Determination of all stabilizing fractional-order PID controllers that satisfy a robust performance constraint

This paper presents a method for finding all stabilizing fractional-order (FO) proportional-integral-derivative (PID) controllers that satisfy a robust performance constraint for a system of integer or non-integer order with a time delay. All the solutions to such FO PID controllers are calculated in the frequency domain and are given in terms of the proportional gain K
_p
, integral gain K
_i
, and derivative gain K
_d
. In this paper, they will be plotted on the (K
_p
, K
_i
), (K
_p
, K
_d
), and (K
_i
, K
_d
) planes. For a robust performance constraint, a weight is selected to bound all multiplicative errors resulting from parametric uncertainty and a sensitivity function weight is selected to set performance specifications. This approach provides all the possible gain parameter values of FO PID controllers that satisfy a robust performance condition. An example is given to illustrate the usefulness and effectiveness of the method.