Performance comparison between PID, PIDD2 and PIDD2α

We evaluate the performance achievable with Proportional-Integral-Derivative fractional Double-Derivative controllers, denoted as PIDD2α, by comparing it with that achievable with standard PID controllers and with Proportional-Integral-Double-Derivative (PIDD or PIDD2) controllers, which are also called Proportional-Integral-Derivative-Acceleration (PIDA) controllers. The cost-effectiveness ratio improvement obtained when augmenting the control law with an element proportional to the fractional second derivative of the control error is assessed by considering different process transfer functions and by tuning the controllers minimizing the integrated absolute error with genetic algorithms. Further, a constraint on the maximum sensitivity is also posed so that the system robustness is taken into account. The set-point following and the load disturbance rejection tasks are evaluated separately.