Tracking control method of piezoelectric actuator

Abstract

The present invention relates to the field of precision motion control technology, and in particular to a tracking control method for a piezoelectric actuator. The tracking control method proposed in the present invention uses a neural network to approximate the nonlinear dynamic terms of the piezoelectric actuator without establishing a nonlinear dynamic model, uses a hysteresis model to describe the hysteresis nonlinearity, and designs an adaptive law to update the density function of the hysteresis model to ensure that the model can accurately describe the rate-dependent hysteresis characteristics. On this basis, a fuzzy approximator is designed, and the fuzzy approximator is used to replace the switching term in the traditional sliding mode control to solve the sliding mode control chattering problem.

Description

Technical Field

[0001] The present invention belongs to the technical field of precision motion control, and in particular relates to a tracking control method of a piezoelectric actuator. Background Art

[0002] Piezoelectric actuators have outstanding advantages such as high control accuracy and fast response speed. As actuators, they play an important role in precision motion control. However, piezoelectric actuators are affected by the characteristics of piezoelectric materials and have nonlinear characteristics such as hysteresis, which makes it more difficult to control piezoelectric actuators and limits their use in precision motion control. In view of the nonlinear characteristics such as hysteresis of piezoelectric actuators, existing methods generally use hysteresis modeling to feedforward compensate for hysteresis terms, but there are problems such as uncertainty in model parameters, and there is a lack of effective modeling methods for other nonlinear characteristics; among the common control methods of piezoelectric actuators, sliding mode control has advantages in terms of rapidity and anti-interference ability, but the jitter problem introduced by the sliding mode switching term limits the practical application of sliding mode control. Summary of the invention

[0003] In view of this, the present invention aims to provide a tracking control method for a piezoelectric actuator, which accurately compensates for the hysteresis nonlinearity of the piezoelectric actuator by designing an adaptive update method of the hysteresis model, compensates for other nonlinear dynamic terms of the piezoelectric actuator by designing a neural network estimator, and realizes high-precision tracking control of a piezoelectric actuator with nonlinearities such as hysteresis by designing a fuzzy approximator to replace the switching terms in traditional sliding mode control.

[0004] To achieve the above object, the technical solution created by the present invention is implemented as follows:

[0005] A tracking control method for a piezoelectric actuator comprises the following steps:

[0006] S1: The piezoelectric actuator is described as a second-order nonlinear system dynamics model;

[0007] S2: Design a neural network estimator for other nonlinear dynamic terms of the piezoelectric actuator ; Among them, the neural network estimator It is expressed as:

[0008]

[0009] in, represents the estimated weight vector between the hidden layer and the output layer, Represents the Sigmoid activation function of the hidden layer nodes, represents the weight matrix between the input layer and the hidden layer, Represents the input data of the neural network estimator;

[0010] S3: Design a fuzzy approximation adaptive neural network sliding mode tracking controller; wherein the fuzzy approximation adaptive neural network sliding mode tracking controller is expressed as:

[0011]

[0012] in, represents the output value of the fuzzy approximation adaptive neural network sliding mode tracking controller, Desired trajectory signal The second derivative of Represents the adaptive density function in the hysteresis model Adaptive update rate; represents a fuzzy approximator; represents the central average defuzzifier; Represents the estimated weight vector The update rate; represents the adaptive weight vector; represents the output vector of the product inference engine; represents the sliding surface; represents a column vector; , , All represent the parameters of the fuzzy approximation adaptive neural network sliding mode tracking controller, and are positive numbers;

[0013] S4: The output value of the fuzzy approximation adaptive neural network sliding mode tracking controller The piezoelectric actuator is input and tracking control is performed on the piezoelectric actuator.

