Direct method implementation for the stability analysis of multiple time delayed systems

A recent procedure, the direct method (DM), is presented for the stability treatment of a class of linear time invariant (LTI) systems with independent multiple time delays (MTDS). Since they appear in many practical applications in the systems and control community, this class of dynamics has attracted considerable interest. However their stability analysis is complicated because of the infinite dimensional nature (due, solely, to each delay) and the multiplicity of these delays. The problem is more challenging to resolve for the stability posture compared to the TDS with commensurate time delays (where time delays have rational relations). It is shown in the earlier publications of the authors that the DM brings an exact and structured methodology for the commensurate time delays to multiple delay cases motivates our study. It is shown that the DM reveals all possible stability regions in the space of multiple time delays. The systems under consideration do not have to possess stable behavior for zero delays. For the stability analysis the method follows the same fundamental paradigm of "cluster treatment of the characteristic roots" as the direct method does. Finally we present a few numerical examples exhibiting the strengths of DM procedure.