A fault-tolerant control method and system for nonlinear systems with quantifiable performance based on learning.

Abstract

This invention discloses a learning-based fault-tolerant control method and system for quantifiable performance of nonlinear systems, belonging to the field of automatic control technology. The method designs a learning-based fault-tolerant control algorithm that embeds an RBF network to compensate for fault dynamics when a fault occurs. Based on deterministic learning theory, an RBF network is constructed by setting neurons along a desired reference trajectory. Secondly, by analyzing the convergence characteristics of the learning-tolerant control system through sampled data, a detailed quantitative analysis of the learning accuracy and learning speed of the proposed learning-based fault-tolerant control method is performed, and calculation formulas for learning accuracy and learning speed are given. This invention establishes quantitative expressions for learning speed and learning accuracy through theoretical derivation, realizing quantitative analysis and pre-setting of fault-tolerant control performance, and achieving quantitative controllability of learning speed and approximation accuracy while ensuring the closed-loop stability of the system.

Description

Technical Field

[0001] This invention belongs to the field of automatic control technology, specifically relating to a fault-tolerant control method and system for nonlinear systems with quantifiable performance based on learning. Background Technology

[0002] The statements herein provide only background information in relation to this invention and do not necessarily constitute prior art.

[0003] Modern control systems are rapidly becoming more complex and sophisticated, significantly increasing the difficulty of accurately locating and quickly handling system faults. This poses a serious challenge to the stable operation of systems—faults, if not handled promptly, can trigger a chain reaction, leading to a sharp drop in system performance or even closed-loop failure. Therefore, improving system reliability and safety has become a core task. Fault-Tolerant Control (FTC), as a key strategy for dealing with faults, has received widespread attention over the past few decades. Scholars have proposed typical FTC methods such as adaptive control, fuzzy control, and reinforcement learning to enhance the robustness and adaptability of systems to faults.

[0004] In recent years, with the development of artificial intelligence, learning-based Free-Control-Cycle (FTC) has become a highly promising solution for improving system fault tolerance. Related research has covered scenarios such as uncertain nonlinear systems, two-dimensional planar systems, and multi-agent systems, and models such as radial basis function (RBF) neural networks have been used for unknown function approximation and control strategy generation. Most existing performance-based FTC methods focus primarily on tracking performance, such as ensuring predefined convergence speed, overshoot, and steady-state error through preset performance controls. However, the learning performance of the neural network itself in the FTC scheme has received relatively little attention. This makes it difficult to predict the learning speed and accuracy that the selected parameters will bring in practical engineering applications, as the adjustment of controller parameters (such as control gain and learning gain) often relies on empirical trial and error. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of the prior art and provide a fault-tolerant control method and system for nonlinear systems with quantifiable performance based on learning. Through theoretical derivation, quantitative expressions for learning speed and learning accuracy are established, realizing quantitative analysis and pre-setting of fault-tolerant control performance. While ensuring the closed-loop stability of the system, quantitative controllability of learning speed and approximation accuracy is achieved.

[0006] To achieve the above objectives, the present invention is implemented through the following technical solution: In a first aspect, the technical solution of the present invention provides a fault-tolerant control method for nonlinear systems with quantifiable performance based on learning, comprising: A nonlinear continuous system model with actuator faults is established, and the system model is discretized based on a set sampling period to obtain a nonlinear sampled data system model. Construct a periodic or quasi-periodic reference model to generate the desired reference trajectory; A localized radial basis function neural network is constructed based on the system state vector of the nonlinear sampled data system model, and its neurons are arranged along the desired reference trajectory; Define the system's filtering tracking error, and design a fault-tolerant controller and corresponding network weight update law based on the filtering tracking error and the localized radial basis function neural network. Based on the principle of uniform complete observability and Lyapunov stability theory, we quantitatively analyze the learning error system consisting of a fault-tolerant controller and the corresponding network weight update law, and derive and establish quantitative expressions for learning speed and learning accuracy. Based on the quantization expression, the controller gain and learning gain matrix are adjusted so that the system's tracking error and neural network learning error can converge to a preset accuracy range at a preset speed after a fault occurs.