[0014] Furthermore, in step S1:

[0015] The piezoelectric actuator model is characterized as:

[0016]

[0017] in, , , , , are all parameters of the piezoelectric actuator model and are constants. represents the output displacement of the piezoelectric actuator, represents the input voltage of the piezoelectric actuator, represents the hysteresis nonlinearity of the piezoelectric actuator;

[0018] The hysteresis model is used to describe the hysteresis nonlinear term of the piezoelectric actuator. The nonlinear dynamic term of the piezoelectric actuator other than the hysteresis nonlinear term is expressed as , then the piezoelectric actuator model is rewritten as:

[0019]

[0020] in, , Both represent the state of the piezoelectric actuator model. represents the displacement, represents the output displacement value of the piezoelectric actuator model, represents the hysteresis nonlinear term of the piezoelectric actuator described by the hysteresis model, represents the external disturbance, represents the input value of the hysteresis model, represents the density function of the hysteresis model, represents the hysteresis model operator and is expressed as:

[0021]

[0022] in, represents the Preisach plane, , represents the coordinates of the hysteresis model operator on the Preisach plane, and represents the saturation turning point of the hysteresis model operator, express The Preisach plane is a real plane. and express The minimum and maximum values ​​of represents the Lipschitz function updated by the hysteresis model operator, represents the hysteresis loop boundary function and is expressed as:

[0023]

[0024] in, represents the independent variable of the boundary function, represents the maximum value of the hysteresis loop displacement output. In formula (3), according to Neither decreasing nor increasing, or , calculate the corresponding boundary function The value of Represents the hysteresis loop boundary function The slope of , represents the dividing line on the Preisach plane;

[0025] The hysteresis model is:

[0026]

[0027] in, Density function representing the adaptive update of the hysteresis model;

[0028] The Preisach plane of the integral region in equation (5) is replaced by a uniformly distributed Horizontal and vertical lines divide grid; for the A grid, the lower left corner node of the grid Value and The values ​​are defined as and , multiplying the two to get , and as the first The compensation value of the grid; The compensation values ​​of the grids are summed to obtain the total compensation term And expressed as:

[0029]

[0030] in, ; express As a column vector consisting of elements, it is used to express equation (6) in the form of vector product, .

[0031] Furthermore, in step S2:

[0032] Defining the Tracking Error of the Piezoelectric Actuator Model and its derivatives for:

[0033]

[0034] in, represents the expected trajectory signal;

[0035] Neural network estimator for tracking error calculation based on piezoelectric actuator model And expressed as:

[0036]

[0037] in, represents the estimated weight vector between the hidden layer and the output layer, Represents the Sigmoid activation function of the hidden layer nodes, , represents the weight matrix between the input layer and the hidden layer, represents the input data of the neural network estimator, .

[0038] Furthermore, in step S3:

[0039] Tracking error according to the piezoelectric actuator model and its derivatives Design sliding surface , sliding surface It is expressed as:

[0040]

[0041] in, represents the sliding surface parameter, which is a positive constant;

[0042] According to the sliding surface Designing a fuzzy approximator , fuzzy approximator It consists of a product inference engine, a Gaussian fuzzifier and a central average defuzzifier, a fuzzy approximator It is expressed as:

[0043]

[0044] in, represents the adaptive weight vector, represents the output vector of the product inference engine, and Multiply to form a central average defuzzifier ; represents adaptive weight; Represents product inference engine Output: and represents the true value parameter vector, , , represents the length of the true value parameter vector;

[0045] represents the Gaussian fuzzer and is expressed as:

[0046]

[0047] in, and are truth value parameters, respectively and The elements in ;

[0048] In formula (10), the adaptive weight vector The adaptive rate is , the adaptive rate calculation formula is ;

[0049] Based on sliding surface , Neural Network Estimator and fuzzy approximator Design a fuzzy approximation adaptive neural network sliding mode tracking controller, which can be expressed as:

[0050]

[0051] in, represents the output value of the fuzzy approximation adaptive neural network sliding mode tracking controller, Desired trajectory signal The second derivative of Represents the adaptive density function in the hysteresis model Adaptive update rate; represents a fuzzy approximator; represents the central average defuzzifier; Represents the estimated weight vector The update rate; represents the adaptive weight vector; represents the output vector of the product inference engine; represents the sliding surface; represents a column vector; , , Both represent the parameters of the fuzzy approximation adaptive neural network sliding mode tracking controller, and both are positive numbers.

