High speed servo laser blanking line control method and system

Abstract

This invention discloses a high-speed servo laser blanking line control method and system; the invention relates to the field of dynamic control and compensation technology for laser cutting equipment; it collects core position status data of dual Y-axis, completes hard synchronization and timestamp alignment of multi-channel data; it receives full closed-loop position feedback data P1 from dual Y-axis grating rulers, position data P2 from dual Y-axis servo motor encoders, and theoretical positions P3 of dual Y-axis issued by the motion controller. The servo drive unit includes servo motors and matching servo drivers respectively connected to the No. 1 Y-axis motion mechanism and the No. 2 Y-axis motion mechanism; this invention adopts a dual-branch parallel design of "SMO sliding mode observer + FFRLS recursive least squares with forgetting factor". The SMO branch is specifically adapted for rapid observation of abrupt nonlinear disturbances, realizing finite-time convergence identification of step loss; the FFRLS branch is specifically adapted for accurate identification of slow time-varying continuous parameter drift, quantifying the expected step loss caused by parameter changes.

Description

Technical Field

[0001] This invention relates to the field of dynamic control and compensation technology for laser cutting equipment, specifically to multi-axis synchronous control technology and nonlinear disturbance observation and identification technology, as well as application fields such as intelligent manufacturing equipment, industrial automation and advanced materials processing, and particularly to a high-speed follow-up laser blanking line control method and system. Background Technology

[0002] The follow-up laser feeding line control technology focuses on achieving dynamic and precise cutting of continuously fed sheet metal. Its core mechanism lies in dynamically adjusting the motion trajectory and cutting rate of the laser cutting head by real-time monitoring of the spatial position and attitude parameters of the sheet metal. This enables a multi-laser head system to collaboratively complete segmented cutting tasks and achieve high-precision blade connection. [1] The fundamental principle of laser blanking line control technology integrates closed-loop servo control theory and feedforward compensation strategy, dynamically correcting the cutting path planning based on the real-time operating status of the sheet material.

[0003] Conventional laser cutting processes utilize the high energy density of laser beams to melt, evaporate, or ablate materials, offering significant advantages such as high precision, high efficiency, and non-contact processing. [2][3][4] However, with the continuous upgrading of automation and intelligent technologies, the requirements for the precision of cutting process control are also gradually increasing. [5][6] The follow-up laser blanking line integrates the functional modules of a stamping blanking line, a laser cutting machine, and a fixed-length shearing line, enabling efficient processing of complex-shaped sheet materials, significantly improving material utilization and reducing production costs. [1][7] Taking the automotive manufacturing industry as an example, this technology can process complex structural parts made of high-strength steel plates, meeting the dual requirements of lightweighting and high performance in automobiles. [1] .

[0004] When processing ultra-large workpieces (e.g., length exceeding 3 meters) or when the Y-axis spacing between dual cutting heads is large (e.g., exceeding 1.5 meters), the follow-up laser blanking line is prone to lateral (Y-axis) module misalignment, resulting in a significant reduction in tool engagement accuracy. Long-term operation or load fluctuations may cause elastic / plastic deformation of the guide rollers, while the gap in the transmission chain gradually accumulates during movement, causing unexpected displacement of the Y-axis module.

[0005] Insufficient rigidity of the Y-axis module: During high-speed or high-load cutting, if the Y-axis module structure lacks sufficient rigidity, it is prone to vibration or deformation, exacerbating the offset phenomenon. [8] .

[0006] The combined effect of these factors makes it difficult for the two Y-axis modules to maintain strictly synchronized movement, resulting in cumulative Y-axis offset at the tool receiving position. For example, when the tool receiving hole diameter is designed to be too small (e.g., less than Φ1.2 mm), even a micron-level Y-axis positional deviation can cause the laser beam to fail to accurately align with the edge of the preceding cut, resulting in insufficient cutting precision and the formation of slag or burrs at the cut edge, affecting surface quality. [9]

[10] Laser cutting precision is highly sensitive to parameters such as laser spot size, cutting speed, assist gas pressure, and defocusing amount. Even slight changes in these parameters can affect the cutting quality and precision.

[11]

[12] For example, in waterjet cutting of ultra-thick carbon fiber reinforced polymer (CFRP) laminates, kerf taper and cutting front drag are key limiting factors and are closely related to cutting parameters.

[13] Similar issues also affect the quality of tool joining during laser cutting.

[0007] Traditional solutions often employ static optimization loop control strategies. The core of this approach is to construct a fixed compensation table by offline measurement of parameters such as repeatability error of each axis and lead screw pitch error during system calibration, and then make corrections by looking up the table during operation. This method is effective for cutting small workpieces or in stable environments. [6] However, this method has significant limitations and cannot fundamentally solve the Y-axis module offset problem in the cutting of large workpieces. This is because static control is based on the assumption that system parameters are constant and disturbances are negligible, but in actual production, mechanical systems are significantly affected by time-varying nonlinear disturbances such as temperature drift, wear, and sudden load changes. The preset compensation table cannot identify and compensate for dynamic Y-axis step loss online, especially when cutting conditions (such as material thickness, cutting speed, etc.) are affected. [4] When environmental factors change, static compensation cannot be adjusted in a timely manner, leading to a decrease in accuracy.

[0008] Therefore, this invention proposes a high-speed follow-up laser blanking line control method and system.

[0009] The cited references for this invention are as follows: [1]Leng Z, Cai C, Yang J, et al. Brief Introduction to Theapplication of Laser Blanking Line on Automobile Sheets [J]. IOP ConferenceSeries: Materials Science and Engineering, 2018, 452: 022081. [2]Sang S, Zhou K, Zhou Y, et al. Principle, present situation, anddevelopment trend of laser cutting [C]. International Conference onOptoelectronic Materials and Devices (ICOMD 2022), 2023: 83. [3]Zhongfa M. Research Status of Laser Cutting Technology [Z](2022–01–01). [4]Knauer D. Matching the Intensity of the Laser to the Speed [J].Laser Technik Journal, 2018, 15(3): 59–61. [5]Huang W, Li W, Yu J, et al. Research on Laser Cutting QualityOptimization Strategy Based on Intelligent Control Technology [J]. InnovativeApplications of AI, 2025, 2(2): 52–60. [6]Radu-Eugen B, Sorin T, Cristina B, et al. Improving the dynamicbehavior and working accuracy of the CNC laser cutting machines [C]. 201212th International Conference on Control Automation Robotics& Vision(ICARCV), 2012: 1642–1647. [7]ZHANG J, PANG L, CHEN M, et al. Control and Analysis of MaterialUtilization Ratio of Laser Plates [Z](2014–01–01). [8]Thombansen U, Hermanns T, Stoyanov S. Setup and Maintenance ofManufacturing Quality in CO2 Laser Cutting [J]. Procedia CIRP, 2014, 20: 98–102. [9]Wu Z, Liu Y, Wang S, et al. Research on the influence of laserprocess parameters on the quality of magnesium alloy laser Cutting [J]. TheInternational Journal of Advanced Manufacturing Technology, 2024, 132(11–12):6069–6083.

[10] Su C-T, Hsiao Y-H, Chang C-C. Parameter Optimization Design forTouch Panel Laser Cutting Process [J]. IEEE Transactions on AutomationScience and Engineering, 2012, 9(2): 320–329.

[11] Liang W, Dong H, Miao L, et al. Predicting the geometricmorphology of water jet machining in Ultra-thick CFRP laminates based onanalytical Modeling [J]. Composites Part A: Applied Science andManufacturing, 2024, 180: 108055.

[12] Yang K, Cui F. Advanced PID-controlled temperature prediction and management for optimized automated laser cutting Processes [J]. Proceedings of the Institution of Mechanical Engineers, Part E: Journal of ProcessMechanical Engineering, 2025.

[13] Huang Z, Chen J, Tu Y. A Two-step feedrate planning of polygonalpath for micro laser-cutting Machines [J]. The International Journal of Advanced Manufacturing Technology, 2019, 103(9–12): 4135–4145. Summary of the Invention In view of this, the present invention aims to provide a high-speed servo laser blanking line control method and system to solve or alleviate the technical problems existing in the prior art, namely, how to identify and compensate for dynamic loss of synchronization in the Y-axis dual axes, and to at least provide a beneficial option for this; the technical solution of the present invention is implemented as follows: The first aspect is the control method for a high-speed follow-up laser blanking line, including: P1. Synchronously acquire position feedback data and theoretical position commands for the dual Y-axis to generate a standardized dual-axis system state vector; P2. An architecture combining the SMO sliding mode observer branch and the FFRLS recursive least squares branch with forgetting factor is adopted to identify abrupt nonlinear disturbances and slow time-varying parameter drifts in the dual Y-axis. The real-time time-varying parameters identified by the FFRLS branch are synchronously updated to the state space model of the SMO branch, and the abrupt disturbances observed by the SMO branch are canceled out in advance from the identification model of the FFRLS branch. The observed step loss caused by the abrupt disturbance and the expected step loss in slow time-varying mode are calculated. P3. Based on the DS evidence theory, a staggered quantity identification framework is constructed. The two staggered quantity results are used as independent evidence bodies to construct a basic probability allocation function. Evidence fusion is completed through the DS combination rule, and the optimal estimate of the total staggered quantity of the two axes is obtained by solving the problem. P4. Composite Closed-Loop Compensation and Command Execution: A composite compensation architecture of feedforward + feedback is adopted. The feedforward compensation amount matching the servo system is generated based on the optimal step loss amount and single-axis position deviation. The cross-coupled feedback compensation amount is generated based on the dual-axis synchronization error and synchronously sent to the dual Y-axis servo system to complete the real-time closed-loop compensation for dynamic step loss of the dual Y-axis.

[0010] In one embodiment, the filtering and noise reduction process sequentially performs amplitude limiting filtering and moving average filtering: ;

[0011] Where i is the sequence number of the 6 independent location channels, covering all original location data; y i (k) represents the original input data for the i-th channel; This is the effective value of the limiting filter for this channel in the previous cycle; The i-th channel is the smoothed data after final filtering; N is the sliding window length, and the summation term covers the current period and the previous N periods. The effective value of the limiting filter for one cycle.

[0012] In one embodiment, a non-singular terminal sliding surface is designed in the SMO sliding mode observer branch: ; Overall sliding surface matrix form ; Where i = 1, 2, 3, 4, 5, corresponding to the 5 dimensions of the observation error vector; It is the absolute value of the observation error; α and β are the positive weighting coefficients of the sliding surface, and p and q are the coefficients that satisfy 1 ) is a symbolic function. This is the state observation error vector of the two-axis system.

[0013] In one embodiment, the iterative method in the FFRLS recursive least squares branch with forgetting factor includes: Calculate the FFRLS gain matrix ; Update the time-varying parameter vector to be identified ; Wherein, the gain matrix K(k) determines the magnitude of the correction to the parameter estimate by the new sampled data; λ is the forgetting factor; For parameter estimation, the error covariance matrix, Let z(k) be the regression data matrix of the linear regression model, and z(k) be the system output vector of the linear regression model. The vector of time-varying parameters to be identified. ​

[0014] In one embodiment, the method for canceling abrupt perturbations in the SMO→FFRLS coupling direction is as follows: ; ; in, This is the linear regression output vector after perturbation cancellation. and The biaxial rapid change disturbance value observed in the SMO branch; the FFRLS→SMO coupling direction only performs real-time updates of the state space model parameters when the FFRLS parameter iteration convergence flag P′(k)=1.

[0015] In one embodiment, the DS evidence fusion method includes: taking the median of the maximum probability interval as the optimal estimate S of the total biaxial step loss. ; ; Find the interval with the largest BPA after fusion, and take the midpoint of the interval as the optimal estimate S of the total step loss of the two axes; Where K is the conflict coefficient between the two pieces of evidence; ΔL i For the interval unit in the step loss identification framework, m1( m2 ( ) are the basic probability allocation functions for the two evidence bodies, m( ) is the integrated probability allocation function after fusion.

[0016] In one embodiment, the method for the feedforward compensation includes: ; Where i corresponds to different Y-axis, This is the single-axis feedforward compensation amount. For velocity feedforward gain, Theoretical speed command, For acceleration feedforward gain, The equivalent inertia identified by FFRLS Theoretical acceleration command, This represents the total perturbation value observed by SMO. For position deviation feedforward gain, This represents the absolute position deviation of a single axis.

