Elsevier

Neural Networks

Volume 22, Issue 7, September 2009, Pages 869-874
Neural Networks

Neural networks letter
Synchronization of nonidentical chaotic neural networks with time delays

https://doi.org/10.1016/j.neunet.2009.06.009Get rights and content

Abstract

The synchronization problem is studied in this paper for nonidentical chaotic neural networks with time delays, where the mismatched parameters are taken into account. An integral sliding mode control approach is proposed to address it. As a first step, a proper sliding surface is constructed. Based on the drive–response concept and Lyapunov stability theory, both delay-independent and delay-dependent conditions are derived under which the resulting error system is globally asymptotically stable in the specified switching surface. The gain matrix of the sliding mode is achieved by means of a linear matrix inequality, which can be solved by resorting to standard numerical algorithm. Then, a sliding mode controller is synthesized to guarantee the reachability of the specified sliding surface. A simulation example is finally exploited to illustrate the effectiveness of the developed approach.

Introduction

It is well recognized that time delay is frequently encountered in electronic implementations of neural networks due to the finite switching speed of the amplifiers and communication time. Delayed neural networks were thus proposed and have received a great deal of attention. Such neural networks have been fruitfully applied in signal and image processing, associative memories, combinatorial optimization, automatic control, and other areas. The stability of a neural network is usually a prerequisite for some applications. However, the existence of time delay may lead to instability and/or deteriorate the performance of the underlying neural networks. As a result, the stability analysis of neural networks with time delays has become an active research topic. Many nice stability results have been reported in the open literature, see, e.g., Arik (2004), Cao and Wang (2005), Forti, Nistri, and Papini (2005), Li, Chen, Zhou, and Qian (2009), Liu, Wang, and Liu (2006) and Wang, Liu, and Liu (2005).

On the other hand, it was found in Gilli (1993) and Lu (2002) that some delayed neural networks can exhibit chaotic dynamics. These kinds of chaotic neural networks were utilized to solve optimization problems in Kwok and Smith (1999). Since the drive–response concept was introduced by Pecora and Carroll in their seminal work (Pecora & Carroll, 1990), the research on chaotic synchronization (Boccaletti et al., 2002, Pecora et al., 1997) has gained rapid development due to its potential applications in secure communications, chemical reactions, information processing, as well as biological systems. A typical characteristic of chaotic systems is their sensitive dependence on initial conditions. It means that it is generally difficult to achieve synchronization between chaotic systems. Meanwhile, in Milanović and Zaghloul (1996) and Tan and Ali (2001), synchronization schemes were proposed for chaotic neural networks and successfully applied in pattern recognition and communications. In recent years, the study on synchronization of chaotic neural networks has attracted considerable attention, see Cao, Chen, and Li (2008), Jankowski, Londei, Lozowski, and Mazur (1996), Liang, Wang, Liu, and Liu (2008a), Liang, Wang, Liu, and Liu (2008b) and Wang, Wang, and Liang (2009) for examples. In Huang and Cao (2006), a coupling scheme with different coupling delays was proposed for chaotic neural networks such that the complete synchronization, anticipating synchronization and lag synchronization can be discussed in a unified framework. The obtained conditions were expressed in terms of linear matrix inequalities (LMIs). In Yan, Lin, Hung, and Liao (2007), a memoryless decentralized control approach was adopted to deal with the synchronization problem for delayed neural networks with sector nonlinearity. Based on the invariant principle of functional differential equations, an adaptive approach was proposed in Cao and Lu (2006) to investigate the synchronization problem for neural networks with or without time delay.

It is worth noting that most of the reported works were concerned with the synchronization problem for identical chaotic neural networks. In practice, the chaotic systems are inevitably subject to some environmental changes, which may render their parameters to be variant. Furthermore, from the point of view of engineering, it is very difficult to keep the two chaotic systems to be identical all the time. It is thus of great significance to study the synchronization problem of nonidentical chaotic neural networks. Obviously, when the considered neural networks are distinct, it becomes more complex and challenging. The synchronization problem of nonidentical chaotic neural networks with time delays has not yet been fully studied and remains challenging, although an adaptive control approach was presented in Zhang, Xie, Wang, and Zheng (2007). This motivates the present study.

The sliding mode control (SMC) theory introduced by Utkin (1992) provides an efficient way to the robust control problem. Its main advantages are fast response, good transient performance and robustness to variations of system parameters. Recently, many applications on the SMC approach were established (Niu et al., 2005, Young et al., 1999). In this paper, an integral SMC approach will be developed to address the synchronization problem of nonidentical chaotic neural networks with time delays. Firstly, an integral sliding surface is properly constructed. By employing the LMI technique, both delay-independent and delay-dependent conditions are derived under which the resulting error system is globally asymptotically stable in the specified switching surface. The sliding mode gain matrix can be obtained by solving an LMI, which is facilitated readily by resorting to standard numerical softwares (Boyd, EI Ghaoui, Feron, & Balakrishnan, 1994). Then, an integral sliding mode controller is designed to guarantee the reachability of the specified sliding surface. The main contributions of this study are to investigate the effect of the mismatched parameters on the synchronization of drive–response systems and to propose an integral SMC approach to solving it.

Section snippets

Problem formulation

The delayed neural network under consideration is described by ẋ(t)=C1x(t)+A1f(x(t))+B1f(x(tτ))+J1, where x(t)=[x1(t),x2(t),,xn(t)]TRn is the state vector of the neural network with n neurons. C1=diag(c11,c12,,c1n) is a diagonal matrix with c1i>0. The matrices A1 and B1 are, respectively, the connection weight matrix and the delayed connection weight matrix. f(x(t))=[f1(x1(t)),f2(x2(t)),,fn(xn(t))]T denotes the neuron activation function. J1 is an external input vector. τ0 is a

Sliding mode controller design

It can be clearly seen from (5) that the dynamic behavior of the error system (5) relies on the chaotic state y(t) of the response system (3) because of the difference between the drive and response systems. Therefore, complete synchronization between nonidentical chaotic neural networks cannot be achieved only by utilizing output feedback control. To overcome the difficulty, an integral SMC approach will be proposed to investigate the synchronization problem of nonidentical chaotic neural

An illustrative example

Let x(t)=[x1(t),x2(t)]T. The chaotic neural network proposed in Gilli (1993) is adopted as the drive system: ẋ(t)=C1x(t)+A1f(x(t))+B1f(x(tτ)), where f(x)=0.5(|x+1||x1|),τ=1 and C1=I, A1=[1+π4200.11+π4],B1=[1.3π240.10.11.3π24]. The parameters of the measured output (2) are given as D=[10],E=[0.50]. Let y(t)=[y1(t),y2(t)]T. The response system is assumed to be of the form (Lu, 2002): ẏ(t)=C2y(t)+A2g(y(t))+B2g(y(tτ))+u(t), with g(y)=tanh(y),τ=1 and C2=I, A2=[20.152],B2=[1.50.10.21.5

Conclusion

In this paper, the synchronization problem has been studied for nonidentical chaotic neural networks with time delays, which is more difficult and challenging than the one for identical chaotic neural networks. An integral SMC approach has been presented to deal with this problem. Based on the LMI technique, both delay-independent and delay-dependent conditions have been derived for the existence of a suitable sliding surface guaranteeing the asymptotical stability of the sliding mode dynamics.

Acknowledgement

This work was partially supported by the Research Grants Council of the Hong Kong Special Administrative Region of China under Project CityU 113708.

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