A Precision Position Tracking Control Algorithm Based on Real-time Iterative Compensation

Abstract

The present invention discloses a precision position tracking control algorithm based on real-time iterative compensation, comprising the following steps: taking a single axis of an X-Y motion platform as a research object, obtaining its transfer function through parameter identification; a position tracker adopts a discrete PID (proportional integral differential) feedback control algorithm to obtain a feedback control output signal; a FIR (finite impulse filter) parameterized fixed structure controller is used for feedforward control; a disturbance observer module is designed to compensate the disturbance observer output to the control input, and a low-pass filter is introduced; and real-time iterative compensation control is performed based on a Kalman filter. The real-time iterative compensation strategy proposed in the present invention introduces a Kalman filter, which can suppress the influence of system noise to a certain extent, thereby improving the system's anti-interference ability and tracking accuracy; combining the prediction results of the gated recursive unit and the recursive prediction results of the model state space equation, by estimating the system state and noise covariance, a better noise processing capability is provided.

Description

Technical Field

[0001] The invention belongs to the technical field of position tracking control, and in particular relates to a precise position tracking control algorithm based on real-time iterative compensation. Background Art

[0002] Deep ultraviolet wafer defect detection requires multiple data and image acquisitions of the wafer, and all acquisition work needs to be completed by controlling the motion stage to move the wafer. The linear scan camera scans the image line by line, requiring the motion platform to move accurately at a constant speed during scanning. Unstable speed will cause image distortion and overlap, affecting the detection results. Each line of scan needs to be aligned in the y direction. Minor displacement errors will cause image stitching errors and affect the detection results. At the same time, wafer inspection requires the scanning and inspection of the entire wafer to be completed within a limited time. By increasing the platform movement speed, the scanning time for each line can be shortened to improve production efficiency. However, due to the influence of the motion stage model and various complex disturbances, precision motion control technology faces huge challenges. Precision motion control algorithms are the key to the motion stage achieving both high dynamic and precision motion goals.

[0003] Precision motion usually adopts a feedforward-feedback two-degree-of-freedom control structure. Feedback control can suppress various interferences while ensuring the closed-loop stability of the system. Without changing the feedback control structure, feedforward compensation can be added to achieve reference trajectory and disturbance compensation, thereby improving system control performance.

[0004] The patent "Online iterative compensation system and method for servo errors" discloses an online iterative compensation algorithm for fast tool servo control. It uses a model to predict the state of the system within the prediction interval, and modifies the reference trajectory through online iteration to perform error compensation, thereby improving the system tracking accuracy.

[0005] The core of the real-time iterative compensation strategy is to use the online prediction model and iterative calculation to generate the optimal feedforward compensation signal in real time, thereby improving the trajectory tracking accuracy and dynamic adjustment capability. The existing real-time iterative compensation control algorithm mainly relies on the iterative calculation of the error at the future moment to update the compensation in advance. However, the noise accumulation in the iterative process and the prediction process will affect the system tracking performance. At the same time, when designing feedforward compensation, it usually depends on the model accuracy of the controlled object. If the model is not accurate enough, the compensation signal may be limited or unstable. Summary of the invention

[0006] In order to make up for the shortcomings of the prior art, the present invention provides a precise position tracking control algorithm based on real-time iterative compensation, which utilizes the GRU (gated recursive unit) neural network to predict the system position output and the model state space equation to recursively predict the position output, introduces the Kalman filter to reduce the influence of noise and improve the tracking accuracy.

[0007] The technical problem solved by the present invention can be achieved through the following specific technical solutions:

[0008] The precise position tracking control algorithm based on real-time iterative compensation comprises the following steps:

[0009] Step 1: Take the single-axis horizontal two-dimensional XY motion platform as the research object, conduct a system frequency sweep experiment or a unit impulse response experiment, record the control input and position output, and obtain the transfer function P of the controlled object by combining the MATLAB (Matrix Laboratory) software system identification box;

[0010] Step 2: The position tracker uses a discrete PID (proportional integral differential) feedback control algorithm to obtain a feedback control output signal ;

[0011] Step 3: Based on the transfer function P obtained by identification, a FIR (finite impulse filter) parameterized fixed structure controller is used for feedforward control;

[0012] Step 4: Use the inverse model of the system obtained by identification Estimate the system lumped disturbance, design the disturbance observer (DOB) module, compensate the disturbance observer output to the control input, and introduce a low-pass filter Q;

[0013] Step 5. Combine the existing FIR (finite impulse filter) feedforward and design a real-time iterative compensation control algorithm based on the Kalman filter to perform feedforward control together. Use the Kalman filter in combination with the GRU (gated recursive unit) and recursive prediction to predict the tracking error that will occur during real-time motion. Perform real-time iterative compensation control by iteratively calculating the online synthesized feedforward compensation signal.

