Compliance estimates for two-dimensional problems with Dirichlet region of prescribed length

1.   Introduction

In recent years, multi-agent consensus control has been recognized as a vital element of distributed collaborative control for applications such as distributed computing, unmanned aerial vehicle formation, and intelligent transportation systems. Researchers have shown significant interest in this area, and a wide range of control mechanisms have been explored in the past few years. These control mechanisms include adaptive ^[1,2,3,4], fault-tolerant ^[5], impulse ^[6,7], and sliding mode ^[8] methods.

In practical systems, the stability of the system state is susceptible to disruption from unexpected factors, including nonlinear dynamics and system uncertainties. Existing research has primarily focused on continuous-time systems, where the state information of intelligent agents is continuously transmitted between nodes, leading to significant network usage and energy consumption. However, the development of event-triggered control solves this problem by avoiding constant communication. Earlier studies on event-triggered control that can be found have investigated centralized, distributed, and self-triggered event-triggered control techniques ^[9,10,11]. One researcher proposed an adaptive event-triggered control scheme for strongly connected networks that dynamically adjusted the triggering time interval on the basis of sampled data ^[12]. Another mechanism using a dynamic event-triggering mechanism was proposed to reduce communication resource wastage compared with traditional event-triggering mechanisms ^[13]. Others have assumed that system parameters such as the efficiency factor of the executor, external disturbances, and precursor control input signals are all unknown and introduced a fault-tolerant control to obtain sufficient conditions for consistent tracking ^[14]. However, these findings primarily investigate traditional event triggering mechanisms. In ^[15,16], researchers proposed a dynamic event-triggered mechanism, which can significantly reduce the number of triggers and conserve communication resources. researchers respectively proposed centralized and distributed dynamic event-triggered mechanisms in ^[15,16], while scholars suggested both centralized and distributed mechanisms in ^[17], verifying their superiority.

However, the rapid growth of network information technologies has also led to a rise in cyber attacks. Among them, denial-of-service (DoS) attacks are the most common, being relatively easy to execute in the attack space. These attacks usually target the controller or exhaust the resources of the target system directly, resulting in the system being unable to provide normal services or communication. In some cases, these attacks cause the system to crash. Therefore, countering DoS attacks has received significant research attention. Researchers studied the multi-agent systems(MASs) under DoS attacks in given attack frequency and upper bounds on attack duration ^[18,19]. Compared with linear systems, nonlinear MASs are more widely used in real life. Among these, a secure controller based on event triggering was proposed to solve the lead-following consensus problem of second-order nonlinear systems ^[20]. This is more common than linear systems. Another proposal was for an event-triggered adaptive fault-tolerant control strategy, which reduced the computational cost of heterogeneity ^[21]. For nonperiodic DoS attacks, the upper bounds of network attacks, actuator failures, attack duration, and frequency are obtained. Another method uses a security mechanism employing a prediction-based switching observer scheme to address the issue of invalidation in event-triggered mechanisms during attack intervals ^[22]. A novel framework for observer-based event-triggered containment control, taking into account the occurrence of DoS attacks, has also been introduced ^[23]. This framework establishes a resilient event-triggered controller, using a specially designed observer. The goal is to achieve consistent control of MASs in the presence of DoS attacks.

Based on these observations, we aim to explore the security consensus problem of nonlinear MAS with external disturbances under DoS attacks in this paper. Our contributions are as follows.

1) In this paper, a nonlinear system with external disturbances is considered, and the effects of the nonlinear dynamics and uncertainty of the system are eliminated by designing an adaptive scheme and state-feedback control gains by updating the laws of the adaptive parameters online.

2) Compared with ^[22,23], a dynamic variable is introduced to adjust the triggering instances under DoS attacks. Therefore, the event-triggered mechanism proposed in this paper is more flexible and can effectively save communication resources. In addition, continuous communication between agents is not required to determine whether a trigger condition satisfies the trigger condition.

