Modeling method and system for frequency-dependent giant magnetostrictive actuator

Abstract

The invention discloses a modeling method and system for a frequency-dependent giant magnetostrictive actuator. The modeling method comprises the following steps: firstly, establishing a classic PI model; then, a symmetric Play operator in the classic PI model is improved into an asymmetric Play operator, and an improved PI model is obtained; further introducing a frequency factor, representing part of parameters as a frequency function, and constructing a frequency-related improved PI model; the input current and output displacement data under different frequencies are utilized, a nonlinear optimization tool box is adopted for parameter identification, and model parameters are optimized. And finally, comparing the root-mean-square error and the mean absolute error of the model output and the actual output, and verifying the fitting precision of the model under different frequencies. According to the method, the asymmetric hysteresis characteristic and the frequency-dependent hysteresis characteristic of the giant magnetostrictive actuator can be accurately described within the frequency range of 1-90 Hz, and a more accurate theoretical basis is provided for modeling and control of the giant magnetostrictive actuator.

Description

Technical Field

[0001] The present invention belongs to the technical field of giant magnetostrictive actuator control, and in particular relates to a modeling method and system for a frequency-dependent giant magnetostrictive actuator. Background Art

[0002] Giant magnetostrictive actuators have the advantages of high precision, strong load capacity, strong durability, and fast response speed. They have been widely used in ultra-precision machining, precision medicine, aerospace and other fields.

[0003] Giant magnetostrictive actuators are essentially made of magnetostrictive materials. Due to the special properties of magnetostrictive materials, the input and output of giant magnetostrictive actuators are not linear. This nonlinear relationship is called hysteresis, and the hysteresis characteristic is frequency-dependent. The frequency-dependent hysteresis characteristic of giant magnetostrictive actuators makes their modeling and control difficult, which is also a pain point in research related to giant magnetostrictive actuators.

[0004] The hysteresis characteristics of giant magnetostrictive actuators are modeled using the classic Prandtl-Ishlinskii (PI) model. This model faces two challenges: asymmetric hysteresis and frequency-dependent hysteresis. The slope of the hysteresis curve changes with increasing frequency and decreasing travel. The classic PI model cannot address these two issues.

[0005] Therefore, how to develop a modeling method that can accurately describe the frequency-related hysteresis characteristics of giant magnetostrictive actuators has become a technical problem that needs to be solved urgently in this field. Summary of the Invention

[0006] The present invention proposes a modeling method and system for a frequency-dependent giant magnetostrictive actuator to solve the problems existing in the above-mentioned prior art.

[0007] To achieve the above object, the present invention provides a modeling method for a frequency-dependent giant magnetostrictive actuator, comprising the following steps:

[0008] Establishing a classic PI model, which is composed of a weighted superposition of several symmetrical play operators;

[0009] The symmetric play operator in the classic PI model is improved to an asymmetric play operator to obtain an improved PI model, wherein the asymmetric play operator describes the relationship between the input current and the output displacement through an envelope function;

[0010] Introducing a frequency factor into the improved PI model to construct a frequency-dependent improved PI model;

[0011] Parameter identification of the frequency-dependent improved PI model is performed using input current and output displacement data at different frequencies;

[0012] The fitting accuracy of the frequency-dependent improved PI model at different frequencies is verified, and after successful verification, the final frequency-dependent improved PI model is obtained.

[0013] Optionally, improving a symmetric Play operator to an asymmetric Play operator includes:

[0014] Determine an input-output relationship of an asymmetric Play operator, where the output of the asymmetric Play operator increases along a curve as the input increases and decreases along the curve as the input decreases;

[0015] The envelope function of the asymmetric Play operator is a strictly monotonic function.

[0016] Optionally, the introduced frequency factor includes:

[0017] Some parameters in the improved PI model are expressed as functions with frequency as the independent variable;

[0018] The functional relationship between the parameters and the frequency is determined by fitting experimental data, and the parameters include coefficients in the envelope function and coefficients in the weight function.

[0019] Optionally, the parameter identification step includes:

[0020] Using a nonlinear optimization tool to optimize the parameters of the frequency-related improved PI model;

[0021] The fitness function of the nonlinear optimization tool is a square error sum function.

