An LMI Formulation of Small-Gain Theorems for 2-Contraction of Nonlinear Interconnected Systems

This paper introduces small-gain sufficient conditions for 2-contraction of feedback interconnected systems, on the basis of individual gains of suitable subsystems arising from a modular decomposition of the second additive compound matrix variational equation. The gains are computed through the solution of suitable LMI conditions. The criterion applies even to cases when individual subsystems might fail to be contractive, thanks to the extra margin of contraction afforded by the second additive compound matrix. One example is provided to illustrate the theory and show its degree of conservatism and scope of applicability.