Iterative learning control optimization method for direct-current electric motor executing varying tasks

Abstract

The present invention relates to the field of direct-current electric motor control. Disclosed is an iterative learning control optimization method for a direct-current electric motor executing varying tasks. In the method, an iterative learning controller is added in parallel on the basis of a closed-loop feedback control system of a direct-current electric motor, and the direct-current electric motor control system is converted into a time-series input / output matrix model on the basis of a lifting technique. An optimal iterative learning control algorithm is designed under a norm optimization framework, and the system basically realizes zero-error tracking of a desired output by combining inter-batch repeated learning with intra-batch real-time feedback. On the basis of an optimal input sequence and an error sequence which are acquired by means of repeatedly executing a certain task, the least squares fitting method is used to integrate a feedback-plus-feedforward controller into a new learning-based feedback controller, which is finally applied to a system for executing a varying-trajectory task. The method transfers historical learning experience to a brand-new task, without limiting the time length thereof, and thus the trajectory tracking for varying tasks of a direct-current electric motor can be realized without the need for re-learning.

Description

An Iterative Learning Control Optimization Method for DC Motors Performing Changing Tasks Technical Field

[0001] This invention relates to the field of DC motor control technology, and in particular to an iterative learning control optimization method for DC motors performing changing tasks. Background Technology

[0002] DC motors are widely used in industrial automation, robotics, and precision motion control due to their excellent speed regulation and stable operation. In these scenarios, many tasks exhibit periodic or repetitive characteristics, making it difficult for classic PID feedback control methods to maintain ideal control performance in the face of dynamic system changes.

[0003] For such tasks, iterative learning control is an effective control method because it can gradually optimize control performance through experience learning between tasks. It can use error information from previous batches of trials to update the control input for the next batch, achieving high-precision tracking of the desired trajectory. Traditional iterative learning control methods typically rely on the fixedness of the reference trajectory, and their control performance degrades significantly as the reference trajectory changes. This limitation makes traditional iterative learning control ill-suited for dynamically changing task requirements, especially in the field of DC motor control, where dynamic changes in the reference trajectory may stem from complex industrial environments, real-time task adjustments, and uncertain external disturbances. Therefore, designing an iterative learning controller to track unknown changing trajectories in DC motor systems with uncertain disturbances and performing changing tasks is a research area of ​​practical significance.

[0004] In practical industrial control systems, iterative learning control, as an open-loop control method, struggles to guarantee system control performance under the influence of uncertain disturbances. Therefore, it is necessary to combine it with appropriate mechanisms to eliminate the impact of non-repetitive disturbances in order to achieve good tracking performance. Existing solutions employ complex control algorithms, and their control effectiveness is difficult to guarantee when faced with unknown changes in the system's reference trajectory.

[0005] Therefore, in this application, based on the historical learning capability of iterative learning control and the real-time disturbance suppression capability of PID feedback control, we consider combining the two to construct feedback-based iterative learning control, so as to gradually optimize the control performance of the system and transfer the learning experience of repeatedly executing known tasks to new tasks to achieve tracking control of changing trajectories. Summary of the Invention

[0006] The purpose of this invention is to solve the trajectory tracking problem of DC motors performing unknown and changing tasks. A feedback-based iterative learning control optimization method for DC motors is proposed. This method designs a feedback-based iterative learning control algorithm to optimize system performance, transferring experience gained from repeatedly operating specific tasks to entirely new tasks without limiting the time duration. The feedback-based iterative learning control adopts a two-dimensional control structure with parallel configuration characteristics. By combining inter-experiment iterative learning and real-time feedback during experiments, the system repeatedly learns to obtain optimal performance. Then, an identification regression algorithm is designed to integrate the feedback and feedforward controllers into a new learning-based feedback controller to track the changing trajectories of different tasks.

