An adaptive friction compensation feedback control method for multi-axis electro-hydraulic system

Abstract

This invention proposes an adaptive friction compensation feedback control method for multi-axis electro-hydraulic systems, comprising: establishing a dynamic model of a hydraulic leveling system based on friction compensation; decoupling the motion relationship in the overdrive coupled leveling system from a four-input two-output model to a two-input two-output model based on the dynamic model, and constructing a dynamic model of the moving beam in the decoupled coordinate system; designing a time-varying parameter update law with a leakage coefficient based on model error and adaptive gain adjustment to obtain estimated parameters of the leveling cylinder friction model; and designing direct and indirect adaptive disturbance rejection controllers for outer-loop displacement control and inner-loop voltage control based on the designed time-varying parameter update law and adaptive gain adjustment, realizing parallel online parameter estimation and precision control. This invention can effectively suppress creep and oscillation in the leveling of a multi-axis hydraulic press leveling control system, significantly improving the smoothness and accuracy of operation.

Description

Technical Field

[0001] This invention belongs to the field of hydraulic control systems, and specifically relates to an adaptive friction compensation feedback control method for multi-axis electro-hydraulic systems. Background Technology

[0002] Hydraulic presses are widely used in processes such as composite material molding, sheet metal stamping, and precision pressing. Especially in the hot pressing of composite materials, hydraulic presses require multiple hydraulic cylinders to work in tandem to maintain high-precision synchronization and stable movement of the press head during the pressing process, ensuring uniform product thickness, dense internal structure, and consistent fiber layering. However, during the low-speed micro-motion stage of the hydraulic cylinders, due to the significant nonlinear frictional characteristics of the hydraulic system, such as static friction and frictional hysteresis caused by the Stribeck effect, the system is prone to creeping, leading to discontinuous press head movement, lag in displacement response, and sudden speed changes. This results in uneven thickness of the pressed parts, interlayer voids, and molding defects, severely affecting product quality and production stability. Existing hydraulic press leveling control methods typically employ proportional-integral-derivative (PID) control or traditional adaptive control methods. PID control algorithms are simple in structure, but their adaptability to system nonlinear characteristics and friction disturbances is weak, making it difficult to effectively eliminate crawling phenomena in low-speed regions. Adaptive control can adjust the control law online according to changes in system parameters, but its effectiveness in friction compensation is limited, especially when friction characteristics change with load, temperature, and speed, easily leading to parameter drift and increased steady-state error. Furthermore, traditional static friction models cannot fully represent the friction state, while using dynamic friction models alone results in limited compensation accuracy under external disturbances. Therefore, a new control method is needed to achieve stable operation and precise control under low-speed micro-motion conditions. Referring to Chinese Patent Application No. CN202210222556.X, a friction compensation control method for an electro-hydraulic servo system is proposed. This method uses a dynamic friction model for friction modeling, combines offline identification and feedforward compensation, and introduces nonlinear state error feedback and adaptive / fuzzy adjustment to improve tracking accuracy and disturbance rejection capability. Referring to Chinese Patent Application No. CN202410799851.0, an adaptive control method for a multi-cylinder leveling system of a hydraulic press based on parallel dual valves is proposed. This method introduces a structure where dual valves work together to control the leveling cylinders, and designs an adaptive disturbance rejection controller to mitigate interference from excess force impacts in the hydraulic press and estimate unknown off-center loads, thereby achieving high-precision synchronous control. While existing patents can achieve friction compensation in electro-hydraulic systems, they still have some shortcomings and limitations, requiring further improvement.

[0003] (1) Existing hydraulic press control methods mostly adopt linear control and traditional adaptive control. They are not sensitive enough in terms of friction compensation, are easily affected by load and speed changes, and are difficult to solve the crawling phenomenon and accuracy problems.

[0004] (2) Most friction modeling methods use static models, which are difficult to accurately describe the dynamic friction characteristics of hydraulic press operation. While dynamic models such as LuGre can characterize the micro-friction dynamic process, they lack the ability to estimate and accurately compensate the linear parameters of the friction model. Summary of the Invention

[0005] The purpose of this invention is to propose an adaptive friction compensation feedback control method for multi-axis electro-hydraulic systems. This method uses a nonlinear dynamic friction model and estimates and compensates for the friction term, thereby suppressing low-speed crawling phenomena.

[0006] To achieve the above objectives, the technical solution of the present invention is: an adaptive friction compensation feedback control method for multi-axis electro-hydraulic systems, comprising the following steps:

[0007] A dynamic model of a hydraulic leveling system based on friction compensation is established, including a dynamic model of the leveling cylinder and a dynamic model of the moving beam.

[0008] Based on the established dynamic model, the motion relationship in the overdriven coupled leveling system is decoupled from a four-input two-output model to a two-input two-output model, and a dynamic model of the moving beam is constructed in the decoupled coordinate system.

[0009] Based on the dynamic model of the moving beam in the decoupled coordinate system, a time-varying parameter update law with leakage coefficient based on model error and adaptive gain adjustment are designed to obtain the estimated parameters of the leveling cylinder friction model.

