A hierarchical, asynchronous, dynamic event-triggered time-varying output formation control method for multi-agent systems under DoS attacks.

Abstract

This invention discloses a time-varying output formation control method for multi-agent systems under DoS attacks, based on hierarchical asynchronous dynamic event triggering. The method includes establishing dynamic models of the leader and followers; establishing constraints; establishing a DoS attack model; designing a Luenberger observer; designing an adaptive state observer; constructing an output feedback formation controller and establishing a corresponding dynamic tracking error system model; analyzing and proving the observation error of the multi-agent system under DoS attacks; analyzing the formation tracking performance of the multi-agent system under DoS attacks; and verifying its effectiveness in time-varying output formation problems. This invention can estimate the leader's state information in real time and effectively avoids the waste of communication resources and the negative impact of DoS attacks. Based on this observer, combined with output adjustment techniques and state feedback control, the multi-agent system can achieve the desired time-varying formation configuration within a certain error bound.

Description

Technical Field

[0001] This invention relates to the field of multi-agent formation control, specifically to a time-varying output formation control method for multi-agent systems under DoS attacks based on hierarchical asynchronous dynamic event triggering. Background Technology

[0002] With the rapid development of technologies such as artificial intelligence, computers, communications, and microelectronics, heterogeneous multi-agent systems (HMAS), as an important branch of artificial intelligence, have received widespread attention and in-depth research in recent years. HMAS consists of multiple agents with different system dynamics models, cooperating or competing with each other, aiming to solve complex problems through collective behavior. Due to the widespread application of distributed sensors, HMAS is driving rapid development and innovation in fields such as intelligent transportation, aircraft cooperative formation, space satellite detection, and maritime unmanned vessel search and rescue. Time-varying output formation, as one of the important research directions of HMAS, aims to achieve coordination and synchronization among agents in a multi-agent system, enabling them to form specific formation structures according to preset time-varying output patterns. This allows the HMAS to efficiently and collaboratively complete complex tasks, such as mobile robot formation, cooperative perception, and target tracking.

[0003] Due to the open nature of communication networks, the communication channels for information exchange between heterogeneous multi-agent systems are vulnerable to malicious attacks. Malicious DoS attackers can disrupt communication links by consuming significant network and system resources with legitimate or forged requests, affecting information exchange between agents, causing data packet loss, and damaging the normal operation of heterogeneous multi-agent systems.

[0004] With the development of digital networks and the increase in distributed network nodes, network communication bandwidth and the computing and communication resources configured by the agents themselves become limited. Furthermore, continuous communication between agents is often difficult to achieve. Clearly, the aforementioned continuous control methods inevitably increase the network communication burden in certain situations. In contrast, the essence of event-triggered control methods is that the system transmits and updates information only after the set event triggering conditions are met, ultimately reducing communication energy consumption. This technology aims to achieve the expected control objectives of multi-agent systems and effectively improve network resource utilization. For event-triggered control protocols, each agent independently receives neighbor state information only at the moment of sampling, thereby reducing the network communication burden. Therefore, adopting event-triggered control protocols will significantly reduce information transmission, helping to reduce the network communication burden and the risk of communication loss.

[0005] However, existing research on time-varying formation control of heterogeneous multi-agent systems under DoS attacks is still insufficient in balancing and optimizing attack resistance with reducing communication resources. Therefore, future research needs to focus on how to effectively address potential cybersecurity threats faced by multi-agent systems while ensuring communication efficiency. Designing attack-resistant control algorithms to ensure the robustness and reliability of the system is of great practical significance. Summary of the Invention

[0006] To address the aforementioned technical shortcomings, the purpose of this invention is to provide a time-varying output formation control method for multi-agent systems under DoS attacks based on hierarchical asynchronous dynamic event triggering.

