Improved Extended Dissipativity Results for T-S Fuzzy Generalized Neural Networks With Mixed Interval Time-Varying Delays

The asymptotic stability and extended dissipativity performance of T-S fuzzy generalized neural networks (GNNs) with mixed interval time-varying delays are investigated in this paper. It is noted that this is the first time that extended dissipativity performance in the T-S fuzzy GNNs has been studied. To obtain the improved results, we construct the Lyapunov-Krasovskii functional (LKF), which consists of single, double, triple, and quadruple integral terms containing full information of the delays and a state variable. Moreover, an improved Wirtinger inequality, a new triple integral inequality, and zero equation, along with a convex combination approach, are used to deal with the derivative of the LKF. By using Matlab’s LMI toolbox and the above methods, the new asymptotic stability and extended dissipativity conditions are gained in the form of linear matrix inequalities (LMIs), which include passivity, $L_{2}-L_{\infty }$ , $H_{\infty }$ , and dissipativity performance. Finally, numerical examples that are less conservative than previous results are presented. Furthermore, we give numerical examples to demonstrate the correctness and efficacy of the proposed method for asymptotic stability and extended dissipativity performance of the T-S fuzzy GNNs, including a particular case of the T-S fuzzy GNNs.


I. INTRODUCTION
Various types of neural networks (NNs) have attracted the attention of researchers in the past few decades because neural networks have a wide range of applications in many fields such as combinatorial optimization, speed detection of moving objects, pattern classification, associative memory design, and other areas [1]- [5]. And we need to first perform a theoretical stability analysis of the equilibrium point to achieve the mentioned applications. Moreover, an essential factor affecting the model of the system to be used in the stability analysis is the time delay. Time delay is a natural phenomenon that always occurs in neural networks. Note that the latency of information processing and the limited speed of information transmission between neurons causes The associate editor coordinating the review of this manuscript and approving it for publication was Qiuye Sun . the discrete time delay [6], [7]. On the other hand, since the variety of sizes and lengths of the axon, nerve impulses are distributed, which causes the distributed time delay [8]. The presence of such delays frequently leads to system instability, oscillation, and decreased performance. Therefore, time delays cannot be avoided in the analysis of stability and performance for NNs, and many researchers have studied NNs with distributed and discrete time delays [9]- [11]. Additionally, mixed interval time-varying delays can occur in many actual industrial systems, such as the reduced-order aggregate model for large-scale converters [12], a multiagentbased consensus algorithm in the energy internet [13], dualpredictive control for AC microgrids [14].
Recently, several researchers have studied the dynamical behaviors of static neural networks (SNNs) [15] or local field neural networks (LFNNs) [16] separately due to differences in neuron states or local field states. Furthermore, these two models are not equivalent but can be combined into a compact model using reasonable assumptions. Thus, a unified system model was first created by Zhang and Han [17] that included both LFNNs and SNNs, called generalized neural networks (GNNs). Analysis of the stability and performance for GNNs with time delay has become increasingly popular in recent years. For example, Chen et al. [18] analyzed the stability of GNNs with time-varying delay by delay-partitioning method; moreover, they obtained improved criteria by using Free-Matrix-based integral inequality, Peng-Park's inequality, and the novel integral inequality. In [19], the problem of stability analysis for GNNs with time-varying delay is examined based on the new proposed LKF and the developed inequality.
It is well known that most dynamic systems in the real world are complex, ambiguous, and nonlinear that are difficult to control or manipulate. The fuzzy logic theory is an interesting and effective method for dealing with analysis and synthesis issues of complex nonlinear systems. Among the various types of fuzzy approaches, Takagi-Sugeno (T-S) fuzzy [20] approach is popular and successful for dealing with complex nonlinear systems using linear sub-systems. These linear sub-systems are combined through fuzzy membership functions. In addition, the neural networks model also has uncertainty or vagueness, so fuzzy logic has been applied to analyze the dynamical behavior of neural networks. For example, Datta et al. [21] used T-S fuzzy logic to describe Hopfield neural networks (HNNs), and novel stability conditions for fuzzy HNNs are obtained by using Wirtinger inequality. The global exponential stability for the T-S fuzzy Cohen-Grossberg neural network is discussed in [22] by considering the effect of non-singular M-matrix properties and the Lyapunov stability technique. Also, in the T-S fuzzy GNNs model, a nonlinear GNNs system can be represented as a weighted sum of some simple linear GNNs subsystems; then, it provides an excellent chance to use the well-established theory of linear GNNs systems to investigate the complex nonlinear GNNs systems. So, it is interesting to study the T-S fuzzy with the GNNs model.
