Control method and system of piezoelectric micro-positioning platform

Abstract

The invention discloses a control method and system for a piezoelectric micro-positioning platform, and relates to the technical field of micro-positioning platforms, and the method comprises the following steps: obtaining an input voltage, an output displacement and a displacement error of the piezoelectric micro-positioning platform at a current moment; inputting the input voltage and the output displacement at the current moment into a Kalman filter to obtain an estimated state quantity at the current moment; the estimated state quantity and the displacement error at the current moment are input into a DLQT controller for optimal control law solving, and the input voltage at the next moment is obtained; and inputting the input voltage at the next moment to the piezoelectric micro-positioning platform to obtain the output displacement at the next moment. According to the method, the nonlinear model of the piezoelectric micro-positioning platform is represented through the Koopman linear model to obtain the Koopman linear model, the design of a hysteresis compensator is avoided, a linear control strategy is directly adopted to design the controller, and the design complexity of the controller is reduced.

Description

Technical Field

[0001] This invention relates to the field of micro-positioning platform technology, and in particular to a control method and system for a piezoelectric micro-positioning platform. Background Technology

[0002] Piezoelectric micro-positioning platforms, driven by piezoelectric actuators, offer advantages such as high driving force, nanometer-level control precision, low power consumption, and high stability, making them widely used in nanometer measurement, microscopic imaging, and precision machining. However, when an electric field is applied, the piezoelectric actuator exhibits hysteresis nonlinearity with multi-valued mapping, memory, and rate-dependent characteristics, severely affecting the positioning accuracy of the piezoelectric micro-positioning platform and limiting its application. The ideal displacement can be achieved by changing the voltage of the piezoelectric micro-positioning platform.

[0003] The inherent hysteresis nonlinearity of piezoelectric micro-positioning platforms severely affects positioning accuracy. To improve this accuracy, a hysteresis model describing the platform's nonlinear characteristics must first be established. Then, control methods can be developed based on this model to reduce the hysteresis nonlinearity and improve control accuracy. Currently, modeling methods for piezoelectric micro-positioning platforms can be categorized into parametric and nonparametric methods. Parametric models typically describe hysteresis characteristics through combinations of different hysteresis operators or differential equations, including the Preisach model, Prandtl-Ishlinskii model, Duhem model, and Bouc-Wen model. Parametric modeling methods can reflect the internal dynamics of the system, but their drawback is the large number of parameters requiring identification, relying heavily on the superiority of the identification algorithm. Hysteresis nonlinearity is rate-dependent, and most parametric hysteresis models cannot describe rate-dependent hysteresis nonlinear characteristics, requiring improvement and increasing the workload. Non-parametric modeling uses data fitting or training methods to characterize the input-output characteristics of a piezoelectric micro-positioning platform without considering the internal properties of the object being described. Commonly used non-parametric models include neural network models, multinomial models, and fuzzy systems. Non-parametric models can flexibly express input-output characteristics, but their drawback is that they require a large amount of data for training.

[0004] Currently, control methods for piezoelectric micro-positioning platforms mainly include model-free control methods and hysteresis-compensated control methods. Model-free control methods are difficult to solve for the controller and do not meet practical requirements. Therefore, most current control methods for piezoelectric micro-positioning platforms are based on hysteresis models. Hysteresis-based control methods require designing a hysteresis compensator based on the hysteresis model, i.e., solving for the inverse model of the hysteresis model. The inverse model is used to reduce nonlinearity before the control algorithm is studied. However, the compensation effect of the hysteresis compensator depends on the accuracy of the hysteresis model. If the accuracy is not high, the nonlinearity of the system after hysteresis compensation will increase the difficulty of further controller design. Summary of the Invention

[0005] This invention provides a control method and system for a piezoelectric micro-positioning platform, which solves the problems of insufficient modeling accuracy, high computational complexity, and difficulty in designing controllers in existing modeling methods when describing the hysteresis nonlinearity of piezoelectric micro-positioning platforms.

[0006] The present invention adopts the following technical solution: In a first aspect, the present invention provides a control method for a piezoelectric micro-positioning platform, comprising the following steps: Obtain the input voltage, output displacement, and target displacement of the piezoelectric micro-positioning platform at the current moment; The current input voltage and output displacement are input to the Kalman filter to obtain the estimated state variables at the current moment; The estimated state variables and target displacement at the current moment are input to the DLQT controller to solve for the optimal control law, thus obtaining the input voltage at the next moment. Specifically, the neural network is trained using the historical input voltages and output displacements of the piezoelectric micro-positioning platform to obtain a nonlinear model. Based on Koopman operator theory, the nonlinear model is upgraded to a Koopman linear model. The feedforward and feedback control laws in the DLQT controller are designed using the constant matrix of the Koopman linear model and the target displacement. The optimal control law is obtained by summing the feedforward and feedback control laws. The input voltage at the next moment is input to the piezoelectric micro-positioning platform to obtain the output displacement at the next moment.

[0007] Preferably, the step of increasing the dimensionality of the nonlinear model based on Koopman operator theory to obtain a Koopman linear model specifically includes the following steps: The hidden layer output based on the nonlinear model constructs a state boosting function vector; Based on the lifting function vector, the Koopman linear model is obtained by approximating the finite-dimensional Koopman operator using the extended dynamic mode decomposition algorithm. The Koopman linear model is shown in detail below: ; In the formula, In higher-dimensional space k The state quantity at any given time. z ( k +1) represents higher-dimensional space. k The state quantity at time +1 For the first i A state promotion function, To output the predicted displacement value, , and It is a constant matrix. u (k () represents the input to the hidden layer of the nonlinear model; A regularization parameter is introduced to solve for multiple constant matrices.

[0008] Preferably, the step of introducing a regularization parameter to solve for multiple constant matrices includes the following steps: Input voltage at different times With output displacement Construct a matrix and The state variables of two adjacent time steps in the nonlinear model constitute a matrix. and ; Based on the state promotion function and The dimensionality is increased to obtain the state matrix after the dimensionality increase. and ; The upgraded state matrix and input matrix Combined into a data matrix Z ; Introducing regularization parameters Calculate the following pseudo-inverse matrix: ; In the formula, For data matrix Z The regularized pseudo-inverse matrix, For matrix L The regularized pseudo-inverse matrix, I It is the identity matrix. L State matrix X 2lift ; Using regularized pseudo-inverse matrix Solve the matrix and : ; Using regularized pseudo-inverse matrix Solve the matrix : .

[0009] Preferably, the optimal control law is as follows: ; In the formula, u * ( k ) represents the optimal control law. For feedforward control law, For feedback control law, R It is the first positive semi-definite matrix. P It is the second positive semi-definite matrix. A and B For matrix A ED and B ED , g ( k )and g ( k +1) is k Time and k Parameter matrix at time +1 The high-dimensional state variables estimated for the Kalman observer.

[0010] Preferably, the neural network is an LSTM-RNN, comprising an input layer, a hidden layer, and an output layer, wherein the activation functions of the hidden layer and the output layer are respectively selected as the tanh activation function and the linear activation function.

[0011] Secondly, the present invention also provides a control system for a piezoelectric micro-positioning platform, comprising: The data acquisition module is used to acquire the input voltage, output displacement, and target displacement of the piezoelectric micro-positioning platform at the current moment. The estimation module is used to input the current input voltage and output displacement to the Kalman filter to obtain the estimated state variables at the current time. The solution module is used to input the estimated state variables and target displacement at the current moment into the DLQT controller to solve for the optimal control law and obtain the input voltage at the next moment. Specifically, the neural network is trained using the historical input voltage and output displacement of the piezoelectric micro-positioning platform to obtain a nonlinear model. Based on Koopman operator theory, the nonlinear model is upgraded to obtain a Koopman linear model. The feedforward control law and feedback control law in the DLQT controller are designed using the constant matrix in the Koopman linear model and the target displacement. The optimal control law is obtained by summing the feedforward and feedback control laws. The control module is used to input the input voltage of the next moment to the piezoelectric micro-positioning platform to obtain the output displacement of the next moment.

