A Constructive Lyapunov Approach to ISS Small-Gain Theorem via State-Dependent Scaling

Abstract

This paper proposes a new approach to the Input-to-State Stable(ISS) small-gain theorem. A new proof of the ISS small-gain theorem is obtained by constructing a Lyapunov function explicitly, which contrasts with existing proofs based on input-output analysis. The construction of Lyapunov functions through the use of state-dependent scaling motivates the author to formulate a new criterion for stability and performance of interconnected nonlinear systems. Furthermore, a nonlinearly-scaled version of the ISS small-gain condition is addressed.