Novel mathematical modeling of IT2 Takagi–Sugeno fuzzy PID controllers with triangular FoUs and their application: A MagLev case study

This study presents two new mathematical models of interval type-2 Takagi–Sugeno fuzzy proportional-integral-derivative (IT2TSFPID) controllers, incorporating triangular footprints of uncertainty (FoUs), specifically the bottom-wide triangular (BWT) and top-wide triangular (TWT) structures. Unlike previous one-dimensional (1D) input-space-based studies that predominantly employed parallelogram-shaped FoUs to model IT2TSFPID controllers, this study is the first to utilize triangular FoUs in their design. The proposed models compute the scaled incremental control effort by algebraically summing the individual scaled incremental contributions from the proportional, integral, and derivative components, each modeled within the interval type-2 (IT2) Takagi–Sugeno (TS) fuzzy framework. A distinctive feature of this work is the use of the modulus of scaled input variables in rule consequents, which reduces the number of tunable parameters and overall modeling complexity. Furthermore, unlike the existing triangular FoU-based IT2 Mamdani fuzzy proportional-integral-derivative (IT2MFPID) controllers, the absence of bias terms in the controller expressions ensures better adherence to the nonlinear PID structure. Notably, the proposed controllers are developed using the centroid defuzzifier, a method not previously applied to IT2TSFPID models. The use of 1D input spaces further allows independent tuning of IT2TSFPID gains, offering improved flexibility in controller design. In addition to modeling, this study provides a detailed analysis of the properties, including bounded-input bounded-output (BIBO) stability, of the proposed controllers, aimed at delivering a comprehensive understanding of their behavior. Finally, rigorous simulation results from a magnetic levitation (MagLev) case study are presented to demonstrate the effectiveness and practical relevance of the proposed controllers, along with extensive comparisons made against the existing work.