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

Authors: Chaoqun Zhu; Xuan Jia; Pan Zhang
Journal: Journal of Control Science and Engineering
 (2022)

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## 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.


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## Body

## 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 Table1.Table 1
Notations.
NotationsExpressionℝnn-dimensional Euclidean spaceℝn×nn×n real matricesNThe sets of non-negative integersNa,bThe set of integers betweena and bℛ+The non-negative real regionGTThe transpose of the matrixGP>0MatrixP is positive definiteP≥0MatrixP is positive semidefinitexThe Euclidean vector normΔThe differential operator⊗The Kronecker product∗Symmetric entryλminTThe smallest eigenvalues of matrixTdiag⋅The diagonal matrixI/0Identity matrix/zero matrixINTheN×N identity matrixWe consider discrete-delayed complex cyber-physical networks consisting ofN identically coupled nodes as follows:(1)xik+1=Axik+fxik+∑j=1Nl˜ijΓxjk+uik−τk+hxik+Eωik,xiθ=ϕiθ,θ∈−∞,0,where xik∈ℝn, uik∈ℝm, ωik∈ℝn, and ϕiθ∈ℝn,i∈1,2,…,N denote the state vector, the control input, the disturbance input, and the initial conditions, respectively. fxik∈ℝn is the nonlinear vector-valued function. hxik∈ℝn is the physical attacks signal injected by the anomalies [35]. A∈ℝn and E∈ℝn are known constant matrices with appropriate dimensions. τk is the time-varying delay, which satisfies τm≤τk≤τM, where τm and τM are the lower and upper bounds of the time delay, respectively. Γ∈ℝn×n denotes the inner-coupling matrix, and L=l˜ijN×N is a matrix representing the outer-coupling configuration with l˜ij≥0,i≠j and l˜ij=−∑j=1,j≠iNl˜ij. The structure of control for networks node i is shown in Figure 1.Figure 1
The structure of control for the network nodei.In the following, the model of cyber-attacks will be constructed for discrete-time delayed complex cyber-physical networks, which is shown in Figure2. 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 πk is used to indicate whether the FDI attacks occur. The data revamped by the FDI attacks can be denoted as follows [36]:(2)u¯ik=πkguik−τk+1−πkuik−τk,where guik−τk∈ℝn is the vector of FDI attacks. The Bernoulli variable πk∈0,1 satisfies Eπk=π¯, and Eπk−π¯=π2. πk=1 indicates that the FDI attacks have contaminated the control data, and πk=0 denotes that the FDI attacks have failed to affect the transmitted data.Figure 2
Illustration of complex cyber-physical networks under mixed attacks.In addition, random DoS attacks in the control channel [37]. Similar to the FDI attacks, another variable λk with the Bernoulli distribution is utilized to describe the DoS attacks signal, which can be expressed as follows:(3)λk=0,others,1,occurs,where the Bernoulli variable λk∈0,1 satisfies Eλk=λ¯ and Eλk−λ¯=λ2. λk=0 represents that the DoS attacks are not occurring, while λk=1 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:(4)u˜ik=1−λku¯ik.Comminating (2) and (4), the control input after networks transmission is obtained as follows:(5)u˜ik−τk=1−λkπkguik−τk+1−πkuik−τk.Then, the model of delayed complex cyber-physical networks (1) under mixed attacks can be expressed as follows:(6)xik+1=Axik+fxik+∑j=1Nl˜ijΓxjk+hxik+Eωk+1−λkπkguik−τk+1−πkuik−τk,xiθ=ϕiθ,θ∈−∞,0.In this paper, the following form of the isolated node is considered:(7)sk+1=Ask+fsk+hsk+Eωk,sθ=ϕθ,θ∈−∞,0,where sk∈ℝn 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 disturbanceω˜k is energy bounded and satisfies the following conditions:(8)∑k+∞ω˜Tkω˜k≤δ,where δ is a given positive scalar.Assumption 2.
For anyv1k∈ℝn and v2k∈ℝn, the nonlinear functions fxik∈ℝn, gxik∈ℝn, and hxik∈ℝn satisfy the following conditions:(9)fv1k−fv2k−ℋ1fv1k−v2kT×fv1k−fv2k−ℋ2fv1k−v2k≤0,gv1k−gv2k−ℋ1gv1k−v2kT×gv1k−gv2k−ℋ2gv1k−v2k≤0,hv1k−hv2k−ℋ1hv1k−v2kT×hv1k−hv2k−ℋ2hv1k−v2k≤0,where ℋ1f, ℋ2f, ℋ1g, ℋ2g, ℋ1h, and ℋ2h∈ℝn×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 Assumption1, 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:(10)eik=xik−sk,ϕ˜iθ=ϕiθ−ϕθ.
Then, the following matrices and notations are introduced:(11)A˜=IN⊗A,E˜=IN⊗E,L˜=L⊗Γ,ℋ˜1f=IN⊗ℋ1f,ℋ˜2f=IN⊗ℋ2f,ℋ˜1g=IN⊗ℋ1g,ℋ˜2g=IN⊗ℋ2g,ℋ˜1h=IN⊗ℋ1h,ℋ˜2h=IN⊗ℋ2h,ek=e1Tke2Tk⋅⋅⋅eNTkT,ϕ˜k=ϕ˜1Tkϕ˜2Tk⋅⋅⋅ϕ˜NTkT,f˜ek=f˜Te1k⋅⋅⋅f˜TeNk,g˜ek=g˜Te1k⋅⋅⋅g˜TeNk,h˜ek=h˜Te1k⋅⋅⋅h˜TeNk,ω˜k=ω˜1Tkω˜2Tk⋅⋅⋅ω˜NTkT,f˜eik=fxik−fsk−ℋ1fek,g˜eik=gxik−gsk−ℋ1gek,h˜eik=hxik−hsk.
According to the above definition and (6), (7), and (9), the synchronization error dynamics can be obtained as follows:(12)ek+1=A˜+L˜+ℋ˜1fek+f˜ek+h˜eik+E˜ω˜k+1−λkπkg˜uk−τk+1−λk1−πkuk−τk,eθ=ϕ˜θ,θ∈−∞,0.
It is from (9) that(13)f˜Tekf˜ek−ℋ˜2f−ℋ˜1fek≤0,g˜Tekg˜ek−ℋ˜2g−ℋ˜1gek≤0,h˜ek−ℋ˜1hekTh˜ek−ℋ˜2hek≤0.
In this paper, to achieve synchronization control of the complex cyber-physical networks (6), an intermittent synchronization controller is employed as follows:(14)uk=K1ek+K2ek−τk,Vk∈ℛ1k,0,Vk∈ℛ2k,uk−1,Vk∈ℛ3k,where Kii=1,2 are parametric intermittent synchronization controller gain matrices. Vk is Lyapunov-like functions. ℛiki=1,2,3 are three subregions of the non-negative real region ℛ+, satisfying ∪i=13ℛi=ℛ+, ℛik∩ℛjk=∅,i,j∈1,2,3,i≠j.Remark 2.
The delayed control termK2ek−τk widens the feasible region of the synchronization control strategy, and we define the synchronization controller (14) running and sleeping states as the work region ℛ1k and the rest region ℛ2k, respectively. In addition, we establish the holding region ℛ3k between the work region ℛ1k and the rest region ℛ2k in order to avoid the controller indefinitely cycling between uk=0 and uk=K1ek+K2ek−τk.
We substitute (14) into (10), which yields the model of the synchronization error dynamics as follows:(15a)ek+1=A˜+L˜+ℋ˜1f+K1ek+f˜ek+K2ℋ˜1g1−λkπkek−τk+K21−λk1−πkek−τk+K21−λkπkg˜ek−τk+h˜ek+E˜ω˜k,Vk∈ℛ1k,(15b)ek+1=A˜+L˜+ℋ˜1fek+f˜ek+h˜ek+E˜ω˜k,Vk∈ℛ2k,(15c)ek+1=A˜+L˜+ℋ˜1fek+f˜ek+h˜ek+E˜ω˜k+uk−1,Vk∈ℛ3k.
In this paper, the relations of Lyapunov-like functionVk and regions ℛik will be constructed for the intermittent synchronization controller, which is shown in Figure 3 where the earthy yellow line represents case Vk∈ℛ1k and the deep green line represents case Vk∈ℛ2k∪ℛ3k, 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 functionVk∈ℛ1k, then the intermittent synchronization controller (14) is activated, which means that the synchronization error dynamics (15a) work.(2)
When Lyapunov-like functionVk∈ℛ2k, then the intermittent synchronization controller (14) is not activated, which means that the synchronization error dynamics (15b) do not work.(3)
Whenk=k−1, if Lyapunov-like function Vk∈ℛ3k and the synchronization error dynamics (15a) work, it means that the intermittent synchronization controller is activated.(4)
Whenk=k−1, if Lyapunov-like function Vk∈ℛ3k 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 ofℛiki=1,2,3:(16)ℛ1k=r∈ℛ+:r≥α1V01−γ1k+β1,ℛ2k=r∈ℛ+:r<α2V01−γ2k+β2,ℛ3k=ℛ+ℛ1k∩ℛ2k,where α1,2>0, β1,2>0, and 0<γ1,2<1 are given positive scalars.
Next, we define the boundary ofℛ1k and ℛ3k as Ψ1k and the boundary of ℛ2k and ℛ3k as Ψ2k, respectively. Then, it is obtained from (16) that(17)Ψ1k=r∈ℛ+:r=α1V01−γ1k+β1,Ψ2k=r∈ℛ+:r=α2V01−γ2k+β2,where ℛ3k≠∅ and ℛik∩ℛjk=∅i,j∈1,2,3,i≠j. It is assumed that the following conditions hold in this paper:(18)α1V01−γ1k+β1>α2V01−γ2k+β2.
To satisfy condition (15a), we can assume that (1) α1>α2, β1>β2, γ1≤γ2; or (2) β1>α2+β2; or (3) α1=α2, β1>β2, γ1<γ2.
Then, the definition of synchronization is introduced as follows.Figure 3
Illustration of the relations of Lyapunov-like functionVk and regions ℛiki=1,2,3.Definition 1.
(see [12]). For the given scalars σ<0 and ς>0, if there exists a scalar δ>0, the following inequality holds:(19)ek2≤1−σς+δλminP.
Then, the synchronization error dynamics (15a)–(15c) is ultimately bounded.


## 3. Main Results

Firstly, we define the piecewise Lyapunov-like function as follows:(20)Vk=V1k+V2k+V3k,where(21)V1k=eTkPek,V2k=∑i=k−τkk−11−ρk−i−1eTiQei,and(22)V3k=∑j=k−τM+1k−τm∑i=jk−11−ρk−i−1eTiQei,and 0<ρ<1, 0<P∈ℝn×n, and 0<Q∈ℝn×n.Theorem 1.
Let the FDI attacks probability0<π<1, the DoS attacks probability 0<λ<1, the scalars δ>0, σ<0, 0<γ1<ρ<1, 0<γ2<ρ<1, and β2>δ/ρ, and the synchronization control gain matrices Ki,i=1,2 be given. If there exist the definite matrices 0<P∈ℝn×n and 0<Q∈ℝn×n and the positive scalars μ1>0, μ2>0, and μ3>0, the following matrices inequalities hold:(23)Ξ1=ΩΠ1T∗−P−1<0,(24)Ξ2=ΩΠ2T∗−P−1<0,where(25)Π1=PA˜+L˜+ℋ˜1f+K1PK2ℋ˜1g1−λπ+K21−λ1−πPPK21−λπPPE˜,Π2=PA˜+L˜+ℋ˜1f0P0PPE˜,Ω=Ω110Ω13000∗−1−ρτMQ0Ω2400∗∗−2μ1I000∗∗∗−2μ2I00∗∗∗∗−2μ3I0∗∗∗∗∗−I,andΩ11=−ρP+1+τM−τmQ+μ3ℋ˜1gTℋ˜2g+ℋ˜2gTℋ˜1g, Ω13=μ1ℋ˜2f−ℋ˜1fT, and Ω24=μ2ℋ˜2g−ℋ˜1gT.Then, the synchronization error converges toek2≤1−σβ1+δ/λminP under the intermittent synchronization control mechanism (12), which means that the synchronization error dynamics are ultimately bounded.Proof.
of Theorem 1. LetΔVk=Vk+1−Vk, and the forward difference along the trajectory of synchronization error dynamics (15a) can be calculated as(26)ΔVk=eTk+1Pek+1−eTkPek+τM−τm+1eTkQek−1−ρτkeTk−τkPek−τk+ρeTkQek−ρVk≤eTk+1Pek+1−eTkPek+τM−τm+1eTkQek−1−ρτMeTk−τkPek−τk+ρeTkQek−ρVk.