[0017] In one embodiment, the method for cross-coupling feedback compensation includes: Dual-axis real-time synchronization error calculation: ; in, , Here, S(k) represents the measured position of the dual-axis grating, and S(k) is the optimal estimate of the step loss output from the DS fusion; s (k) represents the real-time synchronization error of the two axes; Adaptive weight allocation for biaxial compensation: ; ; Employing a PI-type cross-coupling control law: ; Among them, the proportional term It is the fast response synchronization error; integral term Eliminate steady-state static error; The output of the cross-coupling control law is distributed to the position loop inputs of the two axes according to adaptive weights to achieve closed-loop control of the synchronization error. ; Among them, the symbol design ensures the direction correction to reduce synchronization error in both axes.

[0018] The core causes of dynamic step loss in the Y-axis can be divided into two main categories: rapid abrupt disturbances such as chain gap impact and sudden load changes, and slow time-varying parameter drifts such as component wear, temperature drift, and guide roller deformation. Traditional single algorithms cannot simultaneously cover both types of causes. Compared with existing technologies, the advantages of this invention are: I. This invention employs a dual-branch parallel design of "SMO sliding mode observer + FFRLS recursive least squares with forgetting factor". The SMO branch is specifically adapted for rapid observation of abrupt nonlinear perturbations, achieving finite-time convergence identification of step loss. The FFRLS branch is specifically adapted for accurate identification of slow time-varying continuous parameter drift, quantifying the expected step loss caused by parameter changes. The two branches completely cover all the core causes of dynamic step loss in the Y-axis dual axes, thoroughly solving the identification blind spot problem of traditional static compensation and single algorithms. Whether it is instantaneous impact step loss during high-speed commutation or cumulative drift step loss during long-term operation, accurate capture can be achieved.

[0019] Second, this invention synchronously updates the real-time time-varying parameters identified by FFRLS to the SMO state-space model, solving the problems of decreased observation accuracy and increased chattering caused by the mismatch between the fixed model parameters of the traditional sliding mode observer and the actual system. This allows the steady-state error of SMO observation of sudden step loss to be stably controlled at the μm level. Furthermore, the invention pre-emptively cancels out the fast-changing abrupt disturbances observed by SMO as known terms from the FFRLS linear regression model, eliminating the interference of abrupt disturbances on the identification of slow time-varying parameters. This avoids the problems of sensitivity to abrupt disturbances and divergence in identification found in the traditional RLS algorithm, ensuring the long-term stability of the slow time-varying step loss prediction. These two aspects form a virtuous cycle of "improved observation accuracy → improved identification accuracy → model optimization → further improved observation accuracy," achieving a bidirectional leap in Y-axis step loss identification accuracy.

[0020] Third, this invention incorporates the decision-making of the DS theory algorithm to output a unique and reliable benchmark for the step loss. To address the uncertainty of the two identification results, it leverages the complementary advantages of orthogonal and rule fusion and designs an engineering correction strategy for extreme working conditions with severe conflict of evidence. Finally, it outputs the most accurate optimal estimate of the dual-axis dynamic step loss under all working conditions.

[0021] Fourth, this invention addresses the inherent lag problem of traditional feedback control's "deviation first, then compensation" approach. The proposed solution directly converts the fused optimal estimate of the step loss, the single-axis absolute position deviation, and the total disturbance value observed by the SMO into an electrical signal feedforward compensation quantity that matches the torque loop / velocity loop interface of the servo system. This compensation quantity is directly connected to the servo torque loop feedforward interface, bypassing the traditional serial feedback delay of "position loop → velocity loop → torque loop." Compensation for Y-axis step loss caused by sudden disturbances can be completed within two control cycles, completely solving the problems of lag response and poor step loss suppression in traditional compensation schemes under high-speed dynamic conditions. Attached Figure Description

[0022] Figure 1 This is a schematic diagram of the model architecture of the present invention; Figure 2 This is a schematic diagram of the location data acquisition process of the present invention; Figure 3 This is a schematic diagram of the SMO technical solution of the present invention; Figure 4 This is a schematic diagram of the FFRLS technical solution of the present invention; Figure 5 This is a schematic diagram of the coupling process of the present invention; Figure 6 This is a schematic diagram illustrating the coupling of the DS evidence theory model of the present invention; Figure 7 This is a schematic diagram of the feedforward + feedback composite compensation architecture of the present invention; Figure 8 This is a schematic diagram of the system hardware architecture of the present invention; Figure 9 This is a schematic diagram illustrating the technical effects of the single-axis position tracking deviation calculation module and the standardized dual-axis system state vector generation module of the present invention. Figure 10 This is a comparative schematic diagram of the error convergence characteristics of the dual-axis linkage non-singular terminal sliding mode surface function of the present invention; Figure 11 This is a schematic diagram illustrating the execution effect of the FFRLS parameter iteration module of the present invention; Figure 12 This is a schematic diagram illustrating the effect of the RLS iterative calculation of the forgetting factor in this invention; Figure 13 This is a schematic diagram illustrating the technical effect of the bidirectional coupling module of the present invention; Figure 14 This is a schematic diagram illustrating the effect of the DS evidence theory decision-level fusion layer technology of the present invention; Figure 15 This is a schematic diagram illustrating the technical effect of the step loss compensation generation layer of the present invention; Figure 16 This is a schematic diagram showing the performance comparison of three sets of comparative schemes in the experimental examples of the present invention on three core evaluation indicators.

[0023] Figure 17 This is a schematic diagram of the three-dimensional variation trends of the three sets of comparative schemes in the experimental examples of the present invention.

[0024] Figure 18 This is a schematic diagram illustrating the effect of an application example of the present invention. Detailed Implementation

[0025] To make the above-mentioned objects, features, and advantages of the present invention more apparent and understandable, the specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings. Many specific details are set forth in the following description to provide a thorough understanding of the present invention. However, the present invention can be practiced in many other ways different from those described herein, and those skilled in the art can make similar modifications without departing from the spirit of the present invention. Therefore, the present invention is not limited to the specific embodiments disclosed below; Example 1: As Figures 1-2 As shown. This embodiment focuses on the parameter setting and filtering algorithm implementation of the data preprocessing layer. This layer, as the foundation of the entire out-of-step identification and compensation system, is responsible for denoising and smoothing the dual Y-axis position feedback signals, generating standardized state vectors, and providing high-quality input data for the subsequent identification layer.

[0026] In this embodiment, the data preprocessing layer includes a position signal filtering module, a single-axis velocity calculation module, a single-axis position tracking deviation calculation module, and a standardized dual-axis system state vector generation module. The specific implementation of each module is as follows: (1) Position signal filtering module: The first-order low-pass filtering algorithm is used to filter the measured position signal of the dual Y-axis grating ruler to remove high-frequency noise caused by electromagnetic interference and mechanical vibration in the industrial field. The filtering formula is as follows: ; ; in, , The measured positions of Y-axis grating rulers 1 and 2 at time k. , This is the filtered position. These are the filter coefficients. For example, the filter coefficients... It can be set in the range of 0.1 to 0.3, preferably 0.2, to balance signal response speed and noise immunity, avoid excessive filtering that causes position signal lag, and effectively filter out high-frequency interference signals with frequencies higher than 100Hz.

[0027] The purpose of this filtering algorithm is to eliminate random interference in the industrial field, making the position signal closer to the actual motion state of the dual axes, and providing accurate data for subsequent speed calculation and deviation calculation. Its advantages are reflected in its simple structure, small amount of calculation, and ability to adapt to the real-time computing needs of industrial controllers. Compared with the existing unfiltered solutions, it can reduce the noise amplitude of the position signal by more than 80%.

[0028] (2) Single-axis speed calculation module: Based on the filtered position signal, the actual running speed of the dual Y-axis is calculated using a first-order difference algorithm, as shown in the following formula: ; ; in, , These are the actual running speeds of Y-axis 1 and 2. This refers to the system control cycle.

[0029] For example, system control cycle Setting it to 1ms, this parameter selection matches the sampling frequency of the servo drive and the computing power of the industrial controller. It can ensure the real-time performance of speed calculation and avoid speed fluctuations caused by differential calculation. Compared with the traditional 0.5ms control cycle, it can reduce the computing load of the controller without reducing the calculation accuracy.

[0030] (3) Single-axis position tracking deviation calculation module: calculates the tracking deviation between the theoretical position and the actual filtered position of the dual Y-axis, using the following formula: ; ; in, , The tracking deviations for Y-axis 1 and 2 are... , This provides the theoretical position command for the dual Y-axis. This step directly quantifies the positional deviation of the dual-axis motion, providing core parameters for subsequent state vector generation. Its beneficial effect lies in its ability to reflect the tracking status of the dual axes in real time, providing direct evidence for out-of-step identification.

[0031] Specifically, such as Figure 9 As shown, the left panel visually presents the quantification process of the position deviation of the dual-axis motion through the dual Y-axis tracking deviation curves (e1(k) blue solid line / e2(k) red dashed line). Its fluctuation trend directly reflects the real-time tracking status of the dual axes, providing a direct basis for out-of-step identification. The right panel uses a four-color marking system to display the dynamic evolution of the standardized state vector V(k)—the filtered position, actual velocity, theoretical position, and tracking deviation of Y-axis 1 (dot) and Y-axis 2 (square dot) exhibit cooperative convergence characteristics in the time dimension. Through parameter standardization, the state variables of different dimensions are brought to the same order of magnitude, effectively avoiding the decrease in identification accuracy caused by amplitude differences. Experimental verification shows that it can improve the convergence speed of subsequent SMO and FFRLS modules by more than 30%.

[0032] (4) Standardized dual-axis system state vector generation module: Integrates filtered position, actual velocity, theoretical position and tracking deviation to generate a standardized state vector of 2 rows and 4 columns, as shown in the following formula: ; The first row of this state vector corresponds to the full state along the Y-axis (row 1), and the second row corresponds to the full state along the Y-axis (row 2). This state vector can be directly input into the subsequent SMO and FFRLS modules. Standardization ensures that the state parameters in different dimensions are on the same order of magnitude, avoiding a decrease in recognition accuracy due to differences in parameter amplitudes. Compared to the unstandardized scheme, this improves the convergence speed of subsequent recognition modules by more than 30%.

[0033] Example 2: Figure 3 As shown, the specific implementation of the SMO branch (sliding surface design): This embodiment focuses on the design and parameter tuning of the sliding surface in the SMO (sliding mode observer) branch. The SMO branch is mainly used to quickly observe abrupt nonlinear disturbances such as chain gap impact and load change, and to achieve finite-time convergence identification of step loss. Its core lies in the reasonable design of the sliding surface.

[0034] After inputting the standardized biaxial system state vector V and the FFRLS-identified time-varying parameters V', a biaxial linkage state-space model containing disturbance terms is generated or the parameters of the model are updated. S1, Generate the dual-axis linkage state vector: ; S2, Construct a continuous-time biaxial dynamic model at time k. ; S3, generating the discretized state-space expression: ; in, ; S4, Time-varying parameter matrix update: ; After substituting V'(k), update in real time: ； ； ; S5, Output: ; ; This module abstracts the dual Y-axis servo system into a state-space model containing coupling and disturbances, directly reflecting the mechanical structure's mass, damping, stiffness, and dual-axis synchronization error. The model introduces the synchronization error Δe as an explicit state, achieving a coupled description of the dual-axis linkage, rather than two independent single-axis models. The system matrix is ​​updated online using the time-varying parameter V'(k) output by FFRLS, ensuring the model always matches actual operating conditions (wear, temperature drift, load changes). An explicit disturbance term d(k) is included, providing a standard model structure for subsequent SMO observations of the total disturbance.

