Stability analysis for LTI systems with multiple time delays using the bounds of its imaginary spectra

The stability of linear time invariant (LTI) systems with multiple independent time delays and the cluster treatment of characteristic roots (CTCR) paradigm are investigated from a new perspective. Any delay composition that results in an imaginary characteristic root lies either on a small number of kernel hypersurfaces (KH) or their infinitely many offspring hypersurfaces (OH). The complete description of KH is the only prerequisite for the CTCR-based stability assessment procedure. As the number of delays increases, however, the determination of these KH becomes computationally costly. Instead, we present a practical procedure to extract the 2-D cross-sections of the KH set in the domain of the two arbitrarily selected delays. First, we determine the exact upper and lower bounds of the imaginary spectra in this 2-D cross-section of interest. This process starts with a half-angle tangent representation of the characteristic equation followed by the Dixon resultant operation. Based on the full knowledge of KH, the CTCR paradigm creates the complete stability map in the domain of these two arbitrarily selected delays. We demonstrate the effectiveness of this methodology over an example case study.