On the reachability and observability of path and cycle graphs

TL;DR

This paper characterizes the reachability and observability of path and cycle graphs in network systems using algebraic number theory, revealing conditions based on graph size and structure for control and observation capabilities.

Contribution

It provides necessary and sufficient algebraic conditions for reachability and observability in path and cycle graphs, including specific criteria based on graph size and node selection.

Findings

Path graphs are reachable from any single node if and only if n is a power of two.

Cycle graphs are reachable from any pair of nodes if and only if n is prime.

Closed-form expressions for unreachable eigenvalues and eigenvectors are provided.

Abstract

In this paper we investigate the reachability and observability properties of a network system, running a Laplacian based average consensus algorithm, when the communication graph is a path or a cycle. More in detail, we provide necessary and sufficient conditions, based on simple algebraic rules from number theory, to characterize all and only the nodes from which the network system is reachable (respectively observable). Interesting immediate corollaries of our results are: (i) a path graph is reachable (observable) from any single node if and only if the number of nodes of the graph is a power of two, $n=2^i, i\in \natural$, and (ii) a cycle is reachable (observable) from any pair of nodes if and only if $n$ is a prime number. For any set of control (observation) nodes, we provide a closed form expression for the (unreachable) unobservable eigenvalues and for the eigenvectors of the (unreachable) unobservable subsystem.