Linear reluctance electrical probability related hysteresis modeling method based on phase shift rational basis

Abstract

The invention discloses a linear reluctance electrical probability related hysteresis modeling method based on a phase shift rational basis, and relates to the technical field of ultra-precision motion control. Firstly, mechanism analysis is performed on a hysteresis loop between input current and output magnetic flux of the linear reluctance motor, modeling is performed based on a phase shift rational basis, then a hysteresis phenomenon between the input current and the output magnetic flux of the linear reluctance motor is decomposed and divided into an ascending part and a descending part, and the descending part is divided into a descending part and a descending part. And finally, performing parameter estimation by using a least square algorithm to obtain a complete linear reluctance electrical probability related hysteresis model. According to the method, the hysteresis phenomenon at the turning point can be modeled more accurately, the precision of the hysteresis model of the linear reluctance motor is improved, meanwhile, rate correlation factors are added, and the hysteresis rate correlation characteristics are met.

Description

Technical Field

[0001] The invention relates to the technical field of ultra-precision motion control, and in particular to a linear reluctance motor frequency-dependent hysteresis modeling method based on a phase-shift rational basis. Background Art

[0002] In order to meet the needs of high-end lithography machines for high speed, large stroke and nano-level positioning, the mask stage and the workbench both adopt a macro-micro composite structure of a large-stroke macro-motion stage superimposed on a small-stroke micro-motion stage. The large-stroke macro-motion stage is mainly responsible for expanding the stroke and completing the high-speed scanning process, while the small-stroke micro-motion stage is responsible for achieving nano-level positioning. To meet the requirements, the micro-motion stage must have a high-rigidity structure and high-performance drive. For its drive system, an actuator developed based on magnetic resistance, namely a linear reluctance motor, has been proposed. This actuator can provide higher output density and lower energy dissipation. However, the output of the linear reluctance motor has complex nonlinearity, and its complex nonlinearity problem must be solved before it can be effectively applied to nano-precision positioning scenarios.

[0003] The complex nonlinear characteristics of linear reluctance motors include the inherent hysteresis nonlinearity of ferromagnetic materials, the nonlinearity of output force proportional to the square of current, and the position dependence of output force inversely proportional to the square of air gap. Among the various nonlinearities, the inherent hysteresis nonlinearity of soft magnetic materials is the main factor affecting and limiting its performance. Although the influence of this factor can be reduced by using a magnetic core made of a thin layer of soft ferromagnetic material, in high-precision positioning applications, higher output linearity is required, and the influence of hysteresis nonlinearity becomes obvious. Modeling and compensation methods must be used to improve the output linearity of linear reluctance motors. Therefore, it is crucial to accurately model the hysteresis phenomenon.

[0004] Domestic and foreign scholars have studied the hysteresis nonlinearity of reluctance motors. A variety of models have been developed and studied for this type of complex hysteresis nonlinearity. According to different modeling principles, it is mainly divided into two categories: physical hysteresis model and phenomenological hysteresis model. Among them, phenomenological hysteresis model is widely used in practical systems because of its simple modeling, but this modeling method ignores the physical properties and structure of the material itself, such as Duhem model, Bouc-Wen model, Preisach model, etc. At present, these existing models generally have the problem of inaccurate hysteresis model establishment when the current reverses, that is, at the turning point. At the same time, the Bouc-Wen model and other models also have the problem of not being able to characterize rate-related hysteresis characteristics.

[0005] An invention patent application with publication number CN110110380B, publication date July 4, 2023, and titled "A method and application for modeling hysteresis nonlinearity of piezoelectric actuators" provides a piezoelectric actuator modeling method based on an improved chaotic quantum particle swarm, which has the advantages of being able to effectively simulate hysteresis nonlinearity and high parameter identification accuracy. However, the model in this patent is rate-independent and is not suitable for establishing a rate-dependent hysteresis model of a linear reluctance motor.

[0006] The invention patent application with publication number CN117291142A, publication date December 26, 2023, and name “Core dynamic hysteresis modeling method based on permeance-capacitance analogy method and analytical Preisach model” established an improved core dynamic hysteresis permeance model, reducing the error caused by the traditional model. However, this modeling method is not a model established for the hysteresis phenomenon of the linear reluctance motor. The model establishment is not accurate enough at the turning point and is not suitable for hysteresis compensation of the linear reluctance motor. Summary of the invention

[0007] In order to solve the problem that most current hysteresis models are not accurate enough at the turning point and the model is independent of the input frequency, the present invention provides a linear reluctance motor frequency-dependent hysteresis modeling method based on a phase-shift rational basis, which can more accurately model the hysteresis phenomenon at the turning point, improve the accuracy of the linear reluctance motor hysteresis model, and at the same time increase the rate-related factor, which is in line with the hysteresis rate-related characteristics.