[0022] Optionally, the verification step includes:

[0023] Select input and output data of giant magnetostrictive actuators in different frequency ranges;

[0024] Substituting the input and output data into the frequency-dependent improved PI model to calculate the output of the model;

[0025] Compare the output of the model with the actual output to determine the effectiveness of the model.

[0026] Optionally, the frequency ranges from 1 Hz to 90 Hz.

[0027] The present invention also provides a modeling system for a frequency-dependent giant magnetostrictive actuator, comprising:

[0028] A classic PI model building module is used to build a classic PI model, wherein the classic PI model is composed of a weighted superposition of several symmetrical play operators;

[0029] An improved PI model construction module is used to improve the symmetric play operator in the classic PI model into an asymmetric play operator to obtain an improved PI model, wherein the asymmetric play operator describes the relationship between the input current and the output displacement through an envelope function;

[0030] A frequency-dependent improved PI model generation module is used to introduce frequency factors into the improved PI model to construct a frequency-dependent improved PI model;

[0031] A parameter identification module, configured to perform parameter identification on the frequency-dependent improved PI model using input current and output displacement data at different frequencies;

[0032] The model verification module is used to verify the fitting accuracy of the frequency-dependent improved PI model at different frequencies, and obtain the final frequency-dependent improved PI model after successful verification.

[0033] The present invention also provides a computer device, comprising a memory, a processor, and a computer program stored in the memory, wherein the processor executes the computer program to implement the steps of the method.

[0034] The present invention also provides a computer-readable storage medium having a computer program stored thereon, which implements the steps of the method when executed by a processor.

[0035] The present invention also provides a computer program product, comprising a computer program, which implements the steps of the method when executed by a processor.

[0036] Compared with the prior art, the present invention has the following advantages and technical effects:

[0037] The present invention proposes a modeling method and system for a frequency-dependent giant magnetostrictive actuator, significantly improving modeling accuracy and adaptability compared to existing technologies. By introducing a frequency factor to improve the classic PI model, the present invention accurately describes the asymmetric and frequency-dependent hysteresis characteristics of the giant magnetostrictive actuator at different frequencies, effectively addressing the poor modeling performance of traditional models when the frequency varies. Model parameter identification and optimization using a nonlinear optimization toolbox further ensures model accuracy and reliability. This precise modeling provides a reliable theoretical foundation for precise control of the giant magnetostrictive actuator, significantly improving its performance and reliability in practical applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] The accompanying drawings, which constitute part of this application, are intended to provide a further understanding of this application. The exemplary embodiments and descriptions of this application are intended to explain this application and do not constitute an improper limitation on this application. In the accompanying drawings:

[0039] Figure 1 is a graph showing the relationship between input current and output displacement at different frequencies according to an embodiment of the present invention;

[0040] Figure 2 Schematic diagram of a single play operator in an embodiment of the present invention;

[0041] Figure 3 Schematic diagram of an improved play operator according to an embodiment of the present invention;

[0042] Figure 4 This is a diagram showing the fitting of the classic PI model at a frequency of 1 Hz according to an embodiment of the present invention;

[0043] Figure 5 This is the fitting of the frequency-dependent improved PI model at a frequency of 1 Hz according to an embodiment of the present invention;

[0044] Figure 6 This is the fitting of the frequency-dependent improved PI model at a frequency of 50 Hz according to an embodiment of the present invention;

[0045] Figure 7 This is the fitting of the frequency-dependent improved PI model at 60 Hz according to an embodiment of the present invention;

[0046] Figure 8 This is the fitting of the frequency-dependent improved PI model at a frequency of 70 Hz according to an embodiment of the present invention;

[0047] Figure 9 This is the fitting of the frequency-dependent improved PI model at 80 Hz according to an embodiment of the present invention;

[0048] Figure 10 This is the fitting condition of the frequency-dependent improved PI model at a frequency of 90 Hz according to an embodiment of the present invention. DETAILED DESCRIPTION

[0049] It should be noted that, in the absence of conflict, the embodiments and features of the embodiments in this application can be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0050] It should be noted that the steps shown in the flowcharts of the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and that, although a logical order is shown in the flowcharts, in some cases, the steps shown or described can be executed in an order different from that shown here.