[0010] Based on the designed time-varying parameter update law and adaptive gain adjustment, a direct and indirect adaptive disturbance rejection controller is designed. The controller adopts a dual closed-loop structure of outer-loop displacement control and inner-loop voltage control to achieve parallel online parameter estimation and precision control.

[0011] Preferably, the specific implementation method for establishing the dynamic model of the hydraulic leveling system based on friction compensation is as follows:

[0012] The hydraulic leveling system includes a movable beam, four leveling cylinders, and a servo proportional valve. During the descent of the movable beam, the four leveling cylinders act on the four corners of the movable beam. The small chamber of the leveling cylinder is controlled at a constant pressure by a pressure reducing valve, and the large chamber of the leveling cylinder is controlled by a servo proportional valve to output force. When the movable beam falls, the pressure, the pressure controlled by the large chamber, and the constant pressure of the small chamber interact to achieve passive leveling of the movable beam.

[0013] Regarding the hydraulic press leveling system, the following assumptions are made:

[0014] (1) The system oil supply pressure is a constant value, denoted as Ps, and the hydraulic pump output pressure does not change with time, and the return oil pressure is approximately zero;

[0015] (2) Ignore the pressure loss along the pipeline and the local resistance loss of hydraulic oil. Assume that hydraulic oil is an isothermal, incompressible Newtonian fluid with constant density and viscosity. At the same time, do not consider the effect of oil temperature change on system performance.

[0016] (3) Both the servo proportional valve and the flow compensation valve are ideal zero-opening four-sided slide valves, and each throttling port is symmetrically matched.

[0017] (4) The leveling cylinders are arranged symmetrically, and the cylinder movement occurs only in the vertical direction. Ignoring the coupling deformation between the lateral oscillation and the cylinder, the load can be regarded as a rigid body with a constant moment of inertia.

[0018] The dynamic mathematical model of a single leveling cylinder is as follows:

[0019] (1)

[0020] In the formula, , This represents the pressure values ​​of the large chamber and small chamber of the leveling cylinder i. This indicates the effective volumetric elastic modulus of the leveling cylinder; Indicates the volume of the leveling cylinder; Indicates the area of ​​the large cavity of the leveling cylinder; Indicates the area of ​​the small cavity of the leveling cylinder; This indicates the piston displacement of leveling cylinder number i; This indicates the pressure flowing into the large chamber of the i-th leveling cylinder; Indicates the leakage coefficient; Indicates the flow coefficient; This represents the area gradient of the servo proportional valve. This indicates the spool displacement of the servo proportional valve. Indicates the density of hydraulic oil; Indicates the pump source pressure; Indicates the external load force of the leveling cylinder; Express the total mass of the piston and other loads; Indicates the viscous damping coefficient; Indicates the spring stiffness; where It is a symbolic function;

[0021] With the geometric center of the movable beam as the origin, the longer side as the x-direction, the wider side as the y-direction, and the direction perpendicular to the movable beam as the z-direction, the movable beam rotates about the x and y directions during its descent. The dynamic equations of the movable beam are then expressed as:

[0022] (2)

[0023] In the formula, Indicates the mass of the movable beam; This represents the initial displacement of the movable beam in the z-direction; This represents the driving force of the movable beam in the z-direction; Represents gravitational acceleration; These represent the output forces of the four leveling cylinders; This represents the frictional force during the movement of the four leveling cylinders; , These represent the moments of inertia of the movable beam about the x and y directions, respectively. , These represent the deflection angles of the movable beam about the x and y axes, respectively. , These represent half the length of the movable beam in the x and y directions, respectively. , These represent the load moments of the movable beam about the x and y axes, respectively.

[0024] Among them, the friction force during the movement of the i-th leveling cylinder The LuGre friction model is used to represent:

[0025] (3)

[0026] In the formula, Indicates the bristle stiffness coefficient; This indicates the average deformation of the bristles; Indicates the bristle damping coefficient; Indicates the viscous damping coefficient; This represents the Stribeck curve function, which describes the relationship between steady-state frictional characteristics and velocity. Represents Coulomb friction; This represents static friction. Indicates Stribeck's velocity;

[0027] The following geometric relationship exists between the deflection angle of the movable beam and the piston displacement of each leveling cylinder:

[0028] (4)

[0029] in, These are the piston displacement vectors of each leveling cylinder and the deflection angle vectors of the movable beam around the x and y axes, respectively. , , Here is the coefficient matrix:

[0030]

[0031] definition These are the output force vectors of each leveling cylinder, the friction force vector of each leveling cylinder, and the load torque vector of the movable beam, respectively. , The dynamic model of the moving beam is then expressed as:

[0032] (5)

[0033] in and The coefficient matrices are as follows:

[0034] ,

[0035] ,

[0036] in, .

[0037] Preferably, the motion relationship is decoupled from four-input two-output to two-input two-output, as follows:

[0038] By taking the difference between the diagonal cylinders of the four leveling cylinders, the relationship between the displacement and rotation angle on the diagonal of the movable beam is established:

[0039] (6)

[0040] Define intermediate variables The piston displacement of the four leveling cylinders is controlled by an intermediate variable. Angular acceleration of the moving beam about the x and y axes This achieves decoupling from four-input two-output to two-input two-output:

[0041] (7).

