Robust high-bandwidth control method and system with intelligent error prediction and feedforward compensation

Abstract

The present invention provides a robust high-bandwidth control method and system for intelligent error prediction and feedforward compensation, including: treating the model uncertainty and external interference in the fast tool servo system processing process as a lumped disturbance d(t) for estimation and compensation; designing a corresponding dual-loop high-bandwidth controller and applying a positive acceleration-velocity-position feedback damping controller; designing an online intelligent error prediction and feedforward compensation module based on Koopman operator theory to perform online reference trajectory correction; a data-driven method based on Koopman operator theory can predict the output of the closed-loop system online, obtain an estimated tracking error by subtracting the reference input from the predicted output, and implement feedforward compensation online by means of reference trajectory correction. The present invention solves the problem of resisting the lumped disturbance composed of the system's own internal uncertainty and external interference during the fast tool servo processing process by adopting a design disturbance observer structure.

Description

Technical Field

[0001] The present invention relates to the technical field of error prediction, and in particular to a robust high-bandwidth control method and system for intelligent error prediction and feedforward compensation, and more particularly to a robust high-bandwidth control method for online intelligent error prediction and feedforward compensation based on Koopman operator theory. Background Art

[0002] Fast tool servo systems are widely used in ultra-precision machining. However, the hysteresis nonlinearity within the drive and the lightly damped resonance characteristics of the mechanism seriously affect their control accuracy and bandwidth. External interference such as cutting forces during machining further increases the control challenges.

[0003] Patent document CN106707766A discloses a fast mirror feedforward control method based on an error observer. To address the shortcomings of current fast mirror feedforward control, a high-gain observer is implemented by fusing the line of sight error provided by the CCD and the output of the position loop controller to observe and estimate the target position. This is then used to implement feedforward control of the target position, reducing its CCD tracking error.

[0004] Therefore, it is necessary to propose a new technical solution to improve the above technical problems. Summary of the Invention

[0005] In view of the defects in the prior art, the object of the present invention is to provide a robust high-bandwidth control method and system with intelligent error prediction and feedforward compensation.

[0006] According to the present invention, a robust high-bandwidth control method with intelligent error prediction and feedforward compensation is provided, the method comprising the following steps:

[0007] Step S1: The robust disturbance observer regards the model uncertainty and external disturbance in the fast tool servo system during machining as a whole lumped disturbance d(t) and performs estimation and compensation.

[0008] Step S2: Design a corresponding dual-loop high-bandwidth controller, and apply a positive acceleration-velocity-position feedback damping controller in the inner loop;

[0009] Step S3: Design an online intelligent error prediction and feedforward compensation module based on the Koopman operator theory to perform online reference trajectory correction;

[0010] Step S4: The data-driven method based on the Koopman operator theory can predict the output of the closed-loop system online, obtain the estimated tracking error by taking the difference between the reference input and the predicted output, and implement feedforward compensation online by correcting the reference trajectory.

[0011] Preferably, the driving mode of the fast tool servo system in step S1 is piezoelectric drive or normal stress electromagnetic drive, and the entire system is described by the following third-order model in the Laplace continuous domain:

[0012]

[0013] Among them, a3, a2, a1, a0, b2, b1, and b0 are the parameters of the system model. The third-order system is a non-minimum phase system. A slight correction is made to the third-order system to make it a minimum phase system:

[0014]

[0015] Design the corresponding low-pass filter Q(s) so that P n -1 (s)Q(s) is physically achievable using a second-order low-pass filter:

[0016]

[0017] where τ is the parameter that determines the bandwidth of the low-pass filter, which is tuned to achieve a compromise between the observer's disturbance rejection performance and stability; the total disturbance of the system is estimated as And use control methods to compensate.

[0018] Preferably, the expression of the positive acceleration-velocity-position feedback damping controller in step S2 is:

[0019]

[0020] The damping controller has five parameters that can be adjusted: β2, β1, β0, ξ, ω p , the desired positions of the five system poles can be arbitrarily configured; in the outer loop, a high-gain proportional-integral tracking controller is applied, and its expression is:

[0021]

[0022] Among them, k p and k i are the proportional gain and the integral gain; the parameters of the inner-loop damping controller and the outer-loop tracking controller can be synchronously tuned using an intelligent evolutionary algorithm to maximize the control bandwidth.

[0023] Preferably, the step S3 takes into account the nonlinear characteristics between the input and output of the closed-loop system, and the nonlinear dynamics between the input and output of the robust high-bandwidth closed-loop system is expressed by the following discrete nonlinear equation:

[0024] y k+1 =f(y k ,rk ) (6)

[0025] Where k represents the sampling point of the discrete system, r k and y k Represent the input and output of the closed-loop system at the current moment respectively. From (6), we can see that the output of the system at the current moment depends on the input and output of the system at the previous moment. In order to predict the output of the system at the next moment based on the input and output of the system at the current and previous moments, the Koopman operator is used to lift the nonlinear function into the infinite-dimensional linear space, and the Koopman operator is defined as F→F is:

[0026]

[0027] in is an infinite-dimensional linear operator, F is the Hilbert space spanned by observables, and σ∈F is an observable that elevates the system input and output to a high-dimensional space; the dynamic equation of the closed-loop system is written as:

[0028]

[0029] The nonlinear dynamics between input and output are preserved by the infinite-dimensional linear Koopman operator, which can be estimated linearly in finite dimensions by the developed algorithm. Extended Dynamic Mode Decomposition is an efficient data-driven algorithm for the finite-dimensional linear approximation of the Koopman operator. The goal of the algorithm is to find an optimal K that satisfies:

[0030]

[0031] Where K:F q →F q yes Finite-dimensional linear approximation of , γ is a non-zero vector, Φ is a vector consisting of observable quantities, let:

[0032]

[0033] in:

[0034]

[0035] There are N basis functions that need to be designed Composition, i = 1, ..., N, so K is obtained by minimizing the following two-norm matrix:

[0036]

[0037] Where A and B are the parameter matrices obtained by solving the above optimization problem; the following recursive relationship is obtained:

[0038]

[0039] Afterwards, in the initial state of the system When and r0 are known, the following predictor is constructed and the output of the system is estimated by solving the following least squares problem:

[0040]

[0041] where C is the optimal parameter matrix obtained by solving the least squares problem.

[0042] Preferably, step S4 completes error prediction and feedforward compensation within a single sampling period, and performs real-time online compensation during the operation of the control system.

[0043] The present invention also provides a robust high-bandwidth control system with intelligent error prediction and feedforward compensation, the system comprising the following modules:

[0044] Module M1: The robust disturbance observer treats the model uncertainty and external disturbances in the fast tool servo system during machining as a whole lumped disturbance d(t) for estimation and compensation.

[0045] Module M2: Design the corresponding dual-loop high-bandwidth controller. In the inner loop, a positive acceleration-velocity-position feedback damping controller is applied.

[0046] Module M3: Design an online intelligent error prediction and feedforward compensation module based on the Koopman operator theory to perform online reference trajectory correction;

[0047] Module M4: The data-driven system based on the Koopman operator theory can predict the output of the closed-loop system online, obtain the estimated tracking error by taking the difference between the reference input and the predicted output, and implement feedforward compensation online by correcting the reference trajectory.

[0048] Preferably, the driving mode of the fast tool servo system in the module M1 is piezoelectric drive or normal stress electromagnetic drive, and the entire system is described by the following third-order model in the Laplace continuous domain:

[0049]

[0050] Among them, a3, a2, a1, a0, b2, b1, and b0 are the parameters of the system model. The third-order system is a non-minimum phase system. A slight correction is made to the third-order system to make it a minimum phase system:

[0051]

[0052] Design the corresponding low-pass filter Q(s) so that P n-1 (s)Q(s) is physically achievable using a second-order low-pass filter:

[0053]

[0054] where τ is the parameter that determines the bandwidth of the low-pass filter, which is tuned to achieve a compromise between the observer's disturbance rejection performance and stability; the total disturbance of the system is estimated as And use the controlled system to compensate.

[0055] Preferably, the expression of the positive acceleration-velocity-position feedback damping controller in the module M2 is:

[0056]

[0057] The damping controller has five parameters that can be adjusted: β2, β1, β0, ξ, ω p , the desired positions of the five system poles can be arbitrarily configured; in the outer loop, a high-gain proportional-integral tracking controller is applied, and its expression is:

[0058]

[0059] Among them, k p and k i are the proportional gain and the integral gain; the parameters of the inner-loop damping controller and the outer-loop tracking controller can be synchronously tuned using an intelligent evolutionary algorithm to maximize the control bandwidth.

