Adaptive neural network cooperative fault-tolerant control method for a class of multi-agent systems

Abstract

The application discloses a kind of multi-agent system adaptive neural network collaborative fault-tolerant control method, belong to nonlinear multi-agent system control field, including steps: constructing multi-agent system model, for output system state;The multi-agent system includes leader and follower;Design distributed mode estimator, the system state is input into distributed mode estimator, obtains the estimated value of leader trajectory, and calculates the estimation error of leader trajectory;The estimation error is converted into first-order and high-order error variables by error conversion module;Based on the first-order and high-order error variables, an adaptive law and an adaptive fault-tolerant controller are obtained using backstepping recursion technique;The adaptive fault-tolerant controller is used to control the system state output by the multi-agent system model.A kind of adaptive fault-tolerant control strategy is constructed, to ensure the stability and tracking performance of the system, under the condition that state does not violate the constraint, all signals in the closed-loop system remain bounded.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of nonlinear multi-agent system (MASs) control, and particularly relates to an adaptive neural network cooperative fault-tolerant control method for a multi-agent system. BACKGROUND

[0002] With the deep development of artificial intelligence technology and distributed architecture, multi-agent cooperative systems show significant application potential in complex scenarios such as aerospace formation control, intelligent manufacturing pipeline cooperation, and military cluster operations. However, multi-agent systems in actual engineering applications generally face the following technical bottlenecks:

[0003] First, in existing multi-agent system control methods, the case of actuator failure is easily ignored. However, under long-term high-load operation or extreme environmental disturbance, the actuators of intelligent agents are prone to complex faults such as gain attenuation, sticking, and output deviation. Existing fault-tolerant control strategies are mostly based on single fault mode assumptions, and are difficult to effectively cope with the impact of time-varying complex faults on system robustness, resulting in a decrease in cooperative tracking accuracy and even system instability.

[0004] Second, in existing research on multi-agent system control methods, the model of the multi-agent system usually does not consider the constraints in actual engineering. In some actual engineering, due to system physical limits (such as unmanned aerial vehicle flight envelope limits) or safety specifications (such as mechanical arm joint angle thresholds), it is necessary to ensure that the state trajectory of the system is strictly within the range of asymmetric time-varying constraints. Compared with traditional symmetric fixed constraints, the upper and lower boundaries of asymmetric time-varying constraints have independent time-varying characteristics. In addition, since the tracking error of the multi-agent system contains neighbor node information, it is difficult for existing methods to directly construct an asymmetric time-varying barrier Lyapunov function, resulting in a significant increase in the risk of constraint violation.

[0005] Third, in existing research on multi-agent system fault-tolerant control schemes, multiple adaptive parameter estimators are usually configured for each intelligent agent, which will cause a computational dimension explosion problem in large-scale cluster systems. For example, a hundred-node system may need to update thousands of parameters online, causing a sharp decrease in controller real-time performance and seriously restricting the engineering scalability. SUMMARY

[0006] To solve the problems in the prior art, an adaptive neural network fault-tolerant control method is provided, and the specific steps are as follows:

[0007] A multi-agent system model is constructed for outputting system states; the multi-agent system includes a leader and a follower;

[0008] A distributed sliding mode estimator is designed, the system state is input into the distributed sliding mode estimator, an estimated value of the leader trajectory is obtained, and an estimated error of the leader trajectory is calculated;

[0009] transforming the estimation error into first and higher order error variables by an error transformation module;

[0010] obtaining an adaptive law and an adaptive fault-tolerant controller using backstepping recursive technique based on the first and higher order error variables;

[0011] the adaptive fault-tolerant controller is used to control the system state of the multi-agent system model output.

[0012] Further, the dynamics equation of the multi-agent system model is:

[0013]

[0014] where i = 1,..., N, i represents the index of the follower, used to distinguish different individuals in the multi-agent system, and N represents the total number of followers.

[0015] represents the s-th state vector of the i-th follower, x i,s represents the s-th state variable in the state vector, s = 1,..., n-1.

[0016] is the state vector, represents the n-th state vector of the i-th follower, where x i,n represents the n-th state variable in the i-th follower.

