Aiming to compensate for the cross-periodic coupling of piezoelectric nano-stages with measurement delays in raster nano-scanning， particularly in high-speed scanning applications， this paper presents a robust periodic disturbance observer （PDOB） control strategy that does not require solving complex inverse models. First， an electromechanical multi-perturbation model is developed to describe the dynamic behavior of piezoelectric stage systems. Then， a PDOB control structure based on the model is constructed. The problem of optimal controller parameter solution is transformed into a mixed sensitivity optimization problem for time-delay systems with the formulated performance optimization function and robust stability condition， which is solved via the infinite-dimensional H^∞ control method. Finally， comprehensive experimental investigations are conducted on the piezoelectric nano-stage. The results indicate that the developed dynamic model can effectively fit the experimental response of the stage and that the proposed robust periodic disturbance control method has superior anti-disturbance and uncertainty compensation performance to the method without disturbance observer and the conventional PDOB control method， with improvements of >99% and >50%， respectively， in the anti-disturbance performance at the fundamental harmonic frequency.