Non-linear strict feedback system adaptive identification control method based on regression expansion and mixing

Abstract

The invention aims to provide a self-adaptive identification control method for a nonlinear strict feedback system based on regression expansion and mixing, belongs to the field of self-adaptive control, and aims to solve the problem of self-adaptive identification control of the nonlinear strict feedback system. Existing researches are mostly based on an original linear regression equation of a system, and parameter identification performance highly depends on a continuous excitation condition. According to the method, the availability of measurement information is considered, first-order inertial filtering is performed on system state data to obtain an output quantity, an auxiliary matrix is constructed by utilizing historical excitation data to realize expansion reconstruction of a regression vector, and the regression vector is converted into an expansion linear regression equation in a scalar form by adopting a mixing technology. On the basis, system tracking errors and modeling errors are constructed, a gradient descent algorithm is adopted to design a parameter identification updating law, and finally the adaptive control method based on parameter identification is designed in combination with a backstepping method framework. According to the method, historical flight data can be effectively utilized through regression expansion and mixing, excitation conditions needed by parameter identification are relaxed, efficient online identification of model parameters is achieved, and the tracking control performance of an uncertain nonlinear strict feedback system is guaranteed.

Description

Technical Field

[0001] This invention relates to an adaptive identification control method, and more particularly to an adaptive identification control method for nonlinear strict feedback systems based on regression extension and hybrid approaches, belonging to the field of adaptive control. Existing technology

[0002] Nonlinear rigorous feedback systems can effectively describe real-world objects such as aircraft, robotic arms, and spacecraft. Researching adaptive identification control methods for these systems is crucial for addressing model uncertainties in practical physical systems and improving control robustness. Existing research largely relies on the system's original linear regression equations, employing gradient descent algorithms to design parameter identification and update laws and control laws. However, the parameter identification performance of this method is highly dependent on continuous excitation conditions. Due to the requirements of task execution and system stability, these conditions are often difficult to guarantee in practical applications. Therefore, research is urgently needed to achieve online model parameter identification and adaptive control under weak excitation conditions.

[0003] The paper "Adaptive Optimal Switching Control Method for Nonlinear Systems" (Mao Yanling, Fu Yue, *Acta Automatica Sinica*, 2023, 49(10): 2122-2135) proposes a control strategy combining a linear adaptive optimal switching controller and an unmodeled dynamic compensator for continuous-time nonlinear systems with unknown dynamics. This method achieves adaptive optimal switching control even when model parameters are unknown. Its core idea is to accurately estimate model parameters based on sampled data and then construct an adaptive switching control law. However, this method requires applying excitation near the equilibrium point to collect state data, resulting in weak real-time performance and certain limitations in practical applications.

[0004] 3. Purpose of the invention

[0005] This invention addresses the control requirements of uncertain nonlinear strict feedback systems under weak excitation conditions by proposing an adaptive identification control method based on regression extension and hybridization. Considering the availability of measurement information, first-order inertial filtering is applied to the system state data to obtain the output. An auxiliary matrix is ​​constructed using historical excitation data to extend and reconstruct the regression vector, and a hybridization technique is employed to transform it into a scalar-form extended linear regression equation. Based on this, the system tracking error and modeling error are constructed, and a parameter identification update law is designed using the gradient descent algorithm. Finally, an adaptive control method based on parameter identification is designed using a backstepping framework. This method, through regression extension and hybridization, effectively utilizes historical flight data, relaxes the excitation conditions required for parameter identification, achieves efficient online identification of model parameters, and ensures the tracking control performance of uncertain nonlinear strict feedback systems. Summary of the Invention

[0006] The technical solution adopted by this invention to solve its technical problem is: an adaptive identification control method for nonlinear strict feedback systems based on regression extension and hybrid approaches, which is achieved through the following steps:

[0007] (a) Considering the nonlinear strict feedback system model as follows

[0008]

[0009] Where, Let i represent the system state vector, i = 1, ..., n-1, and u represent the control input. and It can be converted into the following parameterized form

[0010]

[0011] Where, and Represents an unknown model parameter vector. and Let p represent a known state vector. i and b i They represent and Dimensions.

