Intermittent Control for Synchronization of Discrete-Delayed Complex Cyber-Physical Networks under Mixed Attacks

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Introduction
As typical massively interconnected complex systems, complex networks are composed of a large number of interacting individuals or nodes, whose dynamics can be described by a single nonlinear vector feld, such as multiagent systems, transportation networks, neural networks, and electric power grids. [1][2][3][4][5]. Over several decades, the synchronization control of complex networks has generally been recognized as one of the most fascinating issues of research, and scholars have carried out increasing research on emergency behaviour and coordinated movement of complex networks [6]. On the one hand, due to the simultaneous transmission signal between a tremendous number of nodes in complex networks and the complex coupling of communication networks, it is inevitable to encounter the problem of time delay, which may lead to the damage of the system performance. Terefore, the synchronization problem of complex networks with time delays has received considerable attention (see, e.g., [7][8][9]). On the other hand, as a result of the complicated network structure, it is always difcult to achieve spontaneous synchronization. To date, various control techniques have been presented to investigate the synchronization problem of complex networks [10][11][12][13][14][15]. Among them, intermittent control methods are widely investigated due to the fact that they are easy implementation and more economic than continuous-time ones. For instance, in [14], the authors propose aperiodically intermittent pinning control methods for dynamical networks, and in [11], an intermittent control strategy is proposed to ensure exponential synchronization of neural networks under actuator saturations. It is worth mentioning that in [11], the control actions are clock-dependent, which means that the controller only works at the prescribed times. To reduce the limits of the preset clock, the authors of [12] design an event-dependent intermittent controller for quasisynchronization control of delayed discrete-time neural networks. In [15], the authors propose intermittent control methods for competitive neural networks.
In practice, due to open network connections between individual nodes, complex networks are often vulnerable to direct or indirect damage from cyber-attacks and resulting in degraded or even missing synchronous performance of complex networks. Specifcally, the availability and integrity performance of the modern system information is at serious risk, as testifed by several example incidents. For instance, the Iran nuclear program has been attacked by the Stuxnet virus in 2010, and the Ukrainian power grid has been attacked by the Black-Energy 3 virus in 2015 [16]. Generally speaking, according to the type of physical implementation, cyber-attacks can be broadly classifed into false data injection (FDI) attacks [4,17,18], replay attacks [19,20], and denial of service (DoS) attacks [21][22][23][24]. To date, a large number of interesting fndings have been reported, which reveal the impact of cyber-attacks on system performance and provide a number of detection and identifcation schemes (see, e.g., [25][26][27][28][29][30]). For example, the adaptive eventtriggered nonfragile state estimation problem is discussed for fractional-order complex networked systems subject to cyber-attacks [30]. Most existing results mainly report on the detection and estimation of attacks under a predesigned controller. However, there are still few explorers on the security synchronization control of complex networks. Compared with replay attacks and DoS attacks, FDI attacks can maliciously tamper with the critical operational data of complex cyber-physical networks and are more concealed and destructive than the other two types of cyber-attacks. To name a few, in [17], the resilient consensus problem is discussed for discrete-time complex cyber-physical networks subject to FDI attacks. Te authors in [18] establish a defense framework for cyber-physical systems under FDI attacks. From the perspective of cyber-attacks' implementation methods, the DoS attacks are most easily put into efect and can block the communication channels or interrupt the communication of the target system. In [21], the pinning-observer-based secure synchronization control problem is investigated for complex dynamical networks under DoS attacks. In [24], the authors propose distributed cooperative control methods for linear multiagent systems subject to DoS attacks. Unfortunately, the above results only focus on cyber-attacks, and few works are involved with physical attacks. As a kind of adversarial disturbance, physical attacks may cause the system components to operate incorrectly by maliciously modifying system inputs and thus lead to system instability [31,32]. In [31], the machine learning method is utilized to detect physical attacks on Internet of Tings applications. In response to the problem of multiple stochastic physical attacks, the robust secure controller is developed to ensure the stability of cyber-physical systems [32]. Up to now, most of the existing results are concerning single malicious attacks in complex cyber-physical networks for the simplicity of analysis and design. However, in control practice, complex cyber-physical systems are often simultaneously attacked by mixed attacks (e.g., FDI attacks, DoS attacks, and physical attacks). Due to the inherent coupling between network node dynamics and the threat of mixed attacks, the issue of synchronization control for cyber-physical networks with mixed attacks remains a technical challenge that needs to be addressed urgently, which is the primary motivation of the current investigation.
Motivated by the aforementioned discussions, this paper is devoted to the investigation of the synchronization control problem for discrete-time complex cyber-physical networks with input delays and mixed attacks. Te main contributions of this paper are summarized as follows: (1) A unifed model with both cyber-attacks and physical attacks is proposed to characterize the pattern feature of mixed attacks, and then the intermittent synchronization controller is proposed for discrete-time complex cyber-physical networks with input delays. (2) Diferent from the periodic intermittent control mechanism in [33,34], in which the time interval must be preset in advance, this paper adopts an event-dependent nonperiodic intermittent control mechanism. In other words, the control input is state dependent. Terefore, the control cost will be reduced fundamentally. (3) An analytical expression of the synchronization error dynamics is developed within the energyconstrained mixed attacks, and sufcient conditions are derived to guarantee the ultimate boundedness of the synchronization control performance.

