Multi-agent system encircling control method triggered by intermittent dynamic event under hybrid attack

Abstract

A multi-agent system encirclement control method triggered by intermittent dynamic events under hybrid attacks comprises the following steps: S1, constructing a directed communication topology and a Laplacian matrix of a multi-agent system according to an actual task; s2, constructing a multi-agent system model including uncertain disturbance, nonlinear dynamics and false data injection attacks; s3, designing a sliding-mode observer and a double-layer attack detection mechanism for denial of service attack and false data injection attack; s4, designing a fixed time encircling controller of the intermittent dynamic event triggered multi-agent system; s5, the controller designed in the step S4 is used for achieving encircling control within fixed time, and the upper bound of convergence time is obtained; and S6, continuously operating until the encircling control of the multi-agent system is completed. The invention aims to solve the problem of safe, efficient and rapid encircling control when a multi-agent system is subjected to DoS attack, FDI attack and external disturbance at the same time.

Description

Technical Field

[0001] This invention belongs to the technical fields of multi-agent system cooperative control, network security and control theory, specifically a method for encirclement control of multi-agent systems triggered by intermittent dynamic events under hybrid attacks. Background Technology

[0002] Multi-agent systems have become a research hotspot due to their wide application in fields such as drone swarms, distributed robotics, and smart grids. Encirclement control, as one of the key issues in cooperative control, requires that the states of all followers eventually enter the convex hull formed by the leader's state, thereby achieving effective constraints on the overall operating area of ​​the system.

[0003] In practical applications, multi-agent systems often exchange information via wireless communication networks, making them highly vulnerable to network attacks, such as denial-of-service (DoS) attacks and fictitious data injection (FDI) attacks. DoS attacks disrupt information exchange between agents by blocking communication channels, potentially damaging the system's topological connectivity. FDI attacks, on the other hand, inject malicious signals into transmitted data to mislead system behavior; while not interrupting communication, they contaminate the data. Furthermore, the system itself is subject to model uncertainties and external disturbances. These factors severely threaten the secure and stable operation of multi-agent systems.

[0004] To address the challenges of resource constraints and cyberattacks, several advanced control strategies have been proposed. Traditional time-triggered control requires periodic communication and control updates, potentially leading to unnecessary consumption of communication and computational resources. Event-triggered control, by setting pre-defined trigger conditions, only performs communication and control updates when system state changes exceed a specific threshold, effectively mitigating the aforementioned problems. Dynamic event-triggered control further introduces dynamic internal variables, enabling more flexible adjustment of the trigger threshold and further reducing the number of triggers compared to static event triggering. Intermittent control divides the control process into working and resting periods, pausing control during these periods, which helps conserve energy and enhances the system's robustness in situations such as control interruptions (e.g., DoS attacks). Fixed-time control guarantees system state convergence within a finite time, with the upper bound of the convergence time independent of the system's initial state, exhibiting performance superior to asymptotic or finite-time convergence.

[0005] The differences between this application and the prior art are as follows:

[0006] Comparison with the technology of patent CN119414748B, "Method for Fixed-Time Encirclement Control of Multi-Agent Systems Triggered by Intermittent Dynamic Events";

[0007] Patent CN119414748B also focuses on fixed-time encirclement control of multi-agent systems, but its implementation relies on idealized communication security assumptions, meaning it doesn't model or address network attacks. Its application focuses on optimizing encirclement performance under internal disturbances and resource constraints. In contrast, this patent explicitly considers hybrid network attacks as the core threat, and its application scenario involves adversarial environments with malicious attackers, unreliable communication links, and the potential for data tampering. The two patents differ fundamentally in whether they consider the crucial premise of external malicious attacks.

[0008] The core mechanism of patent CN119414748B is "resource-adaptive performance optimization." It aims to reduce unnecessary communication and control updates through carefully designed dynamic event triggering conditions and intermittent control laws, and guarantees convergence speed using fixed-time control theory. Its working mechanism is performance- and efficiency-oriented. In contrast, the core mechanism of this patent is "security-oriented situational awareness and decision-making." Building upon the aforementioned performance optimization mechanism, it deeply integrates attack detection and compensation functions. Its controller does not operate consistently but dynamically adjusts its behavior based on attack detection flags, even suspending control output during severe DoS attacks. This is a reactive defense design that prioritizes system survival and task completion. The core working mechanisms of the two are fundamentally different; this patent achieves a leap from a single performance dimension to a dual "performance-security" dimension.

[0009] Patent CN119414748B aims to achieve efficient and rapid encirclement in ideal environments, with its system capabilities manifested in the fine-grained management of computing and communication resources and precise control of convergence time. In contrast, this patent aims to achieve stable and reliable encirclement under hybrid network attacks, with its system capabilities manifested in attack tolerance, fault tolerance, and continuous control. The controller in this patent is essentially a system with an immune system, rather than merely a high-performance machine, thus solving the problem of insufficient adaptability of existing technologies in adversarial environments.

