Observability and reachability of grid graphs via reduction and symmetries

In this paper we investigate the observability and reachability properties of a network system, running a Laplacian based average consensus algorithm, when the communication graph is a grid. More in detail, we characterize the structure of the grid eigenvectors by means of suitable decompositions of the graph. For each eigenvalue, based on its multiplicity and on suitable symmetries of the corresponding eigenvectors, we provide necessary and sufficient conditions to characterize all and only the nodes from which the network system is observable (reachable). We discuss the proposed criteria and show, through suitable examples, how such criteria reduce the complexity of the observability (respectively reachability) analysis of the grid.