Adaptive neural network cooperative fault-tolerant control method for multi-agent system

Abstract

The invention discloses a multi-agent system adaptive neural network cooperative fault-tolerant control method, and belongs to the field of nonlinear multi-agent system control, and the method comprises the steps: constructing a multi-agent system model for outputting a system state; the multi-agent system comprises a leader and a follower; designing a distributed sliding mode estimator, inputting the system state into the distributed sliding mode estimator, obtaining an estimated value of the leader trajectory, and calculating an estimated error of the leader trajectory; converting the estimation error into a first-order error variable and a high-order error variable through an error conversion module; based on the first-order and high-order error variables, a self-adaptive law and a self-adaptive fault-tolerant controller are obtained by adopting a back-stepping recursion technology; and the adaptive fault-tolerant controller is used for controlling the system state output by the multi-agent system model. A self-adaptive fault-tolerant control strategy is constructed, the stability and tracking performance of the system are ensured, and all signals in the closed-loop system are kept bounded under the condition that the state does not violate constraints.

Description

Technical Field

[0001] The present invention belongs to the field of nonlinear multi-agent system (MASs) control, in particular to a multi-agent system adaptive neural network collaborative fault-tolerant control method. Background Art

[0002] With the in-depth development of artificial intelligence technology and distributed architecture, multi-agent collaborative systems have shown significant application potential in complex scenarios such as aerospace formation control, intelligent manufacturing assembly line collaboration, and military swarm operations. However, multi-agent systems in actual engineering applications generally face the following technical bottlenecks:

[0003] First, existing multi-agent system control methods tend to overlook actuator failures. However, under long-term high-load operation or extreme environmental interference, agent actuators are prone to compound failures such as gain attenuation, stuckness, or output deviation. Existing fault-tolerant control strategies, which often assume a single failure mode, struggle to effectively address the impact of time-varying compound failures on system robustness, leading to reduced collaborative tracking accuracy and even system instability.

[0004] Second, existing research on control methods for multi-agent systems often fails to consider constraints encountered in real-world projects. In some real-world projects, physical limitations (e.g., drone flight envelope limitations) or safety regulations (e.g., robotic arm joint angle thresholds) necessitate ensuring that the system's state trajectory strictly falls within asymmetric time-varying constraints. Compared to traditional symmetric fixed constraints, the upper and lower bounds of asymmetric time-varying constraints have independent time-varying characteristics. Furthermore, because the tracking error of a multi-agent system contains information about neighboring nodes, existing methods struggle to directly construct asymmetric time-varying barrier Lyapunov functions, significantly increasing the risk of constraint violations.

[0005] Third, existing research on fault-tolerant control schemes for multi-agent systems typically configures multiple adaptive parameter estimators for each agent. This can lead to a computational dimensionality explosion in large-scale cluster systems. For example, a system with hundreds of nodes may require thousands of parameters to be updated online, resulting in a sharp decline in controller real-time performance and severely restricting project scalability. Summary of the Invention

[0006] To address the deficiencies in the prior art, the present invention proposes an adaptive neural network fault-tolerant control method, the specific steps of which are as follows:

[0007] Constructing a multi-agent system model for outputting a system state; the multi-agent system includes a leader and a follower;

[0008] Designing a distributed sliding mode estimator, inputting the system state into the distributed sliding mode estimator, obtaining an estimated value of the leader trajectory, and calculating an estimation error of the leader trajectory;

[0009] Converting the estimated error into first-order and higher-order error variables through an error conversion module;

[0010] Based on the first-order and higher-order error variables, an adaptive law and an adaptive fault-tolerant controller are obtained by adopting a backstepping recursive technique;

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

[0012] Furthermore, the dynamic equation of the multi-agent system model is:

[0013]

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

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

[0016] is the state vector, represents the n-order state vector of the ith follower, where x i,n represents the nth 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 with error gain fault, d i,s (t) and d i,n (t) represent bounded external disturbances 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 Indicates external interference d i,s (t) is the upper bound and is a constant greater than zero, Indicates external interference d i,n (t) is the upper bound of the value and is a constant greater than zero.