[0010] Comparison with the technology of patent CN120358089B, "A method and system for asynchronous dynamic intermittent security control of coupled networks";

[0011] Patent CN120358089B focuses on providing a secure synchronization control strategy against multiple deception attacks for complexly coupled dynamic networks. Its application scenarios emphasize the mean-square synchronization and consistency of node states in infrastructure such as sensor networks and smart grids. In contrast, this patent targets the specific field of "encirclement control of multi-agent systems," particularly in hybrid threat environments where denial-of-service attacks and spoofing attacks coexist. Its core task is to ensure that the states of all follower agents ultimately enter the convex hull formed by the leader's state. The two patents differ significantly in their application scenarios regarding control objectives and the complexity of the threats they face.

[0012] The core mechanism of patent CN120358089B is "passive tolerance based on probability modeling." It introduces Bernoulli distributed random variables to describe the probability of injection and substitution attacks, and considers their mathematical expectation during the intermittent controller design phase. This aims to weaken the impact of attacks on system synchronization performance from a statistical average perspective—a serial defense approach relying on prior probability models. In contrast, the core mechanism of this patent is "active compensation and reconstruction based on real-time detection." It designs a sliding mode observer and a two-layer attack detection mechanism targeting DoS and FDI attacks. This mechanism can perceive and accurately distinguish the type of attack currently being attacked in real time, triggering highly targeted countermeasures: when a connection-disrupting DoS attack is detected, it intelligently enters a dormant phase of intermittent control to conserve energy; when an FDI attack is detected, it dynamically injects compensation terms into the control law to offset attack deviations. This is a fully parallel, feedforward-oriented active security process that does not rely on the long-term statistical characteristics of attacks, achieving a fundamental shift from "tolerance" to "overcoming."

[0013] The system architecture of patent CN120358089B embeds attack probability as a parameterized module within a standard coupled network synchronization control framework. Its overall design philosophy is to enhance robustness against attacks while ensuring synchronization performance, which can be seen as a security enhancement of traditional controllers. In contrast, the system architecture of this patent constructs an independent "security perception-decision-execution" closed-loop framework. This architecture deeply integrates a sliding mode observer, dual-layer attack detection logic, and an intermittent controller with attack compensation terms, enabling the system to autonomously decide on the control mode based on the real-time network security situation. Its design philosophy is to build an intelligent security system with inherent resilience and adaptability, where control and security are not simply modules stacked but symbiotic. The system construction philosophies and the final security levels achieved by the two differ significantly.

[0014] In summary, by deeply integrating proactive detection, diagnosis, and compensation mechanisms against external network attacks with intermittent control, dynamic event triggering, and fixed-time stability theory, a comprehensive encirclement control framework that combines resource conservation, proactive security, and performance assurance has been constructed. Filling the research gap in existing technologies for dealing with complex network attack scenarios has become an urgent problem to be solved. Summary of the Invention

[0015] The purpose of this invention is to propose a method for encirclement control of multi-agent systems triggered by intermittent dynamic events under hybrid attacks, aiming to solve the problem of safe, efficient and fast encirclement control of multi-agent systems when they are simultaneously subjected to DoS attacks, FDI attacks and external disturbances.

[0016] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0017] A method for encirclement control of a multi-agent system triggered by intermittent dynamic events under hybrid attacks includes the following steps:

[0018] Step S1: Based on the actual task, construct the directed communication topology and Laplace matrix of the multi-agent system;

[0019] Step S2: Construct a multi-agent system model that includes uncertain perturbations, nonlinear dynamics, and spoofed data injection attacks;

[0020] Step S3: Design a sliding mode observer and a two-layer attack detection mechanism for denial-of-service attacks and fake data injection attacks;

[0021] Step S4: Design a fixed-time encirclement controller for an intermittent dynamic event-triggered multi-agent system;

[0022] Step S5: Use the controller designed in step S4 to achieve encirclement control within a fixed time and obtain the upper bound of the convergence time;

[0023] Step S6: Continue running until the encirclement control of the multi-agent system is completed.

[0024] As a further improvement of the present invention, the process of constructing the directed topological graph and Laplace matrix of the multi-agent system in step S1 includes:

[0025] For a multi-agent system consisting of N agents, its communication topology is a directed graph G with N nodes. Let n be the number of followers and m be the number of leader agents. Then, nodes 1 to n represent follower agents, and nodes n+1 to n+m represent leader agents. The edge set is... This means that if node i receives information from node j, then The adjacency matrix is ​​represented as If ,but Conversely ,at the same time The in-degree matrix is ​​defined as follows: Where, diag represents a diagonal matrix, Therefore, the Laplace matrix of the graph is defined as follows: .

