Piezoelectric hysteresis loop characteristic compensation method based on temperature change PI model

Abstract

The invention discloses a piezoelectric hysteresis loop characteristic compensation method based on a temperature change PI model, and aims to solve the problem that the compensation precision is reduced due to the fact that an existing piezoelectric hysteresis loop model cannot track the temperature influence. The method mainly comprises five steps of hysteresis loop characteristic data acquisition, temperature change characteristic data acquisition, temperature change PI model modeling, parameter identification and compensation application and the like. Wherein the temperature change PI model is the core of the method, the material science theory is creatively used as a model transformation basis, and the reasonability of modeling and the accuracy of parameter identification can be fully demonstrated. The method has the compensation precision not lower than that of similar methods under the condition of constant temperature, and has the capability of tracking temperature change and eliminating influence under the condition of temperature change. Therefore, the actual working requirements are met more widely, and optimistic popularization and application prospects are shown.

Description

Technical Field

[0001] The present invention belongs to the technical field of control engineering, and in particular relates to a piezoelectric hysteresis loop characteristic compensation method based on a temperature-dependent PI model. Background Art

[0002] Compensating for piezoelectric hysteresis loop characteristics is the most common method for eliminating nonlinear characteristics in piezoelectric actuators. The feedforward control employed is a pre-set compensation mechanism based on the known behavior of the controlled object. This approach is highly suited to the characteristics of direct-drive piezoelectric devices, such as piezoelectric actuators. These piezoelectric actuators are important actuators in semiconductor, optical, and aerospace precision engineering. They are characterized by their small size, lightweight, fast response, low heat generation, and non-magnetic properties, capable of outputting nanometer-scale displacements. Therefore, the importance of piezoelectric compensation technology is self-evident.

[0003] However, with the continuous improvement of precision requirements, namely sub-nanometer scale requirements, the influence of temperature on piezoelectric actuation ability is significantly amplified; considering that the existing operator model can better solve the common hysteresis loop phenomenon, modifying it and innovating a new compensation method that can be correlated with temperature is the most stable and effective strategy to solve this problem. Summary of the Invention

[0004] The purpose of the present invention is to propose a piezoelectric hysteresis loop characteristic compensation method based on a temperature-dependent PI model, which is used to improve the problem that the existing piezoelectric hysteresis loop model cannot track the influence of temperature and the compensation accuracy is reduced. It mainly includes five steps: hysteresis loop characteristic data acquisition, temperature change characteristic data acquisition, temperature-dependent PI model modeling, parameter identification and compensation application. Among them, the temperature-dependent PI model is the core of the present invention. It innovatively uses material science theory as the basis for model transformation, and can fully demonstrate the rationality of modeling and the accuracy of parameter identification. The present invention has a compensation accuracy that is not lower than that of similar methods under constant temperature conditions, and has the ability to track temperature changes and eliminate the effects under temperature changes. Therefore, the present invention more widely meets the actual work needs and shows an optimistic prospect for popular application.

[0005] Those skilled in the art will understand that before elaborating on the details of the present invention, the following three parts should be considered as basic common sense:

[0006] First, the piezoelectric actuation process uses the inverse piezoelectric effect of the piezoelectric actuating material as an output characteristic. Specifically, when a voltage is applied to the piezoelectric actuating material, a nanometer-scale micro-displacement is output.

[0007] Secondly, the piezoelectric hysteresis loop characteristic is a common feature of piezoelectric actuating materials represented by PZT. Specifically, the voltage-displacement relationship in the rising and falling sections is characterized by hysteresis, forming a counterclockwise hysteresis loop.

[0008] Third, the Prandtl-Ishlinskii (PI) model is the most commonly used operator model in piezoelectric compensation technology, which has the advantages of simple mathematical formula, invertibility, and easy improvement.

[0009] The technical solution of the present invention is specifically described as follows.

