The increasing size of modern wind turbines leads to increased structural loads caused by effects such as turbulence or interactions between the rotor and tower structure. A common approach to alleviate out-of-plane structural loads in wind turbine control is the use of individual pitch control (IPC). IPC is frequently designed based on a linear time-invariant model, which is derived from averaging the dynamics over one rotational period. Wind turbine dynamics are, however, inherently time-periodic. Hence, this paper proposes the direct design of a periodic robust controller. The controller is designed through classical mixed sensitivity synthesis for optimal induced L2 performance. A novel, structured, observer-based approach is used to simplify the synthesis problem by consecutively solving two periodic Riccati differential equations. Additionally, this leads to a highly structured controller that simplifies implementation on the hardware. To provide a realistic application of the proposed method, a controller was designed for a utility-scale 2.5 MW Liberty research turbine. Its stability and performance were verified using a high-fidelity nonlinear simulation and compared to baseline controllers, which were directly obtained from the manufacturer.