[0026] As a further improvement of the present invention, the construction of a multi-agent system model in step S2, which includes uncertain perturbations, nonlinear dynamics, and spoofed data injection attacks, includes:

[0027] The dynamic model of the follower agent is as follows:

[0028] (1)

[0029] The dynamic model of the leader agent is as follows:

[0030] (2)

[0031] in and These represent the set of followers and the set of leaders, respectively. , Let xi (t) ∈ R represent the state of the i-th agent, ui (t) ∈ R represent the control input of the i-th agent, ∆i (t) be the bounded uncertainty perturbation, θi (t) be the bounded bias signal introduced by the fake data injection attack, and f (t) be the nonlinear dynamics.

[0032] As a further improvement of the present invention, the denial-of-service attack in step S3 includes a connection-keeping denial-of-service attack and a connection-disrupting denial-of-service attack, wherein:

[0033] A connection-persistent denial-of-service attack refers to an attack in which, after the attack, a directed path still exists between at least one leader and all followers in the communication topology.

[0034] A connection-disruption denial-of-service attack occurs when, after the attack, there is no directed path between at least one leader and all followers in the communication topology.

[0035] As a further improvement to the present invention, the process of designing the sliding mode observer and the two-layer attack detection mechanism against denial-of-service attacks and fake data injection attacks in step S3 includes:

[0036] S31. Design a sliding mode observer:

[0037] (3)

[0038] in, K1 is the estimated value of state xi(t), and K1 > 0 is the sliding mode gain, which is used to ensure the robustness of the observer to disturbances and attacks.

[0039] θcomp,i(t) is a compensation term designed to counter fake data injection attacks, and it is defined as follows:

[0040] (4)

[0041] Where K2 > 0 is the compensation gain;

[0042] The observer's estimation error ei(t) is defined as:

[0043] (5)

[0044] S32. Based on the output of the sliding mode observer, design a two-layer attack detection mechanism for denial-of-service attacks and spoofed data injection attacks, to identify and distinguish between denial-of-service attacks and spoofed data injection attacks:

[0045] First layer: Denial-of-service attack detection:

[0046] The detection is performed by judging whether the absolute value of the observation error ei(t) exceeds the preset threshold εDoS, and a preliminary denial-of-service attack indication signal giDoS(t) is generated for each agent:

[0047]

[0048] Then, a system-level denial-of-service attack detection flag S(t) is generated:

[0049]

[0050] in, This indicates that the multi-agent system is not subject to denial-of-service attacks or is only subjected to connection-maintaining denial-of-service attacks. This indicates that the multi-agent system has been subjected to a connection-disruption denial-of-service attack.

[0051] Second layer: Detection of fake data injection attacks:

[0052] This layer of detection only occurs when... The system is activated when the attack occurs, meaning it is not completely disconnected due to a denial-of-service attack. Detection is performed by determining whether the absolute value of the observer compensation term θcomp,i(t) exceeds a preset false data injection attack detection threshold ηFDI, and a false data injection attack detection flag is generated for each agent. :

[0053]

[0054] in, This indicates that agent i has been subjected to a fake data injection attack;

[0055] S33. Based on the above analysis, the principles for selecting the attack detection threshold are as follows:

[0056] εDoS should be greater than the steady-state upper bound of the observation error when only the inherent perturbation ∆i(t) exists;

[0057] ηFDI should be greater than the nominal fluctuation range of the compensation term θcomp,i (t) when only disturbances exist, and should be able to respond sensitively to attacks that inject real and fake data.

[0058] As a further improvement of the present invention, the process of designing a fixed-time encirclement controller for an intermittent dynamic event-triggered multi-agent system in step S4 includes:

[0059] S41. Design encirclement error:

[0060]

[0061] S42. Design an intermittent controller:

[0062] (10)

[0063]

[0064] in, It is a ratio of positive odd numbers. For adaptive parameters, the design is as follows:

[0065] (11)

[0066] in, ;

[0067] S43. Design measurement error:

[0068]

[0069]

[0070]

[0071] S44. Design trigger functions, dynamic variables, and trigger conditions:

[0072] The trigger function is designed as follows:

[0073]

[0074] in, and ;

[0075] The dynamic variable is designed as follows:

[0076]

[0077] in, and ;

[0078] The triggering conditions are designed as follows:

[0079] (15).

[0080] As a further improvement of the present invention, the upper bound of the convergence time of the multi-agent system in step S6 is specifically as follows:

[0081] (1) Multi-agent systems need to ensure and All of these hold true, and the following assumptions are satisfied simultaneously:

[0082] Assumption 1: Each leader is a neighbor of only a subset of its followers, and at least one leader is connected to all other followers. Furthermore, there exist positive real numbers... , , making , ;

[0083] Based on assumption 1, the Laplace matrix can be written as:

[0084]

[0085] in, and ;

[0086] Assumption 2: There exists a positive real number ∆ such that the following equation holds:

[0087] (17)

[0088] in, ;

[0089] Assumption 3: There exists a positive real number This makes the following equation true:

[0090] (18)

[0091] Assumption 4: There exists a positive real number This makes the following equation true:

[0092] (19)

[0093] (2) Connection interruption denial-of-service attacks must ensure that: This holds true, and simultaneously satisfies the following assumptions:

[0094] Assumption 5: For any interval ,in The number of connection interruption denial-of-service attacks, n(t1, t2).

