A high-order iterative learning control method for variable-length nonlinear system saturation disturbance

Abstract

The application relates to the technical field of automatic control, in particular to a high-order iterative learning control method for saturation disturbance of a variable-length nonlinear system, which comprises the following steps: for a discrete-time affine nonlinear system simultaneously affected by input saturation, variable iteration length, random initial state offset and external disturbance, Bernoulli random variables are introduced to describe the variable iteration length, an extended control input sequence uniform time scale is constructed, and a modified tracking error is defined; a high-order feedforward and instantaneous feedback compound control law with dynamic time-varying characteristics is constructed: the high-order feedforward synthesizes the saturation extended control input and the modified tracking error of multiple historical iterations, and a saturation residual memory mechanism is introduced for adaptive anti-integral saturation compensation; the feedback term utilizes the current iteration error to suppress local fluctuations; a cooperative convergence condition of a time-varying learning gain matrix is given, the mathematical expectation of a tracking error is proved to converge to a bounded region related to the disturbance amplitude and the initial state offset; and under ideal conditions, the tracking error asymptotically converges to zero.

Description

Technical Field

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

[0002] Iterative Learning Control (ILC) is an advanced data-driven control method specifically designed for dynamic systems with repetitive operation characteristics. ILC corrects the current input by extracting errors and control data from historical iterations, allowing the system output to gradually approximate the target within the iterative domain and achieve high-precision trajectory tracking. Its main advantage lies in ensuring high-precision trajectory tracking without relying on a precise mathematical model of the controlled object. Thanks to this model-free characteristic, ILC has been widely applied in various periodic operation scenarios. Its applications extend beyond electromechanical systems such as industrial robotic arms and high-speed trains, showing broad prospects in biomedical engineering fields such as rehabilitation exoskeletons and functional electrical stimulation.

[0003] In practical engineering applications, the performance of ILC is often constrained by a number of non-ideal factors. First, actuators typically have physical saturation limits, preventing the control input from increasing indefinitely. Ignoring this nonlinear constraint in algorithm design will significantly reduce tracking accuracy and even disrupt closed-loop stability. Second, the runtime may vary randomly between different batches due to equipment protection, task changes, or external events. Furthermore, the initial state is difficult to reset precisely at each startup, resulting in random offsets; and the system is inevitably affected by external disturbances during operation. The simultaneous presence of these factors makes it difficult for traditional ILC methods to guarantee convergence.

[0004] To address issues such as variable iteration length, random initial state shifts, and non-repetitive perturbations, various improvement schemes have been proposed in existing research. For example, Chinese invention patent application CN121500787A discloses a high-order iterative learning control method for nonlinear non-repetitive systems. This method uses a Bernoulli random variable to describe the length of the iterative trajectory and constructs a high-order learning law using control inputs from multiple previous iteration cycles and corrections to the tracking error. This allows the tracking error to converge to a bounded region in a mathematically expected sense. However, this scheme, as well as most existing iterative learning control methods, do not consider the practical constraint of actuator input saturation. When the control command exceeds the amplitude achievable by the actuator, saturation nonlinearity will disrupt the linear update relationship of the control quantity, causing the theoretical convergence condition to no longer hold, and may even lead to system instability.

[0005] Furthermore, directly introducing existing saturation handling methods into variable-length high-order feedback control architectures is problematic. The lack of effective memory and adaptive compensation mechanisms for saturation overflow energy easily leads to severe integral saturation and control signal chattering in nonlinear systems. It also makes it difficult to mathematically establish quantitative convergence conditions that include saturation and disturbance bounds. Therefore, there is an urgent need in this field for a novel high-order composite control architecture capable of actively extracting and compensating for saturation residuals and possessing dynamic adaptive adjustment capabilities to overcome the analytical and control obstacles posed by the aforementioned complex operating conditions. Summary of the Invention

[0006] The purpose of this invention is to provide a high-order iterative learning control method for saturation disturbances in variable-length nonlinear systems. This method can solve the problem of how to design an iterative learning control law that combines high-order feedforward and instantaneous feedback under the coupled effects of four non-ideal factors: input saturation, variable iteration length, random initial state offset, and external disturbances. This allows the tracking error of the nonlinear system to converge to an acceptable bounded range in the mathematical expectation sense.

