Article
Adaptive Event-Triggered Synchronization of Uncertain
Fractional Order Neural Networks with Double Deception
Attacks and Time-Varying Delay
Zhuan Shen, Fan Yang, Jing Chen, Jingxiang Zhang, Aihua Hu and Manfeng Hu *
Citation: Shen, Z.; Yang, F.; Chen, J.;
Zhang, J.; Hu, A.; Hu, M. Adaptive
Event-Triggered Synchronization of
Uncertain Fractional Order Neural
Networks with Double Deception
Attacks and Time-Varying Delay.
Entropy 2021, 23, 1291. https://
doi.org/10.3390/e23101291
Academic Editor: Luis
Hernández-Callejo
Received: 7 September 2021
Accepted: 25 September 2021
Published: 30 September 2021
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School of Science, Jiangnan University, Wuxi 214122, China; 6191204005@stu.jiangnan.edu.cn (Z.S.);
6191204018@stu.jiangnan.edu.cn (F.Y.); 8201703038@jiangnan.edu.cn (J.C.);
zhangjingxiang@jiangnan.edu.cn (J.Z.); aihuahu@jiangnan.edu.cn (A.H.)
* Correspondence: humanfeng@jiangnan.edu.cn; Tel.: +86-510-8591-0233
Abstract:
This paper investigates the problem of adaptive event-triggered synchronization for
uncertain FNNs subject to double deception attacks and time-varying delay. During network
transmission, a practical deception attack phenomenon in FNNs should be considered; that is, we
investigated the situation in which the attack occurs via both communication channels, from S-C and
from C-A simultaneously, rather than considering only one, as in many papers; and the double attacks
are described by high-level Markov processes rather than simple random variables. To further reduce
network load, an advanced AETS with an adaptive threshold coefficient was first used in FNNs to
deal with deception attacks. Moreover, given the engineering background, uncertain parameters and
time-varying delay were also considered, and a feedback control scheme was adopted. Based on the
above, a unique closed-loop synchronization error system was constructed. Sufficient conditions that
guarantee the stability of the closed-loop system are ensured by the Lyapunov-Krasovskii functional
method. Finally, a numerical example is presented to verify the effectiveness of the proposed method.
Keywords:
uncertain fractional order neural network; adaptive event-triggered scheme; double
deception attacks; time-varying delay
1. Introduction
Neural networks, which bridge the micro-world of communications with the physical
world for processing information as mathematical models, widely exist in a broad range of
areas, such as intelligent control, secure communication, and pattern recognition [
1
–
4
]. Due
to the complexity of the dynamic characteristics of some physical systems, a traditional
integer-order neural network model cannot accurately represent their dynamic behaviors.
Fractional order calculus is not only a generalized form of the traditional integer-order
calculus; it also has some irreplaceable properties of integral order calculus, such as the spe-
cial feature of time memory [
4
–
7
]. Based on these features, the fractional order differential
equation has been used to model neural networks [
8
–
12
]. Synchronization, among several
phenomena arising from the complex nonlinear dynamics of neural networks, has gained
lots of attention and has been applied in many integer-order neural networks [
13
–
17
].
However, there are few studies about the synchronization problem of FNNs, which was
the first motivation of this paper.
The event-triggered scheme (ETS) depends on a predefined event-triggered condition
to determine whether the sampled data should be transmitted to the next control unit rather
than a fixed period; therefore, replacing the time-triggered scheme (TTS) to save network
communication resources and guarantee the system’s performance simultaneously was
suggested in [
16
,
18
–
23
]. Although ETS was adopted in the latest three studies of different
fractional order, real-valued systems [
21
–
23
], there was still a common disadvantage: the
threshold coefficients of traditional ETS are all constants and cannot be timely adjusted
Entropy 2021, 23, 1291. https://doi.org/10.3390/e23101291 https://www.mdpi.com/journal/entropy
