Improved delay-dependent stability criteria for discrete-time stochastic neural networks with time-varying delays

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1 Avalable onlne at Proceda Engneerng 5 ( Improved delay-dependent stablty crtera for dscrete-tme stochastc neural networs wth tme-varyng delays Meng-zhuo Luo a Shou-mng Zhong a,b a* a School of Mathematcs and Scence, Unversty Electronc Scence and echnology of Chna, Chengdu 673, PR Chna b Key Laboratory for NeuroInformaton of Mnstry of Educaton, Unversty of Electronc Scence and echnology of Chna, Chengdu 673, PR Chna Abstract hs letter, nvestgates the problem of mean square exponental stablty for a class of dscrete-tme stochastc neural networ wth tme-varyng delays. By constructng a approprate Lyapunov-Krasovs functonal, combnng the stochastc stablty theory, and the convex theory method, a delay-dependent exponental stablty crtera s obtaned n term of LMIs. Fnally, a numercal example s exploted to show the usefulness of the results derved. Keywords: Delay-dependent stablty; Neural networ; me-varyng delay; Lyapunov-Krasovs; Lnear matrx nequaltes.. Introducton Recent years have wtnessed a growng nterest n nvestgatng neural networs, ths s manly to the great potental applcatons n vares areas such as sgnal processng; pattern recoganzaton; statc mage processng; assocatve memory and combnatoral optmzaton []. As s nown to all, dynamcal behavors of neural networs are ey to the applcatons, and the acheved applcatons heavly depend on the dynamc behavors of equlbrum pont for neural networ, therefore, stablty s one of the most mportant ssues related to such behavor. It s worth pontng out that most neural networs are concerned wth contnuous-tme cases. Snce dscrete-tme neural networs play a more mportant role than ther contnuous-tme counterparts n * Correspondng author. el.: E-mal address: zhuozhuohuahua@63.com Publshed by Elsever Ltd. Open access under CC BY-NC-ND lcense. do:.6/j.proeng

2 Meng-zhuo Luo and Shou-mng Zhong / Proceda Engneerng 5 ( today s dgtal lfe, moreover, n mplementng and applcatons of neural networs, dscrete-tme neural networs also tae a more crucal ey than ther contnuous-tme counterparts []. herefore, both analyss and synthess problem for dscrete-tme neural networs have been extensvely studed and a great number of mportant results have been reported n the lterature [3-4]. But t s our observaton that there stll exst room for further mprovement by constructng ratonal Lyapunov functonals whch motvates the present study... Problem formulaton an prelmnares Consder the followng dscrete-tme stochastc neural networs (DSNNs wth tme-varyng delays: x( + = Cx( + Af ( x( + Bf ( x( τ ( + δ (, x(, x( τ( ω( (. where xt ( = x, x,, xn R n s the neuron state vector. C = dag c, c,, cn wth c <, descrbed the rate wth whch the th neuron wll reset ts state n solaton when dsconnected from networs and external nputs. satsfes < τ τ τ. Assumpton. For any x, y R, x y, l Assumpton. here exst a constant matrx f x f y x y τ ( s tme-varyng delay and l + (. G, and s assumed to satsfy τ τ x x δ ( x,, x ( τ ( δ ( x,, x ( τ( G x( x( G G where G =.. * G 3 3. Man result heorem. Suppose that Assumpton (- hold. hen the DSNNs (. s exponental stable n the mean square f there exst postve defnte matrces PQ,, E, E, Z, dagonal matrces D >, =,, K >, L >, and postve scalars ε > such that followng LMIs hold:

3 4458 Meng-zhuo Luo and Shou-mng Zhong / Proceda Engneerng 5 ( Q Q P ρi Z ρi Q= > * Q3 (3.4 Ξ ρg + τ τ ρg Ξ 5 C PB+ τ τ CI ZB * Ξ μz μz Ξ6 * * E μz < * * * E μz * * * * Ξ APB+ * * * * * Ξ66 Ξ = ρg + τ τ ρg Q + Γ K Γ LΓ D Γ 3μZ ( τ τ 55 AZB Ξ ρg + τ τ ρg Ξ 5 C PB+ τ τ CI ZB * Ξ μz μz Ξ6 * * E μz < * * * E μz * * * * Ξ APB+ * * * * * Ξ66 where Ξ = C PC P+ ρg + θq + E + E θγ K + θγ L + τ τ CI Z C I + τ τ ρg Γ DΓ ( Ξ = C PA+ θq + θk θl+ τ τ C I ZA+ D Γ +Γ ( τ τ θ ( τ τ Ξ =Q K + L+ D Γ +Γ Ξ = A PA+ Q + A ZAD Ξ 66 = BPB Q3 + ( τ τ BZB D θ τ τ = + μ = τ τ [ I I ] [ I I ] Π = Π = Proof of heorem. ae the followng L-K functonal canddate as follows (3.5 AZB (3.6

4 Meng-zhuo Luo and Shou-mng Zhong / Proceda Engneerng 5 ( v ( x = x ( Px( ( ( ( ( ( ( τ ( ( ( ( ( ( x x x x v( x = Q + Q = τ( f x f x j= + τ = j f x f x τ+ { ( ( ( ( ( ( } ( v ( x = f x Γ x K + Γ x f x L x 4 j= τ + = + j 5 τ j= τ = + j ( ( ( ( v ( x = η Zη η = x + x ( = ( + ( + E v x E x Px x Px ( ( ( ( ( E Δv x E x θq x + x θq f x + f x θq f x 3 ( τ ( τ 3 ( τ ( τ ( τ ( τ x f x Q f x Q x x Q f x 3( = ( ( + ( τ ( τ ( τ ( τ 4( θ ( Γ ( ( τ ( τ E v x E x E E x x E x x E x { E v x E f x x Kx f x Kx ( τ ( τ θ ( ( ( τ ( τ ( τ τ + x Γ Kx + Γ x f x Lx } + f x Lx x Γ Lx τ E v5( x = E( τ τ η ( Zη( η ( Zη( = τ τ ( τ τ ( ( ( ( ( ( η Zη = η Zη η Zη = τ = τ = τ ξ ( Π ( ( Iτ Π ξ + ξ ΠI τ Πξ ( ξ ( = x (, x ( τ (, x ( τ, x ( τ, f ( x(, f x τ ( Π ( τ ( = ( ( τ ( τ τ τ τ τ Z (

5 446 Meng-zhuo Luo and Shou-mng Zhong / Proceda Engneerng 5 ( Π ( τ( = τ τ ( τ τ ( τ τ ( Z Now combnng above dscusson, we have a upper bound as ( ξ ( ( τ τ ξ E Δv E Ξ+Π I Π +Π I Π Ξ+Π ( Π +Π ( Π < hen f we want to have I τ I τ for τ τ τ, whch are equvalent to handle followng two LMIs by the convex combnaton theory: Ξ+Π I ( τ Π +Π I ( τ Π < and Ξ+Π I ( τ Π +Π I ( τ Π < that are equvalent to (3.5 and (3.6 hold. herefore, f the LMIs ( hold, we utlze the smlar method proposed n the [3], we can now the system (. s mean square exponental stablty. 4. Example Consder the dscrete-tme stochastc neural networ (. wth:.8... C = A B.9 =.5 =.. Γ = he actvaton functon satsfy Assumpton wth dag, Γ =dag.5.5. By the Matlab LMI Control oolbox, we fnd a soluton to the LMIs ( P= D = D = K = Z = References [] A. Ccho, R. Unbehauen, Neural Networs for Optmzaton and Sgnal Processng, Wley, Chchester, 993 [] A. Stuart, A. Humphres, Dynamcal Systems and Numercal Analyss, Cambrdge Unversty, 998. [3] M. Luo, S. Zhong, R. Wang, W. Kang, Robust stablty analyss of dscrete-tme stochastc neural networs wth tme-varyng delays, Appl. Math. Comput 9; 9: [4] Y. ang, J. Fang, M. Xa, D. Yu, Delay-dstrbuton-dependent stablty of stochastc dscrete-tme neural networs wth randomly mxed tme-varyng delays, Neurocomputng 9; 7: