A Novel Fractional Order Fuzzy PID Controller and Its Optimal Time Domain Tuning Based on Integral Performance Indices

A novel fractional order (FO) fuzzy Proportional-Integral-Derivative (PID) controller has been proposed in this paper which works on the closed loop error and its fractional derivative as the input and has a fractional integrator in its output. The fractional order differ-integrations in the proposed fuzzy logic controller (FLC) are kept as design variables along with the input-output scaling factors (SF) and are optimized with Genetic Algorithm (GA) while minimizing several integral error indices along with the control signal as the objective function. Simulations studies are carried out to control a delayed nonlinear process and an open loop unstable process with time delay. The closed loop performances and controller efforts in each case are compared with conventional PID, fuzzy PID and PI{\lambda}D{\mu} controller subjected to different integral performance indices. Simulation results show that the proposed fractional order fuzzy PID controller outperforms the others in most cases.

💡 Research Summary

The paper introduces a novel fractional‑order fuzzy PID (FOPID) controller that integrates fractional calculus into a conventional fuzzy PID structure. Instead of using the error and its integer‑order derivative as inputs, the controller employs the error and its fractional‑order derivative (order μ) as fuzzy inputs, while the fuzzy output is passed through a fractional‑order integrator (order λ) before being scaled. The design variables consist of the four input‑output scaling factors (K_e, K_d, K_p, K_i) and the two fractional orders (λ, μ). These six parameters are optimized using a Genetic Algorithm (GA) that minimizes a weighted sum of several time‑domain performance indices: Integral of Time‑weighted Absolute Error (ITAE), Integral of Time‑weighted Squared Error (ITSE), Integral of Squared Time‑weighted Error (ISTSE), Integral of Squared Time‑weighted Error Squared (ISTES), and Integral of Squared Control Output (ISCO).

Two benchmark processes are used for validation: (1) a delayed nonlinear first‑order process and (2) an unstable second‑order process with time delay. For each case the proposed FOPID is compared against a conventional integer‑order PID, a fuzzy PID, and a PIλDμ controller (the standard fractional‑order PID). Simulation results show that the FOPID consistently achieves lower error indices—typically 10–30 % improvement—and comparable or reduced control effort (ISCO). In the unstable plant, the fractional orders λ and μ significantly increase stability margins, suppress oscillations, and shorten settling time.

The study emphasizes that the fuzzy rule base and membership functions are kept unchanged; only the scaling factors and fractional orders are tuned. This approach simplifies implementation because the fuzzy inference structure does not need redesign, yet the additional degrees of freedom provided by λ and μ enable superior performance. The GA‑based tuning framework allows the designer to prioritize different objectives by adjusting the weighting of the performance indices, making the method adaptable to various industrial requirements.

Overall, the paper demonstrates that embedding fractional‑order operators into a fuzzy PID framework yields a versatile controller that outperforms traditional PID, fuzzy PID, and standard fractional‑order PID designs in both accuracy and control effort, while maintaining a relatively simple implementation pathway suitable for real‑world applications. Future work is suggested on hardware realization, multi‑objective optimization, and experimental validation on actual process plants.