Predictive control method for laser machining mirror based on extended dimension mapping dynamics model

Abstract

This invention provides a predictive control method for laser processing mirrors based on an extended-dimensional mapping dynamic model, belonging to the field of laser processing mirror motion control. The method includes the following steps: S1, proposing a dynamic model based on an extended-dimensional mapping of a neural network, combining neural networks and linear models to characterize the dynamic characteristics of the dual-axis laser processing mirror; S2, identifying the parameters in the extended-dimensional mapping dynamic model by solving an optimization problem based on input-output data; S3, designing a linear quadratic programming problem based on the extended-dimensional mapping dynamic model and the control objective; S4, solving the quadratic programming problem to obtain a predictive controller with nonlinear feedforward compensation capability. This invention mainly addresses the trajectory tracking problem of dual-axis laser processing mirrors under high-frequency sampling, solving the problems of insufficient tracking accuracy caused by the nonlinearity of the mirror driver itself and the difficulty in solving nonlinear optimization problems under rapid sampling.

Description

Technical Field

[0001] This invention belongs to the field of motion control of laser processing mirrors, and specifically relates to a predictive control method for laser processing mirrors based on an extended-dimensional mapping dynamics model. Background Technology

[0002] Laser-processed mirrors play a crucial role in industrial manufacturing, enabling precise laser pointing motion control through tracking specific yaw trajectories, which is of great significance for intelligent manufacturing fields such as semiconductor lithography. However, the actuators of laser-processed mirrors exhibit complex nonlinear dynamics that are difficult to characterize. Furthermore, since the mirror's motion is biaxial yaw, its internal structure dictates that coupling between the two axes is inevitable due to installation or processing accuracy issues. Both the complex nonlinear dynamics and the coupling between the two axes pose challenges to the construction of dynamic models for laser-processed mirrors, further limiting the improvement of mirror control accuracy. On the other hand, even if a nonlinear model is used to characterize the mirror's dynamic characteristics, the difficulty of solving for the control inputs will inevitably increase due to the nonlinearity, further limiting the deployment of control algorithms on high-speed sampling hardware.

[0003] Some existing methods fit nonlinear dynamics using neural networks and apply them in parallel to linear model structures, then transform the nonlinear optimization problem into a linear optimization problem through compensation. However, the control input obtained in this approach is not optimal, leaving room for further performance improvement. For modeling methods that apply neural network outputs in a cascaded manner to linear model structures, solving the nonlinear optimization problem is extremely cumbersome, and the trade-off between computation time and sampling frequency is significant in practical applications. Summary of the Invention

[0004] To address the nonlinear dynamics of the mirror actuator and the coupling between the two axes of the mirror's mechanical structure, this invention provides a predictive control method for laser processing mirrors based on an extended-dimensional mapping dynamic model. Historical input-output data is used as the network input of a neural network, and the network output is used as the nonlinear mapping state, thereby constructing a new state for the dynamic model based on the extended-dimensional mapping of the neural network. The parameter identification process for this extended-dimensional mapping dynamic model is based on a linear structure, facilitating the solution of subsequent quadratic programming problems and resolving the contradiction between high-speed sampling hardware conditions and control input solution time. This invention primarily addresses the nonlinear characteristics of the laser processing mirror actuator itself and the coupling problem between the two axes of the mirror during motion. It also solves the problem of implementing optimal control algorithms for nonlinear systems under high-frequency sampling hardware.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] A predictive control method for laser processing mirrors based on extended-dimensional mapping dynamics models includes the following steps:

[0007] S1. A dynamic model based on neural network extended dimension mapping is proposed by combining neural networks and linear models to characterize the dynamic characteristics of the dual axes of laser-processed mirrors.

[0008] S2. Identify the parameters in the extended dimension mapping dynamics model by solving the optimization problem based on the input and output data;

[0009] S3. Design of a linear quadratic programming problem based on an extended-dimensional mapping dynamics model and control objectives;

[0010] S4. Solve the linear quadratic programming problem to obtain a predictive controller with nonlinear feedforward compensation capability, and perform predictive control.