[0010] A method for compensating piezoelectric hysteresis loop characteristics based on a temperature-dependent PI model includes the following steps:

[0011] Step S1: Acquiring hysteresis loop characteristic data: Acquiring the characteristics of the reciprocating motion during the piezoelectric actuation process;

[0012] Step S2: collecting temperature variation characteristic data: obtaining the monotonic nonlinear characteristics of the piezoelectric actuation process in the operating temperature range;

[0013] Step S3: Temperature-dependent PI model building: First, a Prandtl-Ishlinskii model (PI model for short) is established. Then, the PI model is modified based on the temperature-dependent theory of materials science that conforms to the operating temperature range. The temperature-dependent theory of materials science is simplified into a temperature-dependent coefficient or formula so that the PI model has different vertical heights at different temperatures.

[0014] Step S4: Parameter identification: Identify model parameters. After identification, the error between the model and the characteristic data is no greater than the specified requirement. The temperature-dependent PI model obtained after the parameters are determined is a three-dimensional surface.

[0015] Step S5: Compensation application: A feedforward control method is used to apply the temperature-dependent PI inverse model with determined model parameters to perform compensation application, thereby completing the hysteresis loop characteristic compensation.

[0016] The working principle of this invention is as follows: the PI model is used to compensate for nonlinear characteristics primarily characterized by hysteresis, and a temperature-dependent coefficient or formula is used to achieve temperature change tracking. Materials science analysis indicates that the effect of temperature changes on actuation capacity is monotonic within the general operating temperature range. Therefore, a three-dimensional temperature-dependent PI model surface can be generated, and the model can be intercepted according to actual temperature conditions to implement compensation applications. The PI model has the advantages of simple mathematical formula, reversibility, and ease of improvement. Using the model's feedforward control method is the most direct compensation application method, and these two factors further enhance the stability of the method.

[0017] In the present invention, step S3 is based on the temperature theory of materials science, the data of step S1 and step S2 serve step S4, step S4 obtains accurate parameters for step S3, and step S5 adopts a feedforward control method to apply a temperature-varying PI inverse model with determined model parameters.

[0018] In the present invention, in step S1, the hysteresis loop characteristic data is collected in the form of a ring-shaped voltage-displacement curve with hysteresis as the primary nonlinear characteristic. In the embodiment, the data is collected by collecting the hysteresis loop characteristics of the piezoelectric actuator at room temperature (20°C) using the positive and negative voltage extremes, and plotting the voltage-displacement data.

[0019] In the present invention, in step S2, the operating temperature range is -40-110°C; the temperature change characteristic data collection is presented in the form of a temperature-displacement curve in which the actuation capacity increases with temperature at -40-110°C.

[0020] In the present invention, the test standards of step S1 and step S2 should be the same. Specifically, data collection uses the controlled variable method, and the experimental process strictly follows the IEEE standard.

[0021] In the present invention, in step S3, the temperature-varying PI model is modeled based on a traditional PI model or a rate-dependent PI model.

[0022] In the present invention, in the temperature PI model of step S3, the material science theory is consistent with the operating temperature range. Specifically, the material science theory of the -40-110°C stage is used as the basis of the model.

[0023] In the present invention, in step S3, the method for modifying the PI model based on the temperature-related theory of materials science is as follows:

[0024] In the framework of Ginzburg-Landau theory, the electric field strength E of PZT piezoelectric material is derived. Z and polarization field strength P Z The relationship is:

[0025] E Z =αP Z +βP Z 3

[0026] Since the polarization direction of the piezoelectric ceramic is the same as the actuation direction, α and β are equivalent to the standard Landau coefficients of the Gibbs free energy equation; when E Z = 0, the polarization field strength P Z Equal to the spontaneous polarization intensity P S ; Remove the 0 solution and get the non-zero solution:

[0027]

[0028] Among them, c c is the Curie constant. The piezoelectric ceramics at the working temperature are always in the ferroelectric phase, so the Curie constant c c Constant; T C is the Curie temperature of the piezoelectric ceramic; ε0≈8.854×10 -12F / m is the constant value of the vacuum dielectric constant; then according to the second kind of piezoelectric equation, the electric displacement D related to temperature T is obtained ij (E i ,T) is:

[0029]

[0030] Among them, E i is the electric field strength in the direction of voltage application, d ij is the piezoelectric constant, ξ j is the elastic modulus in the actuation direction, S ij is the piezoelectric ceramic strain; simplified coefficient ρ is the proportional coefficient of the effect of spontaneous polarization on electric displacement;

[0031] By further simplifying the temperature theory of material science into the temperature formula mentioned above, the PI model is transformed to obtain the temperature-dependent PI model.

