网络线性动力系统的箭图半稳定性与结构化卡尔曼分解
Quiver Semistability and Structured Kalman Decompositions for Networked Linear Dynamical Systems
- Tokyo Metropolitan University（东京都立大学）
- The Institute of Statistical Mathematics（统计数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对网络线性时不变系统，基于箭图σ-半稳定性提出可控性与可观性新定义，构建结构化卡尔曼分解并设计相关高效判定与求解算法，为网络动力系统分析提供新方法。
AI中文摘要：
本文基于箭图表示的σ-半稳定性，为网络线性时不变（LTI）系统引入了可控性与可观性的新定义。利用King的σ-半稳定性判据，本文为网络LTI系统定义了卡尔曼分解的网络推广形式，该分解在尊重底层网络结构的同时，系统地分解了局部与互联动力学。此外，本文提出了高效算法，用于判定给定网络LTI系统的可控性与可观性，以及求解该卡尔曼型分解；还提出了高效算法，当带自环顶点的权重σ符号一致时，判定带自环的无环箭图表示的σ-半稳定性，此类箭图表示与权重源自网络LTI系统。
英文摘要：
We introduce new notions of controllability and observability for networked linear time-invariant (LTI) systems based on $σ$-semistability of quiver representations. Utilizing King's criterion for $σ$-semistability, we define a network generalization of the Kalman decomposition for networked LTI systems, which systematically decomposes the local and interconnection dynamics while respecting the underlying network structure. Furthermore, we present efficient algorithms for deciding the proposed controllability and observability of a given networked LTI system and for finding the Kalman-type decomposition. We also show efficient algorithms for deciding the $σ$-semistability of representations of acyclic quivers with self-loops if the weight $σ$ has the same sign for all vertices with self-loops. Such quiver representations and weights arise from networked LTI systems.
