Nonlinear modeling method of piezoelectric ceramic actuator and application

Abstract

The application discloses a nonlinear modeling method of a piezoceramic actuator and application, and belongs to the technical field of modeling and control of electromechanical systems; a fractional order coupling model constructed includes a cascaded fractional order hysteresis model and a fractional order creep model; wherein the fractional order hysteresis model contains a parallel linear module and a nonlinear module, and the nonlinear module represents the nonlinear relationship between displacement and voltage by establishing an equal relationship between displacement and the weighted sum result of multiple voltage fractional calculus operators, so that the fractional order hysteresis model can accurately reflect the rate-independent hysteresis effect and main characteristics of the piezoceramic actuator with fewer parameters; compared with traditional piezoelectric hysteresis and creep models, the model has the advantages of simple form and high accuracy under the condition of meeting the same modeling accuracy, and can simply and accurately describe the hysteresis characteristics and creep characteristics of the piezoceramic actuator.

Description

Technical Field

[0001] This invention belongs to the field of electromechanical system modeling and control technology, and more specifically, relates to a nonlinear modeling method and application of piezoelectric ceramic actuators. Background Technology

[0002] Piezoelectric ceramic actuators are widely used in micro-vibration control and isolation, high-precision positioning, and semiconductor manufacturing due to their excellent characteristics such as high output, high resonant frequency, high precision, and small size. For example, a 3-DOF micro-vibration isolation experimental platform, a nano-positioning stage, and an atomic force microscope all employ piezoelectric ceramic actuators. However, the inherent hysteresis and creep nonlinearity of piezoelectric ceramic actuators often significantly limit their application accuracy, and in severe cases, may even affect system stability, causing instability. To improve the positioning and scanning accuracy of piezoelectric ceramic actuators, it is necessary to model and study the creep and hysteresis effects.

[0003] Due to the complex hysteresis and creep characteristics of piezoelectric ceramics, different types of piezoelectric ceramics exhibit varying nonlinear properties. To accurately model these properties, researchers primarily employ phenomenological and constitutive methods to describe the nonlinear behavior of piezoelectric ceramic actuators. Constitutive methods focus more on the material's intrinsic properties from materials science, thermodynamics, and physics perspectives, utilizing theories of ferroelectric / paraelectric phase transitions, domain orientation, and domain wall motion for model building and analysis. Phenomenon-phenomenological modeling methods, on the other hand, disregard the system's internal structure and material physical properties, focusing solely on input-output characteristics and expressing these features through model information formulas.

[0004] In existing technologies, phenomenon-based modeling methods are often used to describe nonlinear characteristics and are easy to control and compensate for. Commonly used hysteresis models include the Bouc-Wen model, the Prandtl-Ishlinskii (PI) model, and the Preisach model, while creep models include logarithmic models and linear time-invariant models. It is worth noting that although these models can effectively describe the nonlinear characteristics of piezoelectric ceramics, they are designed for a single characteristic and involve a large number of parameters, which increases the difficulty of coupled modeling and parameter identification, and cannot simply and accurately describe the hysteresis and creep characteristics of piezoelectric ceramic actuators. Summary of the Invention

[0005] In view of the above-mentioned defects or improvement needs of the prior art, the present invention provides a modeling method and application for the hysteresis and creep characteristics of piezoelectric ceramic actuators, so as to solve the technical problem that the prior art cannot simply and accurately describe the hysteresis and creep characteristics of piezoelectric ceramic actuators.

[0006] To achieve the above objectives, in a first aspect, the present invention provides a nonlinear modeling method for piezoelectric ceramic actuators, comprising the following steps:

[0007] S1. Establish a fractional-order coupling model for simulating the hysteresis and creep nonlinear characteristics of piezoelectric ceramic actuators;

[0008] The fractional-order coupling model includes a cascaded fractional-order hysteresis model and a fractional-order creep model; the fractional-order hysteresis model includes parallel linear modules and nonlinear modules; the linear module is used to represent the ideal linear relationship between voltage and displacement; the nonlinear module represents the nonlinear relationship between displacement and voltage by establishing the equality relationship between displacement and the weighted summation results of multiple voltage fractional-order differential operators.

[0009] S2. Based on the input voltage and output displacement data of the simulated piezoelectric ceramic actuator, the parameters in the fractional-order coupling model are identified.

