Hammerstein dynamic hysteresis model based on improved asymmetric generalized Prantl-Ishlinskii model

Abstract

The invention relates to a Hammerstein dynamic hysteresis model based on an improved asymmetric generalized Prantl-Ishlinskii model, and relates to the technical field of servo control, and the Hammerstein dynamic hysteresis model comprises the following steps: introducing a biexponential function of a deviation factor, and adding a first term in an envelope function; the method comprises the following steps: constructing an improved asymmetric generalized Prantl-Ishlinskii model according to a formula on the basis of a double-exponential function envelope with a deviation factor; an ARX model structure is adopted, and dynamic linear characteristics of the system are captured; the improved asymmetric generalized Prantl-Ishlinskii model is used as a static nonlinear part of the Hammerstein model, and the ARX dynamic model is used as a dynamic linear part of the Hammerstein model; and outputting a verification result and an application case according to the Hammerstein dynamic hysteresis model. According to the method, an existing asymmetric generalized Prantl-Ishlinskii model is improved, a double-exponential function envelope with a deviation factor is introduced, and asymmetric hysteresis characteristics can be described more accurately.

Description

Technical Field

[0001] The present invention relates to the technical field of servo control, and particularly to a Hammerstein dynamic hysteresis model based on an improved asymmetric generalized Prandtl-Ishlinskii model. Background Art

[0002] Piezoelectric actuators (PZTs) have characteristics such as high precision and fast response speed, and are widely used in fields such as precision positioning and vibration control. However, their inherent hysteretic nonlinearity significantly affects the control accuracy of the system.

[0003] So far, eliminating hysteresis mainly starts from three aspects: physical hysteresis models, operator hysteresis models, and differential hysteresis models. Among them, the classical Prandtl-Ishlinskii (CPI) model, as a kind of operator hysteresis model, has been widely favored due to its simple mathematical form and easy invertibility. However, the CPI model is applicable to symmetric hysteresis models and cannot effectively solve the asymmetry problem of PZTs. At the same time, the hysteretic nonlinearity of PZTs is mainly manifested as rate dependence of the output displacement on the input voltage signal frequency. As a phenomenological static model, the CPI model cannot accurately fit the hysteresis curve of PZTs.

[0004] The main difficulties in hysteresis compensation are as follows:

[0005] 1. Strong nonlinearity: The hysteresis effect usually has a highly nonlinear input-output relationship, and traditional linear methods are difficult to effectively model and compensate.

[0006] 2. Difficulty in inverting the inverse model: For some hysteresis models, there is no analytical inverse solution, and approximate compensation needs to be carried out through methods such as numerical optimization or neural networks.

[0007] 3. Frequency dependence: Piezoelectric materials exhibit rate-dependent characteristics of the hysteresis curve varying with the input frequency, and conventional static hysteresis models cannot accurately describe this phenomenon.

[0008] 4. Difficulty in parameter identification: Hysteresis models usually contain multiple parameters. The experimental measurement, curve fitting, and numerical optimization of these parameters require high-precision data and are greatly affected by noise. Summary of the Invention

[0009] The present invention aims to solve the technical problems in the prior art of strong hysteresis compensation nonlinearity, difficulty in inverting the inverse model, and inability to accurately describe frequency dependence, and provides a Hammerstein dynamic hysteresis model based on an improved asymmetric generalized Prandtl-Ishlinskii model.

[0010] To solve the above technical problems, the technical solution of the present invention is specifically as follows:

[0011] An Hammerstein dynamic hysteresis model based on an improved asymmetric generalized Prandtl-Ishlinskii model, comprising the following steps:

[0012] Step (1): Introduce a double-exponential function with a bias factor and add a first-order term to the envelope function to describe the envelope hysteresis curve;

[0013] Step (2): Based on the double-exponential function envelope with a bias factor, construct an improved asymmetric generalized Prandtl-Ishlinskii model according to the formula;

[0014] Step (3): Adopt an ARX model structure to capture the dynamic linear characteristics of the system and compensate for the deficiency of the hysteresis nonlinear model in describing dynamic behavior;

[0015] Step (4): Take the improved asymmetric generalized Prandtl-Ishlinskii model as the static nonlinear part of the Hammerstein model, and the ARX dynamic model as the dynamic linear part of the Hammerstein model;

[0016] Step (5): According to the output of the Hammerstein dynamic hysteresis model, verify the results and application cases to prove the effectiveness and practicability of the model.

[0017] In the above technical solution, step (1) is specifically as follows:

[0018] First, design a double-exponential function with a bias factor δ to describe the hysteresis envelope curve;

[0019] Then, add a first-order term to the envelope function; the formula of the envelope function is as follows:

[0020]

[0021] where: δ R represents the rising curve, δ L represents the falling curve; v is the input voltage data; a i and b i (i = 0, 1, 2, 3, 4) are the parameters to be identified.

