On-line iterative compensation system and method for servo errors

Abstract

This invention provides an online iterative compensation system and method for servo errors, comprising: a disturbance observer module: treating the internal nonlinearity and external disturbances of the system as a lumped disturbance during the compensation of slow-cut servo errors by a fast-cut servo system, and estimating and compensating for them; a dual-loop high-bandwidth control module: designing a positive acceleration-velocity-position feedback damping controller in the inner loop to suppress lightly damped resonant modes; an online iterative compensation module: predicting the tracking error of the robust high-bandwidth controller and iteratively compensating it through online reference trajectory correction; and an overall control scheme implementation module: including a disturbance observer, a high-bandwidth dual-loop controller, and an online iterative compensation module; predicting the system state within the prediction interval based on a linear model of the discrete system difference equations, and modifying the reference trajectory online for error compensation. This invention enables real-time iterative compensation, improving the tracking accuracy and anti-interference capability of the fast-cut servo system during the compensation of slow-cut servo errors.

Description

Technical Field

[0001] This invention relates to the fields of ultra-precision machining and motion control technology, specifically to an online iterative compensation system and method for servo errors. Background Technology

[0002] Slow-tool servo diamond turning has wide applications in ultra-precision machining; however, its dynamic servo error significantly affects machining accuracy. A master-slave control strategy integrating a fast-tool servo into a slow-tool servo system can effectively compensate for the dynamic servo error of the slow-tool servo. The reference trajectory of the fast-tool servo system is the real-time generated slow-tool servo error. Since this reference trajectory cannot be predicted in advance, the fast-tool servo system can only be controlled through simple feedback, and its tracking accuracy is inevitably affected by phase lag errors and other factors.

[0003] Chinese patent application CN112947310A discloses a method and device for pre-compensation of rotary servo motor trajectory based on a predictive model. This patent uses a linear error prediction model to accurately predict the tracking error of the rotary servo motor and iteratively compensates for the tracking error online through trajectory pre-compensation. However, this method does not incorporate high-bandwidth control and robust control, making it unable to achieve high-frequency trajectory tracking and interference suppression, and thus unsuitable for high-speed motion scenarios such as fast-tool servo machining. Furthermore, this method requires the reference trajectory to be known at future moments to complete error prediction and compensation, making it unsuitable for scenarios where the reference trajectory needs to be generated in real-time based on measurement results, such as fast-tool compensation for slow-tool servo errors.

[0004] The closest paper:

[0005] (1) The paper "Intelligent Tracking Error Prediction and Feedforward Compensation for Nanopositioning Stages with High-bandwidth Control" by Meng Yixuan et al. from Shanghai Jiao Tong University, IEEE Transactions on Industrial Informatics, 2022, DOI:10.1109 / TII.2022.3199263, proposes an intelligent error prediction and feedforward compensation method for a high-bandwidth controller based on Gaussian processes for nanopositioning systems. However, this method can only achieve offline error prediction. It requires predicting the tracking error of each point on the entire reference trajectory when all reference trajectories are known, and then performing compensation. When the amount of data is large, it takes a lot of time to perform error prediction and compensation, and it is not robust to changes in the trajectory. At the same time, the method does not design a disturbance observer, so it lacks the ability to resist disturbances and is not suitable for application scenarios with large disturbances such as cutting forces in fast-tool servo machining.

[0006] (2) Shanghai Jiao Tong University, Wang Xiangyuan et al., proposed "Simultaneous damping and tracking control of a normal-stressed electromagnetic actuated nano-positioning stage", Sensors and Actuators A: Physical, 2022. This paper proposes a dual-loop controller with inner and outer loop synchronous optimization, which can effectively suppress the hysteresis nonlinearity and lightly damped resonance characteristics of the system and increase the control bandwidth to above the first-order resonant frequency of the system. However, this method is a simple feedback controller, which inevitably has phase lag. For high-frequency trajectories, there is a large tracking error caused by the phase lag, and the controller also lacks the ability to resist interference. Summary of the Invention

[0007] To address the shortcomings of existing technologies, this invention provides an online iterative compensation system and method for servo errors.

[0008] According to the present invention, an online iterative compensation system and method for servo errors are provided, the scheme of which is as follows:

[0009] Firstly, an online iterative compensation system for servo errors is provided, the system comprising:

[0010] Disturbance observer module: Treats the internal nonlinearity and external disturbances of the system as a whole lumped disturbance during the process of fast-tool servo compensation for slow-tool servo error, and performs estimation and compensation.

[0011] Dual-loop high-bandwidth control module: In the inner loop of the system, a positive acceleration-velocity-position feedback damping controller is designed to suppress the lightly damped resonant mode;

[0012] Online iterative compensation module: Predicts the tracking error of the robust high-bandwidth controller through the difference equation of the discrete system, and iteratively compensates for it through online reference trajectory correction;

[0013] The overall control scheme implementation module includes a disturbance observer, a high-bandwidth dual-loop controller, and an online iterative compensation module. It predicts the state of the system within the prediction interval based on a linear model of the discrete system difference equation, and then modifies the reference trajectory through online iteration to compensate for errors.

[0014] Preferably, the disturbance observer module includes:

[0015] The fast-tool servo system is driven by either piezoelectric drive or normal stress electromagnetic drive. The entire system is described in the Laplace continuous domain by the following third-order model:

[0016]

[0017] Where a3, a2, a1, a0, b2, b1, b0 are the parameters of the system model, and s is the Laplace operator;

[0018] The third-order model is a non-minimum-phase system. To ensure the stability of the perturbation observer, the third-order model needs to be modified to become a minimum-phase system.

