Second-order multi-agent system periodic event triggering approximate optimal formation control method and system under DoS attack

Abstract

The invention relates to the technical field of multi-agent cooperative control, and particularly discloses a second-order multi-agent system periodic event triggering approximate optimal formation control method and system under a DoS attack. Comprising the following steps: establishing a second-order multi-agent system model with a bimodal Markov switching topology, and embedding DoS attack frequency and duration constraints through a transition probability matrix; a distributed periodic event triggering mechanism based on a fixed sampling period is designed, and an intelligent agent is synchronously awakened only at a discrete moment and detects a triggering condition; event triggering and self-adaptive dynamic planning are fused, the Critic network weight is updated at the triggering moment so as to approach a solution of a Hamilton-Jacobi-Bellman equation, and collaborative optimization of formation tracking errors and control energy consumption is achieved; based on a multi-Lyapunov function and a linear matrix inequality, controller gain, trigger parameters and a sampling period are designed, and it is ensured that the system is semi-globally consistent and finally bounded under DoS attacks. According to the method, the random DoS attack is effectively resisted, and meanwhile, the consumption of communication and computing resources is remarkably reduced.

Description

Technical Field

[0001] This invention relates to the interdisciplinary fields of multi-agent cooperative control, network security, and optimal control, specifically to a method and system for near-optimal formation control of a second-order multi-agent system under DoS attacks, triggered by periodic events. Background Technology

[0002] Driven by the rapid development of collaborative drone swarm detection, intelligent transportation systems, and distributed robot collaboration, multi-agent systems (MASs) have become a key platform for performing complex tasks due to their flexibility, robustness, and efficiency. As the fundamental core of multi-agent collaboration, formation control aims to guide agents to form and maintain a preset spatial configuration through distributed protocols, playing a crucial role in applications such as target tracking, area coverage, and collaborative transportation. Representative methods include the leader-follower method, the behavior method, the virtual structure method, and the artificial potential field method.

[0003] In practical deployments, multi-agent systems heavily rely on wireless links for state interaction, making them vulnerable to DoS attacks. These attacks disrupt communication channels, causing packet loss or delays, potentially damaging formation stability and interfering with cooperative tasks. Existing research typically models DoS behavior by imposing upper bounds on attack frequency and duration, constructing mathematical frameworks to quantify the maximum permissible attack frequency and the longest single continuous interference time. While these models provide a valuable foundation for designing attack-resistant controllers, they still have two significant limitations: difficulty in accurately capturing the random on / off dynamics of attacks, and a lack of explicit association between communication topology switching and attack states.

[0004] To alleviate communication resource pressure, Event Triggered Control (ETC) strategies have been introduced into multi-agent systems. Unlike traditional time-triggered control, ETC allows agents to transmit data and update control inputs only when local state errors (such as the deviation between their own state and the desired state or the state of their neighbors) exceed a preset threshold, thereby significantly reducing communication load and energy consumption. However, continuous-time ETC requires agents to maintain continuous monitoring, which leads to a sharp increase in computational load. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of the prior art and provide a periodic event-triggered near-optimal formation control method for a second-order multi-agent system under DoS attacks. This method can effectively cope with random DoS attacks, significantly reduce communication and computing resource consumption while ensuring formation stability, and achieve synergistic optimization of tracking error and control energy consumption.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: A method for near-optimal formation control of a second-order multi-agent system under DoS attack triggered by periodic events, the control method comprising: A second-order multi-agent system model with a bimodal Markov switching topology is established. By constructing a continuous-time Markov chain, the DoS attack is modeled as a bimodal random switching process. The attack frequency and duration constraints are embedded in the transition probability matrix, and an explicit correlation mechanism between the attack state and the communication topology is established. The design incorporates a distributed periodic event triggering mechanism based on a fixed sampling period. The agent is only synchronously awakened during high-dispersion periods to perform state sampling and event detection. It communicates with its neighbors and updates control inputs only when the triggering conditions are met, thereby reducing communication and computational load and avoiding Zeno behavior. By integrating event triggering and adaptive dynamic programming, the weights of the Critic network are updated at the triggering time to solve the Hamilton-Jacobi-Bellman equation, thereby achieving synergistic optimization of formation tracking error and control energy consumption. The optimal value function is approximated through a neural network, and the control input remains unchanged within the triggering interval. The network weights are updated at the triggering time to achieve synergistic optimization of formation tracking error and control energy consumption. Based on the system models under DoS attack and without DoS attack, design the constraints and controller gain of the switching topology under DoS attack. Based on the multi-Lyapunov function and linear matrix inequality, derive the mode-dependent stability condition, and design the controller gain, trigger parameters and sampling period to ensure that the system is semi-globally consistent and eventually bounded under DoS attack.

