Adaptive control method and device for piezoelectric platform

Abstract

The application is suitable for the technical field of micro-nano motion control, and provides a self-adaptive control method and device for a piezoelectric platform; the control method is modeled based on actual displacement of the piezoelectric platform and equivalent displacement caused by influence of hysteresis nonlinearity characteristics, so that the actual required piezoelectric platform control quantity can be accurately calculated. The scheme can tolerate a certain degree of model uncertainty and external interference, so that the system state converges to zero in a limited time. The method can not only realize high-precision positioning control of the piezoelectric platform, but also improve the dynamic performance of the piezoelectric platform, and has good control precision of the piezoelectric platform. The control device corresponding to the method also has the same technical effects.

Description

Technical Field

[0001] This invention belongs to the field of micro-nano motion control technology, and particularly relates to an adaptive control method and device for a piezoelectric platform. Background Technology

[0002] The inverse piezoelectric effect of piezoelectric ceramics can convert electrical energy into mechanical energy; that is, piezoelectric ceramics can produce minute deformations under an applied voltage. Piezoelectric platforms typically achieve motion of tens or even hundreds of micrometers by stacking piezoelectric ceramics and combining them with a stroke amplification mechanism. However, the inherent hysteresis nonlinearity of piezoelectric ceramics not only affects the command tracking accuracy of the piezoelectric platform but also limits its dynamic performance. Especially under high-dynamic application conditions, this severely restricts its further application and promotion. Existing technologies have shortcomings. Summary of the Invention

[0003] The purpose of this invention is to provide an adaptive control method and apparatus for a piezoelectric platform, aiming to solve the technical problem that existing control methods for piezoelectric platforms cannot adapt to the inherent hysteresis nonlinear characteristics of piezoelectric ceramics.

[0004] On one hand, the present invention provides an adaptive control method for a piezoelectric platform, used to drive the piezoelectric platform to move after inputting a displacement command; including the following steps:

[0005] S1. A descriptive model of the piezoelectric platform is constructed based on the influence of hysteresis nonlinearity on the displacement of the piezoelectric platform and the actual displacement of the piezoelectric platform;

[0006] S2. The sliding surface state variables are calculated based on the sum of the displacement error and its integral and differential.

[0007] S3. Calculate the adaptive sliding mode finite-time control quantity of the piezoelectric platform based on the control quantity of state feedback and the control quantity of adaptive sliding mode;

[0008] S4. The control quantity is sent to the piezoelectric platform drive module via a digital-to-analog converter; if control continues, steps S3 and S4 are repeated, and the piezoelectric platform achieves continuous displacement command tracking.

[0009] Furthermore, step S1 includes:

[0010] S11. Construct a descriptive model of the actual displacement of the piezoelectric platform;

[0011] S12. Construct a descriptive model of the displacement generated when the piezoelectric platform is affected by the hysteresis nonlinear characteristics.

[0012] Furthermore, in step S2, the constants of the proportional, integral, and differential equations are taken as positive numbers. When the integral constant of the sliding surface is 0, the sliding surface is simplified into a PD-type sliding surface with a simpler structure.

[0013] Furthermore, in step S3, the control quantity based on adaptive sliding mode is calculated using the sign function of the sliding surface state quantity and its state in the model.

[0014] Furthermore, in step S3, the control quantity based on adaptive sliding mode is calculated using the saturation function of the sliding surface state quantity and its state in the model.

[0015] Furthermore, in step S3, the adaptive sliding mode fixed-time convergence control quantity of the piezoelectric platform is calculated based on the double power-law approach and the adaptive control law.

[0016] On the other hand, the present invention also provides an adaptive control device for a piezoelectric platform, used to drive the piezoelectric platform to move after inputting a displacement command; including a piezoelectric driver, a piezoelectric platform, and an ARM-based embedded control circuit electrically connected to a DC power supply; the ARM-based embedded control circuit is used to execute the adaptive control method of the piezoelectric platform as described above.

[0017] Furthermore, the piezoelectric actuator is electrically connected to the piezoelectric platform to output a drive signal to drive the piezoelectric platform to perform corresponding displacement.

[0018] Furthermore, the ARM-based embedded control circuit is electrically connected to the piezoelectric driver and outputs control commands to control the piezoelectric driver to output the drive signal.

[0019] Furthermore, the ARM-based embedded control circuit is used to acquire the model parameters of the piezoelectric platform, and to collect the motion displacement signal of the piezoelectric platform in real time through a displacement sensor, while acquiring the displacement, velocity and acceleration commands of the piezoelectric platform; and to further calculate the displacement error and its integral and differential values, calculate the sliding surface, calculate the adaptive sliding mode finite time control quantity; and after determining whether to perform cyclic control, output control commands to control the piezoelectric actuator.

