Iterative learning control optimization method of direct current motor for executing changing task

Abstract

The invention discloses a direct current motor iterative learning control optimization method for executing a changing task, and relates to the field of direct current motor control. According to the method, an iterative learning controller is added in parallel on the basis of a direct-current motor closed-loop feedback control system, and the direct-current motor control system is converted into an input and output matrix model of a time sequence based on a lifting technology. An optimal iterative learning control algorithm is designed under a norm optimization framework, and zero-error tracking of expected output is basically realized by the system through combination of inter-batch repeated learning and in-batch real-time feedback. Based on an optimal input sequence and an error sequence which are obtained by repeatedly executing a certain task, a feedback controller and a feedforward controller are integrated into a new feedback controller based on learning by utilizing a least square fitting method, and finally the new feedback controller is applied to a system for executing a change track task. According to the method, historical learning experience is transferred to a brand new task without limiting the time length, and trajectory tracking of the direct current motor change task can be realized without re-learning.

Description

Technical Field

[0001] The invention relates to the technical field of direct current motor control, and in particular to an iterative learning control optimization method for a direct current motor that performs a variable task. Background Art

[0002] DC motors have been widely used in industrial automation, robotics, precision motion control and other fields due to their excellent speed regulation performance and stable operation. In these scenarios, many tasks have periodic or repetitive characteristics, and the classic PID feedback control method is difficult to maintain ideal control performance in response to the dynamic changes of the system.

[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 the error information of the previous batch of the system to update the control input of the next batch in multiple repeated experiments, and achieve high-precision tracking of the desired trajectory. Traditional iterative learning control methods usually rely on the fixedness of the reference trajectory, and their control performance will significantly decrease as the reference trajectory changes. This limitation makes it difficult for traditional iterative learning control to cope with dynamically changing task requirements, especially in the field of DC motor control, where the dynamic changes of the reference trajectory may come from complex industrial environments, real-time task adjustments, and uncertain external disturbances. Therefore, for DC motor systems with uncertain interference and changing tasks, how to design iterative learning controllers to achieve tracking of unknown changing trajectories is a research with practical significance.

[0004] In actual industrial control systems, iterative learning control as an open-loop control method is difficult to guarantee system control performance under the influence of uncertain interference, so it is necessary to combine corresponding mechanisms to eliminate the influence of non-repetitive interference to achieve good tracking performance. The existing solution control algorithm is complex, and it is difficult to guarantee the control effect for unknown changes in the system reference trajectory.

[0005] Therefore, in this application, based on the historical learning ability of iterative learning control and the real-time interference suppression ability of PID feedback control, it is considered to combine the two to construct a feedback-based iterative learning control to gradually optimize the control performance of the system and transfer the learning experience of repeatedly performing known tasks to new tasks to achieve tracking control of changing trajectories. Summary of the invention

[0006] The purpose of the present invention is to solve the trajectory tracking problem of a DC motor that performs an unknown changing task, and proposes a feedback-based iterative learning control optimization method for a DC motor. In this method, a feedback-based iterative learning control algorithm is designed to optimize the performance of the system, and the experience gained from repeated operations of specific tasks is transferred to a completely new task without limiting its time length. The feedback-based iterative learning control adopts a two-dimensional control structure with the characteristics of parallel configuration. Through the combination of iterative learning between experiments and real-time feedback during the experiment, the system repeats learning to obtain the best performance, and then an identification regression algorithm is designed to integrate the feedback plus feedforward controller 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] A DC motor iterative learning control optimization method for performing a variable task comprises 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 the DC motor position servo control system usually consists of the electrical equations and mechanical equations of the motor, which mainly describe the conversion relationship between the input voltage and the motor speed. Assuming that the system is ideal and taking into account the influence of the motor's rotational inertia, friction torque, etc., the physical model of the DC motor system is established as follows:

[0011]

