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

I. Introduction

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: $ \mathbb{R},\mathbb{N} $ denote the sets of real and natural numbers, respectively, $ \otimes $ denotes the Kronecker product, $ \lambda_{\rm max}(\cdot),\lambda_{\rm min}(\cdot) $ represent the maximum and minimum eigenvalues, respectively, $ {\rm min}\{\cdot\},{\rm max}\{\cdot\} $ represent the minimum and maximum elements, respectively, and $ {\rm He}\{X\} = X+X^T $.

II. Preliminaries and Problem Formulation

A Graph Theory

A directed graph $ \mathcal{G} $ is a pair $ (\nu,\varepsilon,\mathcal{A}) $, where $\nu = \{\nu_1, \nu_2,\ldots,\nu_N\}$ and is a nonempty finite set of nodes, $ \varepsilon\subseteq\nu\times\nu $ is a set of edges, and $ (\nu_i,\nu_j) $ is an edge that denotes an ordered pair of nodes $ \nu_i,\nu_j $. Adjacency matrix of graph $ \mathcal{G} $ is denoted by $ \mathcal{A} = [a_{ij}]\in \mathbb{R}^{N\times N} $, where $ a_{ij} $ is the weight coefficient of edge $ (\nu_i,\nu_j) $ and $ a_{ii} = 0,a_{ij}>0 $ if $ (\nu_i,\nu_j)\in\varepsilon $; otherwise, $ a_{ij} = 0 $. Node $ \nu_j $ is the neighbor of node $ \nu_i $ if $ (\nu_i,\nu_j)\in\varepsilon $. The set of neighbors of node $ \nu_i $ is denoted by $ \mathcal{N}_i = \{\nu_j\in\nu\mid(\nu_i,\nu_j)\in\varepsilon\} $. The laplacian matrix is $\mathcal{L} = \mathcal{D}-\mathcal{A} = [l_{ij}]\in \mathbb{R}^{N\times N}$, where $ \mathcal{D} = [d_{ii}]\in \mathbb{R}^{N\times N} $ is a diagonal matrix with $ d_{ii} = \sum_{j = 1}^{N}a_{ij} $.

The diagonal matrix is denoted by $\mathcal{B} = {\rm diag}\{b_1,\ldots,b_N\}$, where $ b_i $ is the information interaction between node $ \nu_i $ and the leader. If $ b_i = 1 $, then node $ \nu_i $ can reach the leader via a directed path; otherwise, $ b_i = 0 $. An information-exchange matrix is denoted by $ \mathcal{H} = \mathcal{L}+\mathcal{B} $ for notational convenience.

B MASs With Actuator Fault Modeling

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

where $ x_i(t)\in \mathbb{R}^{n},y_i(t)\in \mathbb{R}^{p} $, and $ u_i(t)\in \mathbb{R}^{m} $ are the state, output, and input, respectively, $ \xi(x_i(t),t)\in \mathbb{R}^{n} $ denotes the state-dependent nonlinear perturbation, $ A, B $, and $ C $ are known system matrices with appropriate dimensions, $f_i(t) = [f_{i1}(t),\ldots, f_{im}(t)]^T\in \mathbb{R}^{m}$ is the incipient and abrupt time-varying actuator fault, and $f_{is}(t),s = 1,\ldots,m$ is modeled in the following form:

where $ \bar{f}_{is} $ is the $ s $th row vector of unknown constant actuator fault bound, $ T_s $ is incipient or abrupt actuator fault occurrence instant, and $ \epsilon_s $ is the unknown decay rate. The actuator fault is denoted as incipient actuator fault (slow decay rate) when $ \underline{\epsilon}_{\rm inc}\le\epsilon_s<\bar{\epsilon}_{\rm inc} $ and abrupt actuator fault (quick decay rate) when $ \epsilon_s\ge\bar{\epsilon}_{\rm inc} $, respectively.

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

where $ x_0(t)\in \mathbb{R}^{n},y_0(t)\in \mathbb{R}^{p} $, and $ \xi(x_0(t),t)\in \mathbb{R}^{n} $ are the respective state, output, and nonlinearity of the leader. The leader’s input $ u_0(t)\in \mathbb{R}^{m} $ is devised as $ u_0(t) = -K_xx_0(t) $, where $ K_x\in \mathbb{R}^{m\times 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 $ \underline{\epsilon}_{\rm inc} $ and $ \bar{\epsilon}_{\rm inc} $.

