Interpreting the iISS small-gain theorem as transient plus ISS small-gain regulation

This paper addresses the problem of establishing stability of interconnections of integral input-to-state stable (iISS) systems. Recently, the small-gain theorem for input-to-state stable (ISS) systems has been extended to the class of iISS systems. It has been also proved that at least one of the two iISS subsystems comprising a feedback interconnection needs to be ISS with respect to the state of the other subsystem for guaranteeing the iISS of the overall system. This paper shows that making use of this necessary condition enables to provide more insight on the iISS small gain theorem by giving an alternative proof of this result from the perspective of transient plus ISS small-gain regulation.