A data-driven model-free predictive control approach for piezoelectric microactuators
Through the data-driven model-free prediction control method, the approximation model is constructed using the first-order tensor product polynomial approximation solver, which solves the problem of insufficient control efficiency and accuracy of piezoelectric microdrivers, and realizes efficient precision motion control, which is suitable for multiple A high-precision field.
Patent Information
- Application Number
- CN202210669056.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2022-06-14
- Publication Date
- 2025-01-03
- Estimated Expiration
- 2042-06-14
AI Technical Summary
With the requirements of high accuracy and high response speed, existing piezoelectric microdrivers have poor control efficiency, accuracy and adaptability, and there are modeling errors and model uncertainties in the dynamic model, resulting in poor controller performance.
Using a data-driven model-free prediction control method, the approximation model is constructed through a first-order tensor product polynomial approximation solver, and the data-driven model-free prediction controller is designed to solve the optimal control sequence based on the reference trajectory and approximation model, so as to realize the piezoelectric micro- Precision motion control of the drive.
It improves control efficiency, accuracy and adaptability, reduces the complexity and ease of use of controller design, and is suitable for semiconductor processing and manufacturing, ultra-precision measurement, aerospace and other fields.
Smart Images
Figure CN114839863B_ABST
Abstract
Description
Technical Field
[0001] The disclosed embodiments relate to the field of electrical technology, and in particular to a data-driven model-free predictive control method for a piezoelectric microactuator. Background Art
[0002] At present, piezoelectric micro-actuators are an indispensable part of the modern semiconductor precision manufacturing industry chain. Common piezoelectric micro-actuators include: piezoelectric scanning tables, piezoelectric positioning tables, piezoelectric actuators, piezoelectric deflection tables, etc., which have been widely used in semiconductor processing and manufacturing, ultra-precision measurement, aerospace, biomedicine, nanoscience research and other fields. Especially in lithography technology, one of the "bottleneck" technologies in chip manufacturing, the aberration of the lithography projection lens determines the quality of the final imaging. In order to improve the imaging quality, a piezoelectric micro-actuator is required to precisely fine-tune the lens. Similarly, in lithography technology, the step-and-scan lithography machine needs to accurately and quickly control the position of the projected imaging on the wafer. Generally, the wafer workbench scanning speed is required to be 0.6 to 1 m / s, the mask stage movement speed is usually four times faster than the wafer workbench, and the motion control accuracy reaches the nanometer level. This puts forward strict requirements on the control performance of the piezoelectric micro-actuator of "high precision and high response speed". However, due to the inherent hysteresis characteristics of piezoelectric ceramics and the current lack of understanding of the essential mechanism of hysteresis characteristics, the design of high-performance controllers for piezoelectric micro-actuators has become difficult. In addition, the piezoelectric microactuator dynamics model inevitably has modeling errors and model uncertainties, which further reduces the controller performance. How to use advanced control technology to solve the above problems and achieve high control performance is a problem that researchers in this field need to solve.
[0003] PID control is often used in traditional industrial applications, but considering the existence of the above problems, PID control is far from enough. In order to describe the hysteresis characteristics, scholars have proposed a variety of hysteresis models such as the Preisach model, the Duhem model, and the Bouc–Wen model. And on the basis of the piezoelectric ceramic dynamic system considering the hysteresis characteristics, a variety of advanced control methods such as sliding mode control, adaptive sliding mode control, and model reference adaptive control have been designed. Although this type of model-based control method has achieved good results, due to the large number of dynamic model parameters and the accurate identification of parameters is also a problem, this type of model-based control method is usually cumbersome when applied to actual engineering, requiring technicians to have a high level of theoretical literacy, which greatly reduces the ease of use of this type of method.
[0004] It can be seen that there is an urgent need for an efficient, accurate, and adaptable data-driven model-free predictive control method for piezoelectric microactuators. Summary of the invention
[0005] In view of this, an embodiment of the present disclosure provides a data-driven model-free predictive control method for a piezoelectric microactuator, which at least partially solves the problems of poor control efficiency, accuracy and adaptability in the prior art.
