Synchronous belt driving system position control method based on feedforward compensation

Abstract

The invention provides a synchronous belt driving system position control method based on feed-forward compensation, and belongs to the technical field of mechanical design and manufacturing. The method comprises the following steps: constructing a system model of the synchronous belt driving system, and obtaining a standard transfer function of the system model by using a dynamic model equivalent to the system model; constructing a feedback control framework of the system model; performing parameter identification on the quasi transfer function to obtain a nominal transfer function; on the basis of the feedback control framework, a feedforward control framework is constructed according to the nominal transfer function, and the feedback control framework and the feedforward control framework are combined to form a position control framework of the system model; and the position control frame is used for performing position control on the synchronous belt driving system. The dynamic response performance of the synchronous belt driving system can be effectively improved.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of mechanical design and manufacturing, and particularly relates to a position control method of a synchronous belt driving system based on feedforward compensation. BACKGROUND

[0002] With the acceleration of the new round of global technological revolution and industrial reform, intelligent manufacturing has become the core direction of the transformation and upgrading of the manufacturing industry. High-precision, high-automation and high-reliability transmission control technology is the key foundation to support the development of high-end equipment (such as industrial robots, precision CNC machine tools, 3D printing equipment and laser processing systems, etc.). Synchronous belt driving is increasingly widely used in high-precision automation fields such as industrial robots, precision CNC machine tools, 3D printing equipment and laser processing systems due to its comprehensive advantages of high efficiency, low energy consumption, high precision and low cost.

[0003] Due to the inherent flexible structural characteristics (such as belt elasticity, tooth gap engagement clearance, etc.) and strong nonlinear behavior (including hysteresis, dead zone and friction disturbance, etc.) of the synchronous belt driving system, its dynamic model usually presents high-order uncertainty, which leads to serious challenges for traditional linear strategies, such as insufficient nonlinear dynamic suppression, rigid-flexible coupling dynamic instability, weak anti-disturbance ability, and prominent contradiction between high speed and high precision.

[0004] In view of the above problems, the existing solutions mainly include improvement of mechanical structure and optimization of traditional control algorithm, but both methods have great limitations. Mainly manifested as:

[0005] Firstly, although the mechanical structure of the synchronous belt driving system can be improved to improve its motion precision, but limited by the inevitable nonlinear physical characteristics and the limitations of the application scene, the improvement space of the mechanical structure is close to the ceiling.

[0006] Secondly, the synchronous belt transmission essentially belongs to a multi-rigid-flexible coupling hybrid body dynamic transmission. The traditional linear control strategy cannot effectively cope with factors such as nonlinear dynamics, rigid-flexible coupling, time-varying and disturbance in the synchronous belt transmission, which leads to the fact that the precision and stability of the synchronous belt driving system cannot be fully guaranteed, especially under high-precision and high-transmission requirements, the existing control algorithm is difficult to meet the precision and response speed requirements in industrial manufacturing.

[0007] Therefore, it is necessary to propose a scheme to improve one or more problems existing in the above related technical solutions.

[0008] It should be noted that the information disclosed in the above background section is only used to strengthen the understanding of the background of the present application, and therefore can include information that does not constitute prior art known to those of ordinary skill in the art. SUMMARY

[0009] This application provides a position control method for a synchronous belt drive system based on feedforward compensation, the method comprising the following steps:

[0010] A system model of a synchronous belt drive system is constructed, and the standard transfer function of the system model is obtained using a dynamic model equivalent to the system model.

[0011] A feedback control framework for the system model is constructed, which includes a multi-loop control framework consisting of an input terminal, a feedback controller, a disturbance observer, the synchronous belt drive system, and an output terminal.

[0012] The standard transfer function is subjected to parameter identification to obtain the nominal transfer function;

[0013] Based on the feedback control framework, a feedforward control framework is constructed according to the nominal transfer function. The feedback control framework and the feedforward control framework are combined to form the position control framework of the system model.

[0014] The position control framework is used to perform position control on the synchronous belt drive system.

[0015] Furthermore, the system model includes a loop system consisting of a controller, a driver, an encoder, a motor, a reducer, a synchronous belt structure, and a grating ruler; the grating ruler constitutes the loop system.

[0016] The synchronous band structure includes:

[0017] A driving wheel and a driven wheel are arranged opposite each other, and a synchronous belt is fitted on the driving wheel and the driven wheel. A connecting plate is provided on the synchronous belt.

[0018] A linear guide rail assembly, comprising linear guide rails respectively disposed on both sides of the synchronous belt; wherein the grating ruler is disposed on the outer surface of one of the linear guide rails, and the grating ruler is parallel to the track of the linear guide rail;

[0019] The load is mounted on the linear guide rail assembly and slides along the linear guide rail assembly, and the load is connected to the connecting plate; the load is equipped with a grating head.

[0020] Furthermore, the step of obtaining the standard transfer function of the system model using a dynamic model equivalent to the system model includes:

[0021] Each element in the system model is associated with at least one equivalent parameter, and the dynamic model is constructed using all equivalent parameters.

[0022] The expression for the dynamic model is:

[0023] (1)

[0024] in, Represents the equivalent moment of inertia. The second derivative represents the angular displacement of the driving wheel. This indicates the torque exerted by the motor on the drive wheel. Indicates the synchronization band at the 1st Stiffness when there are multiple scheduling variables Indicates the radius of the driving wheel. This indicates the angular displacement of the driving wheel. Represents the linear displacement of the load. This represents the relative damping coefficient between the driving pulley and the timing belt. The first derivative of the angular displacement of the driving wheel. The first derivative represents the linear displacement of the load. Indicates the quality of the load. This represents the damping coefficient of the linear guide assembly;

[0025] The dynamic model is subjected to a Laplace transform to obtain an intermediate set of equations;

[0026] The expression for the intermediate equation set is:

[0027] (2)

[0028] in, This indicates the torque exerted by the motor on the drive wheel in the [number]th [phase]. Frequency response of a complex variable in the frequency domain This indicates the stiffness of the timing belt. The linear displacement of the load in the th... Frequency response of a complex variable in the frequency domain This indicates the angular displacement of the driving wheel in the th... Frequency response of a complex variable in the frequency domain;

[0029] All of the intermediate equations After eliminating the terms, the standard transfer function is obtained.

[0030] The expression of the standard transfer function includes:

[0031] (3)

[0032] in, Indicates the first Standard transfer function in the frequency domain of a complex variable.

[0033] Furthermore, the input terminal includes an input reference trajectory;

[0034] The feedback controller is Cascaded feedback controller, the The cascaded feedback controller includes a position loop and a velocity loop; the position loop includes... The controller, i.e., the proportional controller; the speed loop includes The controller is a proportional-integral controller.

[0035] A position error feedback node and a speed error feedback node are sequentially arranged on the main line between the input terminal and the synchronous belt drive system. Cascaded feedback control includes output nodes, feedback compensation nodes, and interference nodes.

[0036] The disturbance observer is connected in parallel with the synchronous belt drive system, and the first port of the disturbance observer is connected to the feedback compensation node, the second port of the disturbance observer is connected to the main line between the interference node and the synchronous belt drive system, and the third port of the disturbance observer is connected to the output terminal and the position error feedback node.

