Nonlinear Feedback Control of Surface Roughness Using a Stochastic PDE:  Design and Application to a Sputtering Process

Abstract

In this work, we develop a method for nonlinear feedback control of the roughness of a one-dimensional surface whose evolution is described by the stochastic Kuramoto−Sivashinsky equation (KSE), a fourth-order nonlinear stochastic partial differential equation. We initially formulate the stochastic KSE into a system of infinite nonlinear stochastic ordinary differential equations by using Galerkin's method. A finite-dimensional approximation of the stochastic KSE is then derived that captures the dominant mode contribution to the surface roughness. A nonlinear feedback controller is then designed based on the finite-dimensional approximation to control the surface roughness. An analysis of the closed-loop nonlinear infinite-dimensional system is performed to characterize the closed-loop performance enforced by the nonlinear feedback controller in the closed-loop infinite-dimensional system. The effectiveness of the proposed nonlinear controller and the advantages of the nonlinear controller over a linear controller resulting from the linearization of the nonlinear controller around the zero solution are demonstrated through numerical simulations. Finally, a successful application of a stochastic KSE-based nonlinear feedback controller to the kinetic Monte Carlo model of a sputtering process is also demonstrated.

Model parameter estimation and feedback control of surface roughness in a sputtering process

“…Specifically, m should be equal or larger than the number of unstable modes of the process to ensure closed-loop stability. Furthermore, according to Theorem 1 in Lou and Christofides, 2006, m should be large enough to have a sufficiently small so that the closed-loop surface roughness of the infinitedimensional system is sufficiently close to the set-point value. However, a very large m should be avoided, since it requires a large number of actuators which may not be practical from a practical implementation point of view.…”

“…Specifically, methods for state feedback control of surface roughness based on linear (Lou and Christofides, 2005a, b;Ni and Christofides, 2005b) and nonlinear (Lou and Christofides, 2006) stochastic PDE process models have been developed. The methods involve the reformulation of a stochastic PDE into a system of infinite linear/nonlinear stochastic ordinary differential equations (ODEs) by using modal decomposition, derivation of a finitedimensional approximation that captures the dominant mode contribution to the surface roughness, and state feedback controller design based on the finite-dimensional approximation.…”

Nonlinear Feedback Control of Surface Roughness Using a Stochastic PDE

“…The normalization procedure can be found in the the simulation part of the journal version of this paper [18]. This convenience is adopted to simplify our presentation.…”

A robust nonlinear model predictive controller for a multiscale thin film deposition process

“…Nevertheless, these detailed models can be employed to derive low-order models that are practical for model-based control techniques (Gallivan and Murray, 2004;Raimondeau and Vlachos, 2000;Varshney and Armaou, 2008). In specific deposition processes, closed-form process models describing surface morphology of thin films can be developed in the form of stochastic PDEs (Hu et al, 2008;Lou and Christofides, 2006;Zhang et al, 2010). The construction and validation of the stochastic PDE models are conducted through a set of snapshots obtained from the KMC simulations that cover the complete operating region (Ni and Christofides, 2005).…”