A compliance feedforward scheme for a class of LTV motion systems

Abstract

Abstract-The implementation of lightweight highperformance motion systems in lithography applications imposes among others lower requirements on actuators, amplifiers, and cooling. However, the decreased stiffness of lightweight designs brings the effect of structural flexibilities to the fore especially when the so-called point of interest is not at a fixed location. This is for example the case when exposing a silicon wafer. To deal with structural flexibilities, a feedforward controller is proposed that combines two concepts: (a) continuous compliance compensation control and (b) snap feedforward control. Expanded to a subclass of LTV motion systems, the resulting controller compensates for the position-dependent and time-varying compliance of a flexible structure. The compliance function used will be derived using partial differential equations (PDE). The method is validated by simulation results.

Linear-time-varying feedforward control for position-dependent flexible structures

“…This feedforward controller is an LTV extension of snap feedforward control [1]. Note that the latter cannot compensate for LTV systems [12]. The POI r p = r p (t), is varying with respect to time t and affects the feedforward controller design.…”

“…The feedforward controller should avoid any division by zero, i.e., 1 + ω 2 s mC(t) = 0, see (11), (12) and (13). The resonance frequency ω s ensures the feasibility of the feedforward controller under the following conditions:…”

“…The region specified by vertical dashed lines is the wafer scanning interval which occurs at constant velocity. In order to correct the minor remaining error of the output, an LTI PID feedback controller is tuned via manual loopshaping [12]. Simulation results are illustrated in Fig.…”

“…This method adds the quasi-static behaviour of flexible structures as a correction block to the control structure. Later, in [12], the former compliance feedforward control is tuned by a low-pass filter to an LTV system. It retains the control objectives of snap feedforward control for an LTV system.…”

“…The method is validated for a rotational two-mass-spring-damper system. Following the previous works, specially [12], [13], the main contribution of this work is to design a feedforward controller for an LTV system in which the stable plant inversion is based on an LTV fourth-order model. The model considers the quasi-static deformation of the plant besides the dominant resonant dynamics of the system.…”