Assumption 3: The nonlinear function $ \xi(\cdot) $ is satisfied with the following Lipschitz form:

where $ \rho>0 $ is a Lipschitz constant for $ \forall \omega_1(t),\omega_2(t),t\ge 0 $.

Definition 1 [35]: The exponential consensus tracking control problem of the leader-following MASs in (1) and (3) is solved if $ \forall t\ge t_0 $, there exists an amplitude $ \mu>0 $ and a decay rate $ \lambda>0 $ such that

where $ e_i(t) = x_i(t)-x_0(t) $ denotes the state tracking error.

C Cyber-Attack Modeling

A switching signal $ \gamma(t):\left[0,\infty\right)\rightarrow \Gamma = \{1,2,3,\ldots,r\} $ is established to denote switchings in each individual topology under cyber-attacks. $ t_k,k = 0,1,2,\ldots, $ is the switching sequence over $ [t_0,t) $. Assume that there exists an infinite sequence of uniformly bounded and non-overlapping time intervals $ [t_k,t_{k+1}) $, in which the graph is time-invariant. The set $ \{\mathcal{G}_1,\mathcal{G}_2,\ldots,\mathcal{G}_r\} $ represents switching topologies $ \mathcal{G}_{\gamma(t)} $, and $ \{\mathcal{H}_1, \mathcal{H}_2,\ldots,\mathcal{H}_r\} $ represents information-switching matrices $ \mathcal{H}_{\gamma(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 $ \gamma(t)\in\Gamma_m = \{1,\ldots,r\} $ for switching topologies $ \mathcal{G}_{\gamma(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 $ \gamma(t)\in\Gamma_p = \{1,\ldots,r\} $ for switching topologies $ \mathcal{G}_{\gamma(t)} $.

3) Connectivity-Mixed Attacks: The switching signal $ \gamma(t)\in\Gamma = \Gamma_m\cup\Gamma_p = \{1,\ldots,q,q+1,\ldots,r\},r\ge 2 $ for switching topologies $ \{\mathcal{G}_1,\ldots,\mathcal{G}_q,\mathcal{G}_{q+1},\ldots,\mathcal{G}_r\} $, where $ \Gamma_m $ and $ \Gamma_p $ are sets of $ q $ connectivity-maintained and $ (r-q) $ connectivity-paralyzed attacks, respectively.

Definition 2 [20]: Denote the time instants $ k\in\mathbb{N} $ and $ {N}_{\Gamma}(t_0,t) = N_{\Gamma_m}(t_0,t)+N_{\Gamma_p}(t_0,t),\forall t>t_0\ge 0 $ for the switching signal $ \gamma(t)\in\Gamma = \Gamma_m\cup\Gamma_p $ as the number of $ \gamma(t) $ varies over $ [t_0,t) $, where $ N_{\Gamma_m}(t_0,t) $ and $ N_{\Gamma_p}(t_0,t) $ are the numbers of connectivity-maintained/paralyzed attacks.

Define the total activation durations of connectivity-maintained and connectivity-paralyzed attacks over $ [t_0,t) $ as $ T_m(t_0,t) $ and $ T_p(t_0,t) $, respectively.

Furthermore, if two scalars $ \bar{N}_{\Gamma}>0 $ and $ T_{\Gamma}>0 $ exist, such that $ {N}_{\Gamma}(t_0,t)\le\bar{N}_{\Gamma}+\frac{t-t_0}{T_{\Gamma}} $, then $ T_{\Gamma} $ and $ \bar{N}_{\Gamma} $ represent the ADT and chattering bounds, respectively.

Definition 3 (Connectivity-Paralyzed Attack Frequency): For $ \gamma(t)\in\Gamma_p $, denote $ \mathcal{F}_{\Gamma_p}(t_0,t) = \frac{N_{\Gamma_p}(t_0,t)}{t-t_0} $ as the connectivity-paralyzed attack frequency over $ [t_0,t) $.

