A parameter identification method for piezoelectric actuator hysteresis model based on MDE algorithm

Abstract

The present invention discloses a method for identifying parameters of a piezoelectric actuator hysteresis model based on an MDE algorithm. The parameter identification method introduces an adaptive mechanism into the scaling factor in the mutation operation and the crossover probability in the crossover operation, so that the numerical values ​​of the scaling factor and the crossover probability change with the number of iterations, which helps the algorithm to jump out of the local optimal solution and improves the convergence speed of parameter identification and the quality of the solution. At the same time, the selection operation of the present invention copies the best individuals that appear in the evolutionary process to the next generation of original population for iteration, and gives individuals in the original population and the test population the opportunity to enter the next generation of original population, which makes up for the defect of premature convergence of the traditional DE algorithm and maintains the diversity of the population. In addition, the present invention can accurately identify the parameters in a complex hysteresis model, effectively solves the problem of low parameter prediction accuracy of the piezoelectric actuator hysteresis model, and can improve the output stability of the model parameters.

Description

Technical Field

[0001] The present invention belongs to the field of micro-nano actuation technology, and in particular relates to a piezoelectric actuator hysteresis model parameter identification method based on an MDE algorithm. Background Art

[0002] In recent years, with the rapid development of microelectronics, micro-nano manufacturing, and ultra-precision machining technologies, the demand for precision in manufacturing equipment has become increasingly stringent. Traditional motor drives are no longer able to meet the stringent requirements for precision motion accuracy, leading to the emergence of intelligent material actuators that convert electrical or magnetic energy into mechanical energy. Piezoelectric ceramic actuators offer advantages such as high displacement resolution, fast frequency response, high stiffness, compact size, and high reliability. They have been widely used in micro-nano machining equipment, precision instrumentation, atomic force microscopes, and ultra-precision machine tools.

[0003] Piezoelectric ceramic transducers (PZTs) utilize the inverse piezoelectric properties of piezoelectric materials to produce deformation in response to an input voltage or current, achieving mechanical motion. However, the complex hysteresis nonlinearity between the input signal and the output displacement of PZT actuators significantly impacts their actuation accuracy and hinders their application. Therefore, the rational modeling of PZT actuator hysteresis characteristics and the accurate identification of model parameters remain both research hotspots and challenges.

[0004] At present, scholars at home and abroad have proposed a variety of parameter identification methods for hysteresis models, such as the differential evolution algorithm (DE). The DE algorithm is a global optimization algorithm based on random search of swarm intelligence theory. It has the advantages of simple principle, few parameters and fast search speed. Like other swarm intelligence algorithms, the DE algorithm is prone to fall into local optimal solutions, resulting in premature convergence, and will produce large errors when identifying complex hysteresis models. In addition, when constructing different piezoelectric ceramic actuator hysteresis models, the convergence speed and solution accuracy of nonlinear parameter identification are different, which also determines the accuracy and operating efficiency of the hysteresis model. In order to overcome this shortcoming, the present invention proposes an MDE algorithm (Modified Differential Evolution, MDE) that introduces an elite retention strategy and an adaptive mutation operator. The algorithm can not only effectively improve the convergence speed of parameter identification and the quality of the solution, but also improve the algorithm performance and have better robustness. Therefore, the MDE algorithm is selected as the main tool for PI model parameter identification in the present invention, in order to achieve the goal of high-precision modeling.

[0005] The paper "Modeling and compensation of hysteresis in piezoelectric actuators" proposes an MPI model based on the traditional PI model to describe the hysteresis characteristics of the hysteresis curve. Because the MPI model is constructed by introducing an envelope function, the model is relatively complex and has a large number of nonlinear parameters. Therefore, this paper verifies the effectiveness and superiority of the MDE algorithm by identifying the MPI model. Summary of the Invention

[0006] Aiming at the limitations of existing identification algorithms, the present invention proposes a piezoelectric ceramic driver hysteresis nonlinear model parameter identification method based on an improved differential evolution algorithm.

[0007] In a first aspect, the present invention proposes a method for identifying parameters of a piezoelectric actuator hysteresis model based on an MDE algorithm, the parameter identification method comprising the following steps:

[0008] Step 1: The parameters to be identified in the piezoelectric actuator hysteresis model are used as individuals in the original population; the dimension n of the original population is the number of parameters to be identified; the fitness function is used to calculate the fitness of each individual in the original population, and the individual with the highest fitness in each dimension is selected as the optimal individual x best，j ; where j = 1, 2, ..., n.

