Piezoelectric nanopositioning systems are indispensable in high-precision applications such as Atomic Force Microscopy (AFM), wafer metrology, and medical applications. Enhancing their throughput while maintaining precision presents significant challenges due to lightly damped resonant modes and substantial dynamic variations associated with payload changes. Contemporary approaches, involve the tuning of motion controllers in a dual-closed-loop architecture that incorporates active damping to achieve higher bandwidths. Although these methods function adequately for nominal systems, they fail to meet performance requirements under system variations in resonance modes caused by payload changes and often overlook higher-order dynamics and delays. This research introduces a robust control framework that synthesises H-infinity and Mu synthesis-based controllers by shaping sensitivities within a dual-closed-loop, considering payload variations and prominent higher-order system dynamics. Systematic design guidelines for weighting functions are established to synthesise robust controllers that meet specified performance criteria. The proposed framework is experimentally validated on an industrial nanopositioning system, demonstrating robust performance despite dynamic variations induced by varying payloads.