[0007] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0008] This invention provides a time-varying output formation control method for multi-agent systems under DoS attacks based on hierarchical asynchronous dynamic event triggering, comprising the following steps:

[0009] Step 1: Establish dynamic models of leaders and followers in a heterogeneous linear multi-agent system;

[0010] Step 2: Design the desired formation configuration of the follower agents;

[0011] Step 3: Establish a DoS attack model with limited energy between the agent communication channels;

[0012] Step 4: Design a Luenberger observer for the leader to obtain the leader's state;

[0013] Step 5: Design a hierarchical, asynchronous, dynamic event-triggered adaptive state observer. This observer enables information exchange between neighbors, real-time estimation of the leader's state information, and effective mitigation of the negative impact of DoS attacks. Simultaneously, establish a corresponding dynamic observation error model.

[0014] Step 6: Based on the constraints of the formation, a novel output feedback formation controller is constructed using the observer's observation information, formation vector information, and the agent's own state information, and a corresponding dynamic tracking error system model is established;

[0015] Step 7: Theoretical analysis proves that the observation error of the multi-agent system based on the hierarchical asynchronous dynamic event-triggered observer under DoS attack is eventually uniformly bounded, and Zeno behavior is avoided;

[0016] Step 8: Analyze the formation tracking performance of the multi-agent system under DoS attack, and verify the effectiveness of the algorithm in the time-varying output formation problem through simulation experiments.

[0017] Preferably, step 1 considers the communication topology of a heterogeneous linear multi-agent system:

[0018]

[0019] in It is a set of nodes. (i,j)∈Ξ indicates that agent j can receive information from agent i. Λ=[a ij ]∈R n×n If a matrix is ​​an adjacency matrix, and (i,j)∈Ξ, then a ij =1, otherwise a ij =0; Laplace matrix L = [l ij ]∈R n×n Used to describe undirected graphs When i ≠ j, l ij =-a ij Otherwise l ii =∑ j≠i a ij Furthermore, define the coupling weight a. i0 , where a i0 =1 indicates that agent i can receive leader state information; otherwise, a i0 =0.

[0020] Preferably, the dynamic model of the leader 0 system in the heterogeneous linear multi-agent system in step 1 is as follows:

[0021]

[0022] y0(t)=C0x0(t)

[0023] Where x0(t), y0(t), and u0(t) represent the state, output, and control input of the leader agent, respectively, and u0 is bounded, satisfying ||u0||≤α0, where α0>0. A0, B0, and C0 are constant system matrices with appropriate dimensions.

[0024] Preferably, the system dynamic model of the i-th follower in the heterogeneous linear multi-agent system of step 1 is as follows:

[0025]

[0026] Where x i (t), y i (t) and u i (t) represent the state, output, and control input of the i-th follower agent, respectively. i B i and C i It is a constant system matrix with appropriate dimensions.

[0027] For all agents i, there exists a matrix X. i and U i Makes the following linear matrix equations hold true:

[0028]

[0029] Preferably, step 2, establishing the constraints on the desired formation vector of the follower agent, specifically includes:

[0030] Define the relative position of follower agent i to the leader as h(t) = [h1] T (t),h2 T (t),...h n T (t)] T ,in

[0031] Satisfy constraints:

[0032]

[0033] Where A hi ∈R n and C hi ∈R n It is the system matrix, h i (t)∈R n It is a piecewise, continuously differentiable formation vector, y hi This represents the output formation state information of heterogeneous multi-agent systems.

[0034] For all agents i, the following regulation equation is satisfied:

[0035]

[0036] Where (X) hi U hi ) is the matrix pair solution of the regulation equation.

[0037] For all agents i, the following regulation equation is satisfied:

[0038]

[0039] Where (X) hi U hi ) is the matrix pair solution of the regulation equation.

[0040] Preferably, step 3 establishes a DoS attack model with limited energy between the agent communication channels, as follows:

[0041] A model of an agent's communication channel (i,j) being subjected to a DoS attack can be derived from time series data. This represents the set of durations of the nth DoS attack. for in This indicates the duration of the nth DoS attack.

