a fucio ha saisfies he followig iequaliy for all : ep( q φ( q ( ep( (3. By virue of (3. ad (3. i follows: 9

τ ep( τ s σ (3. ( s ds ep( τ sup q( φ( s s q ds ep( τ sup q( φ( s s s τ τ φ( s (3. s τ for all τ (3.3 By eploiig (3., (3., (3. ad (3.3 we obai he followig esiaio for he soluio ( of syse (. iiiaed a ( = ad correspodig o soe ipu L (, : a [ (3., (3.3 ( ( ep( ( a ( (3. a sup τ ( φ( τ τ sup q( q( φ( τ τ q( a4( R q( ep( τ ( sup q( φ( τ τ sup q( q( φ( τ τ q( a ( q τ τ τ 4 q dτ φ( τ q(r (3.4 We defie: ( a ( s ( q( a ( a ( s : = a s (3.5a 4 ( 4q(R q( s 4( q( s q( q( ζ ( s : s (3.5b = a I follows fro esiaio (3.4 ad defiiios (3.5a,b ha ( ( sup ζ ( φ( τ a, τ τ The las iequaliy assers, by recallig saee (vi of Proposiio 3., ha syse (. saisfies he ISS propery. 4. Applicaios I Sall-Gai Theore We firs provide sufficie codiios for ISS for coposie syses (.4. The e heore cosiues a geeralizaio of he well-kow sall gai heore for auooous syses uder he presece of (uifor i ie ISS (see [3,4]. Theore 4. For he syse (.4 we assue: A Subsyse (.4a saisfies he ISS propery fro he ipu ( y,. Paricularly, here eis fucios σ ( L, ρ (, φ (, β ( such ha for every R (, R ad for every ipu ( y, = ( y(, of class L ([,, he

soluio holds: ( of (.4a wih ( = eiss for all ad he followig propery ( a σ ( β (,, sup σ β( τ ρ φ( τ a τ eis a cosa λ, fucios σ ( L, ( ρ, φ (, β ( k such ha for every ( R, y R ad for every ipu (, = ( (, of class L ([,, he soluio y( of (.4b wih y ( = y eiss for all ad i holds ha y( a σ ( β( y,, sup σ β( τ ρ φ( τ a τ ( φ ( ( s, li β ( ρ σ = (4.3a li β ( ρ ( φ ( λσ( s, = (4.3b K a ( s < s, such ha he followig iequaliies are saisfied for all : supσ ( β( ρ( φ( σ ( β ( ρ ( φ ( λ s,, a( s, s (4.5a ( β ( ρ ( φ ( λ σ ( β ( φ ( s,, a( s ( ρ ( ( { y( τ, τ }, τ (4. A Subsyse (.4b saisfies he ISS propery fro he ipu (. Paricularly, here ( ( { λ ( τ, τ }, τ (4. A3 The followig properies hold for all, s : A4 There eiss a fucio a ( of class wih s > (4.4 supσ, s (4.5b The syse (.4 saisfies he ISS propery fro he ipu u. II Cascade Coecios I is kow ha he cascade coecio of wo auooous idepede ISS subsyses saisfies he ISS propery. This is o i geeral rue for he ie-varyig case uder he assupio of he o-uifor i ie ISS propery. Paricularly, by specializig Theore 4. o he case.5 we obai he followig resul, which cosiues a geeralizaio of rece resuls cocerig ie-varyig syses (see [9, ].

Corollary 4. Cosider he syse (.5 ad suppose ha he followig hold: A A The subsyse (.5a saisfies he (o-uifor i ie ISS propery fro he ipu ( y, ad ha here eis fucios σ ( KL, ρ ( ad φ (, β ( of class K such ha for every (, R R ad for every ipu ( y, = ( y(, of class L ([, he correspodig soluio ( of (.5a wih ( = eiss for all ad iequaliy (4. holds. The subsyse (.5b saisfies he (o-uifor i ie ISS propery fro he ipu u ad here eis fucios σ ( KL, ρ ( ad φ (, β ( of class K such k ha for every ( R, y R ad for every ipu u = of class L [, he correspodig soluio y( of (.5b wih y ( = y eiss for all ad he followig propery holds: ( y( a σ ( β ( y,, sup σ τ ( β ( τ ρ ( φ ( τ τ, τ (4.6 ( R :, s A3 The followig propery holds for all ( ( φ ( σ ( s, li β ( ρ = (4.7 The syse (.5 ehibis he o-uifor i ie ISS propery fro he ipu u. III Oupu Feedback Sabilizaio We derive sufficie codiios for robus oupu saic feedback sabilizaio for syses i feedback for. The followig resul eeds he correspodig resuls i [3,6]. Theore 4.3 Suppose ha: B There eiss a C fucio k : R R R, wih k (, =, such ha syse (.6a wih y = k(, z saisfies he o-uifor i ie ISS propery fro he ipu z for cerai gai fucio γ (. B Syse (.6a wih y = k(, z saisfies he o-uifor i ie IOS propery fro he ipu z wih k (, as oupu fucio ad he sae gai γ ( as i B. B3 There eiss a fucio ϕ of class K such ha ϕ ( d(, y, (, y R R R The for every fucio, : ( R γ ( s, which is C, ally Lipschiz i s, wih ~ ~ γ (, for each, here eiss a C fucio k : R R R wih k (, = such R

