Consistency control method for edge event triggering of second-order multi-agents under network attacks

Abstract

The present invention relates to a method for controlling the consistency of edge event triggering of second-order multi-agents under network attacks, and belongs to the technical field of secure collaborative control of multi-agent systems under network attacks. The present invention takes into account energy-limited denial of service attacks, and designs a sampling-based edge event triggering control scheme based on a second-order multi-agent system with nonlinear terms. This avoids continuous monitoring of the multi-agent system, greatly reduces the consumption of system communication resources, further reduces the number of event triggering, ensures that the second-order multi-agent system remains consistent under different initial conditions, and effectively guarantees the reliability of the system.

Description

Technical Field

[0001] The present invention relates to the technical field of secure collaborative control of multi-agent systems under network attacks, and in particular to a method for edge event-triggered consistency control of second-order multi-agents under network attacks. Background Art

[0002] In recent years, the clustering problem of multi-agent systems has been widely studied, including formation, encirclement,

[0003] Swarming, clustering, tracking, and consensus problems. The goal of the cluster consensus problem in multi-agent systems is to design a control protocol for each follower so that the follower can track the trajectory of the leader.

[0004] However, traditional consistency control methods have shortcomings, which are mainly reflected in the following aspects: multi-agent systems need to rely on communication to transmit information, but communication is open and vulnerable to malicious attacks, so it is very necessary to consider the security consistency control of multi-agent systems; control strategies designed based on traditional methods need to continuously obtain the state of the agents, which not only consumes a large amount of computing resources of the controller, but also increases the possibility of network attacks; many schemes only consider the ideal situation for the modeling of second-order agents, and do not consider the nonlinear dynamic characteristics of second-order multi-agents due to sensor measurement errors and controller accuracy problems. Summary of the Invention

[0005] In view of the above problems, the present invention provides a method for edge event triggering consistency control of second-order multi-agents under network attacks. The present invention takes into account energy-limited denial of service attacks and designs a sampling-based edge event triggering control scheme based on a second-order multi-agent system with nonlinear terms. This avoids continuous monitoring of the multi-agent system, greatly reduces the consumption of system communication resources, further reduces the number of event triggering, ensures that the second-order multi-agent system remains consistent under different initial conditions, and effectively guarantees the reliability of the system.

[0006] The present invention provides a method for controlling the consistency of edge event triggering of second-order multi-agents under network attacks, comprising:

[0007] Step S1: constructing a network topology diagram of the multi-agent system and modeling the second-order multi-agent system to obtain a multi-agent system model; the multi-agent system model includes a leader agent and multiple follower agents;

[0008] Step S2: Design a DoS attack frequency and a DoS attack time, and establish a DoS attack model based on the DoS attack frequency and the DoS attack time;

[0009] Step S3: Design the event triggering conditions of the communication edge, the controller of the multi-agent system, and the sampling interval time to meet the conditions; sample the multi-agent system model based on the denial of service attack model and the sampling interval time to obtain multiple sampling data; update the controller of the multi-agent system based on the multiple sampling data and the event triggering conditions of the communication edge to obtain an updated controller of the multi-agent system; the controller only considers the denial of service attack situation;

[0010] Step S4: Use the update controller of the multi-agent system to obtain a control signal, input the control signal into the multi-agent system model, and achieve safe consistency control of the follower agent and the leader agent under a denial of service attack based on the sampling time interval meeting the conditions.

[0011] Preferably, the specific steps of constructing the network topology diagram of the multi-agent system in step S1 include:

[0012] Use undirected connected graph to represent the network topology of multi-agent system The network topology of the multi-agent system includes a leader agent and multiple follower agents; the leader agent and the follower agents communicate in a one-way manner; and the multiple follower agents communicate in a two-way manner.

[0013] The multiple agents serve as nodes of a network topology graph, each follower agent forms a corresponding edge with the leader agent, and each pair of follower agents forms a corresponding edge;

[0014] Define the adjacency matrix A of the network topology graph of the multi-agent system, where is a real number, n is the number of agents, when a ij =1, it means that the i-th agent can communicate with the j-th agent. ij = 0, it means that the i-th agent and the j-th agent cannot communicate;

[0015] Define the number of edges in the network topology as m, assign an arbitrary direction to each edge, and define the association matrix D and the internal association matrix in d ij is the correlation between the i-th agent and the j-th agent in the correlation matrix,

[0016]

[0017] in, is the correlation matrix d between the i-th agent and ij relevance;

[0018] The Laplace matrix L is obtained based on the inner correlation matrix and the correlation matrix, and the expression is:

[0019]

[0020] Where L is the Laplace matrix.

