The Projection Method for Reaching Consensus and the Regularized Power Limit of a Stochastic Matrix

Abstract

In the coordination/consensus problem for multi-agent systems, a well-known condition of achieving consensus is the presence of a spanning arborescence in the communication digraph. The paper deals with the discrete consensus problem in the case where this condition is not satisfied. A characterization of the subspace $T_P$ of initial opinions (where $P$ is the influence matrix) that \emph{ensure} consensus in the DeGroot model is given. We propose a method of coordination that consists of: (1) the transformation of the vector of initial opinions into a vector belonging to $T_P$ by orthogonal projection and (2) subsequent iterations of the transformation $P.$ The properties of this method are studied. It is shown that for any non-periodic stochastic matrix $P,$ the resulting matrix of the orthogonal projection method can be treated as a regularized power limit of $P.$

19 pages, 2 figures

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Cited by in corpus (3)

• Models of latent consensus
• A Cyclic Representation of Discrete Coordination Procedures
• The projection method for continuous-time consensus seeking