[0007] In at least one embodiment, the nonlinear continuous system model is specifically represented as follows:

[0008] In the formula, x represents the system state vector. ; This indicates continuous system input; Denotes a smooth nonlinear function, where For actuator fault items; Represents the fault switch function. When it is 0, When it is 1, The time when the fault occurred; Indicates continuous bounded disturbance; The sign is known, and there exists a positive constant. , satisfy And by default ; The discretized nonlinear sampled data system model is specifically represented as follows:

[0009] In the formula, To set the sampling period; This represents the discretized system state vector. ; This indicates continuous system input; This indicates discrete interference.

[0010] In at least one embodiment, the localized radial basis function neural network is specifically represented as follows:

[0011] In the formula, This represents the weight vector estimated by the neural network. The localized regression vector is composed of neurons arranged along the trajectory. , The reference state vector; Indicates the number of neurons along the trajectory; Represents the weights of a neural network; This represents the radial basis function.

[0012] In at least one embodiment, the fault-tolerant controller is specifically represented as follows:

[0013] In the formula, Indicates the gain of the fault-tolerant controller. ; A localized radial basis function neural network set up along the desired reference trajectory.

[0014] In at least one embodiment, the network weight update law corresponding to the fault-tolerant controller is calculated based on gradient descent, specifically expressed as follows:

[0015] In the formula, To learn the gain matrix, .

[0016] In at least one embodiment, the learning speed quantization expression is specifically as follows:

[0017] In the formula, Indicates the amplitude coefficient. Indicates the exponential convergence rate; , For weight estimation error, This is the filtering tracking error; Indicates the sampling time; This is the initial time.

[0018] In at least one embodiment, the quantification expression for learning accuracy specifically includes a steady-state convergent compact expression for the filtering tracking error and a convergent compact expression for the weight estimation error of the localized radial basis function neural network. The steady-state convergent compact set of the filter tracking error is specifically represented as:

[0019] The convergence compact set of the weight estimation error in a localized radial basis function neural network is specifically represented as follows:

[0020] In the formula, Indicates the upper limit of the interference amplification factor; Indicates the upper bound of the overall interference; Indicates the overall time step; and The positive constants derived for UCO properties.

[0021] In at least one embodiment, the learning accuracy further includes quantification of the neural network approximation accuracy, which satisfies:

[0022] In the formula, For localized regression vectors The upper bound of the model.

[0023] In at least one embodiment, the controller gain and learning gain matrix are adjusted according to the quantization expression. Specifically, the adjustment range of the controller gain and learning gain is first solved, and the values ​​of the controller gain and learning gain are changed within the solved adjustment range. Then, the convergence speed and convergence accuracy are calculated to obtain the range of controller gain and learning gain values ​​that meet the preset performance requirements. The convergence accuracy includes the accuracy of the filter tracking error and the accuracy of the neural network learning.

[0024] Secondly, the technical solution of the present invention also provides a fault-tolerant control system for nonlinear systems with quantifiable performance based on learning, comprising: The system modeling and discretization module is configured to: establish a nonlinear continuous system model containing actuator faults, and perform discretization processing based on a set sampling period to obtain a nonlinear sampled data system model; The reference model design module is configured to: construct periodic or quasi-periodic reference models and generate the desired reference trajectory; The localized RBF neural network building module is configured to: construct a localized radial basis function neural network based on the system state vector of the nonlinear sampled data system model, with its neurons arranged along the desired reference trajectory; The controller and weight update law design module is configured to: define the system's filtering tracking error, and design a fault-tolerant controller and the corresponding network weight update law based on the filtering tracking error and the localized radial basis function neural network; The learning performance quantification derivation module is configured to: based on the principle of uniform complete observability and Lyapunov stability theory, quantitatively analyze the learning error system composed of the fault-tolerant controller and the corresponding network weight update law, and derive and establish the quantitative expressions for learning speed and learning accuracy. The parameter optimization and closed-loop control module is configured to adjust the controller gain and learning gain matrix according to the quantization expression, so that the tracking error and neural network learning error of the system can converge to a preset accuracy range at a preset speed after a fault occurs.

[0025] The beneficial effects of the above-described technical solution of the present invention are as follows: 1) The fault-tolerant control method for nonlinear systems based on learning and quantifiable performance of the present invention establishes quantitative expressions for learning speed and learning accuracy through theoretical derivation, realizes quantitative analysis and pre-setting of fault-tolerant control performance, and achieves quantitative controllability of learning speed and approximation accuracy while ensuring the closed-loop stability of the system. It effectively solves the problems of blind parameter adjustment and performance analysis remaining at the qualitative level in the prior art, and has important theoretical significance and engineering application value.

