The control of slewing motion flexible structures is important to a number of systems found in engineering and physical sciences applications, such as aerospace, automotive, robotics, and atomic force microscopy. In this kind of system, the controller must provide a stable and well-damped behavior for the flexible structure vibrations, with admissible control signal amplitudes. Recently, many works have used fractional-order derivatives to model complex and nonlinear dynamical behavior present in the mentioned systems. In order to perform digital computer-based control of fractional-order dynamical systems, a time discretization of the equations is necessary. In many cases, the Grünwald–Letnikov method is used, resulting in an implicit integration method. In this work, a nonlinear slewing motion flexible structure is modeled considering a fractional-order viscous damping in the flexible beam motion. To obtain an explicit integration method, based on the Grünwald–Letnikov definition, the discretization of the dynamical equations is performed considering the existence of sample and hold circuits. In addition, real-time suboptimal infinite horizon tracking control system strategies, namely, the linear quadratic tracking and the state-dependent Riccati equation tracking controller, are designed and implemented to control the fractional-order slewing motion flexible system. The general behavior and performance of the control systems are tested for parameter uncertainties related to the order of the fractional derivatives.