Dual-channel event-triggered consensus of multi-agent systems under DoS attacks over switching topologies

2.1. Graph theory

The communication topology between $$ N $$ agents can be represented by a leader-following undirected digraph $$ {\cal G} = \{ {\cal V}, {\cal E}, {\cal A}\} $$, where $$ {\cal V} = \left\{ {{v_1}, {v_2}, \ldots , {v_N}} \right\} $$ denotes the set of nodes, $$ {\cal E} = {\cal V} \times {\cal V} $$ represents the set of edges and an adjacency matrix $$ {\cal A} $$. The edge $$ {e_{ij}} $$ is represented by a pair of vertices $$ \left( {{v_j}, {v_i}} \right) $$. If $$ {a_{ij}} = 1 $$ and $$ {e_{ij}} \in {\cal E} $$, agent $$ j $$ can exchange information with agent $$ i $$; otherwise, $$ {a_{ij}} = 0 $$. If $$ {a_{ij}} = 1 $$, agents $$ i $$ and $$ j $$ are neighboring nodes of each other. The set of neighboring agents $$ i $$ is denoted by $$ {N_i} = \left\{ {j \in {\cal V}:{e_{ij}} \in {\cal E}} \right\} $$. If $$ i \ne j $$, the Laplace matrix of $$ {\cal G} $$ can be defined as $$ {l_{ii}} = \sum\nolimits_{j = 1, j \ne i}^N {{a_{ij}}} $$, $$ {l_{ij}} = - {a_{ij}} $$. The real part of the eigenvalues of the $$ L $$ matrix satisfies $$ 0{\rm{ = }}{\mathit{λ} _1}\left( L \right) \le {\mathit{λ} _2}\left( L \right) \le \cdots \le {\mathit{λ} _N}\left( L \right) $$. In particular, $$ {\mathit{λ} _1} = 0 $$ corresponds to a feature vector $$ {{\bf{1}}_N} $$. If $$ {{m_i}=1} $$, the follower agent $$ i $$ can exchange information with the leader; otherwise, $$ {{m_i}=0} $$. Define $$ H=L+M $$, $$ M = diag\left\{ {{m_1}, {m_2}, ..., {m_N}} \right\} $$.

Existence of all possible communication topologies for multi-agent systems can be represented by a set of connected undirected graphs $$ \mathcal{G}_{\rho}=\begin{Bmatrix}\mathcal{G}_{1}, \mathcal{G}_{2}\cdots\mathcal{G}_{O}\end{Bmatrix} $$, where $$ O>0 $$ is the number of switching topologies for the entire system. Considering the case of random switching topology, there exists a continuous edge probability matrix $$ \boldsymbol{\aleph}=\left(\ell_{ij}\right)_{N\times N} $$ and $$ 0\leq\ell_{ij}\leq1 $$. The probability of a connection between every two points is independent of each other. Since there are only two possibilities of connection or non-connection between two nodes, a sequence of $$ N(N-1) $$ independent Bernoulli trials is used to describe it. Consider a non-empty set of continuous time intervals $$ \{[t_k, t_{k+1})\mid k=0, 1, \ldots, M\} $$, where $$ t_{0}=0, \tau_{*}<t_{k+1}-t_{k}\leq\tau^{*}, \tau_{*}>0 $$ is a positive constant. The topology is invariant within each interval $$ [t_k, t_{k+1}) $$; the switching signals $$ \Upsilon $$ and $$ L_\Upsilon $$ are constant within this interval.

Remark 1According to the definition of the lower bound of switching time $$ \tau_{*} $$, the number of topology switches must be kept within a certain range. If switching occurs too frequently, the excessive switching may negatively affect both the dynamic performance and the stability of the system. This paper utilizes Bernoulli random switching, leveraging multiple iteration step sizes. By modulating the size of these multiples, one can regulate the frequency of random transitions between different communication topologies.

