A control method for a flexible link manipulator based on an adaptive finite-time disturbance observer

Abstract

The application provides a flexible connecting rod mechanical arm control method based on an adaptive finite time disturbance observer. The method comprises the following steps: establishing a single connecting rod flexible mechanical arm dynamic equation according to the structure of a single connecting rod mechanical arm system, mathematically modeling a nonlinear saturation tracking signal in input control in the single connecting rod flexible mechanical arm dynamic equation, and constructing a single connecting rod flexible mechanical arm system model containing actuator saturation; constructing a radial basis function neural network according to state variables of the single connecting rod mechanical arm system; designing an adaptive finite time disturbance observer to estimate external disturbance in the single connecting rod mechanical arm system; constructing a composite anti-interference controller according to the output of the radial basis function neural network and the adaptive finite time disturbance observer, and controlling the single connecting rod flexible mechanical arm system by using the composite anti-interference controller. The application solves the adverse effects caused by actuator saturation, finely compensates for disturbance, and realizes stable control of the single connecting rod flexible mechanical arm system.

Description

Technical Field

[0001] The present invention relates to the technical field of flexible link mechanical control, and in particular to a flexible link mechanical arm control method based on an adaptive finite-time disturbance observer. Background Art

[0002] In recent years, flexible linkage manipulators have played an important role in various fields, including healthcare, aerospace, and civil engineering, due to their advantages such as light weight, high flexibility, high speed, and low energy consumption. With the development of robotics technology and theory, manipulators are expected to operate in more complex environments. System uncertainty and external disturbances are prevalent in the application of flexible manipulators, not only affecting tracking accuracy but also causing system instability. Flexible structures are susceptible to structural vibrations caused by external loads during movement, which seriously affects their performance. Therefore, vibration suppression and precise positioning are key issues for flexible linkage manipulators.

[0003] The existence of vibrations poses a challenge to achieving good convergence performance and vibration suppression for flexible manipulators. Finite-time control methods are a feasible approach to achieve fast convergence while improving the robustness of flexible manipulators.

[0004] Adaptive finite-time disturbance observers (FTDOs) are based on the assumption that the upper bound of the disturbance or the derivative of the disturbance is known. However, in most practical systems, external disturbances are hidden, making accurate prior knowledge of the disturbance difficult to obtain. This significantly limits the application of adaptive finite-time disturbance observers. Based on the above analysis, finite-time control of flexible arm systems under multiple disturbances remains a research topic. Summary of the Invention

[0005] An embodiment of the present invention provides a flexible link manipulator control method based on an adaptive finite-time disturbance observer to achieve stable control of a single-link flexible manipulator system.

[0006] In order to achieve the above-mentioned purpose, the present invention adopts the following technical solutions.

[0007] A control method for a flexible link manipulator based on an adaptive finite-time disturbance observer comprises:

[0008] According to the structure of the single-link flexible manipulator system, the dynamic equation of the single-link flexible manipulator including the unknown nonlinear design radial basis function neural network is established;

[0009] According to the inevitable input saturation problem in the actuator, a mathematical model is performed on the nonlinear saturation tracking signal in the input control in the dynamic equation of the single-link flexible manipulator, and a single-link flexible manipulator system model with actuator saturation is constructed;

[0010] According to the state variables of the single-link flexible manipulator system, an unknown nonlinear radial basis function neural network is estimated in the dynamic equation of the single-link flexible manipulator, and a radial basis function neural network is constructed;

[0011] designing an adaptive finite-time disturbance observer based on the single-link flexible manipulator system with actuator saturation, and estimating external disturbances in the single-link manipulator system using the adaptive finite-time disturbance observer;

[0012] A composite anti-interference controller is constructed based on the output of the radial basis function neural network and the output of the adaptive finite-time disturbance observer. The composite anti-interference controller is used to control a single-link flexible robotic arm system with actuator saturation, and output the angle position and velocity signals of each joint of the single-link flexible robotic arm system.

[0013] Preferably, the said single-link flexible manipulator arm dynamic equations including unknown nonlinear design radial basis function neural network are established according to the structure of the single-link manipulator arm system, including:

[0014] The single-link flexible manipulator system is considered as an Euler-Bernoulli beam rotating around the horizontal plane. According to the energy equation and Hamilton's principle, the boundary conditions and control equations of the single-link flexible manipulator system including the unknown nonlinear design radial basis function neural network are obtained:

[0015] ρω″′(L,t)+m t ω″′(L,t)=0 (1)

[0016] ω(0,t)=ω′(0,t)=ω″(L,t)=0 (2)

[0017]

[0018]

[0019] Where L is the length of the flexible link, ρ is the average mass per unit length, and I h is the central inertia, M is the total mass of the connecting rod, m t is the tip mass, EI is the bending stiffness; θ(t) represents the central angular position; τ(t) is the input torque; ω(x, t) represents the elastic deformation, and ω″ represents the secondary derivative of the flexibility variable;

[0020] Using the assumed modal method, ω(x,t) is expressed as:

[0021]

[0022] Among them, φ i(x) represents a model function, p i (t) is a generalized coordinate, under the premise of small deflection, the dynamic equation of a single-link flexible manipulator derived from Lagrange equation and assumed mode method is:

[0023]

[0024] wherein Q is defined as Q = [θ, p1, … p N ] T , τ d represents an external disturbance, and the matrix M is positive definite and symmetric.

