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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 517604, 11 pages
http://dx.doi.org/10.1155/2013/517604
Research Article

Stability Analysis for Uncertain Neural Networks of Neutral Type with Time-Varying Delay in the Leakage Term and Distributed Delay

1College of Information Science and Technology, Bohai University, Jinzhou, Liaoning 121013, China
2School of Mathematics and Physics, Bohai University, Jinzhou, Liaoning 121013, China
3Department of Engineering, Faculty of Engineering and Science, University of Agder, N-4898 Grimstad, Norway

Received 19 November 2013; Accepted 4 December 2013

Academic Editor: Ming Liu

Copyright © 2013 Qi Zhou 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

The stability problem is investigated for a class of uncertain networks of neutral type with leakage, time-varying discrete, and distributed delays. Both the parameter uncertainty and the generalized activation functions are considered in this paper. New stability results are achieved by constructing an appropriate Lyapunov-Krasovskii functional and employing the free weighting matrices and the linear matrix inequality (LMI) method. Some numerical examples are given to show the effectiveness and less conservatism of the proposed results.

1. Introduction

Neural networks are a complex large-scale power system and have very rich dynamic property. In the past years, neural networks have some applications in areas of associative memory [1], pattern recognition [2], and optimization problems [35]. Recently, the stability problem for neural networks has been widely investigated and some results have been reported in [614].

Time delays are often encountered in various engineering, biological, and economical systems [1523]. Due to the finite speed of information processing, the existence of time delays frequently causes oscillation, divergence, or instability in neural networks. Recently, the stability of neural networks with time delays has drawn considerable attention, and many results on stability of neural networks with time delays have been reported in the literature [2430]. In practice, the time-varying delay often belongs to a given interval, and the lower bound of the delays may not be zero. Recently, some papers investigated the stability conditions for neural networks with interval time-varying delay in the literature, for example [31]. Furthermore, due to the presence of parallel pathways of different axonal sizes and lengths, there may exist a spatial extent in neural networks, which may cause distributed time delays [3235]. Recently, the stability problem of neural networks with neutral-type was studied in [36, 37].

More recently, more and more attention has been paid to time delay in the leakage (or “forgetting”) term [38, 39], since there exist some theoretical and technical difficulties when handling the leakage delay [40]. It has been pointed out that the leakage delay has a great impact on the dynamics of neural networks [41, 42]. The authors in [43] showed that time delay in the stabilizing negative feedback term has a tendency to destabilize a system. More recently, the authors in [44] focused on recurrent neural networks with time delay in the leakage term and showed that all the results mentioned about the existence and uniqueness of the equilibrium point are independent of time delays and initial conditions. Then, it can be seen that time delays in leakage terms do not affect the existence and uniqueness of the equilibrium point. Recently, some results about stability analysis for neural networks with leakage delay have been reported in [38, 39, 41]. To mention a few, the work in [14] investigated the problems of delay-dependent stability analysis and strict ()--dissipativity analysis for cellular neural networks with distributed delay. However, it should be mentioned that there are few results about stability analysis for uncertain neural networks of neutral type with time-varying delay in the leakage term and distributed delay, which motivates this study.

In this paper, the stability problem is investigated for uncertain neural networks of neutral type with time-varying delay in the leakage term and time-varying distributed delay. Firstly, by constructing a new type of Lyapunov functional and developing some novel techniques to handle the delays considered in this paper, some novel robust stability criteria are proposed. Secondly, the proposed conditions can be expressed in terms of linear matrix inequalities (LMIs), which can be easily solved via standard software. Finally, some numerical examples are given to demonstrate the effectiveness and less conservatism of the proposed results.

Notation. Throughout this paper, the notations are standard. and denote the -dimensional Euclidean space and the set of all real matrices, respectively. In this paper, the superscript denotes matrix transposition. For real symmetric matrices and , the natation ( resp.) means that the is positive-semidefinite (positive-definite resp.). The notation stands for a block-diagonal matrix. is the identity matrix with appropriate dimensions. Matrices if not explicitly stated, where the symbol “” stands for the symmetric term in a matrix, are assumed to have compatible dimensions.

2. Problem Formulation

Consider the following uncertain neural network of neutral type with time delay in the leakage term and distributed delay: where is the network state vector at time ; denotes the activation function at time ; is a positive diagonal matrix; , , , , and are known constant matrices; , , , , and , are unknown matrices; is leakage delay, is time-varying discrete delay; is neutral delay; and is time-varying distributed delay. They satisfy the following conditions: where , , , , , and are constants. The time-varying parameter uncertainties , , , , and are assumed to be of the form where , , , , , are known constant matrices and is an unknown time-varying matrix satisfying Throughout this paper, we make the following assumption:

for any , , there exist constants , , for all , with such that

Lemma 1 (see [45]). For any positive symmetric constant matrices , scalar , vector function such that the integrations concerned are well defined; then

Lemma 2 ([46] Schur complement). Given constant matrices , , and with appropriate dimensions, where and , then if and only if
In order to present novel stability criteria for neural networks (1), the following notations are defined:

3. Main Results

In this section, by considering the delay in the leakage term and distributed delay and using some new techniques, novel stability criteria will be proposed for uncertain neural network of neutral-type with time-varying delay in (1). Firstly, considering neutral-type delay, the delay in the leakage term, and distributed delay, we have the following theorem.

Theorem 3. For given scalars , , , , , and , neural network (1) under Assumption is robustly asymptotically stable, if there exist matrices , , , , , , positive diagonal matrices and , and appropriately dimensioned matrices , , such that the following LMIs hold: where

Proof. By using Newton-Leibniz formulation and considering neural network (1), the following equalities hold for appropriately dimensioned matrices and : with For positive diagonal matrices and , based on Assumption (), it can be seen that the following inequalities hold:
Now, choosing the following Lyapunov-Krasovskii functional: where where , and Then, the derivatives of , with time can be obtained as By Lemma 1, one can have where and .
It can be seen from the condition (9) that which means It follows from (21) and (23) that Therefore, it is straight forward to obtain that where . By Schur complement, it can be seen from the condition (10) that , which means that neural network (1) under Assumption is robustly asymptotically stable. This completes this proof.

When neural network (1) without uncertainties, we have

The stability condition can be easily obtained from Theorem 3 in the following corollary.

Corollary 4. Given scalars , , , , , and , neural network (26) is asymptotically stable, if there exist matrices , , , , , , positive diagonal matrices and , and appropriately dimensioned matrices , , such that the following LMIs hold: where , and have been defined in Theorem 3.

Remark 5. For neural network (1), if time delay in the leakage term is not considered in this paper, the following model can be obtained: In order to present stability criterion for neural network (28), we choose the following Lyapunov-Krasovskii functional: where () is defined in (15). From the proof of Theorem 3, the following stability condition for neural network (28) can be obtained.

Corollary 6. For given scalars , , , , and , neural network (1) under Assumption is asymptotically stable, if there exist matrices , , , , , , positive diagonal matrices and , and appropriately dimensioned matrices , , such that the following LMIs hold: where

Remark 7. When neural network (1) without time-varying neutral delay , the following model can be obtained:
In order to present stability criterion for neural network (32), we choose the following Lyapunov-Krasovskii functional: where () are defined in (15). Following the same line of Theorem 3, the following corollary can be presented.

Corollary 8. Given scalars , , , and , the neural network (32) under Assumptions is asymptotically stable, if there exist matrices , , , , , , positive diagonal matrices and , and appropriately dimensioned matrices , such that the following LMIs hold: where
Consider the following neural network with time-varying delay:
By choosing the Lyapunov-Krasovskii functional where () is defined in (15). The novel stability condition for neural network with time-varying delay (36) can be presented from Theorem 3 in the following corollary.

Corollary 9. For given scalars , and , neural network (36) under Assumptions is asymptotically stable, if there exist matrices , , , , , , positive diagonal matrices and , and appropriately dimensioned matrices , such that the following LMIs hold: where

4. Numerical Examples

In this section, two numerical examples are provided to show the effectiveness of the proposed results.

Example 1. Consider uncertain neural network of neutral type with leakage delay time-varying delay, and distributed delay as follows: where

By using Matlab LMI Toolbox, from Theorem 3, it can be found that the uncertain neural network (1) under Assumption is robustly asymptotically stable for . To calculate the maximum allowable for different in this paper, the numerical results in Table 1 illustrate the effectiveness of the proposed results.

tab1
Table 1: Allowable upper bound for different .

Example 2. Consider the neural network with time-varying delay as follows:
By using Matlab LMI Toolbox, it can be found from Corollary 9 that the maximum allowable can be obtained to guarantee the stability of neural network with time-varying delay in (42) for different . Compared with the previous results proposed in [28], it is clear that the new stability condition in Corollary 9 is less conservative than the one in [28] (see Table 2).

tab2
Table 2: Allowable upper bound for different .

5. Conclusion

In this paper, the problem of stability analysis for neural networks of neutral type with time-varying delay in the leakage term and distributed delay has been studied. By constructing appropriate Lyapunov-Krasovskii functional and employing some advanced methods, some novel stability criteria have been proposed in terms of LMIs, which can be easily solved by standard software. Two examples have been given to illustrate the effectiveness and merit of the proposed results. It should be mentioned that the leakage delay handling method proposed in this paper can also be used to investigate the systems with the delay in the leakage term, for example, the fault-tolerant control systems [47, 48].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was partially supported by the National Natural Science Foundation of China (61304003, 11226138, and 61304002).