For any scalarμi>0,i=1,2,3, it follows from (11) that(27)−2μ1f˜ekTf˜ek−ℋ˜2f−ℋ˜1fek≥0,(28)−2μ2g˜ek−τkTg˜ek−τk−ℋ˜2g−ℋ˜1gek−τk≥0,(29)−2μ3h˜ek−ℋ˜1hekTh˜ek−ℋ˜2hek≥0.
Substituting (27)–(29) into (17), one eventually obtains(30)ΔVk+ρVk≤eTk+1Pek+1−eTkPek+τM−τm+1eTkQek−1−ρτMeTk−τkPek−τk+ρeTkQek−2μ1f˜ekTf˜ek−ℋ˜2f−ℋ˜1fek−2μ2g˜ek−τkTg˜ek−τk−ℋ˜2g−ℋ˜1gek−τk−2μ3h˜ek−ℋ˜1hekTh˜ek−ℋ˜2hek≤ζTkΩ+Π1TPΠ1ζk+ω˜Tkω˜k,whereζk=eTkeTk−τkf˜Tekg˜Tek−τkh˜Tekω˜Tk.
Applying Schur complement lemma to (15c), it is clear that Ω+Π1TPΠ1<0. We substitute Ω+Π1TPΠ1<0 and (8) into (23), which yields(31)ΔVk+ρVk≤δ.
By using the similar method, the forward differenceΔVk=Vk+1−ρVk is determined along with the trajectory of synchronization error dynamics (15b).(32)ΔVk=eTk+1Pek+1−eTkPek+τM−τm+1eTkQek−1−ρτkeTk−τkPek−τk+ρeTkQek−ρV1k+V2k≤eTk+1Pek+1−eTkPek+τM−τm+1eTkQek−1−ρτMeTk−τkPek−τk+ρeTkQek−ρV1k+V2k.
Substituting (18)–(20) into (23), one eventually obtains(33)ΔVk+ρV1k+V2k≤eTk+1Pek+1−eTkPek+τM−τm+1eTkQek−1−ρτMeTk−τkPek−τk+ρeTkQek−2μ1f˜ekTf˜ek−ℋ˜2f−ℋ˜1fek−2μ2g˜ek−τkTg˜ek−τk−ℋ˜2g−ℋ˜1gek−τk−2μ3h˜ek−ℋ˜1hekTh˜ek−ℋ˜2hek≤ζTkΩ+Π2TPΠ2ζk+ω˜Tkω˜k.
Applying Schur complement lemma to (16), it is clear that Ω+Π2TPΠ2<0. Substituting Ω+Π2TPΠ2<0 and (8) into (24), one has(34)ΔVk+ρV1k+V2k≤δ.
According to (22), it is obtained that(35)Vk≤1−ρVk−1+δ≤1−ρ2Vk−2+1−ρδ+δ≤⋯≤1−ρk−iVi+1−1−ρk−iδρ,which means that(36)Vk≤Vi−δρ1−ρk−i+δρ.
From (25), it is directly obtained that(37)ΔVk+σVk≤δ−ρV1k+V2k+σV1k+V2k+V3k.
Noting thatσ<0 and 0<ρ<1, we can obtain from (28) that(38)ΔVk+σVk≤δ+σ−ρV1k+V2k≤δ.
Similarly, from (29), it is directly obtained that(39)Vk≤Vi−δσ1−σk−i+δσ.
In this paper, for∀k∈N0,+∞, we assume that there exist switchings between the synchronization error dynamics (15a) and (15b) according to the following procedure:(1)
When Lyapunov-like functionVk∈ℛ1k, the intermittent synchronization controller uk=K1ek+K2ek−τk is activated, which means that the synchronization error dynamics (15a) work at the initial time k=0. At the same time, because of the scalars ρ>γ1, ρ>γ2, and β1>β2>δ/ρ, there must exist a time k1k1≥1 that guarantees the trajectory of Lyapunov-like function Vk evolution to the rest region ℛ2k.(2)
When Lyapunov-like functionVk∈ℛ2k, then the intermittent synchronization controller uk=0 is activated, which means that the synchronization error dynamics (15b) work at the time k1k1≥1. Simultaneously, in order to avoid the trajectory of Lyapunov-like function Vk stay in ℛ2k or ℛ3k for all k>k1, there must exist a time k2k2>k1 that guarantees the trajectory of Lyapunov-like function Vk evolution to the rest region ℛ1k.