[0025] Preferably, the nonlinear saturation tracking signal in the input control of the single-link flexible manipulator dynamic equation is mathematically modeled according to the inevitable input saturation problem in the actuator, and a single-link flexible manipulator system model containing actuator saturation is constructed, comprising:

[0026] The actuator τ with nonlinear saturation in the single-link flexible manipulator dynamic equation is described by the following formula:

[0027]

[0028] wherein u(t) ∈ R is a control law to be designed, u b is a saturation boundary, the nonlinear saturation tracking signal τ is not smooth at u(t) = u b , and the saturation degree of the actuator is approximated as a smooth function, as follows:

[0029]

[0030] Define Δτ as an approximation error, and obtain:

[0031] τ = g(u) + Δτ (9)

[0032] According to the mean value theorem, g(u) is expressed as:

[0033]

[0034] wherein 0 < μ < 1, g(0) is the initial value of g(u), and Λ = (πμu / 2u b ) 2 / (1 + πμu / 2u b ) 2

[0035] Then g(u) is rewritten as:

[0036] g(u) = u - Λu (11)

[0037] Formula (7) represents the actuator saturation, where u(t) is the input of the actuator, τ is the output of the actuator, and τ is the input of the single-link flexible manipulator dynamic equation (6). Formulas (8)-(11) are approximate treatments of formula (7). When formula (7) is substituted into the single-link flexible manipulator dynamic equation (6), the single-link flexible manipulator system model with actuator saturation is represented.

[0038] Preferably, the method of estimating the unknown nonlinear design radial basis function neural network in the dynamic equation of the single-link flexible manipulator according to the state variables of the single-link manipulator system and constructing the radial basis function neural network includes:

[0039] According to the state variables of the single-link manipulator system, the unknown nonlinear radial basis function neural network in the dynamic equation of the single-link flexible manipulator is estimated, and the following radial basis function neural network is constructed:

[0040]

[0041] Where S(x)=[s1(x),…,s n (x)] T for Basis function, where b i =[b i1 ,…,b iq ] T is the center of the Gaussian function, Q is the width of the Gaussian function, W * is the ideal NN weight vector, which is in the form of

[0042] The radial basis function neural network allows Approximate an unknown continuous function f(x) on the surface.

[0043] Preferably, the designing of a disturbance observer based on the single-link flexible manipulator system with actuator saturation and estimating external disturbances in the single-link manipulator system using the disturbance observer comprises:

[0044] S4.1. For the dynamic equation of the single-link flexible manipulator shown in Equation (6), let z1 = QQ d , Where μ is the imaginary control, we get:

[0045]

[0046]

[0047] Since the system (13) contains uncertainty in M ​​and Q, neural networks can approximate the unknown function. Using radial basis function neural networks (RBFNNs), the uncertainty term is expressed as:

[0048] P(Z)=-M -1 Cz1-M -1 CQ d (14)

[0049] get:

[0050] P(Z)=W * S(Z)+ξ(Z) (15)

[0051] Where W * is the ideal weight, ξ(Z) is the error of the radial basis function;

[0052] Formula (14) and Formula (15), Formula (13) is rewritten as:

[0053]

[0054]

[0055] Where H0 is the known constant matrix, ΔH is the error matrix, H0 = H-ΔH, H = M -1 , d = Hτ d +ΔHτ+ξ;

[0056] The sliding surface of the adaptive finite-time disturbance observer is defined as s = z2 - η, where η satisfies the following auxiliary dynamics:

[0057]

[0058] Where, υ=λ1sgn(s), and d and W respectively * The estimated value of , the estimation error is defined as: From (16a) to (17) we have:

[0059]

[0060] definition:

[0061]

[0062] Taking the derivative of V1 we get:

[0063]

[0064] When the design parameters When the sliding variables s and In a finite time T f1 Converges to 0. Let When σ0>0 is the design parameter, then Where V1(0) is the initial value of V1;

[0065] s and It converges to 0 in a finite time. According to the principle of equivalent output injection, it can be known from formula (18) that Equivalent to υ-ξ, that is

[0066] The designed adaptive finite-time disturbance observer is as follows:

[0067]

[0068]

[0069] Among them, λ2 and λ3 are design parameters, is an estimate of the upper bound of the derivative of d, is the derivative of the virtual control, is the estimate of the weight, is the estimate of interference;

[0070] The adaptive law is designed as:

[0071]

[0072] definition According to formula (21), we can get:

[0073]

[0074] design:

[0075]

[0076] Under the adaptive finite-time disturbance observer shown in Equation (21), the auxiliary system shown in Equation (18), and the adaptive estimation law shown in Equation (22), the disturbance d in the system shown in Equation (16) will be f2 It is estimated that when t>T f2 When the error signal and will converge to a compact set Defined as:

[0077]

[0078]

[0079] Where Q = 2(α / (1-θ1)K1) 1 / l .

