Local Rational Modeling for Identification Beyond the Nyquist Frequency: Applied to a Prototype Wafer Stage

TL;DR

This work tackles fast-rate FRF identification beyond the Nyquist limit using slow-rate outputs by introducing Local Rational Modeling (LRM) across multiple frequency bands, which disentangles aliased dynamics in a single experiment. A weighted linear least-squares formulation yields a closed-form solution for local FRFs across $F$ bands within a window of size $2n_w+1$, with uniqueness guaranteed under design conditions and the possibility to estimate variance. Iterative reweighting via Sanathanan-Koerner and Levenberg-Marquardt refinements can further improve accuracy, though the weighted LS solution often suffices. The framework is validated experimentally on a prototype wafer stage, showing accurate identification of lightly damped resonant dynamics beyond the slow Nyquist and outperforming LPM and spectral analysis in both FRF fidelity and variance. Overall, the method enables reliable fast-rate FRF identification from slow-rate measurements, facilitating fast-rate control design for vision-in-the-loop and other multirate systems.

Abstract

Fast-rate models are essential for control design, specifically to address intersample behavior. The aim of this paper is to develop a frequency-domain non-parametric identification technique to estimate fast-rate models of systems that have relevant dynamics and allow for actuation above the Nyquist frequency of a slow-rate output. Examples of such systems include vision-in-the-loop systems. Through local rational models over multiple frequency bands, aliased components are effectively disentangled, particularly for lightly-damped systems. The developed technique accurately determines non-parametric fast-rate models of systems with slow-rate outputs, all within a single identification experiment. Finally, the effectiveness of the technique is demonstrated through experiments conducted on a prototype wafer stage used for semiconductor manufacturing.

Figures (9)

• Figure 1: Photograph of experimental setup, containing the wafer stage.
• Figure 2: Schematic overview of experimental setup, where the fast-rate input $u_h$ is distributed over the four corners of the chuck. The outputs used for feedback $y_i \; \forall i \in \{1,2,3,4\}$ are measured using encoder scales and heads, which is schematically depicted for $y_3$. The performance output $z_l$ is measured using an additional capacitive scanning sensor, which is suspended less than 1 mm above the wafer and sampled at a reduced sampling rate.
• Figure 3: Open-loop identification setting considered. The fast-rate system $G$ with fast-rate input $u_h$ and output $y_h$ have sampling times $T_h$. The slow-rate output $y_l$ is downsampled and disturbed with measurement noise, i.e., $y_l=\mathcal{S}_dy_h+He_l$, and has sampling time $T_l\xspace=F\xspace T_h\xspace$.
• Figure 4: Experimental feedback scheme used, where the equivalent systems in \ref{['eq:EquivalentSystems']} are to be identified.
• Figure 5: (Middle:) The developed local rational modeling method $\widehat{G}_{LRM}(\Omega_k)$ ()identifies the true system $G_{y_4}(\Omega_k)$ ()accurately, even beyond the Nyquist frequency ()and the lightly-damped resonant dynamics (enlarged in top). Both $\widehat{G}_{LPM}(\Omega_k)$ ()and $\widehat{G}_{SA}(\Omega_k)$ ()identify the true system $G(\Omega_k)$ ()significantly less accurate. (Bottom:) The associated standard deviation for $\widehat{G}_{LPM}(\Omega_k)$ ()and $\widehat{G}_{LRM}(\Omega_k)$ (), calculated using the square root of \ref{['eq:Variance']}.

Theorems & Definitions (7)

• Remark 1
• Lemma 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Lemma 2