Dynamic event-triggered fault-tolerant optimal control for flexible arms considering time-varying failures

Abstract

The present application relates to the field of optimal fault-tolerant control of flexible manipulator system, and particularly relates to a dynamic event-triggered optimal fault-tolerant control of flexible manipulator considering time-varying fault. The steps are as follows: firstly, an adaptive observer is designed to estimate the time-varying fault of actuator; secondly, an evaluation neural network considering unknown terms is constructed by using the approximation ability of neural network to approximate the system cost function; then, a Hamilton-Jacobi-Bellman equation is solved to design an optimal fault-tolerant controller of adaptive dynamic programming method, and an approximate optimal control input for ideal trajectory tracking and vibration suppression is realized; at the same time, a dynamic event-triggered mechanism is adopted to determine the triggering time by real-time monitoring of system state and dynamic threshold when the triggering condition is met, so as to reduce the communication bandwidth occupation and computing resource consumption. The present application can ensure the control effect while having the characteristics of fast response and low resource consumption, thereby ensuring the safe and reliable completion of trajectory tracking and vibration suppression tasks of the flexible manipulator.

Description

Technical Field

[0001] This invention relates to the field of flexible robotic arm control technology, specifically a dynamic event-triggered fault-tolerant optimal control for flexible arms that takes into account time-varying actuator failures. Background Technology

[0002] Flexible robotic arms are typically manufactured using lightweight composite materials or thin-walled structures, offering advantages such as low energy consumption, high speed, large workspace, and a good load-to-weight ratio. They are widely used in extravehicular activities on space stations, precision assembly, medical robotics, and flexible manufacturing. However, flexible robotic arms generate significant elastic vibrations and distributed parameter effects during movement, resulting in dynamic models exhibiting strong nonlinearity, high-order coupling, and infinite-dimensional characteristics. Traditional rigid robotic arm control methods (such as robust control and computational torque methods) are not directly applicable. Achieving both trajectory tracking and vibration suppression simultaneously is necessary, placing extremely high demands on the speed, robustness, and resource efficiency of the control system. Therefore, designing effective control methods for flexible robotic arm systems is crucial.

[0003] Currently, control methods for flexible robotic arms mainly include sliding mode control, adaptive control, and robust control. These methods have achieved certain results in trajectory tracking and vibration suppression, but most do not simultaneously consider issues such as time-varying actuator faults and limited communication and computing resources. Adaptive dynamic programming, as an effective means to solve the optimal control problem of nonlinear systems, can avoid the "curse of dimensionality" by approximating the system cost function with neural networks, showing great potential in the control of flexible robotic arms. Existing research on flexible robotic arm control based on adaptive dynamic programming mostly does not consider actuator faults, especially complex time-varying faults. In practical engineering, actuators often experience unpredictable time-varying faults due to temperature changes, wear, or intermittent dynamic disturbances, leading to control input deviations or even system instability.

[0004] Static event triggering mechanisms, using fixed thresholds, are easy to implement but struggle to balance control performance and triggering frequency. Dynamic event triggering control, on the other hand, replaces the periodic sampling of traditional time-triggered systems by setting trigger conditions. It updates the control signal only when the system state deviates from the expected threshold, thus saving communication bandwidth and computational resources. Dynamic event triggering introduces a dynamic variable that evolves with the system state or error, allowing the trigger conditions to adaptively adjust according to real-time operating conditions. However, event-triggered fault-tolerant control methods for flexible robotic arm systems that simultaneously consider time-varying actuator fault tolerance control and adaptive dynamic programming still require further in-depth research.

[0005] Therefore, for flexible robotic arms with time-varying actuator failures, researching optimal fault-tolerant control strategies based on dynamic event triggering mechanisms and adaptive dynamic programming methods has significant theoretical and practical application value. Summary of the Invention

[0006] This invention aims to address the shortcomings of existing flexible robotic arm control methods in terms of actuator failure (especially time-varying failures) and resource consumption. To this end, it proposes a flexible arm dynamic event-triggered fault-tolerant optimal control that takes into account time-varying failures.

