Trajectory tracking method of non-repetitive time-varying system and application thereof

Abstract

The invention discloses a trajectory tracking method of a non-repetitive time-varying system and application thereof. The method comprises the following steps: constructing a state space model for describing dynamic characteristics of the non-repetitive time-varying system; estimating the system state of the state space model by adopting an adaptive Kalman filtering algorithm; constructing a target function of system trajectory tracking according to the system state; and performing optimization processing on the objective function of system trajectory tracking to obtain an optimization result of system trajectory tracking. Compared with the prior art, the method has the advantages that the system state and parameter change can be estimated in real time, the adaptive capacity to dynamic noise and the state estimation precision are improved by adaptively adjusting the process noise covariance and the measurement noise covariance, and trajectory tracking errors are remarkably reduced.

Description

Technical Field

[0001] The present invention relates to the field of automatic control technology, and more specifically, to a trajectory tracking method for a non-repetitive time-varying system and its application. Background Art

[0002] Iterative Learning Control (ILC) is a learning control method that optimizes control inputs based on historical data and is widely applied to systems that need to repeatedly execute the same task. Its basic idea is to use the control error of the previous iteration to correct the control input of the current iteration to improve the tracking accuracy. ILC is particularly suitable for periodic tasks such as robot motion control and precision manufacturing. In these applications, the system gradually optimizes the control strategy through multiple iterations of learning, thereby improving the accuracy of trajectory tracking.

[0003] However, traditional ILC methods usually assume that the system dynamics remain unchanged during the iteration process and use historical trajectory errors to optimize control inputs to improve the tracking accuracy. However, for non-repetitive time-varying systems (NTVSs), the system parameters may gradually change due to factors such as sensor drift, tool wear, and environmental disturbances, which leads to a decrease in the convergence speed of traditional ILC methods, a reduction in tracking accuracy, and even possible system divergence. In addition, model-based methods may have large modeling errors when the system uncertainty is strong, thus affecting the control performance.

[0004] On the other hand, as a classic state estimation algorithm, Kalman Filtering (KF) has high reliability in system state and parameter estimation. However, when traditional Kalman filtering is applied to NTVSs, it usually relies on a fixed noise covariance matrix, which may lead to a decrease in estimation accuracy in the case of system dynamic changes. Therefore, for NTVSs, how to effectively fuse Kalman filtering and ILC to improve the system's adaptability to dynamic environments remains an urgent problem to be solved.

[0005] The prior art discloses a data-driven adaptive optimal iterative learning control method, which combines a non-parametric model identification method with an optimal iterative learning method to form a data-driven adaptive optimal iterative learning control method. The nominal model is identified and the iterative learning controller is continuously updated during the iteration process; the control signal is iteratively updated using the data-driven adaptive optimal iterative learning method, and the updated driving force is sent back to the motion control card again. This iterative learning process is repeated to improve the tracking accuracy of the servo system. The defect of this solution is that it has a poor application effect in non-repetitive time-varying scenarios such as precision manufacturing and robot dynamic path planning.

[0006] Therefore, in combination with the above requirements and the deficiencies of the existing technology, the present application proposes a trajectory tracking method and its application for a non-repetitive time-varying system. Summary of the Invention

[0007] The present invention provides a trajectory tracking method and its application for a non-repetitive time-varying system to improve the trajectory tracking accuracy and system robustness of non-repetitive time-varying systems (NTVSs).

[0008] The primary object of the present invention is to solve the above technical problems, and the technical solution of the present invention is as follows:

[0009] In the first aspect of the present invention, a trajectory tracking method for a non-repetitive time-varying system is provided, and this method includes the following steps:

[0010] S1. Construct a state-space model for describing the dynamic characteristics of the non-repetitive time-varying system.

[0011] S2. Use the adaptive Kalman filter algorithm to estimate the system state of the state-space model.

[0012] S3. Construct an objective function for system trajectory tracking based on the system state.

[0013] S4. Optimize the objective function for system trajectory tracking to obtain the optimized result of system trajectory tracking.

[0014] Further, in step S1, the expression of the state-space model of the non-repetitive time-varying system is:

[0015]

[0016] where this expression describes the dynamic characteristics of the non-repetitive time-varying system in terms of the properties of the state space and is described using the observation equation and the state equation. is the system output of the observation equation. represents the state vector of the state equation. is the control input. and are the current noises in the state equation and the observation equation, respectively.

[0017] Further, the specific process of step S2 is as follows:

[0018] First, calculate the prior state estimate value and the error covariance matrix at the next moment:

[0019]

[0020] where, based on the state equation, this formula uses A i , B i and Q iConsidered as known quantities, substitute the posterior estimate values (X i (k|k), P i (k|k)) into the formula to obtain the prior estimate value at the current time. Denote the predicted state estimate value at time t+1 based on the available information at time t. P i (k+1|k) is the predicted error covariance matrix, which is used to quantify the uncertainty in state estimation. And A i (k)P i (k|k)A i (k) T is the change in uncertainty brought about by the system state propagation.

[0021] Then, using the observation information, perform the update step, calculate the Kalman gain, and correct the predicted value to obtain the posterior state estimate and posterior covariance:

[0022] K i (k) = P i (k+1|k)C i (k) T [C i (k)P i (k+1|k)C i (k) T +R i (k)] -1

[0023] Substitute the predicted covariance matrix P i (k+1|k) and the observation matrix C i (k) into this formula to calculate the Kalman gain K i (t). The Kalman gain K i (t) determines the optimal weight between the predicted state and the actual measurement value. C i represents the observation matrix, and R i represents the observation noise covariance matrix.

[0024] Furthermore, after each iteration of the adaptive Kalman filtering algorithm is completed, based on the current tracking error e i and the observation residual , correct the covariance of the process noise and measurement noise. Substitute the tracking error and the observation residual into the following formula to obtain the updated Q i+1 and R i+1 :

[0025]

[0026] where α and β are adaptive weight parameters, which are used to control the covariance update rate, and their value ranges are 0 < α, β < 1.

[0027] Furthermore, considering the optimal iterative learning control framework of the norm, the control input for each batch is obtained by optimizing a performance index function. The specific objective function described in step S3 is as follows:

[0028]

[0029] where the objective function J consists of a tracking error term a control variable change term and a state estimation error term . Each term represents a different control objective. u i represents the input at each step, and e i represents the tracking error.

[0030] Furthermore, the specific process of step S4 is as follows: The control input is optimized using the objective function. According to the system input-output operator and the bias term, an update law for the control input is generated, and the error norm is made to decrease with the number of iterations through robust convergence analysis. The method of optimizing the control input using the objective function is as follows:

[0031] The subterms of the objective function are respectively expanded as:

[0032] J(u) = J1(u) + J2(u) + J3(u)

[0033]

[0034]

[0035]

[0036] where r is the reference trajectory, G i+1 is the input-output operator of the system at the (i + 1)-th iteration, d i+1 is the bias term. Partial derivative calculations are respectively performed on J1(u) and J2(u) according to the update law of iterative learning control, and the following expressions are obtained:

[0037]

[0038] When taking the partial derivative of u in the state estimation error term, since and X k+1 usually depend on the update process of the Kalman filter and do not directly take u as the independent variable. However, in the actual system, the introduction of the state estimation error term plays a role in correcting the state estimation error, and the influence between the state estimation and the control input can be balanced by adjusting the corresponding weight P. Then, an additional correction term is used to make up for this part of the influence. Therefore, when taking the derivative of u, its partial derivative term can be considered approximately zero, that is,

[0039] Furthermore, the update law of the control input represents the update law of the non-repetitive time-varying system, and its expression is as follows:

[0040]

[0041] Among them, the operator L u , L e and the coefficient E (d,ε) are expressed as follows:

[0042]

[0043] Since both Q and R are symmetric positive definite matrices, R+(G i+1 ) T QG i+1 is a non-singular matrix.

[0044] Furthermore, the specific process of making the error norm decrease with the number of iterations through robust convergence analysis is as follows:

[0045] Since the uncertainty of the state space model of the actual non-repetitive time-varying system satisfies ||ζ||≤1, the relevant operators include the following forms:

[0046]

[0047] The following inequalities are used to analyze the robustness:

[0048]

[0049] When the above two inequalities are both satisfied and the robustness is available, the error convergence condition of the model is satisfied.

[0050] The second aspect of the present invention provides an application of a trajectory tracking method for a non-repetitive time-varying system. The trajectory tracking method for the non-repetitive time-varying system is applied to a precision machining platform for machining trajectory tracking, specifically as follows: Define a non-repetitive time-varying reference trajectory on the precision machining platform; construct an objective function for precision machining platform trajectory tracking based on the adaptive Kalman filtering algorithm; optimize the objective function to iterate the adaptive Kalman filtering algorithm, compare the tracking error between the actual output and the non-repetitive time-varying reference trajectory, verify that the error of the adaptive Kalman filtering algorithm converges to the micron level of accuracy, complete the iteration of the adaptive Kalman filtering algorithm, and calculate the expected reference trajectory of the system motion by using the adaptive Kalman filtering algorithm after iterative convergence.

[0051] The third aspect of the present invention provides a trajectory tracking system for a non-repetitive time-varying system, which is used for the trajectory tracking method of the non-repetitive time-varying system, and includes: a state space modeling module, an adaptive Kalman filter module, an iterative learning controller, and an expected reference trajectory prediction module.