[0007] The technical solution of the present invention is as follows:

[0008] An iterative learning control optimization method for a DC motor performing a changing task includes the following steps:

[0009] The first step is to establish a dynamic model of the DC motor position servo control system, including:

[0010] The dynamic model of a DC motor position servo control system typically consists of the motor's electrical and mechanical equations, primarily describing the conversion relationship between input voltage and motor speed. Assuming the system is ideal and considering the effects of motor inertia, frictional torque, etc., the physical model of the DC motor system is established as follows:

[0011] Among them, R f L f C represents the armature resistance and armature inductance of a DC motor, respectively. M C is the torque constant of the motor. e J is the back electromotive force constant of the motor. r C f V represents the moment of inertia and coefficient of friction of the mechanical load, respectively. f and i f These represent the armature voltage and armature current, respectively, and ω represents the motor speed. Indicates the rotation angle of the motor;

[0012] The second step is to construct the discrete state-space equations of the DC motor position servo control system, including:

[0013] The armature current, speed, and rotation angle of a DC motor are defined as states. Define the motor's rotation angle as the output. Armature voltage is input u(t) = V f If (t), then equation (1) can be transformed into a state equation form:

[0014] Discretize the DC motor continuous system model (2) and select a sampling period T that satisfies Shannon's sampling theorem. s The discrete state-space model of the DC motor is obtained as follows:

[0015] Where the subscript k represents the number of trials, t = 0, 1, 2, ..., N represents the t-th sampling point, and N represents the sampling point length from the start of the system operation to the end of the current batch; u k (t), y k (t) and x k (t) represent the input, output, and state vectors at the t-th sampling point of the k-th batch of the system, respectively; the system matrices A, B, and C have suitable dimensions and satisfy CB≠0 to ensure system controllability; without loss of generality, the system state x k (t) should be reset to the same initial value at the end of each batch, i.e., x k (0) = x0;

[0016] The third step is to establish a feedback-based iterative learning control trajectory tracking model, including:

[0017] For a linear discrete system (3), its state-space expression can be converted into a discrete transfer function in the discrete time domain: H(z)=C(zI-A) -1 B (4)

[0018] Where z is the discrete transfer operator, I is an identity matrix of appropriate dimension; and an appropriate feedback controller C is selected. fb (z) To ensure system stability, an iterative learning controller is connected in parallel to the DC motor position servo control system to optimize system control performance. In this case, the system input consists of two parts: a stable feedback output and a feedforward iterative learning control input.

[0019] On the iteration axis, the iterative learning control input update law is defined as:

[0020] Where P and L are defined as the robust filter and the learning gain, respectively; e k (t+1) is the output error of the system at the (t+1)th sampling point of the kth batch;

[0021] On the time axis, the output of the PID feedback controller is defined as:

[0022] Where K p K i and K dThese are the proportional coefficient, integral coefficient, and derivative coefficient of the PID feedback controller, respectively.

[0023] The output of the system after incorporating iterative learning control can be expressed as:

[0024] Where y d (t) represents the desired trajectory of the system, G. c (z), G s (z) is a known linear discrete transitive operator;

[0025] By solving G c (z) and G s The unit impulse response of (z) yields the corresponding impulse transfer operator G. d (h) and G r (h) is as follows:

[0026] Where g i ,r i ,i=1,2,3,… are the corresponding impulse response coefficients, h -n Represents delay operations in the discrete-time domain;

[0027] Based on equation (8), the system output equation (7) is transformed into a boosting system framework for the iterative domain:

[0028] in: y k =[y k (1),y k (2),…,y k (N)] T (11)

[0029] The input-output transfer matrix G on the time series in each trial d The exogenous signal d and the exogenous signal d are denoted as follows: d = [d(1), d(2), ..., d(N)] T (13)

[0030] The expected output vector of the improved system is defined as: y d =[y d (1),y d (2),…,y d (N)] T (14)

[0031] Based on the output form of system (9), it can be seen that the performance analysis of feedback-based iterative learning control system can be transformed into the traditional iterative learning control design objective for research.

[0032] Step 4: Design a feedback-based iterative learning control trajectory tracking optimization algorithm, including:

[0033] Within the norm-optimal iterative learning control framework, the optimal input and error information of the feedback-based iterative learning control system are obtained by optimizing the multi-objective performance index function for each batch. This performance index function is defined as:

[0034] in:

[0035] The performance index function (15) consists of three parts: the tracking error of each batch of the system, the input change between two adjacent batches, and the control capability. The weight matrices Q, R, and S of these three components indicate their priority in the optimization process. By adjusting the values ​​of the weight matrices Q, R, and S, the optimal balance between error, control input change, and smoothness of the system can be found.

[0036] The optimal control input for the system can be obtained by minimizing the performance index function.

