Adaptive predefined time tracking control method for Euler-Lagrange system

Abstract

The invention relates to the technical field of automatic control, and discloses a self-adaptive predefined time tracking control method for an Euler-Lagrange system, which comprises the following steps: establishing a kinetic model of the Euler-Lagrange system of a double-joint mechanical arm, and constructing a reinforcement learning framework comprising an evaluation neural network and an execution neural network; calculating a position error according to a joint position in the kinetic model and a given reference signal; generating a predefined time virtual control law containing predefined convergence time parameters, and calculating a speed error in combination with the joint speed; generating a practical predefined time control law containing predefined convergence time parameters based on the speed error, the position error and the control compensation amount output by the execution neural network; and constructing a weight updating law, applying a practical predefined time control law to the dynamical model, and updating the weight according to the weight updating law. By introducing explicit predefined convergence time parameters, an error convergence time upper limit can be set according to task requirements.

Description

Technical Field

[0001] This invention relates to the field of automation control technology, specifically to an adaptive predefined time tracking control method for Euler-Lagrange systems. Background Technology

[0002] Euler-Lagrange systems, as a fundamental modeling framework for describing the dynamic behavior of conserved mechanical energy systems such as robotic arms, mobile robots, and aerospace structures, often face model uncertainties (such as changes in inertia matrix parameters) and external disturbances (such as friction and load fluctuations) in practical engineering applications. With the increasing emphasis on strict time constraints in critical scenarios such as precision assembly and emergency response, traditional proportional-integral-derivative (PID) control and asymptotically convergent adaptive sliding mode control are no longer sufficient. To address finite-time control problems, the fixed-time stability (FxTS) method has been proposed, ensuring system convergence within a finite time independent of the initial state. Furthermore, with the development of intelligent control, adaptive methods based on radial basis function neural networks (RBFNN) and reinforcement learning, such as the evaluation neural network (Critic network)-execution neural network (Actor network) framework, have become important pathways for handling unknown nonlinearities and fault tolerance in systems.

[0003] However, existing control schemes still have significant shortcomings when handling time-critical tasks. First, the convergence time of existing fixed-time stability methods is usually indirectly determined by system architecture parameters. Designers cannot directly and explicitly set specific convergence times, resulting in a lack of time flexibility and difficulty in adapting to different scenarios such as precision assembly requiring rapid response and surface polishing requiring slow and high precision. Second, although traditional reinforcement learning control methods can improve the adaptability to uncertainties in Eulerian-Lagrange system models through data-driven approaches, they generally lack predefined time convergence guarantees, making it difficult to balance system adaptability with the time predictability of trajectory tracking. In addition, considering that Eulerian-Lagrange systems inevitably face model uncertainties and external disturbances in actual operation, some existing predefined time control methods, although able to achieve time-constrained objectives, lack robustness under strong disturbances, and control accuracy is easily affected. Summary of the Invention

[0004] In view of the above-mentioned problems, the present invention is proposed.

[0005] To solve the above-mentioned technical problems, the present invention provides the following technical solution: an adaptive predefined time tracking control method for Euler-Lagrange systems, comprising:

[0006] A dynamic model of the Euler-Lagrange system of a dual-joint robotic arm is established, and a reinforcement learning framework including an evaluation neural network and an execution neural network is constructed.

[0007] Obtain the joint positions and joint velocities in the dynamic model, calculate the position error based on the joint positions and a given reference signal; generate a predefined time virtual control law containing a predefined convergence time parameter based on the position error, and calculate the velocity error by combining the joint velocities with the predefined time virtual control law;

[0008] The position error and the velocity error are input into the execution neural network. Based on the velocity error, the position error, and the control compensation quantity output by the execution neural network, a practical predefined time control law containing a predefined convergence time parameter is generated.

[0009] Weight update laws with nonlinear correction terms are constructed for the evaluation neural network and the execution neural network respectively. The practical predefined time control law is applied to the dynamic model, and the weights of the evaluation neural network and the execution neural network are updated according to the weight update law, so that the system trajectory error converges to the residual set within a predefined time, thereby realizing the tracking control of the given reference signal.

[0010] As a preferred embodiment of the adaptive predefined time tracking control method for the Euler-Lagrange system described in this invention, the establishment of the dynamic model of the Euler-Lagrange system of the dual-joint robotic arm includes establishing dynamic equations describing the dynamic behavior of the system based on the principles of Lagrange mechanics; the dynamic equations include a positive definite inertia matrix, a Coriolis-centrifugal force matrix, a gravity vector, control input, and external disturbances;

[0011] Joint position and joint velocity are extracted as system state variables from the dynamic equations.

[0012] As a preferred embodiment of the adaptive predefined time tracking control method for Euler-Lagrange systems described in this invention, the generation of a predefined time virtual control law containing a predefined convergence time parameter based on the position error includes: constructing a virtual control intermediate quantity based on the position error; the virtual control intermediate quantity includes a linear feedback term of the position error, a derivative term of the given reference signal, and a position-layer predefined time convergence term; the position-layer predefined time convergence term is composed of a predefined convergence time parameter, a predefined time stability parameter, and a sign function exponentiation term of the position error;

[0013] The position error and the virtual control intermediate quantity are subjected to a vector inner product operation to obtain an inner product value. The inner product value is then subjected to singularity avoidance normalization processing to obtain a predefined time virtual control law. The predefined convergence time parameter is an explicitly adjustable parameter. By adjusting the predefined convergence time parameter, the upper limit of the convergence time of the system trajectory error is set.

[0014] As a preferred embodiment of the adaptive predefined time tracking control method for Euler-Lagrange systems described in this invention, the generation of a practical predefined time control law containing predefined convergence time parameters includes constructing intermediate control quantities based on the velocity error; the intermediate control quantities include a linear feedback term of the velocity error, a coupling term of the position error, the control compensation quantity, and a velocity-layer predefined time convergence term; the velocity-layer predefined time convergence term is composed of the predefined convergence time parameters, predefined time stability parameters, and a sign function exponentiation term of the velocity error;

[0015] The speed error and the control intermediate quantity are subjected to a vector inner product operation to obtain an inner product value. The inner product value is then subjected to singularity avoidance normalization processing to obtain a practical predefined time control law.

[0016] As a preferred embodiment of the adaptive predefined time tracking control method for Euler-Lagrange systems described in this invention, the evaluation neural network includes: defining an instantaneous cost function; the instantaneous cost function is calculated based on a weighted average of the deviation between the system state and the desired trajectory and the energy of the control input;

[0017] Define a long-run cost function; the long-run cost function is the discounted integral of the instantaneous cost function from the current moment to infinity in the time domain;

[0018] Using the position error as input, a radial basis function neural network is used to approximate the long-run cost function and output an estimated value of the long-run cost function.