[0080]

[0081] Among them, α=λ2(‖v‖‖ξ‖+‖ξ‖ 2 )+λ3‖ξ‖, 0<l<1, 0<θ<1, V(x(0)) is the initial value of V(x(t)).

[0082] Preferably, the composite anti-interference controller is constructed based on the output of the radial basis function neural network and the output of the adaptive finite-time disturbance observer, and the composite anti-interference controller is used to control the single-link flexible manipulator system with actuator saturation, and output the angle position and velocity signal of each joint of the single-link flexible manipulator system, including:

[0083] The potential barrier Lyapunov function BLF is designed as follows:

[0084]

[0085] where k b >0 is the design function used to implement the constraint on the amplitude of z1, indicating k b =ae -δT +b, then Among them, δ, a, and b are positive numbers;

[0086] The virtual control is designed as follows:

[0087]

[0088] in is the derivative of the tracking signal, 0<l<1, k1>0 is the constant parameter of the design, based on the virtual controller shown in formula (27), the following composite anti-interference controller is designed

[0089] u=u f +u s (28)

[0090] in:

[0091]

[0092]

[0093] Where k2>0 is a constant parameter of the design, according to Λ, Λ max =(π / 2) 2 / (1+(π / 2) 2 );

[0094] The adaptive law Designed to:

[0095]

[0096] definition:

[0097]

[0098] definition:

[0099]

[0100] Define L=L1+L2+L3, and analyze and organize to get in δ1,r is the intermediate variable of the design;

[0101] The composite anti-interference controller of the flexible link manipulator system integrates the virtual controller (29) shown in formula (29), the neural network weight update law shown in formula (31) and the adaptive finite-time disturbance observer shown in formula (21). The composite anti-interference controller is used to control the single-link flexible manipulator system with actuator saturation. If the initial condition Bounded, q(0) satisfies z1∈(-k b ,k b ), the composite anti-interference controller is used in a finite time T f3 Internal stability, when t>T f3 When the error signals z1, z2 and Converge to compact sets and

[0102]

[0103]

[0104]

[0105] Where Q2 = (β / (1-θ1)K2) l ,λ min (Λ -1 ) is Λ -1 Minimum eigenvalue;

[0106] The composite anti-interference controller outputs the angle position and speed signal of each joint of the single-link flexible robotic arm system.

[0107] It can be seen from the technical solutions provided by the above embodiments of the present invention that the present invention proposes a composite learning anti-interference control strategy of an adaptive finite-time disturbance observer, which effectively solves the adverse effects of actuator saturation while finely compensating for disturbances.

[0108] Additional aspects and advantages of the present invention will be set forth in part in the following description, will become apparent from the following description, or may be learned by practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0109] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0110] Figure 1 A flowchart of a flexible link manipulator control method based on an adaptive finite-time disturbance observer provided in an embodiment of the present invention;

[0111] Figure 2 A schematic structural diagram of a single-link robotic arm provided in an embodiment of the present invention;

[0112] Figure 3 A schematic diagram of parameters of a flexible robotic arm provided by an embodiment of the present invention;

[0113] Figure 4 A schematic diagram of the AFTDO estimation performance in an adaptive finite-time disturbance observer composite learning anti-disturbance control method based on unknown disturbances of a flexible link manipulator provided by an embodiment of the present invention;

[0114] Figure 5 A schematic diagram of the time response of z1 to BLF in an adaptive finite-time disturbance observer composite learning anti-disturbance control method based on an unknown disturbance of a flexible link manipulator provided by an embodiment of the present invention;

[0115] Figure 6 A schematic diagram of the control torque τ in an adaptive finite-time disturbance observer composite learning anti-disturbance control method based on unknown disturbances of a flexible link manipulator provided by an embodiment of the present invention;

[0116] Figure 7 Comparison of the following: (a) tracking performance; (b) tracking error diagram in an adaptive finite-time disturbance observer composite learning anti-interference control method for an unknown disturbance of a flexible link manipulator provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0117] The embodiments of the present invention are described in detail below, examples of which are shown in the accompanying drawings, wherein the same or similar reference numerals throughout represent the same or similar elements or elements having the same or similar functions. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and are not to be construed as limiting the present invention.