[0012] Among them, R f , L f Respectively represent the armature resistance and armature inductance of the DC motor, C M is the torque constant of the motor, C e is the back electromotive force constant of the motor, J r , C f Respectively represent the moment of inertia and friction coefficient of the mechanical load, V f and i f are the armature voltage and armature current respectively, ω represents the motor speed, and θ represents the motor angle;

[0013] The second step is to construct the discrete state space equation of the DC motor position servo control system, including:

[0014] The armature current, speed and angle of rotation of the DC motor are defined as state x(t) = [i f (t)ω(t)θ(t)] Τ , the motor's rotation angle is defined as output y(t) = θ(t), and the armature voltage is defined as input u(t) = V f(t), then equation (1) can be converted into the state equation form:

[0015]

[0016] Discretize the DC motor continuous system model (2) and select the sampling period T that satisfies Shannon sampling theorem. s , the discrete state space model of the DC motor is obtained as follows:

[0017]

[0018] The subscript k represents the number of tests, t=0, 1, 2, ..., N represents the corresponding t-th sampling point, and N represents the length of sampling points from the start of the system to the end of the current batch; u k (t), y k (t) and x k (t) are the input, output and state vector of the system at the tth sampling point of the kth batch respectively; the system matrices A, B and C have appropriate dimensions and satisfy CB≠0 to ensure the controllability of the system; 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;

[0019] The third step is to establish a feedback-based iterative learning control trajectory tracking model, including:

[0020] For the linear discrete system (3), its state space expression can be converted into a discrete transfer function in the discrete time domain:

[0021] H(z)=C(zI-A) -1 B (4)

[0022] Where z is a discrete transfer operator, I is an identity matrix of appropriate dimension; choose an appropriate feedback controller C fb (z) Ensure the stability of the system, and add the iterative learning controller in parallel to the DC motor position servo control system to optimize the system control performance. At this time, the system input consists of two parts: stable feedback output and feedforward iterative learning control input, that is,

[0023] On the iteration axis, the iterative learning control input update law is defined as:

[0024]

[0025] Among them, P and L are defined as robust filter and learning gain respectively; e k (t+1) is the output error of the system at the t+1th sampling point of the kth batch;

[0026] On the time axis, the PID feedback controller output is defined as:

[0027]

[0028] Where K p , K i and K d They are the proportional coefficient, integral coefficient and differential coefficient of the PID feedback controller respectively;

[0029] After adding iterative learning control, the output of the system can be expressed as:

[0030]

[0031] where y d (t) represents the expected trajectory of the system, G c (z), G s (z) is a known linear discrete transfer operator;

[0032] By solving G c (z) and G s The unit impulse response of (z) gives the corresponding impulse transfer operator G d (h) and G r (h) as follows:

[0033]

[0034] 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;

[0035] According to formula (8), the system output formula (7) is converted into the lifting system framework of the iteration domain:

[0036]

[0037] in:

[0038]

[0039] y k =[y k (1),y k (2),…,y k (N)] T (11)

[0040] The input-output transfer matrix G of the time series in each trial d and the exogenous signal d are recorded as:

[0041]

[0042] d=[d(1),d(2),…,d(N)] T (13)

[0043] The expected output vector of the improved system is defined as:

[0044] y d =[y d (1),y d (2),…,y d (N)] T (14)

[0045] According to the output form of system (9), it can be seen that the performance analysis of feedback-based iterative learning control systems can be transformed into traditional iterative learning control design objectives for research;

[0046] Step 4: Design a feedback-based iterative learning control trajectory tracking optimization algorithm, including:

[0047] In the norm optimal iterative learning control framework, the optimal input and error information of the feedback-based iterative learning control system is obtained by optimizing the multi-objective performance indicator function of each batch. The performance indicator function is defined as:

[0048]

[0049] in:

[0050]

[0051] The performance index function (15) consists of three parts: the tracking error of each batch system, the input change between two adjacent batches and the control ability. The weight matrices Q, R and S of these three components indicate their priorities in the optimization process. By adjusting the values ​​of the weight matrices Q, R and S, the optimal balance between the system error, control input change and smoothness can be found.