[0019] A temporal difference error is constructed based on the instantaneous cost function, the estimated long-term cost function, and the time derivative of the estimated long-term cost function; the temporal difference error is used to guide the weight update of the evaluation neural network.

[0020] As a preferred embodiment of the adaptive predefined time tracking control method for the Euler-Lagrange system described in this invention, the execution neural network includes: taking the position error and the velocity error as inputs, using a radial basis function neural network to approximate the unknown nonlinear function in the dynamic model, and outputting a control compensation quantity; the unknown nonlinear function includes the external disturbance, the gravity vector, and related terms of the positive definite inertia matrix and the Coriolis-centrifugal force matrix;

[0021] The learning error is calculated based on the deviation between the ideal weights and the estimated weights, as well as the output of the radial basis function.

[0022] The learning error is fused with the long-term cost function estimate, and a cost error gain coefficient is introduced for weighted calculation to obtain the approximation error; the approximation error is used to guide the weight update of the execution neural network.

[0023] As a preferred embodiment of the adaptive predefined time tracking control method for Euler-Lagrange systems described in this invention, the construction of the weight update law containing nonlinear correction terms includes constructing an evaluation network weight update law based on the evaluation neural network; the evaluation network weight update law includes a gradient descent term, a linear damping term, and a nonlinear correction term.

[0024] An execution network weight update law is constructed based on the aforementioned execution neural network; the execution network weight update law includes a gradient descent term, a linear damping term, and a nonlinear correction term.

[0025] The nonlinear correction term is used to make the weights of the evaluation neural network and the execution neural network converge within the time set by the predefined convergence time parameter.

[0026] An adaptive predefined time tracking control system for an Euler-Lagrange system, wherein:

[0027] The system modeling module establishes a dynamic model of the Euler-Lagrange system of the dual-joint robotic arm and constructs a reinforcement learning framework that includes an evaluation neural network and an execution neural network.

[0028] The virtual control law module acquires the joint positions and joint velocities in the dynamic model, calculates the position error based on the joint positions and a given reference signal, generates a predefined time virtual control law containing a predefined convergence time parameter based on the position error, and calculates the velocity error by combining the joint velocities with the predefined time virtual control law.

[0029] The time control law module inputs the position error and the velocity error into the execution neural network, and generates a practical predefined time control law containing predefined convergence time parameters based on the velocity error, the position error, and the control compensation quantity output by the execution neural network.

[0030] The tracking control module constructs weight update laws with nonlinear correction terms for the evaluation neural network and the execution neural network, respectively. It applies the practical predefined time control law to the dynamic model and updates the weights of the evaluation neural network and the execution neural network according to the weight update law, so that the system trajectory error converges to the residual set within a predefined time, thereby realizing the tracking control of the given reference signal.

[0031] A computer device includes: a memory and a processor; the memory stores a computer program, wherein: when the processor executes the computer program, it implements the steps of the method described in any one of the present invention.

[0032] A computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the steps of the method described in any one of the present invention.

[0033] The beneficial effects of this invention are as follows: By introducing an explicit predefined convergence time parameter into the control law, this invention overcomes the limitation of the unadjustable convergence time of traditional fixed-time control. It allows for flexible setting of the upper limit of the error convergence time according to task requirements, improving applicability in critical scenarios with varying timeframes. By constructing a reinforcement learning framework that includes evaluation and execution neural networks, the invention utilizes the neural network to approximate model uncertainties and external disturbances online and generate control compensation quantities, effectively enhancing robustness to unstructured environments such as friction and load fluctuations, and solving the accuracy limitation problem caused by the reliance on robustness terms in traditional methods. By constructing a weight update law containing nonlinear correction terms, the invention ensures that the learning convergence process of the neural network meets predefined time constraints, avoiding system transient response overshoot due to learning lag, and achieving a balance between adaptive capability and time predictability. The control law based on the backstepping method drives the system error to converge to the residual set within a set time. Simultaneously, by adjusting the time parameter, a dynamic trade-off between response speed and control energy can be achieved, meeting the energy efficiency and safety requirements of practical engineering. Attached Figure Description

[0034] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the following description of the embodiments will be briefly introduced. 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.

[0035] Figure 1 The above is an overall flowchart of an adaptive predefined time tracking control method for an Euler-Lagrange system provided in an embodiment of the present invention.

[0036] Figure 2 The tracking performance curve is provided for an embodiment of the present invention.

[0037] Figure 3 The control input curve is provided for an embodiment of the present invention.

[0038] Figure 4 This is a schematic diagram of the weights and position errors of the Actor and Critic networks when the predefined convergence time parameter is 5s, as provided in this embodiment of the invention.

[0039] Figure 5 This is a schematic diagram of the weights and position errors of the Actor and Critic networks when the predefined convergence time parameter is 10s, as provided in this embodiment of the invention.

[0040] Figure 6This is a schematic diagram of the weights and position errors of the Actor and Critic networks when the predefined convergence time parameter is 15s, as provided in this embodiment of the invention. Detailed Implementation

[0041] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the protection scope of the present invention.

[0042] Example 1, referring to Figure 1 As an embodiment of the present invention, an adaptive predefined time tracking control method for an Euler-Lagrange system is provided, comprising:

[0043] S1: Establish a dynamic model of the Euler-Lagrange system of the dual-joint robotic arm, and construct a reinforcement learning framework that includes an evaluation neural network and an execution neural network.

[0044] As a fundamental modeling framework for describing the dynamic behavior of conserved mechanical energy systems, Eulerian-Lagrange systems often face model uncertainties and external disturbances in practical engineering applications, leading to a decline in control performance. Meanwhile, many critical scenarios (such as precision assembly, emergency response, and aerospace maneuvers) impose strict time constraints on the system, requiring stable tracking or stabilization within a finite time specified by the user. This invention addresses the problems of non-adjustable convergence time, weak uncertainty adaptability, and insufficient robustness in existing technologies by integrating the Practical Predefined Time Stability (PPTS) concept into the Actor-Critic framework and combining it with a Radial Basis Function Neural Network (RBFNN) controller design and a weight update law containing nonlinear correction terms.

