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This paper considers the robust stability of a discrete-time Lurye systemconsisting of the feedback interconnection between a linear system and abounded and monotone nonlinearity. It has been conjectured that the existenceof a suitable linear time-invariant (LTI) O'Shea-Zames-Falb multiplier is notonly sufficient but also necessary. Roughly speaking, a successful proof of theconjecture would require: (a) a conic parameterization of a set of multipliersthat describes exactly the set of nonlinearities, (b) a lossless S-procedure toshow that the non-existence of a multiplier implies that the Lurye system isnot uniformly robustly stable over the set of nonlinearities, and (c) theexistence of a multiplier in the set of multipliers used in (a) implies theexistence of an LTI multiplier. We investigate these three steps, showing thecurrent bottlenecks for proving this conjecture. In addition, we provide anextension of the class of multipliers which may be used to disprove theconjecture.

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