Piezoelectric system nonlinear compensation control method based on MRAC-PID composite control

Abstract

The invention discloses a piezoelectric system nonlinear compensation control method based on MRAC-PID composite control, and relates to a piezoelectric system nonlinear compensation control method. The method aims at solving the problems that in the prior art, robustness is poor when complex nonlinearity and parameter mutation of a piezoelectric system are processed, and the capacity of eliminating steady-state errors is weak when the piezoelectric system is independently used. The method comprises the following steps: step 1, establishing a kinetic model of a piezoelectric system; and step 2, designing an MRAC-PID algorithm. The invention belongs to the technical field of nonlinear compensation control of a piezoelectric system.

Description

Technical Field

[0001] The invention relates to a piezoelectric system nonlinear compensation control method, belonging to the technical field of piezoelectric system nonlinear compensation control. Background Art

[0002] Piezoelectric systems are widely used in precision positioning and actuation. They are used to achieve high-precision worktable positioning in lithography equipment for semiconductor manufacturing and as microactuators in microelectromechanical systems (MEMS) to drive the movement of tiny structures. However, the nonlinear properties of piezoelectric materials pose a significant challenge to the precise control of piezoelectric systems.

[0003] Piezoelectric systems based on traditional control algorithms often have difficulty accurately achieving the various performance indicators set by the system during actual operation. Due to the nonlinear characteristics of piezoelectric materials such as hysteresis and creep, as well as the interference of complex and changeable working environments, piezoelectric systems controlled by traditional algorithms are prone to problems such as reduced control accuracy and slow response, and cannot effectively meet the actual use requirements of the system. For example, traditional PID control has difficulty dealing with the nonlinear problems of piezoelectric systems. The hysteresis characteristics of piezoelectric materials make the input-output relationship complex. When system parameters change or are subject to external interference, its response speed decreases and control accuracy is difficult to guarantee, which cannot meet the needs of high-precision control. Although model reference adaptive control (MRAC) can adjust parameters according to the reference model, it lacks robustness when dealing with complex nonlinearities and parameter mutations in piezoelectric systems, and its ability to eliminate steady-state errors is weak when used alone.

[0004] The invention patent with publication number CN107544241B and application date September 25, 2017 discloses a nonlinear PID inverse compensation control method for hysteresis of piezoelectric ceramic actuators. The Preisach hysteresis inverse model is established by numerical methods, and the established inverse model is used for cascade compensation. Then, in order to improve the anti-interference ability of the controller, a nonlinear PID controller is designed. This nonlinear PID controller changes the traditional PID method of directly integrating errors and integrates the errors using a nonlinear function with small error amplification and large error saturation. The established hysteresis inverse compensation can better compensate for the hysteresis nonlinearity of piezoelectric ceramics. The nonlinear PID inverse compensation control established on this basis can not only reduce the oscillation caused by integration, but also improve the control accuracy of the controller.

[0005] The invention patent, with publication number CN114253138A and application date December 16, 2021, discloses a compensation control method and system for a nanopositioning platform based on a dynamic delay PI model. This method, belonging to the field of control engineering, includes: collecting the output displacement generated by the piezoelectric actuator in the nanopositioning platform at different driving voltages at multiple frequencies, and generating hysteresis curves of the driving voltage and output displacement at each frequency; establishing a dynamic delay PI model containing a delay play operator, which is used to introduce a rising delay coefficient and a falling delay coefficient into the driving voltage term of the delay play operator; adjusting the rising delay coefficient and the falling delay coefficient until the fit between the dynamic delay PI model and each hysteresis curve is less than a preset threshold; solving the inverse model of the optimal dynamic delay PI model, and compensating the nanopositioning platform based on the inverse model. This method compensates for the asymmetry and rate-related hysteresis nonlinearity of the piezoelectric actuator, achieving precise control of the nanopositioning platform.

