Self-adaptive fault-tolerant control method of multi-agent system based on neural network

Abstract

The invention relates to the technical field of fault-tolerant control of a multi-agent system, in particular to an adaptive fault-tolerant control method of a multi-agent system based on a neural network, which comprises the following steps: establishing a state updating equation of a leader and a follower in the multi-agent system; constructing a communication topology matrix of the multi-agent system; related parameters of the neural network are selected; and calculating an adjacent error and a surrounding error of the current follower. Calculating a control input weight matrix and control input of the current follower based on the neural network related parameters, the current adaptive parameters and the encircling error and the adjacency error of the current follower, and updating the states of the follower and the leader; calculating an estimation error and a control error of the current follower, and updating adaptive parameters according to rules; and the encircling errors of all followers converge to a preset value. The method can ensure that the multi-agent system can still complete the encircling control under the condition that the actuator fault occurs, and also has better universality and environment adaptability.

Description

Technical Field

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

[0002] In actual application scenarios, multi-agent systems are affected by the working environment (temperature, humidity, etc.) or internal factors (wear and tear), and agents often fail. With the increasing requirements for component reliability in modern precision instruments, component failures can easily have a huge impact on the system. At the same time, in the enclosure control, since multiple follower agents need to be driven to converge within the convex hull formed by multiple leader agents, this control relies on the communication network, resulting in the failure of a certain agent's fault being easily propagated in the communication network, thus leading to the failure of the cooperative control task. Therefore, relevant fault-tolerant mechanisms need to be established to ensure the stability of the system. At present, although many fault-tolerant control methods can solve the problems of nonlinear systems, they still tend to be based on the traditional control idea of feedback and require knowing the exact dynamic model of the system; their processing ability for highly nonlinear and uncertain system dynamic models is not ideal enough. Therefore, more intelligent control methods need to be adopted to better handle actuator failures existing in the system.

[0003] Therefore, how to ensure that the multi-agent system can still complete the enclosure control in the case of actuator failures, while also having good versatility and environmental adaptability, is an urgent problem 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 multi-agent systems based on neural networks, which can not only ensure that the multi-agent system can still complete the enclosure control in the case of actuator failures, but also has good versatility and environmental adaptability.

[0005] To achieve the above object, the present invention adopts the following technical solutions:

[0006] An adaptive fault-tolerant control method for multi-agent systems based on neural networks, comprising the following steps:

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

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

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

[0010] S4: Calculate the adjacent error and enclosure error of the current follower.

[0011] S5: Calculate the control input weight matrix and control input of the current follower based on the neural network related parameters, the current adaptive parameters, and the encirclement error and adjacency error 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 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 errors of all followers converge 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] Among them, x on the left side of the formula i Represents the updated state of follower i; x on the right side of the formula i represents the state of follower i before the update; dt represents the update time interval; f(x i (t),t)=f(x i,1 ,x i,2 ,...x i,n ,t)∈R n is a nonlinear continuous differentiable function, representing the follower's own dynamic model and some unknown nonlinear interference, x i,n represents the nth dimension of the follower i state; B represents the controllable state matrix, B∈R n×m , R represents a set of real numbers, m represents the number of matrix columns, corresponding to the control input dimension, n represents the number of matrix rows, corresponding to the agent state dimension, where m < n, that is, the dimension of the multi-agent system input is smaller than the dimension of the follower's own state; u i (t)∈R m , represents the control input of follower i; ρ i (t) represents the control gain of follower i, ρ i (t) = diag(ρ i,l ), ρ i,l represents the lth efficiency factor of follower i, satisfying ρi,l and Represents the parameter ρ i,l The upper and lower bounds of w i (t) represents the time-varying bias input of the follower, satisfying ||w i (t)||≤δ iw , where δ iw is a positive constant 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 of follower i partially fails;

[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] Among them, x on the left side of the formula s Represents the updated state of the leader s; x on the right side of the formula s represents the state of the leader s before the update; dt represents the update time interval; f(x s (t),t)=f(x s,1 ,x s,2 ,...x s,n ,t)∈R n is a nonlinear continuous differentiable function that represents the leader’s own dynamic model and some unknown nonlinear interference, x s,n The nth dimension representing the state of leader s.

