Hysteresis modeling and parameter identification method for piezoelectric ceramic actuator based on generalized circular base B-W model

Abstract

The invention provides a piezoelectric ceramic actuator hysteresis modeling and parameter identification method based on a generalized circle base B-W model, and the method comprises the steps: obtaining an output displacement generated by a piezoelectric ceramic actuator under an input voltage, building a hysteresis model according to the input voltage and the output displacement through employing the generalized circle base B-W model, and describing the hysteresis nonlinear characteristics; the relation between the hysteresis characteristic and the piezoelectric ceramic actuator is better reflected; and parameter identification is carried out on the generalized circular base Bouc-Wen model by adopting a hierarchical genetic algorithm embedded with a particle swarm, so that the function of high-precision modeling of hysteresis of the piezoelectric ceramic actuator is realized. According to the method, the brand-new generalized circle-based Bouc-Wen model is constructed, and the model is simple in structure and high in modeling precision; according to the method, parameter identification is carried out on the generalized circular base Bouc-Wen model by adopting the hierarchical genetic algorithm embedded with the particle swarm, the problem of multi-parameter identification is solved, the observability is high, and the algorithm complexity is reduced while the identification accuracy is not influenced.

Description

Technical Field

[0001] The invention belongs to the technical field of piezoelectric actuators, and in particular relates to a hysteresis modeling and parameter identification method of a piezoelectric ceramic actuator based on a generalized circular basis BW model. Background Art

[0002] Smart materials represented by piezoelectric ceramics, magnetostrictive materials and shape memory alloys are used in nanopositioning systems and inertial sensors in accelerometers due to their excellent properties such as rapid expansion and short response time. Piezoelectric micro-nano platform is a nano-scale manufacturing equipment that achieves high-precision positioning. It has excellent properties such as no wear, high resolution, high efficiency, high strength, and fast response speed. It is widely used in positioning systems of micro-electromechanical systems, high-speed atomic force microscopes, ultra-precision machine tools and micro-machining platforms.

[0003] Piezoelectric materials have nonlinear and dynamic effects in the input-output relationship. The nonlinearity of piezoelectric materials includes creep nonlinearity and hysteresis nonlinearity. Among them, the hysteresis nonlinearity of piezoelectric materials can cause positioning errors of up to 10% to 15% of the stroke in an open loop, which is the main factor limiting the application of piezoelectric ceramics in a wider range of fields.

[0004] At present, hysteresis nonlinear models are mainly divided into three categories: physical models, intelligent computer models and phenomenon models. Among them, physical modeling is to obtain an accurate model based on the specific physical characteristics of the object. Representative models include Jiles-Atherton model and Maxwell model; phenomenon model focuses on the relationship between the input and output of hysteresis, not the internal physical process, and finds a suitable mathematical expression to fit hysteresis. Common phenomenon models include Preisach model and Bouc-Wen model; intelligent computer model is a data-based model that uses neural network or intelligent algorithm to describe nonlinear relationship. Representative models include neural network model and fuzzy logic hysteresis model. The physical model has a complex structure, which brings difficulties to the design of compensator; the complex model calculation and identification process of intelligent computer model brings difficulties to the engineering application of the model; while the phenomenon model has high modeling accuracy, simple structure and easy implementation. Summary of the invention

[0005] The technical problem to be solved by the present invention is to provide a piezoelectric ceramic actuator hysteresis modeling and parameter identification method based on a generalized circular basis BW model, which is used for high-precision modeling of piezoelectric ceramic actuator hysteresis.

[0006] The technical solution adopted by the present invention to solve the above technical problems is: a piezoelectric ceramic actuator hysteresis modeling and parameter identification method based on a generalized circular basis BW model, comprising the following steps:

[0007] S1: Use the piezoelectric positioning platform to collect the input and output data of the piezoelectric actuator and establish a hysteresis model for the nonlinear hysteresis of piezoelectric ceramics;

[0008] S2: The classic Bouc-Wen model is improved by normalization, generalization and reshaping, and a generalized circle-based Bouc-Wen model is established;

[0009] S3: designing a parameter identification strategy of a hierarchical genetic algorithm embedded in a particle swarm according to the hysteresis model, and using a hierarchical genetic algorithm based on an embedded particle swarm to identify parameters of a generalized circular basis Bouc-Wen model to obtain a target model.

[0010] According to the above scheme, in step S1, the specific steps are:

[0011] The input and output data of the piezoelectric actuator are collected on a piezoelectric micro-motion platform composed of a piezoelectric amplifier and a piezoelectric actuator, and a hysteresis model is established using a modeling method based on the Bouc-Wen model.