[0021] Preferably, the steps of modeling the second-order multi-agent system in step S1 to obtain the multi-agent system model include:

[0022] Obtain the leader and follower agents in the network topology of a multi-agent system

[0023] Based on the Lipschitz condition, the dynamics of the follower agent and the system dynamics of the leader agent are obtained respectively;

[0024] A multi-agent system model is established based on the dynamics of the follower agents and the system dynamics of the leader agent.

[0025] Furthermore, the specific steps of obtaining the multi-agent system model include:

[0026] In the network topology diagram of the multi-agent system, the leader agent is marked as x0 and the follower agent is marked as x i , i=1,2,3…n-1, n represents the total number of agents, n-1 represents the number of follower agents;

[0027] Obtain the dynamics of the follower agent based on the Lipschitz condition;

[0028] Obtain the system dynamics of the leader agent based on the Lipschitz condition;

[0029] A multi-agent system model is established based on the dynamics of the follower agents and the system dynamics of the leader agent.

[0030] Furthermore, the dynamic expression of the follower agent is:

[0031]

[0032] in, is the derivative of the position of the i-th follower agent at time t; is the derivative of the velocity of the ith follower agent at time t; x i (t) is the position of the i-th follower agent at time t, v i (t) is the speed of the i-th follower agent at time t, u i(t) is the control input of the ith follower agent at time t, and f(·) is a nonlinear function.

[0033] Furthermore, the expression of the Lipschitz condition is:

[0034] |f(t,x i (t),v i (t))-f(t,x j (t),v j (t))|≤η|x i (t)-x j (t)|+η|v i (t)-v j (t)|

[0035] Where η is the Lipschitz constant, x j (t) is the position of the j-th follower agent at time t, v j (t) is the velocity of the j-th follower agent at time t.

[0036] Furthermore, the system dynamics of the leader agent is expressed as:

[0037]

[0038] in, is the derivative of the position of the leader agent at time t, is the derivative of the velocity of the leader agent at time t, v0(t) is the velocity of the leader agent at time t, and x0(t) is the position of the leader agent at time t.

[0039] Preferably, the denial of service attack frequency expression in step S2 is:

[0040]

[0041] Where n(0,t) represents the number of effective DoS switch transitions in the time period (0,t); η′ represents the jitter bound of the DoS attack frequency, T D represents the average dwell time between on and off transitions of the DoS system,

[0042] Preferably, the denial of service attack time expression is:

[0043]

[0044] Where |D(0,t)| represents the duration of the denial of service attack in the time period (0,t), K represents the regularization term of the denial of service attack time; TF Indicates the duration limit of the denial of service attack;

[0045] Preferably, the specific steps of obtaining the update controller of the multi-agent system in step S3 include:

[0046] Step S31: Obtain the relative state difference of the agents on each edge in the network topology diagram of the multi-agent system;

[0047] Step S32: Set the event triggering time on each edge;

[0048] Step S33: Design edge trigger event parameters, and design the event trigger conditions of the communication edge based on the relative state difference of the agents on each edge, the edge trigger event parameters, and the event trigger time on each edge;

[0049] Step S34: designing a controller of the multi-agent system based on the denial of service attack model;

[0050] Step S35: setting a sampling time interval and a sampling period of a controller of the multi-agent system to meet a condition based on a denial of service attack model;

[0051] Step S36, let s = 1, when s = 1, it represents the first sampling moment in the rth sampling period; let r = 1, when r = 1, it represents the first sampling period; based on the sampling period meeting the condition and the sampling time interval meeting the condition, the multi-agent system is sampled;

[0052] Step S37, obtaining the sampling data of the sth sampling moment in the rth sampling period; judging whether the sampling data of the sth sampling moment meets the sampling period satisfaction condition;

[0053] Step S38: If not satisfied, input the sampling data of the s-th sampling moment into the controller of the multi-agent system at the corresponding moment for updating, and obtain the updated controller at the s-th sampling moment; if satisfied, determine whether s is greater than or equal to S, where S represents the number of sampling moments; if not, set s = s + T, where T represents the sampling time interval, and return to step S37; if so, proceed to the next step;

[0054] Step S39, determine whether r is greater than or equal to R, where R represents the number of sampling cycles. If so, stop updating and update the controller at the sth sampling moment as the update controller of the multi-agent system; if not, set r=r+1 and return to step S37.

[0055] Preferably, the expression for the relative state difference of the agents on each edge is:

[0056] y ij (t) = x i (t)-xj (t),w ij (t) = v i (t)-v j (t)

[0057] Among them, y ij (t) represents the relative position difference between the ith agent and the jth agent at time t, w ij (t) represents the relative speed difference between the i-th agent and the j-th agent at time t.

[0058] Preferably, the event triggering condition of the communication edge is expressed as:

[0059]

[0060] in, is the relative position difference between the kth event triggering time and the previous event triggering time of the i-th agent and the j-th agent on the corresponding edge; is the relative speed difference between the kth event triggering time and the previous event triggering time of the i-th agent and the j-th agent on the corresponding edge, is the triggering moment of the kth event on the corresponding edge between the i-th agent and the j-th agent; y ij (t s ) is the relative position difference between the sth sampling moment and the previous sampling moment of the i-th agent and the j-th agent on the corresponding edge, w ij (t s ) is the relative speed difference between the sth sampling moment and the previous sampling moment of the i-th agent and the j-th agent on the corresponding edge; t s is the sth sampling moment of the corresponding edge between the i-th agent and the j-th agent, b is the edge trigger event parameter, b>1, and n is the number of agents.

[0061] Furthermore, the expression of the update controller is:

[0062]

[0063] Among them, u i (t) is the control input of the i-th agent at time t; N i is the neighbor set of the ith agent; a ij represents the communication status between the i-th agent and the j-th agent, a ij =1 means that the i-th agent communicates with the j-th agent. At this time, the i-th agent is the neighbor of the j-th agent. ij =0 means that the i-th agent and the j-th agent do not communicate; k1 is the parameter related to position in the feedback matrix, and k2 is the parameter related to speed in the feedback matrix; is the position of the i-th agent at the time when the k-th event is triggered on the corresponding edge; is the position of the j-th agent at the time when the k-th event is triggered on the corresponding edge; is the speed of the i-th agent at the time when the k-th event is triggered on the corresponding edge; is the speed of the j-th agent at the time when the k-th event is triggered on the corresponding edge; is the position of the pilot agent at the time of the kth event triggering on the corresponding edge; is the speed of the pilot agent at the time of the kth event on the corresponding edge; b′ i Whether the i-th agent can obtain the leader agent information;

[0064] Furthermore, the expression that the sampling period satisfies the condition is:

[0065] S r+1 -S r ≤R+1;

[0066] Among them, s r is the rth sampling period, R represents the maximum attack time of a single denial of service;

[0067] Furthermore, the expression for the maximum attack time of a single denial of service is:

[0068]

[0069] Where κ represents the regularization term of the DoS attack time, and T is the sampling time interval.

[0070] Furthermore, the sampling time interval satisfies the condition, and the expression is:

[0071]

[0072] Where T is the sampling interval, σ1 is the upper bound of the control input norm, b is the first parameter to be determined, E2 is the second parameter, D T is parameter three, K is the gain matrix; B is parameter four,

[0073] D T =E2H, represents the Kronecker product, ||.|| represents the 2-norm; α is the upper bound of the nonlinear term norm, ln(·) represents the natural logarithm; ρ3 is the performance index constant;

[0074] ρ3=ρ2 / λ(P) max ,λ(P)max represents the maximum eigenvalue of the positive definite matrix P; ρ1 satisfies performance parameters; b2 is the second parameter to be determined, β is the fifth parameter, μ is the gain parameter, A is the positive definite matrix constant, λ2 is the performance parameter 2, is performance parameter three, I2 is performance parameter four, and H is parameter six; σ2 is the upper bound of the perturbation term norm, λ min (P) represents the minimum eigenvalue of the positive definite matrix P, represents the Kronecker product, and ||.|| represents the 2-norm.

