A constructive proof of ISS small-gain theorem using generalized scaling

This paper presents a Lyapunov-type proof of the input-to-state stable (ISS) small-gain theorem. The proof given in this paper demonstrates how to construct a Lyapunov function explicitly, which contrasts with existing proofs based on input-output analysis. The Lyapunov functions are constructed by selecting state-dependent scaling functions properly. The construction of Lyapunov functions motivates the author to formulate a new criterion for stability and performance of interconnected nonlinear systems. Furthermore, an ISS small-gain condition with nonlinearly-scaled supply rates is obtained naturally in order to reduce the conservatism arising in the application of the ISS small-gain condition.