[0045] Specifically, the dynamic model of the Euler-Lagrange system for the dual-joint robotic arm includes: establishing dynamic equations describing the dynamic behavior of the system based on the principles of Lagrange mechanics, wherein the dynamic equations of the Euler-Lagrange system are:

[0046] ;

[0047] in, These are the position, velocity, and acceleration of the robotic arm joints, respectively. It is a positive definite inertial matrix; The Coriolis-centrifugal force matrix; It is the gravity vector; For control input (robotic arm joint torque); The external disturbance is continuous and bounded; this embodiment is a control algorithm for a two-joint robotic arm, therefore the dimension is taken as... .

[0048] Furthermore, joint positions are extracted from the aforementioned dynamic equations. and joint velocity As system state variables, they provide the basis for subsequent calculations of position and velocity errors.

[0049] Secondly, constructing a reinforcement learning framework that includes an evaluation neural network and an execution neural network involves the following steps: In the Actor-Critic framework, the evaluation neural network (Critic network) plays a guiding role by estimating the future costs associated with the current state, thereby influencing the control policy updates of the execution neural network (Actor network); the execution neural network relies on temporal difference errors. Adjust control input This new control action alters the system trajectory, generating updated state and cost signals, which are then fed back to the evaluation neural network. The evaluation neural network optimizes the weights based on this new data, and this process iterates cyclically. This closed-loop interaction forms the core of the Actor-Critic learning cycle, enabling the control strategy to continuously adapt and optimize in response to dynamically changing system characteristics.

[0050] It should be noted that the core difference between Practical Predefined Time Stability (PPTS) and Fixed Time Stability (FxTS) lies in the explicit definition of the convergence time. Fixed Time Stability guarantees that the system converges within a uniform finite time independent of the initial conditions, but the time is determined by the system parameters and cannot be directly adjusted. In contrast, Practical Predefined Time Stability allows designers to explicitly and precisely set the convergence time, thus providing greater flexibility and accuracy in control design. The trade-off is that it allows for a smaller residual error region. This characteristic makes Practical Predefined Time Stability highly attractive in engineering applications that have stringent time accuracy requirements and allow for small steady-state errors.

[0051] By establishing a dynamic model of an Eulerian-Lagrange system that includes a positive definite inertia matrix, a Coriolis-centrifugal force matrix, a gravity vector, control input, and external disturbances, and constructing a reinforcement learning framework that includes an evaluation neural network and an execution neural network, a model foundation and framework support are provided for subsequent acquisition of joint position and joint velocity, calculation of position and velocity errors, generation of predefined time virtual control laws and practical predefined time control laws, and construction of a weight update law with nonlinear correction terms.

[0052] S2: Obtain the joint position and joint velocity in the dynamic model, calculate the position error based on the joint position and the given reference signal; generate a predefined time virtual control law containing a predefined convergence time parameter based on the position error, and calculate the velocity error by combining the joint velocity with the predefined time virtual control law.

[0053] After obtaining the state information of the Euler-Lagrange system dynamic model established in S1, it is necessary to calculate the tracking error of the system and design a predefined time virtual control law. In this embodiment, the backstepping design idea is adopted. First, the position error is calculated based on the joint position and the given reference signal. Then, a predefined time virtual control law is generated based on the position error. Finally, the velocity error is calculated by combining the joint velocity and the virtual control law, which lays the foundation for the subsequent design of a practical predefined time control law.

[0054] Specifically, obtaining the joint positions and joint velocities in the dynamic model, and calculating the position error based on the joint positions and a given reference signal, includes the following steps:

[0055] Extracting joint position and joint velocity as system state variables from the dynamic equations established in S1, the dynamic equations of the Euler-Lagrange system are transformed into the following system equations:

[0056] ;

[0057] in, For predefined time-based virtual control laws; Given a reference signal; This refers to the positional error; This represents the speed error.

[0058] Based on the system equations and position errors Definition of position error The time derivative is calculated as follows:

[0059] ;

[0060] in, Position error The time derivative; The speed of the robotic arm joints; The time derivative of a given reference signal; For speed error; This is a predefined time-based virtual control law.

[0061] Specifically, generating a predefined time virtual control law containing a predefined convergence time parameter based on the position error includes: constructing a virtual control intermediate quantity based on the position error; the virtual control intermediate quantity includes a linear feedback term of the position error, a derivative term of the given reference signal, and a position-layer predefined time convergence term; the position-layer predefined time convergence term is composed of a predefined convergence time parameter, a predefined time stability parameter, and a sign function exponentiation term of the position error. A vector inner product operation is performed on the position error and the virtual control intermediate quantity to obtain the inner product value; the inner product value is then subjected to singularity avoidance normalization processing to obtain the predefined time virtual control law; wherein, the predefined convergence time parameter is an explicitly adjustable parameter, and by adjusting the predefined convergence time parameter, an upper limit for the convergence time of the system trajectory error is set.

[0062] In this embodiment, a predefined time virtual control law is used. The specific formula is as follows:

[0063] ;

[0064] in, For predefined time-based virtual control laws; This refers to the positional error; For virtual control intermediate quantities; To avoid singular parameters; It is a constant; Predefined convergence time parameters; For predefined time adjustment parameters; The time derivative of the given reference signal.

[0065] It should be noted that the predefined convergence time parameter It is an explicitly adjustable parameter, which can be adjusted by adjusting the predefined convergence time parameter. It allows setting an upper limit for the convergence time of the system trajectory error. This feature differs from fixed-time stability (FxTS), where the upper limit of convergence time is usually a complex function of the controller parameters, making it difficult to set the time precisely.

[0066] Furthermore, to verify that the designed predefined time virtual control law can achieve the tracking task... Construct the Lyapunov function of the position error subsystem as follows. :

[0067] ;

[0068] The Lyapunov function of the position error subsystem derivative for:

[0069] ;

[0070] in, This refers to the positional error; For speed error; For predefined time-based virtual control laws; The time derivative of the given reference signal.

[0071] Predefined time virtual control law Substitution Item By applying the bounded property of the normalized function, we can obtain:

[0072] ;

[0073] in, The bounded error of the normalization function.

[0074] Therefore, the Lyapunov function of the position error subsystem derivative for:

[0075] ;

[0076] in, It is a constant; For predefined time adjustment parameters.

[0077] Using the inequality relationship between vector norm and power function, for We can obtain:

[0078] ;

[0079] for We can obtain:

[0080] ;

[0081] Furthermore, we can conclude from this:

[0082] ;

[0083] in, Lyapunov function of the position error subsystem The derivative; It is a constant; This refers to the positional error; For speed error; For predefined time adjustment parameters; Predefined convergence time parameters; The bounded error of the normalization function.

[0084] It should be noted that the derivation of the Lyapunov function derivative above shows that, through the design of the predefined time virtual control law, the Lyapunov function derivative of the position error subsystem includes a predefined time convergence term. The structure of this predefined time convergence term corresponds to the conditions of Practical Predefined Time Stability (PPTS), thus ensuring that the position error can remain within the predefined convergence time parameter. Convergence; in derivatives The term is a coupling term between position error and velocity error, which will be handled in subsequent S3 by designing a practical predefined time control law.

[0085] S3: Input the position error and the velocity error into the execution neural network, and generate a practical predefined time control law containing predefined convergence time parameters based on the velocity error, the position error and the control compensation quantity output by the execution neural network.

[0086] Specifically, the position error and the velocity error are input into the execution neural network. Based on the velocity error, the position error, and the control compensation quantity output by the execution neural network, a practical predefined time control law containing predefined convergence time parameters is generated. The specific process is as follows:

[0087] Consider the second equation in the system equations of S2, namely the velocity error. Derivation of velocity error based on system equations Dynamic characteristics:

[0088] ;

[0089] in, The time derivative of the velocity error; This refers to the acceleration of the robotic arm joints; The time derivative of a predefined time-based virtual control law; It is a positive definite inertial matrix; The Coriolis-centrifugal force matrix; It is the gravity vector; For control input (robotic arm joint torque); For continuous and bounded external disturbances;

[0090] Choose the Lyapunov function of the overall system. :

[0091] ;

[0092] Utilizing the key properties of robot dynamics, namely matrix For an antisymmetric matrix, we can obtain:

[0093] ;

[0094] in, The time derivative of the positive definite inertia matrix; The speed of the robotic arm joints.

[0095] Then the Lyapunov function of the whole system derivative for:

[0096] ;

[0097] To stabilize the system, a desired feedback control is designed:

[0098] ;

[0099] in, For expected feedback control; It is a constant.

[0100] It should be noted that, due to , , and All are unknown smooth functions, expected feedback control It cannot be implemented directly; there are unknown parts. yes , and The smooth function is denoted as:

[0101] ;

[0102] in, For lumped unknown nonlinear functions; This is the input vector of the neural network. .

[0103] Radial basis function neural network Approximating unknown nonlinear functions in a set Then feedback control is expected. It can be represented as:

[0104] ;

[0105] in, It is a constant; This refers to the positional error; For speed error; For ideal nonlinear compensation; Weights are ideal constants; It is a radial basis function vector; To approximate the error, For having constant bound The approximation error; due to the ideal constant weights Unknown, set for The estimated value, defined , This represents the weight estimation error.

[0106] It should be noted that the execution neural network takes position error and velocity error as input, and uses radial basis function neural network to approximate the unknown nonlinear function in the dynamic model, and outputs control compensation quantity; the unknown nonlinear function includes external disturbance, gravity vector and related terms of positive definite inertia matrix and Coriolis-centrifugal force matrix. Through the online learning capability of the execution neural network, these unknown nonlinear terms can be approximated in real time, improving the system's adaptability to model uncertainty and external disturbance.

[0107] Further, the generation of the practical predefined time control law containing predefined convergence time parameters includes constructing a control intermediate quantity based on the velocity error; the control intermediate quantity includes a linear feedback term of the velocity error, a coupling term of the position error, the control compensation quantity, and a velocity-layer predefined time convergence term; the velocity-layer predefined time convergence term is composed of the predefined convergence time parameter, a predefined time stability parameter, and a sign function exponentiation term of the velocity error. A vector inner product operation is performed on the velocity error and the control intermediate quantity to obtain the inner product value; the inner product value is then subjected to singularity avoidance normalization processing to obtain the practical predefined time control law.

[0108] In this embodiment, the specific formula for the generated practical predefined time control law is as follows:

[0109] ;

[0110] in, For control input (robotic arm joint torque); To control intermediate quantities; It is a constant; This refers to the positional error; For speed error; To avoid singular parameters; The control compensation quantity output by the neural network; For predefined time adjustment parameters; This is a predefined convergence time parameter.

[0111] Furthermore, to verify that the generated practical predefined time control law can achieve predefined time convergence, the practical predefined time control law is substituted into the term. Using the bounded property of the normalization function, we can obtain:

[0112] ;

[0113] in, For control input (robotic arm joint torque); To control intermediate quantities; The bounded error of the normalization function.

[0114] Then the Lyapunov function of the overall system derivative The derivation is as follows:

[0115] ;

[0116] Using the inequality relationship between vector norm and power function, for We can obtain:

[0117]

[0118] ;

[0119] in, Let i be the i-th component of the velocity error.

[0120] for We can obtain:

[0121]

[0122] ;

[0123] Furthermore, the Lyapunov function of the entire system is thus derived. derivative The derivative is rewritten as:

[0124] ;

[0125] in, This is the output of an ideal neural network; The bounded error of the normalization function; The bounded error of the normalization function; For predefined time adjustment parameters; Predefined convergence time parameters; This represents the i-th component of the velocity error; For dimensions.

[0126] Through the above control law design process, the position error and velocity error are input into the execution neural network to obtain the control compensation quantity. Based on the velocity error, position error and control compensation quantity, a practical predefined time control law containing predefined convergence time parameters is generated, which provides the control input basis for the subsequent construction of a weight update law with nonlinear correction terms and the realization of predefined time convergence of the closed-loop system.

[0127] S4: Construct weight update laws with nonlinear correction terms for the evaluation neural network and the execution neural network respectively, apply the practical predefined time control law to the dynamic model, and update the weights of the evaluation neural network and the execution neural network according to the weight update law, so that the system trajectory error converges to the residual set within a predefined time, thereby realizing the tracking control of the given reference signal.

[0128] In this embodiment, S2 and S3 respectively design a predefined time virtual control law and a practical predefined time control law. The derivatives of the corresponding Lyapunov functions all contain terms related to the estimation error of the neural network weights. The traditional gradient descent update law can only guarantee asymptotic convergence or exponential convergence, which cannot meet the requirement of predefined time convergence. This invention introduces a nonlinear correction term into the weight update law to explicitly shape the decay rate of the Lyapunov function, so that the neural network weights can complete adaptive learning within the time specified by the user.

[0129] Specifically, the weight update law with nonlinear correction terms for the evaluation neural network includes:

[0130] In reinforcement learning control design, the instantaneous cost function, based on tracking accuracy and input energy, is used to evaluate the system's performance at the current moment. The specific formula is as follows:

[0131] ;

[0132] in, It is the instantaneous cost function; This refers to the system status; For the desired trajectory; For control input (robotic arm joint torque); and It is a positive definite weighted matrix.

[0133] To evaluate the long-term performance of the system, a cost function is introduced. :

[0134] ;

[0135] in, It is the instantaneous cost function; For a specific moment; It is the integral variable.

[0136] Secondly, if it is necessary to discount future costs, factors can also be introduced. :

[0137] ;

[0138] Both of the above forms can characterize the cumulative performance from the current time t to infinity. In this embodiment, the objective of the optimal controller is to minimize the cost function. .

[0139] It should be noted that the goal of evaluating neural networks is to approximate the cost function. ,definition and ,in To evaluate the input to the neural network; This is a vector of radial basis functions, each of which is typically in Gaussian form; Let be the weight vector to be learned online. Assume that an ideal weight vector exists. satisfy ,in For bounded approximation error; This is an estimate of the long-run cost function.

[0140] The cost function plays a crucial role in the Actor-Critic architecture. Firstly, the cost function... The long-term impact of the current state and related control strategies is quantified, capturing the expected future performance; secondly, for nonlinear or uncertain systems, the cost function is directly calculated. This is usually difficult to achieve, so an evaluation neural network is used to approximate the cost function. Or related functions; finally, accurately estimate the cost function. It is crucial because it provides strong learning signals that guide the execution of neural network adjustments and optimization control strategies.

[0141] In order to achieve the goal of not requiring an exact cost function Update the weight vector to be learned online in the following case We introduce time-series difference error, where the specific formula for time-series difference error for continuous-time systems is as follows:

[0142] ;

[0143] in, This refers to timing difference error; It is the instantaneous cost function; This is an estimate of the long-run cost function; This is the cost discount factor; This is the time derivative of the estimate of the long-run cost function.

[0144] If the estimate of the long-run cost function It can perfectly match the real cost function. Then the timing difference error When the time sequence difference error A non-zero value indicates that the current estimate cannot accurately reflect the actual future cost.

[0145] Define the loss function for evaluating a neural network. Weight vectors for online learning Perform gradient descent updates:

[0146] ;

[0147] in, Let be the time derivative of the weight vector to be learned online; To evaluate the learning rate of a neural network; This represents the weight vector to be learned online. This refers to timing difference error.

[0148] according to and It can be deduced that:

[0149]

[0150] in, Let be the time derivative of the weight vector to be learned online; To evaluate the learning rate of a neural network; This represents the weight vector to be learned online. This refers to timing difference error; For gradient-related intermediate variables, ; This is a vector of radial basis functions, each of which is typically in Gaussian form; This is an estimate of the long-run cost function; This is the cost discount factor; This is the time derivative of the estimate of the long-run cost function; To evaluate the time derivative of the input to the neural network; This is the instantaneous cost function.

[0151] In this embodiment, to achieve predefined time convergence of the evaluation neural network update law, the specific formula for the evaluation network weight update law containing a nonlinear correction term is as follows:

[0152] ;

[0153] in, Let be the time derivative of the weight vector to be learned online; To evaluate the learning rate of a neural network; It is the instantaneous cost function; This represents the weight vector to be learned online. For nonlinear correction terms, ; It is a constant; This is an intermediate variable related to the gradient.

[0154] It should be noted that, unlike the traditional gradient update law used in most evaluation neural networks, the weight update law of the evaluation network constructed in this embodiment forms a nonlinear correction term. The invention aims to ensure that the weight estimation error converges within a user-defined finite time, regardless of the initial conditions. In contrast, traditional gradient descent can only guarantee asymptotic or exponential convergence, and the convergence speed depends on the learning gain and the behavior of the regression matrix. Therefore, this invention provides stronger convergence guarantee and higher time predictability, enhancing its value in real-time control scenarios.

[0155] Secondly, there is a coupling effect between the execution neural network and the evaluation neural network. That is, the gradient evaluation of the cumulative cost by the evaluation neural network will affect the update direction of the execution neural network, thereby driving it to find a control strategy that can minimize the future cumulative cost.

[0156] Specifically, the weight update law with nonlinear correction terms for the execution neural network includes:

[0157] First, the specific formula for executing the output of the neural network is as follows:

[0158] ;

[0159] in, The control compensation quantity output by the neural network; This is the weight vector; It is a Gaussian function; This is the input vector of the radial basis function neural network.

[0160] The specific formula for learning error is as follows:

[0161] ;

[0162] in, This is the learning error; For weight estimation error, , For ideal weights, To estimate the weights.

[0163] Furthermore, in order to incorporate the long-term performance evaluation information provided by the evaluation neural network into the weight update process, the learning error is fused with the long-term cost function estimate, and the approximation error of the execution neural network is defined as:

[0164] ;

[0165] in, To approximate the error of the neural network; This is the learning error; This is an estimate of the long-run cost function; To achieve the desired ideal long-term cost; A constant gain coefficient used to balance the weights of short-term errors and long-term costs.

[0166] It should be noted that the goal of executing a neural network is to make the control input... Approximate the optimal control strategy, balancing short-term and long-term performance; when the estimate of the long-term cost function... A larger value indicates higher cumulative costs in the system's future, prompting the neural network to further optimize the control strategy; when the estimated value of the long-term cost function is... When the error rate is relatively low, it indicates that future costs are controllable, and at this time, the neural network can mainly focus on instantaneous error correction.

[0167] After obtaining the estimate of the long-run cost function, the fusion Approximation error Then, a simple quadratic loss function is constructed. :

[0168] ;

[0169] For the quadratic loss function The estimated weights of the neural network are obtained using the negative gradient descent method. The renewal law:

[0170] ;

[0171] in, Let be the time derivative of the weight vector of the i-th dimension; The learning rate for executing the neural network; To approximate the error of the neural network; This represents the weight estimation error. It is a Gaussian function; This is the input vector of the radial basis function neural network; A constant gain coefficient to balance the weights of short-term errors and long-term costs; This is an estimate of the long-run cost function.

[0172] In this embodiment, to ensure that the execution neural network completes adaptation within the user-specified time, a nonlinear term is introduced into the corresponding update law to explicitly shape the decay rate of the Lyapunov function to satisfy a predefined time inequality. The specific formula for the execution network weight update law with the nonlinear correction term is as follows:

[0173] ;

[0174] in, Let be the time derivative of the weight vector of the j-th dimension; The learning rate for executing the neural network; It is a Gaussian function; For nonlinear correction terms, ; A constant gain coefficient to balance the weights of short-term errors and long-term costs; This is an estimate of the long-run cost function; It is a constant.

[0175] It should be noted that the nonlinear correction terms in the evaluation network weight update law and the execution network weight update law explicitly shape the decay rate of the Lyapunov function, making it satisfy the form requirement of the predefined time stability inequality. This ensures that the weights of the evaluation neural network and the execution neural network converge within the time set by the predefined convergence time parameter, thereby providing a guarantee for the predefined time convergence of the entire closed-loop system.

[0176] Further, applying the practical predefined time control law to the dynamic model and updating the weights of the evaluation neural network and the execution neural network according to the weight update law includes the following steps:

[0177] The practical predefined time control law generated in S3 is applied to the Eulerian-Lagrange system dynamics model, and the weights of the evaluation neural network and the execution neural network are updated according to the aforementioned weight update law to form a complete closed-loop control system. To verify the stability of the closed-loop system and the predefined time convergence performance, the global Lyapunov function is selected based on the Actor-Critic structure of reinforcement learning. :

[0178] ;

[0179] in, Lyapunov function of the whole system The derivative; To evaluate the adaptive gain of a neural network; To evaluate the weight estimation error of the neural network; To implement adaptive gain for the neural network; To calculate the weight estimation error of the neural network.

[0180] Considering the proposed predefined time-evaluation neural network control system, due to the bounded initial conditions, the closed-loop system signal... It is bounded, and its initial value is set to Constructing the Lyapunov function for evaluating neural networks for:

[0181] ;

[0182] in, To evaluate the adaptive gain of a neural network; To evaluate the weight estimation error of the neural network.

[0183] Under the evaluation network weight update law, the Lyapunov function for evaluating the neural network can be obtained through Lyapunov stability analysis. derivative satisfy:

[0184] ;

[0185] in, and It is a positive number; It is a bounded constant; For predefined time adjustment parameters.

[0186] For an Euler-Lagrange system with adaptive reinforcement learning control, consider the weight estimation error. The execution neural network, constructing the Lyapunov function of the execution neural network. for:

[0187] ;

[0188] in, To implement adaptive gain for the neural network; To calculate the weight estimation error of the neural network.

[0189] Under the network weight update law, Lyapunov stability analysis yields the following results: derivative satisfy:

[0190] ;

[0191] in, and It is a positive number; It is a bounded constant; For predefined time adjustment parameters.

[0192] Based on the Actor-Critic structure of reinforcement learning, the global Lyapunov function is selected. for:

[0193] ;

[0194] in, Lyapunov function of the whole system The derivative; To evaluate the adaptive gain of a neural network; To evaluate the weight estimation error of the neural network; To implement adaptive gain for the neural network; To calculate the weight estimation error of the neural network.

[0195] Combining the Lyapunov analysis results of the evaluation neural network and the execution neural network, the global Lyapunov function derivative satisfy:

[0196] ;

[0197] in, For PPTS parameters; Predefined convergence time parameters; It is a bounded constant.

[0198] According to the Practical Predefined Time Stability (PPTS) theory, the above inequality leads to the conclusion that the system trajectory error will be within a predefined time. Converging inward to the residual set:

[0199] ;

[0200] in, It is a residual set; This is the system state vector; It is a global Lyapunov function; For PPTS parameters; Predefined convergence time parameters; These are auxiliary parameters.

[0201] On the other hand, this embodiment also provides an adaptive predefined time tracking control system for an Euler-Lagrange system, which includes:

[0202] The system modeling module establishes a dynamic model of the Euler-Lagrange system of the dual-joint robotic arm and constructs a reinforcement learning framework that includes an evaluation neural network and an execution neural network.

[0203] The virtual control law module acquires the joint positions and joint velocities in the dynamic model, calculates the position error based on the joint positions and a given reference signal, generates a predefined time virtual control law containing predefined convergence time parameters based on the position error, and calculates the velocity error by combining the joint velocities with the predefined time virtual control law.

[0204] The time control law module inputs the position error and the velocity error into the execution neural network, and generates a practical predefined time control law containing predefined convergence time parameters based on the velocity error, the position error, and the control compensation quantity output by the execution neural network.

[0205] The tracking control module constructs weight update laws with nonlinear correction terms for the evaluation neural network and the execution neural network, respectively. It applies the practical predefined time control law to the dynamic model and updates the weights of the evaluation neural network and the execution neural network according to the weight update law, so that the system trajectory error converges to the residual set within a predefined time, thereby realizing the tracking control of the given reference signal.

[0206] If the above functions are implemented as software functional units and sold or used as independent products, they can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this invention, or the part that contributes to the prior art, or a part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of this invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0207] The logic and / or steps represented in the flowchart or otherwise described herein, for example, can be considered as a sequenced list of executable instructions for implementing logical functions, and can be embodied in any computer-readable medium for use by, or in conjunction with, an instruction execution system, apparatus, or device (such as a computer-based system, a processor-including system, or other system that can fetch and execute instructions from, an instruction execution system, apparatus, or device). For the purposes of this specification, "computer-readable medium" can be any means that can contain, store, communicate, propagate, or transmit programs for use by, or in conjunction with, an instruction execution system, apparatus, or device.

[0208] More specific examples of computer-readable media (a non-exhaustive list) include: electrical connections (electronic devices) having one or more wires, portable computer disk drives (magnetic devices), random access memory (RAM), read-only memory (ROM), erasable and editable read-only memory (EPROM or flash memory), fiber optic devices, and portable optical disc read-only memory (CDROM). Furthermore, computer-readable media can even be paper or other suitable media on which the program can be printed, because the program can be obtained electronically, for example, by optically scanning the paper or other medium, followed by editing, interpreting, or otherwise processing as necessary, and then stored in computer memory.

[0209] It should be understood that various parts of the present invention can be implemented in hardware, software, firmware, or a combination thereof. In the above embodiments, multiple steps or methods can be implemented in software or firmware stored in memory and executed by a suitable instruction execution system. For example, if implemented in hardware, as in another embodiment, it can be implemented using any or a combination of the following techniques known in the art: discrete logic circuits having logic gates for implementing logical functions on data signals, application-specific integrated circuits (ASICs) having suitable combinational logic gates, programmable gate arrays (PGAs), field-programmable gate arrays (FPGAs), etc.

[0210] Example 2, refer to Figures 2-6 As an embodiment of the present invention, an adaptive predefined time tracking control method for Euler-Lagrange systems is provided. To verify the beneficial effects of the present invention, scientific demonstration is carried out through economic benefit calculations and simulation experiments.

[0211] To verify the effectiveness of the proposed adaptive predefined time tracking control method for Euler-Lagrange systems based on the Actor-Critic reinforcement learning framework, this embodiment conducts simulation experiments on a two-link robotic arm. Specifically, to achieve high-performance control of the Euler-Lagrange system, the simulation settings focus on predefined time characteristics, reinforcement learning adaptive capabilities, and system dynamic characteristics. Regarding the core predefined time parameter, the predefined convergence time is... The time flexibility was tested by setting the time parameters to 5s, 10s, and 15s respectively. Virtual control gain Avoiding singular parameters Backstep controller position error gain Speed ​​error gain Positive definite matrix of valued function , .

[0212] In the Actor-Critic network parameters, the evaluation neural network has an input dimension of 2, with 3 RBF nodes per dimension, a total of 9 nodes, an RBF width of 1.5, and a learning rate of... Cost discount factor Weights are initialized and ideal weights are randomly generated, with an upper limit of the weight norm of 12. The neural network input dimension is 6, with 3 RBF nodes per dimension, for a total of 729 nodes. The RBF width is 0.8, and the learning rate for joint 1 is... Joint 2 learning rate Cost error gain The weights are initialized and the ideal weights are randomly generated. The upper limit of the weight norm for joints 1 and 2 is 5.

[0213] The system dynamic parameters are for a two-link robot, with link 1 having a mass of [missing information]. ,length Location of the center of mass Moment of inertia Mass of connecting rod 2 ,length Location of the center of mass Moment of inertia Gravitational acceleration And by simplifying parameters Describe the dynamic characteristics.

[0214] In the basic simulation settings, the total simulation time is 30 seconds, and the time step is... ,exist Applying an external load perturbation with a Cartesian amplitude of 8N, the desired trajectory of the two joints is:

[0215] ;

[0216] in, The desired trajectory for the two joints is given, and its first and second derivatives are used for trajectory tracking control. Simultaneously, variables such as system state, joint motion parameters, error, control input, network weights, and performance indicators are all initialized to 0. The center point of the RBF network is uniformly sampled and generated in the interval [-1,1]. The output and gradient are calculated using a Gaussian kernel function, and the Sigmoid function is used for nonlinear correction of the predefined time control law.

[0217] Among them, such as Figure 2 The results of the tracking performance curve analysis are shown below. It can be seen that the actual joint trajectory of the robotic arm closely matches the complex time-varying expected trajectory; under different predefined convergence time parameters... Under these conditions, the system exhibits adjustable time-accuracy characteristics: when At that time, the system was Rapid convergence within the inner region, with the position error quickly reduced to [a value missing]. within; when At that time, the convergence process is smoother, and the steady-state error stabilizes at... left and right; when At that time, the system convergence accuracy was further improved, and the steady-state error could be controlled within [a certain range]. The following fully meet the needs of high-precision assembly, precision machining and other scenarios.

[0218] In addition, After applying external perturbation, the joint position error only showed a brief and slight increase (approximately ), and in Rapidly recovers to steady-state range (e.g.) Figure 4 Figure 6 As shown in the sub-figure of position error, this invention has a strong anti-disturbance capability against external disturbances and can effectively offset the impact of disturbances such as load changes and friction fluctuations on system performance.

[0219] Secondly, from Figure 3 The control input curves shown demonstrate that the joint torque (control input) generated by the controller exhibits dynamic adaptability and energy controllability: during the convergence phase ( ): The torque exhibits a phased peak (the maximum torque at joint 1 is approximately... The maximum torque of joint 2 is approximately This provides the power for the system to quickly track the desired trajectory; Steady-state phase: After convergence, the torque rapidly decreases to a small steady-state value (the steady-state torque of joint 1 is approximately...). The steady-state torque of joint 2 is approximately ), significantly reducing energy consumption; disturbance response phase ( ): The torque is adjusted in time to counteract the effects of disturbance (the torque at joint 1 increases instantaneously to approximately The torque at joint 2 drops instantaneously to approximately It then quickly returns to a steady state, demonstrating a rapid dynamic response capability.

[0220] Different predefined convergence time parameters The control input comparison display below, Peak torque at time Reduced by about 30%, The peak torque was further reduced, verifying that the present invention can be improved by adjusting the predefined convergence time parameter. It achieves a flexible balance between response speed and energy consumption, making it suitable for industrial scenarios with different energy consumption requirements.

[0221] like Figures 4-6 The predefined convergence time parameters are displayed respectively. , , The changes in the weight norm of the evaluation neural network and the execution neural network reflect the boundedness and stability of the network weights: in the evaluation neural network, the weight norm changes in the early stage of simulation ( It rises rapidly, then tends to stabilize (the stable value is around 8); even in After the perturbation, the weight norm remains stable, indicating that the evaluation neural network's approximation accuracy of the cost function is stable and can provide a reliable learning signal for the execution neural network; in the execution neural network, the weight norm remains stable. The system gradually converges, with a stable value of approximately 12. There are no significant jumps before and after the disturbance, verifying the stability of the network weight update law and ensuring the continuity and reliability of the control strategy. All network weights remain bounded and do not diverge. Combined with Lyapunov stability analysis, it is proven that all signals, including the state, control input, and network weights of the closed-loop system, satisfy the boundedness requirement, and the system stability is strictly guaranteed.

[0222] Secondly, from Figure 4 Figure 6 As can be seen from the position error subplot, the error convergence rate strictly follows the predefined convergence time parameter. : The time error converges fastest. The time error convergence is the slowest, but the steady-state accuracy is the highest, reflecting the designability of time-accuracy; the error recovery time after disturbance is less than [time value missing]. Furthermore, the recovered error value is basically consistent with the steady-state error before the disturbance, further verifying the disturbance rejection robustness of the present invention; the steady-state error under all operating conditions is less than [value missing]. ( It meets the accuracy requirements of most industrial scenarios; and has predefined convergence time parameters. The larger the value, the smaller the steady-state error, providing a flexible adaptation possibility for scenarios with different accuracy requirements.

[0223] In summary, through simulation verification of the two-joint robotic arm, it can be seen that the present invention achieves accurate trajectory tracking within a predefined time, and the predefined convergence time parameter... The convergence time can be explicitly set, overcoming the limitation of fixed-time stability (FxTS) where the convergence time is not adjustable; the system exhibits strong robustness to external disturbances, with rapid error recovery after disturbances; simultaneously, the input energy is controllable, allowing for a flexible balance between response speed and energy consumption; the Actor-Critic network weights are stable and bounded, and all signals in the closed-loop system satisfy the boundedness requirement, verifying the stability and reliability of the invention; furthermore, different predefined convergence time parameters... Simulation results show that the present invention has good scene adaptability and the parameters can be adjusted according to actual needs, providing an efficient and feasible technical solution for time constraint control of Euler-Lagrange systems.