[0042] Preferably, the construction of the dynamic model of the moving beam in the decoupled coordinate system is as follows:

[0043] Define the hydraulic driving force of the leveling cylinder on the moving beam. ,in, This represents a vector composed of the pressures in the large chambers of the four leveling cylinders. The vector represents the pressure composition of the four leveling cylinder chambers; the dynamic model of the decoupled moving beam containing the LuGre friction model is as follows:

[0044] (8)

[0045] in

[0046] ,

[0047] ,

[0048] ,

[0049] ,

[0050] ,

[0051] ,

[0052] in, , These are the displacement state vector and velocity state vector after decoupling, respectively. For intermediate displacement variable vectors, For the decoupling equivalent coefficient matrix, For the piston rod mass, For the equivalent virtual torque vector, This is the equivalent load torque vector. This is the equivalent frictional torque vector.

[0053] Preferably, the time-varying parameter update law and adaptive gain adjustment are designed as follows:

[0054] The estimated parameters are three parameters in the LuGre model. , , Regression vector , , Processed using a first-order low-pass filter:

[0055] (9)

[0056] (10)

[0057] (11)

[0058] In the formula , , These represent the current regression vector filter values ​​of the leveling cylinder; , , This represents the filtered value of the previous regression vector; Indicates the sampling period; The time constant that affects the strength of the filter;

[0059] Based on the definition of formula (7), the state and output values ​​of the dynamic model of the moving beam are filtered:

[0060] (12)

[0061] (13)

[0062] (14)

[0063] in, , , Representing state variables respectively The filtered values ​​of the intermediate velocity variable at the current moment, the intermediate velocity variable at the previous moment, and the intermediate acceleration variable at the current moment, where the subscript f indicates filtering. , , Representing state variables respectively The intermediate velocity variable filter value at the current moment, the intermediate velocity variable filter value at the previous moment, and the intermediate acceleration variable filter value at the current moment. , These represent the equivalent virtual torque filtering values ​​around the x-axis at the current time and the previous time, respectively. , These represent the equivalent virtual torque filtering values ​​around the y-axis at the current time and the previous time, respectively. , These represent the equivalent virtual torque components about the x-axis and y-axis, respectively, which are the output values ​​of the dynamic model;

[0064] Guided by model error, according to formula (8), the model error is obtained as follows:

[0065] (15)

[0066] in, , Let represent the model prediction errors of the decoupled channels around the x-axis and y-axis, respectively. , These represent the equivalent coefficients of the two channels in the decoupled dynamics model (corresponding to the diagonal elements of the decoupled equivalent coefficient matrix S). , These represent the friction torque filtering values ​​for the two channels, respectively. , These represent the equivalent external load torque of the two channels, respectively.

[0067] Define the error set as :

[0068] (16)

[0069] Among the 12 parameters to be estimated in the LuGre model for the 4 leveling cylinders, the first... Adaptive gain of each parameter for:

[0070] (17)

[0071] In the formula, , These represent the time-varying coefficients, This represents the gain at the previous time step; =1,2…,11,12, which correspond to the three estimated friction parameters of each of the four leveling cylinders; Let m represent the regression variable corresponding to the adaptive gain of the m-th parameter. The values ​​corresponding to the three parameters to be estimated in the update of each leveling cylinder are respectively the bristle deformation filter values. , Derivative filtering value of mane deformation and piston speed filter value ;

[0072] The parameter adaptation law of the LuGre model is:

[0073] (18)

[0074] In the formula, , , These represent the i-th leveling cylinder ( Online estimates of the parameters of bristle stiffness, bristle damping, and viscous damping; , , These represent the adaptive laws for the friction parameters corresponding to the i-th leveling cylinder; Represents the coefficient matrix The geometric distribution coefficient corresponding to the i-th leveling cylinder; These represent the leakage coefficients in the update laws of the three corresponding friction parameters, used to prevent parameter drift and achieve rapid parameter convergence; , , This represents the initial value of the friction parameter corresponding to the i-th leveling cylinder. Preferably, the design of the direct and indirect adaptive disturbance rejection controller is as follows:

[0075] Set the target displacement value to Then the tracking error Construct Lyapunov functions ,make ,have to ,definition ,in Let the velocity target value vector be... , To address the velocity tracking error, the Lyapunov function is reconstructed, resulting in... , , The control gain constants are all greater than zero; the virtual torque is designed according to formula (8) to make the error approach 0:

[0076] (19)

[0077] In the formula , Indicates virtual torque; , This represents the equivalent frictional torque of the decoupled channels about the x-axis and y-axis, respectively. , This represents the equivalent external load torque of the decoupled channels about the x-axis and y-axis, respectively. , This represents the derivative of the virtual velocity target value (i.e., the virtual acceleration target value) of the decoupled channel around the x-axis and y-axis. , This represents the equivalent coefficient of the two channels in the decoupled dynamics model; , This represents the velocity tracking error of the decoupled channels around the x-axis and y-axis, respectively. , Represents a constant greater than zero; , Represents the compensation term in the fast model;

[0078] For virtual torque , Torque distribution is performed to obtain the target pressure of the four leveling cylinders in the actual process. , , , :

[0079] ,

[0080] ,

[0081] ,

[0082] ,

[0083] in Indicates basic pressure;