[0095] satisfy:

[0096]

[0097] in, Indicates the initial offset or tolerance level for the number of attacks. This parameter represents the average interval between attacks.

[0098] Assumption 6: For any interval (in The total duration of a connection-disrupting denial-of-service attack.

[0099] satisfy:

[0100]

[0101] in, The initial offset or tolerance indicating the duration of the attack. A proportional parameter representing the duration of the attack;

[0102] (3) Determine the upper bound of the system's convergence time.

[0103] The beneficial effects of this invention are mainly reflected in:

[0104] This invention employs a fixed-time control strategy, which keeps the system convergence time finite and unaffected by the initial state. It can precisely design the convergence performance through parameter configuration, ensuring stable completion of encirclement control within a preset time, thereby improving the determinism and controllability of the control process.

[0105] This invention combines an intermittent control strategy to address the problem of control signal interruption that may be caused by network attacks. By dynamically adjusting the timing of control action, it enhances the robustness of the system in communication-unstable scenarios and better meets the actual operational needs of multi-agent systems in complex environments.

[0106] This invention introduces a dynamic event-triggered control strategy, which reduces unnecessary control input updates through a dynamic threshold mechanism. While ensuring control accuracy, it significantly reduces the communication frequency between agents, saves communication bandwidth and energy consumption, and improves system resource utilization efficiency.

[0107] The invention presents a two-layer attack detection and compensation mechanism for DoS and FDI attacks. It uses a sliding mode observer to accurately identify DoS and FDI attacks and introduces targeted attack compensation terms to effectively suppress the combined disturbances of the two types of attacks. This ensures that the multi-agent system can still stably achieve the encirclement of the target under attack scenarios, greatly improving the security and reliability of the system. Attached Figure Description

[0108] Figure 1 This is a flowchart of a fixed-time encirclement control method for a multi-agent system under attack disturbance triggered by intermittent dynamic events, showing the complete process including attack detection, control decision-making and encirclement implementation.

[0109] Figure 2 This is a diagram illustrating the control and attack response to intermittent events, demonstrating the update logic of control input during the intervals between DoS attacks and the triggering mechanism under attack scenarios;

[0110] Figure 3 This is a topology diagram of a multi-agent system instance of the present invention, which marks the communication links between the leader and followers and the distribution of potential attack nodes;

[0111] Figure 4 This is a timing diagram of the occurrence of DoS and FDI attacks in the multi-agent system example of the present invention, showing the trigger time, duration and superposition relationship of the two types of attacks, and intuitively reflecting the time distribution characteristics of the attack scenario;

[0112] Figure 5 This is a simulation diagram of the agent trajectory of the multi-agent system instance of the present invention under a combined DoS and FDI attack, showing the dynamic process of the followers encircling the leader through the convex hull;

[0113] Figure 6 This is a diagram showing the triggering time distribution of follower agents under attack and disturbance in a multi-agent system example of the present invention, demonstrating the communication scheduling effect of the event triggering mechanism in an attack scenario.

[0114] Figure 7 This is an input simulation diagram of an example of the multi-agent system of the present invention under attack compensation control, demonstrating the suppressive effect of the control signal on FDI attack deviation;

[0115] Figure 8 is a simulation change diagram of attack detection and dynamic compensation related variables in the multi-agent system example of the present invention, reflecting the response characteristics of the two-layer attack detection logic against DoS attack and FDI attack.

[0116] Figure 9 is a simulation change diagram of attack detection and dynamic compensation related variables in the multi-agent system example of the present invention, reflecting the response characteristics of the two-layer attack detection logic against DoS attack and FDI attack. Detailed Implementation

[0117] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are for illustrative purposes only and are not intended to limit the scope of the invention.

[0118] Implementation Example 1:

[0119] A method for encirclement and control of multi-agent systems triggered by intermittent dynamic events under hybrid attacks, the flowchart of which is as follows: Figure 1 As shown, the method includes the following steps:

[0120] Step S1: Based on the actual task, construct the directed communication topology and Laplace matrix of the multi-agent system.

[0121] The directed communication topology and Laplace matrix described in step S1 are constructed in the following manner:

[0122] For a multi-agent system consisting of N agents, its communication topology is a directed graph G with N nodes. Let n be the number of followers and m be the number of leader agents. Then, nodes 1 to n represent follower agents, and nodes n+1 to n+m represent leader agents. The edge set is... This means that if node i can receive information from node j, then The adjacency matrix is ​​represented as If ,but Conversely ,at the same time The in-degree matrix is ​​defined as follows: Where, diag represents a diagonal matrix, Therefore, the Laplace matrix of the graph is defined as follows: .

