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Research Article | Open Access
Stability Analysis for Delayed Neural Networks: Reciprocally Convex Approach
This paper is concerned with global stability analysis for a class of continuous neural networks with time-varying delay. The lower and upper bounds of the delay and the upper bound of its first derivative are assumed to be known. By introducing a novel Lyapunov-Krasovskii functional, some delay-dependent stability criteria are derived in terms of linear matrix inequality, which guarantee the considered neural networks to be globally stable. When estimating the derivative of the LKF, instead of applying Jensen’s inequality directly, a substep is taken, and a slack variable is introduced by reciprocally convex combination approach, and as a result, conservatism reduction is proved to be more obvious than the available literature. Numerical examples are given to demonstrate the effectiveness and merits of the proposed method.
In recent years, important application in the fields of pattern recognition, signal processing, optimization and associative memories, and so forth has made neural networks (NNs) the eye of attention. Stability analysis of NNs with time-varying delay, as a result, has received much attention ever since, because time delay is frequently encountered in NNs, owing to finite switching speed of amplifiers in the communication and response of neurons, and could cause instability and oscillations in the system. Both delay-independent and delay-dependent stability criteria have been brought up. While delay-independent criteria tend to be more conservative, more attention is given to delay-dependent criteria, because they could make use of the length of the delay.
Global stability of various recurrent neural networks has been investigated in [1–4]. In stability analysis of neural networks, the qualitative properties primarily concerned are uniqueness, global asymptotic stability, and global exponential stability of their equilibria. When the system with time delay is described by the following dynamic equation: exponential and asymptotic stability analysis has been done by [5–10]. The stability criteria in [5, 9, 10] are delay independent, while those in [6, 8, 11] are delay dependent, and  includes both. Also, [8–10] adopted the delay-partitioning approach. After some changes are made on system description, such as setting , adding a new term , and introducing a new activation function to substitute for , asymptotic stability criteria are derived in [13–17], and uniqueness analysis is done in [5, 18].
Various methods have been proposed to reduce conservatism when deriving stability criteria. For example, the free-weighting matrix approach noticed in [6, 7, 19–22] is proved to be very effective since bounding techniques on some cross-product terms are avoided. Stability analyses of NNs with multiple and single time-varying delays are done by [17, 23], respectively. Moreover, a new free-weighting matrix approach is brought up in  to estimate the derivative of Lyapunov functional without missing any negative quadratic terms, and thus, improved delay-dependent stability criteria are established. Along with free-weighting matrix approach, [14, 25, 26] adopted the delay partitioning idea to solve delay-dependent stability problem, and the proposed methods have significantly reduced conservatism.
When deriving stability criteria for delayed systems, two kinds of approaches are usually used, namely, Lyapunov function approaches [27–29] and Lyapunov-Krasovskii functional (LKF) approaches [17, 23, 24, 30–33]. The former makes no restriction on the derivative of time delay and usually gives a simpler stability criterion or delay-independent criterion while the latter, expressed in the form of LMI, takes the derivative of time delay into account, gives a delay-dependent criterion, and thus can be less conservative since LKF makes use of more information about the system. Discretized LKF method developed in  is another method for stability analysis, and  made some necessary adjustments to make the method compatible with robust stability problem for NNs with uncertain delays.
In addition, the range of time-varying delay for NNs is mostly considered to have a lower bound of zero, as seen in [7, 11, 23, 24], while in practice it may not be restricted to 0, and setting to zero would result in increased conservatism. It is the same with the derivative of time delay. In many papers, time delay of NNs is either constant or unknown, as noticed in [14, 25, 26], while an upper or lower bound could be assumed.
In this paper, the stability problem for continuous NNs with time-varying delay is taken into consideration. A novel LKF is brought up, and changes are made to deal with different cases concerning time delay and its derivative. When estimating the derivative of LKF, instead of applying Jensen’s inequality directly, a substep is taken, and a slack variable is introduced, and consequently, conservatism reduction is proved to be more obvious than existing results. Numerical examples are given, and analysis is made to demonstrate the effectiveness and merits of the proposed method.
In Section 1, a brief introduction is presented, and some notations are defined. In Section 2, the stability problem is formulated, and some preliminaries are given. In Section 3, new criteria in the form of one theorem and three corollaries for NNs with time-varying delay are presented. In Section 4, numerical examples are presented, along with results from the other literature. The paper is concluded in Section 5.
Notations. The notations used throughout the paper are standard. The superscript “” stands for matrix transposition; denotes the -dimensional Euclidean space; the notation means that is a real positive definite; and represent the identity matrix and a zero matrix, respectively; stands for a block-diagonal matrix; denotes the minimum (maximum) eigenvalue of symmetric matrix ; denotes the Euclidean norm of a vector and its induced norm of a matrix. In symmetric block matrices, we use an asterisk () to represent a term that is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.
2. Model Descriptions and Preliminaries
The dynamic behavior of a continuous-time neural networks with time delay can be described by the following state equation: where is the state vector of the neural network; is a positive matrix; and are the connection weight and the delayed connection weight matrices, respectively. represents the activation function vector of neurons, and is a constant external bias vector. denotes axonal signal transmission delay, which is nonnegative, bounded, and has , , and will be written as for short throughout the paper. The initial conditions associated with system (2) are of the form where is the Banach space of continuous functions mapping interval into .
The following assumptions are made on system (2) throughout this paper.(H1) The activation functions are bounded and monotonically nondecreasing on .(H2) The activation functions satisfy
It is known that bounded activation functions always guarantee the existence of an equilibrium point for model (2). For convenience of exposition, in the following, we will shift the equilibrium point of model (2) to the origin. The transformation puts system (2) into the following form: where is the state vector of the transformed system, and with , , , . Obviously, the equilibrium point of system (2) is globally stable if and only if the origin of system (5) is globally stable. Assume that , then the functions satisfy Rewriting (6), we can get where and .
Lemma 1 (see ). For positive definite , scalars and , and a vector function , the following inequality holds:
3. Main Results
In this section, some delay-dependent sufficient conditions of the global stability for the neural networks with time-varying delay in (5) are derived. First, we consider the case where the upper bound of is known, and correspondingly, the global stability condition is given as follows.
Theorem 2. Suppose that in reference system (5), the time delay satisfies , . Under the condition given in (6), if there exist matrices , , , , , , , , , , and such that where , , , and are defined as where , , , , , , and are defined as then system (5) is globally stable. Moreover, where is defined as
Proof. We choose an LKF as
The derivatives of , , are given, respectively, by
By Lemma 1, we can get
By (16) and (17), we have
where is defined as
Then by (7), (18), and (19), we have
It is clear that if (9) and (20) hold, then, for any , we have . It follows that
From (15), we get By a similar method in , we have Thus, , where is defined in (13).
Therefore, we have which shows that system (5) is globally stable. This completes the proof.
Remark 3. Theorem 2 presents a stability criterion for the delayed neural network. When coping with , instead of using Jensen’s inequality directly, we use a substep which can make the method less conservative, which can be noticed as reciprocally convex combination approach in . It follows that If no substep is taken, it will follow that
We can see that compared to (27), (26) is relatively free since could be more than nonnegative, and consequently, its LMI could suffer bigger delay. For the same reason, if the middle term could be found between and 0, conservatism would be further reduced.
Remark 4. Based on Theorem 2, we can determine the maximum admissible delay and at a known upper bound of . Moreover, the relationship between and could be specified using and . As to the case when is unknown, we can refer to Corollary 5, where some changes are made on the LKF.
Corollary 5. Suppose that the time delay in reference system (5) satisfies . Under the condition given by (6) and (7), if there exist matrices , , , , , , , , , and such that the following matrix inequalities hold: where , , , and are defined as where , , , , , , and are defined in (11), then system (5) is globally stable. Moreover, where is defined as
Remark 6. Corollary 5 presents the stability criterion when the upper bound of is unknown. Since is unknown and under current conditions, it cannot be estimated or substituted, the first term in of should be changed or eliminated from the LKF. In Corollary 5, the term is eliminated because the other two terms in can serve the same function, and it is unnecessary to keep any extra or . The rest of the terms were reserved because they will not generate any -related terms when estimating the derivative of the LKF.
Corollary 7. Suppose that the time delay in reference system (5) satisfies and . Under the condition given by (6) and (7), if there exist matrices , , , , ,, , , , and such that the following matrix inequalities hold: where , , , and are defined as where , , , , , and are defined as then system (5) is globally stable. Moreover, where is defined as
Remark 8. Corollary 7 presents the stability criterion when the lower bound is zero. If is zero, the second term in of should be changed or eliminated from the LKF. In Corollary 7, the term is eliminated because there is no need to introduce an extra variable , while and other matrices can serve the same function. Moreover, and in can be merged because when is zero, they have the same form. But and can be reserved, because when estimating the upper bound of the LKF, it is still useful to introduce a like Theorem 2.
Corollary 9. Suppose that the time delay in reference system (5) satisfies . Under the condition given by (6) and (7), if there exist matrices , , , , , , , , , , and such that the following matrix inequalities hold: where , , , and are defined as where , , , , , and are defined in (16), then system (5) is globally stable. Moreover, where is defined as
Remark 10. Corollary 9 presents the stability criterion when is zero and the upper bound of time delay’s derivative, or , is unknown. If is zero and is unknown, the first and second terms of and in should be eliminated from the LKF. There is no need to introduce an extra variable or keep any or , and the reason is the same with Corollaries 5 and 7. Moreover, the terms in can be merged, while and should be reserved for the same reason with Corollary 7.
4. Numerical Examples
In this section, examples are provided to demonstrate the advantages of the proposed stability criteria.
Example 11. Consider the delayed neural network in (5) with the following parameters, which has also been investigated by [13, 39]: Our objective is to find the allowable maximum time delay such that the system is stable under different and . The simulation results from the available literature are shown in Table 1, along with results from Theorem 2 and Corollaries 5, 7, and 9. It is clear that the conservatism reduction proves to be more obvious than those in [23, 24, 37, 38].