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C. Liu, B. Jiang, X. F. Wang, H. L. Yang, and S. R. Xie, “Distributed fault-tolerant consensus tracking of multi-agent systems under cyber-attacks,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 6, pp. 1037–1048, Jun. 2022. doi: 10.1109/JAS.2022.105419
Citation: C. Liu, B. Jiang, X. F. Wang, H. L. Yang, and S. R. Xie, “Distributed fault-tolerant consensus tracking of multi-agent systems under cyber-attacks,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 6, pp. 1037–1048, Jun. 2022. doi: 10.1109/JAS.2022.105419

Distributed Fault-Tolerant Consensus Tracking of Multi-Agent Systems Under Cyber-Attacks

doi: 10.1109/JAS.2022.105419
Funds:  This work was supported by the National Key R&D Program of China (2018AAA0102804), National Natural Science Foundation of China (62020106003, 62103250, 61773201), Fundamental Research Funds for the Central Universities (NC2020002, NP2020103), Shanghai Sailing Program (21YF1414000)
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  • This paper investigates the distributed fault-tolerant consensus tracking problem of nonlinear multi-agent systems with general incipient and abrupt time-varying actuator faults under cyber-attacks. First, a decentralized unknown input observer is established to estimate relative states and actuator faults. Second, the estimated and output neighboring information is combined with distributed fault-tolerant consensus tracking controllers. Criteria of reaching leader-following exponential consensus tracking of multi-agent systems under both connectivity-maintained and connectivity-mixed attacks are derived with average dwelling time, attack frequency, and attack activation rate technique, respectively. Simulation example verifies the effectiveness of the fault-tolerant consensus tracking algorithm.

     

  • MULTI-AGENT systems (MASs) tracking has been successfully used in many important civil and military applications, such as rigid and flexible spacecrafts, autonomous unmanned aerial vehicles, and mobile robots [1]-[3]. A brief survey of consensus and coordination of MASs was given in [4]. The distributed consensus of the second-order, high-order, homogeneous, and heterogeneous MASs has attracted considerable interest recently [5]-[8]. State-based consensus of homogeneous MASs and output-based consensus of heterogeneous MASs were investigated with event-triggered strategy [7]. Thus, assigning an effective concept of the distributed consensus tracking of MASs is important but challenging.

    MASs are vulnerable to network security problems, such as hostile cyber-attacks [9], stealthy attack on remote estimation [10], switching and quantization of nonsmooth behaviors [11], and emergence of sensors on lossy and bandwidth-limited channels [12]. The distributed consensus tracking of MASs requires ideal and accurate information interconnection of individual agents through communication topologies, and the essential issue under cyber-attacks is destroyed. Consensus tracking of MASs under cyber-attacks has been extensively investigated [13]-[16]. The observer-based event-triggered consensus control was developed for discrete-time MASs in the presence of loss sensors and network attacks [14]. Specifically, the secure consensus and coordination of cyber-physical MASs under denial-of-service attacks [17], deception attacks [18] and random attacks [19] were investigated. However, for undirected or directed balanced topologies, the consensus tracking problem of MASs under cyber-attacks cannot be directly addressed via the existing methods based on graph theory. Secure consensus tracking was achieved via switching control strategy for linear MASs with the respective paralyzed and maintained networks caused by cyber-attacks [20]. Therefore, developing a novel consensus tracking control protocol for MASs under cyber-attacks via the switching control concept is necessary but challenging.

    The intrinsic state evolution of MASs is affected not only by handover topologies subjected to cyber-attacks but also actuator faults occurring on local agents [21]-[26]. Therefore, MASs need to work safely and healthily, and the fault-tolerant consensus tracking control (FCTC) is an effective method for realizing the desired local tolerance and global synchronization of the entire MASs. Development trends and methodologies of cooperative fault-tolerant control of MASs were briefly summarized in [21]. Partial loss of effectiveness and bias faults of leader-following MASs were addressed by the adaptive robust fault-tolerant control strategy [25], [27]. A distributed observer-based FCTC protocol was developed for linear MASs to achieve the global consensus through local estimations and compensate for multiple heterogeneous actuator faults [28]. However, the majority of studies focus on constant or time-varying abrupt actuator faults [22], [29] and ignore incipient and inchoate actuator faults [30]. Notably, a massive collapse of the entire MASs may be caused by individual agents broadcasting early incipient faults to their neighbors through the communication network and ultimately breaking the synchronization of the MASs. On the one hand, most studies on the FCTC scheme in a distributed manner are either based on neighborhood state information [22], [23], [28] or state-feedback control gains [25]. The FCTC research on the neighboring output information of MASs has been rarely explored. On the other hand, as a result of the existing distributed FCTC methodology, the observer-based strategy is rarely considered from the fault estimation (FE) system to the tolerance system to construct unmeasurable information of each follower [26]. Moreover, the consensus problem of MASs under cyber-attacks has been investigated [31], but few studies focus on the fault-tolerant consensus tracking community, especially for nonlinear MASs in the presence of general incipient and abrupt time-varying actuator faults as well as cyber-attacks. Actuator faults and denial-of-service attacks in the respective linear MASs [32] and nonlinear MASs [33] were simultaneously solved via the distributed fault-tolerant control method with the dwell time technique. The adaptive fault-tolerant control scheme was developed for MASs with deception attacks in the communication layer and actuator faults in the physical layer in a distributed manner [34]. Therefore, developing an observer-based estimation and tolerance algorithm associated with the distributed output measurements for nonlinear MASs with incipient and abrupt actuator faults is crucial for the tolerance problem of MASs under cyber-attacks.

    The distributed FCTC is proposed for nonlinear MASs with incipient and abrupt time-varying actuator faults under cyber-attacks. First, the actuator fault and cyber-attack models (connectivity-maintained, connectivity-paralyzed and connectivity-mixed attacks) are established. Second, the unknown input observer (UIO) [26], [29] in decentralized FE is developed to estimate the unmeasurable state and fault information. Finally, two sufficient conditions with attack frequency, activation rate, and average dwelling time (ADT) technique [17], [20], [33] are given to achieve the exponential consensus tracking of leader-following MASs.