[0007] Preferably, the establishment of a second-order multi-agent system model with a dual-modal Markov switching topology includes: Establish a second-order multi-agent formation model consisting of one leader and N followers, including: The navigator's dynamic equations are:

[0008] in and These represent the navigator's position and speed status, respectively. Represents the nonlinear dynamics of the navigator; Followers The dynamic equation is:

[0009] in and Followers Position and velocity state, To control the input vector, It is a differentiable nonlinear function and satisfies ; If for all followers , If true, then the second-order multi-agent system is considered to achieve formation tracking, in which... For followers The state vector, Let this be the navigator's state vector. Indicates follower A predefined formation structure is specified relative to the navigator's expected offset; A continuous-time Markov chain model is used to model DoS attacks, and its state space is as follows: These correspond to non-attack mode and attack mode, respectively, and are state variables. This represents the operating mode of the system at time t, and its dynamics are determined by the generator matrix.

[0010] Description, generating matrix parameters and The attack frequency and average duration are mapped respectively; when in attack mode, the control input is forced to zero, and the system operates in open loop; when in sleep mode, the control input functions normally; then the global error dynamic equation is:

[0011] in, , To account for measurement error, For auxiliary variable error, For nonlinear terms, , , .

[0012] Preferably, the design based on a distributed periodic event triggering mechanism with a fixed sampling period includes: setting a fixed sampling period. The agent only at time State sampling and event detection are performed only at discrete time points for the i-th follower. Perform state sampling and determine whether to trigger control updates; the sampling period is a constant. The event trigger time sequence is defined as follows: The triggering condition is

[0013] in, For position-velocity combined measurements

[0014]

[0015] For triggering parameters; Introducing auxiliary variables Record the error state at the time of the last event trigger and keep it constant between two triggers. Where K is the control gain. , .

[0016] Preferably, the fusion of event triggering and adaptive dynamic programming, updating the Critic network weights at the triggering time to solve the Hamilton-Jacobi-Bellman equation, and achieving synergistic optimization of formation tracking error and control energy consumption includes: Define the cost function for the i-th agent as:

[0017] in, This is the weight matrix; Define value functions:

[0018] in .

[0019] The Hamiltonian function is defined as:

[0020] in .

[0021] The optimal control law is obtained by solving the Hamilton-Jacobi-Bellman equation:

[0022] The optimal value function is approximated using a Critic neural network.

[0023] in, For the target weight matrix, For activation function, To approximate the error; Given the event-triggered context, the estimated optimal control strategy is:

[0024] in These are estimates of the weights; The Critic network weights are updated only at the trigger time, and the update rule is as follows:

[0025]

[0026] in, For timing difference error, This is the learning rate.

[0027] Preferably, the step of designing the constraints and controller gains for the switching topology under a DoS attack based on the system model under a DoS attack and the system model without a DoS attack includes: Selecting multiple Lyapunov functions:

[0028] in, These are the weighting coefficients for modal dependence; The controller gain K and trigger parameters are solved using linear matrix inequalities. Sampling period And the Critic network parameters, such that the following stability condition holds:

[0029] in, For a Markov chain with a stationary distribution, Let be the drift coefficients of the Lyapunov function. The switching coefficient, This refers to the average length of stay. When the above conditions are met, the system error semi-globally consistent and eventually bounded, thus achieving formation tracking control.

[0030] The present invention also discloses a periodic event-triggered near-optimal formation control system for a second-order multi-agent system under DoS attack, including a computer-readable storage medium and a processor; The computer-readable storage medium is used to store executable instructions; The processor is used to read executable instructions stored in the computer-readable storage medium and execute the control method described above.

