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TECHNICAL PAPERS

Nominal and Robust Feedforward Design With Time Domain Constraints Applied to a Wafer Stage

[+] Author and Article Information
Edgar de Gelder, Carsten Scherer, Camile Hol, Okko Bosgra

 Delft University of Technology, Delft Center for Systems and Control, Mekelweg 2, 2628 CD Delft, The Netherlands

Marc van de Wal1

 Philips Applied Technologies, Mechatronics, High Tech Campus 7, LA036, 5656 AE Eindhoven, The Netherlands

1

Corresponding author. To whom correspondence should be addressed; e-mail: M.M.J.van.de.Wal@philips.com

J. Dyn. Sys., Meas., Control 128(2), 204-215 (Apr 04, 2005) (12 pages) doi:10.1115/1.2192821 History: Received May 19, 2004; Revised April 04, 2005

A new method is proposed to design a feedforward controller for electromechanical servo systems. The settling time is minimized by iteratively solving a linear programming problem. A bound on the amplitude of the feedforward control signal can be imposed and the McMillan degree of the controller can be fixed a priori. We choose Laguerre basis functions for the feedforward filter. Since finding the optimal pole location is very difficult, we present a computationally cheap method to determine the pole location that works well in practice. Furthermore, we show how the method can account for plant and/or reference signal uncertainty. Uncertainty in servo systems can usually be modeled by additive norm-bounded dynamic uncertainty. We will show that, because the feedforward controller is designed for a finite-time interval, we can replace the dynamic uncertainty set by a parametric one. This allows us to design a robust feedforward controller by solving an LMI problem, under the assumption that the transfer functions of the plant, sensitivity, and process sensitivity depend affinely on the uncertainty. If the uncertainty set is a finite set, which is usually the case for uncertainty in the reference profiles, the feedforward design problem reduces to a linear program. These classes of uncertainty sets are well suited to describe variations in the plant and in the reference profile of a wafer stage, which is important for the practical application of the filter. Experimental results for a wafer stage demonstrate the performance improvement compared to a standard inertia feedforward filter.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Standard two DOF control structure for servo systems

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Figure 2

Basic layout of a wafer scanner

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Figure 3

A servo error e as function of time and its amplitude bounds ±w

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Figure 4

Schematic top view of the wafer stage

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Figure 5

The identified frequency response of the wafer stage

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Figure 6

Bode diagram of inertia FF (solid), second order (dashed), and fourth order FDTC FF (dash-dot), and the inverse of the flexible plant dynamics (dotted)

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Figure 7

Simulated servo errors for inertia FF (solid) and second (dashed) and fourth order FDTC FF (dash-dot)

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Figure 8

Measured servo errors for inertia FF (dotted) and with the fourth order FDTC FF (dashed) and a tuned version thereof (solid)

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Figure 9

Left: the simulated (solid) and measured (dotted) response with inertia FF. Right: the difference between the simulated and measured response with the fourth order FF (solid) and the difference between the simulated and measured response with inertia FF (dotted)

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