(437a) Model Predictive Control Of Nonlinear Stochastic Partial Differential Equations

Authors: 
Lou, Y., Advanced Projects Research, Inc
Hu, G., University of California, Los Angeles


Stochastic nonlinear partial differential equations (SPDEs) arise naturally in the modeling of the evolution of the surface height profile of ultra thin films in a variety of material preparation processes [1, 2, 3]. The height fluctuation of thin film surfaces, which can be described by the thin film surface roughness, can significantly affect the quality of such thin films and is subsequently an important variable from a control point of view. The need to improve the quality of thin films of advanced materials by implementing model-based real-time feedback control in industrially important material preparation processes has motivated extensive research over the last five years on the development of advanced control methods for stochastic partial differential equations.

Specifically, feedback control of linear SPDEs initially attracted significant research effort [4, 5, 6]. However, most of the material preparation processes are inherently nonlinear and a linear controller designed based on the linearization of a nonlinear SPDE process model may not be sufficient to successfully control an inherently nonlinear stochastic process under a wide range of process initial conditions and operating conditions. Motivated by this, we recently developed a method for nonlinear control of surface roughness using nonlinear SPDEs [7]. The effectiveness of the controller was demonstrated under the condition that there was no significant model errors in the nonlinear SPDE process model. However, model errors may be present in many SPDE models for thin film growth and sputtering processes due to various approximations made during the process modeling procedure. It is, therefore, necessary to develop controllers for SPDEs that posses good robustness properties against model parameter uncertainties.

In this work, we develop methods for nonlinear model predictive control of surface roughness using nonlinear SPDEs. Both state feedback control and output feedback control are considered. To demonstrate the method, we focus on two nonlinear SPDEs, the stochastic Kuramoto-Sivashinsky equation in a one-dimensional spatial domain, which arises in the modeling of film surface evolution in ion-sputtering processes, and a stochastic diffusion-reaction type PDE in a two-dimensional spatial domain, which arises in the modeling of epitaxial film growth processes. We initially formulate the SPDEs into a system of infinite nonlinear stochastic ordinary differential equations by using modal decomposition. A finite-dimensional approximation is then derived that captures the dominant mode contribution to the surface roughness. A state feedback predictive control problem is first formulated, in which the finite-dimensional model is used to predict the surface roughness and the control action is computed by minimizing an objective function including the distance between the predicted surface roughness and a reference trajectory and a terminal penalty. An analysis of the closed-loop nonlinear in finite-dimensional system is performed to characterize the closed-loop performance enforced by the model predictive controller.

Since state feedback control assumes a full knowledge of all states of the process, which may be restrictive to certain practical applications, we proceed to design output feedback predictive controllers by combining the state feedback control law and a state observer. Both static and dynamic state estimation approaches are proposed to construct the state observers for nonlinear SPDEs. In the static state estimation scheme, the states of the reduced-order model are computed directly from the process output measurements at different spatial locations, while in the dynamic state estimation scheme, a Kalman filter-based estimator is designed to estimate the states, which potentially has the advantage of requiring fewer number of measurement sensors.

Numerical simulations are performed to demonstrate the effectiveness of the nonlinear model predictive controllers. A comparison between the state feedback controller and output feedback controller is made and it is shown that the output feedback controller achieves similar closed-loop performance with an appropriate selected order of the controller. Both the state feedback controller and output feedback controller possess good robustness properties against parameter uncertainties in the SPDE process models.

[1] S. F. Edwards and D. R. Wilkinson. The surface statistics of a granular aggregate. Proc. R. Soc. Lond. A, 381:17-31, 1982.

[2] 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.

[3] K. B. Lauritsen, R. Cuerno, and H. A. Makse. Noisy Kuramote-Sivashinsky equation for an erosion model. Physical Review E, 54:3577-3580, 1996.

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

[5] Y. Lou and P. D. Christofides. Feedback control of surface roughness in sputtering processes using the stochastic Kuramoto-Sivashinsky equation. Computers & 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] 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, 45:7177-7189, 2006.