Adaptive Iterative Learning Control Method for Speed Tracking of the Extrusion Rod of an Extrusion Machine

Abstract

The present invention belongs to the technical field of intelligent control, and discloses a speed tracking adaptive iterative learning control method for the extrusion rod of an extruder, which includes the following steps: determining the dynamic model of the extrusion rod and the control objective according to the physical properties of the extrusion rod and the movement mode of the extrusion rod; establishing a neural network model for the uncertain part in the movement model of the extrusion rod based on the approximation theory, and determining the speed tracking error system of the extrusion rod; designing a virtual control input and an actual control input based on the adaptive iterative learning control theory; and analyzing the stability of the designed controller based on the Lyapunov stability theorem. By adopting the above-mentioned speed tracking adaptive iterative learning control method for the extrusion rod of an extruder, the present invention can effectively control the finite-time speed tracking of the extrusion rod, realize the precise movement control of the extrusion rod, and ensure the dimensional accuracy, surface quality and production efficiency of the product.

Description

Technical Field

[0001] The present invention relates to the technical field of intelligent control, and particularly to a speed tracking adaptive iterative learning control method for an extrusion rod of an extruder. Background Art

[0002] Intelligent manufacturing is an important engine to promote the high-quality development of China's economy. The extruder plays an important role in intelligent manufacturing. It directly affects the dimensional accuracy, surface quality and production efficiency of products. Through precise speed tracking, the uniform flow of materials during the extrusion process can be ensured, defects can be avoided, and thus the overall quality of products can be improved. However, due to various interference factors during the extrusion process, such as temperature fluctuations, changes in material properties, die wear, etc., these factors may all cause fluctuations in the speed of the extrusion rod. Therefore, how to achieve precise speed control is a major difficulty in the speed tracking control of the extrusion rod.

[0003] Finite-time speed tracking control means that within a given time range, the speed of the extrusion rod can quickly and accurately track and maintain at the set reference speed. Therefore, the traditional control methods for the finite-time speed tracking control of the extrusion rod mainly include PID control, open-loop control and simple closed-loop control, etc. The PID control has limited control effects on the non-linear, time-varying and uncertain extrusion process, and may not be able to quickly and accurately track the speed changes of the extrusion rod. The open-loop control has poor accuracy and stability, and is easily affected by factors such as extrusion materials, dies and extrusion temperatures. Although the simple closed-loop control has a feedback mechanism, its control algorithm is relatively simple and may not be able to quickly and accurately respond to the speed changes of the extrusion rod. In the face of complex and changeable working conditions, traditional control schemes often lack intelligence and adaptability, and cannot automatically adjust control parameters and strategies according to the changes in working conditions.

[0004] In view of the above problems, the present invention designs a speed tracking adaptive iterative learning control method for an extrusion rod of an extruder to achieve the speed tracking control of the extrusion rod in a finite time interval. The adaptive iterative learning control technology based on a radial basis function neural network is used to control the speed of the extrusion rod, so that its speed control accuracy is higher, the precise motion control of the extrusion rod is realized, and the dimensional accuracy, surface quality and production efficiency of products are ensured. Summary of the Invention

[0005] The purpose of the present invention is to provide a speed tracking adaptive iterative learning control method for an extrusion rod of an extruder, to provide a finite-time speed tracking control strategy for the extrusion rod, and to achieve the finite-time high-precision speed tracking control of the extrusion rod based on the adaptive iterative learning control theory and in combination with the Fourier series-radial basis function neural network approximation method.

[0006] To achieve the above object, the present invention provides a speed tracking adaptive iterative learning control method for the extrusion rod of an extruder, comprising the following steps:

[0007] Step S1: Determine the dynamic model of the extrusion rod and the control objective according to the physical properties of the extrusion rod of the extruder and the movement mode of the extrusion rod;

[0008] Step S2: Based on the approximation theory, establish a neural network model for the uncertain part in the extrusion rod model and determine the speed tracking error system of the extrusion rod;

[0009] Step S3: Design the virtual control input and the actual control input based on the adaptive iterative learning control theory;

[0010] Step S4: Analyze the stability of the designed controller based on the Lyapunov stability theorem.