[0031] Preferably, the system consists of one leader agent and N follower agents, with all follower agents tracking the leader agent in a pre-defined formation.

[0032] This invention effectively solves the formation control problem of multi-agent systems under DoS attacks through a collaborative design of Markov switching system modeling, periodic event-triggered control, and adaptive dynamic programming. While ensuring system stability and performance, it significantly improves resource utilization efficiency. Compared with traditional periodic control and continuous event-triggered control, this invention significantly reduces communication load and control costs while maintaining formation performance.

[0033] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. By using a bimodal Markov switching model to accurately characterize the random start-stop characteristics of DoS attacks, the limitations of deterministic models are overcome, providing a new approach to stability analysis under random attacks.

[0034] 2. By adopting a periodic event triggering mechanism, the communication frequency and computational overhead are significantly reduced, while avoiding Zeno behavior, making it more suitable for resource-constrained real-world systems.

[0035] 3. By integrating adaptive dynamic programming technology, the system achieves coordinated optimization of formation tracking error and control energy consumption under attack conditions, thereby improving the overall system performance.

[0036] 4. Sufficient conditions for system stability are given through rigorous Lyapunov stability theory and linear matrix inequality method, ensuring the reliability of the theory. Attached Figure Description

[0037] Figure 1 This represents the temporal relationship between mode switching, sampling, and event triggering.

[0038] Figure 2 Here is the communication topology, where Topology 1: Normal communication mode. Description: Corresponds to "Mode 1" or "dormant state" in a Markov chain. In this mode, the system is not subject to DoS attacks, and the communication network remains intact, such as... Figure 2 As shown in (a), the agents can exchange information normally, and the controller works at full capacity to achieve optimal formation tracking. Its function is to represent the ideal operating state of the system under no-attack conditions.

[0039] Topology 2: DoS Attack Mode. Description: Corresponds to "Mode 2" or "Attack State" in a Markov chain. In this mode, a DoS attack is activated, and the communication link between all followers is completely severed, such as... Figure 2 As shown in (b). At this point, the part of the controller that depends on neighbor information fails, the control input is set to zero or depends only on its own information, and the system operates in open loop. Purpose: To accurately characterize the worst-case scenario of a DoS attack—complete paralysis of the communication network. Topology 3: Another normal communication mode (or generalized switching topology).

[0040] Figure 3 The system formation tracking performance is given under PETC-ADP when switching between topology 1 and topology 2, where A represents motion tracking in formation mode and B represents the switching between topology 1 and topology 2.

[0041] Figure 4 shows the evolution of position and velocity tracking errors for all following agents.

[0042] Figure 5 shows the control strategy under PETC-ADP.

[0043] Figure 6 This refers to the event trigger time for all agents under PETC-ADP.

[0044] Figure 7 The error and trigger thresholds for all agents under PETC-ADP.

[0045] Figure 8 To evaluate the network.

[0046] Figure 9 Cumulative costs for all agents: (a) PETC-ADP, (b) PETC.

[0047] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings. Detailed Implementation

[0048] The present invention will be further described below with reference to the embodiments, but the description of the embodiments does not limit the scope of protection of the present invention in any way.

[0049] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used herein in the description of the invention is for the purpose of describing particular embodiments only and is not intended to limit the invention. Furthermore, while this document may provide examples of parameters containing specific values, it should be understood that the parameters need not be exactly equal to the corresponding values, but may approximate the corresponding values ​​within acceptable error tolerances or design constraints.

[0050] The meanings of the terms used in this invention are shown in Table 1.

