A data-driven adaptive sliding mode iterative control method for a piezoelectric ceramic micro-motion platform

Abstract

The present invention discloses a data-driven adaptive sliding mode iterative control method for a piezoelectric ceramic micro-motion platform, belonging to the field of micro-nano control technology. In this method, an adaptive learning rate is introduced in the data-driven adaptive sliding mode iterative control based on compact-form dynamic linearization to design a data-driven controller. Compared with the existing technologies, the present invention does not require any information of other models such as the hysteresis nonlinear model of the piezoelectric ceramic micro-motion platform, and introduces a compact-form dynamic linearization data model, avoiding the complex process of platform modeling and the influence of the accuracy of the established model on the effectiveness of the controller; an adaptive learning rate is adopted in the iterative process, so that the most appropriate learning rate can be selected at each moment to further improve the control performance of the controller.

Description

Technical Field

[0001] The present invention belongs to the technical field of micro-nano control, and more specifically, relates to a data-driven adaptive sliding mode iterative control method for a piezoelectric ceramic micro-motion platform. Background Art

[0002] With the rapid development of a series of industrial directions such as the high-end manufacturing industry and ultra-precision machining, piezoelectric ceramics are widely used in ultra-precision positioning fields such as precision chip manufacturing, medical imaging, ultrasonic motors, and aerospace equipment. However, due to the strong hysteresis non-linearity and creep characteristics of the piezoelectric ceramic material itself, it will seriously affect its control accuracy during actual trajectory tracking experiments, and even lead to divergence. Therefore, solving its inherent non-linearity problem is an important issue for expanding the application range of piezoelectric ceramic micro-motion platforms.

[0003] In order to improve the control accuracy of piezoelectric micro-motion platforms, scholars have proposed various control schemes based on offline mathematical models. Gu Guoying et al. used an improved PI model to perform hysteresis inverse compensation on piezoelectric actuators to achieve precise control. Su Chunyi et al. from Concordia University used a robust adaptive sliding mode control strategy to achieve high-precision tracking control of piezoelectric ceramic micro-positioning platforms; Cheng Long et al. from the University of Chinese Academy of Sciences used a model predictive control method to achieve high-precision trajectory tracking control of piezoelectric ceramic micro-positioning platforms. These control methods have high requirements for the accuracy of offline models. However, when facing many actual control problems, it is often difficult to establish or even impossible to obtain the mathematical model of the controlled object. Therefore, many researchers have started to study data-driven control methods. Hou Zhongsheng et al. proposed a data-driven adaptive control method, which can effectively control the controlled system only relying on the I / O data of the system, thus avoiding a series of problems brought by offline mathematical models. Due to its strong applicability and the advantage of avoiding the establishment of complex offline mathematical models, many scholars have proposed various data-driven control methods on this basis. Zhang Yongchang et al. proposed a data-driven predictive control for driving permanent magnet synchronous motors. Wang Yinsong et al. proposed a data-driven adaptive terminal sliding mode control and achieved high-precision control of water tanks. Among them, data-driven sliding mode control has shown good performance in solving trajectory tracking problems of various non-linear systems due to its strong ability to overcome system uncertainties.

[0004] In order to further improve the control accuracy of data-driven sliding mode control, data-driven sliding mode iterative learning control has been proposed. By taking advantage of the characteristic that iterative learning control can continuously optimize its own control performance, the control accuracy of data-driven sliding mode control is continuously improved during iteration. However, during the iteration process, due to the continuous pursuit of improving control accuracy, the convergence performance of the system is often ignored, resulting in phenomena such as slow system convergence speed and excessive amplitude of chatter during the convergence process, seriously affecting the control performance. Among them, the learning rate plays a crucial role in the control effect of the iterative learning control link. If the learning rate is too large, the system will gradually diverge during the iteration process, while if the learning rate is too small, the system will require too many iterations to reach convergence, and the control accuracy will be too low. Therefore, in the actual operation process, selecting a suitable learning rate often requires a lot of work. Even if a relatively suitable learning rate is finally obtained, since it is a fixed value, it is often impossible to take into account both the convergence performance and the control accuracy of the system during the iteration process.

[0005] Therefore, how to improve the learning rate during the iteration process has strong research significance. In the proposed method, an adaptive learning rate is proposed to replace the previous fixed learning rate, making its iteration process adaptive. Thus, the convergence performance and control accuracy are improved. Summary of the Invention

[0006] Aiming at the defects and improvement requirements of the existing technology, the present invention provides a data-driven adaptive sliding mode iterative control method for a piezoelectric ceramic micro-motion platform. The purpose is to improve the trajectory tracking accuracy of the piezoelectric ceramic micro-motion platform by adopting a data-driven adaptive sliding mode iterative control method for the piezoelectric ceramic micro-motion platform without any model parameter information related to the platform.

[0007] The present invention is implemented by the following scheme:

[0008] A data-driven adaptive sliding mode iterative control method for a piezoelectric ceramic micro-motion platform specifically includes the following steps:

[0009] Step 1: Establish a compact-form dynamic linearization data model for the piezoelectric ceramic micro-motion platform, and obtain a data-driven adaptive controller based on the compact-form dynamic linearization through the control input criterion function.

[0010] The piezoelectric ceramic micro-motion platform is described as the following nonlinear non-affine system:

[0011] y l (k + 1) = f(y l (k), y l (k - 1), …, y l (k - n y ), u l (k), u l(k - 1), …, u l (k - n u )) (1)

[0012] wherein, and are the input voltage and output displacement of the system at time k, respectively, n y , n u ∈Z + is the unknown order of the output and input of the system, l is the number of iterations, and f(·) is an unknown nonlinear function.

[0013] When f(·) satisfies the continuous partial derivative with respect to any variable and satisfies the generalized Lipschitz condition, the piezoelectric ceramic micro - motion platform (1) can be equivalent to the following compact - form dynamic linearization data model:

[0014]

[0015] wherein, is called the pseudo - partial derivative, and the pseudo - partial derivative is bounded at any time.

[0016] Define the pseudo - partial derivative criterion function as:

[0017]

[0018] where μ is the penalty factor, restricting the change range.

[0019] Substitute Equation (2) into Equation (3), and according to obtain the pseudo - partial derivative estimation formula as:

[0020]

[0021] where η is the parameter related to the boundedness of the pseudo - partial derivative; μ is the penalty factor, used to limit the change value of the pseudo - partial derivative.

[0022] Step 2: Under the guidance of the dynamic linearization model, establish a sliding mode surface for the system and select an exponential reaching law to obtain a preliminary control law.

[0023] Define the sliding mode surface as follows:

[0024] ρ l (k) = ce l (k), c > 0 (5)

[0025] where c is the parameter related to the sliding mode surface, participating in the error convergence analysis, e l (k) = y r (k) - y l (k) is the error of the l - th iteration, where y r(k) is the desired trajectory.

[0026] From (5), the sliding mode surface at time k + 1 is:

[0027]

[0028] To further reduce the jitter phenomenon brought by sliding mode control, the saturation function is defined as follows:

[0029]

[0030] Γ is the boundary layer, which can determine the upper and lower bounds of the saturation function.

[0031] To obtain the data-driven sliding mode control reaching law, the exponential reaching law is defined as follows:

[0032] ρ l (k + 1) = (1 - qT)ρ l (k) - λTsat(ρ l (k)) (8)

[0033] Among them, q is the exponential reaching parameter, used to adjust the error convergence speed; T is the sampling period, and λ is the saturation function limit parameter.

[0034] From (6), (7) and (8), the data-driven adaptive sliding mode control law is as follows:

[0035]

[0036] Step 3: Use the information of the previous iteration and combine the proposed adaptive learning rate to continuously optimize the preliminary control law of the sliding mode control on the iteration axis, so as to improve the control performance of the system.

[0037] The iterative control law is as follows:

[0038]

[0039] Among them, is the adaptive learning rate.

