Piezoelectric positioning platform immersion and invariant adaptive control method based on basis function approximation

Abstract

The application provides a piezoelectric positioning platform immersion and invariant adaptive control method based on a base function approximation, and comprises the following steps: S1: constructing a nonlinear model based on base function fitting according to the nonlinear characteristics of the piezoelectric positioning platform; S2: establishing a reference trajectory tracking error model based on the piezoelectric positioning platform model, and designing a controller and a manifold structure; S3: solving a partial differential equation to design the dynamic characteristics of unknown parameter estimation error; and S4: selecting a Lyapunov function to prove the stability of the tracking system, and selecting parameters according to the system performance. The application mainly aims at the problems of strong hysteresis nonlinearity and modeling uncertainty faced by piezoelectric positioning platform tracking control, and can realize online adaptive control of the piezoelectric positioning platform and improve the transient performance of the system.

Description

Technical Field

[0001] This invention belongs to the field of piezoelectric drive modeling and tracking control, specifically involving an immersion and invariant adaptive control method for a piezoelectric positioning platform based on basis function approximation. It is mainly used for fitting the hysteresis nonlinear characteristics of the piezoelectric positioning platform and high-precision tracking control, thereby reducing the system tracking error. Background Technology

[0002] Piezoelectric positioning platforms, made from piezoelectric ceramics and other materials exhibiting the piezoelectric effect, can generate sub-nanometer and nanometer-scale displacements, offering advantages such as fast response speed and high displacement resolution. Therefore, they are widely used in fields requiring high-precision positioning, such as photoelectric tracking platforms, microelectronics fabrication, and atomic force microscopy. However, the inherent hysteresis nonlinearity of piezoelectric positioning platforms severely limits their positioning accuracy, posing a significant challenge to establishing dynamic models and improving tracking control precision. Current methods still cannot accurately characterize the complex dynamic characteristics of hysteresis nonlinearity, exhibiting low fitting accuracy and insufficient generalization ability, and lacking a connection between nonlinearity and system operating states. This makes the hysteresis effect the primary factor limiting performance improvements in piezoelectric positioning platforms.

[0003] To meet the high-precision tracking control performance requirements, an efficient precision control strategy must be designed based on accurate modeling. Adaptive control methods have shown good performance in systems with unknown parameters. However, traditional adaptive control does not always converge to the true value of the unknown parameters. Therefore, the interference caused by parameter estimation errors makes the controller less robust to noise and disturbances, thus limiting the control performance. In addition, some existing control strategies do not make sufficient use of trajectory information, and the controller design is relatively conservative, which limits the overall tracking performance of the piezoelectric positioning platform. Summary of the Invention

[0004] To address the difficulties in fitting hysteresis nonlinearities and insufficient controller robustness in the precision servo control of piezoelectric positioning platforms, this invention proposes an immersion and invariant adaptive control method for piezoelectric positioning platforms based on basis function approximation. First, based on the characteristics of the nonlinearity of the piezoelectric positioning platform, a nonlinear model based on basis function fitting is proposed. Then, a tracking error model for the reference trajectory is established based on the aforementioned nonlinear model, and an adaptive controller and manifold structure are designed. Subsequently, given the designed control law, the dynamic characteristics of the unknown parameter estimation error are designed by solving partial differential equations, causing the function formed by the unknown parameter estimation error and the basis functions to converge to an attracting manifold. Finally, the stability of the tracking system is proven using Lyapunov functions, and parameters are selected based on system performance to achieve high-precision adaptive tracking control of the piezoelectric positioning platform.

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

[0006] An immersion and invariant adaptive control method for a piezoelectric positioning platform based on basis function approximation includes the following steps:

[0007] S1: Construct a nonlinear model based on basis function fitting according to the nonlinear characteristics of the piezoelectric positioning platform;

[0008] S2: Based on the piezoelectric positioning platform model, establish a reference trajectory tracking error model and design the controller and manifold structure;

[0009] S3: Design the dynamic characteristics of unknown parameter estimation error by solving partial differential equations;

[0010] S4: Choose the Lyapunov function to demonstrate the stability of the tracking system and select parameters based on system performance.

