An important subject in cooperative control of autonomous MASs is to design suitable control schemes which utilize a reasonable amount of energy and computation resources. Conventional consensus frameworks are based on continuous-time update of the control protocol which is energy consuming and difficult to implement from the actuator point of view. Recently, event-triggered control (ETC) strategies are employed which enable the control protocol to be updated only if a pre-designed condition is satisfied. Several ETC strategies have been proposed for consensus in MAS (Peng and Li, 2018; Ge et al., 2019). More recently, dynamic event-triggering control (DETC) schemes (Hu et al., 2018; Deng et al., 2020; Zhao and Hua, 2021; Yang R. et al., 2022) have been recognized as one of the most efficient ETC schemes. Unlike conventional ETC schemes, in DETC an auxiliary dynamic variable is designed which helps in reducing the amount of control updates. The advantage of DETC scheme in reducing the amount of events over other simplified schemes is proved in Girard (2014). Recently, there has been a surge of considerable interest on DETC methodologies in different applications as surveyed in Ge et al. (2019). For instance, Meng et al. (2023) focused on DETC fault interval estimation for aeroengine sensors, where to reconstruct sensor fault a DET-based robust augmented state observer is designed. In (Cao et al., 2023) effects of time-varying delay on observer-based DETC mechanisms for MASs have been investigated. Furthermore, design of DETC mechanisms for distributed bipartite consensus in MASs has been considered in Du X. et al. (2023). He et al. (2022) surveys different secure control problems associated with MASs. The intuition behind this survey is the increased attack surface of MASs due to network-enabled information sharing as a consequence of expanded connectivity in practical scenarios. Along a similar path, Wang et al. (2023) targets surveying recent literature on resilient consensus control for MASs. Please refer to this work for a complete treatment of state-of-the-art concerning DoS attacks, spoofing attacks and Byzantine attacks in MASs, where attack model and mechanisms are introduced together with associated resilient consensus control structure. As a final note, when it comes to secure consensus of MASs, Shang (2022) proposed a median-based consensus strategy for resilient consensus control of MAS considering a time-varying directed random network. The proposed approach is superior to Weighted-Mean-Subsequence-Reduced techniques eliminating the need for the number of malicious agents in vicinity of each cooperating agent. In Shang (2021), the problem of resilient coordinated control in MASs is considered in presence o malicious agents, where agents' dynamics can be continuous-time or discrete-time. Furthermore, this work introduces the intriguing concept of heterogeneous robustness, which facilitates convergence analysis, and aims at capturing topological structure of the underlying network. Finally, Shang (2023) focused on resilient tracking consensus with a single leader considering a time-varying random directed graph, where in addition to cooperative agents, Byzantine agents are present.

Generally speaking, the DETC schemes often depend on multiple unknown parameters which should be designed based on the stability of the closed-loop MAS. The capability of the event-triggering schemes in reducing the number of control updates highly depends on the operating values of the design parameters. Regarding the DETC schemes, it is often the case that some feasible regions are derived for the design parameters (Hu et al., 2018; Yi et al., 2018; He et al., 2019; He and Mo, 2022). However, even when the feasible regions are known, selecting proper operating values that efficiently reduces the control updates is still inexplicable and requires trial and error. It is, therefore, desirable to develop a systematic design framework that computes the exact values of the unknown parameters and guarantee a substantial reduction for the control updates. Motivated by Peng and Yang (2013); Abdelrahim et al. (2014); Amini et al. (2022), where the convex optimization techniques are utilized to include some performance objectives (such as H_∞ optimization and inter-event interval maximization), in this article we develop a convex optimized design framework with a focus on reducing the control updates as much as possible.

