On numerical construction of worse-case convergence rates for Discrete-time Lurye systems with odd nonlinearities

This paper considers the asymptotic convergence rate of Lurye systems consisting of a discrete-time, LTI system in feedback with a nonlinearity. This work is motivated by convergence rate analysis of first-order optimization algorithms. Lurye analysis has been used in the literature to provide an upper bound on the worst-case convergence rate. If the nonlinearity is smooth then the asymptotic convergence rate is governed by the linearization near the equilibrium point. In this paper, we present a novel construction of nonlinearities that obtain slower convergence rates than the worst-case over smooth functions. In particular, our constructed nonlinearities are odd and have a fractal construction.