论文标题
使用半限定编程层次结构在状态约束下的lyapunov函数计算 *
Computation of Lyapunov Functions under State Constraints using Semidefinite Programming Hierarchies *
论文作者
论文摘要
我们提供用于计算lyapunov函数的算法，用于一类系统，其中状态轨迹受到限制以在封闭的凸集内进化。我们考虑的动力系统包括一个微分方程，可确保域内连续演变，并确保状态轨迹始终保持在预先指定的集合中。找到这样一个系统的Lyapunov函数归结为找到满足可允许的状态限制集的某些不平等的函数。众所周知，尽管存在凸面，但在计算上还是很困难。对于圆锥约束，我们基于SIM-PLEX的简单分区提供了离散化算法，因此通过构建线性程序的层次结构（与分区中单元格的直径相关联）来解决对所需函数的搜索。我们的第二个算法是针对半代数集量身定制的，其中构建了半决赛程序的层次结构，以计算Lyapunov的功能作为一个平方的多项式。
We provide algorithms for computing a Lyapunov function for a class of systems where the state trajectories are constrained to evolve within a closed convex set. The dynamical systems that we consider comprise a differential equation which ensures continuous evolution within the domain, and a normal cone inclusion which ensures that the state trajectory remains within a prespecified set at all times. Finding a Lyapunov function for such a system boils down to finding a function which satisfies certain inequalities on the admissible set of state constraints. It is well-known that this problem, despite being convex, is computationally difficult. For conic constraints, we provide a discretization algorithm based on simplicial partitioning of a sim-plex, so that the search of desired function is addressed by constructing a hierarchy (associated with the diameter of the cells in the partition) of linear programs. Our second algorithm is tailored to semi-algebraic sets, where a hierarchy of semidefinite programs is constructed to compute Lyapunov functions as a sum-of-squares polynomial.
