Fuzzy fractional quadratic regulator problem under granular fuzzy fractional derivatives

In this paper, a class of uncertain linear dynamical systems called fuzzy fractional linear dynamical systems is investigated. The aim is to find control inputs to keep the states of the fuzzy fractional dynamical systems near the zero in an optimal manner. The optimality criterion is in a form of a granular fuzzy integral whose integrand is a quadratic function of the state variables and control inputs. The fuzzy fractional dynamical system is described using fuzzy fractional differential equations (FFDEs). In order to achieve the aim, an effective approach for solving FFDEs should be at disposal. Due to some restrictions imposed by the previous approaches dealing with FFDEs, a new approach is proposed. The proposed approach is based on the granular derivative and the so-called relative-distance-measure fuzzy interval arithmetic. New definitions of fuzzy fractional derivatives and integral called left and right granular Riemann-Liouville fuzzy fractional derivatives, left and right granular Caputo fuzzy fractional derivatives, and the left and right granular fuzzy fractional integral are also presented. In addition, the concepts of granular fuzzy partial derivative and granular fuzzy chain rule are introduced. By the approximations of the granular fuzzy fractional integral and the granular Caputo fuzzy fractional derivative, the approximation solution to the FFDEs is obtained. Consequently, based on the new concepts and theorems, the solution to the fuzzy fractional quadratic regulator problem is given by a theorem. This paper closes with an example of regulating the motion of Boeing 747 in longitudinal direction with the presence of uncertainty in the initial conditions and the coefficients.