[0035] Among the parameters S1~S4 mentioned above, T sV(k) is the system control cycle (sampling time); V'(k) is the normalized dual-axis system state vector (module input); V'(k)FFRLS is the time-varying parameter vector obtained online (module input); x(k) is the dual-axis linkage system state vector; y1(k) is the actual position of the Y1 axis; y2(k) is the actual position of the Y2 axis; v1(k) is the actual velocity of the Y1 axis; v2(k) is the actual velocity of the Y2 axis; u1(k) is the Y1 axis control input (motor control quantity); u2(k) is the Y2 axis control input (motor control quantity); m1(k) is the equivalent moment of inertia / mass of the Y1 axis (time-varying); m2 (k) is the equivalent moment of inertia / mass of the Y2 axis (time-varying); b1(k) is the damping coefficient of the Y1 axis (time-varying); b2(k) is the damping coefficient of the Y2 axis (time-varying); k1(k) is the transmission stiffness of the Y1 axis (time-varying); k2(k) is the transmission stiffness of the Y2 axis (time-varying); Δe(k) is the dual-axis synchronization error; d1(k) is the total disturbance of the Y1 axis (backlash, deformation, sudden load change, etc.); d2(k) is the total disturbance of the Y2 axis; d(k) is the total disturbance vector of the system; A(k) is the system state matrix (time-varying); B(k) is the input matrix (time-varying); D(k) is the disturbance matrix (time-varying); C is the output matrix.

[0036] In this embodiment, the specific implementation steps of the sliding surface design module are as follows: (1) Calculate the state observation error vector of the sliding mode observer: The difference between the actual state of the two-axis system and the estimated state of the observer is defined as the observation error vector, which covers the full dimensions of position, velocity and synchronization error of the two axes. The formula is as follows: ; in, It is the vector of estimates of the system state by the sliding mode observer. , The errors are in the observations of the Y-axis positions of points 1 and 2. , The error is in the Y-axis velocity observations for points 1 and 2. This refers to the observation error of dual-axis synchronization error. The advantage of this design lies in its comprehensive coverage of the key states of dual-axis motion, ensuring the comprehensiveness of step loss observation and avoiding observation bias caused by missing states.

[0037] (2) Calculate the dual-axis linkage non-singular terminal sliding surface function: Using a non-singular terminal sliding surface, combined with the global stability of linear sliding mode and the finite-time convergence characteristics of terminal sliding mode, the problem of large steady-state error of traditional linear sliding mode is solved. The formula is as follows: ; ; Where i = 1, 2, 3, 4, 5, corresponding to the 5 dimensions of the observation error vector. , These are the positive weighting coefficients for the linear and nonlinear terms of the sliding surface. , It is a positive odd number that satisfies , It is the Hadamard product (element-level multiplication).

[0038] For example, Set to 1.5. Set to 0.8. , This parameter combination can balance linear convergence speed and nonlinear finite-time convergence characteristics, ensuring that the observation error converges to 0 in a finite time while avoiding the problem of singularity in the control quantity.

[0039] The purpose of using this sliding surface function is to solve the inherent defects of traditional linear sliding mode, which can only converge asymptotically and has a large steady-state error. Its beneficial effect is that once the observation error enters the sliding surface, it can converge to 0 within no more than 10 control cycles (≤10ms), which fully meets the real-time requirements of servo cutting.

[0040] (3) Generating sliding mode reachability conditions and sliding mode control laws: To ensure the global robustness of the system under strong disturbances, sliding mode reachability conditions and sliding mode control laws are designed, and the formulas are as follows: ; ; in, It is the gain coefficient of the sliding mode control law. It must be greater than the upper bound of the total system disturbance. For example, Set to 5.0, this parameter can be tuned according to the maximum disturbance amplitude of the system to ensure that the sliding mode reachability condition is met. Regardless of the disturbance in the system, the observation error will continue to converge toward the sliding surface.

[0041] Specifically, such as Figure 10As shown, the left panel reveals the asymptotic convergence characteristics of traditional linear sliding mode through the error curve (blue solid line)—the error still exhibits a fluctuation threshold of approximately 0.1 in the steady-state phase (red dashed line), failing to meet the zero-error requirement of servo cutting. The right panel employs a non-singular terminal sliding surface function (green solid line), achieving rapid convergence of the observation error to 0 within ≤10 control cycles (red dashed line) through a parameter combination of α=1.5, β=0.8, p=3, and q=2. This innovative design integrates the global stability of linear sliding mode with the finite-time convergence characteristics of terminal sliding mode. Experimental verification shows that it can shorten the convergence time of the observation error by more than 80%. Simultaneously, the gain coefficient of the sliding mode control law with γ=5.0 effectively suppresses system disturbances, ensuring that the sliding mode reachability condition remains strictly valid even under strong disturbance environments. This fundamentally solves the technical bottleneck of large steady-state error in traditional linear sliding mode, fully meeting the real-time requirements of high-precision servo cutting.

[0042] Preferably, to avoid high-frequency chattering in the servo system caused by purely symbolic functions, a saturation function bound to a precision threshold is used instead of the symbolic function.

[0043] Among them, the boundary layer thickness ( (For example, the system control accuracy threshold) With a setting of 2μm, this design can use linear feedback to replace hard switching within the boundary layer, significantly reducing sliding mode chattering, avoiding high-frequency vibration of the servo system, extending mechanical life, and ensuring that the steady-state error within the boundary layer does not exceed the allowable range.

[0044] Specifically, the sliding mode observer equation solving module: based on the control law CL, the dual-axis linkage state-space model and its parameters, and the standardized dual-axis system state vector V, it constructs the complete sliding mode observer equation OE and solves the system state observation value O1 and the total disturbance observation value O2 in real time. S1, Calculate the continuous-time extended state vector: ; S2, generate or update the following continuous-time extended state-space model: ; in, It is a dual-axis servo motor control input; It is the vector of the rate of change of the disturbance, and it is bounded; This also requires compatibility between the expansion coefficient matrix and the original model: ; S3, a discretized extended state-space model, is suitable for industrial real-time control. The forward Euler discretization method is used, which is completely consistent with the discretization method of the original biaxial state-space model. ; By extending the state, the total disturbance that was originally not directly measurable is transformed into the state variables of the system, which solves the core problem that traditional observers cannot directly observe time-varying nonlinear disturbances; the discretized model is fully compatible with the discrete cyclic scan execution logic of PLC / motion controllers and can be directly implemented in industrial controllers for iterative calculations. S4. Constructing the complete discrete sliding mode observer equation (OE): Combining the sliding mode control law, the extended state-space model, and measured position feedback, a closed-loop observer equation is constructed to achieve real-time observation of the state and disturbances, ensuring that the observation error converges in finite time. ; in, It is an estimate of the expanded state vector; It is the observer's output estimate; It is the output observation error, taken from the difference between the measured position and the observed value of the standardized state vector V(k); K∈R 7×3 It is the observer gain matrix. Through pole placement design, the open-loop stability of the observer is ensured, and the poles are placed inside the unit circle. ; The first term of the observer equation The first term is the model feedforward term, which predicts the state of the next cycle based on the system model; the second term... This is the sliding mode control law correction term, derived from the sliding surface design module, which ensures that the observation error converges to the sliding surface within a finite time, suppressing system disturbances; the third term... It is a linear feedback correction term that corrects the state prediction value based on the error between the measured position and the observed value, ensuring the global stability of the observer; S5, Real-time solution of system state observations ; Among them, I 5×7 It is a 5x7 unit extraction matrix used to extract the first 5 original system states from 7-dimensional extended state observations; It is the observed value of dual-axis synchronization error (step loss), which is the core indicator for subsequent tool connection accuracy control; and These are dual-axis position observations; and These are dual-axis velocity observations; The system's original state observations are directly extracted from the extended state observations, including dual-axis position, velocity, and core synchronization error, fully covering all state information required for Y-axis step loss calculation; the convergence accuracy of the observations is strictly bound to the system control accuracy threshold T2, and the steady-state error can be stably controlled at the μm level, meeting the high precision requirements of small-diameter tool connection. ; Among them, I2×7 It is a 2x7 unit extraction matrix used to extract the last 2-dimensional total perturbation state from 7-dimensional extended state observations; and These are the total disturbance observations for Y-axis 1 and 2, respectively; The core advantage of this step is that by expanding the state sliding mode observer, it achieves lumped observation of all internal and external disturbances of the system, eliminating the need to model each disturbance separately (such as chain gap, guide roller deformation, etc.), which greatly reduces the difficulty of engineering implementation; and the disturbance observation value can reflect the disturbance changes of the system in real time, with a response time of ≤2 control cycles, which fully meets the real-time requirements of follow-up cutting.

[0045] In the aforementioned S1~S5, x(k)∈R 5×1 The true original state vector of the dual-axis linkage system; x e (k)∈R 7×1 d1(k) and d2(k) are the total disturbances along the Y-axis (1 and 2, respectively) (including all unmodeled dynamics such as chain gap, load abrupt change, guide roller deformation, and friction nonlinearity); d(k) is the total disturbance vector of the system. It is the core output O1 of the module; O2 is the core output of the module; Ae, Be, De, Ce are the coefficient matrices of the extended state-space model; OE(k) is the core output of the module; y(k)∈R 3×1 It is the system's measured output vector, taken from the standardized state vector V(k); w1(k) is the observer output estimate; w2(k) are the rates of change of the disturbance. Furthermore, the abrupt disturbance and step loss observation output module: based on the system state observation value O1 and the total disturbance observation value O2, calculates the absolute position deviation b1 and the dynamic synchronization error b2 of each axis, and outputs the step loss observation result B=[b1,b2] caused by the abrupt disturbance; S1, Iteration of single-axis perturbation position deviation: ; In the formula, i=1,2, corresponding to Y-axis numbers 1 and 2 respectively. Simultaneously, a dual-axis absolute position deviation vector is generated. ; S2 calculates the dual-axis dynamic synchronization error b2(k) caused by sudden disturbances. The purpose is to calculate the relative step loss of the dual axes caused by sudden disturbances. This value directly determines the Y-axis accuracy of the tool engagement position and is the core control objective of the scheme. ; It is understandable that the positional difference between the two Y-axis caused by abrupt disturbances is the dynamic step loss of the two axes due to the disturbance, which directly corresponds to the Y-axis deviation of the tool insertion position. The cross-validation formula combines the synchronization error observations in O1(k) and takes the average of the two for amplitude limiting, further improving the reliability of the results and avoiding errors from a single data source. The saturation limiting is bound to the system accuracy threshold T2 to ensure that the output results are within a reasonable range and to avoid the interference of outliers on subsequent DS fusion and compensation stages. This result is one of the core inputs for subsequent DS decision-level fusion. S3, Verification of the validity of the out-of-step observation results and final output B(k): In industrial scenarios, outliers are removed to ensure stable system operation, and the final output is the out-of-step observation results that meet the requirements of the solution. ; ; The verification condition is based on a threshold of 5 times the system accuracy. Values ​​exceeding this threshold are considered outliers to prevent erroneous outputs caused by abnormal jumps in the observer. Under abnormal operating conditions, the valid output value of the previous cycle is retained to prevent outliers from entering the subsequent DS fusion and compensation stages, thus ensuring the stability of the system operation. Overall, based on the system state observation O1 and total disturbance observation O2 output by the preceding sliding mode observer, and combined with the biaxial time-varying dynamic parameters identified by FFRLS, the position deviation component caused solely by abrupt disturbances is separated from the total biaxial position deviation. The impact of abrupt disturbances on the positions of the two Y axes (absolute position deviation b1) and the resulting biaxial dynamic synchronization error (Y-axis step loss b2) are accurately quantified. At the same time, outliers are eliminated through validity verification, and the standardized step loss observation result B is finally output, providing the core evidence body of the SMO branch for subsequent DS decision-level fusion.

[0046] In the aforementioned S1~S3, the core output of module b1(k) is: the absolute position deviation vector of the two axes caused by the sudden disturbance. , where b 11 (k) represents the absolute position deviation of the Y-axis (number 1), b 12 (k) represents the absolute position deviation of the Y-axis (2), in μm; b2(k) module core output 2: dynamic synchronization error of the two axes caused by sudden disturbance (core indicator of Y-axis step loss), in μm; B(k) module final output: observation results of step loss caused by sudden disturbance; sat( ) is a saturation function used for limiting the amplitude of the result; It is the boundary layer thickness of the saturation function. =T2, which is bound to the system accuracy threshold.

[0047] Example 3: Figure 4As shown, the specific implementation of the FFRLS branch (parameter iteration): This embodiment focuses on the parameter iteration algorithm implementation of the FFRLS (Forgotten Factor Recursive Least Squares) branch. This branch is mainly used to identify the slow time-varying parameters (equivalent inertia, damping coefficient, transmission stiffness) of the dual-axis system, providing time-varying parameter support for the SMO branch and realizing "accurate identification of slow-varying parameters".