[0007] The present invention achieves its objective through the following technical means:

[0008] a. A dynamic model of the flexible robotic arm and a state-space model considering time-varying actuator faults were established based on the hypothetical modal method;

[0009] b. Design an adaptive fault observer to estimate the system state and time-varying faults, and determine the observer design parameters and update law;

[0010] c. Based on the adaptive dynamic programming method, the cost function of the system is defined, and the optimal cost function is approximated by an evaluation neural network composed of radial basis function neural networks.

[0011] d. Based on the approximated optimal cost function, derive the approximate optimal fault-tolerant control law;

[0012] e. Design a dynamic event triggering mechanism. By monitoring the system status and dynamic thresholds in real time, determine the triggering time when the triggering conditions are met, thereby reducing communication bandwidth usage and computing resource consumption.

[0013] Furthermore, in step a, the dynamic model of the flexible robotic arm is established using the Lagrange equation method, and the expression is:

[0014]

[0015] in, Representing a rotating coordinate system The elastic offset within, It is the rotation angle of the servo motor. It is the moment of inertia of the motor. It refers to the bending stiffness of the flexible robotic arm. It is the input torque. Represents the length of the flexible cantilever beam. This indicates the density of the flexible robotic arm. This represents the elastic vibration of the system. for right The first-order partial derivative, for right The second-order partial derivative, for right The third partial derivative of .

[0016] The elastic vibration of the flexible robotic arm is analyzed using the hypothetical modal method. Discretize into,

[0017]

[0018] in For the first A spatially related mode function, For the first A time-related function, This indicates the number of modes. Furthermore, This can be expressed as,

[0019]

[0020] in Represents the vibration frequency. for The second derivative with respect to time.

[0021] For mode function It can be represented as,

[0022]

[0023] Among them are , ,and It is the smallest positive solution to the following equation;

[0024]

[0025] definition Using the Lagrange equation method, the dynamic model of the flexible robotic arm system can be obtained:

[0026]

[0027] in, , The inertia matrix represents positive definite symmetric inertia. Representing the stiffness matrix of the system:

[0028]

[0029]

[0030] in , , , .

[0031] definition Then the dynamic model of the system can be expressed as:

[0032]

[0033] in, , .

[0034] Considering the possibility of a time-varying actuator failure in the robotic arm system, then , , It is the input signal of the actuator. It takes into account the actual input signal after a time-varying actuator failure. It is the time-varying multiplicative fault function of the actuator. It is an additive fault. ,in For ease of calculation, the definition is... Therefore, the dynamic model of the system containing time-varying actuator faults is: .

[0035] Secondly, in step b, to compensate for the fault function, an adaptive fault observer is designed, denoted as follows:

[0036]

[0037] in System status Estimated value It is a time-varying actuator malfunction. The estimated value, Here is the gain matrix of the positive definite observer; the observer parameter update law is... ,in It is the state observation error. These are the parameters to be designed.

[0038] Next, in step c, the cost function of the system is defined as follows:

[0039]

[0040] in and Let be the system error vector and the derivative of the error vector with respect to time, respectively. and These are the expected state and its derivative, respectively. It is a positive constant to be designed. Represents the effect function. and It is a positive definite matrix. The Hamilton-Jacobi-Bellman equation is defined as follows:

[0041]

[0042] in for about The partial derivatives and The optimal cost function of the system is defined as follows:

[0043]

[0044] in It is the optimal cost function. It is the allowable set of control strategies. Applying the Bellman optimality principle to the dynamic programming framework, the optimal performance index function is... This can be obtained by solving the following equation:

[0045]

[0046] in ,if If it is continuously differentiable, then the optimal control scheme is:

[0047]

[0048] The optimal performance index function is usually obtained by solving the Hamilton-Jacobi-Bellman equation. However, since this equation is a nonlinear partial differential equation, it is difficult to obtain an analytical solution. Therefore, a radial basis function neural network is used to approximate the cost function, and its form is as follows:

[0049]

[0050] in It is an unknown ideal weight vector. It is the activation function vector. It is an approximation error. This indicates the number of neurons in the hidden layer. Since the ideal weight vector is unknown, we use... If the ideal weight vector is estimated, then the cost function can be expressed as follows:

[0051]

[0052] but The Hamiltonian equation can be further expressed as,

[0053]

[0054] According to the gradient descent method, for the objective function By taking the partial derivative, we can obtain the weight update rate for evaluating the neural network. ,in It is the learning law for evaluating neural networks.

