论文标题
基于Koopman的神经Lyapunov功能
Koopman-Based Neural Lyapunov Functions for General Attractors
论文作者
论文摘要
在过去的十年中，Koopman光谱理论已成为动态系统分析和控制的强大工具。在本文中，我们展示了如何利用使用神经网络估算Koopman不变子空间的最新数据驱动技术，以提取基础系统的Lyapunov证书。在我们的工作中，我们特别关注具有极限周期的系统，仅仅是一个孤立的平衡点，并使用Koopman eigenfunctions有效地将候选lyapunov函数参数化，以在某些（未知）吸引者动力学下构建前向不变的集合。此外，当动力学是多项式的，并且当神经网络被多项式替换为我们方法中功能近似器的选择时，人们可以进一步利用方形计划和/或非线性程序来产生可证明正确的Lyapunov证书。在这种多项式情况下，与直接制定和解决方案总和优化问题相比，我们基于Koopman的构建Lyapunov函数的方法使用的决策变量明显更少。
Koopman spectral theory has grown in the past decade as a powerful tool for dynamical systems analysis and control. In this paper, we show how recent data-driven techniques for estimating Koopman-Invariant subspaces with neural networks can be leveraged to extract Lyapunov certificates for the underlying system. In our work, we specifically focus on systems with a limit-cycle, beyond just an isolated equilibrium point, and use Koopman eigenfunctions to efficiently parameterize candidate Lyapunov functions to construct forward-invariant sets under some (unknown) attractor dynamics. Additionally, when the dynamics are polynomial and when neural networks are replaced by polynomials as a choice of function approximators in our approach, one can further leverage Sum-of-Squares programs and/or nonlinear programs to yield provably correct Lyapunov certificates. In such a polynomial case, our Koopman-based approach for constructing Lyapunov functions uses significantly fewer decision variables compared to directly formulating and solving a Sum-of-Squares optimization problem.
