On the Discernibility of Topology Changes in Networked LTI Systems

Overview & Synopsis

This article investigates the discernibility of topology changes in networked linear time-invariant systems, with particular focus on edge disconnection. We introduce a new notion of discernibility motivated by a limitation of the definition commonly adopted in the literature. In consensus networks, topology changes are inherently indiscernible for all initial conditions belonging to the synchronization subspace. To overcome this issue, we first introduce the notion of dynamical discernibility of a topology change for elementary consensus networks. This notion explicitly disregards the intrinsic indiscernibility associated with states in the consensus subspace. We characterize this property for directed communication graphs and then specialize the resulting conditions to topology changes arising from either the disconnection of an edge or a variation in its weight. We then extend the analysis to networks of homogeneous agents by introducing discernibility modulo synchronization, which factors out the unavoidable indiscernibility associated with synchronized trajectories. We derive a general characterization showing how discernibility depends jointly on the topology change and on the dynamics of the individual agents. Finally, we specialize the analysis to undirected communication graphs and show that, under mild additional assumptions, discernibility modulo synchronization is equivalent to the dynamical discernibility of the corresponding topology change. This establishes a direct connection between the two notions and extends existing results on the discernibility of topology variations in dynamical networks. The resulting characterizations are then further specialized to the case of edge disconnection.