[0012] Assume u(t) is the input signal of the hysteresis system, H(u,t) is the output signal of the hysteresis system; α, k, D, A, β, γ, n are model parameters, the output of the system includes the elastic term αku(t) composed of the input signal and the pure hysteresis term (1-α)Dkh(t), h(t) is the hysteresis auxiliary variable; the relationship between the amplitude and direction of the output signal is characterized by constructing a differential equation, which is expressed by the standard Bouc-Wen model as follows:

[0013] H(u,t)=αku(t)+(1-α)Dkh(t)

[0014]

[0015] Furthermore, in step S2, the specific steps are:

[0016] S21: Let α, k, D, A, β, γ, and n be the unnormalized model parameters, n is the normalized model parameter, is an equivalent replacement for the pure lag term, is a normalized pure lag term; parameter redundancy is removed by the following transformation:

[0017]

[0018] Assume y(t) is the output signal of the hysteresis system, parameter k is the weight coefficient, B and C are the model parameters of Bouc-Wen; then the expression of the normalized Bouc-Wen model is:

[0019]

[0020] S22: Let h be the pure lag auxiliary variable, u be the input signal, is the derivative of the input signal, is the control function of the Bouc-Wen model. The Bouc-Wen model is based on u, The sign of h divides the image into 6 phases; the normalized Bouc-Wen model is generalized and expanded, and the expression of the generalized Bouc-Wen model is obtained as follows:

[0021]

[0022] S23: Reshape the generalized Bouc-Wen model; let h1 represent the lag term after rotation, and u1 represent the input signal after rotation; introduce circular differential and rotation transformation into the reshape of the Bouc-Wen model, and obtain the elliptic differential and rotation transformation expressions of the generalized circular basis Bouc-Wen model as follows:

[0023]

[0024] Let γ represent the square of the ratio of the major axis to the minor axis of the ellipse; combining the above formula, the differential equation of the rotating ellipse is:

[0025]

[0026] The control function is expanded to 5 dimensions, representing the five parameters of the generalized circular basis Bouc-Wen model:

[0027] ψ (t) =[ψ1ψ2ψ3ψ4ψ5] T ;

[0028] Assume w (t) is an auxiliary variable, h (t) is the pure lag term after replacing the differential; combining the rotational elliptic differential equation and the generalized Bouc-Wen model, the basic formula of the generalized circular basis Bouc-Wen model is:

[0029]

[0030] Furthermore, in step S22, β1, β2, β3, β4, β5, and β6 are six parameters to be determined; then the control function The expression is:

[0031]

[0032] Furthermore, in step S23, according to the division of the expectation phase and the stable phase, the term tT / 2 is introduced into the control function, and the control function expression is:

[0033]

[0034] The control function divides the model space into six phases, the first and second phases act on the model parameters in the startup phase, and the third, fourth, fifth, and sixth phases act on the model parameters in the stable phase.

[0035] Furthermore, in step S23, y0 and y1 are respectively assumed to represent the displacement of the hysteresis curve at time T / 2 and time T, and λ is the amplitude of the input sine wave; in order to highlight the local characteristics of the piezoelectric actuator, k is fixed as a constant to satisfy the expression:

[0036] y0-y1=2λk;

[0037] Introduce axis degeneration to increase the expressive power of the model:

[0038]

[0039] Let x0 represent the horizontal coordinate of the geometric center of the model, y0 represent the vertical coordinate of the geometric center of the model, α represent the rotation deviation angle, r represent the axis ratio, σ represent the axis ratio degradation coefficient; define a linear transformation to enhance the observability of the model:

[0040] Γ=Ωβ T ;

[0041] Γ=[x0 y0 α γ σ];

[0042] The length of each column vector is 6, which corresponds one-to-one to the control parameters of the 6 phases;

[0043] The interpretable Γ matrix is ​​the parameter matrix of the generalized circular basis Bouc-Wen model; the matrix Ω is expressed as:

[0044]

[0045] Furthermore, in step S23, let y (t) represents the output displacement, u (t) Indicates input voltage, h (t) 、w (t) is the hysteresis auxiliary variable; ψ (t) is a 5-dimensional control function, and the matrix Γ is a 6×5 parameter moment; the expression of the generalized circular basis Bouc-Wen model is:

[0046] y(t)=ku (t) +h (t)

[0047]

[0048] Γ=Ωβ T .

[0049] According to the above scheme, in step S3, the specific steps are:

[0050] S31: Divide the parameters into several levels according to their characteristics, and give different constraints and calculation processes to different levels;

[0051] In the hierarchical genetic algorithm embedded in particle swarm, the genetic algorithm is set as the outer algorithm, the particle swarm algorithm with crossover factor is set as the middle algorithm, and the binary search algorithm is set as the inner algorithm;

[0052] The actual input-output relationship of the piezoelectric actuator is the superposition of hysteresis nonlinearity and dynamic effect. The dynamic effect is equivalent to a sine wave with amplitude attenuation to eliminate the dynamic effect.

