In this paper we investigate the 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 observable. Interesting immediate corollaries of our results are: (i) a path graph is 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 ∈ N, and (ii) a cycle is observable from any pair of observation nodes if and only if n is a prime number. For any set of observation nodes, we provide a closed form expression for the unobservable eigenvalues and for the eigenvectors of the unobservable subspace.
On the observability of path and cycle graphs / Gianfranco Parlangeli; Giuseppe Notarstefano. - STAMPA. - (2010), pp. 1492-1497. (Intervento presentato al convegno IEEE Conf. on Decision and Control tenutosi a Atlanta, USA nel December 15-17, 2010) [10.1109/CDC.2010.5717507].
On the observability of path and cycle graphs
Giuseppe Notarstefano
2010
Abstract
In this paper we investigate the 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 observable. Interesting immediate corollaries of our results are: (i) a path graph is 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 ∈ N, and (ii) a cycle is observable from any pair of observation nodes if and only if n is a prime number. For any set of observation nodes, we provide a closed form expression for the unobservable eigenvalues and for the eigenvectors of the unobservable subspace.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
