执行器约束下基于轨迹学习的核正则化最优迭代学习控制

doi:10.3969/j.issn.1004-132X.2025.10.014

摘要/Abstract

摘要：

针对非重复性轨迹跟踪和执行器可能超限的问题，提出了一种基于先前轨迹学习的核正则化最优迭代学习控制算法（KROILC），在迭代过程中利用输入输出的测量值，使用基于核的正则化方法估计系统的脉冲响应，展示了脉冲响应估计领域几种常用核的零均值高斯过程实现，估计得到的脉冲响应被应用于最优迭代学习控制器。通过目标函数加权实现对执行器的约束，迭代过程中参考轨迹变化后的初始前馈力通过轨迹学习得到。在直流无刷电机上的实验验证结果表明，所提出的算法能够在执行器约束下实现非重复性轨迹的全轨迹和稳定段的最优跟踪性能。

关键词: 执行器约束, 数据驱动, 非重复性轨迹, 轨迹学习, 核正则化, 迭代学习控制

Abstract:

To address the issues of non-repetitive trajectories tracking and potential actuator saturation， a kernel regularization optimal iterative learning control （KROILC） algorithm was proposed. The kernel-based regularization method was used to estimate the system's impulse response from input-output data. Several zero-mean Gaussian process kernels were demonstrated for this purpose. The estimated impluse response was applied to the controller， and actuator constraints were weighted in the objective function. Initial feedforward input after trajectory changes was learned iteratively. Experimental results on a brushless DC motor show that the proposed algorithm achieves optimal tracking for non-repetitive trajectories while maintaining actuator stability.

Key words: actuator constraint, data-driven, non-repetitive trajectory, trajectory learning, kernel regularization, iterative learning control

中图分类号:

TP273

杨亮亮, 陈泓, 鲁文其. 执行器约束下基于轨迹学习的核正则化最优迭代学习控制[J]. 中国机械工程, 2025, 36(10): 2274-2283.

Liangliang YANG, Hong CHEN, Wenqi LU. Kernel Regularization Optimal Iterative Learning Control Based on Trajectory Learning under Actuator Constraints[J]. China Mechanical Engineering, 2025, 36(10): 2274-2283.

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链接本文: https://www.cmemo.org.cn/CN/10.3969/j.issn.1004-132X.2025.10.014

https://www.cmemo.org.cn/CN/Y2025/V36/I10/2274

图/表 20

图1 伺服系统控制框图

Fig.1 Control block diagram of the servo system

图2 迭代学习控制框图

Fig.2 Block diagram of ILC

图3 DC核的零均值高斯过程实现

Fig.3 Zero-mean GPR with DC kernel

图4 TC核的零均值高斯过程实现

Fig.4 Zero-mean GPR with TC kernel

图5 SS核的零均值高斯过程实现

Fig.5 Zero-mean GPR with SS kernel

图6 四阶S型点到点轨迹

Fig.6 Fourth-order S-curve point-to-point trajectory

图7 仿真对象伯德图

Fig.7 Bode plot of simulation model

表 1 辨识效果对比

Tab.1 Comparison of identification performance

	DDOILC	KROILC
	DDOILC	DC	TC	SS
MSE	$2.47 × 10 - 13$	$1.75 × 10 - 17$	$8.14 × 10 - 17$	$2.28 × 10 - 16$
F_IT	$91.24$	$99.94$	$99.85$	$99.73$

表 1 辨识效果对比

Tab.1 Comparison of identification performance

	DDOILC	KROILC
	DDOILC	DC	TC	SS
MSE	$2.47 × 10 - 13$	$1.75 × 10 - 17$	$8.14 × 10 - 17$	$2.28 × 10 - 16$
F_IT	$91.24$	$99.94$	$99.85$	$99.73$

图 8 参考轨迹1和参考轨迹2

Fig.8 Reference trajectory 1 and reference trajectory 2

表2 轨迹参数变化程度

Tab.2 Degree of trajectory parameter variation

	$r m a x$	v_max	a_max
轨迹1	1000°	2590.7°/s	43177.9°/s²
轨迹2	1800°	4663.2°/s	77720.2°/s²
变化率	80%	80%	80%

表2 轨迹参数变化程度

Tab.2 Degree of trajectory parameter variation

	$r m a x$	v_max	a_max
轨迹1	1000°	2590.7°/s	43177.9°/s²
轨迹2	1800°	4663.2°/s	77720.2°/s²
变化率	80%	80%	80%

表3 η = 2×10-7时不同 ω 下的最大控制信号

Tab.3 The maximum control signal under different ω when η = 2×10-7

前馈信号加权系数 $ω$	最大控制信号/ $(m N ⋅ m)$
$2 × 10 - 11$	46.11
$2 × 10 - 8$	45.85
$2 × 10 - 7$	42.88

表3 η = 2×10-7时不同 ω 下的最大控制信号

Tab.3 The maximum control signal under different ω when η = 2×10-7

前馈信号加权系数 $ω$	最大控制信号/ $(m N ⋅ m)$
$2 × 10 - 11$	46.11
$2 × 10 - 8$	45.85
$2 × 10 - 7$	42.88

图9 η = 2×10-7时不同 ω 对误差二范数的影响

Fig.9 The influence of different ω on the 2-norm of the error when η = 2×10-7

图10 ω = 2×10-11、 η = 2×10-7时全轨迹迭代误差曲线

Fig.10 Full trajectory iterative error curve when ω = 2×10-11， η = 2×10-7

图11 非重复性轨迹仿真中三种算法的误差二范数

Fig.11 The 2-norm of the error for three algorithms in non-repetitive trajectory simulation

