Convex Characterisation of Frequency-Domain Discrete-Time Zames-Falb Multipliers

Stability of discrete-time Lur'e systems is one of the classical problems within the literature. For nonlinearities characterised by slope restriction, Zames-Falb multipliers provide a sufficient condition for robust stability, which may also be necessary, as posited by the Carrasco conjecture. In this paper, we explore a novel class of non-parametric discrete-time Zames-Falb multipliers with an infinite impulse response. We begin by introducing a new class of discrete-time systems characterised as continuous piecewise affine systems in the frequency domain. These systems exhibit well-defined and l1 bounded impulse responses and include both causal and anticausal components. This enables the specification of the class of non-parametric discrete-time Zames-Falb multipliers through linear constraints, allowing for convex search of the multipliers. Our analysis demonstrates that the proposed class of multipliers yields results similar to the state-of-the-art finite impulse response multipliers, while showing better numerical stability. This brings us closer to addressing the Carrasco conjecture regarding the necessity of Zames-Falb multipliers for the stability analysis of Lur'e systems.