[0009] Step 2: Perform a mutation operation on the original population to generate a mutant population. The scaling factor F in the mutation operation changes adaptively, and its expression is:

[0010] F = F0 × (1 + W × rand (0, 1))

[0011] Among them, F0 is the initial mutation rate; W is the adaptive weight; rand(0, 1) is a random number in the range of [0, 1].

[0012] The scaling factor F is checked for boundaries, and if the scaling factor F exceeds a preset boundary, it is adjusted to be within the boundary range.

[0013] Step 3: Perform a crossover operation on the original population and the mutant population to obtain the test population. The expression of the crossover probability CR in the crossover operation is:

[0014]

[0015] Among them, CR max is the maximum value of the crossover probability CR; CR min is the minimum value of the crossover probability CR; P_CR is the adaptive adjustment probability; CR0 is the initial crossover probability.

[0016] All individuals in the experimental population are checked for boundaries. If an individual in the experimental population exceeds the preset individual boundary, it will be adjusted to within the boundary range.

[0017] Step 4: Perform selection operations on the experimental population and the original population to obtain the next generation original population. Use the fitness function to calculate the fitness of each individual in the next generation original population; compare the highest fitness in each dimension with the fitness of the optimal individual. If the individual fitness in the next generation original population is higher, update the optimal individual x best，j ; On the contrary, the optimal individual x best，j constant.

[0018] Step 5: Repeat steps 2 to 4 until the number of iterations reaches the preset maximum number of iterations G; the optimal individual x obtained after the iteration is completed is best，j As the parameter value that needs to be identified in the hysteresis model of the piezoelectric actuator.

[0019] As a preference, in step 4, the individual x in the next generation original population i，j The method to obtain (g+1) is:

[0020]

[0021] Among them, z i，j (g+1) is the individual after the crossover operation; f(z i，j (g+1)) is individual z i，j The fitness of (g+1); x i，j (g) is the individual in the original population of the gth generation; f(x i，j (g)) is the individual x i，j (g) fitness; x max，j (g) is the individual with the highest fitness in the original population; f(x max，j (g)) is the individual x max，j (g) Fitness.

[0022] Preferably, in step 1, the population size M of each dimension is 2 to 6 times the dimension n; the method for initializing the population is as follows:

[0023]

[0024] Among them, x i，j is the i-th individual in the j-th dimension; is the upper boundary of the individual in the jth dimension; is the lower boundary of the individual in the jth dimension; rand(0,1) is a random number in the range [0,1]; i=1,2,...,M.

[0025] As a preference, in the step 2, the individual y in the mutant population i，j The method to obtain (g+1) is:

[0026]

[0027] in, and For individual y i，j (g+1) corresponds to three random individuals in the dimension and r1≠r2≠r3≠i.

[0028] As a preference, in step 3, the individual z in the test population i，j The method to obtain (g+1) is:

[0029]

[0030] Among them, j rand is a randomly selected dimension.

[0031] Preferably, in step 2, the method for adjusting the adaptive scaling factor F to within the boundary range is as follows:

[0032] F=max{min{F,F max}, F min}

[0033] Among them, F max is the upper bound of the adaptive scaling factor F; F min is the lower bound of the adaptive scaling factor F.

[0034] As a preference, in step 3, the individuals z in the test population are i，j The method for adjusting (g+1) to the boundary range is as follows:

[0035]

[0036] in, is the upper boundary of the individual in the jth dimension; is the lower boundary of the individual in the j-th dimension.

[0037] In second aspect, the present invention provides a method for modeling a piezoelectric actuator hysteresis model, which is as follows: constructing an operator of the piezoelectric actuator hysteresis model, and constructing the piezoelectric actuator hysteresis model according to the operator; obtaining an operator of the hysteresis inverse model based on the operator of the piezoelectric actuator hysteresis model, and constructing the hysteresis inverse model according to the inverse operator, and using the above-mentioned parameter identification method to identify the parameters in the hysteresis inverse model.

[0038] In a third aspect, the present invention provides a feedforward control method for a piezoelectric actuator, wherein the method comprises the following steps: during the control process of the piezoelectric actuator, the input of the piezoelectric actuator feedforward controller is the desired output signal y of the piezoelectric actuator at time t. d (t); the output signal y d (t) Input the piezoelectric actuator hysteresis inverse model obtained above to obtain the output signal ν(t) of the piezoelectric driver feedforward controller; input the output signal ν(t) into the piezoelectric driver to obtain the actual output signal y(t) of the piezoelectric driver at time t, thereby realizing the output displacement control of the piezoelectric driver.