[0007] To achieve the above objectives, the present invention provides the following technical solution: a high-order iterative learning control method for saturation disturbance of a variable-length nonlinear system, comprising the following steps: Step 1: Establish a discrete-time affine nonlinear model that is affected by input saturation and external disturbances and runs repeatedly across multiple iterations; Step 2: Define the reference output trajectory and reference control input; Step 3: Construct an extended control input sequence to handle non-uniform iteration lengths; Step 4: For the case where the iteration time varies randomly, introduce a random variable that follows a Bernoulli distribution to correct the trajectory tracking error; Step 5: Design a high-order feedback iterative learning control law: This high-order feedback iterative learning control law is composed of a high-order ILC feedforward part and a feedback control part that instantaneously compensates for the current iteration tracking error; wherein, the high-order ILC feedforward part integrates past... The saturated extended control input and tracking error correction of each historical iteration are implemented, and the gain matrix is ​​learned through time-varying high-order feedforward. and Adjustments are made, and saturation residuals caused by actuator constraints in historical iterations are extracted. An anti-saturation dynamic compensation matrix is ​​then introduced. Adjusted adaptive anti-integral saturation compensation term; feedback control part controls the gain moment through state-dependent feedback control. The array adjusts the correction tracking error for the current iteration; Step 6: Apply the high-order feedback iterative learning control strategy to the controlled object; use the tracking deviation corrected in each iteration to update the control input for subsequent iterations, ultimately ensuring that the expected tracking error can converge to a bounded region.

[0008] Preferably, in step one, the formula for the discrete-time affine nonlinear model is expressed as: (1); in, and These represent the iteration order and the discrete time step, respectively. This is the actual iteration length; , , These are defined as the system's state, control signals, and output variables, respectively. The range of values ​​is ; , , Represents a nonlinear function; For input saturation function; External disturbances And it is assumed that it exists in all iteration orders. With time step Uniformly bounded, meaning there exists an unknown positive constant. , making .

[0009] Preferably, at the beginning of each iteration, the system state is... The mathematical expectation in the reference state Reset nearby, and meet the following conditions: (2); in, It is a positive constant.

[0010] Preferably, in step two, the definition formula for the reference output trajectory is: (10); in, ; For reference only; The reference iteration length; for feasible reference output trajectories Assuming there is a unique reference control input This makes the reference state equations satisfy: (11); in, , , Represents a nonlinear function; For reference control input.

[0011] Preferably, in step three, the formula for constructing the extended control input sequence is as follows: when hour: (13); in, This is the actual iteration length; The reference iteration length; For system control signals; when hour: (14); Accordingly, the saturation function for the extended control input is defined as follows: when hour: (15); when hour: (16); in, For input saturation function; To extend the control input saturation function.

[0012] Preferably, in step four, the introduced random variable follows a Bernoulli distribution. , ; This indicates that the system continues to run until time step z in the m-th iteration, with a probability of . ; 0 indicates that the system terminated prematurely before time step z, with a probability of 0. Therefore, the expected value of this Bernoulli random variable is calculated as follows: (17); The formula for correcting tracking error is defined as follows: (18); in, ; The formula for calculating the initial tracking error is: ; ; The output trajectory is used as a reference. These are the system's output variables.

[0013] Preferably, the calculation formula for the higher-order feedback iterative learning control law is as follows: (twenty one); (twenty two); (twenty three); in, This represents the feedforward portion of the higher-order ILC. Represents the order of the learning law; To extend the control input saturation function; For actuator saturation residual; To correct tracking errors; and All are time-varying high-order feedforward learning gain matrices; This is the anti-saturation dynamic compensation matrix; For the feedback control section, This is the state-dependent feedback control gain matrix.

[0014] Preferably, the time-varying high-order feedforward learning gain matrix and The following conditions must be met: (twenty four); (25); (26); in, For equivalent Jacobian matrices; For reference only; It is a nonlinear function; Let be the mathematical expectation of the Bernoulli random variable; It is a contraction factor; Defined as the anti-saturation constraint factor. ,in, For actuator saturation residual, To extend the control input error, This represents the ratio of the residual, determined by the characteristics of the saturation function, to the norm limit of the input error.

