Piezoelectric actuator hysteresis modeling and control method based on trdpi model
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- CHINA JILIANG UNIV
- Filing Date
- 2026-05-28
- Publication Date
- 2026-06-26
Smart Images
Figure CN122284352A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of micro-nano drive and precision control technology, specifically relating to a piezoelectric actuator hysteresis modeling method and a feedforward compensation control method based on the model. Background Technology
[0002] Piezoelectric actuators are widely used in precision positioning, micro-nano manipulation, ultra-precision machining, and precision measurement due to their advantages such as high resolution, fast response speed, compact structure, and immunity to electromagnetic interference. In practical applications, piezoelectric actuators typically achieve minute displacement output by adjusting the input voltage, and the accuracy of their displacement control has a significant impact on the overall system performance.
[0003] However, piezoelectric actuators generally exhibit significant hysteresis nonlinearity during operation, meaning the output displacement depends not only on the current input voltage but also on the historical changes in the input voltage. This hysteresis nonlinearity leads to deviations between the input and output, reducing positioning accuracy and even causing instability in the control system. Therefore, accurate modeling and effective compensation of the hysteresis characteristics of piezoelectric actuators have always been a key research focus in the field of precision drive and control.
[0004] To describe the hysteresis characteristics of piezoelectric actuators, researchers have proposed various hysteresis modeling methods. Among them, the Prandtl-Ishlinskii (PI) model is widely used for hysteresis modeling and feedforward control of piezoelectric actuators due to its simple structure, clear physical meaning of parameters, and ease of constructing an inverse model. Addressing the issue of hysteresis characteristics varying with excitation frequency under dynamic conditions, some studies have introduced a rate-dependent threshold into the PI model, forming a rate-dependent Prandtl-Ishlinskii (RDPI) model to characterize the influence of excitation frequency variations on the hysteresis loop width and shape.
[0005] However, experimental studies have shown that the hysteresis characteristics of piezoelectric actuators are not only affected by the excitation frequency but also closely related to the ambient temperature. With temperature changes, the electromechanical coupling characteristics of the piezoelectric material change, leading to significant drift in the output displacement amplitude and hysteresis loop shape. Under high-frequency drive or long-term operating conditions, piezoelectric actuators are often in an electro-thermal coupling environment, and their hysteresis behavior exhibits obvious temperature-frequency coupling characteristics.
[0006] Existing hysteresis modeling methods often focus on single-factor modeling, such as considering only frequency-dependent effects or introducing only temperature correction parameters, making it difficult to accurately describe hysteresis characteristics under conditions where temperature and frequency change simultaneously. Furthermore, traditional models are prone to significant modeling errors in the input signal commutation region, further affecting the feedforward compensation control effect. Therefore, it is necessary to propose a piezoelectric actuator hysteresis modeling and control method that can simultaneously consider the coupling effects of temperature and excitation frequency and has high modeling accuracy. Summary of the Invention
[0007] To address the aforementioned deficiencies in existing technologies, this invention provides a piezoelectric actuator hysteresis modeling and control method based on the TRDPI model, which can simultaneously consider the coupling effects of temperature and excitation frequency, and accurately describe the hysteresis characteristics under conditions of co-changing temperature and frequency.
[0008] To achieve the above objectives, the present invention adopts the following technical solution: This invention provides a piezoelectric actuator hysteresis modeling method based on the TRDPI model, the method comprising the following steps: Step 1: Construct a dynamic threshold function related to the input voltage change rate and ambient temperature to characterize the temperature-frequency coupling effect of the piezoelectric actuator's hysteresis characteristics. The dynamic threshold function is a function that includes the input voltage change rate and temperature. Step 2: Construct the upper and lower envelope functions in the form of Fermi-Dirac distribution to describe the asymmetric characteristics of the piezoelectric actuator hysteresis loop; Step 3: Based on the dynamic threshold function, upper envelope function, and lower envelope function, construct the improved generalized Play operator; Step 4: Construct a memoryless nonlinear input function for nonlinear mapping of the input voltage; Step 5: Construct an inflection point compensation term to correct the modeling error in the hysteresis loop commutation region of the piezoelectric actuator; Step 6: The outputs of each Play operator are superimposed according to preset weights to construct the TRDPI hysteresis model. The TRDPI hysteresis model is used to calculate the output displacement of the piezoelectric actuator based on the input voltage signal.