2.2. DoS attacks model

The DoS attacks usually involve injecting large amounts of data into a communication channel to paralyze the channel, with the main aim of preventing information interaction and affecting monitoring equipment. Some useful assumptions and common lemmas are presented for further analysis.

Assumption 1^[23] The number of DoS attacks in the $$ [{t_0}, t] $$ is $$ N_0({t_0}, t) $$. We assume the constants $$ {N_1}>0 $$ and $$ {T^f} > 0 $$ such that

Assumption 2^[23] The total time of DoS attacks in the $$ [{t_0}, t] $$ is $$ \Xi_a({t_0}, t) $$. We assume the constants $$ {N_2}>0 $$ and $$ {T^d} > 0 $$ such that

Remark 2Assumptions 1 and 2 limit the frequency and duration of attacks to make control meaningful. When the average attack frequency is greater than the sampling frequency of the system, the system is uncontrollable. In addition, these assumptions are justified by the fact that the duration is limited by energy.

2.3. Problem statement

First, consider a MAS that has one leader and $$ N $$ followers. In this system, the state of the leader $$ X_0 $$ is proposed as

and followers’ dynamics is

where $$ \mathbb{S} = 1, 2, ..., N $$; $$ {x_0}(t) \in {\Re ^n} $$ is the states of the leader. The $$ {x_i}(t) \in {\Re ^n} $$ and $$ {u_i}(t) \in {\Re ^p} $$ are the states and control input of agent $$ i $$. The $$ A \in {\Re ^{n \times n}} $$ and $$ B \in {\Re ^{n \times p}} $$ are the constant matrices of the corresponding dimension. The following assumption is considered.

Assumption 3The topology graph of MASs $$ \mathcal{G} $$ is undirected and connected; the pair of $$ (A, B) $$ is stabilisable.

Lemma 1^[24] Under Assumption 3, a unique solution denoted as P which satisfies the algebraic Riccati equation as follows

where $$ \kappa>0 $$.

Definition 1Given initial conditions, the consensus of MASs is realized if

Lemma 2^[25]$$ {\cal G} $$ is structurally balanced; thus, there is a matrix $$ \Delta = diag\left\{ {{\Delta _1}, {\Delta _2}, \ldots , {\Delta _N}} \right\} > 0 $$ such that $$ \Delta H + {H^T}\Delta > 0 $$.

3.1. Controller design

Firstly, the follower's controller is designed as:

where $$ K $$ is a constant gain matrix; $$ \Xi_s ({t_0}, t) $$ is the period when the MASs without DoS attacks. In certain studies, the controller relied on the state value at the triggering instants. However, in this research, the controller employs an estimated value instead. The design of the estimator is outlined below:

where $$ \omega_j>0 $$ is a constant; $$ \hat x_j $$ is the estimated state of $$ j $$th agent in $$ [t_k^j, t_{k + 1}^j) $$. When $$ t=t_k^j $$, there is $$ \hat x_j=x_j $$. The error variable $$ e_i(t) $$ of the $$ i $$th agent is

The time derivative of the error variable $$ e_i(t) $$ is

Consider a positive definite matrix $$ Q $$ that fulfills the given Riccati inequality

where $$ r $$ is minimum eigenvalue of matrix $$ H $$. And matrix $$ K $$ is calculated by $$ K=B^TQ $$ in Equation (7). According to the tracking error $$ \psi_i(t)={x}_i(t)-{x}_0(t) $$. The time derivative of tracking error $$ \psi_i(t) $$ as

Then we can obtain

where $$ \Gamma (t)={(\Gamma _1^T(t), \Gamma _2^T(t), ..., \Gamma _N^T(t))^T} $$. Thus, Equation (13) can be rewritten in Kronecker forms as

where $$ \Psi (t) = {(\Psi _1^T(t), \Psi _2^T(t), ..., \Psi _N^T(t))^T} $$, $$ e(t) = {(e_1^T(t), e_2^T(t), ..., e_N^{\text{T}}(t))^T} $$.

