  In some of the complicated control problems we have to use the controllers
that apply nonlocal operators to the error signal to generate the control.
Currently, the most famous controller with nonlocal operators is the
fractional-order PID (FOPID). Commonly, after tuning the parameters of FOPID
controller, its transfer function is discretized (for realization purposes)
using the so-called generating function. This discretization is the origin of
some errors and unexpected results in feedback systems. It may even happen that
the controller obtained by discretizing a FOPID controller works worse than a
directly-tuned discrete-time classical PID controller. Moreover, FOPID
controllers cannot directly be applied to the processes modeled by, e.g., the
ARMA or ARMAX model. The aim of this paper is to propose a discrete-time
version of the FOPID controller and discuss on its properties and applications.
Similar to the FOPID controller, the proposed structure applies nonlocal
operators (with adjustable memory length) to the error signal. Two methods for
tuning the parameters of the proposed controller are developed and it is shown
that the proposed controller has the capacity of solving complicated control
problems with a high performance.