[0052] The state space modeling module constructs a state space model for describing the dynamic characteristics of the non-repetitive time-varying system; the adaptive Kalman filter module uses the adaptive Kalman filter algorithm to estimate the system state of the state space model; the iterative learning controller constructs an objective function for system trajectory tracking according to the system state; the expected reference trajectory prediction module performs optimization processing on the objective function of system trajectory tracking to obtain an optimized result of system trajectory tracking.

[0053] Compared with the prior art, the beneficial effects of the technical solution of the present invention are:

[0054] The present invention provides a trajectory tracking method and its application for a non-repetitive time-varying system. The designed adaptive Kalman filter (AKF) algorithm can estimate system state and parameter changes in real time, and improve the adaptability to dynamic noise and state estimation accuracy by adaptively adjusting the process noise covariance and measurement noise covariance; by balancing the tracking error, control input smoothness, and state estimation error, the control input is optimized, significantly reducing the trajectory tracking error and improving the robustness of the control input; a compensation term is introduced to offset the system deviation, accelerating the error convergence rate and avoiding system instability caused by sudden changes in the control quantity. Description of the Drawings

[0055] Figure 1 It is a flowchart of a trajectory tracking method for a non-repetitive time-varying system of the present invention.

[0056] Figure 2 It is a schematic diagram of a precision machining experimental platform in an embodiment of the application of a trajectory tracking method for a non-repetitive time-varying system of the present invention.

[0057] Figure 3 It is an output trajectory diagram when executing Algorithm 1 in each iterative learning control experiment in an embodiment of the present invention.

[0058] Figure 4 It is a schematic diagram of the average absolute tracking error under different weight values in an embodiment of the present invention.

[0059] Figure 5 It is a comparison diagram of the average absolute tracking error between PID control, norm-optimal iterative learning control, and the algorithm of the present invention in an embodiment of the present invention.

[0060] Figure 6This is a comparison graph of the output signal of the channel 1 system and the actual reference trajectory in the final experiment of an embodiment of the present invention.

[0061] Figure 7 This is a comparison graph of the output signal of the channel 2 system and the actual reference trajectory in the final experiment of an embodiment of the present invention.

[0062] Figure 8 This is a comparison graph of the reference input signal of the channel 1 system and the actually applied input signal in the final experiment of an embodiment of the present invention.

[0063] Figure 9 This is a comparison graph of the reference input signal of the channel 2 system and the actually applied input signal in the final experiment of an embodiment of the present invention.

[0064] Figure 10 This is a schematic diagram of a trajectory tracking system for a non-repetitive time-varying system of the present invention. Detailed implementation manners

[0065] In order to more clearly understand the above objects, features and advantages of the present invention, the present invention will be further described in detail below with reference to the drawings and specific implementation manners. It should be noted that, without conflict, the embodiments of the present application and the features in the embodiments can be combined with each other.

[0066] In the following description, many specific details are set forth in order to fully understand the present invention. However, the present invention can also be implemented in other ways different from those described herein. Therefore, the protection scope of the present invention is not limited by the specific embodiments disclosed below.

[0067] Embodiment 1

[0068] As Figure 1 shown, the present invention provides a trajectory tracking method for a non-repetitive time-varying system, and this method includes the following steps:

[0069] S1. Construct a state space model for describing the dynamic characteristics of the non-repetitive time-varying system.

[0070] S2. Use an adaptive Kalman filtering algorithm to estimate the system state of the state space model.

[0071] S3. Construct an objective function for system trajectory tracking according to the system state.

[0072] S4. Optimize the objective function for system trajectory tracking to obtain an optimized result of system trajectory tracking.

[0073] The expression of the state space model of the non-repetitive time-varying system in step S1 is:

[0074]

[0075] Among them, the expression describes the dynamic characteristics of a non-repetitive time-varying system in terms of the properties of the state space, and is described by an observation equation and a state equation. is the system output of the observation equation. represents the state vector of the state equation. is the control input. and are the current noises in the state equation and the observation equation respectively.

[0076] The specific process of step S2 is as follows:

[0077] First, calculate the prior state estimate value and the error covariance matrix at the next moment:

[0078]

[0079] Among them, formula (2), as the prediction step of the Kalman filter, is based on the state equation in formula (1), regarding A i , B i and Q i as known quantities, substituting the posterior estimate value (X i (k|k), P i (k|k)) at the previous moment into formula (2) to obtain the prior estimate value at the current moment. represents the predicted state estimate value at time t + 1 based on the available information at time t, and P i (k + 1|k) is the predicted error covariance matrix, which is used to quantify the uncertainty in state estimation, while A i (k)P i (k|k)A i (k) T is the change in uncertainty brought about by the system state propagation.

