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

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 \(P.\)