[0019]

[0020] For the inverse P of the minimum phase system n -1 Given the order constraint of P(s), a corresponding low-pass filter Q(s) needs to be designed to ensure that P n -- Q(s) can be realized in practical applications using a second-order low-pass filter:

[0021]

[0022] Where τ is a parameter that determines the bandwidth of the low-pass filter, and by tuning it, the lumped disturbance estimate of the system is: And use control methods to compensate.

[0023] Preferably, the dual-ring high-bandwidth control module includes:

[0024] The inner loop damping controller expression for the positive acceleration-velocity-position feedback damping controller is:

[0025]

[0026] The inner-loop damping controller has five adjustable parameters: β2, β1, β0, ξ, ω. p By arbitrarily configuring the desired locations of the five system poles to achieve the desired system damping ratio, the transfer function of the inner-loop damping part is:

[0027]

[0028] For inputs in the mid-to-low frequency range, the inner-loop damping component can be approximated as a second-order rigid system, with the following expression:

[0029]

[0030] Where c1, c0, k1, and k0 are parameters of the rigid system; in the outer loop, a high-gain proportional-integral tracking controller is applied to reduce residual tracking error and achieve accurate trajectory tracking. The expression for the outer loop tracking controller is:

[0031]

[0032] Where, k p and k o The proportional gain and integral gain are used; high-bandwidth control is achieved by optimizing the parameters of the inner-loop damping controller and the outer-loop tracking controller.

[0033] The disturbance observer module and the high-bandwidth dual-loop control module together form a robust high-bandwidth controller.

[0034] Preferably, the online iterative compensation module includes:

[0035] Linear error prediction submodule: According to equations (6) and (7), the fast-tool servo system calculates the error from the input y. r To output y out The transfer function between them is written as follows:

[0036]

[0037] The transfer function is a third-order transfer function; in the discrete domain, the transfer function is expressed in the form of a difference equation:

[0038] y out (k+1)=0y out (k)+1y out (k-1)+2y out (k-2)+β0y r(k)+β1y r (k-1)+β2y r (k-2)(9)

[0039] Where α0, α1, α2, β0, β1, and β2 are constants related to the sampling frequency and the coefficients of each term in the overall closed-loop system transfer function; in the state space, equation (9) is written as:

[0040]

[0041] For the above spatial state equation, x(k)=[x1(k)2(k)3(k)] T It is a state vector, where x1(k) = y out (k),x2(k)=1(k-1)+h1y r (k-1),x3(k)=2(k-1)+h2y r (k-1), h0=0, h1=2 / 2, State matrix A = [α 012 [1 0 0; 0 1 0], Input matrix B = [h0; h1; h2], Output matrix C = [1 0 0]; γ d () represents the lumped disturbance of the system, which is estimated by the designed disturbance observer and fed back into the control loop for compensation; therefore, for this linear error prediction model, γ d The nonlinear effect of () is negligible; at the current sampling time kT s All parameters can be obtained, thus enabling the prediction of the state at the next moment:

[0042]

[0043] Right now

[0044]

[0045] Assume N p If the prediction interval is defined, then all states within the prediction interval can be predicted, and the predicted system output can be derived as follows:

[0046]

[0047] The realization of this prediction requires a reference trajectory y at a future time. r (k+i); For the fast tool servo system to compensate for the slow tool servo error, the reference trajectory of the fast tool servo system is generated in real time based on the servo error measured by the slow tool servo system.

[0048] Preferably, the online iterative compensation module further includes:

[0049] The future moment reference trajectory prediction submodule uses a quadratic polynomial for interpolation. It fits the coefficients of the quadratic polynomial to the reference trajectories of ten sampling points (the current sampling moment and the previous nine sampling moments), i.e., the slow-motion servo errors of these ten sampling points; then, y... r (k+i) can be obtained by interpolating the quadratic polynomial, thereby enabling the prediction of the output of the fast-tool servo system within the prediction interval.

[0050] Preferably, the online iterative compensation module further includes:

[0051] Error Iteration Compensation Submodule: The error e(k) is defined as the difference between the reference trajectory and the actual tracking trajectory; without compensation, (k+N) p )T s The tracking error at time step is predicted as:

[0052]

[0053] Assume N c It is a compensation interval, and N c ≤N p -1; when compensation item c r (k+N c Add to reference trajectory y r (k+N p When ), the tracking error e(k+N) p ) changes, due to the system output y out (k+N p When the system output changes, the change is as follows:

[0054] c out (k+Np)=εc r (k+N c (15)

[0055] in, It is a constant; therefore, after compensation, the system error is derived as:

[0056]

[0057] This error is gradually compensated through an iterative process; since the prediction interval N... p The system state within has been accurately predicted, and the iterative process is implemented online during the model prediction process;

[0058] Let the compensation term be c1 in the first iteration, then we have

[0059]

[0060] Where ∈ is the compensation gain; substituting equation (17) into equation (15), the tracking error after the first iteration is derived as:

[0061]

[0062] During the second iteration, e c1 It is also integrated into the compensation items and designed as follows:

[0063] c2(k+N c )=c1(k+N c )+∈e c1 (k+N p (19)

[0064] Therefore, the tracking error after the second iteration is:

[0065]

[0066] The compensation term and tracking error after the nth iteration are derived in the same way, and their expressions are:

[0067] c n (k+n c ) = c (n-1) (k+N c )+∈e c(n-1) (k+N p ) (twenty one)

[0068]

[0069] Therefore, the compensation gain is satisfied when the following conditions are met:

[0070]

[0071] If n approaches infinity, the tracking error of the system approaches 0.

