Multi-Agent Coordination with Lagrangian Measurements*

The paper is devoted to the study of a class of optimization problems with convex objective and constraint functions by using extremum seeking methods. Multi-agent systems with single integrator and unicycle dynamics are considered. The purpose is to develop a control algorithm that steers the agents to the set of saddle points of the associated Lagrangian, assuming that only Lagrangian measurements are available. For solving this problem, extremum seeking loops are introduced both for the states of a system and for Lagrangian multipliers. It is shown that, under certain assumptions, the trajectories of the system obtained approximate the trajectories of an auxiliary system representing a saddle point flow. The main result of the paper states the practical uniform asymptotic stability of the set of saddle points. The algorithm proposed is applied to distributed optimization problems.