[0008] To achieve the above object, the present invention adopts the following technical solution: a linear reluctance motor frequency-dependent hysteresis modeling method based on a phase-shift rational basis, comprising the following steps:

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

[0010] The expression of the output 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 the time. represents the phase difference between the input current and the output magnetic flux related to the angular frequency, Δ represents the error interference term of the output magnetic flux, and I represents the input current signal;

[0013] make c(w)=Δ, the expression of output flux signal B is rewritten as:

[0014]

[0015] Let a(w), b(w) and c(w) all be quintic polynomials as follows:

[0016]

[0017] Step 2: Decompose the hysteresis loop phenomenon between the input current and the output magnetic flux of the linear reluctance motor into a rising part and a falling part;

[0018] The rising and falling parts of the hysteresis loop are symmetrical about the origin. Assuming formula (2) is the output flux expression of the rising part, the output flux expression of the falling part is:

[0019]

[0020] Order I d = -I, B d = -B, then we get a d (w) = -a(w), b d (w) = b(w), c d (w) = -c(w);

[0021] Integrating the ascending and descending parts gives:

[0022]

[0023] in,

[0024]

[0025] Where, sign is the sign function;

[0026] Step 3: Use the least squares algorithm to estimate parameters;

[0027] First, the hysteresis loop model at a single frequency is modeled, and then the hysteresis loop model at multiple frequencies is modeled. According to the least squares method, the estimated values ​​of a(w), b(w), and c(w) are obtained respectively. and Finally, according to formula (6), the hysteresis loop model of the rising part is supplemented:

[0028]

[0029] A complete linear reluctance motor frequency-dependent hysteresis model is obtained.

[0030] Furthermore, when the hysteresis loop model under a single frequency is modeled in step 3, the angular frequency w of the input current signal I is fixed to w1, the input and output data of the linear reluctance motor are collected, and the data is decomposed into an ascending part and a descending part. Assuming that there are n sampling points in the ascending part, the input current is recorded as I = [I1, I2, ..., I n ] T , the output flux is recorded as B = [B1, B2, ... B n ] T , then formula (2) is written in matrix form as follows:

[0031]

[0032] The input matrix Denoted as X, the coefficient matrix [c(w1)b(w1)a(w1)] T Denoted as A, formula (8) is written as:

[0033] XA=B (11)

[0034] According to the least squares method, we get:

[0035]

[0036] In the formula, are the estimated values ​​of a(w1), b(w1), and c(w1) respectively.

[0037] Furthermore, when the hysteresis loop model is built at multiple frequencies in step 3, the angular frequencies w of the input current signal I are fixed to w2, w3, ..., w m , the polynomial coefficient a(w) is converted into matrix form as follows:

[0038]

[0039] make get:

[0040] WD a =H a (16)

[0041] According to the least squares method, we get the estimated value of the coefficient a(w) as follows:

[0042]

[0043] Right now get:

[0044]

[0045] Similarly, we get:

[0046]

[0047] In the formula, are the estimated values ​​of a(w), b(w), and c(w) respectively.

[0048] Furthermore, when the hysteresis loop model is built at multiple frequencies in step three, m≥6.

[0049] Furthermore, the input current signal and the output magnetic flux signal have the same angular frequency, different amplitudes and a phase difference.

[0050] Furthermore, the input current signal I=A0sin(wt), wherein A0 represents the amplitude of the input current.

[0051] Compared with the prior art, the beneficial effects of the present invention are as follows: the present invention first models the hysteresis characteristics with a phase shift rational basis according to the mechanism of the hysteresis loop, and then divides the hysteresis loop into a rising part and a falling part according to the hysteresis loop phenomenon, and models the two parts respectively, and utilizes mechanism analysis and phenomenon decomposition to accurately model the hysteresis nonlinearity of the linear reluctance motor, and estimates the model parameters using the least squares method, which can more accurately model the hysteresis phenomenon at the turning point, improve the accuracy of the hysteresis model of the linear reluctance motor, and at the same time increase the rate-related factor, so that the model output is correlated with the input frequency, which is in line with the hysteresis rate-related characteristics of the linear reluctance motor. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 It is a flowchart of the present invention;

[0053] Figure 2 is a physical picture of the linear reluctance motor motion platform established in the embodiment;

[0054] Figure 3 It is an implementation effect diagram obtained in the embodiment. DETAILED DESCRIPTION

[0055] The technical solution of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0056] like Figure 1 As shown, the linear reluctance motor frequency-dependent hysteresis modeling method based on phase-shift rational basis includes the following steps:

[0057] Step 1: Analyze the mechanism of the hysteresis loop between the input current and output flux of the linear reluctance motor and model it based on the phase shift rational basis. Specifically:

[0058] 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, where A0 represents the amplitude of the input current, A1 represents the amplitude of the output magnetic flux, w=2πf, w represents the angular frequency, f represents the frequency, and t represents the time. It represents the phase difference between the input current and the output flux related to the angular frequency, and Δ represents the error interference term of the output flux.