[0051] Example 1

[0052] This embodiment uses a Model 050-LH giant magnetostrictive actuator with a maximum input AC current of 5A and an output displacement range of ±25 microns. The input current is a sinusoidal signal with a peak-to-peak value of 0-0.7416 at different frequencies. Experiments have shown the relationship between the input current and output displacement at different frequencies. Figure 1 It can be seen that the hysteresis characteristics of the giant magnetostrictive actuator at different frequencies, as the frequency increases, the slope of the hysteresis loop will change, the displacement stroke will become smaller, but the phase difference between current and displacement will not increase significantly.

[0053] Based on this, this embodiment proposes a modeling method for a frequency-dependent giant magnetostrictive actuator, including the following steps:

[0054] Establish a classic PI model, which is composed of a weighted superposition of several symmetrical play operators;

[0055] The symmetric play operator in the classic PI model is improved to an asymmetric play operator, and an improved PI model is obtained. The asymmetric play operator describes the relationship between input current and output displacement through an envelope function.

[0056] Introducing frequency factors into the improved PI model and constructing a frequency-related improved PI model;

[0057] The input current and output displacement data at different frequencies are used to perform parameter identification on the frequency-dependent improved PI model;

[0058] Verify the fitting accuracy of the frequency-dependent improved PI model at different frequencies, and obtain the final frequency-dependent improved PI model after successful verification.

[0059] The specific steps are as follows:

[0060] Step 1:

[0061] Establishing the classic PI model:

[0062] The frequency-dependent improved PI model is an improvement of the PI model. The PI model has been widely used since its introduction. It is composed of weighted superposition of different Play operators, such as Figure 2 shown.

[0063] In the figure, v is the input, w is the output, and r is the threshold. The play operator is shaped like a parallelogram and moves counterclockwise around the parallelogram. The PI model based on the play operator can be expressed as:

[0064]

[0065] Where y(t) is the model output, p0v is the linear part of the PI model, which is a linear function of the input signal, and p0 is the coefficient of the linear part. It is the nonlinear part of the PI model, which is composed of the weighted superposition of play operators. Its meaning is the integral of the threshold r from 0 to R, where R is a finite positive constant, p is the weight, and w is the output of the play operator.

[0066] Establish a frequency-dependent improved PI model:

[0067] The improved PI model is derived from the classic PI model. The classic PI model has a symmetric play operator. By transforming the symmetric play operator into an asymmetric play operator, the improved PI model can be obtained. The input and output relationship of a single improved Play operator is as follows: Figure 3 shown.

[0068] Depend on Figure 2 It can be seen that the output w of the improved play operator will follow the curve γ as the input v increases. r increases, and as the input v decreases along the curve γ l Decrease. Curve γ r and γ l are called envelope functions of the improved play operators, and they are strictly monotonic.

[0069] Let the interval C[0,T] be the domain of the input v. For any v∈C[0,T], the output w of the improved play operator is:

[0070] w(0)=f(v(0),0) (2)

[0071] w(t)=f(v(t),f(v)) (3)

[0072] Assume t i <t<t i+1 , then:

[0073] w t =max(γ r (v)-r,min(γ l (v)+r,w t-1 )) (4)

[0074] Then the expression of the improved PI model is:

[0075]

[0076] Where y G (t) is the output of the improved PI model, H(t) is the function of the input signal, and its form is consistent with the envelope function. p is the weight function, and the rest is consistent with the classic PI model.

[0077] To facilitate the solution, the integral expression (5) can be discretized as:

[0078]

[0079] Where n is the number of improved play operators, p is a normal number, y0=[y 10 ,···,y n0 ] T is the initial value.

[0080] The improved play operator in formula (6) can be expressed as:

[0081] w k =max(γ r (v)-r,min(γ l (v)+r,w k-1 )) (7)

[0082] When the input amount increases monotonically or decreases monotonically, the envelope function in the improved play operator is different, which can be expressed as:

[0083]

[0084] Threshold r i and weight p i Usually selected as:

[0085] r i =αi (9)

[0086]

[0087] According to the input-output hysteresis characteristic curve of GMA, the envelope function can be selected as:

[0088]

[0089] In equations (9), (10), and (11), a0, a1, a2, a3, b0, b1, b2, b3, α, ρ, and τ are the parameters to be identified, where α, ρ, and τ are all greater than zero.

[0090] After extensive experiments, we discovered that some of the 11 parameters identified by the improved PI model exhibit regular variations as the data frequency changes. Therefore, we added a frequency factor to the improved PI model and proposed a rate-dependent improved PI model, successfully resolving the rate-dependent issues that the PI model couldn't address.