References

  1. A. N. Michel, J. A. Farrell, and H.-F. Sun, “Analysis and synthesis techniques for Hopfield type synchronous discrete time neural networks with application to associative memory,” IEEE Transactions on Circuits and Systems, vol. 37, no. 11, pp. 1356–1366, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. M. Galicki, H. Witte, J. Dörschel, M. Eiselt, and G. Griessbach, “Common optimization of adaptive preprocessing units and a neural network during the learning period. Application in EEG pattern recognition,” Neural Networks, vol. 10, no. 6, pp. 1153–1163, 1997. View at Publisher · View at Google Scholar · View at Scopus
  3. Y. Hayakawa, A. Marumoto, and Y. Sawada, “Effects of the chaotic noise on the performance of a neural network model for optimization problems,” Physical Review E, vol. 51, no. 4, pp. R2693–R2696, 1995. View at Publisher · View at Google Scholar · View at Scopus
  4. C. Peterson and B. S. Söderberg, “A new method for mapping optimization problems onto neural networks,” International Journal of Neural Systems, vol. 1, no. 01, pp. 3–22, 1989.
  5. Y.-J. Liu, C. L. P. Chen, G.-X. Wen, and S. Tong, “Adaptive neural output feedback tracking control for a class of uncertain discrete-time nonlinear systems,” IEEE Transactions on Neural Networks, vol. 22, no. 7, pp. 1162–1167, 2011. View at Publisher · View at Google Scholar · View at Scopus
  6. J. Cao, D.-S. Huang, and Y. Qu, “Global robust stability of delayed recurrent neural networks,” Chaos, Solitons & Fractals, vol. 23, no. 1, pp. 221–229, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. X. Liao, G. Chen, and E. N. Sanchez, “Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach,” Neural Networks, vol. 15, no. 7, pp. 855–866, 2002. View at Publisher · View at Google Scholar · View at Scopus
  8. H. Li, H. Gao, and P. Shi, “New passivity analysis for neural networks with discrete and distributed delays,” IEEE Transactions on Neural Networks, vol. 21, no. 11, pp. 1842–1847, 2010. View at Publisher · View at Google Scholar · View at Scopus
  9. H. Li, B. Chen, Q. Zhou, and W. Qian, “Robust stability for uncertain delayed fuzzy Hopfield neural networks with Markovian jumping parameters,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 39, no. 1, pp. 94–102, 2009. View at Publisher · View at Google Scholar · View at Scopus
  10. B. Zhang, S. Xu, G. Zong, and Y. Zou, “Delay-dependent exponential stability for uncertain stochastic hopfield neural networks with time-varying delays,” IEEE Transactions on Circuits and Systems I, vol. 56, no. 6, pp. 1241–1247, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. T. Li, W. X. Zheng, and C. Lin, “Delay-slope-dependent stability results of recurrent neural networks,” IEEE Transactions on Neural Networks, vol. 22, no. 12, pp. 2138–2143, 2011. View at Publisher · View at Google Scholar · View at Scopus
  12. L. Wu, P. Shi, and H. Gao, “State estimation and sliding-mode control of markovian jump singular systems,” IEEE Transactions on Automatic Control, vol. 55, no. 5, pp. 1213–1219, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. H. R. Karimi and P. Maass, “Delay-range-dependent exponential H synchronization of a class of delayed neural networks,” Chaos, Solitons & Fractals, vol. 41, no. 3, pp. 1125–1135, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. Z. Feng and J. Lam, “Stability and dissipativity analysis of distributed delay cellular neural networks,” IEEE Transactions on Neural Networks, vol. 22, no. 6, pp. 976–981, 2011. View at Publisher · View at Google Scholar · View at Scopus
  15. S. C. Tong, Y. M. Li, and H.-G. Zhang, “Adaptive neural network decentralized backstepping output-feedback control for nonlinear large-scale systems with time delays,” IEEE Transactions on Neural Networks, vol. 22, no. 7, pp. 1073–1086, 2011. View at Publisher · View at Google Scholar · View at Scopus
  16. B. Chen, X. Liu, and S. Tong, “New delay-dependent stabilization conditions of T-S fuzzy systems with constant delay,” Fuzzy Sets and Systems, vol. 158, no. 20, pp. 2209–2224, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  17. L. Zhang and H. Gao, “Asynchronously switched control of switched linear systems with average dwell time,” Automatica, vol. 46, no. 5, pp. 953–958, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. H. Gao and T. Chen, “Network-based H output tracking control,” IEEE Transactions on Automatic Control, vol. 53, no. 3, pp. 655–667, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. H. Gao, X. Meng, and T. Chen, “Stabilization of networked control systems with a new delay characterization,” IEEE Transactions on Automatic Control, vol. 53, no. 9, pp. 2142–2148, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. H. Li, X. Jing, and H. R. Karimi, “Output-feedback based H control for active suspension systems with control delay,” IEEE Transactions on Industrial Electronics, vol. 61, no. 1, pp. 436–446, 2014.
  21. H. Li, H. Liu, H. Gao, and P. Shi, “Reliable fuzzy control for active suspension systems with actuator delay and fault,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 2, pp. 342–357, 2012. View at Publisher · View at Google Scholar · View at Scopus