[0084] Define pressure error ,in, This indicates the actual pressure in the large chamber of the i-th leveling cylinder. Let i represent the target pressure of the leveling cylinder. According to formula (1), the voltage of the servo proportional valve is designed as follows:

[0085] (20)

[0086] In the formula, , , , These represent the servo proportional valve control voltages for the 1st to 4th leveling cylinders, respectively. , , , These represent the volumes of the first through fourth leveling cylinders, respectively. This represents the area gradient of the servo proportional valve. Indicates the elastic modulus; , , , These represent the unmodeled, slowly varying standard values ​​of error defined for the four cylinders; This represents the reciprocal of the expected pressure for each cylinder; , represents the nonlinear pressure drop coefficient of the servo proportional valve of the i-th leveling cylinder; , , , Both represent constants greater than zero; , , , This represents the compensation term for the fast model.

[0087] Compared with the prior art, the present invention has the following beneficial effects:

[0088] This invention addresses the crawling and vibration phenomena that occur during the leveling process of a hydraulic press by proposing an adaptive friction compensation feedback control method for multi-axis electro-hydraulic systems. This method can estimate and accurately compensate for friction during leveling, achieving smooth operation. Its significant advantages include:

[0089] (1) To address the creeping vibration phenomenon that occurs during the hydraulic press pressing process, the LuGre friction model is introduced to alleviate the creeping phenomenon during the leveling process and improve stability.

[0090] (2) To address the problem of unknown off-center load in pressing, an indirect adaptive controller was designed. Guided by the filtering model error, a time-varying parameter update law with leakage coefficient and adaptive gain adjustment based on the allocated model error were designed to achieve accurate compensation of friction under different off-center loads, providing a new method for pressing flat and stable composite materials. Attached Figure Description

[0091] Figure 1 This is a flowchart illustrating the method implementation of an embodiment of the present invention;

[0092] Figure 2 This is a framework diagram of the control method described in this invention;

[0093] Figure 3This is a force diagram of the multi-axis leveling system of the hydraulic press according to an embodiment of the present invention;

[0094] Figure 4 This is a hydraulic schematic diagram of the multi-axis leveling system of the hydraulic press according to an embodiment of the present invention;

[0095] Figure 5 This is a simulation result diagram of an embodiment of the present invention. Detailed Implementation

[0096] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0097] It should be noted that the following detailed descriptions are exemplary and intended to provide further explanation of this application. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains.

[0098] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the exemplary embodiments according to this application. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0099] like Figure 1 As shown, this embodiment provides an adaptive friction compensation feedback control method for multi-axis electro-hydraulic systems, including the following steps:

[0100] Step S1: Establish a dynamic model of the hydraulic leveling system based on friction compensation;

[0101] Step S2: Based on the dynamic model, decouple the motion relationship in the overdriven coupled leveling system from a four-input two-output model to a two-input two-output model;

[0102] Step S3: Design a time-varying parameter update law with leakage coefficient based on model error and adaptive gain adjustment to obtain the estimated parameters of the hydraulic press friction model;

[0103] Step S4: Based on the designed time-varying parameter update law and adaptive gain adjustment, design a direct / indirect adaptive disturbance rejection controller for the outer loop displacement and inner loop voltage to achieve parallel online parameter estimation and precision control.

[0104] The hydraulic leveling system mainly consists of a movable beam, four leveling cylinders, and a servo proportional valve. During the descent of the movable beam, the four leveling cylinders act on the four corners of the movable beam to achieve passive leveling. The small chamber of the leveling cylinder is controlled at a constant pressure by a pressure reducing valve, while the large chamber is controlled by a servo proportional valve to output force. When the movable beam falls, the pressure, the constant pressure of the large chamber, and the control pressure of the small chamber interact to achieve leveling of the movable beam.

[0105] In this embodiment, a press leveling system with a movable beam weighing 655kg is used as an example, and the parameters are as follows;

[0106]

[0107] The following section will elaborate on the relevant aspects of this method.

[0108] like Figure 1 As shown, in step S1, considering the influence of friction in the leveling system, a model of the movable beam and four leveling cylinders was established.

[0109] Figure 3 , Figure 4 This is a schematic diagram of the leveling system in this embodiment. To simplify the model, factors with minor influence are ignored, and the following assumptions are made:

[0110] (1) The system oil supply pressure is a constant value, denoted as Ps, and the hydraulic pump output pressure does not change with time, and the return oil pressure is approximately zero;

[0111] (2) Ignore the pressure loss along the pipeline and the local resistance loss of hydraulic oil. Assume that hydraulic oil is an isothermal, incompressible Newtonian fluid with constant density and viscosity. At the same time, do not consider the effect of oil temperature change on system performance.

[0112] (3) Both the servo proportional valve and the flow compensation valve are ideal zero-opening four-sided slide valves, and each throttling port is symmetrically matched.

[0113] (4) The leveling cylinders are arranged symmetrically, and the cylinder movement occurs only in the vertical direction. Ignoring the coupling deformation between the lateral oscillation and the cylinder, the load can be regarded as a rigid body with a constant moment of inertia.

