Lyapunov-Free Analysis for Consensus of Nonlinear Discrete- Time Multi-Agent Systems

In this paper we propose a novel method to establish stability and convergence to a consensus state for a class of nonlinear discrete-time Multi-Agent System (MAS) which is not based on Lyapunov function arguments. In particular, we focus on a class of discrete-time multi-agent systems whose global dynamics can be represented by sub-homogeneous and order-preserving nonlinear maps. The preliminary results of this paper directly generalize results for sub-homogeneous and order-preserving linear maps which are shown to be the counterpart to stochastic matrices thanks to nonlinear Perron-Frobenius theory. We provide sufficient conditions on local interaction rules among agents to establish convergence to a fixed point and study the consensus problem in this generalized framework as a particular case. Examples to show the effectiveness of the method are provided to corroborate the theoretical analysis. In these examples, some nonlinear interaction protocols are proved to converge to the consensus state without the use of Lyapunov functions.