[0037] In Hilbert space, the induced norm of a matrix is ​​defined as follows:

[0038] Substituting equations (16), (17), and (19) into equation (15) respectively, we get:

[0039] make We can obtain:

[0040] Combining like terms yields:

[0041] Since matrices Q, R, and S are positive definite, then It is reversible. After rearranging equation (22), we can obtain the feedback-based iterative learning control input update law:

[0042] in:

[0043] The system combines repeated learning of inter-batch errors with feedback control of intra-batch real-time errors, and iteratively updates the system control input using equation (23) to obtain the optimal control input sequence. and tracking error sequence Where k maxThis represents the maximum number of iterations for the system.

[0044] Step 5: Integrate the feedback and feedforward controllers into a new learning-based feedback controller, including:

[0045] Based on the above two-dimensional parallel control framework, the optimal system control input U opt and tracking error E opt The least squares fitting method is used to calculate the parameters of the learning-based feedback controller; the PID controller is essentially linear, and its linear parameterization and the output at the j-th sampling point in the discrete time domain according to equation (6) are:

[0046] Where θ = [θ1, θ2, θ3] T u is the unknown parameter vector of the controller. fb (j) is the output at each sampling point and the system has N sampling points;

[0047] Define information vector The feedback system output can then be converted to: U fb =Φθ (26)

[0048] Among them: U fb =[u fb (1),u fb (2),…,u fb (N)] T (27)

[0049] By minimizing the model output U fb and actual output U opt The optimal parameter estimate of the model is derived from the sum of squared errors. That is, by minimizing the following quadratic criterion function:

[0050] The estimate of the parameter vector θ is obtained, where and For the observation data of a feedback-based iterative learning control system, J(θ) can be reformulated as: J(θ) = V T V=(U opt -Φθ) T (U opt -Φθ) (30)

[0051] Where V = [v(1), v(2), ... v(N)] Tv(i), i = 1, ..., N is the difference between the actual output and the model output at each time step;

[0052] set up hour, Setting the partial derivative of J(θ) with respect to θ to zero, we get:

[0053] or

[0054] When (Φ T When Φ is a positive definite matrix, it can be obtained from the above equation (32). Obtain parameter estimates for the learning-based feedback controller.

[0055] Step 6: Analyze the convergence of the feedback-based iterative learning control trajectory tracking optimization algorithm;

[0056] Step 7: Implement iterative learning control, guided feedback control, and learning trajectory tracking control, including:

[0057] Using the learning-based parameters obtained in step five above A new feedback controller is obtained and applied to a DC motor position servo control system performing different tasks for trajectory tracking control. Based on the optimal input and error information obtained in the two-dimensional iterative learning control system with feedback and feedforward, the feedback and feedforward controllers are integrated into a new learning-based feedback controller using the least squares fitting method. This controller can fit the input-output characteristics of the original system and can track and control changing trajectories without relearning. Through the above method, the learning experience of repeatedly performing a certain task can be transferred to a completely new task without limiting its time length, and the DC motor position servo control system can track and control changing trajectories.

[0058] Its further technical solution is to analyze the convergence of the feedback-based iterative learning control trajectory tracking optimization algorithm, including:

[0059] For the expected trajectory y d There exists a unique bounded expected control input u. d Satisfy y d =G d u d +d, defines the input error of the k-th trial. for:

[0060] The tracking error of the system in the k-th trial is:

[0061] According to equations (33) and (23), we can obtain:

[0062] Taking the norm of both sides of equation (35), we obtain the inequality:

[0063] From equation (24), the common factor R can be extracted to obtain Because of Q, R, S and G d Since it is a positive definite matrix, it is obvious that... Therefore, we can conclude that:

[0064] Define a positive scalar q that satisfies Equation (36) can be reformulated as:

[0065] Based on the inequality, after k iterations of the system, we can obtain:

[0066] in q represents the input error of the system before iteration. u =q||u d ||;If a suitable weight matrix is ​​chosen so that the constraint condition||L u -L e G d The condition ||≤ξ<1 holds true. When the number of trials k→∞, according to the lemma of contraction mappings, we can obtain... Therefore, equation (40) can be expressed as:

[0067] Combining equations (34) and (41), we can obtain:

[0068] Let ||G d ||=l, further we can obtain:

[0069] That is, the tracking error of the system can converge to a bounded value;

[0070] Its further technical solution is, in particular, when S is a zero matrix, L u =I, at this point the system can achieve perfect tracking, that is

[0071] The beneficial technical effects of this invention are:

[0072] For a DC motor position servo control system performing varying tasks, a two-dimensional parallel control framework is constructed by introducing an iterative learning controller in parallel on top of traditional feedback control. Through a combination of inter-experiment iterative learning and real-time feedback during experiments, the system iteratively learns and changes the control input to achieve high tracking accuracy while ensuring time-domain stability. Then, the feedback and feedforward controllers are integrated into a new learning-based feedback controller, enabling the transfer of learning experience gained from repeatedly executing known tasks to entirely new tasks, without limiting the time duration. Utilizing the real-time disturbance suppression capability of feedback control and the learning capability of iterative learning control, the influence of non-repetitive uncertainty disturbances during repetition can be eliminated. The integrated feedback controller, based on the system's optimal input information and error information, is applied to varying tasks and essentially achieves zero-error tracking of the desired trajectory. Attached Figure Description

[0073] Figure 1 is a block diagram of the parallel feedforward feedback empirical transfer ILC DC motor model in this application.

[0074] Figure 2 is a graph showing the actual output curves of the DC motor system before and after the introduction of iterative feedback control in this application.

[0075] Figure 3 is a convergence plot of the root mean square error of the feedback-based iterative learning control system in this application.

[0076] Figure 4 is a block diagram of the learning-based parameter estimation of the integrated feedback controller in this application.

[0077] Figure 5 is a comparison of the output of the optimized control algorithm proposed in this application and the original feedback control in tracking the changing trajectory on the DC motor.

[0078] Figure 6 is a comparison of the root mean square error of the algorithm proposed in this application and two different iterative optimization algorithms as a function of batch size.

[0079] Figure 7 shows the output response of the proposed algorithm on a DC motor under real-time interference conditions. Detailed Implementation

[0080] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0081] This application provides an iterative learning control optimization method for a DC motor performing a changing task. The specific implementation steps are described in steps one through seven of the invention description. Specifically, for the DC motor physical model shown in equation (1), it is assumed that the armature current and motor angular velocity do not have a direct dynamic coupling relationship. The parameters in the DC motor position control system are set as follows: R f =20Ω, L f =1H,C M =0.5Nm / A, Jr =2Nm / A, C f =2Nm / A. Furthermore, the system test duration is set to T = 10s, and the sampling time interval T... s =0.01s, then the parameter matrices of the discrete state-space equations of the system are as follows: C = [0 0 1]

[0082] The initial state of the system is set to x0 = [0 0 0]. T The initial control input u0 = 0. In this embodiment, the desired trajectory of the DC motor output within a finite time interval t ∈ [0, T] is set as: y d (t) = 1.25 × t(Tt)

[0083] First, Figure 1 shows the block diagram of the DC motor model disclosed in this application. Based on the feedback-based iterative learning control constructed in Figure 1, the performance of the motor system performing a known task is optimized. After obtaining the optimal control input and tracking error, the feedforward and feedback controllers are integrated into a new learning-based feedback controller to cope with the challenges of changing trajectory tasks. Starting from the initial feedback control system, to ensure the normal execution of the DC motor task, the classical ZN adjustment method is usually used to determine the initial feedback controller parameters of the system to ensure system stability. In this application, the initial feedback controller parameters are set as: K p =0.2,K i =0.1,K d =0.05

[0084] The next step combines feedback control and iterative learning control methods. The feedback-based iterative learning control design adopts a two-dimensional control structure with parallel configuration characteristics. The system input for the k-th experiment is: Choose a weight matrix Q = I, R = 0.1I, S = 0.01I, satisfying ||L u -L e G d ||≤ξ<1, according to equation (23), the system input is iteratively updated to act on the DC motor, and the output trajectory is continuously corrected until it tracks the desired trajectory. Figure 2 shows the corresponding DC motor trajectory tracking curve, which shows the learning capability of iterative learning control. Figure 3 is the root mean square error curve of the system tracking error, which shows that after several iterations, the system achieves bounded convergence.

[0085] Then, Figure 4 shows the block diagram of the equivalent feedback controller parameter estimation. Based on the optimal control input and error information obtained by the system executing the known task, the equivalent integrated learning-based feedback controller parameters are obtained according to Equation (32). The output curve of the DC motor system is shown in Figure 5. After iteratively learning several batches of the known trajectory, the desired trajectory can be tracked without relearning. Moreover, the time length is not limited for any changing trajectory, indicating that the method realizes the transfer of learning experience obtained by repeatedly executing a certain task to a new task. In addition, the iterative learning control algorithm of the proposed optimized feedback control system is compared with the other two methods. The root mean square error changes with the batch as shown in Figure 6. The feedback-based iterative learning control proposed in this application can make the system converge quickly. If a disturbance is suddenly added during the system iteration process, as can be seen from Figure 7, the system can maintain the original control performance after several batches of iteration, and the influence of real-time disturbance can be suppressed.

