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This paper proposes a robust design of the time-varying internal modelprinciple-based control (TV-IMPC) for tracking sophisticated referencesgenerated by linear time-varying (LTV) autonomous systems. The existing TV-IMPCdesign usually requires a complete knowledge of the plant I/O (input/output)model, leading to the lack of structural robustness. To tackle this issue, we,in the paper, design a gray-box extended state observer (ESO) to estimate andcompensate unknown model uncertainties and external disturbances. By means ofthe ESO feedback, the plant model is kept as nominal, and hence the structuralrobustness is achieved for the time-varying internal model. It is shown thatthe proposed design has bounded ESO estimation errors, which can be furtheradjusted by modifying the corresponding control gains. To stabilize theESO-based TV-IMPC, a time-varying stabilizer is developed by employing LinearMatrix Inequalities (LMIs). Extensive simulation and experimental studies areconducted on a direct-drive servo stage to validate the proposed robust TV-IMPCwith ultra-precision tracking performance ($\sim 60$nm RMSE out of $\pm80$mmstroke).

A robust design of time-varying internal model principle-based control for ultra-precision tracking in a direct-drive servo stage