[0020] This invention proposes an adaptive control method and apparatus for a piezoelectric platform. Its purpose is to accurately calculate the required control quantities of the piezoelectric platform by modeling the actual displacement of the platform and the equivalent displacement caused by its hysteresis nonlinear characteristics. This scheme can tolerate a certain degree of model uncertainty and external disturbances, allowing the system state to converge to zero within a finite time. This method not only achieves high-precision positioning control of the piezoelectric platform but also improves its dynamic performance, exhibiting good control accuracy. Attached Figure Description

[0021] Figure 1 This is a flowchart illustrating the implementation of the adaptive control method for the piezoelectric platform provided in Embodiment 1 of the present invention.

[0022] Figure 2 This is a schematic diagram of the architecture of the adaptive control device for the piezoelectric platform provided in Embodiment 2 of the present invention;

[0023] Figure 3 This is a schematic diagram of the operation process of the device in Embodiment 2 of the present invention. Detailed Implementation

[0024] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0025] The specific implementation of the present invention will be described in detail below with reference to specific embodiments:

[0026] The term "limited" means that when a step signal command is input, the controller designed based on this invention enables the piezoelectric platform to track the input command within a limited time.

[0027] The sliding surface is a fundamental state variable in sliding mode control. It is a mathematically designed equation that, when the controlled system enters the sliding surface, will exhibit the desired dynamic behavior.

[0028] Example 1:

[0029] Figure 1 The implementation flow of the adaptive control method for the piezoelectric platform provided in Embodiment 1 of the present invention is shown. For ease of explanation, only the parts related to the embodiments of the present invention are shown, and are described in detail below:

[0030] On one hand, the present invention provides an adaptive control method for a piezoelectric platform, used to drive the piezoelectric platform to move after inputting a displacement command; including the following steps:

[0031] S1. A descriptive model of the piezoelectric platform is constructed based on the influence of hysteresis nonlinearity on the displacement of the piezoelectric platform and the actual displacement of the piezoelectric platform;

[0032] S2. The sliding surface state variables are calculated based on the sum of the displacement error and its integral and differential.

[0033] S3. Calculate the adaptive sliding mode finite-time control quantity of the piezoelectric platform based on the control quantity of state feedback and the control quantity of adaptive sliding mode;

[0034] S4. The control quantity is sent to the piezoelectric platform drive module via a digital-to-analog converter; if control continues, steps S3 and S4 are repeated, and the piezoelectric platform achieves continuous displacement command tracking.

[0035] Furthermore, step S1 includes:

[0036] S11. Construct a descriptive model of the actual displacement of the piezoelectric platform;

[0037] S12. Construct a descriptive model of the displacement generated when the piezoelectric platform is affected by the hysteresis nonlinear characteristics.

[0038] In specific implementation of step S1: the model of the piezoelectric platform containing hysteresis nonlinear characteristics is described in the following form, that is, the description model of the piezoelectric platform is constructed based on the nonlinear characteristics of hysteresis in the hysteresis state and the dynamic characteristics of the piezoelectric platform in acceleration:

[0039]

[0040] Where x(t) represents the actual displacement of the piezoelectric platform, Indicates the speed of the piezoelectric platform. Let represent the acceleration of the piezoelectric platform; u(t) represents the total control quantity of the piezoelectric platform. The derivatives of the control quantities of the piezoelectric platform are given; d(t) represents the equivalent disturbance experienced by the piezoelectric platform; h(t) represents the influence of hysteresis nonlinearity on the piezoelectric platform model, which is equivalent to a displacement. The velocity quantities are equivalent to the hysteresis state of the piezoelectric platform; a0, a1, a2, a3 are constants used to describe the dynamic characteristics of the piezoelectric platform; α, β, γ are constants used to describe the hysteresis characteristics of the piezoelectric platform.

[0041] The actual displacement is obtained through a displacement sensor on a piezoelectric platform, while the velocity and acceleration are obtained through displacement signals and a differentiator. The constants a0, a1, a2, a3, α, β, and γ in the model can be obtained online or offline using an iterative optimization algorithm.

[0042] Furthermore, in step S2, the constants of the proportional, integral, and differential equations are taken as positive numbers. When the integral constant of the sliding surface is 0, the sliding surface is simplified into a PD-type sliding surface with a simpler structure.

