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.

Gianfranco Parlangeli, Giuseppe Notarstefano (2010). On the observability of path and cycle graphs. USA : IEEE [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.
2010
IEEE Conference on Decision and Control
1492
1497
Gianfranco Parlangeli, Giuseppe Notarstefano (2010). On the observability of path and cycle graphs. USA : IEEE [10.1109/CDC.2010.5717507].
Gianfranco Parlangeli; Giuseppe Notarstefano
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/671977
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