A Local Small Gain Theorem and Its Use for Robust Stability of Uncertain Feedback Volterra Systems

Abstract

The requirement to evaluate a gain over the whole signal space is one of the restrictions in the well-known small gain theorem. Using the concepts of local gain and strict causality a local form of small gain theorem is proposed, which can be used to analyze input magnitude dependent stability problems of feedback nonlinear systems, such as a Volterra system. Since only finite order Volterra series can be handled in practice, an uncertainly model is derived to address the robustness issue of approximating a nonlinear system by a finite Volterra series in the context of closed-loop control. The local small gain theorem is then used to analyze the feedback properties of the uncertain Volterra system and a sufficient condition for robust stability is obtained.