[0009] Preferably, the TRDPI hysteresis model constructed in step 6 is as follows: The initial output displacement is set to zero. At any time t, the output value of the memoryless nonlinear input function, the output value of the Play operator after each correction weight, and the output value of the inflection point compensation term are summed to obtain the output displacement of the piezoelectric actuator at time t.
[0010] Preferably, the correction weight coefficient of the Play operator exhibits an exponentially decreasing trend as the threshold increases.
[0011] Preferably, the upper envelope function and the lower envelope function are constructed using nonlinear functions based on S-curves, and are used to fit the rising edge boundary and falling edge boundary of the hysteresis loop, respectively.
[0012] Preferably, the memoryless nonlinear input function is configured to use a first nonlinear mapping relationship when the input voltage change rate is positive, and a second nonlinear mapping relationship when the input voltage change rate is negative.
[0013] Preferably, the inflection point compensation term is configured to apply a secondary compensation correction based on the rate of change to the output displacement when the input voltage is above 75% of the maximum voltage and a change occurs, and when the input voltage is below 25% of the maximum voltage and a change occurs.
[0014] Preferably, the model parameters of the TRDPI model include weight threshold, envelope function, memoryless nonlinear input function, and inflection point compensation term, and the model parameters are identified by the dung beetle algorithm.
[0015] This invention also provides a feedforward control method for piezoelectric actuators based on the TRDPI model, the method specifically including the following steps: Step A: Construct the dynamic threshold of the inverse model, which is determined based on the dynamic threshold and weights of the forward model through inverse and recursive relationships; Step B: Construct the inverse Play operator, which uses the desired displacement as the center value of the boundary constraint; Step C: Construct the weights of the inverse TRDPI model; Step D: Construct the TRDPI inverse model based on the inverse Play operator, use the TRDPI inverse model to perform feedforward control on the piezoelectric actuator, and solve for the required input voltage based on the desired displacement.
[0016] Preferably, the model parameters of the inverse TRDPI model include weight threshold parameters, and the model parameters of the inverse TRDPI model are identified by the Dung Beetle Algorithm (DBO).
[0017] Preferably, the specific process of feedforward control in step D is as follows: In the control process of the piezoelectric actuator, the input of the feedforward controller of the piezoelectric actuator is the desired output displacement of the piezoelectric actuator; the desired output displacement is input into the TRDPI inverse model to obtain the output voltage of the feedforward controller of the piezoelectric actuator; the output voltage is input into the piezoelectric actuator to obtain the actual output displacement of the piezoelectric actuator, thus realizing the output displacement control of the piezoelectric actuator.
[0018] The beneficial effects of this invention are: 1. This invention improves the dynamic threshold function in the hysteresis model by introducing a temperature-frequency coupling effect, thus solving the problem of hysteresis rate variation caused by changes in excitation frequency and ambient temperature in existing technologies. This improvement can accurately describe the hysteresis characteristics of piezoelectric actuators under different operating conditions.
[0019] 2. By introducing a nonlinear envelope function based on the Fermi-Dirac distribution, the model's ability to fit asymmetric hysteresis loops is enhanced, avoiding the errors caused by traditional symmetric models, thereby improving the overall modeling accuracy.
[0020] 3. By adopting an inflection point compensation term, the modeling error near the input signal reversal point is significantly reduced, the fitting accuracy of the model in key regions is enhanced, and the overall smoothness of the model is improved.