[0039] The present invention has the following beneficial effects:

[0040] 1. The present invention introduces an adaptive mechanism into the scaling factor in the mutation operation and the crossover probability in the crossover operation, so that the values ​​of the scaling factor and the crossover probability change with the number of iterations, which helps the algorithm to escape from the local optimal solution and improves the convergence speed of parameter identification and the quality of the solution.

[0041] 2. The selection operation of the present invention copies the best individuals that appear in the evolution process to the next generation original population for iteration, which makes up for the defect of premature convergence of the traditional DE algorithm, accelerates the parameter identification convergence speed, and ensures the excellent characteristics of the original population during the search process; at the same time, while retaining excellent individuals, the selection operation also gives individuals in the original population and the test population the opportunity to enter the next generation original population, thereby maintaining the diversity of the population.

[0042] 3. The present invention can accurately identify the parameters in a complex hysteresis model, effectively solving the problem of low prediction accuracy of the hysteresis model parameters of the piezoelectric actuator, and can improve the output stability of the model parameters. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 This is a control flow chart in Example 1 of the present invention.

[0044] Figure 2 These are the iterative convergence graphs of the MDE algorithm and the DE algorithm in Comparative Example 1 of the present invention; wherein, (a) is the iterative convergence graph for testing the Ackley function; (b) is the iterative convergence graph for testing the Alpine function; (c) is the iterative convergence graph for testing the Sphere function; (d) is the iterative convergence graph for testing the Rastrigin function; (e) is the iterative convergence graph for testing the Schwefel's problem 22 function; and (f) is the iterative convergence graph for testing the Griewank function.

[0045] Figure 3It is the spatial optimal position diagram when the MDE algorithm in comparative example 1 of the present invention solves the test function; wherein, (a) is the spatial optimal position diagram when testing the Ackley function; (b) is the spatial optimal position diagram when testing the Alpine function; (c) is the spatial optimal position diagram when testing the Sphere function; (d) is the spatial optimal position diagram when testing the Rastrigin function; (e) is the spatial optimal position diagram when testing the Schwefel's problem 22 function; and (f) is the spatial optimal position diagram when testing the Griewank function.

[0046] Figure 4 This is a fitting effect diagram of the MPI model under the MDE algorithm in comparative example 2 of the present invention.

[0047] Figure 5 This is the fitting error diagram of the MPI model under the MDE algorithm in Comparative Example 2 of the present invention. DETAILED DESCRIPTION

[0048] The present invention will be further described below with reference to the accompanying drawings.

[0049] Example 1

[0050] like Figure 1 As shown, a piezoelectric actuator hysteresis model parameter identification method based on the MDE algorithm includes the following steps:

[0051] Step 1: Initialize the population.

[0052] The parameters that need to be identified in the piezoelectric actuator PI hysteresis model are used as individuals in the original population; the dimension n of the original population is the number of parameters to be identified; an upper and lower bound is defined for the individuals in each dimension Ensure that the generated individuals (solutions) are within the feasible region. Set the number of individuals contained in the original population of each dimension, that is, the population size M, which is twice the dimension n. Initialize the original population:

[0053]

[0054] Among them, x i，j is the i-th individual of the j-th dimension in the original population; rand(0,1) is a random number in the range [0,1]; i=1,2,...,M; j=1,2,...,n.

[0055] Use the fitness function to calculate the fitness of each individual in the original population, and select the individual with the highest fitness in each dimension as the optimal individual x best，j .

[0056] The quality of the initial population has a significant impact on the performance of the algorithm, and the population size M affects the algorithm's search ability and computational burden. A good initial population should be able to evenly cover the solution space, thus providing the algorithm with a wide range of search starting points.

[0057] Step 2: Mutation operation.

[0058] Generate a new population, the mutant population, through mutation operation to increase the diversity of the population and explore the solution space. Randomly select three different individuals in the same dimension from the original population. and And generate mutant individual y based on three randomly selected individuals i，j (g+1); where r1≠r2≠r3≠i. Mutant individual y i，j The expression for (g+1) is:

[0059]

[0060] Where F is the scaling factor.

