Localized high-order consensus destabilizes large-scale networks

Abstract

We study the problem of distributed consensus in networks where the local agents have high-order (n ≥ 3) integrator dynamics, and where all feedback is localized in that each agent has a bounded number of neighbors. We prove that no consensus algorithm based on relative differences between states of neighboring agents can then achieve consensus in networks of any size. That is, while a given algorithm may allow a small network to converge to consensus, the same algorithm will lead to instability if agents are added to the network so that it grows beyond a certain finite size. This holds in classes of network graphs whose algebraic connectivity, that is, the smallest non-zero Laplacian eigenvalue, is decreasing towards zero in network size. This applies, for example, to all planar graphs. Our proof, which relies on Routh-Hurwitz criteria for complex-valued polynomials, holds true for directed graphs with normal graph Laplacians. We survey classes of graphs where this issue arises, and also discuss leaderfollower consensus, where instability will arise in any growing, undirected network as long as the feedback is localized. E. Tegling and H. Sandberg are with the

Scale-free collaborative protocol design for state and regulated state synchronization of multi-agent systems with arbitrary fast convergence

“…• the protocol achieves regulated state synchronization (8) for any μ > 0, any N and any graph G ∈ G N C where G N C is defined as Definition 1 . • for a given desired r * asym , any given graph G ∈ G N C and any x r , by choosing parameter μ sufficiently large, the protocol achieves a convergence rate (for the synchronization error) less than r * asym .…”

“…Most of the proposed protocols in the literature for synchronization of MAS requires some knowledge of the communication network such as bounds on the spectrum of the associated Laplacian matrix or the number of agents. As it is pointed out in [7][8][9] , these protocols suffer from scale fragility wherein stability properties are lost for large-scale networks or when the communication graph changes. Therefore, there exist a current research effort focusing on scalability of consensus laws.…”

Second Order Consensus with Absolute Information

“…Consensus without leaders in higher order systems was studied in [24], where the presence of a spanning tree in the network topology was proved to be a necessary condition for consensus, and in [25], where the problem of developing a control input capable of guiding all nodes to the same location was studied. Finally, in [26] the authors prove that consensus is not guaranteed to be reachable in higher order systems as the network grows infinitely large. Further, they give conditions for consensus in third order systems where all nodes are leaders.…”

Coherence and Concentration in Tightly-Connected Networks

“…Consensus [4], [11]- [16], on one hand, refers to the ability of the network nodes to asymptotically reach a common value over some quantities of interest. Many extensions of this problem include the study of robustness and performance of consensus networks in the presence of noise [12]- [14], timedelay [15], [16], and switching graph topology [16]. Synchronization [5], [8]- [10], [17]- [19], on the other hand, refers to the ability of network nodes to follow a commonly defined trajectory.…”

“…A vast body of work, triggered by the seminal paper [13], has quantitatively studied the role of the network topology in the emergence of coherence. Examples include, directed [14] and undirected [20] consensus networks, transportation networks [13], and power networks [7], [21], [22]. The key technical approach amounts to quantify the level of coherence by computing the H 2 -norm of the system for appropriately defined nodal disturbance and performance signals.…”