[0032] In the present invention, in step S3, after the temperature-varying PI model is built, the weighted superposition term of the PI operators in the PI model composed of n operators is becomes:

[0033]

[0034] Where: η is the temperature state supplementary parameter, satisfying 0<η<1, ω i are the weights of the i-th operator, Ω i (t) is the output value of the i-th operator, T C is the Curie temperature of the piezoelectric ceramic, T is the operating temperature, and the simplified coefficient ρ is the proportional coefficient of the effect of spontaneous polarization on electric displacement, β is the standard Landau coefficient of the Gibbs free energy equation, and c c is the Curie constant, and ε0 is the constant of the vacuum dielectric constant.

[0035] In the present invention, in step S4, a particle swarm algorithm (PSO) or a quadratic programming algorithm is used to identify model parameters of the temperature-dependent PI model, where the model parameters include weight threshold parameters and temperature formula parameters.

[0036] In the present invention, the order of step S1, step S2, step S3, step S4, and step S5 is changeable. Specifically, the three steps of data acquisition and modeling are not in a particular order, and the temperature-varying PI model must exist before the parameter identification of step S4 is performed, and accurate model parameters must be obtained before the compensation application of step S5 is performed. For example: (1) step S2, step S1, step S3, step S4, and step S5, or (2) step S3, step S1, step S2, step S4, and step S5, or (3) step S3, step S2, step S1, step S4, and step S5.

[0037] In the present invention, in step S3, a traditional PI model may be used for modeling, or further modifications may be made based on other types of PI models.

[0038] The first and second embodiments of the present invention provide a method for compensating the piezoelectric hysteresis loop characteristics based on a temperature-dependent PI model: from the perspective of the composition of the embodiments, the first embodiment uses the traditional PI model as the modification object, and implements compensation for the piezoelectric actuator in the order of step S1, step S2, step S3, step S4, and step S5; the second embodiment uses the rate-dependent PI model as the modification object, and implements compensation for the tangential piezoelectric stack in the order of step S3, step S1, step S2, step S4, and step S5. In the first embodiment, the sequence and internal logic of the compensation method of the present invention are illustrated, the application of feedforward control is illustrated, and the results of implementing the technology are illustrated. The second embodiment only gives the sequence of the compensation method, but the control method of the second embodiment is the same as that of the first embodiment.

[0039] Compared with the existing piezoelectric compensation technology, the beneficial effects of the present invention are:

[0040] The core innovation, which utilizes materials science theory to transform the PI model, addresses the critical issue of the inability of current piezoelectric hysteresis loop characteristic models to compensate for temperature changes. The accuracy of feedforward control depends on the precision of the model, and its integration with control engineering makes the compensation method of the present invention highly practical and repeatable. The present invention covers nearly all application scenarios of piezoelectric devices, such as piezoelectric actuators, piezoelectric stacks, and piezoelectric motors, and even some devices used in reluctance motors, demonstrating broad industry adaptability. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 Schematic diagram of a piezoelectric hysteresis loop characteristic compensation method based on a temperature-dependent PI model provided in Example 1 of the present invention.

[0042] Figure 2 Schematic diagram of the feedforward control method provided in the first embodiment of the present invention.

[0043] Figure 3 This is a schematic diagram of the modeling results provided in Example 1 of the present invention.

[0044] Figure 4 Schematic diagram of a piezoelectric hysteresis loop characteristic compensation method based on a temperature-dependent PI model provided in the second embodiment of the present invention. DETAILED DESCRIPTION

[0045] The present invention will be further described in detail below with reference to the accompanying drawings and examples. It will be understood that the specific embodiments described herein are intended only to illustrate the present invention and are not intended to limit the present invention. It should also be noted that, for ease of description, the accompanying drawings only show portions relevant to the present invention, not all details.

[0046] In the description of the embodiments of the present invention, the terms "upper", "lower", "right", "inside", "outside" and other directions or positional relationships are based on the directions or positional relationships shown in the accompanying drawings and are only for the convenience of description and simplification of operation. The terms "first" and "second" are only used to distinguish in the description and have no special meaning.

[0047] Example 1

[0048] Figure 1 This is a method for compensating piezoelectric hysteresis loop characteristics based on a temperature-dependent PI model, provided in Example 1 of the present invention. The method comprises steps S1: collecting hysteresis loop characteristic data; S2: collecting temperature-dependent characteristic data; S3: building a temperature-dependent PI model; S4: parameter identification; and S5: applying compensation. In Example 1, the compensation target is a piezoelectric actuator with a thickness of 5 mm and a travel of approximately 250 nanometers from zero voltage to voltage extremes at room temperature of 20°C. The control system uses a real-time target computer as the controller, and the control signal is amplified by a voltage driver before being applied to the piezoelectric actuator.

[0049] Specifically, in step S1, the hysteresis loop characteristics of the piezoelectric actuator are collected through the positive and negative extremes of the voltage at room temperature (20° C.), and a voltage-displacement data graph is plotted.

[0050] Specifically, in step S2, the positive and negative extreme values ​​of the voltage are kept constant, the loading frequency is kept constant, and the hysteresis loop characteristics of the piezoelectric actuator are collected at different temperatures within the working range, wherein the experimental data at -10°C and 50°C are selected for plotting.

[0051] Specifically, step S3 uses the temperature theory of materials science as a basis to establish a temperature-dependent PI model, including:

[0052] Step S3.1: Establish a traditional Prandtl-Ishlinskii (PI) model. Select the Play operator as the operator of the PI model in Example 1, and its formula is:

[0053] Ω(t)=max{x(t)-r,min[x(t)+r,Ω(t-1)]}

[0054] Where r is the threshold, x(t) is the input at time t, and the value of t satisfies t∈[0,t m ], is the input signal x(t) at 0=t0 <t1<t2<…<t m The division on the interval. When t = 0, Ω(t-1) takes the initial value, which is determined by the initial value of the displacement. Ω(t) is the output corresponding to the input signal x(t). Using a finite number of operators with different thresholds r, multiplying them by weights ω to change the slope, and finally superimposing and summing them to obtain the PI model, the formula is:

[0055]

[0056] Among them, P[x(t)] is the PI model composed of n operators, ω0 is the weight of input x(t), ω i are the weights of the i-th operator, Ω i (t) is the output value of the i-th operator, and each operator contains an independent threshold r i Threshold r i and weight ω i In both cases, the specific values ​​need to be obtained through parameter identification.

[0057] Step S3.2: Modify the PI model based on the temperature-related theory of materials science. Under the framework of the Ginzburg-Landau theory, the electric field intensity E of the PZT piezoelectric material can be derived. Z and polarization field strength P Z The relationship is:

[0058] E Z =αP Z +βP Z 3

[0059] Since the polarization direction of the piezoelectric ceramic is the same as the actuation direction, α and β are equivalent to the standard Landau coefficients of the Gibbs free energy equation. α is directly related to temperature. When E Z = 0, the polarization field strength P Z Equal to the spontaneous polarization intensity P S . Remove the 0 solution and get the non-zero solution:

[0060]

[0061] Among them, c c is the Curie constant. The piezoelectric ceramics at the working temperature are always in the ferroelectric phase, so the Curie constant c c Constant. Cis the Curie temperature of the piezoelectric ceramic. ε0≈8.854×10 -12 F / m is the constant value of the vacuum dielectric constant. According to the second kind of piezoelectric equation, the temperature-dependent electric displacement D is obtained. ij (E i ,T) is:

[0062]

[0063] Among them, E i is the electric field strength in the direction of voltage application, ξ j is the elastic modulus in the actuation direction, S ij is the strain of the piezoelectric ceramic. Simplified coefficient ρ is the proportional coefficient of the effect of spontaneous polarization on electric displacement.

