Scale fragilities in localized consensus dynamics

Input-Output Pseudospectral Bounds for Transient Analysis of Networked and High-Order Systems

“…This result together with Proposition 1 shows that n th order consensus can be achieved with a local relative feedback controller with finite gain and thus achieve scalable stability. This result can be compared with [9] where it is shown that the conventional consensus is not scalably stable for any order larger than n = 3 if a graph Laplacian with vanishing algebraic connectivity is used. We can summarize this fact in the following corollary:…”

“…However when the controllers only depend on local measurements there is a possibility that controllers that manage to coordinate N agents stop stabilizing as the number of agents increases. In [9] it was shown that for the 3 rd and higher order consensus problem with controller A k = a k L N in (4), the closed loop system will become unstable if the algebraic connectivity λ 2 (L N ) → 0 as N → ∞. This motivates the notion of Scalable stability Definition 4 (Scalable stability [9, Def.…”

“…Over the years, it has become clear that the structural constrains imposed by the network topology in consensus problems often lead to fundamentally poor dynamic behaviors in large networks. This concerns controllability [4], performance [5], [6] and disturbance propagation [7], [8], but, as recently highlighted in [9], also stability. The poor stability properties characterized in earlier work [9] (which motivate the present work) apply to higher-order consensus, where the local dynamics of each agent is modeled as an nth order integrator, with n ≥ 2, and the control is a weighted average of neighbors' relative states.…”

“…This concerns controllability [4], performance [5], [6] and disturbance propagation [7], [8], but, as recently highlighted in [9], also stability. The poor stability properties characterized in earlier work [9] (which motivate the present work) apply to higher-order consensus, where the local dynamics of each agent is modeled as an nth order integrator, with n ≥ 2, and the control is a weighted average of neighbors' relative states. This is a theoretical generalization of first-order consensus [10], but is also relevant in practice.…”

“…More specifically, [9] shows that conventional high-order consensus (n ≥ 3) is not scalably stable for many growing graph structures. When the network grows beyond a certain size, stability is lost.…”