Determination of All Stabilizing Constant Feedback Gains for Open- loop Unstable Second-order Systems with Time Delay

In this paper, the problem of stabilizing an open-loop unstable second-order system with time delay is considered. A constant feedback gain is utilized to stabilize the system. Using a modified version of Hermite-Biehler Theorem, which is applicable to quasi-polynomialsthe complete set of stabilizing constant feedback gains for open-loop unstable plants, is determined. The desired closed-loop performance, e.g., overshoot, settling time, rise time, etc. can be achieved by proper selection of the gain within the boundaries of obtained set of stabilizing gains. The results reported here will serve as a stepping stone for tackling the more complicated case of analytic control and stabilization of second order systems with time delay, using a PI or a PID controller. The proposed procedure is studied on a numerical example. Key-Words: Second-order systems, open-loop unstable plants, feedback stabilization, time delay, constant feedback gain.