Adaptive Event-Triggered Control for Complex Dynamical Network with Random Coupling Delay under Stochastic Deception Attacks

This study concentrates on adaptive event-triggered control of complex dynamical networks with unpredictable coupling delays and stochastic deception attacks. The adaptive event-triggered mechanism is used to avoid the wasting of limited bandwidth. The probability of data communicated by the network is established by statistical properties and Bernoulli stochastic variables with an uncertain occurrence probability. Stability analysis based on Lyapunov–Krasovskii functional (LKF) and the stability of the closed-loop system is guaranteed. Using the LMI technique, we obtain triggered parameters. To demonstrate the feasibility and usefulness of the suggested methodology, two examples are shown.


Introduction
Complex dynamical network systems (CDNs) are typically made up of multiple nodes spread across a large area, with each node representing a dynamical system and control signals exchanged via a communication network [1]. CDNs have piqued the interest of many researchers in recent decades due to their wide range of applications in fields such as real-world networks, physics, telephone cell graphs, scientific citation webs, metabolic pathways, electrical power grids, biological networks, and food webs [2][3][4][5]. As a result, academics have spent considerable time studying the topological structure and dynamic behavior of CDNs [6][7][8][9]. [14][15][16][17][18]. To deal with limited communication and processing resources, a learning-based ETM is presented, in which the triggering threshold can be adaptively changed via a vehicle communication network based on the states of the vehicle [19,20]. e time delay is inevitable due to the ampli er's constrained switching speed and the nodes' inherent communication time. Its presence will have an impact on the stability of complex networks by causing oscillation and instability. In the models, the coupling delays of the huge complex systems are deterministic [10,21]. Coupling delay is unavoidable in large-scale coupled nonlinear systems, such as CDNs, due to the limiting transmission speed of information between nodes [22]. Time-varying delays are more regular than constant time delays. e use of time-varying coupling delays in complex dynamical network stability has gained a lot of attention [23,24]. In [25,26], the synchronization of both continuous and discrete time complex dynamical networks are investigated. In [27], the problem of CDNs with sampled-data control and time-varying coupling delay was studied. To construct CDNs with time-varying coupling delay, the event-triggered mechanism and Jensen inequality were employed to estimate portions of the integral terms of the Lyapunov functional [28,29].
Networked embedded signals in networked power systems are frequently transmitted through infrastructures, public networks, and devices that are susceptible to potential cyberattacks, and due to the inherent cyber vulnerability, the transmitted data could be exposed to malicious attacks by adversaries [30][31][32]. Cyberattacks are carried out by malicious attackers, according to several network control system researchers, and di erent cyberattacks attempt to compromise the data's security or availability [33,34]. Deception attacks and denial-of service (DoS) attacks are two of the most popular forms of attacks [35,36]. It is worth noting that DoS attacks can disrupt communication and cause data to become unavailable by interrupting the transmission medium [37,38]. One of the really popular types of network security risks is deception attacks. Deception attacks, in particular, may undermine information integrity by modifying the content of sent data packets to prohibit the achievement of a predetermined performance index [39][40][41][42][43][44]. Some recent results about deception attacks are discussed in [36,45,46].
Based on the previous discussions, the purpose of this research is to build an adaptive event-triggered technique to address the complex dynamical network with random coupling delay and unknown probability under stochastic deception attacks. e following are the major contributions of this study: (1) is study addresses the problem of adaptive eventtriggered control for complex dynamical networks with time-varying coupling delays under stochastic deception attacks. (2) Due to the threat of cyber security, the e ect of deception attacks is considered. e independent Bernoulli variable is used to determine the probability of deception attacks.
(3) By constructing Lyapunov-Krasvosikii functional, novel su cient criteria are established for stochastic stability. (4) e deception attacks damage the actuators and sensor signals, changing their value, delaying them, or doing both.
Notations: throughout the study, the symmetry-induced vector term is denoted by the symbol * . Q > 0 denotes Q is a positive de nite matrix. e superscript T is the transpose. R m signi es the m-dimensional Euclidean space, and R m×n denotes the set of all m × n real matrices. Kronecker product is written in the form ⊗ . e expectation operator is denoted by E and · refers to Euclidean norm. I is an identity matrix with appropriate dimension.

Remark 1.
A coupling delay appears during the transmission of signals or information in many natural systems and also practical systems such as communication channels. e random variable λ(r) satisfies the Bernoulli distributed white sequence, where 0 ≤ λ ≤ 1. If λ � 0, the coupling delay does not happen. If λ � 1, the coupling delay happens, which obeys the following probability distribution laws: e configuration of AETC for the complex dynamical network system with deception attack is given in Figure 1. Adaptive event-triggered device is introduced between the sensor and the communication network in each subsystem.
is device is responsible for selecting some necessary sampling packets for the control system to transmit over the network. It is assumed that the systems' state variables are periodically measured by a set of sensors with a constant sampling period. e measured state variables are transmitted to the AETM which is located near the sensor.
If the following condition is violated, the AETM transmits the instant to the controller via the communication network: where e(iZ) � ϖ(r k Z) − ϖ(iZ). Here, ϖ(r k Z) and ϖ(iZ) represent the released instant and initial instant, respectively. Z is sampling period; r k Z, r 1 , r 2 , . . . ⊆N represents the triggered instant. 0 < Φ ∈ R n ϖ ×n ϖ is a matrix to be designed. e time-dependent function 0 < η(iZ) < 1 is a triggering threshold function which is revised by the adaption law that follows: where ζ > 0 is a design parameter, η and η are chosen as lower and upper bound of the triggering threshold function η(iZ), respectively, and prescribed by the designer with considering the constraint 0 < η ≤ η < 1.