    The major contributions of this study are summarized as follows. 1) In comparison with the cooperative consensus of MASs under independent cyber-attacks [13], [31] or the FCTC of MASs under traditional abrupt bounded faults [28], [29], this study attempts to combine network anti-attack and fault-tolerant control technologies effectively. By virtue of the switching topology theory [7], [20], it is a brand-new attempt to address the different types of constraints of self-dynamics (general incipient and abrupt time-varying actuator faults) in physical hierarchy and maintained/paralyzed links (connectivity-mixed attacks) in networked hierarchy in an integrated FE-based distributed FCTC framework. 2) Unlike local state information [22], [23] and direct state-feedback protocols [25], [33] of neighboring agents, a novel control structure is proposed with the effective combination advantages of local fault/state estimation in the decentralized UIO-based FE and adjacent output information in the distributed FCTC to ensure the exponential consensus tracking of nonlinear leader-following MASs. 3) For connectivity-maintained attacks, dual metrics (attack frequency and attack duration) in [17] are circumvented and replaced by a single constrained metric (ADT) to realize secure consensus tracking with less computing resources. For connectivity-mixed attacks, the ADT switching between stable and unstable sub-topologies [20], [33] is evaded and replaced by multi-constraint metrics (attack frequency, activation rate and upper and lower bounds of FCTC parameters) to deal with faults and cyber-attacks simultaneously.

    The remainder of this study is structured as follows. Preliminaries and problem formulation are presented in Section II. The decentralized FE and distributed FCTC designs for leader-following MASs under cyber-attacks are discussed in Sections III and IV, respectively. Simulations are provided in Sec-tion V to illustrate the effectiveness of the FCTC strategy. Finally, conclusions of this study are drawn in Section VI.

    Notations: R,N denote the sets of real and natural numbers, respectively, denotes the Kronecker product, λmax(),λmin() represent the maximum and minimum eigenvalues, respectively, min{},max{} represent the minimum and maximum elements, respectively, and He{X}=X+XT.

    A directed graph G is a pair (ν,ε,A), where ν={ν1,ν2,,νN} and is a nonempty finite set of nodes, εν×ν is a set of edges, and (νi,νj) is an edge that denotes an ordered pair of nodes νi,νj. Adjacency matrix of graph G is denoted by A=[aij]RN×N, where aij is the weight coefficient of edge (νi,νj) and aii=0,aij>0 if (νi,νj)ε; otherwise, aij=0. Node νj is the neighbor of node νi if (νi,νj)ε. The set of neighbors of node νi is denoted by Ni={νjν(νi,νj)ε}. The laplacian matrix is L=DA=[lij]RN×N, where D=[dii]RN×N is a diagonal matrix with dii=Nj=1aij.

    The diagonal matrix is denoted by B=diag{b1,,bN}, where bi is the information interaction between node νi and the leader. If bi=1, then node νi can reach the leader via a directed path; otherwise, bi=0. An information-exchange matrix is denoted by H=L+B for notational convenience.

    A class of nonlinear MASs with a group of N followers are considered. The dynamics of the ith follower with nonlinearities and actuator faults (i=1,,N) are modeled as

    {˙xi(t)=Axi(t)+B(ui(t)+fi(t))+ξ(xi(t),t)yi(t)=Cxi(t) (1)

    where xi(t)Rn,yi(t)Rp, and ui(t)Rm are the state, output, and input, respectively, ξ(xi(t),t)Rn denotes the state-dependent nonlinear perturbation, A,B, and C are known system matrices with appropriate dimensions, fi(t)=[fi1(t),,fim(t)]TRm is the incipient and abrupt time-varying actuator fault, and fis(t),s=1,,m is modeled in the following form:

    fis(t)={0,tTs(1eϵs(tTs))ˉfis,t>Ts,s=1,,m (2)

    where ˉfis is the sth row vector of unknown constant actuator fault bound, Ts is incipient or abrupt actuator fault occurrence instant, and ϵs is the unknown decay rate. The actuator fault is denoted as incipient actuator fault (slow decay rate) when ϵ_incϵs<ˉϵinc and abrupt actuator fault (quick decay rate) when ϵsˉϵinc, respectively.

    The leader’s dynamics (labeled as 0) are modeled as

    {˙x0(t)=Ax0(t)+Bu0(t)+ξ(x0(t),t)y0(t)=Cx0(t) (3)

    where x0(t)Rn,y0(t)Rp, and ξ(x0(t),t)Rn are the respective state, output, and nonlinearity of the leader. The leader’s input u0(t)Rm is devised as u0(t)=Kxx0(t), where KxRm×n denotes the derived state-feedback matrix.

    Assumption 1: The dynamics (A,B) and (A,C) are controllable and observable, respectively.

    Assumption 2: i) The incipient and abrupt actuator faults are differentiable after each fault occurrence instant. ii) The lower and upper bounds of incipient actuator fault are determined manually with the known positive scalars ϵ_inc and ˉϵinc.

    Assumption 3: The nonlinear function ξ() is satisfied with the following Lipschitz form:

    ξ(ω1(t),t)ξ(ω2(t),t)ρω1(t)ω2(t) (4)

    where ρ>0 is a Lipschitz constant for ω1(t),ω2(t),t0.

    Definition 1 [35]: The exponential consensus tracking control problem of the leader-following MASs in (1) and (3) is solved if tt0, there exists an amplitude μ>0 and a decay rate λ>0 such that

    ei(t)2μeλ(tt0)ei(t0)2 (5)

    where ei(t)=xi(t)x0(t) denotes the state tracking error.

    A switching signal γ(t):[0,)Γ={1,2,3,,r} is established to denote switchings in each individual topology under cyber-attacks. tk,k=0,1,2,, is the switching sequence over [t0,t). Assume that there exists an infinite sequence of uniformly bounded and non-overlapping time intervals [tk,tk+1), in which the graph is time-invariant. The set {G1,G2,,Gr} represents switching topologies Gγ(t), and {H1,H2,,Hr} represents information-switching matrices Hγ(t).

    The following kinds of cyber-attacks are defined:

    1) Connectivity-Maintained Attacks: The connectivity-maintained topology (removing or adding edges) still remains connected and contains a directed spanning tree with the leader as the root although MASs suffer from actuator faults. The switching signal is denoted by γ(t)Γm={1,,r} for switching topologies Gγ(t).

    2) Connectivity-Paralyzed Attacks: The topology under connectivity-paralyzed attacks (removing or adding edges remarkably) becomes disconnected without a directed spanning tree but can be recovered into connectivity with the repair mechanism although actuator faults exist. The switching signal is denoted by γ(t)Γp={1,,r} for switching topologies Gγ(t).