Problem Formulation and Preliminaries
We will model discrete-delayed complex cyber-physical networks under mixed attacks. To improve readability, the notations used in this paper are standard and expressed as Table 1.
We consider discrete-delayed complex cyber-physical networks consisting of N identically coupled nodes as follows: where and ϕ i (θ) ∈ R n , (i ∈ 1, 2, . . . , N { }) denote the state vector, the control input, the disturbance input, and the initial conditions, respectively. f(x i (k)) ∈ R n is the nonlinear vectorvalued function. h(x i (k)) ∈ R n is the physical attacks signal injected by the anomalies [35]. A ∈ R n and E ∈ R n are known constant matrices with appropriate dimensions. τ k is the time-varying delay, which satisfes τ m ≤ τ k ≤ τ M , where τ m and τ M are the lower and upper bounds of the time delay, respectively. Γ ∈ R n×n denotes the inner-coupling matrix, 2 Journal of Control Science and Engineering and L � (l ij ) N×N is a matrix representing the outer-coupling confguration with l ij ≥ 0, (i ≠ j) and l ij � − N j�1,j ≠ i l ij . Te structure of control for networks node i is shown in Figure 1.
In the following, the model of cyber-attacks will be constructed for discrete-time delayed complex cyberphysical networks, which is shown in Figure 2. Firstly, the FDI attacks in the communication network aim to contaminate the control input with false data and thus threaten system security. For the FDI attacks, a Bernoulli variable π(k) is used to indicate whether the FDI attacks occur. Te data revamped by the FDI attacks can be denoted as follows [36]: (2) where g(u i (k − τ(k))) ∈ R n is the vector of FDI attacks. Te π(k) � 1 indicates that the FDI attacks have contaminated the control data, and π(k) � 0 denotes that the FDI attacks have failed to afect the transmitted data.
In addition, random DoS attacks in the control channel [37]. Similar to the FDI attacks, another variable λ(k) with the Bernoulli distribution is utilized to describe the DoS attacks signal, which can be expressed as follows: where the Bernoulli variable λ(k) ∈ 0, 1 represents that the DoS attacks are not occurring, while λ(k) � 1 denotes that the network sufers from the DoS attacks.
Based on the above description, the control signal suffering from the DoS attacks and FDI attacks can be derived as follows: Comminating (2) and (4), the control input after networks transmission is obtained as follows: Ten, the model of delayed complex cyber-physical networks (1) under mixed attacks can be expressed as follows: In this paper, the following form of the isolated node is considered: where s(k) ∈ R n is the state vector of the isolated node.
For the established model of complex cyber-physical networks under mixed attacks (6), the following assumptions are given. Assumption 1. Te disturbance ω(k) is energy bounded and satisfes the following conditions: where δ is a given positive scalar.
Assumption 2. For any v 1 (k) ∈ R n and v 2 (k) ∈ R n , the nonlinear functions f(x i (k)) ∈ R n , g(x i (k)) ∈ R n , and h(x i (k)) ∈ R n satisfy the following conditions: Identity matrix/zero matrix where H 1f , H 2f , H 1g , H 2g , H 1h , and H 2h ∈ R n×n are known constant matrices.
Remark 1. Based on the above presentations, a mixed attacks model is proposed following the FDI attacks, DoS attacks, and physical attacks strategies. According to Assumption 1, the FDI attacks and physical attacks are always constrained by the limited energy. In the cyber layer, when FDI attacks and DoS attacks occur simultaneously, the DoS attacks will lead to the loss of data injected by the FDI attacks. Defne the synchronization error and the initial error as follows: Ten, the following matrices and notations are introduced:

Journal of Control Science and Engineering
According to the above defnition and (6), (7), and (9), the synchronization error dynamics can be obtained as follows: It is from (9) that In this paper, to achieve synchronization control of the complex cyber-physical networks (6), an intermittent synchronization controller is employed as follows: where K i (i � 1, 2) are parametric intermittent synchronization controller gain matrices.
Remark 2. Te delayed control term K 2 e(k − τ k ) widens the feasible region of the synchronization control strategy, and we defne the synchronization controller (14) running and sleeping states as the work region R 1 (k) and the rest region R 2 (k), respectively. In addition, we establish the holding region R 3 (k) between the work region R 1 (k) and the rest region R 2 (k) in order to avoid the controller indefnitely cycling between u(k) � 0 and u(k) We substitute (14) into (10), which yields the model of the synchronization error dynamics as follows:

Journal of Control Science and Engineering
In this paper, the relations of Lyapunov-like function V(k) and regions R i (k) will be constructed for the intermittent synchronization controller, which is shown in Figure 3 where the earthy yellow line represents case V(k) ∈ R 1 (k) and the deep green line represents case According to the intermittent synchronization controller (14) and the synchronization error dynamics (15a)-(15c), the intermittent synchronization control strategy in this paper is given as follows: (1) When Lyapunov-like function V(k) ∈ R 1 (k), then the intermittent synchronization controller (14) is activated, which means that the synchronization error dynamics (15a) work. (2) When Lyapunov-like function V(k) ∈ R 2 (k), then the intermittent synchronization controller (14) is not activated, which means that the synchronization error dynamics (15b) do not work.
and the synchronization error dynamics (15a) work, it means that the intermittent synchronization controller is activated.
and the synchronization error dynamics (15b) work, it means that the intermittent synchronization controller is not activated.
Next, we defne the boundary of R 1 (k) and R 3 (k) as Ψ 1 (k) and the boundary of R 2 (k) and R 3 (k) as Ψ 2 (k), respectively. Ten, it is obtained from (16) that where . It is assumed that the following conditions hold in this paper: To satisfy condition (15a), we can assume that (1) α 1 > α 2 , Ten, the defnition of synchronization is introduced as follows.

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Journal of Control Science and Engineering (2) When Lyapunov-like function V(k) ∈ R 2 (k), then the intermittent synchronization controller u(k) � 0 is activated, which means that the synchronization error dynamics (15b) work at the time k 1 (k 1 ≥ 1). Simultaneously, in order to avoid the trajectory of Lyapunov-like function V(k) stay in R 2 (k) or R 3 (k) for all k > k 1 , there must exist a time k 2 (k 2 > k 1 ) that guarantees the trajectory of Lyapunov-like function V(k) evolution to the rest region R 1 (k).

