A high-order iterative learning control method for nonlinear non-repetitive systems

Abstract

This invention relates to the field of automatic control technology, and in particular to a high-order iterative learning control method for nonlinear, non-repetitive systems. The method includes the following steps: Step 1, establishing a system model: For the i-th iteration, a discrete-time multi-input multi-output system model with multi-source non-repetitive uncertainty is established; Step 2, defining a corrected tracking error: A random variable following a Bernoulli distribution is introduced to correct the tracking error to handle the iteratively changing trajectory length; Step 3, designing a high-order iterative learning control law: The current control signal is updated using control inputs based on multiple previous iteration cycles and the corrected tracking error; Step 4, applying the high-order iterative learning control law to the controlled system, using the corrected tracking error obtained in each iteration to update the control input for the next iteration, so that the system control output tracks the desired trajectory in a mathematically expected sense, and the expected value of the tracking error eventually converges to a bounded region.

Description

Technical Field

[0001] This invention relates to the field of automatic control technology, specifically to a high-order iterative learning control method for nonlinear, non-repetitive systems. Background Technology

[0002] In high-precision control fields such as modern industrial automation, robot navigation, and aerospace, achieving accurate positioning and trajectory tracking of unknown targets has always been a core technological challenge. Traditional control methods mainly rely on single strategies, such as model-based feedforward control (MFC) or iterative learning control (ILC), but these methods face significant limitations in complex dynamic environments.

[0003] Chinese invention patent CN120469245B discloses an automation control method and system based on optimal iterative learning control and feedforward control. While it can optimize control input through historical error data and achieve a certain level of trajectory tracking accuracy in repetitive tasks, this method is highly dependent on the initial conditions of the system and lacks effective handling of actuator physical constraints. When the task trajectory changes or non-repetitive disturbances occur, its tracking performance will significantly decrease, making it difficult to meet the high standards of adaptability and robustness required by modern industrial systems. Combining iterative learning control with feedforward control has significant shortcomings in handling actuator saturation constraints, adapting to non-repetitive tasks, and optimizing control resources. For example, at the moment of trajectory switching, the traditional ILC learning mechanism requires multiple iterations to reconverge, and existing technologies are limited to dealing with only one or two types of non-repetitive factors, which does not fit the complex characteristics of actual engineering systems. Summary of the Invention

[0004] The purpose of this invention is to provide a high-order iterative learning control method for nonlinear non-repetitive systems. This method can incorporate non-repetitive system parameters, iterative trajectory length, random changes in initial state, non-repetitive external disturbances, and system nonlinear dynamics into a unified ILC framework. Based on the high-order iterative learning control law, the boundedness of error convergence in each iteration is derived, and it is proved that under the combined effect of multiple non-repetitive factors and nonlinear coupling, the mathematical expectation of the tracking error eventually converges to the bounded domain.

[0005] To achieve the above objectives, the present invention provides the following technical solution: a high-order iterative learning control method for nonlinear, non-repetitive systems, the method comprising the following steps:

[0006] Step 1, Establish a system model: For the first In the next iteration, a discrete-time multi-input multi-output system model with multi-source non-repetitive uncertainty is established;

[0007] Regarding the first In the next iteration, the system model is defined by the following formula:

[0008] (1);

[0009] in, For the number of iterations, Indicates time, and , For the first The actual trajectory length of each iteration varies randomly within the preset upper and lower bounds of the step size; , , These are system status, control input, and control output, respectively. and These represent system state disturbances and output disturbances, respectively. It is a non-linear vector-valued function that does not have repeatability; , These are the control input matrix and the observation output matrix, respectively.