[0053] Redefine the outer parameters x0, y0, and α; let (x1, y1) represent the starting position of the model; (x2, y2) represent the ending position of the model; (x′0, y′0) represent the ideal zero point of the model after redefinition; (x0, y0) represent the geometric center point of the model before redefinition; (X, Y) represent the geometric center point of the model after redefinition; the redefinition expression is as follows:

[0054]

[0055] After redefinition, the search range of parameters X and Y is:

[0056] S={(X,Y)|X∈[-1,0], Y∈[0,1]};

[0057] S32: adding the crossover module of the genetic algorithm to the particle swarm algorithm to improve the convergence speed of the particle swarm algorithm;

[0058] S33: Parameters with similar independence are divided into a layer. The outermost parameters are x0, y0, and α with the highest independence, and a genetic algorithm is used for global search. The middle layer parameter is γ, which has a high dependence on the first layer parameters, and a particle swarm algorithm is used. The inner layer parameter is σ, which has the worst independence. This parameter has a clear directionality and only has one extreme value.

[0059] The search process of the hierarchical genetic algorithm embedded in the particle swarm is to use the search results of the inner algorithm as the fitness function of the outer algorithm and adjust the parameter size according to the results of the inner algorithm; let y i represents the output displacement of the piezoelectric ceramic actuator measured at the i-th sampling time point; represents the output displacement of the piezoelectric ceramic actuator calculated by the model at the i-th sampling time point; N represents the number of sampling points during the experiment; the fitness functions of the inner, middle and outer algorithms are:

[0060]

[0061] S34: Parameters of the generalized circle-based Bouc-Wen model are identified based on a hierarchical genetic algorithm embedded in a particle swarm.

[0062] Furthermore, in step S31, the specific steps of eliminating the dynamic effect are:

[0063] The difference curves of the startup phase and the stable phase are obtained by subtracting the T-3T / 2 time hysteresis curve affected by the dynamic effect and the 0-T / 2 time hysteresis curve not affected by the dynamic effect; the difference includes the nonlinear bias and dynamic response of the startup phase and the stable phase; the amplitude-frequency characteristic curve is drawn for the difference curve to find the main frequency of the dynamic effect;

[0064] The amplitude-frequency characteristic curve under 10Hz input includes two peaks at 20Hz and 160Hz. The dynamic response compensation formula for 10Hz and 20Hz sine wave input is:

[0065]

[0066] A computer storage medium stores a computer program executable by a computer processor, wherein the computer program executes a hysteresis modeling and parameter identification method of a piezoelectric ceramic actuator based on a generalized circular basis BW model.

[0067] The beneficial effects of the present invention are:

[0068] 1. The hysteresis modeling and parameter identification method of a piezoelectric ceramic actuator based on a generalized circular basis BW model of the present invention obtains the output displacement generated by the piezoelectric ceramic actuator under the input voltage, and uses the generalized circular basis Bouc-Wen model to establish a hysteresis model according to the input voltage and the output displacement to describe the hysteresis nonlinear characteristics, thereby better reflecting the relationship between the hysteresis characteristics and the piezoelectric ceramic actuator; a hierarchical genetic algorithm embedded in a particle swarm is used to perform parameter identification on the generalized circular basis Bouc-Wen model, thereby realizing the function of high-precision modeling of the hysteresis of the piezoelectric ceramic actuator.

[0069] 2. The Bouc-Wen model cannot fit the problem of hardening of the abdomen and softening of both ends. The normalization method eliminates the parameter redundancy of the Bouc-Wen model, so that the model has a relatively simple mathematical expression. The generalized circle-based Bouc-Wen model introduces the circular differential of axial degeneration to replace the original exponential differential, which changes the original shape of the Bouc-Wen model and improves the ability of the generalized circle-based Bouc-Wen model to describe the hysteresis of the piezoelectric actuator. The present invention constructs a new generalized circle-based Bouc-Wen model by normalizing, generalizing and reshaping the Bouc-Wen model. According to the modeling method of the generalized Bouc-Wen model, a control function that distinguishes the initial stage and the stable stage is designed, which enables the model to describe the hysteresis of the piezoelectric actuator in all stages; the model has a simple structure, and the description of the nonlinear characteristics of the hysteresis of the piezoelectric ceramic actuator is more accurate, thereby improving the modeling accuracy.

