Adaptive secure output consensus tracking of linear multi-agent systems via dynamic event-triggered impulsive control under DoS attack

Abstract

In this paper, the adaptive secure output consensus tracking (ASOCT) problem for general linear multi-agent systems (MASs) with denial-of-service (DoS) attacks via novel dynamic event-triggered impulsive control (DETIC) protocols is considered. An observer-based distributed impulsive control scheme is proposed, where the designed control protocols and event-triggered mechanisms (ETMs) use the estimated value of the observer instead of the sampled information. The DoS attack considered in this paper appears aperiodically with an unknown attack strategy, which interrupts all communication links between agents, and further damages the controllers of agents. To handle the ASOCT under the DoS attack, some sufficient conditions for attack duration and attack frequency are derived. Moreover, the ETM considered in this paper is pull-based dynamic event-triggered mechanism (DETM), that is, the controllers of each agent merely are updated at its own triggering time while its neighbors will not be affected, as well each agent dynamically regulates its event-triggering threshold with the help of auxiliary parameters, which can effectively reduce controllers updates and allow larger triggering intervals. Furthermore, rigorous mathematical proofs are given, and some sufficient conditions are obtained to guarantee ASOCT of the linear MASs. The Zeno behavior is also proved theoretically to be precluded. Finally, the validity of the proffered control schemes is demonstrated by provided simulations.

Keywords

Linear multi-agent systems DoS attack adaptive secure output consensus tracking dynamic event-triggered mechanism impulsive control protocol

Introduction

Multi-agent systems (MASs) could model and solve a mass of problems in piratical fields, including the formation flying of spacecrafts, distributed tracking of mobile sensor network Predd et al. (2006) and Olfati-Saber and Sandell (2008), cooperative control of vehicles or robots, and many more. Obviously, it is a special and powerful tool with the rapid evolution of computer science and the communication technology in engineering. The cooperative control of MASs has drawn further attention from different fields.

The systems need to equip with real-time communication and excellent computing power for achieving control objectives. However, the systems need more communication burden and energy in practical applications. The real-time communication is unrealistic and will lead to energy consumption. In order to better apply the research results in practice, researchers always assume the MASs have limited energy. Moreover, use some special control methods to decrease depletion of energy, such as, event-triggered control (ETC) Xiong et al. (2022), Jiang et al. (2023) and Zhi et al. (2023), intermittent control Liu et al. (2020), sampled control Lu et al. (2020), etc. Here, the ETC can drop the frequency of communication between neighbors or the controller update. The mechanism is triggered only when the defined error surpasses the preset value. He et al. (2020) first proposed a distributed DETM. Compared with traditional static ETM, dynamic threshold further reduces triggered instants. So far, the researchers have executed a number of results, like Ruan et al. (2020), Zhao et al. (2022), Hu et al. (2021), and Koike et al. (2021). For linear MASs containing exterior disturbances, Ruan and the other authors investigated a leader–following bounded consensus problem under an observer-based DETM in Ruan et al. (2020). In Koike et al. (2021), DETM design is based on the output feedback, and it is applied to a linear MAS to achieve consensus.

Most of the results use the continuous control signal to address the consensus issue for MASs. However, a mass of controllers of the system in practical applications are discontinuous or even saltant. Therefore, some research results focus on the discontinuous control methods and related controllers. As one of the special discontinuous control methods, the impulsive control signals act on agents at some time instants which makes the states of agents change instantaneously. It is obvious that the instantaneous impulsive jump can enhance the efficiency for solving the consensus problem of MASs. Actually, we notice that impulsive control is not able to curtail the cost of communication and computation. The impulsive frequency needs to be high enough to ensure the convergence rate of systems. Based on this, Tan et al. (2019) employed a distributed ETM to impulsive control to design controllers for achieving consensus. Different from general impulsive control methods, the controller of each agent in Tan et al. (2019) is updated only when the triggered condition is satisfied. It is evident that unnecessary communication can be prevented, and the event-triggered impulsive control (ETIC) method can effectively enhance robustness and resource utilization for system with limited communication bandwidth. Based on the merit of ETIC, it has drawn growing attention, such as Li et al. (2020, 2022, 2023), Li and Li (2022), Zhu et al. (2018), Hu and Mu (2022), and Bao et al. (2024). Zhu et al. (2018) discussed exponentially stable problem for continuous dynamical system based on the ETIC method. The proposed control scheme was applied to the synchronization problem of the slave–master memristive neural networks. Li et al. (2020) researched the Lyapunov stability problem via the ETIC scheme for impulsive system, and it gave some sufficient conditions for globally asymptotic stability and uniform stability. Zhu et al. (2022) studied the input-to-state stability problem via ETIC for the nonlinear system, which is affected by external disturbance and packet loss phenomenon. Recently, Li et al. (2023) and Bao et al. (2024) designed the ETIC protocol to handle the secure problem of MASs. However, according to the authors’ knowledge, the efficiency of the event-triggered mechanism (ETM) can be improved, which leads to this paper.

Cyber attack is one of the major menace for MASs, which cannot always be avoided since the system cannot assure the accuracy of the controller and completeness of data. There exist two categories of major cyber attacks for MASs: injection attack Zhang et al. (2021) and DoS attack Shang et al. (2021). For the DoS attack, the malicious adversary can remove the communication among agents to prevent information transmission or can interrupt the control signal from its own controller to the actuator. It is no doubt that there existed abundant results about the security problems related to MASs (see Deng and Wen, 2020; Feng and Ishii, 2022; Sun et al., 2022; Wen et al., 2020; Xu et al., 2020a). Sun et al. (2022) investigated a random injection attack for leader–following MASs and proposed minimum attacked edges algorithm according to the Max-Flow Min-Cut lemma to prevent the system from reaching consensus. Feng and Ishii (2022) investigated the consensus problem with data constraints and DoS attack, in which the dynamic quantization with Zooming-in and Zooming-out was used to lessen the effect of attack. Pan et al. (2024) and Ma et al. (2022) concerned with the distributed impulsive consensus problem for nonlinear MASs subject to DoS attacks. Although some results about the DoS attack have been published, as far as the authors’ knowledge, there are a small number of results about adaptive secure output consensus tracking (ASOCT) for linear MASs via ETIC under the DoS attack.

Driven by the aforementioned results, this paper focuses on designing dynamic ETIC protocols for the ASOCT problem. We summarize the main contributions as follows. First, using a DETM, a novel observer-based distributed impulsive control scheme is designed to achieve ASOCT under DoS attack. The designed control protocols and ETMs use the estimated value of the observer instead of sampled information and the comparison is given in simulation to show the advantages. Second, the DoS attack is considered to appear aperiodically with an unknown attack strategy, which interrupts all communication links between agents and further damages the controllers of agents. It is a common pattern of malicious adversaries to prevent the systems from achieving goals. For achieving ASOCT under the DoS attack, some sufficient conditions for attack frequency and attack duration are derived. Third, dissimilar to the results in existence, such as Sun et al. (2022) and Xu et al. (2020b), a pull-based ETM is considered in this paper. That is, each agent refreshes its controller merely at its triggering time, while its neighbors will not be affected, which can effectively reduce controller updates. Moreover, the ETM considered in this paper is a DETM, that is, each agent dynamically regulates its event-triggering threshold with the help of auxiliary parameters, which allows larger triggering intervals. Finally, rigorous mathematical proofs are given, and some sufficient conditions are obtained to guarantee ASOCT of the linear MASs, while the precluded Zeno behavior is also proven.

The structure of this paper is described as follows. First, we describe some preliminaries and the problem in section “Preliminaries and problem description.” Second, in section “ASOCT of MAS with dynamic event-triggered impulsive control under DoS attack,” a novel adaptive DETM and impulsive control protocol be designed. The stability analysis of ASOCT is presented. In section “Simulation,” an example is given to illustrate the validity of the offered control protocols method. Finally, conclusions are presented in section “Conclusion.”

Preliminaries and problem description

Graph theory

This paper describes the communication among different agents as a directed graph $G = {V, E, A}$ . Use the node $v_{i}$ to represent the $i th$ agent and denote $V = {v_{1}, v_{2}, \dots, v_{N}}$ is a set of nodes. $E \in V \times V$ is a set of edges. A directed edge $e_{ij} \in E$ represents that the information of $v_{j}$ is available to the $v_{i}$ . $A = [a_{ij}] \in R^{N \times N}$ is a weighted adjacency matrix of the graph $G$ with non-negative entries. $a_{ij} > 0$ means that an edge $e_{ij}$ exists in graph $G$ . Otherwise, $a_{ij} = a_{ji} = 0$ . The Laplace matrix of graph $G$ is $L \in R^{N \times N}$ which is associated with weighted adjacency matrix $A$ . The elements of Laplace matrix are defined as $l_{ij} = - a_{ij}$ for $i \neq j$ and $l_{ii} = \sum_{k = 1, k \neq i}^{N} a_{ik}$ . Define the set of neighbors of $v_{i}$ as $N_{i} = {v_{j} | e_{ij} \in E}$ . Refer to a node as a root node if it can transform information to its neighbors but the neighbors’ information is unavailable for it. And assume node $v_{0}$ is the unique root node in this paper. Define a matrix $Ω = diag {κ_{1}, κ_{2}, \dots, κ_{N}}$ , where $κ_{i}$ is used to describe the communicative link between root node $v_{0}$ and $v_{i}$ . $κ_{i} > 0$ implies the information of root node $v_{0}$ is accessible for $v_{i}$ , otherwise $κ_{i} = 0$ . Define $H = L + Ω$ .