[0015] Preferably, under ideal conditions, i.e., the system is free from external disturbances ( And it meets the ideal initial state learning conditions ( When (=0), the mathematical expectation of the tracking error asymptotically converges to zero, satisfying the following formula: (41); in, The output trajectory is used as a reference. These are the system's output variables.

[0016] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. This invention simultaneously considers four types of non-ideal factors commonly found in practical engineering: input saturation, variable iteration length, random initial state deviation, and external disturbances. Theoretical analysis proves that under this combined constraint, the expected tracking error can gradually converge to a bounded region related to the amplitude of the external disturbance and the degree of initial state fluctuation, rather than relying on idealized assumptions of no disturbance, fixed length, and precise initial state reset.

[0017] 2. This invention addresses the problem of missing historical information caused by variable iteration length by designing a dynamic adaptive high-order feedback iterative learning control law with a saturated residual memory mechanism. On one hand, it utilizes a high-order feedforward part to synthesize data from multiple historical iterations, extracts and memorizes saturated overflow residuals for adaptive reverse compensation, effectively compensating for the lack of global information in the iteration domain. On the other hand, it introduces a state-dependent feedback control part to suppress local fluctuations in the time domain using the tracking error of the current iteration. The two work together to overcome the shortcomings of pure feedforward high-order ILC, which fails to compensate when local errors are persistently large.

[0018] 3. This invention addresses the stochastic uncertainties caused by non-uniform iteration lengths and random initial state shifts by employing convergence analysis in the mathematical expectation sense, combined with... By combining norms and mathematical induction, and effectively decoupling nonlinear saturation functions and random packet loss, we have theoretically proven that the expected tracking error converges to a strictly bounded region, the boundary of which is explicitly determined by the magnitude of the external perturbation and the upper bound of the initial state offset. This convergence framework avoids the strong conditions of existing methods, such as reliance on ergodicity or almost certain convergence, and successfully overcomes the theoretical defect that the traditional linear error update law fails under saturation conditions.

[0019] 4. This invention ensures that the expected tracking error asymptotically converges to zero under ideal conditions where there are no external disturbances and the mathematical expectation of the random initial state offset is zero. This demonstrates that the invention does not sacrifice the accurate tracking performance of traditional ILC under ideal conditions, reflecting the theoretical completeness of the proposed robust control architecture. Attached Figure Description

[0020] Figure 1 This is a flowchart illustrating the high-order iterative learning control method in Embodiment 1 of the present invention. Figure 2 In Embodiment 2 of the present invention, the random iteration length With the number of iterations The curve of change; Figure 3 In Embodiment 2 of the present invention, the actual initial state (Top part) and (Lower part) Curve showing the change with the number of iterations; Figure 4In Embodiment 2 of the present invention, under different feedback learning gains in Case 1, the cumulative tracking error is... (The upper part) and (Lower part) Actual variation curve with iteration number; Figure 5 In Embodiment 2 of the present invention, under the low feedback learning gain of Case 1, in and At the 20th and 37th iterations (The upper part) and (The following section describes the actual dynamic tracking response;) Figure 6 In Embodiment 2 of the present invention, under the benchmark feedback learning gain of Case 1, in and At the 20th and 37th iterations (The upper part) and (The following section describes the actual dynamic tracking response;) Figure 7 In Embodiment 2 of the present invention, under the high feedback learning gain of Case 1, in and During the 20th and 37th iterations (The upper part) and (The following section describes the actual dynamic tracking response;) Figure 8 In case 2 of this invention, the initial state is referenced. (Top part) and (Lower part) Curve showing the change with the number of iterations; Figure 9 In Embodiment 2 of the present invention, under the benchmark feedback learning gain of Case 2, the cumulative tracking error (The upper part) and (Lower part) Reference curve showing the change with the number of iterations; Figure 10 In Embodiment 2 of the present invention, under the benchmark feedback learning gain of Case 2, during the 20th and 37th iterations... (The upper part) and (Lower section) Reference dynamic tracking response; Figure 11 In Embodiment 3 of the present invention, the cumulative tracking error of the two-link planar robotic arm is... (The upper part) and (Lower part) Actual variation curve with iteration number; Figure 12 In Embodiment 3 of the present invention, the two-link planar robotic arm, during the 20th and 37th iterations... (The upper part) and (Lower section) Reference dynamic tracking response. Detailed Implementation

[0021] 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.