[0118] It will be understood by those skilled in the art that, unless expressly stated otherwise, the singular forms "a", "an", "said" and "the" used herein may also include the plural forms. It should be further understood that the term "comprising" used in the description of the present invention refers to the presence of the features, integers, steps, operations, elements and / or components, but does not exclude the presence or addition of one or more other features, integers, steps, operations, elements, components and / or groups thereof. It should be understood that when we refer to an element as being "connected" or "coupled" to another element, it may be directly connected or coupled to the other element, or there may be intermediate elements. In addition, "connected" or "coupled" as used herein may include wireless connections or couplings. The term "and / or" used herein includes any unit and all combinations of one or more associated listed items.

[0119] It will be understood by those skilled in the art that, unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by those skilled in the art in the art to which the present invention pertains. It should also be understood that terms such as those defined in common dictionaries should be understood to have meanings consistent with their meanings in the context of the prior art and, unless defined as such herein, will not be interpreted in an idealized or overly formal sense.

[0120] To facilitate understanding of the embodiments of the present invention, several specific embodiments will be further explained below with reference to the accompanying drawings, and each embodiment does not constitute a limitation on the embodiments of the present invention.

[0121] In order to improve the anti-interference ability of the flexible link manipulator control system, the processing flow of a flexible link manipulator control method based on an adaptive finite time disturbance observer provided in this embodiment is as follows: Figure 1 As shown, the following steps are included:

[0122] Step S1, establishing a single-link flexible robotic arm dynamic equation according to the structure of the single-link robotic arm system;

[0123] The above-mentioned dynamic equations for the single-link flexible manipulator include unknown nonlinearities in the radial basis function neural network. The input data for the dynamic equations are the tracking signal, the system state, and the disturbance observer estimate. The dynamic equations for the single-link flexible manipulator serve as a control-oriented model for the manipulator and are the control targets of the controller designed in the subsequent steps. Specifically, the controller is the input to the dynamic equations, which in turn include the nonlinear saturated tracking signal, the system state, and the disturbance observer estimate.

[0124] Step S2: Based on the inevitable input saturation problem in the actuator, mathematical modeling is performed on the nonlinear saturation tracking signal in the input control in the dynamic equation of the single-link flexible manipulator, and a single-link flexible manipulator system model with actuator saturation is constructed.

[0125] Step S3: According to the state variables of the single-link flexible manipulator system, the unknown nonlinear radial basis function neural network in the power equation of the single-link flexible manipulator is estimated to construct a radial basis function neural network (RBFNN).

[0126] Step S4: designing an adaptive finite-time disturbance observer based on the single-link flexible manipulator system with actuator saturation, and using the adaptive finite-time disturbance observer to estimate external disturbances in the single-link manipulator system, wherein the output data in S3 is required for the estimation.

[0127] Step S5: Construct a composite anti-interference controller based on the output of the radial basis function neural network and the output of the adaptive finite-time disturbance observer. Simulations of the composite anti-interference controller demonstrate that it can further improve the anti-interference capability of controlling a single-link flexible robotic arm system. The composite anti-interference controller is then used to control a single-link flexible robotic arm system with actuator saturation, outputting the angular position and velocity signals for each joint of the system.

[0128] Specifically, step S1 includes: the single-link flexible manipulator is considered as an Euler-Bernoulli beam rotating around a horizontal plane. Based on the energy equation and Hamilton's principle, the boundary conditions and control equations can be obtained:

[0129] ρω″′(L,t)+m t ω″′(L,t)=0 (1)

[0130] ω(0,t)=ω′(0,t)=ω″(L,t)=0 (2)

[0131]

[0132]

[0133] Where L is the length of the flexible link; ρ is the average mass per unit length; I h is the central inertia; M is the total mass of the connecting rod; m t is the tip mass; EI is the bending stiffness; θ(t) represents the central angular position; τ(t) is the input torque; ω(x, t) represents the elastic deformation, and ω″ represents the secondary derivative of the flexibility variable.

[0134] It can be expressed as follows using the Assume mode method (AMM):

[0135]

[0136] Among them, φ i (x) represents the model function, p i (t) is the generalized coordinate. Under the premise of small deflection, the dynamic equation of the single-link flexible manipulator derived from the Lagrange equation and AMM is:

[0137]

[0138] Where Q is defined as Q=[θ,p1,…p N ] T , τ d Represents external disturbance, and the matrix M is positive definite and symmetric.