[0072] Preferably, in the overall control scheme implementation module, to meet the compensation for tracking error at the next sampling time, the compensation interval N is... c The value must be greater than or equal to 1, therefore the prediction interval N p It must be greater than or equal to 2.

[0073] Secondly, an online iterative compensation method for servo errors is provided, the method comprising:

[0074] Disturbance observer steps: Treat the internal nonlinearity and external disturbances of the system as a whole lumped disturbance during the process of fast-tool servo compensation for slow-tool servo error, and estimate and compensate for them.

[0075] Dual-loop high-bandwidth control steps: In the inner loop of the system, a positive acceleration-velocity-position feedback damping controller is designed to suppress the lightly damped resonant mode;

[0076] Online iterative compensation steps: predict the tracking error of the robust high-bandwidth controller through the difference equation of the discrete system, and iteratively compensate by correcting the online reference trajectory;

[0077] The overall control scheme implementation steps include: a disturbance observer, a high-bandwidth dual-loop controller, and an online iterative compensation module; the linear model based on the discrete system difference equation predicts the system state within the prediction interval, and then the reference trajectory is modified online for error compensation.

[0078] Preferably, the disturbance observer step includes:

[0079] The fast-tool servo system is driven by either piezoelectric drive or normal stress electromagnetic drive. The entire system is described in the Laplace continuous domain by the following third-order model:

[0080]

[0081] Where a3, a2, a1, a0, b2, b1, b0 are the parameters of the system model, and s is the Laplace operator;

[0082] The third-order model is a non-minimum-phase system. To ensure the stability of the perturbation observer, the third-order model needs to be modified to become a minimum-phase system.

[0083]

[0084] For the inverse P of the minimum phase system n -1 Given the order constraint of P(s), a corresponding low-pass filter Q(s) needs to be designed to ensure that P n -1 Q(s) can be realized in practical applications using a second-order low-pass filter:

[0085]

[0086] Where τ is a parameter that determines the bandwidth of the low-pass filter, and by tuning it, the lumped disturbance estimate of the system is: And use control methods to compensate.

[0087] Preferably, the dual-ring high-bandwidth control step includes:

[0088] The inner loop damping controller expression for the positive acceleration-velocity-position feedback damping controller is:

[0089]

[0090] The inner-loop damping controller has five adjustable parameters: β2, β1, β0, ξ, ω. p By arbitrarily configuring the desired locations of the five system poles to achieve the desired system damping ratio, the transfer function of the inner-loop damping part is:

[0091]

[0092] For inputs in the mid-to-low frequency range, the inner-loop damping component can be approximated as a second-order rigid system, with the following expression:

[0093]

[0094] Where c1, c0, k1, and k0 are parameters of the rigid system; in the outer loop, a high-gain proportional-integral tracking controller is applied to reduce residual tracking error and achieve accurate trajectory tracking. The expression for the outer loop tracking controller is:

[0095]

[0096] Where, k p and k i The proportional gain and integral gain are used; high-bandwidth control is achieved by optimizing the parameters of the inner-loop damping controller and the outer-loop tracking controller.

[0097] The disturbance observer step and the high-bandwidth dual-loop control step together constitute a robust high-bandwidth controller.

[0098] The online iterative compensation steps include:

[0099] Linear error prediction sub-step: According to equations (6) and (7), the fast-tool servo system calculates the error from the input y. r To output y out The transfer function between them is written as follows:

[0100]

[0101] The transfer function is a third-order transfer function; in the discrete domain, the transfer function is expressed in the form of a difference equation:

[0102] y out (k+1)=0y out (k)+1y out (k-1)+2y out (k-2)+β0y r (k)+β1y r (k-1)+β2y r (k-2)(9)

[0103] Where α0, α1, α2, β0, β1, and β2 are constants related to the sampling frequency and the coefficients of each term in the overall closed-loop system transfer function; in the state space, equation (9) is written as:

[0104]

[0105] For the above spatial state equation, x(k)=[x1(k)2(k)3(k)] T It is a state vector, where x1(k) = y out (k),x2(k)=1(k-1)+h1y r (k-1),x3(k)=2(k-1)+h2y r (k-1), h0=0, h1=2 / 2, State matrix A = [α] 012 [1 0 0; 0 1 0], Input matrix B = [h0; h1; h2], Output matrix C = [1 0 0]; γ d () represents the lumped disturbance of the system, which is estimated by the designed disturbance observer and fed back into the control loop for compensation; therefore, for this linear error prediction model, γ d The nonlinear effect of () is negligible; at the current sampling time kT s All parameters can be obtained, thus enabling the prediction of the state at the next moment:

[0106]

[0107] Right now

[0108]

[0109] Assume N p If the prediction interval is defined, then all states within the prediction interval can be predicted, and the predicted system output can be derived as follows:

[0110]

[0111] The realization of this prediction requires a reference trajectory y at a future time. r (k+i); For the fast tool servo system to compensate for the slow tool servo error, the reference trajectory of the fast tool servo system is generated in real time based on the servo error measured by the slow tool servo system.

