Unified Necessary and Sufficient Conditions for the Robust Stability of Interconnected Sector-Bounded Systems
Provides a unified theoretical framework for robust stability analysis of interconnected systems, benefiting control theorists and engineers by simplifying and generalizing existing results.
The paper unifies classical robust stability conditions (small gain, circle, passivity, conicity) into a single condition based on relations in a semi-inner product space, recovering both sufficient and necessary-and-sufficient versions, and deriving a new necessary and sufficient condition for weighted stability that implies exponential stability.
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.