[0011] Preferably, in step S1, according to the physical properties of the extrusion rod and the movement mode of the extrusion rod, construct the dynamic model of the extrusion rod and determine the control objective. The specific process is as follows:

[0012] Under uncertain working conditions, due to various external disturbances, the basic dynamic equation of the extrusion rod is as follows:

[0013] ;

[0014] Wherein, is the effective mass of the extrusion rod and the metal inside it; is the acceleration of the extrusion rod; is the damping coefficient, representing the energy dissipation during the extrusion process; is the speed of the extrusion rod; is the stiffness coefficient; is the displacement of the extrusion rod; is the extrusion force; is the frictional force; is the external disturbance force;

[0015] Transform the above extrusion rod dynamic model into an uncertain nonlinear parameterized system, then the finite-time high-precision speed tracking control problem of the extrusion rod is regarded as the finite-time high-precision speed tracking control problem of the uncertain nonlinear parameterized system;

[0016] Through the dynamic research on the extrusion rod of the extruder, establish the dynamic model as follows:

[0017] ;

[0018] Wherein, are respectively the two state variables - displacement and speed of the nonlinear parameterized extrusion rod system; is the extrusion pressure control input of the non - linear parametric extrusion rod system; is the unmodeled dynamics with unknown time - varying parameters , including model uncertainties and unknown time - varying disturbances; is the unknown bounded external disturbance; is the displacement output of the system, denotes the number of iterations.

[0019] Preferably, in step S2, based on the approximation theory, a neural network model is established for the uncertain part in the extrusion rod model, and the velocity tracking error system of the extrusion rod and its corresponding displacement tracking error system are determined. The specific process is as follows:

[0020] The velocity tracking error system of the extrusion rod is as follows:

[0021] ;

[0022] where, is the error between the system velocity output at the -th iteration and the target velocity trajectory , that is, the velocity tracking error at the -th iteration;

[0023] Its corresponding displacement tracking error system of the extrusion rod is as follows:

[0024] ;

[0025] where, is the error between the system displacement output at the -th iteration and the target trajectory , that is, the displacement tracking error at the -th iteration;

[0026] Step S21: In the extrusion rod model, process the uncertain time - varying parameter terms;

[0027] On the finite - time interval, the uncertain time - varying parameter term is a periodic signal. Expand into Fourier series respectively as follows:

[0028] ;

[0029] where, is the trigonometric function matrix about time; is the corresponding weight matrix; is the truncation error after expansion, and its upper bound is ;

[0030] Step S22: Establish a new approximator of Fourier series expansion - radial basis function neural network, and model and respectively, as follows:

[0031] ;

[0032] ;

[0033] ;

[0034] where is the non - modeled non - linear parameterized term in the system approximated by the newly constructed approximator ; is the neural network Gaussian basis function vector with respect to parameters and ; the weight matrices and are bounded and satisfy , ; is the unknown corresponding upper bound, and its value is positive; , represents 's upper bound; and are the estimates of the unknown weight vectors and ; and respectively represent and the estimation errors between the estimated values and the actual values; is the approximation error when the simple neural network approximates the unknown non - parameterized term;

[0035] ;

[0036] where is the neural network basis function when the basis function contains the estimated values of unknown parameters; is the derivative of ; represents the sum of the higher - order terms in the Taylor series expansion;

[0037] The total approximation error of the approximator is as follows:

[0038] ;

[0039] ;

[0040] Among them, , , ; The remainder has a boundary of ;

[0041] Step S23, Design of the error function and processing of the unknown bound;

[0042] Introduce a typical convergent series sequence to process the unknown upper bound of each error term as follows:

[0043] , , satisfying ;

[0044] At the beginning of each iteration, the initial error value should satisfy , , among which, is a convergent series sequence;

[0045] Construct a new error function as follows:

[0046] ;

[0047] Among them, the saturation function is:

[0048] ;

[0049] Among them, is an improved time-varying layer boundary.

