Contraction theory based control law design for distributed optimization problem of nonlinear multi-agent systems: beyond high-gain and backstepping

The paper is concerned with the distributed optimization problem for a class of nonlinear multi-agent systems with second relative degrees. Unlike the existing results on control law design that rely on the high-gain or backstepping techniques, novel distributed control law is proposed such that a sum of convex local cost functions is minimized on the consensus value, where the gradients of the convex functions are locally Lipschitz. A contraction theory approach is introduced to show the incremental stability of the closed-loop system. Due to the introduction of the non-Lipschitz nonlinear gradients, the Euclidean metric is not suitable for characterizing the distance between two trajectories of a convergent system on differentiable manifolds. Motivated by this, a Finsler-Lyapunov function is constructed to invoke the Finsler distance that is shrinking along the trajectories. The effectiveness of the proposed methods is validated through some numerical simulations of a group of networked nonlinear tunnel-diode circuit systems.