[0012] Can Written as

[0013]

[0014] Where, Represents a vector of non-zero parameters. Represents an unknown and bounded parameter vector.

[0015] (b) Establish the control-oriented model shown in equation (4) and the parameter identification-oriented model shown in equation (5) as follows:

[0016]

[0017] Where,

[0018] (c) Step 1: Define the tracking error as The error dynamics can be obtained as follows:

[0019]

[0020] The virtual control quantity is designed as follows:

[0021]

[0022] In the formula, k1 > 0 represents the control parameter. and They represent and The estimated value.

[0023] To obtain make The first-order filter shown in equation (8) is

[0024]

[0025] In the formula, ε2>0 represents the design parameter. express The filtered value of the first-order filter is shown in Equation (8).

[0026] To obtain x 1f and Make x1 and The first-order filter shown in equation (9) is

[0027]

[0028] In the formula, κ 11 >0 and κ 12 >0 indicates filter parameters, x 1f and Representing x1 and The filtered value of the first-order filter is shown in equation (9).

[0029] Further, we can obtain

[0030]

[0031] In the formula, Δ T1 This represents the bounded filtering error.

[0032] Based on equation (10), a linear regression equation is established as follows:

[0033]

[0034] Where, Indicates the amount of observation.

[0035] Construct auxiliary matrix M1 and auxiliary vector N1 as follows

[0036]

[0037] In the formula, l1 > 0 represents the forgetting factor, and further, M1 and N1 can be obtained as follows:

[0038]

[0039] Based on equations (10) and (13), the extended linear regression model is constructed as follows:

[0040] N1=M1ω1+v1 (14)

[0041] Where,

[0042] Multiplying both sides of equation (14) by the adjoint matrix adj(M1) of M1 on the left, a scalar extended linear regression model is established as follows:

[0043]

[0044] Where, Indicates extended observations, Let m represent the extended regression matrix after decoupling. 1d Let I represent the determinant of M1. d Represents the identity matrix.

[0045] Design parameter identification and update law

[0046]

[0047] In the formula, γ1>0 and γ z1 >0 indicates a design parameter. This represents the estimated value of ω1.

[0048] Step i, i = 2, 3, ..., n-1: Define the tracking error as... The error dynamics can be obtained as follows:

[0049]

[0050] The virtual control quantity is designed as follows:

[0051]

[0052] In the formula, k i >0 indicates controller parameters, and They represent and The estimated value.

[0053] To obtain make The first-order filter shown in equation (19) is

[0054]

[0055] In the formula, ε i+1 >0 indicates a design parameter. express The filtered value of the first-order filter is shown in equation (19).

[0056] To obtain x if and Make x i and The first-order filter shown in equation (20) is

[0057]

[0058] In the formula, κ i1 >0 and κ i2 >0 indicates filter parameters, x if and They represent x respectively i and The filtered value of the first-order filter is shown in equation (20).

[0059] Further, we can obtain

[0060]

[0061] In the formula, Δ Ti This represents the bounded filtering error.

[0062] Based on equation (21), the linear regression equation is established as follows:

[0063]

[0064] Where, Indicates the amount of observation.

[0065] Construct auxiliary matrix M i and auxiliary vector N i for

[0066]

[0067] In the formula, l i >0 represents the forgetting factor, and further, M can be obtained. i and N i for

[0068]

[0069] Based on equations (21) and (24), the extended linear regression model is established as follows:

[0070] N i =M i ω i +v i (25)

[0071] Where,

[0072] Multiply both sides of equation (25) by M on the left. iThe adjoint matrix adj(M) i To establish a scalar form extended linear regression model as follows:

[0073]

[0074] Where, Indicates extended observations, Let m represent the decoupled regression matrix. id M represents i The determinant,

[0075] The design parameter identification and update law is:

[0076]

[0077] In the formula, γ i >0 and γ zi >0 indicates a design parameter. Represents ω i The estimated value,

[0078] Step n: Define the tracking error as... The error dynamics can be obtained as follows:

[0079]

[0080] The design control quantity is

[0081]

[0082] In the formula, k n >0 indicates controller parameters, and They represent and The estimated value.

