A hamiltonian constrained singular perturbation output distribution fuzzy fault-tolerant control method

Abstract

This invention belongs to the field of fault-tolerant control technology for the output probability density function of nonlinear singular perturbation systems, and discloses a fuzzy fault-tolerant control method for Hamiltonian-constrained singular perturbation output distribution. This method solves the problem of tracking and fault-tolerant control of the output probability density function in fast-slow dynamically coupled systems under faults and disturbances. First, a fuzzy model of the square root form of the output probability density function is established based on T-S fuzzy modeling and B-spline functions. Second, a fuzzy observer is designed as a fault estimator, and the gain is solved based on H∞ performance and linear matrix inequalities to achieve online fault reconstruction. Then, combining the fuzzy model and fault estimation, a gradient descent algorithm based on Hamiltonian constraints is used to optimize the parameters of the fuzzy fault-tolerant controller to maintain tracking performance and closed-loop stability under faults. Finally, the system convergence is analyzed using Lyapunov functions. This method achieves real-time fault estimation and dynamic compensation, and is suitable for industrial processes requiring precise control of the output distribution.

Description

Technical Field

[0001] This invention belongs to the field of fault-tolerant control technology of output probability density function of nonlinear singular perturbation system, and discloses a fuzzy fault-tolerant control method for output distribution of Hamiltonian-constrained singular perturbation. Background Technology

[0002] With the increasing demands for product quality uniformity in high-end manufacturing and process industries, the control of key output variables has evolved from traditional mean and variance indices to precise regulation of the overall probability density function (PDF) distribution. However, for nonlinear singular perturbation systems with widely intertwined fast and slow dynamics, fault-tolerant control of the output PDF faces a series of bottlenecks, making existing methods difficult to apply directly.

[0003] Specifically, this manifests in several ways: First, system modeling is extremely complex. Such systems exhibit both strong coupling between fast and slow time scales and highly nonlinear dynamics, with their outputs being randomly distributed. Traditional methods struggle to accurately establish a coupled model describing the transition from "deterministic nonlinear dynamics" to "random output distribution." Second, faults and disturbances are difficult to distinguish. Actuator faults and external random disturbances couple and affect the output PDF. Existing fault estimators often fail to effectively separate these two factors, leading to insufficient fault reconstruction accuracy and response lag, thus compromising the accuracy of fault-tolerant control. Finally, multi-objective control design is challenging. On the established complex model, output PDF tracking, fault compensation, disturbance robustness, and closed-loop stability must be simultaneously achieved. Traditional fault-tolerant control strategies often struggle to coordinate and optimize these coupled constraints and performance indicators, easily resulting in performance trade-offs or instability.

[0004] Therefore, for fault-tolerant control of the output PDF of nonlinear singular perturbation systems, there is an urgent need for a new systematic approach that can uniformly solve the problems of accurate modeling, high-precision fault estimation, and high-performance fault-tolerant control under multiple constraints. Summary of the Invention

[0005] To address the technical problems existing in the background art, this invention proposes a Hamiltonian-constrained singular perturbation output distribution fuzzy fault-tolerant control method. This method is applicable to singular perturbation systems where the output variable probability density function is measurable. Through fuzzy modeling and accurate estimation of fault signals, based on the fuzzy control strategy, precise control of the output probability density function can be achieved even when the actuator is faulty, such as controlling the ore particle size in grinding processes and controlling the fiber length in papermaking processes.

[0006] To address the aforementioned technical problems, this application proposes a Hamiltonian-constrained singular perturbation output distribution fuzzy fault-tolerant control method, comprising the following steps:

[0007] S1. Establish a fuzzy model of the output probability density function of the nonlinear singular perturbation system: Based on the TS fuzzy modeling method, a fuzzy model of the output probability density function of the nonlinear singular perturbation system is established. Multiple fuzzy rules are used to describe the dynamic evolution relationship of the output probability density function, and the membership function is used to weight and fuse each local probability density function sub-model to obtain a unified fuzzy probability density function model that characterizes the fast and slow dynamic coupling characteristics, fault characteristics and random distribution characteristics of the system.

[0008] S2. Establishing a fault estimator: Based on the TS fuzzy singular perturbation probability density function model established in step S1, the fault term is introduced into the system description as an unknown input to be estimated, and a fuzzy observer matching the system is constructed to form a fault estimator; an augmented error system is established by outputting the probability density function error, state estimation error, and fault reconstruction error; the solution conditions for the gain of the fault estimator are given in combination with the H infinity performance index, so that the augmented error system is asymptotically stable and the fault can be accurately reconstructed, thereby realizing online estimation and tracking of system fault information;

[0009] S3. Design a fault-tolerant controller: Based on the fuzzy model of the output probability density function of the nonlinear singular perturbation system established in step S1 and the fault estimator established in step S2, design the corresponding Hamiltonian constraints and gradient descent algorithm, determine the solution conditions of the control parameters, so that the system can still maintain the tracking performance of the output probability density function and the overall operational stability after a fault occurs, and realize dynamic compensation and fault-tolerant control under fault conditions.

[0010] S4. Analysis of the convergence of the control algorithm: Based on the augmented state error, a closed-loop system is constructed, and the stability of the fuzzy fault-tolerant control algorithm proposed in steps S1 to S3 is analyzed.

