Piezoelectric actuator finite time terminal sliding mode control method and system and medium

Abstract

The invention relates to a finite time terminal sliding mode control method and system for a piezoelectric actuator and a medium, and the method specifically comprises the steps: collecting an operation signal of the piezoelectric actuator, the operation signal comprising input voltage and displacement output data; establishing a fractional order coupling model containing asymmetric hysteresis characteristics; parameter identification is carried out on parameters in the fractional order coupling model, and the parameters in the fractional order coupling model are adjusted; a hysteresis feed-forward compensator is constructed, and the fractional order coupling model after parameter identification is compensated; and for the compensated fractional order coupling model, through a finite time terminal sliding mode control algorithm, realizing tracking control on the track of the piezoelectric actuator. On the basis of asymmetric hysteresis effect compensation, a finite time terminal sliding mode control algorithm is introduced, it is ensured that tracking errors are rapidly converged to zero within finite time, and the trajectory tracking precision and stability of the piezoelectric actuator within the finite time are improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of tracking control of piezoelectric actuators, and specifically, relates to a finite-time terminal sliding mode control method, system and medium for piezoelectric actuators. Background Technique

[0002] With the rapid development of the micro-nano technology field, the demand for high-precision system applications has increased sharply, and the requirements for motion accuracy and resolution have reached unprecedentedly strict standards. Against this background, piezoelectric actuators, with their unique advantages - ultra-high-speed response ability, excellent high-resolution characteristics and high energy density, have played a crucial role in many high-tech fields, such as semiconductor precision manufacturing, biological microscopy, high-precision robot systems, advanced optical imaging and aerospace exploration, etc., significantly promoting the technological progress and development of these fields.

[0003] However, the excellent performance of piezoelectric actuators is not flawless. In practical applications, it is often restricted by nonlinear factors, especially hysteresis and creep phenomena. Among these nonlinear characteristics, the hysteresis effect is particularly significant and complex. The hysteresis characteristic of piezoelectric actuators shows an asymmetric multi-valued mapping relationship closely related to the frequency of the input signal, that is, the relationship between the input voltage and the output displacement is not a simple linear correspondence, but shows a complex nonlinear relationship with the change of frequency. This frequency-dependent hysteresis phenomenon not only significantly reduces the control accuracy of the system, but also weakens the robust stability of the system, becoming the main bottleneck restricting the application of piezoelectric actuators under higher precision requirements.

[0004] In the continuous exploration of high-precision system application fields, piezoelectric actuators, with their unique performance advantages such as high precision, fast response and high energy density, have become an indispensable key component in many precision control systems. However, the inherent nonlinear hysteresis characteristic of piezoelectric actuators, as a major obstacle to improving the trajectory tracking control accuracy, has long been a technical challenge faced by both academia and industry. This characteristic not only increases the complexity of system modeling, but also directly affects the design and implementation effect of control strategies. In order to effectively address the nonlinear hysteresis problem of piezoelectric actuators, researchers have proposed and tried various control methods, among which the optimization research of robust stability control schemes is the most extensive.

[0005] For example, the paper "Bouc–Wen modeling and inverse multiplicative structure to compensate hysteresis nonlinearity in piezoelectric actuators" published in *IEEE Transactions on Automation Science and Engineering* uses the classical Bouc-Wen model to achieve inverse compensation for the hysteresis effect, providing a basis for the design of subsequent control strategies. In addition, the paper "UDE-based trajectory tracking control of piezoelectric stages" published in *IEEE Transactions on Industrial Electronics* introduces a controller based on uncertainty and disturbance estimators, aiming to achieve more precise trajectory tracking by real-time estimating and compensating for internal and external disturbances in the system. The paper "Extended state observer–based fractional order sliding-mode control of piezoelectric actuators" published in *Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering* further combines the Bouc-Wen model, hysteresis compensator and fractional order sliding-mode control strategy, and integrates an extended state observer to improve the comprehensive control performance of the system.

[0006] These research results have improved the control effect of piezoelectric actuators to a certain extent, but they are still limited by the limitations of specific control algorithms themselves and are difficult to fully meet the requirements of high-precision control. Among many control strategies, sliding mode control stands out with its fast response ability and strong robustness characteristics, becoming a research hotspot in the field of trajectory tracking control. Sliding mode control enables the system state to quickly adjust and maintain stability in the face of external disturbances or model uncertainties through a carefully designed sliding surface, ensuring that the system moves along a predetermined trajectory. However, despite the above significant advantages of sliding mode control, its asymptotic convergence property causes the system state to not reach the target value instantaneously, thus restricting the further improvement of control precision to a certain extent. Summary of the Invention

[0007] To this end, the technical problem to be solved by the present invention is to overcome the influence of the non-linear hysteresis characteristics of the piezoelectric actuator on the trajectory tracking control accuracy and system stability in the prior art, as well as the limitations such as slow convergence speed and strong dependence on the disturbance upper bound in the traditional sliding mode control method. A finite-time terminal sliding mode control method, system and medium for piezoelectric actuators are provided. On the basis of compensating for the asymmetric hysteresis effect, a finite-time terminal sliding mode control algorithm is introduced to ensure that the system tracking error quickly converges to zero within a finite time, greatly shortening the convergence time and significantly improving the trajectory tracking accuracy and stability of the piezoelectric actuator within a finite time.