[0055] In step d, using the evaluation neural network obtained in step c, an approximate optimal fault-tolerant control law can be derived, the expression of which is:

[0056]

[0057] In step e, a dynamic event triggering mechanism is designed. By monitoring the system status and dynamic thresholds in real time, the triggering time is determined when the triggering conditions are met, thereby realizing the discrete updating of the control signal.

[0058] Define the order in which events are triggered and executed. , and .when hour, This represents the error in triggering the event. It is defined as follows: The expression for the event triggering error equation can be derived as follows:

[0059]

[0060] in Indicates in Error in time.

[0061] The rules for dynamic event-triggered updates are as follows:

[0062]

[0063] in , yes The smallest eigenvalue, It is used to evaluate the estimated weights of the neural network. It is used to evaluate the activation function of a neural network. The upper realm, It is a nonlinear function of the system The upper realm, , These are the parameters to be designed. satisfy And it is a positive number. , All are normal numbers.

[0064] The beneficial effects of the present invention are as follows:

[0065] To address time-varying actuator faults in flexible robotic arms, an adaptive fault observation mechanism was developed. This mechanism uses an online-adjustable identification algorithm to extract time-varying fault characteristics in real time and perform dynamic compensation. Compared to traditional fault-tolerant methods that are only applicable to constant faults, this scheme can cover the more common time-varying degradation processes in engineering, preventing the decline in trajectory accuracy and amplification of flexible link vibration caused by fault evolution from the source. This ensures that the robotic arm can still accurately follow the desired trajectory and effectively attenuate elastic vibrations even when the fault continues to change, thus balancing operational quality and structural safety.

[0066] This invention integrates adaptive dynamic programming and a dynamic event triggering mechanism into a flexible robotic arm system, constructing a control architecture that balances optimality and resource efficiency. On one hand, an evaluation network approximates the optimal cost function and embeds fault estimation information, enabling the system to maximize energy savings and compensate for fault effects while tracking the desired trajectory and attenuating elastic vibrations, thus overcoming the shortcomings of conventional control that neglect energy optimization. On the other hand, the dynamic event triggering strategy introduces internal dynamic variables to adaptively adjust the trigger threshold, refreshing control commands only when predetermined conditions are met, thereby significantly reducing communication frequency and processor load. This method effectively reduces the number of triggers and computational overhead while ensuring control quality, improving system operating efficiency, and is particularly suitable for operating conditions with limited communication bandwidth and hardware resources. Attached Figure Description

[0067] Other features, objects, and advantages of this application will become more apparent from the following detailed description of non-limiting embodiments with reference to the accompanying drawings:

[0068] Figure 1 This is a flowchart illustrating the dynamic event-triggered fault-tolerant optimal control of a flexible arm that considers time-varying faults according to the present invention.

[0069] Figure 2 This is a desired trajectory tracking curve of a flexible arm dynamic event-triggered fault-tolerant optimal control considering time-varying faults according to the present invention.

[0070] Figure 3 This is a vibration curve diagram of a flexible arm dynamic event-triggered fault-tolerant optimal control considering time-varying faults according to the present invention.

[0071] Figure 4 This is a tracking error curve diagram of a flexible arm dynamic event-triggered fault-tolerant optimal control considering time-varying faults according to the present invention.

[0072] Figure 5 This is a control input curve diagram of a flexible arm dynamic event-triggered fault-tolerant optimal control considering time-varying faults according to the present invention;

[0073] Figure 6This is a fault estimation curve diagram of a flexible arm dynamic event-triggered fault-tolerant optimal control considering time-varying faults according to the present invention.

[0074] Figure 7 This invention relates to an internal execution time interval diagram of a dynamic event triggering mechanism for a flexible arm with dynamic event triggering fault-tolerant optimal control that considers time-varying faults. Detailed Implementation

[0075] The present application will now be described in further detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are for illustrative purposes only and are not intended to limit the invention. Furthermore, it should be noted that, for ease of description, only the parts relevant to the invention are shown in the accompanying drawings.