System dynamical model

Consider the MASs containing one leader and N followers. The $i th$ agent’s dynamics satisfy

{\begin{matrix} {\overset{\cdot}{x}}_{i} (t) = A x_{i} (t) + B u_{i} (t) \\ y_{i} (t) = C x_{i} (t) \end{matrix} i = 0, 2 \dots, N

(1)

where $x_{i} \in R^{n \times 1}$ , $u_{i} (t) \in R^{m \times 1}$ and $y_{i} (t)$ are the state vector, designed control input vector, output vector, respectively. And the leader is labeled as $0$ , we assume the input $u_{0} (t) = 0$ . $A \in R^{n \times n}$ , $B \in R^{n \times m}$ , and $C \in R^{r \times n}$ are the system matrices.

Assumption 1. For the MAS equation (1), the graph $G = {V, E, A}$ is connected and undirected, and each follower has a directed path from leader, that is, there exists at least one node influenced by leader directly.

Assumption 2. The pair (A,B) is stabilizable and the pair (A,C) is detectable. There exist positive-defined matrices $P, S \in R^{n \times n}$ satisfy the following algebraic Riccati equations (AREs)

PA + A^{T} P - PB B^{T} P + Q = 0

(2)

and there exists an observer gain matrix $G$ such that the matrix $A_{e} = A + GC$ is the Hurwitz matrix.

ASOCT of MAS with dynamic event-triggered impulsive control under DoS attack

In this section, first, we design distributed control protocols for achieving ASOCT. Second, we design a DETM. Based on the designed control protocols and DETM, we prove that the system will achieve ASOCT exponentially when some sufficient conditions are satisfied. Furthermore, the Zeno behavior will not present is proven. Third, it is shown that in the case of existing DoS attack, the ASOCT will be also achieved when some conditions about attack frequency and attack duration are satisfied.

Control protocol design

In this section, we construct a closed-loop distributed control protocol for each agent to achieve ASOCT. Suppose that the agent merely obtains the neighbors’ data at the triggered moment. To accomplish this objective, each agent is endowed with a distributed protocol in this section.

In practice, the state $x_{i}$ of $i th$ agent may be unavailable. We design the following observers for $i th$ agents to estimate their states for addressing this problem

\begin{matrix} {\overset{\cdot}{\hat{x}}}_{i} (t) = A {\hat{x}}_{i} (t) + B u_{i} (t) - G (y_{i} - C {\hat{x}}_{i}) \end{matrix}

(3)

where $G \in R^{n \times r}$ is a designed observer gain matrix and satisfies the Assumption 2.

Before designing the control inputs of followers, we define distributed state errors as $p_{i} (t) = \sum_{i = 1}^{N} a_{ij} ({\hat{x}}_{j} (t) - {\hat{x}}_{i} (t)) + κ_{i} ({\hat{x}}_{i} (t) - {\hat{x}}_{0} (t)) .$ Then, we depict the distributed state error of $i th$ agent at $k th$ triggered instant $t_{k}^{i}$ , $k \in N$ as

p_{i} (t_{k}^{i}) = \sum_{i = 1}^{N} a_{ij} ({\hat{x}}_{j} (t_{k}^{i}) - {\hat{x}}_{i} (t_{k}^{i})) + κ_{i} ({\hat{x}}_{i} (t_{k}^{i}) - {\hat{x}}_{0} (t_{k}^{i}))

(4)

Instead of using the factual distributed error, we use the estimated value of the $i th$ distributed state error ${\tilde{p}}_{i} (t) = \exp {A (t - t_{k}^{i})} p_{i} (t_{k}^{i})$ in this part of the work. On the basis of DETM and impulsive control, the control protocol for $i th$ follower is designed as followers

u_{i} (t) = \sum_{k = 1}^{+ \infty} d_{i} (t) K {\tilde{p}}_{i} (t) + δ (t - t_{k}^{i}) ζ [FC {\tilde{p}}_{i} (t) - κ_{i} F (y_{0} (t) - C {\hat{x}}_{0} (t))]

(5)

where $K \in R^{m \times n}$ is the feedback gain matrix, $ζ > 0$ which satisfies some conditions to be given later. $F \in R^{m \times r}$ is a constant matrix, which allows the matrix $BFC$ to be semi-positive definite. $δ (t)$ is the impulsive control function, and it is defined as $δ (t) = + \infty$ for $t = 0$ , $δ (t) = 0$ for $t \neq 0$ , and $\int_{- \infty}^{+ \infty} δ (t) dt = 1$ . For any piecewise continuous real function $u (t)$ , $\int_{- \infty}^{+ \infty} u (t) δ (t - t_{k}^{i}) dt = u (t_{k}^{i}) .$ We design the adaptive parameter $d_{i} (t)$ as

{\overset{\cdot}{d}}_{i} (t) = {\begin{matrix} τ_{i} \exp (m_{i} t) {\tilde{p}}_{i}^{T} (t) Γ {\tilde{p}}_{i} (t), d_{i} (t) \leq {\tilde{d}}_{i} \\ 0 d_{i} (t) > {\tilde{d}}_{i} \end{matrix}

(6)

where $τ_{i}$ and $m_{i}$ are the positive constant gains. $Γ = PB B^{T} P$ , ${\tilde{d}}_{i}$ are the positive constants. And the ${\tilde{d}}_{i}$ need satisfy $\frac{c (1 + ε_{6}) + ε_{4}}{2 - ε_{7}} + \frac{δ_{i}}{(2 - ε_{7}) (1 - ε_{8})} < {\tilde{d}}_{i}$ , $δ_{i}$ , $ε_{4}$ , $ε_{6}$ , $ε_{7}$ , and $ε_{8}$ are the positive constants, and the definitions will be given later.

Remark 1. Compared with the proposed strategy in Li et al. (2023) and Bao et al. (2024) which designed an ETIM to handle the secure problem of MASs, and the proposed strategies in Pan et al. (2024) and Ma et al. (2022) which are concerned with the distributed impulsive consensus problem for nonlinear MASs subject to DoS attacks, this paper designed the DETIM to handle the secure consensus problem of MASs with DOS attack. In this paper, the DETM can decrease the triggered times through a dynamic threshold, which has been proved in He et al. (2020). Compared with the control protocols in Bao et al. (2024), the estimation of distributed state errors ${\tilde{p}}_{i} (t)$ are adopted in control protocol $u_{i} (t)$ of this paper to instead of using sampled data signal $p_{i} (t_{k}^{i})$ in control protocol, which is closer to real value $p_{i} (t)$ . The control protocols and the ETM are more effective and easier to be applied. The control protocol $u_{i} (t)$ resets at the triggering instant ${t_{k}^{i}, k \in N}$ . It is obvious that the control protocol $u_{i} (t)$ of $i th$ follower does not require the continuous signal from its neighbors. The control protocol $u_{i} (t)$ resets only when its event-triggered condition is satisfied and is not influenced by its neighbors’ ETM.

Adaptive DETM design

In this section, we design an adaptive DETM such that the MAS equation (1) achieves ASOCT via impulsive control. The proposed DETM for agent $i$ is as follows.

Define a triggered function of $i th$ agent as

f_{i} (e_{p i} (t), {\tilde{p}}_{i} (t)) = θ_{i} [- δ_{i} π_{i} {\tilde{p}}_{i}^{T} (t) Γ {\tilde{p}}_{i} (t) + e_{p i}^{T} (t) Γ e_{p i} (t)]

(7)

where $θ_{i}$ , $δ_{i}$ , and $π_{i}$ are the positive constants, $e_{pi} (t) = {\tilde{p}}_{i} (t) - p_{i} (t)$ .

