#### Abstract

This paper is concerned with the synchronization control problem for discrete-delayed complex cyber-physical networks under mixed attacks. To handle input delays and mixed attacks, the intermittent control mechanism is employed, which is distinctly different from the traditional control method. By utilizing the Lyapunov stability theorem, a novel synchronization control method is developed for the synchronization control of complex cyber-physical networks with mixed attacks. Then, sufficient conditions are derived to guarantee that the synchronization error dynamics are ultimately bounded. Moreover, the conditions for a special case where the absence of input delays. Subsequently, certain optimization problems are formulated with the aim to minimize the synchronization error. Finally, two numerical examples are given to verify the effectiveness and superiority of the proposed synchronization control strategy.

#### 1. 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 field, such as multiagent systems, transportation networks, neural networks, and electric power grids. [1–5]. 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. Therefore, the synchronization problem of complex networks with time delays has received considerable attention (see, e.g., [7–9]). On the other hand, as a result of the complicated network structure, it is always difficult to achieve spontaneous synchronization. To date, various control techniques have been presented to investigate the synchronization problem of complex networks [10–15]. 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 quasi-synchronization 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. Specifically, the availability and integrity performance of the modern system information is at serious risk, as testified 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 classified into false data injection (FDI) attacks [4, 17, 18], replay attacks [19, 20], and denial of service (DoS) attacks [21–24]. To date, a large number of interesting findings have been reported, which reveal the impact of cyber-attacks on system performance and provide a number of detection and identification schemes (see, e.g., [25–30]). For example, the adaptive event-triggered 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. The 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 effect 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 Things 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. The main contributions of this paper are summarized as follows:(1)A unified 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)Different 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. Therefore, the control cost will be reduced fundamentally.(3)An analytical expression of the synchronization error dynamics is developed within the energy-constrained mixed attacks, and sufficient conditions are derived to guarantee the ultimate boundedness of the synchronization control performance.

#### 2. 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 identically coupled nodes as follows:where , , , and denote the state vector, the control input, the disturbance input, and the initial conditions, respectively. is the nonlinear vector-valued function. is the physical attacks signal injected by the anomalies [35]. and are known constant matrices with appropriate dimensions. is the time-varying delay, which satisfies , where and are the lower and upper bounds of the time delay, respectively. denotes the inner-coupling matrix, and is a matrix representing the outer-coupling configuration with and . The structure of control for networks node is shown in Figure 1.

In the following, the model of cyber-attacks will be constructed for discrete-time delayed complex cyber-physical 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 is used to indicate whether the FDI attacks occur. The data revamped by the FDI attacks can be denoted as follows [36]:where is the vector of FDI attacks. The Bernoulli variable satisfies , and . indicates that the FDI attacks have contaminated the control data, and denotes that the FDI attacks have failed to affect the transmitted data.

In addition, random DoS attacks in the control channel [37]. Similar to the FDI attacks, another variable with the Bernoulli distribution is utilized to describe the DoS attacks signal, which can be expressed as follows:where the Bernoulli variable satisfies and . represents that the DoS attacks are not occurring, while denotes that the network suffers 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:

Then, 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 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. *The disturbance is energy bounded and satisfies the following conditions:where is a given positive scalar.

*Assumption 2. *For any and , the nonlinear functions , , and satisfy the following conditions:where , , , , , and 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.

Define the synchronization error and the initial error as follows:Then, the following matrices and notations are introduced:According to the above definition and (6), (7), and (9), the synchronization error dynamics can be obtained as follows:It is from (9) thatIn this paper, to achieve synchronization control of the complex cyber-physical networks (6), an intermittent synchronization controller is employed as follows:where are parametric intermittent synchronization controller gain matrices. is Lyapunov-like functions. are three subregions of the non-negative real region , satisfying , .

*Remark 2. *The delayed control term widens the feasible region of the synchronization control strategy, and we define the synchronization controller (14) running and sleeping states as the work region and the rest region , respectively. In addition, we establish the holding region between the work region and the rest region in order to avoid the controller indefinitely cycling between and .

We substitute (14) into (10), which yields the model of the synchronization error dynamics as follows:In this paper, the relations of Lyapunov-like function and regions will be constructed for the intermittent synchronization controller, which is shown in Figure 3 where the earthy yellow line represents case and the deep green line represents case , respectively.

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 , then the intermittent synchronization controller (14) is activated, which means that the synchronization error dynamics (15a) work.(2)When Lyapunov-like function , then the intermittent synchronization controller (14) is not activated, which means that the synchronization error dynamics (15b) do not work.(3)When , if Lyapunov-like function and the synchronization error dynamics (15a) work, it means that the intermittent synchronization controller is activated.(4)When , if Lyapunov-like function and the synchronization error dynamics (15b) work, it means that the intermittent synchronization controller is not activated.In this paper, we consider the following forms of :where , , and are given positive scalars.

Next, we define the boundary of and as and the boundary of and as , respectively. Then, it is obtained from (16) thatwhere and . It is assumed that the following conditions hold in this paper:To satisfy condition (15a), we can assume that (1) , , ; or (2) ; or (3) , , .

Then, the definition of synchronization is introduced as follows.

*Definition 1. *(see [12]). For the given scalars and , if there exists a scalar , the following inequality holds:Then, the synchronization error dynamics (15a)–(15c) is ultimately bounded.

#### 3. Main Results

Firstly, we define the piecewise Lyapunov-like function as follows:whereandand , , and .

Theorem 1. *Let the FDI attacks probability , the DoS attacks probability , the scalars , , , , and , and the synchronization control gain matrices be given. If there exist the definite matrices and and the positive scalars , , and , the following matrices inequalities hold:whereand, , and .*

Then, the synchronization error converges to under the intermittent synchronization control mechanism (12), which means that the synchronization error dynamics are ultimately bounded.