[0070] 3. Since there are multiple parameters in the hysteresis model, there are coupling effects between them, which brings difficulties to parameter identification. The present invention adopts a hierarchical genetic algorithm embedded in a particle swarm to identify the parameters of the generalized circular basis Bouc-Wen model. The algorithm has a three-layer structure, which effectively increases the search efficiency of the algorithm, solves the problem of multi-parameter identification, has strong observability, and reduces the complexity of the algorithm without affecting the accuracy of identification. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1 is a flow chart of an embodiment of the present invention.

[0072] Figure 2 It is a flow chart of parameter identification using a hierarchical genetic algorithm embedded with a particle swarm according to an embodiment of the present invention.

[0073] Figure 3 This is a diagram of the adjustment process of adjusting some parameters at an input voltage frequency of 20 Hz in a hierarchical genetic algorithm embedded in a particle swarm according to an embodiment of the present invention.

[0074] Figure 4 This is a diagram of the adjustment process of adjusting other parameters in a hierarchical genetic algorithm embedded in a particle swarm at an input voltage frequency of 20 Hz according to an embodiment of the present invention.

[0075] Figure 5 Graph showing the influence of the model center point position on the modeling effect at an input voltage frequency of 20 Hz according to an embodiment of the present invention.

[0076] Figure 6 It is a comparison diagram of hysteresis curves of a piezoelectric actuator modeled using a generalized circular basis Bouc-Wen model at an input voltage frequency of 10 Hz according to an embodiment of the present invention.

[0077] Figure 7This is a comparison diagram of output displacement tracking of a piezoelectric actuator modeled using a generalized circular basis Bouc-Wen model at an input voltage frequency of 10 Hz according to an embodiment of the present invention.

[0078] Figure 8 This is an output displacement error diagram of a piezoelectric actuator modeled using a generalized circular basis Bouc-Wen model at an input voltage frequency of 10 Hz according to an embodiment of the present invention.

[0079] Fig. 9 It is a comparison diagram of hysteresis curves of a piezoelectric actuator modeled using a generalized circular basis Bouc-Wen model at an input voltage frequency of 20 Hz according to an embodiment of the present invention.

[0080] Fig.10 This is a comparison diagram of output displacement tracking of a piezoelectric actuator modeled using a generalized circular basis Bouc-Wen model at an input voltage frequency of 20 Hz according to an embodiment of the present invention.

[0081] Fig.11 This is an output displacement error diagram of a piezoelectric actuator modeled using a generalized circular basis Bouc-Wen model at an input voltage frequency of 20 Hz according to an embodiment of the present invention. DETAILED DESCRIPTION

[0082] The present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0083] See also Figure 1 , an embodiment of the present invention comprises the following steps:

[0084] Step 1: Use the piezoelectric positioning platform to collect the input and output data of the piezoelectric actuator and establish a hysteresis model for the nonlinear hysteresis of piezoelectric ceramics; the specific steps are:

[0085] The input and output data of the piezoelectric actuator are collected on a piezoelectric micro-motion platform composed of a piezoelectric amplifier and a piezoelectric actuator, and the hysteresis model is established by adopting a modeling method based on the Bouc-Wen model.

[0086] The differential equation is constructed to characterize the relationship between the output signal amplitude and direction, which can be expressed as follows using the standard Bouc-Wen model:

[0087]

[0088] Where u(t) is the input signal of the hysteresis system, and H(u,t) is the output signal of the hysteresis system. The output of the system is divided into the elastic term αku(t) composed of the input signal and the pure hysteresis term (1-α)Dkh(t). h(t) is the hysteresis auxiliary variable, α, k, D, A, β, γ, and n are model parameters. Selecting appropriate model parameters can represent various hysteresis curves.

[0089] Step 2: Normalize, generalize and reshape the classic Bouc-Wen model, and establish a generalized circle-based Bouc-Wen model; the specific steps are:

[0090] Step 21: The basic Bouc-Wen model has 7 parameters that determine the nonlinear hysteresis behavior of the model, but there are redundant parameters. Remove the parameter redundancy through the following transformation:

[0091]

[0092] Among them, α, k, D, A, β, γ, and n are unnormalized model parameters. n is the normalized model parameter, is an equivalent replacement for the pure lag term, is the normalized pure lag term.

[0093] The Bouc-Wen model expression after normalization transformation is:

[0094]

[0095] Where u(t) is the input signal of the hysteresis system, y(t) is the output signal of the hysteresis system, h(t) is the hysteresis auxiliary variable, parameter k is the weight coefficient, and A, B, C, and n are the model parameters of Bouc-Wen.

[0096] Step 22: Based on the normalized Bouc-Wen model described in step 21, a generalized Bouc-Wen model is obtained, and the model expression is:

[0097]

[0098] In the formula, is the control function of the Bouc-Wen model. The Bouc-Wen model is based on u, The sign of h divides the image into 6 phases. The expression of the control function is:

[0099]

[0100] Among them, β1, β2, β3, β4, β5, and β6 are 6 parameters to be determined. h is a pure lag auxiliary variable, u is the input signal, is the derivative of the input signal.

