Single-frequency hysteresis nonlinear modeling method for linear reluctance motor based on phase-shift rational basis

Abstract

This paper presents a nonlinear modeling method for single-frequency hysteresis of linear reluctance motors based on phase-shift rational bases, relating to the field of ultra-precision motion control technology. First, the mechanism of the hysteresis loop between the input current and output flux of the linear reluctance motor is analyzed, and modeling is performed based on phase-shift rational bases. Then, the hysteresis phenomenon between the input current and output flux is decomposed into rising and falling parts. Finally, the least squares algorithm is used for parameter estimation. This method simplifies the complexity of the hysteresis model, reduces the number of parameters to be identified, lowers the difficulty of identifying the hysteresis model, and enables more accurate modeling of the hysteresis phenomenon at inflection points, thereby improving the accuracy of the linear reluctance motor hysteresis model.

Description

Technical Field

[0001] This invention relates to the field of ultra-precision motion control technology, specifically to a single-frequency hysteresis nonlinear modeling method for linear reluctance motors based on phase-shift rational bases. Background Technology

[0002] Following Moore's Law in the semiconductor industry, to meet the stringent requirements of future high-precision, short-stroke industrial motion systems in terms of production volume and accuracy, it is essential to develop actuators that can effectively provide high output density. Currently proposed actuators based on magnetoresistive technology, namely linear reluctance motors, can provide higher output density and lower energy dissipation. However, their output exhibits complex nonlinearity, and this complex nonlinearity must be addressed before they can be effectively applied to nanometer-precision positioning scenarios.

[0003] From a control perspective, the main output nonlinearities include the inherent second-order nonlinearity between current and force, the position dependence of the output force, and the rate-dependent hysteresis nonlinearity of the ferromagnetic core. Therefore, a comprehensive study of control methods is necessary to achieve consistent and accurate nano-positioning forces. Addressing hysteresis in soft ferromagnetic materials becomes crucial as it becomes more prominent. An effective method for compensating for magnetization hysteresis is to utilize the inverse model of the hysteresis model; therefore, accurate modeling of the hysteresis phenomenon is essential.

[0004] Scholars both domestically and internationally have studied the hysteresis nonlinearity of reluctance motors, and various models have been developed and researched to address this complex hysteresis nonlinearity. Based on different modeling principles, these models are mainly divided into two categories: physical hysteresis models and phenomenological hysteresis models. Physical hysteresis models are established using physical laws, describing hysteresis based on the physical properties and structure of the material itself. The Jiles-Atherton model and the Maxwell model are typical physical hysteresis models; however, because models based on physical laws are relatively complex, computationally difficult, and have limited modeling accuracy, they are generally difficult to apply to practical control systems. Phenomenological hysteresis models are established to describe the hysteresis phenomenon of materials. Based on experimentally obtained input-output data, various intelligent algorithms are used to identify the parameters in the model, thus simulating various practical hysteresis systems. However, they neglect the physical properties and structure of the material itself; examples include the Duhem model, the Bouc-Wen model, and the Preisach model. Currently, these existing models generally suffer from drawbacks such as model complexity, a large number of parameters to be identified, and inaccurate hysteresis model establishment when the current reverses, i.e., at the inflection point.

[0005] The invention patent application with publication number CN117130276A, publication date November 28, 2023, and titled "A Hysteresis Compensation Method for Piezoelectric Ceramics Based on Duhem Hysteresis Model" provides a hysteresis compensation method for piezoelectric ceramics. This method uses the inverse model of the Duhem model for feedforward control to compensate for the hysteresis characteristics of piezoelectric ceramics. However, the Duhem model in this patent has many parameters to be identified, making identification difficult.