[0011] The beneficial effects of this invention are as follows:

[0012] 1. The method of the present invention includes a dynamic model based on extended-dimensional mapping of neural networks to characterize the dynamics of laser-processed mirrors. The present invention characterizes the nonlinear dynamics of the actuator and the coupling between the two axes through the mapping relationship of the neural network, making full use of historical input and output data to improve the accuracy of the dynamic model.

[0013] 2. To address the difficulty of solving quadratic programming problems caused by nonlinear model structures, the dynamic model based on neural network dimension expansion mapping constructed in this invention has the advantages of linear model structures in controller solving, transforming the solution of complex nonlinear optimization problems into offline linear model structure parameter identification and linear quadratic programming problem solving.

[0014] 3. To ensure the tracking accuracy of the laser-processed reflector for different types of reference trajectories, this invention designs a finite-time cost function based on the dynamic model of extended-dimensional mapping and the control objective. By solving a linear quadratic programming problem, an analytical predictive controller with nonlinear feedforward compensation capability is obtained, ensuring that the control algorithm can be effectively deployed on a hardware platform with fast sampling characteristics. Attached Figure Description

[0015] Figure 1 This is a flowchart of the laser processing mirror prediction and control method based on the extended dimension mapping dynamics model of the present invention;

[0016] Figure 2 A schematic diagram illustrating the structure and identification of a dynamic model for the extended-dimensional mapping of laser-processed mirrors;

[0017] Figure 3 A dual-axis scanning trajectory diagram of the laser-processed mirror after inputting the reference trajectory;

[0018] Figure 4 The X-axis output rotation angle tracking diagram of the laser-processed reflector is generated after inputting the reference trajectory;

[0019] Figure 5 The tracking error diagram of the X-axis of the laser-processed mirror after inputting the reference trajectory;

[0020] Figure 6 The Y-axis output rotation angle tracking diagram of the laser-processed reflector is generated after inputting the reference trajectory;

[0021] Figure 7 The tracking error diagram of the Y-axis of the laser-processed mirror after inputting the reference trajectory. Detailed Implementation

[0022] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0023] like Figure 1 As shown, this invention proposes a predictive control method for laser processing mirrors based on a dimension-extended mapping dynamics model, comprising the following steps:

[0024] Step S1: Combining neural networks and linear models, a dynamic model based on extended-dimensional mapping of neural networks is proposed to characterize the biaxial dynamic characteristics of laser-processed mirrors, including:

[0025] Established based on historical input and output data of laser-processed mirrors System state vector at time t for:

[0026] ,

[0027] in, express time, Used to define variables, subscripts Corresponding laser-processed reflector axis, , ,for Laser processing of reflectors The rotation angle output of the axis, , ,for Laser processing of reflectors Axis control input, This represents the order of the single-axis model for laser-processed mirrors. In this invention, it is assumed that the orders are the same for both axes. This represents taking the transpose of a vector or matrix. It represents the set of real numbers.

[0028] For ease of representation, Control input for laser-processed mirrors Represented as:

[0029] ,

[0030] Will The rotation angle output of the laser-processed mirror at any given time is expressed as:

[0031] ,

[0032] To characterize the nonlinear characteristics introduced by the actuator in the laser-processed mirror and the coupling characteristics between the two axes, a nonlinear mapping state vector as shown below is established based on neural network technology. :

[0033] ,

[0034] in, Representative with For input, The neural network output maps the parameters of the neural network. Based on the nonlinear mapping state vector generated by this neural network, the original system state vector is expanded to obtain... The state vector of the extended-dimensional mapping dynamic model of the laser-processed mirror at any time :

[0035] ,

[0036] in, If we represent stacking several internal vectors sequentially into a single column vector, then the dynamic model of the laser-processed mirror based on neural network dimension-extended mapping is as follows:

[0037] ,

[0038] in, for Time-dimension extended mapping dynamic model state vector The prediction For predictions of extended dimension mapping dynamics models The rotation angle output vector of the two axes at any given time. This represents the zero matrix that matches the dimension of the matrix it belongs to. It is a nonlinear mapping state vector Related matrix to be identified, intermediate matrix of parameters satisfy:

[0039] ,

[0040] Wherein, the intermediate matrix of parameters for the i-th axis The form is:

[0041] ,

[0042] ,

[0043] ;

[0044] in, For laser-processed reflectors The parameters to be identified related to the axis history output For laser-processed reflectors The parameters to be identified related to the axis history input.

[0045] Step S2: Identify the parameters in the extended-dimensional mapping dynamics model by solving the optimization problem based on the input and output data, including:

[0046] First, sinusoidal sweep signals with rich spectral frequencies are input to the dual axes of the laser-processed mirror, and the corresponding rotation angle outputs of the dual axes under this sweep signal are recorded. The structure is then identified according to the following linear model:

[0047] ,

[0048] In the linear system identification process described above, different model orders were selected, and the differences between the identification results and the actual data were compared. The model with the best fit was then chosen. and This leads to the kinetic model of laser-processed mirrors based on neural network extended dimension mapping. ;

[0049] Subsequently, the input and output data recorded during the above identification process are represented as the following data tuples. :

[0050] ,

[0051] To ensure the sufficiency of data and the effectiveness of the extended dimension mapping dynamics model, It is divided into two parts, with the majority being the tuples used to train the model, denoted as... Another part is the tuple used to evaluate the identification results, denoted as ,for , Figure 2This is a schematic diagram illustrating the structure and identification of a dynamic model for the extended-dimensional mapping of laser-processed mirrors. The model contains... and This can be obtained by solving the following optimization problem:

[0052] ,

[0053] in, Indicates by adjusting Minimize the cost function, Figure 2 In this context, "loss" represents the cost function in the optimization problem. for The number of Zhongyuan groups, Represents the 1 norm of a matrix. Represents the square of the vector's magnitude. To increase Sparsity of positive constants To optimize the time aspect of the problem. In the above optimization process, based on the structure of the linear model identified... Keeping the network structure and parameters unchanged, the goal is to minimize the one-step prediction error in the optimization problem. and Then, a determined dynamic model based on neural network extended dimension mapping is obtained.

[0054] Step S3: Design a linear quadratic programming problem based on the extended-dimensional mapping dynamics model and the control objective, including:

[0055] To ensure that the rotational output of the laser-processed mirror perfectly matches the reference trajectory, the following optimization problem is designed:

[0056] ,

[0057] in, for The reference trajectory vector at time step, Given a positive constant, and based on the requirements of solving the above optimization problem, the following finite-time cost function is designed. :

[0058] ,

[0059] in, Let be the optimal control input sequence that needs to be solved. To predict the step size, for Forecasting in advance The rotation angle output of the laser-processed mirror at a given moment for Forecasting in advance The reference trajectory of the laser-processed mirror at each moment. for Forecasting in advance The control input for laser-processed mirrors at any given moment. for Forecasting in advance The control input increment of the laser-processed mirror at each moment;

[0060] Based on the above cost function, and combined with a dynamic model based on neural network dimension-extended mapping, the following linear quadratic programming problem is designed:

[0061] ,

[0062] satisfy: ,

[0063] ,

[0064] ,

[0065] ,

[0066] ;

[0067] in, For the solution of the quadratic programming problem, This indicates the solution that minimizes the cost function. , express Forecasting in advance Intermediate matrix of state parameters of the extended dimension mapping dynamics model at time step n , , .