[0017] y i is the control output of the nonlinear multi-agent system MASs, is the control input containing the bias gain fault, d i,s (t) and d i,n (t) respectively represent the bounded external disturbance in the nonlinear multi-agent system and satisfy and F i,s (·) and F i,n (·) are unknown smooth nonlinear functions, g i represents an unknown constant, where, represents the upper bound of the external disturbance d i,s (t) and is a constant greater than zero, represents the upper bound of the external disturbance d i,n (t) and is a constant greater than zero.

[0018] Define the bias gain fault of the actuator as where 0 < △ i <1 is an unknown loss of control rate, is a bounded signal and has vi (t) represents the control input.

[0019] Furthermore, the distributed sliding mode estimator is designed as follows:

[0020]

[0021] Where α and β > 0 are design parameters, and sign(·) represents the sign function. The output of the distributed sliding mode estimator represents the estimated leader trajectory obtained by the i-th follower. The estimated value of the leader's trajectory obtained by the j-th follower, a i,j This represents the topology weighting coefficients for communication between i followers and j followers. express The first derivative.

[0022] and y represents the estimated value of the leader's trajectory obtained by the leader. r This indicates the leader's trajectory.

[0023] also, The estimated value of the b-th derivative Represented as:

[0024]

[0025] Define the estimation error of the i-th follower on the leader's trajectory. Represented as:

[0026]

[0027] when From time to time Established, among which, Let y represent the initial estimate of the leader's trajectory obtained by the i-th follower. r (0) represents the initial value of the leader's trajectory.

[0028] Furthermore, the estimation error is transformed into first-order and higher-order error variables through an error transformation module, specifically as follows:

[0029]

[0030] Where z i,1 Let z represent the first-order error variable obtained after error transformation for the i-th follower, i.e., the tracking error. i,s Let v represent the s-th order error variable of the i-th follower after error transformation. i,s-1 It is a virtual controller.

[0031] Further, the nonlinear dynamics in the nonlinear multi-agent system model are completely unknown, and are handled by a neural network module and assumed to be:

[0032]

[0033] where

[0034]

[0035] are known functions;

[0036] p = 1, 2,... s - 1, k is the number of neural network nodes;

[0037] and is the ideal weight vector of the radial basis function neural network, and is given by represents and q represents the order index of the state vector, q = 1,..., n, and is the adaptive parameter and Θ i,s (X i,s ) is the radial basis function of the neural network; and δ i,s (X i,s ) is the approximation error, and satisfies is an unknown constant.

[0038] Further, the adaptive law is designed as:

[0039]

[0040] where ω i > 0 is a positive design parameter, is the estimate of the adaptive parameter μ i > 0 is the design parameter of the asymmetric time-varying barrier Lyapunov function of the i-th follower, τ is an integer parameter satisfying 2τ≥n+2, and q represents the order index of the state vector, taking values in the range of 1, 2,... n.

[0041] w i,q is a parameter defined in the backstepping recursive process:

[0042]

[0043] where z i,qdenotes the q-th order error variable of the i-th follower, q = 1,...,n, τ satisfies 2τ≥n+2 and is an integer.k bi,q (t) and k ci,q (t) satisfies -k bi,q (t)<z i,q <k ci,q (t), r i,q denotes the weight for adjusting the shape of barrier function, ε i,q denotes the intermediate variable in backstepping design. m is the intermediate step of backstepping, m = 2, 3,...n-1, n is the total number of steps of backstepping.

[0044] Further, the adaptive fault-tolerant controller is designed as:

[0045]

[0046] where N i (x i ) denotes the Nussbaum function, x i is an adaptive parameter, and the first derivative satisfies where o i is a positive design parameter, k i,n is a positive design parameter;

[0047] is a positive design parameter.

[0048] Compared with the prior art, the present application has the following beneficial effects:

[0049] Firstly, the present application combines the asymmetric time-varying barrier Lyapunov function, and in the case of bias gain fault of the actuator, an adaptive fault-tolerant control strategy is constructed to ensure the stability and tracking performance of the system, and good fault tolerance and robustness are exhibited.

