发表机构
UCLouvain（天主教鲁汶大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种迭代图提升算法，通过图论分析优化条件并识别瓶颈节点，自动细化路径完备图以逼近切换线性系统的联合谱半径，实验证明优于现有方法。
AI 中文摘要
任意切换下切换线性系统的稳定性是控制理论中的一个基本问题，与联合谱半径（JSR）密切相关，JSR 刻画了系统轨迹的最坏情况增长率。在本文中，我们为逼近 JSR 的路径完备方法做出贡献。该框架使用标记有向图（称为路径完备图）构建代数稳定性证书，这些证书可通过相关的优化问题计算。我们提出一种迭代算法，以高效且简洁的方式细化路径完备图。该算法依赖于对底层优化问题最优性条件的图论分析。特别地，我们推导出一个充分条件，在该条件下，给定路径完备图可达到精确的 JSR。当该条件不满足时，我们通过分析由活动约束诱导的图来识别瓶颈节点。然后利用这些信息通过局部图提升（节点分裂）来细化路径完备图，并重复该过程。数值实验证明了所提方法的有效性和可扩展性，在所有测试的具有挑战性的实例上均优于现有方法。
英文摘要
Stability of switched linear systems under arbitrary switching is a fundamental problem in control theory, closely related to the joint spectral radius (JSR), which characterizes the worst-case growth rate of system trajectories. In this paper, we contribute to the path-complete approach for approximating the JSR. This framework constructs algebraic stability certificates using labeled directed graphs, known as path-complete graphs. These certificates can be computed via an associated optimization problem. We propose an iterative algorithm that refines path-complete graphs in an efficient and parsimonious manner. The algorithm relies on a graph-theoretic analysis of the optimality conditions of the underlying optimization problem. In particular, we derive a sufficient condition under which the exact JSR is attained by a given path-complete graph. When this condition is not satisfied, we identify bottleneck nodes by analyzing the graph induced by the active constraints. We then use this information to refine the path-complete graph via local graph lifting (node splitting), and repeat the procedure. Numerical experiments demonstrate the effectiveness and scalability of the proposed approach, outperforming state-of-the-art methods on all challenging instances tested.
CommentsThis paper is an extended version of a paper accepted for CDC 2026
