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Delay-dependent stability analysis of neural networks with time-varying delay: A generalized free-weighting-matrix approach 在深入分析题目“Delay-dependent stability analysis of neural networks with time-varying delay: A generalized free-weighting-matrix approach”和给定的内容片段后，我们可以提取以下知识点： 1. 神经网络的延迟依赖稳定性问题：文章主要研究的是具有时间变化延迟的神经网络的稳定性问题。时间变化延迟是实际应用中常见的问题，由于放大器有限的切换速度和神经元之间的固有通信时间，都不可避免地会导致延迟的产生。这些延迟可能会导致不希望的动态行为，例如振荡和不稳定。 ***apunov-Krasovskii泛函(LKF)方法：文中提到利用LKF方法来分析神经网络的稳定性问题。LKF方法是一种强有力的数学工具，被广泛应用于时滞系统稳定性分析中，其核心思想是构造一个合适的Lyapunov泛函，通过分析其导数的性质来判断系统的稳定性。 3. 一般化自由加权矩阵(GFWM)方法：GFWM方法是为了更精确估计LKF导数而开发的。这个方法基于几个零值等式，并且文章中理论证明了GFWM方法相比现有的常用方法，在延迟依赖稳定性准则的保守性方面有所降低。 4. 时间变化延迟的神经网络稳定性准则：研究者应用GFWM方法来研究具有延迟的神经网络的稳定性，并推导出若干稳定性准则。这些准则有助于确定使得具有延迟的神经网络保持稳定的可接受的最大延迟界限(AMDB)。 5. 神经网络的应用：文章开头提到了神经网络在图像处理、模式识别、联想记忆、优化问题等领域的成功应用。但为了确保这些应用的成功，神经网络的稳定性是一个重要条件。 6. 数值例子和理论优势的验证：为了验证所提出的稳定准则的优势，文中给出了三个数值例子。这些例子说明了通过GFWM方法得到的稳定性准则在减少稳定性准则的保守性方面的改进。综合上述知识点，可以概括出，本论文的研究重点是发展了一种新的方法来分析具有时间变化延迟的神经网络的稳定性问题，提出了一般化自由加权矩阵(GFWM)方法，并通过理论分析和数值例子证明了该方法在降低稳定性准则保守性方面的优势。这对于进一步优化神经网络的设计和应用具有重要的意义。

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Applied Mathematics and Computation 294 (2017) 102–120

Contents lists available at ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier.com/locate/amc

Delay-dependent stability analysis of neural networks with

time-varying delay: A generalized free-weighting-matrix

approach

Chuan-Ke Zhang

a , b , ∗

, Yong He

, Lin Jiang

, Wen-Juan Lin

, Min Wu

School of Automation, China University of Geosciences, Wuhan 430074, China

Department of Electrical Engineering & Electronics, University of Liverpool, Liverpool L69 3GJ, United Kingdom

a r t i c l e i n f o

Keywords:

Neural networks

Time-varying delay

Generalized free-weighting-matrix approach

Stability

a b s t r a c t

This paper investigates the delay-dependent stability problem of continuous neural net-

works with a bounded time-varying delay via Lyapunov–Krasovskii functional (LKF)

method. This paper focuses on reducing the conservatism of stability criteria by estimating

the derivative of the LKF more accurately. Firstly, based on several zero-value equalities, a

generalized free-weighting-matrix (GFWM) approach is developed for estimating the single

integral term. It is also theoretically proved that the GFWM approach is less conservative

than the existing methods commonly used for the same task. Then, the GFWM approach

is applied to investigate the stability of delayed neural networks, and several stability cri-

teria are derived. Finally, three numerical examples are given to verify the advantages of

the proposed criteria.

Introduction

Neural networks have been successfully applied in image processing, pattern recognition, associative memory, optimiza-

tion problem, etc. [1,2,3] . For those applications, the artiﬁcial neural networks usually must be stable [6] . However, the ﬁnite

switching speed of ampliﬁers and the inherent communication time between neurons inevitably cause time delays during

the implementation of artiﬁcial neural networks [7] . These delays might lead to undesired dynamics like oscillation and in-

stability. Therefore, it is an important job to determine the admissible maximal delay bound (AMDB) such that the delayed

neural networks (DNNs) with a delay less than this bound remain stable. In the past few decades, such delay-dependent

stability analysis problem has become a hot issue in the ﬁeld of neural networks [4] .

