Sets of generalized exponential stability criteria for switched multilayer dynamic neural networks

Ahn and Lee Advances in Difference Equations 2012, 2012:150 http://www.advancesindifferenceequations.com/content/2012/1/150 RESEARCH Open Access Sets of generalized H exponential stability criteria for switched multilayer dynamic neural networks Choon Ki Ahn1* and Young Sam Lee2 * Correspondence: 1 School of Electrical Engineering, Korea University, Anam-dong, Seongbuk-Gu, Seoul, 136-701, Korea Full list of author information is available at the end of the article Abstract This paper investigates new sets of generalized H2 exponential stability criteria for switched multilayer dynamic neural networks. These sets of sufficient stability criteria in forms of linear matrix inequality (LMI) and matrix norm are presented, under which switched multilayer dynamic neural networks reduce the effect of external input to a predefined level. The proposed sets of criteria also guarantee exponential stability for switched multilayer dynamic neural networks without external input. 1 Introduction Switched systems are a class of hybrid systems consisting of a family of continuous (or discrete) time subsystems and a logical rule that orchestrates the switching between these subsystems. Switched systems have been extensively researched, and several efforts have been focused on the analysis of switched systems [, ]. Recently, switched recurrent neural networks were introduced to represent some complex nonlinear systems efficiently by integrating the theory of switched systems with recurrent neural networks [–]. Switched dynamic neural networks have found applications in the field of artificial intelligence and high speed signal processing [, ]. In [–], some stability criteria for switched dynamic neural networks were studied. There always exist noise disturbances and model uncertainties in real physical systems. Recently, this has led to an interest in a generalized H approach [–]. The generalized H approach has been known as a significant concept to examine the stability of various nonlinear dynamical systems. Here, we have the following natural question: Can we obtain a generalized H stability criterion for switched dynamic neural networks. This paper provides an answer to this question. To the best of the authors’ knowledge, the generalized H analysis of switched dynamic neural networks has not yet been studied in the literature. In this paper, we propose new sets of generalized H exponential stability criteria for switched multilayer dynamic neural networks. The sets of conditions proposed in this paper are a new contribution to the stability evaluation of switched neural networks. The proposed sets of sufficient stability criteria in forms of linear matrix inequality (LMI) and matrix norm guarantee that switched multilayer dynamic neural networks reduce the effect of external input to a predefined level. This paper is organized as follows. In Section , new sets of generalized H exponential stability criteria are derived. Finally, conclusions are presented in Section . © 2012 Ahn and Lee; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Ahn and Lee Advances in Difference Equations 2012, 2012:150 http://www.advancesindifferenceequations.com/content/2012/1/150 Page 2 of 8 2 New sets of generalized H2 exponential stability criteria Consider the following model of switched multilayer neural networks []:   ẋ(t) = Aα x(t) + Wα φ Vα x(t) + J(t), () z(t) = Hα x(t), () where x(t) = [x (t) · · · xn (t)]T ∈ Rn is the state vector, z(t) ∈ Rp is a linear combination of the states, A = diag{–a , . . . , –an } ∈ Rn×n (ak > , k = , . . . , n) is the selffeedback matrix, W ∈ Rn×n and V ∈ Rn×n are the connection weight matrices, φ(x(t)) = [φ (x(t)) · · · φn (x(t))]T : Rn → Rn is the nonlinear function vector satisfying the global Lipschitz condition with Lipschitz constant Lφ > , J(t) ∈ Rn is an external input vector, and H ∈ Rp×n is a known constant matrix. α is a switching signal which takes its values in the finite set I = {, , . . . , N}. The matrices (Aα , Wα , Vα , Hα ) are allowed to take values, at an arbitrary time, in the finite set {(A , W , V , H ), . . . , (AN , WN , VN , HN )}. Throughout this paper, we assume that the switching rule α is not known a priori and its instantaneous value is available in real time. Define the indicator function ξ (t) = (ξ (t), ξ (t), . . . , ξN (t))T , where ⎧ ⎨, when the switched system is described by the ith mode (A , W , V , H ), i i i i ξi (t) = ⎩, otherwise, with i = , . . . , N . By using this indicator function, the model of the switched multilayer neural networks ()-() can be written as ẋ(t) = N     ξi (t) Ai x(t) + Wi φ Vi x(t) + J(t) , () ξi (t)Hi x(t), () i= z(t) = N  i= where N i= ξi (t) =  is satisfied under any switching rules. Let γ >  be a predefined level of disturbance attenuation. In this paper, for a given κ > , we derive sets of criteria such that the switched multilayer neural network ()-() with J(t) =  is exponentially stable (x(t) ≤ ϒ exp(–κt), where ϒ > ) and ∞ sup exp(κt)zT (t)z(t) < γ  t≥ exp(κt)J T (t)J(t) dt, ()  under zero-initial conditions for all nonzero J(t) ∈ L [, ∞), where L [, ∞) is the space of square integrable vector functions over [, ∞). A set of generalized H exponential stability criterion of the switched multilayer neural network ()-() is derived in the following theorem. Theorem  For given γ >  and κ > , the switched multilayer neural network ()-() is generalized H exponentially stable if  Wi  < ki – P – κP , Lφ Vi  () Ahn and Lee Advances in Difference Equations 2012, 2012:150 http://www.advancesindifferenceequations.com/content/2012/1/150 Page 3 of 8 Vi  =  ,  () P < () ki , ki > , P = PT > , +κ  Hi  ≤ γ λmin (P), () for i = , . . . , N , where λmin (·) is the minimum eigenvalue of the matrix and P satisfies the Lyapunov inequality ATi P + PAi < –ki I. Proof We consider the Lyapunov function V (t) = exp(κt)xT (t)Px(t). The time derivative of the function along the trajectory of () satisfies V̇ (t) < N    ξi (t) exp(κt) –xT (t)[ki – κP]x(t) + xT (t)PWi φ Vi x(t) + xT (t)PJ(t) . () i= Applying Young’s inequality [], we have xT (t)PWi φ(Vi x(t)) ≤ xT (t)PPx(t) + φ T × (Vi x(t))WiT Wi , φ(Vi x(t)) ≤ P x(t) +Lφ Wi  Vi  x(t) and xT (t)PJ(t) ≤ xT (t)P × PT x(t) + J T (t)J(t) ≤ P x(t) + J(t) . Substituting these inequalities into (), we have V̇ (t) < N       ξi (t) exp(κt) – ki – ( + κ)P – Lφ Wi  Vi  x(t) + J(t) i= =– N     ξi (t) exp(κt) ki – ( + κ)P – Lφ Wi  Vi  x(t) i= + N    ξi (t) exp(κt)J(t) . () i= If the following condition is satisfied (...truncated)