Research Article  Open Access
DelayRangeDependent Global Robust Passivity Analysis of DiscreteTime Uncertain Recurrent Neural Networks with Interval TimeVarying Delay
Abstract
This paper examines a passivity analysis for a class of discretetime recurrent neural networks (DRNNs) with normbounded timevarying parameter uncertainties and interval timevarying delay. The activation functions are assumed to be globally Lipschitz continuous. Based on an appropriate type of Lyapunov functional, sufficient passivity conditions for the DRNNs are derived in terms of a family of linear matrix inequalities (LMIs). Two numerical examples are given to illustrate the effectiveness and applicability.
1. Introduction
Recurrent neural networks have been extensively studied in the past decades. Two popular examples are Hopfield neural networks and cellular neural networks. Increasing attention has been draw to the potential applications of recurrent neural networks in information processing systems such as signal processing, model identification, optimization, pattern recognition, and associative memory. However, these successful applications are greatly dependent on the dynamic behavior of recurrent neural networks (RNNs). On the other hand, time delay is inevitably encountered in RNNs, since the interactions between different neurons are asynchronous. Generally, time delays, both constant and time varying, are often encountered in various engineering, biological, and economic systems due to the finite switching speed of amplifiers in electronic networks, or to the finite signal propagation time in biological networks [1]. The existence of time delay could make delayed RNNs be instable or have poor performance. Therefore, many research interests have been attracted to the stability analysis for delayed RNNs. A lot of results related to this issue have been reported [2–6].
The theory of passivity plays an important role for analyzing the stability of nonlinear system [7] and has received much attention in the literature from the control community since 1970s [8–14]. It is well known that the passivity theory plays an important role in both electrical network and nonlinear control systems, provides a nice tool for analyzing the stability of system [15], and has found applications in diverse areas such as signal processing, chaos control and synchronization, and fuzzy control. The passivity condition for delayed neural networks with or without timevarying parametric uncertainties using LMIs [16] has been proposed in [17]. By constructing proper Lyapunov functionals and using some analytic techniques, sufficient conditions are given to ensure the passivity of the integrodifferential neural networks with timevarying delays in [18]. In [19, 20], the authors studied the delaydependent robust passivity criterion for the delayed cellular neural networks and the delayed recurrent neural networks. It should be pointed out that the aforementioned results are the continuoustime neural networks.
Recently, the stability analysis problems for discretetime neural networks with time delay have received considerable research interests. For instance in [21], global exponential stability of a class of discretetime Hopfield neural networks with variable delays is considered. By making use of a difference inequality, a new global exponential stability result is provided. Under different assumptions on the activation functions, a unified linear matrix inequality (LMI) approach has been developed to establish sufficient conditions for the discretetime recurrent neural networks with interval variable time to be globally exponentially stable in [22]. Delaydependent results on the global exponential stability problem for discretetime neural networks with timevarying delays were presented in [23, 24], respectively. However, no delayrangedependent passivity conditions on discretetime uncertain recurrent neural networks with interval timevarying delay are available in the literature and remain essentially open. The objective of this paper is to address this unsolved problem.
The purpose of this paper is to deal with the problem of passivity conditions for discretetime uncertain recurrent neural networks with interval timevarying delay. The interval timevarying delay includes both lower and upper bounds of delay, and the parameter uncertainties are assumed to be time varying but norm bounded which appear in all the matrices in the state equation. It is then established that the resulting passivity condition can be cast in a linear matrix inequality format which can be conveniently solved by using the numerically effective Matlab LMI Toolbox. In particular, when the interval timedelay factor is known, it is emphasized that delayrangedependent passivity condition yields more general and practical results. Finally, two numerical examples are given to demonstrate the effectiveness.
Throughout this paper, the notation for symmetric matrices and indicates that the matrix is positive and semidefinite (resp., positive definite); represents the transpose of matrix .
2. Preliminaries
Consider a discretetime recurrent neural network with interval timevarying delay described by where is the state vector, with , , is the state feedback coefficient matrix, and are the interconnection matrices representing the weighting coefficients of the neurons, is the neuron activation function with , is the timevarying delay of the system satisfying where are known integers. Let be the output of the neural networks. is the input vector. , , and are unknown matrices representing timevarying parameter uncertainties, which are assumed to be of the form where , , , and are known real constant matrices, and is unknown timevarying matrix function satisfying The uncertain matrices , , and are said to be admissible if both (2.3) and (2.4) hold.
In order to obtain our main results, the activation functions in (2.1) are assumed to be bounded and satisfy the following assumption.
Assumption 1. The activation functions , , are globally Lipschitz and monotone nondecreasing; that is, there exist constant scalars such that for any and ,
Definition 2.1 ([25]). System (2.1) is called passive if there exists a scalar such that for all and for all solutions of (2.1) with .
3. Mathematical Formulation of the Proposed Approach
This section explores the globally robust delayrangedependent passivity conditions of the discretetime recurrent uncertain neural network with interval timevarying delay given in (2.1). Specially, an LMI approach is employed to solve the robust delayrangedependent passivity condition if the system in (2.1) is globally asymptotically stable for all admissible uncertainties , , and satisfying (2.6). The analysis commences by using the LMI approach to develop some results which are essential to introduce the followingLemma 3.1for the development of our main theorem.
Lemma 3.1. Let , , , , and be real matrices of appropriate dimensions with and satisfying . Then the following statements hold.(a)For any and vectors , (b)For vectors , For any matrices , , , and of appropriate dimensions, it follows from null equations that
To study the globally robust delayrangedependent passivity conditions of the discretetime uncertain recurrent neural network with interval timevarying delay, the following theorem reveals that such conditions can be expressed in terms of LMIs.
Theorem 3.2. Under Assumption 1, given scalars , system (2.1) with interval timevarying delay satisfying (2.2) is globally asymptotically robust stability, if there exist matrices , , , , , diagonal matrices , , and matrices , , and of appropriate dimensions, a positive scalar and a scalar such that the following LMI holds: where in which , . Then system (2.1) satisfying (3.8) with interval timevarying delay is robust delayrangedependent passivity condition in the sense of Definition 2.1.
Proof. Choose the LyapunovKrasovskii functional candidate for the system in (2.1) as
Then, the difference of along the solution of (2.1) gives
Defining the following new variables:
and combining null equations (3.3)–(3.7), it yields
Moreover,
Using Assumption 1 and noting that and are diagonal matrices, one has
where .
Following from Lemma 3.1(a) results in
Substituting (3.14)–(3.19) into (3.13), it is not difficult to deduce that
Using the Schur complement to the (3.20) and in view of LMI (3.8), it follows that
It follows from (3.21) that
for , one has , so (2.6) holds, and hence the system is robust delayrangedependent passivity condition in the sense of Definition 2.1. This completes the proof of Theorem 3.2.
Remark 3.3. Theorem 3.2 provides a sufficient passivity condition for the globally robust stability of the discretetime uncertain recurrent neural network with interval timevarying delay given in (2.1) and proposes a delayrangedependent criterion. Even for , the result in Theorem 3.2 may lead to the delaydependent stability criteria. In fact, if , with , being sufficiently small scalar, , , , , , Theorem 3.2 yields the following delaydependent passivity criterion.
Corollary 3.4. Under Assumption 1, given scalars , , system (2.1) with timevarying delay satisfying (2.3) is globally asymptotically robust stability, if there exist matrices , , , , diagonal matrices , , and matrices , , and of appropriate dimensions and , a positive scalar , and a scalar such that the following LMI holds: where Therefore, the discretetime recurrent neural network with timevarying delay in (2.1) (i.e., the lower bounds and the given upper bounds ) approaches globally robustly delaydependent passivity condition in the sense of Definition 2.1.
Remark 3.5. In the stochastic context, robust delaydependent passivity conditions are studied in [26] for discretetime stochastic neural networks with timevarying delays. In this paper, however, robust delayrangedependent passivity conditions are studied in the deterministic context. It should be noted that deterministic systems and stochastic systems have different properties and need to be dealt with separately. The results given in Theorem 3.2 provide an LMI approach to the robust delayrangedependent passivity conditions for deterministic discretetime recurrent neural networks with interval timevarying delay, which is new and represents a contribution to recurrent neural networks systems.
Two numerical examples are now presented to demonstrate the usefulness of the proposed approach.
4. Examples
Example 4.1. Consider the following discretetime uncertain recurrent neural network: where The activation functions in this example are assumed to satisfy Assumption 1 with , . For interval timevarying delay, the best approached values of by Theorem 3.2, for the given upper bound and the various lower bounds , are listed inTable 1by the Matlab LMI Control Toolbox. Therefore, using Theorem 3.2, the discretetime uncertain recurrent neural network with interval timevarying delay (4.1) satisfies robustly delayrangedependent passivity conditions in the sense of Definition 2.1 for various levels.

Example 4.2. Consider the discretetime uncertain recurrent neural network with the following parameters: The activation functions in this example are assumed to satisfy Assumption 1 with , , . By the Matlab LMI Control Toolbox, it can be verified that Corollary 3.4 in this paper is feasible solution for all delays (i.e., the lower bound and the upper bound ) as follows: Thus, by Corollary 3.4, the discretetime uncertain recurrent neural network with timevarying delay in (2.1) (i.e., the lower bound and the given upper bound ) attains globally robustly delaydependent passivity condition in the sense of Definition 2.1.
5. Conclusions
This study has investigated the problem of globally robust passivity conditions for a discretetime recurrent uncertain neural network with interval timevarying delay. A sufficient condition for the solvability of this problem, which takes into account the range for the time delay, has been established that the passivity conditions can be cast in linear matrix inequalities format. It has been shown that the bound for the timevarying delay in a range which ensures that the discretetime recurrent uncertain neural network with interval timevarying delay attains globally robust passivity conditions can be obtained by solving a convex optimization problem. Two numerical examples have been presented to demonstrate the effectiveness of the proposed approach.
References
 S. Arik, “Global asymptotic stability of a larger class of neural networks with constant time delay,” Physics Letters A, vol. 311, no. 6, pp. 504–511, 2003. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 I. Győri and F. Hartung, “Stability analysis of a single neuron model with delay,” Journal of Computational and Applied Mathematics, vol. 157, no. 1, pp. 73–92, 2003. View at: Publisher Site  Google Scholar  MathSciNet
 E. Kaszkurewicz and A. Bhaya, “On a class of globally stable neural circuits,” IEEE Transactions on Circuits and Systems I, vol. 41, no. 2, pp. 171–174, 1994. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. Cao and J. Wang, “Global asymptotic stability of a general class of recurrent neural networks with timevarying delays,” IEEE Transactions on Circuits and Systems I, vol. 50, no. 1, pp. 34–44, 2003. View at: Publisher Site  Google Scholar  MathSciNet
 M. Joy, “Results concerning the absolute stability of delayed neural networks,” Neural Networks, vol. 13, no. 6, pp. 613–616, 2000. View at: Publisher Site  Google Scholar
 S. Xu, Y. Chu, and J. Lu, “New results on global exponential stability of recurrent neural networks with timevarying delays,” Physics Letters A, vol. 352, no. 45, pp. 371–379, 2006. View at: Publisher Site  Google Scholar