[0080] Then, utilize the observation information, execute the update step, calculate the Kalman gain and correct the predicted value to obtain the posterior state estimate value and the posterior covariance:

[0081]

[0082] After completing the prediction step in formula (3), substitute the predicted covariance matrix P i (k + 1|k) and the observation matrix C i (k) into formula (4) to calculate the Kalman gain K i (t). The Kalman gain K i (t) determines the optimal weight between the predicted state and the actual measurement value. C i represents the observation matrix, and R i represents the observation noise covariance matrix.

[0083]

[0084] It should be noted that in the Kalman filter, this process is repeated at each discrete time instant, enabling the filter to perform an optimal estimation of the system state in a noisy environment. Different from the traditional Kalman filter, the adaptive Kalman filter (AKF) adjusts the noise covariance Q i and R i .

[0085] The predicted state is corrected using the Kalman gain in Equation (4) to obtain the posterior estimate X i (k+1|k+1), and its update process is described by Equation (5). Subsequently, the corresponding posterior covariance matrix P i (k+1|k+1) is also updated according to Equation (6), thus completing the Kalman filter step at this time instant.

[0086] P i (k+1∣k+1) = [I - K i (k)C i (k)]P i (k+1∣k) (5)

[0087] After each iteration of the adaptive Kalman filter algorithm is completed, the covariance of the process noise and measurement noise is corrected based on the current tracking error ei and the observation residual . Substituting the tracking error and the observation residual into the following equations gives the updated Q i+1 and R i+1 , so that in the next iteration, the filter can more accurately describe the noise characteristics:

[0088]

[0089] where α and β are adaptive weight parameters used to control the covariance update rate, and their value ranges are 0 < α, β < 1.

[0090] It should be noted that the goal of traditional iterative learning control (ILC) is to design an update law as follows:

[0091] u i+1 = F(u i , e i ) (7)

[0092] This update law utilizes the previous control input signal and the tracking error to continuously adjust the next input signal. Given the system dynamics, an iterative learning control update law is designed:

[0093]

[0094] Equation (8) generates the input sequence {u i} k≥0 and the corresponding tracking error {e i} k≥0 . By continuously performing iterative calculations, it determines the input u i at each step and simultaneously generates the tracking error e i . These sequences can be used to evaluate and improve the control strategy to gradually enhance the tracking performance of the system.

[0095] Considering the norm-optimal iterative learning control framework, the control input for each batch is obtained by optimizing a performance metric function. The specific objective function described in step S3 is as follows:

[0096]

[0097] where the objective function J consists of a tracking error term a control variable change term and a state estimation error term . Each term represents a different control objective. u i represents the input at each step, and e i represents the tracking error.

[0098] The specific process of step S4 is as follows: Optimize the control input using the described objective function, generate an update law for the control input based on the system input-output operator and the bias term, and make the error norm decrease with the number of iterations through robust convergence analysis.

[0099] Specifically, the tracking error term represents "penalizing" (i.e., incorporating into the objective function calculation) the difference between the system output and the reference trajectory, prompting the system to be able to track the desired trajectory and ensuring that the system output meets expectations.

[0100] The control variable change term represents restricting the update of the control input to avoid excessive changes in the control action, thereby enhancing the smoothness of the system operation and preventing adverse effects on the system caused by sudden changes in the control variable.

[0101] The state estimation error term represents ensuring that the state value estimated by the adaptive Kalman filter is close to the true state of the system. In this way, even in the presence of modeling uncertainties, the system can still maintain good performance, enhancing the robustness of the system.

[0102] In addition, the tracking error term in Equation (9) can also be written as e i+1 = r - G i+1 u i+1 - di+1 , where \(r\) is the reference trajectory, and \(G\) i+1 is the input-output operator of the system at the \((i + 1)\)-th iteration, and \(d\) i+1 is the bias term.

[0103] To solve for the optimal \(u\) i+1 , it is necessary to take the partial derivative of \(J\) with respect to \(u\) and set it equal to zero.

[0104] Therefore, the subterms of the objective function can be expanded separately as:

[0105]

[0106]

[0107] where \(r\) is the reference trajectory, and \(G\) i+1 is the input-output operator of the system at the \((i + 1)\)-th iteration, and \(d\) i+1 is the bias term. According to the update law of iterative learning control, take the partial derivatives of \(J_1(u)\) and \(J_2(u)\) with respect to \(u\) (denote \(u = u\) i+1 ), and the following expressions are obtained:

[0108]

[0109]

[0110] When taking the partial derivative of \(u\) in the state estimation error term, since and \(X\) k+1 usually depend on the update process of the Kalman filter and do not directly take \(u\) as the independent variable. However, in an actual system, the introduction of the state estimation error term plays a role in correcting the state estimation error, and the influence between state estimation and control input can be balanced by adjusting the corresponding weight \(P\). Then, through an additional correction term to compensate for this part of the influence. Therefore, when taking the derivative with respect to \(u\), its partial derivative term can be considered approximately zero, that is

[0111]

[0112] Add Equation (11) and Equation (13) and set it equal to 0, and after rearrangement, we get.