[0059] The expression of the output flux signal B is expanded as follows:

[0060]

[0061] Then, let c(w) = Δ, then the expression of the output flux signal B can be rewritten as:

[0062]

[0063] That is, the output flux signal B can be approximately regarded as a polynomial function of the input current signal I, and its polynomial coefficient is a polynomial function related to the angular frequency w. In order to ensure the modeling accuracy and reduce the modeling difficulty, a(w), b(w) and c(w) are all set to quintic polynomials as follows:

[0064]

[0065] In the formula, a0, a1, a2, a3, a4, a5 are the coefficients of a(w), b0, b1, b2, b3, b4, b5 are the coefficients of b(w), and c0, c1, c2, c3, c4, c5 are the coefficients of c(w).

[0066] Step 2: Decompose the hysteresis loop phenomenon between the input current and output magnetic flux of the linear reluctance motor into the rising part and the falling part. Specifically:

[0067] Ideally, the hysteresis loop is centrally symmetrical, so the rising part and the falling part are centrally symmetrical about the origin. If formula (2) is used as the output flux expression of the rising part, the output flux expression of the falling part can be set as:

[0068]

[0069] Since the ascending part and the descending part are symmetrical about the origin, let I d = -I, B d= -B, then we can get a d (w) = -a(w), b d (w) = b(w), c d (w) = -c(w).

[0070] Integrating the ascending and descending parts gives:

[0071]

[0072] in,

[0073]

[0074] Wherein, sign is a sign function, which is used to determine the rise and fall of the input current signal I.

[0075] Step 3: Use the least squares algorithm to estimate parameters, specifically:

[0076] 3.1. Model the hysteresis loop model under a single frequency, fix the angular frequency w of the input current signal I to w1, then collect the input and output data of the linear reluctance motor, and process the data to decompose it into the rising part and the falling part.

[0077] The least squares method is used to estimate the parameters of the rising part of the hysteresis loop under a single frequency. Assuming that there are n sampling points in the rising part, the sampled data is written in matrix form, and the input current is recorded as I = [I1, I2, ..., I n ] T , the output flux is recorded as B = [B1, B2, ... B n ] T , then formula (2) can be written in matrix form as follows:

[0078]

[0079] Right now:

[0080]

[0081] The input matrix Denoted as X, that is:

[0082]

[0083] The coefficient matrix [c(w1)b(w1)a(w1)] T Denoted as A, that is:

[0084]

[0085] Then formula (8) can be written as:

[0086] XA=B (11)

[0087] According to the least squares method, we get the estimated value of the coefficient matrix as follows:

[0088]

[0089] That is, the estimated values ​​of a(w1), b(w1), and c(w1) are:

[0090]

[0091] 3.2. Model the hysteresis loop model at multiple frequencies, and fix the angular frequency w of the input current signal I to w2, w3, ..., w m , where m ≥ 6, and the hysteresis loop model at a single frequency in step 3.1 is used to obtain as well as For the polynomial coefficients a(w), we can get the following system of equations:

[0092]

[0093] It can be converted into matrix form as follows:

[0094]

[0095] make You can get:

[0096] WD a =H a (16)

[0097] According to the least squares method, we get the estimated value of the coefficient a(w) as follows:

[0098]

[0099] Right now Then the estimate of a(w) It is expressed as:

[0100]

[0101] Similarly, we can get an estimate of b(w) and the estimate of c(w) It is expressed as:

[0102]

[0103] 3.3. According to formula (6), the hysteresis loop model of the rising part obtained in step 3.2 is supplemented to obtain the complete linear reluctance motor frequency-related hysteresis model as follows:

[0104]

[0105] Through the above process, a linear reluctance motor frequency-dependent multi-frequency hysteresis nonlinear model based on a phase-shift rational basis can be finally obtained.

[0106] Example

[0107] In this embodiment, a linear reluctance motor motion platform is established. Figure 2 As shown, the motion platform is composed of a voice coil motor, a grating ruler, an air floating guide rail, a reluctance motor and its supporting device. Among them, the linear reluctance motor is the modeling object of the method of the present invention, which is composed of an I-type mover and an E-type stator. The voice coil motor is responsible for positioning the I-type mover of the linear reluctance motor, the optical ruler is used for position signal measurement, the E-type stator is connected and fixed to the supporting device, and the I-type mover is connected to the guide rail and moves with the guide rail.

