A data-driven adaptive control method based on a discrete piezoelectric positioning platform model

Abstract

The application provides a data-driven adaptive control method based on a discrete piezoelectric positioning platform model, comprising the following steps: S1: a discrete nonlinear state space model is proposed according to the nonlinear characteristics of the piezoelectric positioning platform; S2: an adaptive iterative algorithm for parameter estimation is designed according to the nonlinear characteristics of the model; S3: a data-driven algorithm is designed on the basis of the adaptive iterative algorithm; S4: a tracking controller is designed according to the linear part, and an analytical expression of the control input at each moment is obtained; and S5: a parameter design criterion in the data-driven adaptive iterative algorithm is proposed. The application mainly aims at the problems of strong nonlinear dynamics and unknown parameter estimation in piezoelectric positioning platform tracking control, can fully utilize historical data to realize unknown dynamic parameter estimation of the piezoelectric positioning platform, and improve tracking performance.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of piezoelectric driving modeling and adaptive tracking control, and particularly relates to a data-driven adaptive control method based on a discrete piezoelectric positioning platform model, which is mainly used for unknown parameter estimation and robust tracking control of a piezoelectric positioning platform model, and improves the transient performance of the system. BACKGROUND

[0002] The piezoelectric positioning platform plays a key role in the fields of micro-nano material processing, micro-robot and other precision engineering. The piezoelectric actuator, which is the core driving element of the piezoelectric positioning platform, has been widely concerned due to its fast dynamic response, high positioning accuracy and other characteristics. However, in actual application, the piezoelectric actuator has strong hysteresis nonlinearity and dynamic characteristics changing with control input and working state, which brings difficulties to the high-precision servo tracking control of the piezoelectric positioning platform. Most of the existing methods for describing strong hysteresis nonlinearity cannot describe the case that the dynamic characteristics of the piezoelectric actuator change with the control input and the working state. Some models can describe the above case but still rely on offline identification for compensation and suppression, and cannot realize online identification and utilization of unknown parameters, which makes it difficult to improve the transient performance of the piezoelectric positioning platform.

[0003] For the nonlinear model of the piezoelectric positioning platform, an efficient control scheme is needed to realize tracking control. Adaptive control is mainly applied to control nonlinear systems because it can estimate unknown parameters online. Ordinary adaptive control schemes almost cannot make unknown parameters converge to true values, which leads to the existence of disturbance in the closed-loop control system caused by estimation error, and finally only ensures that the system state converges to a bounded range. In addition, the persistent excitation condition in adaptive control is difficult to satisfy and verify, which also increases the difficulty of unknown parameters converging to true values. SUMMARY

[0004] In view of the strong hysteresis nonlinearity of the piezoelectric actuator and the problem that the dynamic characteristics change with the control input and the working state, the application proposes a data-driven adaptive control method based on a discrete piezoelectric positioning platform model. First, a discrete nonlinear state space model is proposed according to the nonlinear characteristics of the piezoelectric positioning platform. Then, an adaptive iterative algorithm for unknown parameter estimation is designed based on the above discrete nonlinear state space model. Thereafter, a data-driven algorithm is designed by fully considering historical data on the basis of the above adaptive iterative algorithm. Then, a tracking controller is designed according to the linear part to obtain an output feedback parameter vector. Finally, the effectiveness of the above data-driven adaptive iterative algorithm for making unknown parameters converge to true values is verified by Lyapunov function theory.

[0005] To achieve the above purpose, the technical scheme adopted by the application is as follows:

[0006] A data-driven adaptive control method based on a discrete piezoelectric positioning platform model includes the following steps:

[0007] S1: A discrete nonlinear state-space model is proposed based on the nonlinear characteristics of the piezoelectric positioning platform;

[0008] S2: Design an adaptive iterative algorithm for parameter estimation based on the nonlinear characteristics of the model;

[0009] S3: Design a data-driven algorithm based on the adaptive iterative algorithm;

[0010] S4: Design a tracking controller based on the linear part to obtain the analytical expression of the control input at each moment;

[0011] S5: Propose parameter design criteria for data-driven adaptive iterative algorithms.