[0018] Define the actuator’s deviation gain fault as Where 0<△ i <1 is the 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] Among them, α, β>0 are design parameters, sign(·) represents the sign function, is the output of the distributed sliding mode estimator, which represents the estimated value of the 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 weight coefficient between i followers and j followers, express The first derivative of .

[0022] and represents the estimated value of the leader trajectory obtained by the leader, y r Represents the leader trajectory.

[0023] also, The estimated value of the b-order derivative of Expressed as:

[0024]

[0025] Define the estimation error of the leader's trajectory by the i-th follower Expressed as:

[0026]

[0027] when Sometimes Established, among which 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.

[0028] Furthermore, the estimated error is converted into first-order and higher-order error variables through an error conversion module, specifically:

[0029]

[0030] where z i,1 represents the first-order error variable obtained by the error conversion of the i-th follower, i.e., the tracking error, z i,s represents the s-order error variable of the i-th follower after error conversion, v i,s-1 For virtual controllers.

[0031] Furthermore, the nonlinear dynamics in the nonlinear multi-agent system model Completely unknown, use the neural network module to process and assume:

[0032]

[0033] in

[0034]

[0035] is a known function;

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

[0038] Furthermore, the adaptive law Designed to:

[0039]

[0040] Among them, ω i >0 is a positive design parameter, is the adaptive parameter The estimated value of 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 that satisfies 2τ≥n+2, and q represents the order index of the state vector, which ranges from 1, 2, ...n.

[0041] w i,q Parameters defined in the backstepping process:

[0042]

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

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

[0045]

[0046] Among them, N i (χ i ) represents the Nussbaum type function, χ i is an adaptive parameter, and the first-order derivative is Among them 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 invention has the following beneficial effects:

[0049] First, the present invention combines the asymmetric time-varying barrier Lyapunov function to construct an adaptive fault-tolerant control strategy in the event of a deviation gain failure in the actuator, ensuring the stability and tracking performance of the system, and showing good fault tolerance and robustness.

[0050] Second, the present invention fully considers the external interference problems faced by the system during operation, and constructs a distributed sliding mode estimator to estimate the leader trajectory, which simplifies the tracking error structure, effectively improves the system's response speed to external interference, and enhances the system's anti-interference ability.

[0051] Third, the present invention incorporates a distributed sliding mode estimator to ensure that the output trajectories of followers and leaders in a multi-agent system remain consistent, and that all signals within the closed-loop system remain bounded, provided that the state does not violate constraints. Compared with existing technologies, this invention offers significant advantages in multiple respects and possesses high practical application value. BRIEF DESCRIPTION OF THE DRAWINGS

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

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

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

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

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

[0057] Figure 6 is the control input signal v of the multi-agent system i Trajectory map. DETAILED DESCRIPTION

[0058] The present invention is further described below with reference to the accompanying drawings. The present invention designs an adaptive neural network fault-tolerant control method, such as Figure 1 As shown. In the nonlinear strict feedback multi-agent system control, 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 follower in the nonlinear strict feedback multi-agent system is input into the distributed sliding mode estimator; the output of the distributed sliding mode estimator is input into the distributed sliding mode estimator. Input to the error conversion module; use the backstepping recursion technique to obtain the adaptive law and adaptive fault-tolerant controller; the output v of the adaptive fault-tolerant controller module i The input is fed into the adaptive law and nonlinear strict feedback multi-agent system. The design goal of the present invention is to keep the output trajectories of all followers and leaders consistent and all signals in the closed-loop system bounded, provided that the controlled system does not violate its asymmetric time-varying state constraints.

[0059] A. Nonlinear Strict Feedback Multi-Agent System Model

[0060] Construct a nonlinear multi-agent system model. In this specific embodiment, the multi-agent system includes two roles: leader and follower. The leader and followers form a collaborative network through a communication topology structure to jointly achieve a specific control goal. The nonlinear multi-agent system model includes actuator failure terms and disturbance terms.

[0061] The dynamic equation of the nonlinear multi-agent system model is:

[0062]

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

[0064] represents the s-order state vector of the i-th agent, x i,s represents the sth state variable in the state vector, s=1,...,n-1;

[0065] is the state vector, represents the n-order state vector of the i-th agent, where x i,n represents the nth state variable in the i-th agent.