[0006] The present disclosure provides a data-driven model-free predictive control method for a piezoelectric microactuator, comprising:
[0007] Step 1, giving the desired reference trajectory of the piezoelectric microactuator control system;
[0008] Step 2, input the initial control signal for initial sampling and construct a first-order tensor product polynomial approximation solver to solve the column vector quantization representation of the first-order tensor and the second-order tensor;
[0009] Step 3, dynamically linearize the piezoelectric microactuator according to the column vector quantization representation of the first-order tensor and the second-order tensor to obtain an approximate model;
[0010] Step 4, design a data-driven model-free predictive controller and solve the predictive controller based on the reference trajectory and the approximation model to obtain the optimal control sequence;
[0011] Step 5, using the first control variable of the optimal control sequence as the optimal control input for the next step to control the piezoelectric micro-actuator control system.
[0012] According to a specific implementation of the embodiment of the present disclosure, step 2 specifically includes:
[0013] Step 2.1, Denoted as a new state variable of the piezoelectric micro-actuator control system, the unknown piezoelectric micro-actuator dynamic model is obtained as
[0014] Step 2.2, obtain the unknown dynamic model based on the tensor product polynomial approximation theory At state point z 0 =[x 0 ,u 0 ] T The first-order tensor product polynomial approximation at is:
[0015]
[0016] Where, δz = zz 0 , are first-order and second-order tensors respectively, × 2 For tensors 2-modular product operator with vector δz;
[0017] Step 2.3, according to the first-order tensor product polynomial approximation, the column vector representation of the first-order tensor and the second-order tensor is obtained. for:
[0018]
[0019]
[0020] in, Δz i =z s+2-i -z s+2 , Δt is the sampling interval of the system, It represents the true value of each state variable of the piezoelectric micro-actuator control system at time t, and s+2 is the number of sampling data groups.
[0021] According to a specific implementation of the embodiment of the present disclosure, step 3 specifically includes:
[0022] Step 3.1, update the data structure of the sampled data;
[0023] Step 3.2, dynamically linearize the column vector representation of the first-order tensor and the second-order tensor in real time to obtain an approximate model.
[0024] According to a specific implementation of the embodiment of the present disclosure, step 3.2 specifically includes:
[0025] Step 3.2.1, converting the first-order tensor product polynomial approximation into a state equation according to the column vector quantization representation of the first-order tensor and the second-order tensor and using the state equation as a continuous linear approximation model;
[0026] Step 3.2.2, discretize the state equation to obtain a discrete linear approximation model.
[0027] According to a specific implementation of the embodiment of the present disclosure, the expression of the continuous linear approximation model is: in, is the system matrix of the continuous linear approximation system, is the input matrix of the continuous linear approximation system, is a constant vector,
[0028] According to a specific implementation method of the embodiment of the present disclosure, the expression of the discrete linear approximation model is: in, is the unit matrix, and Δt is the sampling interval of the system.
[0029] According to a specific implementation of the embodiment of the present disclosure, step 4 specifically includes:
[0030] According to the current system state and reference trajectory of the piezoelectric micro-actuator control system, the continuous linear approximation model / discrete linear approximation model is solved to obtain the optimal control sequence.
[0031] The data-driven model-free predictive control scheme for the piezoelectric micro-actuator in the disclosed embodiment includes: step 1, giving a desired reference trajectory of the piezoelectric micro-actuator control system; step 2, inputting an initial control signal for initial sampling and constructing a first-order tensor product polynomial approximation solver to solve the column vector representation of the first-order tensor and the second-order tensor; step 3, dynamically linearizing the piezoelectric micro-actuator according to the column vector representation of the first-order tensor and the second-order tensor to obtain an approximation model; step 4, designing a data-driven model-free predictive controller and solving the predictive controller according to the reference trajectory and the approximation model to obtain an optimal control sequence; step 5, using the first control quantity of the optimal control sequence as the optimal control input for the next step to control the piezoelectric micro-actuator control system.
[0032] The beneficial effects of the embodiments of the present disclosure are as follows: through the scheme of the present disclosure, the I / O data of the system is firstly collected, and a first-order tensor product polynomial approximation solver is constructed to solve the approximate dynamic system, thereby realizing the discrete dynamic linearization of the unknown dynamic system, and then a model predictive controller is designed based on the linear approximation model to perform precise motion control on the piezoelectric micro-actuator, thereby improving the control efficiency, accuracy and adaptability. BRIEF DESCRIPTION OF THE DRAWINGS
[0033] In order to more clearly illustrate the technical solutions of the embodiments of the present disclosure, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present disclosure. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.