[0037] The position error feedback node is a position error feedback node that is the reference position of the reference trajectory signal and the actual position of the synchronization belt; the velocity error feedback node is a velocity error feedback node that is the reference velocity of the reference trajectory signal and the actual velocity of the synchronization belt.

[0038] Furthermore, the step of parameter identification of the standard transfer function to obtain the nominal transfer function includes:

[0039] The synchronous band drive system in the open-loop state is frequency swept using a sinusoidal sweep frequency signal to obtain a response output signal;

[0040] Performing a Fourier transform on the sinusoidal sweep signal yields the linear displacement of the load at the th ... The actual frequency response of the complex variable in the frequency domain The response output signal is then subjected to a Fourier transform to obtain the torque exerted by the motor on the drive wheel in the [missing information]. The actual frequency response of the complex variable in the frequency domain ;

[0041] Using the actual frequency response and the actual frequency response Calculate the actual transfer function of the system model. ;

[0042] Define the actual transfer function of the nominal model as Then we get: G 0 s = G n s ∙[1+∆( s )] ,in, Represents the actual transfer function and the actual transfer function Uncertainty between them;

[0043] The actual transfer function and the actual transfer function Perform a fitting process to eliminate the aforementioned uncertainties. The nominal transfer function is obtained.

[0044] Furthermore, the expression for the transfer function of the P / PI cascaded feedback controller is:

[0045] (4)

[0046] in, express The transfer function of a cascaded feedback controller. Represents the standard transfer function. This represents the proportional gain of the velocity loop. This represents the integral gain of the velocity loop. Indicates the proportional gain of the position loop;

[0047] The expression for the closed-loop transfer function from the input to the output of the system model is:

[0048] (5)

[0049] in, This represents the closed-loop transfer function from the input to the output of the system model. Represents the actual transfer function The reverse;

[0050] The expression for the closed-loop transfer function of the system model from external disturbances to the output is:

[0051] (6)

[0052] in, This represents the closed-loop transfer function of the system model from external disturbances to the output.

[0053] Furthermore, the actual transfer function and the actual transfer function The fitting process includes:

[0054] The actual transfer function The parameterization process is performed to obtain the parameter transfer function of the nominal model. , ,in, The first term representing the frequency response of the synchronous belt drive system Each frequency point, Indicates the parameter to be identified. This represents the set of coefficients of the numerator polynomial of the parameter to be identified. θ n =[ b 0 , b 1 ,…, b N n ] , This represents the set of coefficients of the denominator polynomial of the parameter to be identified. θ d =[1, a 1 , a 2 ,…, a N d ] , Indicates the first The frequency response of the numerator polynomial function of the parameter to be identified at each frequency point. , Represents the imaginary unit. The first polynomial function of the numerator Order coefficient, Indicates the first The frequency response of the denominator polynomial function of the parameter to be identified at each frequency point. , The function representing the denominator polynomial is the first... Order coefficient;

[0055] Based on the structure of the disturbance observer, we can obtain: ,in, Indicates the first A low-pass filter in the frequency domain of the complex variable. Let represent the closed-loop transfer function of the disturbance observer.

[0056] ,in, express The reverse, express The reverse;

[0057] According to the small gain theorem, when At that time, the closed-loop system of the disturbance observer is stable. Represents the infinite norm, The closed-loop transfer function of the disturbance observer is represented in the first... Frequency response at each frequency point Indicating uncertainty In the Frequency response at each frequency point ,in, The actual transfer function of the system model is represented in the t-th... Frequency response at each frequency point;

[0058] This yields the objective function for parameter identification;

[0059] The expression for the objective function identified by the parameters is:

[0060] (7)

[0061] in, This represents the objective function for parameter identification. Indicates the low-pass filter at the 1st Frequency response at each frequency point;

[0062] The infinity norm is approximated as a L2 norm, and the L2 norm is identified using a weighted least squares iterative algorithm to identify the parameter to be identified. ;

[0063] The parameters to be identified The expression is:

[0064] (8)

[0065] in, Indicates the first At the frequency point, the first The parameters to be identified in the nth iteration are obtained from the first iteration. The weight function for the next iteration , Indicates the first At the frequency point The frequency response of the molecule polynomial function corresponding to the parameter to be identified in the next iteration. No. The parameters to be identified in the next iteration The parameter to be identified when the minimum value is reached. The value of , Indicates the first At the frequency point The frequency response of the denominator polynomial function corresponding to the parameter to be identified in the next iteration. Indicates the first The set of coefficients of the denominator polynomial of the parameter to be identified in the next iteration. Indicates the first At the frequency point The frequency response of the molecule polynomial function corresponding to the parameter to be identified in the next iteration. Indicates the first The set of coefficients of the numerator polynomial of the parameters to be identified in the next iteration; the identification result is substituted into the parameter transfer function of the nominal model. In this process, the nominal transfer function is obtained.

[0066] Furthermore, the step of constructing a feedforward control framework based on the nominal transfer function on the basis of the feedback control framework, and combining the feedback control framework and the feedforward control framework to form the position control framework of the system model includes:

[0067] One end of the feedforward controller is connected to the input terminal, and the other end of the feedforward controller is connected to the feedback compensation node; the feedforward controller is a linear time-varying feedforward controller, and the linear time-varying feedforward controller includes a linear time-varying system;

[0068] The feedforward control framework is obtained by adjusting and setting the feedforward controller according to the nominal transfer function.

[0069] The feedback control framework and the feedforward control framework are combined to form the position control framework of the system model.

[0070] Further, the step of adjusting the feedforward controller according to the nominal transfer function to obtain the feedforward control framework includes:

[0071] The expression of the standard transfer function is transformed into a linear time-varying transfer function, and the inverse dynamic model of the linear time-varying transfer function is derived.

[0072] The expression for the linear time-varying transfer function is:

[0073] (9)

[0074] The expression for the inverse dynamics model is:

[0075] (10)

[0076] in, This indicates the torque exerted by the motor on the drive wheel. Indicates the first One scheduling variable, The fourth-order time-varying coefficients representing the linear displacement of the load. The fourth derivative representing the linear displacement of the load. The fourth-order time-varying term representing the linear displacement of the load. The third-order time-varying coefficients representing the linear displacement of the load. The third derivative represents the linear displacement of the load. The third-order time-varying term representing the linear displacement of the load. The second-order time-varying coefficient represents the linear displacement of the load. The second derivative represents the linear displacement of the load. The second-order time-varying term representing the linear displacement of the load. The first-order time-varying term represents the linear displacement of the load. Indicates redundant time-varying coefficients. Indicates a redundant variable. , Indicates the synchronization band at the 1st The first-order partial derivative of the stiffness when there are multiple scheduling variables. , Indicates the synchronization band at the 1st The second-order partial derivative of stiffness with 1 scheduling variable , Indicates the first The first derivative of each scheduling variable Indicates the first The second derivatives of each scheduling variable;

[0077] The linear time-varying feedforward controller is parameterized to obtain the polynomial feedforward of the linear time-varying feedforward controller, and the polynomial feedforward is solved to obtain the feedforward control framework.