  22. H. Gao, T. Chen, and J. Lam, “A new delay system approach to network-based control,” Automatica, vol. 44, no. 1, pp. 39–52, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. B. Zhang, S. Xu, and Y. Zou, “Improved stability criterion and its applications in delayed controller design for discrete-time systems,” Automatica, vol. 44, no. 11, pp. 2963–2967, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. J. Cao and J. Wang, “Global asymptotic and robust stability of recurrent neural networks with time delays,” IEEE Transactions on Circuits and Systems I, vol. 52, no. 2, pp. 417–426, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. 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
  26. S. Xu, J. Lam, D. W. C. Ho, and Y. Zou, “Delay-dependent exponential stability for a class of neural networks with time delays,” Journal of Computational and Applied Mathematics, vol. 183, no. 1, pp. 16–28, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  27. C. Hua, C. Long, and X. Guan, “New results on stability analysis of neural networks with time-varying delays,” Physics Letters A, vol. 352, no. 4-5, pp. 335–340, 2006. View at Publisher · View at Google Scholar · View at Scopus
  28. Y. He, G. P. Liu, D. Rees, and M. Wu, “Stability analysis for neural networks with time-varying interval delay,” IEEE Transactions on Neural Networks, vol. 18, no. 6, pp. 1850–1854, 2007. View at Publisher · View at Google Scholar · View at Scopus
  29. S. Xu and J. Lam, “A new approach to exponential stability analysis of neural networks with time-varying delays,” Neural Networks, vol. 19, no. 1, pp. 76–83, 2006. View at Publisher · View at Google Scholar · View at Scopus
  30. H. R. Karimi and H. Gao, “New delay-dependent exponential H 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
  31. Y. He, G. Liu, and D. Rees, “New delay-dependent stability criteria for neural networks with yime-varying delay,” IEEE Transactions on Neural Networks, vol. 18, no. 1, pp. 310–314, 2007. View at Publisher · View at Google Scholar · View at Scopus
  32. Z. Wang, H. Shu, Y. Liu, D. W. C. Ho, and X. Liu, “Robust stability analysis of generalized neural networks with discrete and distributed time delays,” Chaos, Solitons & Fractals, vol. 30, no. 4, pp. 886–896, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  33. 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
  34. Z. Wang, Y. Liu, K. Fraser, and X. Liu, “Stochastic stability of uncertain Hopfield neural networks with discrete and distributed delays,” Physics Letters A, vol. 354, no. 4, pp. 288–297, 2006. View at Publisher · View at Google Scholar · View at Scopus
  35. J. Cao, K. Yuan, and H.-X. Li, “Global asymptotical stability of recurrent neural networks with multiple discrete delays and distributed delays,” IEEE Transactions on Neural Networks, vol. 17, no. 6, pp. 1646–1651, 2006. View at Publisher · View at Google Scholar · View at Scopus
  36. J. H. Park and O. M. Kwon, “Analysis on global stability of stochastic neural networks of neutral type,” Modern Physics Letters B, vol. 22, no. 32, pp. 3159–3170, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  37. J. H. Park, O. M. Kwon, and S. M. Lee, “LMI optimization approach on stability for delayed neural networks of neutral-type,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 236–244, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  38. X. Li, X. Fu, P. Balasubramaniam, and R. Rakkiyappan, “Existence, uniqueness and stability analysis of recurrent neural networks with time delay in the leakage term under impulsive perturbations,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 4092–4108, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  39. X. Li, R. Rakkiyappan, and P. Balasubramaniam, “Existence and global stability analysis of equilibrium of fuzzy cellular neural networks with time delay in the leakage term under impulsive perturbations,” Journal of the Franklin Institute, vol. 348, no. 2, pp. 135–155, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  40. K. Gopalsamy, “Leakage delays in BAM,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1117–1132, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  41. X. Li and J. Cao, “Delay-dependent stability of neural networks of neutral type with time delay in the leakage term,” Nonlinearity, vol. 23, no. 7, pp. 1709–1726, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  42. P. Balasubramaniam, M. Kalpana, and R. Rakkiyappan, “Global asymptotic stability of BAM fuzzy cellular neural networks with time delay in the leakage term, discrete and unbounded distributed delays,” Mathematical and Computer Modelling, vol. 53, no. 5-6, pp. 839–853, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  43. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Springer, New York, NY, USA, 1992.
  44. C. Li and T. Huang, “On the stability of nonlinear systems with leakage delay,” Journal of the Franklin Institute, vol. 346, no. 4, pp. 366–377, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  45. J. Sun, G. P. Liu, and J. Chen, “Delay-dependent stability and stabilization of neutral time-delay systems,” International Journal of Robust and Nonlinear Control, vol. 19, no. 12, pp. 1364–1375, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  46. S. P. Boyd, Linear Matrix Inequalities in System and Control Theory, vol. 15, SIAM, Philadelphia, Pa, USA, 1994.
  47. S. Yin, H. Luo, and S. X. Ding, “Real-time implementation of fault-tolerant control systems with performance optimization,” IEEE Transactions on Industrial Electronics, vol. 61, no. 5, pp. 2402–2411, 2013.
  48. S. Yin, S. X. Ding, A. Haghani, H. Hao, and P. Zhang, “A comparison study of basic data-driven fault diagnosis and process monitoring methods on the benchmark tennessee eastman process,” Journal of Process Control, vol. 22, no. 9, pp. 1567–1581, 2012.