Cyber security against malicious attacks is another important issue, which poses new challenges in performance and stability guarantees of the MASs. Generally, there exist three types of attacks targeted at cyber-physical systems, namely replay attacks, false data injection (FDI), and denial of service (DoS). In DoS (De Persis and Tesi, 2015; Zhang et al., 2018; Zhang and Feng, 2019; Liu et al., 2020), the adversary blocks the communication channels, hence the neighboring agents do not receive the transmitted signals. It is clear that DoS can significantly impact the behavior of the agents and, in extreme cases, can destabilize the MAS. In literature, the occurrence of DoS is often modeled either in a periodic (Hu et al., 2019; Xu Y. et al., 2019) or unknown pattern (Xu et al., 2018; Feng and Hu, 2019; Liu et al., 2020). In practice, the periodic scenario may not be able to fully model the pattern of DoS, as the adversary can launch the attacks in non-periodic patterns. A common assumption considered in many related works such as Xu et al. (2018); Feng and Hu (2019); Xu Y. et al. (2019); Zha et al. (2019); Amini et al. (2022); Deng and Wen (2020); Zhang and Ye (2021) is that DoS simultaneously paralyzes all communication channels. In this scenario, the MAS undergoes a binary situation based on the DoS being active or inactive. If DoS is inactive, the MAS operates normally based on the initially designated network. If DoS is active, the communication network is fully paralyzed and all agents are open-loop. In a more complex and more general DoS, which is referred to as the asynchronous (distributed) DoS (Lu and Yang, 2018; Xu W. et al., 2019; Yang Y. et al., 2020; Liu and Wang, 2021; Yang and Ye, 2022), the adversary attacks any arbitrary channel at different instants. Dealing with the asynchronous DoS is more challenging as the MAS may confront numerous connected or disconnected topologies depending on the status of each individual channel being healthy or under attack. Secure event-triggered consensus under asynchronous DoS is an important and challenging topic which, to date, has not been studied proportionately. It should be noted that the asynchronous DoS works (Lu and Yang, 2018; Xu W. et al., 2019; Yang Y. et al., 2020; Liu and Wang, 2021; Yang and Ye, 2022) have practical shortcomings which require further improvement. In particular, Lu and Yang (2018); Yang and Ye (2022); Liu and Wang (2021) are based on time-triggered or sampled-data control protocols, not event-triggered. Additionally, the ETC schemes used in Xu W. et al. (2019); Yang Y. et al. (2020) has lower inter-event interval compared to the more advanced schemes such as DETC. This motivates us to develop a DETC method for consensus under asynchronous DoS attack.

Motivated by the above discussion and following our previous work (Amini et al., 2022), this article studies the dynamic event-triggered control for consensus in general linear MASs under unknown and asynchronous DoS attacks. The main contributions of the article are as follows:

The remaining article is organized as follows. Section 2 introduces notation and discusses the problem to be studied. Section 3 formulates the consensus problem under DoS. Section 4 presents the main results and sufficient conditions to guarantee consensus. Simulation examples are included in Section 5. Finally, Section 6 concludes the paper.

Each agent measures and transmits its state value x_i(t) to its neighbors through an undirected network. As shown in Figure 1, a DETC scheme (which will be presented later) is employed to reduce the amount of control input updates. Only if the DETC condition is fulfilled an event is triggered and u_i(t) is updated. We denote the sequence {tki}k∈ℕ0 as the triggering times for agent i, where u_i(t) is being updated. Using this notation, xi(tki) is the state value at the k-th event for agent i. The inter-event interval for agent i is given by {tk+1i-tki}k∈ℕ0 which shows the time interval between two events in a row. We denote the following disagreement vector for agent i

Note that the neighboring set Ni(t) is time-varying since the DoS attack, as will be discussed later, can block the communication channel between agent i and any of its neighbors. The following control protocol is used for agent i to achieve consensus

where K ∈ ℝ^m×n is the control gain to be designed. Let ei(t)=xi(tki)-xi(t), ∀i∈V, denote the event-triggering error. Initialized by t0i=0, ∀i∈V, the next event instant is triggered by the following DETC condition (Yi et al., 2018)

where matrices Φ₁ ≥ 0, Φ₂ ≥ 0 and scalar ϕ₃ ≥ 0 are design parameters. The auxiliary state η_i(t) follows

where η_i(0) > 0. Scalar ϕ₄ ≥ 0 and matrix Φ₅ ≥ 0 are the other unknown parameters to be designed.