[0010] More preferably, the control equation of the above-mentioned nonlinear module is:

[0011]

[0012] Among them, f nonlinear (u) represents the nonlinear displacement under hysteresis characteristics; h1 is... The nonlinear coefficient; The result is the fractional-order voltage calculus operator at the fractional-order α; u(t) is the voltage signal; h2 is... The nonlinear coefficient; It is the absolute value of the rate of change of the voltage signal; This is the result of the voltage fractional-order calculus operator in the β fractional order.

[0013] More preferably, Where Γ(·) is the gamma function; n is the nearest integer order. More preferably, the governing equation of the above linear module is:

[0014] f linear (u)=l·u(t)

[0015] Among them, f linear (u) represents the linear displacement under hysteresis; l is the linear coefficient; u(t) is the voltage signal.

[0016] More preferably, the time-domain governing equation of the fractional-order creep model is:

[0017]

[0018] Where d(t) is the time-domain displacement under creep characteristics; k c The creep coefficient; f(u) is the intermediate displacement state output of the fractional-order hysteresis model; μ is the fractional-order creep parameter.

[0019] More preferably, the frequency domain governing equation of the fractional-order creep model is:

[0020] G c (s)=k c ·s μ -1 < μ < 0

[0021] Among them, G c (s) represents the Laplace transform of the displacement under creep characteristics; k c is the creep coefficient; s is the Laplace constant; μ is the fractional creep parameter.

[0022] More preferably, the method for identifying the fractional-order creep parameters in a fractional-order creep model includes:

[0023] By controlling the frequency of the voltage signal input to the piezoelectric ceramic actuator, the corresponding displacement frequency domain data can be obtained;

[0024] By using the voltage signal frequency and the corresponding displacement frequency domain data, the parameters in the structural dynamic equation of the piezoelectric ceramic actuator are fitted to obtain the creep fractional-order parameters in the fractional-order creep model.

[0025] More preferably, based on the input voltage and output displacement data of the simulated piezoelectric ceramic actuator, the particle swarm optimization algorithm is used to identify the parameters in the fractional-order coupling model.

[0026] In a second aspect, the present invention also provides a computer-readable storage medium comprising a stored computer program, wherein, when the computer program is run by a processor, it controls the device where the storage medium is located to execute the modeling method for hysteresis and creep characteristics of piezoelectric ceramic actuators provided in the first aspect of the present invention.

[0027] Thirdly, the present invention provides a feedforward control method for a piezoelectric ceramic actuator, comprising:

[0028] An inverse model of the fractional-order coupling model is established, and the inverse model is used to compensate for the hysteresis nonlinearity and creep nonlinearity of the piezoelectric ceramic actuator.

[0029] The fractional-order coupling model is constructed using the nonlinear modeling method for piezoelectric ceramic actuators provided in the first aspect of this invention.

[0030] Fourthly, the present invention provides a feedforward controller for a piezoelectric ceramic actuator, used to execute the feedforward control method provided in the third aspect of the present invention.

[0031] In summary, the above-described technical solutions conceived in this invention can achieve the following beneficial effects:

[0032] 1. This invention provides a modeling method for the hysteresis and creep characteristics of piezoelectric ceramic actuators. The constructed fractional-order coupled model includes a cascaded fractional-order hysteresis model and a fractional-order creep model. The fractional-order hysteresis model contains linear and nonlinear modules. The nonlinear module represents the nonlinear relationship between displacement and voltage by establishing the equality relationship between the weighted summation results of displacement and multiple voltage fractional-order calculus operators. This allows the fractional-order hysteresis model to accurately reflect the rate-independent hysteresis effect and its main characteristics of the piezoelectric ceramic actuator with fewer parameters. Compared with traditional piezoelectric hysteresis and creep models, this invention describes the nonlinearity of the piezoelectric actuation process by simply utilizing the nonlinear characteristics of the fractional-order calculus circuit without introducing other models. Under the condition of meeting the same modeling accuracy, it reduces the number of model parameters required and has the advantages of simple form and accurate model. It can simply and accurately describe the hysteresis and creep characteristics of piezoelectric ceramic actuators.