[0043] In specific implementation of step S2: calculate the sliding surface state (based on the sum of the displacement error and its integral and differential to calculate the sliding surface state quantity), the formula is as follows:

[0044]

[0045] Where σ(t) represents the sliding surface state variable in sliding mode control, and e(t) is the displacement error of the piezoelectric platform. The integral of the displacement error, This is the differential of the displacement error, i.e., the velocity error. kp σ It is the proportionality constant of the sliding surface, ki σThe integral constant of the sliding surface, kd σ These are the differential constants of the sliding surface, and all three are usually positive. In practical applications, kd σ Usually, kp is taken as 1. σ and ki σ The value of determines the speed at which the error converges to zero when the system reaches the sliding surface.

[0046] It should be noted that ki σ When the value is 0, formula (2) degenerates into a simpler PD-type sliding surface, which does not affect the design of the subsequent controller.

[0047] The displacement error is calculated as follows:

[0048] e(t) = x(t) - x d (t) (3)

[0049] Where, x d (t) represents the desired displacement of the piezoelectric platform, i.e., the displacement command.

[0050] The integral value of the displacement error can be obtained by the following calculation:

[0051] e sum (t=k)=e sum (t=k-1)+e(t=k-1)*st (4)

[0052] Among them, e sum (t=k) and e sum (t=k-1) represents the integral value of the displacement error in the k-th and k-1-th periods, respectively, and st represents the sampling time, which is set to 0.001s in this embodiment.

[0053] The differential of the displacement error, i.e., the velocity error, can be obtained through the following calculation:

[0054]

[0055] in, This indicates the desired speed of the piezoelectric platform, i.e., the speed command.

[0056] In specific implementation of step S3: the adaptive sliding mode finite-time control quantity (device) of the piezoelectric platform is calculated based on the adaptive sliding mode control and finite-time convergence characteristics, in the following form:

[0057] u(t)=u e (t)+u n (t) (6)

[0058] Where u(t) represents the total control quantity of the piezoelectric platform, u e (t) represents the control quantity based on state feedback, un (t) represents the control quantity based on adaptive sliding mode.

[0059]

[0060] Furthermore, in step S3, the control quantity based on adaptive sliding mode is calculated using the sign function of the sliding surface state quantity and its state in the model.

[0061] u n (t)=-k1σ(t)-k a (t)|σ(t)| m sign(σ(t)) (8)

[0062] Where k1 > 0 and 0 < m < 1, k1 and m are constants, both of which affect the speed at which the system tends toward the sliding surface. The value of k1 plays a dominant role when the system state is far from the sliding surface, and the value of m plays a dominant role when the system state is close to the sliding surface.

[0063] The absolute value in formula (8) is the design result that guarantees finite-time convergence.

[0064] Furthermore, in step S3, the control quantity based on adaptive sliding mode is calculated using the sign function of the sliding surface state quantity and its state in the model.

[0065] The symbolic function sign(·) is represented as follows:

[0066]

[0067] k a (t) is the adaptive change, and the adaptive control law is as follows:

[0068]

[0069] Among them, g a It is a constant greater than zero.

[0070] Furthermore, in step S3, the control quantity based on adaptive sliding mode is calculated using the saturation function of the sliding surface state quantity and its state in the model. That is, the saturation function can be used to replace the sign function in formula (9).

[0071] In practice, the larger the value of the control parameter k1, the stronger the system's resistance to model uncertainty and external disturbances. At the same time, the chattering effect of sliding mode control is also more severe, so a compromise needs to be made in the value.

[0072] The larger the value of the control parameter m, the greater the approach speed of the system when it is far away from the sliding surface, and the slower the approach speed when it is close to the sliding surface. In practical applications, a value of 0.8-0.9 is better.

[0073] The constant g a It is the steady-state value that the adaptive parameter approaches; generally, a value of 1-10 yields good results.

[0074] In order to achieve the goal that the system state convergence time is independent of the initial state, the adaptive sliding mode fixed-time convergence control quantity of the piezoelectric platform is calculated in step S3 based on the double power reaching law and the adaptive control law.

[0075] Specifically, in step S3, formula (8) is designed as follows to obtain the adaptive sliding mode fixed-time convergence control quantity (device) of the piezoelectric platform.

[0076] u n (t)=-k1σ(t)-k a (t)|σ(t)| m sign(σ(t))-k b (t)|σ(t)| n sign(σ(t)) (11)

[0077] Where, k a The adaptive control law of (t) remains unchanged as in formula (10), k b The adaptive control law design for (t) is as follows:

[0078]

[0079] Where n is a constant greater than 1, g b It is a constant greater than zero.