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

[0026]

[0027] Where A1=[a ij ] N×N represents the sub-communication topology between followers, a ij 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 is the subcommunication topology between follower i and leader l, l=N+1,....N+M, a il Corresponding to the communication strength between leaders 1, 2, ...M and follower i, if leader l can send information 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 of agents in the multi-agent system. Its mathematical expression is as follows:

[0029]

[0030] in, represents the weight matrix, q represents the number of neural network nodes, and R represents the set of real numbers; ε i Represents the neural network estimation error, satisfying is a finite value, S(x i )=[S1(x i ),S2(x i ),...,S q (x i )] T represents the basis function matrix, S q (x i ) represents the qth basis function.

[0031] Furthermore, in S4, the calculation formula for the current follower's encirclement error is:

[0032]

[0033] Among them, x i Indicates the current state of follower i, a ik represents the communication intensity between follower i and agent k. When k=1,2,...,N, x k represents the current state of follower j; when k=N+1,N+2,...,N+M, x k Represents the state of the current leader l; N represents the number of followers, and M represents the number of leaders.

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

[0035]

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

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

[0038] Ψ i = diag(Ψ i,p ) n×n

[0039]

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

[0041] When p = m, there is

[0042]

[0043] [[ID=5l]]

[0044] where e i,p is the i -th element of e p , e i represents the encirclement error of follower i; e ib,p is the p-th element of , e ib represents the control error, represents the adjacency error; {η i,p , ζ i,p}> 1, η i,p and ζ i,p represent constants respectively; τ 1i,p , τ 2i,p are positive constants that satisfy ρ i,p represents the lower bound of the control input efficiency factor; and ​are adaptive parameters respectively; S(x i ) represents the basis function of the neural network; Φ i,p Represents a control parameter related to the neural network estimation result; m represents the control input dimension, and n represents the dimension of the agent state.

[0045] Furthermore, in S5, the control input of the current follower is expressed 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 Represents the updated adaptive parameters, and the right side of the formula and represents the adaptive parameter before updating; dt is the updating time interval; r i ,l i ,g i ,d i are all positive numbers and remain unchanged throughout the control process; e i,l represents the lth element of the encirclement error of agent i; p represents the pth element, n represents the dimension of the agent state, is the selected estimation error, control error

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

[0051] This invention ensures that a multi-agent system can achieve its encirclement control objectives even when its dynamics model is unknown and a follower actuator fails. Because the method provided by this invention does not require the exact form of the system's dynamics model, requiring only that the system's dynamics model be a continuously differentiable function, it is highly versatile. Furthermore, this invention improves the system's reliability and security. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are merely embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on the provided drawings without paying any creative work.

[0053] Figure 1This is a flow chart of the adaptive fault-tolerant control method for a multi-agent system based on a neural network provided by the present invention.

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

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

[0056] Figure 4 It is the variation of the encirclement error of all followers in the multi-agent system provided by the present invention over time. DETAILED DESCRIPTION

[0057] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0058] like Figure 1 As shown, the embodiment of the present 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 follower in the multi-agent system and establish the state update equations of the leader and follower respectively;

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

[0061] S3: Select neural network related parameters;

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

[0063] S5: Calculate the control input weight matrix and control input of the current follower based on the neural network related parameters, the current adaptive parameters, and the encirclement error and adjacency error 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 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 errors of all followers converge to the preset value.

[0067] Next, each step of the present invention will 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] Among them, x i on the left side of the formula represents the state of the updated follower i; x i on the right side of the formula represents the state of the follower i before update; dt represents the update time interval; f(x i (t), t) = f(x i,1 , x i,2 ,... x i,n , t) ∈ R n is a non-linear continuously differentiable function, representing the dynamic model of the follower itself and some unknown non-linear disturbances, which can have a known specific form or be an uncertain continuously differentiable function, x i,n represents the nth dimension of the state of the follower i; B represents a controllable state matrix, with a known specific form, B ∈ R n×m , R represents the set of real numbers, m represents the number of matrix columns, corresponding to the dimension of the control input, n represents the number of matrix rows, corresponding to the dimension of the agent state, where m < n, that is, the dimension of the input of the multi-agent system is less than the dimension of the follower's own state; u i (t) ∈ R m , representing the control input of the follower i; ρ i (t) represents the control gain of the follower i, ρ i (t) = diag(ρ i,l ), ρ i,l represents the lth efficiency factor of the follower i, satisfying ρ i,l and respectively represent the upper and lower bounds of the parameter ρ i,l ; w i (t) represents the time-varying bias input of the follower, satisfying ||w i (t)|| ≤ δ iw , where δ iw is a positive constant, satisfying the following conditions:

[0072] If w i = 0, the actuator of follower i works properly;

[0073] If ρ i,l > 0, w i = 0, the actuator of follower i is partially failed;

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

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

[0076] It can also be expressed as:

[0077]

[0078] represents the change in the state of the follower.

[0079] 2) The state update equation of the leader is:

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

[0081] where, x s on the left side of the formula represents the state of leader s after update; x s on the right side of the formula represents the state of leader s before update; dt represents the update time interval; f(x s (t), t) = f(x s,1 , x s,2 ,... x s,n , t) ∈ R n is a non - linear continuously differentiable function, representing the dynamic model of the leader itself and some unknown non - linear disturbances, and x s,n represents the n - th dimension of the state of leader s.

[0082] It can also be expressed as:

[0083]

[0084] where, represents the change in the state of the leader.

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

[0086]

[0087] in, x s =[x s1 ,x s2 ] T , t is the running time of the multi-agent system, 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 can be understood that, in this embodiment, m=1, 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 states of the four followers are:

[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, and a partial failure occurs in follower 2. The control efficiency factor is ρ2 = 1-0.001t, until the value reaches the lower bound of the efficiency factor. A bias input failure occurs in follower 4. The bias input expression is w4 = 0.1 + 0.1cos(it). Followers 1 and 3 operate normally, with ρ1 = ρ3 = 1 and w1 = w3 = 0.

[0097] It can be seen that the initial states are different, so it is necessary to design a distributed fault-tolerant encirclement control method for each follower so that the states of all followers can still converge to the convex hull of the leader in the event of an actuator failure.

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

[0099]

[0100] where A1 = [a ij N×N represents the sub-communication topology among the followers, and a ij 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 is the sub-communication topology between follower i and leader l, where l = N + 1,....N + M, and a il corresponds to the communication strengths between leaders 1, 2,...M and follower i respectively. If leader l can send information to follower i, then a il > 0; otherwise, a il = 0.

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

[0102]

[0103] where

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

[0105]

[0106] where is the current neural network weight matrix, R represents the set of real numbers, is the current estimation error, which is regarded as an adaptive parameter and will change during the control process. In this embodiment, the initial values of both are q is the dimension of the weight matrix, and its actual meaning is the number of neurons. Here, q = 20 is selected. S(x i ) = [S1(x i ), S2(x i ),..., S q (x i )] T is the basis function, and S q (x i ​​) represents the qth basis function. In this embodiment, the basis function is selected as a Gaussian function, which is in the following form:

[0107]

[0108] Among them, η l ,μ il Respectively represent the width and center of the basis function. In this embodiment, η l =1,μ il =[μ ilk ] 20×1 , where μ ilk is a random number between [-10,10].

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

[0110]

[0111] in, represents the weight matrix, q represents the number of neural network nodes; ε i Represents the neural network estimation error, satisfying is a finite value, S(x i )=[S1(x i ),S2(x i ),...,S q (x i )] T represents the basis function matrix.

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

[0113] The calculation formula for the current follower's encirclement error is:

[0114]

[0115] Among them, x i Indicates the current state of follower i, a ik represents the communication intensity between follower i and agent k. When k=1,2,...,N, x k represents the current state of follower j; when k=N+1,N+2,...,N+M, x k Represents the state of the current leader l; N represents the number of followers, and M represents the number of leaders.

[0116] The calculation formula of the current follower's adjacency error is:

[0117]

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

[0119] 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 control input of the current follower into the follower's state update equation to obtain the updated follower state, and update the leader's state.

