Fractional order PI-PD controller design for stable, unstable, and integrating processes with time delay

In the existing literature, the proposed design techniques that rely on determining the stability region’s center point for non-integer controllers are typically graphical and necessitate iterative drawings. The elimination of these iterative graphical procedures through analytical solutions is crucial for time-saving purposes. That’s why the objective of the article is to introduce a completely analytical approach to adjust the parameters of the fractional order PI-PD (FOPI-PD) controller applicable to time-delayed stable, unstable, and integrating plants. In pursuit of this objective, the analytical weighted geometrical center (AWGC) method, which furnishes analytical equations for deriving stability conditions expressed in terms of plant and controller parameters, has been modified to encompass the tuning of FOPI-PD type controllers. Facilitating the determination of the centroid point through analytical equations, the AWGC method enables the determination of the controller adjusting parameters by eliminating the repetitious drawing process of the stability boundary curves. Furthermore, simple analytical expressions have been given to compute the fractional integral and derivative orders, λ and µ, through the minimization of the Integral of Squared Time Error (ISTE) cost function. Moreover, analytical expressions are also provided for robustness parameters such as gain margin (GM), phase margin (PM), and maximum sensitivity (Ms), allowing the designer to determine if the design would result in adequate closed-loop performance or not. The effectiveness of the presented method has been showcased by the comparisons conducted regarding unit step responses for nominal, perturbed, and measurement noise cases. Moreover, robustness and performance parameters are compared using metrics such as settling time (ts), maximum sensitivity (Ms), integral of squared time error (ISTE), and total variation (TV) values. Also, the applicability of the introduced method in industrial plants has been proven through an application on an inverted pendulum mechanical unit.