[0022] Example 1: Control method.

[0023] Please see Figure 1 A high-order iterative learning control method for saturation disturbances in a variable-length nonlinear system includes the following steps: Step 1: Establish a discrete-time affine nonlinear model that is affected by input saturation and external disturbances and repeats across multiple iterations; the formula for the discrete-time affine nonlinear model is expressed as: (1); in, and These represent the iteration order and the discrete time step, respectively. This is the actual iteration length; , , These are defined as the system's state, control signals, and output variables, respectively. The range of values ​​is ; , , Represents a nonlinear function; For input saturation function; External disturbances And it is assumed that it exists in all iteration orders. With time step Uniformly bounded, meaning there exists an unknown positive constant. , making .

[0024] Assumption 1: At the beginning of each iteration, the system state is... The mathematical expectation in the reference state Reset nearby, and meet the following conditions: (2); in, It is a positive constant.

[0025] Assumption 2: For the discrete-time affine nonlinear model (1), assume the nonlinear function... , , Regarding global Lipschitz continuity of state variables. Specifically, for any of and There exists a nonnegative Lipschitz constant. , and This makes the following relationship hold: ; ; (3); ; The global Lipschitz condition described above is a fundamental requirement for verifying the convergence of tracking errors.

[0026] Lemma 1: Suppose that for all Defined real-valued sequence ,when The following inequality must be satisfied: (4); in, ( Given a sequence of real values, assume non-negative coefficients. The following conditions must be met: (5); Therefore, if Then we can obtain: (6).

[0027] Lemma 2: Consider the following recursive form of the discrete-time difference inequality: (7); in, and The representation is defined at discrete time steps. nonnegative scalar functions on, Let be a given positive constant. Based on this recursive relation, for any time step... 1. Function The following limits must be met: (8).

[0028] Lemma 3: Assume the system's reference control input Strictly limited to the physical saturation boundary, i.e. Located in the interval Internally, based on this, for sets any discrete time step The following inequalities hold: (9).

[0029] Step 2: Define the reference output trajectory and reference control input. The formula for defining the reference output trajectory is: (10); in, ; For reference only; The reference iteration length; for feasible reference output trajectories Assuming there is a unique reference control input This makes the reference state equations satisfy: (11); in, , , Represents a nonlinear function; For reference control input.

[0030] In formula (1), the input saturation function operates element-wise and is defined as follows: (12); Where, vector and These represent the lower and upper limits of input saturation, respectively.

[0031] Step 3: Construct an extended control input sequence to handle non-uniform iteration lengths; To effectively handle input constraints and iterative periods of random fluctuations during ILC synthesis of nonlinear discrete-time systems, this invention constructs an extended control input sequence. Specifically, when When, use the zero-fill method; when When this happens, the sequence is truncated to match the reference length. The formula for constructing the extended control input sequence is as follows: when hour: (13); in, This is the actual iteration length; The reference iteration length; For system control signals; when hour: (14); Accordingly, the saturation function for the extended control input is defined as follows: when hour: (15); when hour: (16); in, For input saturation function; To extend the control input saturation function.

[0032] Step 4: For the case where the iteration time varies randomly, introduce a random variable that follows a Bernoulli distribution to correct the trajectory tracking error; To describe the characteristics of random variable iteration length, a random variable following a Bernoulli distribution is introduced. , ; This indicates that the system continues to run until time step z in the m-th iteration, with a probability of . ; 0 indicates that the system terminated prematurely before time step z, with a probability of 0. Therefore, the expected value of this Bernoulli random variable is calculated as follows: (17); Due to the non-uniformity of the actual running time, the initial tracking error... Only during public time periods It contains a clear definition. The output trajectory is used as a reference. These are the system's output variables.

[0033] To facilitate theoretical analysis on a unified time scale, the formula for correcting tracking error is defined as follows: (18); in, ; Based on the actual iteration length With reference length The relationship between these factors and the correction of tracking error are explicitly defined in the following two cases: when (In case of early termination): (19); when hour: (20).