[0026] 2) This invention provides complete quantitative expressions for learning speed, learning accuracy, and neural network approximation accuracy, and gives clear definitions and calculation methods for each parameter in the expressions, solving the problem of blind parameter adjustment in the prior art.

[0027] 3) This invention optimizes the controller gain and learning gain based on quantization expressions, which can achieve rapid compensation for actuator faults. The learning speed and accuracy can be preset and verified according to theoretical formulas, which significantly improves the performance of fault-tolerant control.

[0028] 4) This invention ensures the validity of the quantization expression and the rigor of its derivation by using the UCO principle, Lyapunov stability theory and some PE conditions. Combined with some PE conditions, it enhances the learning ability of the neural network and ensures that the weights converge to the optimal value.

[0029] 5) The quantitative expression of this invention is directly related to the design parameters and control performance. The required parameter values ​​or ranges can be directly calculated based on the preset performance indicators without the need for a complicated trial and error process, which greatly reduces the difficulty of engineering implementation. Attached Figure Description

[0030] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0031] Figure 1 This is a schematic diagram of the fault-tolerant control method for nonlinear systems with quantifiable performance based on learning disclosed in Embodiment 1 of the present invention; Figure 2 This is the single-link robotic arm system disclosed in Embodiment 1 of the present invention. Schematic diagram of state tracking error trajectory; Figure 3 This is the single-link robotic arm system disclosed in Embodiment 1 of the present invention. Schematic diagram of state tracking error trajectory; Figure 4 This is the single-link robotic arm system disclosed in Embodiment 1 of the present invention. Schematic diagram of state tracking error trajectory; Figure 5 This is the single-link robotic arm system disclosed in Embodiment 1 of the present invention. Schematic diagram of state tracking error trajectory; Figure 6 This is a graph showing the relationship between the control gain and learning gain on the convergence speed as disclosed in Embodiment 1 of the present invention; Figure 7 This is a graph showing the relationship between the control gain and learning gain on the learning accuracy of the neural network, as disclosed in Embodiment 1 of the present invention. Detailed Implementation

[0032] It should be noted that the following detailed description is illustrative and intended to provide further explanation of the invention. Unless otherwise specified, all technical and scientific terms used in this invention have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains.

[0033] As described in the background section, the purpose of this invention is to overcome the shortcomings of the prior art and provide a fault-tolerant control method and system for nonlinear systems with quantifiable performance based on learning. Through theoretical derivation, quantitative expressions for learning speed and learning accuracy are established, realizing quantitative analysis and pre-setting of fault-tolerant control performance. While ensuring the closed-loop stability of the system, quantitative controllability of learning speed and approximation accuracy is achieved.

[0034] Example 1 In a typical embodiment of the present invention, such as Figures 1 to 7 As shown in the figure, this embodiment discloses a fault-tolerant control method for nonlinear systems with quantifiable performance based on learning, specifically including the following steps: S1. Establish a nonlinear continuous system model containing actuator faults, and discretize it based on a set sampling period to obtain a nonlinear sampled data system model; S2. Construct a periodic or quasi-periodic reference model to generate the desired reference trajectory; S3. Construct a localized radial basis function neural network based on the system state vector of the nonlinear sampled data system model, with its neurons arranged along the desired reference trajectory; S4. Define the system's filtering tracking error, and design a fault-tolerant controller and the corresponding network weight update law based on the filtering tracking error and the localized radial basis function neural network. S5. Based on the principle of uniform complete observability and Lyapunov stability theory, we quantitatively analyze the learning error system consisting of a fault-tolerant controller and the corresponding network weight update law, and derive and establish quantitative expressions for learning speed and learning accuracy. S6. Based on the quantization expression, adjust the controller gain and learning gain matrix so that the system's tracking error and neural network learning error can converge to a preset accuracy range at a preset speed after a fault occurs.

[0035] The following detailed description of the above-mentioned fault-tolerant control method for nonlinear systems with quantifiable performance based on learning, with reference to specific implementation methods, is provided below.

[0036] S1. Establish a nonlinear continuous system model containing actuator faults, and discretize it based on a set sampling period to obtain a nonlinear sampled data system model.

[0037] In this step, a nonlinear continuous system model including actuator faults is established considering the continuous system model of a single-link robotic arm. Euler discretization is performed based on a preset sampling period to obtain a nonlinear sampled data system model, and the mathematical expressions and boundary constraints of fault terms and disturbance terms are clarified.

