A data-driven control method for a precision positioning platform based on step-skipping inversion

Abstract

The present invention provides a data-driven control method for a precision positioning platform based on skipped inversion, comprising the following steps: S1, establishing an autoregressive model with time-varying linear parameters and external inputs based on the dynamic characteristics of the precision positioning platform; S2, designing an adaptive parameter estimator based on a dynamic regression extension and hybrid method; S3, constructing a p-step recursive model of the precision positioning platform based on the adaptive parameter estimator; and S4, designing an adaptive controller based on skipped inversion based on the number of zero points outside the unit circle of the autoregressive model and the estimated parameter values. This method primarily addresses the problems of real-time parameter estimation and unstable inversion faced by precision positioning platform tracking control. It can estimate the true parameter values ​​in real time when the parameters do not change drastically, overcome the control divergence problem caused by unstable inversion, and improve the tracking performance of the precision positioning platform.

Description

Technical Field

[0001] The present invention belongs to the field of adaptive tracking control of precision positioning platforms, and specifically relates to a data-driven control method for precision positioning platforms based on step-skipping inversion, which is mainly used for parameter estimation and robust tracking control of the model when the precision positioning platform model has unstable zero points and the model parameters are not fixed constants. Background Art

[0002] Precision positioning platforms have been widely used due to their high positioning accuracy and play a key role in complex robotic processing tasks such as spraying, welding, and laser processing. However, in actual applications, the sensors, actuators, or mechanical parts of the precision positioning platform itself may introduce unstable zero points. In addition, for precision positioning platforms that do not have unstable zero points themselves, their control systems generally use fast sampling, which will cause the appearance of discrete unstable zero points caused by fast sampling. Unstable zero points will cause the divergence of control instructions during the tracking process, which brings great difficulties to the stable and precise tracking control of precision positioning platforms. Most of the existing adaptive control methods are based on the assumption that the controlled object has a stable zero point, which cannot match the actual engineering needs. On the other hand, as the processing task progresses, the dynamic characteristics of the precision positioning platform may change. Most of the existing tracking control methods are based on the model parameters being fixed constants, which cannot meet the requirements of precise tracking control performance.

[0003] Adaptive control can be used to estimate unknown parameters online for the unsteady parameter model of a precision positioning platform and applied to control system design. Conventional adaptive control schemes are rarely able to converge the unknown parameters to their true values, and the parameter convergence process can only guarantee the convergence of the norm of the estimated parameter vector, not the convergence of each element. In addition, for the rapid sampling problem in actual engineering applications, it is necessary to design an adaptive control scheme that does not rely on a stable zero-point model to achieve efficient tracking control. Control methods based on solving constrained optimization problems will cause control instructions to frequently reach the actuator saturation boundary, which is not conducive to further improving tracking accuracy. Summary of the Invention

[0004] Aiming at the problem that the precision positioning platform faces unstable zero points and model parameters are not fixed constants during rapid sampling, the present invention proposes a data-driven control method for the precision positioning platform based on step-skipping inversion. First, according to the dynamic characteristics of the precision positioning platform, an autoregressive model with time-varying linear parameters and external input is proposed; then, an adaptive parameter estimator is designed based on the dynamic regression extension and hybrid method; then, a precision positioning platform is constructed based on the above-mentioned adaptive iterative algorithm. step-recursive model; then, an expected controller is designed according to the number of zero points outside the unit circle of the autoregressive model, and the controller stability condition is given based on the eigenvalues ​​of the update matrix of the expanded state. Then, the fixed model parameters involved in the controller are replaced by the estimated values ​​obtained by the parameter estimator, thereby designing an adaptive controller based on skip-step inversion.

[0005] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0006] A data-driven control method for a precision positioning platform based on step-skipping inversion comprises the following steps:

[0007] S1: Based on the dynamic characteristics of the precision positioning platform, an autoregressive model with time-varying linear parameters and external input is established;

[0008] S2: Design of adaptive parameter estimators based on dynamic regression extension and hybrid methods;

[0009] S3: Building a Precision Positioning Platform Based on Adaptive Parameter Estimation Step recursive model;

[0010] S4: Design an adaptive controller based on skip-step inversion according to the number of zeros outside the unit circle of the autoregressive model and the parameter estimates.