[0091] To increase the frequency factor:

[0092] a1=g1(f) (12)

[0093] a2=g2(f) (13)

[0094] a3=g3(f) (14)

[0095] b1=g4(f) (15)

[0096] b2=g5(f) (16)

[0097] b3=g6(f) (17)

[0098] In formula (12), formula (13), formula (14), formula (15), formula (16), and formula (17), f represents frequency. These four formulas respectively indicate that the parameters a1, a2, a3, b1, b2, and b3 are functions with frequency as the independent variable, thus forming a rate-dependent improved PI model.

[0099] Step 2:

[0100] Frequency-dependent improved PI model parameter identification:

[0101] From Equations (9), (10), and (11), we can see that the parameters that need to be identified for the first time in the model are a0, a1, a2, a3, b0, b1, b2, b3, α, ρ, and τ, a total of 11 parameters. The frequency-dependent improved PI model uses 10 improved play operators. The Fmincon toolbox is used to perform optimization calculations using the input and output data of the GMA at different frequencies to complete parameter identification.

[0102] After that, multiple parameter identification and curve fitting will be performed to transform the parameters a1, a2, a3, b1, b2, and b3 into expressions of functions with frequency as the independent variable.

[0103] The Fmincon toolbox, a native Matlab toolbox, offers powerful solutions for nonlinear optimization problems. Its greatest strength lies in its ability to solve constrained nonlinear optimization problems. For complex nonlinear problems, the Fmincon toolbox supports multiple starting point strategies, maximizing the chance of finding a global optimal solution and preventing a single solution from becoming trapped in a local optimum.

[0104] When using the Fmincon toolbox to identify parameters, the fitness function can be set as the error sum square function:

[0105]

[0106] In the formula, F is the value of the fitness function, N is the number of data, and y G is the theoretical output of the hysteresis model, y t is the actual output of GMA.

[0107] With frequency as the independent variable and parameters a2, a3, b2, b3 as the dependent variables, the function expressed is:

[0108]

[0109] Step 3:

[0110] Improved frequency-dependent PI model validation:

[0111] The validity of the model is tested using the input and output data of GMA in the frequency range of 1 to 90 Hz. Figure 4 The curve fitting of the classical PI model and single frequency experimental data is given. Figure 5 The curve fitting of the frequency-dependent improved PI model and the experimental data is given when the input signal frequency changes. Tables 1 and 2 respectively give the modeling errors of the classic PI model at a single frequency and the frequency-dependent improved PI model at different frequencies.

[0112] The accuracy of the model is determined by the root mean square error (RMSE) and the mean absolute error (MAE), which are expressed as follows:

[0113]

[0114]

[0115] in, is the model output, y(k) is the actual output of GMA, and N is the number of data.

[0116] Table 1

[0117]

[0118] Figure 4 In the figure, the horizontal axis is the input current of GMA, and the vertical axis is the output displacement of GMA. The solid line is the hysteresis curve drawn using the experimental data of GMA, and the dotted line is the fitting curve of PI model. Figure 4 It can be seen that the classic PI model, with its symmetric play operator structure, is not ideal for modeling the hysteresis characteristics of GMA. Furthermore, because the classic PI model is inherently rate-independent, the modeling effect worsens when the input signal frequency changes.

[0119] Table 2

[0120]

[0121] Figure 5-10In the figure, the horizontal axis is the input current of GMA, and the vertical axis is the output displacement of GMA. The solid line part is the hysteresis curve drawn using the experimental data of GMA at different frequencies, and the dotted line part is the fitting curve of the rate-dependent improved PI model. Figure 5 As can be seen from Table 2, the modeling error of the established frequency-dependent improved PI model is very small. The error tends to increase with increasing input frequency, but the error remains small overall. Therefore, the established frequency-dependent improved PI model can accurately describe the hysteresis characteristics of the GMA at different frequencies from 1 to 90 Hz.