[0120] Specifically, the calculation formula of 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 pth element, e i represents the encirclement error of follower i; e ib,p yes The pth element, e ib represents the control error, represents the adjacency error; i,p ,ζ i,p}>1,η i,p and ζ i,p Respectively represent constants; τ 1i,p ,τ 2i,p is a normal number, satisfying ρ i,p represents the lower bound of the control input efficiency factor; and They are adaptive parameters respectively; S(x i) represents the basis function of the neural network; Φ i,p Represents a control parameter related to the neural network estimation result; m represents the control input dimension, and n represents the dimension of the agent state.

[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 parameter

[0128] Afterwards, the following formula is used to obtain the control input of the current follower:

[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] Adaptive parameters are updated according to the following rules:

[0136]

[0137] Among them, the left side of the formula and Represents the updated adaptive parameters, and the right side of the formula and represents the adaptive parameter before updating; dt is the updating time interval; r i ,l i ,g i ,d i are all positive numbers and remain unchanged throughout the control process; e i,l represents the lth element of the encirclement error of agent i; p represents the pth element, n represents the dimension of the agent state, is 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 a preset value.

[0139] The control target expression is:

[0140]

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

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

[0143] like Figure 3 The figure shows how each agent's state changes 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 beginning of control, each agent's state is distinct. By the end of control, the follower states converge to the convex hull of the leader, achieving the encirclement control goal.

[0144] like Figure 4 As shown, the encirclement error of the follower agent changes with 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, which shows that the present invention can make the encirclement error of the multi-agent system converge to a small range in a shorter time, proving the effectiveness of the control protocol provided by the present invention without losing its speed.

[0145] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. Reference can be made to the common and similar parts between the various embodiments. For the devices disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple, and the relevant parts can be referred to the method description.

[0146] The above description of the disclosed embodiments is intended to enable one skilled in the art to implement or use the present invention. Various modifications to these embodiments will be readily apparent to one 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 present invention. Therefore, the present invention is not limited to the embodiments shown herein but is intended to conform to the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. An adaptive fault-tolerant control method for a multi-agent system based on a neural network, characterized in that, Including 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 the relevant parameters of the neural network; S4: Calculate the adjacency error and encirclement error of the current follower. S5: Based on the relevant parameters of the neural network, the current adaptive parameter, 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 control input of the current follower into the state update equation of the follower to obtain the updated state of the follower, and update the state of the leader; S6: Calculate the estimation error and control error of the current follower, and update the adaptive parameter according to the rules; S7: Repeat S4 - S6 until the encirclement errors of all followers converge to a 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 of the follower is: x i = x i + [f(x i (t), t) + B(ρ i (t)u i (t) + w i (t))]dt Among them, \(x\) on the left side of the formula i represents the updated state of follower \(i\); \(x\) on the right side of the formula i represents the state of follower \(i\) before update; \(dt\) represents the update time interval; \(f(x i (t),t)=f(x i,1 ,x i,2 ,...x i,n ,t)\in\mathbb{R} n is a non - linear continuously differentiable function, representing the dynamic model of the follower itself and some unknown non - linear disturbances, \(x i,n represents the \(n\) - th dimension of the state of follower \(i\); \(B\) represents the controllable state matrix, \(B\in\mathbb{R} n×m , \mathbb{R}\) represents the set of real numbers, \(m\) represents the number of columns of the matrix, corresponding to the dimension of the control input, \(n\) represents the number of rows of the matrix, corresponding to the dimension of the agent state, where \(m < n\), that is, the dimension of the input of the multi - agent system is less than the dimension of the follower's own state; \(u i (t)\in\mathbb{R} m , represents the control input of follower \(i\); \(\rho i (t)\) represents the control gain of follower \(i\), \(\rho i (t)=\text{diag}(\rho i,l ), \rho i,l represents the \(l\) - th efficiency factor of follower \(i\), satisfying ρ i,l and respectively represent the upper and lower bounds of the parameter \(\rho i,l ; \(w i (t)\) represents the time - varying bias input of the follower, satisfying \(\|w i (t)\|\leq\delta iw , where \(\delta iw is a positive constant, satisfying the following conditions: If the actuator of follower i works properly; If the actuator part of follower i fails; If the actuator of follower i has a bias fault; If The actuator bias fault and the partial actuator failure fault of follower i 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 state update equation of the leader is: x s = x s + [f(x s (t), t)] dt Among them, \(x\) on the left side of the formula s represents the state of the updated leader \(s\); \(x\) on the right side of the formula s represents the state of the leader \(s\) before the update; \(dt\) represents the update time interval; \(f(x\) s (t),t)=f(x s,1 ,x s,2 ,...x s,n ,t)\(\in\mathbb{R}\) n is a non - linear continuously differentiable function, representing the dynamic model of the leader itself and some unknown non - linear disturbances, and \(x\) s,n represents the \(n\) - th dimension of the state of the leader \(s\).

