Fractional order-linear term iterative learning control method and medium
By adopting a fractional-linear iterative learning control method, the system input is updated by setting the target tracking trajectory and expected accuracy, which solves the problem of slow convergence speed in iterative learning control methods, achieves high-precision tracking and fast convergence, and improves industrial production efficiency.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- RENMIN UNIVERSITY OF CHINA
- Filing Date
- 2023-06-07
- Publication Date
- 2026-05-01
AI Technical Summary
Existing iterative learning control methods have limitations in convergence speed, failing to achieve high-precision tracking within a limited number of iteration batches, thus affecting the efficiency of industrial production.
A fractional-linear iterative learning control method is adopted. By setting the target tracking trajectory and expected accuracy, the initial values of the state variables are set from the first iteration batch, and the system input is updated by linear superposition based on the system output and tracking error until the expected accuracy is achieved.
It achieves high-precision tracking and a significant improvement in convergence speed, thereby enhancing the working efficiency of industrial production control systems.
Smart Images
Figure CN116560218B_ABST
Abstract
Description
A fractional-linear iterative learning control method and medium Technical Field
[0001] This invention relates to the field of automatic control technology, and in particular to a fractional-linear iterative learning control method and a computer-readable storage medium. Background Technology
[0002] In actual production practice, many controlled systems complete one operation within a limited time and repeat continuously. For example, in automobile production, the completion of a car assembly by a robotic arm constitutes one operation of the control system, and subsequent assembly is a continuous repetition of the production process of the first car.
[0003] Traditional control technologies primarily rely on feedback mechanisms, using current information from the control system to manage it. These methods neglect the role of historical information. Iterative learning control algorithms, however, have improved control performance by learning from historical data, enabling the control system to precisely achieve its target control tasks.
[0004] Currently, the most widely used form of iterative learning control method is the P-type algorithm. It has advantages such as simplicity, ease of use, and high tracking accuracy. However, the inventors of this application discovered in their research that the P-type algorithm, precisely because of its simple linear form, limits the improvement of its convergence speed. In actual production practice, it is impossible for the control system to perform an infinite number of iterations; therefore, iterative learning control algorithms with faster convergence speeds are required. Summary of the Invention
[0005] To address the aforementioned problems, the purpose of this application is to provide a fractional-linear iterative learning control method that not only achieves high-precision tracking but also significantly improves the convergence speed, i.e., achieves the required tracking accuracy within a limited number of iteration batches, thereby enhancing the working efficiency of control systems in industrial production practices.
[0006] To achieve the above objectives, this application adopts the following technical solution:
[0007] In a first aspect, this application provides a fractional-linear term iterative learning control method, the method comprising:
[0008] S1, for the controlled system, sets the target tracking trajectory and expected tracking accuracy relative to the system output;
[0009] S2, starting from the first iteration batch, sets the initial values of the state variables of the control system corresponding to the current iteration batch;
[0010] S3. Based on the initial values of the state variables of the current iteration batch and the system input of the current iteration batch, obtain the system output of each discrete time point in the current iteration batch, and calculate the system tracking error of each discrete time point in the current iteration batch based on the system output of each discrete time point in the current iteration batch and the target tracking trajectory.
[0011] S4. Based on the system tracking error and its fractional power at each discrete time point of the current iteration batch, update the system input of the next iteration batch by superimposing the linear terms of the two.
[0012] S5, return to S2 and continue execution until the system tracking error at each discrete time point of the current iteration batch is less than the expected tracking accuracy.
[0013] In one implementation of this application, the controlled system is an integer-order discrete single-input single-output linear system, specifically described as follows:
[0014]
[0015] in, These are the state variables of the control system. It is the system input of the control system. It is system output. It is the system matrix, k is the label of the iteration batch, and t = 0, 1, ..., T are the discrete time points in each iteration batch.
[0016] In one implementation of this application, setting the initial values of the state variables of the control system corresponding to the current iteration batch includes:
[0017] For each iteration batch of the control system, the initial values of the state variables are set to the same values.