[0040] From (9) and (10), the final control law is summarized as follows:

[0041]

[0042] The beneficial effects of the present invention:

[0043] The present invention provides a data-driven adaptive sliding mode iterative control method for a piezoelectric ceramic micro-motion platform, which combines data-driven adaptive sliding mode iterative control based on compact-form dynamic linearization with an adaptive learning rate, designs a data-driven adaptive sliding mode iterative controller by using a compact-form dynamic linearization data model; and improves the control performance of the controller by adopting an adaptive learning rate. It avoids the complex process of platform modeling and the influence of the accuracy of the established model on the effectiveness of the controller; an adaptive learning rate is adopted during the iterative process, greatly improving the control effect of the controller on the piezoelectric ceramic micro-motion platform; the addition of the adaptive learning rate enables the iterative process of the system to find a suitable learning rate at each moment, thereby improving the overall trajectory tracking accuracy of the control system. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 is an experimental device diagram of the piezoelectric ceramic micro-motion platform of the present invention;

[0045] Figure 2 is a working principle diagram of the piezoelectric ceramic micro-motion platform of the present invention;

[0046] Figure 3 is a block diagram of the data-driven adaptive sliding mode iterative control system of the piezoelectric ceramic micro-motion platform of the present invention;

[0047] Figure 4 is a tracking curve of the desired displacement and the actual displacement of the system when the input signal frequency of the present invention is 1 Hz;

[0048] Figure 5 is a tracking curve of the desired displacement and the actual displacement of the system when the input signal frequency of the present invention is 10 Hz;

[0049] Figure 6 is a tracking curve of the desired displacement and the actual displacement of the system when the input signal frequency of the present invention is 20 Hz;

[0050] Figure 7 is a tracking curve of the desired displacement and the actual displacement of the system when the input signal frequency of the present invention is 50 Hz;

[0051] Figure 8 is a tracking curve of the desired displacement and the actual displacement of the system when the input signal frequency of the present invention is 100 Hz;

[0052] Figure 9 is an error curve of the desired displacement and the actual displacement of the system when the input signal frequency of the present invention is 1 Hz;

[0053] Figure 10 is an error curve of the desired displacement and the actual displacement of the system when the input signal frequency of the present invention is 10 Hz;

[0054] Figure 11The error curve between the expected displacement and the actual displacement of the system when the input signal frequency of the present invention is 20 Hz;

[0055] Figure 12 The error curve between the expected displacement and the actual displacement of the system when the input signal frequency of the present invention is 50 Hz;

[0056] Figure 13 The error curve between the expected displacement and the actual displacement of the system when the input signal frequency of the present invention is 100 Hz. Detailed implementation manners

[0057] To make the technical solution proposed by the present invention and the technical problems to be solved clearer, the present invention will be further described by elaborating on a preferred specific embodiment in detail.

[0058] The input and output data required for the data-driven adaptive sliding mode iterative control method of the piezoelectric ceramic micro-motion platform described in the present invention are obtained by on-line acquisition of the piezoelectric ceramic micro-motion platform.

[0059] As Figure 1 shows the main equipment of the piezoelectric ceramic micro-motion platform control system, and its model and uses are:

[0060] High-precision micrometer, model MDSL-0500M6-1A, mainly used to complete high-precision displacement measurement; high-precision vibration isolation table, model J02-1809, mainly used to reduce external vibration interference; precision positioning controller, model PPC-2CR0120, mainly used to amplify signals; multi-functional data acquisition card, model PCI-1716, mainly used to perform A / D and D / A conversions on the acquired signals; piezoelectric ceramic micro-motion platform, model MPT-2MRL102A, as the experimental object, mainly provides experimental data; computer, model Dell Vostro 3900, mainly used to process and complete the control algorithm.