[0123] Step S2: Construct a multi-agent system model that includes uncertain perturbations, nonlinear dynamics, and fake data injection attacks.

[0124] The multi-agent system models constructed in step S2 are categorized as follows:

[0125] In an ideal, attack-free scenario, the nominal dynamics model of the follower agent is:

[0126]

[0127] Furthermore, considering the impact of FDI attacks on the system, the dynamic model of the follower agent is modified as follows:

[0128]

[0129] Where θi(t) represents the FDI attack bias on the i-th agent.

[0130] The attack model is represented as Where θj(t) ∈ R represents fake data generated by the attacker. It is an independent random variable that follows a Bernoulli distribution.

[0131] The definition is as follows:

[0132]

[0133] The probability of a successful FDI attack is:

[0134] ,

[0135] The dynamic model of the leader agent is as follows:

[0136]

[0137] Here, f(t) represents nonlinear dynamics.

[0138] Step S3: Design a sliding mode observer and a two-layer attack detection mechanism for DoS and FDI attacks.

[0139] Before describing the sliding mode observer and the two-layer attack detection mechanism against DoS and FDI attacks, we first define and classify DoS attacks.

[0140] A connection-persistent DoS attack refers to an attack in which, after the attack, there is at least one leader and at least one directed path between the leader and all followers in the communication topology.

[0141] A connection-disrupting DoS attack refers to an attack in which, after the attack, there is no directed path between at least one leader and all followers in the communication topology.

[0142] Among these, persistence-based attacks may slow down the convergence speed but do not significantly affect the convergence result. Under a persistence-based attack, the attacker can target any communication link. In this case, although the topology assumptions still hold, the communication topology will change.

[0143] Change. Therefore, by switching signals Define the switching topology. In the connection...

[0144] The switching topology resulting from a persistence denial-of-service attack can be described as follows: ,in This represents the initial communication topology, and the switching time series is... .

[0145] The sliding mode observer described in step S3, along with the two-layer attack detection mechanism against DoS and FDI attacks, is specifically designed as follows:

[0146] S31. Design a sliding mode observer:

[0147]

[0148] Where, x (t) is the estimated value of state xi(t), and K1 > 0 is the sliding mode gain, used to ensure that the observer is sensitive to disturbances and

[0149] Robustness to attacks.

[0150] The compensation term designed to counter FDI attacks is defined as follows:

[0151]

[0152] Where K2 > 0 is the compensation gain.

[0153] The observer's estimation error ei(t) is defined as:

[0154]

[0155] S32. Based on the output of the sliding mode observer, design a two-layer attack detection mechanism for DoS attacks and FDI attacks to identify and distinguish between denial-of-service attacks and fake data injection attacks:

[0156] It should be noted that the first layer of the above content is: DoS attack detection.

[0157] Detection is performed by judging whether the absolute value of the observation error ei(t) exceeds a preset threshold εDoS. For each agent...

[0158] Generate an initial DoS attack indication signal giDoS(t):

[0159]

[0160] Then, a system-level DoS attack detection flag S(t) is generated:

[0161]

[0162] in, This indicates that the multi-agent system is free from DoS attacks or only suffers from connection-maintaining DoS attacks. This indicates that the multi-agent system has been subjected to a connection disruption DoS attack.

[0163] Second layer: FDI attack detection:

[0164] This layer of detection only occurs when... Start when (i.e., the system is not completely disconnected due to a DoS attack). This is determined by the observed...

[0165] Device compensation item Does the absolute value exceed the preset FDI attack detection threshold? To perform detection, an FDI attack detection flag is generated for each agent. :

[0166]

[0167] in, This indicates that agent i has been subjected to an FDI attack.

[0168] S33. Based on the above analysis, the principles for selecting the attack detection threshold are as follows:

[0169] It should be greater than the presence of only inherent disturbances. The steady-state upper bound of the observation error;

[0170] It should be greater than the compensation term when only disturbances exist. It has a nominal fluctuation range and can respond sensitively to real FDI attacks.

[0171] Step S4: Design a fixed-time encirclement controller for an intermittent dynamic event-triggered multi-agent system.

[0172] The encirclement control objective described in step S4 is: for any bounded initial state of the system, there exists a state independent of the initial state.

[0173] Fixed convergence time upper bound This ensures that for all follower agents, encirclement control is considered achieved when their states satisfy the following condition:

[0174]

[0175] in, A sufficiently small positive constant is used to define the convergence accuracy of the encirclement error; a non-negative constant. satisfy This ensures that the follower's state converges to within the convex hull formed by the leader's state. The convergence time T satisfies... And its upper boundary It is independent of the initial conditions of the system.

[0176] The design of the intermittent dynamic event triggering controller described in step S4 is specifically described as follows:

[0177] Figure 2 shows a schematic diagram of the control process for intermittent control. The overall control process... Divided into control phases and

[0178] rest phase Where k is a positive integer, and dynamic event triggering occurs only during the control phase. For this intermittent control strategy, there exists an average control rate. With elasticity coefficient This makes the following equation true:

[0179]

[0180] in, for Total control time.

