  This paper proposes a robust design of the time-varying internal model
principle-based control (TV-IMPC) for tracking sophisticated references
generated by linear time-varying (LTV) autonomous systems. The existing TV-IMPC
design 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 and
compensate unknown model uncertainties and external disturbances. By means of
the ESO feedback, the plant model is kept as nominal, and hence the structural
robustness is achieved for the time-varying internal model. It is shown that
the proposed design has bounded ESO estimation errors, which can be further
adjusted by modifying the corresponding control gains. To stabilize the
ESO-based TV-IMPC, a time-varying stabilizer is developed by employing Linear
Matrix Inequalities (LMIs). Extensive simulation and experimental studies are
conducted on a direct-drive servo stage to validate the proposed robust TV-IMPC
with ultra-precision tracking performance ($\sim 60$nm RMSE out of $\pm80$mm
stroke).