The triggering instants are determined as follows

{\begin{matrix} t_{0}^{i} = 0 \\ t_{k + 1}^{i} = \inf {t \geq t_{k}^{i} ∥ f_{i} (e_{pi} (t), {\tilde{p}}_{i} (t)) > η_{i} (t)} \end{matrix}

(8)

where $i = 1, 2, \dots, N$ , and $η_{i} (t)$ is a dynamical variable and its dynamics satisfies

{\overset{\cdot}{η}}_{i} (t) = {\begin{matrix} - μ_{i} η_{i} + l_{i} [δ_{i} π_{i} {\tilde{p}}_{i}^{T} (t) Γ {\tilde{p}}_{i} (t) - e_{pi}^{T} (t) Γ e_{pi} (t)] t \neq t_{k}^{i} \\ - μ_{i} η_{i} t = t_{k}^{i} \end{matrix}

(9)

where $η_{i} (0) > 0$ . In fact, $η_{i} (t) > 0$ , which has been proved in Lemma 1 in He et al. (2020) when $t \neq t_{k}^{i}$ , and for the case of $t = t_{k}^{i}$ , it is obvious. Setting the threshold in equation (9) is to ensure the validity of DETM.

Once equation (8) is satisfied, the $i th$ agent will update its controller, while the agents in $N_{i}$ do not need to update their controllers. That is, the controller of agent $i$ updates only when its triggered function is satisfied, which is not related to the triggered functions of its neighbors.

Let $ξ_{01} (t) = x_{0} (t) - {\hat{x}}_{0} (t)$ , $ξ_{i 1} (t) = x_{i} (t) - {\hat{x}}_{i} (t)$ , $ξ_{i 2} (t) = {\hat{x}}_{i} (t) - x_{0} (t)$ , $ξ_{i 3} (t) = {\hat{x}}_{i} (t) - {\hat{x}}_{0} (t)$ , $i = 1, 2, \dots, N$ . And $ξ_{1} (t) = {[ξ_{11}^{T} (t), ξ_{21}^{T} (t), \dots, ξ_{N 1}^{T} (t)]}^{T}$ , $ξ_{2} (t) = {[ξ_{12}^{T} (t), ξ_{22}^{T} (t), \dots, ξ_{N 2}^{T} (t)]}^{T}$ , $ξ_{3} (t) = {[ξ_{13}^{T} (t), ξ_{23}^{T} (t), \dots, ξ_{N 3}^{T} (t)]}^{T}$ , $\tilde{P} (t) = {[{\tilde{p}}_{1}^{T} (t), {\tilde{p}}_{2}^{T} (t), \dots, {\tilde{p}}_{N}^{T} (t)]}^{T}$ , $P (t) = {[p_{1}^{T} (t), p_{2}^{T} (t), \dots, p_{N}^{T} (t)]}^{T}$ , $e_{p} (t) = {[e_{p 1}^{T} (t), e_{p 2}^{T} (t), \dots, e_{pN}^{T} (t)]}^{T}$ .

Combining equations (1) and (3), we have

{\overset{\cdot}{ξ}}_{01} (t) = (A + GC) ξ_{01} (t)

(10)

{\overset{\cdot}{ξ}}_{i 1} (t) = (A + GC) ξ_{i 1} (t)

(11)

{\overset{\cdot}{ξ}}_{i 2} (t) = A ξ_{i 2} (t) + B u_{i} - GC ξ_{i 1} (t)

(12)

Substituting the control protocol (5) into system (12), we have

{\begin{matrix} {\overset{\cdot}{ξ}}_{i 2} (t) = A ξ_{i 2} (t) - GC ξ_{i 1} (t) + d_{i} (t) BK {\tilde{p}}_{i} (t) & t \neq t_{k}^{i} \\ Δ ξ_{i 2} (t_{k}^{i}) = B ζ [FC {\tilde{p}}_{i} (t_{k}^{i -}) - κ_{i} FC ξ_{01} (t_{k}^{i -})] & t = t_{k}^{i} \end{matrix}

(13)

where $Δ ξ_{i 2} (t_{k}^{i}) = ξ_{i 2} (t_{k}^{i +}) - ξ_{i 2} (t_{k}^{i -})$ , $ξ_{i 2} (t_{k}^{i +}) = lim_{l \to 0^{+}} ξ_{i 2} (t_{k}^{i} + l)$ , $ξ_{i 2} (t_{k}^{i -}) = lim_{l \to 0^{+}} ξ_{i 2} (t_{k}^{i} - l)$ . In this paper, we presuppose that the system is left-hand continuous at $t = t_{k}$ , that is, $ξ_{i 2} (t_{k}^{-}) = ξ_{i 2} (t_{k})$ . For $t \neq t_{k}^{i}$ , ${\overset{\cdot}{ξ}}_{i 2} (t) = A ξ_{i 2} (t) - GC ξ_{i 1} (t) + d_{i} (t) BK p_{i} (t) + d_{i} (t) BK e_{pi} (t)$ . When $t = t_{k}^{i}$ , $Δ ξ_{i 2} (t_{k}^{i}) = ζ B [FC p_{i} (t_{k}^{i -}) - κ_{i} FC ξ_{01} (t_{k}^{i -})]$ .

Construct a time sequence $J_{i} = {t_{k}^{i}, k \in N^{+}}$ for $i th$ agent, where $t_{k}^{i}$ is the $k th$ triggered instant. Define a time sequence $J = ⋃_{i = 1}^{N} J_{i}$ which contains all time sequences of all agents, that is, $J = {t_{s}, s \in N^{+}} = ⋃_{i = 1}^{N} {t_{k}^{i}, k \in N^{+}}$ . The time sequence $J = {t_{s}, s \in N^{+}}$ satisfies $0 = t_{0} < t_{1} < \dots < t_{s} < t_{s + 1} < \dots,$ and $lim_{s \to \infty} t_{s} = + \infty$ . Define a function $ϕ_{i} (t_{s})$ satisfying

ϕ_{i} (t_{s}) = {\begin{matrix} 0, & t_{k}^{i} \neq t_{s} \\ 1, & t_{k}^{i} = t_{s} \end{matrix}

(14)

Define $ϕ (t_{s}) = diag {ϕ_{1} (t_{s}), ϕ_{2} (t_{s}), \dots, ϕ_{N} (t_{s})}$ . Then, it follows from equation (13) that

\begin{matrix} ξ_{2} (t_{s}^{+}) = (I_{N} \otimes I_{n} - ζ ϕ (t_{s}) H \otimes BFC) ξ_{2} (t_{s}^{-}) \end{matrix}

(15)

Let $ζ = \frac{1}{λ_{\max} (BFC) λ_{\max} (H)}$ . At the triggering instants $t_{s}$ , $s = 1, 2, \dots,$ there exists one agent $i$ at the minimum, such that $t_{s} = t_{k}^{i}$ and $ϕ_{i} (t_{s}) = 1$ , which means $ϕ (t_{s}) < I_{N}$ . $λ (ζ ϕ (t_{s}) H \otimes BFC)$ is less than one, which means that $0 \leq λ (I_{N} \otimes I_{n} - ζ ϕ (t_{s}) H \otimes BFC) \leq 1$ . There is a positive constant $χ$ so as to satisfy $0 \leq λ (I_{N} \otimes I_{n} - ζ ϕ (s) H \otimes BFC) < \exp {- \frac{χ}{2}}$ .

Theorem 1. Hypothesize Assumptions 1 and 2 are met. Observed-based output consensus tracking of the linear MAS equation (1) would be achieved exponentially under the DETM equations (7) and (8) and the control protocols (5) and (6), if the parameters satisfy $ε_{1}, ε_{3}, ε_{4}, ε_{6}, α_{1}, α_{2}$ $> 0$ , $ε_{8} \in (0, 1)$ , $ε_{7} \in (0, 2)$ , and $c λ_{\min} (H) - ε_{1} \frac{λ_{\max}^{2} (P)}{λ_{\min} (Γ)} > 0$ , $φ_{i} = \frac{1}{π_{i}} = (\frac{1}{ε_{4}} + \frac{1}{ε_{3}}) {\tilde{d}}_{i}^{2} + (\frac{1}{ε_{8}} - 1) [(2 - ε_{7}) {\tilde{d}}_{i} - c (1 + ε_{6}) - ε_{4}] > 0$ , $τ_{i} \leq (2 - ε_{7}) (1 - ε_{8})$ , and the symmetric positive matrices $P, P_{1}, P_{2}$ satisfy the following conditions

PA + A^{T} P - c λ_{\min} (H) Γ + ε_{1} PP + Q = 0

(16)

P_{1} (A + GC) + (A + GC)^{T} P_{1} + \frac{1}{ε_{1}} (GC)^{T} GC + α_{1} P_{1} = 0

(17)

\begin{matrix} P_{2} (A + GC) + (A + GC)^{T} P_{2} + (c (1 + \frac{1}{ε_{6}}) \frac{\tilde{d}}{ε_{7}} + ε_{3} \end{matrix}) ς Γ + α_{2} P_{2} = 0

(18)

\frac{c (1 + ε_{6}) + ε_{4}}{2 - ε_{7}} + \frac{δ_{i}}{(2 - ε_{7}) (1 - ε_{8})} < {\tilde{d}}_{i}

(19)

ρ_{i} = μ_{i} - \frac{φ_{i} - l_{i}}{θ_{i}} > 0

(20)

Proof. Describe the Lyapunov function with

V = V_{1} (t) + V_{2} (t) + V_{3} (t) + V_{4} (t) + V_{5} (t)

(21)

where $V_{1} (t) = ξ_{1}^{T} (t) (H \otimes P_{1}) ξ_{1} (t)$ , $V_{2} (t) = ξ_{2}^{T} (t) (H \otimes P) ξ_{2} (t)$ , $V_{3} (t) = \sum_{i = 1}^{N} \exp (- m_{i} t) \frac{{(d_{i} (t) - {\tilde{d}}_{i})}^{2}}{2}$ , $V_{4} (t) = \sum_{i = 1}^{N} η_{i} (t)$ , and $V_{5} (t) = ξ_{01}^{T} (t) P_{2} ξ_{01} (t)$ .