[0012] Compared with the prior art, the above-mentioned at least one technical solution adopted by the present invention can achieve the following beneficial effects: This invention first trains a neural network using the historical input voltage and output displacement of a piezoelectric micro-positioning platform to obtain a nonlinear model. Then, this nonlinear model is characterized using a Koopman linearization model, resulting in a Koopman linear model, thus avoiding the need for a hysteresis compensator. A DLQT controller is then constructed based on the Koopman linear model. Specifically, the feedforward and feedback control laws in the DLQT controller are designed based on its constant matrix. In specific iterations, the input voltage and output displacement of the piezoelectric micro-positioning platform at the current moment are input to a Kalman filter to obtain the estimated state variables at the current moment. These estimated state variables and the target displacement are then input to the DLQT controller to solve for the optimal control law, yielding the input voltage at the next moment. Finally, the input voltage at the next moment is input to the piezoelectric micro-positioning platform to obtain the output displacement at the next moment. This invention directly employs a linear control strategy to design the controller, reducing its design complexity. Attached Figure Description

[0013] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0014] Figure 1 This is a schematic diagram of a control method for a piezoelectric micro-positioning platform according to the present invention; Figure 2 This is a schematic diagram of data acquisition according to the present invention; Figure 3 The hysteresis characteristic curve of the piezoelectric micro-positioning platform of the present invention is shown below. Figure 4 This is a schematic diagram of the LSTM-RNN structure in an embodiment of the present invention; Figure 5 This is a schematic diagram of the control system of a piezoelectric micro-positioning platform according to the present invention. Detailed Implementation

[0015] 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 embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0016] Example 1 Relevant background: The rapid development of Koopman operator theory has provided a new approach to the study of nonlinear systems. Its basic idea is to elevate the nonlinear system to a higher-dimensional linear space, globally linearizing the system while preserving its original nonlinear characteristics. This method transforms a nonlinear problem into a linear one. Currently, the main finite-dimensional approximation algorithms for the Koopman operator include Dynamic Mode Decomposition (DMD), Extended Dynamic Mode Decomposition (EDMD), and Deep Neural Network (DNN). In recent years, the Koopman operator has been gradually extended to controlled nonlinear systems. Based on the Koopman high-dimensional linear model, linear control design methods can be used to design controllers for nonlinear systems, such as robust, adaptive, and model predictive controllers. Therefore, this invention proposes applying Koopman operator theory to a piezoelectric micro-positioning platform. The EDMD algorithm is simpler and easier to implement than the DNN algorithm, while also achieving better approximation results than the DMD algorithm. Therefore, this invention uses the EDMD algorithm to approximate the finite-dimensional linear operator of the Koopman operator.

[0017] A control method for a piezoelectric micro-positioning platform is proposed. By transforming a nonlinear system into a high-dimensional linear system, global linearization is achieved, providing a new approach for the modeling and control of nonlinear systems in piezoelectric micro-positioning platforms. (Refer to...) Figure 1 Specifically, it includes the following steps:

[0018] S1: Complete the nonlinear experimental test of the piezoelectric micro-positioning platform. Using a data acquisition card, piezoelectric control box, and host computer, collect the input and output data of the piezoelectric micro-positioning platform. Identify the LSTM-RNN (Long Short-Term Memory-Recurrent Neural Network) nonlinear model of the piezoelectric micro-positioning platform, such as... Figure 2 As shown.

[0019] S11: Complete the nonlinear experimental test of the piezoelectric micro-positioning platform and collect its input and output data. Specifically, determine that the input of the piezoelectric micro-positioning platform is the control voltage. The output is displacement. .

[0020] Control voltages with frequencies of 1Hz, 10Hz, 20Hz, and 40Hz were input sequentially. Input and output data from the piezoelectric micro-positioning platform were collected to construct an LSTM-RNN dataset. The relationship between input and output is as follows: Figure 3 The hysteresis characteristic curve of the piezoelectric micro-positioning platform is shown.

[0021] S12: Identify the LSTM-RNN based on the collected dataset.

[0022] Initialize the neural network and determine the structure of the LSTM-RNN as follows: Figure 4 As shown, it includes an input layer, a hidden layer, and an output layer. f ( k (This is the Gate of Oblivion) i ( k ) is the input gate. o ( k () is the output gate. The input voltage signal of the piezoelectric micro-positioning platform at the current moment is selected. As input to the neural network The output displacement of the piezoelectric micro-positioning platform As the output of the neural network Number of input layer nodes in LSTM-RNN p Set to 1, output the number of layer nodes. q Set to 1, the number of hidden layer nodes is set to 30 by trial and error, and the activation functions of the hidden layer and the output layer are selected as tanh activation function and linear activation function, respectively.

