Phase limitations of discrete-time Zames-Falb multipliers

We derive phase limitations of discrete-time Zames-Falb multipliers. In some cases the phase limitations may be used to show there exists no appropriate Zames-Falb multiplier for a Lur´e system with a given LTI plant and a given class of slope-restricted nonlinearities. We conjecture that this in turn implies the Lur´e system is not absolutely stable. We propose an algorithm to obtain, given an LTI plant, the minimum slope restriction where the phase limitation implies there is no appropriate Zames-Falb multiplier. We illustrate the results with some numerical examples and discuss the implications for benchmarking of stability tests.