[0012] Furthermore, in S1, constructing a discrete nonlinear state-space model includes:

[0013] Based on the strong hysteresis nonlinearity and the fact that the dynamic characteristics of the piezoelectric positioning platform change with the control input and operating state, the following discrete nonlinear state-space model of the piezoelectric positioning platform can be established:

[0014] ,

[0015] in, piezoelectric positioning platform Output displacement at time 10:00 piezoelectric positioning platform Time-based control input, For the dynamic vector of the piezoelectric positioning platform, Let be the state vector of the piezoelectric positioning platform. Let be the set of all real numbers. denoted as the order of the nonlinear state-space model; This refers to the input proportional coefficient for the control input of the piezoelectric positioning platform. piezoelectric positioning platform The unmodeled dynamics at time t, including nonlinear characteristics, wherein Φ(k) = Φ(x(k)) = [ φ (x(k)), φ (x(k)), φ (x(k)) ] τ(k) = Ax(τ) + bu(τ) θ^ (k) + β(k) Φ(τ(k)P(k)z(k) - γφ(k) φ - γφ(k) φ (k) ) + γφ(k) φ (k) Ψ Ψ & + γΨ Ψ φ(k) φ (k) - 2ηΨ Ψ η Ψ Ψ z(k) & ≤ -γ(2 - γ λ (k)φ(k) + 2γη λ Ψ Ψ - 2ηΨ Ψ η Ψ Ψ ]z(k) - ηΨ Ψ - ηΨ Ψ - 2γ λ )z(k) & ≤ [1 - η λ (2 - η λ - 2γ λ Figure 1 1 Figure 2 Figure 3 2 Figure 4 ⋯ , Figure 5 m Figure 6 ] ⊤ ∈ R m For the basis function vector, for The Middle a basis function, unknown true weight vector of basis functions; superscript T represents the transpose of a matrix;

[0016] Selecting The piezoelectric positioning platform strong hysteresis nonlinearity and dynamic characteristics with control input and working state change characteristics through Description, accurate modeling.

[0017] Further, in the S2, the adaptive iterative algorithm design process of parameter estimation includes:

[0018] First, define The estimation error of unknown parameter At time t is , which satisfies:

[0019] ,

[0020] Where, is the estimation variable of At time t in the immersion and invariant manifold method, is the auxiliary variable at time t in the immersion and invariant manifold method, and its iterative algorithm satisfies: ,

[0021] Where, is a constant gain,

[0022] for normalization, a is a normal number; According to the form of auxiliary variable The iterative algorithm of

[0023] is: ,

[0024] Further, the estimation error of is obtained as:

[0025]

[0026] .

[0027] Further, in the S3, the data-driven algorithm design process includes:

[0028] Design matrix , , , , satisfies:

[0029] , ​​

[0030] ,

[0031] ,

[0032] ,

[0033] ,

[0034] in, Represents c selected from time 0 to time k. When all eigenvalues ​​are greater than 0, the intermediate parameter ;

[0035] Based on S2 The data-driven adaptive iterative algorithm is represented as follows:

[0036] ,

[0037] in, For a constant gain, for and The output of the constructed model satisfies:

[0038] Q Figure 7 j Figure 8 Figure 1 j Φ(k) = Φ(x(k)) = [ φ (x(k)), φ (x(k)), φ (x(k)) ] τ(k) = Ax(τ) + bu(τ) θ^ (k) + β(k) Φ(τ(k)P(k)z(k) - γφ(k) φ - γφ(k) φ (k) ) + γφ(k) φ (k) Ψ Ψ & + γΨ Ψ φ(k) φ (k) - 2ηΨ Ψ η Ψ Ψ z(k) & ≤ -γ(2 - γ λ (k)φ(k) + 2γη λ Ψ Ψ - 2ηΨ Ψ η Ψ Ψ ]z(k) - ηΨ Ψ - ηΨ Ψ - 2γ λ )z(k) & ≤ [1 - η λ (2 - η λ - 2γ λ Figure 2 j )+ [ θ = Φ(x) = [ θ Figure 3 Figure 4 ⊤ Figure 5 θ = Φ(x) = [ θ j ) ,

[0039] Based on the aforementioned data-driven adaptive iterative algorithm, the estimation error is... The iterative form is updated as follows:

[0040] .

[0041] Furthermore, in S4, the tracking controller design process includes:

[0042] Define piezoelectric positioning platform The tracking error at time is ,satisfy ,in for Reference trajectory at any given moment;

[0043] Since the reference trajectory r is completely known, the design is based on the discrete nonlinear state-space model of the piezoelectric positioning platform in S1. The parsed form of the input is controlled at all times to satisfy:

[0044] ,

[0045] wherein, is a nominal controller designed based on the linear part of the model, which satisfies the condition that the model known linear part can stably track the reference trajectory; is an adaptive controller designed based on the nonlinear part of the model, which satisfies:

[0046] ,

[0047] further, the output of the closed-loop system satisfies:

[0048] ,

[0049] wherein, , the update form of the closed-loop system error is:

[0050] .

