  In this paper a new optimum tuning method of PI controllers in first-order
time-delay systems, based on the deadbeat response to a step setpoint
variation, is presented. The deadbeat performance, already studied for the
plants without delay, consists of a fast achievement of the steady state with
an overshoot included in a narrow band and with minimum rise and settling
times. In the proposed method the rise and settling times are both replaced
with the integral of the squared error and the constraint, which requires a
controller output less than a preset value, is added. The transient behaviour
is strictly evaluated by means of the analytical solutions of the relative
differential difference equations, obtained by the method of steps for the
first time. Moreover the proposed tuning method is compared with other three
methods, selected among the most used, which are the Ziegler-Nichols
time-domain and frequency-domain and the Zhuang and Atherton ISTE ones. Finally
tuning charts, having as coordinates the two PI controller parameters and
provided with the borderlines of the stability region, are introduced.