[0080] The larger the value of the control parameter n, the greater the approach speed of the system when it is far away from the sliding surface, and the slower the approach speed when it is close to the sliding surface. However, an excessively large parameter will affect the process of the system slowing down when it approaches the sliding surface. In practical applications, a value of 1.1-1.2 is better.

[0081] The constant g b It is the steady-state value that the adaptive parameter approaches; generally, a value of 1-10 yields good results.

[0082] In specific implementation of step S4: the control quantity is sent to the piezoelectric platform drive module through the digital-to-analog converter; if control continues, steps S3 and S4 are repeated, and the piezoelectric platform realizes continuous displacement command tracking to obtain the high-precision positioning control effect of the piezoelectric platform.

[0083] The above control method has the following characteristics:

[0084] First, to demonstrate that this invention has a certain robustness to model uncertainty and external disturbances, the effects of model uncertainty and external disturbances are now equated to d, and a positive definite Lyapunov function representing the energy of the system is selected:

[0085]

[0086] The conclusion can be drawn through derivation:

[0087]

[0088] Typically, the equivalent disturbance to a system is bounded, assuming that |d|≤δ<k1+ k a , k a Indicates k a The lower bound of can then be further derived:

[0089]

[0090] When |σ(t)|≥1, based on the premise of bounded perturbation, it is easy to see that the system converges to σ(t)=1; when |σ(t)|<1, further:

[0091]

[0092] or

[0093] It can be seen that when or This guarantees that the system converges to the equilibrium point, that is, the system converges to:

[0094]

[0095] It can be seen that when the model uncertainty and external disturbances are bounded, the energy function of the system can still converge to a small range. Therefore, the piezoelectric platform adaptive controller design method of the present invention has a certain degree of robustness.

[0096] Second, for the finite-time convergent controller (8), a new positive definite Lyapunov function is selected:

[0097]

[0098] The conclusion can be drawn through derivation:

[0099]

[0100] Where A is related to m and g aWith the presence of positive constants, it can be seen that the solution to the above differential equation is related to the initial state of the system, and therefore the controller has finite-time convergence characteristics.

[0101] Third, for the controller (11) with fixed-time convergence, a new Lyapunov function is selected:

[0102]

[0103] The same derivation leads to the following conclusion:

[0104]

[0105] Where B is related to n and g b With the presence of relevant positive constants, it can be seen that the solution of the above differential equation is independent of the initial state of the system. Therefore, the piezoelectric platform adaptive controller of the present invention has fixed-time convergence characteristics.

[0106] In summary, this invention proposes an adaptive sliding mode finite-time control method for a piezoelectric platform, which is robust to model uncertainties and external disturbances, and can improve the positioning accuracy and dynamic performance of the piezoelectric platform.

[0107] Example 2:

[0108] Figure 2 A schematic diagram of the architecture of the adaptive control device for the piezoelectric platform provided in Embodiment 2 of the present invention is shown. For ease of explanation, only the parts related to the embodiments of the present invention are shown.

[0109] The present invention provides an adaptive control device for a piezoelectric platform, used to drive the piezoelectric platform to move after inputting a displacement command; it includes a piezoelectric driver, a piezoelectric platform, and an ARM-based embedded control circuit electrically connected to a DC power supply; the ARM-based embedded control circuit is used to execute the adaptive control method of the piezoelectric platform as described above.

[0110] Furthermore, the piezoelectric actuator is electrically connected to the piezoelectric platform to output a drive signal to drive the piezoelectric platform to perform corresponding displacement.

[0111] Furthermore, the ARM-based embedded control circuit is electrically connected to the piezoelectric driver and outputs control commands to control the piezoelectric driver to output the drive signal.

[0112] Furthermore, such as Figure 3As shown, the ARM-based embedded control circuit is used to acquire the model parameters of the piezoelectric platform, and to collect the motion displacement signal of the piezoelectric platform in real time through a displacement sensor. At the same time, it acquires the displacement, velocity and acceleration commands of the piezoelectric platform; and further calculates the displacement error and its integral and differential values, calculates the sliding surface, calculates the adaptive sliding mode finite time control quantity; and outputs control commands to control the piezoelectric actuator after determining whether to perform cyclic control.

