Innovative tracking control using interval type-2 fuzzy and fractional methods

This paper delves into a unique subset of uncertain linear dynamical systems known as interval type-2 fuzzy fractional linear dynamical systems. The aim is to ensure the system's output trajectory matches a desired reference trajectory. The optimality criterion is characterized by a granular fuzzy integral, where the integrand is a quadratic function that includes both the tracking error and the control effort. The dynamics of the fuzzy fractional system are modeled using Interval Type-2 Fuzzy Fractional Differential Equations (IT2FFDEs). To achieve this goal, an effective method for solving IT2FFDEs is essential. Due to the limitations of previous methods in handling IT2FFDEs, a novel approach is introduced in this study. This new method utilizes the granular derivative and a technique called relative-distance-measure fuzzy interval arithmetic. New definitions for interval type-2 fuzzy fractional derivatives and integrals are proposed. Furthermore, the concepts of interval type-2 granular fuzzy partial derivative and interval type-2 granular fuzzy chain rule are introduced. By approximating the interval type-2 granular Caputo fuzzy fractional derivative, an approximate solution to the IT2FFDEs is derived. Consequently, based on the newly introduced concepts and theorems, a theorem is presented that provides a solution to the interval type-2 fuzzy fractional tracking control problem.