Exponential lag synchronization for delayed fuzzy cellular neural networks via periodically intermittent control

doi:10.1016/j.matcom.2011.11.006

Mathematics and Computers in Simulation

Volume 82, Issue 5, January 2012, Pages 895-908

https://doi.org/10.1016/j.matcom.2011.11.006 Get rights and content

Abstract

In this paper, lag synchronization for a class of delayed fuzzy cellular networks is investigated. By utilizing inequality technique, Lyapunov functional theory and the analysis method, some new and useful criteria of lag synchronization for the addressed networks are derived in terms of p-norm under a periodically intermittent controller. Finally, an example with simulation is given to show the effectiveness of the obtained results.

Introduction

At present, cellular neural networks (CNN) have been extensively investigated both in theory and applications since it was proposed by Chua and Yang [2], [3]. However, in mathematical modeling of real world problems, the uncertainty or vagueness is unavoidable. In order to take vagueness into consideration, fuzzy theory is considered as a suitable tool. Based on traditional CNN, the fuzzy cellular neural networks (FCNN) were first introduced by Yang and Yang [28], [29], which integrate fuzzy logic into the structure of traditional CNN and maintain logic connectedness among cells. Meanwhile, many studies have been revealed that FCNN is a useful paradigm for image processing problems, which is a cornerstone in image processing and pattern recognition. Therefore, the dynamical analysis of FCNN is important and interesting from both theoretical and applied points of view.

In implementation of FCNN such as in the process of moving images, time delays are unavoidable encountered in the signal transmission among the neurons due to the finite switching speed of neurons and amplifiers, which will affect the stability of the neural system and may lead to some complex dynamic behaviors, such as instability, chaos, oscillation or other performance of the neural network. Therefore, in order to make full use of their advantages and restrain even eliminate their disadvantages, the controlling issue with regard to the delayed neural networks seems so crucial for researchers even all people.

In view of the significance of the control for delayed neural networks, in recent years, the stabilization and synchronization of FCNN with delays have been extensively investigated by many researchers, for instance, see [11], [20], [21], [22], [30], [32] and [6], [7], [8], [25]. Ding [6] studied the synchronization of delayed fuzzy cellular neural networks with impulses by using a non-impulsive system to replace the impulsive system and some synchronization criteria were obtained by the well known Lyapunov–Lasall principle. The synchronization for a class of delayed fuzzy cellular neural networks with all the parameters unknown was investigated by Ding et al. [7] by means of the Lyapunov–Lasall principle of functional differential equations and the adaptive control method. In [8], Some criteria of lag synchronization for delayed fuzzy cellular neural networks with impulses were obtained. In [25], the synchronization for a class of coupled identical Yang–Yang type fuzzy cellular neural networks (YYFCNN) with time-varying delays was proposed by Yang et al.

Meanwhile, many control approaches have also been developed to stabilize or synchronize neural networks and nonlinear systems in the past few decades such as adaptive control [4], [10], fuzzy control [9], [23], impulsive control [27], [31] and intermittent control [5], [33] and so on. Intermittent control, which was first introduced to control linear econometric models in [5], has been used for a variety of purposes such as manufacturing, transportation and communication. The main idea of intermittent control is to control systems via discontinuous control inputs at a control period. An extreme case of intermittent control is impulsive control. The prominent characteristic of impulsive control is that the states of the controlled system will “jump” at certain discrete time moments, that is, the control is with zero duration of time. Since the states of controlled systems are changed directly, impulsive control is an effective approach when the states are observable. But it seems to be invalid when the states of controlled systems are unobservable [13]. Moreover, generally, the realization of control needs to a time duration and it may be more reasonable to achieve control process in some time intervals other than some time instants in real control application. Recently, a lot of studies (for example Refs. [1], [12], [13], [14], [15], [16], [17], [24], [26]) have been obtained concerning intermittent control.

In Refs. [12], [16], the stabilization and synchronization of nonlinear systems without delays were studied under the periodically intermittent control based on 2-norm. A class of chaotic systems and networks with constant delay were investigated in Refs. [13], [14], [15], [17] by designing periodically intermittent controllers and some stabilization and synchronization criteria were derived in those papers based on 2-norm. And then, the periodically intermittent control was applied to the complex dynamical networks with constant delay in Refs. [1], [24] based on 2-norm. It is worth noting that the most previous results presented in Refs. [1], [12], [13], [14], [15], [16], [17], [24], [26] were obtained by constructing a Lyapunov function and using two central differential inequalities and the restriction that the control width is greater than the time delay was imposed in Refs. [1], [13], [14], [15], [17], [24], [26]. And another restriction that the non-control width is also greater than the time delay was given in Refs. [1], [24], [26].

From above analysis, many restrictions are imposed in the previous results concerning periodically intermittent control, which is disadvantageous in real applications. On the other hand, to the best of our knowledge, there are few even no results with regard to the lag synchronization for FCNNs with time delays by applying intermittent control based on p-norm. Motivated by above analysis, in this paper, a class of FCNNs with time delays is investigated by using intermittent control and some criteria based on p-norm are derived to ensure the exponential lag synchronization of the addressed FCNNs.

This paper is organized as follows. In Section 2, the problem statements and some useful lemmas and preliminaries are given. In Section 3, the criteria of exponential lag synchronization are rigorously derived. In Section 4, the effectiveness of the developed methods is shown by a numerical example.