[0075] Furthermore, step S3 further includes: designing a control target of an update controller of the multi-agent system; if the designed update controller of the multi-agent system can achieve the control target of the multi-agent system under any initial conditions of the multi-agent system, then the multi-agent system achieves consistency between the leader and the follower;

[0076] The expression of the control objective of the multi-agent system is:

[0077]

[0078] Preferably, the method further includes step S5, designing a Lyapunov function based on the sampling time interval satisfying the conditions and the update controller of the multi-agent system, and proving the secure consistency control of the follower agent and the leader agent under a denial of service attack based on the Lyapunov function and the update controller of the multi-agent system.

[0079] Compared with the prior art, the present invention has at least the following beneficial effects:

[0080] (1) The present invention adopts a sampling-based edge event triggering scheme, which avoids the continuous monitoring of the agent state. It only needs to periodically sample the agent state without relying on the transmission of continuous variables. In some practical application scenarios, the method of the present invention helps to reduce equipment costs;

[0081] (2) The present invention adopts an edge event triggering mechanism, which avoids the situation where two agents are in very close states and still need to trigger communication. This reduces the number of triggers in the multi-agent system and the consumption of communication resources and controller computing resources, which is of great significance for engineering applications.

[0082] (3) This invention takes into account the existence of nonlinear terms in second-order agents. The system model considered is more in line with the actual situation and can meet more application scenarios of second-order multi-agent systems.

[0083] (4) The present invention takes into account the existence of denial of service attacks, enabling the multi-agent system to cope with denial of service attacks with limited energy, thereby enhancing the security and reliability of the multi-agent system. BRIEF DESCRIPTION OF THE DRAWINGS

[0084] The drawings are only for purposes of illustrating particular embodiments and are not to be considered limiting of the invention.

[0085] Figure 1 Flowchart of a method for consistency control of a second-order multi-agent system under a denial of service attack according to an embodiment of the present invention;

[0086] Figure 2 Schematic diagram of the multi-agent system topology in an embodiment of the present invention.

[0087] Figure 3 Schematic diagram of a multi-agent system subjected to a denial of service attack in an embodiment of the present invention;

[0088] Figure 4 A schematic diagram of a position trajectory diagram under a denial of service attack on a multi-agent system according to an embodiment of the present invention;

[0089] Figure 5 Schematic diagram of the speed trajectory diagram of a multi-agent system under a denial of service attack in an embodiment of the present invention. DETAILED DESCRIPTION

[0090] In order to more clearly understand the above-mentioned objects, features and advantages of the present invention, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other. In addition, the present invention can also be implemented in other ways different from those described herein. Therefore, the scope of protection of the present invention is not limited by the specific embodiments disclosed below.

[0091] A specific embodiment of the present invention, as Figure 1-5 , discloses a method for edge event-triggered consistency control of second-order multi-agents under network attacks. In order to illustrate the effectiveness of the method proposed by the present invention, the above technical solution of the present invention is described in detail through a specific embodiment. The specific implementation steps are as follows:

[0092] The present invention provides a method for controlling the consistency of edge event triggering of second-order multi-agents under network attacks, comprising:

[0093] Step S1: constructing a network topology diagram of the multi-agent system and modeling the second-order multi-agent system to obtain a multi-agent system model; the multi-agent system model includes a leader agent and multiple follower agents;

[0094] Step S2: Design a DoS attack frequency and a DoS attack time, and establish a DoS attack model based on the DoS attack frequency and the DoS attack time;

[0095] Step S3: Design the event triggering conditions of the communication edge, the controller of the multi-agent system, and the sampling interval time to meet the conditions; sample the multi-agent system model based on the denial of service attack model and the sampling interval time to obtain multiple sampling data; update the controller of the multi-agent system based on the multiple sampling data and the event triggering conditions of the communication edge to obtain an updated controller of the multi-agent system; the controller only considers the denial of service attack situation;

[0096] Step S4: Use the update controller of the multi-agent system to obtain a control signal, input the control signal into the multi-agent system model, and achieve safe consistency control of the follower agent and the leader agent under a denial of service attack based on the sampling time interval meeting the conditions.

[0097] The specific steps of constructing the network topology diagram of the multi-agent system in step S1 include:

[0098] Use undirected connected graph to represent the network topology of multi-agent system The network topology of the multi-agent system includes a leader agent and multiple follower agents; the leader agent and the follower agents communicate in a one-way manner; and the multiple follower agents communicate in a two-way manner.