[0101] In simulating various physical phenomena with hysteresis characteristics, the BW model is simple and accurate, and can describe a series of softening, hardening, degradation, shrinkage, smooth changes, and near-bilinear hysteresis behaviors. It has been applied to various hysteresis scenarios, but it cannot describe the hysteresis behavior of hardening in the middle and softening at both ends, and often has large errors in hysteresis modeling involving smooth transition requirements of the endpoints.

[0102] Step 23: Based on the generalized Bouc-Wen model described in step 22, the model shape is reshaped to obtain a generalized circle-based Bouc-Wen model.

[0103] The circular differential and rotation transformation are introduced into the shape reshaping of the Bouc-Wen model. The expressions of elliptical differential and rotation transformation are:

[0104]

[0105] Among them, h represents the hysteresis term, u represents the input voltage, h1 represents the hysteresis term after rotation, and u1 represents the input voltage after rotation. Combining the above formula, the rotation ellipse differential equation is:

[0106]

[0107] Among them, γ represents the square of the ratio of the major axis to the minor axis of the ellipse. Considering the division of the expectation phase and the stable phase, the tT / 2 term is introduced into the control function, and the designed control function expression is:

[0108]

[0109] Among them, the control function divides the model space into 6 phases, the first and second phases act on the model parameters in the startup phase, and the third, fourth, fifth, and sixth phases act on the model parameters in the stable phase. The control function is extended to 5 dimensions, which represent the five parameters of the generalized circular basis Bouc-Wen model:

[0110] ψ (t) =[ψ1ψ2ψ3ψ4ψ5] T ;

[0111] Combining the rotational elliptic differential equation and the generalized Bouc-Wen model, the basic formula of the generalized circular basis Bouc-Wen model is obtained:

[0112]

[0113] Among them, w (t) is an auxiliary variable, h (t) is the pure hysteresis term after replacing the differential. In order to highlight the local characteristics of the piezoelectric actuator, k is fixed as a constant to satisfy the expression:

[0114] y0-y1=2λk;

[0115] Where y0 and y1 represent the displacement of the hysteresis curve at time T / 2 and T respectively, and λ is the amplitude of the input sine wave. Axis degeneration is introduced to increase the expressive power of the model:

[0116]

[0117] Define a linear transformation to enhance the observability of the model:

[0118] Γ=Ωβ T ;

[0119] Γ=[x0 y0 α γ σ];

[0120] Among them, x0 represents the horizontal coordinate of the geometric center of the model, y0 represents the vertical coordinate of the geometric center of the model, α represents the rotation deviation angle, r represents the axis ratio, and σ represents the axis ratio degradation coefficient. The length of each column vector is 6, which corresponds to the control parameters of the 6 phases. The interpretable Γ matrix is ​​the parameter matrix of the generalized circular basis Bouc-Wen model. The matrix Ω is expressed as:

[0121]

[0122] The expression of the generalized circle-based Bouc-Wen model is:

[0123] y (t) =ku (t) +h (t)

[0124]

[0125] Γ=Ωβ T ;

[0126] Among them, y (t) represents the output displacement, u (t) Indicates input voltage, h (t) 、w (t) is the hysteresis auxiliary variable. (t) is a 5-dimensional control function, and the matrix Γ is a 6×5 parameter matrix.

[0127] In contrast to the hardened curve of the Bouc-Wen model, the generalized circular-based Bouc-Wen model has been changed in model morphology, and its ability to describe the hysteresis of the piezoelectric actuator has been greatly improved. The model has a strong adhesion to the fast bending part at the end of the hysteresis curve, achieving a smooth transition at the end of the hysteresis curve.

[0128] Step 3: If Figure 2As shown, according to the hysteresis model, a parameter identification strategy of a hierarchical genetic algorithm embedded in a particle swarm is designed, and a parameter identification of a generalized circular basis Bouc-Wen model is performed using a hierarchical genetic algorithm based on an embedded particle swarm to obtain a target model. The specific steps are:

[0129] Step 31: Divide the parameters into several levels according to their characteristics, and give different constraints and calculation processes to different levels. Figure 3 , Figure 4 , Figure 5 As shown, the parameter identification results of different levels and the influence of the model center point are displayed.

[0130] In the hierarchical genetic algorithm embedded in the particle swarm, the genetic algorithm is set as the outer algorithm. This algorithm has a strong overall search capability for the target area and is not easy to fall into local extreme values. The middle-level algorithm is set as a particle swarm algorithm with a crossover factor, which has a faster convergence speed. The fast and high-precision binary search method is set as the inner-level algorithm.