[0011] Furthermore, in S1, constructing an autoregressive model with external input includes:

[0012] When a zero-order holder is used to quickly sample the precision positioning platform, the following time-varying linear parameter autoregressive model with external input is established according to the dynamic characteristics of the precision positioning platform:

[0013] ,

[0014] in, For precision positioning platform The output displacement at time , For precision positioning platform The control input at the moment, Model parameters related to the historical output of the precision positioning platform, are the model parameters of the autoregressive model with external input for the precision positioning platform, is the set of all real numbers, is the order of the model, In the representative model zeros, where the model parameters May be at any time There are changes, but the changes are not drastic, and the model order and the number of zero points remain unchanged;

[0015] Model parameters of the autoregressive model with external input according to the precision positioning platform The zero-point equation is ,in is the model zero point. When there is a zero point outside the unit circle, it means that the precision positioning platform is unstable. The number of zero points outside the unit circle is , that is, the number of unstable zeros is .

[0016] Furthermore, in S2, the adaptive parameter estimator design process includes:

[0017] First, a set of initial values ​​of model parameters are obtained through system identification, where The initial value of , The initial value of ;

[0018] definition The unknown parameter vector in the autoregressive model with external input at time is ,satisfy:

[0019] ,

[0020] superscript Indicates the transpose of the matrix, the superscript Represents the current true value of the parameter, then Moment regression vector ,satisfy:

[0021] ϕ(k ) ⊤ =[-y(k)-y(k - 1)⋯ - y(k - n + 1)u(k)u(k - 1)…u(k - m + 1)] ,

[0022] The autoregressive model of the precision positioning platform with external input is expressed as ;

[0023] definition 、 is an intermediate variable, represents the filter constant, and The iterative algorithm is:

[0024] ,

[0025] ,

[0026] Then the algebraic relationship between the above intermediate variables satisfies , multiply both sides of the equation The adjoint matrix have to:

[0027] ,

[0028] in, , , Representative Matrix The determinant of

[0029] definition Extending and hybrid methods for dynamic regression Time for unknown parameter vector The estimated vector of for No. elements, satisfying , , and its iterative algorithm satisfies:

[0030] ,

[0031] in, is a constant gain, for No. elements;

[0032] definition Extending and hybrid methods for dynamic regression Always No. The estimated error of elements satisfies:

[0033] ,

[0034] when When smaller, It will converge to 0 at a faster speed. is a process variable.

[0035] Furthermore, in said S3, The design process of the step-by-step recursive model includes:

[0036] Defining state variables For the future Step control input, For history Step control input, For the future Step output displacement, For history Step output displacement, satisfying:

[0037] u _ F (k ) ⊤ =[u(k)u(k + 1)…u(k + p - 1)] ,

[0038] u _ P (k ) ⊤ =[u(k - p)u(k - p + 1)…u(k - 1)] ,

[0039] y _ F (k ) ⊤ =[y(k + 1)y(k + 2)…y(k + p)] ,

[0040] y _ P (k ) ⊤ =[y(k - p + 1)y(k - p + 2)…y(k)] ,

[0041] Among them, the step length , is a positive integer, According to the autoregressive model with external input of the precision positioning platform, the above state variables satisfy:

[0042] ,

[0043] in, 、 、 、 is a coefficient matrix that satisfies:

[0044] ,

[0045] ,

[0046] because Full rank, the relationship between state variables is expressed as:

[0047] ,

[0048] in, 、 、 , because each Step 1 updates the above state variables once to satisfy , , so define non-negative integers , representing the first indivual Steps to meet:

[0049] u _ (pw) ⊤ = u _ F (pw) ⊤ =[u(pw)u(pw + 1)…u(pw + p - 1)] ,

[0050] u _ (p(w - 1)) ⊤ = u _ P (pw) ⊤ =[u(pw - p)u(pw - p + 1)…u(pw - 1)] ,

[0051] y _ pw ⊤ = y _ F (pw) ⊤ =[y(pw + 1)y(pw + 2)…y(pw + p)] ,

[0052] y _ p(w - 1) ⊤ = y _ P (pw) ⊤ =[y(pw - p + 1)y(pw - p + 2)…y(pw)] ,

[0053] in, hour and are all 0 vectors, so the precision positioning platform The step recursive model is:

[0054] .

