Hammerstein dynamic hysteresis model based on improved asymmetric generalized prandtl- ishlinskii model

Abstract

The application relates to a Hammerstein dynamic hysteresis model based on an improved asymmetric generalized Prandtl-Ishlinskii model, relates to the technical field of servo control, and comprises the following steps: introducing a double exponential function of a deviation factor and adding a linear term in an envelope function; based on the double exponential function envelope with the deviation factor, an improved asymmetric generalized Prandtl-Ishlinskii model is constructed according to a formula; an ARX model structure is adopted to capture the dynamic linear characteristics of a system; the improved asymmetric generalized Prandtl-Ishlinskii model is taken as a static nonlinear part of a Hammerstein model, and an ARX dynamic model is taken as a dynamic linear part of the Hammerstein model; and the output of the Hammerstein dynamic hysteresis model is used to verify the results and application cases. The application improves the existing asymmetric generalized Prandtl-Ishlinskii model, introduces the double exponential function envelope with the deviation factor, and can more accurately describe the asymmetric hysteresis characteristics.

Description

Technical Field

[0001] This invention relates to the field of servo control technology, and in particular to a Hammerstein dynamic hysteresis model based on an improved asymmetric generalized Prandtl-Ishlinskii model. Background Technology

[0002] Piezoelectric actuators (PZTs) are widely used in precision positioning and vibration control due to their high precision and fast response. However, their inherent hysteresis nonlinearity significantly affects the control accuracy of the system.

[0003] To date, hysteresis elimination has primarily focused on three aspects: physical hysteresis models, operator hysteresis models, and differential hysteresis models. Among these, the classic Prandtl-Ishlinskii (CPI) model, as a type of operator hysteresis model, is widely favored due to its simple mathematical form and ease of inversion. However, the CPI model is suitable for symmetric hysteresis models and cannot effectively address the asymmetry problem of PZTs. Furthermore, the hysteresis nonlinearity of PZTs mainly manifests as a rate-dependent relationship between the output displacement and the frequency of the input voltage signal. As a phenomenological static model, the CPI model cannot accurately fit the hysteresis curve of the PZT.

[0004] The main difficulties in delay compensation are:

[0005] 1. Strong nonlinearity: Hysteresis effects usually have highly nonlinear input-output relationships, which are difficult to effectively model and compensate for using traditional linear methods.

[0006] 2. Difficulty in inverting the inverse model: Some hysteresis models do not have analytical inverse solutions and require approximate compensation through numerical optimization or neural networks.

[0007] 3. Frequency dependence: Piezoelectric materials exhibit rate-dependent characteristics in their hysteresis curves as the input frequency changes. Conventional static hysteresis models cannot accurately describe this phenomenon.

[0008] 4. High 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] This invention aims to solve the technical problems of strong nonlinearity in hysteresis compensation, difficulty in inverting the inverse model, and inability to accurately describe frequency correlation in the prior art, and provides a Hammerstein dynamic hysteresis model based on an improved asymmetric generalized Prandtl-Ishlinskii model.

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

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

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

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

[0014] Step (3): Use the ARX model structure to capture the dynamic linear characteristics of the system and compensate for the shortcomings of the hysteresis nonlinear model in describing dynamic behavior.

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

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

[0017] In the above technical solution, step (1) specifically includes:

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

[0019] Then, add a linear term to the envelope function; the formula for the envelope function is as follows:

[0020]

[0021] Where: δ R Indicates an ascending curve, δ L This represents the descent 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) specifically includes:

[0023] The generalized Play operator with asymmetric envelope function is introduced to enhance the ability to describe asymmetric hysteresis loops;

[0024] The asymmetric generalized Prandtl-Ishlinskii model can be expressed in terms of a finite number of generalized Play operators as follows:

[0025]

[0026] Where: y F y0 represents 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 It outputs the initial value; The output of the generalized Play operator; The reciprocal of the input signal. Indicates the rising curve δ R , The descending curve δ L ;

[0027] threshold r i and weight w i The formula is:

[0028] r i =αi

[0029]

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

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

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

[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) specifically includes:

[0038] First, the parameters in the improved asymmetric generalized Prandtl-Ishlinskii model are identified based on the actual data of the piezoelectric platform. The transfer function of the ARX dynamic model is obtained by using a swept frequency signal or a mixed signal.

