Complete stability assessment of LTI systems with multiple constant-coefficient distributed delays using the improved frequency sweeping framework

The stability problem of linear time-invariant systems with multiple constant-coefficient distributed delays is investigated from a new perspective. We present an equivalent system with multiple lumped delays and then apply the Dixon resultant to determine the exact upper bound of the projected imaginary spectra of the system. We then adopt the improved frequency sweeping framework over the obtained range to get the kernel and offspring hypersurfaces (KOH). Furthermore, the singularity at the zero characteristic root is scrutinised, leading to what we call the stationary root boundary (SRB) in the domain of the delays. With these, we resort to the Cluster Treatment of Characteristic Roots (CTCR) paradigm to determine the complete stability map of the system with the complete knowledge of the KOH and SRB. Finally, the effectiveness of our proposed procedure is shown by two case studies.