| Output Feedback Covariance Control Of Nonlinear Stochastic Partial Differential Equations | ||
| Gangshi Hu, Department of Chemical and Biomolecular Engineering, University of California, Los Angeles, Los Angeles, CA 90095, Yiming Lou, Advanced Projects Research, Inc, 1925 McKinley Ave., Suite B, La Verne, CA 91750 and Panagiotis D. Christofides, Department of Chemical and Biomolecular Engineering, Department of Electrical Engineering, University of California, Los Angeles, Los Angeles, CA 90095 Stochastic partial differential equations (SPDEs) arise frequently in the modeling of evolution of surface micro-configuration in a deposition and sputtering processes. The fact that SPDEs are available in closed-form makes them a natural choice as a basis for feedback controller design. State feedback control of SPDEs, which assumes complete availability of all states for feedback to, has been investigated in the context of linear and nonlinear covariance control of SPDEs [3,4,5]. However, the assumption that the full state vector can be measured in real-time may not be valid for a number of applications, which motivates the study of output feedback control. In this work, we focus on output feedback covariance control of nonlinear SPDEs. First, linear/nonlinear state feedback covariance controllers are designed on the basis of reduced-order finite-dimensional approximations, which are derived from the SPDEs using Galerkin's method. Subsequently, two different approaches to output feedback control are developed: static output feedback control and dynamic output feedback control. In static output feedback control, the states of the reduced-order system are calculated directly from multiple measurements of the SPDE state at different positions and are directly fed to the state feedback controller. While in dynamic output feedback control, a Kalman-filter-based estimator is used to estimate the states of the finite dimensional system used for controller design from a small number of output measurements. The issue of further reduction of the dimension of the finite-dimensional system used for controller/estimator design using the method of approximate inertial manifolds and singular perturbation techniques [7] by taking into account SPDE modes which are not included in the model used for controller design is also studied. Simulations are performed to demonstrate the effectiveness of static/dynamic output feedback covariance control in the context of the stochastic Kuramoto-Sivashinsky equations (SKSE) and of a stochastic second-order parabolic partial differential equation. We demonstrate that the output feedback controller effectively regulates the roughness, which is a direct function of the covariance level of the closed-loop system, to a desired value. [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] K. B. Lauritsen, R. Cuerno, and H. A. Makse. Noisy Kuramote-Sivashinsky equation for an erosion model. Phy. Rev. E, 54:3577-3580, 1996. [3] Y. Lou and P. D. Christofides. Feedback control of surface roughness using stochastic PDEs. A.I.Ch.E. J., 51:345-352, 2005. [4] Y. Lou and P. D. Christofides. Feedback control of surface roughness in sputtering processes using the stochastic Kuramoto-Sivashinsky equation. Comp. & Chem. Eng., 29:741-759, 2005. [5] Y. Lou and P. D. Christofides. Nonlinear feedback control of surface roughness using a stochastic PDE: Design and application to a sputtering process. Ind & Eng. Chem. Res., 45:7177-7189, 2006. [6] A. Hotz and R. E. Skelton. Covariance control theory. Int. J. Contr., 46:13-32, 1987. [7] P. D. Christofides. Nonlinear and Robust Control of Partial Differential Equation Systems: Methods and Applications to Transport-Reaction Processes. Birkhauser, Boston, 2001 Extended Abstract Status: Not Uploaded | ||