[0114] The mathematical model for a single leveling cylinder, derived from the orifice throttling flow equation, is as follows:

[0115] (1)

[0116] In the formula, , This represents the pressure values ​​of the large chamber and small chamber of the leveling cylinder i. This indicates the effective volumetric elastic modulus of the leveling cylinder; Indicates the volume of the leveling cylinder; Indicates the area of ​​the large cavity of the leveling cylinder; Indicates the area of ​​the small cavity of the leveling cylinder; This indicates the piston displacement of leveling cylinder number i; This indicates the pressure flowing into the large chamber of the i-th leveling cylinder; Indicates the leakage coefficient; Indicates the flow coefficient; This represents the area gradient of the servo proportional valve. This indicates the spool displacement of the servo proportional valve. Indicates the density of hydraulic oil; Indicates the pump source pressure; Indicates the external load force of the leveling cylinder; Express the total mass of the piston and other loads; Indicates the viscous damping coefficient; Indicates the spring stiffness; where For symbolic functions

[0117]

[0118] Treating the movable beam as a uniform rigid body, with its geometric center as the origin, its longer side as the x-direction, its wider side as the y-direction, and the direction perpendicular to the beam as the z-direction, a Cartesian coordinate system is established. The movable beam rotates around the x and y directions during its descent. Based on the dynamic equations and the geometric relationship of the movable beam, the dynamic equations of the movable beam are obtained:

[0119] (2)

[0120] In the formula, Indicates the mass of the movable beam; This represents the initial displacement of the movable beam in the z-direction; This represents the driving force of the movable beam in the z-direction; Represents gravitational acceleration; Indicates the first The output force of each leveling cylinder ( =1,2,3,4); Indicates the first Friction during the movement of the leveling cylinder; , These represent the moments of inertia of the movable beam about the x and y directions, respectively. , These represent the deflection angles of the movable beam about the x and y axes, respectively. , These represent half the length of the movable beam in the x and y directions, respectively. , These represent the load moments of the movable beam about the x and y axes, respectively.

[0121] The friction force adopts the LuGre friction model, and the mathematical model corresponding to a single leveling cylinder is as follows:

[0122] (3)

[0123] In the formula, Indicates the bristle stiffness coefficient; This indicates the average deformation of the bristles; Indicates the bristle damping coefficient; Represents the viscous damping coefficient (representing the viscous damping coefficient at the microscopic level); The Stribeck curve function represents the relationship between steady-state frictional characteristics and velocity. Represents Coulomb friction; This represents static friction. Indicates Stribeck's velocity;

[0124] In multi-axis leveling, the movable beam is in close contact with the leveling cylinder. Since the movable beam is a rigid body, its deflection angle is very small, which can be approximated as... ,get

[0125] (4)

[0126] in , , Here is the coefficient matrix:

[0127]

[0128] definition , The dynamic model of the moving beam is obtained by rearranging (2)-(4):

[0129] (5)

[0130] in and The coefficient matrices are as follows:

[0131] ,

[0132] ,

[0133] in, .

[0134] The dynamic model of the leveling cylinder and the dynamic model of the movable beam together constitute the dynamic model of the hydraulic leveling system based on friction compensation.

[0135] In step S2, the motion relationship in the overdrive coupled leveling system is decoupled from a four-input two-output model to a two-input two-output model.

[0136] By taking the difference between the diagonal cylinders of the four leveling cylinders, the relationship between the displacement and rotation angle on the diagonal of the movable beam is established:

[0137] (6)

[0138] definition , can be obtained

[0139] (7)

[0140] Define the hydraulic driving force of the leveling cylinder on the moving beam. ,in, This represents a vector composed of the pressures in the large chambers of the four leveling cylinders. The vector representing the pressure in the small chambers of the four leveling cylinders, after decoupling, has the following dynamic model of the moving beam with the LuGre friction model:

[0141] (8)

[0142] in

[0143] ,

[0144] ,

[0145] ,

[0146] ,

[0147] ,

[0148] ,

[0149] in, , These are the displacement state vector and velocity state vector after decoupling, respectively. For intermediate displacement variable vectors, For the decoupling equivalent coefficient matrix, For the piston rod mass, For the equivalent virtual torque vector, This is the equivalent load torque vector. This is the equivalent frictional torque vector.

[0150] like Figure 2 As shown, in step S3, a time-varying parameter update law with leakage coefficient based on model error and adaptive gain adjustment are designed to obtain the estimated parameters of the hydraulic press friction model.

[0151] The estimated value is in the LuGre model. , , Three parameters, regression vector , , Processed using a first-order low-pass filter:

[0152] (9)

[0153] (10)

[0154] (11)

[0155] In the formula , , This represents the current regression vector filter value of the leveling cylinder; , , This represents the filtered value of the previous regression vector; Indicates the sampling period; This represents the time constant, and its magnitude affects the strength of the filter.