[0078] The expression for the linear time-varying feedforward controller is:

[0079] (11)

[0080] in, To represent a linear time-varying system, The highest-order derivative representing the linear displacement of the load. Represents the scheduling variable. Represents the linear displacement of the load, when hour, The first linear displacement of the load The first derivative, when hour, The first linear displacement of the load Second integral. The highest-order derivative of the second-order integral of the torque exerted by the motor on the drive wheel. ,for They all ;

[0081] According to the basis function parameterization method, the zero dynamics of the system on the right side of equation (9) are ignored, i.e. ,get:

[0082] (12)

[0083] in, This represents a parameterized expression. This represents the number of all basis functions. Indicates the first When the scheduling variable is the first Time-varying coefficients of each basis function Indicates the first The expression for each basis function, Indicates reference input. Indicates the first When the scheduling variable is the first Time-varying coefficients of each basis function Indicates the first When the scheduling variable is the first Time-varying coefficients of each basis function Indicates the first When the scheduling variable is the first Time-varying coefficients of each basis function Indicates the first When the scheduling variable is the first Time-varying coefficients of each basis function The expression representing the first basis function, The expression representing the second basis function. The expression representing the third basis function. The expression representing the fourth basis function;

[0084] By taking the second derivative of both sides of equation (12), the polynomial feedforward is obtained.

[0085] The expression for the polynomial feedforward is:

[0086] (13)

[0087] in, This represents the polynomial feedforward of a linear time-varying feedforward controller. This represents the fourth derivative of the reference input. This represents the third derivative of the reference input. This represents the second derivative of the reference input. This represents the first derivative of the reference input;

[0088] The polynomial feedforward is solved, and the solution process includes:

[0089] A reference signal sequence is input to the synchronous belt drive system in closed-loop state, and the actual linear displacement of the load at each moment is recorded. Actual scheduling variables and the actual torque exerted by the motor on the drive wheel ;

[0090] Calculate the quadratic exponential kernel function for each time step using the actual scheduling variables at each time step; and calculate the actual linear displacement of the load at each time step. and the actual torque exerted by the motor on the drive wheel And the hyperparameters of the squared exponential kernel function at this moment are optimized using the marginal likelihood optimization method;

[0091] The expression for the quadratic exponential kernel function is:

[0092] (14)

[0093] in, Indicates the first The time-varying coefficient and the first time-varying coefficient Squared exponential kernel function between time-varying coefficients The variance of the squared exponential kernel function is represented by . This represents the length scale of the squared exponential kernel function. The first-order partial derivative of the reference scheduling variable is represented; the kernel matrix is ​​calculated using each optimized quadratic exponential kernel function;

[0094] The expression for the kernel matrix is:

[0095] (15)

[0096] in, Represents the kernel matrix;

[0097] The objective function of each time-varying coefficient is solved separately, and the feedforward control framework is obtained using all the solution results.

[0098] Furthermore, the step of solving the objective function for each of the time-varying coefficients and obtaining the feedforward control framework using all the solution results includes: constructing the objective function for the time-varying coefficients using the kernel regularized least squares method and solving it;

[0099] The objective function for solving the time-varying coefficients is expressed as follows:

[0100] (16)

[0101] in, This represents the time-varying coefficients to be solved. Represents the time-varying coefficient when taking the minimum value. The value of , This represents the data vector obtained by the second-order integral of the actual torque exerted by the measuring motor on the drive wheel. , Indicates the first At that moment, Represents the time-varying coefficient parameter vector. , Indicates the first Transpose of the time-varying coefficient vector of the basis functions Indicates transpose. This represents the number of all basis functions. Represents the basis function matrix, , Indicates the penalty factor. Indicates the first A vector of basis functions Denotes the square-induced norm on the reproducing kernel Hilbert space. , Represents the inverse of the kernel matrix. Represents the representation of the kernel Hilbert space;

[0102] The expression for the solution result is:

[0103] (17)

[0104] in, This represents the solution result for the time-varying coefficients. express An identity matrix of order 1;

[0105] Substitute all the solution results into formula (13) to obtain the feedforward control framework;

[0106] The expression for the feedforward control framework is:

[0107] (18)

[0108] in, , Indicates the first The first-order partial derivative of the time-varying coefficient of the fourth basis function when there are scheduling variables. Indicates the first The second-order partial derivative of the time-varying coefficient of the fourth basis function when there are scheduling variables.

[0109] This application provides a position control method for a synchronous belt drive system based on feedforward compensation, which has at least the following advantages:

[0110] (1) This application constructs a system model of a synchronous belt drive system and identifies the parameters of the system model, thereby accurately obtaining the transfer function model of the system model. At the same time, considering the disturbance control performance of the system model, the influence of the system model on nonlinear disturbances is effectively reduced.

[0111] (2) By constructing a linear parameter change feedforward compensation method, this application can effectively solve the problem that the traditional linear time polynomial feedforward control method is difficult to accurately compensate for the dynamic time-varying characteristics of the synchronous belt drive system during motion, thereby effectively improving the tracking performance of the synchronous belt drive system at the trajectory reversal point.

[0112] (3) This application constructs a position control framework that combines feedback control framework and feedforward control framework, which can effectively solve the problems of poor controllability and low motion accuracy of traditional control methods under system model uncertainty and nonlinear disturbance, thereby effectively improving the dynamic response performance of synchronous belt drive system. Attached Figure Description

[0113] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this application and, together with the description, serve to explain the principles of this application. It is obvious that the drawings described below are merely some embodiments of this application, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort.

[0114] Figure 1 This illustration shows the steps of a position control method for a synchronous belt drive system based on feedforward compensation in an exemplary embodiment of this application.

[0115] Figure 2 This diagram illustrates the structural structure of a system model of a synchronous belt drive system in an exemplary embodiment of this application.

[0116] Figure 3 This diagram illustrates the synchronous belt structure in an exemplary embodiment of this application.

[0117] Figure 4 A schematic diagram of the feedback control framework of the synchronous belt drive system in an exemplary embodiment of this application is shown;

[0118] Figure 5 Exemplary embodiments of this application are shown. Figure 4 A schematic diagram illustrating the relationship between the DOB and the synchronous belt drive system;

[0119] Figure 6 A schematic diagram showing the fitting of the transfer function of the synchronous belt drive system in an exemplary embodiment of this application is shown;

[0120] Figure 7This diagram illustrates the position control framework of the synchronous belt drive system in an exemplary embodiment of this application.

[0121] Figure 8 This illustration shows a parameterized schematic diagram of the position control framework of the synchronous belt drive system in an exemplary embodiment of this application;

[0122] Figure 9 The simulation experiment shown in this application employs... A schematic diagram of the trajectory of different parameters of a synchronous belt drive system controlled by a feedback control method;

[0123] Figure 10 This diagram illustrates a comparison of experimental results using different feedforward controls in the simulation experiments of this application.

[0124] In the diagram, 100 is the controller; 200 is the driver; 300 is the encoder; 400 is the motor; 500 is the reducer; 600 is the synchronous belt structure; 601 is the driving pulley; 602 is the driven pulley; 603 is the synchronous belt; 604 is the connecting plate; 605 is the linear guide; 606 is the load; 607 is the grating head; 700 is the grating ruler; 800 is the position error feedback node; 801 is the speed error feedback node; and 802 is the speed error feedback node. Output node of cascaded feedback control; 803, feedback compensation node; 804, interference node. Detailed Implementation

[0125] Exemplary embodiments will now be described more fully with reference to the accompanying drawings. However, these exemplary embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, they are provided to make this application more comprehensive and complete, and to fully convey the concept of the exemplary embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments.