[0052] The optimal control input of the system can be obtained by minimizing the performance index function:

[0053]

[0054] The induced norm of a matrix in Hilbert space is defined as follows:

[0055]

[0056] Substituting equations (16), (17), and (19) into equation (15), we obtain:

[0057]

[0058] make We can get:

[0059]

[0060] Combining like terms gives:

[0061]

[0062] Since the matrices Q, R, and S are positive definite, then Reversible, after rearranging formula (22), we can get the feedback-based iterative learning control input update law:

[0063]

[0064] in:

[0065]

[0066] The system combines repeated learning of inter-batch errors with feedback control of real-time errors within a batch, and iteratively updates the system control input by equation (23) to obtain the optimal control input sequence and tracking error series where k max is the maximum number of iterations of the system;

[0067] Step 5: Integrate the feedback plus feedforward controller into a new learning-based feedback controller, including:

[0068] Based on the optimal control input U of the system under the above two-dimensional parallel control framework opt and tracking error E opt , the least squares fitting method is used to calculate the learning-based feedback controller parameters; the PID controller is linear in nature, and its linear parameterization and the output at the jth sampling point in the discrete time domain can be obtained according to formula (6):

[0069]

[0070] where θ=[θ1,θ2,θ3] T is the unknown parameter vector of the controller, u fb (j) is the output at each sampling point and the system has N sampling points;

[0071] Defining the information vector Then the feedback system output can be converted to:

[0072] U fb =Φθ (26)

[0073] in:

[0074] U fb =[u fb (1),u fb (2),…,u fb (N)] T (27)

[0075]

[0076] By minimizing the model output U fb And the actual output U opt The optimal parameter estimate of the model is derived from the sum of squared errors between That is, by minimizing the following quadratic criterion function:

[0077]

[0078] Get an estimate of the parameter vector θ, where and For the observation data of the feedback-based iterative learning control system, J(θ) can be restated as:

[0079] J(θ)=V T V=(U opt -Φθ) T (U opt -Φθ) (30)

[0080] 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 moment;

[0081] set up hour, Let the partial derivative of J(θ) with respect to θ be zero, and we get:

[0082]

[0083] or

[0084]

[0085] When (Φ T When Φ) is a positive definite matrix, the above formula (32) can be obtained Get parameter estimates for the learning-based feedback controller

[0086] Step 6: Analyze the convergence of the feedback-based iterative learning control trajectory tracking optimization algorithm;

[0087] Step 7: Implement iterative learning control to guide feedback control learning and tracking control of changes, including:

[0088] Using the learning-based parameters obtained in step 5 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 of feedback plus feedforward, the feedback plus feedforward controller is integrated into a new learning-based feedback controller using the least squares fitting method. The controller can fit the input-output characteristics of the original system and can track and control the changing trajectory without re-learning; the above method can realize the transfer of the learning experience of repeatedly performing a task to a new task without limiting its time length, and the DC motor position servo control system can track and control the changing trajectory.

[0089] The further technical solution is to analyze the convergence of the feedback-based iterative learning control trajectory tracking optimization algorithm, including:

[0090] For the expected trajectory y d , there exists a unique bounded expected control input u d Satisfy d =G d u d +d, defines the input error of the kth trial for:

[0091]

[0092] The tracking error of the system in the kth trial is:

[0093]

[0094] According to formula (33) and (23), we can get:

[0095]

[0096] Taking the norms on both sides of equation (35), we can get the inequality:

[0097]

[0098] From formula (24), extracting the common factor R yields Since Q, R, S and G d is a positive definite matrix, obviously Thus we can get:

[0099]

[0100] Define a positive scalar q satisfying Formula (36) can be reformulated as:

[0101] (39) According to the inequality relationship, after the system performs k iterations, we can obtain:

[0102]

[0103] in is the input error when the system is not iterated, q u =q||u d ||; If a suitable weight matrix is ​​chosen so that the constraint condition ||L u -L e G d ||≤ξ<1 holds. When the number of trials k→∞, according to the compression mapping lemma, we can get Therefore, formula (40) can be expressed as:

[0104]

[0105] Combining equations (34) and (41), we can obtain:

[0106]

[0107] Let ||G d ||=l, we can further get:

[0108]

[0109] That is, the tracking error of the system can converge to a bounded value;

[0110] Its further technical solution is, in particular, when S is a zero matrix, L u =I, then the system can achieve perfect tracking, that is,

[0111] The beneficial technical effects of the present invention are:

[0112] For the DC motor position servo control system that performs variable tasks, an iterative learning controller is introduced in parallel on the basis of traditional feedback control to form a two-dimensional parallel control framework. Through the combination of iterative learning between experiments and real-time feedback during the experiment, the system repeatedly learns and iteratively changes the control input to achieve high tracking accuracy while ensuring time domain stability. Then the feedback plus feedforward controller is integrated into a new learning-based feedback controller, which realizes the transfer of learning experience obtained by the system from repeatedly performing known tasks to new tasks without limiting its time length. The real-time interference suppression capability of feedback control and the learning ability of iterative learning control can eliminate the influence of non-repetitive uncertainty disturbances in the repetitive process. The integrated feedback controller based on the acquisition of the system's optimal input information and error information is applied to variable tasks to basically achieve zero-error tracking of the desired trajectory. BRIEF DESCRIPTION OF THE DRAWINGS

[0113] Figure 1 This is the model block diagram of the parallel feedforward feedback experience migration ILC DC motor in this application.

[0114] Figure 2 This is a graph of the actual output of the DC motor system before and after the introduction of iterative feedback control in this application.

[0115] Figure 3 It is the root mean square error convergence diagram of the feedback-based iterative learning control system in this application.

[0116] Figure 4 This is a block diagram of the learning-based parameter estimation of the integrated feedback controller in this application.

[0117] Figure 5 It is a comparison diagram of the output of the optimized control algorithm proposed in this application and the original feedback control tracking the change trajectory on the DC motor.

[0118] Figure 6 It is a comparison chart of the root mean square error of the algorithm proposed in this application and two different iterative optimization algorithms as the batch changes.

[0119] Figure 7 This is the output response diagram of the proposed algorithm on the DC motor in the presence of real-time interference in this application. DETAILED DESCRIPTION

[0120] The specific implementation of the present invention will be further described below in conjunction with the accompanying drawings.

[0121] The present application provides a DC motor iterative learning control optimization method for executing a variable task. The specific implementation steps refer to the contents of the first to seventh steps in the invention content. For the DC motor physical model shown in equation (1), it is assumed that the armature current and the motor angular velocity have no direct dynamic coupling relationship, and the various parameters in the DC motor position control system are set to: R f =20Ω, L f =1 H,C M =0.5Nm / Α,J r =2Nm / Α,C f =2Nm / Α. In addition, the system test time length is set to T=10s, and the sampling time interval is T s =0.01s, then the parameter matrices of the discrete state space equation expressions of the system are:

[0122]

[0123] C=[0 0 1]

[0124] At the same time, set the initial state of the system to x0 =

[000] T , the initial control input u0 = 0. This implementation assumes that the desired trajectory of the DC motor system output within a finite time interval t∈[0,T] is:

[0125] y d (t) = 1.25 × t (Tt)

[0126] first, Figure 1 The DC motor model block diagram disclosed in this application is given. Figure 1 The feedback-based iterative learning control constructed optimizes the performance of the motor system performing known tasks. After obtaining the optimal control input and tracking error, the feedforward plus feedback controller is integrated into a new learning-based feedback controller to meet the challenges of changing trajectory tasks. Starting from the initial feedback control system, in order to ensure that the DC motor performs the task normally, the classic ZN adjustment method is usually used to determine the system's initial feedback controller parameters to ensure the stability of the system. This application sets the initial feedback controller parameters as:

[0127] K p =0.2,K i =0.1,K d =0.05

[0128] In the next step, the feedback control and iterative learning control methods are combined. The feedback-based iterative learning control design adopts a two-dimensional control structure with the characteristics of parallel configuration. The system input for the kth trial is: Select the weight matrix Q = I, R = 0.1I, S = 0.01I, satisfying ||L u -L e G d ||≤ξ<1, according to formula (23), the system input is iteratively updated to act on the DC motor, and the output trajectory is continuously corrected until the desired trajectory is tracked. Figure 2 The corresponding DC motor trajectory tracking curve is given, demonstrating the learning ability of iterative learning control. Figure 3 This is the root mean square error curve of the system tracking error, which shows that after several batches of iterations, the system achieves bounded convergence.