[0050] Preferably, in step S3, based on the adaptive iterative learning control theory, design the virtual control input and the actual control input to obtain the finite-time high-precision displacement tracking control strategy for the extrusion rod. The specific process is as follows:

[0051] Step S31, Design the virtual control input and the parameter adaptation law , ;

[0052] Select new error functions and as follows:

[0053] ;

[0054] ;

[0055] For the first subsystem of the extrusion rod model, as follows:

[0056] ;

[0057] Select the Lyapunov function according to the position tracking error as follows:

[0058] ;

[0059] In order to make negative semi - definite, design the virtual controller as follows:

[0060] ;

[0061] The parameter adaptation law is:

[0062] ;

[0063] ;

[0064] Among them, is the estimation of the neural network weight for approximating the unknown function and the estimation error of the weight ; is the estimation of the unknown parameter , is the gain value to be designed; , , , are the gain matrices to be designed;

[0065] Step S32: Design the actual control input and the parameter adaptation law , ;

[0066] Design a new error function as follows:

[0067] ;

[0068] Select the Lyapunov function as follows:

[0069] ;

[0070] Among them, is the estimation of the neural network weight for approximating the unknown function and the estimation error of the weight ; is the estimation of the unknown parameter ​​is the gain value to be designed; , , , is the gain matrix to be designed;

[0071] Taking the derivative of it gives:

[0072] ;

[0073] wherein, is the estimation of the neural network weights for approximating the unknown function and the estimation error of the weights ; is the estimation of the unknown parameter , is the gain value to be designed; , , , is the gain matrix to be designed;

[0074] Design the actual controller as follows:

[0075] ;

[0076] Select the parameter update law as:

[0077] .

[0078] Therefore, the present invention adopts the above-mentioned speed tracking adaptive iterative learning control method for the extrusion rod of an extruder, and the beneficial effects are as follows:

[0079] (1) Compared with the prior art, the present invention takes the extrusion rod of the extruder as the control object, and its motion is a reciprocating motion. Modeling it as a non-linear parameterized system and combining the adaptive iterative learning control theory, it effectively realizes the high-precision speed tracking control of the extrusion rod in a finite time interval;

[0080] (2) Compared with the traditional control method, the finite-time control method can achieve precise tracking of the speed of the extrusion rod of the extruder, meeting the requirements of high-precision processing;

[0081] (3) The control method proposed in this proposition can cope with various uncertainty factors, such as changes in extrusion pressure, uncertainties in friction, differences in material properties, and possible external disturbances, etc., which is beneficial to maintaining the stability and performance of the system.

[0082] The technical solution of the present invention will be further described in detail below through the drawings and embodiments. Description of the Drawings

[0083] Figure 1 is the overall flowchart of the speed tracking adaptive iterative learning control method for the extrusion rod of an extruder according to the present invention;

[0084] Figure 2 is the curve of the maximum value of the displacement error versus the number of iterations in Embodiment 2 of the present invention;

[0085] Figure 3 is the curve of the maximum value of the speed error versus the number of iterations in Embodiment 2 of the present invention;

[0086] Figure 4 is the curve of the norm of the control input versus the number of iterations in Embodiment 2 of the present invention. Detailed implementation manners

[0087] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0088] As Figure 1 shown, a speed tracking adaptive iterative learning control method for the extrusion rod of an extruder according to the present invention includes the following steps:

[0089] Step S1: Determine the dynamic model of the extrusion rod and the control objective according to the physical properties of the extrusion rod of the extruder and the movement mode of the extrusion rod;

[0090] Step S2: Based on the approximation theory, establish a neural network model for the uncertain part in the extrusion rod model of the extruder, and determine the speed tracking error system of the extrusion rod;

[0091] Step S3: Design a virtual control input and an actual control input based on the adaptive iterative learning control theory;

[0092] Step S4: Analyze the stability of the designed controller based on the Lyapunov stability theorem.

[0093] Embodiment 1

[0094] Step S1: Construct the dynamic model of the extrusion rod and determine the control objective according to the physical properties of the extrusion rod of the extruder and the movement mode of the extrusion rod.

[0095] Under uncertain working conditions, since various factors need to be considered, such as changes in extrusion force, uncertainties in friction, differences in material properties, and possible external disturbances, etc., the basic dynamic equation of the extrusion rod is as follows:

[0096] ;

[0097] Wherein, is the effective mass of the extrusion rod and the metal inside it; is the acceleration of the extrusion rod; is the damping coefficient, representing the energy dissipation during the extrusion process; is the velocity of the extrusion rod; is the stiffness coefficient, reflecting the elasticity of the extrusion rod and its support structure; is the displacement of the extrusion rod; is the extrusion force, which changes with time; is the frictional force, including the friction between the extrusion rod and components such as the die and the extrusion pad, and also changes with time; is the external disturbance force, which may include uncertain factors such as vibration and shock.