[0131] The actual input-output relationship of the piezoelectric actuator is the superposition of hysteresis nonlinearity and dynamic effects. In order to obtain the hysteresis curve, it is necessary to remove the dynamic effect from the input-output curve. The present invention equates the dynamic effect to a sine wave with amplitude attenuation. Steps to eliminate the dynamic effect: subtract the T~3T / 2 time hysteresis curve affected by the dynamic effect from the 0~T / 2 time hysteresis curve not affected by the dynamic effect to obtain the difference curve between the startup phase and the stable phase. The difference includes both the nonlinear bias and the dynamic response of the startup phase and the stable phase. Draw the amplitude-frequency characteristic curve for the difference curve to find the main frequency of the dynamic effect.

[0132] The amplitude-frequency characteristic curve under 10Hz input has two peaks of 20Hz and 160Hz. The 20Hz peak is caused by the nonlinear bias between the startup phase and the stable phase, so the equivalent sinusoidal wave frequency of the dynamic effect is about 160Hz, and the amplitude is about 0.06. The dynamic effect frequency under 20Hz input is about 160Hz, and the amplitude is about 0.39. The dynamic response compensation formula for 10Hz and 20Hz sine wave input is:

[0133]

[0134] Since the geometric meanings of the outer parameters x0, y0, and α are unclear, it is difficult to select the parameter range of the parameter identification system, so it is necessary to redefine the outer parameters. The redefinition expression is as follows:

[0135]

[0136] Among them, (x1, y1) represents the starting position of the model; (x2, y2) represents the ending position of the model; (x′0, y′0) represents the ideal zero point of the model after redefinition; (x0, y0) represents the geometric center point of the model before redefinition; (X, Y) represents the geometric center point of the model after redefinition.

[0137] After redefinition, the search range of parameters X and Y is determined as:

[0138] S={(X,Y)|X∈[-1,0],Y∈[0,1]};

[0139] Step 32: In order to improve the convergence speed of the particle swarm algorithm, the particle swarm algorithm is improved. Adding the crossover module of the genetic algorithm to the particle swarm algorithm can improve the convergence speed of the particle swarm algorithm to a certain extent.

[0140] Step 33: If the effect of a parameter on the fitness value does not depend on other parameters, it means that the parameter is independent, and the parameters with similar independence are divided into one layer. The outermost parameters are x0, y0, and α with the highest independence. A genetic algorithm is used for global search to avoid falling into local extreme values. The middle layer parameter is γ, which is highly dependent on the first layer parameters. When x0, y0, and α are fixed as constraints, γ has fewer local extreme values, and the use of the particle swarm algorithm will not cause errors due to falling into local extreme values. The inner layer parameter is σ with the worst independence. This parameter has a clear directionality and has only one extreme value. Figure 6 , Figure 7 and Figure 8 The modeling results of the basic Bouc-Wen model of the circle identified by the hierarchical genetic algorithm embedded with the particle swarm algorithm at an input voltage frequency of 10Hz are shown respectively, and the root mean square error is 0.0405um and the relative root mean square error is 0.0976%. Fig. 9 , Fig.10 and Fig.11 The modeling results of the basic Bouc-Wen model of the circle identified by the hierarchical genetic algorithm embedded with the particle swarm algorithm at an input voltage frequency of 20Hz are shown respectively, and the root mean square error is 0.0504um and the relative root mean square error is 0.1230%.

[0141] The search process of the hierarchical genetic algorithm embedded in the particle swarm is to use the search results of the inner algorithm as the fitness function of the outer algorithm and adjust the parameter size according to the results of the inner algorithm. The fitness functions of the inner, middle and outer algorithms are:

[0142]

[0143] Among them, y i represents the output displacement of the piezoelectric ceramic actuator measured at the i-th sampling time point; represents the output displacement of the piezoelectric ceramic actuator calculated by the model at the i-th sampling time point; N represents the number of sampling points during the experiment.

[0144] Step 34: Identify the parameters of the generalized circular basis Bouc-Wen model according to the identification method. The number of individuals in the genetic algorithm is set to 100, the maximum genetic generation is set to 10, the generation gap is set to 0.8, the crossover probability is set to 0.7, the mutation probability is set to 0.01, and the crossover method adopts the xovsp method; the number of particles of the particle swarm algorithm is set to 5, the maximum number of iterations is set to 4; the maximum number of iterations of the binary search method is set to 8.

[0145] It should be understood that the size of the serial numbers of the steps in the above embodiments does not mean the order of execution. The execution order of each process should be determined by its function and internal logic, and should not constitute any limitation on the implementation process of the embodiments of the present application.