[0112] The online iterative compensation step also includes:

[0113] The future moment reference trajectory prediction submodule uses a quadratic polynomial for interpolation. It fits the coefficients of the quadratic polynomial to the reference trajectories of ten sampling points (the current sampling moment and the previous nine sampling moments), i.e., the slow-motion servo errors of these ten sampling points; then, y... r (k+i) can be obtained by interpolating the quadratic polynomial, thereby enabling the prediction of the output of the fast-tool servo system within the prediction interval;

[0114] The online iterative compensation step also includes:

[0115] Error Iteration Compensation Sub-step: The error e(k) is defined as the difference between the reference trajectory and the actual tracking trajectory; without compensation, (k+N) p )T s The tracking error at time step is predicted as:

[0116]

[0117] Assume N c It is a compensation interval, and N c ≤N p -1; when compensation item c r (k+N c Add to reference trajectory y r (k+N p When ), the tracking error e(k+N) p ) changes, due to the system output y out (k+N p When the system output changes, the change is as follows:

[0118] c out (k+N p )=εc r (k+N c (15)

[0119] in, It is a constant; therefore, after compensation, the system error is derived as:

[0120]

[0121] This error is gradually compensated through an iterative process; since the prediction interval N... p The system state within has been accurately predicted, and the iterative process is implemented online during the model prediction process;

[0122] Let the compensation term be c1 in the first iteration, then we have

[0123]

[0124] Where ∈ is the compensation gain; substituting equation (17) into equation (15), the tracking error after the first iteration is derived as:

[0125]

[0126] During the second iteration, e c1 It is also integrated into the compensation items and designed as follows:

[0127] c2(k+N c )=c1(k+N c )+∈e c1 (k+N p (19)

[0128] Therefore, the tracking error after the second iteration is:

[0129]

[0130] The compensation term and tracking error after the nth iteration are derived in the same way, and their expressions are:

[0131] c n (k+N c ) = c (n-1) (k+N c )+∈e c(n-1) (k+N p ) (twenty one)

[0132]

[0133] Therefore, the compensation gain is satisfied when the following conditions are met:

[0134]

[0135] If n approaches infinity, the tracking error of the system approaches 0;

[0136] In the overall control scheme implementation module, to compensate for the tracking error at the next sampling time, the compensation interval N is... c The value must be greater than or equal to 1, therefore the prediction interval N p It must be greater than or equal to 2.

[0137] Compared with the prior art, the present invention has the following beneficial effects:

[0138] 1. By adopting a designed disturbance observer structure, the problem of resisting the lumped disturbance composed of internal nonlinearity and external disturbances in the process of fast tool servo compensation for slow tool servo error is solved. The internal nonlinearity of the system includes the nonlinearity of the driver itself, the identification error of the system model, the model changes caused by various internal and external disturbances, and the uncertainty caused by the slight changes of various parameters of the mechanical and electrical system during use. The external disturbances of the system include the external disturbances caused by the cutting force during the machining process, various voltage and electromagnetic disturbances caused by the control circuit, and various mechanical and electronic disturbances in the machine tool and the external environment.

[0139] 2. By designing a dual-loop controller consisting of an inner-loop damping controller and an outer-loop tracking controller, the high-bandwidth control problem of the fast-blade servo system was solved. Fast-blade servo systems are typically driven by piezoelectric or normal stress electromagnetic drives, with displacement transmitted by a flexible guiding mechanism. The inherent hysteresis nonlinearity of the actuator and the slightly damped resonance characteristics of the mechanical structure pose significant challenges to its high-precision, high-bandwidth motion control. The inner-loop damping controller can suppress the slightly damped resonance characteristics of the system and improve the system's damping ratio, thus facilitating the realization of high gain in the tracking controller. The high-gain tracking controller in the outer loop can suppress the effects of hysteresis nonlinearity, improving control accuracy and bandwidth.

[0140] 3. By using a linear error prediction model based on the difference equations of discrete systems, the problem of accurately predicting the tracking error of a robust high-bandwidth controller is solved. When the nonlinear disturbance of the system has been accurately estimated and compensated by the disturbance observer, a linear model that accurately predicts the tracking error of the closed-loop controller can be constructed based on the difference equations of discrete systems. This model can estimate the system state at future times, thereby accurately predicting the tracking error at the next sampling point and using it for online compensation.

[0141] 4. The problem of improving the tracking accuracy of the robust high-bandwidth controller is solved by using an online iterative compensation method. The robust high-bandwidth controller is a pure feedback controller, which inevitably has phase lag error, as well as tracking error caused by disturbance observer compensation and other uncertainties. Based on the error model of accurate prediction, the tracking error is continuously reduced in the iteration through online iterative compensation, thereby continuously improving the tracking accuracy. It is worth mentioning that each compensation term generated in the iteration process is calculated within the sampling interval, so online compensation can be achieved, which has good robustness to various uncertainties such as trajectory changes.

[0142] 5. By using trajectory interpolation, the problem of error prediction when the reference trajectory at future moments is unknown is solved. Accurate prediction of tracking errors at future moments requires the reference trajectory at future moments. However, when the fast-tool system compensates for the slow-tool servo error, the reference trajectory of the fast-tool servo system is the slow-tool servo error measured in real time, which cannot be predicted in advance. Therefore, by using interpolation, the reference trajectory at future moments can be accurately estimated, realizing error prediction and online compensation.

[0143] Other beneficial effects of the present invention will be explained in detail through the introduction of specific technical features and technical solutions in specific embodiments. Those skilled in the art should be able to understand the beneficial technical effects brought about by these technical features and technical solutions through the introduction of these technical features and technical solutions. Attached Figure Description

[0144] Other features, objects, and advantages of the present invention will become more apparent from the following detailed description of non-limiting embodiments with reference to the accompanying drawings:

[0145] Figure 1 This is a diagram of the overall control scheme;

[0146] Figure 2 The flowchart shows the master-slave control process for fast-tool servo compensation of slow-tool servo error. Detailed Implementation

[0147] The present invention will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present invention, but do not limit the invention in any way. It should be noted that those skilled in the art can make several changes and improvements without departing from the concept of the present invention. These all fall within the protection scope of the present invention.

