Fast reflector linear quadratic tracking control method based on non-linear elimination

Abstract

The invention provides a nonlinear elimination-based linear secondary tracking control method for a fast reflector. The method comprises the following steps of S1, establishing a biaxial depth coupling dynamic model structure according to dynamic characteristics of a piezoelectric driving fast reflector and a deep learning technology; s2, training a biaxial depth coupling dynamic model through input and output data; s3, designing a rolling horizon optimized secondary tracking controller based on the biaxial depth coupling dynamics model; s4, based on the structure of the biaxial depth coupling dynamic model, designing an approximate optimal secondary tracking controller based on nonlinear compensation; and high-speed and high-precision tracking control is realized. The method mainly aims at the control problem caused by nonlinear dynamic and double-axis coupling of the fast reflecting mirror driven by the piezoelectric actuator, and can overcome the problem of precision reduction caused by different tracks of double axes of the fast reflecting mirror.

Description

Technical Field

[0001] The present invention belongs to the field of precision motion control of fast mirrors, and specifically relates to a linear quadratic tracking control method for fast mirrors based on nonlinear elimination. The present invention is mainly used to deal with the hysteresis nonlinearity and mutual coupling problems of dual-axis motion faced by fast mirrors driven by piezoelectric materials, and at the same time to deal with the deployment problem of control algorithms under high-speed sampling frequencies. Background Art

[0002] Fast mirrors driven by piezoelectric materials have extremely high swing accuracy and can locate or continuously track specific trajectories. They are widely used in space laser communications, precision laser processing, semiconductor lithography and other fields. However, in practical applications, piezoelectric materials themselves have hysteresis nonlinear characteristics, which brings huge challenges to the overall modeling of fast mirrors, and further makes the design of control systems difficult. In addition, the mechanical structure of the fast mirror determines that it can rotate with two degrees of freedom, but due to the accuracy of processing and assembly, there is coupling between the two axes, resulting in a decrease in control performance during movement. On the other hand, when tracking a fast reference trajectory, the sampling frequency of the system is extremely high, which constrains the computational complexity of the control strategy. The control methods generally used for nonlinear models cannot be effectively deployed in such high-speed sampling systems, limiting the tracking performance of high-frequency reference trajectories.

[0003] The inventors of this application have found that the strong hysteresis nonlinearity and dual-axis coupling of the fast reflector can be fitted using deep learning technology, and the rolling time domain optimization technology can be used to design the control system. The solution process of ordinary nonlinear optimization problems is complicated, the calculation time is long, and it is impossible to guarantee the optimal control input within the specified sampling interval. On the other hand, the only measurable data of the fast reflector is the output angle. The ordinary optimal control method needs to estimate the system state based on the output angle, which further increases the calculation burden of the controller and is not conducive to the application of the control algorithm. Summary of the invention

[0004] In view of the hysteresis nonlinearity and mutual coupling problems of the piezoelectric material driven fast reflector and the biaxial motion, the present invention proposes a linear quadratic tracking control method for the fast reflector based on nonlinear elimination. Firstly, a biaxial deep coupling dynamic model structure is established according to the dynamic characteristics of the piezoelectric driven fast reflector and deep learning technology; then the biaxial deep coupling dynamic model is trained by input and output data; thereafter, a rolling time domain optimized quadratic tracking controller is designed based on the biaxial deep coupling dynamic model; thereafter, an approximate optimal quadratic tracking controller based on nonlinear compensation is designed by relying on the structural advantages of the biaxial deep coupling dynamic model to achieve high-speed and high-precision tracking control. The nonlinear and coupling effects are compensated by decomposing the control input, and the original nonlinear quadratic programming problem is converted into a linear quadratic tracking problem, and an approximate optimal quadratic tracking controller is obtained, thereby realizing the control strategy deployment of the fast reflector under high-speed sampling.

[0005] To achieve the above object, the technical solution adopted by the present invention is as follows: A fast mirror linear quadratic tracking control method based on nonlinear elimination comprises the following steps: S1: Establish a dual-axis deep coupling dynamic model structure based on the dynamic characteristics of the piezoelectric driven fast reflection mirror and deep learning technology; S2: Training the dual-axis deep coupling dynamics model through input and output data; S3: Design of a quadratic tracking controller with rolling time domain optimization based on a dual-axis deep coupling dynamics model; S4: Based on the structure of the dual-axis deeply coupled dynamics model, a near-optimal quadratic tracking controller with nonlinear compensation is designed.