[0113]

[0114] After rearrangement, we get:

[0115]

[0116] Multiply both sides by :

[0117]

[0118] A compensation term related to Δd is added to the control input update law to effectively offset this system deviation that varies with the number of iterations, thereby improving the accuracy of trajectory tracking and the robustness of the system. i

[0119] The update law of the control input represents the update law of a non-repetitive time-varying system, and its expression is:

[0120]

[0121] where the operator L u , L e and the coefficient E (d,ε) are expressed as follows:

[0122]

[0123] Since both Q and R are symmetric positive definite matrices, R+(G i+1 ) T QG i+1 is a non-singular matrix.

[0124] The specific process of making the error norm decrease with the number of iterations through robust convergence analysis in step S3 is as follows:

[0125] Since the uncertainty of the state space model of the actual non-repetitive time-varying system satisfies ||ζ||≤1, the relevant operators include the following forms:

[0126]

[0127] The following inequalities are used to analyze the robustness:

[0128]

[0129]

[0130] When both of the above two inequalities are satisfied and the system has robustness, the error convergence condition of the model is satisfied.

[0131] The following is the process of proving that these two inequalities have robustness: Considering the model uncertainty ζ, from the formula y i =G i u i +d, it can be deduced that the output of the (i + 1)-th trial has the following form.

[0132]

[0133] Substituting formula (14) into the above equation, the error of the (i + 1)-th trial is as follows:

[0134] ​

[0135] Based on the formula and the update law (18), the error at the (i + 1)-th trial can be derived to obtain the following expression:

[0136]

[0137] Taking the norm on both sides, the norm of the error can be rewritten as:

[0138]

[0139] According to the properties of the norm inequality, the inequality can be obtained:

[0140]

[0141] If both formulas (15) and (16) are satisfied, the error converges, that is

[0142] ||e i+1 || < ||e i || (26)

[0143] Thus, the robust convergence is proved.

[0144] According to the above technical features, the present invention adopts an adaptive Kalman filter to improve the real-time estimation ability of the system state and enhance the anti-interference performance of the system; through an adaptive noise covariance update mechanism, the performance of the Kalman filter is optimized to adapt to the dynamic changes of the system; combined with iterative learning control, the historical error information is fully utilized to optimize the control input, accelerate the convergence of the system error, and improve the trajectory tracking accuracy.

[0145] Embodiment 2

[0146] Based on the above Embodiment 1, combined with Figures 2 - 9 , this embodiment elaborates in detail the actual application of the present invention on a precision machining platform.

[0147] The second aspect of the present invention provides an application of a trajectory tracking method for a non-repetitive time-varying system. The trajectory tracking method for the non-repetitive time-varying system is applied to a precision machining platform for machining trajectory tracking, specifically as follows: defining a non-repetitive time-varying reference trajectory on the precision machining platform; constructing an objective function for the trajectory tracking of the precision machining platform based on the adaptive Kalman filter algorithm; optimizing the objective function to iterate the adaptive Kalman filter algorithm, comparing the tracking error between the actual output and the non-repetitive time-varying reference trajectory, verifying that the error of the adaptive Kalman filter algorithm converges to the micron-level accuracy, completing the iteration of the adaptive Kalman filter algorithm, and calculating the expected reference trajectory of the system motion by using the adaptive Kalman filter algorithm after iterative convergence.

[0148] In a specific embodiment, the precision machining experimental platform for verifying the algorithm performance is as follows Figure 2 As shown, it can be seen that the precision machining experimental platform consists of a gantry Cartesian robot, an industrial camera equipped with a telecentric lens, a laser galvanometer, a two-dimensional rotary table, and a two-dimensional precision sliding table. The theoretical motion resolution of the gantry Cartesian robot on the X, Y, and Z axes is approximately 10 microns, which makes it suitable for large-scale workpiece handling and initial alignment. The industrial camera is paired with a telecentric lens and has a constant magnification at a working distance of approximately 25 cm, providing high-precision measurement for subsequent machining path planning. The laser galvanometer achieves rapid scanning and high-precision marking of the laser beam through a high-speed deflecting mirror. The two-dimensional rotary table can perform angular positioning in various scenarios.