[0022] In the above technical solution, step (2) is specifically as follows:

[0023] Introduce the generalized Play operator of the asymmetric envelope function to enhance the description ability of the asymmetric hysteresis loop;

[0024] The asymmetric generalized Prandtl-Ishlinskii model is expressed in the form of a finite number of generalized Play operators as:

[0025]

[0026] Where: y F is the output displacement of the AGPI model; t is the number of sampling points; N is the number of sample points of the input time series; q is a constant; y0 = [0, 0, 0,..., 0] T is the output initial value; is the output of the generalized Play operator; is the reciprocal of the input signal, represents the rising curve δ R , represents the falling curve δ L ;

[0027] Threshold r i and weight w i The formulas are:

[0028] r i = αi

[0029]

[0030] where α, ρ, and κ are parameters to be identified.

[0031] In the above technical solution, step (3) is specifically:

[0032] The dynamic linear module of the Hammerstein model is described by the ARX model as:

[0033] A(z)y(t) = B(z)u(t) + e(t)

[0034] where: z-1 is the unit delay operator; e(t) is the error; A(z) = a0 + a1z -1 + a2z -2 +... + a n z -n ; B(z) = b0 + b1z -1 + b2z -2 +... + b m z -m ; m and n are the orders of the numerator and denominator respectively, and m ≤ n;

[0035] The transfer function is:

[0036]

[0037] In the above technical solution, step (4) is specifically:

[0038] First, identify the parameters in the improved asymmetric generalized Prandtl-Ishlinskii model based on the actual data of the piezoelectric platform. The ARX dynamic model uses a swept-frequency signal or a mixed signal to obtain the transfer function of the ARX model.

[0039] Next, cascade the transfer functions of the identified improved asymmetric generalized Prandtl-Ishlinskii model and the ARX model to complete the construction of the Hammerstein dynamic hysteresis model.

[0040] In the above technical solution, step (5) is specifically as follows:

[0041] The input signal passes through the piezoelectric platform to obtain the actual output, and the input signal passes through the Hammerstein dynamic hysteresis model to obtain the experimental output. Compare the actual output with the experimental output results to prove the effectiveness and practicability of the present invention.

[0042] The present invention has the following beneficial effects:

[0043] The Hammerstein dynamic hysteresis model of the present invention based on the improved asymmetric generalized Prandtl-Ishlinskii model improves the existing asymmetric generalized Prandtl-Ishlinskii model, introduces a double-exponential function envelope with a deviation factor, and can more accurately describe the asymmetric hysteresis characteristics.

[0044] The Hammerstein dynamic hysteresis model of the present invention based on the improved asymmetric generalized Prandtl-Ishlinskii model adds a first-order term to the envelope function, makes up for the problem that the exponential term decays too fast, and improves the fitting accuracy of the approximately linear segment of the hysteresis curve, and is applicable to a wider range of asymmetric hysteresis systems.

[0045] The Hammerstein dynamic hysteresis model of the present invention based on the improved asymmetric generalized Prandtl-Ishlinskii model uses the Hammerstein structure to separate the nonlinear part from the dynamic linear part, making up for the disadvantages of the traditional PI model and its improved models in having low fitting accuracy for the hysteresis phenomenon of rate-dependent characteristics and lacking generality.

[0046] The Hammerstein dynamic hysteresis model of the present invention based on the improved asymmetric generalized Prandtl-Ishlinskii model improves the calculation efficiency of the dynamic characteristics and the overall accuracy of the model through the improved envelope function and the Hammerstein structure, and is especially applicable to application scenarios with high performance requirements such as piezoelectric ceramics. Description of the Drawings

[0047] The present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0048] Figure 1 It is a schematic diagram of the overall framework of the Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model of the present invention.

[0049] Figure 2 It is a schematic diagram of the series flow chart of the Hammerstein model.

[0050] Figure 3 It is a schematic diagram of the comparison between experimental data and actual data. Specific embodiments

[0051] The inventive concept of the present invention is as follows:

[0052] Aiming at the weak description ability of the existing classical CPI model and its improved models for complex non-linear behaviors and the difficulty in more comprehensively and accurately describing the characteristics of complex hysteresis systems, the present invention proposes a Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii (AGPI) model.

[0053] The Hammerstein dynamic hysteresis model of the present invention based on the improved asymmetric generalized Prandtl-Ishlinskii model introduces and designs a double-exponential function with a deviation factor to describe the envelope curve, and adds a first-order term to the envelope function, which solves the asymmetry problem while making up for the problem of too fast decay of the exponential term and improving the fitting accuracy of the approximate linear segment of the hysteresis curve.

