Dixon resultant theory for stability analysis of distributed delay systems and enhancement of delay robustness

This study scrutinizes the stability problem of linear time-invariant feedback control systems with a constant-coefficient, partial delay distribution from a new perspective, which is built on an equivalence between the system of interest and the one with two lumped delays. We aim to determine all the potential stability changing curves (PSCC) of the system in the domain of delays in order to make a non-conservative stability assessment. First, we propose the Dixon resultant-based frequency sweeping procedure to calculate the so-called kernel and offspring hypersurfaces (KOH) of the system. The superiority in the computational efficiency of this Dixon-type method is revealed by comparison with the Sylvester-type one. Second, we specifically tackle the standing root case for the singularity at the zero root, leading to what we call the standing root boundary (SRB). Then, we claim that the union of the KOH and SRB constitutes the PSCC of the system. With these, the stability map of the system is then created using the Cluster Treatment of Characteristic Roots paradigm. Furthermore, we declare the delay robustness is enhanced by the proposed control law. Finally, we demonstrate the effectiveness of the presented procedures over two example case studies by the Quasi-Polynomial mapping-based Root-finder routine as well as the Simulink-based simulation.