Equilibrium of consensus problems for discrete-time multi-agent systems

The equilibrium of consensus problems for discrete-time multi-agent system is studied. For a fixed topology system, based on the spectrum of nonnegative stochastic matrix and its left eigenvector, it is proved that only the initial conditions of the root vertices contribute to the equilibrium value in interaction topology which contains a spanning tree. For a time-varying topology system, by the fitness of style of nonnegative matrices, it is proved that only the nodes that have directed paths to all the others in the union of switching interaction topology have contributions to the final value. Numerical examples are used to prove the correctness of the theorems.