[0010] Step 2, Define the corrected tracking error : Introduce random variables that follow a Bernoulli distribution to correct the tracking error and handle the iteratively changing trajectory length;

[0011] Introduce random variables that follow a Bernoulli distribution. To handle the changing trajectory length The formula for correcting tracking error is defined as follows:

[0012] (6);

[0013] in, For the number of iterations, The length of the desired output trajectory, To track errors, To output the desired trajectory; For handling iterative variable trajectory lengths, its mathematical expectation is:

[0014] (7);

[0015] in, This represents the probability that the control input is effective at time point t;

[0016] Step 3: Design a high-order iterative learning control law: Update the current control signal using control inputs based on multiple previous iteration cycles and correction tracking errors;

[0017] The formula for the higher-order iterative learning control law is expressed as:

[0018] (11);

[0019] in, The order of the control law; To learn the gain matrix, For the control gain matrix, , For the control parameter matrix, Represented as in the first In the next iteration, the system is Control input value at any time, Represented as in the first In the next iteration, the system is The tracking error between the output at a given time and the desired trajectory, and the initial control input. It is typically chosen as the zero vector in correcting tracking errors.

[0020] Step four: Apply the higher-order iterative learning control law to the controlled system, and use the corrected tracking error obtained in each iteration to update the control input for the next iteration, so that the system control output tracks the expected trajectory in the mathematical expectation sense, and the expected value of the tracking error eventually converges to a bounded region.

[0021] Preferably, in step two, when This refers to the system in the first... In this iteration, the control input is effectively applied up to the time point. In this scenario, the probability of the event occurring is... , ; 0 represents the time during which the control input failed to take effect, with a probability of 1- The correction of tracking error is specifically manifested as follows:

[0022] when At that time, the tracking error is corrected as follows:

[0023] (8);

[0024] in, To correct tracking errors, To track errors, The length of the desired output trajectory;

[0025] when At that time, the tracking error is corrected as follows:

[0026] (9);

[0027] in, To correct tracking errors, To track errors, The length of the desired output trajectory.

[0028] Preferably, the initial state of the system It is a random change, and the initial state of the system is set to a fixed state. The deviation fluctuates around the target area, and the expected value of the deviation is bounded (i.e., less than a constant), which can be expressed as:

[0029] (3);

[0030] in, This represents the initial state of the system. It is in a fixed state. It is a constant;

[0031] The nonlinear vector value function Satisfying the iteration-related Lipschitz condition: for any and There exists a constant that varies with the number of iterations. The following inequalities are satisfied:

[0032] (4);

[0033] in, for Nonlinear vector-valued functions, for Nonlinear vector-valued functions, It is a constant that varies with the number of iterations;

[0034] The system satisfies boundedness: that is, the non-repeating system matrix, initial state, and external disturbances are all uniformly bounded, and the control input matrix is ​​also bounded. Observation output matrix Initial state System state interference Output interference and expected control input It is bounded in each run;

[0035] When the number of iterations When the control input matrix approaches infinity, and observation output matrix They will each tend towards a fixed baseline value.

[0036] Preferably, in step three, the learning gain matrix and control gain matrix Convergence conditions must be met:

[0037] (12);

[0038] in, , and exist , and They are respectively in The observation-output matrix and input control matrix at time t. To learn the gain matrix, For the control gain matrix, To control the input The probability of effective action at any given time; under this condition, the number of iterations. When it approaches infinity, the tracking error... The upper limit of the mathematical expectation is a finite value.

[0039] Preferably, when the initial state of the iteration is equal to the desired initial state, that is, when the following conditions are met:

[0040] (twenty two);

[0041] in, This is the initial state. It is in a fixed state at this time. 0,

[0042] When there are no non-repeating interference terms and the number of iterations... As it approaches infinity, we obtain:

[0043] (25);

[0044] That is, when the number of iterations When the tracking error approaches infinity, The mathematical expectation will converge to zero.

[0045] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0046] 1. Synergistic processing of multi-source non-repetitive characteristics and nonlinearity: For the first time, non-repetitive system parameters, iterative variable trajectory length, initial state drift, non-repetitive external disturbances and system nonlinear dynamics are incorporated into a unified ILC framework, breaking through the limitation of existing solutions that can only deal with one or two types of non-repetitive factors, and better matching the complex characteristics of actual engineering systems.

[0047] 2. Lipschitz constant varying with iteration number: For multi-source non-repetitive nonlinear systems, a Lipschitz constant varying with iteration number is proposed. Based on the higher-order iterative learning control law, the boundedness of error convergence in each iteration is derived. This bound changes with the iteration number, proving the robustness of the higher-order iterative learning control law.