[0038] Specifically, the continuous system model of a single-link robotic arm can be represented as:

[0039] In the formula, , , and These represent the joint angles respectively. , , and These represent the corresponding joint angular velocities; the values ​​for each physical parameter are: , , , , , , , , .

[0040] To facilitate controller design, state transitions are performed first, specifically as follows:

[0041] The transformed system is represented as:

[0042] In the formula,

[0043]

[0044] Set the time of fault occurrence to The actuator fault item is Then, the nonlinear continuous system model containing actuator faults can be specifically represented as:

[0045] In the formula, x represents the system state vector. ; Indicates continuous system input. ; Denotes a smooth nonlinear function, where For actuator fault items; Represents the fault switch function. When it is 0, When it is 1, The time when the fault occurred; Indicates continuous bounded disturbance; The sign is known, and there exists a positive constant. , satisfy And by default .

[0046] Set the sampling period to The above nonlinear continuous system model containing actuator faults is discretized using Euler method to obtain the discretized nonlinear sampled data system model, specifically represented as follows:

[0047] In the formula, This represents the discretized system state vector. ; This indicates continuous system input; Represents discrete interference, satisfying , The upper bound of the known disturbance is given.

[0048] S2. Construct a periodic or quasi-periodic reference model to generate the desired reference trajectory.

[0049] In this step, a periodic or quasi-periodic reference model is constructed to generate the desired reference trajectory, providing a target benchmark for system tracking control.

[0050] Specifically, the reference trajectory is selected as follows:

[0051] In the formula, The sampling period; The sampling time.

[0052] Corresponding reference state The trajectory is a periodic trajectory with a period of 1. .

[0053] S3. Construct a localized radial basis function neural network based on the system state vector of the nonlinear sampled data system model, with its neurons arranged along the desired reference trajectory.

[0054] In this step, a localized radial basis function (RBF) neural network is constructed, with neurons configured along a desired reference trajectory to satisfy the partial sustained excitation (PE) condition.

[0055] Specifically, a localized radial basis function (RBF) neural network is constructed, as follows:

[0056] In the formula, This represents the weight vector estimated by the neural network. The localized regression vector is composed of neurons arranged along the trajectory. , The reference state vector; Indicates the number of neurons along the trajectory; Represents the weights of a neural network; This represents the radial basis function.

[0057] In this step, Gaussian radial basis functions are used to balance approximation accuracy and computational complexity. The neuron configuration constraints and related parameter definitions are clearly defined, specifically as follows:

[0058] In the formula, Indicates the center of the neuron; The norm of a vector; This represents the width parameter of the Gaussian function, which determines the size of the receptive field.

[0059] The placement of neuron centers along the system's desired reference trajectory must simultaneously satisfy the following constraints:

[0060] In the formula, for The trajectory; This represents the maximum distance between the neuron's center point and its trajectory. This represents the minimum distance between the centers of neurons; For the first The central point of each neuron.

[0061] Partial Continuous Incentive (PE) condition refers to the existence of a constant. , and integers , so that for any All satisfy:

[0062] In the formula, It is the identity matrix; constant. , , ; .

[0063] S4. Define the system's filtering tracking error, and design a fault-tolerant controller and the corresponding network weight update law based on the filtering tracking error and the localized radial basis function neural network.

[0064] In this step, by defining the filtering tracking error, the fault dynamics and system uncertainty are unified and approximated by the RBF neural network. Then, based on the filtering tracking error and the localized radial basis function neural network, a fault-tolerant controller with neural network compensation terms and a weight update law based on gradient descent are designed, thereby clarifying key parameters such as controller gain and learning gain.

[0065] Specifically, the filtering tracking error is defined as:

[0066] In the formula, Indicates the first The tracking error for each state, ( ); This represents the design parameters that stabilize the error subsystem.

[0067] A fault-tolerant controller with neural network compensation terms is designed based on filtered tracking error and localized radial basis function neural network, specifically expressed as follows:

[0068] In the formula, Indicates the gain of the fault-tolerant controller. ; A localized radial basis function neural network is configured along a desired reference trajectory to approximate unknown fault dynamics.

[0069] The network weight update law corresponding to this fault-tolerant controller is calculated based on gradient descent, and is specifically expressed as follows:

[0070] In the formula, To learn the gain matrix, .