[0083] To obtain x nf and Make x n and The first-order filter shown in equation (30) is

[0084]

[0085] In the formula, κ n1 >0 and κ n2 >0 indicates filter parameters, x nf and They represent x respectively n and The filtered value of the first-order filter is shown in equation (30).

[0086] Further, we can obtain

[0087]

[0088] In the formula, Δ Tn This represents the bounded filtering error.

[0089] Based on equation (31), the linear regression equation is established as follows:

[0090]

[0091] Where, Indicates the amount of observation.

[0092] Construct auxiliary matrix M n and auxiliary vector N n for

[0093]

[0094] In the formula, l n >0 represents the forgetting factor, and further, M can be obtained. n and N n for

[0095]

[0096] Based on equations (31) and (34), the extended linear regression model is established as follows:

[0097] N n =M n ω n +v n (35)

[0098] Where,

[0099] Multiply both sides of equation (35) by M on the left. n The adjoint matrix adj(M) n To establish a scalar form extended linear regression model as follows:

[0100]

[0101] Where, Indicates extended observations, Let m represent the decoupled regression matrix. nd M represents n The determinant,

[0102] The design parameter identification and update law is:

[0103]

[0104] In the formula, γ n>0 and γ zn >0 indicates a design parameter. Represents ω n The estimated value,

[0105] (d) Input the calculated control command u into the system (1) to achieve effective tracking control of the output command.

[0106] 5. Effects of the invention

[0107] The advantages of this invention compared to the prior art are as follows:

[0108] (1) The present invention uses first-order inertial filtering to obtain the output quantity, constructs an auxiliary matrix using historical excitation data, and constructs an extended linear regression equation in scalar form through hybrid technology. Based on this, the continuous excitation conditions required for the original system parameter identification are converted into weak excitation conditions, thus relaxing the requirements for excitation conditions.

[0109] (2) This invention constructs modeling error based on extended linear regression equation, designs gradient descent parameter identification update law in combination with tracking error, and designs an adaptive controller based on parameter identification, which effectively handles system uncertainty.

[0110] (3) By applying regression extension and hybrid technology, this invention makes full use of historical flight data and combines online parameter identification with backstepping control, providing a solution for the effective control of uncertain nonlinear systems under weak excitation conditions. Attached Figure Description

[0111] Figure 1 This is a block diagram of an adaptive identification control method based on regression extension and hybridization. This invention addresses the control requirements of uncertain nonlinear strict feedback systems under weak excitation conditions by proposing an adaptive identification control method based on regression extension and hybridization. Considering the availability of measurement information, first-order inertial filtering is applied to the system state data to obtain the output quantity. An auxiliary matrix is ​​constructed using historical excitation data to achieve the extended reconstruction of the regression vector, and a hybridization technique is used to transform it into a scalar form extended linear regression equation. Based on this, the system tracking error and modeling error are constructed, and a parameter identification update law is designed using the gradient descent algorithm. Finally, an adaptive control method based on parameter identification is designed using a backstepping framework. This method, through regression extension and hybridization, can effectively utilize historical flight data, relax the excitation conditions required for parameter identification, achieve efficient online identification of model parameters, and ensure the tracking control performance of uncertain nonlinear strict feedback systems.

