引用本文:廖玮,梁涛涛,魏小辉.带有运行成本函数的Hamilton-Jacobi可达性分析[J].控制理论与应用,2022,39(6):986~994.[点击复制]
LIAO Wei,LIANG Tao-tao,WEI Xiao-hui.Hamilton-Jacobi reachability analysis with running cost function[J].Control Theory and Technology,2022,39(6):986~994.[点击复制]
带有运行成本函数的Hamilton-Jacobi可达性分析
Hamilton-Jacobi reachability analysis with running cost function
摘要点击 2002  全文点击 587  投稿时间：2021-05-27  修订日期：2022-03-21
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DOI编号  10.7641/CTA.2021.10455
  2022,39(6):986-994
中文关键词  可达集  水平集方法  Hamilton-Jacobi方程  最优控制
英文关键词  reachable set  level set method  Hamilton-Jacobi equation  optimal control
基金项目  国家自然科学基金项目(51905264), 国防科技卓越青年科学基金项目(2018–JCJQ–ZQ–053), 中央高校基本科研业务费专项资金项目(NP2018001)资助.
作者单位E-mail
廖玮 南京航空航天大学 liaowei_nuaa@hotmail.com 
梁涛涛 南京航空航天大学  
魏小辉* 南京航空航天大学 wei_xiaohui@nuaa.edu.cn 
中文摘要
      水平集方法将可达集表示为Hamilton-Jacobi方程解的零水平集, 保存多个不同时间范围的可达集则需要保 存Hamilton-Jacobi方程在多个时刻的解, 这不仅需要消耗大量的存储空间还为控制律的设计造成了困难. 针对这些 局限性, 提出了一种改进的基于Hamilton-Jacobi方程的可达集表示方法. 该方法在Hamilton-Jacobi方程中加入了一 项运行成本函数, 可以用同一个时刻的解的多个非零水平集表示多个不同时间范围的可达集, 极大地节省了存储空 间并为控制律的设计提供了便利. 为了求解所构造的带有运行成本函数的Hamilton-Jacobi方程, 采用了一种基于递 归和插值的方法. 最后, 通过一些数值算例验证了所提出的方法的精确性、在存储空间方面的优越性以及设计的控 制律的有效性.
英文摘要
      The level set method represents the reachable set as the zero level set of the solution to a Hamilton-Jacobi equation. Saving reachable sets with different time horizons requires saving the Hamilton-Jacobi equation solution at different moments. This not only consumes a lot of storage space but also creates difficulties in the design of the control law. To address these limitations, an improved representation of the reachable set based on the Hamilton-Jacobi equation is proposed. The method adds a running cost function to the Hamilton-Jacobi equation so that multiple reachable sets of different time horizons can be represented by multiple non-zero level sets of solutions at the same moment, which greatly saves storage space and facilitates the design of control laws. To solve the constructed Hamilton-Jacobi equation with a running cost function, a recursive and interpolation-based method is used. Finally, some numerical examples are provided to verify not only the accuracy of our method and its superiority in terms of storage space, but also the effectiveness of the designed control law.