[0126] Furthermore, the accompanying drawings are merely illustrative of this application and are not necessarily drawn to scale. The same reference numerals in the drawings denote the same or similar parts, and therefore repeated descriptions of them will be omitted. Some block diagrams shown in the drawings are functional entities and do not necessarily correspond to physically or logically independent entities. These functional entities can be implemented in software, in one or more hardware modules or integrated circuits, or in different network and / or processor devices and / or microcontroller devices.

[0127] This example implementation provides a position control method for a synchronous belt drive system based on feedforward compensation, such as... Figure 1 As shown, the method may include the following steps:

[0128] Step S101 of this embodiment: Construct a system model of the synchronous belt drive system, and obtain the standard transfer function of the system model using a dynamic model equivalent to the system model. Step S101 of this embodiment may include the following sub-steps:

[0129] Sub-step S1011: Construct the system model of the synchronous belt drive system. For example... Figure 2 As shown, the system model includes a loop system consisting of a controller 100, a driver 200, an encoder 300, a motor 400, a reducer 500, a synchronous belt structure 600, and a grating ruler 700.

[0130] Furthermore, such as Figure 3 As shown, the synchronous belt structure 600 includes:

[0131] The driving wheel 601 and the driven wheel 602 are arranged opposite to each other. A timing belt 603 is fitted on the driving wheel 601 and the driven wheel 602. A connecting plate 604 is provided on the timing belt 603.

[0132] The linear guide assembly includes linear guides 605 respectively disposed on both sides of the synchronous belt 603; a grating ruler 700 is disposed on the outer surface of one of the linear guides 605, and the grating ruler 700 is parallel to the track of the linear guide 605.

[0133] The load 606 is mounted on the linear guide rail assembly and slides along the linear guide rail assembly. The load 606 is connected to the connecting plate 604 and a grating head 607 is mounted on the load 606.

[0134] Sub-step S1012: Assign each element in the system model to at least one equivalent parameter, and construct a dynamic model using all equivalent parameters.

[0135] Furthermore, the expression for the dynamic model is:

[0136] (1)

[0137] in, Represents the equivalent moment of inertia. The second derivative represents the angular displacement of the driving wheel. This indicates the torque exerted by the motor on the drive wheel. Indicates the synchronization band at the 1st Stiffness when there are multiple scheduling variables Indicates the radius of the driving wheel. This indicates the angular displacement of the driving wheel. Represents the linear displacement of the load. This represents the relative damping coefficient between the driving pulley and the timing belt. The first derivative of the angular displacement of the driving wheel. The first derivative represents the linear displacement of the load. Indicates the quality of the load. This represents the damping coefficient of the linear guide rail assembly.

[0138] Sub-step S1013: Perform a Laplace transform on the dynamic model to obtain the intermediate equation set.

[0139] Furthermore, the expression for the intermediate equation system is:

[0140] (2)

[0141] in, This indicates the torque exerted by the motor on the drive wheel in the [number]th [phase]. Frequency response of a complex variable in the frequency domain This indicates the stiffness of the timing belt. The linear displacement of the load in the th... Frequency response of a complex variable in the frequency domain This indicates the angular displacement of the driving wheel in the th... Frequency response of a complex variable in the frequency domain.

[0142] Sub-step S1014: All of the intermediate equations in the set of intermediate equations Eliminating the term yields the standard transfer function.

[0143] Furthermore, the expression of the standard transfer function includes:

[0144] (3)

[0145] in, Indicates the first Standard transfer function in the frequency domain of a complex variable.

[0146] Step S102 of this embodiment: as follows Figure 4 As shown, a feedback control framework for constructing the system model is presented. The feedback control framework is a multi-loop control framework consisting of an input terminal, a feedback controller, a disturbance observer, a synchronous belt drive system, and an output terminal.

[0147] Depend on Figure 4 and Figure 5 As can be seen, in this embodiment, the input signal at the input terminal is the reference trajectory. The feedback controller in this embodiment is preferably... Cascaded feedback controller, It refers to the proportion. It refers to points. This refers to the proportional integral. A cascaded feedback controller includes a position loop and a velocity loop. The position loop includes... The controller, namely the proportional controller, includes a speed loop. The controller is a proportional-integral controller.

[0148] Position error feedback node 800 and speed error feedback node 801 are sequentially installed on the main line between the input end and the synchronous belt drive system. The cascaded feedback controller has an output node 802, a feedback compensation node 803, and an interference node 804.

[0149] Furthermore, the position error feedback node is the position error feedback node between the reference position of the reference trajectory and the actual position of the synchronization belt. The velocity error feedback node is the velocity error feedback node between the reference velocity of the reference trajectory and the actual velocity of the synchronization belt.

[0150] Figure 4 and Figure 5 In This represents the proportional gain of the position loop. This represents the proportional gain of the velocity loop. This represents the integral gain of the velocity loop. Indicates the first Frequency domain of a complex variable This indicates a disturbance observer.

[0151] Furthermore, the expression for the transfer function of the P / PI cascaded feedback controller is:

[0152] (4)

[0153] in, express The transfer function of a cascaded feedback controller. Represents the standard transfer function. This represents the proportional gain of the velocity loop. This represents the integral gain of the velocity loop. This represents the proportional gain of the position loop.

[0154] Furthermore, the expression for the closed-loop transfer function from the input to the output of the system model is:

[0155] (5)

[0156] in, This represents the closed-loop transfer function from the input to the output of the system model. Represents the actual transfer function The reverse.

[0157] Furthermore, the expression for the closed-loop transfer function of the system model from external disturbances to the output is:

[0158] (6)

[0159] in, This represents the closed-loop transfer function of the system model from external disturbances to the output.

[0160] The disturbance observer is connected in parallel with the synchronous belt drive system. The first port of the disturbance observer is connected to the feedback compensation node, the second port of the disturbance observer is connected to the interference node, and the third port of the disturbance observer is connected to the position error feedback node.

[0161] Step S103 of this embodiment: Parameter identification is performed on the standard transfer function to obtain the nominal function. Step S103 of this embodiment may include the following steps:

[0162] Sub-step S1031: Use a sinusoidal sweep frequency signal to sweep the frequency of the synchronous belt drive system in the open-loop state to obtain the response output signal.

[0163] Sub-step S1032: Perform a Fourier transform on the sinusoidal sweep frequency signal to obtain the linear displacement of the load at the th ... The actual frequency response of the complex variable in the frequency domain, using To represent; and to perform a Fourier transform on the response output signal to obtain the torque exerted by the motor on the drive wheel in the th order. The actual frequency response of a complex variable in the frequency domain, using To express.

[0164] Sub-step S1033: Utilizing the actual frequency response and actual frequency response Calculate the actual transfer function of the system model using express.

[0165] Sub-step S1034: Define the actual transfer function of the nominal model as... Then we get: G 0 s = G n s ∙[1+∆( s )] ,in, Represents the actual transfer function and actual transfer function The uncertainty between them.

[0166] Figure 6 Sub-step S1035: Transfer the actual transfer function and actual transfer function Perform fitting, and eliminate uncertainty during the fitting process. Thus, the nominal transfer function is obtained.

