On the Necessity and Sufficiency of Discrete-Time O'Shea-Zames-Falb Multipliers

TL;DR

This paper examines the conditions under which discrete-time O'Shea-Zames-Falb multipliers are necessary and sufficient for robust stability of Lurye systems, highlighting current challenges and extending multiplier classes.

Contribution

It analyzes the conjecture on the necessity and sufficiency of LTI multipliers and introduces an extended class of multipliers to challenge the conjecture.

Findings

Identifies key bottlenecks in proving the conjecture.

Provides an extended class of multipliers to disprove the conjecture.

Highlights the need for a lossless S-procedure and conic parameterization.

Abstract

This paper considers the robust stability of a discrete-time Lurye system consisting of the feedback interconnection between a linear system and a bounded and monotone nonlinearity. It has been conjectured that the existence of a suitable linear time-invariant (LTI) O'Shea-Zames-Falb multiplier is not only sufficient but also necessary. Roughly speaking, a successful proof of the conjecture would require: (a) a conic parameterization of a set of multipliers that describes exactly the set of nonlinearities, (b) a lossless S-procedure to show that the non-existence of a multiplier implies that the Lurye system is not uniformly robustly stable over the set of nonlinearities, and (c) the existence of a multiplier in the set of multipliers used in (a) implies the existence of an LTI multiplier. We investigate these three steps, showing the current bottlenecks for proving this conjecture. In addition, we provide an extension of the class of multipliers which may be used to disprove the conjecture.