4. The adaptive fault-tolerant control method for a multi-agent system based on a neural network according to claim 1, wherein In S2, assuming there are N followers and M leaders in the multi-agent system, the communication topology matrix is: Among them, A1 = [a ij N×N represents the sub-communication topology among followers, and a ij 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 is the sub-communication topology between follower i and leader l, where l = N + 1,....N + M, and a il corresponds to the communication strengths between leaders 1, 2,... M and follower i respectively. If leader l can send information to follower i, then a il > 0; otherwise, a il = 0.​​ 5. The adaptive fault-tolerant control method for a multi-agent system based on a neural network according to claim 1, characterized in that In S3, the neural network is used to estimate the maximum boundary of the state difference of the agents in the multi-agent system, and its mathematical expression is as follows: Among them, represents the weight matrix, q represents the number of neural network nodes, and R represents the set of real numbers; ε i represents the neural network estimation error and satisfies is a finite value, S(x i ) = [S1(x i ), S2(x i ),..., S q (x i )] T represents the basis function matrix, and S q (x i ) represents the q-th 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 calculation formula for the encirclement error of the current follower is: where x i represents the state of the current follower i, and a ik represents the communication strength between the follower i and the agent k. When k = 1, 2, ..., N, x k represents the state of the current follower j; when k = N + 1, N + 2, ..., N + M, x k represents the state of the current leader l; 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 1, characterized in that, In S4, the calculation formula for the adjacency error of the current follower is: Among them, l ji Represents the elements in the matrix L, L = D-A1 = [l ij ] N×N , D=diag(d i ), D represents the degree matrix, d i Represents the i-th element of the degree matrix; a jk Represents the elements in the communication topology matrix A; e j represents the encirclement error of follower j; x k represents the state of agent k.

8. The adaptive fault-tolerant control method for a multi-agent system based on a neural network according to claim 1, wherein In S5, the calculation formula for the control input weight matrix of the current follower is: Ψ i = diag(Ψ i,p ) n×n where, p = 1, 2,... m - 1; When p = m, there is Among them, e i,p It is e i The pth element, e i represents the encirclement error of follower i; e ib,p yes The pth element, e ib represents the control error, represents the adjacency error; i,p ,ζ i,p }>1,η i,p and ζ i,p Respectively represent constants; τ 1i,p ,τ 2i,p is a normal number, satisfying ρ i,p represents the lower bound of the control input efficiency factor; and are adaptive parameters respectively; S(x i ) represents the basis function of the neural network; Φ i,p Represents a control parameter related to the neural network estimation result; m represents the control input dimension, and n represents the dimension of the agent state.

9. The adaptive fault-tolerant control method for a multi-agent system based on a neural network according to claim 8, characterized in that, In S5, the control input of the current follower is expressed as:

10. The adaptive fault-tolerant control method for a multi-agent system based on a neural network according to claim 8, wherein The adaptive parameter update formula in S6 is as follows: Among them, the left side of the formula and Represents the updated adaptive parameters, and the right side of the formula and represents the adaptive parameter before updating; dt is the updating time interval; r i ,l i ,g i ,d i are all positive numbers and remain unchanged throughout the control process; e i,l represents the lth element of the encirclement error of agent i; p represents the pth element, n represents the dimension of the agent state, is the selected estimation error, control error

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