[0050] Secondly, the present application fully considers the external interference problem faced by the system during operation, and a distributed sliding mode estimator is constructed to estimate the leader trajectory, which simplifies the tracking error structure, effectively improves the response speed of the system to external interference, and enhances the anti-interference ability of the system.

[0051] Thirdly, in the present application, the distributed sliding mode estimator is combined to ensure that the output trajectories of the followers and the leader of the multi-agent system are consistent, and under the condition that the state does not violate the constraint, all signals in the closed-loop system remain bounded. Compared with the prior art, the present application has obvious advantages in many aspects and has high practical application value. BRIEF DESCRIPTION OF DRAWINGS

[0052] Figure 1This is a diagram of an adaptive neural network collaborative fault-tolerant control method for multi-agent systems.

[0053] Figure 2 This is a schematic diagram of the communication topology of a multi-agent system.

[0054] Figure 3 It is the state variable x i,2 Its asymmetric time-varying constraint boundary trajectory diagram.

[0055] Figure 4 It is a tracking error trajectory diagram of a multi-agent system.

[0056] Figure 5 It is a trajectory comparison diagram of the output trajectories of followers and leaders in a multi-agent system.

[0057] Figure 6 It is the control input signal v of a multi-agent system i Trajectory diagram. Detailed Implementation

[0058] The invention will be further described below with reference to the accompanying drawings. The invention designs an adaptive neural network fault-tolerant control method, such as... Figure 1 As shown. In the control of a nonlinear strict feedback multi-agent system, the state information of the nonlinear strict feedback multi-agent system is input into the neural network module; the state information of the neighbor nodes of the followers in the nonlinear strict feedback multi-agent system is input into the distributed sliding mode estimator; the output of the distributed sliding mode estimator... The input is fed into the error conversion module; the adaptive law and adaptive fault-tolerant controller are obtained using backstepping recursion; the output v of the adaptive fault-tolerant controller module is then converted. i The input is fed into an adaptive law and a nonlinear strict feedback multi-agent system. The design goal of this invention is to ensure that the output trajectories of all followers and leaders remain consistent, and that all signals within the closed-loop system remain bounded, without violating the asymmetric time-varying state constraints of the controlled system.

[0059] A. Nonlinear rigorous feedback multi-agent system model

[0060] A nonlinear multi-agent system model is constructed. In this specific embodiment, the multi-agent system includes two roles: leader and follower. The leader and follower form a cooperative network through a communication topology to jointly achieve a specific control objective. The nonlinear multi-agent system model includes actuator fault terms and disturbance terms.

[0061] The dynamic equations of the nonlinear multi-agent system model are:

[0062]

[0063] where i = 1,..., N, i denotes the index of the agent, which is used to distinguish different individuals in the multi-agent system, and N denotes the total number of agents.

[0064] denotes the s-th state variable in the state vector x i,s denotes the s-th state variable in the state vector x

[0065] denotes the s-th state variable in the state vector x denotes the s-th state variable in the state vector x i,n denotes the s-th state variable in the state vector x

[0066] denotes the s-th state variable in the state vector x i is the control output of the nonlinear multi-agent system MASs, is the control input containing the bias gain fault d i,s (t) and d i,n (t) respectively represent the bounded external disturbance in the nonlinear multi-agent system and satisfy and F i,s (·) and F i,n (·) are unknown smooth nonlinear functions, g i denotes an unknown constant, where denotes the upper bound of the external disturbance d i,s (t) and is a constant greater than zero, denotes the upper bound of the external disturbance d i,n (t) and is a constant greater than zero.

[0067] The bias gain fault of the actuator is defined as where 0 < △ i <1 is an unknown loss rate, is a bounded signal and has v i (t) represents the original control input.

[0068] B. Distributed sliding mode estimator

[0069] The distributed sliding mode estimator obtains an accurate estimation value of the leader trajectory by estimating the leader trajectory, and enables each follower to obtain an accurate estimation value of the leader trajectory, thereby performing tracking.

[0070] The distributed sliding mode estimator is designed as follows:

[0071]

[0072] Where α and β > 0 are design parameters, and sign(·) represents the sign function. The output of the distributed sliding mode estimator represents the estimated leader trajectory obtained by the i-th follower. The estimated value of the leader's trajectory obtained by the j-th follower, a i,j Represents the communication topology weighting coefficient. express The first derivative.