3.2. Dynamic event-triggered design

Considered a dynamic ETC protocol for agent $$ i $$ with a time-varying parameter threshold $$ {\theta _i}(t) $$

where $$ f\left( {{e_i}(t), {\Gamma _i}(t)} \right) = (\frac{{{\varepsilon _2}}}{\tau } + {\varepsilon _3}){\left\| {{e_i}(t)} \right\|^2} - {\sigma _i}({\varepsilon _1} - \tau {\varepsilon _2}){\left\| {{\Gamma _i}(t)} \right\|^2} $$; $$ ({\varepsilon _1} - \tau {\varepsilon _2}) > 0 $$, $$ \tau > 0 $$, $$ \mathit{𝛾} _i > 0 $$ and $$ {\sigma _i} \in (0, 1] $$. Other parameters $$ {\varepsilon _1}, {\varepsilon _2}, {\varepsilon _3} $$ are as follows:

the constant $$ r_{1} > 0 $$ satisfies $$ 0<r_1<r $$ and $$ \varepsilon _2>0 $$, $$ \varepsilon _3>0 $$. $$ {\theta _i}(t) $$ is designed as:

where $$ {\alpha _i}>0 $$. $$ {\theta _i}(t) $$ represents a time-varying parameter that updates are contingent upon self-feedback, measurement discrepancies, neighbor errors, and leader errors. It is noteworthy that $$ {\theta _i}(t) $$ possesses a property crucial to the subsequent process of proving or deriving the conclusion, as outlined in

Lemma 3For predefined scalars $$ {\varepsilon _1}, {\varepsilon _2}, {\varepsilon _3}, {\alpha _i}, {\theta _i}(0) $$ we can obtain

Proof: Consider the system (14) with (16)(18), when $$ t \in \cup _{k = 1}^\infty (t_k^i, t_{k + 1}^i] = (0, \infty ) $$ there exists $$ {k^\prime } $$, $$ t \in (t_{{k^\prime }}^i, t_{{k^\prime } + 1}^i] $$. The following formula holds: when the required parameter is not satisfied, the inequality

applies. When $$ t=t_{k'+1}^i $$, one has

and thus,

from $$ t_1^i=0 $$, one has

It is detailed that $$ {\theta _i}(0)>0 $$. Thus, $$ {\theta _i}(t)>0 $$ holds for $$ [0, \infty ) $$.

Theorem 1Considering the MASs (3)(4), consensus among the MASs can be achieved if there exists

where $$ {\mu _1} = \min \left\{ {\frac{1}{N}\sum\limits_{i = 1}^N {{\alpha _i}} , {r_2}} \right\} $$, $$ {\mu _2}>0 $$ is a constant. $$ {\rho _1} = \frac{{{\mathit{λ} _{\max }}({I_N} \otimes Q)}}{{{\mathit{λ} _{\min }}({I_N} \otimes P)}} $$, and $$ \rho_{2}=\frac{\mathit{λ}_{\mathrm{max}}(I_{N}\otimes P)}{\mathit{λ}_{\mathrm{min}}(I_{N}\otimes Q)} $$, $$ PA+A^TP<\mu_2P $$ and $$ P>0 $$.

Proof: Consider the MASs (3)(4) and design the following Lyapunov function

when $$ t\in\Xi_{s} (t_{0}, t) $$, the system is free from attacks. During this period, design $$ V_s\left(t\right)=V_1\left(t\right)+V_2\left(t\right) $$ and

The time derivative of $$ { V_1}\left( t \right) $$ is

where $$ r = {r_1} + {r_2} $$. According to Young's inequality $$ pq \le \frac{1}{{2k}}{\left\| p \right\|^2} + \frac{k}{2}{\left\| q \right\|^2} $$, we can rewrite $$ {\dot V_1}\left( t \right) $$ that

