Stability criteria for linear time-invariant systems with point delays based on one-dimensional Routh–Hurwitz tests

This brief deals with the asymptotic stability of a class of linear time-invariant systems subject to point constant uncommensurate delays. Results are obtained dependent on and independent of the delays which may be tested be performing a finite set of 1-D Routh–Hurwitz tests on a corresponding set of auxiliary delay-free linear time-invariant systems plus some supplementary conditions related to either commutation pairwise of the associate dynamics matrices or their norm closeness or some simple testable conditions if the matrices have particular forms and there is only one single delay. The obtained results are applicable to the case when a finite set of matrices describing the dynamics of a set of delay-free auxiliary dynamical systems are Hurwitz and either commute pair-wise or their norms are sufficiently close to each other. Some extensions are given to robust stability of point time-delay time invariant systems with commensurate delays parametrized by an uncertain parameter vector whose components are in real intervals of known limits.