Notation $ \mathbb{R} $ is the set of real numbers, and $ \mathbb{R}^{N\times N} $ is the set of $ N\times N $ real matrix. $ \left \| \cdot \right \| $ represents a Euclidean norm of vectors or matrices. The superscripts $ A^{-1} $ and $ A^{T} $ represent the inverse and transpose of matrix $ A $. $ \lambda _{max}(A) $ is the maximum eigenvalue, and $ \lambda _{min}(A) $ is the minimum eigenvalue of matrix $ A $. $ D^{+}(\cdot) $ denotes the righthand derivative of a function, and $ \bigotimes $ is Kronecker product. $ diag\left \{A_{1}, \cdots, A_{n} \right \} $ is the diagonal matrix. $ \cap $ is the intersection of sets, and $ \cup $ denotes the union of sets.

2.   Problem statement and preliminaries

2.1.   Communication graphs

For a given MAS, the digraph $ \mathcal{G} $ is $ \left (\mathcal{V}, \mathcal{E}\right) $, where $ \mathcal{V} = \left \{1, 2, \cdots, N\right \} $ is the set of nodes, and $ \mathcal{E}\subseteq \mathcal{V} \times \mathcal{V} $ represents the edge set of followers. The information exchange between each node can be described by the adjacency matrix $ A $ and the Laplacian matrix $ L $. $ A = \left [ a_{ij}\right ]\in \mathbb{R}^{N\times N} $ if agents $ i $ and $ j $ communicate with one another, $ a_{ij} = 1 $; otherwise $ a_{ij} = 0 $ and $ L = \left [ l_{ij}\right ]\in \mathbb{R}^{n\times n} $ where $ L = D-A $, The degree matrix $ D = diag(d_{i}) $ with $ d_{i} = \sum_{i = 1}^{N}a_{ij} $. In this paper, we assume that the agents are linked by a balancing topology, i.e., $ a_{ij} = a_{ji} $. If the agent $ i $ communicates with the leader, then $ b_{i} = 1 $; otherwise, $ b_{i} = 0 $.

2.2.   MAS modeling

For a leader-following system, the dynamics of the leader are described by the equation

The $ i $th follower system is

In the preceding, $ x(t)\in \mathbb{R}^{n} $ are positions of the agent, $ u_{i}(t)\in \mathbb{R} $ is the control input, $ A\in \mathbb{R}^{n\times n} $ and $ B\in \mathbb{R}^{n\times p} $ are system matrices, $ f(x) $ is a nonlinear function, and $ w_{i} $ is the uncertainty input satisfying

where $ \varsigma _{i1} < 1 $, $ \varsigma _{i2} $ and $ \gamma_{i} $ are unknown constants.

Lemma 1. If the nonlinear function $ f(t, x_{i}(t)), i = 1, 2, ..., $ is continuously differentiable in a region $ S\in R^{2} $ and $ x_{i}(t_{0})\in S $, then for any $ x_{i}(t_{0})\in S $, the following formula is satisfied:

where $ f(\cdot) = f(x_{i}(t_{0}))+\Delta (x_{i}(t)-x_{i}(t_{0})) $, $ 0 < \Delta < 1 $.

Assumption 1. If there is a continuously differentiable function $ f(t, x_{i}(t)) $ and the highest order $ s\in \left \{1, 2, \cdots N\right \} $, there exist bounded positive scalars $ \delta _{ix} $, such that

where $ x_{is} = \left | x_{i}\right |^{s}+\left | x_{i}\right |^{s-1}+\cdots+1, s\geq1 $. We also need some assumptions to ensure that the purpose is achieved.

Assumption 2. $ A, B $ can be stabilized, and the digraph $ \mathcal{G} $ is strongly connected.

Next, we define the position errors $ e_{i}(t) $:

According to Definition 1, we have

We also have

where $ e(t) = [e_{1}(t), e_{2}(t), \cdots, e_{i}(t)] $, $ \frac{\partial \boldsymbol{f}(\boldsymbol{z_{i}})}{\partial \boldsymbol{x}} = diag^{N}_{i = 1}[\frac{\partial {f}(z_{i})}{\partial x_{i}}] $, $ L_{0} = diag^{N}_{i = 1}[b_{i}] $. $ L $ is defined as

, then Eq (2.8) can be expressed as

where $ A = $$\begin{bmatrix} I_{N\times N} & 0 \\ 0 & 0 \end{bmatrix}$$ $, $ \bar{L} = $$\begin{bmatrix} 0\\ L+L_{0} \end{bmatrix}$$ $, $ \Delta _{A} = $$\begin{bmatrix} 0 & 0\\ 0 & \frac{\partial \boldsymbol{f}(\boldsymbol{z_{i}})}{\partial \boldsymbol{x}} \end{bmatrix}$$ . $