[0148] This invention provides an online iterative compensation system for servo errors, referring to... Figure 1 As shown, the system specifically includes the following:

[0149] Disturbance Observer Module: During the process of fast-tool servo compensation for slow-tool servo errors, various internal nonlinear disturbances and external interferences such as cutting forces and machine tool vibrations can severely affect the tracking performance of the controller. To estimate the lumped disturbance of the system during machining and facilitate disturbance compensation using control methods, a robust disturbance observer applicable to non-minimum-phase systems is designed. This observer treats the internal nonlinearities and external disturbances during the fast-tool servo compensation for slow-tool servo errors as a single lumped disturbance d(t), and then estimates and compensates for it. Typically, the driving mode of a fast-tool servo system is piezoelectric drive or normal stress electromagnetic drive, and the entire system can be described in the Laplace continuous domain by the following third-order model:

[0150]

[0151] Where a3, a2, a1, a0, b2, b1, b0 are the parameters of the system model, and s is the Laplace operator;

[0152] The third-order model is a non-minimum-phase system. To ensure the stability of the perturbation observer, the third-order model needs to be modified to become a minimum-phase system.

[0153]

[0154] For the inverse P of the minimum phase system n -1 Given the order constraint of P(s), a corresponding low-pass filter Q(s) needs to be designed to ensure that P n -1 Q(s) is achievable in practical applications, and a second-order low-pass filter is used in this invention:

[0155]

[0156] Here, τ is a parameter that determines the bandwidth of the low-pass filter, and it needs to be tuned to achieve a trade-off between observer disturbance suppression performance and stability. Thus, the lumped disturbance estimate of the system is... And use control methods to compensate.

[0157] Dual-loop high-bandwidth control module: To suppress the lightly damped resonance characteristics and hysteresis nonlinearity of the fast-blade servo system and improve the control bandwidth, a corresponding dual-loop high-bandwidth controller was designed. In the inner loop, a positive acceleration-velocity-position feedback damping controller specifically designed for this third-order system is applied, and its expression is:

[0158]

[0159] The inner-loop damping controller has five adjustable parameters: β2, β1, β0, ξ, ω. p The desired locations of the five system poles can be arbitrarily configured to achieve the desired system damping ratio, ensuring that the outer-loop tracking controller has sufficient stability margin to guarantee system stability even at high gain. Therefore, the transfer function of the inner-loop damping part is:

[0160]

[0161] Since the system's lightly damped resonance characteristics have been effectively suppressed by the damping controller, for inputs in the mid-to-low frequency range, the inner-loop damping component can be approximated as a second-order rigid system, with the following expression:

[0162]

[0163] Where c1, c0, k1, and k0 are parameters of the rigid system; in the outer loop, a high-gain proportional-integral tracking controller is applied to reduce residual tracking error and achieve accurate trajectory tracking. The expression for the outer loop tracking controller is:

[0164]

[0165] Where, k p and k i The proportional gain and integral gain are used; by optimizing the parameters of the inner-loop damping controller and the outer-loop tracking controller, high-bandwidth control is achieved, which is beneficial for tracking the high-frequency components of slow-speed servo errors. The above-mentioned disturbance observer module and high-bandwidth dual-loop control module together constitute a robust high-bandwidth controller.

[0166] Online Iterative Compensation Module: To compensate for the unavoidable phase lag error and residual tracking error caused by various nonlinearities in the closed-loop control system, an online iterative compensation module was designed. This module predicts the tracking error of the robust high-bandwidth controller using the difference equations of the discrete system, and then iteratively compensates for it through online reference trajectory correction, further improving the tracking accuracy of the robust high-bandwidth controller. The online iterative compensation module mainly includes the following sub-modules:

[0167] Linear error prediction submodule: According to equations (6) and (7), the fast-tool servo system calculates the error from the input y. r To output y out The transfer function between them can be written as:

[0168]

[0169] The transfer function is a third-order transfer function; in the discrete domain, the transfer function is expressed in the form of a difference equation:

[0170] y out (k+1)=0y out (k)+1y out (k-1)+2y out (k-2)+β0y r (k)+β1y r (k-1)+β2y r (k-2)(9)

[0171] Where α0, α1, α2, β0, β1, and β2 are constants related to the sampling frequency and the coefficients of each term in the overall closed-loop system transfer function; in the state space, equation (9) is written as:

[0172]

[0173] For the above spatial state equation, x(k)=[x1(k)2(k)3(k)] T It is a state vector, where x1(k) = y out (k),x2(k)=1(k-1)+h1y r (k-1),x3(k)=2(k-1)+h2y r (k-1), h0=0, h1=2 / 2, State matrix A = [α 012 [1 0 0; 0 1 0], Input matrix B = [h0; h1; h2], Output matrix C = [1 0 0]; γ d () represents the lumped disturbance of the system, which is estimated by the designed disturbance observer and fed back into the control loop for compensation; therefore, for this linear error prediction model, γ d The nonlinear effect of () is negligible; at the current sampling time kT s All parameters can be obtained, thus enabling the prediction of the state at the next moment:

[0174]

[0175] Right now

[0176]

[0177] Assume N p If the prediction interval is defined, then all states within the prediction interval can be predicted, and the predicted system output can be derived as follows:

[0178]

[0179] The realization of this prediction requires a reference trajectory y at a future time. r (k+i); For the fast-tool servo system to compensate for the slow-tool servo error, the reference trajectory of the fast-tool servo system is generated in real time based on the servo error measured by the slow-tool servo system. In order to obtain the reference trajectory for future times within the prediction interval, a corresponding module needs to be designed to accurately predict the reference trajectory for future times.

