# Convex searches for discrete-time Zames-Falb multipliers

## TL;DR

This paper introduces convex optimization methods to find discrete-time Zames-Falb multipliers, demonstrating their effectiveness in stability analysis and their potential to approximate continuous-time multipliers.

## Contribution

It develops convex search techniques for discrete-time Zames-Falb multipliers using IIR and FIR approaches, showing FIR's completeness and effectiveness for large orders.

## Key findings

- FIR multipliers can approximate IIR multipliers arbitrarily well.
- Discrete-time searches yield state-of-the-art $\
- $-stability results for slope-restricted nonlinearities.

## Abstract

In this paper we develop and analyse 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 $\ell_{2}$-stability results in the literature for slope-restricted nonlinearities. Finally, we demonstrate that the discrete-time search can provide an effective method to find suitable continuous-time multipliers.