[0112] 7. Implementation Examples

[0113] Reference Figure 1An adaptive identification control method for nonlinear rigorous feedback systems based on regression extension and hybrid approaches is proposed. The specific steps are as follows:

[0114] (a) Considering the model of a second-order nonlinear rigorous feedback system,

[0115]

[0116] Where, θ1=0.5, θ2=-4, θ3=2, θ4=1, and It can be converted into the following parameterized form

[0117]

[0118] Where,

[0119] Can Written as

[0120]

[0121] Where, Δθ4=0.2,

[0122] (b) Establish the control-oriented model shown in equation (4) and the parameter identification-oriented model shown in equation (5) as follows:

[0123]

[0124] In the formula, ω2 = [θ1, θ2, θ3, Δθ4] T ,

[0125] (c) Step 1: Define the tracking error as The error dynamics can be obtained as follows:

[0126]

[0127] The virtual control quantity is designed as follows:

[0128]

[0129] In the formula, k1 = 5 represents the controller parameters.

[0130] To obtain make The first-order filter shown in equation (8) is

[0131]

[0132] In the formula, ε2=0.04 represents the design parameter. express The filtered value of the first-order filter is shown in Equation (8).

[0133] Step 2: Define the tracking error as... The error dynamics can be obtained as follows:

[0134]

[0135] The design control quantity is

[0136]

[0137] In the formula, k2 = 5 represents the controller parameters. and They represent and The estimated value.

[0138] To obtain x 2f and Make x2 and The first-order filter shown in equation (11) is

[0139]

[0140] In the formula, κ 21 =0.006 and κ 22 =0.01 indicates the filter parameter, x 2f and They represent x2 and The filtered value of the first-order filter is shown in equation (11).

[0141] Further, we can obtain

[0142]

[0143] In the formula, Δ T2 This represents the bounded filtering error.

[0144] Based on equation (13), the linear regression equation is established as follows:

[0145]

[0146] Where,

[0147] Construct auxiliary matrix M2 and auxiliary vector N2 as follows

[0148]

[0149] In the formula, l2 = 0.001 represents the forgetting factor, and further, M2 and N2 can be obtained as follows:

[0150]

[0151] Based on equations (12) and (15), the extended linear regression model is established as follows:

[0152] N2=M2ω2+v2 (16)

[0153] Where,

[0154] Multiplying both sides of equation (16) by the adjoint matrix adj(M2) of M2 on the left, we establish a scalar extended linear regression model as follows:

[0155]

[0156] Where, Indicates extended observations, Let m represent the decoupled regression matrix. 2d Represents the determinant of M2.

[0157] The design parameter identification and update law is:

[0158]

[0159] In the formula, γ2=4 and γ z2 =10 9 Indicates design parameters, This represents the estimated value of ω2.

[0160] (d) Input the calculated control command u into the system (1) to achieve effective tracking and control of the system command.

[0161] The parts of this invention not described in detail are common knowledge to those skilled in the art.

Claims

1. An adaptive identification control method for nonlinear rigorous feedback systems based on regression extension and hybrid approaches, characterized in that, The adaptive identification and control method includes: 1) Establish a model for a nonlinear rigorous feedback system; 2) Establish control-oriented models and parameter identification-oriented models; 3) Design an adaptive identification control law based on regression extension and hybrid methods.

2. The adaptive identification control method for a nonlinear strict feedback system according to claim 1, characterized in that, Establish a nonlinear rigorous feedback system model as follows Where, Let i represent the system state vector, i = 1, ..., n-1, and u represent the control input. and It can be converted into the following parameterized form Where, and Represents an unknown model parameter vector. and Let p represent a known state vector. i and b i They represent and Dimensions. Can Written as Where, Represents a vector of non-zero parameters. Represents an unknown and bounded parameter vector.

