State-Dependent Scaling Characterization for Interconnected Nonlinear Systems–Part I : Constructive Generalization of Stability Criteria

This paper is concerned with the problem of global stability and performance of nonlinear interconnected systems. The state-dependent scaling problem is proposed as a unified mathematical formulation whose solutions explicitly provide Lyapunov functions proving stability and dissipative properties of feedback and cascade systems. The framework covers nonlinear systems having diverse and strong nonlinearities represented by general supply rates. Previously existing stability theorems such as the L2 small-gain theorem, the passivity theorems, the circle and Popov criteria and the input-to-state stable(ISS) small-gain theorem are extracted from the framework as special elementary cases. Indeed, those existing stability criteria are sufficient conditions for the existence of solutions to the state-dependent scaling problems, and the solutions can be shown explicitly. A notable benefit of the proposed formulation is that it actually yields useful answers for systems disagreeing with classical and ISS nonlinearities, which is much more than formal applicability. The paper focuses on generalization and unification, while the follow-up paper puts a special emphasis on theoretical demonstration of the effectiveness of the state-dependent scaling characterization