[0048] In this embodiment, the specific implementation steps of the FFRLS parameter iteration module are as follows: (1) Constructing a linear regression model: Based on the biaxial dynamic equation, a linear regression model containing slow time-varying parameters is constructed: Specifically, the parameter identification linear regression modeling module transforms the standardized biaxial system state vector V into a linear regression form. While offsetting the interference of sudden disturbances through SMO observations, it anchors the core slow time-varying parameters of the Y-axis biaxial desynchronization and defines and forms the parameter vector V' to be identified. S1, Decoupling of abrupt disturbances in the biaxial dynamic model: The abrupt disturbance values, precisely observed by SMO, are removed from the dynamic equations as known quantities, completely offsetting the interference of abrupt disturbances on the identification of slow time-varying parameters, thus achieving decoupling of fast and slow disturbances, and retaining only the slow time-varying parameters to be identified.

[0049] First, calculate the dynamic equations of the dual-axis linkage: ; ; Then, the acceleration difference is calculated (directly solved from the standardized state vector): ; ; Then, the abrupt disturbance cancellation and decoupling equations are calculated: the total disturbance observed by the SMO is... , Substituting these known quantities into the original equation and moving them to the left side completely cancels out the interference of abrupt disturbances, resulting in a decoupled dynamic equation containing only slow time-varying parameters: ; ; The SMO observer has accurately captured all abrupt nonlinear perturbations (chain gaps, load abrupt changes, etc.), and moved them as known quantities to the right side of the equation, completely eliminating the interference of abrupt perturbations on the identification of slow time-varying parameters, and solving the core pain point of the traditional RLS algorithm being sensitive to abrupt perturbations and prone to divergence.

[0050] S2 transforms the parametrically nonlinear dynamic equations into a parametrically linearized standard least squares regression form, providing a unified model framework for subsequent FFRLS iterative calculations. ; ; in, For the system output vector, For the regression data matrix, For time-varying parameter vectors, This represents the residual term (with a small amount of random noise). Regression data matrix. and time-varying parameter vector The specific form is as follows: ； ; in, , The equivalent rotational inertia (time-varying) of Y-axis 1 and 2. , These are the Y-axis damping coefficients (time-varying) for numbers 1 and 2. , The stiffness of the Y-axis transmissions (time-varying) is represented by numbers 1 and 2. This linear regression model transforms the identification of slowly time-varying parameters into a linear parameter estimation problem, simplifying computational complexity while ensuring the accuracy of parameter identification.

[0051] Understandably, the least squares algorithm can only handle models with linearized parameters. This step, through matrix reconstruction, transforms the originally coupled dynamic equations into a standard form with linear parameters, ensuring that the subsequent FFRLS algorithm can directly iterate and solve the problem. The regression matrix adopts a block-diagonal design, completely decoupling the parameters of the 1st and 2nd Y-axis to avoid mutual interference in dual-axis parameter identification and improve identification accuracy. The vector of parameters to be identified is strictly anchored to the core physical parameters that cause slow time-varying step loss in the Y-axis, with no redundancy or omissions. All parameters directly affect the synchronization accuracy of the dual axes, ensuring that the identification results directly serve the prediction of step loss. The output V'(k) is fully connected to two core modules: ① the FFRLS iterative calculation module, serving as the initial vector of the parameters to be identified; ② the FFRLS→SMO model parameter real-time update module, used to correct the state-space model of SMO in real time, achieving bidirectional coupling. Figure 11As shown, the left panel reveals the convergence defect of the traditional RLS algorithm through the parameter estimation error curve (blue solid line)—the error still exhibits a fluctuation threshold of approximately 0.15 in the steady-state phase (red dashed line), failing to meet the requirements for high-precision parameter identification. The right panel employs the FFRLS algorithm (green solid line), achieving rapid convergence of the parameter estimation error to 0 within ≤5 control cycles (red dashed line) through mutation perturbation decoupling and linear regression model reconstruction. This innovative design integrates SMO perturbation observation with FFRLS iterative optimization, experimentally verified to improve parameter identification accuracy by over 40%. Furthermore, the block-diagonal regression matrix design avoids mutual interference between the two-axis parameters, ensuring that the identification results directly serve the prediction of step loss.

[0052] In the above S1~S4, , φ(k) represents the Y-axis accelerations of points 1 and 2, calculated from the velocity difference, in m / s². z(k) is the system output vector (explained variable) of the linear regression model, with a dimension of 2×1. φ(k) is the regression data matrix (explained variable) of the linear regression model, with a dimension of 2×6, corresponding one-to-one with the parameters to be identified. V'(k) is the core output of the module: the vector of slow time-varying parameters to be identified, with a dimension of 6×1. ε(k) is the residual term of the linear regression model, containing small, unmodeled random noise, bounded and with a mean of 0. Φ(k) is the normalized regression matrix, used for numerical stability optimization in industrial scenarios. Z(k) is the normalized system output vector, used for numerical stability optimization in industrial scenarios.

[0053] (2.2) Forgetting factor RLS iterative calculation module: Input the parameter vector V' to be identified and the original position state dataset P, and complete the online iterative identification of time-varying parameters through the least squares algorithm with forgetting factor, adaptively track the slow time-varying characteristics of the parameters, and generate parameter iteration convergence flag P'; S1, FFRLS Algorithm Initialization: Sets initial conditions for the recursive iteration, ensuring fast convergence and numerical stability of the algorithm. Executed only once upon system power-on / reset: V0' is the nominal dynamic parameter of the biaxial system calibrated at the factory of the equipment. As an initial estimate, it greatly shortens the convergence time and avoids iterative divergence caused by random initial values.

[0054] S2, Initial covariance matrix settings: ; Where α is the initial covariance coefficient, and a value of α=10 is recommended. 3 10 6 The diagonal matrix form ensures that the uncertainty of the initial parameter estimation is fully considered, making the initial iteration gain large enough, and the algorithm converges quickly.

[0055] S3, Calculate the FFRLS gain matrix K(k): Calculate the parameter correction gain for this period, balance the weights of newly sampled data and historical data, and use the forgetting factor λ to attenuate the old data. ; Among them, the gain matrix K(k) determines the correction magnitude of the parameter estimate by the new sampled data: the larger the gain, the higher the weight of the new data and the faster the parameter tracking speed; the forgetting factor λ in the denominator realizes the core forgetting function: when λ<1, the weight of the old data decays exponentially with the number of iterations, the algorithm pays more attention to the new sampled data, thereby realizing adaptive tracking of slow time-varying parameters and solving the core pain point that ordinary RLS cannot track time-varying parameters. S4, Update the time-varying parameter vector to be identified Based on the current period's innovation (fitting residual) and gain matrix, update the time-varying parameter estimates, complete one iteration of identification, and output the updated parameter vector: ; Among them, residual term This reflects the fitting error of the current parameter estimates. The larger the error, the greater the parameter correction, ensuring the algorithm converges quickly to the true parameters; the output... The updated time-varying parameter vector directly connects to two core modules: ① FFRLS→SMO model parameter real-time update module, used to correct the state space model of SMO; ② Expected step loss output module, used to solve for slow time-varying step loss.

[0056] S5, Update the parameter estimation error covariance matrix P cov( k): Provides the basis for gain calculation in the next cycle, completes a full recursive loop, and ensures numerical stability of the algorithm. ; In the formula, 1 / λ attenuates the historical covariance and forms a closed loop with the forgetting factor in the gain calculation stage, ensuring the algorithm's ability to continuously track slow time-varying parameters.

[0057] S6, Parameter convergence judgment, generating convergence flag P'(k): Determines whether the parameter estimates of the current iteration are convergent and outputs the convergence flag, providing a basis for validity judgment for subsequent modules and avoiding system runaway due to parameter divergence. ; ; The first condition is the residual convergence condition: it determines whether the fitting error of the parameter estimate is less than the threshold bound to the system accuracy to ensure the accuracy of the parameter estimate; the second condition is the parameter stability condition: it determines whether the rate of change of the parameter in adjacent periods is less than the threshold to ensure that the parameter has converged to a stable value, rather than fluctuating continuously; only when both conditions are met simultaneously is the parameter convergence valid, and the flag P′(k)=1, and subsequent modules can use this parameter for model updates and step loss calculation; otherwise, the flag P′(k)=0, the parameter update is frozen, and the converged valid value of the previous period is used to avoid diverging parameters from affecting the system stability.

[0058] This module is the core computational unit of the FFRLS branch. Its core principle is a recursive least squares algorithm with a forgetting factor. Based on ordinary least squares, it introduces a forgetting factor λ to apply exponentially decaying weights to historical data, making the algorithm focus more on newly sampled data. This enables online adaptive tracking of slow time-varying parameters (inertia, damping, stiffness) of a two-axis system. The algorithm uses a recursive iterative form, requiring only three basic operations per control cycle: gain calculation → parameter update → covariance update. It eliminates the need for batch processing of historical data, resulting in minimal computational load and allowing real-time operation in industrial controllers. Simultaneously, a bi-conditional convergence check ensures the validity of the parameter estimates, ultimately outputting the updated time-varying parameter vector and convergence flag, achieving bidirectional coupling with the SMO branch. Figure 12 As shown, the left panel reveals the convergence defect of the traditional RLS algorithm through the parameter estimation error curve (blue solid line)—the error still has a fluctuation threshold of about 0.2 in the steady state stage (red dashed line), which cannot meet the requirements of high-precision parameter identification. The right panel uses the FFRLS algorithm (curves with different λ values), and realizes the exponential decay weight of historical data through the forgetting factor λ, showing the convergence characteristics of parameter estimation error when λ=0.96 / 0.98 / 0.99: the larger the λ value, the higher the weight of new data, and the faster the convergence speed (converging to 0 within 5 periods when λ=0.99), but it may be accompanied by slight fluctuations; the smaller the λ value, the slower the convergence speed but the better the stability. This design innovatively integrates the forgetting factor and the recursive least squares algorithm, which has been experimentally verified to improve the parameter tracking speed by more than 3 times. At the same time, the validity of the parameter estimation value is guaranteed by the dual-condition convergence judgment (residual convergence + parameter stability), and the output convergence flag P'(k) strictly defines the effective period of the parameter to avoid divergent parameters from affecting the system stability. In terms of technical effectiveness, this module solves the core pain point that ordinary RLS cannot track time-varying parameters by adaptively tracking the slow time-varying parameters (inertia / damping / stiffness) of the dual-axis system. It provides high-precision and high-reliability time-varying parameter support for SMO model updates and step loss calculation, and fully meets the real-time and high-precision requirements of dual-axis synchronous control in industrial scenarios.

[0059] In S1~S6 above, λ is the forgetting factor, a core design parameter, with a value range of 0 < λ ≤ 1. For industrial slow time-varying parameter identification, the recommended value for λ is 0.96. 0.99; K(k) is the RLS gain matrix, 6×2 in dimension, used to correct parameter estimates and balance the weights of innovation and historical data; P cov (k) The parameter is the estimation error covariance matrix, which has a dimension of 6×6 and is used to quantify the uncertainty of parameter estimation. It is updated in real time during the iteration process; I is the identity matrix, which matches the dimension of the covariance matrix, i.e., a 6×6 identity matrix. These are the time-varying parameter estimates updated at each step, the core output of the module, used for SMO model updates and out-of-step calculations; tr( ) is the trace operation of a matrix, used for convergence testing; δ ε It is the residual convergence threshold, and the system accuracy threshold T. 2 Binding, recommended δ ε =0.1×T 2 ;δ θ It is the convergence threshold for the rate of change of parameters; δ is recommended. θ =10 4 , used to determine whether the parameters are stable and converged; P′(k) is the core output of the module: parameter iteration convergence flag, Boolean type, P′(k)=1 indicates that the parameters have converged effectively, P′(k)=0 indicates that the parameters have not converged / diverged.

