State-Dependent Scaling Characterization for Interconnected Nonlinear Systems–Part II : Small Gain Theorems for iISS and ISS Properties

This paper is devoted to the problem of stability analysis for interconnected integral input-to-state stable(iISS) systems and input-to-state stable(ISS) systems. In the first part of this two-part paper, the state-dependent scaling problem has been proposed as a general mathematical formulation whose solutions explicitly provide Lyapunov functions proving stability properties of feedback and cascade connection of dissipative nonlinear systems. A main purpose of this second part is to demonstrate that the generalization is surely beyond formal applicability, and it effectively deals with systems having diverse and strong nonlinearities which are not covered by classical and existing advanced stability criteria. In particular, this paper derives small-gain-type theorems for interconnected systems involving iISS systems smoothly from the state-dependent scaling formulation as special cases. This paper also provides explicit solutions to the statedependent scaling problems. The new framework enables us not only to characterize stability of interconnected iISS and ISS systems in a unified manner, but also to extend the ISS small-gain theorem to iISS supply rates seamlessly. The results are considered as some theoretical evidences that the state-dependence of the scaling is crucial to effective treatment of essential nonlinearities