On the Internal Stability of Diffusively Coupled Multi-Agent Systems and the Dangers of Cancel Culture

TL;DR

This paper investigates the limitations of diffusive control in stabilizing multi-agent systems with unstable dynamics, revealing intrinsic instability issues that affect distributed control protocols under disturbances.

Contribution

It establishes a fundamental condition showing diffusive controllers cannot stabilize agents with shared unstable dynamics, highlighting intrinsic cancellation effects.

Findings

Diffusive controllers cannot stabilize agents with common unstable dynamics.

Intrinsic cancellations of unstable dynamics occur even with static controllers.

Internal instability explains issues in distributed control under noise and disturbances.

Abstract

We study internal stability in the context of diffusively-coupled control architectures, common in multi-agent systems (i.e. the celebrated consensus protocol), for linear time-invariant agents. We derive a condition under which the system can not be stabilized by any controller from that class. In the finite-dimensional case the condition states that diffusive controllers cannot stabilize agents that share common unstable dynamics, directions included. This class always contains the group of homogeneous unstable agents, like integrators. We argue that the underlying reason is intrinsic cancellations of unstable agent dynamics by such controllers, even static ones, where directional properties play a key role. The intrinsic lack of internal stability explains the notorious behavior of some distributed control protocols when affected by measurement noise or exogenous disturbances.