[0034] Figure 1 A schematic flow chart of a data-driven model-free predictive control method for a piezoelectric microactuator provided in an embodiment of the present disclosure;
[0035] Figure 2 A closed-loop control structure diagram applied to a piezoelectric micro-actuator and related to a data-driven model-free predictive control method for a piezoelectric micro-actuator provided in an embodiment of the present disclosure;
[0036] Figure 3 An algorithm flow chart of a data-driven piezoelectric micro-actuator model-free predictive control method provided in an embodiment of the present disclosure applied to a piezoelectric micro-actuator;
[0037] Figure 4 A schematic diagram of the control effect of a data-driven piezoelectric micro-actuator model-free predictive control method provided by an embodiment of the present disclosure;
[0038] Figure 5 A schematic diagram of a setting control error of a data-driven piezoelectric micro-actuator model-free predictive control method provided by an embodiment of the present disclosure;
[0039] Figure 6 A schematic diagram of trajectory tracking control effect of a data-driven piezoelectric micro-actuator model-free predictive control method provided in an embodiment of the present disclosure;
[0040] Figure 7 A schematic diagram of trajectory tracking control error of a data-driven piezoelectric microactuator model-free predictive control method provided in an embodiment of the present disclosure. DETAILED DESCRIPTION
[0041] The embodiments of the present disclosure are described in detail below with reference to the accompanying drawings.
[0042] The following describes the embodiments of the present disclosure through specific examples, and those skilled in the art can easily understand other advantages and effects of the present disclosure from the contents disclosed in this specification. Obviously, the described embodiments are only a part of the embodiments of the present disclosure, rather than all of the embodiments. The present disclosure can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present disclosure. It should be noted that the following embodiments and features in the embodiments can be combined with each other without conflict. Based on the embodiments in the present disclosure, all other embodiments obtained by ordinary technicians in the field without making creative work are within the scope of protection of the present disclosure.
[0043] It should be noted that various aspects of the embodiments within the scope of the appended claims are described below. It should be apparent that the aspects described herein may be embodied in a wide variety of forms, and any specific structure and / or function described herein is merely illustrative. Based on the present disclosure, it should be understood by those skilled in the art that an aspect described herein may be implemented independently of any other aspect, and two or more of these aspects may be combined in various ways. For example, any number of aspects described herein may be used to implement the device and / or practice the method. In addition, other structures and / or functionalities other than one or more of the aspects described herein may be used to implement this device and / or practice this method.
[0044] It should also be noted that the illustrations provided in the following embodiments are only schematic illustrations of the basic concept of the present disclosure. The drawings only show components related to the present disclosure rather than being drawn according to the number, shape and size of components in actual implementation. In actual implementation, the type, quantity and proportion of each component may be changed arbitrarily, and the component layout may also be more complicated.
[0045] Additionally, in the following description, specific details are provided to facilitate a thorough understanding of the examples. However, it will be understood by those skilled in the art that the aspects described may be practiced without these specific details.
[0046] The disclosed embodiments provide a data-driven model-free predictive control method for a piezoelectric microactuator, which can be applied to the control process of a piezoelectric microactuator in scenarios such as semiconductor processing and manufacturing, ultra-precision measurement, aerospace, biomedicine, and nanoscience research.
[0047] See also Figure 1 , is a flow chart of a data-driven piezoelectric micro-actuator model-free predictive control method provided by an embodiment of the present disclosure. Figure 1 As shown, the method mainly comprises the following steps:
[0048] Step 1, giving the desired reference trajectory of the piezoelectric microactuator control system;
[0049] In specific implementation, for a piezoelectric micro-actuator control system, assuming it has n state variables and m control inputs, it can be written as is the unknown piezoelectric microactuator dynamic model, where represents the true value of each state variable of the piezoelectric micro-actuator control system at time t, represents the control input of the piezoelectric micro-actuator control system at time t. The reference trajectory of the continuous form of the piezoelectric micro-actuator control system is defined as:
[0050]
[0051] The reference trajectory represents the expected values of each state variable of the piezoelectric microactuator control system at time t.