~ ha syse (.6 wih u = k (, y k(, v gai γ (, s. saisfies he ISS propery fro he ipu v wih Refereces [] Ageli, D. ad E.D. Soag, Forward Copleeess, Ubouded Observabiliy ad heir Lyapuov Characerizaios, Syses ad Corol Leers, 38(4-5, 999, 9-7. [] Ageli, D., E.D. Soag ad Y. Wag, A Characerizaio of Iegral Ipu-o-Sae Sabiliy, IEEE Tras. Auoa. Cor., 45(6,, 8-96. [3] Jiag, Z.P., A. Teel ad L. Praly, "Sall-Gai Theore for ISS Syses ad Applicaios", Maheaics of Corol, Sigals ad Syses, 7, 994, 95-. [4] A. Isidori, Noliear Corol Syses II, Spriger-Verlag, Lodo, 999. [5] Karafyllis, I. ad J. Tsiias, "Global Sabilizaio ad Asypoic Trackig for a Class of Noliear Syses by Meas of Tie-Varyig Feedback ", o appear i he Ieraioal Joural of Robus ad Noliear Corol. [6] Karafyllis, I. ad J. Tsiias, "A Coverse Lyapuov Theore for No-Uifor i Tie Global Asypoic Sabiliy ad is Applicaio o Feedback Sabilizaio", subied o SIAM Joural Corol ad Opiizaio. [7] Karafyllis, I. ad J. Tsiias, "Trackig Corol of Noholooic Syses i Chaied For, subied o IEEE Trasacios o Auoaic Corol. [8] Li, Y., E.D. Soag ad Y. Wag, "A Sooh Coverse Lyapuov Theore for Robus Sabiliy", SIAM J. Corol ad Opiizaio, 34, 996,4-6. [9] Paeley, E. ad A. Loria, "O Global Uifor Asypoic Sabiliy of Noliear Tie- Varyig Syses i Cascade", Syses ad Corol Leers, 33(, 998, 3-38. [] Soag, E.D., "Sooh Sabilizaio Iplies Coprie Facorizaio", IEEE Tras. Auoa. Cor., 34, 989, 435-443. [] Soag, E.D. ad Y. Wag, "O Characerizaios of he Ipu-o-Sae Sabiliy Propery", Syses ad Corol Leers, 4, 995, 35-359. [] Soag, E.D. ad Y. Wag, "Noios of Ipu o Oupu Sabiliy", Syses ad Corol Leers, 38, 999, 35-48. [3] Soag, E.D. ad Y. Wag, "New Characerizaios of Ipu-o-Sae Sabiliy ", IEEE Tras. Auoa. Cor., 4, 996, 83-94. [4] Soag, E.D., "Coes o Iegral Varias of ISS", Syses ad Corol Leers, 34, 998, 93-. 3

[5] Teel, A., A Noliear Sall Gai Theore For The Aalysis of Corol Syses Wih Sauraios, IEEE Tras. Au. Cor., AC-4, 56-7. [6] Tsiias, J., "Versios of Soag's Ipu o Sae Sabiliy Codiio ad Oupu Feedback Sabilizaio", J. Mah. Syses Esiaio ad Corol, 6(, 996, 3-6. [7] Tsiias, J., "Ipu o Sae Sabiliy Properies of Noliear Syses ad Applicaios o Bouded Feedback Sabilizaio usig Sauraio", ESAIM Corol, Opi. Calc. Var.,, 997, 57-85. [8] Tsiias, J. ad I. Karafyllis, "ISS Propery for Tie-Varyig Syses ad Applicaio o Parial-Saic Feedback Sabilizaio ad Asypoic Trackig", IEEE Tras. Auoa. Cor., 44(, 999, 79-85. [9] Tsiias, J., "Backseppig Desig for Tie-Varyig Noliear Syses wih Ukow Paraeers", Syses ad Corol Leers, 39(4,, 9-7. [] Tsiias J., "A Coverse Lyapuov Theore for No-Uifor i Tie, Global Epoeial Robus Sabiliy", Syses ad Corol Leers, 44(5,, 373-384. 4