[0066] y i is the control output of the nonlinear multi-agent system MASs, is the control input with error gain fault, d i,s (t) and d i,n (t) represent bounded external disturbances 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 Indicates external interference d i,s (t) is the upper bound and is a constant greater than zero, Indicates external interference d i,n (t) is the upper bound of the value and is a constant greater than zero.

[0067] Define the actuator’s deviation gain fault as Where 0<△ i <1 is the unknown loss of control 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 estimates the leader's trajectory and obtains an accurate estimate of the leader's trajectory, and enables each follower to obtain an accurate estimate of the leader's trajectory and thus track it.

[0070] Design the distributed sliding mode estimator as follows:

[0071]

[0072] Among them, α, β>0 are design parameters, sign(·) represents the sign function, is the output of the distributed sliding mode estimator, which represents the estimated value of the 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 weight coefficient, express The first derivative of .

[0073] and represents the estimated value of the leader trajectory obtained by the leader, y r Represents the leader trajectory.

[0074] also, The estimated value of the b-order derivative of Expressed as:

[0075]

[0076] Define the estimation error of the leader's trajectory by the i-th follower Expressed as:

[0077]

[0078] when Sometimes Established, among which 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.

[0079] Similarly, define the estimation error The b-order derivative of Expressed as:

[0080]

[0081] Thus, if Then there is

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

[0083] C. Error conversion module

[0084] The error conversion module uses the leader trajectory estimates received by followers in a multi-agent system to simplify the tracking error structure that includes the state information of neighboring nodes, thereby performing error conversion. The converted error is used as input data for the backstepping technique.

[0085] Design the following tracking error:

[0086]

[0087] where e i,1 is the tracking error, e i,s is the error variable, v i,s-1 For virtual controllers. represents the neighbor set of node i.

[0088] Using the output of the distributed sliding mode estimator Performing error conversion yields:

[0089]

[0090] where z i,1 represents the first-order error variable obtained by the error conversion of the i-th agent, that is, the tracking error, z i,s Denotes the s-order error variable of the ith agent after error conversion. Define the asymmetric time-varying state constraint:

[0091] in k ai,1 (t), k ai,s (t), is a known function.

[0092] D. Backstepping technique

[0093] The backstepping recursive technique uses information such as errors in the error conversion module to construct asymmetric time-varying barrier Lyapunov functions (ATVBLFs) to obtain subsequent adaptive fault-tolerant controllers and adaptive laws. The backstepping recursive technique has a total of n steps.

[0094] First define

[0095]

[0096] Among them, z i,q k represents the q-order error variable of the i-th agent, q=1,...,n, τ satisfies 2τ≥n+2 and is an integer. bi,q (t) and k ci,q (t) satisfies -k bi,q (t) <z i,q <k ci,q (t), and k bi,q (t) and k ci,q (t) is defined in the subsequent stability analysis, r i,qRepresents the weight for adjusting the shape of the 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 represents the intermediate variable in the backstepping recursive design and is used to construct the asymmetric time-varying barrier Lyapunov function in steps.

[0097] Step 1: Calculate z i,1 The time derivative of for:

[0098]

[0099] in, Indicates that the first-order state vector of the i-th agent is brought into an unknown smooth nonlinear function;

[0100] An asymmetric time-varying barrier Lyapunov function is constructed to ensure that the system state always satisfies the constraint range. 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 parameters, is the adaptive parameter, is an adaptive parameter estimated value.

[0103] make Then there is

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

[0105]

[0106] Taking the first-order derivative of formula (11), we get The expression:

[0107]

[0108] Substitute formula (9) into From the expression of , we can deduce:

[0109]

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

[0111]

[0112] in

[0113] represents the adaptive law, represents an unknown smooth nonlinear function, which is used to describe the nonlinear dynamic characteristics of the system that cannot be accurately modeled. q represents the order index of the state vector. In the mth step, the range of q is 1, 2, ..., m-1;

[0114]

[0115] Construct an asymmetric time-varying barrier Lyapunov function to ensure that the system state always meets the constraint range. The asymmetric time-varying barrier Lyapunov function ATVBLF in the mth step is:

[0116]

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

[0118]

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

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

[0121]

