This paper proposes a new strategy to design an optimal fractional state feedback control for linear fractional periodic time-delayed (FPTD) systems. Although there exist different techniques to design the state feedback control for linear ordinary periodic time-delayed (OPTD) systems such as discretizing their monodromy matrix, there is no systematic method to design state feedback control for FPTD systems. Moreover, linear OPTD systems have the monodromy operator defined explicitly in a Banach space, and can be discretized in arbitrary basis functions. However, linear FPTD systems do not have any monodromy operator because of the nonlocal properties of fractional operators. In the proposed method, a monodromy matrix is defined for the steady state solution. Then, the efficiency of the proposed control technique is shown by implementing the method to a double inverted pendulum with fractional dampers subject to a periodic retarded follower force.