Input-to-state stability of switched nonlinear systems

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1 Scece Cha Seres F: Iforao Sceces 28 SCIENCE IN CHINA PRESS Sprer fo.sccha.co Ipu-o-sae saby of swched oear syses FENG We,2 & ZHANG JFe 2 Coee of Iforao Sceces ad Eeer, Shado Arcuura Uversy, Ta a 278, Cha; 2 Acadey of aheacs ad Syses Scece, Chese Acadey of Sceces, Be 8, Cha The pu-o-sae saby (ISS probe s suded for swched syses wh fe subsyses. By us upe Lyapuov fuco ehod, a suffce ISS codo s ve based o a quaave reao of he coro ad he vaues of he Lyapuov fucos of he subsyses before ad afer he swch sas. I ers of he averae dwe-e of he swch aws, soe suffce ISS codos are obaed for swched oear syses ad swched ear syses, respecvey. swched syse, pu-o-sae saby, Lyapuov fuco, dwe-e Iroduco Sce he perforace of a rea coro syse s affeced ore or ess by uceraes, such as uodeed dyacs, paraeer perurbaos, exoeous dsurbaces, easuree errors, ec., he research o robusess of coro syses do aways have a va saus he deveope of coro heory ad echooy. A a robusess aayss of oear coro syses, Soa, Wa ad L [ ] deveoped a ew ehod fro he po of vew of pu-o-sae saby (ISS, pu-o-oupu saby (IOS ad era pu-o-sae saby (ISS, ad obaed a seres of fudaea resus by uz ISS-, IOS-Lyapuov fucos. Recey, aca- Auar ad García [2] apped he dea o sudy he robusess of swched oear syses of he for x ( = f ( x(, u( ( Λ, where Λ s he dex se. For swched syses, ahouh os of resus have bee preseed, hey ay focus o he probes of saby, coroaby, observaby ad sabzao coro [3 2]. For he robusess sudy of such syses, he reeva eraure s o rch, ad ref. [2] sees he oy oe o he ISS of swched oear syses, o our kowede. Receved Jauary 22, 27; acceped Auus 5, 28; pubshed oe Ocober, 28 do:.7/s Correspod auhor (ea: fwsakefa@26.co Suppored by he Naoa Naura Scece Foudao of Cha (Gra No Sc Cha Ser F-If Sc Dec. 28 vo. 5 o

2 I hs paper, we vesae he ISS of eera swched oear syses (cud he case where here s o coo Lyapuov fuco. Uke he exs resus, whch ay focus o esabsh ISS coverse heores for oear syses [ ], by ope ou he characersc of swched oear (SNL syses, we a a prese soe suffce ISS codos for SNL syses, cud for sace, he reao of he ISS ad he averae dwe-e of he swch aw. Precsey, we w vesae SNL e-vary syses, whch ay vove fe subsyses. I hs case, swch ao dffere subsyses ay ead o dscouy of he syse fuco, ad dssasfes he couy assupo requred by refs. [ ]. Thus, he resus refs. [ ] cao be eerazed o eera SNL syses drecy. I soe speca cases, for sace, where here exss a coo ISS-Lyapuov fuco (CISSLF, a suffce ad ecessary ISS codo of SNL syses wh arbrary swch aws s ve [2] uder he assupo ha f ( xu, s ufory (wh respec o ocay Lpschz couous o x, u. Here, by us he ehods of upe Lyapuov fuco ad averae dwe-e, soe suffce ISS codos are ve for eera SNL syses, whch ay have o CISSLF. The ISS-Lyapuov fucos of he subsyses are aowed o be dffere fro each oher raher ha spy assu he exsece of a CISSLF. Besdes, he ufory assupo o he oca Lpschz couy of f ( xu, wh respec o s o requred. Thus, our fraework s ore eera ha ha ref. [2]. The reader of hs paper s orazed as foows. Seco 2 descrbes he probe o be vesaed ad roduces soe oaos ad defos. I seco 3, by us he upe Lyapuov fuco ehod, a suffce ISS codo s ve for eera swched oear syses. I seco 4, by us he averae dwe-e ehod, soe suffce ISS codos are preseed for SNL syses ad swched ear syses, respecvey. I seco 5, soe cocud rearks are ve. 2 Noaos ad probe foruao Cosder he foow swched oear syse x ( = f (, x(, u(, x( = x, ( (, x ( where x( R ad u( R are he syse sae ad pu, respecvey; ad (, : [, R I ( I s he dex se, aybe fe s he swch aw ad s rh-had couous ad pecewse cosa o ; for ay I, fuco f ( xu,, : [, R R s couous wh respec o xu,,, ufory ocay Lpschz couous wh respec o x, u, ad sasfes f (,,. Here, be dffere fro ref. [2], f ( xu,, s e vary, ad he ufory of he oca Lpschz couy of f ( xu,, s wh respec o raher ha. Throuhou he paper, R deoes he rea uber se [, ; for a fuco γ ( : R R, γ K eas ha γ s couous ad srcy creas, ad sasfes γ ( = ; γ K eas ha γ K ad γ are ubouded; for a fuco β ( s, : R R R, β KL FENG We e a. Sc Cha Ser F-If Sc Dec. 28 vo. 5 o