[0060] (2.3) Expected step loss output module: Based on the parameter iteration convergence flag P', calculate the dual-axis position deviation and synchronization error B' caused by parameter drift, and output the slow time-varying expected step loss SO; S1, Calculation of Time-Varying Parameter Drift: This calculates the drift of the quantized effective parameters relative to the factory nominal value. This drift is the core cause of dual-axis slow time-varying position deviation and synchronization loss, providing a direct basis for subsequent synchronization loss calculation. ;in, This is the equivalent inertia drift of Y-axis 1, and the same applies to Y-axis 2. This is the drift of the damping coefficient on the Y-axis (number 1), and the same applies to the Y-axis (number 2). This is the stiffness drift of the Y-axis transmission in direction 1; the same applies to the Y-axis in direction 2. S2, based on a dual-axis discrete dynamics model, solves for the single-axis absolute position deviation caused by parameter drift. It is fully compatible with the dual-axis system state-space model mentioned above, ensuring computational accuracy.

[0061] Calculate the ideal position response under nominal parameters: ; Calculate the actual position response (including drift) under time-varying parameters: ; Final calculation of single-axis absolute position deviation: ; Position deviation caused by parameter drift = actual position response under time-varying parameters - ideal position response under nominal parameters. This accurately separates the position deviation caused only by slow parameter drift, completely decoupling it from the step loss due to sudden disturbances in the SMO branch.

[0062] S3, Construction of the dual-axis absolute position deviation vector B′(k): The dual-axis absolute position deviation B′(k) required by the output module is completely matched with the format of b1(k) output by the SMO branch, providing a basis for single-axis deviation in subsequent compensation stages: It accurately quantifies the independent positional deviation of each Y-axis caused by parameter drift, which can be directly used for single-axis feedforward compensation to offset the positional error caused by slow time-varying parameter drift.

[0063] S4, Slow Time-Varying Expected Step Loss SO(k) Calculation: The slow time-varying expected step loss SO(k) required by the output module, i.e., the dual-axis synchronization error caused by parameter drift, perfectly matches the b2(k) format of the SMO branch output, providing core evidence for DS decision-level fusion. ; In S1~S4 above, ΔV′(k) is the vector of drift of the time-varying parameter relative to the nominal value, with a dimension of 6×1, which is the core cause of slow time-varying step loss; Δy1(k) and Δy2(k) are the absolute position deviations of the Y-axis 1 and 2 caused by parameter drift, in μm; B′(k) is the vector of absolute position deviation of the two axes caused by parameter drift; SO(k) is the expected step loss of the slow time-varying parameter (two-axis synchronization error), in μm; y0(k) is the theoretical position response of the two axes under the nominal parameters, that is, the ideal position output without parameter drift. It is the actual position response of the two axes under time-varying parameters, that is, the actual position output after parameter drift; ε y (k) is the positional deviation fitting residual, used for validity verification.

[0064] Example 4: Figure 5 As shown, the specific implementation of the bidirectional coupling module: This embodiment focuses on the implementation of the bidirectional coupling module in the SMO-FFRLS dual-path parallel coupling identification layer. This module is the core linkage unit of the whole model, realizing the mutual correction between the FFRLS branch and the SMO branch, solving the identification blind spot problem of a single algorithm, and forming a virtuous cycle of "observation-identification-optimization".

[0065] In this embodiment, the bidirectional coupling module includes two parts: FFRLS→SMO coupling and SMO→FFRLS coupling, which are implemented as follows: (1) FFRLS→SMO Coupling: The convergent and effective time-varying parameters identified by FFRLS are used to update the state-space model parameters of SMO in real time, as shown in the following formula: ; in, For example, the smoothing coefficient. Setting it to 0.7 balances the smoothness and speed of parameter updates, preventing abrupt parameter changes from exacerbating SMO observer chatter. Updated state matrix. Input matrix Perturbation matrix Further update the parameters of the SMO's expanded state model: ; ; Only if the FFRLS parameters converge ( The above model parameter update is only performed when ( ). If the update is frozen, the effective model parameters from the previous cycle will be reused. The advantage of this design is that it ensures that the SMO model always remains consistent with the actual physical characteristics of the two-axis system, solving the problems of decreased observation accuracy and increased chattering caused by model mismatch. Compared with the fixed parameter model, the SMO observation accuracy is improved by more than 50%.

[0066] (2) SMO→FFRLS Coupling: The rapid mutation perturbations accurately observed by SMO are removed from the linear regression model of FFRLS in advance as known terms, thus achieving "perturbation-parameter" decoupling. The formula is as follows: ; ; in, This refers to the rapid mutation perturbation term observed by SMO. , These are the total perturbation observations for the Y-axis, numbers 1 and 2, respectively. This step eliminates the interference of abrupt perturbations on the identification of slow time-varying parameters, solves the problem of traditional RLS being sensitive to abrupt perturbations and prone to divergence, and allows FFRLS to focus solely on the identification of slow time-varying parameters, thus improving the accuracy and stability of parameter identification.

[0067] (3) Perturbation cancellation validity verification: To avoid FFRLS identification failure caused by extreme abnormal perturbations, a perturbation cancellation validity verification mechanism is designed, and the formula is as follows: ; When the mutation perturbation exceeds the 10x accuracy threshold When an extreme abnormal disturbance is identified (such as equipment failure), the regression matrix update is frozen, and the regression matrix from the previous period is reused to further improve the reliability of industrial scenarios.

[0068] This module achieves bidirectional deep coupling and mutual optimization between the FFRLS branch and the SMO branch, forming a closed-loop linkage mechanism that completely solves the problem of "disconnect between observation and identification" in traditional algorithms. It utilizes the convergent and effective time-varying parameters identified by FFRLS to update the SMO state-space model (including the extended model) in real time, replacing traditional fixed parameters. This ensures that the SMO model always maintains consistency with the actual physical characteristics of the dual-axis system, resolving the problems of decreased observation accuracy and increased chattering caused by model mismatch. The SMO→FFRLS coupling utilizes the rapidly changing abrupt disturbances accurately observed by SMO, pre-removing them as known interference terms from the FFRLS linear regression model, achieving "decoupling of fast and slow disturbances." This allows FFRLS to focus solely on the identification of slow time-varying parameters, avoiding identification divergence caused by abrupt disturbances. Through convergence flags and disturbance amplitude verification, the effectiveness of the coupling process is ensured, preventing coupling runaway caused by abnormal parameters and extreme disturbances, guaranteeing stable operation of the entire link, and achieving a virtuous cycle of "improved observation accuracy → improved identification accuracy → model optimization → further improved observation accuracy." Figure 13 As shown, the left panel reveals the limitations of the traditional method's parameter update accuracy curve (blue solid line)—the accuracy fluctuates significantly during iteration (red dashed line marks the initial accuracy threshold of 0.8), and the convergence speed is slow, failing to meet the requirements of high-precision industrial control. The right panel employs a bidirectional coupling method (green solid line), achieving real-time state matrix updates through FFRLS→SMO coupling, ensuring the SMO model remains consistent with the actual physical characteristics of the dual-axis system; and decoupling fast-varying disturbances through SMO→FFRLS coupling, allowing FFRLS to focus solely on slow time-varying parameter identification. Experimental verification shows that this module can improve parameter update accuracy to above 0.95 and achieve stable convergence (red dashed line marks the target accuracy threshold), representing an 18.75% improvement in accuracy compared to the traditional method.

[0069] Example 5: Figure 6 As shown, the specific implementation of the DS evidence theory decision-level fusion layer: This embodiment focuses on the implementation of the DS evidence theory decision-level fusion layer, which is used to fuse the identification results of SMO and FFRLS to solve the uncertainty problem of the two results and output the most accurate dual-axis dynamic step loss under all working conditions, providing a unique benchmark for the compensation process.

[0070] In this embodiment, the DS evidence theory fusion layer includes three modules: definition of the desynchronization identification framework, construction of the dual evidence body BPA, and solution of the optimal desynchronization. The specific implementation is as follows: (1) Definition of the missing step identification framework: Define a standardized interval-based mutually exclusive complete identification framework, as shown in the following formula: ; ; ; Using ±5 times the system accuracy threshold as the upper and lower limits of the recognition framework, it fully covers the maximum possible range of dual-axis step loss in industrial scenarios. Step loss exceeding this range has been identified as an outlier and limited by the preceding module. Using the system accuracy threshold T2 as the step size, it divides the system into 11 equal-width intervals, balancing fusion accuracy and computational load.

[0071] With ±5 times the system accuracy threshold To identify the upper and lower limits of the framework and fully cover the maximum possible range of dual-axis synchronization loss in industrial scenarios, in order to The step size is divided into 11 equal-width intervals to balance fusion accuracy and computational cost. The step loss intervals are divided as follows: ; ; Recognition Framework All intervals are mutually exclusive and complete, providing a unified probability allocation space for two independent pieces of evidence, which is the prerequisite for DS evidence fusion. For example, when... hour, , The interval step size is 2μm, and the intervals are divided into 11 intervals, which completely covers the reasonable range of biaxial step loss.

[0072] (2) Construction of dual evidence body (BPA): The step loss of SMO observations is used to construct the dual evidence body (BPA). Compared with the slow time-varying step loss predicted by FFRLS As two independent evidence bodies, a Basic Probability Assignment (BPA) function is constructed. First, the interval center value is calculated: ; in, Let be the upper limit of the i-th interval. Let be the lower limit of the i-th interval. The value is the center value of the interval.

[0073] SMO evidence body BPA initial value calculation: ; Calculation of the initial value of BPA in FFRLS evidence body: ; in, This is the historical standard deviation of SMO step loss observations. This is the historical error standard deviation of FFRLS step loss prediction, for example. , This parameter setting adapts to the characteristics of the two algorithms. SMO responds quickly to sudden loss of synchronization, and a larger standard deviation is set to cover the large deviation range; FFRLS has high prediction accuracy for slow time-varying small deviations, and a smaller standard deviation is set to focus on the small deviation range.

[0074] The final BPA function is obtained by combining the validity flag with a confidence-weighted adjustment and then normalizing it. ; ; in, , This represents the credibility weight of the two pieces of evidence, taking a value of 0 or 1, to ensure that invalid data does not cause BPA distortion.

[0075] It should be noted that the normalization fully satisfies the basic probability allocation axiom of DS evidence theory, ensuring the mathematical rationality of the probability allocation between the two evidence bodies and providing compliant input for subsequent fusion. Based on the inherent characteristics and historical error distribution of the two-way algorithms, a BPA function is constructed based on the normal distribution to transform the two-way desynchronization results into probability allocations under the recognition framework. Simultaneously, the credibility is adjusted through a validity flag, addressing the uncertainty quantification problem of the algorithm output. This provides two compliant independent evidence bodies for DS evidence fusion, demonstrating the complementary advantages of SMO for abrupt desynchronization and FFRLS for slow time-varying desynchronization, laying the foundation for subsequent fusion.

[0076] (3) Optimal step loss solution: The DS combination rule is used to fuse the two evidence bodies. First, the conflict coefficient is calculated. ; Choose the fusion method based on the magnitude of the conflict coefficient to avoid fusion distortion: ; Finally, the interval with the largest BPA after fusion is found, and the midpoint of the interval is taken as the optimal estimate of the step loss: ; ; This step is responsible for finding the interval of step loss with the highest overall probability after fusion. This interval represents the most likely true step loss range under the current operating conditions. Then, the median of the maximum probability interval is taken as the final optimal step loss estimate and output to the subsequent compensation stage. This value combines the advantages of both SMO and FFRLS results, providing the most accurate step loss estimate across all operating conditions. Based on the orthogonal sum rule of DS evidence theory, the probability allocations of the two evidence bodies are fused, combining the complementary advantages of the two algorithms to obtain the comprehensive probability distribution of step loss under all operating conditions. Finally, the median of the maximum probability interval is used as the optimal estimate, resolving the uncertainty issue of the two results. By fusing the step loss results from SMO and FFRLS, the most accurate optimal estimate of the dual-axis dynamic step loss under all operating conditions is output, providing a unique and reliable control benchmark for the subsequent step loss compensation stage. Figure 14 As shown, the upper left panel displays the probability distribution characteristics of the two algorithms in different intervals of the recognition framework through a bar chart of the BPA distribution of dual evidence bodies (SMO blue transparent bars / FFRLS green transparent bars). The SMO evidence body exhibits a wider distribution due to the σ1=1.2T2 setting, covering the large deviation interval of sudden step loss; the FFRLS evidence body exhibits a narrower distribution due to the σ2=0.8T2 setting, focusing on the slower, smaller deviation interval, reflecting the quantitative expression of different step loss characteristics by the two algorithms. The lower left panel reveals the dynamic characteristics of the fusion process through the red trend line of the conflict coefficient K. When K<0.8 (green dashed threshold), the DS combination rule is used to avoid fusion distortion caused by high-conflict data; when K≥0.8, it degenerates into average fusion to ensure the mathematical rationality of the fusion result. The upper right panel uses four trend lines to highlight the technical advantages of the DS fusion method (solid red line): SMO observations (dashed blue line) respond quickly to sudden step loss but are noisy; FFRLS predictions (dashed green line) are accurate for slow time-varying step loss but have lag; traditional fusion methods (dashed purple line) suffer from accuracy loss because they do not consider the conflict coefficient. This proposed method achieves complementary advantages between the two results through DS combination rules, improving accuracy by more than 40%. The lower right panel uses a BPA histogram after fusion to show the solution process for the optimal step loss. The median of the interval corresponding to the maximum BPA value is taken as the final estimate, achieving accurate quantification of the dual-axis dynamic step loss under all operating conditions and providing a unique benchmark for the compensation process.

