Keywords: iterative learning control; sliding mode control; predefined-time convergence; time-varying sliding mode surface; robotic arm

This paper will focus on second-order nonlinear systems with repetitive motion, and propose a PD-type closed-loop iterative learning control strategy based on the predefined-time convergence sliding mode surface, aiming to show that the controlled system under any initial value can not only follow a local trajectory for accurate tracking, but also predetermine the local convergence time, Ts, in advance. The main innovations and contribution of this study can be summarized as follows:

• Provides Lyapunov stability criterion for the stability of nonlinear systems within a predefined time and describes the theoretical proof under the given conditions.

• Presents a design for a time-varying sliding mode surface with predefined time convergence characteristics in which the convergence time of the trajectory tracking error located in the sliding mode surface can be preset, bringing the advantage that the convergence time is not affected by the controlling constraints or the initial value of the iteration.

• Converts the trajectory tracking control problem, where the initial value of the trajectory tracking error is not zero, into a sliding mode surface tracking control problem in which the initial value of the sliding mode surface being zero. Establishes a bridge between the iterative learning control theory with an arbitrary iterative initial value and the same iterative initial value.

• The iterative learning control strategy not only solves the problem of arbitrary iterative initial value suppression and simplifies the theoretical proof of the convergence of iterative learning, it also achieves the engineering application of the system output, accurately tracking the desired trajectory within a preset local time.

The remainder of this paper is as follows. Section 2 presents the control problem formulation and also describes several lemmas for iterative learning convergence proof. Section 3 proposes an arbitrary initial value suppression strategy based on the predefined time convergence sliding mode control principle, mentioning its principles. The Lyapunov stability criterion for predefined time convergence of nonlinear systems is given, and a design for a sliding mode surface with the character of predefined time convergence and initial value of zero is presented. The main results of iterative convergence are discussed in Section 4, which also demonstrates the predefined time convergence condition for a PD-type ILC. In Section 5, the effectiveness of the proposed nonlinear control strategy is illustrated by simulations for a robotic system, the results of which are briefly explained. Finally, Section 6 presents the conclusions.

摘要：针对任意迭代初值下二阶非线性系统实现局部时间精确跟踪控制中，局部时间无法预设的实际工程问题，提出一种可预设时间收敛的迭代学习控制策略。构建了零初值时变滑模面，理论证明了位于滑模面内的轨迹跟踪误差可在预设时间内收敛到零；接着将任意迭代初值下的轨迹跟踪迭代控制问题，转换成迭代初值为零的时变滑模面跟踪迭代控制问题，设计关于时变滑模面的PD型闭环迭代学习控制器，使轨迹跟踪误差经有限次迭代学习后收敛并稳定在滑模面内，结合时变滑模面的预设时间收敛特性，达到系统输出在预设时间区间内精确跟踪期望轨迹的控制目的。重复运动机械臂的轨迹跟踪控制数值仿真，验证了本文方法的有效性和对外部干扰的鲁棒性。

关键字：迭代学习控制,滑模控制,预设时间收敛,时变滑模面,机械臂

本文的创新点主要为：给出一种非线性系统在预设时间内稳定的Lyapunov稳定判据，并对其进行了理论证明；构建了一种具有预设时间收敛特性的时变滑模面，使位于滑模面内的轨迹跟踪误差收敛时间可提前预设，且该收敛时间不受控制参数和迭代初值影响；将轨迹跟踪误差初值不为零的轨迹跟踪控制问题转换为滑模面初值恒为零的滑模面跟踪控制问题,建立了任意迭代初值与相同迭代初值的迭代学习控制理论连接桥梁,提出了一种基于预设时间收敛滑模面的PD型闭环迭代学习控制策略，不仅解决了任意迭代初值抑制问题,简化了迭代学习收敛性的理论证明，还实现了系统输出在预设局部时间内精确跟踪期望轨迹的工程应用目的。

本文已经发表在Mathematics 2023, 11(1), 56; https://doi.org/10.3390/math11010056，欢迎引用

MDPI and ACS Style

Yin, C.-W.; Riaz, S.; Zaman, H.; Ullah, N.; Blazek, V.; Prokop, L.; Misak, S. A Novel Predefined Time PD-Type ILC Paradigm for Nonlinear Systems. Mathematics 2023, 11, 56. https://doi.org/10.3390/math11010056

AMA Style

Yin C-W, Riaz S, Zaman H, Ullah N, Blazek V, Prokop L, Misak S. A Novel Predefined Time PD-Type ILC Paradigm for Nonlinear Systems. Mathematics. 2023; 11(1):56. https://doi.org/10.3390/math11010056

Chicago/Turabian Style

Yin, Chun-Wu, Saleem Riaz, Haider Zaman, Nasim Ullah, Vojtech Blazek, Lukas Prokop, and Stanislav Misak. 2023. "A Novel Predefined Time PD-Type ILC Paradigm for Nonlinear Systems" Mathematics 11, no. 1: 56. https://doi.org/10.3390/math11010056