Definition 1. MAS (2.1) and (2.2) are said to have consensus if each agent's position state in the system satisfies

2.3.   Controller design

The distributed adaptive control input is

where $ \bar{e}_{i} = \sum_{j = 1}^{N}a_{ji}e_{ji}, a_{ii} = a_{i0}+\sum_{j = 1, j\neq i}^{N}a_{ij} $. $ k_{i} $ is the control gain determined based on the linear matrix inequality(LMI)

where $ P = $$\begin{bmatrix} P_{0} & 0_{N\times N} \\ 0_{N\times N} & I_{N\times N} \end{bmatrix}$$ > 0, K = diag[k_{i}], and P_{0} $ is a positive definition matrix. $ \Psi _{i}(t) $ is defined as

where $ \hat{\delta} _{ix} $ is the estimate of unknown parameters $ \delta _{ix} $. The following describes the updated laws for the adaptive parameters:

Since $ \delta _{i} $ is an unknown constant, defined as $ \tilde{\delta}_{i}(t) = \hat{\delta }_{i}(t)-\delta _{i} $, the adaptive error systems are described by

It follows from Eq (2.12) that

where $ k(t) = [k_{1}(t), k_{2}(t), \cdots ]^{T} $ and $ \bar{e} = \left [ \bar{e}_{1}, \bar{e}_{2}\cdots \bar{e}_{n}\right ] $. According to Eqs (2.11) and (2.15), we have

Next, we define the event trigger time series as $ \left \{t_{k}^{j}\right \} $ for the $ j $th agent. Therefore, the next triggering time $ t_{k+1}^{i} $ for the $ i $th agent can be expressed as

The function $ H_{i}(\cdot) $ is given by

where $ \theta _{i} > 0 $, $ \alpha _{i}, \eta _{i}\in \mathbb{R}^{n} > 0 $. $ q_{i}(t) $ is defined as the measurement error according to Eq (2.6):

$ \chi _{i}(t) $ satisfies

where $ \beta _{i} > 0 $, initial value $ \chi _{i}(0) > 0 $ could be randomly selected, and $ t_{0}^{i} = 0 $.

Remark 1: The internal dynamic variable updates according to internal variables such as self-feedback, measurement error, and neighborhood error. In comparison with the conventional static triggering strategy ^[22,23], the dynamic event triggered control protocol we proposed can more effectively reduce network communication and save resources.

3.   Cyber attack model

A DoS attack aims to block the communication channels so the targeted system cannot exchange information normally. Communication channels are not the only things affected by DoS attacks because the attacks can damage communication equipment along with hindering data transmission, measurement, and control channels simultaneously. DoS attacks are extremely destructive to the system, but their energy consumption requires attackers to replenish energy supplies after the attack is over, which takes time. Therefore, the time series can be split into two sections based on whether a DoS assault was launched. In the absence of the DoS attack, the system functions and communicates properly. However, in the presence of a DoS attack, communication is cut off, and the controller stops functioning. Here, we assume that the time interval of DoS attacks is $ \left \{t_{m}\right \}_{m\in \mathbb{N}} $, where $ t_{m} $ is the moment of the DoS attack, and $ \left [t_{m}, t_{m}+\Delta _{m}\right ] $ is the $ m $th DoS time interval, and $ \Delta _{m} $ is the time duration of the $ m $th attack. The DoS attack interval is the same for all multi-agents. Thus, the set time instants where communication is blocked (the interval of the DoS attack) are

Similarly, the sequence of time intervals without attacks is given by

Because of the recovery mechanism, the MAS cannot immediately restore communication after the end of a DoS attack, and due to the event-triggering mechanism, there is an upper bound for the time when the two events occur consecutively. We assume that they can exist at the same time. Therefore, the actual DoS attack lasts longer, and consequently, the $ m $th DoS attack's actual time frame may be described as $ \left [t_{m}, t_{m}+\bar{\Delta} _{m} \right ] $. The new time period of the DoS attack is as follows:

Assumption 2. Define $ n_{a}(t_{0}, t) $ as the number of attacks in the period $ \left [ t_{0}, t\right ] $, so the attack frequency $ F_{a}(t_{0}, t) > 0 $ satisfies

Assumption 3. Define $ N_{a}(t_{0}, t) $ as the total time interval of the DoS attack in the period $ \left [ t_{0}, t\right ] $. The constants $ T_{0}\geq 0, F_{0}\geq 0, 0 < \frac{1}{T_{1}} < 1, 0 < \frac{1}{F_{1}} < 1 $ are such that

where $ \frac{1}{T_{1}} $ is the attack strength.