The derivative of $V_{1}$ with respect to time gives

{\overset{\cdot}{V}}_{1} = ξ_{1}^{T} (t) (H \otimes P_{1} (A + GC) + (A + GC)^{T} P_{1}) ξ_{1} (t)

(22)

For $t \neq t_{s}$ , the derivative of $V_{2}$ along error system (13) is given as

\begin{array}{l} {\dot{V}}_{2} = ξ_{2}^{T} (t) (H \otimes P A + A^{T} P) ξ_{2} (t) - 2 ξ_{2}^{T} (t) (H \otimes P G C) ξ_{1} (t) \\ + 2 ξ_{2}^{T} (t) (H \otimes P B) (D (t) \otimes K) P (t) \\ + 2 ξ_{2}^{T} (t) (H \otimes P B) (D (t) \otimes K) e_{p} (t) \end{array}

(23)

where $D (t) = diag {d_{1} (t), d_{2} (t), \dots, d_{N} (t)}$ . According to Young’s inequality

\begin{matrix} - 2 ξ_{2}^{T} (t) (H \otimes PGC) ξ_{1} (t) \leq ε_{1} ξ_{2}^{T} (t) (H \otimes PP) ξ_{2} (t) \\ + \frac{1}{ε_{1}} ξ_{1}^{T} (t) (H \otimes (GC)^{T} GC) ξ_{1} (t) \end{matrix}

(24)

Take $K = B^{T} P$ and then, equation (23) becomes into

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq ξ_{2}^{T} (t) (H \otimes PA + A^{T} P) ξ_{2} (t) + ε_{1} ξ_{2}^{T} (t) (H \otimes PP) ξ_{2} (t) \\ + \frac{1}{ε_{1}} ξ_{1}^{T} (t) (H \otimes (GC)^{T} GC) ξ_{1} (t) \\ + 2 ξ_{2}^{T} (t) (H \otimes PB) (D (t) \otimes B^{T} P) P (t) \\ + 2 ξ_{2}^{T} (t) (H \otimes PB) (D (t) \otimes B^{T} P) e_{p} (t) \end{matrix}

(25)

Take a positive constant $c$ , equation (25) will become

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq ξ_{2}^{T} (t) (H \otimes PA + A^{T} P - c λ_{\min} (H) PB B^{T} P + ε_{1} P^{T} P) ξ_{2} (t) \\ + c ξ_{2}^{T} (t) (HH \otimes PB B^{T} P) ξ_{2} (t) + 2 ξ_{2}^{T} (t) (H \otimes PB) (D (t) \otimes B^{T} P) P (t) \\ + 2 ξ_{2}^{T} (t) (H \otimes PB) (D (t) \otimes B^{T} P) e_{p} (t) \\ + \frac{1}{ε_{1}} ξ_{1}^{T} (t) (H \otimes (GC)^{T} GC) ξ_{1} (t) \end{matrix}

(26)

Since

\begin{matrix} p_{i} (t) = \sum_{i = 1}^{N} a_{ij} ({\hat{x}}_{j} (t) - {\hat{x}}_{i} (t)) + κ_{i} ({\hat{x}}_{i} (t) - {\hat{x}}_{0} (t)) \\ = \sum_{i = 1}^{N} a_{ij} (ξ_{j 2} (t) - ξ_{i 2} (t)) + κ_{i} ξ_{i 2} (t) + κ_{i} ξ_{01} (t) \end{matrix}

(27)

we get the following inequality for a positive constant $ε_{6}$

\begin{matrix} c ξ_{2}^{T} (t) (HH \otimes PB B^{T} P) ξ_{2} (t) \\ = c ξ_{3}^{T} (t) (HH \otimes Γ) ξ_{3} (t) - 2 c ξ_{2}^{T} (t) (H Ω 1_{N} \otimes Γ) ξ_{01} (t) \\ + ξ_{01}^{T} (t) ({(Ω 1_{N})}^{T} Ω 1_{N} \otimes Γ) ξ_{01} (t) \\ \leq c (1 + ε_{6}) ξ_{3}^{T} (t) (HH \otimes Γ) ξ_{3} (t) + c (1 + \frac{1}{ε_{6}}) ξ_{01}^{T} (t) ((Ω 1_{N})^{T} \\ Ω 1_{N} \otimes Γ) ξ_{01} (t) \\ \leq c (1 + ε_{6}) P^{T} (t) (I \otimes Γ) P (t) + c (1 + \frac{1}{ε_{6}}) ξ_{01}^{T} (t) ((Ω 1_{N})^{T} \\ Ω 1_{N} \otimes Γ) ξ_{01} (t) \end{matrix}

(28)

Then taking $\tilde{d} = \max {{\tilde{d}}_{1}, {\tilde{d}}_{2}, \dots, {\tilde{d}}_{N}}$ and for a positive constant $ε_{7}$ , get

\begin{matrix} 2 ξ_{2}^{T} (t) (HD (t) \otimes Γ) P (t) = 2 ξ_{3}^{T} (t) (HD (t) \otimes Γ) P (t) \\ - 2 ξ_{01}^{T} (t) ((Ω 1_{N})^{T} D (t) \otimes Γ) P (t) \\ \leq - (2 - ε_{7}) P^{T} (t) (D (t) \otimes Γ) P (t) + \frac{\tilde{d}}{ε_{7}} ξ_{01}^{T} (t) ((Ω 1_{N})^{T} Ω 1_{N} \otimes Γ) \\ ξ_{01} (t) \end{matrix}

(29)

While

\begin{matrix} 2 ξ_{2}^{T} (t) (HD (t) \otimes Γ) e_{p} (t) = 2 ξ_{3}^{T} (t) (HD (t) \otimes Γ) e_{p} (t) \\ - 2 ξ_{01}^{T} (t) ({(Ω 1_{N})}^{T} D (t) \otimes Γ) e_{p} (t) \\ \leq (\frac{1}{ε_{4}} + \frac{1}{ε_{3}}) e_{p}^{T} (t) (D (t) D (t) \otimes Γ) e_{p} (t) + ε_{4} P^{T} (t) (I_{N} \otimes Γ) P (t) \\ + ε_{3} ξ_{01}^{T} (t) ({(Ω 1_{N})}^{T} Ω 1_{N} \otimes Γ) ξ_{01} (t) \end{matrix}

(30)

where $ε_{4}$ and $ε_{3}$ are the two positive constants. Combining equations (26) and (28)–(30), we have

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq ξ_{2}^{T} (t) (H \otimes PA + A^{T} P - c λ_{\min} (H) Γ + ε_{1} PP) Γ) ξ_{2} (t) \\ + \sum_{i = 1}^{N} (\frac{1}{ε_{4}} + \frac{1}{ε_{3}}) d_{i}^{2} (t) e_{pi}^{T} (t) Γ e_{pi} (t) \\ + \frac{1}{ε_{1}} ξ_{1}^{T} (t) (H \otimes (GC)^{T} GC) ξ_{1} (t) \\ + (c (1 + \frac{1}{ε_{6}}) + \frac{\tilde{d}}{ε_{7}} + ε_{3}) ξ_{01}^{T} (t) ({(Ω 1_{N})}^{T} Ω 1_{N} \otimes Γ) ξ_{01} (t) \\ - \sum_{i = 1}^{N} [(2 - ε_{7}) d_{i} (t) - c (1 + ε_{6}) - ε_{4}] p_{i}^{T} (t) Γ p_{i} (t) \end{matrix}

(31)

Since $e_{pi} (t) = {\tilde{p}}_{i} (t) - p_{i} (t)$ , and take an positive constant $ε_{8} < 1$ , it then follows from Young’s inequality that

\begin{matrix} - p_{i}^{T} (t) Γ p_{i} (t) = ({\tilde{p}}_{i} (t) - e_{pi} (t))^{T} Γ ({\tilde{p}}_{i} (t) - e_{pi} (t)) \\ \leq - (1 - ε_{8}) {\tilde{p}}_{i}^{T} (t) Γ {\tilde{p}}_{i} (t) - (1 - \frac{1}{ε_{8}}) e_{pi}^{T} (t) Γ e_{pi} (t)) \end{matrix}