According to the above procedure, we assume that all work time and rest time of the intermittent synchronization controller areK1=N0,k−1∪∪i=0+∞Nk2i+1,k2i+1−1 and K2=∪i=0+∞Nk2i+1,k2i+1−1, respectively.
For∀k∈K1, the synchronization error dynamics (15a) are active. According to (27), one has(40)Vk≤V0−δρ1−ρk+δρ,∀k∈N0,k1−1,and(41)Vk≤Vk2i+1−δρ1−ρk−k2i+1+δρ,∀k∈Nk2i+1,k2i+3−1.
Note thatVk=ℛ2k∪ℛ3k at k=k2i+1−1, which yields Vk2i+1−1<α1V01−γ1k2i+1−1+β1. Then, it is obtained from (30) that(42)Vk2i+1−1≤Vk2i+1−1−δσ1−σ+δσ≤α1V01−γ1k2i+1−1+β1−δσ1−σ+δσ=α1V01−σ1−γ11−γ1k2i+1−1+β11−σ+δ.
Substituting (33) into (32), for ∀k∈∪i=0+∞Nk2i+1,k2i+3−1, it is obtained that(43)Vk≤α1V01−σ1−γ11−γ1k2i+1−1+β11−σ+δ−δρ1−ρk−k2i+1+δρ=α1V01−σ1−γ11−γ1k2i+1−11−ρk−k2i+1+1−σβ1−δ1−ρρ1−ρk−k2i+1+δρ.
Note thatβ2>δ/ρ and 0<γ1<ρ<1, then ∀k∈Nk2i+1,k2i+3−1. It is obtained from (33) that(44)Vk≤α1V01−σ1−γ11−γ1k+β11−σ+δ.
Furthermore, for∀k∈K2 and Vk=ℛ2k∪ℛ3k, Vk≤α1V101−γ1k+β1 holds. Then, from K1∪K2∈N0,+∞, one has(45)Vk≤V0−δρ1−ρk+δρ,∀k∈N0,k1−1,α1V01−σ1−γ11−γ1k+β11−σ+δ,∀k∈Nk1,+∞.
Similarly, for∀k∈N0,+∞, we assume that there exist switchings between the synchronization error dynamics (15a) and (15b) according to the following procedure:(1)
When Lyapunov-like functionVk∈ℛ2k∪ℛ3k, the intermittent synchronization controller uk=0 is activated, which means that the closed-loop synchronization error dynamics (15b) work at the initial time k=0. Therefore, there must exist a time k1k1≥1 that guarantees the trajectory of Lyapunov-like function Vk evolution to the rest region ℛ1k.(2)
When Lyapunov-like functionVk∈ℛ1k, then the intermittent synchronization controller uk=K1ek+K2ek−τk is activated, which means that the closed-loop synchronization error dynamics (15a) work at time k1k1≥1. Simultaneously, for the scalars ρ>γ1, ρ>γ2, and β1≥β2>δ/ρ, there must exist a time k2k2>k1 that guarantees the trajectory of Lyapunov-like function Vk evolution to the rest region ℛ2k.
Similar to the analysis of the first case, we defineK˜1=∪i=0+∞Nk2i+1,k2i+1−1 and K˜2=N0,k−1∪∪i=0+∞Nk2i+1,k2i+1−1 as the work time and the rest time, respectively.
Then, for∀k∈K˜1 and K˜1∪K˜2∈N0,+∞, according to (35) and noting the fact Vk≤α1V01−γ1k+β1,Vk∈ℛ2k∪ℛ3k, one has(46)Vk≤α1V01−σ1−γ11−γ1k+β11−σ+δ,∀k∈N0,+∞.
From (36) and (37) and noting the facts Vk≥eTkPek≥λminPek2, one has(47)ek2≤α1V01−σ/1−γ11−γ1k+β11−σ+δλminP≤1−σβ1+δλminP.
The proof is thus completed.
For the given parameterized synchronization control gainsKii=1,2, 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 Ki and the synchronization control performance, and the value of control gains Ki would affect the feasibility of the LMI (19). Now, we are in the position of designing the synchronization control gains Ki on the basis of Theorem 1.Theorem 2.