[0060] Preferably, the module M3 takes into account the nonlinear characteristics between the input and output of the closed-loop system. The nonlinear dynamics between the input and output of the robust high-bandwidth closed-loop system is expressed by the following discrete nonlinear equation:

[0061] y k+1 =f(y k ,r k ) (6)

[0062] Where k represents the sampling point of the discrete system, r k and y k Represent the input and output of the closed-loop system at the current moment respectively. From (6), we can see that the output of the system at the current moment depends on the input and output of the system at the previous moment. In order to predict the output of the system at the next moment based on the input and output of the system at the current and previous moments, the Koopman operator is used to lift the nonlinear function into the infinite-dimensional linear space, and the Koopman operator is defined as F→F is:

[0063]

[0064] in is an infinite-dimensional linear operator, F is the Hilbert space spanned by observables, and σ∈F is an observable that elevates the system input and output to a high-dimensional space; the dynamic equation of the closed-loop system is written as:

[0065]

[0066] The nonlinear dynamics between input and output are preserved by the infinite-dimensional linear Koopman operator, which can be estimated linearly in finite dimensions by the developed algorithm. Extended Dynamic Mode Decomposition is an efficient data-driven algorithm for the finite-dimensional linear approximation of the Koopman operator. The goal of the algorithm is to find an optimal K that satisfies:

[0067]

[0068] Where K:F q →F q yes Finite-dimensional linear approximation of , γ is a non-zero vector, Φ is a vector consisting of observable quantities, let:

[0069]

[0070] in:

[0071]

[0072] There are N basis functions that need to be designed Composition, i = 1, ..., N, so K is obtained by minimizing the following two-norm matrix:

[0073]

[0074] Where A and B are the parameter matrices obtained by solving the above optimization problem; the following recursive relationship is obtained:

[0075]

[0076] Afterwards, in the initial state of the system When and r0 are known, the following predictor is constructed and the output of the system is estimated by solving the following least squares problem:

[0077]

[0078] where C is the optimal parameter matrix obtained by solving the least squares problem.

[0079] Preferably, the module M4 completes error prediction and feedforward compensation within a single sampling period, and performs real-time online compensation during the operation of the control system.

[0080] Compared with the prior art, the present invention has the following beneficial effects:

[0081] 1. The present invention solves the problem of resisting the lumped disturbance caused by both the system's internal uncertainty and external interference during fast tool servo machining by adopting a disturbance observer structure. The internal uncertainty of the system includes the identification error of the system model, model changes caused by various internal and external interferences, and the uncertainty caused by slight changes in various parameters of the mechanical and electrical systems during use. The external interference of the system includes external interference caused by cutting forces during machining, various voltage and electromagnetic interference caused by the control circuit, and various mechanical and electronic interferences in the machine tool and the external environment.

[0082] 2. The present invention solves the high-bandwidth control problem of fast tool servo systems by designing a dual-loop controller consisting of an inner-loop damping controller and an outer-loop tracking controller, and by using a method for synchronously optimizing the inner and outer-loop parameters. Fast tool servo systems are typically driven by piezoelectric or normal-stress electromagnetic actuators, with displacement transmitted by flexible guide mechanisms. The hysteresis nonlinearity within the actuator itself and the lightly damped resonance characteristics of the mechanical structure pose significant challenges to high-precision and high-bandwidth motion control. The inner-loop damping controller can suppress the lightly damped resonance characteristics of the system and improve the system damping ratio, thereby facilitating the realization of high gain in the tracking controller. The outer-loop high-gain tracking controller can suppress the influence of hysteresis nonlinearity and improve control accuracy and bandwidth. By using a method for synchronously optimizing the inner and outer-loop parameters, the optimal control bandwidth can be achieved under the set target.

[0083] 3. The present invention solves the tracking error problem of the online prediction closed-loop system through a data-driven intelligent error prediction and feedforward compensation method, thereby suppressing the phase lag of the robust high-bandwidth dual-loop controller and the tracking errors caused by various parameter changes, greatly improving the tracking accuracy of the closed-loop system; the extended dynamic mode decomposition algorithm based on the Koopman operator is a purely data-driven algorithm that can realize nonlinear prediction in a very short time, thereby realizing accurate online estimation of the closed-loop system tracking error, and greatly improving the tracking accuracy of the robust high-bandwidth dual-loop controller through reference trajectory correction. This method can accurately track various types of high-frequency reference trajectories in the presence of strong external interference; the online error prediction and compensation method reduces the time-consuming iteration and prediction process of various offline compensation methods, improves the flexibility to reference trajectory changes, and enhances practicality. BRIEF DESCRIPTION OF THE DRAWINGS