The time derivative of $$ {V_2}\left( t \right) $$ is

To sum up, we can obtain that

For $$ t\in\Xi_{a}\left(t_{0}, t\right) $$, there exist DoS attacks among the agents’ communication. We choose the following Lyapunov function

and its derivative is

From (27) and (29), we obtain

We can calculate that $$ {\mu_1}{\mu_2}>1 $$. For $$ t \in \left[ {{t_{2\left( {l - 1} \right)}}, {t_{2l - 1}}} \right) $$, it has

Under Assumptions 1-2, there is

and

The (31) can be scaled to

Therefore, as $$ t \in \left[ {{t_{2\left( {l - 1} \right)}}, {t_{2l - 1}}} \right) $$, we can obtain $$ V\left( t \right) \le {\varsigma _1}{e^{ - \beta \left( {t - {t_0}} \right)}}V\left( {{t_0}} \right) $$ where $$ \varsigma_1=(\rho_1\rho_2)^{N_1}e^{(\mu_1+\mu_2)N_2}. $$ And as $$ t \in \left[ {{t_{2\left( {l - 1} \right)}}, {t_{2l - 1}}} \right) $$, it has $$ V\left(t\right)\leq\varsigma_{2}e^{-\beta\left(t-t_{0}\right)}V\left(t_{0}\right) $$ as above, where $$ \varsigma_2={\rho_1}^{N(t_0, t)-1}\rho_2^{N(t_0, t)}e^{(\mu_1+\mu_2)N_2} $$. By analyzing the above two situations, we can obtain

where $$ \varsigma=\max{\{\varsigma_1, \varsigma_2\}}. $$ Similar to Theorem 1, the consensus of MASs was realized.

3.3. Zeno behavior analysis

In this chapter, we will analyze and confirm that Zeno behavior is excluded in the systems under two specific scenarios. There exists inequality

where $$ Z_{_k}^{i}=2\|\dot{\hat{x}}_{i}(t)\|+\|(1+\omega_{i})Bu_{i}(t)\| $$. The proof that follows will be presented in two scenarios.

1. If $$ \|A\|\neq0 $$, from Equation (36), one can get

For the designed ETC protocol, the triggering instants $$ t_{k+1}^i $$ satisfies

Then,

Subsequently, we employ proof by contradiction to demonstrate that Zeno behavior does not occur. First, we suppose that the Zeno behavior of agent $$ i $$ will occur, indicating that $$ \sum_{k=0}^{\infty}\Delta_{k}^{i} $$ must converge, where $$ \Delta_{k}^{i}=t_{k+1}^{i}-t_{k}^{i} $$ is a positive sequence, and then $$ \lim_{m\to\infty}\sum_{k=0}^{m}\Delta_{k}^{i}=\lim_{k\to\infty}t_{k+1}^{i}-t_{0}=\infty $$, which is divergent. Thus, Zeno behavior does not occur.

2.if $$ \|A\|=0 $$, then $$ \left\|e^T_i(t)\right\|\leq Z_{_k}^{i}(t-t_k^i) $$ The proof process for this case is analogous to the first cases and therefore will not be reiterated here. It can be seen that the Zeno behavior of agent $$ i $$ will be excluded from the above two situations. If the MASs are subjected to DoS attacks, $$ {u_i}(t) = 0 $$; the Zeno behavior of the agent $$ i $$ also will not occur.

3.4. Dual-channel event-triggered consensus

In this part, based on the idea of dual channels, the ETC protocol between the controller and actuator channel with the communication channel is designed to save resources. The Zeno behavior in the controller and actuator channel can be eliminated by introducing the exponential parameter, and the ETC protocol in the controller and actuator channel can still work stably when the input signal is very small. In addition, the ETC protocol in the controller and actuator channel is designed before the communication channel, and it is designed to depend only on the input signal and does not require neighbor information. First, an equivalent controller $$ {\tilde u_i}\left( t \right) $$ is designed as

