Nonfragile Estimator Design for Fractional-Order Neural Networks under Event-Triggered Mechanism

Shao, Xiaoguang; Lyu, Ming; Zhang, Jie

doi:https://doi.org/10.1155/2021/6695353

Discrete Dynamics in Nature and Society

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

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Performance Analysis and Synthetisation of Complex Systems with Event-triggered Communication Scheduling

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Research Article | Open Access

Volume 2021 | Article ID 6695353 | https://doi.org/10.1155/2021/6695353

Nonfragile Estimator Design for Fractional-Order Neural Networks under Event-Triggered Mechanism

Xiaoguang Shao,¹Ming Lyu,¹and Jie Zhang¹

Academic Editor: Wangyan Li

Received25 Dec 2020

Revised30 Jan 2021

Accepted05 Mar 2021

Published29 Mar 2021

Abstract

This paper is concerned with the nonfragile state estimation for a kind of delayed fractional-order neural network under the event-triggered mechanism (ETM). To reduce the bandwidth occupation of the communication network, the ETM is employed in the sensor-to-estimator channel. Moreover, in order to reflect the reality, the transmission delay is taken into account in the model establishment. Sufficient criteria are supplied to make sure that the augmented system is asymptotically stable by using the fractional-order Lyapunov indirect approach and the linear matrix inequality method. In the end, the theoretical result is shown by means of two numerical examples.

1. Introduction

The past several decades have witnessed that artificial neural network (ANN) has attracted particular research attention. Because of the outstanding performance, ANN has been extensively applied in image recognition, signal processing [1], fault diagnosis [2], and so on. With the rapid developments of artificial intelligence, the ANN has received considerable attention again by the scholars, which relates to synchronization, dissipativity, attractivity, stability, and state estimation (SE) for various kinds of ANNs [3–7].

As we all know, ANNs are composed of plenty of artificial neurons and the SE problem of the neurons plays a vital role in practical applications. As such, quite a lot of results have been reported on the SE issue (see [8–11] and the reference therein). For the practical systems, the parameter uncertainties are often considered. So far, a lot of research studies regarding uncertain systems have been conducted [12–14]. It is worth noting that the existing results assumed that parameter of the estimator is accurate, which, however, is unrealistic. To solve this problem, we aim to design a nonfragile estimator so as to alleviate the effects induced by the uncertainty of the estimator parameter on the system performance. Till now, some initial results have been published on the nonfragile controller design problems [14–17].

In the networked systems, the network bandwidth is always limited which therefore may result in network congestion when a large amount of data is transmitted. Up to now, the network-induced phenomena including transmission delay, packet loss, and quantification have been discussed adequately. In recent years, much attention has been focused on the ETM and many communication protocols, which aims to avoid the occurrence of the network-induced phenomena. Based on the ETM, plenty of literature has been available on stability analysis, event-triggered condition design, controller/filter design, and so on [18]. Noting that compared with time-triggered mechanism, the ETM exhibits better performance because the necessary sampling depends on the “event” rather than the “time” [19, 20].

In addition, by applying the fractional calculus to the ANNs, the researchers have found that the performance of the fractional-order models is better than integer-order ones, especially in the aspect of memory and hereditary. Till now, some novel fractional-order theories and methods concerning the ANNs have been proposed. For example, a nonfragile nonlinear fractional-order observer is designed in [21] and an adaptive event-triggered scheme has been developed in [22]. But these existing fractional-order systems employed ETM are introduced with single delay or without only. Especially, it is a challenge in a fractional system. However, the problem of multiple time delays in real systems is often encountered. Nevertheless, there are few related studies on the nonfragile SE for fractional-order neural network based on ETM with multiple time delays, which motivates us to shorten this gap.

Inspired by the aforementioned lines, a nonfragile state estimator is designed for a class of fractional-order neural networks (FNNs) based on ETM. The advantages in this paper are as follows: (1) compared with the existing estimators, a fractional-order nonfragile estimator is first constructed; (2) to save bandwidth resources, an ETM is applied in the SE problem of the fractional-order neural network; (3) the LMI method and the fractional Lyapunov indirect method are adopted to design the state estimator.

The remaining content is outlined as follows. In Section 2, some preliminary knowledge is recalled. In Section 3, state estimation criteria are voiced. In Section 4, two numerical examples are given with some simulation figures to support the theorems.

Notation. Throughout this paper, and the symbol in matrix represent matrix transposition and the symmetric term, respectively. is the set of integers, and denotes the -dimensional Euclidean space. means -dimensional identity matrix. () is defined as a positive-definite (negative-definite) matrix. is the Euclidean norm of a vector in . () represents the maximum (minimum) eigenvalue of and means .

2. Preliminaries and Problem Formulation

Some fractional definitions and model descriptions are presented firstly. In addition, some important lemmas that will be used in Section 3 are also presented.

Definition 1. (see [23, 24]). For , the fractional integral form is defined aswhere and is a gamma function.

Definition 2. (see [23, 24]). Caputo’s derivative of is denoted bywhere and is a positive integer.
In what follows, stands for for the convenience of presentation. In this paper, let us consider the following FNN model:where is the predetermined fractional order, with , is the connection matrix, the vector stands for the neuron state, denotes the activation function of the neurons, is the measurement output, is the system input, and is a known constant matrix.
In what follows, ETM is introduced in order to reduce the communication burden. The event-triggered condition is predesigned as follows:where , is a given constant, and are the sampling instant and the release instant, respectively, stands for the latest sampled signal, and denotes the release period.

