Adaptive Fault-Tolerant Control Method for Multi-Agent Systems Based on Neural Networks

Abstract

This invention relates to the field of fault-tolerant control technology for multi-agent systems, specifically to an adaptive fault-tolerant control method for multi-agent systems based on neural networks. The method includes: establishing state update equations for the leader and followers in the multi-agent system; constructing the communication topology matrix of the multi-agent system; selecting relevant neural network parameters; and calculating the adjacency error and encirclement error of the current follower. Based on the relevant neural network parameters, the current adaptive parameters, and the encirclement and adjacency errors of the current followers, the method calculates the control input weight matrix and control input of the current follower, updating the states of the followers and leader; calculating the estimation error and control error of the current follower, and updating the adaptive parameters according to rules; until the encirclement errors of all followers converge to a preset value. This invention ensures that the multi-agent system can still complete encirclement control even in the event of actuator failure, and also has good versatility and environmental adaptability.

Description

Technical Field

[0001] This invention relates to the field of fault-tolerant control technology for multi-agent systems, and more specifically to an adaptive fault-tolerant control method for multi-agent systems based on neural networks. Background Technology

[0002] In practical applications, multi-agent systems are frequently affected by their operating environment (temperature, humidity, etc.) or internal factors (wear and tear), leading to frequent agent failures. As the reliability requirements for components in precision instruments increase, component failures can easily have a significant impact on the system. Furthermore, in encirclement control, because multiple follower agents need to converge into the convex hull formed by multiple leader agents, this control relies on communication networks. This means that a failure in one agent can easily propagate through the communication network, leading to the failure of the collaborative control task. Therefore, relevant fault-tolerant mechanisms are needed to ensure system stability. While many fault-tolerant control methods can solve problems in nonlinear systems, they still tend to rely on traditional feedback-based control, requiring knowledge of the system's exact dynamic model. Their ability to handle highly nonlinear and uncertain system dynamic models is insufficient. Therefore, more intelligent control methods are needed to better address actuator failures in the system.

[0003] Therefore, how to ensure that a multi-agent system can still complete encirclement control in the event of actuator failure, while also having good versatility and environmental adaptability, is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0004] In view of this, the present invention provides an adaptive fault-tolerant control method for a multi-agent system based on a neural network, which can not only ensure that the multi-agent system can still complete encirclement control in the event of actuator failure, but also has good versatility and environmental adaptability.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] An adaptive fault-tolerant control method for a multi-agent system based on neural networks includes the following steps:

[0007] S1: Determine the leader and followers in the multi-agent system, and establish the state update equations for the leader and followers respectively;

[0008] S2: Construct the communication topology matrix of the multi-agent system;

[0009] S3: Select relevant parameters for the neural network;

[0010] S4: Calculate the adjacency error and encirclement error of the current follower.

[0011] S5: Based on the neural network related parameters, the current adaptive parameters, and the encirclement error and adjacency error of the current follower, calculate the control input weight matrix and control input of the current follower;

[0012] Substitute the current follower's control input into the follower's state update equation to obtain the updated follower's state, and then update the leader's state.

[0013] S6: Calculate the estimation error and control error of the current follower, and update the adaptive parameters according to the rules;

[0014] S7: Repeat S4-S6 until the encirclement error of all followers converges to the preset value.

[0015] Furthermore, in S1, the follower's state update equation is:

[0016] x i =x i +[f(x i (t),t)+B(ρ i (t)u i (t)+w i (t))]dt

[0017] Where x is on the left side of the formula i This represents the updated state of follower i; x on the right side of the formula i The state of follower i before the update is represented by dt; the update time interval is represented by f(x). i (t),t)=f(x i,1 ,x i,2 ,...x i,n ,t)∈R n It is a nonlinear, continuously differentiable function representing the dynamic model of the follower itself and some unknown nonlinear disturbances, x. i,n B represents the nth dimension of the follower i's state; B represents the controllable state matrix, B∈R n×m R represents the set of real numbers, m represents the number of columns in the matrix, corresponding to the control input dimension, and n represents the number of rows in the matrix, corresponding to the agent's state dimension, where m < n, meaning the input dimension of the multi-agent system is less than the dimension of the follower's own state; u i (t)∈R m ρ represents the control input of follower i; i (t) represents the control gain of follower i, ρ i (t)=diag(ρ i,l ), ρ i,l Let the l-th efficiency factor of follower i satisfy... ρi,l and They represent the parameter ρ respectively i,l upper and lower bounds; w i (t) represents the time-varying bias input of the follower, satisfying ||w i (t)||≤δ iw , where δ iw Let be a positive integer that satisfies the following conditions:

[0018] like w i =0, the actuator of follower i works normally;

[0019] like ρ i,l >0, w i =0, the actuator part of follower i is disabled;

[0020] like w i >0, the actuator of follower i has a bias fault;

[0021] like ρ i,l >0, w i >0, the actuator bias fault and the actuator partial failure fault of follower i occur simultaneously.