[0167] The fitting process includes:

[0168] First, the actual transfer function Parameterization is performed to obtain the parameter transfer function of the nominal model. , ,in, The first term representing the frequency response of the synchronous belt drive system Each frequency point, Indicates the parameter to be identified. This represents the set of coefficients of the numerator polynomial of the parameter to be identified. θ n =[ b 0 , b 1 ,…, b N n ] , This represents the set of coefficients of the denominator polynomial of the parameter to be identified. θ d =[1, a 1 , a 2 ,…, a N d ] , Indicates the first The frequency response of the numerator polynomial function of the parameter to be identified at each frequency point. , Represents the imaginary unit. The first polynomial function of the numerator Order coefficient, Indicates the first The frequency response of the denominator polynomial function of the parameter to be identified at each frequency point. , The function representing the denominator polynomial is the first... Order coefficient.

[0169] Secondly, based on the structure of the perturbation observer, we can obtain: ,in, Indicates the first A low-pass filter in the frequency domain with multiple complex variables. Let represent the closed-loop transfer function of the disturbance observer.

[0170] ,in, express The reverse, express The reverse.

[0171] Next, according to the small gain theorem, when At that time, the closed-loop system of the disturbance observer is stable. Represents the infinite norm, The closed-loop transfer function of the disturbance observer is represented in the first... Frequency response at each frequency point Indicating uncertainty In the Frequency response at each frequency point ,in, The actual transfer function of the system model is represented in the t-th... Frequency response at each frequency point.

[0172] This yields the objective function for parameter identification.

[0173] Furthermore, the expression for the objective function of parameter identification is:

[0174] (7)

[0175] in, This represents the objective function for parameter identification. Indicates the low-pass filter at the 1st Frequency response at each frequency point.

[0176] Finally, the infinity norm is approximated as the L2 norm, and the L2 norm is identified using a weighted least squares iterative algorithm to identify the parameters to be identified. .

[0177] Furthermore, the parameters to be identified The expression is:

[0178] (8)

[0179] in, Indicates the first At the frequency point, the first The parameters to be identified in the nth iteration are obtained from the first iteration. The weight function for the next iteration , Indicates the first At the frequency point The frequency response of the molecule polynomial function corresponding to the parameter to be identified in the next iteration. No. The parameters to be identified in the next iteration The parameter to be identified when the minimum value is reached. The value of , Indicates the first At the frequency point The frequency response of the denominator polynomial function corresponding to the parameter to be identified in the next iteration. Indicates the first The set of coefficients of the denominator polynomial of the parameter to be identified in the next iteration. Indicates the first At the frequency point The frequency response of the molecule polynomial function corresponding to the parameter to be identified in the next iteration. Indicates the first The set of numerator polynomial coefficients of the parameters to be identified in the next iteration; the identification results are substituted into the parameter transfer function of the nominal model. In this process, the nominal transfer function is obtained.

[0180] In this embodiment, by Figure 6 As shown in figures a and b, frequency response analysis reveals that the synchronous belt drive system exhibits a high amplitude response in the low-frequency range, effectively amplifying or tracking low-frequency signals. As the frequency increases, the amplitude gradually decreases, and the response begins to decay around 30Hz. The amplitude roll-off rate in the high-frequency range is approximately -80dB / decade, exhibiting characteristics of a fourth-order low-pass filter. Simultaneously, the synchronous belt drive system shows significant phase lag in the mid-to-high frequency region, with the maximum phase lag approaching -180 degrees. Furthermore, due to the high-order dynamic characteristics of the synchronous belt drive system, a resonance peak exists around 300Hz in the frequency response curve, reflecting its tendency to amplify input signals at specific frequencies. This needs to be considered in controller design. Overall, the synchronous belt drive system demonstrates good response capabilities in the low and mid-frequency ranges, making it suitable for trajectory tracking control requirements. However, the high-frequency characteristics require a well-designed controller to compensate for phase lag and gain roll-off, ensuring the stability and robustness of the synchronous belt drive system.

[0181] Step S104 of this embodiment: as follows Figure 7 and Figure 8 As shown, a feedforward control framework is constructed based on the feedback control framework and the nominal transfer function. The feedback control framework and the feedforward control framework are combined to form the position control framework of the system model. Step S104 in this embodiment may include the following sub-steps:

[0182] Sub-step S1041: Connect one end of the feedforward controller to the input terminal and the other end to the feedback compensation node.

[0183] Furthermore, in this embodiment, the feedforward controller is a linear time-varying feedforward controller, which includes a linear time-varying system.

[0184] Sub-step S1042: Adjust and set the feedforward controller according to the nominal transfer function to obtain the feedforward control framework. The process of sub-step S1042 is as follows:

[0185] The first step is to transform the expression of the standard transfer function into a linear time-varying transfer function and derive the inverse dynamic model of the linear time-varying transfer function.

[0186] Furthermore, the expression for the linear time-varying transfer function is:

[0187] (9)

[0188] The expression for the inverse dynamics model is:

[0189] (10)

[0190] in, This indicates the torque exerted by the motor on the drive wheel. Indicates the first One scheduling variable, The fourth-order time-varying coefficients representing the linear displacement of the load. The fourth derivative representing the linear displacement of the load. The fourth-order time-varying term representing the linear displacement of the load. The third-order time-varying coefficients representing the linear displacement of the load. The third derivative represents the linear displacement of the load. The third-order time-varying term representing the linear displacement of the load. The second-order time-varying coefficient represents the linear displacement of the load. The second derivative represents the linear displacement of the load. The second-order time-varying term representing the linear displacement of the load. The first-order time-varying term represents the linear displacement of the load. Indicates redundant time-varying coefficients. Indicates a redundant variable. , Indicates the synchronization band at the 1st The first-order partial derivative of the stiffness when there are multiple scheduling variables. , Indicates the synchronization band at the 1st The second-order partial derivative of stiffness with 1 scheduling variable , Indicates the first The first derivative of each scheduling variable Indicates the first The second derivative of each scheduling variable.

[0191] The second step is to parameterize the linear time-varying feedforward controller to obtain the polynomial feedforward of the linear time-varying feedforward controller, and then solve the polynomial feedforward to obtain the feedforward control framework.

[0192] Furthermore, the expression for the linear time-varying feedforward controller is:

[0193] (11)

[0194] in, To represent a linear time-varying system, The highest-order derivative representing the linear displacement of the load. Represents the scheduling variable. Represents the linear displacement of the load, when hour, The first linear displacement of the load The first derivative, when hour, The first linear displacement of the load Second integral. The highest-order derivative of the second-order integral of the torque exerted by the motor on the drive wheel. ,for They all .

[0195] Next, according to the basis function parameterization method, the zero dynamics of the system on the right side of equation (9) are ignored, i.e. ,get:

[0196] (12)

[0197] in, This represents a parameterized expression. This represents the number of all basis functions. Indicates the first When the scheduling variable is the first Time-varying coefficients of each basis function Indicates the first The expression for each basis function, Indicates reference input. Indicates the first When the scheduling variable is the first Time-varying coefficients of each basis function Indicates the first When the scheduling variable is the first Time-varying coefficients of each basis function Indicates the first When the scheduling variable is the first Time-varying coefficients of each basis function Indicates the first When the scheduling variable is the first Time-varying coefficients of each basis function The expression representing the first basis function, The expression representing the second basis function. The expression representing the third basis function. The expression representing the fourth basis function.