[0011] Furthermore, S1 specifically includes the following steps:

[0012] S11, Output for singular perturbation systems probability density function By using B-spline basis functions to approximate its square root form in finite dimensions, we obtain: ,in, Indicates control input System output under action The probability density function; Indicates the first A predefined B-spline basis function; Representation and control input The relevant time-varying weights; This represents the number of B-spline basis functions; This represents the bounded approximation error; through the above representation, the random output distribution is transformed into a finite-dimensional weighted value expression for control modeling;

[0013] S12. Based on the finite-dimensional approximation results, construct the first... The singular perturbation local model corresponding to the fuzzy rule: when the fuzzy antecedent variable They belong to fuzzy subsets respectively At that time, the system satisfies: , , ,in, Represents the system state vector; ,in Represents the identity matrix. Indicates the small parameter of the singular perturbation; Indicates control input; Indicates external disturbance; Indicates the first The system state matrix under fuzzy rules; Indicates the first The system input matrix under fuzzy rules; Represents the system disturbance matrix; This indicates the output weight matrix; This represents the weight vector of the output probability density function; Representation and output variables Related weight mapping vector; Represents the weight vector A known nonlinear function; This represents a pre-selected B-spline basis function term; Indicates the first A fuzzy antecedent variable at time 1 The possible values ​​of ; This represents a fuzzy subset corresponding to the antecedent variable; Indicates the number of fuzzy antecedent variables;

[0014] S13. Weighted fusion of each local fuzzy rule yields the overall fuzzy model representing the random output characteristics of the singular perturbation system: , , ,in, , ; Indicates the first Normalized weights of fuzzy rules; Indicates the first The antecedent activation strength of a fuzzy rule; Represents a vector consisting of all the preceding variables; Indicates the total number of fuzzy rules;

[0015] S14, Introducing a fault The local fuzzy rules are further weighted and fused to obtain the overall fault system model: , , ;in, and These represent the fused global system matrix, For the corresponding normalized fuzzy weights, This represents the fault input vector, which has the same dimension as the control input channel.

[0016] Furthermore, S2 specifically includes the following steps:

[0017] S21. Incorporate the actuator fault estimate into the state estimation process to construct the fuzzy estimation model corresponding to the fault estimator; when the fuzzy antecedent variables... They belong to fuzzy subsets respectively At that time, the first Under the fuzzy rule, the following conditions must be met: , , After weighted fusion of the various fuzzy rules, the overall estimation model is obtained: ,in, Represents the state estimation vector. Represents the fault input vector The estimated value, Represents the weight vector The estimated value, Indicates the first The estimator matrix corresponding to the fuzzy rules, This represents the fused estimator matrix. Indicates the first The normalized weights of the fuzzy rules, and These represent the fused system matrices, The matrix representing the mapping from states to weight vectors. Representation and output variables The relevant weight mapping vector, Indicates about The known nonlinear function, Denotes the B-spline basis function term. Indicates the presence of singular perturbation parameters The generalized system matrix, Indicates the small parameter of the singular perturbation;

[0018] S22. Define the state estimation error, fault estimation error, and output probability density tracking error, and establish the correspondence between the output probability density tracking error and the state estimation error; wherein, the state estimation error is defined as... Fault estimation error is defined as The weight estimation error is defined as The output probability density tracking error is defined as follows: ;in, This represents a pre-defined weighting function. This indicates the setting of the target probability density function. Let [a, b] represent the output variable corresponding to the target probability density function, and let [a, b] represent the output variable. Domain range Indicates the lower bound. Indicates the upper bound;

[0019] S23. Based on the aforementioned state estimation error and fault estimation error, the augmented error closed-loop system model is constructed as follows: , ;in, This represents the augmented error state vector. This represents the augmented perturbation vector. Indicates external disturbance. Represents the fault input vector rate of change, This represents the fault detection residual signal. Represents an augmented generalized matrix. Represents the augmented system matrix. Represents the augmented input matrix. This represents the augmented output matrix. Represents the perturbation input matrix. and This represents the gain matrix of the fault estimator to be determined. Represents the identity matrix;

[0020] S24. For the augmented error closed-loop system, a preset value is given. Performance indicators Construct linear matrix inequality constraints; when a symmetric positive definite matrix exists... and Estimator matrix Auxiliary matrix , and scaling matrix and , so that for any All satisfy: ,and ;in, , , ; and Represents Lyapunov matrix variables. and This represents the auxiliary variable used to solve for the gain of the fault estimator. and Represents the scaling matrix. Indicates preset Performance metrics , , and Represents a known system matrix;

[0021] S25. When the linear matrix inequality has a feasible solution, then for any Both can guarantee the asymptotic stability of the augmented error closed-loop system and satisfy the above requirements. Performance indicators Based on this, the fault estimator gain matrix is ​​obtained as follows: , ;in, Represents singular perturbation parameters The allowed upper bound; based on the gain matrix and This enables online estimation of actuator fault inputs and provides fault compensation information for the subsequent construction of fault-tolerant control laws.