[0006] The invention patent application with publication number CN115857311A, publication date March 28, 2023, and titled "A Hysteresis Compensation Control Method for a Micro-vibration Suppression Platform" provides a hysteresis compensation control method for a micro-vibration suppression platform. This method introduces a frequency factor, improves the Bouc-Wen model, and uses it as feedforward control, which can improve the dynamic performance and steady-state accuracy of the micro-vibration suppression platform. This patent is more accurate in modeling than the Bouc-Wen model, but it is not a model specifically for the hysteresis phenomenon of linear reluctance motors. The model is not accurate enough at the turning point and is not suitable for hysteresis compensation of linear reluctance motors. Summary of the Invention

[0007] To address the problems of complex physical hysteresis models with numerous parameters to be identified and inaccurate modeling due to the neglect of mechanisms in phenomenological hysteresis models, this invention provides a single-frequency hysteresis nonlinear modeling method for linear reluctance motors based on phase-shift rational bases. This method simplifies the complexity of hysteresis models, reduces the number of parameters to be identified, lowers the difficulty of identifying hysteresis models, and enables more accurate modeling of hysteresis phenomena at inflection points, thereby improving the accuracy of linear reluctance motor hysteresis models.

[0008] To achieve the above objectives, the present invention adopts the following technical solution: a single-frequency hysteresis nonlinear modeling method for linear reluctance motors based on phase-shift rational bases, comprising the following steps:

[0009] Step 1: Perform mechanism analysis on the hysteresis loop between the input current and output flux of the linear reluctance motor, and model it based on phase-shift rational basis;

[0010] The expression for the output magnetic flux signal B is expanded as follows:

[0011]

[0012] In the formula, A1 represents the amplitude of the output magnetic flux, w represents the angular frequency, and t represents time. The phase difference between the input current and the output magnetic flux is represented by Δ, where Δ represents the error interference term of the output magnetic flux, and I represents the input current signal.

[0013] Among them Taylor expansion of the terms:

[0014]

[0015] The expression for the output magnetic flux signal B is then written as:

[0016]

[0017] Rewritten as:

[0018] B = aI 4 +bI 2 +cI+d (5)

[0019] in,

[0020] Step 2: Decompose the hysteresis phenomenon between the input current and output flux of the linear reluctance motor into rising and falling parts;

[0021] The rising and falling portions of the hysteresis loop are symmetrical about the origin. Let formula (5) be the expression for the output magnetic flux of the rising portion, then the expression for the output magnetic flux of the falling portion is:

[0022]

[0023] Let I d =-I,B d =-B, then we get a d =-a,b d =-b,c d =c,d d =d;

[0024] Step 3: Estimate parameters using the least squares algorithm;

[0025] First, the input and output data of the linear reluctance motor are collected, and then decomposed into rising and falling parts. Assume there are n sampling points, and the input current is denoted as I = [I1, I2, ..., I...]. n ] T The output magnetic flux is denoted as B = [B1, B2, ... B]. n ] T Then, formula (5) can be written in matrix form as follows:

[0026]

[0027] Input matrix [I] 0 I 1 I 2 I 4 Let X be the coefficient matrix [abcd]. T Let A be the abbreviation, then formula (8) can be written as:

[0028] XA=B (11)

[0029] The parameters of the rising portion of the hysteresis loop are estimated using the least squares method, yielding estimates of the coefficient matrix. as follows:

[0030]

[0031] Replace the input matrix X and output matrix B of the rising part with the input matrix X of the falling part, respectively. d and output matrix B d This yields an estimate of the coefficient matrix of the descent portion. This allows us to obtain a single-frequency hysteresis nonlinear model of the linear reluctance motor based on phase-shift rational bases.

[0032] Furthermore, the input current signal and the output magnetic flux signal have the same angular frequency, different amplitudes, and a phase difference.

[0033] Furthermore, the input current signal I = A0sin(wt) is given, where A0 represents the amplitude of the input current.