[0064] The above material science theory is simplified into a temperature formula, and the temperature-dependent PI model formula obtained after modification is:

[0065]

[0066] Wherein, η is a temperature state supplementary parameter, satisfying 0<η<1.

[0067] Specifically, step S4 is used to identify the temperature-dependent PI model parameters in step S3. Example 1 uses the PSO algorithm for identification, and the code toolbox can be called directly. The parameters to be identified include two parts: the PI model weight threshold parameters and the temperature formula parameters. Among them, the weight threshold has a total of 10 groups of 20 parameters, which are too many to be displayed in detail. The two parameters of the temperature formula are μ=1.65 and η=0.89. The identification result is substituted into the temperature-dependent PI model of step S3 to obtain a specific model. The model appears as a curved surface in three-dimensional space, with different longitudinal heights at different temperatures.

[0068] Specifically, step S5 utilizes feedforward control to implement compensation application, including:

[0069] Step S5.1: The temperature sensor is connected to the control system.

[0070] Step S5.2: According to the feedback value of the temperature sensor, a temperature-dependent PI model captured at the current temperature is selected.

[0071] Step S5.3: Convert the intercepted temperature-dependent PI model into an inverse model, and use the compensation voltage to control the piezoelectric actuator in a feedforward control link.

[0072] Figure 2This is a schematic diagram of the feedforward control method provided in Example 1 of the present invention, which further illustrates the implementation of step S5 in Example 1. The specific principle is: temperature information is fed back to the real-time target machine, and the hysteresis loop characteristics intercepted from the temperature-dependent PI model are the expected characteristics of the piezoelectric actuator at the current temperature; the inverse model of the expected characteristics and the actual characteristics cancel each other out, so that the expected trajectory is converted into the required compensation voltage after the inverse model; the compensation voltage acts on the piezoelectric actuator, so that the output of the piezoelectric actuator is linear, completing the hysteresis loop characteristic compensation. The difference between it and feedback control is that feedback control changes the model through the direct displacement result of the piezoelectric actuator, while feedforward control only uses temperature information, has a faster response and does not fluctuate around the expected trajectory.

[0073] Figure 3 This is a schematic diagram of the modeling results provided by Example 1 of the present invention, to further demonstrate the effect of Example 1. Figure 3 The left side shows the modeling results of the PI model, which appears as a two-dimensional circular curve. Because the PI model uses the hysteresis loop characteristics at room temperature for parameter identification, the results are similar to those at 20°C. When the temperature drops to -10°C, the PI model clearly loses its tracking ability, with its maximum error and mean square error increasing by 2.67% and 6.38%, respectively, compared to room temperature. Figure 3 The right side shows the modeling results of the temperature-dependent PI model, which exhibits a three-dimensional annular surface. Even with the addition of a 50°C characteristic curve, the tracking capability of the temperature-dependent PI model is still observed, with overall accuracy improved by 6.4% compared to the traditional PI model. Model fitting experiments demonstrate that the McPI model maintains overall modeling accuracy while also tracking temperature changes. This also indirectly demonstrates that its feedforward compensation results will also achieve higher control accuracy.

[0074] Example 2

[0075] Figure 4 The second embodiment of the present invention provides a method for compensating piezoelectric hysteresis loop characteristics based on a temperature-dependent PI model, comprising step S3: temperature-dependent PI model building, step S1: hysteresis loop characteristic data collection, step S2: temperature-dependent characteristic data collection, step S4: parameter identification, and step S5: compensation application. The compensation target in this second embodiment is a tangential piezoelectric stack with a length of 10 mm and a travel of approximately 0.5 microns from zero voltage to voltage extremes at room temperature (20°C). The controller uses the real-time target machine used in the verification method, applying voltage to the tangential piezoelectric stack through voltage drive amplification.

[0076] Specifically, in step S1 , the hysteresis loop characteristics of the tangential piezoelectric stack are collected at room temperature (20° C.) by using the positive and negative extremes of the voltage, and then the voltage-displacement data are maintained and plotted.