图12 无刷直流电机实验平台

Fig.12 Brushless direct current motor experimental platform

图13 控制方案结构框图

Fig.13 Block diagram of the control scheme

表4 η = 1.6×10-7时不同 ω 下的最大控制信号

Tab.4 The maximum control signal under different ω when η = 1.6×10-7

前馈信号加权系数 $ω$	最大控制信号/（ $m N ⋅ m$ ）
$2 × 10 - 8$	48.56
$4 × 10 - 8$	46.16
$8 × 10 - 8$	42.10

表4 η = 1.6×10-7时不同 ω 下的最大控制信号

Tab.4 The maximum control signal under different ω when η = 1.6×10-7

前馈信号加权系数 $ω$	最大控制信号/（ $m N ⋅ m$ ）
$2 × 10 - 8$	48.56
$4 × 10 - 8$	46.16
$8 × 10 - 8$	42.10

图14 η = 1.6×10-7时，不同 ω 对误差二范数的影响曲线

Fig.14 The influence curve of different ω on the 2-norm of the error when η = 1.6×10-7

图15 ω = 2×10-8、 η = 1.6×10-7时的全轨迹迭代误差

Fig.15 Full trajectory iterative error when ω = 2×10-8， η = 1.6×10-7

图16 非重复轨迹实验中三种算法的误差二范数

Fig.16 The 2-norm of the error for three algorithms in non-repetitive trajectory experiment

参考文献 20

[1]	GOUBEJ M， MEEUSEN S， MOOREN N， et al. Iterative Learning Control in High-performance Motion Systems：from Theory to Implementation［C］∥2019 24th IEEE International Conference on Emerging Technologies and Factory Automation （ETFA）. Zaragoza， 2019：851-856.
[2]	OOMEN T， ROJAS C R. Sparse Iterative Learning Control with Application to a Wafer Stage：Achieving Performance， Resource Efficiency， and Task Flexibility［J］. Mechatronics， 2017， 47：134-147.
[3]	BOEREN F， BAREJA A， KOK T， et al. Frequency-domain ILC Approach for Repeating and Varying Tasks：with Application to Semiconductor Bonding Equipment［J］. IEEE/ASME Transactions on Mechatronics， 2016， 21（6）：2716-2727.
[4]	GU P， TIAN S. P-type Iterative Learning Control with Initial State Learning for One-sided Lipschitz Nonlinear Systems［J］. International Journal of Control， Automation and Systems， 2019， 17：2203-2210.
[5]	FREEMAN C T. Newton-method Based Iterative Learning Control for Robot-assisted Rehabilitation Using FES［J］. Mechatronics， 2014， 24（8）：934-943.
[6]	FIORENTINO A， CERETTI E， FERITI G C， et al. Improving Accuracy in Aluminum Incremental Sheet Forming of Complex Geometries Using Iterative Learning Control［J］. Key Engineering Materials， 2015， 651：1096-1102.
[7]	BALTA E C， BARTON K， TILBURY D M， et al. Learning-based Repetitive Precision Motion Control with Mismatch Compensation［C］∥2021 60th IEEE Conference on Decision and Control （CDC）. Austin， TX， 2021：3605-3610.
[8]	van de WIJDEVEN J， BOSGRA O H. Using Basis Functions in Iterative Learning Control：Analysis and Design Theory［J］. International Journal of Control， 2010， 83（4）：661-675.
[9]	PILLONETTO G， de NICOLAO G. A New Kernel-based Approach for Linear System Identification［J］. Automatica， 2010， 46（1）：81-93.
[10]	BLANKEN L， OOMEN T. Kernel-based Identification of Non-causal Systems with Application to Inverse Model Control［J］. Automatica， 2020， 114：108830.
[11]	PILLONETTO G， DINUZZO F， CHEN T， et al. Kernel Methods in System Identification， Machine Learning and Function Estimation：a Survey［J］. Automatica， 2014， 50（3）：657-682.
[12]	CHEN T， OHLSSON H， LJUNG L. On the Estimation of Transfer Functions， Regularizations and Gaussian Processes—Revisited［J］. Automatica， 2012， 48（8）：1525-1535.
[13]	LJUNG L， CHEN T， MU B. A Shift in Paradigm for System Identification［J］. International Journal of Control， 2020， 93（2）：173-180.
[14]	CHEN T. On Kernel Design for Regularized LTI System Identification［J］. Automatica， 2018， 90：109-122.
[15]	THORPE A J， NEARY C， DJEUMOU F， et al. Physics-informed Kernel Embeddings：Integrating Prior System Knowledge with Data-driven Control［C］∥2024 American Control Conference （ACC）. Toronto， 2024：3130-3137.
[16]	YU X， FANG X， MU B， et al. Kernel-based Regularized Iterative Learning Control of Repetitive Linear Time-varying Systems［J］. Automatica， 2023， 154：111047.
[17]	JANSSENS P， PIPELEERS G， SWEVERS J. A Data-driven Constrained Norm-optimal Iterative Learning Control Framework for LTI Systems［J］. IEEE Transactions on Control Systems Technology， 2012， 21（2）：546-551.
[18]	杨亮亮，袁锐，史伟民，等. 基于数据驱动的自适应最优迭代学习控制研究［J］. 机械工程学报， 2021， 57（17）：207-216.
	YANG Liangliang， YUAN Rui， SHI Weimin， et al. Research on Data Driven Adaptive Optimal Iterative Learning Control［J］. Journal of Mechanical Engineering， 2021， 57（17）：207-216.
[19]	MURPHY K P. Machine Learning：a Probabilistic Perspective ［M］. Cambridge， MA：MIT Press， 2012.
[20]	BOLDER J， OOMEN T， KOEKEBAKKER S， et al. Using Iterative Learning Control with Basis Functions to Compensate Medium Deformation in a Wide-format Inkjet Printer［J］. Mechatronics， 2014， 24（8）：944-953.