Elongated plasmas lead to improved performance in tokamaks but make the plasma prone to vertical instability, which requires active feedback control, a critical issue for future fusion reactors. Vertical control was optimized for the TCV tokamak by applying modern control theory to electromagnetic models for the plasma-vessel-coils dynamics. Two different optimal combinations of poloidal field coils for vertical control actuation are derived from linear plasma response models and used on different timescales for controlling the plasma vertical position. On fast timescales, the priority is input minimization, while on long timescales position control is designed to be compatible with shape control. A structured H-infinity design extending classical H-infinity to fixed-structure control systems was subsequently applied to obtain an optimized controller using all available coils for position control. Closed-loop performance improvement was demonstrated in dedicated TCV experiments, showing a reduction of input requirement for stabilizing the same plasma, thus reducing the risk of power supply saturation and consequent loss of vertical control. This novel algorithm is adaptable to different plasma equilibria as it is designed for model-based automated coil selection and controller tuning, thus avoiding extensive experimental gain scans when performing plasma discharges in TCV. The presented technique is general and can be applied to any present tokamak with independent coils or for the design of future tokamak magnetic control systems.