[0139] Specifically, the above step S2 includes describing the actuator τ with nonlinear saturation in the dynamic equation of the single-link flexible manipulator with the following formula:

[0140]

[0141] Where u(t)∈R is the control law to be designed, u b is the saturation boundary. Since the control input τ is u(t)=u b The saturation of the actuator is approximated as a smooth function as shown below:

[0142]

[0143] Defining Δτ as the approximation error, we get:

[0144] τ=g(u)+Δτ (9)

[0145] According to the mean value theorem, g(u) can be expressed as:

[0146]

[0147] Where 0<μ<1, g(0) is the initial value of g(u). Define Λ=(πμu / 2u b ) 2 / (1+πμu / 2u b ) 2

[0148] Then g(u) can be rewritten as:

[0149] g(u)=u-Λu (11)

[0150] Formula (7) represents the actuator saturation, where u(t) is the input of the actuator, τ is the output of the actuator, and τ is the input of the single-link flexible manipulator dynamic equation (6). Formulas (8)-(11) are approximate treatments of formula (7). When formula (7) is substituted into the single-link flexible manipulator dynamic equation (6), the single-link flexible manipulator system model with actuator saturation is represented.

[0151] Specifically, the above step S3 includes: Radial Basis Function Neural Network (RBFNN) is a powerful tool for modeling control systems, allowing Approximate the unknown continuous function f(x). The form is as follows:

[0152]

[0153] Where S(x)=[s1(x),…,s n (x)] T for Basis function, where b i =[b i1 ,…,b iq ] T is the center of the Gaussian function, and Q is the width of the Gaussian function. * is the ideal NN weight vector, which is in the form of

[0154] Specifically, the above step S4 includes the following sub-steps:

[0155] S4.1. For system (6), let z1 = QQ d , Where μ is the virtual control. We get:

[0156]

[0157]

[0158] Since system (13) contains uncertainty in M ​​and Q, neural networks can approximate the unknown function. Using RBFNNs (Radial Basis Function Neural Networks), the uncertainty term can be expressed as:

[0159] P(Z)=-M -1 Cz1-M -1 CQ d (14)

[0160] get:

[0161] P(Z)=W * S(Z)+ξ(Z) (15)

[0162] Where W * is the ideal weight, ξ(Z) is the error of the radial basis function.

[0163] Therefore, from equations (14) and (15), equation (13) can be written as:

[0164]

[0165]

[0166] Where H0 is the known constant matrix, ΔH is the error matrix, H0 = H-ΔH, H = M -1 , d = Hτ d +ΔHτ+ξ.

[0167] S4.2. Next, we design the adaptive finite-time disturbance observer in detail and define the sliding surface s = z2 - η, where η satisfies the following auxiliary dynamics:

[0168]

[0169] Where, υ=λ1sgn(s), and d and W respectively * The estimated value of . The estimation error is defined as: From (16a) to (17) we have:

[0170]

[0171] definition:

[0172]

[0173] Taking the derivative of V1 we get:

[0174]

[0175] When the design parameters When the sliding variables s and In a finite time T f1 Converges to 0. Let When σ0>0 is the design parameter, then Where V1(0) is the initial value of V1.

[0176] After that, s and Converges to 0 in a finite time. According to the principle of equivalent output injection, it can be seen from (18) that Equivalent to υ-ξ, that is

[0177] The designed adaptive finite-time disturbance observer is as follows:

[0178]

[0179]

[0180] where λ2, λ3 are design parameters, is an estimate of the upper bound of the derivative of d, is the derivative of the virtual control, is an estimate of the weight, is an estimate of the disturbance.

[0181] The adaptive law is designed as:

[0182]

[0183] Definition According to (21), we have:

[0184]

[0185] Design:

[0186]

[0187] Under the disturbance observer (21), the auxiliary system (18) and the adaptive estimation law (22), the disturbance d in the system (16) will be estimated in a finite time T f2 shown in (25). When t > T f2 , the error signals and will converge to the compact set defined as:

[0188]

[0189]

[0190] where Q = 2 (α / (1 - θ1) K1) 1 / l .

[0191]

[0192] where α = λ2 (‖v‖‖ξ‖ + ‖ξ‖ 2 ) + λ3‖ξ‖, 0 < l < 1, 0 < θ < 1, V(x(0)) is the initial value of V(x(t)).

[0193] Specifically, the above step S5 includes: in order to achieve high-precision position control, a barrier Lyapunov function (BLF, barrier Lyapunov function) is designed as follows:

[0194]

[0195] where k b >0 is a design function used to implement the constraint on the amplitude of z1. b =ae -δT +b, then Where δ, a, and b are positive numbers.

[0196] The virtual control is designed as follows:

[0197]

[0198] in is the derivative of the tracking signal, 0<l<1, k1>0 is the constant parameter of the band design. Based on the virtual controller (27), we designed the following composite anti-interference controller

[0199] u=u f +u s (28)

[0200] in:

[0201]

[0202]

[0203] Where k2>0 is a constant parameter of the design, according to Λ, Λ max =(π / 2) 2 / (1+(π / 2) 2 ).

