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    Area of Science:

    • Computational Intelligence
    • Control Systems Engineering
    • Fuzzy Set Theory

    Background:

    • Interval type-2 fuzzy sets (IT2 FS) are defined by upper and lower membership functions, forming a footprint of uncertainty (FOU).
    • The FOU quantifies uncertainty in the defuzzified space, typically calculated using iterative algorithms like Karnik-Mendel or Newton-Raphson.
    • Existing methods for measuring IT2 FS uncertainty can be computationally intensive, limiting real-time applications.

    Purpose of the Study:

    • To develop a closed-form formula for accurately evaluating the span of uncertainty in IT2 FS.
    • To demonstrate the efficacy of this formula in a real-time control application.
    • To compare the performance of the proposed method against existing algorithms and type-1 fuzzy logic systems.

    Main Methods:

    • Derivation of a closed-form formula to calculate the span of uncertainty for IT2 FS.
    • Implementation of the closed-form formula in a real-time control system.
    • Performance evaluation through computer simulations, including comparisons with the Karnik-Mendel algorithm and type-1 fuzzy logic.

    Main Results:

    • The proposed closed-form formula provides a precise measurement of IT2 FS uncertainty with lower runtime complexity than iterative methods.
    • The real-time control application using the closed-form formula achieved reduced root mean square error and computational overhead.
    • Simulations showed superior performance in terms of maximum overshoot, peak overshoot, and root mean square error, especially under noisy conditions and high sampling rates.

    Conclusions:

    • The closed-form formula offers an efficient and precise method for quantifying uncertainty in IT2 FS.
    • The proposed IT2 FS-based control scheme demonstrates significant advantages over existing methods and type-1 fuzzy logic in real-time applications.
    • This approach is particularly effective in handling system uncertainties and measurement noise, leading to improved control performance.