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

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 fourth-order 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.