[0042] For any time interval This represents a set of time intervals during a DoS attack. express The complement, i.e. No attacks occurred. This represents the total duration of a malicious attacker's activity within the range [t0, t]. This represents the total length of the attacker's sleep time in the range [t0, t]. For any T2 > T1 ≥ t0, there exists a positive scalar τ. a If T > 1 and T0 > 0, then the attack duration over [T1, T2) is... Must meet: There exists a scalar F a >0, making the attack frequency F a (T1,T2) satisfy N a (T1,T2)≤F a ·(T2-T1).

[0043] Preferably, step 4 involves designing a Luenberger observer for the leader to obtain the leader's state, and the observer protocol is as follows:

[0044]

[0045] in The observer observes the leader's state vector, A0, B0, C0 are the leader system's state matrices, u0 is the leader's control input, and L0 is the feedback gain matrix of the observer to be designed.

[0046] Preferably, step 5 involves designing a hierarchical, asynchronous, dynamic event-triggered adaptive state observer for each follower. This observer enables information exchange between neighbors, real-time estimation of the leader's state information, and effective mitigation of the negative impact of DoS attacks. The observer protocol is as follows:

[0047]

[0048] Where η i (t) is the estimate of the leader's state by agent i, c ij (t),c i0 (t) is the adaptive coupling gain, where c ij (0)=c ji (0), K∈R n ,Ω∈Rn It is the gain matrix to be designed, π ij ,π i0 ,β ij ,β i0 It is a constant to be selected. These are the compensation values ​​η i (t) is the open-loop estimate of edge (i,j) and edge (0,i).

[0049] estimated value and It is obtained through the following estimator:

[0050]

[0051] Right now: in and Let represent the k-th trigger time of the leader agent, the k-th trigger time of the communication channel (0,i) between the leader and the follower agents, and the k-th trigger time of the communication channel (i,j) between the follower agents, respectively.

[0052] Define error variables

[0053] Preferably, the layered asynchronous dynamic event triggering mechanism designed in step 5 is as follows:

[0054]

[0055]

[0056] in α i0 >1-μ i0 ,μ i0 ∈(0,1),c i0 (0)>0,γ i0 (0)>0,α ij >1-μ ij ,μ ij ∈(0,1)

[0057] μ ij ∈(0,1),c ij (0)>0,γ ij (0) > 0 is the constant to be selected.

[0058] Preferably, in step 5, a corresponding dynamic observation error model is established:

[0059] Define the estimation error of agent i for the leader's state as e. i (t)=η i (t)-x0(t), Therefore, the error dynamic system is as follows:

[0060]

[0061] Preferably, step 6, based on formation constraints, utilizes observer observation information, formation vector information, and the agent's own state information to construct a novel output feedback formation controller. The controller protocol is as follows:

[0062] u i (t)=K 1i x i (t)+K 2i η i (t)+K 3i h i (t)

[0063] Among them, K 1i ,K 2i ,K 3i It is a matrix whose appropriate dimension needs to be selected, x i (t),η i (t),h i (t) represents the state information of the follower agent i, the observation value of the observer, and the formation vector information, respectively.

[0064] Preferably, in step 6, a corresponding dynamic tracking error system model is established;

[0065] Define the estimated value of agent i

[0066]

[0067] Formation tracking error ξ i It can be written as:

[0068]

[0069] Preferably, step 7 proves through theoretical analysis that the observation error of the multi-agent system based on the asynchronous dynamic event-triggered observer under a DoS attack is eventually uniformly bounded, and avoids Zeno behavior;

[0070] Assume the communication topology of the multi-agent system is connected when no DoS attack occurs, and that the formation vectors satisfy the desired formation constraints. If there exists κ... * ∈(0,κ1) such that the duration and frequency of a DoS attack satisfy

[0071]

[0072] in And σ1 satisfies Δ is the time required to restore communication. Choose a suitable L0 such that A0 + L0C0 is Hurwitz, and design K = -P, Ω = P. 2 Let P and Q be solutions to the following algebraic Ricardian equation (ARE):

[0073] PA0+A0 T PP 2 +I q =0.

[0074] Q(A0+L0C0)+(A0+L0C0) T Q+qI n =0.