[0008] To solve the above technical problem, the present invention provides a finite-time terminal sliding mode control method for piezoelectric actuators. The specific steps include:

[0009] S1. Collect the operating signals of the piezoelectric actuator, where the operating signals include input voltage and displacement output data;

[0010] S2. Based on the collected input voltage and output displacement data, establish a fractional-order coupling model including asymmetric hysteresis characteristics;

[0011] S3. Perform parameter identification on the parameters in the fractional-order coupling model, and adjust the parameters in the fractional-order coupling model to minimize the sum of squares of the error between the expected displacement of the model and the actual displacement of the piezoelectric actuator;

[0012] S4. Based on the identified fractional-order coupling model parameters, construct a hysteresis feedforward compensator to compensate the fractional-order coupling model after parameter identification, so that the actual displacement of the piezoelectric actuator approaches the expected displacement of the model;

[0013] S5. Through the finite-time terminal sliding mode control algorithm, quickly converge the tracking error between the actual displacement and the expected displacement of the piezoelectric actuator to zero within a finite time, and realize the trajectory tracking control of the piezoelectric actuator.

[0014] In an embodiment of the present invention, in step S1, the fractional-order coupling model includes:

[0015] A dynamic model for describing the linear dynamic behavior of the piezoelectric actuator;

[0016] An asymmetric Bouc-Wen hysteresis model for capturing and describing the asymmetric hysteresis characteristics of the piezoelectric actuator.

[0017] In an embodiment of the present invention, the expression of the dynamic model is as follows:

[0018]

[0019] The asymmetric Bouc-Wen hysteresis model is respectively as follows:

[0020]

[0021] Among them, the parameter m is the mass of the piezoelectric actuator, c is the damping coefficient of the piezoelectric actuator, k is the piezoelectric driving stiffness, and x is the output displacement; the parameter k d represents the product of the piezoelectric coefficient and the stiffness, is the velocity, is the acceleration; u(t) represents the input voltage of the piezoelectric actuator; h(t) is the hysteresis variable of the piezoelectric actuator; is the first derivative of u(t), is the first derivative of h(t);

[0022] α, β, γ, and n are all parameters of the Bouc-Wen hysteresis model; α is the amplitude of the hysteresis component, β

[0023] is the shape coefficient, γ is used to control the shape of the hysteresis return line, δ is the asymmetry coefficient, and n is the plasticity index, which is used to control the growth rate of plastic deformation and the shape of the hysteresis loop; is the asymmetry formula; sign(.) is the sign function, and p represents the perturbation of the system, including model uncertainty, external perturbation, creep, vibration, and other non-linear factors.

[0024] In an embodiment of the present invention, in step S3, the parameter identification method includes the following steps:

[0025] Define a fitness function to measure the error between the expected displacement and the actual displacement;

[0026] Use the non-linear least squares algorithm to optimize and adjust the parameters of the asymmetric Bouc-Wen model to minimize the sum of the squares of the errors between the model prediction value and the actual displacement;

[0027] Among them, the fitness function is represented by the following formula:

[0028]

[0029] Among them, C is the total number of sample points, and i is each equally spaced sampling time point; x exp is the experimental observation value at the i-th sampling point; x sim is the model prediction value at the i-th sampling point, and the sampling intervals of x exp and x sim are equal;

[0030] Use the minimum value of F(m, b, k, k d , α, β, γ, δ, n) to fit

[0031]

[0032] In one embodiment of the present invention, the compensation method of the hysteresis feedforward compensator includes:

[0033] In step S4, the hysteresis compensation signal u is calculated according to the inverse model of the asymmetric Bouc-Wen hysteresis model c ;

[0034] The calculated hysteresis compensation signal u c is added to the original control signal to form a compensated control signal, and finally the compensated control signal is used for the piezoelectric actuator.

[0035] In one embodiment of the present invention, the expression of the hysteresis compensation signal u c output by the feedforward compensator is as follows:

[0036]

[0037] where k d is the damping coefficient of the piezoelectric actuator, x d (t) is the desired displacement, and h r (t) is the hysteresis compensation term.

