# Conditions for the equivalence between IQC and graph separation stability results

## TL;DR

This paper explores the conditions under which IQC and graph separation stability results are equivalent, highlighting the importance of symmetric factorizations for translating frequency-domain conditions into time-domain ones.

## Contribution

It introduces the concept of 'doubly-hard' factorizations necessary for equivalence and compares IQC and graph separation methods under these conditions.

## Key findings

- Doubly-hard factorizations are essential for equivalence.
- IQC can have advantages under certain conditions.
- A novel comparison between IQC and graph separation results.

## Abstract

This paper provides a link between time-domain and frequency-domain stability results in the literature. Specifically, we focus on the comparison between stability results for a feedback interconnection of two nonlinear systems stated in terms of frequency-domain conditions. While the Integral Quadratic Constrain (IQC) theorem can cope with them via a homotopy argument for the Lurye problem, graph separation results require the transformation of the frequency-domain conditions into truncated time-domain conditions. To date, much of the literature focuses on "hard" factorizations of the multiplier, considering only one of the two frequency-domain conditions. Here it is shown that a symmetric, "doubly-hard" factorization is required to convert both frequency-domain conditions into truncated time-domain conditions. By using the appropriate factorization, a novel comparison between the results obtained by IQC and separation theories is then provided. As a result, we identify under what conditions the IQC theorem may provide some advantage.