[0084] Other features, objects and advantages of the present invention will become more apparent upon reading the detailed description of non-limiting embodiments with reference to the following drawings:

[0085] Figure 1 This is the overall control scheme diagram of the present invention;

[0086] Figure 2 This is a signal flow chart of the control scheme implementation process of the present invention. DETAILED DESCRIPTION

[0087] The present invention will be described in detail below with reference to specific embodiments. The following examples will help those skilled in the art to further understand the present invention, but are not intended to limit the present invention in any form. It should be noted that, for those skilled in the art, several changes and improvements can be made without departing from the scope of the present invention. These all fall within the scope of protection of the present invention.

[0088] Example 1:

[0089] According to the present invention, a robust high-bandwidth control method with intelligent error prediction and feedforward compensation is provided, the method comprising the following steps:

[0090] Step S1: The robust disturbance observer treats the model uncertainty and external disturbances within the fast tool servo system during machining as a lumped disturbance d(t) for estimation and compensation. The fast tool servo system is driven by piezoelectric or normal stress electromagnetic drive, and the entire system is described by the following third-order model in the Laplace continuous domain:

[0091]

[0092] Among them, a3, a2, a1, a0, b2, b1, and b0 are the parameters of the system model. The third-order system is a non-minimum phase system. A slight correction is made to the third-order system to make it a minimum phase system:

[0093]

[0094] Design the corresponding low-pass filter Q(s) so that P n -1 (s)Q(s) is physically achievable using a second-order low-pass filter:

[0095]

[0096] where τ is the parameter that determines the bandwidth of the low-pass filter, which is tuned to achieve a compromise between the observer's disturbance rejection performance and stability; the total disturbance of the system is estimated as And use control methods to compensate.

[0097] Step S2: Design a corresponding dual-loop high-bandwidth controller. In the inner loop, apply a positive acceleration-velocity-position feedback damping controller. The expression of the positive acceleration-velocity-position feedback damping controller is:

[0098]

[0099] The damping controller has five parameters that can be adjusted: β2, β1, β0, ξ, ω p , the desired positions of the five system poles can be arbitrarily configured; in the outer loop, a high-gain proportional-integral tracking controller is applied, and its expression is:

[0100]

[0101] Among them, k p and k i are the proportional gain and the integral gain; the parameters of the inner-loop damping controller and the outer-loop tracking controller can be synchronously tuned using an intelligent evolutionary algorithm to maximize the control bandwidth.

[0102] Step S3: Design an online intelligent error prediction and feedforward compensation module based on the Koopman operator theory to perform online reference trajectory correction. Considering the nonlinear characteristics between the input and output of the closed-loop system, the nonlinear dynamics between the input and output of the robust high-bandwidth closed-loop system is expressed by the following discrete nonlinear equation:

[0103] y k+1 =f(y k ,r k ) (6)

[0104] Where k represents the sampling point of the discrete system, r k and y k Represent the input and output of the closed-loop system at the current moment respectively. From (6), we can see that the output of the system at the current moment depends on the input and output of the system at the previous moment. In order to predict the output of the system at the next moment based on the input and output of the system at the current and previous moments, the Koopman operator is used to lift the nonlinear function into the infinite-dimensional linear space, and the Koopman operator is defined as F→F is:

[0105]

[0106] in is an infinite-dimensional linear operator, F is the Hilbert space spanned by observables, and σ∈F is an observable that elevates the system input and output to a high-dimensional space; the dynamic equation of the closed-loop system is written as:

[0107]

[0108] The nonlinear dynamics between input and output are preserved by the infinite-dimensional linear Koopman operator, which can be estimated linearly in finite dimensions by the developed algorithm. Extended Dynamic Mode Decomposition is an efficient data-driven algorithm for the finite-dimensional linear approximation of the Koopman operator. The goal of the algorithm is to find an optimal K that satisfies:

[0109]

[0110] Where K:F q →F q yes Finite-dimensional linear approximation of , γ is a non-zero vector, Φ is a vector consisting of observable quantities, let:

[0111]

[0112] in:

[0113]

[0114] There are N basis functions that need to be designed Composition, i = 1, ..., N, so K is obtained by minimizing the following two-norm matrix:

[0115]

[0116] Where A and B are the parameter matrices obtained by solving the above optimization problem; the following recursive relationship is obtained:

[0117]

[0118] Afterwards, in the initial state of the system When and r0 are known, the following predictor is constructed and the output of the system is estimated by solving the following least squares problem:

[0119]

[0120] where C is the optimal parameter matrix obtained by solving the least squares problem.