[0048] 3. Rigorous convergence analysis framework: Based on the state stability theory and mathematical induction of discrete system multiple-input multiple-output (MIMO), it is proved that under the action of multiple non-repetitive factors and nonlinear coupling, the mathematical expectation of the tracking error eventually converges to a bounded neighborhood, and the neighborhood range is related to the non-repetitive disturbance and the initial state drift. In particular, when the expectation of the initial state drift is zero and there is no non-repetitive disturbance, the tracking error eventually converges to zero. Attached Figure Description

[0049] Figure 1 This is a flowchart of the high-order iterative learning control method in this invention;

[0050] Figure 2 This is a block diagram illustrating the working principle of the control method in this invention;

[0051] Figure 3 The trajectory length dynamically adjusted on continuous iterations in the computer simulation MIMO system of Embodiment 1 of this invention. ;

[0052] Figure 4 For Embodiment 1 of the present invention Initial state under certain conditions Curves showing the changes in different iteration trajectories;

[0053] Figure 5 For Embodiment 1 of the present invention Relevant iterative learning control tracking error metrics under different iterations in different situations ;

[0054] Figure 6 For Embodiment 1 of the present invention In the case of iteration = 15 and = 32 will output the system output Output to reference trajectory (See above image) and output the system. Output to reference trajectory The actual tracking situation is shown in the image below.

[0055] Figure 7 For Embodiment 1 of the present invention Initial state under certain conditions Curves showing the changes in different iteration trajectories;

[0056] Figure 8 In Embodiment 1 of the present invention Relevant iterative learning control tracking error metrics under different iterations in different situations ;

[0057] Figure 9In Embodiment 1 of the present invention In case 0, during iteration = 15 and = 32 will output the system output Output to reference trajectory (See above image) and output the system. Output to reference trajectory The above (below) image shows the actual tracking situation;

[0058] Figure 10 This is the initial state of Embodiment 2 in this invention. Curves showing the changes in different iteration trajectories;

[0059] Figure 11 The relevant iterative learning control tracking error index for the application of higher-order iterative learning control law in Embodiment 2 of this invention. ;

[0060] Figure 12 For Example 2 of the present invention, in the iteration =20 and =37 system Output to a reference trajectory with a high-order iterative learning control law The actual tracking situation is shown in the figure. Detailed Implementation

[0061] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0062] Please see Figure 1-12 A high-order iterative learning control method for nonlinear, non-repetitive systems, the method comprising the following steps:

[0063] Step 1, Establish a system model: For the first In the next iteration, a discrete-time multi-input multi-output system model with multi-source non-repetitive uncertainty is established, expressed by the formula:

[0064] (1);

[0065] in, For the number of iterations, , Indicates time, and , For the first The trajectory length at the next iteration satisfies This length is unknown and changes randomly and unpredictably with each iteration, and Indicates the lower bound of the step size. Let N represent the upper bound. d It is the length of the expected output trajectory, satisfying ;at the same time, This indicates the state of the system. , These represent the system's control inputs and control outputs, respectively. , These represent system state disturbances and output disturbances, respectively. This represents a class of nonlinear vector-valued functions that do not exhibit repeatability. The control input matrix describes how external control input signals drive the evolution of the system state. The observation output matrix is ​​used to map the internal state to the measurable output. , , These are system status, control input, and control output, respectively.

[0066] For the expected reference trajectory Considering any achievable reference trajectory There is a unique expected control input for each. , The desired state is as follows:

[0067] (2);

[0068] in For the corresponding expected state, , To control the input, For the desired control output, The iterative learning control tracking error of the nonlinear discrete multi-input multi-output system in the l-th iteration is expressed as: .

