Synchronization of a class of uncertain stochastic discrete-time delayed neural networks
© Chen et al.; licensee Springer 2014
Received: 23 March 2014
Accepted: 11 July 2014
Published: 4 August 2014
The global asymptotical synchronization problem is discussed for a general class of uncertain stochastic discrete-time neural networks with time delay in this paper. Time delays include time-varying delay and distributed delay. Based on the drive-response concept and the Lyapunov stability theorem, a linear matrix inequality (LMI) approach is given to establish sufficient conditions under which the considered neural networks are globally asymptotically synchronized in the mean square. Therefore, the global asymptotical synchronization of the stochastic discrete-time neural networks can easily be checked by utilizing the numerically efficient Matlab LMI toolbox. Moreover, the obtained results are dependent not only on the lower bound but also on the upper bound of the time-varying delays, that is, they are delay-dependent. And finally, a simulation example is given to illustrate the effectiveness of the proposed synchronization scheme.
Since Chua and Yang in [1, 2] proposed the theory and applications of cellular neural networks, the dynamical behaviors of neural networks have attracted a great deal of research interest in the past two decades. Those attentions have mainly concentrated on the stability and the synchronization problems of neural networks (see [3–28]). Especially after synchronization problems of chaotic systems had been studied by Pecora and Carroll in [29, 30], in which they proposed the drive-response concept, the control and synchronization problems of chaotic systems have been thoroughly investigated [5–8, 14–18, 31–35]. And many applications of such systems have been found in different areas, particularly in engineering fields such as creating secure communication systems (see [33–35]). As long as we can reasonably design the receiver so that the state evolution of the response system synchronize to that of the driven system, the message obtained by the receiver can be hidden in a chaotic signal, hence, secure communication can be implemented.
It is well known that neural networks, including Hopfield neural networks (HNNs) and cellular neural networks (CNNs), are large-scale and complex nonlinear high-dimensional systems composed of a large number of interconnected neurons. And they have also been found effective applications in many areas such as image processing, optimization problems, pattern recognition, and so on. Therefore, it is not easy to achieve the control and synchronization of these systems. In [8, 16, 19, 22], by drive-response method, some results are given for different type neural networks to guarantee the synchronization of drive system and response system in the models discussed. It is easy to apply those results to real neural networks. And in [5–7, 18], synchronization in an array of linearly or nonlinearly coupled networks has been analyzed in details. The authors studied the global asymptotic or exponential synchronization of a complex dynamical neural networks through constructing a synchronous manifold and showed that it is globally asymptotically or exponentially stable. To the best of our knowledge, up till now, most of the synchronization methods of chaotic systems (especially neural networks) are of drive-response type (which is also called a master-slave system).
At the same time, most of the papers mentioned above are concerned with continuous-time neural networks. When implementing these networks for practical use, discrete-time types of models should be formulated. The readers may refer to [3, 23] for more details as regards the significance of investigating discrete-time neural networks. Therefore, it is important to study the dynamical behaviors of discrete-time neural networks. On the other hand, because the synaptic transmission is probably a noisy process brought about by random fluctuations from the release of neurotransmitters, and a stochastic disturbance must be considered when formulating real artificial neural networks. Recently, the stability and synchronization analysis problems for stochastic or discrete-time neural networks have been investigated; see e.g. [3, 10–13, 19–21, 23], and references therein. So, in this paper, based on drive-response concept and Lyapunov functional method, some different decentralized control laws will be given for global asymptotical synchronization of a general class of discrete-time delayed chaotic neural networks with stochastic disturbance. In the neural network model, the parameter uncertainties are norm-bounded, the neural networks are subjected to stochastic disturbances described in terms of a Brownian motion, and the delay includes time-varying delay and distributed delay. Up to now, to the best of our knowledge, there are few works about the synchronization problem of discrete-time neural networks with distributed delay. And the master-slave system’s synchronization problem for the uncertain stochastic discrete-time neural networks with distributed delay is little investigated.
