Estimation and control of large-scale systems with an application to adaptive optics for EUV lithography

Extreme UltraViolet (EUV) lithography is a new technology for production of integrated circuits. In EUV lithographic machines, optical elements are heated by absorption of exposure energy. Heating induces thermoelastic deformations of optical elements and consequently, it creates wavefront aberrations. These Thermally Induced Wavefront Aberrations (TIWA) can significantly degrade the resolution of EUV lithographic machines. One of the ways to correct TIWA and consequently, to improve the resolution of EUV lithographic machines, is to use the Adaptive Optics (AO) technique and predictive control algorithms. However, the predictive control of TIWA is a challenging problem, mainly because the dynamical behavior of TIWA is described by thermoelastic Partial Differential Equations (PDEs). By discretizing the thermoelastic equations using the finite difference or finite element methods, large-scale state-space models can be obtained. A striking feature of these state-space models is that they have sparse (multi) banded matrices. Consequently, these state-space models can be interpreted as large-scale networks of local subsystems. All this implies that the problem of correcting TIWA can be placed in a much more general context of identifying, estimating and controlling large-scale interconnected (distributed) systems. However, currently used estimation and control algorithms are not computationally feasible for large-scale systems. For this reason, this thesis focuses on the development of computationally efficient identification, estimation and control algorithms for large-scale interconnected systems. In this thesis, we prove that the inverses of (finite-time) Gramians of large-scale interconnected systems, obtained by discretizing PDEs, belong to a class of off-diagonally decaying matrices. Consequently, these inverses can be approximated by sparse (multi) banded matrices with O(N) complexity, where N is the number of local subsystems. To compute the approximate inverse
Delft Center for Systems and Control
Mechanical, Maritime and Materials Engineering