[0198] The third step is to take the second derivative of both sides of the equation (12) to obtain the polynomial feedforward.

[0199] Furthermore, the expression for polynomial feedforward is:

[0200] (13)

[0201] in, This represents the polynomial feedforward of a linear time-varying feedforward controller. This represents the fourth derivative of the reference input. This represents the third derivative of the reference input. This represents the second derivative of the reference input. This represents the first derivative of the reference input.

[0202] The fourth step is to solve the polynomial feedforward, including:

[0203] A reference signal sequence is input to the synchronous belt drive system in closed-loop state, and the actual linear displacement of the load at each moment is recorded. Actual scheduling variables and the actual torque exerted by the motor on the drive wheel ;

[0204] Calculate the quadratic exponential kernel function for each time step using the actual scheduling variables at each time step; and calculate the actual linear displacement of the load at each time step. and the actual torque exerted by the motor on the drive wheel The hyperparameters of the squared exponential kernel function at that moment are optimized using the marginal likelihood optimization method.

[0205] Furthermore, the expression for the quadratic exponential kernel function is:

[0206] (14)

[0207] in, Indicates the first The time-varying coefficient and the first time-varying coefficient Squared exponential kernel function between time-varying coefficients The variance of the squared exponential kernel function is represented by . This represents the length scale of the squared exponential kernel function. Let represent the first-order partial derivative of the reference scheduling variable; calculate the kernel matrix using each optimized quadratic exponential kernel function.

[0208] Furthermore, the expression for the kernel matrix is:

[0209] (15)

[0210] in, Represents the kernel matrix.

[0211] The objective function for each time-varying coefficient is solved separately, and the feedforward control framework is obtained using all the solution results.

[0212] Furthermore, the objective function for solving the time-varying coefficients is expressed as follows:

[0213] (16)

[0214] in, This represents the time-varying coefficients to be solved. Represents the time-varying coefficient when taking the minimum value. The value of , This represents the data vector obtained by the second-order integral of the actual torque exerted by the measuring motor on the drive wheel. , Indicates the first At that moment, Represents the time-varying coefficient parameter vector. , Indicates the first Transpose of the time-varying coefficient vector of the basis functions Indicates transpose. This represents the number of all basis functions. Represents the basis function matrix, , Indicates the penalty factor. Indicates the first A vector of basis functions Denotes the square-induced norm on the reproducing kernel Hilbert space. , Represents the inverse of the kernel matrix. This represents the reproducible kernel Hilbert space.

[0215] Furthermore, the expression for the solution result is:

[0216] (17)

[0217] in, This represents the solution result for the time-varying coefficients. express An identity matrix of order 1.

[0218] Substituting all the solution results into formula (13) respectively, the feedforward control framework is obtained.

[0219] The expression for the feedforward control framework is:

[0220] (18)

[0221] in, , Indicates the first The first-order partial derivative of the time-varying coefficient of the fourth basis function when there are scheduling variables. Indicates the first The second-order partial derivative of the time-varying coefficient of the fourth basis function when there are scheduling variables.

[0222] Sub-step S1043: Combine the feedback control framework and the feedforward control framework to form the position control framework of the system model.

[0223] Step S105 of this embodiment: Use the position control framework to perform position control on the synchronous belt drive system.

[0224] To verify the superiority of the position control method for synchronous belt drive system based on feedforward compensation proposed in this application, the following simulation experiments were conducted.

[0225] The hardware of the synchronous belt drive system experimental platform used in this simulation experiment includes: the dSPACE real-time hardware-in-the-loop simulation system, the host computer, and the synchronous belt drive system. The synchronous belt drive system mainly includes a servo motor, a reducer, a driving pulley, a driven pulley, a synchronous belt, a linear guide, a load, and a linear scale.

[0226] The driver is Tongyi System driver; servo motor is The company produces an EC042B-30M0-803-A22 brushless DC motor with a rated torque of 0.058 Nm; the grating ruler is a RU2LCFN03R steel belt grating ruler manufactured by LAMOTION, with an output resolution of 1 micrometer; in the synchronous belt drive system, the total length of the synchronous belt is 1.8 m, the radius of the synchronous pulley is 0.02 m, and the end load mass is 2 kg; before each experiment, the tension of the synchronous belt is pre-calibrated to ensure that the conditions of each experiment are as consistent as possible.

[0227] The grating ruler is a RU2LCFN03R steel belt grating ruler manufactured by LAMOTION, with an output resolution of 1 micrometer; in the synchronous belt drive system, the total length of the synchronous belt is 1.8m, the radius of the synchronous pulley is 0.02m, and the end load mass is 2kg; before each experiment, the tension of the synchronous belt is pre-calibrated to ensure that the conditions of each experiment are as consistent as possible.

[0228] like Figure 9 As shown, Figure 9In graph 'a', the horizontal axis represents time in seconds; the vertical axis represents the accelerometer limit. . Figure 9 In equation b, the horizontal axis represents time in seconds; the vertical axis represents the accelerometer limit. . Figure 9 In equation c, the horizontal axis represents time in seconds; the vertical axis represents the acceleration limit. . Figure 9 In the graph d, the horizontal axis represents time in seconds; the vertical axis represents the speed limit. . Figure 9 In the graph 'e', ​​the horizontal axis represents time in seconds; the vertical axis represents positional constraints. . Figure 9 In the graph f, the horizontal axis represents time in seconds; the vertical axis represents the reference constraint. This simulation experiment uses 5 sampling periods with a sampling time interval of 0.1 ms. Data acquisition and calculation during the experiment are handled by... Hardware and The software is complete.

[0229] Depend on Figure 9 As can be seen from a, c, and d, the reference trajectory is , will adopt The feedforward control strategy adopted in this application Comparison of feedforward control strategies with the use of alone Compared to the feedforward control strategy, the synchronous belt drive system shows some improvement in the tracking performance of the reference trajectory in the constant-speed segment after the introduction of feedforward control. However, the trajectory tracking performance of the two feedforward compensation methods is almost the same in the constant-speed segment; at the reversing point... The trajectory tracking performance of the feedforward control strategy is better than that of the method using... Feedforward control.

[0230] Depend on Figure 10 Figure 10 Figures b, e, and f show the trajectory tracking error map and a magnified local view. Compared to feedforward control strategies, the proposed method in this paper... The overall trajectory tracking error of the feedforward control strategy is lower than that of the feedforward control strategy. Feedforward control strategy.

[0231] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Therefore, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of the embodiments of this application, "multiple" means two or more, unless otherwise explicitly specified.

[0232] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of this application. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. In addition, those skilled in the art can combine and integrate the different embodiments or examples described in this specification.

[0233] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any person skilled in the art can easily conceive of various equivalent modifications or substitutions within the scope of the technology disclosed in this application, and these modifications or substitutions should all be covered within the scope of protection of this application.

[0234] Other embodiments of this application will readily occur to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of this application that follow the general principles of this application and include common knowledge or customary techniques in the art not disclosed herein.