[0051] Further, in the S5, the parameter design criterion in the data-driven adaptive iterative algorithm comprises:

[0052] Consider the following Lyapunov function:

[0053] ,

[0054] its iterative form is represented as:

[0055] ,

[0056] wherein, , , is a unit matrix of order n, and the following inequality relationship exists:

[0057] V z (k+1)&= z ⊤ Figure 6 &= z ⊤ (k)[ I m Figure 7 Figure 8 ⊤ (k)(2 I m ​ ​ ⊤ ​ ​ ⊤ ​ ​ ⊤ ​ ​ ⊤ ​ ​ ⊤ ​ ​ ⊤ + ​ 2 ​ ​ ⊤ 2 ​ ​ ​ 2 ) z ⊤ ​ 2 + z ⊤ (k)[ I m ​ ​ 2 ​ ​ ⊤ ​ ​ ⊤ + ​ 2 ​ ​ ⊤ 2 ​ ,

[0058] wherein, denotes the maximum eigenvalue of the matrix , when , the iterative relationship is:

[0059] V z (k+1)&≤ z ⊤ (k)[ I m ​ ​ ⊤ (2 I m ​ ​ ⊤ ​ ​ 2 I m ​ ​ ​ 3 ​ ​ 4 ​ ​ 2 )] V z (k) ,

[0060] wherein, denote the minimum eigenvalue and the maximum eigenvalue of the matrix , respectively, and the matrix and the parameter , The value of satisfies the following condition:

[0061] ,

[0062] ,

[0063] Then the iterative form of Lyapunov function satisfies:

[0064] ,

[0065] Where iff , i.e. 0, no longer changes.

[0066] The beneficial effects of the present application are:

[0067] 1. The method of the present application contains a model of a discrete piezoelectric positioning platform, by taking the output displacement and control input at continuous multiple time points as the state vector, and using the inner product of the weight vector and the basis function vector to describe the unmodeled dynamics of the piezoelectric positioning platform, the characteristics of the model are effectively improved with the change of the control input and the working state of the piezoelectric positioning platform, and since the state vector contains the control input at continuous multiple time points, the description of the hysteresis nonlinearity will be more flexible, which lays the foundation for the targeted design of the control strategy.

[0068] 2. Compared with the general control strategy, the control method of the present application converts the processing of unmodeled dynamics into an online form, and through the design of the iterative algorithm of the estimation variable and the auxiliary variable, the iterative form of the unknown parameter estimation error is obtained, so that the closed-loop control system can be affected by the continuous weakening of the nonlinear part during the control process.

[0069] 3. In order to make the unknown parameter estimation error converge to the true value, the present application makes full use of the historical data, and constructs a data storage matrix through the state variable, the output displacement and the control input at multiple time points, thereby providing more flexibility for the satisfaction of the parameter condition, and weakening the continuous excitation condition required by the ordinary adaptive control to the excitation condition at some time points in history, greatly reducing the conservativeness of the system design.

[0070] 4. The present application proves through theoretical analysis that the unknown weight parameter can converge to the true value, ensuring that the system can effectively track the reference trajectory under the condition of discrete sampling, reducing the influence of modeling uncertainty on control performance, and being suitable for deployment in computer control systems controlled by data acquisition cards. BRIEF DESCRIPTION OF DRAWINGS

[0071] ​Flow chart of data-driven adaptive control method based on discrete piezoelectric positioning platform model of the present application.

[0072] ​ Overall control block diagram of data-driven adaptive control method based on discrete piezoelectric positioning platform model of the present application.

[0073] ​ Output displacement and reference trajectory graph of piezoelectric positioning platform when inputting 250Hz sinusoidal reference trajectory.

[0074] ​ Error variation graph of output displacement and reference trajectory of piezoelectric positioning platform when inputting 250Hz sinusoidal reference trajectory.

[0075] ​ Unknown weight parameter variation graph when inputting 250Hz sinusoidal reference trajectory.

[0076] ​ Output displacement and reference trajectory graph of piezoelectric positioning platform when inputting composite reference trajectory.

