(561d) Construction of Nonlinear Stochastic Pdes for Nonlinear Feedback Control of Surface Roughness: Application to a Sputtering Process
It is desirable to control the surface roughness in material preparation processes to produce high quality thin films of advanced materials used in a wide range of applications. Stochastic partial differential equations (PDEs) can be used to describe the evolution of the height profile for surfaces in both deposition and sputtering processes [1, 2]. Recently, model-based feedback control methods have been developed to control the surface roughness of thin films based on both linear [3, 4, 5] and nonlinear  stochastic PDE process models. Also, a method was developed to construct linear stochastic PDE models for thin film growth using first-principles-based microscopic simulations . Many deposition or sputtering processes are, however, inherently nonlinear. Therefore, compared to linear controllers, nonlinear feedback controllers designed directly on the basis of nonlinear process model have the advantages of providing better performance for a wider range of process initial conditions and operating conditions . Consequently, it is desirable that a nonlinear process model is constructed and is directly used as the basis for controller synthesis to achieve improved closed-loop performance. However, the construction of nonlinear stochastic PDE models for thin film preparation processes directly based on microscopic process rules is a very difficult task, which has prohibited the development of nonlinear stochastic PDE models, and subsequently the design of nonlinear feedback control systems.
This work focuses on construction of nonlinear stochastic PDE models for nonlinear feedback control of the surface roughness of thin films. To demonstrate our results, we focus on a sputtering process including two surface micro-processes, diffusion and erosion. A method is developed to construct a nonlinear stochastic PDE model of the sputtering process for feedback control of the surface roughness. The method initially reformulates a general nonlinear stochastic PDE for the evolution of the surface into a system of infinite nonlinear stochastic ordinary differential equations (ODEs). Then, kinetic Monte-Carlo simulations of the sputtering process are performed to generate surface snapshots to determine the state covariance of the stochastic ODE system. The correlations between model parameters and the state covariance are established and the parameters of the nonlinear stochastic PDE model are subsequently computed so that the evolution of the surface roughness computed from the stochastic PDE model is consistent to that computed from kinetic Monte-Carlo simulations. Based on the nonlinear PDE process model, we use the method proposed in  to design a nonlinear feedback controller and apply it to the kinetic Monte-Carlo model of the sputtering process to regulate the surface roughness at a desired level. 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.
 D. D. Vvedensky, A. Zangwill, C. N. Luse, and M. R. Wilby. Stochastic equations of motion for epitaxial growth. Physical Review E, 48:852-862, 1993.
 K. B. Lauritsen, R. Cuerno, and H. A. Makse. Noisy Kuramoto-Sivashinsky equation for an erosion model. Physical Review E, 54:3577-3580, 1996.
 Y. Lou and P. D. Christofides. Feedback control of surface roughness using stochastic PDEs. AIChE Journal, 51:345-352, 2005.
 Y. Lou and P. D. Christofides. Feedback control of surface roughness in a sputtering process using the stochastic Kuramoto-Sivashinsky equation. Computers and Chemical Engineering, 29:741-759, 2005.
 D. Ni and P. D. Christofides. Multivariable predictive control of thin film deposition using a stochastic PDE model. Ind. Eng. Chem. Res., 44:2416-2427, 2005.
 Y. Lou and P. D. Christofides. Nonlinear feedback control of surface roughness using a stochastic PDE: Design and application to a sputtering process. Industrial & Engineering Chemistry Research, submitted, 2006.
 D. Ni and P. D. Christofides. Construction of stochastic PDEs for feedback control of surface roughness in thin film deposition. In Proceedings of American Control Conference, pages 2540-2547, Portland, OR, 2005.