[0146] The above embodiments are only used to illustrate the design ideas and features of the present invention, and their purpose is to enable those skilled in the art to understand the content of the present invention and implement it accordingly. The protection scope of the present invention is not limited to the above embodiments. Therefore, any equivalent changes or modifications made based on the principles and design ideas disclosed by the present invention are within the protection scope of the present invention.

Claims

1. A piezoelectric ceramic actuator hysteresis modeling and parameter identification method based on a generalized circular basis BW model, characterized by: The following steps are involved: S1: Use the piezoelectric positioning platform to collect the input and output data of the piezoelectric actuator and establish a hysteresis model for the nonlinear hysteresis of piezoelectric ceramics; S2: The classic Bouc-Wen model is improved by normalization, generalization and reshaping, and a generalized circle-based Bouc-Wen model is established; S3: designing a parameter identification strategy of a hierarchical genetic algorithm embedded in a particle swarm according to the hysteresis model, and using a hierarchical genetic algorithm based on an embedded particle swarm to identify parameters of a generalized circular basis Bouc-Wen model to obtain a target model.

2. The method for hysteresis modeling and parameter identification of piezoelectric ceramic actuators based on the generalized circle-based BW model according to claim 1, characterized in that: In the step S1, the specific steps are: The input and output data of the piezoelectric actuator are collected on a piezoelectric micro-motion platform composed of a piezoelectric amplifier and a piezoelectric actuator, and a hysteresis model is established using a modeling method based on the Bouc-Wen model. Assume u(t) is the input signal of the hysteresis system, H(u,t) is the output signal of the hysteresis system; α, k, D, A, β, γ, n are model parameters, the output of the system includes the elastic term αku(t) composed of the input signal and the pure hysteresis term (1-α)Dkh(t), h(t) is the hysteresis auxiliary variable; the relationship between the amplitude and direction of the output signal is characterized by constructing a differential equation, which is expressed by the standard Bouc-Wen model as follows:

3. The hysteresis modeling and parameter identification method of piezoelectric ceramic actuator based on the generalized circle basis BW model according to claim 2, characterized in that: In the step S2, the specific steps are: S21: Let α, k, D, A, β, γ, and n be the unnormalized model parameters, n is the normalized model parameter, is an equivalent replacement for the pure lag term, is a normalized pure lag term; parameter redundancy is removed by the following transformation: Assume y(t) is the output signal of the hysteresis system, parameter k is the weight coefficient, B and C are the model parameters of Bouc-Wen; then the expression of the normalized Bouc-Wen model is: S22: Let h be the pure lag auxiliary variable, u be the input signal, is the derivative of the input signal, is the control function of the Bouc-Wen model. The Bouc-Wen model is based on u, The sign of h divides the image into 6 phases; the normalized Bouc-Wen model is generalized and expanded, and the expression of the generalized Bouc-Wen model is obtained as follows: S23: Reshape the generalized Bouc-Wen model; let h1 represent the lag term after rotation, and u1 represent the input signal after rotation; introduce circular differential and rotation transformation into the reshape of the Bouc-Wen model, and obtain the elliptic differential and rotation transformation expressions of the generalized circular basis Bouc-Wen model as follows: Let γ represent the square of the ratio of the major axis to the minor axis of the ellipse; combining the above formula, the differential equation of the rotating ellipse is: The control function is expanded to 5 dimensions, representing the five parameters of the generalized circular basis Bouc-Wen model: ψ (t) =[ψ1 ψ2 ψ3 ψ4 ψ5] T ; Assume w (t) is an auxiliary variable, h (t) is the pure lag term after replacing the differential; combining the rotational elliptic differential equation and the generalized Bouc-Wen model, the basic formula of the generalized circular basis Bouc-Wen model is:

4. The hysteresis modeling and parameter identification method of piezoelectric ceramic actuator based on the generalized circle basis BW model according to claim 3 is characterized in that: In step S22, let β1, β2, β3, β4, β5, and β6 be six parameters to be determined; then the control function The expression is:

5. The method for hysteresis modeling and parameter identification of piezoelectric ceramic actuators based on the generalized circle-based BW model according to claim 4, characterized in that: In step S23, according to the division of the expectation phase and the stable phase, the term tT / 2 is introduced into the control function, and the control function expression is: The control function divides the model space into six phases, the first and second phases act on the model parameters in the startup phase, and the third, fourth, fifth, and sixth phases act on the model parameters in the stable phase.