[0181] Furthermore, this invention is based on the following assumptions:

[0182] Assumption 1: Each leader is only a neighbor of a subset of followers, and there is at least one leader connected to all other followers.

[0183] It is true. Furthermore, there exist positive real numbers. , , making , .

[0184] Based on assumption 1, the Laplace matrix can be written as:

[0185]

[0186] in, and .

[0187] Assumption 2: There exists a positive real number ∆ such that the following equation holds:

[0188]

[0189] in, .

[0190] Assumption 3: There exists a positive real number This makes the following equation true:

[0191]

[0192] Assumption 4: There exists a positive real number This makes the following equation true:

[0193]

[0194] Assumption 5: For any interval (in Number of connection interruption DoS attacks

[0195] satisfy:

[0196]

[0197] in, Indicates the initial offset or tolerance level for the number of attacks. When representing the average interval of attacks

[0198] Inter-parameter.

[0199] Assumption 6: For any interval (in Total duration of a connection disruption DoS attack satisfy:

[0200]

[0201] in, The initial offset or tolerance indicating the duration of the attack. Indicates the duration of the attack.

[0202] Proportional parameters.

[0203] The specific design of the intermittent dynamic event-triggered multi-agent system fixed-time encirclement controller described in step S4 is as follows:

[0204] S41. Design encirclement error:

[0205]

[0206] S42. Design an intermittent controller:

[0207]

[0208] in, It is a ratio of positive odd numbers. For adaptive parameters,

[0209] Designed as follows:

[0210]

[0211] in, .

[0212] S43. Design measurement error:

[0213]

[0214] S44. Design the trigger function, dynamic variables, and trigger conditions:

[0215] The trigger function is designed as follows:

[0216]

[0217] in, and ;

[0218] The dynamic variable is designed as follows:

[0219]

[0220] in, and ;

[0221] The triggering conditions are designed as follows:

[0222]

[0223] Because when hour, This holds true, therefore the following inequality holds true:

[0224]

[0225] Based on the above formula, we can obtain:

[0226]

[0227] therefore:

[0228]

[0229] in, Therefore, dynamic variables can have

[0230] This effectively reduces the number of times the event is triggered.

[0231] Step S5: Use the controller designed in step S4 to achieve encirclement control within a fixed time and obtain the upper bound of the convergence time.

[0232] To prove that the controller designed in step S3 enables the multi-agent system to achieve encirclement control within a fixed time, the following lemma is introduced:

[0233] Lemma 1: For a positive semidefinite Lyapunov function W(t), if it satisfies:

[0234]

[0235] in It is a positive real number, and If satisfied and ,in Then the system can achieve practical fixed-time stability. Simultaneously, W(t) will converge to the following region:

[0236]

[0237] in, and .

[0238] System convergence time The following upper bound estimate is satisfied:

[0239]

[0240] in, .

[0241] Lemma 2: If Then the following inequality relationship exists:

[0242]

[0243] as well as

[0244]

[0245] Lemma 3: For any real number The following inequality exists:

[0246]

[0247] in, For larger positive real numbers, It is a constant.

[0248] Lemma 4: Matrix It is a positive definite matrix, where .

[0249] Lemma 5: Matrix Each element is non-negative, and the sum of the elements in each row is 1.

[0250] Next, we will prove that when all assumptions are true and the system satisfies and Simultaneously, a connection interruption-type DoS attack satisfies At that time, the controller designed in step S4 can enable the multi-agent system to achieve encirclement control within a fixed time, where... .

[0251] Define the desired position state as:

[0252]

[0253] in , , The error vector is defined as follows: Based on the above design, the encirclement control objective is: .

[0254] Design the following composite Lyapunov function:

[0255]

[0256] in,

[0257]

[0258]

[0259]

[0260]

[0261] First of all Differentiating, we get:

[0262]

[0263]

[0264]

[0265] in, .

[0266] Next analysis :

[0267] For intermittent control, the connection interruption DoS attack cycle is considered a rest period. ,at this time The period of a persistent DoS attack or the period without an attack are considered the active period. ,at this time .

[0268] when At that time, Differentiation yields:

[0269]

[0270]

[0271]

[0272]

[0273]

[0274]

[0275]

[0276]

[0277]

[0278] From the above analysis, we can conclude that:

[0279]

[0280]

[0281] in, .

[0282] in, Representation matrix The minimum value, Representation matrix The largest non-singular value, Representation matrix The largest eigenvalue.

[0283] when At that time, Differentiation yields:

[0284]

[0285]

[0286]

[0287]

[0288]

[0289]

[0290] From the above analysis, we can conclude that:

[0291]

[0292]

[0293] in, .