Remark 1. As observed in Equation (5), the updating protocol for η_i(t) is based on the disagreement vector q_i(t) and a negative quadratic self-feedback. Intuitively, η_i(t) can be regarded as a linear first-order filtered value of q_i(t). Compared to the conventional ETC strategy ||e_i(t)|| ≤ α_i∥q_i(t)∥ used in Hu et al. (2015), the utilization of the auxiliary variable η_i(t) helps in regulating the threshold (Equation 4) in a dynamic manner and in a better relationship with disagreement q_i(t). It is also proved in Girard (2014) that the inter-event interval using DETC (Equation 4) is larger than the conventional ETC strategy ||e_i(t)|| ≤ α_i∥q_i(t)∥, meaning less event triggering is proved. It is straightforward to show that the ETC schemes proposed in Qian et al. (2018); Wu et al. (2018); Xu W. et al. (2019); Yi et al. (2019) are all special cases of the DETC (Equation 4).

Next, we show that parameter η_i(t) remains positive over time.

Proposition 2. If Φ₅ > Φ₂ and η_i(0) > 0, ∀i∈V, parameter η_i(t) remains positive over time. In particular the following condition holds

Proof: Based on Equation (4), it holds that eiT(t)Φ1ei(t)-ϕ3ηi2(t)≤qiT(t)Φ2qi(t) for t∈[tki,tk+1i). If Φ₅ > Φ₂, we get that η˙i(t)≥-(ϕ3+ϕ4)ηi2(t)+eiT(t)Φ1ei(t). Since eiT(t)Φ1ei(t) is non-negative, it then follows that

By solving differential inequality (Equation 7) for t∈[tki,t), we obtain 1ηi(t)≤(ϕ3+ϕ4)(t-tki)+1η(tki). Considering one event-triggering interval back, i.e., t∈[tk-1i,tki), one obtains 1ηi(tki)≤(ϕ3+ϕ4)(tki-tk-1i)+1η(tk-1i). By successively moving backward through intervals [tki,t), [tk-1i,tki), …, [0,t1i) and comparing the associated inequalities the following is obtained

The proof is complete, noting that Equation (8) is equivalent to Equation (6).     □

From the implementation point of view, in an event-triggering scheme the time interval between two arbitrary event instants must be strictly positive. Otherwise, the event-detector scheme would potentially detect an infinite number of events in a finite interval (He et al., 2019). This undesirable phenomenon is known as the Zeno-behavior in the context of event-triggering control schemes. Therefore, it is necessary to exclude the possibility of Zeno-behavior in the proposed scheme. The exclusion of the Zeno-behavior is usually accomplished by obtaining a strictly positive lower-bound between two potential event instants. In other words, if we guarantee that the minimum inter-event time (MIET) between two successive event instants in Equation (4) is strictly positive, then the possibility of detecting infinite number of events in a finite period is ruled out. To exclude Zeno-behavior in Equation (4), in what follows we prove that the minimum inter-event interval (MIET) between any two events is strictly positive.

Proposition 3. The minimum inter-event time (MIET) for agent i, (∀i∈V), is strictly positive and lower-bounded by

where

Proof: Consider two consecutive event instants (tki and tk+1i) for agent i. Based on Equation (4), at t=tki it holds that ∥ei(tki)∥=0. For t≥tki, the event-triggering error e_i(t) evolves from zero until Equation 4 is satisfied and the next event is detected. From ei(t)=xi(tki)-xi(t) we obtain that e˙i(t)=-x˙i(t). From Equations (3) and (1), it holds that x˙i(t)=Axi(t)-BKqi(t). Combining the last three equations leads to e˙i(t)=Aei(t)-Axi(tki)+BKqi(t), or ∥e˙i(t)∥≤∥A∥∥ei(t)∥+∥Axi(tki)∥+∥BK∥∥qi(t)∥, t∈[tki,tk+1i). It, then, follows that ‖ei(t) ‖ ≤‖ A−1 ‖Fi(t)(e‖ A ‖(t−ki)−1) or equivalently

where F_i(t) is defined in Equation (10). The next event is detected by Equation (4) at t=tk+1i where ||eiT(tk+1i)Φ112||t2=qiT(tk+1i)Φ2qi(tk+1i)+ϕ3ηi2(tk+1i). Then, from Equation (6), it follows that ||eiT(tk+1i)Φ112|| 2≥qiT(tk+1i)Φ2qi(tk+1i)+ϕ3((ϕ3+ϕ4)tk+1i+1ηi(0))2. By combining the latter inequality with Equation (11), expression (9) is obtained. The lower-bound derived in Equation (9) is strictly positive which implies that tk+1i is strictly greater than tki, i.e., tk+1i>tki. Conceptually speaking, it is guaranteed that the next event instant is always greater than the current one. Therefore, the possibility of infinite event-triggering in finite time is ruled out and DETC (Equation 4) does not exhibit Zeno behavior.