This paper is organized as follows. In Section 2, model formulation and some preliminaries are presented for our main results. In Section 3, based on the drive-response concept and the Lyapunov functional method, we discuss global asymptotical synchronization in mean square for uncertain stochastic discrete-time delayed neural networks with mixed delays. A numerical example is given to illustrate the effectiveness and feasibility of our results in Section 4. And finally, in Section 5, we give the conclusions.
Notations Throughout this paper, ℝ, , and are used to denote, respectively, the real number field, the real vector space of dimension n, and the set of all real matrices. And denotes a n-dimensional identity matrix. The set of all integers on the closed interval is denoted as , where a, b are integers and . We use ℕ to denote the set of all positive integers. Also, we assume that represents the set of all functions . The superscript ‘T’ represents the transpose of a matrix or a vector, and the notation (respectively, ) means that is a positive semi-definite matrix (respectively, a positive definite matrix) where X and Y are symmetric matrices. The notation denotes the Euclidian norm of a vector and refers to an m-dimensional identity matrix. Let be a complete probability space with a natural filtration satisfying the usual conditions (i.e., the filtration contains all -null sets and is right continuous) and generated by Brownian motion . stands for the mathematical expectation operator with respect to the given probability measure . The asterisk ∗ in a matrix is used to denote the term that is induced by symmetry. Usually, if not explicitly specified, matrices are always assumed to have compatible dimensions.
2 Model formulation and preliminaries
It is well known that most of the synchronization methods of chaotic systems are of the master-slave (drive-response) type. The system, which is called a slave system or a response system, can be driven by another system, which is called a master system or drive system, so that the behavior of the slave system can be influenced by the master system, i.e., the master system is independent to the slave system but the slave system is driven by the master system. In this paper, our aim is to design the controller reasonably such that the behavior of the slave system synchronizes to that of the master system.
where and are known positive integers. and are real diagonal constant matrices (corresponding to the state feedback and the delayed state-feedback coefficient matrices, respectively) with , , , , , and are the connection weight matrix, the discretely delayed connection weight matrix and the distributively delayed connection weight matrix, respectively. The functions , and in denote the neuron activation functions. The real vector is the exogenous input.
Obviously, such a distributed delay term will bring about an additional difficulty in our analysis.
For the activation functions in the model (1), we have the following assumptions:
Assumption 1 The activation functions , , and () in model (1) are all continuous and bounded.
where , , , , , are some constants.
Remark 2 The constants , , , , , in Assumption 2 are allowed to be positive, negative or zero. So, the activation functions in this paper are less conservative than the usual sigmoid functions.
where and are known constant scalars.
here denotes the family of all -measurable -valued random variables satisfying .
where , , . Denoted the error state vector as . Correspondingly, from (8) and (12), the initial condition of error system (12) is . Here it is necessary to assume that .
Now, we firstly give the definition of the globally robust asymptotical synchronization in mean square of the master system (1) and the slave system (9) as follows.
That is, if the error system (13) is globally robustly asymptotically stable in the mean square, then system (1) and system (9) are robustly globally asymptotically synchronized in the mean square.
Remark 3 Assumption 1 and Assumption 2 can derive that the error system (13) has at least an equilibrium point. Our main aim is to design the controller reasonably such that the equilibrium point of the error system (13) is robustly globally asymptotically stable in the mean square.
where is a constant gain matrix.
Although a memoryless controller (14) has the advantage of easy implementation, its performance cannot be better than a discretely delayed-feedback controller which utilizes the available information of the size of time-varying delay. Therefore, in this respect, the controller (16) could be considered as a compromise between the performance improvement and the implementation simplicity.
To complete this particular issue, we still need several lemmas to be used later.
holds for any scalar .
Lemma 2 (Schur complement)
holds for any and ().