Let the FDI attacks probability0<π<1, the DoS attacks probability 0<λ<1, and the scalars δ>0, σ<0, 0<γ1<ρ<1, 0<γ2<ρ<1, and β2>δ/ρ, and the synchronization control gain matrices Ki,i=1,2 be given. If there exist the matrices 0<P∈ℝn×n, 0<Q∈ℝn×n, and 0<Xi∈ℝn×ni=1,2 and the positive scalars μ1>0, μ2>0, and μ3>0, the following matrix inequalities hold:(48)Ξ˜1=ΩΠ˜1T∗−P−1<0,(49)Ξ2=ΩΠ2T∗−P−1<0,where(50)Π˜1=PA˜+L˜+ℋ˜1f+X1X2ℋ˜1g1−λπ+X21−λ1−πPX21−λπPPE˜.
Then, the synchronization controller can be given by(51)Ki=P−1Xi,i=1,2,which renders the closed-loop synchronization error dynamics (13) to be bounded, and the synchronization error converges to ek2≤1−σβ1+δ/λminP.Proof.
of Theorem 2. The variable substitution method is employed to prove this theorem. LetPKi=Xi,i=1,2 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 errorδ0, which enables ek2≤δ0. This means that the following inequality holds:(52)ek2≤1−σβ1+δλminP≤δ0.
According to (41), we can obtain(53)1−σβ1+δδ02I≤λminP.
This means is that(54)1−σβ1+δδ02IIT≤P.
Applying Schur complement lemma to (43), it is clear that(55)PI∗δ021−σβ1+δ≥0.
Therefore, we can provide an optimization problem to minimize the synchronization errorδ0 and determine positive definite matrices P∈ℝn×n, Q∈ℝn×n, and synchronization control gains Ki.(56)minδ0,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 π=0.10 and λ=0.20, 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:(57)ek+1=A˜+L˜+ℋ˜1fek+f˜ek+h˜eik+E˜ω˜k+1−λkπkg˜uk,eθ=ϕ˜θ,θ∈−∞,0.
Choose the following intermittent synchronization controller:(58)uk=K1ekVk∈ℛ1k,0Vk∈ℛ2k,uk−1Vk∈ℛ3k,and select the following Lyapunov function:(59)V1k=eTkPek.
Then, it is easy to obtain the following result.Corollary 1.
Let the FDI attacks probability0<π<1, the DoS attacks probability 0<λ<1, and the scalars δ>0, σ<0, 0<γ1<ρ<1, 0<γ2<ρ<1, and β2>δ/ρ, and the synchronization control gain matrices Ki,i=1,2 be given. If there exist the matrices 0<P∈ℝn×n, 0<Q∈ℝn×n, and 0<X∈ℝn×n and the positive scalars μ1>0, μ2>0, and μ3>0, such that the following matrices inequalities hold:(60)Ξ1↔=Ω↔Π1T↔∗−P−1<0,Ξ2↔=Ω↔Π2T↔∗−P−1<0,where(61)Π1↔=PA˜+L˜+ℋ˜1f+ℋ˜1g1−λπ+1−λ1−πI+XP1−λπIPPE˜,Π2↔=PA˜+L˜+ℋ˜1f+ℋ˜1g1−λπ+1−λ1−πIP1−λπIPPE˜,Ω=Ω11↔Ω12↔000∗−2μ1IΩ23↔00∗∗−2μ2I00∗∗∗−2μ3I0∗∗∗∗−I,and Ω11↔=−P+μ3ℋ˜1gTℋ˜2g+ℋ˜2gTℋ˜1g, Ω12↔=μ1ℋ˜2f−ℋ˜1fT, and Ω23↔=μ2ℋ˜2g−ℋ˜1gT.
Then, the synchronization controller can be given by(62)K=P−1X,which renders the closed-loop synchronization error dynamics (47) to be bounded, and the synchronization error converges to ek2≤1−σβ1+δ/λminP.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. Theorems1 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:(63)A=−0.50.200.95,fx=tanhx1tanhx2,E=1001,τk=9+1+−1k2.
The inner-coupling matrix is set asΓ=0.3I, and the outer-coupling matrices L is given as follows:(64)L=−0.100.10−0.10.10.10.1−0.2.
Assumption1 is easily verified by using(65)ℋ1f=000−0.75,ℋ2f=0.2000,ℋ1g=ℋ1h=0.1000.1,ℋ2g=ℋ2h=0.2000.1.
The probabilities of the FDI attacks and DoS attacks are set asπ=0.10 and λ=0.20, respectively, and the FDI attacks and physical attacks have the following forms:(66)gx=tanh0.25x1ktanh0.15x2kT,hx=tanh0.04x1ktanh0.05x2kT.