[0022] Furthermore, in S1, the leader's state update equation is:

[0023] x s =x s +[f(x s (t),t)]dt

[0024] Where x is on the left side of the formula s This represents the state of the updated leader s; x on the right side of the formula s The state of leader s before the update is represented; dt represents the update time interval; f(x) represents the state of leader s before the update .... s (t),t)=f(x s,1 ,x s,2 ,...x s,n ,t)∈R n It is a nonlinear, continuously differentiable function representing the dynamic model of the leader itself and some unknown nonlinear disturbances, x. s,n The nth dimension represents the state of leader s.

[0025] Furthermore, in S2, assuming there are N followers and M leaders in the multi-agent system, the communication topology matrix is ​​as follows:

[0026]

[0027] Where, A1=[a ij ] N×N Describes the sub-communication topology between followers, a ij a represents the communication strength between follower i and follower j. If follower i and follower j can communicate with each other, then a ij >0, otherwise a ij =0; A2=[a il ] N×M It is a sub-communication topology between follower i and leader l, where l = N+1, ..., N+M, a il These correspond to the communication strengths between leaders 1, 2, ..., M and followers i, respectively. If leader l can send a message to follower i, a il >0, otherwise a il =0.

[0028] Furthermore, in S3, a neural network is used to estimate the maximum boundary of the state difference between agents in a multi-agent system, and its mathematical expression is as follows:

[0029]

[0030] in, Let ε represent the weight matrix, q represent the number of nodes in the neural network, and R represent the set of real numbers; i The neural network estimation error is represented by the condition that... For a finite value, S(x) i )=[S1(x i ),S2(x i ),...,S q (x i )] T Let S represent the basis function matrix. q (x i ) represents the q-th basis function.

[0031] Furthermore, in S4, the formula for calculating the encirclement error of the current followers is:

[0032]

[0033] Where, x i a represents the current state of follower i. ik x represents the communication strength between follower i and agent k, when k = 1, 2, ..., N. k This represents the current state of follower j; when k = N+1, N+2, ..., N+M, x k This represents the current state of leader l; N represents the number of followers, and M represents the number of leaders.

[0034] Furthermore, in S4, the formula for calculating the adjacency error of the current follower is:

[0035]

[0036] Among them, l ji Denotes the elements in matrix L, L = D - A1 = [l ij ] N×N D = diag(d i ), D represents the degree matrix, d i a represents the i-th element of the degree matrix; jk Represents an element in the communication topology matrix A; e j x represents the encirclement error of follower j; k This represents the state of agent k.

[0037] Furthermore, in S5, the formula for calculating the control input weight matrix of the current follower is:

[0038] Ψ i =diag(Ψ) i,p ) n×n

[0039]

[0040] Where p = 1, 2, ..., m-1;

[0041] When p = m, we have

[0042]

[0043]

[0044] Among them, e i,p It is e i The p One element, e i e represents the encirclement error of follower i; ib,p yes The p-th element, e ib Indicates control error. Represents the adjacency error; , {η i,p ,ζ i,p}>1,η i,p and ζ i,p τ represents a constant; 1i,p ,τ 2i,p It is a positive constant and satisfies ρ i,p Indicates the lower bound of the control input efficiency factor; and These are the adaptive parameters; S(x) i ) represents the basis functions of the neural network; Φ i,p This represents a control parameter associated with the neural network estimation result; m represents the control input dimension, and n represents the agent state dimension.