Journal of Control Science and Engineering
Similarly, for ∀k ∈ N[0, +∞), we assume that there exist switchings between the synchronization error dynamics (15a) and (15b) according to the following procedure: (1) When Lyapunov-like function V(k) ∈ R 2 (k) ∪ R 3 (k), the intermittent synchronization controller u(k) � 0 is activated, which means that the closed-loop synchronization error dynamics (15b) work at the initial time k � 0. Terefore, there must exist a time k 1 (k 1 ≥ 1) that guarantees the trajectory of Lyapunov-like function V(k) evolution to the rest region R 1 (k).
Similar to the analysis of the frst case, we defne ) as the work time and the rest time, respectively.
Ten, for ∀k ∈ K 1 and K 1 ∪ K 2 ∈ N[0, +∞), according to (35) and noting the fact From (36) and (37) and noting the facts Te proof is thus completed. For the given parameterized synchronization control gains K i (i � 1, 2), Teorem 1 provides sufcient conditions for ensuring the bounded of the synchronization error dynamics (13). Considering the LMI (19), it is easy to fnd that there is certain relationship between the control gains K i and the synchronization control performance, and the value of control gains K i would afect the feasibility of the LMI (19). Now, we are in the position of designing the synchronization control gains K i on the basis of Teorem 1.
□ Remark 3. Up to now, most of the existing results concerning complex networks are only subject to either cyberattacks or time delays for the simplicity of analysis and design (see, e.g., [12,21,38]). Unfortunately, complex cyberphysical networks may be afected by the combined efects of cyber-attacks, physical attacks, and time delay in a control practice. It is worth noting that both FDI attacks and physical attacks may bring signifcant risks to some practical applications (e.g., electric power grids, the Internet of Tings, and connected vehicles [31,39]) due to their concealed characteristics. Furthermore, the time delay is another major constraint for the application of complex cyber-physical networks, which will cause system performance degradation or even instability. Consequently, the proposed synchronous control method is an indispensable supplement to the existing results for complex cyber-physical networks with both time delays and mixed attacks. Teorems 1 and 2 only provide sufcient conditions for ensuring the boundedness of the synchronization error dynamics for the simplicity of analysis. It is worth pointing out, however, that people are interested to obtain the minimized synchronization error as much as possible in the control practice. Terefore, we are providing an optimization strategy to minimize the synchronization error.
Assuming that there exists the minimized synchronization error δ 0 , which enables ‖e(k)‖ 2 ≤ δ 0 . Tis means that the following inequality holds: According to (41), we can obtain Tis means is that Applying Schur complement lemma to (43), it is clear that Terefore, we can provide an optimization problem to minimize the synchronization error δ 0 and determine positive defnite matrices P ∈ R n×n , Q ∈ R n×n , and synchronization control gains K i .
Remark 4. Subject to (38), (39), and (45), the synchronization control gains can be determined by (42). In this optimization problem, we further analyze the efects of the probability of mixed attacks on the synchronization error. Specifcally, the probability of mixed attacks (the probabilities of the FDI attacks and DoS attacks are set as π � 0.10 and λ � 0.20, respectively) directly afects the upper limit of the synchronization error for discrete-delayed complex cyber-physical networks. A complex cyber-physical network that information of malicious attack usually results in a change in the synchronization error, such as the FDI attack and DoS attack (see [17,21]). Note that the purpose of this optimization problem is interested to obtain the minimized synchronization error as much as possible. Let us consider a special case where in the absence of input delay, the corresponding synchronization error dynamics can be written as follows: Choose the following intermittent synchronization controller: and select the following Lyapunov function: Ten, it is easy to obtain the following result.
Proof of Corollary 1. Te proof of this corollary can be obtained directly from that of Teorems 1 and 2.
□ Remark 5. Till now, a systematic study has been conducted on the intermittent synchronization control problem for complex cyber-physical networks under mixed attacks. Teorems 1 and 2 provide sufcient conditions for the synchronization error to be bounded. Ten, we developed an optimization problem to obtain minimize the synchronization error. In addition, for complex cyber-physical networks with constant delay and mixed attacks, the corresponding results can be readily obtained by revising the Lyapunov functional and synchronization error dynamics.

Numerical Simulations
In this section, two numerical examples are given to verify the efectiveness and superiority of the proposed synchronization control strategy.

