Razumikhin-type stability theorems for discrete delay systems

Transcription

1 Automatca 43 (2007) Bref paper Razumkhn-type stablty theorems for dscrete delay systems Bn Lu a, Horaco J. Marquez b, a Department of Informaton and Computaton Scences, Hunan Unversty of Technology, Zhuzhou 42008, Chna b Department of Electrcal and Computer Engneerng, Unversty of Alberta, Edmonton, Alta., Canada T6G 2V4 Receved February 2006; receved n revsed form 30 July 2006; accepted December 2006 Abstract In ths paper, by employng the Razumkhn technque and Lyapunov functons, Razumkhn-type theorems that guarantee the unform stablty, unformly asymptotc stablty and unformly exponental stablty for the general dscrete delay systems are establshed, respectvely. Moreover, Razumkhn-type unformly exponental stablty theorem gves the estmaton of the convergence speed. As theoretc applcaton, the Razumkhn-type unformly exponental stablty result s further studed and used to show some well-known stablty results for some knds of dscrete delay systems. Fnally, examples are also worked through to llustrate our results. Crown Copyrght 2007 Publshed by Elsever Ltd. All rghts reserved. Keywords: Razumkhn-type theorem; Dscrete delay system; Unform stablty; Unformly asymptotc stablty; Unformly exponental stablty; Lyapunov exponent. Introducton Tme delays are commonly encountered n many physcal systems and control schemes due to the fnte swtchng tmes, network traffc congestons, etc. Stablty analyss of tme-delay systems has attracted ncreasng attenton for the last 3 decades (Hale, 977; Hale & Lunel, 993; Lao, 200; Marquez, 2003; Nculescu, 200). It s well known that tme delays often lead to the falure of stablty for a stable system. It s, therefore, very mportant to nvestgate the stablty problem for systems wth tme delays. Among the methods contrbuted to the study of stablty problem for tme-delay systems, the Lyapunov functonal, comparson prncple and Razumkhn technque (Hale, 977; Hale & Lunel, 993) are the three man methods. The Lyapunov functonal method requres constructvely a Lyapunov functon that decreases on the whole state space. The comparson prncple requres fndng an addtonal system, wth known Ths paper was not presented at any IFAC meetng. Ths paper was recommended for publcaton n revsed form by Assocate Edtor Zongl Ln under the drecton of Edtor Hassan Khall. Correspondng author. Tel.: ; fax: E-mal addresses: olverlu78@263.net (B. Lu), marquez@ece.ualberta.ca (H.J. Marquez). stablty propertes, and then compare that to the orgnal tmedelay system. Recently, Knospe and Roozbehan (2003) and Zhang, Knospe, and Tsotras (200, 2003) used the comparson prncple to nvestgate the stablty condtons for lnear contnuous delay systems, n whch they derved many prevously reported stablty crtera by establshng a comparson system free of delays. On the other hand, the Razumkhn technque has advantage that, when dealng wth tme delays, the Lyapunov functon s not requred to be decrescent on the whole state space. The Razumkhn technque has been appled successfully by varous authors to study of several stablty problems for contnuous delay systems, see, for nstance, Hale (977), Hale and Lunel (993), Mao (997), Teel (998) and Teel, Nešć, and Kokotovć (998). Recently, there have appeared several papers devoted to the study of rght-contnuous mpulsve delay system by usng Razumkhn technque (Lu, Lu, Teo, & Wang, 2006; Lu & Ballnger, 200; Shen & Yan, 998; Stamova & Stamov, 200). Razumkhn-type stablty theorems of contnuous delay systems and rght-contnuous delay systems are based on the fact that the soluton of these type of systems s a contnuous or rght-contnuous functon. Unlke contnuous systems and rght-contnuous systems, the soluton of a dscrete-tme system s no longer a contnuous or rght-contnuous functon. Ths brngs dffcultes n the use Razumkhn technque to /$ - see front matter Crown Copyrght 2007 Publshed by Elsever Ltd. All rghts reserved. do:0.06/j.automatca