[0052] Step 2, input the initial control signal for initial sampling and construct a first-order tensor product polynomial approximation solver to solve the column vector quantization representation of the first-order tensor and the second-order tensor;
[0053] Furthermore, the step 2 specifically includes:
[0054] Step 2.1, Denoted as a new state variable of the piezoelectric micro-actuator control system, the unknown piezoelectric micro-actuator dynamic model is obtained as
[0055] Step 2.2, obtain the unknown dynamic model based on the tensor product polynomial approximation theory At state point z 0 =[x 0 ,u 0 ]T The first-order tensor product polynomial approximation at is:
[0056]
[0057] Where, δz = zz 0 , are first-order and second-order tensors respectively, × 2 For tensors 2-modular product operator with vector δz;
[0058] Step 2.3, according to the first-order tensor product polynomial approximation, the column vector representation of the first-order tensor and the second-order tensor is obtained. for:
[0059]
[0060]
[0061] in, Δz i =z s+2-i -z s+2 , Δt is the sampling interval of the system, It represents the true value of each state variable of the piezoelectric micro-actuator control system at time t, and s+2 is the number of sampling data groups.
[0062] In specific implementation, if As a new state variable of the piezoelectric micro-actuator control system, the unknown piezoelectric micro-actuator dynamic model can be expressed as According to the tensor product polynomial approximation theory, the unknown dynamic model at the state point z 0 =[x 0 ,u 0 ] T can be approximated by the following first-order tensor product polynomial:
[0063]
[0064] where δz = zz 0 , are first-order tensors and second-order tensors respectively. 2 For tensors The 2-modular product operator of the vector δz.
[0065] Assume that the piezoelectric microactuator dynamic system is continuously differentiable at a high order, and there are s+2 sets of sampled data z in a small neighborhood of the current state point of the system. 1 =[x 1 ,u 1 ] T ,z 2=[x 2 ,u 2 ] T ,...,z s+2 =[x s+2 ,u s+2 ] T , the sampling interval of the system is Δt. If z 0 =z s+2 , then in z 0 The column vectorization form of the coefficient tensors in the first-order tensor product polynomial approximation (1) is It can be obtained by the following formula:
[0066]
[0067] The definitions of the variable symbols are as follows:
[0068]
[0069] Δz i =z s+2-i -z s+2
[0070] After substituting the coefficients obtained from equation (2) into equation (1), we call equation (1) the piezoelectric microactuator at z 0 =[x s+2 ,u s+2 ] T The first-order tensor polynomial approximation of the dynamic model.
[0071] Furthermore, we will construct a first-order tensor product polynomial approximation solver based on equation (2) to solve The column vectorized form of The specific algorithm implementation of the first-order tensor product polynomial approximation solver is as follows:
[0072]
[0073]
[0074] Step 3, dynamically linearize the piezoelectric microactuator according to the column vector quantization representation of the first-order tensor and the second-order tensor to obtain an approximate model;
[0075] Based on the above embodiment, step 3 specifically includes:
[0076] Step 3.1, update the data structure of the sampled data;
[0077] Step 3.2, dynamically linearize the column vector representation of the first-order tensor and the second-order tensor in real time to obtain an approximate model.
[0078] Furthermore, the step 3.2 specifically includes:
[0079] Step 3.2.1, converting the first-order tensor product polynomial approximation into a state equation according to the column vector quantization representation of the first-order tensor and the second-order tensor and using the state equation as a continuous linear approximation model;
[0080] Step 3.2.2, discretize the state equation to obtain a discrete linear approximation model.
[0081] Furthermore, the expression of the continuous linear approximation model is: in, is the system matrix of the continuous linear approximation system, is the input matrix of the continuous linear approximation system, is a constant vector,
[0082] Furthermore, the expression of the discrete linear approximation model is: in, is the unit matrix, and Δt is the sampling interval of the system.