[0006] However, the above technologies all have poor robustness in dealing with complex nonlinearities and parameter mutations of piezoelectric systems, and their ability to eliminate steady-state errors is weak when used alone. Summary of the Invention

[0007] In order to solve the problems in the prior art of poor robustness in dealing with complex nonlinearities and parameter mutations of piezoelectric systems, and weak ability to eliminate steady-state errors when used alone, the present invention proposes a piezoelectric system nonlinear compensation control method based on MRAC-PID composite control.

[0008] The technical solution adopted by the present invention to solve the above problems is: the steps of the present invention include:

[0009] Step 1: Establish a dynamic model of the piezoelectric system;

[0010] Step 2: Design the MRAC-PID algorithm.

[0011] Furthermore, the process of establishing the piezoelectric system dynamic model in step 1 is as follows:

[0012] The mechanical structure of the piezoelectric system is equivalent to a mass-spring-damper system. Let the mass be m, the spring stiffness be k, the damping coefficient be b, and the external force F acting on the system to produce a displacement x.

[0013] According to Newton's second law, the dynamic equation of the system is:

[0014]

[0015] Perform Laplace transform on formula (1) to obtain the transfer function G of the mechanical structure: m (s) is:

[0016]

[0017] In formula (2), s represents the complex variable in the Laplace transform, X(s) represents the expression of the displacement x(t) in the time domain after Laplace transform in the frequency domain, and F(s) represents the expression of the external force F(t) in the time domain after Laplace transform in the frequency domain.

[0018] Furthermore, the steps of designing the MRAC-PID algorithm in step 2 include:

[0019] Step 201: Establish a second-order system reference model;

[0020] Step 202: Error definition;

[0021] Step 203: Design a PID controller;

[0022] Step 204: Design the MRAC adaptive control rate;

[0023] Step 205: derive the adaptive law based on Lyapunov stability theory.

[0024] Furthermore, step 201 specifically includes:

[0025] The dynamic equation of the reference model is:

[0026]

[0027] In formula (3), m m represents the equivalent mass of the reference model, b m represents the damping coefficient of the reference model, k m represents the spring stiffness of the reference model, x m (t) represents the displacement output of the reference model, r(t) represents the reference input, and α(m) represents the piezoelectric conversion coefficient of the reference model.

[0028] Furthermore, the error definition in step 202 specifically includes:

[0029] The tracking error e(t) is defined as the difference between the actual system output x(t) and the reference model output x m The difference between (t):

[0030] e(t)=x(t)-x m (t)(4),

[0031] Take the first and second derivatives of the error:

[0032]

[0033] Furthermore, designing the PID controller in step 203 includes:

[0034] The PID controller calculates the control quantity u based on the error e(t) PID (t), its expression is:

[0035]

[0036] In formula (7), K p Represents the proportionality coefficient, K i Indicates the integral coefficient, K d represents the differential coefficient.

[0037] Furthermore, designing the MRAC adaptive control rate in step 204 includes:

[0038] Introduce an adaptive control law to integrate the adaptation mechanism:

[0039]

[0040] In formula (8), θ1(t), θ2(t), and θ3(t) represent adaptive parameters.

[0041] Furthermore, in step 205, the adaptive law is derived based on Lyapunov stability theory, including:

[0042] The Lyapunov function V(t) is selected to ensure the stability of the system, and the following is selected:

[0043]

[0044] In formula (9), represents the estimation error of the adaptive parameters, represents the optimal parameter value, γ1, γ2 and γ3 represent positive adaptive gains;

[0045] Find the first-order derivative of V(t):

[0046]

[0047] Substituting the equations of the actual system and the reference model into Then substitute the expression By sorting and selecting the adaptive law, The adaptive law is derived:

[0048]

[0049] The beneficial effects of the present invention are:

[0050] 1. This invention improves control accuracy: By combining the adaptive capabilities of MRAC with the precise adjustment characteristics of PID, this invention can effectively compensate for the nonlinear characteristics of the piezoelectric system and significantly improve control accuracy. Compared with traditional PID control and pure MRAC control, it can significantly reduce steady-state errors and enable more precise control of fast piezoelectric systems.