[0033] 2. Furthermore, in the modeling method for the hysteresis and creep characteristics of piezoelectric ceramic actuators provided by this invention, the nonlinear module represents the nonlinear relationship between voltage and displacement by establishing a linear relationship between displacement and the fractional-order calculus results of two voltages. This allows the fractional-order hysteresis model to accurately reflect the rate-independent hysteresis effect and its main characteristics of the piezoelectric ceramic actuator with only 5 parameters: l, h1, α, h2, and β. It can simply and accurately describe the hysteresis and creep characteristics of the piezoelectric ceramic actuator.

[0034] 3. Furthermore, the modeling method for hysteresis and creep characteristics of piezoelectric ceramic actuators provided by the present invention identifies the fractional-order creep parameters in the fractional-order creep model by introducing the dynamic equation of the piezoelectric ceramic actuator. Since this method takes into account the influence of actuator structural parameters, the identification of structural parameters can further eliminate dynamic interference and improve the identification accuracy of the piezoelectric ceramic creep model. Attached Figure Description

[0035] Figure 1 This is a schematic diagram of the linear module structure of the fractional hysteresis model provided in Embodiment 1 of the present invention;

[0036] Figure 2 This is a schematic diagram of the fractional-order coupling model provided in Embodiment 1 of the present invention;

[0037] Figure 3 A flowchart illustrating the experimental identification of fractional creep parameters provided in Embodiment 1 of the present invention;

[0038] Figure 4 This is a schematic diagram of the piezoelectric ceramic actuator provided in Embodiment 1 of the present invention;

[0039] Figure 5This is a schematic diagram of the equivalent dynamic model of the piezoelectric ceramic actuator provided in Embodiment 1 of the present invention;

[0040] Figure 6 This is a schematic diagram of the voltage-displacement tracking effect of the voltage-displacement experimental data and the identified fractional-order hysteresis model provided in Embodiment 1 of the present invention. Detailed Implementation

[0041] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0042] Example 1

[0043] A nonlinear modeling method for piezoelectric ceramic actuators includes the following steps:

[0044] S1. Establish a fractional-order coupling model for simulating the hysteresis and creep nonlinear characteristics of piezoelectric ceramic actuators;

[0045] The fractional-order coupling model includes a cascaded fractional-order hysteresis model and a fractional-order creep model; the fractional-order hysteresis model includes parallel linear modules and nonlinear modules; the linear module is used to represent the ideal linear relationship between displacement and voltage; the nonlinear module represents the nonlinear relationship between displacement and voltage by establishing the equality relationship between the weighted summation results of displacement and multiple voltage fractional-order calculus operators, that is, by establishing the linear relationship between displacement and multiple voltage fractional-order calculus results to represent the nonlinear relationship between displacement and voltage.

[0046] Specifically, the linear module of the fractional-order hysteresis model is constructed using the linear relationship between the two main physical quantities, voltage and displacement, such as... Figure 1 As shown, it is composed of the linear coefficient l and the voltage input u(u), that is...

[0047] f linear (u)=l·u(t)

[0048] Among them, f linear (u) represents the displacement reference value (intermediate value) after processing by the linear module. The model mainly adjusts the basic amplitude of the hysteresis curve under the input voltage by adjusting the linear coefficient l, and represents the ideal linear relationship between voltage and displacement by establishing a linear model between displacement and voltage. Subsequently, a nonlinear module with four parameters for the fractional hysteresis model is constructed by taking advantage of the characteristics of fractional operators that are different from traditional mapping relationships. The structure is as follows: Figure 1 As shown, its module information is as follows:

[0049]

[0050] Among them, f nonlinear (u) represents the displacement reference value (intermediate value) after processing by the linear module; and These are two fractional operators in the nonlinear module; h1 and h2 are their nonlinear coefficients, respectively. It is the absolute value of the rate of change of the input signal, mainly responsible for controlling the relative magnitude of the hysteresis when the input signal changes suddenly. Parameters α, β, h1, and h2 together determine the direction and shape of the voltage-displacement hysteresis curve; -1 < α < 1, -1 < β < 1. The two modules work together to accurately express the hysteretic nonlinear behavior of the piezoelectric ceramic during the actuation process. The overall structure is as follows: Figure 1 As shown.

