Tuning PI and fractional order PI controllers with an additional fractional order Pole

The PI controller with an additional pole (PI?+?P) already has been proposed to decrease the noise effect on the control signal. In this paper, a fractional order pole is employed to increase the PI?+?P controller performance. The fractional order is obtained by adjusting the Nyquist plot slope in gain crossover frequency. This condition as well as gain crossover frequency and phase margin specifications are utilized to design the PI controller augmented with an additional fractional order pole (PI?+?FO[P]). To design these two controllers, a first-order plus delay time (FOPDT) model is utilized. For plants that could not be described by this model, its fractional order version (FFOPDT) could be utilized. In this case, a FOPI?+?FO[P] controller is obtained that could improve the transient response of the closed-loop system. The numerical simulations accomplished on various plant models (including chemical plants) demonstrate the effectiveness of the proposed controller comparing with the PI?+?P, PID, FOPI, and fractional order proportional-integral-derivative controllers.