[0224] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. An Euler-Lagrange system adaptive pre-defined time tracking control method, characterized in that, include: A dynamic model of the Euler-Lagrange system of a dual-joint robotic arm is established, and a reinforcement learning framework including an evaluation neural network and an execution neural network is constructed. Obtain the joint positions and joint velocities in the dynamic model, calculate the position error based on the joint positions and a given reference signal; generate a predefined time virtual control law containing a predefined convergence time parameter based on the position error, and calculate the velocity error by combining the joint velocities with the predefined time virtual control law; The position error and the velocity error are input into the execution neural network. Based on the velocity error, the position error, and the control compensation quantity output by the execution neural network, a practical predefined time control law containing a predefined convergence time parameter is generated. Weight update laws with nonlinear correction terms are constructed for the evaluation neural network and the execution neural network respectively. The practical predefined time control law is applied to the dynamic model, and the weights of the evaluation neural network and the execution neural network are updated according to the weight update law, so that the system trajectory error converges to the residual set within a predefined time, thereby realizing the tracking control of the given reference signal.

2. The Euler-Lagrange system adaptive pre-defined time tracking control method of claim 1, wherein: The dynamic model of the Euler-Lagrange system for the dual-joint robotic arm includes establishing dynamic equations describing the dynamic behavior of the system based on the principles of Lagrange mechanics; the dynamic equations include a positive definite inertia matrix, a Coriolis-centrifugal force matrix, a gravity vector, control input, and external disturbances; Joint position and joint velocity are extracted as system state variables from the dynamic equations.

3. The Euler-Lagrange system adaptive pre-defined time tracking control method of claim 2, wherein: Generating a predefined time virtual control law based on the position error, including a predefined convergence time parameter, includes constructing a virtual control intermediate quantity based on the position error; the virtual control intermediate quantity includes a linear feedback term of the position error, a derivative term of the given reference signal, and a position-layer predefined time convergence term; the position-layer predefined time convergence term is composed of a predefined convergence time parameter, a predefined time stability parameter, and a sign function exponentiation term of the position error; The position error and the virtual control intermediate quantity are subjected to a vector inner product operation to obtain an inner product value. The inner product value is then subjected to singularity avoidance normalization processing to obtain a predefined time virtual control law. The predefined convergence time parameter is an explicitly adjustable parameter. By adjusting the predefined convergence time parameter, the upper limit of the convergence time of the system trajectory error is set.

4. The Euler-Lagrange system adaptive pre-defined time tracking control method of claim 3, wherein: The generation of a practical predefined time control law containing predefined convergence time parameters includes constructing a control intermediate quantity based on the velocity error; the control intermediate quantity includes a linear feedback term of the velocity error, a coupling term of the position error, the control compensation quantity, and a velocity-layer predefined time convergence term; the velocity-layer predefined time convergence term is composed of the predefined convergence time parameter, the predefined time stability parameter, and the sign function exponentiation term of the velocity error; The speed error and the control intermediate quantity are subjected to a vector inner product operation to obtain an inner product value. The inner product value is then subjected to singularity avoidance normalization processing to obtain a practical predefined time control law.

5. The Euler-Lagrange system adaptive pre-defined time tracking control method of claim 4, wherein: The evaluation neural network comprises defining an instantaneous cost function; the instantaneous cost function is calculated based on the deviation of the system state from the expected trajectory and the energy of the control input; defining a long-term cost function; the long-term cost function is the discounted integral of the instantaneous cost function from the current time to infinity; using a radial basis function neural network to approximate the long-term cost function with the position error as input, outputting a long-term cost function estimate value; constructing a time difference error according to the instantaneous cost function, the long-term cost function estimate value and the time derivative of the long-term cost function estimate value; the time difference error is used to guide the weight update of the evaluation neural network.

6. The Euler-Lagrange system adaptive pre-defined time tracking control method of claim 5, wherein: The execution neural network comprises using a radial basis function neural network to approximate the unknown nonlinear function in the dynamic model with the position error and the velocity error as input, outputting a control compensation; the unknown nonlinear function includes the external disturbance, the gravity vector and the related items of the positive definite inertia matrix and the Coriolis-centrifugal force matrix; Based on the deviation of the ideal weight and the estimated weight and the radial basis function output, the learning error is calculated; fuse the learning error and the long-term cost function estimate value, and introduce a cost error gain coefficient for weighted calculation to obtain an approximation error; the approximation error is used to guide the weight update of the execution neural network.

7. The Euler-Lagrange system adaptive pre-defined time tracking control method of claim 6, wherein: The weight update law with nonlinear correction term comprises constructing an evaluation network weight update law based on the evaluation neural network; the evaluation network weight update law comprises a gradient descent term, a linear damping term and a nonlinear correction term; constructing an execution network weight update law based on the execution neural network; the execution network weight update law comprises a gradient descent term, a linear damping term and a nonlinear correction term; Wherein, the nonlinear correction term is used to make the weights of the evaluation neural network and the execution neural network converge within the time set by the pre-defined convergence time parameter.

8. An Euler-Lagrange system adaptive pre-defined time tracking control system using the method of any one of claims 1-7, characterized by: a system modeling module for establishing a dynamic model of an Euler-Lagrange system of a double-joint robot arm and constructing a reinforcement learning framework comprising an evaluation neural network and an execution neural network; a virtual control law module for obtaining joint position and joint velocity in the dynamic model, calculating position error based on the joint position and given reference signal; generating a pre-defined time virtual control law containing a pre-defined convergence time parameter based on the position error, and calculating velocity error combining the joint velocity and the pre-defined time virtual control law; a time control law module for inputting the position error and the velocity error into the execution neural network, generating a practical pre-defined time control law containing a pre-defined convergence time parameter based on the velocity error, the position error and the control compensation output by the execution neural network; The tracking control module constructs a weight updating law with a nonlinear correction term for the evaluation neural network and the execution neural network respectively, applies the practical predefined time control law to the dynamic model, and updates the weights of the evaluation neural network and the execution neural network according to the weight updating law, so that the system trajectory error converges to a residual set within a predefined time, and the tracking control of the given reference signal is realized.

9. A computer device comprising: A memory and a processor; the memory stores a computer program, and the processor executes the computer program to implement the steps of the method according to any one of claims 1-7.

10. A computer readable storage medium having stored thereon a computer program, characterized in that: The computer program is executed by the processor to implement the steps of the method according to any one of claims 1-7.

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