2 220 B. Lu, H.J. Marquez / Automatca 43 (2007) nvestgate the stablty problem for dscrete delay system. Zhang and Chen (998) studed a class of dscrete delay systems and establshed a backward Razumkhn-type unformly asymptotc stablty theorem. However, the man condton developed n Zhang and Chen (998) s restrctve and dffcult to test. Hence, fndng stablty crtera n whch the condtons can be less restrctve and easly tested s of practcal sgnfcance. The am of ths paper s to establsh Razumkhn-type stablty crtera wth less restrctve assumptons for general dscrete delay systems. In order to overcome the shortcomng n Zhang and Chen (998), we provde the forward Razumkhntype unform stablty theorems, n whch the less restrctve Razumkhn-type unformly asymptotc stablty theorem and the easly tested Razumkhn-type exponental stablty crtera are establshed for dscrete delay systems. To the best of our knowledge, no Razumkhn-type exponental stablty theorem has been prevously reported for dscrete delay systems. The rest of ths paper s organzed as follows. In Secton 2, we ntroduce our notaton and defntons. Then n Secton 3, we develop forward Razumkhn-type unform stablty theorems for dscrete delay systems. In Secton 4, as the theoretc applcaton, the Razumkhn-type unformly exponental stablty result s specalzed to some knds of dscrete delay systems. Fnally, n Secton 5, we dscuss some examples to llustrate our results. 2. Prelmnares In the sequel, R denotes the feld of real numbers, R + the subset of non-negatve elements of R, defned by R + =[0, + ), and R n the n-dmensonal Eucldean space. N represents the natural numbers, N ={0,, 2,...}, N ={0,, 2,...}, and for some postve nteger m, let N m ={ m,...,, 0}. Gven a matrx A, A denotes the norm of A nduced by the Eucldean vector norm,.e., A =[λ max (A T A)] /2.A functon γ : R + R + s of class K (γ K) f t s contnuous, zero at zero and strctly ncreasng. For a gven postve real number r>0, let C([ r, 0],R n ) ={ψ :[ r, 0] R n, ψ s contnuous}. Gven a postve nteger m, we defne φ m = max θ N m { φ(θ) }. Consder the dscrete delay system of the form { x(n + ) = f (n, xn ), n n 0, () x n0 =, where x R n, n 0 N, f C(N C([ m, 0],R n ), R n ), C([ m, 0],R n ), where m N represents the delay n system (), and x n C([ m, 0],R n ) s defned by x n (s) = x(n + s) for any s [ m, 0]. We assume f (n, 0) 0 so that system () admts the trval soluton. We also assume that system () has an unque soluton, denoted by x(n) = x(n, n 0, ), for any gven ntal data: n 0 N and C([ m, 0],R n ). Remark 2.. System () consdered n ths paper s more general than that n Zhang and Chen (998), n whch the functon f( ) needs to satsfy: f (n, φ) L φ, for some postve constant L>0 and any n N. Defnton 2.. The trval soluton of system () s sad to be unformly stable (US) f, for any gven ntal data: n 0 N, x n0 =, and for any ε > 0, there exsts a δ=δ(ε)>0 ndependent of n 0 such that when m δ, the followng nequalty holds: x(n, n 0, ) ε for any n n 0,n N. (2) Defnton 2.2. The trval soluton of system () s sad to be unformly attractve f, for each gven ntal data: n 0 N, x n0 =, and any η > 0, there exst a postve real number σ = σ(η)>0 and a postve nteger K = K(η)>0, where both σ and K are ndependent of n 0, such that when m < σ and n n 0 + K, the followng nequalty holds: x(n, n 0, ) < η, (3).e., lm n x(n, n 0, ) =0. (4) Defnton 2.3. The trval soluton of system () s sad to be unformly asymptotcally stable (UAS) f for any ntal data: n 0 N, x n0 =, the trval soluton of system () s US and unformly attractve. Defnton 2.4. The trval soluton of system () s sad to be unformly exponentally stable (UES) f, for any ntal data: n 0 N, x n0 =, there exst two postve numbers α > 0,M>0, where both α and M are ndependent of n 0, such that for all n n 0,n N, x(n, n 0, ) M m e α(n n 0). (5) Remark 2.2. Obvously, f (5) holds, then the Lyapunov exponent of system () s not greater than α. For smplcty, f the trval soluton of system () s US (UAS, UES), system () s also called as US (UAS, UES). 3. Razumkhn-type theorems for dscrete delay systems In ths secton, three types of stablty (US, UAS and UES) for dscrete delay system () are nvestgated. Theorem 3.. Assume that there exst functons c,c 2 K and a postve defnte functon V (n, x) such that the followng condtons hold: () c ( x ) V (n, x) c 2 ( x ); () for any φ C([ m, 0],R n ) and s N m, then V (n, φ(0)) V(n+ s, φ(s)) mples V(n+, f (n, φ)) V (n, φ(0)); () for any φ C([ m, 0],R n ), and some s N m {0}, then V (n, φ(0)) V(n + s, φ(s)) mples V(n +, f (n, φ)) max θ N m {V(n+ θ, φ(θ))}. Then, system () s US.

