Structured uncertainty modeling of position-dependent model variations for an ultra-precision stage

Abstract

This study presents a systematic approach to modeling the position-dependent model variations of an ultra-precision positioning stage with minimal structured uncertainty in linear fraction transformation (LFT) representation. The position dependent dynamics characterizing by varying resonances, are introduced by the structural flexibility together with the measurement transformations, and should be accurately and non-conservatively modeled to maximize the achievable performance in subsequent controller design. The dynamics of the stage are first estimated by a group of nonparametric frequency response functions (FRF) through close-loop identification. And an iterative global rational fraction polynomial method is applied to fit the FRFs into transfer functions with common denominator. Then the differences in the coefficients of the fitted transfer functions are described by parametric uncertainty, and further converted to structured uncertainty in LFT form. Finally the dimension of the uncertain structure is reduced by principal components analysis (PCA). The reduced model captures the variations of identified FRFs very well, and it is demonstrated that the proposed method could be effectively used to develop the minimal structured uncertain model for position-dependent dynamics with the same characteristic equation.