[0098] Transform the above dynamic model of the extrusion rod of the extruder into an uncertain nonlinear parameterized system. Then, the finite-time high-precision velocity tracking control problem of the extrusion rod can be regarded as the finite-time high-precision velocity tracking control problem of the uncertain nonlinear parameterized system. Through the dynamic research of the extrusion rod, the dynamic model is established as follows:

[0099] ;

[0100] where, are respectively the two state variables - displacement and velocity of the nonlinear parameterized extrusion rod system; is the extrusion pressure control input of the nonlinear parameterized extrusion rod system; is the unmodeled dynamics containing unknown time-varying parameters including model uncertainties and unknown time-varying disturbances; is the unknown bounded external disturbance; is the displacement output of the system, represents the number of iterations.

[0101] Step S2: Based on the approximation theory, establish a neural network model for the uncertain part in the extrusion rod model of the extruder, and determine the velocity tracking error system of the extrusion rod and its corresponding displacement tracking error system of the extrusion rod.

[0102] Among them, the velocity tracking error system of the extrusion rod is as follows:

[0103] ;

[0104] where, is the error between the system velocity output at the th iteration and the target velocity trajectory , that is, the velocity tracking error at the th iteration.

[0105] Its corresponding displacement tracking error system of the extrusion rod is as follows:

[0106] ;

[0107] wherein, is the system displacement output at the th iteration, and the target trajectory , that is, the displacement tracking error at the th iteration.

[0108] Step S21: In the extruder ram model, process the uncertain time-varying parameter terms.

[0109] On a finite time interval, the uncertain time-varying parameter term is a periodic signal. Expand into Fourier series respectively as follows:

[0110] ;

[0111] wherein, is a trigonometric function matrix with respect to time; is the corresponding weight matrix; is the truncated error after expansion, which is bounded, and its upper bound is .

[0112] Step S22: Establish a new Fourier series expansion - radial basis function neural network approximator, and model and respectively as follows:

[0113] ;

[0114] ;

[0115] ;

[0116] wherein, is the newly constructed approximator used to approximate the unmodeled non-linear parameterized term in the system; is the neural network Gaussian basis function vector with respect to the parameters and ; The weight matrices and are bounded and satisfy , ; is the unknown corresponding upper bound, and its value is a positive number; , represents the upper bound of ; and is the unknown weight vector and estimation; and respectively represent and the estimation error between the estimated value and the actual value; is the approximation error when the simple neural network approximates the unknown non-parametric term.

[0117] ;

[0118] wherein, is the basis function the neural network basis function when the estimated value of the unknown parameter is included; is derivative; represents the sum of the high-order terms in the Taylor series expansion.

[0119] The total approximation error of the approximator is as follows:

[0120] ;

[0121] ;

[0122] wherein, , , ; the remainder term the boundary of is .

[0123] Step S23, design of the error function and processing of the unknown bound.

[0124] Introduce a typical convergent series sequence to process the unknown upper bound of each error term as follows:

[0125] , satisfies ;

[0126] At the beginning of each iteration, the initial error value should satisfy , , wherein, is a convergent series sequence.

[0127] Construct a new error function as follows:

[0128] ;

[0129] wherein, the saturation function is:

[0130] ;

[0131] Among them, is an improved time-varying layer boundary.

[0132] Step S3: Based on the adaptive iterative learning control theory, design the virtual control input and the actual control input to obtain the finite-time high-precision speed tracking control strategy for the extrusion rod.

[0133] Step S31: Design the virtual control input and the parameter adaptation law , .

[0134] Select the new error functions and , as follows:

[0135] ;

[0136] ;

[0137] For the first subsystem of the extrusion rod model, as follows:

[0138] ;

[0139] According to the position tracking error, select the Lyapunov function, as follows:

[0140] ;

[0141] In order to make negative semi-definite, design the virtual controller, as follows:

[0142] ;

[0143] The parameter adaptation law is:

[0144] ;

[0145] ;

[0146] Among them, is the estimate of the neural network weight for approximating the unknown function and the estimation error of the weight and the weight ; is the estimate of the unknown parameter , is the gain value to be designed; , , , is the gain matrix that needs to be designed.

[0147] Step S32: Design actual control input And parameter adaptation law , .

