Master–slave synchronization of neural networks subject to mixed-type communication attacks

doi:10.1016/j.ins.2021.01.063

Information Sciences

Volume 560, June 2021, Pages 20-34

https://doi.org/10.1016/j.ins.2021.01.063 Get rights and content

Abstract

This paper concerns the master–slave synchronization issue of neural networks subject to mixed-type communication attacks. The synchronization strategy is based on static output feedback controller followed by an event-triggered scheme. The communication network is assumed to be under various types of cyber-attacks, namely, deception, replay, and denial-of-service attacks. All these attacks are investigated in a unified Markovian jump framework. Using the Lyapunov–Krasovskii theory and stochastic analysis techniques, some design criteria are derived and formulated in terms of matrix inequalities. A convex optimization algorithm is proposed to design the static output feedback controller. Finally, two chaotic examples are presented to demonstrate the effectiveness of the event-triggered static output feedback controller.

Introduction

Neural networks have attracted a great deal of researchers for decades due to their numerous applications including associative memory, optimization, nonlinear systems analysis and control, signal processing, intelligent control, and pattern recognition [1], [2], [3]. Apart from the stability analysis and stabilization design problem of neural networks [4], [5], [6], [7], synchronization on master–slave or multiple interconnected neural networks is one of the hot topics of interest [8], [9], [10], [11], [12].

With the advancement of digital communication and network technology, networked control systems (NCSs) have received much attention and raised problems and requirements are focused by the researchers. One of these issues of interest is the problem of sampled-data control and its effect on NCSs [13], [14]. Communication channel induced delay is another inherent feature of NCSs and has been extensively investigated [15], [16]. The lack of network resources and the essential need to increase network bandwidth for the joint use of equipment from a shared network lead to the necessity to reduce the number of data transmissions. Researchers have responded to this requirement by introducing various event-triggered schemes [17], [18].

In recent years, the issue of cyber-attacks has emerged as one of the major threats to the NCSs and cyber-physical systems [19], [20], [21], [22], [23]. In essence, these attacks fall into three main categories: deception attack, replay attack, and denial-of-service (DoS) attack [24]. Although research in this area is still taking its first steps, remarkable researches have been conducted on controller design in the presence of each of these attacks [25], [26]. On the one hand, there are very few papers that have considered a combination of these attacks. Zhao et al. considered both deception and DoS attacks for control design of a class of stochastic systems [27]. Xu et al. proposed an observer-based synchronization scheme for Markovian jump neural networks with DoS attack on the observer side and deception attack on the controller side [28]. On the other hand, almost all published papers use a Bernoulli stochastic variable to model the occurrence of the attacks [29], [27], [30], [31]. Since the Bernoulli variable has two states, several independent Bernoulli variables are used for cases where more than two states are needed [27], [32]. As a result, the development of this method for modeling hybrid attacks with several different modes is not reasonable.

Based on the above discussion and the fact that little research has been done on the synchronization of neural networks in the presence of cyber-attacks [28], this paper aims to fill this gap and investigate the master–slave synchronization issue of neural networks subject to mixed-type communication attacks. The main contributions of this paper are summarized as follows:

1.
The communication network is assumed to be under mixed-type of cyber-attacks. In this way, the attacker could randomly launch all three types of attacks, that is deception attack, replay attack, and DoS attack.
2.
All these attacks have been investigated in a unified Markovian jump framework. This method has two advantages: 1- Simpler than defining several Bernoulli variables, 2- Markov process is more general than Bernoulli process.
3.
Investigating master–slave synchronization of neural networks in the presence of mixed-type communication attacks.
4.
The control signal is based on discrete output feedback measurements which is more practical than state feedback.

For a matrix A, its transpose and inverse are denoted by $A^{T}$ and $A^{- 1}$ , respectively. Notation $P > 0$ means that P is a real symmetric positive definite matrix. A transpose term in a real symmetric matrix is indicated by $*, col {\dots}$ stands for column vector or matrix, and $diag {\dots}$ denotes a block diagonal matrix. The symbol $N$ represents the set of all natural numbers and $N_{0} = N \cup {0}$ . The symbol $\otimes$ stands for the Kronecker product, and $He [R] = R + R^{T}$ . $E {.}$ denotes the expectation, and $E {x | y}$ represents the expectation of x conditional to y.