[0052] Compared with the prior art, the invention can achieve the following beneficial effects:

[0053] The present invention realizes accurate compensation for the hysteresis nonlinearity of the piezoelectric actuator by designing an adaptive updating method of a hysteresis model, improves the hysteresis compensation effect of the piezoelectric actuator in the control process, compensates for other nonlinear dynamics of the piezoelectric actuator by designing a neural network, avoids complex nonlinear modeling, and avoids sliding mode chattering by designing a fuzzy approximator to replace the switching item in the traditional sliding mode control, thereby forming a fuzzy approximation adaptive neural network sliding mode tracking controller for the hysteresis nonlinearity of the piezoelectric actuator, realizes high-precision tracking control of the piezoelectric actuator with nonlinearities such as hysteresis, thereby improving the tracking control accuracy of the piezoelectric actuator, which is of great significance to expanding the application of piezoelectric actuators in precision motion control. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] The drawings constituting part of the present invention are used to provide a further understanding of the present invention. The exemplary embodiments and descriptions of the present invention are used to explain the present invention and do not constitute an improper limitation on the present invention. In the drawings:

[0055] Figure 1It is a logic block diagram of a fuzzy approximation adaptive neural network sliding mode tracking controller according to an embodiment of the invention;

[0056] Figure 2 is a flow chart of a tracking control method of a piezoelectric actuator according to an embodiment of the present invention;

[0057] Figure 3 It is a logic block diagram of the experimental test hardware according to the embodiment of the invention;

[0058] Figure 4 is a trajectory tracking displacement curve diagram of the tracking control method of the piezoelectric actuator according to the embodiment of the invention;

[0059] Figure 5 It is a trajectory tracking error curve diagram of the tracking control method of the piezoelectric actuator described in the embodiment of the invention.

[0060] The reference numerals therein include: fuzzy approximation adaptive neural network sliding mode tracking controller 1, neural network estimator 11, fuzzy approximator 12, piezoelectric actuator 2, position sensor 3, A / D conversion module 4, D / A conversion module 5, piezoelectric driver 6. DETAILED DESCRIPTION

[0061] In order to make the purpose, technical solution and advantages of the invention more clear, the invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described here are only used to explain the invention and do not constitute a limitation of the invention.

[0062] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.

[0063] In the description of the present invention, it should be understood that the terms "center", "longitudinal", "lateral", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside" and the like indicate positions or positional relationships based on the positions or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation on the present invention. In addition, the terms "first", "second", etc. are only used for descriptive purposes, and cannot be understood as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Thus, features defined as "first", "second", etc. may explicitly or implicitly include one or more of the features. In the description of the present invention, unless otherwise specified, "multiple" means two or more.

[0064] In the description of the invention, it should be noted that, unless otherwise clearly specified and limited, the terms "installation", "connection" and "connection" should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection, or it can be indirectly connected through an intermediate medium, or it can be the internal communication of two components. For ordinary technicians in this field, the specific meanings of the above terms in the invention can be understood according to specific circumstances.

[0065] The following will refer to Figure 1-Figure 5 The invention is described in detail with reference to the embodiments.

[0066] The present invention provides a tracking control method for a piezoelectric actuator, comprising the following steps:

[0067] S1: Considering the nonlinear characteristics of the piezoelectric actuator such as hysteresis nonlinearity, the piezoelectric actuator is described as a second-order nonlinear system dynamics model.

[0068] The piezoelectric actuator model is represented as follows:

[0069]

[0070] in, , , , , are all parameters of the piezoelectric actuator model and are constants. represents the output displacement of the piezoelectric actuator, represents the input voltage of the piezoelectric actuator, Represents the hysteresis nonlinearity of the piezoelectric actuator.

[0071] In order to describe the hysteresis nonlinearity and other nonlinear dynamic terms of the piezoelectric actuator, the hysteresis model (Krasnosel'skii-Pokrovskii model, hereinafter referred to as KP model) is used to describe the hysteresis nonlinear term of the piezoelectric actuator. The nonlinear dynamic term of the piezoelectric actuator other than the hysteresis nonlinear term is expressed as , then the piezoelectric actuator model is rewritten as:

[0072]

[0073] in, , Both represent the state of the piezoelectric actuator model. represents the displacement, represents the output displacement value of the piezoelectric actuator model, represents the hysteresis nonlinear term of the piezoelectric actuator described by the KP model, represents the external disturbance, represents the input value of the KP model, represents the density function of the KP model, represents the KP model operator and is expressed as:

[0074]

[0075] in, represents the Preisach plane, , represents the coordinates of the KP model operator on the Preisach plane, and represents the saturation turning point of the KP model operator, express The Preisach plane is a real plane. and Represents the input value of the KP model The minimum and maximum values ​​of represents the Lipschitz function updated by the KP model operator, represents the hysteresis loop boundary function and is expressed as:

[0076]

[0077] in, represents the independent variable of the boundary function, represents the maximum value of the hysteresis loop displacement output. In formula (3), according to Neither decreasing nor increasing, or , calculate the corresponding boundary function The value of Represents the hysteresis loop boundary function The slope of , represents the dividing line on the Preisach plane;

[0078] The KP model is:

[0079]

[0080] in, Represents the density function of the adaptive update of the KP model.