[0083] In the specific implementation, the piezoelectric micro-actuator is linearized and approximated in real time. This process is called dynamic linearization. In the present invention, we will use the first-order tensor product polynomial approximation solver constructed in Algorithm 1 to dynamically linearize the piezoelectric micro-actuator. The steps are as follows:
[0084] S31. Data structure of sampled data
[0085] In order to realize the dynamic linearization of the system using Algorithm 1, the following queue with a length of s+2 is defined to store the sampled data:
[0086]
[0087] Queue Z follows the first-in, first-out rule when updating the data elements therein: when a new sample data is obtained, the new sample data can only enter from the head end of the queue, and at the same time, an old data is taken out from the tail end of the queue and discarded.
[0088] S32. Real-time solution of first-order tensor product polynomial approximation for dynamic linearization
[0089] Furthermore, equation (1) can be rewritten as the state equation as follows:
[0090]
[0091] Formula (3) is called the piezoelectric microactuator at z 0 =[x s+2 ,u s+2 ]T A continuous linear approximation system at . is the system matrix of the linear approximation system, is the input matrix of the linear approximation system, is a constant vector. And
[0092] Discretizing equation (3) yields the following equation: 0 =[x s+2 ,u s+2 ] T Discrete linear approximation system at :
[0093]
[0094] in is the unit matrix, and Δt is the sampling interval of the system.
[0095] In fact, both equation (3) and equation (4) are essentially approximation systems of piezoelectric micro-actuators, but they are different in form. In actual engineering, whether to use equation (3) or equation (4) should be selected according to the needs of controller design.
[0096] In order to realize the dynamic linearization of the piezoelectric micro-actuator and obtain its linear approximation system, the following algorithm 2 is designed to generate equation (3) or equation (4) in real time.
[0097]
[0098]
[0099] Step 4, design a data-driven model-free predictive controller and solve the predictive controller based on the reference trajectory and the approximation model to obtain the optimal control sequence;
[0100] Furthermore, the step 4 specifically includes:
[0101] According to the current system state and reference trajectory of the piezoelectric micro-actuator control system, the continuous linear approximation model / discrete linear approximation model is solved to obtain the optimal control sequence.
[0102] When implementing it, Figure 2 As shown, the embodiment of the present disclosure obtains an approximation model by dynamically linearizing the real-time acquisition system I / O data, and then solves the model predictive control based on the approximation model. The specific implementation method is introduced below.
[0103] S41. Input the initial control signal I(t) for initial sampling
[0104] Before the system starts running, since the input and output data of the system have not been sampled, a small initial input signal needs to be input into the system to collect s+2 sets of initial input and output data sampling values. The initial sampling value data will be used to obtain an initial linear approximation dynamic system. If the sampling interval of the system is Δt, the initial input signal must last at least for T = (s+2)Δt. If the initial signal is I(t), the external input of the closed-loop control system is recorded as:
[0105]
[0106] Assume x(t) and u(t) represent the real system state and external control input of the piezoelectric micro-actuator at different times. After the initial sampling is completed, an initial queue storing the initial sampling data will be obtained as shown below:
[0107]
[0108] S42. Real-time solution of model predictive controller based on approximation model
[0109] The embodiment of the present disclosure selects to use a discrete dynamic linearization method to solve the approximation model, and assumes that x(k+i|k), u(k+i|k), and r(k+i|k) represent the predicted values of the system state and control input at time k+i and the expected value of the system state obtained based on the current system state x(k) at time k. The following symbols are defined:
[0110] X k =[x(k+1|k),x(k+2|k),...,x(k+p|k)] T
[0111] U k =[u(k+1|k),u(k+2|k),...,u(k+p|k)] T
[0112] R k =[r(k+1|k),r(k+2|k),...,r(k+p|k)] T
[0113] Where p is the control time domain length of the model predictive control. k is the system state prediction sequence with a length of p starting from the kth moment, U k is the control input sequence of length p starting from the kth moment, R k is a reference trajectory sequence of length p starting from the kth moment.
[0114] Suppose the feasible set of control input is Then the model predictive control based on the approximation model can be formalized as the following constrained quadratic programming problem:
[0115]
[0116]
[0117] The definitions of the various parameter symbols are as follows:
[0118] H=2(θQθ+W),f T =2E T Qθ,E=Ψx(k)+Υ-R k
[0119]
[0120] The above quadratic programming problem (5) can be solved by the interior point method to obtain the optimal control sequence U k .