    3) Connectivity-Mixed Attacks: The switching signal γ(t)Γ=ΓmΓp={1,,q,q+1,,r},r2 for switching topologies {G1,,Gq,Gq+1,,Gr}, where Γm and Γp are sets of q connectivity-maintained and (rq) connectivity-paralyzed attacks, respectively.

    Definition 2 [20]: Denote the time instants kN and NΓ(t0,t)=NΓm(t0,t)+NΓp(t0,t),t>t00 for the switching signal γ(t)Γ=ΓmΓp as the number of γ(t) varies over [t0,t), where NΓm(t0,t) and NΓp(t0,t) are the numbers of connectivity-maintained/paralyzed attacks.

    Define the total activation durations of connectivity-maintained and connectivity-paralyzed attacks over [t0,t) as Tm(t0,t) and Tp(t0,t), respectively.

    {Tm(t0,t)=kN,γ(tk)Γm(tk+1tk)Tp(t0,t)=kN,γ(tk)Γp(tk+1tk). (6)

    Furthermore, if two scalars ˉNΓ>0 and TΓ>0 exist, such that NΓ(t0,t)ˉNΓ+tt0TΓ, then TΓ and ˉNΓ represent the ADT and chattering bounds, respectively.

    Definition 3 (Connectivity-Paralyzed Attack Frequency): For γ(t)Γp, denote FΓp(t0,t)=NΓp(t0,t)tt0 as the connectivity-paralyzed attack frequency over [t0,t).

    Definition 4 (Connectivity-Maintained Attack Activation Rate): For γ(t)Γm, denote RΓm(t0,t)=Tm(t0,t)tt0 as the connectivity-maintained attack activation rate over [t0,t).

    Definition 5 (Connectivity-Paralyzed Attack Activation Rate): For γ(t)Γp, denote RΓp(t0,t)=Tp(t0,t)tt0 as the connectivity-paralyzed attack activation rate over [t0,t).

    Lemma 1 [36]: Φγ(t)Hγ(t)+HTγ(t)Φγ(t) is a symmetric positive-definite matrix and Φγ(t)=diag{ϕ1γ(t),1,,ϕ1γ(t),N} is a diagonal positive-definite matrix, where ϕγ(t),i,i=1,,N is the element of matrix ϕγ(t)=HTγ(t)1N with the switching signal γ(t)Γ.

    Remark 1: Communication topologies may be subjected to frequent connectivity-maintained/-paralyzed attacks and eventually lead to poor consensus with destroyed links among agents. Compared with the arbitrary switching [5] or switching networks that satisfy jointly connected condition [7], connectivity-maintained attacks with a directed spanning tree and connectivity-paralyzed attacks without a spanning tree are modeled. It is required that the switching characteristics of dynamic switching topologies Gγ(t) are known and deterministic, and can be detected by some intelligent devices, while assuming that the upper or lower bounds of ADT, attack frequency and attack activation rate need to satisfy specific conditions in order to maintain consensus tracking performance of MASs.

    Remark 2: Compared with the linear state-dependent nonlinear dynamics [24] and quadratic nonlinear constraints [26], the modeled nonlinear perturbation ξ(xi(t),t),i=0,,N in Assumption 3 is constrained within the Lipchitz condition. Moreover, control structures consisting of FE, FCTC designs, and switching topologies caused by cyber-attacks are shown in Fig. 1. This study aims to develop a decentralized FE-based FCTC controller to achieve the leader-following exponential consensus tracking property in a distributed manner in (5) for MASs with general incipient and abrupt time-varying actuator faults in (2) as well as cyber-attacks in two cases.

    Figure  1.  The structure with FE, FCTC, and network hierarchies.

    A UIO-based FE scheme is developed for MASs to obtain the fault and state estimated information with decentralized structure in this section.

    The augmented state is denoted as ˉxi(t)=[xTi(t)fTi(t)]T and the uncertainty is denoted as ˉdi(t)=˙fi(t). Subsequently, the augmented dynamics of the ith follower are rewritten as follows:

    {˙ˉxi(t)=ˉAˉxi(t)+ˉBui(t)+ˉDˉdi(t)+ˉξ(A0ˉxi(t))yi(t)=ˉCˉxi(t) (7)

    where ˉξ(A0ˉxi(t))=[ξT(xi(t),t)01×m]T with A0=[In0n×m], and the system matrices ˉC=[C0p×m],

    ˉA=[AB0m×n0m],ˉB=[B0m],ˉD=[0n×mIm]. (8)

    Decentralized observers only need the output and input information of each agent, but do not require the information of their neighboring observers. Then, each augmented state ˉxi(t) can be obtained in the decentralized UIO-based FE design as follows:

    {˙zi(t)=Mzi(t)+Gui(t)+Jyi(t)+Θˉξ(A0ˆˉxi(t))ˆˉxi(t)=zi(t)+Hyi(t) (9)

    where zi(t)Rn+m is the state of the UIO, ˆˉxi(t)=[ˆxTi(t)ˆfTi(t)]T is the estimation of ˉxi(t) with the estimated state vector ˆxi(t)Rn and the estimated fault vector ˆfi(t)Rm, ˉξ(A0ˆˉxi(t))=[ξT(ˆxi(t),t)01×m]T is the estimated state and fault-dependent nonlinear perturbation, and M,G,J,Θ,H are designed matrices of proper dimensions.

    The estimation error is denoted as ˜ei(t)=ˉxi(t)ˆˉxi(t)=[eTxi(t)eTfi(t)]T with the state estimation error exi(t)=xi(t)ˆxi(t) and the fault estimation error efi(t)=fi(t)ˆfi(t). Next, define Θ=In+mHˉC and J=J1+J2, and the estimation error dynamics are derived as

    ˙˜ei(t)=(ΘˉAJ1ˉC)˜ei(t)+(ΘˉAJ1ˉCM)zi(t)+((ΘˉAJ1ˉC)HJ2)yi(t)+(ΘˉBG)ui(t)+Θ(ˉξ(A0ˉxi(t))ˉξ(A0ˆˉxi(t)))+ΘˉDˉdi(t). (10)

    Subsequently, with the following equality constraints of M,G,J1,J2, and H:

    {M=ΘˉAJ1ˉCJ2=(ΘˉAJ1ˉC)HG=ΘˉB (11)

    where M is a Hurwitz matrix, and the corresponding FE error dynamics are expressed as

    ˙˜ei(t)=M˜ei(t)+ΘˉDˉdi(t)+ΘΔˉξi(t) (12)

    where Δˉξi(t)=ˉξ(A0ˉxi(t),t)ˉξ(A0ˆˉxi(t),t).