[0156] By defining formula (7), the state and output values ​​of the dynamic model of the moving beam are filtered:

[0157] (12)

[0158] (13)

[0159] (14)

[0160] in, , , Representing state variables respectively The filtered values ​​of the intermediate velocity variable at the current moment, the intermediate velocity variable at the previous moment, and the intermediate acceleration variable at the current moment, where the subscript f indicates filtering. , , Representing state variables respectively The intermediate velocity variable filter value at the current moment, the intermediate velocity variable filter value at the previous moment, and the intermediate acceleration variable filter value at the current moment. , These represent the equivalent virtual torque filtering values ​​around the x-axis at the current time and the previous time, respectively. , These represent the equivalent virtual torque filtering values ​​around the y-axis at the current time and the previous time, respectively. , These represent the equivalent virtual torque components about the x-axis and y-axis, respectively, which are the output values ​​of the dynamic model;

[0161] Guided by model error, according to formula (8), the model error can be obtained as follows:

[0162] (15)

[0163] in, , Let represent the model prediction errors of the decoupled channels around the x-axis and y-axis, respectively. , These represent the equivalent coefficients of the two channels in the decoupled dynamics model (corresponding to the diagonal elements of the decoupled equivalent coefficient matrix S). , These represent the friction torque filtering values ​​for the two channels, respectively. , These represent the equivalent external load torque of the two channels, respectively.

[0164] Define the error set as

[0165] (16)

[0166] Among the 12 parameters to be estimated in the LuGre model for the 4 leveling cylinders, the first... The adaptive gain adjustment of each parameter is as follows:

[0167] (17)

[0168] In the formula, , These represent the time-varying coefficients, This represents the gain at the previous time step; =1,2…,11,12, which correspond to the three estimated friction parameters of each of the four leveling cylinders; Let m represent the regression variable corresponding to the adaptive gain of the m-th parameter. The values ​​corresponding to the three parameters to be estimated in the update of each leveling cylinder are: bristle deformation filter value , Derivative filtering value of mane deformation and piston speed filter value ;

[0169] From the above formula, the parameter adaptation law of the LuGre model can be obtained as follows:

[0170] (18)

[0171] In the formula, , , These represent the i-th leveling cylinder ( Online estimates of the parameters of bristle stiffness, bristle damping, and viscous damping; , , These represent the adaptive laws for the friction parameters corresponding to the i-th leveling cylinder; Represents the coefficient matrix The geometric distribution coefficient corresponding to the i-th leveling cylinder; These represent the leakage coefficients in the update laws of the three corresponding friction parameters, used to prevent parameter drift and achieve rapid parameter convergence; , , This represents the initial value of the friction parameter corresponding to the i-th leveling cylinder.

[0172] In step S4, a direct / indirect adaptive disturbance rejection controller for the outer loop displacement and inner loop voltage is designed.

[0173] Set the target displacement value to Then the tracking error ,structure To make Lyapunov functions stable, then ,Right now ,make ,but Similarly, construct the Lyapunov function to obtain... , Take 1 or 2. , The control gain constants are all greater than zero; to make the error as close to 0 as possible, according to formula (8), the virtual torque is designed as follows:

[0174] (19)

[0175] In the formula , Indicates virtual torque; , This represents the equivalent frictional torque of the decoupled channels about the x-axis and y-axis, respectively. , This represents the equivalent external load torque of the decoupled channels about the x-axis and y-axis, respectively. , This represents the derivative of the virtual velocity target value (i.e., the virtual acceleration target value) of the decoupled channel around the x-axis and y-axis. , This represents the equivalent coefficient of the two channels in the decoupled dynamics model; , This represents the velocity tracking error of the decoupled channels around the x-axis and y-axis, respectively. , Represents a constant greater than zero; , Represents the compensation term in the fast model;

[0176] By distributing the obtained virtual torque, the target pressure in the actual process can be obtained:

[0177] ,

[0178] ,

[0179] ,

[0180] ,

[0181] in Indicates basic pressure. Indicates the target pressure of the four leveling cylinders ( =1,2,3,4)

[0182] According to formula (1), the pressure error is defined so that the actual rodless chamber pressure of the four cylinders can track the desired pressure as closely as possible. Similar to the synchronous control approach, the voltage design of the servo proportional valve is as follows:

[0183] (20)

[0184] In the formula, , , , These represent the servo proportional valve control voltages for the 1st to 4th leveling cylinders, respectively. , , , These represent the volumes of the first through fourth leveling cylinders, respectively. This represents the area gradient of the servo proportional valve. Indicates the elastic modulus; , , , These represent the unmodeled, slowly varying standard values ​​of error defined for the four cylinders; This represents the reciprocal of the expected pressure for each cylinder; , represents the nonlinear pressure drop coefficient of the servo proportional valve of the i-th leveling cylinder; , , , Both represent constants greater than zero; , , , This represents the compensation term for the fast model. Figure 5 This is a schematic diagram of the synchronization error of the hydraulic leveling system in this embodiment. According to the adaptive friction compensation feedback control method for multi-axis electro-hydraulic systems described in this invention, M is applied at 3 seconds. x 750 N / m, M y With a torque of 675 N / m, it can be precisely leveled in complex external environments.

