Creep nonlinear dynamics model of piezoelectric actuator and construction method and system thereof

Abstract

The application belongs to the field of piezoelectric driver nonlinear modeling, and particularly discloses a piezoelectric driver creep nonlinear dynamics model and a construction method and system thereof. c (s)=s ‑μ ; wherein s is a Laplace operator; the identification method of the parameter mu includes: constructing input voltage signals with different frequencies omega i , calculating input voltage signal amplitudes U i ; according to the input voltage signals, obtaining output displacement signals of the piezoelectric driver, and simultaneously calculating output displacement signal amplitudes D i ; calculating the amplitude ratio A i of D i and U i ; obtaining a discrete data point set (x i , y i ) which satisfies: according to the linear fitting of the discrete data point set (x i , y i ), determining the slope of the fitted straight line, and determining the parameter mu according to the slope. The model constructed based on the fractional order involves fewer unknown parameters, is easy to identify, and at the same time, the modeling method is not limited by special discrete points in the time / frequency domain, and has higher universality.

Description

Technical Field

[0001] This invention belongs to the field of nonlinear modeling of piezoelectric actuators, and more specifically, relates to a nonlinear dynamic model of creep of a piezoelectric actuator and its construction method and system. Background Technology

[0002] Piezoelectric actuators have advantages such as high precision (nanometer level), rapid response, high stiffness and output force, and are therefore widely used in high-precision positioning and motion control systems, such as micro-vibration isolation systems, lithography machines, atomic force microscopes, scanning tunneling electron microscopes and satellite ultra-quiet platforms.

[0003] However, the inherent creep effect of piezoelectric materials (at low frequencies, when the input voltage is a step input, the output displacement of the piezoelectric actuator exhibits a slow drift in the time domain; or when the input voltage remains constant, the output displacement of the piezoelectric actuator drifts with time) greatly reduces the output accuracy of the piezoelectric actuator. Therefore, establishing an accurate nonlinear dynamic model describing the creep of the piezoelectric actuator is the primary prerequisite for realizing creep effect compensation and control of the piezoelectric actuator.

[0004] Currently, there are two main methods for modeling the nonlinear creep of piezoelectric actuators:

[0005] 1) The logarithmic model in the time domain is as follows:

[0006]

[0007] In the formula: y(t) is the output displacement of the piezoelectric actuator when the input voltage remains constant, t0 is the start time of the creep effect of the piezoelectric actuator, y0 is the output displacement of the piezoelectric actuator at time t0, and γ is the logarithmic response coefficient.

[0008] The parameters of the logarithmic time-domain model are related to the historical input voltage, and its output displacement increases infinitely with time. This means that the logarithmic time-domain model is not applicable at times t→0 and t→+∞.

[0009] 2) Frequency Domain Spring-Damping Model. This model equates the piezoelectric actuator creep dynamics model to multiple spring-damping subsystems, thus establishing a piezoelectric creep model. This method involves many parameters, which are difficult to identify.

[0010] In summary, current methods for modeling the nonlinear creep of piezoelectric actuators all have shortcomings, which can be summarized as either lacking general applicability or having difficulty identifying parameters. Therefore, a new method to describe the nonlinear creep effect of piezoelectric actuators is urgently needed to address these problems. Summary of the Invention

[0011] In view of the above-mentioned defects or improvement needs of the prior art, the present invention provides a piezoelectric actuator creep nonlinear dynamic model and its construction method and system. The purpose is to construct an accurate, universal and parameter-easy-to-identify piezoelectric actuator creep nonlinear dynamic model.

[0012] To achieve the above objectives, according to a first aspect of the present invention, a method for constructing a creep nonlinear dynamic model of a piezoelectric actuator is proposed, comprising the following steps:

[0013] The constructed nonlinear dynamic model of the piezoelectric actuator creep is as follows: G c (s)=s -μ ;

[0014] Among them, G c (s) represents the creep effect of the piezoelectric actuator, where s is the Laplace operator; μ is the parameter to be identified, and its identification methods include:

[0015] S21, Construct structures with different frequencies ω i The input voltage signal is calculated, and the amplitude U of the input voltage signal is also calculated. i i = 1, 2, ..., n, where n is the total number of input voltage signals;

[0016] S22. Based on the input voltage signal, obtain the output displacement signal of the piezoelectric actuator, and simultaneously calculate the amplitude D of the output displacement signal. i ;

[0017] S23. Calculate the amplitude D of the output displacement signal. i With the amplitude U of the input voltage signal i The amplitude ratio of A i ;

[0018] S24. Obtain the discrete data point set (x) i ,y i ), which satisfies:

[0019] S25. Based on the discrete data point set (x) i ,y i Perform linear fitting, determine the slope of the fitted line, and determine the parameter μ based on the slope of the line.

