Allowing vanishing stability margins in preservation of (i)ISS dissipation inequalities by scaling

Scaling of Lyapunov functions is one of effective tools in analyzing and designing dynamical systems from components. Integral input-to-state stability and input-to-state stability are notions which allow one to make use of dissipation inequalities in such modular-based analysis with the help of nonlinear gain which may not be globally defined. Effectively selecting nonlinear scalings is crucial for coping with nonlinearities in components without causing degradation of dissipative properties and their estimates. This paper focuses on scalings for systems which do not admit uniform stability margins and proposes new tools applicable to cases which have not been addressed in the literature. In addition to demonstrating preservation of (i)ISS dissipation inequalities by scaling, this paper illustrates the usefulness of the proposed scaling formulas in analyzing stability of cascaded systems through examples.