[0022] Furthermore, S3 specifically includes the following steps:

[0023] S31. Construct the augmented tracking error of the weight domain of the output probability density function, wherein the augmented tracking error is expressed as: ,in, express Augmented weight error at time step This represents the preset weight selection matrix, used to extract the target control component from the weight vector. This represents the weight vector corresponding to the square root form of the output probability density function. Indicates the reference trajectory. This represents a matrix that maps system states to weight vectors. Represents the system state vector; in the singular perturbation matrix Under reversible conditions, the system is equivalently represented as ,in, Represents the equivalent state matrix. Represents the equivalent input matrix. Represents the equivalent perturbation matrix. This represents the system matrix after fuzzy fusion. This represents the input matrix after fuzzy fusion. This represents the original system disturbance matrix. This represents the set of normalized weights for fuzzy rules. Indicates control input, Indicates external disturbance;

[0024] S32. Based on the augmented tracking error, dynamically organize the error into the input affine form. ,in, This represents the set of normalized weights for fuzzy rules. This represents the rate of change of the augmented tracking error. Represents dynamic variables within the system. This represents a composite error term, which includes at least the nominal state term. Disturbance terms Reference trajectory change term and the B-spline approximation error term , Represents the reference trajectory rate of change, This represents the bounded error in the square root approximation of the output probability density function. This indicates the value of the system output variable. It represents the equivalent input gain, used to characterize the relationship between fuzzy fusion uncertainty and control input;

[0025] S33. Construct a fuzzy fault-tolerant controller, wherein the control input is represented as... ,in, Indicates the number of fuzzy rules in the controller. Indicates the first Normalized weights of the control rules Indicates the first The activation strength of the control rule, Indicates the number of antecedent variables of the controller. Represents the augmented error state vector The One portion, Indicates the first Rule number 1 Gaussian membership functions of the antecedent variables Indicates the center of the membership function. This represents the width parameter of the membership function. This represents the augmented error state vector. This represents the state estimation error. This indicates the fault estimation error. Represents the fault estimation vector The One portion, Indicates the first The control weight vector that acts on the augmented error state in the rule. Indicates the first The compensation weights applied to the fault estimate in the rule;

[0026] S34, using the controller parameter set To optimize the object, construct performance metrics. ,in, This indicates the performance metrics corresponding to the controller parameters. The term represents the operating cost function, and the operating cost function includes at least an output probability density function deviation term and a control energy term. [a,b] represents the output variable. The range of values, Indicates control input The output probability density function under the action, Represents the terminal cost function. Represent the terminal time; further construct the Hamiltonian function. ,in, Represents Hamiltonian functions, Indicates the accompanying variable, The right-hand side of the system state equation is represented by the gradient of the controller parameters. Calculate, and through Update, in which, Represents performance metrics with respect to the set of parameters gradient, Indicates the number of iterations. Indicates the first Controller parameters at the next iteration This represents the learning rate.

[0027] Furthermore, S4 specifically includes the following steps:

[0028] S41. Construct a Lyapunov function for the augmented error system corresponding to the fuzzy fault estimator. ,in, Let Lyapunov function be defined with respect to the augmented error state vector. This represents the augmented error state vector. This represents the state estimation error. This indicates the fault estimation error. Represents the augmented singular perturbation matrix. Represents the singular perturbation matrix. express 3D identity matrix This indicates the dimension of the actuator fault input. Let represent the positive definite matrix to be found, and satisfy . To ensure The augmented error state vector is positive definite;

[0029] S42, with As a performance output, As an augmented perturbation input, design Make matrix inequalities Established, among which, Represents the performance output vector. Represents the performance output matrix. Indicates external disturbance and failure rate The resulting augmented perturbation vector This represents the H∞ disturbance suppression index. This represents the set of normalized weights for fuzzy rules. , This represents the augmented error system matrix after fuzzy fusion. This represents the augmented perturbation matrix after fuzzy fusion. Represents the identity matrix that matches the corresponding block dimension, denoted by [symbol]. If the condition is met by a symmetric term, then the following condition is satisfied. This ensures that the fault estimation error system meets the H∞ performance constraint.

[0030] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. Simplified Modeling: To address the complex problem of modeling the output distribution of singular perturbation systems, this paper integrates TS fuzzy modeling and B-spline function approximation to transform the difficult-to-describe "deterministic dynamics-random distribution" coupling relationship into a unified fuzzy weighted model, which effectively characterizes the fast and slow dynamic coupling and the distribution characteristics of the output probability density function. 2. Accurate Fault Estimation: To address the difficulty in distinguishing between faults and disturbances in the output distribution of singular perturbation systems, a fault estimator based on a fuzzy observer is designed. By constructing an augmented error system and introducing the H∞ performance index, the gain is solved using the linear matrix inequality (LMI), thereby effectively separating the influence of disturbances and faults, achieving online high-precision fault reconstruction, and providing an accurate prerequisite for fault-tolerant control. 3. Cooperative Control under Multiple Constraints: For the problem of solving control inputs under multiple constraints, a gradient descent optimization strategy based on Hamiltonian constraints is designed, building upon the existing model and fault estimation. This strategy integrates multiple objectives, including output PDF tracking, fault estimation, disturbance suppression, and closed-loop stability, into a unified performance index. By optimizing controller parameters, it achieves cooperative optimization across multiple constraints, avoiding performance trade-offs and ensuring dynamic compensation and stable system operation under fault conditions. Attached Figure Description

