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摘要: 2017年, 宋永端教授团队于Automatica发表高阶非线性系统预设时间控制的原创性工作. 不同于传统控制方法, 该策略能够人为预先设定系统收敛的时间, 同时具备出色的扰动抑制性能, 成果面世后迅速成为研究热点, 国内外学者围绕该方向已开展大量深入研究. 本文从系统稳定性相关的基本概念出发, 系统辨析有限时间、固定时间、预定时间与预设时间稳定性之间的联系与区别, 进而聚焦于预设时间控制的实现途径. 以经典的比例制导、终端约束最优控制、滑模控制等为基础, 梳理其理论渊源, 重点阐述状态变换、时间尺度变换、复合反馈及周期时滞反馈等核心设计框架. 通过比较不同方法的收敛性能, 进一步分析增强或削弱反馈信号对系统性能与工程可实现性的影响. 此外, 还讨论预设时间控制中存在的数值奇异和噪声敏感等挑战, 总结多种解决方案, 并对未来在理论融合、智能控制、跨领域应用验证等方面的发展方向进行展望. 预设时间控制在航空航天、机器人、多智能体等对动态性能与时间精度要求严苛的领域中展现出重要应用潜力, 其进一步发展将推动高可靠控制系统的工程实现.Abstract: In 2017, Professor Song Yong-Duan＇s team published an original work on predefined-time control of high-order nonlinear systems in Automatica. Unlike traditional control methods, this strategy enables the system convergence time to be artificially prespecified while exhibiting excellent disturbance rejection performance. Upon its release, it quickly became a research hotspot, prompting extensive follow-up studies worldwide. This paper begins with the fundamental concepts related to system stability, systematically analyzes the connections and distinctions among finite-time, fixed-time, predefined-time, and prescribed-time stability, and then focuses on the implementation approaches of prescribed-time control. Building on classical proportional navigation, optimal control with terminal constraint, and sliding mode control, etc., this paper traces their theoretical origins and elaborates on core design frameworks such as state transformation, time-scale transformation, composite feedback, and periodic time-delay feedback. By comparing the convergence performance of different methods, it further analyzes the impact of enhancing or weakening feedback signals on system performance and engineering feasibility. Additionally, the paper discusses the challenges in prescribed-time control, such as numerical singularities and noise sensitivity, summarizes various solutions, and prospects future development directions, including theoretical integration, intelligent control, and cross-domain application validation. Prescribed-time control demonstrates significant application potential in fields with stringent requirements for dynamic performance and temporal precision, such as aerospace, robotics, and multi-agent systems. Its further development will advance the engineering implementation of high-reliability control systems.
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Key words:
- prescribed-time control /
- stability /
- control theory
1)1 1符号$ \|{\boldsymbol{d}}(\tau)\| $定义为$ \|{\boldsymbol{d}}(\tau)\|=\sup_{0 \leq \tau \leq t_f} |{\boldsymbol{d}}(\tau)| $.2)2 2在有限时间控制相关文献中, “有限时间稳定”通常默认平衡点不仅是稳定的而且是吸引的, 即会在有限时间内收敛到零. 对于平衡点非吸引但系统状态可在有限时间内进入并保持于原点某邻域内的情况, 则多被称为“实用有限时间稳定”. 如注2所述, 相较于数学描述, 纯文字表述易引发歧义, 因此在阅读文献时需结合具体语境仔细辨析.3)3 3为描述简洁, 后文中提到的“稳定”均指“全局一致渐近稳定”. -
图 3 有限时间、固定时间、预定时间、预设时间稳定的关系
Fig. 3 Relationships among finite-time, fixed-time, predefined-time, and prescribed-time stability
图 6 周期时滞反馈控制下的增益和输出信号示意图
Fig. 6 Schematic diagram of control gain and output under periodic delayed feedback control
图 7 有限时间控制下的系统状态与控制输入轨迹
Fig. 7 System state and control input trajectories under finite-time control
图 10 预设时间控制下的系统状态与控制输入轨迹
Fig. 10 System state and control input trajectories under prescribed-time control
图 8 固定时间控制下的系统状态与控制输入轨迹
Fig. 8 System state and control input trajectories under fixed-time control
图 9 预定时间控制下的系统状态与控制输入轨迹
Fig. 9 System state and control input trajectories under predefined-time control
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