LMI searches for anticausal and noncausal rational Zames-Falb multipliers

Given a linear time-invariant plant, the search for a suitable multiplier over the class of Zames-Falb multipliers is a challenging problem which has been studied for several decades. Recently, a new linear matrix inequality search has been proposed over rational and causal Zames-Falb multipliers. This letter analyzes the conservatism of the restriction to causality on the multipliers and presents a complementary search for rational and anticausal multipliers. The addition of a Popov multiplier to the anticausal Zames-Falb multiplier is implemented by analogy with the causal search. As a result, a search over a noncausal subset of Zames-Falb multipliers is obtained. A comparison between all the search methods proposed in the literature is given.