Generalized Dixon Resultant for Strong Delay-Independent Stability of Linear Systems With Multiple Delays

The strong delay-independent stability (DIS) of linear time-invariant-retarded multiple time-delay systems (MTDSs) is studied in the entire delay space. We develop a new one-step discriminant approach utilizing the generalized Dixon resultant, which unveils the necessary and sufficient algebraic condition for the upper frequency bound. The lower frequency bound is rigorously ascertained by further inspecting the zero spectrum. Then, we raise seven postulates concerning the system stability state for the strong DIS analysis, which are examined by four independent stability criteria. Additionally, we integrate the Gaussian-elimination-based rank subdeterminant calculation into our algorithm to deal with the degenerate resultant operator. Finally, we prove that our one-step approach is more uniform, automatic, and efficient compared to the iterated discriminant approach with successive Sylvester resultant computations, especially when tackling a complicated MTDS with more than two delays. Several example cases are provided.