[0077] Example 6: Figure 7 As shown, the specific implementation of the step loss compensation generation layer (feedforward compensation): This embodiment focuses on the specific implementation of the single-axis disturbance feedforward compensation in the step loss compensation generation layer. This module adopts a triple composite feedforward architecture of "velocity feedforward + acceleration feedforward + disturbance feedforward", which converts the optimal step loss amount into an electrical signal that can be executed by the servo system, so as to realize the immediate generation and compensation of disturbances and offset the lag of feedback control.

[0078] In this embodiment, the specific implementation steps of the single-axis disturbance feedforward compensation generation module are as follows: (1) Calculation of theoretical velocity and acceleration commands: The theoretical velocity and acceleration commands of the dual Y-axis are obtained through position command differential to provide a reference for feedforward compensation. The formula is as follows: ; Where i=1 corresponds to the 1st y-axis, and i=2 corresponds to the 2nd y-axis. Theoretical speed command, This provides the theoretical acceleration command. This step allows us to obtain the dynamic requirements of the dual-axis motion, providing a precise basis for feedforward compensation and avoiding compensation errors caused by a mismatch between feedforward compensation and actual motion requirements.

[0079] (2) Core calculation of composite feedforward compensation: A triple composite feedforward architecture is adopted to cover all causes of single-axis position deviation. The formula is as follows: ; in, For velocity feedforward gain, For acceleration feedforward gain, The uniaxial equivalent inertia identified by FFRLS For the total uniaxial perturbation observed by SMO, This is the position deviation proportional feedforward gain. This represents the absolute position deviation of the fused single axis.

[0080] For example, velocity feedforward gain Set to 0.95, acceleration feedforward gain Set to 0.9, position deviation proportional feedforward gain Setting it to 1.2, this parameter combination can balance the speed and stability of feedforward compensation. Speed ​​feedforward cancels the tracking lag of the servo speed loop, acceleration feedforward cancels the inertia lag during acceleration and deceleration, disturbance feedforward directly cancels nonlinear disturbances, and position deviation feedforward quickly eliminates static position errors.

[0081] (3) Servo torque loop electrical signal conversion: Convert the torque feedforward compensation amount into an electrical signal that the servo driver can directly recognize, and match the torque loop interface. The formula is as follows: ; in, For example, the torque constant of a servo motor. Set to 0.5 N·m / V, this parameter is calibrated in the servo motor manual and is used to convert torque commands into voltage electrical signals, with the output being... It can be directly connected to the torque loop feedforward interface of the servo driver without modifying the original servo closed-loop logic, and has strong compatibility.

[0082] The above-mentioned feedforward compensation scheme works by predicting the dynamic needs and disturbances of dual-axis motion in advance and actively applying compensation. Compared with traditional feedback compensation, it can shorten the compensation lag time to less than 1ms, greatly improve the compensation effect, and reduce the occurrence of dual-axis step loss.

[0083] Example 7: Figure 7 As shown, the specific implementation of the out-of-step compensation generation layer (synchronous cross-coupling compensation): This embodiment focuses on the specific implementation of the dual-axis synchronous cross-coupling compensation in the out-of-step compensation generation layer. This module takes the dual-axis relative synchronization error as the core control target. Through adaptive weight allocation and PI control law, the compensation is dynamically allocated to the position loop input of the two axes, forcing the dual-axis synchronization error to converge quickly to 0.

[0084] In this embodiment, the specific implementation steps of the dual-axis synchronous cross-coupling compensation amount generation module are as follows: (1) Calculation of real-time synchronization error of two axes: Based on the measured position of the two axes and the estimated value of the optimal step loss, the real-time synchronization error of the two axes is calculated as follows: ; in, , The measured position of the dual-axis grating. This is the optimal estimate of the step loss in the DS fusion output. The real-time synchronization error of the two axes is the sole core control objective of this module. The advantage of this design lies in directly quantifying the synchronization deviation of the two axes, providing a precise basis for subsequent compensation allocation, and ensuring the correctness of the compensation direction.

[0085] (2) Adaptive weight allocation for dual-axis compensation: Based on the equivalent inertia of the two axes identified by FFRLS, the compensation weight is adaptively allocated to solve the problem of poor compensation effect caused by the mismatch of dual-axis parameters. The formula is as follows: ; in, , The weights are adaptively assigned to the biaxial compensation values, with a range of 0 < , <1, and The core principle of this design is to assign a larger compensation weight to the axis with greater inertia and slower response, ensuring the dynamic consistency of the dual-axis synchronous correction and avoiding compensation overshoot caused by traditional fixed weights.

[0086] For example, when the equivalent inertia of the Y-axis is 1 Equivalent inertia of Y-axis No. 2 hour, , A larger compensation weight is assigned to axis 2 to ensure synchronous convergence of the two axes.

[0087] (3) Design of PI-type cross-coupled control law: The PI-type cross-coupled control law is adopted, which takes into account both speed and zero steady-state error characteristics. The formula is as follows: ; in, For proportional gain, For integral gain, based on system accuracy threshold Adjustment. For example, Set to 8.0. Setting this parameter to 0.5 ensures rapid convergence of synchronization errors while eliminating steady-state error, with overshoot less than [a certain value]. No oscillation. The proportional term responds quickly to synchronization error, forcing the error to converge rapidly; the integral term eliminates steady-state static error, ensuring that the synchronization error converges stably to 0 during long-term operation, meeting the long-term accuracy requirements of small-diameter tool connections.

[0088] (4) Generation of dual-axis position loop compensation: The output of the cross-coupling control law is distributed to the position loop inputs of the two axes according to adaptive weights, as shown in the following formula: ; Among them, the symbol design ensures the direction correction of reducing synchronization error in both axes. Compared with traditional master-slave control, the synchronization accuracy is improved by more than one order of magnitude, effectively solving the problem of insufficient tool connection accuracy caused by excessive synchronization deviation in both axes.

[0089] like Figure 15As shown: The upper left panel uses a comparison of dual error curves (blue for the traditional method / red for the method in this embodiment) to visually demonstrate the significant improvement in synchronization error convergence achieved by this technology. The traditional method suffers from steady-state fluctuations of approximately ±0.05 due to improper matching of fixed weights and PI parameters. The method in this embodiment, through adaptive weights and optimized PI parameters, achieves rapid convergence of the synchronization error to 0 within 10 control cycles, meeting the long-term accuracy requirements for small-diameter tool connections. The lower left panel uses dual weight curves (green w1 / magenta w2) to dynamically demonstrate the adaptive weight allocation process based on the equivalent inertia ratio. When the equivalent inertia m1 of axis 1 increases, w1 automatically decreases, and w2 increases accordingly, ensuring that the axis with larger inertia and slower response receives a larger compensation weight, achieving dynamic consistency in dual-axis synchronous correction. The upper right panel uses a superposition of three curves (cyan proportional term / yellow integral term / black total output) to clearly illustrate the design principle of the PI-type cross-coupling control law. The proportional term provides a rapid response to synchronization errors, forcing rapid error convergence; the integral term eliminates steady-state error, ensuring that the synchronization error converges stably to zero during long-term operation; the total output CLcc(k) balances speed and zero steady-state error, providing a precise control signal for compensation allocation. The lower right panel visually demonstrates the process of compensation being allocated to the dual-axis position loop inputs according to adaptive weights through dual compensation curves (blue dashed line Δycc1 / red dashed line Δycc2). The sign design ensures directional correction to reduce synchronization errors in both axes. Compared to traditional master-slave control, synchronization accuracy is improved by more than one order of magnitude, effectively solving the problem of insufficient tool engagement accuracy caused by excessive dual-axis synchronization deviation.

[0090] Example 8: As Figure 8 As shown, the high-speed follow-up laser blanking line dual Y-axis dynamic out-of-step synchronization control system includes a mechanical body, a feeding belt unit, an X-axis motion mechanism, a Y-axis motion mechanism 1, a Y-axis motion mechanism 2, a laser cutting head 1, a laser cutting head 2, a servo drive unit, a position detection unit, and a motion controller. The feeding belt unit, X-axis motion mechanism, Y-axis motion mechanism 1, and Y-axis motion mechanism 2 are all mounted on the mechanical body. Laser cutting head 1 and laser cutting head 2 are respectively mounted on the slides of Y-axis motion mechanism 1 and Y-axis motion mechanism 2. The servo drive unit includes servo motors and matching servo drivers that are respectively connected to Y-axis motion mechanism 1 and Y-axis motion mechanism 2. The position detection unit includes grating rulers respectively mounted on Y-axis motion mechanism 1 and Y-axis motion mechanism 2, and an encoder integrated into the servo motor. The signal input terminal of the motion controller is electrically connected to the position detection unit, and the signal output terminal of the motion controller is electrically connected to the servo driver of the servo drive unit. The motion controller integrates a position data acquisition layer, an SMO-FFRLS dual-channel parallel coupling identification layer, a DS evidence theory decision-level fusion layer, and a step loss compensation generation layer, all connected sequentially. The position data acquisition layer includes a position data synchronization acquisition module and a position data standardization preprocessing module. The synchronization acquisition module synchronously acquires the dual Y-axis grating ruler full closed-loop position feedback data output from the position detection unit, the servo motor encoder position data, and the dual Y-axis theoretical position commands issued by the motion controller, completing hard synchronization and timestamp alignment of multi-channel data to generate the original position state dataset. The position data standardization preprocessing module performs amplitude limiting filtering and moving average filtering on the original position state dataset, eliminating instantaneous jumps and impulse noise to generate a standardized dual-axis system state vector. The SMO-FFRLS dual-path parallel coupling identification layer includes parallel SMO branches and FFRLS branches, as well as bidirectional coupling core modules that are bidirectionally connected to the SMO and FFRLS branches respectively. The SMO branch is used to quickly observe abrupt nonlinear disturbances such as dual Y-axis chain gap impact, load change, and commutation vibration, and to calculate the biaxial step loss observation results caused by the abrupt disturbance. It includes a biaxial system state space modeling module, a sliding mode surface design module, a sliding mode observer equation solving module, and an abrupt disturbance and step loss observation output module that are sequentially connected by signals. The FFRLS branch is used to accurately identify slow time-varying parameter drifts such as wear, temperature drift, and guide roller deformation of dual Y-axis components, and to calculate the expected step loss caused by parameter drift. It includes a parameter identification linear regression modeling module, a forgetting factor RLS iterative calculation module, and an expected step loss output module connected in sequence. The bidirectional coupling core module is used to synchronously update the real-time time-varying parameters identified by the FFRLS branch to the state space model of the SMO branch, solving the problem of decreased observation accuracy and increased chattering caused by fixed model parameter mismatch. At the same time, it cancels out the known terms of fast-change abrupt disturbances observed by the SMO branch from the linear regression model of the FFRLS branch in advance, eliminating the interference of abrupt disturbances on the identification of slow time-varying parameters, and realizing bidirectional optimization of the two identification paths. The DS evidence theory decision-level fusion layer includes a staggered step identification framework definition module, a dual-evidence body BPA construction module, and an optimal staggered step solution module, which are connected in sequence. The staggered step identification framework definition module is used to define a standardized interval-based mutually exclusive complete staggered step identification framework bound to the system control accuracy threshold. The dual-evidence body BPA construction module is used to treat the staggered step observation results output by the SMO branch and the slow time-varying expected staggered step output by the FFRLS branch as two independent evidence bodies, and construct basic probability allocation BPA functions based on the convergence characteristics and historical error distribution of the two algorithms. The optimal staggered step solution module is used to fuse the two evidence bodies through the DS combination rule to obtain the comprehensive probability of each staggered step interval, and take the median of the maximum probability interval as the optimal estimate of the total staggered step of the two axes. The staggered step compensation generation layer includes a single-axis disturbance feedforward compensation generation module, a dual-axis synchronous cross-coupling compensation generation module, and a compensation command integration and issuance module that is signal-connected to both compensation generation modules. The single-axis disturbance feedforward compensation generation module is used to convert the optimal estimate of the total step loss of the two axes and the absolute position deviation of the single axis into feedforward compensation electrical signals that match the torque loop / speed loop interface of the servo drive unit, so as to realize the immediate generation and compensation of disturbances; the dual-axis synchronization cross-coupling compensation generation module is used to take the relative synchronization error of the two axes as the core control target, and dynamically distribute the compensation amount to the position loop input of the two axes through the cross-coupling control law, so as to force the dual-axis synchronization error to converge to 0 quickly; The compensation command integration and distribution module integrates the feedforward compensation electrical signal and the position command correction amount. After completing the limiting and smoothing processing, it synchronously distributes the command to the corresponding servo driver of the dual Y-axis via the industrial bus to achieve real-time closed-loop compensation for dynamic step loss of the dual Y-axis.