[0045] Furthermore, in S5, the control input of the current follower is represented as:

[0046]

[0047] Furthermore, the adaptive parameter update formula in S6 is as follows:

[0048]

[0049] Among them, the left side of the formula and This represents the updated adaptive parameters, shown on the right side of the formula. and The adaptive parameters before the update are represented; dt is the update time interval; r i ,l i ,g i ,d i All values ​​are positive and remain constant throughout the control process; e i,l The l-th element represents the encirclement error of agent i; p represents the p-th element, and n represents the dimension of the agent's state. For the selected estimation error, control error

[0050] As can be seen from the above technical solution, compared with the prior art, the present invention has the following beneficial effects:

[0051] This invention ensures that a multi-agent system can still achieve encirclement control even when the dynamic model is unknown or when the follower experiences actuator failure. Because the method provided by this invention does not require knowledge of the exact form of the system's dynamic model, only that the model be a continuously differentiable function, it has good versatility. Furthermore, this invention improves the system's reliability and security. Attached Figure Description

[0052] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0053] Figure 1The flowchart shows the adaptive fault-tolerant control method for a multi-agent system based on a neural network provided by this invention.

[0054] Figure 2 The communication topology diagram of the multi-agent system provided by the present invention.

[0055] Figure 3 This describes the changes in the state of all agents over time in the multi-agent system provided by this invention.

[0056] Figure 4 This describes the change in the encirclement error of all followers in the multi-agent system provided by this invention over time. Detailed Implementation

[0057] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0058] like Figure 1 As shown in the figure, this invention discloses an adaptive fault-tolerant control method for a multi-agent system based on a neural network, comprising the following steps:

[0059] S1: Determine the leader and followers in the multi-agent system, and establish the state update equations for the leader and followers respectively;

[0060] S2: Construct the communication topology matrix of the multi-agent system;

[0061] S3: Select relevant parameters for the neural network;

[0062] S4: Calculate the adjacency error and encirclement error of the current follower.

[0063] S5: Based on the neural network related parameters, the current adaptive parameters, and the encirclement error and adjacency error of the current follower, calculate the control input weight matrix and control input of the current follower;

[0064] Substitute the current follower's control input into the follower's state update equation to obtain the updated follower's state, and then update the leader's state.

[0065] S6: Calculate the estimation error and control error of the current follower, and update the adaptive parameters according to the rules;

[0066] S7: Repeat S4-S6 until the encirclement error of all followers converges to the preset value.

[0067] The steps of this invention will now be described in detail.

[0068] S1. Determine the leader and followers in the multi-agent system, and establish the state update equations for the leader and followers respectively.

[0069] 1) The state update equation for the follower is:

[0070] x i =x i +[f(x i (t),t)+B(ρ i (t)u i (t)+w i (t))]dt

[0071] Where x is on the left side of the formula i This represents the updated state of follower i; x on the right side of the formula i The state of follower i before the update is represented by dt; the update time interval is represented by f(x). i (t),t)=f(x i,1 ,x i,2 ,...x i,n ,t)∈R n It is a nonlinear, continuously differentiable function representing the dynamic model of the follower itself and some unknown nonlinear disturbances. It can have a known specific form or be an uncertain, continuously differentiable function, x. i,n B represents the nth dimension of the follower i's state; B represents the controllable state matrix, the specific form of which is known, B∈R n×m R represents the set of real numbers, m represents the number of columns in the matrix, corresponding to the control input dimension, and n represents the number of rows in the matrix, corresponding to the agent's state dimension, where m < n, meaning the input dimension of the multi-agent system is less than the dimension of the follower's own state; u i (t)∈R m ρ represents the control input of follower i; i (t) represents the control gain of follower i, ρ i (t)=diag(ρ i,l ), ρ i,l Let the l-th efficiency factor of follower i satisfy... ρ i,l and They represent the parameter ρ respectively i,l upper and lower bounds; w i (t) represents the time-varying bias input of the follower, satisfying ||w i (t)||≤δ iw , where δ iw Let be a positive integer that satisfies the following conditions:

[0072] like w i =0, the actuator of follower i works normally;

[0073] like ρ i,l >0, w i =0, the actuator part of follower i is disabled;

[0074] like w i >0, the actuator of follower i has a bias fault;

[0075] like ρ i,l >0, w i >0, the actuator bias fault and the actuator partial failure fault of follower i occur simultaneously.

[0076] It can also be expressed as:

[0077]

[0078] This indicates the change in the follower's state.

[0079] 2) The leader's state update equation is:

[0080] x s =x s +[f(x s (t),t)]dt

[0081] Where x is on the left side of the formula s This represents the state of the updated leader s; x on the right side of the formula s The state of leader s before the update is represented; dt represents the update time interval; f(x) represents the state of leader s before the update .... s (t),t)=f(x s,1 ,x s,2 ,...x s,n ,t)∈R n It is a nonlinear, continuously differentiable function representing the dynamic model of the leader itself and some unknown nonlinear disturbances, x. s,n The nth dimension represents the state of leader s.