[0051] In a preferred embodiment, the calculation formula for the above fractional calculus operator is as follows:

[0052]

[0053]

[0054] Where Γ is the gamma function, n is the nearest integer order, and t is the independent variable.

[0055] Furthermore, the time-domain model information of the fractional-order creep stage used in the fractional-order coupling model to describe the drift creep phenomenon of piezoelectric ceramics with increasing time is as follows:

[0056]

[0057] Where, k c Here, f(u) is the creep coefficient; f(u) is the intermediate displacement state output of the fractional-order hysteresis model, that is, the intermediate displacement state after the voltage input signal u(t) is processed by the fractional-order hysteresis model, specifically f(u). linear (u)+f nonlinear (u); μ is the creep fractional-order parameter; the creep coefficient and the creep fractional-order parameter work together to express the nonlinear creep behavior of the electroceramic during the actuation process. Since it mainly exhibits the characteristics of an integral element in the frequency domain, the specific frequency domain expression in the range (-1,0) is:

[0058] G c (s)=k c ·s μ -1 < μ < 0

[0059] Among them, G c (s) represents the Laplace transform of the displacement under creep characteristics; k cis the creep coefficient; s is the Laplace constant; μ is the fractional creep parameter.

[0060] The fractional-order hysteresis model and the fractional-order creep model are cascaded to form the fractional-order hysteresis and creep coupling model of the piezoelectric ceramic actuator. The structure of the cascaded fractional-order coupling model is as follows: Figure 2 As shown.

[0061] It should be noted that the nonlinear characteristics of fractional order are very similar to many nonlinear behaviors in the physical world. This invention constructs fractional order hysteresis model and fractional order creep model based on fractional order, which have the advantages of simple form but high tracking accuracy.

[0062] S2. Based on the input voltage and output displacement data of the simulated piezoelectric ceramic actuator, the parameters in the fractional-order coupling model are identified.

[0063] The parameters in the model are determined through voltage-displacement experiments using piezoelectric ceramic actuators. Displacement data under different input voltage signals are sampled, and then an identification algorithm is used to obtain accurate model parameters, ultimately yielding complete fractional-order coupled model information. In this embodiment, the parameters in the fractional-order hysteresis model and the fractional-order creep model are identified separately. The parameters in the model are identified by minimizing the difference between the simulated displacement output by the model and the actual displacement obtained from the experiment. The displacement output by the fractional-order hysteresis model is the sum of the displacements output by its linear and nonlinear modules. Particle swarm optimization (PSO) algorithms, genetic algorithms, etc., can be used for parameter identification.

[0064] In a preferred embodiment, the identification algorithm used is the Particle Swarm Optimization (PSO) algorithm, which optimizes the parameters of the initially selected fractional-order coupled model to obtain all model information of the piezoelectric ceramic fractional-order coupled model. PSO, as an emerging optimization algorithm, is widely used due to its simple concept, ease of implementation, and fast convergence speed. The basic idea of ​​PSO is to simulate the foraging behavior of a flock of birds randomly searching for food: the flock adjusts its search path through its own experience and communication with other birds to find the location with the most food. The position and path of each bird are combinations of independent variables, and the food density at each location reached is the function value. Each search adjusts its search direction and speed based on its own experience (its own historical optimal search location) and communication with other birds (the population's historical optimal search location) to find the optimal solution. This invention uses PSO to identify the parameters of the proposed piezoelectric fractional-order coupled model. Due to its fast convergence speed and ease of implementation, the PSO algorithm is widely used in various optimization problems. In the parameter identification problem of piezoelectric fractional-order coupled models, the model has relatively few parameters, and PSO is very suitable for application in the parameter identification of piezoelectric fractional-order coupled models.

[0065] The velocity and position iteration formulas for the particle swarm optimization algorithm are as follows:

[0066]

[0067] Among them, v id p represents inertia id Representing its own influence, p gd Represents population influence; x id Represents the initial position, c1 and c2 are the individual weights, and r1 and r2 are the population weights.

