Abstract and Applied Analysis

Volume 2013, Article ID 510903, 7 pages

http://dx.doi.org/10.1155/2013/510903

## Synchronization of Neural Networks with Mixed Time Delays under Information Constraints

^{1}School of Control Science and Engineering, Hebei University of Technology, Tianjin 300130, China^{2}Hebei University of Science and Technology, Shijiazhuang 050018, China

Received 13 October 2013; Accepted 5 December 2013

Academic Editor: Khalil Ezzinbi

Copyright © 2013 Dedong Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper investigates the synchronization problem of neural networks with mixed time delays under information constrains. The designed synchronization scheme is built on the framework of hybrid systems. Besides including nonuniform sampling, some other characteristics, such as quantization, transmission-induced delays, and data packet dropouts, are also considered. The sufficient condition that depended on network characteristics is obtained to guarantee the remote asymptotical synchronization of neural networks with mixed time delays. A numerical example is given to illustrate the validity of the proposed method.

#### 1. Introduction

Recently, neural networks have been widely studied by many scholars due to its potential applications in pattern recognition, image processing, signal processing, biology engineering, and information science [1–3]. Moreover, there exists a broad class of neural networks with mixed time delays. Following the development in this field, many master-slave synchronization schemes for neural networks with mixed time delays have been proposed, such as [4–14].

Generally, some useful approaches can be utilized for the synchronization problem of neural networks, which include passivity analysis [5, 12], impulsive control [15, 16], adaptive control [17, 18], and stochastic method [13]. In recent years, the sampled-data control scheme is utilized for the synchronization of neural networks with mixed time delays, such as [10, 11, 14]. However, these schemes did not consider quantization, transmission-induced delays, and data packet dropouts. Essentially, in the range of deterministic systems, the general framework considering information constrains can be divided into three cases: continuous-time models; discrete-time models; hybrid models. In this paper, the networked controller design for asymptotical synchronization of neural networks with mixed time delays is discussed in the framework of hybrid systems. Besides including nonuniform sampling, we also consider other network characteristics, such as quantization, transmission-induced delays, and data packet dropouts. From the view of helpful technologies, the input delay approach and the free-weighting matrix technology are applied to obtain the less conservative condition.

*Notation.* Throughout this paper, superscripts and mean the transpose and the inverse of a matrix, respectively, denotes natural number, denotes integer number, denotes the -dimensional Euclidean space, is the set of all real matrices, the identity matrices and zero matrices are denoted by and , respectively, the notation always denotes the symmetric block in one symmetric matrix, the standard notation is used to denote the positive (negative)-definite ordering of matrices, and inequality shows that the matrix is positive definite.

#### 2. Preliminaries

Consider the following general master-slave neural network with mixed time delays [11]: where , , and and denote the states of the th neuron of master and slave neural networks, respectively; , , and and denote the neuron activation functions of master and slave neural networks, respectively; denotes a diagonal matrix with positive entries; , , and denote the connection weight matrices; denotes the control input; denotes the external input vector; and denote the discrete delay and the distributed delay, respectively, and satisfy where and are constants.

*Assumption 1 (see [7]). *For , the neuron activation functions satisfy
where and are some constants, , and .

*Assumption 2 (see [7]). *The neuron activation functions are bounded.

Let the error be . Then, the synchronization error system can be represented as where . The networked synchronization controller is designed as where denotes the quantizer, denotes the controller gain matrix, and are available measurements of and at sampling instant , , denotes the serial number of the available data packet such that , denotes the sampling instant of the available data packet, and denotes the network-induced delay calculated from the instant .

The above controller (5) is related with the quantitative function so the following definition is introduced.

*Definition 3 (see [19]). *A quantizer is called logarithmic if the set of quantized levels is characterized by

The quantizer is assumed to be symmetric; that is, , as described in [19]. Selected as the logarithmic quantizer, is given as where and .

Generally, denotes the overall sampling instant and all the data packets are assigned as serial numbers. But, the data packets may be discontinuous due to dropouts such that the sampling is nonuniform. Similar to many existing results, the sensor is clock-driven; the controller and actuator are event-driven. The clocks among all the devices are synchronized.

*Assumption 4 (see [20]). *There exist three constants , , and such that
where denotes the upper bound of the interval between two consecutive sampling instants and and denote the minimum and maximum of network-induced delays, respectively. It is assumed that denotes the admitted maximum of successive data packet dropouts in network transmissions. Considering condition (8) and successive data packet dropouts, the following result can be obtained:

Set and then the initial condition of on is supplemented as , , where is a continuous function on . By the sector bound method as [19], the synchronization error system (4) can be expressed as where , and . According to (8) and (9), and . The control objective is to design the controller gain matrix such that the synchronization error system (10) is asymptotically stable; that is, as .