Remark 1. The sensor is time-driven at discrete instants, which can avoid the Zeno behavior. Moreover, when , ETM becomes a time-triggered one.
In this paper, the transmission delay between sensor and estimator is considered, where is a positive scalar. Therefore, is the arrival time of the transmitted data from sensor to estimator.
In view of [25], the holding interval can be rewritten as , where . For the convenience of analysis, denote , and then we have . Then, the measurement outputs arrived at the estimator can be rewritten aswhere is the error vector.
Design a nonfragile state estimator for system (3) as follows:where stands for the estimate of , is the gain matrix to be determined, and represents the gain variation that satisfies , in which and are known real matrices and is an unknown satisfying .
Defining , the estimation error dynamics can be obtained from (3) and (5) as follows:where .
For notation simplicity, we define . An augmented system model from (3) and (8) is given in the following form:where

Lemma 1 (see [26]). For and and any positive scale , one has

Lemma 2 (see [27]). For and , if is continuous and differential, then

Lemma 3 (see [28]). For matrices , where is symmetric, the inequalityholds if and only ifin which refers to a scalar and .

Lemma 4 (see [29]). Consider a class of fractional-order nonlinear systems:and the initial condition is = . Suppose that are positive functions, and . If there exist two constants and a continuous differential function such taht satisfyingthen the fractional-order system is globally uniformly asymptotically stable.

Lemma 5 (see [30]). The matrixholds if and only if , or .

Lemma 6 (see [31]). is continuous on and bounded on . If there exist such thatwhere , and , then .

3. Main Results

Theorem 1. For the given positive scalars , system (8) is globally asymptotically stable if there exist a symmetric matrix and four scalars satisfying the following LMI:where

Furthermore, the nonfragile estimator gain of (8) is designed as .

Proof. First, we denoteConsidering system (8), design the following Lyapunov function:From Lemma 2, one obtainsIt follows from Lemma 3 thatwhere and .
Combining (24)–(36), we can getBy employing Lemma 5, is equivalent towhereDefining an equation as follows:we haveFurthermore, from (28) and (31), we arrive at .
Based on Lemma 3, we can obtainCombining (27) and (32), we haveIt follows from Lemma 4 that (4) is an asymptotical estimator and system (3) is globally asymptotically stable. The proof is completed.
It is worth noting that the parameter uncertainties are often unavoidable resulting from the inaccuracy of modeling or the changing environment. In addition, the network output is composed of linear and nonlinear parts. Therefore, the following model of FNN is established:where is a time-varying delay satisfying . Here, is a constant. are parameter uncertainties which satisfy the following condition:where are known matrices and is an unknown matrix function which satisfies .

Assumption 1. stands for the nonlinear disturbance which satisfies the Lipschitz condition:where and is a known constant matrix.
The estimator and estimation error dynamics are obtained as follows:The augmented system is derived as follows:whereThe following theorem is given to ensure the above system is asymptotically stable.

Theorem 2. For given positive scalars satisfying , the augmented system (48) is asymptotically stable if there exist a symmetric matrix and seven scalars , satisfying the following LMI:whereFurthermore, the nonfragile estimator gain is designed as .

Proof. Construct the following Lyapunov functional:From Lemma 2, the following inequality is obtained:By using Lemma 1 and Lipschitz condition, one getsFrom event-triggered condition (5), we can obtainCombining (45)–(50), one can getand thereforeBased on the above inequations, we havewhereBy Lemma 5, is equivalent towhereLetThen, one gets . Based on Lemma 5, (55) and (57) imply that . According to (40) and (42), we know that , and . Therefore, we can obtainTherefore, system (37) is an asymptotical estimator of (34) by using Lemma 6. This completes the proof.

4. Numerical Example

To illustrate the theoretical results, two numerical examples are shown in this section.

Example 1. where diag, N = diag, and diag.
The other parameters are given as , . The activation function is given as follows:By using Matlab to solve LMIs (26) and (27), the gain matrix can be obtained asThe simulation results are shown in Figures 1–3, where represent the true states and their estimates and the initial conditions are (): .
Figure 4 shows the estimate error as . According to the simulation results, we can see the effectiveness of the estimator design method. Figure 5 shows the release instants and intervals with the threshold parameter .

Example 2. To verify that the estimator contains uncertain terms in (34), the following fractional-order model is shown, and the corresponding parameters are as follows:where , , , , and . Besides, , and the time delays are set as . The measurement output is . Let the activation function be
By using Matlab to solve the LMI (37), the gain matrix can be obtained asThe simulation results are shown in Figures 6–8, where represent the true states and their estimates are depicted, respectively, with the initial condition (.): .
Figure 9 shows the estimate error , as . In Figure 8, we can see clearly the error states . Figure 10 shows the release instants and intervals with . According to the figures, we can see that the simulation results voiced the effectiveness of the estimator design.

5. Conclusions

This paper has investigated the nonfragile SE issue under the ETM for the FNNs with time delays. Sufficient conditions have been obtained to ensure the asymptotic stability of the considered system by means of the fractional-order Lyapunov functions and the LMI method. The gain matrix of the nonfragile estimator has been characterized by a LMI. At last, two numerical results have confirmed the validity of the designed estimator. In addition, the results could be extended to the SE issue of discrete FNNs with fading measurements and so on.

Data Availability

In this paper, the initial conditions have been given. As a control system, the data in the Numerical Example is suffienct to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by the Natural Science Funds of Jiangsu Province (no. BK20180467).

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Copyright © 2021 Xiaoguang Shao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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