Designing non-overshooting fractional-order PI controllers for a particular case of fractional-order systems

It is crucial to attain a non-overshooting step response in various applications. On the other hand, although various approaches have been presented to design fractional-order proportional-integral (FOPI) controllers in the literature, none of them can ensure a non-overshooting unit step response (USR) for the closed-loop system. This paper designs non-overshooting FOPI controllers for a fractional-order system (FOS) with one and two fractional orders. First, the FOPI controller parameters are chosen to achieve a monotonic magnitude-frequency response (MFR) for the closed-loop system, thereby achieving a non-overshooting or minimum overshoot USR. It is demonstrated that if the sum of the fractional order of the FOPI controller and the maximum order of the plant (for the FOS with one and two fractional orders, where the fractional orders are between one and zero) is equal to 1, a monotonic MFR can be attained. The proportional gain and the integrator time constant are then calculated to attain a desired phase margin (PM) and loop gain crossover frequency (GCF). The main constraints on PM are attained, and the stability of the closed-loop system is proved. Numerical simulations demonstrate the correctness of the presented controller in the presence of disturbance and uncertainty in model parameters. Comparative simulations demonstrate the superiority of the designed FOPI to an already published work in the literature. The performance of the proposed FOPI controller in controlling the ionic polymer-metal composite actuator is also demonstrated.