A novel stability study on multiple time-delay systems (MTDS) using the root clustering paradigm

A novel treatment is presented for the stability of linear time invariant (LTI) systems with rationally independent multiple time delays. The stability analysis of the time-delayed systems (TDS) is quite complex, because they are infinite dimensional. Multiplicity and ´rationally independent´ feature of the delays makes the problem even more challenging compared to the TDS with commensurate time delays (where time delays are rationally related). Recently the authors introduced a new perspective, which brings a unique, exact and structured methodology for the stability analysis of commensurate time delayed cases. The transition from this class of TDS to those with multiple delays using a similar perspective motivates the present study. The new framework is described and the enabling propositions are proven. We show that this procedure reveals all possible stability regions exclusively in the space of independent multiple time delays. As an added strength, it does not require the MTDS under consideration to be stable for zero delays. We present an example numerical case study, which is considered prohibitively difficult to solve using the peer methodologies.