3. The adaptive identification control method for a nonlinear strict feedback system according to claim 1, characterized in that, Establish control-oriented models and parameter identification-oriented models. Where, 4. The adaptive identification control method for a nonlinear strict feedback system according to claim 1, characterized in that, The design is based on an adaptive identification control law that combines regression extension and hybrid approaches. Step 1: Define the tracking error as The error dynamics can be obtained as follows: The virtual control quantity is designed as follows: In the formula, k1>0 represents the control parameter. and They represent and The estimated value. To obtain make The first-order filter shown in equation (8) is In the formula, ε2>0 represents the design parameter. express The filtered value of the first-order filter is shown in Equation (8). To obtain x 1f and Make x1 and The first-order filter shown in equation (9) is In the formula, κ 11 >0 and κ 12 >0 indicates filter parameters, x 1f and Representing x1 and The filtered value of the first-order filter is shown in equation (9). Further, we can obtain In the formula, △ T1 This represents the bounded filtering error. Based on equation (10), a linear regression equation is established as follows: Where, Indicates the amount of observation. Construct auxiliary matrix M1 and auxiliary vector N1 as follows In the formula, l1>0 represents the forgetting factor, and further, M1 and N1 can be obtained as follows: Based on equations (10) and (13), the extended linear regression model is constructed as follows: N1=M1ω1+v1 (14) Where, Multiplying both sides of equation (14) by the adjoint matrix adj(M1) of M1 on the left, a scalar extended linear regression model is established as follows: Where, Indicates extended observations, Let m represent the extended regression matrix after decoupling. 1d Let I represent the determinant of M1. d Represents the identity matrix. Design parameter identification and update law In the formula, γ1>0 and γ z1 >0 indicates a design parameter. This represents the estimated value of ω1. Step i, i = 2, 3, ..., n-1: Define the tracking error as... The error dynamics can be obtained as follows: The virtual control quantity is designed as follows: In the formula, k i >0 indicates controller parameters. and They represent and The estimated value. To obtain make The first-order filter shown in equation (19) is In the formula, ε i+1 >0 indicates a design parameter. express The filtered value of the first-order filter is shown in equation (19). To obtain x if and Make x i and The first-order filter shown in equation (20) is In the formula, κ i1 >0 and κ i2 >0 indicates filter parameters, x if and They represent x respectively i and The filtered value of the first-order filter shown in equation (20) can be further obtained. In the formula, △ Ti This represents the bounded filtering error. Based on equation (21), the linear regression equation is established as follows: Where, Indicates the amount of observation. Construct auxiliary matrix M i and auxiliary vector N i for In the formula, l i >0 represents the forgetting factor, which further gives M i and N i for Based on equations (21) and (24), the extended linear regression model is established as follows: N i =M i ω i +v i (25) Where, Multiply both sides of equation (25) by M on the left. i The adjoint matrix adj(M) i To establish a scalar form extended linear regression model as follows: Where, Indicates extended observations, Let m represent the decoupled regression matrix. id M represents i The determinant, The design parameter identification and update law is: In the formula, γ i >0 and γ zi >0 indicates a design parameter. Represents ω i The estimated value, Step n: Define the tracking error as... The error dynamics can be obtained as follows: The design control quantity is In the formula, k n >0 indicates controller parameters. and They represent and The estimated value. To obtain x nf and Make x n and The first-order filter shown in equation (30) is In the formula, κ n1 >0 and κ n2 >0 indicates filter parameters, x nf and They represent x respectively n and The filtered value of the first-order filter shown in equation (30) can be further obtained. In the formula, △ Tn This represents the bounded filtering error. Based on equation (31), the linear regression equation is established as follows: Where, Indicates the amount of observation. Construct auxiliary matrix M n and auxiliary vector N n for In the formula, l n >0 represents the forgetting factor, which further gives M n and N n for Based on equations (31) and (34), the extended linear regression model is established as follows: N n =M n ω n +v n (35) Where, Multiply both sides of equation (35) by M on the left. n The adjoint matrix adj(M) n To establish a scalar form extended linear regression model as follows: Where, Indicates extended observations, Let m represent the decoupled regression matrix. nd M represents n The determinant, The design parameter identification and update law is: In the formula, γ n >0 and γ zn >0 indicates a design parameter. Represents ω n The estimated value,

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