[0073] and y represents the estimated value of the leader's trajectory obtained by the leader. r This indicates the leader's trajectory.

[0074] also, The estimated value of the b-th derivative Represented as:

[0075]

[0076] Define the estimation error of the i-th follower on the leader's trajectory. Represented as:

[0077]

[0078] when From time to time Established, among which, Let y represent the initial estimate of the leader's trajectory obtained by the i-th follower. r (0) represents the initial value of the leader's trajectory.

[0079] Similarly, define estimation error. b-th derivative Represented as:

[0080]

[0081] Therefore, if Then there is

[0082] Therefore, we can define positive constants B0, B1, ..., B b satisfy

[0083] C. Error Conversion Module

[0084] The error transformation module simplifies the tracking error structure, which includes state information from neighboring nodes, by using the estimated leader trajectory received by the followers in a multi-agent system. The transformed error is then used as input data for the backstepping recursion technique.

[0085] The tracking error is designed as follows:

[0086]

[0087] where e i,1 is the tracking error, e i,s is the error variable, and v i,s-1 is the virtual controller. denotes the neighbor set of node i.

[0088] The output of the distributed sliding mode estimator is utilized The error transformation gives

[0089]

[0090] where z i,1 denotes the first-order error variable of the ith agent after error transformation, i.e., the tracking error, and z i,s denotes the s-th order error variable of the ith agent after error transformation. The asymmetric time-varying state constraint condition is defined as

[0091] where k ai,1 (t), k ai,s (t), are known functions.

[0092] D. Backstepping recursion technique

[0093] The backstepping recursion technique is to construct asymmetric time-varying barrier Lyapunov functions (ATVBLFs) using the error and other information in the error transformation module to obtain the subsequent adaptive fault-tolerant controller and adaptive law. The backstepping recursion technique has a total of n steps.

[0094] First, define

[0095]

[0096] where z i,q denotes the q-th order error variable of the ith agent, q = 1,..., n, and τ satisfies 2τ≥n+2 and is an integer. k bi,q (t) and k ci,q (t) satisfy -k bi,q (t)<z i,q <k ci,q (t), and k bi,q (t) and k ci,q (t) are defined in the subsequent stability analysis. r i,qdenote the weight that adjusts the shape of barrier function, which is used to construct the asymmetric time-varying barrier Lyapunov function to ensure that the system state always satisfies the time-varying asymmetric constraint, ε i,q denote the intermediate variable in the backstepping recursive design, which is used to construct the asymmetric time-varying barrier Lyapunov function step by step.

[0097] First step: calculate z i,1 the time derivative of z is:

[0098]

[0099] where, denote the first-order state vector of the ith agent into the unknown smooth nonlinear function;

[0100] The asymmetric time-varying barrier Lyapunov function is constructed to ensure that the system state always satisfies the constraint range, and the asymmetric time-varying barrier Lyapunov function ATVBLF is:

[0101]

[0102] where μ i > 0 is the design parameter of the asymmetric time-varying barrier Lyapunov function of the ith agent, is the estimation error of the adaptive parameter, is the adaptive parameter, is the estimated value of the adaptive parameter .

[0103] Let then there exists

[0104] According to formulas (6)-(8), the asymmetric time-varying barrier Lyapunov function formula (10) can be rewritten as:

[0105]

[0106] Take the first-order derivative of formula (11) to obtain the expression of

[0107]

[0108] Substitute formula (9) into the expression of , and derive:

[0109]

[0110] Step m (m = 2, 3,..., n-1): calculate the time derivative of z i,m ​​as follows:

[0111]

[0112] in

[0113] This represents the adaptive law. Represents an unknown smooth nonlinear function used to describe the nonlinear dynamic characteristics of a system that cannot be precisely modeled. q represents the order index of the state vector. In the m-th step, q takes the range 1, 2, ..., m-1.