[0069] Assumption 1: In practical engineering, the initial state of the system in each run may have random deviations due to factors such as reset errors and sensor noise. This is considered a common non-repeating uncertainty. The initial state of this system... It is a random change, allowing the initial state of the system to be in a fixed state. The deviation fluctuates around the target area, and the expected value of the deviation is bounded (i.e., less than a constant), which can be expressed as:

[0070] (3);

[0071] Assumption 2: The system addressed by this method may contain complex nonlinear dynamics, such as robot friction and motor magnetic saturation effects. To analyze their impact, we consider a class of nonlinear functions commonly found in engineering, whose dynamic rate of change is finite and allowed to vary across different running batches (iterations). Specifically, nonlinear vector-valued functions... Satisfying the iteration-related Lipschitz condition: for any and There exists a constant that varies with the number of iterations. The following inequalities are satisfied:

[0072] (4);

[0073] Define the difference of functions:

[0074] (5);

[0075] This is defined as the difference between functions with the same internal variables but different function forms, where the difference occurs as the number of iterations approaches infinity. It equals 0.

[0076] Assumption 3: All and The non-repeating system matrix, initial state, and external disturbances are all uniformly bounded, i.e., the system's control input matrix... Observation output matrix System state interference Output interference and expected control input There is an upper limit to the number of times it can be run.

[0077] ;

[0078] ; are all finite unknown boundaries.

[0079] Assumption 4: The core advantage of this method lies in its ability to handle non-repetitiveness during the iteration process. From the perspective of long-term learning effects, many non-repetitive factors (such as equipment aging drift and specific types of random noise) tend to have stable statistical characteristics or long-term averages. Therefore, in convergence analysis, when the number of iterations approaches infinity, the time-varying system matrix... and They will each tend towards a fixed baseline value. , It is an iteratively invariant matrix.

[0080] Step 2, Define the corrected tracking error Introduce random variables that follow a Bernoulli distribution. Correction is applied to tracking errors to handle varying trajectory lengths. The formula for correcting the tracking error is expressed as:

[0081] (6);

[0082] in, For the number of iterations, To track errors, To determine the desired output trajectory, a Bernoulli random variable is introduced. The mathematical expectation of this method, used to correct the length of the iterative variable trajectory, is:

[0083] (7);

[0084] in, This represents the probability that the time step reaches time point t.

[0085] In formula (6), when This refers to the system in the first... In this iteration, the control input is effectively applied up to the time point. In this scenario, the probability of the event occurring is... , ; 0 represents the time during which the control input failed to take effect, with a probability of 1- The correction of tracking error is specifically manifested as follows: when hour:

[0086] (8);

[0087] when hour:

[0088] (9);

[0089] Based on correcting tracking error and From the mathematical expectation, we can obtain the formula for correcting the expected tracking error:

[0090] (10);

[0091] in, for The mathematical expectation, This represents the probability that the control input is effective at time point t.

[0092] Step 3: Design a high-order iterative learning control law: Update the current control signal using control inputs based on multiple previous iteration cycles and correction tracking errors;

[0093] For nonlinear discrete multiple-input multiple-output systems, assuming that assumptions 1, 2, and 3 all hold true, regarding... and the number of iterations In this case, the formula for the higher-order iterative learning control law is expressed as:

[0094] (11);

[0095] in, The order of the control law; , for norm, and The learning gain matrix and the control gain matrix are shown in the following order. , For the control parameter matrix, Represented as in the first In the next iteration, the system is Control input value at any time, Represented as in the first In the next iteration, the system is The tracking error between the output at a given time and the desired trajectory, and the initial control input. It is typically chosen as the zero vector in correcting tracking errors.

[0096] Under assumptions 1, 2, 3, and 4, for a dual-variable non-repetitive nonlinear discrete-time system with iteratively variable trajectory length, initial condition deviation, and disturbance, applying a high-order iterative learning control law, if for and All and It satisfies the following convergence formula:

[0097] (12);

[0098] in, , and exist , and They are respectively in The observation-output matrix and input control matrix at time t. To control the input The probability of being effective at any given moment;

[0099] definition: ;

[0100] in , , These represent the desired state, the desired control input, and the desired control output, respectively. ,get:

[0101] (13);

[0102] use Subtract both sides of equation (11), and use equation (6) and ,in Simplifying, we get:

[0103] (14);

[0104] From equations (4) and (5), we get:

[0105] (15);

[0106] Taking the norm of both sides of equation (11), from (14), we get that for ,have:

[0107] (16);

[0108] Lemma 1: When hour, The real sequence is defined as:

[0109] ;

[0110] in for The specified sequence of real numbers. If the numbers are non-negative. , , ..., It satisfies the following formula:

[0111] (17);

[0112] If it exists This is represented as follows:

[0113] (18);

[0114] By Lemma 1 and From equation (16), we get that in The scope includes:

[0115] (19);

[0116] Using mathematical derivation, it can be proven that, The following exist within the area:

[0117] (20);

[0118] From equations (19) and (20), assumption 1, assumption 2 and (13), we obtain that in Above:

[0119] (twenty one);

[0120] in, , , , All are finite unknown boundaries. It is a constant that varies with the number of iterations. For the observation output matrix, for Iteratively invariant matrix Given the system state, it can be proven through mathematical induction that the system tracking error is bounded, and we can obtain... Bounded, making ,in It is bounded.

[0121] When the initial state of the iteration is equal to the desired initial state, that is, when the following conditions are met:

[0122] (twenty two)

[0123] in, This is the initial state. It is in a fixed state at this time. 0.

[0124] When there are no non-repeating interference terms and the number of iterations... As it approaches infinity, we obtain:

[0125] (twenty three)

[0126] at this time and It can be deduced that:

[0127] (twenty four);

[0128] in, To control the input, for Bounded values, , , All are finite unknown boundaries.

[0129] when At this time, ,available:

[0130] (25);

[0131] Therefore, when the number of iterations As the system approaches infinity, the tracking error... The mathematical expectation will converge to zero.

[0132] When the above convergence condition is satisfied, the higher-order iterative learning control law for both the system and the application can guarantee that the mathematical expectation of the system tracking error is... With the number of iterations The upper bound eventually becomes uniformly bounded due to the increase of the initial state, and this upper bound is consistent with the initial state deviation bound. and non-repetitive interference and Relatedly, in particular, when the system is free from non-repetitive disturbances and the initial state is unbiased, the mathematical expectation of the system tracking error will converge to zero.

[0133] Step four: Apply the higher-order iterative learning control law to the controlled system, and use the corrected tracking error obtained in each iteration to update the control input for the next iteration, so that the system control output tracks the expected trajectory in the mathematical expectation sense, and the expected value of the tracking error eventually converges to a bounded region.

[0134] Example 1: Computer Simulation Example

[0135] To demonstrate the effectiveness of the proposed high-order iterative learning control law in handling the trajectory length of iterative changes, non-repeating random disturbances, and random initial state drift in MIMO systems, this embodiment provides a computer simulation example. Since this system is a bivariate (time-varying and iteratively varying) dynamic system, the system matrix in the system equations is configured in the simulation parameters. To include time t and number of iterations The dependencies are simplified by specifying different expressions for even and odd iterations, which are used to define the iterative changes.

[0136] When the number of iterations When the number is even, the MIMO system formula is:

[0137] (26);

[0138] When the number of iterations When the number is odd, the MIMO system formula (27) is:

[0139] ;

[0140] The initial state of the iteration is: With the number of iterations And random variation, assuming Desired output trajectory for odd or even MIMO systems and It is expressed as, when Less than or equal to 50 (expected trajectory length) When ), the expected output trajectory is as follows:

[0141] (28);

[0142] when When the value is greater than 50, the expected output trajectory is as follows:

[0143] (29);

[0144] in, Actual trajectory length index If a uniform distribution is satisfied in each iteration, then there exists... =80 and = 120; The performance of iterative learning control tracking based on iterative variable trajectory length is evaluated using the following expected absolute error and metrics:

[0145] (30);

[0146] Applying the proposed higher-order iterative learning control law of M = 2, according to the convergence formula, the control parameter matrix of the second-order iterative learning control law is:

[0147] ;

[0148] Please see Figure 3 N represents the trajectory length N dynamically adjusted in continuous iterations in a computer simulation of a MIMO system. Please refer to the diagram. Figure 4 , is the initial state The graph shows the changes in different iteration trajectories, at which point the initial iteration state is shown. satisfy Therefore, it exists Please see Figure 5 ,for In the case of non-repetitive interference and All fall within the range of [-0.4, 0.4], representing the relevant iterative learning control tracking error indices under different iterations. Please see Figure 6 ,for In the case of iteration = 15 and = 32 will output the system output Output to reference trajectory (See above image) and output the system. Output to reference trajectory The actual tracking situation is shown in the figure below.