[0055] Furthermore, in S4, the design process of the adaptive controller based on step-skipping inversion includes:

[0056] According to the number of unstable zeros The step-by-step recursive model is decomposed into:

[0057] ,

[0058] in, , ,satisfy:

[0059] y _ a (pw) ⊤ =[y(pw + v)y(pw + 2v)…y(pw + qv)] ,

[0060] y _ u (pw ) ⊤ =[ y _ u1 (pw) y _ u2 (pw)… y _ u(q - 1) (pw)] ,

[0061] y _ uc (pw)=[y(pw + cv + 1)y(pw + cv + 2)…y(pw + cv + l)], c = 0, 1, …, q - 1 ,

[0062] Where, , Based on and Rearrange the matrix The intermediate variables generated by the rows; , Based on and Rearrange the matrix The intermediate variables generated by the rows; , , , Based on and Rearrange the matrix The line generated , , and then according to and Rearrange the matrix With the matrix The intermediate variables generated by the columns;

[0063] The controller designed based on the true value of the parameters is as follows:

[0064] ,

[0065] in, for The reference trajectory at the corresponding moment of the element, for Moore-Penrose generalized inverse;

[0066] Constructing new state variables [ y _ a (pw) y _ u (pw) u _ pw ] , substitute the controller into The new state variable iteration equation of the step-by-step recursive model satisfies:

[0067] ,

[0068] ,

[0069] definition Represents the intermediate parameter matrix The largest modulus value among all eigenvalues, adjustment satisfy ;

[0070] Since the model parameters of the precision positioning platform are not fixed constants, the parameter estimates obtained by the adaptive parameter estimator are used to replace the coefficient matrix in the controller design process. 、 、 、 The constant parameter elements in replace , replace , respectively The moment coefficient matrix 、 、 、 Estimates 、 、 、 , and obtain the estimated 、 、 、 、 、 、 、 、 、 , design the adaptive controller based on the above matrix estimation:

[0071] ,

[0072] As the parameter estimates obtained by the adaptive parameter estimator converge to the true value, the above adaptive controller will converge to a controller designed based on the true values ​​of the parameters .

[0073] The beneficial effects of the present invention are:

[0074] 1. The method of the present invention includes an autoregressive model with external input for a precision positioning platform. For non-constant model parameters, the present invention makes full use of online input and output information to perform online estimation of the model parameters of the precision positioning platform, and decouples the estimation process of different model parameters by introducing an adjoint matrix. The convergence speed of a certain element in the estimated parameter vector can be adjusted by selecting a specific gain, thereby greatly enhancing the online identification capability of the closed-loop system.

[0075] 2. In the case of unstable zero points in discrete models, ordinary indirect adaptive control will cause the control quantity to gradually diverge when facing the reference trajectory tracking problem of the controlled object with unstable zero points. The present invention avoids the problems caused by unstable zero points through the step-by-step control method, and can achieve zero-error tracking at the desired sampling frequency of the precision positioning platform.

[0076] 3. To ensure the efficient operation of the precision positioning platform, the present invention replaces the constant parameters in the step-by-step control with the estimated values ​​of the parameters, which can incorporate the current platform operation status information into the controller design considerations, reducing the limitations on system performance caused by designing the controller based only on the nominal model. BRIEF DESCRIPTION OF THE DRAWINGS

[0077] Figure 1 This is a flow chart of a data-driven control method for a precision positioning platform based on step-skipping inversion according to the present invention;

[0078] Figure 2 This is a logic control block diagram of the dynamic regression expansion and hybrid adaptive control method for a precision positioning platform based on step-skipping inversion of the present invention;

[0079] Figure 3 The output displacement and reference trajectory of the precision positioning platform when a sinusoidal reference trajectory is input;

[0080] Figure 4 This is the error diagram between the output displacement of the precision positioning platform and the reference trajectory when a sinusoidal reference trajectory is input;

[0081] Figure 5 The output displacement and reference trajectory diagram of the precision positioning platform when the composite reference trajectory is input;

[0082] Figure ⑥ This is the error diagram between the output displacement of the precision positioning platform and the reference trajectory when the composite reference trajectory is input. DETAILED DESCRIPTION

[0083] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific implementation examples described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

[0084] like Figure 1 As shown, the present invention proposes a data-driven control method for a precision positioning platform based on step-skipping inversion, comprising the following steps:

[0085] Step S1: Establishing an autoregressive model with time-varying linear parameters and external inputs based on the dynamic characteristics of the precision positioning platform, including:

[0086] When a zero-order holder is used to quickly sample the precision positioning platform, the following time-varying linear parameter autoregressive model with external input is established according to the dynamic characteristics of the precision positioning platform:

[0087] ,

[0088] in, For precision positioning platform The output displacement at time , For precision positioning platform The control input at the moment, Model parameters related to the historical output of the precision positioning platform, Input model parameters related to the history of the precision positioning platform, is the set of all real numbers, is the order of the model, In the representative model zeros, where the model parameters May be at any time There are changes, but the changes are not drastic, and the model order and the number of zero points remain unchanged;

[0089] Model parameters of the autoregressive model with external input according to the precision positioning platform The zero-point equation is ,in is the model zero point. When there is a zero point outside the unit circle, it means that the precision positioning platform is unstable. The number of zero points outside the unit circle is , that is, the number of unstable zeros is .

[0090] Step S2: Design an adaptive parameter estimator based on the dynamic regression extension and hybrid method. The specific design steps are as follows:

[0091] First, a set of initial values ​​of model parameters are obtained through system identification, where The initial value of , The initial value of ;

[0092] definition The unknown parameter vector in the autoregressive model with external input at time is ,satisfy:

[0093] ,

[0094] superscript Indicates the transpose of the matrix, the superscript Represents the current true value of the parameter, then Moment regression vector ,satisfy:

[0095] ϕ(k ) ⊤ =[-y(k)-y(k - 1)⋯ - y(k - n + 1)u(k)u(k - 1)…u(k - m + 1)] ,

[0096] The autoregressive model of the precision positioning platform with external input is expressed as ;

[0097] definition 、 is an intermediate variable, represents the filter constant, and The iterative algorithm is:

[0098] ,

[0099] ,

[0100] Then the algebraic relationship between the above intermediate variables satisfies , multiply both sides of the equation The adjoint matrix have to:

[0101] ,

[0102] in, , , Representative Matrix The determinant of

[0103] definition Extending and hybrid methods for dynamic regression Time for unknown parameter vector The estimated vector of for No. elements, satisfying , , and its iterative algorithm satisfies:

[0104] ,

[0105] in, is a constant gain, for No. elements;

[0106] definition Extending and hybrid methods for dynamic regression Always No. The estimated error of elements satisfies:

[0107] ,

[0108] when When smaller, It will converge to 0 at a faster speed. is a process variable.

[0109] Step S3: Constructing a precision positioning platform based on the adaptive parameter estimator The specific design steps of the step-by-step recursive model are as follows:

[0110] Defining state variables For the future Step control input, For history Step control input, For the future Step output displacement, For history Step output displacement, satisfying:

[0111] u _ F (k ) ⊤ =[u(k)u(k + 1)…u(k + p - I)] ,

[0112] u _ P (k ) ⊤ =[u(k - p)u(k - p + 1)…u(k - 1)] ,

[0113] y _ F (k ) ⊤ =[y(k + 1)y(k + 2)…y(k + p)] ,

[0114] y _ P (k ) ⊤ =[y(k - p + 1)y(k - p + 2)…y(k)] ,

[0115] Among them, the step length , is a positive integer, According to the autoregressive model with external input of the precision positioning platform, the above state variables satisfy:

[0116] ,

[0117] in, 、 、 、 is a coefficient matrix that satisfies:

[0118] ,

[0119] ,

[0120] because Full rank, the relationship between state variables is expressed as:

[0121] ,

[0122] in, 、 、 , because each Step 1 updates the above state variables once to satisfy , , so define non-negative integers , representing the first indivual Steps to meet:

[0123] u _ (pw) ⊤ = u _ F (pw) ⊤ =[u(pw)u(pw + 1)…u(pw + p - 1)] ,

[0124] u _ (p(w - 1)) ⊤ = u _ P (pw) ⊤ =[u(pw - p)u(pw - p + 1)…u(pw - 1)] ,

[0125] y _ pw ⊤ = y _ F (pw) ⊤ =[y(pw + 1)y(pw + 2)…y(pw + p)] ,

[0126] y _ p(w - 1) ⊤ = y _ P (pw) ⊤ =[y(pw - p + 1)y(pw - p + 2)…y(pw)] ,

[0127] in, hour and are all 0 vectors, so the precision positioning platform The step recursive model is:

[0128] .