#### 3. Main Result

In this section, the stability of the error system (10) will be analyzed by constructing a corresponding Lyapunov functional. Before beginning the proof procedure, two useful lemmas are introduced.

Lemma 5. *Let be any matrix, for any constant and any positive-definite symmetric matrix , such that
**
for all , , and .*

Lemma 6 (see [21]). *For any constant symmetric matrix , , scalar , vector function , such that the integrations in the following are well defined, then
*

Theorem 7. *Given scalars , , and composed of , , , and diagonal matrices , , and , the synchronization error system (10) is global asymptotically stable in network environments, if there exist matrices , , , , , any matrices , with appropriate dimensions, and the controller gain matrix such that the following condition holds:
**
where
*

*Proof. *Construct the following Lyapunov functional as
where , , , , and . Moreover, the following equations hold for any appropriate dimensional matrices , , , , , and:
On the other hand, for any , it follows from (3) such that
Combining (2), (16), and (17), the corresponding time derivative of is given by
Using Lemmas 5 and 6 and the inequality
the following result can be obtained:
where , matrices , , and are defined in (13). It is explicit that if
then for any nonzero . Utilizing Schur complement [22], the condition in Theorem 7 can be obtained, and the proof is completed.

Because the uncertain matrix is involved at the condition (13) in Theorem 7, the following theorem is given to obtain a solvable result.

Theorem 8. *Given scalars , , and composed of , , , and constant diagonal matrices , , and , the synchronization error system (10) is global asymptotically stable in network environments, if there exist matrices , , , , , any matrices , with appropriate dimensions, and the controller gain matrix such that the following condition holds:
**
where
*

*Proof. *Utilizing Schur formula and matrix inequality , condition (22) in Theorem 8 is obtained. The proof is completed.

*Remark 9. *In the above process, the free-weighting matrix technology is applied to complete the proof. Moreover, similar to [23], we can select the new Lyapunov-Krasovskii functional to reduce the conservatism, which utilizes the bounds of the network-induced delay. This will be done in the future works.

#### 4. A Numerical Example

In this section, a numerical example is given to demonstrate the effectiveness of the proposed synchronization scheme.

*Example 1. *Consider the following master-slave neural network with mixed time delays as in [11, 14]:
where
, , , and . Thus, and . It is assumed that , . According to Assumption 1, the following result is satisfied:

The chaotic behaviors of the master and slave systems are given in Figures 1 and 2, with the initial condition chosen as and , respectively. Setting , , and appropriate matrices , the condition (22) in Theorem 8 is feasible for , with the controller gain matrix

Similar to [14], the initial conditions of the master and slave systems are chosen as and , respectively. The response curves of the error system are shown in Figure 3 for the upper bound of sampling interval . It shows that the synchronization error converges to zero asymptotically.

#### 5. Conclusion

In the present works, the networked synchronization scheme for master-slave neural networks with mixed time delays has been proposed. The error system can be stabilized under information constraints. The obtained result depends on network characteristics. In future works, more performance requirements for synchronization of master-slave neural networks with mixed time delays will be considered in a uniform network topological structure.

#### Acknowledgments

This work is supported by the Natural Science Foundation of China under Grant 61203076, the Natural Science Foundation of Tianjin City under Grant 13JCQNJC03500, the Natural Science Foundation of Hebei Province under Grant F2012202100, and the Excellent Young Technological Innovation Foundation Project in Hebei University of Technology under Grant 2011005.