(32)

Define ${\hat{d}}_{i} (t) = d_{i} (t) - \frac{c (1 + ε_{6}) + ε_{4}}{2 - ε_{7}} - \frac{δ_{i}}{(2 - ε_{7}) (1 - ε_{8})}$ . According to equation (16), we have

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq - ξ_{2}^{T} (t) (H \otimes Q) ξ_{2} (t) \\ + \sum_{i = 1}^{N} [(\frac{1}{ε_{4}} + \frac{1}{ε_{3}}) d_{i}^{2} (t) + (\frac{1}{ε_{8}} - 1) [(2 - ε_{7}) d_{i} (t) \\ - c (1 + ε_{6}) - ε_{4}]] e_{pi}^{T} (t) Γ e_{pi} (t) + \frac{1}{ε_{1}} ξ_{1}^{T} (t) (H \otimes (GC)^{T} GC) ξ_{1} (t) \\ - \sum_{i = 1}^{N} δ_{i} {\tilde{p}}_{i}^{T} (t) Γ {\tilde{p}}_{i} (t) \\ + (c (1 + \frac{1}{ε_{6}}) + \frac{\tilde{d}}{ε_{7}} + ε_{3}) ξ_{01}^{T} (t) ({(Ω 1_{N})}^{T} Ω 1_{N} \otimes Γ) ξ_{01} (t) \\ - \sum_{i = 1}^{N} (2 - ε_{7}) (1 - ε_{8}) {\hat{d}}_{i} (t) {\tilde{p}}_{i}^{T} (t) Γ {\tilde{p}}_{i} (t) \end{matrix}

(33)

Now taking the time derivatives of $V_{3}$ , $V_{4}$ , and $V_{5}$ , we have

\begin{matrix} {\overset{\cdot}{V}}_{3} (t) = - \sum_{i = 1}^{N} m_{i} \exp (- m_{i} t) \frac{{(d_{i} (t) - {\tilde{d}}_{i})}^{2}}{2} \\ + \sum_{i = 1}^{N} \exp (- m_{i} t) (d_{i} (t) - {\tilde{d}}_{i}) {\overset{\cdot}{d}}_{i} (t) \end{matrix}

(34)

{\overset{\cdot}{V}}_{4} (t) = \sum_{i = 1}^{N} - μ_{i} η_{i} (t) + l_{i} [δ_{i} π_{i} {\tilde{p}}_{i}^{T} (t) Γ {\tilde{p}}_{i} (t) - e_{pi}^{T} (t) Γ e_{pi} (t)]

(35)

{\overset{\cdot}{V}}_{5} (t) = ξ_{01}^{T} (t) (P_{2} (A + GC) + (A + GC)^{T} P_{2}) ξ_{01} (t)

(36)

Combine equations (22) and (33)–(36). Let $ς = \sum_{i = 1}^{N} κ_{i}^{2}$ , $α = \frac{λ_{\min} (Q)}{λ_{\max} (P)}$ , and $φ_{i} = (\frac{1}{ε_{4}} + \frac{1}{ε_{3}}) {\tilde{d}}_{i}^{2} + (\frac{1}{ε_{8}} - 1) [(2 - ε_{7}) {\tilde{d}}_{i} - c (1 + ε_{6}) - ε_{4}]$ , we have

\begin{matrix} \overset{\cdot}{V} < - α ξ_{2}^{T} (t) (H \otimes P) ξ_{2} (t) + ξ_{01}^{T} (t) (P_{2} (A + GC) \\ + (A + GC)^{T} P_{2} + (c (1 + \frac{1}{ε_{6}}) + \frac{\tilde{d}}{ε_{7}} + ε_{3}) ς Γ) ξ_{01} (t) \\ + \sum_{i = 1}^{N} - μ_{i} η_{i} (t) + l_{i} [δ_{i} π_{i} {\tilde{p}}_{i}^{T} (t) Γ {\tilde{p}}_{i} (t) - e_{pi}^{T} (t) Γ e_{pi} (t)] \\ + \sum_{i = 1}^{N} φ_{i} e_{pi}^{T} (t) Γ e_{pi} (t) \\ - \sum_{i = 1}^{N} (2 - ε_{7}) (1 - ε_{8}) {\hat{d}}_{i} (t) {\tilde{p}}_{i}^{T} (t) Γ {\tilde{p}}_{i} (t) - \sum_{i = 1}^{N} δ_{i} {\tilde{p}}_{i}^{T} (t) Γ {\tilde{p}}_{i} (t) \\ + ξ_{1}^{T} (t) (H \otimes P_{1} (A + GC) + (A + GC)^{T} P_{1} + \frac{1}{ε_{1}} (GC)^{T} GC) ξ_{1} (t) \\ - \sum_{i = 1}^{N} m_{i} \exp (- m_{i} t) \frac{{(d_{i} (t) - {\tilde{d}}_{i})}^{2}}{2} \\ + \sum_{i = 1}^{N} \exp (- m_{i} t) (d_{i} (t) - {\tilde{d}}_{i}) {\overset{\cdot}{d}}_{i} (t) \end{matrix}

(37)

On account of the update law (6) of adaptive parameters $d_{i} (t)$ , we divide the following analysis into two cases:

For the case $d_{i} (t) \leq {\tilde{d}}_{i}$ , based on $τ_{i} \leq (2 - ε_{7}) (1 - ε_{8})$ and equation (19), we have

\begin{matrix} - (2 - ε_{7}) (1 - ε_{8}) {\hat{d}}_{i} (t) {\tilde{p}}_{i}^{T} (t) Γ {\tilde{p}}_{i} (t) + \exp (- m_{i} t) \\ (d_{i} (t) - {\tilde{d}}_{i}) {\overset{\cdot}{d}}_{i} (t) < 0 \end{matrix}

(38)

For the case $d_{i} (t) > {\tilde{d}}_{i}$ , ${\overset{\cdot}{d}}_{i} (t) = 0$ , we have

\begin{matrix} - (2 - ε_{7}) (1 - ε_{8}) {\hat{d}}_{i} (t) {\tilde{p}}_{i}^{T} (t) Γ {\tilde{p}}_{i} (t) + \exp (- m_{i} t) \\ (d_{i} (t) - {\tilde{d}}_{i}) {\overset{\cdot}{d}}_{i} (t) = 0 \end{matrix}

(39)

Finally, according to equations (17) and (18), substituting equations (38) and (39) into equation (37), and let $π_{i} = \frac{1}{φ_{i}}$ , we have

\begin{matrix} \overset{\cdot}{V} < - α ξ_{2}^{T} (t) (H \otimes P) ξ_{2} (t) - α_{2} ξ_{01}^{T} (t) P_{2} ξ_{01} (t) \\ - α_{1} ξ_{1}^{T} (t) (H \otimes P_{1}) ξ_{1} (t) \\ - \sum_{i = 1}^{N} (μ_{i} - \frac{φ_{i} - l_{i}}{θ_{i}}) η_{i} (t) - \sum_{i = 1}^{N} m_{i} \exp (- m_{i} t) \frac{{(d_{i} (t) - {\tilde{d}}_{i})}^{2}}{2} \end{matrix}

(40)

Taking $m = \min {m_{i}, i = 1, 2, \dots, N}$ , $ρ = \min {ρ_{i}, i = 1, 2, \dots, N}$ , equation (40) will satisfy

\overset{\cdot}{V} < - α V_{2} (t) - α_{2} V_{5} (t) - α_{1} V_{1} (t) - m V_{3} (t) - ρ V_{4} (t)

(41)

Taking $σ = \min {α, α_{2}, α_{1}, m, ρ}$ , we have

\overset{\cdot}{V} < - σ V (t)

(42)

On the basis of the above analysis, when $t \in (t_{s}, t_{s + 1}]$

V (t) \leq \exp {- σ (t - t_{s})} V (t_{s}^{+})

(43)

When $t = t_{s}$ , following from equation (39), we have

\begin{matrix} V_{2} (t_{s}^{+}) = ξ_{2}^{T} (t_{s}^{-}) {(I_{N} \otimes I_{n} - ζ ϕ (t_{s}) H \otimes BFC)}^{T} (H \otimes P) \\ (I_{N} \otimes I_{n} - ζ ϕ (s) H \otimes BFC) ξ_{2} (t_{s}^{-}) \\ \leq \exp {- χ} V_{2} (t_{s}^{-}) \end{matrix}

(44)

According to equations (43) and (44), we have

{\begin{matrix} V (t) \leq \exp {- σ (t - t_{s})} V (t_{s}^{+}), & t \in (t_{s}, t_{s + 1}] \\ V (t_{s}^{+}) \leq \exp {- χ} V (t_{s}^{-}), & t = t_{s} \end{matrix}

(45)

We can easily get the fact, when $t \in (t_{s}, t_{s + 1}]$

V (t) \leq \exp {- s χ - σ (t - t_{0})} V (t_{0}), s \in N

(46)

Hence, the MAS equation (1) achieves AOCT exponentially under the adaptive ETIC protocols (5) and (6) and ETM equations (7)–(9).□

Theorem 2. For the linear MAS equation (1), under the dynamic ETM equations (7) and (8) and the control protocols (5) and (6), the Zeno behavior of the MAS would be excluded.

