A scaled small gain theorem with applications to spatially interconnected systems

In this note, a new result that extends the scaled small gain theorem is presented. As is well known, the scaled small gain theorem gives necessary and sufficient conditions for robust stability of a nominal linear time-invariant system in feedback with structured operators of norm less than or equal to unity. We propose alternative linear matrix inequality conditions that give necessary and sufficient conditions for robust stability against the class of structured unitary operators (invertible operators of exactly unit norm). It is shown that this result, besides being a less conservative version of the scaled small gain theorem, has connections to several recent results on the control of spatially interconnected systems and serves to unify and quantify the conservatism of those results.