Example 1.
We consider a delayed complex cyber-physical network of the form (1), which is composed of three identical nodes with the following parameters: Te inner-coupling matrix is set as Γ � 0.3I, and the outer-coupling matrices L is given as follows: Assumption 1 is easily verifed by using Te probabilities of the FDI attacks and DoS attacks are set as π � 0.10 and λ � 0.20, respectively, and the FDI attacks and physical attacks have the following forms: Te initial conditions of complex cyber-physical networks are set as x 0 � − 1.5 1.5 . Ten, the FDI attacks and DoS attacks' times are shown in Figures 4 and 5, respectively, where "0" represents that cyber-attacks are not occurring, while "1" denotes that the network sufers from cyber-attacks. Te energy evolution of physical attacks is given in Figure 6. Figures 7 and 8 plot the synchronization error trajectories of complex cyber-physical networks without control input, which show that the network node cannot be spontaneous synchronization with the unforced isolated node.
Let α 1 � 0.5, α 2 � 0.3, β 1 � 0.12, β 2 � 0.09, c 1 � 0.05, c 2 � 0.06, ρ � 0.1, and σ � − 2.4, respectively. Ten, Equation (15a) is satisfed for any V(k). Applying Teorem 2 and solving LMI (38), the corresponding synchronization control gains matrices can be obtained as follows: According to the intermittent synchronization control mechanism (12), the intermittent synchronization controller u(k) � K 1 e(k) + K 2 e(k − τ k ) is activated when Lyapunov-like function. Te synchronization error trajectories of the network nodes under mixed attacks are given in Figures 9  and 10. It can be found from Figures 9 and 10 that synchronization errors can quickly converge within the limited sampling periods, which implies that the presented synchronization control method is efective for the discretetime delayed complex cyber-physical networks with input delays and mixed attacks. Moreover, we choose an acceptable probability of cyber-attacks, by calculating the optimization problem (45), and then can easily obtain that the minimum upper bound δ 0 � 4.6026 for the synchronization error.
In order to prove the superiority of the proposed synchronization control method, the state feedback synchronization control method [40] is utilized for complex cyber-physical networks (1) in the same conditions. Figures 11 and 12 show the synchronization error trajectories of the network nodes under the control method of [40]. It can be seen from Figures 9-12 that the proposed synchronization control method has far lower synchronization error fuctuations than the method of [40], which shows that our control strategy can efectively reduce the negative impact of the mixed attacks and the input delays compared with the state feedback ones.
Example 2. Consider complex cyber-physical networks (1) consisting of three Chua's chaotic circuits [41] with the following parameters:

FDI Attacks
The FDI attacks time        Te inner-coupling matrix is set as Γ � 0.517I, and the outer-coupling matrices L given as follows:   Assumption 1 is easily verifed by using an improved directed crossover genetic algorithm based on multilayer mutation: We assume that the FDI attacks and physical attacks have the following form: g(x(k)) � Let the other parameters be the same as in Example 1. Ten, (15a) is satisfed for any V(k). Applying Teorem 2 and solving LMI (38), the corresponding synchronization control gains matrices can be obtained as follows: Te initial condition of complex cyber-physical networks is set as x i (0) � 1 − 1 1 T . In this case, the trajectories of synchronization error between the unforced isolated node and the network nodes are shown by Figures 13-15, respectively. It could be found from Figures 13-15 that the synchronization errors converge to zero within the limited sampling periods, which imply that the presented synchronization control method is efective for the discretetime delayed complex cyber-physical networks with input delays and mixed attacks.

Conclusions
In this paper, the synchronization control issue has been investigated for discrete-delayed complex cyber-physical networks under mixed attacks. As a means of reducing the control costs of the complex cyber-physical networks, the intermittent mechanisms described by non-negative real regions have been introduced and applied to the design process of the synchronization control, and then an intermittent synchronization controller has been developed for the corresponding delayed complex cyber-physical networks subject to mixed attacks. By utilizing the appropriate Lyapunov function, sufcient conditions are derived for ensuring that the synchronization error dynamics are ultimately bounded, and the desired synchronization control gain matrices have been obtained by solving a group of LMIs. Subsequently, an optimization method has also been provided with the aim to minimize the synchronization error. Finally, two numerical examples are given to verify the efectiveness and superiority of the proposed synchronization control strategy.
On the other hand, it is worth pointing out that the treatment method of the nonlinear function in this article is somewhat conservative. Specifcally, the sector-like descriptions of the nonlinearities do not relate to the current state of the error dynamics. Further research topics include the extension of our results to complex cyber-physical networks with distributed input delays and mixed attacks. Also, it is more interesting to design the state-dependent treatment method for the nonlinear function.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.