[0020] As a further preferred method, the identification method for parameter μ also includes:

[0021] S26. Determine whether the calculated value of parameter μ is within the predetermined range (a, b). If it is within the range (a, b), then the value of parameter μ is the final value. Otherwise, adjust the points in the discrete data point set and return to step S25 to update the value of parameter μ.

[0022] As a further preferred method, the range (a, b) is determined as follows:

[0023] The time-domain and frequency-domain characteristics of the creep effect of the piezoelectric actuator were obtained through experiments, i.e., the experimental results.

[0024] The range of μ is divided into several non-overlapping intervals, and the transfer function s is simulated and analyzed under different ranges of μ. -μ The time-domain and frequency-domain characteristics, i.e., the simulation results;

[0025] The experimental results are compared with the simulation results to determine the range of values ​​(a, b) to which μ belongs.

[0026] As a further preferred option, when obtaining simulation results, the range of μ is divided into four intervals: (-∞, -1), (-1, 0), (0, 1), and (1, +∞).

[0027] As a further preferred approach, when obtaining experimental results, the step response diagram is used to characterize the time-domain characteristics of the piezoelectric actuator creep effect, and the Bode diagram is used to characterize the frequency-domain characteristics of the piezoelectric actuator creep effect.

[0028] As a further preferred option, based on the discrete data point set (x) i ,y i Perform linear fitting to determine the slope of the fitted line. The formula for calculating the slope k is:

[0029]

[0030] in, x, representing the discrete data point set i y i The mean.

[0031] As a further preferred option, the formula for calculating the parameter μ based on the slope of the straight line is:

[0032] According to a second aspect of the present invention, a piezoelectric actuator creep nonlinear dynamic model is provided, which is constructed using the above-described piezoelectric actuator creep nonlinear dynamic model construction method.

[0033] According to a third aspect of the present invention, a system for constructing a creep nonlinear dynamic model of a piezoelectric actuator is provided, comprising a processor, the processor being used in the above-described method for constructing a creep nonlinear dynamic model of a piezoelectric actuator.

[0034] According to a fourth aspect of the present invention, a computer-readable storage medium is provided having a computer program stored thereon, which, when executed by a processor, implements the above-described method for constructing a creep nonlinear dynamics model of a piezoelectric actuator.

[0035] In summary, compared with the prior art, the above-described technical solutions conceived by this invention mainly possess the following technical advantages:

[0036] 1. This invention proposes a nonlinear creep modeling method for piezoelectric actuators based on fractional order. Since it only involves a single parameter, the model parameter is easy to identify. Therefore, a parameter identification method based on logarithmic linearization is given, which is direct and accurate. At the same time, the modeling method is not limited by special discrete points in the time / frequency domain, and has stronger versatility.

[0037] 2. This invention determines the range of parameters by comparing the experimental and simulation results of the time-domain and frequency-domain characteristics of the piezoelectric actuator in advance, thereby verifying and adjusting the obtained parameter values ​​and improving the accuracy of model parameter identification. Attached Figure Description

[0038] Figure 1 This is a flowchart of the method for constructing a creep nonlinear dynamics model of a piezoelectric actuator according to an embodiment of the present invention;

[0039] Figure 2 This is a creep frequency domain characteristic diagram of the piezoelectric actuator according to an embodiment of the present invention;

[0040] Figure 3 This is a time-domain characteristic diagram of the creep of the piezoelectric actuator according to an embodiment of the present invention;

[0041] Figure 4 This is a creep dynamics model of the piezoelectric actuator according to an embodiment of the present invention;

[0042] Figure 5 This is the creep fractional-order model s of the piezoelectric actuator in the μ∈(-∞,-1) interval of this invention. -μ Frequency domain characteristic diagram;

[0043] Figure 6 This is the creep fractional-order model s of the piezoelectric actuator in the μ∈(-1,0) interval of this invention. -μ Frequency domain characteristic diagram;

[0044] Figure 7 This is the creep fractional-order model s of the piezoelectric actuator in the μ∈(0,1) interval of this invention. -μ Frequency domain characteristic diagram;

[0045] Figure 8 This is the creep fractional-order model s of the piezoelectric actuator in the μ∈(1,+∞) interval of this invention. -μ Frequency domain characteristic diagram;