Definition 4 (Connectivity-Maintained Attack Activation Rate): For $ \gamma(t)\in\Gamma_m $, denote $ \mathcal{R}_{\Gamma_m}(t_0,t) = \frac{T_m(t_0,t)}{t-t_0} $ as the connectivity-maintained attack activation rate over $ [t_0,t) $.

Definition 5 (Connectivity-Paralyzed Attack Activation Rate): For $ \gamma(t)\in\Gamma_p $, denote $ \mathcal{R}_{\Gamma_p}(t_0,t) = \frac{T_p(t_0,t)}{t-t_0} $ as the connectivity-paralyzed attack activation rate over $ [t_0,t) $.

Lemma 1 [36]: $ \Phi_{\gamma(t)}\mathcal{H}_{\gamma(t)}+\mathcal{H}_{\gamma(t)}^T\Phi_{\gamma(t)} $ is a symmetric positive-definite matrix and $ \Phi_{\gamma(t)} = {\rm diag}\{\phi_{\gamma(t),1}^{-1},\ldots,\phi_{\gamma(t),N}^{-1}\} $ is a diagonal positive-definite matrix, where $ \phi_{\gamma(t),i},i = 1,\ldots,N $ is the element of matrix $ \phi_{\gamma(t)} = \mathcal{H}_{\gamma(t)}^{-T}1_N $ with the switching signal $ \gamma(t)\in\Gamma $.

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 $ \mathcal{G}_{\gamma(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 $ \xi(x_i(t),t),i = 0,\ldots,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.

DownLoad: Full-Size Img PowerPoint

III. Decentralized Fault Estimation Design

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 $ {\bar{x}}_i(t) = [x_i^T(t)\;f_i^T(t)]^T $ and the uncertainty is denoted as $ \bar{d}_i(t) = \dot{f}_i(t) $. Subsequently, the augmented dynamics of the $ i $th follower are rewritten as follows:

where $ \bar{\xi}(A_0\bar{x}_i(t)) = [\xi^T(x_i(t),t)\quad 0_{1\times m}]^T $ with $ A_0 = [I_n\quad 0_{n\times m}] $, and the system matrices $ \bar{C} = [C\quad 0_{p\times m}] $,

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 $ {\bar{x}}_i(t) $ can be obtained in the decentralized UIO-based FE design as follows:

where ${\textit{z}}_i(t)\in \mathbb{R}^{n+m}$ is the state of the UIO, $\hat{\bar{x}}_i(t) = [\hat{x}_i^T(t)\; \hat{f}_i^T(t)]^T$ is the estimation of $ \bar{x}_i(t) $ with the estimated state vector $ \hat{x}_i(t)\in \mathbb{R}^{n} $ and the estimated fault vector $ \hat{f}_i(t)\in \mathbb{R}^{m} $, $\bar{\xi}(A_0\hat{\bar{x}}_i(t)) = [\xi^T(\hat{x}_i(t),t)\; 0_{1\times m}]^T$ is the estimated state and fault-dependent nonlinear perturbation, and $ M,G,J,\Theta,H $ are designed matrices of proper dimensions.

The estimation error is denoted as $\tilde{e}_i(t) = \bar{x}_i(t)- \hat{\bar{x}}_i(t) = [e_{xi}^T(t)\; e_{fi}^T(t)]^T$ with the state estimation error $ e_{xi}(t) = x_i(t)-\hat{x}_i(t) $ and the fault estimation error $ e_{fi}(t) = f_i(t)-\hat{f}_i(t) $. Next, define $ \Theta = I_{n+m}-H\bar{C} $ and $ J = J_1+J_2 $, and the estimation error dynamics are derived as

Subsequently, with the following equality constraints of $ M,G,J_1,J_2 $, and H:

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

where $ \Delta\bar{\xi}_i(t) = \bar{\xi}(A_0\bar{x}_i(t),t)-\bar{\xi}(A_0\hat{\bar{x}}_i(t),t) $.

It then follows that:

where $ \tilde{e}(t) = [\tilde{e}_1^T(t),\ldots,\tilde{e}_N^T(t)]^T,\bar{d}(t) = [\bar{d}_1^T(t),\ldots,\bar{d}_N^T(t)]^T $ and $ \Delta\bar{\xi}(t) = [\Delta\bar{\xi}_1^T(t),\ldots,\Delta\bar{\xi}_N^T(t)]^T $.