Remark 2. In addition to excluding the possibility of the Zeno-behavior, another important observation can be made from expression (9). Based on the MIET, i.e., the right hand side of Equation (9), it can be shown that smaller values for ∥K∥, ∥Φ₁∥, and ϕ₄ increase the value of the MIET (i.e., the intensity of control updates is reduced). On the other hand, higher values for ||Φ₂||, ||Φ₃|| increase the MIET and help in reducing the control updates. Note that the impact of DETC parameters ||Φ₁||, ||Φ₂|| and ||Φ₃|| on the intensity of events is intuitive from Equation (4) and also confirmed by Equation (9). We use this observation in the proposed objective function considered in Theorems 1 and 2.

As shown in Figure 1, the DoS attacks, when active, target the communication channels and block the state transmission between the neighboring agents. Unlike many existing works (Xu et al., 2018; Feng and Hu, 2019; Xu Y. et al., 2019; Zha et al., 2019; Deng and Wen, 2020; Zhang and Ye, 2021; Amini et al., 2022), where it is assumed that the DoS attacks simultaneously block all communication channels, in this article we consider a more general and realistic scenario where the adversary attacks any arbitrary link.

Let G0=(V,E0,A0) denote the initially designed communication topology. When the adversary is completely inactive (i.e., none of the communication links are blocked) the MAS operates based on G₀. The associated Laplacian matrix to G₀ is defined by L₀. Let

denote the c-th DoS interval on channel (i, j). Parameter dcij is the time instant when the adversary begins the c-th attack on channel (i, j) and τcij is the duration of attack. Since DoS on channel (i, j) also implies DoS on channel (j, i), condition i < j is mentioned in Equation (12). Note that the first DoS can occur at t = 0, i.e., d0ij=0. Therefore, d0ij does not need to be strictly positive.

For channels (i, j) and (j, i), the state of “being under DoS1" or “being healthy" is a binary variable. Hence, there exist 2|E0|2 possible communication topologies labeled by G₀, G₁, …, G_f, where f=2|E0|2-1. Let Υ and Λ, respectively, denote the set of all possible graphs under DoS attack and their corresponding Laplacian matrices, i.e.,

During consensus iterations, the operating communication topology at time instant t is denoted by G(t)=(V,E(t),A(t)). It is clear that G(t) ∈ Υ, ∀t ≥ 0. We refer to E(t) as the set of healthy edges (i.e., not under attack) at instant t.

The DoS graph is defined by the edges that are blocked by DoS. The DoS graph at instant t is denoted by GD(t)=(V,ED(t),AD(t)), where (i,j)∈ED(t) if and only if the (i, j) communication channel is blocked by DoS at time t. In other words

It is straightforward to verify that the “healthy edges E(t)" and “blocked edges ED(t)" satisfy

In a similar fashion to Equation (13), we define the following sets which include all possible DoS graphs and their corresponding Laplacian matrices

Now, we classify the DoS attack modes based on their impact on the initial communication graph G₀.

Mode I. Connectivity-preserved DoS (CP-DoS): In this type of attack, a subset of the initial transmission links are attacked by DoS. However, the resultant communication topology remains connected.

Mode II. Connectivity-broken DoS (CB-DoS): In this scenario, a subset of the initial links are blocked and the resultant communication topology is disconnected. In the extreme case, all communication channels between the agents can be blocked which is referred to as the “full DoS".

Example: Consider a MAS with five nodes as shown in Figure 2. The initially designed communication topology is labeled with G₀ which consists of 6 bidirectional edges. At t = 1, two of the edges are blocked by DoS. The corresponding DoS graphs are shown above each rightward arrows. The attack at t = 1 is a CP-DoS as the resulting graph is still connected. Then, a CB-DoS occurs at t = 2 which isolates node 4. Note that the DoS graph is always constructed based on G₀, not the previously operating communication topology. Afterwards, the adversary is inactive at t = 3, so the MAS can operate based on G₀. Finally, a full DoS is observed at t = 4 which isolates all the nodes. Note that the cardinality of Υ is 2⁶ = 64 which implies that there exist 63 different topologies under DoS for graph G₀. For ease of reference, Table 1 lists important parameters used to model the communication topology of the MAS and DoS attacks.