[0046] Figure 9 This is the creep fractional-order model s of the piezoelectric actuator in the μ∈(-∞,-1) interval of this invention. -μ Time-domain characteristic diagram;

[0047] Figure 10 This is the creep fractional-order model s of the piezoelectric actuator in the μ∈(-1,0) interval of this invention. -μ Time-domain characteristic diagram;

[0048] Figure 11 This is the creep fractional-order model s of the piezoelectric actuator in the μ∈(0,1) interval of this invention. -μ Time-domain characteristic diagram;

[0049] Figure 12 This is the creep fractional-order model s of the piezoelectric actuator in the μ∈(1,+∞) interval of this invention. -μ Time-domain characteristic diagram;

[0050] Figure 13 This is the parameter identification result of the piezoelectric actuator creep dynamics model based on the fractional-order model in this embodiment of the invention. Detailed Implementation

[0051] 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.

[0052] This invention provides a method for constructing a nonlinear dynamic model of creep in a piezoelectric actuator, such as... Figure 1 As shown, it includes the following steps:

[0053] S1. Determine the form of the nonlinear dynamic model of the piezoelectric actuator creep.

[0054] The present invention is based on a fractional-order piezoelectric actuator creep nonlinear dynamics model as follows:

[0055] G c (s)=s -μ ;

[0056] Among them, G c (s) represents the creep effect of the piezoelectric actuator, where s is the Laplace operator; μ is the parameter to be identified.

[0057] The following is an explanation of the above model concept and its effectiveness:

[0058] (1) A creep dynamics model of the piezoelectric actuator is established using a series spring-damping system, such as Figure 4 As shown, the following Laplace domain equation is obtained:

[0059]

[0060] In the formula: s is the Laplace operator, k0 is the elastic parameter of the piezoelectric actuator at low frequencies, and c i k i Let N represent the spring stiffness and damping coefficient of the i-th subsystem, respectively, and N represent the number of equivalent subsystems. Rewriting the above equation in pole-zero form, we get:

[0061]

[0062] In the formula: K c z i p i These represent the gain, zero, and pole of the creep nonlinear dynamics model of the piezoelectric actuator, respectively.

[0063] (2) From a purely theoretical perspective, a transfer function of a specific form is found that can formally characterize the creep dynamics equation of the piezoelectric actuator.

[0064] The fractional transfer function is now known to be s. -μ (where μ is a fraction). When the frequency bandwidth is (ω) a ,ω b When this is the case, a recursive algorithm can be used to approximate the fractional transfer function, and the result is as follows:

[0065]

[0066] In the formula: N is the approximate order, K, z k p k Let G(s) be the gain, zeros, and poles of the transfer function G(s), respectively, which satisfy the following condition:

[0067]

[0068] Compare two transfer functions G c From G(s) and G(s), we can see that they are identical in form, but the latter is a fractional transfer function s -μ The approximation equation is given. Therefore, a fractional-order nonlinear modeling method for piezoelectric actuator creep is proposed, namely, the nonlinear dynamic model of piezoelectric actuator creep is: G c (s)=s -μ , where μ is a non-integer.

[0069] S2. Identify the parameter μ in the model, including the following steps:

[0070] S21, Construct structures with different frequencies ω i The sinusoidal voltage input signal is used to calculate the amplitude U of the input voltage signal. i i = 1, 2, ..., n, where n is the total number of input voltage signals;

[0071] S22. Based on the input voltage signal, obtain the output displacement signal of the piezoelectric actuator, and simultaneously calculate the amplitude D of the output displacement signal. i ;

[0072] S23. Calculate the amplitude ratio A of the output and input signals. i ,Right now:

[0073]

[0074] S24. Obtain the discrete data point set (x) i ,y i ), where x i y i The following conditions must be met:

[0075]

[0076] S25. Based on the discrete data point set (x) i ,y i Perform linear fitting, determine the slope of the fitted line, and determine the parameter μ based on the slope of the line.

[0077] S26. Determine whether the calculated value of parameter μ is within the predetermined range (a, b). If it is within the range (a, b), then the value of parameter μ is the final value. Otherwise, adjust the points in the discrete data point set and return to step S25 to update the value of parameter μ until the value of parameter μ is within the range (a, b).

[0078] Furthermore, step S25 includes:

[0079] S251, construct the following linear fitting curve:

[0080] That is: log 10 A =klog 10 w +d

[0081] In the formula: k and d are the slope and intercept of the line, respectively.

