Iterative learning control method for non-repetitive time-varying system average operator

Abstract

The invention relates to an iterative learning control method for a non-repetitive time-varying system average operator, and the method comprises the steps: constructing a discrete dynamic model which allows system parameters to change in a non-repetitive manner along with the time and the number of iterations, and generating an expected trajectory and a tracking error of dynamic truncation; performing dynamic truncation processing on the randomly changed track length through random variables of Bernoulli distribution to generate an optimization correction error signal; constructing a variable track length average operator based on weighted average and correction error signals of historical control input, designing an iterative learning control law, and verifying the convergence of the iterative learning control law; and finally, the tracking precision and robust stability of the system under the variable trajectory length and variable initial state are verified through simulation. The method breaks through the limitation of traditional iterative learning control on hypotheses such as a fixed system model and a fixed test length, solves the problem of tracking failure of a non-repetitive time-varying system caused by parameter drift and trajectory abrupt change, does not need to depend on an accurate model or a large amount of data training, and has algorithm conciseness, real-time performance and interpretability.

Description

Technical Field

[0001] The present invention relates to the technical field of iterative learning control, and in particular to an iterative learning control method for an average operator of a non-repetitive time-varying system. Background Art

[0002] Iterative learning control (ILC) is a technique for controlling a system to repeat operations within a finite learning time. This learning strategy iteratively adjusts the current control input based on previous control experience and output error, enabling the system to accurately track its desired trajectory. ILC has a wide range of applications, such as mobile robotic systems, motion control systems, intelligent transportation systems, and multi-agent systems. Traditional ILC itself has six design assumptions, namely a fixed reference signal, a fixed trial length, a fixed initial state, a fixed system dynamics model, a fixed input update rule, and system stability requirements, which limit the potential application range of ILC.

[0003] A non-repeating time-varying system means: "Time-varying system" means that the dynamic characteristics of the system change with time, such as parameters or structure change with time, while "non-repeating time-varying system" means that this change is not periodic, and the parameters or structure also change with the number of iterations.

[0004] Many researchers in other fields have attempted to use neural networks and machine learning to solve problems related to non-repeating time-varying systems. However, neural networks and machine learning require specific, known engineering application data samples and sufficient time for online or offline training. Their black-box nature, coupled with poor interpretability, limits their use in safety-critical fields such as aerospace. In contrast, ILCs have fewer parameters, are simpler to control, and do not require prior knowledge of the precise system function. Summary of the Invention

[0005] Based on this, the purpose of the present invention is to provide an iterative learning control method for an average operator of a non-repetitive time-varying system that can solve the motion trajectory tracking problem of a non-repetitive time-varying discrete system with variable trajectory length and variable initial state.

[0006] The purpose of the present invention is achieved by the following scheme:

[0007] In a first aspect, the present invention provides an iterative learning control method for an average operator of a non-repetitive time-varying system, comprising the following steps:

[0008] S1: Based on discrete time index and iteration index, a discrete dynamic model including state, input and output is constructed;

[0009] The discrete time index is used to indicate the state change of the system at each sampling moment, the iteration index is used to identify different batches of control tasks, and the discrete dynamic model is used to indicate a dynamic model that allows system parameters to change non-repeatedly with time and iteration number and the trajectory length to change randomly.

[0010] S2: Define the trajectory according to the discrete dynamic model to generate the desired trajectory, and perform real-time output calculation of the desired trajectory based on the discrete dynamic model to generate the tracking error of the desired trajectory;

[0011] S3: Dynamically truncate the trajectory length of the desired trajectory based on the Bernoulli distributed random variable and the tracking error to generate an optimized corrected error signal containing only valid time points;

[0012] S4: Process the optimized correction error signal based on the weighted average of historical control inputs to generate an iterative learning control model containing a variable trajectory length averaging operator. Verify the convergence of the iterative learning control model based on norm theory and output the convergence result.

[0013] S5: Based on the iterative learning control model, discrete dynamic model, and convergence results, the simulation generates a system output with tracking results of the desired trajectory with variable trajectory length and variable initial state. The tracking results are used to indicate the tracking accuracy of the desired trajectory and the robust stability verification results.

[0014] In one embodiment, the present invention provides an iterative learning control method for an average operator of a non-repetitive time-varying system, wherein S1 includes a discrete dynamic model of state, input, and output, and is expressed as follows:

[0015]

[0016] Where l∈{0,1,…} represents the iteration index, t represents the discrete time index, and x l (t),u l (t) and y l (t) are the state, input and output of the discrete state model, respectively. l (t)∈R n*n , B l (t)∈R n*m , C l (t)∈R p*n They are the discrete system state matrix, discrete system input matrix and discrete system output matrix respectively.

[0017] In one embodiment, S2 of an iterative learning control method for an average operator of a non-repetitive time-varying system provided by the present invention comprises the following steps:

[0018] S21: Based on the state space technology of the discrete dynamic model, the trajectory planning process is performed on the desired state to generate the desired trajectory that meets the preset convergence conditions. The expression of the desired trajectory is:

[0019]

[0020] Among them, x d (t),u d (t), r(t) are the state parameters, input parameters and expected output trajectory of the discrete state model under the expected state, respectively. A(t), B(t), and C(t) are the discrete system state matrix, discrete system input matrix and discrete system output matrix under the expected state, respectively. The preset convergence condition is:

[0021] ||A l (t)||≤β A ,||B l (t)||≤β B ,||C l (t)||≤β C

[0022]

[0023] E{x i (0)}=x d (0)

[0024] Among them, β A is the preset norm upper bound of the discrete system state matrix, β B is the preset norm upper bound of the discrete system input matrix, β C is the preset norm upper bound of the discrete system output matrix, x d (0) is the initial state of the desired trajectory;

[0025] S22: Based on the output equation of the discrete dynamic model, the system state of the desired trajectory is observed and processed to generate the actual output;

[0026] S23: Based on the error feedback technology, the desired trajectory and the actual output are differentially processed to generate the tracking error of the desired trajectory. The calculation formula of the tracking error is:

[0027] e l (t) = r(t) - y l (t)

[0028] Among them, e l (t) is the tracking error, r(t)∈R p t∈{0,1,…,N d} is the expected trajectory, y l(t) is the actual output of the discrete dynamic model, t∈{0,1,…,min{N l +1,N d +1}},N l is the tracking length of the system at the lth iteration, which is unknown and changes randomly in each iteration, N d is the expected trajectory length of the system.

[0029] In one embodiment, S3 of the iterative learning control method for an average operator of a non-repetitive time-varying system provided by the present invention comprises the following steps:

[0030] S31: Based on the preset probability parameters and the time-varying trajectory length constraint, a Bernoulli random variable sequence is generated. The expression of the Bernoulli random variable sequence is:

[0031] ξ l (t)~Bernoulli(p)

[0032] Among them, p is the preset probability parameter, p∈(0,1];

[0033] S32: Based on the Bernoulli random variable sequence and discrete time index, the tracking error is subjected to dynamic error threshold processing to generate a correction error signal. The calculation formula of the correction error signal is:

[0034]

[0035] in, To correct the error signal, ξ l (t) is the Bernoulli random variable sequence, e l (t) is the tracking error;

[0036] S33: Processing the correction error signal based on the probability normalization compensation and iterative accumulation mechanism to generate an optimized correction error signal containing only valid time points. The expression of the optimized correction error signal is:

[0037]

[0038] in, is the expected value of the optimized corrected error signal containing only valid time points, p(t) is the probability normalized function at time t, E[e l (t)] is the expected value of the tracking error signal.

[0039] In one embodiment, S4 of the iterative learning control method for an average operator of a non-repetitive time-varying system provided by the present invention comprises the following steps:

[0040] S41: The corrected error signal is processed based on the weighted average of the historical control inputs, and an iterative learning control law containing a variable trajectory length averaging operator is constructed. An iterative learning control model containing a variable trajectory length averaging operator is generated based on the iterative learning control law. The expression of the iterative learning control law is:

[0041]

[0042] Among them, u l+1 (t) is the iterative learning control law, which means the control input at time t in the l+1th iteration, L is the gain matrix, L∈Rp*m, To correct the error signal;

[0043] S42: Based on the norm theory, the control model is subjected to convergence constraint design and convergence conditions are generated. The convergence conditions are:

[0044] ||Ip(t+1)LC(t+1)B(t)||≤ε,0<ε<1

[0045] Among them, ε is the preset convergence threshold;

[0046] S43: Based on the iterative learning control model and the convergence conditions, the error is processed through λ-norm analysis and the convergence result is outputted, which shows that the error converges asymptotically with the iteration.

[0047] In a second aspect, the present invention provides an iterative learning control system system of a non-repetitive time-varying system averaging operator, comprising

[0048] Discrete model building module, used to build discrete dynamic models including states, inputs and outputs based on discrete time index and iteration index;

[0049] The discrete time index is used to indicate the state change of the system at each sampling moment, the iteration index is used to identify different batches of control tasks, and the discrete dynamic model is used to indicate a dynamic model that allows system parameters to change non-repeatedly with time and iteration number and the trajectory length to change randomly.

[0050] The expected trajectory definition module is used to define the trajectory according to the discrete dynamic model, generate the expected trajectory, and perform real-time output calculation of the expected trajectory based on the discrete dynamic model to generate the tracking error of the expected trajectory;

[0051] A trajectory length optimization unit is used to dynamically truncate the trajectory length of the desired trajectory based on the random variable of the Bernoulli distribution and the tracking error, and generate an optimized correction error signal containing only valid time points;

[0052] An iterative model building unit is used to process the correction error signal based on the weighted average of historical control inputs, generate an iterative learning control model containing a variable trajectory length averaging operator, verify the convergence of the iterative learning control model based on norm theory, and output the convergence result;

[0053] The simulation tracking verification unit is used to generate the tracking results of the desired trajectory with variable trajectory length and variable initial state through the simulation generation system based on the iterative learning control model, discrete dynamic model and convergence results. The tracking results are used to indicate the tracking accuracy of the desired trajectory and the robust stability verification results.

[0054] In a third aspect, the present application provides a computer device comprising a memory and a processor, wherein the memory stores a computer program, and when the processor executes the computer program, it implements any of the above-mentioned iterative learning control methods for the average operator of a non-repetitive time-varying system.

[0055] In a fourth aspect, the present application provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements any of the above-mentioned iterative learning control methods for the average operator of a non-repetitive time-varying system.

[0056] In summary, the iterative learning control method for an average operator of a non-repeating time-varying system provided by the present invention generates a dynamically truncated desired trajectory and tracking error by constructing a discrete dynamic model that allows system parameters to vary non-repeatingly with time and the number of iterations. The randomly varying trajectory length is dynamically truncated using a Bernoulli-distributed random variable to generate an optimized corrected error signal. A variable trajectory length average operator is constructed based on the weighted average of historical control inputs and the corrected error signal, and an iterative learning control law is designed and its convergence verified. Finally, the tracking accuracy and robust stability of the system under varying trajectory lengths and varying initial states are verified through simulation. This method overcomes the limitations of traditional iterative learning control on assumptions such as fixed system models and fixed test lengths, and addresses tracking failures caused by parameter drift and trajectory mutations in non-repeating time-varying systems. It does not rely on precise models or large amounts of data training, and combines algorithmic simplicity, real-time performance, and interpretability, making it suitable for high-security scenarios such as aerospace and intelligent robotics.

[0057] For better understanding and implementation, the present invention is described in detail below with reference to the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 A flowchart of an iterative learning control method for an average operator of a non-repetitive time-varying system provided in an embodiment of the present application;

[0059] Figure 2In the simulation test of the iterative learning control method of the non-repetitive time-varying system average operator provided in the embodiment of the present application, the discrete system state matrix A is defined as l , discrete system input matrix B l and the discrete system output matrix C l Matrix expression diagram of the value of ;

[0060] Figure 3 The iterative variable tracking length N under different iteration times in the iterative learning control method of the average operator of a non-repeating time-varying system provided in the embodiment of the present application is represented. l Line chart of

[0061] Figure 4 The initial state under different iteration times in the iterative learning control method of the average operator of a non-repeating time-varying system provided in the embodiment of the present application is represented and Line chart of numerical values;

[0062] Figure 5 A graph showing the system output profiles at the 10th and 25th iterations after adopting the average learning control (ILC) law in an iterative learning control method for an average operator of a non-repeating time-varying system provided in an embodiment of the present application;

[0063] Figure 6 A graph showing an overview of ILC tracking error indicators at different numbers of iterations using an averaging operator iterative learning control (ILC) law in an iterative learning control method for a non-repeating time-varying system provided by an embodiment of the present application;

[0064] Figure 7 A structural diagram of an iterative learning control system of an averaging operator for a non-repetitive time-varying system provided in another embodiment of the present application. DETAILED DESCRIPTION

[0065] To facilitate understanding of the present invention, the present invention will be described more fully below with reference to the accompanying drawings. The drawings illustrate preferred embodiments of the present invention. However, the present invention may be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to provide a more thorough and comprehensive understanding of the disclosure.

[0066] Unless otherwise defined, all technical and scientific terms used herein have the same meanings as those commonly understood by those skilled in the art to which this invention pertains. The terms used in this specification are for the purpose of describing specific embodiments only and are not intended to limit the invention. The term "and / or" as used herein includes any and all combinations of one or more of the associated listed items.

[0067] In one embodiment, Figure 1 As shown, an iterative learning control method for an average operator of a non-repeating time-varying system is provided. This embodiment uses the method applied to a terminal as an example for illustration. It is understood that the method can also be applied to a server, or to a system including a terminal and a server, and implemented through interaction between the terminal and the server. In this embodiment, the method includes the following steps:

[0068] S1: Based on the discrete time index and iteration index, a discrete dynamic model including state, input and output is constructed.

[0069] Specifically, in the field of iterative learning control, building an accurate system model is fundamental to achieving precise trajectory tracking. For non-repeating, time-varying systems, their dynamic characteristics exhibit non-periodic variations over time and iteration number, and trajectory lengths can vary randomly. Traditional fixed models are unable to adapt to this complexity, necessitating the construction of a discrete dynamic model that captures the system's time-varying and iterative characteristics while allowing for random variations in trajectory length.

[0070] Among them, the discrete time index t∈{1,2,…,N} is used to indicate the state change of the system at each sampling moment, N is the maximum number of sampling points for each batch, but in actual operation, due to the random variation of trajectory length, the actual number of sampling points may be less than N; the iteration index l∈{0,1,…} is used to identify control tasks of different batches, and R is the total number of iterations.

[0071] Preferably, the discrete dynamic model including state, input and output is expressed as:

[0072]

[0073] Where l∈{0,1,…} represents the iteration index, t represents the discrete time index, and x l (t),u l (t) and y l (t) are the state, input and output of the discrete state model, respectively. l (t)∈R n*n , (t)∈R n*m , C l (t)∈R p*n They are the discrete system state matrix, discrete system input matrix and discrete system output matrix respectively. It should be noted that the system matrix changes with time and the number of iterations, which means it is a non-repeating time-varying system. is the tracking length of the system at the lth iteration, which is unknown and changes randomly in each iteration. N and denote the minimum and maximum tracking lengths respectively. In the problem based on iterative learning control (ILC), their specific information does not need to be known in advance in the design of the ILC scheme.

[0074] This model exhibits non-repeating time-varying behavior and random variations in trajectory length. This allows system parameters to vary non-repeatingly over time and iterations. For example, in a chemical reaction, the heat transfer coefficient of a reactor may be affected by various factors, such as ambient temperature and reactant concentration. These factors can vary across batches (different iterations) and at different time points within a batch (different sampling times), and the patterns of variation are not periodic. In this case, the discrete dynamic model accurately describes this complex dynamic behavior, providing a foundation for subsequent control strategy design. Furthermore, the model allows for random variations in trajectory length, meaning that control tasks in different batches may have different durations. For example, in an automated assembly line, due to differences in workpiece size and complexity, assembling a simple workpiece may require a shorter time (fewer sampling points), while assembling a complex workpiece may require a longer time (more sampling points). In this case, the model can flexibly adapt to these variations, enabling effective control strategies for trajectories of varying lengths.

[0075] S2: Define the trajectory according to the discrete dynamic model to generate the expected trajectory, and perform real-time output calculation of the expected trajectory based on the discrete dynamic model to generate the tracking error of the expected trajectory.

[0076] Specifically, the system determines the desired trajectory based on discrete dynamic models, system performance requirements, and mission objectives. For example, in a robot tracking task, the desired trajectory can be a pre-planned spatial curve that satisfies the robot's kinematic and dynamic constraints within the workspace while efficiently and accurately completing a specific task, such as a welding path or material handling path. The desired trajectory can be represented as a series of desired states and output values ​​at discrete time points.

[0077] Specifically, using the output matrix C in the model l (t) and the input corresponding to the desired trajectory (which may be zero or a specific reference input), according to the output equation y of the model l (t) = C l (t)x l (t), calculate the theoretical output value of the expected trajectory at each sampling time t and iteration number l. This process needs to consider the changing characteristics of system parameters over time and iteration number to ensure the accuracy and real-time performance of the calculation results. At the same time, the system calculates the actual output y l(t) is compared with the theoretical output of the desired trajectory to obtain the tracking error. This error reflects the system's tracking accuracy of the desired trajectory at the current iteration number and sampling time. For example, in drone flight control, the difference between the actual flight altitude and the desired flight altitude is the tracking error. This error serves as feedback information to guide subsequent control input adjustments to reduce the error and enable the drone to more accurately track the desired flight trajectory.

[0078] S3: The trajectory length of the desired trajectory is dynamically truncated based on the random variable of the Bernoulli distribution and the tracking error to generate an optimized corrected error signal that only contains valid time points.

[0079] Specifically, the system introduces a Bernoulli-distributed random variable, whose value is either 0 or 1. This random variable is used to dynamically truncate the desired trajectory length. For example, on a production line with random interference, some steps may not complete the entire scheduled production cycle due to equipment failure or raw material problems. In this case, the Bernoulli-distributed random variable can simulate this random termination, allowing the control strategy to adapt to this uncertainty.

[0080] Specifically, the system uses random variables from the Bernoulli distribution to dynamically truncate the tracking error. Specifically, at each sampling time t and iteration number l, if the random variable is 1, the tracking error at that moment is retained; if the random variable is 0, the tracking error at that moment is truncated to 0. In this way, an optimized corrected error signal is generated that only contains valid time points. It reflects the system's tracking error of the desired trajectory while considering random changes in trajectory length, providing more accurate and practical feedback information for adjusting the control strategy, helping to improve the system's adaptability and robustness. For example, in an automated inspection system with random task termination, when a critical defect is detected, the task may terminate prematurely. At this time, through this truncation process, only the errors generated within the effective inspection time can be considered, and the errors at subsequent invalid time points can be ignored, making the optimization process more in line with actual conditions.

[0081] S4: The optimized correction error signal is processed based on the weighted average of the historical control inputs to generate an iterative learning control model containing a variable trajectory length averaging operator. The convergence of the iterative learning control model is verified based on the norm theory and the convergence result is output.

[0082] Specifically, the system considers the control inputs of the previous k iterations, assigns a weight coefficient to each historical control input, and then calculates the weighted average control input. For example, in a motor speed control system with learning capabilities, in order to fully utilize past control experience, the control inputs of the three most recent iterations can be assigned larger weights (such as 0.5, 0.3, and 0.2), respectively, to place greater emphasis on recent control strategies while taking into account the trends of historical data. By performing weighted averaging on historical control inputs, the system can smooth out changes in control inputs, reduce the impact of random interference and noise, and comprehensively consider the advantages and disadvantages of control strategies under different numbers of iterations, providing a more representative and stable reference input for subsequent correction error signal processing, which helps to improve the performance and reliability of the iterative learning control model.

[0083] Subsequently, the system combines the weighted averaged control input with the optimized corrected error signal to construct an iterative learning control model with a variable trajectory length averaging operator. This model can generally be expressed as:

[0084]

[0085] Where L is the learning gain matrix, which is used to adjust the influence of the corrected error signal on the control input. For example, in a complex chemical process control, the learning gain matrix can be designed based on the system's dynamic characteristics and control objectives to ensure that the control input effectively reduces the tracking error and gradually approaches the desired control performance under varying trajectory lengths.

[0086] By introducing a variable trajectory length averaging operator, this iterative learning control model can adapt to random variations in system trajectory length and fully utilize historical control input information to improve the adaptability and learning efficiency of the control strategy. Compared with traditional iterative learning control models with fixed trajectory length, it offers greater flexibility and practicality for handling non-repeating time-varying systems, and can better cope with the uncertainty, randomness, and complexity inherent in real industrial processes.

[0087] The system also verifies the convergence of the iterative learning control model based on norm theory. The process is as follows: First, an appropriate norm (such as the Euclidean norm or the infinite norm) is selected to measure the magnitude and trend of vectors or matrices such as the tracking error, control input, and learning gain matrix. Then, the convergence of the system is determined by analyzing how these norms change during the iterative process. For example, if the norm of the tracking error gradually approaches 0 as the number of iterations increases, the system is convergent under that norm. This means that the tracking error can be gradually reduced through iterative learning, ultimately achieving accurate tracking of the desired trajectory.

[0088] Specifically, the system derives sufficient conditions to ensure the convergence of the iterative learning control model based on norm theory and the dynamic characteristics of the system. These conditions usually involve the selection of the learning gain matrix, the system matrix A l (t), B l (t), C l (t) and parameters such as the random variable of the Bernoulli distribution. By substituting the actual system parameters into these convergence conditions for verification, it is possible to determine whether the designed iterative learning control model can theoretically guarantee convergence. For example, through methods such as matrix spectral radius analysis or inequality constraint verification, it is possible to ensure that the learning gain matrix and system parameters meet the convergence conditions, thus providing theoretical support for the practical application of the system.

[0089] S5: Based on the iterative learning control model, discrete dynamic model, and convergence results, the simulation generates a system output with tracking results of the desired trajectory with variable trajectory length and variable initial state. The tracking results are used to indicate the tracking accuracy of the desired trajectory and the robust stability verification results.

[0090] Specifically, numerical simulations were performed on the proposed iterative learning control model, discrete dynamic model, and convergence results to verify its effectiveness in a non-repetitive system with variable trajectory length and initial state offset uncertainty. To ensure the comprehensiveness and accuracy of the experiments, a non-repetitive system was constructed as a testbed, and a detailed open- and closed-loop control strategy was designed to evaluate the system's performance.

[0091] In the simulation environment, the system builds a discrete dynamic model based on the structure and characteristics of the actual system and sets the corresponding system parameters, such as the specific values ​​or expressions of the state matrix, input matrix, and output matrix, as well as the system's initial state, input, and output ranges. Furthermore, based on the actual application scenario, the shape, length, and variation of the expected trajectory are defined, and the probability parameters of the Bernoulli distribution random variables are set to create a simulation scenario that is as close to reality as possible.

[0092] Specifically, the system analyzes the tracking error data recorded during simulation and calculates statistical indicators such as the average, maximum, and standard deviation at different iterations to evaluate the system's tracking accuracy. For example, the root mean square (RMS) value of the tracking error in the final iteration is calculated and compared with the RMS value of the initial iteration to intuitively reflect the degree of improvement in the system's tracking accuracy after iterative learning. At the same time, the convergence trend of the tracking error curve is observed to determine whether the system can achieve the expected tracking accuracy requirements within a limited number of iterations.

[0093] To verify the robust stability of a system under varying trajectory lengths and initial states, various disturbances and uncertainties can be introduced into the simulation, such as system parameter perturbations, external interference, and initial state deviations, to observe the system's response and recovery capabilities under these conditions. By analyzing the system's output curves, tracking error curves, and control input curves under different disturbance conditions, it can be determined whether the system can maintain stable operation and meet performance requirements within a certain range. For example, in the simulation of a robotic arm control system with parameter uncertainty, by varying parameters such as the robotic arm's load mass and joint friction coefficient, the trajectory tracking performance at different iterations can be observed to evaluate the robust stability of the system.

[0094] Based on the analysis and evaluation of simulation results, the performance of the iterative learning control model under different operating conditions was summarized, and existing problems and shortcomings were identified. To address these issues, the control model parameters, learning gain matrix, and weight distribution of historical control inputs were optimized to further improve the system's tracking accuracy and robust stability. Through multiple simulation tests and feedback, the control strategy was continuously refined, ultimately achieving a control system design that met the requirements of practical applications.

[0095] In summary, the iterative learning control method for an average operator of a non-repeating time-varying system provided by the present invention generates a dynamically truncated desired trajectory and tracking error by constructing a discrete dynamic model that allows system parameters to vary non-repeatingly with time and the number of iterations. The randomly varying trajectory length is dynamically truncated using a Bernoulli-distributed random variable to generate an optimized corrected error signal. A variable trajectory length average operator is constructed based on the weighted average of historical control inputs and the corrected error signal, and an iterative learning control law is designed and its convergence verified. Finally, the tracking accuracy and robust stability of the system under varying trajectory lengths and varying initial states are verified through simulation. This method overcomes the limitations of traditional iterative learning control on assumptions such as fixed system models and fixed test lengths, and addresses tracking failures caused by parameter drift and trajectory mutations in non-repeating time-varying systems. It does not rely on precise models or large amounts of data training, and combines algorithmic simplicity, real-time performance, and interpretability, making it suitable for high-security scenarios such as aerospace and intelligent robotics.

[0096] In one embodiment, S2 of an iterative learning control method for an average operator of a non-repetitive time-varying system provided by the present invention comprises the following steps:

[0097] S21: Based on the state space technology of the discrete dynamic model, the trajectory planning process is performed on the desired state to generate the desired trajectory that meets the preset convergence conditions.

[0098] Specifically, the reference trajectory r(t)∈R p , t∈{0,1,…N d}, where xd (t) is the desired state. For any achievable trajectory r(t), at t∈{0,1,…N d There is a unique control input u within the range d (t)∈R m , the expression of the expected trajectory is:

[0099]

[0100] Among them, x d (t),u d (t), r(t) are the state parameters, input parameters and expected output trajectory of the discrete state model under the expected state, respectively. A(t), B(t), and C(t) are the discrete system state matrix, discrete system input matrix and discrete system output matrix under the expected state, respectively. The preset convergence conditions include:

[0101] ||A l (t)||≤β A ,||B l (t)||≤β B ,||C l (t)||≤β C

[0102]

[0103] Explanation: The above limit expression almost necessarily converges, which means that A l (t) almost necessarily converges to A(t)

[0104] E{x i (0)}=x d (0)

[0105] Among them, β A is the preset norm upper bound of the discrete system state matrix, β B is the preset norm upper bound of the discrete system input matrix, β C is the preset norm upper bound of the discrete system output matrix, x d (0) is the initial state of the desired trajectory;

[0106] For E{x i (0)}=x d (0), where the strictly identical initial conditions in traditional iterative learning are relaxed:

[0107]

[0108] (Definition 2, this is the definition of norm)

[0109] S22: Based on the output equation of the discrete dynamic model, the system state of the desired trajectory is observed and processed to generate the actual output.

[0110] S23: Based on the error feedback technology, the desired trajectory and the actual output are differentially processed to generate the tracking error of the desired trajectory.

[0111] Specifically, the calculation formula of tracking error is:

[0112] e l (t) = r(t) - y l (t)

[0113] Among them, e l (t) is the tracking error, r(t)∈R p t∈{0,1,…,N d} is the expected trajectory, y l (t) is the actual output of the discrete dynamic model, t∈{0,1,…,min{N l +1,N d +1}},N l is the tracking length of the system at the lth iteration, which is unknown and changes randomly in each iteration, N d is the expected trajectory length.

[0114] In one embodiment, S3 of the iterative learning control method for an average operator of a non-repetitive time-varying system provided by the present invention comprises the following steps:

[0115] S31: Generate a Bernoulli random variable sequence based on preset probability parameters and time-varying trajectory length constraints.

[0116] Specifically, in order to better solve the iterative learning control problem of the linear discrete-time non-repetitive system (1) with iterative time-varying trajectory length, we define ξ l (t) is a random variable that follows a Bernoulli distribution and takes binary values ​​0 and 1.

[0117] The expression of the Bernoulli random variable sequence is:

[0118] ξ l (t)~Bernoulli(p)

[0119] Wherein, p is a preset probability parameter, which takes binary values ​​0 and 1, p∈(0,1];

[0120] S32: Based on the Bernoulli random variable sequence and discrete time index, the tracking error is subjected to dynamic error threshold processing to generate a correction error signal. The calculation formula of the correction error signal is:

[0121]

[0122] When N l <T d hour,

[0123]

[0124] When N l <T d hour,

[0125]

[0126] in, To correct the error signal, ξ l (t) is the Bernoulli random variable sequence, e l (t) is the tracking error;

[0127] S33: Processing the correction error signal based on the probability normalization compensation and iterative accumulation mechanism to generate an optimized correction error signal containing only valid time points. The expression of the optimized correction error signal is:

[0128]

[0129] in, is the expected value of the optimized corrected error signal containing only valid time points, p(t) is the probability normalized function at time t, E[e l (t)] is the expected value of the tracking error signal.

[0130] In one embodiment, S4 of the iterative learning control method for an average operator of a non-repetitive time-varying system provided by the present invention comprises the following steps:

[0131] S41: The corrected error signal is processed based on the weighted average of the historical control inputs, and an iterative learning control law containing a variable trajectory length averaging operator is constructed. An iterative learning control model containing a variable trajectory length averaging operator is generated based on the iterative learning control law. The expression of the iterative learning control law is:

[0132]

[0133] Among them, u l+1 (t) is the iterative learning control law, which means the control input at time t in the l+1th iteration, L is the gain matrix, L∈Rp*m, To correct the error signal;

[0134] S42: Based on the norm theory, the control model is subjected to convergence constraint design and convergence conditions are generated. The convergence conditions are:

[0135] ||Ip(t+1)LC(t+1)B(t)||≤ε,0<ε<1

[0136] Among them, ε is the preset convergence threshold;

[0137] S43: Based on the iterative learning control model and the convergence conditions, the error is processed through λ-norm analysis and the convergence result is outputted, which shows that the error converges asymptotically with the iteration.

[0138] Specifically, the λ-norm is a well-suited norm for analyzing iterative learning control systems, taking into account the error trends at different iteration numbers. Through λ-norm analysis, we can assess the convergence rate and ultimate degree of system error as the number of iterations increases.

[0139] Based on the iterative learning control model and convergence conditions, the λ-norm analysis method is used to quantitatively analyze the system error. The specific steps include:

[0140] S51: Collect the tracking error sequence of the system at different iteration times {e l (t)}, where l represents the number of iterations and t represents the sampling time.

[0141] S52: Calculate the λ-norm of the error sequence. The specific calculation method is: First, for each iteration number l and sampling time t, the tracking error e l (t) is squared to obtain |e l (t)| 2 ; Then, multiply these squared error values ​​by the power of λ, i.e. |e l (t)| 2λ , where λ is a forgetting factor between 0 and 1, which is used to balance the influence of recent and long-term errors; then, the above products of all iteration numbers and sampling moments are summed up to obtain the sum and perform square root operation on the sum to obtain the λ-norm.

[0142] S53: Based on the calculated λ-norm, determine whether the system error meets the asymptotic convergence condition. If the λ-norm gradually decreases and approaches 0 with increasing iterations, it indicates that the system error is asymptotically convergent in the sense of the λ-norm, and the convergence result is output. Otherwise, it is necessary to adjust the gain matrix L or other system parameters and re-analyze until the convergence condition is met.

[0143] The above-mentioned convergence result output process based on λ-norm analysis can provide a reliable theoretical guarantee for the practical application of the system, ensuring that the iterative learning control strategy can effectively track the desired trajectory in a non-repetitive time-varying system with good convergence performance and robust stability.

[0144] In one embodiment, the system numerically simulated the aforementioned iterative learning control law to verify its effectiveness in a non-repetitive system with variable trajectory length and initial state offset uncertainty. To ensure the comprehensiveness and accuracy of the experiment, the system constructed a non-repetitive system as a test platform and designed a detailed open-loop and closed-loop control strategy to evaluate the system's performance.

[0145] like Figure 2 As shown, A l and B l The value of will change with the number of iterations, and the system is a non-repetitive time-varying system with specific dynamic characteristics. It simulates complex situations that may be encountered in actual applications in order to more accurately test the effectiveness of the control law in the face of various challenges.

[0146] In order to evaluate the iterative learning control (ILC) tracking accuracy of the system, this method also constructs the following error function (the specific content of the function is not given in the original paper, but the idea is consistent with the error definition given in the paper). The expressions of error and desired trajectory are:

[0147]

[0148] in, represents the cumulative error value in the i-th case (i=1,2), which is a cumulative measure of the error between the expected trajectory and the actual trajectory over a period of time, and is used to evaluate the tracking performance of the system in this case; To find the expected (mean) operation, is the expected trajectory value at time t in the i-th case.

[0149] Specifically, the function evaluates performance by calculating the absolute difference between the system output and the expected output at each iteration, and summing these differences while taking into account randomness and uncertainty to ensure the robustness of the system.

[0150] Preferably, if Figure 3 、 Figure 4 、 Figure 5 and Figure 6 As shown, in order to simulate the trajectory length change in iterative learning control (ILC), t∈{1,2,...,N l} defines the time variable, N l ∈{90,91,…,115} is randomly assigned in MATLAB to generate random trajectory lengths. The average operator control law is applied to the non-repetitive system for 150 iterations. The tracking robustness of the system under the average operator control rate is shown below.

[0151] in, Figure 3 Shows the length N of random trajectories in the system at each iterationl The situation varies between 90 and 115. Figure 4 It shows how the initial state shift changes in each iteration of the system. Figure 3 and Figure 4 This shows that the system has uncertainty in the variable trajectory length and variable initial state. Under this uncertainty, the system Figure 4 and Figure 5 The effectiveness and robustness of the proposed control rate for tracking target trajectories in non-repetitive systems are demonstrated. Figure 5 By employing the Average Learning Control (ILC) law, the system output profiles at the 10th and 25th iterations compared the expected trajectory with the output trajectory when l = 10 and l = 25. The results show that as the number of iterations increases, the output trajectory gradually converges to the expected trajectory. This highlights the system's ability to effectively reduce tracking error and achieve stable control after multiple iterations.

[0152] Figure 6 By adopting the average operator iterative learning control (ILC) law, the overview of the ILC tracking error indicators under different numbers of iterations shows that the tracking errors CE1 and CE2 decrease rapidly during the iteration process and tend to be stable after about 40 iterations, which indicates that the system has strong learning ability and robustness under open-closed-loop control.

[0153] In one embodiment, Figure 7 The system 600 is an iterative learning control system of a non-repeating time-varying system averaging operator. The system is configured with the following modules:

[0154] A discrete model construction module 610 is used to construct a discrete dynamic model including states, inputs, and outputs based on a discrete time index and an iteration index;

[0155] The discrete time index is used to indicate the state change of the system at each sampling moment, the iteration index is used to identify different batches of control tasks, and the discrete dynamic model is used to indicate a dynamic model that allows system parameters to change non-repeatedly with time and iteration number and the trajectory length to change randomly.

[0156] The desired trajectory definition module 620 is used to define the trajectory according to the discrete dynamic model, generate the desired trajectory, and perform real-time output calculation of the desired trajectory based on the discrete dynamic model to generate the tracking error of the desired trajectory;

[0157] a trajectory length optimization unit 630 for dynamically truncating the trajectory length of the desired trajectory based on a random variable of a Bernoulli distribution and a tracking error, and generating an optimized correction error signal containing only valid time points;

[0158] an iterative model construction unit 640 for processing the corrected error signal based on a weighted average of historical control inputs, generating an iterative learning control model including a variable trajectory length averaging operator, verifying the convergence of the iterative learning control model based on norm theory, and outputting a convergence result;

[0159] The simulation tracking verification unit 650 is used to generate a tracking result of a desired trajectory with a variable trajectory length and a variable initial state through a simulation generation system based on the iterative learning control model, the discrete dynamic model and the convergence result. The tracking result is used to indicate the tracking accuracy of the desired trajectory and the robust stability verification result.

[0160] In summary, the present invention provides an iterative learning control system for a non-repeating time-varying system averaging operator. This system generates a dynamically truncated desired trajectory and tracking error by constructing a discrete dynamic model that allows system parameters to vary non-repeatingly with time and the number of iterations. The randomly varying trajectory length is dynamically truncated using a Bernoulli-distributed random variable to generate an optimized, corrected error signal. A variable trajectory length averaging operator is constructed based on the weighted average of historical control inputs and the corrected error signal, and an iterative learning control law is designed and its convergence verified. Finally, simulations verify the tracking accuracy and robust stability of the system under varying trajectory lengths and initial states. This system overcomes the limitations of traditional iterative learning control on assumptions such as fixed system models and fixed test lengths, addressing tracking failures caused by parameter drift and trajectory mutations in non-repeating time-varying systems. It does not rely on precise models or extensive data training, and combines algorithmic simplicity, real-time performance, and interpretability, making it suitable for high-security scenarios such as aerospace and intelligent robotics.

[0161] Preferably, the desired trajectory definition module 620 is configured with the following units:

[0162] The desired trajectory generating unit 621 is configured to perform trajectory planning processing on the desired state based on the state space technology of the discrete dynamic model, and generate a desired trajectory that meets a preset convergence condition;

[0163] The actual output observation unit 622 is used to observe the system state of the desired trajectory based on the output equation of the discrete dynamic model and generate the actual output;

[0164] The tracking error calculation unit 623 is used to perform differential processing on the desired trajectory and the actual output based on the error feedback technology to generate the tracking error of the desired trajectory.

[0165] Preferably, the trajectory length optimization unit 630 is configured with the following units:

[0166] A random variable sequence generating unit 631 is used to generate a Bernoulli random variable sequence based on a preset probability parameter and a time-varying trajectory length constraint;

[0167] an initial error signal generating unit 632 for performing dynamic error threshold processing on the tracking error based on the Bernoulli random variable sequence and the discrete time index to generate a corrected error signal;

[0168] The optimized error signal generating unit 633 is used to process the corrected error signal based on the probability normalization compensation and iterative accumulation mechanism to generate an optimized corrected error signal containing only valid time points.

[0169] Preferably, the iterative model building unit 640 is configured with the following units:

[0170] a control model construction unit 641 for processing the corrected error signal based on a weighted average of historical control inputs, constructing an iterative learning control law including a variable trajectory length averaging operator, and generating an iterative learning control model including a variable trajectory length averaging operator based on the iterative learning control law;

[0171] A convergence condition generating unit 642 is used to perform convergence constraint design on the control model based on norm theory and generate convergence satisfaction conditions;

[0172] The convergence result output unit 643 is used to process the error through λ-norm analysis based on the iterative learning control model and the convergence condition, and output the convergence result of the error asymptotically converging with the iteration.

[0173] In one embodiment, the present application further provides a computer device including a memory and a processor, wherein the memory stores a computer program, and when the processor executes the computer program, the iterative learning control method of the non-repetitive time-varying system average operator is implemented.

[0174] In one embodiment, the present application further provides a computer-readable storage medium having a computer program stored thereon. When the computer program is executed by a processor, the iterative learning control method of the non-repetitive time-varying system average operator is implemented.

[0175] In the description of this specification, the reference terms "one embodiment," "some embodiments," "example," "specific example," or "some examples" mean that the specific features, structures, materials, or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the present application. Moreover, the specific features, structures, materials, or characteristics described may be combined in any appropriate manner in any one or more embodiments or examples. In addition, those skilled in the art may combine and integrate different embodiments or examples described in this specification, as well as features of different embodiments or examples, unless they are mutually inconsistent.

[0176] For the device embodiments, since they basically correspond to the method embodiments, the relevant parts can be referred to the partial description of the method embodiments. The device embodiments described above are merely illustrative, wherein the components described as separate parts may or may not be physically separated, and the parts displayed as units may or may not be physical units, that is, they may be located in one place, or they may be distributed on multiple network units. Some or all of the modules can be selected according to actual needs to achieve the purpose of the disclosed solution. A person of ordinary skill in the art can understand and implement it without expending creative work.

[0177] The above description is merely a specific embodiment of the present application, but the scope of protection of the present application is not limited thereto. Any person skilled in the art can easily conceive of various modifications or substitutions within the technical scope disclosed in this application, and such modifications or substitutions should be included within the scope of protection of the present application. Therefore, the scope of protection of the present application should be based on the scope of protection of the claims.

Claims

1. An iterative learning control method for an average operator of a non-repeating time-varying system, characterized in that: The following steps are involved: S1: Based on discrete time index and iteration index, a discrete dynamic model including state, input and output is constructed; The discrete time index is used to indicate the state change of the system at each sampling moment, the iteration index is used to identify different batches of control tasks, and the discrete dynamic model is used to indicate a dynamic model that allows system parameters to change non-repeatedly with time and iteration number and the trajectory length to change randomly; S2: defining a trajectory according to the discrete dynamic model to generate a desired trajectory, and performing real-time output calculation on the desired trajectory based on the discrete dynamic model to generate a tracking error of the desired trajectory; S3: dynamically truncating the trajectory length of the desired trajectory based on the random variable of the Bernoulli distribution and the tracking error to generate an optimized corrected error signal containing only valid time points; S4: Processing the optimized corrected error signal based on a weighted average of historical control inputs to generate an iterative learning control model containing a variable trajectory length averaging operator, verifying the convergence of the iterative learning control model based on norm theory, and outputting a convergence result; S5: Based on the iterative learning control model, the discrete dynamic model and the convergence result, a tracking result of the desired trajectory with a variable trajectory length and a variable initial state is output by the simulation generation system, and the tracking result is used to indicate the tracking accuracy and robust stability verification result of the desired trajectory.

2. The iterative learning control method according to claim 1, characterized in that: The discrete dynamic model of S1 including state, input and output is expressed as: Where l∈{0,1,…} represents the iteration index, t represents the discrete time index, and x l (t),u l (t) and y l (t) are the state, input and output of the discrete state model, They are the discrete system state matrix, discrete system input matrix and discrete system output matrix respectively.

3. The iterative learning control method according to claim 2, characterized in that: The S2 includes: S21: Based on the state space technology of the discrete dynamic model, trajectory planning is performed on the desired state to generate a desired trajectory that meets the preset convergence conditions. The expression of the desired trajectory is: Among them, x d (t),u d (t), r(t) are the state parameters, input parameters and expected output trajectory of the discrete state model under the expected state, respectively; A(t), B(t), C(t) are the discrete system state matrix, discrete system input matrix and discrete system output matrix under the expected state, respectively; the preset convergence condition is: ||A l (t)||≤β A ,||B l (t)||≤β B ,||C l (t)||≤β C E{x i (0)}=x d (0) Among them, β A is the preset norm upper bound of the discrete system state matrix, β B is the preset norm upper bound of the discrete system input matrix, β C is the preset norm upper bound of the discrete system output matrix, x d (0) is the initial state of the desired trajectory; S22: Based on the output equation of the discrete dynamic model, observing the system state of the desired trajectory to generate an actual output; S23: Based on the error feedback technology, the desired trajectory and the actual output are differentially processed to generate a tracking error of the desired trajectory. The calculation formula of the tracking error is: e l (t)=r(t)-y l (t) Among them, e l (t) is the tracking error, r(t)∈R p t∈{0,1,…,N d } is the expected trajectory, y l (t) is the actual output of the discrete dynamic model, t∈{0,1,…,min{N l +1,N d +1}},N l is the tracking length of the system at the lth iteration, which is unknown and changes randomly in each iteration, N d is the expected trajectory length of the system.

4. The iterative learning control method according to claim 3, characterized in that: The S3 includes: S31: Based on the preset probability parameters and the time-varying trajectory length constraint, a Bernoulli random variable sequence is generated. The expression of the Bernoulli random variable sequence is: ξ l (t)~Bernoulli(p) Among them, p is the preset probability parameter, p∈(0,1]; S32: Based on the Bernoulli random variable sequence and the discrete time index, perform dynamic error threshold processing on the tracking error to generate a corrected error signal. The calculation formula of the corrected error signal is: in, To correct the error signal, ξ l (t) is the Bernoulli random variable sequence, e l (t) is the tracking error; S33: Processing the corrected error signal based on a probability normalization compensation and iterative accumulation mechanism to generate an optimized corrected error signal containing only valid time points. The expression of the optimized corrected error signal is: in, is the expected value of the optimized corrected error signal containing only valid time points, p(t) is the probability normalized function at time t, E[e l (t)] is the expected value of the tracking error signal.

5. The iterative learning control method according to claim 4, characterized in that: The S4 includes: S41: Processing the corrected error signal based on the weighted average of historical control inputs, constructing an iterative learning control law containing a variable trajectory length averaging operator, and generating an iterative learning control model containing a variable trajectory length averaging operator based on the iterative learning control law. The expression of the iterative learning control law is: Among them, u l+1 (t) is the iterative learning control law, which means the control input at time t during the l+1th iteration, L is the gain matrix, To correct the error signal; S42: Based on the norm theory, a convergence constraint design is performed on the control model to generate a convergence condition. The convergence condition is: ||Ip(t+1)LC(t+1)B(t)||≤ε,0<ε<1 Among them, ε is the preset convergence threshold; S43: Based on the iterative learning control model and the convergence condition, processing is performed through λ-norm analysis, and a convergence result is outputted, in which the error converges asymptotically with iteration.

6. An iterative learning control system for an average operator of a non-repeating time-varying system, characterized in that: include: Discrete model building module, used to build discrete dynamic models including states, inputs and outputs based on discrete time index and iteration index; The discrete time index is used to indicate the state change of the system at each sampling moment, the iteration index is used to identify different batches of control tasks, and the discrete dynamic model is used to indicate a dynamic model that allows system parameters to change non-repeatedly with time and iteration number and the trajectory length to change randomly; a desired trajectory definition module, configured to define a trajectory according to the discrete dynamic model, generate a desired trajectory, and perform real-time output calculation on the desired trajectory based on the discrete dynamic model to generate a tracking error of the desired trajectory; a trajectory length optimization unit, configured to dynamically truncate the trajectory length of the desired trajectory based on a random variable of a Bernoulli distribution and the tracking error, and generate an optimized correction error signal containing only valid time points; an iterative model construction unit, configured to process the corrected error signal based on a weighted average of historical control inputs, generate an iterative learning control model containing a variable trajectory length averaging operator, verify the convergence of the iterative learning control model based on norm theory, and output a convergence result; A simulation tracking verification unit is used to output a tracking result of a desired trajectory with a variable trajectory length and a variable initial state through a simulation generation system based on the iterative learning control model, the discrete dynamic model and the convergence result, wherein the tracking result is used to indicate the tracking accuracy and robust stability verification result of the desired trajectory.

7. A computer device comprising a memory and a processor, wherein the memory stores a computer program, wherein: When the processor executes the computer program, the method according to any one of claims 1 to 5 is implemented.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the method according to any one of claims 1 to 5 is implemented.

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