Unified Necessary and Sufficient Conditions for the Robust Stability of Interconnected Sector-Bounded Systems

TL;DR

This paper introduces a unified framework for the robust stability of interconnected systems with sector-bounded nonlinearities, generalizing classical results and providing necessary and sufficient conditions in a broad semi-inner product space setting.

Contribution

It presents a novel, unified stability condition applicable to a wide class of systems, recovering classical results and deriving new necessary and sufficient conditions for weighted and exponential stability.

Findings

Unified stability condition in semi-inner product spaces

Recovery of classical necessary and sufficient results

New conditions for weighted and exponential stability

Abstract

Classical conditions for ensuring the robust stability of a linear system in feedback with a sector-bounded nonlinearity include small gain, circle, passivity, and conicity theorems. In this work, we present a similar stability condition, but expressed in terms of relations defined on a general semi-inner product space. This increased generality leads to a clean result that can be specialized in a variety of ways. First, we show how to recover both sufficient and necessary-and-sufficient versions of the aforementioned classical results. Second, we show that suitably choosing the semi-inner product space leads to a new necessary and sufficient condition for weighted stability, which is in turn sufficient for exponential stability.