[0038] In one embodiment of the present invention, the finite-time terminal sliding mode control algorithm includes the following steps:

[0039] S51: Define the desired displacement x d (t) of the piezoelectric actuator;

[0040] S52: Measure the actual displacement x(t) of the piezoelectric actuator;

[0041] S53: Calculate the tracking error e and take the derivative of e;

[0042] e(t) = x(t) - x d (t)

[0043]

[0044] S54: Design the sliding surface

[0045]

[0046] Substitute into the above formula to obtain

[0047]

[0048] S55: Design a control law u with finite-time convergence such that s converges to zero in finite time. The expression of the control law u is as follows:

[0049] u = u eq + u sw

[0050]

[0051] S56: Apply the control law u to the piezoelectric actuator to generate the actual displacement x(t);

[0052] S57: At each sampling instant, update the sliding mode surface s and the control law u until s converges to zero.

[0053] where k d is the damping coefficient, c is the damping ratio, A, B, υ, k s , η and λ are all control parameters, x is the actual displacement, x d is the desired displacement, e is the tracking error, is the first derivative of the tracking error, is the second derivative of the tracking error, s is the sliding mode surface, is the derivative of the sliding mode surface, u eq is the equivalent control input, u sw is the switching control input, sign(.) is the sign function.

[0054] The present invention provides a finite-time terminal sliding mode control system for a piezoelectric actuator, comprising:

[0055] A data acquisition module for acquiring the input voltage signal and the output displacement signal of the piezoelectric actuator;

[0056] A model construction module for establishing a fractional-order coupling model of the piezoelectric actuator;

[0057] A parameter identification module for optimizing the parameters of the fractional-order coupling model;

[0058] A hysteresis compensation module for compensating the asymmetric hysteresis effect of the piezoelectric actuator;

[0059] A nonsingular terminal sliding mode controller, which generates a control signal according to the error between the actual displacement and the desired displacement acquired by the data acquisition module and acts on the piezoelectric actuator, and completes the high-precision trajectory tracking control of the piezoelectric actuator through a finite-time terminal sliding mode control algorithm.

[0060] In a second aspect, to solve the above technical problem, the present invention provides a computer-readable storage medium, on which a computer program is stored, and when the computer program is executed, the piezoelectric actuator based on the finite-time terminal sliding mode control method described in the first aspect is implemented.

[0061] In a third aspect, to solve the above technical problems, the present invention further provides an electronic device, including a memory, a processor, and a computer program stored on the memory and running on the processor. When the processor executes the computer program, the steps of the finite-time terminal sliding mode control method for the piezoelectric actuator described in the first aspect are implemented.

[0062] The above technical solution of the present invention has the following beneficial effects compared with the prior art:

[0063] The present invention provides a finite-time terminal sliding mode control method for a piezoelectric actuator, which introduces a fractional-order coupling model including an asymmetric term, can accurately capture the complex hysteresis characteristics of the piezoelectric actuator during dynamic operation, and also compensates for the asymmetric hysteresis effect of the piezoelectric actuator through a hysteresis feedforward compensator, significantly improving the control accuracy and response speed of the piezoelectric actuator. At the same time, by adopting a finite-time terminal sliding mode control algorithm, it ensures that the tracking error quickly converges to zero within a finite time. Compared with the traditional asymptotic convergence method, the convergence time is greatly shortened, the dynamic response ability of the system is improved, the singular problem that may occur in the traditional sliding mode control is effectively avoided, and the stability and reliability of the control system are enhanced. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] In order to make the content of the present invention easier to be clearly understood, the following further describes the present invention in detail according to specific embodiments of the present invention in combination with the drawings, where

[0065] Figure 1 is a flowchart of the finite-time terminal sliding mode control method for the piezoelectric actuator in the preferred embodiment of the present invention;

[0066] Figure 2 is a schematic diagram of the 1Hz sinusoidal driving voltage of the piezoelectric actuator in the present invention;

[0067] Figure 3 is a schematic diagram of the hysteresis curve generated by the piezoelectric actuator in the present invention;

[0068] Figure 4 is the simulation result of the hysteresis when the input voltage is a 1Hz biased sine wave in the present invention;

[0069] Figure 5 is a structural block diagram of the piezoelectric actuator tracking feedforward compensation control in the present invention;

[0070] Figure 6 is the sine tracking feedforward compensation control result when the input voltage is 1Hz in the present invention;

[0071] Figure 7 is a structural block diagram of the piezoelectric actuator tracking feedforward compensation combined with finite-time terminal sliding mode control in the present invention;

[0072] Figure 8 This is the result of sine tracking feedforward compensation combined with feedback control when the input voltage is 1 Hz in the present invention;

[0073] Figure 9 This is the error of sine tracking feedforward compensation combined with feedback control when the input voltage is 1 Hz in the present invention. Specific embodiments

[0074] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, so that those skilled in the art can better understand the present invention and be able to implement it, but the illustrated embodiments are not intended to limit the present invention.