3 B. Lu, H.J. Marquez / Automatca 43 (2007) Proof. Let x(n) = x(n, n 0, ) be a soluton of system (). Defne V (n) = max θ N m {V(n + θ,x(n + θ))}. By condtons () (), we get for any n n 0,n N, V(n+,x(n+ )) V (n), (6) whch mples that V(n+ ) V (n), n n 0, n N. (7) Therefore, for any n n 0,n N, V (n) V(n 0 ). Thus, for any n 0 N, and any ε > 0, let δ = δ(ε)>0 wth 0 < δ < ε and c 2 (δ) c (ε), f m δ, then by condton (), we obtan that c ( x(n) ) V (n, x(n)) V (n) V(n 0 ) c 2 ( m ) c 2 (δ) c (ε). (8) Thus, for any n n 0,n N, x(n) ε. Theorem 3.2. Assume that there exst postve functons c,c 2 K and a postve defnte functon V (n, x) such that the condton () of Theorem 3. holds. The condtons () () of Theorem 3. are replaced by: () there exst postve scalar functons p( ), q( ) satsfyng p(s)>s (s>0) and 0 <q(s)<s (s>0) such that, for any φ C([ m, 0],R n ) and s N m, V(n+ s, φ(s)) p(v (n, φ(0))) mples V(n +, f (n, φ)) V (n, φ(0)) q(v (n, φ(0))); () for any φ C([ m, 0],R n ) and for some s N m {0}, then V(n + s, φ(s)) V (n, φ(0)) mples p(v (n +, f (n, φ))) max s N m {V(n+ s, φ(s))}. Then, system () s UAS. Proof. By Theorem 3., t s easy to see that system () s US. So we just need to prove that system () s also unformly attractve to the trval soluton. Wthout loss of generalty, let n 0 = 0. Denote x(n) = x(n, n 0, ). For any fxed postve number H>0, we choose a postve number 0 < δ H such that c 2 (δ) = c (H ). Thus, from the unform stablty of system (), for any C([ m, 0],R n ),f m δ, then we have x(n) H, n m (9) and V (n, x(n)) < c 2 (δ), n m. (0) It follows from (0) that V (n) c 2 (δ) for any n N. In order to show the unform attracton of trval soluton, we need to prove that, for any postve real number η satsfyng 0 < η H, there exsts a postve nteger K = K(η, δ) ndependent of n 0 such that when m δ and n K + m, wehave x(n) = x(n, n 0, ) η. () If for any n K, wehave V (n, x(n)) c (η), (2) then, by condton (), () can be nduced from (2). Hence, n the followng, we just prove that (2) holds. For any postve real number s wth c (η) s c 2 (δ), then there exsts a postve number a>0 such that p(s) s>a. Let K 0 = mn{u : u N,c (η) + u a c 2 (δ)}, r = mn c (η) s c (H ){q(s)}, and let M (0) = [c 2 (δ)/r], M (j) = M (j ) + m, =, 2,...,K 0,j, where [X] stands for the maxmal nteger less than X. We clam that for =, 2,...,K 0, V (n, x(n)) c (η) + (K 0 )a, n M ( ). (3) When =, we frst show that there exsts a postve nteger n wth n M (0) such that V(n,x(n )) c (η) + (K 0 )a. (4) Otherwse, for any n satsfyng 0 n M (0),wehave V (n, x(n)) > c (η) + (K 0 )a. (5) For any 0 n M (0), s N m, t follows from (0) and (5) that p(v (n, x(n))) > V (n, x(n)) + a>c (η) + K 0 a c 2 (δ) V(n+ s, x(n + s)). Thus, by condton () and c (H ) = c 2 (δ), for any 0 n M (0), we have that V(n+,x(n+ )) V (n, x(n)) q(v (n, x(n))) r, whch mples that V(n+,x(n+ )) V(0,x(0)) r(n + ) c 2 (δ) r(n + ). (6) Let n = M (0). It follows from (6) that V(M (0) +,x(m (0) + ))<c 2 (δ) c 2 (δ) = 0. (7) Ths s a contradcton wth the postve defnteness of V (n, x(n)). Hence, (4) holds for some n : n M (0). Then, we show that for any n n, V (n, x(n)) c (η) + (K 0 )a. (8) Otherwse, there exsts an n n such that V( n +,x( n + ))>c (η)+(k 0 )a V( n,x( n )). Thus, from the fact that V(n+ ) V (n) (see (7) n the proof of Theorem 3.), there exsts an s N m such that V( n + s, x( n + s)) V( n +,x( n + ))>V( n,x( n )), Thus, by condton (),weget V( n ) p(v ( n +,x( n + ))) >V( n +,x( n + )) + a c 2 (η). (9) Ths s a contradcton wth V (n) c 2 (η). Thus, (8) holds for all n n. It follows from (8) and M (0) n that (3) holds when =. Now we assume that > and (3) holds for. Then, t follows from the nducton assumpton that V (n, x(n)) c (η) + (K 0 + )a for any n M ( 2), whch leads to V (n, x(n)) c (η) + (K 0 + )a, for any n M ( ). Smlar to the case =, frst, we show that there s n 2 [M ( ),M( ) ] such that V(n 2,x(n 2 )) c (η) + (K 0 )a. Otherwse, for any n [M ( ),M( ) ], wehave

4 222 B. Lu, H.J. Marquez / Automatca 43 (2007) c (η) + (K 0 )a < V (n, x(n)) c (η) + (K 0 + )a. It leads to, for any n [M ( ),M( ) ] and any s N m, p(v (n, x(n))) > V (n, x(n)) + a c (η) + (K 0 + )a V(n+ s, x(n + s)). Thus, by condton () and c (H ) = c 2 (δ), for any n,m( ) ], we have that [M ( ) V(n+,x(n+ )) V (n, x(n)) r, (20) whch mples that V(M ( ) V(M ( ) +,x(m ( ) + )),x(m( ) )) r(m( ) + M ( ) ) c 2 (δ) r(m (0) + M (0) ) <c 2 (δ) c 2 (δ) = 0. (2) Ths s a contradcton. Hence, there s n 2 [M ( ),M( ) ] such that V(n 2,x(n 2 )) c (η)+(k 0 )a. Then, we show that V (n, x(n)) c (η)+(k 0 )a holds for any n n 2. Otherwse, there exsts an n 2 n 2 such that for any n 2 n n 2, V( n 2 +,x( n 2 +))>c (η)+(k 0 )a V (n, x(n)). (22) From (22) and the fact that V(n + ) V (n), there exsts an s N m such that V( n 2 + s, x( n 2 + s)) V( n 2 +,x( n 2 + ))>V( n 2,x( n 2 )). Thus, by condton () and n 2 n 2 M ( ) = M ( 2) + m, weget V( n 2 ) p(v ( n 2 +,x( n 2 + ))) >c (η) + (K 0 + )a V( n 2 ). (23) Ths s a contradcton. Hence, V (n, x(n)) c (η) + (K 0 )a holds for any n n 2, whch mples that V (n, x(n)) c (η) + (K 0 )a holds for any n M ( ). Thus, (3) holds for the case. Hence, by the nducton prncple, (3) holds for all =, 2,...,K 0. Let K = M (K 0 ) K 0, then, for any n K, t follows from (3) that V (n, x(n)) c (η). Thus, (2) holds and hence () holds. The proof s complete. Remark 3.. It should be notced that, n Zhang and Chen (998), condton () s changed nto: ( ) V(n+ s, x(n + s)) p(v (n, x(n))) mples V (n, x(n)) V(n,x(n )) q( x(n) ). Hence, Razumkhn-type unformly asymptotc stablty theorem s backward,.e., comparng between V (n, x(n)) and ts backward tem V(n,x(n )) after havng compared V (n, x(n)) and V(n,x(n )),...,V(n m, x(n m)). Thus, condton ( ) puts one restrcton on V( ) or p( ) and q( ): V (n, x(n)) p(v (n, x(n))) q( x(n) ) or p(s) s +q(c (s)) for all s R +. In Theorem 3.2, t s forward,.e., comparng between V (n, x(n)) and ts forward tem V(n+,x(n+ )), and hence there s no such knd of restrcton on V( ) or p( ) and q( ). In addton, n Zhang and Chen (998), functons p( ) and q( ) need to satsfy other restrctve condtons: p( ) s needed to be a non-decreasng functon and q( ) K. Here, n Theorem 3.2, t s not necessary for them. Hence, Theorem 3.2 s forward Razumkhn-type theorem beng less restrctve than that n Zhang and Chen (998). Theorem 3.3. Assume that there exst a postve defnte functon V (n, x) and constants r>0,p>,c > 0,c 2 > 0, >λ>0 such that the followng condtons hold: () c x r V (n, x) c 2 x r ; () for any φ C([ m, 0],R n ) and s N