Proof. For $t \in (t_{k}^{i}, t_{k + 1}^{i}]$ , on the basis of the triggered condition equation (8), we get

- δ_{i} π_{i} {\tilde{p}}_{i}^{T} (t) Γ {\tilde{p}}_{i} (t) + e_{pi}^{T} (t) Γ e_{pi} (t) < \frac{η_{i} (0)}{θ_{i}} \exp {- (μ_{i} + \frac{l_{i}}{η_{i}}) t}

(47)

Calculating the right derivative of $∥ e_{pi} (t) ∥$ in $(t_{k}^{i}, t_{k + 1}^{i}]$ gives

\begin{matrix} D^{+} ∥ e_{pi} (t) ∥ = ∥ {\overset{\cdot}{\tilde{p}}}_{i} (t) - {\overset{\cdot}{p}}_{i} (t) ∥ \\ \leq ∥ A e_{pi} (t) ∥ + ∥ BK \sum_{i = 1}^{N} a_{ij} (d_{j} (t) {\tilde{p}}_{j} (t) - d_{i} (t) {\tilde{p}}_{i} (t)) + κ_{i} (d_{i} (t) {\tilde{p}}_{i} (t)) ∥ \\ + ∥ GC \sum_{i = 1}^{N} a_{ij} (ξ_{j 1} (t) - ξ_{i 1} (t)) + κ_{i} (ξ_{i 1} (t) - ξ_{01} (t)) ∥ \\ \leq ∥ A ∥ ∥ e_{pi} (t) ∥ + ∥ BK ∥ ∥ \sum_{i = 1}^{N} a_{ij} (d_{j} (t) {\tilde{p}}_{j} (t) - d_{i} (t) {\tilde{p}}_{i} (t)) \\ + κ_{i} (d_{i} (t) {\tilde{p}}_{i} (t)) ∥ + ∥ GC ∥ ∥ \sum_{i = 1}^{N} a_{ij} (ξ_{j 1} (t) - ξ_{i 1} (t)) \\ + κ_{i} (ξ_{i 1} (t) - ξ_{01} (t)) ∥ \\ = ∥ A ∥ ∥ e_{pi} (t) ∥ + Π_{k}^{i} \end{matrix}

(48)

where $Π_{k}^{i} = ∥ B K ∥ ∥ \sum_{i = 1}^{N} a_{i j} (d_{j} (t) {\tilde{p}}_{j} (t) - d_{i} (t) {\tilde{p}}_{i} (t)) {\tilde{p}}_{i} (t)) ∥ + ∥ G C ∥$ $∥ \sum_{i = 1}^{N} a_{ij} (ξ_{j 1} (t) - ξ_{i 1} (t)) + κ_{i} (ξ_{i 1} (t) - ξ_{01} (t)) ∥ \neq 0$ . If $Π_{k}^{i} = 0$ , it means that output consensus of the MAS is achieved. The following analysis is divided into two parts

$∥ A ∥ \neq 0$ , we have

∥ e_{pi}^{T} (t) PB ∥ \leq \frac{Π_{k}^{i} ∥ PB ∥}{∥ A ∥} (\exp {∥ A ∥ (t - t_{k}^{i})} - 1)

(49)

At the triggered instant $t_{k + 1}^{i}$

\begin{matrix} e_{pi}^{T} (t_{k + 1}^{i}) Γ e_{pi} (t_{k + 1}^{i}) = ∥ e_{pi} (t) PB ∥^{2} > δ_{i} π_{i} ∥ {\tilde{p}}_{i}^{T} (t) PB ∥^{2} \\ + \frac{η_{i} (0)}{θ_{i}} \exp {- (μ_{i} + \frac{l_{i}}{η_{i}}) t_{k + 1}^{i}} \end{matrix}

(50)

Combining with equation (49), we have

\begin{matrix} 0 \leq \frac{η_{i} (0)}{θ_{i}} \exp {- (μ_{i} + \frac{l_{i}}{η_{i}}) t_{k + 1}^{i}} \\ \leq δ_{i} π_{i} ∥ {\tilde{p}}_{i}^{T} (t) PB ∥^{2} + \frac{η_{i} (0)}{θ_{i}} \exp {- (μ_{i} + \frac{l_{i}}{η_{i}}) t_{k + 1}^{i}} \\ \leq ∥ e_{pi} (t_{k + 1}^{i}) PB ∥^{2} \\ \leq {\frac{Π_{k}^{i} ∥ PB ∥}{∥ A ∥} (\exp {∥ A ∥ (t_{k + 1}^{i} - t_{k}^{i})} - 1)}^{2} \end{matrix}

(51)

Then

\begin{matrix} t_{k + 1}^{i} - t_{k}^{i} \geq \frac{1}{∥ A ∥} \ln \\ {1 + \frac{∥ A ∥}{∥ {Pi}_{k}^{i} ∥ PB ∥} (δ_{i} π_{i} ∥ {\tilde{p}}_{i}^{T} (t) PB ∥^{2} + \frac{η_{i} (0)}{θ_{i}} \\ \exp {- (μ_{i} + \frac{l_{i}}{η_{i}}) t_{k + 1}^{i}}^{\frac{1}{2}})} \end{matrix}

(52)

Hence, we find a lower bound which does not equal to zero, that is, the Zeno behavior will not appear.

$∥ A ∥ = 0$ , we have

∥ e_{pi}^{T} (t) PB ∥ \leq Π_{k}^{i} ∥ PB ∥ (t - t_{k}^{i})

(53)

Because the proof is analogous to that for the case of $∥ A ∥ \neq 0$ , we leave out here. The proof is completed.□

ASOCT of linear MAS under DoS attack

In this section, considering that the DoS attack can damage the information transmission in the network, which implies the information of agents’ neighbors is not accessible to the agent when the system is attacked. It indicates that the control inputs of agents will be zero. It is obvious that the system will not be able to achieve stability if the system is equipped with unlimited attack signals. We presuppose that the attacker has limited attack energy in an active varying period, which means that the attacker needs to rest to stockpile power for the next attack. Figure 1 indicates the attack process.

So, we separate the time interval into two areas, the attack area and the communication area, which are displayed in Figure 2.

Denote a attack sequence as ${{\tilde{t}}_{m}, m \in N}$ , where successful DoS attack is initiated at time instant ${\tilde{t}}_{m}$ . Denote the duration of attack as $η_{m}$ . The $m th$ attack interval would be further represented as $ϕ_{m} = [{\tilde{t}}_{m}, {\tilde{t}}_{m} + η_{m})$ , where communication is aborted. Furthermore, denote

Θ_{a} (t_{0}, t) = ⋃_{m \in N} [{\tilde{t}}_{m}, {\tilde{t}}_{m} + η_{m}) ⋂ [t_{0}, t)

Θ_{s} (t_{0}, t) = [t_{0}, t) \ Θ_{a} (t_{0}, t)

Define $t_{m}' = max {t_{k} | t_{k} \in Θ_{s} (t_{0}, t)}$ is a moment in time of the latest propitious communication before the $m th$ successful attack, while ${\bar{t}}_{m}$ is a moment in time of the first propitious communication after the $m th$ successful attack. Define $Δ_{m}$ as the duration between the end moment of $m th$ successful attack and the first event-triggered instant after the $m th$ successful DoS attack as shown in Figure 2. We assume that the $Δ_{m}$ has an upper bound $Δ_{*}$ . So, for any period $[t_{0}, t)$ , we generalize that $[t_{0}, t) = {\hat{Θ}}_{a} (t_{0}, t) ⋃ {\hat{Θ}}_{s} (t_{0}, t)$ , where ${\hat{Θ}}_{a} (t_{0}, t) = ⋃_{m \in N} [{\tilde{t}}_{m}, {\tilde{t}}_{m} + η_{m} + Δ_{m}) ⋂_{[} t_{0}, t)$ , ${\hat{Θ}}_{s} (t_{0}, t) = [t_{0}, t) \ {\hat{Θ}}_{a} (t_{0}, t)$ .

We assume that the attack energy is limited, and the attack frequency and attack duration are restricted, which will be shown in Theorem 3. Before giving the theoretical results, the related definitions are given.

