Stabilization of unstable first-order time-delay systems using fractional-order PD controllers

This paper considers the problem of stabilizing unstable first-order time-delay (FOTD) processes using fractional-order proportional derivative (PD) controllers. It investigates how the fractional derivative order mu in the range (0, 2) affects the stabilizability of unstable FOTD processes. The D-partition technique is used to characterize the boundary of the stability domain in the space of process and controller parameters. The characterization of a stability boundary allows one to describe and compute the maximum stabilizable time delay as a function of derivative gain and/or proportional gain. It is shown that for the the same derivative gain, a fractional-order PD controller with derivative order less than unity has greater ability to stabilize unstable FOTD processes than an integer-order PD controller. Such a fractional-order PD controller can allow the use of higher derivative gain than an integer-order PD controller. However, the setting of derivative gain greater than unity makes the maximum stabilizable time delay decrease drastically. When the derivative order mu is greater than unity, the allowable derivative gain is restricted to less than unity, as in the case of using an integer-order PD controller, and, for a fixed derivative gain, the maximum stabilizable time delay decreases as the derivative order mu is increased.