Model predictive control of nonlinear stochastic PDEs: application to a sputtering process

  • Authors:

  • Yiming Lou;Gangshi Hu;Panagiotis D. Christofides

  • Affiliations:

  • Hamilton Sundstraud United Technologies Corporation, Pomona, CA;Department of Chemical and Biomolecular Engineering, University of California, Los Angeles, CA;Department of Chemical and Biomolecular Engineering and the Department of Electrical Engineering, University of California, Los Angeles, CA

  • Venue:
  • ACC'09 Proceedings of the 2009 conference on American Control Conference
  • Year:
  • 2009

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Abstract

In this work, we develop a method for model predictive control of nonlinear stochastic partial differential equations (PDEs) to regulate the state variance, which physically represents the roughness of a surface in a thin film growth process, to a desired level. We initially formulate a nonlinear stochastic PDE into a system of infinite nonlinear stochastic ordinary differential equations (ODEs) by using Galerkin's method. A finite-dimensional approximation is then derived that captures the dominant mode contribution to the state variance. A model predictive control problem is formulated based on the finite-dimensional approximation so that the future state variance can be predicted in a computationally efficient way. The control action is computed by minimizing an objective function including penalty on the discrepancy between the predicted state variance and a reference trajectory, and a terminal penalty. An analysis of the closed-loop nonlinear infinite-dimensional system is performed to characterize the closed-loop performance enforced by the model predictive controller. The model predictive controller is initially applied to the stochastic Kuramoto-Sivashinsky equation (KSE), a fourthorder nonlinear stochastic PDE. Simulation results demonstrate that the proposed predictive controller can successfully drive the norm of the state variance of the stochastic KSE to a desired level in the presence of significant model parameter uncertainties. In addition, we consider the problem of surface roughness regulation in a one-dimensional ion-sputtering process. The predictive controller is applied to the kinetic Monte Carlo model of the sputtering process to successfully regulate the expected surface roughness to a desired level.