Hence, the objective of decentralized FE design aims at devising H and $ J_1 $ such that the FE error dynamics in (12) and (13) are robustly asymptotically stable.

Remark 3: The $ i $th UIO in (9) exists for the augmented dynamics (7) with $ \Theta\bar{D}\bar{d}_i(t) = 0 $ and $ \Theta\Delta\bar{\xi}_i(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 $ \tilde{e}_i(t) $ approaches to zero, i.e., $ \lim_{t\rightarrow \infty}\tilde{e}_i(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 $ J_2 $ can be calculated by the derived matrices H and $ J_1 $, i.e., $M = (I_{n+m}-H\bar{C})\bar{A}- J_1\bar{C},\;G = (I_{n+m}-H\bar{C})\bar{B}$, and $J_2 = ((I_{n+m}- H\bar{C})\bar{A}-J_1\bar{C})H$.

IV. Distributed Fault-Tolerant Consensus Tracking Control Design

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

where $ K = [K_x\;I_m] $ is the compensation matrix with the state-feedback matrix $ K_x\in \mathbb{R}^{m\times n} $, $ R $ is the constant matrix, and $ \delta $ is a positive scalar. Then, the output information $\Xi_i(y_j(t),\gamma(t)), j = 0,1,\ldots,N,\gamma(t)\in\Gamma = \Gamma_m\cup\Gamma_p$ in a distributed manner is devised as

where $ a_{ij}^{\gamma(t)} $ is the $ (i,j) $th entry of $ \mathcal{A}_{\gamma(t)} $ with the graph $ \mathcal{G}_{\gamma(t)} $, and $ b_i^{\gamma(t)} = 1 $ when the $ i $th agent can access the leader and $ b_i^{\gamma(t)} = 0 $, otherwise.

The state tracking error dynamics can be rewritten as

where $ \Delta\tilde{\xi}_i(t) = {\xi}({x}_i(t),t)-{\xi}({x}_0(t),t) $.

Furthermore, it follows that:

where the global vectors ${e}(t) = [{e}_1^T(t),\dots,{e}_N^T(t)]^T,\;\Delta\tilde{\xi}(t) = [\Delta\tilde{\xi}_1^T(t),\dots,\Delta\tilde{\xi}_N^T(t)]^T$, and $ \mathcal{H}_{\gamma(t)} $ denotes the nonsingular information-exchange matrix.

A Case I: Connectivity-Maintained Attacks

The objective of Case I with proposing the distributed FCTC protocol in (14) and (15) is to determine $ K_x $ 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., $ \gamma(t)\in\Gamma_m $.

Theorem 1: Consider leader-following MASs with incipient and abrupt actuator faults in (2) under connectivity-maintained attacks. Given positive scalars $ \tau_1,\tau_3,\sigma_{{\Gamma}_m} $, and a positive chattering bound $ \bar{N}_{{\Gamma}_m} $, 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 $ P\in \mathbb{R}^{n\times n} $, matrix $K_x\in \mathbb{R}^{m\times n},\; H\in \mathbb{R}^{(n+m)\times p},\;J_1\in \mathbb{R}^{(n+m)\times p}$, and positive scalars $ {c},\tau_2,\tau_4 $ such that

where $ \underline{\phi} = {\rm min}\phi_{\gamma(t),i},\bar{\phi} = {\rm max}\phi_{\gamma(t),i},\gamma(t)\in\Gamma_m,i = 1,\ldots,N $ and $ {\lambda}_1 = \lambda_{\rm max}(\Phi_{\gamma(t)}^2) $ with the known $ \Phi_{\gamma(t)} $ in Lemma 1.