[0082] It can also be expressed as:

[0083]

[0084] in, This indicates the amount of change in the leader's state.

[0085] In one specific embodiment, assuming the multi-agent system has 2 leaders and 4 followers, the leader's state update equation is:

[0086]

[0087] in, x s =[x s1 ,x s2 ] T t is the running time of the multi-agent system, and s = 1, 2.

[0088] The follower state update equation is:

[0089]

[0090] in, x i =[x i1 ,x i2 ] T , i = 1, 2, 3, 4.

[0091] It is understood that in this embodiment, m = 1 and n = 2.

[0092] In this embodiment, the initial states of the two leaders are:

[0093] x1 = [1, 1.3] T x2 = [1.2, 1] T

[0094] The initial state of the four followers is as follows:

[0095] x1 = [0.9, 1] T x2 = [1, 0.8] T x3 = [1, 1.2] T x4 = [0.8, 1] T

[0096] In this embodiment, ρ i,l =0.4, follower 2 experiences a partial failure, and the control efficiency factor is ρ2 = 1 - 0.001t, until the value reaches the lower bound of the efficiency factor. Follower 4 experiences a bias input failure, and the bias input expression is w4 = 0.1 + 0.1cos(it). Followers 1 and 3 work normally, with ρ1 = ρ3 = 1 and w1 = w3 = 0.

[0097] As can be seen, the initial states are all different. Therefore, a distributed fault-tolerant encirclement control method needs to be designed for each follower so that the states of all followers can still converge into the convex hull of the leader even if the actuator fails.

[0098] S2. Assume a multi-agent system has N followers and M leaders, and the communication topology matrix is ​​as follows:

[0099]

[0100] Where, A1=[a ij ] N×N Describes the sub-communication topology between followers, a ij a represents the communication strength between follower i and follower j. If follower i and follower j can communicate with each other, then a ij >0, otherwise a ij =0; A2=[a il ] N×M It is a sub-communication topology between follower i and leader l, where l = N+1, ..., N+M, a il These correspond to the communication strengths between leaders 1, 2, ..., M and followers i, respectively. If leader l can send a message to follower i, a il >0, otherwise a il =0.

[0101] Specifically, the communication topology of a multi-agent system is as follows: Figure 2 As shown, the communication topology matrix is:

[0102]

[0103] in,

[0104] S3. Select relevant parameters for the neural network, including the initial values ​​of the neural network weight matrix, the basis function structure, the parameters related to the basis functions, the number of neural network nodes, and the neural network estimation error. The relevant mathematical expressions for the neural network are as follows:

[0105]

[0106] in, Let R be the weight matrix of the current neural network, and let R represent the set of real numbers. The current estimation error is subsequently treated as an adaptive parameter, which will change during the control process. In this embodiment, both are initially set to... q is the dimension of the weight matrix, which represents the number of neurons; here, q = 20 is chosen. S(x i )=[S1(x i ),S2(x i ),...,S q (x i )] T S is a basis function. q (x i() represents the q-th basis function. In this embodiment, the basis function is chosen as a Gaussian function, which takes the following form:

[0107]

[0108] Where, η l ,μ il η represents the width and center of the basis function, respectively. In this embodiment, η l =1,μ il =[μ ilk ] 20×1 , where μ ilk A random number between [-10, 10].

[0109] In this step, a neural network is used to estimate the maximum boundary of the state difference between agents in a multi-agent system, and its mathematical expression is as follows:

[0110]

[0111] in, ε represents the weight matrix, q represents the number of nodes in the neural network; i The neural network estimation error is represented by the condition that... For a finite value, S(x) i )=[S1(x i ),S2(x i ),...,S q (x i )] T This represents the basis function matrix.

[0112] S4. Calculate the adjacency error and encirclement error of the current follower.

[0113] The formula for calculating the encirclement error of the current followers is:

[0114]

[0115] Where, x i a represents the current state of follower i. ik x represents the communication strength between follower i and agent k, when k = 1, 2, ..., N. k This represents the current state of follower j; when k = N+1, N+2, ..., N+M, x k This represents the current state of leader l; N represents the number of followers, and M represents the number of leaders.