*Proof. * of Theorem 1. Let , and the forward difference along the trajectory of synchronization error dynamics (15a) can be calculated asFor any scalar , it follows from (11) thatSubstituting (27)–(29) into (17), one eventually obtainswhere.

Applying Schur complement lemma to (15c), it is clear that . We substitute and (8) into (23), which yieldsBy using the similar method, the forward difference is determined along with the trajectory of synchronization error dynamics (15b).Substituting (18)–(20) into (23), one eventually obtainsApplying Schur complement lemma to (16), it is clear that . Substituting and (8) into (24), one hasAccording to (22), it is obtained thatwhich means thatFrom (25), it is directly obtained thatNoting that and , we can obtain from (28) thatSimilarly, from (29), it is directly obtained thatIn this paper, for , we assume that there exist switchings between the synchronization error dynamics (15a) and (15b) according to the following procedure:(1)When Lyapunov-like function , the intermittent synchronization controller is activated, which means that the synchronization error dynamics (15a) work at the initial time . At the same time, because of the scalars , , and , there must exist a time that guarantees the trajectory of Lyapunov-like function evolution to the rest region .(2)When Lyapunov-like function , then the intermittent synchronization controller is activated, which means that the synchronization error dynamics (15b) work at the time . Simultaneously, in order to avoid the trajectory of Lyapunov-like function stay in or for all , there must exist a time that guarantees the trajectory of Lyapunov-like function evolution to the rest region .According to the above procedure, we assume that all work time and rest time of the intermittent synchronization controller are and , respectively.

For , the synchronization error dynamics (15a) are active. According to (27), one hasandNote that at , which yields . Then, it is obtained from (30) thatSubstituting (33) into (32), for , it is obtained thatNote that and , then . It is obtained from (33) thatFurthermore, for and , holds. Then, from , one hasSimilarly, for , we assume that there exist switchings between the synchronization error dynamics (15a) and (15b) according to the following procedure:(1)When Lyapunov-like function , the intermittent synchronization controller is activated, which means that the closed-loop synchronization error dynamics (15b) work at the initial time . Therefore, there must exist a time that guarantees the trajectory of Lyapunov-like function evolution to the rest region .(2)When Lyapunov-like function , then the intermittent synchronization controller is activated, which means that the closed-loop synchronization error dynamics (15a) work at time . Simultaneously, for the scalars , , and , there must exist a time that guarantees the trajectory of Lyapunov-like function evolution to the rest region .Similar to the analysis of the first case, we define and as the work time and the rest time, respectively.

Then, for and , according to (35) and noting the fact , one hasFrom (36) and (37) and noting the facts , one hasThe proof is thus completed.

For the given parameterized synchronization control gains , Theorem 1 provides sufficient conditions for ensuring the bounded of the synchronization error dynamics (13). Considering the LMI (19), it is easy to find that there is certain relationship between the control gains and the synchronization control performance, and the value of control gains would affect the feasibility of the LMI (19). Now, we are in the position of designing the synchronization control gains on the basis of Theorem 1.

Theorem 2. *Let the FDI attacks probability , the DoS attacks probability , and the scalars , , , , and , and the synchronization control gain matrices be given. If there exist the matrices , , and and the positive scalars , , and , the following matrix inequalities hold:where**Then, the synchronization controller can be given bywhich renders the closed-loop synchronization error dynamics (13) to be bounded, and the synchronization error converges to .*

*Proof. * of Theorem 2. The variable substitution method is employed to prove this theorem. Let and substitute it into (23)-(24), which yields (48)-(49). This completes the proof.

*Remark 3. *Up to now, most of the existing results concerning complex networks are only subject to either cyber-attacks or time delays for the simplicity of analysis and design (see, e.g., [12, 21, 38]). Unfortunately, complex cyber-physical networks may be affected by the combined effects 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 significant risks to some practical applications (e.g., electric power grids, the Internet of Things, 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.

Theorems 1 and 2 only provide sufficient 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. Therefore, we are providing an optimization strategy to minimize the synchronization error.

Assuming that there exists the minimized synchronization error , which enables . This means that the following inequality holds:According to (41), we can obtainThis means is thatApplying Schur complement lemma to (43), it is clear thatTherefore, we can provide an optimization problem to minimize the synchronization error and determine positive definite matrices , , and synchronization control gains .subject to (38), (39), and (45), and the synchronization control gains can be determined by (44).

*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 effects of the probability of mixed attacks on the synchronization error. Specifically, the probability of mixed attacks (the probabilities of the FDI attacks and DoS attacks are set as and , respectively) directly affects 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:Then, it is easy to obtain the following result.

Corollary 1. *Let the FDI attacks probability , the DoS attacks probability , and the scalars , , , , and , and the synchronization control gain matrices be given. If there exist the matrices , , and and the positive scalars , , and , such that the following matrices inequalities hold:whereand , , and .**Then, the synchronization controller can be given bywhich renders the closed-loop synchronization error dynamics (47) to be bounded, and the synchronization error converges to .*

*Proof. * of Corollary 1. The proof of this corollary can be obtained directly from that of Theorems 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. Theorems 1 and 2 provide sufficient conditions for the synchronization error to be bounded. Then, 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.

#### 4. Numerical Simulations

In this section, two numerical examples are given to verify the effectiveness 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:The inner-coupling matrix is set as , and the outer-coupling matrices is given as follows:Assumption 1 is easily verified by usingThe probabilities of the FDI attacks and DoS attacks are set as and , respectively, and the FDI attacks and physical attacks have the following forms: