Robust Stability Analysis for T-S Fuzzy Neural Networks with Time-varying Delays

In this paper, the robust stability of T-S fuzzy uncertain system for neural networks with time-varying delays is investigated. The constraint on the time-varying delay function is removed, which means that a fast time-varying delay is allowed. Based on the Lyapunov-Krasovskii functional techniques and integral inequality approach (IIA), novel robust stability criteria have been derived in terms of linear matrix inequalities which can be easily solved using the efficient convex optimization algorithm. By taking the relationship among the time-varying delay, its upper bound and their difference into account, some less conservative LMI-based delay-dependent stability criteria are obtained without ignoring any useful terms in the derivative of Lyapunov-Krasovskii functional. Examples are included to illustrate our results. These results are shown to be less conservative than those reported in the literature.


Introduction
There has been a large amount of literature that studies the stability properties of linear time delay systems. Some fundamental results on these topics have been published in the literature [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and the reference therein. Recently, people have paid more and more attention on the robust stability of T-S fuzzy system with time delay [2] and time-varying delay [1,4,5,6,7,8,9,10,11,13,15,16]. Unfortunately, the existing results always assume that the time-varying delay function is continuously differentiable and its derivative is smaller than one, see [7] for example, which is a rigorous constraint. Therefore, it is interesting but challenging to develop the robust stability condition without any constraint on the time-varying delay.
Recently, Fuzzy logic theory has shown to be an appealing and efficient approach to dealing with the analysis and synthesis problems for complex nonlinear systems. The well known T-S fuzzy model [14] is a popular and convenient tool in functional approximations. During the last decade, the problem of stability analysis and control synthesis for systems in T-S fuzzy model with time delay have been studied extensively and lot of research works have been reported in the literature [3, 4, 7-11, 15, 16].
More recently, Neural network (NN) have been extensively studied recurrently and involved in many different applications such as signal processing, optimization, fixed point computations, image processing, and other areas [5]. These applications are built upon the stability of the equilibrium of neural networks. Thus, the stability analysis is a necessary step for the design and applications of neural networks. For example, when a neural network is applied to solve the optimization problem, it must have the unique equilibrium which is globally stable. Therefore, stability analysis of neural networks has received much attention and various stability conditions have been obtained. It has been realized that significant time delays as a source of instability and bad performance may occur in neural processing and signal transmission. In contrast to pure neural networks or fuzzy systems, the fuzzy neural networks possess both their advantages. It combines the capability of fuzzy reasoning in handling uncertain information and the capability of artificial neural networks in learning from process. In the last decade, the concept of incorporating fuzzy logic into neural network has grown into a popular research topic [1,5,6].
Stability criteria for T-S fuzzy systems are generally classified into two types: delay-dependent and delay-independent. Since delay-dependent criteria make use of information on the lengths of delays, they are less conservative than delay-independent ones, especially when the delay is small. Moreover, the stability of uncertain T-S fuzzy systems with a time-varying delay was studied in [7]. However, there is still room for further investigation. For example, Universal Journal of Control and Automation 2(1): 32-41, 2014 33 which may lead to considerable conservativeness. The integral inequality approach (IIA) [12,13] has recently been devised to study the stability of time-delay systems, and less conservative stability criteria have been derived.
In this paper, we discuss the stability problem for T-S fuzzy uncertain system for neural networks with time-varying delays by employing an integral inequality approach (IIA). Under considering the relationship among the time-varying delay, its upper bound and their difference, some improved LMI-based stability criteria for uncertain T-S fuzzy uncertain system for neural networks with time-varying delays are obtained without ignoring any useful terms in the derivative of a Lyapunov functional. Finally, numerical examples are given to demonstrate the effectiveness and merits of the proposed method.

Stability Description and Preliminaries
The recurrent neural networks with time-varying delay in this paper can be described by the following normalized equations: is the state vector with the n neurons; is called an activation function indicating how the j-th neuron responses to its input; is a diagonal matrix with each 0 i c > controlling the rate with which the i-th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs; ( ) , are unknown matrices that represent the time-varying parameter uncertainties and h(t) is the time delay of the system satisfies where h and d h are some positive constants. In this paper, the neuron activation functions are assumed to be bounded and satisfy the following assumption.
where i γ and i k are known constant scalars. Remark 1. If the neuron activation functions satisfy Assumption 1, then they satisfy However, we shall point out that this assumption is much strong and may lead to conservative conditions for the delay-dependent stability analysis of delayed neural networks. For example, if 0, i i k γ < < then the delay-dependent stability result obtained by using (4) is generally less conservative than the one obtained by using (5). This will be shown via numerical examples in Sec. 4 in this paper.
Next, the equilibrium point (1) is shifted to the origin through the transformation ( ) ( ) x t u t u * = − , then system (1) can be equivalently written as the following system where T T 1 ] , n n n are the uncertainties of the system and have the form where , , , i ci ai D E E and bi E are known constant real matrices with appropriate dimensions and ( ) F t is an unknown matrix function with Lebesgue-measurable elements bounded by where I is an appropriately dimensioned identity matrix. The fuzzy model of (5) described by 34 Robust Stability Analysis for T-S Fuzzy Neural Networks with Time-varying Delays φ is a vector-valued initial condition; the scalars r is the number of IF-Then rules. By fuzzy blending, the overall fuzzy model is inferred as follows: is the membership function of the system with respect to the plant rule In the following, we will develop some practically computable stability criteria for the system described (9).The following lemmas are useful in deriving the criteria.
First, we introduce the following integral inequality approach (IIA), which be used in the proof of ours.
Lemma 1 [12,13]. For any positive semi-definite matrices 11 the following integral inequality holds 33 ( ) 11 12 13 Secondary, the following Schur complement result, which is essential in the proofs of Theorem 1, is introduced.
x is equivalent to Finally, the following Lemma 3 will be used to handle the parametrical perturbation. Lemma 3 [2].Given symmetric matrices Ω and , ,