[0108] according to Figure 1 The process shown in the figure is as follows: first, a polynomial hysteresis loop model is established at a single frequency. After positioning the motion platform using a voice coil motor, the input and output data when the input current frequency f1 = 1 Hz is collected, and the rising part is extracted. According to step 3.1, the least squares algorithm is used for parameter identification to obtain the polynomial hysteresis loop model coefficients when the input angular frequency is w1. Complete the parameter identification of the polynomial hysteresis loop model at a single frequency; then, establish the polynomial hysteresis loop model at multiple frequencies, fix the input current frequency to f2 = 5Hz, f3 = 10Hz, f4 = 15Hz, f5 = 20Hz and f6 = 25Hz, collect its input and output data, and obtain the polynomial hysteresis loop model coefficient function at multiple frequencies according to step 3.2 Complete the polynomial hysteresis loop model parameter identification at multiple frequencies; finally, integrate the rising part and the falling part according to step 3.3 to obtain a complete linear reluctance motor frequency-related multi-frequency hysteresis nonlinear model, and implement the effect combined Figure 3 As shown, the obtained polynomial hysteresis model is compared with the linear reluctance motor, and the fitting is good at multiple frequencies such as f1, f2, f3, and f4, and the fitting effect at the turning point is also ideal, which verifies that the modeling method of the present invention has the advantages of rate-related characteristics and high model accuracy.

[0109] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other forms of assembly without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered in all respects as exemplary and non-restrictive, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations within the meaning and range of equivalents of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

[0110] In addition, it should be understood that although the present specification is described according to implementation modes, not every implementation mode contains only one independent technical solution. This description of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment may also be appropriately combined to form other implementation modes that can be understood by those skilled in the art.

Claims

1. A linear reluctance motor frequency-dependent hysteresis modeling method based on phase-shift rational basis, characterized by: The following steps are involved: Step 1: Analyze the mechanism of the hysteresis loop between the input current and output flux of the linear reluctance motor and model it based on the phase shift rational basis; The expression of the output 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 the time. represents the phase difference between the input current and the output magnetic flux related to the angular frequency, Δ represents the error interference term of the output magnetic flux, and I represents the input current signal; make c(w)=Δ, the expression of output flux signal B is rewritten as: Let a(w), b(w) and c(w) all be quintic polynomials as follows: Step 2: Decompose the hysteresis loop phenomenon between the input current and the output magnetic flux of the linear reluctance motor into a rising part and a falling part; The rising and falling parts of the hysteresis loop are symmetrical about the origin. Assuming formula (2) is the output flux expression of the rising part, the output flux expression of the falling part is: Let I d = -I, B d = -B, then we get a d (w) = -a(w), b d (w) = b(w), c d (w) = -c(w); Integrating the ascending and descending parts gives: in, Where, sign is the sign function; Step 3: Use the least squares algorithm to estimate parameters; First, the hysteresis loop model at a single frequency is modeled, and then the hysteresis loop model at multiple frequencies is modeled. According to the least squares method, the estimated values ​​of a(w), b(w), and c(w) are obtained respectively. and Finally, according to formula (6), the hysteresis loop model of the rising part is supplemented: A complete linear reluctance motor frequency-dependent hysteresis model is obtained.

2. The method for modeling linear reluctance motor frequency-dependent hysteresis based on phase-shift rational basis according to claim 1, characterized in that: When the hysteresis loop model under a single frequency is modeled in step 3, the angular frequency w of the input current signal I is fixed to w1, the input and output data of the linear reluctance motor are collected, and the data is decomposed into an ascending part and a descending part. Assuming that there are n sampling points in the ascending part, the input current is recorded as I=[I1,I2,...,I n ] T , the output flux is recorded as B = [B1, B2, ... B n ] T , then formula (2) is written in matrix form as follows: The input matrix Denoted as X, the coefficient matrix [c(w1)b(w1)a(w1)] T Denoted as A, formula (8) is written as: XA=B (11) According to the least squares method, we get: In the formula, are the estimated values ​​of a(w1), b(w1), and c(w1) respectively.

3. The method for modeling linear reluctance motor frequency-dependent hysteresis based on phase-shift rational basis according to claim 2, characterized in that: When the hysteresis loop model is built at multiple frequencies in step 3, the angular frequency w of the input current signal I is fixed to w2, w3, ..., w m , the polynomial coefficient a(w) is converted into matrix form as follows: make get: WD a =H a (16) According to the least squares method, we get the estimated value of the coefficient a(w) as follows: Right now get: Similarly, we get: In the formula, are the estimated values ​​of a(w), b(w), and c(w) respectively.

4. The method for modeling linear reluctance motor frequency-dependent hysteresis based on phase-shift rational basis according to claim 3, characterized in that: When the hysteresis loop model is built at multiple frequencies in step three, m≥6.

5. The method for modeling linear reluctance motor frequency-dependent hysteresis 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.

6. The method for modeling linear reluctance motor frequency-dependent hysteresis based on phase-shift rational basis according to claim 1, characterized in that: The input current signal I=A0sin(wt), wherein A0 represents the amplitude of the input current.

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