3 Main results and proofs
In this section, some sufficient criteria are presented for the globally asymptotically synchronization in the mean square of the neural networks (1) and (9).
where and represents the unit column vector having ‘1’ as the element on its i th row and zeros elsewhere.
The main results are as follows.
The above inequality (41) results by Lemma 3.
with , , .
with , , and .
and , , , , , , , are defined in (30).
According to Definition 1, it can be deduced that the master system (1) and the slave system (9) are globally robustly asymptotically synchronized in the mean square, and the proof is then completed. □
where , , , , , , , are defined in (30).
This corollary is very easily accessible from Theorem 1.
where , , and , , , , , are defined in (30).
with , , and is defined as (30).
Moreover, in this case, if the stochastic disturbance in the response system (59) is (), then we need only rewrite and as and , and the corollary will still be true.
The proofs of Corollary 2 and Corollary 3 are similar to that of Theorem 1 and are therefore omitted.
In this case, we will show that the neural networks (62) and (63) are not only globally, robustly, and asymptotically synchronized in the mean square, but also globally, robustly, and exponentially synchronized in the mean square. The definition of the globally robustly exponentially synchronization in the mean square is given firstly in the following.
holds for and all .
Then we have the following theorem.
where and , , , , , , are defined in (30).
where , , and are similar to (32), (33), and (34).
Now, we are in a position to establish the robust global exponential stability in the mean square of the error system (64).
According to Definition 2, this completes the proof. □
Remark 4 Based on the drive-response concept, synchronization problems of discrete-time neural networks are little investigated. To the best of our knowledge, for master-slave systems, the synchronization analysis problem for stochastic neural networks with parameter uncertainties, especially distributed delay, is for the first time discussed.
4 Numerical example
In this section, an example will be illustrated to show the feasibility of our results.
Then Corollary 1 proves that the response system (9) and the drive system (1) with the given parameters can achieve globally robustly asymptotically synchronization in the mean square.
In this paper, based on Lyapunov stability theorem and drive-response concept, the globally asymptotically synchronization has been discussed for a general class of uncertain stochastic discrete-time neural networks with mixed time delays which consist of time-varying discrete and infinite distributed time delays. The proposed controller is robust to a stochastic disturbance and to the parameter uncertainties. In comparison with previous literature, the distributed delay is taken into account in our models, which are few investigated in the discrete-time complex networks. By using the linear matrix inequality (LMI) approach, several easy-to-verify sufficient criteria have been established to ensure the uncertain stochastic discrete-time neural networks to be globally robustly asymptotically synchronized in the mean square. The LMI-based criteria obtained are dependent not only on the lower bound, but also on the upper bound of the time-varying delay, and they can be solved efficiently via the Matlab LMI Toolbox. Also, the proposed synchronization scheme is easy to implement in practice.
Zhong Chen (1971-), male, native of Xingning, Guangdong, M.S.D., engages in the groups theory and differential equation.
This work was supported by the NSF of Guangdong Province of China under Grant S2013010015944 and by the National Science Foundation of China (10576013; 10871075). The authors wish specially to thank the managing editor and referees for their very helpful comments and useful suggestions.