[0149] First, it is clear that the control design objective is to perform trajectory tracking, and these transfer functions are described as follows:

[0150]

[0151] G x (z) and G y (z) represent the dynamic responses of the system in the x and y axis directions respectively. Their forms belong to the classical expressions of discrete system transfer functions. The values in the denominator determine the pole positions of the system, and the poles have a decisive impact on the dynamic performance indicators such as the stability, response speed, and overshoot of the system.

[0152] In order to be able to deeply analyze and design the system within the theoretical framework of discrete systems, a zero-order hold is adopted in a specific embodiment to convert the originally continuous-time system into a discrete-time representation. The sampling time T s = 0.01 s is selected as the sampling time. The determination of this sampling time is comprehensively considered. While ensuring that the dynamic information of the system can be accurately captured, it will not overly increase the computational complexity. By strictly following this discretization strategy, the continuous-time transfer function is efficiently and accurately mapped into the discrete-time framework, providing convenient conditions for subsequent algorithm development, simulation verification, and actual control implementation based on discrete systems.

[0153] In addition, the desired reference trajectory, as the target guidance for the system motion, is clearly defined as:

[0154]

[0155] From Figure 3It can be observed that as the number of iterations of iterative learning control (ILC) increases, the trajectory tracking error of the system gradually decreases. In the initial few trials (Trajectory 1 to Trajectory 4), there are still relatively obvious deviations in the output trajectory of the system, especially in the parts with larger trajectory curvatures. However, as the ILC iteration progresses, the control input is continuously optimized, and the error is effectively compensated. Finally, in the final trial (Final Trajectory), the output trajectory of the system almost completely coincides with the reference trajectory, verifying the high-precision trajectory tracking ability of the algorithm proposed in the present invention in non-repetitive time-varying systems (NTVSs). Figure 3 The enlarged detail part of Figure 3 further demonstrates the evolution process of the trajectory error. In the initial trial (Trajectory 1), the trajectory error is large, but as the iteration progresses, the error gradually shrinks. In the final trial (Final Trajectory), the trajectory error has been reduced to close to the order of 0.0026 mm, indicating that ILC combined with adaptive Kalman filter (AKF) can effectively adapt to the changes in system parameters and reduce the impact of uncertainties on the trajectory tracking accuracy.

[0156] The experimental results further confirm that the proposed adaptive Kalman filter iterative learning control strategy can significantly improve the trajectory tracking performance, enabling the system to maintain high stability and accuracy in the face of complex dynamic environments. Therefore, this method is particularly suitable for non-repetitive time-varying industrial application scenarios, such as precision manufacturing, robot path planning, and high-precision control of complex mechatronic systems.

[0157] In addition, the choice of the weight matrix Q plays a crucial role in determining the convergence characteristics of the iterative learning control framework. Comparative analysis of different weight values shows that increasing the magnitude of Q will significantly accelerate the reduction of the tracking error. The results show that a larger Q value will cause the mean absolute error (MAE) in successive iterations to decrease more rapidly, thus enhancing the learning efficiency of the iterative process. This observation emphasizes the importance of reasonably adjusting the weight matrix for optimizing control performance, improving the convergence speed, and enhancing the overall system stability. Figure 4 These effects were compared in detail, and the role of different Q values in error attenuation was systematically analyzed.

[0158] Figure 5 Shows the variation of the mean absolute error (MAE) during 100 iterations for different control methods (including PID control, norm-optimal iterative learning control (NO-ILC), and the iterative learning control method with adaptive Kalman filter (AKF-ILC) proposed in this paper). As can be seen from the figure, all methods show relatively large errors in the initial stage due to the influence of system uncertainties and modeling errors. However, as the number of iterations increases, the error of the iterative learning control method with adaptive Kalman filter decreases faster than other methods and finally converges to a stable and lower error level.

[0159] Figure 6 and Figure 7 The results shown illustrate the tracking performance of the system output relative to the reference trajectory in the final experiment. Among them Figure 6 depicts the output u on the x-axis x compared with the tracking performance of the reference trajectory r1, Figure 7 while presenting the output u on the y-axis y and its corresponding tracking results with the reference trajectory r2. The red solid line represents the reference trajectory, and the blue dashed line represents the output of the final experiment. The high coincidence of these curves within the time interval from 0 to 2 seconds verifies the effectiveness of the proposed control strategy in accurately tracking the target trajectory.

[0160] Figure 8 and Figure 9 The results shown illustrate the variation of the system input signal during the final experiment. Among them Figure 8 depicts the evolution of the input u on the x-axis corresponding to channel 1 x , Figure 9 while presenting the variation of the input u on the y-axis related to channel 2 y . In Figure 8 and Figure 9 , the red solid line represents the reference input, and the blue dashed line represents the actual input applied in the experiment. The high coincidence of the actual input and the reference input signal within the time interval from 0 to 2 seconds indicates that the designed control input effectively follows the desired reference trajectory, verifying the accuracy and robustness of the proposed control strategy.