[0014] Furthermore, in step 2, according to the tracking error , refer to the input signal r and the feedback position signal y, and get the feedback control output signal ,Right now:

[0015] ,

[0016] in, , , is the control parameter of PID (proportional integral differential) controller, k is the discrete sampling number, k = 1, 2, ..., N; Ts is the sampling period, is the tracking error at the kth sampling time, is the expectation at the kth sampling time, is the measured output at the kth sampling time, ; is the feedback control output at the kth sampling time.

[0017] Furthermore, in step 3, the feedforward control output signal ,in, , , is the feedforward parameter, s is the complex plane parameter; according to the measured feedback signal and a known input signal , using the least squares method to minimize the error between the feedback control output signal and the ideal feedforward control output signal, that is, Solution , , .

[0018] Furthermore, in step 3, the low-pass filter , is the system bandwidth.

[0019] Furthermore, in step 5, an online prediction model and iterative calculation are used to generate an optimal feedforward compensation signal in real time. At the same time, a Kalman filter is introduced to reduce the impact of noise, and the optimal iterative gain is dynamically adjusted to find the optimal iterative gain when the noise meets a certain probability condition.

[0020] Furthermore, the specific process of step 5 includes:

[0021] Step 51, online prediction based on system model position output;

[0022] Step 52, outputting an estimated prediction based on the Kalman filter predicted position;

[0023] Step 53: Feedforward output real-time iterative compensation.

[0024] Furthermore, in step 51, the online output prediction includes:

[0025] ① First, the online prediction model is established according to the system state space equation as follows:

[0026] + + d(z)——system transfer function (1)

[0027] In the formula, is a feedback controller, P is the transfer function of the controlled object, R(z) represents the desired input, F(z) is the feedforward signal, represents the actual output, d(z) is the lumped disturbance;

[0028] ② Then, establish the discrete system state space equation:

[0029]

[0030] Where k represents the current discrete time step of the system, is the state vector at time k, is the input vector at time k, is the feedforward vector at time k, A is the state matrix, , is the input matrix, C is the output matrix, is the disturbance vector at time k+1, is the predicted state vector at time k+1, is the predicted output vector at time k;

[0031] ③ Given the current time k, from formula (2-1) and formula (2-2), the output recursive prediction at time k+Np is:

[0032] ,

[0033] In the formula, is the number of future sampling periods to be predicted, To predict the output at time k+Np considering disturbance, is the state vector at time k, is the input vector at time k+i, is the feedforward vector at time k+i, A is the state matrix, , is the input matrix, C is the output matrix, i is the number of sampling periods in the prediction range, i = 0, 1, ..., Np-1; is the disturbance vector at time k+i, is the Np-th power of the state matrix.

[0034] Furthermore, in step 52, the specific contents of the output based on the Kalman filter estimation prediction include:

[0035] Trajectory error prediction based on GRU (Gated Recurrent Unit): GRU (Gated Recurrent Unit) neural network is used to predict the system position at time k+Np ,The input of the training data set consists of the position, velocity and acceleration of multiple sampling points, and the output of the training set is the actual position output after passing through a zero-phase low-pass filter;

[0036] Position output prediction based on Kalman filter: Considering the output of the system at time k+Np when the lumped disturbance is considered (3), the a priori estimated predicted output is: ,

[0037] In the formula, is the prior estimated state at time k, for The prior estimate predicts the output at time, is the input vector at time k+i, is the feedforward vector at time k+i, A is the state matrix, , is the input matrix, C is the output matrix, i is the number of sampling periods in the prediction range, i = 0, 1, ..., Np-1; is the Np power of the state matrix;

[0038] Therefore, the system's posterior estimation prediction output is:

[0039] ;

[0040] Where G is the Kalman gain, for The predicted position output of the GRU (Gated Recurrent Unit) at time t, for The posterior estimate predicts the output at time instant, for The moment prior estimate predicts the output.

[0041] Furthermore, the Kalman gain G is solved as follows:

[0042] A priori estimate error: ,in, is the prior estimate prediction error at time k+Np, To predict the output at time k+Np considering disturbance, for The moment prior estimate predicts the output;

[0043] According to formula (3), the system state space equation predicts the results Affected by model uncertainty and other disturbances, the lumped disturbance is assumed to be , is a random disturbance with mean 0, variance: , let the covariance matrix of the prior estimation error be: , Kalman gain , is the trace of the covariance matrix of the prior estimate errors.

[0044] Further, in step 53, define To compensate the time domain, based on the prediction results, The compensation term will be added to the future sampling time of the reference trajectory, and the specific contents of the feedforward output real-time iterative compensation are as follows:

[0045] At the moment , additional control input compensation signal The online feedforward controller Produced The control input signal is becomes ,by Time control input compensation Influence, The position output at the moment is: ; Always increase control input compensation Post-system tracking error: ,in, for The variable after time compensation, To increase control input compensation The position increment after compensation is The relationship expression of For the case where no compensation signal is added The prediction error at the moment, To add compensation signal Moment prediction error;

[0046] first The moment control input compensation signal is: , after the first compensation The tracking error at time is: , where is a learning filter, without iterative calculation Tracking error at time: , for The reference input at the moment, for The posterior estimate at time instant predicts the output;

[0047] After repeated iterative calculations, the compensation signal of the nth step is:

[0048] , the tracking error after the nth iteration compensation is: , where The compensation signal of the nth step, is the tracking error after the nth iteration compensation, for The tracking error after the n-1th iteration compensation at time, To increase control input compensation The position increment after compensation is The relationship expression of is a learning filter, without iterative calculation Tracking error at time: , for The reference input at time, for The posterior estimate at time t predicts the output.

[0049] Compared with the prior art, the present invention has the following advantages:

[0050] (1) The real-time iterative compensation strategy proposed in the present invention introduces a Kalman filter, which can suppress the influence of system noise to a certain extent, thereby improving the system's anti-interference ability and tracking accuracy; combining the GRU (gated recurrent unit) prediction results and the model state space equation recursive prediction results, by estimating the system state and noise covariance, it provides better noise processing capabilities, improves the prediction accuracy when the noise meets a certain probability condition, and thus improves the system tracking performance.

[0051] (2) Traditional real-time iterative compensation algorithms usually rely on the model accuracy of the controlled object when designing feedforward compensation. If the model is not accurate enough, the compensation signal may be limited or unstable. The present invention uses GRU (Gated Recurrent Unit) to predict the system position output, which is a data-based control strategy that does not rely on the system model. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 This is a flow chart of the control algorithm steps of the present invention;

[0053] Figure 2 It is a real-time iterative compensation control framework of the present invention. DETAILED DESCRIPTION

[0054] In order to make the purpose, technical solution and advantages of the present invention clearer, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0055] like Figure 1 As shown, a precise position tracking control algorithm based on real-time iterative compensation includes the following steps:

[0056] A precise position tracking control algorithm based on real-time iterative compensation includes the following steps:

[0057] Step 1: Take the single axis XY (horizontal two-dimensional) motion platform as the research object, conduct a system frequency sweep experiment or a unit pulse response experiment, record the control input and position output, and combine the MATLAB (Matrix Laboratory) software system identification box to obtain the transfer function P of the controlled object.

[0058] Step 2: The position tracker uses a discrete PID (proportional integral differential) feedback control algorithm to obtain a feedback control output signal .

[0059] According to the tracking error , refer to the input signal r, and the feedback position signal y, and get the feedback control output signal: ,in, , , is the control parameter of the PID controller, k = 1, 2, ..., N is the number of discrete sampling times, Ts is the sampling period, , is the tracking error at the kth sampling time, is the expectation at the kth sampling time, is the measured output at the kth sampling time, is the feedback control output at the kth sampling time.

[0060] Step 3: To reduce system tracking, a FIR (finite impulse filter) parameterized fixed structure controller is used for feedforward control based on the transfer function P obtained by identification.

[0061] Feedforward control output signal ,in, , , is the feedforward parameter, F is the feedforward control output signal, and s is the complex plane parameter. In the low frequency range, the feedback control signal can be an approximate ideal feedforward signal. and a known input signal , using the least squares method to minimize the error between the feedback control signal and the ideal feedforward signal, that is, Solution , , .

[0062] Step 4: Use the inverse model of the system obtained by identification Estimate the system lumped disturbance, design the disturbance observer (DOB) module, compensate the disturbance observer output to the control input, and enhance the system's robustness to various unmodeled disturbances including nonlinear disturbances. In order to ensure the robust stability of the closed-loop system, a low-pass filter Q is introduced. , is the system bandwidth.