[0031] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0032] Figure 1 This is a schematic diagram of the implementation steps of the method of the present invention;

[0033] Figure 2 The estimation effect of the fault estimator for the output PDF of the singular perturbation system provided in the embodiments of the present invention;

[0034] Figure 3 This is the output PDF control effect of the grinding and classification process provided in the embodiments of the present invention. Detailed Implementation

[0035] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0036] A Hamiltonian-constrained singular perturbation output distribution fuzzy fault-tolerant control method includes the following steps:

[0037] S1. Establishing a fuzzy model of the output probability density function of a nonlinear singular perturbation system: This step aims to establish an accurate and easily tractable output probability density function (PDF) model for a nonlinear singular perturbation stochastic system exhibiting both fast and slow dynamics. The core of this step lies in utilizing TS fuzzy modeling technology to linearize the complex nonlinear system at multiple local operating points, and then globally fusion using membership functions to ultimately obtain a holistic fuzzy model that can uniformly characterize the system's fast-slow dynamic coupling characteristics, actuator failure characteristics, and output random distribution characteristics. Specifically, this includes the following sub-steps:

[0038] S11, B-spline approximation of the output PDF. For singular perturbation system output... probability density function We employ B-spline basis functions to perform a finite-dimensional approximation of its square root form, thereby transforming the infinite-dimensional distribution control problem into a finite-dimensional weight control problem. The approximate expression is: ,in, Indicates control input System output under action The probability density function; Indicates the first A predefined B-spline basis function; Representation and control input The relevant time-varying weights; This represents the number of B-spline basis functions; This represents the bounded approximation error; through the above representation, the random output distribution is transformed into a finite-dimensional weighted value expression for control modeling;

[0039] S12, Constructing the first The singular perturbation local model corresponding to the fuzzy rule: Based on the approximation result of step S11, the original nonlinear system is fuzzily partitioned using fuzzy antecedent variables. When the fuzzy antecedent variables... They belong to fuzzy subsets respectively At that time, the singular perturbation local model corresponding to the k-th fuzzy rule is: , , ,in, Represents the system state vector; ,in Represents the identity matrix. Indicates the small parameter of the singular perturbation; Indicates control input; Indicates external disturbance; Indicates the first The system state matrix under fuzzy rules; Indicates the first The system input matrix under fuzzy rules; Represents the system disturbance matrix; This indicates the output weight matrix; This represents the weight vector of the output probability density function; Representation and output variables Related weight mapping vector; Represents the weight vector A known nonlinear function; This represents a pre-selected B-spline basis function term; Indicates the first A fuzzy antecedent variable at time 1 The possible values ​​of ; This represents a fuzzy subset corresponding to the antecedent variable; Indicates the number of fuzzy antecedent variables;

[0040] S13. Weighted fusion of each local fuzzy rule yields the overall fuzzy model representing the random output characteristics of the singular perturbation system: , , ,in, , ; Indicates the first Normalized weights of fuzzy rules; Indicates the first The antecedent activation strength of a fuzzy rule; Represents a vector consisting of all the preceding variables; Indicates the total number of fuzzy rules;

[0041] S14, Introducing a fault The local fuzzy rules are further weighted and fused to obtain the overall fault system model, providing a foundation for subsequent fault estimation and fault-tolerant control design. , , ;in, and These represent the fused global system matrix, For the corresponding normalized fuzzy weights, This represents the fault input vector, which has the same dimension as the control input channel.

[0042] S2. Establishing a Fault Estimator: The goal of this step is to design a fuzzy observer for the system model containing faults established in step S1, enabling online joint estimation of the system state and actuator faults. By constructing an augmented error system and satisfying a specific H-infinity performance index, the estimation error is ensured to be dynamically asymptotically stable and robust to disturbances, thereby achieving high-precision fault reconstruction. Specifically, this includes the following sub-steps:

[0043] S21. Incorporate the actuator fault estimate into the state estimation process to construct the fuzzy estimation model corresponding to the fault estimator; when the fuzzy antecedent variables... They belong to fuzzy subsets respectively At that time, the first Under the fuzzy rule, the following conditions must be met: , , After weighted fusion of the various fuzzy rules, the overall estimation model is obtained: ,in, Represents the state estimation vector. Represents the fault input vector The estimated value, Represents the weight vector The estimated value, Indicates the first The estimator matrix corresponding to the fuzzy rules, This represents the fused estimator matrix. Indicates the first The normalized weights of the fuzzy rules, and These represent the fused system matrices, The matrix representing the mapping from states to weight vectors. Representation and output variables The relevant weight mapping vector, Indicates about The known nonlinear function, Denotes the B-spline basis function term. Indicates the presence of singular perturbation parameters The generalized system matrix, Indicates the small parameter of the singular perturbation;

[0044] S22. Define the state estimation error, fault estimation error, and output probability density tracking error, and establish the correspondence between the output probability density tracking error and the state estimation error; wherein, the state estimation error is defined as... Fault estimation error is defined as The weight estimation error is defined as The output probability density tracking error is defined as follows: ;in, This represents a pre-defined weighting function. Represents the target probability density function. Let [a, b] represent the output variable corresponding to the target probability density function, and let [a, b] represent the output variable. Domain range Indicates the lower bound. Indicates the upper bound;