Claims

1. A self-adaptive friction compensation feedback control method for multi-axis electro-hydraulic systems, characterized in that, Includes the following steps: A dynamic model of a hydraulic leveling system based on friction compensation is established, including a dynamic model of the leveling cylinder and a dynamic model of the moving beam. Based on the established dynamic model, the motion relationship in the overdriven coupled leveling system is decoupled from a four-input two-output model to a two-input two-output model, and a dynamic model of the moving beam is constructed in the decoupled coordinate system. Based on the dynamic model of the moving beam in the decoupled coordinate system, a time-varying parameter update law with leakage coefficient based on model error and adaptive gain adjustment are designed to obtain the estimated parameters of the leveling cylinder friction model. Based on the designed time-varying parameter update law and adaptive gain adjustment, a direct and indirect adaptive disturbance rejection controller is designed. The controller adopts a dual closed-loop structure of outer-loop displacement control and inner-loop voltage control to achieve parallel online parameter estimation and precision control.

2. The adaptive friction compensation feedback control method for multi-axis electro-hydraulic system according to claim 1, wherein, The specific implementation method for establishing the dynamic model of the hydraulic leveling system based on friction compensation is as follows: The hydraulic leveling system includes a movable beam, four leveling cylinders, and a servo proportional valve. During the descent of the movable beam, the four leveling cylinders act on the four corners of the movable beam. The small chamber of the leveling cylinder is controlled at a constant pressure by a pressure reducing valve, and the large chamber of the leveling cylinder is controlled by a servo proportional valve to output force. When the movable beam falls, the pressure, the pressure controlled by the large chamber, and the constant pressure of the small chamber interact to achieve passive leveling of the movable beam. The dynamic mathematical model of a single leveling cylinder is as follows: (1) In the formula, , This represents the pressure values ​​of the large chamber and small chamber of the leveling cylinder i. This indicates the effective volumetric elastic modulus of the leveling cylinder; Indicates the volume of the leveling cylinder; Indicates the area of ​​the large cavity of the leveling cylinder; This indicates the area of ​​the small cavity of the leveling cylinder; This indicates the piston displacement of leveling cylinder number i; This indicates the pressure flowing into the large chamber of the i-th leveling cylinder; Indicates the leakage coefficient; Indicates the flow coefficient; This represents the area gradient of the servo proportional valve. This indicates the spool displacement of the servo proportional valve. Indicates the density of hydraulic oil; Indicates the pump source pressure; Indicates the external load force of the leveling cylinder; Express the total mass of the piston and other loads; Indicates the viscous damping coefficient; Indicates the spring stiffness; where It is a symbolic function; With the geometric center of the movable beam as the origin, the longer side as the x-direction, the wider side as the y-direction, and the direction perpendicular to the movable beam as the z-direction, the movable beam rotates about the x and y directions during its descent. The dynamic equations of the movable beam are then expressed as: (2) In the formula, Indicates the mass of the movable beam; This represents the initial displacement of the movable beam in the z-direction; This represents the driving force of the movable beam in the z-direction; Represents gravitational acceleration; These represent the output forces of the four leveling cylinders; This represents the frictional force during the movement of the four leveling cylinders; , These represent the moments of inertia of the movable beam about the x and y directions, respectively. , These represent the deflection angles of the movable beam about the x and y axes, respectively. , These represent half the length of the movable beam in the x and y directions, respectively. , These represent the load moments of the movable beam about the x and y axes, respectively. Among them, the friction force during the movement of the i-th leveling cylinder The LuGre friction model is used to represent: (3) In the formula, Indicates the bristle stiffness coefficient; This indicates the average deformation of the bristles; Indicates the bristle damping coefficient; Indicates the viscous damping coefficient; This represents the Stribeck curve function, which describes the relationship between steady-state frictional characteristics and velocity. Represents Coulomb friction; This represents static friction. Indicates Stribeck's velocity; The following geometric relationship exists between the deflection angle of the movable beam and the piston displacement of each leveling cylinder: (4) in, These are the piston displacement vectors of each leveling cylinder and the deflection angle vectors of the movable beam around the x and y axes, respectively. , , Here is the coefficient matrix: definition These are the output force vectors of each leveling cylinder, the friction force vector of each leveling cylinder, and the load torque vector of the movable beam, respectively. , The dynamic model of the moving beam is then expressed as: (5) in and The coefficient matrices are as follows: ， ， in, .

3. The adaptive friction compensation feedback control method for multi-axis electro-hydraulic systems according to claim 2, characterized in that, The motion relationship is decoupled from four-input two-output to two-input two-output, as follows: By taking the difference between the diagonal cylinders of the four leveling cylinders, the relationship between the displacement and rotation angle on the diagonal of the movable beam is established: (6) Define intermediate variables The piston displacement of the four leveling cylinders is controlled by an intermediate variable. Angular acceleration of the moving beam about the x and y axes This achieves decoupling from four-input two-output to two-input two-output: (7)。 4. The adaptive friction compensation feedback control method for multi-axis electro-hydraulic systems according to claim 3, characterized in that, The specific steps for constructing the dynamic model of the moving beam in the decoupled coordinate system are as follows: Define the hydraulic driving force of the leveling cylinder on the moving beam. ,in, This represents a vector composed of the pressures in the large chambers of the four leveling cylinders. The vector represents the pressure composition of the four leveling cylinder chambers; the dynamic model of the decoupled moving beam containing the LuGre friction model is as follows: (8) in ， ， ， ， ， ， in, , These are the displacement state vector and velocity state vector after decoupling, respectively. For intermediate displacement variable vectors, For the decoupling equivalent coefficient matrix, For the piston rod mass, For the equivalent virtual torque vector, This is the equivalent load torque vector. This is the equivalent frictional torque vector.

