Revisiting the IISS Small‐Gain Theorem through Transient Plus ISS Small‐Gain Regulation

Recently, the small‐gain theorem for input‐to‐state stable (ISS) systems has been extended to the class of integral input‐to‐state stable (iISS) systems. Feedback connections of two iISS systems are robustly stable with respect to disturbance if an extended small‐gain condition is satisfied. It has been proved that at least one of the two iISS subsystems needs to be ISS for guaranteeing globally asymptotic stability and iISS of the overall system. Making use of this necessary condition for the stability, this paper gives a new interpretation to the iISS small gain theorem as transient plus ISS small‐gain regulation. The observation provides useful information for designing and analyzing nonlinear control systems based on the iISS small‐gain theorem.