[0099] The multiple agents serve as nodes of a network topology graph, each follower agent forms a corresponding edge with the leader agent, and each pair of follower agents forms a corresponding edge;

[0100] Define the adjacency matrix A of the network topology graph of the multi-agent system, where is a real number, n is the number of agents, when a ij =1, it means that the i-th agent can communicate with the j-th agent. ij = 0, it means that the i-th agent and the j-th agent cannot communicate;

[0101] Define the number of edges in the network topology as m, assign an arbitrary direction to each edge, and define the association matrix D and the internal association matrix in d ij is the correlation between the i-th agent and the j-th agent in the correlation matrix,

[0102]

[0103] in, is the correlation matrix d between the i-th agent and ij relevance;

[0104] The Laplace matrix L is obtained based on the inner correlation matrix and the correlation matrix, and the expression is:

[0105]

[0106] Where L is the Laplace matrix.

[0107] Preferably, the steps of modeling the second-order multi-agent system in step S1 to obtain the multi-agent system model include:

[0108] Obtain the leader and follower agents in the network topology of a multi-agent system

[0109] Based on the Lipschitz condition, the dynamics of the follower agent and the system dynamics of the leader agent are obtained respectively;

[0110] A multi-agent system model is established based on the dynamics of the follower agents and the system dynamics of the leader agent.

[0111] Furthermore, the specific steps of obtaining the multi-agent system model include:

[0112] In the network topology diagram of the multi-agent system, the leader agent is marked as x0 and the follower agent is marked as x i , i=1,2,3…n-1, n represents the total number of agents, n-1 represents the number of follower agents;

[0113] Obtain the dynamics of the follower agent based on the Lipschitz condition;

[0114] Obtain the system dynamics of the leader agent based on the Lipschitz condition;

[0115] A multi-agent system model is established based on the dynamics of the follower agents and the system dynamics of the leader agent.

[0116] Furthermore, the dynamic expression of the follower agent is:

[0117]

[0118] in, is the derivative of the position of the i-th follower agent at time t; is the derivative of the velocity of the ith follower agent at time t; x i(t) is the position of the i-th follower agent at time t, v i (t) is the speed of the i-th follower agent at time t, u i (t) is the control input of the ith follower agent at time t, and f(·) is a nonlinear function.

[0119] Furthermore, the expression of the Lipschitz condition is:

[0120] |f(t,x i (t),v i (t))-f(t,x j (t),v j (t))|≤η|x i (t)-x j (t)|+η|v i (t)-v j (t)|

[0121] Where η is the Lipschitz constant, x j (t) is the position of the j-th follower agent at time t, v j (t) is the velocity of the j-th follower agent at time t.

[0122] Furthermore, the system dynamics of the leader agent is expressed as:

[0123]

[0124] in, is the derivative of the position of the leader agent at time t, is the derivative of the velocity of the leader agent at time t, v0(t) is the velocity of the leader agent at time t, and x0(t) is the position of the leader agent at time t.

[0125] Preferably, the denial of service attack frequency expression in step S2 is:

[0126]

[0127] Where n(0,t) represents the number of effective DoS switch transitions in the time period (0,t); η′ represents the jitter bound of the DoS attack frequency, T D represents the average dwell time between on and off transitions of the DoS system,

[0128] Preferably, the denial of service attack time expression is:

[0129]

[0130] Where |D(0,t)| represents the duration of the denial of service attack in the time period (0,t), κ represents the regularization term of the denial of service attack time; T F Indicates the duration limit of the denial of service attack;

[0131] Preferably, the specific steps of obtaining the update controller of the multi-agent system in step S3 include:

[0132] Step S31: Obtain the relative state difference of the agents on each edge in the network topology diagram of the multi-agent system;

[0133] Step S32: Set the event triggering time on each edge;

[0134] Step S33: Design edge trigger event parameters, and design the event trigger conditions of the communication edge based on the relative state difference of the agents on each edge, the edge trigger event parameters, and the event trigger time on each edge;

[0135] Step S34: designing a controller of the multi-agent system based on the denial of service attack model;

[0136] Step S35: setting a sampling time interval and a sampling period of a controller of the multi-agent system to meet a condition based on a denial of service attack model;

