Necessary and Sufficient Conditions on Stabilization of Unstable Second-Order Delay Systems under PD Control

Unstable delay processess exist in most of the chemical and biological systems. Stabilization of such a delay plant is of great importance. Despite the great achievements of advanced control technologies, it has domenstrated that proportional-integral-derivative (PID) controller plays high impact in practice. This paper investigates the stabilization of second-order systems with two unstable real poles under PID control. By analyzing the stabilizability via the Nyquist stability criterion, we provide necessary and sufficient conditions on stabilization of unstable second-order plant with a constant delay under PD control. Accordingly, an algorithm on how to determine the feasible parameter region of the stabilizing PD controller is also given. Finally, two examples are proposed to illustrate the main results.