6. The method for hysteresis modeling and parameter identification of piezoelectric ceramic actuators based on the generalized circle-based BW model according to claim 5, characterized in that: In the step S23, y0 and y1 are respectively assumed to represent the displacement of the hysteresis curve at time T / 2 and time T, and λ is the amplitude of the input sine wave; in order to highlight the local characteristics of the piezoelectric actuator, k is fixed as a constant to satisfy the expression: y0-y1=2λk; Introduce axis degeneration to increase the expressive power of the model: Let x0 represent the horizontal coordinate of the geometric center of the model, y0 represent the vertical coordinate of the geometric center of the model, α represent the rotation deviation angle, r represent the axis ratio, σ represent the axis ratio degradation coefficient; define a linear transformation to enhance the observability of the model: C=Ov T ; Γ=[x0 y0 α γ σ]; The length of each column vector is 6, which corresponds one-to-one to the control parameters of the 6 phases; The interpretable Γ matrix is ​​the parameter matrix of the generalized circular basis Bouc-Wen model; the matrix Ω is expressed as:

7. The method for hysteresis modeling and parameter identification of piezoelectric ceramic actuators based on the generalized circle-based BW model according to claim 6, characterized in that: In the step S23, let y ( t ) represents the output displacement, u ( t ) Indicates input voltage, h ( t ) 、w ( t ) is the hysteresis auxiliary variable; ψ ( t ) is a 5-dimensional control function, and the matrix Γ is a 6×5 parameter moment; the expression of the generalized circular basis Bouc-Wen model is: y (t) =stand (t) +h (t) C=Ov T 。 8. The method for hysteresis modeling and parameter identification of piezoelectric ceramic actuators based on the generalized circle-based BW model according to claim 1, characterized in that: In the step S3, the specific steps are: S31: Divide the parameters into several levels according to their characteristics, and give different constraints and calculation processes to different levels; In the hierarchical genetic algorithm embedded in particle swarm, the genetic algorithm is set as the outer algorithm, the particle swarm algorithm with crossover factor is set as the middle algorithm, and the binary search algorithm is set as the inner algorithm; The actual input-output relationship of the piezoelectric actuator is the superposition of hysteresis nonlinearity and dynamic effect. The dynamic effect is equivalent to a sine wave with amplitude attenuation to eliminate the dynamic effect. Redefine the outer parameters x0, y0, and α; let (x1, y1) represent the starting position of the model; (x2, y2) represent the ending position of the model; (x′0, y′0) represent the ideal zero point of the model after redefinition; (x0, y0) represent the geometric center point of the model before redefinition; (X, Y) represent the geometric center point of the model after redefinition; the redefinition expression is as follows: After redefinition, the search range of parameters X and Y is: S={(X,Y)|X∈[-1,0],Y∈[0,1]}; S32: adding the crossover module of the genetic algorithm to the particle swarm algorithm to improve the convergence speed of the particle swarm algorithm; S33: Parameters with similar independence are divided into a layer. The outermost parameters are x0, y0, and α with the highest independence, and a genetic algorithm is used for global search. The middle layer parameter is γ, which has a high dependence on the first layer parameters, and a particle swarm algorithm is used. The inner layer parameter is σ, which has the worst independence. This parameter has a clear directionality and only has one extreme value. The search process of the hierarchical genetic algorithm embedded in the particle swarm is to use the search results of the inner algorithm as the fitness function of the outer algorithm and adjust the parameter size according to the results of the inner algorithm; let y i represents the output displacement of the piezoelectric ceramic actuator measured at the i-th sampling time point; represents the output displacement of the piezoelectric ceramic actuator calculated by the model at the i-th sampling time point; N represents the number of sampling points during the experiment; the fitness functions of the inner, middle and outer algorithms are: S34: Parameters of the generalized circle-based Bouc-Wen model are identified based on a hierarchical genetic algorithm embedded in a particle swarm.

9. The method for hysteresis modeling and parameter identification of piezoelectric ceramic actuators based on the generalized circle-based BW model according to claim 8, characterized in that: In the step S31, the specific steps of eliminating the dynamic effect are: using the T~3T / 2 time hysteresis curve affected by the dynamic effect and the 0~T / 2 time hysteresis curve not affected by the dynamic effect to make a difference, and obtain the difference curve of the startup phase and the stable phase; the difference includes the nonlinear bias and dynamic response of the startup phase and the stable phase; drawing the amplitude-frequency characteristic curve of the difference curve to find the main frequency of the dynamic effect; The amplitude-frequency characteristic curve under 10Hz input includes two peaks at 20Hz and 160Hz. The dynamic response compensation formula for 10Hz and 20Hz sine wave input is:

10. A computer storage medium, characterized in that: A computer program executable by a computer processor is stored therein, and the computer program executes the hysteresis modeling and parameter identification method of a piezoelectric ceramic actuator based on a generalized circular basis BW model as described in any one of claims 1 to 9.

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