On the global robust asymptotic stability of BAM neural networks with time-varying delays

doi:10.1016/j.neucom.2006.02.020

Neurocomputing

Volume 70, Issues 1–3, December 2006, Pages 273-279

https://doi.org/10.1016/j.neucom.2006.02.020 Get rights and content

Abstract

In this paper, the global robust asymptotic stability of a class of delayed bi-directional associative memory (BAM) neural networks, which contain variable uncertain parameters whose values are unknown but bounded, is studied. Some new sufficient conditions are presented for the global stability of BAM neural networks with time-varying delays by constructing Lyapunov functional and using linear matrix inequality (LMI), Halanay's inequality. A numerical example is presented to illustrate the effectiveness of our theoretical results.

Introduction

A series of neural networks related to bi-directional associative memory (BAM) models have been proposed by Kosko in Refs. [10], [14], [15], [16]. These models generalized the single-layer auto-associative Hebbian correlator to a two-layer pattern-matched heteroassociative circuits. Therefore, this class of networks has good application perspective in pattern recognition. Cao et al. have considered the stability of this class of networks and presented criteria for the global stability and periodic oscillatory solution of delayed BAM neural networks (see, for example, [3], [4], [6], [9]). Recently, we studied the global asymptotic stability of BAM neural networks with reaction–diffusion terms or impulses, see [11], [23]. As Kosko considers the global stability for BAM neural networks, his approach requires severe constraint conditions of having symmetric connection weight matrix. To design this kind neural networks, vital data, such as the neurons fire rate, the synaptic interconnection weight and the signal transmission delays, etc., usually need to be measured, acquired and processed by means of statistical estimation which definitely leads to estimation errors. However, parameter fluctuation in neural network implementation on very large-scale integration (VLSI) chips is also unavoidable. It is important to ensure that system be stable or periodic stable with respect to these uncertainties in the design and applications of neural networks. In recent years, the robust stability of delayed neural networks have been investigated by many researchers (e.g. [1], [5], [8], [21], [22], [24]).

Recently, linear matrix inequality (LMI)-based techniques have been successfully used to tackle various stability problems for neural networks with time delays (see, for example, [20], [24], [25]). The main advantage of the LMI-based approaches is that the LMI stability conditions can be solved numerically using the effective interior-point algorithm [2]. In [17], the global robust asymptotical stability is considered for multi-delayed interval neural networks based on LMI approach.

However, to the best of our knowledge, few authors have considered the global robust asymptotic stability of BAM neural networks with time-varying delays and uncertainties. In this paper, by constructing suitable Lyapunov functional based on LMI or Halanay's inequality, we derive some sufficient conditions for the global robust asymptotic stability of BAM neural networks with time-varying delays.

The rest of this paper is organized as follows. In Section 2, the problem to be investigated is stated and some definitions and lemmas are listed. Based on the Lyapunov–Krasovskii stability theory, LMI approach and Halanay's inequality, some robust asymptotic stability criteria for BAM neural networks with time-varying delays are obtained in Section 3. In addition, a numerical example is provided in Section 4, to support the results of the analysis. Finally, some conclusions are drawn in Section 5.

Section snippets

System description

We consider the following BAM model: $\{\begin{matrix} x^{^{'}} (t) = - (C + Δ C (t)) x (t) + (W + Δ W (t)) \\ f (y (t - τ (t))) + I, \\ y^{^{'}} (t) = - (D + Δ D (t)) y (t) + (H + Δ H (t)) \\ g (x (t - σ (t))) + J, \end{matrix}$ where $x = (x_{1}, x_{2}, \dots, x_{m})^{T}$ , $y = (y_{1}, y_{2}, \dots, y_{n})^{T}$ are the neuron state vectors, $C = diag (c_{1}, c_{2}, \dots, c_{m})$ , $D = diag (d_{1}, d_{2}, \dots, d_{n})$ are positive diagonal matrices, $W, H$ are interconnection weight matrices, $τ (t) = τ_{j} (t)_{n \times 1}, σ (t) = σ_{i} (t)_{m \times 1}$ and $0 ⩽ τ (t) ⩽ τ_{0}$ , $0 ⩽ σ (t) ⩽ σ_{0}$ are the time delays, and they are assumed that $0 ⩽ \dot{τ} (t) ⩽ τ^{*} < 1, 0 ⩽ \dot{σ} (t) ⩽ σ^{*} < 1 .$ The $Δ C (t), Δ D (t), Δ W (t), Δ H (t)$ are parametric uncertainties,

Main results

In this section, we will discuss the global robust asymptotic stability of system (5) with the initial conditions (6) and give our main results.