[0051] Table 1. Glossary

[0052] Example 1 A method for near-optimal formation control of a second-order multi-agent system under DoS attack triggered by periodic events, the control method comprising: (1) Establish a second-order multi-agent system model with a dual-modal Markov switching topology. By constructing a continuous-time Markov chain, the DoS attack is modeled as a dual-modal random switching process. The attack frequency and duration constraints are embedded in the transition probability matrix, and an explicit correlation mechanism between the attack state and the communication topology is established. Consider a second-order nonlinear multi-agent system with N followers and one leader. Define the formation tracking error as follows: The navigator's dynamic equations are:

[0053] in and These represent the navigator's position and speed status, respectively. Represents the nonlinear dynamics of the navigator; Followers The dynamic equation is:

[0054] in and Followers Position and velocity state, To control the input vector, It is a differentiable nonlinear function and satisfies ; If for all followers , If true, then the second-order multi-agent system is considered to achieve formation tracking, in which... For followers The state vector, Let this be the navigator's state vector. Indicates follower A predefined formation structure is specified relative to the navigator's expected offset; A continuous-time Markov chain model is used to model DoS attacks, and its state space is as follows: These correspond to non-attack mode and attack mode, respectively, and are state variables. This represents the operating mode of the system at time t, and its dynamics are determined by the generator matrix.

[0055] Description, generating matrix parameters and The attack frequency and average duration are mapped respectively; when in attack mode, the control input is forcibly set to zero, communication is interrupted, and the system operates in open loop; in sleep mode, the control input functions normally; then the global error dynamic equation is:

[0056] in, , To account for measurement error, For auxiliary variable error, For nonlinear terms, , , .

[0057] (2) Design a distributed periodic event triggering mechanism based on a fixed sampling period; set a fixed sampling period τ, and the agent only performs state sampling and event detection at time lτ; design a trigger function, based on the error between the current measurement value and the previous trigger value, and a time-varying threshold, to determine whether to trigger communication and control updates; introduce auxiliary variables to record the state at the trigger time, which are used to calculate the control input within the trigger interval; specifically: For the i-th follower, only at discrete time... Perform state sampling and determine whether to trigger control updates; the sampling period is a constant. Design a trigger function, where the event trigger sequence is defined as follows: The triggering condition is

[0058] in, This is a combined position-velocity measurement. To trigger parameters, an auxiliary variable is introduced. Record the error state at the time of the last event trigger and keep it constant between two triggers. Where K is the control gain. , .

[0059] like Figure 1 As shown, the event is triggered only when... =1 During the sampling time at runtime, due to communication interruption, control updates are performed at... The sequence will fail during the period when the value is 2. Therefore, although these sequences are probabilistically independent, they are behaviorally interconnected.

[0060] (3) Integrating event triggering and adaptive dynamic programming, the Critic network weights are updated at the triggering time to solve the Hamilton-Jacobi-Bellman equation, achieving synergistic optimization of formation tracking error and control energy consumption; the adaptive dynamic programming method includes: defining a local cost function for each agent, including tracking error and control energy consumption; constructing a Critic neural network to approximate the optimal value function, with network input including local and neighbor information; designing a weight update law to update the Critic network weights only at the event triggering time to reduce computational overhead; deriving an approximate optimal control law to keep it constant within the triggering interval; specifically: Define the cost function for the i-th agent as:

[0061] in, This is the weight matrix; Define value functions:

[0062] in .

[0063] The Hamiltonian function is defined as:

[0064] in ; The optimal control law is obtained by solving the Hamilton-Jacobi-Bellman equation:

[0065] The optimal value function is approximated using a Critic neural network:

[0066] in, For the target weight matrix, For activation function, To approximate the error; Given the event-triggered context, the estimated optimal control strategy is:

[0067] in This is an estimate of the weights.

[0068] The Critic network weights are updated only at the trigger time, and the update rule is as follows:

[0069]

[0070] in, For timing difference error, This is the learning rate.

[0071] (4) Based on the system model under DoS attack and the system model without DoS attack, design the constraints and controller gain of the switching topology under DoS attack, construct the mode-dependent Lyapunov function, and use the linear matrix inequality tool to solve for the controller gain, triggering parameter, and upper bound of the sampling period that satisfy the stability condition; specifically: Selecting multiple Lyapunov functions:

[0072] in, These are the weighting coefficients for modal dependence; The controller gain K and trigger parameters are solved using linear matrix inequalities. Sampling period And the Critic network parameters, such that the following stability condition holds:

[0073] in, For a Markov chain with a stationary distribution, Let be the drift coefficients of the Lyapunov function. The switching coefficient, This refers to the average length of stay. When the above conditions are met, the system error semi-globally consistent and eventually bounded, thus achieving formation tracking control.