[0076] This invention provides a dynamic event-triggered fault-tolerant optimal control for a flexible arm that considers time-varying faults, specifically implemented through the following steps:

[0077] a. A dynamic model of the flexible robotic arm and a state-space model considering time-varying actuator faults were established based on the hypothetical modal method;

[0078] b. Design an adaptive fault observer to estimate the system state and time-varying faults, and determine the observer design parameters and update law;

[0079] c. Based on the adaptive dynamic programming method, the cost function of the system is defined, and the optimal cost function is approximated by an evaluation neural network composed of radial basis function neural networks.

[0080] d. Based on the approximated optimal cost function, derive the approximate optimal fault-tolerant control law;

[0081] e. Design a dynamic event triggering mechanism. By monitoring the system status and dynamic thresholds in real time, determine the triggering time when the triggering conditions are met, thereby reducing communication bandwidth usage and computing resource consumption.

[0082] Step a: Establish the dynamic model of the flexible robotic arm using the Lagrange equation method. The expression is:

[0083]

[0084] in, Representing a rotating coordinate system The elastic offset within, It is the rotation angle of the servo motor. It is the moment of inertia of the motor. It refers to the bending stiffness of the flexible robotic arm. It is the input torque. Represents the length of the flexible cantilever beam. This indicates the density of the flexible robotic arm. This represents the elastic vibration of the system. for right The first-order partial derivative, for right The second-order partial derivative, for right The third partial derivative of .

[0085] The elastic vibration of the flexible robotic arm is analyzed using the hypothetical modal method. Discretize into,

[0086]

[0087] in For the first A spatially related mode function, For the first A time-related function, This indicates the number of modes. Furthermore, This can be expressed as,

[0088]

[0089] in Represents the vibration frequency. for The second derivative with respect to time.

[0090] For mode function It can be represented as,

[0091]

[0092] Among them are , ,and It is the smallest positive solution to the following equation;

[0093]

[0094] definition Using the Lagrange equation method, the dynamic model of the flexible robotic arm system can be obtained:

[0095]

[0096] in, , The inertia matrix represents positive definite symmetric inertia. Representing the stiffness matrix of the system:

[0097]

[0098]

[0099] in , , , .

[0100] definition Then the dynamic model of the system can be expressed as:

[0101]

[0102] in, , .

[0103] Considering the possibility of a time-varying actuator failure in the robotic arm system, then , , It is the input signal of the actuator. It takes into account the actual input signal after a time-varying actuator failure. It is the time-varying multiplicative fault function of the actuator. It is an additive fault. ,in For ease of calculation, the definition is... Therefore, the dynamic model of the system containing time-varying actuator faults is: .

[0104] Step b: To compensate for the fault function, an adaptive fault observer is designed, denoted as follows:

[0105]

[0106] in This is the system state estimate. It is a time-varying actuator malfunction. The estimated value, Here is the gain matrix of the positive definite observer; the observer parameter update law is... ,in It is the state observation error. These are the parameters to be designed.

[0107] In step c, the cost function of the system is defined as follows:

[0108]

[0109] in and These are, in order, the system's error vector and the derivative of the error vector with respect to time. and These are the expected state and its derivative, respectively. It is a positive constant to be designed. Represents the effect function. and It is a positive definite matrix. The Hamilton-Jacobi-Bellman equation is defined as follows:

[0110]

[0111] in for about The partial derivatives and The optimal cost function of the system is defined as follows:

[0112]

[0113] in It is the optimal cost function. It is the allowable set of control strategies. Applying the Bellman optimality principle to the dynamic programming framework, the optimal performance index function is... This can be obtained by solving the following equation:

[0114]

[0115] in ,if If it is continuously differentiable, the optimal control law will be obtained:

[0116]

[0117] The optimal performance index function is usually obtained by solving the Hamilton-Jacobi-Bellman equation. However, since this equation is a nonlinear partial differential equation, it is difficult to obtain an analytical solution. Therefore, a radial basis function neural network is used to approximate the cost function, and its form is as follows:

[0118]

[0119] in It is an unknown ideal weight vector. It is the activation function vector. It is an approximation error. This indicates the number of neurons in the hidden layer. Since the ideal weight vector is unknown, we use... If the ideal weight vector is estimated, then the cost function can be expressed as follows:

[0120]

[0121] but The Hamiltonian equation can be further expressed as,

[0122]

[0123] According to the gradient descent method, for the objective function By taking the partial derivative, we can obtain the weight update rate for evaluating the neural network. ,in It is the learning law for evaluating neural networks.