The article addresses the following problems:

Problem 1. As observed earlier, the DETC protocol (Equation 4) and the auxiliary variable η_i(t) which follows Equation (5) depend on the knowledge of multiple unknown gains. These gains can significantly impact the consensus features such as the convergence rate, intensity of the events, and the amount of resilience to DoS. How to efficiently design these gains in a systematic way is a challenging matter. Many references such as Hu et al. (2018); Yi et al. (2018); He et al. (2019) derive some feasible regions for the DETC gains. However, even when the feasible regions are known, selecting the actual operating values that efficiently avoid unnecessary events remains an issue and requires some trial and error. As a more systematic approach, we propose a convex optimization to design the exact values of the unknown control and DETC gains based on an objective function which increases the minimum inter-event time (MIET). Increasing the MIET avoids unnecessary control updates.

Problem 2. In the presence of attack, it is important to develop a secure implementation under all the DoS modes and their different resulting topologies. While some of the 2|E0|2 variations may be homomorphic graphs, the exponential growth of the situations as per the number of edges makes dealing with all the possible scenarios difficult, especially for large networks. How to design proper control and DETC gains in response to CP-DoS and CB-DoS will be investigated. It is clear that if the MAS is subject to DoS attacks with unlimited duration, the control protocol cannot receive sufficient amount of information and consensus may not be achieved. Therefore, it is reasonable to consider an assumption regarding the finiteness of the attack duration and explicitly obtain the tolerable amount of resilience to DoS.

The asynchronous DoS given in Equation (12) can lead to both the CP-DoS and CB-DoS attacks. According to Proposition (1), if the operating graph G(t) becomes disconnected under DoS, it holds that λ₂(L(t)) = 0.

We define the c-th CB-DoS interval as follows

where

with r₋₁+v₋₁ = −1. Conceptually speaking, r_m is the earliest time when DoS on a link (i, j) leads to a disconnected graph [i.e., λ₂(L(t)) = 0]. Also, v_m is the duration of R_m, i.e., the earliest time after r_m when the graph becomes connected again [i.e., λ₂(L(t)) > 0]. Expression r₋₁+v₋₁ = −1 is selected for the initialization of r₀ as the first instance where connectivity of the network is broken. The union of all CB-DoS intervals for t ∈ [t₁, t₂) is

The complement of R_m is W_m, which represents either healthy or CP-DoS intervals. More precisely,

The union of all healthy or CP-DoS intervals for t ∈ [t₁, t₂) is given by

Based on Equations (14)–(19), let |R(t₁, t₂)| and |W(t₁, t₂)|, respectively, denote the accumulative length of corresponding intervals for t ∈ [t₁, t₂). Since W(t₁, t₂) and R(t₁, t₂) are complements of each other, one concludes that

The following assumption holds for the duration of the DoS attacks.

Assumption 1. There exist positive constants T₀, and T₁ such that the following upper-bounds hold (De Persis and Tesi, 2015)

The DoS attacks considered in Hu et al. (2019); Xu Y. et al. (2019) are assumed to follow a periodic pattern. Considering a periodic pattern for DoS may not fully represent the unknown and malicious nature of the adversary. Our considered asynchronous attack is more general, where the DoS is assumed to occur with an unknown pattern. Such a DoS model with unknown pattern can be characterized only by the energy constraints of the adversary. Assumption 1, which is widely used for formulation of unknown DoS attacks, constrains DoS in terms of its average duration. In other words, the strength of the DoS attacks (in terms of duration) is scalable with time. Inequality (Equation 21) expresses property that the DoS intervals satisfy a slow-on-the-average type condition. It implies that the total duration for DoS, on average, should not exceed a certain fraction of time, which is scaled by 1/T₁. Parameter T₀ is included to allow for consideration of DoS at the start time.