[0034] Compared with the prior art, the beneficial effects of the present invention are as follows: First, based on the mechanism of hysteresis loops, the present invention models the hysteresis characteristics using phase-shift rational basis. Then, based on the hysteresis loop phenomenon, the hysteresis loop is divided into an ascending part and a descending part, and these two parts are modeled separately. By using mechanism analysis and phenomenon decomposition, the hysteresis nonlinearity of the linear reluctance motor is accurately modeled, and the model parameters are estimated using the least squares method. This simplifies the complexity of the hysteresis model, reduces the number of parameters to be identified in the hysteresis model, and lowers the difficulty of identifying the hysteresis model. At the same time, by modeling the mechanism and phenomenon of the linear reluctance motor, the hysteresis phenomenon at the inflection point can be modeled more accurately, thus improving the accuracy of the linear reluctance motor hysteresis model. Attached Figure Description

[0035] Figure 1 This is a flowchart of the present invention;

[0036] Figure 2 This is a physical diagram of the linear reluctance motor motion platform established in the embodiment;

[0037] Figure 3 This is a diagram showing the implementation effect obtained in the example. Detailed Implementation

[0038] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the invention, not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0039] like Figure 1 As shown, the single-frequency hysteresis nonlinear modeling method for linear reluctance motors based on phase-shift rational bases includes the following steps:

[0040] Step 1: Perform a mechanism analysis on the hysteresis loop between the input current and output flux of the linear reluctance motor, and model it based on phase-shift rational basis. Specifically:

[0041] Input current signal I = A0sin(wt), output magnetic flux signal The input current signal and the output magnetic flux signal have the same angular frequency, different amplitudes, and a phase difference. Here, A0 represents the amplitude of the input current, A1 represents the amplitude of the output magnetic flux, w represents the angular frequency, and t represents time. This represents the phase difference between the input current and the output magnetic flux, and Δ represents the error interference term of the output magnetic flux.

[0042] The expression for the output magnetic flux signal B is expanded as follows:

[0043]

[0044] Among them Taylor expansion of the terms:

[0045]

[0046] Among them, o(I 6 ) indicates I 6 We can obtain the following by neglecting higher-order infinitesimals:

[0047]

[0048] The expression for the output magnetic flux signal B is then written as:

[0049]

[0050] Then, let The expression for the output magnetic flux signal B can then be rewritten as:

[0051] B = aI 4 +bI 2 +cI+d (5)

[0052] That is, the output magnetic flux signal B can be approximated as a quadratic polynomial of the input current signal I.

[0053] Step 2: Decompose the hysteresis phenomenon between the input current and output flux of the linear reluctance motor into rising and falling parts, specifically:

[0054] Ideally, the hysteresis loop is centrally symmetric, meaning the rising and falling portions are symmetric about the origin. If we assume formula (5) to be the expression for the output magnetic flux of the rising portion, then we can assume the expression for the output magnetic flux of the falling portion is:

[0055]

[0056] Since the rising and falling portions are symmetrical about the origin, let I... d =-I,B d =-B, then we can get a d =-a,b d =-b,c d =c,d d =d.

[0057] Step 3: Perform parameter estimation using the least squares algorithm. Specifically:

[0058] First, the input and output data of the linear reluctance motor are collected. Then, the data is processed and decomposed into rising and falling parts. The rising part of the hysteresis loop is first estimated using the least squares method. Assuming there are n sampling points in the rising part, the sampled data is written in matrix form, and the input current is denoted as I = [I1, I2, ..., I...]. n ] T The output magnetic flux is denoted as B = [B1, B2, ... B]. n ] T Then formula (5) can be written in matrix form as follows:

[0059]

[0060] Right now:

[0061]

[0062] Input matrix [I] 0 I 1 I 2 I 4 Let X be the denoted value, that is:

[0063] X = [I 0 I 1 I 2 I 4 (9)

[0064] The coefficient matrix [abcd] T Let it be denoted as A, that is:

[0065]

[0066] Then formula (8) can be written as:

[0067] XA=B (11)

[0068] The coefficient matrix is ​​estimated using the least squares method. as follows:

[0069]

[0070] Similarly, replace the input matrix X and output matrix B of the rising part with the input matrix X of the falling part, respectively. d and output matrix B d This yields an estimate of the coefficient matrix of the descent portion. The single-frequency hysteresis nonlinear model of the linear reluctance motor based on phase-shift rational basis can be obtained through the above process.