Remark 2.
It is worth noting that the presented eventtriggered mechanism (2) and the adaptation law (3) are completely discrete unlike continuous-time adaptation laws presented in [16]. Hence, the proposed AETM is more practical for implementing on digital hardware. In the event of a disturbance, the value of ρ iZ should be decreased properly to improve performance. Some adaptation mechanisms, such as those described in [16,18], keep the value of ρ iZ constant between two release instants, resulting in a delayed controller response to the disturbance. e suggested event-triggered mechanism (2), in contrast to them, uses an adaptive threshold ρ iZ that is modified at each sampling instant. As a result, the controller may respond quickly to external disturbances. e control signal after the zero-order-hold (ZOH), while considering the effect of deception attacks, is regarded as where Θ(k) � [r k Z + β k , r k+1 Z + β k ), K ∈ R n u ×n ϖ is the state feedback gain matrix to be designed. e time-varying network-induced delay β k ≥ 0, which is satisfying where β a ≥ 0 is known as a constant scalar, and To solve the problem under investigation with all the above challenges, we suggest that the adaptive controller (5) can be written as where f: R n ϖ ⟶ R n ϖ is the function of deception attacks and c(r) is the time delay of the deception attacks.
Remark 3. e objective of a deception attack is to modify or inject data into communication channels. In contrast to the Complexity 3 attack method in [20], the data injection issue in the sensorto-actuator controller channel is specifically examined in this study. e random variable ρ(r k Z) describes the occurrence of deception attacks on the communication network channel. If ρ(r k Z) � 0, the communication network channel suffers from the deception attack, which means the real transmission data are replaced by the deception attack signal. Otherwise, ρ(r k Z) � 1 means that there is no attack and the network is working normally. e sequence of data transmission is from the sampler to the controller. Here, ρ(r k Z) is a random variable with Bernoulli distribution which takes the value 1 with probability ρ and the value 0 Abovementioned ρ is very difficult or impossible in practice. So, it is assumed that this value is accompanied by uncertainty and is described by where ρ 1 and ρ 2 are known nominal value and known constant scaling of the uncertainty, respectively, and y(r k Z) is an unknown function which satisfies the following condition: Noticed that ρ ∈ [0, 1]. If ρ 1 ∈ [0, 1] and ρ 2 ∈ [0, 0.5]; then, the closer ρ to zero, the greater the chances of an attack. e values of ρ 1 and ρ 2 indicate the uncertainty interval on this probability.
Assumption 1 (see [43]). e functions h: R n ϖ ⟶ R n h and f: R n ϖ ⟶ R n ϖ are assumed to satisfy the following conditions: where H and F are known constant matrices.
From (1) and (8), the closed-loop complex dynamical network system can be described by Definition 1 (see [17]). e closed-loop system (12), under AETM and deception attacks, is stochastically stable and satisfies a prescribed H ∞ performance index c ⌣ if the following conditions hold: for any nonzero ω ∈ L 2 [0, ∞) under zero initial condition, where c ⌣ is prescribed performance level.

Main Results
In this section, a sufficient criterion will be established to verify that a complex networked control system with a deception attack is stochastically stable in controlling instants via an adaptive event-triggered mechanism.
is completes the proof. □ Theorem 2. For given positive constants c ⌣ , η, ϱ and ρ, λ ∈ [0, 1], the closed-loop system (12) is stochastically stable, presumed the existence of positive definite matrices V, T, W 1 , W 2 , Q 1 , Q 2 , R 1 , R 2 , R 3 , J 1 , S 1 , and S 2 are any proper dimension matrices. μ be positive scalars and the event-triggered weighting matrix Φ > 0, such that the following conditions hold: Moreover, the controller gain matrix K is given by

Numerical Example
In this section, we provide some numerical information to ensure the method's effectiveness and applicability.
e following solutions are achieved by solving LMI in eorem (2) (2) that adaptive eventtriggered mechanism for a complex dynamical network system subject to deception attack (12) is stochastically stable.

Conclusion
e issue of adaptive event-triggered mechanism for a class of complex dynamical networks with random time-varying coupling delays under stochastic deception attacks has been investigated. We established two sets of random stochastic variables ρ(i k Z) and λ(r), respectively, to represent the probability of data conveyed by the network being subjected to deception attacks and time-varying coupling delays. Based on the Lyapunov-Krasovskii functional theory some sufficient conditions derived for the closed-loop system that can ensure the system is stochastically stable. Two examples are presented to demonstrate the effectiveness of the presented approach.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest.

Authors' Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.