- Chua LO, Yang L: Cellular neural networks: theory. IEEE Trans. Circuits Syst. 1988, 35(10):1257-1272. 10.1109/31.7600MathSciNetView ArticleMATHGoogle Scholar
- Chua LO, Yang L: Cellular neural networks: applications. IEEE Trans. Circuits Syst. 1988, 35(10):1273-1290. 10.1109/31.7601MathSciNetView ArticleGoogle Scholar
- Mohamad S, Gopalsamy K: Exponential stability of continuous-time and discrete-time cellular neural networks with delays. Appl. Math. Comput. 2003, 135(1):17-38. 10.1016/S0096-3003(01)00299-5MathSciNetView ArticleMATHGoogle Scholar
- Lu W, Chen T: Synchronization analysis of linearly coupled networks of discrete time systems. Physica D 2004, 198: 148-168. 10.1016/j.physd.2004.08.024MathSciNetView ArticleMATHGoogle Scholar
- Chen G, Zhou J, Liu Z: Global synchronization of coupled delayed neural networks and applications to chaotic CNN models. Int. J. Bifurc. Chaos 2004, 14(7):2229-2240. 10.1142/S0218127404010655MathSciNetView ArticleMATHGoogle Scholar
- Lu W, Chen T: Synchronization of coupled connected neural networks with delays. IEEE Trans. Circuits Syst. 2004, 51(12):2491-2503. 10.1109/TCSI.2004.838308MathSciNetView ArticleGoogle Scholar
- Wu CW: Synchronization in arrays of coupled nonlinear systems with delay and nonreciprocal time-varying coupling. IEEE Trans. Circuits Syst. 2005, 52(5):282-286.View ArticleGoogle Scholar
- Chen C, Liao T, Hwang C: Exponential synchronization of a class of chaotic neural networks. Chaos Solitons Fractals 2005, 24(2):197-206.MathSciNetView ArticleMATHGoogle Scholar
- Wang Z, Liu Y, Liu X: On global asymptotic stability of neural networks with discrete and distributed delays. Phys. Lett. A 2005, 345: 299-308. 10.1016/j.physleta.2005.07.025View ArticleMATHGoogle Scholar
- Liu Y, Wang Z, Liu X: On global exponential stability of generalized stochastic neural networks with mixed time-delays. Neurocomputing 2006, 70: 314-326. 10.1016/j.neucom.2006.01.031View ArticleGoogle Scholar
- Wang Z, Liu Y, Yu L, Liu X: Exponential stability of delayed recurrent neural networks with Markovian jumping parameters. Phys. Lett. A 2006, 356: 346-352. 10.1016/j.physleta.2006.03.078View ArticleMATHGoogle Scholar
- Wang Z, Shu H, Fang J, Liu X: Robust stability for stochastic Hopfield neural networks with time delays. Nonlinear Anal., Real World Appl. 2006, 7: 1119-1128. 10.1016/j.nonrwa.2005.10.004MathSciNetView ArticleMATHGoogle Scholar
- Wang Z, Liu Y, Fraser K, Liu X: Stochastic stability of uncertain Hopfield neural networks with discrete and distributed delays. Phys. Lett. A 2006, 354: 288-297. 10.1016/j.physleta.2006.01.061View ArticleMATHGoogle Scholar
- Cao J, Li P, Wang W: Global synchronization in arrays of delayed neural networks with constant and delayed coupling. Phys. Lett. A 2006, 353: 318-325. 10.1016/j.physleta.2005.12.092View ArticleGoogle Scholar
- Zhou J, Chen TP: Synchronization in general complex delayed dynamical networks. IEEE Trans. Circuits Syst. 2006, 53(3):733-744.MathSciNetView ArticleGoogle Scholar
- Lou X, Cui B: Asymptotic synchronization of a class of neural networks with reaction-diffusion terms and time-varying delays. Comput. Math. Appl. 2006, 52: 897-904. 10.1016/j.camwa.2006.05.013MathSciNetView ArticleMATHGoogle Scholar
- Cao J, Li P, Wang W: Global synchronization in arrays of delayed neural networks with constant and delayed coupling. Phys. Lett. A 2006, 353: 318-325. 10.1016/j.physleta.2005.12.092View ArticleGoogle Scholar
- Wang W, Cao J: Synchronization in an array of linearly coupled networks with time-varying delay. Physica A 2006, 366: 197-211.View ArticleGoogle Scholar