[0148] Designing a new error function , as shown below:

[0149] ;

[0150] Select the Lyapunov function as follows:

[0151] ;

[0152] in, is an approximation to the unknown function The neural network weights Estimates With weight The estimation error of is an unknown parameter The estimate, is the gain value that needs to be designed; , , , is the gain matrix that needs to be designed.

[0153] Taking its derivative we get:

[0154] ;

[0155] in, is an approximation to the unknown function The neural network weights Estimates With weight The estimation error of is an unknown parameter The estimate, is the gain value that needs to be designed; , , , is the gain matrix that needs to be designed.

[0156] Design the actual controller as follows:

[0157] ;

[0158] Select the parameter update law as:

[0159] ;

[0160] Step S4. Analyze the stability of the designed high-precision speed tracking control system for the extrusion rod based on the Lyapunov stability theorem.

[0161] According to Assumption 1, we have , and take the Lyapunov function as:

[0162] ;

[0163] where ;

[0164] ; ; ;

[0165] ; ; It can be obtained that

[0166] .

[0167] Define , so the above formula can be rewritten as:

[0168] ;

[0169] According to the properties of the convergent series sequence, it can be known that , so is bounded, and there is , so .

[0170] is arbitrary. For , it can be obtained that

[0171] ;

[0172] where is bounded.

[0173] Through the concept of the convergent series sequence, is bounded, and there is , so it can be obtained that is also bounded.

[0174] For , , and , for any , it can be bounded. Therefore, it can be obtained that:

[0175] is bounded.

[0176] Most importantly, for any , is bounded, so it can be obtained that , , , and are all bounded, is also bounded, is bounded; and since is uniformly continuous, it can be proved that in a finite time interval, under uncertain operating conditions, the position tracking error of the extrusion rod system can converge to zero as the number of iterations increases, that is, the task of high-precision position tracking is completed, thus meeting the corresponding requirements of high-precision speed tracking.

[0177] Embodiment 2

[0178] To verify the effectiveness of the method described in the present invention, simulation verification is carried out.

[0179] The effectiveness of the designed control strategy is verified through numerical examples. The model description of the extrusion rod of the extruder is as follows:

[0180] ;

[0181] ;

[0182] ;

[0183] Wherein, ; and are respectively the displacement and velocity variables of the extrusion rod; is the extrusion pressure control input variable. The target output satisfies the following reference output model:

[0184] ;

[0185] ;

[0186] ;

[0187] Wherein, and are state variables. is the ideal tracking trajectory generated by the above formula at .

[0188] Select the following virtual control law and actual control strategy:

[0189] ;

[0190] ;

[0191] ;

[0192] ;

[0193] The parameter update law is as follows:

[0194] ;

[0195] ;

[0196] ;

[0197] The specific parameter selection is as follows: , , , , , , , , , , .

[0198] The curve of the maximum value of the displacement error varying with the number of iterations is as shown in Figure 2 ; the curve of the maximum value of the velocity error varying with the number of iterations is as shown in Figure 3 ; the curve of the norm of the control input varying with the number of iterations is as shown in Figure 4 . It can be seen from Figures 2 - 4 that as the number of iterations increases, the extrusion rod of the extruder can achieve high-precision velocity tracking in a finite time interval.

[0199] Therefore, the present invention adopts the above-mentioned velocity tracking adaptive iterative learning control method for the extrusion rod of an extruder, combines the approximation theory knowledge with the adaptive iterative learning control method, realizes the high-precision velocity tracking control of the extrusion rod in a finite time interval, ensures that the velocity trajectory of the extrusion rod is consistent with the set velocity curve, quickly responds to the process requirements, meets the requirements of the high-precision extrusion process, and improves the production efficiency.

