Feedback control of surface roughness using the linearized stochastic Kuramoto-Sivashinsky equation

In this paper, we control the roughness of a surface described by the linearized stochastic Kuramoto-Sivashinsky equation (KSE), a fourth-order stochastic partial differential equation. We initially reformulate the linearized stochastic KSE into a system of infinite stochastic ordinary differential equations by using modal decomposition. A finite-dimensional approximation of the linearized stochastic KSE is then derived that captures the dominant mode contribution to the surface roughness. A state feedback controller is designed based on the finite-dimensional approximation to control the surface roughness. Analysis of the closed-loop system shows that the controller can drive the surface roughness governed by the infinite-dimensional system to a desired level. The effectiveness of the proposed method is demonstrated by numerical simulations.