Duality bounds for discrete-time Zames–Falb multipliers

This note presents phase conditions under which there is no suitable Zames–Falb multiplier for a given discrete time system. Our conditions can be seen as the discrete-time counterpart of Jönsson’s duality conditions for Zames–Falb multipliers. By contrast with their continuous-time counterparts and other phase limitations in the literature, they lead to numerically efficient results that can be computed either in closed form or via a linear program. The closed-form phase limitations are tight in the sense that we can construct multipliers that meet them with equality. The numerical results allow us to conclude that the current state-of-the-art in searches for Zames–Falb multipliers are not conservative. Moreover, they allow us to show, by construction, that the set of plants for which a suitable Zames– Falb multiplier exists is non-convex.