[0129] Step S4: Design an adaptive controller based on skipped inversion according to the number of zero points outside the unit circle of the autoregressive model and the parameter estimation value. The specific design steps are as follows:

[0130] According to the number of unstable zeros The step-by-step recursive model is decomposed into:

[0131] ,

[0132] in , ,satisfy:

[0133] y _ a (pw) ⊤ =[y(pw + v)y(pw + 2v)…y(pw + qv)] ,

[0134] y _ u (pw ) ⊤ =[ y _ u1 (pw) y _ u2 (pw)… y _ u(q - 1) (pw)] ,

[0135] y _ uc (pw)=[y(pw + cv + 1)y(pw + cv + 2)…y(pw + cv + l)], c = 0, 1, …, q - 1 ,

[0136] Where, , Based on and Rearrange the matrix The intermediate variables generated by the rows; , Based on and Rearrange the matrix The intermediate variables generated by the rows; , , , Based on and Rearrange the matrix The line generated , , and then according to and Rearrange the matrix With the matrix The intermediate variables generated by the columns;

[0137] The controller designed based on the true value of the parameters is as follows:

[0138] ,

[0139] in, for The reference trajectory at the corresponding moment of the element, for Moore-Penrose generalized inverse;

[0140] Constructing new state variables [ y _ a (pw) y _ u (pw) u _ pw ] , substitute the controller into The new state variable iteration equation of the step-by-step recursive model satisfies:

[0141] ,

[0142] ,

[0143] definition Represents the intermediate parameter matrix The largest modulus value among all eigenvalues, adjustment satisfy ;

[0144] Since the model parameters of the precision positioning platform are not fixed constants, the parameter estimates obtained by the adaptive parameter estimator are used to replace the coefficient matrix in the controller design process. 、 、 、 The constant parameter elements in replace , replace , respectively The moment coefficient matrix 、 、 、 Estimates 、 、 、 , and obtain according to the original corresponding relationship 、 、 、 、 、 、 、 、 、 , design the adaptive controller based on the above matrix estimation:

[0145] ,

[0146] As the parameter estimates obtained by the adaptive parameter estimator converge to the true value, the above adaptive controller will converge to a controller designed based on the true values ​​of the parameters , ensuring the stable tracking performance of the precision positioning platform.

[0147] like Figure 2As shown in FIG, according to steps S1 to S4, a logic control block diagram is given. The historical control input and displacement output data are processed by dynamic regression expansion and hybrid technology. The processed data are transmitted to the adaptive parameter estimator. After the parameter estimation is obtained, an adaptive controller based on step-skipping inversion is designed in combination with the displacement output and the reference trajectory to obtain the control input at the current moment.

[0148] Example

[0149] (1) Simulation settings:

[0150] In the present invention, the simulation step size is set to 0.00005s (i.e., the sampling frequency is 20kHz), the total simulation time is 0.02s, and the actual precision positioning platform continuous system model is , is the Laplace operator, then the actual parameters after discretization are , the nominal continuous system model is , then the initial parameter estimate at time 0 after discretization is , the model order is , the number of zero points is 2, and the number of unstable zero points is ,Pick , can be obtained ; In the parameter estimator , ;

[0151] (2) Sine signal tracking test:

[0152] First, a sine wave signal with an amplitude of 1 μm and a frequency of 200 Hz is input as the reference trajectory signal to be tracked. The initial state of the system is ϕ(0)= 000000 ] ⊤ Carry out simulation and record the output displacement and tracking error of the precision positioning platform within 0.02s . Figure 3 The output displacement and reference trajectory of the precision positioning platform when a sinusoidal reference trajectory is input. Figure 4 The error between the output displacement of the precision positioning platform and the reference trajectory when a sinusoidal reference trajectory is input shall not exceed ±0.015μm after stabilization;

[0153] (3) Composite signal tracking test:

[0154] The composite signal composed of the superposition of the sine signals with an amplitude of 1 μm and a frequency of 300 Hz, the amplitude of 2 μm and a frequency of 500 Hz, and the amplitude of 2 μm and a frequency of 10 Hz is used as the reference trajectory signal. The initial state of the system is ϕ(0)= 0 0 0 0 0 0 ] ⊤ . Carry out simulation and record the output displacement and tracking error of the precision positioning platform within 0.02s. Figure 5 The output displacement and reference trajectory of the precision positioning platform when the composite reference trajectory is input. Figure 6 The error between the output displacement of the precision positioning platform and the reference trajectory when the composite reference trajectory is input shall not exceed ±0.065μm after stabilization.