3 eas ha for ay fxed s, β ( s, K, ad for ay fxed, β ( s, s couous ad decreases o zero as s ; deoes he Eucdea or R ad he correspod duced arx or, ad for a oepy subse hods x {} = x whe = {} ; R, x f η xη (obvousy, L deoes he se of a he easurabe ad ocay esseay bouded pu u( R o [, uder he foow or, u = sup{ u(, } < ; (2 for wo fucos ϕ( ad χ(, sybo ϕ χ( deoes he copose fuco ϕ( χ( ; s he rade operaor. For ay ve swch aw (,, a codo x R, u( L, x( x (;, x, u deoes he sae raecory of syse ( wh he axa exs erva [, T, where he cosa T T(, x, u. Defo. Cosder he foow eera oear syse ω( = (, ω(, v(, ω( = ω, (3 where fuco (,, : [, R R sasfes (,,. For ay ω R, L v, f he raecory ω( ω(;, ω, v of (3 s defed we o [,, he he syse s caed forward copee. For a cosed se, L R, f syse (3 s forward copee for ay ω v, ad ω(,, he s caed a cosed vara se of syse (3. Reark. By Defo, f syse ( s forward copee for ay ( x,, he a of he subsyses are forward copee. Reark 2. Obvousy, f s a cosed vara se of a subsyses of syse (, he s aso a cosed vara se of syse (. Defo 2 []. For he forward copee syse (3 ad s cosed vara se R, he syse (3 s caed (obay pu-o-sae sabe (ISS wh respec o, f here exs wo fucos β KL ad γ K such ha for ay ω R \ ad L v, ω( ;, ω, v β( ω, γ( v,. (4 Defo 3 [3]. For he forward copee syse (3 ad s cosed vara se R, a sooh fuco V ( ξ, : R [, R s caed a ISS-Lyapuov fuco of he syse (3 wh respec o ξ R \, R, f here exs fucos αα, K, αχ, K such ha for ay μ R ad, α( ξ V ( ξ, α( ξ, (5 ξ χ ( μ V ( ξ, V ( ξ, (, ξ, μ α ( ξ. (6 For shor, hey w be deoed as ( V ; α, α, α, χ he seque. 994 FENG We e a. Sc Cha Ser F-If Sc Dec. 28 vo. 5 o

4 3 ISS codos based o upe Lyapuov fucos I hs seco, by us he upe Lyapuov fuco ehod, suffce ISS codos are expored for swched oear syses. To hs ed, we eed he foow eas. Lea [3]. For he forward copee syse (3, assue R s s cosed vara se. If syse (3 has a ISS-Lyapuov fuco ( V ; α, α, α, χ such ha (5 ad (6 hod for ay ξ R \, μ R ad, he here exss (, ω, v sasfy such ha he souo ω( ω(;, ω, v of syse (3 has he foow propery: ( ω(, S for ay, ad ( ω(, S for ay <. Here, S = {( ξ, : V ( ξ, α χ( v }. Lea 2 [6]. For ay κ K, here exss a C fuco ρ K such ha ρ( r κ( r ρ(, r r. Lea 3. For he forward copee syse (3 ad s cosed vara se R, f syse (3 has a ISS-Lyapuov fuco ( V ; α, α, α, χ such ha (5 ad (6 hod for ay ξ R \, μ R ad, he here exss a C fuco ρ K deped oy o α ad α such ha W( ξ, χ( μ W( ξ, W( ξ, (, ξ, μ W( ξ,, where W ( ξ, = ρ V ( ξ, ad χ( = ρ α χ( K. Proof. For κ( α α ( K, by Lea 2, α ad α K ca deere a C fuco ρ K such ha ρ( r κ( r ρ(, r r. The, by esfy drecy for W ( ξ, = ρ V ( ξ,, oe ca oba he cocuso. Lea 4. For he forward copee syse (, suppose ha R s a cosed vara se of syse (, ad he swch sas of swch aw ( x, are < 2 < < k <. If for ay ve I, subsyse f ( xu,, has a ISS-Lyapuov fuco ( V ; α, α, α, χ such ha α( sup I α ( ad K ax{ V ( x(,, α χ( u } V ( x(,, (7 x x (, ( (, ( he for se S = {( ξ, : V ( ξ, α χ( v } ( I, here exss a coo e-sa ( x u( such ha,, ( x(, S, [,, (8 (, x ( (, x ( ( x (, S, [,. (9 Proof. For =,, 2,, e (, x( =. The, by Lea, for he subsyse f (, xu, o he erva [,, here exss FENG We e a. Sc Cha Ser F-If Sc Dec. 28 vo. 5 o