[0204] Adaptive Law Proposed as:

[0205]

[0206] definition:

[0207]

[0208] definition:

[0209]

[0210] Define L=L1+L2+L3. in δ1,r is the intermediate variable of the design.

[0211] For the controller (28) of the flexible link manipulator system with the virtual controller (29), the neural network weight update law (31) and AFTDO (21), if the initial condition Bounded, q(0) satisfies z1∈(-k b ,k b ). Then, the closed-loop system is f3 Internal stability. When t>T f3 When the error signals z1, z2 and Converge to compact sets and

[0212]

[0213]

[0214]

[0215] Where Q2 = (β / (1-θ1)K2) l ,λ min (Λ -1 ) is Λ -1 Minimum eigenvalue.

[0216] Next, in order to verify the effectiveness of the adaptive finite-time disturbance observer composite learning anti-disturbance control method based on the unknown disturbance of the flexible link manipulator provided in this embodiment, a simulation experiment is carried out using MATLAB and a detailed description is given.

[0217] The single-link flexible manipulator model provided in this embodiment comprehensively considers the effects of unknown external disturbances, modeling uncertainty, input saturation, and output constraints on the vibration suppression and precise positioning performance of the flexible-link manipulator. The proposed finite-time composite learning control scheme based on an adaptive finite-time disturbance observer (AFTDO) can estimate the unknown upper bound of the disturbance derivative according to the adaptive law, eliminating the requirement for prior knowledge of the disturbance derivative, effectively ensuring the accuracy and timeliness of the disturbance estimation, and improving the robust control performance of the closed-loop system.

[0218] In the simulation experiment, the number of flexible modes designed is 2, and the reference signal of the joint angle is Q d = sin(t), unknown external disturbance τ d = 0.5sin(0.1t), the uncertainty of the system is ΔH = 0.05H0, and C is unknown to the controller. In order to estimate the disturbance d, the parameters of AFTDO (24) are designed as λ0 = 20; λ1 = 20; λ2 = 1; λ3 = 10; and the adaptive update law parameter is δ0 = 2. The output constraint boundary is selected as k d =0.03+0.097e -t , the actuator saturation is u d=5, the design parameters of the finite time controller (38) are k1 = 5; k2 = 5; the weight update law parameters are δ1 = 2; Λ = 0.25. The initial state of the error system is (0.1, 0.1), and the initial value of the weight is set to 0.

[0219] The estimated effect of the perturbation d is as follows Figure 2 As shown in Figure 2, it shows that the designed AFTDO can accurately estimate the disturbance without the need for the upper bound knowledge of the disturbance. The angle tracking error curve is shown in Figure 2. Figure 3 As shown, the error z1 does not violate the boundary conditions and converges quickly under the finite time controller. Figure 4 It can be seen that the maximum control torque is less than the input saturation of 5Nm. Figure 5 A comparison of the angular tracking performance of a flexible-link manipulator using different methods is presented. These include the proposed AFTDO finite-time controller with system lumped disturbance estimation (AFTDO+FTC) and the fuzzy neural network controller (FNNC), "AFTDO+FTNNC." By combining these two disturbance estimation and suppression methods, the proposed controller exhibits superior anti-disturbance performance and trajectory tracking accuracy.

[0220] Figure 6 A schematic diagram of the control torque τ in a composite learning anti-interference control method based on an adaptive finite-time disturbance observer for an unknown disturbance of a flexible link manipulator provided by an embodiment of the present invention. Figure 7 Comparison diagram of the composite learning anti-interference control method of the adaptive finite-time disturbance observer: (a) tracking performance; (b) tracking error diagram.

[0221] The above analysis proves the effectiveness of the adaptive finite-time disturbance observer composite learning anti-disturbance control method strategy based on the unknown disturbance of the flexible link manipulator provided in this embodiment.

[0222] In summary, the embodiments of the present invention eliminate the requirement for prior knowledge of disturbance derivatives and only require that the disturbance derivatives be bounded; introduce an adaptive law to estimate the unknown bounds, effectively ensure the accuracy and timeliness of disturbance estimation, and improve the robust control performance of the closed-loop system; while solving the adverse effects of actuator saturation, finely compensate for the disturbance.

[0223] Those skilled in the art will appreciate that the accompanying drawings are merely schematic diagrams of an embodiment, and the modules or processes in the accompanying drawings are not necessarily required to implement the present invention.

[0224] From the above description of the embodiments, it can be seen that those skilled in the art can clearly understand that the present invention can be implemented by means of software plus the necessary general-purpose hardware platform. Based on this understanding, the technical solution of the present invention, or the portion that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a storage medium such as ROM / RAM, a magnetic disk, or an optical disk, and includes a number of instructions for enabling a computer device (which can be a personal computer, a server, or a network device, etc.) to execute the methods described in various embodiments of the present invention or certain parts of the embodiments.

