Absolute Stability

Abstract

AbstractAbsolute stability is one of the cornerstones of control theory. It was defined by Lurye and Postnikov in 1944 and has attracted much attention since then. Absolute stability concerns the stability of the feedback interconnection between a linear system and a nonlinear and/or uncertain object that belongs to a class of systems and is classically a memoryless nonlinearity defined by its input–output map. This feedback setup is known as the Lurye system. There has been particular focus on the case when the nonlinearity is slope restricted, that is, the maximum incremental gain of the nonlinearity is bounded. Furthermore, absolute stability is linked to the development of several underpinning results in control theory, for example, the Passivity Theorem, the Small Gain Theorem, dissipativity theory, integral quadratic constraint (IQC) theorems, and the Kalman–Yakubovich–Popov (KYP) Lemma. In this article, we cover the fundamental milestones in the development of the field, especially frequency‐domain techniques, from classical results such as the Circle Criterion up to the most recent developments with O'Shea–Zames–Falb multipliers.