[0086] The above descriptions are merely preferred embodiments of this application, and the present invention is not limited to the above embodiments. It is understood that other improvements and variations directly derived or conceived by those skilled in the art without departing from the spirit and concept of the present invention should be considered to be included within the protection scope of the present invention.

Claims

1. An iterative learning control optimization method for a DC motor performing a changing task, characterized in that, The method includes: The first step is to establish a dynamic model of the DC motor position servo control system. Assuming the system is ideal and considering the effects of the motor's moment of inertia and frictional torque, the dynamic model is as follows: Among them, R f L f C represents the armature resistance and armature inductance of a DC motor, respectively. M It is the torque constant of the motor, C e It is the back electromotive force constant of the motor, J r C f V represents the moment of inertia and coefficient of friction of the mechanical load, respectively. f and i f These represent the armature voltage and armature current, respectively, and ω represents the motor speed. Indicates the rotation angle of the motor; The second step is to construct the discrete state-space equations of the DC motor position servo control system, including: The armature current, speed, and rotation angle of a DC motor are defined as states. Define the motor's rotation angle as the output. Armature voltage is input u(t) = V f (t), then equation (1) is transformed into a state equation form: Discretize the continuous state equation (2) of the DC motor and select a sampling period T that satisfies the Shannon sampling theorem. s The discrete state-space equations of the system are obtained as follows: Where the subscript k represents the number of trials, t = 0, 1, 2, ..., N represents the t-th sampling point, and N represents the sampling point length from the start of the system operation to the end of the current batch; u k (t), y k (t) and x k (t) represents the input, output, and state vectors at the t-th sampling point of the k-th batch of the system, respectively; the system matrices A, B, and C have appropriate dimensions and satisfy CB≠0 to ensure the system is controllable; The third step is to establish a feedback-based iterative learning control trajectory tracking model, including: For the discrete state-space equation (3), its state-space expression is converted into a discrete transfer function in the discrete time domain: H(z)=C(zI-A) -1 B (4) Where z is the discrete transfer operator, I is an identity matrix of appropriate dimension; and an appropriate feedback controller C is selected. fb (z) Ensure system stability and add the iterative learning controller in parallel to the DC motor position servo control system to optimize system control performance. At this time, the system input is controlled by stable feedback output. and feedforward iterative learning control input It consists of two parts, namely On the iteration axis, the iterative learning control input update law is defined as: Where P and L are defined as the robust filter and the learning gain, respectively; e k (t+1) is the output error of the system at the (t+1)th sampling point of the kth batch; On the time axis, the output of the PID feedback controller is defined as: Where K p K i and K d These are the proportional coefficient, integral coefficient, and derivative coefficient of the PID feedback controller, respectively. The output of the system after incorporating iterative learning control is represented as follows: Where y d (t) represents the desired trajectory of the system, G. c (z), G s (z) is a known linear discrete transitive operator; By solving G c (z) and G s The unit impulse response of (z) yields the corresponding impulse transfer operator G. d (h) and G r (h) is as follows: Where g i ,r i ,i=1,2,3,… are the corresponding impulse response coefficients, h -n Represents delay operations in the discrete-time domain; Based on equation (8), the output equation (7) of the system is transformed into a lifting system framework for the iterative domain: in: The input-output transfer matrix G on the time series in each trial d The exogenous signal d and the exogenous signal d are denoted as follows: d=[d(1),d(2),…,d(N)] T (13) The expected output vector of the improved system is defined as: and d =[and d (1), and d (2),…,and d (N)] T (14) Step 4: Design a feedback-based iterative learning control trajectory tracking optimization algorithm, including: Within the norm-optimal iterative learning control framework, the optimal input and error information of the feedback-based iterative learning control system are obtained by optimizing the multi-objective performance index function for each batch. This multi-objective performance index function is defined as follows: in: The multi-objective performance index function (15) includes: system tracking error per batch, input variation between two adjacent batches and control capability. The weight matrices Q, R and S of these three components indicate their priority in the optimization process. By adjusting the values ​​of the weight matrices Q, R and S, the optimal balance between error, control input variation and smoothness of the system can be found. The optimal control input of the system is obtained by minimizing the multi-objective performance index function: In Hilbert space, the induced norms of the three components are defined as follows: Substituting equations (16), (17), and (19) into equation (15) respectively, we get: make get: Combining like terms yields: Since matrices Q, R, and S are positive definite, then Reversible, equation (22) can be rearranged to obtain the feedback-based iterative learning control input update law: in: The system combines repeated learning of inter-batch errors with feedback control of intra-batch real-time errors, and iteratively updates the system control input using equation (23) to obtain the optimal control input sequence. and tracking error sequence Where k max This represents the maximum number of iterations for the system. Step 5: Integrate the feedback and feedforward controllers into a new learning-based feedback controller, including: The optimal control input U of the system under a two-dimensional parallel control framework opt and tracking error E opt The least squares fitting method is used to calculate the parameters of the learning-based feedback controller; the PID controller is essentially linear, so its linear parameterization and the output at the j-th sampling point in the discrete time domain according to equation (6) are: Where θ = [θ1, θ2, θ3] T u is the unknown parameter vector of the controller. fb (j) is the output at each sampling point and the system has N sampling points; Define information vector If j∈[1,N], then the output of the feedback system is transformed into: U fb =Fth (26) in: U fb =[u fb (1),u fb (2),…,u fb (N)] T (27) By minimizing the model output U fb and actual output U opt The sum of squared errors between them is used to derive the optimal parameter estimate of the model. That is, the estimate of the parameter vector θ is obtained by minimizing the following quadratic criterion function: in and For observational data of a feedback-based iterative learning control system; J(θ) is restated as follows: J(θ)=V T V=(U opt -Fth) T (U opt -Fth) (30) Where V = [v(1), v(2), ... v(N)] T v(i), i = 1, ..., N is the difference between the actual output and the model output at each time step; set up hour, Setting the partial derivative of J(θ) with respect to θ to zero, we get: or When (Φ T When Φ is a positive definite matrix, it can be obtained from equation (32) above. Obtain parameter estimates for the learning-based feedback controller. Step 6: Analyze the convergence of the feedback-based iterative learning control trajectory tracking optimization algorithm; Step 7: Implement iterative learning control, guided feedback control, and learning trajectory tracking control, including: Using the learning-based parameters obtained in step five above A new feedback controller is obtained and applied to a DC motor position servo control system that performs different tasks for trajectory tracking control; Based on the optimal input and error information obtained in the two-dimensional iterative learning control system with feedback and feedforward, the feedback and feedforward controllers are integrated into a new learning-based feedback controller using the least squares fitting method. This controller fits the input and output characteristics of the original system and can track and control the changing trajectory without relearning. The above method enables the transfer of learning experience from repeatedly performing a certain task to a completely new task without limiting the time length, allowing the DC motor position servo control system to track and control the changing trajectory.