[0023] The performance metric for the neural network is set as the root mean square difference between the predicted output and the actual output, and the training algorithm for the neural network is set as the Adam optimization algorithm.

[0024] After identification is completed, the LSTM-RNN can be represented as a state-space expression: ; In the formula, , This is the activation function of the Sigmoid function, and its output is the amount of information passed through, with a value between 0 and 1. This represents the hidden layer output of the neural network at the current moment. As input to the hidden layer, The current cell state is used to store long-term memory and is updated through a weighted combination of the forget gate and the input gate. , and The weight matrices from the input layer to the output gate, the weight matrices from the hidden layer state to the output gate, and the weight matrices from the output layer to the output gate are respectively. and The bias matrix, for k The output value of LSTM-RNN at time 1.

[0025] S2: Construct a new dataset using the identified LSTM-RNN, increase the dimensionality of the LSTM-RNN model to a higher dimension space based on the Koopman operator theory, and use the extended dynamic mode decomposition algorithm to approximate the Koopman operator to achieve global linearization of the LSTM-RNN model, thus obtaining the Koopman linear model.

[0026] S21: Process the dataset using the LSTM-RNN identified in step S1 to obtain a new dataset; the specific process of constructing the new dataset is as follows: (1) Obtaining input from the piezoelectric micro-positioning platform and output ; (2) Train the LSTM-RNN model using the input and output data to obtain the state variables of the LSTM-RNN at this time. State quantity compared to the previous time step .

[0027] (3) Construct a data matrix, including inputs at different times. With output Construct a matrix and The state variables at two adjacent time points each form a matrix. and .

[0028] S22: Based on Koopman operator theory and the extended dynamic mode decomposition algorithm, the LSTM-RNN model is upgraded to a higher-dimensional space, achieving global linearization of the LSTM-RNN model and obtaining the Koopman linear model. The specific process of obtaining the Koopman linear model is as follows:

[0029] (1) Set the type and number of state promotion functions.

[0030] The choice of state boosting function is directly related to the effectiveness of the extended dynamic mode decomposition algorithm. Only by selecting a relatively complete and suitable state boosting function as the observable can the approximation of the Koopman operator be completed well. Typically, it can be one or a combination of several nonlinear functions such as thinplate radial basis functions, polynomial functions, and Gaussian functions. Through trial and error, the number of state boosting functions was determined to be 70, forming the state boosting function vector as follows: ; In the formula, Let the state promotion function vector be... , For state promotion functions, T This is a transpose.

[0031] To facilitate the state variables after dimensionality upgrade The predicted output displacement is obtained from the data, and the first state boosting function is defined. ; In the above formula, This represents the hidden layer output of the neural network at the current moment. This is the weight matrix between the output gate and the hidden layer.

[0032] The remaining state promotion functions are chosen as 69 Gauss functions: ; In the formula, Set to a random number between [-1, 1] All are set to 1.

[0033] (2) Transform the LSTM-RNN nonlinear model into a Koopman linear model.

[0034] The specific Koopman linear model can be represented as: ; in, For the state variables in the higher-dimensional space after dimensionality upgrade. , For state promotion functions, This represents the predicted output displacement value of the high-dimensional linear model. , and Let be the constant matrix that needs to be solved.

[0035] (3) The following is the solution matrix , and The specific steps.

[0036] Use the defined state promotion function to transform the data matrix and After performing dimensionality increase processing, the state matrix after dimensionality increase is obtained as follows: and .

[0037] Based on the idea of ​​least squares, a regularization parameter is introduced to solve the problem. , and .

[0038] Construct a data matrix.

[0039] The upgraded state matrix Combined with the input matrix U to form a data matrix .

[0040] Introducing regularization parameters Calculate the following pseudo-inverse matrix results: ; ; In the formula, Z pin_reg For data matrix Z The regularized pseudo-inverse matrix, C pin_reg Let L be the regularized pseudo-inverse matrix. L State matrix X 2lift .