[0129] Then, Figure 4 The equivalent feedback controller parameter estimation block diagram is given. Based on the optimal control input and error information obtained by the system performing a known task, the equivalent integrated learning-based feedback controller parameters are obtained according to equation (32) and applied to the DC motor system output curve as shown in Figure 5As shown in the figure, the desired trajectory can be tracked without relearning, and there is no time limit for any change in the trajectory, indicating that this method can transfer the learning experience gained from repeatedly performing a 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 batch as shown in Figure 6 As shown in the figure, the feedback-based iterative learning control proposed in this application can make the system converge quickly. Figure 7 It can be seen that after several batches of iterations, the system can maintain the original control performance and suppress the impact of real-time interference.

[0130] The above is only a preferred embodiment of the present application, and the present invention is not limited to the above embodiments. It is understood that other improvements and changes directly derived or associated by those skilled in the art without departing from the spirit and concept of the present invention should be considered to be included in the protection scope of the present invention.

Claims

1. A DC motor iterative learning control optimization method for performing variable tasks, characterized in that: The method comprises: The first step is to establish a dynamic model of the DC motor position servo control system. Assuming that the system is ideal and taking into account the influence of the motor's rotational inertia and friction torque, the dynamic model is as follows: Among them, R f , L f Respectively represent the armature resistance and armature inductance of the DC motor, C M is the torque constant of the motor, C e is the back EMF constant of the motor, J r , C f Respectively represent the moment of inertia and friction coefficient of the mechanical load, V f and i f are the armature voltage and armature current respectively, ω represents the motor speed, Indicates the rotation angle of the motor; The second step is to construct the discrete state space equation of the DC motor position servo control system, including: The armature current, speed and angle of the DC motor are defined as states Define the motor's rotation angle as output The armature voltage is input u(t) = V f (t), then equation (1) is converted into the state equation form: y=[0 0 1]x Discretize the continuous state equation (2) of the DC motor and select the sampling period T that satisfies Shannon sampling theorem. s , the discrete state space equation of the system is obtained as follows: The subscript k represents the number of tests, t=0, 1, 2, ..., N represents the corresponding t-th sampling point, and N represents the length of sampling points from the start of the system to the end of the current batch; u k (t), y k (t) and x k (t) are the input, output and state vector of the system at the tth sampling point of the kth batch respectively; the system matrices A, B and C have appropriate dimensions and satisfy CB≠0 to ensure the controllability of the system; 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 a discrete transfer operator, I is an identity matrix of appropriate dimension; choose an appropriate feedback controller C fb (z) Ensure the stability of the system, and add the iterative learning controller in parallel to the DC motor position servo control system to optimize the system control performance. At this time, the input of the system is output by the stable feedback. 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: Among them, P and L are defined as robust filter and learning gain respectively; e k (t+1) is the output error of the system at the t+1th sampling point of the kth batch; On the time axis, the PID feedback controller output is defined as: Where K p , K i and K d They are the proportional coefficient, integral coefficient and differential coefficient of the PID feedback controller respectively; After adding iterative learning control, the output of the system is expressed as: where y d (t) represents the expected trajectory of the system, G c (z), G s (z) is a known linear discrete transfer operator; By solving G c (z) and G s The unit impulse response of (z) gives the corresponding impulse transfer operator G d (h) and G r (h) 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; According to formula (8), the output formula (7) of the system is converted into the lifting system framework of the iteration domain: in: and k =[and k (1),and k (2),…,and k (N)] T (11) The input-output transfer matrix G of the time series in each trial d and the exogenous signal d are recorded as: 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: In the norm optimal iterative learning control framework, the optimal input and error information of the feedback-based iterative learning control system is obtained by optimizing the multi-objective