Conditions for the equivalence between IQC and graph separation stability results
For researchers in nonlinear control theory, this work clarifies the relationship between two major stability analysis frameworks, though the contribution is incremental.
This paper establishes conditions under which the IQC theorem and graph separation stability results are equivalent, showing that a symmetric 'doubly-hard' factorization is required to convert both frequency-domain conditions into truncated time-domain conditions. It identifies scenarios where IQC provides an advantage.
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.