[0061] By introducing an adaptive mechanism for the scaling factor in the variation part, the scaling factor F of different dimensions can change randomly with the change of the number of iterations, which helps the algorithm to jump out of the local optimal solution and prevent the algorithm from converging to a suboptimal solution too early. The expression of the scaling factor F is:

[0062] F = F0 × (1 + W × rand (0, 1))

[0063] Among them, F0 is the initial mutation rate; W is the adaptive weight.

[0064] The obtained scaling factor F is checked for boundaries. If the scaling factor F exceeds the boundaries, it is adjusted to within the boundaries according to the following formula:

[0065] F=max{min{F,F max}, F min}

[0066] Among them, F max is the upper bound of the scaling factor F; F min is the lower bound of the scaling factor F.

[0067] When searching the solution space, genetic algorithms need to strike a balance between exploration and exploitation. Exploration means the algorithm tries new possibilities, even if they are inferior to the currently known best solution. Exploitation, on the other hand, means the algorithm searches near known optimal solutions to further optimize them.

[0068] After the mutation operation is completed, the mutant population and the original population are combined to form a temporary population for the next crossover operation.

[0069] Step 3: Cross operation.

[0070] Perform crossover operation on the variant population and the original population to obtain the test population. The individual z in the test population is i，j The expression for (g+1) is:

[0071]

[0072] Among them, j rand is the randomly selected dimension; CR is the crossover probability, which is expressed as:

[0073]

[0074] Among them, CR max is the maximum value of the crossover probability CR; CR min is the minimum value of the crossover probability CR; P_CR is the adaptive adjustment probability; CR0 is the initial crossover probability.

[0075] By adaptively adjusting the crossover probability CR, the probability of losing good genes in the crossover operation is reduced, thereby improving local search capabilities and algorithm performance. Boundary checks are performed on each individual in the test population to ensure that the value of each individual is within the preset upper and lower bounds. If it exceeds the bounds, it is adjusted to the bounds according to the following formula:

[0076]

[0077] Through the crossover operation, features from different individuals are combined to explore new possibilities in the solution space. The crossover operation helps the algorithm maintain the diversity of the population while promoting the convergence of the solution.

[0078] Step 4. Select an operation.

[0079] Through the selection operation, individuals are selected from the experimental population and the original population to be retained in the next generation. In order to avoid premature convergence of the selected offspring and fall into the local optimum problem, the best individuals that appear in the evolution process are directly copied to the next generation. Individual u in the original population of the next generation i，j The expression for (g+1) is:

[0080]

[0081] Among them, z i，j (g+1) is the individual in the experimental population; f(z i，j (g+1)) is individual z i，j The fitness of (g+1); f(x i，j (g)) is the individual x i，j (g) fitness; x max，j(g) is the individual with the highest fitness in the population of the jth dimension; f(x max，j (g)) is the individual x max，j (g) Fitness.

[0082] Use the fitness function to calculate the fitness of each individual in the next generation of the original population, and take the highest fitness f(x max，j (g+1)) and the fitness of the optimal individual f(x best，j ) for comparison, if f(x max，j (g+1))>f(x best，j ), then update the optimal individual x best，j ; On the contrary, the optimal individual x best，j constant.

[0083] The selection operation is the core of the natural selection mechanism in the algorithm, ensuring that the population evolves towards a better solution. The random number is compared with the crossover rate CR to determine whether to retain the new individual z i，j (g+1) or original individual x i，j (g) This probabilistic selection allows the algorithm to retain excellent individuals while also giving new individuals a certain chance to enter the next generation, thereby maintaining the diversity of the population.

[0084] Step 5: Termination conditions.

[0085] Repeat steps 2 to 4 until the number of iterations reaches the preset maximum number of iterations G, and the optimal individual x is obtained. best，j As the parameter value that needs to be identified in the PI hysteresis model of the piezoelectric actuator.

[0086] Step 6. Construct an operator for the piezoelectric actuator PI hysteresis model, and construct the piezoelectric actuator PI hysteresis model based on the operator; obtain the operator for the PI hysteresis inverse model based on the operator of the piezoelectric actuator PI hysteresis model, and construct the PI hysteresis inverse model based on the inverse operator, and use the above-mentioned parameter identification method to identify the parameters in the PI hysteresis inverse model.

[0087] In the control process of the piezoelectric driver, the input of the piezoelectric driver feedforward controller is the expected output signal y of the piezoelectric driver at time t. d (t); the output signal y d (t) is input into the PI hysteresis inverse model to obtain the output signal ν(t) of the piezoelectric driver feedforward controller; the output signal ν(t) is input into the piezoelectric driver to obtain the actual output signal y(t) of the piezoelectric driver at time t, thereby realizing the output displacement control of the piezoelectric driver.