The constant matrix is denoted by $ R = B^TP^{-1}(C^TC)^{-1}C^T $ and the positive scalar is constrained within $ \delta\ge \frac{{c}}{\vert{\lambda}_2\vert} $ and $ {\lambda}_2 = \lambda_{\rm min}(\Phi_{\gamma(t)}\mathcal{H}_{\gamma(t)}+\mathcal{H}_{\gamma(t)}^T\Phi_{\gamma(t)}) $. The switching signal $\gamma(t)\in {\Gamma_m}$ is selected for $ \mathcal{G}_{{\Gamma}_m} $ with the ADT $ T_{{\Gamma}_m} $ satisfying

where $ \tau_m = {\rm min}\{\tau_3,\tau_1\tau_4\} $, and the exponential consensus tracking problem is effectively solved with the following state tracking error expectation:

with the initial maximum amplitude given by

and the decay rate $ {\lambda}_{{\Gamma}_m} $ given by

Proof: Construct a Lyapunov function candidate $ V_1({e}(t),\gamma(t)) $ with a symmetric positive-definite matrix $ P $ and a positive scalar $ \phi_{\gamma(t),i},\gamma(t)\in\Gamma_m,i = 1,\ldots,N $ as follows:

where $ \phi_{\gamma(t),i}^{-1} $ is the element of the diagonal matrix $\Phi_{\gamma(t)} = {\rm diag}\{\phi_{\gamma(t),1}^{-1},\dots,\phi_{\gamma(t),N}^{-1}\}$ from Lemma 1.

Denote $ \varsigma(t) = [\varsigma_1^T(t),\ldots,\varsigma_N^T(t)]^T $ with $ \varsigma_i(t) = P^{-1}{e}_i(t) $. According to the gain matrix $ R = B^TP^{-1}(C^TC)^{-1}C^T $, the derivative of $ V_1({e}(t),\gamma(t)) $ in (26) is obtained as

where the bounds $\underline{\phi} \;=\; {\rm min}\phi_{\gamma(t),i},\;\bar{\phi} \;=\; {\rm max}\phi_{\gamma(t),i},\;{\lambda}_1 \;=$ $\lambda_{\rm max} (\Phi_{\gamma(t)}^2),\; {\lambda}_2 = \lambda_{\rm min}(\Phi_{\gamma(t)}\mathcal{H}_{\gamma(t)}+\mathcal{H}_{\gamma(t)}^T\Phi_{\gamma(t)})$ with $ \gamma(t)\in\Gamma_m $, and the positive scalar $ \delta $ is selected as $ \delta\ge \frac{{c}}{\vert{\lambda}_2\vert} $.

Here, incipient and abrupt actuator faults are considered. The first-order derivative form is obtained with $\bar{d}_i(t) = \dot{f}_i(t) = [\dot{f}_{i1}^T(t),\dots,\dot{f}_{im}^T(t)]^T$.

where $ \dot{\bar{d}}_{is}(t) = -\epsilon_s\bar{d}_{is}(t) $ and the MASs are subject to different time-varying actuator fault when $ \epsilon_s>0 $ is selected.

Subsequently, another Lyapunov function candidate $V_2(\tilde{e}(t),\;\bar{d}(t)) = \frac{1}{\tau_1}\sum_{i = 1}^{N}\tilde{e}_i^T(t)\tilde{e}_i(t) +\frac{1}{\tau_2}\sum_{i = 1}^{N}\bar{d}_i^T(t)\bar{d}_i(t)$ is considered, where $ \tau_1 $ and $ \tau_2 $ are positive constants. The first-order derivative of $ V_2(\tilde{e}(t),\bar{d}(t)) $ is given by

where ${\rm min}\epsilon_s = \mathop{\rm min}\nolimits_{s = 1,\ldots,m}\epsilon_s$.

Finally, consider the Lyapunov function $V({e}(t),\tilde{e}(t)) = $ $V_1 (e(t), \gamma(t))+V_2(\tilde{e}(t),\bar{d}(t))$. On the basis of the inequality constraints in (18) and (19), i.e., ${\rm He}((A-BK_x)P)+({\lambda}_1\bar{\phi}- {c}\underline{\phi})BB^T+\bar{\phi}(\lambda_1I_n+\rho^2P^2)+\tau_3 P < 0$ and $K^TK+\frac{1}{\tau_1}(M+ M^T+ \Theta\bar{D}\bar{D}^T\Theta^T+\Theta\Theta^T+\rho^2A_0^TA_0)+\tau_4I_{n+m} < 0$, and $ {\rm min}\epsilon_s\ge\underline{\epsilon}_{\rm inc} $ in Assumption 2, it is obtained as