[0041] Using regularization pseudo-inverse Solve the matrix , : ,matrix from Extraction of the first part, matrix from Extract the latter part.

[0042] Using regularized pseudo-inverse matrix Solve the matrix : .

[0043] S3: DLQT (Discrete Time Linear Quadratic Tracking) controller design.

[0044] A Kalman observer is designed based on the Koopman linearization model. The high-dimensional state of the system is estimated based on the input voltage and output displacement of the piezoelectric micro-positioning platform. A DLQT controller is designed to realize the tracking control of the piezoelectric micro-positioning platform for the desired displacement.

[0045] The control process of the DLQT controller is as follows: Obtain the current input of the piezoelectric micro-positioning platform u r ( k ) and output y r ( k ).

[0046] Estimating the high-dimensional state variables of a system using a Kalman observer .

[0047] The optimal control law is obtained by using a DLQT controller.

[0048] The process of designing a DLQT controller based on the Koopman linearization model is as follows: Define system tracking error , r ( k )and y ( k ) represent the target displacement and the actual displacement at the current moment, respectively.

[0049] Define system performance metrics: , R It is the first positive semi-definite matrix. Q It is a positive definite matrix. u * ( k This is the optimal tracking control law. n It is a positive integer greater than 1 at the terminal time.

[0050] The DLQT controller includes feedforward control. With feedback control The feedforward term enables the closed-loop system to output a signal that tracks the target displacement, while the feedback term stabilizes the closed-loop system.

[0051] Solve for the optimal tracking control law based on the system's tracking error, performance indicators, and state estimates: ; = , = . For high-dimensional state quantities estimated by Kalman observers, P It is the second positive semi-definite matrix. g ( k ) is a parameter matrix that is related to the target displacement input.

[0052] The following is a matrix P With matrix g ( k Solution method Second matrix P From the Ricardi algebraic equation, we obtain: ; In the formula, A and B The matrix obtained from the above solution A ED and B ED .

[0053] Parameter matrix g ( k The above equation can be solved by reverse iteration: ; In the formula, E It is the identity matrix. C The matrix obtained from the above solution C ED .

[0054] During the iteration process g ( k It will gradually converge, and... Approximation yields the following relationship: ; The optimal tracking control law u * ( k It acts on the piezoelectric micro-positioning platform.

[0055] S4: Obtain the input voltage, output displacement, and target displacement of the piezoelectric micro-positioning platform at the current moment.

[0056] S5: Input the current input voltage and output displacement to the Kalman filter to obtain the estimated state variables at the current time.

[0057] S6: Input the estimated state variables and target displacement at the current moment into the DLQT controller to solve for the optimal control law and obtain the input voltage at the next moment.

[0058] S7: Input the input voltage at the next moment to the piezoelectric micro-positioning platform to obtain the output displacement at the next moment.

[0059] Example 2 Reference Figure 5 The present invention also provides a control system for a piezoelectric micro-positioning platform, including an acquisition module, an estimation module, a solution module and a control module.

[0060] The acquisition module is used to obtain the input voltage, output displacement, and displacement error of the piezoelectric micro-positioning platform at the current moment.

[0061] The estimation module is used to input the current input voltage and output displacement to the Kalman filter to obtain the estimated state variables at the current time.

[0062] The solution module is used to input the estimated state variables and displacement error at the current moment into the DLQT controller to solve for the optimal control law and obtain the input voltage at the next moment. Specifically, the nonlinear model is upgraded based on the Koopman operator theory to obtain the Koopman linear model, and the optimal control law is designed based on the constant matrix in the Koopman linear model. The neural network is trained using the historical input voltage and output displacement of the piezoelectric micro-positioning platform to obtain the nonlinear model.

[0063] The control module is used to input the input voltage of the next moment to the piezoelectric micro-positioning platform to obtain the output displacement of the next moment.

[0064] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of the invention.