[0180] The future reference trajectory prediction submodule of the fast-tool servo system: To obtain the reference trajectory for future moments within the prediction interval, a corresponding interpolation module is designed to achieve this prediction. In this invention, a quadratic polynomial is used for interpolation. The coefficients of the quadratic polynomial used for interpolation are fitted using the reference trajectory of the current sampling moment and the previous nine sampling moments (i.e., the slow-tool servo error of these ten sampling points). Then, y r (k+i) can be obtained by interpolating the quadratic polynomial, thus enabling accurate prediction of the output of the fast-tool servo system within the prediction interval.

[0181] Error Iteration Compensation Submodule: The error e(k) is defined as the difference between the reference trajectory and the actual tracking trajectory; without compensation, (k+N) p )T s The tracking error at time step is predicted as:

[0182]

[0183] Assume N c It is a compensation interval, and N c ≤N p -1; when compensation item c r (k+N c Add to reference trajectory y r (k+N p When ), the tracking error e(k+N) p ) changes, due to the system output y out (k+N p When the system output changes, the change is as follows:

[0184] c out (k+N p )=εc r (k+N c (15)

[0185] in, It is a constant; therefore, after compensation, the system error is derived as:

[0186]

[0187] This error is gradually compensated through an iterative process; since the prediction interval N... p The system state within has been accurately predicted, and the iterative process is implemented online during the model prediction process;

[0188] Let the compensation term be c1 in the first iteration, then we have

[0189]

[0190] Where ∈ is the compensation gain; substituting equation (17) into equation (15), the tracking error after the first iteration is derived as:

[0191]

[0192] During the second iteration, e c1 It is also integrated into the compensation items and designed as follows:

[0193] c2(k+N c )=c1(k+Nc )+∈e c1 (k+N p (19)

[0194] Therefore, the tracking error after the second iteration is:

[0195]

[0196] The compensation term and tracking error after the nth iteration are derived in the same way, and their expressions are:

[0197] c n (k+N c ) = c (n-1) (k+N c )+∈e c(n-1) (k+N p ) (twenty one)

[0198]

[0199] Therefore, the compensation gain is satisfied when the following conditions are met:

[0200]

[0201] If n approaches infinity, the tracking error of the system approaches 0.

[0202] The overall control scheme implementation module includes a disturbance observer, a high-bandwidth dual-loop controller, and an online iterative compensation module. It predicts the system state within the prediction interval based on a linear model of the discrete system difference equations, and then modifies the reference trajectory online for error compensation. To compensate for the tracking error at the next sampling time, the compensation interval N... c The value must be greater than or equal to 1, therefore the prediction interval N p The value must be greater than or equal to 2. The prediction accuracy of the linear model decreases as the prediction interval increases, and the accuracy of the reference trajectory interpolation also decreases as the number of prediction points increases. Therefore, in this invention, the smallest prediction interval and compensation interval, i.e., N, are selected while satisfying the error compensation at the next sampling time. p =2, N c =1. Thus, the quadratic interpolation polynomial only needs to predict the reference trajectories of two points. Theoretically, during the iterative compensation process, as the number of iterations increases, the system's tracking error will continuously decrease and approach 0. However, in practical applications, as iterations proceed, interference such as sensor noise will limit the final achievable minimum tracking error, and increasing the number of iterations will bring additional computational burden, making it impossible to complete the calculation of the compensation term within a single sampling interval. Considering the trade-off between accuracy and efficiency, in this invention, the number of iterations is chosen to be 2.

[0203] This control scheme was implemented on a self-developed fast-tool servo prototype. The fast-tool servo mechanism is piezoelectric driven and transmits displacement through the deformation of a flexible hinge. The control algorithm was implemented in the MATLAB / Simulink environment, achieving real-time motion control via dspace. The control voltage generated in dspace is transmitted to a high-voltage amplifier with a 15x amplification ratio via a 16-bit digital-to-analog converter. The high-voltage amplifier then applies voltage to the piezoelectric actuator to drive its deformation. The displacement of the end effector is captured by a precision capacitive displacement sensor and then transmitted to the dspace controller for feedback via a 16-bit analog-to-digital converter. The sampling frequency of the control system is 20kHz. The master-slave control flowchart for fast-tool servo error compensation and slow-tool servo is shown below. Figure 2 As shown. For slow-tool servo systems, since the tracking error cannot be directly obtained from commercially available machine tools, a spline interpolation-based method is used to estimate the reference trajectory that the fast-tool servo system needs to track during compensation, based on the measured spindle position and the calculated toolpath trajectory.

[0204] This invention provides an online iterative compensation system and method for servo errors. It can accurately predict the tracking error of the next moment by only needing the system state at the current and previous moments, and achieve real-time iterative compensation through reference trajectory correction, which effectively improves the tracking accuracy and anti-interference capability of the fast-tool servo system during the compensation process.

[0205] Those skilled in the art will understand that, besides implementing the system and its various devices, modules, and units provided by this invention in the form of purely computer-readable program code, the same functions can be achieved entirely through logical programming of the method steps, making the system and its various devices, modules, and units of this invention function in the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers, and embedded microcontrollers. Therefore, the system and its various devices, modules, and units provided by this invention can be considered as a hardware component, and the devices, modules, and units included therein for implementing various functions can also be considered as structures within the hardware component; alternatively, the devices, modules, and units for implementing various functions can be considered as both software modules implementing the method and structures within the hardware component.

[0206] Specific embodiments of the present invention have been described above. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art can make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. Unless otherwise specified, the embodiments and features described in this application can be arbitrarily combined with each other.

