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

Saman Cyrus, Laurent Lessard

Published 2018-09-24Version 1

Classical conditions for ensuring the robust stability of a linear system in feedback with a sector-bounded nonlinearity include the circle, small gain, passivity, and conicity theorems. In this work, we present a similar stability condition, but expressed in terms of operators on a 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 exponential stability with a given convergence rate. Finally, in the spirit of classical robust stability analysis, we provide linear matrix inequalities that allow for the efficient verification of the conditions of our theorem.