[0121] Step S4: The data-driven method based on the Koopman operator theory can predict the output of the closed-loop system online, obtain the estimated tracking error by taking the difference between the reference input and the predicted output, and implement feedforward compensation online by correcting the reference trajectory; complete error prediction and feedforward compensation within a single sampling cycle, and perform real-time online compensation during the operation of the control system.

[0122] The present invention also provides a robust high-bandwidth control system with intelligent error prediction and feedforward compensation. The robust high-bandwidth control system with intelligent error prediction and feedforward compensation can be implemented by executing the process steps of the robust high-bandwidth control method with intelligent error prediction and feedforward compensation. That is, those skilled in the art can understand the robust high-bandwidth control method with intelligent error prediction and feedforward compensation as an optimal implementation of the robust high-bandwidth control system with intelligent error prediction and feedforward compensation.

[0123] Example 2:

[0124] The present invention also provides a robust high-bandwidth control system with intelligent error prediction and feedforward compensation, the system comprising the following modules:

[0125] Module M1: The robust disturbance observer treats the model uncertainty and external disturbances within the fast tool servo system during machining as a lumped disturbance d(t) for estimation and compensation. The fast tool servo system is driven by piezoelectric or normal stress electromagnetic drive, and the entire system is described in the Laplace continuous domain by the following third-order model:

[0126]

[0127] Among them, a3, a2, a1, a0, b2, b1, and b0 are the parameters of the system model. The third-order system is a non-minimum phase system. A slight correction is made to the third-order system to make it a minimum phase system:

[0128]

[0129] Design the corresponding low-pass filter Q(s) so that P n -1 (s)Q(s) is physically achievable using a second-order low-pass filter:

[0130]

[0131] where τ is the parameter that determines the bandwidth of the low-pass filter, which is tuned to achieve a compromise between the observer's disturbance rejection performance and stability; the total disturbance of the system is estimated as And use the controlled system to compensate.

[0132] Module M2: Design the corresponding dual-loop high-bandwidth controller. In the inner loop, apply the positive acceleration-velocity-position feedback damping controller. The expression of the positive acceleration-velocity-position feedback damping controller is:

[0133]

[0134] The damping controller has five parameters that can be adjusted: β2, β1, β0, ξ, ω p , the desired positions of the five system poles can be arbitrarily configured; in the outer loop, a high-gain proportional-integral tracking controller is applied, and its expression is:

[0135]

[0136] Among them, k p and k i are the proportional gain and the integral gain; the parameters of the inner-loop damping controller and the outer-loop tracking controller can be synchronously tuned using an intelligent evolutionary algorithm to maximize the control bandwidth.

[0137] Module M3: Design an online intelligent error prediction and feedforward compensation module based on the Koopman operator theory to perform online reference trajectory correction. Considering the nonlinear characteristics between the input and output of the closed-loop system, the nonlinear dynamics between the input and output of the robust high-bandwidth closed-loop system is expressed by the following discrete nonlinear equation:

[0138] y k+1 =f(y k ,r k ) (6)

[0139] Where k represents the sampling point of the discrete system, r k and y k Represent the input and output of the closed-loop system at the current moment respectively. From (6), we can see that the output of the system at the current moment depends on the input and output of the system at the previous moment. In order to predict the output of the system at the next moment based on the input and output of the system at the current and previous moments, the Koopman operator is used to lift the nonlinear function into the infinite-dimensional linear space, and the Koopman operator is defined as F→F is:

[0140]

[0141] in is an infinite-dimensional linear operator, F is the Hilbert space spanned by observables, and σ∈F is an observable that elevates the system input and output to a high-dimensional space; the dynamic equation of the closed-loop system is written as:

[0142]

[0143] The nonlinear dynamics between input and output are preserved by the infinite-dimensional linear Koopman operator, which can be estimated linearly in finite dimensions by the developed algorithm. Extended Dynamic Mode Decomposition is an efficient data-driven algorithm for the finite-dimensional linear approximation of the Koopman operator. The goal of the algorithm is to find an optimal K that satisfies:

[0144]

[0145] Where K:F q →F q yes Finite-dimensional linear approximation of , γ is a non-zero vector, Φ is a vector consisting of observable quantities, let:

[0146]

[0147] in:

[0148]

[0149] There are N basis functions that need to be designed Composition, i = 1, ..., N, so K is obtained by minimizing the following two-norm matrix:

[0150]

[0151] Where A and B are the parameter matrices obtained by solving the above optimization problem; the following recursive relationship is obtained:

[0152]

[0153] Afterwards, in the initial state of the system When and r0 are known, the following predictor is constructed and the output of the system is estimated by solving the following least squares problem:

[0154]

[0155] where C is the optimal parameter matrix obtained by solving the least squares problem.

[0156] Module M4: A data-driven system based on the Koopman operator theory can predict the output of a closed-loop system online. By subtracting the reference input from the predicted output to obtain an estimated tracking error, feedforward compensation can be implemented online through reference trajectory correction. Error prediction and feedforward compensation are completed within a single sampling cycle, allowing real-time online compensation during control system operation.

[0157] Example 3:

[0158] Fast tool servo systems are widely used in ultra-precision machining. However, the hysteresis nonlinearity within the driver and the lightly damped resonance characteristics of the mechanism seriously affect their control accuracy and bandwidth. External interference such as cutting force during machining further increases the control challenges. The present invention provides a robust high-bandwidth control method based on Koopman operator theory, which uses data-driven online intelligent error prediction and feedforward compensation to address the above challenges in the precision motion control of fast tool servo systems and achieve high-precision and high-bandwidth trajectory tracking in the presence of external interference.

[0159] A robust high-bandwidth control method based on online intelligent error prediction and feedforward compensation of Koopman operator theory, such as Figure 1 Shown, including:

[0160] Disturbance observer module: During fast tool servo machining, the system's internal model uncertainty and external disturbances can seriously affect the controller's tracking performance. In order to estimate the system disturbances during machining and facilitate the use of control methods for disturbance compensation, a robust disturbance observer that can be applied to non-minimum phase systems is designed. This observer treats the system's internal model uncertainty and external disturbances during fast tool servo machining as a lumped disturbance d(t) as a whole, and then estimates and compensates for it. Typically, the drive mode of a fast tool servo system is piezoelectric drive or normal stress electromagnetic drive, and the entire system can be described by the following third-order model in the Laplace continuous domain:

[0161]

[0162] Among them, a3, a2, a1, a0, b2, b1, b0 are the parameters of the system model. This third-order system is generally a non-minimum phase system. In order to ensure the stability of the disturbance observer, it is necessary to make a slight correction to the third-order system to make it a minimum phase system:

[0163]

[0164] Since the inverse P of the modified minimum phase system n -1 (s) is usually non-causal, and a corresponding low-pass filter Q(s) needs to be designed so that P n -1 (s)Q(s) is achievable in practical applications. In this invention, a second-order low-pass filter is used:

[0165]

[0166] Where τ is the parameter that determines the bandwidth of the low-pass filter, which needs to be tuned to achieve a compromise between the observer's disturbance rejection performance and stability. In this way, the total disturbance of the system can be estimated as And use control methods to compensate.

[0167] Dual-loop high-bandwidth control module: To suppress the system's lightly damped resonance characteristics and hysteresis nonlinearity and improve the control bandwidth, a corresponding dual-loop high-bandwidth controller is designed. In the inner loop, a positive acceleration-velocity-position feedback damping controller designed specifically for this third-order system is applied, and its expression is:

[0168]

[0169] The damping controller has five parameters that can be adjusted: β2, β1, β0, ξ, ω p , the desired positions of the five system poles can be arbitrarily configured to achieve the desired system damping ratio, so that the outer loop tracking controller can still have sufficient stability margin to ensure system stability at high gain. In the outer loop, a high-gain proportional-integral tracking controller is applied to reduce the residual tracking error and achieve accurate trajectory tracking. Its expression is:

[0170]

[0171] Among them, k p and k i are the proportional gain and integral gain. The parameters of the inner-loop damping controller and the outer-loop tracking controller can be synchronously tuned using an intelligent evolutionary algorithm to achieve optimal control bandwidth. The disturbance observer module and the high-bandwidth dual-loop control module together form a robust high-bandwidth controller.