    It then follows that:

     ˙˜e(t)=(INM)˜e(t)+(INΘˉD)ˉd(t)+(INΘ)Δˉξ(t) (13)

    where ˜e(t)=[˜eT1(t),,˜eTN(t)]T,ˉd(t)=[ˉdT1(t),,ˉdTN(t)]T and Δˉξ(t)=[ΔˉξT1(t),,ΔˉξTN(t)]T.

    Hence, the objective of decentralized FE design aims at devising H and J1 such that the FE error dynamics in (12) and (13) are robustly asymptotically stable.

    Remark 3: The ith UIO in (9) exists for the augmented dynamics (7) with ΘˉDˉdi(t)=0 and ΘΔˉξi(t)=0 if the decoupled FE error dynamics in (12) and (13) are asymptotically stable. Subsequently, since matrix M is Hurwitz, the estimation error dynamics are robustly stable, indicating that ˜ei(t) approaches to zero, i.e., limt˜ei(t)=0. Compared with the fixed-time distributed observer [6] and resilient adaptive distributed observer [32], a novel decentralized UIO-based FE protocol is constructed, under which the FE errors in the presence of cyber-attacks and actuator faults are verified to be asymptotically convergent. In addition, in order to design the decentralized FE hierarchy under the solvable equality constraints in (11), it is guaranteed that the devised UIO matrices M,G, and J2 can be calculated by the derived matrices H and J1, i.e., M=(In+mHˉC)ˉAJ1ˉC,G=(In+mHˉC)ˉB, and J2=((In+mHˉC)ˉAJ1ˉC)H.

    The distributed fault-tolerant consensus tracking controller of the ith agent of MASs under cyber-attacks is designed as

    ui(t)=Kˆˉxi(t)+δRΞi(yj(t),γ(t)),i=1,,N (14)

    where K=[KxIm] is the compensation matrix with the state-feedback matrix KxRm×n, R is the constant matrix, and δ is a positive scalar. Then, the output information Ξi(yj(t),γ(t)),j=0,1,,N,γ(t)Γ=ΓmΓp in a distributed manner is devised as

    Ξi(yj(t),γ(t))=Nj=1aγ(t)ij(yj(t)yi(t))+bγ(t)i(y0(t)yi(t)) (15)

    where aγ(t)ij is the (i,j)th entry of Aγ(t) with the graph Gγ(t), and bγ(t)i=1 when the ith agent can access the leader and bγ(t)i=0, otherwise.

    The state tracking error dynamics can be rewritten as

    ˙ei(t)=(ABKx)ei(t)+BK˜ei(t)+Δ˜ξi(t)+δBRC(Nj=1aγ(t)ij(ej(t)ei(t))bγ(t)iei(t)) (16)

    where Δ˜ξi(t)=ξ(xi(t),t)ξ(x0(t),t).

    Furthermore, it follows that:

    ˙e(t)=(IN(ABKx)δ(Hγ(t)BRC))e(t)+(INBK)˜e(t)+Δ˜ξ(t) (17)

    where the global vectors e(t)=[eT1(t),,eTN(t)]T,Δ˜ξ(t)=[Δ˜ξT1(t),,Δ˜ξTN(t)]T, and Hγ(t) denotes the nonsingular information-exchange matrix.

    The objective of Case I with proposing the distributed FCTC protocol in (14) and (15) is to determine Kx and R such that the leader-following consensus tracking is realized. The sufficient condition of FCTC scheme is proposed for MASs under connectivity-maintained attacks, i.e., γ(t)Γm.

    Theorem 1: Consider leader-following MASs with incipient and abrupt actuator faults in (2) under connectivity-maintained attacks. Given positive scalars τ1,τ3,σΓm, and a positive chattering bound ˉNΓm, the MASs with the distributed FCTC approach in (14) and (15) can realize the exponential consensus tracking performance, if there exists a symmetric positive-definite matrix PRn×n, matrix KxRm×n,HR(n+m)×p,J1R(n+m)×p, and positive scalars c,τ2,τ4 such that

    He((ABKx)P)+(λ1ˉϕcϕ_)BBT+ˉϕ(λ1In+ρ2P2)+τ3P<0 (18)
    1τ1(M+MT+ΘˉDˉDTΘT+ΘΘT+ρ2AT0A0)+KTK+τ4In+m<0 (19)
    τ4<2ϵ_incτ1 (20)
    τ2τ21τ4+2τ1ϵ_inc (21)

    where ϕ_=minϕγ(t),i,ˉϕ=maxϕγ(t),i,γ(t)Γm,i=1,,N and λ1=λmax(Φ2γ(t)) with the known Φγ(t) in Lemma 1.

    The constant matrix is denoted by R=BTP1(CTC)1CT and the positive scalar is constrained within δc|λ2| and λ2=λmin(Φγ(t)Hγ(t)+HTγ(t)Φγ(t)). The switching signal γ(t)Γm is selected for GΓm with the ADT TΓm satisfying

    TΓm>ln(ˉϕϕ_)τm (22)

    where τm=min{τ3,τ1τ4}, and the exponential consensus tracking problem is effectively solved with the following state tracking error expectation:

    E{ei(t)2}μΓmeλΓm(tt0)E{ei(t0)2} (23)

    with the initial maximum amplitude given by

    μΓm=eˉNΓmln(ˉϕϕ_)(λmax(ϕ1γ(t),iP1)+σΓm)λmin(ϕ1γ(t),iP1) (24)

    and the decay rate λΓm given by

    λΓm=τmln(ˉϕϕ_)TΓm. (25)

    Proof: Construct a Lyapunov function candidate V1(e(t),γ(t)) with a symmetric positive-definite matrix P and a positive scalar ϕγ(t),i,γ(t)Γm,i=1,,N as follows:

    V1(e(t),γ(t))=Ni=1eTi(t)ϕ1γ(t),iP1ei(t) (26)

    where ϕ1γ(t),i is the element of the diagonal matrix Φγ(t)=diag{ϕ1γ(t),1,,ϕ1γ(t),N} from Lemma 1.

    Denote ς(t)=[ςT1(t),,ςTN(t)]T with ςi(t)=P1ei(t). According to the gain matrix R=BTP1(CTC)1CT, the derivative of V1(e(t),γ(t)) in (26) is obtained as

    ˙V1(e(t),γ(t))=2Ni=1eTi(t)ϕ1γ(t),iP1BK˜ei(t)+ςT(t)(Φγ(t)((ABKx)P+P(ABKx)T))ς(t)δςT(t)((Φγ(t)Hγ(t)+HTγ(t)Φγ(t))BBT)ς(t)+ςT(t)(Φ2γ(t)In)ς(t)+Ni=1Δ˜ξTi(t)Δ˜ξi(t)ςT(t)(Φγ(t)((ABKx)P+P(ABKx)T+(λ1ˉϕcϕ_)BBT+ˉϕ(λ1In+ρ2P2)))ς(t)+˜eT(t)(INKTK)˜e(t) (27)

    where the bounds ϕ_=minϕγ(t),i,ˉϕ=maxϕγ(t),i,λ1= λmax(Φ2γ(t)),λ2=λmin(Φγ(t)Hγ(t)+HTγ(t)Φγ(t)) with γ(t)Γm, and the positive scalar δ is selected as δc|λ2|.

    Here, incipient and abrupt actuator faults are considered. The first-order derivative form is obtained with ˉdi(t)=˙fi(t)=[˙fTi1(t),,˙fTim(t)]T.

    ˙ˉdi(t)=¨fi(t)=diag(ϵ1,,ϵm)ˉdi(t) (28)

    where ˙ˉdis(t)=ϵsˉdis(t) and the MASs are subject to different time-varying actuator fault when ϵs>0 is selected.

    Subsequently, another Lyapunov function candidate V2(˜e(t),ˉd(t))=1τ1Ni=1˜eTi(t)˜ei(t)+1τ2Ni=1ˉdTi(t)ˉdi(t) is considered, where τ1 and τ2 are positive constants. The first-order derivative of V2(˜e(t),ˉd(t)) is given by

    ˙V2(˜e(t),ˉd(t))1τ1Ni=1˜eTi(t)(M+MT+ΘˉDˉDTΘT+ΘΘT+ρ2AT0A0)˜ei(t)+Ni=1(1τ12minϵsτ2)ˉdTi(t)ˉdi(t) (29)

    where minϵs=mins=1,,mϵs.

    Finally, consider the Lyapunov function V(e(t),˜e(t))= V1(e(t),γ(t))+V2(˜e(t),ˉd(t)). On the basis of the inequality constraints in (18) and (19), i.e., He((ABKx)P)+(λ1ˉϕcϕ_)BBT+ˉϕ(λ1In+ρ2P2)+τ3P<0 and KTK+1τ1(M+MT+ΘˉDˉDTΘT+ΘΘT+ρ2AT0A0)+τ4In+m<0, and minϵsϵ_inc in Assumption 2, it is obtained as

    ˙V(e(t),˜e(t))<τ3ςT(t)(Φγ(t)P)ς(t)τ4˜eT(t)˜e(t)+Ni=1(1τ12τ2ϵ_inc)ˉdTi(t)ˉdi(t)=τ3V1(e(t),γ(t))τ1τ4V2(˜e(t),ˉd(t))+(τ1τ4τ2+1τ12τ2ϵ_inc)Ni=1ˉdTi(t)ˉdi(t). (30)

    According to τ4<2ϵ_incτ1 in (20) and τ2τ21τ4+2τ1ϵ_inc in (21), τ1τ4τ2+1τ12τ2ϵ_inc0 is then derived. Hence, it is proved that ˙V(e(t),˜e(t))<τ3V1(e(t),γ(t))τ1τ4V2(˜e(t),ˉd(t))τmV(e(t),˜e(t)) with τm=min{τ3,τ1τ4}.

    Integrating both sides of (30) over t[tk,tk+1) yields

    V(e(t),˜e(t))eτm(ttk)V(e(tk),˜e(tk)). (31)

    It follows that ϕ_V1Ni=1eTi(t)P1ei(t)ˉϕV1 in (26). At each switching instant tk, it is obtained that V(e(tk),˜e(tk))ˉϕϕ_V(e(tk),˜e(tk)).

    In Definition 2, k=NΓm(t0,t)ˉNΓm+tt0TΓm is calculated with the ADT TΓm and chattering bound ˉNΓm. Then, the expression is given as

    V(e(t),˜e(t))ˉϕϕ_eτm(ttk)V(e(tk),˜e(tk))ˉϕϕ_eτm(ttk1)V(e(tk1),˜e(tk1))(ˉϕϕ_)keτm(tt0)V(e(t0),˜e(t0))eˉNΓmln(ˉϕϕ_)e(τmln(ˉϕϕ_)TΓm)(tt0)V(e(t0),˜e(t0)). (32)

    Next, with the definition of V(e(t),˜e(t)), it is derived as

    V(e(t),˜e(t))}λmin(ϕ1γ(t),iP1)ei(t)2 (33)

    and the expression with the initial time t0 is derived as

    V(e(t0),˜e(t0))(λmax(ϕ1γ(t),iP1)+maxi=1,,N(1τ1˜ei(t0)2+1τ2ˉdi(t0)2)mini=1,,Nei(t0)2)ei(t0)2. (34)

    Denoting ΛΓm=λmax(ϕ1γ(t),iP1)+σΓm with the appropriate positive scalar σΓm, it is finally given by

    ei(t)2μΓmeλΓm(tt0)ei(t0)2 (35)

    with

    μΓm=eˉNΓmln(ˉϕϕ_)ΛΓmλmin(ϕ1γ(t),iP1),λΓm=τmln(ˉϕϕ_)TΓm. (36)

    Hence, both ˙V(e(t),˜e(t))<τmV(e(t),˜e(t))<0 and the inequality in (35) imply that ei(t)0,˜ei(t)0,xi(t)x0(t),ˆxi(t)xi(t) and ˆfi(t)fi(t) as t+ when ADT TΓm is constrained within (22). The FE and state tracking error dynamics of MASs with time-varying faults are asymptotically stable. Moreover, the exponential consensus tracking control issue under connectivity-maintained attacks (γ(t)Γm) is solved through the proposed FCTC scheme.