Definition 1. Feng et al. (2017) and De Persis and Tesi (2015) (Attack Frequency): denote $n_{a} (z_{1}, z_{2})$ as the number of the DoS attack emerging during any time interval $[z_{1}, z_{2})$ , Thus, the attack frequency $F_{a} (\cdot)$ satisfies

F_{a} (z_{1}, z_{2}) = \frac{n_{a} (z_{1}, z_{2})}{z_{2} - z_{1}}

(54)

Definition 2. Feng et al. (2017) and De Persis and Tesi (2015) (Attack Duration): denote $D_{a} (z_{1}, z_{2})$ as the total time interval for system under the DoS attack during any time interval $[z_{1}, z_{2})$ . For existing scalars $z_{0} \geq 0$ and $τ_{a} > 0$ , the attack duration over $[z_{1}, z_{2})$ is defined as

D_{a} (z_{1}, z_{2}) \leq z_{0} + \frac{z_{2} - z_{1}}{τ_{a}}

(55)

The communication link will be interrupted by the attacker in the attack area. Construct Lyapunov function $V_{1 a} (t) = ξ_{1}^{T} (t) (H \otimes R_{1}) ξ_{1} (t)$ , $V_{2 a} (t) = ξ_{2}^{T} (t) (H \otimes R) ξ_{2} (t)$ , $V_{3} (t) = \sum_{i = 1}^{N} τ_{i} \exp (- m_{i} t) \frac{{(d_{i} (t) - {\tilde{d}}_{i})}^{2}}{2}$ , and $V_{4} (t) = \sum_{i = 1}^{N} η_{i} (t)$ , $V_{5 a} (t) = ξ_{01}^{T} (t) R_{2} ξ_{01} (t)$ . Let $V_{a} (t) = V_{1 a} + V_{2 a} + V_{3} + V_{4} + V_{5 a}$ . Take the time derivative of $V_{a}$

{\overset{\cdot}{V}}_{1 a} (t) = ξ_{1}^{T} (t) (H \otimes R_{1} (A + GC) + {(A + GC)}^{T} R_{1}) ξ_{1} (t)

(56)

{\overset{\cdot}{V}}_{2 a} (t) = ξ_{2}^{T} (t) (H \otimes RA + A^{T} R) ξ_{2} (t) + 2 ξ_{2}^{T} (t) (H \otimes RGC) ξ_{1} (t)

(57)

{\overset{\cdot}{V}}_{3} (t) = - \sum_{i = 1}^{N} τ_{i} m_{i} \exp (- m_{i} t) \frac{{(d_{i} (t) - {\tilde{d}}_{i})}^{2}}{2}

(58)

{\overset{\cdot}{V}}_{4} (t) = \sum_{i = 1}^{N} - μ_{i} η_{i} (t)

(59)

{\overset{\cdot}{V}}_{5 a} (t) = ξ_{01}^{T} (t) (R_{2} (A + GC) + (A + GC)^{T} R_{2}) ξ_{01} (t) .

(60)

Based on Young’s inequality, getting $2 ξ_{2}^{T} (t) (H \otimes RGC)$ $ξ_{1} (t) \leq ε_{m} ξ_{2}^{T} (t) (H \otimes RR) ξ_{2} (t) + \frac{1}{ε_{m}} ξ_{1}^{T} (t) (H \otimes (GC)^{T} GC) ξ_{1} (t)$ . Then, we have ${\overset{\cdot}{V}}_{2 a} + {\overset{\cdot}{V}}_{1 a} \leq ξ_{2}^{T} (t) (H \otimes RA + A^{T} R + ε_{m} RR) ξ_{2}$ $(t) + ξ_{1}^{T} (t) (H \otimes R_{1} (A + G C) + {(A + G C)}^{T} R_{1} + \frac{1}{ε_{m}} {(G C)}^{T} (G C)) ξ_{1} (t)$ .

Perceive that for positive constants $β_{m 1}$ and $β_{m 2}$ such that $RA + A^{T} R + ε_{m} RR - β_{m 1} R \leq 0$ and $R_{1} (A + GC) + (A + GC)^{T} R_{1} + \frac{1}{ε_{m}} (GC)^{T} (GC) - β_{m 2} R_{1} \leq 0$ . Hence, ${\overset{\cdot}{V}}_{2 a} + {\overset{\cdot}{V}}_{1 a} \leq β_{m 1} V_{2 a} + β_{m 2} V_{1 a}$ . Take $μ = \min {μ_{i}, i = 1, 2, \dots, N}$ , and we have ${\overset{\cdot}{V}}_{4} (t) \leq - μ V_{4} (t)$ . Because matrix $A_{e} = A + GC$ is Hurwitz, so ${\overset{\cdot}{V}}_{5 a} (t) \leq 0$ .

On the basis of above analysis, taking $β_{m} = max {β_{m 1}, β_{m 2}}$ , we can get

{\overset{\cdot}{V}}_{a} (t) \leq β_{m} V_{a} (t)

(61)

Theorem 3. The linear MAS equation (1) will attain ASOCT under the ETM equations (7) and (8) and adaptive control protocols (5) and (6) if for the whole period, the DoS attack frequency satisfies

F_{a} (z_{1}, z_{2}) = \frac{n_{a} (z_{1}, z_{2})}{z_{2} - z_{1}} \leq \frac{γ^{*}}{(σ + β_{m}) Δ_{*} + 2 \ln (r)}

(62)

where $0 < γ^{*} < σ$ , and the DoS attack duration satisfies

τ_{a} \leq \frac{σ + β_{m}}{σ - γ^{*}}

(63)

Proof. From equations (46) and (61), we effortlessly obtain that

{\begin{matrix} V (t) \leq \exp {- s χ - σ (t - t_{0})} V (t_{0}) t \in {\hat{Θ}}_{s} (t_{0}, t) \\ V_{a} (t) \leq \exp {β_{m} t} V_{a} (t_{0}) t \in {\hat{Θ}}_{a} (t_{0}, t) \end{matrix}

(64)

We assume that the system will be triggered $q_{m}$ times during the communication interval $[{\bar{t}}_{m}, {\tilde{t}}_{m + 1})$ . Thus

V (t) \leq \exp {- q_{m} χ - σ (t - {\bar{t}}_{m})} V ({\bar{t}}_{m}), t \in [{\bar{t}}_{m}, {\tilde{t}}_{m + 1})

(65)

When $t$ belongs to the attack interval, that is, $t \in [{\tilde{t}}_{m}, {\tilde{t}}_{m} + η_{m} + Δ_{m})$ , from equation (61), we have

{\begin{matrix} V_{a} (t) \leq \exp {β_{m} (t - {\tilde{t}}_{m})} V_{a} ({\tilde{t}}_{m}) \\ V_{a} ({\bar{t}}_{m}) \leq \exp {β_{m} ({\bar{t}}_{m} - {\tilde{t}}_{m})} V_{a} ({\tilde{t}}_{m}) \end{matrix}

(66)

Let

r = \max {\frac{λ_{\max} (P)}{λ_{\max} (R)}, \frac{λ_{\max} (P_{1})}{λ_{\max} (R_{1})}, \frac{λ_{\max} (P)}{λ_{\max} (R)}, \frac{λ_{\max} (R_{1})}{λ_{\max} (P_{1})}, 1}

Combining equations (65) and (66), we have

\begin{matrix} V (t) \leq \exp {- q_{m} χ - σ (t - {\bar{t}}_{m})} V ({\bar{t}}_{m}) \\ \leq r \exp {- q_{m} χ - σ (t - {\bar{t}}_{m}) + β_{m} ({\bar{t}}_{m} - {\tilde{t}}_{m})} V_{a} ({\tilde{t}}_{m}) \\ \leq r^{2} \exp {- (q_{m} + q_{m - 1}) χ - σ (t - {\bar{t}}_{m} + {\tilde{t}}_{m} - {\bar{t}}_{m}) \\ + β_{m} ({\bar{t}}_{m} - {\tilde{t}}_{m})} V ({\bar{t}}_{m - 1}) \\ \leq \dots \\ \leq r^{2 m} \exp {- p χ - σ | {\hat{Θ}}_{s} (t_{0}, t) | + β_{m} | {\hat{Θ}}_{a} (t_{0}, t) |} V (t_{0}) \\ \leq r^{2 m} \exp {- σ | {\hat{Θ}}_{s} (t_{0}, t) | + β_{m} | {\hat{Θ}}_{a} (t_{0}, t) |} V (t_{0}) \end{matrix}

(67)

where $p = q_{m} + q_{m - 1} + \dots + q_{0}$ is the triggered times during time interval $[t_{0}, t)$ .