The initial conditions of complex cyber-physical networks are set asx0=−1.51.5. Then, 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 suffers from cyber-attacks. The 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.
Letα1=0.5, α2=0.3, β1=0.12, β2=0.09, γ1=0.05, γ2=0.06, ρ=0.1, and σ=−2.4, respectively. Then, Equation (15a) is satisfied for any Vk. Applying Theorem 2 and solving LMI (38), the corresponding synchronization control gains matrices can be obtained as follows:(67)K11=0.51980.0360−0.17360.1125,K12=1.77720.0741−0.10201.5021,K21=0.34040.0291−0.18220.3764,K22=1.82850.0974−0.06611.4433,K31=0.42670.0206−0.08570.1124,K32=1.78090.0455−0.05511.4872.
According to the intermittent synchronization control mechanism (12), the intermittent synchronization controller uk=K1ek+K2ek−τk is activated when Lyapunov-like function. The 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 effective for the discrete-time 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 δ0=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 fluctuations than the method of [40], which shows that our control strategy can effectively reduce the negative impact of the mixed attacks and the input delays compared with the state feedback ones.Figure 4
Attack time of the DoS attacks.Figure 5
Attack time of the FDI attacks.Figure 6
Energy evolution of the physical attacks.Figure 7
Synchronization errorei1ki=1,2,3 trajectories of the uncontrolled.Figure 8
Synchronization errorei2ki=1,2,3 trajectories of the uncontrolled.Figure 9
Synchronization errorei1ki=1,2,3 trajectories of the controlled.Figure 10
Synchronization errorei2ki=1,2,3 trajectories of the controlled.Figure 11
Synchronization errorei1k,i=1,2,3 trajectories of the control method in [40].Figure 12
Synchronization errorei2k,i=1,2,3 trajectories of the control method in [40].Example 2.
Consider complex cyber-physical networks (1) consisting of three Chua’s chaotic circuits [41] with the following parameters:(68)A=−crc01−11014.280,fxk=0.02sinx100.01tanhx100.02cosx20000.03sinx3,E=I,c=9,r=27.
The inner-coupling matrix is set asΓ=0.517I, and the outer-coupling matrices L given as follows:(69)L=−0.100.10−0.10.10.10.1−0.2,and the time-delay boundaries are τm=3 and τM=4.
Assumption 1 is easily verified by using an improved directed crossover genetic algorithm based on multilayer mutation:(70)ℋ˜1f=00000000−0.75,ℋ˜2f=0.2000−0.20000.2,ℋ˜1g=ℋ˜1h=0.10000.10000.1,ℋ˜2g=ℋ˜2h=0.2000−0.20000.2.
We assume that the FDI attacks and physical attacks have the following form:(71)gxk=0.02sinx10000.02cosx20000.03sinx3,hxk=0.02sinx1−0.01cosx1−0.01tanhx1−0.02sinx200000.01sinx3.
Let the other parameters be the same as in Example1. Then, (15a) is satisfied for any Vk. Applying Theorem 2 and solving LMI (38), the corresponding synchronization control gains matrices can be obtained as follows:(72)K11=2.4698−9.0000−0.0000−1.00001.0000−1.0000−0.000014.28000.3596,K12=0.0214−0.00140.01360.00030.0006−0.0062−0.0290−0.0716−0.0138,K21=2.2055−9.00000.0000−1.00001.0786−1.0000−0.000014.2800−0.0262,K22=0.0213−0.00140.01350.00050.0011−0.0112−0.0216−0.0534−0.0102,K31=2.1922−9.0000−0.0000−1.00001.0811−1.0000−0.000014.2800−0.0567,K32=0.0218−0.00140.01380.00050.0011−0.0114−0.0221−0.0547−0.0105.
The initial condition of complex cyber-physical networks is set asxi0=1−11T. 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 effective for the discrete-time delayed complex cyber-physical networks with input delays and mixed attacks.Figure 13
Synchronization errorei1ki=1,2,3 trajectories of the controlled.Figure 14
Synchronization errorei2ki=1,2,3 trajectories of the controlled.Figure 15
Synchronization errorei3ki=1,2,3 trajectories of the controlled.


## 5. 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, sufficient 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 effectiveness 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. Specifically, 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.



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