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

[0040] In the above technical solution, step (5) specifically includes:

[0041] The input signal is passed through a piezoelectric platform to obtain the actual output, and the input signal is passed through a Hammerstein dynamic hysteresis model to obtain the experimental output. The comparison between the actual output and the experimental output results proves the effectiveness and practicality of the invention.

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

[0043] 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 by introducing a double exponential function envelope with a bias factor, which can more accurately describe the asymmetric hysteresis characteristics.

[0044] The Hammerstein dynamic hysteresis model of this invention, based on the improved asymmetric generalized Prandtl-Ishlinskii model, adds a linear term to the envelope function, which compensates for the problem of the exponential term decaying too quickly and improves the fitting accuracy of the approximate linear segment of the hysteresis curve, making it applicable to a wider range of asymmetric hysteresis systems.

[0045] The Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model of this invention uses the Hammerstein structure to separate the nonlinear part from the dynamic linear part, which makes up for the shortcomings of the traditional PI model and its improved model in that the fitting accuracy of the hysteresis phenomenon of rate correlation characteristics is not high and it lacks universality.

[0046] The Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model of this invention improves the computational efficiency of dynamic characteristics and the overall accuracy of the model through the improved envelope function and Hammerstein structure, and is especially suitable for high-performance applications such as piezoelectric ceramics. Attached Figure Description

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

[0048] Figure 1 This 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 This is a schematic diagram of the Hammerstein model's serial flow.

[0050] Figure 3 This is a diagram showing the comparison between experimental data and actual data. Detailed Implementation

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

[0052] To address the limitations of existing classical CPI models and their improved versions in describing complex nonlinear behavior and their inability to comprehensively and accurately describe the characteristics of complex hysteresis systems, this invention proposes a Hammerstein dynamic hysteresis model based on an improved asymmetric generalized Prandtl-Ishlinskii (AGPI) model.

[0053] The Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model of this invention introduces a double exponential function with a bias factor to describe the envelope curve, and adds a first-order term to the envelope function. This solves the asymmetry problem, compensates for the problem of the exponential term decaying too quickly, and improves the fitting accuracy of the approximate linear segment of the hysteresis curve.

[0054] This invention proposes to concatenate an improved asymmetric generalized Prandtl-Ishlinskii model with an ARX model to form Hammerstein's dynamic hysteresis model. The concatenated model provides a more accurate description of dynamic characteristics compared to the improved asymmetric generalized Prandtl-Ishlinskii model, breaking the limitation of traditional hysteresis models that can only address one aspect of the hysteresis problem and improving fitting accuracy.

[0055] The present invention will now be described in detail with reference to the accompanying drawings.

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

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

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

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

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

[0061] Step (4): The improved asymmetric generalized Prandtl-Ishlinskii model is used as the static nonlinear part of the Hammerstein model, and the ARX dynamic model is used as the dynamic linear part of the Hammerstein model.

[0062] Step (5): Based on 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, such as Figure 1 As shown:

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

[0065] Then, a linear term is added to the envelope function to compensate for the excessively rapid decay of the exponential term, while also improving the fitting accuracy to the approximately linear segment of the hysteresis curve. The formula for the envelope function is as follows:

[0066]

[0067] Where: δ R Indicates an ascending curve, δ L This represents the descent curve; v is the input voltage data; a i and b i (i = 0, 1, 2, 3, 4) are the parameters to be identified.

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

[0069]

[0070] Where: y F y0 represents 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 It outputs the initial value; The output of the generalized Play operator; The reciprocal of the input signal. Indicates the rising curve δ R , The descending curve δ L .