5 (, x(, u, ( such ha for ay, he sae raecory x( x(;, x, u = x(;, x(, u sasfes ( x(, S (ha s, V ( x(, α χ( u ; ad for ay If < for ay,, 2,, case, se = he ( (, x(, S. <, ( x S ( =,,2, for ay. I hs =. Oherwse, here exss a oeave eer such ha. Le { } = :, =. The, by Lea, we have (8, ad ( x(, S S, ha s, Ths pes ha α for [,. Parcuary, ( x (, V ( x(, χ( u. Thus, fro (7, foows ha V ( x(, ax { V ( x(,, α χ( u } α χ( u. (, x(, u. Therefore, V ( x(, α χ( u, [,. 2 Repea he above process for = 2, 3,, oe ca oba (9. Based o he ehod of upe Lyapuov fuco, we have he foow heore. Theore. Cosder he forward copee syse (. Suppose ha R s s cosed vara se, ad he swch sas of he swch aw ( x, are < < k <. If here exss ISS-Lyapuov fuco ( V ; α, α, α, χ of subsyse f ( xu(,, I such ha ( αα, K, αχ, K, where α( f α (, α( sup α (, α( f α ( χ( sup I χ ( ; ( (7 hods a each swch sa ( =,, 2,, he syse ( s pu-o-sae sabe. Proof. Frs, by Defo 3 ad (, for ay ξ R \, I I I ad α( ξ V ( ξ, α( ξ,, μ R ad I, we have ξ χ( μ V( ξ, V( ξ, f(, ξ, μ α( ξ, ; ( ad by Lea 2, we kow ha here exss a α such ha ρ( r κ( r ρ(, r r, where ad χ( = ρ α χ(. The, by Lea 3 we have C fuco ρ K deped oy o α ad κ( α α (. Le W ( ξ, = ρ V ( ξ, ρ α( ξ W ( ξ, ρ α( ξ,, (2 I W( ξ, χ( μ W( ξ, W( ξ, f(, ξ, μ W( ξ,, I. (3 By Lea 4, here exss such ha (8 ad (9 hod. Le be he ares eer such ha. The, fro (8 ad (9 ad he defo of W ( ξ,, foows ha (, x( ( (, χ(, [, W x > u or [,, =,,,, (4 996 FENG We e a. Sc Cha Ser F-If Sc Dec. 28 vo. 5 o

