In this paper, a new simple derivation of all stabilizing proportional controllers, for first order linear time invariant systems with time-delay, is presented. Although several results based on the Hermite-Biehler theorem for finding such a set of controllers exist, the aim of this article is to present a shorter and more instructive derivation, which can be followed easily.

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.

In this paper, a simple derivation for an all-stabilizing proportional controller set for first-order bi-proper systems with time delay is proposed. In contrast to proper systems, an extremely limited number of studies are available in the literature for such bi-proper systems. To fill this gap in the literature, broader aspects of the stabilizing set are taken into consideration. The effect of zero on the stabilizing set is clearly discussed and we also prove that, when their zeros are placed symmetrically to the origin, the stabilizing set of non-minimum phase plant is always smaller than that of the minimum phase one. Moreover, for an open-loop unstable plant, maximum allowable time delay (MATD) is explicitly expressed as a function of the locations of the pole and zero. From that function, it is shown that for a minimum phase plant, the supremum of the MATD is two times that of the time constant of the plant and the infimum of the MATD is the time constant of the plant. We also prove that the supremum is the time constant and the infimum is zero for a non-minimum phase plant.

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