[0074] Example 2 Algorithm: Approximate Optimal Formation Control Based on PETC-ADP for DoS Attack Resistance Input: Sampling period Trigger parameters Markov generating matrix

[0075] Output: Approximate optimal formation control 1: Initialization , ,

[0076] 2: Initialization state , Weight

[0077] 3: Initialize the Markov pattern

[0078] 4: while do 5: Randomly jump to update the Markov model based on the generator matrix.

[0079] 6: if then 7: if then 8:

[0080] 9: Calculate the triggering criteria 10: if then 11:

[0081] 12: Calculate the approximate optimal control 13: Update Critic network weights 14: end if 15: end if 16: Application Control:

[0082] 17: else 18:

[0083] 19: end if 20:

[0084] 21: end while.

[0085] Example 3 Consider a second-order multi-agent system consisting of one leader and eight followers. The dynamic equations of the followers are:

[0086] The dynamic equation of the leader is

[0087] The nonlinear function satisfies the Lipschitz continuity condition.

[0088] A DoS attack is described using Markov chains, and the generated matrix is ​​as follows:

[0089] That is, parameters related to average attack frequency. Average recovery rate .

[0090] Define formation tracking error , ,in For the desired formation offset.

[0091] Step 2: Design a distributed periodic event triggering mechanism based on a fixed sampling period.

[0092] The sampling period is set to τ = 0.01s. The agent only samples at time τ = 0.01s. Detection trigger conditions:

[0093] in, For combined measurements, , , For triggering parameters.

[0094] The control input remains constant during the trigger interval:

[0095] Where K=27 is the control gain.

[0096] Step 3: Integrate event triggering with adaptive dynamic programming to achieve near-optimal control.

[0097] Define the performance index function:

[0098] in, , .

[0099] Constructing a Critic neural network to approximate the optimal value function:

[0100] The network weights are initialized to a normal distribution with a mean of 0 and a standard deviation of 0.2.

[0101] At the trigger time, update the Critic network weights:

[0102] Among them, learning rate .

[0103] The approximate optimal control law is:

[0104] Step 4: Stability analysis and controller parameter design.

[0105] By selecting Lyapunov functions and solving linear matrix inequalities, the stability condition is verified: Verification showed that all parameters met the stability conditions, and the system error was semi-globally consistent and eventually bounded.

[0106] To verify the effectiveness of this invention, the following simulation experiments were conducted: Simulation settings: 8 followers, 1 leader, expected formation as a cuboid structure. Simulation duration: 80 seconds; DoS attacks occur randomly.

[0107] Figure 2 Here is the communication topology, where Topology 1: Normal communication mode. Description: Corresponds to "Mode 1" or "dormant state" in a Markov chain. In this mode, the system is not subject to DoS attacks, and the communication network remains intact, such as... Figure 2 As shown in (a), the agents can exchange information normally, and the controller works at full capacity to achieve optimal formation tracking. Its function is to represent the ideal operating state of the system under no-attack conditions.

[0108] Topology 2: DoS Attack Mode. Description: Corresponds to "Mode 2" or "Attack State" in a Markov chain. In this mode, a DoS attack is activated, and the communication link between all followers is completely severed, such as... Figure 2 As shown in (b). At this point, the part of the controller that depends on neighbor information fails, the control input is set to zero or depends only on its own information, and the system operates in open loop. Purpose: To accurately characterize the worst-case scenario of a DoS attack—complete paralysis of the communication network. Topology 3: Another normal communication mode (or generalized switching topology).

[0109] Comparative experiments and results analysis: Within an 80-second simulation period, the PETC-ADP strategy of this embodiment is compared with the following control strategies: Traditional time-triggered control (TTC)

[13] : sampling period 0.01s; Continuous Event Triggered Control (CETC) [19,22]: Sampling period 0.001s; PETC without exponential decay term (PETC ω 2 =0).