[0034] Step 5: Design a high-order feedback iterative learning control law: This high-order feedback iterative learning control law is composed of a high-order ILC feedforward part and a feedback control part that instantaneously compensates for the current iteration tracking error; wherein, the high-order ILC feedforward part integrates past... The saturated extended control input and tracking error correction of each historical iteration are implemented, and the gain matrix is ​​learned through time-varying high-order feedforward. and Adjustments are made, and the saturation residuals caused by actuator constraints in historical iterations are extracted. A dynamic compensation matrix is ​​then introduced. Adjusted adaptive anti-integral saturation compensation term; feedback control part controls the gain moment through state-dependent feedback control. The array adjusts the tracking error correction for the current iteration.

[0035] To solve the trajectory tracking problem of a nonlinear discrete-time system constrained by input saturation and undergoing stochastically variable iteration lengths, the following high-order feedback iterative learning control law is designed. For the time step... The formula for calculating the higher-order feedback iterative learning control law is: ;(twenty one); ;(twenty two); ;(twenty three); in, This represents the feedforward portion of the higher-order ILC. Represents the order of the learning law; To extend the control input saturation function; For actuator saturation residual; To correct tracking errors; and All are time-varying high-order feedforward learning gain matrices; This is the anti-saturation dynamic compensation matrix; For the feedback control section, This is the state-dependent feedback control gain matrix. This means the controller incorporates past data... Saturated expansion control input and correction tracking error for each historical iteration.

[0036] Under the condition that assumptions 1 and 2 hold, the learning gain matrix and The following conditions must be met: (twenty four); (25); (26); in, For equivalent Jacobian matrices; For reference only; It is a nonlinear function; Let be the mathematical expectation of the Bernoulli random variable; It is a contraction factor; Defined as the anti-saturation constraint factor. ,in, For actuator saturation residual, To extend the control input error, This represents the ratio of the residual, determined by the characteristics of the saturation function, to the norm limit of the input error.

[0037] In this case, the mathematical expectation of the tracking error has an asymptotic bound. This final bound is related to the upper bound of the initial state offset (denoted as ). ) and the boundary of external disturbances (denoted as All of them are related.

[0038] Step 6: Apply the high-order feedback iterative learning control strategy to the controlled object; use the tracking deviation corrected in each iteration to update the control input for subsequent iterations, ultimately ensuring that the expected tracking error can converge to a bounded region.

[0039] To prove that under the above control law and convergence conditions, the mathematical expectation of the tracking error can converge to the bounded region, the key derivation steps are given below: Define the following error variable: State error Feedforward control error and correction of tracking error .

[0040] Subtracting the system model (1) from the reference state equation (11), the tracking error dynamics of the system are obtained as follows: For a nonlinear output function, the tracking error is formulated as follows: ;in, and They represent The expected output and actual output at time t; using the mean value theorem for multivariable functions, this nonlinear difference can be expanded as: ;(28); Define the equivalent Jacobian matrix as ;in, This indicates that it is strictly within the state vector. Compared with reference state Based on Assumption 2 and the definition of state error, the above nonlinear difference is rewritten as the following equivalent method: (29); Simplifying formula (27), we get formula (30): (30); Input the reference control Subtracting the control law (22) yields the feedforward error. Substitute From (29), we can obtain formula (31): (31); Substituting equation (30) into (31) and factoring out the common factor, the feedforward control error can be rewritten as formula (32): (32); By taking the norm of both sides of formula (32), and using the inequality... Lemma 3, and the upper bounds of the saturation function and external perturbation are respectively defined as... and ;in, Using Hypothesis 2 and Jacobian matrix bounds And reduce the residual compensation term to ,in, For actuator saturation residual, To extend the control input error, The ratio of the residual to the norm limit of the input error, determined by the characteristics of the saturation function, can be expressed by formula (33): (33); Taking the mathematical expectation operator E{·} on both sides of formula (33), and considering the convergence condition (25), we can obtain the following formula (34): (34); Taking the norm of both sides of (30) and using Assumption 2 and Lemma 3, and Thus, we obtain formula (35): (35); Taking the mathematical expectation operator E{·} on both sides, and combining it with Assumption 1 and Lemma 2, we can obtain formula (36): (36); The upper bound of the constant is defined as ,and .