[0091] Experimental Example: To verify the effectiveness of the dual Y-axis synchronous out-of-step identification and compensation method and the high-speed follow-up laser blanking line control system of the present invention, a comparative experiment was designed, as follows: (1) Experimental conditions: A high-speed follow-up laser blanking line test platform was built. The dual Y-axis servo system adopted Panasonic MSME series servo motors. The resolution of the grating ruler was 1μm. The system control cycle was System control accuracy threshold The laser cutting speed was set to 10 m / min, the material was 1 mm thick cold-rolled steel plate, and the hole diameter was 5 mm (small hole diameter cutting scenario). The experimental environment was maintained at room temperature of 25℃ to avoid the influence of electromagnetic interference and mechanical vibration on the experimental results.

[0092] (2) Comparison objects: Three comparison schemes are set up as follows: Solution 1: The solution of this invention (SMO-FFRLS bidirectional coupling identification + DS evidence fusion + feedforward + cross-coupling compensation). Option 2: Traditional Option 1 (Single SMO identification + fixed parameter compensation); Option 3: Traditional Option 2 (single FFRLS identification + feedback compensation).

[0093] (3) Evaluation indicators: Three core evaluation indicators are selected, namely: ① Step loss identification accuracy: The maximum deviation between the identified value and the actual step loss (unit: μm). ② Dual-axis synchronization accuracy: The maximum value of dual-axis synchronization error in steady state (unit: μm); ③ Tool connection accuracy: The roundness error of the hole after machining a small diameter hole (unit: μm).

[0094] (4) Experimental procedure: Under the same experimental conditions, the three schemes were run continuously for 2 hours, and the above evaluation indicators were recorded every 10 minutes. The maximum value of each group's indicators was taken as the final experimental result. During the experiment, common disturbances in industrial sites (sudden load changes, chain gap impacts) were simulated to ensure the authenticity and reliability of the experimental results.

[0095] (5) Experimental Results and Conclusions: The experimental results are shown in the table below:

[0096] Conclusion: Experimental results show that the proposed solution significantly outperforms the two traditional solutions in terms of step loss identification accuracy, dual-axis synchronization accuracy, and tool connection accuracy. Specifically, the step loss identification accuracy is improved by over 60%, the dual-axis synchronization accuracy by over 50%, and the tool connection accuracy by over 50%. Figure 16 As shown: (1) The left panel compares the accuracy of the step loss identification of the three schemes using a bar chart: the accuracy of the scheme of this invention (blue bar) is ≤1.5μm, which is more than 60% higher than that of traditional scheme 2 (orange bar) ≤4.2μm and traditional scheme 3 (green bar) ≤3.8μm. This is due to the bidirectional coupling identification mechanism of SMO-FFRLS, which utilizes both the fast response characteristics of SMO to sudden step loss and the accurate identification capability of FFRLS for slow time-varying step loss, thus solving the identification blind spot problem of single algorithms. Figure 17 In this study, the present invention's scheme (Scheme 1) maintains a stable accuracy of 1.5-2.5 μm within the perturbation intensity range of 0-10, forming a flat green low-accuracy region. The accuracy of the traditional scheme 2 (Scheme 2) rapidly deteriorates to 7.2 μm with increasing perturbation intensity, forming a steep orange high-accuracy region. The accuracy of the traditional scheme 3 (Scheme 3) deteriorates to 6.3 μm, forming a yellow transition region. This difference reflects the dual adaptability of SMO-FFRLS bidirectional coupling identification to both abrupt and slow time-varying perturbations.

[0097] (2) The middle panel compares the dual-axis synchronization accuracy using a bar chart: the accuracy of the proposed solution is ≤1.8μm, which is more than 50% higher than that of traditional solutions 2 (≤5.0μm) and 3 (≤4.5μm). This is attributed to the DS evidence fusion layer's optimization of the fusion of the two identification results, and the "feedforward + cross-coupling compensation" strategy's solution to the problems of compensation lag and low synchronization accuracy. Figure 17 In the results, the accuracy of the proposed solution stabilizes in the 1.8-2.3 μm range, forming a stable blue low-precision region; the accuracy of the traditional solution 2 deteriorates to 6.5-8.0 μm, forming a red high-precision region; and the accuracy of the traditional solution 3 deteriorates to 5.5-6.5 μm, forming a pink transition region. This difference reflects the optimization capability of DS evidence fusion for the two-way identification results, as well as the control capability of "feedforward + cross-coupling compensation" for synchronization errors.

[0098] (3) The right panel compares the tool-catching accuracy using a bar chart: the accuracy of the proposed solution is ≤2.0μm, which is more than 50% higher than that of traditional solutions 2 (≤5.5μm) and 3 (≤5.0μm). This verifies the effectiveness of the proposed solution in small-aperture tool-catching scenarios on high-speed servo laser blanking lines, and it is especially suitable for processing scenarios with extremely high synchronization accuracy requirements. Figure 17 In the above, the accuracy of the proposed solution is stable in the range of 2.0-2.8μm, forming a cyan low-precision region; the accuracy of the traditional solution 2 deteriorates to 7.0-8.5μm, forming a brown high-precision region; and the accuracy of the traditional solution 3 deteriorates to 6.0-7.5μm, forming a purple transition region. This difference reflects the accuracy advantage of the proposed solution in the scenario of small-aperture tool receiving on a high-speed follow-up laser blanking line.

[0099] This invention solves the identification blind spot of a single algorithm through SMO-FFRLS bidirectional coupling identification, improves the accuracy of step loss identification through DS evidence fusion, and solves the problems of compensation lag and low synchronization accuracy through "feedforward + cross-coupling compensation". It can effectively meet the high-precision control requirements of high-speed follow-up laser blanking lines, and is especially suitable for scenarios with extremely high synchronization accuracy requirements such as small-diameter tool insertion.

[0100] Application Example: This application example describes the practical application of the present invention in the HSL-1200 high-speed follow-up laser blanking line of a heavy machinery manufacturing enterprise. This equipment is mainly used for precision blanking of automotive parts and engineering machinery parts. Its core requirement is to achieve high-precision synchronous control of dual Y axes to solve problems such as insufficient tooling accuracy for small-diameter holes and machining defects caused by dual-axis step loss.

[0101] (1) Application scenario overview: The dual Y-axis travel of the HSL-1200 high-speed follow-up laser blanking line is 1200mm, the maximum laser cutting speed is 15m / min, the processed materials include cold-rolled steel plate, hot-rolled steel plate, aluminum alloy plate, etc., the processing hole diameter range is 3mm-50mm, of which small hole diameter (3mm-8mm) accounts for 40%, and the requirements for dual-axis synchronization accuracy and tool connection accuracy are extremely high. Traditional control schemes have problems such as excessive tool connection error and frequent dual-axis step loss, which affect the product qualification rate.

[0102] (2) Implementation of the present invention: The dual Y-axis synchronous out-of-step identification and compensation method and the high-speed follow-up laser blanking line control system of the present invention are integrated into the equipment. The specific implementation details are as follows: ① Hardware Integration: A linear encoder (1μm resolution) is added to the dual Y-axis servo system to acquire real-time position signals from both axes; a distance sensor is added to the laser cutting head to assist in detecting the tool contact accuracy; the industrial controller uses a Siemens S7-1500 series PLC to realize real-time algorithm calculation and command issuance; the servo driver uses a Panasonic MINASA6 series, which is compatible with the compensation command output interface of this invention, eliminating the need for hardware modification of the servo driver and reducing integration costs. The linear encoder interacts with the PLC via the Profinet bus, with a data transmission rate set to 100Mbps to ensure the real-time performance and accuracy of position signal transmission; the distance sensor uses a laser ranging type with a measurement accuracy of ±0.5μm, connected to the PLC via an analog interface for real-time feedback of the position deviation at the tool contact point, assisting in calibrating the compensation effect. Simultaneously, a signal isolation module is added between the PLC and the servo driver to avoid command transmission distortion caused by electromagnetic interference in the industrial environment, improving the stability of system operation.

[0103] ② Software Integration: The entire algorithm proposed in this invention—"data preprocessing layer → SMO-FFRLS bidirectional coupling identification layer → DS evidence theory decision-level fusion layer → out-of-step compensation generation layer"—is implemented using the Siemens TIAPortal software platform. A hybrid programming mode combining ladder diagrams and structured text (ST) is employed. The core algorithms (SMO sliding mode observation, FFRLS parameter iteration, and DS evidence fusion) are written in ST language to ensure computational efficiency and adapt to the real-time computational requirements of the PLC (control cycle maintained at 1ms). During software integration, an interface for algorithm parameter tuning is reserved, allowing flexible adjustment of key parameters such as filter coefficients, sliding mode surface parameters, and feedforward gain according to actual processing scenarios (e.g., different materials, different cutting speeds), improving system adaptability. Simultaneously, a data acquisition and monitoring module is integrated to record key data such as dual-axis position, speed, synchronization error, out-of-step identification value, and compensation amount in real time, facilitating later troubleshooting and algorithm optimization.

[0104] ③ Debugging and Optimization: After system integration, phased debugging and optimization are performed to ensure that each module works collaboratively and meets the design accuracy requirements. First, hardware debugging is conducted, checking the connection reliability of the grating ruler, distance sensor, PLC, and servo driver, calibrating the zero point of the grating ruler and the accuracy of the distance sensor to ensure accurate position signal acquisition. Then, software debugging is performed, individually testing the operating effect of each algorithm module, verifying the noise reduction effect of data preprocessing, the accuracy of SMO-FFRLS bidirectional coupling identification, the rationality of DS evidence fusion, and the correctness of compensation quantity generation. Finally, overall machine integration is performed, simulating actual processing scenarios (load sudden changes, chain gap impact), and tuning key parameters. For example, based on the actual characteristics of the HSL-1200 equipment, the filter coefficient λ is adjusted to 0.22, the sliding surface parameter α is adjusted to 1.6, the feedforward gain Kvff is adjusted to 0.96, and the cross-coupling control law PI parameters Kp and Ki are adjusted to 8.5 and 0.55, respectively, to ensure that the dual-axis synchronization accuracy and tool connection accuracy meet the design requirements. During the debugging process, the synchronization error and step loss changes were observed in real time through the monitoring module. For any accuracy deviations that occurred, the smoothing coefficient of the coupling module and the conflict coefficient threshold of the fusion module were fine-tuned to further optimize the system performance.

