LMI searches for discrete-time Zames-Falb multipliers

Recent works have proposed different searches for Zames-Falb multipliers in the continuous-time domain. Although Zames-Falb multipliers are also defined for the discrete-time domain, there is no equivalent literature for their search. In this paper, we propose the discrete-time counterparts of two recently reported algorithms for the search of a Zames-Falb multiplier. The two searches find causal and anticausal multipliers respectively. Several examples show the effectiveness of the method. The examples illustrate the necessity of conducting both causal and anticausal searches to ensure competitiveness over other methods for establishing the stability of loops with slope-restricted nonlinearities.