Tuning of controller parameters using Pythagorean fuzzy similarity measure for stable and time delayed unstable plants

This paper proposes a tuning method based on the Pythagorean fuzzy similarity measure and multi-criteria decision-making to determine the most suitable controller parameters for Fractional-order Proportional Integral Derivative (FOPID) and Integer-order Proportional Integral-Proportional Derivative (PI-PD) controllers. Due to the power of the Pythagorean fuzzy approach to evaluate a phenomenon with two memberships known as membership and non-membership, a multi-objective cost function based on the Pythagorean similarity measure is defined. The transient and steady-state properties of the system output were used for the multi-objective cost function. Thus, the determination of the controller parameters was considered a multi-criteria decision-making problem. Ant colony optimization for continuous domains (ACO(R)) and artificial bee colony (ABC) optimization are utilized to minimize multi-objective cost functions. The proposed method in the study was applied to three different systems: a second-order non-minimum phase stable system, a first-order unstable system with time delay, and a fractional-order unstable system with time delay, to validate its effectiveness. The cost function utilized in the proposed method is compared with the performance measures widely used in the literature based on the integral of the error, such as IAE (Integral Absolute Error), ITAE (Integral Time Absolute Error), ISE (Integral Square Error), and ITSE (Integral Time Square Error). The proposed method provides a more effective control performance by improving the system response characteristics compared to other cost functions. With the proposed method, the undershoot rate could be significantly reduced in the non-minimum phase system. In the other two systems, significant improvements were achieved compared to other methods by reducing the overshoot rate and oscillation. The proposed method does not require knowing the mathematical model of the system and offers a solution that does not require complex calculations. The proposed method can be used alone. Or it can be used as a second and fine-tuning method after a tuning process.