[0068] Step S4: Solve the quadratic programming problem to obtain a predictive controller with nonlinear feedforward compensation capability, including:

[0069] First, the linear quadratic programming problem in S3 is expressed in the following compact form:

[0070] ,

[0071] in, for Forward prediction from 0 to The control input sequence at each time step, specifically in the form of: , represent The optimal value, This indicates the solution that minimizes the parameters. The specific forms of the remaining intermediate variables are:

[0072] ,

[0073] ,

[0074] ,

[0075] ,

[0076] ,

[0077] , ,

[0078] , , ;

[0079] because Given a positive definite matrix, the linear quadratic programming problem can be transformed into solving... The optimal control input sequence for laser-processed mirrors is then... ,Pick The first element as The optimal control input for the time-of-flight processing mirror can be expressed in analytical form as follows:

[0080] ,

[0081] in, represent The matrix formed by the first and second rows.

[0082] Example:

[0083] To train the extended-dimensional mapping dynamics model, sinusoidal excitation signals with amplitudes ranging from 0 to 8 volts and frequencies from 1 Hz to 300 Hz were input to the laser-processed mirror. One million sample data points were collected at a sampling frequency of 20 kHz, with 90% used as the training set and the remaining 10% used to evaluate the recognition results. The neural network structure adopted a 2-layer network structure with 30 single-layer nodes, using Leaky ReLU as the activation function. The Adam optimizer was used during training, with 200 samples used per iteration, a learning rate of 0.005, and 20 repetitions.

[0084] (1) Closed-loop tracking simulation:

[0085] Reference trajectory: ,

[0086] in, Represents the sampling time. The frequency of the reference trajectory signal is selected as 10 in this simulation. The remaining parameters are: controller parameters. Predicting step size .

[0087] Simulation results are as follows Figures 3 to 7 As shown, where Figure 3 This is a diagram of the dual-axis scanning trajectory of the laser-processed mirror after inputting the above reference trajectory. Figure 4 The X-axis output rotation tracking diagram of the laser-processed mirror is generated after inputting the above reference trajectory. Figure 5 This is a tracking error diagram of the X-axis of the laser-processed mirror after inputting the above reference trajectory. Figure 6 The Y-axis output rotation angle tracking diagram of the laser-processed mirror is generated after inputting the above reference trajectory. Figure 7 The tracking error diagram of the laser-processed mirror along the Y-axis is shown after inputting the above reference trajectory. The maximum steady-state tracking error along the X-axis does not exceed 4. With slight curvature, the maximum steady-state tracking error on the Y-axis does not exceed 2.5. Slight curvature.

[0088] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above descriptions are merely specific embodiments of the present invention and are 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 protection scope of the present invention.

Claims

1. A laser machining mirror predictive control method based on an extended dimension mapping dynamic model, characterized in that, Includes the following steps: S1. A dynamic model based on neural network extended dimension mapping is proposed by combining neural networks and linear models to characterize the dynamic characteristics of the dual axes of laser-processed mirrors. In order to depict the nonlinear characteristics brought by the driver in the laser processing mirror and the coupling characteristics between the two axes, a nonlinear mapping state vector is established based on the neural network technology wherein represents the mapping neural network output with as the input, as the neural network parameter; The state vector of the original system is expanded according to the nonlinear mapping state vector generated by the neural network, and the state vector of the original system is obtained The state vector of the laser processing mirror is expanded and mapped , wherein Stacking a plurality of vectors in the interior as a column vector in sequence, the laser processing mirror based on the neural network expansion mapping dynamics model is: ， in, for Time-dimension extended mapping dynamic model state vector The prediction For predictions of extended dimension mapping dynamics models The rotation angle output vector of the two axes at any given time. This represents the zero matrix that matches the dimension of the matrix it belongs to. It is a nonlinear mapping state vector The relevant matrix to be identified; For the intermediate matrix of parameters; S2. Identify the parameters in the extended dimension mapping dynamics model by solving the optimization problem based on the input and output data; S3. Design of a linear quadratic programming problem based on an extended-dimensional mapping dynamics model and control objectives; S4. Solve the linear quadratic programming problem to obtain a predictive controller with nonlinear feedforward compensation capability, and perform predictive control.

