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 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 $P$.