[0110] The comparison between the generated communication load and energy consumption is shown in Table 2. Table 2 Comparison of communication load and energy consumption of several control strategies

[0111] As shown in Table 2, compared with other control strategies, the PETC-ADP strategy of this invention reduces communication load: the total number of triggers for PETC is approximately 9177, a reduction of 85.7% compared to TTC; PETC-ADP further reduces this to 4517, significantly improving communication efficiency. Energy consumption optimization: the cumulative cost of PETC-ADP is 49.41, a reduction of 10.2% compared to CETC and 91.6% compared to PETC, verifying the effectiveness of optimal control. Suppressing control chattering: the exponential decay term (ω²>0) in the trigger function reduces the average trigger frequency of PETC to that of PETC. ω 2 =Approximately 1 / 6 of 0, effectively avoiding control chattering caused by frequent updates.

[0112] Formation tracking performance: Figure 3 This study focuses on formation trajectory tracking during the switching between topology 1 (normal) and topology 2 (attack) in PETC-ADP. During the simulation, mode 2 actually switched 94 times (out of an allowed 114 times), with a total dwell time of 22.857 s, satisfying the stability constraints.

[0113] Figure 4: Position and velocity tracking errors of all followers (pink area is the DoS attack range). The error eventually converges to near zero, verifying the formation tracking effect.

[0114] Figure 5: Control input under PETC-ADP. The input is set to zero in the attack range and the input is smooth and has low energy consumption in the normal range.

[0115] Figure 6: Event triggering times for each agent, with uniform triggering intervals and no Zeno-like behavior.

[0116] Figure 7: Trigger threshold and measurement error. The error is always within the threshold, so the trigger logic is valid.

[0117] Figure 8: Evaluation of network weight updates. The weights eventually converge to stable values, and the network approximation effect is good.

[0118] Figure 9: Cumulative cost comparison. The cumulative cost of PETC-ADP is significantly lower than that of PETC, verifying its optimality.

[0119] 2.3 Robustness Analysis Against Attacks Attack strength was increased by reducing γ2 (attack → dormancy transition rate), and the results are shown in Table 3: Table 3 Robustness under increased attack intensity (K=10.2, γ1=2) (γ1,γ2) <![CDATA[π2]]> <![CDATA[K min ]]> Theoretical dwell time (s) Actual stay time (s) Theoretical number of switching times Actual number of switching times Event trigger count Accumulated costs steady-state error stability （2,5） 0.286 10.12 23 21.97 114 105 4277 147.27 0.0651 yes （2,4） 0.333 10.82 27 27.00 107 102 4321 261.46 0.0656 yes （2,3） 0.400 11.97 32 32.00 96 94 4403 305.16 0.0655 yes （2,2） 0.500 14.28 40 39.30 80 73 4239 1089.44 0.0654 yes （2,1） 0.667 21.21 53 50.52 53 52 3621 4221.11 0.1377 yes （2,0.5） 0.800 35.06 64 54.20 32 32 7050 325339.87 0.7878 yes (The steady-state error is the root mean square value of the deviation of each agent's position from the desired formation in the last 32 seconds (48-80s) of the simulation.) As shown in Table 3, the increase in attack strength (γ2 decreases from 5 to 0.5) causes the attack duty cycle to increase from 28.6% to 80%, and the minimum control gain Kmi required for stability to increase from 10.12 to 35.06. The number of system triggers, cumulative cost, and steady-state error increase with the attack intensity, but all states remain bounded and there is no instability, which verifies the attack robustness of the method. Due to the randomness of Markov mode switching, the specific values ​​for different simulation numbers vary slightly, but the overall trend is consistent.

Claims

1. A method for near-optimal formation control of a second-order multi-agent system under DoS attack triggered by periodic events, characterized in that, The control method includes: Establish a second-order multi-agent system model with a dual-modal Markov switching topology; Design a distributed periodic event triggering mechanism based on a fixed sampling period; By integrating event triggering and adaptive dynamic programming, the weights of the Critic network are updated at the triggering time to solve the Hamilton-Jacobi-Bellman equation, thereby achieving synergistic optimization of formation tracking error and control energy consumption. Design the constraints and controller gain of the switching topology under a DoS attack based on the system model under a DoS attack and the system model without a DoS attack.