[0021] 4. This invention achieves high-precision feedforward compensation control of piezoelectric actuators by constructing an inverse model based on the TRDPI model. This method, by calculating the feedforward voltage input for the desired output displacement, can significantly reduce the hysteresis nonlinearity of the piezoelectric actuator, making the actual output displacement and the desired displacement exhibit a linear relationship, thereby improving displacement control accuracy and having significant engineering application value in high-precision control applications. Attached Figure Description
[0022] Figure 1 This is a schematic diagram of feedforward control of the piezoelectric actuator in this invention; Figure 2 The diagram shows the control effect of the TRDPI inverse model at different frequencies at room temperature in this invention. Figure 3 A comparison of the fitting effects of the traditional RDPI model and the TRDPI model in this invention at 30℃ is shown. Figure 4 A comparison of the fitting effects of the traditional RDPI model and the TRDPI model in this invention at 40℃ is shown. Figure 5 A comparison of the fitting effects of the traditional RDPI model and the TRDPI model in this invention at 50℃ is shown. Figure 6 A comparison of the fitting errors between the traditional RDPI model and the TRDPI model of this invention at 30℃; Figure 7 A comparison of the fitting errors between the traditional RDPI model and the TRDPI model of this invention at 40℃; Figure 8 The graph shows a comparison of the fitting errors between the traditional RDPI model and the TRDPI model in this invention at 50℃. Detailed Implementation
[0023] To further illustrate the technical means and effects of the present invention in achieving its intended purpose, the following detailed description of the specific implementation methods, structures, features, and effects of the present invention, in conjunction with the accompanying drawings and preferred embodiments, is provided below.
[0024] This embodiment provides a piezoelectric actuator hysteresis modeling method based on the TRDPI model, the method including the following steps: Step 1: Construct a dynamic threshold function related to the input voltage change rate and ambient temperature to characterize the temperature-frequency coupling effect of the piezoelectric actuator's hysteresis characteristics. The dynamic threshold function is a function that includes the input voltage change rate and temperature. In this embodiment, the dynamic threshold function... The expression is as follows: in, α , β and λ These are constants that need to be identified. and T These represent the derivative of the input voltage and the temperature, respectively.
[0025] Step 2, construct the upper envelope function in the form of the Fermi-Dirac distribution. and lower envelope function This is used to describe the asymmetric characteristics of the hysteresis loop of a piezoelectric actuator.
[0026] The upper and lower envelope functions are constructed using nonlinear functions based on S-curves, and are used to fit the rising and falling edges of the hysteresis loop, respectively.
[0027] The aforementioned upper envelope function and lower envelope function The expression is: in, a i and b i ( i=1,2,3, () represents the constant to be identified. v This indicates the input voltage.
[0028] Step 3, based on the dynamic threshold function Upper envelope function and lower envelope function Constructing an improved generalized Play operator : ; Step 4: Construct a memoryless nonlinear input function It is used for nonlinear mapping of input voltage.
[0029] The memoryless nonlinear input function is configured to use a first nonlinear mapping relationship when the input voltage change rate is positive, and a second nonlinear mapping relationship when the input voltage change rate is negative.
[0030] The memoryless nonlinear input function The expression is: in, S i , P i and q i ( i =1,2) represents the constants to be identified. v This represents the input voltage.
[0031] Step 5, Construct inflection point compensation items It is used to correct the modeling error in the hysteresis loop commutation region of piezoelectric actuators; The inflection point compensation is configured to apply a secondary compensation correction based on the rate of change to the output displacement when the input voltage is above 75% of the maximum voltage and a change occurs, and when the input voltage is below 25% of the maximum voltage and a change occurs.
[0032] Inflection point compensation item The expression is as follows: in, gi ( i=1,2,...,8 ) is a constant that needs to be identified.
[0033] Step 6: The outputs of each Play operator are superimposed according to preset weights to construct the TRDPI hysteresis model. The TRDPI hysteresis model is used to calculate the output displacement of the piezoelectric actuator based on the input voltage signal.
[0034] The steps for constructing a TRDPI hysteresis model further include: The initial output displacement is set to zero. At any time t, the output value of the memoryless nonlinear input function, the output value of the Play operator after each weight correction, and the output value of the inflection point compensation term are summed to obtain the output displacement of the piezoelectric actuator at time t. The TRDPI hysteresis model constructed in step 6 is as follows: in, This represents the initial output displacement of the piezoelectric actuator. It is a piezoelectric actuator The output displacement of the device at time t, It is a memoryless nonlinear input function. It is the Play operator, where n is the number of operators. P ( r i) is the correction weight coefficient for the i-th operator. It is an inflection point compensation item.
[0035] The corrected weighting coefficient P ( r i The expression for ) is: ,in, ρ Represent a constant. τ It represents a positive integer.
[0036] The model parameters of the TRDPI model include weight threshold parameters, envelope function parameters, memoryless nonlinear input function parameters, and inflection point compensation term parameters, which are identified using the dung beetle algorithm.