[0116] The formula for calculating the adjacency error of the current follower is:

[0117]

[0118] Among them, l ji Denotes the elements in matrix L, L = D - A1 = [l ij ] N×N D = diag(d i ), D represents the degree matrix, d i a represents the i-th element of the degree matrix; jk Represents an element in the communication topology matrix A; e j x represents the adjacency error of follower j; k This represents the state of agent k.

[0119] S5. Based on the relevant parameters of the neural network, the current adaptive parameters, and the encirclement error and adjacency error of the current followers, calculate the control input weight matrix and control input of the current followers; substitute the control input of the current followers into the state update equation of the followers to obtain the updated state of the followers, and update the state of the leader.

[0120] Specifically, the formula for calculating the control input weight matrix of the current follower is:

[0121] Ψ i =diag(Ψ) i,p ) n×n

[0122]

[0123] Where p = 1, 2, ..., m-1;

[0124] When p = m, we have

[0125]

[0126] Among them, e i,p It is e i The p-th element, e i e represents the encirclement error of follower i; ib,p yes The p-th element, e ib Indicates control error. Represents the adjacency error; , {η i,p ,ζ i,p}>1,η i,p and ζ i,p τ represents a constant; 1i,p ,τ 2i,p It is a positive constant and satisfies ρ i,p Indicates the lower bound of the control input efficiency factor; and These are the adaptive parameters; S(x) i) represents the basis functions of the neural network; Φ i,p This represents a control parameter associated with the neural network estimation result; m represents the control input dimension, and n represents the agent state dimension.

[0127] In this embodiment, the constant η i,p =7,τ 1i,p =0.05,τ 2i,p =0.03,ζ i,p =1.1, initial value of adaptive parameters

[0128] Then, the control input of the current follower is obtained using the following formula:

[0129]

[0130] S6. Calculate the estimation error and control error of the current follower, and update the adaptive parameters according to the rules.

[0131] The estimation error ε is calculated using the following formula. i :

[0132]

[0133] The control error e is calculated using the following formula. ib :

[0134]

[0135] Update the adaptive parameters according to the following rules:

[0136]

[0137] Among them, the left side of the formula and This represents the updated adaptive parameters, shown on the right side of the formula. and The adaptive parameters before the update are represented; dt is the update time interval; r i ,l i ,g i ,d i All values ​​are positive and remain constant throughout the control process; e i,l The l-th element represents the encirclement error of agent i; p represents the p-th element, and n represents the dimension of the agent's state. For the selected estimation error, control error In this embodiment, r i =0.003,l i =0.004,g i =0.007,d i=0.001.

[0138] S7. Repeat S4-S6 until the encirclement error converges to the preset value.

[0139] The control objective expression is:

[0140]

[0141] Where ε is a non-negative integer, and in this embodiment it is taken as 0.03.

[0142] Next, the performance of the method of the present invention will be verified.

[0143] like Figure 3 The diagram illustrates the state changes of each agent over time. f1-f4 represent the state changes of the four followers, and l1-l2 represent the state changes of the two leaders. It can be seen that at the start of control, each agent's state is different; at the end of control, the follower states converge to the convex hull of the leaders, achieving the goal of encirclement control.

[0144] like Figure 4 As shown, the encirclement error of the follower agent changes over time. It can be seen that the encirclement control error of the multi-agent system has converged to about 0.03 in about 10 seconds. This shows that the present invention can enable the encirclement error of the multi-agent system to converge to a small range in a short time, proving the effectiveness of the control protocol provided by the present invention without sacrificing speed.

[0145] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0146] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for adaptive fault-tolerant control of a multi-agent system based on neural networks, characterized in that, Includes the following steps: S1: Determine the leader and followers in the multi-agent system, and establish the state update equations for the leader and followers respectively; S2: Construct the communication topology matrix of the multi-agent system; S3: Select relevant parameters for the neural network; S4: Calculate the adjacency error and encirclement error of the current follower; S5: Based on the neural network related parameters, the current adaptive parameters, and the encirclement error and adjacency error of the current follower, calculate the control input weight matrix and control input of the current follower; Substitute the current follower's control input into the follower's state update equation to obtain the updated follower's state, and then update the leader's state. The formula for calculating the control input weight matrix of the current follower is: in, ; when Sometimes, in, yes The One element, Indicates follower i The encirclement error; yes The One element, Indicates control error. Let B represent the adjacency error, and let B represent the controllable state matrix. , and They represent constants respectively; It is a positive constant and satisfies , Indicates the lower bound of the control input efficiency factor; , , and These are adaptive parameters; The basis functions of a neural network; This represents a control parameter related to the estimation result of the neural network; m This indicates the control of the input dimension. n The dimension representing the state of the agent; The current follower's control input is represented as: S6: Calculate the estimation error and control error of the current follower, and update the adaptive parameters according to the rules; S7: Repeat S4-S6 until the encirclement error of all followers converges to the preset value.

