基于多李雅普诺夫函数的一般非线性系统渐近镇定

杨可馨; 李永强; 侯忠生; 冯宇

doi:10.16383/j.aas.c240309

基于多李雅普诺夫函数的一般非线性系统渐近镇定

doi: 10.16383/j.aas.c240309 cstr: 32138.14.j.aas.c240309

1.
浙江工业大学信息工程学院杭州 310000
2.
青岛大学自动化学院青岛 266071

基金项目: 国家自然科学基金(62073294, 62373206, U2341216)资助

详细信息

作者简介:
杨可馨：浙江工业大学信息工程学院硕士研究生. 主要研究方向为非线性控制. E-mail: 211122030042@zjut.edu.cn

李永强：浙江工业大学信息工程学院副教授. 2014年获得北京交通大学控制理论与控制工程专业博士学位. 主要研究方向为非线性控制, 最优控制, 机器人控制和强化学习. 本文通信作者. E-mail: yqli@zjut.edu.cn

侯忠生：青岛大学自动化学院首席教授. 1994年获得东北大学博士学位. 主要研究方向为无模型自适应控制, 数据驱动控制, 学习控制和智能交通系统. E-mail: zshou@qdu.edu.cn

冯宇：浙江工业大学信息工程学院教授. 2011年获得法国南特矿业学院博士学位. 主要研究方向为网络化控制系统, 不确定系统的鲁棒分析与控制, 博弈论与机器学习在决策问题中的应用. E-mail: yfeng@zjut.edu.cn

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出版历程
- 收稿日期: 2024-05-31
- 录用日期: 2024-11-06
- 网络出版日期: 2024-11-26

Asymptotically Stabilization for General Nonlinear Systems Based on Multiple Lyapunov Functions

1.
College of Information Engineering, Zhejiang University of Technology, Hangzhou 310000
2.
College of Automation, Qingdao University, Qingdao 266071

Funds: Supported by National Natural Science Foundation of China (62073294, 62373206, U2341216)

More Information

Author Bio:
YANG Ke-Xin　Master student at the College of Information Engineering, Zhejiang University of Technology. Her main research interest is nonlinear control

LI Yong-Qiang　Associate professor at the College of Information Engineering, Zhejiang University of Technology. He received his Ph.D. degree in control theory and control engineering from Beijing Jiaotong University in 2014. His research interest covers nonlinear control, optimal control, robotic control, and reinforcement learning. Corresponding author of this paper

HOU Zhong-Sheng　Chair professor at the College of Automation, Qingdao University. He received his Ph.D. degree from Northeastern University in 1994. His research interest covers model-free adaptive control, data-driven control, learning control, and intelligent traffic systems

FENG Yu　Professor at the College of Information Engineering, Zhejiang University of Technology. He received his Ph.D. degree from École nationale supérieure des mines de Nantes in 2011. His research interest covers networked control systems, robust analysis and control for uncertainty systems, and applications of game theory and machine learning in decision-making

摘要

摘要: 针对离散时间非线性系统, 提出一种基于多李雅普诺夫(Lyapunov)函数的控制器设计方法. 该方法不仅能够保证闭环系统稳定性, 还能够扩大闭环吸引域(Domain of attraction, DOA). 首先, 给出基于多Lyapunov函数下系统渐近稳定的充分条件. 结果表明, 由多个Lyapunov函数的负定不变集构成的并集是一个稳定的控制集合, 其从控制空间到状态空间的投影是闭环DOA的估计. 随后, 使用区间分析算法求解集合的内近似估计, 基于此算法可以求解多Lyapunov函数的负定不变集的近似值和闭环DOA的估计值, 并给出相应控制器的设计方法. 最后, 通过仿真算例验证本文方法的有效性.
- 非线性系统 /
- 多李雅普诺夫函数 /
- 稳定性 /
- 吸引域 /
- 区间分析
Abstract: In this paper, a controller design method based on multiple Lyapunov functions is proposed for discrete-time nonlinear systems. The proposed method can not only ensure the stabilization of the closed-loop system, but also enlarge the closed-loop domain of attraction (DOA). Firstly, the sufficient conditions for the asymptotically stabilization of the system based on multiple Lyapunov functions are given. It is shown that the union of negative-definite-invariant sets of multiple Lyapunov functions is a stable control set, and its projection from the control space to the state space is an estimate of the closed-loop DOA. Then, the interval analysis algorithm is used to solve the inner approximate extimation of the set. Based on this algorithm, the negative-definite-invariant sets of multiple Lyapunov functions can be approximated and the closed-loop DOA can be estimated, the design method of the corresponding controller is also given. Finally, the simulation example is given to verify the effectiveness of the proposed method.
- Nonlinear systems /
- multiple Lyapunov functions /
- stabilization /
- domain of attraction (DOA) /
- interval analysis

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图 1 引理1渐近稳定的说明

Fig. 1 The illumination of asymptotically stabilization of lemma 1

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图 2 负定集取并集的说明

Fig. 2 Description of the union set of negative-definite sets

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图 3 单个Lyapunov函数生成的近似负定集和负定不变集

Fig. 3 The approximate negative-define set and negative-define-invariant set for a given Lyapunov function

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图 4 粒子群算法寻找最优Lyapunov函数

Fig. 4 Particle swarm optimization algorithm for finding the optimal Lyapunov function

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图 5 不同数量Lyapunov函数负定不变集和闭环DOA对比

Fig. 5 Comparison of negative-definite-invariant sets and closed-loop DOA for different numbers of Lyapunov functions

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图 6 多Lyapunov函数的近似负定集和负定不变集以及收敛轨迹

Fig. 6 The approximate negative-definite sets and negative-definite-invariant sets of multiple Lyapunov functions and convergence trajectories

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