[0225] Each embodiment in this specification is described in a progressive manner. The same or similar parts between the embodiments can be referred to each other. Each embodiment focuses on the differences from other embodiments. In particular, for the device or system embodiments, since they are basically similar to the method embodiments, the description is relatively simple. For the relevant parts, refer to the partial description of the method embodiments. The device and system embodiments described above are merely schematic, wherein the units described as separate components may or may not be physically separated, and the components displayed as units may or may not be physical units, that is, they may be located in one place, or they may be distributed on multiple network units. Some or all of the modules can be selected according to actual needs to achieve the purpose of the scheme of this embodiment. A person of ordinary skill in the art can understand and implement it without making any creative efforts.

[0226] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by a person skilled in the art within the technical scope disclosed in the present invention should be included in the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be based on the scope of protection of the claims.

Claims

1. A flexible link manipulator control method based on an adaptive finite-time disturbance observer, characterized in that: include: According to the structure of the single-link flexible manipulator system, the dynamic equation of the single-link flexible manipulator including the unknown nonlinear design radial basis function neural network is established; According to the inevitable input saturation problem in the actuator, a mathematical model is performed on the nonlinear saturation tracking signal in the input control in the dynamic equation of the single-link flexible manipulator, and a single-link flexible manipulator system model with actuator saturation is constructed; According to the state variables of the single-link flexible manipulator system, an unknown nonlinear radial basis function neural network is estimated in the dynamic equation of the single-link flexible manipulator, and a radial basis function neural network is constructed; designing an adaptive finite-time disturbance observer based on the single-link flexible manipulator system with actuator saturation, and estimating external disturbances in the single-link flexible manipulator system using the adaptive finite-time disturbance observer; constructing a composite anti-interference controller based on the output of the radial basis function neural network and the output of the adaptive finite-time disturbance observer, controlling a single-link flexible manipulator system with actuator saturation using the composite anti-interference controller, and outputting angle position and velocity signals of each joint of the single-link flexible manipulator system; The method of designing an adaptive finite-time disturbance observer based on the single-link flexible manipulator system with actuator saturation and estimating external disturbances in the single-link flexible manipulator system using the adaptive finite-time disturbance observer includes: S4.