[0075] Embodiment 1

[0076] Referring to Figure 1 as shown, the present invention provides a finite-time terminal sliding mode control method for a piezoelectric actuator, including the following steps:

[0077] Step S1: Collect the operating signals of the piezoelectric actuator, where the operating signals include input voltage and displacement output data;

[0078] Step S2: Further, based on the detailed data of the input voltage and output displacement of the piezoelectric actuator in actual operation, establish a fractional-order coupling model including asymmetric hysteresis characteristics; in this embodiment, a Keyence brand laser displacement sensor LK-H020 is used to accurately measure the displacement output of the piezoelectric actuator under different working conditions, and these data are an indispensable basis for establishing an accurate hysteresis model.

[0079] Step 3: Identify the parameters in the fractional-order coupling model, and adjust the parameters in the fractional-order coupling model so that the sum of the squares of the errors between the expected displacement of the model and the actual displacement of the piezoelectric actuator reaches the minimum; by adopting a parameter identification method, accurately adjust the parameters in the fractional-order coupling model to ensure that the model can closely approximate the actual physical process;

[0080] Step S4: Based on the identified fractional-order coupling model parameters, construct a hysteresis feedforward compensator to compensate the fractional-order coupling model after parameter identification, so that the actual displacement of the piezoelectric actuator approaches the expected displacement of the model; this hysteresis feedforward compensator can effectively cancel the asymmetric hysteresis effect in the piezoelectric actuator and provide strong support for subsequent high-precision control;

[0081] Step S5: On the basis of the above modeling and compensation, the present invention introduces a finite-time terminal sliding mode control algorithm, which realizes high-precision trajectory tracking control of the piezoelectric actuator within a finite time. This control strategy not only solves the problem that it is difficult to accurately obtain the upper bound of disturbance in traditional methods, but also significantly improves the convergence speed of the control system, effectively avoids singular phenomena in the control process, ensures the continuity and smoothness of the control quantity, greatly reduces the chattering phenomenon, and thus comprehensively improves the control accuracy and stability.

[0082] Specifically, in step S1, the fractional-order coupling model includes: a dynamic model for describing the linear dynamic behavior of the piezoelectric actuator; and an asymmetric Bouc-Wen hysteresis model for capturing the asymmetric hysteresis characteristics of the piezoelectric actuator. In addition, as part of the model establishment, the dynamic model in the present invention is a second-order differential model, which combines with the asymmetric Bouc-Wen hysteresis model (a second-order nonlinear mathematical model) to jointly describe the dynamic characteristics of the piezoelectric actuator. Its expression specifically shows the linear response basis of the piezoelectric actuator under the action of the input voltage, providing a solid theoretical basis for subsequent asymmetric hysteresis compensation and control strategies.

[0083] Among them, the expression of the dynamic model is as follows:

[0084]

[0085] The asymmetric Bouc-Wen hysteresis models are as follows:

[0086]

[0087] Among them, the parameter m is the mass of the piezoelectric actuator, c is the damping coefficient of the piezoelectric actuator, k is the piezoelectric driving stiffness, and x is the output displacement; the parameter k d represents the product of the piezoelectric coefficient and the stiffness, is the velocity, is the acceleration; u(t) represents the input voltage of the piezoelectric actuator; h(t) is the hysteresis variable of the piezoelectric actuator; is the first derivative of u(t), is the first derivative of h(t); α, β, γ, and n are all parameters of the Bouc-Wen hysteresis model; α is the amplitude of the hysteresis component, β is the shape coefficient, γ is used to control the shape of the hysteresis return line, δ is the asymmetry coefficient, and n is the plasticity index, which is used to control the growth rate of plastic deformation and the shape of the hysteresis loop; is the asymmetric formula; sign(.) is the sign function, and p represents the disturbance of the system, including model uncertainty, external disturbance, creep, vibration, and other nonlinear factors.

[0088] In step S3, the parameters are identified by the nonlinear least squares algorithm. The goal is to minimize the sum of the squares of the errors between the model prediction values and the actual displacements by adjusting the model parameters. The objective function is as follows:

[0089]

[0090] where C represents the total number of sampling points, and respectively represent the experimental output and the simulated output using the asymmetric Bouc-Wen model at each sampling time. The sampling intervals of the two outputs are equal. The system identification principle based on the asymmetric Bouc-Wen model is defined as follows: Using the minimum value of F(m, b, k, k d , α, β, γ, δ, n) to fit

[0091]

[0092] In step S4, assuming the desired displacement is x d , the hysteresis feedforward compensator can be determined by the following formula:

[0093]

[0094] In step S5, based on the above modeling and compensation, the present invention introduces a finite-time terminal sliding mode control algorithm, combined with the hysteresis feedforward compensator, to achieve high-precision trajectory tracking control of the piezoelectric actuator.

