Control of quantized multi-agent systems with linear nearest neighbor rules: A finite field approach

We study the problem of controlling a multi-agent system where each agent is only allowed to be in a discrete and finite set of states. Each agent is capable of updating its state based on the states of its neighbors, and there is a leader agent in the network that is allowed to update its state in arbitrary ways (within the discrete set) in order to put all agents in a desired state. We present a novel solution to this problem by viewing the discrete states of the system as elements of a finite field. Specifically, we develop a theory of structured linear systems over finite fields, and show that such systems will be controllable provided that the size of the finite field is sufficiently large, and that the graph associated with the system satisfies certain properties. We then use these results to show that a multi-agent system with a leader node is controllable via a linear nearest-neighbor update as long as there is a path from the leader to every node, and that the number of discrete states for each node is large enough.