Ensuring localizability of node attacks in consensus networks via feedback graph design

In this paper we consider the problem of localizability of attacks in continuous-time consensus networks. In our previous work [1] we showed that if a consensus network is divided into clusters, then a supervisor can successfully localize the cluster in which an attack may have been launched by simply inspecting the sign patterns of the residues corresponding to the slow poles of its input-output transfer function (TF). A necessary condition for localizability, however, was that the attack must enter through a node that guarantees this TF to be minimal. In case the attacker knows the identity of the so-called zero nodes from where the TFs are non-minimal, and chooses to launch the attack at any of them, then the supervisor cannot localize the attack.We show that this problem can be bypassed by designing a state-feedback controller that equivalently changes the algebraic properties of the underlying network graph, and thereby restores minimality of the TF. We illustrate the approach by simulating a three-area, 30-node graph, and highlighting the performance trade-offs that come as a price of localizability.