[0095] In view of the asymmetric hysteresis characteristics of the piezoelectric actuator, the present invention first conducts a hysteresis experiment by applying sinusoidal voltages with a frequency of 1 Hz and different voltage amplitudes (60 V, 80 V, 100 V), as Figure 2 shown, and uses a laser displacement sensor to record displacement data.

[0096] As Figure 3 shown, the experimental results show that as the voltage increases, the driving displacement increases significantly, and at the same time, the hysteresis rate also increases accordingly.

[0097] To accurately analyze this asymmetric hysteresis phenomenon, the present invention uses the least squares fitting method to decompose the hysteresis curve into a symmetric linear component and an asymmetric hysteresis component.

[0098] Furthermore, based on the experimental data, an improved Bouc-Wen hysteresis model with an asymmetric term is established to more accurately describe the nonlinear hysteresis behavior of the piezoelectric actuator.

[0099] The present invention adopts an efficient parameter identification method, that is, defining the root mean square error between the experimental data and the model output as the fitness function, and applying the nonlinear least squares algorithm to optimize and adjust the hysteresis model parameters.

[0100] This process aims to minimize errors, ensure a high degree of consistency between the simulation results and experimental data, and thus construct a mathematical model that can accurately reflect the hysteresis characteristics of the piezoelectric actuator.

[0101] The fitness function is expressed as follows:

[0102] To accurately identify the piezoelectric actuator system based on the asymmetric Bouc-Wen model, the present invention sets the total number of sampling points as C, and at each equally spaced sampling time point i, the actual displacement of the experimental output and the predicted displacement of the model output are respectively recorded.

[0103] The core principle of parameter identification is to optimize the parameters of the asymmetric Bouc-Wen model by minimizing the least squares error F(m, b, k, k d , α, β, γ, δ, n) between the experimental output and the simulation output so as to achieve the minimum value for accurately fitting the system characteristics.

[0104] The present invention has detailedly evaluated the performance of the established improved asymmetric Bouc-Wen hysteresis model. By setting strict error judgment criteria, it has verified the excellent description ability of this model for the low-frequency asymmetric hysteresis characteristics of the piezoelectric actuator. As Figure 4 shown, the asymmetric Bouc-Wen hysteresis model not only has a simple structure, but also its parameters are easy to identify, showing significant advantages compared with the existing improved models.

[0105] In this embodiment, taking the frequency of 1 Hz as an example to explain the feedforward compensation linearization control process of the hysteresis feedforward compensator in step S4:

[0106] As Figure 5 shown, the desired displacement x d is input into the nominal model of the piezoelectric actuator. The nominal model outputs a signal, which is added to the output of the hysteresis nominal model (asymmetric Bouc-Wen hysteresis model). The added result is used as the control signal and input into the actual piezoelectric actuator. The piezoelectric actuator generates the actual displacement according to the control signal.

[0107] The method for obtaining the control signal acting on the piezoelectric actuator in this embodiment is specifically as follows:

[0108] First, a standard biased sine displacement signal is set as the desired output displacement x d , and the nominal model of the piezoelectric actuator is constructed, and its expression is as follows:

[0109]

[0110] The desired output displacement x of the piezoelectric actuator dAs the input, it is substituted into the constructed nominal model of the piezoelectric actuator, and the corresponding sinusoidal tracking voltage signal u is obtained through model calculation. r .

[0111] The reference input signal u r is introduced and integrated into the differential equation This process not only considers the dynamic characteristics of the system, but also makes full use of the asymmetric Bouc-Wen hysteresis model for accurate representation, as follows:

[0112]

[0113] Through the calculation and optimization of the model, the final sinusoidal tracking voltage signal is finally obtained.

[0114]

[0115] This voltage signal u c is directly applied to drive the piezoelectric actuator to achieve precise control of the actuator displacement. During the experiment, the actual displacement data of the actuator was collected in real time and compared with the initially set desired displacement for analysis. The results are as Figure 6 shown. The consistency between the experimental values and the expected values verifies the effectiveness and accuracy of the constructed nominal model of the piezoelectric actuator, successfully realizing the effective compensation of the hysteresis nonlinearity of the piezoelectric actuator and achieving the purpose of feedforward compensation linearization control.