[0137] Step S36, let s = 1, when s = 1, it represents the first sampling moment in the rth sampling period; let r = 1, when r = 1, it represents the first sampling period; based on the sampling period meeting the condition and the sampling time interval meeting the condition, the multi-agent system is sampled;

[0138] Step S37, obtaining the sampling data of the sth sampling moment in the rth sampling period; judging whether the sampling data of the sth sampling moment meets the sampling period satisfaction condition;

[0139] Step S38: If not satisfied, input the sampling data of the s-th sampling moment into the controller of the multi-agent system at the corresponding moment for updating, and obtain the updated controller at the s-th sampling moment; if satisfied, determine whether s is greater than or equal to S, where S represents the number of sampling moments; if not, set s = s + T, where T represents the sampling time interval, and return to step S37; if so, proceed to the next step;

[0140] Step S39, determine whether r is greater than or equal to R, where R represents the number of sampling cycles. If so, stop updating and update the controller at the sth sampling moment as the update controller of the multi-agent system; if not, set r=r+1 and return to step S37.

[0141] Preferably, the expression for the relative state difference of the agents on each edge is:

[0142] y ij (t) = x i (t)-x j (t),w ij (t) = v i (t)-v j (t)

[0143] Among them, y ij (t) represents the relative position difference between the ith agent and the jth agent at time t, w ij (t) represents the relative speed difference between the i-th agent and the j-th agent at time t.

[0144] Preferably, the event triggering condition of the communication edge is expressed as:

[0145]

[0146] in, is the relative position difference between the kth event triggering time and the previous event triggering time of the i-th agent and the j-th agent on the corresponding edge; is the relative speed difference between the kth event triggering time and the previous event triggering time of the i-th agent and the j-th agent on the corresponding edge, is the triggering moment of the kth event on the corresponding edge between the i-th agent and the j-th agent; y ij (t s ) is the relative position difference between the sth sampling moment and the previous sampling moment of the i-th agent and the j-th agent on the corresponding edge, w ij (t s ) is the relative speed difference between the sth sampling moment and the previous sampling moment of the i-th agent and the j-th agent on the corresponding edge; t s is the sth sampling moment of the corresponding edge between the i-th agent and the j-th agent, b is the edge trigger event parameter, b>1, and n is the number of agents.

[0147] Furthermore, the expression of the update controller is:

[0148]

[0149] Among them, u i (t) is the control input of the i-th agent at time t; N i is the neighbor set of the ith agent; a ij represents the communication status between the i-th agent and the j-th agent, a ij =1 means that the i-th agent communicates with the j-th agent. At this time, the i-th agent is the neighbor of the j-th agent.ij =0 means that the i-th agent and the j-th agent do not communicate; k1 is the parameter related to position in the feedback matrix, and k2 is the parameter related to speed in the feedback matrix; is the position of the i-th agent at the time when the k-th event is triggered on the corresponding edge; is the position of the j-th agent at the time when the k-th event is triggered on the corresponding edge; is the speed of the i-th agent at the time when the k-th event is triggered on the corresponding edge; is the speed of the j-th agent at the time when the k-th event is triggered on the corresponding edge; is the position of the pilot agent at the time of the kth event triggering on the corresponding edge; is the speed of the pilot agent at the time of the kth event on the corresponding edge; b′ i Whether the i-th agent can obtain the leader agent information;

[0150] Furthermore, the expression that the sampling period satisfies the condition is:

[0151] s r+1 -S r ≤R+1;

[0152] Among them, s r is the rth sampling period, R represents the maximum attack time of a single denial of service;

[0153] Furthermore, the expression for the maximum attack time of a single denial of service is:

[0154]

[0155] Where κ represents the regularization term of the DoS attack time, and T is the sampling time interval.