[0121] In summary, the model predictive control algorithm based on the approximation model is as follows:
[0122]
[0123]
[0124] Algorithm 1, Algorithm 2 and Algorithm 3 are as follows Figure 3 Taken together, the results finally form a data-driven model-free predictive controller.
[0125] Step 5, using the first control variable of the optimal control sequence as the optimal control input for the next step to control the piezoelectric micro-actuator control system.
[0126] In specific implementation, after obtaining the optimal control sequence U k Then select U k The first control quantity U k (1) As the optimal control input u for the next step * The piezoelectric micro-actuator of the piezoelectric micro-actuator control system is input for control. At the same time, according to the reference trajectory, the expression of the tracking error of the system can also be obtained as follows:
[0127]
[0128] After controlling the piezoelectric micro-actuator control system according to the optimal control input, the state variables of the piezoelectric micro-actuator control system can be collected in real time, so as to calculate the tracking error according to the expression of the tracking error, so as to obtain whether the performance of the controller meets the use requirements. If not, further adjustments can be made.
[0129] The data-driven model-free predictive control method for piezoelectric micro-actuators provided in this embodiment first collects the I / O data of the system and constructs a first-order tensor product polynomial approximation solver to solve the approximate dynamic system, thereby realizing discrete dynamic linearization of the unknown dynamic system, and then designs a model predictive controller based on the linear approximation model to perform precise motion control on the piezoelectric micro-actuator, thereby improving control efficiency, accuracy and adaptability.
[0130] The present solution will be described below in conjunction with a specific embodiment. In this embodiment, a one-dimensional piezoelectric micro-actuator is selected as the control object. In the simulation experiment of the embodiment, the dynamic model of the one-dimensional piezoelectric micro-actuator is as follows:
[0131]
[0132] in represent the displacement, velocity and acceleration of the piezoelectric microactuator, respectively. represents the control input of the piezoelectric microactuator, Represents the hysteresis term present in the piezoelectric microactuator dynamics. are model parameters, m is the equivalent mass of the piezoelectric micro-actuator, b is the equivalent damping coefficient of the piezoelectric micro-actuator, k is the equivalent elastic coefficient of the piezoelectric micro-actuator, and d is the equivalent damping coefficient of the piezoelectric micro-actuator. e is the equivalent piezoelectric coefficient of the piezoelectric microactuator. In the implementation example, we use the following Bouc-Wen model to describe the dynamics of the hysteresis term h:
[0133]
[0134] in is the Bouc-Wen model parameter, which can be used to adjust the shape of the hysteresis loop. Equations (6) and (7) together constitute the dynamic model of the piezoelectric micro-actuator. In this implementation example, this dynamic model is only used to simulate the real piezoelectric micro-actuator to obtain sampling data. The data-driven model-free predictive control method does not contain any information about the piezoelectric micro-actuator dynamic model. The model parameters set in the simulation experiment of the implementation example are derived from [6]:
[0135] m=0.28kg, b=1302.28N / (m / s), k=53452N / m, de=0.127nm / V,
[0136] α=0.5136, β=0.124, λ=-0.073
[0137] S1. Given the desired reference trajectory r(t)
[0138] The one-dimensional piezoelectric microactuator has two state variables and one control input, so n = 2, m = 1. represents the true value of each state of the system at time t, represents the control input of the system at time t.
[0139] The setting control is a special case of trajectory tracking control, that is, tracking a fixed trajectory. In this embodiment, the desired trajectory r of the setting control is s (t) = 1 μm. The desired trajectory of trajectory tracking control is r t (t) = 1 + 0.1sin(100t) μm.
[0140] S2. Constructing a first-order tensor product polynomial approximation solver
[0141] In this implementation example, the sampling interval is set to Δt=1e-6s, and the number of sampling nodes is set to s+2=10. Then the input of the first-order tensor product polynomial approximation solver (Algorithm 1) is n=2, m=1, Δt=1e-6s, [x 1 ,x 2 ,...,x 10 ],[u 1 ,u 2 ,...,u 10 ].