On the other hand, dissipativity is a widely used and effective tool for analyzing nonlinear systems by describing the energy-related input-output relationship. The concept of dissipativity theory was investigated in 1972 by Willems [23], then attracted considerable attention from researchers as it not only combines passivity and H ∞ performance but can also be applied for control performance analysis, such as power converters [24] and chemical process control [25]. The study of dissipativity problems for NNs and T-S fuzzy NNs are contained in [26]- [28]. Meanwhile, the L 2 − L ∞ method is an effective tool for dealing with external interference or uncertainty in the system. Recently, the L 2 − L ∞ method has been applied to many filtering problems to minimize the maximum value of the estimation error. For example, Choi et al. [29] proposed an L 2 − L ∞ filtering for T-S fuzzy NNs in order to reduce the effect of external disturbances in the state estimation error of T-S fuzzy NNs. In [30], the problems of exponential dissipative and L 2 − L ∞ filtering for the discrete-time switched NNs are investigated by using the discrete-time Wirtinger-type inequality. However, the aforementioned L 2 − L ∞ studies were not linked to dissipativity performance. To accommodate this demand, Zhang et al. [31] devised a novel scheme known as an extended dissipativity performance, which combines all of these performances. Hence, various researches of GNNs with time-varying delay have been examined on extended dissipativity performance. For example, Manivannan et al. [32] examined the exponential stability and extended dissipativity for GNNs with interval time-varying delays by using the appropriate LKFs, reciprocally convex combination (RCC) approach, the Wirtinger single integral inequality (WSII), and the Wirtinger double integral inequality (WDII). The extended dissipativity state estimation problem for GNNs with mixed time-varying delays is studied in [33] by using Jensen's inequality, RCC idea together with the WDII. Furthermore, the problem of extended dissipative for GNNs with interval time-delays via non-fragile control is investigated by using WSII, WDII, and RCC idea [34]. Unfortunately, no studies have been conducted to investigate the extended dissipativity for T-S fuzzy GNNs with interval distributed and discrete time-varying delays.
By the above discussions, the asymptotic stability and extended dissipativity performance problem is studied for the T-S fuzzy GNNs with mixed interval time-varying delays in this work. The major contributions are listed as follows: • We construct the model via T-S fuzzy logic, where linear sub-systems are blended through fuzzy membership functions. Moreover, the model contains different continuous neuron activation functions f , g, h, which are different from [32], [35], [36]. The mixed interval time-varying delays such that do not necessitate being differentiable functions and the lower bound of the time-varying delays does not have to be 0. The output consists of the disturbance, the state vector, and the state vector with interval discrete time-varying delay terms. So, our model is more general and covers others such as [6], [7], [15], [21], [26], [32], [35], [37], [38].
• For the first time, an improved Wirtinger inequality, a new triple integral inequality, and zero equation together with convex combination approach are used in this work; as a result, we obtain more general results VOLUME 10, 2022 and maximum allowable delay bounds greater than in previous literature [6], [7], [15], [36]- [38].
• We gain the new extended dissipative criteria that include passivity, L 2 − L ∞ , H ∞ , and dissipativity performance, and the optimal dissipativity performance less conservative than the performance in [26], [32].
This article is divided into five sections, which are as follows. Section 2 includes preliminaries and problem formulation. In section 3, theorems of asymptotic stability and extended dissipativity performance for T-S fuzzy GNNs and a particular case of T-S fuzzy GNNs are addressed. Numerical examples are given in Section 4 to demonstrate the effectiveness of our results. Section 5 provides conclusions and recommendations for future work.
Notations: R n and R + stand for, respectively, the n-dimensional Euclidean space and the set of non negative real numbers. R a×b is the set of a × b real matrix. . Q T means that the transpose of the matrix Q. Sym{Q} denotes Q + Q T . Q > (≥)0 represents the symmetric matrix Q is positive (semi-positive) definite. e k stands for the unit column vector having one element on its kth row and zeros elsewhere. I represents the identity matrix with appropriate dimensions.
Applying the center-average defuzzifier approach, the system (1) can be expressed as follows: where , ∀t and i = 1, 2, . . . , m, is called the fuzzy weighting function which satisfies The T-S fuzzy GNNs (2) can be expressed compactly aṡ

Remark 1: The T-S fuzzy GNNs (4) is a general type of delay T-S fuzzy GNNs model that includes both T-S fuzzy LFNNs and T-S fuzzy SNNs, and it can be easily modified to each of them by changing the values ofB
• When W = I , the T-S fuzzy GNNs (4) becomes the following model, namely T-S fuzzy LFNNs: • WhenB 0 =B 1 =B 2 = I , the T-S fuzzy GNNs (4) converts to the following model, namely T-S fuzzy SNNs: To achieve the main results in the next section, we need to introduce the definition, lemmas, and assumptions.

A. STABILITY ANALYSIS
In this subsection, we achieve the new sufficient conditions of asymptotic stability for the T-S fuzzy GNNs (4) and a particular case of the T-S fuzzy GNNs (4).
Theorem 1: For given scalars and a positive scalar c 1 such that the following linear matrix inequalities hold for i = 1, 2, . . . , m: T 11 , then, the T-S fuzzy generalized neural networks (4) is asymptotically stable.
Proof: See Appendix A. Next, we study the particular case of the system (4) as follows: and a positive scalar a 1 such that the following linear matrix inequalities hold for i = 1, 2, . . . , m: T 11 , then, the T-S fuzzy GNNs (10) is asymptotically stable.
Proof: See Appendix B.