[0294] Based on the above analysis, the behavior of the Lyapunov function over the entire time domain is obtained as follows:

[0295]

[0296] The system state converges to a compact set within a fixed time:

[0297]

[0298] in, .

[0299] The fixed convergence time satisfies:

[0300]

[0301] And there are:

[0302]

[0303] in, , For matrix The smallest non-zero eigenvalue, For matrix The largest eigenvalue.

[0304] Next, we will further demonstrate that the controller designed in step S4 can be applied to real-world scenarios without producing the Zeno phenomenon. For ,Depend on Differentiation yields:

[0305]

[0306]

[0307]

[0308]

[0309]

[0310]

[0311] in, And there are: .

[0312] because We can obtain:

[0313]

[0314] At the triggering time, the following conditions are met:

[0315]

[0316] Therefore, we can conclude that:

[0317]

[0318] From the above formula, we can obtain:

[0319]

[0320] Therefore, the controller designed in step S4 will not produce the Zeno phenomenon and can be used in practical situations.

[0321] Step S6: Continue running until the encirclement control of the multi-agent system is completed.

[0322] Next, simulations are used to verify the effectiveness of the fixed-time encirclement control method for intermittent dynamic event-triggered multi-agent systems:

[0323] The multi-agent system under consideration, as shown in Figure 3, consists of two leaders and seven followers. , From Figure 3, we can obtain... They are respectively:

[0324]

[0325]

[0326] From the above formula, we can obtain that At the same time, uncertainty

[0327] The expression for the constant disturbance is set as follows This can make .

[0328] Let the leader dynamics model be: This can make Let the follower dynamics model be: ,in Therefore, we can obtain .

[0329] make Let the initial state of the system be... , .

[0330] Figure 4 shows the attack timing diagram, where the total duration of the connection interruption DoS attack is 0.19 seconds. Figure 5 shows the position simulation diagram of the multi-agent system, which shows that the system completed the encirclement control within 0.8 seconds. Figure 6 shows the simulation diagram of the follower encirclement error, which shows that the error approaches zero within 0.8 seconds. Figure 7 shows the event triggering time distribution diagram of the follower, which clearly shows the intermittent control characteristics, and the system enters a rest phase when subjected to a connection interruption DoS attack. Figure 8 shows the simulation diagram of the follower control input, which also clearly shows the intermittent control characteristics, i.e., the system enters a rest phase when subjected to a connection interruption DoS attack, and the control input is zero. Figure 9 shows the dynamic variable change diagram, which shows that the dynamic variable is always greater than 0, indicating that the dynamic triggering method can effectively reduce the number of triggers.

[0331] This invention merely illustrates the technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. The multimodal fusion described in this invention can be extended to all modalities that can be obtained in any scenario, and is not limited to the environmental and device modalities in the examples of this invention. For those skilled in the art, several improvements and modifications can be made without departing from the principle of this invention, and all such improvements and modifications fall within the scope of protection of the claims of this invention.

Claims

1. A method for encirclement control of a multi-agent system triggered by intermittent dynamic events under hybrid attacks, characterized in that, Includes the following steps: Step S1: Based on the actual task, construct the directed communication topology and Laplace matrix of the multi-agent system; Step S2: Construct a multi-agent system model that includes uncertain perturbations, nonlinear dynamics, and spoofed data injection attacks; Step S3: Design a sliding mode observer and a two-layer attack detection mechanism for denial-of-service attacks and fake data injection attacks; Step S4: Design a fixed-time encirclement controller for an intermittent dynamic event-triggered multi-agent system; Step S5: Use the controller designed in step S4 to achieve encirclement control within a fixed time and obtain the upper bound of the convergence time; Step S6: Continue running until the encirclement control of the multi-agent system is completed.

2. The method for encirclement control of a multi-agent system triggered by intermittent dynamic events under hybrid attacks according to claim 1, characterized in that, The process of constructing the directed topological graph and Laplace matrix of the multi-agent system in step S1 includes: For a multi-agent system consisting of N agents, its communication topology is a directed graph G with N nodes. Let n be the number of followers and m be the number of leader agents. Then, nodes 1 to n represent follower agents, and nodes n+1 to n+m represent leader agents. The edge set is... This means that if node i receives information from node j, then The adjacency matrix is ​​represented as If ,but Conversely ,at the same time The in-degree matrix is ​​defined as follows: Where, diag represents a diagonal matrix, Therefore, the Laplace matrix of the graph is defined as follows: .

3. The method for encirclement control of a multi-agent system triggered by intermittent dynamic events under hybrid attacks according to claim 2, characterized in that, The construction of a multi-agent system model in step S2, which includes uncertain perturbations, nonlinear dynamics, and spoofed data injection attacks, includes: The dynamic model of the follower agent is as follows: (1)； The dynamic model of the leader agent is as follows: (2)； in and These represent the set of followers and the set of leaders, respectively. , Let xi (t) ∈ R represent the state of the i-th agent, ui (t) ∈ R represent the control input of the i-th agent, ∆i (t) be the bounded uncertainty perturbation, θi (t) be the bounded bias signal introduced by the fake data injection attack, and f (t) be the nonlinear dynamics.