[0006] The beneficial effects of the present invention are: 1. The method of the present invention includes a dual-axis deep coupling dynamic model of a piezoelectric driven fast mirror. In view of the dual-axis coupling and piezoelectric hysteresis nonlinearity, the present invention constructs the system state based on the historical input and output data of discrete time. On this basis, the deep learning technology is fully utilized to identify the linear part and the nonlinear part in turn through the input and output data, thereby enhancing the model's ability to characterize the dynamic characteristics of the fast mirror in the application.

[0007] 2. Aiming at the problem of precise tracking and control of fast reflectors, the present invention constructs a nonlinear programming problem based on the proposed dual-axis deep coupling dynamic model, and solves the problem online through the rolling time domain optimization technology to achieve high-speed and high-precision tracking of the fast reflector.

[0008] 3. In order to ensure that the piezoelectric driven fast mirror control strategy can adapt to the high-speed sampling hardware platform, the present invention transforms the original nonlinear quadratic tracking problem into a linear quadratic tracking problem through nonlinear compensation, thereby achieving an approximation of the optimal solution to the original nonlinear programming problem and reducing the computational complexity of the control strategy. BRIEF DESCRIPTION OF THE DRAWINGS

[0009] Figure 1 It is a flow chart of a fast reflector linear quadratic tracking control method based on nonlinear elimination of the present invention; Figure 2 Schematic diagram of the structure and identification of the dual-axis deep coupling dynamics model for describing the dual-axis dynamics of the fast mirror; Figure 3 The training loss of the dual-axis deep coupling dynamics model changes with the number of rounds; Figure 4 The fast reflector dual-axis output angle tracking diagram when inputting the reference trajectory; Figure 5 This is the error angle diagram between the dual-axis output angle of the fast reflector and the reference trajectory when the reference trajectory is input. DETAILED DESCRIPTION

[0010] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific implementation examples described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0011] like Figure 1 As shown, the present invention proposes a fast reflector linear quadratic tracking control method based on nonlinear elimination, comprising the following steps: Step S1: Establishing a dual-axis deep coupling dynamic model structure based on the dynamic characteristics of the piezoelectric driven fast reflection mirror and deep learning technology, including: According to the dynamic characteristics of the piezoelectrically driven fast mirror, the following linear autoregressive model with external input is established: , in, The fast reflector axis, Fast reflector Moment The output angle of the shaft, is the present moment, is the number of backtracking time steps corresponding to the output, Fast reflector Moment System control inputs for axes, is the number of backtracking time steps corresponding to the input, For fast reflector Model parameters related to axis history output, For fast reflector Model parameters related to axis history input, is the set of all real numbers, is the order of the model, In the representative model, zero points, the model order and the number of zero points remain unchanged; The fast mirror can be defined according to the autoregressive model with external input The system status at this moment is: , Superscript Represents the transpose of a matrix, a fast mirror The system control input vector at time is: , The structure of the dual-axis deep coupling dynamic model of the fast mirror is as follows: Figure 2 As shown, the specific form is: , in, Based on System state and control input prediction at each moment The system status at the moment, For prediction The output angle vector of the dual axes at the moment, For is the input feedforward neural network, are the weight parameters and bias parameters of the feedforward neural network, satisfy: , in, The specific form is: , , .

[0012] Step S2: Train the dual-axis deep coupling dynamics model through input and output data. The specific design steps are as follows: First, two sets of sinusoidal signals are randomly generated according to a certain amplitude range and frequency range. The two random sinusoidal signals are used as the input of the two axes of the fast reflector respectively, and the output angle vector corresponding to this input is recorded as , the estimated vector of the output angle vector obtained based on the dual-axis deep coupling dynamics model and the input data is , then the data used to identify the biaxial deep coupling dynamics model can be expressed as a tuple , For 90% of the data set selected for network training, for Remove The remaining 10% is used as a data set for network evaluation; Then will The data is loaded into the system identification toolbox to identify the linear part. The identification process is as follows: Figure 2 As shown, by adjusting the system order in the above autoregressive model The linear model with the highest fit was obtained, thus determining the biaxial deep coupling dynamics model. The specific value of is further constructed to identify the nonlinear part of the biaxial deep coupling dynamics model: , in, for The number of tuples in the , the Adam algorithm is used to solve the optimization problem, and different network layers and nodes are selected to minimize the corresponding Loss function to obtain the optimal weight parameters and bias parameters of the feedforward neural network .

