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.