Kernel Regularization Optimal Iterative Learning Control Based on Trajectory Learning under Actuator Constraints

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

Abstract

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

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

CLC Number:

TP273

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.

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.

Figures/Tables 20

Fig.1 Control block diagram of the servo system

Fig.2 Block diagram of ILC

Fig.3 Zero-mean GPR with DC kernel

Fig.4 Zero-mean GPR with TC kernel

Fig.5 Zero-mean GPR with SS kernel

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

Fig.7 Bode plot of simulation model

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	$91.24$	$99.94$	$99.85$	$99.73$

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	$91.24$	$99.94$	$99.85$	$99.73$

Fig.8 Reference trajectory 1 and reference trajectory 2

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%

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%

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

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

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

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

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

Fig.12 Brushless direct current motor experimental platform

Fig.13 Block diagram of the control scheme

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

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

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

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

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

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