Passivity Analysis of Fractional-Order Neutral-Type Fuzzy Cellular BAM Neural Networks with Time-Varying Delays

Ali, M. Syed; Hymavathi, M.; Alsulami, Hamed; Saeed, Tareq; Ahmad, Bashir

doi:https://doi.org/10.1155/2022/9035736

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Numerical Example Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 9035736 | https://doi.org/10.1155/2022/9035736

Passivity Analysis of Fractional-Order Neutral-Type Fuzzy Cellular BAM Neural Networks with Time-Varying Delays

M. Syed Ali,¹M. Hymavathi,¹Hamed Alsulami,²Tareq Saeed,²and Bashir Ahmad²

Academic Editor: Zhiwei Gao

Received14 Nov 2021

Revised20 Jan 2022

Accepted26 Jan 2022

Published10 Mar 2022

Abstract

In this paper, passivity analysis of fractional-order neutral-type fuzzy cellular bidirectional associative memory (BAM) neural networks with time-varying delays is investigated. Based on the Lyapunov–Krasovskii functional, delay-dependent sufficient conditions for solvability of the passive problem are obtained in terms of linear matrix inequalities (LMIs), which can be easily checked by using the MATLAB LMI toolbox. Finally, numerical examples are provided to show the effectiveness of the main results.

1. Introduction

The study of fractional calculus can be dated back to 1695, and the fractional operator concept was put forward by Leibnitz, which did not acquire sufficient attention for a long period since it is complicated. Many actual systems can be described by fractional-order differential equations, making the slowly developed fractional calculus be a renewal of interest [1–5]. Generally speaking, fractional calculus is a generalization of classical calculus and is more accurate to describe reality models compared to the corresponding integer-order calculus in different research communities, such as particle physics, wave mechanics, electrical systems, and computational methods for mathematical physics, and many references cited therein [6–9]. Because of the nonlocality of fractional calculus, it is presented to explain actual systems on the basis of standard integral calculus. Furthermore, fractional calculus is widely used in domains such as statistical mechanics, medical imaging, combinatorial optimization, and viscoelastic systems [10–13]. However, fractional neural networks are still a hot research topic and have been extensively studied in the recent decades [14–16]. Recently, the synchronization of fractional neural networks has attracted increasing interest [17–20].

Fractional-order neural networks with neutral-type delays were also very rarely studied in the available literature. The neural network model having time delays in the time derivatives of states is called delayed neutral-type neural networks. The neutral-type system not only characterizes the dynamic property of the system state but also describes the dynamic varying rule of the delay state of the system. Such a phenomenon constantly encountered in the area of chemical reactors, transmission lines in electrical engineering, heat exchangers, and population dynamic systems. Because of this component, neutral systems have turned into the subject of broad look into by numerous researchers [21–28]. Many neural networks can be regarded as special cases of neutral neural networks, and most of the neural networks can be transformed into neutral neural networks for research [29, 30]. Furthermore, stability criteria for neutral-type BAM neural networks with time delays were derived in [31–34].

In this paper, we would like to integrate fuzzy operations into BAM neural networks. Combining the advantages of the fuzzy operation theory and cellular neural network, T. Yang and L. B. Yang firstly proposed the fuzzy cellular neural networks in 1996. Owing to the contribution of Song, in the establishment of fuzzy logic [35], the fuzzy logic has been paid more and more attention. Studies have shown that fuzzy cellular neural networks have their potential in image processing and pattern recognition, and a few outcomes have been discussed [36–41]. Associative memory is one amongst the foremost significant behaviors of the human brain, which might be applied in the study of brain-like systems, intelligent thinking for intelligent robots, and so on. The bidirectional associative memory (BAM) neural network models were first proposed and studied by Kosko [42], which are made up of two neuron layers, i.e., U-layer and V-layer, which have no interconnection among neurons within the same layer. The neurons in one layer are fully interconnected to the neurons in the second layer and have the features of heteroassociative, content-addressable memory. In real life, BAM neural networks have powerful information processing abilities and some good application fields, such as information associative memory, image processing, and artificial intelligence. Subsequently, different results on the stability and other behaviors of delayed fuzzy BAM neural networks have been derived (see [43–46] and references cited therein).

As pointed out, discrete-time models are more better to describe the dynamical behaviors than continuous ones since the populations have nonoverlapping generations. In different engineering systems, an origin of in stability is time delay exists [47–49]. Based on the dependence of delay, stability criteria have been classified into 2 types: delay-dependent and delay-independent criteria. Thus, delay-dependent criteria play an important role for assuring the addressed neural networks to be stable to get maximum delay bounds. Different methods were imported for these types of problems [50, 51]. The methods were as follows: (i) free-weighting matrix method, (ii) reciprocally convex approach, (iii) augmented Lyapunov–Krasovskii functional.

Passivity theory was first proposed in the circuit analysis [52] and then connected with various distinctive structures, including high-order nonlinear systems and electrical framework [53]. It can be extensively associated with play out the stability analysis, signal processing, fuzzy control, sliding mode control, and networked control. The main aim of passivity theory is that the passive properties of the system can keep the system internally stable. In reality, many systems require to be passive in order to rarefy noise effectively. Recently, several authors have studied the passivity of delay neural networks; see [54–59].