[0116] In step S5, after adopting the finite-time terminal sliding mode control algorithm, the piezoelectric actuator generates a corresponding output signal. This signal is compared with the reference output signal to generate an error signal, and this error signal acts on the nonsingular terminal sliding mode controller to make it generate a corresponding control signal to reduce the error. Thus, the feedback mechanism and the design of the nonsingular terminal sliding mode controller can enable the system to achieve the purpose of tracking the parameter input signal without error for the output signal, that is, the high-precision trajectory tracking control of the piezoelectric actuator.

[0117] The finite-time terminal sliding mode control algorithm described in this embodiment includes the following steps:

[0118] S51: Define the desired displacement x d (t) of the piezoelectric actuator;

[0119] S52: Measure the actual displacement x(t) of the piezoelectric actuator;

[0120] S53: Calculate the tracking error e

[0121] e(t) = x(t) - x d (t)

[0122]

[0123] S54: Design the sliding mode surface

[0124]

[0125] Substitute into the above formula to obtain

[0126]

[0127] S55: Design a control law u with finite-time convergence such that s converges to zero in finite time. The control law u includes:

[0128] u = u eq + u sw

[0129]

[0130] S56: Apply the control law u to the piezoelectric actuator to generate the actual displacement x(t);

[0131] S57: At each sampling moment, update the sliding mode surface s and the control law u until s converges to zero.

[0132] where k d is the damping coefficient, c is the damping ratio, A, B, υ, k s , η and λ are all control parameters, x is the actual displacement, x d is the desired displacement, e is the tracking error, is the first derivative of the tracking error, is the second derivative of the tracking error, s is the sliding mode surface, is the derivative of the sliding mode surface, u eq is the equivalent control input, u sw is the switching control input, and sign(.) is the sign function. In this embodiment, the parameters satisfy A, B > 0, 1 < υ < 2, and the expression of sign(.) is as shown in the following formula:

[0133]

[0134] The sliding mode surface designed by the present invention can ensure that the tracking error of the system converges to zero in finite time, solves the problem of asymptotic convergence in infinite time, has a fast convergence speed, and at the same time, the design of the control law effectively avoids the singular problem that may occur in traditional sliding mode control, enhancing the stability and reliability of the control system.

[0135] Furthermore, in view of the chattering and discontinuity problems existing in traditional sliding mode control, a variable function approaching mechanism is introduced based on the original sliding mode reaching law. This mechanism can dynamically adjust the rate at which the system state approaches the sliding surface:

[0136] When the system state is far from the sliding surface, the approaching rate automatically increases to ensure rapid convergence;

[0137] When the system state is close to the sliding surface, the approaching rate gradually slows down to reduce or eliminate the chattering phenomenon, thereby significantly improving the performance of sliding mode control.

[0138] This improvement not only effectively addresses the inherent problems in traditional sliding mode control but also further enhances the stability and accuracy of the control system.

[0139] In addition, in order to more effectively solve the problem of possible degradation of control performance caused by the discontinuity of the sign function in traditional sliding mode control;

[0140] The hyperbolic tangent function is introduced as an alternative in the present invention. The expression of this function is as follows:

[0141]

[0142] In this improvement, the smooth and continuous characteristics of the tanh function are utilized to replace the traditional sign function as part of the control law. The tanh function has a smooth transition near `s = 0`, which helps to reduce the sudden changes in the control signal, thereby reducing or eliminating the chattering phenomenon caused by discontinuous switching.

[0143] At the same time, by adjusting the input parameters of the tanh function or combining other control strategies, the dynamic response and steady-state accuracy of the control system can be further optimized. This alternative not only retains the robustness and fast response characteristics of sliding mode control but also significantly improves the smoothness of the control signal and the overall performance of the system, providing strong support for the popularization of sliding mode control in practical engineering applications.

[0144] To further verify the stability and efficiency of the control method of the present invention in practical applications, the Lyapunov stability theorem is used for theoretical proof. By constructing a Lyapunov function and analyzing the variation of its derivative with time, it is proved that the control method can achieve the convergence of the system state within a finite time, where the convergence time is expressed by the following formula:

[0145]

[0146] This theoretical result not only supports the good tracking effect observed in the experiment, but also theoretically ensures the stability and reliability of the control method, providing a solid theoretical basis for its wide application in practical engineering. Therefore, the control method of the present invention not only has excellent tracking performance, but also has excellent stability and convergence speed.

[0147] As Figure 8 and Figure 9 shown, by introducing and applying the control method of the present invention, the obtained sine tracking effect has been significantly improved compared with the traditional PID-form sliding mode control.

[0148] It can be clearly observed from the figure that the tracking error is greatly reduced, and the system output can more closely follow the desired displacement trajectory, showing excellent tracking performance.

[0149] This result indicates that the control strategy of the present invention has achieved remarkable results in improving the tracking accuracy and stability of the system, providing new ideas and solutions for the technological progress in related fields.

