Contracting Infinite-Gain Margin Feedback and Synchronization of Nonlinear Systems

In this chapter, we study the problem of synchronization of homogeneous multi-agent systemsMulti-agent system in which the network is described by a connected graphConnected graph. The dynamics of each single agent is described by an input-affine nonlinear systemNonlinearsystem. We propose a feedback controller based on contraction analysis and Riemannian metricsRiemannian metric. The starting point is the existence of a solution to a specific partial differential inequality which generalizes, in the nonlinear context, the well-known algebraic Riccati inequality. Further, we provide new results that allow to relax the Killing vectorKilling vector property so that to obtain a less stringent solution to be approximated with numerical methods. The proposed approach allows to design an infinite-gain margin feedback which is the fundamental ingredient to solve the problem of synchronization. We show that the synchronization problem is solved between two agents and we conjecture that the result holds for any connected graphConnected graph. Simulations for a connected directed graph of Duffing oscillatorsDuffing oscillator corroborate the conjecture.