[0082] S252, Calculate the predicted value Sum of squared errors from the observed value y:

[0083]

[0084] In the formula: S is the sum of squared errors between the predicted and observed values.

[0085] S253, considering the need to minimize S, then:

[0086]

[0087] S254, solve for the slope k and intercept d as follows:

[0088]

[0089] In the formula: x i y i The mean of (i = 1, 2, ..., n).

[0090] S255, the order μ of the piezoelectric actuator creep nonlinear dynamics model is obtained through calculation: According to the characteristics of fractional-order functions, the fractional-order model G... c (s)=s -μ In the Bode plot of the frequency domain response, the relationship between amplitude and frequency is a straight line with a slope of -20μ (dB / sec). Furthermore, -20μ = k. Therefore, the relationship between k and μ can be determined as follows:

[0091]

[0092] In the formula: To identify the system order, the creep dynamics model of the piezoelectric actuator based on fractional order is...

[0093] Furthermore, in step S26, since there may be noise points in the acquired discrete data point set, leading to errors in the parameter identification results, the range (a, b) of parameter μ can be predetermined to verify the parameter identification results. If the obtained parameter value is not within this range, some points in the discrete data point set can be removed (e.g., larger discrete points can be deleted during iteration) before performing linear fitting to solve the problem.

[0094] The method for determining the range of values ​​(a, b) is as follows:

[0095] S261. The time / frequency characteristics of the creep nonlinearity of the piezoelectric actuator were established experimentally. The time-domain characteristics of the creep effect of the piezoelectric actuator were characterized by the step response diagram, and the frequency-domain characteristics of the creep effect of the piezoelectric actuator were characterized by the Bode plot. The experimental results were obtained.

[0096] S262, divide the range of μ into four intervals: (-∞, -1), (-1, 0), (0, 1), and (1, +∞). Write a MATLAB program to simulate and analyze the transfer function s under different ranges of μ. -μ The time-domain and frequency-domain characteristics, i.e., the simulation results;

[0097] S263. Compare the experimental results with the simulation results to determine the range (a,b) of μ. The identification principle is: when μ∈(a,b), the simulation results are consistent with the experimental results for the time / frequency characteristic curve, then (a,b) is the range of μ.

[0098] The following are specific examples:

[0099] (1) Preliminarily determine the range of values ​​for μ.

[0100] Creep is an inherent physical property of piezoelectric materials, and different piezoelectric materials may exhibit different creep characteristics. Therefore, it is necessary to obtain the time / frequency characteristics of the creep nonlinearity of piezoelectric actuators. Bode plots are typically used to characterize the frequency domain characteristics, and step response plots are used to characterize the time domain characteristics, as shown below. Figure 2 and Figure 3 As shown in the figure. From the frequency domain curve, the amplitude has a negative linear correlation with the frequency; from the time domain curve, as the frequency increases, the phase stabilizes at a constant (within ±5% of the error).

[0101] Figures 5-8 The frequency domain characteristics of the order μ in different intervals; Figures 9-12 This is a time-domain characteristic plot of the order μ taking values ​​in different intervals. Figures 5-8 , Figures 9-12 respectively with Figure 2 and Figure 3 By comparing, we can initially identify the range of values ​​for the order, i.e., μ∈(0,1). Therefore:

[0102] G c (s)=s -μ μ∈(0,1)

[0103] (2) Accurately identify the order μ, including:

[0104] S21, construct ω respectively i Given sinusoidal voltage input signals at Hz of 0.001, 0.005, 0.01, 0.05, 0.1, 0.5, and 1 Hz, calculate the amplitude U of the input voltage signal. i (i = 1, 2, ..., 7);

[0105] S22, Based on the input voltage signal, obtain the output displacement signal of the piezoelectric actuator and calculate the amplitude D of the output displacement signal. i (i = 1, 2, ..., 7);

[0106] S23, Calculate the amplitude ratio A of the output and input signals. i (i = 1, 2, ..., 7);

[0107] S24, Obtain the discrete data point set (x i ,yi ), i = 1, 2, ..., 7, where x i y i The following conditions must be met: The obtained dataset is shown in Table 1:

[0108] Table 1 Discrete Data Point Set

[0109]

[0110] S25, based on the discrete data point set (x) i ,y i Linear fitting was performed, and the slope k and intercept d of the fitted line were found to be -0.223 and -0.336, respectively. Therefore, the expression for the fitted curve is as follows:

[0111]

[0112] The order μ of the nonlinear dynamic model of creep in the piezoelectric actuator was obtained through calculation and identification. (See [link]) Figure 2 The amplitude and frequency have a negative linear correlation, and the slope of the straight line in the graph is -1 / 20, therefore:

[0113]

[0114] S26 yields a μ value of 0.01115, which falls within the initially identified range μ∈(0,1). Therefore, the creep dynamics model of the fractional-order piezoelectric actuator is s -0.01115 .

