On the necessity of the small-gain theorem in the performance analysis of nonlinear systems

A simple example illustrates the fact that robust stability need not imply performance for general nonlinear systems, namely, it is possible to have robust stability even when the conditions of the small-gain theorem are violated. This example motivates the introduction of a new performance measure called conditional gain, and it is shown that the conclusion of the classical small-gain theorem remains valid when the gain condition on one of the systems in the feedback connection is replaced with the weaker conditional gain condition. It is then conjectured that the necessity of the small-gain theorem can be recovered for nonlinear systems when this notion of conditional gain is used as the performance measure.