[0077] Specifically, in step S2, the positive and negative voltage extremes are kept constant, the loading frequency is kept constant, and the hysteresis loop characteristics of the tangential piezoelectric stack are collected at different temperatures within the working range, wherein the experimental data at -10°C and 50°C are selected for plotting.

[0078] Specifically, step S3 uses the temperature theory of materials science as a basis to establish a temperature-dependent PI model, including:

[0079] Step S3.1: Establish a rate-dependent Prandtl-Ishlinskii (PI) model. Select the Play operator as the operator of the PI model in Example 2, and its formula is:

[0080]

[0081] in, is the rate correlation coefficient, and z is the rate-related parameter, ω f is the modified initial weight, and the explanations of other parameters are the same as those in the first embodiment.

[0082] Step S3.2: Modify the PI model based on the temperature-related theory of materials science. The temperature-dependent PI model formula obtained after the modification is:

[0083]

[0084] Specifically, step S4 is used to identify the temperature-dependent PI model parameters in step 3. Example 2 uses a quadratic programming algorithm for identification, which can be directly called from the code toolbox. The parameters to be identified include the PI model weight threshold parameter, rate-related parameters, and temperature formula parameters.

[0085] Specifically, step S5 utilizes feedforward control to implement compensation. The following steps are performed: Step S5.1: Connecting a temperature sensor to the control system. Step S5.2: Selecting a temperature-dependent PI model captured at the current temperature based on the temperature sensor's feedback value. Step S5.3: Converting the captured temperature-dependent PI model into an inverse model, and using the compensation voltage in the feedforward control phase to control the piezoelectric actuator.

[0086] The second embodiment of the present invention satisfies the same control method as the first embodiment, so it will not be repeated here. The difference between the second embodiment and the first embodiment is that the second embodiment uses a rate-dependent PI model, which is a more advanced compensation technology. Its advantage is that it can also compensate for frequency effects and can adapt to control requirements at different speeds.

[0087] It should be noted that the original model established in other embodiments may be a generalized PI model or a modified PI model, etc.; the selected temperature-related theoretical basis may also come from the demonstration of formulas in other fields other than material science theory; the control method used in other embodiments may also be a model combined with a forward feedback link or other forms, as long as it meets the characteristics matching the operator model.

[0088] Note that the above are only reasonable embodiments of the present invention and the technical principles used. Those skilled in the art will understand that the present invention is not limited to the specific embodiments described herein, and that various obvious changes, readjustments, combinations, and substitutions can be made by those skilled in the art without departing from the scope of protection of the present invention. Therefore, although the present invention has been described in more detail through the above embodiments, the present invention is not limited to the above embodiments. Without departing from the concept of the present invention, it may also include more other equivalent embodiments, and the scope of the present invention is determined by the scope of the appended claims.

Claims

1. A piezoelectric hysteresis loop characteristic compensation method based on a temperature-dependent PI model, characterized in that: The following steps are involved: Step S1: Acquiring hysteresis loop characteristic data: Acquiring the characteristics of the reciprocating motion during the piezoelectric actuation process; Step S2: collecting temperature variation characteristic data: obtaining the monotonic nonlinear characteristics of the piezoelectric actuation process in the operating temperature range; Step S3: Temperature-dependent PI model building: First, establish a Prandtl-Ishlinskii model, i.e., a PI model. Then, modify the PI model based on the temperature-dependent theory of materials science that conforms to the operating temperature range. The temperature-dependent theory of materials science is simplified into a temperature-dependent coefficient or formula, so that the PI model has different vertical heights at different temperatures. Step S4: parameter identification: identifying model parameters. After determining the parameters, the temperature-dependent PI model obtained presents a three-dimensional surface. Step S5: Compensation application: A feedforward control method is used to apply the temperature-dependent PI inverse model with determined model parameters to perform compensation application, thereby completing the hysteresis loop characteristic compensation.

2. The method for compensating piezoelectric hysteresis loop characteristics based on a temperature-dependent PI model according to claim 1, characterized in that: In step S1 , the hysteresis loop characteristic data is collected in the form of a ring voltage-displacement curve with hysteresis as the main nonlinear characteristic.