m, then V(n+s, φ(s)) pv (n, φ(0)) mples V(n+, f (n, φ)) λv (n, φ(0)); () for any φ C([ m, 0],R n ) and for some s N m {0}, then V(n+ s, φ(s)) > e α V (n, φ(0)) mples V(n+, f (n, φ)) (/p) max s N m {V(n+s, φ(s))}, where α= mn{ln(/λ), (ln p)/(m + )}. Then, system () s UES and ts Lyapunov exponent should not be greater than α/r. Proof. Let x(n) = x(n, n 0, ) be a soluton of system (). Just as n Theorem 3.2, wthout loss of generalty, let n 0 =0. Defne U(n) = max θ N m {e α(n+θ) V(n+ θ,x(n+ θ))}, n N. In the followng, we prove that U(n+ ) U(n), n N. (24) For every fxed n N, we defne θ n = max{θ N m : e α(n+θ) V(n+ θ,x(n+ θ)) = U(n)}. Then, we have U(n) = e α(n+ θ n ) V(n+ θ n,x(n+ θ n )). When θ n, then, for any θ N m {0}, e α(n++θ) V(n+ + θ,x(n++θ)) e α(n+ θ n ) V(n+ θ n,x(n+ θ n )), whch mples that max θ N m {0} {eα(n++θ) V(n++θ,x(n++θ))} U(n). (25) We clam that e α(n+) V(n+,x(n+ )) e α(n+ θ n ) V(n+ θ n,x(n+ θ n )). (26) By the defnton of θ n, we get e α(n+ θ n ) V(n + θ n,x(n + θ n )) > e αn V (n, x(n)), whch mples that V(n + θ n,x(n + θ n )) > e α θ n V (n, x(n)) e α V (n, x(n)). It follows from condton () and θ n that max {V(n+ s, x(n + s))} pv (n +,x(n+ )) s N m e α(m+) V(n+,x(n+)). (27)

5 B. Lu, H.J. Marquez / Automatca 43 (2007) Thus, by (27), we have e α(n+) V(n+,x(n+ )) e αm e αn max {V(n+ s, x(n + s))} s N m max {e α(n m) V(n+ s, x(n + s))} s N m e α(n+ θ n ) V(n+ θ n,x(n+ θ n )). (28) Hence, (26) holds and hence from (25) and (26), we get that U(n+ ) U(n), for θ n. When θ n = 0, then, for any θ N m, e α(n+θ) V(n+ θ,x(n+ θ)) e αn V (n, x(n)), whch mples that V(n + θ,x(n + θ)) e αθ V (n, x(n)) e αm V (n, x(n)) pv (n, x(n)). Thus, by condton (), we get V(n+,x(n+ )) λv (n, x(n)). (29) It follows from (29) that e α(n+) V(n+,x(n+ )) e αn e α λv (n, x(n)) e αn V (n, x(n)) = U(n). (30) Moreover, by θ n = 0, we have, for θ N m {0}, e α(n++θ) V(n+ + θ,x(n+ + θ)) U(n). (3) Hence, we obtan that U(n+ ) U(n), for θ n = 0. Therefore, (24) holds and t leads to, for any n N, U(n) U(0) max θ N m {V(θ,x(θ))}. (32) Thus, by the defnton of U(n) and (32), we get that V (n, x(n)) e αn max θ N m {V(θ,x(θ))}. (33) It follows from (33) and condton () that x(n) ( c2 c ) /r e (α/r)n m, n N. (34) Thus, the concluson of the theorem s true. Remark 3.2. ( ) Theorem 3.3 can easly be used to test exponental stablty. Moreover, for those practcal systems whch have hgh requrement on convergence speed such as synchronzaton of network controlled system (Lu, Lu, Chen, & Wang, 2005), the exponental stablty s more sgnfcant than stablty or asymptotc stablty. Hence, Theorem 3.3 may be of more sgnfcance n practcal applcatons than Theorems 3. and 3.2. (2 ) The convergence speed s estmated n Theorem 3.3. Here, α = mn{ln(/λ), (ln p)/(m + )} s the man factor that affects heavly the convergence speed. From here, one can see that f the maxmum tme delay m s suffcently large such that α = (ln p)/(m + ), then the larger m leads to the slower convergence speed. 