4. The method for encirclement control of a multi-agent system triggered by intermittent dynamic events under hybrid attacks according to claim 3, characterized in that, The denial-of-service attack in step S3 includes connection-persistent denial-of-service attacks and connection-disrupting denial-of-service attacks, wherein: A connection-persistent denial-of-service attack refers to an attack in which, after the attack, a directed path still exists between at least one leader and all followers in the communication topology. A connection-disruption denial-of-service attack occurs when, after the attack, there is no directed path between at least one leader and all followers in the communication topology.

5. The method for encirclement control of a multi-agent system triggered by intermittent dynamic events under hybrid attacks according to claim 4, characterized in that, The process of designing the sliding mode observer and the two-layer attack detection mechanism against denial-of-service attacks and fake data injection attacks in step S3 includes: S31. Design a sliding mode observer: (3)； in, K1 is the estimated value of state xi(t), and K1 > 0 is the sliding mode gain, which is used to ensure the robustness of the observer to disturbances and attacks. θcomp,i(t) is a compensation term designed to counter fake data injection attacks, and it is defined as follows: (4)； Where K2 > 0 is the compensation gain; The observer's estimation error ei(t) is defined as: (5)； S32. Based on the output of the sliding mode observer, design a two-layer attack detection mechanism for denial-of-service attacks and spoofed data injection attacks, to identify and distinguish between denial-of-service attacks and spoofed data injection attacks: First layer: Denial-of-service attack detection: The detection is performed by judging whether the absolute value of the observation error ei(t) exceeds the preset threshold εDoS, and a preliminary denial-of-service attack indication signal giDoS(t) is generated for each agent: ； Then, a system-level denial-of-service attack detection flag S(t) is generated: ； in, This indicates that the multi-agent system is not subject to denial-of-service attacks or is only subjected to connection-maintaining denial-of-service attacks. This indicates that the multi-agent system has been subjected to a connection-disruption denial-of-service attack. Second layer: Detection of fake data injection attacks: This layer of detection only occurs when... The system is activated when the attack occurs, meaning it is not completely disconnected due to a denial-of-service attack. Detection is performed by determining whether the absolute value of the observer compensation term θcomp,i(t) exceeds a preset false data injection attack detection threshold ηFDI, and a false data injection attack detection flag is generated for each agent. : ； in, This indicates that agent i has been subjected to a fake data injection attack; S33. Based on the above analysis, the principles for selecting the attack detection threshold are as follows: εDoS should be greater than the steady-state upper bound of the observation error when only the inherent perturbation ∆i(t) exists; ηFDI should be greater than the nominal fluctuation range of the compensation term θcomp,i (t) when only disturbances exist, and should be able to respond sensitively to attacks that inject real and fake data.

6. The method for encirclement control of a multi-agent system triggered by intermittent dynamic events under hybrid attacks according to claim 5, characterized in that, The process of designing a fixed-time encirclement controller for an intermittent dynamic event-triggered multi-agent system in step S4 includes: S41. Design encirclement error: ； S42. Design an intermittent controller: （10）； in, It is a ratio of positive odd numbers. For adaptive parameters, the design is as follows: (11)； in, ; S43. Design measurement error: （12）； S44. Design trigger functions, dynamic variables, and trigger conditions: The trigger function is designed as follows: ； in, and ; The dynamic variable is designed as follows: ； in, and ; The triggering conditions are designed as follows: (15)。 7. The method according to claim 6, characterized in that, The upper bound of the convergence time of the multi-agent system in step S6 is specifically as follows: (1) Multi-agent systems need to ensure and All of these hold true, and the following assumptions are satisfied simultaneously: Assumption 1: Each leader is a neighbor of only a subset of its followers, and at least one leader is connected to all other followers. Furthermore, there exist positive real numbers... , , making , ; Based on assumption 1, the Laplace matrix can be written as: ； in, and ; Assumption 2: There exists a positive real number ∆ such that the following equation holds: (17)； in, ; Assumption 3: There exists a positive real number This makes the following equation true: (18)； Assumption 4: There exists a positive real number This makes the following equation true: (19)； (2) Connection interruption denial-of-service attacks must ensure that: This holds true, and simultaneously satisfies the following assumptions: Assumption 5: For any interval ,in The number of connection interruption denial-of-service attacks, n(t1, t2). satisfy: ； in, Indicates the initial offset or tolerance level for the number of attacks. This parameter represents the average interval between attacks. Assumption 6: For any interval (in The total duration of a connection-disrupting denial-of-service attack. satisfy: ； in, The initial offset or tolerance indicating the duration of the attack. A proportional parameter representing the duration of the attack; (3) Determine the upper bound of the system's convergence time.

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