[0156] Furthermore, the sampling time interval satisfies the condition, and the expression is:

[0157]

[0158] Where T is the sampling interval, σ1 is the upper bound of the control input norm, b is the first parameter to be determined, E2 is the second parameter, D T is parameter three, K is the gain matrix; B is parameter four,

[0159] D T =E2H, represents the Kronecker product, ||.|| represents the 2-norm; α is the upper bound of the nonlinear term norm, ln(·) represents the natural logarithm; ρ3 is the performance index constant;

[0160] ρ3=ρ2 / λ(P) max ,λ(P) max represents the maximum eigenvalue of the positive definite matrix P; ρ1 satisfies performance parameters; b2 is the second parameter to be determined, β is the fifth parameter, μ is the gain parameter, A is the positive definite matrix constant, λ2 is the performance parameter 2, is performance parameter three, I2 is performance parameter four, and H is parameter six; σ2 is the upper bound of the perturbation term norm, λ min (P) represents the minimum eigenvalue of the positive definite matrix P, represents the Kronecker product, and ||.|| represents the 2-norm.

[0161] Furthermore, step S3 further includes: designing a control target of an update controller of the multi-agent system; if the designed update controller of the multi-agent system can achieve the control target of the multi-agent system under any initial conditions of the multi-agent system, then the multi-agent system achieves consistency between the leader and the follower;

[0162] The expression of the control objective of the multi-agent system is:

[0163]

[0164] Preferably, the method further includes step S5, designing a Lyapunov function based on the sampling time interval satisfying the conditions and the update controller of the multi-agent system, and proving the secure consistency control of the follower agent and the leader agent under a denial of service attack based on the Lyapunov function and the update controller of the multi-agent system.

[0165] Example 1

[0166] Build Figure 2 The multi-agent system consists of five agents shown;

[0167] The parameter for edge event triggering is designed to be b=1.05. According to the communication topology and control protocol, the control parameter can be designed to be μ=0.3,K=[2.97,4.17],and the other parameters after calculation are as follows: η=0.05,β=1.0027,α=1.0707,ρ1=20,ρ2=4.4453,σ1=26.1578,σ2=48.3606,ρ3=0.1333,and finally the complete control scheme can be obtained.

[0168] And select the initial conditions as follows X=

[12345] T , V=[1-10-2-3] T , simulate the multi-agent system under denial of service attack, the simulation time is 10s, the denial of service signal is as follows Figure 3 As shown in , when the signal is 2, the agent state value cannot be transmitted and the edge event cannot be triggered. The state trajectory obtained by Matlab simulation is as follows Figure 4 and Figure 5 shown.

[0169] It can be seen from the schematic diagram that when the second-order multi-agent system is subjected to a denial of service attack, the proposed edge event triggered control scheme can effectively cope with the denial of service attack and achieve consensus under the denial of service attack.

[0170] The above description is only a preferred specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by any technician familiar with this technical field within the technical scope disclosed by the present invention should be covered by the scope of protection of the present invention.

Claims

1. A method for edge event-triggered consistency control of second-order multi-agents under network attacks, characterized by: include: Step S1: constructing a network topology diagram of the multi-agent system and modeling the second-order multi-agent system to obtain a multi-agent system model; the multi-agent system model includes a leader agent and multiple follower agents; Step S2: Design a DoS attack frequency and a DoS attack time, and establish a DoS attack model based on the DoS attack frequency and the DoS attack time; Step S3: Design the event triggering conditions of the communication edge, the controller of the multi-agent system, and the sampling interval time to meet the conditions; Based on the denial of service attack model and the sampling interval time meeting the conditions, the multi-agent system model is sampled to obtain multiple sampling data; updating the controller of the multi-agent system based on the plurality of sampled data and the event triggering condition of the communication edge to obtain an updated controller of the multi-agent system; The expression of the event triggering condition of the communication edge is: in, For the i The agent and j The first agent on the corresponding edge k The relative position difference between the triggering moment of an event and the triggering moment of the previous event; For the i The agent and j The first agent on the corresponding edge k The relative speed difference between the triggering time of an event and the triggering time of the previous event, For the i The agent and j The first agent on the corresponding edge k The event triggering moment; For the i The agent and j The first agent on the corresponding edge s The relative position difference between the sampling moment and the previous sampling moment, For the i The agent and j The first agent on the corresponding edge s The relative speed difference between the sampling moment and the previous sampling moment; For the i The agent and j The first agent on the corresponding edge s Sampling time, b Edge trigger event parameters; n is the number of agents; The expression of the update controller of the multi-agent system is: in, It's time t No. i Control input for each agent; It is i The neighbor set of an agent; Indicates the i The agent and j The communication status of each agent; is the position-related parameter in the feedback matrix; is the speed-related parameter in the feedback matrix; For the i The first agent on the corresponding edge k The location of the event triggering moment; For the j The first agent on the corresponding edge k The location of the event triggering moment; For the i The first agent on the corresponding edge k The speed at which an event is triggered; For the j The first agent on the corresponding edge k The speed at which an event is triggered; is the first position of the pilot agent on the corresponding edge k The location of the event triggering moment; is the first position of the pilot agent on the corresponding edge k The speed at which an event is triggered; For the i Whether individual agents can obtain information about the leader agent; Step S4: Use the update controller of the multi-agent system to obtain a control signal, input the control signal into the multi-agent system model, and achieve safe consistency control of the follower agent and the leader agent under a denial of service attack based on the sampling time interval meeting the conditions.