2. The iterative learning control optimization method for a DC motor performing a changing task according to claim 1, characterized in that, The analysis is based on the convergence of the feedback-based iterative learning control trajectory tracking optimization algorithm, including: For the expected trajectory y d There exists a unique bounded expected control input u. d Satisfy y d =G d u d +d, defines the input error of the k-th trial. for: The tracking error of the system in the k-th trial is: According to equations (33) and (23), we get: Taking the norm of both sides of equation (35), we get the inequality: From equation (24), the common factor R is extracted to obtain Because of Q, R, S and G d For a positive definite matrix, we have Therefore, we get: Define a positive scalar q that satisfies Then equation (36) can be rewritten as: Based on the inequality, after k iterations of the system, the following results are obtained: in q represents the input error of the system before iteration. u =q||u d ||;If a suitable weight matrix is ​​chosen so that the constraint condition||L u -L e G d The condition ||≤ξ<1 holds true when the number of trials k→∞, according to the lemma of compression mappings, we obtain... Therefore, equation (40) can be expressed as: Combining equations (34) and (41), we get: Let ||G d ||=l, further yielding: Therefore, the tracking error of the system converges to a bounded value.

3. The iterative learning control optimization method for DC motors performing changing tasks according to claim 2, characterized in that: When S is a zero matrix, L u =I, at this point the system achieves perfect tracking, that is

Images