iISS and ISS dissipation inequalities: preservation and interconnection by scaling

In analysis and design of nonlinear dynamical systems, (nonlinear) scaling of Lyapunov functions has been a central idea. This paper proposes a set of tools to make use of such scalings and illustrates their benefits in constructing Lyapunov functions for interconnected nonlinear systems. First, the essence of some scaling techniques used extensively in the literature is reformulated in view of preservation of dissipation inequalities of integral input-to-state stability (iISS) and input-to-state stability (ISS). The iISS small-gain theorem is revisited from this viewpoint. Preservation of ISS dissipation inequalities is shown to not always be necessary, while preserving iISS which is weaker than ISS is convenient. By establishing relationships between the Legendre–Fenchel transform and the reformulated scaling techniques, this paper proposes a way to construct less complicated Lyapunov functions for interconnected systems.