Inspired by the above discussions, the problem of passivity-based fractional-order neutral-type fuzzy cellular BAM neural networks with time-varying delays is examined in this paper. More precisely, in this work, by employing a Lyapunov functional with augmented elements and the linear matrix inequality technique, a set of delay-dependent passivity conditions is derived for obtaining the required result. Finally, the numerical example is presented.

2. Preliminaries and Model Description

In this section, we recall definitions of fractional calculus and several lemmas which will be used later.

Definition 1 (see [60]). The Caputo fractional derivative of order for a function f(t) is defined aswhere and n is an integer satisfying .

Definition 2 (see [61]). System (4) is called passive if there exists a scalar such thatfor all solutions of (12) with and and for all .

Lemma 1 (see [62]). Let be a continuous and derivable function. Then, for any , the following inequality holds:

Lemma 2 (see [63]). For any positive definite matrix , scalar , and vector function , the following inequality holds:

Lemma 3 (see [64]). Given constant matrices , , and , where , , and ; then,if and only if

Lemma 4 (see [65]). Suppose that are the two states of system (1). Then, one has

Assumption 1. (t) and are time-varying functions satisfying

3. Main Results

Consider the following fractional-order neutral-type fuzzy cellular BAM neural networks with time-varying delays aswhere is Caputo’s fractional derivative, and are the state of the ith neuron and jth neuron at time t, respectively, and denote the functions of the ith neuron and the jth neuron at time t, respectively, and are the diagonal matrices with positive entries, and represent the rates with which the ith neuron and the jth neuron will reset their potential to the resting state in isolation when disconnected from the networks and external inputs, respectively, denote the connection weight matrices of the feedback template and feedforward template, denote the connection weight matrices of the feedback template and feedforward template, respectively, and denote the connection weights of the delay fuzzy feedback MIN template and the delay fuzzy feedback MAX template, respectively, and denote the inputs of the ith neuron and the jth neuron at time t, respectively, and and denote the fuzzy AND and fuzzy OR operation, respectively.

In this paper, we refer to model (9) as the drive system; the response system is given as follows:where and indicate (the control input, and let and for ; then, from systems (9) and (10), the error dynamical system can be derived as

It is sufficiently rewritten as

Theorem 1. Under Assumption 1, system (4) satisfies the passivity analysis if there exist matrices , , , , , , , , , , , , , and and positive diagonal matrices , , , and such that the following LMI holds:wherewhere .

Proof. LetwhereTaking the time derivative along the trajectories of system (12),From the Cauchy–Schwarz inequality and (23), the following inequality holds:According to Lemma 2, we getIn addition, for any diagonal matrices , , , and , the following inequality holds:From (17)–(28) and passive definition, we havewhereTherefore, we conclude thatBy integrating (31) with respect to t over the time period , we haveFor u(0) = 0 and (0) = 0, we have V (0, u(0), (0)) = 0 and . Thus, it follows from Definition 2 that the neutral-type fuzzy BAM neural network is passive. The proof is completed.

Remark 1. Equation (9) proposes an improved passivity-based fractional-order neutral-type fuzzy cellular BAM neural network with time-varying delays. Based on Theorem 1, we have the following result for neutral delay is not equal to discrete delay of fuzzy cellular BAM neural networks with time-varying delays, without neutral term then system (12) becomes

Theorem 2. The fuzzy BAM neural networks in (20) satisfy the passivity analysis if there exist matrices , , , , , , , , , , , and , positive diagonal matrices , , , and , and positive scalars and such that the following LMI holds:wherewhere (1, 1) = , (2, 2) = , (6, 6) = , and (8, 8) = −.

Proof. LetwhereThe proof of this theorem immediately follows from Theorem 1.

Remark 2. Consider the following BAM neural network (12), and let ; then,

Theorem 3. The fuzzy BAM neural networks in (21) satisfy the passivity analysis if there exist matrices , , , , , , , , , , , and , positive diagonal matrices , , , and , and positive scalars and such that the following LMI holds:where , , , and .

Proof. LetwhereBy Lemma 1, we obtain the time derivative along the trajectories of system (38):The proof of this theorem immediately follows from Theorem 1.

4. Numerical Example

In this section, we provide numerical examples to illustrate the usefulness of our stability results. Consider the fractional-order neutral-type fuzzy cellular BAM neural networkwith the following system parameter values:

Here, we take the activation functions as . Then, let , , , By solving LMI (13), we get the following feasible solutions:

Therefore, it follows from Theorem 1 that the fractional-order neutral-type fuzzy cellular BAM neural network with time-varying delay (43) is passive.

5. Conclusion

The problem of passivity-based fractional-order neutral-type fuzzy cellular BAM neural networks with time-varying delays has been presented in this paper. By constructing suitable LKFs and using Jensen’s inequalities, new delay-dependent conditions are established in terms of LMIs to guarantee the passivity criteria for the neural network. Future work will focus on the fractional-order complex-valued neural networks with delays and fuzzy complex-valued neural networks with delays. Finally, two examples are presented to illustrate the effectiveness of the proposed criteria.

Data Availability

No data were associated in this work.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, Saudi Arabia, funded this project (Grant no. KEP-51-130-42).

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