2. The method for near-optimal formation control of a second-order multi-agent system under DoS attack triggered by periodic events, as described in claim 1, is characterized in that... The establishment of the second-order multi-agent system model with a dual-modal Markov switching topology includes: Establish a second-order multi-agent formation model consisting of one leader and N followers, including: The navigator's dynamic equations are: in and Let f0(t, v0(t)) represent the position and velocity states of the navigator, respectively, and let f0(t, v0(t)) represent the nonlinear dynamics of the navigator. The dynamic equation of follower i is: in and These represent the position and velocity states of follower i, respectively. To control the input vector, It is a differentiable nonlinear function and satisfies f i (t,0)=0; If for all followers i, If true, then the second-order multi-agent system is considered to achieve formation tracking, in which... Let i be the state vector of follower i. Let this be the navigator's state vector. This represents the expected offset of follower i relative to the leader, and specifies a predefined formation structure. A continuous-time Markov chain model is used to model DoS attacks, and its state space is as follows: Corresponding to non-attack mode and attack mode respectively, state variables This represents the operating mode of the system at time t, and its dynamics are determined by the generator matrix. Description: When in an attack state, the control input is forcibly set to zero, and the system operates in open-loop mode; in a sleep state, the control input functions normally; therefore, the global error dynamic equation is: in, e(t) is the measurement error. For auxiliary variable error, For nonlinear terms, Ξ3=[0 Np ,I Np ] T ,I p It is a p-order identity matrix.

3. The method for near-optimal formation control of a second-order multi-agent system under DoS attack triggered by periodic events, as described in claim 2, is characterized in that... The design, based on a distributed periodic event triggering mechanism with a fixed sampling period, includes: For the i-th follower, only at discrete time s l =lτ, Perform state sampling and determine whether control update is triggered. The sampling period is a constant τ>

0. The event trigger time sequence is defined as follows: Triggering condition is in, These are combined position-velocity measurements, where ω1 and ω2 are trigger parameters. Introducing auxiliary variables Record the error state at the time of the last event trigger and keep it constant between two triggers. That is in K is the control gain.

4. The method for near-optimal formation control of a second-order multi-agent system under DoS attack triggered by periodic events, as described in claim 3, is characterized in that... The fusion of event triggering and adaptive dynamic programming updates the Critic network weights at the triggering moment to solve the Hamilton-Jacobi-Bellman equation, achieving coordinated optimization of formation tracking error and control energy consumption, including: Define the cost function for the i-th agent as: in, This is the weight matrix; Define value functions: in The Hamiltonian function is defined as: in The optimal control law is obtained by solving the Hamilton-Jacobi-Bellman equation: The optimal value function is approximated using a Critic neural network: Among them, W mci For the target weight matrix, ε is the activation function. mci To approximate the error; Given the event-triggered context, the estimated optimal control strategy is: in These are estimates of the weights; The Critic network weights are updated only at the trigger time, and the update rule is as follows: Among them, E ci For timing difference error, r i This is the learning rate.

5. The method for near-optimal formation control of a second-order multi-agent system under DoS attack triggered by periodic events, as described in claim 4, is characterized in that... The design of the switching topology under a DoS attack and the controller gain based on the system model under a DoS attack and the system model without a DoS attack includes: Selecting multiple Lyapunov functions: in, These are the weighting coefficients for modal dependence; Solving for the controller gain K, trigger parameters ω1, ω2, sampling period τ, and Critic network parameters using linear matrix inequalities ensures the following stability condition holds: Where, π m For a stationary distribution of a Markov chain, χ m Let ι be the drift coefficient of the Lyapunov function. m v is the switching coefficient. m This refers to the average length of stay. When the above conditions are met, the system error semi-globally consistent and eventually bounded, thus achieving formation tracking control.

6. A periodic event-triggered near-optimal formation control system for a second-order multi-agent system under DoS attack, characterized in that, Includes computer-readable storage media and processors; The computer-readable storage medium is used to store executable instructions; The processor is used to read executable instructions stored in the computer-readable storage medium and execute the control method according to any one of claims 1-5.

7. The system according to claim 6, characterized in that, The system consists of one leader agent and N follower agents, with all follower agents tracking the leader agent in a pre-defined formation.

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