[0045] S23. Based on the aforementioned state estimation error and fault estimation error, the augmented error closed-loop system model is constructed as follows: , ;in, This represents the augmented error state vector. This represents the augmented perturbation vector. Indicates external disturbance. Represents the fault input vector rate of change, This represents the fault detection residual signal. Represents an augmented generalized matrix. Represents the augmented system matrix. Represents the augmented input matrix. This represents the augmented output matrix. Represents the perturbation input matrix. and This represents the gain matrix of the fault estimator to be determined. Represents the identity matrix;

[0046] S24. For the augmented error closed-loop system, a preset value is given. Performance indicators Construct linear matrix inequality constraints; when a symmetric positive definite matrix exists... and Estimator matrix Auxiliary matrix , and scaling matrix and , so that for any All satisfy: ,and ;in, , , ; and Represents the Lyapunov matrix. and This represents the auxiliary variable used to solve for the gain of the fault estimator. and Represents the scaling matrix. Indicates preset Performance metrics , , and Represents a known system matrix;

[0047] S25. When the linear matrix inequality has a feasible solution, then for any Both can guarantee the asymptotic stability of the augmented error closed-loop system and satisfy the above requirements. Performance indicators Based on this, the fault estimator gain matrix is ​​obtained as follows: , ;in, Represents singular perturbation parameters The allowed upper bound; based on the gain matrix and This enables online estimation of actuator fault inputs and provides fault compensation information for the subsequent construction of fault-tolerant control laws.

[0048] S3. Design a fault-tolerant controller: Based on the system model established in step S1 and the fault estimation information obtained in step S2, a fuzzy adaptive fault-tolerant controller is designed. By constructing a Hamiltonian function and using the gradient descent method to optimize the controller parameters online, the system can still accurately track the desired output probability density function even after an actuator failure, ensuring the overall stability of the closed-loop system. Specifically, this includes the following sub-steps:

[0049] S31. Construct the augmented tracking error of the weight domain of the output probability density function, wherein the augmented tracking error is expressed as: ,in, express Augmented weight error at time step This represents the preset weight selection matrix, used to extract the target control component from the weight vector. This represents the weight vector corresponding to the square root form of the output probability density function. Indicates the reference trajectory. This represents a matrix that maps system states to weight vectors. Represents the system state vector; in the singular perturbation matrix Under reversible conditions, the system is equivalently represented as ,in, Represents the equivalent state matrix. Represents the equivalent input matrix. Represents the equivalent perturbation matrix. This represents the system matrix after fuzzy fusion. This represents the input matrix after fuzzy fusion. This represents the original system disturbance matrix. This represents the set of normalized weights for fuzzy rules. Indicates control input, Indicates external disturbance;

[0050] S32. Based on the augmented tracking error, dynamically organize the error into the input affine form. ,in, This represents the rate of change of the augmented tracking error. Represents dynamic variables within the system. This represents a composite error term, which includes at least the nominal state term. Disturbance terms Reference trajectory change term and the B-spline approximation error term , Represents the reference trajectory rate of change, This represents the bounded error in the square root approximation of the output probability density function. This indicates the value of the system output variable. It represents the equivalent input gain, used to characterize the relationship between fuzzy fusion uncertainty and control input;

[0051] S33. Construct a fuzzy fault-tolerant controller, wherein the control input is represented as... ,in, Indicates the number of fuzzy rules in the controller. Indicates the first Normalized weights of the control rules Indicates the first The activation strength of the control rule, Indicates the number of antecedent variables of the controller. Represents the augmented error state vector The One portion, Indicates the first Rule number 1 Gaussian membership functions of the antecedent variables Indicates the center of the membership function. This represents the width parameter of the membership function. This represents the augmented error state vector. This represents the state estimation error. This indicates the fault estimation error. Represents the fault estimation vector The One portion, Indicates the first The control weight vector that acts on the augmented error state in the rule. Indicates the first The compensation weights applied to the fault estimate in the rule;

[0052] S34. Optimize controller parameters based on Hamiltonian framework: using the controller parameter set To optimize the object, construct performance metrics. ,in, This indicates the performance metrics corresponding to the controller parameters. The term represents the operating cost function, and the operating cost function includes at least an output probability density function deviation term and a control energy term. [a,b] represents the output variable. The range of values, Indicates control input The output probability density function under the action, This represents the target output probability density function. This represents the positive definite control weight matrix. Represents the terminal cost function. Represent the terminal time; further construct the Hamiltonian function. ,in, Represents Hamiltonian functions, Indicates the accompanying variable, The right-hand side of the system state equation is represented by the gradient of the controller parameters. Calculate, and through Update, in which, Represents performance metrics with respect to the set of parameters gradient, Indicates the number of iterations. Indicates the first Controller parameters at the next iteration This represents the learning rate.