[0071] Further define the auxiliary parameters as follows: (1) Weight estimation error ,in, This represents the weight vector estimated by the neural network. This represents the ideal weight vector.

[0072] (2) State dependence coefficient ,in, Indicates the gain of the fault-tolerant controller. .

[0073] (3) Comprehensive interference item ,in, Indicates discrete interference. For the neural network approximation error, satisfying , The upper bound of the known approximation error is given.

[0074] (4) Normalized regression vector ,in, This is the localized regression vector.

[0075] S5. Based on the principle of uniform complete observability and Lyapunov stability theory, the learning error system consisting of a fault-tolerant controller and the corresponding network weight update law is quantitatively analyzed, and quantitative expressions for learning speed and learning accuracy are derived and established.

[0076] In this step, for the learning error system consisting of the fault-tolerant controller and the corresponding network weight update law, a quantitative analysis is performed on the learning error system based on the Uniform Complete Observability (UCO) principle and Lyapunov stability theory. The quantitative expressions for learning speed and learning accuracy are derived, and explicit expressions for the exponential convergence speed of the weight estimation error of the localized radial basis function neural network, as well as explicit expressions for the steady-state convergence compact set of the filtering tracking error and the weight estimation error, are obtained. The mathematical form, physical meaning, and calculation method of each parameter in the expressions are defined in detail.

[0077] Specifically, based on the Uniform Complete Observability (UCO) principle and Lyapunov stability theory, the learning speed quantization expression is as follows:

[0078] In the formula, Indicates the amplitude coefficient; Indicates the exponential convergence rate; ; Indicates the sampling time; This is the initial time.

[0079] The amplitude coefficient is specifically expressed as follows:

[0080] The exponential convergence rate is specifically expressed as:

[0081] In the formula, Denotes the minimum eigenvalue of the Lyapunov function matrix. ; This represents the largest eigenvalue of the Lyapunov function matrix. ; Eigenvalues ​​of Lyapunov function matrix , This is a preset constant; Indicates the convergence performance coefficient; This indicates the overall time step.

[0082] Among them, the convergence performance coefficient , and The positive constants derived for UCO properties are specifically represented as follows:

[0083] In the formula, , , and As an intermediate variable, it is jointly determined by controller parameters, learning gain, number of neurons, system state transition matrix, and some PE condition parameters. The specific calculation method is as follows:

[0084] In the formula, for The minimum value of the square. ,in ; As the first auxiliary coefficient, , for The smallest absolute value, ; ; , These are positive constants in the PE condition; , ; in, ; ; ; ; Indicates the overall time step. , This refers to the time step in some PE conditions. The compensation coefficient, determined by the system's dynamic characteristics, is expressed as... ; in, for The maximum absolute value, ; As the second auxiliary coefficient, ; As the third auxiliary coefficient, ; This represents the maximum value of the composite parameter. ; ; ; in, ; .

[0085] The learning accuracy is quantified to obtain the filtered tracking error. The steady-state convergent compact set is specifically represented as:

[0086] Localized radial basis function neural network weight estimation error The specific representation of a convergent compact set is:

[0087] Furthermore, learning accuracy also includes the quantification of the approximation accuracy of the neural network, which is defined as the neural network output. The modulus is specifically represented as:

[0088] In the formula, Indicates the upper limit of the interference amplification factor; Indicates the upper bound of the overall interference; Indicates the overall time step; and Positive numbers derived for UCO properties; This indicates the learning accuracy of the neural network.

[0089] Among them, the upper limit of interference amplification factor , ,in, .

[0090] Upper bound of comprehensive interference ,in, .

[0091] Learning accuracy satisfy:

[0092] In the formula, For localized regression vectors The upper bound of the model, .

[0093] S6. Based on the quantization expression, adjust the controller gain and learning gain matrix so that the system's tracking error and neural network learning error can converge to a preset accuracy range at a preset speed after a fault occurs.

[0094] In this step, the controller gain is adjusted within the range that satisfies the stability condition, based on the quantization expression. Learning gain These parameters allow the tracking error and weight estimation error to converge to a preset compact set, enabling fast and high-precision fault compensation.

[0095] Specifically, the controller gain is first determined by solving a system of inequalities. and learning gain The adjustment range of this system of inequalities is specifically expressed as follows:

[0096] In the formula, .