[0077] ​ Error variation graph of output displacement and reference trajectory of piezoelectric positioning platform when inputting composite reference trajectory.

[0078] ​ Unknown weight parameter variation graph when inputting composite reference trajectory. DETAILED DESCRIPTION

[0079] In order to make the objectives, technical solutions and advantages of the present application clearer, the present application will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific implementation examples described herein are only used to explain the present application and are not used to limit the present application. In addition, the technical features involved in each embodiment of the present application described below can be combined with each other as long as they do not conflict with each other.

[0080] As shown in ​ The present application proposes a data-driven adaptive control method based on discrete piezoelectric positioning platform model, comprising the following steps:

[0081] Step S1: According to the nonlinear characteristics of the piezoelectric positioning platform, a discrete nonlinear state space model is proposed, including:

[0082] According to the strong hysteresis nonlinearity of the piezoelectric positioning platform and the characteristics that the dynamic characteristics will change with the control input and working state, the following discrete nonlinear state space model of the piezoelectric positioning platform can be established:

[0083] ,

[0084] wherein, piezoelectric positioning platform Output displacement at time 10:00 piezoelectric positioning platform Time-based control input, For the dynamic vector of the piezoelectric positioning platform, Let be the state vector of the piezoelectric positioning platform. Let be the set of all real numbers. denoted as the order of the nonlinear state-space model; This refers to the input proportional coefficient for the control input of the piezoelectric positioning platform. piezoelectric positioning platform The unmodeled dynamics at time t, including nonlinear characteristics, wherein ​ ​ 1 ​ ​ 2 ​ ⋯ , ​ m ​ ] ⊤ ∈ R m For the basis function vector, for The Middle basis functions The unknown true weight vector of the basis functions; the superscript T denotes the transpose of the matrix;

[0085] Selected by system identification The strong hysteresis nonlinearity and dynamic characteristics of the piezoelectric positioning platform that vary with control input and operating state are addressed through... Depiction, to achieve accurate modeling.

[0086] Step S2: Design an adaptive iterative algorithm for parameter estimation based on the nonlinear characteristics of the model. The specific design steps are as follows:

[0087] First, define Unknown parameters at time The estimation error is ,satisfy:

[0088] ,

[0089] in, For immersion and invariant manifold methods Always The estimated variables, For immersion and invariant manifold methods The auxiliary variable at time step, whose iterative algorithm satisfies:

[0090] ,

[0091] wherein, is a constant gain, for normalization, a is a normal number;

[0092] According to the auxiliary variable in the form of, The iterative algorithm of

[0093] ,

[0094] Further, the estimation error is The iterative form is:

[0095] .

[0096] Step S3: Design a data-driven algorithm based on the adaptive iterative algorithm, and the specific design steps are as follows:

[0097] Design the matrix , , , , Satisfy:

[0098] ,

[0099] ,

[0100] ,

[0101] ,

[0102] ,

[0103] wherein, representing the selection of c from 0 to k time points All eigenvalues are greater than 0, ;

[0104] Then the data-driven adaptive iterative algorithm based on the iterative algorithm of step S2 can be expressed as:

[0105] ,

[0106] wherein, is a constant gain, is and The model output composed of satisfies:

[0107] Q ​ j ​ ​ j ​ ​ j )+ [ ​ ​ ​ ⊤ ​ ​ j ) ,

[0108] Based on the aforementioned data-driven adaptive iterative algorithm, the estimation error is... The iterative form is updated as follows:

[0109] ,

[0110] Step S4: Design a tracking controller based on the linear part to obtain the analytical expression of the control input at each moment. The specific design steps are as follows:

[0111] Define piezoelectric positioning platform The tracking error at time is ,satisfy ,in for Reference trajectory at any given moment.

[0112] Since the reference trajectory r is completely known, the nonlinear state-space model of the piezoelectric positioning platform in S1 can be used to design... The parsed form of the input is controlled at all times to satisfy:

[0113] ,

[0114] in, For a nominal controller designed based on the linear part of a model, it needs to satisfy the condition that the linear part of the model is known. The conditions for being able to stably track the reference trajectory; For an adaptive controller designed based on the nonlinear part of the model, the following conditions must be met:

[0115] ,

[0116] Furthermore, the closed-loop system output satisfies:

[0117] ,

[0118] in, The update form of the closed-loop system error is:

[0119] .