[0018] In one implementation of this application, the system tracking error at each discrete time point of the current iteration batch is e. k (t)=y d (t)-y k (t),
[0019] Wherein, the target tracking trajectory is y d (t), t=1,2,...,T;y k (t) represents the system output of the current iteration batch.
[0020] In one implementation of this application, updating the system input for the next iteration batch by superimposing linear terms of the system tracking error and its fractional power at each discrete time point of the current iteration batch includes:
[0021] Using formula
[0022] u k+1 (t)=u k (t)+αe k (t+1)+β|e k (t+1)| γ sgn(e k (t+1))
[0023] Update the system input for the next iteration batch;
[0024] Where α and β are preset learning gains; 0 < γ < 1 is the order of the fractional power; sgn(x) is the sign function.
[0025] In one implementation of this application, α and β satisfy the following constraints:
[0026] -1 < 1 - αcb < 0, βcb < 0.
[0027] In one implementation of this application, α and β further satisfy the following constraints:
[0028]
[0029] Where ε represents the expected tracking accuracy.
[0030] In one implementation of this application, the optimal value of γ is
[0031] Secondly, this application provides a computer-readable storage medium storing a computer program, wherein the computer program, when running, controls the device where the computer-readable storage medium is located to execute the fractional-linear term iterative learning control method described in the first aspect.
[0032] The present invention, by adopting the above technical solution, has the following advantages: In the solution of the present invention, firstly, for the controlled system, a target tracking trajectory and expected tracking accuracy relative to the system output are set. Then, starting from the first iteration batch, the initial values of the state variables of the control system corresponding to the current iteration batch are set. Based on the initial values of the state variables of the current iteration batch and the system input of the current iteration batch, the system output of each discrete time point in the current iteration batch is obtained. Based on the system output of each discrete time point in the current iteration batch and the target tracking trajectory, the system tracking error of each discrete time point in the current iteration batch is calculated. Then, based on the system tracking error of each discrete time point in the current iteration batch and its fractional power, the system input of the next iteration batch is updated by superimposing the linear terms of the two, thereby obtaining a new system output again. This process is continuously iterated until the system tracking error of each discrete time point in the iteration batch is less than the expected tracking accuracy. Therefore, compared with the prior art, a significant improvement in convergence speed can be achieved, thereby improving the efficiency of industrial production. Attached Figure Description
[0033] Figure 1 is a block diagram of the fractional-linear term iterative learning control method in an embodiment of the present invention;
[0034] Figure 2 shows the evolution of the tracking error of the system along the iteration axis at the first 20 time points after applying the fractional-linear term control method to the integer-order discrete single-input single-output linear system.
[0035] Figure 3 shows the system tracking error corresponding to different initial values at the first three time points after 50 system iterations, when the fractional-linear term control method is used for an integer-order discrete single-input single-output linear system.
[0036] Figure 4 shows the evolution of the absolute value of the difference between the tracking error and its limit value for different fractional order values along the iteration axis after applying the fractional-linear term control method to an integer-order discrete single-input single-output linear system.
[0037] Figure 5 shows the evolution of the tracking error along the iteration axis for an integer-order discrete single-input single-output linear system when the initial tracking error is small, with the same learning gain selected. The proportional (P-type) algorithm and the fractional-linear term control method are used respectively.
[0038] Figure 6 shows the evolution of the tracking error along the iteration axis for an integer-order discrete single-input single-output linear system when the initial tracking error is large, with the same learning gain selected. The algorithm is proportional (P-type) and the control method is fractional-linear term. Detailed Implementation
[0039] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, 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, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the described embodiments of the present invention are within the scope of protection of the present invention.
