Adaptive synchronization for uncertain chaotic neural networks with mixed time delays using fuzzy disturbance observer
Introduction
In the past few decades, there has been considerable attention in the study of neural networks due to their potential applications in various areas, such as signal processing pattern recognition, static image processing, associative memory and combinatorial optimization [1], [2], [3], [4]. It has been shown that artificial neural network models can exhibit some chaotic behaviors [5], [6], [7], [8]. Since the pioneering works of Pecora and Carroll [9], Synchronization of chaotic neural networks has been intensively investigated in many fields [10], [11], [12], [13]. In the implementation of the neural networks, time delays between neurons in the networks often arise in the processing of information storage and transmission, which may lead to instability, oscillation, and bifurcation of the neural network model [8], [14]. Many studies have been developed for the synchronization problem of delayed chaotic neural networks. Some have considered the networks with time-varying delays [15], [16], [17], [18], [19]. However, there exist various chaotic neural networks with both time-varying delays and distributed delays in realistic network models. Therefore, it is worth taking into account the chaotic neural networks with the mixed time delays including time-varying and distributed delays [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. A control method with two sufficient conditions to ensure the globally exponential stability for the error system has been proposed based on the drive-response concept [20], [21]. In [22], a synchronization problem of the networks with mixed delays has been discussed by using an adaptive feedback control technique. Sufficient conditions for asymptotical or exponential synchronization are derived in terms of Linear matrix inequalities (LMIs) by constructing proper Lyapunov–Krasovskii functional [23], [24], [25]. Sliding mode control technique is proposed to synchronize nonidentical chaotic neural networks with mixed delays [26], [27]. The synchronization problems of stochastic perturbed chaotic neural networks with mixed delays have been investigated in [28], [29].
It is known that the uncertainty and disturbance are unavoidable factors in many practical situations and they can destroy the network stability or can make the synchronization more difficult. Some works for uncertain neural networks have been developed to overcome their effects [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32]. They often require some prior information of the uncertain factors, such as its structure or upper bound. However, the information may not be available due to physical limitations in practical cases. Fuzzy logic system can be a good solution to be used in the situations because it can provide an estimator for a unknown function or value. Fuzzy disturbance observer (FDO) has been proposed to estimate uncertainty and disturbance without requiring any prior information about them [33]. The estimated values have been used to compensate the uncertain factors via state feedback controller. In [34], a robust tracking control approach using a discrete-time FDO has been proposed for nonlinear sampled systems. Recently, a more precise FDO has been constructed by modifying the law used to update the parameter vector and the modified FDO showed better performances, compared with the conventional one [35]. Even though the FDO presented good performances to overcome the unknown factor, applications of the existing research are still limited. Especially, there has been still no research using the technique for uncertain chaotic neural networks with mixed time delays.
In this paper, we propose a robust adaptive synchronization method for uncertain chaotic neural networks with time-varying delays and distributed delays. The uncertain factors including uncertainties and disturbances are estimated by the FDO without requiring any prior knowledge about the factors. The estimated values are used to compensate the factors in the proposed method. Based on Lyapunov–Krasovskii stability theory, the control scheme with adaptive laws is derived and guarantees the globally asymptotical synchronization between the networks. An example is illustrated to show the effectiveness of the proposed method.
Section snippets
Problem statement
Consider the following chaotic neural network with time-varying delay and distributed delay:where is the neuron state vector and is a positive diagonal matrix. , and are the connection weight matrix, the time varying delayed connection weight matrix and distributively delayed connection weight matrix, respectively. is an external input vector,
Adaptive synchronization using fuzzy disturbance observer
In this section, we propose an adaptive synchronization method for the uncertain chaotic neural networks (1), (2). The first step for the synchronization is how well we can overcome the overall disturbance . We will use the fuzzy logic system (FLS) to accomplish that [38]. First, let us briefly describe the basic configuration of the FLS used in this paper. The FLS performs a mapping from a compact set to a compact set . The fuzzy rule base consists of a collection of M
Numerical examples
In this section, a numerical example is presented to illustrate the effectiveness of our scheme proposed in the previous sections. The simulations are conducted in Simulink (MATLAB) using a fixed-step fourth order Runge–Kutta solver with sample period s. We consider a two-dimensional chaotic neural network with the mixed delay as the drive system (1), which is described with
Conclusion
We have proposed a robust adaptive synchronization method for uncertain chaotic neural networks with both time-varying and distributed delays. By using the FDO, the uncertain factors including uncertainties and disturbances have been estimated without requiring any prior information about the factors. The estimated values have been used to compensate the factors. Based on Lyapunov–Krasovskii stability theory, the control scheme with adaptive laws has been derived, guaranteeing the globally
Acknowledgements
The work of J.H. Park was supported by 2012 Yeungnam University Research Grant. Park would like to thank Maureen Seo and E.K. Park for their valuable comments and supports.
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