[0124] Step d: Using the evaluation neural network obtained in step c, the approximate optimal fault-tolerant control law can be derived, with the following expression:

[0125]

[0126] Step e: Design a dynamic event triggering mechanism. By monitoring the system status and dynamic thresholds in real time, determine the triggering time when the triggering conditions are met, and realize the discrete update of the control signal.

[0127] Define the order in which events are triggered and executed. , and .when hour, This represents the error in triggering the event. It is defined as follows: The expression for the event triggering error equation can be derived as follows:

[0128]

[0129] in Indicates in Error in time.

[0130] The rules for dynamic event-triggered updates are as follows:

[0131]

[0132] in , yes The smallest eigenvalue, It is used to evaluate the estimated weights of the neural network. It is used to evaluate the activation function of a neural network. The upper realm, It is a nonlinear function of the system The upper realm, , These are the parameters to be designed. satisfy And it is a positive number. , All are normal numbers.

[0133] To demonstrate that the controller proposed in this invention can stably and effectively control a flexible robotic arm with time-varying actuator failure, this invention employs the Lyapunov direct method, selects a suitable Lyapunov function, and performs system stability analysis, proving that the system error is eventually uniformly bounded. The specific process of stability analysis is as follows:

[0134] For the observer error in step b, the Lyapunov candidate function is selected as follows.

[0135]

[0136] Taking its derivative with respect to time, we obtain the observer's update rate. Substituting this into the equation, we get...

[0137]

[0138] in express The smallest eigenvalue. In the set In addition, it can be guaranteed that the observation error of the fault is eventually uniformly bounded.

[0139] For the evaluation neural network's weight approximation error in step c... The Lyapunov candidate functions are selected as follows:

[0140]

[0141] Taking its derivative with respect to time and substituting the weight update rate, we can obtain...

[0142]

[0143] in In the set Otherwise, Therefore, it can be concluded that the approximation error of the neural network weights is eventually uniformly bounded.

[0144] For the systematic error in step d, the Lyapunov candidate function is selected as follows.

[0145]

[0146] Considering the obtained approximate optimal fault-tolerant control law, the derivative of this candidate function with respect to time is:

[0147]

[0148] Based on the assumption of reasonableness, we have:

[0149]

[0150] in .

[0151] Therefore, we can conclude that in the set outside, And the following conditions must be met:

[0152]

[0153] Therefore, it can be concluded that the trajectory tracking error of the system is eventually uniformly bounded.

[0154] The controller will remain inactive until the event trigger error reaches the specified error threshold. The control actions for the state remain unchanged. Its expression is as follows:

[0155]

[0156] By introducing an event triggering mechanism The system expression can be rewritten as follows:

[0157]

[0158] With the introduction of control input, the control operation is updated only upon triggering an event. Therefore, the event-triggered Hamilton-Jacobi-Bellman equation is restated as follows:

[0159]

[0160] Since directly solving the Hamilton-Jacobi-Bellman equation is quite complex, neural networks are used to approximate its solution. Therefore, an evaluation neural network is used to evaluate the optimal cost function.

[0161]

[0162] For the weights of the neural network, For activation function, This represents the approximation error. Its differential form is:

[0163]

[0164] When neural networks are introduced, the Hamiltonian equation can be described as:

[0165]

[0166] in . This represents the residual generated by the neural network approximation. To obtain the approximate Hamiltonian equation, we first derive an estimate of the cost function.

[0167]

[0168]

[0169] Secondly, the approximate Hamiltonian equation can be described as:

[0170]

[0171] To train the evaluation neural network, a function is defined. , so that the objective function Minimize. Then, using the gradient descent algorithm, the weight update pattern is estimated as follows:

[0172]

[0173] because The weighting error can be further described as follows:

[0174]

[0175] The rules for dynamic event-triggered updates are as follows:

[0176]

[0177] For the proof in step e, the Lyapunov candidate function is selected as follows:

[0178]

[0179] in

[0180] Event triggering mechanisms are generally divided into two cases: non-triggered events and triggered events.

