A small-gain result for orthant monotone systems under mixed feedback

This paper introduces a small-gain result for interconnected orthant-monotone systems for which no matching condition is required between the partial orders in input and output spaces. Previous results assumed that the partial orders adopted would be induced by positivity cones in input and output spaces and that such positivity cones should fulfill a compatibility rule: namely either be coincident or be opposite. Those two configurations correspond to positive feedback or negative feedback cases. We relax those results by allowing arbitrary orthant orders. A linear example is provided showing that the small-gain iteration used for the negative feedback case is not sufficient for global attractivity under mixed feedback. An algebraic characterization is given of the new small-gain condition, generalizing a result known in the negative feedback case. An application is given to nonlinear protein networks with one positive and one negative feedback loop.