Non-linear Feedforward Control for Lithographic Positioning
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Solution Overview
Problem
Conventional lithographic apparatus control systems face performance degradation when dealing with dynamic systems having non-linear characteristics, requiring larger FIR filters and increased calibration time due to the need to map non-linearity into linear feedforward coefficients.
Innovation Solution
A control system that performs a non-linear operation on the set-point signal based on a non-linear functional relationship between the output signal and the parameter, explicitly addressing the non-linear characteristic of the dynamic system, such as using a Taylor series expansion to approximate the non-linear relationship and optimizing feedforward coefficients using a Gauss-Newton method.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If FIR filters with optimized coefficients are used to control a dynamic system with non-linear characteristics, then system performance is improved, but the number of feedforward coefficients increases substantially
Solution Approach 1:
The patent transforms the control approach by changing the parameter representation from linear FIR coefficients to polynomial coefficients that directly model non-linear characteristics. This allows the system to maintain high performance while using fewer coefficients by capturing the essential non-linear behavior through polynomial terms rather than requiring many linear coefficients to approximate the same behavior.
Solution Approach 2:
Instead of mapping non-linear characteristics into linear feedforward coefficients (the conventional approach), the patent inverts the approach by using polynomial coefficients that explicitly represent non-linear relationships. This inversion allows direct modeling of non-linear dynamics without the need for substantial linear approximation.
2Measurement precision
If larger FIR filters are used to account for non-linear characteristics, then control accuracy is maintained, but calibration time increases
Solution Approach 1:
The patent changes the parameter set from linear FIR coefficients to polynomial coefficients, which naturally capture non-linear characteristics with fewer parameters. This reduction in parameter count directly decreases calibration time while maintaining positioning accuracy, as the polynomial form efficiently represents the non-linear system behavior.
3Adaptability or versatility
If non-linear characteristics are mapped to linear feedforward coefficients, then the control system can handle non-linear dynamics, but the number of coefficients required increases
Solution Approach 1:
The patent inverts the conventional approach by not mapping non-linear characteristics into linear coefficients, but rather using polynomial coefficients that inherently represent non-linear relationships. This allows the control system to adapt to non-linear dynamics while maintaining a compact coefficient set.
Solution Approach 2:
The patent substitutes the mechanical approach of linear approximation with a mathematical approach using polynomial representations. This substitution allows direct modeling of non-linear dynamics through polynomial terms, eliminating the need for extensive linear coefficient sets.
Data Source
AI summary
A control system configured to control a parameter of a dynamic system, wherein the parameter depends on an output signal. The control system comprises a set-point generator and a feedforward, wherein the set-point generator is arranged to provide a set-point signal to the feedforward. The feedforward is arranged to provide the output signal based on the set-point signal, wherein the feedforward is arranged to perform a non-linear operation on the set-point signal. The non-linear operation is based on a non-linear functional relationship between the output signal and the parameter.