[0120] Step S5: Propose the parameter design criteria in the data-driven adaptive iterative algorithm. The specific derivation steps are as follows:

[0121] Consider the following Lyapunov function :

[0122] ,

[0123] Its iterative form can then be expressed as:

[0124] ,

[0125] in, , , yes For an identity matrix of order 1, the following inequalities hold:

[0126] V z (k+1)&= z ⊤ ​ &= z ⊤ (k)[ I m ​ ​ ⊤ (k)(2 I m ​ ​ ⊤ ​ ​ ⊤ ​ ​ ⊤ ​ ​ ⊤ ​ ​ ⊤ ​ ​ ⊤ + ​ 2 ​ ​ ⊤ 2 ​ ​ ​ 2 ) z ⊤ ​ 2 + z ⊤ (k)[ I m ​ ​ 2 ​ ​ ⊤ ​ ​ ⊤ + ​ 2 ​ ​ ⊤ 2 ​ ,

[0127] in, Representation matrix The largest eigenvalue, when At that time, the iterative relationship is:

[0128] V z (k+1)&≤ z ⊤ (k)[ I m ​ ​ ⊤ (2 I m ​ ​ ⊤ ​ ​ 2 I m ​ ​ ​ 3 ​ ​ 4 ​ ​ 2 )] V z (k) ,

[0129] in, Represent matrices respectively The minimum and maximum eigenvalues ​​are selected. Matrix and parameters , The value of satisfies the following conditions:

[0130] ,

[0131] ,

[0132] The iterative form of the Lyapunov function then satisfies:

[0133] ,

[0134] in, If and only if ,Right now At 0 o'clock, No further changes. The above process illustrates how to design an adaptive law gain. , and selection Matrix can make As the tracking process converges to 0, the unknown true weight vector of the basis function is estimated, which also means that the effect of the control of the closed-loop tracking system is gradually reduced by the influence of nonlinear uncertainty, and the control effect is improved.

[0135] According to steps S1-S5, the overall control block diagram as shown in ​ .

[0136] The model verification experiment and tracking control simulation of the application include the following steps:

[0137] (1) Simulation setting:

[0138] In the application, the simulation step is set to 0.0001s (i.e. the sampling frequency is 10kHz), the total simulation time is 1s, the data storage space dimension c = 4, the constant gain , the normal number for normalization; the linear part of the model of the piezoelectric nanometer positioning platform is A=[ 1.9565 -1.75180.64010.49770.1155 ],b =0.1443 , the weight of the nonlinear part is ​ [0.1 -0.3 ] ⊤ , the basis function is ​ 1 1+ e -0.1x(2) 1 1+ e -0.4x(4) ] ⊤ , the nominal controller is , wherein is the tracking controller, is the damping controller, and satisfies:

[0139] ,

[0140] ,

[0141] represents a time delay operator, i.e. .

[0142] (2) Sine signal tracking test:

[0143] First, input a sine wave signal with an amplitude of 1um and a frequency of 250Hz as the reference trajectory signal to be tracked. The initial state of the system is x=[ 1 2 1 0 0 ] ⊤ , and the initial value of the nonlinear part weight is ​ 0 =[ 00 ] ⊤ The simulation is carried out, and the output displacement of the piezoelectric positioning platform and the change of the unknown weight parameter within 0.2s are recorded. ​ The output displacement of the piezoelectric positioning platform and the reference trajectory when the 250Hz sinusoidal reference trajectory is input are shown in the figure. ​ The error change of the output displacement of the piezoelectric positioning platform and the reference trajectory when the 250Hz sinusoidal reference trajectory is input is shown in the figure, and the tracking error after stabilization is not more than ±0.05μm. ​ The change of the unknown weight parameter when the 250Hz sinusoidal reference trajectory is input is shown in the figure, and finally converges to the true value.

[0144] (3) Composite signal tracking test:

[0145] The composite signal composed of the sinusoidal signals with an amplitude of 1μm and a frequency of 300Hz, an amplitude of 2μm and a frequency of 50Hz, and an amplitude of 1μm and a frequency of 10Hz is superimposed as a reference trajectory signal. The initial state of the system is x=[ 0 0 0 0 0 ] ⊤ The initial value of the nonlinear part weight is ​ 0 =[ 00 ] ⊤ The simulation is carried out, and the output displacement of the piezoelectric positioning platform within 0.79s-0.85s and the change of the unknown weight parameter within 0.85s are recorded. ​ The output displacement of the piezoelectric positioning platform and the reference trajectory when the composite reference trajectory is input are shown in the figure. ​ The error change of the output displacement of the piezoelectric positioning platform and the reference trajectory when the composite reference trajectory is input is shown in the figure, and the tracking error after stabilization is not more than ±0.07μm. ​ The change of the unknown weight parameter when the composite reference trajectory is input is shown in the figure, and finally converges to the true value.