[0081] In order to calculate the compensation term of the KP model, the integral function form of the KP model equation (5) is transformed into an algebraic summation form. The method is to replace the Preisach plane of the integral region in equation (5) with a uniformly distributed Horizontal and vertical lines divide grid; for the grid, The node at the lower left corner of the grid Value and The values ​​are defined as and , multiplying the two to get , as the first The compensation value of each grid; then, The compensation values ​​of the grids are summed up; finally, the total compensation term is expressed as:

[0082]

[0083] in, ; express As a column vector consisting of elements, it is used to express equation (6) in the form of vector product, .

[0084] S2: For other nonlinear dynamic terms , design neural network estimator11.

[0085] Defining the Tracking Error of the Piezoelectric Actuator Model and its derivatives for:

[0086]

[0087] in, represents the expected trajectory signal;

[0088] The neural network estimator 11 is calculated based on the tracking error of the piezoelectric actuator model, and the neural network estimator 11 is expressed as:

[0089]

[0090] in, represents the neural network estimator 11, represents the estimated weight vector between the hidden layer and the output layer, Represents the Sigmoid activation function of the hidden layer nodes, , represents the weight matrix between the input layer and the hidden layer, represents the input data of the neural network estimator, .

[0091] S3: Design a fuzzy approximation adaptive neural network sliding mode tracking controller1.

[0092] Tracking error according to the piezoelectric actuator model and its derivatives Design sliding surface , sliding surface It is expressed as:

[0093]

[0094] in, Represents the sliding surface parameter, which is a positive constant.

[0095] In order to replace the switching items of traditional sliding mode control and avoid the sliding mode chattering problem, according to the sliding mode surface A fuzzy approximator 12 is designed. The fuzzy approximator 12 is composed of a product inference engine, a single-point fuzzifier and a central average defuzzifier. The fuzzy approximator 12 is expressed as:

[0096]

[0097] in, represents the adaptive weight vector, represents the output vector of the product inference engine, and Multiply to form a central average defuzzifier ; represents adaptive weight; Represents product inference engine Output: and represents the true value parameter vector, , , represents the length of the true value parameter vector;

[0098] represents the Gaussian fuzzer and is expressed as:

[0099]

[0100] in, and are truth value parameters, respectively and The elements in ;

[0101] In formula (10), the adaptive weight vector The adaptive rate is , the adaptive rate calculation formula is .

[0102] Based on sliding surface , a neural network estimator 11 and a fuzzy approximator 12 are used to design a fuzzy approximation adaptive neural network sliding mode tracking controller 1, and the fuzzy approximation adaptive neural network sliding mode tracking controller 1 is expressed as:

[0103]

[0104] in, represents the output value of the fuzzy approximation adaptive neural network sliding mode tracking controller, Desired trajectory signal The second derivative of Represents the adaptive density function in the KP model Adaptive update rate; Represents the estimated weight vector in the neural network estimator The update rate; , , Both represent the parameters of the fuzzy approximation adaptive neural network sliding mode tracking controller, and both are positive numbers.

[0105] S4: The output value of the fuzzy approximation adaptive neural network sliding mode tracking controller 1 The piezoelectric actuator 2 having nonlinearity such as hysteresis is input, and the piezoelectric actuator 2 is subjected to tracking control.

[0106] Example 1

[0107] The tracking control method of the piezoelectric actuator comprises the following steps:

[0108] Step 1: Use the position sensor 3 to measure the initial displacement of the piezoelectric actuator 2, convert the analog signal of the position sensor 3 into a digital quantity through the A / D conversion module 4 and input it into the fuzzy approximation adaptive neural network sliding mode tracking controller 1, and provide initial values ​​for the model variables in the fuzzy approximation adaptive neural network sliding mode tracking controller 1.

[0109] At this time, the model of the piezoelectric actuator 2 is:

[0110]

[0111] Step 2: Obtain the displacement of the piezoelectric actuator 2 at the current moment.