[0150] Embodiment 2

[0151] The present invention provides a finite-time terminal sliding mode control system for a piezoelectric actuator, including:

[0152] A data acquisition module for collecting the input voltage signal and output displacement signal of the piezoelectric actuator;

[0153] A model construction module for establishing a fractional-order coupling model of the piezoelectric actuator;

[0154] A parameter identification module for optimizing the parameters of the fractional-order coupling model;

[0155] A hysteresis compensation module for compensating the asymmetric hysteresis effect of the piezoelectric actuator;

[0156] A nonsingular terminal sliding mode controller generates a control signal according to the error between the actual displacement and the desired displacement collected by the data acquisition module and acts on the piezoelectric actuator, and completes the high-precision trajectory tracking control of the piezoelectric actuator through the finite-time terminal sliding mode control algorithm.

[0157] As Figure 7 shown, the desired displacement x dIt is input into a non-singular terminal sliding mode controller, which generates a control signal. This signal is adjusted by a hysteresis compensation module to compensate for the hysteresis effect of the piezoelectric actuator. The adjusted control signal is sent to the piezoelectric actuator, which generates an actual displacement according to the control signal. The actual displacement is measured by a laser displacement sensor and fed back to the control system for comparison with the desired displacement. The comparison result (error signal) is used to adjust the output of the controller to reduce the difference between the desired displacement and the actual displacement.

[0158] Embodiment III

[0159] The present invention provides a computer-readable storage medium, on which a computer program is stored. When the computer program is executed, it implements the piezoelectric actuator based on the finite-time terminal sliding mode control method described in Embodiment I.

[0160] Those skilled in the art should understand that the embodiments of the present application can be provided as methods, systems, or computer program products. Therefore, the present application can take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk memories, CD-ROMs, optical memories, etc.) containing computer-usable program codes.

[0161] The present application is described with reference to the flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each flow and / or block in the flowcharts and / or block diagrams, as well as the combination of flows and / or blocks in the flowcharts and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to the processors of general-purpose computers, special-purpose computers, embedded processors, or other programmable data processing devices to generate a machine, so that the instructions executed by the processors of the computer or other programmable data processing devices generate means for implementing the functions specified in Figure 1 one or more flows and / or blocks Figure 1 one or more blocks.

[0162] These computer program instructions can also be stored in a computer-readable memory that can direct a computer or other programmable data processing device to work in a specific manner, so that the instructions stored in the computer-readable memory generate a manufactured article including instruction means, and the instruction means implements the functions specified in Figure 1 one or more flows and / or blocks Figure 1 one or more blocks.

[0163] These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus, such that a series of operation steps are performed on the computer or other programmable apparatus to produce a computer-implemented process, so that the instructions executed on the computer or other programmable apparatus provide steps for implementing the functions specified in one process or a plurality of processes and / or blocks Figure 1 one process or a plurality of processes and / or blocks Figure 1 steps for implementing the functions specified in one block or a plurality of blocks.

[0164] Obviously, the above-described embodiments are merely examples for clear illustration and are not limitations on the implementation manners. For those of ordinary skill in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to enumerate all the implementation manners here. The obvious changes or modifications derived therefrom are still within the protection scope of the present invention.

Claims

1. A finite-time terminal sliding mode control method for a piezoelectric actuator, characterized in that: The specific steps include: S1, collecting an operating signal of a piezoelectric driver, wherein the operating signal includes input voltage and output displacement data; S2. Based on the collected input voltage and output displacement data, a fractional-order coupling model including asymmetric hysteresis characteristics is established; S3, performing parameter identification on the parameters in the fractional-order coupling model, and adjusting the parameters in the fractional-order coupling model so that the sum of squares of errors between the expected displacement of the model and the actual displacement of the piezoelectric actuator is minimized; S4. Based on the identified fractional-order coupling model parameters, a hysteresis feedforward compensator is constructed to compensate the fractional-order coupling model after parameter identification, so that the actual displacement of the piezoelectric actuator is close to the expected displacement of the model; S5. Through the finite-time terminal sliding mode control algorithm, the tracking error between the actual displacement and the expected displacement of the piezoelectric actuator is quickly converged to zero within a finite time, thereby realizing the trajectory tracking control of the piezoelectric actuator.

2. A piezoelectric actuator finite time terminal sliding mode control method according to claim 1, characterized in that: In step S1, the fractional-order coupling model includes: Dynamic models to describe the linear dynamic behavior of piezoelectric actuators; The asymmetric Bouc-Wen hysteresis model is used to capture the asymmetric hysteresis characteristics of piezoelectric actuators.