[0075] Therefore, the adaptive coupling gain c can be obtained. ij and dynamic threshold δ ij The state observer estimation error e converges to a certain finite value. i (t) is UUB. Furthermore, this event triggering mechanism does not exhibit Zeno behavior.

[0076] Preferably, step 8 analyzes the formation tracking performance of a multi-agent system under a DoS attack, and verifies the effectiveness of the algorithm in time-varying output formation problems through simulation experiments;

[0077] Assume the elastic distributed formation controller gain K 1i Satisfy A i +B i K 1i It's by Hurwiz, K. 2i =U i -K 1i X i K 3i =U hi -K 1i X hi Therefore, under a DoS attack, a heterogeneous multi-agent system can achieve the desired formation configuration within a certain error margin by using a layered asynchronous dynamic event-triggered observer and an output feedback formation controller.

[0078] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0079] (1) This scheme designs three types of event triggering functions for the non-autonomous leader itself, the leader and followers, and the communication channels between followers. By dynamically adjusting parameters such as adaptive coupling gain and dynamic threshold, the number of triggers is effectively reduced. It is assumed that the control input of the non-autonomous leader is non-zero and bounded to improve the maneuverability and flexibility of the reference trajectory.

[0080] (2) By analyzing the adverse effects of DoS attacks and large-scale distributed network communication, a hierarchical asynchronous dynamic event-triggered (DET) state observer is designed. This observer has the advantages of both security and efficiency, and can effectively resist DoS attacks and reduce network resource utilization. Inspired by output conditioning technology and state feedback control, a new elastic distributed output feedback formation controller is constructed based on follower state information, observer observation information, and formation vector information. The aim is to ensure that each follower can accurately track the leader's expected trajectory, so that the multi-agent system can achieve the desired time-varying formation configuration. Attached Figure Description

[0081] To more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the accompanying drawings used are briefly described below. Obviously, the drawings described below only illustrate some embodiments of the present invention. For those skilled in the art, other related designs can be obtained from these drawings without creative work.

[0082] Figure 1 This is a control structure block diagram of a time-varying output formation control method based on hierarchical asynchronous dynamic event triggering for a multi-agent system under DoS attack, provided by the present invention.

[0083] Figure 2 This is a schematic diagram of the communication topology between four multi-agents in an embodiment of the present invention;

[0084] Figure 3 This is a simulation diagram of the adaptive state observer tracking the leader state according to an embodiment of the present invention;

[0085] Figure 4 These are the event trigger times for each communication channel in this embodiment of the invention;

[0086] Figure 5 This is a graph showing the adaptive coupling gain variation in the observer of an embodiment of the present invention.

[0087] Figure 6 This is a dynamic threshold change curve in the observer of an embodiment of the present invention;

[0088] Figure 7 In this embodiment of the invention, the three follower agents output their formation at the 20th second.

[0089] Figure 8 This is a graph showing the elastic time-varying output formation error of the three intelligent agents in an embodiment of the present invention; Detailed Implementation

[0090] The present invention will be further described in detail below with reference to the accompanying drawings of the embodiments. Obviously, the embodiments described are only some examples of the present invention, and not all of them. Those skilled in the art can make various modifications or alterations to the present invention based on these embodiments, and these changes are equally applicable to the scope of the appended claims.

[0091] like Figures 1 to 5 As shown, this embodiment provides a time-varying output formation control method for multi-agent systems under DoS attacks based on hierarchical asynchronous dynamic event triggering. This example uses a formation of one leader agent and three follower agents to further illustrate and explain the invention.

[0092] The desired outcome is that they can form a triangular formation. To achieve this formation, the following steps are required:

[0093] Step 1: Consider the communication topology and establish a dynamic model of the leader and followers in the heterogeneous linear multi-agent system;

[0094] Consider the following communication topology diagram of a heterogeneous linear multi-agent system:

[0095]

[0096] in It is a set of nodes. (i,j)∈Ξ indicates that agent j can receive information from agent i. Λ=[a ij ]∈R n×n If a matrix is ​​an adjacency matrix, and (i,j)∈Ξ, then a ij =1, otherwise a ij =0; Laplace matrix L = [l ij ]∈R n×n Used to describe undirected graphs When i ≠ j, l ij =-a ij Otherwise l ii =∑ j≠i a ij Furthermore, define the coupling weight a. i0 , where a i0 =1 indicates that agent i can receive leader state information; otherwise, a i0 =0.