6 (, x( χ W ( x(, ( u, [, or [,, =, (5 Ths oeher wh (3 ad (4 ves d W (, ( ( (, (, ( ( (,, [, x x < W x x or [,, =,,,. (6 d Hece, we have ( W(, ( ( (, (, ( ( (,, [,, x x W x x e (7 ( W,,,. (, ( ( (, (, ( ( (,, x x W x x e = Fro (7 ad (8, he defo of W ( ξ,, (4, ad (3 s easy o see The, for ay W ( x(, W ( x(,, =,,,. (8 (, x( (, x( [, or [,, =,,,, by (7 ad (8 we have W ( x(, W ( x(, e (, x( (, x( ( ( ( (, ( (, ( W x e W x e Ths oeher wh (2 ad (5 eads o ( ρ α( x ( ax{ ρ α( x e, χ( u }. Le ad s β(, s = α ρ ( ρ( α( r e ad ( (, ( (,. 2 x 2 x γ( r = α α χ( r. The, β ( s, KL, γ K x( = x ( ;, x, u β( x, γ( u,. Thus, syse ( s pu-o-sae sabe. Reark 3. Codo ( of Theore says ha he eery of he syse shoud o be creas a swch sas. Ths s because ha he ISS s a oba propery hod for a wh respec o x( = x ad u (, raher ha a -sup propery. Oherwse, for sace, f sup x( s cosdered, he he codo ca be reaxed o ha (7 hods afer fe swch sas. Reark 4. Fro he proof of Theore, we see ha β ( s, KL ad γ K are depede of he cocree choce of (,. I oher words, he swched oear syse ( s ISS for a (, sasfy (7. Reark 5. I Theore, sead of codo (, f we assue ha here exss a eer such ha (, (,, x u ad W(, ( ( (, (, ( ( (,,,,,, x x W x x = (9 W (, ( ( (, (,, 2,, x x α χ u = he syse ( s pu-o-sae sabe. I he seque, we w provde soe suffce ISS codos fro aoher po of vew by epoy he cocep of dwe-e of a swch aw. Defo 4. For a swch aw ( x,, suppose s swch sas are < < < k <. The, τ = f k ( k k s caed s dwe-e. FENG We e a. Sc Cha Ser F-If Sc Dec. 28 vo. 5 o

7 Coroary. For he forward copee syse (, suppose ha R s s cosed vara se, ad he swch sas of swch aw ( x, are < < < k <. Uder he codos ad oaos of Theore, sead of codo (, f we assue ha here exss a eer such ha τ ad (9 hods, he syse ( s pu-o-sae sabe. Here, s ve by (. Proof. By Defo 4, we see τ. Hece, (, (,. x u Ths oeher wh Reark 5 eads o he cocuso. Coroary 2. For he forward copee syse (, suppose ha R s s cosed vara se, ad he swch sas of ( x, are < < < k <, ad he dex se I s fe ad deoed wh {,2,, N}( N <. If here exs ISS-Lyapuov fuco ( V ; α, α, α, χ ( I such ha ax{ V ( x(,, α χ( u,, α χ( u } V ( x(,, (, x( N (, x( for =,,, he syse ( s pu-o-sae sabe. Is proof s easy ad oed. 4 ISS codos based o he averae dwe-e I hs seco, we w use he cocep of averae dwe-e o oba soe suffce ISS codos for boh SNL syses ad swched ear syses. Defo 5 [7]. For ay ve cosas τ > ad N, e N ( s, deoe he swch uber of ( x, [,, s > s, ad e [, ] (,: (, s S τ N = N s N, > s. τ The, τ s caed he averae dwe-e of S [, τ N], ad s τ sup sup > s N (, s N s caed he averae dwe-e of ( x,. 4. ISS aayss of swched oear syses Theore 2. For he forward copee syse (, suppose ha R s s cosed vara se, ad swch sas of swch aw ( x, are < < < k <. If here are ISS-Lyapuov fucos ( V; α, α, α, χ ( I ad cosas c >, η, such ha for ay ξ R \, μ R ad I, α( ξ V ( ξ, α( ξ,, (2 ξ χ( μ ( ξ, ( ξ, (, ξ, μ ( ξ,,, V V f cv (2 ax{ η V ( x(,, α χ( u } V ( x(,, =,,, (22 (, x( (, x( 998 FENG We e a. Sc Cha Ser F-If Sc Dec. 28 vo. 5 o

