Coherence in Large-Scale Networks: Dimension-Dependent Limitations of Local Feedback

Abstract

Abstract-We consider distributed consensus and vehicular formation control problems. Specifically we address the question of whether local feedback is sufficient to maintain coherence in large-scale networks subject to stochastic disturbances. We define macroscopic performance measures which are global quantities that capture the notion of coherence; a notion of global order that quantifies how closely the formation resembles a solid object. We consider how these measures scale asymptotically with network size in the topologies of regular lattices in 1, 2 and higher dimensions, with vehicular platoons corresponding to the 1 dimensional case. A common phenomenon appears where a higher spatial dimension implies a more favorable scaling of coherence measures, with a dimensions of 3 being necessary to achieve coherence in consensus and vehicular formations under certain conditions. In particular, we show that it is impossible to have large coherent one dimensional vehicular platoons with only local feedback. We analyze these effects in terms of the underlying energetic modes of motion, showing that they take the form of large temporal and spatial scales resulting in an accordion-like motion of formations. A conclusion can be drawn that in low spatial dimensions, local feedback is unable to regulate largescale disturbances, but it can in higher spatial dimensions. This phenomenon is distinct from, and unrelated to string instability issues which are commonly encountered in control problems for automated highways.

Optimal Distributed Control for Platooning via Sparse Coprime Factorizations

“…The standard analysis for platooning systems must take into account the effects of the disturbance w j (at any j-th vehicle in the platoon or at the leader) on the errors z k and controls u k , for all successors in the string (k > j). We will prove next that, as a bonus feature of leader information controllers, the effect of the disturbances w j on any of its successors k in the string does not formally depend on the number (k − j) of in-between vehicles but only on the following factors: (i) the attenuations obtained at the j-th and ( j + 1)-th local problems respectively (which are optimized by design in (52)); (ii) the stable, minimum phase dynamics Φ j and Φ k particular to the j-th and the k-th vehicle, respectively; and (iii) the constant timeheadway H. In particular, the effect of the disturbances w 0 at the leader on any successor 11 Due to similar arguments as in footnote (8).…”

“…In [9] optimal quadratic regulators for platooning are proposed while showing that for an increasing number of vehicles the resulted LQR problems become ill-posed. It was later proved in [11] that "local" measurements based distributed controllers cannot achieve "coherent" coordination in large-sized platoons, results further extended in [10] as to achieve superior coherence formation via optimal controllers.…”

A distributed solution to the network reconstruction problem

“…This definition of robustness corresponds to the steadystate least-mean-square deviation introduced in [16] for discrete-time consensus dynamics: a related H 2 normbased notion quantifying the deviation from consensus, is called (first-order) network coherence in [19]. Finally, under the same assumptions of Prop.…”

“…Note that the iterates P i (h)m a ynot be invertible for all values of h: hence, estimate (19) at node i can only be computed once P i (h) becomes nonsingular. We also observe here that differently from the iterations in (18a), which have to be performed only once, the iterations in (18b) must be executed at each time instant k ∈{0, 1, ...,S − 1}, which is computationally demanding, especially for large networks.…”

Robust parallel decision-making in neural circuits with nonlinear inhibition

“…We wonder whether it is possible to replace the global sum of excitatory activities by a set of local inhibitory neurons pooling local excitatory inputs. However, local inhibition typically produces pattern formation [15], and consensus formation with local connections can be unstable [4], thus it is an interesting open question whether WTA across a population of neurons can be implemented with purely local connectivity.…”