(598d) Nonlinear Feedback Control of Stochastic Pdes

Stochastic partial differential equations (PDEs) can be used to describe the evolution of the height profile for surfaces in a variety of material preparation processes [1, 2, 3]. The height fluctuation of thin film surfaces, characterized by the surface roughness, can significantly affect the quality of such thin films and is subsequently an important variable for control. This motivated the research on model-based feedback control of surface roughness using stochastic PDEs [4, 5, 6]. Specifically, based on linear stochastic PDE process models, pole-placement controllers were developed to control the surface roughness in a deposition process [4] and sputtering processes [5]; and a predictive controller was developed to control both the growth rate and the surface roughness in thin film growth [6]. A linear stochastic PDE model is obtained by either linearizing a nonlinear stochastic PDE process model derived based on microscopic rules or using a data-driven model construction method recently developed in [7]. Since most of the material preparation processes are inherently nonlinear, it is expected that the controller designed based on the linear stochastic PDE model is only going to provide good closed-loop performance locally (i.e., for initial conditions close to the desired set-point) for the nonlinear closed-loop system. To perform feedback control design for nonlinear stochastic processes, i.e., provide good performance for a wide range of process initial conditions and operating conditions, it is desirable that a nonlinear model is directly used as the basis for controller synthesis.

In this work, we develop a method for nonlinear feedback control of stochastic partial differential equations [8]. To demonstrate the method, we focus on nonlinear control of the stochastic Kuramoto-Sivashinsky equation (KSE). The stochastic KSE is a nonlinear fourth-order stochastic PDE that describes the evolution of the height profile for surfaces in certain deposition and sputterring processes [3,9]. We initially formulate the stochastic KSE into a system of infinite nonlinear stochastic ordinary differential equations by using modal decomposition. 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 of the stochastic KSE are demonstrated through numerical simulations.

References:

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[4] Y. Lou and P. D. Christofides. Feedback control of surface roughness using stochastic PDEs. AIChE Journal, 51:345-352, 2005.

[5] 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.

[6] D. Ni and P. D. Christofides. Multivariable predictive control of thin film deposition using a stochastic PDE model. Industrial & Engineering Chemistry Research, 44:2416-2427, 2005.

[7] 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.

[8] 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.

[9] Z. Insepov, I. Yamada, and M. Sosnowski. Surface smoothing with energetic cluster beams. Journal of Vacuum Science & Technology A - Vacuum Surfaces and Films, 15:981-984, 1997.