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Information structures for distributed k-dimensional agreement
G. Bianchin, M. Vaquero, J. Cortés, E. Dall'Anese
Automatica, submitted

Abstract

Given a network of agents, the agents are said to reach a k-dimensional agreement when the state variables agree within a k-dimensional linear subspace. This problem is a generalization of the well-studied average consensus problem, where the asymptotic states of the agents are not required to coincide, but rather to agree in a generalized sense. In this paper, we investigate what structural properties of the interaction graph are required to enable the agents to reach a k-dimensional agreement. We find that agreement protocols impose the use of communication graphs with a high network connectivity; more precisely, we show that the number of edges in the graph must grow linearly with the size of the agreement space k. We study under what conditions common graph topologies -- such as line and circulant graphs -- can sustain agreement protocols, and provide insights into the relationship between network connectivity and the space dimension k. Our characterization identifies the presence of cycles (precisely, of independent cycle families) in the network as a basic structural property that enables agents to reach an agreement. The applicability of the framework is illustrated via simulations on problems in robotic formation.

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