[0172] An online intelligent error prediction and feedforward compensation module based on Koopman operator theory is designed to compensate for the inevitable phase lag error in closed-loop control systems and the residual tracking error caused by various model uncertainties. This module significantly improves the tracking accuracy of the robust high-bandwidth controller through online reference trajectory correction. Considering the nonlinear characteristics between the input and output of the closed-loop system, the nonlinear dynamics between the input and output of the robust high-bandwidth closed-loop system can be expressed as the following discrete nonlinear equation:

[0173] y k+1 =f(y k ,r k ) (6)

[0174] where r k and y kRepresent the input and output of the closed-loop system respectively. Therefore, the output of the system at the current moment depends on the input and output of the system at the previous moment. In order to predict the output of the system at the next moment based on the input and output of the system at the current and previous moments, the Koopman operator is used to lift the nonlinear function into an infinite-dimensional linear space. The Koopman operator is defined as F→F is:

[0175]

[0176] in is an infinite-dimensional linear operator, F is the Hilbert space spanned by observables, and σ∈F is an observable that elevates the system input and output to a higher-dimensional space. Therefore, the dynamic equation of the closed-loop system can be written as:

[0177]

[0178] The nonlinear dynamics between input and output are preserved by the infinite-dimensional linear Koopman operator, which can be linearly estimated in finite dimensions using the developed algorithm. Extended Dynamic Mode Decomposition is an efficient data-driven algorithm for implementing the finite-dimensional linear approximation of the Koopman operator. The goal of the algorithm is to find an optimal K that satisfies:

[0179]

[0180] Where K:F q →F q yes Finite-dimensional linear approximation of , γ is a non-zero vector, Φ is a vector consisting of observable quantities, let:

[0181]

[0182] in:

[0183]

[0184] There are N basis functions that need to be designed Composition, so K can be obtained by minimizing the following two-norm matrix:

[0185]

[0186] Where A and B are the parameter matrices obtained by solving the above optimization problem. In this way, the following recursive relationship can be obtained:

[0187]

[0188] Afterwards, in the initial state of the system When and r0 are known, the following predictor can be constructed, and the output of the system can be estimated by solving the following least squares problem:

[0189]

[0190] where C is the optimal parameter matrix obtained by solving the least squares problem.

[0191] Overall control scheme implementation module: The overall control scheme includes a disturbance observer, a high-bandwidth dual-loop controller, and an online intelligent error prediction and feedforward compensation module. The data-driven method based on the Koopman operator theory can accurately predict the output of the closed-loop system online. By taking the difference between the reference input and the predicted output, the estimated tracking error can be obtained, so that the feedforward compensation can be realized online by correcting the reference trajectory, greatly improving the tracking accuracy of the robust high-bandwidth dual-loop controller. This data-driven intelligent error prediction method can complete error prediction and feedforward compensation within a single sampling cycle (0.00005s) with extremely high operating efficiency, so real-time online compensation can be realized during the operation of the control system. Since the compensation term only modifies the reference trajectory of the system, the control bandwidth, stability and other characteristics of the robust high-bandwidth controller itself will not change. This control scheme is implemented on a self-developed fast tool servo prototype, and its signal flow chart is shown below. Figure 2 As shown in the figure, this fast tool servo mechanism is piezoelectrically driven, transmitting displacement through the deformation of a flexible hinge. The control algorithm is implemented in the MATLAB / Simulink environment, using DSpace for real-time motion control. The control voltage generated in DSpace is transmitted via a 16-bit digital-to-analog converter to a high-voltage amplifier with a 15x amplification ratio, which 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 control system has a sampling frequency of 20 kHz.

[0192] Those skilled in the art may understand this embodiment as a more specific description of Embodiment 1 and Embodiment 2.

[0193] Those skilled in the art will appreciate that, in addition to implementing the system and its various devices, modules, and units provided by the present invention in purely computer-readable program code, it is entirely possible to implement the same functions of the system and its various devices, modules, and units provided by the present invention in the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers, and embedded microcontrollers by logically programming the method steps. Therefore, the system and its various devices, modules, and units provided by the present invention can be considered 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; the devices, modules, and units for implementing various functions can also be considered as both software modules implementing the method and structures within the hardware component.

[0194] The above describes specific embodiments of the present invention. It should be understood that the present invention is not limited to the specific embodiments described above, and those skilled in the art may make various changes or modifications within the scope of the claims, which do not affect the essence of the present invention. The embodiments of this application and the features in the embodiments may be combined with each other in any manner unless there is a conflict.