[0071] threshold r i and weight w i The formula is:

[0072] r i =αi

[0073]

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

[0075] Step (3) involves establishing an ARX dynamic model to describe the dynamic linearity of the system. The dynamic linear module of the Hammerstein model is generally described using an ARX model:

[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 Let m and n be the orders of the numerator and denominator, respectively, and m ≤ n. The transfer function is:

[0078]

[0079] Step (4) involves constructing the Hammerstein dynamic hysteresis model, using 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. The cascaded process is as follows: Figure 2 As shown, firstly, the parameters in the improved asymmetric generalized Prandtl-Ishlinskii model are identified based on actual data from the piezoelectric platform. The transfer function of the ARX dynamic model is obtained using a swept-frequency signal or a mixed signal. Then, the identified improved asymmetric generalized Prandtl-Ishlinskii model and the transfer function of the ARX model are concatenated to complete the construction of the Hammerstein dynamic hysteresis model.

[0080] Step (5) Based on step (4), the input signal is passed through the piezoelectric platform to obtain the actual output, and the input signal is passed through the Hammerstein dynamic hysteresis model to obtain the experimental output. The actual output is compared with the experimental output to prove the effectiveness and practicality of the invention.

[0081] By comparing experimental data from the piezoelectric platform with actual data, such as Figure 3 As shown, the Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model of this invention has a higher fitting accuracy than 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 by introducing a double exponential function envelope with a bias factor, which can more accurately describe the asymmetric hysteresis characteristics.

[0083] The Hammerstein dynamic hysteresis model of this invention, based on the improved asymmetric generalized Prandtl-Ishlinskii model, adds a linear term to the envelope function, which compensates for the problem of the exponential term decaying too quickly and improves the fitting accuracy of the approximate 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 this invention uses the Hammerstein structure to separate the nonlinear part from the dynamic linear part, which makes up for the shortcomings of the traditional PI model and its improved model in that the fitting accuracy of the hysteresis phenomenon of rate correlation characteristics is not high and it lacks universality.

[0085] The Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model of this invention improves the computational efficiency of dynamic characteristics and the overall accuracy of the model through the improved envelope function and Hammerstein structure, and is especially suitable for high-performance applications such as piezoelectric ceramics.

[0086] Obviously, the above embodiments are merely illustrative examples for clear explanation and are not intended to limit the implementation. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations here. However, obvious variations or modifications derived therefrom are still within the scope of protection of this invention.

Claims

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

2. The Hammerstein dynamic hysteresis model based on the improved asymmetric generalized Prandtl-Ishlinskii model according to claim 1, characterized in that, Step (1) is as follows: First, design with a bias factor The double exponential function is used to describe the hysteresis envelope curve; Then, add a linear term to the envelope function; the formula for the envelope function is as follows: in: Indicates an ascending curve. Indicates a descending curve; Input voltage data; and These 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, characterized in that, Step (2) specifically involves: The generalized Play operator with asymmetric envelope function is introduced to enhance the ability to describe asymmetric hysteresis loops; The asymmetric generalized Prandtl-Ishlinskii model can be expressed in terms of a finite number of generalized Play operators as follows: in: This represents the output displacement of the AGPI model; This represents the number of sampling points; The number of time series sample points input; It is a constant; It outputs the initial value; The output of the generalized Play operator; The reciprocal of the input signal. Indicates an ascending curve , Indicates a descending curve ; threshold and weight The formula is: in, , and These are the 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, Step (3) is as follows: The dynamic linear module of the Hammerstein model is described using the ARX model as follows: in: Delay operator for units; For error; ; ; and Let be the orders of the numerator and denominator, respectively, and ; 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, Step (4) is as follows: First, the parameters in the improved asymmetric generalized Prandtl-Ishlinskii model are identified based on the actual data of the piezoelectric platform. The transfer function of the ARX dynamic model is obtained by using a swept frequency signal or a mixed signal. Next, the improved asymmetric generalized Prandtl-Ishlinskii model and the transfer function of the ARX model are concatenated 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, Step (5) is as follows: The input signal is passed through a piezoelectric platform to obtain the actual output, and the input signal is passed through a Hammerstein dynamic hysteresis model to obtain the experimental output. The actual output is then compared with the experimental output.

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