According to $ \tau_4<\frac{2\underline{\epsilon}_{\rm inc}}{\tau_1} $ in (20) and $ \tau_2\le-\tau_1^2\tau_4+2\tau_1\underline{\epsilon}_{\rm inc} $ in (21), $ \frac{\tau_1\tau_4}{\tau_2}+\frac{1}{\tau_1}-\frac{2}{\tau_2}\underline{\epsilon}_{\rm inc}\le 0 $ is then derived. Hence, it is proved that $\dot{V}({e}(t),\;\tilde{e}(t)) \; < \; -\tau_3V_1({e}(t),\;\gamma(t))\;-\;\tau_1\tau_4V_2(\tilde{e}(t),\;\bar{d}(t))\; \le\; -\tau_m V ({e}(t),\; \tilde{e}(t))$ with $ \tau_m = {\rm min}\{\tau_3,\tau_1\tau_4\} $.

Integrating both sides of (30) over $ t\in\left[t_{{k}},t_{{k}+1}\right) $ yields

It follows that $ \underline{\phi}V_1\le\sum_{i = 1}^{N}{e}_i^T(t)P^{-1}{e}_i(t)\le \bar{\phi}V_1 $ in (26). At each switching instant $ t_{{k}} $, it is obtained that $V(e(t_{{k}}),\tilde{e}(t_{{k}}))\le \frac{\bar{\phi}}{\underline{\phi}}V({e}(t_{{k}}^-),\tilde{e} (t_{{k}}^-))$.

In Definition 2, $ {k} = N_{{\Gamma}_m}(t_0,t)\le \bar{N}_{{\Gamma}_m}+\frac{t-t_0}{T_{{\Gamma}_m}} $ is calculated with the ADT $ T_{{\Gamma}_m} $ and chattering bound $ \bar{N}_{{\Gamma}_m} $. Then, the expression is given as

Next, with the definition of $ V({e}(t),\tilde{e}(t)) $, it is derived as

and the expression with the initial time $ t_0 $ is derived as

Denoting $ \Lambda_{{\Gamma}_m} = \lambda_{\rm max}(\phi_{\gamma(t),i}^{-1}P^{-1})+\sigma_{{\Gamma}_m} $ with the appropriate positive scalar $ \sigma_{{\Gamma}_m} $, it is finally given by

with

Hence, both $ \dot{V}({e}(t),\tilde{e}(t))<-\tau_m V({e}(t),\tilde{e}(t))<0 $ and the inequality in (35) imply that ${e}_i(t)\rightarrow 0,\tilde{e}_i(t)\rightarrow 0, x_i(t)\rightarrow x_0(t), \hat{x}_i(t)\rightarrow {x}_i(t)$ and $ \hat{f}_i(t)\rightarrow {f}_i(t) $ as $ t\rightarrow +\infty $ when ADT $ T_{{\Gamma}_m} $ 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 $ (\gamma(t)\in\Gamma_m) $ 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 $ -K\hat{\bar{x}}_i(t) $ and rely on switching rules $ \gamma(t)\in\Gamma_m $ 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 $ \Xi_i(y_j(t),\gamma(t)),j = 0,1,\ldots,N $.

B Case II: Connectivity-Mixed Attacks

The objective of Case II with designing the fully distributed FCTC protocol in (14) and (15) is to determine $ K_x $ 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., $ \gamma(t)\in\Gamma = \Gamma_m\cup\Gamma_p $.

Theorem 2: Consider leader-following MASs with incipient and abrupt actuator faults in (2) under connectivity-mixed attacks. Given positive scalars $ \tau_1,\tilde{\tau}_3 $, and $ \sigma_{{\Gamma}} $, 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 $ P\in \mathbb{R}^{n\times n} $, matrix $ K_x\in \mathbb{R}^{m\times n},H\in \mathbb{R}^{(n+m)\times p},J_1\in \mathbb{R}^{(n+m)\times p} $, and positive scalars $ \tilde{c},\tau_2,\tilde{\tau}_4 $ such that

and the positive scalar $ \delta $ in (14) satisfies

and for a positive decay rate $ \lambda_{{\Gamma}}\in(0,\lambda_{{\Gamma}}^{*}) $, the connectivity-paralyzed attack frequency $ \mathcal{F}_{\Gamma_p}(t_0,t) $ satisfies