[0053] S4. Analysis of Control Algorithm Convergence: This step theoretically analyzes the closed-loop stability of the overall fuzzy fault-tolerant control algorithm proposed in steps S1 to S3, ensuring that the designed fault estimator and fault-tolerant controller can work together to keep the system stable under faults and disturbances. S4 specifically includes the following steps:

[0054] S41. Construct the Lyapunov function. For the augmented error system corresponding to the fuzzy fault estimator, construct the Lyapunov function. ,in, Let Lyapunov function be defined with respect to the augmented error state vector. This represents the augmented error state vector. This represents the state estimation error. This indicates the fault estimation error. Represents the augmented singular perturbation matrix. Represents the singular perturbation matrix. express 3D identity matrix This indicates the dimension of the actuator fault input. Let represent the positive definite matrix to be found, and satisfy . To ensure The augmented error state vector is positive definite;

[0055] S42. Verify H∞ performance. As a performance output, As an augmented perturbation input, design Make matrix inequalities Established, among which, Represents the performance output vector. Represents the performance output matrix. Indicates external disturbance and failure rate The resulting augmented perturbation vector This represents the H∞ disturbance suppression index. This represents the set of normalized weights for fuzzy rules. , This represents the augmented error system matrix after fuzzy fusion. This represents the augmented perturbation matrix after fuzzy fusion. Represents the identity matrix that matches the corresponding block dimension, denoted by [symbol]. If the condition is met by a symmetric term, then the following condition is satisfied. This indicates that the fault estimation error system is not only internally asymptotically stable but also satisfies the H∞ performance constraint, meaning it has good robustness to external disturbances and fault variations. Combined with the online optimization of controller parameters based on gradient descent in step S3, the tracking error convergence is guaranteed, thus jointly ensuring the stability of the entire closed-loop fuzzy fault-tolerant control system.

[0056] In summary, this invention establishes a TS fuzzy model that accurately describes the fast and slow dynamics and random output characteristics of the system through step S1; designs a fault estimator with H∞ robust performance through step S2, achieving online high-precision reconstruction of actuator faults; designs an adaptive fuzzy fault-tolerant controller by combining fault estimation information with gradient optimization under the Hamiltonian framework through step S3; and finally, through theoretical analysis in step S4, ensures the stable convergence of the entire closed-loop system. This method systematically solves the challenges of modeling, fault estimation, and fault-tolerant control of nonlinear singular perturbation systems in the shape control of the output probability density function. (H∞ performance constraint.)

[0057] like Figure 1As shown, this invention provides a Hamiltonian-constrained singular perturbation output distribution fuzzy fault-tolerant control method. First, a system output probability density function model is constructed based on a T-S fuzzy model, and the square root form of the probability density function is used to handle non-negative constraints, avoiding negative values ​​during model optimization. Then, to address the difficulty in accurately estimating faults, an augmented system is established based on the H∞ performance index, and the LMI method is used to solve for the fault estimator parameters, achieving effective fault reconstruction. Finally, to address the problem of multi-constraint coupling in fault-tolerant control, a closed-loop error propagation relationship is constructed based on the fuzzy model. Using the output probability density function performance index as the objective, a gradient descent algorithm based on Hamiltonian constraints is used to optimize the controller parameters, thereby achieving stable estimation, dynamic compensation, and fault-tolerant control under system faults.

[0058] Figure 2 The real-time estimation performance of the proposed fuzzy fault estimator under periodic slurry pump failures is demonstrated. Clearly, the estimated fault signal tracks the true periodic fault profile with high fidelity and fast transient response. The steady-state estimation error is negligible, and there is no significant jitter during abrupt fault transitions, demonstrating the excellent convergence and robustness of the observer-based diagnostic scheme. This high-precision reconstruction of the fault amplitude provides a reliable basis for active fault-tolerant compensation, ensuring the controller can offset intermittent actuator anomalies during the grinding stage process. Regarding the overall system output, Figure 3 The temporal evolution of particle size distribution (PDF) within the experimental range was depicted. Despite periodic pump failures, the output distribution remained strictly confined within the target range, demonstrating the effectiveness of the fuzzy fault-tolerant controller. Although the morphological peaks of the PDF exhibited slight periodic fluctuations corresponding to the fault intervals, the system avoided divergent behavior and maintained a stable distribution shape. This closed-loop stability and disturbance immunity confirm that the integrated stochastic distribution control strategy can successfully maintain specified product quality and operational reliability even under severe actuator disturbances.

[0059] The above description is merely an example and illustration of the concept of the present invention. Those skilled in the art can make various modifications or additions to the specific embodiments described or use similar methods to replace them, as long as they do not deviate from the concept of the invention or exceed the scope defined in this specification, they should all fall within the protection scope of the present invention.

