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.

中文翻译：

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粒状模糊分数阶导数下的模糊分数二次调节器问题


在本文中，研究了一类称为模糊分数线性动力系统的不确定线性动力系统。目的是找到控制输入，以最佳方式保持模糊分数动力系统的状态接近于零。最优准则采用粒状模糊积分的形式，其被积函数是状态变量和控制输入的二次函数。使用模糊分数阶微分方程（FFDE）描述模糊分数动力系统。为了实现这一目标，应该找到解决FFDE问题的有效方法。由于以前处理 FFDE 的方法施加了一些限制，因此提出了一种新方法。所提出的方法基于粒度导数和所谓的相对距离测量模糊区间算法。还提出了模糊分数阶导数和积分的新定义，称为左右粒状黎曼-刘维尔模糊分数阶导数、左右粒状Caputo模糊分数阶导数以及左右粒状模糊分数阶积分。此外，还介绍了粒模糊偏导数和粒模糊链式法则的概念。通过粒状模糊分数阶积分和粒状Caputo模糊分数阶导数的近似，得到FFDE的近似解。因此，基于新的概念和定理，由定理给出了模糊分数二次调节器问题的解。本文最后以在初始条件和系数存在不确定性的情况下调节波音 747 纵向运动的示例作为结束。