数据驱动的吸引域估计：RKHS上的Zubov–Koopman算子及其谱
Data-Driven Domain of Attraction Estimation via Convergent Koopman-Zubov Approximation
浏览论文内容
中文总结 AI 辅助
本文提出在RKHS上构造Zubov–Koopman算子，实现了吸引域的有效核估计，误差随样本量缩小，数值示例验证了方法的有效性。
中文摘要 AI 辅助
平衡点周围有限大小吸引域（DOA）的存在是非线性动力学的体现，但其计算因需搜索Zubov函数而困难。从非线性系统的算子理论视角，Zubov–Koopman算子概念已被提出，但缺乏理想谱特性使其在Zubov函数估计中的应用难以得到理论保证。本文将Zubov–Koopman算子定义在常数函数空间与再生核希尔伯特空间（RKHS）的直和上，即线性函数空间与Sobolev–Hilbert空间的张量积。通过该构造，算子具有唯一特征值1，其特征函数为表征DOA的Zubov函数，而RKHS上的其余谱完全局限于原点。这一新的RKHS公式化方法允许高效的基于核的估计，其误差至多为扇形有界，且随样本量增大而缩小。数值示例验证了所提方法的有效性。
英文摘要
The computation of a domain of attraction (DOA) around an equilibrium point is a key issue in nonlinear stability analysis, which boils down to the difficult problem of searching for a Zubov function. With an operator-theoretical viewpoint of nonlinear systems, the concept of Zubov--Koopman operator has been introduced. However, due to the lack of convergence guarantee on the infinite-times action of Zubov--Koopman operator, the Zubov function estimate is unamenable to a theoretical bound under data-based learning errors. In this paper, considering a reproducing kernel Hilbert space (RKHS) with a linear--radial product kernel, the operator is proved to have a spectrum inside the unit circle. Hence, by augmenting this RKHS with constant-valued functions, the Zubov function that characterizes the DOA is obtained as the unique invariant element under the operator's action. This new RKHS formulation allows an efficient kernel-based estimation, which has an at most sectorially bounded error that scales down with the sample size. The proposed approach is tested with numerical examples, showing high accuracy of on-DOA/off-DOA classification of states, with two order-of-magnitude faster computation than neural networks.