If $t \in [{\tilde{t}}_{m}, {\bar{t}}_{m})$

\begin{matrix} V (t) \leq \exp {β_{m} (t - {\tilde{t}}_{m})} V_{a} ({\tilde{t}}_{m}) \\ \leq r \exp {β_{m} (t - {\tilde{t}}_{m})} V ({\tilde{t}}_{m}) \\ \leq r \exp {- q_{m - 1} χ - σ ({\tilde{t}}_{m} - {\bar{t}}_{m - 1}) + β_{m} (t - {\tilde{t}}_{m})} V ({\bar{t}}_{m - 1}) \\ \leq \dots \\ \leq r^{2 m - 1} \exp {- p χ - σ | {\hat{Θ}}_{s} (t_{0}, t) | + β_{m} | {\hat{Θ}}_{a} (t_{0}, t) |} V (t_{0}) \\ \leq r^{2 m - 1} \exp {- σ | {\hat{Θ}}_{s} (t_{0}, t) | + β_{m} | {\hat{Θ}}_{a} (t_{0}, t) |} V (t_{0}) \end{matrix}

(68)

We aware the relationship between $| {\hat{Θ}}_{a} (t_{0}, t) |$ and $| {\hat{Θ}}_{s} (t_{0}, t) |$ satisfies

{\begin{matrix} | {\hat{Θ}}_{s} (t_{0}, t) | = (t - t_{0}) - | {\hat{Θ}}_{a} (t_{0}, t) | \\ | {\hat{Θ}}_{a} (t_{0}, t) | = | Θ_{a} (t_{0}, t) | + (1 + n_{a} (t_{0}, t)) Δ_{*} \end{matrix}

(69)

Based on equations (67) and (68)

\begin{matrix} V (t) \leq r^{2 m} \exp {- σ (t - t_{0}) + (σ + β_{m}) | {\hat{Θ}}_{a} (t_{0}, t) |} V (t_{0}) \\ \leq r^{2 m} \exp {- σ (t - t_{0}) + (σ + β_{m}) (| Θ_{a} (t_{0}, t) | + (1 + n_{a} (t_{0}, t)) Δ_{*})} V (t_{0}) \\ \leq r^{2 m} \exp {- σ (t - t_{0}) + (σ + β_{m}) (T_{0} + \frac{t - t_{0}}{τ_{a}} + (1 + n_{a} (t_{0}, t)) Δ_{*}))} V (t_{0}) \\ \leq r^{2 m} \exp {(σ + β_{m}) (T_{0} + Δ_{*}) + [- γ^{*} + (σ + β_{m}) F_{a} (t_{0}, t) Δ_{*}] (t - t_{0}) \\ - [σ - \frac{σ + β_{m}}{τ_{a}} - γ^{*}] (t - t_{0})} V (t_{0}) \end{matrix}

(70)

Then, based on equations (62) and (63), we have

lim_{t \to \infty} V (t) = 0

Thus, the system will achieve secure consensus.□

Simulation

We provide an example in this section to verify the efficacy of designed controllers and DETM under DoS attack.

In the example, we consider that the linear MAS equation (1) contains four followers and one leader. Suppose the dimension of state $x_{i}$ is 3 and the dimension of output $y_{i}$ is 2. Figure 3 display the communication relationship of the MAS equation (1).

Suppose the system matrices $A$ , $B$ , $C$ and the Laplacian matrix $H$ are as follows: $A = [\begin{matrix} - 2.3 & 5.9 & 0 \\ 1 & - 1 \\ 0 & - 7 & 0 \end{matrix}],$ $B = [\begin{matrix} 5.7 & 0 \\ 0 & 1.5 \\ 1 & 0 \end{matrix}],$ $C = [\begin{matrix} 2 & 0.7 & 0 \\ 0 & 2.3 & 1 \end{matrix}],$ $H = [\begin{matrix} 3 & - 1 & 0 & - 1 \\ - 1 & 2 & 0 & - 1 \\ 0 & - 1 & 2 & - 1 \\ - 1 & 0 & - 1 & 2 \end{matrix}]$ , $G = [\begin{matrix} - 4.0842 & - 0.0205 \\ - 1.8653 & - 1.0161 \\ 9.4147 & - 6.8888 \end{matrix}],$ and $F = [\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}] .$

The other parameters are given as, $ε_{1} = 0.5, ε_{3} = 0.2, ε_{4} = 0.3, ε_{6} = 10$ , $ε_{7} = 1.8,$ $ε_{8} = 0.6$ , $c = 80$ , $\tilde{d} = [15, 15, 18, 13]$ , $μ = [12, 15, 14, 18]$ , $l = [1300, 1302, 2130, 832]$ , $θ = [2, 0.5, 2, 5]$ , $ζ = 0.0178$ , $τ = [4, 6, 7, 10]$ , $δ = [9.5, 9, 12, 11]$ , and $π = [0.0008, 0.0008, 0.0005, 0.0012]$ .

In this example, we calculate a group of feasible solutions of satisfying the matrix inequity conditions (16)–(20) in Theorem 1 as: $P = [\begin{matrix} 0.1854 & 0.1097 & - 0.2786 \\ 0.1097 & 0.7305 & - 0.5517 \\ - 0.2786 & - 0.5517 & 1.7541 \end{matrix}],$ $P_{1} = [\begin{matrix} 50.5770 & 0.4803 & - 7.5028 \\ 0.4803 & 102.1255 & - 178.2714 \\ - 7.5028 & - 178.2714 & 534.4691 \end{matrix}],$ and $P_{2} = [\begin{matrix} 18.0284 & 21.4491 & - 38.2952 \\ 21.4491 & 79.2341 & - 123.5012 \\ - 38.2952 & - 123.5012 & 319.1145 \end{matrix}]$ .

Then, based on the above matrix solutions, we chose a group of parameters as: $β_{m 1} = 11.0956, β_{m 2} = 12.6166, r = 3.9816, σ = 0.2286, ε_{m} = 4.32$ . Founded on the conditions (62) and (63) in Theorem 3, we easily get that the DoS attack frequency need to satisfy $F_{a} \leq 0.0726$ and DoS attack duration need to satisfy $τ_{a} \leq 28.0934$ . So, we select the attack interval as $Θ_{a} = [1.2, 1.6] \cup [3, 4] \cup [4.3, 4.7] \cup [6, 6.3] \cup [9.1, 9.5]$ . The gray parts in Figure 5 are the attack intervals.

Figure 4 illustrates the event-triggered instants of different followers. From Figure 4, it is not difficult to get that the Zeno behavior does not exhibit.

Figure 5 shows the output tracking errors of different followers on different dimensions under attack. Figure 6 are the observer errors $ξ_{i 1}, i = 1, 2, 3, 4 .$ of followers on different dimensions. From Figures 5 and 6, the MAS under DoS attack can achieve the ASOCT with designed impulsive control protocol and DETM.

In this paper, we adopt the estimated values of the observers are used to replace the sampled state information in DETM. In Table 1, the authors provide the comparison of the triggered number between different distributed state errors. From the table, we find that the DETM in this paper is more effective.

Table 1.

Triggered number with different distributed state error.

	${\tilde{p}}_{i} (t) = \exp {A (t - t_{k}^{i})} p_{i} (t_{k}^{i})$	${\tilde{p}}_{i} (t) = p_{i} (t_{k}^{i})$
Agent 1	113	153
Agent 2	144	159
Agent 3	124	139
Agent 4	165	166
Sum	546	617

For showing the effect of DoS attack, we give an external example. Figure 7 shows the output tracking errors with higher attack frequency and longer attack interval. Figure 8 shows the output tracking errors without attack. From Figures 7 and 8, it is easy to find that the system under higher attack frequency and longer attack interval has poorer stability compared with the system without attack.

Conclusion

This paper investigates ASOCT for general linear MASs with DoS attacks via novel dynamic ETIC protocols. We propose an observer-based distributed impulsive control scheme, and the estimated values of the observer are used in the designed control protocols and ETM instead of sampled information. The DoS attack interrupts all communication links between agents and further damages the controllers of agents. Some sufficient conditions for attack duration and attack frequency are reduced to handle the ASOCT under the DoS attack. Moreover, this paper considers a pull-based DETM, which means that each agent updates its controller only at its own triggering time while its neighbors will not be affected, as well each agent dynamically regulates its event-triggering threshold with the help of auxiliary parameters, which can effectively reduce controllers’ updates and allows larger triggering intervals. Furthermore, we give rigorous mathematical proofs and derive some sufficient conditions for assuring ASOCT of the linear MASs. We also demonstrate that the Zeno behavior will not exist. Finally, we offer the simulations to show the validity of the ETM and controller. In the future, one possible work is to extend the control approaches to the time-varying formation containment problem for heterogeneous MASs with unknown nonlinear function based on the switching direct graph via DETM.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by China Scholarship Council CSC202007090024 and the National Natural Science Foundation of China under grants 61977004.

ORCID iD

Guoguang Wen

Data availability statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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