Phase limitations of multipliers for nonlinearities with monotone bounds

We consider Lurye systems, whose nonlinear operator is characterized by a nonlinearity that is bounded above and below by monotone functions. Absolute stability can be established using a subclass of the O’Shea-Zames-Falb multipliers. We develop phase conditions for both continuous-time and discrete time systems under which there can be no such suitable multiplier for the transfer function of a given plant. In discrete time the condition can be tested via a linear program, while in continuous time it can be tested efficiently by exploiting convex structure. Results provide useful insight into the dynamic behaviour of such systems and we illustrate the phase limitations with examples from the literature.