4. Applcaton to a class of dscrete delay systems In ths secton, we focus our attenton on a specal class of dscrete delay systems. Ths class of dscrete delay systems s often encountered n practcal applcatons such as networked control systems wth tme delays, see Zhvoglyadov and Mddleton (2003) and Lu and Chen (2004). Consder the dscrete delay systems of the form { x(n + ) = f (n, x(n), x(n h ),...,x(n h m0 )), (35) x 0 =, where f C(N R } n R {{ n },R n ), and h j (n) m 0 + {, 2,...,m}, for any n N, and j =, 2,...,m 0. Theorem 4.. Assume that condton () of Theorem 3.3 holds, whle condtons () () of Theorem 3.3 are replaced by the followng condton () : () there exst postve constants 0 < λ <, 0 < λ <, =, 2,...,m 0, such that V(n+,x(n+)) λv (n, x(n))+ m0 = λ V(n h (n), x(n h (n))). If λ + m 0 = λ <, then system (35) s UES and ts Lyapunov exponent s less than or equal to ln p/r(m+ ), where p> s the unque root of the equaton m 0 λ p λ = ln p m +. (36) = Proof. If λ > m 0 = λ, then we see that Eq. (36) has a unque root satsfyng <p< λ/ m 0 = λ. Thus, for any h (n) N m,fv(n h (n), x(n h (n))) pv (n, x(n)), then, by (),wehave ( ) m 0 V(n+,x(n+ )) λ + p λ V (n, x(n)). = It follows from the fact λ + p m 0 = < that the condton () of Theorem 3.3 s satsfed. Denote λ = λ + p m 0 = λ. From (36), we get that ln p/(m + ) = λ. It s easy to see from the propertes of contnuous functon that for any 0 < λ <, we have λ < ln λ, whch mples that ln p/(m + ) ln λ holds. Hence, α = mn{ln / λ, ln p/(m + )} =ln p/(m + ). Thus, we get that e α = /pe αm < /p. By (36), we have m 0 λ j = j= j= ( λ) ln p/(m + ) p = p λ p ln p p(m + ). (37) It follows from (37) that λe α m 0 + λ j < λ p + p λ p ln p p(m + ) < p. (38)

6 224 B. Lu, H.J. Marquez / Automatca 43 (2007) Thus, f for some h (n) N m {0}, V(n h (n), x(n h (n))) > e α V (n, x(n)), then, by condton () and (38), t follows that V(n+,x(n+)) (λe α + m 0 j= λ j ) max j m0 {V(n h j (n), x(n h j (n)))} (/p) V (n). Therefore, the condton () of Theorem 3.3 also holds. Thus, the concluson of the theorem s true x Theorem 4.2. Assume that condton () of Theorem 3.3 holds, whle condtons () () of Theorem 3.3 are replaced by the followng condton: () there exsts a postve constant 0 < λ < such that V(n+,x(n+ )) λ V (n). Then, system (35) s UES and ts Lyapunov exponent s less than or equal to ln λ/r(m + 2). x(n) x 2 x n Proof. Let p = (/λ) (m+)/(m+2), then, from λ <, we get that <p</λ, and ln p/(m + ) = ln /λp. Thus, for any h (n) N m,fv(n h (n), x(n h (n))) pv (n, x(n)), then, by (), we have V(n +,x(n + )) λ V (n) λpv (n, x(n)). It follows from the fact λp< that the condton () of Theorem 3.3 s satsfed. Let α = ln p/(m + ) = ln /λp. If for some h (n) N m {0}, V(n h (n), x(n h (n))) > e α V (n, x(n)), then, by condton (), t follows that V(n +,x(n + )) λ V(n)<(/p) V (n). Hence, the condton () of Theorem 3.3 also holds. Thus, the concluson of the theorem s true. Remark 4.. Theorem 4. can be seen as the counterpart of the Halanay-type nequalty on contnuous delay systems (Nculescu, 200). And Theorem 4.2 s a well-known stablty result for dscrete delay system (Lao, 200). But ts proof n Lao (200) s complcated. Here, by usng Theorem 3.3, we can easly derve the result. Moreover, the Lyapunov exponent can also be estmated. Furthermore, Theorem 4. can be derved from Theorem Examples To llustrate our theorems obtaned n the prevous secton, we now consder some llustratve examples. Example 5.. Consder the dscrete delay system: { x(n + ) = Ax(n) + F (n, x(n), x(n h(n))), x n0 =, (39) ( ) where m = 2, h(n) =, or 2, A = and F (n, x(n), x(n h(n))) = 4 ((x (n h(n))/( + sn 2 n + x(n) 2 ), x 2 (n h(n)) sn(x 3 (n)), x 3 (n h(n)) cos(x 3 (n))) T. Let V (n, x) = x T x, then V(n +,x(n + )) V (n, x(n)) V(n h(n), x(n h(n))), whch mples that condtons of Theorem 4. hold. Thus, by Theorem 4., Fg.. Numercal smulaton of Example 5.. system (39) s UES wth ts Lyapunov exponent less than or equal to The numercal smulaton s gven n Fg.. Example 5.2. Consder the dscrete delay system n the form of m 0 x(n + ) = A(n)x(n) + B (n)x(n h (n)), (40) = where x R n, A(n), B j (n) R n and h j (n) N m. Let A = sup n N { A(n) }, Bj = sup n N { B j (n) }, j =, 2,...,m 0. Denote λ A + B + B 2 + +B m 0. Let Lyapunov functon be V (n, x(n)) = x(n), then t s easy to get that V(n+,x(n+ )) λ V (n, x(n)). Hence, If λ <, then by Theorem 4.2, we obtan that system (40) s UES wth ts Lyapunov exponent less than or equal to ln λ/(m + ). Remark 5.. Example 5.2 s dscussed n Zhang and Chen (998). However, the UAS of system (40) s not easly tested by the results n Zhang and Chen (998). Here, not only the UAS, but also the UES of system (40) can be easly derved and the convergence speed also can be estmated. 6. Conclusons In ths paper, Razumkhn-type stablty theorems have been nvestgated for the general dscrete delay systems. By usng the Razumkhn technque and Lyapunov functon, forward Razumkhn-type theorems of unform stablty, unformly asymptotc stablty and unformly exponental stablty are derved, respectvely. Moreover, Razumkhn-type unformly exponental stablty theorem gves the estmaton of the convergence speed, n whch the tme delay s one of the man factors that affects the convergence speed. As theoretc applcaton, the Razumkhn-type unformly exponental stablty result s used to derve some well-known stablty results for some classcal dscrete delay systems. Fnally, some relevant examples have

7 B. Lu, H.J. Marquez / Automatca 43 (2007) been solved so as to llustrate the theoretcal results obtaned n ths paper. Acknowledgements Ths work s supported by the NSERC Canada. The authors would lke to thank the referees for ther helpful suggestons and remarks. References Hale, J. K. (977). Theory of functonal dfferental equatons. New York: Sprnger. Hale, J. K., & Lunel, S. M. V. (993). Introducton to functonal dfferental equatons. New York: Sprnger. Knospe, C. R., Roozbehan, M. (2003). Stablty of lnear systems wth nterval tme-delay. In Proceedngs of the Amercan control conference (Vol. 2, pp ). Lao, X. X. (200). Theory and applcaton of stablty for dynamcal systems. Bejng: Natonal Defence Industry Press. Lu, B., Lu, X., Chen, G., & Wang, H. (2005). Robust mpulsve synchronzaton of uncertan dynamcal networks. IEEE Transactons on Crcuts and Systems I: Regular Paper, 52, Lu, B., Lu, X., Teo, K. L., & Wang, Q. (2006). Razumkhn-type theorems on exponental stablty of mpulsve delay systems. IMA Journal of Appled Mathematcs, 7, Lu, X., & Ballnger, G. (200). Unform asymptotc stablty of mpulsve delay dfferental equatons. Computers and Mathematcs wth Applcatons, 4, Lu, W., & Chen, T. (2004). Synchronzaton analyss of lnearly coupled networks of dscrete tme systems. Physca D, 98, Mao, X. (997). Razumkhn-type theorems on exponental stablty of neural