Dixon Resultant for Cluster Treatment of LTI Systems with Multiple Delays

Abstract

The cluster treatment of characteristic roots (CTCR) paradigm, is studied in a new angle for the stability analysis of linear time invariant (LTI) systems with multiple independent delays. For such systems, all the imaginary characteristic roots can be found exactly and exhaustively along a set of hypersurfaces in the domain of the delays. These are called the kernel hypersurfaces (KH). Any delay composition that yields an imaginary root resides either on this kernel or its infinitely many offspring hypersurfaces (or OH). The exhaustive determination of the KH is the only prerequisite for the CTCR paradigm. However, as the number of delays increases, it takes longer to calculate the KH in the entire delay domain. Alternatively, we present a novel procedure to determine the 2–D cross-section of the KH in the space of any two of the delays. This procedure utilizes the half angle tangent substitution method and the Dixon resultant theory to determine KH. With the information of KH, CTCR is then used for the 2–D cross-section of the exact stability map. Finally, an example case with six delays is presented to show the validity of the methodology.