[0065] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

Claims

1. A control method for a piezoelectric micro-positioning platform, characterized in that, Includes the following steps: Obtain the input voltage, output displacement, and target displacement of the piezoelectric micro-positioning platform at the current moment; The current input voltage and output displacement are input to the Kalman filter to obtain the estimated state variables at the current moment; The estimated state variables and target displacement at the current moment are input to the DLQT controller to solve for the optimal control law, thus obtaining the input voltage at the next moment. Specifically, the neural network is trained using the historical input voltages and output displacements of the piezoelectric micro-positioning platform to obtain a nonlinear model. Based on Koopman operator theory, the nonlinear model is upgraded to a Koopman linear model. The feedforward and feedback control laws in the DLQT controller are designed using the constant matrix of the Koopman linear model and the target displacement. The optimal control law is obtained by summing the feedforward and feedback control laws. The input voltage at the next moment is input to the piezoelectric micro-positioning platform to obtain the output displacement at the next moment.

2. The control method for a piezoelectric micro-positioning platform as described in claim 1, characterized in that, The method of increasing the dimensionality of the nonlinear model based on Koopman operator theory to obtain the Koopman linear model specifically includes the following steps: The hidden layer output based on the nonlinear model constructs a state boosting function vector; Based on the lifting function vector, the Koopman linear model is obtained by approximating the finite-dimensional Koopman operator using the extended dynamic mode decomposition algorithm. The Koopman linear model is shown in detail below: ； In the formula, In higher-dimensional space k The state quantity at any given time. z ( k +1) represents higher-dimensional space. k The state quantity at time +1 For the first i A state promotion function, To output the predicted displacement value, , and It is a constant matrix. u ( k () represents the input to the hidden layer of the nonlinear model; A regularization parameter is introduced to solve for multiple constant matrices.

3. The control method for a piezoelectric micro-positioning platform as described in claim 1, characterized in that, The method of introducing regularization parameters to solve multiple constant matrices includes the following steps: Input voltage at different times With output displacement Construct a matrix and The state variables of two adjacent time steps in the nonlinear model constitute a matrix. and ; Based on the state promotion function and By increasing the dimensionality, we obtain the state matrix after the dimensionality increase. and ; The upgraded state matrix and input matrix Combined into a data matrix Z ; Introducing regularization parameters Calculate the following pseudo-inverse matrix: ； In the formula, For data matrix Z The regularized pseudo-inverse matrix, For matrix L The regularized pseudo-inverse matrix, I It is the identity matrix. L State matrix X 2lift ; Using regularized pseudo-inverse matrix Solve the matrix and : ； Using regularized pseudo-inverse matrix Solve the matrix : 。 4. The control method for a piezoelectric micro-positioning platform as described in claim 1, characterized in that, The optimal control law is as follows: ； In the formula, u * ( k ) represents the optimal control law. For feedforward control law, For feedback control law, R It is the first positive semi-definite matrix. P It is the second positive semi-definite matrix. A and B For matrix A ED and B ED , g ( k )and g ( k +1) is k Time and k Parameter matrix at time +1 The high-dimensional state variables estimated for the Kalman observer.

5. The control method for a piezoelectric micro-positioning platform as described in claim 1, characterized in that, The neural network uses LSTM-RNN and includes an input layer, a hidden layer, and an output layer. The activation functions of the hidden layer and the output layer are respectively the tanh activation function and the linear activation function.

6. A control system for a piezoelectric micro-positioning platform, characterized in that, include: The data acquisition module is used to acquire the input voltage, output displacement, and target displacement of the piezoelectric micro-positioning platform at the current moment. The estimation module is used to input the current input voltage and output displacement to the Kalman filter to obtain the estimated state variables at the current time. The solution module is used to input the estimated state variables and target displacement at the current moment into the DLQT controller to solve for the optimal control law and obtain the input voltage at the next moment. Specifically, the neural network is trained using the historical input voltage and output displacement of the piezoelectric micro-positioning platform to obtain a nonlinear model. Based on Koopman operator theory, the nonlinear model is upgraded to obtain a Koopman linear model. The feedforward control law and feedback control law in the DLQT controller are designed using the constant matrix in the Koopman linear model and the target displacement. The optimal control law is obtained by summing the feedforward and feedback control laws. The control module is used to input the input voltage of the next moment to the piezoelectric micro-positioning platform to obtain the output displacement of the next moment.

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