    Remark 4: In the presence of complicated actuator faults and connectivity-maintained attacks, it is difficult for the existing fault-tolerant methods [22], [25], [26] to guarantee the consensus tracking property of MASs, especially with the coupling characteristics between the estimation and tolerance systems. Thus, a decentralized UIO in (9) is developed to estimate the relative states/actuator faults, and the estimated information is then applied to the novel distributed FCTC protocol in (14) to compensate for Lipschitz nonlinearity, actuator faults, and connectivity-maintained attacks and subsequently achieve the overall consensus tracking. Unlike the decentralized adaptive fault-tolerant tracking control via local estimations [28], these FCTC methods, which depend on the offset item Kˆˉxi(t) and rely on switching rules γ(t)Γm in this study, can be utilized in a distributed structure. This distributed structure has advantages in computation, robustness and communication cost, and only requires the adjacent information Ξi(yj(t),γ(t)),j=0,1,,N.

    The objective of Case II with designing the fully distributed FCTC protocol in (14) and (15) is to determine Kx and R such that the leader-following consensus tracking is realized. The following sufficient condition is developed for MASs under connectivity-mixed attacks, i.e., γ(t)Γ=ΓmΓp.

    Theorem 2: Consider leader-following MASs with incipient and abrupt actuator faults in (2) under connectivity-mixed attacks. Given positive scalars τ1,˜τ3, and σΓ, the MASs in (1) and (3) with the distributed FCTC approach in (14), (15) can solve the exponential consensus tracking problem, if there exists a symmetric positive-definite matrix PRn×n, matrix KxRm×n,HR(n+m)×p,J1R(n+m)×p, and positive scalars ˜c,τ2,˜τ4 such that

    He((ABKx)P)+(1+˜c)BBT+In+ρ2P2˜τ3P<0 (37)
    1τ1(M+MT+ΘˉDˉDTΘT+ΘΘT+ρ2AT0A0)+KTK˜τ4In+m<0 (38)
    τ2τ21˜τ4+2τ1ϵ_inc (39)

    and the positive scalar δ in (14) satisfies

    c|λ2|δ˜c|λ3| (40)

    and for a positive decay rate λΓ(0,λΓ), the connectivity-paralyzed attack frequency FΓp(t0,t) satisfies

    FΓp(t0,t)λΓλΓ2ln(ˉϕϕ_) (41)

    and for a scalar λΓ(0,τm), the connectivity-maintained attack activation rate RΓm(t0,t) and the connectivity-paralyzed attack activation rate RΓp(t0,t) satisfy

    RΓm(t0,t)τp+λΓτm+τp,RΓp(t0,t)τmλΓτm+τp (42)

    where λ3=λmax(Hγ(t)+HTγ(t)) and τp=max{˜τ3,τ1˜τ4} with the switching signal γ(t)Γp.

    Thus, the consensus tracking performance is exponentially achieved with the state tracking error expectation

    E{ei(t)2}μΓeλΓ(tt0)E{ei(t0)2} (43)

    with the initial maximum amplitude given by

    μΓ=max{λmax(ϕ1γ(t),iP1),λmax(P1)}+σΓmin{λmin(ϕ1γ(t),iP1),λmin(P1)}. (44)

    Proof: Construct the Lyapunov function candidate ˜V1(e(t),γ(t)) with a positive scalar ϕγ(t),i,γ(t)Γ,

    ˜V1(e(t),γ(t))={eT(t)(Φγ(t)P1)e(t),γ(t)ΓmeT(t)(INP1)e(t),γ(t)Γp. (45)

    Then, the Lyapunov function candidate ˜V(t)=˜V(e(t),˜e(t))=˜V1(e(t),γ(t))+V2(˜e(t),ˉd(t)) is chosen. When the MASs suffer from q connectivity-maintained attacks, i.e., γ(t)Γm. According to the same proof process in Theorem 1, it is obtained that ˙˜V(e(t),˜e(t))<τ3˜V1(e(t),γ(t))τ1τ4V2(˜e(t),ˉd(t))τm˜V(e(t),˜e(t)) with the scalar τm=min{τ3,τ1τ4}.

    When the MASs suffer from (rq) connectivity-paralyzed attacks, i.e., γ(t)Γp, the first-order derivative of the function ˜V1(e(t),γ(t)) in (45) is given by

    ˙˜V1(e(t),γ(t))=ςT(t)(INHe((ABKx)P)δHe(Hγ(t))BBT)ς(t)+2ςT(t)(INBK)˜e(t)+2ςT(t)Δ˜ξ(t)ςT(t)(IN(He((ABKx)P)+(1+˜c)BBT+In+ρ2P2))ς(t)+˜eT(t)(INKTK)˜e(t) (46)

    where δ˜c|λ3| with λ3=λmax(Hγ(t)+HTγ(t)),γ(t)Γp.

    According to the constraints in (37)–(39), i.e., He((ABKx)P)+(1+˜c)BBT+In+ρ2P2˜τ3P<0,KTK+1τ1(M+MT+ΘˉDˉDTΘT+ ΘΘT+ρ2AT0A0)˜τ4In+m<0 and τ2τ21˜τ4+2τ1ϵ_inc, the first-order derivative of ˜V(e(t),˜e(t)) is given as ˙˜V(e(t),˜e(t))<˜τ3טV1(e(t),γ(t))+τ1˜τ4V2(˜e(t),ˉd(t))+ˉdT(t)(IN(τ1˜τ4τ2+1τ12τ2ϵ_inc))ˉd(t). Thus, it follows that ˙˜V(e(t),˜e(t))<˜τ3˜V1(e(t),γ(t))+τ1˜τ4V2(˜e(t),ˉd(t))τp˜V(e(t),˜e(t)) with τp=max{˜τ3,τ1˜τ4}.

    Integrating both sides of ˙˜V(e(t),˜e(t)) with γ(t)Γ=ΓmΓp over t[tk,tk+1) yields

    ˜V(t){eτm(ttk)˜V(e(tk),˜e(tk)),γ(t)Γmeτp(ttk)˜V(e(tk),˜e(tk)),γ(t)Γp. (47)

    It follows that k=NΓ(t0,t)2NΓp(t0,t) is obtained with the number of the connectivity-paralyzed attacks NΓp(t0,t) from Definition 2. Subsequently, ˜V(e(t),˜e(t)) is given by

    ˜V(e(t),˜e(t))(ˉϕϕ_)keτpTp(t0,t)τmTm(t0,t)˜V(e(t0),˜e(t0))e2NΓp(t0,t)ln(ˉϕϕ_)+τpTp(t0,t)τmTm(t0,t)˜V(e(t0),˜e(t0)). (48)

    On the basis of the inequality constraint in (41) of the connectivity-paralyzed attack frequency FΓp in Definition 3, 2NΓp(t0,t)ln(ˉϕϕ_)(λΓλΓ)(tt0) is derived. Furthermore, with the aid of Definitions 4 and 5 of the connectivity-maintained/paralyzed attack activation rates RΓm(t0,t) and RΓp(t0,t) in (42), τpTp(t0,t)τmTm(t0,t)λΓ(tt0) is then obtained with Tm(t0,t)+Tp(t0,t)=tt0.