[0041] Substituting (36) into (34) and performing algebraic operations, we take the upper limit as m→∞, and after further algebraic operations, we can obtain the asymptotic convergence upper bound of the feedforward error: (37); Taking the upper limit of both sides of (36) as m approaches infinity, we can obtain the asymptotic bound formula for the state error (38): (38); Furthermore, utilizing the system output equation and the Lipschitz continuity defined in Hypothesis 2, and combining it with (27), the norm of the output tracking error satisfies the following scaling relation formula (39): (39); Taking the expected value and upper limit of both sides of (39), and substituting them into (38), we get formula (40): (40); Under ideal conditions, i.e., the system has no external disturbances ( And it meets the ideal initial state learning conditions ( When σ = 0, we can obtain σ = 0, and the mathematical expectation of the tracking error asymptotically converges to zero, satisfying the following formula: (41); in, The output trajectory is used as a reference. These are the system's output variables.

[0042] Example 2: Computer simulation case.

[0043] To verify the effectiveness of the high-order feedback iterative learning control law in handling variable iteration length, random initial state shift, input saturation, and external disturbances, an affine nonlinear discrete-time MIMO system with composite constraints is established, and the specific formula (42) is as follows: (42); Among them, time step Please see Figure 2 As shown, the running length of each iteration In sets between different iterations Internal random variation, with the initial state set as a reference. Please see. Figure 3 As shown, in each iteration, the actual initial state Allowed in Random fluctuations within a range. Saturation function. The physical boundaries of the actuator are restricted, with its maximum amplitude threshold set at 15.0.

[0044] In addition, the system's input gain matrix External state perturbation that changes with iteration The specific definitions are as follows: (43); (44); The reference output trajectories of system (42) are denoted as follows: and Its expression is: (45); (46); In the simulation settings, the total number of iterations is set to 80. It is typically assumed that this applies to all time steps. The initial control inputs all satisfy .

[0045] To evaluate the trajectory tracking performance of the system, this embodiment uses cumulative absolute tracking error as a performance indicator: (47); in, Indicates the first The output channel is in the first The cumulative tracking error at the next iteration For reference, output trajectory Indicates the system at the 1st The output response at the next iteration.

[0046] Formulas (21), (22), and (23) of the higher-order feedback iterative learning control law are applied to this system. Let the order of the higher-order controller be... The weight matrix is ​​selected as follows and The feedforward learning gain matrix is ​​set to... and Meanwhile, to investigate the impact of feedback learning gain on convergence performance, the feedback gain matrix was set as follows: , and .

[0047] Scenario 1: Evaluating the system's tracking performance in a real-world complex environment. In this scenario, we examine the system's tracking capability under three different feedback learning gain matrices. First, when using a low feedback learning gain matrix... Please refer to Figure 4 As shown, in the early stage of the iteration process, the accumulated tracking error... They exhibit a consistent downward trend. Please refer to [link / reference]. Figure 5 The figure shows the dynamic tracking response at the 20th and 37th iterations. It can be seen that, under lower feedback strength, the higher-order feedback iterative learning control law can gradually bring the system closer to the reference trajectory. However, due to insufficient suppression of variable iteration lengths and complex perturbations, the output of the actual system fluctuates within a relatively wide residual neighborhood.

[0048] When the feedback gain matrix is ​​increased to the baseline setting At that time, transient tracking performance was significantly improved; please refer to [link / reference]. Figure 4 and Figure 6 As shown, the cumulative tracking error decreases faster compared to the low-gain configuration. Subsequently, with the number of iterations... As the value increases, the actual output trajectory fluctuates within a significantly smaller range. This confirms that an appropriate baseline feedback mechanism can effectively reduce local fluctuations caused by changes in the iterative trajectory, thereby forcing the system to accurately follow the reference trajectory within a narrower residual neighborhood.

[0049] Finally, when examining the high-feedback learning gain matrix For information on system performance, please refer to [link / reference]. Figure 4 and Figure 7 As shown, the high-gain setting not only achieves the fastest transient convergence speed, but also gradually confines the accumulated tracking error to a very small residual region as the number of iterations increases. This indicates that a larger feedback learning gain significantly improves the system's ability to withstand different running lengths and complex external disturbances, thereby minimizing the steady-state tracking error and exhibiting the best overall tracking performance among the three feedback configurations.