[0011] Furthermore, in S1, constructing the nonlinear model based on basis function fitting includes:

[0012] According to Newton's second law, the dynamic characteristics of the piezoelectric positioning platform are as follows:

[0013]

[0014] Where y represents the displacement output of the piezoelectric positioning platform. For the output speed of the piezoelectric positioning platform, Let be the output acceleration of the piezoelectric positioning platform, m be the mass of the moving parts in the positioning platform, d1 be the damping coefficient of the positioning platform, k1 be the stiffness of the positioning platform, T be the proportionality coefficient related to the power amplifier, u(t) be the input voltage, and H(t) be the nonlinear force dominated by hysteresis nonlinearity. The remaining total interference;

[0015] H(t) is expressed as:

[0016]

[0017]

[0018] Where D > 0 represents the constant displacement generated, σ ∈ (0, 1) represents the stiffness coefficient, and h represents the state of the auxiliary system. As the derivative of the auxiliary system state h, α determines the amplitude of the hysteresis loop, while β and λ together determine the shape of the hysteresis loop;

[0019] The state h of the above auxiliary system is represented by a neural network using basis functions and weights, as follows:

[0020]

[0021] in, It is the expected weight vector in the h-neural network representation. Let be the set of all real numbers. This represents the fitting error of the neural network. It is a regression vector composed of basis functions. Let i be the i-th basis function in the regression vector.

[0022] Define x1 = y, Assuming the system state, the model of the piezoelectric positioning platform is further represented as:

[0023]

[0024] in, v=bu, a2=d1 / m, a1=k1 / m, b=T / m, ε is bounded, satisfying 0 ≤ |ε| <D。

[0025] Further, S2 includes:

[0026] First, based on the model of the piezoelectric positioning platform, when tracking the reference trajectory, the reference trajectory is defined as r, and its velocity is... e1 = x1 - r, The error model can then be expressed as:

[0027]

[0028] in, For the error vector, The derivative of e,

[0029] Based on the above error model, the controller is designed with the following structure:

[0030] v = v n +v a

[0031]

[0032]

[0033] Where v represents the total control input generated by the controller, v n v a These represent the nominal controller and the adaptive controller, respectively, where K = [k1 k2] is the linear feedback gain. The acceleration of the reference trajectory, For estimating the unknown parameter θ, β(x) is an auxiliary function, and sgn(·) is the sign function;

[0034] Define the estimation error of the unknown parameter θ as z, and the attracting manifold satisfy:

[0035]

[0036]

[0037] The dynamic characteristics of the estimation error z are:

[0038]

[0039] in, for The derivative, Let be the partial derivative of β(x) with respect to x. Let x be the derivative of x, such that Converging to an attracting manifold

[0040] Further, S3 includes:

[0041] Substituting the total control input v into the piezoelectric positioning platform model, the closed-loop system dynamics can be obtained as follows:

[0042]

[0043] According to the dynamic characteristics of error z in S3, it should be made Converging to the design objective of the attracting manifold M, the partial derivative of β(x) with respect to x is... The derivatives are designed as follows:

[0044]

[0045]

[0046] in, A c =A+BK, Ω=[Ψ(x) 0], For auxiliary functions, satisfying:

[0047]

[0048] Through the above auxiliary functions and The value of β(x) can be further expressed as:

[0049]

[0050] The introduction of the auxiliary function Ψ(x) ensures Given an analytical design, the partial differential equation for β(x) is solvable. The analytical solution of β(x) is obtained by solving the above partial differential equation.

