How network perturbations distort agreement trajectories in LTI multi-agent systems

Abstract

Distributed coordination of multi-agent systems frequently relies on cooperative protocols designed to achieve agreement on a prescribed, non-trivial trajectory. While the robustness of such protocols to various uncertainties is well documented, existing literature universally assumes that the target agreement trajectory itself remains invariant. This assumption may hold in ideal cases, but we prove that network perturbations can vastly modify the asymptotic agreement trajectory. We first investigate the exact trajectories of Linear Time-Invariant (LTI) agents subjected to dynamic coupling uncertainties by establishing a new Laplace-domain criterion that characterizes the specific closed-loop poles governing the perturbed agreement manifold. To formalize our analysis, we introduce the notion of structure-preserving dynamics, perturbations that maintain the null space of the communication graph's Laplacian, and contrast them with transmission only dynamics, affecting only the adjacency matrix. We prove a critical fragility within standard cooperative output regulation schemes: while static consensus is uniquely robust to heterogeneous transmission delays, synchronization to periodic trajectories is destroyed by arbitrarily small transmission delays. Furthermore, we demonstrate that for d-regular topologies, uniform transmission perturbations can easily shift the system to synchronize with an unexpected, entirely new frequency. These findings expose a previously unidentified vulnerability in classical robust synchronization, demonstrating that transmission dynamics necessitate fundamental structural modifications to networked reference generators.