For $$ t \in \Xi_s ({t_0}, t) $$, the controller update time depends on the following ETC protocol

where $$ \wp _i^u\left( t \right) = {{u_i}\left( t \right) - {{\tilde u}_i}\left( t \right)} $$, $$ 0 < {{\text{z}}_i} < 1 $$ and $$ {\upsilon _i} > 0 $$. Then, there exist $$ |\tilde{\alpha}_i(t)|\leq1 $$ and $$ |\tilde{\beta}_{i}(t)|\leq1 $$. Thus, we have

In that case, the tracking error $$ \psi_i(t) $$ is

where $$ {\hat \alpha _i}\left( t \right) = \frac{1}{{1 + {{\tilde \alpha }_i}\left( t \right){z_i}}} $$, $$ {\hat \beta _i}\left( t \right) = \frac{{{{\tilde \beta }_i}\left( t \right){e^{ - {\upsilon_i}t}}}}{{1 + {{\tilde \alpha }_i}\left( t \right){z_i}}} $$ and $$ {\hat \alpha _i}\left( t \right) < \frac{1}{{1 - {z_i}}} = {\bar \alpha _i} $$, $$ {\hat \beta _i}\left( t \right) \leqslant \frac{1}{{1 - {z_i}}}{e^{ - {\upsilon _i}t}} = \bar \beta {e^{ - {\upsilon _i}t}} $$. The tracking error $$ \psi $$ is written in the form as

where $$ \hat \alpha \left( t \right) = diag\left\{ {{{\hat \alpha }_1}\left( t \right), {{\hat \alpha }_2}\left( t \right), \ldots , {{\hat \alpha }_N}\left( t \right)} \right\} $$ and $$ \hat \beta \left( t \right) = {({\hat \beta _1}\left( t \right), {\hat \beta _2}\left( t \right), \ldots , {\hat \beta _N}\left( t \right))^T} $$. In the context of Equation (40), the following ETC protocol can be obtained

where $$ \hat f\left( {{e_i}(t), {{\hat \Gamma }_i}(t)} \right) = \left( {\frac{{({{\hat \varepsilon }_2} + {\delta _1})}}{{{k_1}}} + {{\hat \varepsilon }_3}} \right){\left\| {{\Gamma _i}(t)} \right\|^2} - {{\hat \sigma }_i}({{\hat \varepsilon }_1} - {k_2}{{\hat \varepsilon }_4} - {k_1}({{\hat \varepsilon }_2} + {\delta _1})){\left\| {{{\hat \Gamma }_i}(t)} \right\|^2} + {\delta _3}{e^{ - 2{\upsilon_i}t}} $$, $$ {{\hat \sigma }_i}({{\hat \varepsilon }_1} - {k_2}{{\hat \varepsilon }_4} - {k_1}({{\hat \varepsilon }_2} + {\delta _1})) >0 $$ with $$ {k_1}, {k_2} > 0 $$, $$ {\hat \sigma _i} \in (0, 1] $$. Other parameters are designed as follows:

where $$ \hat W = PB{B^T}P $$, $$ \dot{\hat{\theta}}_{i}(t)=-\widehat{\alpha}_{i}\hat{\theta}_{i}(t)-\hat{f}\left(e_{i}(t), \hat{\Gamma}_{i}(t)\right) $$, $$ \widehat{\alpha}_{i}>0 $$ and $$ \hat{\theta}_i(0)>0 $$.

Remark 3Dual-channel ETC ensures system stability by triggering actions only when necessary and adjusting parameters based on communication topology and agent dynamics to achieve system security consensus. Dual-channel ETC enhances the efficiency of the system by separating communication and control channels, reducing unnecessary communication. It also offers flexible triggering mechanisms, optimizing resource utilization and boosting overall system performance. Thus, while the dual-channel triggering protocol designed here is similar to the triggering form of the existing work, the design idea is different and from a practical point of view, the approach in this section is simpler and does not require additional wiring.

