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Abstract
Absolute stability attracted much attention in the 1960s. Several stability conditions for loops with slope-restricted nonlinearities were developed. Results such as the Circle Criterion and the Popov Criterion form part of the core curriculum for students of control. Moreover, the equivalence of results obtained by different techniques, specifically Lyapunov and Popov׳s stability theories, led to one of the most important results in control engineering: the KYP Lemma.For Lurye11Also written as Lur׳e or Lurie. systems this work culminated in the class of multipliers proposed by O׳Shea in 1966 and formalized by Zames and Falb in 1968. The superiority of this class was quickly and widely accepted. Nevertheless the result was ahead of its time as graphical techniques were preferred in the absence of readily available computer optimization. Its first systematic use as a stability criterion came 20 years after the initial proposal of the class. A further 20 years have been required to develop a proper understanding of the different techniques that can be used. In this long gestation some significant knowledge has been overlooked or forgotten. Most significantly, O׳Shea׳s contribution and insight is no longer acknowledged; his papers are barely cited despite his original parameterization of the class.This tutorial paper aims to provide a clear and comprehensive introduction to the topic from a user׳s viewpoint. We review the main results: the stability theory, the properties of the multipliers (including their phase properties, phase-equivalence results and the issues associated with causality), and convex searches. For clarity of exposition we restrict our attention to continuous time multipliers for single-input single-output results. Nevertheless we include several recent significant developments by the authors and others. We illustrate all these topics using an example proposed by O׳Shea himself.
Highlights
A feedback interconnection of a linear system and a static nonlinearity is said to be absolutely stable if the interconnection is stable for every nonlinearity in a given class
We review the main results: the stability theory, the properties of the multipliers, and convex searches
All other classes of multipliers in the literature have been shown to be phase-equivalent to Zames-Falb multipliers (Carrasco et al, 2013, 2014a). It is an open question whether any possible class of multipliers preserving the positivity of monotone and bounded nonlinearities is phase-contained in the class of Zames–Falb multipliers
Summary
A feedback interconnection of a linear system and a static nonlinearity is said to be absolutely stable if the interconnection is stable (in some sense) for every nonlinearity in a given class. The absolute stability problem can be studied, broadly, from either the perspective of internal stability, or from that of input-output stability The former, and perhaps more common, approach typically involves the search for the parameters of a proposed Lyapunov function which can be used to guarantee asymptotic stability of the origin for as large a class of nonlinearities as possible. Zames and Falb (1968) focus on the relation of the nonlinearity to the monotone and bounded static nonlinearity; O’Shea’s insights into the phase properties of the multipliers have been largely forgotten (with one notable exception: the discussion of Megretski (1995) on phase limitation) In this tutorial paper we re-examine Zames-Falb multipliers and, in particular, use an example of O’Shea (1967) to discuss the phase properties of the Zames-Falb multipliers and how they can be used advantageously in the study of the absolute stability problem. While we emphasise the tutorial aspect of this overview, some mathematical formalism and machinery is inevitable; this is given in the appendix
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