8 η he syse ( s pu-o-sae sabe for ay (, S, N, c where S η η, N = (, [ τ, N]: τ >. c S c Proof. For a ve e sa, assue ha syse ( has swch sas [,, ad deoe he as < 2 < <. Le (, x( = ( =,,, ad ( =,, be he e sa defed accord o (. If f <, he V ( x( s, s > α χ( u for ay s [, Ths oeher wh (2 ves o [,, he by Lea, V ( x(, α χ( u ; (23, ad hece, by (2 we have x( s > χ( u, s [,. (24 Thus, whe d V ( x ( s, s cv ( x ( s, s, s [,. (25 ds <, we have cs ( V ( x( s, s V ( x(, e, s [,. (26 Now, e us cosder he erva [, ( =,2,,. Whe whe whe eads o <, we have, we have V ( x(, α χ( u ; (27 V ( x( s, s > α χ( u, s [,, whch oeher wh (2 x( s ( u, s [,. > χ (28 The, by (2 we have d V ( x ( s, s cv ( x ( s, s, [,. s (29 ds Ths pes ha I parcuar, cs ( V x s s V x e s (3 ( (, ( (,, [,. c ( V x V x e (3 ( (, ( (,. Le π = α χ( u. The, by (23, (26, (27, ad (3, we have c ( { } V ( x(, ax V ( x(, e, π, (32 c ( { π} V ( x(, ax V ( x(, e,, =,2,,. (33 Reca (22 ad subsu (33 o (32 sequeay, we oba { c ( c ( c ( V x V x e e e ( (, ax η ( (,, c ( c ( c ( c ( e e e,, e,. 2 η π π π } FENG We e a. Sc Cha Ser F-If Sc Dec. 28 vo. 5 o

9 Noce ha N (, = ( =,,2,,. The, we have η = e. Thus, { } N (, η N (, η c( N (, η c( V x x e e ( (, ax α( (, π, N (, η c( c(, πe, πe, π. η η Le a = c. The, a >. By Defo 5, for ay ve (, S, N, τ c we s have N (, s N, > s. Ths resus N (, s η c( s Nη τ a ( s. Therefore, by (2 we have N { η a ( N η } α( x ( V ( x (, ax α( x ( e, α χ( u e. Nηas Nη Le β α α γ α α χ (, rs = ( ( re, ( r = ( ( re, r, s. The β KL, γ K ad x( = x ( ;, x, u β( x, γ( u,. Thus, syse ( s pu-o-sae sabe. Reark 6. By η, he codo (22 Theore 2 s obvousy weaker ha he codo (7 Theore. Ths cudes he case where he eery of he subsyse afer a swch sa s reaer ha ha of he subsyse before he swch sa. 4.2 ISS aayss of swched ear syses I hs subseco, we w vesae swched ear u-varabe syses of he for x ( = A x( B u(, x( = x, (34 (, x ( (, x ( where A R, B R are cosa arces for ay I, respecvey. I he seque, for ay arx A, J A deoes he Jordaa ora for of A, η ( A s he ares rea par of he eevaues of A, ad Δ ( A ad Δ ( A are he ares ad saes suar vaues of arx A, respecvey. For a ve arx se A R, A deoes he se of a sabe arces of A, ad A 2 deoes A\ A. Noc ha η ( A depeds couousy upo he paraeer of A, whe A s copac, we have ax A A η( A < ad ax A A η( A = (. A A η A I parcuar, whe he uber of a sabe arces A s fe ad reaer ha zero, we have ax η( A A. A < Lea 5. For ay ve arx se A R, f A ad A are copac, ad A s oepy, he for ay (, A η( A, here exss a cosa ( > such ha where A A a ( A e λ ( A e, A A,, (35 λ ( A ax λ ( A <, (36 A A ( PA ( a( A = η( A, λ( A ( Δ >, ad P( A s osuar a- Δ ( PA ( 2 FENG We e a. Sc Cha Ser F-If Sc Dec. 28 vo. 5 o

10 rx sasfy A= P( A J P( A. A Proof. For ay ve A A, assue ha has p Jorda bocks wh desos of 2 p, ad he rea pars of he correspod eevaues are τ, τ 2,, τ p, respecvey. The, 2 p = ad η( A ax pτ. Noc ha Furher, s easy o see ha J A A e = P( A e P( A, we have = A J ( ( A Δ PA JA e = P( A e P( A e, >. Δ ( PA ( J /2 A e = ( Σ (, where Σ ( = e ( ( 2 e ( ( 2 Sce for we have 4 2( 4 2τ 2 2τ (2! ((! (2! 2( 2( 2 4 p 2 p 2 τ ( ( 2 2 e p p p. 2 2 (( 2! (2! (( p! (, A η( A, A 4 2( ( p Σ 2 2 ( p ( 2 p e, 2 a ( A 2 2 e (2! (( p! Σ( Σ( =, ad hece, ( sup a A [, <, ad 2 a ( A 2 ( e e A JA e Δ( PA ( e Δ( PA ( Σ a ( A a ( A 2 a ( A ( PA ( ( PA ( ( = e Δ e Δ e 2 λ ( A,.e., (35 hods. Furher, by he copacess of A, osuary of P( A o A, ad he couy of Δ ( P ( A ad Δ ( P ( A o A, λ ( A s couous o he copac se A wh respec o A. Thus, (36 hods. Now, we sudy he ISS propery of he swched ear syse (34. For syse (34, e A= { A : I} R ad B= { B : I } R, assue ha A ad B are copac, ad he subse A coss of a he sabe arces of A s oepy ad copac. For ay ve ad se A (, A η( A, defe a ( A ad λ ( A as Lea 5, A a ( A = A a( A, a ( A = ax{,ax A A a( A}, (37 ( N λ ( b ( B = ax B, = e. (38 B B For a ve swch aw ( x, ad a e erva [ s,, e T ( s, A ad T ( s, he oa e of syse (34 ru o sabe subsyses ad usabe subsyses [ s,, respecvey; ad for ay a (, ( A ] ad τ >, defe a be FENG We e a. Sc Cha Ser F-If Sc Dec. 28 vo. 5 o

11 T (, s a ( A a S[ a, τ; A] = (, S[ τ, N ]:sup > s, τ, > τ T (, s a ( A a where he averae dwe-e τ of ( x, s ve by Defo 5. I he seque, for spcy of expresso, we w drop he arues of λ ( A, a ( A, λ a ( A ad b ( B ad deoe he by, a, a ad b, respecvey. Theore 3. For swched ear syse (34, assue ha A ad B are copac, ad he subse A coss of a he sabe arces of A s oepy ad copac. The for ay ve (, A A η( A, ad a (, a ], here exss τ λ such ha a ( for ay (, S[ a, τ; A ], syse (34 s forward copee, ad ( syse (34 s ISS f ad oy f he coro-free syse x ( = A (, x ( x( s asypocay sabe. Proof. Par ( ad he ecessy of par ( are obvous. Thus, beow we eed oy o show he suffcecy. For ay ve e sa, assue ha he e erva [,, syse (34 has swch sas: < 2 < <. Le (, x( = ( =,,,. The, he souo of syse (34 ca be expressed as x( = x (;, x, u =Φ (, x Φ (, sb usds ( Φ(, sb usds (,(39 where (, x( (, x( A ( A ( A ( s Φ = (, ( (, ( (, ( (, x x x s e e e, s [,. We frs show ha for ay ve cosa a (, a ], τ λ, ad swch aw a (, S[ a, τ; A ], here are a > ad > such ha Noc ha we have λ N [ N ( s, ] λ e a ( s Φ( s, e, s. (4, s, =,,2,, ; (, s =, s, = for s [, ; ad [ N ( s, ] λ λ e,. I parcuar, N (, = ad λ = e. Thus, = for s [, ( =,, [ N (, ] λ a ( A(, ( ( ( (, ( ( ( (, ( ( x a A x a A x s Φ( s, λ e e e e [ N ( s, ] λ at ( s, at ( s, e By he defo of S[ a, τ; A ], for ay ve (, S [ a, τ; A ], we have at ( s, s at (, s a( s. Ths oeher wh N (, s N ad. τ τ > τ λ a 22 FENG We e a. Sc Cha Ser F-If Sc Dec. 28 vo. 5 o

12 pes ha a a λ. > The, by soe srahforward cacuaos, we have τ a ( s Φ( s, e, s [, [,, =,,,, ( N λ where = e. Thus, (4 s rue, whch oeher wh (39 ves a ( a ( s a ( s ( ;,, x x u e x e b u ds e b u ds a ( a ( s a ( b e x e b u ds e x u. a as Le β ( rs, = e r ad b γ ( r = r. The, β KL, γ K ad a x ( ;, x, u ( x, ( u, β γ.e., syse (34 s pu-o-sae sabe. Reark 7. Copar Theore 3 wh Theore 2, oe ca see ha for swched ear syse case, soe of he subsyses of syse (34 are aowed o be usabe. However, for swched oear syses, a of s subsyses are requred o be sabe, sce he deree of saby of oear syses s hard o be characerzed. Reark 8. By Theore 3, he ISS of swched ear syse (34 s depede of he cocree choce of (, S[ a, τ; A ]. 5 Cocuso I hs paper, he ISS of swched oear syse ad swched ear syse are vesaed, respecvey. The a resus ca rouhy be dvded o wo casses. Oe s based o upe Lyapuov fuco ehod, ad he oher s based o (averae dwe-e ehod. Frsy, by us he ehod of upe Lyapuov fuco, a suffce ISS codo s ve for eera SNL syses based o a quaave reao of he coro ad he vaues of he Lyapuov fucos of he subsyses before ad afer he swch sas. Here, he ISS-Lyapuov fucos of he subsyses are aowed o be dffere fro each oher raher ha spy assu he exsece of a CISSLF. Thus, he codo s suffce o oy for he swched syses possess a CISSLF, bu aso suffce for he swched syses whou ay CISSLF. Secody, by epoy he ehod of he averae dwe-e, soe ISS suffce codos are ve for swched oear syses ad swched ear syses, respecvey. Ao ohers, he codo o swched oear syses s characerzed by he sze of he dwe-e, ad ha o swched ear syses s characerzed by he averae dwe-e ad he rao of he oa e ha he syse rus o usabe subsyses o he oa e ha he syse rus o sabe subsyses. Soa E D. Sooh sabzao pes copre facorzao. IEEE Tras Auo Coro, 989, 34(4: [DOI] 2 L Y, Soa E D, Wa Y, e a. Ipu o sae sabzaby for paraeerzed faes of syses. I J Robus Noear Cor, 995, 5(3: [DOI] 3 Soa E D, Wa Y. O characerzao of he pu-o-sae saby propery. Sys Cor Le, 995, 24(5: [DOI] 4 L Y, Soa E D, Wa Y. A sooh coverse Lyapuov heore for robus saby. SIA J Coro Op, 996, 34(: FENG We e a. Sc Cha Ser F-If Sc Dec. 28 vo. 5 o

13 24 6 [DOI] 5 Soa E D, Wa Y. New characerzaos of pu o sae saby. IEEE Tras Auo Coro, 996, 4(9: [DOI] 6 Pray L, Wa Y. Sabzao spe of ached uodeed dyacs ad a equvae defo of pu-o-sae saby. ah Coro Sa Sys, 996, 9(: 33 [DOI] 7 Soa E D, Wa Y. A oo of pu o oupu saby. I: Proc of Europea Coro Coferece, Brusses, Ae D, Soa E D, Wa Y. A Lyapuov characerzao of era pu-o-sae saby. IEEE Tras Auo Coro, 2, 45(6: [DOI] 9 Soa E D, Wa Y. Lyapuov characerzaos of pu o oupu saby. SIA J Coro Op, 2, 39(: [DOI] Lberzo D, orse A S, Soa E D. Oupu-pu saby ad u-phase oear syses. IEEE Tras Auo Cor, 22, 47(3: [DOI] Ae D, Soa E D, Wa Y. Ipu-o-sae saby wh respec o pus ad her dervaves. I J Robus Noear Coro, 23, 3(: [DOI] 2 aca-auar J L, García R A. O coverse Lyapuov heores for ISS ad ISS swched oear syses. Sys Cor Le, 2, 42(: [DOI] 3 aro. Jup Lear Syses Auoac Coro. New York: arce Dekker, 99 4 Lberzo D, orse A S. Basc probe saby ad des of swched syses. IEEE Cor Sys, 999, 9(5: 59 7 [DOI] 5 Bracky S. upe Lyapuov fucos ad oher aayss oos for swched ad hybrd syses. IEEE Tras Auo Coro, 998, 43(4: [DOI] 6 Xe G, Wa L. Coroaby ad sabzaby of swched ear syses. Sys Cor Le, 23, 48(2: [DOI] 7 Hespaha J P, orse A S. Saby of swched syses wh averae dwe-e. I: Proc. 38h IEEE Cof. Decso Coro, Zha G, Hu B, Yasuda K, e a. Saby aayss of swched syses wh sabe ad usabe subsyses: a averae dwe e approach. I J Sys Sc, 2, 32(8: Su Z, Ge S S, Lee T H. Coroaby ad reachaby crera for swched ear syses. Auoaca, 22, 38(5: [DOI] 2 Gua A, Seazu C, Va Der ee C. Opa coro of swched auooous ear syses. I: Proc. 4h IEEE Cof. Decso ad Coro, J Y, Chzeck H J. Coroaby, sabzaby ad couous-e arkova up ear quadrac coro. IEEE Tras Auo Coro, 99, 35(7: [DOI] 24 FENG We e a. Sc Cha Ser F-If Sc Dec. 28 vo. 5 o