Claims

1. A position control method for a synchronous belt drive system based on feedforward compensation, characterized in that, The method includes the following steps: A system model of a synchronous belt drive system is constructed, and the standard transfer function of the system model is obtained using a dynamic model equivalent to the system model. A feedback control framework for the system model is constructed, which includes a multi-loop control framework consisting of an input terminal, a feedback controller, a disturbance observer, the synchronous belt drive system, and an output terminal. The standard transfer function is subjected to parameter identification to obtain the nominal transfer function; Based on the feedback control framework, a feedforward control framework is constructed according to the nominal transfer function. The feedback control framework and the feedforward control framework are combined to form the position control framework of the system model. The position control framework is used to control the position of the synchronous belt drive system.

2. The position control method for a synchronous belt drive system based on feedforward compensation according to claim 1, characterized in that, The system model includes a loop system consisting of a controller, a driver, an encoder, a motor, a reducer, a synchronous belt structure, and a grating ruler; the grating ruler itself constitutes the loop system. The synchronous band structure includes: A driving wheel and a driven wheel are arranged opposite each other, and a synchronous belt is fitted on the driving wheel and the driven wheel. A connecting plate is provided on the synchronous belt. A linear guide rail assembly, comprising linear guide rails respectively disposed on both sides of the synchronous belt; wherein the grating ruler is disposed on the outer surface of one of the linear guide rails, and the grating ruler is parallel to the track of the linear guide rail; The load is mounted on the linear guide rail assembly and slides along the linear guide rail assembly, and the load is connected to the connecting plate; the load is equipped with a grating head.

3. The position control method for a synchronous belt drive system based on feedforward compensation according to claim 2, characterized in that, The step of obtaining the standard transfer function of the system model using a dynamic model equivalent to the system model includes: Each element in the system model is associated with at least one equivalent parameter, and the dynamic model is constructed using all equivalent parameters. The expression for the dynamic model is: (1) in, Represents the equivalent moment of inertia. The second derivative represents the angular displacement of the driving wheel. This indicates the torque exerted by the motor on the drive wheel. Indicates the synchronization band at the 1st Stiffness when there are multiple scheduling variables Indicates the radius of the driving wheel. This indicates the angular displacement of the driving wheel. Represents the linear displacement of the load. This represents the relative damping coefficient between the driving pulley and the timing belt. The first derivative of the angular displacement of the driving wheel. The first derivative representing the linear displacement of the load. Indicates the quality of the load. This represents the damping coefficient of the linear guide assembly; The dynamic model is subjected to a Laplace transform to obtain an intermediate set of equations; The expression for the intermediate equation set is: (2) in, This indicates the torque exerted by the motor on the drive wheel in the [number]th [phase]. Frequency response of a complex variable in the frequency domain This indicates the stiffness of the timing belt. The linear displacement of the load in the th... Frequency response of a complex variable in the frequency domain This indicates the angular displacement of the driving wheel in the th... Frequency response of a complex variable in the frequency domain; All of the intermediate equations After eliminating the terms, the standard transfer function is obtained. The expression of the standard transfer function includes: (3) in, Indicates the first Standard transfer function in the frequency domain of a complex variable.

4. The position control method for a synchronous belt drive system with feedforward compensation according to claim 3, characterized in that, The input terminal includes the input reference trajectory; The feedback controller is Cascaded feedback controller, the The cascaded feedback controller includes a position loop and a velocity loop; the position loop includes... The controller, i.e., the proportional controller; the speed loop includes The controller is a proportional-integral controller. A position error feedback node and a speed error feedback node are sequentially arranged on the main line between the input terminal and the synchronous belt drive system. Cascaded feedback control includes output nodes, feedback compensation nodes, and interference nodes. The disturbance observer is connected in parallel with the synchronous belt drive system, and the first port of the disturbance observer is connected to the feedback compensation node, the second port of the disturbance observer is connected to the main line between the interference node and the synchronous belt drive system, and the third port of the disturbance observer is connected to the output terminal and the position error feedback node. The position error feedback node is a position error feedback node that is the reference position of the reference trajectory signal and the actual position of the synchronization band; The speed error feedback node is a speed error feedback node that is the reference speed of the reference trajectory signal and the actual speed of the synchronization belt.

5. The position control method for a synchronous belt drive system with feedforward compensation according to claim 4, characterized in that, The step of parameter identification of the standard transfer function to obtain the nominal transfer function includes: The synchronous band drive system in the open-loop state is frequency swept using a sinusoidal sweep frequency signal to obtain a response output signal; Performing a Fourier transform on the sinusoidal sweep signal yields the linear displacement of the load at the th ... The actual frequency response of the complex variable in the frequency domain The response output signal is then subjected to a Fourier transform to obtain the torque exerted by the motor on the drive wheel in the [missing information]. The actual frequency response of the complex variable in the frequency domain ; Using the actual frequency response and the actual frequency response Calculate the actual transfer function of the system model. ; Define the actual transfer function of the nominal model as Then we get: ,in, Represents the actual transfer function and the actual transfer function Uncertainty between them; The actual transfer function and the actual transfer function Perform a fitting process to eliminate the aforementioned uncertainties. The nominal transfer function is obtained.

6. The position control method for a synchronous belt drive system with feedforward compensation according to claim 5, characterized in that, The expression for the transfer function of the P / PI cascaded feedback controller is: (4) in, express The transfer function of a cascaded feedback controller. Represents the standard transfer function. This represents the proportional gain of the velocity loop. This represents the integral gain of the velocity loop. Indicates the proportional gain of the position loop; The expression for the closed-loop transfer function from the input to the output of the system model is: (5) in, This represents the closed-loop transfer function from the input to the output of the system model. Represents the actual transfer function The reverse; The expression for the closed-loop transfer function of the system model from external disturbances to the output is: (6) in, This represents the closed-loop transfer function of the system model from external disturbances to the output.

7. The position control method for a synchronous belt drive system with feedforward compensation according to claim 5, characterized in that, The actual transfer function and the actual transfer function The fitting process includes: The actual transfer function The parameterization process is performed to obtain the parameter transfer function of the nominal model. , ,in, The first term representing the frequency response of the synchronous belt drive system Each frequency point, Indicates the parameter to be identified. This represents the set of coefficients of the numerator polynomial of the parameter to be identified. , This represents the set of coefficients of the denominator polynomial of the parameter to be identified. , Indicates the first The frequency response of the numerator polynomial function of the parameter to be identified at each frequency point. , Represents the imaginary unit. The first polynomial function of the numerator Order coefficient, Indicates the first The frequency response of the denominator polynomial function of the parameter to be identified at each frequency point. , The function representing the denominator polynomial is the first... Order coefficient; Based on the structure of the disturbance observer, we can obtain: ,in, Indicates the first A low-pass filter in the frequency domain of the complex variable. This represents the closed-loop transfer function of the perturbation observer. ,in, express The reverse, express The reverse; According to the small gain theorem, when At that time, the closed-loop system of the disturbance observer is stable. Represents the infinite norm, The closed-loop transfer function of the disturbance observer is represented in the first... Frequency response at each frequency point Indicating uncertainty In the Frequency response at each frequency point ,in, The actual transfer function of the system model is represented in the t-th... Frequency response at each frequency point; This yields the objective function for parameter identification; The expression for the objective function identified by the parameters is: (7) in, This represents the objective function for parameter identification. Indicates the low-pass filter at the 1st Frequency response at each frequency point; The infinity norm is approximated as a L2 norm, and the L2 norm is identified using a weighted least squares iterative algorithm to identify the parameter to be identified. ; The parameters to be identified The expression is: (8) in, Indicates the first At the frequency point, the first The parameters to be identified in the nth iteration are obtained from the first iteration. The weight function for the next iteration , Indicates the first At the frequency point The frequency response of the molecule polynomial function corresponding to the parameter to be identified in the next iteration. No. The parameters to be identified in the next iteration The parameter to be identified when the minimum value is reached. The value of , Indicates the first At the frequency point The frequency response of the denominator polynomial function corresponding to the parameter to be identified in the next iteration. Indicates the first The set of coefficients of the denominator polynomial of the parameter to be identified in the next iteration. Indicates the first At the frequency point The frequency response of the molecule polynomial function corresponding to the parameter to be identified in the next iteration. Indicates the first The set of coefficients of the numerator polynomial of the parameters to be identified in the next iteration; the identification result is substituted into the parameter transfer function of the nominal model. In this process, the nominal transfer function is obtained.