[0122] in

[0123] Construct an asymmetric time-varying barrier Lyapunov function to ensure that the system state always meets the constraint range. The n-th step asymmetric time-varying barrier Lyapunov function ATVBLF is:

[0124]

[0125] Taking the derivative of formula (19), we get V i,n The derivative of

[0126]

[0127] in

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

[0129] E. Neural Network Module

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

[0131] Due to nonlinear dynamic Completely unknown, use the neural network module to process and assume:

[0132]

[0133] in k is the number of neural network nodes. and is the ideal weight vector of RBF NNs, and the adaptive parameters and Θ i,s (X i,s ) is the neural network radial basis function; and δ i,s (X i,s ) is the approximation error, and satisfies is an unknown constant.

[0134] F. Adaptive Law

[0135] Output based on the state information of multi-agent system and backstepping recursion technology Constructing adaptive laws Their function is to reflect the dynamic changes of adaptive parameters in the controller, and they input the dynamic changes of parameters into the adaptive fault-tolerant controller.

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

[0137]

[0138] Among them, μ i >0 is the design parameter of the asymmetric time-varying barrier Lyapunov function of the i-th agent, ωi >0 is a positive design parameter, is the adaptive parameter The estimated value of , τ is an integer parameter that satisfies 2τ≥n+2, and q is the order index representing the state vector, and its value range is 1, 2, ...n.

[0139] G. Adaptive Fault-Tolerant Controller

[0140] Output based on backstepping recursion technology Adaptive Law The state information of the multi-agent system is used to design an adaptive fault-tolerant controller as follows:

[0141]

[0142] in and k i,n are all positive design parameters, N i (χ i ) represents the Nussbaum type function, χ i is an adaptive parameter, and the first-order derivative is Among them i is a positive design parameter.

[0143] Obviously, there is

[0144] By adjusting the parameters appropriately k i,s-1 , which can make the virtual controller v i,s-1 , meeting the requirements and Then get

[0145] make Can get

[0146] The present invention constructs an asymmetric time-varying barrier Lyapunov function to achieve strict restriction of state variables under dynamic asymmetric constraints; designs a lightweight neural network approximator to significantly reduce the computational complexity of the multi-agent system; combines a distributed sliding mode observer to simplify the tracking error structure and enhance the system's dynamic compensation capability for compound faults and external disturbances, ensuring that the output trajectories of the followers and leaders of the multi-agent system remain consistent, and that all signals in the closed-loop system remain bounded as long as the state does not violate the constraints.

[0147] The communication topology selected for simulation is as follows: Figure 2As shown in , "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 Figures 3 to 6 As shown, it shows that all signals in the multi-agent system remain bounded. Figure 3 Represents the state variable x i,2 (i=1,2,3) and its asymmetric time-varying constraint boundary trajectory, from Figure 3 As can be seen from the figure, the state variable x i,2 (i=1,2,3) is always strictly within the asymmetric time-varying constraint boundary, where k ai,2 (t) represents the lower bound of the asymmetric time-varying state constraint of the second state variable of the second-order multi-agent system selected for simulation, It represents the upper bound of the asymmetric time-varying state constraint of the second state variable, without collision or out-of-bounds risk, ensuring safe operation of the system; Figure 4 Denotes the tracking error z of the multi-agent system i,1 trajectory, z i,1 represents the trajectory of the tracking error of the first follower, z 2,1 represents the trajectory of the tracking error of the second follower, z 3,1 represents the tracking error trajectory of the third follower, given by Figure 4 It can be seen that the tracking error of the system fluctuates within a very small range; Figure 5 Represents the output trajectory of the leader and follower, from Figure 5 It can be seen from the figure that all followers can track the leader’s trajectory with high precision without overshoot or oscillation, where x 1,1 represents the leader trajectory, x 2,1 represents the second follower trajectory, x 3,1 represents the third follower trajectory; Figure 6 represents the control input signal v of the multi-agent system i (t), v1 represents the trajectory of the first follower control input, v2 represents the trajectory of the second follower control input, and v3 represents the trajectory of the third follower control input, which is represented by Figure 6 It can be seen that the input signal generated by the controller when suppressing faults and disturbances is smooth and bounded, which is consistent with the physical limits of the actual actuator.