#### References

- C. M. Bishop,
*Neural Networks for Pattern Recognition*, Oxford University Press, New York, NY, USA, 1996. - M. M. Gupta, L. Jin, and N. Homma,
*Static and Dynamic Neural Networks: From Fundamentals to Advanced Theory*, Wiley-IEEE Press, New York, NY, USA, 2003. - A. N. Michel and D. Liu,
*Qualitative Analysis and Synthesis of Recurrent Neural Networks*, vol. 248 of*Monographs and Textbooks in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 2002. View at MathSciNet - H. R. Karimi and H. Gao, “New delay-dependent exponential ${H}_{\infty}$ synchronization for uncertain neural networks with mixed time delays,”
*IEEE Transactions on Systems, Man, and Cybernetics, Part B*, vol. 40, no. 1, pp. 173–185, 2010. View at Publisher · View at Google Scholar · View at Scopus - H. Li, C. Wang, P. Shi, and H. Gao, “New passivity results for uncertain discrete-time stochastic neural networks with mixed time delays,”
*Neurocomputing*, vol. 73, no. 16–18, pp. 3291–3299, 2010. View at Publisher · View at Google Scholar · View at Scopus - T. Li, S.-M. Fei, Q. Zhu, and S. Cong, “Exponential synchronization of chaotic neural networks with mixed delays,”
*Neurocomputing*, vol. 71, no. 13–15, pp. 3005–3019, 2008. View at Publisher · View at Google Scholar · View at Scopus - Y. Liu, Z. Wang, and X. Liu, “Global exponential stability of generalized recurrent neural networks with discrete and distributed delays,”
*Neural Networks*, vol. 19, no. 5, pp. 667–675, 2006. View at Publisher · View at Google Scholar · View at Scopus - Q. Song, “Design of controller on synchronization of chaotic neural networks with mixed time-varying delays,”
*Neurocomputing*, vol. 72, no. 13–15, pp. 3288–3295, 2009. View at Publisher · View at Google Scholar · View at Scopus - Z. Wang, Y. Liu, M. Li, and X. Liu, “Stability analysis for stochastic Cohen-Grossberg neural networks with mixed time delays,”
*IEEE Transactions on Neural Networks*, vol. 17, no. 3, pp. 814–820, 2006. View at Publisher · View at Google Scholar · View at Scopus - Z.-G. Wu, P. Shi, H. Su, and J. Chu, “Stochastic synchronization of markovian jump neural networks with time-varying delay using sampled data,”
*IEEE Transactions on Cybernectics*, vol. 43, no. 6, pp. 1796–1806, 2013. View at Publisher · View at Google Scholar - Z.-G. Wu, P. Shi, H. Su, and J. Chu, “Exponential synchronization of neural networks with discrete and distributed delays under time-varying sampling,”
*IEEE Transactions on Neural Networks and Learning Systems*, vol. 23, no. 9, pp. 1368–1376, 2012. View at Publisher · View at Google Scholar - Z.-G. Wu, P. Shi, H. Su, and J. Chu, “Passivity analysis for discrete-time stochastic markovian jump neural networks with mixed time delays,”
*IEEE Transactions on Neural Networks*, vol. 22, no. 10, pp. 1566–1575, 2011. View at Publisher · View at Google Scholar · View at Scopus - Y. Chen and W. X. Zheng, “Stability analysis of time-delay neural networks subject to stochastic perturbations,”
*IEEE Transactions on Cybernectics*, vol. 43, no. 6, pp. 2122–2134, 2013. View at Publisher · View at Google Scholar - C.-K. Zhang, Y. He, and M. Wu, “Exponential synchronization of neural networks with time-varying mixed delays and sampled-data,”
*Neurocomputing*, vol. 74, no. 1–3, pp. 265–273, 2010. View at Publisher · View at Google Scholar · View at Scopus - X. Li and R. Rakkiyappan, “Impulse controller design for exponential synchronization of chaotic neural networks with mixed delays,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 18, no. 6, pp. 1515–1523, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - X. Yang, C. Huang, and Q. Zhu, “Synchronization of switched neural networks with mixed delays via impulsive control,”
*Chaos, Solitons and Fractals*, vol. 44, no. 10, pp. 817–826, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - S. C. Jeong, D. H. Ji, J. H. Park, and S. C. Won, “Adaptive synchronization for uncertain chaotic neural networks with mixed time delays using fuzzy disturbance observer,”
*Applied Mathematics and Computation*, vol. 219, no. 11, pp. 5984–5995, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. Zhu, W. Zhou, D. Tong, and J. Fang, “Adaptive synchronization for stochastic neural networks of neutral-type with mixed time-delays,”
*Neurocomputing*, vol. 99, pp. 477–485, 2013. View at Publisher · View at Google Scholar - M. Fu and L. Xie, “The sector bound approach to quantized feedback control,”
*IEEE Transactions on Automatic Control*, vol. 50, no. 11, pp. 1698–1711, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - D. Yang and K.-Y. Cai, “Decentralized networked stabilization for interconnected nonlinear systems: a fuzzy control approach,” in
*Proceedings of the 8th World Congress on Intelligent Control and Automation (WCICA '10)*, pp. 4823–4828, Jinan, China, July 2010. View at Publisher · View at Google Scholar · View at Scopus - K. Gu, V. L. Kharitonov, and J. Chen,
*Stability of Time-Delay Systems*, Control Engineering, Birkhäuser, Boston, Mass, USA, 2003. View at MathSciNet - S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan,
*Linear Matrix Inequalities in System and Control Theory*, vol. 15 of*SIAM Studies in Applied Mathematics*, SIAM, Philadelphia, Pa, USA, 1994. View at Publisher · View at Google Scholar · View at MathSciNet - X. Jiang, Q.-L. Han, S. Liu, and A. Xue, “A new ${H}_{\infty}$ stabilization criterion for networked control systems,”
*IEEE Transactions on Automatic Control*, vol. 53, no. 4, pp. 1025–1032, 2008. View at Publisher · View at Google Scholar · View at Scopus