and for a scalar $ \lambda_{{\Gamma}}^{*}\in(0,\tau_m) $, the connectivity-maintained attack activation rate $ \mathcal{R}_{\Gamma_m}(t_0,t) $ and the connectivity-paralyzed attack activation rate $ \mathcal{R}_{\Gamma_p}(t_0,t) $ satisfy

where $ {\lambda}_3 = \lambda_{\rm max}(\mathcal{H}_{\gamma(t)}+\mathcal{H}_{\gamma(t)}^T) $ and $ \tau_p = {\rm max}\{\tilde{\tau}_3,\tau_1\tilde{\tau}_4\} $ with the switching signal $ \gamma(t)\in\Gamma_p $.

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

with the initial maximum amplitude given by

Proof: Construct the Lyapunov function candidate $ \tilde{V}_1({e}(t),\gamma(t)) $ with a positive scalar $ \phi_{\gamma(t),i},\gamma(t)\in\Gamma $,

Then, the Lyapunov function candidate $\tilde{V}(t) = \tilde{V}({e}(t), \; \tilde{e}(t)) = \tilde{V}_1(e(t),\gamma(t))+V_2(\tilde{e}(t),\bar{d}(t))$ is chosen. When the MASs suffer from $ q $ connectivity-maintained attacks, i.e., $ \gamma(t)\in\Gamma_m $. According to the same proof process in Theorem 1, it is obtained that $\dot{\tilde{V}}({e}(t),\tilde{e}(t)) < -\tau_3\tilde{V}_1({e}(t),\gamma(t))-\tau_1\tau_4V_2(\tilde{e}(t), \bar{d}(t)) \le -\tau_m \tilde{V}({e}(t),\tilde{e}(t))$ with the scalar $ \tau_m = {\rm min}\{\tau_3,\tau_1\tau_4\} $.

When the MASs suffer from $ (r-q) $ connectivity-paralyzed attacks, i.e., $ \gamma(t)\in\Gamma_p $, the first-order derivative of the function $ \tilde{V}_1({e}(t),\gamma(t)) $ in (45) is given by

where $ \delta\le \frac{\tilde{c}}{\vert{\lambda}_3\vert} $ with $ {\lambda}_3 = \lambda_{\rm max}(\mathcal{H}_{\gamma(t)}+\mathcal{H}_{\gamma(t)}^T),\gamma(t)\in\Gamma_p $.

According to the constraints in (37)–(39), i.e., ${\rm He}((A - BK_x )P) +$$ (1+\tilde{c})BB^T+I_n+\rho^2P^2-\tilde{\tau}_3 P < 0,K^TK+\frac{1}{\tau_1}(M+M^T+\Theta\bar{D}\bar{D}^T\Theta^T+ $ $\Theta\Theta^T+ \rho^2A_0^TA_0) - \tilde{\tau}_4I_{n+m}<0$ and $ \tau_2\le\tau_1^2\tilde{\tau}_4+2\tau_1\underline{\epsilon}_{\rm inc} $, the first-order derivative of $ \tilde{V}(e(t),\tilde{e}(t)) $ is given as $\dot{\tilde{V}}(e(t),\tilde{e}(t)) < \tilde{\tau}_3\times \tilde{V}_1(e(t),\gamma(t))+{\tau}_1\tilde{\tau}_4V_2(\tilde{e}(t), \bar{d}(t)) +\bar{d}^T(t)(I_N\otimes(-\frac{\tau_1\tilde{\tau}_4}{\tau_2}+\frac{1}{\tau_1}-\frac{2}{\tau_2} \underline{\epsilon}_{\rm inc}))\bar{d}(t)$. Thus, it follows that $\dot{\tilde{V}}({e}(t),\tilde{e}(t)) < \tilde{\tau}_3\tilde{V}_1({e}(t),\gamma(t))+{\tau}_1\tilde{\tau}_4V_2(\tilde{e}(t), \bar{d}(t)) \le\tau_p \tilde{V}({e}(t),\tilde{e}(t))$ with $ \tau_p = {\rm max}\{\tilde{\tau}_3,\tau_1\tilde{\tau}_4\} $.