[0113] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. An adaptive control method for a piezoelectric platform, used to drive the piezoelectric platform to move after inputting a displacement command; characterized in that, Includes the following steps: S1. A descriptive model of the piezoelectric platform is constructed based on the influence of hysteresis nonlinearity on the displacement of the piezoelectric platform and the actual displacement of the piezoelectric platform; S2. The sliding surface state variables are calculated based on the sum of the displacement error and its integral and differential. S3. Calculate the adaptive sliding mode finite-time control quantity of the piezoelectric platform based on the control quantity of state feedback and the control quantity of adaptive sliding mode; S4. Send the control signal to the piezoelectric platform drive module via a digital-to-analog converter; If control continues, steps S3 and S4 are repeated, and the piezoelectric platform achieves continuous displacement command tracking. Step S3 specifically includes: The adaptive sliding mode finite-time control quantity of the piezoelectric platform is calculated based on adaptive sliding mode control and finite-time convergence characteristics, and its form is as follows: u(t)=u e (t)+u n (t) (1) Where u(t) represents the total control quantity of the piezoelectric platform, u e (t) represents the control quantity based on state feedback, u n (t) represents the control quantity based on adaptive sliding mode; Where a0, a1, a2, and a3 are constants used to describe the dynamic characteristics of the piezoelectric platform; x(t) represents the actual displacement of the piezoelectric platform. Indicates the speed of the piezoelectric platform; x d (t) represents the desired displacement of the piezoelectric platform. This indicates the desired speed of the piezoelectric platform. denoted by , h(t) represents the desired acceleration of the piezoelectric platform; h(t) represents the effect of hysteresis nonlinearity on the piezoelectric platform model, equivalent to a displacement; kp σ It is the proportionality constant of the sliding surface, ki σ The integral constant of the sliding surface, kd σ These are the differential constants of the sliding surface, and all three take positive values; kd σ Take 1, kp σ and ki σ The value of determines the velocity at which the error converges to zero when the system reaches the sliding surface; e(t) is the displacement error of the piezoelectric platform. This is the differential of the displacement error, i.e., the velocity error; In step S3, the control quantity based on adaptive sliding mode is calculated using the sign function of the sliding surface state quantity and its state in the model. u n (t)=-k1σ(t)-k a (t)|σ(t)| m sign(σ(t)) (3) where k a (t) is the adaptive change amount, σ(t) represents the sliding mode surface state quantity in sliding mode control, and sign() is the sign function; k1>0 and 0<m<1, k1 and m are constants, and both affect the speed of the system approaching the sliding mode surface. Among them, the value of k1 plays a dominant role when the system state is far from the sliding mode surface, and the value of m plays a dominant role when the system state is close to the sliding mode surface; the absolute value is the design result to ensure finite-time convergence.

2. The method as described in claim 1, characterized in that, Step S1 includes: S11. Construct a descriptive model of the actual displacement of the piezoelectric platform; S12. Construct a descriptive model of the displacement generated when the piezoelectric platform is affected by the hysteresis nonlinear characteristics.

3. The method as described in claim 1, characterized in that, In step S2, the constants of the proportional, integral, and differential equations are positive. When the integral constant of the sliding surface is 0, the sliding surface is simplified into a PD-type sliding surface with a simpler structure.

4. The method as described in claim 3, characterized in that, In step S3, the control quantity based on adaptive sliding mode is calculated using the sign function of the sliding surface state quantity and its state in the model.

5. The method as described in claim 3, characterized in that, In step S3, the control quantity based on adaptive sliding mode is calculated using the saturation function of the sliding surface state quantity and its state in the model.

6. The method as described in claim 4, characterized in that, In step S3, the adaptive sliding mode fixed-time convergence control quantity of the piezoelectric platform is calculated based on the double power-law approach and the adaptive control law.

7. An adaptive control device for a piezoelectric platform, used to drive the piezoelectric platform to move after inputting a displacement command; characterized in that, It includes a piezoelectric driver, a piezoelectric platform, and an ARM-based embedded control circuit, all electrically connected to a DC power supply; the ARM-based embedded control circuit is used to execute an adaptive control method for the piezoelectric platform as described in any one of claims 1-6.

8. The apparatus as claimed in claim 7, characterized in that, The piezoelectric actuator is electrically connected to the piezoelectric platform to output a drive signal to drive the piezoelectric platform to perform corresponding displacement.

9. The apparatus as claimed in claim 8, characterized in that, The ARM-based embedded control circuit is electrically connected to the piezoelectric driver and outputs control commands to control the piezoelectric driver to output the drive signal.

10. The apparatus as claimed in claim 9, characterized in that, The ARM-based embedded control circuit is used to acquire the model parameters of the piezoelectric platform, and to collect the motion displacement signal of the piezoelectric platform in real time through a displacement sensor. It also acquires the displacement, velocity and acceleration commands of the piezoelectric platform. Furthermore, it calculates the displacement error and its integral and differential values, calculates the sliding surface, and calculates the adaptive sliding mode finite-time control quantity. After determining whether cyclic control is needed, a control command is output to control the piezoelectric actuator.

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