Optimal Consensus of Euler-Lagrangian Systems with Kinematic Constraints

In this paper, we study a distributed constrained consensus problem for heterogeneous Euler-Lagrange (EL) systems with kinematic constraints. Each agent is assigned with a convex function as individual cost, and we design a distributed control law to achieve consensus at the optimum of the aggregate cost under given constraints on the velocity and control input which are required to be bounded. Noticing that an EL system with exact knowledge of nonlinearities can be turned into a double-integrator, we first explore the consensus of double-integrator multi-agent systems by Lyapunov method, then extend the result to the case of EL dynamics by inverse dynamics control. Specifically, with knowledge of the furthest distance from the optimum to initial positions, it is shown that control gains can be properly selected to achieve an exponentially fast convergence while satisfying the bounded kinematic constraints, if the fixed undirected topology is connected. A numeric example is given to illustrate the result.