Claims

1. A Hamiltonian constrained singular perturbation output distribution based fuzzy fault tolerant control method, characterized in that, Includes the following steps: S1. Establish a fuzzy model of the output probability density function of the nonlinear singular perturbation system: Based on the TS fuzzy modeling method, a fuzzy model of the output probability density function of the nonlinear singular perturbation system is established. Multiple fuzzy rules are used to describe the dynamic evolution relationship of the output probability density function, and the membership function is used to weight and fuse each local probability density function sub-model to obtain a unified fuzzy probability density function model that characterizes the fast and slow dynamic coupling characteristics, fault characteristics and random distribution characteristics of the system. S2. Establishing a fault estimator: Based on the TS fuzzy singular perturbation probability density function model established in step S1, the fault term is introduced into the system description as an unknown input to be estimated, and a fuzzy observer matching the system is constructed to form a fault estimator; an augmented error system is established by outputting the probability density function error, state estimation error, and fault reconstruction error; the solution conditions for the gain of the fault estimator are given in combination with the H infinity performance index, so that the augmented error system is asymptotically stable and the fault can be accurately reconstructed, thereby realizing online estimation and tracking of system fault information; S3. Design a fault-tolerant controller: Based on the fuzzy model of the output probability density function of the nonlinear singular perturbation system established in step S1 and the fault estimator established in step S2, design the corresponding Hamiltonian constraints and gradient descent algorithm, determine the solution conditions of the control parameters, so that the system can still maintain the tracking performance of the output probability density function and the overall operational stability after a fault occurs, and realize dynamic compensation and fault-tolerant control under fault conditions. S4. Analysis of the convergence of the control algorithm: Based on the augmented state error, a closed-loop system is constructed, and the stability of the fuzzy fault-tolerant control algorithm proposed in steps S1 to S3 is analyzed.

2. The Hamiltonian constrained singularly perturbed output distribution fuzzy fault-tolerant control method according to claim 1, characterized in that, S1 specifically includes the following steps: S11, Output for singular perturbation systems probability density function By using B-spline basis functions to approximate its square root form in finite dimensions, we obtain: ,in, Indicates control input System output under action The probability density function; Indicates the first A predefined B-spline basis function; Representation and control input The relevant time-varying weights; This represents the number of B-spline basis functions; This represents the bounded approximation error; through the above representation, the random output distribution is transformed into a finite-dimensional weighted value expression for control modeling; S12. Based on the finite-dimensional approximation results, construct the first... The singular perturbation local model corresponding to the fuzzy rule: when the fuzzy antecedent variable They belong to fuzzy subsets respectively At that time, the system satisfies: , , ,in, Represents the system state vector; ,in Represents the identity matrix. Indicates the small parameter of the singular perturbation; Indicates control input; Indicates external disturbance; Indicates the first The system state matrix under fuzzy rules; Indicates the first The system input matrix under fuzzy rules; Represents the system disturbance matrix; This indicates the output weight matrix; This represents the weight vector of the output probability density function; Representation and output variables Related weight mapping vector; Represents the weight vector A known nonlinear function; This represents a pre-selected B-spline basis function term; Indicates the first A fuzzy antecedent variable at time 1 The possible values ​​of ; Represents a fuzzy subset corresponding to the antecedent variable; Indicates the number of fuzzy antecedent variables; S13. Weighted fusion of each local fuzzy rule yields the overall fuzzy model representing the random output characteristics of the singular perturbation system: , , ,in, , ; Indicates the first Normalized weights of fuzzy rules; Indicates the first The antecedent activation strength of a fuzzy rule; Represents a vector consisting of all the preceding variables; Indicates the total number of fuzzy rules; S14, Introducing a fault The local fuzzy rules are further weighted and fused to obtain the overall fault system model: , , ;in, and These represent the fused global system matrix, For the corresponding normalized fuzzy weights, This represents the fault input vector, which has the same dimension as the control input channel.

3. A Hamiltonian-constrained singular perturbation output distribution fuzzy fault-tolerant control method according to claim 1 or 2, characterized in that, S2 specifically includes the following steps: S21. Incorporate the actuator fault estimate into the state estimation process to construct the fuzzy estimation model corresponding to the fault estimator; when the fuzzy antecedent variables... They belong to fuzzy subsets respectively At that time, the first Under the fuzzy rule, the following conditions must be met: , , After weighted fusion of the various fuzzy rules, the overall estimation model is obtained: ,in, Represents the state estimation vector. Represents the fault input vector The estimated value, Represents the weight vector The estimated value, Indicates the first The estimator matrix corresponding to the fuzzy rules, This represents the fused estimator matrix. Indicates the first The normalized weights of the fuzzy rules, and These represent the fused system matrices, The matrix representing the mapping from states to weight vectors. Representation and output variables The relevant weight mapping vector, Indicates about The known nonlinear function, Denotes the B-spline basis function term. Indicates the presence of singular perturbation parameters The generalized system matrix, Indicates the small parameter of the singular perturbation; S22. Define state estimation error, fault estimation error, and output probability density tracking error, and establish the correspondence between output probability density tracking error and state estimation error; wherein, state estimation error is defined as... Fault estimation error is defined as The weight estimation error is defined as The output probability density tracking error is defined as follows: ;in, This represents a pre-defined weighting function. This indicates the setting of the target probability density function. Let [a, b] represent the output variable corresponding to the target probability density function, and let [a, b] represent the output variable. Domain range Indicates the lower bound. Indicates the upper bound; S23. Based on the state estimation error and the fault estimation error, the augmented error closed-loop system model is constructed as follows: , ;in, This represents the augmented error state vector. This represents the augmented perturbation vector. Indicates external disturbance. Represents the fault input vector rate of change, This represents the fault detection residual signal. Represents an augmented generalized matrix. Represents the augmented system matrix. Represents the augmented input matrix. This represents the augmented output matrix. Represents the perturbation input matrix. and This represents the gain matrix of the fault estimator to be determined. Represents the identity matrix; S24. For the augmented error closed-loop system, a preset value is given. Performance indicators Construct linear matrix inequality constraints; when a symmetric positive definite matrix exists... and Estimator matrix Auxiliary matrix , and scaling matrix and , so that for any All satisfy: ,and ;in, , , ; and Represents Lyapunov matrix variables. and This represents the auxiliary variable used to solve for the gain of the fault estimator. and Represents the scaling matrix. Indicates preset Performance metrics , , and Represents a known system matrix; S25. When the linear matrix inequality has a feasible solution, then for any Both can guarantee the asymptotic stability of the augmented error closed-loop system and satisfy... Performance indicators Based on this, the fault estimator gain matrix is ​​obtained as follows: , ;in, Represents singular perturbation parameters The allowed upper bound; based on the gain matrix and This enables online estimation of actuator fault inputs and provides fault compensation information for the subsequent construction of fault-tolerant control laws.