B. EXTENDED DISSIPATIVE ANALYSIS
In this subsection, based on the criteria that were developed in Theorem 1 and 2, we achieve the new sufficient conditions of extended dissipativity for the T-S fuzzy GNNs (4) and a particular case of the T-S fuzzy GNNs (4). Theorem 3: For given scalars η 1 , η 2 , d 1 , d 2 , α 1 , α 2 , and a positive scalar b < 1, if there exist symmetric positive definite matrices P, Q 1 , and a positive scalar c 1 such that the following linear matrix inequalities hold for i = 1, 2, . . . , m:

Remark 3: Recently, extended dissipativity for NNs and
GNNs has received a lot of attention [32], [35], [39] because it not only covers the efficiency of passivity, H ∞ , L 2 − L ∞ , and dissipativity, but it can also be applied in science and engineering fields. In 2015, Choi et al. [29] investigated an L 2 − L ∞ filtering for the T-S fuzzy NNs in order to reduce the effect of external disturbances on the state estimation error of the T-S fuzzy NNs. Furthermore, Datta et al. [21] investigated the asymptotic stability of the fuzzy HNNs with interval discrete time-varying delay. It is well known that the above model is a particular case of the T-S fuzzy GNNs, and distributed delay is unavoidable in the analysis of the delayed T-S fuzzy GNNs systems. Thus, the study of extended dissipativity for the T-S fuzzy GNNs with both interval discrete and interval distributed time-varying delays is a fascinating and challenging problem that we have explored and analyzed in this paper.

y(t)) W i y(t)
≤ H + i . Furthermore, to bound the derivative of the LKF, improved Wirtinger inequality [42], a new triple integral inequality [41], zero equation, and convex combination approach are used. So, the construction of the LKF together with the assistance of the above technique is the main key to improving the results of this work.
Remark 6: In the proof, we use the Lyapunov-Krasovskii functional that is suitable and sufficiently informative. To estimate the derivative of LKF, we use improved Wirtinger inequality [42], a new triple integral inequality [41] with the contribution of zero equation and convex combination approach. These technique are applied to get better results than the others [6], [7], [15], [26], [32], [36]- [38]. However, such complex calculations lead to large LMIs and may be difficult to practical applications. Therefore, in the future, it will be interesting to study and develop methods to achieve results that are easier to use in practical applications.

This section includes seven numerical examples to demonstrate the efficacy of the improved results.
Example 1: Consider the T-S fuzzy generalized neural networks (10) with the following parameters:  [6], [7], [15], [36]- [38].   Table 2 shows that the maximum delay upper bounds for the relevant cases are not documented. Table 2 indicates that the stability criteria in this work give less conservative results when compared to other studies [6], [15], [36], [38]. In addition, we achieve the MAUBs of η 2 for various values of η 1 , as shown in Table 3. The state response solution y(t) is depicted in Figure 1 where u(t) = 0 and the initial function

V. CONCLUSION
In this article, we investigated the extended dissipativity problem for the T-S fuzzy GNNs with mixed interval timevarying delays. Firstly, we gain the novel asymptotic stability conditions for the T-S fuzzy GNNs and a particular case of the T-S fuzzy GNNs by using an appropriate LKF consisting of single, double, triple, and quadruple integral terms, a new triple integral inequality, an improved Wirtinger inequality, zero equation together with a convex combination approach. Next, the asymptotic stability results are developed to the analysis of extended dissipativity performance for the T-S fuzzy GNNs and a particular case of the T-S fuzzy GNNs, which covers L 2 − L ∞ , H ∞ , passivity, and dissipativity performance. Furthermore, we obtain the less conservative results for maximum allowable delay bounds and the optimal dissipativity performance for a particular case of the T-S fuzzy GNNs. Finally, illustrative examples are given to demonstrate the correctness and effectiveness of the proposed method. The proposed results and methods in this work are expected to be extended in the future topic to the exponential projective synchronization problem of T-S fuzzy GNNs, the extended dissipativity analysis of T-S fuzzy memristive GNNs, and so on [43], [45], [50].

APPENDIX A
Proof of Theorem 1: Let us use the Lyapunov-Krasovskii functional candidate for the T-S fuzzy GNNs (4) as follows: ×Sh(Wy(s)) ds dτ.
Since 0 ≤ κ ≤ 1, the term κ( i +c 1 I ) is a convex combination of (1) i + c 1 I and (2) i + c 1 I . The combination is negative definite only if each term is negative; so, (42) is equivalent to (7) and (8). Then, the T-S fuzzy generalized neural networks (4) is asymptotically stable.