[0097] Change the controller gain within the obtained adjustment range. and learning gain The value of the convergence rate and convergence accuracy are then calculated to obtain the controller gain that meets the preset performance requirements. and learning gain The range of values ​​for is defined. Convergence accuracy includes both the accuracy of the filtering tracking error and the accuracy of the neural network learning.

[0098] As an example, the time of failure occurrence is set to... And set the sampling period to When configuring neurons along a desired reference trajectory, the number of neurons... Neuron width parameter minimum intercentral spacing of neurons The maximum distance between the neuron's center point and its trajectory Initial weights .

[0099] Define the fault-tolerant controller and system parameters as: filter tracking error parameters , , Lyapunov function parameters = , The upper bound of g(x); the upper bound of the approximation error. Therefore ; ; The value range is [0.75, 1.15]. , .

[0100] Based on the above parameter values, the tracking error converges compactly. The weight estimation error converges to a compact set. Neural network learning accuracy Learning convergence rate , .

[0101] Based on the above settings, the single-link robotic arm is controlled. to State tracking trajectory such as Figures 2 to 5 As shown in the figure, the relationship between control gain and learning gain on convergence speed and neural network learning accuracy is illustrated in the graph. Figure 6 and Figure 7 As shown. From Figures 2 to 5 It can be seen that after the fault occurs, the system state can quickly and accurately track the expected trajectory, and the tracking error converges to the theoretically calculated compact set range, which verifies the effectiveness of the method.

[0102] This invention provides a deterministic learning fault-tolerant control method for nonlinear sampled data systems to quantify learning performance. Through theoretical derivation, quantitative expressions for learning speed and learning accuracy are established, enabling quantitative analysis and pre-setting of fault-tolerant control performance. This effectively solves the problems of blind parameter adjustment and performance analysis remaining at the qualitative level in existing technologies, and has significant theoretical and engineering application value.

[0103] Example 2 In a typical embodiment of the present invention, this embodiment discloses a fault-tolerant control system for nonlinear systems with quantifiable performance based on learning, comprising: The system modeling and discretization module is configured to: establish a nonlinear continuous system model containing actuator faults, and perform discretization processing based on a set sampling period to obtain a nonlinear sampled data system model; The reference model design module is configured to: construct periodic or quasi-periodic reference models and generate the desired reference trajectory; The localized RBF neural network building module is configured to: construct a localized radial basis function neural network based on the system state vector of the nonlinear sampled data system model, with its neurons arranged along the desired reference trajectory; The controller and weight update law design module is configured to: define the system's filtering tracking error, and design a fault-tolerant controller and the corresponding network weight update law based on the filtering tracking error and the localized radial basis function neural network; The learning performance quantification derivation module is configured to: based on the principle of uniform complete observability and Lyapunov stability theory, quantitatively analyze the learning error system composed of the fault-tolerant controller and the corresponding network weight update law, and derive and establish the quantitative expressions for learning speed and learning accuracy. The parameter optimization and closed-loop control module is configured to adjust the controller gain and learning gain matrix according to the quantization expression, so that the tracking error and neural network learning error of the system can converge to a preset accuracy range at a preset speed after a fault occurs.

[0104] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A fault-tolerant control method for nonlinear systems with quantifiable performance based on learning, characterized in that, include: A nonlinear continuous system model with actuator faults is established, and the system model is discretized based on a set sampling period to obtain a nonlinear sampled data system model. Construct a periodic or quasi-periodic reference model to generate the desired reference trajectory; A localized radial basis function neural network is constructed based on the system state vector of the nonlinear sampled data system model, and its neurons are arranged along the desired reference trajectory; Define the system's filtering tracking error, and design a fault-tolerant controller and corresponding network weight update law based on the filtering tracking error and the localized radial basis function neural network. Based on the principle of uniform complete observability and Lyapunov stability theory, we quantitatively analyze the learning error system consisting of a fault-tolerant controller and the corresponding network weight update law, and derive and establish quantitative expressions for learning speed and learning accuracy. Based on the quantization expression, the controller gain and learning gain matrix are adjusted so that the system's tracking error and neural network learning error can converge to a preset accuracy range at a preset speed after a fault occurs.