[0050] Scenario 2: Evaluating the system's tracking performance under ideal conditions. In this scenario, a benchmark feedback gain matrix is ​​used. The near-perfect tracking performance of the system under ideal conditions is evaluated. In this configuration, by setting... Set to zero to eliminate external disturbances. Simultaneously, retain the variable iteration length and, as... Figure 8 The small initial state fluctuations are shown. Please refer to [link / reference]. Figure 9 As shown, in the absence of strong interference, the cumulative tracking error rapidly approaches zero. Nevertheless, due to the persistent effects of inherent non-uniformity, Figure 10 The output trajectories of the 20th and 37th iterations show that there are still extremely small dynamic deviations in the actual output. This not only verifies the effectiveness of the benchmark control law in suppressing residual oscillations, but also further reveals the objective constraint of the system's inherent uncertainties on steady-state accuracy.

[0051] Example 3: Application of a two-link planar robotic arm system.

[0052] To further verify the application potential of the method of this invention in practical physical systems, this embodiment introduces a two-link planar robotic arm system for verification. The continuous-time nonlinear dynamic equations of this system are described as follows: (48); in, and These represent the angular displacement and angular velocity of the two joints, respectively. The control torque applied to the joint motor; For continuous external disturbances; The inertia matrix; For the Coriolis and centrifugal force matrix, This is the gravity vector.

[0053] Using sampling time The system is discretized using forward Euler discretization. The discrete-time state vector is defined as follows: The system output is its joint angular displacement. After discretization mapping, the physical system can be transformed into the discrete-time affine nonlinear form described by formula (1) in this embodiment: (49); Among them, control input For discrete control torque, the nonlinear system function and input matrix are specifically as follows: (50); (51); The physical parameter settings are as follows: 1. Robotic arm hardware parameters: Link mass Linkage length Distance from center of mass to joint Moment of inertia gravitational acceleration The discretization sampling step size is set to... .

[0054] 2. Reference Trajectory and Variable Iteration Length: The system is expected to track a smooth, periodic joint trajectory within one cycle (1 s). (52); Reference expected time step Due to the influence of actual working conditions, the iteration runtime is within the set. The inner distribution follows a uniform random distribution.

[0055] 3. Actuator saturation threshold: Considering the motor output limit, the saturation boundaries of joint 1 and joint 2 are set as follows: minimum boundary .

[0056] 4. Time-varying external disturbances: The combined external disturbances in the system, such as sudden load changes and unmodeled friction, are defined as follows: (53); in, , for Random white noise that is uniformly distributed between them.

[0057] 5. Initial state random offset: reference initial state The actual initial position and velocity in each iteration have random assembly deviations, which are respectively in... and Uniform fluctuation within.

[0058] For two-link planar robotic arm systems, please refer to Figure 11 The cumulative tracking error of the two-link planar robotic arm (The upper part) and (Lower section) Reference curve showing the change with the number of iterations; please refer to [link / reference]. Figure 12 For the two-link planar robotic arm during the 20th and 37th iterations (The upper part) and (Lower section) Reference dynamic tracking response.

[0059] 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.

[0060] 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 saturation disturbance of a variable-length nonlinear system, comprising the following steps: Step 1: Establish a discrete-time affine nonlinear model that is affected by input saturation and external disturbances and runs repeatedly across multiple iterations; Step 2: Define the reference output trajectory and reference control input; Step 3: Construct an extended control input sequence to handle non-uniform iteration lengths; Step 4: For the case where the iteration time varies randomly, introduce a random variable that follows a Bernoulli distribution to correct the trajectory tracking error; Step 5: Design a high-order feedback iterative learning control law: This high-order feedback iterative learning control law is composed of a high-order ILC feedforward part and a feedback control part that instantaneously compensates for the current iteration tracking error; wherein, the high-order ILC feedforward part integrates past... The saturated extended control input and tracking error correction of each historical iteration are implemented, and the gain matrix is ​​learned through time-varying high-order feedforward. and Adjustments are made, and saturation residuals caused by actuator constraints in historical iterations are extracted. An anti-saturation dynamic compensation matrix is ​​then introduced. Adjusted adaptive anti-integral saturation compensation term; feedback control part controls the gain moment through state-dependent feedback control. The array adjusts the correction tracking error for the current iteration; Step 6: Apply the high-order feedback iterative learning control strategy to the controlled object; use the tracking deviation corrected in each iteration to update the control input for subsequent iterations, ultimately ensuring that the expected tracking error can converge to a bounded region.

