Estimate of parameter-deviation degree for
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wn(01) en
L
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en
⎤ ⎥⎦n×n2
,
and
39
Zhiyuan Xue, Chuandong Li, and Rong Zhang
Estimate of parameter-deviation degree for delayed neural networks
FWT
=
⎡ ⎢⎣
w1(01) e1
Keyword: Neural networks, time delays,
I. Introduction
During recent decades, several neural network models with or without delays have been extensively studied, particularly, regarding their stability analysis [1-8]. However, It is well known that the stability of a given system may often be destroyed by its unavoidable uncertainty due to the existence of modeling error, external disturbance and parameter fluctuation during the implementation. So it is essential to introduce the robust technique to design a system with such uncertainty. If the uncertainty of a system is only due to the deviations and perturbations of its parameters, and if these deviations and perturbations are all bounded, then the system is called an interval system. Recently, several global and robust stability criteria for interval neural networks with constant or time-varying delays have been proposed [9, 10-13]) since the pioneering work of Liao and Yu [10]. In most existing literature about the uncertain neural network model, the maximum permissible deviation degree of the system parameters has not been investigated and estimated. However, this is an unavoidable and important factor to be considered during the implementation. In this paper, we use functional differential equation to describe a neural network model with parameter deviation, and further study its robust stability with respect to the parameter deviation degree.
in the form of
f (xj(t))[f (xj(t))− k xj] ≤ 0
(4)
II. Stability Analysis
In this section, model (3) is first transformed into another equivalent form, which allows
2 College of Computer Science and Engineering, Chongqing University, Chongqing 400030, P.R. China xuezy@
3 College of Economics and Business Administration, Chongqing University, Chongqing 400030, P.R. China zhangrong@
For this purpose, consider a neural network with parameter deviations described by
38
International Journal of Information Technology, Vol. 11 No. 6 2005
Zhiyuan Xue, Chuandong Li, and Rong Zhang
Estimate of parameter-deviation degree for delayed neural networks
Estimate of parameter-deviation degree for delayed neural networks
degree of the system parameters is denoted by α ≥ 0 . Here, if we select the matrices
( ) A0 = diag ai(0) n×n > 0 (throughout this paper, we use Ω > 0 to denote the symmetrical positive ( ) matrix Ω ) and W0 = wi(j0) n×n as the reference matrices of A and W, respectively, then
L
w1(n0) en
L
wn(01) e1 L
wn(0n)
en
⎤ ⎥⎦n×n2
,
where ei denotes the i-th column-vector of the n× n identity matrix. Obviously, we have
EW EWT
=
diag⎜⎜⎛ ⎝
n
∑
j=1
w1(0j)
0
≤
g
j (x) −
x−
g y
j (y)
≤
k
,
for
any
x,
y ∈ R,
j
= 1, 2,L, n.
(2)
Due to the boundedness of the activation functions, there exists at least an equilibrium point u* for system (1). In the following, we always shift this equilibrium point into the origin. By
u&(t) = −Au(t) + W g(u(t − τ )) + I
(1)
[ ] ( ) where u(t) = u1(t), L ,un(t)T is the neuron state vector. A = diag a1, a2,L,an is a positive
diagonal matrix, g(u) = [g1(u1), L ,gn(un)]T denotes the neuron activation functions with
g(0) = 0 , I = [I1, L ,In]T is a constant vector, W ( ) = wij n×n is the connect-weighting matrix,
τ > 0 is transmission delay. Throughout this paper, we assume that the permissible deviation
A∈ N[A0 ,α] and W ∈ N[W0 ,α ], where
{ } N[A0,α] = diag(ai)n×n (1−α )ai(0) ≤ ai ≤ (1+ α )ai(0) ,
and
( ) } N [W0,α ] = ⎧⎨⎩ wij n×n (1−α ) wi(j0) ≤ wij ≤ (1+ α ) wi(j0) or (1+ α ) wi(j0) ≤ wij ≤ (1−α ) wi(j0)
Furthermore, we assume that each activation function in (1) is bounded and satisfies the following sector condition: There exists a positive constant, k > 0 , such that
us to analyze expediently. And then a simple robust stability condition for model (3) will be
derived.
For this purpose, let
EW
=
⎡ ⎢⎣
w1(01) e1
L
w1(n0) e1 L
Abstract
The stability of a neural network model may often be destroyed by the parameter deviations during the implementation. However, few results (if any) for the asymptotical stability of such system with a certain degree of parameter deviations have been reported in the literature. In this paper, we present a simple delayed neural network model, in which each parameter deviates the reference point with the degree no more than a certain value, and further investigate the robust asymptotical stability of this model and estimate the maximum permissible deviation degree.