[0097] The dynamic model of the leader 0 system in the heterogeneous linear multi-agent system is as follows:

[0098]

[0099] y0(t)=C0x0(t)

[0100] Where x0(t), y0(t), and u0(t) represent the state, output, and control input of the leader agent, respectively, and u0 is bounded, satisfying ||u0||≤α0, where α0>0. A0, B0, and C0 are constant system matrices with appropriate dimensions.

[0101] The system dynamics model of the i-th follower is:

[0102]

[0103] Where x i (t), y i (t) and u i (t) represent the state, output, and control input of the i-th follower agent, respectively. i B i and C i It is a constant system matrix with appropriate dimensions.

[0104] For all agents i, there exists a matrix X. i and U i Makes the following linear matrix equations hold true:

[0105]

[0106] Step 2: Establish constraints on the desired formation vector of the follower agents;

[0107] Consider the expected formation vector constraints for multi-agent systems, specifically including:

[0108] Define the relative position of follower agent i to the leader as h(t) = [h1] T (t),h2 T (t),...h n T (t)] T ,in

[0109] Satisfy constraints:

[0110]

[0111] Where A hi ∈R n and C hi ∈R n It is the system matrix, h i (t)∈R n It is a piecewise, continuously differentiable formation vector, y hi This represents the output formation state information of heterogeneous multi-agent systems.

[0112] For all agents i, the following regulation equation is satisfied:

[0113]

[0114] Where (X) hi U hi ) is the matrix pair solution of the regulation equation.

[0115] Step 3: Establish a power-limited DoS attack model between agent communication channels to model spoofing attacks that occur independently on communication channels between agents;

[0116] Complex distributed communication networks in heterogeneous multi-agent systems are vulnerable to malicious attacks. DoS attacks, by consuming a large amount of communication resources in the channel, prevent the normal transmission of necessary interactive information between agents, thus rendering some or all agents in the system unable to communicate with each other. A model of a DoS attack on an agent communication channel (i,j) can be derived from time series... This represents the set of durations of the nth DoS attack. for in This indicates the duration of the nth DoS attack.

[0117] For any time interval This represents a set of time intervals during a DoS attack. express The complement, i.e. No attacks occurred. This represents the total duration of a malicious attacker's activity within the range [t0, t]. This represents the total length of the attacker's sleep time in the range [t0, t]. For any T2 > T1 ≥ t0, there exists a positive scalar τ. a If T > 1 and T0 > 0, then the attack duration over [T1, T2) is... Must meet: There exists a scalar F a >0, making the attack frequency F a (T1,T2) satisfy N a (T1,T2)≤F a ·(T2-T1).

[0118] Step 4: Design a Luenberger observer for the leader to obtain the leader's state;

[0119] Assume the leader's state is unavailable. Therefore, a Luenberger observer is used to obtain the leader's state. The observer protocol is as follows:

[0120]

[0121] in The observer observes the leader's state vector, A0, B0, C0 are the leader system's state matrices, u0 is the leader's control input, and L0 is the feedback gain matrix of the observer to be designed.

[0122] Step 5: Design a hierarchical, asynchronous, dynamic event-triggered adaptive state observer. This observer enables information exchange between neighbors, real-time estimation of the leader's state information, and effective mitigation of the negative impact of DoS attacks. Simultaneously, establish a corresponding dynamic observation error model.

[0123] The observer protocol is as follows:

[0124]

[0125] Where η i (t) is the estimate of the leader's state by agent i, c ij (t),c i0 (t) is the adaptive coupling gain, where c ij (0)=c ji (0), K∈R n ,Ω∈R n It is the gain matrix to be designed, π ij ,π i0 ,β ij ,β i0 It is a constant to be selected. These are the compensation values ​​η i (t) is the open-loop estimate of edge (i,j) and edge (0,i).