[0122] This embodiment further provides a modeling system for a frequency-dependent giant magnetostrictive actuator, including:

[0123] A classic PI model building module is used to build a classic PI model, wherein the classic PI model is composed of a weighted superposition of several symmetrical play operators;

[0124] An improved PI model construction module is used to improve the symmetric play operator in the classic PI model into an asymmetric play operator to obtain an improved PI model, wherein the asymmetric play operator describes the relationship between the input current and the output displacement through an envelope function;

[0125] A frequency-dependent improved PI model generation module is used to introduce frequency factors into the improved PI model to construct a frequency-dependent improved PI model;

[0126] A parameter identification module, configured to perform parameter identification on the frequency-dependent improved PI model using input current and output displacement data at different frequencies;

[0127] The model verification module is used to verify the fitting accuracy of the frequency-dependent improved PI model at different frequencies, and obtain the final frequency-dependent improved PI model after successful verification.

[0128] This embodiment further provides a computer device, including a memory, a processor, and a computer program stored in the memory, wherein the processor executes the computer program to implement the steps of the method.

[0129] This embodiment further provides a computer-readable storage medium having a computer program stored thereon, which implements the steps of the method when executed by a processor.

[0130] This embodiment also provides a computer program product, including a computer program, which implements the steps of the method when executed by a processor.

[0131] The above are merely preferred embodiments of the present application, but the scope of protection of the present application is not limited thereto. Any changes or substitutions that can be easily conceived by a person skilled in the art within the technical scope disclosed in this application should be included in the scope of protection of the present application. Therefore, the scope of protection of the present application should be based on the scope of protection of the claims.

Claims

1. A modeling method for a frequency-dependent giant magnetostrictive actuator, characterized in that: The following steps are involved: Establishing a classic PI model, which is composed of a weighted superposition of several symmetrical play operators; The symmetric play operator in the classic PI model is improved to an asymmetric play operator to obtain an improved PI model, wherein the asymmetric play operator describes the relationship between the input current and the output displacement through an envelope function; Introducing a frequency factor into the improved PI model to construct a frequency-dependent improved PI model; Parameter identification of the frequency-dependent improved PI model is performed using input current and output displacement data at different frequencies; The fitting accuracy of the frequency-dependent improved PI model at different frequencies is verified, and after successful verification, the final frequency-dependent improved PI model is obtained.

2. The method according to claim 1, characterized in that Improving the symmetric Play operator to an asymmetric Play operator includes: Determine an input-output relationship of an asymmetric Play operator, where the output of the asymmetric Play operator increases along a curve as the input increases and decreases along the curve as the input decreases; The envelope function of the asymmetric Play operator is a strictly monotonic function.

3. The method according to claim 1, characterized in that The introduced frequency factors include: Some parameters in the improved PI model are expressed as functions with frequency as the independent variable; The functional relationship between the parameters and the frequency is determined by fitting experimental data, and the parameters include coefficients in the envelope function and coefficients in the weight function.

4. The method according to claim 1, wherein The step of parameter identification includes: Using a nonlinear optimization tool to optimize the parameters of the frequency-related improved PI model; The fitness function of the nonlinear optimization tool is a square error sum function.

5. The method according to claim 1, wherein The verification step includes: Select input and output data of giant magnetostrictive actuators in different frequency ranges; Substituting the input and output data into the frequency-dependent improved PI model to calculate the output of the model; Compare the output of the model with the actual output to determine the effectiveness of the model.

6. The method according to claim 1, characterized in that The frequency range is 1 Hz to 90 Hz.

7. A modeling system for a frequency-dependent giant magnetostrictive actuator, characterized in that: include: A classic PI model building module is used to build a classic PI model, wherein the classic PI model is composed of a weighted superposition of several symmetrical play operators; An improved PI model construction module is used to improve the symmetric play operator in the classic PI model into an asymmetric play operator to obtain an improved PI model, wherein the asymmetric play operator describes the relationship between the input current and the output displacement through an envelope function; A frequency-dependent improved PI model generation module is used to introduce frequency factors into the improved PI model to construct a frequency-dependent improved PI model; A parameter identification module, configured to perform parameter identification on the frequency-dependent improved PI model using input current and output displacement data at different frequencies; The model verification module is used to verify the fitting accuracy of the frequency-dependent improved PI model at different frequencies, and obtain the final frequency-dependent improved PI model after successful verification.

8. A computer device comprising a memory, a processor, and a computer program stored in the memory, wherein: The processor executes the computer program to implement the steps of the method according to any one of claims 1 to 6.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 6 are implemented.

10. A computer program product comprising a computer program, characterized in that When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 6 are implemented.

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