Controlling consensus in networks with symmetries

TL;DR

This paper investigates how symmetries in networks affect controllability and proposes methods to design controllers that achieve desired group consensus states while managing control input placement.

Contribution

It introduces a framework for analyzing controllability in symmetric networks and develops optimal control strategies for steering and stabilizing group consensus.

Findings

Controllability is limited to the group consensus subspace.

Optimal controllers can steer the network to desired states within this subspace.

Additional control inputs can be strategically added to stabilize the consensus subspace.

Abstract

We study networks with linear dynamics where the presence of symmetries of the pair (A,B) induces a partition of the network nodes in clusters and the matrix A is not restricted to be in Laplacian form. For these networks, an invariant group consensus subspace can be defined, in which the nodes in the same cluster evolve along the same trajectory in time. We prove that the network dynamics is uncontrollable in directions orthogonal to this subspace. Under the assumption that the dynamics parallel to this subspace is controllable, we design optimal controllers that drive the group consensus dynamics towards a desired state. Then, we consider the problem of selecting additional control inputs that stabilize the group consensus subspace and obtain bounds on the minimum number of additional inputs and driver nodes needed to this end. Altogether, our results indicate that it is possible to design independently the control action along and transverse to the group consensus subspace.