Theorem 3.1

If there exist symmetric positive matrices $P_{1}$ , $P_{2}$ such that the following LMIs hold: $[\begin{matrix} ϒ_{1} & P_{1} L_{0} & \sqrt{\frac{1}{1 - τ^{*}}} P_{1} L_{2} \\ L_{0}^{T} P_{1} & E & 0 \\ \sqrt{\frac{1}{1 - τ^{*}}} L_{2}^{T} P_{1} & 0 & E \end{matrix}] < 0$ and $[\begin{matrix} ϒ_{2} & P_{2} L_{1} & \sqrt{\frac{1}{1 - σ^{*}}} P_{2} L_{3} \\ L_{1}^{T} P_{2} & E & 0 \\ \sqrt{\frac{1}{1 - σ^{*}}} L_{3}^{T} P_{2} & 0 & E \end{matrix}] < 0,$ where $ϒ_{1} = - (P_{1} C + C^{T} P_{1}) + E_{0}^{T} E_{0} + \frac{1}{1 - σ^{*}} {BQ}_{2} B + \frac{1}{1 - τ^{*}} P_{1}^{2},$ $ϒ_{2} = - (P_{2} D + D^{T} P_{2}) + E_{1}^{T} E_{1} + \frac{1}{1 - τ^{*}} {AQ}_{1} A + \frac{1}{1 - σ^{*}} P_{2}^{2},$ $Q_{1} = W^{T} W + E_{2}^{T} E_{2}, Q_{2} = H^{T} H + E_{3}^{T} E_{3} .$

Then the equilibrium $(x^{*}, y^{*})$ of (1) is globally asymptotically stable dependent of

A numerical example

In this section, we give an illustrative example for our main result.

Example 4.1

Consider system (5) with time-varying delays: $τ (t) = \frac{1}{2} \sin^{2} (t)$ , $σ (t) = \frac{1}{3} \cos^{2} (t)$ with the initial values of the system as follows: $\{\begin{matrix} φ_{p 1} (t) = φ_{p 2} (t) = 0, & t \in [- \frac{1}{3}, 0), \\ φ_{q 1} (t) = φ_{q 1} (t) = 0, & t \in [- \frac{1}{2}, 0), \end{matrix}$ and take $C = [\begin{matrix} 2.6 & 0 \\ 0 & 2.1 \end{matrix}], W = [\begin{matrix} 1.1 & 1 \\ - 0.2 & 0.1 \end{matrix}], D = [\begin{matrix} 3 & 0 \\ 0 & 2 \end{matrix}], H = [\begin{matrix} 0.9 & 0.1 \\ - 0.1 & 0.1 \end{matrix}], L_{0} = [\begin{matrix} - 0.2 & 0.2 \\ 0.2 & 0.2 \end{matrix}], L_{1} = [\begin{matrix} - 0.4 & 0.3 \\ 0.3 & 0.4 \end{matrix}], L_{2} = [\begin{matrix} 0.2 & 0.5 \\ 0.1 & - 0.3 \end{matrix}], L_{3} = [\begin{matrix} - 0.2 & 0.4 \\ 0.4 & 0.2 \end{matrix}], F_{0} (t) = [\begin{matrix} \sin (t) & 0 \\ 0 & \cos (t) \end{matrix}], F_{1} (t) = [\begin{matrix} \sin^{2} (t) & 0 \\ 0 & 0.2 \cos^{2} (t) \end{matrix}], F_{2} (t) = [\begin{matrix} 0.5 \sin^{3} (t) & 0 \\ 0 & \cos^{3} (t) \end{matrix}], F_{3} (t) = [\begin{matrix} 1 - 2 \sin^{2} (t) & 0 \\ 0 & 1 - 2 \cos^{2} (t) \end{matrix}], E_{0} = L_{0}, E_{1} = L_{1}, E_{2} = L_{2}, E_{3} = L_{3}, P$

Conclusion

A robust stability criterion for general delayed BAM neural networks with parametric uncertainties and time-varying delay has been presented. The stability criterion is given in terms of linear matrix inequality (LMI) which can be easily solved by some existing software packages. An example has been provided to illustrate the effectiveness of our theoretical results.

