Interval Consensus for Multiagent Networks

TL;DR

This paper introduces an interval consensus method for multiagent networks with constraints, demonstrating convergence to a common value within intersecting bounds and analyzing equilibrium stability when constraints are disjoint or incompatible.

Contribution

It presents a fully distributed algorithm for interval consensus, proving convergence and stability properties under various intersection conditions of the constraints.

Findings

Consensus converges within the intersection of agent intervals.

Existence of at least one equilibrium when intervals do not intersect.

Unique globally attractive equilibrium for disjoint intervals in sparse graphs.

Abstract

The constrained consensus problem considered in this paper, denoted interval consensus, is characterized by the fact that each agent can impose a lower and upper bound on the achievable consensus value. Such constraints can be encoded in the consensus dynamics by saturating the values that an agent transmits to its neighboring nodes. We show in the paper that when the intersection of the intervals imposed by the agents is nonempty, the resulting constrained consensus problem must converge to a common value inside that intersection. In our algorithm, convergence happens in a fully distributed manner, and without need of sharing any information on the individual constraining intervals. When the intersection of the intervals is an empty set, the intrinsic nonlinearity of the network dynamics raises new challenges in understanding the node state evolution. Using Brouwer fixed-point theorem we prove that in that case there exists at least one equilibrium, and in fact the possible equilibria are locally stable if the constraints are satisfied or dissatisfied at the same time among all nodes. For graphs with sufficient sparsity it is further proven that there is a unique equilibrium that is globally attractive if the constraint intervals are pairwise disjoint.