[0063] Step 5. Combined with the existing FIR (finite impulse filter) feedforward, a real-time iterative compensation control algorithm based on the Kalman filter is designed to perform feedforward control together. The Kalman filter is used in combination with the GRU (gated recursive unit) and recursive prediction to predict the tracking error that will occur during real-time motion. The feedforward compensation signal is iteratively calculated online to perform real-time iterative compensation control. While ensuring high tracking accuracy, it also enhances task flexibility and anti-interference ability.

[0064] The specific contents include the following:

[0065] (1) Online prediction based on system model position output.

[0066] First, an online prediction model is established based on the system state space equation. The formula is as follows:

[0067] + + d(z)——system transfer function (1)

[0068] In the formula, is a feedback controller, P is the transfer function of the controlled object, R(z) represents the desired input, F(z) is the feedforward signal, represents the actual output, and d(z) is the lumped disturbance.

[0069] Then, the discrete system state space equation is established:

[0070]

[0071] Where k represents the current discrete time step of the system, is the state vector at time k, is the input vector at time k, is the feedforward vector at time k, A is the state matrix, , is the input matrix, C is the output matrix, is the disturbance vector at time k+1, is the predicted state vector at time k+1, is the predicted output vector at time k.

[0072] Given the current time k, from formula (2-1) and formula (2-2), we know that the output recursive prediction at time k+Np is:

[0073] ,

[0074] In the formula, is the number of future sampling periods to be predicted, To predict the output at time k+Np considering disturbance, is the state vector at time k, is the input vector at time k+i, is the feedforward vector at time k+i, A is the state matrix, , is the input matrix, C is the output matrix, i is the number of sampling periods in the prediction range, i = 0, 1, ..., Np-1; is the disturbance vector at time k+i, is the Np power of the state matrix.

[0075] (2) Estimated prediction output based on the Kalman filter predicted position.

[0076] First, trajectory error prediction based on GRU (Gated Recurrent Unit): The GRU (Gated Recurrent Unit) neural network is used to predict the system position at time k+Np ,The input of the training data set consists of the position, velocity and acceleration of multiple ,sampling points, and the output of the training set is the actual position output after ,passing a zero-phase low-pass filter.

[0077] Then, based on the Kalman filter position output prediction: Considering the system output at time k+Np when the lumped disturbance is considered (3), the a priori estimated predicted output is: ,

[0078] In the formula, is the prior estimated state at time k, for The prior estimate predicts the output at time, is the input vector at time k+i, is the feedforward vector at time k+i, A is the state matrix, , is the input matrix, C is the output matrix, i is the number of sampling periods in the prediction range, i = 0, 1, ..., Np-1; is the Np power of the state matrix.

[0079] Therefore, the system's posterior estimation prediction output is:

[0080] ;

[0081] Where G is the Kalman gain, for The predicted position output of the GRU (Gated Recurrent Unit) at time t, for The posterior estimate predicts the output at time instant, for The moment prior estimate predicts the output.

[0082] Among them, the Kalman gain G is solved as follows:

[0083] A priori estimate error: ,in, is the prior estimate prediction error at time k+Np, To predict the output at time k+Np considering disturbance, for The moment prior estimate predicts the output.

[0084] According to formula (3), the system state space equation predicts the results Affected by model uncertainty and other disturbances, the lumped disturbance is assumed to be , is a random disturbance with mean 0, variance: , let the covariance matrix of the prior estimation error be: , Kalman gain , is the trace of the covariance matrix of the prior estimate errors.

[0085] (3) Real-time iterative compensation of feedforward output.

[0086] definition To compensate the time domain, based on the prediction results, is the future sample time at which the compensation term will be added to the reference trajectory.

[0087] At the moment , additional control input compensation signal The online feedforward controller Produced The control input signal is becomes ,by Time control input compensation Influence, The position output at the moment is: ; Always increase control input compensation Post-system tracking error: ,in, for The variable after time compensation, To increase control input compensation The position increment after compensation is The relationship expression of For the case where no compensation signal is added The prediction error at the moment, To add compensation signal Time prediction error.

[0088] first The moment control input compensation signal is: , after the first compensation The tracking error at time is: , where is a learning filter, without iterative calculation Tracking error at time: , for The reference input at time, for The posterior estimate at time t predicts the output.

[0089] In order to further reduce the tracking error, after repeated iterative calculations, the compensation signal of the nth step is:

[0090] , the tracking error after the nth iteration compensation is: , where is the compensation signal of the nth step, is the tracking error after the nth iteration compensation, for The tracking error after the n-1th iteration compensation at time, To increase control input compensation The position increment after compensation is The relationship expression of is a learning filter, without iterative calculation Tracking error at time: , for The reference input at time, for The posterior estimate at time t predicts the output.

[0091] The real-time iterative compensation control algorithm based on Kalman filter proposed in the present invention combines the prediction results of GRU (gated recurrent unit) and the recursive prediction results of the model state space equation, and provides better noise processing capability by estimating the system state and noise covariance, thereby improving the prediction accuracy under the condition that the noise meets a certain probability, thereby improving the system tracking performance; based on the GRU (gated recurrent unit) neural network, it does not rely on the system model and predicts the system output.

[0092] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A precise position tracking control algorithm based on real-time iterative compensation, characterized in that: The following steps are involved: Step 1: Take the single-axis horizontal two-dimensional XY motion platform as the research object, conduct a system frequency sweep experiment or a unit pulse response experiment, record the control input and position output, and obtain the transfer function P of the controlled object in combination with the MATLAB software system identification box; Step 2: The position tracker uses a discrete PID feedback control algorithm to obtain the feedback control output signal u fb (kTs); Step 3: Based on the transfer function P obtained by identification, a FIR parameterized fixed structure controller is used for feedforward control; Step 4: Use the system inverse model P obtained by identification -1 Estimate the system lumped disturbance, design the disturbance observer module, compensate the disturbance observer output to the control input, and introduce the low-pass filter Q; Step 5: Combine the existing FIR feedforward and design a real-time iterative compensation control algorithm based on Kalman filter to perform feedforward control together. Use Kalman filter in combination with gated recursive unit and recursive prediction to predict the tracking error that will occur during real-time motion. Perform real-time iterative compensation control by iteratively calculating the online synthesized feedforward compensation signal. The specific process includes: Step 51, online prediction based on system model position output; Step 52, outputting an estimated prediction based on the Kalman filter predicted position; Step 53: Feedforward output real-time iterative compensation; In step 52, the specific contents of the output based on Kalman filter estimation prediction include: Trajectory error prediction based on GRU: GRU neural network is used to predict the position y of the system at time k+Np pred (k), the input of the training data set consists of the position, velocity, and acceleration of multiple sampling points, and the output of the training set is the actual position output after passing through a zero-phase low-pass filter; Position output prediction based on Kalman filter: Considering the output of the system at time k+Np when the lumped disturbance is considered (3), the a priori estimated predicted output is: In the formula, is the prior estimated state at time k, is the a priori estimated prediction output at time k+Np, r(k+i) is the input vector at time k+i, f(k+i) is the feedforward vector at time k+i, A is the state matrix, B r , B f is the input matrix, C is the output matrix, i is the number of sampling periods in the prediction range, i = 0, 1, ..., Np-1; is the Np power of the state matrix; Therefore, the system's posterior estimation prediction output is: Where G is the Kalman gain, is the position output predicted by the gated recurrent unit at time k+Np, is the posterior estimation prediction output at time k+Np, The prior estimate predicts the output at time k+Np.

2. A precise position tracking control algorithm based on real-time iterative compensation according to claim 1, characterized in that: In step 2, the feedback control output signal u is obtained according to the tracking error e, the reference input signal r and the feedback position signal y. fb (kTs), that is: at fb (kTs)=k p e(kTs)+k i ∑e(kTs)+k d {e(kTs)-e[(k-1)Ts]}, Among them, k p , k i , k d is the control parameter of the proportional-integral-differential PID controller, k is the discrete sampling number, k=1, 2, ..., N; Ts is the sampling period, e(kTs) is the tracking error at the kth sampling, r(kTs) is the expectation at the kth sampling, y(kTs) is the measured output at the kth sampling, e(kTs)=r(kTs)-y(kTs); u fb (kTs) is the feedback control output at the kth sampling time.

3. The precise position tracking control algorithm based on real-time iterative compensation according to claim 1 is characterized in that: In step 3, the feedforward control output signal F = m a s 2 +m j s 3 +m s s 4 , where m a 、m j 、m s is the feedforward parameter, s is the complex plane parameter; according to the measured feedback signal u fb and the known input signal r, the least square method is used to minimize the error between the feedback control output signal and the ideal feedforward control output signal, that is, argmin(u fb -F*f) to solve for m a , m j , m s .

