New conditions for the design of robust periodic observer-based controllers for periodic discrete-time linear time-varying (LTV) systems are proposed in this paper. The control system is designed to be robust to external noises, and such robustness is achieved by minimizing the ℋ_∞ norm from the input noise to the error output in the observer design, and the ℋ_∞ norm from the error and disturbance signal to the output in the control design. Concerning the observer, two LMI (Linear Matrix Inequality) conditions are proposed. As part of the main contribution, an LMI condition, based on a dual representation of the LTV system, is also proposed for the synthesis of a robust state-feedback controller that makes use of the observed states rather than the actual states. A numerical example illustrates the validity of the proposed technique.