[0149] Please see Figure 7 , is the initial state The graph shows the changes in different iteration trajectories, at the initial state of the iteration. satisfy Therefore, 0, and there is no non-repetitive interference, i.e. and Please see Figure 8 ,for Under condition 0, the relevant iterative learning control tracking error index under different iterations Please see Figure 9 ,for In case 0, during iteration = 15 and = 32 will output the system output Output to reference trajectory (See above image) and output the system. Output to reference trajectory The above (or below) image shows the actual tracking situation.

[0150] Example 2: Speed ​​control application of a two-link robotic fish

[0151] The aforementioned high-order iterative learning control law is now applied to the speed control of a two-link robotic fish; the mathematical model of the two-link robotic fish performing repetitive tasks is described as follows:

[0152] (31);

[0153] in It's about the quality of the robotic fish. It's the speed of the robotic fish. It is the water resistance coefficient. This is the forward thrust generated by the movement of the robotic fish's tail. In reality, due to various factors, the above parameters may fluctuate within a certain range.

[0154] Consider the following iterative learning control problem for a nonlinear, non-repeating discrete-time MIMO system based on variable trajectory length and non-repeating random disturbances. When the number of iterations is even, the MIMO system is:

[0155] (32);

[0156] When the number of iterations When the number is odd, the MIMO system is:

[0157] (33);

[0158] For the above MIMO systems (32) and (33), the iterative initial state It changes randomly with the number of iterations. Assume the desired output trajectory of the above MIMO systems (32) and (33) is... The expected output trajectory is as follows:

[0159] (34);

[0160] in Actual trajectory length index If the distribution is uniform in each iteration, then there exists... and The performance of iterative learning control based on iterative variable trajectory length is evaluated using the following expected absolute error and metrics:

[0161] (35);

[0162] in, To track error metrics, To control the output, The desired output trajectory.

[0163] For discrete two-link robotic fish MIMO systems, please refer to Figure 10 The image shows the initial state. The curves showing the changes in different iterative trajectories, and satisfying... ,in Applied to the proposed The higher-order iterative learning control law (11). The control parameters of the first-order iterative learning control law are selected as follows: Therefore, please refer to Figure 1 As shown, this is the relevant iterative learning control tracking error index for applying a higher-order iterative learning control law. Please see Figure 12 As shown, this is during the iteration =20 and =37 system Output to a reference trajectory with a high-order iterative learning control law The actual tracking situation is shown in the figure.

[0164] As described above, this system proposes a robust high-order iterative learning control framework for a class of multi-input multi-output nonlinear discrete-time systems with multiple non-repetitive uncertainties. This framework utilizes the tracking error learned from previous iterations. It incorporates a Bernoulli-distributed random variable to describe trajectory changes and uses control inputs from multiple previous iterations and corrected tracking difference information to compensate for missing control information. This ensures that, under multi-source non-repetitive conditions, the tracking error converges to a small neighborhood, including variations in the two-dimensional time iterative model, random trajectory length fluctuations, non-repetitive disturbances, and initial state deviations. As a result, with increasing iterations, the mathematical expectation period of the iterative learning control tracking error at the desired trajectory is driven into a bounded region. This region varies with the number of iterations, but remains bounded for each iteration, with the bounds related to random initial state changes and repetitive disturbance terms. Specifically, when the mathematical expectation of the random initial state drift is zero and there are no non-repetitive disturbances, the tracking error of the iterative learning control can be driven to zero in the mathematical expectation. This control method relaxes the stringent requirements of traditional iterative learning control on system dynamic repeatability, trajectory length fixity, initial state consistency, and disturbance invariance.