performance indicator function of each batch. The multi-objective performance indicator function is defined as: in: The multi-objective performance indicator function (15) includes: the tracking error of each batch system, the input change between two adjacent batches and the control ability. The weight matrices Q, R and S of these three components indicate their priorities in the optimization process. By adjusting the values ​​of the weight matrices Q, R and S, the optimal balance between the system error, control input change and smoothness is found. The optimal control input of the system is obtained by minimizing the multi-objective performance index function: The induced norms of the three components in the Hilbert space are defined as follows: Substituting equations (16), (17), and (19) into equation (15), we obtain: make get: Combining like terms gives: Since the matrices Q, R, and S are positive definite, then Reversible, after rearranging formula (22), we can get the input update law of iterative learning control based on feedback: in: The system combines repeated learning of inter-batch errors with feedback control of real-time errors within a batch, and iteratively updates the system control input by equation (23) to obtain the optimal control input sequence and tracking error series where k max is the maximum number of iterations of the system; Step 5: Integrate the feedback plus feedforward controller into a new learning-based feedback controller, including: The optimal control input U of the system based on the two-dimensional parallel control framework opt and tracking error E opt , the least squares fitting method is used to calculate the learning-based feedback controller parameters; the PID controller is linear in nature, and its linear parameterization and the output at the jth sampling point in the discrete time domain are obtained according to formula (6): where θ=[θ1,θ2,θ3] T is the unknown parameter vector of the controller, u fb (j) is the output at each sampling point and the system has N sampling points; Defining the information vector Then the feedback system output is converted to: 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 the actual output U opt The optimal parameter estimate of the model is derived from the sum of squared errors between That is, the parameter vector θ is estimated by minimizing the following quadratic criterion function: in and Observation data for feedback-based iterative learning control systems; J(θ) can be restated as: 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 moment; set up hour, Let the partial derivative of J(θ) with respect to θ be zero, and we get: or When (Φ T When Φ) is a positive definite matrix, we can obtain Get 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 to guide feedback control learning and tracking control of changes, including: Using the learning-based parameters obtained in step 5 above A new feedback controller is obtained and applied to the DC motor position servo control system for trajectory tracking in different tasks. Based on the optimal input and error information obtained from the two-dimensional iterative learning control system of feedback plus feedforward, the feedback plus feedforward controller is integrated into a new feedback controller based on learning by using the least squares fitting method. This controller fits the input-output characteristics of the original system and tracks the changing trajectory without re-learning. The above method can be used to transfer the learning experience of repeatedly performing a task to a completely new task without limiting its time length, so that the DC motor position servo control system can track and control the changing trajectory.

2. The iterative learning control optimization method for a DC motor that performs a variable 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 expectation control input u d Satisfy d =G d u d +d, defines the input error of the kth trial for: The tracking error of the system in the kth trial is: According to equations (33) and (23), we can obtain: Taking the norms on both sides of equation (35), we get the inequality: From formula (24), we can extract the common factor R and obtain Since Q, R, S and G d is a positive definite matrix, we have Thus we get: Define a positive scalar q satisfying Then equation (36) can be reformulated as: According to the inequality relationship, the system performs k iterations and obtains: in is the input error when the system is not iterated, q u =q||u d ||; If a suitable weight matrix is ​​chosen so that the constraint condition ||L u -L e G d ||≤ξ<1 holds. When the number of trials k→∞, we can get Therefore, formula (40) is expressed as: Combining equations (34) and (41), we obtain: Let ||G d ||=l, further we get: Therefore, the tracking error of the system converges to a bounded value.

3. The iterative learning control optimization method for a DC motor that performs a variable task according to claim 2, characterized in that: When S is a zero matrix, L u =I, then the system achieves perfect tracking, that is,

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