[0114]

[0115] Constructing an asymmetric time-varying barrier Lyapunov function ensures that the system state always satisfies the constraint range. The asymmetric time-varying barrier Lyapunov function ATVBLF in step m is:

[0116]

[0117] Differentiating (14), we get V i,m first derivative The formula, and substitute (13) into get:

[0118]

[0119] in k is the number of nodes in the Radial Basis Function Neural Networks (RBF NNs). This represents the lower bound of the asymmetric time-varying state constraint. It represents the upper bound of the asymmetric time-varying state constraint.

[0120] Step n: Calculate the error variable z i,n time derivative

[0121]

[0122] in

[0123] Constructing an asymmetric time-varying barrier Lyapunov function ensures that the system state always satisfies the constraints. The nth-step asymmetric time-varying barrier Lyapunov function ATVBLF is:

[0124]

[0125] Taking the derivative of formula (19), V i,n is obtained

[0126]

[0127] where

[0128] represents the n-p+1 order derivative of k bi,1 , k is the number of nodes of the neural network.

[0129] E, neural network module

[0130] The neural network module uses the state information and other information in the multi-agent system, uses adaptive parameters and neural network radial basis functions to approximate the unknown nonlinear dynamics, and is used for adaptive law calculation.

[0131] Since the nonlinear dynamics are completely unknown, the neural network module is used to process and assume that:

[0132]

[0133] where k is the number of nodes of the neural network. and are ideal weight vectors of RBF NNs, and adaptive parameters and Θ i,s (X i,s ) are neural network radial basis functions; and δ i,s (X i,s ) are approximation errors, and satisfy is an unknown constant.

[0134] F, adaptive law

[0135] Based on the state information of the multi-agent system and the backstepping recursive technique, the adaptive law is constructed The function is to reflect the dynamic change of the adaptive parameters in the controller, which inputs the parameter dynamic change into the adaptive fault-tolerant controller.

[0136] For the multi-agent system, the following adaptive law is designed:

[0137]

[0138] where, μ i >0 is the design parameter of the asymmetric time-varying obstacle Lyapunov function of the i-th agent, ωi >0 is a positive design parameter, is an estimate of the adaptive parameter , τ is an integer parameter satisfying 2τ≥n+2, q is an index of the order of the state vector, and q∈{1,2,...,n}.

[0139] G, an adaptive fault-tolerant controller

[0140] is output based on backstepping recursive technique adaptive law An adaptive fault-tolerant controller is designed for the state information of a multi-agent system as follows:

[0141]

[0142] wherein and k i,n >0 are positive design parameters, N i (χ i ) represents a Nussbaum-type function, χ i is an adaptive parameter, and satisfies the first-order derivative wherein o i >0 is a positive design parameter.

[0143] Obviously, there exists

[0144] By properly adjusting the parameter k i,s-1 , the virtual controller v i,s-1 can meet the requirement and and then

[0145] Let , and

[0146] The application realizes strict restriction of state variables under dynamic asymmetric constraints by constructing an asymmetric time-varying barrier Lyapunov function, significantly reduces the computational complexity of the multi-agent system by designing a lightweight neural network approximator, simplifies the tracking error structure by combining a distributed sliding mode observer, enhances the dynamic compensation capability of the system to composite faults and external disturbances, ensures that the output trajectories of the followers and leaders of the multi-agent system are consistent, and all signals in the closed-loop system remain bounded under the condition that the state does not violate the constraints.