Claims

1. An online iterative compensation system for servo errors, characterized in that, include: Disturbance observer module: Treats the internal nonlinearity and external disturbances of the system as a whole lumped disturbance during the process of fast-tool servo compensation for slow-tool servo error, and performs estimation and compensation. Dual-loop high-bandwidth control module: In the inner loop of the system, a positive acceleration-velocity-position feedback damping controller is designed to suppress the lightly damped resonant mode; Online iterative compensation module: Predicts the tracking error of the robust high-bandwidth controller through the difference equation of the discrete system, and iteratively compensates for it through online reference trajectory correction; The overall control scheme implementation module includes a disturbance observer, a high-bandwidth dual-loop controller, and an online iterative compensation module. The linear model based on the difference equation of the discrete system predicts the state of the system within the prediction interval, and then the reference trajectory is modified through online iteration to compensate for the error. The disturbance observer module includes: The fast-tool servo system is driven by either piezoelectric drive or normal stress electromagnetic drive. The entire system is described in the Laplace continuous domain by the following third-order model: 1） in, , , , , , , For the parameters of the system model, For the Laplace operator; The third-order model is a non-minimum-phase system. To ensure the stability of the perturbation observer, the third-order model needs to be modified to become a minimum-phase system. 2） To satisfy the inverse of the minimum phase system Given the order constraint, a corresponding low-pass filter needs to be designed. Make In practical applications, this can be achieved using a second-order low-pass filter: 3） in, To determine the parameters of the low-pass filter bandwidth, tuning them can maximize the bandwidth. The lumped disturbance estimate of the system is: And use control methods to compensate; The dual-ring high-bandwidth control module includes: The inner loop damping controller expression for the positive acceleration-velocity-position feedback damping controller is: 4） The inner loop damping controller has five adjustable parameters: , , , , This allows for the arbitrary configuration of the desired locations of the five system poles to achieve the desired system damping ratio. Therefore, the transfer function of the inner-loop damping part is: 5） For inputs in the mid-to-low frequency range, the inner-loop damping component is treated as a second-order rigid system, and its expression is: 6） in, , , , The parameters are those of a rigid system. In the outer loop, a high-gain proportional-integral tracking controller is applied to reduce residual tracking error and achieve accurate trajectory tracking. The expression for the outer loop tracking controller is: 7） in, and The proportional gain and integral gain are used; high-bandwidth control is achieved by optimizing the parameters of the inner-loop damping controller and the outer-loop tracking controller. The disturbance observer module and the high-bandwidth dual-loop control module together form a robust high-bandwidth controller.

2. The online iterative compensation system for servo errors according to claim 1, characterized in that, The online iterative compensation module includes: Linear error prediction submodule: Based on equations 6 and 7), the fast-tool servo system receives input from... To output The transfer function between them is written as follows: 8） The transfer function is a third-order transfer function; in the discrete domain, the transfer function is expressed in the form of a difference equation: + + 9） in, , , , , , It is a constant related to the sampling frequency and the coefficients of each term in the overall closed-loop system transfer function; in state space, Equation 9) is written as: 10） in, These represent the sampling points of the discrete system. For the above spatial state equation, It is a state vector, where , , , , , State matrix Input matrix Output matrix ; The lumped disturbance representing the system is estimated by a designed disturbance observer and fed back into the control loop for compensation; therefore, for a linear error prediction model, The nonlinear effects are ignored; at the current sampling time All parameters can be obtained, thus enabling the prediction of the state at the next moment: 11） Right now 12） Assumption If the prediction interval is defined, then all states within the prediction interval can be predicted, and the predicted system output can be derived as follows: 13） The realization of this prediction requires reference trajectories at future moments. For fast-tool servo systems compensating for slow-tool servo errors, the reference trajectory of the fast-tool servo system is generated in real time based on the servo error measured by the slow-tool servo system. Therefore... It cannot be obtained directly; it needs to be obtained from the trajectory prediction module at future moments.

3. The online iterative compensation system for servo errors according to claim 2, characterized in that, The online iterative compensation module also includes: The future moment reference trajectory prediction submodule uses a quadratic polynomial for interpolation. It fits the coefficients of the quadratic polynomial to the reference trajectories of ten sampling points (the current sampling moment and the previous nine sampling moments), i.e., the slow-motion servo errors of these ten sampling points. Then... It is possible to predict the output of the fast-tool servo system within the prediction interval by interpolating the quadratic polynomial.

4. The online iterative compensation system for servo errors according to claim 3, characterized in that, The online iterative compensation module also includes: Error Iteration Compensation Submodule: Error Defined as the difference between the reference trajectory and the actual tracking trajectory; without compensation, The tracking error at time step is predicted as: 14） Assumption It is the compensation range, and When compensation items Add to reference trajectory During the tracking process, the tracking error Changes occur due to system output The system output changes as follows: 15） in, It is a constant; therefore, after compensation, the system error is derived as: 16） This error is compensated asymptotically through iteration; because it is within the prediction interval The system state within has been accurately predicted, and the iterative process is implemented online during the model prediction process; Let the compensation term be during the first iteration. Then there is 17） in, It is the compensation gain; substituting Equation 17) into Equation 15), the tracking error after the first iteration is derived as: 18） During the second iteration, It is also integrated into the compensation items and designed as follows: 19） Therefore, the tracking error after the second iteration is: 20） The compensation term and tracking error after the nth iteration are derived in the same way, and their expressions are: 21） 22） Therefore, the compensation gain is satisfied when the following conditions are met: 23） If n approaches infinity, the tracking error of the system approaches 0.

