Constrained consensus via logarithmic barrier functions

In this paper, we consider distributed algorithms for consensus of multiple agents in presence of convex state constraints on individual agent state. Each agent´s state is assumed to be constrained in a distinct compact convex set. We show that following the proposed distributed protocol, the agents are guaranteed to reach an agreement on a state that lies at the intersection of individual convex constraint sets. This is accomplished by introducing and sharing auxiliary variables in the network. The auxiliary variable utilizes a logarithmic barrier function to form a convex potential that is augmented to the consensus protocol. The consensus algorithm is then interpreted as a gradient-descent algorithm which operates with the desire to reach consensus while avoiding violation of the constraint sets. This modified consensus algorithm is applicable when each agent is required to satisfy its own constraints while synchronizing with others, e.g., attitude synchronization in presence of attitude constraints. An example is given for two different network topologies to evaluate the effectiveness and the convergence rate of the proposed algorithm.