[0200] Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that: they can still modify or equivalently replace the technical solutions of the present invention, and these modifications or equivalent replacements cannot make the modified technical solutions deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A speed tracking adaptive iterative learning control method for an extruder extrusion rod, characterized in that: The following steps are involved: Step S1: Determine the dynamic model of the extrusion rod and the control target according to the physical properties of the extrusion rod of the extruder and the movement mode of the extrusion rod. The specific process is as follows: Under uncertain working conditions, due to various external disturbances, the basic dynamic equation of the extruded rod is as follows: ； in, is the effective mass of the extruded rod and the metal inside it; is the acceleration of the squeeze rod; is the damping coefficient, representing the energy dissipation during the extrusion process; is the speed of squeezing the rod; is the stiffness coefficient; is the displacement of the extrusion rod; It is the squeezing force; is the friction force; It is an external disturbance force; The above extrusion rod dynamics model is transformed into an uncertain nonlinear parameterized system, and the finite time high precision velocity tracking control problem of the extrusion rod is regarded as the finite time high precision velocity tracking control problem of the uncertain nonlinear parameterized system. Through the dynamics study of the extrusion rod of the extruder, the dynamic model is established as follows: ； in, They are the two state variables of the nonlinear parameterized extruded rod system—displacement and velocity; is the extrusion pressure control input of the nonlinear parameterized extrusion rod system; For unknown time-varying parameters The unmodeled dynamics of the model, including model uncertainties and unknown time-varying disturbances; is an unknown bounded external disturbance; is the displacement output of the system, Indicates the number of iterations; Step S2: Based on the approximation theory, a neural network model is established for the uncertain part of the extrusion rod model, and the speed tracking error system of the extrusion rod is determined. The specific process is as follows: The velocity tracking error system of the extrusion rod is as follows: ； in, For the The system speed output at the iteration With the target speed trajectory The error, that is, Velocity tracking error at the iteration; The corresponding displacement tracking error system of the extrusion rod is as follows: ； in, For the The system displacement output at the iteration With target trajectory The error, that is, Displacement tracking error at iterations; Step S21, in the extruded rod model, processing the uncertain time-varying parameter item; In a finite time interval, the uncertain time-varying parameter term is a periodic signal, They are expanded into Fourier series as follows: ； in, It is a matrix of trigonometric functions about time; is the corresponding weight matrix; is the truncation error after expansion, and its upper bound is ; Step S22: Establish a new Fourier series expansion-radial basis function neural network approximator. and Model them separately as follows: ； ； ； in, is a newly constructed approximator used to approximate the unmodeled nonlinear parameterized terms in the system ; It's about parameters and Neural network Gaussian basis function vector; weight matrix and is bounded, satisfies , ; is the corresponding upper bound of the unknown, and its value is a positive number; , express The upper bound of and is the unknown weight vector and estimates; and Respectively and The estimated error between the estimated value and the actual value; The approximation error when a simple neural network approximates unknown non-parametric terms; ； in, is the basis function Neural network basis functions when contains unknown parameter estimates; yes The derivative of represents the sum of higher-order terms in the Taylor series expansion; The total approximation error of the approximator is as follows: ； ； in, , , ; Remainder The boundary is ; Step S23, design of error function and processing of unknown bounds; A typical convergent series sequence is introduced to deal with the unknown upper bound of each error term as follows: , ,satisfy ; At the beginning of each iteration, the initial error value should satisfy , ,in, is a convergent series sequence; Construct a new error function , as shown below: ； in, The saturation function is: ； in, It is an improved time-varying layer boundary; Step S3: Based on the adaptive iterative learning control theory, the virtual control input and the actual control input are designed. The specific process is as follows: Step S31: Design virtual control input and parameter adaptation law , ; Choose a new error function and , as shown below: ； ； The first subsystem for the extruded rod model is shown below: ； According to the position tracking error, the Lyapunov function is selected as follows: ； In order to make Negative semi-definite, design a virtual controller as follows: ； The parameter adaptation law is: ； ； in, is an approximation to the unknown function The neural network weights Estimates With weight The estimation error of is an unknown parameter The estimate, is the gain value that needs to be designed; , , , is the gain matrix that needs to be designed; Step S32: Design actual control input And parameter adaptation law , ; Designing a new error function , as shown below: ； Select the Lyapunov function as follows: ； in, is an approximation to the unknown function The neural network weights Estimates With weight The estimation error of is an unknown parameter The estimate, is the gain value that needs to be designed; , , , is the gain matrix that needs to be designed; Taking its derivative we get: ； in, is an approximation to the unknown function The neural network weights Estimates With weight The estimation error of is an unknown parameter The estimate, is the gain value that needs to be designed; , , , is the gain matrix that needs to be designed; Design the actual controller as follows: ； Select the parameter update law as: ； Step S4: Analyze the stability of the designed controller based on the Lyapunov stability theorem.

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