[0126] estimated value and It is obtained through the following estimator:

[0127]

[0128] Right now: in and Let represent the k-th trigger time of the leader agent, the k-th trigger time of the communication channel (0,i) between the leader and the follower agents, and the k-th trigger time of the communication channel (i,j) between the follower agents, respectively.

[0129] Define error variables

[0130] The layered asynchronous dynamic event triggering mechanism is designed as follows:

[0131]

[0132] in α i0 >1-μ i0 ,μ i0 ∈(0,1),c i0 (0)>0,γ i0 (0)>0,α ij >1-μ ij ,μ ij ∈(0,1)

[0133] μ ij ∈(0,1),c ij (0)>0,γ ij (0) > 0 is the constant to be selected.

[0134] Establish a corresponding dynamic observation error model based on the observer protocol:

[0135] Define the estimation error of agent i for the leader's state as e. i (t)=η i (t)-x0(t), Therefore, the error dynamic system is as follows:

[0136]

[0137] Step 6: Based on the constraints of the formation, a novel output feedback formation controller is constructed using the observer's observation information, formation vector information, and the agent's own state information, and a corresponding dynamic tracking error system model is established;

[0138] The controller protocol is as follows:

[0139] u i (t)=K 1i x i (t)+K 2i η i (t)+K 3i h i (t)

[0140] Among them, K 1i ,K 2i ,K 3i It is a matrix whose appropriate dimension needs to be selected, x i (t),η i (t),h i (t) represents the state information of the follower agent i, the observation value of the observer, and the formation vector information, respectively.

[0141] A corresponding dynamic tracking error system model is established based on the above formation controller;

[0142] Define the estimated value of agent i

[0143]

[0144] Formation tracking error ξ i It can be written as:

[0145]

[0146] Step 7: Theoretical analysis proves that the observation error of the multi-agent system based on the hierarchical asynchronous dynamic event-triggered observer under DoS attack is eventually uniformly bounded, and Zeno behavior is avoided;

[0147] Assume the communication topology of the multi-agent system is connected when no DoS attack occurs, and that the formation vectors satisfy the desired formation constraints. If there exists κ... * ∈(0,κ1) such that the duration and frequency of a DoS attack satisfy

[0148]

[0149] in q>2||L0C0||, and σ1 satisfies Δ is the time required to restore communication. Choose a suitable L0 such that A0 + L0C0 is Hurwitz, and design K = -P, Ω = P. 2 Let P and Q be solutions to the following algebraic Ricardian equation (ARE):

[0150] PA0+A0 T PP 2 +I q =0

[0151] Q(A0+L0C0)+(A0+L0C0) T Q+qI n =0

[0152] Therefore, the adaptive coupling gain c can be obtained. ij and dynamic threshold δ ij The state observer estimation error e converges to a certain finite value. i (t) is UUB. Furthermore, this event triggering mechanism does not exhibit Zeno behavior.

[0153] Step 8: Analyze the formation tracking performance of the multi-agent system under DoS attack, and verify the effectiveness of the algorithm in the time-varying output formation problem through simulation experiments.

[0154] Assume the elastic distributed formation controller gain K 1i Satisfy A i +B i K 1i It's by Hurwiz, K.2i =U i -K 1i X i K 3i =U hi -K 1i X hi Therefore, under a DoS attack, a heterogeneous multi-agent system can achieve the desired formation configuration within a certain error margin by using a layered asynchronous dynamic event-triggered observer and an output feedback formation controller.

[0155] In this embodiment, four agents are selected as an example, and their communication topology is as follows: Figure 2 As shown, agent 0 is the leader, and agents 1-3 are followers. The system dynamic model parameters for the four agents are:

[0156]

[0157] Then, by selecting the following parameters and calculating using the theorem, we can obtain:

[0158] π i0 =π ij =0.4,β i0 =β ij =1,μ i0 =μ ij =0.1,α i0 =α ij =5, Luenberger observer control gain According to Theorem 1, the control gain of the state observer based on asynchronous dynamic event triggering is obtained. The control input for the leader agent is u0 = [0.1sin(t) 0.1cos(t)] T .