[0061] Combined with Figure 2 , the working process of the piezoelectric ceramic micro-motion platform is as follows:

[0062] A computer installed with MATLAB / Simulink software emits a desired signal. The desired signal output by the computer is converted into a voltage signal through the D / A converter of the multifunctional data acquisition card. Then, the voltage signal is amplified 15 times by the precision positioning controller as the driving voltage signal of the piezoelectric ceramic micro-motion platform to drive the piezoelectric actuator in the piezoelectric ceramic micro-motion platform to generate a displacement signal. The displacement signal is measured by the high-precision micrometer in the piezoelectric ceramic micro-motion platform and input into the A / D converter of the multifunctional data acquisition card and converted into a tracking signal to be transmitted back to the computer, completing the control process of the piezoelectric ceramic micro-motion platform. During the experiment, the piezoelectric ceramic micro-motion platform is always located on the high-precision vibration isolation table. The specific parameters in the experiment are as follows: η = 0.55, μ = 0.3, q = 6000, c = 100, λ = 1×10 -6 , Γ = 50, and the number of iterations is selected as l = 20 during this experimental process.

[0063] Next, design a data-driven adaptive sliding mode iterative controller.

[0064] As Figure 3 shown in the data-driven adaptive sliding mode iterative control block diagram of the piezoelectric ceramic micro-motion platform, according to a data-driven control method for a piezoelectric ceramic micro-motion platform provided by an embodiment of the present invention, it includes a compact-form dynamic linearization data-driven adaptive sliding mode controller and an iterative learning controller.

[0065] Describe the piezoelectric ceramic micro-motion platform as the following non-linear non-affine system:

[0066] y l (k + 1) = f(y l (k), y l (k - 1), …, y l (k - n y ), u l (k), u l (k - 1), …, u l (k - n u )) (1)

[0067] where and are the input voltage and output displacement of the system at time k respectively, n y , n u ∈Z + are the unknown orders of the output, input, and total disturbance of the system, l = 1, 2, 3, … is the number of iterations, and f(·) is an unknown non-linear function.

[0068] When f(·) satisfies the continuous partial derivative with respect to any variable and satisfies the generalized Lipschitz condition, the piezoelectric ceramic micro-motion platform (1) can be equivalent to the following compact-form dynamic linearization data model:

[0069]

[0070] Among them, is called the pseudo partial derivative, and the pseudo partial derivative is bounded at any time.

[0071] Define the pseudo partial derivative criterion function as:

[0072]

[0073] where μ is the penalty factor, restricting the change range of is the estimated value of

[0074] Substitute Equation (2) into Equation (3), and according to obtain the pseudo partial derivative estimation formula as:

[0075]

[0076] where η is the parameter related to the boundedness of the pseudo partial derivative.

[0077] Define the sliding mode surface as follows:

[0078] ρ l (k) = ce l (k), c > 0 (5)

[0079] where c is the parameter related to the sliding mode surface, participating in the error convergence analysis, and e l (k) = y r (k) - y l (k) is the error of the l-th iteration, where y r (k) is the desired trajectory.

[0080] From (5), the sliding mode surface at time k + 1 can be obtained as:

[0081]

[0082] To further reduce the jitter phenomenon brought by the sliding mode control, define the saturation function as follows:

[0083]

[0084] Γ is the boundary layer, which can determine the upper and lower bounds of the saturation function.

[0085] To obtain the data-driven sliding mode control reaching law, define the exponential reaching law as follows:

[0086] ρ l (k + 1) = (1 - qT)ρ l (k) - λTsat(ρ l(k)) (8)

[0087] Among them, q is the exponential approaching parameter used to adjust the error convergence speed, T is the sampling period, and λ is the saturation function limit parameter.

[0088] From (6), (7) and (8), the data-driven adaptive sliding mode control law is obtained as follows:

[0089]

[0090] To further improve the controller performance, iterative learning control is introduced on the basis of the original data-driven adaptive sliding mode control, and an adaptive learning rate is adopted to improve its control performance.

[0091] The iterative control law is as follows:

[0092]

[0093] Among them, is the adaptive learning rate.

[0094] From (9) and (10), the final control law is summarized as follows:

[0095]

[0096] Finally, the effectiveness of the data-driven adaptive sliding mode iterative control method for the piezoelectric ceramic micro-motion platform proposed by the present invention is verified through the piezoelectric ceramic micro-motion platform control system.

[0097] Given that the frequencies g of the desired sine signals of the piezoelectric ceramic micro-motion platform are 1 Hz, 10 Hz, 20 Hz, 50 Hz and 100 Hz, that is, the desired input signal is: y r = 3sin(2πgk - 0.5π) + 3. The range of the trajectory tracking signal is 0 - 36 V.