[0161] This embodiment verifies the effectiveness of the present invention on the precision machining experimental platform through simulation experiments and actual tests, and demonstrates its application value in complex dynamic environments.

[0162] Embodiment 3

[0163] As Figure 10 shown, the present invention also provides a trajectory tracking system for a non-repetitive time-varying system, which is used for the trajectory tracking method of the non-repetitive time-varying system, including: a state space modeling module, an adaptive Kalman filtering module, an iterative learning controller, and an expected reference trajectory prediction module.

[0164] The state space modeling module constructs a state space model for describing the dynamic characteristics of the non-repetitive time-varying system; the adaptive Kalman filtering module uses the adaptive Kalman filtering algorithm to estimate the system state of the state space model; the iterative learning controller constructs an objective function for system trajectory tracking according to the system state; the expected reference trajectory prediction module performs optimization processing on the objective function of system trajectory tracking to obtain an optimized result of system trajectory tracking.

[0165] In the embodiments provided by the present invention, it should be understood that the disclosed systems and methods can be implemented in other ways. Those of ordinary skill in the art can understand that all or part of the steps of implementing the above method embodiments can be completed by hardware related to program instructions. The aforementioned program can be stored in a computer-readable storage medium. When the program is executed, it performs the steps including those of the above method embodiments; and the aforementioned storage medium includes: removable storage devices, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks and other various media that can store program codes.

[0166] Alternatively, if the above embodiments of the present invention are implemented in the form of software function modules and sold or used as independent products, they can also be stored in a computer-readable storage medium. Based on such an understanding, the technical solutions of the embodiments of the present invention essentially or the parts that contribute to the prior art can be embodied in the form of a software product. The computer software product is stored in a storage medium and includes several instructions to enable a computer device to execute all or part of the methods described in the various embodiments of the present invention. And the aforementioned storage medium includes: removable storage devices, ROM, RAM, magnetic disks, or optical disks and other various media that can store program codes.

[0167] Obviously, the above embodiments of the present invention are merely examples for clearly explaining the present invention, rather than limitations on the implementation manners of the present invention. The icons describing the structural and positional relationships in the drawings are only for illustrative purposes and cannot be construed as limitations on the present invention. For those of ordinary skill in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to enumerate all the implementation manners here. Any modifications, equivalent replacements, and improvements made within the spirit and principle of the present invention shall be included in the protection scope of the claims of the present invention.

Claims

1. A trajectory tracking method for a non-repetitive time-varying system, characterized in that Including the following steps: S1. Construct a state - space model for describing the dynamic characteristics of a non - repetitive time - varying system; S2. Use an adaptive Kalman filtering algorithm to estimate the system state of the state - space model; S3. Construct an objective function for system trajectory tracking based on the system state; S4. Optimize the objective function for system trajectory tracking to obtain an optimized result for system trajectory tracking.

2. The trajectory tracking method of a non-repetitive time-varying system according to claim 1, wherein In step S1, the expression of the state - space model for describing the dynamic characteristics of a non - repetitive time - varying system is: where the expression describes the dynamics of a non-repetitive time-varying system in terms of the properties of the state space, and is described by an observation equation and a state equation. is the system output of the observation equation. represents the state vector of the state equation. is the control input. and are the current noises in the state equation and the observation equation, respectively.

3. A trajectory tracking method for a non-repetitive time-varying system according to claim 2, characterized in that The specific process of step S2 is: First, calculate the prior state estimate value and the error covariance matrix at the next moment: Among them, based on the state equation, this formula regards A i , B i and Q i as known quantities, substitutes the posterior estimate values (X i (k|k), P i (k|k)) at the previous moment into the formula to obtain the prior estimate value at the current moment. represents the predicted state estimate value at time t + 1 based on the available information at time t, and P i (k + 1|k) is the predicted error covariance matrix, which is used to quantify the uncertainty in state estimation. And A i (k)P i (k|k)A i (k) T is the change in uncertainty brought about by the system state propagation. Then, use the observation information to execute the update step, calculate the Kalman gain and correct the prediction value to obtain the posterior state estimate value and the posterior covariance: K i (k) = P i (k + 1|k)C i (k) T [C i (k)P i (k + 1|k)C i (k) T + R i (k)] -1 P i (k + 1|k + 1) = [I - K i (k)C i (k)]P i (k + 1|k) Substitute the predicted covariance matrix P i (k + 1|k) and the observation matrix C i (k) into this formula to calculate the Kalman gain K i (t). The Kalman gain K i (t) determines the optimal weight between the predicted state and the actual measurement value. C i represents the observation matrix, R i represents the observation noise covariance matrix, and X i (k + 1|k + 1) represents the posterior estimate, and P i (k + 1|k + 1) represents the posterior covariance matrix.