[0159] Based on the state observer designed for the follower agent described above, a distributed formation controller was subsequently designed. The relevant parameters of the formation controller are as follows:

[0160] The desired formation vector is i = 1, 2, 3.

[0161]

[0162] Figure 3 The diagram describes the two states of the leader agent and the state observer within 20 seconds under a DoS attack. The diagram shows that the state observer based on asynchronous dynamic event triggering is effective, effectively defending against DoS attacks and accurately estimating the leader state. This lays the foundation for the subsequent design of the formation controller.

[0163] Figure 4The values ​​represent the event trigger times of the communication channels of each agent under a DoS attack. These times are adaptively adjusted by the event trigger conditions of each channel, and the shaded areas represent the time series of the DoS attack. As shown in Table 1, the number of event trigger times is much smaller than the sampling time. This further verifies that the hierarchical dynamic event triggering control proposed in this paper effectively saves communication and computing resources between agents.

[0164] Figure 5 and Figure 6 The curves showing the changes in adaptive coupling gain and dynamic threshold respectively verify the conclusion of eventual uniform boundedness in Theorem 1.

[0165] Given the initial states of the three agents, x1(0) =

[10] T x2(0)=

[01] T x3(0)=

[11] T , Figure 7 The formation shape of three follower agents at t=20s under a DoS attack is described.

[0166] Figure 8 Error curves for elastic time-varying output formation tracking of three follower agents are presented. The results show that HMASs achieves time-varying output formation tracking within the allowable error range and effectively resists the adverse effects of DoS attacks.

[0167] This article uses specific examples to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the core ideas and methods of the invention. At the same time, those skilled in the art can make various modifications and variations to the present invention without departing from the spirit and scope of the present invention, and these modifications and variations should also be considered within the protection scope of the present invention.

Claims

1. A time-varying output formation control method for multi-agent systems under DoS attacks based on hierarchical asynchronous dynamic event triggering, characterized in that, Includes the following steps: Step 1: Consider the communication topology and establish a dynamic model of the leader and followers in the heterogeneous linear multi-agent system; Step 2: Establish the desired formation configuration of the follower agents; Step 3: Establish a DoS attack model with limited energy between the agent communication channels; Specifically, this includes: agent communication channels The model subject to DoS attack consists of time series This represents the set of durations of the nth DoS attack. for ,in , This represents the duration of the nth DoS attack. For any time interval , A set representing time intervals during a DoS attack; express The complement, i.e. No attack occurred. Indicates that malicious attackers are Total duration of the event Indicates the attacker is Total sleep duration; for any There exists a positive scalar , ,but Attack duration Must meet: ; scalar exists This increases the attack frequency. satisfy ; Step 4: Design a Luenberger observer for the leader to obtain the leader's state; Step 5: Design an adaptive state observer based on hierarchical asynchronous dynamic event triggering, use the observer to realize information interaction between neighbors, estimate the state information of the leader in real time, effectively resist the negative impact of DoS attacks, and establish a corresponding dynamic observation error model. The observer protocol is as follows: in It is agent i's estimate of the leader's state. It is the adaptive coupling gain, where , It is the gain matrix to be designed. It is a constant to be selected. These are the compensation values. On the side and edge The open-loop estimate; estimated value , and It is obtained through the following estimator: Right now: , , ,in , and Represents the k-th trigger time of the leader agent and the communication channel between the leader and follower agents, respectively. Communication channel between the k-th trigger time and the follower agent The k-th trigger time; define the error variable. , , ; The hierarchical asynchronous dynamic event triggering mechanism designed in step 5 is as follows: in ; These are constants to be selected; Step 6: Based on the constraints of the formation, a novel output feedback formation controller is constructed using the observer's observation information, formation vector information, and the agent's own state information, and a corresponding dynamic tracking error system model is established; The controller protocol is as follows: in, It is a matrix whose appropriate dimension needs to be selected. These represent the state information of the follower agent i, the observation value of the observer, and the formation vector information, respectively. Step 6, which involves establishing the corresponding dynamic tracking error system model, includes: Define the estimated value of agent i: Formation tracking error for: ； Step 7: Theoretical analysis proves that the observation error of the multi-agent system based on the hierarchical asynchronous dynamic event-triggered observer under DoS attack is eventually uniformly bounded, and Zeno behavior is avoided. Assume the communication topology of the multi-agent system is connected when no DoS attack occurs, and that the formation vectors satisfy the desired formation constraints; if there exists The duration and frequency of a DoS attack must satisfy the following conditions: in , , ,and satisfy ; This refers to the time required to restore communication; choose the appropriate time. , making It's by Hurwitz, the design. , , The following are solutions to the algebraic Ricardi equation (ARE): Therefore, the adaptive coupling gain is obtained. and dynamic threshold The state observer estimation error converges to a certain finite value. It's UUB; Step 8: Analyze the formation tracking performance of the multi-agent system under DoS attack, and verify the effectiveness of the hierarchical asynchronous dynamic event-triggered time-varying output formation control method of the multi-agent system under DoS attack in the time-varying output formation problem through simulation experiments.

