Critical Effects of the Polarity Change in Delayed States Within an LTI Dynamics With Multiple Delays

This note concerns the polarity inversion of the delayed states in linear time-invariant multiple time delay systems (LTI-MTDS). To start with, for such systems the assessment of asymptotic stability is complicated due to the infinite dimensionality introduced by the delays. It is shown that the mentioned polarity inversion influences the respective delay-dependent stability maps in interesting ways. There are some intriguing invariant characteristics and correspondences between the original and inverted systems, which form the primary contributions of this note. These findings are shown to be the enabling features to relate the stability maps of the two systems. The Cluster Treatment of Characteristic Roots (CTCR) paradigm is utilized to identify these features. An example case study is provided to illustrate the claims.