[0051] 2. The present invention can enhance anti-interference capabilities: the combination of the rapid response of the PID controller and the adaptive adjustment of the MRAC enables the system to quickly react and adjust the control signal when facing external interference. In the presence of interference such as mechanical vibration and temperature changes, the system can maintain stable operation, effectively suppressing the impact of interference on the piezoelectric system and improving the robustness of the system.

[0052] 3. The present invention can accelerate convergence speed: the MRAC-PID composite control structure optimizes the parameter adjustment process and can make the system output converge to the reference model output more quickly than traditional control methods. When the system starts or switches working states, the piezoelectric system can reach a stable state more quickly, improving the dynamic response performance of the system.

[0053] 4. The present invention is highly adaptable: The MRAC and PID parameters can be flexibly adjusted according to the characteristics and application scenarios of different piezoelectric systems, making it widely applicable. Whether in high-precision astronomical observation or industrial laser processing, it can achieve excellent control effects. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 It is a schematic diagram of the dynamic equivalent model of the piezoelectric system;

[0055] Figure 2 This is the control block diagram of the MRAC-PID compensation method;

[0056] Figure 3 This is the MRAC-PID algorithm flow chart. DETAILED DESCRIPTION

[0057] Specific implementation method 1: Figures 1 to 3 As shown, a nonlinear compensation control method for a piezoelectric system based on MRAC-PID composite control includes the following steps:

[0058] Step 1: Establish a dynamic model of the piezoelectric system; the specific process includes:

[0059] The mechanical structure of the piezoelectric system is equivalent to a mass-spring-damper system. Let the mass be m, the spring stiffness be k, the damping coefficient be b, and the external force F acting on the system to produce a displacement x.

[0060] According to Newton's second law, the dynamic equation of the system is:

[0061]

[0062] Perform Laplace transform on formula (1) to obtain the transfer function G of the mechanical structure: m (s) is:

[0063]

[0064] In formula (2), s represents the complex variable in the Laplace transform, X(s) represents the expression of the displacement x(t) in the time domain after Laplace transform in the frequency domain, and F(s) represents the expression of the external force F(t) in the time domain after Laplace transform in the frequency domain.

[0065] Step 2: Design the MRAC-PID algorithm. The specific steps include:

[0066] Step 201: Establish a second-order system reference model;

[0067] The dynamic equation of the reference model is:

[0068]

[0069] In formula (3), m m represents the equivalent mass of the reference model, b m represents the damping coefficient of the reference model, k m represents the spring stiffness of the reference model, x m (t) represents the displacement output of the reference model, r(t) represents the reference input, and α(m) represents the piezoelectric conversion coefficient of the reference model;

[0070] Step 202: Error definition;

[0071] The tracking error e(t) is defined as the difference between the actual system output x(t) and the reference model output x m The difference between (t):

[0072] e(t)=x(t)-x m (t)(4),

[0073] Take the first and second derivatives of the error:

[0074]

[0075] Step 203: Design a PID controller;

[0076] The PID controller calculates the control quantity u based on the error e(t) PID (t), its expression is:

[0077]

[0078] In formula (7), K p Represents the proportionality coefficient, K i Indicates the integral coefficient, K d represents the differential coefficient;

[0079] Step 204: Design the MRAC adaptive control rate;

[0080] Introduce an adaptive control law to integrate the adaptation mechanism:

[0081]

[0082] In formula (8), θ1(t), θ2(t), and θ3(t) represent adaptive parameters;

[0083] Step 205: derive the adaptive law based on Lyapunov stability theory;

[0084] The Lyapunov function V(t) is selected to ensure the stability of the system, and the following is selected:

[0085]

[0086] In formula (9), represents the estimation error of the adaptive parameters, θ i * represents the optimal parameter value, γ1, γ2 and γ3 represent positive adaptive gains;

[0087] Find the first-order derivative of V(t):

[0088]

[0089] Substituting the equations of the actual system and the reference model into Then substitute the expression By sorting and selecting the adaptive law, The adaptive law is derived:

[0090]

[0091] The above description is merely a preferred embodiment of the present invention and does not constitute any form of limitation to the present invention. Although the present invention has been disclosed as a preferred embodiment as above, it is not intended to limit the present invention. Any technician familiar with the present profession can make some changes or modifications to equivalent embodiments of equivalent changes using the technical content disclosed above without departing from the scope of the technical solution of the present invention. However, any simple modification, equivalent replacement and improvement of the above embodiments made according to the technical essence of the present invention, within the spirit and principles of the present invention, without departing from the content of the technical solution of the present invention, shall still fall within the scope of protection of the technical solution of the present invention.