2. The method for time-varying output formation control of a multi-agent system under DoS attack based on hierarchical asynchronous dynamic event triggering, as described in claim 1, is characterized in that... The communication topology of the heterogeneous linear multi-agent system described in step 1 is as follows: in It is a set of nodes. This means that agent j can receive information from agent i. It is an adjacency matrix, if it satisfies ,but ,otherwise Laplace matrix Used to describe undirected graphs ,when hour ,otherwise ; In addition, define coupling weights ,in This indicates that agent i can receive leader state information; otherwise... ; The leader dynamic model in the heterogeneous linear multi-agent system described in step 1 is as follows: in , and Let these represent the leader agent's state, output, and control input, respectively. It is bounded, satisfying ,in , It is a constant system matrix with appropriate dimensions; In the heterogeneous linear multi-agent system described in step 1, the first... The dynamic model of the follower system is as follows: in , and They represent the first The state, output, and control input of a follower agent. and It is a constant system matrix with appropriate dimensions; For all agents i, there exists a matrix and Makes the following linear matrix equations hold true: 。 3. The method for time-varying output formation control of a multi-agent system under DoS attack based on hierarchical asynchronous dynamic event triggering, as described in claim 1, is characterized in that... The vector constraints for establishing the desired formation configuration of the follower agents in step 2 specifically include: Define the relative positions of follower agent i and leader as follows: Where the following constraints are satisfied: in and It is a system matrix. It is a piecewise, continuously differentiable formation vector. This represents the output formation state information of multiple agents; For all agents i, the following regulation equation is satisfied: in It is the matrix pair solution of the regulating equation.

4. The method for time-varying output formation control of a multi-agent system under a DoS attack based on hierarchical asynchronous dynamic event triggering, as described in claim 1, is characterized in that... Step 4 involves designing a Luenberger observer for the leader to obtain the leader's state. The observer protocol is as follows: in It is the observer that observes the leader's state vector. It is the state matrix of the leader system. It is the leader's control input. It is the feedback gain matrix of the observer to be designed.

5. The method for time-varying output formation control of a multi-agent system under a DoS attack based on hierarchical asynchronous dynamic event triggering, as described in claim 1, is characterized in that... The corresponding dynamic observation error model established in step 5 is as follows: Define the estimation error of agent i for the leader's state as: , Therefore, the error dynamic system is as follows: 。 6. The method for time-varying output formation control of a multi-agent system under DoS attack based on hierarchical asynchronous dynamic event triggering, as described in claim 1, is characterized in that... Step 8 specifically involves: Assuming the elastic distributed formation controller gain satisfy It's by Hurwitz. ; Therefore, under a DoS attack, the multi-agent system can achieve the desired formation configuration within a certain error margin by triggering the observer and the output feedback formation controller through layered asynchronous dynamic events.

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