[0147] The selected communication topology structure in simulation is as follows: Figure 2As shown, where "0" represents the number of the leader node in the multi-agent system, "1" represents the number of the first follower node, "2" represents the number of the second follower node, and "3" represents the number of the third follower node. The simulation results are shown in FIG. 3, which shows that all the signals in the multi-agent system remain bounded. Figures 3 to 6 As shown, it is shown that all the signals in the multi-agent system remain bounded. Figure 3 x0(t) represents the state variable x0 of the leader node, x1(t) represents the state variable x1 of the first follower node, x2(t) represents the state variable x2 of the second follower node, and x3(t) represents the state variable x3 of the third follower node. i,2 x1(t) represents the trajectory of the first follower node, x2(t) represents the trajectory of the second follower node, and x3(t) represents the trajectory of the third follower node. Figure 3 As can be seen from FIG. 2, the state variables x1(t), x2(t), and x3(t) are always strictly within the asymmetric time-varying constraint boundary, where, i,2 x1(t) represents the trajectory of the first follower node, x2(t) represents the trajectory of the second follower node, and x3(t) represents the trajectory of the third follower node. k x1(t) represents the trajectory of the first follower node, x2(t) represents the trajectory of the second follower node, and x3(t) represents the trajectory of the third follower node. ai,2 x1(t) represents the trajectory of the first follower node, x2(t) represents the trajectory of the second follower node, and x3(t) represents the trajectory of the third follower node. x1(t) represents the trajectory of the first follower node, x2(t) represents the trajectory of the second follower node, and x3(t) represents the trajectory of the third follower node. Figure 4 z0(t) represents the tracking error z0 of the leader node, z1(t) represents the tracking error z1 of the first follower node, z2(t) represents the tracking error z2 of the second follower node, and z3(t) represents the tracking error z3 of the third follower node. i,1 z0(t) represents the tracking error z0 of the leader node, z1(t) represents the tracking error z1 of the first follower node, z2(t) represents the tracking error z2 of the second follower node, and z3(t) represents the tracking error z3 of the third follower node. i,1 z0(t) represents the tracking error z0 of the leader node, z1(t) represents the tracking error z1 of the first follower node, z2(t) represents the tracking error z2 of the second follower node, and z3(t) represents the tracking error z3 of the third follower node. 2,1 z0(t) represents the tracking error z0 of the leader node, z1(t) represents the tracking error z1 of the first follower node, z2(t) represents the tracking error z2 of the second follower node, and z3(t) represents the tracking error z3 of the third follower node. 3,1 z0(t) represents the tracking error z0 of the leader node, z1(t) represents the tracking error z1 of the first follower node, z2(t) represents the tracking error z2 of the second follower node, and z3(t) represents the tracking error z3 of the third follower node. Figure 4 As can be seen from FIG. 3, the tracking errors of the system fluctuate within a very small range; Figure 5 u0(t) represents the output trajectory of the leader node, and u1(t), u2(t), and u3(t) represent the output trajectories of the first, second, and third follower nodes, respectively. Figure 5 As can be seen from FIG. 4, all the followers can track the leader trajectory with high precision, and there is no overshoot or oscillation phenomenon, where, 1,1 u0(t) represents the output trajectory of the leader node, and u1(t), u2(t), and u3(t) represent the output trajectories of the first, second, and third follower nodes, respectively. 2,1 u0(t) represents the output trajectory of the leader node, and u1(t), u2(t), and u3(t) represent the output trajectories of the first, second, and third follower nodes, respectively. 3,1 u0(t) represents the output trajectory of the leader node, and u1(t), u2(t), and u3(t) represent the output trajectories of the first, second, and third follower nodes, respectively. Figure 6 v0(t) represents the trajectory of the control input signal v0 of the multi-agent system, v1 represents the trajectory of the control input of the first follower, v2 represents the trajectory of the control input of the second follower, and v3 represents the trajectory of the control input of the third follower. i v0(t) represents the trajectory of the control input signal v0 of the multi-agent system, v1 represents the trajectory of the control input of the first follower, v2 represents the trajectory of the control input of the second follower, and v3 represents the trajectory of the control input of the third follower. Figure 6 As can be seen from FIG. 5, the input signals generated by the controller in suppressing faults and disturbances are smooth and bounded, which meets the physical limits of actual actuators.

[0148] The present application is not limited to the present embodiment, and any equivalent concept or change within the technical scope disclosed in the present application is included in the protection scope of the present application.