[0054] The present invention proposes to connect the improved asymmetric generalized Prandtl-Ishlinskii model in series with the ARX model to form a Hammerstein dynamic hysteresis model. The series-connected model describes the dynamic characteristics more accurately than the improved asymmetric generalized Prandtl-Ishlinskii model, breaks the limitation that the traditional hysteresis model can only be limited to solving one aspect of the hysteresis problem, and improves the fitting accuracy.

[0055] The present invention will be described in detail below in conjunction with the accompanying drawings.

[0056] The Hammerstein dynamic hysteresis model of the present invention based on the improved asymmetric generalized Prandtl-Ishlinskii model has four main components: the envelope design of the double-exponential function with a deviation factor, the improved asymmetric generalized Prandtl-Ishlinskii model, the ARX model, and the Hammerstein dynamic hysteresis model.

[0057] The overall architecture of the Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model of the present invention is as follows Figure 1 shown, and includes the following steps:

[0058] Step (1): Introduce a double-exponential function with a bias factor and add a first-order term to the envelope function to describe the envelope hysteresis curve;

[0059] Step (2): Based on the double-exponential function envelope with a bias factor, construct an improved asymmetric generalized Prandtl-Ishlinskii model according to the formula;

[0060] Step (3): Adopt the ARX model structure to capture the dynamic linear characteristics of the system and compensate for the deficiency of the hysteresis nonlinear model in describing dynamic behavior;

[0061] Step (4): Take the improved asymmetric generalized Prandtl-Ishlinskii model as the static nonlinear part of the Hammerstein model and the ARX dynamic model as the dynamic linear part of the Hammerstein model;

[0062] Step (5): According to the output of the Hammerstein dynamic hysteresis model, verify the results and application cases to prove the effectiveness and practicality of the model.

[0063] Specifically, as Figure 1 shown:

[0064] First of all, in step (1), a double-exponential function with a bias factor δ is designed to describe the hysteresis envelope curve. The introduction of the bias factor can better capture the asymmetric hysteresis characteristics.

[0065] Then, add a first-order term to the envelope function to make up for the problem that the exponential term decays too fast, and at the same time improve the fitting accuracy of the approximately linear segment of the hysteresis curve. The formula of the envelope function is as follows:

[0066]

[0067] where: δ R represents the rising curve, δ L represents the falling curve; v is the input voltage data; a i and b i (i = 0, 1, 2, 3, 4) are parameters to be identified.

[0068] For the construction of the improved asymmetric generalized Prandtl-Ishlinskii model in step (2), the present invention enhances the description ability of asymmetric hysteresis loops by introducing a generalized Play operator with an asymmetric envelope function. The asymmetric generalized Prandtl-Ishlinskii model can be expressed in the form of a finite number of generalized Play operators as follows:

[0069]

[0070] where: y F is the output displacement of the AGPI model; t is the number of sampling points; N is the number of input time series sample points; q is a constant; y0 = [0, 0, 0,..., 0] T is the output initial value; is the output of the generalized Play operator; is the reciprocal of the input signal, represents the rising curve δ R , represents the falling curve δ L .

[0071] The threshold r i and the weight w i The formulas are:

[0072] r i = αi

[0073]

[0074] where α, ρ, and κ are parameters to be identified.

[0075] Step (3) is to establish an ARX dynamic model to describe the dynamic linear characteristics of the system. The dynamic linear module of the Hammerstein model is generally described by the ARX model as:

[0076] A(z)y(t) = B(z)u(t) + e(t)

[0077] where: z-1 is the unit delay operator; e(t) is the error; A(z) = a0 + a1z -1 + a2z -2 +... + a n z -n ; B(z) = b0 + b1z -1 + b2z -2 +... + b m z -m ; m and n are the orders of the numerator and denominator respectively, and m ≤ n. The transfer function is:

[0078]

[0079] Step (4) is to construct a Hammerstein dynamic hysteresis model, taking the improved asymmetric generalized Prandtl-Ishlinskii model as the static nonlinear part of the Hammerstein model and the ARX dynamic model as the dynamic linear part of the Hammerstein model. The series connection process is as Figure 2 shown. First, the parameters in the improved asymmetric generalized Prandtl-Ishlinskii model are identified according to the actual data of the piezoelectric platform. The ARX dynamic model uses a swept-frequency signal or a mixed signal to obtain the transfer function of the ARX model. Then, the identified transfer function of the improved asymmetric generalized Prandtl-Ishlinskii model is connected in series with the transfer function of the ARX model to complete the construction of the Hammerstein dynamic hysteresis model.

[0080] In step (5), based on step (4), the input signal passes through the piezoelectric platform to obtain the actual output, and the input signal passes through the Hammerstein dynamic hysteresis model to obtain the experimental output. By comparing the actual output with the experimental output results, the effectiveness and practicability of the present invention are proved.

