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

Abstract

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.