A nonlinear Perron-Frobenius approach for stability and consensus of discrete-time multi-agent systems

In this paper we propose a novel method to study stability and, in addition, convergence to a consensus state for a class of discrete-time Multi-Agent System (MAS) where agents evolve with nonlinear dynamics, possibly different for each agent (heterogeneous local interaction rules). In particular, we focus on a class of discrete-time MASs whose global dynamics can be represented by positive, sub-homogeneous and type-K order-preserving nonlinear maps. This paper generalizes results that apply to linear MASs to the nonlinear case by exploiting nonlinear Perron–Frobenius theory. We provide sufficient conditions on the structure of the nonlinear local interaction rules to guarantee stability of a MAS and an additional condition on the topology of the network ensuring the achievement of consensus as a particular case. Two examples are provided to corroborate the theoretical analysis. In the first one we consider a susceptible–infected–susceptible (SIS) model while in the second we consider a novel protocol to solve the max-consensus problem.