[0115] The piezoelectric actuator creep nonlinear modeling method proposed in this invention can simultaneously achieve high-precision modeling of creep effects in both the frequency and time domains. Furthermore, the fractional-order creep dynamics model, involving only a single unknown parameter, ensures efficient and reliable identification. Moreover, compared to traditional methods, this piezoelectric creep characteristic modeling method has greater versatility.

[0116] 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 method for constructing a nonlinear dynamic model of creep in a piezoelectric actuator, characterized in that, Includes the following steps: The constructed nonlinear dynamic model of the piezoelectric actuator creep is as follows: G c (s)=s -μ ; Among them, G c (s) represents the creep effect of the piezoelectric actuator, where s is the Laplace operator; μ is the parameter to be identified, and its identification methods include: S21, Construct structures with different frequencies ω i The input voltage signal is calculated, and the amplitude U of the input voltage signal is also calculated. i i = 1, 2, ..., n, where n is the total number of input voltage signals; S22. Based on the input voltage signal, obtain the output displacement signal of the piezoelectric actuator, and simultaneously calculate the amplitude D of the output displacement signal. i ; S23. Calculate the amplitude D of the output displacement signal. i With the amplitude U of the input voltage signal i The amplitude ratio of A i ; S24. Obtain the discrete data point set (x) i ,y i ), which satisfies: S25. Based on the discrete data point set (x) i ,y i Perform linear fitting, determine the slope of the fitted line, and determine the parameter μ based on the slope of the line.

2. The method for constructing a creep nonlinear dynamic model of a piezoelectric actuator as described in claim 1, characterized in that, Methods for identifying parameter μ also include: S26. Determine whether the calculated value of parameter μ is within the predetermined range (a, b). If it is within the range (a, b), then the value of parameter μ is the final value. Otherwise, adjust the points in the discrete data point set and return to step S25 to update the value of parameter μ.

3. The method for constructing a creep nonlinear dynamics model for a piezoelectric actuator as described in claim 2, characterized in that, The method for determining the range of values ​​(a, b) is as follows: The time-domain and frequency-domain characteristics of the creep effect of the piezoelectric actuator were obtained through experiments, i.e., the experimental results. The range of μ is divided into several non-overlapping intervals, and the transfer function s is simulated and analyzed under different ranges of μ. -μ The time-domain and frequency-domain characteristics, i.e., the simulation results; The experimental results are compared with the simulation results to determine the range of values ​​(a, b) to which μ belongs.

4. The method for constructing a nonlinear dynamic model of a piezoelectric actuator as described in claim 3, characterized in that, When obtaining simulation results, the range of μ is divided into four intervals: (-∞, -1), (-1, 0), (0, 1), and (1, +∞).

5. The method for constructing a creep nonlinear dynamic model of a piezoelectric actuator as described in claim 3, characterized in that, When obtaining experimental results, the step response diagram was used to characterize the time-domain characteristics of the creep effect of the piezoelectric actuator, and the Bode plot was used to characterize the frequency-domain characteristics of the creep effect of the piezoelectric actuator.

6. The method for constructing a creep nonlinear dynamic model of a piezoelectric actuator as described in claim 1, characterized in that, Based on the discrete data point set (x) i ,y i Perform linear fitting to determine the slope of the fitted line. The formula for calculating the slope k is: in, x, representing the discrete data point set i y i The mean.

7. The method for constructing a creep nonlinear dynamic model of a piezoelectric actuator as described in any one of claims 1-6, characterized in that, The formula for calculating the parameter μ based on the slope of the straight line is:

8. A nonlinear dynamic model for creep of a piezoelectric actuator, characterized in that, It was constructed using the method for constructing a creep nonlinear dynamic model of a piezoelectric actuator as described in any one of claims 1-7.

9. A system for constructing a nonlinear dynamic model of creep in a piezoelectric actuator, characterized in that, Includes a processor for executing the piezoelectric actuator creep nonlinear dynamics model construction method as described in any one of claims 1-7.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the method for constructing a nonlinear dynamic model of a piezoelectric actuator as described in any one of claims 1-7.

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