Claims

1. A piezoelectric system nonlinear compensation control method based on MRAC-PID composite control, characterized in that: The specific steps include: Step 1: Establish a dynamic model of the piezoelectric system; Step 2: Design the MRAC-PID algorithm.

2. The piezoelectric system nonlinear compensation control method based on MRAC-PID composite control according to claim 1, characterized in that: The process of establishing the piezoelectric system dynamic model in step 1 is: The mechanical structure of the piezoelectric system is equivalent to a mass-spring-damper system. Let the mass be m, the spring stiffness be k, the damping coefficient be b, and the external force F acting on the system to produce a displacement x. According to Newton's second law, the dynamic equation of the system is: Perform Laplace transform on formula (1) to obtain the transfer function G of the mechanical structure: m (s) is: In formula (2), s represents the complex variable in the Laplace transform, X(s) represents the expression of the displacement x(t) in the time domain after Laplace transform in the frequency domain, and F(s) represents the expression of the external force F(t) in the time domain after Laplace transform in the frequency domain.

3. The piezoelectric system nonlinear compensation control method based on MRAC-PID composite control according to claim 1, characterized in that: The steps in designing the MRAC-PID algorithm in step 2 include: Step 201: Establish a second-order system reference model; Step 202: Error definition; Step 203: Design a PID controller; Step 204: Design the MRAC adaptive control rate; Step 205: derive the adaptive law based on Lyapunov stability theory.

4. The piezoelectric system nonlinear compensation control method based on MRAC-PID composite control according to claim 3 is characterized in that: Step 201 specifically includes: The dynamic equation of the reference model is: In formula (3), m m represents the equivalent mass of the reference model, b m represents the damping coefficient of the reference model, k m represents the spring stiffness of the reference model, x m (t) represents the displacement output of the reference model, r(t) represents the reference input, and α(m) represents the piezoelectric conversion coefficient of the reference model.

5. The piezoelectric system nonlinear compensation control method based on MRAC-PID composite control according to claim 3, characterized in that: The error definition in step 202 specifically includes: The tracking error e(t) is defined as the difference between the actual system output x(t) and the reference model output x m The difference between (t): e(t)=x(t)-x m (t)(4), Take the first and second derivatives of the error:

6. The piezoelectric system nonlinear compensation control method based on MRAC-PID composite control according to claim 3, characterized in that: Designing a PID controller in step 203 includes: The PID controller calculates the control quantity u based on the error e(t) PID (t), its expression is: In formula (7), K p Represents the proportionality coefficient, K i Indicates the integral coefficient, K d represents the differential coefficient.

7. The piezoelectric system nonlinear compensation control method based on MRAC-PID composite control according to claim 3, characterized in that: Designing the MRAC adaptive control rate in step 204 includes: Introduce an adaptive control law to integrate the adaptation mechanism: In formula (8), θ1(t), θ2(t), and θ3(t) represent adaptive parameters.

8. The piezoelectric system nonlinear compensation control method based on MRAC-PID composite control according to claim 3 is characterized in that: In step 205, the adaptive law is derived based on Lyapunov's stability theory, including: The Lyapunov function V(t) is selected to ensure the stability of the system, and the following is selected: In formula (9), represents the estimation error of the adaptive parameters, represents the optimal parameter value, γ1, γ2 and γ3 represent positive adaptive gains; Find the first-order derivative of V(t): Substituting the equations of the actual system and the reference model into Then substitute the expression By sorting and selecting the adaptive law, The adaptive law is derived:

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