Convex Conditions on Decentralized Control for Graph Topology Preservation

This technical note focuses on the preservation of a given graph topology which is usually chosen to ensure its connectivity. This is an essential ingredient allowing interconnected systems to accomplish tasks by using decentralized control strategies. We consider a networked system with discrete-time dynamics in which the subsystems are able to communicate if an algebraic relation between their states is satisfied. Each subsystem is called agent and the connected subsystems are called neighbors. The agents update their state in a decentralized manner by taking into account the neighbors´ states. The characterization of the local control feedback gains ensuring topology preservation is provided. The results are based on invariance and set-theory and yield to conditions in Linear Matrix Inequality (LMI) form. The conditions for topology preservation are applied to an illustrative example concerning partial state consensus of agents with double integrator dynamics.