[0013] Step S3: Design a quadratic tracking controller with rolling time domain optimization based on the dual-axis deep coupling dynamics model. The specific design steps are as follows: Design a cost function with limited prediction time according to the control objective: , in, is the control input sequence to be optimized, is the prediction step length, for Always look ahead The output angle vector of the fast mirror at the moment, for Always look ahead The two-axis reference trajectory vector of the fast reflector at the moment, represents the norm of a vector, is an optional constant penalty factor, for Always look ahead The control input vector of the fast mirror at the moment, for Always look ahead The control input increment vector at each moment; In order to solve the optimal control sequence, the following nonlinear programming problem can be designed in combination with the dual-axis deep coupling dynamics model of the fast reflector: , satisfy: , , , ; in, express Always look ahead The state of the dual-axis deep coupling dynamics model at the moment, for Always look ahead The feedforward neural network output of the dual-axis deep-coupled dynamics model of the fast mirror at each moment.

[0014] Step S4: Based on the structure of the dual-axis deep coupling dynamics model, an approximately optimal quadratic tracking controller based on nonlinear compensation is designed. The specific design steps are as follows: First, The control input vector of the fast mirror at time is decomposed into: , in, is the linear quadratic tracking control input vector, is the control input vector for nonlinear compensation, and the predicted The system status at this moment is: , The design process can be expressed as solving the following optimization problem: , The optimal solution is ; Subsequently, the nonlinear programming problem in S3 can be transformed into solving the following linear quadratic tracking problem: , satisfy: , , , ; in, The input sequence of the linear quadratic tracking control to be optimized is: for Always look ahead The linear quadratic tracking control input vector of the fast mirror at the moment; The above linear quadratic tracking control problem is transformed into a compact quadratic programming problem: , in, is the decision vector, represents its optimal value, , , , , , , , , ; in, yes The dual-axis reference trajectory vector of the fast reflector at time instant, Is to The vector formed by the dual-axis reference trajectory vector of the fast reflector at the moment, is the Nth power of the intermediate parameter A, since is a positive definite matrix, and the gradient of the cost function of the quadratic programming problem is set to 0 to obtain ,but The optimal linear quadratic tracking control input vector of the fast reflector at time is ,in represent The first and second rows of the piezoelectrically driven fast mirror are The control input vector at time is .

[0015] Example: Simulation settings: A fully connected deep neural network is constructed as a feedforward neural network in the dual-axis deep coupling dynamics model, namely , the number of network layers is 2, and the number of nodes in each layer is 30. 5 million sample data are collected, and the sampling frequency of the data is 20kHz. 90% of the data are randomly selected to train the dual-axis deep coupling dynamics model, and the remaining 10% of the data are used to evaluate the training model. The model training is carried out in the Pytorch environment. The optimization algorithm selected is Adam, the learning rate is 0.005, the learning rate decrease rate is 0.5, and the number of repeated training is 20.

[0016] Figure 3 The training loss of the dual-axis deep coupling dynamics model changes with the number of rounds. At the end of the 20th round, the training loss of the dual-axis deep coupling dynamics model is , the estimated loss is ; (1) Closed-loop tracking simulation: Reference trajectory: ; Indicates sampling time, weight coefficient , prediction step length .

[0017] result, Figure 4 The fast reflector dual-axis output angle tracking diagram when inputting the reference trajectory, Figure 5 is the error angle between the dual-axis output angle of the fast reflector and the reference trajectory when the reference trajectory is input. The maximum steady-state tracking error of axis 1 is Milliradian, the maximum steady-state tracking error of the two axes is Milliradian.

[0018] The specific embodiments described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific 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 in the scope of protection of the present invention.

Claims

1. A fast mirror linear quadratic tracking control method based on nonlinear elimination, characterized in that: The following steps are involved: S1. Establish a dual-axis deep coupling dynamic model structure based on the dynamic characteristics of the piezoelectric driven fast reflection mirror and deep learning technology; S2, training the dual-axis deep coupling dynamics model through input and output data; S3. Design of a quadratic tracking controller with rolling time domain optimization based on a dual-axis deep coupling dynamics model; S4. Based on the structure of the dual-axis deep coupling dynamic model, an approximate optimal quadratic tracking controller with nonlinear compensation is designed.