[0037] This invention also provides a piezoelectric actuator feedforward control method based on the TRDPI model, characterized in that the method specifically includes the following steps: Step A: Construct the dynamic threshold of the inverse TRDPI model. The expression is as follows: in, This represents the dynamic threshold of the TRDPI model; Step B: Construct the inverse Play operator of the inverse TRDPI model. The expression of the inverse Play operator is: in, η Indicates the desired displacement; Step C: Construct the weights of the inverse TRDPI model The expression is as follows: in, Indicates the weights of the TRDPI model; Step D: Construct the TRDPI inverse model based on the inverse Play operator, and use the TRDPI inverse model to perform feedforward control on the piezoelectric actuator. The expression of the TRDPI inverse model is as follows: in, v(0) This is the initial input voltage of the piezoelectric actuator; v 0 This is the initial input signal for the operator; v ( t The input voltage of the piezoelectric actuator at time t; γ R -1 and γ F -1 It is the envelope function γ R and γ F Inverse function.
[0038] The model parameters of the inverse TRDPI model include weight threshold parameters, which are identified by the Dung Beetle Algorithm (DBO).
[0039] like Figure 1 As shown, in the preferred step D, the specific process of feedforward control is as follows: In the control process of a piezoelectric actuator, the input to the feedforward controller of the piezoelectric actuator is the desired output displacement y of the piezoelectric actuator at time t. d (t); shift the desired output displacement y d (t) Input the TRDPI inverse model to obtain the output voltage v of the piezoelectric actuator feedforward controller. d (t), the output voltage v d (t) Input the piezoelectric actuator to obtain the actual output displacement y(t) of the piezoelectric actuator at time t, and realize the output displacement control of the piezoelectric actuator.
[0040] like Figure 2 The figure shown is a comparison of the feedforward compensation control effects of a piezoelectric actuator based on the TRDPI model according to the present invention, under normal temperature conditions, for four different excitation frequencies of 10Hz, 50Hz, 100Hz and 200Hz.
[0041] As can be seen, before the compensation method of this invention is used, the curve deviates significantly from the ideal linear relationship, that is, the actual displacement does not match the expected displacement, and there is a significant nonlinear deviation. This is a typical result caused by the inherent hysteresis characteristics of piezoelectric actuators. Furthermore, this deviation exists across the entire frequency range, and the nonlinearity becomes more severe as the frequency increases.
[0042] After compensation using the method of this invention, the curve almost completely coincides with the linear trend of the desired displacement, the actual output displacement closely matches the desired displacement, and the deviation is greatly suppressed. Even under high-frequency conditions of 200Hz, the fitting degree of the compensated curve remains excellent with no obvious deviation.
[0043] Depend on Figure 2 It is evident that the feedforward compensation of the TRDPI inverse model significantly improves the positioning control accuracy and is adaptable to different excitation frequencies of piezoelectric actuators, solving the problem that the compensation effect of traditional models may decay at high frequencies.
[0044] The following section compares existing RDPI models to construct a comparative model. Comparative Example 1 The expression for the Play operator related to the proportion of usage is as follows: in, For rate-related thresholds, c Input voltage, u This is the output of the model at the previous time step.
[0045] The dynamic hysteresis model used in this comparative example is the traditional RDPI model, which is based on the rate-related Play operator and is expressed as follows: Where y0 is the initial output displacement of the operator.
[0046] Tables 1, 2, and 3 below are the identification model parameter tables for the TRDPI model of this embodiment and the RDPI model of the comparative example under temperature conditions of 30℃, 40℃, and 50℃, respectively.
[0047] Table 1. Specific parameter values for RDPI and TRDPI models at 30℃ Table 2. Specific parameter values for RDPI and TRDPI models at 40℃ Table 3. Specific parameter values for RDPI and TRDPI models at 50℃ like Figure 3 , Figure 4 , Figure 5 The figure shows a comparison between the RDPI model and the TRDPI model at different temperatures. The experimental data are curves showing the relationship between a given set of voltage data and the corresponding actual output displacement data of the piezoelectric ceramic actuator. Depend on Figure 3 It can be seen that at 30℃, the output displacement obtained by the TRDPI model is closer to the actual output displacement of the piezoelectric ceramic actuator than that obtained by the traditional RDPI model.
[0048] like Figure 4 As shown, at 40℃, the output displacement obtained by the TRDPI model is closer to the actual output displacement of the piezoelectric ceramic actuator than that obtained by the traditional RDPI model.