[0112] Step 3: Output of adaptive neural network sliding mode tracking controller 1 based on the fuzzy approximation of the previous moment Calculate the sliding surface at the current moment with the expected displacement input at the current moment With adaptive update rate , , :

[0113]

[0114]

[0115]

[0116]

[0117] Step 4: Based on adaptive update rate , , Update the parameters of the fuzzy approximation adaptive neural network sliding mode tracking controller 1 at the current moment , , .

[0118] Step 5: The sliding surface at the current moment calculated based on step 3 The parameters of the fuzzy approximation adaptive neural network sliding mode tracking controller 1 at the current moment updated in step 4 are , , , calculate the compensation term of the KP model, the neural network estimator, the fuzzy approximator and the second-order derivative of the expected displacement:

[0119]

[0120]

[0121]

[0122] Step 6: Calculate the output value of the fuzzy approximation adaptive neural network sliding mode tracking controller 1 at the current moment according to the calculation result of step 5 :

[0123]

[0124] The output value of the fuzzy approximation adaptive neural network sliding mode tracking controller 1 The analog quantity is converted into an analog quantity by the A / D conversion module 4 and input to the piezoelectric driver 6, thereby driving the piezoelectric actuator 2 to perform high-precision displacement.

[0125] Step 7: Determine whether the piezoelectric actuator 2 continues to move; if the piezoelectric actuator 2 continues to move, return to step 2 and perform closed-loop control in a cycle; if the piezoelectric actuator 2 does not continue to move, end the control process.

[0126] Figure 4 The trajectory tracking displacement curve of the fuzzy approximation adaptive neural network sliding mode tracking control method for a piezoelectric actuator provided according to an embodiment of the present invention is shown.

[0127] like Figure 4As shown, the piezoelectric actuator 2 tracks a sinusoidal signal with an amplitude of 10 nanometers and a frequency of 1 Hz with high precision, and the fuzzy approximation adaptive neural network sliding mode tracking controller 1 can achieve high-precision tracking control of the piezoelectric actuator 2.

[0128] Figure 5 The figure shows a trajectory tracking error curve of a fuzzy approximation adaptive neural network sliding mode tracking control method for a piezoelectric actuator provided according to an embodiment of the present invention.

[0129] like Figure 5 As shown, under the action of the fuzzy approximation adaptive neural network sliding mode tracking controller 1, when the piezoelectric actuator 2 tracks a sinusoidal signal with an amplitude of 10 nanometers and a frequency of 1 Hz, its trajectory tracking error is less than 0.1 nanometers. It can be seen that the present invention improves the tracking control accuracy of the piezoelectric actuator.

[0130] It should be understood that the various forms of processes shown above can be used to reorder, add or delete steps. For example, the steps described in the disclosure of the present invention can be performed in parallel, sequentially or in different orders, as long as the desired results of the technical solution disclosed in the present invention can be achieved, and this document does not limit this.