2. The fast mirror linear quadratic tracking control method based on nonlinear elimination according to claim 1, characterized in that: The S1 includes: According to the dynamic characteristics of the piezoelectrically driven fast mirror, the following linear autoregressive model with external input is established: ， in, The fast reflector axis, Fast reflector Moment The output angle of the shaft, is the present moment, is the number of backtracking time steps corresponding to the output, Fast reflector Moment System control inputs for axes, is the number of backtracking time steps corresponding to the input, For fast reflector Model parameters related to axis history output, For fast reflector Model parameters related to axis history input, is the set of all real numbers, is the order of the model, In the representative model, zero points, the model order and the number of zero points remain unchanged; According to the autoregressive model with external input, the fast mirror is defined The system status at this moment is: ， Superscript represents the transpose of a matrix, The system control input vector at this moment is: ， The structure of the dual-axis deep coupling dynamic model of the fast mirror is: ， in Based on System state and control input prediction at each moment The system status at the moment, For prediction The output angle vector of the dual axes at the moment, For is the input feedforward neural network, are the weight parameters and bias parameters of the feedforward neural network, satisfy: ， in The specific form is: ， ， 。 3. The fast mirror linear quadratic tracking control method based on nonlinear elimination according to claim 2, characterized in that: The S2 includes: First, two sets of sinusoidal signals are randomly generated according to a certain amplitude range and frequency range. The two random sinusoidal signals are used as the input of the two axes of the fast reflector respectively, and the output angle vector corresponding to this input is recorded as , the estimated vector of the output angle obtained based on the dual-axis deep coupling dynamics model and the input data is , then the data used to identify the biaxial deep coupling dynamics model is represented as a tuple , For 90% of the data set selected for network training, for Remove The remaining 10% is used as a data set for network evaluation; Then will The data is loaded into the system identification toolbox for linear part identification. By adjusting the system order, the linear model with the highest fit is obtained, thus determining the linear model in the dual-axis deep coupling dynamics model. The specific value of is further constructed to identify the nonlinear part of the biaxial deep coupling dynamics model: ， in, for The number of tuples in the , the Adam algorithm is used to solve the optimization problem, and different network layers and nodes are selected to minimize the corresponding The loss function is used to obtain the optimal weight parameters and bias parameters of the feedforward neural network.

4. The fast mirror linear quadratic tracking control method based on nonlinear elimination according to claim 3 is characterized in that: The S3 includes: Design a cost function with limited prediction time according to the control objective: ， in, is the control input sequence to be optimized, is the prediction step length, for Always look ahead The output angle vector of the fast mirror at the moment, for Always look ahead The two-axis reference trajectory vector of the fast reflector at the moment, represents the norm of a vector, is an optional constant penalty factor, for Always look ahead The control input vector of the fast mirror at the moment, for Always look ahead The control input increment vector at each moment; In order to solve the optimal control sequence, the following nonlinear programming problem is designed in combination with the dual-axis deep coupling dynamics model of the fast reflector: ， satisfy: , ， ， ； in, express Always look ahead The state of the dual-axis deep coupling dynamics model at the moment, for Always look ahead The feedforward neural network output of the dual-axis deep-coupled dynamics model of the fast mirror at each moment.

5. The fast mirror linear quadratic tracking control method based on nonlinear elimination according to claim 4, characterized in that: The S4 includes: First, The control input vector of the fast mirror at time is decomposed into: ， in, is the linear quadratic tracking control input vector, is the control input vector for nonlinear compensation, and the predicted The system status at this moment is: ， The design process is expressed as solving the following optimization problem: ， The optimal solution is ; Subsequently, the nonlinear programming problem in S3 is transformed into solving the following linear quadratic tracking problem: ， satisfy: , ， ， ； in, The input sequence of the linear quadratic tracking control to be optimized is: for Always look ahead The linear quadratic tracking control input vector of the fast mirror at the moment; The above linear quadratic tracking control problem is transformed into a compact quadratic programming problem: ， in, is the decision vector, represents its optimal value, ， ， ， ， ， ， ， ， ； in, yes The dual-axis reference trajectory vector of the fast reflector at time instant, Is to The vector formed by the dual-axis reference trajectory vector of the fast reflector at the moment, is the intermediate parameter of power, due to is a positive definite matrix, and the gradient of the cost function of the quadratic programming problem is set to 0 to obtain ,but The optimal linear quadratic tracking control input vector of the fast reflector at time is ,in represent The first and second rows of the piezoelectrically driven fast mirror are obtained in The control input vector at time is .

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