[0051] Based on the above partial derivatives of β(x) with respect to x and The derivative of z gives the dynamic characteristics of the estimation error z as follows:

[0052]

[0053] Further, S4 includes:

[0054] Consider constructing the following Lyapunov stability function for the system:

[0055]

[0056] Its derivative is expressed as:

[0057]

[0058] Some of the terms satisfy the following inequality:

[0059]

[0060]

[0061] The derivative of the Lyapunov function can then be further expressed as:

[0062]

[0063] because The convergence of tracking error in the closed-loop system and Selection related ， Define A c +BB T =-A * The eigenvalues ​​of the correlation matrix satisfy:

[0064] λ min (A c )<0

[0065] λ min (A * )>D 2 / δ>0

[0066] Where, λ min (A c ) represents matrix A c The smallest eigenvalue, λ min (A * ) represents matrix A * The smallest eigenvalue, δ, is a sufficiently small positive constant;

[0067] If matrix A * If the eigenvalues ​​satisfy the above conditions, then when the square of the e-norm is... When the derivative of the Lyapunov function satisfies:

[0068]

[0069] And the equal sign only applies to Established at that time;

[0070] By designing a linear feedback gain K, A c A * The smallest eigenvalue satisfies the eigenvalue condition, causing the closed-loop system e to converge to a compact set, and this compact set is composed of... Definition; at the same time, As the tracking process converges to the attracting manifold To restore the performance of the closed-loop tracking system to its unaffected state The impact of the situation.

[0071] The beneficial effects of this invention are as follows:

[0072] 1. The method of this invention includes a precise characterization of the complex hysteresis nonlinear characteristics of a piezoelectric positioning platform. It accurately approximates these hysteresis nonlinear characteristics by constructing basis functions. The selection of different basis functions effectively improves the generalization ability of the model approximation and reduces the modeling uncertainty of the nonlinear characteristics. This method also provides insights for modeling a series of other objects with complex nonlinear characteristics.

[0073] 2. Compared with traditional control methods, this invention makes full use of existing information in the controller design, enabling the dynamic performance of the closed-loop system to be restored to the state without the influence of basis functions. Furthermore, by introducing tracking trajectory information as a reference for the controller design, the overall tracking performance of the system is significantly improved compared with traditional methods.

[0074] 3. Compared with conventional adaptive control, this invention focuses on solving the performance degradation problem caused by parameter estimation errors. By designing an attractive manifold, the transient performance of the closed-loop system is improved; by introducing an auxiliary function, the solution to the partial differential equations is obtained, leading to an analytical solution for the dynamic estimation of unknown parameters; and the stability of the designed system is proven, thereby eliminating the constant disturbance caused by estimation errors and improving control performance. Attached Figure Description

[0075] Figure 1 This is a flowchart of the immersion and invariant adaptive control method for a piezoelectric positioning platform based on basis function approximation according to the present invention.

[0076] Figure 2 This is an overall control block diagram of the present invention.

[0077] Figure 3The tracking effect of the piezoelectric positioning platform after inputting a 100Hz reference trajectory signal into the closed-loop system.

[0078] Figure 4 The error variation diagram of the output displacement of the piezoelectric positioning platform and the reference trajectory after inputting a 100Hz reference trajectory signal into the closed-loop system.

[0079] Figure 5 The tracking effect of the piezoelectric positioning platform after inputting the composite reference trajectory signal into the closed-loop system is shown in the figure.

[0080] Figure 6 The error variation diagram of the output displacement of the piezoelectric positioning platform and the reference command after inputting the composite reference trajectory signal into the closed-loop system. Detailed Implementation

[0081] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0082] like Figure 1 As shown, this invention proposes an immersion and invariant adaptive control method for a piezoelectric positioning platform based on basis function approximation, comprising the following steps:

[0083] Step S1: Construct a nonlinear model based on basis function fitting according to the nonlinear characteristics of the piezoelectric positioning platform;

[0084] According to Newton's second law, the dynamic characteristics of the piezoelectric positioning platform are as follows:

[0085]

[0086] Where y represents the displacement output of the piezoelectric positioning platform. For the output speed of the piezoelectric positioning platform, Let be the output acceleration of the piezoelectric positioning platform, m be the mass of the moving parts in the positioning platform, d1 be the damping coefficient of the positioning platform, k1 be the stiffness of the positioning platform, T be the proportionality coefficient related to the power amplifier, u(t) be the input voltage, and H(t) be the nonlinear force dominated by hysteresis nonlinearity. The remaining total interference;

