Determination of stabilizing feedback gains for second-order systems with time delay

In this paper we consider the problem of stabilizing a second-order plant with dead-time using a constant gain controller. Due to the presence of the time delay, the number of roots of the characteristic equation of the closed-loop system is infinite, making the problem posed in this paper a difficult one. A complete analytical characterization of all stabilizing feedback gains is provided using a version of the Hermite-Biehler Theorem applicable to quasipolynomials. Both open-loop stable and unstable plants are considered. The results reported in this paper will serve as a stepping stone for tackling the more complicated cases of stabilization using a PI or a PID controller.