Calculation of all gains providing time-delay independent stability via root locus

In this paper, a simple root locus based graphical method is proposed to calculate all stabilizing (destabilizing) proportional controller set which provides time-delay independent stability for single input single output (SISO) systems with time-delay. In other words, for the gains belong to this set, if the delay-free system is stable, the time-delay system controlled with these gains remains stable, regardless of the value of the time-delay. Conversely, if the delay-free system is unstable, stability of the time-delay system is not affected by the value of the time-delay; it remains unstable.