2. The high-order iterative learning control method for saturation disturbance of a variable-length nonlinear system according to claim 1, characterized in that: In step one, the formula for the discrete-time affine nonlinear model is expressed as follows: （1）； in, and These represent the iteration order and the discrete time step, respectively. This is the actual iteration length; , , These are defined as the system's state, control signals, and output variables, respectively. The range of values ​​is ; , , Represents a nonlinear function; For input saturation function; External disturbances And it is assumed that it exists in all iteration orders. With time step Uniformly bounded, meaning there exists an unknown positive constant. , making .

3. The high-order iterative learning control method for saturation disturbance of a variable-length nonlinear system according to claim 2, characterized in that: At the start of each iteration, the system state The mathematical expectation in the reference state Reset nearby, and meet the following conditions: （2）； in, It is a positive constant.

4. The high-order iterative learning control method for saturation disturbance of a variable-length nonlinear system according to claim 1, characterized in that: In step two, the definition formula for the reference output trajectory is: （10）； in, ; For reference only; The reference iteration length; for feasible reference output trajectories Assuming there is a unique reference control input This makes the reference state equations satisfy: （11）； in, , , Represents a nonlinear function; For reference control input.

5. The high-order iterative learning control method for saturation disturbance of a variable-length nonlinear system according to claim 1, characterized in that: In step three, the formula for constructing the extended control input sequence is as follows: when hour: （13）； in, This is the actual iteration length; The reference iteration length; For system control signals; when hour: （14）； Accordingly, the saturation function for the extended control input is defined as follows: when hour: （15）； when hour: （16）； in, For input saturation function; To extend the control input saturation function.

6. The high-order iterative learning control method for saturation disturbance of a variable-length nonlinear system according to claim 1, characterized in that: In step four, a random variable following a Bernoulli distribution is introduced. , ; This indicates that the system continues to run until time step z in the m-th iteration, with a probability of . ; 0 indicates that the system terminated prematurely before time step z, with a probability of 0. Therefore, the expected value of this Bernoulli random variable is calculated as follows: （17）； The formula for correcting tracking error is defined as follows: （18）； in, ; The formula for calculating the initial tracking error is: ; ; The output trajectory is used as a reference. These are the system's output variables.

7. The high-order iterative learning control method for saturation disturbance of a variable-length nonlinear system according to claim 1, characterized in that: The calculation formula for the higher-order feedback iterative learning control law is as follows: （21）； （22）； （23）； in, This represents the feedforward portion of the higher-order ILC. Represents the order of the learning law; To extend the control input saturation function; For actuator saturation residual; To correct tracking errors; and All are time-varying high-order feedforward learning gain matrices; This is the anti-saturation dynamic compensation matrix; For the feedback control section, This is the state-dependent feedback control gain matrix.

8. The high-order iterative learning control method for saturation disturbance of a variable-length nonlinear system according to claim 7, characterized in that: The time-varying high-order feedforward learning gain matrix and The following conditions must be met: （24）； （25）； （26）； in, For equivalent Jacobian matrices; For reference only; It is a nonlinear function; Let be the mathematical expectation of the Bernoulli random variable; It is a contraction factor; Defined as the anti-saturation constraint factor. ,in, For actuator saturation residual, To extend the control input error, This represents the ratio of the residual, determined by the characteristics of the saturation function, to the norm limit of the input error.

9. A high-order iterative learning control method for saturation disturbance of a variable-length nonlinear system according to claim 8, characterized in that: Under ideal conditions, i.e., the system has no external disturbances ( And it meets the ideal initial state learning conditions ( When (=0), the mathematical expectation of the tracking error asymptotically converges to zero, satisfying the following formula: （41）； in, The output trajectory is used as a reference. These are the system's output variables.

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