[0068] In a preferred embodiment, the root mean square trajectory tracking error e is used. RMS As the fitness function of the particle swarm optimization algorithm, e RMS With the goal of minimizing, the parameters in the model are iteratively optimized using the particle swarm optimization algorithm, and its calculation formula is as follows:

[0069]

[0070] Taking the identification of parameters in a fractional-order hysteresis model as an example, Θ=[l,h1,α,h2,β] are the parameters to be optimized, and y i and Here, n represents the experimental displacement and the simulated displacement, while i is the number of data points in the numerical calculation and the sequence number of the measured data. For the identification process, in the unlabeled case, a 100V sinusoidal input voltage signal and a 1Hz actual displacement response are used by default for parameter estimation, simultaneously identifying the model's coefficients and fractional powers.

[0071] This embodiment uses experimentally sampled voltage-displacement data and employs the particle swarm optimization (PSO) algorithm to identify the parameters of the fractional-order model. The effectiveness of the method can be demonstrated experimentally. This embodiment realizes the design, implementation, and optimization of the fractional-order coupling model of piezoelectric hysteresis and creep, proving the effectiveness and simplicity of the piezoelectric coupling model of this invention.

[0072] In a preferred embodiment, during the identification process of the fractional creep model, it is considered that there are s to be identified in the frequency domain expression of the fractional creep model. μ The term, and the same s exists in the structural dynamic equation of the piezoelectric ceramic actuator. μ To eliminate dynamic interference and improve the identification accuracy of the fractional-order creep model, this embodiment introduces the structural dynamic equation of the piezoelectric ceramic actuator to identify the s-axis of the fractional-order creep model. μ Item. Specifically, such as Figure 3Experiments were conducted by controlling the frequency of the voltage signal input to the piezoelectric ceramic actuator (using a sinusoidal voltage signal as an example in this embodiment) to obtain the corresponding displacement frequency domain data, thereby obtaining the amplitude-frequency experimental data points of the fractional-order creep stage. Using the voltage signal frequency and the corresponding displacement frequency domain data, the parameters in the structural dynamic equation of the piezoelectric ceramic actuator were fitted, thus obtaining a series of parameter values, including the creep fractional-order parameters in the fractional-order creep model. In the above process, particle swarm optimization algorithms, genetic algorithms, and other algorithms can also be used to fit the parameters in the structural dynamic equation of the piezoelectric ceramic actuator.

[0073] The structural schematic diagram and equivalent dynamic model schematic diagram of the piezoelectric ceramic actuator are shown below. Figure 4 and Figure 5 As shown. The structural dynamic equations and their Laplace transform forms of the piezoelectric ceramic actuator are as follows:

[0074]

[0075] F p (t)=nd 33 k p U p (t)=k p 'U p (t)

[0076]

[0077]

[0078] Where m is the equivalent mass of the piezoelectric ceramic; d(t) is the output displacement of the piezoelectric ceramic actuator; c and c p These are the damping coefficients of the flexible hinge and piezoelectric stack in the actuating structure, respectively; k and k p These are the equivalent stiffnesses of the actuating flexible hinge and the piezoelectric stack, respectively. F p (t) is the driving force of the piezoelectric stack.

[0079] It should be noted that the term 'a' obtained in the creep parameter identification method involving structural dynamics encompasses the scaling factor k of the traditional creep model. c The identification process no longer requires a separate discussion of the model scaling factor.

[0080] Furthermore, such as Figure 6The diagram shows the voltage-displacement tracking effect of the voltage-displacement experimental data and the identified fractional-order hysteresis model. The FOH model specifically refers to the fractional-order hysteresis model constructed using the nonlinear modeling method for piezoelectric ceramic actuators provided in this invention. The NBW model is a traditional Bouc-Wen model also used to describe hysteresis. Finally, based on the experimental data, the piezoelectric fractional-order hysteresis creep model proposed in this invention achieves [results / results / etc.]. RMS The value is 3.75% higher than that of the traditional coupled model.

[0081] In summary, the fractional-order coupled modeling method for hysteresis and creep characteristics of piezoelectric ceramic actuators provided by this invention can accurately describe the nonlinear phenomena of piezoelectric ceramic actuators. When applied to the simulation modeling of hysteresis and creep nonlinearity of piezoelectric ceramics, it can effectively improve the modeling accuracy, thereby achieving the goal of accurately describing the hysteresis and creep behavior of piezoelectric ceramics and simplifying the modeling complexity.

[0082] Example 2

[0083] A computer-readable storage medium includes a stored computer program, wherein when the computer program is run by a processor, it controls the device where the storage medium is located to execute the modeling method for hysteresis and creep characteristics of piezoelectric ceramic actuators provided in Embodiment 1 of the present invention.