[0071] Example

[0072] In this embodiment, a linear reluctance motor motion platform is established in conjunction with... Figure 2 As shown, the motion platform consists of a voice coil motor, an optical encoder, an air-bearing guide rail, a reluctance motor, and their supporting devices. The linear reluctance motor, which is the modeling object of this invention, consists of a type I mover and an type E stator. The voice coil motor is responsible for positioning the type I mover of the linear reluctance motor. The optical encoder is used for position signal measurement. The type E stator is connected to and fixed to the supporting device, and the type I mover is connected to the guide rail and moves with the guide rail.

[0073] according to Figure 1 The process shown first establishes a polynomial hysteresis model with four parameters to be identified. Then, a voice coil motor is used to position the motion platform, and data with an input current frequency of 10Hz is collected. This data is decomposed into rising and falling parts, and the least squares algorithm is used to identify the parameters separately. The implementation effect is then combined with... Figure 3 As shown in the figure, the identified model fits the measurement data well, and the fitting effect at the inflection point is also ideal, verifying that the modeling method of the present invention has the advantages of fewer model parameters and higher model accuracy.

[0074] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered illustrative and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of the equivalents of the claims are intended to be included within the present invention. No reference numerals in the claims should be construed as limiting the scope of the claims.

[0075] Furthermore, it should be understood that although this specification describes embodiments, not every embodiment contains only one independent technical solution. This narrative style is merely for clarity. Those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. A single-frequency hysteresis nonlinear modeling method for linear reluctance motors based on phase-shift rational bases, characterized in that: Includes the following steps: Step 1: Perform mechanism analysis on the hysteresis loop between the input current and output flux of the linear reluctance motor, and model it based on phase-shift rational basis; The expression for the output magnetic flux signal B is expanded as follows: In the formula, A1 represents the amplitude of the output magnetic flux, w represents the angular frequency, and t represents time. The phase difference between the input current and the output magnetic flux is represented by Δ, where Δ represents the error interference term of the output magnetic flux, and I represents the input current signal. Among them Taylor expansion of the terms: The expression for the output magnetic flux signal B is then written as: Rewritten as: B = aI 4 +bI 2 +cI+d (5) in, Step 2: Decompose the hysteresis phenomenon between the input current and output flux of the linear reluctance motor into rising and falling parts; The rising and falling portions of the hysteresis loop are symmetrical about the origin. Let formula (5) be the expression for the output magnetic flux of the rising portion, then the expression for the output magnetic flux of the falling portion is: Let I d =-I,B d =-B, then we get a d =-a,b d =-b,c d =c,d d =d; Step 3: Estimate parameters using the least squares algorithm; First, the input and output data of the linear reluctance motor are collected, and then decomposed into rising and falling parts. Assume there are n sampling points, and the input current is denoted as I = [I1, I2, ..., I...]. n ] T The output magnetic flux is denoted as B = [B1, B2, ... B]. n ] T Then, formula (5) can be written in matrix form as follows: Input matrix [I] 0 I 1 I 2 I 4 Let X be the coefficient matrix [abcd]. T Let A be the abbreviation, then formula (8) can be written as: XA=B (11) The parameters of the rising portion of the hysteresis loop are estimated using the least squares method, yielding estimates of the coefficient matrix. as follows: Replace the input matrix X and output matrix B of the rising part with the input matrix X of the falling part, respectively. d and output matrix B d This yields an estimate of the coefficient matrix of the descent portion. This allows us to obtain a single-frequency hysteresis nonlinear model of the linear reluctance motor based on phase-shift rational bases.

2. The method for modeling single-frequency hysteresis nonlinearity of a linear reluctance motor based on phase-shift rational basis according to claim 1, characterized in that: The input current signal and the output magnetic flux signal have the same angular frequency, different amplitudes, and a phase difference.

3. The single-frequency hysteresis nonlinear modeling method for linear reluctance motors based on phase-shift rational bases according to claim 2, characterized in that: The input current signal is I = A0sin(wt), where A0 represents the amplitude of the input current.

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