[0142] S3. Dynamic Linearization of Piezoelectric Microactuators
[0143] In this implementation example, we will use the first-order tensor product polynomial approximation solver constructed in Algorithm 1 to dynamically linearize the piezoelectric micro-actuator. The steps are as follows:
[0144] S31. Data structure of sampled data
[0145] In order to use Algorithm 1 to realize the dynamic linearization of the system, the following queue with a length of 10 is defined to store the sampled data:
[0146]
[0147] Queue Z follows the first-in, first-out rule when updating the data elements therein: when a new sample data is obtained, the new sample data can only enter from the head end of the queue, and at the same time, an old data is taken out from the tail end of the queue and discarded.
[0148] S32. Real-time solution of first-order tensor product polynomial approximation for dynamic linearization
[0149] In this embodiment, the discrete dynamic linearization approximation formula (4) is used to perform dynamic linearization of the piezoelectric micro-actuator, and z is recorded as 0 =[x 10 ,u 10 ] T The discrete linear approximation system at is:
[0150]
[0151] In order to realize the dynamic linearization of the piezoelectric micro-actuator and obtain the above-mentioned discrete linear approximation system, Algorithm 2 is called to generate the above formula in real time.
[0152] S4. Design of data-driven model-free predictive controller
[0153] A data-driven model-free predictive controller is designed for the piezoelectric microactuator. The specific implementation method is introduced below.
[0154] S41. Input the initial control signal I(t) for initial sampling
[0155] Before the system starts running, since the input and output data of the system have not been sampled, a small initial input signal needs to be input to the system to collect 10 sets of initial input and output data sampling values. The initial sampling value data will be used to obtain an initial linear approximation dynamic system. If the sampling interval of the system is Δt, the initial input signal needs to last at least T = 10Δt. The initial signal set in this implementation example is Then the external input of the closed-loop control system is recorded as:
[0156]
[0157] Assume x(t) and u(t) represent the real system state and external control input of the piezoelectric micro-actuator at different times. After the initial sampling is completed, an initial queue storing the initial sampling data will be obtained as shown below:
[0158]
[0159] S42. Real-time solution of model predictive controller based on approximation model
[0160] In the present invention, a discrete dynamic linearization method is selected to solve the approximation model, and x(k+i|k), u(k+i|k), and r(k+i|k) are respectively represented by the predicted values of the system state and control input at time k+i obtained according to the current system state x(k) at time k, and the expected value of the system state. In this implementation example, the control time domain length p of the model predictive control is set to 5. Then the following symbols are defined:
[0161] X k =[x(k+1|k),x(k+2|k),...,x(k+5|k)] T
[0162] U k=[u(k+1|k),u(k+2|k),...,u(k+5|k)] T
[0163] Rs k =[r s (k+1|k),r s (k+2|k),...,r s (k+5|k)] T
[0164] R k =[r t (k+1|k),r t (k+2|k),...,r t (k+5|k)] T
[0165] X k is the system prediction state sequence with a length of 5 starting from the kth moment. k is a control input sequence of length 5 starting from the kth moment. k is the reference trajectory sequence of the control setting starting from the kth moment with a length of 5, Rt k is a trajectory tracking control reference trajectory sequence with a length of 5 starting from the kth moment.
[0166] In this implementation example, it is assumed that the minimum value of the control input voltage is 0 V and the maximum value is 50 V. And because in actual application scenarios, the transformation of the control input u cannot be completed instantly, it is assumed that the control input satisfies the Lipschitz continuity condition as follows:
[0167] |u(t+Δt)-u(t)|≤L u |Δt|
[0168] Where L u is the Lipschitz continuity constant. In this embodiment, let L u =100000. Then the feasible set of control input is L u =100000, Δt = 1e-6s, then the model predictive control based on the approximation model can be formalized as the following constrained quadratic programming problem:
[0169]
[0170]
[0171] in:
[0172] H=2(θQθ+W),f T =2E T Qθ,E=Ψx(k)+Υ-Rk
[0173]
[0174] The weighted matrices Q and W in the optimization problem of this implementation example are set as follows:
[0175]
[0176]
[0177] Then, the interior point method is used to solve the optimal control sequence. In this embodiment, the quadprog function in MATLAB is directly called to solve the above quadratic programming problem using the interior point method. k Select the first control variable U k (1) As the optimal control input u for the next step * When R k =Rs k When * is the optimal control input for setting control, when R k =Rt k When * is the optimal control input for trajectory tracking control.