4. The precise position tracking control algorithm based on real-time iterative compensation according to claim 3 is characterized in that: In step 3, the low-pass filter τ is the system bandwidth.

5. The precise position tracking control algorithm based on real-time iterative compensation according to claim 1 is characterized in that: In step 5, the online prediction model and iterative calculation are used to generate the optimal feedforward compensation signal in real time. At the same time, the Kalman filter is introduced to dynamically adjust and find the optimal iterative gain.

6. A precise position tracking control algorithm based on real-time iterative compensation according to claim 1 or 5, characterized in that: In step 51, the online output prediction includes: ① First, the online prediction model is established according to the system state space equation as follows: In the formula, C fb is a feedback controller, P is the transfer function of the controlled object, R(z) represents the desired input, F(z) is the feedforward signal, Y(z) represents the actual output, and d(z) is the lumped disturbance; ②Then, establish the discrete system state space equation: In the formula, k represents the current discrete time step of the system, x(k) is the state vector at time k, r(k) is the input vector at time k, f(k) is the feedforward vector at time k, A is the state matrix, and B is r , B f is the input matrix, C is the output matrix, is the disturbance vector at time k+1, is the predicted state vector at time k+1, is the predicted output vector at time k; ③ Given the current time k, from formula (2-1) and formula (2-2), we know that the output recursive prediction at time k+Np is: Where Np is the number of future sampling periods to be predicted, To predict the output at time k+Np when considering disturbance, x(k) is the state vector at time k, r(k+i) is the input vector at time k+i, f(k+i) is the feedforward vector at time k+i, A is the state matrix, B r , B f is the input matrix, C is the output matrix, i is the number of sampling periods in the prediction range, i = 0, 1, ..., Np-1; is the disturbance vector at time k+i, is the Np power of the state matrix.

7. The precise position tracking control algorithm based on real-time iterative compensation according to claim 6 is characterized in that: The Kalman gain G is solved as follows: A priori estimate error: Among them, e-(k+Np) is the prior estimation prediction error at time k+Np, To predict the output at time k+Np considering disturbance, The a priori estimate predicted output at time k+Np; According to formula (3), the system state space equation predicts the results Affected by model uncertainty and other disturbances, assume that the aggregate disturbance is ψ, ψ is a random disturbance with a mean of 0, and the variance of ψ is: E(ψψ T )=R, let the covariance matrix of the prior estimation error: P=E(e-(k+N p ) - (k+N p ) T ), Kalman gain G = argmin(Tr(P)), Tr(P) is the trace of the covariance matrix of the prior estimation error.

8. The precise position tracking control algorithm based on real-time iterative compensation according to claim 7 is characterized in that: In step 53, Nc≤Np-1 is defined as the compensation time domain. Based on the prediction result, Nc is the future sampling time when the compensation term will be added to the reference trajectory. The specific content of the feedforward output real-time iterative compensation is as follows: At time k+Nc, the additional control input compensation signal u c The online feedforward controller C ff The generated u ff The control input signal is u ff becomes u ff +u c , controlled by input compensation u at time k+Nc c (k+Nc) influence, the position output at time k+Np is: k+Nc time increases the control input compensation u c (k+Nc) system tracking error: in, is the variable after compensation at time k+Np, ε is the increase in control input compensation u c The position increment after (k+Nc) and the control input compensation u c (k+Nc), e(k+Np) is the prediction error at time (k+Np) without adding compensation signal, is the prediction error at the time (k+Np) when the compensation signal is added; First k+N c The moment control input compensation signal is: After the first compensation k+N p The tracking error at time is: Where L(z) is a learning filter. When there is no iterative calculation, k+N p Tracking error at time: r(k+Np) is k+N p The reference input at time, The posterior estimate prediction output at time k+Np; After repeated iterative calculations, the compensation signal of the nth step is: The tracking error after the nth iteration compensation is: In the formula, is the compensation signal of the nth step, is the tracking error after the nth iteration compensation, k+N p The tracking error after the n-1th iteration compensation at time, ε is the increase in control input compensation u c The position increment after (k+Nc) and the control input compensation u c (k+Nc), L(z) is a learning filter, and k+N is p Tracking error at time: r(k+Np) is k+N p The reference input at time, The posterior estimate prediction output at time k+Np.

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