    Thus, when the connectivity-mixed attacks are considered, i.e., γ(t)Γ=ΓmΓp, it follows that:

    ˜V(e(t),˜e(t))eλΓ(tt0)˜V(e(t0),˜e(t0)) (49)

    and the state tracking error is given as

    ei(t)2˜V(e(t),˜e(t))min{λmin(ϕ1γ(t),iP1),λmin(P1)}μΓeλΓ(tt0)ei(t0)2 (50)

    with the chosen positive scalar σΓ and μΓ expressed in (44).

    Hence, the state tracking error expectation in (50) indicates that the consensus tracking performance under connectivity-mixed attacks (γ(t)Γ) is exponentially realized through the proposed FCTC scheme.

    Remark 5: Compared with the distributed FCTC algorithm with ADT technique in Theorem 1, the success of connectivity-maintained and connectivity-paralyzed activation ratio of connectivity-mixed attacks is considered to ensure the tolerance and represent the connectivity and paralyzation of attacks passing through protection or repair devices subject to network fluctuations and limited resources. Unlike the fixed graph [7], [23], the preset switching graph is more general and more extensively applied in addressing network interruption [33]. The proposed distributed FCTC algorithm can be combined with the switching mechanism [20] to ensure that each agent can still realize the consensus tracking of MASs with connectivity-mixed attacks provided that attack frequency and activation rates in Theorem 2 satisfy certain conditions in (41) and (42). Furthermore, compared with existing studies on the FCTC design related to the state measurement error [17], the captured output information is added on the basis of the selected parameters in (40) to avoid high-precision measuring device.

    In this section, a network of five single-link manipulators with revolute joints is put forward to verify the effectiveness of the proposed distributed FCTC design under cyber-attacks.

    The model of the single-link manipulator with a flexible joint actuated by DC motor is given with the following state vector xi=[xTi1xTi2xTi3xTi4]T=[θmiωmiθliωli]T, where {\theta}_{mi},{\omega}_{mi}, {\theta}_{li}, and {\omega}_{li} denote the angular rotation of the motor, the angular velocity of the motor, the angular position of the link, and the angular velocity of the link, respectively [37].

    \left\{\begin{aligned} &\dot{\theta}_{mi} = \omega_{mi}\\ &\dot{\omega}_{mi} = \frac{k_s}{J_m}\left(\theta_{li}-\theta_{mi}\right)-\frac{l_{\rm{link}}}{J_m}\omega_{mi}+\frac{k_{\tau}}{J_m}u_{i}\\ &\dot{\theta}_{li} = \omega_{li}\\ &\dot{\omega}_{li} = -\frac{\eta k_s}{J_l}\left(\theta_{li}-\theta_{mi}\right)-\frac{\eta mgh}{J_l}{\rm sin}\left(\theta_{li}\right). \end{aligned}\right. (51)

    The physical meanings and values of the parameters in the single-link manipulator are illustrated in Table I. Furthermore, the state-dependent nonlinear perturbation \xi(x_i,t) is given as \xi(x_i,t) = [0\;0\;0\;\frac{-qmgh}{J_l}{\rm sin}({\theta}_{li})]^T , and it follows that \xi(x_i,t) satisfies with the Lipschitz condition.

    Table  I.  The Physical Parameters of the Single-Link Manipulator [37]
    Parameter Physical meaning Value/Unit
    J_m Inertia of the motor 0.0037\;{\rm kg.m^2}
    J_l Inertia of the link 0.0093\;{\rm kg.m^2}
    k_s Torsional spring constant 0.18\;{\rm Nm/rad}
    k_{\tau} Amplifier gain 0.08\;{\rm Nm/V}
    \eta Transformation coefficient 0.1
    l_{\rm{link} } Length of the link 0.31\;{\rm m}
    h Center of gravity height 0.015\;{\rm m}
    m Point mass of the arm 0.139\;{\rm kg}
    g Gravity constant 9.8\;{\rm m/s^2}
     | Show Table
    DownLoad: CSV

    To demonstrate the efficiency of the proposed algorithms in Theorems 1 and 2, the incipient and abrupt time-varying actuator faults f_i(t),i = 1,\ldots,5 in the control input channel in each single-link manipulator system are given as

    \begin{split} &f_1(t) = f_3(t) = f_5(t) = 0,f_2(t) = 1-e^{-0.05t}\\ &f_4(t) = \left\{\begin{aligned} &1-e^{-0.05t},&t\le 20\\ &1-e^{-0.5t},& t>20. \end{aligned}\right. \end{split} (52)

    Simulation parameters are set as \epsilon_1 = 0.05,T_1 = 0,\bar{f}_{21} = 1 for the second faulty manipulator and \epsilon_1 = 0.05\;(t\le 20\;{\rm s}), \epsilon_1 = 0.5\;(t > 20\;{\rm s}),T_1 = 20,\bar{f}_{41} = 1 for the fourth faulty one. The lower and upper bounds of the incipient actuator fault in each single-link manipulator are designed as \underline{\epsilon}_{\rm inc} = 0.005, \bar{\epsilon}_{\rm inc} = 0.1, and the Lipschitz constant is given as \, \rho = 0.2197 . The preset scalars are set as \tau_1 = 0.0065,\;\tau_3 = 2.32,\;\sigma_{\Gamma_m} = 1.68 for Theorem 1 and \tau_1 = 0.037,\tilde{\tau}_3 = 5.6,\;\sigma_{\Gamma} = 3.75 for Theorem 2.

    The switching topologies under connectivity-maintained attacks are shown in Fig. 2 and the switching topologies under connectivity-mixed attacks are shown in Fig. 3, in which the second single-link manipulator fails with the incipient actuator fault and the fourth manipulator suffers from the combined incipient and abrupt faults at each fault occurrence time, i.e., T_1 = 0\;{\rm s} and 20\;{\rm s}. The gain matrices are derived by solving the algorithms in Theorems 1 and 2, i.e., K_x = [-2.2463\; -2.7938\;2.2453\;0] in Case I with connectivity-maintained attacks and K_x = [0.2882\;0.1547\;2.39\;-0.9016] in Case II with connectivity-mixed attacks. Furthermore, the switching signal shows the switching of multi-manipulators \mathcal{G}_1,\mathcal{G}_2,\mathcal{G}_3 , and \mathcal{G}_4 at each cyber-attack occurring time instants, i.e., t = 10\;{\rm s},30\;{\rm s}, and 60\;{\rm s} in Fig. 4.

