Minimum Fuel Consensus for Multi-agent System

The problem of achieving consensus using minimum total fuel in a given final time for N identical double-integrator agents with bounded control inputs is considered. Key tool used is the attainable set, which is the collection of all the points that can be attained from an initial condition using admissible input at the given final time. The properties of the attainable set as a function of fuel are studied. The considered problem is equivalent to fnding an allocation of fuels to each agent so that all agent’s attainable sets have non-empty intersection and the total fuel is minimized. Using the attainable set description, a lower bound in terms of the initial conditions of all agents is derived. A new quantity β¯ is introduced and is defined in terms of the intersection of the attainable sets of all agents. This quantity β¯ is used to derive an upper bound on the total minimum fuel required for consensus. The computation of β¯ is made tractable and performed in distributed manner using Helly’s theorem from convex geometry. Finally, an example is used to demonstrate the derived lower and upper bounds.