[0098] As Figure 4 、 Figure 5 、 Figure 6 、 Figure 7 and Figure 8 are the tracking curves of the desired displacement and the actual displacement of the system when the input desired sine signals are 1 Hz, 10 Hz, 20 Hz, 50 Hz and 100 Hz respectively.

[0099] As Figure 9 、 Figure 10 、 Figure 11 、 Figure 12 and Figure 13 are the error curves of the desired displacement and the actual displacement of the system when the input desired sine signals are 1 Hz, 10 Hz, 20 Hz, 50 Hz and 100 Hz respectively.

[0100] Calculate the root mean square error (RMSE) and the maximum absolute error percentage (MAXE) after stabilization at the above different frequencies to better illustrate the control effect of the present invention.

[0101] Table 1 below shows the root mean square error and the maximum absolute error of the piezoelectric ceramic micro-motion platform when the expected sine signals are at different frequencies of 1 Hz, 10 Hz, 20 Hz, 50 Hz, and 100 Hz.

[0102]

[0103] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A data-driven adaptive sliding mode iterative control method for a piezoelectric ceramic micro-motion platform, characterized in that The method specifically includes the following steps: Step 1: Establish a compact-form dynamic linearization data model for the piezoelectric ceramic micro-motion platform, and obtain a data-driven adaptive controller based on the compact-form dynamic linearization through the control input criterion function; The piezoelectric ceramic micro-motion platform is described as the following non-linear non-affine system: y l (k + 1)= f(y l (k), y l (k - 1),…, y l (k - n y ), u l (k), u l (k - 1),…, u l (k - n u )) (1) Among them, and are the input voltage and output displacement of the system at time k, respectively. n y , n u ∈Z + are the unknown orders of the output and input of the system, l is the number of iterations, and f(·) is an unknown nonlinear function; When f(·) satisfies the continuous partial derivative with respect to any variable and satisfies the generalized Lipschitz condition, the piezoelectric ceramic micro-motion platform (1) can be equivalent to the following compact-form dynamic linearization data model: △y l (k + 1) = θ l (k)△u l (k)(2) where, θ l (k) is called the pseudo partial derivative, and the pseudo partial derivative is bounded at any moment; Define the pseudo partial derivative criterion function as: where μ is the penalty factor, which restricts the change range of θ(k); Substitute Equation (2) into Equation (3). According to The pseudo partial derivative estimation formula is obtained as follows: where η is the parameter related to the boundedness of the pseudo partial derivative; μ is the penalty factor, which is used to restrict the change value of the pseudo partial derivative; Step 2: Under the guidance of the dynamic linearization model, establish a sliding mode surface for the system and select an exponential reaching law to obtain a preliminary control law; Define the sliding mode surface as follows: ρ l (k) = ce l (k), c > 0 (5) Among them, c is a parameter related to the sliding surface, which is involved in the analysis of error convergence, and e l (k) = y r (k) - y l (k) is the error of the l-th iteration, where y r (k) is the desired trajectory; From (5), the sliding mode surface at time k+1 can be obtained as: To further reduce the jitter phenomenon brought by the sliding mode control, define the saturation function as follows: Γ is the boundary layer, which can determine the upper and lower limits of the saturation function; To obtain the data-driven sliding mode control reaching law, define the exponential reaching law as follows: ρ l (k + 1) = (1 - qT)ρ l (k) - λTsat(ρ l (k)) (8) where q is the exponential reaching parameter, which is used to adjust the error convergence speed; T is the sampling period, and λ is the saturation function limit parameter; From (6), (7) and (8), the data-driven adaptive sliding mode control law can be obtained as follows: Step 3: Use the information of the previous iteration and combine the proposed adaptive learning rate to continuously optimize the preliminary control law obtained by the sliding mode control on the iteration axis, so as to improve the system control performance. The iterative control law is as follows: Among them, is the adaptive learning rate; From (9) and (10), the final control law is summarized as follows:

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