A Lyapunov-based perspective on absolute stability

Abstract

This article presents a unifying perspective on absolute stability concepts. In particular, it develops a Lyapunov-like explanatory framework for a nonscalar circle criterion with its small-gain and strict-passivity special cases. A general defining inequality for the Lyapunov-like function is proposed that avoids definiteness constraints. We discuss different ways to derive a quadratic solution: via a linear matrix inequality (LMI), an algebraic Riccati equation, and a matrix equation. By exploiting the Kalman–Yakubovich–Popov (KYP) lemma, classical frequency-domain results are recovered. A passivity-index-based result is derived that simplifies the evaluation. Overall, the presented interrelations may be useful for both analysis and teaching.