Observability through matrix-weighted graph

Observability of an array of identical LTI systems with incommensurable output matrices is studied, where an array is called observable when identically zero relative outputs imply synchronized solutions for the individual systems. It is shown that the observability of an array is equivalent to the connectivity of its interconnection graph, whose edges are assigned matrix weights. The interconnection graph is studied by means of a collection of simpler graphs, each of which is associated to an eigenvalue of the system matrix of individual dynamics. It is reported that the interconnection graph is connected if and only if no member of this collection is disconnected. Moreover, to better understand the relative behavior of distant units, pairwise observability which concerns with the synchronization of a certain pair of individual systems in the array is studied. This milder version of observability is shown to be closely related to certain connectivity properties of the interconnection graph as well. Pairwise observability is also analyzed using the circuit theoretic tool effective conductance. The observability of a certain pair of units is proved to be equivalent to the nonsingularity of the (matrix-valued) effective conductance between the associated pair of nodes of a resistive network (with matrix-valued parameters) whose node admittance matrix is the Laplacian of the array's interconnection graph.