Lyapunov functions for discrete-time multivariable Popov criterion with indefinite multipliers

This paper shows the existence of Lur´e-Postkinov type Lyapunov functions for the discrete-time multivariable Popov criterion with indefinite multipliers. The nonlinearities in the Lur´e systems considered here are monotonic, sector- and slope-restricted. We discuss the case where the nonlinearities are diagonal. Our derivation is based on the discrete-time Kalman-Yakubovich-Popov (KYP) lemma and the S-Procedure, and results in Linear Matrix Inequality (LMI) conditions which can be solved using convex optimization methods.