Lemma 1. Previous research considers Eq (2.1) and this DoS attack model under Assumptions 2 and 3 ^[18]. If the Lyapunov function $ V_{1}(t), V_{2}(t) $ satisfies

where $ l_{0}, l_{1}, \tau _{0}, \tau_{1} $ are positive constants. $ T_{1}, F_{1} $ defined in Assumption 3 satisfies

where $ 0 < \eta ^{*} < l_{0} $ is the time to restore communication. $ \rho > 0 $, $ \kappa\geq1 $ is a constant satisfying

Thus, we say that $ V(t) $ are bounded.

Remark 2: Lemma 1 gives an upper bound on DoS attack frequency and duration, ensuring that the Lyapunov function remains stable over the entire time span ^[18].

Remark 3: The DoS attack considered in this paper mainly attacks the communication channels between agents. Thus, when the DoS attack comes, there is no information interaction between neighboring agents, and the event-triggering control is not triggered. In addition, we consider a DETC. Compared with the traditional event-triggering control, we introduce a dynamic variable that uses communication resources more effectively. In the simulation section below, we compare our method with the traditional event-triggering mechanism.

4.   Stability analysis

In this section, we prove system stability. Our presentation has two sections: the stability study of the MAS (2.1) and (2.2) under a DoS attack and the proof of non-Zeno behavior.

Theorem 1. For the MAS (2.1) and (2.2) under DoS attacks, we consider Assumption 1 and the controller (2.12). If the LMI (2.13) satisfies $ (A+\bar{L}\otimes BK)^{T}P+P(A+\bar{L}\otimes BK)\leq \xi _{i}P $, where $ \xi _{i}\in \mathbb{R}^{n} = \sigma _{i}\eta _{i}, \sigma _{i} > 1 $, then a feasible solution exists and the MAS is said to achieve leader-following consensus.

Proof of Theorem 1. The system stability proof is also divided into two parts. The communication of the system is damaged under a DoS attack, but the system is not always in an impassable state. The proof is divided between DoS attacks and non-DoS attacks, as per the prior section. When there are non-DoS attacks in the system, we consider the Lyapunov function

It follows from Eqs (2.20)–(2.22) that

The preceding implies that

which leads to $ W(t) > 0 $.

The derivative of $ W $ is

According to Eq (2.3), the condition in Assumption 1, and the control protocol in {Eq (2.12)}, we have

Choosing $ \Psi _{i}(t) $ as in Eq (2.15), we obtain

On the other hand

Based on the condition in Theorem 1, Eq (2.22), and Eq (4.6), we have

then

where $ l_{0} = \min ([(\sigma _{M}-1)/\lambda _{\max}(P)], 1 $, $ \beta _{m}) > 0 $, $ \sigma _{M} = \max[\sigma _{i}] $, $ \beta _{m} = \min [\beta _{i}] $, $ \tau _{0} = \sum_{i = 1}^{N}\kappa _{i}^{-1}\tilde{\delta _{i}^{2}} $, $ l_{0}, and \tau _{0} $ are positive constants.

When there are DoS attacks in the system, then communication and control channel blockages exist. In this case, the control input becomes 0, $ u_{i}(t) = 0 $, so the Lyapunov function can be expressed as

Similar to (4.4), (4.10) can be written as

where $ l_{1} = 1 $, and $ \tau _{1} = 2\sum_{i = 1}^{N}\delta _{ix}\left | e\right |^{2}x_{is}+2\sum_{i = 1}^{N}\left | e\right |(\varsigma _{i2}\left | x\right |+\delta _{i}) $. According to the conditions of (3) and Assumption 1, we know that $ \tau_{1} $ has an upper bound. From Lemma 1, we know that the system stabilizes in a limited time under a DoS attack. The proof is completed.