[0105] (3) Application effect: After adopting the solution of this invention, the HSL-1200 high-speed follow-up laser blanking line has achieved significant application effect after 3 months of continuous and stable operation. Specifically, the effect is as follows: ① The frequency of dual-axis step loss is significantly reduced, from an average of 1-2 times every 8 hours in the traditional solution to no more than once a month, effectively reducing processing interruptions and product scrap caused by step loss; ② The machining accuracy is significantly improved. The tool connection accuracy for small-diameter holes (3mm-8mm) has been improved from ≤5.0μm in the traditional solution to ≤2.0μm, meeting the machining requirements of precision parts; ③ Improved product qualification rate: The qualification rate of blanking and processing of automotive parts increased from 92.3% to 98.7%, significantly reducing production costs; ④ Enhanced system anti-disturbance capability: Under simulated industrial site load changes (load fluctuation ±10%) and chain gap impact scenarios, the dual-axis synchronization error can still be stably controlled within ≤1.8μm, and the operational stability is significantly better than traditional solutions.

[0106] Specifically, such as Figure 18 As shown, ① The upper left panel's nonlinear surface exhibits a composite trend of "sine wave fluctuation + logarithmic decay" due to the dual influence of processing speed (5-20m / min) and aperture (3-60mm). The synchronization accuracy value stabilizes within the 1.0-3.0μm range, where: Speed ​​dimension: The sinusoidal wave characteristics (amplitude ±0.3μm) demonstrate the adaptability of this invention to high-speed cutting scenarios, which not only avoids the sharp drop in accuracy under high speed in traditional solutions, but also enables a rapid response to speed changes through the SMO-FFRLS bidirectional coupling identification layer. Aperture dimension: The logarithmic decay feature (attenuation coefficient -0.1μm / ten times the aperture) reflects the precise control of large aperture processing in this invention, and the nonlinear compensation of aperture-related step loss is achieved through the DS evidence fusion layer.

[0107] The surface demonstrates that the present invention achieves synchronization accuracy fluctuations within ±0.5μm in a speed range of 5-20m / min and an aperture range of 3-60mm, thus meeting the requirements for high-precision synchronization control.

[0108] ② Upper right panel - Nonlinear surface for tool connection accuracy: Influenced by both aperture (3-60mm) and disturbance intensity (0-15%), the surface exhibits a composite trend of "cosine fluctuation + exponential decay." The tool connection accuracy value stabilizes within the 2.0-3.5μm range, where: Aperture dimension: Cosine wave characteristics (amplitude ±0.2μm) demonstrate the optimization capability of this invention for small aperture tool connection scenarios, and improve tool connection accuracy through the cross-coupling control law of the step loss compensation generation layer; Disturbance Dimension: The exponential decay characteristic (attenuation coefficient -0.1μm / 10% disturbance) reflects the invention's anti-interference capability in disturbed scenarios, achieved through noise reduction processing in the data preprocessing layer and disturbance decoupling in the DS fusion layer. This surface demonstrates that the invention maintains tool connection accuracy fluctuations within ±0.3μm in the 3-60mm aperture range and 0-15% disturbance intensity, meeting the high-precision requirements for small-diameter tool connection.

[0109] ③ Bottom Left Panel - Nonlinear Surface of Step-Out Frequency: This surface is influenced by both disturbance intensity (0-15%) and processing speed (5-20 m / min), exhibiting a composite trend of "exponential decay + logarithmic growth." The step-out frequency value stabilizes in the range of 0.1-0.5 times / month, where: Disturbance dimension: The exponential decay characteristic (decay coefficient -0.3 times / month / 10% disturbance) reflects the strong adaptability of this invention to disturbance scenarios. It achieves accurate acquisition and processing of disturbance signals through hardware integration (grating ruler + distance sensor) and software algorithm (data preprocessing layer). Speed ​​dimension: The logarithmic growth characteristic (growth coefficient 0.05 times / month / 5m / min) reflects the stability control of this invention in high-speed cutting scenarios, which is achieved through adaptive weight allocation of the out-of-step compensation generation layer.

[0110] The surface demonstrates that the present invention controls the out-of-step frequency to within 0.5 times / month within the range of 0-15% disturbance intensity and 5-20m / min speed, which is more than 90% lower than the traditional solution.

[0111] This application example demonstrates that the dual Y-axis synchronous out-of-step identification and compensation method and the high-speed follow-up laser blanking line control system of the present invention can effectively adapt to the actual needs of industrial sites, solve the problems of insufficient accuracy, frequent out-of-step, and weak anti-disturbance ability of traditional solutions, and can be directly applied to various high-speed follow-up laser blanking lines. It is especially suitable for precision machining scenarios with high requirements for synchronization accuracy and tool connection accuracy, and has good industrial application value and promotion prospects.

[0112] All the above embodiments merely illustrate implementation methods for relevant practical applications of the present invention, and their descriptions are relatively specific and detailed, but they should not be construed as limiting the scope of the invention patent. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these all fall within the protection scope of the present invention. Therefore, the protection scope of this invention patent should be determined by the appended claims.

[0113] List of Formula Symbols for this Invention:

Claims

1. A high-speed follow-up laser blanking line control method, characterized in that, include: P1. Synchronously acquire position feedback data and theoretical position commands for the dual Y-axis to generate a standardized dual-axis system state vector; P2. An architecture combining the SMO sliding mode observer branch and the FFRLS recursive least squares branch with forgetting factor is adopted to identify abrupt nonlinear disturbances and slow time-varying parameter drifts in the dual Y-axis. The real-time time-varying parameters identified by the FFRLS branch are synchronously updated to the state space model of the SMO branch, and the abrupt disturbances observed by the SMO branch are canceled out in advance from the identification model of the FFRLS branch. The observed step loss caused by the abrupt disturbance and the expected step loss in slow time-varying mode are calculated. P3. Based on the DS evidence theory, a staggered quantity identification framework is constructed. The two staggered quantity results are used as independent evidence bodies to construct a basic probability allocation function. Evidence fusion is completed through the DS combination rule, and the optimal estimate of the total staggered quantity of the two axes is obtained by solving the problem. P4. Composite Closed-Loop Compensation and Command Execution: A composite compensation architecture of feedforward + feedback is adopted. The feedforward compensation amount matching the servo system is generated based on the optimal step loss amount and single-axis position deviation. The cross-coupled feedback compensation amount is generated based on the dual-axis synchronization error and synchronously sent to the dual Y-axis servo system to complete the real-time closed-loop compensation for dynamic step loss of the dual Y-axis.

2. The control method according to claim 1, characterized in that: The filtering and noise reduction process sequentially performs amplitude limiting filtering and moving average filtering: ； Where i is the sequence number of the 6 independent location channels, covering all original location data; y i (k) represents the original input data for the i-th channel; This is the effective value of the limiting filter for this channel in the previous cycle; The i-th channel is the smoothed data after final filtering; N is the sliding window length, and the summation term covers the current period and the previous N periods. The effective value of the limiting filter for one cycle.

3. The control method according to claim 1, characterized in that: In the SMO sliding mode observer branch, a non-singular terminal sliding surface is designed: ； Overall sliding surface matrix form ; Where i = 1, 2, 3, 4, 5, corresponding to the 5 dimensions of the observation error vector; It is the absolute value of the observation error; α and β are the positive weighting coefficients of the sliding surface, and p and q are the coefficients that satisfy 1 ) is a symbolic function. This is the state observation error vector of the two-axis system.​ 4. The control method according to claim 1, characterized in that: In the FFRLS recursive least squares branch with forgetting factor, the iterative method includes: Calculate the FFRLS gain matrix ; Update the time-varying parameter vector to be identified ; Wherein, the gain matrix K(k) determines the magnitude of the correction to the parameter estimate by the new sampled data; λ is the forgetting factor; For parameter estimation, the error covariance matrix, Let z(k) be the regression data matrix of the linear regression model, and z(k) be the system output vector of the linear regression model. The vector of time-varying parameters to be identified.

5. The control method according to claim 4, characterized in that: The method for canceling abrupt perturbations in the SMO→FFRLS coupling direction is as follows: ； ； in, This is the linear regression output vector after perturbation cancellation. and The biaxial rapid change disturbance value observed in the SMO branch; the FFRLS→SMO coupling direction only performs real-time updates of the state space model parameters when the FFRLS parameter iteration convergence flag P′(k)=1.

6. The control method according to claim 1, characterized in that: The DS evidence fusion method includes: taking the median of the maximum probability interval as the optimal estimate S of the total biaxial step loss. ； ； Find the interval with the largest BPA after fusion, and take the midpoint of the interval as the optimal estimate S of the total step loss of the two axes; Where K is the conflict coefficient between the two pieces of evidence; ΔL i For the interval unit in the step loss identification framework, m1( m2 ( ) are the basic probability allocation functions for the two evidence bodies, m( ) is the integrated probability allocation function after fusion.

7. The control method according to any one of claims 1 to 6, characterized in that: The method for the feedforward compensation amount includes: ； Where i corresponds to different Y-axis, This is the single-axis feedforward compensation amount. For velocity feedforward gain, Theoretical speed command, For acceleration feedforward gain, The equivalent inertia identified by FFRLS Theoretical acceleration command, This represents the total perturbation value observed by SMO. For position deviation feedforward gain, This represents the absolute position deviation of a single axis.

8. The control method according to any one of claims 1 to 6, characterized in that: The method for cross-coupling feedback compensation includes: Dual-axis real-time synchronization error calculation: ; in, , Here, S(k) represents the measured position of the dual-axis grating, and S(k) is the optimal estimate of the step loss output from the DS fusion; s (k) represents the real-time synchronization error of the two axes; Adaptive weight allocation for biaxial compensation: ； ； Employing a PI-type cross-coupling control law: ; Among them, the proportional term It is the fast response synchronization error; integral term Eliminate steady-state static error; The output of the cross-coupling control law is distributed to the position loop inputs of the two axes according to adaptive weights to achieve closed-loop control of the synchronization error. ； Among them, the symbol design ensures the direction correction to reduce synchronization error in both axes.

9. A system for implementing the control method as described in any one of claims 1 to 8, characterized in that, The system includes: Y-axis motion mechanism No. 1 and Y-axis motion mechanism No. 2; The position detection unit includes a grating ruler installed on Y-axis motion mechanism 1 and Y-axis motion mechanism 2 respectively, and an encoder integrated into the servo motor; The signal input terminal of the motion controller is electrically connected to the position detection unit, and the signal output terminal of the motion controller is electrically connected to the servo driver of the servo drive unit.

10. The system according to claim 9, characterized in that, Also includes: The location data acquisition layer includes: Location data synchronization acquisition module: Acquires core position status data of dual Y-axis, and completes hard synchronization and timestamp alignment of multi-channel data; Location data standardization preprocessing module: Performs amplitude limiting filtering and moving average filtering on the original location state dataset P to form a standardized dual-axis system state vector V; The SMO-FFRLS dual-path parallel coupled identification layer completes mutual correction through bidirectional coupling modules, including: SMO branch: Adapted for the observation of abrupt nonlinear disturbances such as chain gap impact, load change and commutation vibration; FFRLS branch: Expected step loss due to changes in quantization parameters; The bidirectional coupling core module synchronously updates the real-time time-varying parameter V' identified by FFRLS to the state-space model of SMO; and preemptively cancels out the rapid mutation perturbations observed by SMO from the linear regression model of FFRLS as known terms. The DS evidence theory decision-making fusion layer provides a unique benchmark for the compensation process; it includes: The module defining the missing step quantity identification framework defines a standardized interval-based mutually exclusive complete identification framework. Dual Evidence Body (BPA) Construction Module: The observed step loss result B and the slow time-varying expected step loss SO are treated as two independent evidence bodies. Based on the convergence characteristics of the two-way algorithm and the historical error distribution, the Basic Probability Allocation (BPA) function is constructed to quantify the probability of each step loss interval. The optimal step loss calculation module uses the DS combination rule to fuse the two evidence sources, obtain the comprehensive probability of each step loss interval, and take the median of the interval with the highest probability as the optimal estimate S of the total step loss of the two axes. Step loss compensation generation layer: This layer converts the fused, precise step loss quantity into control compensation quantities executable by the servo system, including: Single-axis disturbance feedforward compensation generation module: converts the optimal estimated value S and the single-axis absolute position deviation B'' into feedforward compensation, and directly generates an electrical signal E to match the torque loop / speed loop interface of the servo system; Dual-axis synchronous cross-coupling compensation generation module: dynamically distributes the compensation amount to the position loop input terminals of the two axes through the control law CL; Compensation command integration and distribution module: integrates feedforward compensation amount and position command correction amount, and distributes them to the dual Y-axis servo system.

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