[0084] The relevant technical solutions are the same as in Embodiment 1, and will not be repeated here.

[0085] Example 3

[0086] A feedforward control method for a piezoelectric ceramic actuator includes:

[0087] An inverse model of the fractional-order coupling model is established, and the inverse model is used to compensate for the hysteresis nonlinearity and creep nonlinearity of the piezoelectric ceramic actuator.

[0088] The fractional-order coupling model is constructed using the nonlinear modeling method for piezoelectric ceramic actuators provided in Embodiment 1 of this invention.

[0089] The relevant technical solutions are the same as in Embodiment 1, and will not be repeated here.

[0090] Example 4

[0091] A feedforward controller for a piezoelectric ceramic actuator is provided for executing the feedforward control method provided in Example 3.

[0092] The relevant technical solutions are the same as in Embodiment 3, and will not be repeated here.

[0093] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A nonlinear modeling method for piezoelectric ceramic actuators, characterized in that, Includes the following steps: S1. Establish a fractional-order coupling model for simulating the hysteresis and creep nonlinear characteristics of piezoelectric ceramic actuators; The fractional-order coupling model includes a cascaded fractional-order hysteresis model and a fractional-order creep model; the fractional-order hysteresis model includes a parallel linear module and a nonlinear module; the linear module is used to represent the ideal linear relationship between displacement and voltage; the nonlinear module represents the nonlinear relationship between displacement and voltage by establishing an equality relationship between the weighted summation results of displacement and multiple voltage fractional-order differential operators. S2. Based on the input voltage and output displacement data of the simulated piezoelectric ceramic actuator, identify the parameters in the fractional-order coupling model; The control equation for the nonlinear module is: in, This refers to the nonlinear displacement under hysteresis characteristics. for The nonlinear coefficient; for α Results of the fractional-order voltage fractional-order calculus operator; It is a voltage signal; for The nonlinear coefficient; It is the absolute value of the rate of change of the voltage signal; for β Results of the fractional-order voltage fractional-order calculus operator.

2. The nonlinear modeling method according to claim 1, characterized in that, ; ;in, It is the gamma function; n It is the nearest integer order.

3. The nonlinear modeling method according to claim 1, characterized in that, The control equations for the linear module are: in, The linear displacement under hysteresis characteristics; The coefficients are linear. It is a voltage signal.

4. The nonlinear modeling method according to any one of claims 1-3, characterized in that, The time-domain governing equations of the fractional-order creep model are as follows: in, The displacement in the time domain under creep characteristics; The creep coefficient; ; It is the intermediate displacement state output of the fractional-order hysteresis model; For creep fractional-order parameters.

5. The nonlinear modeling method according to any one of claims 1-3, characterized in that, The frequency domain control equation of the fractional creep model is: in, The Laplace transform of the displacement under creep characteristics; The creep coefficient; s It is the Laplace constant; For creep fractional-order parameters.

6. The nonlinear modeling method according to claim 5, characterized in that, The methods for identifying the fractional-order creep parameters in the fractional-order creep model include: By controlling the frequency of the voltage signal input to the piezoelectric ceramic actuator, the corresponding displacement frequency domain data can be obtained; By using the voltage signal frequency and the corresponding displacement frequency domain data, the parameters in the structural dynamic equation of the piezoelectric ceramic actuator are fitted to obtain the creep fractional-order parameters in the fractional-order creep model.

7. A computer-readable storage medium, characterized in that, The computer-readable storage medium includes a stored computer program, wherein, when the computer program is run by a processor, it controls the device containing the storage medium to execute the modeling method for hysteresis and creep characteristics of piezoelectric ceramic actuators as described in any one of claims 1-6.

8. A feedforward control method for a piezoelectric ceramic actuator, characterized in that, include: An inverse model of the fractional-order coupling model is established, and the inverse model is used to compensate for the hysteresis nonlinearity and creep nonlinearity of the piezoelectric ceramic actuator. The fractional-order coupling model is constructed using the nonlinear modeling method described in any one of claims 1-6.

9. A feedforward controller for a piezoelectric ceramic actuator, characterized in that, Used to perform the feedforward control method as described in claim 8.

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