Remark 4In order to avoid continuous communication and further save communication resources, an estimator is introduced in this chapter. However, the use of the estimator leads to the inability to exclude Zeno behavior. A common approach is to use a constant trigger threshold to exclude Zeno behavior. However, this design will make $$ {\tilde u_i}\left( t \right) $$ can not reach $$ {u_i}(t) $$. Inspired by the references^{[26, 27]}, an exponential threshold $$ e^{ - {\upsilon _i}t} $$ is designed to solve problems caused by the estimator in this section. The designed controller update protocol still works properly when the input signal is extremely small.

Theorem 2Considering MASs (3)(4), with the control protocol (45), consensus among the MASs can be achieved if there exist

Proof: for $$ t\in\Xi_s ({t_0}, t) $$, choose $$ \hat{V}_s(t)=\hat{V}_1(t)+\hat{V}_2(t) $$, and we have

Then, applying (48), $$ \hat{V}_{1}(t) $$ designed as

where $$ \hat{K}=-B^TP $$. The time derivative of $$ {\hat V_1}\left( t \right) $$ is

From

we obtain

Applying Young's inequality, we get

Applying the derivatives of $$ \hat{V}_2(t) $$, then

The subsequent process of proving the consensus is similar to Theorem 1 and is not given to avoid redundancy. Next, Zeno behavior exclusion is divided into two categories: the actuator channel and the communication channel. 1. For the controller–actuator channel, we have

where $$ \wp _i^u\left( t \right) = {u_i}\left( t \right) - {\tilde u_i}\left( t \right) $$ then we have

Considering the event-triggered interval $$ [t_k^i, t_{k+1}^i), $$ there exist $$ |\wp _i^u(t_k)|=0 $$ and $$ \lim_{t\to t_{k+1}}|\wp _i^u(t)|=\nu_{i}|\tilde{u}_{i}(t)|+e^{-\upsilon{i}t} $$; then, we can obtain

If Zeno behavior exists, it means infinite triggers within a limited time $$ T_{c} $$, i.e., $$ \sum_{k=0}^\infty\Delta_k^i=T_c, $$ where $$ \Delta_k^i=t_{k+1}^i-t_k^i $$. Then, we have $$ \sum_{k=0}^{\infty}\Delta_{k}^{i}=\lim_{k\to\infty}t_{k+1}^{i}-t_{0}^{i}=T_{c} $$, indicating $$ \lim_{k\to\infty}t_{k+1}^i=T_c $$. Thus, we have $$ \sum_{k=0}^\infty\Delta_k^i=\infty $$ and $$ \lim_{k\to\infty}t_{k+1}^i=\infty $$. This contradicts $$ \operatorname*{lim}_{k\to\infty}t_{k+1}^{i}=T_{c} $$; thus, Zeno behavior will not occur. For dynamic ETC protocol in the communication channel, similar to chapter 3.4., Zeno behavior can be excluded. This proof is finished.

Acknowledgments

We would like to express our great appreciation to the editors and reviewers.

Authors’ contributions

Conceptualization, methodology, numerical simulations, writing: Li, W.; Chen, X.

Formal analysis, review, editing: Chen, X.; Cheng, L.; Ma, J.; Zhao, F.

All authors have read and agreed to the published version of the manuscript.

Availability of data and materials

The authors generated all data and materials used in the research as an integral part of the study, with explicit details provided in the methodology section of the manuscript. These are available from the corresponding author upon reasonable request.

Financial support and sponsorship

This research was supported in part by the National Natural Science Foundation of China under Grants 62173175, 62406132 and 61877033, and in part by the Natural Science Foundation of Shandong Province under Grants ZR2024MF032 and ZR2024QF039.

Conflicts of interest

Chen, X. is the Guest Editor of the Special Issue of “Adaptive Event-Triggered Control and Optimization for Complex Systems”. He is not involved in any steps of editorial processing, notably including reviewer selection, manuscript handling, or decision-making, while the other authors have declared that they have no conflicts of interest.

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Copyright

© The Author(s) 2025.