2. The edge event trigger consistency control method according to claim 1, characterized in that: The steps of modeling the second-order multi-agent system and obtaining the multi-agent system model in step S1 include: Obtain the leader agent and follower agents among multiple agents in the network topology graph of the second-order multi-agent system; Based on the Lipschitz condition, the dynamics of the follower agent and the system dynamics of the leader agent are obtained respectively; A multi-agent system model is established based on the dynamics of the follower agents and the system dynamics of the leader agent.

3. The edge event trigger consistency control method according to claim 1, characterized in that: The expression of the denial of service attack frequency in step S2 is: in, Indicates time period The number of valid denial of service switch transitions within the The jitter bound representing the frequency of denial of service attacks, represents the average dwell time between on and off transitions of the denial of service system; Indicates time.

4. The edge event trigger consistency control method according to claim 3, characterized in that: The denial of service attack time expression in step S2 is: in, Indicates time period The duration of the denial of service attack, Regularization term representing the duration of the denial of service attack; Indicates the duration limit for a denial of service attack.

5. The edge event triggered consistency control method according to claim 1, characterized in that: The specific steps of obtaining the updated controller of the multi-agent system in step S3 include: Step S31: Obtain the relative state difference of the agents on each edge in the network topology diagram of the multi-agent system; Step S32: Set the event triggering time on each edge; Step S33: Design edge trigger event parameters, and design the event trigger conditions of the communication edge based on the relative state difference of the agents on each edge, the edge trigger event parameters, and the event trigger time on each edge; Step S34: designing a controller of the multi-agent system based on the denial of service attack model; Step S35: setting a sampling time interval and a sampling period of a controller of the multi-agent system to meet a condition based on a denial of service attack model; Step S36: s =1, when s =1, indicating the r The first sampling moment in a sampling cycle; r =1, when r =1, indicating the first sampling period; sampling the multi-agent system based on the sampling period meeting the conditions and the sampling time interval meeting the conditions; Step S37, obtain the r In the sampling period s The sampling data of the sampling moment; judge the s Whether the sampling data at each sampling moment meets the sampling period conditions; Step S38: If not satisfied, s The sampling data of the sampling time is input into the controller of the multi-agent system at the corresponding time to update, and the first s Update the controller at each sampling moment; if satisfied, then judge s Is greater than or equal to S , S Indicates the number of sampling moments. If not, let s = s+T , T Indicates the sampling time interval, and returns to step S37; if yes, proceeds to the next step; Step S39, judgment r Is greater than or equal to R , R Indicates the number of sampling cycles. If yes, stop updating. s The updated controller at each sampling moment is used as the updated controller of the multi-agent system; if not, let r = r+ 1. Return to step S37.

6. The edge event triggered consistency control method according to claim 5, characterized in that: The expression that the sampling period satisfies the condition is: ； in, For the r Sampling period; Indicates the r+ 1 sampling period; Indicates the maximum duration of a single DoS attack.

7. The edge event triggered consistency control method according to claim 5, characterized in that: The sampling time interval meets the conditions, and the expression is: ； in, is the sampling interval, is the upper bound of the control input norm; is the upper bound of the nonlinear term norm; represents the natural logarithm; is the performance index constant; is the upper bound of the perturbation term norm.

8. The edge event triggered consistency control method according to claim 1, characterized in that: The method also includes step S5, designing a Lyapunov function based on the sampling time interval satisfying the conditions and the update controller of the multi-agent system, and proving the secure consistency control of the follower agent and the leader agent under a denial of service attack based on the Lyapunov function and the update controller of the multi-agent system.

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