[0165] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0166] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A high-order iterative learning control method for nonlinear, non-repetitive systems, characterized in that, The method includes the following steps: Step 1, Establish a system model: For the first In the next iteration, a discrete-time multi-input multi-output system model with multi-source non-repetitive uncertainty is established; Regarding the first In the next iteration, the system model is defined by the following formula: （1）； in, For the number of iterations, Indicates time, and , For the first The actual trajectory length of each iteration varies randomly within the preset upper and lower bounds of the step size; , , These are system status, control input, and control output, respectively. and These represent system state disturbances and output disturbances, respectively. It is a non-linear vector-valued function that does not have repeatability; , These are the control input matrix and the observation output matrix, respectively. Step 2, Define the corrected tracking error : Introduce random variables that follow a Bernoulli distribution to correct the tracking error and handle the iteratively changing trajectory length; Introduce random variables that follow a Bernoulli distribution. To handle the changing trajectory length The formula for correcting tracking error is defined as follows: （6）； in, For the number of iterations, The length of the desired output trajectory, To track errors, To output the desired trajectory; For handling iterative variable trajectory lengths, its mathematical expectation is: （7）； in, This represents the probability that the control input is effective at time point t; Step 3: Design a high-order iterative learning control law: Update the current control signal using control inputs based on multiple previous iteration cycles and correction tracking errors; The formula for the higher-order iterative learning control law is expressed as: （11）； in, The order of the control law; To learn the gain matrix, For the control gain matrix, , For the control parameter matrix, Represented as in the first In the next iteration, the system is Control input value at any time, Represented as in the first In the next iteration, the system is The tracking error between the output at a given time and the desired trajectory, and the initial control input. It is typically chosen as the zero vector in correcting tracking errors; Step four: Apply the higher-order iterative learning control law to the controlled system, and use the corrected tracking error obtained in each iteration to update the control input for the next iteration, so that the system control output tracks the expected trajectory in the mathematical expectation sense, and the expected value of the tracking error eventually converges to a bounded region.

2. The high-order iterative learning control method for nonlinear, non-repetitive systems according to claim 1, characterized in that: In step two, when This refers to the system in the first... In this iteration, the control input is effectively applied up to the time point. In this case, the probability of the event occurring is... , ; 0 represents the time during which the control input failed to take effect, with a probability of 1- The correction of tracking error is specifically manifested as follows: when At that time, the tracking error is corrected as follows: （8）； in, To correct tracking errors, To track errors, The length of the desired output trajectory; when At that time, the tracking error is corrected as follows: （9）； in, To correct tracking errors, To track errors, The length of the desired output trajectory.

3. The high-order iterative learning control method for nonlinear, non-repetitive systems according to claim 1, characterized in that: initial state of the system It is a random change, and the initial state of the system is set to a fixed state. The deviation fluctuates around the target area, and the expected value of the deviation is bounded (i.e., less than a constant), which can be expressed as: （3）； in, This represents the initial state of the system. It is in a fixed state. It is a constant; The nonlinear vector value function Satisfying the iteration-related Lipschitz condition: for any and There exists a constant that varies with the number of iterations. The following inequalities are satisfied: （4）； in, for Nonlinear vector-valued functions, for Nonlinear vector-valued functions, It is a constant that varies with the number of iterations; The system satisfies boundedness: that is, the non-repeating system matrix, initial state, and external disturbances are all uniformly bounded, and the control input matrix is ​​also bounded. Observation output matrix Initial state System state interference Output interference and expected control input It is bounded in each run; When the number of iterations When the control input matrix approaches infinity, and observation output matrix They will each tend towards a fixed baseline value.

4. The high-order iterative learning control method for nonlinear, non-repetitive systems according to claim 1, characterized in that: In step three, the learning gain matrix and control gain matrix Convergence conditions must be met: （12） in, , and exist , and They are respectively in The observation output matrix and control input matrix at time t. To learn the gain matrix, For the control gain matrix, To control the input The probability of effective action at any given time; under this condition, the number of iterations. When it approaches infinity, the tracking error... The upper limit of the mathematical expectation is a finite value.

5. The high-order iterative learning control method for nonlinear, non-repetitive systems according to claim 4, characterized in that: When the initial state of the iteration is equal to the desired initial state, that is, when the following conditions are met: （22）； in, This is the initial state. It is in a fixed state at this time. 0, When there are no non-repeating interference terms and the number of iterations... As it approaches infinity, we obtain: （25）； That is, when the number of iterations When the tracking error approaches infinity, The mathematical expectation will converge to zero.

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