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.