5. The adaptive friction compensation feedback control method for multi-axis electro-hydraulic systems according to claim 4, characterized in that, The design of the time-varying parameter update law and adaptive gain adjustment is as follows: The estimated parameters are three parameters in the LuGre model. , , Regression vector , , Processed using a first-order low-pass filter: (9) (10) (11) In the formula , , These represent the current regression vector filter values ​​of the leveling cylinder; , , This represents the filtered value of the previous regression vector; Indicates the sampling period; The time constant that affects the strength of the filter; Based on the definition of formula (7), the state and output values ​​of the dynamic model of the moving beam are filtered: (12) (13) (14) in, , , Representing state variables respectively The intermediate velocity variable filter value at the current moment, the intermediate velocity variable filter value at the previous moment, and the intermediate acceleration variable filter value at the current moment. , , Representing state variables respectively The intermediate velocity variable filter value at the current moment, the intermediate velocity variable filter value at the previous moment, and the intermediate acceleration variable filter value at the current moment. , These represent the equivalent virtual torque filtering values ​​around the x-axis at the current time and the previous time, respectively. , These represent the equivalent virtual torque filtering values ​​around the y-axis at the current time and the previous time, respectively. , These represent the equivalent virtual torque components about the x-axis and y-axis, respectively, which are the output values ​​of the dynamic model; Guided by model error, according to formula (8), the model error is obtained as follows: (15) in, , These represent the model prediction errors of the decoupled channels around the x-axis and y-axis, respectively. , These represent the equivalent coefficients of the two channels in the decoupled dynamics model, corresponding to the diagonal elements of the decoupled equivalent coefficient matrix S; , These represent the friction torque filtering values ​​for the two channels, respectively. , These represent the equivalent external load torque of the two channels, respectively. Define the error set as : (16) Among the 12 parameters to be estimated in the LuGre model for the 4 leveling cylinders, the first... Adaptive gain of each parameter for: (17) In the formula, , These represent the time-varying coefficients, This represents the gain at the previous time step; =1,2…,11,12, which correspond to the three estimated friction parameters of each of the four leveling cylinders; Let m represent the regression variable corresponding to the adaptive gain of the m-th parameter. The values ​​corresponding to the three parameters to be estimated in the update of each leveling cylinder are: bristle deformation filter value , Derivative filtering value of mane deformation and piston speed filter value ; The parameter adaptation law of the LuGre model is: (18) In the formula, , , These represent the online estimated values ​​of the bristle stiffness, bristle damping, and viscous damping parameters of the i-th leveling cylinder, respectively. , , These represent the adaptive laws for the friction parameters corresponding to the i-th leveling cylinder; Represents the coefficient matrix The geometric distribution coefficient corresponding to the i-th leveling cylinder; These represent the leakage coefficients in the update laws of the three corresponding friction parameters, used to prevent parameter drift and achieve rapid parameter convergence; , , This represents the initial value of the friction parameter corresponding to the i-th leveling cylinder.

6. The adaptive friction compensation feedback control method for multi-axis electro-hydraulic systems according to claim 5, characterized in that, The design of the direct and indirect adaptive disturbance rejection controllers is as follows: Set the target displacement value to Then the tracking error Construct Lyapunov functions ,make ,have to ,definition ,in Let the velocity target value vector be... , To address the velocity tracking error, the Lyapunov function is reconstructed, resulting in... , , The control gain constants are all greater than zero; the virtual torque is designed according to formula (8) to make the error approach 0: (19) In the formula , Indicates virtual torque; , This represents the equivalent frictional torque of the decoupled channels about the x-axis and y-axis, respectively. , This represents the equivalent external load torque of the decoupled channels about the x-axis and y-axis, respectively. , This represents the derivative of the virtual velocity target value of the decoupled channel around the x-axis and y-axis, i.e., the virtual acceleration target value; , This represents the equivalent coefficient of the two channels in the decoupled dynamics model; , This represents the velocity tracking error of the decoupled channels around the x-axis and y-axis, respectively. , Represents a constant greater than zero; , Represents the compensation term in the fast model; For virtual torque , Torque distribution is performed to obtain the target pressure of the four leveling cylinders in the actual process. , , , : ， ， ， ， in Indicates basic pressure; Define pressure error ,in, This indicates the actual pressure in the large chamber of the i-th leveling cylinder. Let i represent the target pressure of the leveling cylinder. According to formula (1), the voltage of the servo proportional valve is designed as follows: (20) In the formula, , , , These represent the servo proportional valve control voltages for the 1st to 4th leveling cylinders, respectively. , , , These represent the volumes of the first through fourth leveling cylinders, respectively. This represents the area gradient of the servo proportional valve. Indicates the elastic modulus; , , , These represent the unmodeled, slowly varying standard values ​​of error defined for the four cylinders; This represents the reciprocal of the expected pressure for each cylinder; , represents the nonlinear pressure drop coefficient of the servo proportional valve of the i-th leveling cylinder; , , , Both represent constants greater than zero; , , , This represents the compensation term for the fast model.

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