Delay-Independent Stability Test for Systems With Multiple Time-Delays

Delay-independent stability (DIS) of a general class of linear time-invariant (LTI) multiple time-delay system (MTDS) is investigated in the entire delay-parameter space. Stability of such systems may be lost only if their spectrum lies on the imaginary axis for some delays. We build an analytical approach that requires the inspection of the roots of finite number of single-variable polynomials in order to detect if the spectrum ever lies on the imaginary axis for some delays, excluding infinite delays. The approach enables to test the necessary and sufficient conditions of DIS of LTI-MTDS, technically known as weak DIS, as well as the robust stability of single-delay systems against all variations in delay ratios. The proposed approach, which does not require any parameter sweeping and graphical display, becomes possible by establishing a link between the infinite spectrum and algebraic geometry. Case studies are provided to show the effectiveness of the approach.