    Figure  2.  The switching topologies under connectivity-maintained attacks.
    Figure  3.  The switching topologies under connectivity-mixed attacks.
    Figure  4.  Topology switching signal.

    In the presence of the incipient and abrupt actuator faults, the results in Figs. 5-8 under connectivity-maintained attacks (Case I) and in Figs. 9-12 under connectivity-mixed attacks (Case II) validate the effectiveness of the proposed distributed FCTC algorithms in Theorems 1 and 2. The respective angular rotation tracking x_{i1}-x_{01} and the angular velocity tracking x_{i2}-x_{02},i = 1,\ldots,5 of each motor under connectivity-maintained attacks are depicted in Figs. 5 and 6. The respective angular position tracking x_{i3}-x_{03} and the angular velocity tracking x_{i4}-x_{04},i = 1,\ldots,5 of each link in the single-link manipulator system under connectivity-maintained attacks are shown in Figs. 7 and 8. The incipient actuator fault of the second manipulator has influenced the state tracking errors, while the abrupt fault of the fourth manipulator in 20\;{\rm s} has the greatest influence on the angular velocity tracking of the motor and the least influence on the angular position tracking of the link. Due to the connectivity-maintained attacks in 10\;{\rm s},\;30\;{\rm s}, and 60\;{\rm s} of Case I, severe oscillation convergence with different amplitudes is formed with the topological interconnection, and finally the consensus of state tracking errors is achieved with the distributed FCTC algorithm. Furthermore, the respective angular rotation tracking x_{i1}-x_{01} , the angular velocity tracking x_{i2}-x_{02} of each motor, the angular position tracking x_{i3}-x_{03} and the angular velocity tracking x_{i4}-x_{04},i = 1,\ldots,5 of each link under connectivity-mixed attacks are illustrated in Figs. 9-12. All these figures show that the angular position and velocity tracking can converge asymptotically in spite of incipient and abrupt faults in the second and fourth manipulators. It is worth noting that despite the connectivity-mixed attacks in Case II, the dynamic convergence duration time of the single-link manipulator system is prolonged in 10\;{\rm s},30\;{\rm s} , and 60\;{\rm s} under the action of topological recovery mechanism. Multiple switchings in 10\;{\rm s} 30\;{\rm s} and 30\;{\rm s} 60\;{\rm s} also cause angular and angular velocity tracking error fluctuations of each motor and link, which can gradually converge by the proposed distributed FCTC algorithm.

    Figure  5.  The angular rotation tracking of each motor: x_{i1}-x_{01},i = 1,\ldots,5 under connectivity-maintained attacks.
    Figure  6.  The angular velocity tracking of each motor: x_{i2}-x_{02},i = 1,\ldots,5 under connectivity-maintained attacks.
    Figure  7.  The angular position tracking of each link: x_{i3}-x_{03},i = 1,\ldots,5 under connectivity-maintained attacks.
    Figure  8.  The angular velocity tracking of each link: x_{i4}-x_{04},i = 1,\ldots,5 under connectivity-maintained attacks.
    Figure  9.  The angular rotation tracking of each motor: x_{i1}-x_{01},i = 1,\ldots,5 under connectivity-mixed attacks.
    Figure  10.  The angular velocity tracking of each motor: x_{i2}-x_{02},i=1,\ldots,5 under connectivity-mixed attacks.
    Figure  11.  The angular position tracking of each link: x_{i3}-x_{03},i=1,\ldots,5 under connectivity-mixed attacks.
    Figure  12.  The angular velocity tracking of each link: x_{i4}-x_{04},i = 1,\ldots,5 under connectivity-mixed attacks.

    Fig. 13 shows the rated and estimated incipient faults of the second manipulator and the combined incipient and abrupt faults of the fourth manipulator. Compared with our previous hierarchical FE and fault-tolerant control scheme [26], the proposed FE scheme in the distributed FCTC algorithm has smaller estimation error and better robustness at each cyber-attack occurring time instants. Fig. 14 depicts the rated and estimated values under the considered connectivity-maintained attacks without fault, the connectivity-mixed attacks with incipient fault and abrupt fault, respectively. In this case, our previous FE/fault-tolerant algorithm [26] cannot achieve an excellent estimation effect, and the phenomenon of continuous non-dissipation oscillation or deviation occurs under the considered connectivity-mixed attacks.

    Figure  13.  The rated fault, the estimated fault [26], and the estimated fault (FCTC algorithm) under cyber-attacks.
    Figure  14.  The comparisons of the considered connectivity-maintained attacks without fault, connectivity-mixed attacks with incipient fault and abrupt fault.

    A novel UIO-based FE and FCTC method was proposed in this study to ensure that nonlinear leader-following MASs can realize the exponential consensus tracking objective regardless of incipient/abrupt actuator faults and cyber-attacks. Distributed FCTC protocols based on fault and state estimations from the decentralized FE hierarchy and the relative adjacent output information of each individual agent were proposed in connectivity-maintained and connectivity-mixed attack cases. Two sufficient criteria utilizing the ADT, attack frequency, and attack activation rate method were proposed to ensure the prescribed consensus tracking performance. Multistep calculations were demonstrated to derive parameters in decentralized FE and distributed FCTC algorithms for both cases. Simulations proved the effectiveness of the proposed FCTC scheme. Future studies could fruitfully explore the consensus tracking issue of heterogeneous nonlinear MASs subject to complicated time-varying actuator and sensor faults in physical hierarchy and hostile DoS attacks in networked hierarchy further by the improved distributed event-triggered FCTC scheme with a substantial reduction in computational resources when message transmission fails intermittently.

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    Highlights

    • This study attempts to combine network anti-attack and fault-tolerant control technologies effectively
    • It is a brand-new attempt to address the different types of constraints of self-dynamics in physical hierarchy and maintained/paralyzed links in networked hierarchy
    • A novel control structure is proposed with the effective combination of local fault/state estimation in decentralized FE and adjacent output information in distributed FCTC

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