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A Novel Fractional Order Fuzzy PID Controller and Its Optimal Time Domain Tuning Based on Integral Performance Indices
Saptarshi Das, Indranil Pan, Shantanu Das, Amitava Gupta
TL;DR
Tuning fuzzy PID controllers for nonlinear and delayed unstable processes is difficult because analytical stability analysis is limited. The paper proposes a genetic-algorithm-tuned fractional-order fuzzy PID controller and reports superior performance in most cases, especially for set-point tracking.
Problem
Analytical stability analysis using error criteria is limited to linear systems, motivating time-domain tuning for nonlinear processes with controller nonlinearities.
Method
The proposed fractional-order fuzzy PID controller uses error and fractional error-rate inputs, a fractional integrator output, and genetic-algorithm tuning of orders, scaling factors, and performance objectives.
Results
The proposed fuzzy FOPID outperforms the compared controllers for almost all set-point-tracking performance indices, while control-signal advantages vary across cases.
Takeaways & Limitations
Fractional-order fuzzy PID tuning provides a strong option for set-point control of the studied delayed nonlinear and unstable processes.
Takeaways & Limitations
The controllers are optimized for set-point changes rather than load disturbances, so their load-disturbance rejection is not consistently strong.
Abstract
from arXiv · showhide
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 PIλDμ controller subjected to different integral performance indices. Simulation results show that the proposed fractional order fuzzy PID controller outperforms the others in most cases.
3. New fractional order fuzzy PID controller and its time domain optimal tuning:
The paper develops a fractional-order fuzzy PID controller whose fractional differ-integrals and scaling factors are tuned for time-domain performance. Its design uses a compact fuzzy rule base and rational approximations to support simulation and implementation.
- Controller structure: The proposed controller combines fuzzy PI and fuzzy PD structures, using error and fractional error-rate information in its fuzzy control design.Its membership functions and rule bases follow earlier integer-order fuzzy PID structures.
- Controller structure: The tuning emphasizes scaling factors because output scaling changes can affect fuzzy-controller performance more strongly than membership-function shape changes.Therefore, tuning parameters are not equally potent in determining overall controller performance.
- Fractional-order realization: Fractional-order elements are rationalized during optimization with Oustaloup’s 5th-order approximation over the frequency range 10^-2 to 10^2 rad/sec.Band-limited realization is required because fractional differ-integrals are infinite-dimensional filters.
- Optimal tuning: ITAE, ITSE, ISTES, and ISTSE are used as time-domain integral performance indices together with control cost to handle nonlinear process and fuzzy-inference effects.The framework uses optimization because analytical ISE-based stability analysis applies only to linear systems.
4. Simulations and Results:
Simulations compare optimally tuned fuzzy and non-fuzzy PID variants across nonlinear and open-loop unstable delayed processes using multiple integral performance indices. The proposed fuzzy FOPID generally improves tracking and disturbance rejection, but performance and control effort depend on the criterion and process.
- Nonlinear process: For the nonlinear process, fuzzy controllers provide better set-point tracking than corresponding non-fuzzy controllers under ITAE tuning.Under ITSE tuning, fuzzy FOPID achieves the best load-disturbance rejection and set-point tracking, while overshoot is lower than with ITAE tuning.
- Nonlinear process: Under ISTES tuning, fuzzy PID and fuzzy FOPID produce lower peak overshoot and better load-disturbance responses than the other controllers.Under ISTSE tuning, fuzzy PID gives the best disturbance response, closely followed by fuzzy FOPID; both fuzzy controllers have lower initial control signals.
- Open-loop unstable delayed process: For the open-loop unstable delayed process with ITAE tuning, fuzzy FOPID and fuzzy PID have almost no overshoot, while fuzzy FOPID has faster rise time and better disturbance suppression.PID and FOPID controllers require substantially higher initial control outputs than their fuzzy counterparts.
- Open-loop unstable delayed process: Under ITSE tuning for the unstable process, PID and FOPID settle faster but have higher overshoot, whereas fuzzy FOPID is sluggish with relatively poor disturbance suppression.Under ISTES tuning, PID has better disturbance suppression, while fuzzy FOPID rises faster than fuzzy PID and both fuzzy controllers show almost no overshoot.
- Overall comparison and tuning: The proposed fuzzy FOPID outperforms the other controllers for almost all performance indices in set-point tracking, although disturbance attenuation was not optimized directly.The genetic algorithm tunes input-output scaling factors and fractional parameters through a weighted objective including integral indices and control signal.