# Convex searches for discrete-time Zames-Falb multipliers

arXiv: https://arxiv.org/abs/1812.02397 · HTML: https://mlsanity.com/paper/1812.02397/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Dec 2018.

## Built on by (most cited first, 4 of 4)

- [Linear systems with neural network nonlinearities: Improved stability analysis via acausal Zames-Falb multipliers](https://mlsanity.com/paper/2103.17106/) (2021)
- [Stability and Performance Analysis of Discrete-Time ReLU Recurrent Neural Networks](https://mlsanity.com/paper/2405.05236/) (2024)
- [A Complete Set of Quadratic Constraints for Repeated ReLU and Generalizations](https://mlsanity.com/paper/2407.06888/) (2024)
- [Mapping back and forth between model predictive control and neural networks](https://mlsanity.com/paper/2404.12030/) (2024)

## Abstract (arXiv)

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 ℓ2-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.