Main Results
In this section, we use the integral inequality approach (IIA) to obtain stability criterion for a T-S fuzzy neural networks with time-varying delays. First, we take up the case where in system (9) as follows: positive semi-definite matrices 11 Ω Ω Proof: Choose the following fuzzy Lyapunov-Kravoskii functional candidate to be Then, taking the time derivative of ( ) V t with respect to t along the system (13) yield Alternatively, the following equations are true: along solution to (13), gives as follows:  T   T   T  T  T  T  T  T   T  T  T  T  T  T   t

hZx t x Cx t Ag x t Bg x t h t hZ Cx t Ag x t Bg x t h t t h ZCx t t ZAg x t t ZBg x t h t x C x hC x hC x t h ZCx t x t h ZAg x t x t h ZBg x t h t g g g A
A (4), it can be verified that Similarly, there holds 0 ( ( ( ))) ( ( ( ))) ( ( ( ))) ( ( ( )))

t h t Vg x t h t x t h t Vg x t h t g g t h t VKx t h t t h t V K g x t h t x x x t h t Vg x t h t g
Substituting the above equations (17)-(26) into (16), we obtain   T  T  T  T   T  T   T   T  T  T   T , , .
Finally, using the Schur complements of Lemma 2, with some effort we can show that (27) guarantees of Ω which is equivalent to (27). Therefore, if LMIs (14) are feasible, the system (13) is asymptotically stable. This completes the proof.
Based on Theorem 1, we have the following result for uncertain T-S fuzzy system (9). Theorem 2. For given positive scalars h and , d h t the uncertain T-S fuzzy neural network with time-varying delay (9) is asymptotically stable if there exist symmetric positive-definite matrices 0, 0, and positive semi-definite matrices   11  12  13  11  12  13   12  22  23  12  22  23   13  23  33  13  23  33 0, 0,   T  T  T  ci  ci  ci  ai  ci  bi  i  i  i  i   T  T  T  T  ai  ai  ai  ai  ai  bi  i  i  i   T  T  T  T  T  bi  a  bi  ai  bi  bi  i i i where , ( , 1,..., 6; 6) ij i j i j = < ≤ Ω are defined in (14). It is, incidentally, worth noting that the time delay uncertain system (9) is asymptotically stable, that is, the uncertain parts of the nominal system can be tolerated within allowable time delay .
h Proof: Replacing , , respectively, we apply Lemma 2 for system (9) is equivalent to the following condition: Lemma 3,(29) is true if there exist a scalar 0 i ε > such that the following inequality holds 1 0.
Applying the Schur complement shows that (30) is equivalent to (28a). This completes the proof.
If the upper bound of the derivative of time-varying delay d h is unknown, Theorem 2 can be reduced to the result with 0 Q = and 0, Y = we have the following Corollary 1. Corollary 1. Consider system (9) with constant delay. For given a positive scalar , h the system is asymptotically stable if there exist symmetric positive-definite matrices 0, 23 11 13 13 .
Proof. If the matrix 0 Q = is selected in (15).This proof can be completed in a similar formulation to Theorem 1 and Theorem 2.
Remark 1: Theorem 2 provides delay-dependent robust asymptotic stability criterion for the time delay uncertain systems (9) in terms of solvability of LMIs [2]. Based on them, we can obtain the maximum allowable delay bound (MADB) h such that (9) is stable by solving the following convex optimization problem

Maximize
Subject to (28) Inequality (32) is a convex optimization problem and can be obtained efficiently using the MATLAB LMI Toolbox.

Illustrative Examples
In this section, the following examples are provided to show the effectiveness of the proposed methods. Example 1. Consider a time delayed fuzzy system without uncertainty. The T-S fuzzy model of this fuzzy neural network with time-varying delay is of the following form: Plant rules: and the membership function for rule 1 and rule 2 are The neuron activation functions are assumed to satisfy Assumption 1 with is essential in [6], but it is not necessary in our result.
The MADB h that guarantees the system (33) to be asymptotically stable is calculated to be 1.1503 h = in [5], which is h = 2.6853 by using Theorem 1 in this paper. It is seen that our results improve the existing results [5,6]. In case of 2.6853, h = solving Theorem 1 yields the following set of feasible solutions Table 1. From the above results of Table 1, if the d h increases the delay time length decreases. It is shown that when d h is known, our obtained results are better than those in [8,9,11]; when d h is unknown, the results in [7] fail to verify that the system is stable, while Theorem 2 of this paper can obtain the better upper bounds than those in [7,8,9,11].

Conclusion
In this paper, some less conservative LMI-based robust stability criteria are obtained without ignoring any terms in the derivative of Lyapunov-Krasovskii functional for uncertain T-S fuzzy neural networks with time-varying delays. Based on the Lyapunov-Krasovskii functional techniques, novel robust stability criteria have been derived in terms of linear matrix inequalities which can be easily solved using the efficient convex optimization algorithm. The LMI optimization approaches are used to obtain sufficient conditions that are very easy to be checked by using the LMI Toolbox in Matlab. Numerical examples demonstrate that the proposed method is an improvement over the existing ones.