These delays are usually time-varying and Lyapunov–Krasovskii functional (LKF) method can easily handle such DNNs,

thus, the LKF method has become the one of most popular methods for stability analysis and ﬁnding the AMDBs. The DNNs

with bounded delays are asymptotically stable if there exists an LKF, which consists of state vector based quadratic terms, is

positive deﬁnite and has a negative gradient over the time. Linear matrix inequalities (LMIs) based delay-dependent stability

criteria can be easily used to check whether or not such LKF can be found for the DNNs. However, the criteria derived have

some conservatism since they are only suﬃcient conditions. In order to ﬁnd the AMDBs more accurately, one important

issue in the related research is to develop new criteria with less conservatism. This paper also investigates the stability of

DNNs following this direction.

This work is supported partially by the National Natural Science Foundation of China under grant nos. 61503351 , 51428702 , and 61304011 , and the

Hubei Provincial Natural Science Foundation of China under grant 2015CFA010.

∗

Corresponding author. Fax: +86(0)2787175060.

E-mail address: ckzhang@cug.edu.cn (C.-K. Zhang).

http://dx.doi.org/10.1016/j.amc.2016.08.043

C.-K. Zhang et al. / Applied Mathematics and Computation 29 4 (2017) 102–120 103

By constructing a special form of LKF candidate, tractable LMI-based stability criteria are derived using necessary tech-

niques to estimate the LKF and its derivative. Therefore, the construction and the treatment for the LKF are the basic issues

related to how conservative the criteria are.

In the early research, the LKFs for the stability analysis of the DNNs were constructed by introducing delay-based single

and double integral terms into the typical non-integral quadratic form of Lyapunov function for delay-free systems [5,6,8–

14]

. Later, researchers have developed many new LKFs by making the previous ones more general in three aspects.

(1) Firstly, based on several subintervals divided from the whole delay region, some scholars have developed the delay-

partition-based LKFs by replacing the original integral terms with multiple new integral terms with smaller domain

of integration [16–30] .

(2) Secondly, by using various state vectors (current, delayed, and/or integrated state vectors etc. ), some scholars have

developed new LKFs by augmenting the quadratic terms of original LKFs [31–42] .

(3) Thirdly, since the triple integral term was found to be helpful for reducing the conservatism of stability criteria for

linear time-delay systems [47] , similar and/or extended forms have also been widely applied to stability analysis of

various DNNs [49–58] .

Although those LKFs have different forms, they all include a common term with the form of



−h



t+ θ

(s ) Ry (s ) d sd θ

(Here h > 0 is the scalar, y ( s ) is the system state-based vector, and matrix R > 0.). Then its derivative contains the term as

follows

−



t−h

(s ) Ry (s ) ds (1)

This term was directly dropped in the early literature [5] , but such treatment is very conservative. Later, this term was

retained to improve the results, in which case it must be estimated to represent the criterion in the form of tractable LMI.

As mentioned in [43] , the estimation of the above single integral term is strongly related to the conservatism of criteria.

Therefore, the stability criteria of DNNs have been improved gradually by using more effective techniques for this estimation

task.

The basic inequality was used to estimate the single integral term [6] . Since He et al. [44] proposed the free-weighting

matrix (FWM) approach, which is more effective than the basic inequality, the FWM approach has been widely used in the

stability analysis of DNNs [8–13,28–32] . The slack matrices introduced by the FWM approach provide great freedom of the

criteria.

An alternative method that estimates the original integral terms directly was also used in the stability analysis of DNNs.

The criteria from this type of method are strongly linked to the inequalities used. At the beginning, the Jensen inequality

has been wildly applied to analyze the stability of the DNNs [14–26,33,34,51,52] . In 2013, Seuret et al. [43] presented a

Wirtinger-based inequality and proved that it is less conservative than the Jensen inequality. Since then, Wirtinger-based

inequality has become the most popular method to estimate the single integral term during the investigation of DNNs

[27,37–41,53–56] .

Very recently, Zeng et al. proposed a free-matrix-based inequality (FMBI) in [45] and extended it to the research of DNNs

[42,50] . To the best of the authors’ knowledge, this inequality is the least conservative among the existing inequalities for

estimating single integral term. However, there is further room to be investigated using the FMBI. Some slack matrices in

the FMBI do not seem to contribute to a reduction of the conservatism. In [42] , the FMBI was only used to estimate the

single integral term without any augmented vector, while the augmented integral term was still estimated via the Jensen

inequality.

It can be expected that stability criteria with less conservatism will be obtained by developing and using a more effective

approach to estimate the single integral term. This motivates the present research.

This paper further investigates delay-dependent stability of DNNs following the development of a more effective method

to estimate the single integral term (1) . The contributions of the paper are summarized as follows:

1. A general free-weighting-matrix (GFWM) approach is developed to estimate single integral term. Based on several zero-

value equalities, a new estimation method, named as GFWM approach, is developed by following the basic estimation

procedure of the FWM approach. And a new inequality is derived based on the GFWM approach ( Lemma 5 ).