- Yu W, Cao J: Synchronization control of stochastic delayed neural networks. Physica A 2007, 373: 252-260.View ArticleGoogle Scholar
- Wang Z, Lauria S, Fang J, Liu X: Exponential stability of uncertain stochastic neural networks with mixed time-delays. Chaos Solitons Fractals 2007, 32: 62-72. 10.1016/j.chaos.2005.10.061MathSciNetView ArticleMATHGoogle Scholar
- Liu Y, Wang Z, Serrano A, Liu X: Discrete-time recurrent neural networks with time-varying delays: exponential stability analysis. Phys. Lett. A 2007, 362: 480-488. 10.1016/j.physleta.2006.10.073View ArticleGoogle Scholar
- Yan J, Lin J, Hung M, Liao T: On the synchronization of neural networks containing time-varying delays and sector nonlinearity. Phys. Lett. A 2007, 361: 70-77. 10.1016/j.physleta.2006.08.083View ArticleMATHGoogle Scholar
- Liu Y, Wang Z, Liu X: Robust stability of discrete-time stochastic neural networks with time-varying delays. Neurocomputing 2008, 71(4-6):823-833. 10.1016/j.neucom.2007.03.008View ArticleGoogle Scholar
- Chen T, Wu W, Zhou W: Global μ -synchronization of linearly coupled unbounded time-varying delayed neural networks with unbounded delayed coupling. IEEE Trans. Neural Netw. 2008, 19(10):1809-1816.View ArticleGoogle Scholar
- Liu X, Chen T: Robust μ -stability for uncertain stochastic neural networks with unbounded time-varying delays. Physica A 2008, 387: 2952-2962. 10.1016/j.physa.2008.01.068MathSciNetView ArticleGoogle Scholar
- Liu Y, Wang Z, Liu X: On synchronization of coupled neural networks with discrete and unbounded distributed delays. Int. J. Comput. Math. 2008, 85(8):1299-1313. 10.1080/00207160701636436MathSciNetView ArticleMATHGoogle Scholar
- Liu Y, Wang Z, Liang J, Liu X: Synchronization and state estimation for discrete-time complex networks with distributed delays. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 2008, 38(5):1314-1325.MathSciNetView ArticleGoogle Scholar
- Liu Y, Wang Z, Liu X: On synchronization of discrete-time Markovian jumping stochastic complex networks with mode-dependent mixed time-delays. Int. J. Mod. Phys. B 2009, 23(3):411-434. 10.1142/S0217979209049826View ArticleMATHGoogle Scholar
- Pecora LM, Carroll TL: Synchronization chaotic systems. Phys. Rev. Lett. 1990, 64(8):821-824. 10.1103/PhysRevLett.64.821MathSciNetView ArticleMATHGoogle Scholar
- Carroll TL, Pecora LM: Synchronization chaotic circuits. IEEE Trans. Circuits Syst. 1991, 38(4):453-456. 10.1109/31.75404View ArticleGoogle Scholar
- Wu CW, Chua LO: A unified framework for synchronization and control of dynamical systems. Int. J. Bifurc. Chaos 1994, 4(4):979-998. 10.1142/S0218127494000691MathSciNetView ArticleMATHGoogle Scholar
- Wu CW, Chua LO: Synchronization in an array of linearly coupled dynamical systems. IEEE Trans. Circuits Syst. 1995, 42(8):430-447. 10.1109/81.404047MathSciNetView ArticleMATHGoogle Scholar
- Liao T, Tsai S: Adaptive synchronization of chaotic systems and its application to secure communications. Chaos Solitons Fractals 2000, 11(9):1387-1396. 10.1016/S0960-0779(99)00051-XView ArticleMATHGoogle Scholar
- Lu J, Wu X, Lv J: Synchronization of a unified chaotic system and the application in secure communications. Phys. Lett. A 2002, 305(6):365-370. 10.1016/S0375-9601(02)01497-4MathSciNetView ArticleGoogle Scholar
- Feki M: An adaptive chaos synchronization scheme applied to secure communications. Chaos Solitons Fractals 2003, 18: 141-148. 10.1016/S0960-0779(02)00585-4MathSciNetView ArticleMATHGoogle Scholar
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