2. The fault-tolerant control method for nonlinear systems with quantifiable performance based on learning as described in claim 1, characterized in that, The nonlinear continuous system model is specifically represented as follows: In the formula, x represents the system state vector. ; This indicates continuous system input; Denotes a smooth nonlinear function, where For actuator fault items; Represents the fault switch function. When it is 0, When it is 1, The time when the fault occurred; Indicates continuous bounded disturbance; The sign is known, and there exists a positive constant. , satisfy And by default ; The discretized nonlinear sampled data system model is specifically represented as follows: In the formula, To set the sampling period; This represents the discretized system state vector. ; This indicates continuous system input; This indicates discrete interference.

3. The fault-tolerant control method for nonlinear systems with quantifiable performance based on learning as described in claim 1, characterized in that, The localized radial basis function neural network is specifically represented as follows: In the formula, This represents the weight vector estimated by the neural network; The localized regression vector is composed of neurons arranged along the trajectory. , The reference state vector; Indicates the number of neurons along the trajectory; Represents the weights of a neural network; This represents the radial basis function.

4. The fault-tolerant control method for nonlinear systems with quantifiable performance based on learning as described in claim 3, characterized in that, The fault-tolerant controller is specifically represented as: In the formula, Indicates the gain of the fault-tolerant controller. ; A localized radial basis function neural network set up along the desired reference trajectory.

5. The fault-tolerant control method for nonlinear systems with quantifiable performance based on learning as described in claim 4, characterized in that, The network weight update law corresponding to the fault-tolerant controller is calculated based on gradient descent, and is specifically expressed as follows: In the formula, To learn the gain matrix, .

6. The fault-tolerant control method for nonlinear systems with quantifiable performance based on learning as described in claim 5, characterized in that, The specific expression for the learning speed quantization is as follows: In the formula, Indicates the amplitude coefficient. Indicates the exponential convergence rate; , For weight estimation error, This refers to the filtering tracking error; Indicates the sampling time; This is the initial time.

7. The fault-tolerant control method for nonlinear systems with quantifiable performance based on learning as described in claim 6, characterized in that, The quantitative expression for learning accuracy specifically includes the steady-state convergent compact expression for the filtering tracking error and the convergent compact expression for the weight estimation error of the localized radial basis function neural network. The steady-state convergent compact set of the filtering tracking error is specifically represented as: The convergence compact set of the weight estimation error in a localized radial basis function neural network is specifically represented as follows: In the formula, Indicates the upper limit of the interference amplification factor; Indicates the upper bound of the overall interference; Indicates the overall time step; and The positive constants derived for UCO properties.

8. The fault-tolerant control method for nonlinear systems with quantifiable performance based on learning as described in claim 7, characterized in that, Learning accuracy also includes the quantification of the neural network's approximation accuracy, which satisfies: In the formula, For localized regression vectors The upper bound of the model.

9. The fault-tolerant control method for nonlinear systems with quantifiable performance based on learning as described in claim 1, characterized in that, Based on the quantization expression, the controller gain and learning gain matrices are adjusted. Specifically, the adjustment range of the controller gain and learning gain is first solved. Within the obtained adjustment range, the values ​​of the controller gain and learning gain are changed, and then the convergence speed and convergence accuracy are calculated to obtain the range of controller gain and learning gain values ​​that meet the preset performance requirements. The convergence accuracy includes the accuracy of the filter tracking error and the accuracy of the neural network learning.

10. A fault-tolerant control system for nonlinear systems with quantifiable performance based on learning, characterized in that, include: The system modeling and discretization module is configured to: establish a nonlinear continuous system model containing actuator faults, and perform discretization processing based on a set sampling period to obtain a nonlinear sampled data system model; The reference model design module is configured to: construct periodic or quasi-periodic reference models and generate the desired reference trajectory; The localized RBF neural network building module is configured to: construct a localized radial basis function neural network based on the system state vector of the nonlinear sampled data system model, with its neurons arranged along the desired reference trajectory; The controller and weight update law design module is configured to: define the system's filtering tracking error, and design a fault-tolerant controller and the corresponding network weight update law based on the filtering tracking error and the localized radial basis function neural network; The learning performance quantification derivation module is configured to: based on the principle of uniform complete observability and Lyapunov stability theory, quantitatively analyze the learning error system composed of the fault-tolerant controller and the corresponding network weight update law, and derive and establish the quantitative expressions for learning speed and learning accuracy. The parameter optimization and closed-loop control module is configured to adjust the controller gain and learning gain matrix according to the quantization expression, so that the tracking error and neural network learning error of the system can converge to a preset accuracy range at a preset speed after a fault occurs.

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