2. Necessary theoretical studies are carried out to compare the GFWM approach and several previous estimation methods.

It is proved that the inequality obtained from the GFWM approach can encompass the Wirtinger-based inequality and

the FMBI.

3. Several new stability criteria with less conservatism for the DNNs are derived. For generalized neural networks with a

time-varying delay, based on two LKFs (one with delay-product-type terms and the other without similar terms), two

stability criteria are derived by using the GFWM to estimate the single integral term appearing in the derivative of the

LKFs.

The remainder of the paper is organized as follows. Section 2 gives the problem formulation and necessary preliminary.

In Section 3 , the development of the GFWM approach and its comparison to previous methods are discussed in detail. The

GFWM approach is applied to a generalized DNN and several stability criteria are derived in Section 4 . In Section 5 , three

numerical examples are used to demonstrate the beneﬁts of the proposed criteria. Conclusions are given in Section 6 .

104 C.-K. Zhang et al. / Applied Mathematics and Computation 294 (2017) 102–120

Notations : Throughout this paper, the superscripts T and −1 mean the transpose and the inverse of a matrix, respectively;

denotes the n -dimensional Euclidean space; R

n ×m

is the set of all n × m real matrices; · refers to the Euclidean vector

norm; P > 0 ( ≥ 0) means that P is a real symmetric and positive-deﬁnite (semi-positive-deﬁnite) matrix; diag{ } denotes

a block-diagonal matrix; the symmetric term in a symmetric matrix is denoted by ∗; and Sym { X} = X + X

2. Problem formulation and preliminary

This section describes the problem to be investigated and gives related preliminaries.

2.1. Problem formulation

Consider the following generalized DNNs [35] :

y (t) = −Ay (t ) + W

g(W y (t )) + W

g(W y (t − d(t))) + J (2)

where y (t) = [ y

(t) y

(t ) . . . y

(t )]

is the state vector associated with the n neurons; g(·) = [ g

(·) g

(·) . . . g

(·)]

represents

the neuron activation function with g(0) = 0 ; A = diag { a

, a

, . . . , a

}>0 ; W , W

and W

are the connection weight matrices;

J = [ J

. . . J

]

is a vector representing the bias; and d ( t ) is a time-varying delay satisfying

0 ≤ d(t) ≤ h (3)

and

Case I :

d (t) ≤ μ (4)

Case I I : |

d (t) | ≤ μ (5)

where h and μ are constants.

Assumption 1. The neuron activation function g

( ·) is assumed to be bounded and satisﬁes the following condition:

−

≤

) − g

)

− s

≤ σ

, s

 = s

, i = 1 , 2 , . . . , n (6)

where σ

−

and σ

are known real constants.

Remark 1. Constants σ

and σ

−

in Assumption 1 are allowed to be positive, negative, or zero, and many activation func-

tions (monotonic or non-monotonic) satisfy the condition of (6) [15] .

Based on this assumption for the activation function, there exists an equilibrium point y

∗

for the neural network, i.e.,

0 = −Ay

∗

+ W

g(W y

∗

) + W

g(W y

∗

) + J. Using transformation x (t) = y (t) − y

∗

[4] , one can shift the equilibrium point y

∗

(2) to the origin and rewrite system (2) as:

x (t) = −Ax (t) + W

f (W x (t)) + W

f (W x (t − d(t))) (7)

where f (. ) = [ f

(·) f

(·) . . . f

(·)]

and f

2 i

x (t )) = g

2 i

x (t ) + W

2 i

∗

) − g

2 i

∗

) with f

(0) = 0 and W

2 i

denoting the i th

row vector of the matrix W . Then,

it follows from (6) and f

(0) = 0 that [51]

−

≤

) − f

)

− s

≤ σ

, s

 = s

(8)

−

≤

(s )

≤ σ

, s  = 0 (9)

This paper is concerned with the delay-dependent stability of DNN (2) . In order to determine the AMDBs more accurately,

this paper aims to derive new delay-dependent stability criteria with conservatism as small as possible. As mentioned in

Section 1 , the construction of LKFs and the treatment of their derivatives are two aspects related to conservatism. Since

there are many different forms of LKFs, this paper mainly pays attention to the treatment of the single integral term that

appears in the derivative of all LKFs.

2.2. Preliminary

The following lemmas are used in subsequent sections of this paper.

Lemma 1. Wirtinger-based inequality [43] : For symmetric positive deﬁnite matrix R ∈ R

n ×n

, scalars a < b , and vector ω :

[ a, b] → R

such that the integrations concerned are well deﬁned, then the following inequality holds



(s ) Rω(s ) ds≥

b − a







R 0

0 3 R





(10)

where χ



ω(s ) ds and χ

= χ

−

b−a



ω(u ) d ud s=−χ

b−a



ω(u ) d ud s .

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