[0040] Existing iterative learning control methods suffer from slow convergence speeds, limiting their application. This application addresses this issue by providing a fractional-linear iterative learning control method and medium. The method includes: setting a target tracking trajectory relative to the system output and an expected tracking accuracy for the controlled system; starting from the first iteration batch, setting initial values for the state variables of the control system corresponding to the current iteration batch; obtaining the system output at each discrete time point in the current iteration batch based on the initial values of the state variables and the system input of the current iteration batch; calculating the system tracking error at each discrete time point in the current iteration batch based on the system output at each discrete time point and the target tracking trajectory; updating the system input of the next iteration batch by superimposing the system tracking error and its fractional power at each discrete time point in the current iteration batch using a linear term superposition method; and returning to continue execution until the system tracking error at each discrete time point of the current iteration batch is less than the expected tracking accuracy. This technical solution not only achieves high-precision tracking but also significantly improves convergence speed, thereby enhancing the working efficiency of control systems in industrial production practices.
[0041] Referring to the accompanying drawings, in one aspect of an embodiment of this application, a fractional-order linear term iterative learning control method is provided.
[0042] Figure 1 is a control block diagram of the fractional-linear term iterative learning control method of this application. The method of this application specifically includes:
[0043] S1, for the controlled system, sets the target tracking trajectory and expected tracking accuracy relative to the system output;
[0044] Specifically, the controlled system (the controlled system in Figure 1) is an integer-order discrete single-input single-output linear system, specifically described as follows:
[0045]
[0046] in, These are the state variables of the control system. It is the system input of the control system. It is system output. It is the system matrix, k is the label of the iteration batch, and t = 0, 1, ..., T are the discrete time points in each iteration batch.
[0047] In this embodiment, the target tracking trajectory and expected tracking accuracy are set manually according to actual needs.
[0048] For example, based on actual control requirements, the system tracking target trajectory is set as follows: y d (t), t=1,2,...,T. Meanwhile, the tracking accuracy of the system can be set to ε according to actual needs.
[0049] S2, starting from the first iteration batch, sets the initial values of the state variables of the control system corresponding to the current iteration batch;
[0050] Specifically, in this embodiment of the application, for each iteration batch of the control system, the initial value of the state variable is set to the same value, that is:
[0051] x k (0)=x k+1 (0), k = 0, 1, ...
[0052] Setting the state parameters in each iteration batch to the same arbitrary value can eliminate the influence of the initial tracking error on system control.
[0053] S3. Based on the initial values of the state variables of the current iteration batch and the system input of the current iteration batch, obtain the system output of each discrete time point in the current iteration batch, and calculate the system tracking error of each discrete time point in the current iteration batch based on the system output of each discrete time point in the current iteration batch and the target tracking trajectory.
[0054] Specifically, based on the initial values of the state variables of the current iteration batch and applying the system input of the current batch, the system output of each discrete time point of the current iteration batch can be obtained at the system output terminal.
[0055] Furthermore, based on the system output and the target tracking trajectory, the system tracking error can be calculated:
[0056] e k (t)=y d (t)-y k (t)
[0057] Wherein, the target tracking trajectory is y d (t), t=1,2,...,T;y k (t) represents the system output of the current iteration batch.
[0058] S4. Based on the system tracking error and its fractional power at each discrete time point of the current iteration batch, update the system input of the next iteration batch by superimposing the linear terms of the two.
[0059] Specifically, the formula is used.
[0060] u k+1 (t)=u k (t)+αe k (t+1)+β|e k (t+1)| γ sgn(e k (t+1))
[0061] Update the system input for the next iteration batch;
[0062] Where α and β are the preset learning gains; 0 < γ < 1, which is the order of the fractional derivative; sgn(x) is the sign function, defined as follows:
[0063]
[0064] The specific design rules for gains α and β are as follows:
[0065] -1 < 1 - αcb < 0, βcb < 0;
[0066] Applying a fractional-linear iterative learning control method to the controlled system, it can be proven that the tracking error of the system eventually converges to 0. One of these three values; therefore, designing appropriate values for the learning gain and fractional order is crucial to ensure that... This ensures that the tracking error of the system can ultimately meet the given accuracy requirements;
[0067] After completing the parameter design, in order to achieve faster local convergence, a fractional value is chosen. At this point, the algorithm has the fastest local convergence speed and a superlinear convergence speed. In addition, the learning gain value should be adjusted appropriately so that the accuracy requirements of the tracking error are still met. In order to obtain a faster global convergence speed, the value of gain α is set so that the value of αcb-1 is small, which can result in a faster global convergence speed. At this time, attention should be paid to adjusting the value of gain β so that the tracking accuracy requirements are met.