8. The position control method for a synchronous belt drive system based on feedforward compensation according to claim 7, characterized in that, The step of constructing a feedforward control framework based on the nominal transfer function on the basis of the feedback control framework, and combining the feedback control framework and the feedforward control framework to form the position control framework of the system model includes: One end of the feedforward controller is connected to the input terminal, and the other end of the feedforward controller is connected to the feedback compensation node; the feedforward controller is a linear time-varying feedforward controller, and the linear time-varying feedforward controller includes a linear time-varying system; The feedforward control framework is obtained by adjusting and setting the feedforward controller according to the nominal transfer function. The feedback control framework and the feedforward control framework are combined to form the position control framework of the system model.

9. The position control method for a synchronous belt drive system based on feedforward compensation according to claim 8, characterized in that, The step of adjusting the feedforward controller according to the nominal transfer function to obtain the feedforward control framework includes: The expression of the standard transfer function is transformed into a linear time-varying transfer function, and the inverse dynamic model of the linear time-varying transfer function is derived. The expression for the linear time-varying transfer function is: (9) The expression for the inverse dynamics model is: (10) in, This indicates the torque exerted by the motor on the drive wheel. Indicates the first One scheduling variable, The fourth-order time-varying coefficients representing the linear displacement of the load. The fourth derivative representing the linear displacement of the load. The fourth-order time-varying term representing the linear displacement of the load. The third-order time-varying coefficients representing the linear displacement of the load. The third derivative represents the linear displacement of the load. The third-order time-varying term representing the linear displacement of the load. The second-order time-varying coefficient represents the linear displacement of the load. The second derivative represents the linear displacement of the load. The second-order time-varying term representing the linear displacement of the load. The first-order time-varying term represents the linear displacement of the load. Indicates redundant time-varying coefficients. Indicates a redundant variable. , Indicates the synchronization band at the 1st The first-order partial derivative of the stiffness when there are multiple scheduling variables. , Indicates the synchronization band at the 1st The second-order partial derivative of stiffness with 1 scheduling variable , Indicates the first The first derivative of each scheduling variable Indicates the first The second derivatives of each scheduling variable; The linear time-varying feedforward controller is parameterized to obtain the polynomial feedforward of the linear time-varying feedforward controller, and the polynomial feedforward is solved to obtain the feedforward control framework. The expression for the linear time-varying feedforward controller is: (11) in, To represent a linear time-varying system, The highest-order derivative representing the linear displacement of the load. Represents the scheduling variable. Represents the linear displacement of the load, when hour, The first linear displacement of the load The first derivative, when hour, The first linear displacement of the load Second integral. The highest-order derivative of the second-order integral of the torque exerted by the motor on the drive wheel. ,for They all ; According to the basis function parameterization method, the zero dynamics of the system on the right side of equation (9) are ignored, i.e. ,get: (12) in, This represents a parameterized expression. This represents the number of all basis functions. Indicates the first When the scheduling variable is the first Time-varying coefficients of each basis function Indicates the first The expression for each basis function, Indicates reference input. Indicates the first When the scheduling variable is the first Time-varying coefficients of each basis function Indicates the first When the scheduling variable is the first Time-varying coefficients of each basis function Indicates the first When the scheduling variable is the first Time-varying coefficients of each basis function Indicates the first When the scheduling variable is the first Time-varying coefficients of each basis function The expression representing the first basis function, The expression representing the second basis function. The expression representing the third basis function, The expression representing the fourth basis function; By taking the second derivative of both sides of equation (12), the polynomial feedforward is obtained. The expression for the polynomial feedforward is: (13) in, This represents the polynomial feedforward of a linear time-varying feedforward controller. This represents the fourth derivative of the reference input. This represents the third derivative of the reference input. This represents the second derivative of the reference input. This represents the first derivative of the reference input; The polynomial feedforward is solved, and the solution process includes: A reference signal sequence is input to the synchronous belt drive system in closed-loop state, and the actual linear displacement of the load at each moment is recorded. Actual scheduling variables and the actual torque exerted by the motor on the drive wheel ; Calculate the quadratic exponential kernel function for each time step using the actual scheduling variables at each time step; and calculate the actual linear displacement of the load at each time step. and the actual torque exerted by the motor on the drive wheel And the hyperparameters of the squared exponential kernel function at this moment are optimized using the marginal likelihood optimization method; The expression for the quadratic exponential kernel function is: (14) in, Indicates the first The time-varying coefficient and the first time-varying coefficient Squared exponential kernel function between time-varying coefficients The variance of the squared exponential kernel function is represented by . This represents the length scale of the squared exponential kernel function. Let the first-order partial derivative of the reference scheduling variable be used; calculate the kernel matrix using each optimized quadratic exponential kernel function; The expression for the kernel matrix is: (15) in, Represents the kernel matrix; The objective function of each time-varying coefficient is solved separately, and the feedforward control framework is obtained using all the solution results.

10. The position control method for a synchronous belt drive system based on feedforward compensation according to claim 9, characterized in that, The step of solving the objective function for each of the time-varying coefficients and obtaining the feedforward control framework using all the solution results includes: constructing the objective function for the time-varying coefficients using the kernel regularized least squares method and solving it; The objective function for solving the time-varying coefficients is expressed as follows: (16) in, This represents the time-varying coefficients to be solved. Represents the time-varying coefficient when taking the minimum value. The value of , This represents the data vector obtained by the second-order integral of the actual torque exerted by the measuring motor on the drive wheel. , Indicates the first At that moment, Represents the time-varying coefficient parameter vector. , Indicates the first Transpose of the time-varying coefficient vector of the basis functions Indicates transpose. This represents the number of all basis functions. Represents the basis function matrix, , Indicates the penalty factor. Indicates the first A vector of basis functions Denotes the square-induced norm on the reproducing kernel Hilbert space. , Represents the inverse of the kernel matrix. Represents the representation of the kernel Hilbert space; The expression for the solution result is: (17) in, This represents the solution result for the time-varying coefficients. express An identity matrix of order 1; Substituting all the solution results into formula (13) respectively, the feedforward control framework is obtained. The expression for the feedforward control framework is: (18) in, , Indicates the first The first-order partial derivative of the time-varying coefficient of the fourth basis function when there are scheduling variables. Indicates the first The second-order partial derivative of the time-varying coefficient of the fourth basis function when there are scheduling variables.

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