计算对稀疏对角不确定性的鲁棒性

中文摘要 AI 辅助

最近提出了一种新的鲁棒性度量$\nu$，作为结构化奇异值$\mu$的替代，以更好地捕捉对稀疏对角不确定性的鲁棒性，但其计算一直是一个开放问题。在本文中，我们证明了计算$\nu$等价于最大化非负矩阵乘积的谱半径。这一等价性使我们能够将谱半径最大化的现有结果转移到$\nu$上，包括一个精细的上界以及边界重合的条件。然后，我们提供了重构和结构结果，使得一种算法能够使用全局求解器解决非平凡问题的非凸优化问题。这也使得识别系统中最脆弱的部分成为可能。我们的结果尚未提供计算$\nu$的可扩展解决方案，但它们是朝着可计算性和可解释性迈出的重要一步。

英文摘要

A new robustness metric $ν$ was recently proposed as a substitute for the structured singular value $μ$ to better capture robustness to sparse diagonal uncertainty, but its computation has remained an open problem. In this paper, we show that computing $ν$ is equivalent to maximizing the spectral radius of a nonnegative matrix product. This equivalence allows us to transfer existing results on spectral-radius maximization to $ν$, including a refined upper bound and conditions under which the bounds coincide. We then provide reformulations and structural results that enable an algorithm to solve the nonconvex optimization problem for nontrivial problems using global solvers. This also enables identification of the most fragile parts of the system. Our results do not yet provide a scalable solution for computing $ν$, but they are an important step toward computability and interpretability.

发表机构

• Lund University（隆德大学）

机构由 AI 辅助整理，请以论文原文为准。