[0088] Comparative Example 1

[0089] The process of performance testing and analysis of the MDE algorithm is as follows:

[0090] In order to verify the effectiveness and superiority of the MDE algorithm, the MDE algorithm and the DE algorithm are used to test their performance in high-dimensional cases on the benchmark functions in Table 1 under the same variable conditions.

[0091] Table 1 Benchmark functions

[0092]

[0093]

[0094] In order to make the task requirements of the MDE algorithm and the DE algorithm as close as possible to the actual work requirements of identifying the nonlinear parameters of the model, the population size M is set to 30, the dimension n is set to 15, the number of iterations is set to 1000, the initial mutation rate F0 is set to 0.6, the initial crossover probability CR0 is set to 0.9, the adaptive weight W is set to 0.8, and the adaptive adjustment probability P_CR is set to 0.2. The MDE algorithm and the DE algorithm are compared in terms of the best solution Best, the worst solution Worst, the mean Mean, the standard deviation StdDev, the time taken for the algorithm to run 30 times, the convergence speed, and the ability to find the optimal position in space. The test results are shown in Table 2. Figure 2 and Figure 3 shown.

[0095] Table 2 Comparison of evaluation parameters between MDE algorithm and DE algorithm

[0096]

[0097]

[0098] From Table 2, we can see that the MDE algorithm has better performance, stronger robustness and higher solution accuracy than the traditional DE algorithm. Figure 3 It can be seen that in the tests of six different test functions, the convergence speed of the MDE algorithm is much faster than that of the DE algorithm. When actually identifying the parameters of complex nonlinear hysteresis models, it provides theoretical and data support for the MDE algorithm to have a faster identification speed. Figure 4 It can be seen that the spatial position optimization ability of the MDE algorithm based on the present invention is far superior to that of the DE algorithm. When actually identifying the parameters of complex nonlinear hysteresis models, it provides theoretical and data support for the MDE algorithm to have better robustness.

[0099] Comparative Example 2

[0100] The MDE algorithm and DE algorithm are used to identify the parameters of the complex nonlinear hysteresis model MPI model. In the parameter identification of the MPI model, the fitness function F(X) is:

[0101]

[0102] Where X is the MPI model parameter to be identified; N is the total number of samples; y(k) is the actual displacement; Y(k) is the model output displacement; is the mean of the actual displacement.

[0103] from Figure 4 It can be seen that the MDE algorithm has a better identification effect on the MPI model than the DE algorithm, with a significantly improved identification accuracy and a more accurate description of complex hysteresis phenomena. Under the same set of voltage data as the experimental data, the output displacement obtained by the MPI model based on the above-given MDE algorithm is compared with the output displacement obtained by the MPI model based on the above-given DE algorithm. The output displacement of the MPI model identified by the MDE algorithm is closer to the actual output displacement of the piezoelectric ceramic driver; Figure 5 It can be seen that the fluctuation of the curve of the error of the output displacement of the MPI model identified by the MDE algorithm of the present invention as the sampling point position changes is also much smaller than the fluctuation of the curve of the error of the output displacement of the MPI model identified by the traditional DE algorithm as the sampling point position changes.

Claims

1. A piezoelectric actuator hysteresis model parameter identification method based on the MDE algorithm, characterized by: The following steps are involved: Step 1: Parameters to be identified in the piezoelectric actuator hysteresis model are used as individuals in the original population; The dimension n of the original population is the number of identified parameters; the fitness function is used to calculate the fitness of each individual in the original population, and the individual with the highest fitness in each dimension is selected as the optimal individual x best,j ; where j = 1, 2, ..., n; Step 2: Perform a mutation operation on the original population to generate a mutant population. The scaling factor F in the mutation operation changes adaptively, and its expression is: F=F0×(1+W×rand(0,1)) Among them, F0 is the initial mutation rate; W is the adaptive weight; rand(0,1) is a random number in the range of [0,1]; Perform a bounds check on the scaling factor F. If the scaling factor F exceeds a preset bound, adjust it to within the bounds. Step 3: Perform a crossover operation on the original population and the mutant population to obtain the test population. The expression of the crossover probability CR in the crossover operation is: Among them, CR max is the maximum value of the crossover probability CR; CR min is the minimum value of the crossover probability CR; P_CR is the adaptive adjustment probability; CR0 is the initial crossover probability; All individuals in the experimental population are checked for boundaries. If an individual in the experimental population exceeds the preset individual boundary, it is adjusted to the boundary range; Step 4: Perform selection operations on the experimental population and the original population to obtain the next generation original population; use the fitness function to calculate the fitness of each individual in the next generation original population; compare the highest fitness in each dimension with the fitness of the optimal individual. If the individual fitness in the next generation original population is higher, update the optimal individual x best,j ; On the contrary, the optimal individual x best,j constant; Step 5: Repeat steps 2 to 4 until the number of iterations reaches the preset maximum number of iterations G; the optimal individual x obtained after the iteration is completed is best,j As the parameter value that needs to be identified in the hysteresis model of the piezoelectric actuator.

