Robustness in fractional proportionalߝintegralߝderivative-based closed-loop systems

Robustness of a fractional proportional-integral-derivative (PID)-based control system is investigated. At first the largest pathwise connected region subset of a box in three-dimensional space of the parameters of the model is determined such that the closed-loop system is bounded-input bounded-output stable for any point inside it. Then a value that represents the size (in a specified sense) of the calculated region in the first stage and can be considered as a margin for the robustness of the closed-loop system is computed. Furthermore, lower and upper frequency bounds required in depiction of boundaries of the region and computing the mentioned margin are provided. Some special cases in two-dimensional space of the model parameters are investigated as well. To illustrate the results, a numerical example is presented.