State-Dependent Scaling Problems and Stability of Interconnected iISS and ISS Systems

This paper addresses the problem of establishing stability of nonlinear interconnected systems. This paper introduces a mathematical formulation of the state-dependent scaling problems whose solutions directly provide Lyapunov functions proving stability properties of interconnected dissipative systems in a unified manner. Stability criteria are interpreted as sufficient conditions for the existence of solutions to the state-dependent scaling problems. Computing solutions to the problems is straightforward for systems covered by classical stability criteria. It, however, could be too difficult for systems with strong nonlinearity. The main purpose of this paper is to demonstrate the effectiveness beyond formal applicability by focusing on interconnected integral input-to-state stable (iISS) systems and input-to-state stable (ISS) systems. This paper derives small-gain-type theorems for interconnected systems involving iISS systems from the state-dependent scaling formulation. This paper provides solutions and Lyapunov functions explicitly. The new framework seamlessly generalizes the ISS small-gain theorem and classical stability criteria such as thecal Lₚsmall-gain theorem, the passivity theorems, the circle, and Popov criteria. State-dependence of the scaling is crucial for effective treatment of essential nonlinearities, while constants are sufficient for classical nonlinearities.