1. For the dynamic equation of the single-link flexible manipulator shown in equation (6), Let z1 = QQ d , Where μ is the imaginary control, we get: Since Equations (13a) and (13b) contain uncertainty in M ​​and Q, neural networks can approximate unknown functions. Using radial basis function neural networks (RBFNNs), the uncertainty term is expressed as: P(Z)=-M -1 Cz1-M -1 CQ d (14) get: Where W * is the ideal neural network weight, ξ is the error of the radial basis function; From equations (14) and (15), equations (13a) and (13b) can be rewritten as: Where H0 is the known constant matrix, ΔH is the error matrix, H0 = H-ΔH, H = M -1 , d = Hτ d +ΔHτ+ξ; The sliding surface of the adaptive finite-time disturbance observer is defined as s = z2 - η, where η satisfies the following auxiliary dynamics: Where, υ=λ1sgn(s), and d and W respectively * The estimated value of , the estimation error is defined as: From (16a), (16b) to (17), we have: definition: Taking the derivative of V1 we get: When the design parameters When the sliding variables s and In a finite time T f1 Converges to 0, let σ0>0 is the design parameter, then Where V1(0) is the initial value of V1; s and It converges to 0 in a finite time. According to the principle of equivalent output injection, it can be known from formula (18) that Equivalent to υ-ξ, that is The designed adaptive finite-time disturbance observer is as follows: Among them, λ2 and λ3 are design parameters, is an estimate of the upper bound of the derivative of d, is the derivative of the virtual control, is the estimate of the weight, is the estimate of interference; The adaptive law is designed as: definition According to formula (21a) and formula (21b), we can get: design: Under the adaptive finite-time disturbance observers shown in Equations (21a) and (21b), the auxiliary system shown in Equation (18), and the adaptive law shown in Equation (22), the disturbance d in the system shown in Equation (16b) will be f2 It is estimated that when t>T f2 When the error signal and will converge to a compact set Defined as: Where x1=2(α / (1―θ1)K1) 1 / l ; Where, α=λ2(‖V2‖‖ξ‖+‖ξ‖ 2 )+λ3‖ξ‖,0<l<1,0<θ1<1,V2(0) is the initial value of V2; The single-link flexible manipulator system structure is used to establish a single-link flexible manipulator dynamic equation including an unknown nonlinear design radial basis function neural network, including: The single-link flexible manipulator system is considered as an Euler-Bernoulli beam rotating around the horizontal plane. According to the energy equation and Hamilton's principle, the boundary conditions and control equations of the single-link flexible manipulator system including the unknown nonlinear design radial basis function neural network are obtained: ρω″′(L,t)+m t ω″′(L,t)=0(1) ω(0,t)=ω′(0,t)=ω″(L,t)=0(2) Where L is the length of the flexible link, ρ is the average mass per unit length, and I h is the central inertia, m t is the tip mass, EI is the bending stiffness; θ(t) represents the central angular position; τ(t) is the input torque; ω(x, t) represents the elastic deformation, and ω″ represents the secondary derivative of the flexibility variable; Using the assumed modal method, ω(x,t) is expressed as: Among them, φ i (x) represents the model function, p i (t) is the generalized coordinate. Under the premise of small deflection, the dynamic equation of the single-link flexible manipulator derived from the Lagrange equation and the assumed modal method is: Where Q is defined as Q=[θ,p1,…p N ] T , τ d represents external disturbance, and the matrix M is positive definite and symmetric; According to the inevitable input saturation problem in the actuator, a mathematical model is performed on the nonlinear saturation tracking signal in the input control in the dynamic equation of the single-link flexible manipulator, and a single-link flexible manipulator system model with actuator saturation is constructed, including: The dynamic equation of the single-link flexible manipulator contains a nonlinear saturation tracking signal τ(t), which is described by the following formula: Where u(t)∈R is the control law to be designed, u b is the saturation boundary, the nonlinear saturation tracking signal τ(t) is at u(t) = u b The saturation of the actuator is approximated as a smooth function as shown below: Defining Δτ as the approximation error, we get: τ(t)=g(u)+Δτ (9) According to the mean value theorem, g(u) is expressed as: where \(0 < \nu < 1\), \(g(0)\) is the initial value of \(g(u)\), and \(\Lambda=\frac{\frac{\pi\nu u}{2u}}{1 + \frac{\pi\nu u}{2u}}\) b ) 2 / (1+πνu / 2u b ) 2 Then g(u) can be rewritten as: g(u)=u-Λu (11) Formula (7) represents the actuator saturation, where u(t) is the input of the actuator, τ(t) is the output of the actuator, and τ(t) is the input of the single-link flexible manipulator dynamic equation (6). Formulas (8)-(11) are approximate treatments of formula (7). When formula (7) is substituted into the single-link flexible manipulator dynamic equation (6), the single-link flexible manipulator system model with actuator saturation is represented. The method of estimating the unknown nonlinear radial basis function neural network in the dynamic equation of the single-link flexible manipulator according to the state variables of the single-link flexible manipulator system and constructing the radial basis function neural network includes: According to the state variables of the single-link flexible manipulator system, the unknown nonlinear radial basis function neural network in the dynamic equation of the single-link flexible manipulator is estimated, and the following radial basis function neural network is constructed: Where S(Z)=[s1(Z),…,s n (Z)] T for Basis functions with 1≤i≤n, where b i =[b i1 ,…,b iq ] T is the center of the Gaussian function, W * is the ideal neural network weight, which is of the form The radial basis function neural network allows Approximate the unknown continuous function f(Z) on the surface.

2. The method according to claim 1, characterized in that The method comprises: constructing a composite anti-interference controller based on the output of the radial basis function neural network and the output of the adaptive finite-time disturbance observer, controlling a single-link flexible manipulator system with actuator saturation by using the composite anti-interference controller, and outputting the angle position and velocity signals of each joint of the single-link flexible manipulator system. The potential barrier Lyapunov function BLF is designed as follows: where k b >0 is the design function used to implement the constraint on the amplitude of z1, indicating k b =ae ―δt +b, then Among them, δ, a, and b are positive numbers; The virtual controller is designed as follows: in is the derivative of the tracking signal, 0<l<1, k1>0 is the constant parameter to be designed. Based on the virtual controller shown in formula (27), the following composite anti-interference controller is designed: in=in f +in s (28) in: Where k2>0 is the constant parameter to be designed, according to Λ, Λ max =(π / 2) 2 / (1+(π / 2) 2 ); Adaptive Law Designed to: definition: definition: Define L=L1+L2+L3, and analyze and organize to get in δ1,r is the intermediate variable of the design; The composite anti-interference controller of the single-link flexible manipulator system integrates the virtual controller shown in Equation (27), the adaptive law shown in Equation (31), and the adaptive finite-time disturbance observers shown in Equations (21a) and (21b). The composite anti-interference controller is used to control the single-link flexible manipulator system with actuator saturation. If the initial condition is Bounded, q(0) satisfies z1∈(-k b ,k b ), the composite anti-interference controller is used in a finite time T f3 Internal stability, when t>T f3 When the error signals z1, z2 and Converge to compact sets and Where Q2 = (β / (1-θ1)K2) l ,λ min (Λ -1 ) is Λ -1 Minimum eigenvalue; The composite anti-interference controller outputs the angle position and speed signal of each joint of the single-link flexible robotic arm system.

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