Hara et al. derive bounds for robust instability radius in uncertain networks

A new paper from Shinji Hara, Yutaka Hori, Tetsuya Iwasaki, Chung-Yao Kao, and Sei Zhen Khong tackles a fundamental question in networked control: how much uncertainty can destabilize an already-unstable system? The work, posted on arXiv (2608.18561), focuses on multi-agent systems composed of identical single-input-single-output agents whose dynamics are nominally unstable. Each agent is subject to heterogeneous perturbations, and the network is modeled as a feedback interconnection of a diagonal uncertainty block, the nominal agents, and a static interconnection matrix. Assuming the nominal network is unstable, the authors define the robust instability radius (RIR) as the smallest norm of a stable uncertainty that actually stabilizes the entire network.

The core contribution is a set of conditions for network stability and tight upper and lower bounds on the RIR. In the special case where the network connectivity matrix is rank one and all diagonal entries share the same sign or are zero, the team proves that the RIR can be exactly characterized using a small-gain argument—a clean, classical result for an otherwise complex problem. This theoretical advance is relevant for any distributed control setting where uncertainty is unavoidable, from power grid stabilization to coordinated motion of robotic swarms. By quantifying the distance to stability, the RIR gives engineers a rigorous safety margin and helps predict when adding uncertainty to a nominally unstable system will cause an unexpected transition to stable behavior.