Integrating both sides of $ \dot{\tilde{V}}({e}(t),\tilde{e}(t)) $ with $ \gamma(t)\in\Gamma = \Gamma_m\cup\Gamma_p $ over $ t\in\left[t_{{k}},t_{{k}+1}\right) $ yields

It follows that $ {k} = N_{{\Gamma}}(t_0,t)\le 2N_{\Gamma_p}(t_0,t) $ is obtained with the number of the connectivity-paralyzed attacks $ N_{\Gamma_p}(t_0,t) $ from Definition 2. Subsequently, $ \tilde{V}({e}(t),\tilde{e}(t)) $ is given by

On the basis of the inequality constraint in (41) of the connectivity-paralyzed attack frequency $ \mathcal{F}_{\Gamma_p} $ in Definition 3, $ 2N_{\Gamma_p}(t_0,t)\ln(\frac{\bar{\phi}}{\underline{\phi}})\le(\lambda_\Gamma^{*}-\lambda_\Gamma)(t-t_0) $ is derived. Furthermore, with the aid of Definitions 4 and 5 of the connectivity-maintained/paralyzed attack activation rates $ \mathcal{R}_{\Gamma_m}(t_0,t) $ and $ \mathcal{R}_{\Gamma_p}(t_0,t) $ in (42), $ \tau_pT_p(t_0,t)-\tau_mT_m(t_0,t)\le-\lambda_\Gamma^{*}(t-t_0) $ is then obtained with $ T_m(t_0,t)+T_p(t_0,t) = t-t_0 $.

Thus, when the connectivity-mixed attacks are considered, i.e., $ \gamma(t)\in\Gamma = \Gamma_m\cup\Gamma_p $, it follows that:

and the state tracking error is given as

with the chosen positive scalar $ \sigma_{{\Gamma}} $ and $ \mu_{{\Gamma}} $ expressed in (44).

Hence, the state tracking error expectation in (50) indicates that the consensus tracking performance under connectivity-mixed attacks $ (\gamma(t)\in\Gamma) $ 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.

V. Simulation Results

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 $ x_i = [x_{i1}^T\;x_{i2}^T\;x_{i3}^T\;x_{i4}^T]^T = [{\theta}_{mi}\;{\omega}_{mi}\;{\theta}_{li}\;{\omega}_{li}]^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].

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

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.

DownLoad: Full-Size Img PowerPoint

Figure  3.  The switching topologies under connectivity-mixed attacks.

DownLoad: Full-Size Img PowerPoint

Figure  4.  Topology switching signal.

DownLoad: Full-Size Img PowerPoint

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.

DownLoad: Full-Size Img PowerPoint

Figure  6.  The angular velocity tracking of each motor: $x_{i2}-x_{02},i = 1,\ldots,5$ under connectivity-maintained attacks.

DownLoad: Full-Size Img PowerPoint

Figure  7.  The angular position tracking of each link: $x_{i3}-x_{03},i = 1,\ldots,5$ under connectivity-maintained attacks.

DownLoad: Full-Size Img PowerPoint

Figure  8.  The angular velocity tracking of each link: $x_{i4}-x_{04},i = 1,\ldots,5$ under connectivity-maintained attacks.

DownLoad: Full-Size Img PowerPoint

Figure  9.  The angular rotation tracking of each motor: $x_{i1}-x_{01},i = 1,\ldots,5$ under connectivity-mixed attacks.

DownLoad: Full-Size Img PowerPoint

Figure  10.  The angular velocity tracking of each motor: $x_{i2}-x_{02},i=1,\ldots,5$ under connectivity-mixed attacks.

DownLoad: Full-Size Img PowerPoint

Figure  11.  The angular position tracking of each link: $x_{i3}-x_{03},i=1,\ldots,5$ under connectivity-mixed attacks.

DownLoad: Full-Size Img PowerPoint

Figure  12.  The angular velocity tracking of each link: $x_{i4}-x_{04},i = 1,\ldots,5$ under connectivity-mixed attacks.

DownLoad: Full-Size Img PowerPoint

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.

DownLoad: Full-Size Img PowerPoint

Figure  14.  The comparisons of the considered connectivity-maintained attacks without fault, connectivity-mixed attacks with incipient fault and abrupt fault.

DownLoad: Full-Size Img PowerPoint

VI. Conclusion

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.