[0155] The specific embodiments described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above are only specific embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A data driven control method for a precision positioning platform based on step-skipping inversion, characterized in that: The following steps are involved: S1. Establish an autoregressive model with time-varying linear parameters and external input based on the dynamic characteristics of the precision positioning platform; S2. Design of adaptive parameter estimators based on dynamic regression extension and hybrid methods; S3, Construction of Precision Positioning Platform Based on Adaptive Parameter Estimator Step recursive model; S4. Design an adaptive controller based on step-by-step inversion according to the number of zero points outside the unit circle of the autoregressive model and the estimated values ​​of the parameters; Wherein, the S1 includes: When a zero-order holder is used to quickly sample the precision positioning platform, the following time-varying linear parameter autoregressive model with external input is established according to the dynamic characteristics of the precision positioning platform: ， in, For precision positioning platform The output displacement at time , For precision positioning platform The control input at the moment, Model parameters related to the historical output of the precision positioning platform, are the model parameters of the autoregressive model with external input for the precision positioning platform, is the set of all real numbers, is the order of the model, In the representative model zero points, the model order and the number of zero points remain unchanged; Model parameters of the autoregressive model with external input according to the precision positioning platform The zero-point equation is ,in is the model zero point. When there is a zero point outside the unit circle, it means that the precision positioning platform is unstable. The number of zero points outside the unit circle is , that is, the number of unstable zeros is ; The S2 includes: First, a set of initial values ​​of model parameters are obtained through system identification, where The initial value of , The initial value of ; definition The unknown parameter vector in the autoregressive model with external input at time is ,satisfy: ， superscript Indicates the transpose of the matrix, the superscript Represents the current true value of the parameter, then Moment regression vector ,satisfy: ， The autoregressive model of the precision positioning platform with external input is expressed as ; definition 、 is an intermediate variable, represents the filter constant, and The iterative algorithm is: ， ， Then the algebraic relationship between the above intermediate variables satisfies , multiply both sides of the equation The adjoint matrix have to: ， in, , , Representative Matrix The determinant of definition Extending and hybrid methods for dynamic regression Time for unknown parameter vector The estimated vector of for No. elements, satisfying , , and its iterative algorithm satisfies: ， in, is a constant gain, for No. elements; definition Extending and hybrid methods for dynamic regression Always No. The estimated error of elements satisfies: ， is the process variable; The S3 includes: Defining state variables For the future Step control input, For history Step control input, For the future Step output displacement, For history Step output displacement, satisfying: ， ， ， ， Among them, the step length , is a positive integer, According to the autoregressive model with external input of the precision positioning platform, the above state variables satisfy: ， in 、 、 、 is a coefficient matrix that satisfies: because Full rank, the relationship between state variables is expressed as: ， in, 、 、 , because each Step 1 updates the above state variables once to satisfy , , so define non-negative integers , representing the first indivual Steps to meet: ， ， ， ， in, hour and are all 0 vectors, so the precision positioning platform The step recursive model is: ； The S4 includes: According to the number of unstable zeros The step-by-step recursive model is decomposed into: ， in, , ,satisfy: ， ， ， Where, , Based on and Rearrange the matrix The intermediate variables generated by the rows; , Based on and Rearrange the matrix The intermediate variables generated by the rows; , , , Based on and Rearrange the matrix The line generated , , and then according to and Rearrange the matrix With the matrix The intermediate variables generated by the columns; The controller designed based on the true value of the parameters is as follows: ， in, for The reference trajectory at the corresponding moment of the element, for Moore-Penrose generalized inverse; Constructing new state variables , substitute the controller into The new state variable iteration equation of the step-by-step recursive model satisfies: ， ， definition Represents the intermediate parameter matrix The largest modulus value among all eigenvalues, adjustment satisfy , Replacing the coefficient matrix in the controller design process with parameter estimates obtained from an adaptive parameter estimator 、 、 、 The constant parameter elements in replace , replace , respectively The moment coefficient matrix 、 、 、 Estimates 、 、 、 , and obtain the estimated 、 、 、 、 、 、 、 、 、 , design the adaptive controller based on the above matrix estimation: ， As the parameter estimates obtained by the adaptive parameter estimator converge to the true value, the above adaptive controller will converge to a controller designed based on the true values ​​of the parameters .

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