Claims

1. A self-adapting neural network fault-tolerant control method, characterized in that, The method comprises the following steps: A multi-agent system model is constructed for outputting a system state; the multi-agent system comprises a leader and followers; a dynamic equation of the multi-agent system model is: Wherein i=1,....,N, i represents an index of the followers, used for distinguishing different individuals in the multi-agent system, and N represents a total number of the followers; x s (i) denotes the s-th state vector of the i-th follower, s = 1,..., n - 1 i,s x s (i) denotes the s-th state variable in the state vector, s = 1,..., n - 1 is the state vector, is the n-th order state vector of the i-th follower, where x i,n is the n-th state variable in the i-th follower; y i is the control output of the nonlinear multi-agent system MASs, is the control input with bias gain fault, d i,s (t) and d i,n (t) represent the bounded external disturbance in the nonlinear multi-agent system, respectively, and satisfy and F i,s (·) and F i,n (·) are unknown smooth nonlinear functions, g i denotes unknown constants, where, denotes the upper bound of the external disturbance d i,s (t) and is a constant greater than zero, denotes the upper bound of the external disturbance d i,n (t) and is a constant greater than zero; A bias gain fault of the actuator is defined as where 0 < Δ < 1 i <1 is the unknown runaway rate, is a bounded signal and has v i (t) denotes the control input; A distributed sliding mode estimator is designed, the system state is input into the distributed sliding mode estimator, an estimated value of a leader trajectory is obtained, and an estimated error of the leader trajectory is calculated; The distributed sliding mode estimator is designed as follows: where a, b > 0 are design parameters, sign(·) denotes the sign function, is the output of the distributed sliding mode estimator, and denotes the estimate of the leader trajectory obtained by the ith follower, denotes the estimate of the leader trajectory obtained by the jth follower, a i,j denotes the communication topology weight coefficient between the ith follower and the jth follower, denotes the first derivative of denotes the first derivative of and denotes an estimate of the leader trajectory obtained by the leader, y r denotes the leader trajectory; Furthermore, the estimated value of the b-th derivative is expressed as: Define the estimation error of the i-th follower in following the leader trajectory is represented as: When occurs, holds, where, represents the initial estimate of the leader trajectory obtained by the i-th follower, y r (0) represents the initial value of the leader trajectory; The estimated error is converted into first-order and high-order error variables through an error conversion module; Based on the first-order and high-order error variables, an adaptive law and an adaptive fault-tolerant controller are obtained by using a backstepping recursion technique; The adaptive fault-tolerant controller is used for controlling the system state output by the multi-agent system model.

2. The adaptive neural network fault-tolerant control method according to claim 1, wherein, The estimated error is converted into first-order and high-order error variables through an error conversion module, specifically as follows: where z i,1 represents the first order error variable, i.e. the tracking error, z i,s represents the s-th order error variable, v i,s-1 is a virtual controller, in the s-th order error variable z i,s in the calculation formula, s = 2,..., n.

3. The adaptive neural network fault-tolerant control method according to claim 2, wherein, The nonlinear dynamics in the nonlinear multi-agent system model completely unknown, take neural network module processing and assume: wherein k ai,s (t), is a known function; k is the number of neural network nodes; and are ideal weight vectors of the radial basis function neural network, and adaptive parameters q represents the order index of the state vector, q = 1,..., n, and Θ i,s (X i,s ) is a neural network radial basis function; and δ i,s (X i,s ) is an approximation error, and satisfies is an unknown constant.

4. The adaptive neural network fault-tolerant control method according to claim 3, characterized in that, Adaptive law Designed to: where ω i > 0 is a positive design parameter, is an estimate of the adaptive parameter , μ i > 0 is a design parameter of the asymmetric time-varying barrier Lyapunov function for the i-th follower, τ is an integer parameter satisfying 2τ≥n+2, q represents the index of the order of the state vector, and takes values in the range 1, 2,....n; w i,q Parameter defined for the backstepping procedure: where z i,q denotes the qth order error variable of the ith follower, q = 1,..., n, τ satisfies 2τ≥n+2 and is an integer; k bi,q (t) and k ci,q (t) satisfy -k bi,q (t)<z i,q <k ci,q (t), r i,q denotes the weight to adjust the shape of barrier function, ε i,q denotes the intermediate variable in the design of backstepping recursion; m is the intermediate step of backstepping recursion, m = 2, 3,... n - 1, n is the total number of steps of backstepping recursion.

5. The adaptive neural network fault-tolerant control method according to claim 4, characterized in that, The adaptive fault-tolerant controller is designed as follows: where N i (χ i ) denotes a Nussbaum-type function, χ i is an adaptive parameter, and satisfies the first derivative where o i is a positive design parameter, k i,n is a positive design parameter; is a positive design parameter.

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