stochastc functonal dfferental equatons. SIAM Journal on Mathematcal Analyss, 28, Marquez, H. J. (2003). Nonlnear control systems: Analyss and desgn. New Jersey: Wley. Nculescu, S. (200). Delay effects on stablty: A robust control approach. Lecture notes n control and nformaton scence. New York, Berln Hedelberg: Sprnger. Shen, J. H., & Yan, J. (998). Razumkhn type stablty theorems for mpulsve functonal dfferental equatons. Nonlnear Analyss, 33, Stamova, I. M., & Stamov, G. T. (200). Lyapunov Razumkhn method for mpulsve functonal equatons and applcatons to the populaton dynamcs. Journal of Computatonal and Appled Mathematcs, 30, Teel, A. R. (998). Connectons between Razumkhn-type theorems and the ISS nonlnear small gan theorem. IEEE Transactons on Automatc Control, 43, Teel, A. R., Nešć, D., & Kokotovć, P. V. (998). A note on nput-to state stablty of sampled-data nonlnear systems. In Proceedngs of the 37th IEEE conference on decson and control (pp ), Tampa, FL. Zhang, J., Knospe, C. R., & Tsotras, P. (200). Stablty of tme-delay systems: Equvalence between Lyapunov and scaled small gan condtons. IEEE Transactons on Automatc Control, 46, Zhang, J., Knospe, C. R., & Tsotras, P. (2003). New results for the analyss of lnear systems wth tme-nvarant delays. Internatonal Journal of Robust and Nonlnear Control, 3, Zhang, S., & Chen, M. P. (998). A new Razumkhn theorem for delay dfference equatons. Computers and Mathematcs wth Applcatons, 36, Zhvoglyadov, P. V., & Mddleton, R. H. (2003). Networked control desgn for lnear systems. Automatca, 39, Bn Lu receved the M.Sc. degree from the Department of Mathematcs, East Chna Normal Unversty, Shangha, Chna, n 993, and Ph.D. degree from the Department of Control Scence and Engneerng, Huazhong Unversty of Scence and Technology, Wuhan, Chna, n June He was a Post-Doctoral Fellow at the Huazhong Unversty of Scence and Technology from July 2003 to July 2005, a Post-Doctoral Fellow at the Unversty of Alberta, Edmonton, Alta., Canada, from August 2005 to October 2006 and a vstng Research Fellow at the Hong Kong Polytechnc Unversty, Hong Kong, Chna, n Snce July 993, he has been wth the Department of Informaton and Computaton Scence, Hunan Unversty of Technology, Hunan, Chna, where he became an Assocate Professor n 200, and has been a Professor snce Hs research nterests nclude stablty analyss and applcatons of nonlnear systems and hybrd systems, chaos synchronzaton and control and Le algebra. Horaco J. Marquez receved the B.Sc. degree from the Insttuto Tecnologco de Buenos Ares, Argentna, and the M.Sc.E and Ph.D. degrees n electrcal engneerng from the Unversty of New Brunswck, Fredercton, Canada, n 987, 990 and 993, respectvely. From 993 to 996 he held vstng appontments at the Royal Roads Mltary College, and the Unversty of Vctora, Vctora, BC. Snce 996 he has been wth the Department of Electrcal and Computer Engneerng, Unversty of Alberta, where he s currently a Professor and Department Char. He s the author of Nonlnear Control Systems: Analyss and Desgn (Wley, 2003). He receved the 2003/2004 Unversty of Alberta McCalla Research Professorshp. Hs current research nterests nclude nonlnear dynamcal systems and control, nonlnear observer desgn, robust control and applcatons.