2. The adaptive fault-tolerant control method for a multi-agent system based on a neural network according to claim 1, characterized in that, In S1, the state update equation for the follower is: Among them, the left side of the formula Indicates the updated followers i The state; the right side of the formula Indicates followers before the update i The state; Indicates the update time interval; It is a nonlinear, continuously differentiable function representing the dynamic model of the follower itself and some unknown nonlinear disturbances. x i,n Indicates follower i The state of n One dimension; , R Represents the set of real numbers. m This indicates the number of columns in the matrix, corresponding to the control input dimension. n This represents the number of rows in the matrix, corresponding to the agent's state dimension, where That is, the dimension of the input to a multi-agent system is smaller than the dimension of the follower's own state; , representing the control input of follower i; Indicates follower i The control gain, , Indicates follower i The One efficiency factor, satisfying , and They represent parameters respectively The upper and lower bounds; The time-varying bias input of the follower satisfies ,in Let be a positive integer that satisfies the following conditions: like followers i The actuator is working normally; like followers i The actuator part failed; like followers i The actuator has a bias fault; like followers i The actuator bias fault and the actuator partial failure fault occur simultaneously.

3. The adaptive fault-tolerant control method for a multi-agent system based on a neural network according to claim 1, characterized in that, In S1, the leader's state update equation is: Among them, the left side of the formula This represents the updated state of leader s; the right side of the formula... Indicates the leader before the update s The state; Indicates the update time interval; It is a nonlinear, continuously differentiable function, representing the dynamic model of the leader itself and some unknown nonlinear disturbances. x s,n Indicates the leader s The state of n Each dimension.

4. The adaptive fault-tolerant control method for a multi-agent system based on a neural network according to claim 1, characterized in that, In S2, assuming there are N followers and M leaders in the multi-agent system, the communication topology matrix is ​​as follows: in, This represents the sub-communication topology between followers. This represents the communication strength between follower i and follower j. i With followers j If they can communicate with each other, then ,otherwise ; Followers i and leaders l Sub-communication topology between , Corresponding to leaders With followers i The strength of communication between leaders, if the leader l Can give followers i Send message, ,otherwise .

5. The adaptive fault-tolerant control method for a multi-agent system based on a neural network according to claim 2, characterized in that, In S3, a neural network is used to estimate the maximum boundary of the state difference between agents in a multi-agent system. Its mathematical expression is as follows: in, Represents the weight matrix, q Represents the number of nodes in a neural network. R Represents the set of real numbers; The neural network estimation error is represented by the condition that... , For a finite value, Represents the basis function matrix, S q ( x i ) indicates the first q One basis function.

6. The adaptive fault-tolerant control method for a multi-agent system based on a neural network according to claim 1, characterized in that, In S4, the formula for calculating the encirclement error of the current followers is: in, Indicates the current follower i state, Indicates follower i With intelligent agents k The communication strength between them, when hour, Indicates the current follower j The state; when hour, Indicates the current leader l The state; N represents the number of followers, and M represents the number of leaders.

7. The adaptive fault-tolerant control method for a multi-agent system based on a neural network according to claim 4, characterized in that, In S4, the formula for calculating the adjacency error of the current follower is: in, Represents the elements in matrix L. , , D Degree matrix, d i The degree matrix represents the first degree. i One element; a jk Representing the communication topology matrix A Middle element; e j Indicates follower j The encirclement error; x k Represents intelligent agents k The state.

8. The adaptive fault-tolerant control method for a multi-agent system based on a neural network according to claim 1, characterized in that, The adaptive parameter update formula in S6 is as follows: Among them, the left side of the formula , , and This represents the updated adaptive parameters, shown on the right side of the formula. , , and This represents the adaptive parameters before the update; dt This is the update time interval; All values ​​were normal and remained constant throughout the control process; Represents intelligent agents i The first of the encirclement error One element; , p Indicates the first p One element, n The dimension representing the state of an agent. , where is the selected estimation error, and is the control error. .

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