[0068] S5, return to S2 and continue execution until the system tracking error at each discrete time point of the current iteration batch is less than the expected tracking accuracy.
[0069] Specifically, based on the above parameter design and the control method of correcting the input of the fractional-linear term superposition correction system, the convergence effect can be improved quickly.
[0070] In summary, the present invention first sets the target tracking trajectory and expected tracking accuracy relative to the system output for the controlled system. Then, starting from the first iteration batch, it sets the initial values of the state variables of the control system corresponding to the current iteration batch. Based on the initial values of the state variables of the current iteration batch and the system input of the current iteration batch, it obtains the system output at each discrete time point in the current iteration batch and calculates the system tracking error at each discrete time point in the current iteration batch. Then, based on the system tracking error at each discrete time point in the current iteration batch and its fractional power, it updates the system input of the next iteration batch by superimposing the linear terms of the two, thereby obtaining a new system output again. This process is repeated until the system tracking error at each discrete time point in the iteration batch is less than the expected tracking accuracy. Compared with the prior art, this method can significantly improve the convergence speed, thereby improving the efficiency of industrial production.
[0071] In another aspect of the embodiments of this application, a computer storage medium is also provided, which stores a computer program that, when executed by a computer, implements the aforementioned method.
[0072] The method provided by the embodiments of this application will now be described in a specific application scenario.
[0073] Consider the following permanent magnet motor system:
[0074]
[0075] The system matrix is:
[0076]
[0077] Determine the desired tracking trajectory of the system: Determine the desired tracking trajectory as y d (t), t = 1, 2, ..., T. Where, for any time point t, y d (t) is a random number between -10 and 10, meaning that the expected tracking trajectory is a set of 20 random values (T).
[0078] Determine the tracking error requirement: Assume the tracking error requirement is ε = 0.2, meaning that after a finite number of iterations, the tracking error needs to satisfy |e k (t)|<0.2.
[0079] Determine the initial value of the system: Set the initial value of the system to x.k (0) = [0 0] T The system's operating cycle is T = 20, and the initial system input is u0(t) = 0, t = 0, 1, ..., T-1.
[0080] Get the system input u in the current batch k (t), t=0,1,...,T-1, system output y k (t), t=1,2,...,T, and the corresponding tracking error e k (t)=y d (t)-y k (t), t=1,2,...,T.
[0081] Based on the selection rules for the learning gain and the accuracy requirements of the error, the values of the learning gain and fractional order are determined to be α = 70, β = -30, and γ = 0.5, respectively. At this point, the limit value of the tracking error is no greater than... Therefore, the given accuracy requirement is met.
[0082] Based on the system input u for the current batch k (t), the calculated system tracking error e k Given (t), the determined learning gain and fractional order, the control input for the next batch of systems is determined according to the following relationship:
[0083] u k+1 (t)=u k (t)+70e k (t+1)-30β|e k (t+1)| 0.5 sgn(e k (t+1)).
[0084] Record the system input for the next batch. k+1 (t), t = 0, 1, ..., T-1 and y k+1 (t), t=1,2,...,T, store the corresponding values in memory.
[0085] Return and continue until the tracking task of the desired target trajectory is completed.
[0086] Figure 2 illustrates the variation of the tracking error along the iteration axis of the system at the first 20 time points after applying the fractional-linear-term iterative learning control method to the integer-order discrete linear system. As can be seen from Figure 2, the fractional-linear-term control method proposed in this invention has a very fast convergence speed. After six iterations, the system's tracking error has already reached the given accuracy requirement.
[0087] Figure 3 shows the system tracking error for different experiments (i.e., different target tracking trajectories) after 50 iterations of the proposed fractional-linear algorithm at the first three time points. As can be seen from the figure, the system tracking error converges to 0 for any target tracking trajectory. One of these three values.