Let x=[x1T(t),…,xNT(t)]T, e=[e1T(t),…,eNT(t)]T, x~=[x1T(tk1),…,xNT(tkN)]T, η=[η1(t),…,ηN(t)]T. From Equations (1) and (3), the closed-loop MAS under DoS is given below

Next, we transform system (21) through the eigenvalue decomposition of L₀. It is straightforward to show that

where J̄0=diag(0,λ2(L0),…,λN(L0)) is a diagonal matrix consisting the eigenvalues of L₀ and matrix V̄=[v̄i,j]∈ℝN×N includes the normalized eigenvectors of L₀. We construct the (N−1) × N dimensional matrix V₀ which includes rows 2 to N of matrix V̄0T. In other words, matrix V₀ is obtained by removing the first row of matrix V̄0T (the corresponding eigenvector to eigenvalue zero). With this definition, it holds that L0=V0TJ0V0. Now, consider the following transformation

It is proved in Ge and Han (2017) that consensus is achieved in Equation (21) iff limt→∞z(t)=0. Using Equation 22, system (21) is converted to

where W(t)=V0VDT(t)JD(t)VD(t)V0T and J_D(t) = diag(λ₂(L_D(t)), …, λ_N(L_D(t))). Matrix V_D(t) with unity norm includes the eigenvectors of L_D(t).

In this section, we propose a theorem that guarantees consensus under the situation where MAS (Equation 23) is only subjected to CP-DoS. Section 4.2 extends the framework to both the DoS cases.

Theorem 1. Consider MAS (Equation 23) with the initially designed communication topology G0=(V,E0,A0) under CP-DoS attacks. Given a desired consensus convergence rate ω₁, if there exist positive definite matrices P_n×n > 0, M_1n×n > 0, M_2n×n > 0, M_5n×n > 0, free matrix Ω_m×n, positive scalars m₃ > 0, m₄ > 0, ϵ₁ > 0, and θ_c > 0, (1 ≤ c ≤ 7), such that the following convex minimization problem is feasible

subject to:

where

then the unknown parameters for control protocol (Equation 2) and DETC scheme (Equation 4) are designed as follows

The following bounds are guaranteed for the designed parameters associated with the convex minimization problem defined through Equations (24)–(30)

Using Equation (31), the convergence rate of z(t) satisfies

where μ=λmax(P-1)zT(0)z(0)+η(0).

Proof. For the sake of readability, we remove the time argument t in the proof. Consider the following expression

where V = V₁+V₂ and

If (34) is guaranteed, condition (33) is satisfied and ω₁ determines the exponential consensus convergence rate. We compute the time derivative for V₁ as follows

where Ξ=IN-1⊗(ATP-1+P-1A)-2(J0-JD)⊗P-1BK. Remind that in this theorem we assume that the MAS is either in healthy intervals or under CP-DoS. In this situation, all the diagonal elements of J₀−J_D are non-zero. Under condition (P⁻¹BK)^T + P⁻¹BK > 0, the following holds

We expand V˙2+ω1V2 based on Equation (5)

Since L_D ≥ 0, it holds that xT(L0-LD)2⊗Φ5x≤xTL02⊗Φ5x. Recalling that L0=V0TJ0V0, V0V0T=I, and using transformation (22), it is straightforward to show

Considering Equations (38, 39), and inequality ω1η≤ω12η2+ω122, we conclude that

As for the DETC (Equation 4), it holds that eiT(t)Φ1ei(t)≤qiT(t)Φ2qi(t)+ϕ3ηi2(t), t∈[tki,tk+1i). In a collective sense, we obtain eT(IN⊗Φ1)e≤qT(IN⊗Φ2)q+ϕ3ηTη. Similar to Equation (39), we can derive that qT(IN⊗Φ2)q≤zT(J02⊗Φ2)z. Therefore, the following condition is obtained

Let ν = [z^T, e^T, η^T]T. Based on Equations (36, 37, 40), and (41), we re-arrange Equation (34) as follows

Inequality (Equation 34) is guaranteed if Ψ̄1<0. We pre- and post multiply Ψ̄1 by ℙ = diag(I_N−1⊗P, I_N⊗P, I_N), which results in ℙΨ̄1ℙ. Denote the following alternative variables

Now, re-arrange ℙΨ̄1ℙ as follows

where

According to Young's inequality, condition (44) is guaranteed if there exists a positive scalar ϵ₁ such that

The only non-zero element of Y^TY is (IN⊗BΩ)T(V0TWTWV0⊗In)(IN⊗BΩ). Now, we consider the following upper-bound