Finite-time scaled consensus through parametric linear iterations

This paper deals with finite-time scaled consensus problems over undirected and directed topologies, wherein agents' states reach prescribed ratios in a finite time. We develop distributed linear iterations as a function of a linear operator on the underlying network and present necessary and sufficient conditions guaranteeing scaled consensus in a fixed number of steps equal to the number of distinct eigenvalues of a related linear operator. We identify the dependence of the final consensus states on the initial state condition, which can be conveniently and freely tuned by designing suitable parameters. Our results extend the recently developed approach on successive nulling of eigenvalues from complete consensus to scaled consensus, and from undirected topologies to directed topologies. Numerical examples and comparison studies are provided to illustrate the effectiveness of our theoretical results.