Section snippets

Preliminaries

In this paper, we consider a class of fuzzy cellular neural network with discrete delays described by $\dot{x_{i}} (t) = - c_{i} x_{i} (t) + \sum_{j = 1}^{n} a_{ij} f_{j} (x_{j} (t)) + \sum_{j = 1}^{n} b_{ij} v_{j} + ⋀_{j = 1}^{n} α_{ij} f_{j} (x_{j} (t - τ_{j})) + ⋀_{j = 1}^{n} T_{ij} v_{j} + ⋁_{j = 1}^{n} β_{ij} f_{j} (x_{j} (t - τ_{j})) + ⋁_{j = 1}^{n} S_{ij} v_{j} + I_{i},$ where $i \in I = {1, 2, \dots, n}$ , n corresponds to the number of units in a neural network, x_i(t) is the state variable of the ith neuron at time t; α_ij, β_ij, T_ij and S_ij are elements of fuzzy feedback MIN template, fuzzy feedback MAX template, fuzzy feed-forward MIN template, fuzzy feed-forward

Lag synchronization

In this section, we will derive some criteria to guarantee the exponentially lag synchronization of systems (1), (3).

Let $λ_{i} = {pc}_{i} - {pH}_{i} - \sum_{j = 1, j \neq i}^{n} \sum_{ℓ = 1}^{p - 1} | a_{ij} |^{p γ_{ℓ ij}} L_{j}^{p δ_{ℓ ij}} - \sum_{j = 1}^{n} \sum_{ℓ = 1}^{p - 1} (| α_{ij} |^{p ϑ_{ℓ ij}} L_{j}^{p δ_{ℓ ij}} + | β_{ij} |^{p η_{ℓ ij}} L_{j}^{p δ_{ℓ ij}}) - \sum_{j = 1, j \neq i}^{n} \frac{μ_{j}}{μ_{i}} | a_{ji} |^{p γ_{pji}} L_{i}^{p δ_{pji}},$ $ω_{i} = {pk}_{ii} + \sum_{j = 1, j \neq i}^{n} \sum_{ℓ = 1}^{p - 1} | k_{ij} |^{{ph}_{ℓ ij}} + \sum_{j = 1, j \neq i}^{n} \frac{μ_{j}}{μ_{i}} | k_{ji} |^{{ph}_{pji}},$ $ζ_{i} = \sum_{j = 1}^{n} \frac{μ_{j}}{μ_{i}} (| α_{ji} |^{p ϑ_{pji}} L_{i}^{p δ_{pji}} + | β_{ji} |^{p η_{pji}} L_{i}^{p δ_{pji}}),$ where $L_{i} = max {| {\underline{F}}_{i} |, {\bar{F}}_{i}}$ , $H_{i} = \{\begin{matrix} a_{ii} {\bar{F}}_{i}, & a_{ii} \geq 0, \\ a_{ii} {\underline{F}}_{i}, & a_{ii} < 0, \end{matrix}$ and nonnegative real numbers γ_ℓij, δ_ℓij, h_ℓij η_ℓij and ϑ_ℓij satisfy $\sum_{ℓ = 1}^{p} γ_{ℓ ij} = 1$ , $\sum_{ℓ = 1}^{p} δ_{ℓ ij} = 1$

Numerical simulations

In this section, a chaotic fuzzy network is given to show the effectiveness of our results obtained in this paper.

Consider the following fuzzy Hopfied neural networks with discrete delays: $\dot{x_{i}} (t) = - c_{i} x_{i} (t) + \sum_{j = 1}^{2} a_{ij} f_{j} (x_{j} (t)) + ⋀_{j = 1}^{2} α_{ij} f_{j} (x_{j} (t - τ_{j})) + ⋁_{j = 1}^{2} β_{ij} f_{j} (x_{j} (t - τ_{j})) + I_{i}, i = 1, 2,$ for t ≥ 0, where c₁ = c₂ = 1, a₁₁ = 1.75, a₁₂ = − 0.1, a₂₁ = − 1.6, a₂₂ = 2, α₁₁ = β₁₁ = − 1.7, α₁₂ = β₁₂ = − 0.1, α₂₁ = β₂₁ = 0.1, α₂₂ = β₂₂ = − 1.6, f_i(x_i) = tanh x_i, τ_i = 1 and I_i = 0 for i = 1, 2.

System (21) exists a chaotic behavior as shown in Fig. 1 with initial

Conclusion

In this paper, the exponential lag synchronization for a class of fuzzy cellular neural networks with discrete delays under the periodical intermittent control was discussed for the first time. And then, some new and useful lag synchronization criteria in terms of p-norm were obtained by introducing some parameters and using Lyapunov functional technique. Particularly, the restriction that the control width is greater than the time delay imposed in Refs. [1], [13], [14], [15], [17], [24], [26]

Acknowledgements

This work was supported by the National Natural Science Foundation of People's Republic of China (Grant Nos. 10961022, 10901130 and 61164004), the Natural Science Foundation of Xinjiang (Grant No. 2010211A07), and the Excellent Doctor Innovation Program of Xinjiang University (Grant No. XJUBSCX-2011003).

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Original articleExponential lag synchronization for delayed fuzzy cellular neural networks via periodically intermittent control

Abstract

Introduction

Section snippets

Preliminaries

Lag synchronization

Numerical simulations

Conclusion

Acknowledgements

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Communications in Nonlinear Science and Numerical Simulation

Physics Letters A

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Nonlinear Analysis: RWA

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Journal of Computational and Applied Mathematics

Physics Letters A

Neurocomputing

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Physical D

Original article
Exponential lag synchronization for delayed fuzzy cellular neural networks via periodically intermittent control