Claims

1. A robust high-bandwidth control method with intelligent error prediction and feedforward compensation, characterized in that: The method comprises the following steps: Step S1: The robust disturbance observer regards the model uncertainty and external disturbance in the fast tool servo system during machining as a whole lumped disturbance d(t) and performs estimation and compensation. Step S2: Design a corresponding dual-loop high-bandwidth controller, and apply a positive acceleration-velocity-position feedback damping controller in the inner loop; Step S3: Design an online intelligent error prediction and feedforward compensation module based on the Koopman operator theory to perform online reference trajectory correction; Step S4: The data-driven method based on the Koopman operator theory can predict the output of the closed-loop system online. The estimated tracking error is obtained by subtracting the reference input from the predicted output, and feedforward compensation is implemented online by correcting the reference trajectory. The driving mode of the fast tool servo system in step S1 is piezoelectric drive or normal stress electromagnetic drive, and the entire system is described by the following third-order model in the Laplace continuous domain: Among them, a3, a2, a1, a0, b2, b1, and b0 are the parameters of the system model. The third-order model is a non-minimum phase system. A slight modification is made to the third-order model to make it a minimum phase system: Design the corresponding low-pass filter Q(s) so that P n -1 (s)Q(s) is physically achievable using a second-order low-pass filter: where τ is the parameter that determines the bandwidth of the low-pass filter, which is tuned to achieve a compromise between the observer's disturbance rejection performance and stability; the total disturbance of the system is estimated as and use control methods to compensate; The expression of the positive acceleration-velocity-position feedback damping controller in step S2 is: The damping controller has five parameters that can be adjusted: β2, β1, β0, ξ, ω p , the desired positions of the five system poles can be arbitrarily configured; in the outer loop, a high-gain proportional-integral tracking controller is applied, and its expression is: Among them, k p and k i are the proportional gain and the integral gain; the parameters of the inner-loop damping controller and the outer-loop tracking controller can be synchronously tuned using an intelligent evolutionary algorithm to maximize the control bandwidth.

2. The robust high-bandwidth control method with intelligent error prediction and feedforward compensation according to claim 1, characterized in that: In step S3, the nonlinear characteristics between the input and output of the closed-loop system are taken into consideration. The nonlinear dynamics between the input and output of the robust high-bandwidth closed-loop system is expressed by the following discrete nonlinear equation: y k+1 =f(y k ,r k ) (6) Where k represents the sampling point of the discrete system, r k and y k Represent the input and output of the closed-loop system at the current moment, respectively. From (6), we can see that the output of the system at the current moment depends on the input and output of the system at the previous moment. In order to predict the output of the system at the next moment based on the input and output of the system at the current and previous moments, the Koopman operator is used to lift the discrete nonlinear equation into the infinite-dimensional linear space. The Koopman operator κ: F→F is defined as: Kσ(y k r k )=σ(f(y k r k ),r k+1 ) (7) Where K is an infinite-dimensional linear operator, F is the Hilbert space spanned by observables, and σ∈F is an observable that elevates the system input and output to a high-dimensional space. The dynamic equation of the closed-loop system is written as: Kσ(y k r k )=σ(y k+1 r l+1 ) (8) The nonlinear dynamics between input and output are preserved by the infinite-dimensional linear Koopman operator, which can be estimated linearly in finite dimensions by the developed algorithm. Extended Dynamic Mode Decomposition is an efficient data-driven algorithm for the finite-dimensional linear approximation of the Koopman operator. The goal of the algorithm is to find an optimal K that satisfies: where K:F q →F q is a finite-dimensional linear approximation of K, γ is a non-zero vector, Φ is a vector consisting of observable quantities, let: in: There are N basis functions that need to be designed Composition, i = 1, ..., N, so K is obtained by minimizing the following two-norm matrix: Where A and B are parameter matrices obtained by solving formula (12); the following recursive relationship is obtained: Afterwards, in the initial state of the system When and r0 are known, the following predictor is constructed and the output of the system is estimated by solving the following least squares problem: where C is the optimal parameter matrix obtained by solving the least squares problem.

3. The robust high-bandwidth control method with intelligent error prediction and feedforward compensation according to claim 1, characterized in that: The step S4 completes error prediction and feedforward compensation within a single sampling period, and performs real-time online compensation during the operation of the control system.

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