[0088] Figure 4 illustrates the impact of different fractional order values on the convergence speed of the algorithm under a fixed learning gain. The vertical axis represents the maximum absolute value of the difference between the system tracking error and the error convergence limit over the first 20 time points. Based on the method for calculating the optimal fractional order value, in this example, the optimal fractional order value is... As can be seen from the figure, when At that time, the algorithm had the fastest convergence speed and achieved superlinear convergence.
[0089] Figure 5 shows the relationship between the convergence speed of the P-type algorithm and the fractional-linear term algorithm when the initial error is small. The P-type algorithm is used as follows:
[0090] u k+1 (t)=u k (t)+pe k (t+1).
[0091] At this point, we fix the target tracking trajectory as y d (t) = 1, t = 1, 2, ..., T. For comparison purposes, p = 30. At this point, we have |1 - pcb| = |1 - αcb|, therefore, the two algorithms have the same compression effect, thus providing a better comparison. Furthermore, we choose a fractional order γ = 0.9, resulting in a smaller convergence error for the fractional-linear term algorithm.
[0092] As shown in Figure 5, the fractional-linear term algorithm has a faster convergence speed, and its tracking accuracy is also within the acceptable error range. Therefore, for a given accuracy requirement, to achieve fast convergence within a finite number of batches, the fractional-linear term algorithm is superior to the traditional P-type algorithm.
[0093] Figure 6 shows the relationship between the convergence speed of the P-type algorithm and the fractional-linear term algorithm when the initial error is large.
[0094] At this point, we fix the target tracking trajectory as y d (t) = 10, t = 1, 2, ..., T. Take p = 30 and fractional order γ = 0.5.
[0095] As shown in Figure 6, the fractional-linear term control algorithm exhibits a faster convergence speed, and its tracking accuracy remains within acceptable limits. In summary, regardless of whether the initial error is large or small, the fractional-linear term iterative learning control method can achieve rapid convergence while maintaining the required tracking accuracy.
[0096] In the several embodiments provided by this invention, it should be understood that the disclosed methods can be implemented in other ways. For example, the embodiments described above are merely illustrative.
[0097] The integrated units implemented as software functional units described above can be stored in a computer-readable storage medium. These software functional units, stored in a storage medium, include several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) or processor to execute some steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.
[0098] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A fractional-linear term iterative learning control method, characterized in that, The method includes: S1, setting a target tracking trajectory and expected tracking accuracy relative to the system output for the controlled system; S2, setting initial values for the state variables of the control system corresponding to the current iteration batch, starting from the first iteration batch; S3, obtaining the initial values of the state variables and the system input of the current iteration batch, calculating the system output at each discrete time point in the current iteration batch, and calculating the system tracking error at each discrete time point in the current iteration batch based on the system output at each discrete time point in the current iteration batch and the target tracking trajectory; S4, updating the system input of the next iteration batch by superimposing the linear terms of the system tracking error at each discrete time point in the current iteration batch and its fractional power; S5, returning to S2 and continuing execution until the system tracking error at each discrete time point in the current iteration batch is less than the expected tracking accuracy; the controlled system is an integer-order discrete single-input single-output linear system, specifically described as: in, These are the state variables of the control system. It is the system input of the control system. It is system output. It is a system matrix. This is the label for the iteration batch. For each discrete time point in the iterative batch; the system tracking error for each discrete time point in the current iterative batch is: The target tracking trajectory is as follows: ； The system output for the current iteration batch; the step of updating the system input for the next iteration batch by superimposing linear terms of the system tracking error and its fractional power at each discrete time point of the current iteration batch includes: using the formula Update the system input for the next iteration batch; where, and The preset learning gain; , where is the order of the fractional power; For symbolic functions; the stated and The following constraints must be met: , The and It also satisfies the following constraints: ;in, This represents the expected tracking accuracy.
2. The fractional-linear term iterative learning control method according to claim 1, characterized in that, The parameters One possible value 。 3. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program, which, when executed, controls the device containing the computer-readable storage medium to perform the fractional-linear iterative learning control method according to any one of claims 1 to 2.