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We consider interconnected nonlinear systems with external inputs, where eachof the subsystems is assumed to be input-to-state stable (ISS). Sufficientconditions of small gain type are provided guaranteeing that theinterconnection is ISS with respect to the external input. To this end weextend recently obtained small gain theorems to a more general type ofinterconnections. The small gain theorem provided here is applicable tosituations where the ISS conditions are formulated differently for eachsubsystem and are either given in the maximization or the summation sense.Furthermore it is shown that the conditions are compatible in the sense that itis always possible to transform sum formulations to maximum formulationswithout destroying a given small gain condition. An example shows theadvantages of our results in comparison with the known ones.

A small gain condition for interconnections of ISS systems with mixed ISS characterizations