[0181] Scenario 1: When The event will not be triggered, and the state remains unchanged. .but

[0182]

[0183] in Based on the assumption of rationality, we can conclude that...

[0184]

[0185] in , .

[0186]

[0187] According to Young's inequality:

[0188]

[0189] in Therefore, it can be deduced that... The expression:

[0190]

[0191] When the weight update rule satisfies the requirements described in the above formula, and the weight error can be resolved as described in the above formula, it can be proven that... The Lyapunov stability criterion is satisfied, thus proving the stability of case 1.

[0192] Scenario 2: When Triggering event

[0193]

[0194] in ,

[0195] ,

[0196] From case 1, we can obtain continuous... and Make .for ,have , represent Function-like.

[0197] Therefore, the Lyapunov function The system decreases in both cases 1 and 2, proving its stability. The Lyapunov candidate function is selected as follows:

[0198]

[0199] Differentiating with respect to time, we have:

[0200]

[0201] If the weight inequality satisfies , can get .therefore and It asymptotically converges to zero. Furthermore, because... ,therefore It is also convergent.

[0202] Minimum interval for dynamic event triggering:

[0203]

[0204] in

[0205] for Its derivative satisfies:

[0206]

[0207] This relationship is involved. , It is a very small positive number. We can obtain... .therefore According to the dynamic event triggering mechanism, there are ,when from Change to , can get Therefore, its minimum dynamic event trigger interval is always greater than Therefore, Zeno's behavior can be effectively avoided.

[0208] This concludes the proof, demonstrating that the dynamic event-triggered fault-tolerant optimal control for flexible arms that considers time-varying faults proposed in this invention is stable and reliable.

[0209] The invention will now be further described and illustrated through a specific implementation example. In this example, the parameters for the control design are given as follows: , , , and Set the desired trajectory for the trajectory tracking task. Initial state of the flexible robotic arm system The value is 0, the activation function is chosen as the Gaussian function and Actuator time-varying fault .

[0210] In this section, for the case of time-varying actuator faults, the simulation results are as follows: Figure 2-7 As shown. Figure 2 The trajectory tracking performance of the proposed control strategy is demonstrated, showing that the proposed control scheme can still smoothly track the desired trajectory even in the event of a fault. From Figure 3 It can be clearly seen that the vibration energy of the proposed control scheme is effectively suppressed to near 0. Figure 4 The trajectory tracking error of the proposed control scheme is described, and the error of the proposed control scheme is relatively small. Figure 5 The simulation control input was plotted, and it can be seen that the control input amplitude of the proposed control scheme is small, which meets the requirement of low energy consumption in optimal control. Furthermore, due to the addition of a dynamic event triggering mechanism, the control input presents a square wave form. Figure 6 The curves showing the actual fault and the observed fault are presented. The fault occurs at 5 seconds. Despite the extreme complexity of the fault function, the observer is still able to effectively estimate the fault value. From Figure 7 As can be seen, the minimum internal execution time and trigger interval of the simulation are different and non-zero, thus avoiding the occurrence of Zeno's behavior.

[0211] This invention addresses the control requirements of flexible robotic arms under time-varying actuator failure conditions. It organically integrates adaptive dynamic programming with a dynamic event-triggered fault-tolerant mechanism to construct an optimal control strategy. Compared to existing methods, the beneficial effects of this technical solution are as follows:

[0212] This invention is the first to employ a dynamic event triggering mechanism in a flexible robotic arm system. This mechanism monitors the rate of change of the system's state norm and dynamically adjusts the trigger threshold, maximizing the control signal update time interval while ensuring system stability and tracking accuracy. This provides a more convenient and economical solution for practical applications. Compared to traditional adaptive dynamic programming, this invention constructs a novel cost function that integrates trajectory tracking error, link vibration state, control energy consumption, and fault estimation. By solving the Hamilton-Jacobi-Bellman equations, an approximately optimal control law that dynamically balances tracking accuracy, vibration suppression, and fault compensation is obtained, thus solving the multi-objective coordination optimization problem unique to flexible systems. This scheme allows the flexible robotic arm to simultaneously consider multiple performance indicators during operation, effectively improving the overall system performance.