3. The method for compensating piezoelectric hysteresis loop characteristics based on a temperature-dependent PI model according to claim 1, wherein: In step S2, the operating temperature range is -40-110°C; the temperature change characteristic data collection is presented in the form of a temperature-displacement curve in which the actuation capacity increases with temperature at -40-110°C.

4. The method for compensating piezoelectric hysteresis loop characteristics based on a temperature-dependent PI model according to claim 1, wherein: In step S3, the temperature-dependent PI model is modeled based on a traditional PI model or a rate-dependent PI model.

5. The method for compensating piezoelectric hysteresis loop characteristics based on a temperature-dependent PI model according to claim 1, wherein: In step S3, the method for modifying the PI model based on the temperature-related theory of materials science is as follows: In the framework of Ginzburg-Landau theory, the electric field strength E of PZT piezoelectric material is derived. Z and polarization field strength P Z The relationship is: E Z =αP Z +βP Z 3 Since the polarization direction of the piezoelectric ceramic is the same as the actuation direction, α and β are equivalent to the standard Landau coefficients of the Gibbs free energy equation; when E Z = 0, the polarization field strength P Z Equal to the spontaneous polarization intensity P S ; Remove the 0 solution and get the non-zero solution: Among them, c c is the Curie constant. The piezoelectric ceramics at the working temperature are always in the ferroelectric phase, so the Curie constant c c Constant; T C is the Curie temperature of the piezoelectric ceramic; ε0≈8.854×10 -12 F / m is the constant value of the vacuum dielectric constant; then according to the second kind of piezoelectric equation, the electric displacement D related to the working temperature T is obtained ij (E i ,T) is: Among them, E i is the electric field strength in the direction of voltage application, d ij is the piezoelectric constant, ξ j is the elastic modulus in the actuation direction, S ij is the piezoelectric ceramic strain; simplified coefficient ρ is the proportional coefficient of the effect of spontaneous polarization on electric displacement; By further simplifying the temperature theory of material science into the temperature formula mentioned above, the PI model is transformed to obtain the temperature-dependent PI model.

6. The method for compensating piezoelectric hysteresis loop characteristics based on a temperature-dependent PI model according to claim 1, wherein: In step S3, after the temperature-varying PI model is built, the weighted superposition term of the PI operators in the PI model composed of n operators is becomes: Where: η is the temperature state supplementary parameter, satisfying 0<η<1, ω i are the weights of the i-th operator, Ω i (t) is the output value of the i-th operator, T C is the Curie temperature of the piezoelectric ceramic, T is the operating temperature, and the simplified coefficient ρ is the proportional coefficient of the effect of spontaneous polarization on electric displacement, β is the standard Landau coefficient of the Gibbs free energy equation, and c c is the Curie constant, and ε0 is the constant of the vacuum dielectric constant.

7. The method for compensating piezoelectric hysteresis loop characteristics based on a temperature-dependent PI model according to claim 1, wherein: In step S4, a particle swarm algorithm (PSO) or a quadratic programming algorithm is used to identify model parameters of the temperature-dependent PI model, where the model parameters include weight threshold parameters and temperature formula parameters.

8. The method for compensating piezoelectric hysteresis loop characteristics based on a temperature-dependent PI model according to claim 1, characterized in that: Step S5 specifically includes: Step S5.1: The temperature sensor is connected to the control system; Step S5.2: Based on the feedback value of the temperature sensor, a temperature-dependent PI model captured at the current temperature is selected as the model used in the feedforward control; Step S5.3: converting the intercepted temperature-dependent PI model into an inverse model and using it as a feedforward link to compensate the controlled object.

9. The method for compensating piezoelectric hysteresis loop characteristics based on a temperature-dependent PI model according to claim 1, characterized in that: The order of step S1, step S2, step S3, step S4, and step S5 is changeable; specifically, the three steps of data acquisition and modeling are not prioritized, and the temperature-dependent PI model must exist before the parameter identification of step S4 is performed, and accurate model parameters must be obtained before the compensation application of step S5 is performed.

Images