2. The method for identifying hysteresis model parameters of a piezoelectric actuator based on an MDE algorithm according to claim 1, wherein: In step 4, the individual x in the next generation original population i,j The method to obtain (g+1) is: Among them, z i,j (g+1) is the individual after the crossover operation; f(z i,j (g+1)) is individual z i,j The fitness of (g+1); x i,j (g) is the individual in the original population of the gth generation; f(x i,j (g)) is the individual x i,j (g) fitness; x max,j (g) is the individual with the highest fitness in the original population; f(x max,j (g)) is the individual x max,j (g) Fitness.

3. The method for identifying hysteresis model parameters of a piezoelectric actuator based on the MDE algorithm according to claim 1, characterized in that: In step 1, the population size M of each dimension is 2 to 6 times the dimension n; the method for initializing the population is as follows: Among them, x i,j is the i-th individual in the j-th dimension; is the upper boundary of the individual in the jth dimension; is the lower boundary of the individual in the jth dimension; rand(0,1) is a random number in the range [0,1]; i=1,2,...,M.

4. The method for identifying hysteresis model parameters of a piezoelectric actuator based on an MDE algorithm according to claim 1, wherein: In the step 2, the individual y in the mutant population i,j The method to obtain (g+1) is: y i,j (g+1)=x r1,j (g)+F×(x r3,j (g)-x r2,j (g)) Among them, x r1,h (g), x r2,j (g) and x r3,h (g) is individual y i,j (g+1) corresponds to three random individuals in the dimension and r1≠r2≠r3≠i.

5. The method for identifying hysteresis model parameters of a piezoelectric actuator based on an MDE algorithm according to claim 1, wherein: In the step 3, the individual z in the test population i,j The method to obtain (g+1) is: Among them, j rand is a randomly selected dimension.

6. The method for identifying parameters of a piezoelectric actuator hysteresis model based on an MDE algorithm according to claim 1, wherein: In step 2, the method for adjusting the adaptive scaling factor F to within the boundary range is as follows: F=max{min{F,F max },F min } Among them, F max is the upper bound of the adaptive scaling factor F; F min is the lower bound of the adaptive scaling factor F.

7. The method for identifying parameters of a piezoelectric actuator hysteresis model based on an MDE algorithm according to claim 1, wherein: In the step 3, the individual z in the test population is i,j The method for adjusting (g+1) to the boundary range is as follows: in, is the upper boundary of the individual in the jth dimension; is the lower boundary of the individual in the j-th dimension.

8. A piezoelectric actuator hysteresis modeling method based on the MDE algorithm, characterized by: An operator of a piezoelectric actuator hysteresis model is constructed, and the piezoelectric actuator hysteresis model is constructed according to the operator; an operator of a hysteresis inverse model is obtained based on the operator of the piezoelectric actuator hysteresis model, and the hysteresis inverse model is constructed according to the inverse operator, and the parameters in the hysteresis inverse model are identified using the piezoelectric actuator hysteresis model parameter identification method based on the MDE algorithm described in claim 1.

9. A feedforward control method for a piezoelectric actuator, characterized in that: In the control process of the piezoelectric driver, the input of the piezoelectric driver feedforward controller is the expected output signal y of the piezoelectric driver at time t. d (t); the output signal y d (t) is input into the piezoelectric actuator hysteresis inverse model described in claim 8 to obtain the output signal ν(t) of the piezoelectric driver feedforward controller; the output signal ν(t) is input into the piezoelectric driver to obtain the actual output signal y(t) of the piezoelectric driver at time t, thereby realizing the output displacement control of the piezoelectric driver.

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