[0081] By comparing the experimental data of the piezoelectric platform with the actual data, as Figure 3 shown, the Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model of the present invention has a higher fitting accuracy than the existing methods.

[0082] The Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model of the present invention improves the existing asymmetric generalized Prandtl-Ishlinskii model, introduces a double-exponential function envelope with a deviation factor, and can more accurately describe the asymmetric hysteresis characteristics.

[0083] The Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model of the present invention adds a first-order term to the envelope function, compensates for the problem of too fast decay of the exponential term, and improves the fitting accuracy of the approximately linear segment of the hysteresis curve, making it applicable to a wider range of asymmetric hysteresis systems.

[0084] The Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model of the present invention uses the Hammerstein structure to separate the nonlinear part from the dynamic linear part, compensating for the disadvantages of the traditional PI model and its improved models, such as low fitting accuracy for the hysteresis phenomenon of rate-dependent characteristics and lack of generality.

[0085] The Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model of the present invention improves the calculation efficiency of dynamic characteristics and the overall accuracy of the model through the improved envelope function and Hammerstein structure, and is particularly suitable for application scenarios with high performance requirements such as piezoelectric ceramics.

[0086] Obviously, the above embodiments are merely examples given for clear illustration and are not limitations on the implementation manners. For those of ordinary skill in the art, other different forms of changes or variations can be made based on the above description. It is not necessary and impossible to exhaustively list all the implementation manners here. And the obvious changes or variations derived therefrom still fall within the protection scope of the present invention.

Claims

1. A Hammerstein dynamic hysteresis model based on an improved asymmetric generalized Prandtl-Ishlinskii model, characterized in that, It includes the following steps: Step (1): Introduce a double-exponential function with a deviation factor and add a first-order term to the envelope function to describe the envelope hysteresis curve; Step (2): Based on the double-exponential function envelope with a deviation factor, construct an improved asymmetric generalized Prandtl-Ishlinskii model according to the formula; Step (3): Adopt the ARX model structure to capture the dynamic linear characteristics of the system and compensate for the deficiency of the hysteresis nonlinear model in describing dynamic behavior; Step (4): Take the improved asymmetric generalized Prandtl-Ishlinskii model as the static nonlinear part of the Hammerstein model, and the ARX dynamic model as the dynamic linear part of the Hammerstein model; Step (5): According to the output of the Hammerstein dynamic hysteresis model, verify the results and application cases to prove the effectiveness and practicability of the model.

2. The Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model according to claim 1, wherein Specifically, step (1) is as follows: First, design a double-exponential function with a deviation factor δ to describe the hysteresis envelope curve; Then, add a first-order term to the envelope function; the formula of the envelope function is as follows: Where: δ R represents the rising curve, and δ L represents the falling curve; v is the input voltage data; a i and b i (i = 0, 1, 2, 3, 4) are the parameters to be identified.

3. The Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model according to claim 1, wherein Specifically, step (2) is as follows: Introduce the generalized Play operator of the asymmetric envelope function to enhance the description ability of the asymmetric hysteresis loop; The asymmetric generalized Prandtl-Ishlinskii model is expressed in the form of a finite number of generalized Play operators as: where: y F is the output displacement of the AGPI model; t is the number of sampling points; N is the number of sample points of the input time series; q is a constant; y0 = [0, 0, 0,..., 0] T is the initial output value; is the output of the generalized Play operator; is the reciprocal of the input signal, represents the rising curve δ R , represents the falling curve δ L ; Threshold r i and weight w i The formula is: r i ＝αi where α, ρ, and κ are parameters to be identified.

4. The Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model according to claim 1, characterized in that, Specifically, step (3) is as follows: The dynamic linear module of the Hammerstein model is described by the ARX model as: A(z)y(t)=B(z)u(t)+e(t) where: z-1 is a unit delay operator; e(t) is the error; A(z) = a0 + a1z -1 + a2z -2 +... + a n z -n ; B(z) = b0 + b1z -1 + b2z -2 +... + b m z -m ; m and n are the orders of the numerator and denominator respectively, and m ≤ n; The transfer function is:

5. The Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model according to claim 1, characterized in that, Specifically, step (4) is as follows: First, identify the parameters in the improved asymmetric generalized Prandtl-Ishlinskii model according to the actual data of the piezoelectric platform, and the ARX dynamic model uses a swept-frequency signal or a mixed signal to obtain the transfer function of the ARX model; Next, connect the transfer functions of the identified improved asymmetric generalized Prandtl-Ishlinskii model and the ARX model in series to complete the construction of the Hammerstein dynamic hysteresis model.

6. The Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model according to claim 1, characterized in that, Specifically, step (5) is as follows: The input signal passes through the piezoelectric platform to obtain the actual output, and the input signal passes through the Hammerstein dynamic hysteresis model to obtain the experimental output. Compare the actual output with the experimental output results to prove the effectiveness and practicability of the invention.

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