Recovering the Structural Observability of Composite Networks via Cartesian Product

Observability is a fundamental concept in system inference and estimation. This paper is focused on structural observability analysis of Cartesian product networks. Cartesian product networks emerge in variety of applications including in parallel and distributed systems. We provide a structural approach to extend the structural observability of the constituent networks (referred as the factor networks) to that of the Cartesian product network. The structural approach is based on graph theory and is generic. We introduce certain structures which are tightly related to structural observability of networks, namely parent Strongly-Connected-Component (parent SCC), parent node, and contractions. The results show that for particular type of networks (e.g. the networks containing contractions) the structural observability of the factor network can be recovered via Cartesian product. In other words, if one of the factor networks is structurally rank-deficient, using the other factor network containing a spanning cycle family, then the Cartesian product of the two nwtworks is structurally full-rank. We define certain network structures for structural observability recovery. On the other hand, we derive the number of observer nodes--the node whose state is measured by an output-- in the Cartesian product network based on the number of observer nodes in the factor networks. An example illustrates the graph-theoretic analysis in the paper.