3. A piezoelectric actuator finite time terminal sliding mode control method according to claim 2, characterized in that: The expression of the dynamic model is as follows: The asymmetric Bouc-Wen hysteresis models are as follows: Among them, parameter m is the mass of the piezoelectric actuator, c is the damping coefficient of the piezoelectric actuator, k is the piezoelectric actuator stiffness, and x is the output displacement; parameter k d represents the product of the piezoelectric coefficient and the stiffness, For speed, is the acceleration; u(t) represents the input voltage of the piezoelectric actuator; h(t) is the hysteresis variable of the piezoelectric actuator; is the first-order derivative of u(t), is the first-order derivative of h(t); α, β, γ and n are all parameters of the Bouc-Wen hysteresis model; α is the amplitude of the hysteresis component, β is the shape coefficient, γ controls the shape of the hysteresis return line, δ is the asymmetry coefficient, and n is the plasticity index, which is used to control the growth rate of plastic deformation and the shape of the hysteresis loop; is an asymmetric formula; sign(.) is a sign function, and p represents the disturbance of the system, including model uncertainty, external disturbance, creep, vibration and other nonlinear factors.

4. A piezoelectric actuator finite time terminal sliding mode control method according to claim 3, characterized in that: In step S3, the parameter identification method includes the following steps: Define a fitness function to measure the error between the expected displacement and the actual displacement; The nonlinear least squares algorithm is used to optimize the parameters of the asymmetric Bouc-Wen model to minimize the sum of squares of the errors between the model prediction value and the actual displacement.

5. A piezoelectric actuator finite time terminal sliding mode control method according to claim 4, characterized in that: The fitness function is expressed by the following formula: Where C is the total number of sample points, i is each equally spaced sampling time point; x exp is the experimental observation value of the i-th sampling point; x sim is the model prediction value of the i-th sampling point, x exp and x sim The sampling intervals are equal; The method of optimizing the parameters of the asymmetric Bouc-Wen model using the nonlinear least squares algorithm is as follows: F(m,b,k,k d ,α,β,γ,δ,n) to fit 6. A piezoelectric actuator finite time terminal sliding mode control method according to claim 1, characterized in that: In step S4, the compensation method of the hysteresis feedforward compensator includes: The hysteresis compensation signal u is calculated based on the inverse model of the asymmetric Bouc-Wen hysteresis model. c ; The calculated hysteresis compensation signal u c The compensated control signal is added to the original control signal to form the compensated control signal, and finally the compensated control signal is used for the piezoelectric driver.

7. A piezoelectric actuator finite time terminal sliding mode control method according to claim 6, characterized in that: The hysteresis compensation signal u output by the feedforward compensator c The expression is as follows: Among them, k d is the damping coefficient of the piezoelectric actuator, x d (t) is the expected displacement, h r (t) is the hysteresis compensation term.

8. A piezoelectric actuator finite time terminal sliding mode control method according to claim 1, characterized in that: The finite time terminal sliding mode control algorithm comprises the following steps: S51: Define the desired displacement x of the piezoelectric actuator d (t); S52: measuring the actual displacement x(t) of the piezoelectric actuator; S53: Calculate the tracking error e and derive e; e(t)=x(t)-x d (t) S54: Design of sliding surface Will Substituting into the above formula, we get S55: Design a finite-time convergent control law u so that s converges to zero within a finite time. The expression of the control law u is as follows: in=in eq +in sw S56: Applying the control law u to the piezoelectric actuator to generate an actual displacement x(t); S57: at each sampling moment, update the sliding surface s and the control law u until s converges to zero; Among them, k d is the damping coefficient, c is the damping ratio, A, B, υ, k s , η and λ are control parameters, x is the actual displacement, x d is the expected displacement, e is the tracking error, is the first-order derivative of the tracking error, is the second-order derivative of the tracking error, s is the sliding surface, is the derivative of the sliding surface, u eq is the equivalent control input, u sw is the switching control input, and sign(.) is the sign function.

9. A piezoelectric actuator finite time terminal sliding mode control system, characterized in that: include: A data acquisition module, used for acquiring an input voltage signal and an output displacement signal of the piezoelectric driver; Model building module for building fractional-order coupled models of piezoelectric actuators; A parameter identification module, used for optimizing the parameters of the fractional-order coupling model; A hysteresis compensation module is used to compensate for the asymmetric hysteresis effect of the piezoelectric actuator; The non-singular terminal sliding mode controller generates a control signal based on the error between the actual displacement collected by the data acquisition module and the expected displacement, and acts on the piezoelectric driver, and completes high-precision trajectory tracking control of the piezoelectric driver through a finite-time terminal sliding mode control algorithm.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed, the piezoelectric drive finite-time terminal sliding mode control method according to any one of claims 1 to 8 is implemented.

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