On the necessity and sufficiency of the small phase theorem for nonlinear fading memory systems

Abstract

This paper is concerned with the phases of sectorial systems to closed-loop stability of discrete-time time-varying (TV) nonlinear systems with fading memory. The framework considers a nonlinear system to be a causal nonlinear operator on signal spaces. For the class of stable nonlinear systems with fading memory, a small phase theorem is established for closed-loop stability. When the closed-loop system is uniformly stable over the phase bounded uncertainty set, we show that the small phase theorem is necessary.