带有运行成本函数的Hamilton-Jacobi可达性分析

Hamilton-Jacobi reachability analysis with running cost function

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.

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.