Abstract: The problem of robust stability for a class of neural networks with mixed time-varying delays and nonlinear perturbations is investigated. The mixed delays contain discrete and neutral-type time-varying delays. By constructing a general form of Lyapunov-Krasovskii functional, using some free-weighting matrices, two delay-dependent stability criteria are derived. In particular, the proposed stability conditions are presented in terms of LMI which can be easily solved by some standard numerical packages. In addition, the nonlinear perturbations (or norm-bounded uncertainty) which are more general than those discussed in the previous literature.
INTRODUCTION
In the past several decades, artificial neural networks have been widely investigated and employed to solve many practical engineering problems in various fields. In the implementation of networks, time delay is often encountered and its existence is frequently a source of oscillations, chaos and instability (Liao et al., 2002; Cao and Wang, 2003, 2005). Thus, the stability analysis of various delayed neural networks models has been discussed by many authors and some stability criteria have been deduced (Li et al., 2007; Zhang et al., 2007; Zhang and Chen, 2008; Xiong and Xu, 2008; Yang et al., 2009). Recently, the stability of neural networks with neutral-type delays also has been deeply studied via the LMI technique and Lyapunov functions and several important stability conditions have been introduced, see (Qiu and Ren, 2006; Park et al., 2008; Lee et al., 2010) and their references.
On the other hand, the nonlinearities are the important sources of delayed system instability. Therefore, it is of great importance to consider nonlinear perturbations on the stability of delayed system (or neural network with time delays), (Xie et al., 2006; Wang et al., 2009; Kwon et al., 2008; Qiu et al., 2010). Based on the Lyapunov method, a sufficient delay-dependent criterion for asymptotic stability of dynamic systems with time-varying delays and nonlinear perturbations is derived by Kwon et al. (2008). However, neutral-type delays were not considered by Kwon et al. (2008). Using the Lyapunov functional technique combined with matrix inequality technique, Qiu et al. (2010) present a novel asymptotic stability criterion for neutral systems with nonlinear perturbations. Very recently, the stability of neural networks with norm-bounded uncertainties and neutral-type delays has received much attention (Lien et al., 2008; Zhu et al., 2009; Zhang et al., 2010). For example, Zhu et al. (2009) have investigated the stability analysis of neutral-type neural networks with parameter uncertainties via LMI approach. In Zhang et al. (2010), the robust stability of neural networks with neutral-type delays and parameter uncertainties is studied by employing a Lyapunov-Krasovskii functional combined with the LMI approach. Most of the obtained results are based on restricting norm-bounded for parameter uncertainty. However, to the best of the authors’ knowledge, the robust stability analysis for a class of neural networks with nonlinear perturbations, discrete and neutral-type and time-varying delays has not been investigated which motives our research.
Present study is concerned with the robust stability analysis problem for a class of neural networks with both nonlinear perturbations and mixed time-varying delays, which comprise discrete, time-varying and neutral-type time delays. Based on the Lyapunov stability theory and the free-weighting technique, the stability conditions are obtained in terms of LMI. In addition, the proposed stability conditions are delay-dependent. The proposed model of neural networks is quite general since many factors such as nonlinear perturbations, discrete time-varying delays and neutral-type time-varying delays are considered in this study.
Notations: Throughout this study, for symmetric matrices A and B, A
> B (respectively, A≥B) means that A-B≥0 (A-B>0) is a positive semi-definite
(respectively, positive definite) matrix. The superscripts T and -1 stand for
matrix transposition and matrix inverse, respectively;
PROBLEM DESCRIPTION AND PRELIMINARIES
Consider the following a class of neural networks with mixed time-varying delays and nonlinear perturbations:
| (1) |
where, x (t)
| (2) |
where, τ, τd, h, hd are constants,
| (3) |
where, α ≥ 0,β ≥ 0,γ≥0 and π≥0 are given constants.
Eq. 3 can be rewritten as follow:
| (4) |
For the sake of simplicity, the following notations are adopted:
Then, system (1) can be rewritten as:
| (5) |
The following lemmas will be used in the development of main results in sequel.
Lemma 1(Schur complement): Given constant S1, S2
and S3 with appropriate dimensions where, ST1
= S1 and
| (6) |
Lemma 2: For any constant matrix
| (7) |
Lemma 3: Letandbe real constant matrices with appropriate dimensions, matrix F (t) satisfies FT (t) F (t) ≤I. Then we have:
| (8) |
MAIN RESULTS
The main results of this study are given in the following theorems.
New stability criterion
Theorem 1: Assume time-varying delays τ (t) and h (t) satisfy
(2), system (1) is robustly stable, if there exist matrices:
Ni, Mi, Ki, i = 1, 2, ..., 12 and positive scalars εi > 0, i = 1, 2, 3, 4 such that the following LMI hold:
| (9) |
where,
with:
Proof: First, Let us introduce the lyapunov-krasoskill functional as follows:
| (10) |
where,
| (11) |
The time derivative ofalong the trajectory of system (1) is given by:
| (12) |
where,
| (13) |
| (14) |
| (15) |
| (16) |
| (17) |
By applying Lemma 2, it follows that:
| (18) |
By using the Newton-Leibniz formula and Eq. 5, one has:
| (19) |
| (20) |
| (21) |
By applying Lemma 3, it follows that:
| (22) |
From Eq. 3, one can obtain for any scalars ε1 > 0, ε2 > 0, ε3 > 0 and ε4 > 0:
| (23) |
Combining Eq. 13-23, it is easy to verify that:
| (24) |
where,
It is obvious that for Π < 0, by using the Lyapunov-Krasovskii stability theorem, one can conclude that the neural network of neutral-type (1) is globally asymptotically stable if (9) holds. This completes the proof of Theorem 1.
Norm-bounded uncertainty: Here, we will present a delay-dependent robust
criterion for the system (1) that
| (25) |
The time-varying uncertainties are of the form:
| (26) |
where, Ti, i = 1, 2, 3, 4, L are constant matrices of appropriate dimensions. F (t) is an unknown and possibly time-varying real matrix with Lebesgue measurable elements and satisfies:
| (27) |
Then, system (1) becomes the following system:
| (28) |
Theorem 2: Assume time-varying delays τ (t) and h (t) satisfy (2), system (28) is robustly stable, if there exist matrices:
i = 1, 2, Ni, Mi, Ki, I = 1, 2, ..., 12 and positive scalar ε > 0 such that the following LMI holds:
| (29) |
where,
with:
Proof: Using Lemma 2(Schur complement),
implies that:
| (30) |
where,
Then, noting that Eq. 26 and 27, using Lemma 3, one has:
| (31) |
Where:
Therefore, one obtains:
| (32) |
Where:
Define a Lyapunov functional candidate for system (28):
| (33) |
where,
From Eq. 20, the following equations hold:
| (34) |
| (35) |
| (36) |
where,
Then, substituting Eq. 11-18, 22, 34-36 into Eq. 12, one obtains:
| (37) |
If
CONCLUSION
In present study, the robust stability of a class of neural networks with mixed time-varying delays and nonlinear perturbations has been studied. The mixed time delays comprise both the discrete and neutral-type time-varying delays. By constructing a new Lyapunov-Krasovskii functional and introducing some free-weighting matrices, sufficient delay-dependent stability conditions are derived in the form of a standard LMI. In addition, present results are shown to be generalizations of some previously published results. More importantly, the present results are also applicable to neural networks of neutral-type with multiple time delays.
ACKNOWLEDGMENT
This work was supported by the National Natural Science Foundation (No. 60974090) and by the Fundamental Research Funds for the Central Universities (No. CDJXS11172237).