Next is the proof of no Zeno behavior. We assume that there is a positive constant $ T_{0} $ such that $ \lim_{k \to\infty }t_{k}^{i} = T_{0} $. Based on the property of limit, we know that for any $ \varepsilon _{0} > 0 $, there exists $ N(\varepsilon _{0}) $ such that $ t^{i}_{k}\in (T_{0}-\varepsilon _{0}, T_{0}+\varepsilon _{0}), \forall k\geq N(\varepsilon _{0}) $. This means $ t_{N(\varepsilon _{0}+1)}^{i}-t_{N(\varepsilon _{0})}^{i} < 2\varepsilon _{0} $.

According to (4.11), $ W(t) $ gradually decreases to 0, Then $ \xi _{m}\lambda _{\min }(P)\left \| e_{i}(t)\right \|^{2}\leq V(t) < W(t) $. Therefore, we have

Because $ \left \| e_{i}(t)\right \| $ and $ \left \| q_{i}(t)\right \| $ are bounded, the Dini derivative of $ \left \| q_{i}(t)\right \| $ is

where $ \bar{A} = A+\Delta_{A} $. According to Eqs (2.3), (2.12), and (2.14), we obtain $ u_{i}(t) $. $ w_{i}(t) $ has an upper bound, and $ M_{1}, M_{2} $ is their upper bound.

Since only the trigger condition in {Eq (2.19)} is met and the event is triggered when $ \left \| q_{i}(t)\right \| $ is reset to 0, then $ \left \| q_{i}(t)\right \|\geq \sqrt{\frac{\eta _{i}}{\alpha _{i}}\left \| e_{i}(t)\right \|^{2}+\frac{\theta _{i}}{\alpha _{i}}\chi _{i}}\geq \sqrt{\frac{\theta _{i}}{\alpha _{i}}\chi _{i}}, t_{k}^{i-}, k = 1, 2\cdots $, which implies that

then, we can obtain

If $ \varepsilon _{0} > 0 $ is a solution of

then

As a result, the aforementioned assumption is false, concluding the evidence that the agent $ i $ does not have Zeno behavior. □

5.   Simulation

To show the efficacy of the proposed control strategy, we present a simulation example in this section. Our simulation uses MASs composed of six agents as shown in Figure 1, where agent 1 is the leader, and others are followers. The system is

The system parameters are set as

In this example, we consider the flight of an aircraft, $ \phi = 0.002 $ is the angular velocity of the aircraft, and $ I_{3} $ represents the identity matrix of 3 × 3.

Figures 2 and 3 show the response curves and consistency errors of the system state for all agents. They show that the followers' states converge toward those of the leader as time progresses. Figures 3 and 4 show the control input curves and event trigger time instant for all agents. There are four times DoS attacks, with $ T_{0} = 3, F_{0} = 4 $. The duration of the DoS attack is $ \left | \Xi _{a}(0, 40)\right | = 3.5 $. In Table 1, we can see that the dynamic event-triggered mechanism proposed in this paper has far fewer triggering instances in the same time than the other two literatures ^[22,23], which can effectively save communication resources. In addition, continuous communication between agents is not required to determine whether a trigger condition satisfies a trigger condition. Considering the static event-triggered control protocol, we have

where $ \rho $ and $ \varrho $ are positive constants. Our DETC effectively reduces communication frequency.

6.   Conclusions

In this paper, we propose a dynamic event-triggered adaptive control approach to address the leader-following consensus problem for nonlinear MASs experiencing DoS attacks. We have presented a distributed control strategy and adaptive update laws to ensure system stability in the presence of uncertainties. The Lyapunov stability theory is used to derive conditions for achieving consensus. The DoS attacks considered here mainly target the MASs' communication channels. In reality, there are other types, scales, and levels of DoS attacks. Formulating mathematical models of these other types of DoS attacks and solving these models is the direction of our future research.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was funded in part by National Natural Science Foundation of China under Grant 62273109, in part by the Guangdong Basic and Applied Basic Research Foundation under Grant 2023A1515010168, Grant 2019A1515010830, in part by the Key Special Foundation for General Universities in Guangdong Province under Grant 2022ZDZX1018, and in part by the Maoming Science and Technology Plan Foundation under Grant 2022S043.

Conflict of interest

The authors declare there is no conflict of interest.