 L. O. Chua, “Passivity and complexity,” IEEE Transactions on Circuits and Systems I, vol. 46, no. 1, pp. 71–82, 1999. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 L. Xie, M. Fu, and H. Li, “Passivity analysis for uncertain signal processing systems,” in Proceedings of the IEEE International Conference on Signal Processing, vol. 46, pp. 2394–2403, 1998. View at: Google Scholar
 M. S. Mahmoud and A. Ismail, “Passivity and passification of timedelay systems,” Journal of Mathematical Analysis and Applications, vol. 292, no. 1, pp. 247–258, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 E. Fridman and U. Shaked, “On delaydependent passivity,” IEEE Transactions on Automatic Control, vol. 47, no. 4, pp. 664–669, 2002. View at: Publisher Site  Google Scholar  MathSciNet
 T. Hayakawa, W. M. Haddad, J. M. Bailey, and N. Hovakimyan, “Passivitybased neural network adaptive output feedback control for nonlinear nonnegative dynamical systems,” IEEE Transactions on Neural Networks, vol. 16, no. 2, pp. 387–398, 2005. View at: Publisher Site  Google Scholar
 S.I. Niculescu and R. Lozano, “On the passivity of linear delay systems,” IEEE Transactions on Automatic Control, vol. 46, no. 3, pp. 460–464, 2001. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. C. TraviesoTorres, M. A. DuarteMermoud, and J. L. Estrada, “Tracking control of cascade systems based on passivity: the nonadaptive and adaptive cases,” ISA Transactions, vol. 45, no. 3, pp. 435–445, 2006. View at: Publisher Site  Google Scholar
 L. Keviczky and Cs. Bányász, “Robust stability and performance of timedelay control systems,” ISA Transactions, vol. 46, no. 2, pp. 233–237, 2007. View at: Publisher Site  Google Scholar
 R. Lozano, B. Brogliato, O. Egeland, and B. Maschke, Dissipative Systems Analysis and Control: Theory and Application, Communications and Control Engineering Series, Springer, London, UK, 2000. 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: MathSciNet
 C. Li and X. Liao, “Passivity analysis of neural networks with time delay,” IEEE Transactions on Circuits and Systems II, vol. 52, no. 8, pp. 471–475, 2005. View at: Google Scholar
 X. Lou and B. Cui, “Passivity analysis of integrodifferential neural networks with timevarying delays,” Neurocomputing, vol. 70, no. 4–6, pp. 1071–1078, 2007. View at: Google Scholar
 J. H. Park, “Further results on passivity analysis of delayed cellular neural networks,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1546–1551, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Q. Song and Z. Wang, “New results on passivity analysis of uncertain neural networks with timevarying delays,” International Journal of Computer Mathematics. In press. View at: Publisher Site  Google Scholar
 Q. Zhang, X. Wei, and J. Xu, “On global exponential stability of discretetime Hopfield neural networks with variable delays,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 67675, 9 pages, 2007. View at: Google Scholar  MathSciNet
 Y. Liu, Z. Wang, A. Serrano, and X. Liu, “Discretetime recurrent neural networks with timevarying delays: exponential stability analysis,” Physics Letters A, vol. 362, no. 56, pp. 480–488, 2007. View at: Publisher Site  Google Scholar
 W.H. Chen, X. Lu, and D.Y. Liang, “Global exponential stability for discretetime neural networks with variable delays,” Physics Letters A, vol. 358, no. 3, pp. 186–198, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. Liang, J. Cao, and D. W. C. Ho, “Discretetime bidirectional associative memory neural networks with variable delays,” Physics Letters A, vol. 335, no. 23, pp. 226–234, 2005. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. S. Mahmoud, Robust Control and Filtering for TimeDelay Systems, vol. 5 of Control Engineering, Marcel Dekker, New York, NY, USA, 2000. View at: MathSciNet
 Q. Song, J. Liang, and Z. Wang, “Passivity analysis of discretetime stochastic neural networks with timevarying delays,” Neurocomputing, vol. 72, no. 7–9, pp. 1782–1788, 2009. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2009 ChienYu Lu 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.