[0049] like Figure 5As shown, at 50℃, the output displacement obtained by the TRDPI model is closer to the actual output displacement of the piezoelectric ceramic actuator than that obtained by the traditional RDPI model.
[0050] Furthermore, as can be seen from the enlarged view in the figure, the inflection point compensation term in this embodiment has a very significant effect, which greatly reduces the modeling error near the input signal reversal point, enhances the model's fitting accuracy in key regions, and improves the overall model smoothness.
[0051] like Figure 6 As shown, at 30℃, the relative root mean square error and relative maximum error of the TRDPI model are both smaller than those of the RDPI model.
[0052] like Figure 7 As shown, at 40℃, the relative root mean square error and relative maximum error of the TRDPI model are both smaller than those of the RDPI model.
[0053] like Figure 8 As shown, at 50℃, the relative root mean square error and relative maximum error of the TRDPI model are both smaller than those of the RDPI model.
[0054] The relative maximum error (emax) and root mean square error (erms) were used to evaluate the simulation performance of the TRDPI and RDPI models at different temperatures (30℃, 40℃, 50℃) and different frequencies (10Hz, 50Hz, 100Hz, 200Hz), respectively. The results are shown in Tables 4, 5, and 6. Table 4. Comparison of evaluation parameters between the TRDPI and RDPI models at 30℃ Table 5. Comparison of evaluation parameters between the TRDPI and RDPI models at 40℃ Table 6. Comparison of evaluation parameters between the TRDPI and RDPI models at 50℃ The comparison shows that the relative maximum error (emax) and root mean square error (erms) of the TRDPI model in this embodiment are significantly lower than those of the RDPI model under all temperature-frequency combinations, and the difference in error between the two is more obvious when the frequency is higher and the temperature changes.
[0055] Both the TRDPI and RDPI models in this embodiment and the comparative embodiment show a common trend of slightly increasing error with increasing excitation frequency. However, the error increase of TRDPI in this embodiment is more gradual, while the error increase of RDPI gradually intensifies with increasing temperature. This demonstrates that the TRDPI model effectively represents the temperature-frequency coupling effect and fundamentally suppresses the amplification of modeling errors caused by temperature-frequency changes.
[0056] From 30℃ to 50℃, the error curve of TRDPI only shifted slightly upward, and its overall position remained stable; while the error curve of RDPI shifted significantly upward with increasing temperature, and the steep upward trend in the high-frequency band was more obvious. This proves that the TRDPI model can effectively suppress the modeling error caused by changes in ambient temperature and solve the core defect of the traditional RDPI model that only considers frequency and does not consider temperature-frequency coupling.
[0057] Especially under conditions of 40℃ (Table 5, Figure 7 The error of the RDPI model is significantly amplified, especially in the high-frequency range where it exhibits obvious instability. At 200Hz, the erms of RDPI surge to 12.458% and the emax reaches 19.194%, while the erms of TRDPI are only 0.61% and the emax is only 1.784%, with an error difference of more than 20 times. This is the most significant difference between the two in the temperature-frequency combination.
[0058] The above comparison clearly demonstrates that: The TRDPI model in this embodiment constructs a dynamic threshold function for temperature-frequency coupling, taking into account the influence of frequency and ambient temperature, and accurately characterizes the temperature-frequency coupling hysteresis characteristics of the piezoelectric actuator, avoiding the error of single-factor modeling from the bottom layer of the model.
[0059] By introducing the asymmetric envelope function and inflection point compensation term of the Fermi-Dirac distribution, the asymmetric error of the hysteresis loop and the modeling error of the reversing region are corrected respectively, further reducing the error amplification of local operating conditions; By using the dung beetle algorithm to accurately identify all parameters of the model, the adaptability of the model parameters under different temperature frequencies is ensured, avoiding high-frequency / temperature-varying errors caused by fixed parameters.
[0060] Traditional RDPI models only consider the rate-dependent effect of the excitation frequency and do not introduce temperature correction and temperature-frequency coupling characterization. Therefore, under high-frequency operating conditions with temperature changes, especially temperature-frequency coupling, the modeling error is greatly amplified and cannot be adapted to the actual working scenarios of piezoelectric actuators.
[0061] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent changes and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the scope of the present invention.