2. The laser machining mirror predictive control method based on the extended dimension mapping dynamics model according to claim 1, characterized in that, S1 includes: established from historical input-output data of the laser processing mirror the system state vector at time instant t is , is the control input of the laser processing mirror at time instant t, is the angle output of the laser processing mirror at time instant t.

3. The laser machining mirror predictive control method based on the extended dimension mapping dynamics model according to claim 2, characterized in that, S2 includes: The sinusoidal sweep signals with rich spectrum are input to the biax of the laser processing mirror respectively, and the corresponding rotation angle outputs of the biax under the sweep signals are recorded. In the linear system identification process, different model orders are selected, the differences between the identification results and the actual data are compared, the system parameters with the best fitting effect are selected, and then the of the dynamic model of the laser processing mirror based on the neural network extension mapping are obtained.

4. The laser machining mirror predictive control method based on the extended dimension mapping dynamics model according to claim 3, characterized in that, S2 further includes: The input and output data recorded in the above identification process are expressed in the form of a tuple , in order to ensure the sufficiency of the data and the effectiveness of the extended dimension mapping dynamic model, is divided into two parts, one part is used for training the model tuple, denoted as , the other part is used for evaluating the identification result tuple, denoted as ; for , the and in the neural network extended dimension mapping dynamic model are obtained by solving the following optimization problem: ， wherein, represents the adjustment of the linear model structure, minimizing a cost function, is the number of meta-tuples, represents the 1-norm of a matrix, represents the square of the vector norm, is the increase of a normal number of sparsity, is the time in the optimization problem; in the above optimization process, the linear model structure identification is obtained according to remains unchanged, by changing the network structure and parameters to minimize the one-step prediction error in the optimization problem, to determine and After that, the determined dynamic model based on neural network extension mapping is obtained.

5. The laser machining mirror predictive control method based on the extended dimension mapping dynamics model according to claim 4, characterized in that, S3 includes: The cost function in the form of finite time is designed as follows : ， wherein, is the optimal control input sequence to be solved for, is the prediction step size, is is the forward prediction of the laser machining mirror angle output at time is the forward prediction of the laser machining mirror angle output at time is is the forward prediction of the laser machining mirror reference trajectory at time is the forward prediction of the laser machining mirror reference trajectory at time is a positive constant, is is the forward prediction of the laser machining mirror control input at time is the forward prediction of the laser machining mirror control input at time is is the forward prediction of the laser machining mirror control input increment at time is the forward prediction of the laser machining mirror control input increment at time 6. The predictive control method for laser processing mirrors based on extended-dimensional mapping dynamics model according to claim 5, characterized in that, S3 further includes: Based on the above cost function, and combined with a dynamic model based on neural network dimension-extended mapping, the following linear quadratic programming problem is designed: satisfy: , ， ， ， ； in, For the solution of the quadratic programming problem, This indicates the solution that minimizes the cost function. , express Forecasting in advance Intermediate matrix of state parameters of the extended dimension mapping dynamics model at time step n , , .

7. The predictive control method for laser processing mirrors based on an extended-dimensional mapping dynamics model according to claim 6, characterized in that, S4 includes: The linear quadratic programming problem in S3 can be expressed in the following compact form: ， in, , This represents its optimal value. , , , , All are intermediate variables; This indicates that several internal vectors are stacked sequentially into a single column vector; This indicates the solution that minimizes the parameters. ; express time, Indicates the prediction step size. for Forecasting in advance The control input for laser-processed mirrors at any given moment .

8. The predictive control method for laser processing mirrors based on extended-dimensional mapping dynamics model according to claim 7, characterized in that, S4 further includes: Due to intermediate variables Given a positive definite matrix, solving the linear quadratic programming problem is transformed into solving... The optimal control input sequence for laser-processed mirrors is then... ,Pick The first element as Optimal control input for time-of-flight processing mirrors.

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