[0213] Unlike existing fault-tolerant control strategies that only consider common faults, the controller constructed in this invention ensures that the flexible robotic arm can still track the desired trajectory and suppress vibrations even in the event of time-varying actuator failures. Even under more complex and severe faults and uncertainties, the system can still maintain good overall performance, thereby significantly improving the reliability and operational stability of the robotic arm in complex working environments and demonstrating important engineering practical value.

Claims

1. A flexible arm dynamic event-triggered fault-tolerant optimal control considering time-varying failures, characterized in that, Includes the following steps, a. A dynamic model of the flexible robotic arm and a state-space model considering time-varying actuator faults were established based on the hypothetical modal method. in, Representing a rotating coordinate system The elastic offset within, It is the rotation angle of the servo motor. It is the moment of inertia of the motor. It refers to the bending stiffness of the flexible robotic arm. It is the input torque. Represents the length of the flexible cantilever beam. This indicates the density of the flexible robotic arm. This represents the elastic vibration of the system. for right The first-order partial derivative, for right The second-order partial derivative, for right The third partial derivative, Using the assumed modes method to model the elastic vibrations of a flexible robot arm is discretized into, in For the first A spatially related mode function, For the first A time-related function, Indicates the number of modes, in addition, This can be expressed as, in Represents the vibration frequency. for The second derivative with respect to time, For mode function It can be represented as, Among them are , ,and It is the smallest positive solution to the following equation. definition By using the Lagrange equation method, a dynamic model of the flexible robotic arm system can be obtained. in, , The inertia matrix represents positive definite symmetric inertia. Represents the stiffness matrix of the system. in , , , , definition Then the dynamic model of the system can be expressed as, in, , , Considering the possibility of a time-varying actuator failure in the robotic arm system, then , , It is the input signal of the actuator. It takes into account the actual input signal after a time-varying actuator failure. It is the time-varying multiplicative fault function of the actuator. It is an additive fault. ,in For ease of calculation, the definition is as follows Therefore, the dynamic model of the system containing time-varying actuator faults is: ; b. Design an adaptive fault observer to estimate the system state and time-varying faults, and determine the observer design parameters and update law. in System status Estimated value It is a time-varying actuator malfunction. The estimated value, Here is the gain matrix of the positive definite observer; the observer parameter update law is... ,in It is the state observation error. These are the parameters to be designed; c. Based on the adaptive dynamic programming method, the cost function of the system is defined, and the optimal cost function is approximated by an evaluation neural network composed of radial basis function neural networks. d. Based on the approximated optimal cost function, derive the approximate optimal fault-tolerant control law; e. Design a dynamic event triggering mechanism. By monitoring the system status and dynamic thresholds in real time, determine the triggering time when the triggering conditions are met, thereby reducing communication bandwidth usage and computing resource consumption.

2. The flexible arm dynamic event-triggered fault-tolerant optimal control considering time-varying faults according to claim 1, characterized in that, In step c, the system cost function is defined as follows: in and These are, in order, the system's error vector and the derivative of the error vector with respect to time. and These are the expected state and its derivative, respectively. It is a positive constant to be designed. Represents the effect function. and It is a positive definite matrix, and the optimal cost function is approximated using a radial basis function neural network, in the form of: ,in These are weight estimates. It is the activation function matrix related to the error.

3. The flexible arm dynamic event-triggered fault-tolerant optimal control considering time-varying faults according to claim 1, characterized in that, In step d, based on the Hamilton-Jacobi-Bellman equations, we have: Combining step c, the approximate optimal control law can be derived as follows: in This indicates taking the partial derivative with respect to the error.

4. The flexible arm dynamic event-triggered fault-tolerant optimal control considering time-varying faults according to claim 1, characterized in that, In step e, a dynamic event triggering mechanism is designed. By monitoring the system status and dynamic thresholds in real time, the triggering time is determined when the triggering conditions are met. The dynamic event triggering update rule is as follows: in , yes The smallest eigenvalue, It is used to evaluate the estimated weights of the neural network. It is used to evaluate the activation function of a neural network. The upper realm, It is a nonlinear function of the system The upper realm, , These are the parameters to be designed. satisfy And it is a positive number. , All are normal numbers.

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