LFT-Structured Uncertainty State-Space Modeling for State Feedback Robust Control of the Ultra-Precision Wafer Stage

Abstract

This paper presents linear-fractional-trans-formation (LFT)-structured uncertainty state-space modeling for robust control of the multiple-input multiple-output (MIMO) ultra-precision wafer stage with flexible structures to meet the challenges of vibration-dependent model uncertainties. Specifically, based on the modal MIMO state-space model, a parametric uncertainty model related with flexible modes is presented, and the uncertainty parameters are structured to a diagonal matrix with the LFT method. Meanwhile, a multiplicative uncertainty model is employed to express the uncertainty in high frequency with respect to ignored high-order modes. Thus, the final uncertainty model is structured with uncertainty parameters and high-frequency uncertainty. Based on the proposed uncertainty model, a comprehensive robust control scheme is proposed. Specifically, the weighting function matrixes are designed by the loop-shaping approach, and a $\mu$ analysis is employed to achieve a nonconservative solution with respect to the structured uncertainty. A standard stability criterion of a $\bf{M} \boldsymbol{\Delta}$ structure system is then utilized to determine the system stability by the largest structure singular value. Comparative experiments on a developed ultra-precision wafer stage are finally conducted, and the results validate that the proposed method achieves significant improvements on control performance, robustness, and robust stability in all exposure fields.