A fully analytical controller design is proposed to tackle a periodic control problem for stable linear systems with an input delay. Applying the internal model control scheme, the controller design reduces to designing a filter, which is done through placement of poles and zeros. The zeros are placed to compensate for the harmonics and to gain the filter properness. For placing the poles, a quasi-optimal procedure is proposed utilizing the standard LQR method. Supposing high-dimensionality of the filter due to targeting large number of harmonics, the design as well as controller implementation is performed over a state space representation. A thorough experimental case study is included to demonstrate both the practical feasibility and effectiveness of the proposed control design. The experimental validation is performed on a physical system where the goal is to reject periodic vibrations acting on a mass-spring-damper setup where the sensor and the actuator are non-collocated.