4. The Hamiltonian-constrained singular perturbation output distribution fuzzy fault-tolerant control method according to claim 1, characterized in that, S3 specifically includes the following steps: S31. Construct the augmented tracking error of the weight domain of the output probability density function, wherein the augmented tracking error is expressed as: ,in, express Augmented weight error at time step This represents the preset weight selection matrix, used to extract the target control component from the weight vector. This represents the weight vector corresponding to the square root form of the output probability density function. Indicates the reference trajectory. This represents a matrix that maps system states to weight vectors. Represents the system state vector; in the singular perturbation matrix Under reversible conditions, the system is equivalently represented as ,in, Represents the equivalent state matrix. Represents the equivalent input matrix. Represents the equivalent perturbation matrix. This represents the system matrix after fuzzy fusion. This represents the input matrix after fuzzy fusion. This represents the original system disturbance matrix. This represents the set of normalized weights for fuzzy rules. Indicates control input, Indicates external disturbance; S32. Based on the augmented tracking error, dynamically organize the error into the input affine form. ,in, This represents the set of normalized weights for fuzzy rules. This represents the rate of change of the augmented tracking error. Represents dynamic variables within the system. This represents a composite error term, which includes at least the nominal state term. Disturbance terms Reference trajectory change term and the B-spline approximation error term , Represents the reference trajectory rate of change, This represents the bounded error in the square root approximation of the output probability density function. This indicates the value of the system output variable. It represents the equivalent input gain, used to characterize the relationship between fuzzy fusion uncertainty and control input; S33. Construct a fuzzy fault-tolerant controller, wherein the control input is represented as... ,in, Indicates the number of fuzzy rules in the controller. Indicates the first Normalized weights of the control rules Indicates the first The activation strength of the control rule, Indicates the number of antecedent variables of the controller. Represents the augmented error state vector The One portion, Indicates the first Rule number 1 Gaussian membership functions of the antecedent variables Indicates the center of the membership function. This represents the width parameter of the membership function. This represents the augmented error state vector. This represents the state estimation error. This indicates the fault estimation error. Represents the fault estimation vector The One portion, Indicates the first The control weight vector that acts on the augmented error state in the rule. Indicates the first The compensation weights applied to the fault estimate in the rule; S34, using the controller parameter set To optimize the object, construct performance metrics. ,in, This indicates the performance metrics corresponding to the controller parameters. The term represents the operating cost function, and the operating cost function includes at least an output probability density function deviation term and a control energy term. [a,b] represents the output variable. The range of values, Indicates control input The output probability density function under the action, Represents the terminal cost function. Represent the terminal time; further construct the Hamiltonian function. ,in, Represents Hamiltonian functions, Indicates the accompanying variable, The right-hand side of the system state equation is represented by the gradient of the controller parameters. Calculate, and through Update, in which, Represents performance metrics with respect to the set of parameters gradient, Indicates the number of iterations. Indicates the first Controller parameters at the next iteration This represents the learning rate.

5. The Hamiltonian-constrained singular perturbation output distribution fuzzy fault-tolerant control method according to claim 1, characterized in that, S4 specifically includes the following steps: S41. Construct a Lyapunov function for the augmented error system corresponding to the fuzzy fault estimator. ,in, Let Lyapunov function be defined with respect to the augmented error state vector. This represents the augmented error state vector. This represents the state estimation error. This indicates the fault estimation error. Represents the augmented singular perturbation matrix. Represents the singular perturbation matrix. express 3D identity matrix This indicates the dimension of the actuator fault input. Let represent the positive definite matrix to be found, and satisfy . To ensure The augmented error state vector is positive definite; S42, with As a performance output, As an augmented perturbation input, design Make matrix inequalities Established, among which, Represents the performance output vector. Represents the performance output matrix. Indicates external disturbance and failure rate The resulting augmented perturbation vector This represents the H∞ disturbance suppression index. This represents the set of normalized weights for fuzzy rules. , This represents the augmented error system matrix after fuzzy fusion. This represents the augmented perturbation matrix after fuzzy fusion. Represents the identity matrix that matches the corresponding block dimension, denoted by [symbol]. If the condition is met by a symmetric term, then the following condition is satisfied. This ensures that the fault estimation error system meets the H∞ performance constraint.

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