Acknowledgments

The authors would like to thank the anonymous reviewers and the associate editor for their comments on improving the overall quality of the paper. This work is supported by the National Natural Science Foundation of China (No. 10371072) and the Science Foundation of Southern Yangtze University.

Xuyang Lou was born in 1982. He is presently a Ph.D. candidate in Research Center of Control Science and Engineering, Southern Yangtze University, China and received the B.S. degree from the Zhejiang Ocean University, China in 2004. His current research interests include nonlinear systems, neural networks and stability theory.

References (25)

J. Cao
Global asymptotic stability of delayed bidirectional associative memory neural network
Appl. Math. Comput.
(2003)
J. Cao et al.
Exponential stability of delayed bi-directional associative memory networks
Appl. Math. Comput.
(2003)
J. Cao et al.
Global robust stability of delayed recurrent neural networks
Chaos, Solitons Fractals
(2005)
J. Cao et al.
Exponential stability of high-order bidirectional associative memory neural networks with time delays
Physica D
(2004)
J. Cao et al.
Absolute exponential stability of recurrent neural networks with Lipschitz-continuous activation functions and time delays
Neural Networks
(2004)
A. Chen et al.
Exponential stability of BAM neural networks with transmission delays
Neurocomputing
(2004)
B.T. Cui et al.
Global asymptotic stability of BAM neural networks with distributed delays and reaction–diffusion terms
Chaos, Solitons Fractals
(2006)
C. Li et al.
Global robust asymptotical stability of multi-delayed interval neural networks: an LMI approach
Phys. Lett. A
(2004)
C. Li et al.
Delay-dependent exponential stability analysis of bi-directional associative memory neural networks with time delay: an LMI approach
Chaos, Solitons Fractals
(2005)
J.L. Liang et al.
Global asymptotic stability of bi-directional associative memory networks with distributed delays
Appl. Math. Comput.
(2004)

S. Arik

Global robust stability of delayed neural networks

IEEE Trans. Circuits Syst. I

(2003)

S. Boyd et al.

Linear Matrix Inequalities in System and Control Theory

(1994)

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Citation Excerpt :
Since stability is one of the most important behaviors of the neural networks, the analysis of stability for BAM networks has attracted considerable attentions. A great deal of results for BAM neural networks concerning the existence of equilibrium point, global asymptotic or exponential stability have been derived (see, for example [3–11,30–32] and the references therein). In [21], under the conditions that the activation functions are supposed to be bounded and globally Lipschitz continuous, by introducing some new integral inequalities and constructing a Lyapunov–Krasovskii functional, new LMI conditions are established on global asymptotic stability for the neural networks system (1.2).
In this paper, we discuss global asymptotic stability to a BAM neural networks of neutral type with delays. Under the assumptions that the activation functions only satisfy global Lipschitz conditions, a new and complicated LMI condition is established on global asymptotic stability for the above neutral neural networks by means of using Homeomorphism theory, matrix and Lyapunov functional. In our result, the hypotheses for boundedness in [20], [21] and monotonicity in [20] on the activation functions are removed. On the other hand, the LMI condition is also different from those in [20], [21]. Finally, an example is given to show the effectiveness of the theoretical result.

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Baotong Cui was born in 1960. He received the M.S. degree in applied mathematics from the Department of Mathematics, Anhui University in 1989 and the Ph.D. degree in control theory and control engineering from the College of Automation Science and Engineering, South China University of Technology, China in 2003. He was a post-doctoral fellow at Shanghai Jiao Tong University, China from 2003 to 2005. He joined the Department of Mathematics, Binzhou University, Shangdong, China in 1982, where he became an associate professor in 1993, and a full professor in 1995. In 2003, he moved to the Southern Yangtze University, China where he is a full professor and vice-director for Research Center of Control Science and Engineering and vice-dean for College of Communication and Control Engineering. His current research interests include systems analysis, stability theory, artificial neural networks impulsive and chaos synchronization.

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On the global robust asymptotic stability of BAM neural networks with time-varying delays

Abstract

Introduction

Section snippets

System description

Main results

A numerical example

Conclusion

Acknowledgments

Appl. Math. Comput.

Appl. Math. Comput.

Chaos, Solitons Fractals

Physica D

Neural Networks

Neurocomputing

Chaos, Solitons Fractals

Phys. Lett. A

Chaos, Solitons Fractals

Appl. Math. Comput.

Global robust stability of delayed neural networks

IEEE Trans. Circuits Syst. I

Linear Matrix Inequalities in System and Control Theory