Convex Searches for Discrete-Time Zames–Falb Multipliers

In this article, we develop and analyze convex searches for Zames-Falb multipliers. We present two different approaches: infinite impulse response (IIR) and finite impulse response (FIR) multipliers. The set of FIR multipliers is complete in that any IIR multipliers can be phase-substituted by an arbitrarily large-order FIR multiplier. We show that searches in discrete time for FIR multipliers are effective even for large orders. As expected, the numerical results provide the best ℓ
₂
-stability results in the literature for slope-restricted nonlinearities. In particular, we establish the equivalence between the state-of-the-art Lyapunov results for slope-restricted nonlinearities and a subset of the FIR multipliers. Finally, we demonstrate that the discrete-time search can provide an effective method to find suitable continuous-time multipliers.