Section snippets

Problem statement

Consider the Master–Slave neural networks $M : \{\begin{matrix} {\dot{x}}_{m} (t) = - {Ax}_{m} (t) + Bf (x_{m} (t)) + Df (x_{m} (t - μ (t))) + I (t), \\ y_{m} (t) = C_{y} x_{m} (t), \\ z_{m} (t) = C_{z} x_{m} (t), \\ x_{m} (t) = φ_{m} (t), \forall t \in [- \bar{μ}, 0], \end{matrix}$ $S : \{\begin{matrix} {\dot{x}}_{s} (t) = - {Ax}_{s} (t) + Bf (x_{s} (t)) + Df (x_{s} (t - μ (t))) + I (t) + H ω (t) + u (t), \\ y_{s} (t) = C_{y} x_{s} (t), \\ z_{s} (t) = C_{z} x_{s} (t), \\ x_{s} (t) = φ_{s} (t), \forall t \in [- \bar{μ}, 0], \end{matrix}$ where $x_{m} (t) (x_{s} (t)) \in R^{n_{x}}, y_{m} (t) (y_{s} (t)) \in R^{n_{x}}$ and $z_{m} (t) (z_{s} (t)) \in R^{n_{z}}$ are the state, output, and control objective vectors of the master (slave) neural network, respectively, $ω (t) \in R^{n_{ω}}$ is the exogenous disturbance vector belonging to energy bounded signals,

Main results

The following theorem provides the stochastic stabilization conditions of the synchronization error system (11).

Theorem 1

The synchronization error dynamics (11) is stochastically stable with $ω (t) = 0$ , according to Definition 1, and satisfies the $H_{\infty}$ performance index (12) for any nonzero $ω \in L_{2} [0, \infty)$ , if there exist positive definite matrices P, $Q_{i}, R_{i}, J, L, Q_{m, i}, R_{m, i}, J_{m}, L_{m} \in R^{n_{x} \times n_{x}}, m = 1, 2, Ω \in R^{n_{y} \times n_{y}}$ , any matrices $M_{i}, M_{m, i} \in R^{n_{x} \times n_{x}}, m = 1, 2$ , $Y \in R^{n_{x} \times n_{y}}$ , positive definite diagonal matrices $Λ_{1}, Λ_{2} \in R^{n_{x} \times n_{x}}, Λ_{3} \in R^{n_{y} \times n_{y}}$ , such that

Numerical examples

Example 1. Consider the neural networks (1), (2) with the following parameters [38]: $\begin{matrix} A = I_{3}, C_{z} = I_{3}, C_{y} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}], H = [\begin{matrix} 0.1 \\ 0.1 \\ 0.1 \end{matrix}], \\ B = [\begin{matrix} 1.2 & - 1.6 & 0 \\ 1.25 & 1 & 0.9 \\ 0 & 2.2 & 1.5 \end{matrix}], D = 10^{- 3} \times [\begin{matrix} - 9 & 2 & 1 \\ 2 & 1 & 3 \\ 1 & 2 & - 1 \end{matrix}], \end{matrix}$ and $J (t) = 0_{3 \times 1}$ , $\begin{matrix} f_{i} (x_{i} (t)) = \frac{1}{2} (| x_{i} (t) + 1 | - | x_{i} (t) - 1 |), i = 1, 2, 3 \\ g_{i} (y_{i} (t)) = \tanh (0.2 y_{i} (t)), i = 1, 2 . \end{matrix}$ and the time-varying delay is assumed to be $μ (t) = 0.1 e^{t} / (e^{t} + 1)$ . The chaotic behaviour of the presented neural network is shown in Fig. 2 with $x_{m} (0) = {[0.2, 0.1, 0.2]}^{T}$ . According to the given information, it can be calculated that $F = I_{3}, G = 0.2 I_{2}, \bar{μ} = 0.1$ .

The

Conclusion

The master–slave synchronization of neural networks subject to mixed-type communication attacks has been investigated. It was assumed that the communication network is under mixed-type of deception, replay, and DoS attacks. The discrete output feedback measurement has been considered instead of state feedback to increase the applicability of the synchronization strategy. All the attacks have been investigated in a unified Markovian jump framework. Some design criteria have been derived and

Funding

The work was partially supported by General Research Fund 17201219.

CRediT authorship contribution statement

Ali Kazemy: Conceptualization, Methodology, Software, Writing - original draft. Ramasamy Saravanakumar: Validation, Writing - review & editing. James Lam: Supervision, Project administration.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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View full text

Master–slave synchronization of neural networks subject to mixed-type communication attacks

Abstract

Introduction

Section snippets

Problem statement

Main results

Numerical examples

Conclusion

Funding

CRediT authorship contribution statement

Declaration of Competing Interest

Ocean Eng.

Neurocomputing

Inf. Sci.

Neurocomputing

Neurocomputing

J. Franklin Inst.

J. Franklin Inst.

ISA Trans.

Nonlinear Anal. Hybrid Syst.

Inf. Sci.

Annu. Rev. Control

Neurocomputing

Inf. Sci.

Inf. Sci.

Inf. Sci.

Neurocomputing

Nonlinear Anal. Hybrid Syst.

ISA Trans.

J. Franklin Inst.

J. Franklin Inst.

Neurocomputing

J. Franklin Inst.

Appl. Math. Comput.

J. Franklin Inst.