5. The online iterative compensation system for servo errors according to claim 4, characterized in that, In the overall control scheme implementation module, to compensate for the tracking error at the next sampling time, the compensation interval is... The value must be greater than or equal to 1, therefore the prediction interval is... It must be greater than or equal to 2.

6. An online iterative compensation method for servo errors, characterized in that, include: Disturbance observer steps: Treat the internal nonlinearity and external disturbances of the system as a whole lumped disturbance during the process of fast-tool servo compensation for slow-tool servo error, and estimate and compensate for them. Dual-loop high-bandwidth control steps: In the inner loop of the system, a positive acceleration-velocity-position feedback damping controller is designed to suppress the lightly damped resonant mode; Online iterative compensation steps: predict the tracking error of the robust high-bandwidth controller through the difference equation of the discrete system, and iteratively compensate by correcting the online reference trajectory; The overall control scheme implementation steps include: a disturbance observer, a high-bandwidth dual-loop controller, and an online iterative compensation module; The linear model based on the difference equation of the discrete system predicts the state of the system within the prediction interval, and then the reference trajectory is modified through online iteration to compensate for the error. The disturbance observer steps include: The fast-tool servo system is driven by either piezoelectric drive or normal stress electromagnetic drive. The entire system is described in the Laplace continuous domain by the following third-order model: 1） in, , , , , , , For the parameters of the system model, For the Laplace operator; The third-order model is a non-minimum-phase system. To ensure the stability of the perturbation observer, the third-order model needs to be modified to become a minimum-phase system. 2） For the minimum phase system inverse Given the order constraint, a corresponding low-pass filter needs to be designed. Make In practical applications, this can be achieved using a second-order low-pass filter: 3） in, To determine the parameters of the low-pass filter bandwidth, it is tuned, and the lumped disturbance estimate of the system is: And use control methods to compensate; The dual-ring high-bandwidth control steps include: The inner loop damping controller expression for the positive acceleration-velocity-position feedback damping controller is: 4） The inner loop damping controller has five adjustable parameters: , , , , By arbitrarily configuring the desired locations of the five system poles to achieve the desired system damping ratio, the transfer function of the inner-loop damping part is: 5） For inputs in the mid-to-low frequency range, the inner-loop damping component is treated as a second-order rigid system, and its expression is: 6） in, , , , The parameters are those of a rigid system. In the outer loop, a high-gain proportional-integral tracking controller is applied to reduce residual tracking error and achieve accurate trajectory tracking. The expression for the outer loop tracking controller is: 7） in, and The proportional gain and integral gain are used; high-bandwidth control is achieved by optimizing the parameters of the inner-loop damping controller and the outer-loop tracking controller. The disturbance observer step and the high-bandwidth dual-loop control step together constitute a robust high-bandwidth controller.

7. The online iterative compensation method for servo errors according to claim 6, characterized in that, The online iterative compensation steps include: Linear error prediction sub-step: According to equations (6) and (7), the fast knife servo system from the input To output The transfer function between them is written as follows: 8） The transfer function is a third-order transfer function; in the discrete domain, the transfer function is expressed in the form of a difference equation: + + 9） in, , , , , , It is a constant related to the sampling frequency and the coefficients of each term in the overall closed-loop system transfer function; in state space, Equation 9) is written as: 10） For the above spatial state equation It is a state vector, where , , , , , State matrix Input matrix Output matrix ; The lumped disturbance representing the system is estimated by a designed disturbance observer and fed back into the control loop for compensation; therefore, for a linear error prediction model, The nonlinear effects are ignored; at the current sampling time All parameters can be obtained, thus enabling the prediction of the state at the next moment: 11） Right now 12） Assumption If the prediction interval is defined, then all states within the prediction interval can be predicted, and the predicted system output can be derived as follows: 13） The realization of this prediction requires reference trajectories at future moments. For the fast-tool servo system to compensate for the slow-tool servo error, the reference trajectory of the fast-tool servo system is generated in real time based on the servo error measured by the slow-tool servo system. The online iterative compensation step also includes: The future moment reference trajectory prediction submodule uses a quadratic polynomial for interpolation. It fits the coefficients of the quadratic polynomial to the reference trajectories of ten sampling points (the current sampling moment and the previous nine sampling moments), i.e., the slow-motion servo errors of these ten sampling points. Then... It is possible to predict the output of the fast-tool servo system within the prediction interval by interpolating the quadratic polynomial. The online iterative compensation step also includes: Error Iteration Compensation Sub-step: Error Defined as the difference between the reference trajectory and the actual tracking trajectory; without compensation, The tracking error at time step is predicted as: 14） Assumption It is the compensation range, and When compensation items Add to reference trajectory During the tracking process, the tracking error Changes occur due to system output The system output changes as follows: 15） in, It is a constant; therefore, after compensation, the system error is derived as: 16） This error is compensated asymptotically through iteration; because it is within the prediction interval The system state within has been accurately predicted, and the iterative process is implemented online during the model prediction process; Let the compensation term be during the first iteration. Then there is 17） in, It is the compensation gain; substituting Equation 17) into Equation 15), the tracking error after the first iteration is derived as: 18） During the second iteration, It is also integrated into the compensation items and designed as follows: 19） Therefore, the tracking error after the second iteration is: 20） The compensation term and tracking error after the nth iteration are derived in the same way, and their expressions are: 21） 22） Therefore, the compensation gain is satisfied when the following conditions are met: 23） If n approaches infinity, the tracking error of the system approaches 0; In the overall control scheme implementation module, to compensate for the tracking error at the next sampling time, the compensation interval is... The value must be greater than or equal to 1, therefore the prediction interval is... It must be greater than or equal to 2.

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