Small gain and scaling for stability and performance robustness of nonlinear systems

This paper focuses on robust stability and performance of uncertain nonlinear systems with norm-bounded structured and parameter uncertainty. It is shown that, in a general setting under a considerably mild assumption, a small gain condition with constant or dynamic scaling implies robust performance. The proof involves neither Riccati inequality nor Hamilton-Jacobi inequality nor destabilizing perturbation. It is demonstrated that by contrast with uncertain LTI systems, dynamic scaling gives a less conservative stability condition for parameter uncertainty, even when uncertainty is unstructured. A sufficient test for robust local performance is also obtained in terms of a local small gain condition