Claims
1. A method for modeling hysteresis in piezoelectric actuators based on the TRDPI model, characterized in that, The method includes the following steps: Step 1: Construct a dynamic threshold function related to the input voltage change rate and ambient temperature to characterize the temperature-frequency coupling effect of the piezoelectric actuator's hysteresis characteristics. The dynamic threshold function is a function that includes the input voltage change rate and temperature. Step 2: Construct the upper and lower envelope functions in the form of Fermi-Dirac distribution to describe the asymmetric characteristics of the piezoelectric actuator hysteresis loop; Step 3: Based on the dynamic threshold function, upper envelope function, and lower envelope function, construct the improved generalized Play operator; Step 4: Construct a memoryless nonlinear input function for nonlinear mapping of the input voltage; Step 5: Construct an inflection point compensation term to correct the modeling error in the hysteresis loop commutation region of the piezoelectric actuator; Step 6: The outputs of each Play operator are superimposed according to preset weights to construct the TRDPI hysteresis model. The TRDPI hysteresis model is used to calculate the output displacement of the piezoelectric actuator based on the input voltage signal.
2. The piezoelectric actuator hysteresis modeling method based on the TRDPI model according to claim 1, characterized in that, The TRDPI hysteresis model constructed in step 6 is as follows: The initial output displacement is set to zero. At any time t, the output value of the memoryless nonlinear input function, the output value of the Play operator after each correction weight, and the output value of the inflection point compensation term are summed to obtain the output displacement of the piezoelectric actuator at time t.
3. The piezoelectric actuator hysteresis modeling method based on the TRDPI model according to claim 2, characterized in that, The correction weight coefficient of the Play operator exhibits an exponential decay trend as the threshold increases.
4. The piezoelectric actuator hysteresis modeling method based on the TRDPI model according to claim 1, characterized in that, The upper and lower envelope functions are constructed using nonlinear functions based on S-curves, and are used to fit the rising and falling edges of the hysteresis loop, respectively.
5. The piezoelectric actuator hysteresis modeling method based on the TRDPI model according to claim 1, characterized in that, The memoryless nonlinear input function is configured to use a first nonlinear mapping relationship when the input voltage change rate is positive, and a second nonlinear mapping relationship when the input voltage change rate is negative.
6. The piezoelectric actuator hysteresis modeling method based on the TRDPI model according to claim 1, characterized in that, The inflection point compensation is configured to apply a secondary compensation correction based on the rate of change to the output displacement when the input voltage is above 75% of the maximum voltage and a change occurs, and when the input voltage is below 25% of the maximum voltage and a change occurs.
7. The piezoelectric actuator hysteresis modeling method based on the TRDPI model according to claim 1, characterized in that, The model parameters of the TRDPI model include weight threshold, envelope function, memoryless nonlinear input function, and inflection point compensation term, which are identified using the dung beetle algorithm.
8. A feedforward control method for piezoelectric actuators based on the TRDPI model, characterized in that, The method specifically includes the following steps: Step A: Construct the dynamic threshold of the inverse model, which is determined based on the dynamic threshold and weights of the forward model through inverse and recursive relationships; Step B: Construct the inverse Play operator, which uses the desired displacement as the center value of the boundary constraint; Step C: Construct the weights of the inverse TRDPI model; Step D: Construct the TRDPI inverse model based on the inverse Play operator, use the TRDPI inverse model to perform feedforward control on the piezoelectric actuator, and solve for the required input voltage based on the desired displacement.
9. The piezoelectric actuator feedforward control method based on the TRDPI model according to claim 8, characterized in that, The model parameters of the inverse TRDPI model include weight threshold parameters, which are identified by the Dung Beetle Algorithm (DBO).
10. The piezoelectric actuator feedforward control method based on the TRDPI model according to claim 8, characterized in that, In step D, the specific process of feedforward control is as follows: In the control process of the piezoelectric actuator, the input of the feedforward controller of the piezoelectric actuator is the desired output displacement of the piezoelectric actuator; the desired output displacement is input into the TRDPI inverse model to obtain the output voltage of the feedforward controller of the piezoelectric actuator; the output voltage is input into the piezoelectric actuator to obtain the actual output displacement of the piezoelectric actuator, thus realizing the output displacement control of the piezoelectric actuator.