  The response to a setpoint change in PID controlled feedback systems plays an
important role for the tuning methods. This response may be easily evaluated in
linear systems without delay by solving the related ordinary differential
equations. These equations are also obtained for system with delay, if the
delay exponential term is approximated to a polynomial by one of the Pade
expressions. If this simplification is avoided, one must deal with differential
difference equations. In this paper their explicit solution, obtained with the
method of steps, is presented. Moreover a second order equation is investigated
as example and the complete set of the coefficients formulas is given.