[0146] Those skilled in the art will easily understand that the above description is only the preferred embodiment of the present application, and is not used to limit the present application, and any modification, equivalent replacement and improvement made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A data-driven adaptive control method based on a discrete piezoelectric positioning platform model, characterized in that, Includes the following steps: S1: A discrete nonlinear state-space model is proposed based on the nonlinear characteristics of the piezoelectric positioning platform; Constructing a discrete nonlinear state-space model includes: Based on the strong hysteresis nonlinearity and the dynamic characteristics of the piezoelectric positioning platform that vary with control input and operating state, the following discrete nonlinear state-space model of the piezoelectric positioning platform is established: in, piezoelectric positioning platform Output displacement at time 10:00 piezoelectric positioning platform Time-based control input, For the dynamic vector of the piezoelectric positioning platform, Let be the state vector of the piezoelectric positioning platform. Let be the set of all real numbers. denoted as the order of the nonlinear state-space model; This refers to the input proportional coefficient for the control input of the piezoelectric positioning platform. piezoelectric positioning platform The unmodeled dynamics at time t, including nonlinear characteristics, wherein For the basis function vector, for The Middle basis functions The unknown true weight vector of the basis functions; the superscript T denotes the transpose of the matrix; Selected by system identification The strong hysteresis nonlinearity and dynamic characteristics of the piezoelectric positioning platform that vary with control input and operating state are addressed through... Depiction, to achieve accurate modeling; S2: Design an adaptive iterative algorithm for parameter estimation based on the nonlinear characteristics of the model; the design process of the adaptive iterative algorithm for parameter estimation includes: First, define Unknown parameters at time The estimation error is ,satisfy: in For immersion and invariant manifold methods Always The estimated variables, For immersion and invariant manifold methods The auxiliary variable at time step, whose iterative algorithm satisfies: in For a constant gain, Used for normalization, It is a positive number; According to auxiliary variables Form, design The iterative algorithm is as follows: The estimation error is further obtained as follows: The iterative form is: ； S3: Design a data-driven algorithm based on the adaptive iterative algorithm; the process of designing a data-driven algorithm includes: Design Matrix , , , , satisfy: in Represents from time 0 to Time selection individual When all eigenvalues ​​are greater than 0, the intermediate parameter ; Based on S2 The data-driven adaptive iterative algorithm is represented as follows: in For a constant gain, for and The output of the constructed model satisfies: Based on the aforementioned data-driven adaptive iterative algorithm, the estimation error is... The iterative form is updated as follows: ； S4: Design a tracking controller based on the linear part to obtain the analytical expression of the control input at each moment; the tracking controller design process includes: Define piezoelectric positioning platform The tracking error at time is ,satisfy ,in for Reference trajectory at any given moment; Due to the reference trajectory Given that everything is completely known, then based on the discrete nonlinear state-space model of the piezoelectric positioning platform in S1, design... The parsed form of the input is controlled at all times to satisfy: in A nominal controller designed based on the linear part of a model, which satisfies the condition that the linear part of the model is known. The conditions for being able to stably track the reference trajectory; For an adaptive controller designed based on the nonlinear part of the model, the following conditions must be met: Furthermore, the closed-loop system output satisfies: in, The update form of the closed-loop system error is: ； S5: Propose parameter design criteria for data-driven adaptive iterative algorithms.

2. The data-driven adaptive control method based on a discrete piezoelectric positioning platform model according to claim 1, characterized in that, The derivation process of the parameter design criteria for S5 includes: Consider the following Lyapunov function: Its iterative form is then expressed as: in , , yes For an identity matrix of order 1, the following inequalities hold: in, Representation matrix The largest eigenvalue, when At that time, the iterative relationship is: in, Represent matrices respectively The minimum and maximum eigenvalues ​​are selected. Matrix and parameters , The value of satisfies the following conditions: The iterative form of the Lyapunov function then satisfies: in If and only if ,Right now At 0 o'clock, It will not change.

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