4. A trajectory tracking method for a non-repetitive time-varying system according to claim 3, characterized in that, After each iteration of the adaptive Kalman filtering algorithm is completed, it will be based on the current tracking error e i and the observation residual to correct the covariance of the process noise and the measurement noise. After substituting the tracking error and the observation residual into the following formula, the updated Q i+1 and R i+1 are obtained as follows: Where α and β are adaptive weight parameters used to control the covariance update rate, and their value ranges are 0 < α, β < 1.

5. A trajectory tracking method for a non-repetitive time-varying system according to claim 4, characterized in that Considering the norm - optimal iterative learning control framework, the control input for each batch is obtained by optimizing a performance index function. The specific objective function described in step S3 is: Among them, the objective function J consists of a tracking error term a control variable change term and a state estimation error term . Each term represents a different control objective. u i represents the input at each step, and e i represents the tracking error.

6. A trajectory tracking method for a non-repetitive time-varying system according to claim 5, characterized in that The specific process of step S4 is: Use the objective function to optimize the control input. According to the system input - output operator and the bias term, generate an update law for the control input, and make the error norm decrease with the number of iterations through robust convergence analysis; The way to optimize the control input using the objective function is: Expand the sub - terms of the objective function respectively as: J(u)=J1(u)+J2(u)+J3(u) where r is the reference trajectory, G i+1 is the input-output operator of the system at the (i + 1)-th iteration, d i+1 is the bias term. The partial derivatives of J1(u) and J2(u) are calculated respectively according to the update law of iterative learning control, and the following expressions are obtained: When taking the partial derivative of \(u\) in the state estimation error term, since and \(X\) k+1 usually depend on the update process of the Kalman filter and do not directly take \(u\) as the independent variable. However, in an actual system, the introduction of the state estimation error term plays a role in correcting the state estimation error, and the influence between the state estimation and the control input can be balanced by adjusting the corresponding weight \(P\). Then, through the additional correction term to make up for this part of the influence. Therefore, when taking the derivative of \(u\), its partial derivative term can be considered approximately zero, that is 7. A trajectory tracking method for a non-repetitive time-varying system according to claim 6, characterized in that, The update law of the control input represents the update rule of the non - repetitive time - varying system, and its expression is: Among them, the operator L u , L e and the coefficient E (d,ε) are expressed as follows: E (d,e) = e i -Δd i Since both Q and R are symmetric positive definite matrices, R+(G i+1 ) T QG i+1 is a non-singular matrix.

8. A trajectory tracking method for a non-repetitive time-varying system according to claim 7, characterized in that The specific process of making the error norm decrease with the number of iterations through robust convergence analysis is: Since the uncertainty of the state - space model of the actual non - repetitive time - varying system satisfies ||ζ||≤1, the relevant operators include the following forms: Use the following inequalities to analyze the robustness: When both of the above two inequalities are satisfied and the robustness is achieved, the error convergence condition of the model is satisfied.

9. An application of a trajectory tracking method for a non-repetitive time-varying system, characterized in that: Applied to the machining trajectory tracking on a precision machining platform, specifically as follows: Define a non - repetitive time - varying reference trajectory on the precision machining platform; Based on the adaptive Kalman filtering algorithm, construct an objective function for the precision machining platform trajectory tracking; Optimize the objective function to iterate the adaptive Kalman filtering algorithm, compare the tracking error between the actual output and the non - repetitive time - varying reference trajectory, verify that the error of the adaptive Kalman filtering algorithm converges to the micron - level accuracy, complete the iteration of the adaptive Kalman filtering algorithm, and use the adaptive Kalman filtering algorithm after iterative convergence to calculate the expected reference trajectory of the system motion.

10. A trajectory tracking system for a non-repetitive time-varying system, which is used for the trajectory tracking method of a non-repetitive time-varying system according to any one of claims 1-8, characterized in that, Including: a state - space modeling module, an adaptive Kalman filtering module, an iterative learning controller, and an expected reference trajectory prediction module; The state space modeling module constructs a state space model for describing the dynamic characteristics of a non-repetitive time-varying system; the adaptive Kalman filter module estimates the system state of the state space model by using an adaptive Kalman filter algorithm; the iterative learning controller constructs an objective function for system trajectory tracking according to the system state; the desired reference trajectory prediction module performs an optimization process on the objective function for system trajectory tracking to obtain an optimized result of system trajectory tracking.

Images