[0148] The present invention is not limited to this embodiment, and any equivalent concepts or modifications within the technical scope disclosed by the present invention are included in the protection scope of the present invention.

Claims

1. An adaptive neural network fault-tolerant control method, characterized in that: The following steps are involved: Constructing a multi-agent system model for outputting a system state; the multi-agent system includes a leader and a follower; Designing a distributed sliding mode estimator, inputting the system state into the distributed sliding mode estimator, obtaining an estimated value of the leader trajectory, and calculating an estimation error of the leader trajectory; Converting the estimated error into first-order and higher-order error variables through an error conversion module; Based on the first-order and higher-order error variables, an adaptive law and an adaptive fault-tolerant controller are obtained by adopting a backstepping recursive technique; The adaptive fault-tolerant controller is used to control the system state output by the multi-agent system model.

2. The adaptive neural network fault-tolerant control method according to claim 1, characterized in that: The dynamic equation of the multi-agent system model is: Where i = 1,…,N, i represents the index of the follower, which is used to distinguish different individuals in the multi-agent system, and N represents the total number of followers; represents the s-order state vector of the i-th follower, x i,s represents the sth state variable in the state vector, s=1,…,n-1; is the state vector, represents the n-order state vector of the ith follower, where x i,n represents the nth 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 error gain fault, d i,s (t) and d i,n (t) represent bounded external disturbances 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 Indicates external interference d i,s (t) is the upper bound and is a constant greater than zero, Indicates external interference d i,n (t) is the upper bound of the value and is a constant greater than zero; Define the actuator’s deviation gain fault as Where 0<△ i <1 is the unknown loss of control rate, is a bounded signal and has v i (t) represents the control input.

3. The adaptive neural network fault-tolerant control method according to claim 2, characterized in that: Design the distributed sliding mode estimator as follows: Among them, α, β>0 are design parameters, sign(·) represents the sign function, is the output of the distributed sliding mode estimator, which represents the estimated value of the 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 weight coefficient between i followers and j followers, express The first derivative of ; and represents the estimated value of the leader trajectory obtained by the leader, y r represents the leader trajectory; also, The estimated value of the b-order derivative of Expressed as: Define the estimation error of the leader's trajectory by the i-th follower Expressed as: when Sometimes Established, among which 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.

4. The adaptive neural network fault-tolerant control method according to claim 3, characterized in that: The estimated error is converted into first-order and higher-order error variables through the error conversion module, specifically: where z i,1 represents the first-order error variable obtained by the error conversion of the i-th follower, i.e., the tracking error, z i,s represents the s-order error variable of the i-th follower after error conversion, v i,s-1 is a virtual controller, with the s-order error variable z i,s In the calculation formula, s = 2,…,n.

5. The adaptive neural network fault-tolerant control method according to claim 2, characterized in that: The nonlinear dynamics of the nonlinear multi-agent system model Completely unknown, use neural network module to process and fake set up: in k ai,s (t), is a known function; k is the number of neural network nodes; and is the ideal weight vector of the radial basis function neural network, and the adaptive parameter q represents the order index of the state vector, q = 1,…,n; and Θ i,s (X i,s ) is the neural network radial basis function; and δ i,s (X i,s ) is the approximation error, and satisfies is an unknown constant.

6. The adaptive neural network fault-tolerant control method according to claim 5, characterized in that: Adaptive Law Designed to: Among them, ω i >0 is a positive design parameter, is the adaptive parameter The estimated value of 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 that satisfies 2τ≥n+2, and q represents the order index of the state vector, which ranges from 1, 2, …n; w i,q Parameters defined in the backstepping process: Among them, z i,q represents the q-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 represents the weight for adjusting the shape of the barrier function, ε i,q Represents the intermediate variable in the backstepping recursive design; m is the middle step of the backstepping recursion, m=2,3,...n-1, n is the total number of steps of the backstepping recursion.

7. The adaptive neural network fault-tolerant control method according to claim 6, characterized in that: The adaptive fault-tolerant controller is designed as follows: Among them, N i (χ i ) represents the Nussbaum type function, χ i is an adaptive parameter, and the first-order derivative is Among them i is a positive design parameter, k i,n is a positive design parameter; is a positive design parameter.

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