[0178] S5. Apply data-driven model-free predictive control method to piezoelectric microactuator and conduct simulation experiments
[0179] The data-driven model-free predictive control method is applied to the piezoelectric microactuator. The mathematical software MATLAB is used to perform the setting control and trajectory tracking control, and the experimental data are collected and plotted. The simulation time is set to 0.1 seconds. Figure 4 The setting control effect when the present invention is applied to the implementation example, Figure 6 The figure shows the trajectory tracking control effect when the present invention is applied to an implementation example. Figure 5 is the setting control error when the present invention is applied to the implementation example, Figure 7 is the trajectory tracking control error when the present invention is applied to the implementation example. From a macroscopic point of view, under data-driven model-free predictive control, the displacement x of the piezoelectric microactuator quickly converges to the given reference trajectory. From a microscopic point of view, it can be seen that the steady-state accuracy of the set tracking reaches |ε|≤3e-3μm, and the steady-state mean square error of the trajectory tracking control is 3.3e-3μm. This shows that when the present invention is applied to the piezoelectric microactuator, both the set control and the trajectory tracking control have very high control accuracy.
[0180] It should be understood that various parts of the present disclosure may be implemented in hardware, software, firmware, or a combination thereof.
[0181] The above is only a specific implementation of the present disclosure, but the protection scope of the present disclosure is not limited thereto. Any changes or substitutions that can be easily thought of by a person skilled in the art within the technical scope disclosed in the present disclosure should be included in the protection scope of the present disclosure. Therefore, the protection scope of the present disclosure should be based on the protection scope of the claims.
Claims
1. A data-driven model-free predictive control method for a piezoelectric microactuator, characterized in that: include: Step 1, giving the desired reference trajectory of the piezoelectric microactuator control system; Step 2, input the initial control signal for initial sampling and construct a first-order tensor product polynomial approximation solver to solve the column vector quantization representation of the first-order tensor and the second-order tensor; The step 2 specifically includes: Step 2.1, Denoted as a new state variable of the piezoelectric micro-actuator control system, the unknown piezoelectric micro-actuator dynamic model is obtained as Step 2.2: Obtain the unknown piezoelectric microactuator dynamics model based on the tensor product polynomial approximation theory. At the state point z0=[x0,u0] T The first-order tensor product polynomial approximation at is: Among them, δz=z-z0, are first-order tensors and second-order tensors respectively, ×2 is a tensor 2-modular product operator with the vector δz; Step 2.3, according to the first-order tensor product polynomial approximation, the column vector representation of the first-order tensor and the second-order tensor is obtained. for: in, Δz i =z s+2-i -z s+2 , Δt is the sampling interval of the system, represents the true value of each state variable of the piezoelectric micro-actuator control system at time t, and s+2 is the number of sampling data groups; Step 3, dynamically linearize the piezoelectric microactuator according to the column vector quantization representation of the first-order tensor and the second-order tensor to obtain an approximate model; The step 3 specifically includes: Step 3.1, update the data structure of the sampled data; Step 3.2, dynamically linearize the column vector representation of the first-order tensor and the second-order tensor in real time to obtain an approximation model; Step 4, design a data-driven model-free predictive controller and solve the predictive controller based on the reference trajectory and the approximation model to obtain the optimal control sequence; Step 5, using the first control variable of the optimal control sequence as the optimal control input for the next step to control the piezoelectric micro-actuator control system.
2. The method according to claim 1, characterized in that , the step 3.2 specifically includes: Step 3.2.1, converting the first-order tensor product polynomial approximation into a state equation according to the column vector quantization representation of the first-order tensor and the second-order tensor and using the state equation as a continuous linear approximation model; Step 3.2.2, discretize the state equation to obtain a discrete linear approximation model.
3. The method according to claim 2, characterized in that , the expression of the continuous linear approximation model is in, is the system matrix of the continuous linear approximation system, is the input matrix of the continuous linear approximation system, is a constant vector, 4. The method according to claim 3, characterized in that , the expression of the discrete linear approximation model is in, is the unit matrix, and Δt is the sampling interval of the system.
5. The method according to claim 4, characterized in that , the step 4 specifically includes: According to the current system state and reference trajectory of the piezoelectric micro-actuator control system, the continuous linear approximation model / discrete linear approximation model is solved to obtain the optimal control sequence.