Title: Conditions for the equivalence between IQC and graph separation stability 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.