Graph reduction based observability conditions for network systems running average consensus algorithms

This paper represents a preliminary step toward the solution of the observability problem for a network system running an average consensus algorithm by use of algebraic graph tools. We introduce a mathematical framework that, in our opinion, is suitable to model a wide class of estimation and diagnostic problems. We provide preliminary results to reduce the complexity of the observability problem by looking at suitable subgraphs of the original communication graph. In particular we provide a sufficient condition to obtain observability of a tree. Finally, we show examples of graphs that are guaranteed to be observable for suitable choices of the observation nodes and unobservable graphs that may inspire future research directions.