[0131] The above specific implementations do not constitute a limitation on the protection scope of the present invention. It should be understood by those skilled in the art that various modifications, combinations, sub-combinations and substitutions can be made according to design requirements and other factors. Any modification, equivalent substitution and improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A tracking control method for a piezoelectric actuator, characterized in that: The steps include: S1: Describe the piezoelectric actuator as a second-order nonlinear system dynamics model; In step S1: The piezoelectric actuator model is characterized as: in, , , , , are all parameters of the piezoelectric actuator model and are constants. represents the output displacement of the piezoelectric actuator, represents the input voltage of the piezoelectric actuator, represents the hysteresis nonlinearity of the piezoelectric actuator; The hysteresis model is used to describe the hysteresis nonlinear term of the piezoelectric actuator. The nonlinear dynamic term of the piezoelectric actuator other than the hysteresis nonlinear term is expressed as , then the piezoelectric actuator model is rewritten as: in, , Both represent the state of the piezoelectric actuator model. represents the displacement, represents the output displacement value of the piezoelectric actuator model, represents the hysteresis nonlinear term of the piezoelectric actuator described by the hysteresis model, represents the external disturbance, represents the input value of the hysteresis model, represents the density function of the hysteresis model, represents the hysteresis model operator and is expressed as: in, represents the Preisach plane, , represents the coordinates of the hysteresis model operator on the Preisach plane, and represents the saturation turning point of the hysteresis model operator, express The Preisach plane is a real plane. and express The minimum and maximum values ​​of represents the Lipschitz function updated by the hysteresis model operator, represents the hysteresis loop boundary function and is expressed as: in, represents the independent variable of the boundary function, Indicates the maximum value of the hysteresis loop displacement output, according to Neither decreasing nor increasing, or , calculate the corresponding boundary function The value of Represents the hysteresis loop boundary function The slope of , represents the dividing line on the Preisach plane; The hysteresis model is: in, Density function representing the adaptive update of the hysteresis model; The Preisach plane of the integration region is divided into uniformly distributed Horizontal and vertical lines divide grid; for the A grid, the lower left corner node of the grid Value and The values ​​are defined as and , multiplying the two to get , and as the first The compensation value of the grid; The compensation values ​​of the grids are summed to obtain the total compensation term And expressed as: in, ; express As a column vector of elements, ; S2: Design a neural network estimator for the nonlinear dynamic terms of the piezoelectric actuator other than the hysteresis nonlinear term ; Among them, the neural network estimator It is expressed as: in, represents the estimated weight vector between the hidden layer and the output layer, Represents the Sigmoid activation function of the hidden layer nodes, represents the weight matrix between the input layer and the hidden layer, Represents the input data of the neural network estimator; S3: Design a fuzzy approximation adaptive neural network sliding mode tracking controller; wherein the fuzzy approximation adaptive neural network sliding mode tracking controller is expressed as: in, represents the output value of the fuzzy approximation adaptive neural network sliding mode tracking controller, Desired trajectory signal The second derivative of Represents the adaptive density function in the hysteresis model Adaptive update rate; represents a fuzzy approximator; represents the central average defuzzifier; Represents the estimated weight vector The update rate; represents the adaptive weight vector; represents the output vector of the product inference engine; represents the sliding surface; represents a column vector; , , All represent the parameters of the fuzzy approximation adaptive neural network sliding mode tracking controller, and are positive numbers; S4: The output value of the fuzzy approximation adaptive neural network sliding mode tracking controller The piezoelectric actuator is input and tracked. The output value of the fuzzy approximation adaptive neural network sliding mode tracking controller is It is expressed as: 。 2. The tracking control method of a piezoelectric actuator according to claim 1, characterized in that: In step S2: Defining the Tracking Error of the Piezoelectric Actuator Model and its derivatives for: in, represents the expected trajectory signal; Neural network estimator for tracking error calculation based on piezoelectric actuator model And expressed as: in, represents the estimated weight vector between the hidden layer and the output layer, Represents the Sigmoid activation function of the hidden layer nodes, , represents the weight matrix between the input layer and the hidden layer, represents the input data of the neural network estimator, .

3. The tracking control method of a piezoelectric actuator according to claim 1, characterized in that: In step S3: Tracking error according to the piezoelectric actuator model and its derivatives Design sliding surface , sliding surface It is expressed as: in, represents the sliding surface parameter, which is a positive constant; According to the sliding surface Designing a fuzzy approximator , fuzzy approximator It consists of a product inference engine, a Gaussian fuzzifier and a central average defuzzifier, a fuzzy approximator It is expressed as: in, represents the adaptive weight vector, represents the output vector of the product inference engine, and Multiply to form a central average defuzzifier ; represents adaptive weight; Represents product inference engine Output: and represents the true value parameter vector, , , represents the length of the true value parameter vector; represents the Gaussian fuzzer and is expressed as: in, and are truth value parameters, respectively and The elements in ; Adaptive Weight Vector The adaptive rate is , the adaptive rate is calculated as ; Based on sliding surface , Neural Network Estimator and fuzzy approximator Design a fuzzy approximation adaptive neural network sliding mode tracking controller, which can be expressed as: in, represents the output value of the fuzzy approximation adaptive neural network sliding mode tracking controller, Desired trajectory signal The second derivative of Represents the adaptive density function in the hysteresis model Adaptive update rate; represents a fuzzy approximator; represents the central average defuzzifier; Represents the estimated weight vector The update rate; represents the adaptive weight vector; represents the output vector of the product inference engine; represents the sliding surface; represents a column vector; , , Both represent the parameters of the fuzzy approximation adaptive neural network sliding mode tracking controller, and both are positive numbers.

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