[0087] H(t) is expressed as:

[0088]

[0089]

[0090] Where D > 0 represents the constant displacement generated, σ ∈ (0, 1) represents the stiffness coefficient, and h represents the state of the auxiliary system. As the derivative of the auxiliary system state h, α determines the amplitude of the hysteresis loop, while β and λ together determine the shape of the hysteresis loop;

[0091] The state h of the above auxiliary system is represented by a neural network using basis functions and weights, as follows:

[0092]

[0093] in, It is the expected weight vector in the h-neural network representation, where the superscript T denotes the transpose of the vector or matrix. Let be the set of all real numbers. This represents the fitting error of the neural network. It is a regression vector composed of basis functions. Let i be the i-th basis function in the regression vector.

[0094] Define x1 = y, Assuming the system state, the model of the piezoelectric positioning platform is further represented as:

[0095]

[0096] in, v=bu, a2=d1 / m, a1=k1 / m, b=T / m, ε is bounded, satisfying 0 ≤ |ε| <D。

[0097] The system is identified using the input and output data of the piezoelectric positioning platform. Based on the obtained model state variables, a basis function form is designed so that the piezoelectric positioning platform has only two unknown parameters, θ and ε. In this invention, [the following is selected]... ε=0.09sin(1000πt), D=0.1.

[0098] Step S2: Establish a reference trajectory tracking error model based on the piezoelectric positioning platform model, and design the controller and manifold structure, including:

[0099] First, based on the model of the piezoelectric positioning platform, when tracking the reference trajectory, the reference trajectory is defined as r, and its velocity is... e1 = x1 - r, The error model can then be expressed as:

[0100]

[0101] in, For the error vector, The derivative of e,

[0102] Based on the above error model, the controller is designed with the following structure:

[0103] v = v n +v a

[0104]

[0105]

[0106] Where v represents the total control input generated by the controller, v n v a These represent the nominal controller and the adaptive controller, respectively. K = [k1 k2] is the linear feedback gain, which is taken as [-3480 -7000] in this invention. The acceleration of the reference trajectory, For estimating the unknown parameter θ, β(x) is an auxiliary function, and sgn(·) is the sign function, which has the following form:

[0107]

[0108] in

[0109] Define the estimation error of the unknown parameter θ as z, and the attracting manifold satisfy:

[0110]

[0111]

[0112] The dynamic characteristics of the estimation error z are:

[0113]

[0114] in, for The derivative, Let be the partial derivative of β(x) with respect to x. The derivative of x should be designed to make Converging to an attracting manifold

[0115] Step S3: Design the dynamic characteristics of the unknown parameter estimation error by solving the partial differential equation, including:

[0116] Substituting the total control input v into the piezoelectric positioning platform model, the closed-loop system dynamics can be obtained as follows:

[0117]

[0118] According to the dynamic characteristics of error z in S3, it should be made Converging to the design objective of the attracting manifold M, the partial derivative of β(x) with respect to x is... The derivatives are designed as follows:

[0119]

[0120]

[0121] in, A c =A+BK, Ω=[Ψ(x) 0], For auxiliary functions, satisfying:

[0122]

[0123] Through the above auxiliary functions and The value of β(x) can be further expressed as:

[0124]

[0125] Where Φ(x) is the regression vector in S1, and the introduction of the auxiliary function ψ(x) ensures... Given an analytical design, the partial differential equation for β(x) has a solution. By solving the above partial differential equation, the analytical solution of β(x) can be obtained. In this invention, the solution can be obtained as follows:

[0126] Based on the above partial derivatives of β(x) with respect to x and The derivative of z gives the dynamic characteristics of the estimation error z as follows:

[0127]

[0128] Step S4: Select a Lyapunov function to demonstrate the stability of the tracking system, and select parameters based on system performance, including:

[0129] Consider constructing the following system stability Lyapunov function:

[0130]

[0131] Its derivative is expressed as:

[0132]

[0133] Some of the terms satisfy the following inequality:

[0134]

[0135]

[0136] The derivative of the Lyapunov function can then be further expressed as:

[0137]

[0138] because The convergence of tracking error in the closed-loop system and Choice related, define A c +BB T =-A * The eigenvalues ​​of the correlation matrix satisfy:

[0139] λ min (A c )<0

[0140] λ min (A * )>D 2 / δ>0

[0141] Where, λ min (A c ) represents matrix A c The smallest eigenvalue, λ min (A * ) represents matrix A * The smallest eigenvalue of matrix A is δ, where δ is a sufficiently small positive constant. * If the eigenvalues ​​satisfy the above conditions, then when the square of the e-norm is... When the Lyapunov function is at a certain time, the differential satisfies:

[0142]

[0143] And the equal sign only applies to This is true at that time. The above process illustrates that by designing a linear feedback gain K, A is made... c A * The smallest eigenvalue satisfies the eigenvalue condition, which allows the closed-loop system's e to converge to a compact set, and this compact set is composed of... Definition. On the other hand, the above inequality also illustrates... It will converge to an attracting manifold during the tracking process. To restore the performance of the closed-loop tracking system to its unaffected state The impact of the situation.

[0144] Based on steps S1-S4, we can obtain the following: Figure 2The overall control block diagram shown demonstrates how an adaptive controller, adjusted online by immersion and invariant adaptive laws, generates control inputs to enable the piezoelectric nanopositioning platform to track the reference trajectory.

[0145] The model verification experiment and tracking control simulation of this invention include the following steps:

[0146] Step (1) Simulation settings:

[0147] In this invention, the simulation step size is set to 0.00005s (i.e., the sampling frequency is 20kHz), and the total simulation time is 0.1s; the linear part of the piezoelectric nanopositioning platform model is... The weight of the nonlinear part is The basis functions are The tail term is ε = 0.09sin(1000πt), and its upper bound is D = 0.1. The linear feedback gain is K = [-3480 -7000].

[0148] Step (2) Sine wave tracking test:

[0149] First, a sinusoidal signal with an amplitude of 0.1 μm and a frequency of 100 Hz is input as the reference trajectory signal to be tracked. The initial state of the system is as follows: The initial values ​​of the nonlinear part weights are: Simulations were conducted to obtain the tracking performance of the piezoelectric nanopositioning platform under these conditions. Figure 3 The tracking effect of the piezoelectric positioning platform after inputting the reference trajectory signal into the closed-loop system. Figure 4 After inputting the reference trajectory signal into the closed-loop system, the piezoelectric positioning platform outputs the displacement and the error change of the reference trajectory. After stabilization, the tracking error does not exceed ±0.1nm.

[0150] Step (3) Composite signal tracking test:

[0151] A composite signal, formed by superimposing sinusoidal signals with an amplitude of 0.04 μm and a frequency of 100 Hz, an amplitude of 0.02 μm and a frequency of 50 Hz, and an amplitude of 0.05 μm and a frequency of 30 Hz, is used as the reference trajectory signal. The initial state of the system is... The initial values ​​of the nonlinear part weights are: Simulations were conducted to obtain the tracking performance of the piezoelectric positioning platform under these conditions. Figure 5 The tracking effect of the piezoelectric positioning platform after inputting a composite reference trajectory signal into the closed-loop system. Figure 6 After inputting a composite reference trajectory signal into the closed-loop system, the piezoelectric positioning platform outputs a displacement with an error variation relative to the reference trajectory. Once stabilized, the tracking error does not exceed ±0.1 nm.

[0152] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A piezoelectric positioning platform immersion and invariant adaptive control method based on basis function approximation, characterized in that, Includes the following steps: S1: Construct a nonlinear model based on basis function fitting according to the nonlinear characteristics of the piezoelectric positioning platform; Constructing a nonlinear model based on basis function fitting includes: According to Newton's second law, the dynamic characteristics of the piezoelectric positioning platform are as follows: ； in, For displacement output of the piezoelectric positioning platform For the output speed of the piezoelectric positioning platform, To output acceleration for the piezoelectric positioning platform, To determine the mass of the moving parts in the positioning platform, To determine the damping coefficient of the positioning platform, To determine the platform's rigidity, This is a scaling factor related to the power amplifier. Input voltage The nonlinear force is dominated by hysteresis nonlinearity. The remaining total interference; Represented as: ； in, To produce a constant displacement, Represents the stiffness coefficient. To assist in the status of the auxiliary system, For auxiliary system status The derivative, The amplitude of the hysteresis loop is determined. and Together they determine the shape of the hysteresis loop; The state of the aforementioned auxiliary system is represented by a neural network using basis functions and weights. , is represented as: ； in, yes The expected weight vector in the neural network representation, Let be the set of all real numbers. This represents the fitting error of the neural network. It is a regression vector composed of basis functions; For the regression vector, the first... One basis function; definition , , Assuming the system state, the model of the piezoelectric positioning platform is further represented as: ； in, , , , , , , , , , It is bounded, satisfying ; S2: Based on the piezoelectric positioning platform model, establish a reference trajectory tracking error model, and design the controller and manifold structure. include: First, based on the model of the piezoelectric positioning platform, when tracking the reference trajectory, the reference trajectory is defined as... Its speed is , , The error model is then expressed as: ； in, For the error vector, for The derivative, , ; Based on the above error model, the controller is designed with the following structure: ； in, This represents the total control input generated by the controller. , These represent the nominal controller and the adaptive controller, respectively. For linear feedback gain, The acceleration of the reference trajectory, Unknown parameters The estimate, For auxiliary functions, It is a symbolic function; Define unknown parameters The estimation error is Attracting manifold ,satisfy: ； ； estimation error The dynamic characteristics are: ； in, for The derivative, for right The partial derivatives, for The derivative of makes Converging to an attracting manifold ; S3: Design the dynamic characteristics of unknown parameter estimation errors by solving partial differential equations, including: Transmit the master control input Substituting the piezoelectric positioning platform model, the closed-loop system dynamics are obtained as follows: ； According to the error in S3 The dynamic characteristics should make Converging to an attracting manifold The design goal is to right partial derivatives and The derivatives are designed as follows: ； ； in, , , , For auxiliary functions, satisfying: ； Through the above auxiliary functions and The value will right The partial derivatives are further expressed as: ； Auxiliary functions The introduction of [something] ensures Given the premise of analytical design, to target The partial differential equation has a solution, which is obtained by solving the above partial differential equation. The analytical solution; Based on the above right partial derivatives and The derivative of the estimation error The dynamic characteristics are: ； S4: Choose the Lyapunov function to demonstrate the stability of the tracking system and select parameters based on system performance.

2. The immersion and invariant adaptive control method for a piezoelectric positioning platform based on basis function approximation according to claim 1, characterized in that, S4 includes: Consider constructing the following Lyapunov stability function for the system: ； Its derivative is expressed as: ； Some of the terms satisfy the following inequality: ； The derivative of the Lyapunov function can then be further expressed as: ； because The convergence of the tracking error of the closed-loop system and Selection related, definition The eigenvalues ​​of the correlation matrix satisfy: ； ； in, Representation matrix The smallest eigenvalue, Representation matrix The smallest eigenvalue, It is a sufficiently small positive number; If matrix If the eigenvalues ​​satisfy the above conditions, then when The square of the second norm When the derivative of the Lyapunov function satisfies: ； And the equal sign only applies to Established at that time; By designing linear feedback gain make , The minimum eigenvalue of the closed-loop system satisfies the eigenvalue condition. It converges to a compact set, and that compact set is composed of Definition; at the same time, As the tracking process converges to the attracting manifold This restores the performance of the closed-loop tracking system to a level unaffected by [other factors]. The impact of the situation.

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