Observability in Connected Strongly Regular Graphs and Distance Regular Graphs

Alain Kibangou (1) , Christian Commault (2)

1 NECS - Networked Controlled Systems
2 GIPSA-SLR - GIPSA - Systèmes linéaires et robustesse

This paper concerns the study of observability in consensus networks modeled with strongly regular graphs or distance regular graphs. We first give a Kalman-like simple algebraic criterion for observability in distance regular graphs. This criterion consists in evaluating the rank of a matrix built with the components of the Bose-Mesner algebra associated with the considered graph. Then, we define some bipartite graphs that capture the observability properties of the graph to be studied. In particular, we show that necessary and sufficient observability conditions are given by the nullity of the so-called local bipartite observability graph (resp. local unfolded bipartite observability graph) for strongly regular graphs (resp. distance regular graphs). When the nullity cannot be derived directly from the structure of these bipartite graphs, the rank of the associated bi-adjacency matrix allows evaluating observability. Eventually, as a by-product of the main results we show that non-observability can be stated just by comparing the valency of the graph to be studied with a bound computed from the number of vertices of the graph and its diameter. Similarly nonobservability can also be stated by evaluating the size of the maximum matching in the above mentioned bipartite graphs.

Mots clés: Consensus networks; Graph; Networked systems; Observability; Association scheme; Distance regular graphs; Strongly regular graphs; Nul- lity of graphs; Bipartite graph

Domaines: Automatique / Robotique; Automatique; Systèmes dynamiques [math.DS]; Système multi-agents [cs.MA]

Alain Kibangou, Christian Commault. Observability in Connected Strongly Regular Graphs and Distance Regular Graphs. IEEE Transactions on Control of Network Systems, 2014, 1 (4), pp.360-369. ⟨10.1109/TCNS.2014.2357532⟩. ⟨hal-01092954⟩