small-gain theorem

A formalism for the study of random dynamical systems with inputs and outputs (RDSIO) is introduced. An axiomatic framework and basic properties of RDSIO are developed, and a theorem is shown that guarantees the stability of interconnected systems.

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.

This note introduces a small-gain result for interconnected MIMO orthant-monotone systems for which no matching condition is required between the partial orders in input and output spaces of the considered subsystems. 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 corresponded to positive-feedback or negative feedback cases. We relax those results by allowing arbitrary orthant orders.

A general ISS-type small-gain result is presented. It specializes to a small-gain theorem for ISS operators, and it also recovers the classical statement for ISS systems in state-space form. In addition, we highlight applications to incrementally stable systems, detectable systems, and to interconnections of stable systems.